34 THE JOURNAL OF RISK AND INSURANCE the insured purchases medical care, the insured has an incentive to select a supraopti- mal amount of medical care no matter what his health status is. The insurer recognizes this ex post moral hazard problem and therefore (4)is imposed as a constraint in the consumers expected utility maximization. This can be considered as a form of self- selection constraint which guarantees that an individual facing o will,for a glven s, voluntarily choose the combination [o, m(o, s)] corresponding to the values in the optimal solution In the above principal-agent framework, moral hazard leads to a nonconvexity in the set of feasible contracts, which is a well-known problem with opm on the vertical imization under moral hazard. To see this, consider the contract space(o, R), with R axis and a on the horizontal axis. The set of zero-profit insurance policies given by (7)is curved inward toward the axes; i.e., it is concave to the origin. This is because as a falls, R increases more than proportionately (since as o falls, not only does the insurers portion of medical expenses increase, but also total expenditure increases due to the moral hazard problem(Phelps, 2003, P. 335). The feasible sets of contracts (i.e., those making a nonnegative profit), however, are those contracts to the left of this curve, including the curve. The set is clearly nonconvex. This problem affects the interpretation of our comparative statics analysis of the impact of medical care price on the choice of insurance contract. Since the implicit function theorem is based on e first-order conditions, which do not uniquely determine the optimum when there is a nonconvexity problem, the expression in( 13 )does not define the global impact We assume that the first-order conditions of the health insurance demand decision do characterize the local impact. I The first-order condition for the maximization of (6)subject to(7)is given by (1-a) -U1dF(s)-UimpdF(s)+/[-U1op+U2Hilo-dF(s +Pm-(1-0)5-4P+HydF()=0 whereY=Y-R,-pEIm-(1-a)a0]=aa, and the E[ operator refers to the ex pectation across S Using(4), the optimum condition for health insurance choice is mpU1dF (s)=PE m-(1-a) UdF(s), I This also can be seen by analyzing the first and second derivatives of r with respect to o m(a,s)dF(s)+(1-a)p/dF(s) -2p/adF(s)+(1-a)p dF(s) if 2-m>0. i.e., if increasing rate as o falls. We thank a referee for helping us to understand this problem and for his guidance on how to deal with it134 THE JOURNAL OF RISK AND INSURANCE the insured purchases medical care, the insured has an incentive to select a supraopti￾mal amount of medical care no matter what his health status is. The insurer recognizes this ex post moral hazard problem and therefore (4) is imposed as a constraint in the consumer’s expected utility maximization. This can be considered as a form of self￾selection constraint which guarantees that an individual facing σ will, for a given s, voluntarily choose the combination [σ, m(σ,s)] corresponding to the values in the optimal solution. In the above principal–agent framework, moral hazard leads to a nonconvexity in the set of feasible contracts, which is a well-known problem with optimization under moral hazard. To see this, consider the contract space (σ, R), with R on the vertical axis and σ on the horizontal axis. The set of zero-profit insurance policies given by (7) is curved inward toward the axes; i.e., it is concave to the origin. This is because as σ falls, R increases more than proportionately (since as σ falls, not only does the insurer’s portion of medical expenses increase, but also total expenditure increases due to the moral hazard problem (Phelps, 2003, p. 335)).13 The feasible sets of contracts (i.e., those making a nonnegative profit), however, are those contracts to the left of this curve, including the curve. The set is clearly nonconvex. This problem affects the interpretation of our comparative statics analysis of the impact of medical care price on the choice of insurance contract. Since the implicit function theorem is based on the first-order conditions, which do not uniquely determine the optimum when there is a nonconvexity problem, the expression in (13) does not define the global impact. We assume that the first-order conditions of the health insurance demand decision do characterize the local impact.14 The first-order condition for the maximization of (6) subject to (7) is given by −pE m − (1 − σ) ∂m ∂σ  S −U1dF (s) − S U1mpdF (s) + S [−U1σ p + U2H1] ∂m ∂σ dF (s) +pE m − (1 − σ) ∂m ∂σ  S [−U1σ p + U2H1] ∂m ∂Y dF (s) = 0, (8) where Y = Y − R, −pE[m − (1 − σ) ∂m ∂σ ] = ∂R ∂σ , and the E[] operator refers to the ex￾pectation across S. Using (4), the optimum condition for health insurance choice is S mpU1dF (s) = pE m − (1 − σ) ∂m ∂σ  S U1dF (s), (9) 13 This also can be seen by analyzing the first and second derivatives of R with respect to σ: ∂R ∂σ =  −p S m(σ,s)dF (s) + (1 − σ)p S ∂m ∂σ dF (s)  < 0. ∂2R ∂σ2 = −2p S ∂m ∂σ dF (s) + (1 − σ)p S ∂2m ∂σ2 dF (s) is positive if ∂2m ∂σ2 > 0, i.e., if m increases at an increasing rate as σ falls. 14 We thank a referee for helping us to understand this problem and for his guidance on how to deal with it.