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

Posted: 20 Mar 2012

See all articles by Maria Chiarolla

Maria Chiarolla

Università del Salento - Dipartimento di Scienze dell'Economia (DSE); University of Rome I - Department of Methods and Models for Economics, Territory and Finance (MEMOTEF)

Giorgio Ferrari

Bielefeld University - Center for Mathematical Economics

Frank Riedel

Bielefeld University - Center for Mathematical Economics

Date Written: March 16, 2012

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. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank [SIAM Journal on Control and Optimization 44 (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 [Annals of Probability 32 (2004)].

Keywords: stochastic irreversible investment, optimal stopping, the Bank and El Karoui Representation Theorem, base capacity, Lagrange multiplier optional measure

JEL Classification: C02, E22, D92, G31

Suggested Citation

Chiarolla, Maria and Ferrari, Giorgio and Riedel, Frank, Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment Under Limited Resources (March 16, 2012). Institute of Mathematical Economics Working Paper No. 463, Available at SSRN: https://ssrn.com/abstract=2026279

Maria Chiarolla

Università del Salento - Dipartimento di Scienze dell'Economia (DSE) ( email )

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University of Rome I - Department of Methods and Models for Economics, Territory and Finance (MEMOTEF) ( email )

Via del Castro Laurenziano 9
Roma, Roma 00161
Italy

Giorgio Ferrari

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501
Germany

Frank Riedel (Contact Author)

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501
Germany

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