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

Individual and Collective Welfare in Risk Sharing with Many States