Hammond [6] de nes two kinds of non-paternalistic social welfare function that can be employed under these circumstances.2 The ex ante social welfare function is a function of the expected utilities, while the ex post social welfare function is the expectation of a function of the realized utilities. [...] Under each system the health care of the agents in each group (the rich or the poor) is covered by an insurance program that is nanced by a levy on the incomes of the agents in that group, implying ci = yi mi i = R;P (3) The health care system is not permitted to redistribute income between the two groups. [...] The function r shows the relative value of increases in the utility of the healthy and the utility of the sick in each group under the social welfare function W: Speci cally, an increase in the utility of the healthy raises social welfare by only r times as much as an increase in the utility of the sick. [...] The bene t of an additional unit of consumption is given by the rst two terms of f. The increase in consumption bene ts both the healthy, who have measure 1 , and the sick, who have measure : The gains of the healthy are weighted by the factor r because their gains have a smaller impact on social welfare than the gains of the sick. [...] The sign of the slope of the locus is the negative of the sign of the derivative f1 = (1 )r 0 1u + (1 )ru00 + v11 v12 Inspection of this expression shows that the sign of the rst term is indetermi- nate, while the remaining terms the adjustments to the marginal utilities are negative.