Private information may limit insurance possibilities when two agents get together to pool idiosyncratic risk. However, if there is capital accumulation, bilateral insurance possibilities may improve because misreporting distorts investment. We show that if one of the Pareto weights is sufficiently large, that agent does not have incentives to misreport. This implies that, under some conditions, the full information allocation is incentive compatible when agents have equal Pareto weights. In the long run, either one of the agents goes to immiseration, or both agents’ lifetime utilities are approximately equal. The second case is only possible with capital accumulation.

Introduction

Private information limits insurance possibilities when two agents get together to pool idiosyncratic risk. We show that if their total income depends on capital accumulation and production, bilateral insurance possibilities improve because misreporting distorts investment. This occurs because cheating implies that the misrepresenting agent will receive higher utility today at the expense of reducing the other agent’s current consumption and of reducing investment. The latter reduces both agents’ continuation values, and it therefore provides incentives to prevent cheating. This mechanism is more significant when the agent’s Pareto weight is larger, as he cares more about the total level of capital in the economy. In the long run, either one of the agents is driven to immiseration, or both agents’ lifetime utilities are approximately equal. We show that the second case is only possible in economies with capital accumulation. We study the optimal bilateral insurance arrangement between two risk-averse agents facing preference shocks.1 We compare two alternative setups: (i) Agents share the ownership of production and investment (Capital Accumulation Economy, CAE), and (ii) Total income is constant and exogenous (Endowment Economy, EE). The optimal contract maximizes the weighted sum of the agents’ lifetime utility. Ideally, each agent’s consumption should depend on his preference shock. However, if shocks are private information, the optimal arrangement must also deal with incentives to misreport. Thus, private information may limit insurance capabilities. To understand the role of capital accumulation for incentives under private information, we show that there is a clear difference between the CAE and EE. In the EE, incentives to cheat exist because reporting a high-preference shock increases consumption of the reporting agent at the expense of reducing consumption of the other agent. Thus, unless promises about future consumption are modified as a function of the report, agents always want to report the highest value of the preference shock. In contrast, in the CAE, if an agent chooses to cheat, she will consume more this period at the cost (at least partially) of reducing investment, which will reduce her future consumption. We show that this force is more important for providing incentives for agents with larger Pareto weights, as they more strongly internalize the effect on future available resources. Thus, in addition to incorporating capital accumulation, it is crucial to consider a small number of agents. To the best of our knowledge, this result was not previously demonstrated or understood, because past work looked at only a continuum of agents or an endowment economy.2 We made two additional assumptions that help simplify the analysis: Agents are ex-ante identical, and the preference shock can be either high or low. Under these conditions we derive three key results as summarized by Propositions 1, 2, and 3. First, Proposition 1 shows that in the CAE an agent has no incentives to cheat under the full information allocation when his weight is above a threshold smaller than one. To understand this result, note that if an agent’s weight is equal to one, all the extra funds she receives after cheating are only “financed” by disinvestment because the consumption of the other agent is already zero. She also fully internalizes the effect of the investment distortion because given that her weight is one, she will receive all future output. Thus, she would be strictly worse off misrepresenting her preference shock when her weight is exactly one. Now, as a small change in the weight changes consumption and investment only slightly (by continuity of the full information allocation), cheating will not be desirable even if the weight were slightly smaller than one. Although this result demonstrates that at some point the incentives to cheat vanish completely, the numerical solution of the model suggests a more general interpretation. In the CAE, there is a force created by the behavior of investment in the full information allocation that helps provide incentives for truthful revelation, and this force is increasing in the value of the agent’s Pareto weight. In sharp contrast, Proposition 1 also shows that in the EE, if both agents have positive Pareto weights, their incentive compatibility constraints must be binding.