We study a continuous-time problem of public good contribution under uncertainty for an economy with a finite number of agents. Each agent aims to maximize his expected utility allocating his initial wealth over a given time period between private consumption and repeated but irreversible contributions to increase the stock of some public good. We study the corresponding social planner problem and the case of strategic interaction between the agents . These problems are set up as stochastic control problems with both monotone and classical controls representing the cumulative contribution into the public good and the consumption of the private good, respectively. We characterize the optimal investment policies by a set of necessary and sufficient stochastic Kuhn-Tucker conditions , which in turn allow to identify a universal signal process that triggers the public good investments. Further we show that our model exhibits a dynamic free rider effect. We explicitly evaluate it in a symmetric Black-Scholes setting with Cobb-Douglas utilities and we show that uncertainty and irreversibility of public good provisions need not affect the degree of free-riding .
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, although the latter is not related to a meaningful control problem. We finally also provide a detailed analysis of the so-called free rider effect .
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stochastic control problems with both monotone and classical controls motivated by the so-called
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under uncertainty for an economy with a finite number of agents. Each agent aims to maximize his
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repeated but
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investigate
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. These problems are set up as stochastic control problems with both monotone and classical controls representing the cumulative contribution into the public good and the consumption of the private good, respectively. We characterize the optimal investment policies by a set of
, i.e. the public good contribution game. We show existence and uniqueness of the social planner's optimal policy, we characterize it by
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, which in turn allow to identify a universal signal process that triggers
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
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investments. Further we show that our model exhibits a dynamic free rider effect. We explicitly evaluate it in a symmetric Black-Scholes setting with Cobb-Douglas utilities and we show that uncertainty and irreversibility of public good provisions need not affect the degree of free-riding
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, although the latter is not related to a meaningful control problem. We finally also provide a detailed analysis of the so-called free rider effect