New Economic Windows Series Editor MASSIMO SALZANO
Series Editorial Board Series Editorial Board
Jaime Gil Aluja Departament d’Economia i Organització d’Empreses, Universitat de Barcelona, Spain Jaime Gil Aluja Departament d’Economia i Organització d’Empreses, Universitat de Barcelona, Spain Fortunato Arecchi Dipartimento di Fisica, Università di Firenze and INOA, Italy Fortunato Arecchi Dipartimento di Fisica, Università di Firenze and INOA, Italy David Colander Department of Economics, Middlebury College, Middlebury, VT, USA David Colander Richard H. of Day Department Economics, Middlebury College, Middlebury, VT, USA Department of Economics, University of Southern California, Los Angeles, USA Richard H. Day Mauro Gallegati Department of Economics, University of Southern California, Los Angeles, USA Dipartimento di Economia, Università di Ancona, Italy Mauro Gallegati Steve Keen di Economia, Università di Ancona, Italy Dipartimento School of Economics and Finance, University of Western Sydney, Australia Steve Keen Alan SchoolKirman of Economics and Finance, University of Western Sydney, Australia GREQAM/EHESS, Université d’Aix-Marseille III, France Giulia Iori Marji Linesof Mathematics, King’s College, London, UK Department Dipartimento di Science Statistiche, Università di Udine, Italy
Alan Kirman Thomas Lux GREQAM/EHESS, Université d’Aix-Marseille III, France Department of Economics, University of Kiel, Germany
Marji Lines Alfredo Medio Dipartimento di Science Statistiche, Università di Udine, Italy Dipartimento di Scienze Statistiche, Università di Udine, Italy
Alfredo Medio Paul Ormerod Dipartimento di Scienze Statistiche, Università di Udine, Italy
Directors of Environment Business-Volterra Consulting, London, UK
Paul Ormerod Peter Richmond Directors of Environment Business-Volterra Consulting, London, UK School of Physics, Trinity College, Dublin 2, Ireland
J. Barkley Rosser J.Department Barkley Rosser of Economics, James Madison University, Harrisonburg, VA, USA Department of Economics, James Madison University, Harrisonburg, VA, USA
Sorin Solomon Sorin Solomon Racah Institute of Physics, The Hebrew University of Jerusalem, Israel Racah Institute of Physics, The Hebrew University of Jerusalem, Israel
Kumaraswamy Pietro Terna (Vela) Velupillai Department of Economics, National University of Ireland, Ireland
Dipartimento di Scienze Economiche e Finanziarie, Università di Torino, Italy
Nicolas Vriend (Vela) Velupillai Kumaraswamy
Department of Economics, Queen Mary University of London, UK Department of Economics, National University of Ireland, Ireland
Lotfi Zadeh Nicolas Vriend
Computer Science Division, University of California Berkeley, USA Department of Economics, Queen Mary University of London, UK
Lotfi Zadeh
EditorialScience Assistants Computer Division, University of California Berkeley, USA Maria Rosaria Marisa FagginiAlfano Marisa Faggini Dipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy Editorial Assistants Dipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy
Marisa Faggini
Dipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy
Massimo Salzano • David Colander
Complexity Hints for Economic Policy
IV
A. Achiron et al.
MASSIMO SALZANO Dipartimento di Scienze Economiche e Statistiche Università degli Studi di Salerno, Italy David Colander Middlebury College, Middlebury, VT, USA
The publication of this book has been made possible thanks to the financial support of the MIUR-FIRB RBAU01 B49F
Library of Congress Control Number: 2006930687 ISBN-978-88-470-0533-4 Springer Milan Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole of part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in databanks. Duplication of this pubblication or parts thereof is only permitted under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Italian Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Italia 2007 Printed in Italy Cover design: Simona Colombo, Milano Typeset by the authors using a Springer Macro package Printing and binding: Grafiche Porpora, Segrate (MI) Printed on acid-free paper
Preface
To do science is to find patterns, and scientists are always looking for patterns that they can use to structure their thinking about the world around them. Patterns are found in data, which is why science is inevitably a quantitative study. But there is a difficulty in finding stable patterns in the data since many patterns are temporary phenomena that have occurred randomly, and highly sophisticated empirical methods are necessary to distinguish stable patterns from temporary or random patterns. When a scientist thinks he has found a stable pattern, he will generally try to capture that pattern in a model or theory. A theory is essentially a pattern, and thus theory is a central part of science. It would be nice to have a single pattern - a unified theory - that could serve as a map relating our understanding with the physical world around us. But the physical world has proven far too complicated for a single map, and instead we have had to develop smaller sub maps that relate to small areas of the physical world around us. This multiple theory approach presents the problem of deciding not only what the appropriate map for the particular issue is, but also of handling the map overlays where different maps relate to overlapping areas of reality. It is not only science that is focused on finding patterns; so too are most individuals. In fact, as pointed out by Andy Clark (1993), human brains are ‘associative engines’ that can be thought of as fast pattern completers - the human brain has evolved to see aspects of the physical world and to create a pattern that places that aspect in context and allows individuals to draw broader implications for very limited data1. Brian W. Arthur gives the following example ‘If I see a tail going around a corner, and it’s a black swishy tail, I say, “There’s a cat”! There are patterns in music,
—————— 1
This discussion is based on observations by Brian W. Arthur (2000).
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art, religion, business; indeed humans find patterns in just about everything that they do2. Determining when a pattern fits, when there are multiple patterns mapping to the same physical phenomena, and which pattern is the appropriate pattern, is a difficult task that is the subject of much debate. Science differentiates itself from other areas of inquiry by setting rules about when a pattern can be assumed to fit, and when not, and what the structure of the map can be. It essentially is a limit on the fast pattern completion nature of humans. For example, that swishy tail could be a small boy with who is playing a trick with a tail on one end of a stick. To prevent too-fast pattern completion, and hence mistakes, standard science requires the map to be a formal model that could be specified in a set of equations, determined independently of the data sequence for which it is a pattern, and that the patterns match the physical reality to a certain degree of precision. That approach places standard logic at the center of science. Arthur tells the story of Bertrand Russell’s to make this point. A schoolboy, a parson, and a mathematician are crossing from England into Scotland in a train. The schoolboy looks out and sees a black sheep and says, ‘Oh! Look! Sheep in Scotland are black!’ The parson, who is learned, but who represents a low level of science, says, ‘No. Strictly speaking, all we can say is that there is one sheep in Scotland that is black’. The mathematician, who might well represent a skilled theoretical scientist, says, ‘No, that is still not correct. All we can really say is that we know that in Scotland there exists at least one sheep, at least one side of which is black’. The point is that science, and the formal models that underlie it, works as a brake on our natural proclivity to complete patterns. In many areas the standard science approach has served us well, and has added enormous insight. By insisting on precision, scientists have built an understanding that fits not just the hastily observed phenomena, but the carefully observed phenomena, thereby developing much closer patterns. Once individuals learn those patterns, the patterns become obvious to them - yes, that table is actually nothing but a set of atoms - but without science, what is obvious to us now would never have become obvious. In other areas, though, we have been unable to find a precise set of equations or models that match the data representing the physical world, leaving large areas of physical reality outside the realm of science. For many scientists, —————— 2
Human’s ‘fast pattern completer’ brains have been shown today to be less rational than previously supposed by economists. The new “neuroeconomics”, in fact, has demonstrated that many choices are often made more from an emotional than a rational approach.
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a group of areas that have proven impervious to existing mappings includes most of the social sciences; these areas are simply too hard for traditional science. The difficulty of finding a precise map has not stopped social scientists from developing precise formal models and attempting to fit those precise models to the data, and much of the methodological debate in the social sciences concerns what implications we can draw from the vague imprecise mappings that one can get with existing analytic theories and data series. Critics of economics have called it almost useless - the celestial mechanics of a nonexistent universe. Until recently, the complaints of critics have been ignored, not because standard economists did not recognize the problems, but because the way we were doing economics was the best we could do. But science, like all fields, is subject to technological change, and recently there have been significant changes in analytic and computational technology that are allowing new theories to develop. Similar advances are occurring in empirical measurement, providing scientists with much more data, and hence many more areas to place finer patterns on, and in the analytics of empirical measurement, which allows the patterns developed by theories to be brought to the data better3. Such technological change has been occurring in the economics profession over the last decade, and that technological change is modifying the way economics is done. This book captures some of that change. The new work is sometimes described as complexity theory, and in many ways that is a helpful and descriptive term that we have both used (Colander 2000a, b; Salzano and Kirman 2004). But it is also a term that is often misused by the popular press and conveys to people that the complexity approach is a whole new way of doing economics, and that it is a replacement for existing economics. It is neither of those things. The complexity approach is simply the integration of some new analytic and computational techniques into economists’ bag of tools. We see the new work as providing some alternative pattern generators, which can supplement existing approaches by providing an alternative way of finding patterns than can be obtained by the traditional scientific approach. The problem with the use of the complexity moniker comes about because, as discussed above, individuals, by nature, are fast pattern completers. This has led some scientific reporters, and some scientists in their non—————— 3
As usual the reality is more complicated than can be presented in a brief introduction. The problem is that measurement does not stand alone, but is based on theory. Discontent about traditional theories can lead to a search for new ways of measurement, and improvements in the quality of data.
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scientific hats, to speculate about possible patterns that can follow from the new models and techniques. As this speculation develops, the terms get away from the scientists, and are no longer seen as simply descriptions of certain properties of specific mathematical models, but instead as grand new visions of how science is to be done, and of our understanding of reality. Such a grandiose vision makes it seem that complexity is an alternative to standard science, when it is actually simply a continuation of science as usual4. The popular press has picked up on a number of ideas associated with these models - ‘emergent structure’, ‘edge of order’, ‘chaos’, ‘hierarchy, ‘self organized criticality’, ‘butterfly effect’, ‘path dependency’, ‘histeresis’, etc. - and, using its fast-pattern completer skills, has conveyed many of these ideas to the general population with a sense that they offer a whole new way of understanding reality, and a replacement for standard science. Scientists have shied away from such characterizations and have emphasized that while each of these terms has meaning, that meaning is in the explicit content in the mathematical model from which they derive, not in some general idea that is everywhere appropriate. The terms reflect a pattern that economic scientists are beginning to develop into a theory that may prove useful in understanding the economy and in developing policies. But the work is still in the beginning stages, and it is far too early to declare it a science and a whole new way of looking at something. Scientists are slow, precise, pattern completers and they recognize that the new work has a long way be go before it will describe a meaningful pattern, and an even longer way to go before it can be determined whether those patterns are useful5. Thus, even if the complexity approach to economics is successful, it will be a complement to, not a substitute for, existing approaches in economics. That said, there are some hopeful signs and research in “complexity” is some of the most exciting research going on in economics. The most hopeful work is occurring in analysis of the financial sector, where —————— 4
This is not dissimilar from what has already happened for the “biological evolution” work, which was a more general, but not fundamentally different, approach from the mechanical physical biological model. 5 In agent-based modeling (ABM), the model consists of a set of agents that encapsulate the behaviors of the various individuals that make up the system, and execution consists of emulating these behaviors. In equation-based modeling (EBM), the model is a set of equations, and execution consists of evaluating them. Thus, “simulation” is the general term that applies to both methods, which are distinguished as (agent-based) emulation and (equation-based) evaluation. See Parunak & A. (1998).
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enormous amounts of quality data are available. But even where high quality data is not available, such as in questions of industrial organization, the approach is changing the way the questions are conceptualized, with markets being conceptualized as dynamic rather than static as in the more traditional approach.
How the Complexity Approach Relates to the Standard Approach The standard theoretical approach used in economics is one loosely based on a vision of rational agents optimizing, and is a consideration of how a system composed of such optimizing agents would operate. The complexity approach retains that same vision, and thus is simply an extension of the current analysis. Where complexity differs is in the assumptions it allows to close the model. The complexity approach stresses more local, rather than global, optimization by agents than is done in the traditional approach. Agent heterogeneity and interaction are key elements of the complexity approach. The standard approach, which developed over the last 100 years, was limited in the methods available to it by the existing evolving analytic and empirical technology. That meant that it had to focus on the solvable aspects of the model, and to structure assumptions of the model to fit the analytics, not the problem. The analysis evolved from simple static constrained optimization to nonstochastic control theory to dynamic stochastic control theory, but the general structure of the analysis - the analytic pattern generating mechanism - remained the same, only more jazzed up. In the search to focus on the solvable aspects of the model, the standard approach had to strongly simplify the assumptions of the model using the representative agent simplification, and the consequent gaussianeity in the heterogeneity of agents. Somehow, models the economy without any consideration of agent heterogeneity were relied upon to find the patterns that could exist. The complexity approach does not accept that, and proposes a different vision, which is a generalization of the traditional analysis. In fact, it considers the representative agent hypothesis that characterizes much of modern macro as a possible, but highly unlikely, case and thus does not find it a useful reference point. But these changes in assumptions are not without cost. The cost of making agent heterogeneity central is that the complexity model is not analytically solvable. To gain insight into it, researchers must make use of simulations, and generally, with simulations, results are neither univocally nor probabilistically determined.
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The standard approach offered enormous insights, and proved highly useful in generating patterns for understanding and applying policy. It led to understanding of rationing situations, shadow pricing, and underlay a whole variety of applied policy developments: cost benefit analysis, modern management techniques, linear programming, non-linear programming, operations research, options pricing models, index funds … the list could be extended enormously. It suggested that problems would develop if prices were constrained in certain ways, and outlined what those problems would be; it led to an understanding of second order effectsexternalities, and how those second order effects could be dealt with. It also led to actual policies - such as marketable permits as a way of reducing pollution - and underlies an entire approach to the law. Moreover, the standard approach is far from moribund; there are many more areas where the patterns generated by the standard approach will lead to insights and new policies over the coming decades. Standard economics remains strong. Despite its academic successes there are other areas in which standard economics has not been so helpful in generating useful patterns for understanding. These include, paradoxically, areas where it would seem that the standard theory directly applies - areas such as stock market pricing, foreign exchange pricing, and understanding the macro economy more generally. In matching the predictions of standard theory to observed phenomena, there seem to be too many movements uncorrelated with underlying fundamentals, and some patterns, such as the arch and garch movement in stock price data, that don’t fit the standard model. The problem is twofold the first is the simplicity of the model assumptions do not allow the complexity of the common sense interactions that one would expect; the second is the failure of the models to fit the data in an acceptable way. It is those two problems that are the starting points for the complexity approach. The reason why the complexity approach is taking hold now in economics is because the computing technology has advanced. This advance allows consideration of analytical systems that could not previously be considered by economists. Consideration of these systems suggested that the results of the ‘control-based’ models might not extend easily to more complicated systems, and that we now have a method - piggybacking computer assisted analysis onto analytic methods - to start generating patterns that might provide a supplement to the standard approach. It is that approach that we consider the complexity approach. It is generally felt that these unexplained observations have something to do with interdependent decisions of agents that the standard model assumes away. Moreover, when the dynamics are non-linear, local variations from the averages can lead to significant deviations in the overall system
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behavior. Individuals interact not only within the market; they also interact with other individuals outside the market. Nowadays, theorists are trying to incorporate dynamic interdependencies into the models. The problem is that doing so is enormously complex and difficult, and there are an almost infinite number of possibilities. It is that complexity that the papers in this volume deal with. In terms of policy the papers in this volume suggest that when economists take complexity seriously, they become less certain in their policy conclusions, and that they expand their bag of tools by supplementing their standard model with some additional models including (1) agent-based models, in which one does not use analytics to develop the pattern, but instead one uses computational power to deal with specification of models that are far beyond analytic solution; and (2) non-linear dynamic stochastic models many of which are beyond analytic solution, but whose nature can be discovered by a combination of analytics and computer simulations. It is elements of these models that are the source of the popular terms that develop. Developments in this new approach will occur on two dimensions. The first is in further development of these modeling techniques, understanding when systems exhibit certain tendencies. It is one thing to say that butterfly effects are possible. It is quite another to say here are the precise characteristics that predict that we are near a shift point. Until we arrive at such an understanding, the models will be little help in applied policy. Similarly with agent-based models. It is one thing to find an agent-based model that has certain elements. It is quite another to say that it, rather than one of the almost infinite number of agent based models that we could have chosen, is the appropriate model to use as our theory. The second development is in fitting the patterns developed to the data. Is there a subset of aspects of reality that better fit these models than the standard models? Are there characteristics of reality that tell us what aspects they are? Both these developments involve enormous amounts of slogging through the analytics and data. The papers in this volume are elements of that slogging through. They are divided into four sections: general issues, modeling issues, applications, and policy issues. Each struggles with complicated ideas related to our general theme, and a number of them try out new techniques. In doing so, they are part of science as usual. The choice of papers highlights the necessity to consider a multifaceted methodology and not a single methodology in isolation. Our goal is to give the reader a sense of the different approaches that researchers are following, so as to provide a sense of the different lines of work in the complexity approach.
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It is commonly said that science progresses one funeral at a time; the papers in this volume suggest that there is another method of progression one technique at a time, and as various of these techniques prove fruitful, eventually the sum of them will lead economics to be something different than it currently is, but it is a change that can only be seen looking back from the future.
Part I: General Issues The first two papers deal with broad definitional and ontological issues, the cornerstone of economic thinking. One of the ways economists have arrived at patterns from theory is to carefully delineate their conception of agents, restricting the analyst to what Herbert Simon called sub rationality, and which Vercelli, in the first paper, “Rationality, Learning, and Complexity: from the Homo Economicus to the Homo Sapiens”, calls ‘very restrictive notions of expectations formation and learning that deny any role for cognitive psychology’. He argues that the approach of standard economics succeeds in simplifying the complexity of the economic system only at the cost of restricting its theoretical and empirical scope. He states, ‘If we want to face the problems raised by the irreducible complexity of the real world we are compelled to introduce an adequate level of epistemic complexity in our concepts and models’, (p 15) concluding, ‘epistemic complexity is not a virtue, but a necessity’. In the second paper, “The Confused State of Complexity Economics: An Ontological Explanation”, Perona addresses three issues: the coexistence of multiple conceptions and definitions of complexity, the contradictions manifest in the writings of economists who alternate between treating complexity as a feature of the economy or as a feature of economic models, and finally the unclear status of complexity economics as an orthodox/heterodox response to the failures in traditional theory. He argues that economists supporting the complexity ideas tend to move alternatively between different conceptions of complexity, which makes it hard to follow what their argument is. Using Tony Lawson’s ontic/theoretietic distinction, he argues that the plurality of complexity definitions makes sense when we observe that most of the definitions are theoretic notions, but that the apparent agreement between heterodox and orthodox traditions over complexity ideas is fictitious since the ‘two sides represent in fact quite different and even opposite responses to the problems perceived in traditional theory’. (p 14).
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Part II: Modeling Issues I - Modeling Economic Complexity The next set of papers enters into the issues of modeling. The first of these, “The Complex Problem of Modeling Economic Complexity” by Day, develops the pattern sense of science that we emphasized in the introduction, and provides a general framework for thinking about the problem of modeling complexity. Day argues that rapid progress only began in science when repetitive patterns, which could be expressed in relatively simple mathematical formulas, were discovered, and that the patterns developed could provide simplification in our understanding of reality. However, this approach left out many aspects that could not be described by simple equations. Day argues that Walrasian general equilibrium is a theory of mind over matter, that has shown “how, in principle a system of prices could coordination the implied flow of goods and services among the arbitrarily many heterogeneous individual decision-makers”. The problem with the theory is that it is about perfect coordination, while the interesting questions are to be found in less than perfect coordination - how the process of coordination works out of equilibrium. He then goes through some of the key literature that approached the problem along these lines. He concludes with an admonition to complexity researchers not be become intrigued with the mathematics and models, but to concentrate on the task of explaining such things as how governments and central banks interact with private households and business sectors, with the hope of identifying “policies that improve the stability and distributional properties of the system as a whole”. Day’s paper is the perfect introduction to the modeling sections of the book because it provides the organizing belief of the contributors to this volume that the work in complexity on chaotic dynamics and statistical mechanics “provides potential templates for models of variables in any domain whose behavior is governed by nonlinear, interacting causal forces and characterized by nonperiodic, highly irregular, essentially unpredictable behavior beyond a few periods into the future”. Above we argued that the complexity revolution was not be a revolution in the sense of an abandonment of previous work, but will be a set of developments that occur one technique at a time, which when looked back upon from a long enough perspective, will look like a revolution. The second paper in this section, “Visual Recurrence Analysis: Application to Economic Time Series” by Faggini, suggests one of those new techniques. In the paper Faggini argues that the existing linear and non-linear techniques of time series analysis are inadequate when considering chaotic phenomena. As a supplement to standard time series techniques he argues
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that Recurrence Plots can be a useful starting point for analyzing nonstationary sequences. In the paper he compares recurrence plots with classical approaches for analyzing chaotic data and detecting bifurcation. Such detection is necessary for deciding whether the standard pattern or one of the new complexity patterns is the pattern appropriate for policy analysis. The third paper in this part, “Complexity of Out-of-equilibrium Play in a Tax Evasion Game” by Lipatov, uses the same evolutionary framework that characterizes work in complexity, but combines it with more traditional economic models, specifically evolutionary game theory with learning, to model interactions among taxpayers in a tax evasion game. The paper expands on previous models of tax evasion, but adds an explicitly characterization of taxpayer interaction, which is achieved by using an evolutionary approach allowing for learning of individuals from each other. He argues that the dynamic approach to tax compliance games reopens many policy issues, and that as a building block for more general models, the evolutionary approach he provides can be used to consider policy issues.
Part III: Modeling Issues II - Using Models from Physics to Understand Economic Phenomena Part III is also about modeling, but is focus is on adapting models from physics to explain economic phenomena. The papers in this section explore various issues of modeling, demonstrating various mathematical techniques that are available to develop patterns that can then be related to observation to see if they provide a useful guide. The first paper in this section, “A New Stochastic Framework for Macroeconomics: Some Illustrative Examples” by Aoki, extends the Day’s argument, starting with the proposition that that the standard approach to microfoundations of macroeconomics is misguided, and that therefore we need a new stochastic approach to study macroeconomics. He discusses the stochastic equilibria and ultra metrics approaches that might be used to get at the problems, and how those approaches can provide insight into real world problems. He gives an example of how these approaches can explain Okun’s Law and Beveridge curves, and can demonstrate how they shift in response to macroeconomic demand policies. He also shows how these new tools can reveal some unexpected consequences of certain macroeconomic demand policies. The second paper in this part, “Probability of Traffic Violations and Risk of Crime: A Model of Economic Agent Behavior” by Mimkes, shows the relationship between models in physics and economic models. Specifi-
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cally he relates the system of traffic agents under constraint of traffic laws to correspond to atomic systems under constraint of energy laws, and to economic models of criminal behavior. He finds that the statistical Lagrange LeChatalier principle agrees with the results, and concludes by suggesting that “similar behavior of agents may be expected in all corresponding economic and social systems (situations) such as stock markets, financial markets or social communities” (p. 11). Muchnik and Solomon’s paper, “Markov Nets and the Nat Lab Platform: Application to Continuous Double Auction”, considers Markov Nets, which preserve the exclusive dependence of an effect even on the event directly causing it but makes no assumption on the time lapse separating them. These authors present a simulation platform (NatLab) that uses the Markov Net formalisms to make simulations that preserve the causal timing of events and consider it in the context of a continuous double auction market. Specifically, they collect preferred trading strategies from various subjects and simulate their interactions, determining the success of each trading strategy within the simulation. Arecchi, Meucci, Allaria, and Boccaletti’s paper “Synchronization in Coupled and Free Chaotic Systems”, examines the features of a system affected by heteroclinic chaos when subjected to small perturbations. They explore the mathematics of what happens when there is an intersection point between a stable and unstable manifold, which can create homoclinic tangles, causing systemic erratic behavior and sensitivity to initial conditions. The possibilities of such behavior should serve as a cautionary tale to policy makers who make policy relying on the existence of systemic stability.
Part IV: Agent Based Models Part IV considers agent based models, which differ from the models in the previous section in that they begin with specification of agent strategies and do not have any analytic solution, even in principle. Instead one allows agents with various strategies to interact in computer simulations and from those interactions one gains knowledge about the aggregate system. This allows the exploration of models in which the agents are assumed to have far less information and information processing capabilities than is generally required for analytic models. The first paper in this section, “Explaining Social and Economic Phenomena by Models with Low or Zero Cognition Agents” by Omerod, Trabatti, Glass, and Colbaugh, examines two socio-economic phenomena, the distribution of the cumulative size of economic recessions in the US
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and the distribution of the number of crimes committed by individuals. They show that the key macro-phenomena of these two systems can be accounted for by models in which agents have low or zero cognitive ability. They conclude by arguing that the likely explanation for the success of such simple models of low cognition agents in describing actual phenomena is the existence of institutions, which have evolved to enable even low cognition agents to arrive at good outcomes. The second paper, by Delre and Parisi, entitled “Information and Unionization in a Simulated Labor Market: A Computation Model of the Evolution for the Evolution of Workers and Firms”, uses a genetic algorithm in an agent-based model to consider the bargaining behavior of workers and firms, and how the behavior is likely to evolve over time. They consider the social interaction among workers when they are looking for a new job. They find that the value of information about job offers is higher when both firms and workers evolve than when only workers evolve. They also find that when workers are connected in social networks the value of information about job offers increases.
Part V: Applications Part V deals with applications of the broad complexity ideas to empirical evidence issues. The first paper entitled “Income Inequality, Corruption, and the Non-Observed Economy: A Global Perspective” by Ahmed, Rosser, and Rosser,extents their previous work that found a strong and robust positive relationship between income inequality and the size of the non-observed economy in a global data set. In this paper they find that the relationship between income inequality and the size of the non-observed economy carries over, and that the corruption index is also significantly related. However, the relationship between inflation and growth of the nonobserved economy does not carry over, nor does the relationship between higher taxes and the size of the nonobserved economy. The second paper, by McNelis and McAdam, entitled “Forecasting Inflation with Forecast Combinations: Using Neural Networks in Policy”, discusses the use of non linear models, which Granger and Jeon have called “thick models”. These non-linear models use split sample and bootstrap methods in inflation forecasting. They find that in many cases the thick models perform better than linear models for out-of-sample forecasting. They recognize that the approach is ad hoc, and advocate the thick model “as a sensible way to utilize alternative neural network specifications and ‘training methods”‘ in a ‘learning context.’” (p. 2)
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Part VI: Policy Issues The final three papers in the book deal with complexity and policy. In the first, “The Impossibility of an Effective Theory of Policy in a Complex Economy”, Velupillai discusses the problem of drawing policy from the complexity work. His starting point is papers by Duncan Foley (2000) and Brock and Colander (2000), who argue that complexity changes the way in which economists should think about applied policy. Velupillai agrees with much of their argument, but argues that it can be extended into a formal impossibility theorem - that ultimately there is an undecidablity of policy in a complex economy. In the second paper of this section, “Implications of Scaling Laws for Policy Makers”, Gallegati, Kirman, and Palestrini integrate some of the insights from Alan Kirman’s representative agent approach for economic policies with recent work discovering that some phenomena in economics can be described by scaling laws, and draw policy implications from that work. They hold that in order to stabilize in power law related processes in which a main source of aggregate fluctuations is idiosyncratic volatility, policies that control individual volatility can reduce aggregate fluctuations. They also argue that since firm size is descriptive of a power law relationship (Axtell), economic policies that reduce legal protection of intellectual property can help increase growth, that economic policies to reduce the concentration in the market can reduce aggregate fluctuations and that economic policies can be useful to control financial crises. The final paper, “Robust Control and Monetary Policy Delegation” by Diana and Sidiropoulos, considers “the robust control method of Hansen and Sargent” as a way of getting around model misspecification. They begin with Rogoff’s article on classic game theoretical model of monetary policy delegation, and put the argument into a robust control framework, which takes into account the uncertainty of policy, and hence the need to make sure the model is robust to different specifications. Doing so, they suggest that governments should be very hesitant to delegate monetary policy to a “conservative” central banker who would be concerned only with inflation.
June, 2006
David Colander Massimo Salzano
Contents
Part I General Issues Rationality, learning and complexity: from the Homo economicus to the Homo sapiens A. Vercelli ..………………………………………………………………..…… The Confused State of Complexity Economics: An Ontological Explanation E. Perona ...………………………………………………………………..…..
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Part II Modeling Issues I Modeling Economic Complexity The Complex Problem of Modeling Economic Complexity R. H. Day ...………………………………………………………………..…..
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Visual Recurrence Analysis: Application to Economic Time series M. Faggini ..……………………………………………………………………
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Complexity of Out-of-Equilibrium Play in Tax Evasion Game V. Lipatov .…………………………………………………………………….
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Part III Modeling Issues II Using Models from Physics to Understand Economic Phenomena A New Stochastic Framework for Macroeconomics: Some Illustrative Examples M. Aoki ..………………………………………………………...……………
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Probability of Traffic Violations and Risk of Crime - A Model of Economic Agent Behavior J. Mimkes ..………………………………………………………………….. 145
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Markov Nets and the NetLab Platform: Application to Continuous Double Auction L. Muchnik and S. Solomon ..……………………………………………... 157 Synchronization in Coupled and Free Chaotic Systems F.T. Arecchi, R. Meucci, E. Allaria, and S. Boccaletti ..…………………
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Part IV Agent Based Models Explaining Social and Economic Phenomena by Models with Low or Zero Cognition Agents P. Ormerod, M. Trabatti, K. Glass, and R. Colbaugh ..………………… 201 Information and Cooperation in a Simulated Labor Market: A Computational Model for the Evolution of Workers and Firms S. A. Delre and D. Parisi ..…………………………………………………
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Part V Applications Income Inequality, Corruption, and the Non-Observed Economy: A Global Perspective E. Ahmed, J. B. Rosser, Jr., and M. V. Rosser ..……………………….. 233 Forecasting Inflation with Forecast Combinations: Using Neural Networks in Policy P. McNelis and P. McAdam ..…………………………………………….. 253
Part VI Policy Issues The Impossibility of an Effective Theory of Policy in a Complex Economy K. Vela Velupillai …………………………………………………………… 273 Implications of Scaling Laws for Policy-Makers M. Gallegati, A. Kirman, and A. Palestrini ..………………………………
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Robust Control and Monetary Policy Delegation G. Diana and M. Sidiropoulos ..……………………………………………
303
List of Contributors
E. Ahmed
R. Colbaugh
James Madison University, Harrisonburg, USA
[email protected] Department of Defense, Washington, USA
[email protected] E. Allaria
R. H. Day
Dept. of Physics, Univ. of Firenze INOA Italy
[email protected] University of Southern California, USA
[email protected] M. Aoki
S. A. Delre
Department of Economics, University of California USA
[email protected] Faculty of Management and Organization, University of Groningen, The Netherlands.
[email protected] F.T. Arecchi
G. Diana
Dept. of Physics, Univ. of Firenze INOA Italy
[email protected] BETA-Theme, University Louis Pasteur of Strasbourg France
[email protected] S. Boccaletti
V. Lipatov
Dept. of Physics, Univ. of Firenze INOA Italy
[email protected] Economics Department, European University Institute Italy
[email protected] XXII
List of Contributors
M. Faggini
L. Muchnik
Dipartimento di Scienze Economiche e Statistiche Università degli Studi di Salerno, Italy
[email protected] Department of Physics, Bar Ilan University, Israel
[email protected] M. Gallegati
P. Ormerod
Università Politecnica delle Marche Dipartimento di Economia, Italy
[email protected] Volterra Consulting Ltd., London UK
[email protected] K. Glass
A. Palestrini
National Center for Genome Resources, USA
[email protected] University of Teramo Italy
[email protected] A. Kirman
D. Parisi
Université Paul Cézanne (AixMarseille III), EHESS, Greqam-Idep, France
[email protected] Institute of Cognitive Sciences and Technologies, National Research Council, Italy
[email protected] P. McAdam
E. Perona
DG-Research, European Central Bank Frankfurt am Main
[email protected] Departamento de Economía, Universidad Nacional de Córdoba Espain
[email protected] P. McNelis
J. B. Rosser, Jr.
Fordham University, USA
[email protected] MSC 0204, James Madison University, USA
[email protected] J. Mimkes
M. V. Rosser
Physics Department, University of Paderborn - Germany
[email protected] MSC 0204, James Madison University, USA
[email protected] R. Meucci
M. Sidiropoulos
Dept. of Physics, Univ. of Firenze – INOA, Italy riccardo.meucci&inoa.it
BETA-Theme, University Louis Pasteur of Strasbourg France
[email protected] List of Contributors
XXIII
S. Solomon
K. Vela Velupillai
Racah Institute of Physics., The Hebrew University, Israel and Lagrange Interdisciplinary. Lab. for Excellence in Complexity, ISI, Italy
[email protected] National University of Ireland, and Department of Economics, University of Trento,
[email protected] M. Trabatti
A. Vercelli
National Center for Genome Resources, USA
[email protected] Dipartimento di Politica Economica, Finanza e Sviluppo Italy
[email protected] 1
Part I
General Issues
3
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Rationality, learning and complexity: from the Homo economicus to the Homo sapiens1 A. Vercelli
1. Introduction Standard economics is based on methodological individualism but this does not imply that the individuals play a crucial role in its models. On the contrary, in such a theory the individual is deprived of authentic subjective characteristics and plays no sizeable role as genuine subject. The so called Homo economicus is just a signpost for given preferences that, however, are generally conceived as exogenous and invariant through time. Therefore, the genuine psychological features of the economic agent do not matter. In this chapter we intend to concentrate on the impact of cognitive psychology upon the economic behavior. Casual observation and experimental research suggest that cognitive psychology significantly affects expectations and learning that play a crucial role in many economic decisions. In standard economics, however, expectations and learning are conceived in such a way to make cognitive psychology altogether irrelevant. This essay —————— 1
A preliminary version of the ideas discussed in this chapter was presented at the Conference “Keynes, Knowledge and Uncertainty” held at the University of Leeds in 1996, and eventually published in its Proceedings (Vercelli, 2002). Successive versions have been presented on many occasions, including the annual Conference of Anpac (San Salvador de Bahia, December 2001), the annual conference of EAEPE (Siena, November 2001), the first meeting of the Italian Society of “Cognitive sciences” held in Rovereto in September 2002, the International World Congress of IEA held in Lisbon in September 2002, the conference on “Complex behavior in economics” held in Aix-en-Provence in May 2003. I thank very warmly the audiences of my presentations for the constructive comments.
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intends to clarify the reasons of this neglect and aims to suggest a way to reduce the chasm between economics and cognitive psychology. The crucial obstacle to be removed in order to close this gap is the very narrow notion of rationality entertained by standard economics, generally called “substantive rationality” following Simon (1982), that implies very restrictive notions of expectations formation and learning that deny any role for cognitive psychology. The straitjacket of substantive rationality is felt as suffocating by an increasing number of economists. Many of them are actively exploring new research avenues. There are basically two strategies: the progressive relaxation of some of the most constrictive assumptions of the standard approach in order to generalize it, or the search for a more satisfactory alternative approach2. As a consequence of this effort, recent advances in economic theory brought about many important insights in different specialized subfields of economics that go beyond the limits of substantive rationality. What is still missing is the coordination of these new insights in a fully-fledged economic paradigm able to replace the traditional paradigm based on substantive rationality. Having in mind this goal, this essay aims to individuate the main assumptions underlying the traditional paradigm and to classify the deviations from it in coherent alternative paradigms. Therefore, the conceptual backbone of the argument is a taxonomy of different notions of a few basic concepts that play a crucial role in economics: rationality, learning, expectations, uncertainty, time, and a few crucial ontological assumptions on the real economic world. This essay intends to show that between these notions there are relations of semantic congruence or incongruence having interpretive and normative implications. In particular, whenever it is necessary to take account of the complexity of economic behavior, and in particular of the impact of cognitive psychology on it, we cannot rely on the standard paradigm but we have to refer to alternative paradigms whose characteristics are specified in some detail in the following sections. The definitions of complexity vary according to the epistemic and ontological context of the analysis. Since the scope of the analysis that follows is rather broad, the argument is not based upon a unifying definition —————— 2
The dividing line between the two strategies is not always clear and is often more a matter of language than of substance. An apparently small change in an axiom of a theory may be enough to evoke a completely different conceptual world as in some recent contributions to decision theory under uncertainty, while what appears to be a radical change may turn out to be of secondary importance for the empirical application as in the case of the distinction between objectiveprobability and subjective-probability decision theory (see section 6).
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of complexity. The following discussion is developed at an intermediate level of abstraction relying on a taxonomy of different basic notions of a few crucial concepts that characterize economic analysis according to their degree of complexity. There is a clear relationship, however, between the concept of complexity here utilized and the analytic literature on complexity. The usual measures of complexity are syntactic, or computational, measures concerning the minimal number of logical steps, or amount of information, necessary to describe a certain system or to obtain a particular class of its transformations. We recall that the most popular measures of complexity refer to properties of linguistic systems, or dynamical systems, or computing machines and that there are quite strict correspondences between them (for a critical survey with special reference to economics, see Foley, 2001 and Albin, 2001). In each of these cases four basic levels of complexity are distinguished. In terms of dynamical systems the degree 1 of minimal complexity corresponds to stable linear systems having a unique equilibrium; the degree 2 to stable systems having a regular periodic attractor (such as a limit cycle); the level 3 to nonlinear systems characterized by chaotic behavior and monotonic propagation of a disturbance, and the level 4 by chaotic behavior and irregular propagation of a disturbance (ibidem). In this chapter, however, complexity is discussed from the semantic point of view, i.e. from the point of view of the correspondence and the interaction between the degree of complexity of the objective system that we intend to describe or forecast and their subjective representation through models or expectations. This semantic approach clarifies that standard economics applies, under further limiting conditions, only if the degree of complexity does not exceed the level 2 of complexity. In order to take account of higher levels of complexity more advanced approaches based on more sophisticated concepts of rationality and learning are required. The structure of the paper is as follows. Section 2 introduces the basic framework of the analysis centered on the interaction between subject and object in economics that is characterized by different concepts of rationality, learning and expectations. In this section the usual specification of standard economics based on substantive rationality is clarified in its inner logic. In section 3 more general concepts of rationality are introduced (procedural and designing rationality) in order to overcome the strictures of substantive rationality. In section 4 the relationship between expectations, learning and rationality is further spelled out. In section 5 more general concepts of learning are introduced to cope with the more general concepts of rationality introduced before, and their economic value is assessed and compared in qualitative terms. In section 6 the implications of different ontological assumptions concerning the nature of uncertainty, the
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irreversibility of time, and the nature of the “world” are clarified. In section 7 the correspondences between the different versions of the keyconcepts examined in the preceding sections are discussed and their normative implications are made explicit. Concluding remarks follow.
2. The interaction between subject and object and the strictures of substantive rationality The subject may be defined only in relation to the object. Therefore the analysis of the subjective features of economic behavior must start from a representation of the interaction between subject and object in economics, i.e. the interaction between the economic agent and the economic system (that for the sake of brevity is called in this essay just the Interaction). According to the usual conceptualization of the Interaction, that is shared by most economists, the economic agent is characterized by a set of preferences, an initial endowment of resources and a set of options each of which leads to well-specified outcomes for each possible state of the economic system (i.e. the “world” under investigation). Whenever the economic agent has to face an inter-temporal decision problem she chooses the best possible option according to her preferences and an objective function on the basis of her preferences and her expectations on the future values of the key variables that affect the value of the objective function. These expectations are based on a “model”, i.e. on a representation as accurate as possible, of the part of the “world” that is relevant for the decision problem. In this chapter the epistemic and ontological properties of the Interaction are sharply distinguished in order to study their mutual impact. The ontological properties of the “world” are defined as properties of the economic system that are considered “true” regardless of their epistemic representation, while the epistemic properties of the “model” are provisionally entertained by a subject as representation of the corresponding ontological properties but are liable to be revised and corrected to represent them better. The divergence between an ontological property and its epistemic representation is here defined as stochastic error when it depends exclusively on exogenous shocks, and systematic error when it depends on some intrinsic bias of the representation (e.g. a different value of one parameter of the probability distribution). Within this conceptual framework, learning is defined as the epistemic process directed to reduce, and possibly to eliminate, the systematic errors, while rationality may be defined essentially as ability to learn. The feed-back between rationality and learning is crucial in determining the economic behavior of decision makers and is therefore
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at the center of what follows. The Interaction as here represented is deceivingly simple since it involves a crucial self-referential loop of the economic agent whose behavior is part of the world that has to described and predicted and affects it through its representations and expectations (see Fig.1). This awkward circle partially overlaps with what the philosophers call the “hermeneutic circle” (see, e.g., Gadamer, 1960) whose solution is very controversial and divides the analytic school from the continental school. Also in economics the way in which the Interaction is conceived and modeled is crucial in classifying the economic theories and assessing their capability to represent, predict and control complex economic behavior. The dynamic decision problem, as defined above, may give determinate results only by imposing some more structure to it. This role is played by the assumption that the agent is rational from both the epistemic (or cognitive) and pragmatic (or practical) point of view in a specific sense that must be accurately defined and modeled. According to the standard point of view, the epistemic rationality is specified as ability to avoid systematic errors. In a deterministic decision problem, this implies that the agent has perfect foresight. In a stochastic decision problem the above condition implies that the agent has rational expectations which by definition avoid any sort of systematic errors. In a deterministic model the value expected is the correct one, while in a stochastic model the subjective probability distribution of the forecast coincides with the “objective” or “true” probability distribution. As for pragmatic rationality, it is assumed that no systematic errors are made in the execution of the decision so that, under these assumptions that involve the stationarity of the real world, the maximization of the objective function translates in its effective maximization so that the absence of systematic errors ex ante translates in the absence of systematic mistakes ex post. In this view the subjective characteristics of the economic agent are irrelevant, since the model (and therefore the expectations based on it) is by definition “true” in the sense that the subjective representation of the world “coincides” with its objective features. It is true that the preferences of the decision maker play a crucial role in the choice of the option that maximizes the objective function, but this does not involve any role for the psychological characteristics of the agent. Preferences should express, by definition, subjective features of the individual, i.e. her tastes, but in the standard approach they are exogenous and given, exactly as the initial endowment of resources. Moreover, self-interest is the only motivation taken into account, completely neglecting different motivations such as altruism, or equity or solidarity, whose role in the real world is confirmed by massive empirical evidence and extensive experimental work. Finally, even
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cognitive psychology has no role to play within the assumptions of substantive rationality, since the economic agent is assumed to base her choices on the “true” model of the real world and is endowed of perfect foresight or its equivalent in an uncertain world: rational expectations. Learning does not completely disappear but in this context it keeps only a very trivial role: the updating of the information set as soon as new observed values of the relevant variables occur. This role does not involve, however, an influence of the subjective cognitive features of the agent as it could be performed by a computer that collects in real time all the relevant news to update the information set that is relevant for the decision problem, exactly as a seismograph does with earthquakes. We may conclude that within the framework of standard economics the psychological features of the agent are altogether neglected and this is in particular true with its cognitive aspects. SUBJECT EXPECTATIONS: predictive systematic errors
MODEL:
DECISIONS:
cognitive systematic errors
ex ante pragmatic systematic errors
CONSEQUENCES: ex post pragmatic systematic errors
OBJECT Fig. 1: The Interaction between rationality and learning
Under the assumptions of substantive rationality, thus, the Interaction is described by a dynamic process that is assumed to be always in equilibrium. In the absence of systematic errors ex post there is no reason to revise the model of the world and the expectations based on it. Since the axiom of intertemporal coherence that is crucial in the definition of substantive rationality assures that the preferences of the agent do not change through time, the agent makes the same choices in the same circumstances and the Interaction settles in a stationary equilibrium.
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3. More general concepts of rationality The advantage of substantive rationality is that within its assumptions we may, indeed we have to, assume equilibrium and this greatly simplifies the analysis of the economic behavior permitting the systematic application of sophisticated formalization that leads to determinate results (Vercelli, 1991). The trouble is that the same axioms restrict the scope of legitimate application to a closed, simple and familiar world, i.e. to problems that are well known and are characterized by stationarity, trivial uncertainty, and irrelevance of the psychological and cognitive features of the agent. This has been recognized long ago by economists dissatisfied with the standard approach. Schumpeter (1921) clarified that the standard approach applies to the circular flow characterized by routine economic behavior but is unable to analyze development characterized by non-stationarity, innovation and radical uncertainty. Keynes (1936) clarified that the axioms of the standard approach are like the axioms of Euclidean geometry that cannot be legitimately applied to a non-Euclidean world, i.e. whenever hard uncertainty and structural instability play a crucial role as is typical in a monetary economy. Simon (1982) further clarified the crucial role played by the concept of rationality of standard economics, called by him “substantive rationality”, in narrowing its empirical scope and suggested a more general concept, called “procedural rationality”, borrowed from cognitive psychology. This concept became immediately so successful that we do not need to expand on it. What is important to emphasize is that the concept of procedural rationality allows the analysis of the dynamic process that describes the Interaction also from the point of view of disequilibrium. Whenever a systematic error ex post is detected by the agent, the model of the world is revised in order to correct the expectations and to eliminate, or at least to reduce, the size of the error. From this more general point of view the equilibrium of the process does not need to imply a complete absence of systematic errors. According to the criterion suggested by Simon, the dynamic process of learning may stop as soon as the size of errors becomes small enough to be satisficing. A further attempt to eliminate the residual errors in a situation characterized by limited information and bounded rationality could in the average imply more economic and psychological costs than benefits. The dynamic process of learning so conceived does not need to be linear and may be complex. In particular the equilibrium may depend on the initial conditions and on the path actually followed. In this view substantive rationality may be seen as a special case of procedural rationality as it focuses exclusively on a particular equilibrium characterized by the absence of systematic errors to which the process of learning could
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converge under a series of restrictive assumptions: linearity of the process of learning, no cost of transaction and of learning itself, weak or “soft” uncertainty (see section 6), and unbounded rationality. This generalization, however, is still limited if it expresses the process of passive adaptation of the agent to a given environment. In this view the agent is option-taker in the sense that cannot alter the set of options of which she is endowed to begin with. Biologists tell us that the adaptation of the Homo sapiens to the environment is obtained also by adapting the environment to her needs. This is almost altogether absent in the other living species and, in any case, was progressively developed in the course of evolution of this particular species to degrees unimaginable by other living beings, including the other primates. While animal rationality is purely adaptive, the Homo sapiens developed a peculiar kind of rationality that became progressively more and more proactive. The reciprocal adaptation of man to the environment is seeked on the basis of a project or design that may be updated or revised according to its success. For this reason we may call “designing” this more general concept of rationality (Vercelli, 1991). The agent is in this case also option-maker in the sense that she is endowed of creativity that allows her to introduce new options made possible by advances in scientific research and technology. Human rationality therefore re-acquires the specific features that distinguish our species from the others, features that are completely neglected by substantive rationality and, to some extent, also by adaptive procedural rationality3. The latter may be considered as a limiting case of designing rationality when the set of options is given and invariant.
Expectations, learning and rationality Coming back to the Interaction, after the implementation of the option chosen the economic agent observes its consequences. Whenever she ascertains the absence of systematic errors ex post, as by definition in the case of substantive rationality, there is no reason to revise the model and the expectations based on it apart from their updating. Therefore, in this case, the Interaction repeats itself unchanged apart from the exogenous drift induced by updating. In other words, substantive rationality focuses exclusively on the equilibrium path of the Interaction. This point of view is —————— 3
Some advocates of procedural rationality give a broader definition that encompasses what we have called here designing rationality. In this case, to avoid confusion, we may distinguish between adaptive and designing procedural rationality.
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reasonable and productive to the extent that we are able to provide dynamic foundations to it by showing that such an equilibrium is stable and that the convergence to it is sufficiently rapid, otherwise the scope of the theory and of its applications would be very limited. For this reason an extensive literature started long ago at least from the study of the tâtonnement by Walras (1774), in order to study the stability of the general equilibrium. The results have been on the whole quite discomforting. As is well known, the stability of a Walrasian general equilibrium requires quite demanding conditions4. In particular the convergence to the substantiverationality equilibrium crucially depends on the process of learning of selfinterested rational agents pursuing the maximization of the objective function that is assumed to obtain in the position of equilibrium. In order to clarify the relation between rationality, learning and expectations let us express their analytical relation. We start from the deterministic case that prevailed in economic theory until the 1960s. In this case, the prevailing paradigm may be expressed through the following difference equation: xt – xt-1 = a [ T(xt-1) – xt-1 ] + kt
(1)
where xt is the expectations of a certain variable that describes the behavior of the “world”, T(xt) designates the effective behavior of such a variable that is affected by its expectations, while kt is an exogenous variable that affects the dynamics of the endogenous variable and its expectations5. The dynamics of the expectations depends on two terms. The first one, a [T(xt-1) – x t-1 ], describes the endogenous dynamics of the interaction between the subject who formulates the expectations and the economic system. The endogenous dynamics depends on what we have called the systematic error observed ex post: T(xt-1) – xt-1. Whenever the endogenous dynamics is stable, i.e. the systematic error tends to zero, we have genuine learning and convergence towards the equilibrium path of the interaction. The exogenous dynamics determined by the second term kt specifies the —————— 4
This difficulty was already fairly clear in Samuelson (1947). Later on Scarf (1960) provided precise examples of plausible trading processes failing to exhibit global stability. Sonnenschein (1973) proved that under the usual assumptions about consumer preferences and behaviour even quasi-global stability cannot be assured. This result was confirmed and extended by subsequent papers. A recent brief survey on this literature may be found in Mas-Colell, Whinston, and Green (1995). 5 This equation may be easily generalized by assuming that the variables represented are vectors of variables.
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nature of the equilibrium path that is stationary when the exogenous term is stationary and mobile whenever it is a function of time. When the first term describing the endogenous dynamics of the interaction is systematically zero we have the case of perfect foresight that in a deterministic environment, by definition, is the only case fully consistent with substantive rationality. However, since perfect foresight is a very demanding assumption for applied economics, in the 1960s an alternative hypothesis of expectations formations, called adaptive expectations, became hegemonic (Cagan, 1956). The reason of its success rested on its capacity to deal, to some extent, with the existence of systematic errors and genuine learning. Notwithstanding the high sophistication reached by this literature, the idea was very simple and was constrained in such a way to avoid to challenge the paradigm of substantive rationality but only to add realism to it. It was assumed that a rational agent is able to learn so that the parameter a is always in the range of values that assure dynamic stability. The endogenous part, moreover, is assumed to be relevant for the short period (business cycle) while the exogenous part, fully consistent with the axioms of substantive rationality, was considered to be the center of attraction of the system always prevailing in the long period (growth). In addition, the linear specification of equation (1) that excludes complex dynamics is crucial as it assures that the equilibrium path is not affected by the disequilibrium dynamics. Therefore a sort of hierarchy was established between equilibrium dynamics that describes, so to say, the “essence” of the economic dynamics and the study of the deviations from it that are assumed to be fairly small and rapidly vanishing in consequence of the capacity of genuine learning attributed to rational agents. Therefore the substantive-rationality paradigm was somehow saved as gravity center of the economy, while the introduction of disequilibrium dynamics was just meant to confer to it some more realism. The paradigm of adaptive expectations broke down at the end of the 1960s, for a series of reasons that have to do both with the history of facts (stagflation) and the history of ideas. On this occasion we only mention, very briefly, a few crucial reasons related to the latter6. The adaptive expectations paradigm was consistent with the idea, prevailing until the late 1960s, that economic equilibrium needed sound dynamic foundations. This was fully recognized by Walras (1774) who tried himself to solve the problem by studying the process of tâtonnement. An important breakthrough came with the Ph.d thesis of Samuelson who for the first time was able to suggest quite comprehensive dynamic foundations to general equi—————— 6
For some hints to the role of the history of facts see Vercelli (2003).
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librium theory (see Samuelson, 1947). This approach inspired many economists in the two following decades and is clearly at the background of the success of adaptive expectations. In the meantime, in consequence of the axiomatization suggested by Debreu (1959), a new point of view emerged in economic theory according to which (general) equilibrium theory did not need dynamic foundations but only axiomatic foundations (Ingrao-Israel, 1990). In this view, the latter assure in the best possible way the logical consistency of the theory while its empirical validity is best assessed by statistic and econometric tests so that dynamic foundations seem unnecessary. From the point of view of the axiomatic foundations of general equilibrium theory the adaptive expectations hypothesis is clearly unacceptable because the existence of systematic mistakes involved by such hypothesis are logically inconsistent with the axioms of the theory (in particular that of intertemporal independence or “sure thing principle”). From the logical point of view, their alleged small size and vanishing nature are hardly an acceptable excuse. In the meantime, in the same circle of economists and mathematicians, a fully-fledged theory of general equilibrium under uncertainty was elaborated (Arrow, 1953; Arrow-Hahn, 1971). This permitted a stochastic interpretation of equation (1) that seemed to provide a different and better solution. The equilibrium was then described in terms of probability distributions of the relevant variables so that it does not exclude the possibility of ex post errors provided that they are not systematic (see, e.g., Begg, 1982). This adds to the realism of the substantiverationality interpretation of the Interaction. In the meantime a new hypothesis of expectations formations was suggested in the literature that turned out to be fully consistent with this view: the rational expectations hypothesis (Muth, 1961). According to this hypothesis, the rational agent is assumed to be always able to avoid systematic errors by predicting their expected value, in the mathematical sense, given the probability distribution of the variable under the assumption that the subjective distribution of this variable always coincides with its objective distribution (Begg, 1982). Clearly in this view there is no room for genuine learning. The only kind of learning consistent with, indeed necessary to, this view is the “real-time updating” of the information set that assures the persistent and continuous absence of systematic mistakes. The issue of equilibrium stability, however, did not disappear because if the logical possibility of disequilibrium is not excluded, one needs to know if there are sound reasons for focusing on it. In particular, a literature developed on the stability of rational expectations equilibrium. We may distinguish two basic streams based on the study of expectational stability or on the study of learning rules. The first stream based on contributions by Lucas (1978, section 6), DeCanio (1979), and Evans (1983), ultimately
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(see Evans and Honkapohja, 1990) refer to the following difference equation xt – xt-1 = a [ T(xt-1) – xt-1 ]
(2)
that is nothing but the equation (1) without considering the exogenous factor that by definition does not affect the stability of the system. This equation is meant to describe a stylized process of learning which occurs in ‘notional’ time t, that determines a progressive reduction in the gap between the dynamics of expectations xt and the effective dynamics T (xt), which is a function of perceived dynamics. The dynamics of expectations is a function of the systematic ex post errors defined by T (xt) - xt. Actual learning implies a process of convergence towards the rational expectations equilibrium. Notice that, as soon as this equilibrium is effectively reached, the systematic errors vanish and the process of genuine learning stops. The second stream focuses on the study of the learning rules in real time and is based on contributions by Bray (1982), Bray and Savin (1986), Fourgeaud, Gourieroux, and Pradel (1986), Marcet and Sargent (1989 a and b), and Woodford (1990). Learning rules are expressed in terms of approximation algorithms (in particular recursive least squares, and recursive ARMA estimations). Also in this case, the learning process may be expressed through a dynamic equation which is a function of the systematic ex post mistakes. For example, the seminal model by Bray (1982) may be expressed by the following stochastic approximation algorithm: ßt = ßt-1 + (1/t)[pt-1 - ßt-1]
(3)
where pt is the effective price at time t and ßt is the expectation of pt+1 which is equal by hypothesis to the average of realized prices. Also in this case, the dynamics of expectations depends exclusively on the ex post systematic errors, which are expressed in equation (3) by the term inside square brackets (i.e. by the deviation of the last observation from the average of past values)7. Also in this case, the dynamic process of learning stops only if the ex post systematic errors are fully corrected, that is as soon as the rational expectations equilibrium is reached. The rational expectations hypothesis implies that the economic agent is not allowed to make systematic errors ex post: therefore genuine learning, i.e. correction —————— 7
Marcet and Sargent (1989a) proved that the limiting behavior of ß can be analyzed by the following differential equation: (d/dt)ß = a + (b - 1)ß which is stable for b < 1. The right-hand side of the differential equation can be redefined by the difference T(ß)-ß, where T(ß)=a+b(ß), which expresses the ex post systematic error in expectations (see Sargent, 1993, n.2, p. 88).
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of systematic errors is inconceivable in any theory or model based upon this hypothesis. On the contrary, the process of convergence towards the rational expectations equilibrium, which can be considered as a process of genuine learning, would imply that, while the agent is learning, she is not allowed to form her expectations on the basis of the rational expectations hypothesis. Therefore, whether the learning process converges or not towards a rational expectations equilibrium, this literature proves the inconsistency between rational expectations and genuine learning. The only kind of learning really consistent with rational expectations is the trivial process of real-time updating of the realizations of the relevant stochastic variables, which by hypothesis does not affect the parameters of the stochastic processes involved. The way out from the above dilemma should be seeked by elaborating alternative hypothesis of expectations formation that do not rule out the ubiquity of systematic errors and the crucial role of genuine learning in correcting them. To proceed in this direction, the tradition of adaptive expectations could be rescued and consistently developed within the assumptions of broader concepts of rationality. This may be done by eliminating the constraints introduced to comply with the axioms of substantive rationality. In this different framework, the criticisms leveled against adaptive expectations from the point of view of substantive rationality loose their logical strength. In particular, the resting points of the dynamical system do not need to be unique, may be path-dependent and could be reached in a finite, possibly short, time as soon as the perceived marginal costs of learning (plus transition costs) exceeds its perceived marginal advantages in the light of the awareness of bounded rationality. In this case the possible persistence of systematic errors becomes fully consistent with economic logic. In addition, the process of revision of expectations does not need to be backward looking as in the traditional specification of the adaptive expectations hypotheses. The announcement of a change in the policy environment leads the bounded-rational agent to re-define instantaneously the resting points of the system complying with the criterion of satisficing. This may determine a shift of behavior similar to that occurring in the hypothesis of rational expectations that, however, is in general unlikely to eliminate the systematic errors and to stop the process of genuine learning. In addition, from the point of view of designing rationality, expectations may be pro-active in the sense that their influence on future events may be taken into account in order to realize a pro-active adaptation to the environment modified according to a rational design.
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4. Concepts of learning and their economic value In the light of the preceding analysis we may explore which is the economic value of the different concepts of learning within the different concepts of rationality. To this end a preliminary clarification of the concept of learning is needed. There are different concepts of learning in the economic literature. Their common denominator may be expressed in the following way. Let’s define Ωt as the information set at time t, t-nΩ t as the ‘information flow’ in the time spell going from t-n to t, so that: Ωt-n ∪
t-nΩt
= Ωt
(4)
For the sake of simplicity we assume, as is usual, that no loss of information is possible (for a memory failure or a breakdown in the systems of information storage, and so on). Therefore: Ωt-n ⊆ Ωt,
(5)
that is the stock of information, or information set, cannot shrink throughout time. On the basis of these premises, it is possible to say that there has been effective learning in the relevant period, whenever Ωt-n ⊂ Ωt,
(6)
that is the information set at the beginning of the period turns out to be a proper subset of the information set at the end of the period. The simplest concept of learning that complies with the general definition (6) is the updating of the information set that is obtained simply by adding the most recent values of the relevant deterministic variables, or realizations of the relevant stochastic variables, to the information set: Ωt-1 ∪
t-1Ωt
= Ωt,
t-1Ωt
≠ Ø.
(7)
We define “real-time updating” the instantaneous updating that occurs without involving any sort of time lag between the occurrence of the new events and their registration in the information set. This simplistic and demanding concept of learning is the only concept of learning consistent with substantive rationality. In fact the other concepts of learning, including noninstantaneous updating, would involve the existence of systematic errors that are excluded by definition under the assumptions of substantive rationality. A lag in updating would induce the rational agent to formulate mistaken expectations and therefore to make sub-optimal choices. This is excluded by the assumption of unbounded rationality intrinsic in substantive rationality that implies a complete set of relevant information always updated in real time. Therefore real-time updating is not only consistent with substantive rationality but also a necessary condition of it. Real-time
Rationality, learning and complexity
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updating, however, is just assumed not explained. In order to provide behavioral foundations to it, we have to consider the full dynamics of the Interaction without restricting its analysis to its equilibrium values. The new events shift, generally speaking, the equilibrium of the process. Therefore a lag in the updating of the information set would induce a pragmatic error that would lead to lower than expected wealth. There is an economic incentive, thus, to update immediately the information set by eliminating the cognitive error and the ensuing pragmatic error. The value v of updating from t-1 to t, t-1Ωt , is given by: v ( t-1Ωt ) = max u (xi|Ωt) – max u (xi|Ωt-1)
(8)
where xi ε X is an option belonging to the set of options X. This value is in general positive, unless the new information is redundant, as in the case of the average values of stationary variables. This implies the paradox that real-time updating is a necessary condition of substantive rationality but has no value since it applies legitimately only to stationary processes. Therefore substantive rationality remains without any economic foundations within its own assumptions. As we have recalled before, a possible way out has been explored since long and became hegemonic in the 1960s and early 1970s, under the name of “adaptive expectations”, just before the rational expectations revolution (Cagan, 1956). In this case, the existence of systematic errors is admitted and their correction mechanism is formalized in such a way to assume convergence, although only asymptotic, towards the substantive-rationality equilibrium. The hypothesis of adaptive expectations is patently inconsistent with substantive rationality as it describes a mechanism of correction of systematic errors the very existence of which is denied by the assumptions underlying substantive rationality. This hypothesis, however, was utilized in such a way to conceal this contradiction by assuming that the dynamics in disequilibrium (adaptive expectations learning) could not affect the equilibrium dynamics, being relevant only in the short term (business cycles) but not in the long term (growth). This compromise gave a role, though limited to the short period, to genuine learning, i.e. correction of systematic errors, without challenging the ultimate validity of substantive rationality (at least in the long period), but only at the expense of logical consistency. The systematic errors converge only asymptotically to substantive rationality equilibrium and therefore do not disappear in the long period as required by the above compromise. Eventually in the 1970s the requirement of logical consistency and analytic simplicity prevailed sweeping away the hypothesis of adaptive expectations in favor of the rational expectations hypothesis that is fully consistent with the hypothesis of substantive rationality (Muth, 1961). In the case of rational expectations, by definition, the economic agent does
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not make systematic errors, neither ex ante nor ex post. Systematic ex ante errors are excluded by the assumption that the agent makes an optimal forecast conditional to the information set. Ex post errors are by definition non-systematic, since they are neither correlated nor auto-correlated, nor do they show any bias, as is implied by the assumption that the subjective probability distribution coincides with the ‘objective’ probability distribution (see e.g. Begg, 1982). However logical consistency is obtained at the cost of giving up the analysis of genuine learning and leaving the hypothesis of rational expectations and the assumptions of substantive rationality without any economic foundations, since this sort of issues can be discussed only within the assumptions of a more general concept of rationality. The assumption of adaptive expectations itself may be rescued and properly developed within the assumptions of procedural rationality. In this case there is no reason to assume that the process of learning converges towards the substantive-rationality equilibrium. It may stop before, whenever the psychological and economic costs of learning exceed the benefits of the expected improvement. In this context, there is no logical reason of neglecting the psychological factors. The assumptions of procedural rationality allow a serious analysis of genuine learning, i.e. of the effective mechanism of correction of systematic errors, taking account of cognitive psychology and other aspects of human subjectivity. A lot of experimental work was pursued from this perspective confirming that economic agents often do not comply with the tenets of substantive rationality, but so far no general theory of learning emerged from this research. In this essay we limit ourselves to show that, only by assuming a concept of rationality more general than that of substantive rationality we may analyze the economic role of genuine learning. As we have seen learning, conceived as updating, has an economic value as it assures optimization through time. However, this very weak concept of learning, the only one consistent with substantive rationality, cannot justify any change in the intertemporal strategy of the decision maker, that may be justified only by the discovery and elimination of systematic errors. We may conclude that updating does not have any strategic value. As was hinted at before, learning may have a strategic economic value because it permits the exploitation of new information in order to substitute a more profitable strategy for the existing strategy8. In slightly more formal terms, it is possible that, in the light of the new, larger, information set induced by strategic learning, at time t+n a new strategy is discovered —————— 8
This definition of strategic learning intends to ‘capture’ its economic motivations and not to exclude further, and perhaps more important, motivations.
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19
whose expected value v(t+nst+h | Ωt+n) exceeds that of the old strategy, chosen at time t, recalculated in the light of the new information set: v(tst+h | Ω t+n) . Therefore the value of strategic learning from t to t+n, tVt+n, as assessed at time t+n, may be defined as the difference between the value of the optimal strategy at time t+n, in the light of the new enlarged information set, and the value of the optimal strategy chosen at time t and reassessed at time t+n: tVt+n = v(t+nst+h | Ωt+n) - v(tst+h | Ωt+n)
≥ 0.
(9)
Generally speaking, this value is not negative because the updated information set may offer new opportunities which were non-existent or unclear before. However, in order to calculate the net value of strategic learning, it is necessary to take into account the costs cl associated with learning (such as the cost for the acquisition of new information). Therefore, the net value of strategic learning V’, neglecting the subscripts for simplicity, may be defined in the following way: V’ = V-cl
(10)
A positive net value of strategic learning in an uncertain and open world, is a sufficient economic motivation for implementing it. However, in order to justify a change in strategy, we have to take into account also the transition costs ci (such as the transaction costs) associated with it. Generally speaking, the new optimal strategy will be implemented only when V’ > ci.
(11)
The reason for distinguishing (10) from (11) depends on the fact that, by assumption, the outcome of strategic learning has been defined as a permanent acquisition while the transition costs are contingent; therefore, when the (11) is not currently satisfied, it cannot be excluded that a fall in the transition costs will justify a change in strategy in the future. It is important to emphasize that strategic learning implies the possibility of systematic errors ex post (not necessarily ex ante, whenever the existing information is efficiently utilized). In the absence of systematic errors ex post, learning would be deprived of any strategic value and would become meaningless, at least from the economic point of view. This is the case of perfect foresight and rational expectations. So far we have considered strategic learning from the point of view of procedural rationality as adaptation to a world that is open since it is characterized by the emergence of new states which, however, does not depend by a conscious design of the agents. In this case learning is a genuine addition of knowledge on the world and its evolution, but the set of options is
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not modified by the conscious will of the decision maker who is assumed to be, so to say, option-taker. From the point of view of designing rationality the creativity of the Homo sapiens, missing in the Homo economicus of the standard approach, becomes crucial. The biologists maintain that the Homo sapiens is the only living species that adapts to the environment not only passively but also pro-actively, i.e. by modifying it. Therefore in this view the economic agent has to be considered as option-maker in the sense that she is able to add new elements to the option set based on a design of modification of the world. In this case the value of strategic learning is potentially higher since it permits the discovery of a better strategy in the light not only of a better knowledge of the world and its evolution but also of the growing extension of the option set Xt: tV’t+n =
v(t+nst+h | Xt+n U Ωt+n) - v(tst+h | XtU Ωt+n) – ci.
(12)
This relation clarifies that from the point of view of designing rationality the strategic value of learning is higher than from the point of view of procedural rationality since the Homo sapiens learns not only to adapt better to a changing world but also, for a given world, to extend the set of options in order to find a better one. This reflects, and motivates, the increasing capacity of the economic agent to innovate the technological, organizational and institutional processes in which she is involved.
5. Ontological assumptions and decision theory under uncetainty The subject, in our case the economic agent, defines herself and her objectives in relation to the perceived features of the environment, or “world”, in which she lives and happens to operate. Therefore the assumptions formulated on the relevant characteristics of the world are crucial in determining the behavior o the economic agents and should be accurately analyzed by the economists who observe, interpret and forecast it. We intend to examine these neglected aspects of economic methodology from the point of view of decision theory. We may say that the ultimate foundations of standard economic theory, that adheres to methodological individualism, rely on decision theory. This is self-evident in microeconomics that is often nothing but decision theory applied to specific economic problems of individuals, or – by extension – families or firms. When the strategic interaction between single decisionmakers is studied, the foundations are based on game theory that, however, is nothing but an extension of decision theory to this field. As for macroeconomic theory, a branch refers to a representative agent whose behavior
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21
is explained and predicted on the basis of standard microeconomic theory and thus ultimately on standard decision theory. Another branch that emphasizes the importance of disaggregation, stresses the crucial role of foundations in terms of general equilibrium theory. However general equilibrium models need in their turn microeconomic foundations. These have been suggested in terms of game theory and therefore, ultimately, of decision theory. Even the branches of macroeconomics that do not accept methodological individualism often tried to provide foundations in terms of a different variety of decision theory whose different assumptions falsify the assertions of standard theory and justify alternative theoretical assertions (a case in point is Shackle who elaborated a non-standard decision theory as foundations of a non-standard interpretation of Keynesian economics: see Shackle, 1952). Mainstream economic theory may claim solid foundations in standard decision theory. The analysis of these foundations is illuminating since the axioms of standard decision theory clarify under which conditions mainstream macroeconomic theory may be considered as well founded. We may thus assess its empirical scope and limitations. As is well known, there are basically two varieties of standard decision theory, the objectivist theory suggested by Morgenstern and Von Neumann (1944), and the subjectivist theory, often called Bayesian, suggested by Savage (1954) who built upon previous work by De Finetti9. Since these theories are axiomatized, it is possible to study in a rigorous way their theoretical and empirical scope. They are apparently very different, since the first one is based on a frequentist concept of probability while the second one on a personalist (or subjectivist) concept. As for their scope, textbooks typically claim that the objectivist theory applies when the probabilities are “known” as in the case of roulette, while the Bayesian theory when they are “unknown” as in the case of a horse race. However their axioms and ontological implications are almost identical (Vercelli, 1999). Both theories refer to a world that is familiar to the decision maker in the sense that the optimal adaptation to it has already happened (Lucas, 1981). In addition, such a world is closed in the sense that the decision maker knows the complete list of its possible states, and of her possible options, and she knows which are exactly the consequences of each of her choices for each possible state of the world. These assumptions make sense only if the world is stationary and time is unimportant so that genuine innovations and unexpected structural change are excluded. In both cases the crucial —————— 9
Anscombe and Aumann (1963) suggested a sort of synthesis of the two basic varieties. The discussion of the latter applies also to this stream of literature.
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axiom, called “axiom of independence” in the Morgenstern-von Neumann case or “sure-thing principle” in Bayesian theory, ensures intertemporal coherence and absence of systematic errors, guarantying the substantive rationality of the decision maker through time. Moreover, both theories may consider only a very weak kind of uncertainty, which may be called “soft uncertainty”, such that the beliefs of the decision maker may be expressed by a unique fully-reliable additive probability distribution. Under these assumptions it is natural to adopt a decision criterion based on the maximization, respectively, of expected utility or subjective expected utility. On the contrary, the more general concepts of rationality and learning sketched above are inconsistent with the tenets of standard decision theory. However, alternative decision theories have been suggested in recent times in order to explain non-standard behavior in an open and non-stationary world. These theories assume more general measures of uncertainty such as non-additive probabilities (Schmeidler, 1982, Gilboa, 1987), multiple probabilities (Ellsberg, 1961, Gärdenfors and Sahlin, 1982, Gilboa and Schmeidler, 1989), fuzzy measures (Zadeh, 1965; Ponsard, 1986), and so on (see Vercelli, 1999a, for a more detailed survey). They may be called theories of decision under “hard uncertainty” to stress that the beliefs of decision makers may be represented only by a non-additive probability distribution or by a plurality of additive distributions none of which may be considered as fully reliable. The non-additivity of the probability distribution reflects uncertainty aversion, i.e. the awareness that in an open and non-stationary world relevant unforeseen, and unforeseeable, contingencies may occur. Although both standard decision theories have a well developed intertemporal version, even in these versions time is substantially unimportant. The decision maker chooses in the first period an optimal intertemporal strategy contingent to future states of the world (Kreps, 1988, p.190). In order to comply with the axioms of the theory that assume intertemporal coherence, this strategy cannot be revised in the subsequent periods (Epstein and Le Breton, 1993). Therefore these theories can be applied only to a closed and stationary world. In fact the standard approach is unable, by definition, to take into account the influence that a certain choice may have on the future ‘states of nature’ (which is forbidden by the Savage definition of states of nature)10, on future uncertainty (which would imply the analysis of ‘endogenous uncertainty’, while only exogenous uncertainty is —————— 10
According to Savage, the states of nature describe the evolution of the environment and are independent of the actions of the agents.
Rationality, learning and complexity
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considered)11 and future choice sets (to analyze intertemporal flexibility preference which cannot even be defined in the standard approach)12. For the reasons mentioned above, the Von Neumann-Morgenstern and Savage theories of decision-making under uncertainty cannot be applied to a genuine process of learning, since in this case many of their crucial assumptions and conclusions become implausible. In particular the existence of systematic errors is inconsistent with the axiom of independence (or “sure-thing” in Bayesian theory). In addition, in order to violate the ‘surething principle’ and the ‘compound lottery axiom’, each of which is necessary for a rigorous use of the expected utility approach (see Machina and Schmeidler, 1992, pp.748, 756; Segal, 1987, p.177), it is sufficient to assume non-instantaneous learning in the sense that some delay may occur between the time a choice is made and the time the uncertainty is actually resolved. The intertemporal analysis of decisions under uncertainty introduces a new dimension of the utmost importance for economic analysis: the degree of irreversibility which characterizes the consequences of sequential decisions. It is irreversibility which makes uncertainty such an important issue in many fields of economics. Any kind of uncertainty, even soft uncertainty, implies unavoidable ex post errors even for the most rational decision maker. If these errors were easily remedied (promptly and at a low cost) the value of a normative theory of decision under uncertainty would be quite limited. Irreversibility implies that the consequences of an error have a much higher value. Unfortunately, while irreversibility greatly increases the practical importance of normative decision theory under uncertainty, it also prevents the use of standard theories: neither objectivist theories nor subjectivist theories may be satisfactorily applied to irreversible events. In fact it is generally agreed that the objectivist decision theories apply only to stationary processes with stable frequencies. In the case of Bayesian theory it is possible to show that the requisite of exchangeability implies stationarity. This is not the case in many fields of economics which are often characterized by irreversible structural changes. In addition, it may be proved that, whenever decision-makers believe that the economic system might be non-stationary, their behavior would become nonstationary even assuming that the exogenous environment is in fact stationary (Kurz, 1991b, p.10). For the same reason in the standard theories of decision under uncertainty the value of strategic learning, involving genuine or ‘creative’ learning, is nil. In fact, as was established before, the —————— 11 12
See Kurz, 1974. See Kreps, 1988.
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value of strategic learning is necessarily zero whenever complete irreversibility is postulated; the postulate of independence which implies dynamic coherence (Epstein and Le Breton, 1993) also implies strict irreversibility of the strategy chosen conditional upon future information. This ‘axiomatic’ irreversibility implies that strategic learning is without economic value.
6. Semantic congruence We are now in a position to summarize the results obtained so far. We have to distinguish different notions of a few key concepts (rationality, learning, expectations, uncertainty, time reversibility and a few crucial ontological assumptions on the real world) that may be semantically congruent or incongruent. In fact we have argued that there are strict semantic correspondences between different notions of the above concepts, according to a gradient of ontological and epistemic complexity. These correspondences may be summarized in a Synoptic Table. Each column lists different notions of the concept written at its top according to an order of growing complexity going down. The notions of the different concepts which stay on the same line are characterized by semantic congruence while different notions on different lines are incongruent. We wish to emphasize that semantic congruence is a neglected, but important, requisite of a sound theory or model. Its normative strength does not refer to logic: incongruence, differently from logical inconsistence, does not imply logical contradictions but semantic incompatibilities. There are two kinds of semantic incongruence. The first kind is between a crucial characteristic of a theoretical construct (concept, model or theory), and a different connotation of a certain empirical field that forbids a sound application to it of the incongruent construct. For example, the rational expectations hypothesis implies stationarity of the relevant stochastic processes and cannot be safely applied to empirical evidence that does not exhibit this property. The second kind of semantic incongruence is between concepts having different semantic scope so that they cannot be integrated in the same theory or in the same model because they cannot be applied to the same set of data. For example rational expectations and genuine learning cannot be part of the same theory or model because the first concept implies stationarity while the second implies non-stationarity. Semantic incongruence is a crucial source of misguided application of theory to the empirical evidence. Its consideration is crucial in order to define the empirical scope of concepts, theories or models. In what follows we utilize the information on semantic congruence summarized by the synoptic table
Rationality, learning and complexity
25
in order to discuss the empirical scope of different approaches that characterize economics. RATIONALITY
LEARNING
EXPECTATIONS
UNCERTAINTY
TIME
WORLD
substantive
updating
perfect
certainty
reversible
deterministic
substantive
updating
rational
soft
reversible
closed
procedural
adaptive
unconstrained
hard
irreversible
open
designing
creative
proactive
hard
irreversible
open
SYNOPTIC TABLE
Reading down the table the degree of generality increases with the degree of complexity taken into account. According to the point of view here advocated, designing rationality is the most comprehensive conception of rationality that may better take into account the complexity of the real world. Designing rationality is applicable in cases of hard uncertainty as well as in the limiting case of soft uncertainty, whatever is the degree of irreversibility. Therefore it encompasses as special case procedural rationality, when the environment is considered as given and substantive rationality under the further restrictive assumption of equilibrium. Of course, whenever the decision problem is characterized by full reversibility or irreversibility and uncertainty is soft, it is possible to use substantive rationality (the Morgenstern-von Neumann theory for roulette-wheel problems, or Bayesian theory, also for horse race problems) but their use is intelligible and justified only within the broader framework of designing rationality, structural learning, hard uncertainty and time irreversibility. It is important to emphasize that the correspondences summarized in the Synoptic Table are compelling from the conceptual and methodological point of view. For example, genuine structural learning implies a certain degree of irreversibility and non-stationarity, so that it cannot be analyzed within the traditional decision theories under (soft) uncertainty. This must be kept well in mind because, reading down in the table, the awareness of the complexity of the object analyzed increases, making more and more difficult the application of rigorous methods and sophisticated formal languages. We have to resist the temptation to which economists too often succumb of applying to complex phenomena powerful formal approaches that, however, are fit only for simple phenomena. In order to study consistently the complex phenomena that characterize the behavior of modern
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A. Vercelli
economies we have to develop, diligently but patiently, specific methods. A case in point in the analytical field are the recent advances in decision theory under hard uncertainty mentioned in this paper that offer new opportunities for a rigorous analysis of structural learning and creative rationality in an open and evolutionary environment. Another promising example is the development of simulation methods that allow a precise, formal, analysis of complex economic behavior without requiring excessive simplifications. Finally, another growing stream of literature is based on experimental methods that try to reconstruct the effective economic behavior without irrealistic normative constraints.
7. Concluding remarks In this essay we have introduced a sharp distinction between ontological complexity that refers to the properties of the economic system and epistemic complexity that refers to the formal properties of the model that represents it. On the basis of this distinction we have emphasized the limitations of the approach of standard economics that succeeds in simplifying the complexity of the economic system only at the cost of restricting its theoretical and empirical scope. If we want to face the problems raised by the irreducible complexity of the real world we are compelled to introduce an adequate level of epistemic complexity in our concepts and models. Examples of epistemic complexity are the degree of non-linearity of a dynamic system, and the degree of non-additivity of a probability distribution. Conversely, there are ontological sources of complexity that prevent simplification. To keep in touch with the examples of epistemic complexity just mentioned, it may be cited structural change as a source of nonlinearity in dynamic systems, while we can cite non-stationarity and openness of the world that we have to analyze as a source of both. In particular evolutionary phenomena, that are by definition non-stationary and open processes and are often characterized by essential non-linearities, imply irreducible complexity. Complexity has been discussed in this essay from the semantic point of view, i.e. from the point of view of the correspondence and the interaction between the degree of complexity of the objective system that we intend to describe or forecast and their subjective representation through models or expectations. The semantic approach to complexity here outlined has clear implications also from the point of view of the usual formal definitions of complexity recalled in the Introduction. In particular, focusing on the dynamic definition of complexity, it may be inferred from the preceding analysis that standard economics applies, under further limiting conditions,
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only if the degree of complexity does not exceed the level 2. In other words it may be applied only to stable linear systems having a unique equilibrium (level 1) or to dynamic systems having a regular periodic attractor, such as a limit cycle (level 2). In order to take account of higher levels of complexity, more advanced approaches based on more sophisticated concepts of rationality and learning are required. Epistemic complexity is not a virtue, but a necessity. We have to introduce in our theoretical constructs and models the minimal level of complexity sufficient to take account of ontological complexity to the extent that it is essential for understanding and controlling empirical phenomena. All the definitions of complexity have in common the emergence of properties that cannot be defined within a simpler context. If these properties are considered essential for the purposes of the analysis we cannot ignore them. If we insist to do so, as too often happens in economic analysis, we are liable to incur in systematic errors that may lead us astray. Therefore we have to introduce in economic models not less, and possibly no more, than the minimal degree of complexity required by the object. This establishes compelling correspondences between the degree of ontological complexity ascertained in the object and the degree of epistemic complexity to be attributed to the concepts utilized in its analysis. We may conclude by endorsing the advice of a famous scientist: “make things as simple as possible - but no simpler”13.
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—————— 13
The original advice is by Einstein according to Phillips, Freeman, and Wicks (2003, p.486).
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Ponsard, C., (1986): Foundations of soft decision theory, in J. Kacprzyk, and R.R. Yeger, eds., Management Decision Support Systems Using Fuzzy Sets and Possibility Theory, Verlag TUV, Rheinland, Köln. Runde, J., (1990): Keynesian uncertainty and the weight of arguments, Economics and Philosophy, 6, pp. 275-292. Runde, J., (1994): Keynesian Uncertainty and liquidity preference, Cambridge Journal of Economics, 18, pp.129-144. Samuelson, P.A., (1947): Foundations of Economic Analysis, Cambridge, Mass.: Harvard University Press Sargent, T.J., (1993): Bounded Rationality in Macroeconomics, MIT Press, Boston. Savage, L.J., (1954): The Foundations of Statistics, John Wiley and Sons, New York. Revised and enlarged edition, Dover, New York, 1972. Scarf, H., (1960): Some examples of Global Instability of Competitive Equilibrium, International Economic Review, vol.1, 3, pp.157-72 Schmeidler, D., (1982): Subjective probability without additivity, Working Paper, Foerder Institute for Economic Research, Tel Aviv University. Schmeidler, D., (1986): Integral representation without additivity, Proceedings of the American Mathematical Society, 97, 2, pp.255-261. Schmeidler, D., (1989): Subjective probability and expected utility without additivity, Econometrica, 57, pp.571-87. Schumpeter, J.A., (1934): The theory of Economic Development, Oxford, Oxford University Press. Segal, U., (1987): The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach, International Economic Review, 28(1), 175-202. Shackle, G.L.S., 1952, Expectations in Economics, Cambridge, Cambridge University Press. Simon, H.A., (1982): Models of Bounded Rationality, MIT Press, Cambridge (Mass.). Simonsen, M.H., Werlang S.R.C., 1991, Subadditive probabilities and portfolio inertia, Revista de Econometria, 11, pp.1-19. Sonnenschein, H., (1973): Do Walras’ Identity and Continuity Characterize the Class of Community Excess Demand Functions?, Journal of Economic Theory, vol.6, pp.345-354. Tobin, J., (1958): Liquidity Preference as Behaviour Toward Risk, Review of Economic Studies, 25, 65-86. Vercelli, A., (1991): Methodological foundations of macroeconomics. Keynes and Lucas, Cambridge, Cambridge University Press. Vercelli, A., (1992): Probabilistic causality and economic analysis: a survey, in A.Vercelli, and N.Dimitri, eds. 1992, Macroeconomics: A Survey of research Strategies, Oxford, Oxford University Press. Vercelli, A., (1995): From soft uncertainty to hard environmental uncertainty, Economie Appliquée, 48, pp.251-269. Vercelli, A., (1999): The recent advances in decision theory under uncertainty: a non-technical introduction, in L.Luini, ed., Uncertain Decisions: Bridging Theory and Experiments, Kluwer, Dordrecht.
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Vercelli, A., (2000): Financial Fragility and Cyclical Fluctuations, Structural Change and Economic Dynamics, 1, 139-156. Vercelli, A., (2002): Uncertainty, rationality and learning: a Keynesian perspective, in S.Dow and J.Hillard, eds., 2002, Keynes, Uncertainty and the Global Economy, vol.II, Edward Elgar, Cheltenham, Glos. (U.K.), pp.88-105. Vercelli, A., (2003): Updated Liberalism vs. Neo-liberalism: Policy Paradigms and the Structural Evolution of Western Industrial Economies after W.W. II, in Arena-Salvadori, eds., Money, Credit and the State: Essays in Honour of Augusto Graziani, Aldershot, Ashgate, 2003 Walras, L., (1874): Elément d’économie politique pure ou théorie de la richesse sociale Wakker, P., (1989): Continuous Subjective Expected Utility With Non-Additive Probabilities, Journal of Mathematical Economics, 18, pp.1-27 Willinger, M., (1990): Irréversibilité et cohérence dynamique des choix, Revue d’Economie Politique, 100, 6, pp. 808-832. Woodford, M., (1990): Learning to Believe in Sunspots, Econometrica, 58, pp.277-307. Zadeh, L.A., (1965): Fuzzy sets, Information and Control, 8, pp.338-353.
Acknowledgements The Editors are very grateful to Palgrave MacMillan Publisher for the permission to publish the paper by A. Vercelli.
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The Confused State of Complexity Economics: An Ontological Explanation E. Perona
1. Introduction1 ‘Complexity’ ideas are becoming a hot issue in economics. Originally developed within the natural sciences, complexity theory is now believed by many to be a novel and powerful framework of thought, capable of challenging the fundamental principles sustained by mainstream economics for more than a century. One of its main advocates, David Colander (2000a:31), has written that “Complexity changes everything; well, maybe not everything, but it does change quite a bit in economics”. Also, in a recent issue of a popular scientific magazine, it is suggested that complexity ideas “are beginning to map out a radical and long-overdue revision of economic theory’ (Buchanan, New Scientist, 2004:35). Indeed, the proliferation of publications, journals, summer schools, and conferences on complexity economics during the past few years can hardly pass unnoticed. Amidst such a generalized optimism, a more careful reflection on the economic literature on complexity theorizing reveals various prima facie problematic features: confusions or at least curiosities. Some of them are widely acknowledged and have been matters of discussion within the complexity community. Others are more subtle and need clarification. In this paper I will discuss, as well as seek to remedy and/or explain, some of —————— 1
I am grateful to my former supervisor, Dr Tony Lawson, for his suggestions on the original structure of this paper. Thanks also to Gustavo Marqués and several colleagues from the Department of Economics for their useful comments and to Universidad de Córdoba for providing financial support. Any errors remain mine.
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the most significant problems and confusions subjacent to complexity economics. This exercise is helpful to carry out a more accurate assessment of the potential benefits attributed to the complexity paradigm in relation to traditional economic theories.
2. Three puzzling features of complexity economics The problems with complexity economics I will focus upon here comprise three broad issues: (a) the coexistence of multiple conceptions and definitions of complexity, (b) the contradictions manifest in the writings of economists who alternate between treating complexity as a feature of the economy or as a feature of economic models, and (c) the unclear status of complexity economics as an orthodox/heterodox response to the failures in traditional theory. The first source of confusion is that there are various competing conceptions of complexity, with the range of notions apparently resistant to successful systematization. In other words, the range and types of notions of complexity appear not to have been successfully clarified or explained. This fact is admitted quite openly by researchers, not just in economics but in the sciences at large. For example, according to the statement on the website of the ‘Complexity Group’ at the Faculty of Engineering in Cambridge2. Complexity is difficult to define. Many people have tried to define it based on the characteristics of complexity in the context of their own research fields. But not a single definition seems comprehensive enough to suit all the situations where complexity exists. It seems that a general definition of complexity can only be approximated by adding different definitions. Other comments pointing out the non-existence of a unique complexity conception and the difficulty of systematizing the various existing definitions include Cowan (1997:56), Dent (1999:5), Edmonds (1999:210), Fontana and Ballati (1999:16), Heylighen (1999:17,19), Kilpatrick (2001:17) and Lissack (1999:12). Specifically within economics, Brian Arthur et al. have echoed the general trend. These authors affirm in their introduction to the proceedings of the conference held at the Santa Fe Institute, The economy as an evolving complex system II: —————— 2
http://www-mmd.eng.cam.ac.uk/complexity/complexity.htm
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But just what is the complexity perspective in economics? That is not an easy question to answer... Indeed, the authors of the essays in this volume by no means share a single, coherent vision of the meaning and significance of complexity in economics. What we will find instead is a family resemblance, based upon a set of interrelated themes that together constitute the current meaning of the complexity perspective in economics (1997:2). Is it possible to explain the coexistence of a variety of complexity definitions? Can a commonly accepted conception of complexity (in general or at least in economics) be found at all? These questions, which transcend disciplinary boundaries, are central to the vision of complexity I will be defending. A second notable feature of complexity theorizing in economics is that many authors provide what seems to be an inconsistent account of the nature of their project. In particular most economists suppose that complexity exists as a feature of economic reality; for example Wible comments, “Economic phenomena are complex social phenomena. In today’s intellectual milieu, this is hardly a novel statement” (2000:15). At the same time, however, many among them affirm and/or believe that such a feature is, or act as if it were, irrelevant for the sort of complexity analysis they support: “[The complexity and simplicity visions]... have little to do with how complex one believes the economy is; all agree that the economy is very complex” (Colander 2000b:2). In other words, many researchers take for granted that the economy, or some aspects of it, are complex (a contentious matter in itself), whilst in their opinion the complexity project is independent of how complex one believes the economy really is. At first sight, this looks like a paradox indeed. For if their ideas of complexity are not connected to the (admittedly) complex nature of the social realm, i.e. if complexity economics is not about the complex economy, what is it about? An analysis of this matter, focusing in particular on the connection between models and reality, is useful to expose the extent to which complexologists are prone to methodological inconsistencies. A third rather remarkable feature of modern complexity theorizing in economics is that, unlike other methodological conceptions, this one seems to be embraced by mainstream and heterodox economists alike. For instance, from an evolutionary perspective, Potts writes: My sincere hope is that the approach developed herein, and the strength of the platform it expresses, might then further encourage students and researchers to make stronger connections between a unified heterodox economics - an evolution-
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ary microeconomics - and the burgeoning research fields of complex systems theory (2000:xii). Austrian economists also seem to be pleased by the prospects of complexity theorizing: Economists should recognize that complexity theory is not an ‘import’ from the natural sciences. Its two fundamental principles of self-organization and evolution can be traced to seventeenth and eighteenth-century philosophers of society. The view of the economy as a complex adaptive system is entirely congruent with the methodological approach of Menger and the Austrian school (Simpson 2000:211). This is surprising in that currently the distinct nature of mainstream and heterodox traditions is highlighted by emphasizing their irreconcilable methodological differences (Lawson 2003b). Does the rise of complexity theory thus mean that we need to theorize the mainstream/heterodox projects all over again? Or is it the case that the different traditions mean something different by the idea of complexity? In the following sections I will outline one way of rendering intelligible the phenomena expressed in these observations. That is, I will endeavor to advance and defend an interpretation of what is going on that can account for the: • various competing conceptions of complexity; • recurrent incoherences that arise in complexity theorizing; • apparent agreement between otherwise opposed mainstream and heterodox traditions that complexity is an acceptable and indeed seemingly potentially fruitful category of analysis.
3. Explanatory background: Lawson’s ontic/theoretic distinction My explanation for the phenomena noticed above follows from a broad thesis about the nature of modern economics advanced by Lawson (2003a, 1997). Lawson starts from a premise about the object of study (i.e. the economy), asserting that economic systems are typically open and structured systems. Then he adheres to the observation - also subscribed to by
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many others - that economics, as a discipline, is not in a healthy state3. He concludes that the main reason for such a failure is that the methods largely employed by the dominant tradition in economics - essentially mathematical-deductive modeling - entail ontological presuppositions that are at odds with the actual nature of the object of study. In short, mathematical models are appropriate for dealing with a reality consisting of closed and atomistic systems rather than open and structured ones. To put it differently, the problem can be phrased in terms of a conflict between the nature of economic reality and the methods employed to apprehend it. This idea is advanced by Lawson in the form of an ontic/theoretic distinction: If I can use the term theoretic to denote the quality of being a feature of a model and the term ontic to denote the quality of being features of the world the economists presumes to illuminate, a more succinct way of describing the problem that arises through the prioritization of the modeling orientation is a conflation of the theoretic and ontic, with the latter reduced to the former (2003c:4). As Lawson argues, the limited power of formalistic methods to illuminate social reality, i.e. the lack of fit of the former to the latter, necessarily results in mainstream economists inventing ‘a reality’ of a form that their modeling methods can address - for example, a world of isolated atomistic individuals possessing perfect foresight, rational expectations, pure greed, and so on, rather than real-world categories whose meaning is driven by their more usual historical or intuitive understandings. That is, the ontic dimension is determined by, and hence reduced to, the theoretic dimension, and this happens in ways that are often unappreciated, if ultimately explicable. Any viable alternative should then be one that takes into account the open and structured nature of the real economy. In Lawson’s opinion, this is the case of heterodox approaches in economics. He contends, however, that such a commitment to ontology goes to a large extent unnoticed by heterodox practitioners (2003b), who have tended to take the mainstream constructs at face value, and thereby to counterpoise alternative conceptions at the same level, mostly failing to appreciate that in the two sides to the discussion, the respective proponents are talking about entirely different worlds. —————— 3
According to Lawson (2003a, 2003c), economics: (a) achieves few explanatory or predictive successes, (b) is plagued by theory-practice inconsistencies, (c) relies on constructs recognized as quite fictitious, and (d) generally lacks direction.
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Therefore, in addition to the mainstream not fully realizing their method-driven orientation together with the (restrictive) ontological presuppositions entailed by deductivist mathematical modeling, heterodoxy has failed to fully acknowledge their ontological foundations. In Lawson’s view, it is only through the recognition of the ontic/theoretic divide that the fundamentally distinct nature of the orthodox and heterodox projects can be appreciated. In the following sections I will make use of the ontic/theoretic distinction in a rather different fashion. Essentially I will be arguing that such a dichotomy is also fundamental to help to explain the conflict that appears when some categories or system properties are, inadvertently, dealt with at both levels. And I will show that complexity is indeed a category that suffers such a fate4. In particular, my claim is that complexity admits two different, not necessarily related, meanings: first, an ontic meaning with complexity being a property of real-systems, and second, a theoretic meaning with complexity being a property of model-systems. When these two distinct aspects or dimensions of complexity fail to be recognized by its practitioners, as is often the case in economics, the problematic features noted at the beginning of this essay easily follow. In other words, I will be suggesting that it is only through sustaining the ontic/theoretic distinction that we can ultimately comprehensively explain the conundrums surrounding complexity ideas, that is: (a) the variety of complexity notions on offer, (b) the number of methodological inconsistencies that arise, and (c) the apparent agreement between the two implacably opposed traditions of mainstream and heterodox economics. Let me now defend this claim in detail.
4. The ontic/theoretic distinction in the context of complexity analysis 4.1 Various competing conceptions of complexity The first puzzling feature noticed above is the existence of a variety of notions of complexity, with most supporters of the complexity project explic—————— 4
Further examples of interest, but not considered here, are those of equilibrium (Lawson 2003c) or the econometric idea of a data generation process or DGP (Pratten 2005).
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itly acknowledging the lack of agreement about what complexity actually means. I would like to suggest that a plausible way of making sense of the confusion is to observe that some conceptions of complexity found in the literature (the majority in fact) are theoretic notions, whilst others are ontic. Moreover, this difference in the nature of the competing conceptions goes largely unnoticed. This I argue is indeed a fundamental distinction to draw in any attempt to systematize in a meaningful way the various conceptions of complexity. Most if not all complexity definitions (see below) agree with viewing complexity as a property of systems, i.e. a set of elements/aspects/parts that interact giving rise to wholes with their own system-level properties. There is often no indication, however, as to whether the system to which the property of complexity is attributed is the object of study itself (a realsystem), or whether it is a representation of the object of study (a modelsystem)5. Complexity would be an ontic feature in the former case, and a theoretic feature in the latter. Let me now review some of the most common conceptions to establish their ontic/theoretic character. A first group of complexity notions is given by those basic (dictionarylike) definitions, which emphasize the systemic nature of complex objects but are otherwise ambiguous in relation to their ontic/theoretic status. For example6: Dict. 1) Complexity is the state of being marked by an involvement of many parts, aspects, details,and notions, and necessitating earnest study or examination to understand or cope with. Dict. 2) Complexity has many varied interrelated parts, patterns, or elements and consequently is hard to understand fully. Other notions are more precise than the previous general ones. Some of them take as central the idea that complexity is fundamentally related to certain mathematical methods, especially nonlinear dynamic analysis —————— 5
A detailed defense of complexity as a property of both real-systems and modelsystems (with a different meaning in each case) has been made elsewhere (Perona 2004). There it is claimed that together with the systemic form, another key characteristic of complex systems is the existence of unknowable aspects within the system. In the case of ontic complexity, these unknowable aspects or ‘difficulties of understanding’ refer to actual human cognitive limitations, whereas in the case of theoretic complexity such difficulties are merely a consequence of model design (e.g. a model involving nonlinearities). 6 Reproduced in the website of the Complexity Group at the Faculty of Engineering (Cambridge University): http://www-mmd.eng.cam.ac.uk/complexity/complexity.htm
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(among other features, such as self-organization, adaptability, and chaos). The following definitions provide good examples: Nld. 1)... the study of complexity concerns those systems that allegedly cannot be understood when using a linear reductionist approach (McIntyre 1998:27). Nld. 2) Complexity is a broad term for describing and understanding a wide range of chaotic, dissipative, adaptive, non-linear and complex systems and phenomena7. Nld. 3) [Complexity]... includes such ideas as phase changes, fitness landscapes, self-organization, emergence, attractors, symmetry and symmetry breaking, chaos, quanta, the edge of chaos, self-organized criticality, generative relationships, and increasing returns to scale (Lissack 1999:112). It is apparent that these (mathematically-oriented) definitions convey theoretic conceptions of complexity, for nonlinearity and its potential consequences are features of (systems of) equations, a particular type of reality representation or model. The notion of complexity as a property associated with nonlinear models is widely accepted among complexologists; proof of this is that the leading institutions for the advancement of complexity adhere to this type of notion8. A third group of complexity definitions or conceptions is given by the numerous ‘measures of complexity’ that abound in the literature (see e.g. Simon 1976; Edmonds 1999: Appendix 1; Lloyd 1999). These measures are also theoretic in nature, regarding complexity as quantifiable aspects of systems - for example the system’s size, its degree of variability, or the effort involved in solving a problem or performing a certain task. Although the definitions/measures above clearly regard complexity as a property of representations of reality (hence being theoretic), most commentators do not spend time arguing that this is the case. An exception is Edmonds, whose efforts to systematize complexity ideas led him to reflect upon the ontic/theoretic dichotomy. In his opinion, although complexity may be seen as a property of real systems, it is not a useful notion as such. The problem, Edmonds argues, is that the complexity of real systems is
—————— 7
This notion can be found under the title ‘What is Complexity?’, defining the position of the interdisciplinary Center for Complexity Research at the University of Liverpool: http://www.liv.ac.uk/ccr/ 8 For example, the Santa Fe Institute (www.santafe.edu) or (http://necsi.org/) Necsi.
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unmanageably large9; in consequence, he strives to develop what he sees as a general notion of model (theoretic) complexity: Complexity is that property of a model which makes it difficult to formulate its overall behavior in a given language, even when given reasonably complete information about its atomic components and their inter-relations (1999:72; emphasis added). Now with most complexity notions being (deliberately or not) theoretic in nature, the confusion arising from the plethora of competing conceptions currently available can be easily explained. For where complexity is merely a model concept, or more precisely, a property of a system of equations, there can clearly be as many definitions of complexity as there are possibilities for model-system construction. And scope for the latter seems limitless. Such a generalized emphasis on theoretic complexity can also be observed within economics. In effect, the dominant strand in complexity economics corresponds to the vision advanced by the Santa Fe Institute. According to this general vision, the economy is seen as an adaptive system, consisting of many agents who interact continuously among themselves, whilst being immersed in an overall ‘evolutionary’ environment (Arthur et al. 1997). ‘Agents’ interactions are usually described by means of (predetermined nonlinear) rules, which are repeated through several steps, thus accounting for the system’s dynamic nature. This is clearly a theoretic perspective, where complexity is a property attributed to models, i.e. simulation models. Within this broad theoretic framework, several approaches to complexity economics can and do coexist. To repeat, when theoretic notions of complexity predominate, that is, when complexity is understood as a property of reality representations, the existence of multiple complexity conceptions poses no problems or contradictions. Since modeling possibilities are unlimited, so are the potential complexity definitions and measures. This seemingly problematic feature has, after all, a simple explanation. What about ontic complexity, i.e. complexity as a property of real systems or phenomena? Does a unique or universal notion make any sense in this case? In comparison to theoretic complexity, there are not many examples of ontological elaboration regarding complexity analysis. Within —————— 9
I do not claim that this position is correct; In fact I disagree with Edmonds’s defense of a theoretic approach to complexity. It is, however, a position worth remarking because, as I will discuss later, many economists believe that model complexity is to a large extent independent of the complexity of the real economy.
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the natural sciences, researchers usually admit that the complexity of models reflects (to a greater or lesser extent) the complexity found in reality itself, due to the fact that models are typically constructed taking into account the underlying reality in the first instance. The situation is more complicated in the case of the social sciences. Not only are social systems the result of an unaccountable number of facts that converge to produce a state of the system10, but also they often involve elements that are beyond the researcher’s understanding capabilities (one such element is the subjective nature of human beings). In the social realm there is no easy association between reality and its representations, and consequently the ontic/theoretic divide becomes a fundamental issue in this case. Among the few attempts acknowledging an ontic dimension of complexity within economics and the social sciences, we can mention the work by Schenk (2003) and Amin and Hausner (1997), from an institutional perspective: Ont. 1) Complexity of economic interactions will be taken head-on in a concept for composite economic systems - their structure, and, in turn, the impact of structural patterns on behavior and economic performance. It will be the variety of coordinating interactions which is understood here as a definition for complexity... (Schenk 2003:2-3; his emphasis). Ont. 2) The complex economy, consequently, is a microcosm of criss-crossing organizational and institutional forms, logics and rationalities, norms and governance structures. It is difficult to grasp in anything like its entirety by individuals and it escapes the reach of a central organization... (Amin and Hausner 1997:6). Also Potts (2000) engages in ontological elaboration, this time from an evolutionary point of view. His position emphasizes the systemic nature of economic phenomena, i.e. economic systems consisting of elements which are interrelated by means of multiple connections giving rise to networks. Essentially Potts sustains what he calls an ‘ontology of connections’, followed by certain epistemological considerations that lead him to conclude that systems theory (or graph theory) might be appropriate to deal with complexity at the representational level. —————— 10
As early as 1967 this problem was pointed out by Hayek in his Theory of complex phenomena: “The chief difficulty in [those disciplines concerned with complex phenomena]... becomes one of in fact ascertaining all the data determining a particular manifestation of the phenomenon in question, a difficulty which is often insurmountable in practice and sometimes even an absolute one” (1967:27).
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Finally, Delorme (2001, 1997) focuses on the problem of understanding, characteristic of complex events, which necessarily arises between the (complex) object and the subject of knowledge. For Delorme, “complexity is a property of both the world and the process of inquiry into the world” (2001:81), where the former “originates in the properties of the subject matter that the analytical capacities of the scientist cannot reach” (1997:36). A discussion and criticism of these visions, together with an attempt to derive a generalized conception of ontic complexity which takes into account both the systemic form and the limitations in knowledge typical of complex systems, has been carried out elsewhere (Perona 2004). My point here is to illustrate the distinct nature of ontic vs. theoretic conceptions of complexity: whilst the former focus on the difficulty in understanding real systems, the latter center on the representational level, understanding complexity as the complexity of models and, more precisely, mathematical models. In this case models are seen as complex because of, for example, the nonlinearities involved in their construction, which make prediction difficult. To summarize, I have been arguing that when the theoretic vision predominates among complexity supporters - which is clearly the case in economics and the sciences at large - the existence of many different (and sometimes incompatible) complexity notions, definitions and/or measures, should not be an unexpected result. The reason why complexologists regard it as a problem - in contrast to, for instance, philosophers of science (for whom this problematic feature is not problematic at all) - is their general lack of concern with methodological issues. In particular, were complexity researchers aware of the ontic/theoretic distinction, they would not be preoccupied with the multiplicity of definitions on offer. The mere existence of such a preoccupation reveals that at least some of them tend to confuse and/or to conflate complexity as a property of models with complexity as a property of the real world. 4.2 Some recurrent incoherences of complexity theorizing in economics Whilst the problem of several coexisting definitions belongs to complexity analysis in general, the second problematic feature, namely the recurrent incoherences that arise in complexity theorizing, is specific to economics. These incoherences start with the fact that most economists take for granted that complexity is a feature of economic reality; i.e. they accept, as
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a matter of common sense, that economic phenomena are complex phenomena11. However, it has been pointed out earlier that most complexity conceptions in economics are, in effect, theoretic accounts of complexity. In particular, the so-called ‘complexity theory’ project may be regarded as an attempt to extend traditional economics by means of incorporating more flexible modeling features such as agents’ interactions, increasing returns to scale, non-equilibrium outcomes, emergence of behavioral patterns, and so forth. Now what is the connection between these theoretic complexity conceptions and the allegedly complex nature of economic phenomena explicitly accepted? Actually the majority of contributions seem to imply that there is none; in other words, it does not matter “how complex one believes the economy is”. Papers that portray nonlinear models constructed using ad-hoc rules lacking empirical foundations or any other type of ‘real’ connection to the object or phenomena being represented do indeed support this sort of position12. Complexity is, from the point of view of most contributors, a property of models totally independent of the complex nature of the actual economy. This is clearly an instrumentalist methodology whose main drawbacks (especially in the case of economics and the social sciences) have been pointed out by Lawson (2001). Can economists assert the complexity of the real economy on the one hand, and complexity as a modeling feature on the other, as two distinct and unrelated concepts? Can such a separation between models and reality (which presupposes an instrumentalist stance) be sustainable at all? In spite of the criticisms cast upon instrumentalism, there are no a priori reasons to rule it out as a valid methodology so long as its advocates stick to it consistently. What I want to suggest is that, when it comes to complexity, some economists maintain instrumentalist positions both unwarily and inconsistently. In effect, instrumentalist methodologies presuppose the possibility of prediction. Quantitative prediction is, however, ruled out in the case of complexity modeling, whose outcome are usually ‘likely scenarios’ or ‘patterns’ associated with specific rules and particular —————— 11
See the quotations by Wibble and Colander in Section 2. Edmonds (2001) suggests that this situation is generalized in the case of complexity modeling: “What I am criticizing is the use of such algorithms without either any justification of their appropriateness or modification to make them appropriate. Thus many agent-based models fail to escape the problems of more traditional models. They attempt to use some ensemble of interacting agents to reproduce some global outcome without knowing if the behavior of the individual agents is at all realistic. The wish for the ‘magic’ short-cut is still there”.
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values of initial conditions. Faced with this dilemma, some authors (e.g. Day 1999:354-355) have stressed the importance of qualitative prediction, namely the possibility of forecasting ‘general tendencies’ rather than specific values. In any case it is hard to see how theory assessment could be seriously carried out in those circumstances. The only plausible justification for the employment of such methods is, then, for complexologists to posit that the problem setting, including the rules and parameter values chosen, has been selected bearing in mind, even if in an imperfect fashion, realistic properties in the first instance. To put it differently, rules have been chosen to be reasonable and to ‘make sense’. But this clearly contradicts the assumed independence of models from reality. In sum, this type of methodological incoherence, which I shall call an ‘instrumental inconsistency’, is displayed by researchers who: (a) believe that the economy is complex in a fundamental way and (b) affirm that and/or act as if models are independent from reality, whilst they (c) defend complexity theory methods on the grounds that they are better representations of reality than traditional theory provides. There are other views on the relationship between reality and representations (i.e. the economy and economic models) that coexist with the previous one, contributing to aggravate the general state of ontic/theoretic incoherence. On occasion, contributions to complexity economics seem to imply that it might be possible to describe reality in its full complexity (this is the case, for example, with some econometric works attempting to retrieve nonlinear data generation processes from seemingly chaotic series). To suppose that reality is ultimately explainable is, however, inconsistent with the purported belief that economic phenomena are inherently complex. In addition, a number of works on complexity economics reveal that their authors do often tend to use the same terminology to speak indiscriminately of the complexity of models, and the complexity of the real economy, thus showing an erroneous choice to conflate theoretic and ontic features in the analysis. This is noticeable in the frequent use of expressions such as the ‘inherent nonlinearity of social systems’ (e.g. Elliot and Kiel 1998) or the ‘occurrence of chaos within the [human] realm’ (e.g. Day 1999:Chap.26). My explanatory thesis for the second problematic feature affecting complexity economics is, then, that the virtual absence of methodological elaboration in modern economics - in particular regarding the ontic/theoretic divide, or the connection between reality and models - leads economists to adopt positions that are ultimately inconsistent, both within themselves and also relative to each other. Sometimes researchers act as if
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methods could be improved to the point of uncovering the ‘true model’, although this is inconsistent with asserting the existence of complexity as a feature of reality, something that, in general, they do. Alternatively, they maintain (explicitly or implicitly) an instrumentalist position which is not sustainable due to the typical (nonlinear) nature of complexity methods. In addition, they frequently confuse, at a semantic level, the complexity of models with the complexity of the real economy13. In short, complexity theory - as advocated by most leading institutions and practised by most economists - is a theoretic project, an attempt to extend traditional economics to incorporate more flexible modeling features. But it is one thing to suppose that models and reality are the same thing, quite another to hold that reality is something different from models but ultimately attainable, and still another that they are different and independent from each other. Economists supporting complexity ideas often tend to alternate, without realizing it, between these positions. 4.3 Apparent agreement between mainstream and heterodox traditions The third problem noted at the outset is the apparent agreement between mainstream and heterodox economists about the relevance of a complexity notion. In addition to the ‘complexity revolution’ taking place in science, the growing interest in complexity within economics seems to be caused by an increasing discontent with the performance of traditional (neoclassical) theories, whose problems are well known and extensively highlighted in the literature14. In effect, many economists - ranging from the more conservative to the more heterodox - claim to have found ideas in complexity —————— 13
A thesis by Glenn Fox asserting the coexistence of several methodologies within economics supports my argument: “Philosophically, economics is a house divided. [Economists]... have combined elements from various methodological doctrines in an eclectic manner. But these hybrid methodologies are not coherent. Methodological discussion among economists is made more difficult by the variations in meaning attached to certain fundamental terms” (1997:122). 14 Depending on the author’s methodological position, the most common criticisms of traditional economic theory include the construction of unrealistic assumptions, the neglect of institutions, the lack of behavioral foundations, the systematic failure to obtain accurate predictions, the difficulties in carrying out empirical tests, and the inadequate treatment of aspects such as time, evolution, uncertainty, and the formation of expectations.
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that provide a way out of the trap in which much of today’s economics seems to have been caught. Researchers coming from a more mainstream orientation have centered primarily on the advancement of nonlinear dynamic modeling, in an attempt to provide a formal description of the non-equilibrium and evolutionary nature of the economy as well as the unpredictability of aggregate outcomes. This is the theoretic vision outlined in previous sections. Concepts such as self-organization and emergence, technological lock-in and path dependence, increasing returns to scale, and interaction among learning agents are frequently emphasized in this literature (see, for example, the classic papers in Arthur et al. 1997, or the more recent volumes on Complexity in Economics by Rosser 2004). From a non-mainstream point of view, representatives of several heterodox schools including the (new) Austrian, institutionalist, and evolutionary approaches, are also, to different degrees, supporting the complexity paradigm15. Even some critical realists have become strong advocates of complexity theory, viewing it as a ‘scientific ontology’ compatible with the critical realist ‘philosophical ontology’ (Reed and Harvey 1992; Harvey and Reed 1998). Heterodox contributions on complexity do, in general, see it as promising that the new approach has been able to incorporate a number of notions hitherto neglected by traditional theory, such as evolution, self-organization, uncertainty, historical contexts, and so on. Does this mean that the two types of reaction to the neoclassical shortcomings have something in common? Is it the case that a sort of convergence is taking place between mainstream and non-mainstream schools, with complexity ideas providing the missing link between them? To be blunt, the answer is ‘no’, and I can once again use the ontic/theoretic distinction to provide a plausible explanation. Essentially, my claim is that the two groups - i.e. mainstream and heterodox supporters of complexity - are employing the category in systematically different ways, without this difference being recognized. For the mainstream, complexity is mainly a feature of their models. For the heterodox, on the other hand, complexity is a feature of the reality being studied. To put it differently, the orthodox response to the failures in traditional theory is a theoretic response: the goal of the researcher is to develop more powerful modeling frameworks, including fictional agents capable of displaying non-equilibrium, adaptive, evolutionary-like behavior. The heterodox response is in turn an ontic response: in this case, the purpose of the —————— 15
See e.g. the quotations in Section 2.
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researcher is to investigate the nature of social reality, to provide (partial and fallible) explanations of inherently unknowable (and hence complex) economic phenomena. To some extent orthodox complexologists acknowledge their theoretic orientation when they remark on the importance of formal modeling, drawing a line between their project and the one sustained by their nonmainstream colleagues16: Complexity economics differs from heterodox economics in that it is highly formal; it is a science that involves simplification and the search for efficient means of data compression. Thus, complexity economics will be more acceptable to standard economists because it shares the same focus on maintaining a formal scientific framework, and less acceptable to many heterodox economists who otherwise accept its general vision (Colander 2000b:5). In the case of heterodox advocates of complexity, it not so clear that they perceive their ideas to be fundamentally different from the ones postulated by the mainstream side. Many heterodox complexologists are not only non-skeptical about theoretic developments in complexity economics but they actually seem to welcome them17. The fact that certain economists within the heterodox traditions, largely opposed to any sort of formal modeling, appear to be interested in complexity theory at all, is indeed surprising. A crucial factor explaining such a bizarre situation has been cogently pointed out by Lawson (2003b). In his view, although currently the distinct nature of mainstream and heterodox visions is highlighted by emphasizing their methodological differences, and in particular the ontological commitment of the latter, the ontic/theoretic distinction is never explicitly formulated and/or systematized. It therefore goes largely unrecognized by both orthodox and heterodox traditions, whilst sloppy use of language serves to camouflage the fact that the two traditions are indeed concerned with significantly different endeavors: ... I do not claim that the ontological orientation of the heterodox opposition has always been, or is always, recognized. To the contrary I believe that one reason that the heterodox tradi—————— 16
Mainstream complexologists do not acknowledge the ontic orientation of heterodox approaches, though. In other words, they are not aware of the ontic/theoretic divide. 17 For instance, Lavoie believes that “Computational modeling, the use of computer programs to create artificial, complex evolutionary systems, is an approach to modeling that a Hayekian would find interesting” (1994:550).
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tions have been less effective than their case appears to warrant is precisely that the ontological nature of their opposition has not been made sufficiently clear (2003b:19). The rise of complexity within economics is not thus merely a case of two projects coming to the same response from different angles. On the one hand, the theoretic response advanced by complexity theory advocates, centered on the ‘complexification’ of models, serves to confirm the mainstream reliance on mathematical methods. On the other hand, the heterodox traditions are concerned about understanding a world of real or inherent complexity; an open, structured, and intrinsically dynamic world, comprising totalities of internally related parts. The two projects could not be more different. The failure to recognize this comes down to a failure to see that one term, ‘complexity’, is being used in quite different ways ways that are distinct from one another. In addition to the confusion arising from the lack of proper recognition of the ontic/theoretic dichotomy, it is possible to conjecture another reason explaining the recent interest of a number of heterodox researchers in complexity analysis. This reason is of a more pragmatic order and relates to the necessary bi-directionality characterizing the relationship of models to reality. If the mainstream has been often criticized by heterodoxy for lacking ontological elaboration and, more specifically, for endorsing modeling frameworks underpinned by ontological foundations at odds with the actual nature of the socio-economic realm, it is also true that the heterodox paramount concern with ontology has seldom given room to epistemological elaboration, namely to the discussion about research guidelines and/or methods consistent with the posited nature of economic phenomena18. In this context it is not unreasonable to hypothesize that the rare attempts by heterodox economists to clarify the means by which (complex) economic phenomena can be studied in practice might have induced some of them to regard complexity modeling as a representational strategy compatible with their claims at an ontological level, i.e. a complex world characterized by uncertainty, evolution, structures, institutions, and so on. This is actually what Reed and Harvey (1992, 1998) mean when they argue that complexity theory provides a ‘scientific ontology’ consistent with a critical realist ‘philosophical ontology.’ In other words, the relative lack of concern of heterodox studies with practical matters might have led some supporters of heterodoxy to see in —————— 18
This is changing, though. In recent years, heterodox economists have become increasingly concerned with epistemological issues. A good example is Downward (2003).
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complexity theory a plausible way to make heterodox claims operational. This feature, together with the lack of awareness about the ontic/theoretic divide separating the mainstream from the non-mainstream, and the resulting confusion arising from common terminology, explains the apparent interest of heterodoxy in complexity economics19. Our last problematic feature is thus rendered intelligible.
5. Concluding remarks: ontology and methods matter In this paper I have discussed and elucidated three puzzling and/or problematic features of complexity analysis in economics, aided by Lawson’s insightful ontic/theoretic distinction. First, the existence of a plurality of complexity definitions makes sense when we observe that most of them are theoretic notions. Second, the methodological inconsistencies pervading complexity economics are brought to light when the tacit connections between models and reality are investigated. Finally, the apparent agreement between heterodox and orthodox traditions about complexity ideas is shown to be fictitious when considered from the ontic/theoretic perspective: the two sides represent in fact quite different and even opposite responses to the problems perceived in traditional theory. Complexity is thus a category that illustrates quite forcefully the necessity of ontological elaboration in economic theory. The real question, the question of social import, and the one that is ultimately relevant for economic policy, is how can we understand a world of complex, evolving socio-economic structures. And this question can only start with a debate of what is to be identified as complex (i.e. what complexity actually means), followed by a discussion on how to deal with such complex objects (i.e. a discussion about methods). This is precisely the ontic/theoretic dichotomy I have been defending. The ontological implications of the various complexity projects in economics, and in particular the strand characterized by a quantitative orientation, are rarely explored, though. Mainstream complexity theory, including the development of more flexible and comprehensive mathematical models and the reliance on simulation methods, may bring about important insights for both theoretical and applied economics. It is often the case, how—————— 19
There are also heterodox economists who reject complexity ideas at once, arguing that complexity is merely a theoretic project revolving around simulation modeling. These researchers do not acknowledge the possibility of an ontic dimension of complexity.
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ever, that these more flexible modeling strategies are not the result of inferential processes trying to uncover the mechanisms in operation in actual economic systems, but simply more elaborate deductive constructions. A turn to ontology may be of help here because, as Leijonhufvud has warned us, “If all one is doing is playing with the equations that specify the interactions among pixels, the result, obviously, is just another Nintendo game” (1995:1500). In short, ontological elaboration, the question of methods, and the link between the two, are inevitable issues in the complexity debate, if the new complexity economics vision intends to be fruitful.
References Amin A., Hausner J. (1997): Beyond market and hierarchy. Interactive governance and social complexity. Edward Elgar, Cheltenham Arthur B., Durlauf S.,Lane D. (1997): The economy as an evolving complex system II. Perseus Books, Reading Buchanan M. (2004): It’s the economy, stupid. New Scientist 182:34-37 Colander D. (2000a): Complexity and the history of economic thought. Routledge, London Colander D. (2000b): The complexity vision and the teaching of economics. Edward Elgar, Cheltenham Northampton Cowan G. (1997): Complexity? Who ordered that? Complexity 2:56-57 Delorme R. (1997): The foundational bearing of complexity. In: Amin A., Hausner J. (eds):, Beyond market and hierarchy: Interactive governance and social complexity. Edward Elgar, Cheltenham, pp 32-56 Delorme R. (2001): Theorizing complexity. In: Foster J., Metcalfe J. S. (eds) Frontiers of evolutionary economics. Competition, self-organization and innovation policy. Edward Elgar, Cheltenham Northampton, pp 80-108 Dent E. B .1999): Complexity science: a worldview shift. Emergence 1:5-19 Downward P. (2003): Applied economics and the critical realist critique. Routledge, London New York Edmonds B. (1999): Syntactic measures of complexity. Ph.D. thesis, University of Manchester (http://bruce.edmonds.name) Edmonds B. (2001): Against: a priori theory. For: descriptively adequate computational modeling. Post-Autistic Economics Review 10:article 2 (http://www.paecon.net) Elliott E., Kiel L. D. (1998): Introduction. In: Kiel L. D., Elliott E. (eds) Chaos theory in the social sciences. Foundations and applications. The University of Michigan Press, Ann Arbor, pp 1-15 Fontana W., Ballati S. (1999): Complexity. Why the sudden fuss? Complexity 4:14-16 Fox G. (1997): Reason and reality in the methodologies of economics. Edward Elgar, Cheltenham Lyme
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Harvey D. L., Reed M. (1998): Social science as the study of complex systems. In: Kiel L. D., Elliott E. (eds): Chaos theory in the social sciences. Foundations and applications. The University of Michigan Press, Ann Arbor, pp 295-323 Hayek F. A. (1967): The theory of complex phenomena. In: Hayek FA, Studies in philosophy, politics and economics. Routledge and Kegan Paul, London Heylighen F. (1999): The growth of structural and functional complexity during evolution. In: Heylighen F., Bollen J., Riegler A. (eds) The evolution of complexity. The violet book of “Einstein meets Magrite”, VUB University Press and Kluwer Academic Publishers, Brussels Dordrecht, pp 17-44 Kilpatrick H. E. (2001): Complexity, spontaneous order, and Friedrich Hayek: are spontaneous order and complexity essentially the same thing? Complexity 6:16-20 Lavoie D. (1994): Austrian models? Possibilities of evolutionary computation. In: Boettke P. J. (ed) The Elgar companion to Austrian Economics. Edward Elgar, Cheltenham, pp 549-555 Lawson T. (2001): Two responses to the failings of modern economics: the instrumentalist and the realist. Review of Population and Social Policy 10:155181 Lawson T. (2003a): Reorienting economics. Routledge, London Lawson T. (2003b): Heterodox economics: what is it? (Paper presented at the Annual Meeting of the Association for Heterodox Economics, Nottingham Trent University-UK) Lawson T. (2003c): The (confused) state of equilibrium analysis in modern economics: an (ontological) explanation. (Paper presented at the London History and Philosophy of Science Meetings, LSE-CPNSS, London-UK) Leijonhufvud A. (1995): Adaptive behavior, market processes and the computable approach. Revue Economique 46:1497-1510 Lissack M. (1999): Complexity: the science, its vocabulary, and its relation to organizations. Emergence 1:110-126 Lloyd S. (1999): Complexity: plain and simple. Complexity 4:72 McIntyre L. (1998): Complexity: a philosopher’s reflections. Complexity 3:26-32 Perona E. (2004): Conceptualising complexity in economic analysis: a philosophical, including ontological, study. Ph.D. thesis, University of Cambridge Potts J. (2000): The new evolutionary microeconomics. Complexity, competence and adaptive behavior. Edward Elgar, Cheltenham Northampton Pratten S. (2005): Economics as progress: the LSE approach to econometric modeling and Critical Realism as programmes for research. Cambridge Journal of Economics 29:179-205) Reed M., Harvey D. L. (1992): The new science and the old: complexity and realism in the social sciences. Journal for the Theory of Social Behavior 22:353380 Rosser J. B. (2004): Complexity in economics. Edward Elgar, Cheltenham Northampton Schenk K. E. (2003): Economic institutions and complexity. Structures, interactions and emergent properties. Edward Elgar, Cheltenham Northampton
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Simon H. (1976): How complex are complex systems? PSA: Proceedings of the Biennal Meeting of the Philosophy of Science Association, Vol 1976, Issue Two: Symposia and Invited Papers, The University of Chicago Press, pp 507522 Simpson D. (2000): Rethinking economic behavior. How the economy really works. Macmillan, Basingstoke Wible J. (2000): What is complexity? In: Colander D. (ed) Complexity and the history of economic thought. Routledge, London, pp 15-30
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Part II
Modeling Issues I: Modeling Economic Complexity
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The Complex Problem of Modeling Economic Complexity R. H. Day
Complexity: a group of interrelated or entangled relationships. The New Shorter Oxford English Dictionary (1993)
... history... where everything is in a state of flux, of perpetual transition and combinations... 1 Jacob Burckhardt, Reflections on History (1979)
1. Some Scientific Background Theoretical science generates testable, logical (mathematical) systems of thought that explain or comprehend observations and empirical data that themselves only reflect some properties of some realm of experience. Progress began to occur rapidly when stable, repetitive patterns were discovered, such as the motion of stellar bodies that could be carefully observed and their distances, masses, and velocities measured, and such as the ratios in which various pure substances interacted to form different compound substances. At the basis of all such efforts is an elemental faculty of the brain, one that enables the mind to discriminate, to discern the differences and similarities of things and happenings; in the former case, between complex and simple motions isolated by telescopic observations, in the latter, between simple and complex interactions isolated in test tubes. In general, this fundamental faculty permits classification of the manifold complexity of indi—————— 1
Published in German posthumously in 1905. Burckhardt died in 1897. Translated by M.D. HoHinger in 1943. See Burckhardt (1979).
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vidual things and events into categories relatively small in number with measurable, quantifiable, or qualitatively discernible attributes. It was the brilliant insight of Newton to show how a quantifiable index of matter in motion, the mass of objects, could play a central role in explaining and predicting the behavior of macro physical objects - as long as one wanted only to explain their positions relative to one another and not all the things going on inside or immediately around them. So the earth, moon, sun, and other planets become mass points. There never has been a more radical simplification of experience than that! But the single number, used to characterize an infinitely complex object like the earth, is not a contradiction of experience. It is not the earth, it is of the earth, an abstract representation of a quantifiable, emergent macro property - and useful: among other things, men landed on the moon. For a long time the principal of cause and effect that is central to the kind of theories just exemplified seemed to be wholly inappropriate for understanding complex patterns and irregular trajectories of physical, biological, and cultural events. The Nineteenth Century, however, began to contribute analytical advances that greatly extended the applicability of the kind of mathematics already in use to explain regular relationships and predictable events. Among the advances, one attributed to Poincaré is the discovery that the nonlinear equations of mechanics could generate nonperiodic, random-like trajectories. It was a crucial step in the development of chaotic dynamics that isolated the conditions that imply irregular, essentially unpredictable behavior and showed how to characterize it with the tools of probability theory. These developments provided the foundation for modeling complex dynamics that arise in many fields where the nonlinear causal relationships involved are already known and derivable from standard theoretical constructs. A rather different development that had a somewhat earlier origin was the finding that the average location of independently and dynamically interacting but constrained objects could be characterized by stable probability distributions, even though their individual trajectories could not be followed. Together, chaotic dynamics and statistical mechanics provide potential templates for models of variables in any domain whose behavior is governed by nonlinear, interacting causal forces and characterized by nonperiodic, highly irregular, essentially unpredictable behavior beyond a few periods into the future. Although widely applied now, including extensively in economics, the theories mentioned so far originated in physical science. A second, more general, analytically less tractable category of dynamical concepts had long been recognized. Eighteenth Century explorers and scientists had already provided the classificatory foundations for geological and biological
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theories of transformation. Then in the next century a comprehensive theory of evolution in the biological world was set forth by Darwin and Wallace based on variation and selection2. Even broader ideas of structural transformation and evolution were emerging from speculation about ecological and archaeological observations and the study of written records and histories produced in the ancient world. Differing production technologies and forms of socio-political organization and their development from one into another could be discerned. Adam Smith, later exponents of the German Historical School, and subsequent researches in archaeology, as for example described in V. Gordon Childe’s (1951) Man Makes Himself, explained how economic life passed through stages of successively advancing development3. Ancient historians were well aware of the development of peoples and their periods of growth and decay. The widespread revival of Greek and Roman literature led inescapably to the idea of Civilization’s Rise and Fall already enshrined in masterful prose by Edward Gibbon in just the same year that Adam Smith’s own masterpiece appeared. These broad descriptive works describe and explain technological, economic, political, and cultural evolution, but for the most part lack the mathematical precision of the physical sciences. The possibility, however, that a first rate scientific mind could construct a rigorous conceptual system that clarifies and helps understand human history was produced in Carol Quigley’s (1979) Evolution of Civilizations. In any case, human experience as the object of inquiry poses a unique problem for the development of scientific theory, one that convinces many that a science of human cultural evolution is an oxymoron. In common parlance that problem is “mind over matter”.
2. Mind Over Matter: On Rationality and Equilibrium Philosophers, scientists, and mystics have argued over the issue: Can the human mind change matter or energy? Most people do not ask the question. They just live their lives, move themselves about, fabricate things with their hands, or control the use of a machine or instrument that transforms a part of the material world. Obviously, therefore, the question —————— 2
For a beautifully insightful survey into the development of physical and biological sciences, see Toulmin and Goodfield (1961, 1962). 3 For a brief survey of the German Historical School, see Schumpeter, pp. 808882.
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makes sense only if it is about the world in its ultimately elemental aspects of material and energy. If it is about human action, the mind is elemental. Thus, is it not true that all the dynamics of agriculture and industry are initiated by the force of human thought? Is it not true that humans even cause atoms to be split and can guide electrons along pre-designed and constricted pathways? Somehow, out of the matter and energy that make up a brain, emerge thoughts that live in a realm of mind that cannot be seen, that can be communicated only imperfectly, but that move men and mountains. It is the discerning, analyzing, and problem solving rational mind we are now talking about. It is this aspect of mind that brings us to economics. Formalizing their predecessors’ verbally expressed theories of economic systems and behavior, Cournot and later Walras constructed the mathematical theories of production, consumption, exchange, and competition based on an idealized disentanglement of two of the basic components of a market economy, the Consumer or Household and the Producer or Business Firm. The general mathematical conditions required to demonstrate the existence and properties of equilibrium were provided many decades later by Wald, von Neumann, Arrow, Debreu, Dorfman, Samuelson, and Solow. The first empirical implementation (still more radically simplified compared to the pure theory) was the inter-industry economics of Leontief. Iconoclasts of the present era, in our eagerness to gain attention for an explicitly dynamic, adaptive, evolutionary point of view, tirelessly point out the deficiencies of that new classical school: its dependence primarily on statics; its exclusion of common attributes of economizing behavior such as imitation, trial and error search, learning by doing, and so on. What we often forget is that this general equilibrium theory is a theory of complexity based on mind over matter, not on the random encounters of thoughtless objects, but on the choices of thinking consumers and producers. These theoretical, thinking, heterogeneous agents, an arbitrary number of them (none necessarily alike) exercise the rational capacity of the mind to set the flow of commodities in motion. The general equilibrium theorists showed how - in principle - a system of prices could coordinate the implied flow of goods and services among the arbitrarily many heterogeneous individual decision-makers. Of course, that is not good enough. We must have a theory that goes beyond the equilibrium property, that directly confronts the more fundamental problem of existence: how to model the mechanisms that mimic the dynamic process of coordination out of equilibrium. Before touching on a few contributions to the struggle to reach this objective, let me make an important point about the logical content of the
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neoclassical concept of equilibrium among heterogeneous, rational agents. The rationality is based on the same faculty of discrimination that permits scientific thought. We discriminate among things and actions and compare their desirability according to some kind of attribute or set of attributes, and order those alternatives, picking one among those highest (best or optimal) in the order. The mathematical theorist gives the set of alternatives being considered and the ordering of elements of that set symbolic representations amenable to a logical characterization and further analytical investigation. The exact conditions that permit feasible exchanges among all the individuals so that the agents are able to execute what they consider to be their best choices is the theoretical problem Walras formulated and his followers solved. Agents in this theory need not be assumed to have perfect knowledge of all possible alternatives. It need only be assumed that they recognize distinct well ordered opportunities. For this reason, in a competitive equilibrium so defined, Pareto superior choices could exist if only the agents knew more. Contrastingly, for a social welfare equilibrium total knowledge of all possible alternatives would be required. But aside from that requirement, in pure logic the theory is not about perfect knowledge or unbounded rationality! If neoclassical economics - interpreted correctly - is not about perfect rationality, what is it about? It is about perfect coordination. It identifies those conditions on the alternatives and orderings for which a price system exists - in principle - that could - if it prevailed - permit perfect decentralized coordination: agents only know their own minds - to the extent that they do know them - and among perceived feasible alternatives choose one they prefer. If they base their choice on equilibrium prices, they all get to carry out their chosen actions. Thoughtful economists understand the questions begged by this theory of perfect coordination among boundedly rational minds: Where do the perfectly coordinating prices come from? Supposing they are found, what do agents do until they appear? And, once the dust settles, wouldn’t the agents, those with any mammal’s curiosity, observe something not noticed before, and learn something from the choices made? The simplest way to deal with these questions is to call them ‘some other kind of economics’ and ignore them all together, or instead - in contradiction of the boundedly rational foundations of the theory - endow agents with equilibrium knowledge and call it ‘rational expectations’. Now there is an oxymoron! The more complicated way to deal with the problems not yet solved is to acknowledge their existence and do something about them. That brings us to consider some steps in the right direction.
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3. Steps in the Right Direction First, we should remember that the seminal founders of neoclassical economics themselves initiated steps in this direction: Cournot with his adaptive or recursive best response strategies as a model of competition in oligopolistic markets; Walras with his consumers’ and producers’ tâtonnement mechanism with prices adjusting to demand/supply disequilibrium and quantities adjusting to profit disequilibrium; Marshall, more or less analogously, with his quasi-rent theory of investment and marginal adjustment of consumer choices at their present operating point. Some readers may be surprised that these steps had already been taken well over a century ago and may wonder why no one has told you about them. All of us should wonder why these early disequilibrium theories of market dynamics are not being exposed to graduate students today. Quite distinct from the neoclassical microeconomics and interindustry messoeconomics of Leontief was the construction of a macroeconomics theory on an abstract basis as daring as Newton’s, namely Keynes’ equations of aggregate consumption, investment, government spending, demand, output and income, all represented by statistical indexes constructed from various observational data, but, like Newton’s mass, not observable themselves. At about the same time, Tinbergen constructed the first macroeconometric model of a national economy, encompassing more detail than did Keynes but still basing the statistical estimates on more or less highly aggregated variables. A decade later the Klein-Goldberger model initiated a worldwide explosion of research in macroeconometric models. This general approach came to form a separate sub-discipline just as optimal growth theory has done in more recent decades. But out-ofequilibrium macro - still being extended in substantial works - has been defamed on grounds it is ad hoc. Rather, it is a tractable approach for modeling out-of-equilibrium behavior. Modern investigators are continuing to improve on the approach. (I think here, for example, of recent studies by Flaschel, Franke, and Semmler (1997) and Chiarella, Groh, Flaschel, and Franke (2000). The frequent criticism that Keynesian macro theory does not have a micro foundation is also mistaken. Indeed, Morishima (1994) provided a quite explicit micro foundation of macro based on an elaborate extension of general equilibrium theory. At the other extreme, Simon and his then-young colleagues, March and Cyert, began a serious reconstruction of microeconomics from an adaptive out-of-equilibrium point of view. Simon’s original work and his later collaboration with March was concerned with the internal structure of decision-making and administration in large business and government organizations. Cyert and March elaborated these ideas in an explicit attempt to
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model and simulate the adaptive microeconomic processes on computers. A more formal effort to develop an adaptive theory of business behavior was that of Modigliani and Cohen who focused on the formulation of expectations and partial adjustment mechanisms. Hard on the heels of these works came two studies, each attempting to build a coherent dynamic out-of-equilibrium theory. Forgive me if I mention my own work at this point, the first applied, empirical version of which was published in 1963. It is perhaps à propos on this occasion because I gave my first lectures in Italy about it in the Spring of 1968. This recursive programming (RP) or adaptive economizing approach is a more general and explicitly behavioral restatement of Cournot and Walras’ models of recursively optimizing agents who adapt to disequilibrium signals. Agents form alternative possibilities subject to technical constraints and choose a best alternative on the basis of adaptively adjusted price expectations and updated resource constraints - and here a specific behavioral element is introduced - they choose in the neighborhood of their current operating point. This neighborhood is their zone of flexible response or ZFR. In the specific form implemented in empirical work, the ZFR expands in the direction of a successful past choice and contracts in the direction of less successful past choices. Behavior then follows a local economizing, learning-by-doing sequence of adjustments to experience as it unfolds. In all of this early work, I represented choice alternatives as processes which allow one to represent families of distinct technological activities, to follow their paths of adoption and abandonment, and to derive the implied demands for labor and other resources. These applied studies and the basic ideas on which they are based, together with some broad inferences about economic development in general, are described in a recent Cambridge University Press volume, The Divergent Dynamics of Economic Growth (Day, 2004). Sidney Winter’s (1964) dissertation, “Economic ‘Natural’ Selection and the Theory of the Firm”, in the Yale Economic Essays, advanced the Simon-March-Cyert behavioral viewpoint in an evolutionary framework built up on the real primitives of decision-making in complex organizations: rules, strategies, or routines. The fundamental selection process driving businesses in his system is the innovation, adoption, and abandonment of alternative rules and procedures or routines, the work was subsequently elaborated and developed in collaboration with Nelson, culminating in their influential book in 1984, An Evolutionary Theory of Economic Change, which marks a true milestone in the development of the discipline.
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Clearly, any selection among alternatives involving an explicit comparison according to an ordering criterion is an example of the economic principle of rationality, the simplest being a choice between two objects such as two apples, between two or more alternative production activities in my RP models, or between two routines that could be carried out to accomplish some task within a complex production or administrative process, as in Nelson and Winter. Moreover, all operational optimizing models of any complexity require algorithms for their solution, which consist of sequences of simple computational routines that involve at each step a locally improving direction of search according to an explicit criterion. In my terms they constitute recursive programming systems based on local search within prescribed zones of flexible response. In this way, we find our way back to a foundation in neoclassical theory based on rational choice. We are not developing ‘some other kind of theory’ but extending the theory of our intellectual fathers in the same direction as their own faltering steps guided them, but, hopefully, well beyond. And it is possible to derive analytical results: the condition under which trajectories exist and their dynamic properties such as their multiple phase switching (evolutionary) character and possible non-convergence, periodic, chaotic, or convergent properties. To derive such properties, one must construct models that simplify the approach with one (or at least a very small number) of variables, or one must formulate the model abstractly as the Day-Kennedy paper did (1970), or as Nelson and Winter did more concretely in their “Economic Growth as a Pure Selection Process” (Chapter 10 in their 1984 book). I also want to remind you of a very early collection of papers, Adaptive Economic Models, based on a 1974 conference at the University of Wisconsin’s Mathematics Research Center, which made significant advances in outlining the methodological underpinnings of the adaptive evolutionary approach and in exemplifying various aspects of the theory in specific theoretical and empirical models. Coincidentally, two other participants in the present volume in addition to myself contributed to that collection: Massanao Aoki, who wrote about dual (adaptive) control theory, and Alan Kirman, whose paper, “Learning by Firms About Demand Conditions”, illustrated an especially important insight: (i) what can potentially be learned depends on how you learn; and (ii) a learning procedure may converge when there is still more to be learned if only the way one is learning could be suitably reconstituted. As Alan put it, “the firm... may develop an inaccurate picture of the world which nevertheless generates exactly the information it does in fact observe. The firm may well be satisfied with
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this situation and there is little evidence to suggest that such unjustified complacency is uncommon”. (Kirman, 1975, p. 138)
4. Disequilibrium Existence Most of what has been reviewed in the previous section is concerned with how agents behave out-of-equilibrium given that they have and find feasible opportunities. Yet to be dealt with is the disequilibrium existence problem. How does an agent get along in a world it does not fully understand, when it is influenced by other agents whose actions it does not control, when one may find it difficult to ‘fit in’, and when one’s existence can be threatened by unpredictable physical or financial disasters? Indeed, how does any living thing survive the vicissitudes of its life? The answer is immediately available on every hand, just by observing the living systems around us, especially our own. The physician, Cannon, in an exceptionally readable book, The Wisdom of the Body, articulated the solution precisely in physiological terms: the stock/flow mechanisms in living beings that stores the various fats, proteins, and carbohydrates in various forms throughout the organism, retaining their quantities sufficient for all but the most extreme and prolonged demands, replenishing them at more or less regular intervals as they are drawn down. Cannon coined the term ‘homeostasis’ for these mechanisms. He also suggested explicitly that such homeostatic mechanisms operate within economic systems, and where they do not, troubles can arise that might be overcome by introducing such systems. The stocks maintained in economic homeostasis are inventories of goods in process, goods in transit, goods in warehouses and stores, and goods maintained in household cupboards, pantries, and refrigerators. Building on papers by D.H. Robertson and Eric Lundberg, Lloyd Metzler showed how, at the macroeconomic level, the stock/flow inventory mechanisms led to fluctuations, which, given certain stability properties on the marginal propensity to consume and invest, are bound within limits, just as physiological balances adjust within limits consistent with survival. The study of such stock/flow mechanisms at the micro level took place in the literature on inventory control, but as far as I know has never been incorporated into general economic interdependence theory. Metzler himself emphasized the mathematical difficulties of further complications. But progress might be made by adapting the Bourboki-Arrow-Debreu-like level of abstraction, an obvious task, it seems to me, that should be undertaken.
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But that will not be enough to solve the disequilibrium existence problem. Monetary economies run on financial assets and debts, which define intended future flows of monetary payments. The creation of such assets and debts by means of bank credit has played a crucial role in the development of market economies and in the nonperiodic incidence of prosperity and depression. The flow of income payments that the stock of debts is supposed to generate is based on expectations that may not be realized. When such failures become widespread and sufficiently grave to produce widespread bankruptcy, homeostasis is threatened. Economywide breakdowns can occur. The non-periodic incidence of alternating prosperity and recession has led to further innovation in the use of debt instruments. Hence, we have the refinancing of consumer and business debt, with central banks relaxing or restricting the requirements for credit expansion, and with ad hoc government loans and grants as saviors of last resort. My old friend, John Burr Williams, said that, “To leave money out of a book on the private enterprise market economy was like leaving blood out of a book on physiology”. It was Keynes’ most important contribution to have devised a way to introduce that financial blood into economics at the macroeconomic level. In spite of some brave efforts to overcome this lacuna at the microeconomic level and despite the flourishing development of finance, the task remains, for there is no area of economic theory so badly in need of development as the disequilibrium dynamics of financial stocks and flows. I have not mentioned in these remarks the creative faculty of mind responsible for generating the design for things that did not exist before and scenarios of action one has never previously experienced. Nor have I touched on the social aspect of mind that leads people to cohere in groups, to believe in similar thoughts, to follow a leader, and to adhere to the dictates of some kind of group mentality. But perhaps the basic problem of disequilibrium existence has to be solved first. My impression is that many of us go out of our way to avoid that task, becoming intrigued with all manner of mathematics, models, and mechanisms only peripherally related to the task of explaining how governments and central banks interact with the private household and business sectors. Of course, there is the view that market systems must inevitably selfdestruct. Believers in this point of view argue that the best course of action is to help them do so and get it over with. It is evident that experiments carried out in the last century with that intent proved unpleasant. They produced long, miserable interludes in the general process of economic evolution. I would rather continue our basic mission, that of focusing on how democratic market systems work, how market institutions and banks
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use out-of-equilibrium stock/flow intermediation and adjustment processes to maintain viability for the system of interacting households, firms, government institutions. In the process of this research we might then identify policies that improve the stability and distributional properties of the system as a whole.
References Burckhardt J., (1979): Reflections on History, Liberty Classics, Indianapolis. Cannon W., (1963): The Wisdom of the Body, W.W. Norton and Co., New York. Chiarella C., Flaschel P., Groh G., Semmler W., (2000): Disequilibrium Growth and Labor Market Dynamics, Springer-Verlag, Berlin. Childe V.G., (1951): Man Makes Himself, The New American Library of World Literature, Inc., New York. Originally published in 1931. Cyert R., March, J., (1992): A Behavioral Theory of the Firm, 2nd edition, Blackwell Publishers, Cambridge, MA. Day R., (1963): Recursive Programming and Production Response, NorthHolland Publishing Co., Amsterdam. Day R., (1994): Complex Economic Dynamics, Volume I, An Introduction to Dynamical Systems and Market Mechanisms, The MIT Press, Cambridge, MA. Day R., 2000, Complex Economic Dynamics, Volume II, An Introduction to Macroeconomic Dynamics, The MIT Press, Cambridge, MA. Day R., (2000): The Divergent Dynamics of Economic Growth, Cambridge University Press, Cambridge, England. Day R., Kennedy P., (1970): Recursive decision systems: an existence analysis, Econometrica 38, 666-681. Flaschel, P., Franke, R. and Semmler, W., (1997): Dynamic Macroeconomics: Instability, Fluctuations and Growth in Monetary Economics, The MIT Press, Cambridge, MA. Kirman A., (1975): Learning by firms about demand conditions, Day, R and Groves, T. (eds), Adaptive Economic Models, Academic Press, New York. March J., Simon H., (1958): Organizations, John Wiley and Sons, Inc., New York. Mezler L., (1941): The nature and stability of inventory cycles, Review of Economic Statistics 23, 100-129. Modigliani, F. and Cohen, K. J.: 1963, The Role of Anticipations and Plans in Economic Behavior, University of Illinois, Urbana. Morishima M., (1994): Dynamic Economic Theory, Cambridge University Press, Cambridge. Nelson, R. and Winter, S.: 1982, An Evolutionary Theory of Economic Change, Harvard University Press, Cambridge, MA. Quigley C., (1979): The Evolution of Civilization, Liberty Press, Indianapolis. Schumpeter J., (1954): History of Economic Analysis, Oxford University Press, New York.
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Simon H., (1983): Reason in Human Affairs, Stanford University Press, Stanford. Toulmin S., Goodfield J., (1961): The Fabric of the Heavens, University of Chicago Press, Chicago. Toulmin S., Goodfield J., (1962): The Architecture of Matter University of Chicago Press, Chicago. Winter S., (1964): Economic ‘natural’ selection and the theory of the firm, Yale Economic Essays, Yale University Press, New Haven.
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Visual Recurrence Analysis: Application to Economic Time series M. Faggini
1. Introduction The existing linear and non-linear techniques of time series analysis (Casdagli, 1997), long dominant within applied mathematics, the natural sciences, and economics, are inadequate when considering chaotic phenomena. In fact, on their basis, the irregular behaviour of some non-linear deterministic systems is not appreciated and when such behaviour is manifested in observations, often it is considered to be stochastic. This has driven the search for more powerful tools to detect, analyse and cope with non-stationarity and chaotic behaviour (Kantz and Schreiber 2000). Nevertheless the typical tools used to analyse chaotic data fail when there are short time series. Therefore the typical tools are unable to detect chaotic behaviour, in particular in economic time series. For this goal different tools are identified. Some of them, such as Recurrence Plots (RPs), rely on the presence of deterministic structure underlying the data. The RPs can be a useful starting point for analysing the non-stationarity sequences. It seems especially useful for cases in which there is modest data availability. RPs can be compared to classical approaches for analysing chaotic data, especially in the ability to detect bifurcation (Zbilut, Giuliani, and Webber 2000). Starting from this point of view we will apply a topological tool (Visual Recurrence Analysis) to macroeconomic data that has already been analysed with metric tools (correlation dimension, Lyapunov exponent) by Frank., Gencay, and Stengos (1988) and we will compare the different results obtained when using one method or the other.
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In the last few decades, there has been an increasing interest in nonlinear dynamic models, in all scientific fields (mathematics, chemistry, physics, and so on)1. The discovery that simple models can show complex and chaotic dynamics has also propelled some economists to be interested in those fields2. In fact, in the literature there are many examples of nonlinear economic models that exhibit chaotic dynamics. Linear models are inadequate when we deal-with such phenomena and particularly, when we want to analyse economic and financial time series3. Irregular behaviour of some non-linear deterministic systems is not appreciated and when such behaviour is manifest in observations, it is typically explained as stochastic. Almost all existing linear and non-linear techniques for analysis of time series assume some kind of stationarity, that is, in the time series there is no systematic change in mean (no trend) in variance, and strictly periodic variations have been removed. In fact, the probability approach to stationary time series requires that a non-stationary time series be turned into a stationary time series4. However, these techniques, long dominant within applied mathematics, the natural sciences, and economics, are inadequate when considering chaotic phenomena. Moreover, in order to perform a more reliable analysis it is important to consider the time series in a non-linear framework because nonlinearities provide information about the structure of time series and insights about the nature of the process governing that structure. The last point is crucial because in the absence of information about the structure of time series it is difficult to distinguish the stochastic from the chaotic process5.
—————— 1
Paragraphs 4 and 5 are parts of an article by Marisa Faggini Un approccio di teoria del Caos all’analisi delle serie storiche in Rivista di Politica Economica, n. 7-8 (luglio-agosto) 2005, Editore SIPI SpA, Roma. 2 Interest in non-linear dynamics models in economics is not new, however, and dates back to the time before economists had learned about chaos. Kaldor (1940), Hicks (1950), and Goodwin (1951) have already tried to model economic fluctuations by nonlinear deterministic business cycle models. At that time, attention was focused on regular periodic dynamic behaviour rather than on irregularity and chaos” Hommes (1995). 3 Ramsey (1989). 4 Strozzi et al. (2002). 5 Panas and Ninni (2000).
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This awareness has led to powerful new concepts and tools to detect, analyze, and control apparently random phenomena that at a deeper level could present complex dynamics6. The paper is organised as follows. In section 2 the traditional tests for chaos are described, highlighting their positive and negative features. In section 3 an alternative tool to overcome the problems of metric tools is presented. In section 4 we describe the application of visual recurrence analysis (VRA) to macroeconomic time series, highlighting the different results obtained by applying a topological tool versus a metric tool. In section 5 we report the conclusion of our work: the implications of chaos control theory as a new perspective for modeling economic phenomena showing what implications for policy implementation can be derived by using a chaotic approach to model economic systems and chaos control techniques to control them. The presence of chaos offers attractive possibilities for control strategies7 and this point seems particularly interesting for insights into economic policies. Detecting chaos in economic data is the first condition to apply a chaotic control to phenomena that generate them. In this case the goal is to select economic policies that allow a regular and efficient dynamic by choosing among different aperiodic behaviours: the attractor’s orbits, thus maintaining the natural structure of the system.
2. Tools for detecting non-linearity and chaos in economic time series The discovery of chaotic behaviour in economic models led to the search for such behaviour in data. The first work that took this tack is that of Brock (1986), Sayers (1986), Barnett e Chen (1988), Ramsey (1989), Chen (1993), LeBaron (1994). After these pioneering works, it was realized that chaos in economics had a broad range of potential applications. From forecasting movements in foreign exchange and stock markets to understanding international business cycles there was an explosion of empirical work searching for possible chaos in all types of economic and financial time series. —————— 6
“[...] term ‘complex economics dynamics’ to designate deterministic economic models whose trajectories exhibit irregular (non periodic) fluctuations or endogenous phase switching. The first properties includes chaotic trajectories [...] the second [...] change in the systems states [...] according to intrinsic rules”, Day (1992). 7 Boccaletti et al. (2000).
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The most likely candidates for non-linear dynamic structures in economics were macroeconomic time series. These series, which exhibit a large amount of structure through business cycle fluctuations, seemed like a natural place to look for unseen determinism. Chaotic dynamics are also an appealing model for financial time series. Sometimes apparently random fluctuations that can be characterized like chaotic behaviours are seen in the stock market. While the stochastic models can explain many of these fluctuations as due to external random shocks, in a chaotic system these fluctuations are internally generated as part of the deterministic process. The analysis of macroeconomic time series has not yet led to particularly encouraging results: these first studies have found little or no evidence for chaos in any economic time series. That is due to the small samples and high noise levels for most macroeconomic series, since they are usually aggregated time series coming from a system whose dynamics and measurement probes may be changing over time. The analysis of financial time series has led to results that are as a whole more interesting and reliable even if an important obstacle in applying chaos theory has been the lack of tests, such as autocorrelation functions, that can distinguish between chaotic and random behaviour8. Therefore, to examine the chaotic behaviour of time series a variety of methods have been invented. These are widely used in physics to detect chaos. However, not all of the approaches developed by physicists are applicable in economics because most of those methods require a large amount of data to ensure sufficient precision. There are, therefore, advantages and disadvantages to each of these criteria. Their importance in examining the chaotic structure lies not only in their usefulness in analysing the non-linear structure but also in their relevance and potential utility to distinguish between stochastic behaviour and deterministic chaos9. The main and more frequently used tests for chaos in economic time series are correlation dimension, Lyapunov exponent, and the BDS statistic test. The correlation dimension measures the degree of instability of a dynamic system while being interested in the long term evolution of the system. The Lyapunov exponents allow researchers to estimate the rate of separation between two initially close trajectories. Chaotic dynamics show two fundamental characteristics: a local instability and a global stability. All the trajectories diverge on the attractor (lo—————— 8 9
Sakai and Tokumaru (1980). Pasanu and Ninni (2000).
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cal instability); thus two points as close as possible will have two different trajectories. While the sensitive dependence of initial conditions is measured by means of the Lyapunov exponent, the convergence of all the trajectories towards the attractor (global stability) is measured using the dimension of attractor; that is the correlation dimension. The correlation dimension test (CD) was developed in physics by Grassberger and Procaccia, (1983). This test is based on measuring the dimension of a strange attractor. A pure stochastic process will spread all space as evolving, but the movements of a chaotic system will be restricted by an attractor. If the integer dimension for an attractor is provided, it is not clear whether the system is chaos; on the other hand, a fractal dimension means the system is chaotic because a chaotic attractor has a fractal dimension. The correlation dimension is generally used to estimate the dimension of attractor in an economy because it has the major advantage of being very easy to calculate compared to other dimensions whose estimate algorithms are very difficult. Correlation dimension can be calculated using the distances between each pair of points in the set of N number of points s (i, j ) = X i − X j .
Having defined this distance we can calculate the correlation function, C(r) that will be C ( r ) =
1 x (number of pairs (i, j) with s (i, j ) < r ) or better N2
C (r ) = lim N →∞
1 N2
N
∑θ (r − X
i
−Xj .
(1)
i, j i≠ j
Therefore, we can find Dcorr with the formula: Dcorr = lim r →0
log(C ( r ) . log(r )
Therefore, CD analysis provides necessary but not sufficient conditions for testing the presence of chaos. This test, designed for very large, clean data sets, was found to be problematical when applied to financial and economic series. Data sets with only a few hundred or even a few thousand observations may be inadequate for this procedure10. As indicated above sensitivity to the initial conditions is measured by the value of the first Lyapunov exponent. The Lyapunov exponent is based on the evolution of the variation in the course of time between two —————— 10
Ruelle (1991).
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very close points initially; it measures the rate of separation between trajectories starting from those points. This variation in the course of time diverges in an exponential way: δ xT = δx0 exp(λT ) 11. Here λ is the largest Lyapunov exponent12 and δ xT the difference between the two trajectories after T iterations and is equal to
λ = lim T →∞
1 T
T
∆x ( X 0 , t )
t =1
∆x 0
∑ ln
.
(2)
So if λ < 0 the orbit attracts to a stable fixed point or stable periodic orbit; λ = 0 the orbit is a neutral fixed point; and λ > 0 the orbit is unstable and chaotic. The positive Lyapunov exponent is generally regarded as necessary but not sufficient to the presence of chaos. Therefore, as for correlation dimension, the estimate of Lyapunov exponents requires a large number of observations. Since few economic series of a sufficiently large size are available, Lyapunov exponent estimates of economic data may not be so reliable. One of the most commonly applied tests for nonlinearity is the BDS13 test of Brock, Dechert, and Scheinkman (1996) which, using the correlation dimensions, constructed a variable that had a limiting standard normal distribution, thus enabling a test of the hypothesis of randomness of a time series. This tests the much more restrictive null hypothesis that the series is independent and identically distributed. It is not a test for chaos14. It is useful because it is well defined and easy to apply test and has power against any type of structure in a series. It is sensitive to both linear and nonlinear departures from IID and has been used most widely to examine the adequacy of fit of a variety of time series models with economic and financial data15. —————— 11
Filoll (2001). For a formal explanation of Lyapunov exponent see Wolf A., Swift J., Swinney W., and Vastano J., (1985). 13 ”Details of which may be found in Dechert (1996). Subsequent to its introduction, the BBS test has been generalised by Savit and Green (1991) and Wu, Savit, and Brock (1993) and more recently, DeLima (1998) introduced an iterative version of the BBS test” McKenzie (2001). 14 LeBaron (1994). 15 Brock and Sayers (1986); Frank and Stengos (1989); Scheinkman and Le Baron, (1989); Hsieh, (1989), Brock et al. (1991). 12
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The BDS test was developed for testing time series in the physical sciences where long, clean data sets are the norm. In economic time series, however, small noisy data sets are more common and the application of the BBS test to such data presents a number of problems. The first problem is that noise may render any dimension calculation useless16; to obtain a reliable analysis, large data sets are required. A rapid excursus among the techniques used to test the presence of chaos in time series shows that data quantity and data quality are crucial in applying them. The main obstacle in economic empirical analysis is limited data sources. In order to facilitate the testing of deterministic chaos and to improve our understanding of modern economies, it is worthwhile to develop numerical algorithms that work with moderate data sets and are robust against noise.
3. Recurrence Analysis: VRA An alternative to the technique problems discussed above could be applications of topological methods to detect chaos in economic time series. Generally, methods to analyse time series for detecting chaos could be classified in metric, dynamical, and topological tools17. For example, the correlation dimension is a metric method because it is based on the computation of distances on the system’s attractor; the Lyapunov exponent, on the other hand, is a dynamical method because it is based on computing the way near to where orbits diverge. Topological methods are characterized by the study of the organisation of the strange attractor, and they include close returns plots and recurrence plots. Topological methods exploit an essential property of a chaotic system, i.e. the tendency of the time series nearly, although never exactly, to repeat itself over time. This property is known as the recurrence property. In non-linear dynamical systems, the relation between closeness in time and in phase space is the most relevant manifestation of non-stationarity. The topological approach provides the basis for a new way to test time series data for chaotic behaviour18. In fact it has been successfully applied in the sciences to detect chaos in experimental data19, and it is particularly applicable to economic and financial data since it works well on relatively —————— 16
Brock and Sayers (1986). Belaire-Franch et al. (2001). 18 Mindlin et al. (1990). 19 Mindlin et al. (1991) and Mindlin and Gilmore (1992). 17
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small data sets, is robust against noise, and preserves time-ordering information20. Topological tests may not only detect the presence of chaos (the only information provided by the metric class of tests), but can also provide information about the underlying system responsible for chaotic behaviour. As the topological method preserves time ordering of the data, where evidence of chaos is found, the researcher may proceed to characterise the underlying process in a quantitative way. Thus, one is able to reconstruct the stretching and compressing mechanisms responsible for generating the strange attractor. For this, Recurrence Analysis can represent a useful methodology to detect non-stationarity21 and chaotic behaviours in time series22. At first, this approach was used to show recurring patterns and nonstationarity in time series; then Recurrence Analysis was applied to study chaotic systems because recurring patterns are among the most important features of chaotic systems23. This methodology makes it possible to reveal correlation in the data that is not possible to detect in the original time series. It does not require either assumptions on the stationarity of time series or the underlying equations of motions. It seems especially useful for cases in which there is modest data availability and it can efficiently compare to classical approaches for analysing chaotic data, especially in its ability to detect bifurcation24. Recurrence Analysis is particularly suitable to investigate the economic time series that are characterized by noise and lack of data and are an output of high dimensional systems25. In the literature the tools based on the topological approach are Close Return Test26 and Recurrence Plot27.
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Gilmore (1993). According Holyst and Zebrowska (2000), “system properties that cannot be observed using other linear and non-linear approaches and is specially useful for analysis of non stationarity systems with high dimensional and /or noisy dynamics”. 22 Zbilut et al. (2000). 23 Cao and Cai (2000). 24 Kononov (http://home.netcom.com/~eugenek/download.html), Ekmann et al. (1987); Zbilut et al. (2000). 25 Trulla et al. (1996) pp. 255. 26 Gilmore (1993) 27 Ekmann et al. (1987) 21
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3.1 Recurrence Plot The graphical tool that evaluates the temporal and phase space distance is the Recurrence Plot based on the Space State Reconstruction28. How the information can be recovered from time series was first suggested by Packard et al. (1980) and Takens (1981). They found that a phase space analogous to that of the underlying dynamical system could be reconstructed from time derivatives formed from the data29. The goal of such a reconstruction is to capture the original system states each time we have an observation of that system output. The idea is to expand a one-dimensional signal into an M-dimensional phase space; one substitutes each observation in the original signal X (t ) with vector 30 Yi = {xi , xi − d , xi − 2 d ,..., xi − ( m −1) d }. As a result, we have a series of vectors: Y = {y (1), y (2), y (3), ..., y ( N − (m − 1)d }. The key issue of state space reconstruction is the choice of the embedding parameters31 (delay and dimension) because in practical analysis it —————— 28
This approach is founded on flow of information from unobserved variables to observed variables and is widely used to reduce multivariate data to a few significant variables. The basic idea is that the effect of all the other (unobserved) variables is already reflected in the series of the observed output and the rules that govern the behaviour of the original system can be recovered from its output. Kantz and Schreiber (2000). 29 [...] From this picture we can then obtain the asymptotic properties of the systems, such the positive Lyapunov characteristics like the attractor’s topological dimension”, see Packard et al. (1980). 30 i is the time index, m is the embedding dimension, d is the time delay, N is the length of the original series. 31 The determination of embeddings parameters is a crucial point to obtain a significant state space reconstruction. In the literature there is still controversy regarding this point. For example, some researchers (Giuliani et al. 2000) have established for discrete data analysis a value delay time equal to 1, while for continuous data this value can be over 1. This rule of thumb is very limiting because in economics it is obvious that a value of variable today depends on the value of variable yesterday. The same rule is used for determining a dimension, in fact the typical dimension value for the economic time series state space reconstruction is 10. Other researchers (Stengos 1989, Scheinkman and LeBaron 1989, Hiemstra 1992, Mizrach 1992, Hsieh 1989, Guillaume 1994) of determination of dimension have found a value varying between 3 and 7. The consideration of Eugene Kononov (VRA author) after my request about this argument is the following: “I think the estimates of correlation dimensions in economic data are too optimistic. The presence of serial correlations, noise, and the relatively short length of data make it almost impossible to estimate a dimension reliably. And if there is such a thing as a low-dimensional structure in economic data, I’d like to
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has been demonstrated that different embedding parameters affect the predictions result and the quality of reconstruction. Nevertheless, there is not agreement about the procedures that calculate them. The techniques used for determining the embedding parameters (embedology32) are Mutual Information Function (MIF) and False Nearest Neighbours (FNN)33. The mutual information function provides important information about reasonable delay times while the false nearest neighbours can give guidance about the proper embedding dimension. Recurrence Plot is a graphical method designed to locate hidden recurring patterns, non-stationarity, and structural changes, introduced in Eckmann et al. (1987). The RP is a two-dimensional representation of a single trajectory. It is formed by a 2-dimensional M x M (matrix) where M is the number of embedding vectors Y(i) obtained from the delay co-ordinates of the input signal. In the matrix the point value of coordinates (i, j) is the Euclidean distance between vectors Y(i) and Y(j). In this matrix, the horizontal axis represents the time index Y(i) while the vertical axis represents the time shift Y(j). A point is placed in the array (i, j) if Y(i) is sufficiently close to Y(j). There are two types of RP: thresholded (also known as recurrence matrix) and unthresholded34. The thresholded RPs are symmetric35 around the main diagonal (45° axis).
————— ask the authors why they can’t predict their economic series any better than a random guess. I would suggest that you do your own research and don’t take the published results for granted. From my perspective of applied analysis, the best dimension is the one that gives you the best prediction results when used in your prediction model”. 32 The delay co-ordinates method has negative and positive properties. A positive one is that the “signal to noise ratio on each component is the same; a negative one is that their use implies the choice of delay parameters. If the parameter is too small, redundance, the co-ordinates are about the same and the trajectories of reconstructed space are compressed along main diagonal. If the parameter is too large, irrelevance, and the time series is characterised by chaos and noise, the dynamics may become unrelated and the reconstruction is no longer representative of true dynamics”. See Casdagli, et al. 1991. 33 Kennel and Abarbanel (1999); Cao (1997); Fraser and Swinney (1986). 34 “[...] In an unthresholded PR the pixel lying at (i,j) is color-coded according to the distance, while in a thresholded RP the pixel lying at (i,j) is black if the distance falls within a specified threshold corridor and white otherwise”. Iwanski and Bradley (1988). 35 The recurrence matrix is symmetric across its diagonal if ||Y(i)-Y(j)||=||Y(j)Y(i)||, McGuire et al. (1997).
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3.2 How to interpret RP In the Recurrence Plot it is quite usual to establish a critical radius and to plot a point as a darkened pixel only if the corresponding distance is below or equal to this radius. The points in this array are coloured according to the i − j vectors distance. Usually a dark colour shows the short distances and a light colour the long distance. We start by considering a random time series (default from VRA: White noise). The plot (Fig. 1c) has been built using delay 1 and dimension 12 as selected respectively from mutual information function and false nearest neighbours. As we can see in Fig. 1c, the plot of random time series shows a recurrent point distributed in homogenous random patterns. That means random variable lack of deterministic structures. These visual features are confirmed by the ratio calculated with RQA. We can see that the REC and DET assume values equal to zero, so in time series there are no recurrent points and no deterministic structure. These features are more evident if we compare REC and DET of time series with one of sine function (Fig. 1a). The plots of sine function are more regular and REC shows not only the recurrent point in each epoch but also that this value is the same. DET values are high, meaning a strong structure in the time series confirmed by the Maxline values which are also high, so deterministic rules are present in the dynamics. Comparing the Fig. 1a and Fig. 1b it is possible to see that if the analyzed series is generated from a determinist process in the RP there are long segments parallel to the main diagonal. If the data are chaotic these segments are short (Fig. 1b time series generated from a Lorenz equation). In Fig.1 it is possible to characterise stationary and non-stationary processes. If the coloration of the diagram is homogenous (Fig. 1c) the processes are stationary. The non-stationary, instead, causes changes in the distribution of the recurrent points for which the coloration of the matrix is not homogenous. If the texture of the pattern within such a block is homogeneous, stationarity can be assumed for the given signal within the corresponding period of time; non-stationary systems cause changes in the distribution of recurrence points in the plot, which is visible by brightened areas. Diagonal structures show the range in which a piece of the trajectory is rather close to another piece of the trajectory at different times. The diagonal length is the length of time they will be close to each other and can be interpreted as the mean prediction time. From the occurrence of lines parallel to the diagonal in the recurrence plot, it can be seen how fast neighboured trajectories diverge in phase space. The line segments parallel to the main diagonal are points close to each other successively forward in time and would not occur in a random,
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as opposed to deterministic, process. So if the analysed time series is deterministic, then the recurrence plot shows short line segments parallel to the main diagonal; on the other hand, if the series is white noise, then the recurrence plot does not show any structure. Chaotic behavior causes very short diagonals, whereas deterministic behaviour causes longer diagonals. Therefore, the average length of these lines is a measure of the reciprocal of the largest positive Lyapunov exponent. 3.3 Recurrence Quantification Analysis The RP approach has not gained much popularity because its graphical output is not easy to interpret. The set of lines parallel to the main diagonal is the signature of determinism. That set, however, might not be so clear. In fact, the recurrence plot could contain subtle patterns not easily ascertained by visual inspection. As a consequence Zbilut and Webber (1998) proposed statistical quantification of RPs, well-known as Recurrence Quantification analysis (RQA). RQA methodology is independent of limiting constraints such as data set size, data stationarity, and assumptions regarding statistical distributions of data, so that it seems ideally suited for dynamical systems characterized by state changes, non-linearity, non-stationarity, and chaos (Zbilut et al. 2000). RQA defines measures for diagonal segments in a recurrence plot. These measures are recurrence rate, determinism, averaged length of diagonal structures, entropy, and trend. Recurrence rate (REC) is the ratio of all recurrent states (recurrence points percentage) to all possible states and is the probability of recurrence of a special state. REC is simply what is used to compute the correlation dimension of data. Determinism (DET) is the ratio of recurrence points forming diagonal structures to all recurrence points. DET36 measures the percentage of recurrent points forming line segments that are parallel to the main diagonal. A line segment is a points sequence equal to or longer than a predetermined threshold (Giuliani et al. 1998). —————— 36
“This is a crucial point: a recurrence can, in principle, be observed by chance whenever the system explores two nearby points of its state space. On the contrary, the observation of recurrent points consecutive in time (and then forming lines parallel to the main diagonal) is an important signature of deterministic structuring” Manetti et al. (1999)
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These line segments show the existence of deterministic structures, their absence indicate randomness. Maxline (MAXLINE) represents the averaged length of diagonal structures and indicates the longest line segments that are parallel to the main diagonal. They are claimed to be proportional to the inverse of the largest positive Lyapunov exponent. A periodic signal produces long line segments, while the noise doesn’t produce any segments. Short segments indicate chaos.
(a)
(b)
(c) (d) Fig. 1. Examples by VRA. (a) Periodic time series; (b) Henon equation; (c) White Noise; (d) Lorenz equation.
Entropy (ENT) (Shannon entropy) measures the distribution of those line segments that are parallel to the main diagonal and reflects the complexity of the deterministic structure in the system. This ratio indicates the time series structureness, so high values of ENT are typical of periodic behaviours, while low values of ENT are typical of chaotic behaviours. The high value of ENT means a large diversity in diagonal line lengths; low
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values, on the other hand, mean a small diversity in diagonal line lengths37; “[S]hort line max values therefore are indicative of chaotic behaviours”38. The value trend (TREND), used for detecting drift and non-stationarity in a time series, measures the slope of the least squares regression of %local recurrence as a function of the orthogonal displacement from the main diagonal.
4. Macroeconomic data analysis using VRA: comparison of results In their paper “International Chaos?” (1988) Frank, Gencay, and Stengos analyzed the quarterly macroeconomic data from 1960 to 1988 for West Germany, Italy, Japan, and the United Kingdom. The goal was to check for the presence of deterministic chaos. To ensure that the data analysed was stationary they used a first difference and then tried a linear fit. Using a reasonable AR specification for each time series, their conclusion was that time series showed different structures. In particular, the non-linear structure was present in Japan’s time series. Nevertheless, the application of typical tools for detecting chaos (correlation dimension and Lyapunov exponent) didn’t show the presence of chaos in any time series. Therefore none of the countries’ income appeared to be well interpreted as being chaotic. Starting from this conclusion, we applied VRA to these time series to verify these researchers’ results. The purpose was to see if the analysis performed by a topological tool could give results different from those obtained using a metric tool (Lyapunov exponent and correlation dimension). The time series39 chosen are GNP of Japan and GDP of the United Kingdom. The choice is based on the fact that Japan is considered the most dissimilar one of the four countries studied. In fact, in order to filter this series, the authors used an Ar-4, while for the others an Ar-2 was used. For this series they reject the hypothesis IID, and the correlation dimension value calculated for various values of M, (the embedding dimension) is less compared to the growth of the value of the embedding dimension. There is a saturation point. —————— 37
Trulla et al. 1996. Iwanski and Bradley (1998), Atay and Altintas (1999). 39 T analysis performed by Frank et al. (1998) the data are for the Japan Real GNP seasonally adjusted, quarterly from 1960 to 1988 and for United Kingdom from 1960 to 1988. Source Datastream. 38
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In fact, for calculated values of M, 5, 10, 15, the dimension of correlation is respectively, 1.3, 1.6, 2.1, against values of 1.2, 3.8, 6.8 of the series shuffled.The rejection of the IID hypothesis, the value of correlation dimension, the comparison with the value of shuffled series, and the presence of non-linearity inclined the authors to suspect that the time series could be chaotic. However, tested with the Lyapunov exponent, the conclusion was that the data didn’t manifest chaotic behaviour. In fact the value of the Lyapunov exponent test was negative40. Moreover, it was shown that Japan’s economy is the most stable of the economies in the analysed countries. Therefore, although the presence of nonlinearity and correlation dimension values seemed to admit chaotic behaviour in the time series, the Lyapunov exponent test has not supported this hypothesis. Probably, as admitted by these same authors, this conclusion could be the result of the shortness of the series. With longer time series, matters could change41. The GDP time series of the United Kingdom was chosen because, as emphasized by the authors, for the UK as for Germany, time series is not rejected by the hypothesis IID. The behaviour of correlation dimension is the same for all three European countries in the study: Italy, West Germany, and the UK42. The increase of the embedding dimension corresponds to sustained increase of the dimension of correlation. Such increase is also characterised in shuffled time series obtained from the time series fits with Ar-2. From this conclusion and considering that the values of Lyapunov exponent test were negative, the authors conclude that European time series didn’t show non-linearity and in particular didn’t exhibit chaotic behaviour. 4.1 Macroeconomic time series of Japan’s GNP: data from 1960 to 1988 Fig. 2a shows the Recurrence Plot (RP) of Japan’s GNP. This was built using a delay-time and embedding dimension respectively equal to 2 and 7. Comparison between RP of the original time series (Fig. 2a) and RP of the shuffled series (Fig. 2b) allows us to see that the first is non-stationary. The different and diversified coloration allows us to conclude that the more homogenous coloration we can see from the shuffled series (Fig. 2b) is typical of stationary data. —————— 40
See “Table 4” p. 1580, in Frank et al. (1988). Frank et al. (1988), p. 1581. 42 Table 2, p. 1579 in Frank et al. (1988). 41
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(a)
(b)
Fig. 2. (a) RP of Japan GNP; (b) shuffled time series
Japan
GDP Delay Dimension REC DET ENT MAXL TREND
1960-1988 2 7 2.314 48.485 1.00 28 -87.39
Shuffled 2 8 0.0 -1 0.0 -1 0.0
Table 1. RQA Statistics of original and shuffled time series
In Table 1, RQA results are indicated for both time series. REC is positive, meaning that the data are correlated. DET is also positive, indicating that roughly 43% of the recurrent points are consecutive in the time; that is, form segments are parallel to the main diagonal. This indicates that in the data there is some type of structure. As we can see in Fig. 1a, long segments indicate that the series is periodic; in Fig. 1b we can see that short segments indicate that the series is chaotic. The value of MAXL is 28. This value indicates length of the longer segment in terms of recurrent points of the longer segment and allows us to say that the data are nonlinear and it is not possible to exclude the presence of chaotic behaviour. The analysis of VRA using the shuffled series of Japan is described in Fig. 2b where we find the RP and in Table 1 where the statistics of RQA are indicated. The RP of Fig. 2b shows a homogenous coloration that al-
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lows us to say that the series is stationary. The statistics of the RQA (Table 1) indicate that the shuffled series has lost all information: there are no recurrent points (REC, or segments parallel to the main diagonal (DET). Therefore, no type of determinist structure is present. This consideration is confirmed by the fact that the value of the MAXL is negative. From the comparison between original time series and its shuffling we can conclude that the data of Japan’s GDP are characterised by nonlinearity, confirming the result obtained by Frank et al. (1988), and they are non-stationary. Our conclusion regarding the presence of chaotic behaviour is different43 from that of Frank et al. (1988), who refuse the hypothesis that the data can be chaotic. The authors ascribe the result of their analysis to the shortness of the time series. With longer time series it could be possible to reach a contrary result; that is, not to reject the hypothesis of chaotic data44. 4.2 Macroeconomic time series of United Kingdom GDP: data from 1960 to 1988 In Fig. 3 we can see the RP of the United Kingdom GDP. This was built with delay-time and embedding dimension respectively equal to 1 and 8. By comparing the RP of the original time series (Fig. 3a) and its shuffling (Fig. 3b), we deduce that the time series is non-stationary: the economy of the United Kingdom is characterized by a period of structural change. Table 2 summarises the statistics of RQA for the original time series and its shuffling. The statistics of original time series indicate that in the data there are recurrent points (REC positive), that is, more than 8% of the points that compose the area of the RP’s triangle are correlated. Of this 8%, 32% (DET) shapes segments are parallel to the main diagonal, indicating the presence of determinist structures. This conclusion is confirmed by the presence of a positive value of the MAXL. The same ratios of shuffled series are characterised by zero or negative values.
—————— 43
“[…] None of these countries’ national income would appear to be well interpreted as being chaotic”. Frank et al. (1988), p. 1581. 44 “[…] When interpreting the findings one must be cautions given the shortness of the series. With longer time series matters could change”, Frank et al. (1988), p. 1581.
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(a)
(b)
Fig. 3. (a) RP of UK GDP; (b) shuffled time series
United Kingdom
GDP Delay Dimension REC DET ENT MAXL TREND
1960-1988 1 8 8.458 32.009 1.972 26 77.803
Shuffled 1 9 0.0 -1 0.0 -1 0.0
Table 2. RQA Statistics of original and shuffled time series
Comparing our analysis with that performed by Frank et al. (1988), it is possible to identify some points of difference. While they do not reject hypothesis IID, the analysis led with VRA does cause us to us to reject this hypothesis and to emphasize the presence of structure. The United Kingdom data are non-linear and this non-linearity can be interpreted as chaos. The MAXL and DET value, in fact, allow us not to reject the hypothesis of chaos in these data. For the United Kingdom as for Japan, Frank et al. (1988) emphasized that the analysis carried out on longer series could have obtained different results. In our case the analysis, which was also on short time series, has led to different conclusions. The different conclusions concern the fact that, while for Frank et al. the Japan’s economy seems more stable than that of the European countries (Italy, West Germany, and the UK), our analysis is performed from a different point of view; and this is also true if our analysis is limited to the UK. While in
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Frank et al. the comparison was made between stable economies, our analysis is based on unstable economies. Japan’s economy in these years (1960-1988) was less unstable than that of the UK. The different value of MAXL (Table 1 and Table 2) and the different non-stationary values allow us to conclude that the differences are very small. For Japan (Fig. 2a) these changes are more evident in the data from a later period (years 1970-1980). The same conclusions can be drawn from the relative data of the UK for the 1970-1980 period. From our analysis compared with the more conventional one by Frank et al. (1988), it is possible to conclude that the topological approach can be more useful for economic analysis performed on short time series typical of an economy.
5. Conclusions Among some economists there has been an interest in detecting chaos in economic data. The applications of empirical chaos testing have been either in the context of macroeconomics or of finance. Several chaos tests have been developed to try to distinguish between data generated by a deterministic system and data generated by a random system. In general, it has been stressed that an accurate empirical testing of chaos requires the availability of high quality, high frequency data, which makes financial time-series a good candidate for analyzing chaotic behaviour. In fact, the analysis of financial time series has led to results that are, as a whole, more reliable than those of macroeconomic series. The reason is the much larger sample sizes available with financial data and the superior quality of that financial data. Traditionally the tests used for chaos include correlation dimension, Lyapunov exponent, and BDS. These algorithms were developed for use with experimental data. Since physicists often can generate very large samples of high quality data, their application to these data has been successful. Unfortunately the application to economic data has been very difficult because high noise exists in most aggregated economic data. None of the tests has, however, delivered solid evidence of chaos in economics data. Investigators have found substantial evidence for nonlinearity but relatively weak evidence for chaos per se. A recent development in the literature has been the introduction of tools based on topological invariant testing procedure (close return test and recurrence plot). Compared to the existing metric class of testing proce-
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dures including the BBS test and Lyapunov exponent, these tools could be better suited to testing for chaos in financial and economic time series. Therefore, after a description and comparison between metric and topological tools, we tested some macroeconomic time series that had already been analysed (Fran et al.) with traditional tests for chaos. The application of typical tools for detecting chaos (correlation dimension and Lyapunov exponent) didn’t show the presence of chaos in any time series. The conclusion of the authors is that none of the countries’ income appeared to be well interpreted as being chaotic. Testing these time series with Visual Recurrence Analysis based on the topological approach has provided different conclusions. Our analysis, although performed using a short time series (112 data, indicates the presence of chaotic behaviour. From our analysis compared with the more conventional one by Frank et al. (1988), it is possible to conclude that the topological approach can be more useful for economic analysis performed on short time series, typical of an economy. The presence of chaos offers attractive possibilities for control strategies45 and this point seems particularly relevant for the insights of economic policies. Detecting chaos in economic data is the first condition to apply a chaotic control to phenomena that generate them. In this case, the goal is to select economic policies that allow us to perform a regular and efficient dynamic choosing among different aperiodic behaviours: that is the attractor’s orbits, thus maintaining the natural structure of the system. The economic application of chaos control considering both stabilization46 and targeting47 procedures or the combined48 use of them has highlighted the possibility of modifying the behaviour of a system to achieve the best performance of that system49, choosing among many different behaviours (orbits) that characterize the system because it is chaotic. The main criticism when the analysis is moved to the linear models arises from the fact that the models mislead a real understanding of the economic phenomenon and can induce inadequate and erroneous economic policies50. An alternative that could lead to the formulation of ade—————— 45
Boccaletti et al. (2000). Holyst (1996). 47 Kopel (1997). 48 Kopel (1997). 49 Ott et al. (1990). 50 Bullard and Butler (1993). 46
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quate policies could be the use of non-linear models and, in particular, chaotic models. Two attributes of chaos are relevant for its control: (a) An ergodic chaotic region has embedded within it an infinite dense set of unstable periodic orbits and may have embedded fixed points. (b) Orbits in a chaotic system are exponentially sensitive in that small perturbations to the orbit typically grow exponentially with time. To control a chaotic system means that we have to move from one orbit to another; that is, from a behaviour of systems to another behaviour, exploiting the typical properties of chaotic systems such as aperiodic orbits, sensitivity, and ergodicity. The methods developed for chaos control consider a non-linear system set at particular parameters that result in chaotic motion. Then, using only tiny control adjustments, a chaotic trajectory is stabilised in an unstable periodic solution embedded within the chaotic motion. When the system has been pushed into the appropriate linearized regime by nonlinear aspects of the problem, we stabilize it in a periodic orbit from aperiodic orbits that characterise the chaotic attractor. Small parameter changes and the presence of many aperiodic orbits are characteristics that are attractive for economic policy insights. Using sensitivity to initial conditions to move from given orbits to other orbits of attractors means to choose different behaviour of the systems, that is, different trade-off of economic policy. Moreover the employment of an instrument of control in terms of resources in order to achieve a specific goal of economic policy will be smaller if compared to the use of traditional techniques of control. Therefore the salient feature of applying of chaotic control is the strong “energy saving”, that is resources, to perform economic policy goals. If the system is non-chaotic the effect of an input on the output is proportional to the latter. Vice versa when the system is chaotic, the relation between input and output are made exponential by the sensitivity to initial conditions. We can obtain a relatively large improvement in system performance by using of small controls. Therefore if the system is chaotic limited resources don’t reduce the possibility by policy-makers to catch up prefixed goals of economic policies. Resource saving and choosing among different trade-offs of economic policies (many orbits) could be significant motivations to use chaotic models in the economic analysis. To use them we need to discover chaos in the data and we have demonstrated hat VRA is a useful tool for this end.
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References Atay, F. M., Y. Altintas (1999): Recovering smooth dynamics from time series with the aid of recurrence plots, Physical Review E, 59 pp. 6593-6598. Barnett W. A., Chen P., (1988): The aggregation-theoretic monetary aggregates are chaotic and have strange attractors: an econometric application of mathematical of chaos, in Barnett W. A., Berndt E. R., White A., (eds.) Dynamic Econometric Modeling, Cambridge University Press. Barnett W. A., Chen P., (1988): The aggregation-theoretic monetary aggregates are chaotic and have strange attractors: an econometric application of mathematical of chaos, in Barnett W. A., Berndt E. R., White A., (eds.) Dynamic Econometric Modeling, Cambridge University Press. Belaire-Franch J., Contreras-Bayarri D., and Tordera-Lledó L., (2001): Assessing Non-linear Structures In Real Exchange Rates Using Recurrence Plot Strategies, WP DT 01-03, Departamento de Análisis Económico. Boccaletti S., Grebogi C., Lai Y. C., Mancini H., Maza D., (2000): The Control of Chaos: Theory and Applications, Physics Reports, vol. 329, pp. 103-197. Brock W. A., Sayers C. L., (1986): Is the business cycle characterised by deterministic chaos? Journal of Monetary Economics, 22, pp. 71-90. Brock W.A., Dechert W.D., Scheinkman J., (1996): Econometric Review, 15, pp 197-235. Brock W. A., D. A. Hsieh D. A., LeBaron B., (1991): Non-linear dynamics, chaos, and instability: statistical theory and economic evidence, (MIT Press, Cambridge, MA. Brock W.A., (1986): Distinguishing random and deterministic systems. Abridged version, Journal of Economic Theory, vol. 40, pp. 168-195. Bullard J., Butler A., (1993): Nonlinearity and chaos in economic models: implications for policy decisions, Economic Journal, vol. 103, pp. 849-867. Cao L., Cai H., (2000): On the structure and quantification of recurrence plots, Physics Letters A 270, pp. 75-87. Cao L., (1997): Practical methods for determining the minimum embedding dimension of scalar time series, Physics Letters D 110, pp. 43-50. Casdagli M., Eubank S., Farmer J. D., Gibson J., (1991): State space reconstruction in the presence of noise, Physica D 51, pp. 52-98. Casdagli M.C., (1997): Recurrence plots revisited, Physica D 108 pp. 12-44. Chen P., (1993): Searching for Economic Chaos: A Challenge to Econometric Practice and Nonlinear Tests, Day R. H., Chen P., (eds), Nonlinear Dynamics and Evolutionary Economics, pp. 217-253, Oxford University Press. Day R. H. (1992): Complex Economic Dynamics: Obvious in History, Generic in Theory, Elusive in Data, Journal of Applied Econometrics, 7, pp. 10-23 Ekmann J. P., Kamphorst S. O., Ruelle D., (1987): Recurrence Plots of Dynamical Systems, Europhysics Letters, 4 (9) pp. 973-977. Faggini M., (2005): Un approccio di teoria del caos all’analisi delle serie storiche economiche, Rivista di Politica Economica, 7-8, luglio-agosto.
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Complexity of Out-of-Equilibrium Play in Tax Evasion Game V. Lipatov
1. Introduction In this paper, interaction among taxpayers in the tax evasion game is considered by means of game theory with learning. This interaction has been largely neglected in the modeling of tax evasion so far, although it brings about results significantly different from those of a conventional model. In case of tax authority commitment to a certain auditing probability, nonzero cheating equilibrium becomes possible. In case of tax authority with no commitment to a certain auditing probability, cycling may occur instead of long-run equilibrium, allowing for explanation of the fluctuations in “honesty” of taxpayers within the model rather than by exogenous parameter shifts. The paper reflects complexity of the issue in the sense of generating nontrivial dynamics. In presence of agents not maximizing expected payoff, the equilibrium play does not actually occur. Therefore, the evolutionary dynamics must be taken into account in order to draw accurate inference about the welfare effects of various policies. Without considering the complexity of dynamics explicitly, the policymakers are in danger of making wrong evaluation of tax reforms. The magnitude and importance of the shadow sector is hard to overestimate1. For a long time it has been realized that many classical results of —————— 1
A summary of recent attempts to estimate the size of tax evasion, avoidance and other informal activities is given in Schneider and Enste (2000). The results vary a
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public economics (optimal taxation, optimal provision of public goods, etc.) should be modified if the informal sector is an issue. Therefore, policy recommendations worked out without explicit accounting for the shadow economy are not likely to bring about desired outcomes in reality. A fundamental aspect of informal activities is tax evasion, which is usually defined as an effort to lower one’s tax liability in a way prohibited by law. This paper considers only this phenomenon, leaving tax avoidance and criminal activities aside. Specifically, it is devoted to income tax evasion, which has received the most attention in the theoretical modeling of evasion. This can be partially attributed to the existence of relatively reliable data on this matter (Tax Compliance Measurement Program (TCMP) in the US). Another reason might be tradition founded in 1972 by Allingham and Sandomo’s seminal model. A detailed survey of the models of income tax evasion has been carried out by Andreoni, Erard and Feinstein (1998). They show that there are two directions in modeling of strategic interaction between taxpayers and tax authorities: principal-agent (commitment) approach (for example, Vasin and Vasina 2002) and game theoretic (no commitment) approach (for example, Reinganum and Wilde, 1986, Erard and Feinstein, 1994, Peter Bardsley, 1997, Waly Wane, 2000). Both approaches are valid under the assumption that the parties first think about the strategies of each other, and then proceed to play equilibrium outcome. This, however, does not seem to be realistic when there are many taxpayers, although all the literature to date uses this (implicit) assumption. My paper is an attempt to relax this assumption, hence its main feature is explicit characterization of taxpayers’ interaction. This is achieved by using the evolutionary approach allowing for the learning of individuals from each other. Apart from this, there are a number of properties that distinguish my approach and allow one to match stylized facts about evasion. First, in reality taxpayers possess poor knowledge of audit function, usually overestimating the probability of being audited (Andreoni et al. 1998). In the model they do not know it, but rather make an implicit inference about this probability by observing other individuals’ actions. This is a case of “rational ignorance”, when the perceived benefits of acquiring additional information are lower than its costs. Second, because the agents have poor knowledge of the audit function they do not know the expected payoff of cheating. Their heterogenous beliefs are reflected in the initial distribution of strategies between cheating ————— lot with method and country considered; one general finding is that the shadow sector is growing over time.
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and not cheating. So the second feature of reality - different informational endowments of taxpayers - is implicitly accounted for. Third, instead of keeping in mind and updating perceived probability distributions of audit probability, agents just choose their actions according to a simple rule that compares their behavior with the behavior of the others. In this way another feature of reality is taken into consideration - substantial asymmetry in the reaction of tax authority and taxpayers. Indeed, the tax authority is supposed to explicitly conjecture the distribution of agents and make its decision based on this conjecture. In jargon, the authority is more “rational” than the taxpayers2. The fourth feature is the intertemporal nature of tax evasion decision, which is supported by Engel and Hines’ study (1999). In our framework the individuals have one period memory that allows them to choose a strategy tomorrow on the base of today’s observation of the behavior of the others and their own. As income reporting is a rare (annual) event, short memory can be a not plausible assumption. The game considered is a simple version of tax evasion relation with two levels of income and homogenous population of taxpayers. The learning rule in this population is a simple imitation of a better-performing strategy. Elsewhere I show that the qualitative features of the dynamics are robust to modification of the rule. The dynamics of the game are derived from the microfoundations in Lipatov 2004. The result here is quite intuitive: in the steady state the share of cheaters is the same as in the Nash equilibrium, whereas the auditing probability is not related to its Nash equilibrium value. The dynamics generated in the presence of agents not maximizing expected payoff show that the equilibrium play does not actually occur. But it is necessary to take these into account in order to draw accurate inference about the welfare effects of various policies. The estimation of such effects, however, requires calibration of the parameters of the model, which is a separate issue. The model produces results that are in line with stylized facts. Firstly, non-zero cheating of audited taxpayers is obtained for the commitment case, which is certainly more plausible than absolute honesty found in most of the conventional principle-agent models (for example, Sanchez and Sobel, 1993; Andreoni et al., 1998). Secondly, in the non-commitment case the following features of dynamics are explained: decreasing compli—————— 2
In my opinion the fact that the decision making agents consider more factors than researchers assign to them is not a sign of lack of rationality.
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ance (Graetz, Reinganum and Wilde, 1986) and auditing probability (Dubin, Graetz and Wilde, 1990; Adreoni at al., 1998, p. 820) observed in the US in the second half of the twentieth century, as well as the recent increase in auditing probability with continuing decrease of compliance. Additionally, an alternative explanation for the puzzle of why too much compliance is offered: it might not be the presence of intrinsically honest taxpayers, but the fact that the system is far from equilibrium. This is best illustrated in the commitment case: if the share of cheating taxpayers is below its equilibrium value, it stays so forever, and it looks as if taxpayers are cheating too little. The rest of the paper is organized in the following way. Section 2 outlines the simple static model, which is then played repeatedly. Dynamics of such play is analyzed in section 3, where different learning rules are considered. Underlying the average payoff rule is the norm of high tolerance of evasion and the assumption that taxpayers share among themselves a lot of information about evasion. The conclusion stresses limitations of the model and pictures its possible extensions and applications.
2. Classical game theory: a simple model As a starting point for modeling dynamics I take a simple one-shot game of tax evasion, based on Graetz, Reinganum and Wilde (1986). Honest taxpayers are eliminated from that model, as their presence does not change the results in the given setup. The timing is as follows: 1. nature chooses income for each individual from two levels, high H with probability γ and low L with probability 1-γ; 2. taxpayers report their income, choosing whether to evade or not; 3. tax agency decides whether to audit or not. It is obvious that low income people never choose to evade, because they are audited for sure if they report anything lower than L. At the same time, the high income people can evade, since with a report L the tax agency does not know whether it faces a truthful report by the lower income people or cheating from the higher income people. Then the game simplifies to one between higher income people and the tax agency:
cheat
not cheat
audit (1 − t ) H − st ( H − L), tH + st ( H − L) − c
not audit
(1 − t ) H , tH − c
(1 − t ) H , tH
H − tL, tL
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where t is an income tax rate; s is a surcharge rate, determining the fine for the given amount of tax evaded; c is audit cost. All of them are assumed to be constant and exogenously given for the tax-raising body3. Then the tax authority maximizes its expected revenue choosing the probability to audit p given probability of cheating q: ⎛ qγ p⎜⎜ (tH + st (H − L )) + 1 − γ tL − c ⎞⎟⎟ + (1 − p )tL → max, qγ + 1 − γ ⎝ (qγ + 1 − γ ) ⎠
which is linear in probability due to linearity of audit cost function. The multipliers for the payoffs are probabilities to come across high income cheaters (qγ) or low income honest taxpayers (1-γ) given that only low income reports are audited. First order condition holds with equality for the value of q=
1− γ
γ
c t (1 + s )(H − L ) − c
This is the Nash equilibrium value of cheating probability. A high income individual maximizes its expected the payoff given probability of audit: pq ((1 − t )H − st (H − L )) + (1 − p )q (H − tL ) + (1 − q )(1 − t )H → max ,
which is also linear in probability because of the nature of expected utility. The equilibrium value of p is 1/ (1 + s ) . Simple comparative statics shows that auditing probability is decreasing with the amount of a fine; and the extent of evasion is increasing with costs of auditing and decreasing with amount of fine, tax rate, income differential and share of high income people. Among others, we implicitly assumed here linear tax and penalty schemes, risk neutral individuals, and linear cost function for the tax authority. Even with these strong assumptions, considering the dynamics generated by the game allows us to get some non-trivial insights of what could be going on in reality.
—————— 3
Endogenous determination of tax and penalty rate is an interesting task, but it constitutes the problem of a government rather than a tax authority. Moreover, it has been largely discussed in the literature, see, for example, Cowell (1990).
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3. Dynamics Now consider the game presented in the previous section played every period from 0 to infinity. As before, the populations of high income and low income taxpayers are infinite with measures of γ and 1-γ respectively, and this is common knowledge. The proportion q(τ) of the high income population is cheating by reporting low income at time τ; the agency is auditing the low income reports with probability p(τ). With only two levels of income, it is possible to assume that the people know the income of those with whom they interact. Although the precise amount is not known, the aggregate level is easily guessed from observable characteristics. But the authority does not observe them, so the type of income is private knowledge of the agents who meet. On the other hand, the probability of auditing p is a private knowledge of the authority. Between the rounds the tax agency updates its belief about the distribution of taxpayers between cheating and not cheating, while the high income agents are learning whose strategy performs better. Irrespective of the rule, at a given time there are the following types of high income taxpayers: - honest, comprising proportion 1-q(τ) of population and receiving payoff (1-t)H; - caught cheating, q (τ ) p (τ ) of population with payoff (1-t)H-st(H-L); - not caught cheating, q (τ )(1 − p (τ )) of population with payoff H-tL. Note that the payoff when not cheating is bigger than when cheating and caught, but smaller than when cheating and not caught. In every period the tax agency is maximizing its expected revenue given its belief of distribution of taxpayers (assume fictitious play Eq(τ+1)=q(τ)). γ (1− q)tH + p(qγ (tH + st(H − L)) + (1− γ )tL) − c( p(qγ +1− γ )) + (1− p)(qγ +1− γ )tL → max,
which has an interior solution c′(.)=qt(1+s)(H-L) only if c(.) is a convex function4. This expression yields intuitive results: the tax agency chooses higher probability of audit for higher given values of fine, share of cheating population, and share of high income population (all these factors make auditing more profitable). The problem with this specification is that actual cost function may be concave. Then the optimal auditing probability is either 0 —————— 4
Convex cost function is assumed, for example, in Reinganum and Wilde (1986). In my view, a function best describing reality would be concave for p close to 0, and convex for p close to 1.
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or 1, since there is no budget constraint for auditors. I consider a linear case for the sake of tractability. As has already been mentioned, we cannot take a ready aggregate dynamics for the population of taxpayers because of the asymmetric nature of the players: the tax authority is making fictitious play, whereas taxpayers imitate each other. Without such asymmetry, our game resembles emulation dynamics as it is defined by Fudenberg and Levine (1998), which is known to converge to replicator dynamics under some assumptions. However, these assumptions are not satisfied in our setup: most strikingly, each individual communicates with more than one other. Due to this feature, even payoff monotonicity of the aggregate dynamics cannot be established. The aggregate dynamics corresponding to the rules considered is, however, derived in Lipatov (2004). In order to proceed we have to specify these learning rules.
3.1 Meeting two others: Average payoff principle An agent A meets agents B and C. Each of them has played strategies s _{A}, s _{B}, s _{C}. If s _{A} ≠ s _{B} = s _{C}, and average payoff of B and C is greater than the payoff of A, he/she switches to their strategy. If s _ {C} = s _{A} ≠ s _{B}, A switches in case the payoff of B is greater. The average of caught and not caught payoffs is bigger than the payoff of honesty (1 − t )H − st (H − L ) + H − tL > 2(1 − t )H for plausible value of fine s < 1. As a result, an honest taxpayer remains honest, if facing either of: - two other honest taxpayers
- two caught taxpayers - a caught and an honest taxpayer A caught taxpayer switches to honest, if he/she observes either of - two honest taxpayers - an honest and another caught taxpayer. A not caught taxpayer never switches. Thus:
[
]
1 − q(τ +1) = (1 − q)(1 − q + qp) + pq(1 − q + 2 pq) 2
where q=q(τ ), p=p(τ ). This equation actually defines the aggregate dynamics of the population we were interested in. As can be seen, the proportion of honest taxpayers tomorrow is completely determined by the proportion of honest taxpayers today and the probability of auditing today. We want to see what features this dynamics possesses, namely, we want to know whether the proportion of honest taxpayers is shrinking or expanding as time passes.
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This turns out to be crucially dependent on the level of auditing probability: for p < 1/ 3 the share of honest taxpayers is decreasing; for p > (2 / 3) it is increasing. Unexpectedly, in the small middle interval, change in the proportion of honest taxpayers is negatively related to the number of honest taxpayers. This “anti-scale” effect is explained by the high-enough detection probability, for which the caught cheaters contribute more to the increase of proportion of the honest, than the honest themselves. Further, we consider two cases for the behavior of the tax authority. If it is unable to announce its auditing probability and keep it forever, we are in the “game theoretic” framework, and the dynamics has two dimensions: one already derived for q and another one for p. We start, however, with a more simple case, when the auditors can credibly commit to a certain constant in time strategy (probability), and hence the dynamics is collapsing to one dimension.
3.2 Commitment The authority commits to a certain auditing probability once and forever (this corresponds to the principle-agent framework defined by Andreoni et al. 1998). From the population dynamics it can be seen that the long run predictors in this game are q=1 when p ( 2 / 3) , and q=
2 − 3p 1 − 3 p + 3 p2
when p is in between. Keeping this in mind, the authority chooses p to maximize equilibrium payoff. γ (1 − q( p ))tH + p(q( p )γ (tH + st (H − L)) + (1 − γ )tL ) − cp(q( p )γ + 1 − γ )tL → max
The optimal p can either be zero or fall in the small middle interval. This is true for any learning rule, since the argument does not use the particular form of q( p ). For high values of c and low values of s,t,H “no auditing - all cheating” equilibrium is chosen. Notice that the interior solution does not arise, if γ is small relative to µ = c/(t(H-L)). This is easy to interpret: with small share of high income people (γ ) and low benefit from auditing (stH ) it is better not to audit anybody, given that auditing is costly (c). The comparative statics exercise for the interior solution gives the following relations (they hold for any other learning rule as well):
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dp q + pq ' ; = '' ds q − (1 + s − µ )(2q ' + pq '' )
dp q + pq ' + (1 γ ) − 1 ; = − '' dµ q − (1 + s − µ )(2q ' + pq '' ) dp µ γ2 . = '' dγ q − (1 + s − µ )(2q ' + pq '' ) The sign of any of these derivatives is ambiguous, and to say something definite we have to take advantage of the knowledge of the function q(p). For the average payoff principle, the relation between share of cheaters and auditing probability in the interior steady state is a decreasing and strictly convex function. Under the best average principle (dp/dγ > 0) for our baseline parameter values s = 0.8, t = 0.3, (c /(H-L)) = 0.06 ( µ = 0.2), γ = 0.5. The fine is usually up to the amount of tax evaded, and I take 20% less than the whole. The income tax rate ranges from 0.1 to 0.5 across developed countries; the measures for both (c /(H-L)) and γ are bound to be arbitrary, since in reality the auditing function depends on many more variables than just income, and there is a continuum of income levels rather than two. A convenient way to think of the first measure is to consider what share of audited income has to be foregone for the auditing itself. Andreoni et al. (1998, p. 834) take 0.05 as an example; I think of 0.01 to 0.1 as a possible range. Finally, γ to a certain extent reflects the income distribution, and 0.5 gives an extreme case where there is an equal number of rich and poor. The resulting optimal p = 0.62 brings about (dp/dγ ) = 0.02 > 0 states that with increase of the share of high income taxpayers (the only ones who can cheat!) the auditing probability in steady state rises, since marginal revenue from auditing goes up, whereas marginal costs stay the same. For baseline parameter values auditing is also increasing in the cost tax bill ratio (dp/dµ = 0.15 > 0) and decreasing in the amount of fine (dp/ds = -0.18 < 0), which is contrary to what was expected. Faced with higher fine or lower auditing costs, the taxpayers will cheat less in steady state, hence there is no need for the tax authority to commit to a higher auditing probability. In fact, this stems from the strong asymmetry in the behavior of tax authority and individuals: the authority is very “smart” in the sense that it can predict the level to which the cheating converges for given auditing probability; the individuals are, on the contrary, very naive, since they just imitate a strategy with higher payoff.
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Fig. 1. Optimal auditing probability depending on s, baseline parameter values (µ = 0.2).
To get a quantitative feeling about the influence of parameters, I plot the auditing probability as a function s for the baseline parameter values t = 0.3, c /H=0.06, γ = 0.5.
3.3 Comparison with classical game theory The solution obtained can be compared with the Stackelberg-like equilibrium of the classical evasion game, when the tax authority moves first (much weaker asymmetry). Recall that in this setup q = 1 if p < (1/1+s ), q = 0 if p > (1/1+s ), and undetermined for the equality. Since auditing is costly, the authority will choose either p = 0, q = 1, or p = (1/1+s ), q = 0. The latter is preferred whenever the auditing is not too costly. Comparative statics is trivial in this setup: zero cheating result is independent of parameter changes as long as they do not violate rather mild condition of relatively not too expensive auditing. Auditing probability is decreasing in the surcharge rate, just as in the previous model. The solution of the static model is discrete, and the probability of audit jumps to zero for high enough µ or low s. The prediction of the dynamic model appears to be more plausible, since non-zero cheating is not observed in reality. As is known from the literature, the result of zero cheating in the commitment case generalizes for more complicated models with continuum of taxpayers and presence of intrinsically honest taxpayers. Moreover, the commitment models are usually criticized on the basis of this unrealistic prediction. The model presented eliminates this fault, and allows us to reconsider the view of commitment as something implausible. Then it just boils down to the classical
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case of dynamic inconsistency, and the willingness to commit is equivalent to the planning horizon of the authorities. The comparison of the payoffs of the tax authority in evolutionary and classical settings is ambiguous. For the parameter values chosen (γ = 0.5, µ = 1/3), it is better off imitating taxpayers for the magnitude of fine smaller than 0.5 and worse off for the magnitude larger than 0.5. This is quite intuitive, since low (high) values of s result in large (small) auditing probability of static no cheating equilibrium; auditing, in turn, is costly to implement. In a dynamic setting the auditing probability for given parameter values hits the upper boundary of (2/3), and hence is independent of the surcharge rate, except for the values of s close to 1. Consequently, “static” revenue is increasing with the fine, whereas “dynamic” revenue is staying constant. For the high values of γ the picture remains the same, except that now for very large values of fine the “dynamic” revenue rises so much that it exceeds the “static” one. Finally, with decrease in µ the solution with p strictly less than (2/3) is obtained for larger and larger set of s values, approaching the interval ((1/2), 1]. Correspondingly, the superiority of “static” revenue is preserved only at s = (1/2) in the limit (µ close to 0). An average taxpayer with high income in a Stackelberg setting can only cheat or not cheat with probability one; in the dynamic case there is a possibility of a mixed equilibrium: ⎛ ⎛ 1 ⎞⎞ I ⎜⎜ 0, ⎜ ⎟ ⎟⎟ = (1 − t )H ; ⎝ ⎝1+ s ⎠⎠ I (q ( p ), p ) = (1 − q )(1 − t )H + pq ((1 − t )H − st (H − L )) + (1 − p )q (H − tL ).
This brings about higher “dynamic” payoff for the individual, if p < (1/1+s), and lower payoff otherwise. This simple result is straightforward: in the classical setup the equilibrium payoff of taxpayers does not depend on the auditing probability or the magnitude of fine. Hence, the expected payoff in the dynamic model is greater if the audit probability is lower than in the static model, and vice versa. Note that this does not depend on the learning rule.
3.4 Convergence We talk here about a dynamic model without explicitly considering the dynamics itself. It is important that even in steady state the picture is very different from Nash (Stackelberg) equilibrium. The question of how long it takes to converge to a steady state has escaped our attention so far. As
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could be expected, the speed of convergence depends on the particularity of the imitation rule. In the present case, the learning procedure is in a sense favorable to the cheaters: it takes a long time to approach no cheating equilibrium, and a relatively short time to reach an all cheating equilibrium. Starting from the middle (q = 1/2), getting as close as 0.001 to the steady state takes 597 periods for honesty case and only 14 periods for the cheating case.
3.5 No commitment The authority decides on the optimal auditing rule in every period, assuming that the distribution of the taxpayers has not changed from the last period (fictitious play) Eq(τ + 1)=q(τ ). Then the best response strategy is
γ q (t ) t (1 + s )(H − L ) ⎧ ⎪0, if c > c = 1 − γ + qγ BR = ⎨ ⎪1, if c < c ⎩ As the tax authority is very unlikely to jump from not auditing anybody to auditing everybody and back, we explicitly augment the choice of tax agency with inertia variable: p(τ + 1) = αBR(Eq (τ + 1)) + (1 − α ) p (τ ) ,
where α determines speed of adjustment; BR is the best response function, which is defined above as revenue maximizing p given the belief about the distribution of taxpayers. With α→1, we are back to the case of jumping from 0 to 1 probability; with α→0, the probability of audit stays very close to an initial level forever. The dynamics is best seen in the picture (Fig. 2 and Fig. 3), where q(c) is the level of cheating that induces switch of best response from zero to one or back. It is interesting whether this dynamics brings about convergence of the system to a steady state with q = q(c ) ≡
1− γ
γ
c ≡ µ1 t (1 + s )(H − L ) − c
and
p=
(
)
⎛ 1 2 ⎞ µ1 − 2µ1 − 3 ⎟ 3 ⎝ ⎠ 2µ1
µ1 − 1 + ⎜ −
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or the cycling around this point is possible. Simulation results show that in discrete time setup the cycles are observed, whereas in the continuous time the system converges. A comparative static result for the steady state is possible to obtain because all the parameters are indexed to single µ1, which is bounded by unit interval: dp < 0. dµ 1
Hence, probability of audit in steady state is decreasing in costs of auditing and increasing in the share of high-income taxpayers, the tax rate, the magnitude of fine, and the income differential. Compared to the Nash equilibrium, where probability to audit depends only on the surcharge rate, our result looks more plausible. Still, for all parameters but s and γ the effects are the opposite of those in the commitment model. Whereas it is an open question what horizon a particular tax authority has, we can compare predictions of the two models by their conformability with stylized facts. First, it is almost uniformly accepted that evasion is increasing as the tax rate increases5, so here the commitment model seems to do a better job. Second, there is also a weak evidence that evasion is rising with the income (Witte and Woodbury 1985), and in this sense the long horizon authority is also superior. There is no convincing evidence on the influence of auditing costs on the auditing probability, and it is really difficult to say which model is closer to reality on this point. 1
0.75
0.5
0.25
0 0
0.25
0.5
0.75
1
p
Fig. 2. Phase diagram
—————— 5
See, for example, Clotfelter (1983), Poterba (1987), Giles and Caragata (1999).
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Fig. 3. Discrete time dynamics, µ 1 = 0.125
3.6 Comparison with Nash The rest point of the evolutionary game q is the same as in the Nash equilibrium of one shot game, since it is derived from the same maximizing revenue decision of tax authority. Auditing probability p can be greater or smaller depending on the parameter values, because it is determined by the behavior of the individuals, which is modelled differently. The variation in steady state p is very small: from (1 / 3 ) to (2/3), compared to ((1/2) ,1) in static case for s < 1. Hence, the difference in p for these two models is primarily dependent on s: for large values of fine Nash equilibrium gives less intensive auditing, and for small fines our model results in lower auditing. As for the payoffs, since q is set so that the tax authority is indifferent between auditing and not auditing, its revenue is exactly the same in static and dynamic setups. With fixed q the payoff of the average high-income taxpayer is unambiguously decreasing with p, so that for high penalties the average income is larger in the static model.
3.7 Note on the dynamics feature It is worth noting that the southwest and northwest parts of the picture are consistent with stylized facts presented in the introduction: both audit probability and the proportion of honest taxpayers decrease (second part of the twentieth century), and then eventually audit probability starts increasing, while non-compliance is still increasing (recent years). According to
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this explanation, the observed behavior is out-of-equilibrium adjustment, and sooner or later the tax evasion will have to go down. The southwest of generated dynamics also produces values of noncompliance significantly lower than the Nash equilibrium. This can be taken as an alternative explanation to the puzzle of too high compliance, usually resolved by introduction of intrinsically honest taxpayers (Andreoni et al., 1998, Slemrod, 2002). To see whether this kind of dynamics is not idiosyncratic for the learning rule under consideration, let us proceed to the other specifications of interactions between the taxpayers.
3.8 Meeting two others: Effective punishment principle An agent meets two others, and the following procedure (more favorable to the flourishing of honesty) in which we go even further away from the expected utility maximizing agents, takes place: - if 3 honest people meet, they will all stay honest for the next round; - if 2 honest and 1 caught cheater (or 1 honest and 2 caught cheaters) meet, they will all play honest next time; - if 2 honest and 1 not caught cheater (or 1 honest and 2 not caught cheaters) meet, they are all cheating next time; - if 3 not caught people meet, they play cheater next time; - if 2 caught cheaters and 1 not caught cheater (or 2 not caught cheaters and 1 caught cheater) meet, they all play honest in the next round; - if all three types meet (or 3 caught people), they play honest next round. The first four rules are standard; the last two result from the assumption that to observe punished people (or to be punished) is enough to deter one from cheating for the next year, and that the cheaters are aware of the option to be honest. As a result, the law of motion for q is given by
(
)
2 2 q(τ + 1) = q(1 − p ) 3(1 − q ) + 3(1 − q )(1 − p )q + (1 − p ) q 2 .
This is aggregate population dynamics. Notice that it does not differ qualitatively from the previous learning rule; that is, for p < (2/3) the share of cheating taxpayers is increasing or decreasing depending on q. If p ≥ (2/3), cheating is decreasing. We do not make special subsections for the cases of commitment and absence of commitment, nor for comparison with Nash equilibrium. The
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logic of the exposition is the same as for the average payoff learning principle, and the results are similar. We start with the commitment case: the dynamic converges to an optimal p in the interval [0, (2/3)] compared with the previously discussed Stackelberg outcome. The comparative statics is very much the same as for the previous learning rule, since the relation between p and q is still negative. The effective punishment rule is contributing more to the honest reporting, and it is of no surprise that the optimal probability of auditing is lower here for the same parameter values:
Fig. 4. Optimal auditing probability depending on s, baseline parameter values.
For the baseline parameter values in case of commitment, the effective punishment principle results in the same comparative statics as the average payoff principle, e.g. the auditing probability is decreasing in the magnitude of the fine, and increasing in the cost-tax bill ratio and share of high income taxpayers. From this we can conclude that the one-dimensional dynamics generated by two rules do not qualitatively differ. Comparison with the static equilibrium is exactly the same as before. The payoffs of the tax authority for the effective punishment are increasing in the magnitude of fine more slowly than for the best average. As a result, the interval of s for which the tax revenue of the static game exceeds that of the dynamic game is larger for the effective punishment rule, holding all the parameters constant. The result of the payoffs of individuals does not change, as it does not depend on the learning principle. Convergence features are not altered either: to reach no cheating state from the middle takes 594 periods now (compared with 597 before); to get to all cheating takes 3 periods (14 before). The latter, however, cannot be compared directly, as for the best average all cheating was attainable at
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any p in [0, (1 / 3 ) ] and computed for p = (1 / 3 ) ; for the present rule it can only happen for p=0. In the no commitment case the system converges (in continuous time) to the steady state with lower probability of auditing than with the previous rule. The discrete time cycling has very small amplitude, so that steady state is actually a very good approximation in this case. The steady state value for q certainly remains the same, as it does not depend on the learning principle and actually coincides with the Nash equilibrium value. The steady state value of p comes from as a solution of the third-order polynomial − µ12 p 3 + 3µ1 p 2 − 3 p + µ 12 − 3µ1 + 2 = 0 . From the phase portrait it is clear that this value is lower than for the best average principle. 1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
Fig. 5. Phase diagram
For the baseline parameter values (µ1 = 0.07) the steady state value of p is equal to 0.625 (compared with 0.659 for the previous rule). The values of dq / dx , where x is any parameter of the model (s,γ ,t,H,L), are completely unchanged, and the sign of dp / dx = ( dp / dq )( dq / dx) is dp / dq unchanged, since ( dp / dq ) is non-positive for both rules. As far as the comparison with the Nash equilibrium of the static game is concerned, everything said for the best average rule remains valid. Among others, the “dynamic” auditing probability is normally smaller than the “static” one ( p belongs (0, (2/3)) in dynamic case and p belongs ((1/2), 1) in the static case). Finally, the dynamic feature of cycling is also very similar and is consistent with the evidence.
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Fig. 6. Discrete time dynamics
In total, all the main conclusions of best average imitation are preserved under the effective punishment principle. There is more averse attitude towards the risk of being punished embodied in this rule. This results in lower cheating in the commitment case, and lower auditing probability for both cases.
3.9 Meeting m others: Popularity principle When m others are observed (and m is substantially larger than 2), we can specify an imitation rule that requires minimal information about the individual, only whether he/she was caught cheating. Assume that the availability of this information is assured by the tax authority in order to deter the others6. The rule is then: - for a not caught taxpayer: if more than k* caught individuals are observed, play honest in the next round; if less or equal - play cheat; - for a caught taxpayer: play honest Then the cheating is evolving according to
(
q(τ + 1) = 1 − q(τ ) p(τ )P k ≤ k *
)
The problem with this dynamic is that once the system comes close to extreme values of q (0 or 1), it is jumping between “almost all cheating” —————— 6
This seems plausible in the self-employment sector, especially for the professionals like doctors, auditors, etc.
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and “almost all honest” states in every period. This problem obviously arises from the ‘epidemic’ nature of the specified principle: once there are very many cheaters, almost everybody meets a caught cheater, and then all those switch to playing honest. But once almost everybody is playing honest, almost nobody meets a caught cheater, and then almost everybody is playing cheat. To make the dynamics smoother, the usual method is to introduce some kind of inertia into the system, just as was already done from the tax authority’s side. So let us say that with probability β every unpunished individual changes his/her strategy according to the already specified rule, and, correspondingly, with probability 1-β every unpunished individual plays the same strategy as in the previous period. As before, punished people switch to no cheating with probability 1. Then in every period the dynamics is described by
(
q(τ + 1) = q(1 − p )(1 − β ) + β (1 − qp )P k ≤ k *
)
For small enough values of β it converges to a steady state (cycle in discrete time) rather than jumps between two extreme values. The weakness of this formulation is that steady state value of p depends on the inertia parameter, and this gives an additional ‘degree of arbitrariness’ to our model. For baseline parameter values p is increasing in β , which is understandable: tax authority has to control more, if a larger number of individuals are reconsidering their decision at every period. In general, (dp/dβ ) has an arbitrary sign. It is very hard to analyze the m-dynamic analytically, since for variable k* it involves operating with sums of variable length. That is why I for the moment restrict my attention to the case where observing one caught individual is enough to deter (k* = 0) from evasion, just as was specified in the effective punishment rule. The dynamics is then q(τ + 1) = q(1 − p )(1 − β ) + β (1 − qp ) m + 1
For obvious reasons the closed form solution is impossible to obtain even for this simplified problem. So I simulate steady state for β = 0.1 and m = 19. Obviously, the other line is still q = µ1. In the no commitment case we again observe small cycles around the steady state with implications similar to the previous rules. Compared to the previous imitation rules, the line q(τ+1)=q(τ) is shifted to the low cheating - low auditing corner, meaning that steady state is more likely to have low probability of auditing. This comes from two factors: inertia in decision making and number of people to meet m. Notice, however, that even for p→1 cheating is not eliminated completely.
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Indeed, for p=1 q(τ+1)=β (1-q)m+1, so that q=0 only for β =0, which is impossible. Hence, for large auditing probabilities m-rule results in larger cheating than 2-rules. This seemingly strange result stems from the poor information set of the individuals: if nobody is cheating, nobody is caught, so in the next period of individuals will cheat.
Fig. 7. Steady state line for m-rule, q(τ+1)=q(τ)
Fig. 8. Discrete time dynamics
In the commitment case, then, cheating can be decreasing or increasing depending on whether q(τ)>qss or the opposite. This is true for any value of p chosen by the tax authority. Comparative statics is again similar to the previous rules, since the relation between p and q is negative. Since
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honest reporting is favored even more by this m-rule, we expect optimal auditing to be lower for the same parameters. The magnitude of fine is almost irrelevant under the present imitation rule, since there is no information about payoffs, and people are deterred from evasion by observing caught cheaters regardless of the financial costs of being caught. This intuition is supported by simulation results: for our parameter values the change of surcharge rate is not changing optimal p, and the equilibrium auditing is lower than before: 0.43 compared with 0.65 and 0.64 for the first two rules (the difference between rules is increasing with the cost of auditing). The auditing probability is rising with the proportion of high-income taxpayers, just as in two previous cases. However, it is decreasing in the cost of auditing, and hence in µ. So with the minimum information learning rule the difference in comparative statics between commitment and no commitment cases disappears. Such astounding difference in comparison with the first two learning rules is fully attributed to the form of steady state relation between q and p. In total, the poor information rule shows that the unusual results for the commitment case are due to high informativeness of the taxpayers about each other. When there is no information about payoffs contained in communication, the individuals abstain from evasion on even more “irrational” grounds than before. Then, for not too high auditing, more honesty results. Increasing the number of people met in this rule also brings about less cheating, because seeing more people means a higher chance of observing a caught one.
Fig. 9. Optimal probability depending on γ.
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Fig. 10. Optimal probability depending on c.
The main conclusions from previous imitation rule are still valid for m people meeting. Namely, there is still non-zero cheating with commitment and the dynamics in the western part, which is consistent with observations.
4. Conclusions The model presented in the paper is designed to capture two basic features of reality, which have been largely neglected in the literature on tax evasion, and completely so in the game-theoretic approach to the problem. These features are costly acquisition and processing of information (about probability of auditing), and interaction of taxpayers with each other. The interaction in the model is learning each others’ strategies and payoffs; this also allows individuals to obtain some information without getting involved in the costly process of acquisition. Moreover, with simple imitation rules specified in the game, people also avoid costs of processing information, as they effectively know what decision to take without solving complicated maximization problems. The model rationalizes the decreasing in time auditing probability and compliance observed in the US over past decades as out-of-equilibrium dynamics. The same is true for the recent continuing increase of evasion along with tightening auditing. The model can also explain “too little” cheating by taxpayers: having upward biased initial beliefs about auditing probability, they “undercheat” for a long time due to the inertia and imperfections of the learning rules. All these results are robust to the change of specification of the learning rule (my rules differ in how much people are
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afraid of being caught and how much information they can learn from each other). When I allow the tax agency to commit to a certain probability of auditing, positive cheating may arise in equilibrium. This is strikingly different (and obviously more plausible) from the result of zero cheating of most of the static commitment models. Moreover, the comparative statics with respect to tax rate does not contradict empirical evidence (cheating is increasing with tax), as opposed to the models in the literature. However, the model has its obvious limitations. For instance, nothing can be said about the extent of inertia in auditing decisions, though this could probably be empirically testable. Without good feeling about the inertia parameter and the learning rule we cannot say much about the precise form the dynamics takes. The possible extensions of the model follow directly from its simplified nature. First of all, a continuum of income levels could be considered, as well as a continuum of morality levels, thus fully characterizing an individual in two dimensions. Second, the costs of social stigma could also be incorporated, making it even more difficult to cheat when everybody is honest, and be hones, when everybody is cheating. In general, the dynamic approach to tax compliance games reopens a whole bunch of policy issues. Are the recommendations of equilibrium theory valid, if the system never comes to equilibrium? Are some changes in the existing taxation worth undertaking, if we take into consideration not only difference in benefits between initial and final states, but also the costs of transition? Can the decision rules of the tax authorities and the learning mechanisms governing taxpayers’ behavior be manipulated in a way to achieve maximal social welfare? As a building block for more general models, the evolutionary approach can be employed in studies on how the government can ensure a higher degree of trust in society (and less evasion as a result); how it can provide an optimal (from the point of view of social welfare) level of public goods; how it can bring about faster growth of an economy. For this it would be necessary to consider more complicated government (and hence tax authority) strategies, involving more than one period memory, and possibly heterogenous taxpayers. Finally, the approach taken by no means limits us to consideration of income tax evasion. An even more interesting and exciting task would be to look at all other taxes, especially those levied on enterprises. In this case learning, as well as interaction with tax authorities, is probably more intensive. Moreover, the absolute size of evasion is very likely to be higher than in case with individuals. The modeling of enterprise cheating would proba-
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bly allow us to understand better how the shadow sector in general is functioning.
4.1 Complexity hints for taxation and enforcement policy Summing up the results of the modeling, the following hints for the policy can be derived from the out-of-equilibrium dynamics: The reaction to a drop in a tax rate can be a rise in tax evasion, as the system does not go to the steady state in the short term. How valuable is a proposed tax system in terms of welfare is not stateindependent. The more radical is proposed change, the larger static gains should it provide in order to be justified. Stricter enforcement does not necessarily lead to higher compliance. Indeed, it is the effort of the tax authority that goes down, rather than the evasion amount. It probably makes more sense to target the learning rules of taxpayers rather than the attractiveness of cheating. Whereas intensity of punishment changes equilibrium behaviour, which never actually occurs, the better learning rules (the attitude of people towards tax evasion) may help in minimizing cheating and enhancing welfare.
References Andreoni, J., B. Erard and J. Feinstein (1998): Tax Compliance. Journal of Economic Literature, June, pp. 818-860. Bardsley, P. (1997): Tax compliance games with imperfect auditing. The University of Melbourne Department of Economics research paper no. 548. Clotfelter, C. (1983): Tax Evasion and Tax Rates: an Analysis of Individual Returns. Review of Economics and Statistics, 65(3), pp. 363-73. Cowell, F. (1990): The Economics of Tax Evasion, MIT Press, Cambridge, MA. Dubin, J., M. Graetz and L. Wilde (1990): The Effect of Audit Rates on the Federal Individual Income Tax, 1977-1986. National Tax Journal, 43(4), pp. 395409. Engel, E., and J. Hines (1999): Understanding Tax Evasion Dynamics. NBER Working Paper No. 6903. Erard, B., and J. Feinstein (1994): Honesty and Evasion in the Tax Compliance Game. Rand Journal of Economics, 25(1), pp. 1-19. Fudenberg, D., and D. Levine (1998): The theory of learning in games. MIT Press, Cambridge, MA. Giles D., and P. Caragata (1999): The learning path of the hidden economy: the tax burden and the tax evasion in New Zealand. Econometrics Working Paper 9904, Department of Economics, University of Victoria, Canada.
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Graetz, M., J. Reinganum and L. Wilde (1986): The Tax Compliance Game: Towards an Interactive Theory of Law Enforcement. Journal of Law, Economics and Organization, 2(1), pp. 1-32. Lipatov, V. (2004): Evolution of Personal Tax Evasion. Unpublished manuscript, IUE. Poterba, J. (1987): Tax Evasion and Capital Gains Taxation. American Economic Review, Papers and Proceedings, 77, pp. 234-39. Reinganum, J., and L. Wilde (1986): Equilibrium Verification and Reporting Policies in a Model of Tax Compliance. International Economic Review, 27(3), pp. 739-60. Sanchez, I., and J. Sobel (1993): Hierarchical Design and Enforcement of Income Tax Policies. Journal of Public Economics, 50(3), pp. 345-69. Slemrod, J., and S. Yitzhaki (2002): Tax Avoidance, Evasion and Administration. Handbook of Public Economics, volume 3, 1423-1470, North Holland. Schneider, F., and D. Enste (2000): Shadow Economies: Size, Causes, and Consequences. Journal of Economic Literature, March, pp.77-114. Vasin, A. and P. Vasina (2002): Tax Optimization under Tax Evasion: The Role of Penalty Constraints. EERC Working Paper 01/09E. Wane, W. (2000): Tax evasion, corruption, and the remuneration of heterogeneous inspectors. World Bank Working Paper 2394. Witte, A., and D. Woodbury (1985): The Effect of Tax Laws and Tax Administration on Tax Compliance. National Tax Journal, 38, pp.1-13.
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A New Stochastic Framework for Macroeconomics: Some Illustrative Examples M. Aoki1
1. Introduction We need a new stochastic approach to study macroeconomy composed of a large number of stochastically interacting heterogeneous agents. We reject the standard approach to microfoundation of macroeconomics as misguided, mainly because the framework of intertemporal optimization formulation for representative agents is entirely inadequate to serve as microfoundations of macroeconomics of stochastically interacting microeconomic units. Given that economies are composed of many agents of different types, fundamentally different approaches are needed. This paper illustrates our proposed approaches by brief summaries of examples drawn from four separate problems areas: Stochastic equilibria, uncertainty trap and policy ineffectiveness, stochastic business cycle models, and a new approach to labor market dynamics. In cooperation with some like-minded macroeconomists and physicists, we have advocated modeling approaches to macroeconomics based on continuous-time Markov chains, coupled with random combinatorial analysis. The proposed approaches differ substantially from those commonly used by the macroeconomics profession. Briefly put, we construct continuous-time Markov chains for several types of interacting economic agents to study macroeconomic problems. —————— 1
The author gratefully acknowledges H. Yoshikawa, Faculty of Economics, University of Tokyo, for several important insights on macroeconomics.
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Stochastic dynamics are described by master (i.e., backward ChapmanKolmogorov) equations, and the stochastic dynamic behavior of clusters of agents of various types is examined to describe macroeconomic phenomena and policy implications. Instead of the usual notion of equilibrium as a deterministic concept we use stationary distributions on a set of states as our definition of stochastic equilibria. Needs for stochastic analysis of models composed of a large number of interacting agents of different types have been slow to be recognized by macroeconomists. For example, the notion of power-law distributions has been extensively applied to model financial events, but not to model macroeconomic phenomena. By now we have obtained several new results, and gained new insights into macroeconomic behavior that are different, more informative, or not available in the mainstream macroeconomic literature. We also have some new perspectives on some macroeconomic policy effectiveness questions. This paper surveys four topics. They are stochastic equilibria, uncertainty trap and policy ineffectiveness, stochastic business cycle models, and new approaches to labor market dynamics. More detailed accounts are available in Aoki (1996, 2002); Aoki and Yoshikawa (2002, 2003, 2005); and Aoki, Yoshikawa, and Shimizu (2003). Here are short summaries of these topics: 1. Stochastic equilibrium as probability distributions, and probability distribution of productivities: In a stochastic framework we need a notion of equilibrium that is broader than that described in standard economics textbooks. In the standard framework demand is determined by technology and factor endowments. There are no microeconomic fluctuations. We give simple examples to contradict these statements. In standard arguments, unequalized productivities across sectors imply unexploited profit opportunities, and contradict the notion of equilibrium. Heterogeneous stochastic agents behave differently. All agents and production factors do not move instantaneously and simultaneously to the sector with the highest productivity. Their moves are governed by the transition rates of the continuous-time Markov chains. Value marginal products are not equalized across sectors and productivities of sectors have a Boltzman distribution. Section 2 covers this area. 2. Uncertainty and policy ineffectiveness: A new insight into macroeconomic policy ineffectiveness is described in section 3. We focus on effects of uncertainty on decision-making processes, and show that policy actions become less effective as the degree of uncertainty facing agents increases. 3. Stochastic model of business cycles: Business cycles are often explained as direct outcomes of the behavior of individual rational agents.
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The stronger is the desire to interpret aggregate fluctuations as something rational or optimal, the more likely is this microeconomic approach is to be adopted. We claim that microeconomic fluctuations persist because thresholds for actions among microeconomic agents differ across sectors/firms, and their actions alter the macroeconomic environment, and new, and different, microeconomic fluctuations result; and this process continues. We use a simple quantity adjustment model to illustrate these points in Section 4. 4. A new approach to labor market dynamics: Without using the notion of matching function, we model labor market dynamics in section 5 to explain Okun’s law and Beveridge curves, and how they shift in response to macroeconomic demand policies. We use a simple quantity adjustment model as a Markov chain, and apply the notion of the minimal holding times to select the sectors that jump first. The notion of holding time of Markov chain processes is a natural device for this random selection. This model is extended, then, to include labor dynamics. Pools of laid-off workers are heterogeneous, because their human capital, types of job experience, duration of unemployment, and so on are different. These pools form hierarchical trees. A new notion of distance, called ultrametric distance, is used to model different probabilities for unemployed workers being rehired as functions of ultrametric distance between the sector that is hiring and the unemployed in some pool of unemployed. In addition to the uncertainty trap mentioned above, our analysis of multi-sector models reveals rather unexpected consequences of certain macroeconomic demand policies. This example illustrates an aspect of demand policies that is not discussed in the existing macroeconomic literature.
2. Stochastic Equilibria and Probability Distributions of Sectoral Productivity According to Richard Bellman, the inventor of dynamic programming, state is a collection of information sufficient to determine the future evolution of the model, given whatever exogenous disturbances or control of state vector. In stochastic context, probability distributions are the states in the sense that Bellman describes. Deterministic equilibrium points are now replaced by stationary probability distributions.
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2.1 Stochastic Equilibria As a simple example, we follow Yosikawa (2003), and consider an economy composed of K sectors producing K types of goods. Taking the goods of the least productive sector as numeraire, let Pi be the relative price and Ci be the productivity of sector i, i.e., the amount of goods i produced by one unit of labor of sector / Let Di be the demand for good i. The total output of the economy is K-l
K-l
1=1
i=l A'-l
= LCK +^{Pi-
CK/Ci)Di,
where L is the total amount of labor. In the neoclassical equilibrium the value marginal products are equal across sectors, and the above equation reveals that the total output is independent of demand: Y = LCK. It is only the function of productivity factors and labor endowment. In reality, productivities differ across sectors. We have inequalities cipi > ••• > CiPi > • • -Cfc, Hence increase in Di increases Y, and 7 depends on demands.
2.2 A Boltzman Distribution for Productivity Coefficients The aggregate demand affects distribution of productivities of the economy and the level of total output. This can be seen by a simple entropy maximization argument, Aoki (1996, Sec. 3.2). Suppose that sector i employs Hi workers, and thati = I]j»^i. There are L\/ni\---nK^. numbers of configurations compatible with this constraint. The total output is y = Y^iCiUi. Assuming that all allocations are equi-probable, we define the entropy in analogy to the Boltzman entropy 5 = — y j In Uil ^ — y j Hi (In rii — 1) i
i
See Aoki (1996, p.47). The N\ factor disappear because the allocations are exchangeable in the technical sense of random combinatorial analysis.
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We maximize this subject to the constraints that Ui sum to L and D is the sum of CiUi over / if = 5 + a(L - ^ ni) + /?(D - ^ c^ni) i
i
where a and ^are Lagrange multipHers. Maximization of this expression with respect to ni yields: n* = exp(-a — ^Ci) and the expression for the two parameters as I
where we define the partition function by
and £) = e-"5]cie-^"' = - e From the equality
we obtain
^'^-m In 3(0) - - —0 + ronst., .Li
or where constant K is the number of sectors in the model. The average productivity of the economy is c = ^ Cifii/L = D/L i
To obtain the relation between ^^and c we measure productivities in unit of a sufficiently small positive unit 9, and assume that Ci = iO, i = 1,2,..., iC. Then we have an explicit expression for ^-60
^-60
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and
In other words, we have
The fraction of workers of sector i is distributed as
We note that D/L, the average GDP per sector, plays the role of “economic temperature” in analogy with the Boltzman statistics in statistical mechanics.
3. Policy Ineffectiveness: Uncertainty Trap By 2004 the Japanese economy has apparently come out of a long period of stagnation. Many explanations have been offered as to the causes of this long period of stagnation. Similarly, many suggestions have been offered to bring the economy out of the stagnation, such as by Krugman (1998), Blanchard (2000), and Girardin and Horsewood (2001), among others. In this section we describe one probable source for macroeconomic sluggishness which has not received the attention of economists in general, and those who specialize in Japanese economic performance in particular3. We call this source uncertainty traps. This effect has been pointed out in Aoki, Yoshikawa, and Shimizu (2003)4. The other source of sluggish behavior by the Japanese economy is discussed in Section 4.
3.1 Uncertainty Trap To explain this notion concretely, suppose that there are N agents (firms) in the economy. We keep N fixed for simpler exposition. Each agent has —————— 3
The sources we discuss in this paper are not necessarily specific to the Japanese economy and could affect other macroeconomies when conditions described in this paper are present. 4 Yoshikawa coined the term “uncertainty trap” to distinguish it from the liquidity trap.
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two choices, choice 1 and choice 2, in selecting its production level. Choice i means production at the rate yi, i = 1,2, where 2/1 > 2/2 > 0 Let n be the number of firms with choice 1. The number of firms with choice 2 is then A^ - n. The total output of the economy, GDP, is Y = ny^ + {N-n)y2- We express this in terms of the fraction x = n/N of agents with choice 1 as
Y = N[y,x+{y,{\-x))]. Note that a; is a random variable between 0 and 1, since the number of firms with choice 1 is random. We analyze how agents change their choices over time by modeling the process of changes in n as a continuous-time Markov chain (also called jump Markov chain). Firms can change their mind any time. They constantly evaluate the two present values, Vi,i = 1,2 associated with the two choices, where Vi is the random discounted present value for a firm with choice i, conditional on the number of fraction x. During a small time interval only one firm may change its production rate. Agents' stochastic switching between the two choices is described in terms of two transition rates which uniquely determine the stochastic process involved here. See Breiman (1968, Chapter 15) for example. Denote the probability that choice 1 is better than choice 2, given fraction X by r?(x) := Pr(Vi - V2 > 0|a;). When the random variable T?(X) is approximated by normal distribution with mean 9{x):^E[Vi-V2\x] and variance cr^ which is assumed constant for simpler explanation, it can be approximately expressed as^ r]{x)
X
where /3 = \/2/7ro-, and X = e^9
See Aoki (2002, p. 62) for Ingber's approximate expression of the error function which leads to the expression given here.
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M. Aoki ^ = (1 - 0, then this ^ is moved by [(!-0 with large /?, but is nearly zero with small
4. A Stochastic Model of Business Cycies In standard economic explanations of business cycles are direct consequences of individual agents' choices in changing economic environments such as consumers' intertemporal substitutions. We have three main objectives in this section. First, we demonstrate that aggregate fluctuations arise as an outcome of interactions of many sectors/agents in a simple model. Second, we show that the average level of aggregate output depends on the patterns of demand across sectors. Third, our simple quantity adjustment model clearly shows how some (actually only one in our model) sectors are randomly selected to act first, and their
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actions alter aggregate outputs and the interaction patterns among sectors/agents, thus starting the stochastic cycles all over again.
4.1 A Quantity Adjustment Model Consider an economy composed of K sectors, and sector i employs Ui workers^, i=l,...K. To present a simple model we assume K and prices are fixed in this section^ The output is assumed to be given by a linear production function Yi = CiUi for i = 1,2,...,K, where c, is the productivity coefficient. The total output (GDP) is given by the sum of all sectors K
y-E^i1=1
Demand for good i is given by SiY, where Sj is a positive share of the total output Y which falls on sector i goods, with J2i^i = 1- Here they are treated as exogenously fixed. In the next section we let them depend on the total output (GDP) in explaining Okun's law. Each sector has the excess demand defined by fi = SiY-Yi
(1)
f0Ti=l,2,...,K Changes in Y due to changes in any one of the sector outputs affect the excess demands of all sectors. That is, there exists an externaUty between aggregate output and demands for goods of sectors. Changes in the patterns of s's also affect these sets of excess demands. The time evolution of the model is given by a time-time Markov chain, as described in Aoki (2002, Sec. 8.6). At each point in time, the sectors of economy belong to one of two subgroups; one composed of sectors with positive excess demands for their products, and the other of sectors with negative excess demands. We denote the sets of sectors with positive and negative excess demands by 1+ = {i •• fi >0}, and / - = {i: fi < 0}, respectively. These two groups are used as proxies for groups of profitable and unprofitable secThe variable n, need not be the number of employees in a hteral sense. It should be a variable that represents 'size' of the sector in some sense. For example, it may be the number of lines in assembly lines. Actually, K can change as sectors enter and exit. See Aoki (2002, Sec. 8).
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tors, respectively. All profitable sectors wish to expand their production. All unprofitable sectors wish to contract their production. A novel feature of our model is that only one sector succeeds in adjusting its production up or down by one unit of labor at any given time. We use the notion of shortest holding time as a random selection mechanism of the sectors. That is, the sector with the shortest holding or sojourn time is the sector that jumps first. Only the sector that jumps first succeeds in implementing its desired adjustment. See Lawler (1995) or Aoki (2002, p. 28) for the notion of holding or sojourn time of a continuous-time Markov chain. We call that sector that jumps first as the active sector. Variables of the active sector are denoted with subscript a.
4.2 Transition Rates It is well known that dynamics of this time-time Markov chain are determined uniquely by the transition rates. See Breiman (1968, Chapter 15). We assume that the economy has initially enough numbers of unemployed workers so that sectors incur zero costs of firing or hiring, and do not hoard workers. We also assume no search on the job by workers. To increase outputs the active sector calls back one (unit of) worker from the pool of workers who were earlier laid-off by various sectors^. When fa < 0, Ua is reduced by one, and the number of unemployed pool of sector a, Ua, is increased by one, that is one worker who is immediately laid off. When /„ is positive, Ua is increased by one. See the next section for more detailed explanation.
4.3 Continuum of Equilibria The equilibrium states of this model are such that all excess demands are zero, that is, SiYe = Cin\, i = l,2,. ..,K. where subscript e of Y, and superscript e to Tii denote equilibrium values. Denoting the total equilibrium employment by Le = S i ^^f, we have liYe = Le
(2)
The actual rehired worker is determined by a probabilistic mechanism that involves measuring distances among clusters of heterogeneous laid-off workers by ultra-metrics. See Sec. 5. We note merely that our model can incorporate idiosyncratic variations in profitabihty of sectors andfrictionsin hiring and firing.
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where . This equation is the relation between the equilibrium level of GDP and that of employment. We see that this model has a continuum of equilibria. In equilibria, the sizes of the sectors are distributed as being proportional to the ratio for all ,
(3)
In the next section we see that the parameter in the model behavior.
plays an important role
4.4 Model Behavior Aoki (2002, Sec. 8.6) analyzes a simple version with K = 2 and show that as are changed, so are the resulting aggregate output levels. In simula, and several different demand share patterns: tions we have used some with more demand shares among more productive sectors and others more demand shares on less productive sectors. Simulations verify the sector size distribution formula given above. The aggregate outputs for all demand patterns initially decrease when we start the model with initial too large for equilibrium values. The model quickly conditions with sheds excess labors and settles down to oscillate around equilibrum level, i.e., business cycles. Loosely speaking, the more demands are concentrated among more productive sectors, the more quickly the model settles into business cycles. The more demands are concentrated on more productive sectors, the higher are the average levels of aggregate output. An interesting phenomenon is observed when the demand patterns are switched from more productive to less productive demand patterns and conversely. See Aoki and Yoshikawa (2005) for details.
5. New Model of Labor Dynamics This section discusses Okun’s law and the Beveridge curve by augmenting the model in the previous section by a mechanism for hiring and firing, while keeping the basic model structure the same.
5.1 A New State Vector Consider, as before, an economy composed of K sectors, and sector i employs workers,
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Sectors are now in one of two statuses; either in normal time or in overtime. That is, each sector has two capacity utilization regimes. The output of the sector is now given by
,
where Vi take the value of 0 in normal time, and 1 in overtime. More explicitly, in normal time
for , where is the productivity coefficient, and denotes the number of employees of sector i. In overtime, indicated by variable , workers produce output equal to
In overtime, note that the labor productivity is higher than in normal time because . This setup may be justified due to possible underutilization of labor. The total output (GDP) is given by the sum of all sectors, as before, Recall that demand for good i is given by as in the previous section, where is a positive share of the total output Y which falls on sector i goods, with
5.2 Transition Rates To implement a simple model dynamics we assume the following. Other arrangements of the detail of the model behavior are of course possible. Each sector has three state vector components: the number of employed, , the number of laid off workers, , and a binary variable , where means that sector i is in overtime status producing output with employees. Sectors in overtime status all post one vacancy sign during overtime status. When one of the sectors in overtime status becomes active with positive excess demand, then it actually hires one additional unit of labor and cancels the overtime sign. When a sector in overtime becomes active with negative excess demand, then it cancels the overtime and returns to normal time and the vacancy sign is removed. When , sector i is in normal time producing output with workers. When one of the sectors, sector i say, in normal time becomes active with positive excess demand, then it posts one vacancy sign and changes into one. If this sector has negative excess demand when it becomes active, it fires one unit of labor. To summarize: when , is reduced by one, and is increased by one, that means one worker is immediately laid off. We also assume
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that Va is reset to zero. When fa is positive, we assume that it takes a while for the sector to hire one worker if it has not been in overtime status, i.e., Va is not 1. If sector a had previously posted a vacancy sign, then sector a now hires one worker and cancels the vacancy sign; i.e., resets Va to zero. If it has not previously posted a vacancy sign, then it now posts a vacancy sign, i.e., sets Va to 1, and increases its production with existing number n,, of workers by going into over-utilization state. The transition path may be stated as z to z', where {na,Ua,Va
=0)
—> {na,Ua,Va
= I)_
and ( n o , Wo, 1^0 = 1) —» {na + l,Ua~
1, Va = 0)
In either case the output of the active sector changes into Y^ = K + Co. We next describe the variations in the outputs and employment in business cycles near one of the equilibria.
5.3 Hierarchical tree of unemployment pools In our model jobs are created or destroyed by changes in excess demand patterns. Pools of unemployed are heterogeneous because of geographical location s of sectors, human capital, length of unemployed periods, and so on. A given sector i, say, has associated with it a pool of unemployed who are the laid-off workers of sector i. They have the highest probability of being called back if sector i is active and can hire one worker. Pools of workers who are laid of from sector j , j ^ i have lower probability of being hired by sector i, depending on the distance d{i,j), called ultrametrics. These pools are organized into hierarchical trees with the pool of the laidoff workers from sector eat the root. The probability of a worker who is outside the pool i is a decreasing function of d{i,j). The ultrametric distance between pool i and j is symmetric d{i,j) = d{j,i) and satisfies what is called ultrametric condition d{i,j) < max{{d{i,k),d{k,j)}. See Aoki (1996, p. 36, Chapter 7) for further explanation^".
This ultrametric notion is used also in numerical taxonomy. See Jardine and Sibson (1971).For spin glasses and other physics, see Mezard and Virasoro (1985). See Schikhof (1984) for the mathematics involved. Feigelman and loffe (1991) have an example to show why the usual correlation coefficients between patterns do not work in hierarchical organization.
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5.4 Okun's Law Okun's law is an empirical relationship between changes in GDP, Y, and the unemployment rate u. We define Okun' s law by AY
^=-fii
,jr^U,
— h
(4)
where N = L + U is the total population of which L is employed and U is unemployed. In this paper we keep A^ fixed for simpler presentation. This numerical value of /? is much larger than what one expects under the standard neoclassical framework. Take, for example, the CobbDouglas production function with no technical progress factor. Then, GDP is given hyY = K^-°'L°' with a of about 0.7. We have .MI = ~AL, where AK and AN are assumed to be negligible in the short run. The production function implies then that AY/Y = aAL/L in the short run. That is, one percent decrease in F corresponds to an increase of AU/N = -{1/a){AY/Y){1 -U/N), i.e., an increase of a little over 1 percent of the unemployment rate. To obtain the number 4, as in the Okun's law, we need some other effects, such as increasing marginal product of labor or some other nonlinear effects. See Yoshikawa (2000). We assume that economies fluctuate about their equilibrium state, and refer to the relation (4) as Okun's law, where Ye is the equilibrium level of GDP, approximated by the central value of the variations in Y in simulation. Similarly, AU^ is the amplitude of the business cycle oscillation in the unemployed labor force. Ue is approximated by the central value of the oscillations in U, and Ye and Le are related by the equilibrium relation (2). The changes AY/Y and AUjU are read off from the scatter diagrams in simulation after allowing for a sufficient number of times to ensure that the model is in "stationary" state. In simulations we note that after a sufficient number of time steps have elapsed, the model is in or near the equilibrium distribution. Then, Fand U are nearly linearly related with a negative slope, which can be read off from scatter diagrams i.e., AY — -xAU, and we derive the expression for 0 Ue/N' We next see that the situation changes as the demand shares are made to depend on Y. We now assume that demand shares depend on Y, hence K depends on Y.
A New Stochastic Framework for Macroeconomics
Differentiate the continuum of equilibrium relation superscript e from now on) with respect to Y to obtain
137
(dropping
with
so that
with
Then the coefficient of the Okun’s law becomes
(5)
Okun’s law in the economics literature usually refers to changes in gross domestic products (GDP) and unemployment rates measured at two different time instants, such as one year apart. There may therefore be growth or decline in the economies. To avoid confusing the issues about the relations between GDP and unemployment rates during stationary business cycle fluctuations, that is, those without growth of GDP and those with growth, we run our simulations in stationary states assuming no change in the numbers of sectors, productivity coefficients, or the total numbers of labor force in the model. Okun’s law refers to a stable empirical relation between unemployment rates and rate of changes in GDP: one percent increase (decrease) in GDP corresponds to percent decrease (increase) in unemployment, where is about 4 in the United States.
Example Let the shares vary according to
for , where is the equilibrium value , and to satisfy the condition that the sum equals 1. Here are some numbers. With
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{S1S2) = (1 - s,s) where soi = 0.1, i = l , 2 , ( l / c - 1) = 10^, 71 = 10"^ we obtain /3 = 2.1. With 7 - 1.5 x IQ-^, 0 = 4.3. With 7 = 3 x lO^^^ /3= 3.6,50 =4x10-2 l / c - l = 102. The next two figures are the Okun's coefficients derived from simulation with iiT = 10". UY Curve and Okun Law for D1
UY Curve and Okun Law for D2
(xt OLS «lopa, «^ O k u n cc«f., Convarganea at 1S00)
(x: OLS • l o p o , a : Okun oool., Convergence a l 1 5 M )
x=2a& kx=4.41 a =3.09
8
196
%194 R
#
192
K g
iao
1 •>
^
17S 176
125
Agregates uromplo/ees after convergence
130
1M
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14S
unemployees after convergence
Fig. 1 Examples of Okun's law
5.5 Beveridge Curves In the real world unemployment and vacancies coexist. The relation between the two is called the Beveridge curve. It is usually assumed that its position on the -u - v plane is independent of aggregate demand. In simulations, however, we observe that their loci will shift with Y. Our model has a distribution of productivities. Demand affects not only Fbut also the relation between unemployment and vacancy loci. This result is significant because it means that structural unemployment cannot be separated from cyclical unemployment due to demand deficiency. This implies that the notion of a natural rate of unemployment is not well defined.
The original Matlab program was written by an economics graduate student in the Department of Economics at the University of California at Los Angeles, L.Kalesnikov. It was later revised by two graduate students at the Faculty of Economics, University of Tokyo, Ikeda, and Hamazaki.
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Fig. 2. Examples of Beveridge Curves
5.6 Simulation Studies Since the model is nonlinear and possibly possesses multiple equilibria, we use simulations to deduce some of the properties of the models. We pay attention to the phenomena of trade-offs between GDP and unemployment, and the scatter diagrams of GDP vs. unemployment to gather information on business cycle behaviors. Our model behaves randomly because the jumping sectors are random due to holding times being randomly distributed. This is different from the models in the literature, which behave randomly due to technology shocks that are exogenously imposed. As we indicate below, the state spaces of the model have many basins of attractions, each with nearly equal output levels. Simulations are used to gather information on model behavior12. Various cases with K = 4, K = 8, and K = 10 have been run. Four hundred Monte Carlo runs of duration 7000 elementary time steps each have been run. Fig. 2 is the average GDP of P1. It shows that after 700 time steps the model is in the closed set.
—————— 12
Simulation programs were written originally by V. Kalesnik, a graduate student at the University of California at Los Angeles, and later modified by F. Ikeda and M. Suda, graduate students at the Faculty of Economics, University of Tokyo.
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5.7 Effects of Demand Management on Sector Sizes With some demand patterns such that the low productivity sectors share a major portion of the aggregate demand, and as (2) shows suppose that the least productive sector has the largest equilibrium size to meet the demand. Suppose that the model has entered the closed set and exhibits stationary business cycle, and suppose that the demand pattern is switched so that the high productivity sectors now receive the major portion of the demand. One would conjecture that the stationary Y values will increase and the model will reach a new stationary state. When the size of the least productive sector is very large, the model will start by shrinking the size of the least productive sector more often than increasing the size of the more productive sector. Under some conditions it is easy to show that the probability of size reduction by the least productive sector is much larger than that of the size increase of the productive sector, at least immediately after the switch of demand pattern. When the productivity coefficients demand shares satisfy certain conditions, net reduction of Y is permanent, contrary to our expectation. See Aoki and Yoshikawa (2004) for details.
5.8 Summary of Findings from Simulations Simulation results may be summarized as follows: 1. Larger shares of demand on more productive sectors result in the higher average values of GDP. 2. The relationship between unemployment and vacancy depends on demand. Our simulations show that Beveridge curves shift up or down when Y goes down or up, respectively. In other words, when Y declines (goes up) the Beveridge curve shifts outward (downward). 3. The relationship between unemployment and the growth rate of GDP is described by a relation similar to Okun’s law. 4. The economy reaches the ‘equilibrium’ faster with larger shares of demand falling on more productive sectors. This indicates that demand affects not only the level of GDP but also adjustment speed toward equilibrium. In other words, our simulations show that higher percentages of demands falling on more productive sectors produce four new results: (1) average GDPs are higher; (2) the Okun’s coefficients are larger; (3) transient responses are faster; and (4) timing of the demand pattern switches matter in changing GDP. Unlike the Cobb-Douglas production or linear production function that lead to values of less than one, we obtain values from 2 to 4 depending on as the Okun’s coefficient in our simulation.
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It is remarkable that we can deduce these results from models with linear constant coefficient production functions. This indicates the importance of stochastic interactions among sectors introduced through the device of stochastic holding times. Using a related model, Aoki and Yoshikawa (2003) allow the positive excess demand sectors to go into overtime until they can fill the vacancy. This model produces the Beveridge curse shifts. The details are to appear in Aoki and Yoshikawa (2005).
6. Summing Up We have advanced the following propositions and perspectives by our new stochastic approach to macroeconomics. 1. Equilibria of a macroeconomy are better described as probability distribution. Master equations describe time evolutions of a macroeconomy. 2. Sectoral reallocations of resources generate aggregate fluctuations or business cycles. Given heterogeneous microeconomic objectives and constraints, thresholds for change in strategies differ across sectors/firms. It takes time for productivities across sectors to equalize. In the meantime, in responding to excess demands or supplies, the level of resource inputs in at least one sector changes, and macroeconomic situations also change, initiating another rounds of changes.
References Aoki, M. (1996): New Approaches to Macroeconomic Modeling, Cambridge University Press, New York. Aoki, M. (1998): A simple model of asymmetrical business cycles: Interactive dynamics of large number of agents with discrete choices, Macroeconomic Dynamics 2 427-442. Aoki, M. (2002): Modeling Aggregate Behavior and Fluctuations in Economics, Cambridge University Press, New York. Aoki, M., and H. Yoshikawa (2002): Demand saturation-creation and economic growth, Journal of Economic Behaviour and Organization, 48, 127-154. Aoki, M., and H. Yoshikawa (2002): Modeling Aggregate Behavior and Fluctuations in Economics, Cambridge University Press, New York. Aoki, M., and H. Yoshikawa (2003): A Simple Quantity Adjustment Model of Economic Fluctuation and Growth, in Heterogeneous Agents, Interaction and Economic Peroformance, R. Cowan and N Jonurd (eds). Springer, Berlin.
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Aoki, M., and H. Yoshikawa (2003): A Simple Quantity Adjustment Model of Economic Fluctuation and Growth, in Heterogeneous Agents, Interaction and Economic Performance, R. Cowan and N Jonurd (eds). Springer, Berlin. Aoki, M., and H. Yoshikawa (2003): Uncertainty, Policy Ineffectiveness, and Long Stagnation of the Macroeconomy, Working Paper No. 316, Stern School of Business, New York University. Aoki, M., and H. Yoshikawa (2004): Effects of Demand Management on Sector Sizes and Okun’s Law, presented at 2004 Wild@ace conference, Torino, Italy. Forthcoming in the conference proceeding, and in the special issue of Computational Economics. Aoki, M., and H. Yoshikawa (2004): Stochastic Approach to Macroeconomics and Financial Markets, under preparation for Japan-US Center UFJ monograph series on international financial markets. Aoki, M., and H. Yoshikawa (2005): A New Model of Labor Market Dynamics: Ultrametrics, Okun’s Law, and Transient Dynamics. pp. 204-219 Nonlinear Dynamics and Heterogenous Interacting Agents, Thomas Lux, Stefan Reitz, and Eleni Samanidou (eds), N° 550 Lecture Notes in Economics and Mathematical Systems, Springer –Verlag Berlin Heidelberg 2005. Aoki, M., and H. Yoshikawa (2005): Reconstructing Macroeconomics: A Perspective from Statistical Physics and Combinatorial Stochastic Processes Cambridge University Press, New York, forthcoming 2006. Aoki, M., H. Yoshikawa, and T. Shimizu (2003): The long stagnation and monetary policy in Japan: A theoretical explanation. Conference in Honor of James Tobin; Unemployment: The US, Euro-area, and Japan, W. Semmler (ed). Routledge, New York. Blanchard, O. (2000): Discussions of the Monetary Response-Bubbles, Liquidity Traps, and Monetary Policy, in R. Mikitani and A. S. Posen (eds), Japan’s Financial Crisis and its Parallel to U. S. Experience, Inst. Int. Econ., Washington D.C. Blanchard, O., and P. Diamond (1989): Beveridge Curve, Brookings Papers on Economic Activity 1, 1-60. Blanchard, O., and P. Diamond (1992): The flow approach to labor markets, American Economic Review 82, 354-59. Davis, S. J., J. C. Haltwanger, and S. Schuh (1996): Job Creations and Destructions, MIT Press, Cambridge MA. Davis, S.J., and J. C. Haltiwanger (1992): Gross job creations, gross job destructions, and employment reallocation, Quart. J. Econ. 107, 819-63. Feigelman, M. V., and C. B. Ioffe (1991): Hierarchical organization of memory in models of neural networks, in E. Dommany, J. L. van Hemmen, and K. Schulter (eds). Springer, Berlin. Girardin, E., and N. Horsewood (2001): Regime switching and transmission mechanisms of monetary policy in Japan at low interest rates, Working paper, Univ. de la Mediterrainee, Aix-Marseille II, France. Hamada, K., and Y. Kurosaka (1984): The relationship between production and unemployment in Japan: Okun’s law in comparative perspective. European Economic Review, June.
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Ingber, L. (1982): Statistical Mechanics of Neocortical Interactions, Physica D 5, 83-107. Jardine, N., and R. Sibson (1971): Mathematical Taxonomy, John Wiley, London. Krugman, P. (1998): It’s baaack: Japan’s Slump and the Return of the Liquidity Trap, Brookings Papers on Economic Activities 2, 137-203. Lawler, G. (1995): Introduction to Stochastic Processes, Chapman & Hall, London. Mezard, M., and M. A. Virasoro (1985): The microstructure of ultrametrics, J. Phys. bf 46,1293-1307. Mortensen, D. (1989): The persisitence and indeterminacy of unemployment in search equilibrium, Scand. J. Econ. 91, 347-60. Ogielski, A. T., and D. L. Stein, (1985): Dynamics on ultrametric spaces, Phy. Rev. 55, 1634-1637. Okun, A. M. (1983): Economics for Policymaking: Selected Essays of Arthur M. Okun, ed. A Pechman, MIT Press, Cambridge MA. Schikhof, W. H. (1984): Ultrametric Calculus: An Introduction to p-adic Analysis. Cambridge Univ. Press, London. Taylor, J.B. (1980): Aggregate Dynamics and Staggered Contracts, J. Pol. Econ. 88, 1-23. Yoshikawa, H. (2003): Stochastic Equilibria. 2003 Presidential address, the Japanese Economic Association. The Japanese Econ. Rev. 54, 1-27.
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Probability of Traffic Violations and Risk of Crime: A Model of Economic Agent Behavior J. Mimkes
1. Introduction The behavior of traffic agents is an important topic of recent discussions in social and economic sciences (Helbing, 2002). The methods are generally based on the Focker Planck equation or master equations (Weidlich, 1972, 2000). The present investigations are based on the statistics of binary decisions with constraints. This method is known as the Lagrange LeChatelier principle of least pressure in many-decisions systems (Mimkes, 1995, 2000). The results are compared to data for traffic violations and other criminal acts like shop lifting, theft and murder. The Lagrange LeChatelier principle is well known in thermodynamics (Fowler and Guggenheim, 1960) and is one method to support economists by concepts of physics (Stanley, 1996, 1999).
2. Probability with constraints The distribution of N cars parked on the two sides of a street is easily calculated from the laws of combinations. The probability of N l cars parking on the left side and N r cars parking on the right side of the street is given by P( N l ; N r ) =
N! 1 N l ! N r! 2 N
(1)
In Fig. 1 the cars are evenly parked on both sides of the street. The probability of this even distribution has always the highest probability. According to Eq.(1) P(2;2) = 37,5 %.
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Fig. 1 The distribution of one half of the cars on each side of the street is most probable. According to Eq.(1) the probability for Nl = 2 and Nr = 2 is given by P(2;2) = 37,5 %.
Fig. 2 The distribution of all cars on one side and none on the other side has always the least probability. For Nl = 0 and Nr = 4 we find P = 6,25 %.
3. Constraints In Fig. 3 the no parking sign on the left side forces the cars to park on the right side, only. The “no parking” sign is a constraint, that enforces the least probable distribution of cars and makes the very improbable distribution in Fig. 2 most probable! Laws are constraints that will completely change the distribution in any system.
Probability of Traffic Violations and Risk of Crime 147
[
Fig. 3 The “no parking” sign enforces the least probable distribution of cars.
In Fig. 4 we find one driver ignoring the “no parking” sign. What is the probability of this unlawful distribution?
[
Fig. 4 One car is ignoring the “no parking” sign. This is an unlawful act and may be called a defect. In solids we expect the number of defects to depend on the energy (E) of formation in relation to the temperature (T ). The number of wrong parkers will depend on the amount (E) of the fine in relation to normal income (T ).
Fig. 4 shows an unlawful act. In physics the occupation of a forbidden site is called a defect, like an atom on a forbidden lattice site. The probability of this traffic defect may be solved by looking at the laws of statistics of many decisions with certain constraints, the Lagrange principle.
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4. Probability with constraint (Lagrange principle) The laws of stochastic systems with constraints as introduced by Joseph de Lagrange (1736 – 1813), L = E + T ln P maximum! (2) L is the Lagrange function and a sum of the functions E and T ln P. The function p is the probability of distribution of N cars according to Eq.(1). The function E stands for the constraint [, the fine for wrong parking in Fig. 4. The parameter T is called Lagrange factor and will be a mean amount of money. The natural logarithm of probability ln p is called entropy. In the calculation of ln p according to Eq.(1) we may use the Stirling formula, ln p = ln (N! / (Nw ! Nr ! N ) (2a) = N ln N - Nw ln Nw - (N - Nw) ln(N - Nw) - ln 2N Nw cars parked on the wrong side have to pay Nw parking ticket (-E0). Introducing the relative amount x of wrong parked cars x = N w / N we obtain the Lagrange function of the system of parked cars: L(T, x) = N Σ x (-E0 ) + (2b) + TN { - x ln x - (1 - x) ln (1 - x) - ln 2} → max! At maximum the derivative of L with respect to x will be zero, ∂L / ∂ x = - N {E0 + T { ln x + ln (1- x) } = 0
(2c)
This leads to the relative number x = Nw /N of wrong parked cars as a function of the fine (-E0) at a mean amount of money T, Nw 1 = N 1 + exp( E o / T )
(2d)
Distribution of traffic offenders in Flensburg, 2000, Fig. 5, shows a Boltzmann function. Apparently this distribution is also valid for “social defects” like traffic violations, as indicated by the good agreement of calculations and data.
5. Probability with two constraints (LeChatelier principle) In Fig. 6 again a car is parked on the forbidden side. In contrast to Fig. 4 there is no space on the legal side. Due to the missing space the driver is
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forced to park his car on the illegal side, if there is an important reason or stress for the driver to park in this lot.
2. Traffic offence (A=4000, R = 0, T = 2,2) 2500
Number of offenders
2000 1500 1000 500 0 0
5
10
15
20
-500 Number of points
Fig. 5. Distribution of traffic offenders in Flensburg, 2000. Total number of all offenders: 2869. Red line: Calculation according to (2d)
In Fig. 6 we have two constraints in the decision of the driver to park, the “no parking” sign (E) and a limited space (V). The Lagrange principle now has two constraints and two order parameters, (T) and (p): L=E - pV + T ln p → maximum! L E T V p P
(3)
: Lagrange function : 1. contraint, e.g. ([), fine for parking violation : 1. Lagrange factor: e.g. mean amount of parking costs : 2. constraint, e.g. space, freedom : 2. Lagrange factor, e.g. stress, pressure : probability, eq.(1)
Equation (3) is also called the LeChatelier principle of least pressure. It is applied to all situations where people (or atoms) try to avoid external pressure.
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[
Fig. 6. One cars is forced to violate the “no parking “ sign due to an external stress or pressure (p), that reduces the freedom of choice of the roadside.
6. Risk, stress and punishment We may calculate the probability of the situation in Fig. 6. For Nr cars parked on the right side and Nw cars parked on the wrong side with a parking ricket (E P) and a parking space (v) needed per car we obtain L = Nw (E p - p v) +T{N ln N - Nw ln Nw - Nr ln Nr -ln 2} → max!
(3a)
Relative number x of violators
Risk and fine 0,5 0,4
R= 0 R = 30% R = 60% R = 100%
0,3 0,2 0,1 0 0
1
2
3
4
Relative fine E/ T
Fig. 7. Distribution of traffic violators under various risks of punishment. The probability of violations drops with rising risk and a rising relative fine E / T.
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At equilibrium the maximum is reached and we may again differentiate with respect to the number of delinquents, Nw.
∂L / ∂ Nw = (Ep - p v ) - T { ln (Nw / N ) + ln [(N - Nw) / N ] } = 0
(4)
Solving for the ratio Nw / N we obtain the relative number of agents violating the law as a function of the relative fine (Ep / T) :
Nw 1− R = N 1 − R + exp( E P / T )
(5)
Nw / N is the relative number of cars parked on the wrong side. EP is the fine (or punishment) for the parking violation in relation to a mean value T. Risk: In Eq.(5) the function (1 - R) has been introduced to replace the stress finding a parking space (v): 1 - R = exp ( - p v / T )
(6)
R = 1 - exp ( - p v / T )
(6a)
or The parameter R may be regarded as the risk that is taken into account under external pressure (p): R (p = 0) = 0
(6b)
R (p → ∞) → 1
(6c)
without stress (p=0) no risk is taken. And at very high stress (p →∞) a risk close to 100 % is taken into account. This is an important result: Risk is caused by internal or external pressure. Only people under stress will take into account the risk of punishment. Risk (R) and punishment (EP) of Eq.(5) have been plotted in Fig. 7. At no risk the distribution of right and wrong is 50:50. Doing right or wrong does not matter, if the risk of being caught is zero. With growing risk the number of violators is reduced, if the punishment is high enough. At a risk of 60 % the relative number of violators, Nw/N is close to the Boltzmann distribution observed in Fig. 5. At 100 % risk nobody violates the law, even if the relative fine (E/T) is not very high. In Fig. 8 the percentage of shop lifters is shown as a function of the stolen value (E). The data are given for Germany 2002 and have been taken from the German criminal statistics (BKA). The risk for being caught in shop lifting is given by rate of solved to reported cases. For shop lifting the risk is R = 0,95 according to the statistics of the BKA. Appar-
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ently the generally juvenile shop lifters link the value of the loot (E) to the amount of punishment (EP), fear dominates over greed. Shoplifting (R = 95 %, T = 150 €)
percentage of shop lifters
60 50 40 30 20 10 0 0
200
400
600
800
1000
value in Euro
Fig. 8. Distribution of shop lifters as a function of stolen values at 95% risk in shop lifting in Germany 2002, (www.bka.de). The probability of violations drops with rising stolen value E.
Relative number x of criminals
Probability of crime at risk R and fine Ep /T = 5 1
R=0%
0,8
R = 30 % R = 50 %
0,6 0,4
R = 90 % R = 99 % R = 100 %
0,2 0 0
2
4
6
8
10
Relative amount of gain E / T
Fig. 9. Distribution of criminal violators under various risks (R) at relative fine Ep/T = 5 according to Eq.(7). At low risk nearly all agents will become criminals, if the greed or the expected relative gain (E) is sufficiently high.
7. Criminal acts In contrast to parking problems in criminal acts the greed for expected profit (E) reduces the fear of punishment (EP), and Eq.(5) has to be replaced by
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(7)
rel number of criminal acts
Criminal acts under growing risk at constant E/T 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.3
0.4
0.5
0.6
0.7
risk
Fig. 10. Decreasing criminal violators under growing risks (R) according to Eq.(7) at constant value of (Ep - E) / T = 5 according to Eq.(7).
Rel. number of thefts, risk
Calculation and data on theft and risk in D 2002 70 60
Risk data
50 40 30 20
Theft data and calculations
10 0 . lz lt s. rn nd en W Pf ha der ye n rl a ess n d l e a a e A i n B d H N ei Sa S. Ba Rh
g W ur NR mb Ha
Fig. 11. Distribution of theft (square data points) under various risks (diamond data points) in the 15 federal states of Germany in 2002. There were 1,5 mill. Thefts reported of a total amount of 692 Mill. €, or a mean amount of 461 € per theft [www.bka.de] . Calculations (solid line) according to Eq.(7) with a negative exponent, (Ep - E) / T = -3 (greed dominating fear).
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rel. numbers
Rate of murder D 2002, A = 1.9, E/T = -3
1,5
murder data and calculations
1,25 1 0,75 0,5 1993
Risk data
1995
1997
1999
year murders per 100 risk
2001
calculation
Fig. 12. Distribution of murder (blue data points) under various risks (red data points) in Germany between 1003 and 2001, [www.bka.de]. Calculations (solid line) according to Eq.(7) with a negative exponent, (Ep - E) / T =-3 (greed dominating fear).
The risk (R) is again the same as in Eq.(6). The relative number of criminals Nw/N expected according to Eq.(7) has been plotted in Fig. 9. In Fig. 9 the relative number of criminals Nw/N grows with the expected illegal profit (E/T). According to the figure all agents will become criminals, if the risk is very low and the expected profit is high enough: nearly everybody may be bribed!! In countries with a low standard of living the probability of corruption is much higher than in countries with high standard of living. Unfortunately, no data have been found to match the calculations.
8. Conclusions The statistics of agent behavior under the constraint of laws has been applied to data in traffic and in crime. In both systems the statistical Lagrange LeChatelier principle agrees with the well recorded results. Behavior is apparently determined by risk and punishment (economic losses). A similar behavior of agents may be expected in all corresponding economic and social systems (situations) like in stock markets, financial markets or social communities.
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References BKA (2002): Polizeiliche Kriminalstaistik 2002 der Bundesrepublik Deutschland, http://www.bka.de. Fowler, R. and Guggenheim, E. A., (1960): Statistical Thermodynamics Cambridge University Press. Helbing, D., (2002): Volatile decision dynamics: experiments, stochastic description, intermittency control and traffic optimization, New J. Phys. 4 33. KBA (2002): Kraftfahrt Bundesamt Flensburg 2003 http://www.kba.de. Mimkes, J., (1995): Binary Alloys as a Model for Multicultural Society J. Thermal Anal. 43: 521-537. Mimkes, J., (2000): Die familiale Integration von Zuwanderern und Konfessionsgruppen – zur Bedeutung von Toleranz und Heiratsmarkt in Partnerwahl und Heiratsmuster, Klein,Th., Heidelberg, Editor, Verlag Leske und Budrich, Leverkusen. Mimkes, J., (2000): Society as a many-particle System, J. Thermal Analysis 60: 1055 – 1069. Stanley, H. E.; L. A. N. Amarala, D. Canningb, P. Gopikrishnana, Y. Leea and Y. Liua, (1999): Econophysics: Can physicists contribute to the science of economics? Physica A, 269: 156 – 169. Stanley, M. H. R.; L. A. N. Amaral, S. V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M. A. Salinger, and H. E. Stanley, (1996): Scaling Behavior in the Growth of Companies, Nature 379, 804-806. Weidlich, W., (1972): The use of statistical models in sociology, Collective Phenomena 1, 51. Weidlich, W., (2000): Sociodynamics, Amsterdam : Harwood Acad. Publ.
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Markov Nets and the NetLab Platform: Application to Continuous Double Auction L. Muchnik and S. Solomon
1. Introduction and background 1.1 What’s the problem? In describing dynamics of classical bodies one uses systems of differential equations (Newton laws). Increasing the number of interacting bodies requires finer time scales and heavier computations. Thus one often takes a statistical approach (e.g. Statistical Mechanics, Markov Chains, Monte Carlo Simulations) which sacrifices the details of the event-by-event causality. The main assumption is that each event is determined only by events immediately preceding it rather than events in the arbitrary past. Moreover, time is often divided in slices and the various cause and effect events are assumed to take place in accordance with this arbitrary slicing. The dynamics of certain economic systems can be expressed similarly. However, in many economic systems, the dynamics is dominated by specific events and specific reactions of the agents to those events. Thus, to keep the model meaningful, causality and in particular the correct ordering of events has to be preserved rigorously down to the lowest time scale. We introduce the concept of Markov Nets (MN), which allows one to represent exactly the causal structure of events in natural systems composed of many interacting agents. The Markov Nets preserve the exclusive dependence of an effect event on the event directly causing it but makes no assumption on the time lapse separating them. Moreover, in a Markov Net the possibility exists that an event is affected if another event happens in the meantime between its causation and its expected occurrence. We present a simulation platform (NatLab) that uses the MN formalism to make
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simulations that preserve exactly the causal timing of events without paying an impossible computational cost.
Fig. 1. The figure represents two scenarios, (a) and (b), in which two traders react to the market’s having reached the price 10, by a buy and respectively a sell order. This moment is represented by the leftmost order book in both (a) and (b) sequences. The current price is marked by the two arrow heads at 10. We took, for definiteness, both offer and demand stacks occupied with equal size limit orders (represented by horizontal lines) at integer values around 10. The (a) sequence illustrates the case where the seller is faster and saturates the buy order at the top of the buy stack: 9 (this is shown by the order book in the middle). Then, the buyer saturates the lowest offer order: 11 (right-most column). Thus the final price is 11. The (b) figure illustrates the case where the buyer is faster. In this case the final price is 9. Thus an arbitrary small difference in the traders’ reaction times leads to a totally different subsequent evolution of the market. The problem is to devise a method that insures arbitrary precision in treating the timing of the various events (see discussion of Fig. 13 below).
The present paper describes the application of the “Markov Net” (MN) concept on a generic continuous time asynchronous platform (NatLab) (Muchnik, Louzoun, and Solomon 2005) to the study of continuous double-auction markets. The continuous double-auction market is a fundamental problem in economics. Yet most of the results in economics were obtained for markets that are discrete time, synchronous, or both. The markets evolve in a reality in which time is continuous and traders act asynchronously by sending bid and ask orders independently. A change in the sequence of arrival to the market of two orders may change the actual price by a significant value and eventually lead to completely different subsequent market development (even if their arrival time difference is arbitrarily small). Thus, to describe the market dynamics faithfully one has to insure arbitrary precision in the timing of each event (Fig. 1).
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The insistence on preserving in great detail the causal order of the various events might seem pedantic and irrelevant for large disordered systems of many independent traders. After all, statistical mechanics is able to reproduce equilibrium thermodynamic results of some Ising-like models without even considering the details of their dynamics (McCoy and Wu 2005). However, there are certain systems in which this requirement seems unavoidable. Economic Markets is one of them. The classical theoretical framework for understanding market price formation is discrete and synchronous. It is based on the equilibrium price concept introduced by Walras in 1874. Walras’ equilibrium price did not address the issue of market time evolution. He assumed that in order to compute the market price it is sufficient to aggregate the offer and demand of all the traders. The equilibrium price will then be the one that insures that there is no outstanding demand or offer left. The extension of the Walrasian equilibrium-price mechanism to time evolving markets was suggested only later by Hicks (1939, 1946), Grandmont (1977), and Radner (1972. At each time, the demand curves of all the traders were collected and aggregated synchronously. The intuition that ignoring the asynchronous character of the market may miss its very causality occurred to a number of economists in the past. In fact, Lindahl (1929) suggested that economic phenomena like the business cycle may be due exactly to the iterative process of firms / individuals reacting to each other’s positions. This insight did not have wide impact because the methods used in modeling markets with perfect competition (General Equilibrium Theory) and markets with strategic interaction (Game Theory) are synchronous and, arguably, even timeless1. If the extension of the Walrasian paradigm (Walras 1874, 1909) to time evolving market price would hold, the sequence in which the orders arrive would be irrelevant and the insistence on absolute respect of the time order of the various market events would be unnecessary. However, in spite of the conceptual elegance of the Walrasian construction, its application to time-dependent markets is in stark contrast with what one experiences in the real market. In reality the demand and offer order stacks include at each time only a negligible fraction of the shares and of the share holders. Moreover, only the orders at the top of the stack have a direct influence on the price changes. One may hope that the local fluctuations due to the particular set of current traders would amount to only some noise on top of —————— 1
We are greatly indebt to Martin Hohnisch for very illuminating and informative correspondence on these points.
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a more fundamental dynamics closer to the Walrasian paradigm. However, there are many indications that this is not the case. First, the multi-fractal structure of the market implies that there is no real separation between the dynamics at the shortest time scales and the largest time scales (Mandelbrot, 1963, Liu et al., 1999). Thus one cannot indicate a time beyond which the market is bound to revert to the fundamental/equilibrium price. Second, the motivations, evaluations, and decisions of most of the traders are not in terms of an optimal long range investment but rather in terms of exploiting the short term fluctuations whether warranted or not by changes in the “real” or “fundamental” value of the equity. Third, the most dramatic changes in the price are taking place at time scales of maximum days, so aggregating over longer periods misses their causality. Fourth, the Walrasian auction requires each trader to define a personal demand function, i.e. defining his conditional bid/offer for any eventual price. In practice, not only do the traders not bother with, and do not have the necessary notions to make, their decisions for arbitrary improbable prices, but also the market microstructure just does not have the instruments capable of collecting these kinds of orders. All the arguments above can be summed in a more formal tautological form: the variation of the price for large time periods is the sum (in fact product) of the (relative) variations of the market at the single transaction level. A full understanding of the market dynamics is therefore included in the understanding of the single-trade dynamics. This idea has been investigated in models of high frequency financial data (Scalas 2005; Scalas, Gorenflow, Luckock et al. 2004; Scalas, Gorenflow, and Mainardi, 2000). The fact that Walras did not emphasize this point is due not only to the lack at the time of detailed single-trade data. Ideologically he had no way to be primed in this direction: the program of Statistical Mechanics of deducing global average thermodynamic properties from the aggregation of the binary interactions of individual molecules was not accepted even in physics in his time. In recent times, however, there have been attempts to reformulate Walrasian ideas in a statistical mechanical framework (Smith and Foley 2004).
1.2 The Background of the “Markov Net” concept One may be surprised at the long time the natural continuous time asynchronous reality has been represented in computers in terms of discrete time synchronous approximations. Even after this became unfashionable, the lack of consideration for the precise causal order of events continued to
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prevail through the use of random event ordering. Alternatively, in many applications the exact order in which the various events take place is left to uncontrollable fortuitous details such as arbitrary choices by the operating system or the relative work load on the computer processors. This is even more surprising given the fact that some fundamental scientific problems fall outside the limits imposed by these approximations. An outstanding example is the continuous time double auction market that we present in the second part of this paper. A possible explanation for this state of the art is the influence that the Turing discrete linear machine paradigm has on the thinking about computers. This influence led to the implicit (thus automatic) misconception that discrete machines can only represent discrete-time evolution. The continuous time is then viewed as a limit of arbitrary small time steps that can be achieved only by painstaking efforts. We will show that this is not necessarily the case. We also are breaking away from another implicit traditional assumption in computer simulations: the insistence on seeing the time evolution of the computer state as somewhat related to the time evolution of the simulated system. Even when the computer time intervals are not considered proportional to the simulated system time, the order in which the events are computed is assumed to be the one of the simulated world. As we shall see, in our implementation of the Markov Nets, there is occasional lack of correspondence between the two times. For instance, one may compute the time of a later (or even eventually un-happened) event before implementing the event that happens next according to the simulated world causality. The “Markov Net” (MN) idea is easier to understand if one disabuses oneself of the psychological tendency to assign to the simulated system (1) the discreteness and (2) the causality of the computer process simulating it. The mental effort to overcome this barrier is worthwhile: we can (without computational load penalty) go easily to any time precision (e.g. if desired, one can use time resolution of the order of the time quanta - the time necessary for light to traverse a Plank length ~ 10−44 sec in simulations that involve time scales relevant to daily trading activities). The only requirement would be to use double – or if, desired, arbitrary – real variable bit-length (Koren, 2002). Using the NatLab framework, one can apply the MN concept to applications in heterogeneous agent based macroeconomic models (Gallegatti and Giulioni 2003).
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2. Markov Net Definition and NatLab Description 2.1 Definition, Basic Rules, and Examples In this chapter we describe sequences of events in the Markov Net formalism and the way the NatLab platform implements them. While ostensibly we just describe a sequence of simulated events, this will allow us to make explicit the way the MN treats the scheduling and re-scheduling of future events as the causal flow is generated by the advancing of the process from one present event to the next. A Markov Net is a set of events that happen in time to a set of agents. The events cause one another in time with a certain lag between the cause and effect. Thus at the time of causation the effect event is only potential. Its ultimate happening may be affected by other events happening in the meantime. To take this into account, after executing the current event of the Markov Net, the putative happening times of all its directly caused potential effects are computed. The process then jumps to the earliest yet unhappened putative event, which becomes thereby the current event. From there, the procedure is iterated as above. Note that with the present definition, Markov Nets are deterministic structures. We leave the probabilistic Markov Nets study for future research. The name “Markov Nets” has been chosen in order to emphasize the fact that the present events are always only the result of their immediate cause events (and not of the entire history of the Net). However, as opposed to a Markov Chain (Bharucha-Reid 1960, Papoulis 1984), the current state is not dependent only on the state of the system immediately preceding it. For instance, as seen below, the current event could be directly caused by an event that took place arbitrarily far in the past. This is to be contrasted with even the n-order generalizations of the Markov Chains where the present can only be affected by a limited time band immediately preceding it. To recover continuum time in an n-order Markov Chain, the time step is taken to 0 and the dynamics collapses into an norder stochastic differential equation (continuous Markov Chain) (Wiener 1923). In contrast, a Markov Net exists already in continuum time and the past time-strip that influences the present is undefined depending on the system dynamics. In this, it shares properties with the Poisson process (except that it allows for a rather intricate causal structure). Thus, only in special regimes can it be approximated by differential equations. In order to explain the functioning of MN, one represents the various agents’ time evolution as directed horizontal lines going from some initial time (at the left) towards the future (the right) (Fig.2).
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The events of receiving some input (being caused) or sending some output (causing) will be represented by points on these horizontal lines. Thus each event is associated with an agent line and with a specific time. Its position on the agent axis will correspond to the Markov Net time of the event (Fig. 3). A vertical line will indicate the current time. Note that in a Markov Net, one jumps from the time of one event to the next one without passing through or even considering any intermediate times. The actual transmission/causation will then be represented by a directed line starting on the sending (causing) agent’s axis at the sending (cause event) time and ending on the receiving agent axis at a (later) reception (effect event) time (Fig. 3). Note that while the Markov Net is at a given time (event), the process is (re)calculating and (re)scheduling all the future potential events directly caused (or modified) by it. These events may be later modified or invalidated by “meantime events” (events that have taken place in the meantime), so we represent them by open dots pointed to by grey dotted (rather than full black) lines. Those lines are only to represent that the events’ timings were computed and the events were scheduled, but in fact their timing or even occurrence can be affected by “meantime secondary effects” (secondary effects that have happened in the meantime) of the current state (Figs. 5-7). Incidentally, this procedure, which is explained here for deterministic events, can be extended to probabilistic events. The actual time that the transmission/causation takes is computed based on the state of the sending (causing) and receiving (affected) agents. Other time delays, e.g. between the arrival of an order to the market and its execution, are similarly represented. Since the transmission / causation takes time, it often happens that as a signal travels from one event (agent) to another, another event on the receiver site affects the reception (delaying it, cancelling it ,or changing its effect). The entire dynamics of the system can then be expressed in terms of its Markov Net: the communication/causation lines and their departure (cause) and arrival (effect) event times. We consider the internal processes of an agent (e.g. thinking about a particular input to produce a decision as output) as a special case in which an arrow starts and ends on events belonging to the same agent. The beginning of the arrow represents the time at which the process started and the end of the arrow represents the time the process ended (typically by a decision and sending a communication). The computation of the thinking time and the scheduling of the issue time for the decision are made immediately after the execution of the event that triggers the decision process.
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Of course, as with the evaluation of other future events, if other meantime events affect the agent, the thinking process and the decision can be affected or even cancelled. Thus the scheduling of an event arrival is always provisory / tentative because the relevant agent can undergo another “unexpected” event before the scheduled event. Such an intervening event will modify the state of the agent and in particular the timing or even the actual happening of the event(s) that the agent is otherwise scheduled to undergo. For instance, as shown in Fig. 8, as a result of certain “news”, at 15:00, the agent is tentatively computing a certain internal deliberation that ends up with a sell order at 17:00. If, however, at 16:00 the agent receives a message cancelling the “news”, the ordering event tentatively scheduled for 17:00 is cancelled (Fig. 9) and some other event (going home at 16:30) can be tentatively scheduled (Fig. 9) and executed (Fig. 10). Another example of a Markov Net, involving three agents, is shown in Fig. 11. Their interaction sequence is explicitated through the MN scheduling and re-scheduling mechanisms. At the beginning of the process two events are pre-scheduled to take place at times t1 (affecting the middle agent) and t2 (affecting the upper agent). These events cause the two agents to enter internal deliberation states to prepare some reaction. The middle agent succeeds in making a very fast decision - the decision at t3-and causing the lower agent to send (at t4) a new signal to the upper agent. This new signal, (received at t5) interrupts the internal deliberation state in which the upper agent existed. Thus, a scheduled decision event that was expected to happen at t6 as a result of t2 is cancelled (this is shown by the dashed arrow). Instead, as a result of the signal received from the lower agent, the upper agent enters a new internal deliberation, which concludes into an answer being sent (at t7) to the lower agent.
3. Applications to Continuous Double Auction 3.1 Simple Example of a Double Auction Markov Net The simplest MN diagram of a continuous double-auction is shown in Fig. 12. In this diagram we show only one trader (the upper horizontal line) and the market (the lower horizontal line). The market price evolution is shown below the market line. The process described by this diagram starts with the agent setting P0 (represented on the price graph by a *) as an attention trigger. This is represented by a formal (0-time delay) message sent to the market (at time t1). When the price reaches P0 (at time t2), the agent notices it (t3), considers it (t4), and makes the decision to send a buy order (t5)
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that will be executed at t6. At the same time (t5), the agent also sets a new attention trigger price P1. Returning to the example in the introduction, we can now see how the Markov Net formalism takes care of the details of the traders’ behavior (Tversky and Kahneman, 1974; Kahneman, Knetch, and Thaler, 1986; Akerlof, 1984) in order to represent faithfully the sequence of market events that their behavior engenders. In Fig. 12 one can see that at certain past times (t1 and t2), each of the traders has fixed 10 as an action trigger price threshold. This is represented in the MN conventions by messages that each of them sent to the market. This is only an internal NatLab platform technicality such that the messages are considered as not taking any transmission time in the simulated world. As a consequence of these triggers, when the market reaches 10 at time t3 (following some action by another trader), each of them receives a “message” that is the representation on the platform of the fact that their attention is being awakened by the view (presumably on their computer screen) of the new price display equal to 10. Due to the delays in their perception (and other factors), their reception times (t4 and t5) are slightly different. Once they perceive the new situation, they proceed to think about it and reach a decision. They have different times as among each other until they reach the decision. Thus in the end they send their market orders at different times (t6 and t7). After taking into account the time each order takes to reach the market, one is in a position to determine the arrival times of each of the orders (t8 and t9). Thus, for given parameters (perception times, decision times, and transmission lags), those times are reproducibly computable and unambiguous market scenarios can be run. Fig. 13 A and B illustrates the two possible outcomes discussed in Fig. 1. Incidentally, upon seeing the effect of Agent 2’s order, o, Agent 1 may wish to take fast action to cancel his own order. Of course this possibility depends on the existence of a second communication channel to the market that is faster than the one on which the initial order has been sent.
3.2 Application of Markov Nets to Realistic Continuous DoubleAuction Market We have performed computer-based experiments of a continuous doubleaction market. As seen below (e.g. the spectacular loss of the professional strategy 7), the details of the timing of the various events at the lowest time scale (single trade) have an overwhelming influence on the market outcome.
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Agent 1 Time Fig. 2. The line representing an agent time evolution in a Markov Net. Below it, the time axis is plotted.
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The NatLab experiments were based on the “Avatars” method that we have introduced lately (Daniel, Muchnik, and Solomon. 2005) in order to insure that our simulations/experiments are realistic. The “Avatars”
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method allows us to capture in the computer the strategies of real traders (including professionals). By doing so we are crossing the line between computer simulations and experimental economics (Daniel, Muchnik, and Solomon 2005). In fact we have found that NatLab is a convenient medium to elicit, record, analyze, and validate economic behavior. Not only does NatLab operate in a most motivating and realistic environment but it also provides a co-evolution arena for individual strategies dynamics and market emergent behavior. Casting the subjects’ strategies within computer algorithms (Avatars) allows us to use them subsequently in various conditions: with and without dividend distribution, with and without interest on money in the presence or absence of destabilizing strategies. We present below some experiments in detail. We have collected the preferred trading strategies from seven subjects and have created (by varying the individual strategies parameters) a set of 7000 trading agents. We have verified that the subjects do agree that the traders behave according to their intentions. We then made runs long enough to have a number of trades per trader of the order 1000. In addition, we introduced 10,000 random traders to provide liquidity and stability. One can also consider their influence on the market as a surrogate for market makers (which we intend to introduce in future experiments). In this particular experiment we verified the influence on the market of having 1000 of the traders respecting a daily cycle. We did so by first running the market simulation in the absence of the periodic traders and then repeating the runs in their presence. Let us first describe some of the relevant traders, their algorithms, and their reaction to the introduction of the periodic daily trends (the numbers on the curves in the graphs in Fig 15 to Fig. 18 correspond to the numbers in the following list). 1 Random We introduced random traders. Each random trader extracts his belief about the future price from a flat, relatively narrow distribution around the present price and then offers or bids accordingly with a limit order set to be executed at the price just below (offer) or above (bid) the current market price. The order volume is proportional to the number of the agent’s shares (in case of offer) or to his cash (in case of bid) and to the relative distance between the current market price and the price the agent believes is right. As seen in Fig. 15, the random traders are performing poorly; their average (Fig. 16) is very smooth because their guesses at each moment are uncorrelated. Their performance and behavior are not affected by the pres-
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ence or absence of daily trends, due to their inability to adjust to them (Fig. 17, Fig. 18). 2 Short range mean reverters They compute the average over a certain - very short - previous time period and they assume that the price will return to it in the near future. On this basis they buy or sell a large fraction of their shares. When executed on the market with no periodic daily trends (Fig. 15), this strategy took advantage of the observed short-range negative autocorrelation of price returns. In fact, it was the winning strategy in that case. Agents following this strategy had their behavior strongly correlated among themselves. They were able to adjust to the continuously changing fluctuations (Fig. 16). When periodic trends (and hence short-range correlations in returns annihilating the natural anti-correlations) were introduced (Fig. 17), the performance of the strategy dropped dramatically. However, even in this case, positive returns were recorded. In the case of the periodic market, one can clearly identify the emergence of correlation between the actions of all agents in this group (Fig. 18). 3 Evolutionary extrapolation Each agent that follows this strategy tries to continuously evaluate its performance and occasionally switches between trend following and mean reverting behavior. Due to relatively long memory horizon, any trend and correlation that could have been exploited otherwise was averaged out and the strategy shows moderate performance (Fig. 15 and Fig. 17) with almost no spikes of consistent behavior (Fig. 16, Fig. 18). 4 Long range mean reverters Long range mean reverters compare long range average to short term average in order to identify large fluctuations and submit moderate limit orders to exploit them. This strategy does not have any advantage in the case when no well-defined fluctuations exist, and it cannot outperform other strategies (Fig. 15). The actions of the 1000 agents following this strategy are uncorrelated, and their performance is uncorrelated and in general is averaged to 0 (Fig. 16). However, when long trends do appear in the case when periodic fluctuations of the price are induced, agents following this strategy discover and exploit them (though with some delay) (Fig. 18) to their benefit. In fact, they expect that the price fluctuation will end and price will eventually revert to the average. Unlike the short-range mean reverters, they are not disturbed by the small fluctuations around the - otherwise consistent – longer cycle. However, due to these agents’ high risk aversion, their investment volumes are small which prevents them from outperforming the best strategy in the daily periodic case (Fig. 17).
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5 Rebalancers and Trend Followers They keep 70% of their wealth in the riskless asset. The remaining 30% is used in large order volumes in a trend following manner: if the short-range average price exceeds the long range average by more than one standard deviation, they buy. If the price falls by more than one standard deviation, they sell. Agents practicing this strategy use market orders to ensure that their intentions are executed. In general this strategy is relatively immune to sudden price fluctuations since it causes its agents to put, at most, 30% of their wealth at risk. It attempts to ride moderate trends. This does not produce remarkable performance in the first case when trends are nor regular. The strategy performance is average in this case. However, it is the best possible strategy in the case of periodic fluctuations. The agents following this strategy discover the trends quickly and expect them to continue for as long as the price evolution is consistent with them. In fact, they consistently manage to predict the price evolution and make big earnings (Fig. 17). 6 Conservative mean reverter Similar to long range mean reverters (subsection 5 above) but with slightly shorter range average and less funds in the market. This does not make any difference when no fluctuations are present (Fig. 15). However, when the periodic fluctuations are introduced (Fig. 17), they expect the price to return to the average sooner than it actually does. Thus the periodic fluctuations are mistaken for rather long intervals. 7 Professional volume watcher Professional volume watchers watch the aggregated volumes of the buy and sell limit orders. If a large difference towards offer is discovered, they sell, and vice versa. Their intention is to predict starting dynamics and respond immediately upon the smallest possible indicators by issuing market orders so that the deal is executed ASAP. This strategy, though suggested by a professional trader (and making theoretically some sense), fails to perform well in both cases. The reason for that is, in our opinion, wrong interpretation of the misbalance in the orders. Agents of this type expect the volume of the limit orders in the book to indicate the excess demand or supply and hence give a clue to the immediate price change. However, in our case, the volumes of limit orders on both sides of the book are highly influenced by big market orders (for example, those issued by rebalancers + followers) and in fact are the result of very recent past large deals rather than a sign of an incoming trend. Another way to express this effect is to recall that the prices are strongly anticorrelated at one tick. Thus this strategy is consistently wrong in its expectations. Interestingly enough, its disastrous outcome would be hidden by
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any blurring of the microscopic causal order (that would miss the one-tick anticorrelation). 8 The daily traders Daily traders gradually sell all their holdings in the “morning” and then buy them back in the “evening hours”. This causes the market price to fluctuate accordingly. What is more important, the behavior of all the other strategies is greatly affected as well as their relative success. Some of these strategies adapt and exploit the periodic trends while others are losing systematically from those fluctuations. Some of the results (specifically the time evolution of the average relative wealth of the traders representing each subject) are plotted in Fig. 15 and Fig. 17. The unit of time corresponds to the time necessary to have, on average, a deal per each agent.
4. Conclusions and Outlook For a long while, the discrete time Turing machine concept and the tendency to see computers as digital emulations of the continuous reality led to simulation algorithms that mis-represented the causal evolution of systems at the single event time scale. Thus the events took place only at certain fixed or random times. The decision of how to act at the current time was usually by picking up (systematically or randomly) agents and letting them act based on their current view of the system (in some simulations, event ordering was even left to the arbitrary decision of the operating system!). This neglected the lags between cause, decision, action, and effect. In the “Markov Net” representation it is possible for an event to be affected by other arbitrary events that were caused in the meantime between its causation and its happening. This is achieved exactly and without having to pay the usual price of taking a finer simulation mesh. We have constructed a platform (NatLab) based on the Markov Net (MN) principle and performed a series of numerical and real-subject experiments in behavioral economics. In the present paper we have experimented with the interactions and emergent behavior of real subjects’ preferred strategies in a continuous double-auction market. In the future, we propose to extend the use the of NatLab platform in a few additional directions: • Experiment with the effect on the market of various features and events. • Compare the efficiency of different trading strategies. • Isolate the influence of (groups of) traders’ strategies on the market.
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• Study the co-evolution of traders’ behavior. • Find ways to improve market efficiency and stability. • Study how people depart from rationality. • Study how out-of-equilibrium markets achieve or do not achieve efficiency. • Anticipate and respond to extreme events. • Treat spatially extended and multi-items markets (e.g. real estate), firms’ dynamics, economic development, novelty emergence, and propagation. Last but not least, the mathematical properties of Markov Nets are begging to be analyzed.
Acknowledgments We are very grateful to Martin Hohnisch for sharing with us his extensive and deep understanding of the economic literature on price formation mechanisms. We thank David Bree, Gilles Daniel, Diana Mangalagiu, Andrzej Nowak, and Enrico Scalas for very useful comments. We thank the users of the NatLab platform for their enthusiastic cooperation. This research is supported in part by a grant from the Israeli Academy of Sciences. The research of L.M is supported by Yeshaya Horowitz Association through The Center for Complexity Science. The present research should be viewed as part of a wider interdisciplinary effort in complexity that includes economists, physicists, biologists, computer scientists, psychologists, social scientists and even humanistic fields (http://shum.huji.ac.il/-sorin/ccs/ giacsannex12.doc; http://shum.huji.ac.il/-sorin/ccs/co3_050413.pdf, and http://europa.eu.int/comm/research/fp6/nest/pdf/nest_pathfider_projects_en pdf)
References Akerlof G A (1984): An Economic Theorist’s Book of Tales, Cambridge, UK: Cambridge University Press. Bharucha-Reid A T (1960): Elements of the Theory of Markov Processes and Their Applications, New York: McGraw-Hill. ISBN: 0486695395. Daniel G, Muchnik L, and Solomon S (2005): “Traders imprint themselves by adaptively updating their own avatar”, Proceedings of the 1st Symposium on Artificial Economics, Lille, France, Sept. Erez T, Hohnisch M, and Solomon S (2005): “Statistical Economics on MultiVariable Layered Networks”, in Economics: Complex Windows, M. Salzano and A Kirman eds., Springer, p. 201.
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Synchronization in Coupled and Free Chaotic Systems F.T. Arecchi, R. Meucci, E. Allaria, and S. Boccaletti
1. Introduction Global bifurcations in dynamical systems are of considerable interest because they can lead to the creation of chaotic behaviour [Hilborn, 1994]. Global bifurcations are to be distinguished from local bifurcations around an unstable periodic solution. Typically, they occur when a homoclinic point is created. A homoclinic point is an intersection point between the stable and the unstable manifold of a steady state saddle point p on the Poincaré section of a, at least, 3D flow. The presence of a homoclinic point implies a complicated geometrical structure of both the stable and the unstable manifolds usally referred to as a homoclinic tangle. When a homoclinic tangle has developed, a trajfectory that comes close to the saddle point behaves in an erratic way, showing sensitivity to initial conditions. Homoclinic chaos (Arneodo et al., 1985) represents a class of selfsustained chaotic oscillations that exhibit quite different behaviour as compared to phase coherent chaotic oscillations. Typically, these chaotic oscillators possess the structure of a saddle point S embedded in the chaotic attractor. In a more complex system, the orbit may go close to a second singularity which stabilizes the trajectory away from the first one, even though the chaotic wandering around the first singularity remains unaltered. We are thus in presence of a “heteroclinic” connection and will show experimentally some interesting peculiarities of it. Homoclinic and heteroclinic chaos (HC) have received large consideration in many physical (Arecchi, Meucci, and Gadomski, 1987), chemical (Argoul, Arneodo, and Richetti, 1987), and biological systems (Hodgkin and Huxley, 1952; Feudel et al., 2000).
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Homoclinic chaos has also been found in economic models, that is, in heterogeneous market models (Brock and Hommes, 1997, 1998; Foroni and Gardini, 2003; Chiarella, Dieci, and Gardini, 2001). The heterogeneity of expectations among traders introduces the key nonlinearity into these models. The physical system investigated is characterized by the presence, in its phase space, of a saddle focus SF and a heteroclinic connection to a saddle node SN. Interesting features have been found when an external periodic perturbation or a small amount of noise is added. Noise plays a crucial role, affecting the dynamics near the saddle focus and inducing a regularization of the chaotic behaviour. This peculiar feature leads to the occurrence of a stochastic resonance effect when a noise term is added to a periodic modulation of a system parameter. Besides the behaviour of a single chaotic oscillator, we have investigated the collective behaviour of a linear array of identical HC systems. Evidence of spatial synchronization over the whole length of the spatial domain is shown when the coupling strength is above a given threshold. The synchronization phenomena here investigated are relevant for their similarities and possible implications in different fields such as financial market synchronization, which is a complex issue not yet completely understood.
2. The physical system The physical system consists of a single mode CO2 laser with a feedback proportional to the output intensity. Precisely, a detector converts the laser output intensity into a voltage signal, which is fed back through an amplifier to an intracavity electro-optic modulator, in order to control the amount of cavity losses. The average voltage on the modulator and the ripple around it are controlled by adjusting the bias and gain of the amplifier (Fig. 1). These two control parameters can be adjusted in a range where the laser displays a regime of heteroclinic chaos characterized by the presence of large spikes almost equal in shape but erratically distributed in their time separation T. The chaotic trajectories starting from a neighbourhood of SF leave it slowly along the unstable manifold and have a fast and close return along the stable manifold after a large excursion (spike) which approaches the stabilizing fixed point SN (Fig. 2). Thus a significant contraction region exists close to the stable manifold. Such a structure underlies spiking behaviour in many neuron (Hodgkin and Huxley, 1952; Feudel et al. 2000),
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chemical (Argoul, Arneodo, and Richetti, 1987) and laser (Arecchi, Meucci, and Gadomski, 1987) systems. It is important to note that these HC systems has intrinsically highly nonuniform dynamics and the sensitivity to small perturbations is high only in the vicinity of SF, along the unstable directions. A weak noise thus can influence T significantly.
Fig. 1. Experimental setup. M1 and M2: mirrors; EOM: electro-optical modulator; D: diode detector; WG: arbitrary waveform and noise generator; R: amplifier; B0: applied bias voltage
Fig. 2. Time evolution of the laser with feedback and the reconstructed attractor
3. Synchronization to an external forcing The HC spikes can be easily synchronized with respect to a small periodic signal applied to a control parameter. Here, we provide evidence of such a synchronization when the modulation frequency is close to the ‘‘natural frequency”, that is, to the average frequency of the return intervals. The required modulation is below 1%. An increased modulation amplitude up to 2% provides a wider synchronization domain which attracts a frequency range of 30% around the natural frequency. However, as we move away from the natural frequency, the synchronization is imperfect insofar as phase slips; that is, phase jumps of ± 2π , appear. Furthermore, applying a large negative detuning with respect to the natural frequency gives rise to synchronized bursts of homoclinic spikes
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separated by approximately the average period, but occurring in groups of 2, 3, etc. within the same modulation period (locking 1:2, 1:3 etc.) and with a wide inter-group separation. A similar phenomenon occurs for large positive detuning; this time the spikes repeat regularly every 2, 3 etc. periods (locking 2:1, 3:1 etc.) (Fig. 3).
Fig. 3. Different response of the chaotic laser to external periodic modulation with different frequency with respect to the natural one of the system
The 1:1 synchronization regime has been characterized in the parameter space (amplitude and frequency) where synchronization occurs, Fig. 4. The boundary of the synchronization region is characterized by the occurrence of phase slips, where the phase difference from the laser signal and the external modulation presents a jump of 2π (grey dots).
4. Noise induced synchronization Let us now consider the effects induced by noise. White noise is added as an additive perturbation to the bias voltage. From a general point of view, two identical systems which are not coupled, but are subjected to a common noise, can synchronize, as has been reported both in the periodic (Pikovsky, 1984; Jensen, 1998) and in the chaotic (Matsumoto and Tsuda, 1983; Pikovsky, 1992; Yu, Ott, and Chen, 1990) cases. For noise-induced synchronization (NIS) to occur, the largest Lyapunov exponent (LLE) ( λ1 > 0 in a chaotic system) has to become negative (Matsumoto and Tsuda, 1983; Pikovsky, 1992; Yu, Ott, and Chen 1990). However, whether noise can induce synchronization of chaotic systems has been a subject of intense controversy (Mritan and Banavaar, 1994; Pikovsky, 1994; Longa,
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Curado, and Oliveira, 1996; Herzel and Freund, 1995; Malescio, 1996; Gade and Basu, 1996; Sanchez, Matias, and Perez-Munuzuri, 1997).
Fig.4 Experimental reconstruction of the synchronization region for the parameters amplitude and frequency for the 1:1 synchronization regime
As introduced before, noise changes the competition between contraction and expansion, and synchronization ( λ1 < 0 ) occurs if the contraction becomes dominant. We first carry out numerical simulations on the model which reproduces the dynamics of our HC system
(
x&1 = k 0 x1 x2 − 1 − k1 sin 2 x6
)
(1)
x& 2 = −γ 1 x2 − 2k 0 x1 x2 + gx3 + x4 + p0
(2)
x&3 = −γ 1 x3 + gx2 + x5 + p0
(3)
x& 4 = −γ 2 x4 + zx2 + gx5 + zp0 x&5 = −γ 2 x5 + zx3 + gx4 + zp0
(4) (5)
⎛ rx1 ⎞ ⎟ + Dξ (t ) x&6 = − β ⎜⎜ x6 − b0 + (6) 1 + αx1 ⎟⎠ ⎝ In our case, x1 represents the laser output intensity, x2 the population inversion between the two resonant levels, x6 the feedback voltage signal which controls the cavity losses, while x3 , x4 and x5 account for molecular exchanges between the two levels resonant with the radiation field and the other rotational levels of the same vibrational band. Furthermore, k 0 is the
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unperturbed cavity loss parameter, k1 determines the modulation strength,
g is a coupling constant, γ 1 , γ 2 are population relaxation rates, p0 is the pump parameter, z accounts for the number of rotational levels, and β ,r ,α are respectively the bandwidth, the amplification and the saturation factors of the feedback loop. With the following parameters k0 = 28.5714, k1 = 4.5556, γ1 = 10.0643, γ2 = 1.0643, g = 0.05, p0 = 0.016, z = 10, β = 0.4286, α = 32.8767, r = 160 and b0 = 0.1031, the model reproduces the regime of homoclinic chaos observed experimentally (Pisarchik, Meucci, and Arecchi, 2001). The model is integrated using a Heun algorithm (Garcia-Ojalvo and Sancho, 1999) with a small time step ∆t = 5 ▪ 10-5ms (note that typical T ≈ 0.5 ms). Since noise spreads apart those points of the flow which were close to the unstable manifold, the degree of expansion is reduced. This changes the competition between contraction and expansion, and contraction may become dominant at large enough noise intensity D. To measure these changes, we calculate the largest Lyapunov exponent (LLE) λ1 in the model as a function of D (Fig. 5a, dotted line). λ1 undergoes a transition from a positive to a negative value at Dc ≈ 0.0031. Beyond Dc, two identical laser models x and y with different initial conditions but the same noisy driving Dξ(t) achieve complete synchronization after a transient, as shown by the vanishing normalized synchronization error (Fig.5a, solid line).
E=
x1 − y1 x1 − x1
(7)
At larger noise intensities, expansion becomes again significant; the LLE increases and synchronization is lost when λ1 becomes positive for D=0.052. Notice that even when λ1 < 0, the trajectories still have access to the expansion region where small distances between them grow temporally. As a result, when the systems are subjected to additional perturbations, synchronization is lost intermittently, especially for D close to the critical values. Actually, in the experimental laser system, there exists also a small intrinsic noise source. To take into account this intrinsic noise in real systems, we introduce into the equations x6 an equivalent amount of independent noise (with intensity D=0.0005) in addition to the common one Dξ(t). By comparison, it is evident that the sharp transition to a synchronized regime in fully identical HC systems (Fig. 5a, solid line) is smeared out (Fig. 5a, closed circles).
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In experimental study of NIS, for each noise intensity D we repeat the experiment twice with the same external noise. Consistently with numerical results with a small independent noise, E does not reach zero due to the intrinsic noise, and it increases slightly at large D (Fig. 5b).
Fig. 5. Lyapunov exponents calculated from the model of the laser system and the synchronization error form the simulations and experimental showing the range of noise providing the synchronization between two systems
Experimental evidence of the noise induced synchronization is also provided by the data reported in Fig. 6.
Fig. 6. Experimental evidence of noise induced synchronization, the action of the common noise signal starts at time t=0
5. Coherence Resonance and Stochastic Resonance Coherence effects have been investigated by considering the distribution of the return time T which is strongly affected by the presence of noise. These effects are more evident in the model than in the experiment.
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The model displays a broad range of time scales. There are many peaks in the distribution P(T) of T, as shown in Fig. 7a. In the presence of noise, the trajectory on average cannot come closer to SF than the noise level. As a consequence the system spends a shorter time close to the unstable manifold. A small noise (D=0.0005) changes significantly the time scales of the model: P(T) is now characterized by a dominant peak followed by a few exponentially decaying ones (Fig. 7b).
Fig. 7. Noise-induced changes in time-scales. (a) D=0; (b) D=0.0005; and (c) D=0.01. The dotted lines show the mean interspike interval T0(D), which decreases with increasing D
This distribution of T is typical for small D in the range D = 0.00005÷ 0.002. The experimental system with only intrinsic noise (equivalent to D = 0.0005 in the model) has a very similar distribution P(T ) (not shown). At larger noise intensity D = 0.01, the fine structure of the peaks is smeared out and P(T ) becomes an unimodal peak in a smaller range (Fig. 7c). Note that the mean value T0(D)=〈T 〉t (Fig. 7, dotted lines) decreases with D. When the noise is rather large, it affects the dynamics not only close to S but also during the spiking, so that the spike sequence becomes fairly noisy. We observe the most coherent spike sequences at a certain intermediate noise intensity, where the system takes a much shorter time to escape from S after the fast reinjection, the main structure of the spike being preserved. As a result of noise-induced changes in time scales, the system displays a different response to a weak signal (A = 0.01) with a frequency
f e = f 0 (D ) =
1 , T0 (D )
i.e., equal to the average spiking rate of the unforced model. At D = 0, P(T ) of the forced model still has many peaks (Fig. 8a), while at D = 0.0005, T is sharply distributed around the signal period Te = T0(D) (Fig. 8b). However, at larger intensity D=0.01, P(T ) becomes lower and broader again (Fig. 8c).
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In the model and experimental systems, the pump parameter p0 is now modulated as
p(t ) = p0 [1 + A ⋅ sin (2πf e t )]
(8)
by a periodic signal with a small amplitude A and a frequency ƒe. First we focus on the constructive effects of noise on phase synchronization. To examine phase synchronization due to the driving signal, we compute the phase difference θ (t)=φ (t)-2πƒet. Here the phase φ (t) of the laser spike sequence is simply defined as [Pikovsky et al. ]
⎛
φ (t ) = 2π ⎜⎜ k + ⎝
t −τ k τ k +1 − τ k
⎞ ⎟⎟, (τ k ≤ t ≤ τ k +1 ) ⎠
(9)
where τ k is the spiking time of the kth spike.
Fig. 8. Response of the laser model to a weak signal (A=0.01) at various noise intensities. (a) D=0; (b) D=0.0005; and (c) D=0.01. The signal period Te in (a), (b), and (c) corresponds to the mean interspike interval T0(D) of the unforced model (A=0)
We have investigated the synchronization region (1:1 response) of the laser model in the parameter space of the driving amplitude A and the relative initial frequency difference
∆ω =
f e − f o (D ) f o (D )
, where the average frequency
f o (D ) of the unforced laser model is an increasing function of D. The ac-
tual relative frequency difference in the presence of the signal is calculated as
∆Ω =
f − fe f o (D )
where
f =
1 T t
is the average spiking frequency of the forced
laser model. The synchronization behaviour of the noise-free model is quite complicated and featureless at weak forcing amplitudes (about A 0.012), the system becomes periodic at a large frequency range. The addition of a small noise, D = 0.0005, drastically changes the response: a tonguelike region Fig. 9b, where effective frequency locking ( ∆Ω ≤ 0.003 ) occurs, can be observed similar to that in usual noisy phase-coherent oscillators. The synchronization region shrinks at a stronger noise intensity D = 0.005 (Fig. 9c). The very complicated and unusual response to a weak driving signal in the noise-free model has not been observed in the experimental system due to the intrinsic noise whose intensity is equivalent to D = 0.0005 in the model.
Fig. 9. Synchronization region of the laser model at various noise intensities. A dot is plotted when ∆Ω ≤ 0.003 . (a) D=0; (b) D=0.0005; and (c) D=0.005
We now study how the response is affected by noise intensity D for a fixed signal period Te. Here, in the chaotic lasers without the periodic forcing the average interspike interval T0(D) decreases with increasing D, and stochastic resonance (SR) similar to that in bistable or excitable can also be observed. We employ the following measure of coherence as an indicator of SR (Marino et al., 2002).
I=
Te
σT
(1+α )Te
∫α )P(T )dT ,
(1−
(10)
Te
where 0 < α < 0.25 is a free parameter. This indicator takes into account both the fraction of spikes with an interval roughly equal to the forcing period Te and the jitter between spikes [Marino et al. 2002]. SR of the 1:1 response to the driving signal has been demonstrated both in the model and in the experimental systems by the ratio T T and I in Fig. 10. Again, the t
e
behaviour agrees well in the two systems. For Te < T0(0), there exists a
Synchronization in Coupled and Free Chaotic Systems T
191
≈ 1 . The noise intensity optimizing the synchronization region where T coherence I is smaller than the one that induces coincidence of T0(D) and Te (dashed lines in Fig. 10a, c). It turns out that maximal I occurs when the dominant peak of P(T) (Fig. 9b) is located at Te. This kind of noiseinduced synchronization has not been reported in usual stochastic resonance systems, where at large Te numerous firings occur randomly per signal period and result in an exponential background in P(T) of the forced system, while at small Te a 1:n response may occur which means an aperiodic firing sequence with one spike for n driving periods on average (Marino et al. 2002; Benzi, Sutera, and Vulpiani 1981; Wiesenfeld and Moses 1995; Gammaitoni et al. 1998; Longtin and Chiaivo 1998). t
e
Fig. 10. Stochastic resonance for a fixed driving period. Left panel: model, A = 0.01, Te = 0.3 ms. Right panel: experiments: forcing amplitude 10 mV (A = 0.01) and period , Te = 1.12 ms; here the intensity D is of the added external noise. Upper panel: noise-induced coincidence of average time scales (dashed line, A = 0) and synchronization region. Lower panel: coherence of the laser output. α = 0.1 in Eq. (12)
6. Collective behaviour Synchronization among coupled oscillators is one of the fundamental phenomena occurring in nonlinear dynamics. It has been largely investigated in biological and chemical systems. Here we consider a chain of nearest neighbour coupled HC systems with the aim of investigating the emergence of collective behaviour by increasing the strength of the coupling [Leyva et al. 2003]. Precisely, going back to the model Eqs. (1)-(6) we add a superscript i to denote the site ini dex. Then in the last equation for x6 , we replace x1i with
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(1 + ε )x1i + ε (x1i+1 + x1i−1 − 2 ⋅
x1i
) where ε
is a mutual coupling coeffi-
i
cient and x1 denotes a running time average of x1i . We report in Fig. 11 the transition from unsynchronized to synchronized regimes by showing the space time representation of the evolution of the array. The transition to phase synchronization is anticipated by regimes where clusters of oscillators spike quasi-simultaneously (Leyva et al., 2003; Zheng, G. Hu, and B. Hu, 1998). The number of oscillators in the clusters increases with ε extending eventually to the whole system (see Fig. 11c, d).
Fig. 11. Space-time representation of a chain of coupled chaotic oscillators for different values of ε : (a) 0.0 , (b) 0.05, (c) 0.1 (d) 0.12 (e) 0.25
A better characterization of the synchronized pattern can be gathered by studying the emerging “defects” (Leyva et al., 2003). Each defect consists of a phase slip; that is a spike that does not induce another spike in its immediate neighbourhood. In order to detect the presence of defects, we map the phase in the interval between two successive spikes of a same oscillator as a closed circle ( 0,2π ), as usual in pulsing systems (For a review of the subject, see Pikovsky, Rosenblum, and Kurths 2001; Boccaletti et al.2002). Notice that since a suitable observer threshold isolates the spikes getting rid of the chaotic small inter-spike background, we care only for spike correlations. With this notation, we will consider that a defect has occurred when the phase of an oscillator has change by 2π while the phase of an immediate neighbour has changed by 4π . Here, our “phase synchronization” term denotes a connected line from left to right, which does not necessarily imply equal time occurrence of spikes at different
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sites; indeed, the wavy unbroken lines which represent the prevailing trend as we increase ε are what we call “phase synchronization”.
7. Evidence of homoclinic chaos in financial market problems with heterogeneous agents Important changes have influenced economic modelling during the last decades. Nonlinearities and noise effects have played a crucial role to explain the occasionally high irregular movements in economic and financial time series (Day 1994; Mantegna and Stanley, 1999). One of the peculiar differences between economics and other sciences is the role of expectations or beliefs in the decisions of agents operating on markets. In rational expectation models, largely used in classic economy [Muth 1961], agents evaluate their expectations on the knowledge of the linear market equilibrium equations. Nowadays the rational equilibrium hypothesis is considered unrealistic and a growing interest is devoted to bounded rationality where agents are using different learning algorithms to predict their beliefs. In the last decade several heterogeneous market models have been introduced where at least two types of agents coexist (Brock and Hommes, 1997; 1998; Arthur, Lane, and Durlauf, 1997; Lux and Marchesi, 1999; Farmer and Joshi, 2002). The first group is composed of fundamentalists who believe that the asset prices are completely determined by economic fundamentals. The other group is composed of chartists or technical traders believing that the asset prices are not determined by their fundamentals but can be derived using trading rules based on previous prices. Brock and Hommes first discovered that a heterogeneous agent model with rational versus naive expectation can exhibit complicated price dynamics when the intensity of choice to switch between strategies is high (Brock and Hommes, 1997, 1998). Agents can either adopt a sophisticated predictor H1 (rational expectation) or another simple and low cost predictor H2 (adaptive short memory or naive expectation). Near the steady state or equilibrium price most of the agents use the naive predictor. Prices are driven far from the equilibrium. However, when prices diverge from their equilibrium, forecasting errors tend to increase and as a consequence it becomes more convenient to switch to the sophisticated predictor. The prices will move back to the equilibrium price. According to Brock and Hommes a heterogeneous market can be considered as a complex adaptive system whose dynamics are ruled by a two dimensional map for the variables pt and mt :
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pt +1 =
− b(1 − mt ) pt = f ( p t , mt ) 2 B + b(1 − mt )
2 ⎛ β ⎛ b ⎛ b(1 − m ) ⎞⎞ ⎞ 2 ⎜ t ⎜ ⎜⎜ mt +1 = tanh + 1⎟⎟ pt − C ⎟ ⎟ = g ( pt , mt ) ⎜ 2 ⎜ 2 2 B + b(mt + 1) ⎟⎟ ⎠ ⎠⎠ ⎝ ⎝ ⎝
(11)
(12)
pt represents the deviations from the steady state price p determined by the intersection of demand and supply. mt is the difference between the two fractions of agents, β is the intensity of choice indicating how fast agents switch predictors. The parameter C is the price difference between the two predictors. The parameter b is related to a linear supply curve derived from q2
a quadratic cost function c(q ) = 2b where q is the quantity of a given nonstorable good. The temporal evolution of the variables m and p at the onset of chaos is reported in Fig. 12. The corresponding chaotic attractor is shown in Fig. 13. This attractor occurs after the merging of the two coexisting 4 piece chaotic attractors. The importance of homoclinic bifurcations leading to chaotic dynamics in economic models with heterogeneous agents has been recently emphasized by I. Foroni and L. Gardini, who extended the theoretical investigations also to noninvertible maps (Foroni and Gardini, 2003; Chiarella, Dieci, and Gardini, 2001).
Fig. 12. Simulation of the model showing the chaotic evolution of the price p and the agents difference m. The used parameters are b = 1.35, B = 0.5, C = 1 and β = 5.3
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8. Conclusions Heterogeneous markets models have received much attention during the last decade for the richness of their dynamical behaviours including homoclinic bifurcations to chaos as pointed out by the simple evolutionary economic model proposed by Brock and Hommes. This is similar to the behaviour of a laser with electro-optic feedback. The feedback process, acting on the same time scale of the other two relevant variables of the system, that is, laser intensity and population inversion, is the crucial element leading to heteroclinic chaos. In such a system, chaos shows interesting features. It can be easily synchronized with respect to small periodic perturbations and, in the presence of noise, it displays several constructive effects such as stochastic resonance. As we have stressed in this paper, homoclinic/heteroclinic chaos is a common feature in many different disciplines including economics. Quite frequently, chaos is harmful, so controlling or suppressing its presence has been widely considered in recent years. However, in some situations chaos is a beneficial behaviour. Typically a chaotic regime disappears as a result of a crisis which is an abrupt change from chaos to a periodic orbit or a steady state solution. This occurrence finds its analogy in microeconomics when a firm is near bankruptcy. In such a critical condition, suitable and careful interventions are necessary in order to recover the usual cycle of the firm.
Fig. 13. Attractor of the dynamics of the model. The values of the parameter are the same reported in Fig. 12
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Other important aspects concern the occurrence of synchronization phenomena in economics. Currently, the attention is on synchronization among the world’s economies considering the greater financial openness. In this way, globalization effects lead to an increase of the links between the world’s economies, in particular by means of capital markets and trade flows.
Authors acknowledge MIUR-FIRB contract n. RBAU01B49F\_002
References Arecchi, F.T., R. Meucci, and W. Gadomski, (1987): “Laser Dynamics with Competing Instabilities”, Phys. Rev. Lett. 58, 2205. Argoul, F., A. Arneodo, and P. Richetti, (1987): “A propensity criterion for networking in an array of coupled chaotic systems”, Phys. Lett. 120A, 269. Arneodo, A., P.H. Coullet, E.A. Spiegel, and C. Tresser, (1985): “Asymptotic chaos”, Physica D 14, 327. Arthur, W., D. Lane, and S. Durlauf (eds.), (1997): The Economy as an Evolving Complex System II, Addison-Wesley, Redwood City, CA. Benzi, R., A. Sutera, and A. Vulpiani, (1981): “The mechanism of stochastic resonance”, J. Phys. A 14, L453. Boccaletti, S., J. Kurths, G. Osipov, D. Valladares, and C. Zhou, (2002): “The synchronization of chaotic systems”, Phys. Rep. 366, 1. Brock, W.A., and C.H. Hommes, (1997): “A rational route to randomness”, Econometrica, 65, 1059. Brock, W.A., and C.H. Hommes, (1998): “Heterogeneous beliefs and routes to chaos in a simple asset pricing model”, Journ. of Econom. Dynam. and Control, 22, 1235. Chiarella, C., R. Dieci, and L. Gardini, (2001): “Asset price dynamics in a financial market with fundamentalists and chartists”, Discrete Dyn. Nat. Soc. 6, 69. Day, R.H., (1994): Complex Economics Dynamics, MIT Press, Cambridge, MA. Farmer, J.D., and S. Joshi, (2002): “The Price Dynamics of Common Trading Strategies”, Journ. of Econ. Behavior and Organization 49, 149. Feudel, U. et al., (2000): “Homoclinic bifurcation in a Hodgkin-Huxley Model of Thermally Sensitive Neurons”, Chaos 10, 231. Foroni, I., and L. Gardini, (2003): “Homoclinic bifurcations in Heterogeneous Market Models”, Chaos, Solitons and Fractals 15, 743-760. Gade, P.M., and C. Basu, (1996): “The origin of non-chaotic behavior in identically driven systems”, Phys. Lett. A 217, 21. Gammaitoni L., P. Hanggi, P. Jung, and F. Marchesoni, (1998): “Stochastic Resonance”, Rev. Mod. Phys. 70, 223. Garcia-Ojalvo, J., and J.M. Sancho, (1999): Noise in Spatially Extended System, Springer, New York.
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Herzel, H., and J. Freund, (1995): “Chaos, noise, and synchronization reconsidered”, Phys. Rev. E 52, 3238. Hilborn, R.C., (1994): Chaos and Nonlinear Dynamics, Oxford University Press, Oxford. Hodgkin, A.L., and A.F. Huxley, (1952): “A quantitative description of membrane current and its application to conduction and excitation in nerve”, J. Physiol. 117, 500. Jensen, R.V., (1998): “Synchronization of randomly driven nonlinear oscillators”, Phys. Rev. E 58, R6907. Leyva, I., E. Allaria, S. Boccaletti, and F. T. Arecchi, (2003): “Competition of synchronization domains in arrays of chaotic homoclinic systems“, Phys. Rev. E 68, 066209. Longa, L., E.M.F. Curado, and A. Oliveira, (1996): “Roundoff-induced coalescence of chaotic trajectories”, Phys. Rev. E 54, R2201. Longtin, A., and D. Chialvo, (1998): “Stochastic and Deterministic Resonances for Excitable Systems”, Phys. Rev. Lett. 81, 4012. Lux, T., and M. Marchesi, (1999): “Scaling and criticality in a stochastic multiagent model of a financial market”, Nature 397, 498. Malescio, G., (1996): “Noise and Synchronization in chaotic systems”, Phys. Rev. 53, 6551. Marino, F. et al., (2002): “Experimental Evidence of Stochastic Resonance in an Excitable Optical System”, Phys. Rev. Lett. 88, 040601. Mantegna, R.N., and H.E. Stanley, (1999): An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press. Matsumoto, K., and I. Tsuda, (1983): “Noise-induced Order”, J. Stat. Phys. 31, 87. Maritan, A., and J.R. Banavar, (1994): “Chaos, Noise and Synchronization”, Phys. Rev. Lett. 72, 1451. Muth, J.F, (1961): “Rational Expectations and the Theory of Price Movements” Econometrica 29, 315. Pikovsky, A., (1994): “Comment on “Chaos, Noise and Synchronization’’”, Phys. Rev. Lett. 73, 2931 . Pikovsky, A., M. Rosenblum, and J. Kurths, (2001): Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge, University Press. Pikovsky, A.S., Radiophys, (1984): “Synchronization and stochastization of an ensemble of self-oscillators by external noise”, Quantum Electron. 27, 576. Pikovsky, A.S., M.G. Rosenblum, G.V. Osipov, and J. Kurths, (1997): “Phase Synchronization of Chaotic Oscillators by External Driving”, Physica D 104. Pisarchik, A.N., R. Meucci, and F.T. Arecchi, (2001): “Theoretical and experimental study of discrete behavior of Shilnikov chaos in a CO2 laser”, Eur. Phys. J. D 13, 385. Sanchez, E., M.A. Matias, and V. Perez-Munuzuri, (1997): “Analysis of synchronization of chaotic systems by noise: An experimental study”, Phys. Rev. E 56, 4068. Wiesenfeld, K., and F. Moss, (1995): “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs”, Nature 373, 33.
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Zheng, Z., G. Hu, and B. Hu, (1998): “Phase Slips and Phase Synchronization of Coupled Oscillators”, Phys. Rev. Lett. 81, 5318.
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Agent Based Models
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Explaining Social and Economic Phenomena by Models with Low or Zero Cognition Agents P. Ormerod, M. Trabatti, K. Glass, and R. Colbaugh
1. Introduction We set up agent based models in which agents have low or zero cognitive ability. We examine two quite diverse socio-economic phenomena, namely the distribution of the cumulative size of economic recessions in the United States and the distribution of the number of crimes carried out by individuals. We show that the key macro-phenomena of the two systems can be shown to emerge from the behaviour of these agents. In other words, both the distribution of economic recessions and the distribution of the number of crimes can be accounted for by models in which agents have low or zero cognitive ability. We suggest that the utility of these low cognition models is a consequence of social systems evolving the “institutions” (e.g., topology and protocols governing agent interaction) that provide robustness and evolvability in the face of wide variations in agent information resources and agent strategies and capabilities for information processing. The standard socio-economic science model (SSSM) assumes very considerable cognitive powers on behalf of its individual agents. Agents are able both to gather a large amount of information and to process this efficiently so that they can carry out maximising behaviour. Criticisms of this approach are widespread, even within the discipline of economics itself. For example, general equilibrium theory is a major intellectual strand within mainstream economics. A key research task in the twentieth century was to establish the conditions under which the existence of general equilibrium could be proved. In other words, the conditions under which it could be guaranteed that a set of prices can be found under
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which all markets clear. Radner (1968) established that when agents held different views about the future, the existence proof appears to require the assumption that all agents have access to an infinite amount of computing power. A different criticism is made by one of the two 2002 Nobel prize winners. One of them, Kahneman (2002), describes his empirical findings contradicting the SSSM model as having been easy, because of the implausibility of the SSSM to any psychologist. An alternative approach is to ascribe very low or even zero cognitive ability to agents. The second 2002 Nobel laureate, Smith (2002), describes results obtained by Gode and Sunder. An agent based model is set up for a single commodity, operating under a continuous double auction. The agents choose bids or asks completely at random from all those that do not impose a loss on the agent. They use no updating or learning algorithms. Yet, as Smith notes, these agents “achieve most of the possible social gains from trade”. In practice, agents may face environments that are both high dimensional and not time-invariant. Colbaugh and Glass (2003) set out a general model to describe the conditions under which agents can and cannot learn to behave in a more efficient manner, suggesting that the capability of agents to learn about their environment under these conditions is in general low. Ormerod and Rosewell (2003) explain key stylised facts about the extinction of firms by an agent based model in which firms are unable to acquire knowledge about either the true impact of other firms’ strategies on its own fitness, or the true impact of changes to its own strategies on its fitness. Even when relatively low levels of cognitive ability are ascribed to agents, the model ceases to have properties that are compatible with the key stylised facts. In this paper, we examine two quite diverse socio-economic phenomena, namely the distribution of the cumulative size of economic recessions in the United States and the distribution of the number of crimes carried out by individuals. We set up agent based models in which agents have low or zero cognitive ability. We show that the key macro-phenomena of the two systems can be shown to emerge from the behaviour of these agents. In other words, both the distribution of economic recessions and the distribution of the number of crimes can be accounted for by models in which agents have low or zero cognitive ability. Section 2 describes the model of the cumulative size of economic recessions, and section 3 sets out the crime model. Section 4 concludes by offering an explanation for the efficacy of these simple models for complex socio-economic phenomena.
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2. Economic recessions The cumulative size of economic recessions, the percentage fall in real output from peak to trough, is analysed for 17 capitalist economies by Ormerod (2004). We consider here the specific case of the United States 1900-2004. There are in total 19 observations, ranging in size from the fall of barely one fifth of one per cent in GDP in the recession of 1982, to the fall of some 30 per cent in the Great Depression of 1930-33. Fig. 1 plots the histogram of the data, using absolute values. On a Kolmogorov-Smirnov test, the null hypothesis that the data are described by an exponential distribution with rate parameter = 0.3 is not rejected at the standard levels of statistical significance (rejected at p = 0.59). The observation for the Great Depression is an outlier, suggesting the possibility of a bi-modal distribution, but technically the data follow an exponential distribution. An agent based model of the business cycle that accounts for a range of key features of American cycles in the twentieth century is given in Ormerod (2002). These include positive cross-correlation of output growth between individual agents, and the autocorrelation and power spectrum properties of aggregate output growth. The cumulative size distribution of recession was not, however, considered. The model is populated by firms, which differ in size. These agents operate under uncertainty, are myopic, and follow simple rules of thumb behaviour rather than attempting to maximise anything. In other words, in terms of cognition they are at the opposite end of the spectrum to firms in the standard SSSM. The model evolves in a series of steps, or periods. In each period, each agent decides its rate of growth of output for that period, and its level of sentiment (optimism/pessimism) about the future. Firms choose their rate of growth of output according to:
xi (t ) = ∑ wi y i (t − 1) + ε i (t )
(1)
i
where xi(t) is the rate of growth of output of agent i in period t, yi(t) is the sentiment about the future of the ith agent, and wi is the size of each individual firm (the size of firms is drawn from a power law distribution). Information concerning the sentiment of firms’ about the future can be obtained readily by reading, for example, the Wall Street Journal or the Financial Times. The variable εi(t) plays a crucial role in the model. This is a random variable drawn separately for each agent in each period from a normal distribution with mean zero and standard deviation sd1. Its role is to reflect
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4 0
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Fig. 1. Histogram of cumulative absolute percentage fall in real US GDP, peak to trough, during recessions 1900-2002
The implications of any given level of overall sentiment for the growth rate of output of a firm differs both across the N agents and over time. Firms are uncertain about the precise implications of a given level of sentiment for the exact amount of output that they should produce. Further, the weighted sum of the sentiment of firms is based upon an interpretation of a range of information that is in the public domain. Agents again differ at a point in time and over time in how they interpret this information and in consequence differ in the value that they attach to this overall level of sentiment. The model is completed by the decision rule on how firms decide their level of sentiment: ⎡ ⎤ y i (t ) = (1 − β ) y i (t − 1) − β ⎢∑ wi x i (t − 1) + η i (t )⎥ ⎣ i ⎦
(2)
where ηi is drawn from a normal distribution with mean zero and standard deviation sd2. The coefficient on the weighted sum of firms’ output growth in the previous period, β, has a negative sign, reflecting the Keynesian basis of
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the model. The variable ηi(t) again reflects agent heterogeneity and uncertainty. At any point in time, each agent is uncertain about the implications of any given level of overall output growth in the previous period for its own level of sentiment. In this model, it is as if agents operate on a fully connected network. Each agent takes account of the previous level of overall sentiment and output. In other words, it takes account of the decisions of all other agents in the model. In this context, this seems a reasonable approximation to reality. Information that enables firms to form a view on the previous overall levels of output and sentiment is readily available in the pubic domain, either from official economic statistics or from more general comment in the media. In practice, of course, whilst taking account of the overall picture, firms are likely to give particular weight to prospects in their own sector or sectors, so the actual network across which information is transmitted will contain a certain amount of sparsity. But the assumption of a fully connected network to transmit information seems reasonable. This apparently simple model is able to replicate many underlying properties of time series data on annual real output growth in the United States during the twentieth century. Fig. 2 below compares the cumulative distribution functions of total cumulative fall in real output in recessions generated by the theoretical model, and of the actual cumulative falls of real output in America in the twentieth century. The model is run for 5,000 periods; the number of agents is 100; β = 0.4, sd(ε) = 0.04, sd(η) = 0.5. On a Kolmogorov-Smirnov test, the null hypothesis that the two cumulative distribution functions are the same is only rejected at p = 0.254. In other words, equations (1) and (2) are able to generate economic recessions whose cumulative size distribution is similar to that of the actual data for the US.
3. Crimes by individuals Cook et al. (2004) examine the distribution of the extent of criminal activity by individuals in two widely cited data bases. The Pittsburgh Youth Study measures self-reported criminal acts over intervals of six months or a year in three groups of boys in the public school system in Pittsburgh, PA. The Cambridge Study in Delinquent Development records criminal convictions amongst a group of working class youths in the UK over a 14year period.
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The range of the data is substantially different between these two measures of criminal activity, since one is based on convictions and the other on self-reported acts. However, there are similarities in characteristics of the data sets. Excluding the frequency with which zero crimes are committed or reported, a power law relationship between the frequency and rank of the number of criminal acts per individual describes the data well in both cases, and fits the data better than an exponential relationship. The exponent is virtually identical in both cases. A better fit is obtained for the tail of the distribution. The data point indicating the number of boys not committing any crime does not fit with the power law that characterizes the rest of the data; perhaps a crucial step in the criminal progress of an individual is committing the first act. Once this is done, the number of criminal acts committed by an individual can take place on all scales.
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Fig. 2. Cumulative distribution functions of the percentage fall in real output during economic recessions. The solid line is the empirical distribution function of real US GDP, peak to trough, during recessions 1900-2004. The dotted line is the distribution function of total output series generated by equations (1) and (2)
The stylised facts that characterize both data sets can be described as: • Approximately two-thirds of all boys in the sample commit no crime during a given period of time. • The distribution of the number of crimes committed by individuals who did commit at least one crime fits a power law with exponent of –1.
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We aim to build a model that reconstructs these stylized facts. We imagine that a cohort of youths arrives at the age at which they start to commit crime. These youths are from relatively low income backgrounds, and are themselves relatively unskilled. This particular social type is responsible for a great deal of the total amount of crime that is committed. Of course, in reality, different individuals begin committing crimes at different ages with different motivations. The idea of this model is that the opportunity to commit a crime presents itself sequentially to a cohort over time. We use preferential attachment to represent the process by which a crime opportunity becomes attached to a particular individual, and so becomes an actual rather than a potential crime. Preferential attachment is widely used to describe how certain agents become extremely well connected in a network of interpersonal relationships as existing connections provide opportunities to establish new connections (Newman 2003, Albert and Barabasi 2002). Briefly, to grow a network of this type, new connections are added to the nodes of a graph with a probability equal to the proportion of the total number of connections that any given node has at any particular time. At the outset of our model, none of the agents have committed a crime. During each step a crime opportunity arises and is presented to an agent for attachment with a probability that increases with the number of crimes the agent has already committed. In the beginning, each agent has the same probability to experience a crime opportunity and at each step the opportunity arises for the jth individual to commit a crime with the following probability:
(
Pj = (n j + ε ) / ∑ j=1 n j + εN N
)
(3)
where 1/ε represents approximately how many times the number of past crimes committed is more important than just belonging to the cohort in order to receive a new crime opportunity. Once a crime opportunity is attached, the individual decides whether to commit the crime with a probability that increases with the number of past crimes committed. This is a reasonable hypothesis since, as more crimes are committed, an individual progressively loses scruples about committing new crimes. Initially agents have a heterogeneous attitude towards committing a crime (i.e. probability to commit a crime once they have the opportunity) that increases every time they commit a crime. The initial probability for each agent is drawn from a Gaussian distribution (µ,σ) and then increased by a factor of α for every crime committed.
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The model is run for 200 agents with ε = 0.4, µ = σ = 0.2, α = 0.1 and the results closely match the stylized facts found for the data set mentioned above. Out of 200 agents, 70 commit at least one crime and the cumulative distribution of the number of crimes committed by those with at least one crime follows a power law with exponent of -0.2 (so that the probability density has an exponent of −1.2), in excellent agreement with the stylized facts from the empirical study. It should be observed that the ratio of parameters ε and µ influence the number of individuals not committing crime and this may have a reasonable explanation in the fact that in a community the value of having committed past crimes decreases as the facility to commit crimes arises.
Fig. 3. Cumulative Distribution function of number of crimes committed per criminal. Logarithmic scale
4. Discussion The preceding examples demonstrate that agent-based models in which agents have zero or low cognitive ability are able to capture important aspects of “real world” social systems. We suggest that the utility of these low cognition models may be a consequence of social systems evolving the “institutions” (e.g., topology and protocols governing agent interaction) that provide robustness and evolvability in the face of wide variations in
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agent information resources and agent strategies and capabilities for information processing. More precisely, suppose that the socio-economic system of interest evolves to realize some “objective”, using finite resources in an uncertain world, and by incrementally: (1) correcting observed defects, and (2) adopting innovations. In this case we can prove that such systems are “robust, yet fragile” and allow “deep information from limited data”, and that these properties increase as system maturity increases (Colbaugh and Glass, 2003). Further, if robustness and evolvability are both important, we can show that: (1) system sensitivity to variations in topology increases as the system matures, and (2) system sensitivity to variations in vertex characteristics decreases as the system matures (Colbaugh and Glass, 2004). In other words, these results suggest that as complex social systems (of this class) “mature”, the importance of accurately modeling their agent characteristics decreases. The implication is the degree of cognition that is assigned to agents in socio-economic networks decreases as the system matures. Further, the importance of characterizing the network or institutional structure by which the agents are connected increases. The implication is that successful institutional structures evolve to enable even low cognition agents to arrive at “good” outcomes, and that in such situations the system behaves as if all agent information processing strategies are simple.
References Albert, R., and A. Barabasi (2002): Statistical mechanics of complex networks, Rev. Mod. Physics, Vol. 74, pp. 47-97. Colbaugh, R., and K. Glass (2003): Information extraction in complex systems, Proc. 2003 NAACSOS Conference, Pittsburgh, PA, June (plenary talk). Colbaugh, R., and K. Glass (2004): Simple models for complex social systems, Technical Report, U.S. Department of Defense, February. Cook W., P. Ormerod, and E. Cooper (2004): Scaling behaviour in the number of criminal acts committed by individuals, Journal of Statistical Mechanics: Theory and Experiment, 2004, P07003. Kahneman, D. (2002): Nobel Foundation interview, http://www.nobel.se/economics/ laureates/2002/khaneman-interview.html Newman, M. (2003): The structure and function of complex networks, SIAM Review, Vol. 45, No. 2, pp. 167-256. Ormerod, P. (2002): The US business cycle: power law scaling for interacting units with complex internal structure, Physica A, 314, pp.774-785. Ormerod, P. (2004): Information cascades and the distribution of economic recessions in capitalist economies, Physica A, 341, 556-568, 2004
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Ormerod, P., and B. Rosewell (2003): What can firms know? Proc. 2003 NAACSOS Conference, Pittsburgh, PA, June (plenary talk). Radner, R. (1968): Competitive Equilibrium Under Uncertainty, Econometrica, 36. Smith, V. L. (2003): Constructivist and ecological rationality in economics, American Economic Review, 93, pp. 465-508.
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Information and Cooperation in a Simulated Labor Market: A Computational Model for the Evolution of Workers and Firms S. A. Delre and D. Parisi
1. Introduction In free markets workers and firms exchange work for salaries and they have both competing and mutual interests. Workers need to work to get a salary but they are interested in getting as high a salary as possible. Firms need to hire workers but they are interested in paying them as low a salary as possible. At the same time both categories need each other. Neoclassical microeconomics formalizes the labor market as any other goods market. Workers represent the supply of labor and firms represent the demand. On one hand workers are assumed to have perfect knowledge about wages and the marginal rate of substitution between leisure and income so that they can decide how to allocate their hours: work in order to increase the income and leisure in order to rest. On the other hand, firms know the level of wages, the price of the market, and the marginal physical product of labor (the increasing production resulting from the increase of one unit of labor) so that they can compute the quantity of labor hours they need. At the macro level, workers’ supplies and firms’ demands are aggregated and they intersect in the equilibrium point determining the level of wage and the level of employment. Neoclassical formalization of the labor market is the point of departure for many other works about labor markets (Ehrenberg and Smith, 1997; Bresnahan, 1989). However, the micro foundation of neoclassical economics is a strong simplification of reality and it does not take into consideration many other important factors like fairness (Rees, 1993; Fehr and Schmidt, 1999), bargaining power (Holt, 1995), information matching and social contacts (Rees and Shultz, 1966; Mont-
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gomery, 1991; Granovetter, 1995). In order to include these phenomena in the analysis of labor markets, neoclassical assumptions need to be changed or at least relaxed and extended. Moreover, neoclassical economics provides a static explanation of the equilibrium in which macro-variables should be if workers and firms were always rational and information were always completely available. In the last three decades, alternative approaches have been born and have flourished, such as evolutionary economics (Nelson and Winter, 1982; Dosi and Nelson, 1994; Arthur et al., 1997) and agent-based computational economics (Epstein and Axtell, 1996; Tesfatsion, 2002a). Both approaches have in common the idea that economic agents have bounded rationality. The former focuses on the fitness of agents’ behaviors in the environment and how these behaviors adapt and evolve under the pressure of selection rules. The latter uses computational models in order to simulate economic agents’ behaviors and it shows how macro-regularities of the economy emerge from the micro-rules of the interactions of economic agents. These approaches have introduced new interests also in the field of labor markets and many new works have appeared in order to explain crucial stylized facts like job concentration (Tesfatsion, 2001), the Beveridge curve, the Phillips curve vs the wage curve, Okun’s curve (Fagiolo et al., 2004), and equality and segregation (Tassier and Menczer, 2002). In this paper we present an agent-based model in order to study (a) the evolution of different labor markets, (b) the effects of unions on the labor market; and (c) the effects of social network structures on the value of information for workers. First, labor markets differ according to the constraints firms have when hiring new workers. Either employers are completely free to have a one-toone bargaining process with the worker or they have to respect some lows like minimum wage, some bargaining norms like fairness, or some policy like a wage indexation scale. In this paper we present a comparison of the two extremes of this continuum: in the first scenario firms are completely constrained and they cannot change and adjust their behaviors; in the second scenario firms are able to change, evolve, and enter and leave the market according to their performance. Here we present the effects of these different situations on total production of the market, levels of salaries, firms’ profits, and value of information about a job. Second, in labor markets labor unions increase workers’ bargaining power. Unions limit competition among workers and they try to defend unionized employees’ interests by means of collective bargaining. However, unions’ influences can also affect non-unionized workers because they strongly influence the evolution of firms’ strategies. The model allows us to introduce a collective bargaining where a fraction of worker agents is
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unionized and it shows how this unionization affects the average level of salaries. Third, many studies have shown that informal hiring methods like employee referrals and direct applications are very relevant when workers are looking for a new job (Rees and Schultz, 1970; Granovetter, 1995). Friends and social contacts are big sources of information about employment information. Montgomery (1991) has analyzed four databases about alternative job-finding methods and he found that approximately half of all employed workers found the job through social contacts like friends and relatives. We have connected the worker agents of our multi-agent model in different network structures and we have studied the effects of such structures on the value of information workers have about a job.
2. The model The categories of workers and firms are in a relation of reciprocal activation or mutualism (Epstein, 1997). The two categories need each other and when a category increases in number, it feeds back to the other category. Workers need firms in order to get a salary and firms need workers to develop, to produce, and to sell goods. On the one hand, in any given market, if workers were to disappear, firms also would disappear and vice versa. On the other hand, if firms increased in number because of more resources of the environment, also workers would increase in number in order to use those resources. In such an artificial world, assuming also infinite resources in the environment, workers’ and firms’ populations would simply increase exponentially, but if we assume a limit for available resources (carrying capacity), the process is blocked and at a certain point workers and firms will stop increasing in number. However, although their mutual relationship induces both workers and firms to collaborate in order to increase in number, they compete with each other in that they have opposite interests: workers aim at getting higher salaries from firms and firms aim at paying lower salaries to workers because this enables them to compete more successfully with other firms in the market. Workers want many firms offering high salaries and firms want many workers accepting low salaries. We have reproduced such a situation in an agent-based simulation where worker agents and entrepreneur agents are two separate categories and they live together in an environment with a given carrying capacity. The simulation proceeds with discrete time-steps (cycles). A local selection algorithm (Menczer and Relew, 1996) has been used to evolve the behavior of worker and entrepreneur agents. Both worker agents and entre-
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preneur agents are born, live, reproduce, and die. At birth each agent is endowed with a certain energy of which a constant quantity (individual consumption) is consumed at each time step in order for the agent to remain alive. If an agent’s energy goes to zero, the agent dies. To survive, the agent must procure other energy to reintegrate the consumed energy. As long as an agent succeeds in remaining alive, the agent periodically (every K1 cycles of the simulation) generates an offspring, i.e., a new agent of the same category, that inherits the genotype of its single parent. All agents die at a certain maximum age (K2 cycles of the simulation). The agents that are better able to procure energy live longer and have more offspring. Consequently the two populations of agents vary in number during the simulation run according to the strategies of the agents, and those agents that adopt the best strategies will evolve better then the others. At each time-step agents have costs of energy and gathering of energy: worker agents gain new energy getting salaries and entrepreneurs getting profits. Equations 1 and 2 describe respectively the evolution of energy for worker and entrepreneur agents: ewi , t = ewi , t − 1 − Cp − hi * Cr + si , t eej , t = eej , t − 1 − Cp + πj , t
(1) (2)
where ewi,t indicates the energy of worker agent i at time t, Cp are fixed costs for individual consumption in order to survive, hi, Cr are fixed costs for job research (hi indicates the number of offers evaluated by the worker agent i and Cr indicates fixed costs for each offer), si,t indicates the salary the worker agent i gets for its job, eej,t indicates the energy of entrepreneur agent j at time t, and πj,t indicates its profits at time t. Equation 3 describes profits of entrepreneur agent j:
πj , t = P * yi , t − Cf − ∑ sq, t
(3)
q
revenue P*yj,t (where p is the fixed price and yj,t is the production) minus fixed costs of the entrepreneur’s firm Cf and salaries paid to the employed worker agents. The genetic algorithm permits the evolution of both worker and entrepreneur agents’ behaviors. Each agent, both worker and entrepreneur, possesses a different “genotype” that determines its behavior. For worker agents, the genotype specifies the minimum salary (minS) that the agent is ready to accept from an entrepreneur agent and the number of different entrepreneur agents (h) that it contacts when, being unemployed, it is looking for a job. For entrepreneur agents, the genotype only specifies the maximum salary (maxS) that it is ready to pay to a worker agent when the entrepreneur agent hires it. When a worker agent i is jobless, i contacts h en-
Information and cooperation in a simulated labor market
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trepreneur agents and it takes into consideration only the highest offer of entrepreneur agent j. Worker agent’s genotype (minSi) and entrepreneur agent’s genotype (maxSj) are compared and the worker agent i starts working for the entrepreneur agent j only if maxSj is equal to or higher than minSi. When a contract is set up, the salary of the worker agent will be set to the mean of maxSj and minSi as indicated in equation (4). ⎛1 ⎞ si , t = min Si + ⎜ (max Sj − min Si )⎟ ⎝2 ⎠
(4)
All working contracts have the same length (K3 cycles of the simulation) and, when a contract expires, the worker agent becomes jobless and it looks again for a new job. For both worker agents and entrepreneur agents it is convenient that maxS and minS evolve towards stable values in order to increase the chances of contracts to start. At the same time, for worker agents it is not convenient to have a very low minS because that means low salaries and for entrepreneur agents it is not convenient to have a high maxS because that means high costs and less profits. The evolution of min S and max S (respectively average among worker agents and average among entrepreneur agents) indicates how the relation between worker agents and entrepreneur agents changes during the time of the simulation. Finally, the second part of workers’ genotype measures for a worker agent i how important information about a job is when it is jobless. The higher hi, the higher the chances to find a more remunerative job. However, also hi has a given cost (Cr) so that the evolution of h (average among worker agents) indicates how much worker agents are ready to pay to have more information about job offers. Entrepreneur agents continue hiring worker agents till the marginal revenue (the increase in production from hiring one more worker agent) is higher than or equal to the marginal cost (the salary paid to the new worker agent). We assume that the production curve is convex; that is, the marginal physical product of labor is declining (i.e., the more worker agents are assumed by the entrepreneur agent, the less production will increase from hiring a new worker agent). Equation 5 describe the production curve of the firm of entrepreneur agent j: yj = αNz
(5)
where yj is the production of the firm of entrepreneur agent j, N is the number of workers employed by j, α is a constant indicating how much production an additional worker agent guaranties, and 0
