Long-run investment under uncertain demand

In the literature investigating the impact of uncertainty on short-run and long-run investment, most authors have used a log linear profit function. This functional form has been generally considered a reasonable approximation for a more general one and has the advantage of providing closed form solutions for both short-run investment rule and long-run rate of capital accumulation. In this paper, we consider the profit function for the case of a monopolistic firm facing a linear demand function with additive shocks. Under this assumption, analytical solutions, for both short-run investment rule and long-run rate of capital accumulation, are not available. We then 1) propose an analytical approximation of the short-run investment rule and 2) show how such approximation can be used in order to derive the corresponding i) steady-state distribution of the optimal stock of capital and ii) the long-run average rate of capital accumulation. Finally, we compare the long-run rates of capital accumulation calculated under both profit function specifications. We find that, within a plausible range of parameter values, the two rates are significantly different. Hence, we conclude that the choice of a log linear functional form has a non-trivial impact on the magnitude of the long run rate of capital accumulation.