The optimal subsidy problem always has a solution (a H*, aL*). The optimal high-risk contract aH* will always entail complete in- surance so that V(pH, aH*) = U(W - pHd + a), where a is the per capita subsidy of the high risk by the low risk. This subsidy decreases income for each low-risk person by zya (where My = X/(1 - X)) in each state. Net of this charge aL* breaks even when low-risk individuals buy it. Thus, aL* = (a, + ya, a2 - ya), where a1 = a2pL/(l - pL). To find the optimal contract, one solves the following problem: Choose a and a2 to maximize