Introduction to Modern Economic Growth different from finite-dimensional maximization, and when such infinite-dimensional maximization problems may fail to have solutions 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 example provided in Section 7.4 shows that the stronger transversality condition, which is very useful in many applications, does not always hold This example is a variant of the famous example by Halkin (1974) The interested reader should look at Michel (1982), which contains the original result on the transversality condition of (7.44) for discounted infinite horizon optimal control problems and also a discussion of when the stronger condition (7.55) holds The results presented here are closely related to Michel’s (1982) results, but are stated under assumptions that are more relevant in economic situations Michel assumes that the objective function is nonnegative, which is violated by many of the common utility functions used in economic growth 362