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
7 - Necessary and Sufficient Conditions for Isoperimetric Problems
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
Mathematically, isoperimetric problems are a class of optimal control problems that involve finding an extremum of one integral, subject to another integral having a prescribed value. One such economic problem of this class was presented in Example 3.5 and Mental Exercise 4.21, videlicet, the nonrenewable resource–extracting model of the firm. In each instance, we transformed the given isoperimetric problem into an optimal control problem in standard form so that the control problem could be solved with the theorems developed to that point. The goal of this chapter is to develop theorems, both necessary and sufficient, that will help us solve isoperimetric problems directly.
An important reason for studying this class of control problems is that they provide a unified view of principal-agent problems as well as a general method for their solution. We demonstrate this in Example 7.3 for the optimal contracting problem when the effort (or action) of the agent is observable by the principal. A byproduct of solving the principal-agent problem via this method is that the independent variable t, which we have heretofore always referred to as time, is now the realized value of the random variable profit. As remarked at the end of Chapter 6, one may safely skip this chapter and the next on a first read without loss of continuity. If, however, one wishes to read the chapter, it is recommended that Mental Exercise 4.22 be worked at this juncture, as it introduces basic ideas and concepts for what follows.
- Type
- Chapter
- Information
- Foundations of Dynamic Economic AnalysisOptimal Control Theory and Applications, pp. 174 - 210Publisher: Cambridge University PressPrint publication year: 2005