Book contents
- Frontmatter
- Contents
- Foreword by Eugene Silberberg
- Preface
- 1 Essential Elements of Continuous Time Dynamic Optimization
- 2 Necessary Conditions for a Simplified Control Problem
- 3 Concavity and Sufficiency in Optimal Control Problems
- 4 The Maximum Principle and Economic Interpretations
- 5 Linear Optimal Control Problems
- 6 Necessary and Sufficient Conditions for a General Class of Control Problems
- 7 Necessary and Sufficient Conditions for Isoperimetric Problems
- 8 Economic Characterization of Reciprocal Isoperimetric Problems
- 9 The Dynamic Envelope Theorem and Economic Interpretations
- 10 The Dynamic Envelope Theorem and Transversality Conditions
- 11 Comparative Dynamics via Envelope Methods
- 12 Discounting, Current Values, and Time Consistency
- 13 Local Stability and Phase Portraits of Autonomous Differential Equations
- 14 Necessary and Sufficient Conditions for Infinite Horizon Control Problems
- 15 The Neoclassical Optimal Economic Growth Model
- 16 A Dynamic Limit Pricing Model of the Firm
- 17 The Adjustment Cost Model of the Firm
- 18 Qualitative Properties of Infinite Horizon Optimal Control Problems with One State Variable and One Control Variable
- 19 Dynamic Programming and the Hamilton-Jacobi-Bellman Equation
- 20 Intertemporal Duality in the Adjustment Cost Model of the Firm
- Index
- References
11 - Comparative Dynamics via Envelope Methods
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword by Eugene Silberberg
- Preface
- 1 Essential Elements of Continuous Time Dynamic Optimization
- 2 Necessary Conditions for a Simplified Control Problem
- 3 Concavity and Sufficiency in Optimal Control Problems
- 4 The Maximum Principle and Economic Interpretations
- 5 Linear Optimal Control Problems
- 6 Necessary and Sufficient Conditions for a General Class of Control Problems
- 7 Necessary and Sufficient Conditions for Isoperimetric Problems
- 8 Economic Characterization of Reciprocal Isoperimetric Problems
- 9 The Dynamic Envelope Theorem and Economic Interpretations
- 10 The Dynamic Envelope Theorem and Transversality Conditions
- 11 Comparative Dynamics via Envelope Methods
- 12 Discounting, Current Values, and Time Consistency
- 13 Local Stability and Phase Portraits of Autonomous Differential Equations
- 14 Necessary and Sufficient Conditions for Infinite Horizon Control Problems
- 15 The Neoclassical Optimal Economic Growth Model
- 16 A Dynamic Limit Pricing Model of the Firm
- 17 The Adjustment Cost Model of the Firm
- 18 Qualitative Properties of Infinite Horizon Optimal Control Problems with One State Variable and One Control Variable
- 19 Dynamic Programming and the Hamilton-Jacobi-Bellman Equation
- 20 Intertemporal Duality in the Adjustment Cost Model of the Firm
- Index
- References
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.
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- Foundations of Dynamic Economic AnalysisOptimal Control Theory and Applications, pp. 287 - 311Publisher: Cambridge University PressPrint publication year: 2005