Irreversible investment in oligopoly

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.