References and Literature

The main material covered in this chapter is the topic of many excellent applied mathe­matics and engineering books. The purpose here has been to provide a review of the results that are most relevant for economists, together with simplified versions of the most important proofs.

The theory of optimal control was originally developed by Pontryagin et al. (1962). For this reason, the main necessary condition is also referred to as the Pontryagin’s (Maximum) Principle. The type of problem considered here (and in economics more generally) is referred to as the Lagrange problem in optimal control theory. The Maximum Principle is generally stated either for the somewhat simpler Meyer problem or the more general Bolza problem, though all of these problems are essentially equivalent, and when the problem is formulated in vector form, one can easily go back and forth between these different problems by simple transformations.

There are several books with varying levels of difficulty dealing with optimal control. Many of these books are not easy to read, but are also not entirely rigorous in their proofs. An excellent source that provides an advanced and complete treatment is Fleming and Rishel (1975). The first part of this book provides a complete (but rather different) proof of Pontrya­gin’s Maximum Principle and various applications.

For economists, books that develop the theory of optimal control with economic applica­tions may be more appropriate. Here the best reference is Seierstad and Sydsaeter (1987). While not as rigorous as Fleming and Rishel (1975), this book also has a complete proof of the main results and is also easier and more interesting to read for economists. It also shows how the results can be applied to economic problems. Other references in economics are Kamien and Schwartz (1991) and Leonard and Van Long (1992). Another classic is Arrow and Kurz’s (1970) book, which covers the same material and also presents rich economic insight on growth theory and related problems. This book also states and provides a proof of Arrow’s sufficiency theorem, which also appears in Arrow (1968).

Two recent books on applications of optimal control in economics, Caputo (2005) and Weitzman (2003), might be more readable. My treatment of the sufficiency results here is very similar to Caputo (2005). Weitzman (2003) provides a lively discussion of the applications of the Maximum Principle, especially in the context of environmental economics and depletion of natural resources.

There is some confusion in the literature over the role of the transversality condition. As commented in the previous chapter, in general there need not be a single transversality condition, since the transversality condition represents the necessary conditions obtained from specific types of variations.

The original economic interpretation of the Maximum Principle appeared in Dorfman (1969). The interpretation here builds on the discussion by Dorfman, but extends this based on the no-arbitrage interpretation of asset values in the Hamilton-Jacobi-Bellman equation. This interpretation of Hamilton-Jacobi-Bellman is well known in many areas of macroeco­nomics and labor economics, but is not often used to provide a general economic interpreta­tion for the Maximum Principle. Weitzman (2003) also provides an economic interpretation for the Maximum Principle related to the Hamilton-Jacobi-Bellman equation.

The classic reference for exploitation of a non-renewable resource is Hotelling (1931). Weitzman (2003) provides a detailed treatment and a very insightful discussion. Dasgupta and Heal (1979) and Conrad (1999) are also useful references for applications of similar ideas to sustainability and environmental economics. Classic references on investment with costs of adjustment and the q-theory of investment include Eisner and Strotz (1963), Lucas (1967), Tobin (1969) and Hayashi (1982). Detailed treatments of the q-theory of investment can be found in any graduate-level economics textbook, for example, Blanchard and Fisher (1989) or Romer (1996), as well as in Dixit and Pindyck’s (1994) book on investment under uncertainty and Caballero’s (1999) survey. Caballero (1999) also includes a critique of the q-theory.

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