A unified, comprehensive, and up-to-date introduction to the analytical and numerical tools for solving dynamic economic problems.

This book offers a unified, comprehensive, and up-to-date treatment of analytical and numerical tools for solving dynamic economic problems. The focus is on introducing recursive methods?an important part of every economist's set of tools?and readers will learn to apply recursive methods to a variety of dynamic economic problems. The book is notable for its combination of theoretical foundations and numerical methods. Each topic is first described in theoretical terms, with explicit definitions and rigorous proofs; numerical methods and computer codes to implement these methods follow. Drawing on the latest research, the book covers such cutting-edge topics as asset price bubbles, recursive utility, robust control, policy analysis in dynamic New Keynesian models with the zero lower bound on interest rates, and Bayesian estimation of dynamic stochastic general equilibrium (DSGE) models.

The book first introduces the theory of dynamical systems and numerical methods for solving dynamical systems, and then discusses the theory and applications of dynamic optimization. The book goes on to treat equilibrium analysis, covering a variety of core macroeconomic models, and such additional topics as recursive utility (increasingly used in finance and macroeconomics), dynamic games, and recursive contracts. The book introduces Dynare, a widely used software platform for handling a range of economic models; readers will learn to use Dynare for numerically solving DSGE models and performing Bayesian estimation of DSGE models. Mathematical appendixes present all the necessary mathematical concepts and results. Matlab codes used to solve examples are indexed and downloadable from the book's website. A solutions manual for students is available for sale from the MIT Press; a downloadable instructor's manual is available to qualified instructors.

A unified, comprehensive, and up-to-date introduction to the analytical and numerical tools for solving dynamic economic problems.

I Dynamical Systems 1 Deterministic Difference Equations 1.1 Scalar First-Order Linear Equations 1.2 Lag Operators 1.3 Scalar Second-Order Linear Equations 1.4 First-Order Linear Systems 1.4.1 Nonsingular System 1.4.2 Singular System 1.5 Phase Diagrams 1.6 Nonlinear Systems 1.7 Numerical Solutions Using Dynare 1.8 Exercises 2 Stochastic Difference Equations 2.1 First-Order Linear Systems 2.2 Scalar Linear Rational Expectations Models 2.2.1 Lag Operators 2.2.2 The Method of Undetermined Coefficients 2.3 Multivariate Linear Rational Expectations Models 2.3.1 The Blanchard and Kahn Method 2.3.2 The Klein Method 2.3.3 The Sims Method 2.4 Nonlinear Rational Expectations Models 2.5 Numerical Solutions Using Dynare 2.6 Exercises 3 Markov Processes 3.1 Markov Chains 3.1.1 Classification of States 3.1.2 Stationary Distribution 3.1.3 Countable-State Markov Chains 3.2 General Markov Processes 3.3 Convergence 3.3.1 Strong Convergence 3.3.2 Weak Convergence 3.4 Exercises 4 Ergodic Theory and Stationary Processes 4.1 Ergodic Theorem 4.2 Application to Stationary Processes 4.3 Application to Stationary Markov Processes 4.4 Exercises II Dynamic Optimization 5 Markov Decision Process Model 5.1 Model Setup 5.2 Examples 5.2.1 Discrete Choice 5.2.2 Optimal Stopping 5.2.3 Bandit Model 5.2.4 Optimal Control 5.3 Exercises 6 Finite-Horizon Dynamic Programming 6.1 A Motivating Example 6.2 Measurability Problem 6.3 The Principle of Optimality 6.4 Optimal Control 6.5 The Maximum Principle 6.6 Applications 6.6.1 The Secretary Problem 6.6.2 A Consumption-Saving Problem 6.7 Exercises 7 Infinite-Horizon Dynamic Programming 7.1 The Principle of Optimality 7.2 Bounded Rewards 7.3 Optimal Control 7.3.1 Bounded Rewards 7.3.2 Unbounded Rewards 7.4 The Maximum Principle and Transversality Conditions 7.5 Euler Equations and Transversality Condition 7.6 Exercises 8 Applications 8.1 Option Exercise 8.2 Discrete Choice 8.3 Consumption and Saving 8.3.1 Deterministic income 8.3.2 Stochastic income 8.4 Consumption/Portfolio Choice 8.5 Inventory 8.6 Investment 8.6.1 Neoclassical Theory 8.6.2 Q Theory 8.6.3 Augmented Adjustment Costs 8.7 Exercises 9 Linear-Quadratic Models 9.1 Controlled Linear State-Space System 9.2 Finite-Horizon Problems 9.3 Infinite-Horizon Limits 9.4 Optimal Policy under Commitment 9.5 Optimal Discretional Policy 9.6 Robust Control 9.7 Exercises 10 Control under Partial Information 10.1 Filters 10.1.1 Kalman Filter 10.1.2 Hidden Markov Chain 10.1.3 Hidden Markov-Switching Model 10.2 Control Problems 10.3 Linear-Quadratic Control 10.4 Exercises 11 Numerical Methods 11.1 Numerical Integration 11.1.1 Gaussian Quadrature 11.1.2 Multidimensional Quadrature 11.2 Discretizing AR(1) Processes 11.3 Interpolation 11.3.1 Orthogonal Polynomials 11.3.2 Splines 11.3.3 Multidimensional Approximation 11.4 Perturbation Methods 11.5 Projection Methods 11.6 Numerical Dynamic Programming 11.6.1 Discrete Approximation Methods 11.6.2 Smooth Approximation Methods 11.7 Exercises 12 Structural Estimation 12.1 Generalized Method of Moments 12.2 Maximum Likelihood 12.3 Simulation-Based Methods 12.3.1 Simulated Method of Moments 12.3.2 Simulated Maximum Likelihood 12.3.3 Indirect Inference 12.4 Exercises III Equilibrium Analysis 13 Complete Markets Exchange Economies 13.1 Uncertainty, Preferences, and Endowments 13.2 Pareto Optimum 13.3 Time 0 Trading 13.4 Sequential Trading 13.5 Equivalence of Equilibria 13.6 Asset Price Bubbles 13.7 Recursive Formulation 13.8 Asset Pricing 13.9 Exercises 14 Neoclassical Growth Models 14.1 Deterministic Models 14.1.1 A Basic Ramsey Model 14.1.2 Incorporating Fiscal Policy 14.2 A Basic RBC Model 14.3 Extensions of the Basic RBC Model 14.3.1 Various Utility Functions 14.3.2 Capacity Utilization 14.3.3 Capital or Investment Adjustment Costs 14.3.4 Stochastic Trends 14.4 Exercise 15 Bayesian Estimation of DSGE Models Using Dynare 15.1 Principles of Bayesian Estimation 15.2 Bayesian Estimation of DSGE Models 15.2.1 Numerical Solution and State Space Representation 15.2.2 Evaluating the Likelihood Function 15.2.3 Computing the Posterior 15.3 An Example 15.4 Exercises 16 Overlapping Generations Models 16.1 Exchange Economies 16.2 Production Economies 16.3 Asset Price Bubbles 16.4 Exercises 17 Incomplete Markets Models 17.1 Production Economies 17.1.1 Income Fluctuation Problem 17.1.2 Production 17.1.3 Stationary Recursive Equilibrium 17.1.4 Computation and Implications 17.2 Endowment Economies 17.2.1 Riskfree Rate 17.2.2 Fiat Money 17.2.3 Interest on Currency 17.2.4 Seigniorage 17.3 Aggregate Shocks 17.3.1 Recursive Equilibrium 17.3.2 The Krusell-Smith Method 17.4 Exercises 18 Search and Matching Models of Unemployment 18.1 A Basic DMP Model 18.2 Endogenous Job Destruction 18.3 Unemployment and Business Cycles 18.4 Exercises 19 Dynamic New Keynesian Models 19.1 A Basic DNK Model 19.1.1 Households 19.1.2 Final Goods Firms 19.1.3 Intermediate Goods Firms 19.1.4 Central Bank 19.1.5 Sticky Price Equilibrium 19.1.6 Flexible Price Equilibrium 19.1.7 Log-Linearized System 19.2 Monetary Policy Design 19.2.1 Efficient Allocation 19.2.2 Quadratic Approximation to Utility 19.2.3 Commitment versus Discretion 19.3 Fiscal Stimulus 19.4 A Medium-Scale DSGE Model 19.5 Exercises IV Further Topics 20 Recursive Utility 20.1 Deterministic Case 20.1.1 Koopmans’s Utility 20.1.2 Construction 20.2 Stochastic Case 20.2.1 Epstein-Zin Preferences 20.2.2 Ambiguity Aversion 20.2.3 Temporal Resolution of Uncertainty 20.3 Properties of Recursive Utility 20.3.1 Concavity 20.3.2 Risk Aversion 20.3.3 Utility Gradients and Pricing Kernels 20.4 Portfolio Choice and Asset Pricing 20.4.1 Optimality and Equilibrium 20.4.2 Log-Linear Approximation 20.4.3 Long-Run Risk 20.5 Pareto Optimality 20.5.1 The Lucas-Stokey Approach 20.5.2 The Dumas-Wang-Uppal Approach 20.6 Exercises 21 Dynamic Games 21.1 Repeated Games 21.1.1 Perfect Monitoring 21.1.2 Equilibrium Payoff Set 21.1.3 Computation 21.1.4 Simple Strategies 21.1.5 Imperfect Public Monitoring 21.2 Dynamic Stochastic Games 21.3 Application: The Great Fish War 21.4 Credible Government Policies 21.4.1 The One-Period Economy 21.4.2 The Infinitely Repeated Economy 21.4.3 Equilibrium Value Set 21.4.4 The Best and the Worst SPE Values 21.4.5 Recursive Strategies 21.5 Exercises 22 Recursive Contracts 22.1 Limited Commitment 22.1.1 A Dynamic Programming Method 22.1.2 A Lagrangian Method 22.1.3 An Alternative Characterization 22.2 Hidden Action 22.3 Hidden Information 22.3.1 Characterizations 22.3.2 Long-Run Poverty 22.4 Exercises V Mathematical Appendixes A Linear Algebra B Real and Functional Analysis C Convex Analysis D Measure and Probability Theory