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

• Published in: Applied mathematics & optimization Vol. 75; no. 3; pp. 429 - 470
• Main Authors: Ferrari, Giorgio, Riedel, Frank, Steg, Jan-Henrik
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2017; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Consumption; CONTROL; Economics; EQUATIONS; EQUILIBRIUM; Expected utility; Game theory; Games; INTERACTIONS; Investment policy; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; MATHEMATICAL SOLUTIONS; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Optimization; Public good; Riders; Simulation; STOCHASTIC PROCESSES; Stochasticity; SYMMETRY; Systems Theory; Theoretical; UNCERTAINTY PRINCIPLE; Uniqueness; C02; 60G51; First order conditions for optimality; Nash equilibrium; Irreversible investment; Stochastic games; 93E20; 91A25; 91A15; 91B70; Lévy processes; Free-riding; Singular stochastic control; C62; C73; Public good contribution; C61
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9337-5

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.