Explicit Solution of a Stochastic, Irreversible Investment Problem and Its Moving Threshold

Abstract

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 lto 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 Rela tive Risk Aversion CRRA 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

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

“…Although these details are important, they are not necessary for the understanding of Sections 4 and 5 and could be skipped at a first reading. 14) so that (3.13) can be written as…”

Irreversible Capital Accumulation with Economic Impact

“…Finally, we can use the fact that m, n are the solutions to the quadratic equation (11) and straightforward calculations to obtain …”

“…More relevant to this paper models have been studied by several authors in the economics literature: see Dixit and Pindyck [17,Chapter 11] and references therein. Related models that have been studied in the mathematics literature include Davis, Dempster, Sethi and Vermes [13], Arntzen [4], Øksendal [42], Wang [48], Chiarolla and Haussmann [11], Bank [6], Alvarez [2,3], Løkka and Zervos [35], Steg [45], Chiarolla and Ferrari [9], De Angelis, Federico and Ferrari [15], and references therein. Furthermore, capacity expansion models with costly reversibility were introduced by Abel and Eberly [1], and were further studied by Guo and Pham [22], Merhi and Zervos [40], Guo and Tomecek [23,24], Guo, Kaminsky, Tomecek and Yuen [21], Løkka and Zervos [36], De Angelis and Ferrari [16], and Federico and Pham [19].…”

Solving Singular Control from Optimal Switching

“…In mathematical economics, a typical (ir)reversible investment problem can be formulated as a singular control problem in which a company, by adjusting its production capacity through expansion and contraction according to market fluctuations, wishes to maximize its overall expected net profit over an infinite horizon. This problem has been investigated by numerous authors (See for instance Davis et al (1987); Kobila (1993); Abel and Eberly (1997); Baldursson and Karatzas (1997); Øksendal (2000); Scheinkman and Zariphopoulou (2001); Wang (2003); Chiarolla and Haussmann (2005); Bank (2005); Guo and Pham (2005), and Merhi and Zervos (2007)). For a standard reference on irreversible investment, see Dixit and Pindyck (1994).…”