We provide a quantitative assessment of welfare in the classical model of risk-sharing and exchange under uncertainty. We prove three kinds of results. First, that in an equilibrium allocation, the scope for improving individual welfare by a given margin (an $\ve$-improvement) vanishes as the number of states increases. Second, that the scope for a change in aggregate resources that may be distributed to enhance individual welfare by a given margin also vanishes. Equivalently: in an inefficient allocation, for a given level of resource sub-optimality (as measured by the coefficient of resource under-utilization), the possibilities for enhancing welfare by perturbing aggregate resources decrease exponentially to zero with the number of states. Finally, we consider efficient risk-sharing in standard models of uncertainty aversion with multiple priors, and show that, in an inefficient allocation, certain sets of priors shrink with the size of the state space.

翻译：我们为经典不确定性下风险分担与交换模型中的福利量化评估提供了分析。我们证明了三类结果。首先，在均衡配置中，通过给定幅度（$\ve$-改进）改善个体福利的空间随状态数量增加而消失。其次，可用于提升个体福利的总体资源调整空间同样随状态数量增加而消失。等价地：在非有效配置中，对于给定的资源配置次优程度（以资源利用不足系数衡量），通过扰动总体资源改善福利的可能性随状态数量增加呈指数衰减至零。最后，我们考虑了具有多重先验的标准不确定性厌恶模型中的有效风险分担，并证明在非有效配置中，某些先验集随状态空间规模扩大而收缩。