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11 - Comparative Dynamics via Envelope Methods

Published online by Cambridge University Press:  05 June 2012

Michael R. Caputo
Affiliation:
University of Central Florida
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Summary

We continue our development of the dynamic envelope theorem in this chapter, but do so with a different purpose in mind as well as from a different point of view. The purpose herewith is the development of a general method of comparative dynamics, applicable to any sufficiently smooth optimal control problem. The point of view we take is that the parameters of the optimal control problem, rather than the control variables themselves, are viewed as the choice or decision variables. This dual point of view is fundamental to our development of a general method of comparative dynamics, in that without it, we would not be able to achieve our goal. We will see that by adopting a dual view of an optimal control problem, we can succeed in providing a one-line proof of the dynamic envelope theorem and, at the same time, more simply reveal the envelope nature of the result. More importantly, however, we will show that the comparative dynamics properties of all sufficiently smooth optimal control problems are contained in a symmetric and semidefinite matrix, typically subject to constraint. This matrix, in effect, is a generalized Slutsky-type matrix in integral form, and is shown to characterize the effects that parameter perturbations have on the entire time path of the optimal trajectories. We will also provide sufficient conditions for the optimal value function to be convex in the parameters. Let us now turn to the detailed development of these important results.

Type
Chapter
Information
Foundations of Dynamic Economic Analysis
Optimal Control Theory and Applications
, pp. 287 - 311
Publisher: Cambridge University Press
Print publication year: 2005

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References

Caputo, M. R. (1990a), “Comparative Dynamics via Envelope Methods in Variational Calculus,” Review of Economics Studies, 57, 689–697CrossRefGoogle Scholar
Caputo, M. R. (1990b), “How to do Comparative Dynamics on the Back of an Envelope in Optimal Control Theory,” Journal of Economic Dynamics and Control, 14, 655–683CrossRefGoogle Scholar
Caputo, M. R. (1990c), “New Qualitative Properties in the Competitive Nonrenewable Resource Extracting Model of the Firm,” International Economic Review, 31, 829–839CrossRefGoogle Scholar
Caputo, M. R. (1992a), “A Primal-Dual Approach to Comparative Dynamics with Time-Dependent Parameters in Variational Calculus,” Optimal Control Applications and Methods, 13, 73–86CrossRefGoogle Scholar
Caputo, M. R. (1992b), “Fundamental Symmetries and Qualitative Properties in the Adjustment Cost Model of the Firm,” Journal of Mathematical Economics, 21, 99–112CrossRefGoogle Scholar
Caputo, M. R. (1992c), “Comparative Dynamics in the Labor-Managed Model of the Firm,” Journal of Comparative Economics, 16, 272–286CrossRefGoogle Scholar
Caputo, M. R. (1994), “The Slutsky Matrix and Homogeneity in Intertemporal Consumer Theory,” Journal of Economics, 60, 255–279CrossRefGoogle Scholar
Silberberg, E. (1974), “A Revision of Comparative Statics Methodology in Economics, or How to Do Comparative Statics on the Back of an Envelope,” Journal of Economic Theory, 7, 159–172CrossRefGoogle Scholar
Silberberg, E. and Suen, W. (2001, 3rd Ed.), The Structure of Economics: A Mathematical Analysis (New York: Irwin/McGraw Hill)
Simon, C. P. and Blume, L. (1994), Mathematics for Economists (New York: WW Norton & Company)

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