CHAPTER 1

Theory and Econometrics

Complete market economies are all alike....— Robert E. Lucas, Jr. (1989)

1.1. Introduction

Economic theory identifies patterns that unite apparently diverse subjects. Considerthe 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 incometheory 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 and Prescott's (1971) model of investment under uncertainty.

These models and many more have identical structures because all describecompetitive equilibria with complete markets. This is the meaning of words ofRobert E. Lucas, Jr., with which we have chosen to begin this chapter. Lucasrefers to the fact that complete markets models are cast in terms of a commonset of objects and a common set of assumptions about how those objects fittogether, namely:

1. Descriptions of flows of information over time, of endowments of resources,and of commodities that can be traded

2. A technology for transforming endowments into commodities and an associatedset of feasible allocations

3. A list of people and their preferences over feasible allocations

4. An assignment of endowments to people, a price system, and a single budgetconstraint for each person

5. An equilibrium concept that uses prices to reconcile decisions of diverseprice-taking agents

This book is about constructing and applying competitive equilibria for aclass of linear-quadratic-Gaussian dynamic economies with complete markets.For us, an economy will consist of a list of matrices that describe people's householdtechnologies, their preferences over consumption services, their productiontechnologies, and their information sets. Competitive equilibrium allocationsand prices satisfy some equations that are easy to write down and solve. Thesecompetitive equilibrium outcomes have representations that are convenient torepresent and estimate econometrically.

Practical and analytical advantages flow from identifying an underlyingstructure that unites a class of economies. Practical advantages come from recognizingthat apparently different applications can be formulated and estimatedusing the same tools simply by replacing one list of matrices with another. Analyticaladvantages and deeper understandings come from appreciating the rolesplayed by key assumptions such as completeness of markets and structures ofheterogeneity.

1.2. A Class of Economies

We constructed our class of economies by using (1) a theory of recursive dynamiccompetitive economies, (2) linear optimal control theory, (3) methods for estimatingand interpreting vector autoregressions, and (4) a computer languagefor rapidly manipulating linear systems. Our economies have competitive equilibriawith representations in terms of vector autoregressions that can be swiftlycomputed, simulated, and estimated econometrically. The models thus mergeeconomic theory with dynamic econometrics. The computer language MATLABimplements the computations. It has a structure and vocabulary that economizetime and effort. Better yet, dynare has immensely improved, accelerated, andeased practical applications.

We formulated this class of models because practical difficulties of computingand estimating more general recursive competitive equilibrium modelscontinue to limit their use as tools for thinking about applied problems. Recursivecompetitive equilibria were developed as useful special cases of the Arrow-Debreucompetitive equilibrium model. Relative to the more general Arrow-Debreusetting, the great advantage of recursive competitive equilibria is thatthey can be computed by solving discounted dynamic programming problems.Furthermore, under some additional conditions, a competitive equilibrium canbe represented as a Markov process. When that Markov process has a uniqueinvariant distribution, there exists a vector autoregressive representation. Thus,the theory of recursive competitive equilibria holds out the promise of makingeasier contact with econometric theory than did previous formulations ofequilibrium theory.

Two computational difficulties continue to leave some of this promise unrealized.The first is a "curse of dimensionality" that makes dynamic programminga costly procedure with even small numbers of state variables. The second isthat after a dynamic program has been solved and an equilibrium Markov processcomputed, an implied vector autoregression has to be computed by applyingleast-squares projection formulas involving a large number of moments from themodel's invariant probability distribution. Typically, each of these computationalsteps can be solved only approximately. Good research along several lineshas been directed at improving these approximations.

The need to approximate originates in the fact that for general functionalforms for objective functions and constraints, even one iteration on the keyfunctional equation of dynamic programming (named the "Bellman equation"after Richard Bellman) cannot be performed analytically. It so happens thatthe functional forms economists would most like to use are ones for which theBellman equation cannot be iterated on analytically.

Linear control theory studies the most important special class of problemsfor which iterations on the Bellman equation can be performed analytically,namely, problems having a quadratic objective function and a linear transitionfunction. Application of dynamic programming leads to a system of well understoodand rapidly solvable equations known as the matrix Riccati differenceequation.

The philosophy of this book is to swallow hard and to accept up frontprimitive descriptions of tastes, technology, and information that satisfy the assumptionsof linear optimal control theory. This approach facilitates computingcompetitive equilibria that automatically take the form of a vector autoregression,albeit often cast in terms of some states unobserved to the econometrician.A cost of the approach is that it does not accommodate specifications that wesometimes prefer.

A purpose of this book is to display the versatility and tractability of ourclass of models. Versions of a wide range of models from modern capital theoryand asset pricing theory can be represented within our framework. Competitiveequilibria can be computed so easily that we hope that the reader will soonbe thinking of new models. We provide formulas and software for the reader toexperiment; and for many of our calculations, dynare offers even better software.

1.3. Computer Programs

In writing this book, we put ourselves under a restriction that we should supplythe reader with a computer program that implements every equilibrium conceptand mathematical representation. The programs are written in MATLAB, andare described throughout the book. When a MATLAB program is referredto in the text, we place it in typewriter font. Similarly, all computer codesappear in typewriter font. You will get much more out of this book if youuse and modify our programs as you read.

1.4. Organization

This book is organized as follows. Chapter 2 describes the first-order linear vectorstochastic difference equation and shows how special cases of it can representa variety of models of time series processes popular with economists. We usethis difference equation to represent the information flowing to economic agentsand also to represent competitive equilibria.

Chapter 3 is a catalogue of useful computational tricks that can be skippedon first reading. It describes fast ways to compute equilibria via doublingalgorithms that accelerate computation of expectations of geometric sums ofquadratic forms and solve dynamic programming problems. On first reading,it is good that the reader just knows that these fast methods are available andthat they are implemented both in our programs and in dynare.

Chapter 4 defines an economic environment in terms of a household technologyfor producing consumption services, preferences of a representative agent,a technology for producing consumption and investment goods, stochastic processesof shocks to preferences and technologies, and an information structure.The stochastic processes fit into the model introduced in chapter 2, while thepreferences, technology, and information structure are specified with an eye towardmaking competitive equilibria computable with linear control theory.

Chapter 5 describes a planning problem that generates competitive equilibriumallocations. We formulate the planning problem in two ways, first asa variational problem using stochastic Lagrange multipliers, then as a dynamicprogramming problem. We describe how to solve the dynamic programmingproblem with formulas from linear control theory. The solution of the planningproblem is a first-order vector stochastic difference equation of the form studiedin chapter 2. We also show how to use the value function for the planningproblem to compute Lagrange multipliers associated with constraints on theplanning problem.

Chapter 6 describes a commodity space and a price system that support acompetitive equilibrium. We use a formulation that lets the values to appear inagents' budget constraints and objective functions be represented as conditionalexpectations of geometric sums of streams of future prices times quantities.Chapter 6 relates these prices to Arrow-Debreu state-contingent prices.

Chapter 7 describes a decentralized economy and its competitive equilibrium.Competitive equilibrium quantities solve the chapter 5 planning problem.The price system can be deduced from the stochastic Lagrange multipliers associatedwith the chapter 5 planning problem.

Chapter 8 describes links between competitive equilibria and autoregressiverepresentations. We show how to obtain an autoregressive representationfor observable variables that are error-ridden linear functions of state variables.In describing how to deduce an autoregressive representation from a competitiveequilibrium and parameters of measurement error processes, we completea key step that facilitates econometric estimation of free parameters. An autoregressiverepresentation is naturally affiliated with a recursive representationof a likelihood function for the observable variables. More precisely, a vectorautoregressive representation implements a convenient factorization of the jointdensity of a complete history of observables (i.e., the likelihood function) intoa product of densities of time t observables conditioned on histories of thoseobservables up to time t - 1. Chapter 8 also treats two other topics intimatelyrelated to econometric implementation: aggregation over time and the theoryof approximation of one model by another.

Chapter 9 describes household technologies that describe the same preferencesand dynamic demand functions. It characterizes a special subset ofthem as canonical. Canonical household technologies are useful for describingeconomies with heterogeneity among households' preferences because of howthey align linear spaces consisting of histories of consumption services, on theone hand, and histories of consumption rates, on the other.

Chapter 10 describes some applications in the form of versions of severaldynamic models that fit easily within our class of models. These include modelsof markets for housing, cattle, and occupational choice.

Chapter 11 uses our model of preferences to represent multiple goods versionsof permanent income models. We retain Robert Hall's (1978) specificationof a "storage" technology for accumulating physical capital and also a restrictionon the discount factor, depreciation rate, and gross return on capital thatin Hall's simple setting made the marginal utility of consumption a martingale.In more general settings, adopting Hall's specification of the storage technologyimparts a martingale to outcomes, but it is concealed in an "index" whose incrementsdrive demands for multiple consumption goods that themselves are notmartingales. This permanent income model forms a convenient laboratory forthinking about sources in economic theory of "unit roots" and "co-integratingvectors."

Chapter 12 describes a type of heterogeneity among households that allowsus to aggregate preferences in a sense introduced by W. M. Gorman. Linear Engelcurves of common slopes across agents give rise to a representative consumer.This representative consumer is "easy to find," and, from the point of view ofcomputing equilibrium prices and aggregate quantities, adequately stands in forthe representative household of chapters 4–7. Finding competitive equilibriumallocations to individual consumers requires additional computations that thischapter also describes.

Chapter 13 outlines a setting with heterogeneity among households' preferencesof a kind that violates the conditions for Gorman aggregation. Households'Engel curves are still affine, but dispersion of their slopes prevents Gorman aggregation.However, there is another sense in which there is a representativehousehold whose preferences are a peculiar kind of average over the preferencesof different types of households. We show how to compute and interpret thispreference ordering over economy-wide aggregate consumption. This completemarkets aggregate preference ordering cannot be computed until one knows thedistribution of wealth evaluated at equilibrium prices, so it is less useful thanthe one produced by Gorman aggregation.

Chapter 14 adapts our setups to include features of periodic models ofseasonality studied by Osborn (1988, 1991a, 1991b) and Todd (1983, 1990).

Appendix A is a manual of the MATLAB programs that we have preparedto implement the calculations described in this book.

1.5. Recurring Mathematical Ideas

Duality between control problems and filtering problems underlies the findingthat recursive filtering problems have the same mathematical structure as recursiveformulations of linear optimal control problems. Both problems ultimatelylead to matrix Riccati equations. We use the duality of recursive linear optimalcontrol and linear filtering repeatedly both in chapter 8 (for representingequilibria econometrically) and in chapters 9, 12, and 13 (for representing andaggregating preferences).

In chapter 8, we state a spectral factorization identity that characterizesthe link between the state-space representation for a competitive equilibriumand the vector autoregression for observables. This is by way of obtaining the"innovations representation" that achieves a recursive representation of a Gaussianlikelihood function or quasi-likelihood function. In another guise, the samefactorization identity is also a key tool in constructing what we call a canonicalrepresentation of a household technology in chapter 9.

In more detail:

1. We use a linear state-space system to represent information flows that driveshocks to preferences and technologies (chapter 2).

2. We use a linear state-space system to represent observable quantities andscaled Arrow-Debreu prices associated with competitive equilibria (chapters5 and 7).

3. We coax scaled Arrow-Debreu prices from Lagrange multipliers associatedwith a planning problem (chapters 5 and 7).

4. We derive formulas for scaled Arrow-Debreu prices from gradients of thevalue function for a planning problem (chapters 5 and 7).

5. We use another linear state-space system called an innovations representationto deduce a recursive representation of a Gaussian likelihood functionor quasi-likelihood function associated with competitive equilibrium quantitiesand scaled Arrow-Debreu prices (chapter 8).

a. We use a Kalman filter to deduce an innovations representation associatedwith competitive equilibrium quantities and scaled Arrow-Debreuprices. In particular, we use the Kalman filter to construct a sequenceof densities of time t observables conditional on a history of the observablesup to time t-1. This sequence of conditional densities is anessential ingredient of a recursive representation of the likelihood function(also known as the joint density of the observables over a historyof length T).

b. The innovations in the innovation representation are square summablelinear functions of the history of the observables. Thus, the innovationsrepresentation is said to be "invertible," while the original state-spacerepresentation is in general not invertible.

c. The limiting time-invariant innovations representation associated witha fixed point of the Kalman filtering equations implements a spectralfactorization identity.