# Recursive Macroeconomic Theory

Lars Ljungqvist
Thomas J. Sargent
Pages: 1360
https://www.jstor.org/stable/j.ctt5vjq05

1. Front Matter
(pp. i-iv)
(pp. v-xviii)
3. Acknowledgments
(pp. xix-xix)
4. Preface to the third edition
(pp. xx-xxxvi)
5. Part I: The imperialism of recursive methods
• Chapter 1 Overview
(pp. 3-26)

This chapter provides a nontechnical summary of some themes of this book. We debated whether to put this chapter first or last. A way to use this chapter is to read it twice, once before reading anything else in the book, then again after having mastered the techniques presented in the rest of the book. That second time, this chapter will be easy and enjoyable reading, and it will remind you of connections that transcend a variety of apparently disparate topics. But on first reading, this chapter will be difficult, partly because the discussion is mainly literary and therefore incomplete....

6. Part II: Tools
• Chapter 2 Time Series
(pp. 29-102)

This chapter describes two tractable models of time series: Markov chains and first-order stochastic linear difference equations. These models are organizing devices that put restrictions on a sequence of random vectors. They are useful because they describe a time series with parsimony. In later chapters, we shall make two uses each of Markov chains and stochastic linear difference equations: (1) to represent the exogenous information flows impinging on an agent or an economy, and (2) to represent an optimum or equilibrium outcome of agents’ decision making. The Markov chain and the first-order stochastic linear difference both use a sharp notion...

• Chapter 3 Dynamic Programming
(pp. 103-112)

This chapter introduces basic ideas and methods of dynamic programming.¹ It sets out the basic elements of a recursive optimization problem, describes a key functional equation called the Bellman equation, presents three methods for solving the Bellman equation, and gives the Benveniste-Scheinkman formula for the derivative of the optimal value function. Let’s dive in.

Letβ∈ (0, 1) be a discount factor. We want to choose an infinite sequence of “controls”$\left\{{{u}_{t}} \right\}_{t=0}^{\infty}$to maximize$\sum\limits_{t=0}^{\infty }{{{\beta }^{t}}}r({{x}_{t}},{{u}_{t}}),\caption {(3.1.1)}$subject toxt+1=g(xt,ut), with${{x}_{0}}\in {{IR}^{n}}$given. We assume thatr(xt,ut) is a concave function and that the set$\left\{ ({{x}_{t+1}},{{x}_{t}}):{{x}_{t+1}}\le g({{x}_{t}},{{u}_{t}}),{{u}_{t}}\in {{IR}^{k}} \right\}$is...

• Chapter 4 Practical Dynamic Programming
(pp. 113-126)

We often encounter problems where it is impossible to attain closed forms for iterating on the Bellman equation. Then we have to adopt numerical approximations. This chapter describes two popular methods for obtaining numerical approximations. The first method replaces the original problem with another problem that forces the state vector to live on a finite and discrete grid of points, then applies discrete-state dynamic programming to this problem. The “curse of dimensionality” impels us to keep the number of points in the discrete state space small. The second approach uses polynomials to approximate the value function. Judd (1998) is a...

• Chapter 5 Linear Quadratic Dynamic Programming
(pp. 127-158)

This chapter describes the class of dynamic programming problems in which the return function is quadratic and the transition function is linear. This specification leads to the widely used optimal linear regulator problem, for which the Bellman equation can be solved quickly using linear algebra. We consider the special case in which the return function and transition function are both time invariant, though the mathematics is almost identical when they are permitted to be deterministic functions of time.

Linear quadratic dynamic programming has two uses for us. A first is to study optimum and equilibrium problems arising for linear rational...

• Chapter 6 Search, Matching, and Unemployment
(pp. 159-224)

This chapter applies dynamic programming to a choice between only two actions, to accept or reject a take-it-or-leave-it job offer. An unemployed worker faces a probability distribution of wage offers or job characteristics, from which a limited number of offers are drawn each period. Given his perception of the probability distribution of offers, the worker must devise a strategy for deciding when to accept an offer.

The theory of search is a tool for studying unemployment. Search theory puts unemployed workers in a setting where sometimes they choose to reject available offers and to remain unemployed now because they prefer...

7. Part III: Competitive equilibria and applications
• Chapter 7 Recursive (Partial) Equilibrium
(pp. 227-250)

This chapter formulates competitive and oligopolistic equilibria in some dynamic settings. Up to now, we have studied single-agent problems where components of the state vector not under the control of the agent were taken as given. In this chapter, we describe multiple-agent settings in which components of the state vector that one agent takes as exogenous are determined by the decisions of other agents. We study partial equilibrium models of a kind applied in microeconomics.¹ We describe two closely related equilibrium concepts for such models: a rational expectations or recursive competitive equilibrium, and a Markov perfect equilibrium. The first equilibrium...

• Chapter 8 Equilibrium with Complete Markets
(pp. 251-314)

This chapter describes competitive equilibria of a pure exchange infinite horizon economy with stochastic endowments. These are useful for studying risk sharing, asset pricing, and consumption. We describe two systems of markets: anArrow-Debreustructure with complete markets in dated contingent claims all traded at time 0, and a sequential-trading structure with complete one-periodArrow securities. These two entail different assets and timings of trades, but have identical consumption allocations. Both are referred to as complete markets economies. They allow more comprehensive sharing of risks than do the incomplete markets economies to be studied in chapters 17 and 18, or...

• Chapter 9 Overlapping Generations Models
(pp. 315-362)

This chapter describes the pure exchange overlapping generations model of Paul Samuelson (1958). We begin with an abstract presentation that treats the overlapping generations model as a special case of the chapter 8 general equilibrium model with complete markets and all trades occurring at time 0. A peculiar type of heterogeneity across agents distinguishes the model. Each individual cares about consumption only at two adjacent dates, and the set of individuals who care about consumption at a particular date includes some who care about consumption one period earlier and others who care about consumption one period later. We shall study...

• Chapter 10 Ricardian Equivalence
(pp. 363-374)

This chapter studies whether the timing of taxes matters. Under some assumptions it does and under others it does not. The Ricardian doctrine describes assumptions under which the timing of lump taxes does not matter. In this chapter, we will study how the timing of taxes interacts with restrictions on the ability of households to borrow. We study the issue in two equivalent settings: (1) an infinite horizon economy with an infinitely lived representative agent; and (2) an infinite horizon economy with a sequence of one-period-lived agents, each of whom cares about its immediate descendant. We assume that the interest...

• Chapter 11 Fiscal Policies in a Growth Model
(pp. 375-454)

This chapter studies effects of technology and fiscal shocks on equilibrium outcomes in a nonstochastic growth model. We use the model to state some classic doctrines about the effects of various types of taxes and also as a laboratory to exhibit numerical techniques for approximating equilibria and to display the structure of dynamic models in which decision makers have perfect foresight about future government decisions. Foresight imparts effects on prices and allocations that precede government actions that cause them.

Following Hall (1971), we augment a nonstochastic version of the standard growth model with a government that purchases a stream of...

• Chapter 12 Recursive Competitive Equilibria
(pp. 455-480)

For pure endowment stochastic economies, chapter 8 described two types of competitive equilibria, one in the style of Arrow and Debreu with markets that convene at time 0 and trade a complete set of history-contingent securities, another with markets that meet each period and trade a complete set of one-period-ahead state-contingent securities called Arrow securities. Though their price systems and trading protocols differ, both types of equilibria support identical equilibrium allocations. Chapter 8 described how to transform the Arrow-Debreu price system into one for pricing Arrow securities. The key step in transforming an equilibrium with time 0 trading into one...

• Chapter 13 Asset Pricing Theory
(pp. 481-514)

Chapter 8 showed how an equilibrium price system for an economy with a complete markets model could be used to determine the price of any redundant asset. That approach allowed us to price any asset whose payoff could be synthesized as a measurable function of the economy’s state. We could use either the Arrow-Debreu time 0 prices or the prices of one-period Arrow securities to price redundant assets.

We shall use this complete markets approach again later in this chapter and in chapter 14. However, we begin with another frequently used approach, one that does not require the assumption that...

• Chapter 14 Asset Pricing Empirics
(pp. 515-582)

In chapter 13, we repeatedly encountered an object that in this chapter we shall call a stochastic discount factormt+1, namely${{m}_{t+1}}=\beta {{\left( \frac{{{C}_{t+1}}}{{{C}_{t}}} \right)}^{-\text{ }\!\!\gamma\!\!\text{ }}},\caption {(14.1.1)}$whereβis a discount factor,γis a coefficient of relative risk aversion, andCtis the consumption of a representative consumer. The asset pricing theories in chapter 13 can be summarized in a nutshell as asserting that for any assetjtraded by a representative consumer, its one period gross returnRj,t+1must satisfy${{E}_{t}}({{m}_{t+1}}{{R}_{j,t+1}})=1.\caption {(14.1.2)}$

Empirically, for the stochastic discount factor (14.1.1), restriction (14.1.2) fails to work well when applied to data on returns of stocks...

• Chapter 15 Economic Growth
(pp. 583-612)

This chapter describes basic nonstochastic models of sustained economic growth. We begin by describing a benchmark exogenous growth model where sustained growth is driven by exogenous growth in labor productivity. Then we turn our attention to several endogenous growth models where sustained growth of labor productivity is somehowchosenby the households in the economy. We describe several models that differ in whether the equilibrium market economy matches what a benevolent planner would choose. Where the market outcome doesn’t match the planner’s outcome, there can be room for welfare-improving government interventions. The objective of the chapter is to shed light...

• Chapter 16 Optimal Taxation with Commitment
(pp. 613-696)

This chapter formulates a dynamic optimal taxation problem called a Ramsey problem with a solution called a Ramsey plan. The government’s goal is to maximize households’ welfare subject to raising set revenues through distortionary taxation. When designing an optimal policy, the government takes into account the competitive equilibrium reactions by consumers and firms to the tax system. We first study a nonstochastic economy, then a stochastic economy.

The model is a competitive equilibrium version of the basic neoclassical growth model with a government that finances an exogenous stream of government purchases. In the simplest version, the production factors are raw...

8. Part IV: The savings problem and Bewley models
• Chapter 17 Self-Insurance
(pp. 699-724)

This chapter describes a version of what is sometimes called a savings problem (e.g., Chamberlain and Wilson, 2000). A consumer wants to maximize the expected discounted sum of a concave function of one-period consumption rates, as in chapter 8. However, the consumer is cut off from all insurance markets and almost all asset markets. The consumer can purchase only nonnegative amounts of a single risk-free asset. The absence of insurance opportunities induces the consumer to use variations over time in his asset holdings to acquire “self-insurance.”

This model is interesting to us partly as a benchmark to compare with the...

• Chapter 18 Incomplete Markets Models
(pp. 725-772)

In the complete markets model of chapter 8, the optimal consumption allocation is not history dependent; the allocation depends on the current value of the Markov state variable only. This outcome reflects the comprehensive opportunities to insure risks that markets provide. This chapter and chapter 20 describe settings with more impediments to exchanging risks. These reduced opportunities make allocations history dependent. In this chapter, the history dependence is encoded in the dependence of a household’s consumption on the household’s current asset holdings. In chapter 20, history dependence is encoded in the dependence of the consumption allocation on a continuation value...

9. Part V: Recursive contracts
• Chapter 19 Dynamic Stackelberg Problems
(pp. 775-796)

Previous chapters described decision problems that are recursive in what we can call “natural” state variables, i.e., state variables that describe stocks of capital, wealth, and information that helps forecast future values of prices and quantities that impinge on future utilities or profits. In problems that are recursive in the natural state variables, optimal decision rules are functions of the natural state variables.

This chapter is our first encounter with a class of problems that are not recursive in the natural state variables. Kydland and Prescott (1977), Prescott (1977), and Calvo (1978) gave macroeconomic examples of decision problems whose solutions...

• Chapter 20 Insurance Versus Incentives
(pp. 797-858)

This chapter studies a planner who designs an efficient contract to supply insurance in the presence of incentive constraints imposed by his limited ability either to enforce contracts or to observe households’ actions or incomes. We pursue two themes, one substantive, the other technical. The substantive theme is a tension that exists between offering insurance and providing incentives. A planner can overcome incentive problems by offering “sticks and carrots” that adjust an agent’s future consumption and thereby provide less insurance. Balancing incentives against insurance shapes the evolution of distributions of wealth and consumption.

The technical theme is how memory can...

• Chapter 21 Equilibrium without Commitment
(pp. 859-912)

In section 20.3 of the previous chapter, we studied insurance without commitment. That was a partial equilibrium analysis since the moneylender could borrow or lend resources outside of the village at a given interest rate. Recall also the asymmetry in the environment where villagers could not make any commitments while the moneylender was assumed to be able to commit. We will now study a closed system without access to an outside credit market. Any household’s consumption in excess of its own endowment must then come from the endowments of the other households in the economy. We will also adopt the...

• Chapter 22 Optimal Unemployment Insurance
(pp. 913-936)

This chapter applies the recursive contract machinery studied in chapters 20, 21, and 23 in contexts that are simple enough that we can go a long way toward computing optimal contracts by hand. The contracts encode history dependence by mapping an initial promised value and a random timetobservation into a timetconsumption allocation and a continuation value to bring into next period. We use recursive contracts to study good ways of providing consumption insurance when incentive problems come from the insurance authority’s inability to observe the effort that an unemployed person exerts searching for a job. We...

• Chapter 23 Credible Government Policies, I
(pp. 937-984)

Kydland and Prescott (1977) opened the modern discussion of time consistency in macroeconomics with some examples that show how outcomes differ in otherwise identical economies when the assumptions about the timing of government policy choices are altered.¹ In particular, they compared a timing protocol in which a government chooses its (possibly history-contingent) policies once and for all at the beginning of time with one in which the government chooses sequentially. Because outcomes are worse when the government chooses sequentially, Kydland and Prescott’s examples illustrate the value to a government of having a “commitment technology” that requires it not to choose...

• Chapter 24 Credible Government Policies, II
(pp. 985-1004)

Chapter 23 adopted a simple setting designed to isolate opportunities that confront the government when the private sector’s forecasting problem is theonlysource of dynamics. We studied dynamics that come exclusively from a benevolent government’s incentives to confirm or disappoint private agents’ forecasts of timetgovernment actions on the basis of histories of outcomes observed through timet− 1. To focus attention solely on the government’s incentives to confirm or disappoint expectations, we analyzed credible public policies in simplified settings with competitive equilibria in which households and firms face a sequence of static problems. In particular, we...

• Chapter 25 Two Topics in International Trade
(pp. 1005-1042)

This chapter studies two models in which recursive contracts are used to overcome incentive problems commonly thought to occur in international trade. The first is Andrew Atkeson’s model of lending in the context of a dynamic setting that contains both a moral hazard problem due to asymmetric informationandan enforcement problem due to borrowers’ option to disregard the contract. It is a considerable technical achievement that Atkeson managed to include both of these elements in his contract design problem. But this substantial technical accomplishment is not just showing off. As we shall see,boththe moral hazardandthe...

10. Part VI: Classical monetary and labor economics
• Chapter 26 Fiscal-Monetary Theories of Inflation
(pp. 1045-1092)

This chapter introduces some issues in monetary theory that mostly revolve around coordinating monetary and fiscal policies. We start from the observation that complete markets models have no role for inconvertible currency, and therefore assign zero value to it.¹ We describe one way to alter a complete markets economy so that a positive value is assigned to an inconvertible currency: we impose a transaction technology with shopping time and real money balances as inputs.² We use the model to illustrate 10 doctrines in monetary economics. Most of these doctrines transcend many of the details of the model. The important thing...

• Chapter 27 Credit and Currency
(pp. 1093-1128)

This chapter describes Townsend’s (1980) turnpike model of money and puts it to work. The model uses a particular pattern of heterogeneity of endowments and locations to create a demand for currency. The model is more primitive than the shopping time model of chapter 26. As with the overlapping generations model, the turnpike model starts from a setting in which diverse intertemporal endowment patterns across agents prompt borrowing and lending. If something prevents loan markets from operating, it is possible that an unbacked currency can play a role in helping agents smooth their consumption over time. Following Townsend, we shall...

• Chapter 28 Equilibrium Search and Matching
(pp. 1129-1202)

This chapter presents various equilibrium models of search and matching. We describe (1) Lucas and Prescott’s version of an island model; (2) some matching models in the style of Mortensen, Pissarides, and Diamond; and (3) a search model of money along the lines of Kiyotaki and Wright.

Chapter 6 studied the optimization problem of a single unemployed agent who searched for a job by drawing from an exogenous wage offer distribution. We now turn to a model with a continuum of agents who interact across a large number of spatially separated labor markets. Phelps (1970, introductory chapter) describes such an...

• Chapter 29 Foundations of Aggregate Labor Supply
(pp. 1203-1254)

The section 28.6 employment lotteries model for years served as the foundation of the high aggregate labor supply elasticity that generates big employment fluctuations in real business cycle models. In the original version of his Nobel prize lecture, Prescott (2005a) highlighted the central role of employment lotteries for real business cycle models when he asserted that “Rogerson’s aggregation result is every bit as important as the one giving rise to the aggregate production function.” But Prescott’s enthusiasm for employment lotteries has not been shared universally, especially by researchers who have studied labor market experiences of individual workers. For example, Browning,...

11. Part VII: Technical appendices
• Appendix A. Functional Analysis
(pp. 1257-1268)
• Appendix B. Linear projections and hidden Markov models
(pp. 1269-1274)
12. 1. References
(pp. 1275-1308)
13. 2. Subject Index
(pp. 1309-1314)
14. 3. Author Index
(pp. 1315-1320)
15. 4. Matlab Index
(pp. 1321-1321)