32 THE JOURNAL OF RISK AND INSURANCE where Y is income, R is the cost to the consumer for participating in the health insurance plan(i.e, health insurance premium), c is a composite consumption good whose price is normalized to one without any loss of generality, and m is composite medical care services with gross price p. Since a is the fraction of p that the consumer must pay out of pocket, o p is the net price for a unit of medical care service to the The consumer's utility function, which is assumed to be continuous and bounded, 0 is represented by u=u(, H(m, s)), where u1=au>0,U= ≤0,andU22 <0. We assume y does nah 20; i.e., the consumer's marginal utility of consumption rises(tech- at lh ot fall)as health improves. This ensures that the second-order condition with respect to the ex post decision problem about purchases of medical care and other goods conditional on the ex ante choice of insurance plan is satisfied. H(,)is the health production function with H1 五= as in Dardanoni and Wagstaff (1990). It determines the amount of health produced measured as income equivalent with input m in state s The model is solved by backward induction. First, the consumer solves for the opti mizing values of m and c conditional upon the choice of insurance policy o for each possible realization of the unknown health state s. Second, the consumer formulates a prior probability measure F of the future health states denoted by F(s). Next, substi tuting these optimal values of m and c into the utility function and integrating over s gives the indirect conditional expected utility associated with insurance policy o. This indirect conditional expected utility function forms the basis of choice of insurance policy o. Assuming a perfectly competitive health insurance market, the consumer maximizes this function over the set of zero-profit insurance policies, yielding the For any arbitrary health status, given a choice of insurance policy, our consumer chooses the level of medical care that maximizes u(r-R-opm, H(m, s)), where we have used the budget constraint to rewrite the first argument of u. The first-order condition uap+u2H1=0 Rearranging the first-order condition gives the optimality condition for this second stage problem 10 If the consumers utility functionis continuous and bounded, the expected utility ofinsurance is well defined This assumption appears elsewhere in the literature. See, for example, Wagstaff (1986), Blomqvist(1997), and Jack(2002)132 THE JOURNAL OF RISK AND INSURANCE where Y is income, R is the cost to the consumer for participating in the health insurance plan (i.e., health insurance premium), c is a composite consumption good whose price is normalized to one without any loss of generality, and m is composite medical care services with gross price p. Since σ is the fraction of p that the consumer must pay out of pocket, σ p is the net price for a unit of medical care service to the consumer. The consumer’s utility function, which is assumed to be continuous and bounded,10 is represented by U = U(c, H(m,s)), (2) where U1 = ∂U ∂c > 0,U2 = ∂U ∂ H > 0, U11 = ∂2U ∂c2 ≤ 0, and U22 = ∂2U ∂ H2 ≤ 0. We assume that U12 = ∂2U ∂c∂ H ≥ 0; i.e., the consumer’s marginal utility of consumption rises (tech￾nically does not fall) as health improves. This ensures that the second-order condition with respect to the ex post decision problem about purchases of medical care and other goods conditional on the ex ante choice of insurance plan is satisfied.11 H(., .) is the health production function with H1 = ∂ H ∂m > 0 , H11 = ∂2H ∂m2 ≤ 0, and H2 = ∂ H ∂s > 0, as in Dardanoni and Wagstaff (1990). It determines the amount of health produced measured as income equivalent with input m in state s. The model is solved by backward induction. First, the consumer solves for the opti￾mizing values of m and c conditional upon the choice of insurance policy σ for each possible realization of the unknown health state s. Second, the consumer formulates a prior probability measure F of the future health states denoted by F (s). Next, substi￾tuting these optimal values of m and c into the utility function and integrating over s gives the indirect conditional expected utility associated with insurance policy σ. This indirect conditional expected utility function forms the basis of choice of insurance policy σ. Assuming a perfectly competitive health insurance market, the consumer maximizes this function over the set of zero-profit insurance policies, yielding the optimal insurance plan choice. For any arbitrary health status, given a choice of insurance policy, our consumer chooses the level of medical care that maximizes U(Y − R − σ pm, H(m,s)), (3) where we have used the budget constraint to rewrite the first argument of U. The first-order condition is −U1σ p + U2H1 = 0. (4) Rearranging the first-order condition gives the optimality condition for this second￾stage problem: 10 If the consumer’s utility function is continuous and bounded, the expected utility of insurance is well defined. 11 This assumption appears elsewhere in the literature. See, for example, Wagstaff (1986), Blomqvist (1997), and Jack (2002)