Modeling Aggregate Behavior and Fluctuations in Economics: Stochastic Views of Interacting Agents

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This book analyzes how a large but finite number of agents interact, and what sorts of macroeconomic statistical regularities or patterns may evolve from these interactions. By keeping the number of agents finite, the book examines situations such as fluctuations about equilibria, multiple equilibria and asymmetrical cycles of models which are caused by model states stochastically moving from one basin of attraction to another. All of these are not tractable using traditional deterministic modeling approaches. The book also discusses how agents may form clusters with stationary distributions of cluster sizes. These have important applications in analyzing volatilities of asset returns.
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Editorial Reviews

Discussing both stochastic dynamics and stochastic random combinatorial analysis, Aoki (emeritus, economics, U. of California in Los Angeles) attempts to reformulate macroeconomic models employed by mainstream economists. He argues that models with finite numbers of agents in stochastic contexts reveal economic phenomena that fail to show in deterministic models with infinite number of agents. His approach emphasizes that an outcome of interactions of a large number of agents facing idiosyncratic shocks cannot be described by traditional approaches. He formulates and analyzes a large but finite number of interacting economic agents as continuous-time Markov chains with discrete spaces. Annotation c. Book News, Inc., Portland, OR (
From the Publisher
'This is a most welcome treatment of stochastic dynamics in economic models, and its significance for theoretical and applied economics … Professor Aoki has provided a self-contained description of all necessary and relevant concepts of the modelling toolbox, including key stochastic dynamics concepts such as the Master Equation (the backward Kolmogorov-Chapman Equation), the Fokker Planck equation, Jump Markov Processes, Random Partitions and Combinatorial Randomness.' Asia Pacific Journal of Management

'Once again Masanao Aoki has written a book ahead of its time. This time he applies techniques popular in the hard sciences, but alien to most economists, to analyse the dynamic interaction of a large (finite) number of agents (or entities.) He characterises types of agents by their decisions and the movement between types as endogenous transition probabilities. Aoki uses random combinatorial analysis to describe the distribution of types, and the backward Chapman-Kolmogorov equations to describe the aggregate dynamics. This is an innovative approach to characterise a complicated dynamic stochastic economic environment.' Roger Craine, University of California, Berkeley

'In recent years, the economics profession has experienced a growing interest in modelling economic phenomena as the outcome of the interactions of heterogeneous individuals. Although interaction and heterogeneity appear to be natural and pervasive features of modern economies, the formerly dominating tradition of modelling representative agents has eschewed both to a large extent. However, the increasing awareness of the importance of these features also requires the adaptation or development of appropriate technical tools. Professor Aoki has done a great service to the profession by providing a unified treatment of methods for stochastic modelling of large collections of heterogeneous agents. As a companion and sequel to his seminal New Approaches to Macroeconomic Modeling, the present volume both adds advanced material and demonstrates the power of the statistical approach by applying it to important classes of economic models. With its unique coverage and rigourous presentation, the book will immediately become a standard reference in the exciting new area of interacting agent modelling in economics.' Thomas Lux, University of Kiel

'Economists have long taken optimization as first principles in economics. This belief has lead many to the Walrasian or the intertemporal utility maximization model, and has fundamentally changed macroeconomics. Unfortunately, the standard model such as real business cycle theory looks hopelessly unrealistic. It may sound like a paradox that the model which is based on explicit maximization of the representative agent is so unsatisfactory as a macro model. Professor Aoki's book gives us a good answer to this question, and at the same time, provides us with a new methodology in macroeconomics. He convincingly argues that Jump Markov Process is a powerful and fruitful tool in macroeconomics. Professor Aoki explains the method, and also provides examples. The book will be remembered as a landmark.' Hiroshi Yoshikawa, University of Tokyo

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Product Details

  • ISBN-13: 9780521606196
  • Publisher: Cambridge University Press
  • Publication date: 9/28/2004
  • Edition description: New Edition
  • Pages: 263
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.71 (d)

Meet the Author

Masanao Aoki is Professor Emeritus in the Department of Economics at the University of California, Los Angeles. He also held professorial appointments at the Institute for Social and Economic Research at Osaka University, Tokyo Institute of Technology, and the University of Illinois. Professor Aoki is a past President of the Society for Economic Dynamics and Control, a Fellow of the Econometric Society, and a Fellow of the IEEE Control Systems Society. Currently Associate Editor of the journal Macroeconomic Dynamics published by Cambridge University Press, he also served as Editor of the Journal of Economic Dynamics and Control and the International Economic Review, and Associate Editor of the IEEE's Transaction of Automatic Control, Information Sciences, and the Journal of Mathematical Analysis and Application. Professor Aoki is the author or editor of ten books, including New Approaches to Macroeconomic Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects (Cambridge University Press, 1996).

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Table of Contents

1 Overviews 1
1.1 Our Objectives and Approaches 1
1.2 Partial List of Applications 2
1.3 States: Vectors of Fractions of Types and Partition Vectors 3
1.4 Jump Markov Processes 6
1.5 The Master Equation 7
1.6 Decomposable Random Combinatorial Structures 8
1.7 Sizes and Limit Behavior of Large Fractions 8
2 Setting Up Dynamic Models 9
2.1 Two Kinds of State Vectors 10
2.2 Empirical Distributions 11
2.3 Exchangeable Random Sequences 12
2.4 Partition Exchangeability 13
2.5 Transition Rates 16
2.6 Detailed-Balance Conditions and Stationary Distributions 17
3 The Master Equation 19
3.1 Continuous-Time Dynamics 19
3.2 Power-Series Expansion 23
3.3 Aggregate Dynamics and Fokker-Planck Equation 25
3.4 Discrete-Time Dynamics 25
4 Introductory Simple and Simplified Models 27
4.1 A Two-Sector Model of Fluctuations 27
4.2 Closed Binary Choice Models 30
4.3 Open Binary Models 32
4.4 Two Logistic Process Models 35
4.5 An Example: A Deterministic Analysis of Nonlinear Effects May Mislead! 40
5 Aggregate Dynamics and Fluctuations of Simple Models 41
5.1 Dynamics of Binary Choice Models 41
5.2 Dynamics for the Aggregate Variable 43
5.3 Potentials 45
5.4 Critical Points and Hazard Function 47
5.5 Multiplicity - An Aspect of Random Combinatorial Features 49
6 Evaluating Alternatives 52
6.1 Representation of Relative Merits of Alternatives 53
6.2 Value Functions 54
6.3 Extreme Distributions and Gibbs Distributions 57
6.4 Approximate Evaluations of Value Functions with a Large Number of Alternatives 60
6.5 Case of Small Entry and Exit Probabilities: An Example 60
6.6 Approximate Evaluation of Sums of a Large Number of Terms 61
6.7 Approximations of Error Functions 62
7 Solving Nonstationary Master Equations 66
7.1 Example: Open Models with Two Types of Agents 66
7.2 Example: A Birth-Death-with-Immigration Process 69
7.3 Models for Market Shares by Imitation or Innovation 75
7.4 A Stochastic Model with Innovators and Imitators 80
7.5 Symmetric Interactions 84
8 Growth and Fluctuations 85
8.1 Two Simple Models for the Emergence of New Goods 87
8.2 Disappearance of Goods from Markets 90
8.3 Shares of Dated Final Goods Among Households 93
8.4 Deterministic Share Dynamics 95
8.5 Stochastic Business-Cycle Model 96
8.6 A New Model of Fluctuations and Growth: Case with Underutilized Factor of Production 99
8.7 Langevin-Equation Approach 117
8.8 Time-Dependent Density and Heat Equation 121
8.9 Size Distribution for Old and New Goods 122
9 A New Look at the Diamond Search Model 127
9.1 Model 129
9.2 Transition Rates 129
9.3 Aggregate Dynamics: Dynamics for the Mean of the Fraction 130
9.4 Dynamics for the Fluctuations 131
9.5 Value Functions 132
9.6 Multiple Equilibria and Cycles: An Example 134
9.7 Equilibrium Selection 138
9.8 Possible Extensions of the Model 139
10 Interaction Patterns and Cluster Size Distributions 141
10.1 Clustering Processes 141
10.2 Three Classes of Transition Rates 144
10.3 Transition-Rate Specifications in a Partition Vector 153
10.4 Logarithmic Series Distribution 153
10.5 Dynamics of Clustering Processes 157
10.6 Large Clusters 165
10.7 Moment Calculations 171
10.8 Frequency Spectrum 172
10.9 Parameter Estimation 178
11 Share Market with Two Dominant Groups of Traders 180
11.1 Transition Rates 181
11.2 Ewens Distribution 183
11.3 Market Volatility 187
11.4 Behavior of Market Excess Demand 188
A.1 Deriving Generating Functions via Characteristic Curves 195
A.2 Urn Models and Associated Markov Chains 197
A.3 Conditional Probabilities for Entries, Exits, and Changes of Type 200
A.4 Holding Times and Skeletal Markov Chains 202
A.5 Stirling Numbers 206
A.6 Order Statistics 213
A.7 Poisson Random Variables and the Ewens Sampling Formula 214
A.8 Exchangeable Random Partitions 219
A.9 Random Partitions and Permutations 224
A.10 Dirichlet Distributions 229
A.11 Residual Allocation Models 234
A.12 GEM and Size-Biased Distributions 235
A.13 Stochastic Difference Equations 240
A.14 Random Growth Processes 242
A.15 Diffusion Approximation to Growth Processes 243
References 245
Index 253
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