Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment Under Limited Resources

In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with $N$ firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding social planner optimal policy. Our conditions are a stochastic infinite-dimensional generalization of the Kuhn-Tucker theorem. The Lagrange multiplier takes the form of a nonnegative optional random measure on $[0,T]$ which is flat off the set of times for which the constraint is binding, i.e., when all the fuel is spent. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank [SIAM J. Control Optim., 44 (2005), pp. 1529-1541]. In the infinite-horizon case, with operating profit functions of Cobb-Douglas type, our method allows the explicit calculation of the optimal policy in terms of the “base capacity” process, i.e., the unique solution of the Bank and El Karoui representation problem [Ann. Probab., 32 (2004), pp. 1030-1067]