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 $\varepsilon$-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：本文对不确定性下风险分担与交换的经典模型中的福利进行了定量评估。我们证明了三类结果。首先，在均衡配置中，通过给定边际改进（即ε改进）提升个体福利的空间会随状态数量的增加而消失。其次，可用于按给定边际提升个体福利的总体资源变动空间同样会消失。等价地：在非效率配置中，对于给定的资源次优程度（以资源利用率不足系数衡量），通过扰动总体资源来提升福利的可能性会随状态数量呈指数级衰减至零。最后，我们考察了具有多重先验的标准不确定性厌恶模型中的效率风险分担，并证明在非效率配置中，某些先验集合会随状态空间规模的扩大而收缩。