An equilibrium model of risk and investment

Abstract

The object of this paper is to outline a simple equilibrium theory of investment. The firms in the economy operate in an environment of uncertainty and draw up plans extending over an uncertain future horizon. Firms’ plans consist of choices of rates of investment and degrees of risk for their capital projects. The firms purchase new capital equipment on one set of markets and sell their output on another. To simplify the equilibrium aspect of the model, to focus attention on the investment behaviour of firms, and to avoid entering into an extended discussion of the behaviour of consumers and their lifetime budget constraints, I use a device which reduces the analysis of equilibrium to the analysis of a simple maximum problem. This device is a natural generalisation of the single market consumers’ surplus approach used by Lucas-Prescott [7], Brock-Magi11 [3], Magi11 [8], and Scheinkman [ 121 in the study of a theory of investment. This reduction of the equilibrium problem to a maximum problem can be justified by a more extensive analysis which I shall not enter into in this paper. The paper is arranged as follows. Section 2 introduces the basic class of random processes on which the subsequent analysis is based. The analysis here draws on the framework of Bismut [2] and Brock-Magi11 [3]. Given an underlying Brownian motion process, once velocity and variance (risk) processes are given, lying in a suitable space of integrable functions, the state becomes a well-defined process. A class of concave maximum problems over a random finite horizon is introduced in which the integrand of the basic variational problem depends on the current state, velocity and variance. A class of dual imputed price processes is introduced. I recall a generalisation of the classical sufficiency theorem: a random process which is imputed price supported and satisfies a transversality condition is optimal. Section 3 introduces two models of the underlying economy. The first I call an extensive form model, the second a reduced form model. The