On a Stochastic, Irreversible Investment Problem

The productive sector of the economy, represented by a single firm employing labour to produce the consumption good, is studied in a stochastic continuous time model on a finite time interval. The firm must choose the optimal level of employment and capital investment in order to maximize its expected total profits. In this stochastic control problem the firm’s capacity is modelled as an Itô process controlled by a monotone process, possibly singular, that represents the cumulative real investment. It is optimal to invest when the shadow value of installed capital exceeds the capital’s replacement cost; this threshold is the free boundary of a related optimal stopping problem which we recast as a stopping problem without integral cost, similar to the American Option Problem. Then, under a regularity condition, we characterize the free boundary as the unique solution of a nonlinear integral equation.