Recursive methods in economic dynamics

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Recursive methods in economic dynamics

Book•

Recursive methods in economic dynamics

Nancy L. Stokey, Robert E. Lucas, Edward C. Prescott

01 Jan 1989-

TL;DR: In this article, a deterministic model of optimal growth is proposed, and a stochastic model is proposed for optimal growth with linear utility and linear systems and linear approximations.

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Abstract: I. THE RECURSIVE APPROACH 1. Introduction 2. An Overview 2.1 A Deterministic Model of Optimal Growth 2.2 A Stochastic Model of Optimal Growth 2.3 Competitive Equilibrium Growth 2.4 Conclusions and Plans II. DETERMINISTIC MODELS 3. Mathematical Preliminaries 3.1 Metric Spaces and Normed Vector Spaces 3.2 The Contraction Mapping Theorem 3.3 The Theorem of the Maximum 4. Dynamic Programming under Certainty 4.1 The Principle of Optimality 4.2 Bounded Returns 4.3 Constant Returns to Scale 4.4 Unbounded Returns 4.5 Euler Equations 5. Applications of Dynamic Programming under Certainty 5.1 The One-Sector Model of Optimal Growth 5.2 A "Cake-Eating" Problem 5.3 Optimal Growth with Linear Utility 5.4 Growth with Technical Progress 5.5 A Tree-Cutting Problem 5.6 Learning by Doing 5.7 Human Capital Accumulation 5.8 Growth with Human Capital 5.9 Investment with Convex Costs 5.10 Investment with Constant Returns 5.11 Recursive Preferences 5.12 Theory of the Consumer with Recursive Preferences 5.13 A Pareto Problem with Recursive Preferences 5.14 An (s, S) Inventory Problem 5.15 The Inventory Problem in Continuous Time 5.16 A Seller with Unknown Demand 5.17 A Consumption-Savings Problem 6. Deterministic Dynamics 6.1 One-Dimensional Examples 6.2 Global Stability: Liapounov Functions 6.3 Linear Systems and Linear Approximations 6.4 Euler Equations 6.5 Applications III. STOCHASTIC MODELS 7. Measure Theory and Integration 7.1 Measurable Spaces 7.2 Measures 7.3 Measurable Functions 7.4 Integration 7.5 Product Spaces 7.6 The Monotone Class Lemma

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Posted Content•

The Impact of Uncertainty Shocks

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Nicholas Bloom¹•Institutions (1)

London School of Economics and Political Science¹

01 Sep 2007-Research Papers in Economics

TL;DR: In this paper, a model with a time varying second moment is proposed to simulate a macro uncertainty shock, which produces a rapid drop and rebound in aggregate output and employment, which occurs because higher uncertainty causes firms to temporarily pause their investment and hiring.

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Abstract: Uncertainty appears to jump up after major shocks like the Cuban Missile crisis, the assassination of JFK, the OPEC I oil-price shock and the 9/11 terrorist attack This paper offers a structural framework to analyze the impact of these uncertainty shocks I build a model with a time varying second moment, which is numerically solved and estimated using firm level data The parameterized model is then used to simulate a macro uncertainty shock, which produces a rapid drop and rebound in aggregate output and employment This occurs because higher uncertainty causes firms to temporarily pause their investment and hiring Productivity growth also falls because this pause in activity freezes reallocation across units In the medium term the increased volatility from the shock induces an overshoot in output, employment and productivity Thus, second moment shocks generate short sharp recessions and recoveries This simulated impact of an uncertainty shock is compared to VAR estimations on actual data, showing a good match in both magnitude and timing The paper also jointly estimates labor and capital convex and non-convex adjustment costs Ignoring capital adjustment costs is shown to lead to substantial bias while ignoring labor adjustment costs does not

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3,405 citations

Cites methods from "Recursive methods in economic dynam..."

...23The application of Stokey and Lucas (1989) for the continuous, concave and almost surely bounded normalized returns and cost function in (3.9) for quadratic adjustment costs and partial irreversibilities, and Caballero and Leahy 11 numerical methods knowing that this solution is convergent with…...
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Journal Article•DOI•

Occupational Choice and the Process of Development

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Abhijit Banerjee, Andrew F. Newman

01 Apr 1993-Journal of Political Economy

TL;DR: In this paper, the authors model economic development as a process of institutional transformation by focusing on the interplay between agents' occupational decisions and the distribution of wealth, and demonstrate the robustness of this result by extending the model dynamically and studying examples in which initial wealth distributions have long-run effects.

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Abstract: This paper models economic development as a process of institutional transformation by focusing on the interplay between agents' occupational decisions and the distribution of wealth. Because of capital market imperfections, poor agents choose working for a wage over self-employment, and wealthy agents become entrepreneurs who monitor workers. Only with sufficient inequality, however, will there be employment contracts; otherwise, there is either subsistence or self-employment. Thus, in static equilibrium, the occupational structure depends on distribution. Since the latter is itself endogenous, we demonstrate the robustness of this result by extending the model dynamically and studying examples in which initial wealth distributions have long-run effects. In one case the economy develops either widespread cottage industry (self-employment) or factory production (employment contracts), depending on the initial distribution; in the other example, it develops into prosperity or stagnation.

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2,906 citations

Book•

Numerical methods in economics

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Kenneth L. Judd

01 Jan 1998

TL;DR: In this article, the authors present techniques from the numerical analysis and applied mathematics literatures and show how to use them in economic analyses, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods.

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Abstract: To harness the full power of computer technology, economists need to use a broad range of mathematical techniques. In this book, Kenneth Judd presents techniques from the numerical analysis and applied mathematics literatures and shows how to use them in economic analyses. The book is divided into five parts. Part I provides a general introduction. Part II presents basics from numerical analysis on R^n, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods. Part III covers methods for dynamic problems, including finite difference methods, projection methods, and numerical dynamic programming. Part IV covers perturbation and asymptotic solution methods. Finally, Part V covers applications to dynamic equilibrium analysis, including solution methods for perfect foresight models and rational expectation models. A web site contains supplementary material including programs and answers to exercises.

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2,880 citations

Journal Article•DOI•

Uninsured Idiosyncratic Risk and Aggregate Saving

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S. Rao Aiyagari¹•Institutions (1)

Federal Reserve Bank of Minneapolis¹

01 Aug 1994-Quarterly Journal of Economics

TL;DR: In this article, the authors present a qualitative and quantitative analysis of the standard growth model modified to include precautionary saving motives and liquidity constraints, and address the impact on the aggregate saving rate, the importance of asset trading to individuals, and the relative inequality of wealth and income distributions.

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Abstract: We present a qualitative and quantitative analysis of the standard growth model modified to include precautionary saving motives and liquidity constraints. We address the impact on the aggregate saving rate, the importance of asset trading to individuals, and the relative inequality of wealth and income distributions.

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2,738 citations

Report•DOI•

Financial Development, Growth, and the Distribution of Income

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Jeremy Greenwood¹, Boyan Jovanovic¹•Institutions (1)

University of Rochester¹

01 Oct 1990-Journal of Political Economy

TL;DR: In this paper, a paradigm is presented in which both the extent of financial intermediation and the rate of economic growth are endogenously determined, and the model also generates a development cycle reminiscent of the Kuznet hypothesis.

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Abstract: A paradigm is presented in which both the extent of financial intermediation and the rate of economic growth are endogenously determined. Financial intermediation promotes growth because it allows a higher rate of return to be earned on capital, and growth in turn provides the means to implement costly financial structures. Thus financial intermediation and economic growth are inextricably linked in accord with the Goldsmith-McKinnon-Shaw view on economic development. The model also generates a development cycle reminiscent of the Kuznet hypothesis. In particular, in the transition from a primitive slow-growing economy to a developed fast-growing one, a nation passes through a stage in which the distribution of wealth across the rich and poor widens.

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2,570 citations

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Book•

Stochastic processes

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J. L. Doob, Joseph L. Doob

01 Jan 1953

10,512 citations

"Recursive methods in economic dynam..." refers background in this paper

...1 is based on Doob (1953, sect. V.2). Kemeny and Snell (1960), Chung (1967), and Kemeny, Snell, and Knapp (1976) are also excellent sources. Green (1976) contains a result related to Theorem 11.4. Some writers use the term Markov chain to refer to any Markov process with a discrete time parameter, regardless of the nature of the state space. What we call a Markov chain is then referred to as afinite state Markov chain. Condition D is due to Doeblin; it is discussed in detail in Doob (1953, sect. V.5), where a proof ofTheorem 1 1.9 can also be found. An alternative line of proof uses the fact that Doeblin’s condition holds if and only if the operator associated with the transition function is quasi-compact. A proof of this fact is available in Futia (1982, Theorem 4.9). Neveu (1965, sect. V.3) contains a proof of Theorem 11.9 based on this fact. Both lines of proof also establish that convergence is at a uniform geometric rate. Tweedie (1975) provides an alternative—and quite different—set of sufficient conditions for convergence, conditions that may hold when the rate of convergence is not uniform. We are grateful to C. Ionescu Tulcea for bringing to our attention Condition M. It is discussed in Onicescu (1969), where Theorem 11.12 also appears. The proof offered here, based on the Contraction Mapping Theorem, is new. See Dynkin (1965), Rosenblatt (1971), and Gihman and Skorohod (1974) for more extensive treatments of general Markov processes....
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...1 is based on Doob (1953, sect. V.2). Kemeny and Snell (1960), Chung (1967), and Kemeny, Snell, and Knapp (1976) are also excellent sources....
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...1 is based on Doob (1953, sect. V.2). Kemeny and Snell (1960), Chung (1967), and Kemeny, Snell, and Knapp (1976) are also excellent sources. Green (1976) contains a result related to Theorem 11....
[...]
...1 is based on Doob (1953, sect. V.2). Kemeny and Snell (1960), Chung (1967), and Kemeny, Snell, and Knapp (1976) are also excellent sources. Green (1976) contains a result related to Theorem 11.4. Some writers use the term Markov chain to refer to any Markov process with a discrete time parameter, regardless of the nature of the state space. What we call a Markov chain is then referred to as afinite state Markov chain. Condition D is due to Doeblin; it is discussed in detail in Doob (1953, sect. V.5), where a proof ofTheorem 1 1.9 can also be found. An alternative line of proof uses the fact that Doeblin’s condition holds if and only if the operator associated with the transition function is quasi-compact. A proof of this fact is available in Futia (1982, Theorem 4.9). Neveu (1965, sect. V.3) contains a proof of Theorem 11.9 based on this fact. Both lines of proof also establish that convergence is at a uniform geometric rate. Tweedie (1975) provides an alternative—and quite different—set of sufficient conditions for convergence, conditions that may hold when the rate of convergence is not uniform....
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Journal Article•DOI•

Rational Expectations and the Theory of Price Movements

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John F. Muth

01 Jul 1961-Econometrica

TL;DR: In this article, the Stockholm School hypothesis is used to explain how expectations are formed in the context of an isolated market with a fixed production lag, and commodity speculation is introduced into the system.

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Abstract: In order to explain fairly simply how expectations are formed, we advance the hypothesis that they are essentially the same as the predictions of the relevant economic theory. In particular, the hypothesis asserts that the economy generally does not waste information, and that expectations depend specifically on the structure of the entire system. Methods of analysis, which are appropriate under special conditions, are described in the context of an isolated market with a fixed production lag. The interpretative value of the hypothesis is illustrated by introducing commodity speculation into the system. 1. INTRODUCTION THAT EXPECTATIONS of economic variables may be subject to error has, for some time, been recognized as an important part of most explanations of changes in the level of business activity. The "ex ante" analysis of the Stockholm School-although it has created its fair share of confusion-is a highly suggestive approach to short-run problems. It has undoubtedly been a

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4,984 citations

Journal Article•DOI•

ECONOMIC GROWTH and CAPITAL ACCUMULATION

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T. W. Swan¹•Institutions (1)

Australian National University¹

01 Nov 1956-Economic Record

3,961 citations

Book•

Finite Markov chains

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John G. Kemeny, J. Laurie Snell

01 Jul 1976

TL;DR: This lecture reviews the theory of Markov chains and introduces some of the high quality routines for working with Markov Chains available in QuantEcon.jl.

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Abstract: Markov chains are one of the most useful classes of stochastic processes, being • simple, flexible and supported by many elegant theoretical results • valuable for building intuition about random dynamic models • central to quantitative modeling in their own right You will find them in many of the workhorse models of economics and finance. In this lecture we review some of the theory of Markov chains. We will also introduce some of the high quality routines for working with Markov chains available in QuantEcon.jl. Prerequisite knowledge is basic probability and linear algebra.

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3,255 citations

Posted Content•

Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence

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Robert E. Hall¹, Robert E. Hall²•Institutions (2)

Stanford University¹, National Bureau of Economic Research²

01 Apr 1979-Social Science Research Network

TL;DR: In this paper, the marginal utility of consumption evolves according to a random walk with trend, and consumption itself should evolve in the same way, and the evidence supports a modified version of the life cycle permanent income hypothesis.

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Abstract: Optimization of the part of consumers is shown to imply that the marginal utility of consumption evolves according to a random walk with trend. To a reasonable approximation, consumption itself should evolve in the same way. In particular, no variable apart from current consumption should be of any value in predicting future consumption. This implication is tested with time-series data for the postwar United States. It is confirmed for real disposable income, which has no predictive power for consumption, but rejected for an index of stock prices. The paper concludes that the evidence supports a modified version of the life cycle-permanent income hypothesis.

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2,957 citations