Recursive Models of Dynamic Linear Economies
424
Recursive Models of Dynamic Linear Economies
424Hardcover
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Overview
A common set of mathematical tools underlies dynamic optimization, dynamic estimation, and filtering. In Recursive Models of Dynamic Linear Economies, Lars Peter Hansen and Thomas Sargent use these tools to create a class of econometrically tractable models of prices and quantities. They present examples from microeconomics, macroeconomics, and asset pricing. The models are cast in terms of a representative consumer. While Hansen and Sargent demonstrate the analytical benefits acquired when an analysis with a representative consumer is possible, they also characterize the restrictiveness of assumptions under which a representative household justifies a purely aggregative analysis.
Hansen and Sargent unite economic theory with a workable econometrics while going beyond and beneath demand and supply curves for dynamic economies. They construct and apply competitive equilibria for a class of linear-quadratic-Gaussian dynamic economies with complete markets. Their book, based on the 2012 Gorman lectures, stresses heterogeneity, aggregation, and how a common structure unites what superficially appear to be diverse applications. An appendix describes MATLAB programs that apply to the book's calculations.
Product Details
| ISBN-13: | 9780691042770 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 12/26/2013 |
| Series: | The Gorman Lectures in Economics , #6 |
| Pages: | 424 |
| Sales rank: | 978,233 |
| Product dimensions: | 7.10(w) x 10.10(h) x 1.30(d) |
About the Author
Table of Contents
Preface xiii
Acknowledgments xv
Part I Overview
1 Theory and Econometrics 3
1.1 Introduction.
1.2 A Class of Economies.
1.3 Computer Programs.
1.4 Organization.
1.5 Recurring Mathematical Ideas.
Part II Tools
2 Linear Stochastic Difference Equations 15
2.1 Introduction.
2.2 Notation and Basic Assumptions.
2.3 Prediction Theory.
2.4 Transforming Variables to Uncouple Dynamics.
2.4.1 Deterministic Seasonals.
2.4.2 Indeterministic Seasonals.
2.4.3 Univariate Autoregressive Processes.
2.4.4 Vector Autoregressions.
2.4.5 Polynomial Time Trends.
2.4.6 Martingales with Drift.
2.4.7 Covariance Stationary Processes.
2.4.8 Multivariate ARMA Processes.
2.4.9 Prediction of a Univariate First-Order ARMA.
2.4.10 Growth.
2.4.11 A Rational Expectations Model.
2.4.12 Method of Undetermined Coefficients.
2.5 Concluding Remarks.
3 Efficient Computations 33
3.1 Introduction.
3.2 The Optimal Linear Regulator Problem.
3.3 Transformations to Eliminate Discounting and Cross-Products.
3.4 Stability Conditions.
3.5 Invariant Subspace Methods.
3.5.1 Px as Lagrange Multiplier.
3.5.2 Invariant Subspace Methods.
3.6 Doubling Algorithm.
3.7 Partitioning the State Vector.
3.8 Periodic Optimal Linear Regulator.
3.9 A Periodic Doubling Algorithm.
3.9.1 Partitioning the State Vector.
3.10 Linear Exponential Quadratic Gaussian Control.
3.10.1 Doubling Algorithm for a Risk-Sensitive Problem.
A Concepts of Linear Control Theory.
B Symplectic Matrices.
C Alternative Forms of the Riccati Equation.
Part III Components of Economies
4 Economic Environments 61
4.1 Information.
4.2 Taste and Technology Shocks.
4.3 Production Technologies.
4.4 Examples of Production Technologies.
4.4.1 Other Technologies.
4.5 Household Technologies.
4.6 Examples of Household Technologies.
4.7 Square Summability.
4.8 Summary.
5 Optimal Resource Allocations 79
5.1 Planning Problem.
5.2 Lagrange Multipliers.
5.3 Dynamic Programming.
5.4 Lagrange Multipliers as Gradients of Value Function.
5.5 Planning Problem as Linear Regulator.
5.6 Allocations for Five Economies.
5.6.1 Brock-Mirman (1972) or Hall (1978) Model.
5.6.2 A Growth Economy Fueled by Habit Persistence.
5.6.3 Lucas's Pure Exchange Economy.
5.6.4 An Economy with a Durable Consumption Good.
5.6.5 Computed Examples.
5.7 Hall's Model.
5.8 Higher Adjustment Costs.
5.9 Altered Growth Condition.
5.10 A Jones-Manuelli (1990) Economy.
5.11 Durable Consumption Goods.
5.12 Summary.
A Synthesizing a Linear Regulator.
B A Brock-Mirman (1972) or Hall (1978) Model.
5 .B.1. Uncertainty.
5 .B.2. Optimal Stationary States.
6 A Commodity Space 125
6.1 Valuation.
6.2 Price Systems as Linear Functionals.
6.3 A One-Period Model under Certainty.
6.4 One Period under Uncertainty.
6.5 An Infinite Number of Periods and Uncertainty.
6.5.1 Conditioning Information.
6.6 Lagrange Multipliers.
6.7 Summary.
A Mathematical Details.
7 Competitive Economies 131
7.1 Introduction.
7.2 Households.
7.3 Type I Firms.
7.4 Type II Firms.
7.5 Competitive Equilibrium.
7.6 Lagrangians.
7.6.1 Household Lagrangian.
7.6.2 Type I Firm Lagrangian.
7.6.3 Type II Firm Lagrangian.
7.7 Equilibrium Price System.
7.8 Asset Pricing.
7.9 Term Structure of Interest Rates.
7.10 Reopening Markets.
7.11 Non-Gaussian Asset Prices.
7.12 Asset Pricing Example.
Part IV Representations and Properties
8 Statistical Representations 153
8.1 The Kalman Filter.
8.2 Innovations Representation.
8.3 Convergence.
8.3.1 Computation of Time-Invariant Kalman Filter.
8.4 Factorization of Likelihood Function.
8.4.1 Initialization Assumptions.
8.4.2 Possible Nonexistence of Stationary Distribution.
8.5 Spectral Factorization Identity.
8.6 Wold and Autoregressive Representations.
8.7 Frequency Domain Estimation.
8.8 Approximation Theory.
8.9 Aggregation over Time.
8.10 Simulation Estimators.
A Initialization of Kalman Filter.
B Zeros of Characteristic Polynomial.
C Serially Correlated Measurement Errors.
D Innovations in yt+1 as Functions of wt+1 and ηt+1
E Innovations in a Permanent Income Model.
9 Canonical Household Technologies 191
9.1 Introduction.
9.2 Definition of a Canonical Household Technology.
9.3 Dynamic Demand Functions.
9.3.1 Wealth and the Multiplier μw0
9.3.2 Dynamic Demand System.
9.3.3 Gorman Aggregation and Engel Curves.
9.3.4 Reopened Markets.
9.4 Computing Canonical Representations.
9.4.1 Basic Idea.
9.4.2 An Auxiliary Problem Induces a Canonical Representation.
9.5 An Operator Identity.
9.6 Becker-Murphy Model of Rational Addiction.
A Fourier Transforms.
9.A.1 Primer on z-Transforms.
9.A.2 Time Reversal and Parseval's Formula.
9.A.3 One-Sided Sequences.
9.A.4 Useful Properties.
9.A.5 One-Sided Transforms.
9.A.6 Discounting.
9.A.7 Fourier Transforms.
9.A.8 Verifying Equivalent Valuations.
9.A.9 Equivalent Representations of Preferences.
9.A.10 First Term: Factorization Identity.
9.A.11 Second Term.
9.A.12 Third Term.
10 Examples 217
10.1 Partial Equilibrium.
10.2 The Setup.
10.3 Equilibrium Investment under Uncertainty.
10.4 A Housing Model.
10.4.1 Demand.
10.4.2 House Producers.
10.5 Cattle Cycles.
10.5.1 Mapping Cattle Farms into Our Framework.
10.5.2 Preferences.
10.5.3 Technology.
10.6 Models of Occupational Choice and Pay.
10.6.1 A One-Occupation Model.
10.6.2 Skilled and Unskilled Workers. A. Decentralizing the Household.
11 Permanent Income Models 233
11.1 Technology.
11.2 Two Implications.
11.3 Allocation Rules.
11.4 Deterministic Steady States.
11.5 Cointegration.
11.6 Constant Marginal Utility of Income.
11.7 Consumption Externalities. A. Exotic Tax Smoothing Models.
12 Gorman Heterogeneous Households 253
12.1 Introduction.
12.2 Gorman Aggregation (Static).
12.3 An Economy with Heterogeneous Consumers.
12.4 Allocations.
12.4.1 Consumption Sharing Rules.
12.5 Risk Sharing.
12.6 Implementing the Allocation Rule with Limited Markets.
A Computer Example.
13 Complete Markets Aggregation 269
13.1 Introduction.
13.2 Preferences and Household Technologies.
13.2.1 Production Technology.
13.3 A Pareto Problem.
13.4 Competitive Equilibrium.
13.4.1 Households.
13.4.2 Firms of Types I and II.
13.5 Computation of Equilibrium.
13.5.1 Candidate Equilibrium Prices.
13.5.2 A Negishi Algorithm.
13.6 Complete Markets Aggregation.
13.6.1 Static Demand.
13.6.2 Frequency Domain Representation of Preferences.
13.7 A Programming Problem for Complete Markets Aggregation.
13.7.1 Factoring S'S.
13.8 Summary of Findings.
13.9 The Aggregate Preference Shock Process.
13.9.1 Interpretation of st Component.
13.10 Initial Conditions.
14 Periodic Models of Seasonality 291
14.1 Three Models of Seasonality.
14.2 A Periodic Economy.
14.3 Asset Pricing.
14.4 Prediction Theory.
14.5 Term Structure of Interest Rates.
14.6 Conditional Covariograms.
14.7 A Stacked and Skip-Sampled System.
14.8 Covariances of the Stacked and Skip-Sampled Process.
14.9 Tiao-Grupe Formula.
14.9.1 State-Space Realization of Tiao-Grupe Formula.
14.10 Periodic Hall Model.
14.11 Periodic Innovations Representations for a Periodic Model.
A Disguised Periodicity.
A. MATLAB Programs 327
References 379
Subject Index 393
Author Index 397
MATLAB Index 399
What People are Saying About This
"This is the ideal book for those who want to study, understand, and work with linear-quadratic dynamic economies. Providing a thorough, authoritative, yet accessible treatment, it contains a superb analysis of the connections between various linear-quadratic dynamic programming problems, the general equilibrium properties of these economies, the type of aggregation applicable to them, and the time-series implications for quantities and prices. A great book by two giants of the field."—Fernando Alvarez, University of Chicago"In this tour-de-force of modern macroeconomics, Hansen and Sargent have written the definitive text on linear-quadratic economies that illustrate the connection between preferences and technology and the appropriate time-series representation. This gem of a book not only provides a thorough review of mathematical methods and related computational issues, but also includes cutting-edge economic models. It will be the required reference for anybody who works in modern dynamic macroeconomic problems."—Rodolfo E. Manuelli, Washington University in St. Louis"Modern macroeconomics relies on dynamic equilibrium modeling and the statistical analysis of time-series data. This superb book teaches both techniques hands-on. It guides readers towards mastering a library of computer programs that work for many practical problems, a library that readers will then build on in their own macroeconomic research."—Martin Schneider, Stanford University"It is nearly impossible to think of a better set of coauthors for this subject. I read their superior book with great pleasure and learned much from it."—Jesus Fernandez-Villaverde, University of Pennsylvania"Drawing strong connections between mathematics and economic intuition, this rigorous and insightful book contains an extremely broad set of applications, treated from the same consistent framework. The exposition of the benchmark model is outstanding and unique."—John Stachurski, Australian National University