A. Appendix

Proof of Proposition 1: We consider a policy x that maintains the ordering of individuals’ willingness-to-pay and thus the corresponding surplus from buying insurance conditional on the share of insured individuals Q (x). We denote by w̃ (x) an individual’s net willingness-to-pay and the corresponding density by g. For the marginal individual, the net willingness-to-pay equals w̃ (x) = P (x) = D−1 (Q (x) ;x), while the original willingness-to-pay equals w̃ (0) = D−1 (Q (x) ; 0). Equilibrium welfare for policy x (not accounting for the budgetary cost) equals