Optimal Economic Growth and Turnpike Theorems

This chapter is concerned with the long-term tendencies of paths of capital accumulation that maximize, in some sense, a utility sum for society over an unbounded time span. However, the structure of the problem is characteristic of all economizing over time whether on the social scale or the scale of the individual or the firm. The mathematical methods that are used are closely allied to the old mathematical discipline, calculus of variations. The chapter discusses that the utility function depends on time, as in the standard theory of the calculus of variations. Also the function to be maximized is the sum of utility functions for each period over the future. It is described as a separable utility function over the sequence of future capital stocks and corresponds to the integral of calculus of variations. As the consumption of one period influences the utility of later consumption, the separability assumption is not exact. The treatment of utility in a period as dependent on initial and terminal stocks is not a restriction because the usual assumptions that make utility depend on consumption and consumption on production and terminal stocks implies that an equivalent utility depending on capital stocks exists. The chapter also discusses that the primary sources of the optimal growth model are aggregate savings programs and capital accumulation programs for an economy, the theorems, and methods of the subject find applications in other areas with increasing frequency.