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

	Preface
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