Applications of the Expansion Method

APPLICATIONS OF THE EXPANSION METHOD The social sciences are currently engaged in a critical selfscrutiny regarding th...

Author: John Jones III

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APPLICATIONS OF THE EXPANSION METHOD

The social sciences are currently engaged in a critical selfscrutiny regarding the universality of their theories and models. While there is increasing recognition that the social counterparts of invariant natural laws such as the law of gravity will not be discovered, many social scientists nevertheless construct and estimate models under implicit assumptions of invariance and universality. The central tenets of the expansion method paradigm are that models are likely to hold differently across environments and that the model-context nexus should be theorized and investigated. The expansion method provides a means for introducing the complexities of real world contexts into the decontextualized models, conceptual frameworks, and theories of the social sciences. As a research paradigm, the expansion method provides a systematic methodology appropriate for the investigation of contextual variability in virtually any empirical research setting. This is the first book to bring together researchers with interest in the expansion method. The authors examine the theoretical implications of the paradigm, contribute methodological advances, and offer variety of applications in substantive areas, including population, urban systems, social policy analysis, economic development, and remote sensing. The book will be of interest to those whose substantive research interests involve modelling, whether in geography or in any other social science. John Paul Jones III is Associate Professor of Geography at the University of Kentucky. He is the author of a number of articles and co-edited (with Janet E.Kodras) Geographic Dimensions of US Social Policy (Edward Arnold, 1990). Emilio Casetti is Professor of Geography at Ohio State University. He is editor of Geographical Analysis, and is the author of more than 100 articles.

APPLICATIONS OF THE EXPANSION METHOD Edited by John Paul Jones, III and Emilio Casetti

London and New York

First published 1992 by Routledge 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Routledge a division of Routledge, Chapman and Hall, Inc. 29 West 35th Street, New York, NY 10001 Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2005. “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.” © 1992 J.P.Jones III and E.Casetti All rights reserved. No part of this book may be reprinted or reproduced or utilized 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 Applications of the Expansion Method 1. Human geography I. Jones, John Paul 1955– II. Casetti, Emilio 1928– 304.20724 ISBN 0-203-40538-2 Master e-book ISBN

ISBN 0-203-71362-1 (Adobe eReader Format) ISBN 0-415-03494-9 (Print Edition) Library of Congress Cataloging in Publication Data Applications of the Expansion Method/Edited by John Paul Jones III and Emilio Casetti p. cm. Includes bibliographical references and index. ISBN 0-415-03494-9 1. Social sciences—Mathematical models 2. Population geography— Mathematical models 3. Economic geography—Mathematical models I. Jones, John Paul, 1955– II. Casetti, Emilio, 1928– III. Title: Expansion method H61.25.A63 1992 300′.01′5118–dc20

CONTENTS

List of figures

vi

List of tables

ix

Contributors

xi

Acknowledgments

xiii

1

AN INTRODUCTION TO THE EXPANSION METHOD AND TO ITS APPLICATIONS Emilio Casetti and John Paul Jones, III

1

2

THE DUAL EXPANSION METHOD: AN APPLICATION FOR EVALUATING THE EFFECTS OF POPULATION GROWTH ON DEVELOPMENT Emilio Casetti

8

3

PARADIGMATIC DIMENSIONS OF THE EXPANSION METHOD John Paul Jones, III

32

4

A CONTEXTUAL EXPANSION OF THE WELFARE MODEL Janet E.Kodras

47

5

A COMPARISON OF DRIFT ANALYSES AND THE EXPANSION METHOD: THE EVALUATION OF FEDERAL POLICIES ON THE SUPPLY OF PHYSICIANS Stuart A.Foster, Wilpen L.Gorr and Francis C. Wimberly

71

6

PERSONAL CHARACTERISTICS IN MODELS OF MIGRATION DECISIONS: AN ANALYSIS OF DESTINATION CHOICE IN ECUADOR Mark EllisJohn Odland

88

7

ALTERNATIVE APPROACHES TO THE STUDY OF METROPOLITAN DECENTRALIZATION Shaul Krakover

101

8

LONG-WAVE SPATIAL AND ECONOMIC RELATIONSHIPS IN URBAN DEVELOPMENT

125

v

Shaul Krakover and Richard L.Morrill 9

AN INVESTIGATION INTO THE DYNAMICS OF DEVELOPMENT INEQUALITIES VIA EXPANDED RANKSIZE FUNCTIONS C.Cindy Fan

144

10

IDENTIFYING HIERARCHICAL DEVELOPMENT TRENDS IN THE HUNGARIAN URBAN SYSTEM USING THE EXPANSION METHOD Darrick R.Danta

166

11

AN EXPLORATION OF THE RELATIONSHIP BETWEEN SECTORAL LABOR SHARES AND ECONOMIC DEVELOPMENT Kavita Pandit

179

12

PRODUCTION FUNCTION ESTIMATION AND THE SPATIAL STRUCTURE OF AGRICULTURE Sent Visser

197

13

INCORPORATING THE EXPANSION METHOD INTO REMOTE SENSING-BASED WATER QUALITY ANALYSES Martin Miles, Douglas A.Stow and John Paul Jones, III

219

14

INNOVATION DIFFUSION THEORY AND THE EXPANSION METHOD Michael Sonis

234

15

SPATIAL DEPENDENCE AND SPATIAL HETEROGENEITY: MODEL SPECIFICATION ISSUES IN THE SPATIAL EXPANSION PARADIGM Luc Anselin

264

16

GENERATING VARYING PARAMETER MODELS USING CUBIC SPLINE FUNCTIONS Robert Q.Hanham

280

Index

287

FIGURES

2.1 Effect of population growth on the rate of development 23 2.2 Phase diagram of the relationship between rates and levels of economic 23 development (corresponding to population growth rates of 1, 2, and 3 percent per year) 4.1 Interstate variations in the work-disincentive effect, 1979 59 5.1 Parameter paths for LPRIMR in the GPRIM model 82 5.2 Parameter paths for GPOP in the GPRIM model 83 5.3 Parameter paths for LSPECR in the GSPEC model 83 5.4 Parameter paths for GPOP in the GSPEC model 83 7.1 Urban settlements in the urban region of Tel Aviv 108 7.2 Population development in the Tel Aviv urban region, 1961–80 109 7.3 Shares of population, central city versus suburbs, Tel Aviv urban region,110 1961–80 7.4 Index of population growth, central city versus suburbs, Tel Aviv urban 110 region, 1961–80 7.5 Growth profiles obtained via two different distance bands delineations, 113 Tel Aviv urban region, 1961–80 7.6 Distance-temporal structure of population growth in the Tel Aviv urban 114 region, 1961–80 7.7 Spatio-temporal structure of population growth in the urban region of 118 Tel Aviv, 1970 and 1980 7.8 Distribution of population growth in the urban region of Tel Aviv, raw 118 data, 1961–80 7.9 Southward cross-section of the spatio-temporal population growth 119 structure, urban region of Tel Aviv, 1961–80 7.10 Analysis of southward cross-section from Tel Aviv to Qiryat Eqron, 120 1965–80 8.1 Selected study areas 131 8.2 Estimated spatio-temporal growth structure for the urban region of 135 Philadelphia 8.3 Estimated spatio-temporal growth structure for the urban region of 137 Chicago 8.4 Estimated spatio-temporal growth structure for the urban region of 139 Atlanta 9.1 Zipf’s ideal rank-size distribution 146

vii

9.2 (a) Perfect equality rank-size distribution; (b) perfect inequality rank- 148 size distribution 9.3 Nonlinear rank-size curves 151 9.4 (a) Scatter diagram of ln y and ln r, 1913; (b) scatter diagram of ln y and 156 ln r, 1929; (c) scatter diagram of ln y and ln r, 1950; (d) scatter diagram of ln y and ln r, 1960; (e) scatter diagram of ln y and ln r, 1970; (f) scatter diagram of ln y and ln r, 1980 10.1 The urban turnaround model 167 10.2 Polarized growth 168 10.3 Hungarian rank-size distributions 169 10.4 Estimates of q by rank 174 10.5 Timing of switch of q from increasing to decreasing 174 11.1 Agricultural and manufacturing labor allocation during development 186 11.2 Effect of population size on sectoral labor allocation relations 192 11.3 Effect of resource flows on sectoral labor allocation relations 192 11.4 Temporal variation in sectoral labor allocation relations 193 11.5 Sectoral labor allocation relations for (a) Latin America, (b) Africa, (c) 193 Asia, and (d) more developed countries 12.1 Observations of the relationship between inputs and outputs relative to 198 the actual production function defined for optimal combinations of inputs 12.2 Observations of the relationship between inputs and outputs for farmers 199 responding to variation in real factor and output prices 12.3 Hypothesized shape of agricultural production functions with increasing 204 marginal returns to intensity at low levels of intensity 12.4 Production function shapes and the location allocation of output types 205 12.5 Empirical observations of yield and intensity and the underlying 206 production functions 12.6 Unique linear-log production functions for individual agricultural types 214 generate a production function envelope that is estimated as a CobbDouglas function 12.7 The effect of varying annual prices of output on estimation of 215 production functions measured in terms of value of yield 13.1 Spatial distribution of sampling sites in Neuse Estuary 223 13.2 Distribution of b parameter from turbidity model 226 13.3 Distribution of b parameter from salinity model 227 13.4 Distribution of c parameter from salinity model 227 13.5 Distribution of b parameter from total suspended solids model 229 13.6 Distribution of c parameter from total suspended solids model 229 13.7 Distribution of b parameter from chlorophyll-a model 230 14.1 The operational stages of the expansion method 237 14.2 Cumulative temporal S-shaped growth of the relative portion of adopters239 of an innovation: (a) innovation diffusion within an indifferent environment; (b) innovation diffusion within an active environment

viii

14.3 Scheme of the redistribution of an innovation between adopters and nonadopters caused by the intervention of an active environment 14.4 Construction of the level curves for general spatio-temporal innovation spread 14.5 Qualitative description of innovation diffusion dynamics with asymptotically stable initial and final equilibria 14.6 Interconnections between diffusion of competitive innovations and individual utility choice within an active environment 16.1 Moving window regression time plot of β for Pittsburgh 16.2 Quadratic and cubic spline time plot of β for Pittsburgh

241 245 249 255 283 285

TABLES

2.1 A compilation of correlations between rates of growth of population and 15 product per capita 2.2 Tabulation of PRC1(P'), PRC2(P'), and g[y*(P')] evaluated at a range of 24 values of P' 4.1 Major income maintenance programs, 1979 50 4.2 Varimax rotated factor matrix 55 4.3 Results of the initial model 57 5.1 Descriptive statistics: 1963–83 annual data for the contiguous forty76 eight states 5.2 Ordinary least squares estimates of the terminal model: quadratic 76 expansions in time 5.3 Annual regressions for GPRIM, model (5.5): estimated coefficients and 77 p values 5.4 Annual regression estimates for GSPEC, model (5.6): estimated 78 coefficients and p values 5.5 Three-year moving-average window regressions for GPRIM, model (5. 79 5): estimated coefficients and p values (n=144) 5.6 Three-year moving-average window regressions for GSPEC, model (5. 80 6): estimated coefficients and p values (n=144) 6.1 Coefficients of the terminal model for age categories of males 93 6.2 (a) Parameter estimates for origins in the Costa region; (b) parameter 96 estimates for origins in the Sierra region; (c) parameter estimates for origins in the Oriente region 7.1 Components of decentralization as treated by the four methods 111 7.2 Population and population growth in the Tel Aviv urban region by rings 111 7.3 Location and shifts of the peak point of growth: Tel Aviv, 1961–80 115 7.4 Cross-section to the south from Tel Aviv (128, 36) to Qiryat Eqron 119 (133, 60) 8.1 Summary of regression results 132 8.2 Conformity of results with hypotheses 140 9.1 (a) Estimates for linear rank-size functions; (b) estimates for linear rank-154 size function expanded in time 9.2 Estimates for terminal model: In y'=a+ [b=f(r)] In r 158 9.3 Estimates for terminal models (9.20) and (9.21) 161 9.4 Values of db/dt for selected ranks 162 10.1 Results of rank and time expansion analysis: Hungary’s urban system 172

x

11.1 Structural changes during economic development 11.2 Regression results for the Chenery and Syrquin model 11.3 Regression results for the exponential model 11.4 Regression results—expansion by population 11.5 Regression results—expansion by resource flow 11.6 Regression results—temporal expansion 11.7 Regression results—spatial expansion 12.1 Key to regression variables 12.2 Cobb-Douglas production function estimates 12.3 Linear-log model production function estimates 13.1 Range of observed water quality values 13.2 Range of mean Landsat multispectral scanner digital numbers 13.3 Summary of results

179 184 185 188 189 190 190 210 211 212 223 224 225

CONTRIBUTORS

Luc Anselin is Associate Director, National Center for Geographic Information and Analysis and Professor, Departments of Geography and Economics, University of California, Santa Barbara, California. Emilio Casetti is Professor, Department of Geography, The Ohio State University, Columbus, Ohio. Darrick R.Danta is Associate Professor, Department of Geography, California State University, Northridge, California. Mark Ellis is Assistant Professor, Department of Geography, Florida State University, Tallahassee, Florida. C.Cindy Fan is Assistant Professor, Department of Geography, University of California, Los Angeles, California. Stuart A.Foster is Assistant Professor, Department of Geography and Geology, Western Kentucky University, Bowling Green, Kentucky. Wilpen L.Gorr is Professor, School of Urban and Public Affairs, Carnegie Mellon University, Pittsburgh, Pennsylvania. Robert Q.Hanham is Associate Professor, Department of Geology and Geography, West Virginia University, Morgantown, West Virginia. John Paul Jones, III is Associate Professor, Department of Geography, University of Kentucky, Lexington, Kentucky. Janet E.Kodras is Associate Professor, Department of Geography, Florida State University, Tallahassee, Florida. Shaul Krakover is Senior Lecturer, Department of Geography, Ben Gurion University of the Negev, Beer Sheva, Israel. Martin Miles is a Ph.D. candidate, Department of Geography, University of Colorado, Boulder, Colorado. Richard L.Morrill is Professor, Department of Geography, University of Washington, Seattle, Washington. John Odland is Professor, Department of Geography, Indiana University, Bloomington, Indiana.

xii

Kavita Pandit is Assistant Professor, Department of Geography, University of Georgia, Athens, Georgia. Michael Sonis is Associate Professor, Department of Geography, Bar-Han University, Ramat-Gan, Israel, and Adjunct Professor, Department of Geography, University of Illinois, Urbana-Champaign, Illinois. Douglas A.Stow is Professor, Department of Geography, San Diego State University, San Diego, California. Sent Visser is Associate Professor, Department of Geography and Planning, Southwest Texas State University, San Marcos, Texas. Francis C.Wimberly is Senior Computer Scientist, Pittsburgh Supercomputing Center, Carnegie Mellon University, Pittsburgh, Pennsylvania.

ACKNOWLEDGMENTS

We would like to express thanks to the editors of the IEEE Transactions on Systems, Man, and Cybernetics, the Annals, Association of American Geographers, and Modeling and Simulation, for permission to reprint chapters 2, 4, and 16, respectively. In addition, we would like to thank Steve Grant, Department of Geography, University of Kentucky, for assistance in preparing some of the graphics appearing in this book.

1 AN INTRODUCTION TO THE EXPANSION METHOD AND TO ITS APPLICATIONS Emilio Casetti and John Paul Jones, III

In this introduction we present an overview of the applications of the expansion methodology appearing in this book. First, however, it is useful to outline what the expansion method is, and why you, the reader, might, or should be, interested in it. Often, the processes of scientific inquiry identify critical variables and ‘important relationships’ among them. These relationships are likely to reflect and incorporate theoretical presuppositions, and are eventually formalized into mathematical models and estimated. Production functions, demand functions, the rank-size rule, and spatial interaction models are examples of such ‘important’ or ‘special status’ relationships. Relationships such as these play a central role in the contemporary social sciences. Disciplines such as economics, psychology, or political science grew by carving from a common matrix certain ‘proprietary’ clusters of important relationships. The standing of individual scientific disciplines tends to be related to a major degree to their success in identifying, theorizing, modeling, and estimating such distinctive relationships. There is no question that the abstraction of simple important relations from complex contexts can provide very significant additions to knowledge. Nevertheless, many limitations and shortcomings of the social sciences and their models can be traced to the same processes of abstraction that are also responsible for these advances. Simple and elegant models can yield important insights into naturam rerum (the nature of things), but they are in all likelihood inadequate for understanding complex realities and for intervening to change them. There is a need to reintroduce the complexities of the real world into simple theoretically grounded mathematical models without destroying these models in the process. In fact, the simple models and the simplified important relationships prevalent in the contemporary social sciences should be regarded as early steps in the growth of knowledge, rather than the end point and culmination of it. The expansion methodology combines a technique and a research philosophy that is especially well suited to bring together simple models and complex realities.

2 E.CASETTI AND J.P.JONES, III

The expansion method is both a technique for creating or modifying mathematical models and a research paradigm. As a technique, it consists of the following well-defined operational steps: (a) an ‘initial model’ is specified; the model is made of variables and/or random variables and at least some of its parameters are in letter form; (b) at least some of the letter parameters in the initial model are redefined by ‘expansion equations’ into functions of variables and/or random variables; in many cases these are substantively significant indices representing a context; (c) the expanded parameters are replaced into the initial model to create a ‘terminal model’; and (d) the expansions can be iterated, since the terminal model produced by one expansion can become the initial model of a subsequent one. Suppose that we take as initial model an important relationship with strong theoretical grounding, and that the expansion equations model the contextual variation of this relationship. Then the terminal model obtained from the two will encompass in the same entity both the model and its contextual drift. Thus, the identity of the initial model is preserved, but at the same time the initial model is rendered capable of addressing complex contextual realities that were previously not part of it. The expansion methodology is also a research philosophy which carries within itself the suggestion that important theoretically grounded relationships should be regarded as building blocks of more complex theoretical structures encompassing both them and their contexts or environments. Specifically, these higher structures should reflect both the theory behind the initial model and the theory about the nexus between the initial model and its contexts. Clearly, the expansion paradigm has major implications as regards estimation. The theoretically grounded relationships from individual disciplines tend to be investigated and estimated under the implicit presupposition that they possess some form of quasi universal validity (i.e. invariance). Certainly, in most cases, they are presumed to be invariant over the data sets from which they are estimated. In contrast, the expansion methodology suggests that presuppositions of invariance are almost always unwarranted. Instead, the variation of relationships across contexts should be presumed, investigated, tested for, and theorized. The ‘invariance’ or ‘universal validity’ of a relation should be a conclusion arising from an extensive, protracted, and unsuccessful search for contextual variation, rather than a presupposition. Furthermore, the contextual variation of relationships should not be regarded as a nuisance or an aberration, as is currently the case. On the contrary, the theoretical and empirical investigation of the variation of important relations across contexts should be regarded as the obvious second phase of any scientific effort that has brought these relationships into focus. In this next phase, potentially relevant contexts and environments should be focused upon to determine whether a relationship drifts across them, and to theorize why we should expect such drift to occur.

AN INTRODUCTION TO THE EXPANSION METHOD 3

The word paradigm has diverse meanings. However, it is often used to denote an intertwined cluster of research questions and operational approaches/ techniques to obtain answers to these questions. In this sense, the expansion methodology is a paradigm, since it suggests that researchers ask questions about the contextual variation of relations while at the same time it provides the operational routines to model this variation and to test for its occurrence. Research involving mathematical models involves diverse activities and is carried out within diverse schools of thought. To exemplify, let us consider some cases. One class of model-oriented research aims at determining the optimum states or optimum time paths of systems by techniques such as mathematical programming, optimum control, and others. Other activities are concerned with extracting the implications of models. Examples include research on systems of equations (as in input-output studies), the solving of differential equations, the execution of simulations (as in the System Dynamics tradition), and the investigation of the qualitative properties of dynamic systems. Other types of model-oriented research are concerned with the estimation of a model’s parameters using empirical data. Estimation work is carried out by practitioners and theoreticians such as engineers, econometricians, statisticians, geographers, and physicists, to name a few, all of whom are very different in their objectives, concerns, and preferences. Their approaches may differ in the extent to which a researcher is committed to a specific model or, alternatively, is willing to consider variants or alternatives to it; in the emphasis on substantive modeling vis-à-vis the specification of error terms; in the manner and extent to which prior information is brought to bear upon the estimation process; and so on. The vast diversity of mathematical modeling is placed into focus here in order to make the point that the expansion methodology can be applied to modeloriented research of any kind and within an open-ended spectrum of research approaches. For instance, it can be used to construct and modify very abstract models within a frame of reference encompassing the qualitative study of differential equations in which no estimation is contemplated, or to construct or modify models within, say, an econometric perspective. Indeed, the expansion method has a far greater potential, both methodologically and substantively, than is represented by the diversity of papers in this volume—most of which have been written by scholars with primarily substantive interests. These papers, introduced in the paragraphs that follow, reflect their authors’ perceptions of the expansion method and correspond to their diverse substantive and methodological preferences. Casetti’s paper is a reprint of a 1986 statement on the expansion method that appeared in the IEEE Transactions on Systems, Man, and Cybernetics. The paper provides a guide to diverse applications of the expansion method. It also introduces ‘dual expansions’, a methodology which enables the researcher to investigate the duality between model and context using the expansion method. Casetti shows that when a model is expanded with respect to contextual


variables, an implicit second model becomes defined in which the primal context becomes the dual model and the primal model becomes the dual context. The paper illustrates the model-context duality in an empirical study of economic development and population growth. The next contribution, by Jones, discusses the paradigmatic aspects of the expansion methodology. He explores the implications of the expansion method for ‘open’ research, for altering research trajectories, for testing alternative theoretical frameworks, and for micro and macro level analyses. Jones then uses the expansion method to undercut the distinction between regional and systematic geography. The paper ends by drawing some parallels between the expansion method and analyses employing scientific realism. Kodras’ paper focuses upon the spatial variation of the relationship between participation in welfare programs and welfare benefits. Traditionally, debates over welfare have been dominated by opposing theories reflecting liberal and conservative perspectives. The liberal theory views welfare provision as a policy response to social needs, with the corollary that greater welfare participation is the counterpart of greater social need. The conservatives’ work-disincentive theory formalizes the notion that high welfare benefits discourage participation in the labor force and encourage participation in welfare programs. The research in this area has been largely aimed at determining the validity of these theories, or at most their comparative ability to explain reality. Kodras, instead, investigates spatial variation in the explanatory power of these theories. Her paper is concerned with participation in the Aid to Families with Dependent Children program. In a capsule, Kodras expands an initial model relating program participation and benefits into varimax rotated factors extracted from a number of relevant contextual variables. Her conclusions are that ‘each position in the welfare debate is more valid in some places than in others because the programs have different impacts in different contexts’, and that the spatial pattern of welfare provision is characterized by a mismatch between welfare services and welfare needs. Foster, Gorr, and Wimberly address the comparative merits and the complementarities of drift analyses versus expansion approaches in the study of parametric variation. Drift analyses involve multiple estimations of an initial model, for instance at different points/regions in geographic space, or for different points/intervals in time. In this paper several specifications of moving window regressions are estimated and their results are contrasted with those produced by expansions. The initial model is a functional relation between the growth rates of physicians, the dependent variable, versus density of physicians and population growth, the independent variables. State level data from the American Medical Association master files are employed. The empirical analyses in the Foster et al. paper are suggested by the literature on the locational behavior of physicians, and are designed to estimate the effects of federal programs intended to end physician shortages and maldistribution.


Using random utility theory, Ellis and Odland specify a model of destination choice functionally similar to an originspecific gravity model. This model is expanded first to allow for the distinction between urban and rural destinations and then in order to examine the effects of age and gender on destination choices. All the variables in the model are categorical or categorized. The ages of migrants are categorized into age classes, and the migration distances are categorized into distance bands. The Ellis and Odland formulation yields a 240cell contingency table; is characterized by a binomial sampling error; involves heteroskedasticity; and requires generalized least squares for its estimation. The sampling zeros in the contingency table are removed using pseudo-Bayesian estimates. The empirical analysis presented is based on an Ecuador data set with about 78,000 observations. This paper brings together concepts and formalisms from ANOVA, categorical data analysis, and the expansion method. Krakover discusses four approaches to the investigation of metropolitan decentralization, of which two are applications of the expansion method. The latter approaches involve expanding into time polynomials, respectively, the parameters of a polynomial in distance from the CBD and the parameters of a trend surface. Population growth is the dependent variable in both formulations. The methods discussed in the paper are demonstrated and contrasted by a case study for the urban region of Tel Aviv. The paper by Krakover and Morrill investigates a cluster of hypotheses concerning the dynamics of urban centralization and decentralization. Namely, they hypothesize (a) that the third Kondratieff cycle (1896–1933) coincided with metropolitan centralization; (b) that the fourth Kondratieff cycle (1933–72) coincided with metropolitan deconcentration; and that during both cycles, (c) periods of prosperity were characterized by the growth of more central counties of metropolitan areas and by decline in less central counties, while (d) recessionary years tended to exhibit inner county decline and outer county growth. These hypotheses are tested using county data for the metropolitan areas of Philadelphia, Chicago, and Atlanta. The analyses are based on a model obtained by expanding into time the parameters of an initial formulation relating population growth to distance. Both Danta’s and Fan’s contributions employ expanded rank-size models. Danta analyzes the temporal and structural changes in the Hungarian urban hierarchy between 1870 and 1986. He starts from the classical formulation relating the sizes of urban centers to their rank, and argues that the parameter associated with the rank variable is a measure of hierarchical concentration. In the conventional unexpanded formulation this measure refers to one point in time, and to the entire system. By expanding the parameters of the rank-size model with respect to time and rank, Danta generates a terminal model that can portray agglomerative and deglomerative tendencies over time, and at various levels of an urban hierarchy. His empirical analyses estimate the temporal shifts of agglomerative and deglomerative trends of the Hungarian urban system, and test the effectiveness of that country’s policies aimed at reducing urban primacy.


Fan extends the potential use of the rank-size functions to the study of inequalities. She argues that the slope parameter of a log transformed rank-size relationship is a systemic measure of inequality in any system. Fan proposes using expanded rank-size relationships to investigate the change of inequality of a system across any context, as well as within any system. This approach is applied to investigate the dynamics of development inequalities for thirty-eight countries between 1913 and 1980. A crucial suggestion arising from the expansion method paradigm is the stability of social science ‘laws’. Theoretically grounded empirical regularities with a law or quasi-law status are usually estimated under an implicit presupposition of invariance. As Pandit’s paper demonstrates, the drift of such laws across contexts is likely. She investigates the contextual drift of the law-like country level relationship between labor shares in agriculture, manufacturing, and services on one hand, and gross national product per capita on the other. Pandit’s starting point is a classical study by Chenery and Syrquin in which these relationships are estimated under an implicit assumption of invariance. Using a virtually identical data set, she is able to show that the relationships display a statistically significant drift over time and across space, meaning that they are not invariant and are thus not laws. Visser shows that ‘expansions’ are required to estimate agricultural production functions from areal data in a manner that is consistent with location theory. His argument runs as follows. Location theory tells us that under competitive market equilibrium, agricultural types and intensities are distributed in space so as to maximize rents. For a single type of agriculture the spatial pattern thus produced is one of intensities decreasing with distance from markets. For multiple agricultural types, at each point in space, only one agricultural type will have an optimum intensity that maximizes rent. However, decreasing intensities with distance from market will still prevail for each agricultural type. These propositions imply that when the parameters of an aggregate agricultural production function are estimated from areal data, they should be expanded into indices of the strength of various agricultural types in order to come to grips with the fact that each areal aggregate includes a mix of agricultural types. A successful empirical analysis concludes Visser’s paper. The remote sensing application of the expansion methodology presented by Miles, Stow, and Jones is an effort that opens up a wide vista of similar applications. In remote sensing, measurements of phenomena taken at a distance, for instance from satellites, need to be functionally related to measurements taken at the surface of the earth. These relationships provide the basis for securing high resolution, inexpensive, and reliable information on earth surface phenomena. Miles et al. argue that initial models relating satellite measurements to surface measurements can be usefully expanded in terms of substantively relevant variables for the purpose of improving our ability to make accurate inferences about earth phenomena from space. In their application, trend surface expansions of a model relating satellites and surface measurements of


several estuary water properties were tested, with results ranging from encouraging to very good. The book ends with three papers centering upon theoretical and methodological themes. Sonis employs a generalization of the expansion methodology to link geographic diffusion theory, economic utility theory, and ecological competition theory. His is an example of expansion method applications that are not directly oriented toward estimation. Anselin, on the other hand, addresses estimation themes from a spatial econometrics point of view. His paper focuses upon the issues that arise when the error terms are spatially autocorrelated and/or heteroskedastic (possibly because of stochastic expansion equations). The classes of spatial estimation issues including the ones addressed by Anselin are attracting a growing interest in geography and regional science. In the final paper Hanham discusses the expansions into ‘splines’, thus integrating the expansion methodology with a class of techniques that has been diffusing from engineering into the social sciences. His paper includes an application of spline expansions to regional unemployment response functions. In closing this introduction, we would like to express the hope that the readers of this book will experiment with expansion method techniques and themes in their own research.

2 THE DUAL EXPANSION METHOD: AN APPLICATION FOR EVALUATING THE EFFECTS OF POPULATION GROWTH ON DEVELOPMENT Emilio Casetti The progress of scientific research involves a recursive process in which two logically distinct phases can be identified. One phase is concerned with formulating or revising the disciplinary conceptual frameworks that define (a) issues and phenomena to be studied, (b) what constitutes pertinent information and data, (c) the procedures through which information and data are obtained, and finally (d) the ‘important’ relations among phenomena that are singled out for investigation. The second phase of the process is concerned with modeling. It involves constructing alternative mathematical models, assessing their consistency with empirical data, estimating their parameters, and extracting their quantitative and qualitative implications. The conceptual and modeling phases of this process interact with one another. In fact, research involves recursive iterations between the two. The results of the modeling phase, in the form of implications of models or tests of hypotheses, modifies the previous conceptual framework, which in turn generates new tasks to be carried out during a subsequent modeling phase, and so on. Most of the attention of the scholars working with models tends to focus on parameter estimation, on extracting the implications of propositions formalized into specific mathematical models, or on producing the models required by specific research leads. The logical processes by which models are arrived at tend to remain in the background and generally are not viewed as a distinct object of enquiry. And yet, an awareness of the mental and logical routines by which models are generated has the potential to render modeling simpler, more efficient, and more likely to produce better results. THE EXPANSION METHOD Casetti (1972, 1982) outlined a routine for creating or modifying models made of a sequence of clearly identified logical steps, and called it the expansion method. The expansion method is a technique for generating models. It involves the following: (a) an ‘Initial’ model in which some or all of the parameters are in non-numerical form is selected; (b) some or all of the letter parameters are ‘expanded’ by expansions equations that redefine them as functions of variables

E.CASETTI 9

and/or of random variables, that may or may not appear in the initial model; (c) a ‘terminal’ model is generated by replacing the expanded parameters into the initial model. Let us illustrate the operation of the method at the formal level and in a purely deterministic context. Denote by Y a dependent variable and by X and Z two sets of variables X1, X2,…, Xp and Z1, Z2,…, Zq. For simplicity in the discussion that follows p=q=2. Assume an initial model Y=f(X) represented by a linear relation between a dependent variable Y and the X variables, (2.1) and expansion equations defining the parameters of this initial model into linear functions of the Z variables: (2.2) (2.3) (2.4) By substituting the right-hand side of (2.2), (2.3), and (2.4) for the corresponding coefficients in (2.1) the following terminal model is obtained: (2.5) The first and second subscripts of the c denote respectively which X and Z variables are represented in the expression to which the coefficient is attached. For instance, c12 identifies the presence of variables X1 and Z2. A zero subscript denotes the absence of the corresponding variable. For instance, c01 is associated with an expression containing the Z variable Z1 but no X variable. The usefulness of this notation will become apparent later. Most mathematical models can be related to simpler ones by viewing them as the result of an expansion of the simpler models’ parameters. For instance, quadratic polynomials can be regarded as expansions of linear polynomials. To show this, take as initial model the polynomial and expand n into a linear function of t by the expansion equation Then, the terminal model is a quadratic. As a technique to relate models to one another, the expansion method is quite general, since any model can be expanded, and in turn just about any model can be thought of as the result of the expansion of a simpler model. The primitive within this frame of reference is a model consisting of a variable set equal to a constant value, because no expansion can generate it. On the other hand, any primitive Y=a can be converted by repeated expansions into a wide variety of different functional relations. However, the same terminal model may be arrived at from different initial models. In order to place into focus the generality of the expansion method let us consider the variety of initial models and expansion equations it can encompass. Mathematical models can be purely deterministic, purely stochastic, or they may include both deterministic and stochastic components. A functional relationship between a dependent variable and one or more independent variables exemplifies

10 THE DUAL EXPANSION METHOD

the first case. A mathematical structure involving jointly distributed random variables constitutes an instance of the second type of model. The usual econometric model of the type in which W and e are random variables and the Vs are variables typifies the third situation. Letter parameters may appear in these models within a functional relation involving variables and/or random variables, or as parameters of stochastic processes or of probability distributions. Models from any one of the classes indicated above can be used as initial models, and any of the letter parameters appearing in a model may be redefined by expansion equations as functions of variables or of random variables, or of both. For example, the parameters of a functional relation can be expanded into functions of other variables and/or random variables. Or a probability distribution’s parameters can be expanded into functions of variables, or of random variables, or both. Specialized literatures, often centered on estimation issues, have developed constructs that can easily be conceptualized in terms of the expansion method. Here are a few examples. The random coefficients models (Wald 1947; Swamy 1971; Judge et al. 1980: 352ff.) can be thought of as expansions of the parameters of functional relations or of the deterministic portion of linear models into random variables. Autoregressive or lagged dependent variables models (Johnston 1972: 292ff.; Pindyck and Rubinfeld 1976:485ff.) are easily produced by expanding the parameters of a nonautoregressive initial model into functions of the dependent variable lagged in time. The empirical Bayes inference (Robbins 1955, 1964, 1980; Morris 1983; DuMouchel and Harris 1983) postulates that observed data are produced by a stochastic process the parameters of which result from a second stochastic process. Related to these are the hierarchical statistical models (Lindley and Smith 1972; Good 1980). They involve nested probability distributions in which the parameters of the highest order distribution are random variables with parameters that are also random variables. Within the expansion method’s frame of reference these constructs involve stochastic initial models with parameters expanded into random variables. The expansion method capability to routinize the creation of mathematical models renders it potentially useful along several dimensions. Specifically, it can be used 1 to test hypotheses concerning the drift and/or stability of a model’s parameters, and to obtain functional portraits of this drift; 2 to create complex models from simpler ones for specific research purposes; 3 as an organizing scheme within the context of which mathematical models can be classified and related to one another; 4 to interpret complex models in terms of simpler initial model(s) and related expansion equations; and 5 as an artificial intelligence technique allowing model creation by computer programs.

E.CASETTI 11

These five classes of applications of the expansion method are interrelated and cannot be entirely separated from one another. Samples of the expansion method literature are reviewed in the paragraphs that follow. A number of applications of the expansion method test hypotheses of spatial parameter variation or estimate the structural parameters of models expanded to incorporate spatial dimensions. In one of the earliest applications (Casetti and Demko 1969, 1973; Demko and Casetti 1970) fertility and mortality decline was modeled using a generalized logistic. Then, the model’s parameters specifying the rate and timing of the decline were expanded into the distances from the urban centers where these declines had originated. The terminal model’s parameters were estimated using areally disaggregated data for the USSR for the years 1940, 1950, 1960, and 1965. Strongly significant spatial temporal patterns were observed. Other demographic applications of the expansion method exist in the literature (Casetti 1973; Hanham 1974; Bretschneider et al. 1981; Ying 1982; Zdorkowski and Hanham 1983). Another early application of the expansion method (Casetti and Semple 1969) involved testing hypotheses on the diffusion of tractors in the USA. In order to do this, the parameters of a logistic relating percentage adopters and time were expanded into linear functions of distance from the adoption leader (North Dakota). The terminal model obtained was estimated using data on the percentage of farms using tractors for twenty-five states in the central portion of the USA and for nine time periods spanning the 1920–64 time horizon. The investigation revealed a diffusion lag proportional to distance from the adoption leader. Instead, it did not support the hypothesis that the diffusion rates would differ with distance from the adoption leader. A number of other papers employed the expansion method to investigate diffusion phenomena (Casetti et al. 1972b; Hanham and Brown 1976; Casetti and Gauthier 1977; Zdorkowski and Hanham 1983). Some spatial applications of the expansion method have been concerned with testing for the occurrence of polarized growth (Casetti et al. 1970; Odland et al. 1973; Gaile 1977; Uyanga 1977; Krakover 1983). Recently trend surface expansions have been experimented with. They involve expanding an initial model’s parameter into spatial coordinate polynomials of the type used in trend surface regressions. Trend surface expansions are well suited to produce geographical maps of parameter variation. Also they appear to be one possible tool for removing spatial autocorrelation in regression residuals (Jones 1983). Trend surface expansions have been discussed by Jones (1984), Casetti and Jones (1983), Brown and Jones (1985), and Selwood (1984). In the Brown and Jones application trend surface expansions of a conventional migration model were employed to pinpoint the relations between migration modalities and economic development in Costa Rica. Some applications of the expansion method focused on the temporal drift of the parameters of the initial models. For example, the expansion method was used by Malecki (1975, 1980) to test hypotheses concerning the change over


time of rank-size functions, by Bretschneider and Gorr (1983) to develop models suited for forecasting and policy evaluation, and by Bretschneider et al. (1981) to evaluate natural gas conservation policies in Ohio. In some cases the expansion method has been used to construct models not necessarily or immediately employed in the context of data analyses. For instance, Casetti (1972) showed that expanding some coefficients of a class of mathematical programs in terms of their dual variables generates the spatial equilibrium formulations ordinarily predicated upon the maximization of a social pay-off function (Samuelson 1952; Takayama and Judge 1964). Sonis (1983) used the expansion method to construct very general Volterra-Lotka type vectorial differential equations of diffusion processes. Visser (1980a, b, 1981, 1982) started from optimum relations between agricultural intensity and distance to market and used the expansion method to incorporate technological and other changes into them. Other expansion method applications are disconnected from the clusters of themes referred to above. Here are some examples. Briggs (1974) expanded differential equations relating aggregate CBD sales to urban population. Thrall (1979) used the expansion method to investigate spatial inequities in tax assessments. Thrall and Tsitanidis (1983) employed it to study the changes in the locational patterns of physicians, and Kodras (1984) used it to evaluate the regional variation of the determinants of food stamps program participation. In this paper the expansion method and its dual expansions extension will be applied to the analysis of parameter drift and parameter stability within the framework of a linear model. The population models assumed throughout this paper involve on the left-hand side of the equality sign a dependent random variable and on the right-hand side a linear combination of variables plus a stochastic error term. The expansions dealt with concern the parameters of the independent variables. Let us outline how the expansion method can be used for investigating parameter stability. Suppose we wish to test the hypothesis that the parameters of the initial model (2.1) drift ‘linearly’ with respect to the Z variables. The expansion equations (2.2), (2.3), and (2.4) are a formalization of this linear drift. ‘Population’ models are arrived at by adding random disturbances to (2.1) and (2. 5). If these disturbances satisfy independence and homoskedasticity assumptions the parameters of the terminal model can be estimated by an ordinary regression. If none of the coefficients associated with the terms in which the Z variables appear is statistically significant the initial model is ‘stable’, or at least does not have the type of instability that can be detected by linear expansions. It could still have, though, an instability that could be revealed by some other expansion. If instead one or more of the cs associated with Z variables are significant, this indicates that (2.1) changes with the Z variables in the manner that can be made explicit by replacing estimated cs into (2.2), (2.3), and (2.4). The estimated expansion equations associate each and every point in Z space with a ‘realization’ of the initial model. Since values of Z are given for each

E.CASETTI 13

observation, they can associate every observation with a numerical realization of the initial model. The estimated expansion equations also provide the basis for a sensitivity analysis. To show this, suppose we define a neighborhood in Z space and investigate the realizations of the initial model associated with the points in this neighborhood. This investigation constitutes indeed a sensitivity analysis of the initial model. DUAL EXPANSION METHOD If a terminal model has been generated from a linear initial model by linear expansion equations, there is a second linear initial model and associated linear expansion equation(s) that will yield the same terminal model. If a terminal model is given, as soon as the linear initial model and linear expansion equations capable of producing it are defined, then a second linear initial model and associated linear expansion equations which will produce the same terminal model become defined. Let us call the two initial models and associated expansion equations respectively primal and dual. Which expansion is primal is arbitrary, but when an expansion is defined as primal, the second one becomes the dual of the first one. In the next few paragraphs the dual expansions will be demonstrated using the example introduced earlier. A more formal presentation will follow. The intrinsic duality of the linear expansions is illustrated by the fact that the same terminal model (2.5) can be arrived at from an initial model relating the Y and Z variables by expanding its coefficients in terms of the X variables. To show this assume the initial model Y=g(Z), (2.6) and the expansion equations (2.7) (2.8) (2.9) that will indeed produce again the terminal model (2.5). In fact, if the parameters of (2.5) are estimated using empirical data, the resulting equation can be interpreted either in terms of the initial model (2.1) and of the expansion equations (2.2), (2.3), and (2.4), or in terms of the initial model (2.6) taken in conjunction with the expansion (2.7), (2.8), and (2.9), The estimated terminal model tests simultaneously the hypothesis that the parameters of (2.1) drift with respect to the Z variables, and the hypothesis that the parameters of (2.6) drift with respect to the X variables. Also, upon estimation the terminal model will yield two sets of expansion equations with numerical parameters that specify the nature of the drift of each initial model’s parameters in the dual space represented by the ‘other’ set of variables. This in turn implies that every point in


Z space is associated with a realization of the initial model f(X), and every point in X space is associated with a realization of the dual initial model g(Z). Consequently, every observation can be associated with a realization of the f(X) and g(Z) models. It was noted that as soon as a model is conceptualized and interpreted in terms of a linear initial model and associated expansion equations, a unique linear dual initial model and associated expansion equations are implicitly defined. This situation is reminiscent of the primal dual relationships in mathematical programming. The dual expansion tableau scheme that follows (a) shows that a linear initial model and expansion equations are associated with a unique dual linear initial model and associated expansion equations; and (b) provides a simple routine for extricating primal and dual initial models and expansion equations from cumbersome terminal models. This dual expansion tableau is reminiscent of the Tucker tableau used to relate primal and dual linear programming formulations. The expansions in the previous example as specified by (2.1) – (2.9) are summarized, condensed, and portrayed in the following tableau: The tableau consists of a central region filled by the c coefficients of the expanded model, and surrounded by stubs. The coefficients of the two initial models appear on the left stub and on the top stub. The X and Z variables are located on the stubs opposite to these. The dependent variable Y is in the top left corner. By relating the tableau to (2.1)–(2.9) it will become apparent that any of these equations can be readily arrived at by dropping the X variables sidewise and the Z variables upward onto the appropriate coefficients, and by adding the required plus and equality signs. The tableau clearly shows that a unique linear dual expansion becomes defined as soon as a primal linear expansion is specified. As soon as the as of the primal initial model, the cs of its expansions, and the X and Z variables are given and placed in the appropriate locations in the tableau, the dual initial model and expansions are defined by the vertical entries in it. The tableau above has been compacted and generalized in the vector and matrix formulation below: The entries in this second tableau are the vector and matrix equivalents of the entries in the previous one. For instance, and so on. The relations between initial models, expansion equations, and terminal model are easily represented using a, b, C, and Y. Specifically, is the primal initial model, represents the expansion equations related to it, and the bilinear form is the terminal model. The dual

E.CASETTI 15

initial model is The expansion equations related to it are which upon substitution yield again the terminal model The intrinsic duality of the linear expansions has practical research implications. Let us bring into focus three of them. 1 An awareness of the expansions’ duality carries within itself the suggestion to identify and then to interpret dual initial models and expansion equations. This is reminiscent of the practice of interpreting dual variables in mathematical programming. As in mathematical programming, the attempt to interpret duals can produce interesting results in some cases, and uninteresting ones in others. 2 If the X and Z variables are distinct and competing potential explanators of Y that correspond to alternative theoretical frames of references, the dual expansions formalism allows investigation of their comparative efficacy and whether they influence each other’s efficacy. 3 Finally, the expansions’ duality can be put to use by taking a complex model as a starting point and then interpreting it in terms of alternative primal/dual initial models and associated expansion equations. These three approaches to using the intrinsic duality of the expansion method will be referred to as the dual expansions method. The practical usefulness of the dual expansions method can best be demonstrated by an example. AN EXAMPLE In the sections that follow the dual expansions method will be employed to investigate the effects of population growth on development. The issue of whether population growth has a positive or a negative influence on econom ic growth has been the object of heated ideological debates that often Table 2.1 A compilation of correlations between rates of growth of population and product per capita Source

Number of countries

Type of countries

Time span

R

R2

1

16

LDC

1952–58

−0.680

0.462

63

ALL

1950–64

−0.309

0.095

2

Stockwell 1962 Kuznets 1967


Source

Number of countries

Type of countries

Time span

R

R2

3 4 5 6 7

21 40 35 20 26

MDC LDC LDC MDC LDC

1950–64 1950–64 1959–69 1959–69 1963–68

–0.434 0.111 –0.120 –0.300 –0.370

0.188 0.012 0.014 0.090 0.137

49

ALL

1950–66

–0.125

0.016

32

LDC

1950–66

–0.032

0.001

17

MDC

1950–66

–0.447

0.200

16

MDC

1960–70

–0.010

0.000

76

LDC

1960–70

–0.040

0.002

52

LDC

8 9 10 11 12 13

Kuznets 1967 Kuznets 1967 Sauvy 1972 Sauvy 1972 Stockwell 1972 Thirlwall 1972 Thirlwall 1972 Thirlwail 1972 Chesnais and Sauvy 1973 Chesnais and Sauvy 1973 Chesnais and Sauvy 1973 Simon 1977

1959–61 to 0.110 0.012 1969–71 14 11 MDC 90 to 115 –0.359 0.129 years 15 Simon 1977 10 MDC 40 to 70 years −0.122 0.015 Note: LDC, less developed countries; MDC, more developed countries; ALL, both types of countries.

emphasized the mechanisms through which these influences are exercised. Extensive reviews of this literature are available (United Nations 1953, 1973; Easterlin 1967; Kuznets 1967; Simon 1977, 1981; Cassen 1976; McNicoll 1984). Positive effects of population growth have been credited to scale economies; external economies; increased division of labor and specialization (Chenery 1960; Robinson, 1960; Maizels 1963; Glover and Simon 1975; Simon 1977:275); induced innovation effects (Hirschman 1958; Boserup 1965, 1981; Clark 1967; Binswanger 1978); the easier adjustments to change by younger populations; the greater pool of talent in larger populations; and finally the climate of buoyancy and optimism associated with growth (Kuznets 1965; Simon 1981:197ff.). On the negative side it has been maintained that high rates of population growth produce pressures on limited nonrenewable resources (Fisher and Potter 1971), bring about negative externalities of various kinds as well as external and scale diseconomies, reduce private and public capital formation, and channel investments to maintaining rather than increasing capital intensity (Coale and Hoover 1958; Kelly and Williamson 1974; Cassen 1978; Bilsborrow 1979).

E.CASETTI 17

No consensus has been reached as to which effect prevails. The empirical investigations regarding the relations between population growth and rate of development yielded results that are generally regarded as inconclusive. The compilation given in Table 2.1 shows that the correlations between the two rates are generally low, and are positive in some cases while negative in others. Inconclusive results could be produced by instability or drift of the relation between population growth and rate of development. The expansion method renders easy the testing of hypotheses concerning parameter drift. An awareness of the method renders the researcher wary of assuming parameter stability, and heightens instead his/her propensity to ask questions as to which variables, mechanisms, and theoretical propositions might be related to parameter drift. In the case in point, this line of enquiry suggests level of development as the single most likely cause of drift here. First, the contemporary countries differ to a very major extent in development levels. Second, several considerations suggest that the relation between population growth and rate of development may be different depending on a country’s level of development. The mechanisms through which population growth would exercise positive and negative effects on development can be presumed to operate with different intensity at different development levels. Specifically: 1 High rates of population growth are said to reduce capital formation and to divert savings toward less productive demographic investments. However, this effect will be stronger at the initial stages of the development process that tend to be characterized by a greater shortage of capital. 2 High rates of population growth compound the pressure that diseconomies of scale and negative externalities such as congestion and pollution place on economic growth. However, diseconomies of scale and negative externalities are more likely to be problems confronting mature economies located at the upper end of the development continuum. 3 A number of authors suggested that population growth renders change, and adjustment to change, easier. The bulk of the dislocations associated with the transfer of population from the countryside into evolving hierarchies of urban centers materializes at intermediate stages of the development process, which in turn suggests that the positive influence of population growth in facilitating these transformations will be felt more strongly by countries in an intermediate position on the development scale. These considerations suggest that the effects of population growth on rate of development should be more negative at low and high levels of development. Let us proceed to test this hypothesis. Denote population by POP, product per capita by PRC, and their natural logarithms by respectively P and y. Both the product per capita variable PRC and its logarithm y are indices of level of development. Assume that POP and PRC are continuous and differentiable


functions of time t. Then the time derivatives of P and y, which are identified by primes, denote the instantaneous percentage rates of change of POP and PRC over time: (2.10)

(2.11) A linear relationship between y′ and P′ provides a convenient starting point for investigating the effect of population growth on rate of development: (2.12) The parameters a0 and a1 of this initial model are expanded in terms of development levels, i.e. they are redefined as functions of development levels. Which functions should be selected is dictated by considerations of simplicity and by the need to use a function that for appropriate values of its parameters can produce the type of change of the initial models’ parameters that theoretical presuppositions lead us to expect. In this case, we hypothesize that a1 will be lower at low and high development levels, which suggests expanding a1 into a quadratic of a development index. Considerations of symmetry suggest to expand also a0 in a similar fashion. Both expansions into PRC and y were tried out. The one based on y was eventually selected because it performed better in the empirical analyses described later. The expansion equations employed are: (2.13) (2.14) By replacing the right-hand side of the expansion equations (2.13) and (2.14) into the initial model (2.12) the following terminal model is obtained: (2.15) The dual tableau for this expansion is as follows: We can easily extract from this tableau the dual initial model (2.16) and the dual expansion equations (2.17)

E.CASETTI 19

(2.18) (2.19) Clearly, the dual initial model and expansion equations produce the same terminal model (2.15) generated by the primal formulations. The dual initial model links up to a literature distinct from the one related to the primal expansions. This literature is concerned with the tendency for contemporary countries in an intermediate position on the development scale to experience rates of economic growth higher than those of the countries closer to the end points of this scale. The occurrence of this tendency was suggested by Russett (1964:309ff.) who reported a parabolic relation between annual rates of growth of gross national product (GNP) per capita and levels of GNP per capita. The validity of the finding was later questioned by Hagen and Havrylyshyn (1969:88). The occurrence of the trend suggested by Russett was reaffirmed by Horvat (1972) on the basis of spline regressions relating growth rates to levels of GNP per capita, and then by Kristensen (1974). Kristensen’s conclusions were based on tabular data on average annual growth rates and levels of GNP per capita for seven groups of countries at consecutive development levels. Casetti (1979) fitted statistically significant quadratic and cubic polynomials to data on rates and levels of development for ninety-two countries. These data spanned the time interval 1950–73. The polynomials estimated did yield a maximum rate of development at intermediate development levels. Several interesting theories can be related to the tendency for countries at an intermediate development level to grow faster. The literature on the acceleration in the rates of economic growth associated with the onset of the transition from premodern stagnation to modern exponential growth (Rostow 1960; Kahn et al. 1976) is relevant to explain the increase in rates of growth of GNP per capita associated with the transition from low to intermediate development levels. The theories suggesting that countries modernizing later develop faster also contribute to justifying why the highest rates of development tend to occur at intermediate development levels. The thesis that latecomers to development have potential advantages with respect to the countries that modernize earlier—and consequently at comparative levels of development are likely to progress faster than the countries that preceded them—was proposed by Gershenkron (1962:5–30). A proposition related but not identical to Gershenkron’s has been articulated by Kahn (1976: 34ff.) who contended that the gap between the developed and developing nations, often deplored as the source and cause of underdevelopment, may instead very well be the reason why many nations can develop much more rapidly today than western countries did. A similar statement has also been put forward by Kristensen (1974: 28ff.). Both Kahn and Kristensen suggest that in today’s world the higher rates of growth of developing countries are due to the very existence of the higher level of development of the most developed countries.


Finally, the theoretical statements suggesting that mature economies will tend to experience a retardation in economic growth (Matthews 1982; Olson 1982) are consistent with the occurrence of lower growth rates of GNP per capita in countries characterized by higher levels of GNP per capita. The explanations as to why middle-income modernizing countries experience the highest rates of development have been articulated within three frames of reference, which pertain respectively to (a) the rates of application of the existing stock of scientific and technological knowledge; (b) the dynamics of capital formation and investments; and (c) changes in scale economies. A large stock of scientific and technological knowledge has been generated during the modern economic growth of the more developed countries. The leastdeveloped societies, because of their socio-economic structure and their lack of capital and trained manpower, have difficulty in applying it. As they develop, their ability to use the existing technology grows, while at the same time the stock of unused applicable technologies dwindles and slows down their growth. In the limit, their economic growth becomes dependent upon the creation of new scientific and technological knowledge, as is the case for the developed countries (Kristensen 1974:16). Capital formation tends to be inadequate in pre-modern societies. It has been maintained that self-sustained modern economic growth only begins when the net capital formation exceeds a threshold of about 5 to 6 percent of GNP. As societies develop, their ability to save increases. However, the capital-to-output ratio tends to be low for less developed countries and increases with development. As a country develops, its increasingly large ability to generate savings and its decreasing capital-to-output ratio concur to produce a phase of accelerating economic growth. Then, in the more developed countries, capital formation may tend to be reduced by a growing preference for leisure, and also savings may be increasingly diverted toward consumption and welfare expenditures. These tendencies would of course be associated with a deceleration of economic growth. Finally, the expansion of markets that goes together with development brings about at first increasing returns to scale and external economies and then decreasing returns to scale and external diseconomies. Each of these mechanisms is capable by itself of accounting for higher growth rates at intermediate development levels. However, these mechanisms are also linked to each other and are mutually self-reinforcing. The dual initial model is a differential equation relating the rate of change of development over time to levels of development. Cross-sectional estimates of this equation’s parameters can identify tendencies prevailing over the time interval given. However, these estimates should not be used for predicting development paths over time unless one is prepared to assume that the dynamics of these countries is time invariant (‘stationary’), and that the same differential equation with the same parameters is applicable to all.

E.CASETTI 21

Nevertheless, the estimate of differential equation (2.16) provides insightful information about the tendencies of countries at various levels of development to grow more or less rapidly, or not at all, over a given time interval. Stable or unstable equilibria of the equation can be easily interpreted within this perspective. Equation (2.16) is capable of topologically distinct types of behavior depending upon the values of its parameters. The dual expansion equations (2. 17) – (2.19) express these parameters as functions of population growth rates. Consequently, the estimated expansion equations allow investigation of the qualitative and quantitative changes in behavior of (2.16) that are associated with different values of Pβ. It should be noted that these dual expansions provide an illustration of the use of the expansion method for the type of investigation falling within the scope of ‘nonlinear dynamics’ and ‘catastrophe theory’. However, they also provide an alternative approach to investigating the effects of population growth on development, since if the parameters of yβ(y) drift in the dual Pβ space, this drift represents an effect of population growth on the rate of development. This discussion has demonstrated how the dual expansion method operationalizes questions regarding the effects of population growth on rates of development into tests of hypotheses regarding the parameter drift of complementary models that correspond to distinct theoretical propositions. Let us now consider briefly how the estimated parameters of the terminal equation (2. 15) can shed light on the effects of population growth on the rate of development. Conclusions regarding the effects of population growth on the rates of development that are supported by the data will depend on which nonzero coefficients appear in this final regression equation. As an illustration let us consider a few possibilities. If only c00 and c10 are different from zero the data support the proposition that P′ affects y′ in a manner that is unaffected by development levels. If all the coefficients of the terms in which P′ appear are nonsignificant we would have to conclude that P′ does not affect y′. If, however, c11 or c12 are different from zero, then P′ affects y′ in different manners at different development levels. In this case a portrait of the drift of the two initial models’ parameters can be obtained by replacing the estimated c coefficients into the expansion equations. Let us postpone further consideration of these points until after the empirical analysis described in the next section. AN EMPIRICAL ANALYSIS This section presents an empirical test of the hypothesis that the effect of population growth on development is a function of development levels. The y', P β, and y variables were operationalized as (a) annual percentage rates of change of population and product per capita over the time interval 1950–73 and (b) the logarithm of product per capita at the midpoint of the interval. Variables (a) and (b) were computed using country level data on population and product per capita


for 1973 and on the growth rates of these over the 1950–73 time horizon, published by the World Bank (1976). The sample used (92 data points) included all the countries with market economies for which source data were available and which had a 1973 population greater than 2 million people. The parameters of (2.15) were estimated using a backward selection procedure: first a regression including all the variables was computed; then, at each step of the procedure, the variable with the lowest t value was removed until all the coefficients of the variables still in the equation were significant at the 5 percent level or better. The procedure produced the following regression equation:

where the t values are in parentheses under their respective coefficients. The coefficients of the estimated terminal model were placed in the appropriate locations in the dual expansion tableau. The primal initial model , the dual initial model , and the estimated expansion equations can be read directly from the tableau. They are as follows: (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) The interpretation of the primal results is straightforward. The parameter a1 of the initial model specifies the change in rate of development associated with a unit increase in the rate of population growth. Consequently a1 represents the ‘effect’ of population growth on the rate of development. The estimated expansion equation (2.22) shows that the a1(y) function is a parabola with a maximum at y=ln (801), and with negative values throughout the range of y spanned by the data. This suggests that the effect of population growth on the rate of development is

E.CASETTI 23

Figure 2.1 Effect of population growth on the rate of development: a1(y) indicates by how much the rate of development is reduced by a population growth rate of 1 percent per year; y=ln(PRC), and PRC is product per capita in 1973 US dollars

Figure 2.2 Phase diagram of the relationship between rates and levels of economic development (corresponding to population growth rates of 1, 2, and 3 percent per year)

always negative, and that it is more so at very low and very high development levels, as hypothesized. A sketch of a1(y) is shown in Figure 2.1. The rate of development a0 in the absence of population growth is 4.473 percent per year. The a1(y) function indicates by how much this rate of development is reduced for each 1 percent of population growth, at specified levels of GNP per capita. Figure 2.1 shows that the least negative impact of P′ on


y′ occurs at intermediate development levels, and the highest at very low levels of y. Let us consider the dual expansions next. Recall that the dual initial model is a differential equation relating rates and levels of product per capita, and that the dual expansion equations specify the manner in which its coefficients change as a function of population growth. For any given value of P′ the estimated terminal model yields a realization of . Figure 2.2 shows the phase diagram of three such realizations, that correspond to P' values of 1, 2, and 3 percent. The figure shows that each of these realizations of g(y) attains a maximum at y*(P') and intersects the y axis in two points that are the roots of . The values y1(P β) and y2(Pβ) of y corresponding to these roots are equilibrium values of (2.23). Specifically, y1 corresponds to an unstable equilibrium and y2 to a stable equilibrium, since dg/dy is respectively greater than zero at y1 and smaller than zero at y2. The P′ values appearing in the data set are greater than zero and smaller than 4. Over this range of P′, g(y) is topologically invariant in the sense that it is characterized by a finite maximum at y* and by unstable/stable equilibria y1 and y2 where y1< y*< y2. The effects of P′ on the relationship between rates and levels of product per capita are best understood by expressing y*, g(y*), y1, and y2 as functions of P′. After a few manipulations we obtain the following: (2.27) (2.28) (2.29) (2.30) The value of y at which y′ attains a maximum y* is a ‘constant’ function of P′. Instead, the maximum rate of growth of product per capita g(y*) declines at larger values of P′. The equilibrium values y1 and y2 tend respectively to minus and plus infinity as P′ tends to zero. If P′ is increased past realistic values toward a threshold located approximately at P′=11.6, y1, y*, and y2 collapse into a single value, and g(y*) tends to zero. This means that the topology of g(y) changes at P′ =0, at approximately P′=11.6, and is invariant between these. Let . A tabulation of PRC1, PRC2, and g(y*) for realistic values of P′ is given in Table 2.2. The table shows that g(y) is remarkably sensitive to the changes in P′ that can be found in the contemporary world. The rates of development tend Table 2.2 Tabulation of PRC1(Pβ), PRC2(Pβ), and g[y* (Pβ)] evaluated at a range of values of Pβ Pβ

PRC1

PRC2

g(y*)

0.5 1.0 1.5

1 7 19

721,569 88,200 33,973

4.28 4.09 3.90

E.CASETTI 25

Pβ

PRC1

PRC2

g(y*)

2.0 34 18,962 3.70 2.5 51 12,603 3.51 3.0 69 9,244 3.31 3.5 89 7,213 3.13 Notes: P′, percentage growth rate of population; PRC1, exp(y1) where y1 is the smaller root of g(y)=0; PRC2, exp (y2) where y2 is the larger root of g(y)=0; g(y*), maximum value of y′, y′, rate of development. PRC values are in 1973 US dollars.

to be highest at intermediate levels of development. However, for large rates of growth of population the maximum rates of development y* are lower, and the rates of development associated with any level of development are smaller. Also, the stable equilibrium level of product per capita PRC2 occurs at lower development levels when P′ is higher. Summing up, the dual expansions results reveal perverse effects of higher rates of population growth within a theoretical frame of reference different from the one associated with the primal expansions. DISCUSSION AND CONCLUSIONS At this point, it seems appropriate to place the expansion method and the dual expansion method in perspective. The expansion method is a procedure made of a clearly identified sequence of operational steps for generating or modifying mathematical models. Two aspects of the expansion method will be highlighted here. First, the expansion method is concerned with the formal aspects of the process by which mathematical models are created. By abstracting the manipulation of models from the rationale for and the contexts of such manipulations, a class of useful abstractions is arrived at much in the same way as, say, numbers became useful abstractions when they acquired a separate existence from objects counted. Second, the expansion method has a generality of scope that can be traced to the very fact that it addresses formal aspects of model manipulation. Because of this generality, it is applicable to any model, and any variables. In the literature there have been instances of what can be regarded as special purpose expansions. The technique discussed by Gujarati (1970a, b) can be thought of as involving expansions in terms of dummy variables for the purpose of determining whether regressions are significantly different over data subsets. Some phases of the time-varying parameters literature (Bennett 1979:323ff.) can also be conceptualized as involving expansions in terms of time and/or timelagged variables and variates. Also, some substantively oriented contributions suggest themselves as the product of ‘expansions’. For instance, the introduction of neutral technological


progress into a Cobb-Douglas production function involves what in the expansion method terminology would be called the expansion of a coefficient into a linear function of time. Possibly, researchers may go through the sequence of steps of the expansion method to solve specific substantive research problems without recognizing in these mental processes the application of a general methodology. It can be argued that an awareness of the expansion method as a methodology general in scope can save the effort of redeveloping aspects of this methodology as specific solutions to specific research problems, and consequently can contribute to render research easier, which is a major role of methodological contributions. Also, such awareness can contribute to using the expansion method in situations in which it should be applied but is not. This is an important point that deserves to be developed at some length. Let us concentrate on the issues of parameter drift and stability, in which it is easy to show that an awareness of the expansion method can make a substantial difference. Social scientists assume more often than not that parameters they estimated are stable. Such stability tends to be an untested assumption rather than a conclusion accepted only after the efforts to reveal parameter instability or drift have proven unsuccessful. Indeed, instances of research focusing on the possible occurrence of parameter instability do exist. However, the bias is toward presupposing parameter stability, while instead the opposite should be the rule. Whenever functional relations are estimated, the default presupposition should be that their parameters are likely to vary over space, over time, and over different environments and circumstances. The expansion method provides a simple and easily applicable technique for testing hypotheses of parameter drift. Also, because of its very nature it can contribute to making researchers self-conscious concerning the possible occurrence of parameter drift and the theoretical reasons suggesting it. In a capsule, the expansion method can routinize the asking and answering questions concerning parameter drift and parameter stability. The social scientists’ propensity to assume parameter stability can be traced to a quest for ‘laws’. The mathematical formulation and estimation of invariant ‘laws’ has had a pivotal role in the physical sciences. The importation of mathematical modeling and techniques from the hard sciences into the social sciences tended perhaps to be associated with importing also an aspiration to discover laws. However, in the social sciences, functional relations are likely to represent subsystems that will perform differently in different environments and circumstances rather than invariant laws. The duality principles discussed in this paper state that, whenever a linear initial model and linear expansion equations are defined, also a dual linear initial model and associated expansion equations are implicitly defined. Each of the two initial models relates the dependent variable Y to a set of variables that are potential causes of drift of the ‘other’ initial model. The dual expansion method

E.CASETTI 27

involves investigating whether one set of alternative potential explanators of Y produces effects on Y to an extent determined by the other set of explanators. Both the expansion method and the dual expansion method can perform a most useful function. They can prompt the asking and facilitate the answering of questions, such as the ones concerning parameter drift, that need to be addressed to a much greater extent than has been the case so far. NOTE Reprinted with permission from IEEE Transactions on Systems, Man, and Cybernetics SMC–16 (1), January-February 1986, pp. 29–39. REFERENCES Bennett, R.J. (1979) Spatial Time Series: Analysis, Forecasting and Control, London: Pion. Bilsborrow, R.E. (1979) ‘Age distribution and saving rates in less developed countries’, Economic Development and Cultural Change 28:23–45. Binswanger, H.P. (1978) ‘Induced technical change: evolution of thought’, in H.P.Binswanger and V.W.Ruttan (eds) Induced Innovation; Technology, Institutions, and Development, Baltimore, MD: Johns Hopkins University Press. Boserup, E. (1965) The Conditions of Agricultural Growth, London: Allen & Unwin. ——(1981) Population and Technological Change: A Study of Long-term Trends, Chicago, IL: University of Chicago Press. Bretschneider, S.I. and Gorr, W.L. (1983) ‘Ad hoc model building using time-varying parameter models’, Decision Science 14:221– 39. Bretschneider, S.I., Gorr, W.L. and Roblee, P.R. (1981) An Evaluation of Natural Gas Conservation in the Residential Sector of Ohio using Time-varying Parameter Models, Columbus, OH: National Regulatory Research Institute. Briggs, R. (1974) ‘A model to relate the size of the central business district to the population of a city’, Geographical Analysis 6:265–79. Brown, L.A. and Jones, J.P. (1985) ‘Spatial variation in migration processes and development: a Costa Rican example of conventional modeling augmented by the expansion method’, Demography 22:327–52. Casetti, E. (1972) ‘Generating models by the expansion method: applications to geographic research’, Geographical Analysis 4: 81–91. ——(1973) ‘Testing for spatial temporal trends: an application to urban population density trends using the expansion method’, Canadian Geographer 17:127–37. ——(1979) ‘A class of differential equations relating rates and levels of economic development’, Modeling and Simulation 10: 1419–23. ——(1982) ‘Mathematical modeling and the expansion method’, in R.B.Mandal (ed.) Statistics for Geographers and Social Scientists, pp. 81–95, New Delhi: Concept Publishing. Casetti, E. and Demko, G.J. (1969) ‘A diffusion model of fertility decline: an application to selected Soviet data: 1940–1965’, Discussion Paper 5, Department of Geography, Ohio State University.


——and——(1973) ‘A diffusion model of fertility decline: an application to selected Soviet data: 1940–1965’, Acta Geographica 12:53–67. Casetti, E. and Gauthier, H.L. (1977) ‘A formalization and test of the “Hollow Frontier” hypothesis’, Economic Geography 53:70–8. Casetti, E. and Jones, J.P. (1983) ‘Regional shifts in the manufacturing productivity response to output growth: sunbelt versus snowbelt’, Urban Geography 4:285–301. Casetti, E., and Semple, R.K. (1969) ‘Concerning the testing of spatial diffusion hypothesis’, Geographical Analysis 1:254–9. Casetti, E., King, L.J. and Odland, J. (1970) ‘On the formal identification of growth poles in a spatial temporal context’, Proceedings of the Association of Canadian Geographers 1:39–43. ——,——and——(1972a) ‘The formalization and testing of concepts of growth poles in spatial context’, Environment and Planning 3:377–82. Casetti, E., King, L.J. and Williams, F. (1972b) ‘Concerning the spatial spread of economic development’, in W.P.Adams and F.M.Helleiner (eds) International Geography, pp. 897–9, Toronto: University of Toronto Press. Cassen, R.H. (1976) ‘Population and development: a survey’, World Development 4: 785–830. ——(1978) India: Population, Economy, Society, New York: Holmes and Meier. Chenery, H.B. (1960) ‘Patterns of industrial growth’, American Economic Review 50: 624–54. Chesnais, J.C. and Sauvy, A. (1973) ‘Progres economique et accroissement de la population: une experience commentee’, Population 28:843–57. Clark, C. (1967) Population Growth and Land Use, New York: St Martin’s Press. Coale, A.J. and Hoover, E.M. (1958) Population Growth and Economic Development in Low-income Countries, Princeton, NJ: Princeton University Press. Demko, G.D. and Casetti, E. (1970) ‘A diffusion model for selected demographic variables: an application to Soviet data’, Annals, Association of American Geographers 60:533–9. DuMouchel, W.H. and Harris, J.E. (1983) ‘Bayes methods for combining the results of cancer studies in human and other species’, Journal of the American Statistical Association 78:293– 308. Easterlin, R.A. (1967) ‘Effects of population growth on the economic development of developing countries’, Annals of the Academy of Political and Social Sciences 369: 98–108. Fisher, J.L. and Potter, N. (1971) ‘The effects of population growth on resource adequacy and quality’, in Rapid Population Growth, National Academy of Science, vol. 2, Baltimore, MD: Johns Hopkins University Press. Gaile, G.L. (1977) ‘Toward a strategy of growth paths’, Environment and Planning A 9: 675–9. Gershenkron, A. (1962) Economic Backwardness in Historical Perspective, New York, Washington, and London: Praeger. Glover, D. and Simon, J.L. (1975) ‘The effects of population density upon infra-structure: the case of road building’, Economic Development and Cultural Change 23:453–68. Good, I.J. (1980) ‘Some history of the hierarchical Bayesian methodology’, in J.M.Bernardo et al. (eds) Bayesian Statistics, Valencia University Press. Gujarati, D. (1970a) ‘Use of dummy variables in testing for equality between sets of coefficients in two linear regressions: a note’, American Statistician 24:50–2.

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——(1970b) ‘Use of dummy variables in testing for equality between sets of coefficients in linear regressions: a generalization’, American Statistician 24:18–22. Hagen, E.E. and Havrylyshyn, 0. (1969) ‘Analysis of world income and growth 1955– 1965’, Economic Development and Cultural Change 18:1–96. Hanham, R.Q. (1974) ‘The diffusion of birth control and space-time trends in the decline of fertility’, Proceedings of the Association of the American Geographers 8:80–3. Hanham, R.Q. and Brown, L.A. (1976) ‘Diffusion waves within the context of regional economic development’, Journal of Regional Science 16:65–71. Hirschman, A.O. (1958) The Strategy of Economic Development, New Haven, CT: Yale University Press. Horvat, B. (1972) ‘Relation between the rate of growth and the level of development’, Working Paper 13, International Development Research Center, Indiana University, Bloomington, IN. Johnston, J. (1972) Econometric Methods, New York: McGraw-Hill. Jones, J.P. (1983) ‘Parameter variation via the expansion method with tests for autocorrelation’, Modeling and Simulation 17:853– 7. ——(1984) ‘A spatially varying parameter model of AFDC participation: empirical analysis using the expansion method’, Professional Geographer 36:455–61. Judge, G.G., Griffiths, W.E., Hill, R.C. and Lee, T.C. (1980) The Theory and Practice of Econometrics, New York: Wiley. Kahn, H., Brown, W. and Martel, L. (1976) The Next 200 Years: A Scenario for America and the World, New York: William Morrow. Kelly, A.C. and Williamson, J.G. (1974) Lessons from Japanese Development: An Analytical Economic History, Chicago, IL: University of Chicago Press. Kodras, J.E. (1984) ‘Regional variation in the determinants of food stamp program participation’, Environment and Planning C: Government and Policy 2:67–78. Krakover, S. (1983) ‘Identification of spatio temporal paths of spread and backwash,’ Geographical Analysis 15:318–29. Kristensen, T. (1974) Development in Rich and Poor Countries, New York, Washington, and London: Praeger. Kuznets, S. (1965) Economic Growth and Structure: Selected Essays, New York: Norton. ——(1967) ‘Population and economic growth’, Proceedings of the American Philosophical Society 111:170–93. Lindley, D.W. and Smith, A.F.M. (1972) ‘Bayes estimates for the linear model (with discussion)’, Journal of the Royal Statistical Society, Ser. B 34:1–41. McNicoll, G. (1984) ‘Consequences of rapid population growth: an overview and assessment’, Population and Development Review 10:177–240. Maizels, A. (1963) Industrial Growth and World Trade, Cambridge: Cambridge University Press. Malecki, E.J. (1975) ‘Examining change in rank-size systems of cities’, Professional Geographer 27:43–7. ——(1980) ‘Growth and change in the analysis of rank-size distribution: empirical findings’, Environment and Planning A 12:41–52. Matthews, R.C.O. (ed.) (1982) Slower Growth in the Western World, London: Heinemann. Morris, C.N. (1983) ‘Parametric empirical Bayes inference: theory and applications’, Journal of the American Statistical Association 78:47–55.


Odland, J., Casetti, E. and King, L.J. (1973) ‘Testing hypotheses of polarized growth within a central place hierarchy’, Economic Geography 49:74–9. Olson, M. (1982) The Rise and Decline of Nations: Economic Growth, Stagflation and Social Rigidities, New Haven, CT, and London: Yale University Press. Pindyck, R.S. and Rubinfeld, D.L. (1976) Econometric Models and Economic Forecasts, New York: McGraw-Hill. Robbins, H. (1955) ‘An empirical Bayes approach to statistics’, Proceedings of the Third Berkeley Symposium on Mathematics, Statistics and Probability 1:157–64. ——(1964) ‘The empirical Bayes approach to statistical decision problems’, Annals of Mathematical Statistics 35:1–20. ——(1980) ‘An empirical Bayes estimation problem’, Proceedings of the National Academy of Science, USA 77:6988–9. Robinson, E.A.G. (ed.) (1960) Economic Consequences of the Size of Nations, New York: St Martin’s Press. Rostow, W.W. (1960) The Stages of Economic Growth: A Non Communist Manifesto, Cambridge: Cambridge University Press. Russett, B., Alker Jr, H.R., Deutsch, K.W. and Lasswell, H.D. (1964) World Handbook of Political and Social Indicators, New Haven, CT: Yale University Press. Samuelson, P.A. (1952) ‘Spatial price equilibrium and linear programming’, American Economic Review 42:283–303. Sauvy, A. (1972) ‘Les charges économiques et les avantages de la croissance de la population’, Population 27:9–26. Selwood, D. (1984) ‘Office employment in the U.S. urban system, 1910–1970’, Modeling and Simulation 15:277–82. Simon, J.L. (1977) The Economics of Population Growth, Princeton, NJ: Princeton University Press. ——(1981) The Ultimate Resource, Princeton, NJ: Princeton University Press. Sonis, M. (1983) ‘Spatio-temporal spread of competitive innovations: an ecological approach’, Papers of the Regional Science Association 52:159–74. Stockwell, E.G. (1962) ‘The relationship between population growth and economic development’, American Sociological Review 27:250–2. ——(1972) ‘Some observations on the relationship between population growth and economic development during the 1960s’, Rural Sociology 37:628–32. Swamy, P.A.V.B. (1971) Statistical Inference In Random Coefficients Regression Models, Heidelberg: Springer Verlag. Takayama, T. and Judge, G.G. (1964) ‘Spatial equilibrium and quadratic programming’, Journal of Farm Economics 46:67–93. Thirlwall, A.P. (1972) ‘A cross section study of population growth and the growth of output and per capita income in a production function framework’, Manchester School of Economic and Social Studies 40:339–56. Thrall, G.I. (1979) ‘Spatial inequities in tax assessment: a case study of Hamilton, Ontario’, Economic Geography 55:123–34. Thrall, G.I. and Tsitanidis, J.G. (1983) ‘A model of the change, attributable to government health insurance plans, in location patterns of physicians, with supporting evidence from Ontario, Canada’, Environment and Planning C: Government and Policy I: 45–55. United Nations (1953) The Determinants and Consequences of Population Trends, New York.

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——(1973) The Determinants and Consequences of Population Trends, New York. Uyanga, J. (1977) ‘Testing formal polarization of growth in spatial context’, Nigerian Geographical Journal 20:145–51. Visser, S. (1980a) ‘Technological change and the spatial structure of agriculture’, Economic Geography 56:311–19. ——(1980b) ‘Modeling the spatial structure of agricultural intensity: illustration of the Casetti expansion method’, Modeling and Simulation 11:1393–8. ——(1981) ‘Estimation of agricultural production functions’, Modeling and Simulation 12:833–9. ——(1982) ‘On agricultural location theory’, Geographical Analysis 14:167–76. Wald, A. (1947) ‘A note on regression analysis’, Annals of Mathematical Statistics 18: 586–9. World Bank (1976) World Tables 1976, Baltimore, MD: Johns Hopkins University Press. Ying, K. (1982) ‘A method for projecting urban population by census tracts using the analysis of spatial-temporal trends of urban population density’, Modeling and Simulation 13:1163–7. Zdorkowski, R.T. and Hanham, R.Q. (1983) ‘Two views of the city as a source of spacetime trends and the decline of human fertility’, Urban Geography 4:54–62.

3 PARADIGMATIC DIMENSIONS OF THE EXPANSION METHOD John Paul Jones, III

The introduction of new analytical methodologies in the social sciences tends to follow a common historical pattern. They are typically transported from more analytically oriented disciplines by a small cadre of researchers. If the new approaches have merit in providing solutions to the types of questions addressed by the discipline, they may gain a larger following and through their use achieve a certain degree of acceptance. In the end, however, many newly introduced techniques ultimately become passé—which is to say, they wear out their welcome in the discipline’s premier journal pages, only to be superseded by newer methodologies or ones that are better suited to solving emerging disciplinary questions. The above description perhaps applies best to factor analysis, but it characterizes the life span of many other methodologies as well. One reason for the eventual abandonment of analytical methodologies is that they seldom contain the seeds for rethinking larger disciplinary issues. This is especially true when techniques fail to shed light on historic disciplinary problematics, dualisms, and debates. If the techniques fail to say anything about what constitutes a discipline’s identity search, i.e. for its objects of inquiry, its fundamental questions, and, broadly speaking, its research agendas, then they remain only techniques. Such methodologies will also usually fail to create a paradigmatic space large enough to engage emerging perspectives and critiques. Caught between saying little to resolve historically significant debates, and being superseded by new problems and frameworks, they can become analytical blips in the time path of a discipline. In contrast to the above sketch, the expansion method has for nearly twenty years bypassed the stage of abandonment in methodological history. Indeed, as the papers in this volume attest, it continues to attract a diverse following, in terms of both research style and substantive application area. In this paper I consider some of the reasons behind the expansion method’s continued development, maturation, and use. I direct my comments specifically to the issues raised above, namely, that the method’s paradigm is useful not only for rethinking larger issues of central concern to a discipline, in this case geography, but that it is also capable of intersecting with themes found in newly emerging conceptual frameworks. While these comments are directed specifically to the

J.P.JONES, III 33

question of the expansion method’s role in geography, it will become clear that the methodology has much to contribute to the larger social science effort in general. The remainder of the paper is organized into four sections. In the first I describe the fundamental characteristics of the expansion method paradigm, working from specific characteristics of the methodology to more general aspects and implications of its paradigmatic message. The section describes the consequences of the paradigm for social science laws, the academic division of labor, the evaluation of theories, and the relationship between micro arid macro level research. In the subsequent section I employ the paradigm as a lens to shed new light on the long debated and seemingly intractable distinctions between regional/idiographic and systematic/nomothetic geography. I also discuss shared themes existing between localities research and the expansion method. The penultimate section examines the relationship between the expansion method and realist methodology, a perspective which has been highly critical of the post– 1950s scientific endeavor in geography, while a final section offers conclusions. THE EXPANSION METHOD PARADIGM In a restricted sense, the expansion method (Casetti 1972, 1982, 1986) suggests that we pose questions concerning the manner in which functional relations perform in wider contexts. (It is also of course a methodology for answering such questions.) Functional relations are significant in that the social sciences typically address questions of variation. The variations of interest obviously differ across disciplines, from the behavior of persons in psychology, to the growth of local economies in regional science, to policy outputs among governmental bodies in political science. Nevertheless, the theme of variation across relevant units of interest is prevalent in all social sciences, and the empirical estimation of functional relations is one accepted way of understanding both the hows and the whys of such variation. The expansion method most commonly intervenes at the point where a wellestablished, theoretically grounded, empirical regularity has been identified through the estimation of functional relations. Questions pertaining to the contextual variation of these established relations can then lead to myriad avenues of inquiry. For example, context might include gender, race, cultural attributes, position in the global economy, type of economic or political system, religion, density, environment, hierarchy, demographic structure, level of development, etc. (Examples of such effects in this volume can be found in Chapters 2, 4, 6, 9, 10, 11, and 12. Determining how established functional relations vary with respect to these attributes may both provide clues to the original variation of interest and suggest new inquiries concerning the theoretical and conceptual premises upon which the relationships are predicated. In addition to the above listing, the expansion method’s contexts of inquiry include time and/or space. In the former, we are enjoined to ask questions

34 PARADIGMATIC DIMENSIONS OF THE EXPANSION METHOD

concerning the variability of relationships from one time period to another, in either discrete or continuous form (e.g. see Chapters 5 and 16 in this volume). In the case of the latter we are concerned with the manner in which established functional relations, and by implication the conceptual frameworks which underlie them, vary from place to place. As in the case of time, space may be treated as discrete (e.g. regions or specific places; see Chapter 11) or continuous (e.g. distance from a point of significance or two-dimensional Cartesian space; see Chapter 13). Finally, one aspect of the methodology’s flexibility is that it enables researchers to reflect upon and apprehend the simultaneous variation of relationships in space-time (see Chapters 7 and 8). The above comments provide us with one explanation of the expansion method’s popularity in geography. The discipline, like other social sciences, is concerned with variation. Yet because geography analyzes spatial variations (patterns) and their causes (processes), it is particularly sensitive to contextual variability. For geography, the expansion method contributes an additional question to the fundamental pattern-process problematic, namely, is there a pattern to the process? In other words, is the pattern-process nexus itself contextually mutable? This question—which represents a second tier of inquiry— encourages researchers to question the geography of pattern-process relationships. In other words, it implores us not to be satisfied by the estimation of functional relations, but instead to ascertain how, where, when, and why these relations vary from context to context. Such issues have been raised in other disciplines,1 yet they are especially meaningful in geography, which is arguably the most context-dependent social science. Seldom are analytical approaches so closely matched to an established disciplinary agenda. The paradigmatic aspects of the expansion method are not restricted to the redefinition of the parameters of functional relations by contextual variables, however. It also carries a broader message, i.e. that research should be ‘opened’, or continually reassessed for truth value. An open research agenda actively questions the answers that are obtained at every stage of the research process. It never leaves closed questions regarding the applicability of hypotheses, the universality of models and relationships, or the generality of conclusions. It suggests an interrogation of conceptual premises vis-à-vis empirical settings and thus views conclusions as contextually dependent instead of final. By insisting upon open research procedures, the paradigm disavows the notion of research as a truth finding mission. The quest for laws or universal models is not consistent with open research. Instead, the expansion method poses research as an interactive process in which answers to questions lead to new questions, and new answers to still newer questions. The expansion method can be conceptualized as an umbrella of whats, whens, wheres, and whys that query every round of the question-answer nexus. The result is a paradigm in which explanation, to the extent that it can be achieved, depends upon context. As a consequence of the above, the expansion method is incompatible with a conception of the world as an abstract, nontextured, ahistoric, and aspatial

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isotropic plane. Rather, the world, when viewed from the vantage point of the expansion method, is a complex setting which is temporally and spatially diverse and laden with contextual layers. From an analytical standpoint this diversity may be problematic since contextual information tends to be redundant, overdetermined, or analytically inseparable, but this does not inhibit the asking of questions. Identifying complexity is not an end in itself, however. It does not suggest that we revert to a Passargian search which makes sense of contexts by mapping them one at a time. Instead, context is always fundamentally keyed to theory, to process, and to substantively important questions. This has rather dire consequences for research undertaken in some quarters of the social sciences. It means, for example, that we can no longer treat social science models as expressions of universals, but instead must treat them as mathematical or descriptive portraits of empirical regularities or of subsystems which, if the last twenty years of its use are an indication (see Chapter 2), vary quite frequently across contexts. It also implies limitations on the extent to which the social sciences can or should mirror the procedures and goals of the physical sciences. Instead of seeing social science laws as long-term goals which await the further development of theory, the expansion method views laws themselves as unexamined assumptions about how the world is ordered. An example may clarify this claim. A great deal of empirical research has been concerned with estimating distance decay parameters (for a review see Sheppard 1984). Many studies have revealed systematic spatial variation in the parameter. This has led some (e.g. Fotheringham 1984) to suggest that the models employed to estimate distance decay parameters have been misspecified. If correctly specified, the reasoning follows, distance decay would attain a constant value across space. The expansion method intersects with this line of research at two levels. At one level, it suggests that parameter stability must be proven rather than assumed; thus it is sympathetic to research which questions the spatial stability of the parameter in particular research contexts. At another, more conceptual level, however, the paradigm challenges the very presupposition of order that underpins the search for a constant parameter. It suggests that spatial context cannot be ‘equated’ or ‘controlled for’ through ceteris paribus conditionality. The search for a constant parameter becomes futile as a result (Eldridge and Jones 1991). To summarize, the examination of the contextual variation of relationships in an open and interactive research program is consistent with a complex reality. Those whose goal it is to search for immutable processes might argue that such complexity can be overlooked or couched within distinctions between ‘general’ and ‘particular’. In contrast, the expansion method makes the relationship between explanation and complexity the central question. This does not imply that research merely celebrates the unique, nor does it suggest that tendencies do not exist or that we must reject the modeling enterprise entirely. It requires instead that we open our questioning to the varied contextual settings in which


research is undertaken. In the research process we cannot afford to stop at ‘initial models’—whether they be mathematical, empirically derived, or qualitative. These assertions have several important consequences. First, the method affirms a strong role for geography in the academic division of labor. Economists are often criticized for developing models ‘on the head of a pin’. Much the same criticism can be leveled against other social sciences that do not share geography’s concern for context. By arguing the significance of a textured ‘real world’ upon the research process, and by providing a methodology for the identification of parameter variation, the expansion method is uniquely positioned to contextualize the theories and models of its sister disciplines. This opens considerable room for cross-disciplinary interaction as the general statements of one discipline are molded to particular settings, which are of course the primary concern of geography. The aspatial and de-contextualized models of other social sciences can be improved and modified as a result, while at the same time geography’s traditional concern for the effects of context is placed within a strong theoretical frame of reference. Substantive areas in which this claim is demonstrated include Verdoorn’s law (Casetti and Jones 1983), the sectoral allocation of labor in development (this volume, Chapter 11), the estimation of production functions (Chapter 12), and human capital theory (Chapter 6). Second, application of the paradigm has consequences for the development of theory in the social sciences. It has already proven its potential for influencing the trajectories of research agendas through its ability to open up new lines of inquiry where tired and worn questions and answers now prevail. Questions of the how and why variability of hypotheses, theories, and relationships, a fundamental part of the research process under the expansion method, is absent in a surprising number of literatures. Many such areas have been invigorated and enhanced through its application, as new angles are suggested and fresh connections are made with previously unrelated conceptual frameworks. The current volume provides many such examples, ranging from agricultural location theory (Chapter 12), migration (Chapter 6), development (Chapters 2 and 11), and social policy (Chapters 4 and 5), to urban systems theory (Chapters 9 and 10) and spatial demography (Chapter 7). A further theoretical implication involves the role of the method in the evaluation of competing theoretical frameworks. Many multivariate analyses are carried out in an effort to assess the ‘correctness’ of alternative frameworks. Variables consistent with theoretical propositions are measured and evaluated vis-à-vis one another in an effort to establish theoretical ascendancy. The expansion method suggests that this type of research may be asking limited, if not incorrect, questions. If conceptual frameworks are found to be context dependent, then the search for ‘correct’ positions becomes futile. An example illustrating this point may be found in the work of Jones and Kodras (1986). They evaluated two perspectives on welfare participation growth, one focusing on the supply of welfare programs (e.g. growth in benefit levels) and one

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focusing on the demand for support (e.g. growth in unemployment). In previous work of this type, the issue had been demand versus supply. The contextual expansions carried out in their paper revealed that the demand perspective better characterized the growth of participation in the northeast and north central states, while the supply perspective was more suited to explanation in the south and west. Emerging from their analyses was a concern for the contextual validity of the competing theoretical frameworks, rather than an explicit rejection of either perspective. It should be emphasized that, although the expansion method has primarily been employed to investigate analytical models such as production functions, its paradigmatic message is equally valid for theoretical statements of a purely descriptive nature. Such qualitative models are equally likely to be posed as singular versions of reality without sufficient recognition of contextual dependence. Examples include society versus state-centered theories of the state (Clark and Dear 1984) and the various models of decisionmaking in management science (March 1978). Although sometimes lacking the formal structures that would enable a researcher to test their validity in a quantitative sense, such frameworks are nevertheless sufficiently rigorous to invite empirical investigation. In many cases, however, researchers may be content to assess the validity of such models in a particular research context, without explicit recognition that within that context there may be subcontexts that justifiably call for the type of critical examination undertaken in the expansion method. In assessing society versus state-centered theories of the state, for example, we might well be advised to examine whether evidence in one context points toward one theoretical framework, whereas an opposing perspective is suggested in a different context. In models of decisionmaking the same logic applies: what rational argument can be marshaled to support a research agenda aimed at the ‘discovery’ of a correct theory of decisionmaking? In spite of the fact that such models do not readily lend themselves to functional relations, the paradigm may still be employed as a guide to open research. Finally, the expansion method has implications for questions concerning the appropriate scale of empirical research. An issue of paramount importance in the social sciences is whether research should be directed toward the investigation of the actions of individuals or instead upon the aggregate characteristics, or macro level structures, embodied in societies, institutions, and places (Alexander et al. 1987). In geography this issue is central to the distinction between behavioral research at the individual level, on the one hand, and spatial analyses employing geographic aggregates, on the other hand (Golledge et al. 1972; Bunting and Guelke 1979). In recent years some geographers (e.g. Gregory 1981; Thrift 1983; Pred 1984) have attempted to overcome this dichotomy by adopting a spatialized version of Gidden’s (1984) structuration theory. Giddens situates human agency within larger societal structures which not only enable and constrain agency but are at the same time transformed by it. As a result, the


distinction between the micro and macro levels is overcome, but in such a way as to preserve the tension between both. The expansion method can be employed in such a way as similarly to forge a linkage between micro and macro level research. We can, for example, contextualize most micro level models by expanding their parameters in terms of the contextual settings in which the behavior under investigation takes place. Operationally, this involves merging micro level data with information pertinent to the context experienced by the individual. It is apparent that this approach is consistent with at least part of Giddens’ structure-agency dialectic, namely, that individuals are both enabled and constrained by structures. To illustrate, assume that we adopt human capital theory as the basis for a model of migration that will be estimated with individual level data. The expansion method suggests that we raise questions concerning the variability of parameters for individual characteristics such as education, age, sex, race, etc. That is, are these effects contingent upon space, time, or other contextual characteristics? Identifying how the parameters of individual level variables vary with respect to such contexts not only informs us about the effect of context upon migration (as Ellis and Odland demonstrate in Chapter 6), it provides at the same time a critique of an overly voluntaristic human capital theory. THE REGIONAL GEOGRAPHY QUESTION The debate between the regional and systematic schools in geography has a long and somewhat tendentious history (James and Martin 1981; Johnston 1987; Entrikin and Brunn 1989). The essential outlines of these two perspectives can be described as follows. Regional geography directs us to examine the interrelationships of various earth features, both physical and man-made, that exist within a geographically defined area. The end product of a regional analysis is a coherent and organized description (Hart 1982) and understanding (Hartshorne 1939, 1959) of the area under investigation. In contrast, systematic geography focuses on the spatial aspects of a small number of phenomena, with a greater concern for understanding their spatial variability. Thus, while regional geography takes as an operating framework the constellation of interrelationships existing within a specific place, systematic geography focuses upon the phenomena per se, without regional geography’s concern for providing accurate descriptions and analyses of a particular part of the earth’s surface. Hartshorne (1959) resolved the tension between regional and systematic geography by placing them on a continuum, in which, he suggested, one could examine a single phenomenon across the entire surface of the earth at one extreme, or examine all interrelated phenomena in a very small area at the other extreme. Along this continuum one could move to include more areas but examine fewer phenomena, or, alternatively, examine more interrelationships among phenomena but with less geographic coverage.

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However compelling this compromise may seem, it did not resolve the tensions between the competing schools. The regional perspective grew in disfavor during the late 1950s and 1960s as the discipline underwent a quantitative revolution (Burton 1963) that favored systematic analyses over regional ones (Taaffe 1974; Amedeo and Golledge 1975). One dimension of this disciplinary debate concerned the failure, in the opinion of systematic geographers, of regional geography to provide lasting statements of a causal scientific nature (Shaeffer 1953; Gould 1979). Regional geography was accused of being ‘mere’ description, a term denigrating decades of careful mapping of innumerable phenomena. On the other hand, members of the regional school recounted that systematic geographers were ignorant of the ‘craft’ of the discipline, that they were more concerned with mathematical and statistical models than with real places, and that such research would not lead to the development of geographic laws (e.g. Hart 1982). Interconnected with the debates between regional and systematic geography is the idiographic/nomothetic dualism in human geography. The idiographic views places as fundamentally unique and incapable of being related by general laws and theories, while the nomothetic searches for such laws. Characteristic of research in the former are descriptions of spatial variations that celebrate the unique aspects of the phenomena or region under investigation. Nomothetic research, on the other hand, tends to view the spatial variation of all phenomena as causally related to a set of general processes. In this view, any failure to provide accurate explanations of the real world is related to the current state of theoretical development in the discipline, not to a limitation in the ultimate purpose of general explanation. During the 1960s, regional geographers were accused of being idiographic by systematic geographers who adopted Shaeffer’s (1953) definition of geography—the development of ‘spatial laws’. The expansion method can be used to recast this debate, albeit in a different form from that favored by Hartshorne. It does so by initially aligning itself with those who favor systematic analyses of spatial patterns. At the same time, however, it is consistent with regional geography in that it suggests that researchers would be foolish to ignore the contexts within which the examination of these phenomena takes place. The paradigm does not favor adopting a middle ground; rather, it accepts as a first stage of the research process the empirical assessment of hypotheses emanating from theoretically meaningful propositions. Beyond this, however, it accepts the premises of the regional school, i.e. that the world is composed of diverse layers of phenomena interrelated in such a way as to provide infinite variability and uniqueness. This variation renders impossible the identification of meaningful general statements of a global nature. However, at the same time the expansion method suggests that such variation may have a systematic quality in its effect upon general processes. This is in fact what is accomplished by modifying the parameters of an initial model in an effort to identify regularity in the impact of contexts which are themselves unique. Thus, the expansion method accepts first the identification of causal processes, but, in


recognition of the limitations of lawseeking enterprises imposed by the diversity of the real world, seeks second to identify the variation in general processes resulting from such contexts, while, third, maintaining a concern for the orderly investigation of the mechanisms governing the variation in the general processes. As a consequence, the issue is no longer the correctness of either side of the idiographic/nomothetic dualism, but rather whether one can identify regularities in the effects that contexts have upon general processes. The paradigm is thus consistent with neither of the perspectives as described above. Instead, it views the debates in fundamentally new ways, with general processes and explanations as starting points, their modifications in certain circumstances an inevitable result of the structure of the real world, and the systematic identification of these modifications possible through the analytical procedures of the method. The above paragraphs have sketched the debate largely in terms of polar oppositions in empirical research. The programmatic statements of some regional geographers do show considerable sympathy with the expansion method paradigm, however. In fact, a close reading of these programmatic statements suggests that regional geographers had the expansion method’s paradigmatic structure in mind as the definition of the discipline from the outset. To justify this claim one need only investigate the works of two prominent regional geographers, Preston James and Richard Hartshorne. First, it would be wise to establish that Hartshorne, the acknowledged leader of the regional school, favored scientific work over idiographic treatises. Following Hettner, he writes: scientific advance in geography depends on the development of generic concepts and the establishment and application of principles of generic relationships. (Hartshorne 1959:160) But concerning the possibility of determining general laws, he states: The application of any general principle to a particular case depends on generic concepts which fit the particular case only approximately. The attainment of the maximum degree of accuracy requires determination of the degree to which the particular conditions depart from the ‘norm’ represented in each generic concept involved, and the consequences, in the process relationship, of those minor differences. (Hartshorne 1959:158) Hartshorne is not content, however, to speak simply of anomalies (i.e. residuals) from a general norm due to place uniqueness. Instead, he recognizes that one reason for deviations from the general case is that relationships in one area may differ from those in another. The investigation of spatial variations in relationships, or ‘areal patterns of covariance’, as Hartshorne framed it, was to be

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carried out by ‘comparative regional geography’, a term which never gained much favor in geographic writing. Operationally, comparative regional geography called for controlling variability in process relationships by dividing the world into regions distinguished not on the basis of one or more variables, but on the basis of one or more constant relationships: The purpose in dividing the area is to secure areal sections, or ‘regions,’ such that within each region the elements… under study will demonstrate nearly constant interrelations. (Hartshorne 1959:129, emphasis added) Hartshorne felt a need to construct regions on the basis of constancy in relationships because, in his words: In any part of the world in which man is included, we can be sure before we start…that we will not find the same integration [interrelation] repeated in areas of different culture, or in areas of different climate… (Hartshorne 1959:128, brackets added) Hartshorne’s comparative regional geography was thus an attempt to control for instability in relationships among variables so that investigations would not be contaminated by shifting parameters. The message is that context matters and that geography is not only the study of place variations but also the study of different integrations and interrelations over space. James, writing before the appearance of the Shaeffer paper, expresses the issue even more concisely. He states that geography contributes to an understanding of the operation of processes in particular places. It focuses attention on the modifications in the operation of processes by the other things that are not equal, by noting the actual operation of processes in particular places modified by the presence of the other things unsystematically associated there. (James 1952:222) This sentiment, that processes are modified by other factors, and that these can be clarified by examining them within particular places, is a central notion in the expansion method. A more contemporary perspective on the relationship between geography and the expansion method can be found in Taaffe’s (1974) effort to integrate the regional and systematic schools. He urged scientific geographers to develop general laws and theories that could then be examined for their place-to-place applicability by regional geographers. In other words, he saw regional and systematic inquiry as mutually reinforcing and beneficial. Who better to


contextualize general models and theories of systematic geographers than regional geographers with an intimate knowledge of places? The issues surrounding debates between regional/idiographic and systematic/ nomothetic geography continue to attract the attention of contemporary writers, though in a form and language somewhat different than past accounts. Specifically, they have arisen within the context of the ‘new regional geography’ (Pudup 1988) and ‘localities research’ (Duncan 1989). Both are ventures which have spatial differentiation as the central theoretical question. They arose in response to an overly functionalist radical social science which treated space more as a setting or stage for social processes than as an integral component of their operation. The theoretical background for rethinking the role of space and society derives from Soja’s work on the socio-spatial dialectic (1980, 1989), Harvey’s (1982) effort to integrate space into an understanding of capitalism, and Massey’s (1984) work on spatial divisions of labor. The claim that ‘space matters’ in social processes has led geographers and sociologists to undertake both theoretical work (e.g. Gregory and Urry 1985) and empirical work (e.g. Murgatroyd et al. 1985) which addresses how local social circumstances (e.g. of class, politics, gender, and race) influence the operation of larger social processes (e.g. the international division of labor). In addition, claims have been made on behalf of a constellation of purely local causal processes that define unique localities (Duncan 1989). These developments raise questions of the general and the particular, the idiographic and the nomothetic, and the status of place in theoretical geography. Smith (1987), for example, has sounded a cautionary note against the ‘empirical turn’ of localities research, a trend that would return geography to descriptive, atheoretical accounts of local variations and responses. One consensus, however, is that localities research should seek to investigate how large-scale social processes are affected by the constellation of circumstances in particular places. Duncan (1989) argues that identifying such ‘spatial contingencies’ is an important dimension of localities research. Clearly these issues intersect with themes that are central to the expansion method. In particular, there is a correspondence between the paradigm’s concern for the ways in which processes are molded to particular circumstances in local areas (Jones 1984), and locality research’s interest in the spatial specificity of social processes (Duncan 1989). Why these connections have not been previously illuminated, however, is another question. I believe this results from a disjuncture in methodologies rather than conceptual frameworks. Localities researchers have tended, for various reasons, to adopt a realist methodology in their empirical investigations, while expansion method applications are most often associated with modeling endeavors. In the next section I explore the interconnections between expansion and realist methods with an eye toward synthesizing some of the communalities and differences between these two approaches.

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THE EXPANSION METHOD AND THE REALIST CRITIQUE OF SCIENCE In recent years spatial science has come under increasing attack from those who have adopted realist methodologies (Bhaskar 1978; Sayer 1984). Realism shares with traditional social scientific methods a concern for explanation, but it diverges considerably from them in terms of how that explanation is derived. First, realism offers a critique of extensive research methods that rely upon the identification of common properties and general patterns to establish causal relations. It argues that extensive methodologies fail to tell us what underlying processes have produced the co-extensive patterns under investigation. It maintains that these processes can only be ascertained by intensive research which places in focus a concern for how causal processes work out in a particular case or in a limited number of cases (Sayer 1984, 1985). This requires that the researcher engage in a series of abstractions which ultimately lead to the identification of various necessary and contingent relations that are specific to the case or cases under investigation. Necessary relations are associated with a domain of real structures whose causal powers account for, but are separate from, the empirical events they produce. Interceding between these generative mechanisms and the empirical level of events are a host of contingent relations that give form to concrete events. The purpose of realism is to identify the ordering and operation of necessary and contingent relations. Sayer (1985) offers an example of realist methodology in the area of housing. A necessary relation significant to understanding the rental market is the existence of private property relations. In the absence of this necessary relation the landlord-tenant distinction would not be possible. The degree of conflict between landlord and tenant, however, may be affected by a host of contingent circumstances that can only be ascertained for a set of concrete circumstances. Examples of such relations might include the role of the local state in protecting renters or the race/gender/class position of the landlord and tenant. Intensive research seeks to identify how such relations impinge upon the particular circumstances under investigation. Although the generality of such research is necessarily limited by the case study approach, realists argue that such methods are necessary to make any theoretical claims, since the world is seen by realists to be an open system by virtue of contingent relations which may or may not have effects in particular circumstances. Sayer (1984) argues that closed systems are only encountered in laboratory situations, and that it is only in such contexts that regularities (or laws) can be identified, since the mechanisms under investigation are invariant and the relationships between the mechanisms and the conditions in which they occur are constant. In social systems, which are open, this is not the case, so actual events must be explained by the interplay of contingently related conditions. Herein lies the basis of the realist claim that modelling regularities tell us nothing about causation.


Irrespective of the criticisms that realists offer for traditional scientific methods, there are clearly strong similarities between the world view of the expansion method and realist methodology. In the language of the realist, the expansion method prompts us to engage in investigations of the varying mechanisms of fundamentally open systems. It shares realism’s view of the world as differentiated and stratified. And like realism, the expansion method is concerned with the investigation of contingent effects. Realist analyses of contingencies are presently carried out in detailed case studies, but some empirical work does bear a strong resemblance to the variation in outcomes based on context which is central to expansion method. These messages conjoin in an article by Mark-Lawson et al. (1985). The authors examine the effect of variations in Labour Party strength upon local social welfare provision in the UK. They uncover a contingency which determines whether the Labour Party’s state apparatus gets translated into increased sovial welfare—specifically, they find that gender equality in the workplace results in a stronger political effect. This article specifies hypotheses in nearly the same terminology that one might use in writing an expansion method paper, though it is firmly rooted in the realist tradition. CONCLUSION Science, Whitehead wrote, ‘is to see what is general in what is particular’ (quoted in Gould 1979). The expansion method adds to Whitehead’s homily the following: science should equally not ignore ‘what is particular in what is general’. Thus, the paradigm turns our attention to the contextual specificity of general social scientific statements. It not only leads us to question such statements, but it also provides a means by which such questions may be answered. It is within this spirit that the expansion method was originally proposed. A far more radical interpretation of the paradigm remains to be written, however. In particular, the expansion method is not inconsistent with a world view which argues that there are no ‘general processes’. Social scientific abstraction, however useful it may be as a heuristic device, is still abstraction. In the ‘real world’, which is, after all, what social science aims to understand, processes operate in space-time contexts. While such processes may be generalizable, they need not be conceptualized as general, at least not in the sense of operating in a state of suspension over space-time contexts. Accepting the embeddedness of processes in context does not, however, require that we abandon the aims of the social disciplines qua science. It will require, however, that these fields re-conceptualize the nature of context in the construction of their models and in their procedures of abstraction. For these endeavors, the expansion method will remain an indispensable component of social scientific methodology.

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NOTE 1 Consider, for example, the following warning by political scientists Forbes and Tufte concerning the perils of cross-sectional research: different units can have different causal processes’ (1968:1261).

REFERENCES Alexander, J., Giesen, B., Munch, R. and Smelser, N. (1987) The Micro-Macro Link, Berkeley, CA: University of California Press. Amedeo, D. and Golledge, R.G. (1975) An Introduction to Scientific Reasoning in Geography, New York: Wiley. Bhaskar, R. (1978) A Realist Theory of Science, Brighton: Harvester. Bunting, T.E. and Guelke, E. (1979) ‘Behavioral and perception geography: a critical appraisal’, Annals, Association of American Geographers 69:448–62. Burton, I. (1963) ‘The quantitative revolution and theoretical geography’, Canadian Geographer 7:151–62. Casetti, E. (1972) ‘Generating models by the expansion method: applications to geographical research’, Geographical Analysis 4: 81–91. ——(1982) ‘Mathematical modeling and the expansion method’, in R.B.Mandel (ed.) Statistics for Geographers and Social Scientists, pp. 81–95, New Delhi: Concept Publishing. ——(1986) ‘The dual expansion method: an application for evaluating the effects of population growth on development’, IEEE Transactions on Systems, Man, and Cybernetics SMC–16:29– 39. Casetti, E. and Jones, J.P. (1983) ‘Regional shifts in the manufacturing productivity response to output growth: sunbelt vs. snowbelt’, Urban Geography 4:285–301. Clark, G. and Dear, M. (1984) State Apparatus: Structures and Language of Legitimacy, Boston, MA: Allen & Unwin. Duncan, S. (1989) ‘What is a locality?’, in R.Peet and N.Thrift (eds) New Models in Geography, vol. 2, pp. 221–52, London: Unwin Hyman. Eldridge, J.D. and Jones, J.P. (1991) ‘Warped space: toward a geography of distancedecay’, Professional Geographer 41 (in press). Entrikin, J.N. and Brunn, S. (eds) (1989) Reflections on Richard Hartshorne’s The Nature of Geography, Washington, DC: Association of American Geographers. Forbes, H. and Tufte, E. (1968) ‘A note of caution in causal modeling’, American Political Science Review 62:1258–64. Fotheringham, S. (1984) ‘Spatial flows and spatial patterns’, Environment and Planning A 16:529–43. Giddens, A. (1984) The Constitution of Society, Oxford: Polity Press. Golledge, R., Brown, L.A. and Williamson, F. (1972) ‘Behavioral approaches in geography: an overview’, Australian Geographer 12:59–79. Gould, P. (1979) ‘Geography 1957–77: the Augean period’, Annals, Association of American Geographers 69:139–51. Gregory, D. (1981) ‘Human agency and human geography’, Transactions, Institute of British Geographers NS6:1–18.


Gregory, D. and Urry, J. (eds) (1985) Social Relations and Spatial Structures, New York: St Martin’s Press. Hart, J.F. (1982) ‘The highest form of a geographer’s art’, Annals, Association of American Geographers 72:1–29. Hartshorne, R. (1939) The Nature of Geography: A Critical Survey of Current Thought in Light of the Past, Lancaster, PA: Association of American Geographers. ——(1959) Perspectives on the Nature of Geography, Washington, DC: Association of American Geographers. Harvey, D.W. (1982) Limits to Capital, Chicago, IL: University of Chicago Press. James, P.E. (1952) ‘Towards a fuller understanding of the regional concept’, Annals, Association of American Geographers 42:195– 222. James, P.E. and Martin, G.J. (1981) All Possible Worlds: A History of Geographic Ideas, 2nd edn, New York: Wiley. Johnston, R.J. (1987) Geography and Geographers, 3rd edn. London: Edward Arnold. Jones, J.P. (1984) ‘A spatially-varying parameter model of AFDC participation: empirical analyses using the expansion method’, Professional Geographer 36:455–61. Jones, J.P. and Kodras, J.E. (1986) ‘The policy context of the welfare debate’, Environment and Planning A 18:63–72. March, J.G. (1978) ‘Bounded rationality, ambiguity, and the engineering of choice’, Bell Journal of Economics 9:587–608. Mark-Lawson, J., Savage, M. and Warde, A. (1985) ‘Gender and local politics: struggles over welfare policies, 1918–1939’, in L. Murgatroyd et al. (eds) Localities, Class, and Gender, pp. 195– 215, London: Pion. Massey, D. (1984) Spatial Divisions of Labor: Social Structures and the Geography of Production, London: Methuen. Murgatroyd, L. et al. (eds) (1985) Localities, Class, and Gender, London: Pion. Pred, A.R. (1984) ‘Place as historically contingent process: structuration and the timegeography of becoming places’, Annals, Association of American Geographers 74: 279–97. Pudup, M.B. (1988) ‘Arguments within regional geography’, Progress in Human Geography 12:369–90. Sayer, A. (1984) Method in Social Science: A Realist Approach, London: Hutchinson. ——(1985) ‘Realism and geography’, in R.J.Johnston (ed.) The Future of Geography, pp. 159–73, London: Methuen. Shaeffer, F.K. (1953) ‘Exceptionalism in geography: a methodological examination’, Annals, Association of American Geographers 43:226–49. Sheppard, E. (1984) ‘The distance-decay gravity model debate’, in G.L.Gaile and C.J.Wilmott (eds) Spatial Statistics and Models, Boston, MA: Reidel. Smith, N. (1987) ‘Dangers of the empirical turn: some comments on the CURS initiative’, Antipode 19:59–68. Soja, E. (1980) ‘The socio-spatial dialectic’, Annals, Association of American Geographers 70:207–25. ——(1989) Postmodern Geographies, London: Verso. Taaffe, E.J. (1974) ‘The spatial view in context’, Annals, Association of American Geographers 64:1–16. Thrift, N.J. (1983) ‘On the determination of social action in space and time’, Environment and Planning D: Society and Space 1:23– 57.

4 A CONTEXTUAL EXPANSION OF THE WELFARE MODEL Janet E.Kodras

Nomothetic and idiographic approaches to social science research have traditionally been viewed as antagonistic. The former seeks to develop generalizations about human conduct, culling out idiosyncrasies of behavior and thought, and abstracting to the level of theoretical understanding; the latter rejects the representation of human action in terms of theory and seeks instead to probe deeply into the single place and event. In response to this tension and the acknowledged strengths and weaknesses of each approach, a number of researchers have begun the search for a middle ground. Working from the nomothetic toward the idiographic, they seek ‘more flexible, extroverted, and combinatorial theorizations… theorizations that do not dogmatically project themselves onto the empirical world but instead are informed and open to diversity, uniqueness, and conjuncture—the distinctiveness of time and place, event and locality’ (Soja 1987:293). If this middle ground can be found, the ‘loosening’ of theories, such that their terms can vary across specific contexts, will represent a maturing of social science thought (Taaffe and Casetti 1990). This new formulation thus holds promise for researchers across the spectrum, and a fervent debate has arisen as to how we might best proceed. Most problematical is the issue of how empirical research can be conducted, given that the methodologies used in nomothetic and idiographic research differ fundamentally. The purpose of this paper is to demonstrate that the expansion method can contribute to this new research agenda, since it is a means for empirically testing how relationships posited by theory can vary according to the contingencies of place. This point is made by example. The first section reviews labor-leisure theory, the fundamental model underlying the design and implementation of US welfare policy, as well as popular attitudes toward it. Briefly, welfare assistance, by offering an alternative to employment, is seen to reduce the motivation to work, which decreases the labor supply and lowers output. Based on this argument, public assistance programs are designed such that they minimize competition with the private sector. The second section describes a set of geographically-specific conditions, ignored by labor-leisure theory, which are argued to have an effect upon the decision between work and welfare.

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Specifically, in places with insufficient labor market opportunities, poverty households are effectively prevented from choosing employment no matter how strong their desire to work. Where welfare practices are particularly restrictive, or the cultural stigma attached to the use of welfare is great, the decision to choose public assistance is inhibited. It is argued that the decision between work and welfare is not free-choice, nor is it aspatial, as assumed by theory. In the third section, the expansion method is used to test for variations in the work-disincentive effect of welfare across labor market and policy settings. Such variations are found, and in the following section they are interpreted in the context of regional political cultures, private sector influences, and state fiscal conditions. A concluding section makes the point that labor-leisure theory is overaggregated and underspecified, ignoring the complexities of the locale in which the decision between work and welfare is made. As a result, policy based upon it is misdirected and popular attitudes toward welfare are distorted. The flexible testing procedure of the expansion method allows these complexities to be brought to light. THE THEORY Economic theory represents the decision between work and welfare as a special case of the labor-leisure tradeoff (see Brehm and Saving (1964) for a formal graphical presentation of the labor-leisure model applied to welfare choice). Assuming that the marginal utility of labor is negative and that of leisure is positive, some utility-maximizing individuals will opt for leisure over labor, so long as income support programs guarantee a minimum income level. In the classic Economics of Welfare (1952:728), Pigou describes this process: this type of transference is involved in all Poor Law systems that fix a state of minimum fortune below which they will not allow any citizen to fall. For, in so far as they raise to this level the real income of all citizens whose provision for themselves falls below it, they implicitly promise that any reduction in private provision shall be made good by an equivalent addition to state provision. It is plain that the expectation of these differential transferences will greatly weaken the motive of many poor persons to make provision for themselves. In particular, two financial aspects of welfare programs influence work effort: the guarantee and the marginal tax rate (Masters and Garfinkel 1977; Danziger et al. 1980). The guarantee, which varies by family size, is the payment to a family with no other income. The marginal tax rate is the percentage by which welfare payments decline as earnings increase. For example, a tax rate of 60 percent means that benefits are reduced by 60¢ for each additional dollar earned. Guarantees and tax rates are positive for most welfare programs, including Aid to

A CONTEXTUAL EXPANSION OF THE WELFARE MODEL 49

Families with Dependent Children (AFDC), the program examined here. Thus, benefits are highest for families with no income and fall as income rises. Positive guarantees and tax rates both reduce work effort. The guarantee offers an alternative to employment and, the higher the guarantee is, the more attractive is the prospect of not working. The tax rate effectively reduces the wage rate by which the worker is rewarded (Danziger et al. 1980). Financial analyst Louis Rukeyser demonstrates the impact of the tax rate on the decision to work: ‘A typical family of four living in Los Angeles, with $4, 800 annual earnings will garner $810, 49 in net monthly spendable income if they use all available welfare assistance to which they are entitled. For each additional dollar earned, the family loses so many benefits that real income increases only 5.2 cents, an effective marginal tax rate of 94.8 percent. If family earnings double to $9, 600 a year, net spendable income would decline from $810.49 per month to $773.82’ (quoted in Weil 1978:48–9). Several researchers have sought to quantify the amount of work effort which is lost due to social assistance nationwide. Lampman (1978) estimates that the expansion of all social spending (including housing, education, manpower, and welfare programs) between 1950 and 1976 reduced hours worked by 7 percent more than if the system had not expanded. Danziger et al. (1980) find a work reduction of 3 percent attributable to increases in income support programs only. In addition, a number of studies, explicitly or implicitly derived from laborleisure tradeoff theory, support the contention that benefit levels influence the decision between work and welfare, which then affects the magnitude of public assistance rolls and the labor supply (Brehm and Saving 1964; Cain and Watts 1973; Spall and McGoughran 1974; Hamermesh 1977; Keeley et al. 1978a, 1978b; Bieker 1981; Menefee et al. 1981; Jones 1987). THE CONSTRAINTS Welfare payments which are sufficiently high to compete with earned income do indeed have the potential to reduce work motivation. But poverty is not simply the result of individual failure, the ‘inadequacy of human nature’ (Reagan 1968: 122), and welfare dependence is not merely the result of a free choice decision by the indolent in favor of leisure over labor. The choice between work and welfare is constrained by barriers to employment in the labor market and obstacles to assistance in the welfare system. First, the structure of the labor market makes work and self-sufficiency unattainable for many low-income Americans. That segment of the labor market available to the poor is often typified by high unemployment, seasonal jobs, discriminatory practices, and wage levels so low that even fulltime employment does not allow one to rise above the poverty level (Morrill and Wohlenberg 1971; Levitan and Johnson 1984). As the nation shifts from its traditional manufacturing base to a service-oriented economy, the nature of labor demand (its location, skill requirements, and wage structure) is rearranged. Whereas

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manufacturing typically paid middle-income wages to blue collar labor, the expanding service sector is highly polarized, offering high compensation to a few but very low wages to the growing legions of cashiers, food service personnel, computer terminal operators, etc. (Rumberger 1981; Kuttner 1983). These structural shifts leave technologically displaced labor, the unskilled, and many new entrants to the labor force little option but to accept service positions with low wages, sporadic employment, and little opportunity for self-advancement (Howes and Markusen 1981; Seninger and Smeeding 1981; Kuttner 1983). Furthermore, these labor market constraints are spatially variable. Different areas of the country produce distinctive sets of economic opportunities, earnings potentials, and types of poverty as a result of their position in the core/periphery of the national space economy, their intersectoral patterns of employment concentration, and the attendant wage structures. Those areas with long agrarian histories and recent industrialization, resistance against unionization, and traditions of racial or gender discrimination, as well as the regions leading the current restructuring toward a service economy, may provide many jobs but at low compensation. In addition to these labor market factors which impede the decision to work, there exist many obstacles to choosing welfare. The popular image is that welfare is a nationwide public dole, available to all who choose not to work. Yet the great majority of all income maintenance programs in the USA are confined to the elderly and the disabled, who are not expected to work. In 1979, fully 70 percent of major maintenance program funds were allocated to OASDI (Old Age, Survivors, and Disability Insurance, commonly called Social Security), Medicare, Supplementary Security Income, and Veterans’ Compensation (Table 4.1). Programs available to the able-bodied, working age population are primarily limited to Unemployment Insurance (UI) and Workers’ Compensation, AFDC, General Assistance, and a set of basic need packages, such as food stamps, Medicaid, and housing assistance. UI provides temporary income substitution for the involuntarily unemployed. Table 4.1 Major income maintenance programs, 1979 Program

Expenditure (in $ billion)

Total Social insurance Cash benefits Old Age, Survivors, and Disability Insurance (OASDI) Unemployment Insurance Workers’ Compensation Veterans’ Pensions and Compensation In-kind benefits

252.7 194.2 131.7 11.3 11.5 10.6


Program

Expenditure (in $ billion)

Medicare Welfare Cash benefits Aid to Families with Dependent Children (AFDC) Supplementary Security Income (SSI) General Assistance In-kind benefits Medicaid Food stamps Housing assistance Source: US Bureau of the Census 1984: Tables 605 and 608

29.1 58.5 11.7 7.5 2.2 21.8 7.3 8.0

Because it requires previous attachment to the formal labor market, UI is unavailable to those with poor work histories or employment in the informal sector. As a result, many low-income Americans are excluded. General Assistance provides emergency cash to individuals in need. Funded entirely by the states, the program accounts for less than 1 percent of all income maintenance expenditures and, as such, cannot be regarded as a viable alternative to employment nationwide. Of greater importance are the Food Stamp Program, Medicaid, and housing assistance, which provide the basic needs of food, medical attention, and shelter to poverty households. Before the establishment of the Food Stamp Plan in 1964, no national program existed to assist low-income families headed by able-bodied, employable males (Masters and Garfinkel 1977: 7). Finally, AFDC is a very large and expensive program, which provides financial assistance to low-income families with children under 18 years. Approximately 98.5 percent of all recipient households are headed by females (US Social Security Administration 1982). When first established in 1935, the program was designed to assist women who had lost their husbands and needed to care for children in the home. As societal norms have shifted and program costs have risen, these women are increasingly expected to work. By providing an alternative to employment, AFDC is now regarded as a female work disincentive. In summary, the welfare options available to the working age population are far more limited than generally perceived. They are also spatially variable. The federalized public assistance system, which grants control of program operation largely to the states, ensures that variations in welfare provision will conform primarily to state boundaries. For example, AFDC is decentralized, and the states are responsible for setting eligibility criteria, benefit levels, and the location and operation of welfare offices. Because each state bears part of the financial responsibility, its legislature must devise a welfare budget in accordance with the

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political priorities of public assistance, vis-à-vis other competing concerns, in that state. The state welfare bureaucracy is then responsible for program administration, within the budget constraints sent down from the legislature, and in accordance with local attitudes concerning the viability of public assistance. Thus, where there exists substantial local resistance against welfare, as, for instance, where the private sector requires a low-wage, dependent labor force, public assistance can be made most unattractive to potential recipients by understaffing welfare offices, creating administrative confusion, or explicit harassment (Elman 1966; Albin and Stein 1968; Piven and Cloward 1971, 1981; Masters and Garfinkel 1977). Federalism gives spatial form to the US public assistance system, allowing the states to modify welfare provision to their needs and political priorities. Considering the spatially varying barriers in the labor market and the welfare system together, it is evident that the ‘decision’ between work and welfare is strongly conditioned by the context in which one lives. As Albin and Stein (1968: 301) note: ‘The choice available to the actual or potential recipient is rarely more than a choice between a single, narrowly constrained relief option and the available market opportunities for wage income.’ Accordingly, a number of empirical studies have examined the effect of labor market and/or welfare constraints on the use of public assistance (Kasper 1968; Albin and Stein 1971; Winegarden 1973; Thrall 1981; Kodras 1982, 1984; Jones 1984a; Jones and Kodras 1984; Kodras 1984; Johnston 1990). AN EMPIRICAL TEST Based upon the two positions identified above, this study works from the perspective that welfare assistance can, and does, influence work motivation, but that the extent of this effect is conditioned by spatially varying opportunities in the labor market and in the public assistance system. Having discussed the general mechanisms by which welfare acts as a work disincentive, the study now turns to an empirical investigation, specific to AFDC. As noted above, AFDC is currently regarded as a work disincentive. Established by the Social Security Act of 1935, it has evolved into one of the largest, most costly, and most controversial forms of public assistance. It provides federal grants to states to partially defray the costs of providing financial assistance to needy children. Assistance is granted to households whose children are under the age of 18 and deprived of support because of the death, absence, or incapacity of a parent (Platky 1977). In addition, roughly one-half of the states elect to provide payments to households with an unemployed parent under AFDC-UP, an optional program passed by Congress in 1961. Although the federal government does provide guidelines, states are given considerable discretion in AFDC operation. Decentralized control has resulted in substantial interstate disparities in administrative restrictiveness (Wohlenberg 1976a), eligibility criteria (Chief 1979), benefit levels (Wohlenberg 1976b),


optional program riders (Hosek 1982), and the overall effectiveness of state programs (Wohlenberg 1976c). The analysis consists of a two-step procedure. An initial model applies laborleisure theory to AFDC. It specifies the response, or elasticity, of state level AFDC recipient rates to corresponding work-disincentive levels. The latter are measured as ratios of welfare payment to earned income which estimate the incentive to substitute welfare for work. This model represents the free-choice situation, as no constraints are introduced as controlling variables. In the second step, the initial model is expanded to allow the relationship to vary across different labor market and public assistance contexts. The importance of these constraints upon the work-disincentive effect is judged by parametric shifts in the relationships. The study does not specifically address the individual decisionmaking process between work and welfare. Rather, we examine the aggregate effect of such decisions upon welfare participation. It is the aggregate effect, the magnitude of program participation, which is the concern of welfare officials, private entrepreneurs, and the tax-paying public. State level aggregation is appropriate, since the states exert most control over program administration and regulations. Full definitions and sources of all data are presented in Appendix 4.1. Initial model The magnitude of state AFDC participation is measured here as the proportion of poverty families who are program recipients:1 PR is the number of AFDC recipient families divided by the number of poverty families. The workdisincentive effect, WD, is measured as the mean AFDC income of recipient families divided by the mean earned income of poverty families. Large values of WD indicate high welfare payments relative to that which is gained from employment, and therefore a strong economic motivation to substitute welfare for work.2 Large work disincentives, according to theory, should translate into greater welfare participation. Thus, we anticipate (4.1) The parameters of a double log specification were estimated by ordinary least squares (OLS) regression, using state level data for 1979. The results are as follows: (4.2) where t values indicating the strength of relationships appear in parentheses. The elasticity of the relationship is positive, as expected. As the welfare-to-work income ratio of poverty families increases by 10 percent, their participation in AFDC increases by 6.5 percent. Thus, we do see some evidence of a work

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disincentive. Where states set AFDC benefits high relative to the earned income of the poor, the use of welfare is also high. The constraints Next, a set of labor market and welfare variables, hypothesized to have an effect upon the initial relationship, is examined. Labor market barriers, which are relevant to the predominantly female AFDC population, include state data on the following: 1 FUNEMP—the female unemployment rate; 2 NOJOB—the proportion of state employment which is in occupations other than retail trade, nonprofessional services, or nondurable manufacturing, jobs most open to the AFDC population (Bieker 1981); 3 WAGEDIFF—the ratio of male to female mean annual earnings;3 4 PFWORK—the proportion of female-headed households whose head is employed full time (35 + hours per week) and full year (50 + weeks per year) yet remain below the poverty level; 5 SEVERITY—the median dollar amount by which poverty families fall below the poverty level; 6 FEMPOV—the proportion of female-headed households below the poverty level. These variables are selected as labor market barriers because the decision to work is constrained by high unemployment, the unavailability of jobs for which one is qualified, wage discrimination, meager earnings, and the severity of poverty (in general and specific to the AFDC target group), respectively. Welfare constraints affecting AFDC participation include state data on the following: 1 NEEDS—the state welfare administration’s determination of the amount necessary to meet basic needs; 2 ADMIN—an index of state welfare administration leniency (Jones 1984b); 3 SERVICE—the ratio of state and local welfare employees to poverty families; 4 TRANSFER—the proportion of persons receiving public assistance who are classified poor excluding public assistance and nonpoor including public assistance. These variables represent program regulation (as evidenced by the needs standard and administrative practices), the ability of the welfare bureaucracy to serve potential demand, and the effectiveness of welfare in raising groups above the poverty level, respectively.


Factor analysis was used to group these labor market and welfare variables into contextual factors. There exist both methodological and conceptual rationales for this course of analysis. First, using a small number of factors in the terminal model presented below, rather than the nine variables separately, avoids problems of multicollinearity and ensures a lesser reduction in degrees of freedom. Second, factor analysis allows us to identify spatial patterns in the combinations of constraints rather than considering them as independent forces. As noted above, labor market opportunities vary throughout the country, because of different mixtures in economic base, historical patterns of labor compensation, and regional economic cycling. Given the flexibility of a federalized public assistance system, welfare may be spatially adjusted to these labor market conditions. Thus, different groupings of variables are possible. If liberal welfare provision is positively associated with substantial poverty, for example, welfare services are addressing the need for assistance. The analysis, using state data for 1979, resulted in three principal components attaining eigenvalues greater than 1.0; together they account for 65 percent of the total variation. Table 4.2 Varimax rotated factor matrix Variable

Factor 1

Factor 2

Factor 3

Proportion of variation accounted for by the factors

FUMEMP –0.030 0.716a –0.014 0.514 NOJOB 0.214 –0.118 0.813a 0.721 a WAGEDIFF –0.142 0.034 0.862 0.764 PFWORK –0.653a 0.545a –0.021 0.724 SEVERITY –0.755a 0.008 0.017 0.570 FEMPOV –0.412 0.752a 0.066 0.740 a NEEDS 0.830 –0.075 0.137 0.713 ADMIN –0.200 –0.616a 0.093 0.428 SERVICE 0.546a –0.439 0.151 0.514 TRANSFER 0.880a 0.118 –0.100 0.798 a Note: Factor loadings greater than 0.50 indicate that more than one-quarter of the variation in the variable is accounted for by the factor.

Upon varimax rotation they yielded the factors shown in Table 4.2. Note, from the rightmost column, that the majority of variables are well represented by the grouping procedure. Briefly, factor 1 represents conditions of little poverty and effective welfare provision; factor 2 reflects high female unemployment and poverty and restrictive public assistance; and factor 3 represents labor market constraints, such as the unavailability of jobs open to the AFDC population and female wage discrimination. Taken together, the three factors tend to group

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favorable labor market conditions with liberal welfare provision and substantial poverty with restrictive public assistance. Thus, it does not appear that interstate variations in welfare provision are due to a process of matching services with needs. Considering each in detail, factor 1 exhibits strong negative loadings on the severity of poverty (SEVERITY) and the percentage of female-headed households who work full time yet remain below the poverty level (PFWORK). High positive loadings are shown for the effectiveness of transfer payments in raising families above the poverty level (TRANSFER), liberal need standards set by the state (NEEDS), and welfare bureaucracies which are sufficiently large to serve potential demand (SERVICE). Thus, the factor separates states with little poverty and effective welfare provision from those where poverty is substantial but public assistance restrictive. In general, New England states, the upper Midwest, and the west coast have high positive factor scores while the southeast exhibits strong negative scores (factor scores are listed in Appendix 4.2). Factor 2 shows high positive loadings for the percentage of female-headed households below the poverty level (FEMPOV), the female unemployment rate (FUNEMP), and the percentage of female-headed households who work full time yet remain below the poverty level (PFWORK) and a strong negative loading for leniency in AFDC administration (ADMIN). Thus, this factor represents conditions specific to female poverty and its association with welfare constraints. Factor scores are strongly positive throughout the south but negative in the upper Midwest and interior western states. Finally, factor 3 exhibits strong positive loadings on male-female wage differentials (WAGEDIFF) and the unavailability of jobs for which the majority of the AFDC population are qualified (NOJOB). These labor market constraints are reflected in positive factor scores in the industrial core and Midwest. Negative factor scores are found primarily in the deep south, where wages for males and females are both low, given four years of education, and where lowearning occupations are a greater proportion of all employment than is the case elsewhere. The terminal model To this point, the analysis has demonstrated that the states vary in AFDC use by the poverty population, in accordance with differences in the relative financial advantage of participating. Additionally, the states exhibit variable combinations of labor market and welfare barriers, which may affect the extent to which work disincentives are translated into program use. To test this proposition, the constraints are introduced into the model via the expansion method. Beginning with the initial model, (4.3)


the b coefficient represents the national average in the response of AFDC participation to work disincentives. To incorporate spatially varying conditions in the labor market and welfare system, b can be expanded from a constant to a function of the factor scores: (4.4) The factor scores measure the extent to which conditions identified by the factors exist in each state. Substituting (4.4) into (4.3) yields the terminal model: (4.5) Equivalently, (4.6) The magnitude and direction of the parameters have conceptual meaning. First, b0 represents the independent effect of work disincentives on program participation. Parameters b1 through b3 measure the interactive effect of work disincentives, under conditions imposed by the labor market and welfare system. Where substantial labor market constraints are in effect, we expect greater use of welfare, since employment is difficult to obtain or nonremunerative. Thus, for example, the parameter b3 should be positive, since factor score 3, which represents labor market constraints, should raise the effect of a given WD upon PR. Alternatively, where substantial welfare constraints are in effect, we expect less welfare participation response to a given work-disincentive level. The parameters of equation (4.6) were estimated using OLS stepwise regression. The stepwise procedure was used to build a model which retains only those variables accounting for significant variation. If b1 through b3 are not significant, and b0 retains its significance, the work-disincentive effect is unaffected by labor market and welfare conditions in the states (see Stonecash and Hayes (1981) for a discussion of the stepwise procedure and decision rules regarding significance for interactive models of this kind). Table 4.3 Results of the initial model ln PR=4.144+1.049 ln WD+0.290FS1 ln WD–0.203FS2 ln WD (85.921)

(8.310)

(2.582)

(–3.004)

+0.177FS3 ln WD (2.288) Step

Variable

R2 change

t-to-enter

1 ln WD 0.4697 6.520 2 FS1 ln WD 0.0661 2.587 3 FS2 ln WD 0.0513 –2.390 4 FS3 ln WD 0.0430 2.288 Note: t values appear in parentheses; R2 = 0.63; adj. R2=0.60.

Significance 0.000 0.013 0.021 0.027

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Results are reported in Table 4.3. All variables contribute significantly to the model; the change in R2 is greater than 4 percent at each step and the t-to-enter is significant at the 0.03 level or better. Multicollinearity does not appear to be a problem, since coefficients remain stable as the model builds and zero-order correlations among independent variables are on the order of 0.3 or 0.4. Substituting the estimated parameters into equation (4.5) yields (4.7) The parameter b0 (1.049) represents the overall relationship between WD and PR. As was the case for the initial model, the parameter is positive. In this case, however, the work-disincentive effect on welfare use is modified by the labor market and welfare contexts, represented by the factor scores and their corresponding parameters. The interactive effect with factor score 1 is positive (0.290). Recall that states with positive factor scores on factor 1 have low poverty and generally effective welfare provision. In this context, the relationship between the work-to-welfare income ratio and AFDC use is accentuated, as represented by the positive parameter. For a given ratio of welfare-to-work income, AFDC use is relatively greater in states with little poverty and effective welfare. In other words, where welfare is sufficiently provided and the economic incentive to participate is great, the response is high welfare use. On the other hand, the relationship between AFDC use and the work-to-welfare ratio is diminished in states with substantial poverty and welfare constraints, as represented by their negative factor scores. The interaction with factor 2 is negative (–0.203). States with positive factor scores have substantial female poverty and unemployment, yet restrictive welfare. In this context, then, the work-disincentive effect on AFDC is diminished. For a given ratio of welfare to work income, there is relatively less welfare use, where there exist constraints imposed in the welfare system or substantial female poverty and unemployment. The interaction with factor 3 is positive (0.177). States with positive factor scores exhibit labor market constraints, such as wage discrimination and a small proportion of all jobs which are available to the majority of AFDC heads. In this context, the response of AFDC participation to work disincentives is accentuated. Where jobs are difficult to obtain or non-remunerative, welfare use is relatively high, for a given welfare-to-work income ratio. Individual state parameters, representing the sensitivity of AFDC use to work disincentives in the context of their labor market and welfare constraints, are calculated by substituting parameter estimates and state factor scores into equation (4.4): For example, Minnesota’s parameter is calculated as


Figure 4.1 Interstate variations in the work-disincentive effect, 1979. States in striped pattern have the strongest relationship between work disincentives and welfare participation. Southeastern states, in white, show the lowest response in state welfare use to work disincentives

(4.8)

Compare the magnitude of Minnesota’s work-disincentive effect with that of Mississippi: (4.9)

The state parameters are mapped in Figure 4.1 and listed in Appendix 4.2. The work-disincentive effect is greatest on the west coast, in the upper Midwest, and in lower New England. A given welfare-to-work income ratio results in greater AFDC use in these states, which tend to have negative factor scores on factor 2 and positive on factors 1 and 3. Therefore, welfare use response is high where there is relatively little poverty and welfare is sufficiently provided but labor market constraints are in effect. The group with the highest work-disincentive effect is not entirely as expected. To the extent that welfare critics take a spatial

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perspective, the accusing finger points to the industrial northeast, particularly New York, New Jersey, Ohio, and Illinois, as the area where potential recipients are enticed into welfare dependence, and out of the labor market, with liberal program benefits. These are not the states, however, which show the greatest work-disincentive effects. The response of AFDC participation to the disincentive is lowest in the southeast, which has negative factor scores on factors 1 and 3 but positive scores on factor 2. Thus, in the south, where there is substantial poverty, restrictive welfare, and a greater proportion of jobs available to the AFDC population, welfare participation is low for a given welfare-to-work income ratio. Finally, it is informative to compare situations with and without constraints. Massachusetts represents the case relatively free of labor market and welfare barriers (a positive score for factor 1 and negative scores on factors 2 and 3). In this context, the work-disincentive effect is strong, as predicted by labor-leisure theory. Only three states fulfill the assumption of no constraints made by laborleisure theory. Where these barriers are substantial, the work-disincentive effect is much diminished. In Louisiana, for example, the existence of all constraints (a negative score for factor 1 and positive scores on factors 2 and 3) largely negates the economic motivation to substitute welfare for work. For the great majority of states, one or more barriers limit the work-disincentive effect of welfare.4 INTERPRETATION OF THE SPATIALLY VARYING PARAMETERS The state parameters mapped in Figure 4.1 illustrate that work disincentives are differentially translated into welfare participation across the nation. As estimated in equation (4.7), these variations in the work-disincentive effect derive from different sets of labor market and welfare conditions existing in the states. However, such conditions are but proximate causes, the quantified surrogates, of underlying political and economic forces which affect operation of the welfare system within the larger economy. By extension, the state parameters, representing the severity of the work-disincentive effect in each jurisdiction, are calculated estimates of the variable role played by these forces. For example, the Mississippi state parameter, calculated in equation (4.9), is the lowest in the nation. The motivation to substitute welfare for work is minimal in the state since there exist substantial barriers to the use of welfare (low basic needs standards, restrictive administration, small bureaucracies to serve potential demand, etc.) but relatively inconsequential labor market constraints (e.g. low-skill job availability). These proximate variables of the principal components analysis reflect fundamental political and economic processes, rooted in the historical traditions of the state. First, Mississippi’s punitive welfare system is a function of its small tax base and a traditionalist political culture, which assigns a low priority to public assistance. Second, the state’s labor structure is oriented toward low-skill, small-compensation jobs due


to its long agrarian history and recent industrialization, the role it has played in the periphery of the nation’s economy, and its racial traditions of servitude and slavery. The work-disincentive effect varies by states because they are the physical manifestations of a federalized public assistance system, which operates within the larger, spatially variable economy. First, federalism directly affects the political process of welfare provision. In fact, several of the welfare variables in the principal components analysis would not be variable if it were not for state control. Additionally, federalism allows the states to modify welfare in accordance with their particular economic circumstances. Hence, different combinations of welfare system and labor market conditions are possible. The rationale for decentralized control is that state and local governments are best able to identify and address the economic problems in their jurisdictions, since ‘no nationally uniform system can do justice to the infinite variety of types of need, individual problems, and potentials’ (Freeman 1981:27). Thus, in matching services to needs, ‘the closer the level of government is to the people, the more efficient and effective our social welfare programs are apt to be’ (Anderson 1978: 166). This study indicates that services do not match need, however. Recall from the principal components analysis that states with the most severe poverty, in general and specific to female-headed households, are associated with punitive welfare policies, as evidenced by low basic needs standards, restrictive administration, and small bureaucracies relative to potential demand. Three alternative perspectives may be advanced as to why interstate variations in welfare provision do not accord with a map of need for assistance. First, states with substantial poverty may lack the fiscal ability to support welfare since a small tax base must be allocated among many competing concerns. Thus, states with the greatest need for assistance will have the most punitive welfare programs. Second, state welfare systems may be used to address the needs of capital rather than the needs of the poor. For instance, economies based on low-wage structures cannot be sustained without a restrictive welfare system. The poor states may provide only minimal assistance to prevent welfare benefits from competing with wages in the private sector. Finally, states may modify welfare packages in accordance with local political attitudes toward public assistance. Elazar (1972) has examined the geography of American political ideologies and its effect upon the variable provision of government programs: Many differences in state responses within the federal system appear to be stimulated by differences in political culture among the states…the particular pattern of orientation to political action in which each political system is imbedded. Political culture, like all culture, is rooted in the cumulative historical experiences of particular groups of people…[Thus] every state has certain dominant traditions about what constitutes proper

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government action and every state is generally predisposed toward the federal programs it can accept as consistent with those traditions. (Elazar 1972:88–9) Interstate variations in welfare provision are associated with political cultures dominant in different regions. For example, Elazar (1972) defines three major cultures and their linkages to public policy: the moralist tradition, which prevails in the upper Midwest, views government intervention as a positive force which is necessary to insure the common good of all citizens; the individualist tradition, strongest in the industrial core, views government as an allocation mechanism between individuals bargaining for rewards; and in the traditionalist culture, dominant in the deep south, government is oriented toward the interests of a governing elite. With regard to welfare, the moralist tradition is most supportive, since it addresses the issue of social well-being, while the traditionalist culture is most resistant, since it is a redistributive mechanism between income groups, which threatens the position of elites. The results of this study support these associations. As shown in Appendix 4.2, states with liberal welfare programs (positive scores on factor 1 and negative on factor 2) and the resultant large work-disincentive effects (high parameter values) are predominantly moralist. On the other hand, those with restrictive welfare and minimal work-disincentive effects are primarily traditionalist states in the southeast. None of the three perspectives cited above is alone a sufficient explanation for why welfare provision does not match needs. In fact, state fiscal abilities, private sector influences, and political ideologies are not separable. Moralist political cultures tend to exist in wealthier states, which have the financial ability to support welfare and a private sector not resistant to liberal public assistance. Traditionalist cultures are primarily found in poorer states, which lack the tax base necessary to fund the programs and must be sensitive to the needs of private sectors with low wage structures. Thus, it is not possible to identify first causes among these conditions. The point is that the states are able to adjust welfare to their fiscal abilities, labor market conditions, and dominant political ideologies, given the flexibility of decentralized control. Federalism gives geographic expression to the US public assistance system and is an underlying mechanism which allows spatial variation in the political and economic forces influencing the work-disincentive effect of welfare. THE EXPANSION METHOD AND SPATIALLY VARIABLE SOCIAL POLICY If this study had used a purely nomothetic approach, the empirical test of laborleisure theory would have been confined to the initial model, which found general support for the thesis that welfare engenders a work disincentive. The expansion method was used to extend beyond this initial finding, incorporating


into the terminal model a test for the differential operation of work disincentives, according to the context in which the decision between work and welfare is made. Thus, the two-step procedure can be seen as a movement from a purely nomothetic study in the direction of the idiographic. The validity of the labor-leisure model was found to vary spatially, as the work-disincentive effect of welfare varied across labor market and policy contexts. This finding accords with conclusions drawn from the study of public assistance in particular locales. The point is perhaps best made by reference to the statements of leading critics and advocates of the welfare system. For example, when President Reagan wished to stress a point on welfare fraud, he often used anecdotes from large Northeastern cities, where welfare bureaucracies are often so overloaded that some of the undeserving no doubt receive benefits. An example is the account of Chicago’s Welfare Queen, a woman accused of misrepresenting herself as the widow of several deceased Navy men and collecting welfare and widow’s grants, which Reagan often included in presidential campaign speeches (Hannaford 1983:90–1). When Piven and Cloward (1971) portrayed welfare as a tool manipulated in the interests of capital, they drew most of their examples from the rural south, where, during the 1950s and 1960s, welfare was quite stringently controlled to prevent it from drawing workers away from southern agriculture. With regard to the most recent and famous welfare anecdote (Murray 1984), in which Phyllis and Harold decide between work and marriage versus welfare and separation, Murray and Greenstein (1985) and Greenstein (1985) have discussed whether Phyllis and Harold’s decision was affected by the fact that they lived in the liberal welfare state of Pennsylvania rather than a more conservative state such as Mississippi. It is evident in all these examples that the advocates of different positions use regions that support their cases most strongly. Each position in the welfare debate is more valid in some places than in others because the programs have different impacts in different contexts. Therefore, national welfare programs whose design is based solely on the idiographic study of particular places or on the nomothetic analysis of abstract theory will be misguided. Further, societal perceptions of the welfare system which are derived from only one of these approaches will be distorted. The expansion method is a flexible procedure which tests theoretical notions according to the contingencies of place. As such, it is an appropriate method for researchers seeking a middle ground between the abstractions of nomothetic approaches and the parochialism of idiographic ones. NOTES This paper is a modified version of an article appearing in the Annals, Association of American Geographers 26 (2), 1986, pp. 228– 46. Permission to incorporate it here was granted by the Annals editor.

64 J.E.KODRAS

1 The number of poverty families is used as a surrogate of the number eligible. Due to program regulations, not all poverty families are in fact eligible, but this measure was judged to be more representative than the number of all families, households, or persons, which are commonly used in studies of AFDC participation (Brehm and Saving 1964; Albin and Stein 1971; Winegarden 1973; Bieker 1981; Jones 1984b). These studies incorporate variations in need and eligibility as independent variables. The present study incorporates eligibility variations into the dependent variable. 2 The work-disincentive variable is composed of actual levels in AFDC benefits and earnings rather than potential levels, such as the ratio of the AFDC guarantee to wage rates. As such, it measures the average AFDC payment relative to average earnings of poverty families, rather than the potential which might be acquired from the welfare system or through employment. The ratio of actual levels is the more appropriate measure for several reasons. First, studies have shown that the informal calculus of the work-welfare decision tends to be based on the experiences of acquaintances in similar circumstances and the amounts they actually receive for work or on welfare. Only rarely are AFDC recipients fully aware of the potential amounts they might receive, given the myriad regulations influencing welfare payment levels and the complexity of the employment market (Opton 1971; Chrissinger 1980). Second, potential welfare benefits would be very difficult to estimate for the numerator of the ratio. The states vary not only in the AFDC guarantee (the maximum benefit to a family with no income), but also in the marginal tax rate, by which the benefit decreases as earnings increase. The complexities of these state sliding scales prevent the marginal tax rate from being incorporated with the guarantee as a measure of potential welfare benefits. 3 This measure of wage discrimination is calculated for males and females older than 18 years with four years of high school education, and with full-time (35 + hours per week) and full-year (40 + weeks per year) employment. It does not explicitly control for age differences but state ratios for various age groups were, in most cases, quite similar to the overall adult ratios. Four years of high school education is appropriate for the AFDC population. The measure does not control for job tenure, however, since such data are unavailable (see Rytina (1982) for a discussion of job tenure effects on male-female earnings differentials). 4 Previous research has identified a number of economic, political, and demographic conditions which are associated with aggregate welfare participation (see Kodras (1982) and Jones (1984b) for reviews of these studies). To test whether the workdisincentive effect remains stable after controlling for these variables, and thus represents a truly significant correlate with AFDC use, a number of models were run, with sets of independent variables added to equation (4.6). Demographic variables included percent urban, percent Black, percent Hispanic, and the divorce rate. The economic and political variables were those used in the factor analysis, entered independently into the equation. Because of problems with multicollinearity, all variables could not be incorporated into the same equation. In each model the magnitudes of parameters b0 through b3 were altered, but the workdisincentive effect remained significant and state parameter maps showed similar regional patterns. In fact, zero-order correlations of state parameter values between the various models were in all cases greater than 0.90. These subsidiary models


lend credence to the results reported in the text, demonstrating the importance of spatially varying work-disincentive effects on welfare participation.

REFERENCES Albin, P. and Stein, B. (1968) ‘The constrained demand for public assistance’, Journal of Human Resources 3:300–11. ——and——(1971) ‘Determinants of relief policy at the subfederal level’, Southern Economic Journal 37:445–57. Anderson, M. (1978) Welfare, Stanford, CA: Hoover Institution Press. Bieker, R. (1981) ‘Work and welfare: an analysis of AFDC participation rates in Delaware’, Social Science Quarterly 62:169–76. Brehm, C. and Saving, T. (1964) ‘The demand for general assistance payments’, American Economic Review 54:1002–18. Cain, G. and Watts, H. (1973) Income Maintenance and Labor Supply, Chicago, IL: Markham. Chief, E. (1979) ‘Need determination in the AFDC program’, Social Security Bulletin 42: 11–21. Chrissinger, M. (1980) ‘Factors affecting employment of welfare mothers’, Social Work 25:52–6. Danziger, S., Haveman, R. and Plotnick, R. (1980) ‘Retrenchment or reorientation: options for income support policy’, Public Policy 28:473–90. Elazar, D.J. (1972) American Federalism: A View from the States, New York: Thomas Crowell. Elman, R. (1966) The Poorhouse State: The American Way of Life on Public Assistance, New York: Random House. Freeman, R.A. (1981) The Wayward Welfare State, Stanford, CA: Hoover Institution Press. FSU Census Access System, 1980 Census of Population and Housing. Greenstein, R. (1985) ‘Losing face in “Losing Ground”’, New Republic 192:12–19. Hamermesh, D. (1977) Jobless Pay and the Economy, Baltimore, MD: Johns Hopkins University Press. Hannaford, P. (1983) The Reagans: A Political Portrait, New York: Coward-McCann. Hosek, J. (1982) ‘The AFDC-unemployed fathers program: determinants of participation and implications for welfare reform’, in P. Sommers (ed.) Welfare Reform in America, Boston, MA: Kluwer Nijhoff. Howes, C. and Markusen, A. (1981) ‘Poverty: a regional political economy perspective’, in A.Hawley and S.Mazie (eds) Non-metropolitan America in Transition, ch. 11, Chapel Hill, NC: University of North Carolina Press. Johnston, R.J. (1990) ‘Economic and social policy implementation and outputs: an exploration of two contrasting geographies’, in J.Kodras and J.P.Jones III (eds) Geographic Dimensions of United States Social Policy, pp. 37–58, London: Edward Arnold. Jones, J.P. III (1984a) ‘A spatially-varying parameter model of AFDC participation: empirical analysis using the expansion method’, Professional Geographer 36: 455–61.

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——(1984b) ‘Spatial parameter variation in models of AFDC participation: analyses using the expansion method’, Ph.D. dissertation, Ohio State University. ——(1987) ‘Work, welfare, and poverty among black female-headed families’, Economic Geography 63:20–34. Jones, J.P. III and Kodras, J. (1984) ‘AFDC participation dynamics and policies’, Modeling and Simulation 15:41–5. Kasper, H. (1968) ‘Welfare payments and work incentive: some determinants of the rate of general assistance payments’, Journal of Human Resources 3:86–110. Keeley, M., Robins, P., Spiegelman, R. and West, R. (1978a) ‘The labor-supply effects and costs of alternative negative income tax programs’, Journal of Human Resources 13:3–36. ——, ——, —— and ——(1978b) ‘The estimation of labor supply models using experimental data’, American Economic Review 68:873–87. Kodras, J.E. (1982) ‘The geographic perspective in social policy evaluation: a conceptual approach with application to the U.S. Food Stamp Program’, Ph.D. thesis, Ohio State University. ——(1984) ‘Regional variation in the determinants of food stamp program participation’, Environment and Planning C: Government and Policy 2:67–78. Kuttner, B. (1983) ‘The declining middle’, The Atlantic Monthly July: 60–72. Lampman, R. (1978) ‘Labor supply and social welfare benefits in the United States’, Institute for Research on Poverty Special Report 22, Madison, WI. Levitan, S. and Johnson, C. (1984) Beyond the Safety Net: Reviving the Promise of Opportunity in America, Cambridge, MA: Ballinger. Masters, S. and Garfinkel, 1. (1977) Estimating the Labor Supply Effects of Income Maintenance Alternatives, New York: Academic. Menefee, J., Edwards, B. and Schieber, S. (1981) ‘Analysis of non participation in the SS1 program’, Social Security Bulletin 44:3– 21. Morrill, R. and Wohlenberg, E. (1971) The Geography of Poverty in the United States, New York: McGraw-Hill. Murray, C. (1984) Losing Ground: American Social Policy 1950–1980, New York: Basic Books. Murray, C. and Greenstein, R. (1985) ‘The greed society: an exchange’, New Republic 192:21–3. Opton, E. (1971) Factors Associated with Employment among Welfare Mothers, Berkeley, CA: Wright Institute. Pigou, A. (1952) The Economics of Welfare, London: Macmillan. Piven, F.F. and Cloward, R. (1971) Regulating the Poor: The Functions of Public Welfare, New York: Vintage. —— and ——(1981) ‘Keeping labor lean and hungry’, The Nation 233:466–7. Platky, L. (1977) ‘Aid to families with dependent children: an overview’, Social Security Bulletin 40:17–22. Reagan, R. (1968) The Creative Society, New York: Devin-Adair. Rumberger, R. (1981) ‘The changing skill requirements of jobs in the US economy’, Industrial and Labor Relations Review 34:578–90. Rytina, N. (1982) ‘Tenure as a factor in the male-female earnings gap’, Monthly Labor Review 105:32–4.


Seninger, S. and Smeeding, T. (1981) ‘Poverty: a human resource— income maintenance perspective’, in A.Hawley and S.Mazie (eds) Nonmetropolitan America in Transition, ch. 10, Chapel Hill, NC: University of North Carolina Press. Soja, E.W. (1987) ‘The postmodernization of geography: a review’, Annals, Association of American Geographers 77 (2):289–94. Spall, H. and McGoughran, E. (1974) ‘AFDC in Michigan during the twentieth century’, Review of Social Economy 32:70–85. Stonecash, J. and Hayes, S. (1981) ‘The sources of public policy: welfare policy in the American states’, Policy Studies Journal 9: 681–98. Taaffe, E.J. and Casetti, E. (1990) ‘The model context problem and the expansion method’, Paper presented at the annual meeting of the Association of American Geographers, Toronto, Canada. Thrall, G.I. (1981) ‘Regional dynamics of local government welfare expenditures’, Urban Geography 2:255–68. US Bureau of the Census (1982) State and Metropolitan Area Data Book, Washington, DC: USGPO. ——(1983) Census of Population, 1980, Washington, DC: USGPO. ——(1984) Statistical Abstract of the United States, Washington, DC: USGPO. US Social Security Administration (1980) Annual Statistical Supplement to the Social Security Bulletin, Washington, DC: USGPO. ——(1981) Characteristics of State Plans for AFDC, Washington, DC: USGPO. ——(1982) 1979 Recipient Characteristics Study, Washington, DC: USGPO. Weil, G. (1978) The Welfare Debate of 1978, White Plains, NY: Institute for Socioeconomic Studies. Winegarden, C. (1973) ‘The welfare explosion: determinants of the size and recent growth of the AFDC population’, American Journal of Economics and Sociology 32: 245–56. Wohlenberg, E. (1976a) ‘An index of eligibility standards for welfare benefits’, Professional Geographer 28:381–4. ——(1976b) ‘Interstate variations in AFDC programs’, Economic Geography 52:254–66. ——(1976c) ‘Public assistance effectiveness by states’, Annals, Association of American Geographers 66:440–50.

APPENDIX 4.1: VARIABLE DEFINITIONS AND SOURCES 1 PR—AFDC participation rate:

Source: US Social Security Administration 1980 and US Bureau of the Census 1982: Table C–1002 2 WD—work disincentive:

Source: US Bureau of the Census 1983: Table 248

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3 FUNEMP—female unemployment rate:

Source: US Bureau of the Census 1982: Table C–855 4 NOJOB—proportion of all jobs not in retail trade, nondurable manufacturing, or nonprofessional services:

(RT, retail trade; NDM, nondurable manufacturing; NFS, nonprofessional services). Source: FSU Census Access System: Table 65 5 WAGED1FF—ratio of male to female earnings:

calculated for males and females older than 18 years, with four years of high school education, who are employed full time (35+hours per week) and full year (40+weeks per year). Source: US Bureau of the Census 1983: Table 237 6 PFWORK—the proportion of female-headed households whose head works full time (35+hours per week) and full year (50+weeks per year) who remain below the poverty level:

Source: US Bureau of the Census 1983: Table 246 7 SEVERITY—the median dollar amount by which poverty families fall below the poverty level. Source: US Bureau of the Census 1983: Table 251 8 FEMPOV—the proportion of female-headed households below the poverty level:

Source: US Bureau of the Census 1982: Table C–1007 9 NEEDS—state welfare administration’s determination of the monthly amount necessary to meet basic needs. Source: US Social Security Administration 1981: Table C 10 ADMIN—index of state administrative leniency in welfare provision. Source: Jones 1984b:74 11 SERVICE—ratio of state and local welfare employees to the poverty population:


Source: US Bureau of the Census 1982: Tables C–1184 and C–1012 12 TRANSFER—the proportion of persons receiving public assistance who are classified poor excluding public assistance and nonpoor including public assistance:

Source: US Bureau of the Census 1983: Table 249 APPENDIX 4.2: STATES RANKED BY WORK-DISINCENTIVE EFFECT, WITH FACTOR SCORES AND POLITICAL CULTURE CATEGORY

MN WI NH VT IA WY NB CA CT ND MA WA UT MI OR OK RI PA KS NY MD OH

Parameter value (Figure 4.1)

Factor scores

1 1.735 1.689 1.600 1.589 1.501 1.496 1.474 1.456 1.434 1.426 1.417 1.400 1.390 1.372 1.359 1.322 1.267 1.254 1.185 1.167 1.144 1.128

2 1.250 2.210 0.589 2.243 0.535 –0.262 –0.333 1.830 0.146 0.152 1.328 1.043 0.391 1.816 1.077 0.649 0.392 0.649 –0.481 0.847 –0.678 –0.236

3 –1.253 0.327 –1.866 0.219 –0.953 –0.753 –2.084 0.119 –1.319 –0.777 –0.357 0.228 –0.109 1.675 0.634 –0.037 –0.512 0.024 –1.066 0.356 –1.682 –0.039

Political culturea

0.392 0.365 0.009 –0.380 0.583 2.089 0.553 –0.567 0.422 0.986 –0.508 0.518 1.154 0.764 0.709 0.436 –0.001 0.118 0.333 –0.320 –0.280 0.786

M M MI M MI IM IM MI IM M IM MI M M M TI IM I MI IM I I

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Parameter value (Figure 4.1) 1 NJ 1.109 CO 1.107 WV 1.101 IL 1.092 AK 1.089 MO 1.086 MT 1.077 IN 1.013 HA 0.990 ID 0.968 ME 0.961 SD 0.949 AZ 0.931 VA 0.845 KY 0.752 TX 0.734 NV 0.654 NM LA DE FL AR TN NC AL GA SC MS

2 0.259 –0.096 –0.120 –1.239 0.853 –0.247 –0.516 –0.427 1.557 0.165 0.928 –0.941 –1.028 –0.770 –0.841 –1.748 –0.703 0.631 0.627 0.617 0.615 0.605 0.527 0.464 0.412 0.268 0.221 0.209

Political culturea

Factor scores

–0.986 –1.430 –0.522 –0.971 –0.480 –0.903 –0.611 –0.878 –1.121 –1.254 –1.085

3 –0.069 –0.438 0.768 –1.315 0.680 –0.456 0.149 0.454 0.348 0.950 0.687 –0.445 –0.766 –0.948 0.515 –0.696 –1.350

1.130 1.365 0.156 0.047 1.375 0.777 0.178 1.836 1.105 0.302 2.885

–0.166 –0.016 1.366 0.768 –0.394 0.089 1.171 1.016 –2.483 0.361 –1.232 0.469 0.142 –0.975 0.291 0.291 –2.618 0.552 1.523 –1.401 –0.804 –0.146 –0.576 –2.098 –0.051 –1.305 –2.269 0.336

I M TI I I IT MI I IT MI M MI TM T TI TI I TI T I TI T T TM T T T T

Note: aM, moralist; I, individualist; T, traditionalist. Two-letter code indicates a combination of cultures, with the first culture dominant. Source: Elazar 1972:117

5 A COMPARISON OF DRIFT ANALYSES AND THE EXPANSION METHOD: THE EVALUATION OF FEDERAL POLICIES ON THE SUPPLY OF PHYSICIANS Stuart A.Foster, Wilpen L.Gorr, and Francis C.Wimberly In most cases, the values of social and economic variables vary as a function of time and location. However, it is likely that the relationships among such variables are also context dependent—i.e. vary with time and/or location. For instance, the relationship between the market value of a residential property and the area of the lot is likely to be different for San Francisco than for Oklahoma City and, for a given location, is likely to be different now than it was ten years ago. Contextual variation in functional relationships can be investigated by drift analyses and by the expansion method. The purpose of this paper is to contrast the two approaches and to illustrate the comparative advantages of each in a case study involving the locational behavior of physicians. In the section below we compare the expansion method and drift analyses and discuss their respective merits. Next, we provide background information on the locational behavior of physicians from the 1960s through the 1980s and justify the specification of an initial model which serves as a basis for developing both polynomial and drift analysis models of physician location behavior. We then describe the data to be used in comparing the two approaches, present the results of the comparison, and offer an interpretation of our findings. CONTRASTING METHODOLOGIES The expansion method (Casetti 1972, 1973, 1982a, 1986) is designed to develop models with parameters that vary over contextual domains. To this effect, ‘a “terminal” model is generated from an “initial” one by making some of the parameters of the latter a function of some variables. The expansion method can be used for constructing models meeting requirements that an initial model does not satisfy or for removing inadequacies of an initial model, in such a fashion that whatever validity or usefulness the initial model possesses is not disregarded but rather built upon’ (Casetti 1972:82). For an example, assume the following initial model: (5.1)

72 DRIFT ANALYSIS AND THE EXPANSION METHOD

in which each observation i is also associated with a context vector . This vector may comprise substantive variables, spatial coordinates, time coordinates, or some combination of the three. In the expansion method, parameters are permitted to vary over the contextual domain. For example, expanding β 1 from (5.1) into Z, we obtain (5.2) which, when replaced into initial model (5.1), yields the terminal model (5.3) An example where the function is so expressed is provided by polynomial expansions. For example, the z variables may be powers of time from degree 1 to k. Functional forms other than polynomials have been used; for instance, logistic functions were used in Casetti (1972). Polynomial expansions allow direct testing of whether a parameter is varying over a context. Individual t tests on interaction terms (the approach adopted here) or F tests on the interaction terms of the full polynomial with a causal variable (except for the intercept of the polynomial) determine the significance of the parameter path. Varying parameters can also be investigated by ‘drift analysis’. This approach uses estimators such as DARP (Casetti 1982b) or adaptive filters which estimate the β s directly without the explicit use of pre-specified functional forms. Drift methods for estimation of time-varying parameters include the adaptive estimation procedure (Carbone and Longini 1977) and generalized adaptive filtering (Makridakis and Wheelwright 1977). For spatially varying parameters, two methods have been proposed: DARP (Casetti 1982b) and spatial adaptive filtering (Foster and Gorr 1986). All such methods provide smoothed parameter estimates for each observation point. They are able to accommodate unusual patterns of parameter variation such as step jumps and other discontinuous functions, splines, multi-modalities, etc. Drift analysis methods can be used in an exploratory fashion, such as when plots of estimated parameters are used to suggest functional forms (such plots are called ‘paths’ when the contextual domain is time and ‘maps’ in spatial problems). An example of this is found in Bretschneider and Gorr (1983). In addition, they can be employed as diagnostics for the adequacy of functional expansions through the construction of overlay plots of parameter paths or maps which compare parameter estimates produced by expansion analyses and by drift analyses. We include such a comparison in our empirical analyses. One class of drift analysis is represented by moving window regressions. A limited form of moving windows, moving averages, has long been used in time series analysis and forecasting. For example, classical decomposition and the Census Bureau’s X-ll methods are based on moving averages. For seasonal time series data, a moving average is calculated with the window length equal to the length of a season—for instance, four data points for quarterly data. The first four data points in the time series are averaged with the resulting average

S.A.FOSTER, W.L.GORR, AND F.C.WIMBERLY 73

associated with the center of the window. Then the fifth data point is added while the first data point is dropped as the window is moved forward one step. A new average is then calculated and the process continues in this way until the last four data points have been averaged. The resulting moving averages are capable of smoothing and tracking any pattern in the data. Moving window analyses employ regressions along with the moving average approach of adding and deleting observations to arrive at time paths or maps of parameters. The models evaluated may be multivariate and, unlike polynomial expansions, the regressions provide for significance tests along the path defined by the included observations. In the analyses reported here, each window includes data over a spatial region—the forty-eight contiguous states of the USA. The longer the time window, the smoother the resulting parameter paths. Of course, moving windows collapse to annual regressions (for annual data) as the length of the window is decreased. Tradeoffs related to the choice of the window length are discussed in the context of the results of our case study. SUPPLY OF PHYSICIANS AND FEDERAL POLICIES To explore the comparative advantages of the methods discussed above we have selected an application in which varying parameter models have considerable merit in evaluating policy questions: the supply and geographic distribution of physicians. Physicians are pivotal in the health care industry in the United States since they serve as the entry point and primary providers of health care services. Thus, their supply and geographic distribution are important determinants of the public’s access to health care. Growth in demand for health care services after the Second World War, which resulted largely from rapid population growth and increasing prosperity, led to the widespread perception of a physician shortage. Consequently, the federal government passed several pieces of legislation aimed at increasing physician supply and reducing spatial inequalities in access to physicians—i.e. increasing the relative density of physicians in rural areas. The result of the federal legislation was rapid growth in the number of physicians, commencing in the middle 1960s and continuing throughout the 1970s and early 1980s. By 1981, however, the pendulum had swung from a perception of shortage to one of oversupply of physicians. Subsequently, there has been a greatly reduced federal role in physician manpower planning. Besides the federal initiatives on increasing physician supplies, other important trends and events during this period contributed to major changes in medical practice. Technological innovations within the health care field led to increased specialization and diversity in the availability and provision of health care services. The amendments to the Social Security Act creating the Medicare and Medicaid programs for the elderly and poor restructured the sources and increased the level of demand for health care services. Furthermore, the


corporate movement in the health care industry has radically altered the nature and variety of practice alternatives for physicians. In this study we examine the aggregate locational behavior of physicians during the transitional period of the 1960s through the 1980s. Several federal programs were designed to deal directly with the spatial maldistribution of physicians. An example is the National Health Service Corps, which provided incentives for physicians to locate in physician-poor areas. Moreover, federal policy-makers had intended to increase the supply of physicians to the point of saturating physician-rich areas under the assumption that the resulting oversupply would cause a flow of physicians into the poorly supplied areas. Here we compare methods which can be used to evaluate these policies by seeking evidence of the intended spatial diffusion of physicians. A limited number of previous studies have sought evidence of this diffusion process. Schwartz et al. (1980) observed diffusion of board-certified specialist physicians into increasingly smaller communities over the period from 1960 through 1977. Fruen and Cantwell (1982) examined time trends in physician per capita ratios for different sizes of urban and rural areas. They noted evidence of diffusion into smaller communities, with the exception of the most rural areas where comparatively little improvement in the ratio was observed. This study extends this small body of research. Of the previous studies in this area, only Foster (1988) makes use of the expansion method. PHYSICIAN LOCATION FACTORS AND INITIAL MODEL SPECIFICATION The attributes of a given site contribute to three broad factors in physicians’ location decisions: the professional climate, social amenities, and market factors. Professional concerns generally fall into the categories of sufficient access to hospital and other support facilities and interaction and support of colleagues, including, for example, the availability of continuing education programs. Hence, the attractiveness of the professional environment is a positive function of the supply of physicians in an area. Attitudinal surveys of physicians and residents indicate the relative importance attributed to professional concerns in the choice of a practice location (e.g. Parker and Tuxill 1967; Champion and Olson 1971; Cooper et al. 1975; Steinwald and Steinwald 1975; Diseker and Chappell 1976; Parker and Sorensen 1978, 1979). Social amenities include the social and cultural aspects of a community, such as educational facilities, entertainment and recreational opportunities, and shopping facilities— attributes which are a function of community size. Indeed, the primary focus of research into the effects of social amenities on location choice has been on community-size preferences. Numerous studies from a wide variety of geographic contexts have shown that physicians exercise preferences for community size in choosing a practice location (e.g. Parker and Tuxill 1967; Cooper et al. 1975; Steinwald and Steinwald 1975; Diseker and Chappell 1976;


Parker and Sorensen 1978; Coombs et al. 1985), and these preferences are largely determined by a physician’s previous life experiences. Hence, physicians from large cities tend to prefer practice locations in large cities, and physicians from small communities are more likely to practice in small communities. Nevertheless, population is by far the greatest determinant of physician distribution, and physicians are quite responsive to variations in population growth. Another simple indicator for market characteristics is income potential. The role of income potential in location choice, however, does not appear to be important (e.g. Cooper et al. 1975; Diseker and Chappell 1976; Parker and Sorensen 1978; Foster 1988). Lave et al. (1975) argue that physicians’ incomes are so high that variations in income potential among places are not perceived as important. Given the factors just elucidated, it is evident that our initial model needs to contain measures of population growth and physician density within a geographic area. Furthermore, the federal programs mentioned above, leading from a physician shortage to an oversupply, indicate that physician density should have a varying impact on physicians’ locational decisions. An appropriate initial model is thus: (5.4) where t is time, GPHYSPOP is the percentage growth in the physician population, LDENSITY is the physician-to-population ratio (per 100,000 general population) at the start of the period, GPOP is the percentage growth in general population, and β is a classical disturbance term. The parameter estimate for LDENSITY at any given time is the result of the two opposing locational tendencies. To the extent that LDENSITY reflects the professional climate and social amenities, it should enter into the model with a positive sign, indicating an overall agglomerative trend. In contrast, as far as it is related to the economic competition for practice, its parameter should be negative, consistent with a general deglomerative trend. The estimate of this parameter at any given time thus reflects these opposing tendencies. The parameter estimate associated with GPOP provides an indication of the responsiveness of physician supply to changing market conditions. A parameter estimate of 1.0 indicates that population growth has a proportional effect on physician supply; an estimate exceeding 1.0 indicates a more than proportional response (so that population growth contributes to an increase in physician density); and an estimate of less than 1.0 indicates the opposite. SCOPE AND DATA The scope of our analysis is defined along three dimensions: space, time, and physician specialty. From a spatial perspective the analysis focuses on the distribution of physicians at the national level, with the forty-eight contiguous states


Table 5.1 Descriptive statistics: 1963–83 annual data for the contiguous forty-eight states Variable

N

Mean

Standard deviation

Minimum

Maximum

GPRIM GSPEC LPRIMR LSPECR GPOP

959 959 960 960 960

2.87 4.61 56.99 60.69 1.21

3.38 3.96 12.42 20.25 1.25

–7.65 –6.53 37.26 20.65 –3.25

19.26 18.81 108.03 138.35 8.67

comprising the set of spatial observational units. The temporal frame of the analysis involves annual data from 1963 through 1983. Separate analyses are conducted for primary care and specialty care physicians. Primary care physicians include all those whose primary specialization is in general and family practice, internal medicine, obstetrics and gynecology, or pediatrics. All other physicians are classified as specialists. The physician data are from the American Medical Association (AMA) master file, as made available in the AMA’s series of annual publications regarding the geographic and specialty distributions of physicians. Table 5.1 lists the set of variables used in the analyses and includes descriptive statistics. GPRIM and GSPEC represent the annual percentage growth in the supply of primary care and specialty care physicians, respectively, in a given state and time period. LPRIMR and LSPECR identify the number of physicians per 100,000 persons for primary and specialty care, respectively, calculated at the beginning of the year in which growth in physician supply is measured. THE MODELS This section first presents the functional expansions and drift analyses used to evaluate physician manpower policies and then compares them through overlay plots of parameter time paths. Table 5.2 Ordinary least squares estimates of the terminal model: quadratic expansions in time GPRIM (0.735) (0.158) (0.001) (0.511) (0.049) –0.000812t2LPRIMR+0.578GPOP+0. 052tGPOP (0.002) (0.019) (0.333) –0.00236t2GPOP (0.354) adj. R2=0.332 F=59.9, p=0.0001 n=959 GSPEC

=0.588–0.497t+0.0510t2–0.0216LPRIMR +0.0121tLPRIMR

=4.17+0.153t–0.00188t2–0. 00797LSPECR–0.00585tLSPECR


(0.002) (0.613) (0.896) (0.779) (0.267) +0.000258t2LSPECR+0.507GPOP+0. 109tGPOP (0.248) (0.160) (0.150) –0.00537t2GPOP (0.129) adj. R2=0.097 F=13.7, p=0.0001 n=959 Note: Standard errors are shown in parentheses below the estimates.

Polynomial expansions We investigated functional expansions of initial model (5.4) using polynomials in time interacted with all components of the model. The expansions were computed using GPRIM and GSPEC as the dependent variables for initial model (5.4). Polynomials up through the full fourth order were attempted. Table 5.2, providing ordinary least squares (OLS) estimates of a full quadratic expansion, is included here as an example; standard errors are listed below each co-efficient. The table illustrates a feature of polynomial-based functional expansions which is somewhat problematic. Although both models are highly significant overall (p < 0.0001), many of the individual coefficients are not significant or are marginally so. Indeed, in the specialty physician model none of the coefficients appears to be significant. This results from collinearity among the terms in the polynomial expansions, for instance t, t2, tLPRIMR, t2LPRIMR, etc. The explanatory efficacy is ‘spread’ over these redundant terms with the result that it is lowered for each of them. Table 5.3 Annual regressions for GPRIM, model (5.5): estimated coefficients and p values Year

R2

F

Intercept

LPRIMR

GPOP

1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975

0.45 0.17 0.14 0.07 0.27 0.06 0.32 0.13 0.17 0.32 0.05 –0.02

20.2 4 5.9 2 4.7 1 2.8 0 9.7 3 2.5 0 12.3 4 4.6 1 5.8 2 12.1 4 2.3 0 0.4 0

2.26 0 –0.74 0 –0.59 0 –0.15 0 0.91 0 –3.57 0 0.63 0 –3.34 0 –2.79 0 –0.35 0 2.45 0 4.24 0

–0.0341 0 0.0129 0 –0.0046 0 0.0100 0 –0.0169 0 0.0456 0 0.0127 0 0.0998 1 0.0639 0 –0.0009 0 –0.0050 0 –0.0256 0

0.78 4 0.74 2 0.63 2 0.40 0 1.47 4 0.69 0 1.65 4 0.80 1 0.79 2 1.45 4 0.38 0 0.17 0


Year

R2

F

Intercept

LPRIMR

GPOP

1976 –0.03 0.2 0 5.46 0 –0.0102 0 0.24 0 1977 0.32 11.9 4 12.67 4 –0.0126 2 0.96 1 1978 0.07 2.9 0 6.57 2 –0.0685 1 –0.11 0 1979 0.24 8.5 3 6.58 2 –0.0159 0 0.96 3 1980 0.40 16.6 4 7.16 2 –0.0510 0 1.46 4 1981 0.28 10.3 3 5.80 2 –0.0686 1 0.94 2 1982 0.02 1.4 0 4.38 2 0.0182 0 0.33 0 1983 0.32 11.8 4 6.95 4 –0.0640 2 0.74 1 Notes: (1) Shown is adjusted R2. (2) Statistical significance (shown after each coefficient estimate): 0, not significant; 1, 0. 05 level; 2, 0.01 level; 3, 0.001 level; 4, 0.0001 level.

Moving window regressions As a basis of comparison, Tables 5.3 and 5.4 contain annual estimates of the GPRIM and GSPEC models. Note that while they provide time-varying estimates, the results obtained are ‘noisy’, as can be readily discerned in the figures discussed below. They nevertheless serve as a basis for comparing estimates of other, more sophisticated methods, i.e. polynomial expansions and moving window regression parameter paths. Moving window regressions are analogous to the moving averages described above, except that multivariate models are estimated and each window includes a time series of cross-sections for the forty-eight contiguous states. The longer the time window is, the smoother are the resulting Table 5.4 Annual regression estimates for GSPEC, model (5.6): estimated coefficients and p values Year

R2

F

Intercept

LSPECR

GPOP

1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975

–0.01 0.17 0.00 0.13 0.12 0.16 0.22 0.29 0.50 0.04 0.53 0.03

0.7 0 5.9 2 0.9 0 4.5 1 4.1 1 5.6 2 7.5 2 10.7 3 24.8 4 2.1 0 27.3 4 1.8 0

3.49 1 4.59 3 3.56 3 5.28 3 3.86 2 5.87 3 4.17 3 5.57 3 2.84 1 2.02 0 2.60 1 1.46 0

0.0228 0 –0.0401 0 –0.0092 0 –0.0321 0 0.0020 0 0.0005 0 –0.0490 1 –0.0538 1 –0.0305 0 –0.0047 0 –0.0258 0 0.0162 0

0.14 0 1.23 2 0.31 0 0.93 2 0.96 1 1.25 2 1.65 3 1.02 2 1.67 3 0.58 1 1.36 3 1.52 1


R2

Year

F

Intercept

LSPECR

GPOP

1976 0.13 4.6 1 17.71 2 –0.0223 2 –0.40 0 1977 0.50 24.9 4 14.30 3 –0.0995 3 1.85 3 1978 0.01 1.2 0 2.76 1 0.0226 0 –0.02 0 1979 0.42 17.9 4 3.20 1 –0.0156 0 1.12 3 1980 0.33 12.4 4 9.68 3 –0.0420 1 0.91 3 1981 –0.03 0.4 0 4.44 2 –0.0092 0 0.21 0 1982 0.18 6.0 2 4.05 3 0.0027 0 0.45 2 1983 0.13 4.5 1 2.99 1 0.0038 0 0.90 2 Notes: (1) Shown is adjusted R2. (2) Statistical significance (shown after each coefficient estimate): 0, not significant; 1, 0. 05 level; 2, 0.01 level; 3, 0.001 level; 4, 0.0001 level.

parameter paths. Of course, moving windows collapse to annual regressions (as in Tables 5.3 and 5.4) as the length of the window is decreased. The annual regressions, besides being too noisy, also lack power in individual parameter t tests because of fewer degrees of freedom. While longer windows are therefore desirable to increase smoothing and power, at the other extreme a single regression for all cross-sections pooled together no longer permits any time parameter variation. Here we chose to use three-year moving averages to balance smoothing and responsiveness. These directly estimate the more general timevarying parameter specification of model (5.4), e.g. for primary care and specialty physicians (5.5)

Table 5.5 Three-year moving-average window regressions for GPRIM, model (5.5): estimated coefficients and p values (n=144) Year

R2

F

Intercep Time t

Time2

LPRIM R

GPOP

RHO

1965

0.28

14.8 4

10

00

–0.2 0

0.65 4

0.161 0

1966

0.20

9.9 4

72

–7 3

1.0 3

0.59 4

0.118 0

1967

0.26

13.4 4

–6 0

30

–0.0 0

0.80 4

0.108 0

1968

0.25

13.0 4

–30 3

13 3

–1.4 4

0.79 4

0.116 0

1969

0.47

32.3 4

103 4

–36 4

3.1 4

1.19 4

1970

0.45

29.7 4

–123 4

33 4

–2.3 4

–0. 0047 0 0.0040 0 –0. 0069 0 –0. 0023 0 0.0018 0 0.0597 1

–0.124 0 –0.090 0

0.81 4


Year

R2

F

Intercep Time t

Time2

LPRIM R

GPOP

RHO

1971

0.22

10.8 4

–26 0

70

–0.5 0

0.74 4

0.162 0

1972

0.21

10.5 4

61 0

–13 0

0.7 0

0.95 4

1973

0.16

8.0 4

55 0

–12 0

0.6 0

1974

0.15

7.4 4

–55 0

10 0

–0.4 0

1975

0.15

7.5 4

93 0

–16 0

0.7 0

–0.001 0 –0.133 0 –0.007 0 0.021 0

1976

0.27

13.9 4

–14 0

10

0.0 0

1977

0.37

22.1 4

–591 4

87 4

–3.2 4

1978

0.46

31.6 4

–138 4

4.6 4

1979

0.47

39.8 4

1, 038 4 –766 4

95 4

–2.9 4

1980

0.52

30.3 4

–309 1

39 1

–1.2 2

1981

0.45

30.3 4

0.0752 3 0.0477 1 0.0166 0 –0. 0050 0 –0. 0095 0 –0. 0533 1 –0. 0624 2 –0. 0658 2 –0. 0372 1 –0. 0447 1 –0. 0322 1 –0. 0381 2

0.78 4 0.74 4 0.36 1 0.57 1 0.45 1 0.65 3 0.88 4 1.05 4

–0.117 0 –0.124 0 –0.094 0 –0.084 0 –0.069 0 –0.057 0 0.037 0

1, 200 –133 4 3.7 4 0.87 4 4 1982 0.48 39.9 4 –1, 203 127 4 –3.3 4 0.43 2 4 Notes: (1) Shown is adjusted R2. (2) Statistical significance (shown after each coefficient estimate): 0, not significant; 1, 0. 05 level; 2, 0.01 level; 3, 0.001 level; 4, 0.0001 level. (3) RHO is an estimate of serial autocorrelation. Table 5.6 Three-year moving-average window regressions for GSPEC, model (5.6): estimated coefficients and p values (n=144) Year

R2

F

Intercep Time t

Time2

LSPEC R

GPOP

RHO

1965

0.07

3.6 2

52

–1 0

0.08 0

0.44 2

1966

0.15

7.3 4

10 2

–4 0

0.75 1

1967

0.16

7.2 4

–4 0

40

–0.37 0

0.73 4

–0.044 0 –0.053 0 0.146 0

1968

0.31

16.8 4

23 1

–9 1

0.99 1

1.01 4

0.055 0

1969

0.37

22.4 4

–99 4

36 4

–3.07 4

–0. 0080 0 –0. 0291 1 –0. 0130 0 –0. 0092 0 –0. 0170 0

1.24 4

0.053 0

0.87 4


Year

R2

F

Intercep Time t

Time2

LSPEC R

GPOP

RHO

1970

0.41

25.6 4

116 4

–30 4

2.03 4

1.26 4

0.083 0

1971

0.37

21.7 4

–8 0

30

–0.20 0

1.41 4

0.092 0

1972

0.31

16.9 4

–13 0

40

–0.28 0

1.09 4

1973

0.36

20.8 4

67 0

–13 0

0.62 0

1974

0.14

6.7 4

69 0

–13 0

0.64 0

1975

0.08

4.0 2

–190 0

33 0

–1.35 0

1976

0.26

13.6 4

666 3

–104 3

4.14 3

1977

0.68

75.1 4

206 4

–7.32 4

1978

0.62

55.6 4

–1, 439 4 685 4

–87 4

2.80 4

1979

0.49

35.0 4

595 4

–76 4

2.43 4

1980

0.51

38.2 4

135 4

–3.95 4

1981

0.46

31.7 4

–1, 139 4 765 4

–83 4

2.27 4

1982

0.08

4.0 2

–239 1

26 1

–0.67 1

–0. 0328 2 –0. 0432 3 –0. 0303 1 –0. 0213 0 –0. 0078 0 –0. 0791 1 –0. 1027 2 –0. 0498 3 –0. 0314 1 –0. 0125 0 –0. 0218 1 –0. 0149 0 –0. 0005 0

–0.184 0 –0.246 2 –0.204 0 –1.245 4 –0.418 4 –0.128 0 –0.197 0 –0.299 2 –0.218 1 –0.018 0 –0.072 0

1.21 4 1.11 4 0.94 1 1.08 1 0.79 4 0.95 4 0.71 4 0.84 4 0.59 4 0.46 2

Notes: (1) Shown is adjusted R2. (2) Statistical significance (shown after each coefficient estimate): 0, not significant; 1, 0. 05 level; 2, 0.01 level; 3, 0.001 level; 4, 0.0001 level. (3) RHO is an estimate of serial autocorrelation.

(5.6) Table 5.5 provides time-specific window regression estimates and statistical significance levels for model (5.5). Table 5.6 supplies the same results for model (5.6). Both tables indicate that population growth, GPOP, is significant in nearly every window, while the lagged physician density levels frequently are. Serial correlation in moving window regressions Time series model estimates often have serial correlation in their residuals. To evaluate serial correlation for the pooled cross-sectional time series model of this


Figure 5.1 Parameter paths for LPRIMR in the GPRIM model

paper (i.e. time series for each state with resulting data pooled), we estimated the autoregressive AR(1) model (5.7) from the residuals of each window regression. These analyses reveal that thirteen of the thirty-six regressions had significant negative serial correlation (modal p value 0.0001), with seven in the primary care regressions and six in the specialty regressions. While the resulting parameter estimates for models (5.5) and (5.6) are unbiased for windows with serial correlation, they are inefficient. Thus we decided to remove the serial correlation. We applied the Cochrane-Orcutt and Durbin procedures without success; there were practically no changes in the results for models (5.5) and (5.6) after transformation. Closer analysis of the negative correlation revealed it to be an artifact of the three-year window regressions and nationwide deviations in time trends. All cases of negative serial correlation involved windows in which the second year deviated relatively widely from the ‘time trend’ defined by the first and third years, either higher or lower than the trend. Thus constant parameters estimated by regression analysis for such a window generally had alternating signs in residuals over time. To provide a naive accounting for missing variables and model structure, we applied a functional expansion, but In the small’, with polynomials in time for each window. This expansion required either a local maximum or minimum possible at year two of a window, so we employed a second-order polynomial in time to expand the intercept of each model and window. This works well for primary care physicians, as seen in Table 5.5, as all seven cases of serial correlation are eliminated. For specialty physicians, as seen in Table 5.6, one case of serial correlation is eliminated, three are reduced, and two cases stubbornly remain. (Prior to the expansions, four cases of serial correlation


Figure 5.2 Parameter paths for GPOP in the GPRIM model

Figure 5.3 Parameter paths for LSPECR in the GSPEC model

were significant at the 0.0001 level, one was significant at the 0.0007 level, and one was significant at the 0.007 level for specialty physicians.) Comparison of expansion and drift analysis parameter paths Figures 5.1–5.4 are overlays of parameter paths from the reference annual regressions, a fourth-order polynomial expansion, and the three-year moving window regressions. Some of the polynomial estimates of parameter paths did


Figure 5.4 Parameter paths for GPOP in the GSPEC model

not fit the annual regression estimates of parameters very well. For example, Figure 5.1 contains the estimates of LPRIMR’s parameter from model (5.5). The polynomial, while qualitatively correct in shape, is unable to attain the clear maximum in 1971. The window regression estimates track the annual regression estimates more closely than the polynomial expansion while the noise is considerably reduced relative to the annual regressions. This is a good example of the benefits of a drift analysis estimator. Figure 5.3 depicts an interesting result in that the drift and polynomial expansion models both differ markedly from the annual regressions but agree with each other, more or less, in the vicinity of 1975. This is because they both represent smoothed data in which the high value in 1975 is combined with the low value in 1977. INTERPRETATION OF RESULTS Both the drift analyses and the polynomial expansions combine to yield insights into the locational dynamics of physicians. Let us first examine the estimated parameters and parameter time paths in model (5.5) for primary care physicians. Table 5.3 provides time-specific parameter estimates and statistical significance levels for the intercept as well as for the coefficients of LPRIMR and GPOP. Time paths for the latter two parameters are found in Figures 5.1 and 5.2 respectively. Focusing on the path of LPRIMR as determined by the window regressions, we see that the parameter is initially stable around zero. In 1970 it becomes strongly positive with a peak in 1971 and it remains positive through 1973, after which it is negative.


On the basis of federal health manpower policy, we did not expect the positive maximum in the early 1970s. However, this period does coincide with two potentially important factors. The Medicare and Medicaid programs began in 1966 and federal expenditures for these programs increased rapidly in the years following, resulting in a dramatic increase in demand for physician services. Also, the relative shortage of primary care physicians peaked at about this time. These two factors acted together to increase the practice opportunities for physicians everywhere. One might thus expect physicians to concentrate in those areas attractive to physicians, i.e. those that already had a high density of physicians. After 1971, the physician density parameter estimates follow a decreasing trend, suggesting an increase in the relative importance of deglomerative market forces. This trend persists through 1976, with the sign becoming negative in 1974, which is consistent with expectations associated with an approaching saturation of physicians in physician-rich areas. The negative sign suggests a tendency for the proportional distribution of physicians to become more equitable. During the remainder of the time frame, the estimates are stable at a negative level, indicative of a new equilibrium behavior of primary care physicians in response to LPRIMR. Turning to model (5.6), which describes the behavior of specialty care physicians, and in particular to the time path of the parameter of LSPECR as seen in Figure 5.3, we see that the significant result is the stability of the estimates with respect to time. There is no evidence of a systematic shift in the locational behavior of specialists relative to physician density and, consequently, no evidence that tighter markets for specialists are affecting locational behavior. On the other hand, the systematic variation in the parameter for population growth, GPOP, is quite clear in Figure 5.4 (and Table 5.6). While the parameter is initially quite low, the responsiveness of specialists to GPOP increases dramatically. The GPOP parameter exceeds 1.0 in 1968 and continues to climb through 1971, after which it begins a decreasing trend until it drops below 1.0 in 1977. The GPOP parameter time path in model (5.6) is consistent with a lagged response of specialists to practice opportunities materializing before specialist production permitted them to be filled. Hence, as supply began to grow, specialists responded to current shifts in population as well as to population growth from, say, a decade earlier. As the supply continued to increase the response to population returned toward an equilibrium value. CONCLUSION In this paper we have compared drift and expansion models as policy evaluation tools. We began with an initial model which incorporated factors which, we argued, influence physicians’ choice of practice location by relating percentage growth in primary and specialty care physicians to physician density and growth


in the general population. We then explored parameter drift in these models using a variety of approaches. Each approach has comparative advantages which make it useful for evaluating such policies. In particular, the polynomial-based expansion method supports extensive hypothesis testing and is applicable when interpolation or extrapolation is indicated. On the other hand, moving window regressions provide an effective and simple means for detecting parameter drift, especially for pooled cross-sectional time series data such as the state level physician data used in this paper. In terms of substantive results, we found clear impacts of federal policies on the supply and demand relationship for physicians. The results presented for primary care physicians are initially consistent with locational behavior dominated by professional and social amenity factors, but as the supply of primary physicians increases, market forces become dominant. Meanwhile, the results for specialty care physicians are consistent with the domination of economic concerns. ACKNOWLEDGMENTS This research was funded by NSF Grant SES–8700910 and draws on work by Foster (1988). REFERENCES Bretschneider, S.I. and Gorr, W.L. (1983) ‘Ad hoc model building using time-varying parameter models’, Decision Sciences 14:221– 39. Carbone, R. and Longini, R. (1977) ‘A feedback approach for automated real estate assessment’, Management Science 24:241–8. Casetti, E. (1972) ‘Generating models by the expansion method: applications to geographic research’ , Geographic Analysis 4:81– 91. ——(1973) ‘Testing for spatial-temporal trends: an application of urban population density trends using the expansion method’, Canadian Geographer 17:127–36. ——(1982a) ‘Mathematical modeling and the expansion method’, in R.B.Mandal (ed.) Statistics for Geographers and Social Scientists, pp. 81–95, New Delhi: Concept Publishing. ——(1982b) ‘Drift analysis of regression parameters: an application to the investigation of fertility development relations’, Modeling and Simulation 13:961–6. ——(1986) ‘The dual expansion method: an application for evaluating the effects of population growth on development’, IEEE Transactions on Systems, Man, and Cybernetics SMC–16:29– 39. Champion, D.J. and Olson, D.B. (1971) ‘Physician behavior in southern Appalachia: some recruitment factors’, Journal of Health and Social Behavior 12:245–52. Coombs, D.W., Miller, H.L. and Roberts, R.W. (1985) ‘Practice location preferences of Alabama medical students’, Journal of Medical Education 60:697–706.


Cooper, J.K., Heald, K., Samuels, M. and Coleman, S. (1975) ‘Rural or urban practice: factors influencing the location decision of primary care physicians’, Inquiry 12: 18–25. Diseker, R.A. and Chappell, J.A. (1976) ‘Relative importance of variables in determination of practice location: a pilot study’, Social Science and Medicine 10: 559–63. Foster, S.A. (1988) ‘Analyses of the changing geographic distribution of physicians in the United States from 1950 through 1985’, Ph.D. dissertation, Ohio State University. Foster, S.A. and Gorr, W.L. (1986) ‘An adaptive filter for estimating spatially-varying parameters: application to modeling police hours spent in response to calls for service’, Management Science 32:878–89. Fruen, M.A. and Cantwell, J.R. (1982) ‘Geographic distribution of physicians: past trends and future influences’, Inquiry 19:44–50. Lave, J.R., Lave, L.B. and Leinhardt, S. (1975) ‘Medical manpower models: need, demand, and supply’, Inquiry 12:97–125. Makridakis, S. and Wheelwright, S.C. (1977) ‘Adaptive filtering: an integrated autoregressive/moving average filter for time series forecasting’, Operations Research Quarterly 28:425–37. Parker, R.C. and Sorensen, A.A. (1978) ‘The tide of rural physicians: the ebb and flow, or why physicians move out of and into small communities’, Medical Care 16:152–66. —— and ——(1979) ‘Physician attitudes toward rural practice: answers and questions that were not asked’, Forum on Medicine 2:411–16. Parker, R.C. and Tuxill, T.G. (1967) ‘The attitudes of physicians toward small-community practice’, Journal of Medical Education 42:327–44. Schwartz, W.B., Newhouse, J.P., Bennett, B.W. and Williams, A.P. (1980) ‘The changing geographic distribution of board certified physicians’, New England Journal of Medicine 303:1032–8. Steinwald, B. and Steinwald, C. (1975) ‘The effect of preceptorship and rural training programs on physicians’ practice location decisions’, Medical Care 13:219.

6 PERSONAL CHARACTERISTICS IN MODELS OF MIGRATION DECISIONS: AN ANALYSIS OF DESTINATION CHOICE IN ECUADOR Mark Ellis and John Odland The destination choice component of migration behavior has generally been analyzed by means of aggregate models, most often gravity models in which the volumes of place-to-place migration flows depend on the population sizes of origins and destinations and on the distances separating them. The human capital perspective on migration (Sjaastad 1962; Molho 1986) indicates, however, that heterogeneity in the migrant population may be associated with heterogeneity in destination choices. Human capital models treat migration behavior as a form of investment undertaken by individuals in order to improve their long-term returns to participation in localized labor markets, or their returns from accessibility to other localized opportunities. Conditions for this kind of investment behavior are likely to vary across members of a heterogeneous population, because of differences in liquidity, because place-to-place variations in returns to labor market participation may differ across individuals with different qualifications, and because interregional differences in returns are calculated over different time horizons for persons of different ages. Most analyses within the human capital framework have concentrated on the decision to leave an origin (Nakosteen and Zimmer 1982; Schaeffer 1985) rather than the choice among an array of destinations. Analyses of the effects of individual characteristics on migration decisions in developing country contexts have also emphasized the effects on outmigration decisions rather than the choice of destination (Brown and Goetz 1987). We analyze the effects of some personal characteristics on destination choice in this paper, by fitting a series of origin-specific destination choice models using disaggregate data for migration flows among the cantones of Ecuador during the 1971–4 period. Our model of destination choice is constructed by applying the expansion method (Casetti 1972, 1982) to an initial model of destination choice which has the same functional form as the origin-specific gravity model. This model, however, is derived on the basis of random utility theory and is one component of a general model of migration decisions which includes the decision to move as well as the choice of destination (Moss 1979; Odland and Ellis 1987). The functional equivalence of the destination choice component of the model with the gravity model has been demonstrated by Anas (1983).

ANALYSIS OF DESTINATION CHOICE IN ECUADOR 89

We expand this model in two stages in order to examine individual differentiation in the destination choices of Ecuadorian migrants. A distinction between the attractions of urban and rural destinations has been central in models of migration in developing countries including the Harris-Todaro model (Harris and Todaro 1970; Todaro 1976) and the first stage of the expansion incorporates the distinction between rural and urban destinations. This expansion leads to particular problems in the case of Ecuador because urbanization and population size are strongly collinear over the upper part of the range of regional population sizes, thus making it impossible to estimate separate parameters for the effects of urbanization and population size. Complications of this kind are not infrequent in analyses of data from restricted geographic contexts and we adapt the expansion method in a way which makes it possible to estimate distinct effects for the population sizes of urban and rural destinations for the lower part of the range of population sizes. The upper part of the range, which only contains destinations that are urbanized, is represented by a single categorical variable. Finally, the coefficients of this model are expanded in order to examine the effects of age and gender on destination choice. A DISAGGREGATE MODEL OF MIGRATION BEHAVIOR A general model of disaggregate migration behavior can be established within the random utility formalism by specifying the utilities for alternative destination regions as random variables. Residents of an origin region assign utilities to the members of a choice set Ri, which includes the set of possible destination regions as well as the origin. This assignment of utilities may depend on the objective characteristics of the regions, such as their distance from the origin, or characteristics of the decision-makers, such as their ages. The assignment of utilities also includes stochastic components which leave some uncertainty about the utility assigned to each region. The utility of destination j, for a resident of origin i, may be written as (6.1) where the vectors zj and zij contain observed characteristics of the regions and the decision-makers, and the utility uij of region j depends on functions of those characteristics, v(zj) and v(zij), and also on the corresponding error terms ej and eij. The term v(zj) is a component of the utility of destination j which is independent of the origin of the decision-maker (and ej is a corresponding error term), while v(zij) is an origin-specific component of the utility of the destination. The arguments of this component, zij, are likely to include the distance between the origin and destination j. The utility of the origin, as one of the possible locational choices, reduces to

90 M.ELLIS AND J.ODLAND

Residents of origin i presumably make a locational decision by selecting the region in Ri where utility is maximized, but that choice is uncertain because the error terms make the uij a set of random variables. Consequently the choice of destinations is analyzed in terms of the probability, or the odds, of choosing alternative j from the choice set Ri. This is merely the probability that the value of the random variable uij exceeds the values of all other random variables corresponding to the utilities of regions in Ri. That probability depends on values in zi and zij as well as an associated set of parameters which measures the importance of those variables in the assignment of utilities; but it also depends on the distributions of the error terms ei and eij. Alternative functional forms for models of the probabilities of selecting alternatives can be derived from alternative assumptions about the distributions of these error terms (McFadden 1981). It is useful to notice that the probability of migrating from origin i and selecting destination j can be written as the product of two probabilities, for leaving the origin and for choosing destination j: where the probability p(m|i) of outmigration may depend on the set of utilities for the origin and all destinations, and the probabilities p(j|m, i) for destination choice given outmigration from origin i depend on the utilities of the set of destinations (excluding the origin). A tractable functional form, the nested logit model, can be derived by assuming that ej and eij are independent; that the eij are independently and identically Gumbel-distributed; that the variance of the ej is zero; and that the maximum of ui and uij is also Gumbel-distributed (Ben-Akiva and Lerman 1985:286–7). These assumptions lead to (6.2) for the probability of outmigration, where the summations are over destinations and ø is a parameter whose range of values is 0′ ø

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