Generalized Kuhn-Tucker conditions for \(N\)-firm stochastic irreversible investment under limited resources (Q2873857)

The authors generalize Bank's single firm irreversible investment problem to the case of a social planner in a market with \(N\) firms in which the total investment is bounded by a stochastic, time-dependent, increasing, adapted finite fuel. The social planner's objective is to pursue a vector of efficient irreversible investment processes that maximizes the aggregate expected profit. The operating profit function depends directly on the cumulative control exercised since dynamics of the productive capacity is not allowed. The uncertain status of the economy is modeled by an exogenous economic shock. To overcome some technical difficulties in the analysis of the model, a new approach is developed that is based on a stochastic generalization of the classical Kuhn-Tucker method. The concavity of the profit functional is used to characterize the optimal social planner policy as the unique solution of the stochastic Kuhn-Tucker problem. In the infinite-horizon case, with operating functions of Cobb-Douglas type, the method allows the explicit calculation of the optimal policy in terms of the ``base capacity'' process.