Recursive Models of Dynamic Linear Economies

Lars Peter Hansen
Thomas J. Sargent
Pages: 304
https://www.jstor.org/stable/j.ctt5hhndv

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xiv)
4. Acknowledgments
(pp. xv-xvi)
5. Part I: Overview

• Chapter 1 Theory and Econometrics
(pp. 3-12)

Economic theory identifies patterns that unite apparently diverse subjects. Consider the following models:

1. Ryoo and Rosen’s (2004) partial equilibrium model of the market for engineers;

2. Rosen, Murphy, and Scheinkman’s (1994) model of cattle cycles;

3. Lucas’s (1978) model of asset prices;

4. Brock and Mirman’s (1972) and Hall’s (1978) model of the permanent income theory of consumption;

5. Time-to-build models of business cycles;

6. Siow’s (1984) model of occupational choice;

7. Topel and Rosen’s (1988) model of the dynamics of house prices and quantities;

8. Theories of dynamic demand curves;

9. Theories of dynamic supply curves;

10. Lucas...

6. Part II: Tools

• Chapter 2 Linear Stochastic Difference Equations
(pp. 15-32)

This chapter introduces the vector first-order linear stochastic difference equation.¹ We use it first to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted us to adopt economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations.

Because it expresses next period’s vector of state variables as a linear function of this period’s state vector and a vector of random disturbances, a vector...

• Chapter 3 Efficient Computations
(pp. 33-58)

This chapter describes fast algorithms for computing the value function and optimal decision rule for the type of social planning problem to be described in chapter 5.¹ This same decision rule determines the competitive equilibrium allocation to be described in chapter 7, while the optimal value function for the planning problem contains all of the information needed to determine competitive equilibrium prices. The optimal value function and optimal decision rule can be computed by iterating to convergence on a functional equation called the Bellman equation. These iterations can be accelerated by using ideas from linear optimal control theory. We want...

7. Part III: Components of Economies

• Chapter 4 Economic Environments
(pp. 61-78)

This chapter describes an economic environment with five components: a sequence of information sets, laws of motion for taste and technology shocks, a technology for producing consumption goods, a technology for producing services from consumer durables and consumption purchases, and a preference ordering over consumption services. A particular economy is described by a set of matrices${A_{22}},{C_2},{U_b}$and${U_d}$that characterize the motion of information sets and of taste and technology shocks; matrices${\Phi _c},{\Phi _g},{\Phi _i},\Gamma ,{\Delta _k}$and${\Theta _k}$that determine the technology for producing consumption goods; matrices${\Delta _h},{\Theta _h}, \wedge$and$\prod$that determine the technology for producing consumption services from consumer goods; and...

• Chapter 5 Optimal Resource Allocations
(pp. 79-124)

We eventually want to use our models to study aspects of competitive equilibria, including time series properties of various quantities, spot market prices, asset prices, and rates of return. The first welfare theorem asserts that competitive equilibrium allocations solve a particular resource allocation problem, which in our setting is a linear-quadratic optimal control problem.

In this chapter, we state the optimal resource allocation problem and compare two methods for solving it. The first method uses state-and date-contingent Lagrange multipliers; the second uses dynamic programming. The first method reveals a direct connection between the Lagrange multipliers and the equilibrium prices in...

• Chapter 6 A Commodity Space
(pp. 125-130)

This chapter describes a concept of value that we shall later use to formulate a model in which the decisions of agents are reconciled in a competitive equilibrium. We describe a commodity space in which quantities and prices both will reside. The stochastic Lagrange multipliers of chapter 4 are closely related to equilibrium prices and live in the same mathematical space.

The planning problem studied in chapter 4 produces an outcome in which the process for consumption$\left\{ {{c_t}} \right\}$is an$n$-dimensional stochastic process that belongs to$L_0^2$. To calculate the value$\pi (c)$of a particular consumption plan$c = \left\{ {{c_t}} \right\}$from the...

• Chapter 7 Competitive Economies
(pp. 131-150)

This chapter describes a decentralized economy. We assign ownership and decision making to three distinct economic entities, a household and two kinds of firms. We define acompetitive equilibrium. Two fundamental theorems of welfare economics connect a competitive equilibrium to a planning problem. A price system supports the competitive equilibrium and implies interest rates and prices for derivative assets.

The representative household can be interpreted as a single household drawn from a population that is homogeneous in all respects. Alternatively, the representative household can be interpreted along lines to be described in chapters 12 and 13, as an artificial or...

8. Part IV: Representations and Properties

• Chapter 8 Statistical Representations
(pp. 153-190)

This chapter shows how models restrict observed prices and quantities, and how observations can be used to make inferences about parameters. Earlier chapters have prepared a state-space representation that expresses states${x_t}$and observables${y_t}$as linear functions of an initial state${x_0}$and histories of martingale difference sequences${w_t}$. The${w_t}$’s are shocks to endowments and preferences whose histories are observed by the agents in the economy. The econometrician does not directly observe those shocks but instead observes a history of observables${y_s},s \le t.$. Therefore, to prepare a model for estimation we obtain another representation that is cast in terms...

• Chapter 9 Canonical Household Technologies
(pp. 191-216)

This chapter derives dynamic demand schedules from a household service technology

${h_t} = {\Delta _h}{h_{t - 1}} - {\Theta _h}{c_t}$

${s_t} = \Delta {h_{t - 1}} + \Pi {c_t}$(9.1.1)

with preference shock${b_t} - {U_b}{z_t}$. An equivalence class of household technologies$\left( {{\Delta _h},{\Theta _h},\Pi ,\Lambda ,{U_b}} \right)$give rise to identical demand schedules. Among such household technologies, particular ones that we callcanonicalare convenient for reasons that we explain in this chapter.

We apply the concept of canonical representation of household technologies to a version of Becker and Murphy’s model of rational addiction. The chapter sets the stage for the chapter 10 use of demand curves to construct partial equilibrium interpretations of our models. This chapter also sets the stage for the...

• Chapter 10 Examples
(pp. 217-232)

Some of the general equilibrium models in this book can be reinterpreted as partial equilibrium models that employ the notion of arepresentative firm, and that generalize the preference and technology specifications of Lucas and Prescott (1971). The idea is that there is a large number of identical firms that produce the same goods and sell them in competitive markets. Because they are identical, we carry along only one of these firms, and let it produce the entire output in the industry (which is harmless under constant returns to scale). But we have to be careful in our analysis because...

• Chapter 11 Permanent Income Models
(pp. 233-252)

This chapter describes a class of permanent income models of consumption. These models stress connections between consumption and income implied by present-value budget balance and generate interesting predictions about responses of components of consumption to shocks to consumers’ information sets. The models allow us to characterize how consumption of durables acts as a form of savings and how habit persistence alters consumption-savings profiles.

To focus on dynamics induced by a household technology, it serves our purposes to adopt the following specification of a production technology:

${\phi _c} \cdot {c_t} + {i_t} = {\rm{\gamma }}{k_{t - 1}} + {e_t}$

${k_t} = {k_{t - 1}} + {i_t}$(11.1.1)

where${\phi _c}$is a vector of positive real numbers with${n _c}$elements,${e _t}$is a...

• Chapter 12 Gorman Heterogeneous Households
(pp. 253-268)

This chapter and the next describe methods for computing equilibria of economies with consumers who have heterogeneous preferences and endowments. In both chapters, we adopt simplifications that facilitate coping with heterogeneity. In the present chapter, we describe a class of heterogeneous consumer economies that satisfy M. W. Gorman’s (1953) conditions for aggregation, which lets us compute equilibriumaggregateallocations and pricesbeforecomputing allocations to individuals.¹

In chapter 13, we adopt a more general kind of heterogeneity that causes us to depart from the framework of Gorman. In particular, we adapt an idea of Negishi (1960), who described a social...

• Chapter 13 Complete Markets Aggregation
(pp. 269-290)

Chapter 12 studied a setting in which households have heterogeneous endowments and preference shocks, but otherwise have identical preferences and household technologies, implying that all households share linear Engel curves with the same slopes. The property of identically sloped linear Engel curves delivers a tidy and tractable theory of aggregation that ensures the existence of a representative household. This theory applies when different households share the same household technology$\left( {\Lambda ,\Pi ,{\Delta _h},{\Theta _h}} \right)$.

In this chapter, we maintain the linearity of households’ Engel curves, but permit their slopes to vary across classes of households. In particular, we now allow the households technology matrices...

• Chapter 14 Periodic Models of Seasonality
(pp. 291-326)

Until now, each of the matrices defining preferences, technologies, and information flows has been specified to be constant over time. In this chapter, we relax this assumption and let the matrices be strictly periodic functions of time. Our interest is to apply and extend an idea of Denise Osborn (1988) and Richard Todd (1983, 1990) to arrive at a model of seasonality as a hidden periodicity. Seasonality can be characterized in terms of a spectral density. A variable is said to “have a seasonal” if its spectral density displays peaks at or in the vicinity of the frequencies commonly associated...

9. Appendix A. MATLAB Programs
(pp. 327-378)
10. References
(pp. 379-392)
11. Subject Index
(pp. 393-396)
12. Author Index
(pp. 397-398)
13. MATLAB Index
(pp. 399-400)
14. Back Matter
(pp. 401-402)