Irreversible investment in oligopoly

Abstract

We take a general perspective on capital accumulation games with open loop strategies, as they have been formalized by Back and Paulsen (Rev. Financ. Stud. 22, 4531-4552, 2009). With such strategies, the optimization problems of the individual players are of the monotone follower type. Consequently, one can adapt available methods, in particular the approach of Bank (SIAM J. Control Optim. 44, 1529-1541, 2005. We obtain consistency in equilibrium by proving that with common assumptions from the oligopoly literature on instantaneous revenue, equilibrium determination is equivalent to solving a single monotone follower problem. In the unique open loop equilibrium, only the currently smallest firms invest. This result is valid for arbitrary initial capital levels and general stochastic shock processes, which may be non-Markovian and include jumps. We explicitly solve an example, the specification of Grenadier (Rev. Financ. Stud. 15, 691-721, 2002) with a Lévy process.Keywords Irreversible investment · Oligopoly · Equilibrium · Singular stochastic control Mathematics Subject Classification (2010) 60G40 · 91A15 · 91A25 · 93E20 JEL Classification C73 · D43 · D92

Irreversible Capital Accumulation with Economic Impact

“…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].…”

Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach

“…This approach is very powerful in solving general singular control problems as it has been shown in a quite recent literature. We refer to Bank andRiedel (2001, 2003) for an intertemporal utility maximization problem with Hindy, Huang and Kreps preferences; to Bank (2005), Chiarolla and Ferrari (2014), Ferrari (2015) and Riedel and Su (2011) for the irreversible investment problem of a monopolistic firm with both limited and unlimited resources; to Chiarolla et al (2013) for the social planner problem in a market with N firms and limited resources; to Steg (2012) for a capital accumulation game.…”

“…In equilibrium, the agents act optimally, taking the contribution processes of others as given. In this sense we restrict our attention to open-loop strategies (see also Back andPaulsen, 2009, andSteg, 2012) without explicit reactions to deviations from announced (equilibrium) play. Indeed, as pointed out by Back and Paulsen (2009), there are serious conceptual problems defining a stochastic continuous-time game of singular controls as ours with more explicit feedback (closed-loop) strategies.…”

Continuous-Time Public Good Contribution Under Uncertainty

“…In the latest years several papers tackled singular stochastic control problems by means of such an approach. We refer to Bank and Riedel [10], [11] for an intertemporal utility maximization problem with Hindy, Huang and Kreps preferences; to Bank [6], Chiarolla and Ferrari [16], Ferrari [25] and Riedel and Su [45] for the irreversible investment problem of a monopolistic firm with both limited and unlimited resources; to Chiarolla, Ferrari and Riedel [17] for the social planner problem in a market with N firms and limited resources; to Steg [49] for a general capital accumulation game with open loop strategies.…”

“…This approach has been used in various instances to solve singular stochastic control problems of the monotone follower type (see Bank [6], Bank and Riedel [10], Chiarolla, Ferrari and Riedel [17], Riedel and Su [45] and Steg [49], among others), and it may be thought of as a stochastic, infinite dimensional generalization of the classical Kuhn-Tucker method. In the previous papers the optimal policy is constructed as the running supremum of a desirable value.…”