# Recursive Models of Dynamic Linear Economies.

Recursive Models of Dynamic Linear Economies, by Lars Peter Hansen and Thomas J. Sargent, Princeton, NJ: Princeton University Press, 2014.The recursive competitive equilibrium theory, in which the equilibrium is depicted as a set of stochastic processes with stationary transition probabilities, is the key development that has led to the revolution in macroeconomics, as Prescott (2006) states. The recursive framework implicated in various economic decision problems requires dynamic programming, which is then, in combination with modern general equilibrium theory, utilized to developed tractable models of dynamic economic systems, as exemplified by Stokey, Lucas and Prescott (1989).

The precedence and the value of this book is that it is one of the acclaimed benchmarks on the dynamic economic models, but is also the culmination of the progressive research agenda of the Nobel- Laureate authors on the recursive competitive equilibria. In this book, the authors construct a class of linear-quadratic-Gaussian economies with complete markets by using the theory of competitive economies, linear optimal control theory, and methods for estimating vector autoregressions. They include Matlab codes for manipulating their linear systems. The economies illustrated essentially consist of a list of matrices that describe agents' preference structures, information flows and productions, and technologies. The competitive equilibrium allocations and prices satisfy different forms of solvable equations.

Hansen and Sargent proclaim the practical and analytical advantage of identifying an underlying structure that unites a class of economies, albeit with key assumptions. The difficulties of computing and estimating more general recursive competitive equilibrium models, they add, limit the use in applied problems. Specifications arise as a need to develop tractable and well-defined systems that can provide a basis for econometric applications.

However, Hansen and Sargent further note that the inescapable computational difficulties continue to persevere in recursive competitive equilibrium models with additional conditions. They indicate that the curse of dimensionality, and estimating an implied vector autoregressive representation from an equilibrium Markov process obtained by the solution of a dynamic program, continue to pose challenges against establishing links with the econometric theory. The primitive descriptions of preferences, technology and information that satisfy the assumptions of linear control theory are acknowledged inevitably as the philosophy of recursive competitive equilibria.

The book begins with a short overview and an explanation of recurring mathematical ideas. Chapters Two and Three comprise the second part of the book, namely the Tools. These include describing first-order linear vector stochastic difference equations as the building block for a class of economic structures with competitive equilibrium prices and quantities; and explaining fast algorithms, like the doubling algorithm, for computing the value function and optimal decision rule of social planning problems. The first-order vector stochastic difference equation expresses the next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances that form a martingale difference sequence, to the system. The state of the economy is represented as a process constructed recursively using an initial random vector and a time-invariant law of motion. This representation is used to model the information upon which economic agents base their decisions. It characterizes the competitive equilibrium that accommodates the economic problems faced by agents and the economic arrangements that originate from agents' decisions. In the third part, from Chapters Four through Seven, the Components of Economies are defined upon this representation.

Chapter Four describes the economic environment in terms of five components; a sequence of information sets, laws of motions for taste and technology shocks, production technology for producing consumption goods, household technology for producing services from consumer durables and consumption purchases, and a preference ordering over consumption services. In the examples of production technologies, a pure consumption endowment is utilized as the required specification to create a linear-quadratic version of Lucas's asset pricing model.

Chapter Five deals with the optimal resource allocation problem. It explains the Lagrange multipliers and dynamic programming methods used in solving this problem and demonstrates the relation of Lagrange multipliers with the value function. In this chapter, the authors illustrate step-by-step solutions of planning problems with Matlab scripts. In addition to combining different technologies and preference specifications, the scripts include a rich variety of forms and explanations for specializations, thus making it easier for us to follow transformations of specifications between models, e.g. obtaining Lucas's pure endowment asset pricing economy from the consumption model of Hall, or the growth technology of Jones and Manuelli. Chapter Six introduces a decentralized version of the economy presented in Chapter Five, and Chapter Seven defines competitive equilibrium and the equilibrium price system.

The fourth part of the book starts with Chapter Eight and is categorized as Representations and Properties. This Chapter deals with the interplay between competitive equilibria and autoregressive representations. The above-mentioned martingale difference sequences are shocks to endowments and preferences, whose histories are observed by agents in the economy. Since the shocks are not directly observed in state-space representations of the economy with states and observables, a Kalman filter is implemented to obtain a representation of innovations. Chapter Nine derives the concept of canonical representation of household technologies, and applies it to a version of Becker and Murphy's model of rational addiction.

Chapter Ten exemplifies some models -including models of markets for housing, cattle, and occupational choice- that conform to a general equilibrium framework. The rational expectations equilibrium, or a partial equilibrium, as presented in Lucas and Prescott's (1971) model of investment under uncertainty that employs the notion of a representative firm, can be equivalently presented in the form of general equilibrium models, as defined by the authors. Chapter Eleven describes a class of permanent income models of consumption. Chapters Twelve and Thirteen explain methods for computing equilibria of economies with consumers that have heterogeneous preferences and endowments. The class of heterogeneous consumer economies described in Chapter Twelve satisfies the Gorman conditions for aggregation. That is, households have identical preferences and technologies, implying linear Engel curves with the same slopes, allowing a tractable aggregation trajectory. Chapter Thirteen expands the concept by investigating varying slopes across classes of households, and thus allows the existence of a representative household.

Finally, Chapter Fourteen relaxes the assumption of constant preferences, technologies and information flows by introducing seasonality as hidden periodicity, as an extension of Osborn and Todd. Appendix A provides a helpful manual for Matlab programs.

The parts of the book that deal with the Lucas endowment economies were particularly pleasing to this reviewer. We observe empirical problems, such as equity premium being much larger in the data than is implied by a representative agent asset pricing model with reasonable risk-aversion parameters, or the risk free rates being much lower relative to aggregate rates of consumption growth, and we still return to versions of Lucas's asset pricing model as a benchmark.

While this book illustrates the underpinnings of the recursive competitive equilibrium theory, the abstractions oblige us to refer to the relevant literature. The development of recursive competitive equilibrium theory can be traced back to works of Lucas and Prescott (1971), Lucas (1972), Mehra and Prescott (1977), and Prescott and Mehra (1980). As a prequel, and complementary to this book, Harris (1987), Stokey, Lucas and Prescott (1989) and Ljungqvist and Sargent (2004) should also be mentioned, as an absolute minimum of references to grasp the technical concepts. Furthermore, in a parallel study, Hansen and Sargent (2008) elaborate more on the model misspecification issues in robust control theory applied to economic problems.

It must be acknowledged that recursive competitive equilibrium theory established by these seminal works has brought a Copemican Revolution to macroeconomics; a paradigm shift in aggregate economics that allows developing tractable macro models for drawing scientific inference. Still, it is highly probable that the reader might lose the sense of reality somewhere along the line, since we are operating in a highly idealized world of abstractions.

Frankly, to the vast majority of readers, this book offers blood, toil, tears and sweat. On a more general note, the authors define the world we live in (or at least what we theorize), lay the mathematical foundations of aggregate economics, and they rest on the seventh day.

Yet, if we are to understand the universe created by recursive equilibrium, we need to absorb all the ideas upon which the theory has been established. This includes a visit to the concept of Arrow-Debreu general equilibrium structures in modeling uncertain dynamic economic phenomena, as the main line of reasoning extends from it. Recursive competitive theory, by establishing time-invariant equilibrium decision rules - including a pricing function, a value function, period allocation policies specifying decisions, and a function for law of motion - characterizes the effects of past decisions and current information. This reasoning has been widely used in exploring a wide range of economic issues ranging from business-cycle fluctuations to monetary and fiscal policy riddles. One needs to only pick the appropriate world from the set of worlds, i.e. applications of recursive methods, derived in this work.

Aytac Erdemir Department of Economics Jaume I University, Spain and

Chair of Monetary Economics and International Finance Kiel University, Germany

References

Hansen, L. P., and Sargent, T. J. 2008. Robustness. Princeton, NJ: Princeton University Press.

Harris, M. 1987. Dynamic Economic Analysis. New York, NY: Oxford University Press.

Ljungqvist, L., and Sargent, T. J. 2004. Recursive Macroeconomic Theory. Cambridge, MA: MIT Press.

Lucas Jr, R. E. 1972. "Expectations and the Neutrality of Money." Journal of Economic Theory 4: 108-124.

Lucas Jr, R. E., and Prescott, E. C. 1971. "Investment under Uncertainty." Econometrica: Journal of the Econometric Society, 39(5): 659-81.

Mehra, R., and Prescott, E. C. 1977. Recursive Competitive Equilibria and Capital Asset Pricing: Essays in Financial Economics. Doctoral Dissertation. Pittsburgh, PA: Carnegie Mellon University.

Prescott, E. C., and Mehra, R. 1980. "Recursive Competitive Equilibrium: The Case of Homogeneous Households." Econometrica: Journal of the Econometric Society, 48(6): 1365-79.

Prescott, E. C. 2006. "The Transformation of Macroeconomic Policy and Research." The American Economist, 50(1): 3-20.

Stokey, N. L., and Lucas Jr, R. E., with Prescott, E. C. 1989. Recursive Methods in Economic Dynamics. Cambridge, MA: Harvard University Press.

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Author: | Erdemir, Aytac |
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Publication: | American Economist |

Article Type: | Book review |

Date: | Sep 22, 2014 |

Words: | 1669 |

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