On irreversible investment

This paper develops a general theory of sequential irreversible investments in capital where a firm has the option to expand its current capacity or just wait for better time. Facing economic uncertainty, the firm has an operating function of the current capacity and an exogenous stochastic factor modelled by semimartingale. This general model encompasses all previously studied models, including the deterministic case as well as the stochastic case with Geometric Brownian motions, Lévy processes and even with regime shift. In this paper, general existence and uniqueness results are first provided for irreversible investments with finite and infinite horizon, respectively. As the main contribution of this paper, a new method is proposed to characterize the optimal investment policy, the base capacity policy. Under the policy, the capacity is kept always at or above the base capacity which is characterized by a stochastic backward equation. This new method gives a number of new qualitative insights into the nature of the irreversible investment. It is demonstrated that the optimal policy equals the marginal operating profit and the user cost of capital in those free intervals when the irreversibility constraint does not bind. While, the equality holds true only on average in block intervals when no investment occurs. Besides, this method easily leads to some general comparative statics results: When the operating profit function is supermodular, the base capacity increases monotonically with the exogenous shock; and the firm size always declines with the user cost of capital. Finally, explicit solutions are derived when the exogenous economic shocks are modelled by Lévy processes and the operating profit function is of Cobb–Douglas style.