This paper examines the problem of a company that aims to expand its stochastic production capacity. Investments in capital for expanding capacity are irreversible in the sense that the company cannot recover the investment by reducing capacity. The company faces fixed and proportional costs for purchasing capital and aims to maximize its expected discounted profits over an infinite horizon.
Dixit and Pindick (1994) provide a review of similar investment problems. Davis et al. (1987) where among the first ones to address the issue of optimally determining the timing and size of capacity increases that can be associated with the operation of an investment project in the presence of random economic fluctuations. Kobila (1993) analyzed a model with deterministic capacity in an uncertain market without transaction costs on buying capital. Chiarolla and Haussmann (2003) studied an irreversible investment model in a finite horizon and obtained an explicit solution for a power type production function. Other important contributions include Oksendal (2000), Wang (2003) and Bank (2005). Capacity expansion models in which the installed capacity level can be reduced as well as increased, that is reversible capacity expansion models have been examined by Abel and Eberly (1996) and Guo and Pham (2005).
The presence of fixed costs for purchasing capital requires stochastic impulse control techniques for solving the problem. Here, we adopt the methodological framework presented in Cadenillas and Zapatero (1999), which characterizes the value function as a solution to a system of quasi-variational inequalities (see also Cadenillas, 1999, Suzuki and Pliska, 2004, Cadenillas et al., 2006). Alternatively the problem could have been approached via combining stochastic calculus with standard nonlinear programming techniques as in Alvarez and Virtanen (2004) (see also Alvarez 2004, Dayanik and Egami, 2004). Pham (2005) solved the problem by assuming only proportional costs for purchasing capital and thus relying on singular stochastic control methods. The optimal strategy in that model involves doing infinitesimal small transactions to avoid that the capacity production process leaves a no transaction region. However, transactions in the real world involve not only proportional but also fixed costs. In the presence of fixed investment costs, these strategies would lead to ruin.