Chapter
2.2 Entropy: A Measure of Economic Activity
2.2.2 Laplace Transform and Moment-Generating Functions
2.2.3 Replacing the Sum with the Maximum Term
2.3 Empirical Distributions
2.3.2 Multiplicity of Microstates
2.3.4 Conditional-Limit Theorem
2.4 Stochastic Dynamics and Processes
2.4.1 Mean First-Passage Times
2.4.2 Dynamics with Multiple Equilibria
2.4.3 Random Partition of Agents by Types
2.4.3.2 Generalized Pólya Urn Model
2.5 Hierarchical State Spaces
2.5.1 Examples of Hierarchically Structured State Spaces
2.5.1.1 Coin-Tossing Spaces
2.5.1.2 K-Level Classification
2.5.1.3 Pattern Classification
2.5.2 Tree Metric and Martingales
2.5.3 Markov Chains on Binary Trees
2.5.4 Aggregation of Dynamics on Trees
3 Empirical Distributions: Statistical Laws in Macroeconomics
3.1.1 Micro and Macro Descriptions of Models
3.1.2 Multiplicity of Microstates
3.2 Entropy and Relative Entropy
3.2.1 Kullback-Leibler Divergence Measure
3.2.2 Boltzmann and Shannon Entropies
3.3.1 Discrete-Choice Models
3.3.2 Detailed Balance and Gibbs Distributions
3.3.3 Conditional-Limit Theorems: Gibbs Conditioning Principle
3.3.3.1 Example of Asymptotic Independence
3.4 Maximizing Equilibrium Probabilities or Minimizing Potential
3.4.1 Optimization and Gibbs Distributions
3.5 Finite-State Markov Chains
3.5.1 Entropy Maximization
3.5.2 Cost Minimization with Markov-Chain Dynamics
3.6.1 Example of Asset Returns
3.6.3 Tilted Distribution and Lower Chernoff Bound
3.6.4 Example of Large-Deviation Analysis
3.6.4.1 Sample Mean Greater than the Expected Value
3.6.5 Gärtner-Ellis Theorem
3.6.5.1 Examples of Dependent Asset Returns
3.7.1 From Sanov's Theorem to Cramér's Theorem
3.8 Conditional-Limit Theorem
Part II Modeling Interactions
4 Modeling Interactions I: Jump Markov Processes
4.1 Market Participation and Other Discrete Adjustment Behavior
4.2 Construction and Infinitesimal Parameters
4.3.1 Birth-and-Death Processes
4.4 Equations for Averages: Aggregate Dynamics
4.5 Multidimensional Birth-and-Death Processes
4.5.1 Epidemic Model Interpreted Economically
4.5.2 Open Air Market Linear Model
4.5.3 Open Air Market Linear Model II
4.5.4 Nonlinear Birth-and-Death Models
4.5.5 Birth-and-Death Processes For Partition Patterns
4.6 Discrete Adjustment Behavior
4.6.1 Example of Employment Adjustment Processes I
4.6.2 Example of Employment Adjustment Processes II
4.6.3 Example of Inferring Microstate Distribution
4.7.1 Hazard Functions: Age-Specific Transition Rates
5 Modeling Interactions II: Master Equations and Field Effects
5.1.1 A Collection of Independent Agents
5.2 Structure of Transition Rates
5.3 Approximate Solutions of the Master Equations
5.3.1 Power-Series Expansion
5.4 Macroeconomic Equation
5.5 Specifying Transition Rates: Examples
5.5.1 Two-Sector Capital Reallocation Dynamics
5.5.2 Exchange-Rate Pass-Through
5.6 Field Effects: Stochastic Nonlocal and Diffuse Externalities
5.7 Generalized Birth-and-Death Models
5.7.1 Mean Field Approximation of Transition Rates
5.8 Expressing Relative Merits of Alternative Decisions
5.9 Equilibrium Probability Distributions
5.10 Example of Multiple Equilibria
5.10.3 Some Simulation Results
5.11.1 First-Passage Times of Unstable Dynamics
5.12 The Master Equation for Hierarchical Dynamics
5.13 The Fokker-Planck Equation
5.13.1 Power Series Expansion
5.14 The Diffusion-Type Master Equation
5.14.1 Ornstein-Uhlenbeck Model
5.14.2 Wright-Fisher Model: A Binary-Choice Model
5.14.2.1 Unemployment-Rate Model
6 Modeling Interactions III: Pairwise and Multiple-Pair Interactions
6.1 Pairwise or Multiple-Pair Interactions
6.1.1.1 Long-Range Pairwise Interactions
6.2 A Model of Pairwise Externality
6.2.1 Potentials and Equilibrium Probability Distributions
6.2.2 Analogy Between Economic Agents and Neurons
6.3 Example of Information-Exchange Equilibrium Distribution
6.4 Time Evolution of Patterns of Interaction
Part III Hierarchical Dynamics and Critical Phenomena
7 Sluggish Dynamics and Hierarchical State Spaces
7.1 Examples of Hierarchically Structured State Spaces
7.1.1 Correlated Patterns of Macroeconomic Activity
7.1.2 Vectors in an Infinite Dimensional Space
7.1.3 Cost Barriers as Ultrametrics
7.1.4 Voter Preference Patterns Among Alternatives
7.1.5 Champernowne's Income Distribution Model
7.1.6 Random Walks on a Logarithmic Scale
7.1.7 Random Multicomponent Cost
7.1.7.1 Example of a Two-Level Tree
7.1.9 Branching Processes
7.2 Dynamics on Hierarchical State Spaces
7.2.1 Ultrametrics: Hierarchical Distance
7.2.2 Ogielski-Stein Model
7.2.2.1 An Example of Aggregation of Hierarchical Dynamics
7.2.2.2 A Nonsymmetric Tree
7.2.3 Schreckenberg's Model
7.3 Pruning Trees: Aggregation of Hierarchical Dynamics
7.3.1 Renormalization Group Theory
7.3.2 Collet-Eckman Model
7.3.3 Idiart-Theumann Model
8 Self-organizing and Other Critical Phenomena in Economic Models
8.1 Sudden Structural Changes
8.2 Piling-Up, or Dense Occupancy, of a Common State
8.3 Phase Transitions in the Ising Tree Model
8.3.1 Phase Transitions in a Random Cost Model
8.4 Example of a Two-Level Model
Elaborations and Future Directions of Research
E. 1 Zipf 's Distribution in Economic Models
E. 1.1 Bose-Einstein Allocations
E. 1.1.1 Two-Level Hierarchy
E. 1.1.2 Three-Level Hierarchical Classification
E. 1.2 Dirichlet-Multinomial Model of Chen
E.2 Residual Fraction Model
E.4 Statistical Distribution of Sizes of Attractive Basins
E.5 Transient Distributions
A. 1 The Method of Laplace
A. 1.0.1 Example of Error Integral
A. 1.0.2 Example of Stirling's Formula
A. 1.0.3 Example of the Partition-Function Evaluation
A. 1.1.1 Example of Partition Function Evaluation
A.2.1 Discrete-Time Markov Chains
A.2.2 Example of a Random-Walk Model
A.2.3 Continuous-Time Markov Chains
A.2.3.1 Simple Random Walks
A.2.4 A Standard Wiener Process
A.2.5 First-Passage Times to Absorbing Barrier
A.3 Exchangeable Processes
A.4 Low-frequency Behavior
A.6 Fokker-Planck Equations and Detailed Balance
A.6.1 Fokker-Planck Equations and Stochastic Dynamics
A.6.2 Detailed Balance and the Fokker-Planck Equation
New Approaches to Macroeconomic Modeling
：Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects
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