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Recursive Models of Dynamic Linear Economies

Book Description

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.

Based on the 2012 Gorman lectures, the authors 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 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.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Acknowledgments
  7. Part I: Overview
    1. 1. Theory and Econometrics
      1. 1.1. Introduction.
      2. 1.2. A Class of Economies.
      3. 1.3. Computer Programs.
      4. 1.4. Organization.
      5. 1.5. Recurring Mathematical Ideas.
  8. Part II: Tools
    1. 2. Linear Stochastic Difference Equations
      1. 2.1. Introduction.
      2. 2.2. Notation and Basic Assumptions.
      3. 2.3. Prediction Theory.
      4. 2.4. Transforming Variables to Uncouple Dynamics.
        1. 2.4.1. Deterministic Seasonals.
        2. 2.4.2. Indeterministic Seasonals.
        3. 2.4.3. Univariate Autoregressive Processes.
        4. 2.4.4. Vector Autoregressions.
        5. 2.4.5. Polynomial Time Trends.
        6. 2.4.6. Martingales with Drift.
        7. 2.4.7. Covariance Stationary Processes.
        8. 2.4.8. Multivariate ARMA Processes.
        9. 2.4.9. Prediction of a Univariate First-Order ARMA.
        10. 2.4.10. Growth.
        11. 2.4.11. A Rational Expectations Model.
        12. 2.4.12. Method of Undetermined Coefficients.
      5. 2.5. Concluding Remarks.
    2. 3. Efficient Computations
      1. 3.1. Introduction.
      2. 3.2. The Optimal Linear Regulator Problem.
      3. 3.3. Transformations to Eliminate Discounting and Cross-Products.
      4. 3.4. Stability Conditions.
      5. 3.5. Invariant Subspace Methods.
        1. 3.5.1. Px as Lagrange Multiplier.
        2. 3.5.2. Invariant Subspace Methods.
      6. 3.6. Doubling Algorithm.
      7. 3.7. Partitioning the State Vector.
      8. 3.8. Periodic Optimal Linear Regulator.
      9. 3.9. A Periodic Doubling Algorithm.
        1. 3.9.1. Partitioning the State Vector.
      10. 3.10. Linear Exponential Quadratic Gaussian Control.
        1. 3.10.1. Doubling Algorithm for a Risk-Sensitive Problem.
      11. A. Concepts of Linear Control Theory.
      12. B. Symplectic Matrices.
      13. C. Alternative Forms of the Riccati Equation.
  9. Part III: Components of Economies
    1. 4. Economic Environments
      1. 4.1. Information.
      2. 4.2. Taste and Technology Shocks.
      3. 4.3. Production Technologies.
      4. 4.4. Examples of Production Technologies.
        1. 4.4.1. Other Technologies.
      5. 4.5. Household Technologies.
      6. 4.6. Examples of Household Technologies.
      7. 4.7. Square Summability.
      8. 4.8. Summary.
    2. 5. Optimal Resource Allocations
      1. 5.1. Planning Problem.
      2. 5.2. Lagrange Multipliers.
      3. 5.3. Dynamic Programming.
      4. 5.4. Lagrange Multipliers as Gradients of Value Function.
      5. 5.5. Planning Problem as Linear Regulator.
      6. 5.6. Allocations for Five Economies.
        1. 5.6.1. Brock-Mirman (1972) or Hall (1978) Model.
        2. 5.6.2. A Growth Economy Fueled by Habit Persistence.
        3. 5.6.3. Lucas’s Pure Exchange Economy.
        4. 5.6.4. An Economy with a Durable Consumption Good.
        5. 5.6.5. Computed Examples.
      7. 5.7. Hall’s Model.
      8. 5.8. Higher Adjustment Costs.
      9. 5.9. Altered Growth Condition.
      10. 5.10. A Jones-Manuelli (1990) Economy.
      11. 5.11. Durable Consumption Goods.
      12. 5.12. Summary.
      13. A. Synthesizing a Linear Regulator.
      14. B. A Brock-Mirman (1972) or Hall (1978) Model.
        1. 5.B.1. Uncertainty.
        2. 5.B.2. Optimal Stationary States.
    3. 6. A Commodity Space
      1. 6.1. Valuation.
      2. 6.2. Price Systems as Linear Functionals.
      3. 6.3. A One-Period Model under Certainty.
      4. 6.4. One Period under Uncertainty.
      5. 6.5. An Infinite Number of Periods and Uncertainty.
        1. 6.5.1. Conditioning Information.
      6. 6.6. Lagrange Multipliers.
      7. 6.7. Summary.
      8. A. Mathematical Details.
    4. 7. Competitive Economies
      1. 7.1. Introduction.
      2. 7.2. Households.
      3. 7.3. Type I Firms.
      4. 7.4. Type II Firms.
      5. 7.5. Competitive Equilibrium.
      6. 7.6. Lagrangians.
        1. 7.6.1. Household Lagrangian.
        2. 7.6.2. Type I Firm Lagrangian.
        3. 7.6.3. Type II Firm Lagrangian.
      7. 7.7. Equilibrium Price System.
      8. 7.8. Asset Pricing.
      9. 7.9. Term Structure of Interest Rates.
      10. 7.10. Reopening Markets.
      11. 7.11. Non-Gaussian Asset Prices.
      12. 7.12. Asset Pricing Example.
  10. Part IV: Representations and Properties
    1. 8. Statistical Representations
      1. 8.1. The Kalman Filter.
      2. 8.2. Innovations Representation.
      3. 8.3. Convergence.
        1. 8.3.1. Computation of Time-Invariant Kalman Filter.
      4. 8.4. Factorization of Likelihood Function.
        1. 8.4.1. Initialization Assumptions.
        2. 8.4.2. Possible Nonexistence of Stationary Distribution.
      5. 8.5. Spectral Factorization Identity.
      6. 8.6. Wold and Autoregressive Representations.
      7. 8.7. Frequency Domain Estimation.
      8. 8.8. Approximation Theory.
      9. 8.9. Aggregation over Time.
      10. 8.10. Simulation Estimators.
      11. A. Initialization of Kalman Filter.
      12. B. Zeros of Characteristic Polynomial.
      13. C. Serially Correlated Measurement Errors.
      14. D. Innovations in yt+1 as Functions of wt+1 and ηt+1.
      15. E. Innovations in a Permanent Income Model.
    2. 9. Canonical Household Technologies
      1. 9.1. Introduction.
      2. 9.2. Definition of a Canonical Household Technology.
      3. 9.3. Dynamic Demand Functions.
        1. 9.3.1. Wealth and the Multiplier μ0w.
        2. 9.3.2. Dynamic Demand System.
        3. 9.3.3. Gorman Aggregation and Engel Curves.
        4. 9.3.4. Reopened Markets.
      4. 9.4. Computing Canonical Representations.
        1. 9.4.1. Basic Idea.
        2. 9.4.2. An Auxiliary Problem Induces a Canonical Representation.
      5. 9.5. An Operator Identity.
      6. 9.6. Becker-Murphy Model of Rational Addiction.
      7. A. Fourier Transforms.
        1. 9.A.1. Primer on z-Transforms.
        2. 9.A.2. Time Reversal and Parseval’s Formula.
        3. 9.A.3. One-Sided Sequences.
        4. 9.A.4. Useful Properties.
        5. 9.A.5. One-Sided Transforms.
        6. 9.A.6. Discounting.
        7. 9.A.7. Fourier Transforms.
        8. 9.A.8. Verifying Equivalent Valuations.
        9. 9.A.9. Equivalent Representations of Preferences.
        10. 9.A.10. First Term: Factorization Identity.
        11. 9.A.11. Second Term.
        12. 9.A.12. Third Term.
    3. 10. Examples
      1. 10.1. Partial Equilibrium.
      2. 10.2. The Setup.
      3. 10.3. Equilibrium Investment under Uncertainty.
      4. 10.4. A Housing Model.
        1. 10.4.1. Demand.
        2. 10.4.2. House Producers.
      5. 10.5. Cattle Cycles.
        1. 10.5.1. Mapping Cattle Farms into Our Framework.
        2. 10.5.2. Preferences.
        3. 10.5.3. Technology.
      6. 10.6. Models of Occupational Choice and Pay.
        1. 10.6.1. A One-Occupation Model.
        2. 10.6.2. Skilled and Unskilled Workers.
      7. A. Decentralizing the Household.
    4. 11. Permanent Income Models
      1. 11.1. Technology.
      2. 11.2. Two Implications.
      3. 11.3. Allocation Rules.
      4. 11.4. Deterministic Steady States.
      5. 11.5. Cointegration.
      6. 11.6. Constant Marginal Utility of Income.
      7. 11.7. Consumption Externalities.
      8. A. Exotic Tax Smoothing Models.
    5. 12. Gorman Heterogeneous Households
      1. 12.1. Introduction.
      2. 12.2. Gorman Aggregation (Static).
      3. 12.3. An Economy with Heterogeneous Consumers.
      4. 12.4. Allocations.
        1. 12.4.1. Consumption Sharing Rules.
      5. 12.5. Risk Sharing.
      6. 12.6. Implementing the Allocation Rule with Limited Markets.
      7. A. Computer Example.
    6. 13. Complete Markets Aggregation
      1. 13.1. Introduction.
      2. 13.2. Preferences and Household Technologies.
        1. 13.2.1. Production Technology.
      3. 13.3. A Pareto Problem.
      4. 13.4. Competitive Equilibrium.
        1. 13.4.1. Households.
        2. 13.4.2. Firms of Types I and II.
      5. 13.5. Computation of Equilibrium.
        1. 13.5.1. Candidate Equilibrium Prices.
        2. 13.5.2. A Negishi Algorithm.
      6. 13.6. Complete Markets Aggregation.
        1. 13.6.1. Static Demand.
        2. 13.6.2. Frequency Domain Representation of Preferences.
      7. 13.7. A Programming Problem for Complete Markets Aggregation.
        1. 13.7.1. Factoring S′S.
      8. 13.8. Summary of Findings.
      9. 13.9. The Aggregate Preference Shock Process.
        1. 13.9.1. Interpretation of ŝt Component.
      10. 13.10. Initial Conditions.
    7. 14. Periodic Models of Seasonality
      1. 14.1. Three Models of Seasonality.
      2. 14.2. A Periodic Economy.
      3. 14.3. Asset Pricing.
      4. 14.4. Prediction Theory.
      5. 14.5. Term Structure of Interest Rates.
      6. 14.6. Conditional Covariograms.
      7. 14.7. A Stacked and Skip-Sampled System.
      8. 14.8. Covariances of the Stacked and Skip-Sampled Process.
      9. 14.9. Tiao-Grupe Formula.
        1. 14.9.1. State-Space Realization of Tiao-Grupe Formula.
      10. 14.10. Periodic Hall Model.
      11. 14.11. Periodic Innovations Representations for a Periodic Model.
      12. A. Disguised Periodicity.
  11. A. MATLAB Programs
  12. References
  13. Subject Index
  14. Author Index
  15. MATLAB Index