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

SIAM Journal on Control and Optimization 2013 51:5, 3863-3885

Posted: 11 Feb 2014

See all articles by Frank Riedel

Frank Riedel

Bielefeld University - Center for Mathematical Economics

Giorgio Ferrari

Bielefeld University - Center for Mathematical Economics

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)

Date Written: 2013

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 [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].

Suggested Citation

Riedel, Frank and Ferrari, Giorgio and Chiarolla, Maria, Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment Under Limited Resources (2013). SIAM Journal on Control and Optimization 2013 51:5, 3863-3885 , Available at SSRN: https://ssrn.com/abstract=2393374

Frank Riedel (Contact Author)

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501
Germany

Giorgio Ferrari

Bielefeld University - Center for Mathematical Economics ( email )

Postfach 10 01 31
Bielefeld, D-33501
Germany

Maria Chiarolla

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

via Taranto 35
Piazza Tancredi, N.7
Lecce, Lecce 73100
Italy

HOME PAGE: http://https://www.unisalento.it/scheda-utente/-/people/maria.chiarolla?inheritRedirect=true

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

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