Preface to the Third Edition
Acknowledgements
Contents
List of Figures
List of Tables
Acronyms
List of Symbols
List of Programs
Part I Representative Agent Models
Chapter 1 Basic Models
1.1 Introduction
1.2 The Deterministic Finite-Horizon Ramsey Model
1.2.1 The Ramsey Problem
1.2.2 The Karush-Kuhn-Tucker Theorem
1.3 The Deterministic Infinite-Horizon Ramsey Model
1.3.1 Recursive Utility
1.3.2 Euler Equations
1.3.3 Dynamic Programming
1.3.4 The Saddle Path
1.3.5 Models with Analytical Solution
1.4 The Stochastic Ramsey Model
1.4.1 Stochastic Output
1.4.2 Stochastic Euler Equations
1.4.3 Stochastic Dynamic Programming
1.5 Labor Supply, Growth, and the Decentralized Economy
1.5.1 Substitution of Leisure
1.5.2 Growth and Restrictions on Technology and Preferences
1.5.3 Parameterizations of Utility and Important Elasticities
1.5.4 The Decentralized Economy
1.6 Model Calibration and Evaluation
1.6.1 The Benchmark Business Cycle Model
1.6.2 Calibration
1.6.3 Model Evaluation
1.7 Numerical Solution Methods
1.7.1 Overview
1.7.2 Accuracy of Solutions
A.1 Solution to Example 1.3.1
A.2 Restrictions on Technology and Preferences
Chapter 2 Perturbation Methods: Framework and Tools
2.1 Introduction
2.2 Order of Approximation
2.3 Tools
2.3.1 A Brief List
2.3.2 Application to the Deterministic Ramsey Model
2.4 The Stochastic Linear-Quadratic Model
2.4.1 The Model
2.4.2 Policy Functions
2.4.3 Certainty Equivalence
2.5 A Canonical DSGE Model
2.5.1 Example
2.5.2 Generalization
2.6 More Tools and First Results
2.6.1 Computer Algebra versus Paper and Pencil
2.6.2 Derivatives of Composite Functions and Tensor Notation
2.6.3 Derivatives of Composite Functions and Matrix Chain Rules
2.6.4 Computation of Partial Derivatives
A.3 Solution of the Stochastic LQ Problem
A.4 Third-Order Effects
Chapter 3 Perturbation Methods: Solutions
3.1 Introduction
3.2 First-Order Solution
3.2.1 First-Order Policy Functions
3.2.2 BA Model
3.2.3 System Reduction
3.2.4 Digression: Solving Separately for the Deterministic and Stochastic Components
3.3 Second-Order Solution
3.3.1 Second-Order Policy Functions
3.3.2 Coefficients of the State Variables
3.3.3 Coefficients of the Perturbation Parameter
3.4 Third-Order Solution
3.4.1 Third-Order Policy Functions
3.4.2 Coefficients of the State Variables
3.4.3 Coefficients of the State-Dependent Uncertainty
3.4.4 Coefficients of the Perturbation Parameter
3.5 Implementation
A.5 Coefficients of the State-Dependent Uncertainty
A.6 Coefficients of the Perturbation Parameter
Chapter 4 Perturbation Methods: Model Evaluation and Applications
4.1 Introduction
4.2 Second Moments
4.2.1 Analytic Second Moments: Time Domain
4.2.2 Digression: Unconditional Means
4.2.3 Analytical Second Moments: Frequency Domain
4.2.4 Second Moments: Monte-Carlo Approach
4.3 Impulse Responses
4.4 The Benchmark Business Cycle Model
4.5 Time-to-Build Model
4.6 A New Keynesian Model
4.6.1 The Monopolistically Competitive Economy
4.6.2 Price Staggering
4.6.3 Wage Staggering
4.6.4 Nominal Frictions and Interest Rate Shocks
4.6.5 Habits and Adjustment Costs
A.7 Derivation of the Demand Function
A.8 Price Phillips Curve
A.9 Wage Phillips Curve
Chapter 5 Weighted Residuals Methods
5.1 Introduction
5.2 Analytical Framework
5.2.1 Motivation
5.2.2 Residual, Test, and Weight Function
5.2.3 Common Test Functions
5.2.4 Spectral and Finite Element Functions
5.2.5 Illustration
5.2.6 General Procedure
5.3 Implementation
5.3.1 State Space
5.3.2 Basis Functions
5.3.3 Residual Function
5.3.4 Projection and Solution
5.3.5 Accuracy
5.4 The Deterministic Growth Model
5.5 The Benchmark Business Cycle Model
5.6 The Benchmark Search and Matching Model
5.6.1 Motivation
5.6.2 The Model
5.6.3 Galerkin Solution
5.6.4 Results
5.7 Disaster Risk Models
5.7.1 Motivation
5.7.2 The Benchmark Business Cycle Model with Disaster Risk
5.7.3 Generalized Expected Utility
5.7.4 Adjustment Costs of Capital
5.7.5 Variable Disaster Size and Conditional Disaster Probability
5.7.6 The Full Model
Chapter 6 Simulation-Based Methods
6.1 Introduction
6.2 Extended Path Method
6.2.1 Motivation
6.2.2 The General Algorithm
6.2.3 Application: The Benchmark Business Cycle Model
6.2.4 Application: The Model of a Small Open Economy
6.2.5 Conclusion
6.3 Simulation and Function Approximation
6.3.1 Motivation
6.3.2 The General Algorithm
6.3.3 Application: The Benchmark Business Cycle Model
6.3.4 Application: The Limited Participation Model of Money
6.3.5 Conclusion
Chapter 7 Discrete State Space Value Function Iteration
7.1 Introduction
7.2 Solution of Deterministic Models
7.3 Solution of Stochastic Models
7.3.1 Framework
7.3.2 Approximations of Conditional Expectations
7.3.3 Basic Algorithm
7.3.4 Initialization
7.3.5 Interpolation
7.3.6 Acceleration
7.3.7 Value Function Iteration and Linear Programming
7.3.8 Evaluation
7.4 Further Applications
7.4.1 Nonnegative Investment
7.4.2 The Benchmark Model
Part II Heterogenous Agent Models
Chapter 8
Computation of Stationary Distributions
8.1 Introduction
8.2 Easy Aggregation and Gorman Preferences
8.2.1 A Numerical Example
8.2.2 Gorman Preferences
8.3 A Simple Heterogeneous Agent Model with Aggregate Certainty
8.4 The Stationary Equilibrium of a Heterogeneous Agent Economy
8.4.1 Discretization of the Distribution Function
8.4.2 Discretization of the Density Function
8.4.3 Monte-Carlo Simulation
8.4.4 Function Approximation
8.5 The Risk-Free Rate
8.5.1 The Exchange Economy
8.5.2 Computation
8.5.3 Results
8.6 Heterogeneous Productivity and Income Distribution
8.6.1 Empirical Facts on the Income and Wealth Distribution and Income Dynamics
8.6.2 The Model
8.6.3 Computation
8.6.4 Results
Chapter 9 Dynamics of the Distribution Function
9.1 Introduction
9.2 Motivation
9.3 Transition Dynamics
9.3.1 Partial Information
9.3.2 Guessing a Finite Time Path for the Factor Prices
9.4 Aggregate Uncertainty: The Krusell-Smith Algorithm
9.4.1 The Economy
9.4.2 Computation
9.4.3 Calibration and Numerical Results
9.5 Applications
9.5.1 Costs of Business Cycles with Indivisibilities and Liquidity Constraints
9.5.2 Income Distribution and the Business Cycle
Chapter 10 Overlapping Generations Models with Perfect Foresight
10.1 Introduction
10.2 The Steady State in OLG Models
10.2.1 An Elementary Model
10.2.2 Computational Methods
10.2.3 Direct Computation
10.2.4 Computation of the Policy Functions
10.3 The Laffer Curve
10.4 The Transition Path
10.4.1 A Stylized 6-Period OLG Model
10.4.2 Computation of the Transition Path
10.5 The Demographic Transition
10.5.1 The Model
10.5.2 Calibration
10.5.3 Computation
10.5.4 Results
10.6 Conclusion
A.10 Derivation of Aggregate Bequests in (10.29)
Chapter 11 OLG Models with Uncertainty
11.1 Introduction
11.2 Overlapping Generations Models with Individual Uncertainty
11.2.1 The Model
11.2.2 Computation of the Stationary Equilibrium
11.2.3 Multi-Dimensional Individual State Space
11.3 Overlapping Generations with Aggregate Uncertainty
11.3.1 Perturbation Methods
11.3.2 The OLG Model with Quarterly Periods
11.3.3 Business Cycle Dynamics of Aggregates and Inequality
11.3.4 The Krusell-Smith Algorithm and Overlapping Generations
A.11 Derivation of the Stationary Dynamic Program of the Household
A.12 First-Order Conditions of the Stationary Dynamic Program (11.13)
A.13 Derivation of the Parameters of the AR(1)-Process with Annual Periods
Part III Numerical Methods
Chapter 12 Linear Algebra
12.1 Introduction
12.2 Complex Numbers
12.3 Vectors
12.4 Norms
12.5 Linear Independence
12.6 Matrices
12.7 Linear and Quadratic Forms
12.8 Eigenvalues and Eigenvectors
12.9 Matrix Factorization
12.9.1 Jordan Factorization
12.9.2 Schur Factorization
12.9.3 QZ Factorization
12.9.4 LU and Cholesky Factorization
12.9.5 QR Factorization
12.9.6 Singular Value Decomposition
Chapter 13 Function Approximation
13.1 Introduction
13.2 Function Spaces
13.3 Taylorβs Theorem
13.4 Implicit Function Theorem
13.5 Lagrange Interpolation
13.5.1 Polynomials and the Weierstrass Approximation Theorem
13.5.2 Lagrange Interpolating Polynomial
13.5.3 Drawbacks
13.6 Spline Interpolation
13.6.1 Linear Splines
13.6.2 Cubic Splines
13.7 Orthogonal Polynomials
13.7.1 Orthogonality in Euclidean Space
13.7.2 Orthogonality in Function Spaces
13.7.3 Orthogonal Interpolation
13.7.4 Families of Orthogonal Polynomials
13.8 Chebyshev Polynomials
13.8.1 Definition
13.8.2 Zeros and Extrema
13.8.3 Orthogonality
13.8.4 Chebyshev Regression
13.8.5 Chebyshev Evaluation
13.8.6 Examples
13.9 Multivariate Extensions
13.9.1 Tensor Product and Complete Polynomials
13.9.2 Multidimensional Splines
13.9.3 Multidimensional Chebyshev Regression
13.9.4 The Smolyak Polynomial
13.9.5 Neural Networks
Chapter 14 Differentiation and Integration
14.1 Introduction
14.2 Differentiation
14.2.1 First-Order Derivatives
14.2.2 Second-Order Derivatives
14.3 Numerical Integration
14.3.1 Newton-Cotes Formulas
14.3.2 Gaussian Formulas
14.3.3 Monomial Integration Formula
14.4 Approximation of Expectations
14.4.1 Expectation of a Function of Gaussian Random Variables
14.4.2 Gauss-Hermite Integration
14.4.3 Monomial Rules for Expectations
Chapter 15 Nonlinear Equations and Optimization
15.1 Introduction
15.2 Stopping Criteria for Iterative Algorithms
15.3 Nonlinear Equations
15.3.1 Single Equations
15.3.2 Multiple Equations
15.4 Numerical Optimization
15.4.1 Golden Section Search
15.4.2 Gauss-Newton Method
15.4.3 Quasi-Newton
15.4.4 Genetic Search Algorithms
Chapter 16 Difference Equations and Stochastic Processes
16.1 Introduction
16.2 Difference Equations
16.2.1 Linear Difference Equations
16.2.2 Nonlinear Difference Equations
16.2.3 Boundary Value Problems and Shooting
16.3 Stochastic Processes
16.3.1 Univariate Processes
16.3.2 Trends
16.3.3 Multivariate Processes
16.4 Markov Processes
16.4.1 The First-Order Autoregressive Process
16.4.2 Markov Chains
16.5 Linear Filters
16.5.1 Definitions
16.5.2 The HP-Filter
Bibliography
Name Index
Subject Index