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

TL;DR

This paper models continuous-time public good contribution as a stochastic control problem, analyzing optimal policies, Nash equilibria, and free rider effects in a stochastic framework with economic agents.

Contribution

It introduces a stochastic control approach to public good contribution, characterizes optimal policies and equilibria, and analyzes free rider effects in a novel continuous-time setting.

Findings

Existence and uniqueness of the social planner's optimal policy.

Characterization of Nash equilibria via stochastic backward equations.

Analysis of free rider effects in the model.

Abstract

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.