Economics and the Price Index
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S.N. Afriat

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Economics and the Price Index

A

theft amounting to £1 was a capital offence in 1260, and a judge in 1610 afﬁrmed the law could not then be applied since £1 was no longer what it was. Such association of money with a date is well recognized for its importance in very many different connections. Thus arises the need to know how to convert an amount at one date into an equivalent amount at another date. In other words, a price index. The longstanding question concerning how such an index should be constructed is known as ‘The Index Number Problem’. A pervasive long established institution for economics, it is a number issued by the Statistical Ofﬁce that should tell anyone the ratio of costs of maintaining a given standard of living in two periods where prices differ. For a succession of three periods, the product of the ratios for successive pairs must coincide with the ratio for the endpoints. This is the chain consistency required of price indices. A usual supposition is that the index is determined by an algebraical formula involving price and quantity data for the two reference periods, as with the one or two hundred formulae in the collection of Irving Fisher, always joined with the question of which one to choose, and the perplexity that chain consistency is not obtained with any. Hence ﬁnally they should all be abandoned. This is the reality of ‘The Index Number Problem’. Now in this book consistent prices indices for any number of periods are all computed together to make a resolution of the ‘Problem’, proved unique hence never to be joined by others to make a Fisher-like proliferation. That brings to a conclusion an issue giving rise to extensive thought and theory to which over the decades a remarkable number of economists have each contributed a word, or volume. A product of attention to the ‘Problem’ over a half-century, this book should be of interest to all those for whom ‘The Index Number Problem’ remains, beside of perpetual practical concern, a source of fascination. S. N. Afriat, resident of Siena, intermittent adjunct at the University, Mathematics and Economics, permanent Visiting Professor. Carlo Milana, Research Director at ISAE (Istituto di Studi e Analisi Economica), Rome, a public research institution supported by the Italian Ministry of Economy and Finance.

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friat is the guru of the price index … his work is a classic illustration of how much we learn from new ways of thinking. Angus Deaton Dwight D. Eisenhower Professor of International Affairs Professor of Economics Princeton University, USA

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esearchers interested in price indexes, inter-area comparisons, revealed preference theory, or productivity measurement will ﬁnd this book an enlightening, entertaining, and highly creative treatise on where the theory should go from here. Marshall Reinsdorf Senior Research Economist US Bureau of Economic Analysis

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ydney Afriat is famous for his unique and penetrating insights, often very unsettling to those who have worked long and hard in a ﬁeld—without ever seeing what is obvious to Sydney, who then formulates it neatly in very compact mathematics. In this case he has a very good co-author who has independently developed a critical examination of many current writings on the subject, and has helped him develop his insights in practical directions. This monograph will very likely render many current discussions of index numbers obsolete. Edward Nell Malcolm B. Smith Professor of Economics New School University, NY

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ydney Afriat belongs to that select group of economic theorists who have become a legend in their own times. There is such a thing as “Afriat’s Theorem”, which has become part of the staple for students of microeconomic theory … Moreover, he may belong to another select group of prose stylists who are also masters of some aspects of the mathematical method and its philosophy. K. Vela Velupillai Journal of Economics March 2004

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ew generation price index workers can have pleasure to escape stagnation in the Irving Fisher trap and catch on to the new approach with its conceptual ﬁdelity and computational elegance. Preface

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Economics and the Price Index

S.N. Afriat Carlo Milana foreword Angus Deaton

First published 2009 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Ave, New York, NY 10016 Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2008. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

© 2009 by S. N. Afriat and Carlo Milana Typeset in Times New Roman by Keyword Group Ltd Printed and bound in Great Britain by MPG Books Ltd, Bodmin All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Afriat, S. N., 1925– Economics and the price index / S.N. Afriat and Carlo Milana. p. cm. Includes bibliographical references and index. 1. Price index. 2. Index numbers (Economics) HB225.A326 2008 338.5 28–dc22 2008009726 ISBN 0-203-89148-1 Master e-book ISBN

ISBN10: 0-415-47181-8 (hbk) ISBN10: 0-203-89148-1 (ebk) ISBN13: 978-0-415-47181-7 (hbk) ISBN13: 978-0-203-89148-3 (ebk)

Contents

Foreword Preface Acknowledgements

xv xvii xxi

PART I

Concept and Method

1

1

The Super Price Index: Irving Fisher, and after

3

2

The Price Level Computation Method

25

3

Price Level Computation: Illustrations

69

Bibliography

101

PART II

Precursor 1

The system of inequalities ars > Xs − Xr

109 111

S. N. AFR I AT, Research Memorandum No. 18 (October 1960)

Econometric Research Program, Princeton University Proc. Cambridge Phil. Soc. 59 (1963)

2

On the constructibility of consistent price indices between several periods simultaneously S. N. AFR I AT, in Essays in Theory and Measurement of Demand:

in honour of Sir Richard Stone edited by Angus Deaton, Cambridge University Press, 1981

127

xiv Contents

3

The theory of exact and superlative index numbers revisited

157

C AR L O M I L ANA, Istituto di Studi e Analisi Economica, Roma

EUKLEMS Working Paper No. 3, 2005 http://www.euklems.net.

Appendices

221

1

223

The price index as a utility based concept 1.1 1.2

2

Original approach 225 New approach 230

Terminology

235

Conical v. homogeneous &c

3

Notation

237

4

BASIC computer program

241

BBC BASIC for Windows developed by Richard Russell [email protected].

249

Notes 1

Multilateral indices

2

Chain indices

250

254

Postscript

259

Index

261

Foreword

Sydney Afriat is the guru of the price index. As a young mathematician, he arrived at Richard Stone’s Department of Applied Economics in Cambridge in the early 1950s, then the great center of research on theoretical and applied consumer behaviour. He soon realized that neither he nor anyone else knew very much about what was meant by “the price index”, in spite of being part of the everyday discourse of economics. In the half century since then, he has been exploring the foundations of the topic. Over the years, he has produced beautiful theorems on the topic, many of them completely unexpected even by the cognoscenti of the topic. And where he has led, the profession has followed, often many years later. Along the way, his work has fathered important incidental areas in economics, perhaps most notably the non-parametric analysis of demand, that has carried into other areas, like the analysis of efﬁciency with his frontier production function that extends Farrell’s method and the stochastic frontier. Afriat is one of those rare and rarely gifted individuals who think differently from everyone else. What is obvious to him can sometimes seem bizarre to others, especially at ﬁrst, and his vision of “the price index” often differs sharply from those that dominate the profession. Yet his work is a classic illustration of how much we learn from new ways of thinking, and how the bizarre and unfamiliar conceptions and results of an idiosyncratic leader can become the orthodoxy of the next generation. The last book presented a coherent discussion of ﬁfty years of astonishingly creative work on the price index. Some of the analysis had appeared before but much had not. Afriat’s many friends may have welcomed the deﬁnitive account, as also those who had not previously had the opportunity to understand and enjoy the work. So much is said in my foreword to that book. Now comes the surprise, that the story is not there ended. At the heart of this new book is a paper of twenty-ﬁve years ago in a volume in honour of Sir Richard Stone which I edited, reproduced in Part II. Yes very original, quite unlike anything seen before, and then what? Read on. Angus Deaton Dwight D. Eisenhower Professor of International Affairs Professor of Economics Princeton University, USA

Preface

Though the mathematics of the method, its theoretical rationalization and computations, require an account, the scheme for applications is simple, and conveys an idea of what could be meant by an answer to “The Index Number Problem”. A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of an entirely different type and are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, for the hypothetical underlying utility which in any case there is no need to actually construct. The theory of the price index, proper, starts with the Utility Cost Factorization Theorem going back to 1950’s. By itself it represents no resolution of the Index Number Problem, nor had there even been a real idea of what could be meant by such a resolution. The method now proposed does convey some idea of what could be meant by such a resolution. It even represents such a resolution itself. The method has been available in the main for more than twenty ﬁve years, apart from ampliﬁcations made just now. But only recently has it been recognized as a proper resolution of the Index Number Problem. The here provided ﬁrst exercises with the arithmetic go to convey the practicality of it. The needs of dealing with the EUKLEMS Project data when it became available, joined with chance encounter with data of use for practising the new computations and development of needed software, have stirred into life that almost forgotten work. Here provided in Part I are three papers by present authors: “The Super Price Index: Irving Fisher, and after”

xviii Preface has more to do with history, “The Price Level Computation Method” is an exposition of the mathematics, and. “Price Level Computation: Illustrations” offers ﬁrst exercises with the arithmetic of the method that go to convey the practicality of it. It is submitted that, for the very large number of different traditional type formulae to determine price indices associated with a pair of periods, which are joined with the longstanding question of which one to choose, they should all be abandoned. For the method proposed instead, price levels associated with periods are ﬁrst all computed together, subject to a consistency of the data, and then price indices that are all true taken together are determined from their ratios. An approximation method can apply in the case of inconsistency. At the heart of this new book is a paper of Afriat of twenty ﬁve years ago in a volume in honour of Sir Richard Stone edited by Angus Deaton, reproduced in Part II. It took being reminded of the paper in a ﬁrst meeting with Carlo Milana last Christmas (2006) to see what it represents. It is true, a ﬁrst discovery was agreement about perplexed latter-day phases of the old subject that may deserve notice, perhaps as grim ‘lessons from history’, where one of us, Milana, has a special interest and reports on this subject in Part II. But from then to April this year (2007) three papers were written that take the new method important steps further, and form Part I of this book. The 1960 paper reproduced in Part II is a step for the proof of a Utility Construction Theorem. This, a half century later, is the “Theorem” of “Two New Proofs of Afriat’s Theorem” by A. Fostel, H. E. Scarf and M. J. Todd. According to the Abstract, it’s “celebrated”! It happens this paper also serves for another construction theorem where the utility is ‘constant returns’ (already ‘rediscovered’ as if one of us had fallen asleep a few years, even for ever, before being wakened last Christmas) as required by the Utility Cost Factorization Theorem at the start of price index theory proper from a half-century ago, set forth again in Appendix 1. There it is also noted how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of common sense. In that way it is not just one possible answer to ‘The Index Number Problem’, but the only answer. Out of this simpler and altogether less well appreciated theorem the new price index method emerged, maybe to be “celebrated” in the next half century, or tomorrow—or, with this publication, even today. The method should take its place as the unique step that showed the way out of decades of stagnation in the Irving Fisher mind-set.

Preface xix Tracing linkages to the ‘non-parametric’ approach, that since its origin with utility construction theorems has so many inﬂuences elsewhere, also makes the basis for the present work.1 Though all points in the Paasche-Laspeyres interval are true, all equally, none more true than another, absurd pretentions offer a not merely “true” price index but an extra distinguished “superlative” one, as if it was an important discovery that had escaped everyone, even Irving Fisher, and thicken the quagmire with perambulations about functional form which has no welfare signiﬁcance at all. A basis for the method may be the 1981 paper, but only up to a point. That paper was inconsequential for application without the needed computational apparatus, but now we have that. Then there was appeal to the ‘extension property’ of solutions to prove existence theoretically, but now certain solutions are actually computed, the basic price level solutions that come in pairs with sides as it were associated with Laspeyres and Paasche, from which, it is conjectured, all solutions can be derived. A main basis for this book is not altogether in the remote beginnings but in itself. A precursor is the paper “The Power Algorithm for Generalized Laspeyres and Paasche Indices” at the Athens Meeting of the Econometric Society, September 1979. This deals with a feature in the beginnings of the computational method where a matrix of Laspeyres indices is raised to powers in a bizarre arithmetic where plus means min, maybe rather much for some but now we have the sofware for it. We usually have no dealings with Fisher-type bilateral formulae and the vast literature related there, but for elements of history that include added Notes about chain and multilateral index methods. There is as ever an imbalance since one of us reads everything and the other nothing. New generation price index workers can have pleasure to escape stagnation in the Irving Fisher trap and catch on to the new approach with its conceptual ﬁdelity and computational elegance.

1 A bibliography of works with the non-parametric connection lists about two hundred items, and quite likely there are many others.

Acknowledgements

For materials reproduced, thanks are due to the Cambridge University Press for permissions extending to the following S. N. Afriat, The system of inequalities ars > Xs − Xr Research Memorandum No. 18 (October 1960) Econometric Research Program, Princeton University. Proc. Cambridge Phil. Soc. 59 (1963) Reprinted with permission. S. N. Afriat, On the Constructibility of Consistent Price Indices Between Several Periods Simultaneously. In Essays in Theory and Measurement of Demand: in honour of Sir Richard Stone, edited by Angus Deaton. Cambridge University Press, 1981. 133–161. Reprinted with permission.

We acknowledge with thanks the guidance received from Richard Russell, longtime developer of presently used rendering of BASIC http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html http://www.rtrussell.co.uk/, [email protected]. We also acknowledge with thanks the support of the European Commission within the EUKLEMS research project of the Sixth Framework Programme. Thanks are also due to the European University Institute, San Domenico di Fiesole/Firenze, for support provided by a Jean Monnet Research Fellowship. Special thanks are due to Angus Deaton, witness of this work from the beginning who wrote the Foreword; and to Michael Allingham, Robert Aumann,

xxii Acknowledgements Ali Dogramaci, Nuri Jazairi, Michael Kuczynski, Axel Leijonhufvud, Luigi Luini, Robin Marris, Edward Nell, Marshall Reinsdorf, Paul Sammelson, Herbert Scarf, Amartya Sen, Martin Shubik, Paul Streeten, Vela Velupillai and Giulio Zanella; and those at Routledge who had to do with the publication; and ﬁnally, missing but still in mind Memorial Acknowledgements.

Part I

Concept and Method

Chapter 1 The Super Price Index: Irving Fisher, and after 1 Introduction—or genesis 2 Fisher, and after 3 Data and formulae 4 Utility 5 The True Index Appendix Fisher’s “Superlative” Index Numbers

1 Introduction—or genesis Prices change and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. Reference may be made to this ﬁrst paragraph for the basis of the price index idea. There is of course no absolute reason for having a price index in the ﬁrst place. It is an institution, even if a well established, traditional, and perhaps even today still needed institution, affecting many aspects of economic life. The Price Index issued from the Statistical Ofﬁce is a number that tells everyone how to answer the mentioned question, the index being the multiplier of old expenditure to determine the new. They must change expenditure in proportion to the index to keep the old standard of living. All have submissively to accept the authority of the number produced no doubt on some basis quite likely beyond them. The question of how to produce such an extraordinary number is called The Index Number Problem. It stands out as a fascinating question that has appealed to and occupied a great number of economists. To proceed, there are primitive points to be added. Let Prs denote the price index from base period s to current period r. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. This point seems not to be explicitly represented among Irving Fisher’s famous price index formula “Tests”1 , but the next points are—though we are not now considering formulae but the basic idea of a price index. Related to this description of the practical idea of a price index, reference can be made now and subsequently to Appendix 1, where it is put in utility terms,

1 See Fisher (1922), Allen (1975), Section 1.8, or Afriat and Jazairi (1988).

The Super Price Index 5 and a start is made with development of the price index as a theoretical concept having this genesis. Attention is also drawn to Appendix 2 on Terminology and Appendix 3 on Notation. For the Identity Test, there is the statement Ptt = 1, that is, “when one year is compared with itself, the index shows ‘no change’.” Most formulae go along with this. For the next, if the price change is reversed, so the new prices becoming the old and vice-versa, then the price index, the ratio that turns old expenditure into new, is replaced by the reciprocal. That is, Pts = (Pst )−1 which is the Time Reversal Test. Fisher’s “ideal” index is just about the only formula that satisﬁes this. No wonder it is “ideal”. This thinking seems to be as if the price index was derived as a ratio of price levels, expressing purchasing power of money for obtaining a standard of living by purchase of consumption. That has in fact already been offered as a way to go with the ‘Index Number Problem’ by Afriat (1981), with what may be regarded as a ‘New Formula’ though it is not at all a formula of the type dealt with by Fisher and associates. Rather, with it, ﬁrst determine price levels, and then determine price indices as their ratios. That is, instead of the price index being given directly by some algebraical formula in terms of demand data for the reference periods themselves, among those produced seemingly endlessly and as it were out of nothing (Irving Fisher, joined by others, deals with one or two hundred of them). True, there are many prices each with its own separate level, so it may seem fair to ask what sense can there be to their having a single level when they are taken together. None the less there is a somewhat restricted but nevertheless outstanding sense for it. But ﬁrst there will be account of the early maker of index numbers, Irving Fisher, who knew well about the primitive price index idea but apparently not exactly this sense. For a distinction and the language for it: price level has reference to a single period, while price index has reference to two, and is in principle the ratio of new level to old, so it is the multiplier of old expenditure to produce the new that will currently purchase the same living standard. The second primitive principle mentioned, expressed by Fisher’s Time Reversal Test, would also be an immediate consequence of taking price indices having the form of ratios of price levels. When dealing with more than just two periods, beside the Time Reversal (the Fisher Index is a distinguished case among formulae for satisfying this) there can be introduction of the Chain Test, Prs Pst = Prt

6 Concept and Method (just about never satisﬁed by any of the one or two hundred usual price index formulae) which implies Time Reversal again, and moreover implies, and obviously is implied by, price indices being expressible as the ratios of a set of numbers associated with the periods—the ‘price levels’. For, bringing in the Identity Test, Ptt = 1 we have Pts Pst = Ptt = 1 so Pts = (Pst )−1 which is Time Reversal, and now, for any ﬁxed r, Pst = Psr Prt = Psr (Ptr )−1 = Psr /Ptr so price indices determined relative to a ﬁxed base can serve as ‘price levels’ from which all price indices can be determined as their ratios. Evidently now the Chain Test, from ﬁrst implying Reversal, is equivalent to Fisher’s Circularity Test, Prs Pst Ptr = 1. While there has been invariably no prior determination of price levels from which to obtain price indices as their ratios, the 1981 formula excepted, usually formulae (a great number) are proposed that go directly to the index without a background of levels. In that culture the great headache is to know what formula—as may be good, or true, or … or what? A missing test, not named before and which implies all these others, and which can be called the Ratio Test, is simply that the price index be expressed as a ratio of a set of numbers, maybe price levels. Among formulae, as such, nowhere is that satisﬁed, unless the 1981 formula be allowed, or another, of Bishop William Fleetwood in 1706, mysteriously neglected, Pst = pt a/ps a, the inﬂation rate for a ﬁxed bundle of goods a, in this case over a gap of two or three centuries. “It could be wondered why this formula was not “ideal” for Fisher. Possibly it was too dull but if he was looking for a more interesting one which meets the proper qualiﬁcation, … ”2

2 The quotation is from Afriat (1977, 37–8) and continues “… it does not exist, as Eichhorn (1976) has shown.” Unfortunately our friend Wolfgang Eichhorn, and for that matter Afriat, had not known the 1981- formula.

The Super Price Index 7 This is a moment also to quote Sir Roy Allen (1978, p. 418) where he quotes one of us: “Formerly, it was as if an answer was proposed without ﬁrst having had a question, and it was wondered if it could be an answer to a question, any question at all. Even many answers to no question in particular were proposed—the names of their proposers are attached to some of them”. (Afriat 1977, p. 102) That has to do with the formula approach dominated by Fisher and, as if caught in a trap, still prevalent, for the truck with index formulae does not end with Fisher.

2 Fisher, and after Irving Fisher (1922, 244–8) ranked 134 index number formulas (according to one of us, 126, and 145 by the count of Yrjo Vartia) according to their numerical distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them into several classes, as it were in increasing order of merit. The ﬁrst twelve index numbers constituting the ﬁrst of these classes are labeled as “worthless”, to designate that they are the worst in his ranking. The other six classes are labeled as poor, fair, good, very good, excellent, and superlative (p. 244). Fisher (1922, 244–48) classiﬁed ten (or eleven to include the yardstick Fisher “ideal”) index number formulas in the “superlative” group because, in his numerical example, they performed very closely to formula 353, which is the “ideal” geometric mean of the Laspeyres and Paasche indexes, which was itself put at the top rank in this group. He claimed that all these formulas correspond to combinations of the Laspeyres and Paasche, including the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Reference can always be made to our Appendix for Fisher’s list, and an exhibition of his “superlative” and other formulas. In the numerical example worked by Fisher, the closest formula to 353 is 8053, which is the arithmetic average of 53 and 54, where 53 is Laspeyres index number (p. 471) and 54 is Paasche index number (p. 471). Fisher claimed that all the other “superlative” index numbers are combinations of 53 and 54. As remarked again later, we do not here regard Laspeyres and Paasche as formulae like the others that are offered as price index formulae, but in a category of their own. Of late there has been new venture towards the super, and beyond … Evidently, except the implicit Walsh index number and Fisher’s “ideal” itself, Diewert’s (1976) “superlative” index numbers are not superlative in Fisher’s (1922, 244–48) sense. The implicit Walsh index number, which is given by formula 1154 in Fisher’s list, corresponds to Diewert’s quadratic mean of order-1 index number, whereas Fisher’s “ideal” index number corresponds to Diewert’s quadratic mean

8 Concept and Method of order-2 index number. As set forth by Milana (2005); Diewert’s and Fisher’s indexes cannot understandably be deﬁned as, in some way, approximating the ACTUAL but UNKNOWN true index number up to the second order, in order to admit them as “superlative” in the language of Diewert (1976, 1978). With all that, incredible super-calculus aside, such a formula seems to be essentially no new departure but a determined adherence to the old formula world, only bringing to it an added unconstructive impenetrable burden of complexity and mystery. Though all points in the Paasche-Laspeyres interval are true, all equally, none more true than another, as proved by one of us, here are absurd pretentions about a not just true but a distinguished “Superlative Index Number” as if some important discovery had been made that had escaped everyone, even Irving Fisher, then thickened the quagmire with perambulations about functional form which has no welfare signiﬁcance at all. After the casting around for the super coming from Irving Fisher, there is entry into an entirely new kind of territory beckoning exploration. A question is whether here is a breakthrough to a new, highest quality formula, as the language would suggest—here and there acclaimed as such—or a neo-Fisherian casting around without new guidance. In order not to get lost in this enquiry, a start should be made at the beginning. One of us had a dedicated immersion in the entire super literature while the other, saved by reductio ad absurdum, had never before our encounter ever put eyes on a single item of the later phase.

3 Data and formulae Reference is made to the budget and commodity spaces B and C, spaces of nonnegative row vectors and of column vectors. With the non-negative numbers, B = n , C = n , with p ∈ B, x ∈ C we have M = px ∈ as the money cost of the bundle of goods x at the prices p. With such a purchase, making the demand (p, x) ∈ B × C of commodities x at the prices p, the budget vector is u = M −1 p ∈ B, for which ux = 1. There is the revealed preference of x over every bundle y which, being such that uy ≤ 1, is also attainable at no greater cost. A fundamental area of discussion involves data consisting of a pair of demand observations (pt , xt ) ∈ B × C (t = 0, 1), which is associated with the Laspeyres index

P st = ps xt /pt xt and Paasche index

P st = ps xs /pt xs = P −1 ts . There is also the Fisher index 1 P¯ st = P st P st 2

The Super Price Index 9 which is the geometric mean of both, lying between them. An important condition on the demand data is the Laspeyres-Paasche inequality (LP)

P st ≤ P st ,

equivalently

P st P ts ≥ 1 or again ps xs pt xt ≤ ps xt pt xs . Or with Lst = ps xt /pt xt , again the Laspeyres index, this condition is equivalent to the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt

for price levels Pt (t = 0, 1). Put this way, the usual Laspeyres-Paasche inequality condition for a pair of demands with t = 0,1 is immediately generalized for any number t = 0, 1, 2, . . . , m as in the 1981-formula.3 The system L with s, t = 0,1 requires

P 01 ≤ P1 /P0 ≤ P 01 so its consistency, deﬁning L-consistency, or the existence of a solution, is equivalent to the LP-inequality. Here it is convenient to insert, though we have no immediate concern there, that the counterpart for more than two demands is the cyclical Laspeyres product test Lti Lij · · · Lkt ≥ 1 for every cycle t, i, j, . . . , k, t or brieﬂy Lt...t ≥ 1, which is a strengthening of the revealed preference test for utility construction that serves for conical utility. Introducing the chain Laspeyres and Paasche indices

P sij...kt = P si P ij · · · P kt ,

P sij...kt = P si P ij · · · P kt ,

3 It is interesting that a simple step like this should open the door to a fundamentally new approach. The start was at least a quarter-century ago, but the signiﬁcance of it was recognized more recently.

10 Concept and Method this test is equivalent to (chain-LP)

P s...t ≤ P s...t

for all possible chains … taken separately. Hence introducing the new Laspeyres and Paasche indices

st = minij...k P si P ij · · · P kt ,

st = maxij...k P si P ij · · · P kt ,

this is now equivalent to

(new-LP) st ≤ st . The chain-LP and new-LP provide a narrowing of the basic LP bounds for a single index. This has been exploited for application quite apart from the main use of the method for constructing a complete set of consistent true indices over any number of periods. The matter of bounds there has complications where every commitment to the location of one index has implications for the location of others. After chain-LP, the approach to new-LP appears prevented by the computation involved, though this is already resolved in our method. This extended approach is dealt with more fully in the papers Afriat (1981), (1982), and Afriat and Milana (2007). But we should not now get sidetracked and should return to the basic case of two demand observations having touched brieﬂy on this beginning of the extension. However ﬁrst let it be pointed out that nowhere so far has there been any dependence on utility, let alone conical or constant returns utulity, so there can be no accusation that some such assumption has been made. But then it should be mentioned again that the cyclical condition Lt...t ≥ 1 that turned up is exactly the condition on the data for the constructibility of a conical utility. Surely some use of that opportunity can be expected. Again Appendix 1 should be consulted though there has already been the interest to quote part of it here. Going back to the monumental ﬁrst paragraph, one obvious way of maintaining, at new prices, the old standard of living obtained from the old consumption x0 at a money cost p0 x0 (hypothetically the minimum cost for the utility of it) is to simply buy the old consumption with new cost p1 x0 . So certainly the current cost of the old standard is a most that amount, and its comparison with the old cost, which might at ﬁrst thought serve as a price index, is at most p1 x0 /p0 x0 , which is the ever-pervasive Laspeyres index. But here common sense breaks in with the proposal that one would not necessarily buy the old bundle at the new prices, one could buy instead another bundle that provides at least the old standard perhaps at a lesser minimum cost, making a comparison ratio with the old cost, the price index P10 , not exceeding the Laspeyres index, P10 ≤ p1 x0 /p0 x0 ,

The Super Price Index 11 and by the same principle P01 ≤ p0 x1 /p1 x1 . But in recollection of the second primitive point which asserts P01 = (P10 )−1 this second is equivalent to P10 ≥ p1 x1 /p0 x1 so common sense together with the primitive reversal principle provides both Laspeyres and Paasche, and shows them not only as tied together with the same source in principles, but as upper and lower bounds of the price index, if there is one. In a way, they are not price index formulae like all the others, but fundamental and irrefutable limits for the price index. In that case, of course, the lower bound could not exceed the upper bound. J. R. Hicks (without proving anything) calls that “The Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is confusing, but perhaps Hicks was just being fashionable. But still, a case where Paasche turned out to be greater than Laspeyres could be occasion for a pause. Hicks declares the conclusion that Laspeyres “tends to be greater than” Paasche. Before that is an argument starting with indifference between x0 and x1 implying, by denial of revealed preference, p0 x0 ≤ p0 x1

and p1 x1 ≤ p1 x0

from which it follows that p0 x1 p1 x0 ≥ p0 x0 p1 x1 , equivalent to assertion that Laspeyres is greater than Paasche.When was that freed of indifference of the x’s? In fact never, because the PL-relation is not invariable but just a test, true or false, for the constructibility of a constant returns utility. But when was it proved (before today) that P01 ≤ L01 ? It might have been good sense (corresponding to practice of the practical) to abandon the entire theoretical subject after arrival at this point, where the unavoidable primitive has been joined with common sense. But uncountable formulae for the index, good, true, better, super, &c have been proposed. To go further, as it seems we must, more form needs to be given to ideas we have so far.

12 Concept and Method

4 Utility There are two main things about a consumption bundle x ∈ C. The simple part is that it has a money cost M = px ∈ when the prices are p ∈ B. The other part is that it is the basis—and having it is the objective, use, use-value, or utility— for obtaining a standard of living, with the sacriﬁce of cost. Hence there is a link between cost and standard of living, where prices enter. For this link a gap remains between consumption and its utility, made good hypothetically by introduction of the utility function. A utility function is any numerical-valued function φ deﬁned on the commodity space B, φ:B→ so φ (x) ∈ (x ∈ B) is the utility level of any commodity bundle x. A utility function φ determines a utility order R ⊂ C × C where xRy ≡ φ (x) ≥ φ (y) A utility function φ, with order R, ﬁts a demand element (p, x), with budget vector u, or the demand is governed by the utility, if the revealed preferences of it belong to the utility order, uy ≤ 1 ⇒ xRy (y ∈ C) . In other words, if x has at least the utility level of every bundle y attainable at no greater expenditure with the prices, or x provides the maximum utility φ (x) for all those bundles y under the budget constraint uy ≤ 1, that is py ≤ px ⇒ φ (x) ≥ φ (y) . The utility system is hypothetical and admitted to the extent that it ﬁts available demand observations. The cost of a standard of living is determined as the minimum cost at prevailing prices of getting a consumption that provides it. In terms of a utility function φ, this is gathered from the utility cost function ρ (p, x) = min {py : φ (y) ≥ φ (x)} which tells the minimum cost at given prices p of obtaining a consumption y that has at least the utility of a given consumption x. Since x itself, with cost px, is a possible such y, necessarily ρ (p, x) ≤ px

for all p, x

while ρ (p, x) = px signiﬁes the admissibility, under government by the utility system, of the demand of x at the prices p. It shows the demand is cost effective, getting the maximum of

The Super Price Index 13 utility available for the cost, and cost efﬁcient, getting at minimum cost the utility obtained, which conditions would here be equivalent. A case where admissibility does not hold could be attributed to consumption error, described as failure of efﬁciency, where ρ (p, x) ≥ epx,

0≤e≤1

would show attainment of cost efﬁciency to a level e. This idea has some use in dealing with demand data inconsistent with government by a utility by ﬁtting it to a utility that serves only approximately, as in Afriat (1973). For the service of a price index this utility cost should have the property ρ (pr , x) /ρ (ps , x) is independent of x, which represents the special condition on the utility by which it has the price index property and has this ratio as an associated price index. An equivalent condition is that utility cost be factorize into a product ρ (p, x) = θ (p) φ (x) , of price level P = θ (p) depending on p alone and quantity level X = φ (x) depending on x alone. As proved in Appendix 1: Theorem 1 Utility cost factorization is necessary and sufﬁcient for a utility to have the price index property, This factorization is immediately assured if φ is conical, but also the converse is true, showing the following4 also proved in Appendix 1: Theorem 2 Utility cost factorization it is necessary and sufﬁcient for the utility to be conical. In consequence we have: Theorem 3 For a price index to be based on a utility it is necessary and sufﬁcient that the utility be conical

4 A ray is a half-line with vertex the origin and a cone is a set described by a set of rays. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xl) = φ (x) l. With demand governed by conical utility, the expansion paths, or loci of demand when expenditure varies while prices remain ﬁxed, are rays.

14 Concept and Method This theorem from long ago5 may be a good candidate for the title “The Index Number Theorem” secured by Hicks for another purpose as already noted. The question now is: what utility? A price index being wanted, by the theorem it must be conical, and with given demand data (pt , xt ) ∈ B × C (t = 0, 1, . . .) , and adoption of the efﬁciency criterion, any utility to be entertained would, to ﬁt the data, have to be such that Pt Xt = pt xt (t = 0, 1, . . .) , where Pt = θ (pt ) , Xt = φ (xt ) , so in any case Ps Xt ≤ ps xt and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 0, 1). A question is whether a solution exists. If one does, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from Pt Xt = pt xt . Thus, introduce

φ i (x) = Pi−1 pi x and

φ (x) = min φ i (x) i

so this is a concave conical polyhedral utility function that ﬁts the demand data, with associated price indices as required, to make those prices indices true.

5 Samuelson and Swamy (1974) p. 570 attribute theorem and proof to Afriat (1972) though it comes from the 1950s.

The Super Price Index 15 Another such function, concave conical, which ﬁts the demand data, again with required values and the same associated price indices, is the polytope type function given by

φ (x) = max

Xi ti :

i

6 xi ti ≤ x, ti > 0

i

and if φ is any other concave conical utility that ﬁts the demands and takes the values Xi at the points xi then

φ (x) ≤ φ (x) ≤ φ (x) for all x. Included in the above is the simple conical precursor of the general theorem on utility construction put in service speciﬁcally for price index theory.

5 The True Index Every conical or constant returnsutility has associated with it a price index, derived from the utility cost factorization applicable to such a function. A price index is termed true if it is connected with such a utility that ﬁts the demand data. Every solution for price levels determines true price indices given by their ratios, the existence of a solution requiring the cyclical Laspeyres product test, that requires the cyclical Laspeyres products to be all at least 1. It should be seen what all this has to say in reduction to the classical case of just two periods. The existence of a solution for price levels implies the LP-inequality, and then any point in the LP-interval is expressible as a price index, obtained as the ratio of the price levels, which is true from being associated with a conical utility that ﬁts the data. Hence, as values for the price index, all points in the LP-interval are true—all equally, no one more true than another.

6 The function of this form introduced by Afriat (1971) is the constant-returns ‘frontier production function’ that gives a function representation, and at the same time a computational algorithm, for the production efﬁciency measurement method of Farrell (1957) (Afriat’s colleague at the Department of Applied Economics (DAE), Cambridge, whose work, from after he had left, he at ﬁrst missed) that marks the beginning of ‘data envelope analysis’ (DEA). The Afriat comment attached to Finn R.Førsund, and Nikias Sarafoglou (2005) gives a report. While Afriat is usually given credit for ﬁrst introduction of the ‘non-parametric’ approach, here now is opportunity to give credit to Farrell who made such an introduction for this case as it were implicitly even if not explicitly in the way here exhibited. The same type of function without constant-returns, worth knowing about, is used for the utility construction in Afriat (1961) but arbitrarily, or for simplicity, left aside in the account of (1964) where instead a polyhedral type function is used. It also served for the 1971 extension of Farrell’s method.

16 Concept and Method When this was mentioned a few decades ago, possibly at the Helsinki Meeting of the Econometric Society, August 1976, it was received with complete disbelief.7 Here is a formula to add to Fisher’s collection, a bit different from the others: PRICE INDEX FORMULA: Any point in the LP-interval, if any. This paper draws attention to the generalization of this Formula for any number of periods, the 1981-Formula, where price and quantity levels for the periods are all computed simultaneously, associated with a conical utility that ﬁts the demand data for all the periods. The computations are by solution of a system of inequalities, so with a tolerance like in the above Formula, subject to a consistency condition that is a strengthening of ordinary revealed preference consistency. In the absence of consistency, an approach is made by an approximation method. This amounts to a complete, elegant and practical treatment of the ‘Index Number Problem’ to which Afriat’s attention was brought by Alan Brown the day he arrived at the Department of Applied Economics (DAE), Cambridge, in 1953, and concludes the assignment. In the 1982 paper on “The True Index”, where there are elaborations about algorithms for consistency test and price level computations, there is mention about normalizing price levels so they sum to 1 and are therefore represented by a point in the simplex of reference. Then the set of all price level solutions, based on given data, is represented by a region in this simplex. In the case of three periods, when the reference simplex is simply a triangle, this opens the way to illuminating graphics, as presented in the paper. In the case of two periods the simplex of reference is a line segment and price level solutions describe a subsegment corresponding to the PL-interval range of the price index given by ratios of price levels. Irving Fisher favoured one particular formula out of the many (one or two hundred) he considered. This is his “ideal index” given by the geometric mean of the Laspeyres and Paasche indices. This favouritism is understandable because the formula is impressive and just about alone in meeting the Time Reversal requirement, one of his “tests” basic to the very idea of a price index. And moreover it is comfortably well into the interior of the interval bounded by the limiting Laspeyres and Paasche indices. Here is a proposal for picking one true point out of many, as may be described by the LP interval and can apply just as well when the segment becomes a polyhedron in many dimensions, the point should be: true with minimum distance to non-true a maximum This is one path to the Fisher index and generalizations. With our approach it is necessary to deal directly with price levels instead of price indices which are their ratios.

7 Proof in Afriat (1977) 129–30.

The Super Price Index 17 S. S. Byushgens (1925) submitted that if it could be taken that demand was governed by a homogeneous quadratic utility then the value of the associated price index would be given by Fisher’s “ideal” formula. This result, which seemed at ﬁrst, with initial optimism, to make the Fisher index “true”, must have seemed spectacular. Possibly it is here that the idea of a true index made its ﬁrst entry. Of course now we have the Fisher index as true simply from belonging to the LP-interval, with countless ﬁtting utilities, along with all the other points and no more true than any of them—even though lately there has been a cult, on the part of some, to load function form with a signiﬁcance it cannot possibly have. However, with Byushgens, the Fisher index could be distinguished from the other points as being, not more true than the others—anyway impossible—but simply quadratically true—maybe. And maybe not! For if no quadratic utility can ﬁt the data, Byushgens’ Theorem, while still undoubtedly true, and still pretty, would be vacuous. This is a danger about which quite likely there had not been awareness at ﬁrst. In Afriat (1972), (1977), (1982) and (2004)8 there are accounts of issues to do with Fisher and Byushgens. These include an attempt to save Byushgens from the vacuity hazard by a weakening of his proposal, replacing quadratic by locally quadratic, rendering it with less interest except that it has been taken up by Diewert for doctrine about a formula to which—echoing Fisher with his impressive terminology—he gives the name superlative, that in ordinary language must promise at least something good, while what that good should be is not known at all. Though our proposal for outright abandonment of all price index formulae would of course include Diewert’s promotion, it can still have attention for notable features while celebrated as the Swan Song of the Irving Fisher Price Index Formula era. A notable feature is the citation style, that can be encountered with some frequency, where an item is connected with a string of individuals that includes the originator with others not deserving or wanting credit. This lumping together of originator with readers in a quixotic generous distribution of credit can be joined at the same time with a scatter of compensatory withholdings. But after all, we have been told: If we should ever encounter a case where a theory is named for the correct man, it will be noted. George J. Stigler The Theory of Price (3rd edition), 1966, p. 77 Stigler may have marked, if not just ignorance, a prevalence of ritual citations that amount to nothing more than greetings to friends. Such a phenomenon is consistent with “The Tale of Two Research Communities” and the tendency observed there to “stick to ‘own camp’ references” (good if you happen to have a camp) discovered

8 The 2004 volume contains earlier accounts beside an additional one.

18 Concept and Method by those early pioneers Finn R. Førsund and Nikias Sarafoglou (2005) in the ﬁeld of Citation Analysis, a new ﬁeld for which the way has been opened by advent of the computer whose emergence Stigler might have been happy about. We have been sidetracked, delivering the message to abandon useless formulae and enter a new path, to temper the abruptness and bring in the ﬂavour of history.

Appendix: Fisher’s “Superlative” Index Numbers Irving Fisher (1922, 244–48) ranked 134 index number formulas according to their numerical distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them “arbitrarily into several classes in increasing order of merit. The ﬁrst twelve index numbers, constituting the ﬁrst of these classes, are labeled, rather harshly perhaps, as ‘worthless’ index numbers to designate the fact that they are the worst. The other six classes are labeled as poor, fair, good, very good, excellent, and superlative” (p. 244). He then, classiﬁed ten index number formulas in the superlative group because, in his numerical example, they performed very closely to the “ideal” geometric mean of the Laspeyres and Paasche indexes. He claimed that most of them are the various combinations of these two index number formulas (p. 248). They include the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Using Fisher’s notation, let pti and qit represent, respectively, the price 1and p q1 th quantity of the i commodity during the observed period t and V ≡ i i0 i0 . i pi q i Fisher’s “superlative” index numbers are presented here in the original inverse order of remoteness from the “ideal” index (353) (the last one is included in the list at the ﬁrst ranking position with all formulas being identiﬁed with the original Fisher’s code, and the page of The Making of Index Numbers where they are deﬁned is indicated in parenthesis9 ): (11)

5323 ≡

√ 323 × 325

(geometric mean of two Törnqvist-based index

numbers) (p. 486), √ (10) 1323 ≡ 1123 × 1124 (p. 484), 0 1 1/2 1 (q q ) p (9) 1153 ≡ i i0 i1 1/2 i0 (Walsh index number) (p. 483), pi i (qi qi ) √ (8) 1353 ≡ 1153 × 1154 (p. 484),

9 See Fisher (1922, Appendix V) for the full list of the 134 formulas.

(7)

0 1 1/2 1 (p p ) q 1154 ≡ V i i0 i1 1/2 i0 qi i (pi pi )

The Super Price Index 19 (implicit Walsh index number) (p. 483),

1 0 i pi q i p0 + p1 p0 + p1 1 + p0 q0 i i 1 i i 0 qi qi = (6) 2154 ≡ V i 0i i1 i i 2 2 p q 1 + i i1 i1 i pi q i (p. 484 and p. 488), √ (5) 2353 ≡ 2153 × 2154 (p. 484), 0 qi0 + qi1 qi0 + qi1 (q + qi1 )p1i 1 0 (4) 2153 = pi pi = i i0 1 0 i i 2 2 i (qi + qi )pi (p. 484 and p. 488), p0 q 0 p0 q 1 (3) 8054 ≡ 2 (p. 487), i 1i i0 + i i1 i1 i pi qi i pi q i 1 0 1 1 53 + 54 p q p q (2) 8053 ≡ 2 (487), = i i0 i0 + i i0 i1 2 p q i i i i pi q i 1 0 1 1 √ p q i pi q i (1) 353 ≡ 53 × 54 = 0 0 · i i0 i1 (Fisher “ideal” index number) i pi q i i pi q i (p. 482) 0 1 1 1 p q p q =V i i0 i0 · i i1 i0 , i pi q i i pi q i with √

p1i 1123 ≡ i p0 i

p0i qi0 p1i qi1 /

0 0 1 1 j pj q j pj q j

(p. 483)

√ q1 p0i qi0 p1i qi1 / j p0j qj0 p1j qj1 i 1124 ≡ V (p. 483) i q0 i √ √ √ 4 323 ≡ 23 × 24 × 29 × 30 = 123 × 124 = 223 × 229 √ √ √ 4 325 ≡ 25 × 26 × 27 × 28 = 125 × 126 = 225 × 227 1 p1i 2 √ 123 ≡ 23 × 29 = i p0 i (p. 473),

p0 q0 p1 q1 i 0i 0 + i 1i 1 j pj qj j pj qj

(p. 480), (p. 480),

(Törnqvist index number)

20 Concept and Method q1 12 √ i 124 ≡ 24 × 30 = V i q0 i number) (p. 473), √ 125 ≡ 25 × 27 (p. 473), √ 126 ≡ 26 × 28 (p. 473), √ 223 ≡ 23 × 24 (p. 477), √ 225 ≡ 25 × 26 (p. 477), √ 227 ≡ 27 × 28 (p. 477), √ 229 ≡ 29 × 30 (p. 477),

p0 q0 p1 q1 i 0i 0 + i 1i 1 j pj qj j pj qj

(Implicit Törnqvist index

0 0 0 0 p1i pi qi / j pj qj 23 ≡ (p. 468), i p0 i q1 p0i qi0 / j p0j qj0 i 24 ≡ V (p. 468), i q0 i 0 1 0 1 p1i pi qi / j pj qj (p. 468), 25 ≡ i p0 i q1 p1i qi0 / j p1j qj0 i 26 ≡ V (p. 468), i q0 i 1 0 1 0 p1i pi qi / j pj qj 27 ≡ (p. 468), i p0 i q1 p0i qi1 / j p0j qj1 i (p. 468), 28 ≡ V i q0 i 1 1 1 1 p1i pi qi / j pj qj 29 ≡ (p. 468), i p0 i q1 p1i qi1 / j p1j qj1 i (p. 468), 30 ≡ V i q0 i 1 0 p1 q 1 i pi q i 53 ≡ 0 0 (Laspeyres index number) (p. 471) = V i i1 i0 , i pi q i i pi q i 1 1 0 1 p q p q 54 ≡ i i0 i1 (Paasche index number) (p. 471) = V i i0 i0 . i pi q i i pi q i

The Super Price Index 21 Moreover, formulas numbered 103, 104, 105, 106, 153, 154, 203, 205, 217, 219, 253, 259, 303, 305 reduce to 353. In the numerical example performed by Fisher, the closest formula to 353 is 8053, which is the arithmetic average of 53 (Laspeyres) and 54 (Paasche) index numbers. Fisher claimed that most of “superlative” index numbers are various combinations of 53 and 54. These index numbers should be contrasted with the quadratic mean-of-order-r index numbers by Diewert (1976, pp. 130–131) and called “superlative” by him using Fisher’s (1922, p. 247) terminology (but in a different sense). These are encompassed by the following general formula: 0 1 0 r/2 1r s (p /p ) PD ≡ i i1 i0 i1 r/2 i si (pi /pi )

with r = 0,

pt q t where sit ≡ i ti t for t = 0, 1 and with PD reducing to the Fisher “ideal” index j pj q j with r = 2, to the implicit Walsh index (Fisher’s formula 1154) with r = 1, and to the Törnqvist index number (Fisher’s formula 123) at the limit as r → 0. It can be shown (Diewert, 1976) that, if the shares sit are those derivable exactly from the true utility cost function, then this may have the following quadratic mean-of-order-r functional form: ct (p) = form

i

ln ct (p) = at0 +

t r/2 j αij (pi pj )

t i ai ln pi

1/r

+

with r = 0, which tends to the translog functional

i

t j aij ln pi ln pj

as r → 0.

Following Milana (2005) and generalizing Caves et al. (1982), it can be shown that 0 1 l 1 1 1−l c (p ) c (p ) the equality PD = 0 0 is obtained with some real value of l, c (p ) c1 (p0 ) which can be not necessarily within the range between 0 and 1 and is a function not only of technology but also of prices. Caves et. al. (1982) have considered the case of a translog functional form where the parameters of the second order terms are constant, that is atij = a¯ ij . In this case, l = 1/2. However, in view of the above-mentioned results, Diewert functional forms for superlative index numbers are not necessarily the geometric average of the two ‘true’ indexes. Alternatively, it is possible to deﬁne a quadratic mean-of-order-r quantity index number 0 1 0 r/2 1r s (q /q ) QD ≡ i i1 i0 i1 r/2 with r = 0, which reduces to the Fisher “ideal” i si (qi /qi ) quantity index with r = 2, to the implicit Walsh quantity index with r = 1, and to the Törnqvist quantity index number at the limit as r → 0. If the shares sit are those derivable exactly from the true utility function, then this index number is exact for a utility-based distance function δ t (q) (Deaton (1979b), Deaton and

22 Concept and Method Muellbauer (1980, pp. 53–57)) in the sense that it is a weighted (geometric) average of two Malmquist (1953) index numbers deﬁned as distance function 0 1 l 1 1 1−l δ (q ) δ (q ) ratios, that is QD = 0 0 for some real value of l, where δ (q ) δ 1 (q0 ) 1/r t r/2 δ t (q) = with r = 0, which tends to the translog functional i j βij (qi qj ) form ln δ t (q) = bt0 + i bti ln qi + i j btij ln qi ln qj as r → 0. (δ t (q) is a scalar by which the original quantities q have to be multiplied in order to reach the ﬁnal utility level given the preferences indexed with t). The weight l can be not necessarily between 0 and 1 and is a function not only of technology but also of quantities. In particular, l = 1/2 when δ 0 and δ 1 are quadratic functions with the same parameters in the second order terms. Therefore, here again Diewert functional forms for superlative index numbers are not necessarily the geometric average of two ‘true’ indexes. Moreover, it is tempting to derive the implicit quadratic mean-of-order-r price index as the ratio P˜ D ≡ V /QD , but as pointed out by Samuelson and Swamy (1974, p. 570) echoing Afriat’s (1972) factorization theorem, the homotheticity of the underlying economic function is a necessity as well as a sufﬁciency for the existence (and invariance) of such an index. If this is not the case, the obtained implicit price index is spurious and may not even respect the required degree-one homogeneity property. It is immediate to see also that, except the implicit Walsh index number and Fisher’s “ideal” itself, Diewert’s “superlative” index numbers are not “superlative” in Fisher’s (1922, 244–48) sense10 . The implicit Walsh index number, which is

10 The Törnqvist index, for example, (corresponding to formula 123 in Fisher, 1922, p. 473) is seen as the most superlative by Caves, Christensen, and Diewert (1982, p. 41) and Diewert (2004, p. 450) whereas it was not deemed “superlative” by Fisher (1922, p. 247), who classiﬁed it, in a descending order of merit, below the classes of “excellent” and “superlative” index numbers, with the last group ranked at the top position. Note from CM to SA: This is not the only critical remark that we may make. The Törnqvist index number is exact for a particular quadratic function (the translog) just as the Fisher “ideal” is exact for a simple quadratic function. It does not matter that the Törnqvist index is exact for a translog function that (as shown by Caves et al., 1982) is not necessarily homogeneous (Milana, 2005 has shown that also the other index numbers that are “superlative” in Diewert’s sense are exact for speciﬁc quadratic functions that are not necessarily homogeneous). The main point is that it is not true that: (1) these index numbers are second-order approximations to the “true” index and to each other, (2) this “true” index does not exist at all in the non-homothetic (or non-homogeneous) case (see your LP-inequality theorem) and, therefore, the Caves et al. theorem is vacuous implying that the obtained index number is, in more familiar terms, “path dependent”. Under these circumstances, the Caves, et al. results fall under the criticism raised by your earlier contributions and our present paper. Note from SA to CM: Very interesting for experts like yourself and some others. But this paper, as you are aware, proposes the abandonment of all price-index formulae and whatever anyone may have had to say about any of them, and offers an entirely different approach to “The Index Number Problem”.

The Super Price Index 23 given by formula 1154 in Fisher’s list, corresponds to Diewert’s quadratic mean of order-1 index number, whereas Fisher’s “ideal” index number corresponds to Diewert’s quadratic mean of order-2 index number. For the reasons given in Milana (2005), however, all Diewert’s and Fisher’s indexes cannot be deﬁned as approximating the unknown true index number (if this exists) up to the second order and cannot be considered to be “superlative” also in Diewert’s (1976)(1978) sense.

Chapter 2 The Price Level Computation Method 1 2 3 4 5 6 7 8 9 10 11 12 13

Introduction Data and formulae Minimal chains System and derived system Triangle inequality Extension and exhaustion property Consistency Utility model for the method Solution structure Basic solutions Inconsistency and approximation Old and new: an illustration Conclusion

1 Introduction Prices change Š and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. Reference may be made to this ﬁrst paragraph for the basis of the price index idea. Appendix 1 formulates the idea as a theoretical concept based on utility. The Price Index issued from the Statistical Ofﬁce is a number that tells how to deal with this cost-of-living question, the index being the multiplier of old expenditure to determine the new. The question of how such a number should be produced is known as The Index Number Problem. To proceed about it there are primitive points to be added. Let Pst denote the price index from period s to period t. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. This point seems not to be explicitly represented among Irving Fisher’s well known “Tests”, but the next points are, though we are not now considering applications to actual formulae, as usual, but rather to the basic idea of a price index itself. For the Identity Test, there is the statement Ptt = 1, that is, “when one year is compared with itself, the index shows ‘no change’.” Most formulae go along with this. For the next, if the price change is reversed, so the new prices becoming the old and vice-versa, then the price index, the ratio that turns old expenditure into new, is replaced by the reciprocal. That is, Pts = (Pst )−1 which is the Time Reversal Test. Fisher’s “ideal index” is just about the only formula that satisﬁes this. No wonder it is “ideal.”

The Price Level 27 This thinking seems to be as if the price index was derived as a ratio of price levels, expressing purchasing power of money for obtaining a standard of living by purchase of consumption. For a distinction and the language for it: price level has reference to a single period, while price index has reference to two, and is in principle the ratio of new level to old, so it is the multiplier of old expenditure to produce the new that will currently purchase the same living standard. The second primitive point mentioned, expressed by Fisher’s Time Reversal Test, would also be an immediate consequence of taking price indices having the form of ratios of price levels, or anyway of some numbers. For if Pst = Ps /Pt then Pts = Pt /Ps = (Ps /Pt )−1 = (Pst )−1 . When dealing with more than just two periods, beside the Time Reversal (the Fisher “Ideal Index” is a distinguished case among formulae for satisfying this) there can be introduction of the Chain Test, Prs Pst = Prt (just about never satisﬁed by any of the one or two hundred usual price index formulae) which implies Time Reversal again, and moreover implies, and obviously is implied by, price indices being expressible as the ratios of a set of numbers associated with the periods—the ‘price levels’ or whatever. For, from the Chain Test directly, Prs = Prt /Pst or, bringing in the Identity Test, Ptt = 1 we have Pts Pst = Ptt = 1 so Pts = (Pst )−1 which is Time Reversal, and now, for any ﬁxed r, Pst = Psr Prt = Psr (Ptr )−1 = Psr /Ptr so price indices determined relative to a ﬁxed base can serve as “price levels” from which all price indices can be determined as their ratios.

28 Concept and Method Evidently now the Chain Test, from ﬁrst implying Reversal, is equivalent to Fisher’s Circularity Test, Prs Pst Ptr = 1. While there has been invariably no prior determination of price levels from which to obtain price indices as their ratios, usually formulae, plain algebraic involving demand data just for the reference periods themselves, and a great number of them, are proposed that go directly to the index without a background of levels. In that approach the great problem is to know what formula to use. A missing test, in Fisher’s list, perhaps not before named and which implies all these others, and which could be called the Ratio Test, though it is equivalent, as just seen, to the Chain Test, is simply that the price index be expressed as a ratio of a set of numbers. Among formulae, as such, nowhere is that satisﬁed, unless the now to be considered method, the 1981 formula, be allowed, or another proposed by Bishop William Fleetwood in 1707 and mysteriouly neglected, at least in usual theory if not actual practice, Pts = pt a/ps a, the inﬂation rate for a ﬁxed, perhaps democratically chosen, bundle of goods a.

2 Data and formulae1 Reference is made to two spaces, the budget space B and commodity space C, one the space of non-negative row vectors and the other column vectors, so with as the non-negative numbers, B = n ,

C = n ,

and any p ∈ B, x ∈ C provide M = px ∈ as the money cost of the bundle of goods x at the prices p. With such a purchase, making the demand element (p, x) ∈ B × C of commodities x at the prices p, the associated budget vector is u = M −1 p ∈ B, for which ux = 1. (We follow the rule that a scalar, as if it were a 1 × 1-matrix, multiplies a row-vector on the left and a column-vector on the right.) Any collection of demand elements makes a demand correspondence. A budget element is any (u, x) ∈ B × C such that ux = 1, and an expenditure correspondence consists in any collection of these. With any demand correspondence D there is an associated expenditure correspondence E, obtained by taking the associated budget elements. A fundamental area of discussion involves data provided by a ﬁnite demand correspondence D consisting of a series of demand observations (pt , xt ) ∈ B × C (t = 1, 2, . . . , m) ,

1 It is useful here to make reference to Appendix 3 on Notation.

The Price Level 29 as may be associated with different periods described by the index t. Price levels Pt to be associated with the periods are elements of a vector P in the price level space = m . Without altering the price indices determined from their ratios, they may be normalized to sum to 1, in which case they become barycentric coordinates for a point in the simplex of reference , available for graphic representations in case m = 3. Any pair of periods s, t is associated with the Laspeyres index Lts = pt xs /ps xs with s distinguished as the base and t the current period, so this is simply the inﬂation rate between the periods for the base period bundle of goods. There is also the Paasche index Kts = pt xt /ps xt = (Lst )−1 , which is the inﬂation rate for the current bundle. With any chain described by a series of periods s, i, j, . . . , k, t there is associated the Laspeyes chain product Lsij...kt = Lsi Lij . . . Lkt termed the coefﬁcient on the chain. Obviously Lr...s...t = Lr...s Ls...t A simple chain is one without repeated elements, or loops. There are m (m − 1) · · · (m − r + 1) = m!/r! simple chains of length r ≤ m and therefore altogether the ﬁnite number m! (1 + 1/1! + 1/2! + · · · + 1/ (m − 1)!) of simple chains from among m elements. A chain s, i, j, . . . , k, t whose extremities are the same, that is, s = t, deﬁnes a cycle. It is associated with the Laspeyres cyclical product Ltij...kt = Lti Lij . . . Lkt

30 Concept and Method which is basis for the important Laspeyres cyclical product test, or simply the cycle test, Lt...t ≥ 1 for all cycles t . . . t A simple cycle is one without loops. There are (m − 1) . . . (m − r + 1) = (m − 1)!/r! simple cycles of r ≤ m elements, and the total number of simple cycles from among m elements is the ﬁnite number made up accordingly. The coefﬁcients Lst Lts = Lsts on the cycles of two elements deﬁne the intervals of the system. The interval test Lst Lts ≥ 1 is equivalent to (LP)

Kts ≤ Lts

that is, the Paasche index does not exceed the Laspeyres, or the LP-inequality, a condition very well known from index number theory based on data for just two periods. Here therefore, with the cyclical test, is a generalization of that condition for any number of periods. J. R. Hicks (without proving anything) calls the LPinequality “The Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is confusing, but perhaps Hicks was just being fashionable. Another way of stating this condition, of signiﬁcance since it gives the form for a statement of a direct extension for many periods, is that the 2 × 2 L-matrix

1 Lst Lts 1 be idempotent, or reproduced when multiplied by itself, in the modiﬁed arithmetic where + means min. In fact, as to be shown, raising the general m × m L-matrix to powers in this modiﬁed arithmetic is a basic process in the price level computation method. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

the cycle test Ls...t...s ≥ 1 is equivalently to (chain LP) Ks...t ≤ Ls...t for all possible chains … and … taken separately. Hence introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

The Price Level 31 subject to the now to be considered conditions required for their existence, for which Hst = (Mts )−1 , this is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst showing the relation of the LP-interval and the narrower derived version that involves more data. Here it has been recognized that from Kts = (Lst )−1 follows −1 Kt...s = Ls...t and therefore −1 Hts = max... Kt...s = min... Ls...t = (Mst )−1 , where in each case … is understood so the chain s. . .t is the reverse of t. . .s.

3 Minimal chains Any chain can be represented uniquely as a simple chain, with loops at certain of its elements, given by cycles through those elements; and the coefﬁcient on it is then expressed as the product of coefﬁcients on the simple chain and on the cycles. Also, any cycle can be represented uniquely as a simple cycle, looping in simple cycles at certain of its elements, which loop in cycles at certain of their elements, and so forth, with termination in simple cycles. The coefﬁcient on the cycle is then expressed as a product of coefﬁcients on simple cycles. Thus out of these generating elements of simple chains and cycles, ﬁnite in number, is formed the inﬁnite set of all possible chains. Theorem 3.1 For the chains with ﬁxed extremities to have a minimum the cycle test is necessary and sufﬁcient.

32 Concept and Method If any cycle should be less than 1, then by taking chains which loop repeatedly round that cycle, chains which have decreasing coefﬁcients are obtained without limit; and so no minimum exists. However, should every cycle be at least 1, then by cancelling the loops on any chain, there can be no increase in the coefﬁcient, so no chain coefﬁcient will be smaller than the coefﬁcient for some simple chain. But there is only a ﬁnite number of simple chains on a ﬁnite number of elements, and the coefﬁcients on these have a minimum. Theorem 3.2 For the cycle test the simple cycle test is necessary and sufﬁcient. For the coefﬁcient on any cycle can be expressed as a product of coefﬁcients on simple cycles. Theorem 3.3 The cycle test implies that a minimal chain with given extremities exists and can be chosen simple. For then any chain is then not less than the chain obtained from it by cancelling loops, since the cancelling is then division by a product of numbers all at least 1.

4 System and derived system The computation of price levels Pt (t = 1, . . . , m) depends on solution of the system of inequalities (L)

Lst ≥ Ps /Pt .

Subject to the cyclical product test Lt...t ≥ 1 for every cycle, or equivalently every simple cycle, by Theorem 3.2, it is, by Theorem 3.3, possible to introduce Mst = minij...k Lsi Lij · · · Lkt , attained for a simple chain. Then Lsij...kt ≥ Mst for every chain and, by Theorem 3.3, the equality is attained for some simple chain. In particular, Lst ≥ Mst . The number Mtt is the minimum coefﬁcient for the cycles through t, so that Ltij...kt ≥ Mtt

The Price Level 33 for every cycle, the equality being attained for some simple cycle. In particular, for a cycle of two elements, Lts Lst ≥ Mtt . The cyclical product test that is the hypothesis now has the statement Mtt ≥ 1. With the numbers Mst so constructed, subject to this hypothesis, it is possible to consider with system L also the derived system (M )

Mst ≥ Ps /Pt .

The two systems are said to be equivalent if any solution of one is also a solution of the other. Theorem 4.1 The system L and its derived system M, when this exists, are equivalent. Let system L have a solution Pt . Then, for any chain of elements s, i, j, . . . , k, t there are the relations Lsi ≥ Ps /Pi ,

Lij ≥ Pi /Pj , . . . , Lkt ≥ Pk /Pt ,

from which, by multiplication, there follows the relation Lsij...kt ≥ Ps /Pt . This implies that the derived coefﬁcients Mst exist, and Mst ≥ Ps /Pt . That is, Pt is a solution of system M . Now suppose the derived coefﬁcients for system M are deﬁned, in which case Lst ≥ Mst . and let Pt be any solution of system M , so that Mst ≥ Ps /Pt . Then it follows immediately that Lst ≥ Ps /Pt . or that Pt is a solution of system L. Thus Land M have the same solutions, and are equivalent.

34 Concept and Method Theorem 4.2 If the cycle test holds for L then the interval test holds for the derived system M. Since Mst is the coefﬁcient of some chain with extremities s, t it appears that the interval coefﬁcient Mts Mst of M is the coefﬁcient of some cycle of L through t, and therefore if the cycle test holds for L then so does the interval test hold for the derived system M . Given any solution for system L, and equivalently system M , necessarily Kst ≤ Hst ≤ Ps /Pt ≤ Mst ≤ Lst , showing how price indices, which on the basis of data just for the reference period are conﬁned to the ordinary Laspeyres-Paasche interval, become conﬁned to the narrower derived Laspeyres-Paasche interval when based on the more extended data.

5 Triangle inequality2 From the relation Lr...s Ls...t = Lr...t it follows that the derived coefﬁcients satisfy the multiplicative triangle inequality Mrs Mst ≥ Mrt the one side being the minimum for chains connecting r, t restricted to include s, and the other side being the minimum without this restriction. Theorem 5.1 Any system subject to the cycle test is equivalent to a system which satisﬁes the triangle inequality given by its derived system. This is true in view of Theorems 3.1, 4.1 and 4.2. Theorem 5.2 The interval test holds for any system that satisﬁes the triangle inequality. Thus, from the triangle inequalities applied to any system M , Mtr Mrs ≥ Mts ,

Mts Msr ≥ Mtr

2 It is most useful here to refer to Appendix 3 on Notation, which also contains an elaboration of terminology applicable to the “canonical solutions” introduced in this section.

The Price Level 35 there follows, by multiplication, the relation Mrs Msr ≥ 1 or what is the same Hst ≤ Mst or that the derived LP-interval be non-empty. Theorem 5.3 If a system satisﬁes the triangle inequality then its derived system exists and moreover the two systems are identical. From the triangle inequality, it follows by induction that Msi Mij . . . Mkt ≥ Mst that is Msij...kt ≥ Mst from which it appears that the derived system N exists, with coefﬁcients Nst ≥ Mst so that now Nst = Mst This shows, what is otherwise evident, that no new system is obtained by repeating the operation of derivation, since the ﬁrst derived system satisﬁes the triangle inequality. Theorem 5.4 For any system the triangle inequality is equivalent to idempotence of the matrix in the arithmetic where + means min That is, the matrix is reproduced in multiplication by itself. For, simply, Nij = mink Nik Nkj if and only if Nij ≤ Nik Nkj . The triangle inequality Mrs Mst ≥ Mrt

36 Concept and Method has the restatement Mrs ≥ Mrt /Mst from which it appears that, for any ﬁxed t, taken as base, a solution of the system (M )

Mrs ≥ Pr /Ps

for price levels Pr is given by Pr = Mrt . Similarly, another solution is Pr = 1/Mtr . These solutions may be distinguished as determinations for the ﬁrst and second basic price level systems, with node t as base. Since, by Theorem 5.2, Mtr Mrt ≥ Mtt ≥ 1 they always have the relation 1/Mtr ≤ Mrt , which is the derived LP-relation. However, these are not now price indices, as in that original relation, but here they are price levels from which to derive price indices. Finding these solutions depends directly on the triangle inequality that is characteristic of the derived sysyem (M ), and not on the solution extension property that is a consquence, to which there is appeal in the construction method dealt with in the next Section. Now established, for every t, are two price level solutions Pr from which to derive systems of true price indices Prs = Pr /Ps . The two systems, of basic price indices with base t, are in a way counterparts of the Laspeyres and Paasche endpoints of the PL-interval that describes the range of true price indices for the classical case that involves just two periods. The determinations have reference to periods associated with the data without any dependence on the order 1, . . ., m in which they are taken. This is unlike where there is dependence on the solution extension property for ﬁnding solutions, of the next Section. However, they do depend on which period, corresponding to t in the given order, is taken as base. Coming in pairs there are now 2m determinations, whose pairwise connections and base references are essential.

The Price Level 37 When price level solutions are normalized so as to provide barycentric coordinates for a point in the simplex of reference, the set of all solutions is a convex polydron for which these 2m solutions are a complete set of vertices from which, it is conjectured, all solutions may be obtained by taking convex combinations of them. Note that the ﬁndings of this section apply just a well to the approximation method accounted in Section 10, based on relaxing exact cost efﬁciency, for the ﬁt of utility to demands, to some degree of partial efﬁciency. From the above the following is proved.

Theorem 5.5 The derived system (M), when it exists, admits the solutions given by the basic price levels, so it is always consistent.

Corollary 5.1 In that case also the original system (L) is consistent, and admits those same solutions.

For the system and derived system, when this exists, are equivalent, admitting the same solutions, by Theorem 3.1.

Corollary 5.2

The cycle test is necessary and sufﬁcient for consistency

For, by Theorem 3.1, the test applied to system (L) is necessary and sufﬁcient for the existence of the derived system (M ), which is always consistent when it exists, by the present Theorem, and by Theorem 4.2 it is equivalent to system (L), which therefore also is consistent.

6 Extension and exhaustion property A subsystem Mh of order h ≤ m of a system M of order m is deﬁned by (Mh )

Mst ≥ Ps /Pt

(s, t = 1, . . . , h) .

Then the systems Mh (h = 2, . . . , m) form a nested sequence of subsystems of system M , each being a subsystem of its successor, and Mm = M . Any solution of a system reduces to a solution of any subsystem. But it is not generally true that any solution of a subsystem can be extended to a solution of the original. However, should this be the case, then the system will be said to have the extension property.

38 Concept and Method Theorem 6.1 Any system which satisﬁes the triangle inequality has the extension property. Let P1 , P2 , . . . , Ph−1 be a solution of Mh−1 , so that

Mh−1

Mst ≥ Ps /Pt

(s, t = 1, . . . , h − 1) .

It will be shown that, under the hypothesis of the triangle inequality, it can be extended by an element Ph to a solution of Mh . Thus, there is to be found a number Ph such that Mhs ≥ Ph /Ps ,

Mth ≥ Pt /Ph

(s, t = 1, . . . , h − 1)

that is Mhs Ps ≥ Ph ≥ Pt /Mth So the condition that such a Ph can be found is Mhq Pq ≥ Pp /Mph where Pp /Mph = max Pi /Mih , i

Pq Mhq = min Pj Mhj . j

But if p = q this is equivalent to Mph Mhp ≥ 1 which is veriﬁed by Theorem 5.2, and if p = q it is equivalent to Mph Mhq ≥ Pp /Pq which is veriﬁed since by hypothesis Mph Mhq ≥ Mpq ,

Mpq ≥ Pp /Pq .

Therefore, under the hypothesis, the considered extension is always possible. It follows now by induction that any solution of Mh (h < m) can be extended to a solution of Mm = M . This theorem shows how solutions of any system can be practically constructed, step-by-step, by extending the solutions of subsystems of its derived system. Theorem 6.2 Any system which satisﬁes the triangle inequality is consistent.

The Price Level 39 For, by Theorem 5.2, M12 M21 ≥ 1; and this implies that the system M2 has a solution, which, by Theorem 6.1, can be extended to a solution of M . Therefore M has a solution, and is consistent. However, this result has already been obtained in Theorem 5.4 without appeal to the extension property, but by direct appeal to the triangle inequality instead of to this consequence. It also appears that, subject to the triangle inequality, any solution of M2 , since it can be extended to a solution of M , is a reduction of some solution of M , in other words, reductions of solutions of M exhaust the solutions of M2 , or the system has the exhaustion property. Theorem 6.3 Any system which satisﬁes the triangle inequality has the exhaustion property.

7 Consistency Theorem 7.1 The cyclical product test is necessary and sufﬁcient for consistency of L, and either Lm = M , in the modiﬁed algebra where + means min, is the equivalent derived system with the solution extension property, or system L is inconsistent. If system L is consistent, let Pt be a solution. Then, for any cycle t, i, j, . . . , k, t there are the relations Lti ≥ Pt /Pi ,

Lij ≥ Pi /Pj , . . . , Lkt ≥ Pk /Pt ,

from which it follows, by multiplication, that Ltij...kt = Lti Lij . . . Lkt ≥ (Pt /Pi ) Pi /Pj . . . (Pk /Pt ) =1 and hence Lt...t ≥ 1. Therefore, if L is consistent, all its cycles are at least 1 and the cyclical product test holds. Conversely, let this test be assumed for L. Then the derived system M is deﬁned, satisﬁes the triangle inequality, and has the interval test. Hence, by Theorem 6.3, M is consistent. But, by Theorem 4.1, M is equivalent to L. Therefore, L is consistent. This shows the converse, so the Theorem is proved. Now let L denote the actual m × m-matrix of Laspeyres indices for the system, and Lr its r-th power in a modiﬁed arithmetic where + means min, so L1 = L,

Lr+1 = Lr L

(r = 1, 2, . . .) ,

40 Concept and Method making r Lijr+1 = mink Lik Lkj ,

where it is seen, since Ljj = 1 affecting the possibility k = j, that Lijr+1 ≤ Lijr , which shows what may be termed the monotonicity of the process. In any case, for any r and i, j r Lik = Lis...tk ,

for some chain s. . .t. Subject to the cyclical test, it is proposed that, for r ≤ m the chain is. . .tj is simple. For otherwise a loop with coefﬁcient at least 1, by hypothesis, can be cancelled, and we have an element from an earlier power which is less, violating the process monotonicity. Then the series of powers either terminates in one not later than the mth, when a simple chain cannot be extended further, that is therefore repeated by its successors, or does not terminate. In the ﬁrst case, L = L1 ≥ L2 ≥ . . . ≥ Lt = M = Lt+1 = . . .

(t ≤ m) ,

with ≥ as between elements, where the terminating matrix M is the matrix of the derived system for L. In the second case it is concluded the cyclical product test is violated, system L is inconsistent, and there is no derived system. This follows Afriat (1981), Section 13 on “The power algorithm”, involving matrix powers in a modiﬁed arithmetic where × means + and + means min. There are debts to Jack Edmunds (1973) and S. Bainbridge (1978), for the connection with minimum paths, elaborated in Afriat (1987) where there is also a BASIC computer program pp. 464 ff. applied to “Getting around Berkeley in minimum time”. Here is how it could go: 0 x = L, t = 1 1

y = x, x = yL, t = t + 1

2

if x = y then M = x end

3

if t = m then end else 1.

So it appears that either L is inconsistent, or Lm = M , for which, as is equivalent to the triangle inequality, there is the idempotence M 2 = M where M is reproduced in multiplication by itself, and which is equivalent to L and has the extension property, so individual price level solutions can be constructed step-by-step, starting with any point in any derived LP-interval, which is narrower, because of additional constraints associated with additional data, than the basic or classical LP-interval that involves data just for a pair of periods, the reference periods themselves.

The Price Level 41 Of course, having the basic price levels of Section 4 available as solutions, there is no need to appeal to the extension property for the existence of solutions. However, with that property it is possible to construct other solutions, step-by-step, beside by taking convex combinations of the basic solutions. With any solution for price levels Pt there is, from their ratios, an associated determination of price indices Pst = Ps /Pt , all true, together, by reference to the same utility, better than merely true separately by reference to different utilities, as in the sense of true usually entertained. Then Prs Pst = (Pr /Ps ) (Ps /Pt ) = Pr /Pt = Prt , so that Prs Pst = Prt , which is Fisher’s Chain Test, not satisﬁed by any of the one or two hundred formulae he dealt with, and so forth with other Tests. This is a point for the observation that such price indices, any one for a pair of periods involving data from all the periods, and together giving a realization of all the “Tests” Irving Fisher proposed as proper for price indices from their nature as such, make a sharp contrast with the established tradition of algebraical formulae involving data just for the reference periods themselves, without proper compliance with such basic “Tests”, or guidance about which of the one or two hundred proposed formulae to use, despite his rankings to decide some as better than others, even “superlative”. After the procedure for ﬁnding individual solutions, the further interest is in the collection of all solutions. The solutions describe a polyhedral convex cone in the price level space of dimension m, and the normalized solutions describe a bounded polyhedral convex region in the simplex of reference, with faces or vertices to be determined, the m simplex vertices being in correspondence with the m data periods, and price levels. Then there are approximation methods to serve for the case of inconsistency. But ﬁrst notice will be taken of the price-quantity symmetry inherent in the method, and the utility background that enables all the price indices so determined to be represented as altogether true, that is, all true simultaneously on the basis of the same utility. With any determination of price levels Pt , there is an associated determination of quantity levels Xt , where Pt Xt = pt xt

(t = 1, . . . , m) .

While for price levels, pt xs /ps xs ≥ Pt /Ps ,

42 Concept and Method for quantity levels, equivalently, pt xs /pt xt ≥ Xs /Xt , and one could just as well have solved for the quantity levels ﬁrst, by the same method as for price levels, and then determined the price levels from these. Whichever way, Ps Xt ≤ ps xt

(s, t = 1, . . . , m) ,

with equality for s = t. The introduction of cost efﬁciency up to a level e, where 0 ≤ e ≤ 1, would require Pt Xt ≥ ept xt

(t = 1, . . . , m) .

good also for any lower level, and highest level 1 imposing the equality.

8 Utility model for the method First some remarks about terminology (see Appendix 2). A ray is a half-line with vertex the origin, and every point lies on just one ray, the ray through it, so a = {at : t ∈ } ⊂ C is the ray through any a ∈ C. A cone is a set described by a set of rays, and every set has a conical closure, or cone through it, or projecting it, described by the set of rays through its points. Hence

= {xt : x ∈ A, t ∈ } ⊂ C A is the cone through any A ⊂ C. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xl) = φ (x) l. With a demand element (p, x) ∈ B × C, with expenditure M = px and budget vector u = M −1 p so that ux = 1, there is the revealed preference of x over every bundle y which, being such that uy ≤ 1, is also attainable at no greater cost, as described by the relation R ⊂ C × C given by R = {(x, y) : py ≤ px} = {(x, y) : uy ≤ 1} . Then there would be the transitive closure of a collection of such relations, and a revealed preference consistency Samuelson-Houthakker type condition which excludes conﬂicting preferences.

The Price Level 43 It may be remembered that originally py ≤ px, y = x

⇒

¯ xRy, yRx

going with belief that, in a choice, presumed a maximum and so revealing preferences, it must be more than a mere maximum but moreover a unique maximum—an extra that may be hard to “reveal”. Instead, in the way of revelation without the unsuitable insistence on uniqueness which does not in any way add to preferences, simply py ≤ px

⇒

xRy

has better standing. We take liberty to conﬁne the “revelation” language to this restricted use. For conical revealed preference there would be instead the conical closure of R. Then there would be the transitive closure of a collection of such relations, and a conical revealed preference consistency which excludes conﬂicting preferences. The Laspeyres cyclical product test is exactly such a condition (a part of the version of the utility construction theorem of Afriat (1961) and (1964) then for general utility construction and now instead for conical utility). There are two attributes for a consumption bundle x ∈ C. One is that it has a money cost M = px ∈ when the prices are p ∈ B. The other, its use-value or utility, is that it is the basis for obtaining a standard of living. Hence there is a link between cost and standard of living, where prices enter. For this link a gap remains between consumption and its utility, made good hypothetically by introduction of the utility function, or utility order. A utility function is any numerical valued function φ deﬁned on the commodity space B, φ:B→ so φ (x) ∈ (x ∈ B)is the utility level of any commodity bundle x. A utility function φ determines a utility order R ⊂ C × C where xRy ≡ φ (x) ≥ φ (y) A utility function φ, with order R, ﬁts a demand element (p, x), with budget vector u, or the demand is governed by the utility, if the revealed preferences of it belong to the utility order, uy ≤ 1 ⇒ xRy (y ∈ C) . In other words, if x has at least the utility level of every bundle y (we do not insist y = x, see remark above) attainable at no greater expenditure with the prices,

44 Concept and Method or x provides the maximum utility φ (x) for all those bundles y under the budget constraint uy ≤ 1, that is py ≤ px ⇒ φ (x) ≥ φ (y) . The utility system is hypothetical and admitted to the extent that it ﬁts available demand observations. The cost of a standard of living is determined as the minimum cost at prevailing prices of getting a consumption that provides it. In terms of a utility function φ, this is gathered from the utility cost function ρ (p, x) = min {py : φ (y) ≥ φ (x)} which tells the minimum cost at given prices p of obtaining a consumption y that has at least the utility of a given consumption x. Since x itself, with cost px, is a possible such y, necessarily ρ (p, x) ≤ px

for all p, x

while ρ (p, x) = px signiﬁes the admissibility, under government by the utility system, of the demand of x at the prices p. It shows the demand is cost effective, getting the maximum of utility available for the cost, and cost efﬁcient, getting at minimum cost the utility obtained, which conditions would here be equivalent. A case where admissibility does not hold could be attributed to consumption error, described as failure of efﬁciency, where ρ (p, x) ≥ epx,

0≤e≤1

would show attainment of cost efﬁciency to a level e. This idea has use in dealing with demand data inconsistent with government by a utility, by ﬁtting it to a utility that serves only approximately, as reported below, after the account of Afriat (1973). For the service of a price index this utility cost should factorize into a product ρ (p, x) = θ (p) φ (x) , of price level P = θ (p) depending on p alone and quantity level X = φ (x) depending on x alone. This immediately is assured if φ is conical, but also the converse is true, showing the following, which we are going to prove, if it was not already, probably long ago. (Samuelson and Swamy 1974, p. 570, attribute theorem and proof to Afriat 1972).

The Price Level 45 Theorem (Utility Cost Factorization) For factorization of the utility cost function it is necessary and sufﬁcient that the utility be conical. Given φ conical, ρ (p, x) = min {py : φ (y) ≥ φ (x)} = min py (φ (x))−1 : φ y (φ (x))−1 ≥ 1 φ (x) = θ (p) φ (x) where θ (p) = min {pz : φ (z) ≥ 1} That shows the sufﬁciency. Since, for all p, θ (p) φ (x) ≤ px for all x with equality for some x, as assured with continuous φ, it follows that θ (p) = minx px (φ (x))−1 showing θ to be concave conical semi-increasing. Also for x demandable at some prices, as would be the case for any x if φ is concave, the inequality holds for all p with equality for some p, showing φ (x) = minp (θ (p))−1 px which, in case every x is demandable at some prices, requires φ to be concave conical semi-increasing. But even when not all x are demandable, because they lie in caves and are without a supporting hyperplane, here is a conical function deﬁned for all x that is effectively the same as the actual φ as far as any observable demand behaviour is concerned. So it appears that for the cost function factorization the utility function being conical is also necessary, beside being sufﬁcient, as already remarked. Hence, with some details taken for granted, the Theorem is proved. A pair of functions connected by θ (p) = minx px (φ (x))−1 φ (x) = minp (θ (p))−1 px deﬁne a conjugate pair of price and quantity functions, such that θ (p) φ (x) ≤ px

46 Concept and Method for all p, x and θ (p) φ (x) = px signiﬁes efﬁciency of the demand (p, x), of x at prices p, obtaining maximum utility for the cost and minimum cost for the utility. Instead, θ (p) φ (x) ≥ epx, where 0 ≤ e ≤ 1, will signify cost efﬁciency to a level e, as will serve for development of a utility approximation method applicable in case of inconsistency. The question now is: what utility? A price index being wanted, by the factorization theorem it must be conical, and with given demand data (pt , xt ) ∈ B × C

(t = 1, . . . , m) ,

and belief in efﬁciency, any utility to be entertained would, to ﬁt the data, have to be such that Pt Xt = pt xt , where Pt = θ (pt ) ,

Xt = φ (xt ) .

so in any case Ps Xt ≤ ps xt and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 0, 1). A question is whether a solution exists. If a solution of the system of inequalities (L) does exist, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from the efﬁciency condition Pt Xt = pt xt . A worthwhile observation is that these values Xt = φ (xt ) of the underlying utility φ are determined without ever having to actually construct the utility.

The Price Level 47 If a solution does not exist, and so there is no conical utility that ﬁts the demand data subject to the requirement of strict cost efﬁcincy, that is Pt Xt = et pt xt . where et = 1, this requirement can be relaxed to partial efﬁciency bounded to be at least a certain lower level, so pt xt ≥ Pt Xt = et pt xt . for some et ≤ 1. From this with Ps Xt ≤ ps xt we have Ps /Pt ≤ ps xt /et pt xt so, to replace system (L), we have the efﬁciency modiﬁcation Lst /et ≥ Ps /Pt , or with all et = e we have the system (L/e)

Lst /e ≥ Ps /Pt .

While with solutions of system (L) we deal with utilities that ﬁt the demand data exactly, with solutions of the efﬁciency modiﬁcation they ﬁt approximately, and the closer the et ≤ 1 to 1 the closer the ﬁt. A special case is where we take the et all equal to some e ≤ 1, as with system (L/e), and detemine the critical cost efﬁciency e∗ which is such that this system is then consistent if and only if e ≤ e∗ . With any solution of system (L), introduce

φ (x) = mini Pi−1 pi x so this is a concave conical polyhedral utility function that ﬁts the demand data, with associated price indices as required, to make those prices indices true. Another such function, concave conical, which ﬁts the demand data, again with required values and the same associated price indices, is the polytope type function given by 3 φ (x) = max Xi ti : xi ti ≤ x, ti > 0 i

i

and if φ is any other concave conical utility that ﬁts the demands and takes the values Xi at the points xi then

φ (x) ≤ φ (x) ≤ φ (x) for all x. 3 The function of this form introduced by Afriat (1971) is the constant-returns “frontier production function” that gives a function representation, and at the same time a computational algorithm, for the

48 Concept and Method Included in the above is the simple conical precursor of the general theorem on utility construction put in service speciﬁcally for price index theory. Thus, the concave polyhedral function

φ (x) = mini pi x/Pi = max t : t ≤ pi x/Pi and the concave polytope function θ (p) = min px : pi x ≥ Pi for all i = max vi P i : vi pi ≤ p by LP duality

i

i

are a conjugate pair of quantity and price functions such that

φ (xt ) = Xt

θ (pt ) = Pt ,

where, with ast = ps xt /pt xt ,

bst = pt xs /pt xt

P’s and X ’s connected by Pt Xt = pt xt are, equivalently, such that ast ≥ Ps /Pt ,

bst ≥ Xs /Xt .

For another such conjugate pair, instead, φ (x) = max wi Xi : wi x i ≤ x i

i

θ (p) = mini pxi /Xi .

production efﬁciency measurement method of Farrell (1957) (Afriat’s colleague at DAE Cambridge whose work, done after he left, he at ﬁrst missed) that marks the beginning of “data envelope analysis” (DEA). The comment by Afriat attached to Finn R.Førsund and Nikias Sarafoglou (2005) gives a report. The same type of function but without constant-returns is used for the utility construction in Afriat (1961) but arbitrarily—or for simplicity!, or for the reasons in remarks already made here about over-stringent “revealed preference”—left aside in the account of (1964), where a modiﬁed revealed preference condition to avoid the excess of the original and a polyhedral type function are used instead, as again in accounts such as Fostel et al. (2003). It also served for the 1971 extension of Farrell’s method by an accidental transfer of ideas from demand analysis.

The Price Level 49 These pairs of conjugate functions are such that

θ (p) ≥ θ (p) ,

φ (x) ≤ φ (x) ,

and any other pair for which θ (pt ) = Pt ,

φ (xt ) = Xt

are such that

θ (p) ≥ θ (p) ≥ θ (p) ,

φ (x) ≤ φ (x) ≤ φ (x) .

9 Solution structure The price levels are determined as solutions of the system (M ) Mst ≥ Ps /Pt , derived from and equivalent to the system L, subject to the Laspeyres cyclical product test required for consistency. For a restatement of the inequalities affecting Pt , (Mst )

Mst Pt ≥ Ps ,

and equivalently (Kts )

Pt ≥ Kts Ps .

Any positive solution Pr of system M deﬁnes a permissible system of price levels, represented by a point P in the price level space = m of dimension equal to the number of periods m. The set C of solutions is immediately a polyhedral convex cone in this space. When price levels are normalised to have sum 1 they describe a simplex in the space . This simplex is cut by the cone C in a bounded convex polyhedron, or polytope, D. The cone C is recoverable from its section D, as the cone through that section projecting it from the origin. Taking price levels to be normalised and so represented by points in the simplex is convenient for computation, and for geometrical representation. Only ratios of price levels are signiﬁcant and these are unaltered by normalisation. Every point in the normalised solution set D of the system M is a convex combination of a ﬁnite set of basic solutions, and so the computational problem requires ﬁnding just these. Given any solution Pr we form the matrix of price indices Pst = Ps /Pt , depending only on the price level ratios.

50 Concept and Method Z (

r

S Mrs

Figure 1

Now there will be explorations for a geometrical and diagrammatic understanding of the system M . Dealing with any three periods r, s, t is illustrative of essential features. While the associated solution cone Crst may be hard to visualise, the normalised solution polytope Drst in the simplex rst is much easier, and can be represented graphically. We can refer to any constraint of the system M by the two periods involved, so, as already above, let (Mrs ) denote the general constraint. There has already been some discussion of the case with two periods, in dealing with the P-L interval. Vectors of price levels for any subset of periods r, s, . . ., understood as representing only the ratios, can be denoted Pr:s:... = (Pr : Ps : . . .) . Any period r corresponds to the vertex of the simplex where Pr = 1, and vertices can all be labelled by the corresponding periods. Any point on the edge rs of the simplex corresponds to a ratio Pr : Ps , that is, Pr:s in the notation just introduced. Similarly any point in a simplex face rst speciﬁes the ratios Pr:s:t and so forth for any dimension. The constraint (Mrs ) cuts the edge rs in a point Z and requires Pr:s to lie in the segment Zs, where (rZ : Zs) = (1 : Mrs ) = (Ps : Pr ) Without ambiguity, we can refer to the segment Zs on the edge rs as the segment Mrs , as in Figure 1. At the same time, the constraint (Mrs ) requires Pr:s:t to lie in the simplex Zst, and so forth to any dimension. Considering now a pair of constraints (Mrs ) and (Msr ), we have two segments Mrs and Msr on the edge rs, and they have a nonempty intersection Drs shown in Figure 2. This lies within the Paasche-Laspeyres interval, and is a generalisation of that for when data from other periods are involved. It is generally narrower because any effect of extra data must be to reduce indeterminacy. Msr r

(

Drs

) Mrs

Figure 2

S

The Price Level 51 S

Mrs

Mst

r

Mrt

Y

t

Figure 3

Now consider three constraints associated with the triangle inequality as shown in Figure 3. Two of them produce intervals Mrs and Msr on rs and st and, as it were with the triangle equality instead, jointly produce the interval Yt on rt,. The triangle inequality requires Mrt to be a subinterval of this. If instead of Mrt we take Mtr (see Figure 4) cyclically related to the other two, the resulting joint constraint determines a triangle lying within rst. The other three cyclically related constraints, associated with the opposite cyclic order, determine S

Mrs

Mst

r

Figure 4

Mtr

t

52 Concept and Method S

Drs

Dst

Drst

Drt

r

t

Figure 5

another triangle, so conﬁgured with the ﬁrst that their intersection is a hexagon, Drst , as in Figure 5, by the triangle inequality assured non-empty. It is seen in this ﬁgure that Drs is exactly the projection of Drst from t on to rs. In other words, as Pr:s:t describes Drst , Pr:s describes Drs . Or again, for any point in Drs , there exists a point in Drst that extends it, in the sense of giving the same ratios concerning r and s. That is the extension property described earlier, a consequence of the triangle inequality, and it continues into higher dimensions indeﬁnitely: Drs...t is the projection of Drs...tv from the vertex v of the simplex rs. . .tv onto the opposite face rs. . .t That shows how price levels for the periods can be determined sequentially, one further one at a time. Having found any that satisfy the constraints that concern only them, they can be joined by another so that is true again. Starting with two periods and continuing in this way, ﬁnally a system of price levels will have been found for all the periods. For when the data for a price index between two periods involves data also from other periods, and moreover indices for any subset of periods are to be constructed consistently, these D-polytopes constitute a twofold generalisation of the PaascheLaspeyres range of indeterminacy of a price index between two periods taken alone. For a comment on the triangle inequality and equality, along with Z on rs where (rZ : Zs) = (Ps : Pr ),

The Price Level 53 now introduce X on st where (sX : Xt) = (Pt : Ps ). Let rX and tZ meet in P. Then sP meets tr in Y where (tY : Yr) = (Pr : Pt ). So it appears that by choosing the points Z and X for ratios z and x, we arrive at point Y for a ratio y where y = zx. In other words, we have here a geometricalmechanical multiplication machine, also good for division since from Y and Z for y and z we can arrive at P and so determine X and x for which y = zx, that is, x = y/z.

10 Basic solutions Taking price levels to be normalised and so represented by points in the simplex is convenient for computation, as for geometrical representation, when that is possible. Only the ratios of price levels are signiﬁcant and these are unaltered by normalisation. The normalised solution set of the system M is a convex polyhedron D in the simplex , every point of which is a convex combination of a ﬁnite set of basic solutions, or vertices. The computational problem requires ﬁnding just these. The cases with two periods, or three and four, can serve for a start. Every conical utility has associated with it a price index, derived from the utility cost factorization applicable to such a function. A price index is termed true if it is connected with a conical utility that ﬁts the demand data. Every solution for price levels determines true price indices given by their ratios, the existence of a solution requiring the cyclical Laspeyres product test, that requires the cyclical Laspeyres products to be all at least 1. It should be seen what all this has to say in reduction to the classical case of just two periods. In this case the existence of a solution for price levels is equivalent to the LPinequality, and then any point in the LP-interval is representable as a price index, obtained as the ratio of the price levels, which is a true price index from being associated with a conical utility that ﬁts the data. Hence, as values for the price index, all points in the LP-interval are true—all equally, no one more than another (this should dim the aura of extra truth given to Fisher’s Ideal Index, especially after it became connected with a—possibly nonexistant—quadratic utility). When this was submitted a few decades ago, possibly at the Helsinki Meeting of the Econometric Society, August 1976, it was received with complete disbelief (a proof is in Afriat (1977), 129–30). Here is a formula to add to Fisher’s collection, a bit different from the others: PRICE INDEX FORMULA: Any point in the LP-interval, if any. However, now we deal rather with price levels and should put this formula in such terms. Now the simplex is a line segment, so with two vertices. Each point

54 Concept and Method of the segment corresponds to a ratio of price levels in a solution, and so to a price index. A segment in it, corresponding exactly to the PL-interval, is the normalized price level solution set, with vertices for L and P. These are the basic solutions from which all other solutions are determined. There is not much more that can be said about this case, except that it is a generalization of it that makes the present subject. The case of three periods is already more complex and substantially more interesting, and evocative of the shape of things to come. Already a start was made with that in the last section. Having the picture there obtained, of the hexagonal boundary of the normalized solution set, the immediate task is to obtain formulae for the six vertices. The treatment for system (L) consists mainly in the power algorithm for testing consistency and forming the derived system (M ), equivalent to (L), with the triangle inequality and solution extension property that enables solutions to be constructed step-by-step, starting with two variables and following a path for adding variables, to conclude with an individual solution. At each stage the choice to be made can keep the solution as a vertex of the current solution set, so ﬁnally there will be arrival at a vertex, making a basic solution. To construct a complete basic solution set this way could be laborious. Firstly the path for adding variables has m! possibilities, and with any one path there is a choice between two possibilites at every extension stage. It seems, therefore, there may be about m! × 2m−1 basic solutions, if any, or fewer distinct ones to allow coincidences, with the symbolic description (t1 t2 − v2 , t3 − v3 , . . . , tm − vm ) where vi = 1 or 2. For this discussion, the extension path will simply be 1, . . . , m in that order, though we may not get very far along it. For P1 and P2 referring to periods 1 and 2 (reference denoted 12) there are two basic (non-normalized) solutions (12 − a)

P1 = 1,

(12 − b)

P1 = M12 ,

P2 = M21 . P2 = 1.

Were we dealing with system (L) these would correspond to the L and P bounds of the LP-interval. For (12-a) there is the veriﬁcation M21 ≥ P2 /P1 = M21 M12 ≥ P1 /P2 = (M21 )−1 the second line providing conﬁrmation because M12 M21 ≥ 1. For (12-b) similarly. One of these solutions has to be chosen initially, say (12-a). This can be extended to include a third variable, for period 3, relying on the triangle inequality and the solution extension property that follows from it. Consider (12 − a, 3 − a)

P1 = 1,

P2 = M21 ,

P3 = M31 .

The Price Level 55 This is a solution that extends the solution (12-a), as may be veriﬁed with appeal to M13 M31 ≥ 1, and appeals to the triangle inequality, M32 M21 ≥ M31 and M23 M31 ≥ M21 . Similarly (12 − a, 3 − b)

P1 = 1,

P2 = M21 ,

P3 = 1/M13

is another solution that extends (12-a). If we identify s, t, r of the last section with 1, 2, 3 in this, we have (12-a,3-a), when normalized, corresponds to the lower of the middle pair of vertices of the hexagon, associated with simplex vertex 1, just as (12-a,3-b) is the upper of the pair. Or something like that. Similarly there are pairs of solution vertices similarly associated with the other two simplex vertices 2 and 3. That makes the six vertices of the hexagon. Consider (12 − a, 3 − a, 4 − a)

P1 = 1,

P2 = M21 ,

P3 = M31 ,

P4 = M41 .

This is a solution that extends (12-a, 3-a). And so forth. There may be more to say but for now it may be suitable to submit going further with this approach to the brute computer. However, there is reassurance to be gained from the circumstance that we already have the basic solutions, of Section 5, obtained without tedious stepby-step extension but immediate and complete from a reference to the triangle inequality. None the less there is interest in the determination of all basic solutions, or vertices of the convex polyhedron in the simplex of reference that describes all normalized solutions, illustrated graphically for the case m = 3 in Section 8. The 2m solutions provided by pairs of basic solutions in respect to the m possible bases should be the vertices of the convex polyhedron of all price level solutions normalized to make them points in the simplex of reference. For instance in Section 9 we have 2 × 3 = 6 vertices of the hexagonal region. This would be, once again, as with the basic price levels themselves, a providential ready made solution for what might otherwise have seemed a burdensome abstruse computation.

11 Inconsistency and approximation A demand correspondence being deﬁned as a correspondence between budget constraints and admitted commodity bundles, here the concern is with a ﬁnite correspondence. The approach to constructing a utility that ﬁts such data is most familiar, and now there has been account of the matter where the utility is restricted to be conical, as suits treatment of price indices. When the demand data does not have the consistency required for exact admission of a utility, there arises the question of how to admit a utility approximately. Here the impossibility of exactness is treated as due to error, represented as a failure of efﬁciency.

56 Concept and Method A theorem will be proved on the existence of a positive solution for a certain system of homogeneous linear inequalities. Such a system can be associated with any ﬁnite demand correspondence, together with a number e between 0 and 1 interpreted as a level of cost efﬁciency. The existence of a solution is equivalent to the admissibility of the hypothesis that the consumer, whose behavior is represented by the correspondence, (i) has a deﬁnite structure of wants, represented by an order in the commodity space, as is essential in dealing with price indices, and (ii) programs at a level of cost efﬁciency e. Any solution permits the immediate construction of a utility function which realizes the hypothesis. When e = 1 the utility function ﬁts the data exactly, in the usual sense that its maximum under any budget constraint is at the corresponding commodity point, and when e < 1 it can be considered to ﬁt it approximately, to an extent indicated by e. A determination is required for the critical cost efﬁciency, deﬁned as the upper limit of possible e. Demand analysis which ordinarily knows nothing of approximation and also treats not just a maximum but a strict maximum under the budget constraint, as expressed by the original “revealed preference” idea, is put in perspective with this approach. A utility relation is any order in the commodity space n , that is any R ⊂ n ×n which is reﬂexive and transitive, xRx,

xRyR . . . Rz ⇒ xRz

A utility function is any φ : n → . It represents a utility relation R if xRy ⇔ φ (x) ≥ φ (y) . Such representation for R implies it is complete, xRy ∨ yRx. Consider a utility relation R and a demand element (p, x) with px > 0. A relation between them is deﬁned by the condition (H ∗ )

¯ py ≤ px, y = x ⇒ xRy, yRx

which is to say x is strictly preferred to every other y which costs no more at the prices p. If R is represented by a utility function this condition is equivalent to (H ∗ )

py ≤ px, y = x ⇒ φ (x) > φ (y)

With u = M −1 p where M = px, an equivalent statement, in terms of the associated budget element (u, x), is (H ∗ )

¯ uy ≤ 1, y = x ⇒ xRy, yRx.

The Price Level 57 This can be called the relation of strict compatibility between a utility relation, or function, and a demand, or its associated budget. A demand correspondence being a set D of demand elements, the condition HD∗ (R) of strict compatibility of R with D is deﬁned by simultaneous compatibility of R with all the elements of D. The existence of an order R such that this holds deﬁnes the strict consistency of D. The original “revealed preference” theory deals with this condition. Now let further relations between a utility relation R and an demand correspondence D be deﬁned by HD (R) ≡ xDp, py ≤ px ⇒ xRy HD (R) ≡ xDu, yRx ⇒ py ≥ px with conjunction HD (R) ≡ HD (R) ∧ HD (R) by which R and D can be said to be compatible. Thus H signiﬁes that x is as good as any y which costs no more at the prices p, or that maximum utility is obtained for the cost, and H signiﬁes any y which is as good as x costs as much, or that the utility has been obtained at minimum cost. In the language of cost-beneﬁt analysis, these are conditions of cost efﬁciency and cost efﬁcacy. Evidently HD∗ (R) ⇒ HD (R) that is, compatibility is implied by strict compatibility. Let HD be deﬁned for H in the same way as the similar conditions for H ∗ , and similarly with H and H . Then HD asserts the consistency of D. It is noticed that HD (R) derives from HD∗ (R) just by replacing the requirement for an absolute maximum of original “revealed preference” by a requirement for a maximum. But while HD∗ , and similarly HD , is a proper condition, that is there exist D for which it can be asserted and other D for which it can be denied, HD is vacuous, since it is always validated by a constant utility function. It can be remarked, incidentally, that if R is semi-increasing, x > y ⇒ xRy then H ⇒ H . Also if R is lower-continuous, that is the sets xR = [y: xRy] are closed, then H ⇒ H . Accordingly if, for instance, R is represented by a continuous increasing utility function then H and H are equivalent, so in their conjunction one is redundant,

58 Concept and Method that is mathematically but not economically. But there is no need here to make any assumptions whatsoever about the order R. It can be granted that as a basic principle H ∗ requiring an absolute maximum is unwarranted in place of the more standard H which requires just a maximum. However, while H ∗ produces the well known discussion of Samuelson (1948) and Houthakker (1950), described as revealed preference theory—more suitably revealed preference plus revealed non-preference—that discussion is not generalized but its entire basis evaporates when H ∗ becomes H . From this circumstance there is a hint that the nature of that theory is not properly gathered in its usual description. The critical feature of it is not that it deals with maxima under budget constraints but that it deals especially with absolute maxima. This might have intrinsic suitability, by mathematical accident, for dealing with continuous demand functions. But it is not a direct expression of normal economic principles, which recognize signiﬁcance only for a maximum—not that the maximum under the budget should moreover be unique so revealing an additional non-preference signiﬁcance. If the matter is to be reinitiated, then H is admitted as such a principle and so equally is H , so their conjunction H comes into view as an inevitable basis required by normal economic principles. The question of HD for an expenditure correspondence is proper, that is, capable of being true and false, unlike HD which is always true. Also, since H ∗ ⇒ H , this provides a generalization of the usual theory with H ∗ . It happens, as the mathematical accident just mentioned, that if D is a continuous demand function then HD∗ ⇔ HD . Thus the distinctive revealed preference theory is not lost in this generalization but it just receives a reformulation which puts it in perspective with a normal and broader economic theory not admitting description as revealed preference theory, which moreover is capable of a further simple and necessary extension now to be considered. With a demand correspondence D interpreted as representing the behavior of the consumer, there is the hypothesis that the consumer (i) has a deﬁnite structure of wants, represented by a utility relation R, and (ii) is an efﬁcient programmer. Then HD is the condition of the consistency of the data D with that hypothesis. If it is not satisﬁed, so the data reject the hypothesis, the hypothesis can be modiﬁed. If (i) is not to be modiﬁed, either because there is no way of doing this systematically or because it is a necessary basic assumption, as it is for instance in economic index number theory, then (ii) must be modiﬁed. Instead of requiring exact efﬁciency, a form of partial efﬁciency, signiﬁed by a certain level of cost efﬁciency e where 0 ≤ e ≤ 1, will be considered. When e = 1 there is return to the original, exact efﬁciency model. Thus consider a relation H between a demand (p, x) and a utility relation R together with a number e given by the conjunction of conditions

H py ≤ Me ⇒ xRy H yRx ⇒ py ≥ Me

The Price Level 59 where M = px. They assert x is as good as any y which costs no more than the fraction eM of the cost M of x, at the prices p, and also any y as good as x costs at least that fraction. In the language of cost-beneﬁt analysis these are conditions of cost efﬁcacy and cost efﬁciency, but modiﬁed to allow a margin of waste, which is the fraction (1 − e) M of the outlay M . It is noticed that if H is not to be satisﬁed vacuously then e > 0; and then from H ", with R reﬂexive necessarily e ≤ 1. With R given, for simplicity of illustration say by a continuous increasing strictly quasiconcave function φ, and with p > 0 and M ﬁxed, it can be seen what varying tolerance this condition gives to x as e increases from 0 to 1. When e = 0, x is permitted to be any point in the budget simplex B described by px = M , x ≥ 0. When e = 1, x is required to be the unique point x on B for which φ (x) = max {φ (y) : py = M } . For 0 ≤ e ≤ 1 let xe be the unique point in the set Be described by px = Me for which φ (xe ) = max {φ (y) : py = Me} .

S0=B1

Se

x1

S1

xe Be

B0

x0

Then x is required to be in the convex set Se ⊂ B deﬁned by φ (x) ≥ φ (xe ) , px = M . Evidently, if 0 ≤ e ≤ e ≤ 1

60 Concept and Method then B = S0 ⊃ Se ⊃ Se ⊃ S1 = x1 That is, the tolerance regions Se for x form a nested family of convex sets, starting at the entire budget simplex B when e = 0 and, as e increases to 1, shrinking to the single point x1 attained when e = 1. The higher the level of cost efﬁciency the less the tolerance, and when cost efﬁciency is at its maximum 1 all tolerance is removed: the consumer is required, as usual, to purchase just that point which gives the absolute maximum of utility. For a demand correspondence D, now deﬁne compatibility of D with R at the level of cost efﬁciency e to mean this holds for every element of D.Then e-consistency of D, or consistency at the level of cost efﬁciency e, stated HD (e), will mean this holds for some R. Immediately HD (1) ⇔ HD so l-consistency of E is identical with the formerly deﬁned consistency. Also 0-consistency is valid for every E. Further HD (e) , e ≤ e ⇒ HD e that is, consistency at any level of cost efﬁciency implies it at every lower level. Hence with eD = sup e : HD (e) deﬁning the critical cost efﬁciency of any expenditure correspondence D it follows that 0 ≤ eD ≤ 1, e < eD ⇒ HD (e) ,

e > eD ⇒ H¯ D (e)

The condition HD (e) will now be investigated on the basis of a ﬁnite demand correspondence D with elements (pt , xt ) ∈ B × C (t = 1, . . . , m) , and belief in perfect efﬁciency, any utility to be entertained would, to ﬁt the data, have to be such that (PX =)

Pt Xt = pt xt ,

where Pt = θ (pt ) , Xt = φ (xt ) .

The Price Level 61 so in any case (PX ≤)

Ps Xt ≤ ps xt

and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 1, . . . , m). A question is whether a solution exists. If one does, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from (PX =). If instead of perfect efﬁciency there is to be allowance of partial efﬁciency, at some level e, then (PX =) would be replaced by (PXe) Pt Xt ≥ ept xt , where 0 ≤ e ≤ 1, which for the perfect efﬁciency case e = 1, in view of (PX ≤), becomes again (PX =). Now from (PXe), with (PX ≤), follow the systems (a)

ast /e ≥ Ps /Pt ,

(b)

bst /e ≥ Xs /Xt ,

where ast = ps xt /pt xt ,

bst = pt xs /pt xt

with P’s and X ’s connected by Pt Xt = pt xt . These systems, even if not consistent for e = 1, are always consistent for sufﬁciently small e. From any solution there is obtained a utility that shows demand elements as efﬁcient within the level e. Thus, with φ (x) = mini Xi pi x/pi xi and antithetical θ (p) = min px : Xt pt x ≥ Pt

62 Concept and Method it appears that pt xt ≥ θ (pt ) φ (xt ) ≥ ept xt as required for compatibility at a level of cost efﬁciency e. In case e = 1, then moreover φ (xt ) = Xt ,

θ (pt ) = Pt .

Since ast = ps xt /pt xt is just the Laspeyres index Lst , a restatement of system (a) is the system (L/e)

Lst /e ≥ Ps /Pt .

This can be dealt with following exactly the treatment given to the system (L), by replacing the Laspeyres index Lst by Lste = Lst /e. Then e = Ls...t /e . . . e Ls...t

so that e ≥ 1 ⇔ Lt...t ≥ e . . . e. Lt...t

So it appears that either system (L) is consistent, in which case also system (L/e) is consistent with e = 1, or critical cost efﬁciency e∗ can be determined so that e ≥ 1 ⇔ e ≤ e∗ . Lt...t

Introducing Lst∗ = Lst /e∗ , the system (L∗ )

Lst∗ ≥ Ps /Pt ,

is consistent and determines price levels associated with a utility that represents the given demands as together within a cost efﬁciency at the highest level, in that sense a best approximation to a utility that ﬁts the data, coinciding with a utility that ﬁts the data exactly when that exists. The treatment of (L∗ ) follows exactly the treatment already accounted for the system (L). At this point it can be remarked that, with all additional discussion about it put aside, the system (L∗ ) is the embodiment of the entire method now proposed for the computation of price levels Pt and then price indices Pst = Ps /Pt always available and together true in the exact or approximate sense on the basis of demand data for any number of periods.

The Price Level 63

12 Old and new: an illustration Some illumination is provided by what this method provides for the classical case of two periods, worked for so long by so many authorities that it may seem unlikely there is anything to add there. The data consists in a pair of demands (pt , xt ) ∈ B × C (t = 1, 2) in terms of which there are conventional algebraical (not fancy combinatorial) formulae for price indices, especially those associated with Paasche, Laspeyres and Fisher, beside the one or two hundred in Fisher’s list. The Laspeyres is Lst = ps xt /pt xt , Paasche Kst = (Lts )−1 , and Fisher 1

1

Fst = (Kst Lts ) 2 = (Lst /Lts ) 2 . For the consistency case L12 L21 ≥ 1, where Paasche does not exceed Laspeyres, the PL-interval is non-empty and all points in it are accepted as true price indices, all equally true, no one truer than another. In the contrary case, the data does not admit the existence of true price indices at all, at least not exactly, the PL-interval is empty, and now instead for the critical cost efﬁcincy e∗ , that makes the system Lst /e ≥ Ps /Pt consistent if and only if e ≤ e∗ , which requires L12 L21 = e∗ e∗ there is the determination 1

e∗ = (L12 L21 ) 2 and now 1

∗ L12 = L12 /e∗ = (L12 /L21 ) 2 ,

so that, for the Paasche index ∗ ∗ −1 ∗ ) = L12 K12 = (L21

1

∗ L21 = L21 /e∗ = (L21 /L12 ) 2

64 Concept and Method and the system (L∗ )

Lst∗ ≥ Ps /Pt

(s, t = 1, 2) ,

for determination of approximate price levels, is equivalent to (L∗ )

∗ ∗ , K12 ≤ P1 /P2 ≤ L12

is consistent, but here the limits are coincident and the only price index obtained from a solution is the value 1

P1 /P2 = (L12 /L21 ) 2 —incidentally, usually known as Fisher “Ideal Index”. If the critical e∗ is replaced by a more tolerant lower level e, the system is still consistent, with limits now no longer coincident but admitting a range of values, again including the Fisher index but now not unique but just one of its many points. Hence here we have a New Comment about the Fisher index. For the Old Comment, in the consistency case, Fisher, being the geometric mean of Laspeyres and Paache, is a point of the now non-empty interval, and so is a true index like any other, and no truer than another. This gives a value to Fisher as being a true index, but also it is deﬂating from making it no more distinguished than the others. There was a moment of distinction when Fisher became associated with a quadratic utility, which then became put aside, though recently there may have been what may seem to some to be something of a renaissance, even a cult, see Afriat and Milana (2006). For the New Comment, in the case of inconsistency, when the LP-interval is empty and there are no true indices at all, at least not exactly, at which point in the absence approximation ideas the matter is usually abandoned, Fisher now stands out from being alone associated with a utility that ﬁts the data as closely as possible, in the way here approximation is understood that has reference to cost efﬁciency criteria. After the ﬁrst deﬂation this gives a real distinction to the Fisher “Ideal” index, and a good reason for the term Fisher gave to it even though not one he had in mind. If one does not want to always trouble about consistency and still have an in some way signiﬁcantly “true” price index, surely this is it—as “superlative” as can be, in the language Irving Fisher invented and has had a perplexed persistence in echoings since. Have latter day pedlars of the superlative ever promoted such a quality in their fancy? Fisher’s index having this new status, its generalization invites consideration. Every point in the entire interval between Laspeyres and Paasche is the possible value for a true index. In this unacceptable indecision the Fisher index, as the geometric mean of the limits, at least picks out one value. Now with the new method there is again the unfortunate indecision, even expanded since the line segment is now replaced by a multi-dimensional

The Price Level 65 polyhedron. For a fair remedy such as was found before, it may be fair to try some manner of immitation of the original Fisher index. The points in the LP-interval are “true” merely in the theoretical sense related to utility, without being overruled by some further possibly more real sense perhaps like “actual” which, for want of an approach, cannot be grasped further, or estimated. But it is contradictory for a method to produce price indices to conclude with an unresolved choice. Since “true” based on utility is the outstanding qualiﬁcation in current logic, though there is no further qualiﬁcation available, it is still important to have a proposal for picking a true point out of many, such as may be provided by the PL-interval and can apply just as well when the interval becomes a polyhedron in many dimensions: True with minimum distance from non-true maximum. This is one path to the Fisher index and generalizations. Our approach deals directly with price levels instead of price indices which are their ratios. True just means a solution of the inequalities to determine price levels. Rather than being on the true boundary and so almost non-true it is better to be comfortably in the interior surrounded by true and far away from non-true. The way we adopted to produce a single true outcome is thrown up by basic solutions and judgement about form giving them equal weight. That is all. No single point can be viewed as anything like an “estimate” of the “actual” from among the many “true”, or “actually true” from among the “possibly true”. Though there may be words for that sort of thing there is absolutely no sense. Here the derived system M may just as well be replaced by M ∗ is the case of inconsistency, requiring approximation. Everything that follows now applies equally well in either case. The basic price levels, base t, are Pi = Mit . and Pi = 1/Mti , with geometric mean 1

Pi = (Mit /Mti ) 2 which is also a price level solution, determine systems of basic price indices Pij = Mit /Mjt and Pij = Mtj /Mti

66 Concept and Method with geometric mean 1 Pij = Mit Mtj /Mti Mjt 2 But this geometric mean price index is identical with the price index determined from the geometric mean price levels, 1 1 Pij = (Mit /Mti ) 2 / Mjt /Mtj 2 . Going further, similarly, the geometric mean of all the basic price levels, for all bases, is again a price level solution, the basic mean price level solution, and the price indices derived from it is a price index system where each price index is the geometric mean of the basic price indices, the basic mean price index system. Any price index in this unique last system is a generalized counterpart of the Fisher index, and in the classical case of just two periods it becomes exactly the Fisher index. Thus though the price level solutions, and so also price indices they determine, are many, the geometric mean, element by element, of the basic solutions is again a solution which determines unique price indices that are geometric means of the basic price indices. Here is a fair conclusion in the quest for elimination of indecision, a multi-period generalization of the Fisher index that even has no conﬂict with Fisher’s own “Tests”.

13 Conclusion Though the mathematics of the method, its theoretical rationalization and computations, require an account, the scheme for applications is simple, and conveys an idea of what could be meant by an answer to “The Index Number Problem”. A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of an entirely different type and are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, for the hypothetical underlying utility which in any case there is no need to actually construct. With some m periods listed as 1, . . ., m and demand data (pi , xi ) (i = 1, . . . , m)

The Price Level 67 giving row and column vectors of prices and quantities for some n goods, the ﬁrst step is to compute the matrix L of Laspeyres indices Lij = pi xj /pj xj and raise it to the mth power M = Lm in a modiﬁed arithmetic where + means min. Diagonal elements Mii ≥ 1 tell the consistency of the system (L)

Lij ≥ Pi /Pj

for the determination of price levels Pi , and provide the ﬁrst and second basic price level solutions, with any t as base, given by Pi = Mit , and Pi = 1/Mti , from which are derived two systems of basic price indices Pij = Pi /Pj . The price indices in either system, with any base, will all be true together in respect to a utility that ﬁts the data by criterion of cost efﬁciency of demand in each period i, so the cost pi xi is the minimum cost, at the prices pi , of the utility of xi . Diagonal elements Mii < 1 tell the inconsistency of the system, and enable determination of a critical cost efﬁciency e∗ so that the system (L/e)

Lij /e ≥ Pi /Pj

is consistent if and only if e ≤ e∗ (features in the computation of e∗ remain to be clariﬁed). Then with Lij∗ = Lij /e∗ the system (L∗ )

Lij∗ ≥ Pi /Pj

is consistent, and with M ∗ = (L∗ )m there may be obtained basic price levels and price indices from M ∗ , as before from M . Now instead the price levels of a basic system are together true in respect

68 Concept and Method to a utility that ﬁts the data now not exactly, but approximately in the sense of partial cost efﬁciency at the level e∗ in each period, meaning that the fraction e∗ of the cost, in the period, is at most the minimum cost at the prices of gaining at least the utility. Hence in the case e∗ = 1 that goes with ordinary consistency, the ﬁt would be exact as before.

Chapter 3 Price Level Computation: Illustrations Introduction I Outline of the method 1 2 3 4 5

Original data Consistency of the data Price-quantity duality The power algorithm Cost efﬁciency and approximation II Illustrations

1 2 3 4 5

Three references with consistency, and graphics Four references, with consistency Case of inconsistency and approximation Inconsistency and approximation again EUKLEMS data Notes

1 2 3

Price levels The triangle equality Ratio matrix

Introduction

The theory of the Price Index, proper, starts with the Utility Cost Factorization Theorem, going back to early1960’s. By itself it represents no resolution of the Index Number Problem, nor had there even been a real idea of what could be meant by such a resolution. However, the method now proposed does convey some idea of what could be meant by such a resolution. It even represents such a resolution itself. The method has been available in the main for more than twenty ﬁve years, apart from ampliﬁcations made just now. But only recently has it been recognized as a proper resolution of the Index Number Problem. These ﬁrst exercises with the arithmetic go to convey the practicality of it. Needs of dealing with the EUKLEMS Project data when it becomes availalbe have stirred into life that almost forgotten work and exposed its value. This is the third of three papers by present authors making Part I. The ﬁrst “The Super Price Index: Irving Fisher, and after” has more to do with history, and “The Price Level Computation Method” is an exposition of the mathematics. We start with arbitrary ﬁgures for the Laspeyres matrix L related to quantity indexes of per capita GDP in an inter-country comparison based on the International Comparison Project (ICP) data for 1980 published by the United Nations and the Commission of the European Communities (1987). This accidental source gives prices and quantities for some 38 components of GDP expenditure for 60 countries. In these applications we take the data for various countries, for instance in the ﬁrst illustration just for US, France, and Italy, to form the matrix L. Since prices and quantities can be interchanged symmetrically in the method, and the data is arbitrarily in use just to illustrate computational procedure, there is liberty now to say “price” for afﬁnity with the more usual subject even when “quantity” may ﬁt the data source better, but any reader may always for “price” read “quantity”.

Price Level Computation 71 I OUTLINE OF THE METHOD

1 Original data A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of a completely different type, beside being “nonparametric” rather than conventionally algebraical, are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, in the ﬁt to the data of the hypothetical underlying utility, which in any case there is no need to actually construct. With some m time periods, or countries, or nodes, in any case references— perhaps most typically time periods—listed as 1,…,m, the initial data has the form of some m demand elements (pt , xt )

(t = 1, . . . , m)

giving row and column vectors of prices and quantities for some n goods demanded at the prices. Hence for the initial data scheme: m number of references n number of goods p m × n price matrix, rows pi x n × m quantity matrix, columns xj c = px m × m cross-cost matrix, elements pi xj The ﬁrst step is to compute the matrix L of Laspeyres indices Lij = pi xj /pj xj i being index for the current period and j for the base period. Hence divide column j of c by diagonal element pj xj to form the m × m Laspeyres matrix L with these elements. The Paasche indices are given by Kij = 1/Lji = pi xi /pj xi , forming the elements of an m × m matrix K, obtained by transposition of L and replacing each element by its reciprocal. The Laspeyres-Paasche (LP) inequality Kij ≤ Lij

72 Concept and Method has signiﬁcance for Laspeyres and Paasche indices as price index bounds, and for data consistency. Another well known construction that may have comment is the Fisher index which is the geometric mean of the Laspeyres and Paasche indices, 1 1 Fij = Lij Kij 2 = pi xj pi xi /pj xj pj xi 2 Central to the proposed method is the system of inequalities (L)

Lij ≥ Pi /Pj .

This serves to determine price levels Pi from which the matrix P of price indices Pij = Pi /Pj is derived, and which enter into the construction of an underlying utility which ﬁts the given demand data and represents all these indices together as true. By the geometric mean of two vectors is here meant the vector whose elements correspondingly are geometric means of their elements, and there is a similar understanding about matrices. The same understanding can apply just as well for several vectors, or matrices, also in application of the more general weighted geometric mean. Any two price level solutions Pia and Pib have a geometric mean with elements which are geometric means 1 Pic = Pia Pib 2 of their elements, which also is a price level solution. For from Lij ≥ Pia /Pja

and

Lij ≥ Pib /Pjb

follows Lij Lij ≥ Pia /Pja Pib /Pjb = Pia Pib / Pja Pjb 2 = (Pic )2 / Pjc and hence Lij ≥ Pic /Pjc . There is a similar conclusion in dealing with the geometric means of several price level solutions. It can be added that the price index matrix obtained from the geometric mean of the price level solutions, which is the matrix of ratios of its elements, is the geometric mean of the price index matrices obtained from them.

Price Level Computation 73

2 Consistency of the data The solubility of the system (L) imposes a condition on the given data, deﬁning its consistency, equivalent to the existence of the appropriate underlying utility. With any chain described by a series of periods, or references, s, i, j, . . . , k, t there is associated the Laspeyes chain product Lsij...kt = Lsi Lij . . . Lkt termed the coefﬁcient on the chain. Obviously Lr...s...t = Lr...s Ls...t A chain t, i, j, . . . , k, t whose extremeties are the same deﬁnes a cycle. It is associated with the Laspeyres cyclical product Ltij...kt = Lti Lij . . . Lkt which is basis for the important Laspeyres cyclical product test, or simply the cycle test, Lt...t ≥ 1 for all cycles t . . . t which is necessary and sufﬁcient for consistency of the given data, and is an extension of the PL-inequality. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

the cycle test Ls...t...s ≥ 1 is equivalently to (chain LP) Ks...t ≤ Ls...t for all possible chains … the two occurrences here being taken separately. Hence, introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

subject to the now to be considered conditions required for their existence, where Hst = 1/Mts ,

74 Concept and Method this condition is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst , showing the relation of bounds for the LP-interval and the narrower bounds for the derived version that involves more data. The matrix M , and the matrix H constructed from it, in exactly the same way as the Paasche matrix K is constructed from the Laspeyres matrix L, is important in that their columns, currently as a matter of conjecture, provide a complete set of basic solutions of the system of inequalities (L), the basic price level solutions, from which all other solutions may be derived as combinations.

3 Price-quantity duality With any determination of price levels Pt , there is an associated determination of quantity levels Xt , where Pt Xt = pt xt

(t = 1, . . . , m) .

While for price levels, pt xs /ps xs ≥ Pt /Ps , for quantity levels equivalently in a dual fashion, pt xs /pt xt ≥ Xs /Xt , and one could just as well have solved for the quantity levels ﬁrst, by the same method as for price levels, and then determined the price levels from these. Whichever way, Ps Xt ≤ ps xt

(s, t = 1, . . . , m) ,

with equality for s = t. The introduction of cost efﬁciency up to a level e, where 0 ≤ e ≤ 1, would require Pt Xt ≥ ept xt

(t = 1, . . . , m) .

good also for any lower level, and highest level 1 imposing the equality.

Price Level Computation 75

4 The power algorithm For the main step in the proposed method, matrix L is raised to the mth power in the modiﬁed arithmetic where + means min, to determine M = Lm . Diagonal elements Mii = 1 tell the consistency of the system of inequalities (L) for the determination of price levels Pi , and provide the ﬁrst and second basic price level solutions, with any t as base, given by Pi = Mit , and Pi = Hit , that is, by columns of the matrices M and H. From these are derived the two systems of basic price indices Pij = Pi /Pj . The price indices in either system, with any base, will all be true together in respect to a utility that ﬁts the data by criterion of cost efﬁciency of demand in each period i, so the cost pi xi is the minimum cost, at the prices pi , of the utility of xi .

5 Cost efﬁciency and approximation Diagonal elements Mii < 1 tell the inconsistency of the system, and enable determination of a critical cost efﬁciency e∗ so that the system (L/e)

Lij /e ≥ Pi /Pj

(i = j)

is consistent if and only if e ≤ e∗ . Then with Lij∗ = Lij /e∗

(i = j)

as the elements of the ajusted Laspeyres matrix, the system (L∗ )

Lij∗ ≥ Pi /Pj

is consistent, and with M ∗ = (L∗ )m

76 Concept and Method there may be obtained basic price levels and price indices from M ∗ , as before from M . Now, instead, the price levels of a basic system are together true in respect to a utility that ﬁts the data not exactly, but approximately in the sense of partial cost efﬁciency at the level e∗ in each period, meaning that the fraction e∗ of the cost, in the period, is at most the minimum cost at the prices of gaining at least the utility. Hence in the case e∗ = 1 that goes with ordinary consistency, the ﬁt would be exact as before. For any element Mii < 1 determine the number di of nodes in the path i…i and 1

ei = (Mii ) di giving this the value 1 in case Mii ≥ 1 and then e∗ = min ei i

is the critical cost efﬁciency. Consistency requires Mii = 1, in this case compute the 2m basic price level solutions Pr = Mrt and Pr = Hrt a pair determined for every node t and compute the basic mean price level solution P r and with this the matrix of basic mean price indices P rs = P r /P s . In the other case, of inconsistency, with the critical cost efﬁciency e∗ form the adjusted Laspeyres matrix and proceed exactly as before with this in place of original L. An alternative procedure for the critical cost efﬁciency is available, especially if the path i…i for elements Mii < 1 is not known: Critical cost efﬁciency crude approximation method TEST e: if L/e consistent then YES 0 HIGH = 1 LOW = 0 D = 1/n (for n steps, eg 10) 1 e = (HIGH + LOW)/2 TEST e 2 if YES then LOW = e else HIGH = e 3 if HIGH - LOW < D then e∗ = LOW end else 1 It should be reminded that the following illustrations are not intended for communications of any kind of actual economic information. They are the ﬁrst calculations made following the method, just to assist understanding of it and show the shape of its arithmetic, beside being stimulus for the software development.

Price Level Computation 77 II ILLUSTRATIONS

1 Three references with consistency, and graphics We start with the arbitrary Laspeyres matrix L already described, related to bilateral quantity indexes of per capita GDP in an inter-country comparison based on the International Comparison Project (ICP) data for 1980 published by the United Nations and the Commission of the European Communities (1987). This source gives prices and quantities for some 38 components of GDP expenditure for 60 countries. In the following application, we take the data for the US, France, and Italy to form the matrix L. By raising the matrix L to powers in a modiﬁed arithmetic where + means min, we have Illustration 1 L Laspeyres 1 1.182937 0.913018 1 0.747516 0.813833

1.500803 1.266174 1

L power 2 1 0.913018 0.743044178

1.49780407 1.266174 1

1.182937 1 0.813833

L power 3 = M derived Laspeyres 1 1.182937 1.49780407 0.913018 1 1.266174 0.743044178 0.813833 1 Paths 1,1,1,1 2,1,1,1 3,2,1,1

1,1,1,2 2,2,2,2 3,2,2,2

1,2,2,3 2,2,2,3 3,3,3,3

Consistency case: all diagonal elements = 1

Note that L ≥ L2 = L3 . and at this point one could add “ = …” because after one equality only others can follow. Now we have the derived Laspeyres matrix M = L3

78 Concept and Method The Paasche matrix K is 1 0.8453536 0.6663100

1.0952687 1 0.7897809

1.3377640 1.2287533 1

and the derived Paasche matrix is H derived Paasche 1 0.845353557 0.667644065

1.09526866 1 0.789780867

1.345815 1.22875332 1

Note that Kst ≤ Hst ≤ Mst ≤ Lst , showing the relation of the original LP-interval and the narrower bounds that involve more data. The 6 basic price level systems - the 6 columns of M and H The geometric mean of the matrices H and M , element by element, is the matrix F, whose columns coincide with the geometric means of their corresponding columns: F derived Fisher - mean of derived Laspeyres M and derived Paasche H 1 0.878534583 0.704335883

1.13825912 1 0.801716741

1.41977716 1.24732334 1

The columns of M and H are all solutions of system (L). These are the 6 basic price level solutions, from which all other solutions can be derived, being the 6 vertices of the convex hexagonal region described by solutions normalized to sum 1 each determining a point in the simplex of reference. The columns of F are geometric means of opposite pairs of vertices of the hexagon. The 6 basic solutions, a basis for all solutions, are given by columns of M and H , and basic geometric mean solution has elements given by the geometric means of their columns, or of columns of the matrix F, so it is P mean basic price level system - mean of columns of F 1.17351085 1.03096986 0.826545798 The matrix of basic mean price indices obtained from this, by taking ratios of the elements, is

Price Level Computation 79 S

Drs

Dst

Drst

Drt

r

t

Figure 1 The 6 basic solutions as vertices for the solution set

P/P mean basic price index system 1 0.878534583 0.704335883

1.13825912 1 0.801716741

1.41977716 1.24732334 1

and coincides with the mean of individual basic price index matrices. Notice that this matrix P/P coincides with the matrix F used to obtain it, and see End-note No. 2.

2 Four references, with consistency The start with the Laspeyres matrix L and raising it to powers in a modiﬁed arithmetic where + means min, using CM’s FORTRAN program, we have ⎡

1.000 ⎢0.898 L≡⎢ ⎣0.913 0.747

1.122 1.183 1.000 1.042 0.979 1.000 0.812 0.814

⎤ 1.501 1.350⎥ ⎥ 1.266⎦ 1.000

(elements that change are in bold). and then ⎤ ⎡ L12 L123 L1234 L11 ⎢ L21 L22 L23 L234 ⎥ ⎥ L4 ≡ ⎢ ⎣ L321 L32 L33 L34 ⎦ L4321 L432 L43 L44

80 Concept and Method where Lrij...ks = Lri Lij . . . Lks . with M = L4 , therefore ⎡

1.000 ⎢0.898 M =⎢ ⎣0.879 0.715

1.122 1.000 0.979 0.797

1.169 1.042 1.000 0.814

⎤ 1.480 1.319⎥ ⎥ 1.266⎦ 1.000

Note the triangle inequality Mrs Mst ≥ Mrt From the matrix L, the Paasche matrix K is derive by Kij = 1/Lji , so that ⎡

⎤ 1.000 1.114 1.095 1.338 ⎢0.891 1.000 1.021 1.231⎥ ⎥ K =⎢ ⎣0.845 0.960 1.000 1.229⎦ 0.666 0.741 0.790 1.000 from which, by similar procedure, Hij = 1/Mji , we have ⎡

1.000 ⎢0.891 H ≡⎢ ⎣0.856 0.676

1.114 1.138 1.000 1.021 0.960 1.000 0.758 0.790

⎤ 1.398 1.255⎥ ⎥ 1.229⎦ 1.000

Alternatively, just as M = Lm so similarly H = K m where the arithmetic for powers now has + meaning max instead of min. Note K ≤ L for original bounds, and moreover K ≤H ≤M ≤L showing tighter bounds obtained with additional data. With any Pi which are a price level solution being such that Lij ≥ Pi /Pj there is associated a price index matrix with elements Pij = Pi /Pj The 8 basic solutions, a basis for all solutions, are given by columns of M and H , and basic geometric mean, which has elements given by the geometric means

Price Level Computation 81 of their elements, is also a solution. It is [1.167 1.044 1.012 0.811]. The matrix of basic mean price indices obtained from this, by taking ratios of the basic mean price levels, is ⎡ ⎤ 1.000 1.118 1.153 1.438 ⎢0.894 1.000 1.031 1.287⎥ ⎢ ⎥ ⎣0.867 0.969 1.000 1.247⎦ 0.695 0.777 0.802 1.000 and coincides with the mean of individual basic price index matrices, derived from the individual basic price level solution elements. By taking weighted geometric means instead of the simple geometric mean, it is possible to arrive at all possible price level solutions, and consequently all possible systems of true price indices, without any guidance for choosing just one from among them. Here we have, for want of that guidance and to that extent arbitrarily, like the Fisher “Ideal Index”, adopted one, with weights all equal and no reason for making them different, as a standard, in order to eliminate that residual indecision. Following the above report done using FORTRAN, we include the routine output from another program using BBC BASIC for Windows1 . This deals with two text ﬁles kept in folder c:\0\ as here indicated: REM input from C:0?-input.txt output to C:0?-output.txt REM change the ? *SPOOL "C:02-output.txt" F%=OPENIN "C:02-input.txt"

For instance 2-input.txt looks like 4, 4 1.0000000, 0.8976280, 0.9130180, 0.7475160,

1.1218730, 1.0000000, 0.9792190, 0.8122070,

1.1829370, 1.0418520, 1.0000000, 0.8138330,

1.5008030 1.3498590 1.2661740 1.0000000

which tells it is a 4 x 4 matrix and then tells the elements, the comma “,” being the delimiter. As for 2-output.txt, which need not even exist initially and if it does any contents will be overwritten, it recieves the output when the program is run,

1 We acknowledge with thanks the guidance received from Richard Russell, longtime developer of this rendering of BASIC, http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html,http:// www.rtrussell.co.uk/,[email protected].

82 Concept and Method which with this input is as follows—showing reassuring agreement with earlier ﬁgures. When this program is compiled and so not available for alteration, it will refer to two ﬁles called input.txt and output.txt always with these names though they can have different applications. This compiled version will be supplied to anyone wishing to use it. Illustration 2 L Laspeyres 1 0.897628 0.913018 0.747516

1.121873 1 0.979219 0.812207

1.182937 1.041852 1 0.813833

1.500803 1.349859 1.266174 1

L power 2 1 0.897628 0.878974393 0.729059745

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

L power 3 1 0.897628 0.878974393 0.715338367

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

(Elements that change are in bold. This is after the last power that changes so generation of powers could have stopped here. Note that L ≥ L2 ≥ L3 = L4 . and at this point one could add “ = …” because after one equality only others can follow.) L power 4 = M derived Laspeyres 1 0.897628 0.878974393 0.715338367

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

1,1,1,1,2 2,2,2,2,2 3,2,2,2,2 4,3,2,2,2

1,2,2,2,3 2,2,2,2,3 3,3,3,3,3 4,3,3,3,3

1,3,3,3,4 2,3,3,3,4 3,3,3,3,4 4,4,4,4,4

Paths 1,1,1,1,1 2,1,1,1,1 3,2,1,1,1 4,2,2,1,1

Price Level Computation 83 Consistency case: all diagonal elements = 1 H derived Paasche 1 1.11404725 0.891366491 1 0.855559611 0.959829227 0.675704611 0.758054759

1.13768957 1.02122202 1 0.789780867

1.39793984 1.25482994 1.22875332 1

The 8 basic price level systems - the 8 columns of M and H F derived Fisher - mean of derived Laspeyres 1 1.11795328 1.15315252 0.894491767 1 1.03148543 0.867187978 0.96947564 1 0.695239119 0.77724485 0.801716741

M and derived Paasche H 1.43835405 1.28659585 1.24732334 1

P mean basic price level - mean of columns of F 1.16692797 1.04380747 1.01194591 0.811293977 P/P mean basic 1 0.894491767 0.867187978 0.69523912

price index 1.11795328 1 0.96947564 0.77724485

1.15315252 1.03148543 1 0.801716741

1.43835405 1.28659585 1.24732334 1

As with Illustration No. 1, notice that this matrix P coincides with the matrix F used to obtain it, see end-note No. 2.

3 Case of inconsistency and approximation Starting with the Laspeyres matrix L for the countries Canada, U.S., Norway, Luxembourg, Germany in the year 1980, and raising it to powers in the (+ = min)-arithmetic using the FORTRAN program: L POWER 1 1.0000000 0.9139310 0.9685060 0.9389430 0.8886960

1.0171450 1.0000000 1.0171450 0.9398820 0.8976270

1.1252440 1.1274960 1.0000000 1.0345840 1.0030040

1.2008140 1.1411080 1.1207520 1.0000000 1.0387310

1.1537290 1.1218730 1.0650260 1.0222430 1.0000000

1.0376916 0.9860962

0.9970028 0.9694742

··· ··· ··· ··· ··· ··· ··· ··· ··· M = L POWER 5 0.8641568 0.7897797

0.8789728 0.8641568

0.9723873 0.9559967

84 Concept and Method 0.8122053 0.7795783 0.7626154

0.8789728 0.8122054 0.7756905

0.9723873 0.8985241 0.8581284

1.0140962 0.9588679 0.9157593

0.9970021 0.9212698 0.8798515

Inconsistency case since some diagonal elements < 1 diagonal elements < 1 (in this case all) associated cost efﬁciencies ei critical cost efﬁciency is minimum of these 1

i

Mii

path

di

ei = (Mii ) di

1 2 3 4 5

0.8641568 0.8641568 0.9723873 0.9588679 0.8798515

12121 21212 321213 421214 521215

3 3 4 4 4

0.952498 0.952498 0.990710 0.986097 0.968506

critical cost efﬁciency e∗ = mini ei = e1 = e2 = 0.952498 used to determine the adjusted Laspeyres matrix= L∗ Being near to the value 1, associated with the consistency case where ﬁt of data to the hypothetical underlying utility is exact, this represents a high level of cost efﬁciency, and a closeness of ﬁt for the approximating utility. Note: By computer error the degree di associated with a path is 1 less than the correct count. The effect is to make the cost efﬁciency less than critical, resulting in allowance of a looser ﬁt for the approximate utility. A revision could provide the correction, and moreover a redevelopment of the approach to cost efﬁciency where the critical uniform bound is replaced by discrimination, but for the time being the error does not damage, even enhances, the value of the illustration. L* POWER 1 - adjusted L 1.0000000 1.0678707 1.1813607 0.9595094 1.0000000 1.1837250 1.0168061 1.0678707 1.0000000 0.9857687 0.9867546 1.0861794 0.9330159 0.9423923 1.0530245

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1778216 1.1181396 1.0732230 1.0000000

POWER 2 1.0000000 0.9595094 1.0168061 0.9468003 0.9042342

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1622213 1.1181396 1.0732230 1.0000000

1.0678707 1.0000000 1.0537261 0.9867546 0.9423923

1.1813607 1.1335267 1.0000000 1.0861794 1.0530245

Price Level Computation 85 POWER … …

3 (no change after this power) … … … … … … …

M* = L* POWER 5 1.0000000 1.0678707 0.9595094 1.0000000 1.0110601 1.0537261 0.9468003 0.9867546 0.9042342 0.9423923

1.1813607 1.1335267 1.0000000 1.0861794 1.0530245

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1622213 1.1181396 1.0732230 1.0000000

1.0561890 1.0134232 0.9206582 1.0000000 0.9317728

1.1059081 1.0611292 0.9496455 0.9169826 1.0000000

Consistent, all diagonal elements = 1 From derived Laspeyres M * determine by transposition and element inversion the derived Paasche H* 1.0000000 0.9364429 0.8464815 0.7932105 0.8255823

1.0421993 1.0000000 0.8822024 0.8347135 0.8604213

0.9890609 0.9490132 1.0000000 0.8498742 0.8943427

(Alternatively, just as M = Lm so similarly H = K m where now the arithmetic for powers has + meaning max instead of min, and same here for adjusted *-versions.) The columns of M* and H* provide the 10 basic price level solutions in 5 opposite pairs. Then determine F* the matrix geometric mean of M* and H* 1.0000000 0.9479060 0.9251182 0.8666094 0.8640138

1.0549569 1.0000000 0.9641575 0.9075557 0.9004745

1.0809430 1.0371749 1.0000000 0.9607892 0.9704457

1.1539224 1.1018607 1.0408110 1.0000000 1.0080324

1.1573890 1.1105256 1.0304544 0.9920316 1.0000000

The columns are geometric means of opposite pairs of basic solutions. Now determine the geometric mean of the columns of F* Mean basic price level solution 1.087774 1.037659 0.991172 0.943996 0.946864

86 Concept and Method Coincides with the geometric mean of all 10 of the basic price level solutions. Finally form the mean basic price index matrix P, given by ratios of elements of the mean basic price level solution. Mean basic price index matrix 1.00000 0.94791 0.92512 0.86661 0.86401

1.05496 1.00000 0.96416 0.90756 0.90047

1.08094 1.03717 1.00000 0.96079 0.97045

1.15392 1.10186 1.04081 1.00000 1.00803

1.15739 1.11053 1.03045 0.99203 1.00000

Coincides with the mean of the 10 basic price index matrices obtained from the 10 individual basic price level solutions. The BASIC program produces the following. Illustration 3 L Laspeyres 1 0.913931 0.968506 0.938943 0.888696

1.017145 1 1.017145 0.939882 0.897627

1.125244 1.127496 1 1.034584 1.003004

1.200814 1.141108 1.120752 1 1.038731

1.153729 1.121873 1.065026 1.022243 1

L power 2 0.929600347 0.913931 0.929600347 0.858987296 0.820369142

0.945538345 0.929600347 0.945538345 0.873714633 0.834434371

1.04602721 1.02839537 1 0.966570301 0.923115455

1.07896137 1.06077439 1.07896137 0.997002758 0.952179736

1.06077394 1.04289353 1.06077394 0.980196857 0.936129391

L power 3 0.864156805 0.849590575 0.864156805 0.798514889 0.762615439

0.878972774 0.864156805 0.878972774 0.812205427 0.775690481

0.97238726 0.955996697 0.97238726 0.898524088 0.858128447

1.00300286 0.986096244 1.00300286 0.92681411 0.885146613

0.986095823 0.969474188 0.986095823 0.911191339 0.870226207

L power 4 0.803320466 0.789779693 0.803320466 0.742299718 0.708927577

0.817093396 0.803320466 0.817093396 0.755026446 0.72108214

0.903931535 0.888694861 0.903931535 0.835268304 0.797716502

0.932391811 0.91667541 0.932391811 0.861566718 0.822832599

0.916675019 0.901223541 0.916675019 0.847043784 0.808962584

L power 5 = M derived Laspeyres 0.746766984 0.759570304 0.840295068 0.734179477 0.746766984 0.826131051 0.746766984 0.759570304 0.840295068 0.690042075 0.701872846 0.776465705 0.659019321 0.670318208 0.741557537

0.866751751 0.85214178 0.866751751 0.80091272 0.764905469

0.852141416 0.837777717 0.852141416 0.787412196 0.752011899

Price Level Computation 87 Paths 1,2,2,2,2,1 2,1,2,2,2,1 3,2,2,2,2,1 4,2,2,2,2,1 5,2,2,2,2,1

1,1,1,1,1,2 2,1,1,1,1,2 3,1,1,1,1,2 4,1,1,1,1,2 5,1,1,1,1,2

1,1,1,1,1,3 2,1,1,1,1,3 3,3,1,1,1,3 4,1,1,1,1,3 5,1,1,1,1,3

1,2,2,2,2,4 2,2,2,2,2,4 3,2,2,2,2,4 4,2,2,2,2,4 5,2,2,2,2,4

1,2,2,2,2,5 2,2,2,2,2,5 3,2,2,2,2,5 4,2,2,2,2,5 5,2,2,2,2,5

Inconsistency case: some diagonal elements < 1 1 0.746766984 2 0.746766984 3 0.840295068 4 0.80091272 5 0.752011899 Effective paths - Factor counts - Efficiencies 1 1,2,1 2 0.864156805 2 2,1,2 2 0.864156805 3 3,1,3 2 0.916676098 4 4,2,4 2 0.894937272 5 5,2,5 2 0.867186196 Critical cost efficiency

0.864156805

L adjusted Laspeyres to replace original L 1 1.17703754 1.30212942 1.05759857 1 1.30473543 1.12075262 1.17703754 1 1.08654239 1.087629 1.19721791 1.02839669 1.03873162 1.16067361 L power 2 … …

1.38957883 1.3204872 1.29693129 1 1.2020168

1.33509219 1.29822851 1.23244531 1.18293693 1

1.38957883 1.3204872 1.29693129 1 1.2020168

1.33509219 1.29822851 1.23244531 1.18293693 1

1,1,1,1,1,4 2,2,2,2,2,4 3,3,3,3,3,4 4,4,4,4,4,4 5,4,4,4,4,4

1,1,1,1,1,5 2,2,2,2,2,5 3,3,3,3,3,5 4,4,4,4,4,5 5,5,5,5,5,5

0.920350655 0.919431168 0.835269834 1 0.845353605

0.972387414 0.96271258 0.861568653 0.831935126 1

… L power 4

L power 5 = M derived Laspeyres 1 1.17703754 1.30212942 1.05759857 1 1.30473543 1.12075262 1.17703754 1 1.08654239 1.087629 1.19721791 1.02839669 1.03873162 1.16067361 Paths 1,1,1,1,1,1 2,1,1,1,1,1 3,1,1,1,1,1 4,1,1,1,1,1 5,1,1,1,1,1

1,1,1,1,1,2 2,2,2,2,2,2 3,2,2,2,2,2 4,2,2,2,2,2 5,2,2,2,2,2

1,1,1,1,1,3 2,2,2,2,2,3 3,3,3,3,3,3 4,3,3,3,3,3 5,3,3,3,3,3

Consistency case: all diagonal elements = 1 H derived Paasche 1 0.945538345 0.849590575 1 0.767972818 0.76643891 0.719642514 0.757296246 0.749011948 0.77028042

0.892257565 0.849590575 1 0.771050871 0.811395032

88 Concept and Method The 10 basic price level systems - columns of M and H F derived Fisher - mean of derived Laspeyres 1 1.05495693 1.07788442 0.947905995 1 1.05284896 0.927743253 0.949803858 1 0.88426359 0.907555705 0.960789211 0.877656772 0.894491269 0.970445672

M and derived Paasche H 1.13088451 1.13939758 1.10186074 1.1179539 1.04081102 1.03045439 1 0.992031646 1.00803236 1

P mean basic price level - mean of columns of F 1.07939424 1.04216477 0.988763618 0.94781098 0.94856982 P/P mean basic 1 0.965508928 0.916035665 0.878095276 0.8787983

price index 1.0357232 1 0.948759394 0.909463653 0.910191791

1.09166055 1.05400801 1 0.958581974 0.959349437

1.13882858 1.09954917 1.0432076 1 1.00080062

1.13791754 1.09866955 1.04237305 0.999200017 1

4 Inconsistency and approximation again CAN, US, NOR, LUX, GER, DEN, FRA, BEL, NED, AUT, JPN, UK, ITA, SPN, IRL, GRC, PRT (17 COUNTRIES) Illustration 4 L Laspeyres 1 1.017145 1.125244 1.200814 1.153729 1.193631 1.204422 1.228753 1.24982 1.382647 1.496306 1.541876 1.55893 1.853359 2.325651 2.637944 3.625528 0.913931 1 1.127496 1.141108 1.121873 1.172337 1.182936 1.208041 1.232445 1.367521 1.460823 1.463747 1.500802 1.823941 2.241174 2.585709 3.45907 0.968506 1.017145 1 1.120752 1.065026 1.081122 1.111821 1.124119 1.159512 1.294338 1.388189 1.420487 1.474029 1.73846 2.192406 2.464527 3.293661 0.938943 0.939882 1.034584 1 1.022243 1.069295 1.070365 1.088717 1.136553 1.261119 1.368889 1.392359 1.37163 1.670294 2.083397 2.391689 3.326763 0.888696 0.897627 1.003004 1.038731 1 1.043937 1.041852 1.059715 1.087628 1.208041 1.30604 1.321807 1.349858 1.604801 2.027898 2.30712 3.183559 0.969475 0.963676 1.006018 1.05654 1.012072 1 1.021222 1.042894 1.082204 1.227525 1.30604 1.239861 1.247323 1.617691 1.873859 2.190215 2.857651 0.887807 0.913017 0.97824 1.00904 0.979219 1.00904 1 1.02429 1.063962 1.177036 1.286596 1.243587 1.266174 1.569881 1.902178 2.166255 2.880604 0.878095 0.887807 0.959829 0.98906 0.960789 0.986097 0.984127 1 1.041852 1.16649 1.263644 1.226298 1.234912 1.543418 1.847807 2.104336 2.903741 0.848742 0.875465 0.928671 0.963676 0.935195 0.965605 0.969475 0.976285 1 1.135417 1.218962 1.18649 1.233678 1.508325 1.847807 2.100131 2.869104 0.73565 0.773368 0.837779 0.880733 0.847046 0.876341 0.875465 0.895834 0.917594 1 1.11071 1.102962 1.09308 1.355269 1.647073 1.845961 2.567672

Price Level Computation 89 0.729059 0.768741 0.802518 0.939882 0.842821 0.840296 0.855559 0.877217 1.121873 1.030454 1 1.091988 1.072508 1.321807 1.574598 1.782469 2.539583 0.79692 0.799315 0.826959 0.88692 0.838618 0.842821 0.854704 0.865887 0.889585 1.004008 1.055484 1 1.041852 1.341783 1.551155 1.829421 2.325651 0.758054 0.747515 0.805735 0.869358 0.812207 0.825306 0.813833 0.834435 0.858988 0.960789 1.078962 0.998002 1 1.273794 1.506817 1.709156 2.37976 0.563831 0.562704 0.631915 0.67977 0.631283 0.662324 0.64856 0.66365 0.673006 0.754273 0.863294 0.812207 0.827786 1 1.231213 1.375751 1.950332 0.493121 0.50258 0.547167 0.551011 0.547715 0.562704 0.55046 0.564395 0.593926 0.664978 0.718923 0.675028 0.67368 0.875465 1 1.133148 1.630684 0.478547 0.476637 0.523614 0.695586 0.555992 0.528876 0.52782 0.537944 0.544438 0.635082 0.71177 0.653769 0.619402 0.824482 0.979219 1 1.55426 0.370093 0.374186 0.396531 0.439551 0.39812 0.39812 0.409016 0.47001 0.54335 0.461626 0.462088 0.62688

0.39812 0.408199 0.689354 0.80493 1

L power 2 …… L power 16 L power 17 = M derived Laspeyres 0.310985711 0.316317561 0.349934805 0.360952499 0.354868131 0.370459972 0.369720072 0.376059081 0.385964515 0.428695252 0.462083968 0.457943038 0.462083244 0.569492731 0.694189753 0.783480195 1.05864531 0.305743734 0.310985711 0.344036303 0.354868282 0.348886472 0.364215497 0.363488069 0.369720228 0.379458696 0.421469163 0.454295079 0.450223948 0.454294367 0.55989336 0.682488487 0.770273849 1.04080078 0.310985711 0.316317561 0.349934805 0.360952499 0.354868131 0.370459972 0.369720072 0.376059081 0.385964515 0.428695252 0.462083968 0.457943038 0.462083244 0.569492731 0.694189753 0.783480195 1.05864531 0.287363033 0.292289872 0.323353528 0.333534311 0.327912116 0.34231959 0.341635893 0.347493387 0.356646398 0.39613128 0.426983767 0.423157385 0.426983098 0.526233691 0.641458645 0.723966526 0.978229919 0.274443831 0.279149171 0.308816275 0.318539352 0.313169918 0.326929665 0.326276705 0.331870859 0.340612371 0.378322101 0.407787529 0.404133173 0.40778689 0.502575397 0.612620094 0.691418606 0.934250883 0.294637899 0.299689466 0.331539528 0.341978047 0.33621352 0.350985734 0.350284729 0.356290511 0.365675239 0.406159717 0.437793265 0.433870014 0.437792578 0.539555794 0.657697776 0.742294423 1.00299473 0.279149227 0.283935241 0.314110993 0.324000775 0.31853928 0.332534941 0.331870786 0.337560854 0.34645224 0.384808511 0.41477913 0.411062119 0.41477848 0.511192156 0.623123591 0.703273119 0.950268806 0.271441428 0.276095291 0.305437838 0.315054545 0.309743853 0.323353068 0.322707252 0.328240207 0.336886087 0.374183273 0.403326351 0.399711973 0.403325719 0.497077244 0.605918057 0.683854516 0.924030218 0.267667939 0.272257105 0.301191742 0.310674761 0.305437896 0.31885792 0.318221082 0.32367712 0.332202808 0.368981501 0.397719441 0.394155309 0.397718818 0.49016704 0.597494784 0.674347796 0.911184655 0.23645242 0.240506397 0.266066667 0.274443774 0.269817633 0.281672611 0.281110041 0.285929793 0.293461213 0.325950764 0.351337276 0.348188795 0.351336726 0.433003608 0.527814756 0.595705146 0.804922017

90 Concept and Method 0.235037744 0.239067466 0.264474811 0.272801798 0.268203336 0.279987386 0.279428182 0.284219098 0.291705458 0.324000626 0.349235253 0.346105609 0.349234706 0.430412981 0.524656883 0.59214109 0.800106233 0.244385553 0.248575544 0.274993378 0.283651542 0.278870191 0.291122911 0.290541466 0.295522924 0.303307028 0.336886625 0.363124871 0.359870756 0.363124302 0.447531161 0.545523286 0.615691443 0.831927677 0.228548028 0.232466483 0.257172297 0.265269364 0.260797871 0.272256547 0.271712784 0.276371416 0.283651067 0.315054521 0.339592386 0.336549155 0.339591853 0.418528684 0.510170381 0.575791256 0.778014195 0.172043222 0.174992903 0.193590604 0.199685802 0.196319813 0.204945517 0.20453619 0.208043051 0.213522926 0.237162384 0.255633658 0.253342817 0.255633257 0.315054233 0.384039002 0.433436176 0.585662762 0.153660686 0.156295198 0.172905765 0.178349701 0.175343363 0.183047425 0.182681834 0.185813992 0.190708352 0.211821972 0.228319621 0.226273552 0.228319263 0.281391205 0.343005064 0.387124231 0.523085656 0.145728776 0.148227296 0.163980431 0.169143354 0.166292202 0.173598582 0.173251863 0.17622234 0.180864055 0.200887798 0.216533844 0.214593392 0.216533504 0.266865891 0.325299265 0.367141017 0.496084161 0.113607209 0.115555005 0.127835831 0.13186074 0.12963804 0.135333946 0.135063651 0.137379375 0.140997962 0.156608067 0.168805409 0.167292672 0.168805144 0.208043256 0.253596733 0.286215717 0.386737187

Inconsistency case: some diagonal elements < 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.310985711 0.310985711 0.349934805 0.333534311 0.313169918 0.350985734 0.331870786 0.328240207 0.332202808 0.325950764 0.349235253 0.359870756 0.339591853 0.315054233 0.343005064 0.367141017 0.386737187

Effective paths - Factor counts - Efﬁciencies 1 2 3 4

1,2,1 2,1,2 3,1,3 4,2,4

2 2 2 2

0.557660928 0.557660928 0.591552876 0.577524295

Price Level Computation 91 5 6 7 8 9 10 11 12 13 14 15 16 17

5,2,5 2 0.559615866 6,5,6 2 0.59244049 7,5,7 2 0.576082274 8,5,8 2 0.572922514 9,5,9 2 0.576370374 10,5,10 2 0.570920978 11,2,11 2 0.590961296 12,9,12 2 0.599892287 13,6,13 2 0.582745102 14,5,14 2 0.56129692 15,6,15 2 0.585666342 16,14,16 2 0.605921626 17,6,17 2 0.621881972

Critical cost efﬁciency 0.557660928 L adjusted Laspeyres to replace original L 1 2.15977477 2.79548005

1.82394884 2.20340522 3.32345142

2.01779243 2.24118266 4.17036748

2.15330488 2.47936861 4.73037265

2.06887186 2.68318242 6.50131257

2.1404243 2.76489875

1.63886504 2.12124598 2.69124467

1 2.16626437 3.27069893

2.02183073 2.21002573 4.01888296

2.04623983 2.4522446 4.63670462

2.01174754 2.61955415 6.20281936

2.1022398 2.62479748

1.73672917 1.99372225 2.64323521

1.82394884 2.01577508 3.11741403

1 2.07924196 3.93143197

2.00973736 2.32101253 4.41940053

1.90980925 2.48930655 5.90620722

1.93867267 2.54722346

1.68371667 1.91938317 2.45961288

1.68540049 1.95229206 2.99517846

1.85522053 2.03807178 3.73595656

1 2.26144407 4.28878711

1.83309059 2.45469771 5.96556587

1.91746444 2.49678421

1.59361353 1.86825354 2.42057123

1.60962864 1.90028554 2.87773613

1.79859113 1.95033926 3.63643551

1.86265694 2.16626437 4.13713761

1 2.34199661 5.70877184

1.87199237 2.37027006

1.73846678 1.83125973 2.23670503

1.72806799 1.87012205 2.90085053

1.80399585 1.94061292 3.3602121

1.89459212 2.20120317 3.92750306

1.81485191 2.34199661 5.12435219

1 2.22332413

1.59201937 1 2.27050872

1.63722605 1.83676128 2.81511743

1.75418422 1.90790128 3.41099386

1.80941491 2.11066607 3.88453788

1.75593977 2.30712954 5.16551161

1.80941491 2.23000561

1.57460377 1.76827343 2.1990029

1.59201937 1.76474081 2.21444957

1.72116954 1 2.76766387

1.77358669 1.86825354 3.31349554

1.72289101 2.09175494 3.77350446

1.52196784 1.73846678 2.21223675

1.56988764 1.75067851 2.70473495

1.66529687 1 3.31349554

1.72806799 2.0360347 3.76596404

1.67699574 2.18584796 5.14488976

1.31917078 1.57145849 1.97783625

1.38680686 1.56988764 1.96011581

1.50230894 1.60641342 2.43027426

1.57933424 1.64543355 2.95353846

1.51892657 1 3.31018529

2.26597191 5.20700098 1.73152708 2.12761903

1.99173 4.60436059

92 Concept and Method 1.30735177 1.53419212 1.92322601

1.3785097 1.57302934 2.37027006

1.43907877 2.01174754 2.82357598

1.68540049 1.84781459 3.19633116

1.51135028 1 4.55399128

1.50682244 1.95815763

1.42904041 1.53265893 1.86825354

1.43333513 1.55271233 2.40609111

1.48290647 1.59520769 2.78153789

1.5904288 1.80039151 3.28052569

1.50381344 1.8926985 4.17036748

1.51135028 1

1.35934573 1.45936887 1

1.34044715 1.49631247 2.28417294

1.4448475 1.54034102 2.70203079

1.55893655 1.72289101 3.06486597

1.45645312 1.93479935 4.26739598

1.47994231 1.78962152

1.0110642 1.16300061 1.48438945

1.00904326 1.19006007 1

1.13315273 1.20683729 2.2078165

1.21896652 1.35256562 2.46700267

1.13201942 1.54806255 3.49734382

1.18768228 1.45645312

0.884266721 0.987087265 1.2080459

0.901228641 1.01207557 1.56988764

0.981182243 1.06503069 1

0.988075321 1.19244144 2.03196592

0.982164919 1.28917585 2.92414964

1.00904326 1.21046314

0.858132561 0.946489118 1.11071436

0.85470754 0.964643519 1.47846471

0.938946901 0.976288588 1.75593977

1.24732784 1.1388318 1

0.997007271 1.27634906 2.78710579

0.948382742 1.17234141

0.663652376 0.713910515 0.82861821

0.670991962 0.731984221 1.12412394

0.711061113 0.733449269 1.23615259

0.788204764 0.842823975 1.44340398

0.713910515 0.974337582 1

0.713910515 0.82778975

L power 2 …… L power 16 L power 17 = M derived Laspeyres 1 2.15977477 2.79548005

1.82394884 2.20340522 3.32345142

2.01779243 2.24118266 4.17036748

2.15330488 2.47936861 4.73037265

2.06887186 2.68318242 6.50131257

2.1404243 2.76489875

1.63886504 2.12124598 2.69124467

1 2.16626437 3.27069893

2.02183073 2.21002573 4.01888296

2.04623983 2.4522446 4.63670462

2.01174754 2.61955415 6.20281936

2.1022398 2.62479748

1.73672917 1.99372225 2.64323521

1.82394884 2.01577508 3.11741403

1 2.07924196 3.93143197

2.00973736 2.32101253 4.41940053

1.90980925 2.48930655 5.90620722

1.93867267 2.54722346

1.68371667 1.91938317 2.45961288

1.68540049 1.95229206 2.99517846

1.85522053 2.03807178 3.73595656

1 2.26144407 4.28878711

1.83309059 2.45469771 5.96556587

1.91746444 2.49678421

1.59361353 1.86825354 2.42057123

1.60962864 1.90028554 2.87773613

1.79859113 1.95033926 3.63643551

1.86265694 2.16626437 4.13713761

1 2.34199661 5.70877184

1.87199237 2.37027006

1.73846678 1.83125973 2.23670503

1.72806799 1.87012205 2.90085053

1.80399585 1.94061292 3.3602121

1.89459212 2.20120317 3.92750306

1.81485191 2.34199661 5.12435219

1 2.22332413

1.59201937 1 2.27050872

1.63722605 1.83676128 2.81511743

1.75418422 1.90790128 3.41099386

1.80941491 2.11066607 3.88453788

1.75593977 2.30712954 5.16551161

1.80941491 2.23000561

1.57460377 1.76474081 2.21444957

1.59201937 1 2.76766387

1.72116954 1.86825354 3.31349554

1.77358669 2.09175494 3.77350446

1.72289101 2.26597191 5.20700098

1.76827343 2.1990029

Price Level Computation 93 1.52196784 1.73846678 2.21223675

1.56988764 1.75067851 2.70473495

1.66529687 1 3.31349554

1.72806799 2.0360347 3.76596404

1.67699574 2.18584796 5.14488976

1.73152708 2.12761903

1.31917078 1.56988764 1.96011581

1.38680686 1.60641342 2.43027426

1.50230894 1.64543355 2.95353846

1.57933424 1 3.31018529

1.51892657 1.99173 4.60436059

1.57145849 1.97783625

1.30735177 1.53419212 1.92322601

1.3785097 1.57302934 2.37027006

1.43907877 2.01174754 2.82357598

1.68540049 1.84781459 3.19633116

1.51135028 1 4.55399128

1.50682244 1.95815763

1.42904041 1.53265893 1.86825354

1.43333513 1.55271233 2.40609111

1.48290647 1.59520769 2.78153789

1.5904288 1.80039151 3.28052569

1.50381344 1.8926985 4.17036748

1.51135028 1

1.35934573 1.45936887 1

1.34044715 1.49631247 2.28417294

1.4448475 1.54034102 2.70203079

1.55893655 1.72289101 3.06486597

1.45645312 1.93479935 4.26739598

1.47994231 1.78962152

1.0110642 1.16300061 1.48438945

1.00904326 1.19006007 1

1.13315273 1.20683729 2.2078165

1.21896652 1.35256562 2.46700267

1.13201942 1.54806255 3.49734382

1.18768228 1.45645312

0.884266721 0.987087265 1.2080459

0.901228641 1.01207557 1.56988764

0.981182243 1.06503069 1

0.988075321 1.19244144 2.03196592

0.982164919 1.28917585 2.92414964

1.00904326 1.21046314

0.858132561 0.946489118 1.11071436

0.85470754 0.964643519 1.47846471

0.938946901 0.976288588 1.75593977

1.24732784 1.1388318 1

0.997007271 1.27634906 2.78710579

0.948382742 1.17234141

0.663652376 0.713910515 0.82861821

0.670991962 0.731984221 1.12412394

0.711061113 0.733449269 1.23615259

0.788204764 0.842823975 1.44340398

0.713910515 0.974337582 1

0.713910515 0.82778975

Consistency case: all diagonal elements = 1 Hence immediately the wanted ﬁnal answer: P/P mean basic price index 1 1.16241544 1.43721337

1.02510404 1.1847125 1.80377676

1.0858391 1.2219059 2.14583724

1.14268694 1.35642562 2.32196113

1.12972093 1.41731595 3.16721761

1.14936063 1.40970785

0.975510742 1.13394875 1.40201709

1 1.15569977 1.7596036

1.0592477 1.19198233 2.09328728

1.11470339 1.32320776 2.26509802

1.1020549 1.38260694 3.0896548

1.12121364 1.37518515

0.920946763 1.07052274 1.32359701

0.944066245 1.09105714 1.66118237

1 1.12531028 1.97620186

1.05235384 1.24919578 2.13840258

1.04041283 1.30527254 2.91683881

1.05849995 1.29826588

0.87513033 1.01726501 1.25774901

0.897099634 1.03677784 1.57853975

0.950250725 1.06932691 1.87788725

1 1.1870492 2.03201861

0.988653048 1.24033618 2.77172819

1.00584035 1.2336781

0.885174361 1.02894035 1.27218443

0.907395811 1.04867713 1.59665694

0.961156927 1.08159977 1.89944011

1.01147718 1.20067318 2.05534046

1 1.25457174 2.80353982

1.01738456 1.24783725

94 Concept and Method 0.870048942 1.01135833 1.25044598

0.891890682 1.03075786 1.56937406

0.944733155 1.06311793 1.86698342

0.994193564 1.18015667 2.02021982

0.982912497 1.23313424 2.75563433

1 1.22651483

0.860277629 1 1.23640251

0.88187407 1.01918166 1.55174879

0.934123082 1.05117831 1.84601577

0.983028013 1.16690262 1.99753121

0.971873641 1.21928521 2.72468646

0.988769238 1.21274013

0.844086644 0.98117935 1.21313261

0.865276626 1 1.52254387

0.916542278 1.03139445 1.81127255

0.964526786 1.14494075 1.95993638

0.953582347 1.19633747 2.67340608

0.970159958 1.18991557

0.818393628 0.951313391 1.17620627

0.838938612 0.969561163 1.4761994

0.888643797 1 1.75613952

0.935167713 1.11009008 1.90027819

0.924556409 1.15992234 2.59203071

0.940629417 1.15369592

0.737231726 0.856969542 1.0595593

0.755739218 0.873407642 1.32980145

0.80051503 0.900827793 1.58197929

0.842425066 1 1.71182341

0.832866109 1.04489028 2.33497331

0.847345121 1.03928135

0.705558983 0.820152657 1.01403881

0.723271361 0.835884548 1.27267089

0.766123527 0.862126681 1.51401474

0.806233036 0.95703828 1.63828053

0.797084748 1 2.23465884

0.810941717 0.994632038

0.709366837 0.824578965 1.01951151

0.727174808 0.84039576 1.27953941

0.770258244 0.866779521 1.52218577

0.810584221 0.962203351 1.64712222

0.801386561 1.00539693 2.24671914

0.815318315 1

0.695790909 0.808798097 1

0.71325807 0.824312189 1.25505147

0.755516971 0.850191015 1.49305404

0.795071186 0.943788615 1.61559944

0.786049551 0.986155554 2.20372122

0.799714678 0.980861908

0.554392331 0.644434206 0.796780072

0.568309816 0.656795525 1

0.601980867 0.677415258 1.18963571

0.633496877 0.751991961 1.28727744

0.626308618 0.785749094 1.75588115

0.637196719 0.781531222

0.46601857 0.541707182 0.669768121

0.477717517 0.552098026 0.840593464

0.506021182 0.569430838 1

0.532513334 0.632119527 1.082077

0.526470931 0.660495552 1.47598222

0.535623397 0.656950037

0.430670431 0.50061796 0.618965304

0.441481998 0.510220645 0.776833314

0.467638792 0.52623874 0.924148649

0.492121478 0.584172407 1

0.486537399 0.610396072 1.36402698

0.494995639 0.607119489

0.315734542 0.367014707 0.453777906

0.323660753 0.374054659 0.569514627

0.342836909 0.385797898 0.677514936

0.360785738 0.428270421 0.733123332

0.356691919 0.447495602 1

0.362892852 0.445093462

5 EUKLEMS data Household Consumption in Italy 1992–2004, from ISTAT’s tables. Data collected for the EUKLEMS database concerning Italy. This includes only Italy because at this level of detail the EUKLEMS project does not provide the data collected from the national statistical institutes. Initial treatment just for the ﬁve years 1999–2004. Other years dealt with elsewhere together with inputs of production concerning other countries.

Price Level Computation 95 Illustration 5: 1999-2004 L Laspeyres 1 1.02659 1.05674 1.08751 1.1173

0.9741 1 1.02939 1.0593 1.08823

0.94679 0.97171 1 1.02895 1.0572

0.9206 0.94472 0.97205 1 1.02735

0.89744 0.92068 0.94724 0.97412 1

L power 2 1 1.02659 1.05674 1.08733262 1.11716604

0.9741 1 1.02937043 1.05917071 1.08823

0.946542711 0.97171 1 1.02895 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 3 - final power followed by repetitions 1 0.9741 0.946542711 0.920086842 1.02659 1 0.97171 0.944550706 1.05674 1.02937043 1 0.97205 1.08733262 1.05917071 1.02895 1 1.11707117 1.08813903 1.05709178 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 4 1 1.02659 1.05674 1.08733262 1.11707117

0.9741 1 1.02937043 1.05917071 1.08813903

0.946542711 0.97171 1 1.02895 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 5 = M derived Laspeyres 1 0.9741 0.946542711 1.02659 1 0.97171 1.05674 1.02937043 1 1.08733262 1.05917071 1.02895 1.11707117 1.08813903 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

Paths 1,1,1,1,1,1 2,1,1,1,1,1 3,1,1,1,1,1 4,3,1,1,1,1 5,2,3,1,1,1

1,3,3,3,3,4 2,3,3,3,3,4 3,3,3,3,3,4 4,4,4,4,4,4 5,4,4,4,4,4

1,4,4,4,4,5 2,4,4,4,4,5 3,4,4,4,4,5 4,4,4,4,4,5 5,5,5,5,5,5

0.919681778 0.944134871 0.971864522 1 1.02656757

0.89519811 0.919000215 0.94599165 0.973378109 1

1,1,1,1,1,2 2,2,2,2,2,2 3,1,1,1,1,2 4,1,1,1,1,2 5,2,1,1,1,2

1,2,2,2,2,3 2,2,2,2,2,3 3,3,3,3,3,3 4,3,3,3,3,3 5,4,3,3,3,3

Consistency case: all diagonal elements = 1 H derived Paasche 1 0.974098715 1.02658865 1 1.05647636 1.02911363 1.08685393 1.05870441 1.115729 1.08683162

0.946306566 0.971467576 1 1.02875367 1.05608515

The 10 basic price level systems - the 10 columns of M and H

96 Concept and Method F derived Fisher - mean of derived Laspeyres 1 0.974099358 0.946424631 1.02658932 1 0.97158878 1.05660817 1.02924202 1 1.08709325 1.05893754 1.02885183 1.11639988 1.08748513 1.05658835

M and derived Paasche H 0.919884288 0.89573639 0.944342766 0.919552808 0.971957257 0.946442391 1 0.973748984 1.02695871 1

P mean basic price level - mean of columns of F 0.946499834 0.971666239 1.00007959 1.02893372 1.05667244 P/P mean basic 1 1.02658892 1.05660831 1.08709339 1.11640003

price index 0.974099743 1 1.02924189 1.0589374 1.08748498

0.946424506 0.971588908 1 1.02885183 1.05658835

0.919884167 0.94434289 0.971957257 1 1.02695871

0.895736272 0.91955293 0.946442391 0.973748983 1

0.990092751 0.995005771 0.993835084 1 1.00644396

0.983753485 0.988635049 0.987471858 0.993597301 1

X mean basic quantity level: PX = px 768307.584 772120.065 771211.618 775995.565 780996.047 X/X mean basic 1 1.00496218 1.00377978 1.01000639 1.01651482

quantity index 0.99506232 1 0.998823437 1.0050193 1.0114956

0.996234453 1.00117795 1 1.00620316 1.01268709

Price Level Computation 97 NOTES

1 Price levels Starting with the Laspeyres matrix ⎤ ⎡ 1 L12 L13 L14 ⎢L21 1 L23 L24 ⎥ ⎥ L=⎢ ⎣L31 L32 1 L34 ⎦ . L41 L42 L43 1 in the case of 4 periods, we can always redeﬁne the order of periods so that raising the L matrix to power m = 4 in the arithmetic where + means min yields, in the case of consistency of the data, ⎡ ⎤ 1 L12 L123 L1234 ⎢ L21 1 L23 L234 ⎥ ⎥ M =⎢ ⎣ L321 L32 1 L34 ⎦ L4321 L432 L43 1 where Lrij...ks = Lri Lij . . . Lks . By normalising each column of the M matrix by the respective ﬁrst element, we obtain ⎡ ⎤ 1 1 1 1 ⎢ L21 K21 K21 K21 ⎥ ⎥ A=⎢ ⎣ L321 L32 K21 K321 K321 ⎦ L4321 L432 K21 L43 K321 K4321 where Kij = 1/Lji is the Paasche index of i over j. In the matrix A, the Afriat upper bound for the price level solution is given by the ﬁrst column so that P1 = 1

P2 = L21

P3 = L321

P4 = L4321 ,

which is also equal to the last column of the matrix B deﬁned below. In the case of consistency of the data, raising the Paasche matrix ⎡ ⎤ 1 K12 K13 K14 ⎢K21 1 K23 K24 ⎥ ⎥ K ≡⎢ ⎣K31 K32 1 K34 ⎦ K41 K42 K43 1 to the power m =4 in the arithmetic where + now means max yields ⎡ ⎤ 1 K12 K123 K1234 ⎢ K21 1 K23 K234 ⎥ ⎥ H ≡⎢ ⎣ K321 K32 1 K43 ⎦ K4321 K432 K34 1

98 Concept and Method By normalising each column of the H matrix by the respective ﬁrst element, we obtain ⎡ ⎤ 1 1 1 1 ⎢ K21 L21 L21 L21 ⎥ ⎥ B=⎢ ⎣ K321 K32 L21 L321 L321 ⎦ K4321 K432 L21 K43 L321 L4321 In the matrix B, the Afriat lower bound for the price level solution is given by the ﬁrst column so that P1 = 1

P2 = K21

P3 = K321

P4 = K4321 ,

which coincides with the last column of the matrix A.

2 The triangle equality From the m × m Laspeyres matrix L we obtain the derived Laspeyres matrix M = Lm (where + means min), in the consistency case necessarily having the triangle inequality property Mij Mjk ≥ Mik . But this is without need for the triangle equality property Mij Mjk = Mik which corresponds to Fisher’s price index Chain Test, equivalent to the elements having the form of ratios of some numbers, for instance, for any k, Mij = Mik /Mjk for all i, j. We then obtain the derived Paasche matrix H where Hij = 1/Mji , and then F (for Fisher), the geometric mean of M and H , with elements 1 Fij = Hij Mij 2 1 = Mij /Mji 2 The geometric mean of the columns of F has elements 1

Fi = (k Fik ) m

Price Level Computation 99 with ratios 1 1 Fi /Fj = (k Fik ) m / k Fjk m 1 = k Fik /Fjk m

1 (Mik /Mki ) / Mjk /Mkj 2

= k

1 2

1 = k Mik Mkj / Mjk Mki 2

m1

m1

Hence subject to the triangle equality,

Fi /Fj = k Mij /Mji

12

m1

1 = Mij /Mji 2 = Fij . This shows how, subject to the triangle equality for M , from F we obtain the price levels Fi representing the geometric mean of the columns of F and hence of all the columns of M and H , and then from the price indices which are their ratios we just get back to F. Our Illustrations nos. 1 and 2 are examples.

3 Ratio matrix A matrix M is a ratio matrix if its elements Mij have the form of ratios of some numbers Xi , (i)

Mij = Xi /Xj

forming the elements of a vector X , the base vector from which it is derived. A test for M being a ratio matrix is the triangle equality (ii) Mij Mjk = Mik . Where M a matrix of price indices this would correspond to Fisher’s Chain Test. In case M is a derived Laspeyres matrix one would just have the triangle inequality (iii) Mij Mjk ≥ Mik , with ordinary data without the further imposition.

100 Concept and Method From (i) obviously (ii). And from (ii), for any k, we have (iv)

Mij = Mik /Mjk

for all i, j which exhibits (i) with Xi = Mik making base vector X coincide with column k of M .Hence: For a given matrix to be a ratio matrix the triangle equality is necessary and sufﬁcient and then any column of it is a base vector from which it can be derived. Hence given that M is a ratio matrix its columns, though different, would all be base vectors for the same ratio matrix, coinciding with the matrix M itself. The test for the triangle inequality is that the matrix be idempotent, or reproduced when multiplied by itself (with + = min). The test for the triangle equality is that the matrix coincide with the ratio matrix derived from any of its columns.

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Part II

Precursor

1 The system of inequalities ars > Xs − Xr S.N. Afriat, Research Memorandum No. 18 (October 1960) Econometric Research Program, Princeton University. Proc. Cambridge Phil. Soc. 59 (1963)

Proc. Camb. Phil. Soc. (1963), 59, 125

The system of inequalities ars > Xs − Xr ∗ Princeton University, Princeton, New Jersey, and Rice University, Houston, Texas. (Received 30 November 1961

Introduction In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, . . ., k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr , such that gr = g(xr ) has the direction of pr .† Let ur = pr /er , where er = pr xr so that ur xr = 1. Then gr = ur λr , for some multipliers λr > 0, since φ is increasing. Also, if φr = φ(xr ) then (xr − xs ) gs > φr − φs , since φ is convex. Accordingly λr > 0,

λs Dsr > φr − φs

(r = s),

(I)

where Dsr = usxr − 1. With the number Drs given, there has to be considered all solutions = λr , = φr of the system of inequalities (I). With this system, there is involved a consideration of systems of the form ars > Xr − Xs (where ars = λs Dsr , Xr = φr ), the theory of which is going to be developed here. It is remarked, incidentally, that the existence of , satisfying (I) is equivalent to the existence of satisfying λr > 0,

λr Drs + λs Dst + · · · + λq Dqr > 0

(II)

for all distinct r, s, t, . . . , q taken from 1, . . . , k. ∗ Revised version of Research Memorandum no. 18, Econometric Research Program, Princeton University, October 1960, issued with the partial support of the U.S. Ofﬁce of Naval Research. † S. N. Afriat. Preference analysis: a general method with application to the cost of living index. (Research Memorandum, no. 29, Econometric Research Program, Princeton University, August 1961.)

114 Precursor Further, satisﬁes (II) if and only if there exists a such that , satisfy (I). Moreover, these two equivalent conditions on the number Drs which are provided by the consistency of the system of inequalities (I) and (II) are equivalent to a condition, applying directly to these numbers, which is given by the familiar Houthakker‡ ‘revealed preference’ axiom, which can be stated Drs 0, Dst 0, . . . , Dqr 0

(III)

impossible, for all distinct r, s, t, . . . , q taken from 1, . . . , k. Apart from any independent interest, the results which are now going to be obtained can be applied to a demonstration of these propositions, which are of fundamental importance for a method of empirical preference analysis in economics.

1 Open and closed systems Let n(n − 1) numbers ars (r = s; r, s = 1, . . . , n) be given; and consider the system of simultaneous inequalities S(a): ars > Xr − Xs

(r = s; r, s = 1, . . . , n)

deﬁning the open system S(a), of order n, with coefﬁcients ars . Any set of n numbers Xr (r = 1, . . ., n), forming a vector X , which satisfy these inequalities, deﬁne a solution X of the system S(a); and the system is said to be consistent if it has solutions. With the open system S(a), there may also be considered the closed system S(a), deﬁned by ¯ S(a): ars Xr − Xs

(r = s; r, s = 1, . . . , n).

¯ Obviously, solutions of S(a) are solutions of S(a), and the consistency of S(a) ¯ implies the consistency of S(a), but not conversely.

2 Chain coefﬁcients Let r, l, m, . . ., p, s denote any chain, that is a sequence of elements taken from 1, . . ., n with every successive pair distinct. Now from the coefﬁcients ars of a system there can be formed the chain coefﬁcient arlm···ps , determined on any chain, by the deﬁnition arlm···ps = arl + alm + · · · + aps .

‡ H. S. Houthakker. Revealed preference and the utility function, Economica, 17 (1950), 159–74.

The system of inequalities ars > Xs − Xr 115 Obviously ar···s···t = ar···s + as···t . Chains are considered associated with their coefﬁcients, so that by a positive chain is meant one with positive coefﬁcient, and so on similarly. A simple chain is one without loops, that is one in which no elements is repeated. There are n(n − 1) · · · (n − r + 1) = n!/r! simple chains of length r n, and therefore altogether

1 1 1 n! 1 + + + · · · + (n − 1)! 1! 2!

simple chains. A chain r, l, m, . . .p, s whose extremities are the same, that is, with r = s, deﬁnes a cycle. A simple cycle is one without loops. There are (n − 1) · · · (n − r + 1) = (n − 1)!/r! simple cycles of r n elements, and the total number of simple cycles is made up accordingly. The coefﬁcients ars + asr on the cycles of two elements deﬁne the intervals of the system. Any chain can be represented uniquely as a simple chain, with loops at certain of its elements, given by cycles through those elements; and the coefﬁcient on it is then expressed as the sum of coefﬁcients on the simple chain and on the cycles. Also, any cycles can be represented uniquely as a simple cycle, looping in simple cycles at certain of its elements, which loop in simple cycles at certain of their elements, and so forth, with termination in simple cycles. The coefﬁcient on the cycle is then expressed as a sum of coefﬁcients on simple cycles. Thus out of these generating elements of simple chains and cycles, ﬁnite in number, is formed the inﬁnite set of all possible chains.

3 Minimal chains Theorem 3.1 For the chains to have a minimum it is necessary and sufﬁcient that the cycles be non-negative. If any cycle total should be negative, then by taking chains which loop repeatedly round that cycle, chains which have increasingly negative coefﬁcients are obtained without limit; and so no minimum exists. However, should every cycle coefﬁcient be non-negative, then by cancelling the loops on any chain, there can be no increase in the coefﬁcient, so no chain coefﬁcient will be smaller than the coefﬁcient for some simple chain. But there is only a ﬁnite number of simple chains on a ﬁnite number of elements, and the coefﬁcients on these have a minimum.

116 Precursor Theorem 3.2 For the cycles to be non-negative it is necessary and sufﬁcient that the simple cycles be non-negative. For the coefﬁcient on any cycle can be expressed as a sum of coefﬁcients on simple cycles. Theorem 3.3 If the cycles are non-negative then a minimal chain with given extremities always exists and can be chosen to be simple. For any chain is then not less than the chain obtained from it by cancelling loops, since the cancelling is then the subtraction of a sum of non-negative numbers.

4 Derived systems According to Theorem 3.3, if the cycles of S(a) are non-negative, that is arlm···pr 0 for every cycle r, l, m, . . ., p, r, or equivalently for every simple cycle, by Theorem 3.2, then the coefﬁcients αrlm···ps on the chains with given extremities r, s have a minimum, and it is possible to deﬁne Ars = min arlm···ps l,m,...,p

(r, s = 1, . . . , n).

Then arlm···ps Ars for every chain and, by Theorem 3.3, the equality is attained for some simple chain. In particular, ars Ars . The number Arr is the minimum coefﬁcient for the cycle through r, so that arlm···pr Arr for every cycle, the equality being attained for some simple cycle. In particular, for a chain of two elements, ars + asr Arr . The hypothesis of non-negative cycles now has the statement Arr 0. The numbers Ars (r = s), thus constructed from the coefﬁcient of S(a), deﬁne the coefﬁcients of a system S(A), which will be called the derived system of S(a).

The system of inequalities ars > Xs − Xr 117 Any two systems will be said to be equivalent if any solution of one is also a solution of the other. Theorem 4.1 Any system and its derived system, when it exists, are equivalent. Let a system S(a) have a solution X . Then, for any chain of elements r, l, m, . . ., p, s there are the relations arl > Xr − Xl ,

alm > Xl − Xm ,

...,

aps > Xp − Xs

from which, by addition, there follows the relation arlm···ps > Xr − Xs . This implies that the derived coefﬁcients Ars exist, and Ars > Xr − Xs . That is, X is a solution of S(A). Now suppose the derived coefﬁcients Ars of S(a) are deﬁned, in which case ars Ars and let X be any solution of S(A), so that Ars > Xr − Xs . Then it follows immediately that ars > Xr − Xs or that X is a solution of S(a). Thus S(a) and S(A) have the same solutions, and are equivalent. Theorem 4.2 If the cycles of a system are non-negative or positive , then so correspondingly are the intervals of the derived system . Since Ars is the coefﬁcient of some chain with extremities r, s it appears that Ars + Asr is the coefﬁcient of some cycle through r, and therefore if the cycles of S(a) are non-negative, or positive, so correspondingly are the intervals Ars + Asr of the derived system S(A).

118 Precursor

5 Triangle inequality From the relation ar···s + as···t = ar···t it follows that the derived coefﬁcients Ars (r = s) satisfy the triangle inequality Ars + Ast Art the one side being the minimum for chains connecting r, t restricted to include s, and the other side being the minimum without this restriction. Theorem 5.1 Any system non-negative cycles is equivalent to a system which satisﬁes the triangle inequality , given by its derived system. This is true in view of Theorems 3.1, 4.1 and 4.2. Theorem 5.2 Any system which satisﬁes the triangle inequality has all its intervals non - negative. Thus, from the triangle inequalities applied to any system S(a), atr + ars ats ,

ats + asr atr

there follows, by addition, the relation ars + asr 0. Theorem 5.3 If a system satisﬁes the triangle inequality, then its derived system exists, and, moreover, the two systems are identical. From the triangle inequality, it follows by induction that arl + alm + · · · + aps ars . That is, arlm···ps ars , from which it appears that the derived system exists, with coefﬁcients Ars ars ,

The system of inequalities ars > Xs − Xr 119 so that now Ars = ars . This shows, what is otherwise evident, that no new system is obtained by repeating the operation of derivation, since the ﬁrst derived system satisﬁes the triangle inequality.

6 Extension property of solutions A subsystem Sm (a) of order m n of a system S(a) of order n is deﬁned by Sm (a): ars > Xr − Xs

(r, s = 1, . . . , m).

Then the systems Sm (a) (m = 2, 3, . . ., n) form a nested sequence of subsystems of S(a), each being a subsystem of its successor; and Sn (a) = S(a). Any solution (X1 , . . ., Xn ) of S(a) reduces to a solution (X1 , . . ., Xm ) of the subsystem Sm (a). But it is not generally true that any solution of a subsystem of S(a) can be extended to a solution of S(a). However, should this be the case, then the system S(a) will be said to have the extension property. Theorem 6.1 Any closed system which satisﬁes the closed triangle inequality has the extension property. Let X1 , . . ., Xm−1 be a solution of S¯ m−1 (a), so that ars Xr − Xs

(r, s = 1, . . . , m − 1).

It will be shown that, under the hypothesis of the triangle inequality, it can be extended by an element Xm to a solution of S¯ m (a). Thus, there is to be found a number Xm such that arm Xr − Xm ,

ams Xm − Xs

(r, s = 1, . . . , m − 1),

that is ams + Xs Xm Xr − arm . So the condition that such an Xm can be found is amq + Xq Xp − apm , where Xp − apm = max Xr − arm , r

amq + Xq = min amq + Xq . r

120 Precursor But if p = q, this is equivalent to amq + aqm 0, which is veriﬁed, by Theorem 5.2, and if p = q, it is equivalent to apm + amq Xp − Xq , which is veriﬁed, since, by hypothesis apm + amq apq , apq Xp − Xq . Therefore, under the hypothesis, the considered extension is always possible. It follows now by induction that any solution of Sm (a) can be extended to a solution of Sn (a) = S(a). This theorem shows how solutions of any system can be practically constructed, step-by-step, by extending the solutions of subsystems of its derived system. Theorem 6.2 Any closed system which satisﬁes the closed triangle inequality is consistent. For, by Theorem 5.2, a12 + a21 0; and this implies that the system S¯ 2 (a): a12 X1 − X2 ,

a21 X2 − X1

¯ has a solution, which, by Theorem 6.1, can be extended to a solution of S(a). ¯ Therefore S(a) has a solution, and is thus consistent. Theorem 6.3 Any open system which satisﬁes the triangle inequality and has positive intervals has the extension property, and is consistent. The lines of proof follow those of Theorems 6.1 and 6.2. A system is deﬁned to satisfy the triangle equality if ars + ast = art . Theorem 6.4 If a system satisﬁes the triangle inequality and has null intervals then it also satisﬁes the triangle equality and has null cycles. For, from ars + asr = 0,

ars + ast art

The system of inequalities ars > Xs − Xr 121 follows also ars + ast art so that ars + ast = art . By induction, arl + alm + · · · + aqp = arp and then arl + alm + · · · + apr = 0, that is, the cycles are null.

7 Consistency Theorem 7.1 A necessary and sufﬁcient condition that an open system be consistent is that its cycles by positive. If S(a) is consistent, let X be a solution. Then, for any cycle r, l, m, . . ., p, r there are the relations arl > Xr − Xl ,

alm > Xl − Xm ,

...,

apr > Xp − Xr ,

from which it follows, by addition, that arlm···pr > 0. Therefore, if S(a) is consistent, all its cycles must be positive. Conversely, let the cycles of S(a) be positive. Then the derived system S(A) is deﬁned, satisﬁes the triangle inequality, and has positive intervals. Hence, by Theorem 6.3, S(A) is consistent. But, by Theorem 4.1, S(A) is equivalent to S(a). Therefore, S(a) is consistent. Similarly: Theorem 7.2 A necessary and sufﬁcient condition that a closed system be consistent is that its cycles be non-negative.

122 Precursor

8 Cycle reversibility A cycle is deﬁned to be reversible in a system if the reverse cycle has the same coefﬁcient, thus arl···pr = arp···lr . The condition of k-cycle reversibility for a system is that all cycles of k element be reversible with regard to it; and the general condition of cycle reversibility is the reversibility condition taken unrestrictedly, in respect to all cycles of any number of elements. Theorem 8.1 For the reversibility of cycles in a system, the reversibility of 3-cycles is necessary and sufﬁcient. The proof is by induction, by showing that, given 3-cycle reversibility, the k-cycle condition is implied by that for (k − 1)-cycles. Thus, from al···k + akl = ak···l + alk with aol + alk + ako = aok + akl + alo , by addition, there follows aol + al···k + ako = aok + ak···l + alo . Theorem 8.2 If a system has positive intervals and reversible cycles, then it is consistent. Thus, if ars···pr = arp···sr and ars + asr > 0, then 2ars···pr = ars···pr + arp···sr = (ars + asr ) + · · · + (apr + arp ) > 0, so the cycles are positive, and hence, by Theorem 7.1, the system is consistent.

The system of inequalities ars > Xs − Xr 123 For any system S(a), deﬁne Crs···t = ars···tr − art···sr then Crs···t is an antisymmetric cyclic function of the indices r, s, . . ., t, depending just on the cyclic order of the indices and changing its sign when the cyclic order is reversed. The cycle reversibility condition for the system now has the statement Crs···t = 0 and it has been shown to be necessary and sufﬁcient just that Crst = 0. Thus the reversibility conditions are not all independent, but are implied by those for the 3-cycles. Moreover, not all the 3-cycle reversibility conditions are independent; but, as appears in the following theorem, the reversibility of three of the four 3-cycles in any four elements implies that for the fourth. Theorem 8.3 For any four elements α, β, γ , δ there is the identity Cβγ δ + Cαδγ + Cδαβ + Cγβα = 0. This can be veriﬁed directly. By the dependencies shown in this Theorem, the 16 n(n − 1)(n − 2) conditions for 3-cycle reversibility, contained in and implying a much larger set of general reversibility conditions, reduce to a set of 12 (n − 1)(n − 2) independent conditions. Theorem 8.4 There are 12 (n − 1)(n − 2) independent cycle reversibility conditions in a system of order n.

9 Median solutions Any solution X of a system S(a) must satisfy the condition ars > Xr − Xs > −asr , that is, the differences Xr − Xs must lie in the intervals [−asr , ars ], which are nonempty provided ars +asr > 0. In particular, a solution X such that these differences lie at the mid-points of these intervals will be called a median of the system. Thus, if X is a median of S(a) then Xr − Xs = 12 (ars − asr ). The condition that a system admit a median is decidedly stronger than that of consistency alone.

124 Precursor Theorem 9.1 A necessary and sufﬁcient condition that any system S(a) admit a median is that ars + ast + atr = ats + asr + art , ars + asr > 0. The condition is necessary, since a median is a particular solution of the system, the existence of which implies that the intervals ars + asr of the system are positive. Moreover, addition of the relations 1 Xr − Xs = (ars − asr ), 2 1 Xs − Xt = (ast − ats ), 2 1 Xt − Xr = (atr − art ), 2 gives 0 = ars − asr + ast − ats + atr − art . Also it is sufﬁcient. For it provides that, for any k, and all r, s ars − asr = (ark − aks ) − (ask − aks ), from which it follows that the numbers 1 Xr = (ark − akr ) 2 satisfy 1 Xr − Xs = (ars − asr ); 2 and, then since ars + asr > 0 they must be a solution of the system, which is, moreover, a median. Now, combining with Theorem 8.1, we obtain Theorem 9.2 A necessary and sufﬁcient condition that a system admit a median solution is that its intervals be positive and its cycles reversible.

The system of inequalities ars > Xs − Xr 125

10 Simple systems A system S(a) which is such that, for some k, s ark + akr > 0, ark + aks ars , for all r, s, will be called simple, with respect to the index k. Theorem 10.1 If S(a) is simple, with respect to k, then it is consistent, and admits as solution all sets of number Xr such that ark > Xr > −akr . For then, from relations ark > Xr ,

aks > −Xs ,

there follows ars ark + aks > Xr − Xs .

2 On the constructibility of consistent price indices between several periods simultaneously S.N. Afriat In Essays in Theory and Measurement of Demand in honour of Sir Richard Stone edited by Angus Cambridge University Press, 1981

Deaton.

On the constructibility of consistent price indices between several periods simultaneously

Introduction A price index refers to a pair of consumption periods, and price-index formulae usually involve demand data from the reference periods alone. When there are many periods, a price index can be determined from any one period to any other, in each case using the data from just those two periods. But then consistency questions arise for the set of price indices so obtained. Especially, they must have the consistency that would follow from their being ratios of ‘price levels’. The well-known tests of Irving Fisher have their origin in such questions. When these tests are regarded as giving identities to be satisﬁed by a standard formula and are taken in combination, it is impossible to satisfy them. Such impossibility remains even with partial combinations. Eichhorn and Voeller (1976) have given a full account of the inconsistencies between Fisher’s tests. Reference is made there for their results and for the history of the matter. Fisher recognizes the consistency question also in his idea of the ‘rectiﬁcation of pair comparisons’. In this the price indices are all calculated, as usual, separately and regardless of any consistency they should have together, and then they are all adjusted in some manner so that they can form a consistent set. For instance, by ‘crossing’ a formula with its ‘antithesis’ you got one that satisﬁed the ‘reversal’ test. Here he takes one of the tests separately as if any one could mean anything on its own, and contrives a formula to satisfy it. This is how he arrived at his ‘ideal’ index. It is ‘ideal’ because it satisﬁes the ‘reversal’ test but not so when those other tests are brought in. The search for a really ideal index seemed a hopeless task. In any case these tests are just negative criteria for index-number-making, showing how a formula can be rejected and telling nothing of how one should be arrived at. Something is to be measured and it is not yet considered what, but whatever it is it must ﬁt a certain mould. Here is not measurement but a ritual with form. In the background thought, what is to be measured is the price level, though prices are many so no one quite knows what that means, and a price index is a ratio of price levels. Therefore the set m2 price indices Prs (r, s = 1, . . . , m) between m periods 1, . . . , m must at least have the consistency required by their being ratios −1 Prs = Ps /Pr of m ‘price levels’ Pr (r = 1, . . . , m). Therefore Prr = 1, Prs = Psr ,

130 Precursor Prs Pst = Prt and so forth. There are other parts to Fisher’s tests and here we have the part that touches just the ratio aspect. In a seemingly more coherent approach, utility makes the base for what is being measured. There would be no problem there at all if only the utility function or order to be used could be known. But it is not known and therefore it is dealt with hypothetically. Its existence is entertained and inferences are made from that position. With utility in the picture the natural object of measurement is the ‘cost of living’, and at ﬁrst we know nothing of the price-level or of a price-index. Giving intelligibility to the price index in the utility framework involves imposing a special restriction on utility. Let M0 be any income in a period 0 when the prices are given by a vector p0 . Hypothetically, the bundle of goods x0 consumed with this income has the highest utility among all those which might have been consumed instead. Then it is asked what income M1 in a period 1 when the prices are p1 provides the standard of living, or utility, attained with the income M0 in period 0. With p0 , p1 ﬁxed and the utility order given, M1 is determined as a function M1 = F10 (M0 ) of M0 , where the function F10 depends on the prices p0 , p1 and the utility order. Without making any forbidding extra assumptions it can be allowed that this is a continuous increasing function, and that is all. However, turning to practice with price-indices, we ﬁnd that to offer a relationship between M0 and M1 is the typical use given to a price index. The relationship in this case has the form M1 = P10 M0 , P10 being the price index. In other words, using a price-index corresponds to the idea that there is a homogeneous linear relation between M0 , M1 or that the relation is a line through the origin, the price-index being the slope. To give the function F10 , just this form has implications about the utility from which it is derived. That utility must have a conical structure that is a counterpart of linear homogeneity of that function: if any commodity bundle x has at least the utility of another y then the same holds when x and y are replaced by their multiples xt and yt by any positive number t. To talk about a price-index and at the same time about utility, this assumption about the utility must be made outright. If a conical utility is given, relative to it a price-index P10 can be computed for any prices p0 , p1 . Then, as explained further in section 11, it has the form P10 = P1 /P0 where P0 = θ (p0 ), P1 = θ (p1 ) are the values of a concave conical function θ depending on prices alone. Price indices so computed for many periods automatically satisfy various tests of Fisher. The issue about those tests therefore becomes empty in this context, and pair-comparisons so obtained need no ‘rectiﬁcation’. But a remaining issue come from the circumstance that a utility usually is not given. Should one be proposed arbitrarily as a basis for constructing price indices, there can be no objection to it merely on the basis of the tests, at least with those that concern the ratio-aspect of price indices. With each price-index formula P10 of the very many he surveyed, Fisher associated a quantity-index formula X10 in such a way that the product is the ratio of consumption expenditures M0 = p0 x0 , M1 = p1 x1 in the two periods. For instance

On the constructibility of consistent price indices 131 with the Laspeyres price index P10 = p1 x0 /p0 x0 , the corresponding quantity-index is X10 = p1 x1 /p1 x0 , and then P10 X10 = ( p1 x0 /p0 x0 )( p1 x1 /p1 x0 ) = p1 x1 /p0 x0 = M1 /M0 . As a possible sense to this scheme, it is as if, beside the price-index being a ratio P10 =P1 /P0 of price-levels, also the quantity-index is a ratio X10 =X 1 /X 0 of quantity levels, and price-level multiplied with quantity level is the same as price-vector multiplied with quantity-vector, that is P0 X0 = p0 x0 = M0 ,

P1 X1 = p1 x1 = M1

to give P10 X10 = ( P1 /P0 ) (X1 /X0 ) = ( P1 X1 )/( P0 X0 ) = M1 /M0 . Here there is the simple result that all prices are effectively summarized by a single number and all quantities by a single quantity number, and instead of doing accounts by dealing with each price and quantity separately, and also with their product that gives the cost of the quantity at the price, the entire account can be carried on just as well in terms of these two summary price and quantity numbers, or levels, whose product is, miraculously, the cost of that quantity level at that price level. Though there are many goods and so-many prices and quantities, still it is just as if there was effectively just one good with a price and obtainable in any quantity at a cost which is simply, as with a simple goods, the product of price times quantity. Any mystery about the meaning of a price index vanishes, because it becomes simply a price. Were this scheme valid we could ask for so much utility, enquire the price and pay the right amount by the usual multiplication. When applied to income M 0 in period 0 when the price level is P0 , the level of utility is purchases is X0 =M0 /P0 . Then the income that purchases the same level of utility in period 1 when the price level is P1 is given by M1 =P 1 X 0 . Hence, by division, M1 /M0 =P1 /P0 =P10 , giving the relation M1 =P10 M0 as usual. Whether or not this scheme has serious plausibilities, it is implicit whenever a price index based on utility is in view. However, though such a scheme has here been imputed as belonging to Fisher’s system, or conjured up as though that seems to belong to it or at least gives it an intelligibility, it cannot be considered to have clear presence there. For Fisher’s system does not have a basis in utility and this scheme does. While this circumstance is not evidence of a union it still might not seem to force a separation. However, a symptom of a decided separation is that, even when many periods are involved, Fisher still followed standard custom in regarding an index formula as one involving the demand data just from its pair of reference periods, and really his system is about such formulae. Then he worried about the incoherence of the set of price indices for many periods so obtained. The utility formulation cares nothing about the form of the formula. When many periods are involved and all the price indices between them are to be

132 Precursor calculated, the calculation of one and all should involve the data for all periods simultaneously. In the utility approach immediate thought is not of the demand data and of formulae in these at all, but rather it is of the utility order which gives the basis of the calculations, and necessarily gives coherent results. Instead, Fisher forces incoherence by rigidly following the standard idea of what constitutes an index formula. The main issue with the utility approach is about the utility function or order. When that utility is settled all that remains to be dealt with is a well-deﬁned objective of calculation based on that utility. The role of the demand data is just to put constraints on the permitted utility order, and consequently price indices based on utility become based on that data. Having such constraints, the ﬁrst question then is about the existence of a utility order that satisﬁes them. If none exists then no price indices exist and there the matter ends. Though that is so in the present treatment, by making the constraints more tolerant it is possible to go further (see Afriat, 1972b and 1973). In the standard model of the consumer, choice is governed by utility, to the effect that any bundle of goods consumed has greater utility than any other attainable with the same income at the prevailing prices. With this model, the obvious constraint on a utility for it to be permitted by given data is, ﬁrstly, that it validate the model for the consumer on the evidences provided by that data. Then further, since price-indices are to be dealt with, the conical property of utility should be required. With this method of constraint and the other deﬁnitions that have been outlined, everything is available for developing the questions that are in view. But ﬁrst there will be a change in formulation that has advantages. Instead of requiring that a chosen bundle of goods be represented as being the unique best among all those attainable for no greater cost at the prevailing prices, or as being deﬁnitely better than any others in utility, it will be required that it be just one among the possibly many best, or one at as least as good as any other. This alters nothing if certain prior assumptions are made about the utility order, for instance that it is representable by an increasing strictly quasiconcave utility functions. With the latter assumption a utility maximum under a budget constraint must in any case be a unique maximum, and so adding that the maximum is unique just makes a redundance. But we do not want to introduce additional assumptions about utility. A utility is wanted that ﬁts the data in a certain way, and if all that is now wanted in such a ﬁt is that some commodity bundles be represented as having at least the utility of certain others then we can always count on a utility function that is constant everywhere to do that service. In making what could at ﬁrst seem a slight change in the original formulation of the constraint on permitted utilities, the result is no constraint at all: whatever the data there always exists a permitted utility, for instance the one mentioned which will give zero as the cost of attaining any given standard of living. That change is drastic and no such change is sought. All that is in view is a change that alters nothing important in the results, the effect being something like replacing an open interval by a closed one, while it is better to work with and in any case is conceptually ﬁtting. One possibility is to add a monotonicity condition as an assumption about utility expressing that

On the constructibility of consistent price indices 133 ‘more is better’. But, as said, we do not want any such prior assumptions. Instead consider again the original strict condition, that the chosen bundle be the unique best attainable at no greater cost. It implies the considered weaker condition in which the uniqueness has been dropped. But also it implies a second condition: the cost of the bundle is the minimum cost for obtaining a bundle that is as good as it. These two conditions are generally independent, even though relations between them can be produced from prior assumptions about utility, of which we have none, and their combination is implied by the stricter and analytically more cumbersome original condition. They are just what is wanted. They have equal warrant as economic principles. In the context of cost–beneﬁt analysis they are familiar as constituting the two main criteria about a project, that it be cost-effective or gives best value for the cost, and cost-efﬁcient or the same value is unattainable at lower cost. Now the wanted constraint on an admissible utility can be stated by the requirement that every bundle of goods given in the demand data be represented by it as cost-effective and cost-efﬁcient. Such a utility can be said to be compatible with the demand data. Then that data is consistent if there exists a compatible utility. It is homogeneously consistent if there exists a compatible utility that moreover has the property of being conical, or homogeneous, required whenever dealing with price-indices. A compatible price-index, or a ‘true’ one, is one derived on the basis of a homogeneous utility that is compatible with the given data. The ﬁrst problem therefore is to ﬁnd a test for the homogeneous consistency of the data. In the case where there are just two periods, the test found reduces to a relation that is quite familiar, in a context where it is not at all connected with this test but is offered as a ‘theorem’, though certainly it is not that. The relation is simply that the Paasche index from one period to the other does not exceed that of Laspeyres. The relation is symmetrical between the data in the two periods, and so there is no need to put in this unsymmetrical form where one period is distinguished as the base. But this is the form in which it is familiar and known as the ‘Index-Number Theorem’. That the ‘Theorem’, or relation, is necessary and sufﬁcient for homogeneous consistency of the demands in the two periods is a theorem in the ordinary sense. It is going to be generalized for any number of periods. Related to the Index-Number Theorem is the proposition that the Laspeyres and Paasche indices are upper and lower ‘limits’ for the ‘true index’. From the foregoing consistency considerations it is recognized that even the existence of a price index, at least in the sense entertained here, can be contradicted by the data, so certainly some additional qualiﬁcation is needed in the ‘limits’ proposition. Also, what makes an index ‘true’ has obscurities in early literature. An interpretation emerging in later discussions is that a true index is simply one derived on the basis of utility. This could be accepted to mean one that, in present terms, is compatible with the given demand data. With demand data given for any number of periods and satisfying the homogeneous consistency test, a price index compatible with those data can be constructed from any period to any other. It has many possible values

134 Precursor corresponding to the generally many compatible homogeneous utilities. These values describe a closed interval whose endpoints are given by certain formulae in the given demand data. A special case of this result applies to the situation usually assumed in indexnumber discussions. In this, the only data involved in a price-index construction between two periods are the data from the two reference periods themselves. For this case the formulae for the endpoints of the interval of values for the price index reduce to the Paasche and Laspeyres formulae. Here therefore is a generalization of those well-known formulae for when demand data from any number of periods can be permitted to enter the calculation of a price-index between any two. The values of these generalized Paasche and Laspeyres formulae are well deﬁned just in the case of homogeneous consistency of the data, under which condition they have the price-index signiﬁcance just stated. Then a counterpart of the ‘IndexNumber Theorem’ condition in the context of many periods is that the generalized Paasche formula does not exceed the generalized Laspeyres formula. There seems to be one such condition for each ordered pair of periods, making a collection of conditions. However, all are redundant because they are automatically satisﬁed whenever the formulae have well-deﬁned values, as they do just in the case of homogeneous consistency of the data. For price-indices Prs between many periods to be consistent they should have the form Prs = Ps /Pr for some Pr . Let Pˆ rs , Pˇ rs be the generalized Laspeyres and Paasche formulae. These, when they have well-deﬁned values, are connected by the relation Pˆ rs Pˇ sr = 1 and have the properties Pˆ rs Pˆ st ≤ Pˆ rt , Pˇ rs Pˇ s t ≤ Pˇ rt . Then it is possible to solve the system of simultaneous inequalities Pˆ rs ≥ Ps /Pr for the Pr . The system Pˇ rs ≤ Ps /Pr is identical with this, so solutions automatically satisfy Pˇ rs ≤ Ps /Pr ≤ Pˆ rs Now it is possible to describe all the price-indices Prs between periods that are compatible with the data and form a consistent set: they are exactly those having the form Prs = Ps /Pr where Pr is any solution of the above system of inequalities. The condition for their existence is just the homogeneous consistency of the data. For any solution Pr there exists a homogeneous utility compatible with the data on the basis of which Prs = Ps /Pr is the price-index from period r to periods s. This will be shown by actual construction of such a utility. Now to be remarked is the extension property of any given price-indices for a subset of the periods that are compatible with the data and together are consistent: it is always possible to determine further price-indices involving all the other periods so that the collection price-indices so obtained between all periods are both compatible with the data and together are consistent. There is an ambiguity here about a set of price indices being compatible with the data: they could be that with each taken separately, or in another stricter sense where they are taken simultaneously together. But in the conjunction with consistency the

On the constructibility of consistent price indices 135 ambiguity loses effect. For price-indices Prs that are all independently compatible with the data, the compatibility of each one with the data being established by means of a possibly different utility, if they are all consistent there also they are jointly compatible in that also there exists a single utility, homogeneous and compatible with the data, that establishes their compatibility with the data simultaneously. This completes a description of the main concepts and results dealt with in this paper. Further remarks concern computation of the m × m—matrix of generalized −1 Laspeyres indices Pˆ rs , and hence also the generalized Paasche indices Pˇ rs = Pˆ sr , from the matrix of ordinary Laspeyres indices Lrs = ps xr /pr xr . An algorithm proposed goes as follows. The matrix L with elements Lrs is raised to powers in a sense that is a variation of the usual, in which a + b means min [a, b]. With the modiﬁcation of matrix addition and multiplication that results associativity and distributivity laws are preserved, and matrix ‘powers’ can be deﬁned in the usual way by repeated ‘multiplication’. The condition for the powers L, L2 , L3 , . . . to converge is simply the homogeneous consistency of the data. Then for some k ≤ m, Lk−1 = Lk , and in that case also Lk = Lk+1 = . . . so the calculation of powers can be broken-off as soon as one is found that is identical with its predecessor. Finding such a k ≤ m by this procedure is a test for homogeneous consistency; ﬁnding a diagonal element less than unity denies this condition and terminates the procedure. With such a k let Pˆ = Lk . The elements Pˆ rs of Pˆ are the generalized Laspeyres indices. A programme for this algorithm is available for the TI-59 programmable calculator applicable to m ≤ 6, and another in Standard BASIC for a microcomputer.

1 Demand With n commodities, n is the commodity space and n is the price or budget space. These are described by non-negative column and row vectors with n elements, being the non-negative numbers. Then x ∈ n , p ∈ n have a product px ∈ , giving the value of the commodity bundle x at the prices p. Any (x, p) ∈ n × n with px >0 deﬁnes a demand, of quantities x at prices p, with expenditure given by M = px. Associated with it is the budget vector given by u = M −1 p and such that ux = 1. Then (x, u) is the normal demand associated with (x, p)(px > 0). Some m periods of consumption are considered, and it is supposed that demands (xt , pt )(t = 1, . . . , m) are given for these. With expenditures Mt = pt xt and budgets ut = Mt−1 pt , so that ut xt = 1, the associated normal demands are the (xt , ut ). Then Lrs = ur xs = pr xs /pr xr is the Laspeyres quantity index from r to s, or with r, s as base and current periods. It is such that Lrr = 1. Then the coefﬁcients Drs = Lrs − 1 are such that Drr = 0. To be used also are the Laspeyres chain-coefﬁcients Lrij . . .ks = Lri Lij . . .Lks . Any collection D ⊆ n × n of demands is a demand relation. Here we have a ﬁnite demand relation D with elements (xt , pt ). For any demand (x, p), the collection

136 Precursor of demands of the form (xt, p), where t > 0 is its homogeneous extension, and the homogeneous extension of a demand relation is the union of the homogeneous extension of its elements. A homogeneous demand relation has the property x Dp, t > 0 → xt Dp making it identical with its homogeneous extension. For a normal demand relation E, or one such that xEu → ux = 1, homogeneity is expressed by the condition x Eu, t > 0 → xt Et −1 u If E is the normal demand relation associated with D, or the normalization of D, then this is the condition for D to be homogeneous.

2 Utility A utility relation is any binary relation R ⊆ n × n that is reﬂexive, x Rx, and transitive, x RyRz→x Rz, by which properties it is an order. Then the symmetric part E = R ∩ R , for which xEy ↔ xRy ∧ xR y ↔ xRy ∧ yRx is a symmetric order, or an equivalence, and the antisymmetric part P = R ∩ R , for which

xPy ↔ xRy ∧ xR Y ↔ xRy ∧ yRx ↔ xRy∧ ∼ yRx is irreﬂexive and transitive, or a strict order. A homogeneous, or conical, utility relation is one that is a cone in n × n , that is x Ry, t > 0 → xt Ryt. Any φ : n → is a utility function, and it is homogeneous or conical if its graph is a cone, the condition for this being φ(xt) = φ(x)t (t > 0). A utility function φ represents a utility relation R if xRy ↔ φ(x) ≥ φ(y). If φ is conical so is R.

3 Demand and utility A demand (x, p) and a utility R are compatible if (i)

py ≤ px → xRy

(ii)

y Rx → py ≥ px

(3.1)

They are homogeneously compatible if (xt, p) and R are compatible for all t > 0, in other words if the homogeneous extension of (x, p) is compatible with R. If R is homogeneous, compatibility is equivalent to homogeneous compatibility.

On the constructibility of consistent price indices 137 Demand and utility relations D and R are compatible if the elements of D are all compatible with R, and homogeneously compatible if the homogeneous extension of D is compatible with R. A demand relation D is consistent if it is compatible with some utility relation, and homogeneously consistent if moreover that utility relation can be chosen to be homogeneous. Homogeneous consistency of any demand relation is equivalent to consistency of its homogeneous extension. That the former implies the latter is seen immediately, and the converse will be shown later. It should be noted that (3.1 (ii)) in contrapositive is py < px → yRx, and, with the deﬁnition of P in section 2, this with (3.1 (i)) gives py < px → xPy

(3.2)

so this is a consequence of (3.1).

4 Revealed preference A relation W ⊂ n × n is deﬁned by x Wu ≡ ux ≤ 1. Then x Wu, that is ux ≤1, means the commodity bundle x is within the budget u, and Wu = [x : x Wu] = [x : ux ≤ 1] is the budget set for u, whose elements are the commodity bundles within u. The revealed preference relation of a demand (x, p) is [(x, y) : py ≤ px]. For a normal demand (x, u) (ux = 1) it is (x, Wu) = [(x, y) : y ∈ Wu] = [(x, y) : uy ≤ 1]. The revealed homogeneous preference relation is the conical closure of the revealed preference relation, so it is [(xt, y) : py ≤ pxt, t > 0] for a demand and ∪t>0 (xt, Wt −1 u) for the normal demand. The condition (3.1), which is a part of the requirement for compatibility between a demand and utility (x, p) and R, asserts simply that the utility relation contains the revealed preference relation. If the utility relation is homogeneous, this is equivalent to its containing the revealed homogeneous, this is equivalent to its containing the revealed homogeneous preference relation. Let Rt be the revealed preference relation of the demand (xt , pt ), and R˙ t the revealed homogeneous preference relation. Then, as remarked, compatibility of that demand with a utility R requires that Rt ⊆ R

(4.1)

and if R is homogeneous this is equivalent to R˙ t ⊆ R

(4.2)

Now let RD , the revealed preference relation of the given demand relation D, be deﬁned as the transitive closure of the union of the revealed preference relations

138 Precursor

t Rt . The compatibility of R with D requires (4.1) for all Rt of its elements, RD = ∪

t R˙ t , the t, and because R is transitive this is equivalent to RD ⊂ R. Also let R˙ D = ∪ transitive closure of the union of the revealed homogeneous preference relation R˙ t of the elements of D, deﬁne the revealed homogeneous preference relation of D. Then, by similar argument, with (4.2), compatibility of D with a homogeneous R implies R˙ D ⊆ R. While RD is transitive from its construction, and reﬂexive at the points xt , because Rt is reﬂexive at xt , R˙ D is both transitive and conical, and reﬂexive on the cone through the xt .

5 Revealed contradictions A demand relation D with elements (xr , pr ) is compatible with a utility relation R if (i)

pt x ≤ pt xt → xt R x

(ii)

xRxt → pt x ≥ pt xt

(5.1)

and D is consistent if some compatible order R exists. It has been seen that (5.1) is equivalent to RD ⊆ R

(5.2)

Therefore, if D is compatible with R, x RD xt → x R xt for any t and x, and also pt x < pt xt → xRxt . Therefore, on the hypothesis that D is compatible with some, R, the condition xRD xt , pt x < pt xt

(5.3)

implies xRxt , xRxt making a contradiction, so the hypothesis is impossible and D is inconsistent. The condition (5.3) for any t and x is a revealed contradiction, denying the consistency of D. Thus: The existence of a revealed contradiction is sufﬁcient for D to be inconsistent.

(5.4)

Now it will be seen to be also necessary. The condition for there to be no revealed contradictions is the denial of (5.3), for all t and x; equivalently x RD xt → pt x ≥ pt xt

(5.5)

But this is just the condition (5.1 (ii)) with R = RD . Because (5.1 (i)) is equivalent to (5.2), and because in any case RD ⊆ RD so (5.2) is satisﬁed with R = RD , it is

On the constructibility of consistent price indices 139 seen that (5.4) is necessary and sufﬁcient for (5.1 (i) and (ii)) to be satisﬁed with R = RD , in other words for D to be compatible with RD . Thus: The absence of revealed contradictions is necessary and sufﬁcient for D to be compatible with RD .

(5.6)

As a corollary: The absence of revealed contradictions implies the consistency of D. (5.7) For consistency means the existence of some compatible order, and by (5.5) under this hypothesis RD is one such order. Now with (5.5): The absence of revealed contradictions is necessary and sufﬁcient for the consistency of D and implies compatibility with RD .

(5.8)

By exactly similar argument, xR˙ D xt θ and pr x < pr xr θ, for any r, x and θ > 0, make a homogeneously revealed contradiction denying the homogeneous consistency of D, or the existence of a compatible homogeneous utility. Then xR˙ D xt θ → pr x ≥ pr xr θ for all r, x and θ > 0 asserts the absence of homogeneously revealed contradictions. Then there is the following: Theorem For a demand relation to be compatible with some homogeneous utility relation, and so homogeneously consistent, it is necessary and sufﬁcient that its revealed homogenous preference relation be one such relation, and for this the absence of homogeneously revealed contradictions is necessary and sufﬁcient. This theorem holds unconditionally, regardless of whether or not D is ﬁnite. However, when D is ﬁnite the homogeneous consistency condition has a ﬁnite test, developed in the next two sections. Because the revealed preference relation of the homogeneous extension of a demand relation is identical with its revealed homogeneous preference relation, it appears now, as remarked in section 3, that homogeneous consistency of a demand relation is equivalent to consistency of its homogeneous extension. For, as just seen, the ﬁrst stated condition on D is equivalent to compatibility with R˙ D and the second with RD˙ , so this conclusion follows from R˙ D = RD˙ .

6 Consistency Though RD , R˙ D and the Lrs which give the base for the following work are derived from D, they are also derivable from the normal demand relation E derived from D. Therefore there would be no loss in generality if only normal demand relations were considered.

140 Precursor As a preliminary, the deﬁnition of the revealed homogeneous preference relation R˙ D will be put in a more explicit from. This requires identiﬁcation of the transitive

An R-chain is any sequence closure of any relation R with its chain-extension R.

to its successor, that of elements x, y, . . . , z in which each has the relation R is x Ry R. . . Rz . Then the chain-extension R is the relation that holds between

z ≡ (∨y, . . . )x Ry R. . . R z. The relation R

so deﬁned extremities of R-chains, so x R

can be identiﬁed with the transitive closure of R, that is as the smallest transitive relation containing R, it being such that it is transitive, contains R and is contained in every transitive relation that contains R. Therefore xR˙ D y means x, y are extremities of a chain in the relation ∪t R˙ t . This means there exist r, i, . . . , k and zi , . . . , zk such that xRr zi Ri . . . zk Rk y. But, considering the form of the elements of the Rt , now we must have xr = xr θr , zi = xi θi , . . . , zk = xk θk for some θr , θi , . . . , θk > 0, and uk y ≤ θk . Accordingly, the condition xR˙ D xs θs → us x ≥ θs for all x, s and θs > 0, for the absence of homogeneously revealed contradictions, can be restated as the condition ˙ · · · xk θk xs θs → Us xr θr ≥ θs xr θr R˙ r xi θi Ri

(6.1)

for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0. From the form of the elements of the R˙ t , that is ur xi θi ≤ θr , ..., uk xs θs ≤ θk → us xr θr ≥ θs or, in terms of the Laspeyres coefﬁcients, Lri ≤ θr /θi , . . . , Lks ≤ θk /θs → Lsr ≥ θs /θr

(6.2)

Another way of stating this condition is that (Lri , . . . , Lks , Lsr ) ≤ (θr /θi , . . . , θk /θs , θs /θr )

(6.3)

is impossible for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0. This condition will be ˙ denoted K. While theory based on homogeneity is here the main object, in the background is the further theory without that restriction. Some account of that is given here, but it is mainly given elsewhere as already indicated. The dots used in the notation are to distinguish features in this homogeneous theory from their counterparts without homogeneity. The homogeneous theory is required in dealing with price indices, but still it has its source in the more general theory. It is useful in this section and later to bring counterparts of the two theories together for recognition of the connections and the differences. Condition (6.1) has been identiﬁed with the condition for the absence of homogeneously revealed contradictions that in the last section was shown necessary and sufﬁcient for the consistency of the given demands, that is, for the existence of a homogeneous utility compatible with them all simultaneously.

On the constructibility of consistent price indices 141 The weaker condition that is the counterpart without homogeneity, put in a form that assists comparison, is the condition K given by (Lri , . . . , Lks , Lsr ) ≤ (1, . . . , 1, 1)

(6.4)

is impossible for all r, i, . . . , k, s. By taking the θ s all unity in (6.3), (6.4) is obtained, so (6.3) implies (6.4) as should be expected. If compatibility between demand and utility is replaced by strict compatibility, by replacing cost-efﬁcacy or cost-efﬁciency by their strict counterparts, which conditions in fact are equivalent to each other, and strict consistency of any demands means the existence of a strictly compatible utility, then the test for this condition which is a counterpart of (6.4) is the condition K ∗ given by (Lri , . . . , Lks , Lsr ) ≤ (1, . . . , 1, 1)

(6.5)

is impossible for all r, i, . . . , k, s unless xr = xi = · · · = xk = xs , in which case the equality holds. This is just a way of stating the condition of Houthakker (1950), know as the Strong Axiom of Revealed Preference, for when that condition is applied to a ﬁnite set of demand instead of to the inﬁnite set associated with a demand function. Here the ﬁniteness is not essential and is just a matter of notation, though in later results it does have an essential part. Corresponding to the results obtained for the less strict consistency, and for homogeneous consistency, as in the Theorem or section 5, Houthakker’s condition is necessary and sufﬁcient for the existence of strictly compatible utility, and for the revealed preference relation to be one such utility. While (6.5) is the ‘strict’ counterpart of (6.4), the corresponding counterpart for (6.3) is the condition K given by (Lri , . . . , Lks , Lsr ) ≤ (θr /θi . . . , θk /θs , θs /θr )

(6.6)

is impossible for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0 unless xr θr = xi θi = · · · = xs θs in which case the equality holds. Just as a dot signiﬁes a condition associated which homogeneity, a star signiﬁes belonging to the ‘strict’ theory. The various conditions that have been stated have the relations K˙ → K ↑

↑

K˙ ∗ → K ∗ The main result of this section, which is about K˙ in (6.3) being a consistency condition, will be part of a theorem in the next section where it is developed into another form.

142 Precursor It can be noted that (6.3) is equivalent to the same condition which r, i, . . . , k, s restricted to be all distinct. For the second condition is part of the ﬁrst. Also, the inequalities stated in the ﬁrst, involving a cycle of elements, can be partitioned into groups of inequalities involving simple cycles, each without repeated elements, showing that also the ﬁrst follows from the second. A ﬁnite consistency test is wanted, one that can be decided in a known ﬁnite number of steps. The last conclusion goes a step towards ﬁnding such a test, though it does not give one. That will be left to the next section. However, (6.4) and (6.5) taken with the indices all distinct already represent ﬁnite tests. But still this is not the case for Houthakker’s condition (6.5), or for (6.4), when these are regarded as applying to a demand function (6.5), or for (6.4), when these are regarded as applying to a demand function, for which the number of cycles of distinct demands is unlimited.

7 Finite test Theorem 7.1 For any ﬁnite demand relation D the following conditions are equivalent (H˙ ) D is homogeneously consistent, that is, there exists a compatible homogeneous utility relation; ˙ D is compatible with its own revealed homogeneous utility relation R˙ D ; (R) ˙ (Lrs , Lst , . . . , Lqr ) ≤ (θr /θs , θs /θt , . . . , θq /θr ) is impossible for all distinct (K) r, s, . . . , q and θr , θs , . . . , θq > 0; ˙ Lrst... qr ≥ 1 for all distinct r, s, . . . , q. (L) Arguments for the equivalences between H˙ , R˙ and K˙ have already been given in the last two sections. It is enough now to show K˙ and L˙ are equivalent. By multiplying the inequalities stated for any case where K˙ is denied, it follows that Lrst... qr < (θr /θs )(θs /θt ) · · · (θq /θr ) = 1 ˙ Thus L˙ → K. ˙ Now, contrary to L, ˙ suppose Lrst . . . qr < 1, contrary to L. and let θr = Lrst···qr ,

θs = Lst···qr,··· ,

θq = Lqr

Then Lrs = θr /θs ,

Lst = θs /θt , . . .

and ﬁnally, θr < 1 and θq = Lqr so that Lqr < θq /θr , showing a denial of K. ˙ and the two conditions are now equivalent. Thus K˙ → L,

On the constructibility of consistent price indices 143 Because the number of simple cycles that can be formed from m elements is ﬁnite and given by

m m m (r − 1)! = (r − 1)!m!/r!(m − r)! r r=1

r−1

=

m

(m − r + 1) · · · m/r

r−1

L˙ is a ﬁnite test. The counterpart of K˙ for the general non-homogeneous theory has already been stated, and that for L˙ is (L) There exist positive λs such that for all distinct r, s, t, . . . , q (λr Lrs + λs Lst + · · · + λq Lqr )/(λr + λs + · · · + λq ) ≥ 1 It can be noted that while L˙ shows a ﬁnite test and K˙ does not, L does not and K does. There are several routes for proving the equivalence of K and L, all of some length. From that equivalence it is known that L˙ → L. But this can be seen also directly. From the theorem that the geometric mean does not exceed the arithmetic, 1/k

(Lrs + Lst + · · · + Lqr )/k ≥ Lrst···qr k being the number of elements in the cycle. Therefore L˙ implies (Lrs + Lst + · · · + Lqr )/k ≥ 1 But this validates L with all the λs equal to unity. The counterpart of L for the ‘strict’ theory, equivalent to Houthakker’s revealed preference axiom, is (L*) There exist positive λs such that, for all distinct r, s, t, . . . , q (λr Lrs + λs Lst + · · · + λq Lqr )/(λr + λs + · · · + λq ) ≥ 1 the equality holding just when xr = xs = xt = · · · = xq .

8 A system of inequalities The test Lrs . . . qr ≥ 1 for all distinct r, s, . . . , q, that was found for the homogeneous consistency of a demand relation is also the test for solubility of the system of inequalities Lrs ≥ φs /φr

for all r, s

(8.1)

for numbers φr (r = 1, . . . , m). Such numbers obtained by solving the inequalities will be identiﬁed as utility-levels for the demand periods, because for any demand

144 Precursor period there exists a homogeneous utility compatible with the demands that identify them all as such, in that the numbers Xrs = φs /φr are identiﬁed as quantityindices, compatible with the data, and price indices correspond to these. By taking logarithms the system (7.1) comes into the form ars ≥ xs − xr

(8.2)

where xr = log φr and ars = log Lrs . An account of the system in this from has been given in Afriat (1960), and in the last section here it is developed to suit needs of the present application. The same system, in the additive from, arises also in the version of this theory unrestricted by homogeneity. It is required to ﬁnd a positive solution of the system of homogeneous linear inequalities λr (Lrs − 1) ≥ φs − φr

(8.3)

That the φs be positive is inessential because they enter through their differences, and so a constant can always be added to make them so, but the restriction is essential. The λs occurring in solutions of (7.3) are identical with the λs that are solutions of (6.4), so they can be determined separately. With any λs so determined, and ars = λr (Lrs − 1), (7.3) is in the from (7.2) for determining the φs. The φs and λs in any solution become utilities and marginal utilities at the demanded xs with a compatible utility that is constructed by means of the solution. In the case of homogeneous utility, λr = φr , and with this substitution (7.3) reduces to (7.1). An entirely different connection for the system (7.2) is with minimum paths in networks. With the coefﬁcient ars as direct path-distances, a solution of (7.2) corresponds to the concept of a subpotential for the network, as described by Fiedler and Ptak (1967). Whereas there it is an auxiliary that came in later, here it is a principal objective and a starting point. Then there is the linear programming formula Aij = min[xj − xi : ars ≥ xr − xs ] expressing the minimum path-distance Aij as the minimum subpotential difference, as learnt from Edmunds (1973). It is familiar under the assumption ars > 0, and in the integer programming context. Close to hand in the 1960 account is this formula without the non-negativity restriction on the coefﬁcients and a quite different method of proof.

9 Utility construction Now to be considered is how, for any number φt > 0 such that us xt ≥ φt /φs

(9.1)

it is possible to construct a linearly homogeneous, or conical, utility that is compatible with the given demands Dt and such that φ(xt ) = φt

(9.2)

The utility constructed will moreover be semi-increasing, x < y → φ(x) < φ(y), and, being both conical and superadditive , φ(x +y) ≥ φ(x)+(y); also it is concave.

On the constructibility of consistent price indices 145 The existence of numbers φt satisfying (9.1) is necessary and sufﬁcient that there should exist any compatible homogeneous utility R at all, without further qualiﬁcation. But here it is seen that if there exists one then also there exists one with these additional classical properties. A conclusion is that these classical properties are unobservable in the observational framework of choice under linear budgets, or are without empirical test or meaning and are just a property of the framework. The consistency condition generally becomes more restrictive as additional restrictions are put on utility. Thus homogeneous consistency is more restrictive that the more general consistency that is free of the homogeneity. Then classical consistency, where utility is required to be representable by a utility function with the classical properties, might seem to be more restricted than general consistency, and also the same might be supposed for when homogeneity is added to both these conditions. But the contrary is a theorem: the imposition of the classical properties make no difference whatsoever. Let φ(x) = min φt ut x

(9.3)

t

so, for all x, φ(x) ≤ φt ut x for all t,

φ(x) = φt ut x

for some t

(9.4)

Then, with x = xt , so ut x = 1, we have φ(xt ) < φt . But from (9.1), φs us xt ≥ φt for all s. Hence, with (9.3), φ(xt ) = mins φs us xt ≥ φt , Thus (9.2) is shown. Now further, from (9.4) with φt > 0, ut x < 1 → φt ut x < φt → φ(x) < φt

(9.5)

Hence, with (9.2), ut x < 1 → φ(x) < φ(xt ), and similarly, or from here by continuity, ut x ≤ 1 → φ(x) ≤ φ(xt ), showing that the utility φ(x) and the normal demand (xt , ut ) are compatible.

10 Utility cost Because ut = Mt−1 pt

(10.1)

where Mt = pt xt

(10.2)

and because Xt = φt (xt ), another statement of (9.1), in view of (9.2), is that. ps xt /ps xs ≥ Xt /Xs

(10.3)

146 Precursor Then introducing Pt = Mt /Xt

(10.4)

so that, as a parallel to (10.2) Mt = Pt Xt , (10.3) and (10.4) give ps xt /ps xs ≥ ( pt xt /Pt )/( ps xs /Ps ) = ( pt xt /ps xs )( ps /Pt ) and consequently ps xt /pt xt ≥ Ps /Pt

(10.5)

Or again, introducing Ut = Mt−1 Pt

(10.6)

in analogy with (10.1), so that Ut Xt = 1, this being, in analogy with the normalized budget identity ut xt = 1, an equivalent of (10.3), and also of (10.5), is that us xt≥ Us /Ut . Let θ( p) be the cost function associated with the classical homogeneous utility function φ(x), so that θ ( p) = min [ px : φ(x) ≥ 1]

(10.7)

this again being classical homogeneous, that is semi-increasing, concave and conical. Then by taking x in the form xt −1 , where t>0, θ ( p) = min pxt −1 : φ(xt −1 ) ≥ 1 = min pxt −1 : φ(x) ≥ t because φ is conical. Then by taking t = φ(x), θ ( p) = minx px(φ(x))−1 is obtained as an alternative formula for θ . From this formula the functions θ and φ are such that θ( p)φ(x) ≤ px

(10.8)

for all p, x with equality just in the case of compatibility between the demand (x, p) and the utility φ. For the equality signiﬁes cost efﬁciency, and because φ is continuous this implies also cost effectiveness, and hence also the compatibility. Because φ is concave it is recovered from θ by the same formula by which θ is derived from it, with an exchange of roles between θ and φ; that is φ(x) = min [ px : θ ( p) ≥ 1] = min(θ ( p))−1 px p

(10.9)

In the case of a normal demand (x, u), that is one for which ux = 1, (10.8) becomes θ(u)φ(x) ≤ 1 with equality just in the case of a demand that is compatible with φ.

(10.10)

On the constructibility of consistent price indices 147 In section 9 it was shown that the function φ(x) constructed there is compatible with the given normal demands (xt , ut ). Therefore θ (ut )φ(xt ) = 1 for all t, while, by (10.10), θ(us )φ(xt ) ≤ 1 for all s, t. Also it was shown that φ(xt ) = φt

(10.11)

Hence, introducing θt = φt−1

(10.12)

it is shown that θ(ut ) = θt . It is possible to verify that also directly by inspection of the cost function. Thus, with φ(x) = mint φt ut x, so that φ(x) ≥ 1 is equivalent to φt ut x ≥ 1 for all t, which, with (10.12), is equivalent to ut x ≥ θt for all t, the cost function in (10.7) is also θ(u) = min ux : ut x ≥ θt

(10.13)

so that θ (ut ) ≥ θt . Therefore, by (10.11) and (10.12), θ (ut )φ(xt ) ≥ 1. Then ut xt =1 with (10.10) shows that θ (ut )φ(xt ) = 1 and hence, again with (10.11) and (10.12), that θ (ut ) = θt . By the linear programming duality theorem (Dantzig 1963) applied to (10.13), another formula for the cost function is θ (u) = max s t θt : st ut ≤ u (10.14) Then, as known from the theory of linear programming, for any x, θ (u) ≤ ux for all u if and only if x solves (10.13). Similarly, with the θt now variable while u is ﬁxed, for any st , θ(u) ≥ st θt for all θt if and only if the st solve (10.14). If the θt are a strict solution of (9.1), that is us xt > φt /φs (s = t), then x = xt θt is the unique solution of (10.13) when u = ut . In just that case θ (u) is differentiable at the point u = ut . In that case θ is locally linear, and has a unique support gradient, and the differential gradient which now exists coincides with it. Thus in this case θ (u) ≤ ux for all u, and θ (ut ) = ut x if and only if x = xt θt , so θ (u) has gradient xt θt at u = ut . It can be added that this entire argument could have gone just as well with an interchange of roles between u and x. By solving us xt ≥ θs /θt for the θt , a cost function θ could be constructed ﬁrst, with the form originally given to φ, and then φ could have been derived. Also, φ need not have been given the polyhedral form (9.5). It could have been given the polytope form (10.13) or ( 10.14). Then θ would have had the polyhedral form (9.5).

11 Price and quantity The method established for the determination of index numbers can be stated in a way that treats price and quantity both simultaneously and in a symmetrical fashion.

148 Precursor With the given demands (xt , pt ), numbers (Xt , Pt ) should satisfy ps xt ≥ Ps Xt

for all s, t

(11.1)

Then, in particular, pt xt = Pt Xt

(11.2)

Then division of (11.1) by (11.2) gives ps xt /pt xt ≥ Ps /Pt

(11.3)

as a condition for the ‘price levels’, and also ps xt /ps xs ≥ Xt /Xs

(11.4)

for ‘quantity levels’. Reversely, starting with a solution Pt of (11.3), let Xt be determined from (11.2). Then Xt is a solution of (11.4) and the Pt and Xt together make a solution of (11.1). Just as well, the procedure could start with a solution Xt of (11.4) and go on similarly. It has been established that the existence of solutions to these inequalities is necessary and sufﬁcient for homogeneous consistency of the demand data. The investigation now concerns the identiﬁcation of the numbers Prs = Ps /Pr obtained from solutions with all possible price indices that are compatible with the data, that is, derivable on the basis of compatible homogeneous utilities. Then it will be possible to go further with a description of all possible price indices in terms of closed intervals speciﬁed by formulae for their end-points, or limits. For any utility order R, the derived utility–cost function ρ (p, x) = min[ py : yRx] is deﬁned for all p, x if the sets Rx are closed. If R is a complete order and the sets Rx, xR are closed then, for any p, ρ( p, x) is a utility function representing R (see Afriat, 1979). It follows that, for any p and q, there exists an increasing function w(t), independent of x and carrying p, q as parameters, such that ρ( p, x) = w(ρ(q, x)) for all x. If R is conical then so is ρ( p, x) as a function of x, for any p. In that case so is w(t) as a function of t. But a conical function of one variable must be homogeneous linear, so w(t) has the form wt where w is a function of p, q independent of x. That is, ρ( p, x)/ρ(q, x) = w is independent of x. Then w must have the form w = θ( p)/θ (q) where θ ( p) is a function of p alone, so it follows that ρ( p, x)/(θ( p) must be a function of x alone, and so ρ( p, x) = θ ( p)φ(x), where θ, φ are functions p, x alone. Now φ must be a utility that represents R, and be conical because p is conical in x. Then the condition for φ to be quasi-concave is that the sets Rx be convex. But because φ is conical this is also the condition that φ be concave (Berge, 1963). Moreover, because in any case ρ( p, x) is concave conical in p, so also is θ( p). The reﬂexivity of R gives in any case ρ(p, x) ≤ px, so now θ(p)φ(x) ≤ px for all x. Hence, for all p, θ(p) ≤ minx px(φ(x))−1 , while ρ(p, x) = px for some x

On the constructibility of consistent price indices 149 gives θ (p)φ(x) = px some x, so that now θ( p) = min px(φ(x))−1 . Similarly φ(x) ≤ minp (θ ( p))−1 px. Then let φ(x) = minp (θ ( p))−1 px, so φ(x) ≤ φ(x) for all x and φ is concave conical. Then, for any x, φ(x) = φ(x) is equivalent to θ ( p)φ(x) = px for some p. The condition for this to be so for any x is that φ(x) be quasi-concave, having a quasi-support p at x, for which py < px → φ(y) < φ(x),

py ≤ px → φ(y) ≤ φ(x)

in other words the demand (x, p) is compatible with x for some p. This condition, which means that, with choice governed by φ, x could be demanded at some prices, can deﬁne compatibility between φ and x. The condition that φ be compatible with all x is just that it be quasi-concave, and that now is equivalent to φ(x) = φ(x) for all x. In any case, any x compatible with φ is also compatible with φ. Hence, if utility R is constrained by compatibility with given demands, if φ is acceptable then so is φ, and moreover φ and φ have the same conjugate price function θ . This suggests that, instead of constructing a utility φ compatible with the data having a concave form that is a generally unwarranted restriction on utility and, for all we know now, might make some added restriction on price-index values, it is both possible and advantageous to construct the price function θ ﬁrst instead, and so be free of such a suspicion. It has already been remarked that this might have been done instead, following an identical procedure as that for the φ, so in fact that issue is already disposed of. Form any compatible homogeneous utility a price function θ is derived, giving the Pr = θ (pr ) as a system of ‘price-levels’ compatible with the data, and determining the Prs = Ps /Pr as compatible prices. But the possible such Pr are already identiﬁed with the possible solutions of (11.3). Also, for any solution Pr and the θ that must exist, the Xr determined from (11.2) have the identiﬁcation Xr = φ(Xr) where φ(x) = minp (θ ( p))−1 px. This φ is concave conical. But also any other φ ∗ , not necessary concave but having the same conjugate θ , would do, so there is no inherent restriction to concave utilities here. For such a φ ∗ , generally φ ∗ (x) ≤ φ(x), while φ ∗ (xr ) = φ(xr ) for all r, and all that is required of φ ∗ is that θ(x) = minp px((φ ∗ (xr ))−1 , and there are many φ ∗ for which this is so, that given being just one. The argument in this section permits by-passing complications of the argument involving ‘critical cost functions’ that was used formerly, such as in the exposition of Afriat (1977b) for the special case of just two demand-periods. An interesting point is that the care taken in both arguments to avoid imposing on a compatible utility the requirement that also it be concave makes no difference at all to the range of possible values for a price-index.

12 Extension and exhaustion properties For any coefﬁcients ars (r, s = 1, . . . , k), consider the system S(a) of simultaneous linear inequalities ars ≥ xs − xr

(12.1)

150 Precursor to be solved for numbers xr . This is an alternative form for the system (7.1), and the form that applies directly to the system (7.3). Introduce chain-coefﬁcients arij···ks = ari + aij + · · · + aks

(12.2)

so that ar···s···t = ar···s + as···t

(12.3)

If the system S(a) has a solution x then ari ≥ xi − xr , xij ≥ xj − xi , . . . , aks ≥ xs − xk so by addition, ar···s ≥ xs − xr

(12.4)

In particular ar . . . r ≥ xr − xr = 0, so that ar . . . r ≥ 0, that is, ari + aij + · · · + akr ≥ 0

(12.5)

for every cyclic sequence of elements r, i, j, . . . , k, r. This can be called the cyclical non-negativity condition C on the system S(a), and it has been seen necessary for the existence of a solution. Because ar···s···s···r = ar···s···r + as···s the coefﬁcient on a cycle with a repeated element s can be expressed as a sum of terms that are coefﬁcients on cycles where the repetition multiplicity is reduced, and this decomposition can be performed on those terms and so forth until an expression is obtained with only simple cycles, without repeated elements. From this it follows that the condition C is equivalent to the same condition on cycles that are restricted to be simple, or have elements all distinct. Under the condition C, ar···s···s···t = ar···s···t + as···s ≥ ar···s···t so the cancellation of a loop in a chain does not increase the coefﬁcient along it. It follows that the derived coefﬁcient Ars = min arij···s ij···

(12.6)

exists for any r, s and moreover Ars = arij···s

(12.7)

for some simple chain rij · · · s from r to s. Thus C is sufﬁcient for the existence of the derived coefﬁcients. Also it is necessary. For if as · · ·s < 0 then for any r, t by

On the constructibility of consistent price indices 151 taking the chain that goes from r to t, following the loop s · · · s any number K of times and then going from s to t, we have Art ≤ ars + Kas···s + ast → −∞(K → ∞)

(12.8)

so Art cannot exists. Therefore also C is sufﬁcient for the existence of the derived coefﬁcients. Evidently then either the derived coefﬁcients all exist or none do. Given that they exist, from (12.3) it follows that Ars + Ast ≥ Art , so they satisfy the triangle inequality. The system S(a) and the derived system S(A), of inequalities Ars ≥ xs − xr

(12.9)

have the same solutions. For from (12.4), any solution of (12.1) is a solution of (12.9). Also from Ars ≤ ars , that follows from the deﬁnition (12.6) of the Ars , it is seen that any solution of (12.9) is a solution of (12.1). The triangle inequality is necessary and sufﬁcient for a system to be identical with its derived system, that is for Ars = ars for all r, s. It is necessary because any derived system has that property. Also it is sufﬁcient. For the triangle inequality on S(a) is equivalent to arij . . . ks ≥ ars , but (12.9) implies both that the derived coefﬁcients Ars exist and that Ars ≥ ars which because of (12.8) is equivalent to (12.9). Now the extension property for the solutions of a system that satisﬁes the triangle inequality will be proved. Let S(A) be any such system, so if this is the derived system of some other system then this hypothesis must be valid. A subsystem of S(A) is obtained when the indices are restricted to any subset of 1,… , n. Without loss in generality consider the subsystem Sm−1 (A) on the subset of 1, . . . , m − 1. Let xr (r < m) be any solution for this subsystem, so that Ars ≥ xs − xr

for r, s < m

(12.10)

Now consider any larger system obtained by adjoining a further element to the set of indices. Without loss in generality, let m be that element and Sm (A) the system obtained. It will be shown that there exists xm so that the xr (r ≤ m) that extend the solution xr (r < m) of Sm−1 (A) are a solution of Sm (A), that is Ars ≥ xs − xr

for r, s < m

(12.11)

With the xr (r < m) satisfying (12.10), xm has to satisfy Arm ≥ xm − xr for r < m,

Ams ≥ xs − xm for s < m

(12.12)

Equivalent, xs − Ams ≤ xm ≤ xr + Arm for r, s < m. But a necessary and sufﬁcient condition for the existence of such xm is that xs − Ams ≤ xr + Arm for r, s < m, equivalently Arm + Ams ≥ xs − xr for r, s < m. By the triangle inequality, (12.11)

152 Precursor implies this, so the existence of such xm is now proved. Thus any solution of Sm−1 (A) can be extended to a solution of Sm (A). Then by an inductive argument it follows that, for any m ≤ n, any solution xr (r ≤ m) of Sm (A) can be extended to a solution of Sn (A) = S(A), by adjunction of further elements xr (r > m). It can be concluded that any system with the triangle inequality has a solution, because the triangle inequality requires in particular that A11 + A11 ≥ A11 , or equivalently A11 ≥ 0. This assures that S1 (A) has a solution x1 and then any such solution x1 can be extended to a solution xr (r ≤ n) of S(A). From the foregoing, each of the conditions in the following sequence implies its successor: (i) The existence of a solution. (ii) The cyclical non-negativity test, (iii) The existence of the derived system. (iv) The existence of a solution for the derived system. (v) The existence of a solution. It was shown ﬁrst that (i) → (ii) → (iii). Then because any derived system satisﬁes the triangle inequality and any system with that property has a solution, (iii) ↔ (iv) is shown. Now the identity between the solutions of a system and its derived system shows (iv) ↔ (i) and establishes equivalence between all the conditions, in particular between (i) and (ii). The derived system S(A) can be stated in the form −Asr ≤ xs − xr ≤ Ars

(r ≤ s)

requiring the differences xs − xr (r ≤ s) to belong to the intervals (−Asr , Ars ). The extension property of solutions assures also the interval exhaustion property, that every point in these intervals is taken by some solution. Whenever the derived system exists these intervals automatically are all non-empty. An order U of the indices determined from the coefﬁcients Ars is given by the → transitive closure U = A of the relation A given by rAs = Ars ≤ 0. Also, any solution x determines an order V (x) of the indices, where rV (x)s ≡ xs ≤ xr . Whatever the solution, this is always a reﬁnements of the order U, that is V (x) ⊂ U for every solution x. Moreover, for any order V that is a reﬁnement of U, there always exists a solution x such that V (x) = V . This order exhaustion property can be seen from the interval exhaustion property and also by means of the proof of the extension property of solutions by taking the extensions in the required order. These results can all be translated to apply to a system in the form ars ≥ xs /xr , now with multiplicative chain coefﬁcients, arij···ks = ari aij · · · aks and derived coefﬁcients Ars deﬁned from these as before and satisfying multiplicative triangle inequality Ars Ast ≥ Art . The cyclical non-negativity test becomes ars ast · · · aqr ≥ 1. For any solution x, the ratios xs /xr are required to lie in the intervals Irs = (1/Asr , Ars ). From their form these intervals remind of the Paasche–Laspeyres (PL) interval (1/Lsr , Lrs ). Also, the multiplicative chain coefﬁcients correspond to the familiar procedure of multiplying chains of price indices, except that there are many chains with given extremities and here one is taken on which the coefﬁcient is minimum. While the non-emptiness of the PL-intervals, whether or not the one index exceeds the other, is a well-known issue, there is no such issue at all with the intervals Irs because whenever they are deﬁned they are non-empty, this following

On the constructibility of consistent price indices 153 from the multiplicative triangle inequality that gives Ars Asr ≥ Arr ≥ 1.

13 The power algorithm For a system with coefﬁcients ars , and any k ≤ m ≤ n, let a[k] rs = min (ari2 + ai2 i3 + · · · + aik s ), i2 ik

[k] a(m) rs = min ars k≤m

(13.1)

According to (12.6), if the derived coefﬁcients exist any one has the form Ars = ari + aij + · · · + aks

(13.2)

for some i, j, . . . , k making r, i, j, . . . , s all distinct except possibly for the coincidence of r and s. Because there are just n possible values 1, . . . , n for the indices, it follows from (13.1) and (13.2) that Ars ≥ a(n) rs . But from the deﬁnition (n) of the Ars in (12.6) and from (13.1) again, also Ars ≤ a(n) rs , so now Ars = ars , that is (n) A = a . Now writing+as · and min as+, (13.1) becomes ari1 · ai1 i2 · · · aik−1 s a[k] rs = i,... im−1

= (a · a · · · ·a)rs

(k factors)

= (ak )rs that is a[k] = ak , where the ‘power’ ak so deﬁned is unambiguous because of associativity of ‘multiplication’ and ‘addition’ and the distributivity of ‘multiplication’ over ‘addition’ and is determined recurrently from a1 = a and ak = a · ak−1 Then (13.2) becomes a(m) =

(13.3) m

1a

k

, and is determined from

a(1) = a and a(m) = aa(m−1) + a

(13.4)

This algorithm with powers in the context of minimum paths in networks is from Bainbridge (1978). Observed now is a simpliﬁcation that is applicable to the special case of importance here where arr = 0, or where arr = 1 in the multiplicative formulation. If arr = 0, in which case chains of any length include all those of lesser length, we have a ≥ a2 ≥ · · · ≥ ak ≥ · · · ≥ am where the matrix relation ≥ means that relation simultaneously for all elements. Therefore in this case a(m) = am . This with A = a(n) together with (13.3), shows

154 Precursor that the matrix A of derived coefﬁcients can be calculated by raising the matrix a of the original coefﬁcients to successive powers, the nth power being A. Should am = am−1 for any m ≤ n then also am = am+1 = · · · = an = · · · so that A = am . But in any case the formula A = an is valid. Then evidently A = A2 = · · · , so the derived matrix is idempotent. This property is characteristic of any matrix having the triangle inequality. By taking exponentials, these procedures can be translated to apply to the system in the multiplicative form (7.1), Lrs ≥ φs /φr with Lrr = 1. Matrix powers have been deﬁned in a sense where + means min and · means +. Taking exponentials turns + into · and leaves min as min, so we are back with · meaning ·. This makes Lˆ = Ln a formula for the derived coefﬁcient matrix, where powers now are deﬁned in the ordinary sense except that + now means min. As before, n here can be replaced by any m ≤ n for which Lm = Lm+1 , in particular by the ﬁrst such m found.

References

Afriat, S.N. 1956. ‘Theory of Economic Index Numbers’. Mimeographed. Department of Applied Economics, Cambridge. — 1960. ‘The system of inequalities ars >Xs –Xr .Research Memorandum No. 18, Econometric Research Program, Princeton University, Princeton, NJ. Proc. Cambridge Phil. Soc., 59 (1963), 125–33. — 1964. ‘The construction of utility functions from expenditure data’. Cowles Foundation Discussion Paper No. 144, Yale University. Paper presented at the First World Congress of the Econometric Society, Rome, September 1965. International Economic Review, 8 (1967), 67–77. — 1967a. ‘The Cost of Living Index’. In Studies in Mathematical Economics: In Honor of Oskar Morgenstern. Edited by M. Shubik. Princeton, NJ: Princeton University Press. — 1970a. ‘The Concept of a Price Index and Its Extension’. Paper presented at the Second World Congress of the Econometric Society, Cambridge, August 1970. — 1970b. ‘The Cost and Utility of Consumption’. Mimeographed. Department of Economics, University of North Carolina at Chapel Hill, North Carolina. — 1970c. ‘Direct and Indirect Utility’. Mimeographed. Department of Economics, University of North Carolina at Chapel Hill, North Carolina. — 1972a. ‘Revealed Preference Revealed’. Waterloo Economic Series No. 60, University of Waterloo, Ontario. — 1972b. ‘The Theory of International Comparisons of Real Income and Prices’. In International Comparisons of Prices and Output, Proceedings of the Conference at York University, Toronto, 1970. Edited by D. J. Daly, 13–84. Studies in Income and Wealth, vol. 37. New York: National Bureau of Economic Research. — 1973. ‘On a System of Inequalities in Demand Analysis: an Extension of the Classical Method’. International Economic Review, 14, 460–72. — 1974. ‘Sum-symmetric matrices’. Linear Algebra and its Applications, 8, 129–140. — 1976. Combinatorial Theory of Demand. London: Input-Output Publishing Co. — 1977a. ‘Minimum paths and subpotentials in a valuated network’. Research Paper 7704, Department of Economics, University of Ottawa. — 1977b. The Price Index. Cambridge and New York: Cambridge University Press. — 1978a. ‘Index Numbers in Theory and Practice by R. G. D. Allen’. Canadian Journal of Economics, 11, 367–9. — 1978b. ‘Theory of the Price Index by Wolfgang Eichhorn and Joachim Voeller’. Journal of Economic Literature, 16, 129–30. — 1979. Demand Functions and the Slutsky Matrix. Princeton, NJ: Princeton University Press.

156 Precursor — 1980. ‘Matrix Powers: Classical and Variations’. Paper presented at the Matrix Theory Conference, Auburn University, Auburn, Alabama, 19–22 March 1980. Allen, R. G. D. 1975. Index Numbers in Theory and Practice. London: Macmillan. Bainbridge, S. 1978. ‘Power algorithm for minimum paths’, private communication. Department of Mathematics, University of Ottawa, Ontario. Berge, C. 1963. Topological Space. New York: Macmillan. Berge, C. and Ghouila-Houri, A. 1965. Programming, Games and Transportation Networks. London: Methuen & Co. New York: John Wiley and Sons. Dantzig, G. 1963. ‘Linear Programming and Its Extensions’. Princeton, NJ: Princeton University Press. Edmunds, J. 1973. ‘Linear programming formula for minimum paths’, private communication. Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario. Eichhorn, W. and Voeller, J. 1976. Theory of the Price Index: Fisher’s Test Approach and Generalizations. Berlin, Heidelburg and New York: Springer Verlag. Fiedler, M. and Ptak, V. 1967. ‘Diagonally dominant matrices’. Czech. Math J ., 17, 420–33. Fisher, I. 1927. The Making of Index Numbers. Third edition. Boston and New York: Houghton Mifﬂin Company. Geary, R. C. 1958. ‘A note on the comparison of exchange rates and purchasing power between countries’. J . Roy Stat. Soc. A, 121, 97–9. Hicks, J. R. 1956. A Revision of Demand Theory. Oxford: Clarendon Press. Houthakker, H. S. 1950. ‘Revealed Preference and the Utility Function’, Economica, New Series, 17, 159–74. Samuelson, P. A. 1947. The Foundations of Economic Analysis. Cambridge, Mass.: Harvard University Press. Samuelson, P. A. and Swamy, S. 1974. ‘Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis’. American Economic Review, 64, 566–93. Shephard, R. W. 1953. Cost and Production Functions. Princeton, NJ: Princeton University Press. Theil, H. 1960. ‘Best linear index numbers of prices and quantities’. Econometrica, 28, 464–80. 1975–1976. Theory and Measurement of Consumer Demand. Two volumes. Amsterdam: North-Holland Publishing Company. — 1979. The System-wide Approach to Microeconomics. The University of Chicago Press. — Yen, J. Y. 1975. Shortest Path Network Problems. Meisenheim am Glan: Verlag Anton Hain.

3 The Theory of Exact and Superlative Index Numbers Revisited Carlo Milana Istituto di Studi e Analisi Economica, Roma EUKLEMS Working Paper No. 3, 2005 http://www.euklems.net

THE THEORY OF EXACT AND SUPERLATIVE INDEX NUMBERS REVISITED1 By Carlo Milana

1 Introduction This chapter contains a critical review of the theory of “exact” and “superlative” index numbers that is still dominating the ﬁeld of economic index numbers. An index number is said to be “exact” for a function if it is identically equal to the ratio of numerical values of that function at any pair of points taken into comparison. The ﬁrst use of the concept of “exact” index numbers appeared in the discovery of Byushgens (1925) that Irving Fisher’s ‘Ideal Index’ is an exact formula when demand is governed by a homogeneous quadratic utility (see also Konüs and Byushgens, 1926). This concept became more widely known with Schultz (1939) and its discussion in the Foundations of Paul A. Samuelson (1947, p. 155). In the introduction to the enlarged edition, Samuelson (1983, p. xx) later wrote: “Thirty-ﬁve years after that [revealed preference] analysis appeared there has been but one major advance in index number theory—namely W. E. Diewert’s formalizing concept of a ‘superlative index number’, which is a formula based upon two periods (pj , qj ) data that will be exactly correct as an ordinal indicator of utility for some speciﬁed family of indifference contours.(Only a few different ‘superlative’ formulas are known; perhaps the set of simple superlative formulas is a limited set.)”2 Diewert (1976, p. 117) used Irving Fisher’s (1922, p. 247) terminology of “superlative” index numbers to deﬁne index numbers that are exact for functions that provide a second-order differential approximation to an unknown true function. In another paper, Diewert (1978) pointed out that these index numbers approximate each other up to the second-order at the point where the compared

1 Revised version of a review prepared at ISAE, in Rome, for the Speciﬁc Targeted Research Project “EU KLEMS. Productivity in the European Union: A Comparative Industry Approach” supported by the European Commission with Contract No. 502049 (SCS8) within the Sixth Framework Programme (see http://www.euklems.net).Comments received from Pirkko Aulin-Ahmavaara, Alberto Heimler, and Dale W. Jorgenson are thankfully acknowledged. 2 Diewert’s superlative index numbers are, however, only the subset of known superlative index numbers that are exact for ﬂexible functions expressed in closed algebraic forms. Other superlative index numbers, as for example that deﬁned by Sato (1976) and Vartia (1976), do not appear to be exact for any function in closed form (see Barnett and Choi, 2006).

160 Precursor base and current points of observations are equal and are numerically very close if these two points do not vary very much3 . Quoting Diewert (2004, p. 450) words, “Diewert (1978, p. 888) showed that the three superlative index number formulas listed approximate each other to the second order around any point where the two price vectors [in the calculation of the price index], p0 and p1 ,are equal and where the two quantity vectors [in the calculation of the quantity index], q0 and q1 , are equal. He concluded that ’all superlative indices closely approximate each other’(Diewert, 1978, p. 884)”. This deﬁnition can be contrasted with that given by Irving Fisher (1922, pp. 244–48), who had singled out eleven index number formulas out of 134 examined formulas and called them “superlative” because, in his numerical example, they performed very closely to the “ideal” geometric mean of Laspeyres and Paasche indexes4 . He claimed that all these superlative formulas correspond to combinations of the Laspeyres and Paasche including the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Apart from the Fisher “ideal” and the implicit Walsh indexes, Diewert’s superlative index numbers differ widely in nature and, potentially, in numerical values from those deﬁned superlative by Fisher (1922)5 . Allen and Diewert (1981, p. 430) had recognized that “in many applications involving the use of cross section data or decennial census data, there can be a tremendous amount of variation in prices or in quantities between the two periods so that alternative superlative index number can generate quite different results”. In fact, more recently, Hill (2006) has found a large spread in numerical values of alternative Diewert’s superlative index numbers, with the largest and the smallest ones differing by more than 100 per cent using a standard US national data set and by about 300 per cent in a cross-section comparison of countries using an OECD data set. However, it has been surprising to ﬁnd empirically that the spread between the largest and the smallest Diewert’s superlative index numbers may exceed

3 See also Vartia (1978) and Diewert (2004, p. 450). However, Diewert (1978, p. 890, fn. 8), citing the discussion in Lau (1974, p. 183), recognized that, without performing extensive computations involving the third order partial derivatives of the index number formulae, we cannot specify exactly how small should be the change between the two compared points in order for the superlative indexes to be all very close to each other. 4 Irving Fisher (1922, pp. 244–248) ranked the examined 134 index numbers according to their distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them “arbitrarily into several classes in increasing order of merit. The ﬁrst twelve index numbers, constituting the ﬁrst of these classes, are labeled, rather harshly perhaps, as ‘worthless’ index numbers (to designate the fact that they are the worst). The other six classes are labeled as poor, fair, good, very good, excellent, and superlative” (p. 244). 5 The Törnqvist index, for example, (corresponding to formula 123 in Fisher’s, 1922, p. 473) is seen as the most superlative by Caves, Christensen, and Diewert (1982, p. 41) and Diewert (2004, p. 450) whereas it was not deemed “superlative” by Fisher (1922, p. 247), who classiﬁed it, in a descending order of merit, below the classes of “excellent” and “superlative” index numbers, with the last group ranked at the top position.

The Theory of Exact and Superlative Index Numbers Revisited 161 that between the Laspeyres and Paasche indexes, which are usually considered to be the bounds of the interval of possible values of economic index numbers, at least in the homothetic case. This performance is clearly in contrast with that considered originally by Irving Fisher in identifying his own superlative index numbers. The present paper starts by reformulating, in a general framework, the Quadratic Identity and other propositions on which Diewert’s superlative index numbers are based. A uniﬁed approach is developed, where the underlying unknown true functions may differ not only in parameters, but also in functional forms. It is also shown that, in practical applications, any index number that is constructed using the observed data, in general, cannot be really superlative in Diewert’s sense. The weights used are, in fact, derivable from the unknown true function and not (at least at one point of observation) from the approximating quadratic function for which the index number is meant to be exact. Consequently, using the observed data, Diewert’s superlative index numbers become hybrid formulas that may be quite far from providing the expected second-order approximation to the unknown true values. The rest of the text is organized as follows. Section 2 re-examines the meaning of the Quadratic Identity within a general framework of accounting for functional value changes of an arbitrary differentiable function. Section 3 extends this analytical approach and generalizes the results obtained in the literature up till now. Section 4 extends the analysis further by using transformed functions. Section 5 presents a uniﬁed approach to index numbers and indicators of absolute change by using a general transformed quadratic function with no a priori separability and homogeneity restrictions. Section 6 deals with the approximation properties of index numbers. Section 7 concludes.

2 A general formulation of the quadratic approximation lemma In the general case of an arbitrary differentiable function, the following result is obtained: Lemma 2.1 Accounting for Functional Value Changes. Let z be a vector of N real valued variables and let us assume that an arbitrary function f (z) is continuously differentiable at least once. Then, for all z 0 and z 1 , f z 1 − f z 0 = (1 − θ ) ∇f z 0 + θ ∇f z 1 · z 1 − z 0

(2.1)

where ∇f z t ≡ ∂f /∂z1t ∂f /∂z1t . . . ∂f /∂zNt is the gradient vector of f evaluated at z t and, denoting with R01 (z 0 , z 1 ) and R11 (z 0 , z 1 ) the remainder terms associated respectively with the polynomials of order one in the Taylor series expansion for f around z 0 and z 1 , θ is a scalar that takes the particular value R0 ( z 0 , z 1 ) θ ∗ z 0 , z 1 ≡ R0 z0 , z11 +R1 z0 , z1 when R01 z 0 , z 1 + R11 z 0 , z 1 = 0, or θ may take ( ) ( ) 1 1

162 Precursor

any real number as a value if f is linear in z (in which case R01 z 0 , z 1 = R11 0 1 z , z = 0)6 . Proofs of propositions are given in Appendix B. It is straightforward to show that 0 ≤ θ ≤ 1 if f (z) is quasiconcave or quasiconvex. Lemma (2.1) can be complemented with the following corollaries. Corollary 2.1 Decomposition of Functional Value Changes. The change in the value of an arbitrary differentiable function f(z) can be further decomposed as follows, for all z 0 and z 1 , (2.2) f z 1 − f z 0 = JZ + JS 0 0 1 f z IZ · f z (1 − θ ) si∗0 0 + θ si∗1 where JZ ≡ zi − zi0 1 zi zi 0 0 ∗0 f z 1 ∗1 IZ · f z 0 (1 − θ ) ξ − 1 si JS ≡ + θ ξ − 1 si zi0 zi1 · zi1 − zi0 [∂f (zt )/∂zi ] · zit with θ being deﬁned by Lemma (2.1), si∗t ≡ sit / Nj=1 sjt = N t, t i=1 [∂f (z )/∂zi ] · zi t ∂f (z ) for t = 0, 1 where sit ≡ ∂zt · zit /f z t (the superscript ∗ denoting here ∂f (z0 ) ∂f (z1 ) normalized shares), ξ 0 ≡ Ni=1 ∂z0 · zi0 /f z 0 , and ξ 1 ≡ Ni=1 ∂z1 · zi1 / i i IZ · f z 0 , representing the scale effects, and I Z is the index of functional value change net of scale effects, so that f z1 0 = I Z · IS f z

with

IZ ≡

θ + (1 − θ) (1 − θ) + θ

(2.3) N

1

∗0 zi i=1 si z 0 i 0 N ∗1 zi i=1 i z 1 i

s

and

f z1 I S ≡ 0 IZ . f z

It may be tempting to interpret the terms JZ and JS in (2.2) as contributions to the absolute functional value change arising, respectively, from changes in z“directly” and through the returns to scale. Similarly, the terms IZ and IS in (2.3) might be interpreted, respectively, as the “aggregating” index numbers of relative changes in z and the scale effects. The index IZ is in fact homogeneous of degree one in z.

6 The usual vector or matrix transposition sign is ommitted to simplify notation.

The Theory of Exact and Superlative Index Numbers Revisited 163 It is equal to a Laspeyres-type index number if θ = 0, whereas it is equal to a Paasche-type index number if θ = 1. However, the following remark deﬁnes the limits of such interpretation: Remark 2.1 on the existence of the aggregator index: If and only if the function f is homothetic, so that f (z) = F[φ(z)], where F (•) is a well behaved transformation function (real valued, continuously differentiable, monotonically increasing and quasiconcave) and φ(z) is also well behaved and linearly homogeneous7 , the weights si∗t can be expressed only in terms of the function φ. In fact,

si∗t

∂f z t /∂zi · zit d [F] /dφ · ∂φ z t /∂zit = N t t = d [F] /dφ · Nj=1 ∂φ z t /∂zj · zjt j=1 ∂f z /∂zj · zj ∂φ z t /∂zit · zit = φ zt

(2.4)

since φ (z) = Nj=1 ∂φ (z) /∂zj · zj for Euler’s theorem on linearly homogeneous functions. In the homothetic case, the indexes JZ and IZ are exact for the aggregator or index function8 φ. In this case, these indexes are isolated from the scale effects and are clearly deﬁned as aggregating indexes. This is a well-known result due to the homogeneity price and quantity theorems of Shephard (1953) in the theory of production and Samuelson (1953) in the theory of direct and indirect utility (see also the survey and synthesis of Samuelson and Swamy, 1974, pp. 568–569). It is straightforward to show that, if the function φ is well behaved, and in particular quasiconvex or quasiconcave, then the Laspeyres- and Paasche-type index numbers are the bounds of the interval of all possible numerical values that the ratio φ(z 1 )/φ(z 0 ) may take. By contrast, when the function f is not homothetic, the z-change component is constructed using weights that are also affected by the scale of the function in a non-homothetic way9 . In the more general non-homothetic case, no true indicator and index number of changes in z really exist and any attempt to construct them using JZ and IZ deﬁned respectively in (2.2) and (2.3) ends up to spurious magnitudes.

7 The concept of homotheticity was explicitly spelled out by Shephard (1953) and Malmquist (1953). Earlier researchers as Frisch (1936, p. 25) and Samuelson (1950, p. 24) had dealt with it implicitly. 8 The terms “index function” and “aggregator function” are used here interchangeably. The former was used by Shephard (1953, pp. 47–49) and Solow (1956, pp. 102–106), whereas the latter was used by Diewert (1976). 9 This remark is equivalent to the well-known conclusion on path-independency of Di-visia indexes in the case of homothetic functions (see Hulten, 1973 and Samuelson and Swamy, 1974).

164 Precursor Let us deﬁne the quadratic function: 1 1 fQ (z) ≡ a0 + a z + zA z = a0 + a zz, ai zi + 2 i=1 j=1 ij i j 2 i=1 N

N

N

(2.5)

where a0 , a, and A are, respectively, a scalar, a vector, and a matrix of constant parameters and it is assumed that A is symmetric, that is aij = aji for all i, j, to ensure the symmetry of the Hessian matrix of second derivatives by Young’s theorem. The Bernstein-Weierstrauss approximation theorem states that, on a closed bounded domain, a continuous function can be approximated uniformly by polynomials. The function fQ (z) deﬁned by (2.5) can be seen as providing a second-order approximation to an arbitrary function f (z) around the point z ∗ if its parameters are “calibrated” to particular numerical values such that fQ (z ∗ ) = f (z ∗ ), ∇fQ (z ∗ ) = ∇f (z ∗ ), and ∇ 2 fQ (z ∗ ) = ∇ 2 f (z ∗ )10 . The “problem of accuracy of approximation” was clearly seen and described by Fuss, McFadden, and Mundlak (1978, pp. 233–234) in the following terms: “If a ﬂexible form is calibrated to provide a second-order approximation at a point, then the approximation is of this order only in a small neighborhood of this point. In other regions of interest, the form may be a poor approximation to the true function. […] Further, the qualitative implications of the calibrated approximation may depend on the point of approximation; this is true, for example, of separability, which involves properties of the true function beyond second-order”. This is the general case presented in the standard microeconomics textbooks, where production functions and utility functions are in fact presented in non-linearly homogeneous forms involving non-zero third derivatives due to non-homothetic effects and externalities11 . The following well-known result can be derived as another corollary of Lemma 2.1: Corollary 2.2 Diewert’s (1976, p. 118) Quadratic Identity. If and only if f (z) has the functional form of the quadratic function f Q (z) deﬁned by (2.5), then the weight θ in (2.1) is equal to 1/2 for all z 0 and z 1 , so that 1 fQ z 1 − fQ z 0 = ∇fQ z 0 + ∇fQ z 1 · z 1 − z 0 2

(2.6)

The Quadratic Identity established by Corollary (2.2) can be explained geometrically as in Figure 1, where the quadratic function fQ (z) is represented

10 Two alternative concepts of second-order approximation based, respectively, on the Taylor expansion series and on the equality of ﬁrst- and second-order partial derivatives of the two functions valued at a base point have been discussed by Lau (1974, 183). Barnett (1983) has shown that they are equivalent. 11 A good example of a non-homothetic utility function implying non-zero third derivatives for the integrand of a Divisia. price index is given by Feenstra and Reinsdorf (2000).

The Theory of Exact and Superlative Index Numbers Revisited 165 f 0L(z)

B’

f 1L(z) fQ(z)

B A’ A

z O

z0

z1

Figure 1 Geometrical representation of Diewert’s (1976) Quadratic Identity with a concave function

in one single variable and the functions fL0 (z) and fL1 (z) are, respectively, the ﬁrst-order polynomials of the Taylor’s expansions of fQ (z) around z 0 and z 1 . With the quadratic function, the remainder terms AA and BB of two ﬁrst-order approximations are equal in absolute size with all z 0 and z 1 and, therefore, their difference is always equal to zero. In fact, fQ (z 1 ) − fQ (z 0 ) = AB is equal to 12 fL0 z 1 − fL0 z 0 + fL1 z 1 − fL1 z 0 = 12 fL0 z 0 + fL1 z 1 · z 1 − z 0 = 1 AB + A B = AB + 12 AA − BB , where AA − BB = 0 with all z 0 and z 1 . 2 Some remarks concerning this result are in order: (i)If ∇z fQ z 0 = ∇z fQ z 1 , then fQ is non-linear in z (however, the converse is not true since a non-linear function may have equal ﬁrst derivatives at some different points). (ii)Diewert (1976, p. 138) gave his necessity proofs under the assumption that the examined function fQ is thrice continuously differentiable. The existence of non-zero third derivatives implies, in particular, that fQ is non linear, at least locally. He showed that, since equation (2.6) implies that all the third derivatives vanish for all z 0 = z 1 , the polynomial form of fQ is quadratic (see the necessity proof in Appendix B). (iii)Lau (1979), noted that “generalizing Diewert’s proof to the twice continuously differentiable case is straightforward” (p.74, fn. 1). He also gave an alternative proof of the Quadratic Identity by assuming a once differentiable function and observed that this “widens considerably the applicability of the lemma and consequently of the results which depend on its validity”(p.74). (iv)Under the general differentiability assumption, the quadratic function (2.5) can in fact be regarded as having the most general functional form with which identity (2.6) exactly holds. The relation (2.6) still holds in special (limit) cases

166 Precursor

where ∇z fQ z 0 = ∇z fQ z 1 including the linear function fL (z) ≡ a0 + az, corresponding to fQ with A → 0N ×N , and the constant-value fC (z) ≡ a0 , corresponding to an fQ with a → 0N and A →0N ×N (however, by Lemma 2.1, in these two last cases the weight θ may also take other real numbers as a value other than 1/2). This result weakens somewhat the meaning of name of “Quadratic Identity” that Caves, Christensen, and Diewert (1982) have given to Diewert’s (1976, p. 118) lemma. (v)The necessity part of Corollary (2.2) means that, when a function is indeed non-linear (as implied by the thrice differentiability condition), then the relation (2.6) should represent a “quadratic identity” since, in this case, it is compatible only with all third derivatives equal to zero, thus implying a quadratic functional form of fQ . (vi)Diewert (1976, p. 118) called his Quadratic Identity under the name of “Quadratic approximation lemma” on the implicit assumption that all parameters of the quadratic polynomial function (2.5) are “calibrated” consistently with the second-order approximation to the function f (z) around z 0 . This means that the two functions f (z) and fQ (z) should have the same numerical value and the same ﬁrst and second derivatives at z 0 .The quadratic function fQ (z), however, does not provide a second-order approximation to an arbitrary function without imposing additional conditions requiring that its parameters be “calibrated” to speciﬁc numerical values such that fQ (z ∗ ) = f (z ∗ ), ∇fQ (z ∗ ) = ∇f (z ∗ ), and ∇ 2 fQ (z ∗ ) = ∇ 2 f (z ∗ ) ,with z usually being equal to z 0 .The number of equations deﬁned by these conditions is equal to 1 + N + N (N +1)/212 . By contrast, the total number of the equations fQ (z 0 ) = f (z 0 ), ∇fQ z 0 = ∇f z 0 and ∇fQ z 1 , whose numerical values are the data to be used with (2.6), amounts to 1 + 2N . If 1 + N + N (N + 1)/2 > 1 + 2N , that is if N > 1, then the system of equations consistent with the second-order approximation is underdetermined. This means that, unless N = 1, the accounting equation (2.6) is consistent with an inﬁnite number of alternative tangent functions, of which only one would provide a second-order approximation to f (z). Contrary to what is commonly believed, Diewert’s (1976, p. 118) lemma (represented here by Corollary (2.2)) loses its “superlativeness” character in the multivariate case13 . (vii)The main problem with equation (2.6) is that it cannot be fully empirically implemented in the case of approximation. In fact, in this case, at least one of the two vectors of weights ∇fQ z 0 and ∇fQ z 1 are not “observable”.

12 Homogeneity properties imposed on (transformed) quadratic production functions reduce the number of equations to N (N +1)/2 (see, for example, Fuss, McFadden, and Mundlak, 1978, p. 232, fn. 4) and section 6 below. 13 A similar remark was advanced by Uebe (1978). Diewert (1978) re-deﬁned the superlative index numbers as those index number formulas that are exact for functions that provide a second-order approximation around any point where the two vectors of data under comparison are equal. This change in deﬁnition leaves the problem outlined here unresolved.

The Theory of Exact and Superlative Index Numbers Revisited 167 By contrast, the ﬁrst derivatives ∇f z 0 and ∇f z 1 of the unknown function f are usually “observable” in the economic reality (at least when “ﬁrst-best” optimal choices actually take place). In fact, if the point of approximation occurs at z 0 , then ∇f z 0 = ∇fQ z 0 , but in general the inequality ∇f z 1 = ∇fQ z 1 remains. The usual procedure is to use the observed ﬁrst derivatives ∇f z 1 in place of ∇fQ z 1 in equation (2.6), thus resulting in a hybrid approximation that is not “exact” for the quadratic polynomial form (2.5), being very different from it in size and algebraic sign. It should be stressed that the error of approximation is obtained by that applying formula (2.6) using the “observed” weights ∇f z 0 and ∇f z 1 rather than the weights ∇fQ z 0 and ∇fQ z 1 may be different in size and algebraic sign from the expected error of quadratic approximation (for an explanation of how the ﬁrst derivatives ∇f z 0 and ∇f z 1 are “observed” in the economic reality, see Appendix A). What is actually obtained with such a procedure is a hybrid measurement. It can be assessed more clearly by means of the following result:

Lemma 2.2 General Quadratic Approximation Lemma. Let f (z) be an arbitrary once differentiable function. If the value change of the arbitrary f (z) is accounted for by using (2.1) with θ set equal to 1/2 as in (2.6), then the obtained approximation error is equal to the difference of two ﬁrst-order approximations multiplied by (1/2 − θ ), that is 1 f z 1 − f z 0 = ∇f z 0 + ∇f z 1 · z 1 − z 0 2

1 1 + −θ fL z − fL z 0 − fL z 1 − fL z 0 2

(2.9)

and, for r= 0, 1, fLr (z) is a ﬁrst-order approximating (linear) function that is tangent to f (z) at z r , that is fLr (z) ≡ ar + br z = f (z r ) + ∇f (z r ) (z − z r ), with ar = f (xr ) − ∇f (xr ) xr and br = ∇f (xr ) . The averaging of the ﬁrst order derivatives is often expected to increase by one the order of approximation (see Theil, 1971, 1975, pp. 37–38). Here, instead, the approximation error represented by the second line of (2.9) is not strictly 1 of second-order, 1 1 and the difference with the latter is equal to 1 0 − z ∇f z − ∇f z z . The size of this difference depends on the Q 2 functional form of f and the distance of z 1 from z 0 . This is shown in the following numerical example.

168 Precursor f(z1)−f(z ?) 1.8 1.6 1.4 1.2 Quadratic approximation 1 0.8 Value change of the "true" (cubic) function

0.6 0.4

Hybrid approximation

0.2 0

1

z 1.5

−0.2

Figure 2 Quadratic approximation versus “hybrid” approximation

Example 2.1

Let f (z) be the following cubic function of one single variable:

1 1 f (z) ≡ a + b z + c z 2 + d z 3 2 6

(2.10)

where a = 15.5008794, b = −38.2764683, c = 65.4734033, d = −53.7666768 so that f (z) = 1.0 with z 0 = 1.0 and f (z) = 1.5 with z 1 = 1.5. A second order differential approximation to f (z) is given by the following quadratic function 1 fQ (z) ≡ α + β z + γ z 2 2

(2.11)

If α = 6.5397667, β = −11.3931299, and γ = 11.7067265, the second–order approximation is around z 0 = 1, that is f (z 0 ) = fQ (z 0 ), f z 0 = fQ z 0 , and f z 0 = fQ z 0 . Table 1 compares, for different changes in z, the actual value changes in f , which can be computed exactly using formula (2.1) (see column 6) with their “hybrid” approximation obtained by imposing θ = 1/2 in (2.1) 1 (see column 7). Figure 1 compares, for different values of z ,this “hybrid” 1 approximation with the quadratic approximation given by 2 fQ z 0 + fQ z 1 · 1 z − z 0 ,which is the right-hand side of identity (2.6). Moreover, given that only the ﬁrst derivatives of the unknown true function are normally “observed”(see Appendix A), Diewert’s superlative index numbers are usually constructed by using, as weights, these data rather than the unknown

The Theory of Exact and Superlative Index Numbers Revisited 169

z0

z1

f’(z 0) =f’Q (z 0)

f’(z 1)

(1) 1.0 1.0 1.0 1.0

(2) 1.125 1.250 1.375 1.5

(3) 0.3136 0.3136 0.3136 0.3136

(4) 1.3569 1.5601 0.9232 −0.5539

q*

f(z 1)−f(z 0)

(6) 0.567104 0.724664 1.533673 −0.791270

(7) 0.113156 0.304217 0.468169 0.5

f’Q (z 1)

(5) 1.7769 3.2403 4.7036 6.1670

Figure 3 Table 1–Quadratic and hybrid approximations of functional value changes

fQ(z1) − fQ(z 0) quadratic approximation using (2.6) (Diewert’s superlative exact formula)

Error of approximation

Hybrid approximation of Error of f(z1) − f(z 0) approximation using (2.1) with q = 1/2 (Diewert’s superlative hybrid formula)

(8)

(9) = (8) – (7)

(10)

(11) = (10) – (7)

0.130658

0.017502

0.044051

–0.069105

0.444234 0.940728 1.620139

0.140017 0.472559 1.1201399

0.235982 0.231890 –0.060070

–0.068235 –0.236279 –0.560070

Figure 4 Table 1 (continued)

ﬁrst derivatives of the quadratic approximating function. This procedure may have consequences that have been overlooked in the literature so far, although Diewert (1978) himself has ended to recognize that only at the tangency point where z 1 = z 0 these index numbers can be exact for a quadratic approximation. If z 1 = z 0 (which is the common case), then in general they are not exact for a quadratic approximation and turn out to be hybrid formulas. In the example given in Table 1, the resulting Diewert’s hybrid formula corresponding to the right-hand side of (2.1) with θ = 1/2 (or the right-hand side of the ﬁrst line of (2.9)) gives rise to an absolute error of approximation that is signiﬁcantly higher than that of the exact quadratic approximation in the neighbourhood of z 0 (about a factor of 4 with z 1 differing from z 0 by 12.5%; see column (11) and column (9) in Table 1). The two types of approximation revert their relative size as z 1 increases its distance from z 0 . However, in both cases in the given example, the errors of approximation become macroscopic (more than 50% and more than 100%, respectively) as z 1 and z 0 differ by 50%. This should warn us against constructing this type of indices with very different points of observation as in the interspatial and international comparisons.

170 Precursor

3 Functions differing in parameters or functional forms Let us consider two functions f 0 and f 1 that may differ in functional forms and/or parameters. Moreover, if some parameters are twice continuously differentiable functions of other variables subject to change over t, say k t , then f t (with t = 0, 1) can be redeﬁned as f t (z) ≡ ht z, k t

(3.1)

where k t is an M -dimensional vector of variables. Let us now consider the following result: Lemma 3.1. Accounting for Value Differences of Two Functions with Different Parameters or Functional Forms. If f 0 (z) and f 1 (z) are two arbitrary functions differentiable at least once and characterized by different parameter values or functional forms, then, f 1 z 1 − f 0 z 0 = (1 − θ) ∇f 0 z 0 + θ ∇f 1 z 1 · z 1 − z 0

where

+ parameter-change component (PC) 1 0 PC ≡ θ f z − f 0 z 0 + (1 − θ) f 1 z 1 − f 0 z 1

(3.2) (3.3)

and, by deﬁning R01 and R11 as the remainder terms associated with the ﬁrst-order polynomials in the Taylor series expansion of f 1 (z 1 ) and f 0 (z 0 ) around z 1 and z 0 , respectively, θ = θ ∗ ≡

R01 0 R1 +R11

if R01 + R11 = 0 or θ is a real number that takes any

value if R01 = R11 = 0 (that is when both f 0 (z) and f 1 (z) are linear in z). It is worth noting that the weight θ in (3.2) falls within the interval 0 ≤ θ ≤ 1 if f 0 (z) and f 0 (z) are quasiconcave or quasiconvex. If ht is separable14 in z, then (3.1) can be rewritten as: ht (z, k) = H t [φ (z) , k]

(3.4)

If k is empty, then the function ht is said to be homothetic. If φ is linearly homogeneous, then ht is said to be homothetically separable. If the function ht

14 The concept of separability was independently proposed by Sono (1945) and Leontief (1947a, 1947b) in the consumer and the producer contexts, respectively. The terminology of “weak” and “strong” separability was introduced by Strotz (1959) after a paper by Afriat (1953) and with a 1 2 departure amended by Gorman (1959a)(1959b). More speciﬁcally, a function F(x) = F(x , x , . . . , xM ) is “weakly” separable in the partition (x1 , x2 ,… , xM ) if there exist functions F ∗ , F 1 , … , F M such that F(x) = F ∗ [F 1 (x1 ), F 2 (x2 ), … F M (xM )], and is “strongly” separable if there exist functions F ∗∗ , F 1 , … , FM such that F(x) = F ∗∗ [F 1 (x1 ) + F 2 (x2 ) + … + F M (xM )]. If M =2, “weak” and “strong” separability concepts are indistinguishable. See Blackorby, Primont, and Russell (1978) for an extended treatment of the subject.

The Theory of Exact and Superlative Index Numbers Revisited 171 is homothetically separable in the two groups of variables z and k, then it can be written as ht (z, k) = H t [φ (z) , ψ (k)], with both functions φ and ψ being homogeneous of degree one. Moreover, if also the function ht is homothetically separable in φ as well as in ψ, then ht (z, k) = H t [φ (z) , (ψ (k))]

(3.5)

The concepts of homothetic functions and homothetic separability have been explicitly developed by Shephard (1953, p. 43) as a general condition for aggregation in the producer context. An aggregate of separable variables exists if the index function of those variables is separable and homogeneous of degree one. In our case, we should have λφ (z) = φ (λz) for all z and λψ (k) = ψ (λk) for all k in order for φ(z) and ψ(k) to be aggregator functions. Shephard (1953, p. 63) was the ﬁrst to realize the necessity of imposing the linear homogeneity property on index functions of separable quantity variables in order for the respective dual price index functions to be independent of those quantities15 . Others, as for example Blackorby, Lady, Nissen, and Russell (1970), used this concept in the consumer context. This deﬁnition is related to the concept of homogeneous separability, which is obtained under linear homogeneity restrictions on separable functions, regarded also as aggregability conditions (see, for example, Green, 1964, p. 25)16 . Under the homothetic (homogeneous) separability assumption, Lemma (3.1) can be complemented with the following corollaries. Corollary 3.1. Decomposition of Functional Value Differences Between Two Arbitrary Differentiable Functions. The difference in value of two arbitrary continuously differentiable functions f 0 (z) and f 1 (z) can be further decomposed as follows, for all z 0 and z 1 , f 1 z 1 −f 0 z 0 = JZ +JS +JT N 0 0 0 0 1 0 z z ∗0 f ∗1 IZ · f (1−θ)si where JZ ≡ zi −zi , +θ si 0 1 zi zi i=1

(3.6)

15 Aggregation requires, in fact, separability and homogeneity conditions. Solow (1956, p. 104n), however, attributed to Samuelson the proof – received from him in private correspondence – that, in a two stage maximization procedure, the aggregating quantity index functions must be linearly homogenous. Gorman (1959a, pp. 476–478) found, in some special cases, conditions weaker than homogeneity of degree one for a function of separable variables to be an aggregator function (see also Blackorby, Primont, and Russell, 1978, Ch. 5, Blackorby, Schworm, and Fisher, 1986, and Blackorby and Schworm, 1988). 16 More speciﬁcally, this condition is referred to as homogeneous weak separability if, under the homogeneity condition, a function is weakly separable as in (3.4) (see, for example, Diewert, 1993, pp. 12–13 and p. 28, and Diewert and Wales, 1995, pp. 260–261).

172 Precursor N ∗1 IZ · f 0 z 0 ∗0 f 0 z 0 1 0 (1 − θ ) ξ − 1 si JS ≡ + θ ξ − 1 si zi1 zi0 i=1 · zi1 − zi0 , JT ≡ Parameter − change component ∂f t (zt ) ∂f t (z t ) with θ deﬁned in Lemma (2.1), si∗t ≡ ∂zt · zit / Ni=1 ∂zt · zit = sit / Ni=1 sjt , for ∂f t (z t ) t = 0, 1 where sit ≡ ∂zt · zit /f t z t (the superscript * denoting here normalized ∂f 0 (z0 ) ∂f 1 (z1 ) shares), ξ 0 ≡ Ni=1 ∂z0 · zi0 /f 0 z 0 , ξ 1 ≡ Ni=1 ∂z1 · zi1 / IZ · f 0 z 0 , i

i

representing the scale effects, and IZ is the index of functional value change net of scale effects, so that f 1 z1 = IZ · IS · IT (3.7) f 0 z0

where

IZ ≡

θ + (1 − θ ) (1 − θ) + θ

N

1

∗0 zi i=1 si z 0 i 0 N ∗1 zi i=1 i z 1 i

s

,

f 1 z1 IS · IT = 0 0 /Iz . f z

As in (2.2)–(2.3), the terms JZ and JS in (3.7) might be seen as the contributions to the absolute functional value change arising, respectively, from changes in z “directly” and through the returns to scale, whereas J T is the contribution arising from the changes in parameters or functional form. Similarly, the index numbers IZ , IS , and IT might have been interpreted, respectively, as the aggregator index of z, the index of scale effects, and the index of changes in parameters or functional form. The index IZ is a Laspeyres-type index number if θ = 0, whereas it is a Paasche-type index number if θ = 1. However, the following remark deﬁnes Remark 3.1 on the existence of the aggregator index: If and only if the two functions f 0 (z) and f 0 (z) are homothetically (homogeneously) separable in z, so that f t (z) = F t [φ(z)], for t = 0, 1, where F t (•) is a well behaved transformation function and φ is linearly homogeneous, then for all z 0 and z 1 , in view of (2.4), ∂f t (zt ) ∂f t (z t ) ∂φ (z t )/∂z t si∗t ≡ ∂zt · zit / Nj=1 ∂zt · zjt = φ zt i · zit , for t=0, 1, JZ = φ z 1 − φ z 0 , ( ) i j i JS + JT = f 0 z 0 − f 1 z 1 − JZ , IZ = φ z 1 /φ z 0 , IS · IT = f 1 z 1 /f 0 z 0 /IZ . In the case of homothetic separability of f 0 and f 1 in z, the indexes JZ and IZ are totally independent of the scale and technological change effects and are exact for the function φ. In this case, the functions f 0 and f 1 can be expressed as homothetic transformations of the constant-parameter degreeone homogeneous function φ(z), so that JZ and IZ are functions of z alone and are exact for φ(z). This is a well-known result obtained by Hulten (1973) with the Divisia indexes, which turn out to be “path independent” if and only

The Theory of Exact and Superlative Index Numbers Revisited 173 if the integrand function is a homothetic transformation of a homogeneous function. If this is not the case, then there is no such an index number aggregating only the variables z. The functions f 0 and f 1 should be homothetic and φ should remain the same in both data points, otherwise any attempt to construct an aggregating index number will inevitable end up to a spurious measure17 . Let us deﬁne the quadratic function: 1 t 1 fQt (z) ≡ at0 + at z + z At z = at0 + ati zi + a zz 2 i=1 j=1 ij i i 2 i=1 N

N

N

(3.8)

where all parameters are variable. Moreover, let us assume that the parameters of fQt are function themselves of other variables, that is α0t ≡ α0t +k t · β t + 21 k t · Bt · k t and α t ≡ α t + k t · t so that fQt (z) ≡ htQ z, k t , where 1 1 htQ (z, k) ≡ α0t + α t z + z At z + kβ t + kBt k + k t z. 2 2

(3.9)

The function htQ (z, k) can be expressed also as a quadratic function in k: 1 htQ (z, k) ≡ ψQt (k) ≡ bt0 + kbt + kBt k 2

(3.10)

where bt0 ≡ α0t + α t z + 12 z t At z, bt ≡ β t + t z. The quadratic functional form (3.8)–(3.9)–(3.10) may be reformulated in order to represent special separability cases. The early literature on separability of the so-called “ﬂexible” functions has indicated the required parameter restrictions18 . The “weak” separability of fQt in z with respect to parameter changes occurs if (3.8) can be rewritten as: 1 fQt [ζ (z)] ≡ α0t + α1t ζ (z) + ζ (z) at2 ζ (z) 2

(3.11)

where at1 and at2 are variable scalar parameters and ζ (z) must be the linear function αz in order for the function fQt to have the functional form (3.8). In this case, htQ

17 Samuelson and Swamy (1974, p. 568) comment the non-existence of an aggregating index number in these terms: “we cannot hope for one ideal formula for the index number: if it works for the tastes of Jack Spratt, it won’t work for his wife’s tastes; if, say, a Cobb-Douglas function can be found that works for him with one set of parameters and for her with another set, their daughter will in general require a non-Cobb-Douglas formula! Just as there is an uncountable inﬁnity of different indifference contours–there is no counting tastes–there is an uncountable inﬁnity of different indexnumber formulas, which dooms Fisher’s search for the ideal one. It dos not exist even in Plato’s heaven”. In a footnote, they add “This point, forcefully made in Samuelson (1947, p. 154) has served to intimidate many (but alas, not all) searchers for the nonexistent”. 18 See Diewert (1993, p. 15) for references to this literature.

174 Precursor can be rewritten as 1 htQ [ς (z) , k] = α0t + α1t ς (z) + ς (z) at2 ς (z) 2 1 1 + kβ t + kBt k + kγ t ς (z) 2 2

(3.12)

where γ t is an M -dimensional (column) vector of variable parameters. A similar condition can be imposed on the parameters of the ﬁrst- and second-order terms in k by deﬁning the linear function κ(k) ≡ βk in order to represent the case of “non-linear” separability of htQ in k. Therefore, by imposing the “weak” separability conditions for z and k, htQ becomes a non-linear (quadratic) function whose arguments are the linear functions ζ (z) and κ(k), that is htQ [ς (z) , κ (k)] ≡ at0 + at1 ς (z) + 12 ς (z) at2 ς (z) + κ (k) β1t + 12 κ (k) β2t κ (k) + 12 κ (k) γ t ς (z). The “strong” or “linear” separability of htQ in z, k, and parameter changes occurs when t = 0M ×N in (3.9) so that htQ (z, k) = σ t ςQ (z) + κQ (k)

(3.13)

where ζQ (z) = α0 + αz + 12 z Az, κQ (k) = kβ + 12 kBk, and σ t is a factor of proportionality. Under the conditions of homogeneity and the necessary separability restrictions on parameter values, the form of (3.13) is equivalent to (see section 5) htQ (z, k) = σ t · ςQ∗ (z) · κQ∗ (k)

(3.14)

which is indistinguishable from the “weak separability” case with only two separable groups of variables19 . The following results regarding the accounting for functional value differences can be obtained: Corollary 3.2 Accounting for Value Differences of Two Quadratic Func tions Differing in Parameters. If two functions fQ0 z 0 ≡ h0Q z, k 0 and fQ1 z 1 ≡ h1Q z, k 1 have quadratic functional forms as those deﬁned by (3.8)–(3.9) which may differ in all parameters, then, for all z 0 and z 1 ,

19 This is a well known result (see Gorman, 1959a, p. 471). The functional form (3.13), which is additive in ζ Q and κ q , is identically equal to (3.14), which is multiplicative in ζQ∗ and κQ∗ .

The Theory of Exact and Superlative Index Numbers Revisited 175 1 (3.15) fQ1 z 1 − fQ0 z 0 = ∇fQ0 z 0 + ∇fQ1 z 1 · z 1 − z 0 2 1 1 1 + a10 − a00 + a1 − a0 z + z 0 + z 0 A1 − A0 z 1 2 2 (3.16) = h1Q z 1 , k 1 − h0Q z 0 , k 0 =

1 ∇z h0Q z 0 , k 0 + ∇z h1Q z 1 , k 1 · z 1 − z 0 2 + Residual component

with the residual component being equal to the parameter change component deﬁned by the second line of (3.15), in which the sum of the “zero-” and “ﬁrstorder” terms can be, in turn, decomposed as follows 1 0 1 0 1 0 1 1 a0 −a0 + a −a z +z = ∇k h0 z 0 ,k 0 + ∇k h1 z 1 ,k 1 · k 1 −k 0 2 2 1 0 1 + α01 −α00 + α 1 −α 0 z +z 2 1 1 + k 0 +k 1 β 1 −β 0 + k 0 B1 −B0 k 1 2 2 (3.17) Differently from the Quadratic Identity established by Corollary (2.2), Corollary (3.2) is not an “if and only if” result: the decomposition of value difference between two quadratic functions implies (3.15)–(3.17), but the converse is not true. In fact, equations (3.15)–(3.18) may even hold in the particular case of a functional form that is linear in z.Note that the inequality ∇f 0 z 0 = ∇f 1 z 1 for z 0 = z 1 may occur even with linear functions when these have different parameters. Furthermore, we observe that the theory of exact and superlative index numbers has traditionally considered quadratic functional forms where only the parameters of the zero- and ﬁrst-order terms in z may differ. By contrast, Corollary (3.2) shows that, also when the parameters of second-order terms differ, the difference in functional values of two quadratic functions can be still split into two separate components due to differences in z and differences in parameters. Lemma 3.2. Quadratic Approximation of Value Differences between Two Arbitrary Functions with Different or Functional Forms. Let Parameters f 1 (z) ≡ h1 z, k 1 and f 0 (z) ≡ h0 z, k 0 be two arbitrary once differentiable functions with different parameters or functional forms. If their value difference is accounted for by using (3.2) where θ is set equal to 1/2, then , for all z 0 and z 1 ,

176 Precursor 1 f 1 z 1 −f 0 z 0 = ∇f 0 z 0 +∇f 1 z 1 · z 1 −z 0 2 +θ f 1 z 0 −f 0 z 0 +(1−θ ) f 1 z 1 −f 0 z 1 +error of approximation (3.18)

0 1 0 0 1 1 1 0 1 error of approximation ≡ −θ fL z −fL z − fL z −fL z 2 (3.19) R01 z 0 , z 1 if R01 z 0 , z 1 + R11 z 0 , z 1 = 0 with θ = θ z , z ≡ R1 z 0 , z 1 + R11 z 0 , z 1 or θ may take any value if R01 z 0 , z 1 = R11 z 0 , z 1 = 0, and fLt (z) is a ﬁrstt t is fLt (z) order (linear) that is tangent approximating function t to ft (z) att z (that ≡ t t t t t t t t t t t a +b z = f z +∇f z z − z , with a = f x −∇f x x and b = ∇f t xt , with t = 0, 1). Let us, now, construct the following general decomposition of the absolute value difference between h1 (z 1 , k 1 ) and h0 (z 0 , k 0 ): h1 z 1 , k 1 − h0 z 0 , k 0 = (1 − λ) h0 z 1 , k 0 − h0 z 0 , k 0 + λ h1 z 1 , k 1 − h1 z 0 , k 1 + λ h1 z 0 , k 1 − h0 z 0 , k 0 + (1 − λ) h1 z 1 , k 1 − h0 z 1 , k 0 (3.20) ∗

0

1

0

The following technical result is also obtained: Corollary 3.3. Accounting for the Sum of Value Differences between Two Quadratic Functions with Different “Zero-order” and “First-order” Parameters (Caves, Christensen, and Diewert’s, 1982, pp.1412–1413 Translog Identity). If two quadratic functions are both deﬁned by (3.8)–(3.9) differing in parameters of the “zero-order” and “ﬁrst-order” terms but having equal parameters in the same “second-order” terms in z (that is A1 = A0 = A), then, for all z 0 , z 1 , k 0 , k 1 , 0 1 0 0 1 1 1 0 fQ z −fQ z + fQ z −fQ z = ∇fQ0 z 0 +∇fQ1 z 1 · z 1 −z 0 (3.21) 0 0 0 0 0 0 1 1 1 1 1 1 and hQ z ,k −hQ z ,k + hQ z ,k −hQ z ,k = ∇z h0Q z 0 ,k 0 +∇z h1Q z 1 ,k 1 ·z z 1 −z 0 (3.22) This result is known under the name of Translog Identity, because it was formulated by Caves, Christensen, and Diewert (1982, pp. 1412–1413) in terms of translog functions.

The Theory of Exact and Superlative Index Numbers Revisited 177 In the more general case of quadratic functions with no restrictions on variable parameters, we have the following result: Corollary 3.4. Accounting for the Sum of Value Differences of Two Quadratic Functions with Different Parameters (Caves, Christensen, and Diewert, 1982, pp. 1412–1413). If two quadratic functions are deﬁned by (3.8)–(3.9), differing in all parameters, including those of the “second-order” terms in z (A1 may differ from A0 ), then, for all z 0 , z 1 , k 0 , k 1 ,

fQ0 z 1 −fQ0 z 0 + fQ1 z 1 −fQ1 z 0 = ∇fQ0 z 0 +∇fQ1 z 1 · z 1 −z 0

1 − z 1 −z 0 A1 −A0 z 1 −z 0 2 0 1 0 0 0 0 1 1 1 1 0 1 and hQ z , k −hQ z , k + hQ z , k −hQ z , k = ∇z h0Q z 0 , k 0 +∇z h1Q z 1 , k 1 · z 1 −z 0 −

1 1 0 1 z −z A −A0 z 1 −z 0 2

(3.23)

(3.24)

If A1 = A0 , then the term 12 z 1 − z 0 A1 − A0 z 1 − z 0 is equal to zero and the result obtained with Corollary (3.4) is the same as that obtained with Corollary (3.3). It is evident that Caves, Christensen, and Diewert (1982, pp. 1412–1413) introduced the restriction A1 = A0 in order to obtain the equivalence between an arithmetic average of the two functions (evaluated at z 0 and z 1 ) and the component of weighted changes in z. This equivalence no longer holds when A1 = A0 , but the following result is useful to show that, even in this general case, the overall functional value difference can be still decomposed into separate components of differences in z, k, and variable parameters. Corollary 3.5 If two functions fQ0 (z) ≡ h0Q (z, k 0 ) and fQ1 (z) ≡ h1Q (z, k 1 ) are deﬁned by (3.8)–(3.9), then, for all z 0 , z 1 , k 0 , k 1 ,

fQ1 (z 0 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 ) = 2 a10 − a00 + a1 − a0 z 0 + z 1 + z 0 A1 − A0 z 1 +

1 1 z − z 0 A1 − A0 z 1 − z 0 2

(3.25)

178 Precursor and h1Q z 0 ,k 1 −h0Q z 0 ,k 0 + h1Q z 1 ,k 1 −h0Q z 1 ,k 0 = h1Q z 1 ,k 1 −h1Q z 1 ,k 0 + h0Q z 0 ,k 1 −h0Q z 0 ,k 0 + h1Q z 1 ,k 1 −h0Q z 1 ,k 1 + h1Q z 0 ,k 0 −h0Q z 0 ,k 0 (3.26)

where 1 1 1 hQ z , k − h1Q z 1 , k 0 + h0Q z 0 , k 1 − h0Q z 0 , k 0 = ∇k h0Q z 0 , k 0 + ∇k h1Q z 1 , k 1 k 1 − k 0 −

1 1 k − k 0 B1 − B0 k 1 − k 0 , 2

(3.27)

1 1 1 hQ z , k − h0Q z 1 , k 1 + h1Q z 0 , k 0 − h0Q z 0 , k 0 = 2 α01 − α00 + α 1 − α 0 z 1 + z 0 1 + z 0 A1 − A0 z 1 + z 1 − z 0 A1 − A0 z 1 − z 0 2 1 1 0 + k + k β − β 0 + k 0 B1 − B0 k 1 +

1 1 k − k 0 B1 − B0 k 1 − k 0 2

(3.28)

Adding (3.24) to (3.25)–(3.28) and dividing the resulting expression through by 2 yield the same result obtained by Corollary (3.2). It is straightforward to show that, in order to obtain 12 ∇fQ0 (z 0 ) + ∇fQ1 (z 1 ) · 1 z − z 0 = (1 − γ ) fQ0 (z 1 ) − fQ0 (z 0 ) + γ fQ1 (z 1 ) − fQ1 (z 0 ) , γ must take the particular value 12 F 1 − F 0 + 14 z 1 − z 0 A1 − A0 z 1 − z 0 ∗ 0 1 1 (3.29) γ z ,z ≡ F − F0 where F t ≡ fQt (z 1 ) − fQt (z 0 ) for t = 0, 1. If A1 = A0 , then γ ∗ (z 0 , z 1 ) = 1/2). Even in the more general case where A1 = A0 , it is possible to decompose the difference between two quadratic functions deﬁned by (3.8) into a component due to (weighted) changes in z and a component due to (weighted) differences in parameters. These two components turn out to be a weighted average of two functional value changes evaluated at the respective reference variables. Similarly, it is possible to decompose the functional value changes in htQ (z, k) deﬁned by (3.9) into three separate components, due respectively to (weighted) changes in z, k, and differences in parameters. In the particular case of the function hQ (z, k) where A1 = A0 and B1 = B0 , equations (3.16) and (3.17) reduce to the decomposition

The Theory of Exact and Superlative Index Numbers Revisited 179 procedure used by Diewert and Morrison (1986, Theorem 1, p. 663) in terms of translog functions. The results obtained here widen considerably the scope of applicability of the decomposition of differences in functional values of quadratic functions where also the “second-order” parameter are assumed to be variable. Corollary 3.6. Accounting for Functional Value Differences of Two Quadratic Homothetic Functions. If two continuously once-differentiable quadratic functions fQ0 (z) and fQ1 (z) deﬁned by (3.8) are homothetic transformation of linearly homogeneous functions, so that ai = 0 for i = 0, 1, . . . , N , and 2 1 2 fQ1 (z) = σ t φQ (z) for t = 0, 1, where φQ (z) ≡ 12 zAz / , then, for all z 0 and z 1 . fQ1 (z 1 ) − fQ0 (z 0 ) = JZ + JS + JT N 0 0 fQ1 z 1 1 1 0 fQ z 1 where JZ = JS ≡ sQ,i zi − zi0 + sQ,i t t 2 i=1 zi zi

(3.30)

JT = Parameter change component t with JZ and JS having the same meaning established above, sQ,i ≡ t t t ! N ∂fQ (z ) t ∂φQ (z ) t ! t zit · zi φQ z (t = 0, 1) , and i=1 ∂z t · zi = ∂z t i

∂zit

.

i

fQ1 z 1 = IZ · IS · Iσ fQ0 z 0

(3.31)

⎡ where

∂fQt (z t )

⎢ IZ = IS ≡ ⎣

1 N 0 zi i=1 sQ,i z 0 i 0 N 1 zi i=1 Q,i z 1 i

s

⎤ 12

φQ z 1 σ1 ⎥ 0 and IT ≡ 0 . ⎦ = φQ z σ

We note that IZ and IS in (3.31) are Fisher “ideal”-type index numbers, which are “exact” for φQ (z) (identically equal to the ratio φQ z 1 /φQ z 0 = 1 1 1 1/2 1 0 0 1/2 z Az z Az for any z 0 and z 1 ). 2 2

4 Accounting for value changes of transformed functions In the previous sections, we have seen that the accounting formula deﬁned with the Quadratic Identity cannot be applied to arbitrary functions without incurring a possible non-negligible error of approximation. In the general case, a way to reduce this error is to use a transformed quadratic function. With appropriate parameter values, it is this function rather than a quadratic function that provides a second-order differential approximation to an arbitrary function.

180 Precursor Suppose that the arbitrary function f t (z) considered in section 3 can be deﬁned as f t [Z(q)], where Z(q) = z, with the ith element zi = z(qi ), so that f t [Z(q)] = f t (q). t Furthermore, suppose that a general quadratic function fGQ (q) can be transformed into a quadratic function as follows: t 1 g fGQ (q) = at0 + at Z t (q) + Z t (q)At Z t (q) (4.1) 2 where Z t (q) ≡ z t (q1 ) z t (q2) . . . z t (qN ) so that

1 t t t t −1 t t t fGQ (q) = g a0 + a Z (q) + Z (q)A Z (q) (4.2) 2 Let us deﬁne at0 ≡ α0t + β t K xt + 12 K xt Bt K xt and at = α t + K xt t where t t t K(x) ≡ k (x1 ) k (x2 ) . . . k (xM ) so that g fGQ (q) ≡ g hGQ q, x , with 1 g htGQ (q, x) = α0t + αit Z(q) + Z(q)At Z(q) 2 1 + K(x)β t + K(x)Bt K(x) 2 + K(x) t Z(q)

(4.3)

where zi ≡ z (qi ) and km = K (xm ) or in vector notation z ≡ Z(q) and k ≡ K(x) The values of all parameters may change as t changes, and g, z, and k are continuous and monotonic functions of one single variable with non-zero derivatives. Since the functions z and k are continuous and monotonic, it is possible to invert them in order to obtain qi = z −1 (zi ) or, in vector form, xm = k

−1

(km ) or, in vector form,

q = Z −1 (z) x=k

so that t t −1 g fGQ (q) = g fGQ Z (z) ≡ fQt (z)

−1

(k)

deﬁned by (3.11)

(4.4) (4.5)

(4.6)

and g htGQ (q,x) = g htGQ Z −1 (z),K −1 (k) ≡ htQ (z,k) deﬁned by (3.12)

(4.7)

t If the functions fQt and htQ have speciﬁc parameter values such that fGQ ∗ t ∗ t ∗ t ∗ 2 t ∗ 2 t ∗ (q ) = f (q ) , ∇fGQ (q ) = ∇f (q ) and ∇ fGQ (q ) = ∇ f (q ) , then the function t fGQ = g −1 fQt provides a second-order differential approximation to f t around q∗ . This should be contrasted with the assumption in section 3 that it is fQt rather than g −1 fQt that provides a second-order differential approximation to f t around q∗ .

The Theory of Exact and Superlative Index Numbers Revisited 181 We can now obtain the following result: Lemma 4.1. Accounting for Value Differences of Two General Quadratic t (q) is deﬁned by (4.2), then Functions (Diewert, 2002, p. 67). If the function fGQ

g

1 fGQ

" 1 0 0 1 0 0 0 −1 0 0 g fGQ q · Zˆ q · ∇q fGQ q − g fGQ q = q 2 +g

1 fGQ

# 1 1 −1 1 −1 1 ˆ · ∇q fGQ q · Z q q

· Z q1 − Z q0 + T

(4.8)

where 1 0 T ≡ a10 −a00 + a1 −a0 Z q +Z q1 +Z q0 A1 −A0 Z q0 2

(4.9)

−1 and Zˆ (q) is a diagonal matrix in which the (i, i)th element is equal to ! dqi dz −1 (zi ) = = 1 z (qi ) dzi dzi

(4.10)

(with z (qi ) = 0, by assumption).

5 A general transformed quadratic function The transformed quadratic function (4.1)–(4.2) can be fully speciﬁed by choosing the functional form of the transformation functions g, z, and k. Among the many candidates, if we deﬁne the following functions as suggested in the original work of Box and Cox (1964):

g(y) ≡

yρ − 1 ρ

(5.1)

z (qi ) ≡

qiλ − 1 λ

(5.2)

k (xi ) ≡

xi − 1 μ

μ

(5.3)

182 Precursor and replace the functions (5.1) and (5.2) in (4.1), then we obtain the following quadratic Box-Cox function20 :

ρ

fQt ρ,λ (q)

−1

ρ

N

= at0 +

qλ − 1 ati i λ

i=1

N N λ 1 t qiλ − 1 qj − 1 aij + 2 i=1 j=1 λ λ

(5.4)

which, by replacing (5.1)–(5.3) in (4.3), turns out to be equivalent to

ρ htQρ,λ (q, x) − 1 ρ

= α0t +

N

αit

i=1

+

M

βmt

m=1

+

N N λ qiλ − 1 1 t qiλ − 1 qj − 1 aij + λ λ λ 2 i=1 j=1

M M xmμ − 1 1 t xmμ − 1 xnμ − 1 + b μ 2 m=1 n=1 mn μ μ

M N i=1 m=1

t γmi

xmμ − 1 qiλ − 1 μ λ

(5.5)

by setting

at0 ≡ α0t +

M

βmt

M M xmμ − 1 1 t xmμ − 1 xnμ − 1 + b μ 2 m=1 n=1 mn μ μ

(5.6)

t γmi

xmμ − 1 μ

(5.7)

m=1

ati ≡ αit +

M m=1

With ρ → 0, and λ → 0, the function (5.4) reduces to the following translog functional form:

t In fTrg (q) = at0 +

N i=1

1 t a In qi In qj 2 i=1 j=1 ij N

ati In qi +

N

(5.8)

20 The explicit use of the quadratic Box-Cox function can be dated back at least to the works of Khaled (1977), Kiefer (1977), Appelbaum (1979), and Berndt and Khaled (1979). The special case of a quadratic Box-Cox aggregator function has also been derived by Diewert (1980, pp. 450–451) from a quadratic mean-of-order-r aggregator function.

The Theory of Exact and Superlative Index Numbers Revisited 183 and, when ρ → 0, λ → 0, and μ → 0 the function (5.5) reduces to In

htTrg (q, x) = α0t +

N i=1

+

M

1 t qi + a In qi In qj 2 i=1 j=1 ij 1 t b In xm In xn 2 m=1 n=1 mn M

M

βmt In xm +

m=1

+

N

N

αit In

N M

t γmi In xm In qi

(5.9)

m=1 i=1

t is homogeneous of degree δ if Ni=1 ati = δ and Ni=1 atij = The function fTrg N t t j=1 aij = 0. The function hTrg is homogeneous of degree δ in q under the N t N t following conditions: (i) i=1 αi = δ, (ii) i=1 aij = 0 for j = 1, 2, . . . , N , N t (iii) a = 0 for i = 1, 2, . . . , N (condition (ii) implies (iii) under the j=1 ij N symmetry of matrix A), and (iv) i=1 γmi = 0, for m = 1, . . . , M . Moreover, the function htTrg is homogeneous of degree η in x under the following con M M t ditions: (v) M m=1 βm = η, (vi) m=1 bmn = 0 for j = 1, 2, . . . , N , (vii) n=1 bmn = 0 for i = 1, 2, . . . , N (condition (vi) implies (vii) under the symmetry of matrix B), and (viii) M m=1 γmi = 0, for n = 1, 2, . . . , N . t is homothetically separable in q from parameter changes if The function fTrg the parameters of the ﬁrst- and second-order terms are constant, that is ati = ai and atij = aij for i = 1, 2, . . . , N and j = 1, 2, . . . , N . Moreover, the function htTrg is homothetically separable in q from changes in x and parameters if αit = αi , atij = aij and γmi = 0 for i = 1, 2, . . . , N and j = 1, 2, . . . , N , and m = 1, 2, . . . , M . With ρ = 0, and λ = 0, the general quadratic Box-Cox function (5.4) can be rewritten in the following form: ⎡ fQt ρ,λ (q) = ⎣a¯ t0 +

N

a¯ ti qiλ +

i=1

where

a¯ t0

≡

1 + ρat0 − ρλ

ρ t a . λ2 ij

N i=1

ati

N N 1

2

+

⎤ ρ1 a¯ tij qiλ qjλ ⎦

(5.10)

i=1 j=1

ρ 2λ2

N N i=1 j=1

atij

,

a¯ ti

≡

ρ t a λ i

−

ρ λ

N

atij

and a¯ tij ≡

j=1

The function fQt ρ,λ is homogeneous of degree 2λ/ρ if a¯ t0 = 0 and a¯ ti = 0 for all i’s implying: 1 + ρat0 ρ = N N t 2λ2 i=1 j=1 aij

(5.11)

184 Precursor Therefore, in the case of homogeneity of degree 2λ/ρ, using (5.11), the function (5.10) can be re-written as ⎤ ρ1 ⎡ ⎤ ρ1 ⎡ N N N N ⎢ 1 + ρat ⎥ ⎥ ⎢ fQt ρ,λ (q) = ⎣ 2λρ2 atij qiλ qjλ ⎦ = ⎣ atij qiλ qjλ ⎦ ⎢ N N 0 ⎥ ⎣ t ⎦ i=1 j=1 i=1 j=1 aij ⎤ ρ1

⎡

i=1 j=1

(5.12) Hence ⎤ ρ1 ⎡ N N 1 fQt ρ,λ (q) = ⎣ αijt qiλ qjλ ⎦ 1 + ρat0 ρ

(5.13)

i=1 j=1

where αijt ≡ atij /

N N

atij . Note that

i=1 j=1

N N

αijt qiλ qjλ = 1 when all qi s are equal to 1.

i=1 j=1

Moreover, if all parameters atij in (5.12) change proportionally, so that αijt in (5.13) remain constant (αijt = αij , then the effects of parameter changes are separable from q. Under these conditions, this function can be seen as a homothetic transformation of a linearly homogeneous function, that is r ρ fQt ρ,λ (q) = σ t fQr (q) /

(5.14) 1

Setting r = 2λ, σ t ≡ (1 + ρat0 ) ρ , and ⎤1 ⎡ r N N r r fQr (q) ≡ ⎣ αij qi2 qj2 ⎦

(5.15)

i=1 j=1

which is the quadratic mean-of-order-r aggregator function used by Diewert’s (1976, pp. 129–130)21 . In fact, fQr is homogenous of degree one in q and separable from parameter changes in f tρ,λ . Therefore, fQr has the required properties of an Q

aggregator function of q within f tρ,λ . Q Equation (5.15) reduces to well-known functions for particular values of r. Denny (1972, 1974) noted that, if r = 1, then it reduces to the generalized linear functional form proposed by Diewert (1969, 1971). In an unpublished memorandum, Lau (1973) showed that, at the limit as r tends to zero, it reduces to

21 This functional form is due to McCarthy (1967), Kadiyala (1972), Denny (1972, 1974), and Hasenkamp (1973).

The Theory of Exact and Superlative Index Numbers Revisited 185 the homogeneous translog aggregator function (Lau’s proof is reported in Diewert, 1980, p. 451). Diewert (1976, p. 130) also noted that, if r = 2, then it reduces to the Konüs-Byushgens (1926) functional form. Furthermore, if all αij = 0 for i = j, then it reduces to a CES functional form. If the parameters of the ﬁrst-order terms of the function (5.10) are functions of x as follows a¯ t0 ≡ α¯ 0t +

M

β¯mt xmμ +

m=1

a¯ ti ≡ α¯ it +

M

1 ¯t μ μ b x x 2 m=1 n=1 mn m n M

M

(5.16)

t μ γ¯mi xm ,

(5.17)

m=1

then (5.10) is identically equal to htQρ,λ,μ (q, x) = [α¯ 0t +

N

1 t λ λ α¯ q q 2 i=1 j=1 ij i j

i=1

+

M

β¯mt xmμ +

m=1

+

N M

N

N

α¯ it qiλ +

1 ¯t μ μ b x x 2 m=1 n=1 mn m n M

M

1

t μ λ ρ γ¯mi xm qi ]

(5.18)

m=1 i=1

$ where α¯ 0t ≡ 1 + ρα0t − ρ μλ

M M

N

ρ λ

$ % t γmi ; α¯ it ≡ ρλ αit −

m=1 i=1 % M ρ t γmi ;b¯ m μλ i=1

≡

ρ t b ; γ¯ t μ2 mn mi

ρ 2λ2

αit +

i=1 ρ λ2

≡

N

atij −

j=1

ρ t γ μλ mi

N N

atij −

ρ μ

M

i=1 j=1 m=1 % M ρ t γmi ; β¯mt ≡ μλ m=1

βmt + 2μρ 2

M M

btmn m=1 n=1 $ M ρ t β − μρ2 btmn μ m n=1

+ −

.

$ M The function htQρ,λ,μ is homogeneous of degree 2λ/ρ in q if α¯ 0t + β¯mt xmμ +

1 2

M M

b¯ tmn xmμ xnμ

%

m=1 n=1

degree 2μ/ρ in x if

m=1

$ % M t μ γ¯mi xm = 0 for all i’s. It is homogeneous of = 0 and α¯ it + $

α¯ 0t +

N i=1

m=1

α¯ it qiλ + 12

N N

i=1 j=1

% $ % N t λ a¯ tij qiλ qjλ = 0 and β¯mt + γ¯mi qi = 0 i=1

for all m’s. t Moreover, the function htQρ,λ,μ is additively separable in q and x if γmi = 0 for all m’s and i’s. If it is homothetic and, at the same time, additively separable in q (and/or x), then it is homothetically separable in q (and/or x). If, as a particular case of homothetically separable function, it is separable and homogeneous of degree one in q (and/or x), then it is said to be homogeneously separable in q (and/or x).

186 Precursor Under the hypothesis of homothetic separability, using (5.6) and (5.13) with the restriction of homogeneity of degree 2λ/ρ and separability in q, equation (5.18) becomes ⎤1 ⎡ ρ M M N N M 1 ¯t μ μ t λ λ⎦ t ∗t μ t ⎣ ¯ ¯ hQρ,λ,μ (q, x) = βm xm + b x x β0 + αij qi qj 2 m=1 n=1 mn m n m=1 i=1 j=1

1 ρ

(5.19) M M t bmn ;

M βmt + 2μρ 2 where β¯0t ≡ 1 + ρα0t − μρ m=1 m=1 n=1

N M ρ ρ ρ t t ∗t t t t β¯m ≡ μ βm − μ2 bmn ≡ β¯m + μλ γmi = β¯mt for all m’s (since γmi = 0 for n=1

i=1

all m’s and i’s, as a condition imposed for the additive separability in q and x); b¯ tmn =

ρ t b . μ2 mn

The function (5.19) is also homogeneous of degree 2μ/ρ in x, if β¯0t = 0 and β¯m∗ t = 0 (and, therefore, β¯mt = 0) for all m’s, implying 1 + ρα0t ρ = M M (5.20) t 2μ2 bmn m=1 n=1

Therefore, in the case of homogeneity of degree 2μ/ρ in x, the function (5.19) can be re-written as ⎡ htQρ,λ,μ (q, x) = ⎣

⎤1 ρ M M ρ t μ μ αijt qiλ qjλ ⎦ bmn xm xn 2μ2 m=1 n=1 j=1

N N i=1

1 ρ

using (5.20) ∗

r ⎡⎛ ⎤r ⎞ r ⎤ 2 ⎡ r1∗ ρ 2 ∗ ∗ N M N M r r r r ⎢ t 2 2 ⎠ ⎥ ⎢ t 2 2 ⎥ = ⎣⎝ αij qi qj βmn xm xn ⎦ ⎣ ⎦ × i=1 j=1

m=1 n=1

1

(1 + ρα0t ) ρ setting r =2λ, r

t =2μ, and βmn ≡btmn /

(5.21) M M m=1 n=1

btmn . Note, that

M M

r∗ r∗ t βmn xm2 xn2 =1

m=1 n=1

when all xi ’s and xj ’s are equal to 122 . 22 We may note that, with r=r*=ρ=2 and all “second-order” parameters being constant (αijt =αij t

andβmn =β mn for every value of t), (5.21) reduces to the quadratic mean-of-order-2 functional form used by Diewert (1992, p. 231, eq. (56)).

The Theory of Exact and Superlative Index Numbers Revisited 187 t t If all αijt ’s and βmn ’s are constant over t, that is αijt = αij and βmn = βmn for t all i, j, m, and n, then the function hQρ,λ,μ (q, x) is homothetically separable in q, x, and parameter changes. The last ones are captured only by the changes in the 1 r r r 1 N N proportionality factor (1 + ρα0t ) ρ . The resulting functions αij qi2 qj2 and i=1 j=1

M M

r∗

r∗

1 r∗

βmn xm2 xn2

are homogeneous of degree one and can be considered as

m=1 n=1

“aggregator functions” of q and x, respectively, within the function htQρ,λ,μ 23 .

6 Translog and quadratic mean-of-order-r functions The general quadratic mean-of-order-r function, which is algebraically derivable from the quadratic Box-Cox function deﬁned by (5.4)–(5.5), can be used as a second-order approximation to an arbitrary unknown function. Using the results obtained thus far, it is possible to account for differences in functional values, either in terms of differences or in terms of ratios, into aggregating components of changes in the arguments and parameters of the function. We shall follow the tradition of calling indicators the aggregating components deﬁned in terms of differences and index numbers those that are deﬁned in terms of ratios (see, for example, Diewert, 1998, 2000). In order to save space, we shall deal with only the function fQt ρ,λ (q) deﬁned by (5.10) and leave to the reader the exercise of deriving the corresponding results with the more explicit function htQρ,λ,μ (q, x) deﬁned by (5.18). The following theorems are in order. Theorem 6.1 If two quadratic Box-Cox functions deﬁned by (5.4) with different parameters reduce to the “translog” functional form (5.8) (with p → 0 and λ → 0), then for all q0 and q1 1 0 1 (s + sTrg,i ) (ln qi1 − ln qi0 ) 2 i=1 Trg,i N

1 0 In fTrg − ln fTrg =

+ Parameter − change component

t where sTrg,i ≡

t qit fTrgi t fTrg

t with t = 0, 1 and fTrgi ≡

t ∂fTrgi

∂qit

(6.1)

(6.2)

23 In section 3, we have recalled that an aggregator function must be degree-one homogeneous as well as separable.

188 Precursor The Törnqvist index number is said to be “exact” for the translog function because, taking the antilogarithms, it gives the same result as the ratio of two translog functions. Following (3.29), it is straightforward to show that

0 fTrgi (q1 ) 0 fTrgi (q0 )

γ

1 fTrgi (q1 ) 1 fTrgi (q0 )

1−γ

1 0 1 S In qi1 − In qi0 + STrg,i 2 i=1 Trg,i N

=

with γ taking the particular value given by (3.28), where fQt and z t are, respectively, t replaced with ln fTrg and ln qt . N t 1 0 We note that i=1 sTrg,i = 1 if fTrg and fTrg are homogeneous of degree one in q, N t t t t since in this case fTrg = i=1 qi fTrg,i by Euler’s theorem. If Ni=1 sTrg,i = 1, then N ∗t t t we can deﬁne sTrg,i ≡ sTrg,i / i=1 sTrg,i and rewrite equation (6.1) as follows: 1 0 In fTrg − In fTrg = ln Iq + ln IS + ln IT

(6.3)

1 ∗0 ∗1 where In Iq ≡ + sTrg,i s In qi1 − In qi0 2 i=1 Trg,i N

In IS ≡

N 1 ∗0 ∗1 sTrg,i (ξ − 1) + sTrg,i (ξ − 1) In qi1 − In qi0 2 i=1

In IT ≡ Parameter − change component t t where ξ ≡ Ni=1 qit fTrg,i /fTrg , which represents the degree of returns to scale. The ﬁrst additive term in the right-hand side of (6.3) is the contribution of the change in q to the functional value difference, whereas the second additive term is the contribution of the scale effects. This result is obtained without imposing separability and homogeneity restrictions as it has been observed by Diewert (1980a, p. 463) (1980b, p. 538)(2004, p. 450), and Denny (1980, p. 535–37). Caves, Christensen, and Diewert (1982, p. 41) contended that only the translog-based indicator of relative changes does not rely necessarily on this restrictions. However, based on the developments in section 2 and 3, the following remark is noteworthy: 0 Remark 6.1 on the existence of the aggregator index: If and only if fTrg 0 t t t and fTrg are homothetically separable in q so that fTrg ≡ F [φTrg (q)], where F (·) is a well behaved transformation function and φTrg is a linearly homogeneous translog function with constant parameters, then for all q0 and q1 , si∗t ≡ t (qt ) t (qt ) N ∂fTrg ∂fTrg ∂φ (qt )/∂qit t · q / · qjt = Trg · qit for t=0, 1, in view of (2.4), t i j=1 φ (qt ) ∂q ∂qt i

j

Trg

The Theory of Exact and Superlative Index Numbers Revisited 189 1 1 0 0 Iq = φTrg (q1 )/φTrg (q0 ), IS ·IT = fTrg (q )/fTrg (q ) /Iq (in other words, the translogbased index Iq is independent from the scale and parameter changes since these ∗t do not affect the weights sTrg ). Similarly to the remarks 2.1 and 3.1, this remark warns us against interpreting the Törnqvist (translog-based) index number Iq as an aggregate of relative changes in q in the non homothetic case, where no such aggregate really exists and any attempt to construct it ends up to a spurious magnitude. By contrast, in the special case of homothetic separability, the aggregating indexes are clearly deﬁned and isolated from the effects of other components of the functional value changes. The results obtained with the Törnqvist index number parallel those concerning the general quadratic Box-Cox function. Theorem 6.2 If two quadratic Box-Cox functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 differ in parameters in all terms, then

$

fQ1ρ,λ

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧ N ⎪ ⎨ 1 2 ·

%ρ $ %ρ ⎫ ⎬ fQ0ρ,λ fQ1ρ,λ ⎪ 0 λ + 1 λ ⎪ ⎭ qi qi

$ sQ0 ρ,λ i

⎪ i=1 ⎩

λ λ qi1 − qi0 + parameter − changecomponent (6.4)

t where sQρ,λ ≡ i

ρ λ

t qit fQρ,λ i t fQρ,λ

t with t = 0, 1 and fQρ,λ ≡ i

t ∂fQρ,λ

∂qit

The homothetic separability restrictions are not imposed on (6.4) and (6.5) so that the same caveats expressed for the translog case apply here. The weights are not affected by the particular path taken by technical change and returns to scale only if the two functions are homothetically separable in q. Only in this case, is the q-change component fully independent from parameter changes. N t t We note that i=1 sQρ,λ ,i = 1 if fQρ,λ homogeneous of degree one in q, since in this case fQt ρ,λ = Ni=1 qit fQt ρ,λ ,i by Euler’s theorem. If Ni=1 sQt ρ,λ ,i = 1, then we can deﬁne sQ∗tρ,λ ,i ≡ sQt ρ,λ ,i / Ni=1 sQt ρ,λ ,i and rewrite equation (6.4) as follows:

(fQ1ρ,λ )ρ −(fQ0ρ,λ )ρ = Jq +JS +JT

(6.5)

190 Precursor

ρ 0 N (Iq ·fQ0ρ,λ )ρ 1 ∗0 (fQρ,λ ) ∗1 where Jq ≡ sQρ,λi λ +sQρ,λi λ · (qi1 )λ −(qi0 )λ , 0 1 2 i=1 qi qi N (fQ0ρ,λ )ρ Iq ·(fQ0ρ,λ )ρ 1 0 ∗0 1 ∗1 JS ≡ (ξ −1)sQρ,λi λ +(ξ −1)sQρ,λi λ 2 i=1 qi0 qi1 · (qi1 )λ −(qi0 )λ , JT ≡ Parameter −changecomponent

where Iq is the index of functional value change caused by changes in q, ξ 0 ≡ %ρ ! 0 ! 1 $ 1 %ρ ! $ ρ N ρ N 0 0 1 1 1 0 q f and ξ ≡ q f · f · f . f f I q Qρ,λ i=1 i Qρ,λ,i Qρ,λ i=1 i Qρ,λ,i Qρ,λ λ λ Qρ,λ The ﬁrst line in the right-hand side of (6.5) is the contribution of the change in q to the functional value difference, whereas the second line is the contribution of the scale effects. Let us consider, in particular, the following two special cases. First case: General quadratic linear indicators, which are exact for the general quadratic linear function (5.10) with ρ = 1 and λ = ρ/2. Using the decomposition 1

1

1

(qi1 ) 2 − (qi0 ) 2 = (qi1 − qi0 ) ·

1 (qi0 ) 2

1

+ (qi1 ) 2

,

(6.6)

from (6.4), by deﬁning fQt 1 ≡ fQt ρ,λ , we obtain

fQ11 − fQ01 =

⎧ N ⎪ ⎨

1

(qi0 ) 2 fQ01 i

1

(qi1 ) 2 fQ11 i

⎫ ⎪ ⎬

+ (q1 − qi0 ) 1 1 1 1 ⎪ i ⎪ i=1 ⎩ (qi0 ) 2 + (qi1 ) 2 (qi0 ) 2 + (qi1 ) 2 ⎭

+ Parameter − changecomponent

(6.7)

We can call general quadratic linear indicators of absolute functional value differences the indicator given by the right-hand side of (6.7), which is exact for the general quadratic linear function corresponding to (5.10), where ρ = 1 and λ = 1/2, and does not rely on homogeneity and separability restrictions. This is a rather useful result, since it widens the applicability of this type of indicators. This can be contrasted with the same case that was examined under linear homogeneity and separability restrictions by Diewert (2002, pp. 77–80), who showed that the counterpart index number, expressed in ratio terms is the implicit Walsh index number, which in turn is exact for a homogeneous quadratic linear function. Second case: General Konüs-Byushgens indicator, which is exact for a general Konüs-Byushgens function given by (5.10) with ρ = 2 and λ = ρ/2.

The Theory of Exact and Superlative Index Numbers Revisited 191 Using the general decomposition

fQ12

2

% $ % 2 $ − fQ02 = fQ12 − fQ02 · fQ12 + fQ02 ,

(6.8)

dividing through (6.4) by ( fQ12 + fQ02 ) and rearranging terms, we obtain fQ12

− fQ02

=

N

fQ02 i · fQ02 fQ12 + fQ02

i=1

+

fQ12 i · fQ12

(qi1 − qi0 )

fQ12 + fQ02

+ Parameter − changecomponent

(6.9)

where fQt 2 i ≡ ∂fQt 2 /∂qit for t = 0, 1. We can call general Konüs-Byushgens indicators of absolute and relative functional value differences the indicator given by the right-hand side of (6.9), which is exact for a general Konüs-Byushgens quadratic function corresponding to (5.10) where ρ = 2 and λ = 1 and does not rely on homogeneity and separability restrictions. Also this result is useful to widen the applicability of this type of indicators. This can be contrasted with the same case that was examined under linear homogeneity and separability restrictions by Reinsdorf, Diewert, and Ehemann (2000, pp. 4–6) and Diewert (2002, pp. 72–76), who showed that the counterpart index number (expressed in ratio terms) is the Fisher “ideal” index number, which in turn is exact for a homogeneous Konüs-Byushgens quadratic function. Theorem 6.3 If two quadratic Box-Cox functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 (or, equivalently, by (5.10)) differ in parameters in all terms, then, for all qi0 and qi1 , fQ1ρ,λ fQ0ρ,λ

= Iq · IR

(6.13) 1

where, setting r = 2λ and σ t = (1 + ρat0 ) ρ for t = 0, 1, ⎡

r (qi1 ) 2 ∗0 s r ⎢ i=1 Qρ,λ i ⎢ (qi0 ) 2 Iq ≡ ⎢ r ⎢ ⎣ N ∗1 (qi0 ) 2 s r i=1 Qρ,λ i (qi1 ) 2

with

sQ∗tρ,λ i

N

sQt ρ,λ i

≡ N

t i=1 sQρ,λ i

where

⎤ 2r ⎥ ⎥ ⎥ ⎥ ⎦

sQt ρ,λ i

(6.14)

≡

qit (∂fQt ρ,λ /∂qit ) fQt ρ,λ

and IR ≡ [ fQ1ρ,λ /fQ0ρ,λ ]/Iq is a residual index number representing the combined scale and parameter change effects.

192 Precursor We note that no a priori homogeneity and separability conditions are imposed on (6.13). In general, following (3.29), it is straightforward to show that Iq =

fQ0ρ,λ (q1 )

γ

fQ0ρ,λ (q0 )

fQ1ρ,λ (q1 )

1−γ

(6.15)

fQ1ρ,λ (q0 )

with γ taking the particular value of the parameter given by (3.29), where fQt and z t are, respectively, replaced with fQt ρ,λ and qt . If homothetic separability is assumed, then we obtain the following result: Corollary 6.1 If two functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 (or, equivalently, by (5.10)) are homothetic transformations of the same linearly homogeneous function fQr deﬁned by (5.15) with r = 2λ, then sQ∗tρ,λ i = sQ∗tρ,λ i ≡ (qi ∂fQr /∂qi )/fQr , which are not affected by the particular path taken by parameter changes and returns to scale, and Iq (6.14) is identically equal to fQr (q1 )/fQr (q0 ) for every q 0 and q 1 . Corollary 6.1 is a global version of Byushgens’ (1925) theorem, which was formulated with ρ=2 and λ=ρ/2 and was further discussed by Konüs and Byushgens (1926), Frisch (1936, p. 30), Wald (1939 p. 331), Samuelson (1947, p. 155), Pollak (1971), and Afriat (1972, p. 45) (1977, pp. 141–143) (2005, pp. 177–178). Remark 6.2 on the existence of the aggregator index: If and only if fQ0ρ,λ and fQ0ρ,λ are homothetically separable in q so that fQt ρ,λ ≡ F t [ fQr (q)], where F t (·) is a well behaved transformation function and fQr is the linearly homogeneous function (5.15) with constant parameters, then for all q0 and q1 , si∗t ≡ qit /

1

∂f t ρ,λ (qt ) N Q j=1 ∂qjt 0

∂fQr (qt )/∂qit · qjt for t = 0, 1, in view of (2.4), fQr (qt ) IS · IT = [ fQ1ρ,λ (q1 )/fQ0ρ,λ (q0 )]/Iq (in other words, the

· qjt =

∂f t ρ,λ (qt ) Q

∂qit

·

then Iq =

quadratic fQr (q )/fQr (q ), mean-of-order-r index Iq is independent from the scale and parameter changes since these do not affect the weights sQ∗tρ,λ ). Similarly to the remarks 2.1, 3.1, and 6.1, this remark warns us against interpreting the quadratic mean-of-order-r indexes Jq and Iq as aggregates of absolute and relative changes in q in the non-homothetic case, where no such aggregates really exist and any attempt to construct them ends up to a spurious magnitude. By contrast, in the special case of homothetic separability, the aggregating indexes are clearly deﬁned and isolated from the effects of other components of the functional value changes.

The Theory of Exact and Superlative Index Numbers Revisited 193 Moreover, the quadratic mean-of-order-r index number deﬁned by (6.14) can be contrasted with the following quadratic mean-of-order-r index number deﬁned by Diewert (1976, pp. 130–131): ⎡

N

1

r

0 (qi ) 2 ⎢ si (q0 ) 2r ⎢ i=1 i

IqD ≡ ⎢ N ⎣

r 0 1 (qi ) 2 i 1 r i=1 (qi ) 2

s

⎤ 1r ⎥ ⎥ ⎥ ⎦

(6.16)

where s0 and s1 are actually “observed”. If the unknown true function f is approximated at q0 , then s0 = sQ0 σ,λ , but generally s1 = sQ1 σ,λ . By deﬁning these weights as value shares, formula (6.16) specializes to wellknown index numbers with particular values of r. Diewert (1976, p. 135) (2004, Eq. (17.26), pp. 447–448) noted that, if r = 1, then (6.16) reduces to the implicit Walsh (1901, p. 105) index number, and, if r = 2, then (6.16) reduces to Fisher’s (1922) “ideal” index number. The indexes Iq and IqD differ in the weights used. Even if these two weight systems are numerically equal at the point of approximation, say at t = 0, so that sQ0 r = s0 , they may be substantially different at other points under comparison. This is sufﬁcient to make the two index numbers (6.14) and (6.16) rather different. In order to ensure that sQ0 r be equal to s0 at all values of r, the parameters of the underlying aggregator function should consistently adjust to the changes in r. Consequently, sQ1 r is also a function of r. This means that, using the same s1 with different values of r contradicts the assumption of second-order approximation that is supposed to be provided by IqD . It is, therefore, Iq , rather than IqD , that must be called “superlative” according to the meaning assigned by Diewert (1976, p. 117) to this term. This will be shown more analytically and illustrated by a numerical example in the remainder of this section. The function fQr (q) deﬁned by (5.15) provides a second-order differential approximation to an arbitrary function f (q) around q0 if f (q0 ) = fQr (q0 )

(6.17)

∇f (q0 ) = ∇fQr (q0 )

(6.18)

∇ 2 f (q0 ) = ∇ 2 fQr (q0 )

(6.19)

The equations (6.17)-(6.19) are satisﬁed for certain values of the parameters of fQr for a given r, as established by the following result: Theorem 6.4 If the aggregator function fQr (q) deﬁned by (5.15) provides a second-order approximation to the arbitrary aggregator function f (q) around q0 ,

194 Precursor then its parameters are equal to: αij (r, q0 ) =

ff fij − 1−r f i j r 2

r

f 1−r (qi0 qj0 ) 2 −1 r

fi − (qi0 ) 2 −1 αij (r, q0 ) =

f

N

for i = j,

i, j = 1, 2, . . . N

(6.20)

for i = 1, 2, . . . N

(6.21)

r

f 1−r αij (qj0 ) 2

j =i 1−r 0 r−1 (qi )

where f , fi , and fij denote, respectively, f (q), ∂f (q)/∂qi , and ∂ 2 f (q)/∂qi ∂qj evaluated at q0 . Remark 6.3 For each value r = r ∗ at q0 , there is a set of particular parameters αij = αij (r ∗ , q0 ) such that the function deﬁned by (5.15) satisﬁes (6.17)–(6.19). The system (6.17)–(6.19) is made of 1 + N + N 2 equations with N (N + 1)/2 independent parameters αij ’s of fQr . However, since the Hessian matrices are typically symmetric by Young’s theorem (stating that the values of second-order derivatives are not affected by the sequence of derivation) there are N (N − 1)/2 redundant equations in the subsystem (6.19). Moreover, both f and fQr are assumed to be homogeneous of degree one in q, hence f (q) = q∇f (q) and fQr (q) = q∇fQr (q) by Euler’s theorem. Then, the condition f (q) = q∇fQr (q)

(6.22)

makes one of the equations (6.18) redundant which adds up to the total number of redundant equations by raising it to 1 + N (N − 1)/2. The linear homogeneity of f and fQr brings about also, by Euler’s theorem, q∇ 2 f (q) = 0 and q∇ 2 fQr (q) = 0 and the following additional N conditions q∇ 2 f (q) = 0 = q∇ 2 fQr (q) = 0

(6.23)

which raise, in turn, the number of the redundant equations in the system (6.17)– (6.19) by a further N bringing it to a total of 1+N (N +1)/2. Therefore, the number of independent equations is N (N + 1)/2, which is the same of the independent parameters aij ’s in the function fQr (q)24 . The resulting functions (6.20) and (6.21) determining the coefﬁcients αij ’s are homogeneous of degree zero since, by linear homogeneity, μf (q) = f (μq), fi (q) = fi (μq), and μ1 fij (q) = fij (μq). 24 The obtained coefﬁcients αij satisfy the restriction i j αij = 1 if, together with the system (B.78)–(B.80) under the conditions (B.81) and (B.82), the values of the arguments are normalized appropriately.

The Theory of Exact and Superlative Index Numbers Revisited 195 It is immediate to note that all the functions fQr (q) deﬁned over the domain of r approximate each other up to the second order. Consequently, the corresponding “exact” index numbers deﬁned by (6.14) also approximate each other up to the second order. This is ensured by the adjustment of the shares sQt r to the value of r. It is important to remark, however, that, while the value changes of a fully ﬂexible quadratic function can be accounted for exactly with the index number (6.14), the converse is not true since also the value changes of a “rigid” CES-type functional form or even a polynomial function having all the second derivatives equal to zero may be accounted for with (6.14). The “superlativeness” of the index number (6.14) is, therefore, related only to its potential agreement up to the second derivatives of the unknown true function through the approximating function for which it is exact. In fact, nothing guarantees that the approximating function actually provides a second-order approximation if the second-order derivatives fij remain unobserved and may consistently have any numerical value (including zero) (see also Uebe, 1978 for a similar remark). Furthermore, in general, f (q1 ) = fQr (q1 )

(6.24)

∇f (q1 ) = ∇fQr (q1 )

(6.25)

∇ 2 f (q1 ) = ∇ 2 fQr (q1 )

(6.26)

Therefore, s1 = sQ1 r

(6.27) 1

where si1 ≡ qi1 ∂f∂q(q1 ) /f (q1 ) and sQ1 r i ≡ qi1 i

∂fQr (q1 ) ∂qi1

/fQr (q1 ).

When r changes, say from r ∗ to r ∗∗ , then ∇fQr∗ (q1 ) = ∇fQr∗∗ (q1 )

(6.28)

where fQr (q) is deﬁned by (5.15), and, consequently, s1 = sQ1 r∗ = sQ1 r∗∗

(6.29)

If the same observed shares st are used with different values of r, as in (6.16) instead of the shares sQt r required for the second-order approximation, then the resulting formulas are hybrid and cannot be interpreted as superlative index numbers providing second-order differential approximations to an arbitrary function. These last index numbers are instead deﬁned, in the homothetic case, by ⎡

N

⎤ 1r

r

1

(q ) 2 si0 i0 r 2 (q i=1 i)

⎢ ⎢ Iqs ≡ ⎢ N ⎣

0

r

(q ) 2 sQI r i i1 r (qi ) 2 i=1

⎥ ⎥ ⎥ ⎦

(6.30)

196 Precursor 0)

where si0 ≡ qi0 ∂f∂q(qi

!

f (q0 ) = sQ0 ri ≡ qi0

∂fQr (q0 ) ! fQr (q0 ) and sQ1 ri ∂qi

≡ qi1

∂fQr (q1 ) ! ∂qi1

fQr (q1 )

with fQr (q) being deﬁned by (5.15). The following example is a numerical representation of the differences between the two types of index numbers. Example 6.1. Let us consider the case of three elements (N =3) with qi0 = 1.0 for i = 1, 2, 3 and q11 = 2.0, q21 = 1.7, and q31 = 1.05. Moreover, let us assume that f (q0 ) = 1.0

(6.31)

and the ﬁrst derivatives of the unknown function f (q0 ) are the following: f1 (q0 ) ≡

∂f (q0 ) ∂f (q0 ) ∂f (q0 ) = 0.53, f2 (q0 ) ≡ = 0.37, f3 (q0 ) ≡ = 0.10 ∂q1 ∂q2 ∂q3 (6.32)

At the same observation point q0 , let us assume that the Hessian matrix is the following: ⎡ ⎤ 0.0376 − 0.0088 − 0.0288 (6.33) ∇ 2 f (q0 ) = ⎣ −0.0088 0.0344 − 0.0256 ⎦ −0.0288 − 0.0256 0.0544 At the point t = 0 and t = 1, the weights sit are assumed to be s10 ≡ (q10 · f1 )/f = 0.53

s11 ≡ (q11 · f1 )/f = 0.69

(6.34)

s20 ≡ (q20 · f2 )/f = 0.37

s21 ≡ (q21 · f2 )/f = 0.30

(6.35)

s30

≡ (q30 · f3 )/f

= 0.10

s31

≡ (q31 · f3 )/f

= 0.01

(6.36)

Let us now assume also that a quadratic mean-of-order-r aggregator function of the type deﬁned by (5.15) has such parameters αij∗ for a given r* that it provides a second-order differential approximation to f at q0 . Table 2 tabulates the values of these parameters in correspondence of certain values of r, so that the quadratic mean-of-order-r aggregator functions fQr approximate f at q0 up to the second order. Note that sQ0 r = s0 by assumption and sQ1 r is derived using the speciﬁc parameter values. The corresponding “exact” index numbers Iqs are compared with the hybrid Diewert’s index numbers IqD , which are constructed using the same “observed” shares for different values of r. In this numerical example, the two index numbers diverge increasingly from one another as r increases. The “really” superlative index numbers tend to increase as r increases after a certain minimum turning point, whereas the “hybrid” index numbers tend to decrease as r increases after a certain maximum turning point. Both the “exact” and “hybrid”

The Theory of Exact and Superlative Index Numbers Revisited 197 r=1

r=2

r=4

r = 10

r = 20

r = 40

r = 100

r = 1000

r=∞

Parameters of the underlying 2nd-order approximating “quadratic mean-of-order-r ” function (eqs. (6.20) and (6.21)) a 11 0.6052 0.3185 0.1751 0.0891 0.0605 0.0461 0.0375 0.0324 0.0318 a 22

0.4388

0.1713

0.0376

–0.0427

–0.0694

–0.0828

–0.0908

–0.0957

–0.0962

a 33

0.2088

0.0644

–0.0078

–0.0511

–0.0656

–0.0728

–0.0071

–0.0797

–0.0800

a 12 = a 21

–0.0176

0.1873

0.2897

0.3512

0.3717

0.3820

0.3881

0.3918

0.3922

a 13 = a 31

–0.0576

0.0242

0.0651

0.0896

0.0978

0.1019

0.1044

0.1058

0.1060

a 23 = a 32

–0.0512

0.0114

0.0427

0.0615

0.0677

0.0709

0.0727

0.0739

0.0740

1

sQ r,1

Current period weights with the 2nd-order approximating “quadratic mean-of-order-r” function (eqn. 6.27)) 0.9970 0.6057 0.6025 0.6045 0.6188 0.6544 0.8058 1.0000 1.0000

s1Q r,2

0.3570

0.3339

0.3567

0.3698

0.3448

0.1942

0.0030

0.0000

0.0000

s1Q r,3

0.0373

0.0437

0.0388

0.0114

0.0008

0.0000

0.0000

0.0000

0.0000

I sq

Index number which is exact for a 2nd-order approximating “quadratic mean-of-order-r ” function (6.30) 1.8069 1.8044 1.8066 1.8256 1.8474 1.8750 1.9355 1.9932 2.0000

Current period weights “observed” on the true unknown function (6.34)–(6.36) 0.6900 0.6900 0.6900 0.6900 0.6900 0.6900 0.6900

s11

0.6900

s1 2

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

s1

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

Diewert’s (1976, pp. 130-131) “quadratic mean-of-order-r ” index number (6.16) 1.8336 1.8381 1.8440 1.8389 1.7522 1.6013 1.5078 1.4549

1.4491

3

D

Iq

0.6900

Figure 1 Table 2 – Parameters, weights, and numerical values of quadratic mean-of-order-r index numbers

index numbers have asymptotic values as r → ∞. The former tends to a numerical value equal to 2.0, which is higher than that obtained at r = 1 by more than 10 per cent, whereas the latter tends to a value equal to 1.44914, which is lower than that obtained at r = 1 by more than 20 per cent (see Table 2). These results show that the theory of exact and superlative index numbers has to be widely reconsidered and shed a new light on the apparently paradoxical ﬁndings recently obtained by Hill (2006).

7 Conclusion The theory of exact and superlative index numbers has been critically re-examined here. It has turned out that superlative index numbers are rarely applicable in practice. In fact, these are deﬁned by using weights derived from functions for which they are “exact”. If, instead, the weights are derived from observed data, the result is that of hybrid formulas that may be far from being really superlative. We conclude that, with the “observed” data normally available, it is not possible to rely on the second-order approximation paradigm. Therefore, it would be more appropriate to construct a range of alternative index numbers that are all valid

198 Precursor candidates as measures of the unknown true index number, when this exists, rather than search for only one optimal formula. One promising method would be to test the observed data for consistency with an aggregator function and, in the positive case, to proceed to calculate upper and lower bounds of the unknown true index.

Appendix A: Observable economic variables The propositions presented thus far are theorems in numerical analysis rather than economics. The numerical values of the variables involved are assumed to be either known or derivable from some sources of information. In economics, the contexts where index numbers and indicators are usually applied are those regarding production and consumption activities. The index numbers and indicators are therefore deﬁned with reference to production or transformation and utility functions and their dual counterparts represented by value functions such as cost, revenue and proﬁt functions. All these functions are characterized by certain properties. The following cases are examined. (i) The case of the production function If the function f t (x) is homogeneous of degree 1 or less than 1 and represent a transformation or production function characterized by the usual regularity properties so that y = f t (x)

(A.1)

where y is a scalar measure of the output quantity, and x is a vector of input quantities, then the ﬁrst-order conditions for proﬁt maximization imply (dropping the superscript t to simplify notation) ∂f w = i ∂xi p

for all i s

(A.2)

where p is a scalar measure of the output price. Therefore,

si ≡

∂f xi wi · xi = ∂xi y p·y

(A.3)

Under constant and perfect competition, p · y = Ni=1 wi · xi , so that returns to scale N N si = wi · xi i=1 wi · xi and i=1 si = 1. N In the general case where p · y > i=1 wi · xi , if we want to construct sep< arately index numbers or indicators of differences in quantities and returns

The Theory of Exact and Superlative Index Numbers Revisited 199 to scale, then it might be useful to decompose the foregoing weights as follows: w ·x w ·x si = N i i + N i i · (ξ − 1) i=1 wi · xi i=1 wi · xi

(A.4)

! where ξ ≡ Ni=1 wi · xi p · y represents the degree of the returns to scale. It is obtained using the additional information on output prices and quantities. The second term of the right-hand side of equation (A.4) represents the weight capturing the effect of returns to scale (see Caves, Christensen, and Diewert, 1982, pp. 1405–1406 and p. 1408 for a similar decomposition25 ). Since prices and quantities of inputs and outputs are usually observable, all the necessary information for the calculation of index numbers and related indicators is available. (ii) The case of the utility function Let the function f t (x) represent a utility function characterized by the usual regularity properties so that u = f t (x)

(A.5)

where u represents utility, and x is a vector of the quantities consumed. Utility is typically unobserved, so that at least this important variable is not available for the index number construction. However, if the consumer has a utility-maximizing behavior subject to a budget constraint, then the ﬁrst-order conditions imply (dropping the superscript t to simplify notation)] ∂f = λwi for all i s ∂xi

(A.6)

We sum these conditions multiplied by quantities in order to obtain: N N ∂f xi = λ wi x i ∂xi i=1 i=1

(A.7)

which can be solved for the Lagrange multiplier as follows: N

λ=

i=1 N

∂f x ∂xi i

(A.8) w i xi

i=1

25 The variable parameter ξ (in our notation) is obtained with equation (47) in Caves et. al. (1982, p. 1406), where it is, however, misprinted (the correct form is reported at page 1408 of the same article).

200 Precursor Substituting the Lagrange multiplier back into the ﬁrst-order conditions yields: ⎞ ⎛ N ∂f ⎜ ∂xi xi ⎟ ∂f ⎟ ⎜ i=1 =⎜ n (A.9) ⎟ wi for all i s ⎠ ∂xi ⎝ w i xi i=1

from which we derive the Hotelling (1935, p.71)-Wold (1944, pp. 69–71; 1953, p. 145) identity ∂f 1 wi for all i s N ∂f = N ∂xi i=1 xi w x i=1 i i ∂xi

(A.10)

If we assume linear homogeneity of the utility function marginal (implying ∂f utility of income equal to 1), then, by Euler’s theorem, Ni=1 ∂x x = f (x) and, i i consequently, the foregoing equation becomes ∂f x ∂xi i

si ≡

f (x)

wx − N i i i=1 wi xi

for all i s

(A.11)

(iii) The case of the value functions (cost, revenue, and proﬁt functions) Let a value function (a cost, or revenue, or proﬁt function) be represented by a differentiable function et (p) , where p is a vector of prices. By Hicks (1946, p. 331)-Samuelson (1947, p. 68)-Shephard (1953, p. 11)-Hotelling’s lemma (1932, p. 594), we obtain directly the optimal levels of quantities through differentiation (dropping the superscript t to simplify notation): qi =

∂e ∂pi

(A.12)

where qi is the ith element of an N -dimensional vector of quantities. The value function is always linearly homogeneous in prices by construction. By Euler’s theorem, e (p) = Ni=1 pi qi . Dividing both sides of equation (A.12) by e(p) yields N

qi

i=1 pi qi

=

∂e ∂pi

e (p)

for all i s

(A.13)

Multiplying the foregoing equations by the respective prices pi ’s yields ∂e pi ∂p pi qi i = = si N (p) e p q i=1 i i

for all i s

(A.14)

Here, again, si and the numerical value of e(p) can be calculated using observed data on prices and quantities.

The Theory of Exact and Superlative Index Numbers Revisited 201

Appendix B: Proofs of theorems Proof of Lemma (2.1) (Accounting for Functional Value Differences). Let us consider an arbitrary function f (z) of one single variable. From the Taylor series expansion for f around z 0 , the following equation can be obtained: 2 1 f z 1 − f z 0 = f z 0 z 1 − z 0 + f z 0 z 1 − z 0 + . . . 2! n 1 (n) 0 1 + f z z − z 0 + R0n z 0 , z 1 n! n 1 m = f (m) z 0 z 1 − z 0 + R0n z 0 , z 1 m! m=1

(B.1)

where R0n z 0 , z 1 is the remainder term. Similarly, from the Taylor series expansion for f around z 1 , the following equation can be obtained: 1 2 f z 0 − f z 1 = f z 1 z 0 − z 1 + f z 1 z 0 − z 1 + . . . 2! n 1 (n) 1 0 z z − z 1 + R1n z 0 , z 1 + f n! n m 1 (m) 1 0 = z z − z 1 + R1n z 0 , z 1 f m! m=1

(B.2)

Multiplying through the foregoing equation by –1 and rearranging terms yield n m 1 (−1)m f (m) z 1 z 1 − z 0 − R1n z 0 , z 1 f z1 − f z0 = − m! m=1

(B.3)

Using (1 − θ) and θ as weights, the weighted average of (B.1) and (B.3) is given by

n 1 m 1 1 (1 − θ ) f (m) z 0 − (−1)m θ f (m) z 1 f z1 − f z0 = z − z0 m! m! m=1 (B.4) + (1 − θ ) R0n z 0 , z 1 − θ R1n z 0 , z 1 From (B.4), it follows that f z 1 − f z 0 = (1 − θ ) f z 0 + θf z 1 z 1 − z 0 + (1 − θ ) R01 z 0 , z 1 − θ R11 z 0 , z 1

(B.5)

Let us choose a value of θ, say θ ∗ , that minimizes the squared term (1 − θ ) 2 R01 z 0 , z 1 − θ R11 z 0 , z 1 . Since this term is convex in θ , the necessary and

202 Precursor sufﬁcient condition for its minimization is that its ﬁrst derivative with respect to θ vanishes26 , so that −2 (1 − θ ) R01 z 0 , z 1 − θ R11 z 0 , z 1 R01 z 0 , z 1 + R11 z 0 , z 1 = 0

(B.6)

from which, provided that R01 z 0 , z 1 + R11 z 0 , z 1 = 0, θ =θ

∗

z ,z 0

1

R01 z 0 , z 1 ≡ 0 0 1 R1 z , z + R11 z 0 , z 1

(B.7)

The case of many variables follows in a similar manner using the directional derivatives. In particular, f z 1 − f z 0 = (1 − θ ) ∇f z 0 + θ ∇f z 1 z 1 − z 0

(B.8)

where, if R01 z 0 , z 1 + R11 z 0 , z 1 = 0, then θ = θ ∗ (so that (1 − θ ∗ ) R01 z 0 , z 1 − (z 1 ) (z 0 ) θ ∗ R11 z 0 , z 1 = 0), or, if R1 = R1 = 0, then θ may take any value as a number. Proof of Corollary (2.1) (Decomposition of Functional Value Changes). Adding and subtracting JZ from (2.1) yield directly the decomposition (2.2). Since i si∗t = 1, the meaning of JZ is that of a candidate aggregate of absolute changes in the vector z. Let us deﬁne the functional value IZ · f z 0 = f z 0 + JZ . From (2.2) it is possible to derive IZ · f z 0 − f z 0 = JZ =

f (1 − θ ) si∗0

0 z zi0

IZ + θ si∗1

· f z0 · zi1 − zi0 1 zi (B.9)

Dividing through (B.9) by f (z 0 ) yields IZ − 1 = (1 − θ) + θ IZ

N

z1 si∗0 i0 zi i=1

N i=1

− (1 − θ )

si∗1 − θ IZ

N

si∗0

i=1 N i=1

si∗1

zi0 zi1

(B.10)

$ % 26 The second-order condition for minimization is always respected since ∂ 2 (1 − θ ) R01 z 0 , z 1 − $ % 2 % $ % 2 $ θ R11 z 0 , z 1 /∂θ = 2 R01 z 0 , z 1 + R11 z 0 , z 1 > 0.

The Theory of Exact and Superlative Index Numbers Revisited 203 = (1 − θ )

N

si∗0

i=1

+ θ IZ − θ IZ

zi1 − (1 − θ ) zi0

N

si∗1

i=1

given that

IZ =

N

∗t i=1 si

zi0 zi1

(B.11)

= 1. Rearranging (B.11) and solving for IZ yield

θ + (1 − θ) (1 − θ) + θ

N

1

∗0 zi i=1 si z 0

N

i 0

∗1 zi i=1 si z 1

(B.12)

i

Proof of Corollary (2.2) (Diewert’s, 1976, p. 117, Quadratic Identity). Sufﬁciency: From the deﬁnition (2.5) of the quadratic function fQ , the ﬁrst difference of this function is obtained as follows: 1 fQ z 1 − fQ z 0 = a0 + az 1 + z 1 Az 1 2 1 − a0 − az 0 − z 0 Az 0 2

(B.13)

By adding 12 z 0 Az 1 and subtracting 12 z 1 Az 0 (recall that z 0 Az 1 = z 1 Az 0 since aij = aji for all i, j in A) and substituting 12 z 0 Az 1 with z 0 Az 1 − 12 z 0 Az 1 and 12 z 0 Az 0 with 0 0 1 0 0 z Az − 2 z Az , equation (B.13) can be rearranged into the following fQ z 1 − fQ z 0 = a + z 0 A z 1 − z 0 +

1 1 z − z0 A z1 − z0 2

(B.14)

which can also be derived directly from the Taylor series expansion of fQ (z) around the point z 0 , since ∇fQ z 0 = a + z 0 A ∇ 2 fQ z 0 = A

(B.15) (B.16)

Similarly, from 1 1 fQ z 0 − fQ z 1 = a0 + az 0 + z 0 Az 0 − a0 − az 1 − z 1 Az 1 2 2

(B.17)

by adding 12 z 1 Az 0 and subtracting 12 z 0 Az 1 (recall, again, that z 1 Az 0 =z 0 Az 1 since aij =aji for all i, j in A) and substituting 12 z 1 Az 0 with z 1 Az 0 − 12 z 1 Az 0 and

204 Precursor 1 1 z Az 1 with z 1 Az 1 − 12 z 1 Az 1 , equation (B. 17) can be rearranged into the 2 following 1 fQ z 0 − fQ z 1 = a + z 1 A z 0 − z 1 + z 0 − z 1 A z 0 − z 1 2

(B.18)

which can also be derived directly from the Taylor series expansion of fQ (z) around the point z 1 , since ∇fQ z 1 = a + z 1 A (B.19) (B.20) ∇ 2 fQ z 1 = A Multiplying through (B.18) by –1 and rearranging terms yields 1 fQ z 1 − fQ z 0 = a + z 1 A z 1 − z 0 − z 1 − z 0 A z 1 − z 0 2

(B.21)

The arithmetic average of (B.14) and (B.21) is given by 1 fQ z 1 − fQ z 0 = a + z0 A + a + z1 A z1 − z0 2 1 = ∇fQ z 0 + ∇fQ z 1 z 1 − z 0 2

(B.22)

Necessity: Let us start from the following equation, f (x) − f (y) =

1 [∇f (x) + ∇f (y)] (x − y) 2

(B.23)

Assume f (x) is thrice-differentiable and satisﬁes the functional equation f (x)–f (y) for all x and y, in an open neighbourhood. If f (x) is a function of one variable, the functional equation becomes f (x) − f (y) =

1 f (x) + f (y) (x − y) 2

(B.23 bis)

Let us differentiate the foregoing equation with respect to x in order to obtain f (x) =

1 1 f (x) + f (y) + f (x)(x − y) 2 2

(B.24)

The second derivative of f (x) with respect to x is then 1 1 1 f (x) = f (x) + f (x) + f (x)(x − y) 2 2 2

(B.25)

which means that 12 f (x)(x − y) = 0 and, therefore, f (x) = 0 for every x = y, implying that f (x) can be traced perfectly by a polynomial of degree two.

The Theory of Exact and Superlative Index Numbers Revisited 205 The general multivariate case follows in a similar manner using the partial derivatives. Proof of Lemma (2.2) (General Quadratic Approximation Lemma). Taking into account identity (2.1), from (2.9) we obtain Error of approximation = (1 − θ) [∇f (z 0 ) + θ∇f z 1 ] (z 1 − z 0 ) 1 − [∇f (z 0 ) + ∇f (z 1 )](z 1 − z 0 ) 2

1 = − θ [∇f (z 0 ) − ∇f (z 1 )](z 1 − z 0 ) 2

(B.26)

The ﬁrst-order approximating (linear) function that is tangent to f (z) at z t is given by fLt (z) = f (z t ) + ∇z f (z t )(z − z t ) = at + bt z (with at = f (xt ) − ∇z f (xt )xt and bt = ∇z f (xt )). Therefore fLt (z) − fLt (z t ) = ∇f (z t )(z − z t )

(B.27)

Using (B.27) with t = 0, 1 and z = z 0 , z 1 , equation (B.26) becomes

0 1 1 Error of approximation = −θ fL (z ) − fL0 (z 0 ) − fL1 (z 1 ) − fL1 (z 0 ) 2 (B.28) Proof of Lemma (3.1) (Accounting for Value Differences of Two Functions with Different Parameters or Functional Forms). Let us two arbitrary functions of one single variable, f 0 (z) and f 1 (z). These functions may differ in parameter values or even in their functional forms. From the Taylor series expansion for f 0 (z 1 ) around z 0 , the following equation may be obtained: f 0 (z 1 ) − f 0 (z 0 ) = f 0 (z 0 )(z 1 − z 0 ) + R01

(B.29)

where R01 = R01 (z 0 , z 1 ) is the remainder term of the ﬁrst-order approximation. Adding and subtracting f 1 (z 1 ) and rearranging terms, the foregoing equation becomes: f 1 (z 1 ) − f 0 (z 0 ) = f 0 (z 0 )(z 1 − z 0 ) + f 1 (z 1 ) − f 0 (z 1 ) + R01 (B.30) Similarly, from the Taylor series expansion for f 1 (z 0 ) around z 1 , the following equation may be obtained: f 1 (z 0 ) − f 1 (z 1 ) = f 1 (z 1 )(z 0 − z 1 ) + R11

(B.31)

where R11 = R11 (z 0 , z 1 ) is the remainder term of the ﬁrst-order approximation. By multiplying both sides by –1 and rearranging terms, equation (B.31) becomes: f 1 (z 1 ) − f 1 (z 0 ) = f 1 (z 1 )(z 1 − z 0 ) − R11

(B.32)

206 Precursor Adding and subtracting f 0 (z 0 ) and rearranging terms, the foregoing equation becomes: f 1 (z 1 ) − f 0 (z 0 ) = f 1 (z 0 )(z 1 − z 0 ) + [ f 1´(z 0 ) − f 0 (z 0 )] − R11

(B.33)

Using (1 − θ ) and θ as weights, where θ may take any real number as a value, the weighted average of (B.30) and (B.33) is given by f 1 (z 1 ) − f 0 (z 0 ) = (1 − θ ) f 0 (z 0 ) + θf 1 (z 0 ) (z 1 − z 0 ) + θ f 1 (z 0 ) − f 0 (z 0 ) + (1 − θ ) f 1 (z 1 ) − f 0 (z 1 ) + (1 − θ) R01 − θ R11 (B.34) If θ = θ ∗ , with θ ∗ = θ ∗ (z 0 , z 1 ) ≡

R01

R01 +R11

, then

f1 (z 1 ) − f 0 (z 0 ) = (1 − θ)f 0 (z 0 ) + θ f 1 (z 0 ) (z 1 − z 0 ) + Parameter − change componenr (PC)

(B.35)

where PC ≡ θ ∗ f 1 (z 0 ) − f 0 (z 0 ) + (1 − θ ∗ ) f 1 (z 1 ) − f 0 (z 1 )

(B.36)

The case of many variables follows in a similar manner using the directional derivatives, thus obtaining (3.2)–(3.3). Proof of Corollary (3.1) (Decomposition of Functional Value Differences Between Two Arbitrary Differentiable Functions). The proof of Corollary (3.1) is analogous to that of Corollary (2.1). Proof of Corollary (3.2) (Accounting for Value Differences of Two Quadratic Functions Differing in Parameters). The proof of Corollary (3.2) follows the footsteps of the proof of the “sufﬁciency” part of Corollary (2.2). From the deﬁnition (3.8) of the quadratic function fQt , it is straightforward to obtain: fQ1 (z 1 ) − fQ0 (z 0 ) = (a0 + z 0 A0 )(z 1 − z 0 ) + (a10 − a00 ) + (a1 − a0 )z 1 1 − z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.37)

since fQ1 (z 1 ) − fQ0 (z 0 ) = fQ0 (z 1 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 )

(B.38)

The Theory of Exact and Superlative Index Numbers Revisited 207 where 1 fQ0 (z 1 ) − fQ0 (z 0 ) = (a0 + z 0 A0 )(z 1 − z 0 ) + (z 1 − z 0 )A0 (z 1 − z 0 ), 2

(B.39)

using the Taylor series expansion for f 0 around z 0 , 1 fQ1 (z 1 ) − fQ0 (z 1 ) = (a10 − a00 ) + (a1 − a0 )z 1 + z 1 (A1 − A0 )z 1 2

(B.40)

and 1 1 1 1 (z − z 0 )A0 (z 1 − z 0 ) + z 1 (A1 − A0 )z 1 = −z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2 2 2 (B.41) Similarly, fQ0 (z 0 ) − fQ1 (z 1 ) = (a1 + z 1 A1 )(z 0 − z 1 ) + (a00 − a10 ) + (a0 − a1 )z 0 1 − z 0 A1 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.42)

Since fQ0 (z 0 ) − fQ1 (z 1 ) = fQ1 (z 0 ) − fQ1 (z 1 ) + fQ0 (z 0 ) − fQ1 (z 0 )

(B.43)

where 1 fQ1 (z 0 ) − fQ1 (z 1 ) = (a1 + z 1 A1 )(z 0 − z 1 ) + (z 0 − z 1 )A1 (z 0 − z 1 ) 2

(B.44)

using the Taylor series expansion for f 1 around z 1 1 fQ0 (z 0 ) − fQ1 (z 0 ) = (a00 − a10 ) + (a0 − a1 )z 0 + z 0 (A0 − A1 )z 0 2

(B.45)

and 1 1 1 1 (z − z 0 )A0 (z 1 − z 0 ) + z 1 (A1 − A0 )z 1 = −z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2 2 2 (B.46) Multiplying both sides of (B.42) by –1 and rearranging terms yield fQ1 (z 1 ) − fQ0 (z 0 ) = (a1 + z 1 A1 )(z 1 − z 0 ) + (a10 − a00 ) + (a1 − a0 )z 0 1 + z 0 A1 z 1 − (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.47)

208 Precursor The arithmetic average of (B.37) and (B.47) is therefore fQ1 (z 1 ) − fQ0 (z 0 ) =

1 0 0 ∇fQ (z ) + ∇fQ1 (z 1 ) (z 1 − z 0 ) 2 1 1 + (a00 − a10 ) + (a0 − a1 ) (z 0 + z 1 ) + z 0 (A1 − A0 )z 1 2 2 (B.48)

since ∇fQ0 (z 0 ) = a0 + z 0 A0 , ∇fQ1 (z 1 ) = a1 + z 1 A1 . Moreover, by deﬁnition, 1 1 a00 − a10 = (α01 + k 1 β 1 + k 1 B1 k 1 ) − (α00 + k 0 β 0 + k 0 B0 k 0 ) 2 2 (B.49) 1 1 (a0 − a1 ) (z 0 + z 1 ) = (α 1 + k 1 1 − α 0 − k 0 0 ) (z 0 + z 1 ) 2 2

(B.50)

The sum of (B.49) and (B.50) is 1 1 (a00 − a10 ) + (a0 − a1 ) (z 0 + z 1 ) = (α00 + α01 ) + (α 0 − α 1 ) (z 0 + z 1 ) 2 2 1 + (k 1 β 1 + k 1 B1 k 1 + k 1 1 z 1 + k 1 1 z 0 ) 2 − k 0 β 0 − k 0 B0 k 0 − k 0 0 z 1 − k 0 0 z 0 1 + (k 1 β 1 − k 0 β 0 ) 2

(B.51)

Adding and subtracting 12 k 0 β 1 , 12 k 1 β 0 , 12 k 0 B1 k 1 , and 12 k 0 B0 k 1 to the right-hand side of (B.51) yield (3.17). Proof of Lemma (3.2) (Quadratic Approximation of Value Differences between Two Arbitrary Functions with Different Parameters or Functional Forms). The proof of Lemma (3.2) follows the proof of Lemma (2.2). Proof of Corollary (3.3) (Accounting for the Sum of Value Differences between Two Quadratic Functions with Different “Zero-order” and “First-order” Parameters) (Caves, Christensen, and Diewert’s, 1982, pp.1412–1413 Translog Identity). Let the quadratic function htQ (z, k t ) be deﬁned by (3.9). By the Quadratic Identity (Corollary 2.2), hQt (z 1 , k t ) − hQt (z 0 , k t ) =

1 ∇ h t (z 0 , k t ) + ∇z hQt (z 1 , k t ) · (z 1 − z 0 ) (B.52) 2 z Q

The Theory of Exact and Superlative Index Numbers Revisited 209 and, therefore, the ﬁrst line of the right-hand side of equation (3.20) becomes, setting λ = 1/2, for t = 0 and t = 1, 1 1 0 1 0 h z , k − h0Q z 0 , k 0 + h1Q z 1 , k 1 − h1Q z 0 , k 1 2 Q 2 1 1 z − z0 = ∇z h0Q z 0 , k 0 + ∇z h0Q z 1 , k 0 2 1 1 + ∇z h1Q z 0 , k 1 + ∇z h1Q z 1 , k 1 z − z0 2

(B.53)

Since ∇z h0Q (z 0 , k 0 ) = a0 + z 0 A

(B.54)

∇z h0Q (z 1 , k 0 ) = a0 + z 1 A

(B.55)

∇z h1Q (z 0 , k 1 ) = a1 + z 0 A

(B.56)

∇z h1Q (z 1 , k 1 ) = a1 + z 1 A

(B.57)

where at = α t + k t t , for t = 0, 1, ∇z h0Q (z 1 , k 0 ) + ∇z h1Q (z 0 , k 1 ) = (a0 + z 1 A) + (a1 + z 0 A) = (a0 + z 0 A) + (a1 + z 1 A) = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 )

(B.58)

Using (B.58), equation (B.53) becomes 1 1 0 1 0 hQ (z , k ) − h0Q (z 0 , k 0 ) + h1Q (z 1 , k 1 ) − h1Q (z 0 , k 1 ) 2 2 1 = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 ) (z 1 − z 0 ) 2

(B.59)

which, multiplied through by 2, is equation (3.22). Equation (3.21) follows directly using deﬁnitions (3.8) and (3.9) Proof of Corollary (3.4) (Accounting for the Sum of Value Differences of Two Quadratic Functions with Different Parameters) (Caves, Christensen, and Diewert, 1982, pp. 1412–1413). By the Quadratic Identity

1 fQ0 (z 1 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ1 (z 0 ) = ∇z fQ0 z 0 ) + ∇2 fQ0 (z 1 ) (z 1 − z 0 ) 2 1 + ∇z fQ1 z 0 ) + ∇2 fQ1 (z 1 ) (z 1 − z 0 ) 2 (B.60)

210 Precursor

Adding and subtracting 12 ∇z fQ0 (z 0 ) + ∇z fQ1 (z 1 ) (z 1 −z 0 ) to the foregoing equation and taking into account the deﬁnition of fQt yield 0 1 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ1 (z 0 ) = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) 1 + (a0 + z 1 A0 + a1 + z 0 A1 − a0 − z 0 A0 − a1 − z 1 A1 )(z 1 − z 0 ) 2 = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) 1 + (z 1 A0 z 1 + z 0 A1 z 1 − z 0 A0 z 1 − z 1 A1 z 1 − z 1 A0 z 0 − z 0 A1 z 0 2 + z 0 A0 z 0 + z 1 A 1 z 0 ) 1 = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) − (z 1 − z 0 )(A1 − A0 )(z 1 − z 0 ) 2 (B.61) or

h0Q (z 1 , k 0 ) − h0Q (z 0 , k 0 ) + h1Q (z 1 , k 1 ) − h1Q (z 0 , k 1 ) = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 ) (z 1 − z 0 ) 1 − (z 1 − z 0 )(A1 − A0 )(z 1 − z 0 ) 2

which is equation (3.24). Proof of Corollary (3.5). By deﬁnition, 1 0 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 )

1 1 = a10 + a1 z 0 + z 0 A1 z 0 − a00 − a0 z 0 − z 0 A0 z 0 2 2

1 1 + a10 + a1 z 1 + z 1 A1 z 1 − a00 − a0 z 1 − z 1 A0 z 1 2 2

(B.62)

(B.63)

Adding and subtracting 12 z 0 A0 z 1 and 12 z 0 A1 z 1 and noting that the symmetry of At (that is At = At ) implies z 0 At z 1 = z 1 A1 z 0 , the foregoing equation becomes 1 0 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 ) = 2 a10 − a00 − (a0 − a0 ) z 0 + z 1 + z 0 A1 − A0 z 1 +

1 1 z − z 0 A1 − A0 z 1 − z 0 2

which is equation (3.25).

(B.64)

The Theory of Exact and Superlative Index Numbers Revisited 211 The derivation of equations (3.27) and (3.28) follows the footsteps of that of equations (3.24), (3.25), and (3.26). Proof of Corollary (3.6) (Accounting for Functional Value Ratios of Two Different Quadratic Homothetic Functions). The proof follows directly that of Corollary (2.1). Proof of Lemma (4.1) (Accounting for Value Differences of Two General Quadratic Functions). 1 1 0 0 The difference g fGQ (q ) − g fGQ (q ) and g h1GQ (q1 , x1 ) − g h0GQ (q0 , x0 ) can be decomposed by applying Lemma (3.1) and Corollary (3.1), respectively, thus obtaining: 0 −1 0 0 0 1 −1 1 1 1 Z (z ) −g fGQ Z (z ) (q ) −g fGQ (q ) = g fGQ g fGQ = fQ1 (z 1 )−fQ0 (z 0 ) =

1 ∇z fQ0 (z 0 )+∇z fQ1 (z 1 ) (z 1 −z 0 ) 2

+Parameter change component (PC) " 1 0 0 ˆ 0 −1 0 = (q0 ) g fGQ (q ) · Z (q ) ·∇q fGQ 2 #

1 1 1 −1 1 +g fGQ (q1 ) (q ) · Zˆ (q ) ·∇q fGQ · Z(q1 )−Z(q0 ) +PC

(B.65)

where 1 0 PC ≡ a10 − a00 + a1 − a0 Z(q ) + Z(q1 ) + Z(q0 ) A1 − A0 Z(q0 ), 2 (B.66) −1 is equal to the (i, i)th element of the diagonal matrix Zˆ (q) dqi dz −1 (zi ) = = 1/z (qi ) dzi dzi (with z (qi ) = 0, by assumption). t t −1 Z (z) , Since fQ (z) ≡ g fGQ (q) = g fGQ

(B.67)

212 Precursor

t −1 −1 −1 t ∇z fQ (z) = g fGQ Z (z) · Zˆ (z) · ∇z fGQ Z (z) where the (i, i)th element of the diagonal matrix Zˆ −1 (z) is given by

dz −1 (zi ) dzi

t t (q) · Zˆ −1 (z) · ∇z fGQ (q) using (4.3) = g fGQ −1 t t = g fGQ (q) · Zˆ (q) · ∇z fGQ (q) using (4.9) equation (B.65) can be rewritten as 1 1 0 0 g fGQ (q ) − g fGQ (q ) " 1 1 0 0 0 −1 1 0 (q ) (q0 ) + g fGQ = (q ) · Zˆ (q ) · ∇q fGQ g fGQ 2 −1 3 1 · Zˆ (q1 ) ·∇q fGQ (q1 ) · Z(q1 ) − Z(q0 ) + Pc

(B.68)

(B.69)

Proof of Theorem (6.1). If function g and z in (4.1) are, respectively, logarithmic transformation of functional values and variables, that is g(y) ≡ ln y and Z(q) ≡ ln q, then (6.1) follows directly from (4.9). Proof of Theorem (6.2). Let us apply the decomposition procedure (4.9) to the quadratic Box-Cox function (5.4), thus obtaining: $ %ρ %ρ $ N 0 1−λ fQ1r,λ −1 fQ0r,λ −1 1 qi 0 − = $ %1−ρ fQρ,λ ,i ρ ρ 2 i=1 0 fQρ,λ 1 1−λ q1 λ −1 q0 λ −1 qi i i 1 − +$ %1−r fQρ,λ,i λ λ 1 fQρ,λ +Parameter −change component 0 0 N 1 qi fQρ,λ ,i $ 0 %ρ 1 = fQρ,λ λ + 2 i=1 fQ0ρ,λ q0

+

qi1 fQ1ρ,λ, i $ fQ1ρ,λ

fQ1ρ,λ

%ρ

λ 0 λ qi 1 qi1 − 1 λ λ λ q

+Parameter −change component where fQt ρ,λ ≡ fQt ρ,λ (qt ), fQt ρ,λ ,i ≡ ∂fQt ρ,λ /∂qit .

(B.70)

The Theory of Exact and Superlative Index Numbers Revisited 213 By deﬁning sQt ρ,λ ≡

qit f t ρ,λ Q

f t ρ,λ

,i

with t=0, 1, multiplying equation (B.70) by ρ and

Q

rearranging terms we obtain $

fQ1ρ,λ

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧

N ⎪ ⎨

1ρ 2λ

⎪ i=1 ⎩

%ρ ⎫ %ρ $ ⎬ fQ0ρ,λ fQ1ρ,λ ⎪ 1 0 λ + sQρ,λ i 1 λ ⎪ ⎭ qi qi

$ sQ0 ρ,λ i

λ λ qi1 − qi0 + Parameter − change component (B.71) which is equation (6.4). Proof of Theorem (6.3). Equation (6.4) can be re-written as follows: ⎧ $ %ρ $ %ρ ⎫ N ⎪ ⎬ ⎨ $ %ρ $ %ρ fQ0ρ,λ IQ · fQ0ρ,λ ⎪ fQ1ρ,λ − fQ0ρ,λ = sQ∗0ρ,λ i λ + sQ∗1ρ,λ i 1 ⎪ qi λ ⎪ ⎭ qi0 i=1 ⎩ ⎧ $ %ρ 0 N ⎪ ⎨ λ λ f ρ,λ Q · qi1 − qi0 ξ 0 − 1 sQ∗0ρ,λ i λ + ⎪ qi0 i=1 ⎩ $ + ξ 1 − 1 sQ∗1ρ,λ i

Iq · fQ0ρ,λ 1 λ qi

%ρ ⎫ ⎪ ⎬ λ λ · qi1 − qi0 ⎪ ⎭

+ Parameter − change component where sQ∗tρ,λ i ≡ ρ 2λ

N

1 1 i=1 qi f ρ,λ Q

i

f 1ρ,λ

qit f t ρ,λ Q i N t t j=1 qj fQρ,λ j ρ

f 1ρ,λ Q

IQ ·f 0ρ,λ

Q

IQ · fQ0ρ,λ

ρ 2λ

N

0 0 i=1 qi fQρ,λ i f 0ρ,λ Q

, and ξ 1 ≡

ρ .

Q

Setting $

for t = 0, 1, ξ 0 ≡

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧ N ⎪ ⎨ ⎪ i=1 ⎩ ·

%ρ $ %ρ ⎫ ⎬ fQ0ρ,λ IQ · fQ0ρ,λ ⎪ ∗1 0 λ + sQρ,λ i 1 λ ⎪ ⎭ qi qi

$ sQ∗0ρ,λ i

λ λ qi1 − qi0

$ %ρ dividing through by fQ0ρ,λ , rearranging terms and solving for IQ , we have IQ ≡ Iq · IY

214 Precursor where, setting r = 2λ, ⎡

N

1

r

( qi ) 2 ∗0 ⎢ sQρ,λ i q0 2r ( i) ⎢ i=1

⎤ 1r

⎥ ⎥ r ⎥ and 0 2 ⎦ q ( ) sQ∗1ρ,λ i i1 r ( qi ) 2 i=1

1−1 r

IY = Iqρ

Iq = ⎢ N ⎣

where Iq is the aggregating index number of the elements of q. Proof of Corollary (6.1). Using the deﬁnition (5.15) for fQr and denoting fQr (qt ) with fQt r to simplify notation, from the accounting equation (B.70), we derive: 0 0 r 1 r N qi1 fQ1r i fQ1r fQr q i fQ r i 1 1 2r 0 2r 1 0 r −1 = 0 2r + 1 0 r 1 2r qi − qi 0 fQ r q fQr fQr fQ r q i=1 i

i

! (where fQt r i ≡ ∂fQt r ∂qit ) 1 r N 1 r N 1 2r 0 2r N f f r r q qi Q Q i = sQ0 r i r − sQ0 r i + 0 r sQ1 r i − 0 r sQ1 r i r 0 2 fQr i=1 fQr i=1 qi qi1 2 i=1 i=1 ! N t t t where sQt ρ,λ i = sQt r i = qit fQt r i fQt r = qit fQt ρ,λ i ρr fQt ρ,λ = qit fQt ρ,λ i i=1 qi qi fQρ,λ i , $ % ρr $ % ρr −1 $ % since fQt r = σ1t fQt ρ,λ and fQt r i = σ1t ρr fQt ρ,λ fQt ρ,λ i and, by Euler’s theorem, ρr fQt ρ,λ = Ni=1 qit fQt ρ,λ i , N

=

N

r

qi1 2 sQ0 ρ,λ i r qi0 2 i=1

−1+

fQ1r

fQ0r

r r −

r

1 − 2r qi 1 sQρ,λ i − r 0 r fQr i=1 qi0 2 fQ1r

N

(B.75)

N since fQt r is homogeneous of degree one in q and, by Euler’s theorem, sQt r i = 1. i=1 ! 1. By rearranging (B.75) and solving for fQ1r fQ0r , the following index is obtained:

⎡ fQ1r fQ0r

N

1

r

( qi ) 2 0 ⎢ sQρ,λ i q0 2r ( i) ⎢ i=1

= Iq ≡ ⎢ N ⎣

⎤ 1r

⎥ ⎥ ⎥ ( ) ⎦ ( )

r qi0 2 1 ρ,λ Q i 1 2r qi i=1

s

Let us deﬁne IT ≡ σ 1 /σ 0 , and, taking account of (B.74), % $ 1−1 Iq · IT = IY ≡ Iqρ r fQ1ρ,λ fQ0ρ,λ

(B.76)

(B.77)

The Theory of Exact and Superlative Index Numbers Revisited 215 Proof of Theorem (6.4). The arbitrary twice differentiable function f (q) and fQr (q) deﬁned by (5.15) are assumed to be second-order approximation at q0 , that is f (q0 ) = fQr (q0 )

(B.78)

∇f (q0 ) = ∇fQr (q0 )

(B.79)

∇ 2 f (q0 ) = ∇ 2 fQr (q0 )

(B.80)

With f positive over its domain of deﬁnition, qi0 > 0, and r = 0, the coefﬁcients αij ’s can be found by solving the two systems of equations obtained by rearranging the differential equations resulting from differentiating fQr as deﬁned by (5.15) and replacing the obtained derivatives with those of f , that is fi =

N

r −1 0 2r qj for i = 1, 2, . . . , N f ∗(1−r) αij qi0 2

(B.81)

j=1

fij =

r 1−r) 0 0 2r −1 1−r qi qj , for i = j, and i.j = 1, 2, . . . , N ∗ fi fj + αij f f 2 (B.82)

thus obtaining from (B.81) and (B.82), respectively, N r r −1 f (1−r) αij qj0 2 fi − qi0 2

αii (r, q0 ) = αij (r, q0 ) =

j =i

f

(1−r)

0 r−1 qi

,

for i = 1, 2, . . . , N

ff fij − 1−r f∗ i j r −1 for i = j, and i, j = 1, 2, . . . , N r 1−r f qi0 qj0 2 2

(B.83)

(B.84)

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Appendices 1

The price index as a utility based concept 1.1 1.2

2

Original approach New approach

Terminology Conical v. homogeneous &c

3

Notation

4

BASIC computer program BBC BASIC for Windows developed by Richard Russell [email protected].

1

The price index as a utility based concept Answer to complaints that conical or constant-returns utility for dealing with the price index is an objectionable imposition, instead of being unavoidable as was settled long ago.

“… the artiﬁcial ‘homogeneity’ which underlies most index thinking and usages, can escape mention entirely.” S. N. Afriat Index Numbers in Theory and Practice by R.G.D. Allen Canadian J. Econ. 15, 2 (May 1978)

1.1

Original approach

1.2

New approach

1.1 Original approach Prices change and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. This is the cost of living problem. The Price Index issued from the Statistical Ofﬁce is a number that tells how to deal with the question in a highly restrictive artiﬁcially imposed simple away, the index being the multiplier of old expenditure to determine the new. There is of course no absolute reason for having a price index in the ﬁrst place. It is an institution, even if a well established, traditional, and perhaps even today still needed institution, affecting many aspects of economic life. The question of how to produce such a number is known as The Index-Number Problem. Let Pst denote the price index from period s to period t. The number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by (P)

Mr = Prs Ms

in period r to maintain the same standard of living. This scheme has now to be put in utility terms. Given a utility function φ : n → , such as may govern demand, the cost at prices p ∈ n of attaining the the utility of consumption x ∈ n is (R)

ρ (p, x) = min {py : φ (y) ≥ φ (x)}

ρ being theutility-cost function, depending on prices p and on consumption x through its utility value φ (x) that is representative of standard of living. This is the cost when prices are p of living at the standard represented by consumption x. The cost of living question, put in utility terms, is concerned with how this cost changes when prices change, for a given ﬁxed standard of living. By the condition (S)

ρ (p, x) = px,

demand of x is supported by prices p, its cost being the minimum cost of obtaining its utility, and x is supported if supported by some p. For concave utility, every x is supported since there is a supporting hyperplane to the graph at every point. For a continuous utility every p supports some x. By deﬁnition, for all p, x (a1) φ (y) ≥ φ (x) ⇒ py ≥ ρ (p, x) for all y, and, continuity provided, (a2) φ (y) = φ (x) , py = ρ (p, x) for some y.

226 Appendices In particular, (b1) ρ (p, x) ≤ px for all p, x. and, for all p, (b2) ρ (p, x) = px for some x. To be found now are implications of admissibility of the special resolution of the cost of living question by means of a price index. Let pt be prices in period t, so ρ (pt , x) is the cost at those prices of living at the standard represented by x. In transition from period s to period r the cost changes from ρ (ps , x) to ρ (pr , x), in the ratio ρ (pr , x) /ρ (ps , x) in general depending on x. In case this ratio is independent of x, a price index Prs based on the utility is deﬁned and given by this ratio, so giving satisfaction for all x of the conventional money conversion relation (P) depending on a price index, where now Mr = ρ (pr , x) , Ms = ρ (ps , x) . By this condition the utility has the price index property, and Prs is the associated price index, which provides (Px) Mr = Prs Ms for all x. Here represented in utility terms is the deﬁning image from the form of its use of the price index of ordinary practice. But to have this representation, the ratio to determine the price index has to be independent of the variable consumption bundle x the utility of which is the measure of standard of living, that is, (i)

ρ (pr , x) /ρ (ps , x) is independent of x

This independence represents the special condition on the utility by which it has the price index property and has a price index associated with it. This condition on utility, stated in terms of the utility cost function ρ, will be seen equivalent to the condition of utility cost factorization that requires (ii) ρ (p, x) = θ (p) φ (x) or that the utility cost function ρ factorizes into a product of a function θ of prices alone with a function φ of quantities alone1 . In this case the associated price index, 1 Touched on in Afriat 1972, 1977 pp 101 ff or 2005 pp 87 ff; also Deaton 1979 JASA 74, 365. The equivalence of (i) and (ii) is stated by Samuelson and Swamy (1974) p. 570.

The price index as a utility based concept 227 or the price index based on the utility, is given immediately by Prs = θ (pr ) /θ (ps ) . Theorem 1 For a utility to have the price index property, and so to have an associated price index, utility cost factorization is necessary and sufﬁcient. We have to show (i) ⇔ (ii). Since (ii) ⇒ (i) is immediate, it remains to prove (i) ⇒ (ii). Take any ﬁxed a ∈ C. Then by the price index property (i), ρ (pr , x) /ρ (ps , x) = ρ (pr , a) /ρ (ps , a) for all x. Let θ (p) = ρ (p, a) , so now ρ (pr , x) /ρ (ps , x) = ρ (pr , a) /ρ (ps , a) = θ (pr ) /θ (ps ) , and let φ (x) = ρ (p, x) /θ (p) . Then we have the utility cost factorization required by (ii), completing the proof. Theorem 2 For factorization of the utility cost function it is necessary and sufﬁcient that the utility be conical. This Theorem may be a good candidate for the title “The Index Number Theorem” secured by Hicks for another purpose, and we are going to prove it as happened in 1950s or 60s, anyway long ago2 . Given φ conical, ρ (p, x) = min {py : φ (y) ≥ φ (x)} = min py (φ (x))−1 : φ y (φ (x))−1 ≥ 1 φ (x) = θ (p) φ (x)

2 Samuelson and Swamy (1974) p. 570 cite Afriat (1972).

228 Appendices where θ (p) = min {pz : φ (z) ≥ 1} That shows the sufﬁciency. Since, for all p, θ (p) φ (x) ≤ px, for all x, with equality for some x, as assured with continuous φ, it follows that θ (p) = minx px/φ (x) showing θ to be concave conical semi-increasing. Also for x demandable at some prices, as would be the case for any x if φ is concave, the inequality holds for all p with equality for some p, showing φ (x) = minp px/θ (p) which, in case every x is demandable at some prices, requires φ to be concave conical semi-increasing. But even when not all x are demandable, because they lie in caves and are without a supporting hyperplane, here is a conical function deﬁned for all x that is effectively the same as the actual φ as far as any observable demand behaviour is concerned. So it appears that for the cost function factorization the utility function being conical is also necessary, beside being sufﬁcient, as already remarked. Hence, with some details taken for granted, the Theorem is proved. In consequence we have: Theorem 3 For a price index to be based on a utility it is necessary and sufﬁcient that the utility be conical This approach to the beginning of price index theory a half century ago is one way of bringing forward the inevitablility of the association of a price index with conical utility. There is another approach based on common sense to follow now, where it appears that as soon as you start in any way about a price index, to do with utility, in the ﬁrst moment you have constant returns utility. That should wipe out protests and of course there is bound to be a penalty somewhere in dealing with such a restricted concept as the price index in the ﬁrst place. If there is an assumption anywhere, it is the price index itself. There is, contrary to complaints, no additional assumption about utility being constant returns, only the implication. Will that placate the bitter opponents? Before leaving about factorization there should be notice about the factors. From (ii) joined with (b1) and (b2) we have (c1) θ (p) = minx px/φ (x)

The price index as a utility based concept 229 which shows that θ (p), being the minimum of a family of homogeneous linear functions px/φ (x), is concave conical. Also, for any x that is supported, (c2) φ (x) = minp px/θ (x) . Hence if φ, beside being anyway conical to have the factorization, is also concave, so every x is supported, then this holds unconditionally. Functions that satisfy (c1) and (c2), both necessarily concave conical, deﬁne a conjugate pair of price and quantity functions. The symmetry here reﬂects a perfect symmetry throughout between price and quantity, having various manifestations. See Appendix 3 on Notation.

230 Appendices

1.2 New approach To keep with familiarities, we ﬁrst put it more or less the way it is put in the text. One obvious way of maintaining, at new prices, the old standard of living obtained from the old consumption x0 at a money cost p0 x0 is to simply buy the old consumption x0 with new cost p1 x0 . So certainly the current cost of the old standard is at most that amount, and its comparison ratio with the old cost, which might at ﬁrst thought serve as a price index, is at most p1 x0 /p0 x0 , which is the ever-pervasive Laspeyres index. But here common sense breaks in with the proposal that one would not necessarily buy the old bundle at the new prices, one could buy instead another bundle that provides at least the old standard perhaps at a lesser minimum cost, making a comparison ratio with the old cost, the price index P10 , not exceeding the Laspeyres index, P10 ≤ p1 x0 /p0 x0 , and by the same principle P01 ≤ p0 x1 /p1 x1 . But in recollection of primitive price index reversibility which requires P01 = (P10 )−1 this second is equivalent to P10 ≥ p1 x1 /p0 x1 , so common sense together with the primitive reversal provides both Laspeyres and Paasche, and shows them not only as tied together with the same source in principles, but as upper and lower bounds of the price index. In a way, they are not price index formulae like all the others, but fundamental and irrefutable limits for the price index. In that case, of course, the lower bound could not exceed the upper bound, p1 x1 /p0 x1 ≤ P10 ≤ p1 x0 /p0 x0 . Hence, if there is a price index, (PL) p1 x1 /p0 x1 ≤ p1 x0 /p0 x0 . J. R. Hicks (without proving anything) calls the PL-inequality the “Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is

The price index as a utility based concept 231 confusing, but perhaps Hicks was just being fashionable. A case where Paasche turned out to be greater than Laspeyres could anyway be occasion for a pause. It might have been good sense (corresponding to practice of the practical) to abandon the entire theoretical subject after arrival at this point, where the unavoidable primitive has been joined with common sense. But uncountable formulae for the index, good, true, better, super, &c have been proposed. To go further, as it seems we must … Here we conclude with the observation that the PL-inequality is just the condition on the data for it to admit construction of a conical or constant returns utility. Without having even thought of such a constant returns utility at ﬁrst, here it ominously rears … The protesters about such utility should now have difﬁculty to know where to stick their protest. One may wonder if Hicks ever got the scent of constant returns with his “Index Number Theorem”. Now we start again in a similar fashion to take it all further, even to encompasse the entire index subject, or at least the new method. From Ms = ps xs for data to ﬁt the utility with minimum cost as actual cost, and Mr ≤ pr xs because minimum cost at most actual, follows (P)

Prs = Mr /Ms ≤ pr xs /ps xs = Lrs ,

so (M)

Lrs ≥ Mr /Ms ,

and hence (L)

Lr...r ≥ 1

which is just the condition on the data to admit construction of a conical or constant returns utility, and for the solubility of system (M) for price levels, with price indices given by (P) as their ratios. That is the whole story for this book told in one sentence. Or yet again, by common sense, Prs ≤ Lrs . But by the chain property, for any k, Prs = Prk /Psk ,

232 Appendices so Lrs ≥ Prk /Psk , and hence Lr...r ≥ 1 for all r. . .r, which is, again, just the condition on the data to admit construction of a conical or constant returns utility. For a conclusion, ﬁnally, in dealing with a price index on the basis of utility there is no getting away from constant returns. Now another venture to expose the nature of our approach. The undertood objective is a formula for a system of m(m − 1) price indices Prs (r, s = 1, . . . , m) for consistent cost-of-living conversions between m periods. From that consistency, by which conversion from r to s and then s to t coincides with that from r to t, they must be expressible in the form of a system of ratios (P) Prs = Pr /Ps between numbers which, from this representation, have the role of price levels. In other words, equivalently, they have the property Prs Pst = Pst listed among Fisher’s well known “Tests” as the “Chain Test”. For a point of origin let us start with the already considered property (O)

Prs ≤ Lrs

which was termed “common sense”. By substitution of price indices with their expressions as ratios of price levels, we have (L)

Lrs ≥ Pr /Ps

as a system of inequalities for price levels, from which follows the extended chain system (chain L) Lr...s ≥ Pr /Ps and then the cyclical products consistency test (cycle L) Lr...r ≥ 1 is a conclusion.

The price index as a utility based concept 233 For our price index system Formula we now have: price indices given by (P) for price levels that are a solution of (L) There it is, produced out of a hat! As were the one or two hundred formulae in Irving Fisher’s collection. Rather, just the opposite. For here it is seen how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of ‘common sense’. The main concern of Chapter 2 in Part I is with theory of the system of inequalities (L) for the price levels, and computational method. The cyclical product test, that here immediately appears necessary for the existence of a solution, is there (or from the 1960 paper in Part II) seen to be also sufﬁcient. For this formula there has, apparently, been been no dealing at all with utility, let alone constant returns, or at least no mention. But still it might have, or must have, crept in somewhere somehow. Since nowhere else is available the only place to look is the property (O) in the disguise of “common sense”. There is nothing wrong with the common sense part where it is argued that the cost at the new prices pr of having the standard of living provided by consumption xs is at most the actual cost pr xs at those prices, so Mr ≤ pr xs . After all, at that new cost one could even go on with exactly the same consumption as before. Where there is the ﬂavour of utility is the additional submission that Ms = ps xs , to give Prs = Mr /Ms ≤ pr xs /ps xs = Lrs and hence Prs ≤ Lrs , which is (O). For that submission signiﬁes the cost of the consumption coincides with the minimum cost for obtaining the standard of living it provides, or its utility. It is incidental then, since the price index now should be based on the utility, that it must have constant returns, as already argued. This opens the way to intense criticism from some who ignore the complete inseparability of the price index with constant returns that was before just now thoroughly settled a half century ago, in a decadent abandonment of elementary beginnings fundamental to the notion of a price index. For a last word, our multilateral index formula is not one among a possible many that could come to mind, like bilateral formulae that in Irving Fisher’s collection number one or two hundred, but the only one that can properly be entertained. In that way it is not just one possible answer to the Index Number Problem, but the only answer. For here it has been seen, as already said, how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of ‘common sense’.

2

Terminology

Conical v. homogeneous &c Answer to complaints about use of the term conical when everyone uses some other term A ray is a half-line with vertex the origin, and every point lies on just one ray, the ray through it, so a = {at : t ∈ } ⊂ C is the ray through any a ∈ C(in the commodity space). A cone is a set described by a set of rays, and every set has a conical closure, or cone through it, or projecting it, described by the set of rays through its points. Hence

= {xt : x ∈ A, t ∈ } ⊂ C A is the cone through any A ⊂ C. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xλ) = φ (x) λ. Now there will be argument that this term conical for a function has merit in its favour over other terms in use instead, here below in order of preference, ﬁrst two being quite close together. (1) conical, for a function, refers to the graph saying it is a cone, brief, and essential, does not invoke any spurious extra framework (2) constant returns, peculiar language of economics, double output with double input, very good, much preferable to 3-5 but a mouthfull. (3) linearly homogeneous, a mouthfull, also brings into view the spurious original mathematical framework for homogeneous in some degree n, joined with additional information that n = 1, not good (4) homogeneous, means homogeneous for some unspeciﬁed degree n, without telling that actually n = 1, bad (5) homothetic, used sometimes indiscriminately in place of all the above, terrible idea, had some currency in debates decades ago, such as where whether or not utility could be measured had been an issue; and we do measure utility; use and meaning apparently become unsettled, mostly had to do with production originally, for properties spurious to dealings with price indices; some take it to mean simply any monotonic function of a linearly homogeneous function; see for instance Finn R. Forsund, “The homothetic production function”, Swed. J. of Economics 77, 2 (1975). 234-44. Of course were one determined not to be pedantic one could perhaps use (4) but this has one more syllable than (1). We, or at least one of us, will persevere with (1), with random lapses into (2) or even (4). So much for the disapproval dispensed by ‘homothetic’ adherents …

3

Notation

non-negative numbers

n C = x, y, . . . ∈ C n

n B = n p, q, . . . ∈ B

non-negative column vectors commodity space commodity bundles non-negative row vectors budget space price vectors

p ∈ B, x ∈ C ⇒ px ∈ px cost of commodity bundle x at prices p p, x ∈ B × C demand element showing commodity bundle x demanded at prices p φ:C → utility function ρ p, x = min py : φ y ≥ φ x utility cost ρ pr , x / ρ ps , x independent of x price index property 1 ρ p, x = θ p φ x utility cost factorization 2 φ xl = φ x l constant returns 3 1⇔2⇔3 theorem 1⇔3 consequence θ p = minx px / φ x φ x = minp px / θ p θ p φ x ≤ px for all θ p φ x = px for all

conjugate concave conical price and quantity functions p, x x some p and all p some x

pt , xt ∈ B × C, t = 1, . . . , m demand data

pr xs cross-costs ps xs direct-costs Lrs = pr xs / ps xs Lrij...ks = Lri Lij . . . Lks

ratio = Laspeyres index

Lrij...kr = Lri Lij . . . Lkr

chain Laspeyres cycle Laspeyres

Lrs ≥ Pr / Ps

price levels Pt , t = 1, . . . , m

Lr...r ≥ 1

cycle existence test price index = ratio of price levels

Prs = Pr / Ps

Mrs = minij...k Lri Lij · · · Lks M = Lm

derived Laspeyres plus = min

Krs = pr xr / ps xr Paasche −1 = Lsr Hrs = maxij...k Kri Kij · · · Kks derived Paasche = Msr−1 H = Km plus = max Krs ≤ Lrs Hrs ≤ Mrs Krs ≤ Hrs ≤ Mrs ≤ Lrs

Laspeyres-Paasche inequality derived Laspeyres-Paasche inequality

Mrs ≥ Pr / Ps Mrs Msk ≥ Mrk Mrs ≥ Mrk / Msk

price levels Pt , t = 1, . . . , m triangle inequality basic-Laspeyres price levels Pt = Mtk columns of derived Laspeyres M basic-Paasche price levels Pt = Htk columns of derived Paasche H

Mrs ≥ Hrk / Hsk 1 Frs = Hrs Mrs 2 Mrs ≥ Frk / Fsk

Ft =

m

m1

Fisher analogue Fgeometric mean of H and M basic-Fisher price levels Pt = Ftk columns of Fisher analogue F

Ftk

k=1

Mrs ≥ Fr / Fs

mean-basic price levels Pt = Ft , t = 1, . . . , m geomeric mean of m columns of F and of 2m columns of H and M which are a basis for all true price levels and hence all true price indices

Prs = Fr / Fs

basic price indices

Pr Xs ≤ pr xs

for all r, s

Pt Xt = pt xt

determines correspondence between price levels Pt and quantity levels Xt

Jrs = ps xr / ps xs Jrs ≥ Xr / Xs

quantity-Laspeyres quantity levels Xt , t = 1, . . . , m

Xrs = Xr / Xs

quantity indices

4

BASIC computer program BBC BASIC for Windows developed by Richard Russell [email protected].

eside reports done using FORTRAN we include routine outputs from another program, still undergoing development, using BBC BASIC for Windows. This program deals with two text ﬁles kept in folder c:\0\ as here indicated:

B

REM input from C:\0\?-input.txt output to REM C:\0\?-output.txt REM change the ? *SPOOL "C:\0\2-output.txt" F%=OPENIN "C:\0\2-input.txt" (Richard Russell contributed)

must

understand

these

lines

which

he

For instance 2-input.txt looks like 4, 4 1.0000000, 0.8976280, 0.9130180, 0.7475160,

1.1218730, 1.0000000, 0.9792190, 0.8122070,

1.1829370, 1.0418520, 1.0000000, 0.8138330,

1.5008030 1.3498590 1.2661740 1.0000000

which tells it is a 4×4 matrix and then tells the elements, the comma “,” being the delimiter. As for 2-output.txt, which need not even exist initially and if it does any contents will be overwritten, it recieves the output when the program is run, which with this input is as follows—showing reassuring agreement with earlier FORTRAN ﬁgures. When this program is compiled and so not available for alteration, it will refer to two ﬁles called input.txt and output.txt always with these names though they can have different applications. This compiled version will be supplied to anyone wishing to use it. A later version will start instead with two m×n and n×m matrices P0 and X0 and then form the m×m cross-cost matrix C0=P0.X0 and then, dividing columns by diagonal elements, the m×m Laspeyres matrix L. These diagonal elements have to be retained in or to convert price levels to quantity levels, from the equation PX=px. We acknowledge with thanks the guidance received from Richard Russell, longtime developer of this rendering of BASIC. http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html, http://www.rtrussell.co.uk/, [email protected].

244 Appendices REM Price-Level Computation: PLC.bbc REM BBC BASIC for Windows - guidance from Richard Russell REM http://www.compulink.co.uk/~rrussell/bbcwin/bbcwin.html REM The Power Algorithm: + means min @% = &90F :

REM Set numeric formatting

REM input from C:\0\?-input.txt output to C:\0\?-output.txt REM change the ? *SPOOL "C:\0\?-output.txt" F%=OPENIN "C:\0\?-input.txt" m=FNread(F%) REM n=FNread(F%) REM DIM P0(m, n), X0(n, m) DIM L(m, m), C0(m), M(m, m), N(m, m, m), H(m, m), F(m, m) DIM G(m), P(m, m), C(m), D(m), E(m, m), V(m) DIM XL(m), X(m, m) REM Read P0 and X0 REM Determine C0=P0*X0 PRINT "Read C0(m)" FOR i=1 TO m C0(i)=FNread(F%) NEXT i REM Read L (or Form L from C0) and make M=L FOR i=1 TO m FOR j=1 TO m L(i, j)= FNread(F%) : M(i, j)=L(i, j) NEXT j NEXT i REM Illustration - change the ? PRINT "Illustration ?" : PRINT REM Print L PRINT "L Laspeyres" FOR i=1 TO m FOR j=1 TO m PRINT ;L(i, j),; NEXT j : PRINT NEXT i

1

REM M=M*L print M and paths FOR g=2 TO m fd=0 FOR i=1 TO m FOR j=1 TO m: h=1 : N(i,j, 1)=i FOR k=1 TO m IF M(i, k)*L(k,j)<M(i, h)*L(h, j) THEN h=k ENDIF NEXT k : M(i, j)=M(i, h)*L(h,j): N(i,j, g)=h IF hj THEN fd=1 ENDIF NEXT j

BASIC computer program 245 NEXT i : PRINT REM PRINT "fd="; fd IF fd = 0 THEN PRINT "Final power" PRINT "L power " ; g; IF g=m THEN PRINT " = M derived Laspeyres" ELSE PRINT FOR i=1 TO m FOR j=1 TO m PRINT ;M(i, j),; NEXT j: PRINT NEXT i NEXT g: PRINT PRINT "Paths" FOR i=1 TO m FOR j=1 TO m FOR g=1 TO m PRINT ;N(i,j,g) ","; NEXT g : PRINT ;j ,; NEXT j : PRINT NEXT i: PRINT fi=0 FOR i=1 TO m IF M(i,i)V(g) THEN e=V(g) NEXT g: PRINT PRINT "Critical cost-efficiency " ; e

ENDIF

246 Appendices PRINT PRINT "L adjusted Laspeyres to replace original L" FOR i=1 TO m FOR j=1 TO m IF i j THEN L(i,j)=L(i,j)/e ENDIF M(i,j)=L(i,j): PRINT ;L(i, j),; NEXT j: PRINT NEXT i: PRINT ENDIF IF fi=1 THEN GOTO 1 PRINT "H derived Paasche" FOR i=1 TO m FOR j=1 TO m H(i,j)=1/M(j,i): PRINT NEXT j: PRINT NEXT i :PRINT

;H(i, j),;

PRINT"The " ;2*m;" canonical price-level systems - the " ; 2*m; "columns of M and H" :PRINT PRINT "F derived Fisher - mean of derived Laspeyres M and derived Paasche H" FOR i=1 TO m FOR j=1 TO m F(i,j)= SQR(H(i,j)*M(i,j)): NEXT j: PRINT NEXT i :PRINT

PRINT ;F(i, j),;

PRINT "P mean canonical price-level: mean of columns of F" FOR i=1 TO m: g=1 FOR j=1 TO m: g=g*F(i,j) NEXT j: g=EXP(LN(g)/m): :G (i)=g: PRINT ;g NEXT i :PRINT PRINT "P/P mean canonical price-index" FOR i=1 TO m FOR j=1 TO m P(i,j)=G(i)/G(j): PRINT ;P(i,j),; NEXT j: PRINT NEXT i :PRINT PRINT "X mean canonical quantity-level: PX = px" FOR i=1 TO m XL(i)=C0(i)/G(i): PRINT ;XL(i) NEXT i: PRINT PRINT "X/X mean canonical quantity-index" FOR i=1 TO m FOR j=1 TO m X(i,j)=XL(i)/XL(j): PRINT ;X(i,j),; NEXT j: PRINT NEXT i :PRINT *SPOOL

BASIC computer program 247 END DEF FNread(F%) PRIVATE A$ LOCAL C% C%=INSTR (A$,",") IF C%=0 THEN INPUT#F%, A$ IF ASC (A$)=10 A$=MID$ (A$,2) ELSE A$=MID$ (A$,C%+1) ENDIF =VAL (A$)

Notes 1

Multilateral indices

2

Chain indices

1 Multilateral indices Matthijs van Veelen (2002), the opening paragraph: “Comparing wealth of nations—or across years—is something an economist naturally wants to do. The classic problem we then face, and which the standard GDP ignores, is that prices are not necessarily the same in different countries. Trying to correct for this, quite a few methods of comparison have been designed, but index theorists have always been aware that all of them have their drawbacks, which for instance is reﬂected in overviews of Balk (1996) and Hill (1997). As long as we only want to compare two countries we do have fairly acceptable index numbers, but all known multilateral indices suffer from unwanted peculiarities. A natural question is then whether we can expect a perfect method ever to be discovered. The statement of this paper is that this infallible method does not exist and that the apples and oranges character of multilateral comparison is more fundamental.” With reference to the part in italics we should want to know what is the essential difference betweeen two and many. We of course deal with the classical index problem where difference between dealing with times or countries has no part. What is meant by “infallible”? Here is the continuation following remarks to do with axiomatics: “… it is generally acknowledged that the step from bilateral to multilateral comparison creates a serious problem. In spite of this general awareness, I do think that this impossibility theorem is a contribution, since so far it has never been proved that there is no method of multilateral comparison behaving in a way we can reasonably expect it to do.” What could be that “serious problem” beside failure to make the step? We have made the step as for the classical index problem, with a conclusion about uniqueness and inevitability, stated at the end of Appendix 1: “… our multilateral index formula is not one among a possible many that could come to mind, like bilateral formulae that in Irving Fisher’s collection number one or two hundred, but the only one that can properly be entertained.” If there is a problem with the axiomatics its too bad for the axiomatics (with all respect to the author). “What we are looking for is a method to be applied to a group of countries that gives each of those countries a single number, compressing the information contained in the [price and quantity data] matrices to a vector, entries of which can easily be compared.”

Notes 251 Concerning what they seek see Appendix 3 on Notation, where coming after original “demand data” there is arrival at nothing other than “quantity levels”, representing just what is needed. We have done the impossible! Anything wrong? From unsympathetic reviewers it appears unbearable, but apart from that. John Quiggin and Matthijs van Veelen (2007) state: “The revealed preference approach starts from the assumption that the preferences of the representative conumer are the same in all […] countries we consider. […] Consequently it is natural that data from a third country should affect whether country 1 ranks higher than country 2 or vice versa. […] The aximoatic approach, on the other hand, allows for all countries to have a representative utility function of their own. […] When comparing consumption bundles in two countries, prices and quantities from a third country are therefore not considered informative and it is considered undesirable if they make a difference for how the ﬁrst two countries rank relative to each other.” Whereas with the former approach all the data points are chained together in order to form a multilateral index, with the latter approach the single pairs of data points are used independently from the others in constructing bilateral indices, which generally turn out to be inconsistent among each other in a multilateral setting. The current bilateral-multilateral terminology is not doing good service for understanding what is being said. A bilateral index must be a price index for two points taken in isolation from others, depending on data just for those points. A multilateral index would refer to some m points and consist of a system of m(m − 1) price indices, one associated with each pair. The system can depend on the data for all m points. That at ﬁrst may seem to allow the individual indices of the system to be ordinary bilateral indices. That would be easy, any one of the available one or two hundred fromulae being, as far as one may know, admissible. Then what has all the trouble been about? If there is still a question then a return to beginnings would be suitable. Let Prs denote the system price index from base period s to current period r. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. Consider three periods r, s, t where appropriate expenditures maintain the same standard of living, say expenditure Mi in period i. It is of course required that transitions from t to s followed by s to r must have the same outcome as the direct

252 Notes transition from t to r. That requires (1)

Prs Pst = Prt

which, were we dealing with a bilateral formula, which we are not, reﬂects the “Chain Test” of Irving Fisher, but now merely deﬁnes the consistency of the price index system. Alternatively, by ﬁxing any one value of Mi all the others are determined, and so we have a system of numbers in the role of price levels from which the prices indices of the system are derived as their ratios, (2)

Prs = Mr /Ms .

Obviously (2) ⇒ (1), also from (1), for any k, Prs = Prk /Psk so we have (2) with Mi = Pik , showing that (1) ⇒ (2), and hence (1) ⇔ (2). Here then is the notion for a multilateral price index system: just a consistent system of price indices. The problem had been in the matter of consistency. The celebrated one or two hundred formulae in Irving Fisher’s collection have been notorious for refusing the Chain Test; it would be quite unsuitable to have anything to do with those. Since consistent price indices can be derived as ratios of price levels, to be safe (and more economical) it would be better ﬁrst to deal directly with those (m in number instead of m(m − 1)). That is our method, argued as the one and only method for producing a multilateral price index system.

Bibliography Balk, B. M. (1996). A Comparison of Ten Methods for Multilateral International Price and Volume Comparison. Journal of Ofﬁcial Statistics 12, 199–222. Diewert, W. E. (1986). Microeconomic Approaches to the Theory of International Comparisons. TechnicalWorking Paper No. 53, NBER, Cambridge, MA. — (1999). Axiomatic and Economic Approaches to International Comparisons. In A. Heston and R. E. Lipsey (eds.), Internationaland Interarea Comparisons of Income, Output and Prices, 13–87. University of Chicago Press. Drechsler, L. (1973). Weighing of Index Numbers in Multilateral International Comparisons. Review of Income and Wealth 19, 17–34. Fisher, I. (1922). The Making of Index Numbers. Boston: Houghton Mifﬂin. Geary, R. C. (1958). A Note on the Comparison of Exchange Rates and Purchasing Power Between Countries. Journal of the Royal Statistical Society, Series A, 121, 97–9. Hill, R. J. (1997). A Taxonomy of Multilateral Methods for Making International Comparisons of Prices and Quantities. Review of Income and Wealth, 43, 49–69. Khamis, S. H. (1972). A New System of Index Numbers for National and International Purposes. Journal of the Royal Statistical Society, Series A, 135, 96–121.

Notes 253 Neary, J. P. (1999). True Multilateral Indexes for International Comparisons of Real Income: Theory and Empirics. Mimeo, University College Dublin. Quiggin, John and Matthijs van Veelen (2007). Multilateral indices: conﬂicting approaches? Review of Income and Wealth 53, 2 (June), 372–8. Sen, A. K. (1982). Real National Income. In Choice, Welfare and Measurement. Oxford: Basil Blackwell, 388–415. van IJzeren, J. (1956). Three Methods of Comparing the Purchasing Power of Currencies. Statistical Studies No. 7. Zeist: Netherlands Central Bureau of Statistics/De Haan. van Veelen, Matthijs (2002). An impossibility theorem concerning multilateral international comparison of volumes. Econometrica 70, 1 (January), 369–75.

2 Chain indices The method of chaining index numbers dates back at least to Lehr (1885) and Marshall (1887), who dealt with the problem of new and disappearing commodities within the examined time period. In contemporary debate (see, for example, von der Lippe, 2001 and Balk, 2004), chaining has been also considered to reduce the spread between the Laspeyres and Paasche indices, which are expected to delimit the interval of possible numerical values of the, as it were, unknown true index— whatever that should mean. Dale W. Jorgenson and Zvi Griliches (1971, p. 228), with reference to their previous work (1967), remarked: “The main advantage of a chain index is the reduction of errors of approximation as the economy moves from one production conﬁguration to another. […] The Laspeyres approximation to the Divisia index of total factor productivity was employed in our original study of productivity change”. Samuelson and Swamy (1974, p. 576), say “Fisher missed the point made in Samuelson (1947, p. 151) that knowledge of a third situation can add information relevant to the comparison of two given situations.” (See also Samuelson 1984.) The “Minimum Spanning Tree” method, although it treats minimum distance in the structure of weights, does not pursue minimum chain Laspeyeres (or maximum chain Paasche) indices. See Robert J. Hill (1999a), (1999b), (2001), (2004). Here is an outline of the already accounted chain index method. The most rudimentary idea is that if we have Prs ≤ Lrs and Pst ≤ Lst then, by multiplication, Prs Pst ≤ Lrs Lst , so that, subject to the consistency requirement Prs Pst = Prt , it follows that Prt ≤ Lrs Lst , and similarly Prs ≤ Lr...s , where … is any chain. As bearing on some applications, this may be a summary of the method. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

Notes 255 the cycle test Ls...t...s ≥ 1 is equivalent to (chain LP) Ks...t ≤ Ls...t for all possible chains … and … taken independently. Hence, introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

subject to the conditions required for their existence, for which Hst = (Mts )−1 , this is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst showing the relation of the LP-interval and the narrower derived version that involves more data. Here it has been recognized that from Kts = (Lst )−1 follows −1 Kt...s = Ls...t and therefore −1 Hts = max... Kt...s = min... Ls...t = (Mst )−1 , where in each case … is understood so the chain s . . . t is the reverse of t . . . s. The following is to expose the nature of our approach and the part that chain indices have in it. The objective is a formula for a system of price indices Prs (r, s = 1, . . . , m) for consistent cost-of-living conversions between m periods. From that consistency, by which conversion from r to s and then s to t coincides with that from r to t, they must be expressible in the form of a system of ratios (P)

Prs = Pr /Ps

between numbers which, from this representation, have the role of price levels. In other words, equivalently, they have the property Prs Pst = Pst listed among Fisher’s well known “Tests” as the “Chain Test”.

256 Notes For a point of origin let us start with the already considered property (O)

Prs ≤ Lrs ,

which is the beginning, and at the same time not far from the end, of price index theory. By substitution of price indices with their expressions as ratios of price levels, we have (L)

Lrs ≥ Pr /Ps

as a system of inequalities for price levels, from which follows the extended chain system (chain L)

Lr...s ≥ Pr /Ps

and then the cyclical products consistency test (cycle L)

Lr...r ≥ 1

is a conclusion. For our formula we now have: price indices given by (P) for price levels that are a solution of (L) For any solution of (L) for price levels, and hence for the price indices given by (P), we now have Kst ≤ Hst ≤ Ps /Pt ≤ Mst ≤ Lst directly, and hence Kst ≤ Hst ≤ Pst ≤ Mst ≤ Lst showing the relation of the LP-interval for bounding the price index and the narrower derived version that involves more data. It is the idea represented here, of narrowing the price index bounds, that has been taken up with interest, lifted out of the context in our approach. Firstly, in that context, it depends on the cyclical product test Lr...r ≥ 1, that assures the new bounds, hopefully narrower, are not so narrow as to leave an empty set, or even not narrower at all, but wider. A further point is that the price index to be bounded is not any price index, but one from a chain consistent system, as obtained by our approach. That may be discouragement enough, but there are further points relevant to some applications, where the chains are not min or max but seemingly arbitrary, leaving signiﬁcance even more unanchored.

Notes 257

Bibliography Balk, Bert (2004). Direct and Chained Indices: A Review of Two Paradigms. Presented at the SSHRC International Conference on Index Number Theory and Measurement of Prices and Productivity, Vancouver, 30 June-3 July 2004. Dowrick, Steve and John Quiggin (1994) International Comparisons of Living Standards and Tastes: A Revealed-Preference Analysis. American Economic Review 84 (1), 332–41. — (1997) True Measures of GDP and Convergence. American Economic Review 87 (1), 41–64. Hill, Robert J. (1999a), Comparing Price Levels across Countries Using Minimum Spanning Trees. Review of Economics and Statistics 81 (1), 135–42. — (1999b). International Comparisons using Spanning Trees. In A. Heston and R.E. Lipsey (eds.), International and Interarea Comparisons of Income, Output, and Prices, Studies in Income and Wealth Volume 61, NBER, University of Chicago Press. — (2001a). Measuring Inﬂation and Growth Using Spanning Trees. International Economic Review 44 (1), 145–62. — (2001b). Linking Countries and Regions using Chaining Methods and Spanning Trees. Joint World Bank-OECD Seminar on Purchasing Power Parities Recent Advances in Methods and Applications, Washington, D.C., 30 January–2 February 2001. — (2004). Constructing Price Indexes across Space and Time: The Case of the European Union. American Economic Review 94 (5), 1379–1410. — (2006). When Does Chaining Reduce the Paasche-Laspeyres Spread? An Application to Scanner Data. Review of Income and Wealth 52 (2), 309–25. Jorgenson, Dale W. and Zvi Griliches (1967). The Explanation of Productivity Change. Review of Economics Studies 34, 249–83. — (1971). Divisia Index Numbers and Productivity Measurement. Review of Income and Wealth 17 (2), 53–55. Lehr, J. (1885). Beiträge zur Statistik der Preise. Frankfurt: J.D. Sauerlander. Lippe, P. von der (2001). Chain Indices: A Study in Price Index Theory. Spectrum of Federal Statistics, Volume 16. Stuttgart: Metzler-Peschel. Manser, Marilyn E. and Richard J. McDonald (1988). An Analysis of Substitution Bias in Measuring Inﬂation, 1959–85. Econometrica 56 (4), 909–30. Marshall, A. (1887). Remedies for Fluctuations of General Prices. Contemporary Review 51: 355–75. Neary, Peter (2004). Rationalizing the Penn World Table: True Multilateral Indices for International Comparisons of Real Income, American Economic Review 94 (5), 1411–28. Samuelson, P. A. (1947). Foundations of Economic Analysis. Cambridge, MA: Harvard University Press. — (1984). Second Thoughts on Analytical Income Comparisons. Economic Journal 94: 267–78. — and S. Swammy (1974), Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis, American Economic Review 64 (4): 566–93.

Postscript

review document that has reference to our Chapter 2, the main chapter in this book, does service in provoking—what seems suitable—a review of the review. But now it seems perhaps good practice anyway to give deserved illumination to a review. According to this document: the method “shows in essence that imposing additional information improves the precision of estimated cost-of-living indices.” Hardly the essence. Our method is to simultaneously construct the many price indices between many periods that are both consistent and true. The matter of improved bounds using chain indices is a simple by-product, and a useless one when taken apart from the proper framework, as it usually is. These indices are consistent, in the chain sense, in that exchanging money from a to b and then b to c should amount to the same as from a to c. That is required by the nature of price indices. Fisher had something to do with all that, but he had a ﬁxation, like everyone else, that a single algebraical formula was to apply to all the price indices in the usual way and obey his Chain Test—a hopeless undertaking. They are true, in the utility sense, taken together, meaning they are all associated with a single utility that ﬁts the data. The facile statement with use of terms ‘precision’ and ‘estimated’, as if we should know what they should now mean, would in another setting be interestingly problematic but here it is completely empty. The last paragraph in the document asserts we confuse two meaning of “true”. One is the usual meaning of the theoretical subject where the index derives from a utility that ﬁts the data. The one good word is replace by a rambling rigmarole to make their different meaning different. For the other meaning, if the utility is based on a certain function form then a certain index is true, in the ﬁrst sense. But function form has no welfare signiﬁcance whatsoever, so an index with this sense of true is quite undistinguished. The confusion is certainly ours, and we welcome it. This document, beside a mentioned AER paper, which they complain is cited but not discussed, is classic in ignoring our method, and disregard of the discipline required for their main interest, the formation of chain indices. Discussion of that paper seems after all not now out of place.

A

260 Postscript Here are comments about the mentioned “Ideal Afriat Index” in the AER paper. Of course any way there should not be the association of an individual with a formula completely unknown to them. This formula goes backwards to the Irving Fisher way. Like the Fisher “Ideal”, when computed between several periods, produces price indices in no way having chain consistency, as stated by Fisher’s “Chain Test”, let alone be true together. In our method we give account of a formula for determining simultaneously all the price indices between many periods, both consistent and true as usual, which reduce to Fisher’s index in the case of two periods. We use this result, picked out of the normal indeterminacy range for many periods, as a ﬁnal answer, just as one might pick the Fisher index out of the indeterminacy range represented by the Laspeyres-Paasche interval. This could be, maybe, a sort-of “Ideal Afriat Index”, but now going forwards instead of backwards. We used to cite those authors properly out of respect for having used their data to play with during software development, but that turned out to be a great mistake. For it brought them to attention and caused their advice to be sought about publication of our work. That those before others more qualiﬁed should be chosen, who bring out the ﬂavour of a decadence as its quintessential embodiment, would be a disgrace to professional economics were not our particular subject unfamiliar to most, so they can be easy victims of the overloaded hostile posturing of these ignorant and arrogant would-be authorities. Everywhere in this document there is something wrong, but this is attention enough, though it has been useful for provoking corrective emphasis about interesting points. It is as well to conclude with observation that an understanding of the very ﬁrst things about price indices is manifestly absent—very odd for someone being consulted as an authority. It was settled a half-century ago that dealing with a price index in terms of utility requires that utility to be conical, or linearly homogeneous (see Appendix 2 about terminology). Our Appendix 1 deals with that point. Yet the received document goes on preaching about this being an unwarranted extra assumption. It also presses for use of the entirely unsuitable term “homothetic”— because others use it! Our Appendix 2 deals with that matter. Here it is suitable to express appreciation and thanks to our publisher who— unlike a quite likely well-meaning journal editor—put aside another out-to-kill ‘review’ in favour of others we have been very happy to receive and to quote; our thanks go to those also.

Index

Afriat, S.N. xviii, 5, 7, 10, 13, 16–17, 22, 40, 43–4, 111 et seq, 121 et seq, 144, 149, 223 Allen, Robert C. 160 Allen, Sir Roy 7 associated price indices 13, 226–7 Bainbridge, S. 40, 153 barycentric coordinates 37 “basic” price indices 36, 54–5, 65–7, 74, 81 Bernstein-Weierstrauss approximation 164 Blackorby, Charles 171 Box, G.E.P. 181–2 Box-Cox function 182–3, 187–91 Brown, Alan 16 budget constraints 44, 56, 58, 132, 199 Byushgens, S.S. 17, 159 Caves, Douglas W. 21, 166, 176–7, 188 chain coefﬁcients 114–15, 135 chain indices 9–10, 29–32, 73, 254–6 chain test 5–6, 27–8, 41, 98, 232, 252, 255–6, 259–60 Christensen, Laurits R.166, 176–7, 188 circularity test 6, 28 citation analysis 17–18 compatibility 133–8, 146–9; homogeneous 136–7, 148; strict 141 cones and conical functions 42, 235 conical revealed preference 43 conical utility 9–10, 13–16, 43, 46–7, 53–5, 130, 132, 136, 228, 231, 260 consistency 38–42, 57–8, 73, 76–7, 121–2, 125, 129, 133–42, 252, 259; classical 145; homogeneous 133–5, 139, 142 constant returns to scale 233 consumption error 44 cost effective demand 12–13, 44, 133, 146 cost efﬁcacy 57, 59

cost efﬁcient demand 12–13, 37, 42–7, 56–64, 75–6, 84, 133, 146 cost-of-living indices 12, 259 cost-of-living problem 225–6 Cox, D.R. 181–2 critical cost efﬁciency 47, 56, 60, 75–6 critical cost functions 149 cycle reversibility 122–4 cyclical non-negativity condition 150, 152 cyclical product test (or cycle test) 30–4, 37–40, 43, 49, 73 Deaton, Angus 21–2 demand correspondence 55–8 Denny, Michael 184, 188 derived systems 33–9, 54, 116–17, 120, 151–2 Diewert, W.E. 7–8, 17, 21–3, 159–69, 176–9, 184–96 Divisia indices 172–3, 254 Edmunds, J. 40, 144 efﬁciency modiﬁcation 47 Ehemann, Christian 191 Eichhora, W. 129 EUKLEMS Project xvii, 70, 94 Euler’s theorem 163, 188–9, 194, 200 “exact” index numbers 159, 175, 196–7 exhaustion property 39 extension property 37–41, 54, 119–21, 134, 151 ﬁnite consistency tests 142–3 Fisher, Irving xviii–xix, 7–8, 16, 18, 21, 23, 132, 159–61, 259–60; tests for index number formulae xvii, 4–6, 16, 26–34, 41, 66, 71, 98, 129–30, 232–3, 252, 255; see also “ideal” index Fleetwood, William 6, 28

262 Index Førsund, Finn R. 17–18 Fostel, A. xviii Fusa, Melvyn 164

Nissen, David 171 non-parametric formulae xix, 71 normal demand 135 notation 238–9

Griliches, Zvi 254 order exhaustion property 152 Hicks, J.R. 11, 14, 30, 227, 230–1 Hill, Robert J. 160, 197 homothetic separability 170–2, 185–6, 192 Hotelling, Harold S. 200 Houthakker, H.S. 42, 58, 114, 141–3 Hulten, Charles R. 172 “ideal” index 5–9, 16–22, 26–7, 53, 63–6, 72, 78, 81, 129, 159–60, 191–3, 260 idempotence 35–6, 40, 100 identity test 5–6, 26 index number problem xvii, xviii, 4–5, 16, 26, 66, 70, 233 index number theorem 11, 14, 30, 133–4, 227, 230–1 indicators 187 International Comparison Project 70, 77 interval exhaustion property 152 interval test 34 Jorgenson, Dale W. 254 Konüs-Byushgens indicator 185, 190–1 Lady, George 171 Lagrange multiplier 199–200 Laspeyres indices xix, 7–11, 14–15, 20, 29–30, 46, 62–3, 71, 77–9, 97, 131–5, 163, 172, 230 Lau, Lawrence J. 184–5 Lehr, J. 254 linear programming methods 144, 147 McFadden, Daniel 164 Malmquist, S. 21–2 Marshall, A. 254 median solutions 123–4 Milana, Carlo xviii, 8, 10, 21, 23, 157 et seq minimal chains 31–2, 115–16 “minimum spanning tree” methodology 254 monotonicity 40, 132–3, 180 Morrison, Catherine J. 178–9 Muellbauer, J. 21–2 multilateral indices 250–2 Mundlak, Yair 164

Paasche indices xix, 7–11, 20, 29–30, 63, 71, 78, 133–4, 163, 172, 230 perfect efﬁciency 60 power algorithm 40, 54, 75, 153–4 preference orders 113–14 production functions 198–9 purchasing power of money 5, 27 quadratic approximation 166–9 quadratic utility 17, 22–3, 64, 159 quantity and price, duality between 74, 229 Quiggin, John 251 ratio matrices 99–100 ratio test 6 rays 42 rectiﬁcation of pair comparisons 129 Reinsdorf, Marshall B. 191 returns to scale 199, 233 revealed contradictions 138–40 revealed preference 9, 11, 16, 42–3, 56–8, 114, 137–8, 143, 159, 251; strong axiom of 141 reversal test 129 Russell, R. Robert 171 Russell, Richard 241 et seq Samuelson, Paul A. 22, 42, 44, 58, 159, 163, 254 Sarafoglou, Nikias 17–18 Scarf, H.E. xviii SchuItz, H. 159 separability: strong 174; weak 173–4; see also homothetic separability Shephard, Ronald W. 163, 171 simple systems 125 simplex of reference 16, 29, 37, 55 standard of living 4–5, 10, 12, 26–7, 43–4, 130, 225–6, 230, 233, 251 Stigler, George J. 17–18 strict compatibility 57 “superlative” index number formulae 7–8, 17–23, 64, 159–61, 166–9, 175, 193–7 Swamy, S. 22, 44, 254

Index 263 time reversal test 5–6, 16, 26–7 Todd, M.J. xviii Törnqvist-type indices 18–21, 160, 188–9 translog identity 176 triangle equality 99–100, 120 triangle inequality 34–40, 51–5, 98, 118–20, 151–3 “true” price indices 14–18, 23, 47, 53, 64–5, 72, 133, 198, 259–60 utility 12–15, 43–7, 56, 67–8, 130–6, 199–200, 225–8, 233, 259; construction of 144–5; and demand 136–7; see also conical utility; quadratic utility

utility cost factorization theorem xvii–xviii, 45–9, 53, 70, 226 utility cost functions 145–8, 225, 227 van Veelen, Matthijs 250–1 Vartia, Yrjo 7 Voeller, J. 129 Walsh index numbers 7, 18–23, 160, 190 Wold, Herman 200 Young’s theorem 164, 194

A

theft amounting to £1 was a capital offence in 1260, and a judge in 1610 afﬁrmed the law could not then be applied since £1 was no longer what it was. Such association of money with a date is well recognized for its importance in very many different connections. Thus arises the need to know how to convert an amount at one date into an equivalent amount at another date. In other words, a price index. The longstanding question concerning how such an index should be constructed is known as ‘The Index Number Problem’. A pervasive long established institution for economics, it is a number issued by the Statistical Ofﬁce that should tell anyone the ratio of costs of maintaining a given standard of living in two periods where prices differ. For a succession of three periods, the product of the ratios for successive pairs must coincide with the ratio for the endpoints. This is the chain consistency required of price indices. A usual supposition is that the index is determined by an algebraical formula involving price and quantity data for the two reference periods, as with the one or two hundred formulae in the collection of Irving Fisher, always joined with the question of which one to choose, and the perplexity that chain consistency is not obtained with any. Hence ﬁnally they should all be abandoned. This is the reality of ‘The Index Number Problem’. Now in this book consistent prices indices for any number of periods are all computed together to make a resolution of the ‘Problem’, proved unique hence never to be joined by others to make a Fisher-like proliferation. That brings to a conclusion an issue giving rise to extensive thought and theory to which over the decades a remarkable number of economists have each contributed a word, or volume. A product of attention to the ‘Problem’ over a half-century, this book should be of interest to all those for whom ‘The Index Number Problem’ remains, beside of perpetual practical concern, a source of fascination. S. N. Afriat, resident of Siena, intermittent adjunct at the University, Mathematics and Economics, permanent Visiting Professor. Carlo Milana, Research Director at ISAE (Istituto di Studi e Analisi Economica), Rome, a public research institution supported by the Italian Ministry of Economy and Finance.

A

friat is the guru of the price index … his work is a classic illustration of how much we learn from new ways of thinking. Angus Deaton Dwight D. Eisenhower Professor of International Affairs Professor of Economics Princeton University, USA

R

esearchers interested in price indexes, inter-area comparisons, revealed preference theory, or productivity measurement will ﬁnd this book an enlightening, entertaining, and highly creative treatise on where the theory should go from here. Marshall Reinsdorf Senior Research Economist US Bureau of Economic Analysis

S

ydney Afriat is famous for his unique and penetrating insights, often very unsettling to those who have worked long and hard in a ﬁeld—without ever seeing what is obvious to Sydney, who then formulates it neatly in very compact mathematics. In this case he has a very good co-author who has independently developed a critical examination of many current writings on the subject, and has helped him develop his insights in practical directions. This monograph will very likely render many current discussions of index numbers obsolete. Edward Nell Malcolm B. Smith Professor of Economics New School University, NY

S

ydney Afriat belongs to that select group of economic theorists who have become a legend in their own times. There is such a thing as “Afriat’s Theorem”, which has become part of the staple for students of microeconomic theory … Moreover, he may belong to another select group of prose stylists who are also masters of some aspects of the mathematical method and its philosophy. K. Vela Velupillai Journal of Economics March 2004

N

ew generation price index workers can have pleasure to escape stagnation in the Irving Fisher trap and catch on to the new approach with its conceptual ﬁdelity and computational elegance. Preface

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70 Labour Theory of Value Peter C. Dooley 71 Capitalism Victor D. Lippit 72 Macroeconomic Foundations of Macroeconomics Alvaro Cencini 73 Marx for the 21st Century Edited by Hiroshi Uchida

82 Globalization and the Myths of Free Trade History, theory and empirical evidence Edited by Anwar Shaikh 83 Equilibrium in Economics: Scope and Limits Edited by Valeria Mosini

84 Globalization State of the art and perspectives Edited by Stefan A. Schirm 85 Neoliberalism National and regional experiments with global ideas Edited by Ravi K. Roy, Arthur T. Denzau, Thomas D. Willett 86 Post-Keynesian Macroeconomics Economics Essays in honour of Ingrid Rima Edited by Mathew Forstater, Gary Mongiovi and Steven Pressman 87 Consumer Capitalism Anastasios S. Korkotsides 88 Remapping Gender in the New Global Order Edited by Marjorie Grifﬁn Cohen and Janine Brodie 89 Hayek and Natural Law Eric Angner 90 Race and Economic Opportunity in the Twenty-First Century Edited by Marlene Kim 91 Renaissance in Behavioural Economics Harvey Leibenstein’s impact on contemporary economic analysis Edited by Roger Frantz 92 Human Ecology Economics A new framework for global sustainability Edited by Roy E. Allen 93 Imagining Economics Otherwise Encounters with identity/difference Nitasha Kaul 94 Reigniting the Labor Movement Restoring means to ends in a democratic labor movement Gerald Friedman

95 The Spatial Model of Politics Norman Schoﬁeld 96 The Economics of American Judaism Carmel Ullman Chiswick 97 Critical Political Economy Christian Arnsperger 98 Culture and Economic Explanation Economics in the US and Japan Donald W. Katzner 99 Feminism, Economics and Utopia Time travelling through paradigms Karin Schönpﬂug 100 Risk in International Finance Vikash Yadav 101 Economic Policy and Performance in Industrial Democracies Party governments, central banks and the Fiscal-Monetary policy mix Takayuki Sakamoto 102 Advances on Income Inequality and Concentration Measures Edited by Gianni Betti and Achille Lemmi 103 Economic Representations Academic and everyday Edited by David F. Ruccio 104 Mathematical Economics and the Dynamics of Capitalism Goodwin’s legacy continued Edited by Peter Flaschel and Michael Landesmann 105 The Keynesian Multiplier Edited by Claude Gnos and Louis-Philippe Rochon 106 Money, Enterprise and Income Distribution Towards a macroeconomic theory of capitalism John Smithin

108 Fiscal Decentralization and Local Public Finance in Japan Nobuki Mochida 109 The ‘Uncertain’ Foundations of Post-Keynesian Economics Essays in exploration Stephen P. Dunn

110 Karl Marx’s Grundrisse Foundations of the critique of political economy 150 years later Edited by Marcello Musto 111 Economics and the Price Index S.N. Afriat and Carlo Milana

Economics and the Price Index

S.N. Afriat Carlo Milana foreword Angus Deaton

First published 2009 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Ave, New York, NY 10016 Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2008. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”

© 2009 by S. N. Afriat and Carlo Milana Typeset in Times New Roman by Keyword Group Ltd Printed and bound in Great Britain by MPG Books Ltd, Bodmin All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Afriat, S. N., 1925– Economics and the price index / S.N. Afriat and Carlo Milana. p. cm. Includes bibliographical references and index. 1. Price index. 2. Index numbers (Economics) HB225.A326 2008 338.5 28–dc22 2008009726 ISBN 0-203-89148-1 Master e-book ISBN

ISBN10: 0-415-47181-8 (hbk) ISBN10: 0-203-89148-1 (ebk) ISBN13: 978-0-415-47181-7 (hbk) ISBN13: 978-0-203-89148-3 (ebk)

Contents

Foreword Preface Acknowledgements

xv xvii xxi

PART I

Concept and Method

1

1

The Super Price Index: Irving Fisher, and after

3

2

The Price Level Computation Method

25

3

Price Level Computation: Illustrations

69

Bibliography

101

PART II

Precursor 1

The system of inequalities ars > Xs − Xr

109 111

S. N. AFR I AT, Research Memorandum No. 18 (October 1960)

Econometric Research Program, Princeton University Proc. Cambridge Phil. Soc. 59 (1963)

2

On the constructibility of consistent price indices between several periods simultaneously S. N. AFR I AT, in Essays in Theory and Measurement of Demand:

in honour of Sir Richard Stone edited by Angus Deaton, Cambridge University Press, 1981

127

xiv Contents

3

The theory of exact and superlative index numbers revisited

157

C AR L O M I L ANA, Istituto di Studi e Analisi Economica, Roma

EUKLEMS Working Paper No. 3, 2005 http://www.euklems.net.

Appendices

221

1

223

The price index as a utility based concept 1.1 1.2

2

Original approach 225 New approach 230

Terminology

235

Conical v. homogeneous &c

3

Notation

237

4

BASIC computer program

241

BBC BASIC for Windows developed by Richard Russell [email protected].

249

Notes 1

Multilateral indices

2

Chain indices

250

254

Postscript

259

Index

261

Foreword

Sydney Afriat is the guru of the price index. As a young mathematician, he arrived at Richard Stone’s Department of Applied Economics in Cambridge in the early 1950s, then the great center of research on theoretical and applied consumer behaviour. He soon realized that neither he nor anyone else knew very much about what was meant by “the price index”, in spite of being part of the everyday discourse of economics. In the half century since then, he has been exploring the foundations of the topic. Over the years, he has produced beautiful theorems on the topic, many of them completely unexpected even by the cognoscenti of the topic. And where he has led, the profession has followed, often many years later. Along the way, his work has fathered important incidental areas in economics, perhaps most notably the non-parametric analysis of demand, that has carried into other areas, like the analysis of efﬁciency with his frontier production function that extends Farrell’s method and the stochastic frontier. Afriat is one of those rare and rarely gifted individuals who think differently from everyone else. What is obvious to him can sometimes seem bizarre to others, especially at ﬁrst, and his vision of “the price index” often differs sharply from those that dominate the profession. Yet his work is a classic illustration of how much we learn from new ways of thinking, and how the bizarre and unfamiliar conceptions and results of an idiosyncratic leader can become the orthodoxy of the next generation. The last book presented a coherent discussion of ﬁfty years of astonishingly creative work on the price index. Some of the analysis had appeared before but much had not. Afriat’s many friends may have welcomed the deﬁnitive account, as also those who had not previously had the opportunity to understand and enjoy the work. So much is said in my foreword to that book. Now comes the surprise, that the story is not there ended. At the heart of this new book is a paper of twenty-ﬁve years ago in a volume in honour of Sir Richard Stone which I edited, reproduced in Part II. Yes very original, quite unlike anything seen before, and then what? Read on. Angus Deaton Dwight D. Eisenhower Professor of International Affairs Professor of Economics Princeton University, USA

Preface

Though the mathematics of the method, its theoretical rationalization and computations, require an account, the scheme for applications is simple, and conveys an idea of what could be meant by an answer to “The Index Number Problem”. A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of an entirely different type and are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, for the hypothetical underlying utility which in any case there is no need to actually construct. The theory of the price index, proper, starts with the Utility Cost Factorization Theorem going back to 1950’s. By itself it represents no resolution of the Index Number Problem, nor had there even been a real idea of what could be meant by such a resolution. The method now proposed does convey some idea of what could be meant by such a resolution. It even represents such a resolution itself. The method has been available in the main for more than twenty ﬁve years, apart from ampliﬁcations made just now. But only recently has it been recognized as a proper resolution of the Index Number Problem. The here provided ﬁrst exercises with the arithmetic go to convey the practicality of it. The needs of dealing with the EUKLEMS Project data when it became available, joined with chance encounter with data of use for practising the new computations and development of needed software, have stirred into life that almost forgotten work. Here provided in Part I are three papers by present authors: “The Super Price Index: Irving Fisher, and after”

xviii Preface has more to do with history, “The Price Level Computation Method” is an exposition of the mathematics, and. “Price Level Computation: Illustrations” offers ﬁrst exercises with the arithmetic of the method that go to convey the practicality of it. It is submitted that, for the very large number of different traditional type formulae to determine price indices associated with a pair of periods, which are joined with the longstanding question of which one to choose, they should all be abandoned. For the method proposed instead, price levels associated with periods are ﬁrst all computed together, subject to a consistency of the data, and then price indices that are all true taken together are determined from their ratios. An approximation method can apply in the case of inconsistency. At the heart of this new book is a paper of Afriat of twenty ﬁve years ago in a volume in honour of Sir Richard Stone edited by Angus Deaton, reproduced in Part II. It took being reminded of the paper in a ﬁrst meeting with Carlo Milana last Christmas (2006) to see what it represents. It is true, a ﬁrst discovery was agreement about perplexed latter-day phases of the old subject that may deserve notice, perhaps as grim ‘lessons from history’, where one of us, Milana, has a special interest and reports on this subject in Part II. But from then to April this year (2007) three papers were written that take the new method important steps further, and form Part I of this book. The 1960 paper reproduced in Part II is a step for the proof of a Utility Construction Theorem. This, a half century later, is the “Theorem” of “Two New Proofs of Afriat’s Theorem” by A. Fostel, H. E. Scarf and M. J. Todd. According to the Abstract, it’s “celebrated”! It happens this paper also serves for another construction theorem where the utility is ‘constant returns’ (already ‘rediscovered’ as if one of us had fallen asleep a few years, even for ever, before being wakened last Christmas) as required by the Utility Cost Factorization Theorem at the start of price index theory proper from a half-century ago, set forth again in Appendix 1. There it is also noted how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of common sense. In that way it is not just one possible answer to ‘The Index Number Problem’, but the only answer. Out of this simpler and altogether less well appreciated theorem the new price index method emerged, maybe to be “celebrated” in the next half century, or tomorrow—or, with this publication, even today. The method should take its place as the unique step that showed the way out of decades of stagnation in the Irving Fisher mind-set.

Preface xix Tracing linkages to the ‘non-parametric’ approach, that since its origin with utility construction theorems has so many inﬂuences elsewhere, also makes the basis for the present work.1 Though all points in the Paasche-Laspeyres interval are true, all equally, none more true than another, absurd pretentions offer a not merely “true” price index but an extra distinguished “superlative” one, as if it was an important discovery that had escaped everyone, even Irving Fisher, and thicken the quagmire with perambulations about functional form which has no welfare signiﬁcance at all. A basis for the method may be the 1981 paper, but only up to a point. That paper was inconsequential for application without the needed computational apparatus, but now we have that. Then there was appeal to the ‘extension property’ of solutions to prove existence theoretically, but now certain solutions are actually computed, the basic price level solutions that come in pairs with sides as it were associated with Laspeyres and Paasche, from which, it is conjectured, all solutions can be derived. A main basis for this book is not altogether in the remote beginnings but in itself. A precursor is the paper “The Power Algorithm for Generalized Laspeyres and Paasche Indices” at the Athens Meeting of the Econometric Society, September 1979. This deals with a feature in the beginnings of the computational method where a matrix of Laspeyres indices is raised to powers in a bizarre arithmetic where plus means min, maybe rather much for some but now we have the sofware for it. We usually have no dealings with Fisher-type bilateral formulae and the vast literature related there, but for elements of history that include added Notes about chain and multilateral index methods. There is as ever an imbalance since one of us reads everything and the other nothing. New generation price index workers can have pleasure to escape stagnation in the Irving Fisher trap and catch on to the new approach with its conceptual ﬁdelity and computational elegance.

1 A bibliography of works with the non-parametric connection lists about two hundred items, and quite likely there are many others.

Acknowledgements

For materials reproduced, thanks are due to the Cambridge University Press for permissions extending to the following S. N. Afriat, The system of inequalities ars > Xs − Xr Research Memorandum No. 18 (October 1960) Econometric Research Program, Princeton University. Proc. Cambridge Phil. Soc. 59 (1963) Reprinted with permission. S. N. Afriat, On the Constructibility of Consistent Price Indices Between Several Periods Simultaneously. In Essays in Theory and Measurement of Demand: in honour of Sir Richard Stone, edited by Angus Deaton. Cambridge University Press, 1981. 133–161. Reprinted with permission.

We acknowledge with thanks the guidance received from Richard Russell, longtime developer of presently used rendering of BASIC http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html http://www.rtrussell.co.uk/, [email protected]. We also acknowledge with thanks the support of the European Commission within the EUKLEMS research project of the Sixth Framework Programme. Thanks are also due to the European University Institute, San Domenico di Fiesole/Firenze, for support provided by a Jean Monnet Research Fellowship. Special thanks are due to Angus Deaton, witness of this work from the beginning who wrote the Foreword; and to Michael Allingham, Robert Aumann,

xxii Acknowledgements Ali Dogramaci, Nuri Jazairi, Michael Kuczynski, Axel Leijonhufvud, Luigi Luini, Robin Marris, Edward Nell, Marshall Reinsdorf, Paul Sammelson, Herbert Scarf, Amartya Sen, Martin Shubik, Paul Streeten, Vela Velupillai and Giulio Zanella; and those at Routledge who had to do with the publication; and ﬁnally, missing but still in mind Memorial Acknowledgements.

Part I

Concept and Method

Chapter 1 The Super Price Index: Irving Fisher, and after 1 Introduction—or genesis 2 Fisher, and after 3 Data and formulae 4 Utility 5 The True Index Appendix Fisher’s “Superlative” Index Numbers

1 Introduction—or genesis Prices change and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. Reference may be made to this ﬁrst paragraph for the basis of the price index idea. There is of course no absolute reason for having a price index in the ﬁrst place. It is an institution, even if a well established, traditional, and perhaps even today still needed institution, affecting many aspects of economic life. The Price Index issued from the Statistical Ofﬁce is a number that tells everyone how to answer the mentioned question, the index being the multiplier of old expenditure to determine the new. They must change expenditure in proportion to the index to keep the old standard of living. All have submissively to accept the authority of the number produced no doubt on some basis quite likely beyond them. The question of how to produce such an extraordinary number is called The Index Number Problem. It stands out as a fascinating question that has appealed to and occupied a great number of economists. To proceed, there are primitive points to be added. Let Prs denote the price index from base period s to current period r. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. This point seems not to be explicitly represented among Irving Fisher’s famous price index formula “Tests”1 , but the next points are—though we are not now considering formulae but the basic idea of a price index. Related to this description of the practical idea of a price index, reference can be made now and subsequently to Appendix 1, where it is put in utility terms,

1 See Fisher (1922), Allen (1975), Section 1.8, or Afriat and Jazairi (1988).

The Super Price Index 5 and a start is made with development of the price index as a theoretical concept having this genesis. Attention is also drawn to Appendix 2 on Terminology and Appendix 3 on Notation. For the Identity Test, there is the statement Ptt = 1, that is, “when one year is compared with itself, the index shows ‘no change’.” Most formulae go along with this. For the next, if the price change is reversed, so the new prices becoming the old and vice-versa, then the price index, the ratio that turns old expenditure into new, is replaced by the reciprocal. That is, Pts = (Pst )−1 which is the Time Reversal Test. Fisher’s “ideal” index is just about the only formula that satisﬁes this. No wonder it is “ideal”. This thinking seems to be as if the price index was derived as a ratio of price levels, expressing purchasing power of money for obtaining a standard of living by purchase of consumption. That has in fact already been offered as a way to go with the ‘Index Number Problem’ by Afriat (1981), with what may be regarded as a ‘New Formula’ though it is not at all a formula of the type dealt with by Fisher and associates. Rather, with it, ﬁrst determine price levels, and then determine price indices as their ratios. That is, instead of the price index being given directly by some algebraical formula in terms of demand data for the reference periods themselves, among those produced seemingly endlessly and as it were out of nothing (Irving Fisher, joined by others, deals with one or two hundred of them). True, there are many prices each with its own separate level, so it may seem fair to ask what sense can there be to their having a single level when they are taken together. None the less there is a somewhat restricted but nevertheless outstanding sense for it. But ﬁrst there will be account of the early maker of index numbers, Irving Fisher, who knew well about the primitive price index idea but apparently not exactly this sense. For a distinction and the language for it: price level has reference to a single period, while price index has reference to two, and is in principle the ratio of new level to old, so it is the multiplier of old expenditure to produce the new that will currently purchase the same living standard. The second primitive principle mentioned, expressed by Fisher’s Time Reversal Test, would also be an immediate consequence of taking price indices having the form of ratios of price levels. When dealing with more than just two periods, beside the Time Reversal (the Fisher Index is a distinguished case among formulae for satisfying this) there can be introduction of the Chain Test, Prs Pst = Prt

6 Concept and Method (just about never satisﬁed by any of the one or two hundred usual price index formulae) which implies Time Reversal again, and moreover implies, and obviously is implied by, price indices being expressible as the ratios of a set of numbers associated with the periods—the ‘price levels’. For, bringing in the Identity Test, Ptt = 1 we have Pts Pst = Ptt = 1 so Pts = (Pst )−1 which is Time Reversal, and now, for any ﬁxed r, Pst = Psr Prt = Psr (Ptr )−1 = Psr /Ptr so price indices determined relative to a ﬁxed base can serve as ‘price levels’ from which all price indices can be determined as their ratios. Evidently now the Chain Test, from ﬁrst implying Reversal, is equivalent to Fisher’s Circularity Test, Prs Pst Ptr = 1. While there has been invariably no prior determination of price levels from which to obtain price indices as their ratios, the 1981 formula excepted, usually formulae (a great number) are proposed that go directly to the index without a background of levels. In that culture the great headache is to know what formula—as may be good, or true, or … or what? A missing test, not named before and which implies all these others, and which can be called the Ratio Test, is simply that the price index be expressed as a ratio of a set of numbers, maybe price levels. Among formulae, as such, nowhere is that satisﬁed, unless the 1981 formula be allowed, or another, of Bishop William Fleetwood in 1706, mysteriously neglected, Pst = pt a/ps a, the inﬂation rate for a ﬁxed bundle of goods a, in this case over a gap of two or three centuries. “It could be wondered why this formula was not “ideal” for Fisher. Possibly it was too dull but if he was looking for a more interesting one which meets the proper qualiﬁcation, … ”2

2 The quotation is from Afriat (1977, 37–8) and continues “… it does not exist, as Eichhorn (1976) has shown.” Unfortunately our friend Wolfgang Eichhorn, and for that matter Afriat, had not known the 1981- formula.

The Super Price Index 7 This is a moment also to quote Sir Roy Allen (1978, p. 418) where he quotes one of us: “Formerly, it was as if an answer was proposed without ﬁrst having had a question, and it was wondered if it could be an answer to a question, any question at all. Even many answers to no question in particular were proposed—the names of their proposers are attached to some of them”. (Afriat 1977, p. 102) That has to do with the formula approach dominated by Fisher and, as if caught in a trap, still prevalent, for the truck with index formulae does not end with Fisher.

2 Fisher, and after Irving Fisher (1922, 244–8) ranked 134 index number formulas (according to one of us, 126, and 145 by the count of Yrjo Vartia) according to their numerical distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them into several classes, as it were in increasing order of merit. The ﬁrst twelve index numbers constituting the ﬁrst of these classes are labeled as “worthless”, to designate that they are the worst in his ranking. The other six classes are labeled as poor, fair, good, very good, excellent, and superlative (p. 244). Fisher (1922, 244–48) classiﬁed ten (or eleven to include the yardstick Fisher “ideal”) index number formulas in the “superlative” group because, in his numerical example, they performed very closely to formula 353, which is the “ideal” geometric mean of the Laspeyres and Paasche indexes, which was itself put at the top rank in this group. He claimed that all these formulas correspond to combinations of the Laspeyres and Paasche, including the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Reference can always be made to our Appendix for Fisher’s list, and an exhibition of his “superlative” and other formulas. In the numerical example worked by Fisher, the closest formula to 353 is 8053, which is the arithmetic average of 53 and 54, where 53 is Laspeyres index number (p. 471) and 54 is Paasche index number (p. 471). Fisher claimed that all the other “superlative” index numbers are combinations of 53 and 54. As remarked again later, we do not here regard Laspeyres and Paasche as formulae like the others that are offered as price index formulae, but in a category of their own. Of late there has been new venture towards the super, and beyond … Evidently, except the implicit Walsh index number and Fisher’s “ideal” itself, Diewert’s (1976) “superlative” index numbers are not superlative in Fisher’s (1922, 244–48) sense. The implicit Walsh index number, which is given by formula 1154 in Fisher’s list, corresponds to Diewert’s quadratic mean of order-1 index number, whereas Fisher’s “ideal” index number corresponds to Diewert’s quadratic mean

8 Concept and Method of order-2 index number. As set forth by Milana (2005); Diewert’s and Fisher’s indexes cannot understandably be deﬁned as, in some way, approximating the ACTUAL but UNKNOWN true index number up to the second order, in order to admit them as “superlative” in the language of Diewert (1976, 1978). With all that, incredible super-calculus aside, such a formula seems to be essentially no new departure but a determined adherence to the old formula world, only bringing to it an added unconstructive impenetrable burden of complexity and mystery. Though all points in the Paasche-Laspeyres interval are true, all equally, none more true than another, as proved by one of us, here are absurd pretentions about a not just true but a distinguished “Superlative Index Number” as if some important discovery had been made that had escaped everyone, even Irving Fisher, then thickened the quagmire with perambulations about functional form which has no welfare signiﬁcance at all. After the casting around for the super coming from Irving Fisher, there is entry into an entirely new kind of territory beckoning exploration. A question is whether here is a breakthrough to a new, highest quality formula, as the language would suggest—here and there acclaimed as such—or a neo-Fisherian casting around without new guidance. In order not to get lost in this enquiry, a start should be made at the beginning. One of us had a dedicated immersion in the entire super literature while the other, saved by reductio ad absurdum, had never before our encounter ever put eyes on a single item of the later phase.

3 Data and formulae Reference is made to the budget and commodity spaces B and C, spaces of nonnegative row vectors and of column vectors. With the non-negative numbers, B = n , C = n , with p ∈ B, x ∈ C we have M = px ∈ as the money cost of the bundle of goods x at the prices p. With such a purchase, making the demand (p, x) ∈ B × C of commodities x at the prices p, the budget vector is u = M −1 p ∈ B, for which ux = 1. There is the revealed preference of x over every bundle y which, being such that uy ≤ 1, is also attainable at no greater cost. A fundamental area of discussion involves data consisting of a pair of demand observations (pt , xt ) ∈ B × C (t = 0, 1), which is associated with the Laspeyres index

P st = ps xt /pt xt and Paasche index

P st = ps xs /pt xs = P −1 ts . There is also the Fisher index 1 P¯ st = P st P st 2

The Super Price Index 9 which is the geometric mean of both, lying between them. An important condition on the demand data is the Laspeyres-Paasche inequality (LP)

P st ≤ P st ,

equivalently

P st P ts ≥ 1 or again ps xs pt xt ≤ ps xt pt xs . Or with Lst = ps xt /pt xt , again the Laspeyres index, this condition is equivalent to the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt

for price levels Pt (t = 0, 1). Put this way, the usual Laspeyres-Paasche inequality condition for a pair of demands with t = 0,1 is immediately generalized for any number t = 0, 1, 2, . . . , m as in the 1981-formula.3 The system L with s, t = 0,1 requires

P 01 ≤ P1 /P0 ≤ P 01 so its consistency, deﬁning L-consistency, or the existence of a solution, is equivalent to the LP-inequality. Here it is convenient to insert, though we have no immediate concern there, that the counterpart for more than two demands is the cyclical Laspeyres product test Lti Lij · · · Lkt ≥ 1 for every cycle t, i, j, . . . , k, t or brieﬂy Lt...t ≥ 1, which is a strengthening of the revealed preference test for utility construction that serves for conical utility. Introducing the chain Laspeyres and Paasche indices

P sij...kt = P si P ij · · · P kt ,

P sij...kt = P si P ij · · · P kt ,

3 It is interesting that a simple step like this should open the door to a fundamentally new approach. The start was at least a quarter-century ago, but the signiﬁcance of it was recognized more recently.

10 Concept and Method this test is equivalent to (chain-LP)

P s...t ≤ P s...t

for all possible chains … taken separately. Hence introducing the new Laspeyres and Paasche indices

st = minij...k P si P ij · · · P kt ,

st = maxij...k P si P ij · · · P kt ,

this is now equivalent to

(new-LP) st ≤ st . The chain-LP and new-LP provide a narrowing of the basic LP bounds for a single index. This has been exploited for application quite apart from the main use of the method for constructing a complete set of consistent true indices over any number of periods. The matter of bounds there has complications where every commitment to the location of one index has implications for the location of others. After chain-LP, the approach to new-LP appears prevented by the computation involved, though this is already resolved in our method. This extended approach is dealt with more fully in the papers Afriat (1981), (1982), and Afriat and Milana (2007). But we should not now get sidetracked and should return to the basic case of two demand observations having touched brieﬂy on this beginning of the extension. However ﬁrst let it be pointed out that nowhere so far has there been any dependence on utility, let alone conical or constant returns utulity, so there can be no accusation that some such assumption has been made. But then it should be mentioned again that the cyclical condition Lt...t ≥ 1 that turned up is exactly the condition on the data for the constructibility of a conical utility. Surely some use of that opportunity can be expected. Again Appendix 1 should be consulted though there has already been the interest to quote part of it here. Going back to the monumental ﬁrst paragraph, one obvious way of maintaining, at new prices, the old standard of living obtained from the old consumption x0 at a money cost p0 x0 (hypothetically the minimum cost for the utility of it) is to simply buy the old consumption with new cost p1 x0 . So certainly the current cost of the old standard is a most that amount, and its comparison with the old cost, which might at ﬁrst thought serve as a price index, is at most p1 x0 /p0 x0 , which is the ever-pervasive Laspeyres index. But here common sense breaks in with the proposal that one would not necessarily buy the old bundle at the new prices, one could buy instead another bundle that provides at least the old standard perhaps at a lesser minimum cost, making a comparison ratio with the old cost, the price index P10 , not exceeding the Laspeyres index, P10 ≤ p1 x0 /p0 x0 ,

The Super Price Index 11 and by the same principle P01 ≤ p0 x1 /p1 x1 . But in recollection of the second primitive point which asserts P01 = (P10 )−1 this second is equivalent to P10 ≥ p1 x1 /p0 x1 so common sense together with the primitive reversal principle provides both Laspeyres and Paasche, and shows them not only as tied together with the same source in principles, but as upper and lower bounds of the price index, if there is one. In a way, they are not price index formulae like all the others, but fundamental and irrefutable limits for the price index. In that case, of course, the lower bound could not exceed the upper bound. J. R. Hicks (without proving anything) calls that “The Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is confusing, but perhaps Hicks was just being fashionable. But still, a case where Paasche turned out to be greater than Laspeyres could be occasion for a pause. Hicks declares the conclusion that Laspeyres “tends to be greater than” Paasche. Before that is an argument starting with indifference between x0 and x1 implying, by denial of revealed preference, p0 x0 ≤ p0 x1

and p1 x1 ≤ p1 x0

from which it follows that p0 x1 p1 x0 ≥ p0 x0 p1 x1 , equivalent to assertion that Laspeyres is greater than Paasche.When was that freed of indifference of the x’s? In fact never, because the PL-relation is not invariable but just a test, true or false, for the constructibility of a constant returns utility. But when was it proved (before today) that P01 ≤ L01 ? It might have been good sense (corresponding to practice of the practical) to abandon the entire theoretical subject after arrival at this point, where the unavoidable primitive has been joined with common sense. But uncountable formulae for the index, good, true, better, super, &c have been proposed. To go further, as it seems we must, more form needs to be given to ideas we have so far.

12 Concept and Method

4 Utility There are two main things about a consumption bundle x ∈ C. The simple part is that it has a money cost M = px ∈ when the prices are p ∈ B. The other part is that it is the basis—and having it is the objective, use, use-value, or utility— for obtaining a standard of living, with the sacriﬁce of cost. Hence there is a link between cost and standard of living, where prices enter. For this link a gap remains between consumption and its utility, made good hypothetically by introduction of the utility function. A utility function is any numerical-valued function φ deﬁned on the commodity space B, φ:B→ so φ (x) ∈ (x ∈ B) is the utility level of any commodity bundle x. A utility function φ determines a utility order R ⊂ C × C where xRy ≡ φ (x) ≥ φ (y) A utility function φ, with order R, ﬁts a demand element (p, x), with budget vector u, or the demand is governed by the utility, if the revealed preferences of it belong to the utility order, uy ≤ 1 ⇒ xRy (y ∈ C) . In other words, if x has at least the utility level of every bundle y attainable at no greater expenditure with the prices, or x provides the maximum utility φ (x) for all those bundles y under the budget constraint uy ≤ 1, that is py ≤ px ⇒ φ (x) ≥ φ (y) . The utility system is hypothetical and admitted to the extent that it ﬁts available demand observations. The cost of a standard of living is determined as the minimum cost at prevailing prices of getting a consumption that provides it. In terms of a utility function φ, this is gathered from the utility cost function ρ (p, x) = min {py : φ (y) ≥ φ (x)} which tells the minimum cost at given prices p of obtaining a consumption y that has at least the utility of a given consumption x. Since x itself, with cost px, is a possible such y, necessarily ρ (p, x) ≤ px

for all p, x

while ρ (p, x) = px signiﬁes the admissibility, under government by the utility system, of the demand of x at the prices p. It shows the demand is cost effective, getting the maximum of

The Super Price Index 13 utility available for the cost, and cost efﬁcient, getting at minimum cost the utility obtained, which conditions would here be equivalent. A case where admissibility does not hold could be attributed to consumption error, described as failure of efﬁciency, where ρ (p, x) ≥ epx,

0≤e≤1

would show attainment of cost efﬁciency to a level e. This idea has some use in dealing with demand data inconsistent with government by a utility by ﬁtting it to a utility that serves only approximately, as in Afriat (1973). For the service of a price index this utility cost should have the property ρ (pr , x) /ρ (ps , x) is independent of x, which represents the special condition on the utility by which it has the price index property and has this ratio as an associated price index. An equivalent condition is that utility cost be factorize into a product ρ (p, x) = θ (p) φ (x) , of price level P = θ (p) depending on p alone and quantity level X = φ (x) depending on x alone. As proved in Appendix 1: Theorem 1 Utility cost factorization is necessary and sufﬁcient for a utility to have the price index property, This factorization is immediately assured if φ is conical, but also the converse is true, showing the following4 also proved in Appendix 1: Theorem 2 Utility cost factorization it is necessary and sufﬁcient for the utility to be conical. In consequence we have: Theorem 3 For a price index to be based on a utility it is necessary and sufﬁcient that the utility be conical

4 A ray is a half-line with vertex the origin and a cone is a set described by a set of rays. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xl) = φ (x) l. With demand governed by conical utility, the expansion paths, or loci of demand when expenditure varies while prices remain ﬁxed, are rays.

14 Concept and Method This theorem from long ago5 may be a good candidate for the title “The Index Number Theorem” secured by Hicks for another purpose as already noted. The question now is: what utility? A price index being wanted, by the theorem it must be conical, and with given demand data (pt , xt ) ∈ B × C (t = 0, 1, . . .) , and adoption of the efﬁciency criterion, any utility to be entertained would, to ﬁt the data, have to be such that Pt Xt = pt xt (t = 0, 1, . . .) , where Pt = θ (pt ) , Xt = φ (xt ) , so in any case Ps Xt ≤ ps xt and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 0, 1). A question is whether a solution exists. If one does, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from Pt Xt = pt xt . Thus, introduce

φ i (x) = Pi−1 pi x and

φ (x) = min φ i (x) i

so this is a concave conical polyhedral utility function that ﬁts the demand data, with associated price indices as required, to make those prices indices true.

5 Samuelson and Swamy (1974) p. 570 attribute theorem and proof to Afriat (1972) though it comes from the 1950s.

The Super Price Index 15 Another such function, concave conical, which ﬁts the demand data, again with required values and the same associated price indices, is the polytope type function given by

φ (x) = max

Xi ti :

i

6 xi ti ≤ x, ti > 0

i

and if φ is any other concave conical utility that ﬁts the demands and takes the values Xi at the points xi then

φ (x) ≤ φ (x) ≤ φ (x) for all x. Included in the above is the simple conical precursor of the general theorem on utility construction put in service speciﬁcally for price index theory.

5 The True Index Every conical or constant returnsutility has associated with it a price index, derived from the utility cost factorization applicable to such a function. A price index is termed true if it is connected with such a utility that ﬁts the demand data. Every solution for price levels determines true price indices given by their ratios, the existence of a solution requiring the cyclical Laspeyres product test, that requires the cyclical Laspeyres products to be all at least 1. It should be seen what all this has to say in reduction to the classical case of just two periods. The existence of a solution for price levels implies the LP-inequality, and then any point in the LP-interval is expressible as a price index, obtained as the ratio of the price levels, which is true from being associated with a conical utility that ﬁts the data. Hence, as values for the price index, all points in the LP-interval are true—all equally, no one more true than another.

6 The function of this form introduced by Afriat (1971) is the constant-returns ‘frontier production function’ that gives a function representation, and at the same time a computational algorithm, for the production efﬁciency measurement method of Farrell (1957) (Afriat’s colleague at the Department of Applied Economics (DAE), Cambridge, whose work, from after he had left, he at ﬁrst missed) that marks the beginning of ‘data envelope analysis’ (DEA). The Afriat comment attached to Finn R.Førsund, and Nikias Sarafoglou (2005) gives a report. While Afriat is usually given credit for ﬁrst introduction of the ‘non-parametric’ approach, here now is opportunity to give credit to Farrell who made such an introduction for this case as it were implicitly even if not explicitly in the way here exhibited. The same type of function without constant-returns, worth knowing about, is used for the utility construction in Afriat (1961) but arbitrarily, or for simplicity, left aside in the account of (1964) where instead a polyhedral type function is used. It also served for the 1971 extension of Farrell’s method.

16 Concept and Method When this was mentioned a few decades ago, possibly at the Helsinki Meeting of the Econometric Society, August 1976, it was received with complete disbelief.7 Here is a formula to add to Fisher’s collection, a bit different from the others: PRICE INDEX FORMULA: Any point in the LP-interval, if any. This paper draws attention to the generalization of this Formula for any number of periods, the 1981-Formula, where price and quantity levels for the periods are all computed simultaneously, associated with a conical utility that ﬁts the demand data for all the periods. The computations are by solution of a system of inequalities, so with a tolerance like in the above Formula, subject to a consistency condition that is a strengthening of ordinary revealed preference consistency. In the absence of consistency, an approach is made by an approximation method. This amounts to a complete, elegant and practical treatment of the ‘Index Number Problem’ to which Afriat’s attention was brought by Alan Brown the day he arrived at the Department of Applied Economics (DAE), Cambridge, in 1953, and concludes the assignment. In the 1982 paper on “The True Index”, where there are elaborations about algorithms for consistency test and price level computations, there is mention about normalizing price levels so they sum to 1 and are therefore represented by a point in the simplex of reference. Then the set of all price level solutions, based on given data, is represented by a region in this simplex. In the case of three periods, when the reference simplex is simply a triangle, this opens the way to illuminating graphics, as presented in the paper. In the case of two periods the simplex of reference is a line segment and price level solutions describe a subsegment corresponding to the PL-interval range of the price index given by ratios of price levels. Irving Fisher favoured one particular formula out of the many (one or two hundred) he considered. This is his “ideal index” given by the geometric mean of the Laspeyres and Paasche indices. This favouritism is understandable because the formula is impressive and just about alone in meeting the Time Reversal requirement, one of his “tests” basic to the very idea of a price index. And moreover it is comfortably well into the interior of the interval bounded by the limiting Laspeyres and Paasche indices. Here is a proposal for picking one true point out of many, as may be described by the LP interval and can apply just as well when the segment becomes a polyhedron in many dimensions, the point should be: true with minimum distance to non-true a maximum This is one path to the Fisher index and generalizations. With our approach it is necessary to deal directly with price levels instead of price indices which are their ratios.

7 Proof in Afriat (1977) 129–30.

The Super Price Index 17 S. S. Byushgens (1925) submitted that if it could be taken that demand was governed by a homogeneous quadratic utility then the value of the associated price index would be given by Fisher’s “ideal” formula. This result, which seemed at ﬁrst, with initial optimism, to make the Fisher index “true”, must have seemed spectacular. Possibly it is here that the idea of a true index made its ﬁrst entry. Of course now we have the Fisher index as true simply from belonging to the LP-interval, with countless ﬁtting utilities, along with all the other points and no more true than any of them—even though lately there has been a cult, on the part of some, to load function form with a signiﬁcance it cannot possibly have. However, with Byushgens, the Fisher index could be distinguished from the other points as being, not more true than the others—anyway impossible—but simply quadratically true—maybe. And maybe not! For if no quadratic utility can ﬁt the data, Byushgens’ Theorem, while still undoubtedly true, and still pretty, would be vacuous. This is a danger about which quite likely there had not been awareness at ﬁrst. In Afriat (1972), (1977), (1982) and (2004)8 there are accounts of issues to do with Fisher and Byushgens. These include an attempt to save Byushgens from the vacuity hazard by a weakening of his proposal, replacing quadratic by locally quadratic, rendering it with less interest except that it has been taken up by Diewert for doctrine about a formula to which—echoing Fisher with his impressive terminology—he gives the name superlative, that in ordinary language must promise at least something good, while what that good should be is not known at all. Though our proposal for outright abandonment of all price index formulae would of course include Diewert’s promotion, it can still have attention for notable features while celebrated as the Swan Song of the Irving Fisher Price Index Formula era. A notable feature is the citation style, that can be encountered with some frequency, where an item is connected with a string of individuals that includes the originator with others not deserving or wanting credit. This lumping together of originator with readers in a quixotic generous distribution of credit can be joined at the same time with a scatter of compensatory withholdings. But after all, we have been told: If we should ever encounter a case where a theory is named for the correct man, it will be noted. George J. Stigler The Theory of Price (3rd edition), 1966, p. 77 Stigler may have marked, if not just ignorance, a prevalence of ritual citations that amount to nothing more than greetings to friends. Such a phenomenon is consistent with “The Tale of Two Research Communities” and the tendency observed there to “stick to ‘own camp’ references” (good if you happen to have a camp) discovered

8 The 2004 volume contains earlier accounts beside an additional one.

18 Concept and Method by those early pioneers Finn R. Førsund and Nikias Sarafoglou (2005) in the ﬁeld of Citation Analysis, a new ﬁeld for which the way has been opened by advent of the computer whose emergence Stigler might have been happy about. We have been sidetracked, delivering the message to abandon useless formulae and enter a new path, to temper the abruptness and bring in the ﬂavour of history.

Appendix: Fisher’s “Superlative” Index Numbers Irving Fisher (1922, 244–48) ranked 134 index number formulas according to their numerical distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them “arbitrarily into several classes in increasing order of merit. The ﬁrst twelve index numbers, constituting the ﬁrst of these classes, are labeled, rather harshly perhaps, as ‘worthless’ index numbers to designate the fact that they are the worst. The other six classes are labeled as poor, fair, good, very good, excellent, and superlative” (p. 244). He then, classiﬁed ten index number formulas in the superlative group because, in his numerical example, they performed very closely to the “ideal” geometric mean of the Laspeyres and Paasche indexes. He claimed that most of them are the various combinations of these two index number formulas (p. 248). They include the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Using Fisher’s notation, let pti and qit represent, respectively, the price 1and p q1 th quantity of the i commodity during the observed period t and V ≡ i i0 i0 . i pi q i Fisher’s “superlative” index numbers are presented here in the original inverse order of remoteness from the “ideal” index (353) (the last one is included in the list at the ﬁrst ranking position with all formulas being identiﬁed with the original Fisher’s code, and the page of The Making of Index Numbers where they are deﬁned is indicated in parenthesis9 ): (11)

5323 ≡

√ 323 × 325

(geometric mean of two Törnqvist-based index

numbers) (p. 486), √ (10) 1323 ≡ 1123 × 1124 (p. 484), 0 1 1/2 1 (q q ) p (9) 1153 ≡ i i0 i1 1/2 i0 (Walsh index number) (p. 483), pi i (qi qi ) √ (8) 1353 ≡ 1153 × 1154 (p. 484),

9 See Fisher (1922, Appendix V) for the full list of the 134 formulas.

(7)

0 1 1/2 1 (p p ) q 1154 ≡ V i i0 i1 1/2 i0 qi i (pi pi )

The Super Price Index 19 (implicit Walsh index number) (p. 483),

1 0 i pi q i p0 + p1 p0 + p1 1 + p0 q0 i i 1 i i 0 qi qi = (6) 2154 ≡ V i 0i i1 i i 2 2 p q 1 + i i1 i1 i pi q i (p. 484 and p. 488), √ (5) 2353 ≡ 2153 × 2154 (p. 484), 0 qi0 + qi1 qi0 + qi1 (q + qi1 )p1i 1 0 (4) 2153 = pi pi = i i0 1 0 i i 2 2 i (qi + qi )pi (p. 484 and p. 488), p0 q 0 p0 q 1 (3) 8054 ≡ 2 (p. 487), i 1i i0 + i i1 i1 i pi qi i pi q i 1 0 1 1 53 + 54 p q p q (2) 8053 ≡ 2 (487), = i i0 i0 + i i0 i1 2 p q i i i i pi q i 1 0 1 1 √ p q i pi q i (1) 353 ≡ 53 × 54 = 0 0 · i i0 i1 (Fisher “ideal” index number) i pi q i i pi q i (p. 482) 0 1 1 1 p q p q =V i i0 i0 · i i1 i0 , i pi q i i pi q i with √

p1i 1123 ≡ i p0 i

p0i qi0 p1i qi1 /

0 0 1 1 j pj q j pj q j

(p. 483)

√ q1 p0i qi0 p1i qi1 / j p0j qj0 p1j qj1 i 1124 ≡ V (p. 483) i q0 i √ √ √ 4 323 ≡ 23 × 24 × 29 × 30 = 123 × 124 = 223 × 229 √ √ √ 4 325 ≡ 25 × 26 × 27 × 28 = 125 × 126 = 225 × 227 1 p1i 2 √ 123 ≡ 23 × 29 = i p0 i (p. 473),

p0 q0 p1 q1 i 0i 0 + i 1i 1 j pj qj j pj qj

(p. 480), (p. 480),

(Törnqvist index number)

20 Concept and Method q1 12 √ i 124 ≡ 24 × 30 = V i q0 i number) (p. 473), √ 125 ≡ 25 × 27 (p. 473), √ 126 ≡ 26 × 28 (p. 473), √ 223 ≡ 23 × 24 (p. 477), √ 225 ≡ 25 × 26 (p. 477), √ 227 ≡ 27 × 28 (p. 477), √ 229 ≡ 29 × 30 (p. 477),

p0 q0 p1 q1 i 0i 0 + i 1i 1 j pj qj j pj qj

(Implicit Törnqvist index

0 0 0 0 p1i pi qi / j pj qj 23 ≡ (p. 468), i p0 i q1 p0i qi0 / j p0j qj0 i 24 ≡ V (p. 468), i q0 i 0 1 0 1 p1i pi qi / j pj qj (p. 468), 25 ≡ i p0 i q1 p1i qi0 / j p1j qj0 i 26 ≡ V (p. 468), i q0 i 1 0 1 0 p1i pi qi / j pj qj 27 ≡ (p. 468), i p0 i q1 p0i qi1 / j p0j qj1 i (p. 468), 28 ≡ V i q0 i 1 1 1 1 p1i pi qi / j pj qj 29 ≡ (p. 468), i p0 i q1 p1i qi1 / j p1j qj1 i (p. 468), 30 ≡ V i q0 i 1 0 p1 q 1 i pi q i 53 ≡ 0 0 (Laspeyres index number) (p. 471) = V i i1 i0 , i pi q i i pi q i 1 1 0 1 p q p q 54 ≡ i i0 i1 (Paasche index number) (p. 471) = V i i0 i0 . i pi q i i pi q i

The Super Price Index 21 Moreover, formulas numbered 103, 104, 105, 106, 153, 154, 203, 205, 217, 219, 253, 259, 303, 305 reduce to 353. In the numerical example performed by Fisher, the closest formula to 353 is 8053, which is the arithmetic average of 53 (Laspeyres) and 54 (Paasche) index numbers. Fisher claimed that most of “superlative” index numbers are various combinations of 53 and 54. These index numbers should be contrasted with the quadratic mean-of-order-r index numbers by Diewert (1976, pp. 130–131) and called “superlative” by him using Fisher’s (1922, p. 247) terminology (but in a different sense). These are encompassed by the following general formula: 0 1 0 r/2 1r s (p /p ) PD ≡ i i1 i0 i1 r/2 i si (pi /pi )

with r = 0,

pt q t where sit ≡ i ti t for t = 0, 1 and with PD reducing to the Fisher “ideal” index j pj q j with r = 2, to the implicit Walsh index (Fisher’s formula 1154) with r = 1, and to the Törnqvist index number (Fisher’s formula 123) at the limit as r → 0. It can be shown (Diewert, 1976) that, if the shares sit are those derivable exactly from the true utility cost function, then this may have the following quadratic mean-of-order-r functional form: ct (p) = form

i

ln ct (p) = at0 +

t r/2 j αij (pi pj )

t i ai ln pi

1/r

+

with r = 0, which tends to the translog functional

i

t j aij ln pi ln pj

as r → 0.

Following Milana (2005) and generalizing Caves et al. (1982), it can be shown that 0 1 l 1 1 1−l c (p ) c (p ) the equality PD = 0 0 is obtained with some real value of l, c (p ) c1 (p0 ) which can be not necessarily within the range between 0 and 1 and is a function not only of technology but also of prices. Caves et. al. (1982) have considered the case of a translog functional form where the parameters of the second order terms are constant, that is atij = a¯ ij . In this case, l = 1/2. However, in view of the above-mentioned results, Diewert functional forms for superlative index numbers are not necessarily the geometric average of the two ‘true’ indexes. Alternatively, it is possible to deﬁne a quadratic mean-of-order-r quantity index number 0 1 0 r/2 1r s (q /q ) QD ≡ i i1 i0 i1 r/2 with r = 0, which reduces to the Fisher “ideal” i si (qi /qi ) quantity index with r = 2, to the implicit Walsh quantity index with r = 1, and to the Törnqvist quantity index number at the limit as r → 0. If the shares sit are those derivable exactly from the true utility function, then this index number is exact for a utility-based distance function δ t (q) (Deaton (1979b), Deaton and

22 Concept and Method Muellbauer (1980, pp. 53–57)) in the sense that it is a weighted (geometric) average of two Malmquist (1953) index numbers deﬁned as distance function 0 1 l 1 1 1−l δ (q ) δ (q ) ratios, that is QD = 0 0 for some real value of l, where δ (q ) δ 1 (q0 ) 1/r t r/2 δ t (q) = with r = 0, which tends to the translog functional i j βij (qi qj ) form ln δ t (q) = bt0 + i bti ln qi + i j btij ln qi ln qj as r → 0. (δ t (q) is a scalar by which the original quantities q have to be multiplied in order to reach the ﬁnal utility level given the preferences indexed with t). The weight l can be not necessarily between 0 and 1 and is a function not only of technology but also of quantities. In particular, l = 1/2 when δ 0 and δ 1 are quadratic functions with the same parameters in the second order terms. Therefore, here again Diewert functional forms for superlative index numbers are not necessarily the geometric average of two ‘true’ indexes. Moreover, it is tempting to derive the implicit quadratic mean-of-order-r price index as the ratio P˜ D ≡ V /QD , but as pointed out by Samuelson and Swamy (1974, p. 570) echoing Afriat’s (1972) factorization theorem, the homotheticity of the underlying economic function is a necessity as well as a sufﬁciency for the existence (and invariance) of such an index. If this is not the case, the obtained implicit price index is spurious and may not even respect the required degree-one homogeneity property. It is immediate to see also that, except the implicit Walsh index number and Fisher’s “ideal” itself, Diewert’s “superlative” index numbers are not “superlative” in Fisher’s (1922, 244–48) sense10 . The implicit Walsh index number, which is

10 The Törnqvist index, for example, (corresponding to formula 123 in Fisher, 1922, p. 473) is seen as the most superlative by Caves, Christensen, and Diewert (1982, p. 41) and Diewert (2004, p. 450) whereas it was not deemed “superlative” by Fisher (1922, p. 247), who classiﬁed it, in a descending order of merit, below the classes of “excellent” and “superlative” index numbers, with the last group ranked at the top position. Note from CM to SA: This is not the only critical remark that we may make. The Törnqvist index number is exact for a particular quadratic function (the translog) just as the Fisher “ideal” is exact for a simple quadratic function. It does not matter that the Törnqvist index is exact for a translog function that (as shown by Caves et al., 1982) is not necessarily homogeneous (Milana, 2005 has shown that also the other index numbers that are “superlative” in Diewert’s sense are exact for speciﬁc quadratic functions that are not necessarily homogeneous). The main point is that it is not true that: (1) these index numbers are second-order approximations to the “true” index and to each other, (2) this “true” index does not exist at all in the non-homothetic (or non-homogeneous) case (see your LP-inequality theorem) and, therefore, the Caves et al. theorem is vacuous implying that the obtained index number is, in more familiar terms, “path dependent”. Under these circumstances, the Caves, et al. results fall under the criticism raised by your earlier contributions and our present paper. Note from SA to CM: Very interesting for experts like yourself and some others. But this paper, as you are aware, proposes the abandonment of all price-index formulae and whatever anyone may have had to say about any of them, and offers an entirely different approach to “The Index Number Problem”.

The Super Price Index 23 given by formula 1154 in Fisher’s list, corresponds to Diewert’s quadratic mean of order-1 index number, whereas Fisher’s “ideal” index number corresponds to Diewert’s quadratic mean of order-2 index number. For the reasons given in Milana (2005), however, all Diewert’s and Fisher’s indexes cannot be deﬁned as approximating the unknown true index number (if this exists) up to the second order and cannot be considered to be “superlative” also in Diewert’s (1976)(1978) sense.

Chapter 2 The Price Level Computation Method 1 2 3 4 5 6 7 8 9 10 11 12 13

Introduction Data and formulae Minimal chains System and derived system Triangle inequality Extension and exhaustion property Consistency Utility model for the method Solution structure Basic solutions Inconsistency and approximation Old and new: an illustration Conclusion

1 Introduction Prices change Š and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. Reference may be made to this ﬁrst paragraph for the basis of the price index idea. Appendix 1 formulates the idea as a theoretical concept based on utility. The Price Index issued from the Statistical Ofﬁce is a number that tells how to deal with this cost-of-living question, the index being the multiplier of old expenditure to determine the new. The question of how such a number should be produced is known as The Index Number Problem. To proceed about it there are primitive points to be added. Let Pst denote the price index from period s to period t. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. This point seems not to be explicitly represented among Irving Fisher’s well known “Tests”, but the next points are, though we are not now considering applications to actual formulae, as usual, but rather to the basic idea of a price index itself. For the Identity Test, there is the statement Ptt = 1, that is, “when one year is compared with itself, the index shows ‘no change’.” Most formulae go along with this. For the next, if the price change is reversed, so the new prices becoming the old and vice-versa, then the price index, the ratio that turns old expenditure into new, is replaced by the reciprocal. That is, Pts = (Pst )−1 which is the Time Reversal Test. Fisher’s “ideal index” is just about the only formula that satisﬁes this. No wonder it is “ideal.”

The Price Level 27 This thinking seems to be as if the price index was derived as a ratio of price levels, expressing purchasing power of money for obtaining a standard of living by purchase of consumption. For a distinction and the language for it: price level has reference to a single period, while price index has reference to two, and is in principle the ratio of new level to old, so it is the multiplier of old expenditure to produce the new that will currently purchase the same living standard. The second primitive point mentioned, expressed by Fisher’s Time Reversal Test, would also be an immediate consequence of taking price indices having the form of ratios of price levels, or anyway of some numbers. For if Pst = Ps /Pt then Pts = Pt /Ps = (Ps /Pt )−1 = (Pst )−1 . When dealing with more than just two periods, beside the Time Reversal (the Fisher “Ideal Index” is a distinguished case among formulae for satisfying this) there can be introduction of the Chain Test, Prs Pst = Prt (just about never satisﬁed by any of the one or two hundred usual price index formulae) which implies Time Reversal again, and moreover implies, and obviously is implied by, price indices being expressible as the ratios of a set of numbers associated with the periods—the ‘price levels’ or whatever. For, from the Chain Test directly, Prs = Prt /Pst or, bringing in the Identity Test, Ptt = 1 we have Pts Pst = Ptt = 1 so Pts = (Pst )−1 which is Time Reversal, and now, for any ﬁxed r, Pst = Psr Prt = Psr (Ptr )−1 = Psr /Ptr so price indices determined relative to a ﬁxed base can serve as “price levels” from which all price indices can be determined as their ratios.

28 Concept and Method Evidently now the Chain Test, from ﬁrst implying Reversal, is equivalent to Fisher’s Circularity Test, Prs Pst Ptr = 1. While there has been invariably no prior determination of price levels from which to obtain price indices as their ratios, usually formulae, plain algebraic involving demand data just for the reference periods themselves, and a great number of them, are proposed that go directly to the index without a background of levels. In that approach the great problem is to know what formula to use. A missing test, in Fisher’s list, perhaps not before named and which implies all these others, and which could be called the Ratio Test, though it is equivalent, as just seen, to the Chain Test, is simply that the price index be expressed as a ratio of a set of numbers. Among formulae, as such, nowhere is that satisﬁed, unless the now to be considered method, the 1981 formula, be allowed, or another proposed by Bishop William Fleetwood in 1707 and mysteriouly neglected, at least in usual theory if not actual practice, Pts = pt a/ps a, the inﬂation rate for a ﬁxed, perhaps democratically chosen, bundle of goods a.

2 Data and formulae1 Reference is made to two spaces, the budget space B and commodity space C, one the space of non-negative row vectors and the other column vectors, so with as the non-negative numbers, B = n ,

C = n ,

and any p ∈ B, x ∈ C provide M = px ∈ as the money cost of the bundle of goods x at the prices p. With such a purchase, making the demand element (p, x) ∈ B × C of commodities x at the prices p, the associated budget vector is u = M −1 p ∈ B, for which ux = 1. (We follow the rule that a scalar, as if it were a 1 × 1-matrix, multiplies a row-vector on the left and a column-vector on the right.) Any collection of demand elements makes a demand correspondence. A budget element is any (u, x) ∈ B × C such that ux = 1, and an expenditure correspondence consists in any collection of these. With any demand correspondence D there is an associated expenditure correspondence E, obtained by taking the associated budget elements. A fundamental area of discussion involves data provided by a ﬁnite demand correspondence D consisting of a series of demand observations (pt , xt ) ∈ B × C (t = 1, 2, . . . , m) ,

1 It is useful here to make reference to Appendix 3 on Notation.

The Price Level 29 as may be associated with different periods described by the index t. Price levels Pt to be associated with the periods are elements of a vector P in the price level space = m . Without altering the price indices determined from their ratios, they may be normalized to sum to 1, in which case they become barycentric coordinates for a point in the simplex of reference , available for graphic representations in case m = 3. Any pair of periods s, t is associated with the Laspeyres index Lts = pt xs /ps xs with s distinguished as the base and t the current period, so this is simply the inﬂation rate between the periods for the base period bundle of goods. There is also the Paasche index Kts = pt xt /ps xt = (Lst )−1 , which is the inﬂation rate for the current bundle. With any chain described by a series of periods s, i, j, . . . , k, t there is associated the Laspeyes chain product Lsij...kt = Lsi Lij . . . Lkt termed the coefﬁcient on the chain. Obviously Lr...s...t = Lr...s Ls...t A simple chain is one without repeated elements, or loops. There are m (m − 1) · · · (m − r + 1) = m!/r! simple chains of length r ≤ m and therefore altogether the ﬁnite number m! (1 + 1/1! + 1/2! + · · · + 1/ (m − 1)!) of simple chains from among m elements. A chain s, i, j, . . . , k, t whose extremities are the same, that is, s = t, deﬁnes a cycle. It is associated with the Laspeyres cyclical product Ltij...kt = Lti Lij . . . Lkt

30 Concept and Method which is basis for the important Laspeyres cyclical product test, or simply the cycle test, Lt...t ≥ 1 for all cycles t . . . t A simple cycle is one without loops. There are (m − 1) . . . (m − r + 1) = (m − 1)!/r! simple cycles of r ≤ m elements, and the total number of simple cycles from among m elements is the ﬁnite number made up accordingly. The coefﬁcients Lst Lts = Lsts on the cycles of two elements deﬁne the intervals of the system. The interval test Lst Lts ≥ 1 is equivalent to (LP)

Kts ≤ Lts

that is, the Paasche index does not exceed the Laspeyres, or the LP-inequality, a condition very well known from index number theory based on data for just two periods. Here therefore, with the cyclical test, is a generalization of that condition for any number of periods. J. R. Hicks (without proving anything) calls the LPinequality “The Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is confusing, but perhaps Hicks was just being fashionable. Another way of stating this condition, of signiﬁcance since it gives the form for a statement of a direct extension for many periods, is that the 2 × 2 L-matrix

1 Lst Lts 1 be idempotent, or reproduced when multiplied by itself, in the modiﬁed arithmetic where + means min. In fact, as to be shown, raising the general m × m L-matrix to powers in this modiﬁed arithmetic is a basic process in the price level computation method. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

the cycle test Ls...t...s ≥ 1 is equivalently to (chain LP) Ks...t ≤ Ls...t for all possible chains … and … taken separately. Hence introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

The Price Level 31 subject to the now to be considered conditions required for their existence, for which Hst = (Mts )−1 , this is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst showing the relation of the LP-interval and the narrower derived version that involves more data. Here it has been recognized that from Kts = (Lst )−1 follows −1 Kt...s = Ls...t and therefore −1 Hts = max... Kt...s = min... Ls...t = (Mst )−1 , where in each case … is understood so the chain s. . .t is the reverse of t. . .s.

3 Minimal chains Any chain can be represented uniquely as a simple chain, with loops at certain of its elements, given by cycles through those elements; and the coefﬁcient on it is then expressed as the product of coefﬁcients on the simple chain and on the cycles. Also, any cycle can be represented uniquely as a simple cycle, looping in simple cycles at certain of its elements, which loop in cycles at certain of their elements, and so forth, with termination in simple cycles. The coefﬁcient on the cycle is then expressed as a product of coefﬁcients on simple cycles. Thus out of these generating elements of simple chains and cycles, ﬁnite in number, is formed the inﬁnite set of all possible chains. Theorem 3.1 For the chains with ﬁxed extremities to have a minimum the cycle test is necessary and sufﬁcient.

32 Concept and Method If any cycle should be less than 1, then by taking chains which loop repeatedly round that cycle, chains which have decreasing coefﬁcients are obtained without limit; and so no minimum exists. However, should every cycle be at least 1, then by cancelling the loops on any chain, there can be no increase in the coefﬁcient, so no chain coefﬁcient will be smaller than the coefﬁcient for some simple chain. But there is only a ﬁnite number of simple chains on a ﬁnite number of elements, and the coefﬁcients on these have a minimum. Theorem 3.2 For the cycle test the simple cycle test is necessary and sufﬁcient. For the coefﬁcient on any cycle can be expressed as a product of coefﬁcients on simple cycles. Theorem 3.3 The cycle test implies that a minimal chain with given extremities exists and can be chosen simple. For then any chain is then not less than the chain obtained from it by cancelling loops, since the cancelling is then division by a product of numbers all at least 1.

4 System and derived system The computation of price levels Pt (t = 1, . . . , m) depends on solution of the system of inequalities (L)

Lst ≥ Ps /Pt .

Subject to the cyclical product test Lt...t ≥ 1 for every cycle, or equivalently every simple cycle, by Theorem 3.2, it is, by Theorem 3.3, possible to introduce Mst = minij...k Lsi Lij · · · Lkt , attained for a simple chain. Then Lsij...kt ≥ Mst for every chain and, by Theorem 3.3, the equality is attained for some simple chain. In particular, Lst ≥ Mst . The number Mtt is the minimum coefﬁcient for the cycles through t, so that Ltij...kt ≥ Mtt

The Price Level 33 for every cycle, the equality being attained for some simple cycle. In particular, for a cycle of two elements, Lts Lst ≥ Mtt . The cyclical product test that is the hypothesis now has the statement Mtt ≥ 1. With the numbers Mst so constructed, subject to this hypothesis, it is possible to consider with system L also the derived system (M )

Mst ≥ Ps /Pt .

The two systems are said to be equivalent if any solution of one is also a solution of the other. Theorem 4.1 The system L and its derived system M, when this exists, are equivalent. Let system L have a solution Pt . Then, for any chain of elements s, i, j, . . . , k, t there are the relations Lsi ≥ Ps /Pi ,

Lij ≥ Pi /Pj , . . . , Lkt ≥ Pk /Pt ,

from which, by multiplication, there follows the relation Lsij...kt ≥ Ps /Pt . This implies that the derived coefﬁcients Mst exist, and Mst ≥ Ps /Pt . That is, Pt is a solution of system M . Now suppose the derived coefﬁcients for system M are deﬁned, in which case Lst ≥ Mst . and let Pt be any solution of system M , so that Mst ≥ Ps /Pt . Then it follows immediately that Lst ≥ Ps /Pt . or that Pt is a solution of system L. Thus Land M have the same solutions, and are equivalent.

34 Concept and Method Theorem 4.2 If the cycle test holds for L then the interval test holds for the derived system M. Since Mst is the coefﬁcient of some chain with extremities s, t it appears that the interval coefﬁcient Mts Mst of M is the coefﬁcient of some cycle of L through t, and therefore if the cycle test holds for L then so does the interval test hold for the derived system M . Given any solution for system L, and equivalently system M , necessarily Kst ≤ Hst ≤ Ps /Pt ≤ Mst ≤ Lst , showing how price indices, which on the basis of data just for the reference period are conﬁned to the ordinary Laspeyres-Paasche interval, become conﬁned to the narrower derived Laspeyres-Paasche interval when based on the more extended data.

5 Triangle inequality2 From the relation Lr...s Ls...t = Lr...t it follows that the derived coefﬁcients satisfy the multiplicative triangle inequality Mrs Mst ≥ Mrt the one side being the minimum for chains connecting r, t restricted to include s, and the other side being the minimum without this restriction. Theorem 5.1 Any system subject to the cycle test is equivalent to a system which satisﬁes the triangle inequality given by its derived system. This is true in view of Theorems 3.1, 4.1 and 4.2. Theorem 5.2 The interval test holds for any system that satisﬁes the triangle inequality. Thus, from the triangle inequalities applied to any system M , Mtr Mrs ≥ Mts ,

Mts Msr ≥ Mtr

2 It is most useful here to refer to Appendix 3 on Notation, which also contains an elaboration of terminology applicable to the “canonical solutions” introduced in this section.

The Price Level 35 there follows, by multiplication, the relation Mrs Msr ≥ 1 or what is the same Hst ≤ Mst or that the derived LP-interval be non-empty. Theorem 5.3 If a system satisﬁes the triangle inequality then its derived system exists and moreover the two systems are identical. From the triangle inequality, it follows by induction that Msi Mij . . . Mkt ≥ Mst that is Msij...kt ≥ Mst from which it appears that the derived system N exists, with coefﬁcients Nst ≥ Mst so that now Nst = Mst This shows, what is otherwise evident, that no new system is obtained by repeating the operation of derivation, since the ﬁrst derived system satisﬁes the triangle inequality. Theorem 5.4 For any system the triangle inequality is equivalent to idempotence of the matrix in the arithmetic where + means min That is, the matrix is reproduced in multiplication by itself. For, simply, Nij = mink Nik Nkj if and only if Nij ≤ Nik Nkj . The triangle inequality Mrs Mst ≥ Mrt

36 Concept and Method has the restatement Mrs ≥ Mrt /Mst from which it appears that, for any ﬁxed t, taken as base, a solution of the system (M )

Mrs ≥ Pr /Ps

for price levels Pr is given by Pr = Mrt . Similarly, another solution is Pr = 1/Mtr . These solutions may be distinguished as determinations for the ﬁrst and second basic price level systems, with node t as base. Since, by Theorem 5.2, Mtr Mrt ≥ Mtt ≥ 1 they always have the relation 1/Mtr ≤ Mrt , which is the derived LP-relation. However, these are not now price indices, as in that original relation, but here they are price levels from which to derive price indices. Finding these solutions depends directly on the triangle inequality that is characteristic of the derived sysyem (M ), and not on the solution extension property that is a consquence, to which there is appeal in the construction method dealt with in the next Section. Now established, for every t, are two price level solutions Pr from which to derive systems of true price indices Prs = Pr /Ps . The two systems, of basic price indices with base t, are in a way counterparts of the Laspeyres and Paasche endpoints of the PL-interval that describes the range of true price indices for the classical case that involves just two periods. The determinations have reference to periods associated with the data without any dependence on the order 1, . . ., m in which they are taken. This is unlike where there is dependence on the solution extension property for ﬁnding solutions, of the next Section. However, they do depend on which period, corresponding to t in the given order, is taken as base. Coming in pairs there are now 2m determinations, whose pairwise connections and base references are essential.

The Price Level 37 When price level solutions are normalized so as to provide barycentric coordinates for a point in the simplex of reference, the set of all solutions is a convex polydron for which these 2m solutions are a complete set of vertices from which, it is conjectured, all solutions may be obtained by taking convex combinations of them. Note that the ﬁndings of this section apply just a well to the approximation method accounted in Section 10, based on relaxing exact cost efﬁciency, for the ﬁt of utility to demands, to some degree of partial efﬁciency. From the above the following is proved.

Theorem 5.5 The derived system (M), when it exists, admits the solutions given by the basic price levels, so it is always consistent.

Corollary 5.1 In that case also the original system (L) is consistent, and admits those same solutions.

For the system and derived system, when this exists, are equivalent, admitting the same solutions, by Theorem 3.1.

Corollary 5.2

The cycle test is necessary and sufﬁcient for consistency

For, by Theorem 3.1, the test applied to system (L) is necessary and sufﬁcient for the existence of the derived system (M ), which is always consistent when it exists, by the present Theorem, and by Theorem 4.2 it is equivalent to system (L), which therefore also is consistent.

6 Extension and exhaustion property A subsystem Mh of order h ≤ m of a system M of order m is deﬁned by (Mh )

Mst ≥ Ps /Pt

(s, t = 1, . . . , h) .

Then the systems Mh (h = 2, . . . , m) form a nested sequence of subsystems of system M , each being a subsystem of its successor, and Mm = M . Any solution of a system reduces to a solution of any subsystem. But it is not generally true that any solution of a subsystem can be extended to a solution of the original. However, should this be the case, then the system will be said to have the extension property.

38 Concept and Method Theorem 6.1 Any system which satisﬁes the triangle inequality has the extension property. Let P1 , P2 , . . . , Ph−1 be a solution of Mh−1 , so that

Mh−1

Mst ≥ Ps /Pt

(s, t = 1, . . . , h − 1) .

It will be shown that, under the hypothesis of the triangle inequality, it can be extended by an element Ph to a solution of Mh . Thus, there is to be found a number Ph such that Mhs ≥ Ph /Ps ,

Mth ≥ Pt /Ph

(s, t = 1, . . . , h − 1)

that is Mhs Ps ≥ Ph ≥ Pt /Mth So the condition that such a Ph can be found is Mhq Pq ≥ Pp /Mph where Pp /Mph = max Pi /Mih , i

Pq Mhq = min Pj Mhj . j

But if p = q this is equivalent to Mph Mhp ≥ 1 which is veriﬁed by Theorem 5.2, and if p = q it is equivalent to Mph Mhq ≥ Pp /Pq which is veriﬁed since by hypothesis Mph Mhq ≥ Mpq ,

Mpq ≥ Pp /Pq .

Therefore, under the hypothesis, the considered extension is always possible. It follows now by induction that any solution of Mh (h < m) can be extended to a solution of Mm = M . This theorem shows how solutions of any system can be practically constructed, step-by-step, by extending the solutions of subsystems of its derived system. Theorem 6.2 Any system which satisﬁes the triangle inequality is consistent.

The Price Level 39 For, by Theorem 5.2, M12 M21 ≥ 1; and this implies that the system M2 has a solution, which, by Theorem 6.1, can be extended to a solution of M . Therefore M has a solution, and is consistent. However, this result has already been obtained in Theorem 5.4 without appeal to the extension property, but by direct appeal to the triangle inequality instead of to this consequence. It also appears that, subject to the triangle inequality, any solution of M2 , since it can be extended to a solution of M , is a reduction of some solution of M , in other words, reductions of solutions of M exhaust the solutions of M2 , or the system has the exhaustion property. Theorem 6.3 Any system which satisﬁes the triangle inequality has the exhaustion property.

7 Consistency Theorem 7.1 The cyclical product test is necessary and sufﬁcient for consistency of L, and either Lm = M , in the modiﬁed algebra where + means min, is the equivalent derived system with the solution extension property, or system L is inconsistent. If system L is consistent, let Pt be a solution. Then, for any cycle t, i, j, . . . , k, t there are the relations Lti ≥ Pt /Pi ,

Lij ≥ Pi /Pj , . . . , Lkt ≥ Pk /Pt ,

from which it follows, by multiplication, that Ltij...kt = Lti Lij . . . Lkt ≥ (Pt /Pi ) Pi /Pj . . . (Pk /Pt ) =1 and hence Lt...t ≥ 1. Therefore, if L is consistent, all its cycles are at least 1 and the cyclical product test holds. Conversely, let this test be assumed for L. Then the derived system M is deﬁned, satisﬁes the triangle inequality, and has the interval test. Hence, by Theorem 6.3, M is consistent. But, by Theorem 4.1, M is equivalent to L. Therefore, L is consistent. This shows the converse, so the Theorem is proved. Now let L denote the actual m × m-matrix of Laspeyres indices for the system, and Lr its r-th power in a modiﬁed arithmetic where + means min, so L1 = L,

Lr+1 = Lr L

(r = 1, 2, . . .) ,

40 Concept and Method making r Lijr+1 = mink Lik Lkj ,

where it is seen, since Ljj = 1 affecting the possibility k = j, that Lijr+1 ≤ Lijr , which shows what may be termed the monotonicity of the process. In any case, for any r and i, j r Lik = Lis...tk ,

for some chain s. . .t. Subject to the cyclical test, it is proposed that, for r ≤ m the chain is. . .tj is simple. For otherwise a loop with coefﬁcient at least 1, by hypothesis, can be cancelled, and we have an element from an earlier power which is less, violating the process monotonicity. Then the series of powers either terminates in one not later than the mth, when a simple chain cannot be extended further, that is therefore repeated by its successors, or does not terminate. In the ﬁrst case, L = L1 ≥ L2 ≥ . . . ≥ Lt = M = Lt+1 = . . .

(t ≤ m) ,

with ≥ as between elements, where the terminating matrix M is the matrix of the derived system for L. In the second case it is concluded the cyclical product test is violated, system L is inconsistent, and there is no derived system. This follows Afriat (1981), Section 13 on “The power algorithm”, involving matrix powers in a modiﬁed arithmetic where × means + and + means min. There are debts to Jack Edmunds (1973) and S. Bainbridge (1978), for the connection with minimum paths, elaborated in Afriat (1987) where there is also a BASIC computer program pp. 464 ff. applied to “Getting around Berkeley in minimum time”. Here is how it could go: 0 x = L, t = 1 1

y = x, x = yL, t = t + 1

2

if x = y then M = x end

3

if t = m then end else 1.

So it appears that either L is inconsistent, or Lm = M , for which, as is equivalent to the triangle inequality, there is the idempotence M 2 = M where M is reproduced in multiplication by itself, and which is equivalent to L and has the extension property, so individual price level solutions can be constructed step-by-step, starting with any point in any derived LP-interval, which is narrower, because of additional constraints associated with additional data, than the basic or classical LP-interval that involves data just for a pair of periods, the reference periods themselves.

The Price Level 41 Of course, having the basic price levels of Section 4 available as solutions, there is no need to appeal to the extension property for the existence of solutions. However, with that property it is possible to construct other solutions, step-by-step, beside by taking convex combinations of the basic solutions. With any solution for price levels Pt there is, from their ratios, an associated determination of price indices Pst = Ps /Pt , all true, together, by reference to the same utility, better than merely true separately by reference to different utilities, as in the sense of true usually entertained. Then Prs Pst = (Pr /Ps ) (Ps /Pt ) = Pr /Pt = Prt , so that Prs Pst = Prt , which is Fisher’s Chain Test, not satisﬁed by any of the one or two hundred formulae he dealt with, and so forth with other Tests. This is a point for the observation that such price indices, any one for a pair of periods involving data from all the periods, and together giving a realization of all the “Tests” Irving Fisher proposed as proper for price indices from their nature as such, make a sharp contrast with the established tradition of algebraical formulae involving data just for the reference periods themselves, without proper compliance with such basic “Tests”, or guidance about which of the one or two hundred proposed formulae to use, despite his rankings to decide some as better than others, even “superlative”. After the procedure for ﬁnding individual solutions, the further interest is in the collection of all solutions. The solutions describe a polyhedral convex cone in the price level space of dimension m, and the normalized solutions describe a bounded polyhedral convex region in the simplex of reference, with faces or vertices to be determined, the m simplex vertices being in correspondence with the m data periods, and price levels. Then there are approximation methods to serve for the case of inconsistency. But ﬁrst notice will be taken of the price-quantity symmetry inherent in the method, and the utility background that enables all the price indices so determined to be represented as altogether true, that is, all true simultaneously on the basis of the same utility. With any determination of price levels Pt , there is an associated determination of quantity levels Xt , where Pt Xt = pt xt

(t = 1, . . . , m) .

While for price levels, pt xs /ps xs ≥ Pt /Ps ,

42 Concept and Method for quantity levels, equivalently, pt xs /pt xt ≥ Xs /Xt , and one could just as well have solved for the quantity levels ﬁrst, by the same method as for price levels, and then determined the price levels from these. Whichever way, Ps Xt ≤ ps xt

(s, t = 1, . . . , m) ,

with equality for s = t. The introduction of cost efﬁciency up to a level e, where 0 ≤ e ≤ 1, would require Pt Xt ≥ ept xt

(t = 1, . . . , m) .

good also for any lower level, and highest level 1 imposing the equality.

8 Utility model for the method First some remarks about terminology (see Appendix 2). A ray is a half-line with vertex the origin, and every point lies on just one ray, the ray through it, so a = {at : t ∈ } ⊂ C is the ray through any a ∈ C. A cone is a set described by a set of rays, and every set has a conical closure, or cone through it, or projecting it, described by the set of rays through its points. Hence

= {xt : x ∈ A, t ∈ } ⊂ C A is the cone through any A ⊂ C. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xl) = φ (x) l. With a demand element (p, x) ∈ B × C, with expenditure M = px and budget vector u = M −1 p so that ux = 1, there is the revealed preference of x over every bundle y which, being such that uy ≤ 1, is also attainable at no greater cost, as described by the relation R ⊂ C × C given by R = {(x, y) : py ≤ px} = {(x, y) : uy ≤ 1} . Then there would be the transitive closure of a collection of such relations, and a revealed preference consistency Samuelson-Houthakker type condition which excludes conﬂicting preferences.

The Price Level 43 It may be remembered that originally py ≤ px, y = x

⇒

¯ xRy, yRx

going with belief that, in a choice, presumed a maximum and so revealing preferences, it must be more than a mere maximum but moreover a unique maximum—an extra that may be hard to “reveal”. Instead, in the way of revelation without the unsuitable insistence on uniqueness which does not in any way add to preferences, simply py ≤ px

⇒

xRy

has better standing. We take liberty to conﬁne the “revelation” language to this restricted use. For conical revealed preference there would be instead the conical closure of R. Then there would be the transitive closure of a collection of such relations, and a conical revealed preference consistency which excludes conﬂicting preferences. The Laspeyres cyclical product test is exactly such a condition (a part of the version of the utility construction theorem of Afriat (1961) and (1964) then for general utility construction and now instead for conical utility). There are two attributes for a consumption bundle x ∈ C. One is that it has a money cost M = px ∈ when the prices are p ∈ B. The other, its use-value or utility, is that it is the basis for obtaining a standard of living. Hence there is a link between cost and standard of living, where prices enter. For this link a gap remains between consumption and its utility, made good hypothetically by introduction of the utility function, or utility order. A utility function is any numerical valued function φ deﬁned on the commodity space B, φ:B→ so φ (x) ∈ (x ∈ B)is the utility level of any commodity bundle x. A utility function φ determines a utility order R ⊂ C × C where xRy ≡ φ (x) ≥ φ (y) A utility function φ, with order R, ﬁts a demand element (p, x), with budget vector u, or the demand is governed by the utility, if the revealed preferences of it belong to the utility order, uy ≤ 1 ⇒ xRy (y ∈ C) . In other words, if x has at least the utility level of every bundle y (we do not insist y = x, see remark above) attainable at no greater expenditure with the prices,

44 Concept and Method or x provides the maximum utility φ (x) for all those bundles y under the budget constraint uy ≤ 1, that is py ≤ px ⇒ φ (x) ≥ φ (y) . The utility system is hypothetical and admitted to the extent that it ﬁts available demand observations. The cost of a standard of living is determined as the minimum cost at prevailing prices of getting a consumption that provides it. In terms of a utility function φ, this is gathered from the utility cost function ρ (p, x) = min {py : φ (y) ≥ φ (x)} which tells the minimum cost at given prices p of obtaining a consumption y that has at least the utility of a given consumption x. Since x itself, with cost px, is a possible such y, necessarily ρ (p, x) ≤ px

for all p, x

while ρ (p, x) = px signiﬁes the admissibility, under government by the utility system, of the demand of x at the prices p. It shows the demand is cost effective, getting the maximum of utility available for the cost, and cost efﬁcient, getting at minimum cost the utility obtained, which conditions would here be equivalent. A case where admissibility does not hold could be attributed to consumption error, described as failure of efﬁciency, where ρ (p, x) ≥ epx,

0≤e≤1

would show attainment of cost efﬁciency to a level e. This idea has use in dealing with demand data inconsistent with government by a utility, by ﬁtting it to a utility that serves only approximately, as reported below, after the account of Afriat (1973). For the service of a price index this utility cost should factorize into a product ρ (p, x) = θ (p) φ (x) , of price level P = θ (p) depending on p alone and quantity level X = φ (x) depending on x alone. This immediately is assured if φ is conical, but also the converse is true, showing the following, which we are going to prove, if it was not already, probably long ago. (Samuelson and Swamy 1974, p. 570, attribute theorem and proof to Afriat 1972).

The Price Level 45 Theorem (Utility Cost Factorization) For factorization of the utility cost function it is necessary and sufﬁcient that the utility be conical. Given φ conical, ρ (p, x) = min {py : φ (y) ≥ φ (x)} = min py (φ (x))−1 : φ y (φ (x))−1 ≥ 1 φ (x) = θ (p) φ (x) where θ (p) = min {pz : φ (z) ≥ 1} That shows the sufﬁciency. Since, for all p, θ (p) φ (x) ≤ px for all x with equality for some x, as assured with continuous φ, it follows that θ (p) = minx px (φ (x))−1 showing θ to be concave conical semi-increasing. Also for x demandable at some prices, as would be the case for any x if φ is concave, the inequality holds for all p with equality for some p, showing φ (x) = minp (θ (p))−1 px which, in case every x is demandable at some prices, requires φ to be concave conical semi-increasing. But even when not all x are demandable, because they lie in caves and are without a supporting hyperplane, here is a conical function deﬁned for all x that is effectively the same as the actual φ as far as any observable demand behaviour is concerned. So it appears that for the cost function factorization the utility function being conical is also necessary, beside being sufﬁcient, as already remarked. Hence, with some details taken for granted, the Theorem is proved. A pair of functions connected by θ (p) = minx px (φ (x))−1 φ (x) = minp (θ (p))−1 px deﬁne a conjugate pair of price and quantity functions, such that θ (p) φ (x) ≤ px

46 Concept and Method for all p, x and θ (p) φ (x) = px signiﬁes efﬁciency of the demand (p, x), of x at prices p, obtaining maximum utility for the cost and minimum cost for the utility. Instead, θ (p) φ (x) ≥ epx, where 0 ≤ e ≤ 1, will signify cost efﬁciency to a level e, as will serve for development of a utility approximation method applicable in case of inconsistency. The question now is: what utility? A price index being wanted, by the factorization theorem it must be conical, and with given demand data (pt , xt ) ∈ B × C

(t = 1, . . . , m) ,

and belief in efﬁciency, any utility to be entertained would, to ﬁt the data, have to be such that Pt Xt = pt xt , where Pt = θ (pt ) ,

Xt = φ (xt ) .

so in any case Ps Xt ≤ ps xt and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 0, 1). A question is whether a solution exists. If a solution of the system of inequalities (L) does exist, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from the efﬁciency condition Pt Xt = pt xt . A worthwhile observation is that these values Xt = φ (xt ) of the underlying utility φ are determined without ever having to actually construct the utility.

The Price Level 47 If a solution does not exist, and so there is no conical utility that ﬁts the demand data subject to the requirement of strict cost efﬁcincy, that is Pt Xt = et pt xt . where et = 1, this requirement can be relaxed to partial efﬁciency bounded to be at least a certain lower level, so pt xt ≥ Pt Xt = et pt xt . for some et ≤ 1. From this with Ps Xt ≤ ps xt we have Ps /Pt ≤ ps xt /et pt xt so, to replace system (L), we have the efﬁciency modiﬁcation Lst /et ≥ Ps /Pt , or with all et = e we have the system (L/e)

Lst /e ≥ Ps /Pt .

While with solutions of system (L) we deal with utilities that ﬁt the demand data exactly, with solutions of the efﬁciency modiﬁcation they ﬁt approximately, and the closer the et ≤ 1 to 1 the closer the ﬁt. A special case is where we take the et all equal to some e ≤ 1, as with system (L/e), and detemine the critical cost efﬁciency e∗ which is such that this system is then consistent if and only if e ≤ e∗ . With any solution of system (L), introduce

φ (x) = mini Pi−1 pi x so this is a concave conical polyhedral utility function that ﬁts the demand data, with associated price indices as required, to make those prices indices true. Another such function, concave conical, which ﬁts the demand data, again with required values and the same associated price indices, is the polytope type function given by 3 φ (x) = max Xi ti : xi ti ≤ x, ti > 0 i

i

and if φ is any other concave conical utility that ﬁts the demands and takes the values Xi at the points xi then

φ (x) ≤ φ (x) ≤ φ (x) for all x. 3 The function of this form introduced by Afriat (1971) is the constant-returns “frontier production function” that gives a function representation, and at the same time a computational algorithm, for the

48 Concept and Method Included in the above is the simple conical precursor of the general theorem on utility construction put in service speciﬁcally for price index theory. Thus, the concave polyhedral function

φ (x) = mini pi x/Pi = max t : t ≤ pi x/Pi and the concave polytope function θ (p) = min px : pi x ≥ Pi for all i = max vi P i : vi pi ≤ p by LP duality

i

i

are a conjugate pair of quantity and price functions such that

φ (xt ) = Xt

θ (pt ) = Pt ,

where, with ast = ps xt /pt xt ,

bst = pt xs /pt xt

P’s and X ’s connected by Pt Xt = pt xt are, equivalently, such that ast ≥ Ps /Pt ,

bst ≥ Xs /Xt .

For another such conjugate pair, instead, φ (x) = max wi Xi : wi x i ≤ x i

i

θ (p) = mini pxi /Xi .

production efﬁciency measurement method of Farrell (1957) (Afriat’s colleague at DAE Cambridge whose work, done after he left, he at ﬁrst missed) that marks the beginning of “data envelope analysis” (DEA). The comment by Afriat attached to Finn R.Førsund and Nikias Sarafoglou (2005) gives a report. The same type of function but without constant-returns is used for the utility construction in Afriat (1961) but arbitrarily—or for simplicity!, or for the reasons in remarks already made here about over-stringent “revealed preference”—left aside in the account of (1964), where a modiﬁed revealed preference condition to avoid the excess of the original and a polyhedral type function are used instead, as again in accounts such as Fostel et al. (2003). It also served for the 1971 extension of Farrell’s method by an accidental transfer of ideas from demand analysis.

The Price Level 49 These pairs of conjugate functions are such that

θ (p) ≥ θ (p) ,

φ (x) ≤ φ (x) ,

and any other pair for which θ (pt ) = Pt ,

φ (xt ) = Xt

are such that

θ (p) ≥ θ (p) ≥ θ (p) ,

φ (x) ≤ φ (x) ≤ φ (x) .

9 Solution structure The price levels are determined as solutions of the system (M ) Mst ≥ Ps /Pt , derived from and equivalent to the system L, subject to the Laspeyres cyclical product test required for consistency. For a restatement of the inequalities affecting Pt , (Mst )

Mst Pt ≥ Ps ,

and equivalently (Kts )

Pt ≥ Kts Ps .

Any positive solution Pr of system M deﬁnes a permissible system of price levels, represented by a point P in the price level space = m of dimension equal to the number of periods m. The set C of solutions is immediately a polyhedral convex cone in this space. When price levels are normalised to have sum 1 they describe a simplex in the space . This simplex is cut by the cone C in a bounded convex polyhedron, or polytope, D. The cone C is recoverable from its section D, as the cone through that section projecting it from the origin. Taking price levels to be normalised and so represented by points in the simplex is convenient for computation, and for geometrical representation. Only ratios of price levels are signiﬁcant and these are unaltered by normalisation. Every point in the normalised solution set D of the system M is a convex combination of a ﬁnite set of basic solutions, and so the computational problem requires ﬁnding just these. Given any solution Pr we form the matrix of price indices Pst = Ps /Pt , depending only on the price level ratios.

50 Concept and Method Z (

r

S Mrs

Figure 1

Now there will be explorations for a geometrical and diagrammatic understanding of the system M . Dealing with any three periods r, s, t is illustrative of essential features. While the associated solution cone Crst may be hard to visualise, the normalised solution polytope Drst in the simplex rst is much easier, and can be represented graphically. We can refer to any constraint of the system M by the two periods involved, so, as already above, let (Mrs ) denote the general constraint. There has already been some discussion of the case with two periods, in dealing with the P-L interval. Vectors of price levels for any subset of periods r, s, . . ., understood as representing only the ratios, can be denoted Pr:s:... = (Pr : Ps : . . .) . Any period r corresponds to the vertex of the simplex where Pr = 1, and vertices can all be labelled by the corresponding periods. Any point on the edge rs of the simplex corresponds to a ratio Pr : Ps , that is, Pr:s in the notation just introduced. Similarly any point in a simplex face rst speciﬁes the ratios Pr:s:t and so forth for any dimension. The constraint (Mrs ) cuts the edge rs in a point Z and requires Pr:s to lie in the segment Zs, where (rZ : Zs) = (1 : Mrs ) = (Ps : Pr ) Without ambiguity, we can refer to the segment Zs on the edge rs as the segment Mrs , as in Figure 1. At the same time, the constraint (Mrs ) requires Pr:s:t to lie in the simplex Zst, and so forth to any dimension. Considering now a pair of constraints (Mrs ) and (Msr ), we have two segments Mrs and Msr on the edge rs, and they have a nonempty intersection Drs shown in Figure 2. This lies within the Paasche-Laspeyres interval, and is a generalisation of that for when data from other periods are involved. It is generally narrower because any effect of extra data must be to reduce indeterminacy. Msr r

(

Drs

) Mrs

Figure 2

S

The Price Level 51 S

Mrs

Mst

r

Mrt

Y

t

Figure 3

Now consider three constraints associated with the triangle inequality as shown in Figure 3. Two of them produce intervals Mrs and Msr on rs and st and, as it were with the triangle equality instead, jointly produce the interval Yt on rt,. The triangle inequality requires Mrt to be a subinterval of this. If instead of Mrt we take Mtr (see Figure 4) cyclically related to the other two, the resulting joint constraint determines a triangle lying within rst. The other three cyclically related constraints, associated with the opposite cyclic order, determine S

Mrs

Mst

r

Figure 4

Mtr

t

52 Concept and Method S

Drs

Dst

Drst

Drt

r

t

Figure 5

another triangle, so conﬁgured with the ﬁrst that their intersection is a hexagon, Drst , as in Figure 5, by the triangle inequality assured non-empty. It is seen in this ﬁgure that Drs is exactly the projection of Drst from t on to rs. In other words, as Pr:s:t describes Drst , Pr:s describes Drs . Or again, for any point in Drs , there exists a point in Drst that extends it, in the sense of giving the same ratios concerning r and s. That is the extension property described earlier, a consequence of the triangle inequality, and it continues into higher dimensions indeﬁnitely: Drs...t is the projection of Drs...tv from the vertex v of the simplex rs. . .tv onto the opposite face rs. . .t That shows how price levels for the periods can be determined sequentially, one further one at a time. Having found any that satisfy the constraints that concern only them, they can be joined by another so that is true again. Starting with two periods and continuing in this way, ﬁnally a system of price levels will have been found for all the periods. For when the data for a price index between two periods involves data also from other periods, and moreover indices for any subset of periods are to be constructed consistently, these D-polytopes constitute a twofold generalisation of the PaascheLaspeyres range of indeterminacy of a price index between two periods taken alone. For a comment on the triangle inequality and equality, along with Z on rs where (rZ : Zs) = (Ps : Pr ),

The Price Level 53 now introduce X on st where (sX : Xt) = (Pt : Ps ). Let rX and tZ meet in P. Then sP meets tr in Y where (tY : Yr) = (Pr : Pt ). So it appears that by choosing the points Z and X for ratios z and x, we arrive at point Y for a ratio y where y = zx. In other words, we have here a geometricalmechanical multiplication machine, also good for division since from Y and Z for y and z we can arrive at P and so determine X and x for which y = zx, that is, x = y/z.

10 Basic solutions Taking price levels to be normalised and so represented by points in the simplex is convenient for computation, as for geometrical representation, when that is possible. Only the ratios of price levels are signiﬁcant and these are unaltered by normalisation. The normalised solution set of the system M is a convex polyhedron D in the simplex , every point of which is a convex combination of a ﬁnite set of basic solutions, or vertices. The computational problem requires ﬁnding just these. The cases with two periods, or three and four, can serve for a start. Every conical utility has associated with it a price index, derived from the utility cost factorization applicable to such a function. A price index is termed true if it is connected with a conical utility that ﬁts the demand data. Every solution for price levels determines true price indices given by their ratios, the existence of a solution requiring the cyclical Laspeyres product test, that requires the cyclical Laspeyres products to be all at least 1. It should be seen what all this has to say in reduction to the classical case of just two periods. In this case the existence of a solution for price levels is equivalent to the LPinequality, and then any point in the LP-interval is representable as a price index, obtained as the ratio of the price levels, which is a true price index from being associated with a conical utility that ﬁts the data. Hence, as values for the price index, all points in the LP-interval are true—all equally, no one more than another (this should dim the aura of extra truth given to Fisher’s Ideal Index, especially after it became connected with a—possibly nonexistant—quadratic utility). When this was submitted a few decades ago, possibly at the Helsinki Meeting of the Econometric Society, August 1976, it was received with complete disbelief (a proof is in Afriat (1977), 129–30). Here is a formula to add to Fisher’s collection, a bit different from the others: PRICE INDEX FORMULA: Any point in the LP-interval, if any. However, now we deal rather with price levels and should put this formula in such terms. Now the simplex is a line segment, so with two vertices. Each point

54 Concept and Method of the segment corresponds to a ratio of price levels in a solution, and so to a price index. A segment in it, corresponding exactly to the PL-interval, is the normalized price level solution set, with vertices for L and P. These are the basic solutions from which all other solutions are determined. There is not much more that can be said about this case, except that it is a generalization of it that makes the present subject. The case of three periods is already more complex and substantially more interesting, and evocative of the shape of things to come. Already a start was made with that in the last section. Having the picture there obtained, of the hexagonal boundary of the normalized solution set, the immediate task is to obtain formulae for the six vertices. The treatment for system (L) consists mainly in the power algorithm for testing consistency and forming the derived system (M ), equivalent to (L), with the triangle inequality and solution extension property that enables solutions to be constructed step-by-step, starting with two variables and following a path for adding variables, to conclude with an individual solution. At each stage the choice to be made can keep the solution as a vertex of the current solution set, so ﬁnally there will be arrival at a vertex, making a basic solution. To construct a complete basic solution set this way could be laborious. Firstly the path for adding variables has m! possibilities, and with any one path there is a choice between two possibilites at every extension stage. It seems, therefore, there may be about m! × 2m−1 basic solutions, if any, or fewer distinct ones to allow coincidences, with the symbolic description (t1 t2 − v2 , t3 − v3 , . . . , tm − vm ) where vi = 1 or 2. For this discussion, the extension path will simply be 1, . . . , m in that order, though we may not get very far along it. For P1 and P2 referring to periods 1 and 2 (reference denoted 12) there are two basic (non-normalized) solutions (12 − a)

P1 = 1,

(12 − b)

P1 = M12 ,

P2 = M21 . P2 = 1.

Were we dealing with system (L) these would correspond to the L and P bounds of the LP-interval. For (12-a) there is the veriﬁcation M21 ≥ P2 /P1 = M21 M12 ≥ P1 /P2 = (M21 )−1 the second line providing conﬁrmation because M12 M21 ≥ 1. For (12-b) similarly. One of these solutions has to be chosen initially, say (12-a). This can be extended to include a third variable, for period 3, relying on the triangle inequality and the solution extension property that follows from it. Consider (12 − a, 3 − a)

P1 = 1,

P2 = M21 ,

P3 = M31 .

The Price Level 55 This is a solution that extends the solution (12-a), as may be veriﬁed with appeal to M13 M31 ≥ 1, and appeals to the triangle inequality, M32 M21 ≥ M31 and M23 M31 ≥ M21 . Similarly (12 − a, 3 − b)

P1 = 1,

P2 = M21 ,

P3 = 1/M13

is another solution that extends (12-a). If we identify s, t, r of the last section with 1, 2, 3 in this, we have (12-a,3-a), when normalized, corresponds to the lower of the middle pair of vertices of the hexagon, associated with simplex vertex 1, just as (12-a,3-b) is the upper of the pair. Or something like that. Similarly there are pairs of solution vertices similarly associated with the other two simplex vertices 2 and 3. That makes the six vertices of the hexagon. Consider (12 − a, 3 − a, 4 − a)

P1 = 1,

P2 = M21 ,

P3 = M31 ,

P4 = M41 .

This is a solution that extends (12-a, 3-a). And so forth. There may be more to say but for now it may be suitable to submit going further with this approach to the brute computer. However, there is reassurance to be gained from the circumstance that we already have the basic solutions, of Section 5, obtained without tedious stepby-step extension but immediate and complete from a reference to the triangle inequality. None the less there is interest in the determination of all basic solutions, or vertices of the convex polyhedron in the simplex of reference that describes all normalized solutions, illustrated graphically for the case m = 3 in Section 8. The 2m solutions provided by pairs of basic solutions in respect to the m possible bases should be the vertices of the convex polyhedron of all price level solutions normalized to make them points in the simplex of reference. For instance in Section 9 we have 2 × 3 = 6 vertices of the hexagonal region. This would be, once again, as with the basic price levels themselves, a providential ready made solution for what might otherwise have seemed a burdensome abstruse computation.

11 Inconsistency and approximation A demand correspondence being deﬁned as a correspondence between budget constraints and admitted commodity bundles, here the concern is with a ﬁnite correspondence. The approach to constructing a utility that ﬁts such data is most familiar, and now there has been account of the matter where the utility is restricted to be conical, as suits treatment of price indices. When the demand data does not have the consistency required for exact admission of a utility, there arises the question of how to admit a utility approximately. Here the impossibility of exactness is treated as due to error, represented as a failure of efﬁciency.

56 Concept and Method A theorem will be proved on the existence of a positive solution for a certain system of homogeneous linear inequalities. Such a system can be associated with any ﬁnite demand correspondence, together with a number e between 0 and 1 interpreted as a level of cost efﬁciency. The existence of a solution is equivalent to the admissibility of the hypothesis that the consumer, whose behavior is represented by the correspondence, (i) has a deﬁnite structure of wants, represented by an order in the commodity space, as is essential in dealing with price indices, and (ii) programs at a level of cost efﬁciency e. Any solution permits the immediate construction of a utility function which realizes the hypothesis. When e = 1 the utility function ﬁts the data exactly, in the usual sense that its maximum under any budget constraint is at the corresponding commodity point, and when e < 1 it can be considered to ﬁt it approximately, to an extent indicated by e. A determination is required for the critical cost efﬁciency, deﬁned as the upper limit of possible e. Demand analysis which ordinarily knows nothing of approximation and also treats not just a maximum but a strict maximum under the budget constraint, as expressed by the original “revealed preference” idea, is put in perspective with this approach. A utility relation is any order in the commodity space n , that is any R ⊂ n ×n which is reﬂexive and transitive, xRx,

xRyR . . . Rz ⇒ xRz

A utility function is any φ : n → . It represents a utility relation R if xRy ⇔ φ (x) ≥ φ (y) . Such representation for R implies it is complete, xRy ∨ yRx. Consider a utility relation R and a demand element (p, x) with px > 0. A relation between them is deﬁned by the condition (H ∗ )

¯ py ≤ px, y = x ⇒ xRy, yRx

which is to say x is strictly preferred to every other y which costs no more at the prices p. If R is represented by a utility function this condition is equivalent to (H ∗ )

py ≤ px, y = x ⇒ φ (x) > φ (y)

With u = M −1 p where M = px, an equivalent statement, in terms of the associated budget element (u, x), is (H ∗ )

¯ uy ≤ 1, y = x ⇒ xRy, yRx.

The Price Level 57 This can be called the relation of strict compatibility between a utility relation, or function, and a demand, or its associated budget. A demand correspondence being a set D of demand elements, the condition HD∗ (R) of strict compatibility of R with D is deﬁned by simultaneous compatibility of R with all the elements of D. The existence of an order R such that this holds deﬁnes the strict consistency of D. The original “revealed preference” theory deals with this condition. Now let further relations between a utility relation R and an demand correspondence D be deﬁned by HD (R) ≡ xDp, py ≤ px ⇒ xRy HD (R) ≡ xDu, yRx ⇒ py ≥ px with conjunction HD (R) ≡ HD (R) ∧ HD (R) by which R and D can be said to be compatible. Thus H signiﬁes that x is as good as any y which costs no more at the prices p, or that maximum utility is obtained for the cost, and H signiﬁes any y which is as good as x costs as much, or that the utility has been obtained at minimum cost. In the language of cost-beneﬁt analysis, these are conditions of cost efﬁciency and cost efﬁcacy. Evidently HD∗ (R) ⇒ HD (R) that is, compatibility is implied by strict compatibility. Let HD be deﬁned for H in the same way as the similar conditions for H ∗ , and similarly with H and H . Then HD asserts the consistency of D. It is noticed that HD (R) derives from HD∗ (R) just by replacing the requirement for an absolute maximum of original “revealed preference” by a requirement for a maximum. But while HD∗ , and similarly HD , is a proper condition, that is there exist D for which it can be asserted and other D for which it can be denied, HD is vacuous, since it is always validated by a constant utility function. It can be remarked, incidentally, that if R is semi-increasing, x > y ⇒ xRy then H ⇒ H . Also if R is lower-continuous, that is the sets xR = [y: xRy] are closed, then H ⇒ H . Accordingly if, for instance, R is represented by a continuous increasing utility function then H and H are equivalent, so in their conjunction one is redundant,

58 Concept and Method that is mathematically but not economically. But there is no need here to make any assumptions whatsoever about the order R. It can be granted that as a basic principle H ∗ requiring an absolute maximum is unwarranted in place of the more standard H which requires just a maximum. However, while H ∗ produces the well known discussion of Samuelson (1948) and Houthakker (1950), described as revealed preference theory—more suitably revealed preference plus revealed non-preference—that discussion is not generalized but its entire basis evaporates when H ∗ becomes H . From this circumstance there is a hint that the nature of that theory is not properly gathered in its usual description. The critical feature of it is not that it deals with maxima under budget constraints but that it deals especially with absolute maxima. This might have intrinsic suitability, by mathematical accident, for dealing with continuous demand functions. But it is not a direct expression of normal economic principles, which recognize signiﬁcance only for a maximum—not that the maximum under the budget should moreover be unique so revealing an additional non-preference signiﬁcance. If the matter is to be reinitiated, then H is admitted as such a principle and so equally is H , so their conjunction H comes into view as an inevitable basis required by normal economic principles. The question of HD for an expenditure correspondence is proper, that is, capable of being true and false, unlike HD which is always true. Also, since H ∗ ⇒ H , this provides a generalization of the usual theory with H ∗ . It happens, as the mathematical accident just mentioned, that if D is a continuous demand function then HD∗ ⇔ HD . Thus the distinctive revealed preference theory is not lost in this generalization but it just receives a reformulation which puts it in perspective with a normal and broader economic theory not admitting description as revealed preference theory, which moreover is capable of a further simple and necessary extension now to be considered. With a demand correspondence D interpreted as representing the behavior of the consumer, there is the hypothesis that the consumer (i) has a deﬁnite structure of wants, represented by a utility relation R, and (ii) is an efﬁcient programmer. Then HD is the condition of the consistency of the data D with that hypothesis. If it is not satisﬁed, so the data reject the hypothesis, the hypothesis can be modiﬁed. If (i) is not to be modiﬁed, either because there is no way of doing this systematically or because it is a necessary basic assumption, as it is for instance in economic index number theory, then (ii) must be modiﬁed. Instead of requiring exact efﬁciency, a form of partial efﬁciency, signiﬁed by a certain level of cost efﬁciency e where 0 ≤ e ≤ 1, will be considered. When e = 1 there is return to the original, exact efﬁciency model. Thus consider a relation H between a demand (p, x) and a utility relation R together with a number e given by the conjunction of conditions

H py ≤ Me ⇒ xRy H yRx ⇒ py ≥ Me

The Price Level 59 where M = px. They assert x is as good as any y which costs no more than the fraction eM of the cost M of x, at the prices p, and also any y as good as x costs at least that fraction. In the language of cost-beneﬁt analysis these are conditions of cost efﬁcacy and cost efﬁciency, but modiﬁed to allow a margin of waste, which is the fraction (1 − e) M of the outlay M . It is noticed that if H is not to be satisﬁed vacuously then e > 0; and then from H ", with R reﬂexive necessarily e ≤ 1. With R given, for simplicity of illustration say by a continuous increasing strictly quasiconcave function φ, and with p > 0 and M ﬁxed, it can be seen what varying tolerance this condition gives to x as e increases from 0 to 1. When e = 0, x is permitted to be any point in the budget simplex B described by px = M , x ≥ 0. When e = 1, x is required to be the unique point x on B for which φ (x) = max {φ (y) : py = M } . For 0 ≤ e ≤ 1 let xe be the unique point in the set Be described by px = Me for which φ (xe ) = max {φ (y) : py = Me} .

S0=B1

Se

x1

S1

xe Be

B0

x0

Then x is required to be in the convex set Se ⊂ B deﬁned by φ (x) ≥ φ (xe ) , px = M . Evidently, if 0 ≤ e ≤ e ≤ 1

60 Concept and Method then B = S0 ⊃ Se ⊃ Se ⊃ S1 = x1 That is, the tolerance regions Se for x form a nested family of convex sets, starting at the entire budget simplex B when e = 0 and, as e increases to 1, shrinking to the single point x1 attained when e = 1. The higher the level of cost efﬁciency the less the tolerance, and when cost efﬁciency is at its maximum 1 all tolerance is removed: the consumer is required, as usual, to purchase just that point which gives the absolute maximum of utility. For a demand correspondence D, now deﬁne compatibility of D with R at the level of cost efﬁciency e to mean this holds for every element of D.Then e-consistency of D, or consistency at the level of cost efﬁciency e, stated HD (e), will mean this holds for some R. Immediately HD (1) ⇔ HD so l-consistency of E is identical with the formerly deﬁned consistency. Also 0-consistency is valid for every E. Further HD (e) , e ≤ e ⇒ HD e that is, consistency at any level of cost efﬁciency implies it at every lower level. Hence with eD = sup e : HD (e) deﬁning the critical cost efﬁciency of any expenditure correspondence D it follows that 0 ≤ eD ≤ 1, e < eD ⇒ HD (e) ,

e > eD ⇒ H¯ D (e)

The condition HD (e) will now be investigated on the basis of a ﬁnite demand correspondence D with elements (pt , xt ) ∈ B × C (t = 1, . . . , m) , and belief in perfect efﬁciency, any utility to be entertained would, to ﬁt the data, have to be such that (PX =)

Pt Xt = pt xt ,

where Pt = θ (pt ) , Xt = φ (xt ) .

The Price Level 61 so in any case (PX ≤)

Ps Xt ≤ ps xt

and now, with Lst = ps xt /pt xt , the Laspeyres index, this condition requires the solubility of the system of inequalities (L)

Lst ≥ Ps /Pt ,

for price levels Pt (t = 1, . . . , m). A question is whether a solution exists. If one does, a conical utility can immediately be constructed that ﬁts the given demand data and provides price levels, and consequently also quantity levels Xt , as required, where the Xt are determined from (PX =). If instead of perfect efﬁciency there is to be allowance of partial efﬁciency, at some level e, then (PX =) would be replaced by (PXe) Pt Xt ≥ ept xt , where 0 ≤ e ≤ 1, which for the perfect efﬁciency case e = 1, in view of (PX ≤), becomes again (PX =). Now from (PXe), with (PX ≤), follow the systems (a)

ast /e ≥ Ps /Pt ,

(b)

bst /e ≥ Xs /Xt ,

where ast = ps xt /pt xt ,

bst = pt xs /pt xt

with P’s and X ’s connected by Pt Xt = pt xt . These systems, even if not consistent for e = 1, are always consistent for sufﬁciently small e. From any solution there is obtained a utility that shows demand elements as efﬁcient within the level e. Thus, with φ (x) = mini Xi pi x/pi xi and antithetical θ (p) = min px : Xt pt x ≥ Pt

62 Concept and Method it appears that pt xt ≥ θ (pt ) φ (xt ) ≥ ept xt as required for compatibility at a level of cost efﬁciency e. In case e = 1, then moreover φ (xt ) = Xt ,

θ (pt ) = Pt .

Since ast = ps xt /pt xt is just the Laspeyres index Lst , a restatement of system (a) is the system (L/e)

Lst /e ≥ Ps /Pt .

This can be dealt with following exactly the treatment given to the system (L), by replacing the Laspeyres index Lst by Lste = Lst /e. Then e = Ls...t /e . . . e Ls...t

so that e ≥ 1 ⇔ Lt...t ≥ e . . . e. Lt...t

So it appears that either system (L) is consistent, in which case also system (L/e) is consistent with e = 1, or critical cost efﬁciency e∗ can be determined so that e ≥ 1 ⇔ e ≤ e∗ . Lt...t

Introducing Lst∗ = Lst /e∗ , the system (L∗ )

Lst∗ ≥ Ps /Pt ,

is consistent and determines price levels associated with a utility that represents the given demands as together within a cost efﬁciency at the highest level, in that sense a best approximation to a utility that ﬁts the data, coinciding with a utility that ﬁts the data exactly when that exists. The treatment of (L∗ ) follows exactly the treatment already accounted for the system (L). At this point it can be remarked that, with all additional discussion about it put aside, the system (L∗ ) is the embodiment of the entire method now proposed for the computation of price levels Pt and then price indices Pst = Ps /Pt always available and together true in the exact or approximate sense on the basis of demand data for any number of periods.

The Price Level 63

12 Old and new: an illustration Some illumination is provided by what this method provides for the classical case of two periods, worked for so long by so many authorities that it may seem unlikely there is anything to add there. The data consists in a pair of demands (pt , xt ) ∈ B × C (t = 1, 2) in terms of which there are conventional algebraical (not fancy combinatorial) formulae for price indices, especially those associated with Paasche, Laspeyres and Fisher, beside the one or two hundred in Fisher’s list. The Laspeyres is Lst = ps xt /pt xt , Paasche Kst = (Lts )−1 , and Fisher 1

1

Fst = (Kst Lts ) 2 = (Lst /Lts ) 2 . For the consistency case L12 L21 ≥ 1, where Paasche does not exceed Laspeyres, the PL-interval is non-empty and all points in it are accepted as true price indices, all equally true, no one truer than another. In the contrary case, the data does not admit the existence of true price indices at all, at least not exactly, the PL-interval is empty, and now instead for the critical cost efﬁcincy e∗ , that makes the system Lst /e ≥ Ps /Pt consistent if and only if e ≤ e∗ , which requires L12 L21 = e∗ e∗ there is the determination 1

e∗ = (L12 L21 ) 2 and now 1

∗ L12 = L12 /e∗ = (L12 /L21 ) 2 ,

so that, for the Paasche index ∗ ∗ −1 ∗ ) = L12 K12 = (L21

1

∗ L21 = L21 /e∗ = (L21 /L12 ) 2

64 Concept and Method and the system (L∗ )

Lst∗ ≥ Ps /Pt

(s, t = 1, 2) ,

for determination of approximate price levels, is equivalent to (L∗ )

∗ ∗ , K12 ≤ P1 /P2 ≤ L12

is consistent, but here the limits are coincident and the only price index obtained from a solution is the value 1

P1 /P2 = (L12 /L21 ) 2 —incidentally, usually known as Fisher “Ideal Index”. If the critical e∗ is replaced by a more tolerant lower level e, the system is still consistent, with limits now no longer coincident but admitting a range of values, again including the Fisher index but now not unique but just one of its many points. Hence here we have a New Comment about the Fisher index. For the Old Comment, in the consistency case, Fisher, being the geometric mean of Laspeyres and Paache, is a point of the now non-empty interval, and so is a true index like any other, and no truer than another. This gives a value to Fisher as being a true index, but also it is deﬂating from making it no more distinguished than the others. There was a moment of distinction when Fisher became associated with a quadratic utility, which then became put aside, though recently there may have been what may seem to some to be something of a renaissance, even a cult, see Afriat and Milana (2006). For the New Comment, in the case of inconsistency, when the LP-interval is empty and there are no true indices at all, at least not exactly, at which point in the absence approximation ideas the matter is usually abandoned, Fisher now stands out from being alone associated with a utility that ﬁts the data as closely as possible, in the way here approximation is understood that has reference to cost efﬁciency criteria. After the ﬁrst deﬂation this gives a real distinction to the Fisher “Ideal” index, and a good reason for the term Fisher gave to it even though not one he had in mind. If one does not want to always trouble about consistency and still have an in some way signiﬁcantly “true” price index, surely this is it—as “superlative” as can be, in the language Irving Fisher invented and has had a perplexed persistence in echoings since. Have latter day pedlars of the superlative ever promoted such a quality in their fancy? Fisher’s index having this new status, its generalization invites consideration. Every point in the entire interval between Laspeyres and Paasche is the possible value for a true index. In this unacceptable indecision the Fisher index, as the geometric mean of the limits, at least picks out one value. Now with the new method there is again the unfortunate indecision, even expanded since the line segment is now replaced by a multi-dimensional

The Price Level 65 polyhedron. For a fair remedy such as was found before, it may be fair to try some manner of immitation of the original Fisher index. The points in the LP-interval are “true” merely in the theoretical sense related to utility, without being overruled by some further possibly more real sense perhaps like “actual” which, for want of an approach, cannot be grasped further, or estimated. But it is contradictory for a method to produce price indices to conclude with an unresolved choice. Since “true” based on utility is the outstanding qualiﬁcation in current logic, though there is no further qualiﬁcation available, it is still important to have a proposal for picking a true point out of many, such as may be provided by the PL-interval and can apply just as well when the interval becomes a polyhedron in many dimensions: True with minimum distance from non-true maximum. This is one path to the Fisher index and generalizations. Our approach deals directly with price levels instead of price indices which are their ratios. True just means a solution of the inequalities to determine price levels. Rather than being on the true boundary and so almost non-true it is better to be comfortably in the interior surrounded by true and far away from non-true. The way we adopted to produce a single true outcome is thrown up by basic solutions and judgement about form giving them equal weight. That is all. No single point can be viewed as anything like an “estimate” of the “actual” from among the many “true”, or “actually true” from among the “possibly true”. Though there may be words for that sort of thing there is absolutely no sense. Here the derived system M may just as well be replaced by M ∗ is the case of inconsistency, requiring approximation. Everything that follows now applies equally well in either case. The basic price levels, base t, are Pi = Mit . and Pi = 1/Mti , with geometric mean 1

Pi = (Mit /Mti ) 2 which is also a price level solution, determine systems of basic price indices Pij = Mit /Mjt and Pij = Mtj /Mti

66 Concept and Method with geometric mean 1 Pij = Mit Mtj /Mti Mjt 2 But this geometric mean price index is identical with the price index determined from the geometric mean price levels, 1 1 Pij = (Mit /Mti ) 2 / Mjt /Mtj 2 . Going further, similarly, the geometric mean of all the basic price levels, for all bases, is again a price level solution, the basic mean price level solution, and the price indices derived from it is a price index system where each price index is the geometric mean of the basic price indices, the basic mean price index system. Any price index in this unique last system is a generalized counterpart of the Fisher index, and in the classical case of just two periods it becomes exactly the Fisher index. Thus though the price level solutions, and so also price indices they determine, are many, the geometric mean, element by element, of the basic solutions is again a solution which determines unique price indices that are geometric means of the basic price indices. Here is a fair conclusion in the quest for elimination of indecision, a multi-period generalization of the Fisher index that even has no conﬂict with Fisher’s own “Tests”.

13 Conclusion Though the mathematics of the method, its theoretical rationalization and computations, require an account, the scheme for applications is simple, and conveys an idea of what could be meant by an answer to “The Index Number Problem”. A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of an entirely different type and are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, for the hypothetical underlying utility which in any case there is no need to actually construct. With some m periods listed as 1, . . ., m and demand data (pi , xi ) (i = 1, . . . , m)

The Price Level 67 giving row and column vectors of prices and quantities for some n goods, the ﬁrst step is to compute the matrix L of Laspeyres indices Lij = pi xj /pj xj and raise it to the mth power M = Lm in a modiﬁed arithmetic where + means min. Diagonal elements Mii ≥ 1 tell the consistency of the system (L)

Lij ≥ Pi /Pj

for the determination of price levels Pi , and provide the ﬁrst and second basic price level solutions, with any t as base, given by Pi = Mit , and Pi = 1/Mti , from which are derived two systems of basic price indices Pij = Pi /Pj . The price indices in either system, with any base, will all be true together in respect to a utility that ﬁts the data by criterion of cost efﬁciency of demand in each period i, so the cost pi xi is the minimum cost, at the prices pi , of the utility of xi . Diagonal elements Mii < 1 tell the inconsistency of the system, and enable determination of a critical cost efﬁciency e∗ so that the system (L/e)

Lij /e ≥ Pi /Pj

is consistent if and only if e ≤ e∗ (features in the computation of e∗ remain to be clariﬁed). Then with Lij∗ = Lij /e∗ the system (L∗ )

Lij∗ ≥ Pi /Pj

is consistent, and with M ∗ = (L∗ )m there may be obtained basic price levels and price indices from M ∗ , as before from M . Now instead the price levels of a basic system are together true in respect

68 Concept and Method to a utility that ﬁts the data now not exactly, but approximately in the sense of partial cost efﬁciency at the level e∗ in each period, meaning that the fraction e∗ of the cost, in the period, is at most the minimum cost at the prices of gaining at least the utility. Hence in the case e∗ = 1 that goes with ordinary consistency, the ﬁt would be exact as before.

Chapter 3 Price Level Computation: Illustrations Introduction I Outline of the method 1 2 3 4 5

Original data Consistency of the data Price-quantity duality The power algorithm Cost efﬁciency and approximation II Illustrations

1 2 3 4 5

Three references with consistency, and graphics Four references, with consistency Case of inconsistency and approximation Inconsistency and approximation again EUKLEMS data Notes

1 2 3

Price levels The triangle equality Ratio matrix

Introduction

The theory of the Price Index, proper, starts with the Utility Cost Factorization Theorem, going back to early1960’s. By itself it represents no resolution of the Index Number Problem, nor had there even been a real idea of what could be meant by such a resolution. However, the method now proposed does convey some idea of what could be meant by such a resolution. It even represents such a resolution itself. The method has been available in the main for more than twenty ﬁve years, apart from ampliﬁcations made just now. But only recently has it been recognized as a proper resolution of the Index Number Problem. These ﬁrst exercises with the arithmetic go to convey the practicality of it. Needs of dealing with the EUKLEMS Project data when it becomes availalbe have stirred into life that almost forgotten work and exposed its value. This is the third of three papers by present authors making Part I. The ﬁrst “The Super Price Index: Irving Fisher, and after” has more to do with history, and “The Price Level Computation Method” is an exposition of the mathematics. We start with arbitrary ﬁgures for the Laspeyres matrix L related to quantity indexes of per capita GDP in an inter-country comparison based on the International Comparison Project (ICP) data for 1980 published by the United Nations and the Commission of the European Communities (1987). This accidental source gives prices and quantities for some 38 components of GDP expenditure for 60 countries. In these applications we take the data for various countries, for instance in the ﬁrst illustration just for US, France, and Italy, to form the matrix L. Since prices and quantities can be interchanged symmetrically in the method, and the data is arbitrarily in use just to illustrate computational procedure, there is liberty now to say “price” for afﬁnity with the more usual subject even when “quantity” may ﬁt the data source better, but any reader may always for “price” read “quantity”.

Price Level Computation 71 I OUTLINE OF THE METHOD

1 Original data A price index formula based on a pair of reference periods has conventionally been algebraical and involved data for those periods alone. Then there are inconsistencies between formulae in the treatment of more than two periods, conﬂicting with the nature of price indices as such, as gathered by Irving Fisher’s “Tests”. Formulae proposed now are of a completely different type, beside being “nonparametric” rather than conventionally algebraical, are computed simultaneously for any number of periods, involving the data for all of them, without any of the multi-period consistency problems that go with the conventional formulae. There is either exactness, subject to a condition on the data, or approximation, in the ﬁt to the data of the hypothetical underlying utility, which in any case there is no need to actually construct. With some m time periods, or countries, or nodes, in any case references— perhaps most typically time periods—listed as 1,…,m, the initial data has the form of some m demand elements (pt , xt )

(t = 1, . . . , m)

giving row and column vectors of prices and quantities for some n goods demanded at the prices. Hence for the initial data scheme: m number of references n number of goods p m × n price matrix, rows pi x n × m quantity matrix, columns xj c = px m × m cross-cost matrix, elements pi xj The ﬁrst step is to compute the matrix L of Laspeyres indices Lij = pi xj /pj xj i being index for the current period and j for the base period. Hence divide column j of c by diagonal element pj xj to form the m × m Laspeyres matrix L with these elements. The Paasche indices are given by Kij = 1/Lji = pi xi /pj xi , forming the elements of an m × m matrix K, obtained by transposition of L and replacing each element by its reciprocal. The Laspeyres-Paasche (LP) inequality Kij ≤ Lij

72 Concept and Method has signiﬁcance for Laspeyres and Paasche indices as price index bounds, and for data consistency. Another well known construction that may have comment is the Fisher index which is the geometric mean of the Laspeyres and Paasche indices, 1 1 Fij = Lij Kij 2 = pi xj pi xi /pj xj pj xi 2 Central to the proposed method is the system of inequalities (L)

Lij ≥ Pi /Pj .

This serves to determine price levels Pi from which the matrix P of price indices Pij = Pi /Pj is derived, and which enter into the construction of an underlying utility which ﬁts the given demand data and represents all these indices together as true. By the geometric mean of two vectors is here meant the vector whose elements correspondingly are geometric means of their elements, and there is a similar understanding about matrices. The same understanding can apply just as well for several vectors, or matrices, also in application of the more general weighted geometric mean. Any two price level solutions Pia and Pib have a geometric mean with elements which are geometric means 1 Pic = Pia Pib 2 of their elements, which also is a price level solution. For from Lij ≥ Pia /Pja

and

Lij ≥ Pib /Pjb

follows Lij Lij ≥ Pia /Pja Pib /Pjb = Pia Pib / Pja Pjb 2 = (Pic )2 / Pjc and hence Lij ≥ Pic /Pjc . There is a similar conclusion in dealing with the geometric means of several price level solutions. It can be added that the price index matrix obtained from the geometric mean of the price level solutions, which is the matrix of ratios of its elements, is the geometric mean of the price index matrices obtained from them.

Price Level Computation 73

2 Consistency of the data The solubility of the system (L) imposes a condition on the given data, deﬁning its consistency, equivalent to the existence of the appropriate underlying utility. With any chain described by a series of periods, or references, s, i, j, . . . , k, t there is associated the Laspeyes chain product Lsij...kt = Lsi Lij . . . Lkt termed the coefﬁcient on the chain. Obviously Lr...s...t = Lr...s Ls...t A chain t, i, j, . . . , k, t whose extremeties are the same deﬁnes a cycle. It is associated with the Laspeyres cyclical product Ltij...kt = Lti Lij . . . Lkt which is basis for the important Laspeyres cyclical product test, or simply the cycle test, Lt...t ≥ 1 for all cycles t . . . t which is necessary and sufﬁcient for consistency of the given data, and is an extension of the PL-inequality. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

the cycle test Ls...t...s ≥ 1 is equivalently to (chain LP) Ks...t ≤ Ls...t for all possible chains … the two occurrences here being taken separately. Hence, introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

subject to the now to be considered conditions required for their existence, where Hst = 1/Mts ,

74 Concept and Method this condition is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst , showing the relation of bounds for the LP-interval and the narrower bounds for the derived version that involves more data. The matrix M , and the matrix H constructed from it, in exactly the same way as the Paasche matrix K is constructed from the Laspeyres matrix L, is important in that their columns, currently as a matter of conjecture, provide a complete set of basic solutions of the system of inequalities (L), the basic price level solutions, from which all other solutions may be derived as combinations.

3 Price-quantity duality With any determination of price levels Pt , there is an associated determination of quantity levels Xt , where Pt Xt = pt xt

(t = 1, . . . , m) .

While for price levels, pt xs /ps xs ≥ Pt /Ps , for quantity levels equivalently in a dual fashion, pt xs /pt xt ≥ Xs /Xt , and one could just as well have solved for the quantity levels ﬁrst, by the same method as for price levels, and then determined the price levels from these. Whichever way, Ps Xt ≤ ps xt

(s, t = 1, . . . , m) ,

with equality for s = t. The introduction of cost efﬁciency up to a level e, where 0 ≤ e ≤ 1, would require Pt Xt ≥ ept xt

(t = 1, . . . , m) .

good also for any lower level, and highest level 1 imposing the equality.

Price Level Computation 75

4 The power algorithm For the main step in the proposed method, matrix L is raised to the mth power in the modiﬁed arithmetic where + means min, to determine M = Lm . Diagonal elements Mii = 1 tell the consistency of the system of inequalities (L) for the determination of price levels Pi , and provide the ﬁrst and second basic price level solutions, with any t as base, given by Pi = Mit , and Pi = Hit , that is, by columns of the matrices M and H. From these are derived the two systems of basic price indices Pij = Pi /Pj . The price indices in either system, with any base, will all be true together in respect to a utility that ﬁts the data by criterion of cost efﬁciency of demand in each period i, so the cost pi xi is the minimum cost, at the prices pi , of the utility of xi .

5 Cost efﬁciency and approximation Diagonal elements Mii < 1 tell the inconsistency of the system, and enable determination of a critical cost efﬁciency e∗ so that the system (L/e)

Lij /e ≥ Pi /Pj

(i = j)

is consistent if and only if e ≤ e∗ . Then with Lij∗ = Lij /e∗

(i = j)

as the elements of the ajusted Laspeyres matrix, the system (L∗ )

Lij∗ ≥ Pi /Pj

is consistent, and with M ∗ = (L∗ )m

76 Concept and Method there may be obtained basic price levels and price indices from M ∗ , as before from M . Now, instead, the price levels of a basic system are together true in respect to a utility that ﬁts the data not exactly, but approximately in the sense of partial cost efﬁciency at the level e∗ in each period, meaning that the fraction e∗ of the cost, in the period, is at most the minimum cost at the prices of gaining at least the utility. Hence in the case e∗ = 1 that goes with ordinary consistency, the ﬁt would be exact as before. For any element Mii < 1 determine the number di of nodes in the path i…i and 1

ei = (Mii ) di giving this the value 1 in case Mii ≥ 1 and then e∗ = min ei i

is the critical cost efﬁciency. Consistency requires Mii = 1, in this case compute the 2m basic price level solutions Pr = Mrt and Pr = Hrt a pair determined for every node t and compute the basic mean price level solution P r and with this the matrix of basic mean price indices P rs = P r /P s . In the other case, of inconsistency, with the critical cost efﬁciency e∗ form the adjusted Laspeyres matrix and proceed exactly as before with this in place of original L. An alternative procedure for the critical cost efﬁciency is available, especially if the path i…i for elements Mii < 1 is not known: Critical cost efﬁciency crude approximation method TEST e: if L/e consistent then YES 0 HIGH = 1 LOW = 0 D = 1/n (for n steps, eg 10) 1 e = (HIGH + LOW)/2 TEST e 2 if YES then LOW = e else HIGH = e 3 if HIGH - LOW < D then e∗ = LOW end else 1 It should be reminded that the following illustrations are not intended for communications of any kind of actual economic information. They are the ﬁrst calculations made following the method, just to assist understanding of it and show the shape of its arithmetic, beside being stimulus for the software development.

Price Level Computation 77 II ILLUSTRATIONS

1 Three references with consistency, and graphics We start with the arbitrary Laspeyres matrix L already described, related to bilateral quantity indexes of per capita GDP in an inter-country comparison based on the International Comparison Project (ICP) data for 1980 published by the United Nations and the Commission of the European Communities (1987). This source gives prices and quantities for some 38 components of GDP expenditure for 60 countries. In the following application, we take the data for the US, France, and Italy to form the matrix L. By raising the matrix L to powers in a modiﬁed arithmetic where + means min, we have Illustration 1 L Laspeyres 1 1.182937 0.913018 1 0.747516 0.813833

1.500803 1.266174 1

L power 2 1 0.913018 0.743044178

1.49780407 1.266174 1

1.182937 1 0.813833

L power 3 = M derived Laspeyres 1 1.182937 1.49780407 0.913018 1 1.266174 0.743044178 0.813833 1 Paths 1,1,1,1 2,1,1,1 3,2,1,1

1,1,1,2 2,2,2,2 3,2,2,2

1,2,2,3 2,2,2,3 3,3,3,3

Consistency case: all diagonal elements = 1

Note that L ≥ L2 = L3 . and at this point one could add “ = …” because after one equality only others can follow. Now we have the derived Laspeyres matrix M = L3

78 Concept and Method The Paasche matrix K is 1 0.8453536 0.6663100

1.0952687 1 0.7897809

1.3377640 1.2287533 1

and the derived Paasche matrix is H derived Paasche 1 0.845353557 0.667644065

1.09526866 1 0.789780867

1.345815 1.22875332 1

Note that Kst ≤ Hst ≤ Mst ≤ Lst , showing the relation of the original LP-interval and the narrower bounds that involve more data. The 6 basic price level systems - the 6 columns of M and H The geometric mean of the matrices H and M , element by element, is the matrix F, whose columns coincide with the geometric means of their corresponding columns: F derived Fisher - mean of derived Laspeyres M and derived Paasche H 1 0.878534583 0.704335883

1.13825912 1 0.801716741

1.41977716 1.24732334 1

The columns of M and H are all solutions of system (L). These are the 6 basic price level solutions, from which all other solutions can be derived, being the 6 vertices of the convex hexagonal region described by solutions normalized to sum 1 each determining a point in the simplex of reference. The columns of F are geometric means of opposite pairs of vertices of the hexagon. The 6 basic solutions, a basis for all solutions, are given by columns of M and H , and basic geometric mean solution has elements given by the geometric means of their columns, or of columns of the matrix F, so it is P mean basic price level system - mean of columns of F 1.17351085 1.03096986 0.826545798 The matrix of basic mean price indices obtained from this, by taking ratios of the elements, is

Price Level Computation 79 S

Drs

Dst

Drst

Drt

r

t

Figure 1 The 6 basic solutions as vertices for the solution set

P/P mean basic price index system 1 0.878534583 0.704335883

1.13825912 1 0.801716741

1.41977716 1.24732334 1

and coincides with the mean of individual basic price index matrices. Notice that this matrix P/P coincides with the matrix F used to obtain it, and see End-note No. 2.

2 Four references, with consistency The start with the Laspeyres matrix L and raising it to powers in a modiﬁed arithmetic where + means min, using CM’s FORTRAN program, we have ⎡

1.000 ⎢0.898 L≡⎢ ⎣0.913 0.747

1.122 1.183 1.000 1.042 0.979 1.000 0.812 0.814

⎤ 1.501 1.350⎥ ⎥ 1.266⎦ 1.000

(elements that change are in bold). and then ⎤ ⎡ L12 L123 L1234 L11 ⎢ L21 L22 L23 L234 ⎥ ⎥ L4 ≡ ⎢ ⎣ L321 L32 L33 L34 ⎦ L4321 L432 L43 L44

80 Concept and Method where Lrij...ks = Lri Lij . . . Lks . with M = L4 , therefore ⎡

1.000 ⎢0.898 M =⎢ ⎣0.879 0.715

1.122 1.000 0.979 0.797

1.169 1.042 1.000 0.814

⎤ 1.480 1.319⎥ ⎥ 1.266⎦ 1.000

Note the triangle inequality Mrs Mst ≥ Mrt From the matrix L, the Paasche matrix K is derive by Kij = 1/Lji , so that ⎡

⎤ 1.000 1.114 1.095 1.338 ⎢0.891 1.000 1.021 1.231⎥ ⎥ K =⎢ ⎣0.845 0.960 1.000 1.229⎦ 0.666 0.741 0.790 1.000 from which, by similar procedure, Hij = 1/Mji , we have ⎡

1.000 ⎢0.891 H ≡⎢ ⎣0.856 0.676

1.114 1.138 1.000 1.021 0.960 1.000 0.758 0.790

⎤ 1.398 1.255⎥ ⎥ 1.229⎦ 1.000

Alternatively, just as M = Lm so similarly H = K m where the arithmetic for powers now has + meaning max instead of min. Note K ≤ L for original bounds, and moreover K ≤H ≤M ≤L showing tighter bounds obtained with additional data. With any Pi which are a price level solution being such that Lij ≥ Pi /Pj there is associated a price index matrix with elements Pij = Pi /Pj The 8 basic solutions, a basis for all solutions, are given by columns of M and H , and basic geometric mean, which has elements given by the geometric means

Price Level Computation 81 of their elements, is also a solution. It is [1.167 1.044 1.012 0.811]. The matrix of basic mean price indices obtained from this, by taking ratios of the basic mean price levels, is ⎡ ⎤ 1.000 1.118 1.153 1.438 ⎢0.894 1.000 1.031 1.287⎥ ⎢ ⎥ ⎣0.867 0.969 1.000 1.247⎦ 0.695 0.777 0.802 1.000 and coincides with the mean of individual basic price index matrices, derived from the individual basic price level solution elements. By taking weighted geometric means instead of the simple geometric mean, it is possible to arrive at all possible price level solutions, and consequently all possible systems of true price indices, without any guidance for choosing just one from among them. Here we have, for want of that guidance and to that extent arbitrarily, like the Fisher “Ideal Index”, adopted one, with weights all equal and no reason for making them different, as a standard, in order to eliminate that residual indecision. Following the above report done using FORTRAN, we include the routine output from another program using BBC BASIC for Windows1 . This deals with two text ﬁles kept in folder c:\0\ as here indicated: REM input from C:0?-input.txt output to C:0?-output.txt REM change the ? *SPOOL "C:02-output.txt" F%=OPENIN "C:02-input.txt"

For instance 2-input.txt looks like 4, 4 1.0000000, 0.8976280, 0.9130180, 0.7475160,

1.1218730, 1.0000000, 0.9792190, 0.8122070,

1.1829370, 1.0418520, 1.0000000, 0.8138330,

1.5008030 1.3498590 1.2661740 1.0000000

which tells it is a 4 x 4 matrix and then tells the elements, the comma “,” being the delimiter. As for 2-output.txt, which need not even exist initially and if it does any contents will be overwritten, it recieves the output when the program is run,

1 We acknowledge with thanks the guidance received from Richard Russell, longtime developer of this rendering of BASIC, http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html,http:// www.rtrussell.co.uk/,[email protected].

82 Concept and Method which with this input is as follows—showing reassuring agreement with earlier ﬁgures. When this program is compiled and so not available for alteration, it will refer to two ﬁles called input.txt and output.txt always with these names though they can have different applications. This compiled version will be supplied to anyone wishing to use it. Illustration 2 L Laspeyres 1 0.897628 0.913018 0.747516

1.121873 1 0.979219 0.812207

1.182937 1.041852 1 0.813833

1.500803 1.349859 1.266174 1

L power 2 1 0.897628 0.878974393 0.729059745

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

L power 3 1 0.897628 0.878974393 0.715338367

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

(Elements that change are in bold. This is after the last power that changes so generation of powers could have stopped here. Note that L ≥ L2 ≥ L3 = L4 . and at this point one could add “ = …” because after one equality only others can follow.) L power 4 = M derived Laspeyres 1 0.897628 0.878974393 0.715338367

1.121873 1 0.979219 0.796920736

1.16882563 1.041852 1 0.813833

1.47993662 1.31916592 1.266174 1

1,1,1,1,2 2,2,2,2,2 3,2,2,2,2 4,3,2,2,2

1,2,2,2,3 2,2,2,2,3 3,3,3,3,3 4,3,3,3,3

1,3,3,3,4 2,3,3,3,4 3,3,3,3,4 4,4,4,4,4

Paths 1,1,1,1,1 2,1,1,1,1 3,2,1,1,1 4,2,2,1,1

Price Level Computation 83 Consistency case: all diagonal elements = 1 H derived Paasche 1 1.11404725 0.891366491 1 0.855559611 0.959829227 0.675704611 0.758054759

1.13768957 1.02122202 1 0.789780867

1.39793984 1.25482994 1.22875332 1

The 8 basic price level systems - the 8 columns of M and H F derived Fisher - mean of derived Laspeyres 1 1.11795328 1.15315252 0.894491767 1 1.03148543 0.867187978 0.96947564 1 0.695239119 0.77724485 0.801716741

M and derived Paasche H 1.43835405 1.28659585 1.24732334 1

P mean basic price level - mean of columns of F 1.16692797 1.04380747 1.01194591 0.811293977 P/P mean basic 1 0.894491767 0.867187978 0.69523912

price index 1.11795328 1 0.96947564 0.77724485

1.15315252 1.03148543 1 0.801716741

1.43835405 1.28659585 1.24732334 1

As with Illustration No. 1, notice that this matrix P coincides with the matrix F used to obtain it, see end-note No. 2.

3 Case of inconsistency and approximation Starting with the Laspeyres matrix L for the countries Canada, U.S., Norway, Luxembourg, Germany in the year 1980, and raising it to powers in the (+ = min)-arithmetic using the FORTRAN program: L POWER 1 1.0000000 0.9139310 0.9685060 0.9389430 0.8886960

1.0171450 1.0000000 1.0171450 0.9398820 0.8976270

1.1252440 1.1274960 1.0000000 1.0345840 1.0030040

1.2008140 1.1411080 1.1207520 1.0000000 1.0387310

1.1537290 1.1218730 1.0650260 1.0222430 1.0000000

1.0376916 0.9860962

0.9970028 0.9694742

··· ··· ··· ··· ··· ··· ··· ··· ··· M = L POWER 5 0.8641568 0.7897797

0.8789728 0.8641568

0.9723873 0.9559967

84 Concept and Method 0.8122053 0.7795783 0.7626154

0.8789728 0.8122054 0.7756905

0.9723873 0.8985241 0.8581284

1.0140962 0.9588679 0.9157593

0.9970021 0.9212698 0.8798515

Inconsistency case since some diagonal elements < 1 diagonal elements < 1 (in this case all) associated cost efﬁciencies ei critical cost efﬁciency is minimum of these 1

i

Mii

path

di

ei = (Mii ) di

1 2 3 4 5

0.8641568 0.8641568 0.9723873 0.9588679 0.8798515

12121 21212 321213 421214 521215

3 3 4 4 4

0.952498 0.952498 0.990710 0.986097 0.968506

critical cost efﬁciency e∗ = mini ei = e1 = e2 = 0.952498 used to determine the adjusted Laspeyres matrix= L∗ Being near to the value 1, associated with the consistency case where ﬁt of data to the hypothetical underlying utility is exact, this represents a high level of cost efﬁciency, and a closeness of ﬁt for the approximating utility. Note: By computer error the degree di associated with a path is 1 less than the correct count. The effect is to make the cost efﬁciency less than critical, resulting in allowance of a looser ﬁt for the approximate utility. A revision could provide the correction, and moreover a redevelopment of the approach to cost efﬁciency where the critical uniform bound is replaced by discrimination, but for the time being the error does not damage, even enhances, the value of the illustration. L* POWER 1 - adjusted L 1.0000000 1.0678707 1.1813607 0.9595094 1.0000000 1.1837250 1.0168061 1.0678707 1.0000000 0.9857687 0.9867546 1.0861794 0.9330159 0.9423923 1.0530245

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1778216 1.1181396 1.0732230 1.0000000

POWER 2 1.0000000 0.9595094 1.0168061 0.9468003 0.9042342

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1622213 1.1181396 1.0732230 1.0000000

1.0678707 1.0000000 1.0537261 0.9867546 0.9423923

1.1813607 1.1335267 1.0000000 1.0861794 1.0530245

Price Level Computation 85 POWER … …

3 (no change after this power) … … … … … … …

M* = L* POWER 5 1.0000000 1.0678707 0.9595094 1.0000000 1.0110601 1.0537261 0.9468003 0.9867546 0.9042342 0.9423923

1.1813607 1.1335267 1.0000000 1.0861794 1.0530245

1.2606994 1.1980159 1.1766447 1.0000000 1.0905332

1.2112663 1.1622213 1.1181396 1.0732230 1.0000000

1.0561890 1.0134232 0.9206582 1.0000000 0.9317728

1.1059081 1.0611292 0.9496455 0.9169826 1.0000000

Consistent, all diagonal elements = 1 From derived Laspeyres M * determine by transposition and element inversion the derived Paasche H* 1.0000000 0.9364429 0.8464815 0.7932105 0.8255823

1.0421993 1.0000000 0.8822024 0.8347135 0.8604213

0.9890609 0.9490132 1.0000000 0.8498742 0.8943427

(Alternatively, just as M = Lm so similarly H = K m where now the arithmetic for powers has + meaning max instead of min, and same here for adjusted *-versions.) The columns of M* and H* provide the 10 basic price level solutions in 5 opposite pairs. Then determine F* the matrix geometric mean of M* and H* 1.0000000 0.9479060 0.9251182 0.8666094 0.8640138

1.0549569 1.0000000 0.9641575 0.9075557 0.9004745

1.0809430 1.0371749 1.0000000 0.9607892 0.9704457

1.1539224 1.1018607 1.0408110 1.0000000 1.0080324

1.1573890 1.1105256 1.0304544 0.9920316 1.0000000

The columns are geometric means of opposite pairs of basic solutions. Now determine the geometric mean of the columns of F* Mean basic price level solution 1.087774 1.037659 0.991172 0.943996 0.946864

86 Concept and Method Coincides with the geometric mean of all 10 of the basic price level solutions. Finally form the mean basic price index matrix P, given by ratios of elements of the mean basic price level solution. Mean basic price index matrix 1.00000 0.94791 0.92512 0.86661 0.86401

1.05496 1.00000 0.96416 0.90756 0.90047

1.08094 1.03717 1.00000 0.96079 0.97045

1.15392 1.10186 1.04081 1.00000 1.00803

1.15739 1.11053 1.03045 0.99203 1.00000

Coincides with the mean of the 10 basic price index matrices obtained from the 10 individual basic price level solutions. The BASIC program produces the following. Illustration 3 L Laspeyres 1 0.913931 0.968506 0.938943 0.888696

1.017145 1 1.017145 0.939882 0.897627

1.125244 1.127496 1 1.034584 1.003004

1.200814 1.141108 1.120752 1 1.038731

1.153729 1.121873 1.065026 1.022243 1

L power 2 0.929600347 0.913931 0.929600347 0.858987296 0.820369142

0.945538345 0.929600347 0.945538345 0.873714633 0.834434371

1.04602721 1.02839537 1 0.966570301 0.923115455

1.07896137 1.06077439 1.07896137 0.997002758 0.952179736

1.06077394 1.04289353 1.06077394 0.980196857 0.936129391

L power 3 0.864156805 0.849590575 0.864156805 0.798514889 0.762615439

0.878972774 0.864156805 0.878972774 0.812205427 0.775690481

0.97238726 0.955996697 0.97238726 0.898524088 0.858128447

1.00300286 0.986096244 1.00300286 0.92681411 0.885146613

0.986095823 0.969474188 0.986095823 0.911191339 0.870226207

L power 4 0.803320466 0.789779693 0.803320466 0.742299718 0.708927577

0.817093396 0.803320466 0.817093396 0.755026446 0.72108214

0.903931535 0.888694861 0.903931535 0.835268304 0.797716502

0.932391811 0.91667541 0.932391811 0.861566718 0.822832599

0.916675019 0.901223541 0.916675019 0.847043784 0.808962584

L power 5 = M derived Laspeyres 0.746766984 0.759570304 0.840295068 0.734179477 0.746766984 0.826131051 0.746766984 0.759570304 0.840295068 0.690042075 0.701872846 0.776465705 0.659019321 0.670318208 0.741557537

0.866751751 0.85214178 0.866751751 0.80091272 0.764905469

0.852141416 0.837777717 0.852141416 0.787412196 0.752011899

Price Level Computation 87 Paths 1,2,2,2,2,1 2,1,2,2,2,1 3,2,2,2,2,1 4,2,2,2,2,1 5,2,2,2,2,1

1,1,1,1,1,2 2,1,1,1,1,2 3,1,1,1,1,2 4,1,1,1,1,2 5,1,1,1,1,2

1,1,1,1,1,3 2,1,1,1,1,3 3,3,1,1,1,3 4,1,1,1,1,3 5,1,1,1,1,3

1,2,2,2,2,4 2,2,2,2,2,4 3,2,2,2,2,4 4,2,2,2,2,4 5,2,2,2,2,4

1,2,2,2,2,5 2,2,2,2,2,5 3,2,2,2,2,5 4,2,2,2,2,5 5,2,2,2,2,5

Inconsistency case: some diagonal elements < 1 1 0.746766984 2 0.746766984 3 0.840295068 4 0.80091272 5 0.752011899 Effective paths - Factor counts - Efficiencies 1 1,2,1 2 0.864156805 2 2,1,2 2 0.864156805 3 3,1,3 2 0.916676098 4 4,2,4 2 0.894937272 5 5,2,5 2 0.867186196 Critical cost efficiency

0.864156805

L adjusted Laspeyres to replace original L 1 1.17703754 1.30212942 1.05759857 1 1.30473543 1.12075262 1.17703754 1 1.08654239 1.087629 1.19721791 1.02839669 1.03873162 1.16067361 L power 2 … …

1.38957883 1.3204872 1.29693129 1 1.2020168

1.33509219 1.29822851 1.23244531 1.18293693 1

1.38957883 1.3204872 1.29693129 1 1.2020168

1.33509219 1.29822851 1.23244531 1.18293693 1

1,1,1,1,1,4 2,2,2,2,2,4 3,3,3,3,3,4 4,4,4,4,4,4 5,4,4,4,4,4

1,1,1,1,1,5 2,2,2,2,2,5 3,3,3,3,3,5 4,4,4,4,4,5 5,5,5,5,5,5

0.920350655 0.919431168 0.835269834 1 0.845353605

0.972387414 0.96271258 0.861568653 0.831935126 1

… L power 4

L power 5 = M derived Laspeyres 1 1.17703754 1.30212942 1.05759857 1 1.30473543 1.12075262 1.17703754 1 1.08654239 1.087629 1.19721791 1.02839669 1.03873162 1.16067361 Paths 1,1,1,1,1,1 2,1,1,1,1,1 3,1,1,1,1,1 4,1,1,1,1,1 5,1,1,1,1,1

1,1,1,1,1,2 2,2,2,2,2,2 3,2,2,2,2,2 4,2,2,2,2,2 5,2,2,2,2,2

1,1,1,1,1,3 2,2,2,2,2,3 3,3,3,3,3,3 4,3,3,3,3,3 5,3,3,3,3,3

Consistency case: all diagonal elements = 1 H derived Paasche 1 0.945538345 0.849590575 1 0.767972818 0.76643891 0.719642514 0.757296246 0.749011948 0.77028042

0.892257565 0.849590575 1 0.771050871 0.811395032

88 Concept and Method The 10 basic price level systems - columns of M and H F derived Fisher - mean of derived Laspeyres 1 1.05495693 1.07788442 0.947905995 1 1.05284896 0.927743253 0.949803858 1 0.88426359 0.907555705 0.960789211 0.877656772 0.894491269 0.970445672

M and derived Paasche H 1.13088451 1.13939758 1.10186074 1.1179539 1.04081102 1.03045439 1 0.992031646 1.00803236 1

P mean basic price level - mean of columns of F 1.07939424 1.04216477 0.988763618 0.94781098 0.94856982 P/P mean basic 1 0.965508928 0.916035665 0.878095276 0.8787983

price index 1.0357232 1 0.948759394 0.909463653 0.910191791

1.09166055 1.05400801 1 0.958581974 0.959349437

1.13882858 1.09954917 1.0432076 1 1.00080062

1.13791754 1.09866955 1.04237305 0.999200017 1

4 Inconsistency and approximation again CAN, US, NOR, LUX, GER, DEN, FRA, BEL, NED, AUT, JPN, UK, ITA, SPN, IRL, GRC, PRT (17 COUNTRIES) Illustration 4 L Laspeyres 1 1.017145 1.125244 1.200814 1.153729 1.193631 1.204422 1.228753 1.24982 1.382647 1.496306 1.541876 1.55893 1.853359 2.325651 2.637944 3.625528 0.913931 1 1.127496 1.141108 1.121873 1.172337 1.182936 1.208041 1.232445 1.367521 1.460823 1.463747 1.500802 1.823941 2.241174 2.585709 3.45907 0.968506 1.017145 1 1.120752 1.065026 1.081122 1.111821 1.124119 1.159512 1.294338 1.388189 1.420487 1.474029 1.73846 2.192406 2.464527 3.293661 0.938943 0.939882 1.034584 1 1.022243 1.069295 1.070365 1.088717 1.136553 1.261119 1.368889 1.392359 1.37163 1.670294 2.083397 2.391689 3.326763 0.888696 0.897627 1.003004 1.038731 1 1.043937 1.041852 1.059715 1.087628 1.208041 1.30604 1.321807 1.349858 1.604801 2.027898 2.30712 3.183559 0.969475 0.963676 1.006018 1.05654 1.012072 1 1.021222 1.042894 1.082204 1.227525 1.30604 1.239861 1.247323 1.617691 1.873859 2.190215 2.857651 0.887807 0.913017 0.97824 1.00904 0.979219 1.00904 1 1.02429 1.063962 1.177036 1.286596 1.243587 1.266174 1.569881 1.902178 2.166255 2.880604 0.878095 0.887807 0.959829 0.98906 0.960789 0.986097 0.984127 1 1.041852 1.16649 1.263644 1.226298 1.234912 1.543418 1.847807 2.104336 2.903741 0.848742 0.875465 0.928671 0.963676 0.935195 0.965605 0.969475 0.976285 1 1.135417 1.218962 1.18649 1.233678 1.508325 1.847807 2.100131 2.869104 0.73565 0.773368 0.837779 0.880733 0.847046 0.876341 0.875465 0.895834 0.917594 1 1.11071 1.102962 1.09308 1.355269 1.647073 1.845961 2.567672

Price Level Computation 89 0.729059 0.768741 0.802518 0.939882 0.842821 0.840296 0.855559 0.877217 1.121873 1.030454 1 1.091988 1.072508 1.321807 1.574598 1.782469 2.539583 0.79692 0.799315 0.826959 0.88692 0.838618 0.842821 0.854704 0.865887 0.889585 1.004008 1.055484 1 1.041852 1.341783 1.551155 1.829421 2.325651 0.758054 0.747515 0.805735 0.869358 0.812207 0.825306 0.813833 0.834435 0.858988 0.960789 1.078962 0.998002 1 1.273794 1.506817 1.709156 2.37976 0.563831 0.562704 0.631915 0.67977 0.631283 0.662324 0.64856 0.66365 0.673006 0.754273 0.863294 0.812207 0.827786 1 1.231213 1.375751 1.950332 0.493121 0.50258 0.547167 0.551011 0.547715 0.562704 0.55046 0.564395 0.593926 0.664978 0.718923 0.675028 0.67368 0.875465 1 1.133148 1.630684 0.478547 0.476637 0.523614 0.695586 0.555992 0.528876 0.52782 0.537944 0.544438 0.635082 0.71177 0.653769 0.619402 0.824482 0.979219 1 1.55426 0.370093 0.374186 0.396531 0.439551 0.39812 0.39812 0.409016 0.47001 0.54335 0.461626 0.462088 0.62688

0.39812 0.408199 0.689354 0.80493 1

L power 2 …… L power 16 L power 17 = M derived Laspeyres 0.310985711 0.316317561 0.349934805 0.360952499 0.354868131 0.370459972 0.369720072 0.376059081 0.385964515 0.428695252 0.462083968 0.457943038 0.462083244 0.569492731 0.694189753 0.783480195 1.05864531 0.305743734 0.310985711 0.344036303 0.354868282 0.348886472 0.364215497 0.363488069 0.369720228 0.379458696 0.421469163 0.454295079 0.450223948 0.454294367 0.55989336 0.682488487 0.770273849 1.04080078 0.310985711 0.316317561 0.349934805 0.360952499 0.354868131 0.370459972 0.369720072 0.376059081 0.385964515 0.428695252 0.462083968 0.457943038 0.462083244 0.569492731 0.694189753 0.783480195 1.05864531 0.287363033 0.292289872 0.323353528 0.333534311 0.327912116 0.34231959 0.341635893 0.347493387 0.356646398 0.39613128 0.426983767 0.423157385 0.426983098 0.526233691 0.641458645 0.723966526 0.978229919 0.274443831 0.279149171 0.308816275 0.318539352 0.313169918 0.326929665 0.326276705 0.331870859 0.340612371 0.378322101 0.407787529 0.404133173 0.40778689 0.502575397 0.612620094 0.691418606 0.934250883 0.294637899 0.299689466 0.331539528 0.341978047 0.33621352 0.350985734 0.350284729 0.356290511 0.365675239 0.406159717 0.437793265 0.433870014 0.437792578 0.539555794 0.657697776 0.742294423 1.00299473 0.279149227 0.283935241 0.314110993 0.324000775 0.31853928 0.332534941 0.331870786 0.337560854 0.34645224 0.384808511 0.41477913 0.411062119 0.41477848 0.511192156 0.623123591 0.703273119 0.950268806 0.271441428 0.276095291 0.305437838 0.315054545 0.309743853 0.323353068 0.322707252 0.328240207 0.336886087 0.374183273 0.403326351 0.399711973 0.403325719 0.497077244 0.605918057 0.683854516 0.924030218 0.267667939 0.272257105 0.301191742 0.310674761 0.305437896 0.31885792 0.318221082 0.32367712 0.332202808 0.368981501 0.397719441 0.394155309 0.397718818 0.49016704 0.597494784 0.674347796 0.911184655 0.23645242 0.240506397 0.266066667 0.274443774 0.269817633 0.281672611 0.281110041 0.285929793 0.293461213 0.325950764 0.351337276 0.348188795 0.351336726 0.433003608 0.527814756 0.595705146 0.804922017

90 Concept and Method 0.235037744 0.239067466 0.264474811 0.272801798 0.268203336 0.279987386 0.279428182 0.284219098 0.291705458 0.324000626 0.349235253 0.346105609 0.349234706 0.430412981 0.524656883 0.59214109 0.800106233 0.244385553 0.248575544 0.274993378 0.283651542 0.278870191 0.291122911 0.290541466 0.295522924 0.303307028 0.336886625 0.363124871 0.359870756 0.363124302 0.447531161 0.545523286 0.615691443 0.831927677 0.228548028 0.232466483 0.257172297 0.265269364 0.260797871 0.272256547 0.271712784 0.276371416 0.283651067 0.315054521 0.339592386 0.336549155 0.339591853 0.418528684 0.510170381 0.575791256 0.778014195 0.172043222 0.174992903 0.193590604 0.199685802 0.196319813 0.204945517 0.20453619 0.208043051 0.213522926 0.237162384 0.255633658 0.253342817 0.255633257 0.315054233 0.384039002 0.433436176 0.585662762 0.153660686 0.156295198 0.172905765 0.178349701 0.175343363 0.183047425 0.182681834 0.185813992 0.190708352 0.211821972 0.228319621 0.226273552 0.228319263 0.281391205 0.343005064 0.387124231 0.523085656 0.145728776 0.148227296 0.163980431 0.169143354 0.166292202 0.173598582 0.173251863 0.17622234 0.180864055 0.200887798 0.216533844 0.214593392 0.216533504 0.266865891 0.325299265 0.367141017 0.496084161 0.113607209 0.115555005 0.127835831 0.13186074 0.12963804 0.135333946 0.135063651 0.137379375 0.140997962 0.156608067 0.168805409 0.167292672 0.168805144 0.208043256 0.253596733 0.286215717 0.386737187

Inconsistency case: some diagonal elements < 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.310985711 0.310985711 0.349934805 0.333534311 0.313169918 0.350985734 0.331870786 0.328240207 0.332202808 0.325950764 0.349235253 0.359870756 0.339591853 0.315054233 0.343005064 0.367141017 0.386737187

Effective paths - Factor counts - Efﬁciencies 1 2 3 4

1,2,1 2,1,2 3,1,3 4,2,4

2 2 2 2

0.557660928 0.557660928 0.591552876 0.577524295

Price Level Computation 91 5 6 7 8 9 10 11 12 13 14 15 16 17

5,2,5 2 0.559615866 6,5,6 2 0.59244049 7,5,7 2 0.576082274 8,5,8 2 0.572922514 9,5,9 2 0.576370374 10,5,10 2 0.570920978 11,2,11 2 0.590961296 12,9,12 2 0.599892287 13,6,13 2 0.582745102 14,5,14 2 0.56129692 15,6,15 2 0.585666342 16,14,16 2 0.605921626 17,6,17 2 0.621881972

Critical cost efﬁciency 0.557660928 L adjusted Laspeyres to replace original L 1 2.15977477 2.79548005

1.82394884 2.20340522 3.32345142

2.01779243 2.24118266 4.17036748

2.15330488 2.47936861 4.73037265

2.06887186 2.68318242 6.50131257

2.1404243 2.76489875

1.63886504 2.12124598 2.69124467

1 2.16626437 3.27069893

2.02183073 2.21002573 4.01888296

2.04623983 2.4522446 4.63670462

2.01174754 2.61955415 6.20281936

2.1022398 2.62479748

1.73672917 1.99372225 2.64323521

1.82394884 2.01577508 3.11741403

1 2.07924196 3.93143197

2.00973736 2.32101253 4.41940053

1.90980925 2.48930655 5.90620722

1.93867267 2.54722346

1.68371667 1.91938317 2.45961288

1.68540049 1.95229206 2.99517846

1.85522053 2.03807178 3.73595656

1 2.26144407 4.28878711

1.83309059 2.45469771 5.96556587

1.91746444 2.49678421

1.59361353 1.86825354 2.42057123

1.60962864 1.90028554 2.87773613

1.79859113 1.95033926 3.63643551

1.86265694 2.16626437 4.13713761

1 2.34199661 5.70877184

1.87199237 2.37027006

1.73846678 1.83125973 2.23670503

1.72806799 1.87012205 2.90085053

1.80399585 1.94061292 3.3602121

1.89459212 2.20120317 3.92750306

1.81485191 2.34199661 5.12435219

1 2.22332413

1.59201937 1 2.27050872

1.63722605 1.83676128 2.81511743

1.75418422 1.90790128 3.41099386

1.80941491 2.11066607 3.88453788

1.75593977 2.30712954 5.16551161

1.80941491 2.23000561

1.57460377 1.76827343 2.1990029

1.59201937 1.76474081 2.21444957

1.72116954 1 2.76766387

1.77358669 1.86825354 3.31349554

1.72289101 2.09175494 3.77350446

1.52196784 1.73846678 2.21223675

1.56988764 1.75067851 2.70473495

1.66529687 1 3.31349554

1.72806799 2.0360347 3.76596404

1.67699574 2.18584796 5.14488976

1.31917078 1.57145849 1.97783625

1.38680686 1.56988764 1.96011581

1.50230894 1.60641342 2.43027426

1.57933424 1.64543355 2.95353846

1.51892657 1 3.31018529

2.26597191 5.20700098 1.73152708 2.12761903

1.99173 4.60436059

92 Concept and Method 1.30735177 1.53419212 1.92322601

1.3785097 1.57302934 2.37027006

1.43907877 2.01174754 2.82357598

1.68540049 1.84781459 3.19633116

1.51135028 1 4.55399128

1.50682244 1.95815763

1.42904041 1.53265893 1.86825354

1.43333513 1.55271233 2.40609111

1.48290647 1.59520769 2.78153789

1.5904288 1.80039151 3.28052569

1.50381344 1.8926985 4.17036748

1.51135028 1

1.35934573 1.45936887 1

1.34044715 1.49631247 2.28417294

1.4448475 1.54034102 2.70203079

1.55893655 1.72289101 3.06486597

1.45645312 1.93479935 4.26739598

1.47994231 1.78962152

1.0110642 1.16300061 1.48438945

1.00904326 1.19006007 1

1.13315273 1.20683729 2.2078165

1.21896652 1.35256562 2.46700267

1.13201942 1.54806255 3.49734382

1.18768228 1.45645312

0.884266721 0.987087265 1.2080459

0.901228641 1.01207557 1.56988764

0.981182243 1.06503069 1

0.988075321 1.19244144 2.03196592

0.982164919 1.28917585 2.92414964

1.00904326 1.21046314

0.858132561 0.946489118 1.11071436

0.85470754 0.964643519 1.47846471

0.938946901 0.976288588 1.75593977

1.24732784 1.1388318 1

0.997007271 1.27634906 2.78710579

0.948382742 1.17234141

0.663652376 0.713910515 0.82861821

0.670991962 0.731984221 1.12412394

0.711061113 0.733449269 1.23615259

0.788204764 0.842823975 1.44340398

0.713910515 0.974337582 1

0.713910515 0.82778975

L power 2 …… L power 16 L power 17 = M derived Laspeyres 1 2.15977477 2.79548005

1.82394884 2.20340522 3.32345142

2.01779243 2.24118266 4.17036748

2.15330488 2.47936861 4.73037265

2.06887186 2.68318242 6.50131257

2.1404243 2.76489875

1.63886504 2.12124598 2.69124467

1 2.16626437 3.27069893

2.02183073 2.21002573 4.01888296

2.04623983 2.4522446 4.63670462

2.01174754 2.61955415 6.20281936

2.1022398 2.62479748

1.73672917 1.99372225 2.64323521

1.82394884 2.01577508 3.11741403

1 2.07924196 3.93143197

2.00973736 2.32101253 4.41940053

1.90980925 2.48930655 5.90620722

1.93867267 2.54722346

1.68371667 1.91938317 2.45961288

1.68540049 1.95229206 2.99517846

1.85522053 2.03807178 3.73595656

1 2.26144407 4.28878711

1.83309059 2.45469771 5.96556587

1.91746444 2.49678421

1.59361353 1.86825354 2.42057123

1.60962864 1.90028554 2.87773613

1.79859113 1.95033926 3.63643551

1.86265694 2.16626437 4.13713761

1 2.34199661 5.70877184

1.87199237 2.37027006

1.73846678 1.83125973 2.23670503

1.72806799 1.87012205 2.90085053

1.80399585 1.94061292 3.3602121

1.89459212 2.20120317 3.92750306

1.81485191 2.34199661 5.12435219

1 2.22332413

1.59201937 1 2.27050872

1.63722605 1.83676128 2.81511743

1.75418422 1.90790128 3.41099386

1.80941491 2.11066607 3.88453788

1.75593977 2.30712954 5.16551161

1.80941491 2.23000561

1.57460377 1.76474081 2.21444957

1.59201937 1 2.76766387

1.72116954 1.86825354 3.31349554

1.77358669 2.09175494 3.77350446

1.72289101 2.26597191 5.20700098

1.76827343 2.1990029

Price Level Computation 93 1.52196784 1.73846678 2.21223675

1.56988764 1.75067851 2.70473495

1.66529687 1 3.31349554

1.72806799 2.0360347 3.76596404

1.67699574 2.18584796 5.14488976

1.73152708 2.12761903

1.31917078 1.56988764 1.96011581

1.38680686 1.60641342 2.43027426

1.50230894 1.64543355 2.95353846

1.57933424 1 3.31018529

1.51892657 1.99173 4.60436059

1.57145849 1.97783625

1.30735177 1.53419212 1.92322601

1.3785097 1.57302934 2.37027006

1.43907877 2.01174754 2.82357598

1.68540049 1.84781459 3.19633116

1.51135028 1 4.55399128

1.50682244 1.95815763

1.42904041 1.53265893 1.86825354

1.43333513 1.55271233 2.40609111

1.48290647 1.59520769 2.78153789

1.5904288 1.80039151 3.28052569

1.50381344 1.8926985 4.17036748

1.51135028 1

1.35934573 1.45936887 1

1.34044715 1.49631247 2.28417294

1.4448475 1.54034102 2.70203079

1.55893655 1.72289101 3.06486597

1.45645312 1.93479935 4.26739598

1.47994231 1.78962152

1.0110642 1.16300061 1.48438945

1.00904326 1.19006007 1

1.13315273 1.20683729 2.2078165

1.21896652 1.35256562 2.46700267

1.13201942 1.54806255 3.49734382

1.18768228 1.45645312

0.884266721 0.987087265 1.2080459

0.901228641 1.01207557 1.56988764

0.981182243 1.06503069 1

0.988075321 1.19244144 2.03196592

0.982164919 1.28917585 2.92414964

1.00904326 1.21046314

0.858132561 0.946489118 1.11071436

0.85470754 0.964643519 1.47846471

0.938946901 0.976288588 1.75593977

1.24732784 1.1388318 1

0.997007271 1.27634906 2.78710579

0.948382742 1.17234141

0.663652376 0.713910515 0.82861821

0.670991962 0.731984221 1.12412394

0.711061113 0.733449269 1.23615259

0.788204764 0.842823975 1.44340398

0.713910515 0.974337582 1

0.713910515 0.82778975

Consistency case: all diagonal elements = 1 Hence immediately the wanted ﬁnal answer: P/P mean basic price index 1 1.16241544 1.43721337

1.02510404 1.1847125 1.80377676

1.0858391 1.2219059 2.14583724

1.14268694 1.35642562 2.32196113

1.12972093 1.41731595 3.16721761

1.14936063 1.40970785

0.975510742 1.13394875 1.40201709

1 1.15569977 1.7596036

1.0592477 1.19198233 2.09328728

1.11470339 1.32320776 2.26509802

1.1020549 1.38260694 3.0896548

1.12121364 1.37518515

0.920946763 1.07052274 1.32359701

0.944066245 1.09105714 1.66118237

1 1.12531028 1.97620186

1.05235384 1.24919578 2.13840258

1.04041283 1.30527254 2.91683881

1.05849995 1.29826588

0.87513033 1.01726501 1.25774901

0.897099634 1.03677784 1.57853975

0.950250725 1.06932691 1.87788725

1 1.1870492 2.03201861

0.988653048 1.24033618 2.77172819

1.00584035 1.2336781

0.885174361 1.02894035 1.27218443

0.907395811 1.04867713 1.59665694

0.961156927 1.08159977 1.89944011

1.01147718 1.20067318 2.05534046

1 1.25457174 2.80353982

1.01738456 1.24783725

94 Concept and Method 0.870048942 1.01135833 1.25044598

0.891890682 1.03075786 1.56937406

0.944733155 1.06311793 1.86698342

0.994193564 1.18015667 2.02021982

0.982912497 1.23313424 2.75563433

1 1.22651483

0.860277629 1 1.23640251

0.88187407 1.01918166 1.55174879

0.934123082 1.05117831 1.84601577

0.983028013 1.16690262 1.99753121

0.971873641 1.21928521 2.72468646

0.988769238 1.21274013

0.844086644 0.98117935 1.21313261

0.865276626 1 1.52254387

0.916542278 1.03139445 1.81127255

0.964526786 1.14494075 1.95993638

0.953582347 1.19633747 2.67340608

0.970159958 1.18991557

0.818393628 0.951313391 1.17620627

0.838938612 0.969561163 1.4761994

0.888643797 1 1.75613952

0.935167713 1.11009008 1.90027819

0.924556409 1.15992234 2.59203071

0.940629417 1.15369592

0.737231726 0.856969542 1.0595593

0.755739218 0.873407642 1.32980145

0.80051503 0.900827793 1.58197929

0.842425066 1 1.71182341

0.832866109 1.04489028 2.33497331

0.847345121 1.03928135

0.705558983 0.820152657 1.01403881

0.723271361 0.835884548 1.27267089

0.766123527 0.862126681 1.51401474

0.806233036 0.95703828 1.63828053

0.797084748 1 2.23465884

0.810941717 0.994632038

0.709366837 0.824578965 1.01951151

0.727174808 0.84039576 1.27953941

0.770258244 0.866779521 1.52218577

0.810584221 0.962203351 1.64712222

0.801386561 1.00539693 2.24671914

0.815318315 1

0.695790909 0.808798097 1

0.71325807 0.824312189 1.25505147

0.755516971 0.850191015 1.49305404

0.795071186 0.943788615 1.61559944

0.786049551 0.986155554 2.20372122

0.799714678 0.980861908

0.554392331 0.644434206 0.796780072

0.568309816 0.656795525 1

0.601980867 0.677415258 1.18963571

0.633496877 0.751991961 1.28727744

0.626308618 0.785749094 1.75588115

0.637196719 0.781531222

0.46601857 0.541707182 0.669768121

0.477717517 0.552098026 0.840593464

0.506021182 0.569430838 1

0.532513334 0.632119527 1.082077

0.526470931 0.660495552 1.47598222

0.535623397 0.656950037

0.430670431 0.50061796 0.618965304

0.441481998 0.510220645 0.776833314

0.467638792 0.52623874 0.924148649

0.492121478 0.584172407 1

0.486537399 0.610396072 1.36402698

0.494995639 0.607119489

0.315734542 0.367014707 0.453777906

0.323660753 0.374054659 0.569514627

0.342836909 0.385797898 0.677514936

0.360785738 0.428270421 0.733123332

0.356691919 0.447495602 1

0.362892852 0.445093462

5 EUKLEMS data Household Consumption in Italy 1992–2004, from ISTAT’s tables. Data collected for the EUKLEMS database concerning Italy. This includes only Italy because at this level of detail the EUKLEMS project does not provide the data collected from the national statistical institutes. Initial treatment just for the ﬁve years 1999–2004. Other years dealt with elsewhere together with inputs of production concerning other countries.

Price Level Computation 95 Illustration 5: 1999-2004 L Laspeyres 1 1.02659 1.05674 1.08751 1.1173

0.9741 1 1.02939 1.0593 1.08823

0.94679 0.97171 1 1.02895 1.0572

0.9206 0.94472 0.97205 1 1.02735

0.89744 0.92068 0.94724 0.97412 1

L power 2 1 1.02659 1.05674 1.08733262 1.11716604

0.9741 1 1.02937043 1.05917071 1.08823

0.946542711 0.97171 1 1.02895 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 3 - final power followed by repetitions 1 0.9741 0.946542711 0.920086842 1.02659 1 0.97171 0.944550706 1.05674 1.02937043 1 0.97205 1.08733262 1.05917071 1.02895 1 1.11707117 1.08813903 1.05709178 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 4 1 1.02659 1.05674 1.08733262 1.11707117

0.9741 1 1.02937043 1.05917071 1.08813903

0.946542711 0.97171 1 1.02895 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

L power 5 = M derived Laspeyres 1 0.9741 0.946542711 1.02659 1 0.97171 1.05674 1.02937043 1 1.08733262 1.05917071 1.02895 1.11707117 1.08813903 1.05709178

0.920086842 0.944550706 0.97205 1 1.02735

0.896274995 0.920105733 0.946893346 0.97412 1

Paths 1,1,1,1,1,1 2,1,1,1,1,1 3,1,1,1,1,1 4,3,1,1,1,1 5,2,3,1,1,1

1,3,3,3,3,4 2,3,3,3,3,4 3,3,3,3,3,4 4,4,4,4,4,4 5,4,4,4,4,4

1,4,4,4,4,5 2,4,4,4,4,5 3,4,4,4,4,5 4,4,4,4,4,5 5,5,5,5,5,5

0.919681778 0.944134871 0.971864522 1 1.02656757

0.89519811 0.919000215 0.94599165 0.973378109 1

1,1,1,1,1,2 2,2,2,2,2,2 3,1,1,1,1,2 4,1,1,1,1,2 5,2,1,1,1,2

1,2,2,2,2,3 2,2,2,2,2,3 3,3,3,3,3,3 4,3,3,3,3,3 5,4,3,3,3,3

Consistency case: all diagonal elements = 1 H derived Paasche 1 0.974098715 1.02658865 1 1.05647636 1.02911363 1.08685393 1.05870441 1.115729 1.08683162

0.946306566 0.971467576 1 1.02875367 1.05608515

The 10 basic price level systems - the 10 columns of M and H

96 Concept and Method F derived Fisher - mean of derived Laspeyres 1 0.974099358 0.946424631 1.02658932 1 0.97158878 1.05660817 1.02924202 1 1.08709325 1.05893754 1.02885183 1.11639988 1.08748513 1.05658835

M and derived Paasche H 0.919884288 0.89573639 0.944342766 0.919552808 0.971957257 0.946442391 1 0.973748984 1.02695871 1

P mean basic price level - mean of columns of F 0.946499834 0.971666239 1.00007959 1.02893372 1.05667244 P/P mean basic 1 1.02658892 1.05660831 1.08709339 1.11640003

price index 0.974099743 1 1.02924189 1.0589374 1.08748498

0.946424506 0.971588908 1 1.02885183 1.05658835

0.919884167 0.94434289 0.971957257 1 1.02695871

0.895736272 0.91955293 0.946442391 0.973748983 1

0.990092751 0.995005771 0.993835084 1 1.00644396

0.983753485 0.988635049 0.987471858 0.993597301 1

X mean basic quantity level: PX = px 768307.584 772120.065 771211.618 775995.565 780996.047 X/X mean basic 1 1.00496218 1.00377978 1.01000639 1.01651482

quantity index 0.99506232 1 0.998823437 1.0050193 1.0114956

0.996234453 1.00117795 1 1.00620316 1.01268709

Price Level Computation 97 NOTES

1 Price levels Starting with the Laspeyres matrix ⎤ ⎡ 1 L12 L13 L14 ⎢L21 1 L23 L24 ⎥ ⎥ L=⎢ ⎣L31 L32 1 L34 ⎦ . L41 L42 L43 1 in the case of 4 periods, we can always redeﬁne the order of periods so that raising the L matrix to power m = 4 in the arithmetic where + means min yields, in the case of consistency of the data, ⎡ ⎤ 1 L12 L123 L1234 ⎢ L21 1 L23 L234 ⎥ ⎥ M =⎢ ⎣ L321 L32 1 L34 ⎦ L4321 L432 L43 1 where Lrij...ks = Lri Lij . . . Lks . By normalising each column of the M matrix by the respective ﬁrst element, we obtain ⎡ ⎤ 1 1 1 1 ⎢ L21 K21 K21 K21 ⎥ ⎥ A=⎢ ⎣ L321 L32 K21 K321 K321 ⎦ L4321 L432 K21 L43 K321 K4321 where Kij = 1/Lji is the Paasche index of i over j. In the matrix A, the Afriat upper bound for the price level solution is given by the ﬁrst column so that P1 = 1

P2 = L21

P3 = L321

P4 = L4321 ,

which is also equal to the last column of the matrix B deﬁned below. In the case of consistency of the data, raising the Paasche matrix ⎡ ⎤ 1 K12 K13 K14 ⎢K21 1 K23 K24 ⎥ ⎥ K ≡⎢ ⎣K31 K32 1 K34 ⎦ K41 K42 K43 1 to the power m =4 in the arithmetic where + now means max yields ⎡ ⎤ 1 K12 K123 K1234 ⎢ K21 1 K23 K234 ⎥ ⎥ H ≡⎢ ⎣ K321 K32 1 K43 ⎦ K4321 K432 K34 1

98 Concept and Method By normalising each column of the H matrix by the respective ﬁrst element, we obtain ⎡ ⎤ 1 1 1 1 ⎢ K21 L21 L21 L21 ⎥ ⎥ B=⎢ ⎣ K321 K32 L21 L321 L321 ⎦ K4321 K432 L21 K43 L321 L4321 In the matrix B, the Afriat lower bound for the price level solution is given by the ﬁrst column so that P1 = 1

P2 = K21

P3 = K321

P4 = K4321 ,

which coincides with the last column of the matrix A.

2 The triangle equality From the m × m Laspeyres matrix L we obtain the derived Laspeyres matrix M = Lm (where + means min), in the consistency case necessarily having the triangle inequality property Mij Mjk ≥ Mik . But this is without need for the triangle equality property Mij Mjk = Mik which corresponds to Fisher’s price index Chain Test, equivalent to the elements having the form of ratios of some numbers, for instance, for any k, Mij = Mik /Mjk for all i, j. We then obtain the derived Paasche matrix H where Hij = 1/Mji , and then F (for Fisher), the geometric mean of M and H , with elements 1 Fij = Hij Mij 2 1 = Mij /Mji 2 The geometric mean of the columns of F has elements 1

Fi = (k Fik ) m

Price Level Computation 99 with ratios 1 1 Fi /Fj = (k Fik ) m / k Fjk m 1 = k Fik /Fjk m

1 (Mik /Mki ) / Mjk /Mkj 2

= k

1 2

1 = k Mik Mkj / Mjk Mki 2

m1

m1

Hence subject to the triangle equality,

Fi /Fj = k Mij /Mji

12

m1

1 = Mij /Mji 2 = Fij . This shows how, subject to the triangle equality for M , from F we obtain the price levels Fi representing the geometric mean of the columns of F and hence of all the columns of M and H , and then from the price indices which are their ratios we just get back to F. Our Illustrations nos. 1 and 2 are examples.

3 Ratio matrix A matrix M is a ratio matrix if its elements Mij have the form of ratios of some numbers Xi , (i)

Mij = Xi /Xj

forming the elements of a vector X , the base vector from which it is derived. A test for M being a ratio matrix is the triangle equality (ii) Mij Mjk = Mik . Where M a matrix of price indices this would correspond to Fisher’s Chain Test. In case M is a derived Laspeyres matrix one would just have the triangle inequality (iii) Mij Mjk ≥ Mik , with ordinary data without the further imposition.

100 Concept and Method From (i) obviously (ii). And from (ii), for any k, we have (iv)

Mij = Mik /Mjk

for all i, j which exhibits (i) with Xi = Mik making base vector X coincide with column k of M .Hence: For a given matrix to be a ratio matrix the triangle equality is necessary and sufﬁcient and then any column of it is a base vector from which it can be derived. Hence given that M is a ratio matrix its columns, though different, would all be base vectors for the same ratio matrix, coinciding with the matrix M itself. The test for the triangle inequality is that the matrix be idempotent, or reproduced when multiplied by itself (with + = min). The test for the triangle equality is that the matrix coincide with the ratio matrix derived from any of its columns.

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Part II

Precursor

1 The system of inequalities ars > Xs − Xr S.N. Afriat, Research Memorandum No. 18 (October 1960) Econometric Research Program, Princeton University. Proc. Cambridge Phil. Soc. 59 (1963)

Proc. Camb. Phil. Soc. (1963), 59, 125

The system of inequalities ars > Xs − Xr ∗ Princeton University, Princeton, New Jersey, and Rice University, Houston, Texas. (Received 30 November 1961

Introduction In the investigation of preference orders which are explanations of expenditure data which associates a quantity vector xr with a price vector pr (r = 1, . . ., k), in respect to some n goods, there is considered the class of functions φ, with gradient g, which are increasing and convex in some convex region containing the points xr , such that gr = g(xr ) has the direction of pr .† Let ur = pr /er , where er = pr xr so that ur xr = 1. Then gr = ur λr , for some multipliers λr > 0, since φ is increasing. Also, if φr = φ(xr ) then (xr − xs ) gs > φr − φs , since φ is convex. Accordingly λr > 0,

λs Dsr > φr − φs

(r = s),

(I)

where Dsr = usxr − 1. With the number Drs given, there has to be considered all solutions = λr , = φr of the system of inequalities (I). With this system, there is involved a consideration of systems of the form ars > Xr − Xs (where ars = λs Dsr , Xr = φr ), the theory of which is going to be developed here. It is remarked, incidentally, that the existence of , satisfying (I) is equivalent to the existence of satisfying λr > 0,

λr Drs + λs Dst + · · · + λq Dqr > 0

(II)

for all distinct r, s, t, . . . , q taken from 1, . . . , k. ∗ Revised version of Research Memorandum no. 18, Econometric Research Program, Princeton University, October 1960, issued with the partial support of the U.S. Ofﬁce of Naval Research. † S. N. Afriat. Preference analysis: a general method with application to the cost of living index. (Research Memorandum, no. 29, Econometric Research Program, Princeton University, August 1961.)

114 Precursor Further, satisﬁes (II) if and only if there exists a such that , satisfy (I). Moreover, these two equivalent conditions on the number Drs which are provided by the consistency of the system of inequalities (I) and (II) are equivalent to a condition, applying directly to these numbers, which is given by the familiar Houthakker‡ ‘revealed preference’ axiom, which can be stated Drs 0, Dst 0, . . . , Dqr 0

(III)

impossible, for all distinct r, s, t, . . . , q taken from 1, . . . , k. Apart from any independent interest, the results which are now going to be obtained can be applied to a demonstration of these propositions, which are of fundamental importance for a method of empirical preference analysis in economics.

1 Open and closed systems Let n(n − 1) numbers ars (r = s; r, s = 1, . . . , n) be given; and consider the system of simultaneous inequalities S(a): ars > Xr − Xs

(r = s; r, s = 1, . . . , n)

deﬁning the open system S(a), of order n, with coefﬁcients ars . Any set of n numbers Xr (r = 1, . . ., n), forming a vector X , which satisfy these inequalities, deﬁne a solution X of the system S(a); and the system is said to be consistent if it has solutions. With the open system S(a), there may also be considered the closed system S(a), deﬁned by ¯ S(a): ars Xr − Xs

(r = s; r, s = 1, . . . , n).

¯ Obviously, solutions of S(a) are solutions of S(a), and the consistency of S(a) ¯ implies the consistency of S(a), but not conversely.

2 Chain coefﬁcients Let r, l, m, . . ., p, s denote any chain, that is a sequence of elements taken from 1, . . ., n with every successive pair distinct. Now from the coefﬁcients ars of a system there can be formed the chain coefﬁcient arlm···ps , determined on any chain, by the deﬁnition arlm···ps = arl + alm + · · · + aps .

‡ H. S. Houthakker. Revealed preference and the utility function, Economica, 17 (1950), 159–74.

The system of inequalities ars > Xs − Xr 115 Obviously ar···s···t = ar···s + as···t . Chains are considered associated with their coefﬁcients, so that by a positive chain is meant one with positive coefﬁcient, and so on similarly. A simple chain is one without loops, that is one in which no elements is repeated. There are n(n − 1) · · · (n − r + 1) = n!/r! simple chains of length r n, and therefore altogether

1 1 1 n! 1 + + + · · · + (n − 1)! 1! 2!

simple chains. A chain r, l, m, . . .p, s whose extremities are the same, that is, with r = s, deﬁnes a cycle. A simple cycle is one without loops. There are (n − 1) · · · (n − r + 1) = (n − 1)!/r! simple cycles of r n elements, and the total number of simple cycles is made up accordingly. The coefﬁcients ars + asr on the cycles of two elements deﬁne the intervals of the system. Any chain can be represented uniquely as a simple chain, with loops at certain of its elements, given by cycles through those elements; and the coefﬁcient on it is then expressed as the sum of coefﬁcients on the simple chain and on the cycles. Also, any cycles can be represented uniquely as a simple cycle, looping in simple cycles at certain of its elements, which loop in simple cycles at certain of their elements, and so forth, with termination in simple cycles. The coefﬁcient on the cycle is then expressed as a sum of coefﬁcients on simple cycles. Thus out of these generating elements of simple chains and cycles, ﬁnite in number, is formed the inﬁnite set of all possible chains.

3 Minimal chains Theorem 3.1 For the chains to have a minimum it is necessary and sufﬁcient that the cycles be non-negative. If any cycle total should be negative, then by taking chains which loop repeatedly round that cycle, chains which have increasingly negative coefﬁcients are obtained without limit; and so no minimum exists. However, should every cycle coefﬁcient be non-negative, then by cancelling the loops on any chain, there can be no increase in the coefﬁcient, so no chain coefﬁcient will be smaller than the coefﬁcient for some simple chain. But there is only a ﬁnite number of simple chains on a ﬁnite number of elements, and the coefﬁcients on these have a minimum.

116 Precursor Theorem 3.2 For the cycles to be non-negative it is necessary and sufﬁcient that the simple cycles be non-negative. For the coefﬁcient on any cycle can be expressed as a sum of coefﬁcients on simple cycles. Theorem 3.3 If the cycles are non-negative then a minimal chain with given extremities always exists and can be chosen to be simple. For any chain is then not less than the chain obtained from it by cancelling loops, since the cancelling is then the subtraction of a sum of non-negative numbers.

4 Derived systems According to Theorem 3.3, if the cycles of S(a) are non-negative, that is arlm···pr 0 for every cycle r, l, m, . . ., p, r, or equivalently for every simple cycle, by Theorem 3.2, then the coefﬁcients αrlm···ps on the chains with given extremities r, s have a minimum, and it is possible to deﬁne Ars = min arlm···ps l,m,...,p

(r, s = 1, . . . , n).

Then arlm···ps Ars for every chain and, by Theorem 3.3, the equality is attained for some simple chain. In particular, ars Ars . The number Arr is the minimum coefﬁcient for the cycle through r, so that arlm···pr Arr for every cycle, the equality being attained for some simple cycle. In particular, for a chain of two elements, ars + asr Arr . The hypothesis of non-negative cycles now has the statement Arr 0. The numbers Ars (r = s), thus constructed from the coefﬁcient of S(a), deﬁne the coefﬁcients of a system S(A), which will be called the derived system of S(a).

The system of inequalities ars > Xs − Xr 117 Any two systems will be said to be equivalent if any solution of one is also a solution of the other. Theorem 4.1 Any system and its derived system, when it exists, are equivalent. Let a system S(a) have a solution X . Then, for any chain of elements r, l, m, . . ., p, s there are the relations arl > Xr − Xl ,

alm > Xl − Xm ,

...,

aps > Xp − Xs

from which, by addition, there follows the relation arlm···ps > Xr − Xs . This implies that the derived coefﬁcients Ars exist, and Ars > Xr − Xs . That is, X is a solution of S(A). Now suppose the derived coefﬁcients Ars of S(a) are deﬁned, in which case ars Ars and let X be any solution of S(A), so that Ars > Xr − Xs . Then it follows immediately that ars > Xr − Xs or that X is a solution of S(a). Thus S(a) and S(A) have the same solutions, and are equivalent. Theorem 4.2 If the cycles of a system are non-negative or positive , then so correspondingly are the intervals of the derived system . Since Ars is the coefﬁcient of some chain with extremities r, s it appears that Ars + Asr is the coefﬁcient of some cycle through r, and therefore if the cycles of S(a) are non-negative, or positive, so correspondingly are the intervals Ars + Asr of the derived system S(A).

118 Precursor

5 Triangle inequality From the relation ar···s + as···t = ar···t it follows that the derived coefﬁcients Ars (r = s) satisfy the triangle inequality Ars + Ast Art the one side being the minimum for chains connecting r, t restricted to include s, and the other side being the minimum without this restriction. Theorem 5.1 Any system non-negative cycles is equivalent to a system which satisﬁes the triangle inequality , given by its derived system. This is true in view of Theorems 3.1, 4.1 and 4.2. Theorem 5.2 Any system which satisﬁes the triangle inequality has all its intervals non - negative. Thus, from the triangle inequalities applied to any system S(a), atr + ars ats ,

ats + asr atr

there follows, by addition, the relation ars + asr 0. Theorem 5.3 If a system satisﬁes the triangle inequality, then its derived system exists, and, moreover, the two systems are identical. From the triangle inequality, it follows by induction that arl + alm + · · · + aps ars . That is, arlm···ps ars , from which it appears that the derived system exists, with coefﬁcients Ars ars ,

The system of inequalities ars > Xs − Xr 119 so that now Ars = ars . This shows, what is otherwise evident, that no new system is obtained by repeating the operation of derivation, since the ﬁrst derived system satisﬁes the triangle inequality.

6 Extension property of solutions A subsystem Sm (a) of order m n of a system S(a) of order n is deﬁned by Sm (a): ars > Xr − Xs

(r, s = 1, . . . , m).

Then the systems Sm (a) (m = 2, 3, . . ., n) form a nested sequence of subsystems of S(a), each being a subsystem of its successor; and Sn (a) = S(a). Any solution (X1 , . . ., Xn ) of S(a) reduces to a solution (X1 , . . ., Xm ) of the subsystem Sm (a). But it is not generally true that any solution of a subsystem of S(a) can be extended to a solution of S(a). However, should this be the case, then the system S(a) will be said to have the extension property. Theorem 6.1 Any closed system which satisﬁes the closed triangle inequality has the extension property. Let X1 , . . ., Xm−1 be a solution of S¯ m−1 (a), so that ars Xr − Xs

(r, s = 1, . . . , m − 1).

It will be shown that, under the hypothesis of the triangle inequality, it can be extended by an element Xm to a solution of S¯ m (a). Thus, there is to be found a number Xm such that arm Xr − Xm ,

ams Xm − Xs

(r, s = 1, . . . , m − 1),

that is ams + Xs Xm Xr − arm . So the condition that such an Xm can be found is amq + Xq Xp − apm , where Xp − apm = max Xr − arm , r

amq + Xq = min amq + Xq . r

120 Precursor But if p = q, this is equivalent to amq + aqm 0, which is veriﬁed, by Theorem 5.2, and if p = q, it is equivalent to apm + amq Xp − Xq , which is veriﬁed, since, by hypothesis apm + amq apq , apq Xp − Xq . Therefore, under the hypothesis, the considered extension is always possible. It follows now by induction that any solution of Sm (a) can be extended to a solution of Sn (a) = S(a). This theorem shows how solutions of any system can be practically constructed, step-by-step, by extending the solutions of subsystems of its derived system. Theorem 6.2 Any closed system which satisﬁes the closed triangle inequality is consistent. For, by Theorem 5.2, a12 + a21 0; and this implies that the system S¯ 2 (a): a12 X1 − X2 ,

a21 X2 − X1

¯ has a solution, which, by Theorem 6.1, can be extended to a solution of S(a). ¯ Therefore S(a) has a solution, and is thus consistent. Theorem 6.3 Any open system which satisﬁes the triangle inequality and has positive intervals has the extension property, and is consistent. The lines of proof follow those of Theorems 6.1 and 6.2. A system is deﬁned to satisfy the triangle equality if ars + ast = art . Theorem 6.4 If a system satisﬁes the triangle inequality and has null intervals then it also satisﬁes the triangle equality and has null cycles. For, from ars + asr = 0,

ars + ast art

The system of inequalities ars > Xs − Xr 121 follows also ars + ast art so that ars + ast = art . By induction, arl + alm + · · · + aqp = arp and then arl + alm + · · · + apr = 0, that is, the cycles are null.

7 Consistency Theorem 7.1 A necessary and sufﬁcient condition that an open system be consistent is that its cycles by positive. If S(a) is consistent, let X be a solution. Then, for any cycle r, l, m, . . ., p, r there are the relations arl > Xr − Xl ,

alm > Xl − Xm ,

...,

apr > Xp − Xr ,

from which it follows, by addition, that arlm···pr > 0. Therefore, if S(a) is consistent, all its cycles must be positive. Conversely, let the cycles of S(a) be positive. Then the derived system S(A) is deﬁned, satisﬁes the triangle inequality, and has positive intervals. Hence, by Theorem 6.3, S(A) is consistent. But, by Theorem 4.1, S(A) is equivalent to S(a). Therefore, S(a) is consistent. Similarly: Theorem 7.2 A necessary and sufﬁcient condition that a closed system be consistent is that its cycles be non-negative.

122 Precursor

8 Cycle reversibility A cycle is deﬁned to be reversible in a system if the reverse cycle has the same coefﬁcient, thus arl···pr = arp···lr . The condition of k-cycle reversibility for a system is that all cycles of k element be reversible with regard to it; and the general condition of cycle reversibility is the reversibility condition taken unrestrictedly, in respect to all cycles of any number of elements. Theorem 8.1 For the reversibility of cycles in a system, the reversibility of 3-cycles is necessary and sufﬁcient. The proof is by induction, by showing that, given 3-cycle reversibility, the k-cycle condition is implied by that for (k − 1)-cycles. Thus, from al···k + akl = ak···l + alk with aol + alk + ako = aok + akl + alo , by addition, there follows aol + al···k + ako = aok + ak···l + alo . Theorem 8.2 If a system has positive intervals and reversible cycles, then it is consistent. Thus, if ars···pr = arp···sr and ars + asr > 0, then 2ars···pr = ars···pr + arp···sr = (ars + asr ) + · · · + (apr + arp ) > 0, so the cycles are positive, and hence, by Theorem 7.1, the system is consistent.

The system of inequalities ars > Xs − Xr 123 For any system S(a), deﬁne Crs···t = ars···tr − art···sr then Crs···t is an antisymmetric cyclic function of the indices r, s, . . ., t, depending just on the cyclic order of the indices and changing its sign when the cyclic order is reversed. The cycle reversibility condition for the system now has the statement Crs···t = 0 and it has been shown to be necessary and sufﬁcient just that Crst = 0. Thus the reversibility conditions are not all independent, but are implied by those for the 3-cycles. Moreover, not all the 3-cycle reversibility conditions are independent; but, as appears in the following theorem, the reversibility of three of the four 3-cycles in any four elements implies that for the fourth. Theorem 8.3 For any four elements α, β, γ , δ there is the identity Cβγ δ + Cαδγ + Cδαβ + Cγβα = 0. This can be veriﬁed directly. By the dependencies shown in this Theorem, the 16 n(n − 1)(n − 2) conditions for 3-cycle reversibility, contained in and implying a much larger set of general reversibility conditions, reduce to a set of 12 (n − 1)(n − 2) independent conditions. Theorem 8.4 There are 12 (n − 1)(n − 2) independent cycle reversibility conditions in a system of order n.

9 Median solutions Any solution X of a system S(a) must satisfy the condition ars > Xr − Xs > −asr , that is, the differences Xr − Xs must lie in the intervals [−asr , ars ], which are nonempty provided ars +asr > 0. In particular, a solution X such that these differences lie at the mid-points of these intervals will be called a median of the system. Thus, if X is a median of S(a) then Xr − Xs = 12 (ars − asr ). The condition that a system admit a median is decidedly stronger than that of consistency alone.

124 Precursor Theorem 9.1 A necessary and sufﬁcient condition that any system S(a) admit a median is that ars + ast + atr = ats + asr + art , ars + asr > 0. The condition is necessary, since a median is a particular solution of the system, the existence of which implies that the intervals ars + asr of the system are positive. Moreover, addition of the relations 1 Xr − Xs = (ars − asr ), 2 1 Xs − Xt = (ast − ats ), 2 1 Xt − Xr = (atr − art ), 2 gives 0 = ars − asr + ast − ats + atr − art . Also it is sufﬁcient. For it provides that, for any k, and all r, s ars − asr = (ark − aks ) − (ask − aks ), from which it follows that the numbers 1 Xr = (ark − akr ) 2 satisfy 1 Xr − Xs = (ars − asr ); 2 and, then since ars + asr > 0 they must be a solution of the system, which is, moreover, a median. Now, combining with Theorem 8.1, we obtain Theorem 9.2 A necessary and sufﬁcient condition that a system admit a median solution is that its intervals be positive and its cycles reversible.

The system of inequalities ars > Xs − Xr 125

10 Simple systems A system S(a) which is such that, for some k, s ark + akr > 0, ark + aks ars , for all r, s, will be called simple, with respect to the index k. Theorem 10.1 If S(a) is simple, with respect to k, then it is consistent, and admits as solution all sets of number Xr such that ark > Xr > −akr . For then, from relations ark > Xr ,

aks > −Xs ,

there follows ars ark + aks > Xr − Xs .

2 On the constructibility of consistent price indices between several periods simultaneously S.N. Afriat In Essays in Theory and Measurement of Demand in honour of Sir Richard Stone edited by Angus Cambridge University Press, 1981

Deaton.

On the constructibility of consistent price indices between several periods simultaneously

Introduction A price index refers to a pair of consumption periods, and price-index formulae usually involve demand data from the reference periods alone. When there are many periods, a price index can be determined from any one period to any other, in each case using the data from just those two periods. But then consistency questions arise for the set of price indices so obtained. Especially, they must have the consistency that would follow from their being ratios of ‘price levels’. The well-known tests of Irving Fisher have their origin in such questions. When these tests are regarded as giving identities to be satisﬁed by a standard formula and are taken in combination, it is impossible to satisfy them. Such impossibility remains even with partial combinations. Eichhorn and Voeller (1976) have given a full account of the inconsistencies between Fisher’s tests. Reference is made there for their results and for the history of the matter. Fisher recognizes the consistency question also in his idea of the ‘rectiﬁcation of pair comparisons’. In this the price indices are all calculated, as usual, separately and regardless of any consistency they should have together, and then they are all adjusted in some manner so that they can form a consistent set. For instance, by ‘crossing’ a formula with its ‘antithesis’ you got one that satisﬁed the ‘reversal’ test. Here he takes one of the tests separately as if any one could mean anything on its own, and contrives a formula to satisfy it. This is how he arrived at his ‘ideal’ index. It is ‘ideal’ because it satisﬁes the ‘reversal’ test but not so when those other tests are brought in. The search for a really ideal index seemed a hopeless task. In any case these tests are just negative criteria for index-number-making, showing how a formula can be rejected and telling nothing of how one should be arrived at. Something is to be measured and it is not yet considered what, but whatever it is it must ﬁt a certain mould. Here is not measurement but a ritual with form. In the background thought, what is to be measured is the price level, though prices are many so no one quite knows what that means, and a price index is a ratio of price levels. Therefore the set m2 price indices Prs (r, s = 1, . . . , m) between m periods 1, . . . , m must at least have the consistency required by their being ratios −1 Prs = Ps /Pr of m ‘price levels’ Pr (r = 1, . . . , m). Therefore Prr = 1, Prs = Psr ,

130 Precursor Prs Pst = Prt and so forth. There are other parts to Fisher’s tests and here we have the part that touches just the ratio aspect. In a seemingly more coherent approach, utility makes the base for what is being measured. There would be no problem there at all if only the utility function or order to be used could be known. But it is not known and therefore it is dealt with hypothetically. Its existence is entertained and inferences are made from that position. With utility in the picture the natural object of measurement is the ‘cost of living’, and at ﬁrst we know nothing of the price-level or of a price-index. Giving intelligibility to the price index in the utility framework involves imposing a special restriction on utility. Let M0 be any income in a period 0 when the prices are given by a vector p0 . Hypothetically, the bundle of goods x0 consumed with this income has the highest utility among all those which might have been consumed instead. Then it is asked what income M1 in a period 1 when the prices are p1 provides the standard of living, or utility, attained with the income M0 in period 0. With p0 , p1 ﬁxed and the utility order given, M1 is determined as a function M1 = F10 (M0 ) of M0 , where the function F10 depends on the prices p0 , p1 and the utility order. Without making any forbidding extra assumptions it can be allowed that this is a continuous increasing function, and that is all. However, turning to practice with price-indices, we ﬁnd that to offer a relationship between M0 and M1 is the typical use given to a price index. The relationship in this case has the form M1 = P10 M0 , P10 being the price index. In other words, using a price-index corresponds to the idea that there is a homogeneous linear relation between M0 , M1 or that the relation is a line through the origin, the price-index being the slope. To give the function F10 , just this form has implications about the utility from which it is derived. That utility must have a conical structure that is a counterpart of linear homogeneity of that function: if any commodity bundle x has at least the utility of another y then the same holds when x and y are replaced by their multiples xt and yt by any positive number t. To talk about a price-index and at the same time about utility, this assumption about the utility must be made outright. If a conical utility is given, relative to it a price-index P10 can be computed for any prices p0 , p1 . Then, as explained further in section 11, it has the form P10 = P1 /P0 where P0 = θ (p0 ), P1 = θ (p1 ) are the values of a concave conical function θ depending on prices alone. Price indices so computed for many periods automatically satisfy various tests of Fisher. The issue about those tests therefore becomes empty in this context, and pair-comparisons so obtained need no ‘rectiﬁcation’. But a remaining issue come from the circumstance that a utility usually is not given. Should one be proposed arbitrarily as a basis for constructing price indices, there can be no objection to it merely on the basis of the tests, at least with those that concern the ratio-aspect of price indices. With each price-index formula P10 of the very many he surveyed, Fisher associated a quantity-index formula X10 in such a way that the product is the ratio of consumption expenditures M0 = p0 x0 , M1 = p1 x1 in the two periods. For instance

On the constructibility of consistent price indices 131 with the Laspeyres price index P10 = p1 x0 /p0 x0 , the corresponding quantity-index is X10 = p1 x1 /p1 x0 , and then P10 X10 = ( p1 x0 /p0 x0 )( p1 x1 /p1 x0 ) = p1 x1 /p0 x0 = M1 /M0 . As a possible sense to this scheme, it is as if, beside the price-index being a ratio P10 =P1 /P0 of price-levels, also the quantity-index is a ratio X10 =X 1 /X 0 of quantity levels, and price-level multiplied with quantity level is the same as price-vector multiplied with quantity-vector, that is P0 X0 = p0 x0 = M0 ,

P1 X1 = p1 x1 = M1

to give P10 X10 = ( P1 /P0 ) (X1 /X0 ) = ( P1 X1 )/( P0 X0 ) = M1 /M0 . Here there is the simple result that all prices are effectively summarized by a single number and all quantities by a single quantity number, and instead of doing accounts by dealing with each price and quantity separately, and also with their product that gives the cost of the quantity at the price, the entire account can be carried on just as well in terms of these two summary price and quantity numbers, or levels, whose product is, miraculously, the cost of that quantity level at that price level. Though there are many goods and so-many prices and quantities, still it is just as if there was effectively just one good with a price and obtainable in any quantity at a cost which is simply, as with a simple goods, the product of price times quantity. Any mystery about the meaning of a price index vanishes, because it becomes simply a price. Were this scheme valid we could ask for so much utility, enquire the price and pay the right amount by the usual multiplication. When applied to income M 0 in period 0 when the price level is P0 , the level of utility is purchases is X0 =M0 /P0 . Then the income that purchases the same level of utility in period 1 when the price level is P1 is given by M1 =P 1 X 0 . Hence, by division, M1 /M0 =P1 /P0 =P10 , giving the relation M1 =P10 M0 as usual. Whether or not this scheme has serious plausibilities, it is implicit whenever a price index based on utility is in view. However, though such a scheme has here been imputed as belonging to Fisher’s system, or conjured up as though that seems to belong to it or at least gives it an intelligibility, it cannot be considered to have clear presence there. For Fisher’s system does not have a basis in utility and this scheme does. While this circumstance is not evidence of a union it still might not seem to force a separation. However, a symptom of a decided separation is that, even when many periods are involved, Fisher still followed standard custom in regarding an index formula as one involving the demand data just from its pair of reference periods, and really his system is about such formulae. Then he worried about the incoherence of the set of price indices for many periods so obtained. The utility formulation cares nothing about the form of the formula. When many periods are involved and all the price indices between them are to be

132 Precursor calculated, the calculation of one and all should involve the data for all periods simultaneously. In the utility approach immediate thought is not of the demand data and of formulae in these at all, but rather it is of the utility order which gives the basis of the calculations, and necessarily gives coherent results. Instead, Fisher forces incoherence by rigidly following the standard idea of what constitutes an index formula. The main issue with the utility approach is about the utility function or order. When that utility is settled all that remains to be dealt with is a well-deﬁned objective of calculation based on that utility. The role of the demand data is just to put constraints on the permitted utility order, and consequently price indices based on utility become based on that data. Having such constraints, the ﬁrst question then is about the existence of a utility order that satisﬁes them. If none exists then no price indices exist and there the matter ends. Though that is so in the present treatment, by making the constraints more tolerant it is possible to go further (see Afriat, 1972b and 1973). In the standard model of the consumer, choice is governed by utility, to the effect that any bundle of goods consumed has greater utility than any other attainable with the same income at the prevailing prices. With this model, the obvious constraint on a utility for it to be permitted by given data is, ﬁrstly, that it validate the model for the consumer on the evidences provided by that data. Then further, since price-indices are to be dealt with, the conical property of utility should be required. With this method of constraint and the other deﬁnitions that have been outlined, everything is available for developing the questions that are in view. But ﬁrst there will be a change in formulation that has advantages. Instead of requiring that a chosen bundle of goods be represented as being the unique best among all those attainable for no greater cost at the prevailing prices, or as being deﬁnitely better than any others in utility, it will be required that it be just one among the possibly many best, or one at as least as good as any other. This alters nothing if certain prior assumptions are made about the utility order, for instance that it is representable by an increasing strictly quasiconcave utility functions. With the latter assumption a utility maximum under a budget constraint must in any case be a unique maximum, and so adding that the maximum is unique just makes a redundance. But we do not want to introduce additional assumptions about utility. A utility is wanted that ﬁts the data in a certain way, and if all that is now wanted in such a ﬁt is that some commodity bundles be represented as having at least the utility of certain others then we can always count on a utility function that is constant everywhere to do that service. In making what could at ﬁrst seem a slight change in the original formulation of the constraint on permitted utilities, the result is no constraint at all: whatever the data there always exists a permitted utility, for instance the one mentioned which will give zero as the cost of attaining any given standard of living. That change is drastic and no such change is sought. All that is in view is a change that alters nothing important in the results, the effect being something like replacing an open interval by a closed one, while it is better to work with and in any case is conceptually ﬁtting. One possibility is to add a monotonicity condition as an assumption about utility expressing that

On the constructibility of consistent price indices 133 ‘more is better’. But, as said, we do not want any such prior assumptions. Instead consider again the original strict condition, that the chosen bundle be the unique best attainable at no greater cost. It implies the considered weaker condition in which the uniqueness has been dropped. But also it implies a second condition: the cost of the bundle is the minimum cost for obtaining a bundle that is as good as it. These two conditions are generally independent, even though relations between them can be produced from prior assumptions about utility, of which we have none, and their combination is implied by the stricter and analytically more cumbersome original condition. They are just what is wanted. They have equal warrant as economic principles. In the context of cost–beneﬁt analysis they are familiar as constituting the two main criteria about a project, that it be cost-effective or gives best value for the cost, and cost-efﬁcient or the same value is unattainable at lower cost. Now the wanted constraint on an admissible utility can be stated by the requirement that every bundle of goods given in the demand data be represented by it as cost-effective and cost-efﬁcient. Such a utility can be said to be compatible with the demand data. Then that data is consistent if there exists a compatible utility. It is homogeneously consistent if there exists a compatible utility that moreover has the property of being conical, or homogeneous, required whenever dealing with price-indices. A compatible price-index, or a ‘true’ one, is one derived on the basis of a homogeneous utility that is compatible with the given data. The ﬁrst problem therefore is to ﬁnd a test for the homogeneous consistency of the data. In the case where there are just two periods, the test found reduces to a relation that is quite familiar, in a context where it is not at all connected with this test but is offered as a ‘theorem’, though certainly it is not that. The relation is simply that the Paasche index from one period to the other does not exceed that of Laspeyres. The relation is symmetrical between the data in the two periods, and so there is no need to put in this unsymmetrical form where one period is distinguished as the base. But this is the form in which it is familiar and known as the ‘Index-Number Theorem’. That the ‘Theorem’, or relation, is necessary and sufﬁcient for homogeneous consistency of the demands in the two periods is a theorem in the ordinary sense. It is going to be generalized for any number of periods. Related to the Index-Number Theorem is the proposition that the Laspeyres and Paasche indices are upper and lower ‘limits’ for the ‘true index’. From the foregoing consistency considerations it is recognized that even the existence of a price index, at least in the sense entertained here, can be contradicted by the data, so certainly some additional qualiﬁcation is needed in the ‘limits’ proposition. Also, what makes an index ‘true’ has obscurities in early literature. An interpretation emerging in later discussions is that a true index is simply one derived on the basis of utility. This could be accepted to mean one that, in present terms, is compatible with the given demand data. With demand data given for any number of periods and satisfying the homogeneous consistency test, a price index compatible with those data can be constructed from any period to any other. It has many possible values

134 Precursor corresponding to the generally many compatible homogeneous utilities. These values describe a closed interval whose endpoints are given by certain formulae in the given demand data. A special case of this result applies to the situation usually assumed in indexnumber discussions. In this, the only data involved in a price-index construction between two periods are the data from the two reference periods themselves. For this case the formulae for the endpoints of the interval of values for the price index reduce to the Paasche and Laspeyres formulae. Here therefore is a generalization of those well-known formulae for when demand data from any number of periods can be permitted to enter the calculation of a price-index between any two. The values of these generalized Paasche and Laspeyres formulae are well deﬁned just in the case of homogeneous consistency of the data, under which condition they have the price-index signiﬁcance just stated. Then a counterpart of the ‘IndexNumber Theorem’ condition in the context of many periods is that the generalized Paasche formula does not exceed the generalized Laspeyres formula. There seems to be one such condition for each ordered pair of periods, making a collection of conditions. However, all are redundant because they are automatically satisﬁed whenever the formulae have well-deﬁned values, as they do just in the case of homogeneous consistency of the data. For price-indices Prs between many periods to be consistent they should have the form Prs = Ps /Pr for some Pr . Let Pˆ rs , Pˇ rs be the generalized Laspeyres and Paasche formulae. These, when they have well-deﬁned values, are connected by the relation Pˆ rs Pˇ sr = 1 and have the properties Pˆ rs Pˆ st ≤ Pˆ rt , Pˇ rs Pˇ s t ≤ Pˇ rt . Then it is possible to solve the system of simultaneous inequalities Pˆ rs ≥ Ps /Pr for the Pr . The system Pˇ rs ≤ Ps /Pr is identical with this, so solutions automatically satisfy Pˇ rs ≤ Ps /Pr ≤ Pˆ rs Now it is possible to describe all the price-indices Prs between periods that are compatible with the data and form a consistent set: they are exactly those having the form Prs = Ps /Pr where Pr is any solution of the above system of inequalities. The condition for their existence is just the homogeneous consistency of the data. For any solution Pr there exists a homogeneous utility compatible with the data on the basis of which Prs = Ps /Pr is the price-index from period r to periods s. This will be shown by actual construction of such a utility. Now to be remarked is the extension property of any given price-indices for a subset of the periods that are compatible with the data and together are consistent: it is always possible to determine further price-indices involving all the other periods so that the collection price-indices so obtained between all periods are both compatible with the data and together are consistent. There is an ambiguity here about a set of price indices being compatible with the data: they could be that with each taken separately, or in another stricter sense where they are taken simultaneously together. But in the conjunction with consistency the

On the constructibility of consistent price indices 135 ambiguity loses effect. For price-indices Prs that are all independently compatible with the data, the compatibility of each one with the data being established by means of a possibly different utility, if they are all consistent there also they are jointly compatible in that also there exists a single utility, homogeneous and compatible with the data, that establishes their compatibility with the data simultaneously. This completes a description of the main concepts and results dealt with in this paper. Further remarks concern computation of the m × m—matrix of generalized −1 Laspeyres indices Pˆ rs , and hence also the generalized Paasche indices Pˇ rs = Pˆ sr , from the matrix of ordinary Laspeyres indices Lrs = ps xr /pr xr . An algorithm proposed goes as follows. The matrix L with elements Lrs is raised to powers in a sense that is a variation of the usual, in which a + b means min [a, b]. With the modiﬁcation of matrix addition and multiplication that results associativity and distributivity laws are preserved, and matrix ‘powers’ can be deﬁned in the usual way by repeated ‘multiplication’. The condition for the powers L, L2 , L3 , . . . to converge is simply the homogeneous consistency of the data. Then for some k ≤ m, Lk−1 = Lk , and in that case also Lk = Lk+1 = . . . so the calculation of powers can be broken-off as soon as one is found that is identical with its predecessor. Finding such a k ≤ m by this procedure is a test for homogeneous consistency; ﬁnding a diagonal element less than unity denies this condition and terminates the procedure. With such a k let Pˆ = Lk . The elements Pˆ rs of Pˆ are the generalized Laspeyres indices. A programme for this algorithm is available for the TI-59 programmable calculator applicable to m ≤ 6, and another in Standard BASIC for a microcomputer.

1 Demand With n commodities, n is the commodity space and n is the price or budget space. These are described by non-negative column and row vectors with n elements, being the non-negative numbers. Then x ∈ n , p ∈ n have a product px ∈ , giving the value of the commodity bundle x at the prices p. Any (x, p) ∈ n × n with px >0 deﬁnes a demand, of quantities x at prices p, with expenditure given by M = px. Associated with it is the budget vector given by u = M −1 p and such that ux = 1. Then (x, u) is the normal demand associated with (x, p)(px > 0). Some m periods of consumption are considered, and it is supposed that demands (xt , pt )(t = 1, . . . , m) are given for these. With expenditures Mt = pt xt and budgets ut = Mt−1 pt , so that ut xt = 1, the associated normal demands are the (xt , ut ). Then Lrs = ur xs = pr xs /pr xr is the Laspeyres quantity index from r to s, or with r, s as base and current periods. It is such that Lrr = 1. Then the coefﬁcients Drs = Lrs − 1 are such that Drr = 0. To be used also are the Laspeyres chain-coefﬁcients Lrij . . .ks = Lri Lij . . .Lks . Any collection D ⊆ n × n of demands is a demand relation. Here we have a ﬁnite demand relation D with elements (xt , pt ). For any demand (x, p), the collection

136 Precursor of demands of the form (xt, p), where t > 0 is its homogeneous extension, and the homogeneous extension of a demand relation is the union of the homogeneous extension of its elements. A homogeneous demand relation has the property x Dp, t > 0 → xt Dp making it identical with its homogeneous extension. For a normal demand relation E, or one such that xEu → ux = 1, homogeneity is expressed by the condition x Eu, t > 0 → xt Et −1 u If E is the normal demand relation associated with D, or the normalization of D, then this is the condition for D to be homogeneous.

2 Utility A utility relation is any binary relation R ⊆ n × n that is reﬂexive, x Rx, and transitive, x RyRz→x Rz, by which properties it is an order. Then the symmetric part E = R ∩ R , for which xEy ↔ xRy ∧ xR y ↔ xRy ∧ yRx is a symmetric order, or an equivalence, and the antisymmetric part P = R ∩ R , for which

xPy ↔ xRy ∧ xR Y ↔ xRy ∧ yRx ↔ xRy∧ ∼ yRx is irreﬂexive and transitive, or a strict order. A homogeneous, or conical, utility relation is one that is a cone in n × n , that is x Ry, t > 0 → xt Ryt. Any φ : n → is a utility function, and it is homogeneous or conical if its graph is a cone, the condition for this being φ(xt) = φ(x)t (t > 0). A utility function φ represents a utility relation R if xRy ↔ φ(x) ≥ φ(y). If φ is conical so is R.

3 Demand and utility A demand (x, p) and a utility R are compatible if (i)

py ≤ px → xRy

(ii)

y Rx → py ≥ px

(3.1)

They are homogeneously compatible if (xt, p) and R are compatible for all t > 0, in other words if the homogeneous extension of (x, p) is compatible with R. If R is homogeneous, compatibility is equivalent to homogeneous compatibility.

On the constructibility of consistent price indices 137 Demand and utility relations D and R are compatible if the elements of D are all compatible with R, and homogeneously compatible if the homogeneous extension of D is compatible with R. A demand relation D is consistent if it is compatible with some utility relation, and homogeneously consistent if moreover that utility relation can be chosen to be homogeneous. Homogeneous consistency of any demand relation is equivalent to consistency of its homogeneous extension. That the former implies the latter is seen immediately, and the converse will be shown later. It should be noted that (3.1 (ii)) in contrapositive is py < px → yRx, and, with the deﬁnition of P in section 2, this with (3.1 (i)) gives py < px → xPy

(3.2)

so this is a consequence of (3.1).

4 Revealed preference A relation W ⊂ n × n is deﬁned by x Wu ≡ ux ≤ 1. Then x Wu, that is ux ≤1, means the commodity bundle x is within the budget u, and Wu = [x : x Wu] = [x : ux ≤ 1] is the budget set for u, whose elements are the commodity bundles within u. The revealed preference relation of a demand (x, p) is [(x, y) : py ≤ px]. For a normal demand (x, u) (ux = 1) it is (x, Wu) = [(x, y) : y ∈ Wu] = [(x, y) : uy ≤ 1]. The revealed homogeneous preference relation is the conical closure of the revealed preference relation, so it is [(xt, y) : py ≤ pxt, t > 0] for a demand and ∪t>0 (xt, Wt −1 u) for the normal demand. The condition (3.1), which is a part of the requirement for compatibility between a demand and utility (x, p) and R, asserts simply that the utility relation contains the revealed preference relation. If the utility relation is homogeneous, this is equivalent to its containing the revealed homogeneous, this is equivalent to its containing the revealed homogeneous preference relation. Let Rt be the revealed preference relation of the demand (xt , pt ), and R˙ t the revealed homogeneous preference relation. Then, as remarked, compatibility of that demand with a utility R requires that Rt ⊆ R

(4.1)

and if R is homogeneous this is equivalent to R˙ t ⊆ R

(4.2)

Now let RD , the revealed preference relation of the given demand relation D, be deﬁned as the transitive closure of the union of the revealed preference relations

138 Precursor

t Rt . The compatibility of R with D requires (4.1) for all Rt of its elements, RD = ∪

t R˙ t , the t, and because R is transitive this is equivalent to RD ⊂ R. Also let R˙ D = ∪ transitive closure of the union of the revealed homogeneous preference relation R˙ t of the elements of D, deﬁne the revealed homogeneous preference relation of D. Then, by similar argument, with (4.2), compatibility of D with a homogeneous R implies R˙ D ⊆ R. While RD is transitive from its construction, and reﬂexive at the points xt , because Rt is reﬂexive at xt , R˙ D is both transitive and conical, and reﬂexive on the cone through the xt .

5 Revealed contradictions A demand relation D with elements (xr , pr ) is compatible with a utility relation R if (i)

pt x ≤ pt xt → xt R x

(ii)

xRxt → pt x ≥ pt xt

(5.1)

and D is consistent if some compatible order R exists. It has been seen that (5.1) is equivalent to RD ⊆ R

(5.2)

Therefore, if D is compatible with R, x RD xt → x R xt for any t and x, and also pt x < pt xt → xRxt . Therefore, on the hypothesis that D is compatible with some, R, the condition xRD xt , pt x < pt xt

(5.3)

implies xRxt , xRxt making a contradiction, so the hypothesis is impossible and D is inconsistent. The condition (5.3) for any t and x is a revealed contradiction, denying the consistency of D. Thus: The existence of a revealed contradiction is sufﬁcient for D to be inconsistent.

(5.4)

Now it will be seen to be also necessary. The condition for there to be no revealed contradictions is the denial of (5.3), for all t and x; equivalently x RD xt → pt x ≥ pt xt

(5.5)

But this is just the condition (5.1 (ii)) with R = RD . Because (5.1 (i)) is equivalent to (5.2), and because in any case RD ⊆ RD so (5.2) is satisﬁed with R = RD , it is

On the constructibility of consistent price indices 139 seen that (5.4) is necessary and sufﬁcient for (5.1 (i) and (ii)) to be satisﬁed with R = RD , in other words for D to be compatible with RD . Thus: The absence of revealed contradictions is necessary and sufﬁcient for D to be compatible with RD .

(5.6)

As a corollary: The absence of revealed contradictions implies the consistency of D. (5.7) For consistency means the existence of some compatible order, and by (5.5) under this hypothesis RD is one such order. Now with (5.5): The absence of revealed contradictions is necessary and sufﬁcient for the consistency of D and implies compatibility with RD .

(5.8)

By exactly similar argument, xR˙ D xt θ and pr x < pr xr θ, for any r, x and θ > 0, make a homogeneously revealed contradiction denying the homogeneous consistency of D, or the existence of a compatible homogeneous utility. Then xR˙ D xt θ → pr x ≥ pr xr θ for all r, x and θ > 0 asserts the absence of homogeneously revealed contradictions. Then there is the following: Theorem For a demand relation to be compatible with some homogeneous utility relation, and so homogeneously consistent, it is necessary and sufﬁcient that its revealed homogenous preference relation be one such relation, and for this the absence of homogeneously revealed contradictions is necessary and sufﬁcient. This theorem holds unconditionally, regardless of whether or not D is ﬁnite. However, when D is ﬁnite the homogeneous consistency condition has a ﬁnite test, developed in the next two sections. Because the revealed preference relation of the homogeneous extension of a demand relation is identical with its revealed homogeneous preference relation, it appears now, as remarked in section 3, that homogeneous consistency of a demand relation is equivalent to consistency of its homogeneous extension. For, as just seen, the ﬁrst stated condition on D is equivalent to compatibility with R˙ D and the second with RD˙ , so this conclusion follows from R˙ D = RD˙ .

6 Consistency Though RD , R˙ D and the Lrs which give the base for the following work are derived from D, they are also derivable from the normal demand relation E derived from D. Therefore there would be no loss in generality if only normal demand relations were considered.

140 Precursor As a preliminary, the deﬁnition of the revealed homogeneous preference relation R˙ D will be put in a more explicit from. This requires identiﬁcation of the transitive

An R-chain is any sequence closure of any relation R with its chain-extension R.

to its successor, that of elements x, y, . . . , z in which each has the relation R is x Ry R. . . Rz . Then the chain-extension R is the relation that holds between

z ≡ (∨y, . . . )x Ry R. . . R z. The relation R

so deﬁned extremities of R-chains, so x R

can be identiﬁed with the transitive closure of R, that is as the smallest transitive relation containing R, it being such that it is transitive, contains R and is contained in every transitive relation that contains R. Therefore xR˙ D y means x, y are extremities of a chain in the relation ∪t R˙ t . This means there exist r, i, . . . , k and zi , . . . , zk such that xRr zi Ri . . . zk Rk y. But, considering the form of the elements of the Rt , now we must have xr = xr θr , zi = xi θi , . . . , zk = xk θk for some θr , θi , . . . , θk > 0, and uk y ≤ θk . Accordingly, the condition xR˙ D xs θs → us x ≥ θs for all x, s and θs > 0, for the absence of homogeneously revealed contradictions, can be restated as the condition ˙ · · · xk θk xs θs → Us xr θr ≥ θs xr θr R˙ r xi θi Ri

(6.1)

for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0. From the form of the elements of the R˙ t , that is ur xi θi ≤ θr , ..., uk xs θs ≤ θk → us xr θr ≥ θs or, in terms of the Laspeyres coefﬁcients, Lri ≤ θr /θi , . . . , Lks ≤ θk /θs → Lsr ≥ θs /θr

(6.2)

Another way of stating this condition is that (Lri , . . . , Lks , Lsr ) ≤ (θr /θi , . . . , θk /θs , θs /θr )

(6.3)

is impossible for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0. This condition will be ˙ denoted K. While theory based on homogeneity is here the main object, in the background is the further theory without that restriction. Some account of that is given here, but it is mainly given elsewhere as already indicated. The dots used in the notation are to distinguish features in this homogeneous theory from their counterparts without homogeneity. The homogeneous theory is required in dealing with price indices, but still it has its source in the more general theory. It is useful in this section and later to bring counterparts of the two theories together for recognition of the connections and the differences. Condition (6.1) has been identiﬁed with the condition for the absence of homogeneously revealed contradictions that in the last section was shown necessary and sufﬁcient for the consistency of the given demands, that is, for the existence of a homogeneous utility compatible with them all simultaneously.

On the constructibility of consistent price indices 141 The weaker condition that is the counterpart without homogeneity, put in a form that assists comparison, is the condition K given by (Lri , . . . , Lks , Lsr ) ≤ (1, . . . , 1, 1)

(6.4)

is impossible for all r, i, . . . , k, s. By taking the θ s all unity in (6.3), (6.4) is obtained, so (6.3) implies (6.4) as should be expected. If compatibility between demand and utility is replaced by strict compatibility, by replacing cost-efﬁcacy or cost-efﬁciency by their strict counterparts, which conditions in fact are equivalent to each other, and strict consistency of any demands means the existence of a strictly compatible utility, then the test for this condition which is a counterpart of (6.4) is the condition K ∗ given by (Lri , . . . , Lks , Lsr ) ≤ (1, . . . , 1, 1)

(6.5)

is impossible for all r, i, . . . , k, s unless xr = xi = · · · = xk = xs , in which case the equality holds. This is just a way of stating the condition of Houthakker (1950), know as the Strong Axiom of Revealed Preference, for when that condition is applied to a ﬁnite set of demand instead of to the inﬁnite set associated with a demand function. Here the ﬁniteness is not essential and is just a matter of notation, though in later results it does have an essential part. Corresponding to the results obtained for the less strict consistency, and for homogeneous consistency, as in the Theorem or section 5, Houthakker’s condition is necessary and sufﬁcient for the existence of strictly compatible utility, and for the revealed preference relation to be one such utility. While (6.5) is the ‘strict’ counterpart of (6.4), the corresponding counterpart for (6.3) is the condition K given by (Lri , . . . , Lks , Lsr ) ≤ (θr /θi . . . , θk /θs , θs /θr )

(6.6)

is impossible for all r, i, . . . , k, s and θr , θi , . . . , θk , θs > 0 unless xr θr = xi θi = · · · = xs θs in which case the equality holds. Just as a dot signiﬁes a condition associated which homogeneity, a star signiﬁes belonging to the ‘strict’ theory. The various conditions that have been stated have the relations K˙ → K ↑

↑

K˙ ∗ → K ∗ The main result of this section, which is about K˙ in (6.3) being a consistency condition, will be part of a theorem in the next section where it is developed into another form.

142 Precursor It can be noted that (6.3) is equivalent to the same condition which r, i, . . . , k, s restricted to be all distinct. For the second condition is part of the ﬁrst. Also, the inequalities stated in the ﬁrst, involving a cycle of elements, can be partitioned into groups of inequalities involving simple cycles, each without repeated elements, showing that also the ﬁrst follows from the second. A ﬁnite consistency test is wanted, one that can be decided in a known ﬁnite number of steps. The last conclusion goes a step towards ﬁnding such a test, though it does not give one. That will be left to the next section. However, (6.4) and (6.5) taken with the indices all distinct already represent ﬁnite tests. But still this is not the case for Houthakker’s condition (6.5), or for (6.4), when these are regarded as applying to a demand function (6.5), or for (6.4), when these are regarded as applying to a demand function, for which the number of cycles of distinct demands is unlimited.

7 Finite test Theorem 7.1 For any ﬁnite demand relation D the following conditions are equivalent (H˙ ) D is homogeneously consistent, that is, there exists a compatible homogeneous utility relation; ˙ D is compatible with its own revealed homogeneous utility relation R˙ D ; (R) ˙ (Lrs , Lst , . . . , Lqr ) ≤ (θr /θs , θs /θt , . . . , θq /θr ) is impossible for all distinct (K) r, s, . . . , q and θr , θs , . . . , θq > 0; ˙ Lrst... qr ≥ 1 for all distinct r, s, . . . , q. (L) Arguments for the equivalences between H˙ , R˙ and K˙ have already been given in the last two sections. It is enough now to show K˙ and L˙ are equivalent. By multiplying the inequalities stated for any case where K˙ is denied, it follows that Lrst... qr < (θr /θs )(θs /θt ) · · · (θq /θr ) = 1 ˙ Thus L˙ → K. ˙ Now, contrary to L, ˙ suppose Lrst . . . qr < 1, contrary to L. and let θr = Lrst···qr ,

θs = Lst···qr,··· ,

θq = Lqr

Then Lrs = θr /θs ,

Lst = θs /θt , . . .

and ﬁnally, θr < 1 and θq = Lqr so that Lqr < θq /θr , showing a denial of K. ˙ and the two conditions are now equivalent. Thus K˙ → L,

On the constructibility of consistent price indices 143 Because the number of simple cycles that can be formed from m elements is ﬁnite and given by

m m m (r − 1)! = (r − 1)!m!/r!(m − r)! r r=1

r−1

=

m

(m − r + 1) · · · m/r

r−1

L˙ is a ﬁnite test. The counterpart of K˙ for the general non-homogeneous theory has already been stated, and that for L˙ is (L) There exist positive λs such that for all distinct r, s, t, . . . , q (λr Lrs + λs Lst + · · · + λq Lqr )/(λr + λs + · · · + λq ) ≥ 1 It can be noted that while L˙ shows a ﬁnite test and K˙ does not, L does not and K does. There are several routes for proving the equivalence of K and L, all of some length. From that equivalence it is known that L˙ → L. But this can be seen also directly. From the theorem that the geometric mean does not exceed the arithmetic, 1/k

(Lrs + Lst + · · · + Lqr )/k ≥ Lrst···qr k being the number of elements in the cycle. Therefore L˙ implies (Lrs + Lst + · · · + Lqr )/k ≥ 1 But this validates L with all the λs equal to unity. The counterpart of L for the ‘strict’ theory, equivalent to Houthakker’s revealed preference axiom, is (L*) There exist positive λs such that, for all distinct r, s, t, . . . , q (λr Lrs + λs Lst + · · · + λq Lqr )/(λr + λs + · · · + λq ) ≥ 1 the equality holding just when xr = xs = xt = · · · = xq .

8 A system of inequalities The test Lrs . . . qr ≥ 1 for all distinct r, s, . . . , q, that was found for the homogeneous consistency of a demand relation is also the test for solubility of the system of inequalities Lrs ≥ φs /φr

for all r, s

(8.1)

for numbers φr (r = 1, . . . , m). Such numbers obtained by solving the inequalities will be identiﬁed as utility-levels for the demand periods, because for any demand

144 Precursor period there exists a homogeneous utility compatible with the demands that identify them all as such, in that the numbers Xrs = φs /φr are identiﬁed as quantityindices, compatible with the data, and price indices correspond to these. By taking logarithms the system (7.1) comes into the form ars ≥ xs − xr

(8.2)

where xr = log φr and ars = log Lrs . An account of the system in this from has been given in Afriat (1960), and in the last section here it is developed to suit needs of the present application. The same system, in the additive from, arises also in the version of this theory unrestricted by homogeneity. It is required to ﬁnd a positive solution of the system of homogeneous linear inequalities λr (Lrs − 1) ≥ φs − φr

(8.3)

That the φs be positive is inessential because they enter through their differences, and so a constant can always be added to make them so, but the restriction is essential. The λs occurring in solutions of (7.3) are identical with the λs that are solutions of (6.4), so they can be determined separately. With any λs so determined, and ars = λr (Lrs − 1), (7.3) is in the from (7.2) for determining the φs. The φs and λs in any solution become utilities and marginal utilities at the demanded xs with a compatible utility that is constructed by means of the solution. In the case of homogeneous utility, λr = φr , and with this substitution (7.3) reduces to (7.1). An entirely different connection for the system (7.2) is with minimum paths in networks. With the coefﬁcient ars as direct path-distances, a solution of (7.2) corresponds to the concept of a subpotential for the network, as described by Fiedler and Ptak (1967). Whereas there it is an auxiliary that came in later, here it is a principal objective and a starting point. Then there is the linear programming formula Aij = min[xj − xi : ars ≥ xr − xs ] expressing the minimum path-distance Aij as the minimum subpotential difference, as learnt from Edmunds (1973). It is familiar under the assumption ars > 0, and in the integer programming context. Close to hand in the 1960 account is this formula without the non-negativity restriction on the coefﬁcients and a quite different method of proof.

9 Utility construction Now to be considered is how, for any number φt > 0 such that us xt ≥ φt /φs

(9.1)

it is possible to construct a linearly homogeneous, or conical, utility that is compatible with the given demands Dt and such that φ(xt ) = φt

(9.2)

The utility constructed will moreover be semi-increasing, x < y → φ(x) < φ(y), and, being both conical and superadditive , φ(x +y) ≥ φ(x)+(y); also it is concave.

On the constructibility of consistent price indices 145 The existence of numbers φt satisfying (9.1) is necessary and sufﬁcient that there should exist any compatible homogeneous utility R at all, without further qualiﬁcation. But here it is seen that if there exists one then also there exists one with these additional classical properties. A conclusion is that these classical properties are unobservable in the observational framework of choice under linear budgets, or are without empirical test or meaning and are just a property of the framework. The consistency condition generally becomes more restrictive as additional restrictions are put on utility. Thus homogeneous consistency is more restrictive that the more general consistency that is free of the homogeneity. Then classical consistency, where utility is required to be representable by a utility function with the classical properties, might seem to be more restricted than general consistency, and also the same might be supposed for when homogeneity is added to both these conditions. But the contrary is a theorem: the imposition of the classical properties make no difference whatsoever. Let φ(x) = min φt ut x

(9.3)

t

so, for all x, φ(x) ≤ φt ut x for all t,

φ(x) = φt ut x

for some t

(9.4)

Then, with x = xt , so ut x = 1, we have φ(xt ) < φt . But from (9.1), φs us xt ≥ φt for all s. Hence, with (9.3), φ(xt ) = mins φs us xt ≥ φt , Thus (9.2) is shown. Now further, from (9.4) with φt > 0, ut x < 1 → φt ut x < φt → φ(x) < φt

(9.5)

Hence, with (9.2), ut x < 1 → φ(x) < φ(xt ), and similarly, or from here by continuity, ut x ≤ 1 → φ(x) ≤ φ(xt ), showing that the utility φ(x) and the normal demand (xt , ut ) are compatible.

10 Utility cost Because ut = Mt−1 pt

(10.1)

where Mt = pt xt

(10.2)

and because Xt = φt (xt ), another statement of (9.1), in view of (9.2), is that. ps xt /ps xs ≥ Xt /Xs

(10.3)

146 Precursor Then introducing Pt = Mt /Xt

(10.4)

so that, as a parallel to (10.2) Mt = Pt Xt , (10.3) and (10.4) give ps xt /ps xs ≥ ( pt xt /Pt )/( ps xs /Ps ) = ( pt xt /ps xs )( ps /Pt ) and consequently ps xt /pt xt ≥ Ps /Pt

(10.5)

Or again, introducing Ut = Mt−1 Pt

(10.6)

in analogy with (10.1), so that Ut Xt = 1, this being, in analogy with the normalized budget identity ut xt = 1, an equivalent of (10.3), and also of (10.5), is that us xt≥ Us /Ut . Let θ( p) be the cost function associated with the classical homogeneous utility function φ(x), so that θ ( p) = min [ px : φ(x) ≥ 1]

(10.7)

this again being classical homogeneous, that is semi-increasing, concave and conical. Then by taking x in the form xt −1 , where t>0, θ ( p) = min pxt −1 : φ(xt −1 ) ≥ 1 = min pxt −1 : φ(x) ≥ t because φ is conical. Then by taking t = φ(x), θ ( p) = minx px(φ(x))−1 is obtained as an alternative formula for θ . From this formula the functions θ and φ are such that θ( p)φ(x) ≤ px

(10.8)

for all p, x with equality just in the case of compatibility between the demand (x, p) and the utility φ. For the equality signiﬁes cost efﬁciency, and because φ is continuous this implies also cost effectiveness, and hence also the compatibility. Because φ is concave it is recovered from θ by the same formula by which θ is derived from it, with an exchange of roles between θ and φ; that is φ(x) = min [ px : θ ( p) ≥ 1] = min(θ ( p))−1 px p

(10.9)

In the case of a normal demand (x, u), that is one for which ux = 1, (10.8) becomes θ(u)φ(x) ≤ 1 with equality just in the case of a demand that is compatible with φ.

(10.10)

On the constructibility of consistent price indices 147 In section 9 it was shown that the function φ(x) constructed there is compatible with the given normal demands (xt , ut ). Therefore θ (ut )φ(xt ) = 1 for all t, while, by (10.10), θ(us )φ(xt ) ≤ 1 for all s, t. Also it was shown that φ(xt ) = φt

(10.11)

Hence, introducing θt = φt−1

(10.12)

it is shown that θ(ut ) = θt . It is possible to verify that also directly by inspection of the cost function. Thus, with φ(x) = mint φt ut x, so that φ(x) ≥ 1 is equivalent to φt ut x ≥ 1 for all t, which, with (10.12), is equivalent to ut x ≥ θt for all t, the cost function in (10.7) is also θ(u) = min ux : ut x ≥ θt

(10.13)

so that θ (ut ) ≥ θt . Therefore, by (10.11) and (10.12), θ (ut )φ(xt ) ≥ 1. Then ut xt =1 with (10.10) shows that θ (ut )φ(xt ) = 1 and hence, again with (10.11) and (10.12), that θ (ut ) = θt . By the linear programming duality theorem (Dantzig 1963) applied to (10.13), another formula for the cost function is θ (u) = max s t θt : st ut ≤ u (10.14) Then, as known from the theory of linear programming, for any x, θ (u) ≤ ux for all u if and only if x solves (10.13). Similarly, with the θt now variable while u is ﬁxed, for any st , θ(u) ≥ st θt for all θt if and only if the st solve (10.14). If the θt are a strict solution of (9.1), that is us xt > φt /φs (s = t), then x = xt θt is the unique solution of (10.13) when u = ut . In just that case θ (u) is differentiable at the point u = ut . In that case θ is locally linear, and has a unique support gradient, and the differential gradient which now exists coincides with it. Thus in this case θ (u) ≤ ux for all u, and θ (ut ) = ut x if and only if x = xt θt , so θ (u) has gradient xt θt at u = ut . It can be added that this entire argument could have gone just as well with an interchange of roles between u and x. By solving us xt ≥ θs /θt for the θt , a cost function θ could be constructed ﬁrst, with the form originally given to φ, and then φ could have been derived. Also, φ need not have been given the polyhedral form (9.5). It could have been given the polytope form (10.13) or ( 10.14). Then θ would have had the polyhedral form (9.5).

11 Price and quantity The method established for the determination of index numbers can be stated in a way that treats price and quantity both simultaneously and in a symmetrical fashion.

148 Precursor With the given demands (xt , pt ), numbers (Xt , Pt ) should satisfy ps xt ≥ Ps Xt

for all s, t

(11.1)

Then, in particular, pt xt = Pt Xt

(11.2)

Then division of (11.1) by (11.2) gives ps xt /pt xt ≥ Ps /Pt

(11.3)

as a condition for the ‘price levels’, and also ps xt /ps xs ≥ Xt /Xs

(11.4)

for ‘quantity levels’. Reversely, starting with a solution Pt of (11.3), let Xt be determined from (11.2). Then Xt is a solution of (11.4) and the Pt and Xt together make a solution of (11.1). Just as well, the procedure could start with a solution Xt of (11.4) and go on similarly. It has been established that the existence of solutions to these inequalities is necessary and sufﬁcient for homogeneous consistency of the demand data. The investigation now concerns the identiﬁcation of the numbers Prs = Ps /Pr obtained from solutions with all possible price indices that are compatible with the data, that is, derivable on the basis of compatible homogeneous utilities. Then it will be possible to go further with a description of all possible price indices in terms of closed intervals speciﬁed by formulae for their end-points, or limits. For any utility order R, the derived utility–cost function ρ (p, x) = min[ py : yRx] is deﬁned for all p, x if the sets Rx are closed. If R is a complete order and the sets Rx, xR are closed then, for any p, ρ( p, x) is a utility function representing R (see Afriat, 1979). It follows that, for any p and q, there exists an increasing function w(t), independent of x and carrying p, q as parameters, such that ρ( p, x) = w(ρ(q, x)) for all x. If R is conical then so is ρ( p, x) as a function of x, for any p. In that case so is w(t) as a function of t. But a conical function of one variable must be homogeneous linear, so w(t) has the form wt where w is a function of p, q independent of x. That is, ρ( p, x)/ρ(q, x) = w is independent of x. Then w must have the form w = θ( p)/θ (q) where θ ( p) is a function of p alone, so it follows that ρ( p, x)/(θ( p) must be a function of x alone, and so ρ( p, x) = θ ( p)φ(x), where θ, φ are functions p, x alone. Now φ must be a utility that represents R, and be conical because p is conical in x. Then the condition for φ to be quasi-concave is that the sets Rx be convex. But because φ is conical this is also the condition that φ be concave (Berge, 1963). Moreover, because in any case ρ( p, x) is concave conical in p, so also is θ( p). The reﬂexivity of R gives in any case ρ(p, x) ≤ px, so now θ(p)φ(x) ≤ px for all x. Hence, for all p, θ(p) ≤ minx px(φ(x))−1 , while ρ(p, x) = px for some x

On the constructibility of consistent price indices 149 gives θ (p)φ(x) = px some x, so that now θ( p) = min px(φ(x))−1 . Similarly φ(x) ≤ minp (θ ( p))−1 px. Then let φ(x) = minp (θ ( p))−1 px, so φ(x) ≤ φ(x) for all x and φ is concave conical. Then, for any x, φ(x) = φ(x) is equivalent to θ ( p)φ(x) = px for some p. The condition for this to be so for any x is that φ(x) be quasi-concave, having a quasi-support p at x, for which py < px → φ(y) < φ(x),

py ≤ px → φ(y) ≤ φ(x)

in other words the demand (x, p) is compatible with x for some p. This condition, which means that, with choice governed by φ, x could be demanded at some prices, can deﬁne compatibility between φ and x. The condition that φ be compatible with all x is just that it be quasi-concave, and that now is equivalent to φ(x) = φ(x) for all x. In any case, any x compatible with φ is also compatible with φ. Hence, if utility R is constrained by compatibility with given demands, if φ is acceptable then so is φ, and moreover φ and φ have the same conjugate price function θ . This suggests that, instead of constructing a utility φ compatible with the data having a concave form that is a generally unwarranted restriction on utility and, for all we know now, might make some added restriction on price-index values, it is both possible and advantageous to construct the price function θ ﬁrst instead, and so be free of such a suspicion. It has already been remarked that this might have been done instead, following an identical procedure as that for the φ, so in fact that issue is already disposed of. Form any compatible homogeneous utility a price function θ is derived, giving the Pr = θ (pr ) as a system of ‘price-levels’ compatible with the data, and determining the Prs = Ps /Pr as compatible prices. But the possible such Pr are already identiﬁed with the possible solutions of (11.3). Also, for any solution Pr and the θ that must exist, the Xr determined from (11.2) have the identiﬁcation Xr = φ(Xr) where φ(x) = minp (θ ( p))−1 px. This φ is concave conical. But also any other φ ∗ , not necessary concave but having the same conjugate θ , would do, so there is no inherent restriction to concave utilities here. For such a φ ∗ , generally φ ∗ (x) ≤ φ(x), while φ ∗ (xr ) = φ(xr ) for all r, and all that is required of φ ∗ is that θ(x) = minp px((φ ∗ (xr ))−1 , and there are many φ ∗ for which this is so, that given being just one. The argument in this section permits by-passing complications of the argument involving ‘critical cost functions’ that was used formerly, such as in the exposition of Afriat (1977b) for the special case of just two demand-periods. An interesting point is that the care taken in both arguments to avoid imposing on a compatible utility the requirement that also it be concave makes no difference at all to the range of possible values for a price-index.

12 Extension and exhaustion properties For any coefﬁcients ars (r, s = 1, . . . , k), consider the system S(a) of simultaneous linear inequalities ars ≥ xs − xr

(12.1)

150 Precursor to be solved for numbers xr . This is an alternative form for the system (7.1), and the form that applies directly to the system (7.3). Introduce chain-coefﬁcients arij···ks = ari + aij + · · · + aks

(12.2)

so that ar···s···t = ar···s + as···t

(12.3)

If the system S(a) has a solution x then ari ≥ xi − xr , xij ≥ xj − xi , . . . , aks ≥ xs − xk so by addition, ar···s ≥ xs − xr

(12.4)

In particular ar . . . r ≥ xr − xr = 0, so that ar . . . r ≥ 0, that is, ari + aij + · · · + akr ≥ 0

(12.5)

for every cyclic sequence of elements r, i, j, . . . , k, r. This can be called the cyclical non-negativity condition C on the system S(a), and it has been seen necessary for the existence of a solution. Because ar···s···s···r = ar···s···r + as···s the coefﬁcient on a cycle with a repeated element s can be expressed as a sum of terms that are coefﬁcients on cycles where the repetition multiplicity is reduced, and this decomposition can be performed on those terms and so forth until an expression is obtained with only simple cycles, without repeated elements. From this it follows that the condition C is equivalent to the same condition on cycles that are restricted to be simple, or have elements all distinct. Under the condition C, ar···s···s···t = ar···s···t + as···s ≥ ar···s···t so the cancellation of a loop in a chain does not increase the coefﬁcient along it. It follows that the derived coefﬁcient Ars = min arij···s ij···

(12.6)

exists for any r, s and moreover Ars = arij···s

(12.7)

for some simple chain rij · · · s from r to s. Thus C is sufﬁcient for the existence of the derived coefﬁcients. Also it is necessary. For if as · · ·s < 0 then for any r, t by

On the constructibility of consistent price indices 151 taking the chain that goes from r to t, following the loop s · · · s any number K of times and then going from s to t, we have Art ≤ ars + Kas···s + ast → −∞(K → ∞)

(12.8)

so Art cannot exists. Therefore also C is sufﬁcient for the existence of the derived coefﬁcients. Evidently then either the derived coefﬁcients all exist or none do. Given that they exist, from (12.3) it follows that Ars + Ast ≥ Art , so they satisfy the triangle inequality. The system S(a) and the derived system S(A), of inequalities Ars ≥ xs − xr

(12.9)

have the same solutions. For from (12.4), any solution of (12.1) is a solution of (12.9). Also from Ars ≤ ars , that follows from the deﬁnition (12.6) of the Ars , it is seen that any solution of (12.9) is a solution of (12.1). The triangle inequality is necessary and sufﬁcient for a system to be identical with its derived system, that is for Ars = ars for all r, s. It is necessary because any derived system has that property. Also it is sufﬁcient. For the triangle inequality on S(a) is equivalent to arij . . . ks ≥ ars , but (12.9) implies both that the derived coefﬁcients Ars exist and that Ars ≥ ars which because of (12.8) is equivalent to (12.9). Now the extension property for the solutions of a system that satisﬁes the triangle inequality will be proved. Let S(A) be any such system, so if this is the derived system of some other system then this hypothesis must be valid. A subsystem of S(A) is obtained when the indices are restricted to any subset of 1,… , n. Without loss in generality consider the subsystem Sm−1 (A) on the subset of 1, . . . , m − 1. Let xr (r < m) be any solution for this subsystem, so that Ars ≥ xs − xr

for r, s < m

(12.10)

Now consider any larger system obtained by adjoining a further element to the set of indices. Without loss in generality, let m be that element and Sm (A) the system obtained. It will be shown that there exists xm so that the xr (r ≤ m) that extend the solution xr (r < m) of Sm−1 (A) are a solution of Sm (A), that is Ars ≥ xs − xr

for r, s < m

(12.11)

With the xr (r < m) satisfying (12.10), xm has to satisfy Arm ≥ xm − xr for r < m,

Ams ≥ xs − xm for s < m

(12.12)

Equivalent, xs − Ams ≤ xm ≤ xr + Arm for r, s < m. But a necessary and sufﬁcient condition for the existence of such xm is that xs − Ams ≤ xr + Arm for r, s < m, equivalently Arm + Ams ≥ xs − xr for r, s < m. By the triangle inequality, (12.11)

152 Precursor implies this, so the existence of such xm is now proved. Thus any solution of Sm−1 (A) can be extended to a solution of Sm (A). Then by an inductive argument it follows that, for any m ≤ n, any solution xr (r ≤ m) of Sm (A) can be extended to a solution of Sn (A) = S(A), by adjunction of further elements xr (r > m). It can be concluded that any system with the triangle inequality has a solution, because the triangle inequality requires in particular that A11 + A11 ≥ A11 , or equivalently A11 ≥ 0. This assures that S1 (A) has a solution x1 and then any such solution x1 can be extended to a solution xr (r ≤ n) of S(A). From the foregoing, each of the conditions in the following sequence implies its successor: (i) The existence of a solution. (ii) The cyclical non-negativity test, (iii) The existence of the derived system. (iv) The existence of a solution for the derived system. (v) The existence of a solution. It was shown ﬁrst that (i) → (ii) → (iii). Then because any derived system satisﬁes the triangle inequality and any system with that property has a solution, (iii) ↔ (iv) is shown. Now the identity between the solutions of a system and its derived system shows (iv) ↔ (i) and establishes equivalence between all the conditions, in particular between (i) and (ii). The derived system S(A) can be stated in the form −Asr ≤ xs − xr ≤ Ars

(r ≤ s)

requiring the differences xs − xr (r ≤ s) to belong to the intervals (−Asr , Ars ). The extension property of solutions assures also the interval exhaustion property, that every point in these intervals is taken by some solution. Whenever the derived system exists these intervals automatically are all non-empty. An order U of the indices determined from the coefﬁcients Ars is given by the → transitive closure U = A of the relation A given by rAs = Ars ≤ 0. Also, any solution x determines an order V (x) of the indices, where rV (x)s ≡ xs ≤ xr . Whatever the solution, this is always a reﬁnements of the order U, that is V (x) ⊂ U for every solution x. Moreover, for any order V that is a reﬁnement of U, there always exists a solution x such that V (x) = V . This order exhaustion property can be seen from the interval exhaustion property and also by means of the proof of the extension property of solutions by taking the extensions in the required order. These results can all be translated to apply to a system in the form ars ≥ xs /xr , now with multiplicative chain coefﬁcients, arij···ks = ari aij · · · aks and derived coefﬁcients Ars deﬁned from these as before and satisfying multiplicative triangle inequality Ars Ast ≥ Art . The cyclical non-negativity test becomes ars ast · · · aqr ≥ 1. For any solution x, the ratios xs /xr are required to lie in the intervals Irs = (1/Asr , Ars ). From their form these intervals remind of the Paasche–Laspeyres (PL) interval (1/Lsr , Lrs ). Also, the multiplicative chain coefﬁcients correspond to the familiar procedure of multiplying chains of price indices, except that there are many chains with given extremities and here one is taken on which the coefﬁcient is minimum. While the non-emptiness of the PL-intervals, whether or not the one index exceeds the other, is a well-known issue, there is no such issue at all with the intervals Irs because whenever they are deﬁned they are non-empty, this following

On the constructibility of consistent price indices 153 from the multiplicative triangle inequality that gives Ars Asr ≥ Arr ≥ 1.

13 The power algorithm For a system with coefﬁcients ars , and any k ≤ m ≤ n, let a[k] rs = min (ari2 + ai2 i3 + · · · + aik s ), i2 ik

[k] a(m) rs = min ars k≤m

(13.1)

According to (12.6), if the derived coefﬁcients exist any one has the form Ars = ari + aij + · · · + aks

(13.2)

for some i, j, . . . , k making r, i, j, . . . , s all distinct except possibly for the coincidence of r and s. Because there are just n possible values 1, . . . , n for the indices, it follows from (13.1) and (13.2) that Ars ≥ a(n) rs . But from the deﬁnition (n) of the Ars in (12.6) and from (13.1) again, also Ars ≤ a(n) rs , so now Ars = ars , that is (n) A = a . Now writing+as · and min as+, (13.1) becomes ari1 · ai1 i2 · · · aik−1 s a[k] rs = i,... im−1

= (a · a · · · ·a)rs

(k factors)

= (ak )rs that is a[k] = ak , where the ‘power’ ak so deﬁned is unambiguous because of associativity of ‘multiplication’ and ‘addition’ and the distributivity of ‘multiplication’ over ‘addition’ and is determined recurrently from a1 = a and ak = a · ak−1 Then (13.2) becomes a(m) =

(13.3) m

1a

k

, and is determined from

a(1) = a and a(m) = aa(m−1) + a

(13.4)

This algorithm with powers in the context of minimum paths in networks is from Bainbridge (1978). Observed now is a simpliﬁcation that is applicable to the special case of importance here where arr = 0, or where arr = 1 in the multiplicative formulation. If arr = 0, in which case chains of any length include all those of lesser length, we have a ≥ a2 ≥ · · · ≥ ak ≥ · · · ≥ am where the matrix relation ≥ means that relation simultaneously for all elements. Therefore in this case a(m) = am . This with A = a(n) together with (13.3), shows

154 Precursor that the matrix A of derived coefﬁcients can be calculated by raising the matrix a of the original coefﬁcients to successive powers, the nth power being A. Should am = am−1 for any m ≤ n then also am = am+1 = · · · = an = · · · so that A = am . But in any case the formula A = an is valid. Then evidently A = A2 = · · · , so the derived matrix is idempotent. This property is characteristic of any matrix having the triangle inequality. By taking exponentials, these procedures can be translated to apply to the system in the multiplicative form (7.1), Lrs ≥ φs /φr with Lrr = 1. Matrix powers have been deﬁned in a sense where + means min and · means +. Taking exponentials turns + into · and leaves min as min, so we are back with · meaning ·. This makes Lˆ = Ln a formula for the derived coefﬁcient matrix, where powers now are deﬁned in the ordinary sense except that + now means min. As before, n here can be replaced by any m ≤ n for which Lm = Lm+1 , in particular by the ﬁrst such m found.

References

Afriat, S.N. 1956. ‘Theory of Economic Index Numbers’. Mimeographed. Department of Applied Economics, Cambridge. — 1960. ‘The system of inequalities ars >Xs –Xr .Research Memorandum No. 18, Econometric Research Program, Princeton University, Princeton, NJ. Proc. Cambridge Phil. Soc., 59 (1963), 125–33. — 1964. ‘The construction of utility functions from expenditure data’. Cowles Foundation Discussion Paper No. 144, Yale University. Paper presented at the First World Congress of the Econometric Society, Rome, September 1965. International Economic Review, 8 (1967), 67–77. — 1967a. ‘The Cost of Living Index’. In Studies in Mathematical Economics: In Honor of Oskar Morgenstern. Edited by M. Shubik. Princeton, NJ: Princeton University Press. — 1970a. ‘The Concept of a Price Index and Its Extension’. Paper presented at the Second World Congress of the Econometric Society, Cambridge, August 1970. — 1970b. ‘The Cost and Utility of Consumption’. Mimeographed. Department of Economics, University of North Carolina at Chapel Hill, North Carolina. — 1970c. ‘Direct and Indirect Utility’. Mimeographed. Department of Economics, University of North Carolina at Chapel Hill, North Carolina. — 1972a. ‘Revealed Preference Revealed’. Waterloo Economic Series No. 60, University of Waterloo, Ontario. — 1972b. ‘The Theory of International Comparisons of Real Income and Prices’. In International Comparisons of Prices and Output, Proceedings of the Conference at York University, Toronto, 1970. Edited by D. J. Daly, 13–84. Studies in Income and Wealth, vol. 37. New York: National Bureau of Economic Research. — 1973. ‘On a System of Inequalities in Demand Analysis: an Extension of the Classical Method’. International Economic Review, 14, 460–72. — 1974. ‘Sum-symmetric matrices’. Linear Algebra and its Applications, 8, 129–140. — 1976. Combinatorial Theory of Demand. London: Input-Output Publishing Co. — 1977a. ‘Minimum paths and subpotentials in a valuated network’. Research Paper 7704, Department of Economics, University of Ottawa. — 1977b. The Price Index. Cambridge and New York: Cambridge University Press. — 1978a. ‘Index Numbers in Theory and Practice by R. G. D. Allen’. Canadian Journal of Economics, 11, 367–9. — 1978b. ‘Theory of the Price Index by Wolfgang Eichhorn and Joachim Voeller’. Journal of Economic Literature, 16, 129–30. — 1979. Demand Functions and the Slutsky Matrix. Princeton, NJ: Princeton University Press.

156 Precursor — 1980. ‘Matrix Powers: Classical and Variations’. Paper presented at the Matrix Theory Conference, Auburn University, Auburn, Alabama, 19–22 March 1980. Allen, R. G. D. 1975. Index Numbers in Theory and Practice. London: Macmillan. Bainbridge, S. 1978. ‘Power algorithm for minimum paths’, private communication. Department of Mathematics, University of Ottawa, Ontario. Berge, C. 1963. Topological Space. New York: Macmillan. Berge, C. and Ghouila-Houri, A. 1965. Programming, Games and Transportation Networks. London: Methuen & Co. New York: John Wiley and Sons. Dantzig, G. 1963. ‘Linear Programming and Its Extensions’. Princeton, NJ: Princeton University Press. Edmunds, J. 1973. ‘Linear programming formula for minimum paths’, private communication. Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario. Eichhorn, W. and Voeller, J. 1976. Theory of the Price Index: Fisher’s Test Approach and Generalizations. Berlin, Heidelburg and New York: Springer Verlag. Fiedler, M. and Ptak, V. 1967. ‘Diagonally dominant matrices’. Czech. Math J ., 17, 420–33. Fisher, I. 1927. The Making of Index Numbers. Third edition. Boston and New York: Houghton Mifﬂin Company. Geary, R. C. 1958. ‘A note on the comparison of exchange rates and purchasing power between countries’. J . Roy Stat. Soc. A, 121, 97–9. Hicks, J. R. 1956. A Revision of Demand Theory. Oxford: Clarendon Press. Houthakker, H. S. 1950. ‘Revealed Preference and the Utility Function’, Economica, New Series, 17, 159–74. Samuelson, P. A. 1947. The Foundations of Economic Analysis. Cambridge, Mass.: Harvard University Press. Samuelson, P. A. and Swamy, S. 1974. ‘Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis’. American Economic Review, 64, 566–93. Shephard, R. W. 1953. Cost and Production Functions. Princeton, NJ: Princeton University Press. Theil, H. 1960. ‘Best linear index numbers of prices and quantities’. Econometrica, 28, 464–80. 1975–1976. Theory and Measurement of Consumer Demand. Two volumes. Amsterdam: North-Holland Publishing Company. — 1979. The System-wide Approach to Microeconomics. The University of Chicago Press. — Yen, J. Y. 1975. Shortest Path Network Problems. Meisenheim am Glan: Verlag Anton Hain.

3 The Theory of Exact and Superlative Index Numbers Revisited Carlo Milana Istituto di Studi e Analisi Economica, Roma EUKLEMS Working Paper No. 3, 2005 http://www.euklems.net

THE THEORY OF EXACT AND SUPERLATIVE INDEX NUMBERS REVISITED1 By Carlo Milana

1 Introduction This chapter contains a critical review of the theory of “exact” and “superlative” index numbers that is still dominating the ﬁeld of economic index numbers. An index number is said to be “exact” for a function if it is identically equal to the ratio of numerical values of that function at any pair of points taken into comparison. The ﬁrst use of the concept of “exact” index numbers appeared in the discovery of Byushgens (1925) that Irving Fisher’s ‘Ideal Index’ is an exact formula when demand is governed by a homogeneous quadratic utility (see also Konüs and Byushgens, 1926). This concept became more widely known with Schultz (1939) and its discussion in the Foundations of Paul A. Samuelson (1947, p. 155). In the introduction to the enlarged edition, Samuelson (1983, p. xx) later wrote: “Thirty-ﬁve years after that [revealed preference] analysis appeared there has been but one major advance in index number theory—namely W. E. Diewert’s formalizing concept of a ‘superlative index number’, which is a formula based upon two periods (pj , qj ) data that will be exactly correct as an ordinal indicator of utility for some speciﬁed family of indifference contours.(Only a few different ‘superlative’ formulas are known; perhaps the set of simple superlative formulas is a limited set.)”2 Diewert (1976, p. 117) used Irving Fisher’s (1922, p. 247) terminology of “superlative” index numbers to deﬁne index numbers that are exact for functions that provide a second-order differential approximation to an unknown true function. In another paper, Diewert (1978) pointed out that these index numbers approximate each other up to the second-order at the point where the compared

1 Revised version of a review prepared at ISAE, in Rome, for the Speciﬁc Targeted Research Project “EU KLEMS. Productivity in the European Union: A Comparative Industry Approach” supported by the European Commission with Contract No. 502049 (SCS8) within the Sixth Framework Programme (see http://www.euklems.net).Comments received from Pirkko Aulin-Ahmavaara, Alberto Heimler, and Dale W. Jorgenson are thankfully acknowledged. 2 Diewert’s superlative index numbers are, however, only the subset of known superlative index numbers that are exact for ﬂexible functions expressed in closed algebraic forms. Other superlative index numbers, as for example that deﬁned by Sato (1976) and Vartia (1976), do not appear to be exact for any function in closed form (see Barnett and Choi, 2006).

160 Precursor base and current points of observations are equal and are numerically very close if these two points do not vary very much3 . Quoting Diewert (2004, p. 450) words, “Diewert (1978, p. 888) showed that the three superlative index number formulas listed approximate each other to the second order around any point where the two price vectors [in the calculation of the price index], p0 and p1 ,are equal and where the two quantity vectors [in the calculation of the quantity index], q0 and q1 , are equal. He concluded that ’all superlative indices closely approximate each other’(Diewert, 1978, p. 884)”. This deﬁnition can be contrasted with that given by Irving Fisher (1922, pp. 244–48), who had singled out eleven index number formulas out of 134 examined formulas and called them “superlative” because, in his numerical example, they performed very closely to the “ideal” geometric mean of Laspeyres and Paasche indexes4 . He claimed that all these superlative formulas correspond to combinations of the Laspeyres and Paasche including the direct and implicit Walsh index numbers, one combination of these last two formulas, and a couple of combinations of direct and implicit Törnqvist-type index numbers. Apart from the Fisher “ideal” and the implicit Walsh indexes, Diewert’s superlative index numbers differ widely in nature and, potentially, in numerical values from those deﬁned superlative by Fisher (1922)5 . Allen and Diewert (1981, p. 430) had recognized that “in many applications involving the use of cross section data or decennial census data, there can be a tremendous amount of variation in prices or in quantities between the two periods so that alternative superlative index number can generate quite different results”. In fact, more recently, Hill (2006) has found a large spread in numerical values of alternative Diewert’s superlative index numbers, with the largest and the smallest ones differing by more than 100 per cent using a standard US national data set and by about 300 per cent in a cross-section comparison of countries using an OECD data set. However, it has been surprising to ﬁnd empirically that the spread between the largest and the smallest Diewert’s superlative index numbers may exceed

3 See also Vartia (1978) and Diewert (2004, p. 450). However, Diewert (1978, p. 890, fn. 8), citing the discussion in Lau (1974, p. 183), recognized that, without performing extensive computations involving the third order partial derivatives of the index number formulae, we cannot specify exactly how small should be the change between the two compared points in order for the superlative indexes to be all very close to each other. 4 Irving Fisher (1922, pp. 244–248) ranked the examined 134 index numbers according to their distance from the “ideal” geometric mean of the Laspeyres and Paasche indices and separated them “arbitrarily into several classes in increasing order of merit. The ﬁrst twelve index numbers, constituting the ﬁrst of these classes, are labeled, rather harshly perhaps, as ‘worthless’ index numbers (to designate the fact that they are the worst). The other six classes are labeled as poor, fair, good, very good, excellent, and superlative” (p. 244). 5 The Törnqvist index, for example, (corresponding to formula 123 in Fisher’s, 1922, p. 473) is seen as the most superlative by Caves, Christensen, and Diewert (1982, p. 41) and Diewert (2004, p. 450) whereas it was not deemed “superlative” by Fisher (1922, p. 247), who classiﬁed it, in a descending order of merit, below the classes of “excellent” and “superlative” index numbers, with the last group ranked at the top position.

The Theory of Exact and Superlative Index Numbers Revisited 161 that between the Laspeyres and Paasche indexes, which are usually considered to be the bounds of the interval of possible values of economic index numbers, at least in the homothetic case. This performance is clearly in contrast with that considered originally by Irving Fisher in identifying his own superlative index numbers. The present paper starts by reformulating, in a general framework, the Quadratic Identity and other propositions on which Diewert’s superlative index numbers are based. A uniﬁed approach is developed, where the underlying unknown true functions may differ not only in parameters, but also in functional forms. It is also shown that, in practical applications, any index number that is constructed using the observed data, in general, cannot be really superlative in Diewert’s sense. The weights used are, in fact, derivable from the unknown true function and not (at least at one point of observation) from the approximating quadratic function for which the index number is meant to be exact. Consequently, using the observed data, Diewert’s superlative index numbers become hybrid formulas that may be quite far from providing the expected second-order approximation to the unknown true values. The rest of the text is organized as follows. Section 2 re-examines the meaning of the Quadratic Identity within a general framework of accounting for functional value changes of an arbitrary differentiable function. Section 3 extends this analytical approach and generalizes the results obtained in the literature up till now. Section 4 extends the analysis further by using transformed functions. Section 5 presents a uniﬁed approach to index numbers and indicators of absolute change by using a general transformed quadratic function with no a priori separability and homogeneity restrictions. Section 6 deals with the approximation properties of index numbers. Section 7 concludes.

2 A general formulation of the quadratic approximation lemma In the general case of an arbitrary differentiable function, the following result is obtained: Lemma 2.1 Accounting for Functional Value Changes. Let z be a vector of N real valued variables and let us assume that an arbitrary function f (z) is continuously differentiable at least once. Then, for all z 0 and z 1 , f z 1 − f z 0 = (1 − θ ) ∇f z 0 + θ ∇f z 1 · z 1 − z 0

(2.1)

where ∇f z t ≡ ∂f /∂z1t ∂f /∂z1t . . . ∂f /∂zNt is the gradient vector of f evaluated at z t and, denoting with R01 (z 0 , z 1 ) and R11 (z 0 , z 1 ) the remainder terms associated respectively with the polynomials of order one in the Taylor series expansion for f around z 0 and z 1 , θ is a scalar that takes the particular value R0 ( z 0 , z 1 ) θ ∗ z 0 , z 1 ≡ R0 z0 , z11 +R1 z0 , z1 when R01 z 0 , z 1 + R11 z 0 , z 1 = 0, or θ may take ( ) ( ) 1 1

162 Precursor

any real number as a value if f is linear in z (in which case R01 z 0 , z 1 = R11 0 1 z , z = 0)6 . Proofs of propositions are given in Appendix B. It is straightforward to show that 0 ≤ θ ≤ 1 if f (z) is quasiconcave or quasiconvex. Lemma (2.1) can be complemented with the following corollaries. Corollary 2.1 Decomposition of Functional Value Changes. The change in the value of an arbitrary differentiable function f(z) can be further decomposed as follows, for all z 0 and z 1 , (2.2) f z 1 − f z 0 = JZ + JS 0 0 1 f z IZ · f z (1 − θ ) si∗0 0 + θ si∗1 where JZ ≡ zi − zi0 1 zi zi 0 0 ∗0 f z 1 ∗1 IZ · f z 0 (1 − θ ) ξ − 1 si JS ≡ + θ ξ − 1 si zi0 zi1 · zi1 − zi0 [∂f (zt )/∂zi ] · zit with θ being deﬁned by Lemma (2.1), si∗t ≡ sit / Nj=1 sjt = N t, t i=1 [∂f (z )/∂zi ] · zi t ∂f (z ) for t = 0, 1 where sit ≡ ∂zt · zit /f z t (the superscript ∗ denoting here ∂f (z0 ) ∂f (z1 ) normalized shares), ξ 0 ≡ Ni=1 ∂z0 · zi0 /f z 0 , and ξ 1 ≡ Ni=1 ∂z1 · zi1 / i i IZ · f z 0 , representing the scale effects, and I Z is the index of functional value change net of scale effects, so that f z1 0 = I Z · IS f z

with

IZ ≡

θ + (1 − θ) (1 − θ) + θ

(2.3) N

1

∗0 zi i=1 si z 0 i 0 N ∗1 zi i=1 i z 1 i

s

and

f z1 I S ≡ 0 IZ . f z

It may be tempting to interpret the terms JZ and JS in (2.2) as contributions to the absolute functional value change arising, respectively, from changes in z“directly” and through the returns to scale. Similarly, the terms IZ and IS in (2.3) might be interpreted, respectively, as the “aggregating” index numbers of relative changes in z and the scale effects. The index IZ is in fact homogeneous of degree one in z.

6 The usual vector or matrix transposition sign is ommitted to simplify notation.

The Theory of Exact and Superlative Index Numbers Revisited 163 It is equal to a Laspeyres-type index number if θ = 0, whereas it is equal to a Paasche-type index number if θ = 1. However, the following remark deﬁnes the limits of such interpretation: Remark 2.1 on the existence of the aggregator index: If and only if the function f is homothetic, so that f (z) = F[φ(z)], where F (•) is a well behaved transformation function (real valued, continuously differentiable, monotonically increasing and quasiconcave) and φ(z) is also well behaved and linearly homogeneous7 , the weights si∗t can be expressed only in terms of the function φ. In fact,

si∗t

∂f z t /∂zi · zit d [F] /dφ · ∂φ z t /∂zit = N t t = d [F] /dφ · Nj=1 ∂φ z t /∂zj · zjt j=1 ∂f z /∂zj · zj ∂φ z t /∂zit · zit = φ zt

(2.4)

since φ (z) = Nj=1 ∂φ (z) /∂zj · zj for Euler’s theorem on linearly homogeneous functions. In the homothetic case, the indexes JZ and IZ are exact for the aggregator or index function8 φ. In this case, these indexes are isolated from the scale effects and are clearly deﬁned as aggregating indexes. This is a well-known result due to the homogeneity price and quantity theorems of Shephard (1953) in the theory of production and Samuelson (1953) in the theory of direct and indirect utility (see also the survey and synthesis of Samuelson and Swamy, 1974, pp. 568–569). It is straightforward to show that, if the function φ is well behaved, and in particular quasiconvex or quasiconcave, then the Laspeyres- and Paasche-type index numbers are the bounds of the interval of all possible numerical values that the ratio φ(z 1 )/φ(z 0 ) may take. By contrast, when the function f is not homothetic, the z-change component is constructed using weights that are also affected by the scale of the function in a non-homothetic way9 . In the more general non-homothetic case, no true indicator and index number of changes in z really exist and any attempt to construct them using JZ and IZ deﬁned respectively in (2.2) and (2.3) ends up to spurious magnitudes.

7 The concept of homotheticity was explicitly spelled out by Shephard (1953) and Malmquist (1953). Earlier researchers as Frisch (1936, p. 25) and Samuelson (1950, p. 24) had dealt with it implicitly. 8 The terms “index function” and “aggregator function” are used here interchangeably. The former was used by Shephard (1953, pp. 47–49) and Solow (1956, pp. 102–106), whereas the latter was used by Diewert (1976). 9 This remark is equivalent to the well-known conclusion on path-independency of Di-visia indexes in the case of homothetic functions (see Hulten, 1973 and Samuelson and Swamy, 1974).

164 Precursor Let us deﬁne the quadratic function: 1 1 fQ (z) ≡ a0 + a z + zA z = a0 + a zz, ai zi + 2 i=1 j=1 ij i j 2 i=1 N

N

N

(2.5)

where a0 , a, and A are, respectively, a scalar, a vector, and a matrix of constant parameters and it is assumed that A is symmetric, that is aij = aji for all i, j, to ensure the symmetry of the Hessian matrix of second derivatives by Young’s theorem. The Bernstein-Weierstrauss approximation theorem states that, on a closed bounded domain, a continuous function can be approximated uniformly by polynomials. The function fQ (z) deﬁned by (2.5) can be seen as providing a second-order approximation to an arbitrary function f (z) around the point z ∗ if its parameters are “calibrated” to particular numerical values such that fQ (z ∗ ) = f (z ∗ ), ∇fQ (z ∗ ) = ∇f (z ∗ ), and ∇ 2 fQ (z ∗ ) = ∇ 2 f (z ∗ )10 . The “problem of accuracy of approximation” was clearly seen and described by Fuss, McFadden, and Mundlak (1978, pp. 233–234) in the following terms: “If a ﬂexible form is calibrated to provide a second-order approximation at a point, then the approximation is of this order only in a small neighborhood of this point. In other regions of interest, the form may be a poor approximation to the true function. […] Further, the qualitative implications of the calibrated approximation may depend on the point of approximation; this is true, for example, of separability, which involves properties of the true function beyond second-order”. This is the general case presented in the standard microeconomics textbooks, where production functions and utility functions are in fact presented in non-linearly homogeneous forms involving non-zero third derivatives due to non-homothetic effects and externalities11 . The following well-known result can be derived as another corollary of Lemma 2.1: Corollary 2.2 Diewert’s (1976, p. 118) Quadratic Identity. If and only if f (z) has the functional form of the quadratic function f Q (z) deﬁned by (2.5), then the weight θ in (2.1) is equal to 1/2 for all z 0 and z 1 , so that 1 fQ z 1 − fQ z 0 = ∇fQ z 0 + ∇fQ z 1 · z 1 − z 0 2

(2.6)

The Quadratic Identity established by Corollary (2.2) can be explained geometrically as in Figure 1, where the quadratic function fQ (z) is represented

10 Two alternative concepts of second-order approximation based, respectively, on the Taylor expansion series and on the equality of ﬁrst- and second-order partial derivatives of the two functions valued at a base point have been discussed by Lau (1974, 183). Barnett (1983) has shown that they are equivalent. 11 A good example of a non-homothetic utility function implying non-zero third derivatives for the integrand of a Divisia. price index is given by Feenstra and Reinsdorf (2000).

The Theory of Exact and Superlative Index Numbers Revisited 165 f 0L(z)

B’

f 1L(z) fQ(z)

B A’ A

z O

z0

z1

Figure 1 Geometrical representation of Diewert’s (1976) Quadratic Identity with a concave function

in one single variable and the functions fL0 (z) and fL1 (z) are, respectively, the ﬁrst-order polynomials of the Taylor’s expansions of fQ (z) around z 0 and z 1 . With the quadratic function, the remainder terms AA and BB of two ﬁrst-order approximations are equal in absolute size with all z 0 and z 1 and, therefore, their difference is always equal to zero. In fact, fQ (z 1 ) − fQ (z 0 ) = AB is equal to 12 fL0 z 1 − fL0 z 0 + fL1 z 1 − fL1 z 0 = 12 fL0 z 0 + fL1 z 1 · z 1 − z 0 = 1 AB + A B = AB + 12 AA − BB , where AA − BB = 0 with all z 0 and z 1 . 2 Some remarks concerning this result are in order: (i)If ∇z fQ z 0 = ∇z fQ z 1 , then fQ is non-linear in z (however, the converse is not true since a non-linear function may have equal ﬁrst derivatives at some different points). (ii)Diewert (1976, p. 138) gave his necessity proofs under the assumption that the examined function fQ is thrice continuously differentiable. The existence of non-zero third derivatives implies, in particular, that fQ is non linear, at least locally. He showed that, since equation (2.6) implies that all the third derivatives vanish for all z 0 = z 1 , the polynomial form of fQ is quadratic (see the necessity proof in Appendix B). (iii)Lau (1979), noted that “generalizing Diewert’s proof to the twice continuously differentiable case is straightforward” (p.74, fn. 1). He also gave an alternative proof of the Quadratic Identity by assuming a once differentiable function and observed that this “widens considerably the applicability of the lemma and consequently of the results which depend on its validity”(p.74). (iv)Under the general differentiability assumption, the quadratic function (2.5) can in fact be regarded as having the most general functional form with which identity (2.6) exactly holds. The relation (2.6) still holds in special (limit) cases

166 Precursor

where ∇z fQ z 0 = ∇z fQ z 1 including the linear function fL (z) ≡ a0 + az, corresponding to fQ with A → 0N ×N , and the constant-value fC (z) ≡ a0 , corresponding to an fQ with a → 0N and A →0N ×N (however, by Lemma 2.1, in these two last cases the weight θ may also take other real numbers as a value other than 1/2). This result weakens somewhat the meaning of name of “Quadratic Identity” that Caves, Christensen, and Diewert (1982) have given to Diewert’s (1976, p. 118) lemma. (v)The necessity part of Corollary (2.2) means that, when a function is indeed non-linear (as implied by the thrice differentiability condition), then the relation (2.6) should represent a “quadratic identity” since, in this case, it is compatible only with all third derivatives equal to zero, thus implying a quadratic functional form of fQ . (vi)Diewert (1976, p. 118) called his Quadratic Identity under the name of “Quadratic approximation lemma” on the implicit assumption that all parameters of the quadratic polynomial function (2.5) are “calibrated” consistently with the second-order approximation to the function f (z) around z 0 . This means that the two functions f (z) and fQ (z) should have the same numerical value and the same ﬁrst and second derivatives at z 0 .The quadratic function fQ (z), however, does not provide a second-order approximation to an arbitrary function without imposing additional conditions requiring that its parameters be “calibrated” to speciﬁc numerical values such that fQ (z ∗ ) = f (z ∗ ), ∇fQ (z ∗ ) = ∇f (z ∗ ), and ∇ 2 fQ (z ∗ ) = ∇ 2 f (z ∗ ) ,with z usually being equal to z 0 .The number of equations deﬁned by these conditions is equal to 1 + N + N (N +1)/212 . By contrast, the total number of the equations fQ (z 0 ) = f (z 0 ), ∇fQ z 0 = ∇f z 0 and ∇fQ z 1 , whose numerical values are the data to be used with (2.6), amounts to 1 + 2N . If 1 + N + N (N + 1)/2 > 1 + 2N , that is if N > 1, then the system of equations consistent with the second-order approximation is underdetermined. This means that, unless N = 1, the accounting equation (2.6) is consistent with an inﬁnite number of alternative tangent functions, of which only one would provide a second-order approximation to f (z). Contrary to what is commonly believed, Diewert’s (1976, p. 118) lemma (represented here by Corollary (2.2)) loses its “superlativeness” character in the multivariate case13 . (vii)The main problem with equation (2.6) is that it cannot be fully empirically implemented in the case of approximation. In fact, in this case, at least one of the two vectors of weights ∇fQ z 0 and ∇fQ z 1 are not “observable”.

12 Homogeneity properties imposed on (transformed) quadratic production functions reduce the number of equations to N (N +1)/2 (see, for example, Fuss, McFadden, and Mundlak, 1978, p. 232, fn. 4) and section 6 below. 13 A similar remark was advanced by Uebe (1978). Diewert (1978) re-deﬁned the superlative index numbers as those index number formulas that are exact for functions that provide a second-order approximation around any point where the two vectors of data under comparison are equal. This change in deﬁnition leaves the problem outlined here unresolved.

The Theory of Exact and Superlative Index Numbers Revisited 167 By contrast, the ﬁrst derivatives ∇f z 0 and ∇f z 1 of the unknown function f are usually “observable” in the economic reality (at least when “ﬁrst-best” optimal choices actually take place). In fact, if the point of approximation occurs at z 0 , then ∇f z 0 = ∇fQ z 0 , but in general the inequality ∇f z 1 = ∇fQ z 1 remains. The usual procedure is to use the observed ﬁrst derivatives ∇f z 1 in place of ∇fQ z 1 in equation (2.6), thus resulting in a hybrid approximation that is not “exact” for the quadratic polynomial form (2.5), being very different from it in size and algebraic sign. It should be stressed that the error of approximation is obtained by that applying formula (2.6) using the “observed” weights ∇f z 0 and ∇f z 1 rather than the weights ∇fQ z 0 and ∇fQ z 1 may be different in size and algebraic sign from the expected error of quadratic approximation (for an explanation of how the ﬁrst derivatives ∇f z 0 and ∇f z 1 are “observed” in the economic reality, see Appendix A). What is actually obtained with such a procedure is a hybrid measurement. It can be assessed more clearly by means of the following result:

Lemma 2.2 General Quadratic Approximation Lemma. Let f (z) be an arbitrary once differentiable function. If the value change of the arbitrary f (z) is accounted for by using (2.1) with θ set equal to 1/2 as in (2.6), then the obtained approximation error is equal to the difference of two ﬁrst-order approximations multiplied by (1/2 − θ ), that is 1 f z 1 − f z 0 = ∇f z 0 + ∇f z 1 · z 1 − z 0 2

1 1 + −θ fL z − fL z 0 − fL z 1 − fL z 0 2

(2.9)

and, for r= 0, 1, fLr (z) is a ﬁrst-order approximating (linear) function that is tangent to f (z) at z r , that is fLr (z) ≡ ar + br z = f (z r ) + ∇f (z r ) (z − z r ), with ar = f (xr ) − ∇f (xr ) xr and br = ∇f (xr ) . The averaging of the ﬁrst order derivatives is often expected to increase by one the order of approximation (see Theil, 1971, 1975, pp. 37–38). Here, instead, the approximation error represented by the second line of (2.9) is not strictly 1 of second-order, 1 1 and the difference with the latter is equal to 1 0 − z ∇f z − ∇f z z . The size of this difference depends on the Q 2 functional form of f and the distance of z 1 from z 0 . This is shown in the following numerical example.

168 Precursor f(z1)−f(z ?) 1.8 1.6 1.4 1.2 Quadratic approximation 1 0.8 Value change of the "true" (cubic) function

0.6 0.4

Hybrid approximation

0.2 0

1

z 1.5

−0.2

Figure 2 Quadratic approximation versus “hybrid” approximation

Example 2.1

Let f (z) be the following cubic function of one single variable:

1 1 f (z) ≡ a + b z + c z 2 + d z 3 2 6

(2.10)

where a = 15.5008794, b = −38.2764683, c = 65.4734033, d = −53.7666768 so that f (z) = 1.0 with z 0 = 1.0 and f (z) = 1.5 with z 1 = 1.5. A second order differential approximation to f (z) is given by the following quadratic function 1 fQ (z) ≡ α + β z + γ z 2 2

(2.11)

If α = 6.5397667, β = −11.3931299, and γ = 11.7067265, the second–order approximation is around z 0 = 1, that is f (z 0 ) = fQ (z 0 ), f z 0 = fQ z 0 , and f z 0 = fQ z 0 . Table 1 compares, for different changes in z, the actual value changes in f , which can be computed exactly using formula (2.1) (see column 6) with their “hybrid” approximation obtained by imposing θ = 1/2 in (2.1) 1 (see column 7). Figure 1 compares, for different values of z ,this “hybrid” 1 approximation with the quadratic approximation given by 2 fQ z 0 + fQ z 1 · 1 z − z 0 ,which is the right-hand side of identity (2.6). Moreover, given that only the ﬁrst derivatives of the unknown true function are normally “observed”(see Appendix A), Diewert’s superlative index numbers are usually constructed by using, as weights, these data rather than the unknown

The Theory of Exact and Superlative Index Numbers Revisited 169

z0

z1

f’(z 0) =f’Q (z 0)

f’(z 1)

(1) 1.0 1.0 1.0 1.0

(2) 1.125 1.250 1.375 1.5

(3) 0.3136 0.3136 0.3136 0.3136

(4) 1.3569 1.5601 0.9232 −0.5539

q*

f(z 1)−f(z 0)

(6) 0.567104 0.724664 1.533673 −0.791270

(7) 0.113156 0.304217 0.468169 0.5

f’Q (z 1)

(5) 1.7769 3.2403 4.7036 6.1670

Figure 3 Table 1–Quadratic and hybrid approximations of functional value changes

fQ(z1) − fQ(z 0) quadratic approximation using (2.6) (Diewert’s superlative exact formula)

Error of approximation

Hybrid approximation of Error of f(z1) − f(z 0) approximation using (2.1) with q = 1/2 (Diewert’s superlative hybrid formula)

(8)

(9) = (8) – (7)

(10)

(11) = (10) – (7)

0.130658

0.017502

0.044051

–0.069105

0.444234 0.940728 1.620139

0.140017 0.472559 1.1201399

0.235982 0.231890 –0.060070

–0.068235 –0.236279 –0.560070

Figure 4 Table 1 (continued)

ﬁrst derivatives of the quadratic approximating function. This procedure may have consequences that have been overlooked in the literature so far, although Diewert (1978) himself has ended to recognize that only at the tangency point where z 1 = z 0 these index numbers can be exact for a quadratic approximation. If z 1 = z 0 (which is the common case), then in general they are not exact for a quadratic approximation and turn out to be hybrid formulas. In the example given in Table 1, the resulting Diewert’s hybrid formula corresponding to the right-hand side of (2.1) with θ = 1/2 (or the right-hand side of the ﬁrst line of (2.9)) gives rise to an absolute error of approximation that is signiﬁcantly higher than that of the exact quadratic approximation in the neighbourhood of z 0 (about a factor of 4 with z 1 differing from z 0 by 12.5%; see column (11) and column (9) in Table 1). The two types of approximation revert their relative size as z 1 increases its distance from z 0 . However, in both cases in the given example, the errors of approximation become macroscopic (more than 50% and more than 100%, respectively) as z 1 and z 0 differ by 50%. This should warn us against constructing this type of indices with very different points of observation as in the interspatial and international comparisons.

170 Precursor

3 Functions differing in parameters or functional forms Let us consider two functions f 0 and f 1 that may differ in functional forms and/or parameters. Moreover, if some parameters are twice continuously differentiable functions of other variables subject to change over t, say k t , then f t (with t = 0, 1) can be redeﬁned as f t (z) ≡ ht z, k t

(3.1)

where k t is an M -dimensional vector of variables. Let us now consider the following result: Lemma 3.1. Accounting for Value Differences of Two Functions with Different Parameters or Functional Forms. If f 0 (z) and f 1 (z) are two arbitrary functions differentiable at least once and characterized by different parameter values or functional forms, then, f 1 z 1 − f 0 z 0 = (1 − θ) ∇f 0 z 0 + θ ∇f 1 z 1 · z 1 − z 0

where

+ parameter-change component (PC) 1 0 PC ≡ θ f z − f 0 z 0 + (1 − θ) f 1 z 1 − f 0 z 1

(3.2) (3.3)

and, by deﬁning R01 and R11 as the remainder terms associated with the ﬁrst-order polynomials in the Taylor series expansion of f 1 (z 1 ) and f 0 (z 0 ) around z 1 and z 0 , respectively, θ = θ ∗ ≡

R01 0 R1 +R11

if R01 + R11 = 0 or θ is a real number that takes any

value if R01 = R11 = 0 (that is when both f 0 (z) and f 1 (z) are linear in z). It is worth noting that the weight θ in (3.2) falls within the interval 0 ≤ θ ≤ 1 if f 0 (z) and f 0 (z) are quasiconcave or quasiconvex. If ht is separable14 in z, then (3.1) can be rewritten as: ht (z, k) = H t [φ (z) , k]

(3.4)

If k is empty, then the function ht is said to be homothetic. If φ is linearly homogeneous, then ht is said to be homothetically separable. If the function ht

14 The concept of separability was independently proposed by Sono (1945) and Leontief (1947a, 1947b) in the consumer and the producer contexts, respectively. The terminology of “weak” and “strong” separability was introduced by Strotz (1959) after a paper by Afriat (1953) and with a 1 2 departure amended by Gorman (1959a)(1959b). More speciﬁcally, a function F(x) = F(x , x , . . . , xM ) is “weakly” separable in the partition (x1 , x2 ,… , xM ) if there exist functions F ∗ , F 1 , … , F M such that F(x) = F ∗ [F 1 (x1 ), F 2 (x2 ), … F M (xM )], and is “strongly” separable if there exist functions F ∗∗ , F 1 , … , FM such that F(x) = F ∗∗ [F 1 (x1 ) + F 2 (x2 ) + … + F M (xM )]. If M =2, “weak” and “strong” separability concepts are indistinguishable. See Blackorby, Primont, and Russell (1978) for an extended treatment of the subject.

The Theory of Exact and Superlative Index Numbers Revisited 171 is homothetically separable in the two groups of variables z and k, then it can be written as ht (z, k) = H t [φ (z) , ψ (k)], with both functions φ and ψ being homogeneous of degree one. Moreover, if also the function ht is homothetically separable in φ as well as in ψ, then ht (z, k) = H t [φ (z) , (ψ (k))]

(3.5)

The concepts of homothetic functions and homothetic separability have been explicitly developed by Shephard (1953, p. 43) as a general condition for aggregation in the producer context. An aggregate of separable variables exists if the index function of those variables is separable and homogeneous of degree one. In our case, we should have λφ (z) = φ (λz) for all z and λψ (k) = ψ (λk) for all k in order for φ(z) and ψ(k) to be aggregator functions. Shephard (1953, p. 63) was the ﬁrst to realize the necessity of imposing the linear homogeneity property on index functions of separable quantity variables in order for the respective dual price index functions to be independent of those quantities15 . Others, as for example Blackorby, Lady, Nissen, and Russell (1970), used this concept in the consumer context. This deﬁnition is related to the concept of homogeneous separability, which is obtained under linear homogeneity restrictions on separable functions, regarded also as aggregability conditions (see, for example, Green, 1964, p. 25)16 . Under the homothetic (homogeneous) separability assumption, Lemma (3.1) can be complemented with the following corollaries. Corollary 3.1. Decomposition of Functional Value Differences Between Two Arbitrary Differentiable Functions. The difference in value of two arbitrary continuously differentiable functions f 0 (z) and f 1 (z) can be further decomposed as follows, for all z 0 and z 1 , f 1 z 1 −f 0 z 0 = JZ +JS +JT N 0 0 0 0 1 0 z z ∗0 f ∗1 IZ · f (1−θ)si where JZ ≡ zi −zi , +θ si 0 1 zi zi i=1

(3.6)

15 Aggregation requires, in fact, separability and homogeneity conditions. Solow (1956, p. 104n), however, attributed to Samuelson the proof – received from him in private correspondence – that, in a two stage maximization procedure, the aggregating quantity index functions must be linearly homogenous. Gorman (1959a, pp. 476–478) found, in some special cases, conditions weaker than homogeneity of degree one for a function of separable variables to be an aggregator function (see also Blackorby, Primont, and Russell, 1978, Ch. 5, Blackorby, Schworm, and Fisher, 1986, and Blackorby and Schworm, 1988). 16 More speciﬁcally, this condition is referred to as homogeneous weak separability if, under the homogeneity condition, a function is weakly separable as in (3.4) (see, for example, Diewert, 1993, pp. 12–13 and p. 28, and Diewert and Wales, 1995, pp. 260–261).

172 Precursor N ∗1 IZ · f 0 z 0 ∗0 f 0 z 0 1 0 (1 − θ ) ξ − 1 si JS ≡ + θ ξ − 1 si zi1 zi0 i=1 · zi1 − zi0 , JT ≡ Parameter − change component ∂f t (zt ) ∂f t (z t ) with θ deﬁned in Lemma (2.1), si∗t ≡ ∂zt · zit / Ni=1 ∂zt · zit = sit / Ni=1 sjt , for ∂f t (z t ) t = 0, 1 where sit ≡ ∂zt · zit /f t z t (the superscript * denoting here normalized ∂f 0 (z0 ) ∂f 1 (z1 ) shares), ξ 0 ≡ Ni=1 ∂z0 · zi0 /f 0 z 0 , ξ 1 ≡ Ni=1 ∂z1 · zi1 / IZ · f 0 z 0 , i

i

representing the scale effects, and IZ is the index of functional value change net of scale effects, so that f 1 z1 = IZ · IS · IT (3.7) f 0 z0

where

IZ ≡

θ + (1 − θ ) (1 − θ) + θ

N

1

∗0 zi i=1 si z 0 i 0 N ∗1 zi i=1 i z 1 i

s

,

f 1 z1 IS · IT = 0 0 /Iz . f z

As in (2.2)–(2.3), the terms JZ and JS in (3.7) might be seen as the contributions to the absolute functional value change arising, respectively, from changes in z “directly” and through the returns to scale, whereas J T is the contribution arising from the changes in parameters or functional form. Similarly, the index numbers IZ , IS , and IT might have been interpreted, respectively, as the aggregator index of z, the index of scale effects, and the index of changes in parameters or functional form. The index IZ is a Laspeyres-type index number if θ = 0, whereas it is a Paasche-type index number if θ = 1. However, the following remark deﬁnes Remark 3.1 on the existence of the aggregator index: If and only if the two functions f 0 (z) and f 0 (z) are homothetically (homogeneously) separable in z, so that f t (z) = F t [φ(z)], for t = 0, 1, where F t (•) is a well behaved transformation function and φ is linearly homogeneous, then for all z 0 and z 1 , in view of (2.4), ∂f t (zt ) ∂f t (z t ) ∂φ (z t )/∂z t si∗t ≡ ∂zt · zit / Nj=1 ∂zt · zjt = φ zt i · zit , for t=0, 1, JZ = φ z 1 − φ z 0 , ( ) i j i JS + JT = f 0 z 0 − f 1 z 1 − JZ , IZ = φ z 1 /φ z 0 , IS · IT = f 1 z 1 /f 0 z 0 /IZ . In the case of homothetic separability of f 0 and f 1 in z, the indexes JZ and IZ are totally independent of the scale and technological change effects and are exact for the function φ. In this case, the functions f 0 and f 1 can be expressed as homothetic transformations of the constant-parameter degreeone homogeneous function φ(z), so that JZ and IZ are functions of z alone and are exact for φ(z). This is a well-known result obtained by Hulten (1973) with the Divisia indexes, which turn out to be “path independent” if and only

The Theory of Exact and Superlative Index Numbers Revisited 173 if the integrand function is a homothetic transformation of a homogeneous function. If this is not the case, then there is no such an index number aggregating only the variables z. The functions f 0 and f 1 should be homothetic and φ should remain the same in both data points, otherwise any attempt to construct an aggregating index number will inevitable end up to a spurious measure17 . Let us deﬁne the quadratic function: 1 t 1 fQt (z) ≡ at0 + at z + z At z = at0 + ati zi + a zz 2 i=1 j=1 ij i i 2 i=1 N

N

N

(3.8)

where all parameters are variable. Moreover, let us assume that the parameters of fQt are function themselves of other variables, that is α0t ≡ α0t +k t · β t + 21 k t · Bt · k t and α t ≡ α t + k t · t so that fQt (z) ≡ htQ z, k t , where 1 1 htQ (z, k) ≡ α0t + α t z + z At z + kβ t + kBt k + k t z. 2 2

(3.9)

The function htQ (z, k) can be expressed also as a quadratic function in k: 1 htQ (z, k) ≡ ψQt (k) ≡ bt0 + kbt + kBt k 2

(3.10)

where bt0 ≡ α0t + α t z + 12 z t At z, bt ≡ β t + t z. The quadratic functional form (3.8)–(3.9)–(3.10) may be reformulated in order to represent special separability cases. The early literature on separability of the so-called “ﬂexible” functions has indicated the required parameter restrictions18 . The “weak” separability of fQt in z with respect to parameter changes occurs if (3.8) can be rewritten as: 1 fQt [ζ (z)] ≡ α0t + α1t ζ (z) + ζ (z) at2 ζ (z) 2

(3.11)

where at1 and at2 are variable scalar parameters and ζ (z) must be the linear function αz in order for the function fQt to have the functional form (3.8). In this case, htQ

17 Samuelson and Swamy (1974, p. 568) comment the non-existence of an aggregating index number in these terms: “we cannot hope for one ideal formula for the index number: if it works for the tastes of Jack Spratt, it won’t work for his wife’s tastes; if, say, a Cobb-Douglas function can be found that works for him with one set of parameters and for her with another set, their daughter will in general require a non-Cobb-Douglas formula! Just as there is an uncountable inﬁnity of different indifference contours–there is no counting tastes–there is an uncountable inﬁnity of different indexnumber formulas, which dooms Fisher’s search for the ideal one. It dos not exist even in Plato’s heaven”. In a footnote, they add “This point, forcefully made in Samuelson (1947, p. 154) has served to intimidate many (but alas, not all) searchers for the nonexistent”. 18 See Diewert (1993, p. 15) for references to this literature.

174 Precursor can be rewritten as 1 htQ [ς (z) , k] = α0t + α1t ς (z) + ς (z) at2 ς (z) 2 1 1 + kβ t + kBt k + kγ t ς (z) 2 2

(3.12)

where γ t is an M -dimensional (column) vector of variable parameters. A similar condition can be imposed on the parameters of the ﬁrst- and second-order terms in k by deﬁning the linear function κ(k) ≡ βk in order to represent the case of “non-linear” separability of htQ in k. Therefore, by imposing the “weak” separability conditions for z and k, htQ becomes a non-linear (quadratic) function whose arguments are the linear functions ζ (z) and κ(k), that is htQ [ς (z) , κ (k)] ≡ at0 + at1 ς (z) + 12 ς (z) at2 ς (z) + κ (k) β1t + 12 κ (k) β2t κ (k) + 12 κ (k) γ t ς (z). The “strong” or “linear” separability of htQ in z, k, and parameter changes occurs when t = 0M ×N in (3.9) so that htQ (z, k) = σ t ςQ (z) + κQ (k)

(3.13)

where ζQ (z) = α0 + αz + 12 z Az, κQ (k) = kβ + 12 kBk, and σ t is a factor of proportionality. Under the conditions of homogeneity and the necessary separability restrictions on parameter values, the form of (3.13) is equivalent to (see section 5) htQ (z, k) = σ t · ςQ∗ (z) · κQ∗ (k)

(3.14)

which is indistinguishable from the “weak separability” case with only two separable groups of variables19 . The following results regarding the accounting for functional value differences can be obtained: Corollary 3.2 Accounting for Value Differences of Two Quadratic Func tions Differing in Parameters. If two functions fQ0 z 0 ≡ h0Q z, k 0 and fQ1 z 1 ≡ h1Q z, k 1 have quadratic functional forms as those deﬁned by (3.8)–(3.9) which may differ in all parameters, then, for all z 0 and z 1 ,

19 This is a well known result (see Gorman, 1959a, p. 471). The functional form (3.13), which is additive in ζ Q and κ q , is identically equal to (3.14), which is multiplicative in ζQ∗ and κQ∗ .

The Theory of Exact and Superlative Index Numbers Revisited 175 1 (3.15) fQ1 z 1 − fQ0 z 0 = ∇fQ0 z 0 + ∇fQ1 z 1 · z 1 − z 0 2 1 1 1 + a10 − a00 + a1 − a0 z + z 0 + z 0 A1 − A0 z 1 2 2 (3.16) = h1Q z 1 , k 1 − h0Q z 0 , k 0 =

1 ∇z h0Q z 0 , k 0 + ∇z h1Q z 1 , k 1 · z 1 − z 0 2 + Residual component

with the residual component being equal to the parameter change component deﬁned by the second line of (3.15), in which the sum of the “zero-” and “ﬁrstorder” terms can be, in turn, decomposed as follows 1 0 1 0 1 0 1 1 a0 −a0 + a −a z +z = ∇k h0 z 0 ,k 0 + ∇k h1 z 1 ,k 1 · k 1 −k 0 2 2 1 0 1 + α01 −α00 + α 1 −α 0 z +z 2 1 1 + k 0 +k 1 β 1 −β 0 + k 0 B1 −B0 k 1 2 2 (3.17) Differently from the Quadratic Identity established by Corollary (2.2), Corollary (3.2) is not an “if and only if” result: the decomposition of value difference between two quadratic functions implies (3.15)–(3.17), but the converse is not true. In fact, equations (3.15)–(3.18) may even hold in the particular case of a functional form that is linear in z.Note that the inequality ∇f 0 z 0 = ∇f 1 z 1 for z 0 = z 1 may occur even with linear functions when these have different parameters. Furthermore, we observe that the theory of exact and superlative index numbers has traditionally considered quadratic functional forms where only the parameters of the zero- and ﬁrst-order terms in z may differ. By contrast, Corollary (3.2) shows that, also when the parameters of second-order terms differ, the difference in functional values of two quadratic functions can be still split into two separate components due to differences in z and differences in parameters. Lemma 3.2. Quadratic Approximation of Value Differences between Two Arbitrary Functions with Different or Functional Forms. Let Parameters f 1 (z) ≡ h1 z, k 1 and f 0 (z) ≡ h0 z, k 0 be two arbitrary once differentiable functions with different parameters or functional forms. If their value difference is accounted for by using (3.2) where θ is set equal to 1/2, then , for all z 0 and z 1 ,

176 Precursor 1 f 1 z 1 −f 0 z 0 = ∇f 0 z 0 +∇f 1 z 1 · z 1 −z 0 2 +θ f 1 z 0 −f 0 z 0 +(1−θ ) f 1 z 1 −f 0 z 1 +error of approximation (3.18)

0 1 0 0 1 1 1 0 1 error of approximation ≡ −θ fL z −fL z − fL z −fL z 2 (3.19) R01 z 0 , z 1 if R01 z 0 , z 1 + R11 z 0 , z 1 = 0 with θ = θ z , z ≡ R1 z 0 , z 1 + R11 z 0 , z 1 or θ may take any value if R01 z 0 , z 1 = R11 z 0 , z 1 = 0, and fLt (z) is a ﬁrstt t is fLt (z) order (linear) that is tangent approximating function t to ft (z) att z (that ≡ t t t t t t t t t t t a +b z = f z +∇f z z − z , with a = f x −∇f x x and b = ∇f t xt , with t = 0, 1). Let us, now, construct the following general decomposition of the absolute value difference between h1 (z 1 , k 1 ) and h0 (z 0 , k 0 ): h1 z 1 , k 1 − h0 z 0 , k 0 = (1 − λ) h0 z 1 , k 0 − h0 z 0 , k 0 + λ h1 z 1 , k 1 − h1 z 0 , k 1 + λ h1 z 0 , k 1 − h0 z 0 , k 0 + (1 − λ) h1 z 1 , k 1 − h0 z 1 , k 0 (3.20) ∗

0

1

0

The following technical result is also obtained: Corollary 3.3. Accounting for the Sum of Value Differences between Two Quadratic Functions with Different “Zero-order” and “First-order” Parameters (Caves, Christensen, and Diewert’s, 1982, pp.1412–1413 Translog Identity). If two quadratic functions are both deﬁned by (3.8)–(3.9) differing in parameters of the “zero-order” and “ﬁrst-order” terms but having equal parameters in the same “second-order” terms in z (that is A1 = A0 = A), then, for all z 0 , z 1 , k 0 , k 1 , 0 1 0 0 1 1 1 0 fQ z −fQ z + fQ z −fQ z = ∇fQ0 z 0 +∇fQ1 z 1 · z 1 −z 0 (3.21) 0 0 0 0 0 0 1 1 1 1 1 1 and hQ z ,k −hQ z ,k + hQ z ,k −hQ z ,k = ∇z h0Q z 0 ,k 0 +∇z h1Q z 1 ,k 1 ·z z 1 −z 0 (3.22) This result is known under the name of Translog Identity, because it was formulated by Caves, Christensen, and Diewert (1982, pp. 1412–1413) in terms of translog functions.

The Theory of Exact and Superlative Index Numbers Revisited 177 In the more general case of quadratic functions with no restrictions on variable parameters, we have the following result: Corollary 3.4. Accounting for the Sum of Value Differences of Two Quadratic Functions with Different Parameters (Caves, Christensen, and Diewert, 1982, pp. 1412–1413). If two quadratic functions are deﬁned by (3.8)–(3.9), differing in all parameters, including those of the “second-order” terms in z (A1 may differ from A0 ), then, for all z 0 , z 1 , k 0 , k 1 ,

fQ0 z 1 −fQ0 z 0 + fQ1 z 1 −fQ1 z 0 = ∇fQ0 z 0 +∇fQ1 z 1 · z 1 −z 0

1 − z 1 −z 0 A1 −A0 z 1 −z 0 2 0 1 0 0 0 0 1 1 1 1 0 1 and hQ z , k −hQ z , k + hQ z , k −hQ z , k = ∇z h0Q z 0 , k 0 +∇z h1Q z 1 , k 1 · z 1 −z 0 −

1 1 0 1 z −z A −A0 z 1 −z 0 2

(3.23)

(3.24)

If A1 = A0 , then the term 12 z 1 − z 0 A1 − A0 z 1 − z 0 is equal to zero and the result obtained with Corollary (3.4) is the same as that obtained with Corollary (3.3). It is evident that Caves, Christensen, and Diewert (1982, pp. 1412–1413) introduced the restriction A1 = A0 in order to obtain the equivalence between an arithmetic average of the two functions (evaluated at z 0 and z 1 ) and the component of weighted changes in z. This equivalence no longer holds when A1 = A0 , but the following result is useful to show that, even in this general case, the overall functional value difference can be still decomposed into separate components of differences in z, k, and variable parameters. Corollary 3.5 If two functions fQ0 (z) ≡ h0Q (z, k 0 ) and fQ1 (z) ≡ h1Q (z, k 1 ) are deﬁned by (3.8)–(3.9), then, for all z 0 , z 1 , k 0 , k 1 ,

fQ1 (z 0 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 ) = 2 a10 − a00 + a1 − a0 z 0 + z 1 + z 0 A1 − A0 z 1 +

1 1 z − z 0 A1 − A0 z 1 − z 0 2

(3.25)

178 Precursor and h1Q z 0 ,k 1 −h0Q z 0 ,k 0 + h1Q z 1 ,k 1 −h0Q z 1 ,k 0 = h1Q z 1 ,k 1 −h1Q z 1 ,k 0 + h0Q z 0 ,k 1 −h0Q z 0 ,k 0 + h1Q z 1 ,k 1 −h0Q z 1 ,k 1 + h1Q z 0 ,k 0 −h0Q z 0 ,k 0 (3.26)

where 1 1 1 hQ z , k − h1Q z 1 , k 0 + h0Q z 0 , k 1 − h0Q z 0 , k 0 = ∇k h0Q z 0 , k 0 + ∇k h1Q z 1 , k 1 k 1 − k 0 −

1 1 k − k 0 B1 − B0 k 1 − k 0 , 2

(3.27)

1 1 1 hQ z , k − h0Q z 1 , k 1 + h1Q z 0 , k 0 − h0Q z 0 , k 0 = 2 α01 − α00 + α 1 − α 0 z 1 + z 0 1 + z 0 A1 − A0 z 1 + z 1 − z 0 A1 − A0 z 1 − z 0 2 1 1 0 + k + k β − β 0 + k 0 B1 − B0 k 1 +

1 1 k − k 0 B1 − B0 k 1 − k 0 2

(3.28)

Adding (3.24) to (3.25)–(3.28) and dividing the resulting expression through by 2 yield the same result obtained by Corollary (3.2). It is straightforward to show that, in order to obtain 12 ∇fQ0 (z 0 ) + ∇fQ1 (z 1 ) · 1 z − z 0 = (1 − γ ) fQ0 (z 1 ) − fQ0 (z 0 ) + γ fQ1 (z 1 ) − fQ1 (z 0 ) , γ must take the particular value 12 F 1 − F 0 + 14 z 1 − z 0 A1 − A0 z 1 − z 0 ∗ 0 1 1 (3.29) γ z ,z ≡ F − F0 where F t ≡ fQt (z 1 ) − fQt (z 0 ) for t = 0, 1. If A1 = A0 , then γ ∗ (z 0 , z 1 ) = 1/2). Even in the more general case where A1 = A0 , it is possible to decompose the difference between two quadratic functions deﬁned by (3.8) into a component due to (weighted) changes in z and a component due to (weighted) differences in parameters. These two components turn out to be a weighted average of two functional value changes evaluated at the respective reference variables. Similarly, it is possible to decompose the functional value changes in htQ (z, k) deﬁned by (3.9) into three separate components, due respectively to (weighted) changes in z, k, and differences in parameters. In the particular case of the function hQ (z, k) where A1 = A0 and B1 = B0 , equations (3.16) and (3.17) reduce to the decomposition

The Theory of Exact and Superlative Index Numbers Revisited 179 procedure used by Diewert and Morrison (1986, Theorem 1, p. 663) in terms of translog functions. The results obtained here widen considerably the scope of applicability of the decomposition of differences in functional values of quadratic functions where also the “second-order” parameter are assumed to be variable. Corollary 3.6. Accounting for Functional Value Differences of Two Quadratic Homothetic Functions. If two continuously once-differentiable quadratic functions fQ0 (z) and fQ1 (z) deﬁned by (3.8) are homothetic transformation of linearly homogeneous functions, so that ai = 0 for i = 0, 1, . . . , N , and 2 1 2 fQ1 (z) = σ t φQ (z) for t = 0, 1, where φQ (z) ≡ 12 zAz / , then, for all z 0 and z 1 . fQ1 (z 1 ) − fQ0 (z 0 ) = JZ + JS + JT N 0 0 fQ1 z 1 1 1 0 fQ z 1 where JZ = JS ≡ sQ,i zi − zi0 + sQ,i t t 2 i=1 zi zi

(3.30)

JT = Parameter change component t with JZ and JS having the same meaning established above, sQ,i ≡ t t t ! N ∂fQ (z ) t ∂φQ (z ) t ! t zit · zi φQ z (t = 0, 1) , and i=1 ∂z t · zi = ∂z t i

∂zit

.

i

fQ1 z 1 = IZ · IS · Iσ fQ0 z 0

(3.31)

⎡ where

∂fQt (z t )

⎢ IZ = IS ≡ ⎣

1 N 0 zi i=1 sQ,i z 0 i 0 N 1 zi i=1 Q,i z 1 i

s

⎤ 12

φQ z 1 σ1 ⎥ 0 and IT ≡ 0 . ⎦ = φQ z σ

We note that IZ and IS in (3.31) are Fisher “ideal”-type index numbers, which are “exact” for φQ (z) (identically equal to the ratio φQ z 1 /φQ z 0 = 1 1 1 1/2 1 0 0 1/2 z Az z Az for any z 0 and z 1 ). 2 2

4 Accounting for value changes of transformed functions In the previous sections, we have seen that the accounting formula deﬁned with the Quadratic Identity cannot be applied to arbitrary functions without incurring a possible non-negligible error of approximation. In the general case, a way to reduce this error is to use a transformed quadratic function. With appropriate parameter values, it is this function rather than a quadratic function that provides a second-order differential approximation to an arbitrary function.

180 Precursor Suppose that the arbitrary function f t (z) considered in section 3 can be deﬁned as f t [Z(q)], where Z(q) = z, with the ith element zi = z(qi ), so that f t [Z(q)] = f t (q). t Furthermore, suppose that a general quadratic function fGQ (q) can be transformed into a quadratic function as follows: t 1 g fGQ (q) = at0 + at Z t (q) + Z t (q)At Z t (q) (4.1) 2 where Z t (q) ≡ z t (q1 ) z t (q2) . . . z t (qN ) so that

1 t t t t −1 t t t fGQ (q) = g a0 + a Z (q) + Z (q)A Z (q) (4.2) 2 Let us deﬁne at0 ≡ α0t + β t K xt + 12 K xt Bt K xt and at = α t + K xt t where t t t K(x) ≡ k (x1 ) k (x2 ) . . . k (xM ) so that g fGQ (q) ≡ g hGQ q, x , with 1 g htGQ (q, x) = α0t + αit Z(q) + Z(q)At Z(q) 2 1 + K(x)β t + K(x)Bt K(x) 2 + K(x) t Z(q)

(4.3)

where zi ≡ z (qi ) and km = K (xm ) or in vector notation z ≡ Z(q) and k ≡ K(x) The values of all parameters may change as t changes, and g, z, and k are continuous and monotonic functions of one single variable with non-zero derivatives. Since the functions z and k are continuous and monotonic, it is possible to invert them in order to obtain qi = z −1 (zi ) or, in vector form, xm = k

−1

(km ) or, in vector form,

q = Z −1 (z) x=k

so that t t −1 g fGQ (q) = g fGQ Z (z) ≡ fQt (z)

−1

(k)

deﬁned by (3.11)

(4.4) (4.5)

(4.6)

and g htGQ (q,x) = g htGQ Z −1 (z),K −1 (k) ≡ htQ (z,k) deﬁned by (3.12)

(4.7)

t If the functions fQt and htQ have speciﬁc parameter values such that fGQ ∗ t ∗ t ∗ t ∗ 2 t ∗ 2 t ∗ (q ) = f (q ) , ∇fGQ (q ) = ∇f (q ) and ∇ fGQ (q ) = ∇ f (q ) , then the function t fGQ = g −1 fQt provides a second-order differential approximation to f t around q∗ . This should be contrasted with the assumption in section 3 that it is fQt rather than g −1 fQt that provides a second-order differential approximation to f t around q∗ .

The Theory of Exact and Superlative Index Numbers Revisited 181 We can now obtain the following result: Lemma 4.1. Accounting for Value Differences of Two General Quadratic t (q) is deﬁned by (4.2), then Functions (Diewert, 2002, p. 67). If the function fGQ

g

1 fGQ

" 1 0 0 1 0 0 0 −1 0 0 g fGQ q · Zˆ q · ∇q fGQ q − g fGQ q = q 2 +g

1 fGQ

# 1 1 −1 1 −1 1 ˆ · ∇q fGQ q · Z q q

· Z q1 − Z q0 + T

(4.8)

where 1 0 T ≡ a10 −a00 + a1 −a0 Z q +Z q1 +Z q0 A1 −A0 Z q0 2

(4.9)

−1 and Zˆ (q) is a diagonal matrix in which the (i, i)th element is equal to ! dqi dz −1 (zi ) = = 1 z (qi ) dzi dzi

(4.10)

(with z (qi ) = 0, by assumption).

5 A general transformed quadratic function The transformed quadratic function (4.1)–(4.2) can be fully speciﬁed by choosing the functional form of the transformation functions g, z, and k. Among the many candidates, if we deﬁne the following functions as suggested in the original work of Box and Cox (1964):

g(y) ≡

yρ − 1 ρ

(5.1)

z (qi ) ≡

qiλ − 1 λ

(5.2)

k (xi ) ≡

xi − 1 μ

μ

(5.3)

182 Precursor and replace the functions (5.1) and (5.2) in (4.1), then we obtain the following quadratic Box-Cox function20 :

ρ

fQt ρ,λ (q)

−1

ρ

N

= at0 +

qλ − 1 ati i λ

i=1

N N λ 1 t qiλ − 1 qj − 1 aij + 2 i=1 j=1 λ λ

(5.4)

which, by replacing (5.1)–(5.3) in (4.3), turns out to be equivalent to

ρ htQρ,λ (q, x) − 1 ρ

= α0t +

N

αit

i=1

+

M

βmt

m=1

+

N N λ qiλ − 1 1 t qiλ − 1 qj − 1 aij + λ λ λ 2 i=1 j=1

M M xmμ − 1 1 t xmμ − 1 xnμ − 1 + b μ 2 m=1 n=1 mn μ μ

M N i=1 m=1

t γmi

xmμ − 1 qiλ − 1 μ λ

(5.5)

by setting

at0 ≡ α0t +

M

βmt

M M xmμ − 1 1 t xmμ − 1 xnμ − 1 + b μ 2 m=1 n=1 mn μ μ

(5.6)

t γmi

xmμ − 1 μ

(5.7)

m=1

ati ≡ αit +

M m=1

With ρ → 0, and λ → 0, the function (5.4) reduces to the following translog functional form:

t In fTrg (q) = at0 +

N i=1

1 t a In qi In qj 2 i=1 j=1 ij N

ati In qi +

N

(5.8)

20 The explicit use of the quadratic Box-Cox function can be dated back at least to the works of Khaled (1977), Kiefer (1977), Appelbaum (1979), and Berndt and Khaled (1979). The special case of a quadratic Box-Cox aggregator function has also been derived by Diewert (1980, pp. 450–451) from a quadratic mean-of-order-r aggregator function.

The Theory of Exact and Superlative Index Numbers Revisited 183 and, when ρ → 0, λ → 0, and μ → 0 the function (5.5) reduces to In

htTrg (q, x) = α0t +

N i=1

+

M

1 t qi + a In qi In qj 2 i=1 j=1 ij 1 t b In xm In xn 2 m=1 n=1 mn M

M

βmt In xm +

m=1

+

N

N

αit In

N M

t γmi In xm In qi

(5.9)

m=1 i=1

t is homogeneous of degree δ if Ni=1 ati = δ and Ni=1 atij = The function fTrg N t t j=1 aij = 0. The function hTrg is homogeneous of degree δ in q under the N t N t following conditions: (i) i=1 αi = δ, (ii) i=1 aij = 0 for j = 1, 2, . . . , N , N t (iii) a = 0 for i = 1, 2, . . . , N (condition (ii) implies (iii) under the j=1 ij N symmetry of matrix A), and (iv) i=1 γmi = 0, for m = 1, . . . , M . Moreover, the function htTrg is homogeneous of degree η in x under the following con M M t ditions: (v) M m=1 βm = η, (vi) m=1 bmn = 0 for j = 1, 2, . . . , N , (vii) n=1 bmn = 0 for i = 1, 2, . . . , N (condition (vi) implies (vii) under the symmetry of matrix B), and (viii) M m=1 γmi = 0, for n = 1, 2, . . . , N . t is homothetically separable in q from parameter changes if The function fTrg the parameters of the ﬁrst- and second-order terms are constant, that is ati = ai and atij = aij for i = 1, 2, . . . , N and j = 1, 2, . . . , N . Moreover, the function htTrg is homothetically separable in q from changes in x and parameters if αit = αi , atij = aij and γmi = 0 for i = 1, 2, . . . , N and j = 1, 2, . . . , N , and m = 1, 2, . . . , M . With ρ = 0, and λ = 0, the general quadratic Box-Cox function (5.4) can be rewritten in the following form: ⎡ fQt ρ,λ (q) = ⎣a¯ t0 +

N

a¯ ti qiλ +

i=1

where

a¯ t0

≡

1 + ρat0 − ρλ

ρ t a . λ2 ij

N i=1

ati

N N 1

2

+

⎤ ρ1 a¯ tij qiλ qjλ ⎦

(5.10)

i=1 j=1

ρ 2λ2

N N i=1 j=1

atij

,

a¯ ti

≡

ρ t a λ i

−

ρ λ

N

atij

and a¯ tij ≡

j=1

The function fQt ρ,λ is homogeneous of degree 2λ/ρ if a¯ t0 = 0 and a¯ ti = 0 for all i’s implying: 1 + ρat0 ρ = N N t 2λ2 i=1 j=1 aij

(5.11)

184 Precursor Therefore, in the case of homogeneity of degree 2λ/ρ, using (5.11), the function (5.10) can be re-written as ⎤ ρ1 ⎡ ⎤ ρ1 ⎡ N N N N ⎢ 1 + ρat ⎥ ⎥ ⎢ fQt ρ,λ (q) = ⎣ 2λρ2 atij qiλ qjλ ⎦ = ⎣ atij qiλ qjλ ⎦ ⎢ N N 0 ⎥ ⎣ t ⎦ i=1 j=1 i=1 j=1 aij ⎤ ρ1

⎡

i=1 j=1

(5.12) Hence ⎤ ρ1 ⎡ N N 1 fQt ρ,λ (q) = ⎣ αijt qiλ qjλ ⎦ 1 + ρat0 ρ

(5.13)

i=1 j=1

where αijt ≡ atij /

N N

atij . Note that

i=1 j=1

N N

αijt qiλ qjλ = 1 when all qi s are equal to 1.

i=1 j=1

Moreover, if all parameters atij in (5.12) change proportionally, so that αijt in (5.13) remain constant (αijt = αij , then the effects of parameter changes are separable from q. Under these conditions, this function can be seen as a homothetic transformation of a linearly homogeneous function, that is r ρ fQt ρ,λ (q) = σ t fQr (q) /

(5.14) 1

Setting r = 2λ, σ t ≡ (1 + ρat0 ) ρ , and ⎤1 ⎡ r N N r r fQr (q) ≡ ⎣ αij qi2 qj2 ⎦

(5.15)

i=1 j=1

which is the quadratic mean-of-order-r aggregator function used by Diewert’s (1976, pp. 129–130)21 . In fact, fQr is homogenous of degree one in q and separable from parameter changes in f tρ,λ . Therefore, fQr has the required properties of an Q

aggregator function of q within f tρ,λ . Q Equation (5.15) reduces to well-known functions for particular values of r. Denny (1972, 1974) noted that, if r = 1, then it reduces to the generalized linear functional form proposed by Diewert (1969, 1971). In an unpublished memorandum, Lau (1973) showed that, at the limit as r tends to zero, it reduces to

21 This functional form is due to McCarthy (1967), Kadiyala (1972), Denny (1972, 1974), and Hasenkamp (1973).

The Theory of Exact and Superlative Index Numbers Revisited 185 the homogeneous translog aggregator function (Lau’s proof is reported in Diewert, 1980, p. 451). Diewert (1976, p. 130) also noted that, if r = 2, then it reduces to the Konüs-Byushgens (1926) functional form. Furthermore, if all αij = 0 for i = j, then it reduces to a CES functional form. If the parameters of the ﬁrst-order terms of the function (5.10) are functions of x as follows a¯ t0 ≡ α¯ 0t +

M

β¯mt xmμ +

m=1

a¯ ti ≡ α¯ it +

M

1 ¯t μ μ b x x 2 m=1 n=1 mn m n M

M

(5.16)

t μ γ¯mi xm ,

(5.17)

m=1

then (5.10) is identically equal to htQρ,λ,μ (q, x) = [α¯ 0t +

N

1 t λ λ α¯ q q 2 i=1 j=1 ij i j

i=1

+

M

β¯mt xmμ +

m=1

+

N M

N

N

α¯ it qiλ +

1 ¯t μ μ b x x 2 m=1 n=1 mn m n M

M

1

t μ λ ρ γ¯mi xm qi ]

(5.18)

m=1 i=1

$ where α¯ 0t ≡ 1 + ρα0t − ρ μλ

M M

N

ρ λ

$ % t γmi ; α¯ it ≡ ρλ αit −

m=1 i=1 % M ρ t γmi ;b¯ m μλ i=1

≡

ρ t b ; γ¯ t μ2 mn mi

ρ 2λ2

αit +

i=1 ρ λ2

≡

N

atij −

j=1

ρ t γ μλ mi

N N

atij −

ρ μ

M

i=1 j=1 m=1 % M ρ t γmi ; β¯mt ≡ μλ m=1

βmt + 2μρ 2

M M

btmn m=1 n=1 $ M ρ t β − μρ2 btmn μ m n=1

+ −

.

$ M The function htQρ,λ,μ is homogeneous of degree 2λ/ρ in q if α¯ 0t + β¯mt xmμ +

1 2

M M

b¯ tmn xmμ xnμ

%

m=1 n=1

degree 2μ/ρ in x if

m=1

$ % M t μ γ¯mi xm = 0 for all i’s. It is homogeneous of = 0 and α¯ it + $

α¯ 0t +

N i=1

m=1

α¯ it qiλ + 12

N N

i=1 j=1

% $ % N t λ a¯ tij qiλ qjλ = 0 and β¯mt + γ¯mi qi = 0 i=1

for all m’s. t Moreover, the function htQρ,λ,μ is additively separable in q and x if γmi = 0 for all m’s and i’s. If it is homothetic and, at the same time, additively separable in q (and/or x), then it is homothetically separable in q (and/or x). If, as a particular case of homothetically separable function, it is separable and homogeneous of degree one in q (and/or x), then it is said to be homogeneously separable in q (and/or x).

186 Precursor Under the hypothesis of homothetic separability, using (5.6) and (5.13) with the restriction of homogeneity of degree 2λ/ρ and separability in q, equation (5.18) becomes ⎤1 ⎡ ρ M M N N M 1 ¯t μ μ t λ λ⎦ t ∗t μ t ⎣ ¯ ¯ hQρ,λ,μ (q, x) = βm xm + b x x β0 + αij qi qj 2 m=1 n=1 mn m n m=1 i=1 j=1

1 ρ

(5.19) M M t bmn ;

M βmt + 2μρ 2 where β¯0t ≡ 1 + ρα0t − μρ m=1 m=1 n=1

N M ρ ρ ρ t t ∗t t t t β¯m ≡ μ βm − μ2 bmn ≡ β¯m + μλ γmi = β¯mt for all m’s (since γmi = 0 for n=1

i=1

all m’s and i’s, as a condition imposed for the additive separability in q and x); b¯ tmn =

ρ t b . μ2 mn

The function (5.19) is also homogeneous of degree 2μ/ρ in x, if β¯0t = 0 and β¯m∗ t = 0 (and, therefore, β¯mt = 0) for all m’s, implying 1 + ρα0t ρ = M M (5.20) t 2μ2 bmn m=1 n=1

Therefore, in the case of homogeneity of degree 2μ/ρ in x, the function (5.19) can be re-written as ⎡ htQρ,λ,μ (q, x) = ⎣

⎤1 ρ M M ρ t μ μ αijt qiλ qjλ ⎦ bmn xm xn 2μ2 m=1 n=1 j=1

N N i=1

1 ρ

using (5.20) ∗

r ⎡⎛ ⎤r ⎞ r ⎤ 2 ⎡ r1∗ ρ 2 ∗ ∗ N M N M r r r r ⎢ t 2 2 ⎠ ⎥ ⎢ t 2 2 ⎥ = ⎣⎝ αij qi qj βmn xm xn ⎦ ⎣ ⎦ × i=1 j=1

m=1 n=1

1

(1 + ρα0t ) ρ setting r =2λ, r

t =2μ, and βmn ≡btmn /

(5.21) M M m=1 n=1

btmn . Note, that

M M

r∗ r∗ t βmn xm2 xn2 =1

m=1 n=1

when all xi ’s and xj ’s are equal to 122 . 22 We may note that, with r=r*=ρ=2 and all “second-order” parameters being constant (αijt =αij t

andβmn =β mn for every value of t), (5.21) reduces to the quadratic mean-of-order-2 functional form used by Diewert (1992, p. 231, eq. (56)).

The Theory of Exact and Superlative Index Numbers Revisited 187 t t If all αijt ’s and βmn ’s are constant over t, that is αijt = αij and βmn = βmn for t all i, j, m, and n, then the function hQρ,λ,μ (q, x) is homothetically separable in q, x, and parameter changes. The last ones are captured only by the changes in the 1 r r r 1 N N proportionality factor (1 + ρα0t ) ρ . The resulting functions αij qi2 qj2 and i=1 j=1

M M

r∗

r∗

1 r∗

βmn xm2 xn2

are homogeneous of degree one and can be considered as

m=1 n=1

“aggregator functions” of q and x, respectively, within the function htQρ,λ,μ 23 .

6 Translog and quadratic mean-of-order-r functions The general quadratic mean-of-order-r function, which is algebraically derivable from the quadratic Box-Cox function deﬁned by (5.4)–(5.5), can be used as a second-order approximation to an arbitrary unknown function. Using the results obtained thus far, it is possible to account for differences in functional values, either in terms of differences or in terms of ratios, into aggregating components of changes in the arguments and parameters of the function. We shall follow the tradition of calling indicators the aggregating components deﬁned in terms of differences and index numbers those that are deﬁned in terms of ratios (see, for example, Diewert, 1998, 2000). In order to save space, we shall deal with only the function fQt ρ,λ (q) deﬁned by (5.10) and leave to the reader the exercise of deriving the corresponding results with the more explicit function htQρ,λ,μ (q, x) deﬁned by (5.18). The following theorems are in order. Theorem 6.1 If two quadratic Box-Cox functions deﬁned by (5.4) with different parameters reduce to the “translog” functional form (5.8) (with p → 0 and λ → 0), then for all q0 and q1 1 0 1 (s + sTrg,i ) (ln qi1 − ln qi0 ) 2 i=1 Trg,i N

1 0 In fTrg − ln fTrg =

+ Parameter − change component

t where sTrg,i ≡

t qit fTrgi t fTrg

t with t = 0, 1 and fTrgi ≡

t ∂fTrgi

∂qit

(6.1)

(6.2)

23 In section 3, we have recalled that an aggregator function must be degree-one homogeneous as well as separable.

188 Precursor The Törnqvist index number is said to be “exact” for the translog function because, taking the antilogarithms, it gives the same result as the ratio of two translog functions. Following (3.29), it is straightforward to show that

0 fTrgi (q1 ) 0 fTrgi (q0 )

γ

1 fTrgi (q1 ) 1 fTrgi (q0 )

1−γ

1 0 1 S In qi1 − In qi0 + STrg,i 2 i=1 Trg,i N

=

with γ taking the particular value given by (3.28), where fQt and z t are, respectively, t replaced with ln fTrg and ln qt . N t 1 0 We note that i=1 sTrg,i = 1 if fTrg and fTrg are homogeneous of degree one in q, N t t t t since in this case fTrg = i=1 qi fTrg,i by Euler’s theorem. If Ni=1 sTrg,i = 1, then N ∗t t t we can deﬁne sTrg,i ≡ sTrg,i / i=1 sTrg,i and rewrite equation (6.1) as follows: 1 0 In fTrg − In fTrg = ln Iq + ln IS + ln IT

(6.3)

1 ∗0 ∗1 where In Iq ≡ + sTrg,i s In qi1 − In qi0 2 i=1 Trg,i N

In IS ≡

N 1 ∗0 ∗1 sTrg,i (ξ − 1) + sTrg,i (ξ − 1) In qi1 − In qi0 2 i=1

In IT ≡ Parameter − change component t t where ξ ≡ Ni=1 qit fTrg,i /fTrg , which represents the degree of returns to scale. The ﬁrst additive term in the right-hand side of (6.3) is the contribution of the change in q to the functional value difference, whereas the second additive term is the contribution of the scale effects. This result is obtained without imposing separability and homogeneity restrictions as it has been observed by Diewert (1980a, p. 463) (1980b, p. 538)(2004, p. 450), and Denny (1980, p. 535–37). Caves, Christensen, and Diewert (1982, p. 41) contended that only the translog-based indicator of relative changes does not rely necessarily on this restrictions. However, based on the developments in section 2 and 3, the following remark is noteworthy: 0 Remark 6.1 on the existence of the aggregator index: If and only if fTrg 0 t t t and fTrg are homothetically separable in q so that fTrg ≡ F [φTrg (q)], where F (·) is a well behaved transformation function and φTrg is a linearly homogeneous translog function with constant parameters, then for all q0 and q1 , si∗t ≡ t (qt ) t (qt ) N ∂fTrg ∂fTrg ∂φ (qt )/∂qit t · q / · qjt = Trg · qit for t=0, 1, in view of (2.4), t i j=1 φ (qt ) ∂q ∂qt i

j

Trg

The Theory of Exact and Superlative Index Numbers Revisited 189 1 1 0 0 Iq = φTrg (q1 )/φTrg (q0 ), IS ·IT = fTrg (q )/fTrg (q ) /Iq (in other words, the translogbased index Iq is independent from the scale and parameter changes since these ∗t do not affect the weights sTrg ). Similarly to the remarks 2.1 and 3.1, this remark warns us against interpreting the Törnqvist (translog-based) index number Iq as an aggregate of relative changes in q in the non homothetic case, where no such aggregate really exists and any attempt to construct it ends up to a spurious magnitude. By contrast, in the special case of homothetic separability, the aggregating indexes are clearly deﬁned and isolated from the effects of other components of the functional value changes. The results obtained with the Törnqvist index number parallel those concerning the general quadratic Box-Cox function. Theorem 6.2 If two quadratic Box-Cox functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 differ in parameters in all terms, then

$

fQ1ρ,λ

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧ N ⎪ ⎨ 1 2 ·

%ρ $ %ρ ⎫ ⎬ fQ0ρ,λ fQ1ρ,λ ⎪ 0 λ + 1 λ ⎪ ⎭ qi qi

$ sQ0 ρ,λ i

⎪ i=1 ⎩

λ λ qi1 − qi0 + parameter − changecomponent (6.4)

t where sQρ,λ ≡ i

ρ λ

t qit fQρ,λ i t fQρ,λ

t with t = 0, 1 and fQρ,λ ≡ i

t ∂fQρ,λ

∂qit

The homothetic separability restrictions are not imposed on (6.4) and (6.5) so that the same caveats expressed for the translog case apply here. The weights are not affected by the particular path taken by technical change and returns to scale only if the two functions are homothetically separable in q. Only in this case, is the q-change component fully independent from parameter changes. N t t We note that i=1 sQρ,λ ,i = 1 if fQρ,λ homogeneous of degree one in q, since in this case fQt ρ,λ = Ni=1 qit fQt ρ,λ ,i by Euler’s theorem. If Ni=1 sQt ρ,λ ,i = 1, then we can deﬁne sQ∗tρ,λ ,i ≡ sQt ρ,λ ,i / Ni=1 sQt ρ,λ ,i and rewrite equation (6.4) as follows:

(fQ1ρ,λ )ρ −(fQ0ρ,λ )ρ = Jq +JS +JT

(6.5)

190 Precursor

ρ 0 N (Iq ·fQ0ρ,λ )ρ 1 ∗0 (fQρ,λ ) ∗1 where Jq ≡ sQρ,λi λ +sQρ,λi λ · (qi1 )λ −(qi0 )λ , 0 1 2 i=1 qi qi N (fQ0ρ,λ )ρ Iq ·(fQ0ρ,λ )ρ 1 0 ∗0 1 ∗1 JS ≡ (ξ −1)sQρ,λi λ +(ξ −1)sQρ,λi λ 2 i=1 qi0 qi1 · (qi1 )λ −(qi0 )λ , JT ≡ Parameter −changecomponent

where Iq is the index of functional value change caused by changes in q, ξ 0 ≡ %ρ ! 0 ! 1 $ 1 %ρ ! $ ρ N ρ N 0 0 1 1 1 0 q f and ξ ≡ q f · f · f . f f I q Qρ,λ i=1 i Qρ,λ,i Qρ,λ i=1 i Qρ,λ,i Qρ,λ λ λ Qρ,λ The ﬁrst line in the right-hand side of (6.5) is the contribution of the change in q to the functional value difference, whereas the second line is the contribution of the scale effects. Let us consider, in particular, the following two special cases. First case: General quadratic linear indicators, which are exact for the general quadratic linear function (5.10) with ρ = 1 and λ = ρ/2. Using the decomposition 1

1

1

(qi1 ) 2 − (qi0 ) 2 = (qi1 − qi0 ) ·

1 (qi0 ) 2

1

+ (qi1 ) 2

,

(6.6)

from (6.4), by deﬁning fQt 1 ≡ fQt ρ,λ , we obtain

fQ11 − fQ01 =

⎧ N ⎪ ⎨

1

(qi0 ) 2 fQ01 i

1

(qi1 ) 2 fQ11 i

⎫ ⎪ ⎬

+ (q1 − qi0 ) 1 1 1 1 ⎪ i ⎪ i=1 ⎩ (qi0 ) 2 + (qi1 ) 2 (qi0 ) 2 + (qi1 ) 2 ⎭

+ Parameter − changecomponent

(6.7)

We can call general quadratic linear indicators of absolute functional value differences the indicator given by the right-hand side of (6.7), which is exact for the general quadratic linear function corresponding to (5.10), where ρ = 1 and λ = 1/2, and does not rely on homogeneity and separability restrictions. This is a rather useful result, since it widens the applicability of this type of indicators. This can be contrasted with the same case that was examined under linear homogeneity and separability restrictions by Diewert (2002, pp. 77–80), who showed that the counterpart index number, expressed in ratio terms is the implicit Walsh index number, which in turn is exact for a homogeneous quadratic linear function. Second case: General Konüs-Byushgens indicator, which is exact for a general Konüs-Byushgens function given by (5.10) with ρ = 2 and λ = ρ/2.

The Theory of Exact and Superlative Index Numbers Revisited 191 Using the general decomposition

fQ12

2

% $ % 2 $ − fQ02 = fQ12 − fQ02 · fQ12 + fQ02 ,

(6.8)

dividing through (6.4) by ( fQ12 + fQ02 ) and rearranging terms, we obtain fQ12

− fQ02

=

N

fQ02 i · fQ02 fQ12 + fQ02

i=1

+

fQ12 i · fQ12

(qi1 − qi0 )

fQ12 + fQ02

+ Parameter − changecomponent

(6.9)

where fQt 2 i ≡ ∂fQt 2 /∂qit for t = 0, 1. We can call general Konüs-Byushgens indicators of absolute and relative functional value differences the indicator given by the right-hand side of (6.9), which is exact for a general Konüs-Byushgens quadratic function corresponding to (5.10) where ρ = 2 and λ = 1 and does not rely on homogeneity and separability restrictions. Also this result is useful to widen the applicability of this type of indicators. This can be contrasted with the same case that was examined under linear homogeneity and separability restrictions by Reinsdorf, Diewert, and Ehemann (2000, pp. 4–6) and Diewert (2002, pp. 72–76), who showed that the counterpart index number (expressed in ratio terms) is the Fisher “ideal” index number, which in turn is exact for a homogeneous Konüs-Byushgens quadratic function. Theorem 6.3 If two quadratic Box-Cox functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 (or, equivalently, by (5.10)) differ in parameters in all terms, then, for all qi0 and qi1 , fQ1ρ,λ fQ0ρ,λ

= Iq · IR

(6.13) 1

where, setting r = 2λ and σ t = (1 + ρat0 ) ρ for t = 0, 1, ⎡

r (qi1 ) 2 ∗0 s r ⎢ i=1 Qρ,λ i ⎢ (qi0 ) 2 Iq ≡ ⎢ r ⎢ ⎣ N ∗1 (qi0 ) 2 s r i=1 Qρ,λ i (qi1 ) 2

with

sQ∗tρ,λ i

N

sQt ρ,λ i

≡ N

t i=1 sQρ,λ i

where

⎤ 2r ⎥ ⎥ ⎥ ⎥ ⎦

sQt ρ,λ i

(6.14)

≡

qit (∂fQt ρ,λ /∂qit ) fQt ρ,λ

and IR ≡ [ fQ1ρ,λ /fQ0ρ,λ ]/Iq is a residual index number representing the combined scale and parameter change effects.

192 Precursor We note that no a priori homogeneity and separability conditions are imposed on (6.13). In general, following (3.29), it is straightforward to show that Iq =

fQ0ρ,λ (q1 )

γ

fQ0ρ,λ (q0 )

fQ1ρ,λ (q1 )

1−γ

(6.15)

fQ1ρ,λ (q0 )

with γ taking the particular value of the parameter given by (3.29), where fQt and z t are, respectively, replaced with fQt ρ,λ and qt . If homothetic separability is assumed, then we obtain the following result: Corollary 6.1 If two functions fQ0ρ,λ and fQ1ρ,λ deﬁned by (5.4) with ρ = 0 and λ = 0 (or, equivalently, by (5.10)) are homothetic transformations of the same linearly homogeneous function fQr deﬁned by (5.15) with r = 2λ, then sQ∗tρ,λ i = sQ∗tρ,λ i ≡ (qi ∂fQr /∂qi )/fQr , which are not affected by the particular path taken by parameter changes and returns to scale, and Iq (6.14) is identically equal to fQr (q1 )/fQr (q0 ) for every q 0 and q 1 . Corollary 6.1 is a global version of Byushgens’ (1925) theorem, which was formulated with ρ=2 and λ=ρ/2 and was further discussed by Konüs and Byushgens (1926), Frisch (1936, p. 30), Wald (1939 p. 331), Samuelson (1947, p. 155), Pollak (1971), and Afriat (1972, p. 45) (1977, pp. 141–143) (2005, pp. 177–178). Remark 6.2 on the existence of the aggregator index: If and only if fQ0ρ,λ and fQ0ρ,λ are homothetically separable in q so that fQt ρ,λ ≡ F t [ fQr (q)], where F t (·) is a well behaved transformation function and fQr is the linearly homogeneous function (5.15) with constant parameters, then for all q0 and q1 , si∗t ≡ qit /

1

∂f t ρ,λ (qt ) N Q j=1 ∂qjt 0

∂fQr (qt )/∂qit · qjt for t = 0, 1, in view of (2.4), fQr (qt ) IS · IT = [ fQ1ρ,λ (q1 )/fQ0ρ,λ (q0 )]/Iq (in other words, the

· qjt =

∂f t ρ,λ (qt ) Q

∂qit

·

then Iq =

quadratic fQr (q )/fQr (q ), mean-of-order-r index Iq is independent from the scale and parameter changes since these do not affect the weights sQ∗tρ,λ ). Similarly to the remarks 2.1, 3.1, and 6.1, this remark warns us against interpreting the quadratic mean-of-order-r indexes Jq and Iq as aggregates of absolute and relative changes in q in the non-homothetic case, where no such aggregates really exist and any attempt to construct them ends up to a spurious magnitude. By contrast, in the special case of homothetic separability, the aggregating indexes are clearly deﬁned and isolated from the effects of other components of the functional value changes.

The Theory of Exact and Superlative Index Numbers Revisited 193 Moreover, the quadratic mean-of-order-r index number deﬁned by (6.14) can be contrasted with the following quadratic mean-of-order-r index number deﬁned by Diewert (1976, pp. 130–131): ⎡

N

1

r

0 (qi ) 2 ⎢ si (q0 ) 2r ⎢ i=1 i

IqD ≡ ⎢ N ⎣

r 0 1 (qi ) 2 i 1 r i=1 (qi ) 2

s

⎤ 1r ⎥ ⎥ ⎥ ⎦

(6.16)

where s0 and s1 are actually “observed”. If the unknown true function f is approximated at q0 , then s0 = sQ0 σ,λ , but generally s1 = sQ1 σ,λ . By deﬁning these weights as value shares, formula (6.16) specializes to wellknown index numbers with particular values of r. Diewert (1976, p. 135) (2004, Eq. (17.26), pp. 447–448) noted that, if r = 1, then (6.16) reduces to the implicit Walsh (1901, p. 105) index number, and, if r = 2, then (6.16) reduces to Fisher’s (1922) “ideal” index number. The indexes Iq and IqD differ in the weights used. Even if these two weight systems are numerically equal at the point of approximation, say at t = 0, so that sQ0 r = s0 , they may be substantially different at other points under comparison. This is sufﬁcient to make the two index numbers (6.14) and (6.16) rather different. In order to ensure that sQ0 r be equal to s0 at all values of r, the parameters of the underlying aggregator function should consistently adjust to the changes in r. Consequently, sQ1 r is also a function of r. This means that, using the same s1 with different values of r contradicts the assumption of second-order approximation that is supposed to be provided by IqD . It is, therefore, Iq , rather than IqD , that must be called “superlative” according to the meaning assigned by Diewert (1976, p. 117) to this term. This will be shown more analytically and illustrated by a numerical example in the remainder of this section. The function fQr (q) deﬁned by (5.15) provides a second-order differential approximation to an arbitrary function f (q) around q0 if f (q0 ) = fQr (q0 )

(6.17)

∇f (q0 ) = ∇fQr (q0 )

(6.18)

∇ 2 f (q0 ) = ∇ 2 fQr (q0 )

(6.19)

The equations (6.17)-(6.19) are satisﬁed for certain values of the parameters of fQr for a given r, as established by the following result: Theorem 6.4 If the aggregator function fQr (q) deﬁned by (5.15) provides a second-order approximation to the arbitrary aggregator function f (q) around q0 ,

194 Precursor then its parameters are equal to: αij (r, q0 ) =

ff fij − 1−r f i j r 2

r

f 1−r (qi0 qj0 ) 2 −1 r

fi − (qi0 ) 2 −1 αij (r, q0 ) =

f

N

for i = j,

i, j = 1, 2, . . . N

(6.20)

for i = 1, 2, . . . N

(6.21)

r

f 1−r αij (qj0 ) 2

j =i 1−r 0 r−1 (qi )

where f , fi , and fij denote, respectively, f (q), ∂f (q)/∂qi , and ∂ 2 f (q)/∂qi ∂qj evaluated at q0 . Remark 6.3 For each value r = r ∗ at q0 , there is a set of particular parameters αij = αij (r ∗ , q0 ) such that the function deﬁned by (5.15) satisﬁes (6.17)–(6.19). The system (6.17)–(6.19) is made of 1 + N + N 2 equations with N (N + 1)/2 independent parameters αij ’s of fQr . However, since the Hessian matrices are typically symmetric by Young’s theorem (stating that the values of second-order derivatives are not affected by the sequence of derivation) there are N (N − 1)/2 redundant equations in the subsystem (6.19). Moreover, both f and fQr are assumed to be homogeneous of degree one in q, hence f (q) = q∇f (q) and fQr (q) = q∇fQr (q) by Euler’s theorem. Then, the condition f (q) = q∇fQr (q)

(6.22)

makes one of the equations (6.18) redundant which adds up to the total number of redundant equations by raising it to 1 + N (N − 1)/2. The linear homogeneity of f and fQr brings about also, by Euler’s theorem, q∇ 2 f (q) = 0 and q∇ 2 fQr (q) = 0 and the following additional N conditions q∇ 2 f (q) = 0 = q∇ 2 fQr (q) = 0

(6.23)

which raise, in turn, the number of the redundant equations in the system (6.17)– (6.19) by a further N bringing it to a total of 1+N (N +1)/2. Therefore, the number of independent equations is N (N + 1)/2, which is the same of the independent parameters aij ’s in the function fQr (q)24 . The resulting functions (6.20) and (6.21) determining the coefﬁcients αij ’s are homogeneous of degree zero since, by linear homogeneity, μf (q) = f (μq), fi (q) = fi (μq), and μ1 fij (q) = fij (μq). 24 The obtained coefﬁcients αij satisfy the restriction i j αij = 1 if, together with the system (B.78)–(B.80) under the conditions (B.81) and (B.82), the values of the arguments are normalized appropriately.

The Theory of Exact and Superlative Index Numbers Revisited 195 It is immediate to note that all the functions fQr (q) deﬁned over the domain of r approximate each other up to the second order. Consequently, the corresponding “exact” index numbers deﬁned by (6.14) also approximate each other up to the second order. This is ensured by the adjustment of the shares sQt r to the value of r. It is important to remark, however, that, while the value changes of a fully ﬂexible quadratic function can be accounted for exactly with the index number (6.14), the converse is not true since also the value changes of a “rigid” CES-type functional form or even a polynomial function having all the second derivatives equal to zero may be accounted for with (6.14). The “superlativeness” of the index number (6.14) is, therefore, related only to its potential agreement up to the second derivatives of the unknown true function through the approximating function for which it is exact. In fact, nothing guarantees that the approximating function actually provides a second-order approximation if the second-order derivatives fij remain unobserved and may consistently have any numerical value (including zero) (see also Uebe, 1978 for a similar remark). Furthermore, in general, f (q1 ) = fQr (q1 )

(6.24)

∇f (q1 ) = ∇fQr (q1 )

(6.25)

∇ 2 f (q1 ) = ∇ 2 fQr (q1 )

(6.26)

Therefore, s1 = sQ1 r

(6.27) 1

where si1 ≡ qi1 ∂f∂q(q1 ) /f (q1 ) and sQ1 r i ≡ qi1 i

∂fQr (q1 ) ∂qi1

/fQr (q1 ).

When r changes, say from r ∗ to r ∗∗ , then ∇fQr∗ (q1 ) = ∇fQr∗∗ (q1 )

(6.28)

where fQr (q) is deﬁned by (5.15), and, consequently, s1 = sQ1 r∗ = sQ1 r∗∗

(6.29)

If the same observed shares st are used with different values of r, as in (6.16) instead of the shares sQt r required for the second-order approximation, then the resulting formulas are hybrid and cannot be interpreted as superlative index numbers providing second-order differential approximations to an arbitrary function. These last index numbers are instead deﬁned, in the homothetic case, by ⎡

N

⎤ 1r

r

1

(q ) 2 si0 i0 r 2 (q i=1 i)

⎢ ⎢ Iqs ≡ ⎢ N ⎣

0

r

(q ) 2 sQI r i i1 r (qi ) 2 i=1

⎥ ⎥ ⎥ ⎦

(6.30)

196 Precursor 0)

where si0 ≡ qi0 ∂f∂q(qi

!

f (q0 ) = sQ0 ri ≡ qi0

∂fQr (q0 ) ! fQr (q0 ) and sQ1 ri ∂qi

≡ qi1

∂fQr (q1 ) ! ∂qi1

fQr (q1 )

with fQr (q) being deﬁned by (5.15). The following example is a numerical representation of the differences between the two types of index numbers. Example 6.1. Let us consider the case of three elements (N =3) with qi0 = 1.0 for i = 1, 2, 3 and q11 = 2.0, q21 = 1.7, and q31 = 1.05. Moreover, let us assume that f (q0 ) = 1.0

(6.31)

and the ﬁrst derivatives of the unknown function f (q0 ) are the following: f1 (q0 ) ≡

∂f (q0 ) ∂f (q0 ) ∂f (q0 ) = 0.53, f2 (q0 ) ≡ = 0.37, f3 (q0 ) ≡ = 0.10 ∂q1 ∂q2 ∂q3 (6.32)

At the same observation point q0 , let us assume that the Hessian matrix is the following: ⎡ ⎤ 0.0376 − 0.0088 − 0.0288 (6.33) ∇ 2 f (q0 ) = ⎣ −0.0088 0.0344 − 0.0256 ⎦ −0.0288 − 0.0256 0.0544 At the point t = 0 and t = 1, the weights sit are assumed to be s10 ≡ (q10 · f1 )/f = 0.53

s11 ≡ (q11 · f1 )/f = 0.69

(6.34)

s20 ≡ (q20 · f2 )/f = 0.37

s21 ≡ (q21 · f2 )/f = 0.30

(6.35)

s30

≡ (q30 · f3 )/f

= 0.10

s31

≡ (q31 · f3 )/f

= 0.01

(6.36)

Let us now assume also that a quadratic mean-of-order-r aggregator function of the type deﬁned by (5.15) has such parameters αij∗ for a given r* that it provides a second-order differential approximation to f at q0 . Table 2 tabulates the values of these parameters in correspondence of certain values of r, so that the quadratic mean-of-order-r aggregator functions fQr approximate f at q0 up to the second order. Note that sQ0 r = s0 by assumption and sQ1 r is derived using the speciﬁc parameter values. The corresponding “exact” index numbers Iqs are compared with the hybrid Diewert’s index numbers IqD , which are constructed using the same “observed” shares for different values of r. In this numerical example, the two index numbers diverge increasingly from one another as r increases. The “really” superlative index numbers tend to increase as r increases after a certain minimum turning point, whereas the “hybrid” index numbers tend to decrease as r increases after a certain maximum turning point. Both the “exact” and “hybrid”

The Theory of Exact and Superlative Index Numbers Revisited 197 r=1

r=2

r=4

r = 10

r = 20

r = 40

r = 100

r = 1000

r=∞

Parameters of the underlying 2nd-order approximating “quadratic mean-of-order-r ” function (eqs. (6.20) and (6.21)) a 11 0.6052 0.3185 0.1751 0.0891 0.0605 0.0461 0.0375 0.0324 0.0318 a 22

0.4388

0.1713

0.0376

–0.0427

–0.0694

–0.0828

–0.0908

–0.0957

–0.0962

a 33

0.2088

0.0644

–0.0078

–0.0511

–0.0656

–0.0728

–0.0071

–0.0797

–0.0800

a 12 = a 21

–0.0176

0.1873

0.2897

0.3512

0.3717

0.3820

0.3881

0.3918

0.3922

a 13 = a 31

–0.0576

0.0242

0.0651

0.0896

0.0978

0.1019

0.1044

0.1058

0.1060

a 23 = a 32

–0.0512

0.0114

0.0427

0.0615

0.0677

0.0709

0.0727

0.0739

0.0740

1

sQ r,1

Current period weights with the 2nd-order approximating “quadratic mean-of-order-r” function (eqn. 6.27)) 0.9970 0.6057 0.6025 0.6045 0.6188 0.6544 0.8058 1.0000 1.0000

s1Q r,2

0.3570

0.3339

0.3567

0.3698

0.3448

0.1942

0.0030

0.0000

0.0000

s1Q r,3

0.0373

0.0437

0.0388

0.0114

0.0008

0.0000

0.0000

0.0000

0.0000

I sq

Index number which is exact for a 2nd-order approximating “quadratic mean-of-order-r ” function (6.30) 1.8069 1.8044 1.8066 1.8256 1.8474 1.8750 1.9355 1.9932 2.0000

Current period weights “observed” on the true unknown function (6.34)–(6.36) 0.6900 0.6900 0.6900 0.6900 0.6900 0.6900 0.6900

s11

0.6900

s1 2

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

0.3000

s1

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

0.0100

Diewert’s (1976, pp. 130-131) “quadratic mean-of-order-r ” index number (6.16) 1.8336 1.8381 1.8440 1.8389 1.7522 1.6013 1.5078 1.4549

1.4491

3

D

Iq

0.6900

Figure 1 Table 2 – Parameters, weights, and numerical values of quadratic mean-of-order-r index numbers

index numbers have asymptotic values as r → ∞. The former tends to a numerical value equal to 2.0, which is higher than that obtained at r = 1 by more than 10 per cent, whereas the latter tends to a value equal to 1.44914, which is lower than that obtained at r = 1 by more than 20 per cent (see Table 2). These results show that the theory of exact and superlative index numbers has to be widely reconsidered and shed a new light on the apparently paradoxical ﬁndings recently obtained by Hill (2006).

7 Conclusion The theory of exact and superlative index numbers has been critically re-examined here. It has turned out that superlative index numbers are rarely applicable in practice. In fact, these are deﬁned by using weights derived from functions for which they are “exact”. If, instead, the weights are derived from observed data, the result is that of hybrid formulas that may be far from being really superlative. We conclude that, with the “observed” data normally available, it is not possible to rely on the second-order approximation paradigm. Therefore, it would be more appropriate to construct a range of alternative index numbers that are all valid

198 Precursor candidates as measures of the unknown true index number, when this exists, rather than search for only one optimal formula. One promising method would be to test the observed data for consistency with an aggregator function and, in the positive case, to proceed to calculate upper and lower bounds of the unknown true index.

Appendix A: Observable economic variables The propositions presented thus far are theorems in numerical analysis rather than economics. The numerical values of the variables involved are assumed to be either known or derivable from some sources of information. In economics, the contexts where index numbers and indicators are usually applied are those regarding production and consumption activities. The index numbers and indicators are therefore deﬁned with reference to production or transformation and utility functions and their dual counterparts represented by value functions such as cost, revenue and proﬁt functions. All these functions are characterized by certain properties. The following cases are examined. (i) The case of the production function If the function f t (x) is homogeneous of degree 1 or less than 1 and represent a transformation or production function characterized by the usual regularity properties so that y = f t (x)

(A.1)

where y is a scalar measure of the output quantity, and x is a vector of input quantities, then the ﬁrst-order conditions for proﬁt maximization imply (dropping the superscript t to simplify notation) ∂f w = i ∂xi p

for all i s

(A.2)

where p is a scalar measure of the output price. Therefore,

si ≡

∂f xi wi · xi = ∂xi y p·y

(A.3)

Under constant and perfect competition, p · y = Ni=1 wi · xi , so that returns to scale N N si = wi · xi i=1 wi · xi and i=1 si = 1. N In the general case where p · y > i=1 wi · xi , if we want to construct sep< arately index numbers or indicators of differences in quantities and returns

The Theory of Exact and Superlative Index Numbers Revisited 199 to scale, then it might be useful to decompose the foregoing weights as follows: w ·x w ·x si = N i i + N i i · (ξ − 1) i=1 wi · xi i=1 wi · xi

(A.4)

! where ξ ≡ Ni=1 wi · xi p · y represents the degree of the returns to scale. It is obtained using the additional information on output prices and quantities. The second term of the right-hand side of equation (A.4) represents the weight capturing the effect of returns to scale (see Caves, Christensen, and Diewert, 1982, pp. 1405–1406 and p. 1408 for a similar decomposition25 ). Since prices and quantities of inputs and outputs are usually observable, all the necessary information for the calculation of index numbers and related indicators is available. (ii) The case of the utility function Let the function f t (x) represent a utility function characterized by the usual regularity properties so that u = f t (x)

(A.5)

where u represents utility, and x is a vector of the quantities consumed. Utility is typically unobserved, so that at least this important variable is not available for the index number construction. However, if the consumer has a utility-maximizing behavior subject to a budget constraint, then the ﬁrst-order conditions imply (dropping the superscript t to simplify notation)] ∂f = λwi for all i s ∂xi

(A.6)

We sum these conditions multiplied by quantities in order to obtain: N N ∂f xi = λ wi x i ∂xi i=1 i=1

(A.7)

which can be solved for the Lagrange multiplier as follows: N

λ=

i=1 N

∂f x ∂xi i

(A.8) w i xi

i=1

25 The variable parameter ξ (in our notation) is obtained with equation (47) in Caves et. al. (1982, p. 1406), where it is, however, misprinted (the correct form is reported at page 1408 of the same article).

200 Precursor Substituting the Lagrange multiplier back into the ﬁrst-order conditions yields: ⎞ ⎛ N ∂f ⎜ ∂xi xi ⎟ ∂f ⎟ ⎜ i=1 =⎜ n (A.9) ⎟ wi for all i s ⎠ ∂xi ⎝ w i xi i=1

from which we derive the Hotelling (1935, p.71)-Wold (1944, pp. 69–71; 1953, p. 145) identity ∂f 1 wi for all i s N ∂f = N ∂xi i=1 xi w x i=1 i i ∂xi

(A.10)

If we assume linear homogeneity of the utility function marginal (implying ∂f utility of income equal to 1), then, by Euler’s theorem, Ni=1 ∂x x = f (x) and, i i consequently, the foregoing equation becomes ∂f x ∂xi i

si ≡

f (x)

wx − N i i i=1 wi xi

for all i s

(A.11)

(iii) The case of the value functions (cost, revenue, and proﬁt functions) Let a value function (a cost, or revenue, or proﬁt function) be represented by a differentiable function et (p) , where p is a vector of prices. By Hicks (1946, p. 331)-Samuelson (1947, p. 68)-Shephard (1953, p. 11)-Hotelling’s lemma (1932, p. 594), we obtain directly the optimal levels of quantities through differentiation (dropping the superscript t to simplify notation): qi =

∂e ∂pi

(A.12)

where qi is the ith element of an N -dimensional vector of quantities. The value function is always linearly homogeneous in prices by construction. By Euler’s theorem, e (p) = Ni=1 pi qi . Dividing both sides of equation (A.12) by e(p) yields N

qi

i=1 pi qi

=

∂e ∂pi

e (p)

for all i s

(A.13)

Multiplying the foregoing equations by the respective prices pi ’s yields ∂e pi ∂p pi qi i = = si N (p) e p q i=1 i i

for all i s

(A.14)

Here, again, si and the numerical value of e(p) can be calculated using observed data on prices and quantities.

The Theory of Exact and Superlative Index Numbers Revisited 201

Appendix B: Proofs of theorems Proof of Lemma (2.1) (Accounting for Functional Value Differences). Let us consider an arbitrary function f (z) of one single variable. From the Taylor series expansion for f around z 0 , the following equation can be obtained: 2 1 f z 1 − f z 0 = f z 0 z 1 − z 0 + f z 0 z 1 − z 0 + . . . 2! n 1 (n) 0 1 + f z z − z 0 + R0n z 0 , z 1 n! n 1 m = f (m) z 0 z 1 − z 0 + R0n z 0 , z 1 m! m=1

(B.1)

where R0n z 0 , z 1 is the remainder term. Similarly, from the Taylor series expansion for f around z 1 , the following equation can be obtained: 1 2 f z 0 − f z 1 = f z 1 z 0 − z 1 + f z 1 z 0 − z 1 + . . . 2! n 1 (n) 1 0 z z − z 1 + R1n z 0 , z 1 + f n! n m 1 (m) 1 0 = z z − z 1 + R1n z 0 , z 1 f m! m=1

(B.2)

Multiplying through the foregoing equation by –1 and rearranging terms yield n m 1 (−1)m f (m) z 1 z 1 − z 0 − R1n z 0 , z 1 f z1 − f z0 = − m! m=1

(B.3)

Using (1 − θ) and θ as weights, the weighted average of (B.1) and (B.3) is given by

n 1 m 1 1 (1 − θ ) f (m) z 0 − (−1)m θ f (m) z 1 f z1 − f z0 = z − z0 m! m! m=1 (B.4) + (1 − θ ) R0n z 0 , z 1 − θ R1n z 0 , z 1 From (B.4), it follows that f z 1 − f z 0 = (1 − θ ) f z 0 + θf z 1 z 1 − z 0 + (1 − θ ) R01 z 0 , z 1 − θ R11 z 0 , z 1

(B.5)

Let us choose a value of θ, say θ ∗ , that minimizes the squared term (1 − θ ) 2 R01 z 0 , z 1 − θ R11 z 0 , z 1 . Since this term is convex in θ , the necessary and

202 Precursor sufﬁcient condition for its minimization is that its ﬁrst derivative with respect to θ vanishes26 , so that −2 (1 − θ ) R01 z 0 , z 1 − θ R11 z 0 , z 1 R01 z 0 , z 1 + R11 z 0 , z 1 = 0

(B.6)

from which, provided that R01 z 0 , z 1 + R11 z 0 , z 1 = 0, θ =θ

∗

z ,z 0

1

R01 z 0 , z 1 ≡ 0 0 1 R1 z , z + R11 z 0 , z 1

(B.7)

The case of many variables follows in a similar manner using the directional derivatives. In particular, f z 1 − f z 0 = (1 − θ ) ∇f z 0 + θ ∇f z 1 z 1 − z 0

(B.8)

where, if R01 z 0 , z 1 + R11 z 0 , z 1 = 0, then θ = θ ∗ (so that (1 − θ ∗ ) R01 z 0 , z 1 − (z 1 ) (z 0 ) θ ∗ R11 z 0 , z 1 = 0), or, if R1 = R1 = 0, then θ may take any value as a number. Proof of Corollary (2.1) (Decomposition of Functional Value Changes). Adding and subtracting JZ from (2.1) yield directly the decomposition (2.2). Since i si∗t = 1, the meaning of JZ is that of a candidate aggregate of absolute changes in the vector z. Let us deﬁne the functional value IZ · f z 0 = f z 0 + JZ . From (2.2) it is possible to derive IZ · f z 0 − f z 0 = JZ =

f (1 − θ ) si∗0

0 z zi0

IZ + θ si∗1

· f z0 · zi1 − zi0 1 zi (B.9)

Dividing through (B.9) by f (z 0 ) yields IZ − 1 = (1 − θ) + θ IZ

N

z1 si∗0 i0 zi i=1

N i=1

− (1 − θ )

si∗1 − θ IZ

N

si∗0

i=1 N i=1

si∗1

zi0 zi1

(B.10)

$ % 26 The second-order condition for minimization is always respected since ∂ 2 (1 − θ ) R01 z 0 , z 1 − $ % 2 % $ % 2 $ θ R11 z 0 , z 1 /∂θ = 2 R01 z 0 , z 1 + R11 z 0 , z 1 > 0.

The Theory of Exact and Superlative Index Numbers Revisited 203 = (1 − θ )

N

si∗0

i=1

+ θ IZ − θ IZ

zi1 − (1 − θ ) zi0

N

si∗1

i=1

given that

IZ =

N

∗t i=1 si

zi0 zi1

(B.11)

= 1. Rearranging (B.11) and solving for IZ yield

θ + (1 − θ) (1 − θ) + θ

N

1

∗0 zi i=1 si z 0

N

i 0

∗1 zi i=1 si z 1

(B.12)

i

Proof of Corollary (2.2) (Diewert’s, 1976, p. 117, Quadratic Identity). Sufﬁciency: From the deﬁnition (2.5) of the quadratic function fQ , the ﬁrst difference of this function is obtained as follows: 1 fQ z 1 − fQ z 0 = a0 + az 1 + z 1 Az 1 2 1 − a0 − az 0 − z 0 Az 0 2

(B.13)

By adding 12 z 0 Az 1 and subtracting 12 z 1 Az 0 (recall that z 0 Az 1 = z 1 Az 0 since aij = aji for all i, j in A) and substituting 12 z 0 Az 1 with z 0 Az 1 − 12 z 0 Az 1 and 12 z 0 Az 0 with 0 0 1 0 0 z Az − 2 z Az , equation (B.13) can be rearranged into the following fQ z 1 − fQ z 0 = a + z 0 A z 1 − z 0 +

1 1 z − z0 A z1 − z0 2

(B.14)

which can also be derived directly from the Taylor series expansion of fQ (z) around the point z 0 , since ∇fQ z 0 = a + z 0 A ∇ 2 fQ z 0 = A

(B.15) (B.16)

Similarly, from 1 1 fQ z 0 − fQ z 1 = a0 + az 0 + z 0 Az 0 − a0 − az 1 − z 1 Az 1 2 2

(B.17)

by adding 12 z 1 Az 0 and subtracting 12 z 0 Az 1 (recall, again, that z 1 Az 0 =z 0 Az 1 since aij =aji for all i, j in A) and substituting 12 z 1 Az 0 with z 1 Az 0 − 12 z 1 Az 0 and

204 Precursor 1 1 z Az 1 with z 1 Az 1 − 12 z 1 Az 1 , equation (B. 17) can be rearranged into the 2 following 1 fQ z 0 − fQ z 1 = a + z 1 A z 0 − z 1 + z 0 − z 1 A z 0 − z 1 2

(B.18)

which can also be derived directly from the Taylor series expansion of fQ (z) around the point z 1 , since ∇fQ z 1 = a + z 1 A (B.19) (B.20) ∇ 2 fQ z 1 = A Multiplying through (B.18) by –1 and rearranging terms yields 1 fQ z 1 − fQ z 0 = a + z 1 A z 1 − z 0 − z 1 − z 0 A z 1 − z 0 2

(B.21)

The arithmetic average of (B.14) and (B.21) is given by 1 fQ z 1 − fQ z 0 = a + z0 A + a + z1 A z1 − z0 2 1 = ∇fQ z 0 + ∇fQ z 1 z 1 − z 0 2

(B.22)

Necessity: Let us start from the following equation, f (x) − f (y) =

1 [∇f (x) + ∇f (y)] (x − y) 2

(B.23)

Assume f (x) is thrice-differentiable and satisﬁes the functional equation f (x)–f (y) for all x and y, in an open neighbourhood. If f (x) is a function of one variable, the functional equation becomes f (x) − f (y) =

1 f (x) + f (y) (x − y) 2

(B.23 bis)

Let us differentiate the foregoing equation with respect to x in order to obtain f (x) =

1 1 f (x) + f (y) + f (x)(x − y) 2 2

(B.24)

The second derivative of f (x) with respect to x is then 1 1 1 f (x) = f (x) + f (x) + f (x)(x − y) 2 2 2

(B.25)

which means that 12 f (x)(x − y) = 0 and, therefore, f (x) = 0 for every x = y, implying that f (x) can be traced perfectly by a polynomial of degree two.

The Theory of Exact and Superlative Index Numbers Revisited 205 The general multivariate case follows in a similar manner using the partial derivatives. Proof of Lemma (2.2) (General Quadratic Approximation Lemma). Taking into account identity (2.1), from (2.9) we obtain Error of approximation = (1 − θ) [∇f (z 0 ) + θ∇f z 1 ] (z 1 − z 0 ) 1 − [∇f (z 0 ) + ∇f (z 1 )](z 1 − z 0 ) 2

1 = − θ [∇f (z 0 ) − ∇f (z 1 )](z 1 − z 0 ) 2

(B.26)

The ﬁrst-order approximating (linear) function that is tangent to f (z) at z t is given by fLt (z) = f (z t ) + ∇z f (z t )(z − z t ) = at + bt z (with at = f (xt ) − ∇z f (xt )xt and bt = ∇z f (xt )). Therefore fLt (z) − fLt (z t ) = ∇f (z t )(z − z t )

(B.27)

Using (B.27) with t = 0, 1 and z = z 0 , z 1 , equation (B.26) becomes

0 1 1 Error of approximation = −θ fL (z ) − fL0 (z 0 ) − fL1 (z 1 ) − fL1 (z 0 ) 2 (B.28) Proof of Lemma (3.1) (Accounting for Value Differences of Two Functions with Different Parameters or Functional Forms). Let us two arbitrary functions of one single variable, f 0 (z) and f 1 (z). These functions may differ in parameter values or even in their functional forms. From the Taylor series expansion for f 0 (z 1 ) around z 0 , the following equation may be obtained: f 0 (z 1 ) − f 0 (z 0 ) = f 0 (z 0 )(z 1 − z 0 ) + R01

(B.29)

where R01 = R01 (z 0 , z 1 ) is the remainder term of the ﬁrst-order approximation. Adding and subtracting f 1 (z 1 ) and rearranging terms, the foregoing equation becomes: f 1 (z 1 ) − f 0 (z 0 ) = f 0 (z 0 )(z 1 − z 0 ) + f 1 (z 1 ) − f 0 (z 1 ) + R01 (B.30) Similarly, from the Taylor series expansion for f 1 (z 0 ) around z 1 , the following equation may be obtained: f 1 (z 0 ) − f 1 (z 1 ) = f 1 (z 1 )(z 0 − z 1 ) + R11

(B.31)

where R11 = R11 (z 0 , z 1 ) is the remainder term of the ﬁrst-order approximation. By multiplying both sides by –1 and rearranging terms, equation (B.31) becomes: f 1 (z 1 ) − f 1 (z 0 ) = f 1 (z 1 )(z 1 − z 0 ) − R11

(B.32)

206 Precursor Adding and subtracting f 0 (z 0 ) and rearranging terms, the foregoing equation becomes: f 1 (z 1 ) − f 0 (z 0 ) = f 1 (z 0 )(z 1 − z 0 ) + [ f 1´(z 0 ) − f 0 (z 0 )] − R11

(B.33)

Using (1 − θ ) and θ as weights, where θ may take any real number as a value, the weighted average of (B.30) and (B.33) is given by f 1 (z 1 ) − f 0 (z 0 ) = (1 − θ ) f 0 (z 0 ) + θf 1 (z 0 ) (z 1 − z 0 ) + θ f 1 (z 0 ) − f 0 (z 0 ) + (1 − θ ) f 1 (z 1 ) − f 0 (z 1 ) + (1 − θ) R01 − θ R11 (B.34) If θ = θ ∗ , with θ ∗ = θ ∗ (z 0 , z 1 ) ≡

R01

R01 +R11

, then

f1 (z 1 ) − f 0 (z 0 ) = (1 − θ)f 0 (z 0 ) + θ f 1 (z 0 ) (z 1 − z 0 ) + Parameter − change componenr (PC)

(B.35)

where PC ≡ θ ∗ f 1 (z 0 ) − f 0 (z 0 ) + (1 − θ ∗ ) f 1 (z 1 ) − f 0 (z 1 )

(B.36)

The case of many variables follows in a similar manner using the directional derivatives, thus obtaining (3.2)–(3.3). Proof of Corollary (3.1) (Decomposition of Functional Value Differences Between Two Arbitrary Differentiable Functions). The proof of Corollary (3.1) is analogous to that of Corollary (2.1). Proof of Corollary (3.2) (Accounting for Value Differences of Two Quadratic Functions Differing in Parameters). The proof of Corollary (3.2) follows the footsteps of the proof of the “sufﬁciency” part of Corollary (2.2). From the deﬁnition (3.8) of the quadratic function fQt , it is straightforward to obtain: fQ1 (z 1 ) − fQ0 (z 0 ) = (a0 + z 0 A0 )(z 1 − z 0 ) + (a10 − a00 ) + (a1 − a0 )z 1 1 − z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.37)

since fQ1 (z 1 ) − fQ0 (z 0 ) = fQ0 (z 1 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 )

(B.38)

The Theory of Exact and Superlative Index Numbers Revisited 207 where 1 fQ0 (z 1 ) − fQ0 (z 0 ) = (a0 + z 0 A0 )(z 1 − z 0 ) + (z 1 − z 0 )A0 (z 1 − z 0 ), 2

(B.39)

using the Taylor series expansion for f 0 around z 0 , 1 fQ1 (z 1 ) − fQ0 (z 1 ) = (a10 − a00 ) + (a1 − a0 )z 1 + z 1 (A1 − A0 )z 1 2

(B.40)

and 1 1 1 1 (z − z 0 )A0 (z 1 − z 0 ) + z 1 (A1 − A0 )z 1 = −z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2 2 2 (B.41) Similarly, fQ0 (z 0 ) − fQ1 (z 1 ) = (a1 + z 1 A1 )(z 0 − z 1 ) + (a00 − a10 ) + (a0 − a1 )z 0 1 − z 0 A1 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.42)

Since fQ0 (z 0 ) − fQ1 (z 1 ) = fQ1 (z 0 ) − fQ1 (z 1 ) + fQ0 (z 0 ) − fQ1 (z 0 )

(B.43)

where 1 fQ1 (z 0 ) − fQ1 (z 1 ) = (a1 + z 1 A1 )(z 0 − z 1 ) + (z 0 − z 1 )A1 (z 0 − z 1 ) 2

(B.44)

using the Taylor series expansion for f 1 around z 1 1 fQ0 (z 0 ) − fQ1 (z 0 ) = (a00 − a10 ) + (a0 − a1 )z 0 + z 0 (A0 − A1 )z 0 2

(B.45)

and 1 1 1 1 (z − z 0 )A0 (z 1 − z 0 ) + z 1 (A1 − A0 )z 1 = −z 0 A0 z 1 + (z 0 A0 z 0 + z 1 A1 z 1 ) 2 2 2 (B.46) Multiplying both sides of (B.42) by –1 and rearranging terms yield fQ1 (z 1 ) − fQ0 (z 0 ) = (a1 + z 1 A1 )(z 1 − z 0 ) + (a10 − a00 ) + (a1 − a0 )z 0 1 + z 0 A1 z 1 − (z 0 A0 z 0 + z 1 A1 z 1 ) 2

(B.47)

208 Precursor The arithmetic average of (B.37) and (B.47) is therefore fQ1 (z 1 ) − fQ0 (z 0 ) =

1 0 0 ∇fQ (z ) + ∇fQ1 (z 1 ) (z 1 − z 0 ) 2 1 1 + (a00 − a10 ) + (a0 − a1 ) (z 0 + z 1 ) + z 0 (A1 − A0 )z 1 2 2 (B.48)

since ∇fQ0 (z 0 ) = a0 + z 0 A0 , ∇fQ1 (z 1 ) = a1 + z 1 A1 . Moreover, by deﬁnition, 1 1 a00 − a10 = (α01 + k 1 β 1 + k 1 B1 k 1 ) − (α00 + k 0 β 0 + k 0 B0 k 0 ) 2 2 (B.49) 1 1 (a0 − a1 ) (z 0 + z 1 ) = (α 1 + k 1 1 − α 0 − k 0 0 ) (z 0 + z 1 ) 2 2

(B.50)

The sum of (B.49) and (B.50) is 1 1 (a00 − a10 ) + (a0 − a1 ) (z 0 + z 1 ) = (α00 + α01 ) + (α 0 − α 1 ) (z 0 + z 1 ) 2 2 1 + (k 1 β 1 + k 1 B1 k 1 + k 1 1 z 1 + k 1 1 z 0 ) 2 − k 0 β 0 − k 0 B0 k 0 − k 0 0 z 1 − k 0 0 z 0 1 + (k 1 β 1 − k 0 β 0 ) 2

(B.51)

Adding and subtracting 12 k 0 β 1 , 12 k 1 β 0 , 12 k 0 B1 k 1 , and 12 k 0 B0 k 1 to the right-hand side of (B.51) yield (3.17). Proof of Lemma (3.2) (Quadratic Approximation of Value Differences between Two Arbitrary Functions with Different Parameters or Functional Forms). The proof of Lemma (3.2) follows the proof of Lemma (2.2). Proof of Corollary (3.3) (Accounting for the Sum of Value Differences between Two Quadratic Functions with Different “Zero-order” and “First-order” Parameters) (Caves, Christensen, and Diewert’s, 1982, pp.1412–1413 Translog Identity). Let the quadratic function htQ (z, k t ) be deﬁned by (3.9). By the Quadratic Identity (Corollary 2.2), hQt (z 1 , k t ) − hQt (z 0 , k t ) =

1 ∇ h t (z 0 , k t ) + ∇z hQt (z 1 , k t ) · (z 1 − z 0 ) (B.52) 2 z Q

The Theory of Exact and Superlative Index Numbers Revisited 209 and, therefore, the ﬁrst line of the right-hand side of equation (3.20) becomes, setting λ = 1/2, for t = 0 and t = 1, 1 1 0 1 0 h z , k − h0Q z 0 , k 0 + h1Q z 1 , k 1 − h1Q z 0 , k 1 2 Q 2 1 1 z − z0 = ∇z h0Q z 0 , k 0 + ∇z h0Q z 1 , k 0 2 1 1 + ∇z h1Q z 0 , k 1 + ∇z h1Q z 1 , k 1 z − z0 2

(B.53)

Since ∇z h0Q (z 0 , k 0 ) = a0 + z 0 A

(B.54)

∇z h0Q (z 1 , k 0 ) = a0 + z 1 A

(B.55)

∇z h1Q (z 0 , k 1 ) = a1 + z 0 A

(B.56)

∇z h1Q (z 1 , k 1 ) = a1 + z 1 A

(B.57)

where at = α t + k t t , for t = 0, 1, ∇z h0Q (z 1 , k 0 ) + ∇z h1Q (z 0 , k 1 ) = (a0 + z 1 A) + (a1 + z 0 A) = (a0 + z 0 A) + (a1 + z 1 A) = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 )

(B.58)

Using (B.58), equation (B.53) becomes 1 1 0 1 0 hQ (z , k ) − h0Q (z 0 , k 0 ) + h1Q (z 1 , k 1 ) − h1Q (z 0 , k 1 ) 2 2 1 = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 ) (z 1 − z 0 ) 2

(B.59)

which, multiplied through by 2, is equation (3.22). Equation (3.21) follows directly using deﬁnitions (3.8) and (3.9) Proof of Corollary (3.4) (Accounting for the Sum of Value Differences of Two Quadratic Functions with Different Parameters) (Caves, Christensen, and Diewert, 1982, pp. 1412–1413). By the Quadratic Identity

1 fQ0 (z 1 ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ1 (z 0 ) = ∇z fQ0 z 0 ) + ∇2 fQ0 (z 1 ) (z 1 − z 0 ) 2 1 + ∇z fQ1 z 0 ) + ∇2 fQ1 (z 1 ) (z 1 − z 0 ) 2 (B.60)

210 Precursor

Adding and subtracting 12 ∇z fQ0 (z 0 ) + ∇z fQ1 (z 1 ) (z 1 −z 0 ) to the foregoing equation and taking into account the deﬁnition of fQt yield 0 1 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ1 (z 0 ) = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) 1 + (a0 + z 1 A0 + a1 + z 0 A1 − a0 − z 0 A0 − a1 − z 1 A1 )(z 1 − z 0 ) 2 = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) 1 + (z 1 A0 z 1 + z 0 A1 z 1 − z 0 A0 z 1 − z 1 A1 z 1 − z 1 A0 z 0 − z 0 A1 z 0 2 + z 0 A0 z 0 + z 1 A 1 z 0 ) 1 = ∇z fQ0 (z 0 ) + ∇z fQ0 (z 1 ) (z 1 − z 0 ) − (z 1 − z 0 )(A1 − A0 )(z 1 − z 0 ) 2 (B.61) or

h0Q (z 1 , k 0 ) − h0Q (z 0 , k 0 ) + h1Q (z 1 , k 1 ) − h1Q (z 0 , k 1 ) = ∇z h0Q (z 0 , k 0 ) + ∇z h1Q (z 1 , k 1 ) (z 1 − z 0 ) 1 − (z 1 − z 0 )(A1 − A0 )(z 1 − z 0 ) 2

which is equation (3.24). Proof of Corollary (3.5). By deﬁnition, 1 0 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 )

1 1 = a10 + a1 z 0 + z 0 A1 z 0 − a00 − a0 z 0 − z 0 A0 z 0 2 2

1 1 + a10 + a1 z 1 + z 1 A1 z 1 − a00 − a0 z 1 − z 1 A0 z 1 2 2

(B.62)

(B.63)

Adding and subtracting 12 z 0 A0 z 1 and 12 z 0 A1 z 1 and noting that the symmetry of At (that is At = At ) implies z 0 At z 1 = z 1 A1 z 0 , the foregoing equation becomes 1 0 fQ (z ) − fQ0 (z 0 ) + fQ1 (z 1 ) − fQ0 (z 1 ) = 2 a10 − a00 − (a0 − a0 ) z 0 + z 1 + z 0 A1 − A0 z 1 +

1 1 z − z 0 A1 − A0 z 1 − z 0 2

which is equation (3.25).

(B.64)

The Theory of Exact and Superlative Index Numbers Revisited 211 The derivation of equations (3.27) and (3.28) follows the footsteps of that of equations (3.24), (3.25), and (3.26). Proof of Corollary (3.6) (Accounting for Functional Value Ratios of Two Different Quadratic Homothetic Functions). The proof follows directly that of Corollary (2.1). Proof of Lemma (4.1) (Accounting for Value Differences of Two General Quadratic Functions). 1 1 0 0 The difference g fGQ (q ) − g fGQ (q ) and g h1GQ (q1 , x1 ) − g h0GQ (q0 , x0 ) can be decomposed by applying Lemma (3.1) and Corollary (3.1), respectively, thus obtaining: 0 −1 0 0 0 1 −1 1 1 1 Z (z ) −g fGQ Z (z ) (q ) −g fGQ (q ) = g fGQ g fGQ = fQ1 (z 1 )−fQ0 (z 0 ) =

1 ∇z fQ0 (z 0 )+∇z fQ1 (z 1 ) (z 1 −z 0 ) 2

+Parameter change component (PC) " 1 0 0 ˆ 0 −1 0 = (q0 ) g fGQ (q ) · Z (q ) ·∇q fGQ 2 #

1 1 1 −1 1 +g fGQ (q1 ) (q ) · Zˆ (q ) ·∇q fGQ · Z(q1 )−Z(q0 ) +PC

(B.65)

where 1 0 PC ≡ a10 − a00 + a1 − a0 Z(q ) + Z(q1 ) + Z(q0 ) A1 − A0 Z(q0 ), 2 (B.66) −1 is equal to the (i, i)th element of the diagonal matrix Zˆ (q) dqi dz −1 (zi ) = = 1/z (qi ) dzi dzi (with z (qi ) = 0, by assumption). t t −1 Z (z) , Since fQ (z) ≡ g fGQ (q) = g fGQ

(B.67)

212 Precursor

t −1 −1 −1 t ∇z fQ (z) = g fGQ Z (z) · Zˆ (z) · ∇z fGQ Z (z) where the (i, i)th element of the diagonal matrix Zˆ −1 (z) is given by

dz −1 (zi ) dzi

t t (q) · Zˆ −1 (z) · ∇z fGQ (q) using (4.3) = g fGQ −1 t t = g fGQ (q) · Zˆ (q) · ∇z fGQ (q) using (4.9) equation (B.65) can be rewritten as 1 1 0 0 g fGQ (q ) − g fGQ (q ) " 1 1 0 0 0 −1 1 0 (q ) (q0 ) + g fGQ = (q ) · Zˆ (q ) · ∇q fGQ g fGQ 2 −1 3 1 · Zˆ (q1 ) ·∇q fGQ (q1 ) · Z(q1 ) − Z(q0 ) + Pc

(B.68)

(B.69)

Proof of Theorem (6.1). If function g and z in (4.1) are, respectively, logarithmic transformation of functional values and variables, that is g(y) ≡ ln y and Z(q) ≡ ln q, then (6.1) follows directly from (4.9). Proof of Theorem (6.2). Let us apply the decomposition procedure (4.9) to the quadratic Box-Cox function (5.4), thus obtaining: $ %ρ %ρ $ N 0 1−λ fQ1r,λ −1 fQ0r,λ −1 1 qi 0 − = $ %1−ρ fQρ,λ ,i ρ ρ 2 i=1 0 fQρ,λ 1 1−λ q1 λ −1 q0 λ −1 qi i i 1 − +$ %1−r fQρ,λ,i λ λ 1 fQρ,λ +Parameter −change component 0 0 N 1 qi fQρ,λ ,i $ 0 %ρ 1 = fQρ,λ λ + 2 i=1 fQ0ρ,λ q0

+

qi1 fQ1ρ,λ, i $ fQ1ρ,λ

fQ1ρ,λ

%ρ

λ 0 λ qi 1 qi1 − 1 λ λ λ q

+Parameter −change component where fQt ρ,λ ≡ fQt ρ,λ (qt ), fQt ρ,λ ,i ≡ ∂fQt ρ,λ /∂qit .

(B.70)

The Theory of Exact and Superlative Index Numbers Revisited 213 By deﬁning sQt ρ,λ ≡

qit f t ρ,λ Q

f t ρ,λ

,i

with t=0, 1, multiplying equation (B.70) by ρ and

Q

rearranging terms we obtain $

fQ1ρ,λ

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧

N ⎪ ⎨

1ρ 2λ

⎪ i=1 ⎩

%ρ ⎫ %ρ $ ⎬ fQ0ρ,λ fQ1ρ,λ ⎪ 1 0 λ + sQρ,λ i 1 λ ⎪ ⎭ qi qi

$ sQ0 ρ,λ i

λ λ qi1 − qi0 + Parameter − change component (B.71) which is equation (6.4). Proof of Theorem (6.3). Equation (6.4) can be re-written as follows: ⎧ $ %ρ $ %ρ ⎫ N ⎪ ⎬ ⎨ $ %ρ $ %ρ fQ0ρ,λ IQ · fQ0ρ,λ ⎪ fQ1ρ,λ − fQ0ρ,λ = sQ∗0ρ,λ i λ + sQ∗1ρ,λ i 1 ⎪ qi λ ⎪ ⎭ qi0 i=1 ⎩ ⎧ $ %ρ 0 N ⎪ ⎨ λ λ f ρ,λ Q · qi1 − qi0 ξ 0 − 1 sQ∗0ρ,λ i λ + ⎪ qi0 i=1 ⎩ $ + ξ 1 − 1 sQ∗1ρ,λ i

Iq · fQ0ρ,λ 1 λ qi

%ρ ⎫ ⎪ ⎬ λ λ · qi1 − qi0 ⎪ ⎭

+ Parameter − change component where sQ∗tρ,λ i ≡ ρ 2λ

N

1 1 i=1 qi f ρ,λ Q

i

f 1ρ,λ

qit f t ρ,λ Q i N t t j=1 qj fQρ,λ j ρ

f 1ρ,λ Q

IQ ·f 0ρ,λ

Q

IQ · fQ0ρ,λ

ρ 2λ

N

0 0 i=1 qi fQρ,λ i f 0ρ,λ Q

, and ξ 1 ≡

ρ .

Q

Setting $

for t = 0, 1, ξ 0 ≡

%ρ

$

− fQ0ρ,λ

%ρ

=

⎧ N ⎪ ⎨ ⎪ i=1 ⎩ ·

%ρ $ %ρ ⎫ ⎬ fQ0ρ,λ IQ · fQ0ρ,λ ⎪ ∗1 0 λ + sQρ,λ i 1 λ ⎪ ⎭ qi qi

$ sQ∗0ρ,λ i

λ λ qi1 − qi0

$ %ρ dividing through by fQ0ρ,λ , rearranging terms and solving for IQ , we have IQ ≡ Iq · IY

214 Precursor where, setting r = 2λ, ⎡

N

1

r

( qi ) 2 ∗0 ⎢ sQρ,λ i q0 2r ( i) ⎢ i=1

⎤ 1r

⎥ ⎥ r ⎥ and 0 2 ⎦ q ( ) sQ∗1ρ,λ i i1 r ( qi ) 2 i=1

1−1 r

IY = Iqρ

Iq = ⎢ N ⎣

where Iq is the aggregating index number of the elements of q. Proof of Corollary (6.1). Using the deﬁnition (5.15) for fQr and denoting fQr (qt ) with fQt r to simplify notation, from the accounting equation (B.70), we derive: 0 0 r 1 r N qi1 fQ1r i fQ1r fQr q i fQ r i 1 1 2r 0 2r 1 0 r −1 = 0 2r + 1 0 r 1 2r qi − qi 0 fQ r q fQr fQr fQ r q i=1 i

i

! (where fQt r i ≡ ∂fQt r ∂qit ) 1 r N 1 r N 1 2r 0 2r N f f r r q qi Q Q i = sQ0 r i r − sQ0 r i + 0 r sQ1 r i − 0 r sQ1 r i r 0 2 fQr i=1 fQr i=1 qi qi1 2 i=1 i=1 ! N t t t where sQt ρ,λ i = sQt r i = qit fQt r i fQt r = qit fQt ρ,λ i ρr fQt ρ,λ = qit fQt ρ,λ i i=1 qi qi fQρ,λ i , $ % ρr $ % ρr −1 $ % since fQt r = σ1t fQt ρ,λ and fQt r i = σ1t ρr fQt ρ,λ fQt ρ,λ i and, by Euler’s theorem, ρr fQt ρ,λ = Ni=1 qit fQt ρ,λ i , N

=

N

r

qi1 2 sQ0 ρ,λ i r qi0 2 i=1

−1+

fQ1r

fQ0r

r r −

r

1 − 2r qi 1 sQρ,λ i − r 0 r fQr i=1 qi0 2 fQ1r

N

(B.75)

N since fQt r is homogeneous of degree one in q and, by Euler’s theorem, sQt r i = 1. i=1 ! 1. By rearranging (B.75) and solving for fQ1r fQ0r , the following index is obtained:

⎡ fQ1r fQ0r

N

1

r

( qi ) 2 0 ⎢ sQρ,λ i q0 2r ( i) ⎢ i=1

= Iq ≡ ⎢ N ⎣

⎤ 1r

⎥ ⎥ ⎥ ( ) ⎦ ( )

r qi0 2 1 ρ,λ Q i 1 2r qi i=1

s

Let us deﬁne IT ≡ σ 1 /σ 0 , and, taking account of (B.74), % $ 1−1 Iq · IT = IY ≡ Iqρ r fQ1ρ,λ fQ0ρ,λ

(B.76)

(B.77)

The Theory of Exact and Superlative Index Numbers Revisited 215 Proof of Theorem (6.4). The arbitrary twice differentiable function f (q) and fQr (q) deﬁned by (5.15) are assumed to be second-order approximation at q0 , that is f (q0 ) = fQr (q0 )

(B.78)

∇f (q0 ) = ∇fQr (q0 )

(B.79)

∇ 2 f (q0 ) = ∇ 2 fQr (q0 )

(B.80)

With f positive over its domain of deﬁnition, qi0 > 0, and r = 0, the coefﬁcients αij ’s can be found by solving the two systems of equations obtained by rearranging the differential equations resulting from differentiating fQr as deﬁned by (5.15) and replacing the obtained derivatives with those of f , that is fi =

N

r −1 0 2r qj for i = 1, 2, . . . , N f ∗(1−r) αij qi0 2

(B.81)

j=1

fij =

r 1−r) 0 0 2r −1 1−r qi qj , for i = j, and i.j = 1, 2, . . . , N ∗ fi fj + αij f f 2 (B.82)

thus obtaining from (B.81) and (B.82), respectively, N r r −1 f (1−r) αij qj0 2 fi − qi0 2

αii (r, q0 ) = αij (r, q0 ) =

j =i

f

(1−r)

0 r−1 qi

,

for i = 1, 2, . . . , N

ff fij − 1−r f∗ i j r −1 for i = j, and i, j = 1, 2, . . . , N r 1−r f qi0 qj0 2 2

(B.83)

(B.84)

References

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Appendices 1

The price index as a utility based concept 1.1 1.2

2

Original approach New approach

Terminology Conical v. homogeneous &c

3

Notation

4

BASIC computer program BBC BASIC for Windows developed by Richard Russell [email protected].

1

The price index as a utility based concept Answer to complaints that conical or constant-returns utility for dealing with the price index is an objectionable imposition, instead of being unavoidable as was settled long ago.

“… the artiﬁcial ‘homogeneity’ which underlies most index thinking and usages, can escape mention entirely.” S. N. Afriat Index Numbers in Theory and Practice by R.G.D. Allen Canadian J. Econ. 15, 2 (May 1978)

1.1

Original approach

1.2

New approach

1.1 Original approach Prices change and an individual who enjoys a consumption that provides a certain standard of living at a certain money cost would like to know how much it will cost to maintain the same standard at the new prices. This is the cost of living problem. The Price Index issued from the Statistical Ofﬁce is a number that tells how to deal with the question in a highly restrictive artiﬁcially imposed simple away, the index being the multiplier of old expenditure to determine the new. There is of course no absolute reason for having a price index in the ﬁrst place. It is an institution, even if a well established, traditional, and perhaps even today still needed institution, affecting many aspects of economic life. The question of how to produce such a number is known as The Index-Number Problem. Let Pst denote the price index from period s to period t. The number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by (P)

Mr = Prs Ms

in period r to maintain the same standard of living. This scheme has now to be put in utility terms. Given a utility function φ : n → , such as may govern demand, the cost at prices p ∈ n of attaining the the utility of consumption x ∈ n is (R)

ρ (p, x) = min {py : φ (y) ≥ φ (x)}

ρ being theutility-cost function, depending on prices p and on consumption x through its utility value φ (x) that is representative of standard of living. This is the cost when prices are p of living at the standard represented by consumption x. The cost of living question, put in utility terms, is concerned with how this cost changes when prices change, for a given ﬁxed standard of living. By the condition (S)

ρ (p, x) = px,

demand of x is supported by prices p, its cost being the minimum cost of obtaining its utility, and x is supported if supported by some p. For concave utility, every x is supported since there is a supporting hyperplane to the graph at every point. For a continuous utility every p supports some x. By deﬁnition, for all p, x (a1) φ (y) ≥ φ (x) ⇒ py ≥ ρ (p, x) for all y, and, continuity provided, (a2) φ (y) = φ (x) , py = ρ (p, x) for some y.

226 Appendices In particular, (b1) ρ (p, x) ≤ px for all p, x. and, for all p, (b2) ρ (p, x) = px for some x. To be found now are implications of admissibility of the special resolution of the cost of living question by means of a price index. Let pt be prices in period t, so ρ (pt , x) is the cost at those prices of living at the standard represented by x. In transition from period s to period r the cost changes from ρ (ps , x) to ρ (pr , x), in the ratio ρ (pr , x) /ρ (ps , x) in general depending on x. In case this ratio is independent of x, a price index Prs based on the utility is deﬁned and given by this ratio, so giving satisfaction for all x of the conventional money conversion relation (P) depending on a price index, where now Mr = ρ (pr , x) , Ms = ρ (ps , x) . By this condition the utility has the price index property, and Prs is the associated price index, which provides (Px) Mr = Prs Ms for all x. Here represented in utility terms is the deﬁning image from the form of its use of the price index of ordinary practice. But to have this representation, the ratio to determine the price index has to be independent of the variable consumption bundle x the utility of which is the measure of standard of living, that is, (i)

ρ (pr , x) /ρ (ps , x) is independent of x

This independence represents the special condition on the utility by which it has the price index property and has a price index associated with it. This condition on utility, stated in terms of the utility cost function ρ, will be seen equivalent to the condition of utility cost factorization that requires (ii) ρ (p, x) = θ (p) φ (x) or that the utility cost function ρ factorizes into a product of a function θ of prices alone with a function φ of quantities alone1 . In this case the associated price index, 1 Touched on in Afriat 1972, 1977 pp 101 ff or 2005 pp 87 ff; also Deaton 1979 JASA 74, 365. The equivalence of (i) and (ii) is stated by Samuelson and Swamy (1974) p. 570.

The price index as a utility based concept 227 or the price index based on the utility, is given immediately by Prs = θ (pr ) /θ (ps ) . Theorem 1 For a utility to have the price index property, and so to have an associated price index, utility cost factorization is necessary and sufﬁcient. We have to show (i) ⇔ (ii). Since (ii) ⇒ (i) is immediate, it remains to prove (i) ⇒ (ii). Take any ﬁxed a ∈ C. Then by the price index property (i), ρ (pr , x) /ρ (ps , x) = ρ (pr , a) /ρ (ps , a) for all x. Let θ (p) = ρ (p, a) , so now ρ (pr , x) /ρ (ps , x) = ρ (pr , a) /ρ (ps , a) = θ (pr ) /θ (ps ) , and let φ (x) = ρ (p, x) /θ (p) . Then we have the utility cost factorization required by (ii), completing the proof. Theorem 2 For factorization of the utility cost function it is necessary and sufﬁcient that the utility be conical. This Theorem may be a good candidate for the title “The Index Number Theorem” secured by Hicks for another purpose, and we are going to prove it as happened in 1950s or 60s, anyway long ago2 . Given φ conical, ρ (p, x) = min {py : φ (y) ≥ φ (x)} = min py (φ (x))−1 : φ y (φ (x))−1 ≥ 1 φ (x) = θ (p) φ (x)

2 Samuelson and Swamy (1974) p. 570 cite Afriat (1972).

228 Appendices where θ (p) = min {pz : φ (z) ≥ 1} That shows the sufﬁciency. Since, for all p, θ (p) φ (x) ≤ px, for all x, with equality for some x, as assured with continuous φ, it follows that θ (p) = minx px/φ (x) showing θ to be concave conical semi-increasing. Also for x demandable at some prices, as would be the case for any x if φ is concave, the inequality holds for all p with equality for some p, showing φ (x) = minp px/θ (p) which, in case every x is demandable at some prices, requires φ to be concave conical semi-increasing. But even when not all x are demandable, because they lie in caves and are without a supporting hyperplane, here is a conical function deﬁned for all x that is effectively the same as the actual φ as far as any observable demand behaviour is concerned. So it appears that for the cost function factorization the utility function being conical is also necessary, beside being sufﬁcient, as already remarked. Hence, with some details taken for granted, the Theorem is proved. In consequence we have: Theorem 3 For a price index to be based on a utility it is necessary and sufﬁcient that the utility be conical This approach to the beginning of price index theory a half century ago is one way of bringing forward the inevitablility of the association of a price index with conical utility. There is another approach based on common sense to follow now, where it appears that as soon as you start in any way about a price index, to do with utility, in the ﬁrst moment you have constant returns utility. That should wipe out protests and of course there is bound to be a penalty somewhere in dealing with such a restricted concept as the price index in the ﬁrst place. If there is an assumption anywhere, it is the price index itself. There is, contrary to complaints, no additional assumption about utility being constant returns, only the implication. Will that placate the bitter opponents? Before leaving about factorization there should be notice about the factors. From (ii) joined with (b1) and (b2) we have (c1) θ (p) = minx px/φ (x)

The price index as a utility based concept 229 which shows that θ (p), being the minimum of a family of homogeneous linear functions px/φ (x), is concave conical. Also, for any x that is supported, (c2) φ (x) = minp px/θ (x) . Hence if φ, beside being anyway conical to have the factorization, is also concave, so every x is supported, then this holds unconditionally. Functions that satisfy (c1) and (c2), both necessarily concave conical, deﬁne a conjugate pair of price and quantity functions. The symmetry here reﬂects a perfect symmetry throughout between price and quantity, having various manifestations. See Appendix 3 on Notation.

230 Appendices

1.2 New approach To keep with familiarities, we ﬁrst put it more or less the way it is put in the text. One obvious way of maintaining, at new prices, the old standard of living obtained from the old consumption x0 at a money cost p0 x0 is to simply buy the old consumption x0 with new cost p1 x0 . So certainly the current cost of the old standard is at most that amount, and its comparison ratio with the old cost, which might at ﬁrst thought serve as a price index, is at most p1 x0 /p0 x0 , which is the ever-pervasive Laspeyres index. But here common sense breaks in with the proposal that one would not necessarily buy the old bundle at the new prices, one could buy instead another bundle that provides at least the old standard perhaps at a lesser minimum cost, making a comparison ratio with the old cost, the price index P10 , not exceeding the Laspeyres index, P10 ≤ p1 x0 /p0 x0 , and by the same principle P01 ≤ p0 x1 /p1 x1 . But in recollection of primitive price index reversibility which requires P01 = (P10 )−1 this second is equivalent to P10 ≥ p1 x1 /p0 x1 , so common sense together with the primitive reversal provides both Laspeyres and Paasche, and shows them not only as tied together with the same source in principles, but as upper and lower bounds of the price index. In a way, they are not price index formulae like all the others, but fundamental and irrefutable limits for the price index. In that case, of course, the lower bound could not exceed the upper bound, p1 x1 /p0 x1 ≤ P10 ≤ p1 x0 /p0 x0 . Hence, if there is a price index, (PL) p1 x1 /p0 x1 ≤ p1 x0 /p0 x0 . J. R. Hicks (without proving anything) calls the PL-inequality the “Index Number Theorem” (Revision, 1956, p. 181.) One should remember there was a time when there was, brieﬂy, something of a fashion to call almost anything a “Theorem”. It is

The price index as a utility based concept 231 confusing, but perhaps Hicks was just being fashionable. A case where Paasche turned out to be greater than Laspeyres could anyway be occasion for a pause. It might have been good sense (corresponding to practice of the practical) to abandon the entire theoretical subject after arrival at this point, where the unavoidable primitive has been joined with common sense. But uncountable formulae for the index, good, true, better, super, &c have been proposed. To go further, as it seems we must … Here we conclude with the observation that the PL-inequality is just the condition on the data for it to admit construction of a conical or constant returns utility. Without having even thought of such a constant returns utility at ﬁrst, here it ominously rears … The protesters about such utility should now have difﬁculty to know where to stick their protest. One may wonder if Hicks ever got the scent of constant returns with his “Index Number Theorem”. Now we start again in a similar fashion to take it all further, even to encompasse the entire index subject, or at least the new method. From Ms = ps xs for data to ﬁt the utility with minimum cost as actual cost, and Mr ≤ pr xs because minimum cost at most actual, follows (P)

Prs = Mr /Ms ≤ pr xs /ps xs = Lrs ,

so (M)

Lrs ≥ Mr /Ms ,

and hence (L)

Lr...r ≥ 1

which is just the condition on the data to admit construction of a conical or constant returns utility, and for the solubility of system (M) for price levels, with price indices given by (P) as their ratios. That is the whole story for this book told in one sentence. Or yet again, by common sense, Prs ≤ Lrs . But by the chain property, for any k, Prs = Prk /Psk ,

232 Appendices so Lrs ≥ Prk /Psk , and hence Lr...r ≥ 1 for all r. . .r, which is, again, just the condition on the data to admit construction of a conical or constant returns utility. For a conclusion, ﬁnally, in dealing with a price index on the basis of utility there is no getting away from constant returns. Now another venture to expose the nature of our approach. The undertood objective is a formula for a system of m(m − 1) price indices Prs (r, s = 1, . . . , m) for consistent cost-of-living conversions between m periods. From that consistency, by which conversion from r to s and then s to t coincides with that from r to t, they must be expressible in the form of a system of ratios (P) Prs = Pr /Ps between numbers which, from this representation, have the role of price levels. In other words, equivalently, they have the property Prs Pst = Pst listed among Fisher’s well known “Tests” as the “Chain Test”. For a point of origin let us start with the already considered property (O)

Prs ≤ Lrs

which was termed “common sense”. By substitution of price indices with their expressions as ratios of price levels, we have (L)

Lrs ≥ Pr /Ps

as a system of inequalities for price levels, from which follows the extended chain system (chain L) Lr...s ≥ Pr /Ps and then the cyclical products consistency test (cycle L) Lr...r ≥ 1 is a conclusion.

The price index as a utility based concept 233 For our price index system Formula we now have: price indices given by (P) for price levels that are a solution of (L) There it is, produced out of a hat! As were the one or two hundred formulae in Irving Fisher’s collection. Rather, just the opposite. For here it is seen how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of ‘common sense’. The main concern of Chapter 2 in Part I is with theory of the system of inequalities (L) for the price levels, and computational method. The cyclical product test, that here immediately appears necessary for the existence of a solution, is there (or from the 1960 paper in Part II) seen to be also sufﬁcient. For this formula there has, apparently, been been no dealing at all with utility, let alone constant returns, or at least no mention. But still it might have, or must have, crept in somewhere somehow. Since nowhere else is available the only place to look is the property (O) in the disguise of “common sense”. There is nothing wrong with the common sense part where it is argued that the cost at the new prices pr of having the standard of living provided by consumption xs is at most the actual cost pr xs at those prices, so Mr ≤ pr xs . After all, at that new cost one could even go on with exactly the same consumption as before. Where there is the ﬂavour of utility is the additional submission that Ms = ps xs , to give Prs = Mr /Ms ≤ pr xs /ps xs = Lrs and hence Prs ≤ Lrs , which is (O). For that submission signiﬁes the cost of the consumption coincides with the minimum cost for obtaining the standard of living it provides, or its utility. It is incidental then, since the price index now should be based on the utility, that it must have constant returns, as already argued. This opens the way to intense criticism from some who ignore the complete inseparability of the price index with constant returns that was before just now thoroughly settled a half century ago, in a decadent abandonment of elementary beginnings fundamental to the notion of a price index. For a last word, our multilateral index formula is not one among a possible many that could come to mind, like bilateral formulae that in Irving Fisher’s collection number one or two hundred, but the only one that can properly be entertained. In that way it is not just one possible answer to the Index Number Problem, but the only answer. For here it has been seen, as already said, how the new method with all its simplicity comes uniquely and unavoidably implied just one step after an application of ‘common sense’.

2

Terminology

Conical v. homogeneous &c Answer to complaints about use of the term conical when everyone uses some other term A ray is a half-line with vertex the origin, and every point lies on just one ray, the ray through it, so a = {at : t ∈ } ⊂ C is the ray through any a ∈ C(in the commodity space). A cone is a set described by a set of rays, and every set has a conical closure, or cone through it, or projecting it, described by the set of rays through its points. Hence

= {xt : x ∈ A, t ∈ } ⊂ C A is the cone through any A ⊂ C. A function is conical if its graph is a cone, or what is the same (just more syllables), linearly homogeneous, being such that φ (xλ) = φ (x) λ. Now there will be argument that this term conical for a function has merit in its favour over other terms in use instead, here below in order of preference, ﬁrst two being quite close together. (1) conical, for a function, refers to the graph saying it is a cone, brief, and essential, does not invoke any spurious extra framework (2) constant returns, peculiar language of economics, double output with double input, very good, much preferable to 3-5 but a mouthfull. (3) linearly homogeneous, a mouthfull, also brings into view the spurious original mathematical framework for homogeneous in some degree n, joined with additional information that n = 1, not good (4) homogeneous, means homogeneous for some unspeciﬁed degree n, without telling that actually n = 1, bad (5) homothetic, used sometimes indiscriminately in place of all the above, terrible idea, had some currency in debates decades ago, such as where whether or not utility could be measured had been an issue; and we do measure utility; use and meaning apparently become unsettled, mostly had to do with production originally, for properties spurious to dealings with price indices; some take it to mean simply any monotonic function of a linearly homogeneous function; see for instance Finn R. Forsund, “The homothetic production function”, Swed. J. of Economics 77, 2 (1975). 234-44. Of course were one determined not to be pedantic one could perhaps use (4) but this has one more syllable than (1). We, or at least one of us, will persevere with (1), with random lapses into (2) or even (4). So much for the disapproval dispensed by ‘homothetic’ adherents …

3

Notation

non-negative numbers

n C = x, y, . . . ∈ C n

n B = n p, q, . . . ∈ B

non-negative column vectors commodity space commodity bundles non-negative row vectors budget space price vectors

p ∈ B, x ∈ C ⇒ px ∈ px cost of commodity bundle x at prices p p, x ∈ B × C demand element showing commodity bundle x demanded at prices p φ:C → utility function ρ p, x = min py : φ y ≥ φ x utility cost ρ pr , x / ρ ps , x independent of x price index property 1 ρ p, x = θ p φ x utility cost factorization 2 φ xl = φ x l constant returns 3 1⇔2⇔3 theorem 1⇔3 consequence θ p = minx px / φ x φ x = minp px / θ p θ p φ x ≤ px for all θ p φ x = px for all

conjugate concave conical price and quantity functions p, x x some p and all p some x

pt , xt ∈ B × C, t = 1, . . . , m demand data

pr xs cross-costs ps xs direct-costs Lrs = pr xs / ps xs Lrij...ks = Lri Lij . . . Lks

ratio = Laspeyres index

Lrij...kr = Lri Lij . . . Lkr

chain Laspeyres cycle Laspeyres

Lrs ≥ Pr / Ps

price levels Pt , t = 1, . . . , m

Lr...r ≥ 1

cycle existence test price index = ratio of price levels

Prs = Pr / Ps

Mrs = minij...k Lri Lij · · · Lks M = Lm

derived Laspeyres plus = min

Krs = pr xr / ps xr Paasche −1 = Lsr Hrs = maxij...k Kri Kij · · · Kks derived Paasche = Msr−1 H = Km plus = max Krs ≤ Lrs Hrs ≤ Mrs Krs ≤ Hrs ≤ Mrs ≤ Lrs

Laspeyres-Paasche inequality derived Laspeyres-Paasche inequality

Mrs ≥ Pr / Ps Mrs Msk ≥ Mrk Mrs ≥ Mrk / Msk

price levels Pt , t = 1, . . . , m triangle inequality basic-Laspeyres price levels Pt = Mtk columns of derived Laspeyres M basic-Paasche price levels Pt = Htk columns of derived Paasche H

Mrs ≥ Hrk / Hsk 1 Frs = Hrs Mrs 2 Mrs ≥ Frk / Fsk

Ft =

m

m1

Fisher analogue Fgeometric mean of H and M basic-Fisher price levels Pt = Ftk columns of Fisher analogue F

Ftk

k=1

Mrs ≥ Fr / Fs

mean-basic price levels Pt = Ft , t = 1, . . . , m geomeric mean of m columns of F and of 2m columns of H and M which are a basis for all true price levels and hence all true price indices

Prs = Fr / Fs

basic price indices

Pr Xs ≤ pr xs

for all r, s

Pt Xt = pt xt

determines correspondence between price levels Pt and quantity levels Xt

Jrs = ps xr / ps xs Jrs ≥ Xr / Xs

quantity-Laspeyres quantity levels Xt , t = 1, . . . , m

Xrs = Xr / Xs

quantity indices

4

BASIC computer program BBC BASIC for Windows developed by Richard Russell [email protected].

eside reports done using FORTRAN we include routine outputs from another program, still undergoing development, using BBC BASIC for Windows. This program deals with two text ﬁles kept in folder c:\0\ as here indicated:

B

REM input from C:\0\?-input.txt output to REM C:\0\?-output.txt REM change the ? *SPOOL "C:\0\2-output.txt" F%=OPENIN "C:\0\2-input.txt" (Richard Russell contributed)

must

understand

these

lines

which

he

For instance 2-input.txt looks like 4, 4 1.0000000, 0.8976280, 0.9130180, 0.7475160,

1.1218730, 1.0000000, 0.9792190, 0.8122070,

1.1829370, 1.0418520, 1.0000000, 0.8138330,

1.5008030 1.3498590 1.2661740 1.0000000

which tells it is a 4×4 matrix and then tells the elements, the comma “,” being the delimiter. As for 2-output.txt, which need not even exist initially and if it does any contents will be overwritten, it recieves the output when the program is run, which with this input is as follows—showing reassuring agreement with earlier FORTRAN ﬁgures. When this program is compiled and so not available for alteration, it will refer to two ﬁles called input.txt and output.txt always with these names though they can have different applications. This compiled version will be supplied to anyone wishing to use it. A later version will start instead with two m×n and n×m matrices P0 and X0 and then form the m×m cross-cost matrix C0=P0.X0 and then, dividing columns by diagonal elements, the m×m Laspeyres matrix L. These diagonal elements have to be retained in or to convert price levels to quantity levels, from the equation PX=px. We acknowledge with thanks the guidance received from Richard Russell, longtime developer of this rendering of BASIC. http://www.compulink.co.uk/∼rrussell/bbcwin/bbcwin.html, http://www.rtrussell.co.uk/, [email protected].

244 Appendices REM Price-Level Computation: PLC.bbc REM BBC BASIC for Windows - guidance from Richard Russell REM http://www.compulink.co.uk/~rrussell/bbcwin/bbcwin.html REM The Power Algorithm: + means min @% = &90F :

REM Set numeric formatting

REM input from C:\0\?-input.txt output to C:\0\?-output.txt REM change the ? *SPOOL "C:\0\?-output.txt" F%=OPENIN "C:\0\?-input.txt" m=FNread(F%) REM n=FNread(F%) REM DIM P0(m, n), X0(n, m) DIM L(m, m), C0(m), M(m, m), N(m, m, m), H(m, m), F(m, m) DIM G(m), P(m, m), C(m), D(m), E(m, m), V(m) DIM XL(m), X(m, m) REM Read P0 and X0 REM Determine C0=P0*X0 PRINT "Read C0(m)" FOR i=1 TO m C0(i)=FNread(F%) NEXT i REM Read L (or Form L from C0) and make M=L FOR i=1 TO m FOR j=1 TO m L(i, j)= FNread(F%) : M(i, j)=L(i, j) NEXT j NEXT i REM Illustration - change the ? PRINT "Illustration ?" : PRINT REM Print L PRINT "L Laspeyres" FOR i=1 TO m FOR j=1 TO m PRINT ;L(i, j),; NEXT j : PRINT NEXT i

1

REM M=M*L print M and paths FOR g=2 TO m fd=0 FOR i=1 TO m FOR j=1 TO m: h=1 : N(i,j, 1)=i FOR k=1 TO m IF M(i, k)*L(k,j)<M(i, h)*L(h, j) THEN h=k ENDIF NEXT k : M(i, j)=M(i, h)*L(h,j): N(i,j, g)=h IF hj THEN fd=1 ENDIF NEXT j

BASIC computer program 245 NEXT i : PRINT REM PRINT "fd="; fd IF fd = 0 THEN PRINT "Final power" PRINT "L power " ; g; IF g=m THEN PRINT " = M derived Laspeyres" ELSE PRINT FOR i=1 TO m FOR j=1 TO m PRINT ;M(i, j),; NEXT j: PRINT NEXT i NEXT g: PRINT PRINT "Paths" FOR i=1 TO m FOR j=1 TO m FOR g=1 TO m PRINT ;N(i,j,g) ","; NEXT g : PRINT ;j ,; NEXT j : PRINT NEXT i: PRINT fi=0 FOR i=1 TO m IF M(i,i)V(g) THEN e=V(g) NEXT g: PRINT PRINT "Critical cost-efficiency " ; e

ENDIF

246 Appendices PRINT PRINT "L adjusted Laspeyres to replace original L" FOR i=1 TO m FOR j=1 TO m IF i j THEN L(i,j)=L(i,j)/e ENDIF M(i,j)=L(i,j): PRINT ;L(i, j),; NEXT j: PRINT NEXT i: PRINT ENDIF IF fi=1 THEN GOTO 1 PRINT "H derived Paasche" FOR i=1 TO m FOR j=1 TO m H(i,j)=1/M(j,i): PRINT NEXT j: PRINT NEXT i :PRINT

;H(i, j),;

PRINT"The " ;2*m;" canonical price-level systems - the " ; 2*m; "columns of M and H" :PRINT PRINT "F derived Fisher - mean of derived Laspeyres M and derived Paasche H" FOR i=1 TO m FOR j=1 TO m F(i,j)= SQR(H(i,j)*M(i,j)): NEXT j: PRINT NEXT i :PRINT

PRINT ;F(i, j),;

PRINT "P mean canonical price-level: mean of columns of F" FOR i=1 TO m: g=1 FOR j=1 TO m: g=g*F(i,j) NEXT j: g=EXP(LN(g)/m): :G (i)=g: PRINT ;g NEXT i :PRINT PRINT "P/P mean canonical price-index" FOR i=1 TO m FOR j=1 TO m P(i,j)=G(i)/G(j): PRINT ;P(i,j),; NEXT j: PRINT NEXT i :PRINT PRINT "X mean canonical quantity-level: PX = px" FOR i=1 TO m XL(i)=C0(i)/G(i): PRINT ;XL(i) NEXT i: PRINT PRINT "X/X mean canonical quantity-index" FOR i=1 TO m FOR j=1 TO m X(i,j)=XL(i)/XL(j): PRINT ;X(i,j),; NEXT j: PRINT NEXT i :PRINT *SPOOL

BASIC computer program 247 END DEF FNread(F%) PRIVATE A$ LOCAL C% C%=INSTR (A$,",") IF C%=0 THEN INPUT#F%, A$ IF ASC (A$)=10 A$=MID$ (A$,2) ELSE A$=MID$ (A$,C%+1) ENDIF =VAL (A$)

Notes 1

Multilateral indices

2

Chain indices

1 Multilateral indices Matthijs van Veelen (2002), the opening paragraph: “Comparing wealth of nations—or across years—is something an economist naturally wants to do. The classic problem we then face, and which the standard GDP ignores, is that prices are not necessarily the same in different countries. Trying to correct for this, quite a few methods of comparison have been designed, but index theorists have always been aware that all of them have their drawbacks, which for instance is reﬂected in overviews of Balk (1996) and Hill (1997). As long as we only want to compare two countries we do have fairly acceptable index numbers, but all known multilateral indices suffer from unwanted peculiarities. A natural question is then whether we can expect a perfect method ever to be discovered. The statement of this paper is that this infallible method does not exist and that the apples and oranges character of multilateral comparison is more fundamental.” With reference to the part in italics we should want to know what is the essential difference betweeen two and many. We of course deal with the classical index problem where difference between dealing with times or countries has no part. What is meant by “infallible”? Here is the continuation following remarks to do with axiomatics: “… it is generally acknowledged that the step from bilateral to multilateral comparison creates a serious problem. In spite of this general awareness, I do think that this impossibility theorem is a contribution, since so far it has never been proved that there is no method of multilateral comparison behaving in a way we can reasonably expect it to do.” What could be that “serious problem” beside failure to make the step? We have made the step as for the classical index problem, with a conclusion about uniqueness and inevitability, stated at the end of Appendix 1: “… our multilateral index formula is not one among a possible many that could come to mind, like bilateral formulae that in Irving Fisher’s collection number one or two hundred, but the only one that can properly be entertained.” If there is a problem with the axiomatics its too bad for the axiomatics (with all respect to the author). “What we are looking for is a method to be applied to a group of countries that gives each of those countries a single number, compressing the information contained in the [price and quantity data] matrices to a vector, entries of which can easily be compared.”

Notes 251 Concerning what they seek see Appendix 3 on Notation, where coming after original “demand data” there is arrival at nothing other than “quantity levels”, representing just what is needed. We have done the impossible! Anything wrong? From unsympathetic reviewers it appears unbearable, but apart from that. John Quiggin and Matthijs van Veelen (2007) state: “The revealed preference approach starts from the assumption that the preferences of the representative conumer are the same in all […] countries we consider. […] Consequently it is natural that data from a third country should affect whether country 1 ranks higher than country 2 or vice versa. […] The aximoatic approach, on the other hand, allows for all countries to have a representative utility function of their own. […] When comparing consumption bundles in two countries, prices and quantities from a third country are therefore not considered informative and it is considered undesirable if they make a difference for how the ﬁrst two countries rank relative to each other.” Whereas with the former approach all the data points are chained together in order to form a multilateral index, with the latter approach the single pairs of data points are used independently from the others in constructing bilateral indices, which generally turn out to be inconsistent among each other in a multilateral setting. The current bilateral-multilateral terminology is not doing good service for understanding what is being said. A bilateral index must be a price index for two points taken in isolation from others, depending on data just for those points. A multilateral index would refer to some m points and consist of a system of m(m − 1) price indices, one associated with each pair. The system can depend on the data for all m points. That at ﬁrst may seem to allow the individual indices of the system to be ordinary bilateral indices. That would be easy, any one of the available one or two hundred fromulae being, as far as one may know, admissible. Then what has all the trouble been about? If there is still a question then a return to beginnings would be suitable. Let Prs denote the system price index from base period s to current period r. For a ﬁrst point, the number must apply equally well to everyone experiencing the price change, whatever their standard of living. Hence an expenditure Ms in period s, at whatever level, must be replaced by Mr = Prs Ms in period r to maintain the same standard of living. Consider three periods r, s, t where appropriate expenditures maintain the same standard of living, say expenditure Mi in period i. It is of course required that transitions from t to s followed by s to r must have the same outcome as the direct

252 Notes transition from t to r. That requires (1)

Prs Pst = Prt

which, were we dealing with a bilateral formula, which we are not, reﬂects the “Chain Test” of Irving Fisher, but now merely deﬁnes the consistency of the price index system. Alternatively, by ﬁxing any one value of Mi all the others are determined, and so we have a system of numbers in the role of price levels from which the prices indices of the system are derived as their ratios, (2)

Prs = Mr /Ms .

Obviously (2) ⇒ (1), also from (1), for any k, Prs = Prk /Psk so we have (2) with Mi = Pik , showing that (1) ⇒ (2), and hence (1) ⇔ (2). Here then is the notion for a multilateral price index system: just a consistent system of price indices. The problem had been in the matter of consistency. The celebrated one or two hundred formulae in Irving Fisher’s collection have been notorious for refusing the Chain Test; it would be quite unsuitable to have anything to do with those. Since consistent price indices can be derived as ratios of price levels, to be safe (and more economical) it would be better ﬁrst to deal directly with those (m in number instead of m(m − 1)). That is our method, argued as the one and only method for producing a multilateral price index system.

Bibliography Balk, B. M. (1996). A Comparison of Ten Methods for Multilateral International Price and Volume Comparison. Journal of Ofﬁcial Statistics 12, 199–222. Diewert, W. E. (1986). Microeconomic Approaches to the Theory of International Comparisons. TechnicalWorking Paper No. 53, NBER, Cambridge, MA. — (1999). Axiomatic and Economic Approaches to International Comparisons. In A. Heston and R. E. Lipsey (eds.), Internationaland Interarea Comparisons of Income, Output and Prices, 13–87. University of Chicago Press. Drechsler, L. (1973). Weighing of Index Numbers in Multilateral International Comparisons. Review of Income and Wealth 19, 17–34. Fisher, I. (1922). The Making of Index Numbers. Boston: Houghton Mifﬂin. Geary, R. C. (1958). A Note on the Comparison of Exchange Rates and Purchasing Power Between Countries. Journal of the Royal Statistical Society, Series A, 121, 97–9. Hill, R. J. (1997). A Taxonomy of Multilateral Methods for Making International Comparisons of Prices and Quantities. Review of Income and Wealth, 43, 49–69. Khamis, S. H. (1972). A New System of Index Numbers for National and International Purposes. Journal of the Royal Statistical Society, Series A, 135, 96–121.

Notes 253 Neary, J. P. (1999). True Multilateral Indexes for International Comparisons of Real Income: Theory and Empirics. Mimeo, University College Dublin. Quiggin, John and Matthijs van Veelen (2007). Multilateral indices: conﬂicting approaches? Review of Income and Wealth 53, 2 (June), 372–8. Sen, A. K. (1982). Real National Income. In Choice, Welfare and Measurement. Oxford: Basil Blackwell, 388–415. van IJzeren, J. (1956). Three Methods of Comparing the Purchasing Power of Currencies. Statistical Studies No. 7. Zeist: Netherlands Central Bureau of Statistics/De Haan. van Veelen, Matthijs (2002). An impossibility theorem concerning multilateral international comparison of volumes. Econometrica 70, 1 (January), 369–75.

2 Chain indices The method of chaining index numbers dates back at least to Lehr (1885) and Marshall (1887), who dealt with the problem of new and disappearing commodities within the examined time period. In contemporary debate (see, for example, von der Lippe, 2001 and Balk, 2004), chaining has been also considered to reduce the spread between the Laspeyres and Paasche indices, which are expected to delimit the interval of possible numerical values of the, as it were, unknown true index— whatever that should mean. Dale W. Jorgenson and Zvi Griliches (1971, p. 228), with reference to their previous work (1967), remarked: “The main advantage of a chain index is the reduction of errors of approximation as the economy moves from one production conﬁguration to another. […] The Laspeyres approximation to the Divisia index of total factor productivity was employed in our original study of productivity change”. Samuelson and Swamy (1974, p. 576), say “Fisher missed the point made in Samuelson (1947, p. 151) that knowledge of a third situation can add information relevant to the comparison of two given situations.” (See also Samuelson 1984.) The “Minimum Spanning Tree” method, although it treats minimum distance in the structure of weights, does not pursue minimum chain Laspeyeres (or maximum chain Paasche) indices. See Robert J. Hill (1999a), (1999b), (2001), (2004). Here is an outline of the already accounted chain index method. The most rudimentary idea is that if we have Prs ≤ Lrs and Pst ≤ Lst then, by multiplication, Prs Pst ≤ Lrs Lst , so that, subject to the consistency requirement Prs Pst = Prt , it follows that Prt ≤ Lrs Lst , and similarly Prs ≤ Lr...s , where … is any chain. As bearing on some applications, this may be a summary of the method. Introducing the chain Laspeyres and Paasche indices Lsij...kt = Lsi Lij · · · Lkt ,

Ksij...kt = Ksi Kij · · · Kkt ,

Notes 255 the cycle test Ls...t...s ≥ 1 is equivalent to (chain LP) Ks...t ≤ Ls...t for all possible chains … and … taken independently. Hence, introducing the derived Laspeyres and Paasche indices Mst = minij...k Lsi Lij · · · Lkt ,

Hst = maxij...k Ksi Kij · · · Kkt ,

subject to the conditions required for their existence, for which Hst = (Mts )−1 , this is equivalent to (derived LP) Hst ≤ Mst . In this case Kst ≤ Hst ≤ Mst ≤ Lst showing the relation of the LP-interval and the narrower derived version that involves more data. Here it has been recognized that from Kts = (Lst )−1 follows −1 Kt...s = Ls...t and therefore −1 Hts = max... Kt...s = min... Ls...t = (Mst )−1 , where in each case … is understood so the chain s . . . t is the reverse of t . . . s. The following is to expose the nature of our approach and the part that chain indices have in it. The objective is a formula for a system of price indices Prs (r, s = 1, . . . , m) for consistent cost-of-living conversions between m periods. From that consistency, by which conversion from r to s and then s to t coincides with that from r to t, they must be expressible in the form of a system of ratios (P)

Prs = Pr /Ps

between numbers which, from this representation, have the role of price levels. In other words, equivalently, they have the property Prs Pst = Pst listed among Fisher’s well known “Tests” as the “Chain Test”.

256 Notes For a point of origin let us start with the already considered property (O)

Prs ≤ Lrs ,

which is the beginning, and at the same time not far from the end, of price index theory. By substitution of price indices with their expressions as ratios of price levels, we have (L)

Lrs ≥ Pr /Ps

as a system of inequalities for price levels, from which follows the extended chain system (chain L)

Lr...s ≥ Pr /Ps

and then the cyclical products consistency test (cycle L)

Lr...r ≥ 1

is a conclusion. For our formula we now have: price indices given by (P) for price levels that are a solution of (L) For any solution of (L) for price levels, and hence for the price indices given by (P), we now have Kst ≤ Hst ≤ Ps /Pt ≤ Mst ≤ Lst directly, and hence Kst ≤ Hst ≤ Pst ≤ Mst ≤ Lst showing the relation of the LP-interval for bounding the price index and the narrower derived version that involves more data. It is the idea represented here, of narrowing the price index bounds, that has been taken up with interest, lifted out of the context in our approach. Firstly, in that context, it depends on the cyclical product test Lr...r ≥ 1, that assures the new bounds, hopefully narrower, are not so narrow as to leave an empty set, or even not narrower at all, but wider. A further point is that the price index to be bounded is not any price index, but one from a chain consistent system, as obtained by our approach. That may be discouragement enough, but there are further points relevant to some applications, where the chains are not min or max but seemingly arbitrary, leaving signiﬁcance even more unanchored.

Notes 257

Bibliography Balk, Bert (2004). Direct and Chained Indices: A Review of Two Paradigms. Presented at the SSHRC International Conference on Index Number Theory and Measurement of Prices and Productivity, Vancouver, 30 June-3 July 2004. Dowrick, Steve and John Quiggin (1994) International Comparisons of Living Standards and Tastes: A Revealed-Preference Analysis. American Economic Review 84 (1), 332–41. — (1997) True Measures of GDP and Convergence. American Economic Review 87 (1), 41–64. Hill, Robert J. (1999a), Comparing Price Levels across Countries Using Minimum Spanning Trees. Review of Economics and Statistics 81 (1), 135–42. — (1999b). International Comparisons using Spanning Trees. In A. Heston and R.E. Lipsey (eds.), International and Interarea Comparisons of Income, Output, and Prices, Studies in Income and Wealth Volume 61, NBER, University of Chicago Press. — (2001a). Measuring Inﬂation and Growth Using Spanning Trees. International Economic Review 44 (1), 145–62. — (2001b). Linking Countries and Regions using Chaining Methods and Spanning Trees. Joint World Bank-OECD Seminar on Purchasing Power Parities Recent Advances in Methods and Applications, Washington, D.C., 30 January–2 February 2001. — (2004). Constructing Price Indexes across Space and Time: The Case of the European Union. American Economic Review 94 (5), 1379–1410. — (2006). When Does Chaining Reduce the Paasche-Laspeyres Spread? An Application to Scanner Data. Review of Income and Wealth 52 (2), 309–25. Jorgenson, Dale W. and Zvi Griliches (1967). The Explanation of Productivity Change. Review of Economics Studies 34, 249–83. — (1971). Divisia Index Numbers and Productivity Measurement. Review of Income and Wealth 17 (2), 53–55. Lehr, J. (1885). Beiträge zur Statistik der Preise. Frankfurt: J.D. Sauerlander. Lippe, P. von der (2001). Chain Indices: A Study in Price Index Theory. Spectrum of Federal Statistics, Volume 16. Stuttgart: Metzler-Peschel. Manser, Marilyn E. and Richard J. McDonald (1988). An Analysis of Substitution Bias in Measuring Inﬂation, 1959–85. Econometrica 56 (4), 909–30. Marshall, A. (1887). Remedies for Fluctuations of General Prices. Contemporary Review 51: 355–75. Neary, Peter (2004). Rationalizing the Penn World Table: True Multilateral Indices for International Comparisons of Real Income, American Economic Review 94 (5), 1411–28. Samuelson, P. A. (1947). Foundations of Economic Analysis. Cambridge, MA: Harvard University Press. — (1984). Second Thoughts on Analytical Income Comparisons. Economic Journal 94: 267–78. — and S. Swammy (1974), Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis, American Economic Review 64 (4): 566–93.

Postscript

review document that has reference to our Chapter 2, the main chapter in this book, does service in provoking—what seems suitable—a review of the review. But now it seems perhaps good practice anyway to give deserved illumination to a review. According to this document: the method “shows in essence that imposing additional information improves the precision of estimated cost-of-living indices.” Hardly the essence. Our method is to simultaneously construct the many price indices between many periods that are both consistent and true. The matter of improved bounds using chain indices is a simple by-product, and a useless one when taken apart from the proper framework, as it usually is. These indices are consistent, in the chain sense, in that exchanging money from a to b and then b to c should amount to the same as from a to c. That is required by the nature of price indices. Fisher had something to do with all that, but he had a ﬁxation, like everyone else, that a single algebraical formula was to apply to all the price indices in the usual way and obey his Chain Test—a hopeless undertaking. They are true, in the utility sense, taken together, meaning they are all associated with a single utility that ﬁts the data. The facile statement with use of terms ‘precision’ and ‘estimated’, as if we should know what they should now mean, would in another setting be interestingly problematic but here it is completely empty. The last paragraph in the document asserts we confuse two meaning of “true”. One is the usual meaning of the theoretical subject where the index derives from a utility that ﬁts the data. The one good word is replace by a rambling rigmarole to make their different meaning different. For the other meaning, if the utility is based on a certain function form then a certain index is true, in the ﬁrst sense. But function form has no welfare signiﬁcance whatsoever, so an index with this sense of true is quite undistinguished. The confusion is certainly ours, and we welcome it. This document, beside a mentioned AER paper, which they complain is cited but not discussed, is classic in ignoring our method, and disregard of the discipline required for their main interest, the formation of chain indices. Discussion of that paper seems after all not now out of place.

A

260 Postscript Here are comments about the mentioned “Ideal Afriat Index” in the AER paper. Of course any way there should not be the association of an individual with a formula completely unknown to them. This formula goes backwards to the Irving Fisher way. Like the Fisher “Ideal”, when computed between several periods, produces price indices in no way having chain consistency, as stated by Fisher’s “Chain Test”, let alone be true together. In our method we give account of a formula for determining simultaneously all the price indices between many periods, both consistent and true as usual, which reduce to Fisher’s index in the case of two periods. We use this result, picked out of the normal indeterminacy range for many periods, as a ﬁnal answer, just as one might pick the Fisher index out of the indeterminacy range represented by the Laspeyres-Paasche interval. This could be, maybe, a sort-of “Ideal Afriat Index”, but now going forwards instead of backwards. We used to cite those authors properly out of respect for having used their data to play with during software development, but that turned out to be a great mistake. For it brought them to attention and caused their advice to be sought about publication of our work. That those before others more qualiﬁed should be chosen, who bring out the ﬂavour of a decadence as its quintessential embodiment, would be a disgrace to professional economics were not our particular subject unfamiliar to most, so they can be easy victims of the overloaded hostile posturing of these ignorant and arrogant would-be authorities. Everywhere in this document there is something wrong, but this is attention enough, though it has been useful for provoking corrective emphasis about interesting points. It is as well to conclude with observation that an understanding of the very ﬁrst things about price indices is manifestly absent—very odd for someone being consulted as an authority. It was settled a half-century ago that dealing with a price index in terms of utility requires that utility to be conical, or linearly homogeneous (see Appendix 2 about terminology). Our Appendix 1 deals with that point. Yet the received document goes on preaching about this being an unwarranted extra assumption. It also presses for use of the entirely unsuitable term “homothetic”— because others use it! Our Appendix 2 deals with that matter. Here it is suitable to express appreciation and thanks to our publisher who— unlike a quite likely well-meaning journal editor—put aside another out-to-kill ‘review’ in favour of others we have been very happy to receive and to quote; our thanks go to those also.

Index

Afriat, S.N. xviii, 5, 7, 10, 13, 16–17, 22, 40, 43–4, 111 et seq, 121 et seq, 144, 149, 223 Allen, Robert C. 160 Allen, Sir Roy 7 associated price indices 13, 226–7 Bainbridge, S. 40, 153 barycentric coordinates 37 “basic” price indices 36, 54–5, 65–7, 74, 81 Bernstein-Weierstrauss approximation 164 Blackorby, Charles 171 Box, G.E.P. 181–2 Box-Cox function 182–3, 187–91 Brown, Alan 16 budget constraints 44, 56, 58, 132, 199 Byushgens, S.S. 17, 159 Caves, Douglas W. 21, 166, 176–7, 188 chain coefﬁcients 114–15, 135 chain indices 9–10, 29–32, 73, 254–6 chain test 5–6, 27–8, 41, 98, 232, 252, 255–6, 259–60 Christensen, Laurits R.166, 176–7, 188 circularity test 6, 28 citation analysis 17–18 compatibility 133–8, 146–9; homogeneous 136–7, 148; strict 141 cones and conical functions 42, 235 conical revealed preference 43 conical utility 9–10, 13–16, 43, 46–7, 53–5, 130, 132, 136, 228, 231, 260 consistency 38–42, 57–8, 73, 76–7, 121–2, 125, 129, 133–42, 252, 259; classical 145; homogeneous 133–5, 139, 142 constant returns to scale 233 consumption error 44 cost effective demand 12–13, 44, 133, 146 cost efﬁcacy 57, 59

cost efﬁcient demand 12–13, 37, 42–7, 56–64, 75–6, 84, 133, 146 cost-of-living indices 12, 259 cost-of-living problem 225–6 Cox, D.R. 181–2 critical cost efﬁciency 47, 56, 60, 75–6 critical cost functions 149 cycle reversibility 122–4 cyclical non-negativity condition 150, 152 cyclical product test (or cycle test) 30–4, 37–40, 43, 49, 73 Deaton, Angus 21–2 demand correspondence 55–8 Denny, Michael 184, 188 derived systems 33–9, 54, 116–17, 120, 151–2 Diewert, W.E. 7–8, 17, 21–3, 159–69, 176–9, 184–96 Divisia indices 172–3, 254 Edmunds, J. 40, 144 efﬁciency modiﬁcation 47 Ehemann, Christian 191 Eichhora, W. 129 EUKLEMS Project xvii, 70, 94 Euler’s theorem 163, 188–9, 194, 200 “exact” index numbers 159, 175, 196–7 exhaustion property 39 extension property 37–41, 54, 119–21, 134, 151 ﬁnite consistency tests 142–3 Fisher, Irving xviii–xix, 7–8, 16, 18, 21, 23, 132, 159–61, 259–60; tests for index number formulae xvii, 4–6, 16, 26–34, 41, 66, 71, 98, 129–30, 232–3, 252, 255; see also “ideal” index Fleetwood, William 6, 28

262 Index Førsund, Finn R. 17–18 Fostel, A. xviii Fusa, Melvyn 164

Nissen, David 171 non-parametric formulae xix, 71 normal demand 135 notation 238–9

Griliches, Zvi 254 order exhaustion property 152 Hicks, J.R. 11, 14, 30, 227, 230–1 Hill, Robert J. 160, 197 homothetic separability 170–2, 185–6, 192 Hotelling, Harold S. 200 Houthakker, H.S. 42, 58, 114, 141–3 Hulten, Charles R. 172 “ideal” index 5–9, 16–22, 26–7, 53, 63–6, 72, 78, 81, 129, 159–60, 191–3, 260 idempotence 35–6, 40, 100 identity test 5–6, 26 index number problem xvii, xviii, 4–5, 16, 26, 66, 70, 233 index number theorem 11, 14, 30, 133–4, 227, 230–1 indicators 187 International Comparison Project 70, 77 interval exhaustion property 152 interval test 34 Jorgenson, Dale W. 254 Konüs-Byushgens indicator 185, 190–1 Lady, George 171 Lagrange multiplier 199–200 Laspeyres indices xix, 7–11, 14–15, 20, 29–30, 46, 62–3, 71, 77–9, 97, 131–5, 163, 172, 230 Lau, Lawrence J. 184–5 Lehr, J. 254 linear programming methods 144, 147 McFadden, Daniel 164 Malmquist, S. 21–2 Marshall, A. 254 median solutions 123–4 Milana, Carlo xviii, 8, 10, 21, 23, 157 et seq minimal chains 31–2, 115–16 “minimum spanning tree” methodology 254 monotonicity 40, 132–3, 180 Morrison, Catherine J. 178–9 Muellbauer, J. 21–2 multilateral indices 250–2 Mundlak, Yair 164

Paasche indices xix, 7–11, 20, 29–30, 63, 71, 78, 133–4, 163, 172, 230 perfect efﬁciency 60 power algorithm 40, 54, 75, 153–4 preference orders 113–14 production functions 198–9 purchasing power of money 5, 27 quadratic approximation 166–9 quadratic utility 17, 22–3, 64, 159 quantity and price, duality between 74, 229 Quiggin, John 251 ratio matrices 99–100 ratio test 6 rays 42 rectiﬁcation of pair comparisons 129 Reinsdorf, Marshall B. 191 returns to scale 199, 233 revealed contradictions 138–40 revealed preference 9, 11, 16, 42–3, 56–8, 114, 137–8, 143, 159, 251; strong axiom of 141 reversal test 129 Russell, R. Robert 171 Russell, Richard 241 et seq Samuelson, Paul A. 22, 42, 44, 58, 159, 163, 254 Sarafoglou, Nikias 17–18 Scarf, H.E. xviii SchuItz, H. 159 separability: strong 174; weak 173–4; see also homothetic separability Shephard, Ronald W. 163, 171 simple systems 125 simplex of reference 16, 29, 37, 55 standard of living 4–5, 10, 12, 26–7, 43–4, 130, 225–6, 230, 233, 251 Stigler, George J. 17–18 strict compatibility 57 “superlative” index number formulae 7–8, 17–23, 64, 159–61, 166–9, 175, 193–7 Swamy, S. 22, 44, 254

Index 263 time reversal test 5–6, 16, 26–7 Todd, M.J. xviii Törnqvist-type indices 18–21, 160, 188–9 translog identity 176 triangle equality 99–100, 120 triangle inequality 34–40, 51–5, 98, 118–20, 151–3 “true” price indices 14–18, 23, 47, 53, 64–5, 72, 133, 198, 259–60 utility 12–15, 43–7, 56, 67–8, 130–6, 199–200, 225–8, 233, 259; construction of 144–5; and demand 136–7; see also conical utility; quadratic utility

utility cost factorization theorem xvii–xviii, 45–9, 53, 70, 226 utility cost functions 145–8, 225, 227 van Veelen, Matthijs 250–1 Vartia, Yrjo 7 Voeller, J. 129 Walsh index numbers 7, 18–23, 160, 190 Wold, Herman 200 Young’s theorem 164, 194

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