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 it
134 THE JOURNAL OF RISK AND INSURANCE the insured purchases medical care, the insured has an incentive to select a supraoptimal 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 selfselection 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 expectation 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.
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 135 where the right-hand side represents the cost of additional insurance in terms of expected marginal utility. This cost appears in the form of foregone consumption when the consumer uses some of his income to purchase more insurance. The left hand side represents the benefit of additional insurance in terms of expected marginal utility. Since additional insurance reduces the cost of medical care, the value of the additional insurance appears as the benefit of extra medical care service. Given the construction of the model and the backward induction solution process expected utility maximization yields optimal solutions for a, c, and m COMPARATIVE STATICS IMPLICATIONS FOR EMPIRICAL WORK In this section, we analyze how the insured consumer reacts to a change in the gross price of medical care p. One example of a change in p would be a change n nonnegotiated supply prices that are applicable to a percentage cost share rule Such a change in p changes the net price, for given o One should note here that it would be inappropriate to consider an exogenous change in o because o is not an exogenous parameter in the consumer choice model. The choice of health insurance plan(i.e, a)is endogenous; once chosen, a is set for the period (i.e, deductible,co- pays, expenditure caps, etc become fixed). However, a can change if the consumer changes the intended choice of insurance plan in response to a change in the gross price p. In this case, the net price changes twice, first exogenously when p changes and second endogenously when o changes. Our theoretical model captures this reality and allows for the insurance policy choice response and its subsequent impact on net price and medical care demand. We now turn to the comparative statics analysis of a change n p and examine its implications for demand behavior and measurement of demand The change in individual medical care demand caused by an alteration in the gross price comprises two separate behavioral responses. The first is the standard neoclas sical demand response, depicted by an for a given cost sharing, a change in gross price directly impacts demand. Totally differentiating the first-order condition in(4) with respect to p, we get am U10+U12H10m-U110-pm (10) where 0=U1102p2-2U12H1ap+U2H11+U22(H1)2<0 is the second-order condi tion.Since Lh220,the second-order condition is satisfied; thus, ap<0if medical care is noninferior The second response is indirect. A change in the price of medical care can lead to a change in the intended choice of health insurance. The consumer may now choose to 15 There are two extreme values for g. If g =l. the deductible has not been reached, and the net price of medical care is the same as the gross price. If a =0, then the consumer expenditure cap has been reached and the net price is 0. For analytical convenience, we will assume 0 <a<1. This is the most common case in reality and allows us to emphasize the situation of a positive net price of medical care that differs from the gross price
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 135 where the right-hand side represents the cost of additional insurance in terms of expected marginal utility. This cost appears in the form of foregone consumption when the consumer uses some of his income to purchase more insurance. The lefthand side represents the benefit of additional insurance in terms of expected marginal utility. Since additional insurance reduces the cost of medical care, the value of the additional insurance appears as the benefit of extra medical care service. Given the construction of the model and the backward induction solution process, expected utility maximization yields optimal solutions for σ, c, and m. COMPARATIVE STATICS IMPLICATIONS FOR EMPIRICAL WORK In this section, we analyze how the insured consumer reacts to a change in the gross price of medical care p. One example of a change in p would be a change in nonnegotiated supply prices that are applicable to a percentage cost share rule. Such a change in p changes the net price, for given σ. One should note here that it would be inappropriate to consider an exogenous change in σ because σ is not an exogenous parameter in the consumer choice model. The choice of health insurance plan (i.e., σ) is endogenous; once chosen, σ is set for the period (i.e., deductible, copays, expenditure caps, etc. become fixed). However, σ can change if the consumer changes the intended choice of insurance plan in response to a change in the gross price p. In this case, the net price changes twice, first exogenously when p changes and second endogenously when σ changes. Our theoretical model captures this reality and allows for the insurance policy choice response and its subsequent impact on net price and medical care demand. We now turn to the comparative statics analysis of a change in p and examine its implications for demand behavior and measurement of demand elasticities.15 The change in individual medical care demand caused by an alteration in the gross price comprises two separate behavioral responses. The first is the standard neoclassical demand response, depicted by ∂m ∂p ; for a given cost sharing, a change in gross price directly impacts demand. Totally differentiating the first-order condition in (4) with respect to p, we get ∂m ∂p = U1σ + U12H1σm − U11σ2 pm , (10) where = U11σ2 p2 − 2U12H1σ p + U2H11 + U22(H1) 2 < 0 is the second-order condition. Since U12 ≥ 0, the second-order condition is satisfied; thus, ∂m ∂p < 0 if medical care is noninferior. The second response is indirect. A change in the price of medical care can lead to a change in the intended choice of health insurance. The consumer may now choose to 15 There are two extreme values for σ. If σ = 1, the deductible has not been reached, and the net price of medical care is the same as the gross price. If σ = 0, then the consumer expenditure cap has been reached and the net price is 0. For analytical convenience, we will assume 0 <σ < 1. This is the most common case in reality and allows us to emphasize the situation of a positive net price of medical care that differs from the gross price