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

TL;DR

This paper develops a generalized set of Kuhn-Tucker conditions for optimal stochastic investment in a multi-firm setting with resource constraints, using a concavity approach instead of dynamic programming, and provides explicit solutions in certain cases.

Contribution

It introduces a stochastic infinite-dimensional Kuhn-Tucker framework for multi-firm investment with resource limits, extending classical optimality conditions.

Findings

Derived necessary and sufficient first order conditions for the problem.

Provided an explicit solution for the infinite-horizon Cobb-Douglas case.

Interpreted the conditions for a single firm in existing literature.

Abstract

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 (2005). 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 (2004).