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

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  • Maria B. Chiarolla
  • Giorgio Ferrari
  • Frank Riedel

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).

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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1203.3757.

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Date of creation: Mar 2012
Date of revision: Aug 2013
Handle: RePEc:arx:papers:1203.3757

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  1. Peter Bank & Frank Riedel, 2003. "Optimal Dynamic Choice of Durable and Perishable Goods," Bonn Econ Discussion Papers bgse29_2003, University of Bonn, Germany.
  2. Xia Su & Frank Riedel, 2006. "On Irreversible Investment," Bonn Econ Discussion Papers bgse13_2006, University of Bonn, Germany.
  3. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
  4. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
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Cited by:
  1. Giorgio Ferrari & Jan-Henrik Steg & Frank Riedel, 2013. "Continuous-Time Public Good Contribution under Uncertainty," Working Papers 485, Bielefeld University, Center for Mathematical Economics.

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