Applied Mathematics & Optimization

, Volume 75, Issue 3, pp 429–470 | Cite as

Continuous-Time Public Good Contribution Under Uncertainty: A Stochastic Control Approach



In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner’s optimal policy, we characterize it by necessary and sufficient stochastic Kuhn–Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect.


Singular stochastic control Stochastic games First order conditions for optimality Nash equilibrium Lévy processes Irreversible investment Public good contribution Free-riding 

JEL Classification

C02 C61 C62 C73 

Mathematics Subject Classification

93E20  91B70  91A15  91A25  60G51 



The authors wish to thank two anonymous referees for their pertinent and useful comments. Financial support by the German Research Foundation (DFG) via Grant Ri 1142-4-2 is gratefully acknowledged.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Giorgio Ferrari
    • 1
  • Frank Riedel
    • 1
    • 2
  • Jan-Henrik Steg
    • 1
  1. 1.Center for Mathematical EconomicsBielefeld UniversityBielefeldGermany
  2. 2.Department of Economic and Financial ServicesUniversity of JohannesburgJohannesburgRepublic of South Africa

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