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.