This paper develops a general theory of irreversible investment of a single firm that chooses a dynamic capacity expansion plan in an uncertain environment. The model is set up free of any distributional or any parametric assumptions and hence encompasses all the existing models. As the first contribution, a general existence and uniqueness result is provided for the optimal investment policy. Based upon an alternative approach developed previously to dynamic programming problems, we derive the optimal base capacity policy such that the firm always keeps the capacity at or above the base capacity. The critical base capacity is explicitly constructed and characterized via a stochastic backward equation. This method allows qualitative insights into the nature of the optimal investment under irreversibility. (It is demonstrated that the marginal profit is indeed equal to the user cost of capital in free intervals where investment occurs in an absolutely continuous way at strictly positive rates. However, the equality is maintained only in expectation on average in blocked intervals where no investment occurs. Whenever the uncertainty is generated by a diffusion, the investment is singular with respect to Lebesgue measure. In contrast to the deterministic and Brownian motion case where lump sum investment takes place only at time zero, the firm responses in general more frequently in jumps to shocks. Nevertheless, lump sum investments are shown to be possible only at information surprises which is defined as unpredictable stopping time or unanticipated information jump even at the predictable time.) Furthermore, general monotone comparative statics results are derived for the relevant ingredients of the model. Finally, explicit solutions are derived for infinite time horizon, a separable operating profit function of Cobb?Douglas type and an exponential L´evy process modelled economic shock.
Sequential Irreversible Investment Capacity Expansion Singular Control Problem Lévy Processes