CONTENTS

Preface xiii

1 Overviews 1

1.1 Our Objectives and Approaches 1

1.2 Partial List of Applications

1.3 States: Vectors of Fractions of Types and Partition Vectors

1.3.1 VectorsofFractions 4

1.3.2 PartitionVectors 5

1.4 Jump Markov Processes 6

1.5 The Master Equation

1.6 DecomposableRandomCombinatorialStructures 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 PartitionExchangeability 13

2.5 TransitionRates 16

2.6 Detailed-Balance Conditions and Stationary Distributions

3 The Master Equation 19

3.1 Continuous-TimeDynamics 19

3.2 Power-Series Expansion 23

3.3 Aggregate Dynamics and Fokker-Planck Equation 25

3.4 Discrete-TimeDynamics

4 Introductory Simple and Simplified Models 27

4.1 ATwo-SectorModelofFluctuations 27

4.2 Closed Binary Choice Models 30

4.2.1 A POlya Distribution Model 31

4.3 Open Binary Models 32

4.3.1 Examples 35

4.4 Two Logistic Process Models 35

4.4.1 Model 1: The Aggregate Dynamics and Associated

Fluctuations 35

4.4.2 Model 2: Nonlinear Exit Rate

4.4.3 ANonstationaryPcSlyaModel 38

4.5 An Example: A Deterministic Analysis of Nonlinear Effects

May Mislead! 40

5 Aggregate Dynamics and Fluctuations of Simple Models 41

5.1 DynamicsofBinaryChoiceModels 41

5.2 Dynamics for the Aggregate Variable 43

5.3 Potentials 45

5.4 CriticalPointsandHazardFunction 47

5.5 Multiplicity - An Aspect of Random Combinatorial

Features 49

6 Evaluating Alternatives 52

6.1 Representation of Relative Merits of Alternatives

6.2 Value Functions 54

6.3 Extreme Distributions and Gibbs Distributions 57

6.3.1 Type I: Extreme Distribution 58

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

6.7.1 Generalization 64

6.7.2 Example 65

7 Solving Nonstationary Master Equations 66

7.1 Example: Open Models with Two Types of Agents 66

7.1.1 Equilibrium Distribution 67

7.1.2 Probability-Generating-FunctionMethod 67

7.1.3 Cumulant Generating Functions 68

7.2 Example: A Birth-Death-with-Immigration Process 69

7.2.1 StationaryProbabilityDistribution 70

7.2.2 Generating Function 70

7.2.3 Time-Inhomogeneous Transition Rates 73

7.2.4 The Cumulant-Generating-Function 74

7.3 Models for Market Shares by Imitation or Innovation 75

7.3.1 DeterministicInnovationProcess 76

7.3.2 DeterministicImitationProcess 77

7.3.3 AJointDeterministicProcess 78

7.3.4 AStochasticDynamicModel 79

7.4 A Stochastic Model with Innovators and Imitators 80

7.4.1 Case of a Finite Total Number of Firms 83

7.5 Symmetric Interactions 84

7.5.1 StationaryStateDistribution 84

7.5.2 Nonstationary Distributions 84

8 Growth and Fluctuations 85

8.1 TwoSimpleModelsfortheEmergenceofNewGoods 87

8.1.1 Poisson Growth Model 87

8.1.2 An Um Model for Growth 88

8.2 Disappearance of Goods from Markets 90

8.2.1 Model 91

8.2.2 StabilityAnalysis 92

8.3 Shares of Dated Final Goods Among Households 93

8.3.1 Model 94

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

8.6.1 TheModel 100

8.6.2 Transition-Rate Specifications 101

8.6.3 Holding Times 102

8.6.4 AggregateOutputsandDemands 103

8.6.5 Equilibrium Sizes of the Sectors (Excess

Demand Conditions) 104

8.6.6 Behavior Out of Equilibrium: Two-Sector Model 105

8.6.7 Stationary Probability Distribution:

The Two-Sector Model 107

8.6.8 Emergence of New Sectors 110

8.6.9 SimulationRunsforMulti-SectorModel 111

8.6.10 Discussion 112

8.7 Langevin-EquationApproach 117

8.7.1 Stationary Density Function 119

8.7.2 The Exponential Distribution of the Growth

RatesofFirms 119

8.8 Time-DependentDensityandHeatEquation 121

8.9 SizeDistributionforOldandNewGoods 122

8.9.1 DiffUsion-EquationApproximation 122

8.9.2 Lines of Product Developments and Inventions 123

9 A New Look at the Diamond Search Model 127

9.1 Model 129

9.2 TransitionRates 129

9.3 Aggregate Dynamics: Dynamics for the Mean of

the Fraction 130

9.4 DynamicsfortheFluctuations 131

9.5 Value Functions 132

9.5.1 Expected-ValueFunctions 133

9.6 Multiple Equilibria and Cycles: An Example 134

9.6.1 AsymmetricalCycles 136

9.6.1.1 Approximate Analysis 136

9.6.1.2 Example 137

9.7 Equilibrium Selection 138

9.8 Possible Extensions of the Model 139

10 Interaction Patterns and Cluster Size Distributions 141

10.1 ClusteringProcesses 141

10.2 Three Classes of Transition Rates 144

10.2.1 Selections 144

10.2.2 Multisets 146

10.2.2.1 Capacity-LimitedProcesses 150

10.2.3 Assemblies 150

10.2.3.1 InternalConfigurationsofAssemblies 151

10.3 Transition-RateSpecificationsinaPartitionVector 153

10.4 LogarithmicSeriesDistribution 153

10.4.1 Frequency Spectrum of the Logarithmic

SeriesDistribution 156

10.5 DynamicsofClusteringProcesses 157

10.5.1 ExamplesofClustering-ProcessDistributions 157

10.5.2 Example: Ewens Sampling Formula 162

10.5.3 Dynamics of Partition Patterns: Example 164

10.6 Large Clusters 165

10.6.1 Expected Value of the Largest Cluster Size 166

10.6.2 JointProbabilityDensityfortheLargestr Fractions 169

10.7 MomentCalculations 171

10.8 Frequency Spectrum 172

10.8.1 Definition 173

10.8.2 HerfindahllndexofConcentration 173

10.8.3 AHeuristicDerivation 174

10.8.4 RecursionRelations 176

10.8.5 ExamplesofApplications 177

10.8.6 DiscreteFrequencySpectrum 177

10.9 ParameterEstimation 178

11 Share Market with Two Dominant Groups of Traders 180

11.1 TransitionRates 181

11.2 Ewens Distribution 183

11.2.1 The Number of Clusters and Value of θ 184

11.2.2 Expected Values of the Fractions 185

11.2.3 TheLargestTwoShares 186

11.3 MarketVolatility 187

11.4 BehaviorofMarketExcessDemand 188

11.4.1 Conditions for Zero Excess Demand 188

11.4.2 Volatility of the Market Excess Demand 189

11.4.3 Approximate Dynamics for Price Differences

and Power Law 190

11.4.3.1 Simulation Experiments 192

Appendix 195

A.1 DerivingGeneratingFunctionsviaCharacteristicCurves 195

A.2 Urn Models and Associated Markov Chains 197

A.2.1 Polya’s Urn 197

A.2.2 Hoppe’s Urn 197

A.2.3 Markov Chain Associated with the Urn 199

A.3 Conditional Probabilities for Entries, Exits, and Changes

of Type 200

A.3.1 TransitionProbabilities 200

A.3.2 TransitionRates 202

A.4 Holding Times and Skeletal Markov Chains 202

A.4.1 Sojourn-Time Models 205

A.5 Stirling Numbers 206

A.5.1 Introduction 206

A.5.2 Recursions 207

A.5.3 RelationswithCombinatorics 209

A.5.4 ExplicitExpressionsandAsymptoticRelations 210

A.5.5 Asymptotics 212

A.6 Order Statistics 213

A.7 Poisson Random Variables and the Ewens Sampling

Formula 214

A.7.1 Approximations by Poisson Random Variables 214

A.7.2 ConditionalPoissonRandomVariables 216

A.8 ExchangeableRandomPartitions 219

A.8.1 Exchangeable Random Sequences 219

A.8.2 PartitionExchangeability 221

A.9 Random Partitions and Permutations 224

A.9.1 Permutations 224

A.9.2 Random Partitions 225

A.9.3 Noninterference of Partitions 228

A.9.4 Consistency 229

A.10 DirichletDistributions 229

A.10.1 BetaDistribution 229

A.10.2 DirichletDistribution 230

A.10.3 MarginalDirichletDistributions 232

A.10.4 Poisson-DirichletDistribution 232

A.10.5 Size-BiasedSampling 233

A.11 Residual Allocation Models 234

A.12 GEMandSize-BiasedDistributions 235

A.13 StochasticDifferenceEquations 240

A.14 Random Growth Processes 242

A.15 DiffusionApproximationtoGrowthProcesses 243

References 245

Index

253