Explicit solution of a stochastic, irreversible investment problem and its moving threshold

We consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity. The firm aims to maximize its expected total discounted real profit net of investment on a finite horizon T. The capacity is modeled as a controlled Itô process where the control is the real investment, which is not necessarily a rate, but more generally a monotone process. The result is a singular stochastic control problem. We introduce the associated optimal stopping problem, that is “the optimal cost of not investing.” Its variational formulation turns out to be a parabolic obstacle problem, which we explicitly solve in the case of Constant Relative Risk Aversion production functions. The moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital's replacement cost. Then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits.