The Journal of Risk and Insurance O The Journal of risk and Insurance, 2010, Vol 77, No 1, 129-144 DO:10.11/1539-6975200901335X IMPLICATIONS OF THE NTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND Richard dusansky Ko ABSTRACT The gross price elasticity of demand for medical care is decomposed into wo separate observable components: the medical care gross price elasticity of insurance choice and the cost-sharing elasticity of medical care. When con- sumers alter their choice of health-care plans, the price elasticity of medical care is no longer equivalent to the cost-sharing elasticity; using the latter as a proxy for the former may produce misleading results. We present condi tions under which the medical care price elasticity is positive, the case of a quasi-Giffen good, and provide a theoretical foundation for extant empirical findings of a positive medical care price elasticity of insurance demand INTRODUCTION Among the developed, higher-income countries, medical care expenditures have been rising at a pace that is distinctly above overall growth rates. In the United States, for example, total medical cost outlays have increased at an average rate of 8.9 percent since 1980, which is about 2.6 percentage points above the average for the aggregate economy, as measured by gross domestic product(GDP). Similar patterns prevail when one examines medical expenditures in both per capita and budget share terms As a share of the U.S. economy, medical care has nearly doubled over the 25-year period 1980 to 2005, rising from 9 percent of GDP in 1980 to 16 percent in 2005.2 Medical care spending per ca has also increased dramatically during this period, expenditure leader, similar trends exist in most developed countries, ?T the medical increasing to $6,697 from $1, 102. Although the United States is by These observations prompt interest in the forces underlying medical care demand and in how individuals respond to its rising costs. In this article, we analyze con sumer medical care demand response to a change in the gross price of medical care, paying special attention to the interaction between the consumer's demand for Richard dusansky is with the Department of Economics, University of Texas at Austin. Cagatay Koc is with the Department of Economics, University of Texas at Arlington. Dusansky can be contacted via e-mail: dusansky @eco. utexas. edu 1 Centers for Medicare and Medicaid Services 2 Centers for Medicare and Medicaid Services 3 Centers for Medicare and Medicaid Services 4 Kaiser Family Foundation 129
C The Journal of Risk and Insurance, 2010, Vol. 77, No. 1, 129-144 DOI: 10.1111/j.1539-6975.2009.01335.x IMPLICATIONS OF THE INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND Richard Dusansky C¸ agatay Koc ˘ ¸ ABSTRACT The gross price elasticity of demand for medical care is decomposed into two separate observable components: the medical care gross price elasticity of insurance choice and the cost-sharing elasticity of medical care. When consumers alter their choice of health-care plans, the price elasticity of medical care is no longer equivalent to the cost-sharing elasticity; using the latter as a proxy for the former may produce misleading results. We present conditions under which the medical care price elasticity is positive, the case of a quasi-Giffen good, and provide a theoretical foundation for extant empirical findings of a positive medical care price elasticity of insurance demand. INTRODUCTION Among the developed, higher-income countries, medical care expenditures have been rising at a pace that is distinctly above overall growth rates. In the United States, for example, total medical cost outlays have increased at an average rate of 8.9 percent since 1980, which is about 2.6 percentage points above the average for the aggregate economy, as measured by gross domestic product (GDP).1 Similar patterns prevail when one examines medical expenditures in both per capita and budget share terms. As a share of the U.S. economy, medical care has nearly doubled over the 25-year period 1980 to 2005, rising from 9 percent of GDP in 1980 to 16 percent in 2005.2 Medical care spending per capita has also increased dramatically during this period, increasing to $6,697 from $1,102.3 Although the United States is by far the medical expenditure leader, similar trends exist in most developed countries.4 These observations prompt interest in the forces underlying medical care demand and in how individuals respond to its rising costs. In this article, we analyze consumer medical care demand response to a change in the gross price of medical care, paying special attention to the interaction between the consumer’s demand for Richard Dusansky is with the Department of Economics, University of Texas at Austin. C¸ agatay ˘ Koc¸ is with the Department of Economics, University of Texas at Arlington. Dusansky can be contacted via e-mail: dusansky@eco.utexas.edu 1 Centers for Medicare and Medicaid Services. 2 Centers for Medicare and Medicaid Services. 3 Centers for Medicare and Medicaid Services. 4 Kaiser Family Foundation. 129
30 THE JOURNAL OF RISK AND INSURANCE medical care and his demand for health insurance, and the ensuing implications for the measurement of demand elasticities Private health insurance contracts provide for a price discount at the time the insured purchases medical care; therefore, the more generous an individual's insurance cov erage is, the higher might be his demand for medical care. This effect of insurance on medical care demand is known as the moral hazard effect(Arrow, 1963; Pauly, 1968) It refers to the effect of insurance on the net price of medical care (i.e, percentage of cost sharing times the gross price)and to the consequent incentive effects on medical 6are consumption In medical care, moral hazard combines with the interdependence tween utilization and the insurance contract decision. The choice of type of insur ance is influenced by expected utilization of medical care services, whereas utilization itself is affected by the consumer's insurance contract. This interaction complicates the analysis of the consumer s response to changes in medical care prices. We present a principal-agent model of individual consumer medical care demand that highlights this interaction, so that the consumer is allowed to change both his demand for med ical care and his selection of insurance contract in response to changes in medical oare prices. A change in the gross price of medical care has two effects on demand: a rect effect, for fixed cost sharing, and an indirect effect through a moral hazard re- sponse that results from cost-sharing changes associated with price-induced changes in the choice of insurance plan. Much of the extant literature studies the demand for medical care by examining responses to changes in cost sharing, with the gross price held constant and the choice of health insurance contract remaining fixed. However, if one wants to compare medical care demand with the demand for other goods or categories of goods-or if one simply recognizes the fact that most insured consumers choose from among several insurance plans and can alter their intended choice-then the response to a change in gross price is important. Our analysis reveals that the indirect demand effect through the moral hazard effect of insurance can have a major role in determining the consumers medical care demand response to a price change. Indeed it is possible for the own-demand curve for medical care to be upward sloping, even though medical care is a noninferior good For this possibility to occur, it turns out, the demand for health insurance must be increasing in the price of medical care. Although there is empirical support for this it lacks a theoretical foundation. We provide one by developing a formal analytical condition, which embodies a dominant expenditure risk effect under which it holds There has been some work done on the possibility that in response to an increase in the premium rate, consumers may choose to increase their insurance coverage, i.e This article concerns the case where individuals have some choice about the extent or generos- ity of their insurance coverage. It does not apply to systems where coverage is set exogenously as in some national health insurance systems. However, even in the presence of national health insurance participation such as Medicare, what follows applies if the individual can privately upplement the mandated coverage 6 This interaction in the medical care industry has received recent attention. For example, Mougeot and Naegelen(2009)analyze the optimal outlier payment policy in a prospective yment system for hospitals under both adverse selection and moral hazard Housing is another example of a noninferior good whose own demand can be upward sloping See Dusansky and Kos (2007)
130 THE JOURNAL OF RISK AND INSURANCE medical care and his demand for health insurance, and the ensuing implications for the measurement of demand elasticities.5 Private health insurance contracts provide for a price discount at the time the insured purchases medical care; therefore, the more generous an individual’s insurance coverage is, the higher might be his demand for medical care. This effect of insurance on medical care demand is known as the moral hazard effect (Arrow, 1963; Pauly, 1968). It refers to the effect of insurance on the net price of medical care (i.e., percentage of cost sharing times the gross price) and to the consequent incentive effects on medical care consumption. In medical care, moral hazard combines with the interdependence between utilization and the insurance contract decision. The choice of type of insurance is influenced by expected utilization of medical care services, whereas utilization itself is affected by the consumer’s insurance contract. This interaction complicates the analysis of the consumer’s response to changes in medical care prices.6 We present a principal–agent model of individual consumer medical care demand that highlights this interaction, so that the consumer is allowed to change both his demand for medical care and his selection of insurance contract in response to changes in medical care prices. A change in the gross price of medical care has two effects on demand: a direct effect, for fixed cost sharing, and an indirect effect through a moral hazard response that results from cost-sharing changes associated with price-induced changes in the choice of insurance plan. Much of the extant literature studies the demand for medical care by examining responses to changes in cost sharing, with the gross price held constant and the choice of health insurance contract remaining fixed. However, if one wants to compare medical care demand with the demand for other goods or categories of goods—or if one simply recognizes the fact that most insured consumers choose from among several insurance plans and can alter their intended choice—then the response to a change in gross price is important. Our analysis reveals that the indirect demand effect through the moral hazard effect of insurance can have a major role in determining the consumer’s medical care demand response to a price change. Indeed, it is possible for the own-demand curve for medical care to be upward sloping, even though medical care is a noninferior good.7 For this possibility to occur, it turns out, the demand for health insurance must be increasing in the price of medical care. Although there is empirical support for this, it lacks a theoretical foundation. We provide one by developing a formal analytical condition, which embodies a dominant expenditure risk effect under which it holds. There has been some work done on the possibility that in response to an increase in the premium rate, consumers may choose to increase their insurance coverage, i.e., 5 This article concerns the case where individuals have some choice about the extent or generosity of their insurance coverage. It does not apply to systems where coverage is set exogenously, as in some national health insurance systems. However, even in the presence of national health insurance participation such as Medicare, what follows applies if the individual can privately supplement the mandated coverage. 6 This interaction in the medical care industry has received recent attention. For example, Mougeot and Naegelen (2009) analyze the optimal outlier payment policy in a prospective payment system for hospitals under both adverse selection and moral hazard. 7 Housing is another example of a noninferior good whose own demand can be upward sloping. See Dusansky and Koc¸ (2007)
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 131 the demand for insurance may be upward sloping(Hoy and Robson, 1981; Briys, Dionne, and Eeckhoudt, 1989: Hau, 2008). Our analysis complements this literature nd suggests that health insurance, via its moral hazard effect, can make the demand for medical care upward sloping We also examine medical care's price elasticity of demand and demonstrate that the gross price elasticity can be decomposed into two separate and observable compo- lents, the medical care gross price elasticity of insurance choice and the cost-sharing mine a measure of the gross price elasticity. When one allows for the contract choice response, the gross price elasticity of demand for medical care and the cost-sharing demand elasticity diverge. Consequently, using cost-sharing elasticities as a proxy for gross price elasticities would be unwise, if not misleading THE MODEL We employ a principal-agent model in which the agent(consumer)first choose an insurance plan, when there is uncertainty about his future health status, and then makes optimal choices of medical care and the standard consumer goods, after the health status uncertainty is resolved. The insurance contracts among which the consumer chooses can vary considerably across the basic health characteristics. For example, there can be differences in the annual deductible per person and /or per family, in the physician co-pays per physician type, in the percentage cost shares for hospital stays, for in-network versus out-of-network services, etc. We follow Feldstein (1971)in assuming that the characteristics of any insurance policy can be represented y asingle metric We designate this by o, the percentage of cost sharing. An increase in the deductible or co-pay, for example, increases a. For simplicity, we assume that our consumer faces a continuum of policies, arrayed by percentages of cost sharing, and for a given gross price of medical care chooses a policy with a particular percentage of cost sharing, say a =a*. Here we acknowledge the reality that the majority nsured consumers choose from among several insurance policies. In 1995, 62 percent of insured U.S. workers were offered two or more health plans For firms with 200 or more workers, this figure was 84 percent The consumers ex post budget constraint can be represented by Y一R=C+pm s Hoy and Robson( 1981)show that for the class of constant relative risk-aversion utility func- tions, the coefficient of relative risk aversion must be greater than 1 for insurance to be a Giffen good. Briys, Dionne, and Eeckhoudt(1989)demonstrate that insurance is not a Giffen good if and only if the variation of risk aversion is lower than a minimal bound, which is satisfied with increasing and constant absolute risk-aversion utility functions. Assuming that there are two states of the world (loss and no loss), they confirm Hoy and robsons result by reversing their necessary and sufficient condition. Hau(2008)shows that the necessary condition for insurance to be a giffen good can be made less restrictive than that proposed by the above studies by extending their result to the case with a continuum of states of the world and without the class of constant relative risk-aversion utility functions See Jensen et al. (1997)
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 131 the demand for insurance may be upward sloping (Hoy and Robson, 1981; Briys, Dionne, and Eeckhoudt, 1989; Hau, 2008).8 Our analysis complements this literature and suggests that health insurance, via its moral hazard effect, can make the demand for medical care upward sloping. We also examine medical care’s price elasticity of demand and demonstrate that the gross price elasticity can be decomposed into two separate and observable components, the medical care gross price elasticity of insurance choice and the cost-sharing elasticity of medical care. We show how the two can be combined in order to determine a measure of the gross price elasticity. When one allows for the contract choice response, the gross price elasticity of demand for medical care and the cost-sharing demand elasticity diverge. Consequently, using cost-sharing elasticities as a proxy for gross price elasticities would be unwise, if not misleading. THE MODEL We employ a principal–agent model in which the agent (consumer) first chooses an insurance plan, when there is uncertainty about his future health status, and then makes optimal choices of medical care and the standard consumer goods, after the health status uncertainty is resolved. The insurance contracts among which the consumer chooses can vary considerably across the basic health characteristics. For example, there can be differences in the annual deductible per person and/or per family, in the physician co-pays per physician type, in the percentage cost shares for hospital stays, for in-network versus out-of-network services, etc. We follow Feldstein (1971) in assuming that the characteristics of any insurance policy can be represented by a single metric. We designate this by σ, the percentage of cost sharing. An increase in the deductible or co-pay, for example, increases σ. For simplicity, we assume that our consumer faces a continuum of policies, arrayed by percentages of cost sharing, and for a given gross price of medical care chooses a policy with a particular percentage of cost sharing, say σ = σ∗. Here we acknowledge the reality that the majority of insured consumers choose from among several insurance policies. In 1995, 62 percent of insured U.S. workers were offered two or more health plans. For firms with 200 or more workers, this figure was 84 percent.9 The consumer’s ex post budget constraint can be represented by Y − R = c + σ pm, (1) 8 Hoy and Robson (1981) show that for the class of constant relative risk-aversion utility functions, the coefficient of relative risk aversion must be greater than 1 for insurance to be a Giffen good. Briys, Dionne, and Eeckhoudt (1989) demonstrate that insurance is not a Giffen good if and only if the variation of risk aversion is lower than a minimal bound, which is satisfied with increasing and constant absolute risk-aversion utility functions. Assuming that there are two states of the world (loss and no loss), they confirm Hoy and Robson’s result by reversing their necessary and sufficient condition. Hau (2008) shows that the necessary condition for insurance to be a Giffen good can be made less restrictive than that proposed by the above studies by extending their result to the case with a continuum of states of the world and without the class of constant relative risk-aversion utility functions. 9 See Jensen et al. (1997)
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 (technically 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 optimizing 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, substituting 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 secondstage 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)
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 133 u2 where I=#, the shadow price of health, is the insurance adjusted(net)cost to the consumer of producing an extra unit of health. Since the price of the composite consumption good is normalized to one, the consumers marginal rate of substitution between health and the composite consumption good, at the optimum, equals the The presence of uncertainty complicates planning. Although all prices and the current health state are known when the consumer chooses his insurance policy, the future health state s is unknown. Hence, the consequences of the ex ante action o depend crucially on s, which is revealed in the second stage. When making his choice, the consumer must therefore forecast his ex post consumption stream according to the future health state s, given that he makes some insurance choice a. The consumer is assumed to have a prior subjective probability distribution of the future health states denoted by F(s) In choosing a health insurance plan, the consumer is assumed to confront a per fectly competitive health insurance market. Given the utility-maximizing me care schedule m(o, s)characterized by (4), the consumers expected utility function for a particular insurance plan as represented by its cost sharing o and premium r is V(o, R)=U(Y-R-opm(o, s), H(m(o, s), s)dF(s) The competitive equilibrium health insurance policy maximizes(6)subject to the zero-profit insurance policy constraint R=/(1-0)pm(o, s)dF(s) Our model is similar to those presented in Zeckhauser (1970), Spence and Zeckhauser (1971), Feldstein and Friedman (1977), Baumgardner (1991),and Blomqvist(1997). It is a principal-agent model where the agent(insured consumer observes his future health status(s)and decides on the level of composite medical care services(i. e, his effort). Given a level of medical care chosen by the insured according to(4)and assuming a perfectly competitive health insurance market, the principal (the insurer) maximizes the expected utility of the insured subject to the zero-profit insurance policy constraint, which also incorporates the utility-maximizing medical care schedule. Since health insurance policies provide for a price discount at the time 12 Zeckhauser( 1970), Feldstein and Friedman( 1977), and Baumgardner(1991)consider linear insurance schedules, i.e. a constant coinsurance parameter where health insurance pays the same fraction of the cost of the consumers medical care services, regardless of the tota amount spent. Spence and Zeckhauser (1971)and Blomqvist (1997)analyze the properties of nonlinear health insurance policies that depend on the amount of medical expenditures pent
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 133 U2 U1 = σ p H1 = π, (5) where π = σ p H1 , the shadow price of health, is the insurance adjusted (net) cost to the consumer of producing an extra unit of health. Since the price of the composite consumption good is normalized to one, the consumer’s marginal rate of substitution between health and the composite consumption good, at the optimum, equals the shadow price. The presence of uncertainty complicates planning. Although all prices and the current health state are known when the consumer chooses his insurance policy, the future health state s is unknown. Hence, the consequences of the ex ante action σ depend crucially on s, which is revealed in the second stage. When making his choice, the consumer must therefore forecast his ex post consumption stream according to the future health state s, given that he makes some insurance choice σ. The consumer is assumed to have a prior subjective probability distribution of the future health states denoted by F (s). In choosing a health insurance plan, the consumer is assumed to confront a perfectly competitive health insurance market. Given the utility-maximizing medical care schedule m(σ,s) characterized by (4), the consumer’s expected utility function for a particular insurance plan as represented by its cost sharing σ and premium R is V(σ, R) = S U(Y − R − σ pm(σ,s), H(m(σ,s),s)) dF (s). (6) The competitive equilibrium health insurance policy maximizes (6) subject to the zero-profit insurance policy constraint R = S (1 − σ)pm(σ,s)dF (s). (7) Our model is similar to those presented in Zeckhauser (1970), Spence and Zeckhauser (1971), Feldstein and Friedman (1977), Baumgardner (1991), and Blomqvist (1997).12 It is a principal–agent model where the agent (insured consumer) observes his future health status (s) and decides on the level of composite medical care services (i.e., his effort). Given a level of medical care chosen by the insured according to (4) and assuming a perfectly competitive health insurance market, the principal (the insurer) maximizes the expected utility of the insured subject to the zero-profit insurance policy constraint, which also incorporates the utility-maximizing medical care schedule. Since health insurance policies provide for a price discount at the time 12 Zeckhauser (1970), Feldstein and Friedman (1977), and Baumgardner (1991) consider linear insurance schedules, i.e., a constant coinsurance parameter where health insurance pays the same fraction of the cost of the consumer’s medical care services, regardless of the total amount spent. Spence and Zeckhauser (1971) and Blomqvist (1997) analyze the properties of nonlinear health insurance policies that depend on the amount of medical expenditures spent.
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, 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
36 THE JOURNAL OF RISK AND INSURANCE purchase a policy characterized by a lower degree of overall cost sharing (i.e, a lower a). This lower a can induce an increase in medical care demand. This second response is represented by ao ap, where ap captures the impact of a changing medical care price on the choice of insurance contract (i. e, the insurance contract choice effect)and dd captures the medical care response to a change in the degree of overall cost sharing (i.e, the moral hazard effect). From comparative statics of the optimality condition in (4)with respect to o, we have mU1p+U12H1pm Rewriting the optimality condition in Equation(9)as U1 dF(s)-ump dF(s)=0 (12) and differentiating it with respect to p, we have 人mn-cm-mdr a2R Uhom1-∈ n,pdf() UidF(s) ar/aR U1ldF(s)+Umpd(s L12H1 dF(s) U12H1a-dF(s), where Em, p=-ap H is the standard neoclassical price elasticity of demand for med. ical care. From Equation(11), we learn that medical care demand is decreasing in the degree of overall cost sharing (i.e. ad <o), since medical care is noninferior Equation(13)is key to evaluating the sign of do, the effect of a gross price change on insurance choice. From the implicit function theorem we know that do Since the sign of the insurance contract choice effect is the same as the sign of since the latter is ambiguous, the effect of a change in the gross price of medical care on insurance noice is also ambiguous The two behavioral responses are captured in the expression for the change in medical care demand with respect to a change in its price that is generated by a perturbation of the optimum conditions and can be expressed in elasticity form by multiplying by m and by making use of the comparative statics result op=pa0. Adding the 6 Later we present conditions under which this ambiguity is resol 17a change in P also has two income effects associated with a chang policy premium Ron medical care demand. The first income effect is induced by p's effect on R, given by
136 THE JOURNAL OF RISK AND INSURANCE purchase a policy characterized by a lower degree of overall cost sharing (i.e., a lower σ). This lower σ can induce an increase in medical care demand. This second response is represented by ∂m ∂σ ∂σ ∂p , where ∂σ ∂p captures the impact of a changing medical care price on the choice of insurance contract (i.e., the insurance contract choice effect) and ∂m ∂σ captures the medical care response to a change in the degree of overall cost sharing (i.e., the moral hazard effect). From comparative statics of the optimality condition in (4) with respect to σ, we have ∂m ∂σ = U1 p + U12H1 pm − U11 p2σm . (11) Rewriting the optimality condition in Equation (9) as = −∂R ∂σ S U1 dF (s) − S U1mp dF (s) = 0 (12) and differentiating it with respect to p, we have ∂ ∂p = − S m[1 − m,p] (U1 − U11σmp) dF (s) + ∂R ∂σ S U11σm[1 − m,p]dF (s) − ∂2R ∂σ∂p S U1dF (s) + ∂R ∂p ∂R ∂σ S U11dF (s) + S U11mpdF (s) − S U12H1 ∂m ∂p mpdF (s) − ∂R ∂σ S U12H1 ∂m ∂p dF (s), (13) where m,p = −∂m ∂p p m is the standard neoclassical price elasticity of demand for medical care. From Equation (11), we learn that medical care demand is decreasing in the degree of overall cost sharing (i.e., ∂m ∂σ < 0), since medical care is noninferior. Equation (13) is key to evaluating the sign of ∂σ ∂p , the effect of a gross price change on insurance choice. From the implicit function theorem, we know that ∂σ ∂p = − ∂/∂p ∂/∂σ . Since the sign of the insurance contract choice effect is the same as the sign of ∂ ∂p and since the latter is ambiguous, the effect of a change in the gross price of medical care on insurance choice is also ambiguous.16 The two behavioral responses are captured in the expression for the change in medical care demand with respect to a change in its price that is generated by a perturbation of the optimum conditions and can be expressed in elasticity form by multiplying by p m and by making use of the comparative statics result ∂m ∂p = σ p ∂m ∂σ . 17 Adding the 16 Later we present conditions under which this ambiguity is resolved. 17 A change in p also has two income effects associated with a change in policy premium R on medical care demand. The first income effect is induced by p’s direct effect on R, given by
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 137 two responses together provides an expression for the total gross price elasticity of demand for medical care (1 (14) Equation(14)shows that the gross price elasticity of demand for medical care consists of two observable components: the cost-sharing elasticity of medical care(Em,o)and the medical care gross price elasticity of demand for health insurance (Ea, p). Many studies that analyze the price elasticity of medical care explicitly take the medical care gross price elasticity of demand for health insurance to be 0. As a result, the term in parentheses is unity and the cost-sharing elasticity of medical care demand Em,a is tantamount to the gross price elasticity Em, p. However, if one acknowledges the insurance contract choice effect, then ea, p is not O, the term in parentheses is not unity and Em, a would not be the same as the gross price elasticity of demand for medical It is not the case that the role of health insurance has been ignored. Quite the contrary is true. The importance of insurance in medical care demand is widely recognized There is the pioneering RAND study, in which insurance plans are randomly assigned and plan change is precluded, and other studies both theoretical and empirical. Ex- mples of these are Phelps(1976), Cameron et al.(1988), Strombom, Buchmueller and Feldstein(2002), and Ahking, Giaccotto, and Santerre(2009). Phelps presents a theoretical model of insurance demand response to a change in the price of medical care(the response is ambiguous). Cameron et al. allow for a theoretical interaction between insurance choice and utilization decisions and then estimate an insurance lemand equation, subsequently using the estimated insurance variable as an input to a medical care demand equation. Strombom, Buchmueller, and Feldstein estimate health plan choice as a function of premiums, health plan characteristics, and con sumercharacteristics. Ahking, Giaccotto, and Santerre estimate the aggregate demand for private health insurance coverage in the United States for the period 1966 throug 1999.However, no study takes account of the effect of a change in the price of med ical care on the demand for health insurance and the resulting moral hazard effect of this change in intended insurance coverage on the demand for medical care. The health insurance contract choice effect can play an important role in determining the ay ar aR. The second effect is induced by p's effect on a and its subsequent effect onR,given by票器部 We follow others in eschewing the income effects associated with a change in premium, which is generally considered to be negligible. For examples, see Pauly (1968), Feldstein(1973), Feldman and Dowd(1991), and Manning and Marquis(1996) 18 A selection of papers include Rosett and Huang(1973), Newhouse et al.(1980), Wedig (1988), and Hughes and McGuire(1995). For a complete list of these papers see Cutler and eckhauser(2000)and Zweifel and Manning (2000) 19 There is also an empirical literature that deals with the econometric problems arising from the endogeneity of contract choice. See, for example, Lee(1983), Hosek, Marquis, and Wells (1990), Dowd et al. (1991), Vera-Hernandez(1999), and Deb et al. (2006) The objective of the RANd study was to analyze the impacts of alternative cost-sharing regimes on medical care demand. Therefore, individuals were randomly assigned to insur ance plans and were not permitted to change plans in order to avoid the endogenous contract selection. If the objective of the RANd study were to estimate the gross price elasticity of
INTERACTION BETWEEN INSURANCE CHOICE AND MEDICAL CARE DEMAND 137 two responses together provides an expression for the total gross price elasticity of demand for medical care: T m,p = m,σ (1 + σ,p). (14) Equation (14) shows that the gross price elasticity of demand for medical care consists of two observable components: the cost-sharing elasticity of medical care (m,σ ) and the medical care gross price elasticity of demand for health insurance (σ,p). Many studies that analyze the price elasticity of medical care explicitly take the medical care gross price elasticity of demand for health insurance to be 0.18 As a result, the term in parentheses is unity and the cost-sharing elasticity of medical care demand m,σ is tantamount to the gross price elasticity m,p. However, if one acknowledges the insurance contract choice effect, then σ,p is not 0, the term in parentheses is not unity and m,σ would not be the same as the gross price elasticity of demand for medical care. It is not the case that the role of health insurance has been ignored. Quite the contrary is true. The importance of insurance in medical care demand is widely recognized. There is the pioneering RAND study, in which insurance plans are randomly assigned and plan change is precluded, and other studies both theoretical and empirical. Examples of these are Phelps (1976), Cameron et al. (1988), Strombom, Buchmueller, and Feldstein (2002), and Ahking, Giaccotto, and Santerre (2009). Phelps presents a theoretical model of insurance demand response to a change in the price of medical care (the response is ambiguous). Cameron et al. allow for a theoretical interaction between insurance choice and utilization decisions and then estimate an insurance demand equation, subsequently using the estimated insurance variable as an input to a medical care demand equation. Strombom, Buchmueller, and Feldstein estimate health plan choice as a function of premiums, health plan characteristics, and consumer characteristics. Ahking, Giaccotto, and Santerre estimate the aggregate demand for private health insurance coverage in the United States for the period 1966 through 1999.19 However, no study takes account of the effect of a change in the price of medical care on the demand for health insurance and the resulting moral hazard effect of this change in intended insurance coverage on the demand for medical care.20 The health insurance contract choice effect can play an important role in determining the ∂m ∂Y ∂Y ∂R ∂R ∂p . The second effect is induced by p’s effect on σ and its subsequent effect on R, given by ∂m ∂Y ∂Y ∂R ∂R ∂σ ∂σ ∂p . We follow others in eschewing the income effects associated with a change in premium, which is generally considered to be negligible. For examples, see Pauly (1968), Feldstein (1973), Feldman and Dowd (1991), and Manning and Marquis (1996). 18 A selection of papers include Rosett and Huang (1973), Newhouse et al. (1980), Wedig (1988), and Hughes and McGuire (1995). For a complete list of these papers see Cutler and Zeckhauser (2000) and Zweifel and Manning (2000). 19 There is also an empirical literature that deals with the econometric problems arising from the endogeneity of contract choice. See, for example, Lee (1983), Hosek, Marquis, and Wells (1990), Dowd et al. (1991), Vera-Hernandez (1999), and Deb et al. (2006). 20 The objective of the RAND study was to analyze the impacts of alternative cost-sharing regimes on medical care demand. Therefore, individuals were randomly assigned to insurance plans and were not permitted to change plans in order to avoid the endogenous contract selection. If the objective of the RAND study were to estimate the gross price elasticity of
38 THE JOURNAL OF RISK AND INSURANCE consumer's observable demand response to a change in the gross price of medical care. The medical care demand curve could be upward sloping. Econometric studies of the price elasticity of medical care that ignore it-which is typically the case in the literature-would produce estimates that are erroneous. We now turn our attention to analyzing the implications of the contract choice effect for the slope of the medical care demand curve and for the price elasticity of medical care MEDICAL CARE AS A QUASI-GIFFEN GOoD If the contract choice effect is positive(ie, ap 0),in opposition to the neoclassical response path(i.e, ap) And if the standard neoclassical response path is dominated by the moral hazard induced effect of the change in insurance coverage on medical care demand, then Em, p >0. The qualitative properties of the medical care demand curve are now distinctly different from those of the traditional response. Even though medical care is noninferior, an increase in the gross price of medical care causes an increase in medical care demand. Once health insurance choice is endogenized, it is possible that the demand for medical care is The key to understanding this outcome is to note that the consumer responds to thenet price, given by ap. Suppose that p rises For a fixed a, the net price increases. However, because of the contract choice effect, o falls. If the decline in g more than offsets the rise in P, the final outcome is a decline in the net price. Therefore, the demand for medical care can be upward sloping with regard to the gross price whereas at the same time be downward sloping with regard to its net price. We can characterize the condition under which an upward sloping medical care demand occurs. From(14), we know that the overall price effect is positive if and only if 1 that is, if and only if the demand for health insurance is medical care price elastic Feldstein's(1973)empirical work on health insurance demand in response to changes in the price of medical care is relevant here. Using a variable akin to our o, he found that an increase in the price of hospital care had a positive and statistically significant ffect on insurance demand. This points to the viability of the contract choice effect Feldstein also reported an elasticity of 1.2, which satisfies the condition in(15).This suggests that, in some times and places, there may be some categories of medical services for which own-price demand elasticity is positive medical care demand, another randomized experiment could have been carried out to as- certain the medical care gross price elasticity of demand for health insurance and one could have calculated the gross price elasticity of medical care demand by combining estimates from two sets of randomized experiments. I Note that if the contract choice effect is positive, da<0 according to our model setup, so that Ea,p should be taken as.2
138 THE JOURNAL OF RISK AND INSURANCE consumer’s observable demand response to a change in the gross price of medical care. The medical care demand curve could be upward sloping. Econometric studies of the price elasticity of medical care that ignore it—which is typically the case in the literature—would produce estimates that are erroneous. We now turn our attention to analyzing the implications of the contract choice effect for the slope of the medical care demand curve and for the price elasticity of medical care. MEDICAL CARE AS A QUASI-GIFFEN GOOD If the contract choice effect is positive (i.e., ∂σ ∂p 0), in opposition to the neoclassical response path (i.e., ∂m ∂p ). And if the standard neoclassical response path is dominated by the moral hazard induced effect of the change in insurance coverage on medical care demand, then T m,p > 0. The qualitative properties of the medical care demand curve are now distinctly different from those of the traditional response. Even though medical care is noninferior, an increase in the gross price of medical care causes an increase in medical care demand. Once health insurance choice is endogenized, it is possible that the demand for medical care is upward sloping. The key to understanding this outcome is to note that the consumer responds to the net price, given by σ p. Suppose that p rises. For a fixed σ, the net price increases. However, because of the contract choice effect, σ falls. If the decline in σ more than offsets the rise in p, the final outcome is a decline in the net price. Therefore, the demand for medical care can be upward sloping with regard to the gross price whereas at the same time be downward sloping with regard to its net price. We can characterize the condition under which an upward sloping medical care demand occurs. From (14), we know that the overall price effect is positive if and only if 1 + σ,p 1, that is, if and only if the demand for health insurance is medical care price elastic. Feldstein’s (1973) empirical work on health insurance demand in response to changes in the price of medical care is relevant here. Using a variable akin to our σ, he found that an increase in the price of hospital care had a positive and statistically significant effect on insurance demand. This points to the viability of the contract choice effect. Feldstein also reported an elasticity of 1.2, which satisfies the condition in (15).21 This suggests that, in some times and places, there may be some categories of medical services for which own-price demand elasticity is positive. medical care demand, another randomized experiment could have been carried out to ascertain the medical care gross price elasticity of demand for health insurance and one could have calculated the gross price elasticity of medical care demand by combining estimates from two sets of randomized experiments. 21 Note that if the contract choice effect is positive, ∂σ ∂p < 0 according to our model setup, so that σ,p should be taken as −1.2
