We employ a duality approach and martingale techniques to characterize the solutions to a stochastic optimal control problem modeling the choices available to an economic agent seeking to maximize the expected utility derived from consumption, terminal wealth, and life insurance coverage, while facing both investment risks and mortality uncertainty. The agent dynamically allocates wealth between a financial market composed of one risk-free asset and multiple risky assets, and term life insurance premiums subject, respectively, to uncertainty associated with the market conditions and the agent uncertain lifespan. Our results provide insights into the trade-offs between consumption, wealth accumulation, and life insurance demand in the presence of financial and mortality risks.

Baliga and Maskin introduced a model of contributions for the provisions of public goods such as contributing for the reduction of air pollution. In this work we consider an extended version of Baliga and Maskin's model with a parameter α which is the elasticity of the benefit function, and with heterogeneous agents, each with their own preferences for the good. For this generalized version of the model, we consider the formation of stable coalitions which are absorbing states of a bargaining Markov chain, where agents join or leave coalitions according to their cooperation and free-riding incentives. We show that there is a stable high coalition consisting of the set of agents most preferring/valuing the public good. The increase of the elasticity parameter α increases the size of the stable high coalition that changes from a single member (called the competitive coalition as appearing in Baliga and Maskin's paper) to the grand coalition involving all agents. However, the utility of members of the stable coalition can be very small when compared to the utility of the free-riders, rendering the formation of stable coalitions difficult. We show that a variant of the coalition folk theorem holds, meaning that member heterogeneity will tend to decrease the size of stable coalitions. We show that the formation of stable coalitions is subject to the paradox of cooperation, since even when stable coalitions are large and free-riders have not very low preferences for the public good, the utility of the stable coalition may still be low when compared to the full cooperation scenario of the grand coalition. However, the paradox does not hold when the free-riders have a very low preference for the public good, which also facilitates the spontaneous formation of stable coalitions, or when there are no free-riders and the grand coalition is stable, which is always the case when the elasticity α is large enough.