Non-Convexities in Continuous-Time Investment Theory

Abstract

That non-convex preferences or technologies cause problems for economic theory is well-known. In recent years there have been numerous contributions to the nonconvexity in the context of models with a finite-dimensional commodity space. In the capital theory context non-convexities have received explicit treatment in a few papers, in two distinguishable ways. The problem is cast either in a discrete-time framework (Aoki (1971), Rothschild (1971)) or in a continuous-time framework in which there are implicitly unbounded economies of scale in investment. The latter formulation leads to jumps in the state variable or lumpy investment, the standard problem treated is one of decreasing costs in the installation of new capacity (contributions include Chenery (1952), Dixit, Mirrlees and Stern (1975), Hartwick (1976), Manne (1967), Srinivasan (1967), Starrett (1978), and Weitzman (1970)). These types of analysis unfortunately ignore some of the most interesting problems associated with non-convexities in investment theory. In a discrete-time framework the important distinction between stocks and flows is blurred, while the capacity-installation problem, by admitting unbounded scale economies, excludes the smoothing of state variables that follows by allowing for eventually increasing costs of adjustment. In this paper we undertake a systematic examination of non-convexities' (or sometimes non-concavities) in the context of a classic problem in continuous-time investment theory-the optimal accumulation of capital by a firm maximizing its present value over an infinite horizon. While the results obtained are specific to this particular model we feel that they can be qualitatively extended to any capital-theory problem which can be cast in a continuous-time optimal-control-theory framework. There are various sorts of non2 convexities which can arise in these problems. First, there can be either stock or flow non-convexities; that is, non-convexities which are introduced into the net benefit function (the measure of the flow of benefits at each moment) through changes in the capital stock (state variable) on the one hand or changes in the flow of investment (control variable) on the other. If the benefit and transition functions are not concave in both these variables then necessary conditions are no longer sufficient for optimality and conventional marginal analysis fails. Non-convexities can also be introduced into the value functional, which has a stock dimension. In order to examine these different sorts of non-convexity problems we shall now present the basic economic problem. A firm is faced with the problem of maximizing the present value of discounted cash flow over an infinite horizon. Hence it chooses an investment programme I*(t) to maximize 00