Health Inurance and Social Welfare Run Liang China Center for Economic Reearch, Peking Univerity, Beijing 100871, China, Email: rliang@ccer.edu.cn and Hao Wang China Center for Economic Reearch, Peking Univerity, Beijing 100871, China, Email: hwang@ccer.edu.cn Augut, 2010 The author thank Ake Blomqvit, Simon Chang, Yuk-fai Fong, Huanxing Yang, Lixin Ye, Zhong Zhao, and participant of eminar at Renmin Univerity (2010, Beijing), Central Univerity of Finance and Economic (2010, Beijing), and Summer Workhop in Indutrial Organization and Management Strategy (2010, Shanghai) for helpful comment. The author alo thank the financial upport of the Minitry of Education, P. R. China (Project No. 07JJD840182). 1
Health Inurance and Social Welfare Abtract It i often uggeted in the literature that conumer purchae too much health inurance. But Wigger and Anlauf (2007) find that conumer are actually under-inured againt health expene in market equilibrium from the perpective of ocial welfare. We generalize Wigger and Anlauf (2007) two-tate model to one with a continuum of health tate. We reaffirm their finding that health inurance may leave conumer wore off, but leave health care provider better off. The equilibrium inurance coverage i below the ocial optimal level. Furthermore, we find that the conventional health inurance lead to a econd-bet outcome that i better than that without health inurance. Keyword: Health Inurance; Health care; Welfare JEL code: D6, I1, L1 2
1. Introduction In hi eminal paper about medical care indutry, Arrow (1963) ugget that becaue of the irregular and unpredictable nature of demand, the welfare cae for inurance policie of all ort i overwhelming. It follow that the government hould undertake inurance in thoe cae where thi market, for whatever reaon, ha failed to emerge (page 961). Health inurance uually make conumer face uage rate lower than the marginal cot. Therefore it induce exce demand for health care. Thi phenomenon i refereed to a a moral hazard problem in health economic literature. 1 The firt theoretical analyi of the moral hazard i Pauly (1968), which ugget that medical care expene may be inurable when conumer demand i elatic. There i a prioner dilemma in the market: Each individual may well recognize that exce ue of medical care make the premium he mut pay rie. No individual will be motivated to retrain hi own ue. The direct welfare lo from the moral hazard problem i through exce conumption. A natural conjecture i that imperfect competition in health care market, which lead to relatively high ervice price, could be more efficient than perfect competition, becaue the higher ervice price may drive the uage rate faced by conumer cloer to the marginal cot. Crew (1969) illutrate thi view with a imple diagram. Depite the lack of rigorou analye, thi idea ha extenive influence in health economic. The notion that competition in health care market may be harmful to ocial welfare repreent a failure of the inviible hand, which i phenomenal. Neverthele, Gaynor, Haa-Wilon and Vogt (2000) raie 1 The term moral hazard in health economic often differ from that in information economic. Pauly (1968, p.535) find the repone of eeking more medical care with inurance than in it abence i a reult not of moral perfidy, but of rational economic behavior. But Arrow (1968) diagree with thi point. Indeed, their definition of moral hazard might be a little different. 3
doubt on Crew finding. They ugget that competition in health care market actually improve ocial welfare. Crew (1969) mitake i that he emphaize the ex pot welfare lo from exce conumption, but ignore the ex ante gain from reduced rik. The moral hazard problem may create further efficiency lo when the health care market i not perfectly competitive. Given the tructure of a health care market, health inurance boot conumer demand for health care and thu puh the ervice price up. The higher price, in turn, increae the ex ante financial rik faced by conumer, and drive them purchaing more inurance. Therefore the equilibrium inurance coverage tend to be exceive from the perpective of conumer or even ociety. Uing a panel data for individual tate of the US for the year 1959-1965, Feldtein (1973) find that American familie are generally over-inured againt health expene. If inurance coverage were reduced, the utility lo from increaed rik would be outweighed by the gain due to lower price and the reduced purchae of exce care. Feldman and Dowd (1991) offer a new etimate of the welfare lo of exce health inurance. They ue the Rand Health Inurance Experiment reult regarding price elaticity and conumer rik-averion. According to Feldtein (1973) mot likely parameter value, raiing the average coinurance rate from 0.33 to 0.67 would produce a net gain of $27.8 billion in 1984 price in the private hopital ector. But according to the Rand parameter value, the range of welfare lo i from 33.4 to 109.3 billion dollar in 1984 price. In theoretical tudie, Chiu (1997) and Vaithianathan (2006) how that conventional inurance reduce conumer welfare when the upply of health care i ufficiently 4
price-inelatic. 2 Their policy implication i that government hould intervene in health care market to regulate price o a to prevent inurance-induced price inflation. Wigger and Anlauf (2007) oberve that thee welfare analye of health inurance ignore the profit of health care indutry. They reviit the quetion of whether conumer purchae too much inurance from the perpective of ocial welfare. Conider a health indutry with a monopolitic health care provider and competitive health inurer. Conumer can be either healthy or ick. The paper evaluate the market equilibrium in term of conumer welfare and ocial welfare. It i hown that the conumer welfare criterion ugget that in the market equilibrium conumer purchae too much health inurance coverage. But the ocial welfare criterion ugget that becaue the profit of the health care indutry are properly accounted for, conumer hould purchae more inurance coverage. We generalize Wigger and Anlauf (2007) model with two health tate to one with a continuum of health tate. The model i baed on Png and Wang (2010), which offer a pretty general theory of two-part pricing in market with individual demand uncertainty. In our model health care partially compenate conumer lo from illne. The demand for health care depend on conumer health tate that follow a general probability ditribution. There i a monopolitic health care provider owned by the conumer. The health inurance market i perfectly competitive. Health care ervice i old with linear pricing, while an inurance policy take the form of a two-part pricing. We how that Wigger and Anlauf (2007) reult till hold in the general model. Health inurance tend to reduce conumer welfare, but increae the provider profit. The net effect to the ociety i poitive. Another 2 Thi paper only concern the ituation with ymmetric information. See Pauly (1974) and many other for the dicuion of health inurance under aymmetric information. 5
contribution of the current paper i howing that although the market outcome with health inurance i not firt-bet, it i better than that without health inurance. The ret of the paper proceed a follow. Section 2 depict the model. Section 3 examine how health inurance affect conumer urplu and the health care provider profit. Section 4 evaluate the ocial welfare effect of health inurance. Section 5 conclude the paper. 2. A Health Care Market In a market there i a continuum of conumer that are ex ante homogenou. A conumer health tate, denoted by, follow a cumulative ditribution function G () over the interval[, ], with larger repreenting more eriou illne. A conumer in illne uffer a utility lo, which can be partially compenated by health care. If a conumer in tate conume q unit of health care, the conumer net benefit i gq (, ), which i typically negative ince health care cannot fully compenate conumer lo from illne. Let I denote the conumer initial wealth and m denote her expenditure on non-health good. Like Gaynor et al. (2000), we aume the health care expenditure ha no income effect, hence the ex pot wellbeing i linear with regard to money. It can be written a We aume that gq (, ) atifie: x( qm,, ) gq (, ) + m. (1) g (, ) 0 q q >, gqq ( q, ) < 0, g ( q, ) < 0, and g ( q, ) = g ( q, ) > 0, for (, ], q 0. q q A conumer Bernoulli utility function i denoted by ux ( ), defined on the ex pot 6
wellbeing x( qm.,, ) The Bernoulli utility function i trictly increaing and concave, which mean conumer are rik avere. 3 Health care i provided by a monopolitic provider, which i owned by the conumer. The provider ha a cot function of Cq ( ) = F+ cq, where F i the fixed cot and c i the marginal cot. Suppoe the provider can only et linear price. The inurer are perfectly competitive and rik-neutral. 4 They have zero cot. An inurance policy i written a ( Tk, ), where T repreent the inurance premium and k the coinurance rate. If there i no inurance firm, the producer chooe the health care price p, and conumer chooe the amount of conumption q( p, ) after the realization of health tate. If there are inurer in the market, the health care provider and the inurer move imultaneouly. The health care provider chooe price p and the inurer chooe inurance policy ( Tk, ). Conumer decide whether to get inured before the realization of health tate. 5 After health tate realize, a conumer chooe the amount of conumption qkp (, ) if he hold an inurance policy, or q( p, ) if he doe not. 3. Conumer Surplu and Provider Profit Auming conumer endowment i exogenouly given, thi ection examine how health inurance affect conumer urplu and health care provider profit. We firt characterize the equilibria without and with inurance, and then conduct a welfare comparion. 3 The term Bernoulli utility function follow the textbook of Ma-Colell, Whinton and Green (1995, page 184). It i alo referred a von Neumann-Morgentern utility function, e.g. Wigger and Anlauf (2007). 4 If the health hock of every conumer i i.i.d., the inurer will not encounter ignificant rik. In thi cae, the rik-neutral hypothei i no longer neceary. 5 We aume inurance companie can write enforceable contract that allow conumer purchae at mot one policy, a called by Arnott and Stiglitz (1991) excluive contract. 7
(1) Without inurance In the abence of inurance, the provider provide health care directly to conumer at a linear price p. After the realization of tate, a conumer olve problem: max gq (, ) + m, (2) q 0, m t.. pq+ m I. (3) Condition (3) hould be binding. Hence the conumer ex pot demand for health care in tate i implicitly given by the firt-order condition: gq ( q, ) = p. (4) Let q( p, ) denote conumer ex pot demand function given by (4). We have The monopolitic health care provider then olve problem: q 1 = < (, ) 0. (5) p gqq q max ( p c) q( p, ) dg( ) F. (6) p We aume thi optimization problem i trictly quai-concave in order to avoid the cae with multiple olution. Denote * p a the optimal price. If a long run equilibrium exit, we mut have p * > c, that i, the equilibrium price hould exceed the marginal cot of the health care. The health care provider profit can be expreed a: * * * π = ( p c) q( p, ) dg( ) F. (7) In equilibrium, the conumer urplu can be meaured by expected utility: ( ) * V = u g q p + I p q p dg ( (, ), ) (, ) ( ). (8) (2) With inurance Now there are competitive inurer in the market. Given the price p et by the health 8
care provider and inurance policy ( Tk, ) et by inurer, a conumer problem become: max gq (, ) + m, (9) q 0, m t.. kpq + m+ T I. (10) Condition (10) mut be binding. Hence the conumer ex pot demand i given by the firt-order condition: g (, ) q q = kp. (11) Let qkp (, ) denote the ex pot demand function given by (11). Comparing (11) with (4), one can ee that qkp (, ) = qkp (, ). Note that the conumer ex pot demand depend on the out-of-pocket price kp and the health tate but not the inurance premium T, becaue the income effect ha been aumed away. The conumer ex pot wellbeing, which i expreed in dollar, i: ( ) x() = g q, + I T kp q. (12) The profit-maximizing inurer mut be making zero profit in equilibrium becaue of the perfect competition. Furthermore, an equilibrium inurance policy mut maximize conumer expected utilitie. Otherwie it would be replaced by a more competitive policy. Therefore given health care price p, an equilibrium inurance policy mut be a olution of following optimization problem: max ux ( ( )) dg ( ), (13) Tk, t.. T (1 k) p q( kp, ) dg( ) 0. (14) To eliminate the cae of multiple olution, we again aume the problem i trictly quai-concave. 6 6 If there are multiple olution, only the one with the lowet copayment rate i ocially deirable, becaue it reult in the highet health care provider profit. 9
Given inurance policy ( Tk,, ) the monopolitic provider problem i: max ( p c) q( kp, ) dg( ) F. (15) p Still aume the problem i trictly quai-concave, thu there i a unique olution. Let p and ( T, k ) be the equilibrium price and inurance policy repectively. We hould have p > c. Then the provider profit i: * * ( ) (, ) ( ) = p c q k p dg F. (16) π The conumer expected utility i: * * * * * * ( ) V = u g( q( k p, ), ) + I T k p q( k p, ) dg( ). (17) (3) Comparion In the abence of health inurance, the provider price hould be ignificantly higher than the marginal cot. In the preence of inurance, conumer hould face lower uage rate for health care. Firt we have the following lemma. Lemma 1: Given health care price p > 0, an optimal inurance policy ( T, k ) atifie k < 1. Proof: (See Appendix) But inurance alo increae conumer demand for health care, a a reult puhe the health care provider to raie it price. The higher price to ome extent offet conumer gain from inurance, or even leave them trictly wore off. The price increae tend to be more ignificant when the marginal cot of health care i lower. Vaithianathan (2006) two-tate model indeed find that when the marginal cot of health care i ufficiently low, 10
health inurance reduce conumer urplu. Thi reult remain valid in the current model. Propoition 1: There exit c > 0 uch that when the marginal cot of health care c [0, c), health inurance leave conumer trictly wore off, i.e., V < V. Proof: When there i no inurance and c = 0, the health care provider problem i: max p q( p, ) dg( ) F. (18) p The optimal price p i determined by the firt-order condition: q qpdg (, ) ( ) + p dg ( ) = 0 p. (19) When inurance i available, for given inurance policy * * ( T, k ), the health care price p olve problem: The optimal price p atifie the firt-order condition: max p qk ( pdg, ) ( ) F. (20) p * q (, ) ( ) + ( ) = 0 qk pdg k p dg. (21) p Compared with Equation (19), we have k p = p. Therefore health inurance doe not affect the out-of-pocket price faced by conumer when c = 0. On the other hand the inurance premium charged by inurance companie i given by: * = (1 ) (, ) ( ) T k p q k p dg. (22) Since k < 1 by Lemma 1, we have T > 0. Becaue conumer pay the ame out-of-pocket price but an extra T > 0 a inurance premium, they mut be wore off when inurance i available and c = 0. By the continuality of conumer urplu with repect to c, we have V < V for mall enough marginal cot of health care. Q.E.D. 11
In the real world, the marginal cot of health care i often low. Hence health inurance may decreae conumer urplu, a found by Feldtein (1973) and other. But one hall alo recognize the welfare effect of health inurance on the profit of health care provider, which tend to be poitive a uggeted by the following propoition. Propoition 2: Health inurance increae the profit of the health care provider. Proof: When inurance i available, if the health care provider chooe price p that i the ame a in the abence of inurance, and the inurer offer the optimal policy ( T, k ) accordingly, the health care provider profit π can be written a: π = ( p c) q( kp, ) dg( ) F. (23) Since q( p, ) i decreaing in p and k < 1, for any tate we have Hence qkp (, ) > qp (, ). (26) ( p c ) q ( kp, ) dg ( ) F ( p c ) q ( p π, ) dg ( ) F = > = π. (27) Offering p may not be the health care provider optimal choice. Hence at the optimal health care price, the provider profit π π > π. Q.E.D. 4. Social Welfare From Propoition 1 and 2, we ee that with low enough marginal cot, health inurance lead to oppoite welfare effect on conumer and the health care provider. A natural quetion i how inurance affect ocial welfare. In order to evaluate the variation of ocial 12
welfare, we go back to the aumption that the health care provider i owned by the conumer. Therefore the provider profit become part of conumer income. Thi approach follow Gaynor et al. (2000) and Wigger and Anlauf (2007). (1) Without inurance In the abence of health inurance, given the health care price p, the conumer problem in tate i: max gq (, ) + m, (29) q 0, m t.. pq+ m I+ π. (30) where π i the profit of the health care provider. 7 Since contraint (30) i binding, the conumer ex pot demand for health care i implicitly given by: g (, ) q q = p. (31) Hence the conumer ex pot demand function i the ame at that in the previou ection. The optimization problem of the health care provider i till (6). The profit π * i given by (7). Therefore the ocial welfare, which equal to the overall conumer urplu, can be expreed a a function of the health care price: ( π ) W = u g( q( p, ), ) + I p q( p, ) + dg( ) ( ) = u g( q( p, ), ) + I p q( p, ) + ( p c) q( p, ) dg( ) F dg( ) W( p ). (32) (2) With inurance When there are perfect competitive inurer in the market, given health care price p and 7 Although the value of π depend on the quantity of conumption, each individual conumer take π a exogenou. Since a conumer only take a negligible portion of the profit, he ha no incentive to purchae more in order to boot the provider profit. 13
inurance policy ( Tk,, ) an inured conumer in tate olve problem: max gq (, ) + m, (33) q 0, m t.. kpq + m+ T I+. π (34) where π denote the health care provider profit when there i inurance. It i eay to ee that conumer ex pot demand i till qkp (, ) = qkp (, ). The inurer optimization problem i till given by (13) and (14), but now a conumer ex pot wellbeing x( ) i: ( ) ( (, ), ) x = g q kp + I T kpq + π. (35) The provider optimization problem and profit are till given by (15) and (16) repectively. Since inurer are perfect competitive, contraint (14) mut be binding. Hence in equilibrium the inurance premium i determined by Equation (22). Denote conumer out-of-pocket price in equilibrium a ( π ) τ = k p, the ocial welfare can be expreed a: W = u g( q( k p, ), ) + I T k p q( k p, ) + dg( ) * ( π ) * ( ( ( τ, ), ) (, ) ( ) (, ) ( ) ) τ τ τ τ ( ) = u gqk ( ( p, ), ) + I (1 k) p qk ( p, dg ) ( ) k pqk ( p, ) + dg ( ) = u g q + I q + c q dg F dg = W ( τ ). (36) Since the inurance premium only repreent a tranfer of wealth, only the uage rate affect the reource allocation or ocial welfare. Thi point i important in undertanding the welfare effect of health inurance. (3) Comparion From (32) and (36), we ee that function W (.) can be viewed a a ocial welfare function of the ociety. By comparing the value of W( p ) and W ( τ ), we can evaluate 14
how health inurance affect the ocial welfare. Firt note that Lemma 1 till hold when conumer own the health care provider. We alo have the following lemma about the out-of-pocket price. Lemma 2: The conumer out-of-pocket price τ i trictly increaing in health care price p. It i alo trictly increaing in coinurance rate k when the marginal cot of health care c > 0. Proof: (See Appendix) Lemma 3: When the marginal cot of health care c > 0, the ocial welfare W ( τ ) i decreaing for τ τ. Proof: (See Appendix) Since the ocial welfare only depend on the out-of-pocket price and inurance reduce the out-of-pocket price, one might concern that inurance lead to exce conumption of health care in the equilibrium (Crew, 1969). Following Lemma 4 ugget that the out-of-pocket price would not be too low. Thi i conitent with Gaynor, Haa-Wilon and Vogt (2000) finding that the inviible hand are good hand in a health care market with inurance. Denote τ a the conumer out-of-pocket price when the health care price i given at p = c. Propoition 3: When c > 0, the equilibrium out-of-pocket price τ i higher than the 15
ocially optimal level τ, and lower than the health care price p in the abence of inurance. We have τ = p when c = 0. Proof: Since p > c, by Lemma 2 we have τ > τ. Since k < 1, by Lemma 3 we have τ < p when c > 0, and τ = p when c = 0. Q.E.D. Propoition 3 ugget that the market outcome with inurer i not the firt bet. There are two reaon that lead to thi outcome. Firt, the health care price tend to be too high becaue of the monopoly power of the provider. Second, there i a coordination failure between the health care provider and inurer. The health care provider profit decreae with the out-of-pocket price pecified in the inurance policie. But the inurer do not take thi into conideration, which caue a coordination failure. A a reult, the out-of-pocket price tend to be too high from the perpective of the ociety. Wang Hao (2010) point out that if health care provider are allowed to apply two-part pricing, which mean the health care and inurance integrate, the ocial welfare would be improved. Regarding how health inurance affect the ocial welfare, we have the following reult. Propoition 4: When the marginal cot of health care c > 0, health inurance improve the ocial welfare. When c = 0, the ocial welfare i unchanged. Proof: The firt part i by Lemma 3 and Propoition 3. We have hown in the proof of Propoition 1 that health inurance doe not affect the out-of-pocket price when c = 0. Hence the econd part follow immediately. Q.E.D. 16
Chiu (1997) model aume that the upply of health care i completely inelatic. Therefore health inurance doe not change the equilibrium out-of-pocket price. Hence the ocial welfare i unaffected by the inurance. The ame thing happen when the marginal cot of health care i zero, a dicued in Vaithianathan (2006). The general cae i that the upply of health care i elatic and/or the marginal cot of health care i poitive, a dicued in Wigger and Anlauf (2007) and the current paper. The ocial welfare i improved by the inurance in the general cae. An example: Suppoe conumer benefit from health care i 4 gq (, ) =. The 1 + 2q Bernoulli utility function i ux ( ) = exp( 0.8 x). The health tate follow a uniform ditribution on the interval [0,1]. The initial wealth I = ln(100000) and health care provider fixed cot F = ln(10). Table 1 how the equilibrium outcome under two different marginal cot c = 5.8 or 6.7, where p i the health care price in the abence of inurance, p i the health care price with inurance, k p i the out-of-pocket price pecified in an inurance policy, and τ i the ocial optimal out-of-pocket price. Table 1: Equilibria under Two Different Marginal Cot c p p k p τ 5.8 6.503 6.664 5.954 5.053 6.7 7.123 7.194 6.593 5.996 Figure 1 how how the health care provider profit (the upper two diagram) and 17
ocial welfare (the lower two diagram) change with the health care price under the two different marginal cot. When c = 5.8, health care provider chooe price p = 6.503 in the abence of inurance, and obtain the profit π = 0.007. When there i inurance, health care provider chooe price p = 6.664, and obtain profit π = 0.0165. The out-of-pocket price faced by conumer i k p = 5.954 > c, which i higher than the ocial optimal level τ = 5.053. When c = 6.7, health care provider chooe p = 7.123 in the abence of inurer, and obtain profit 0.0013. When there i inurance, the health care provider chooe price p = 7.194, and obtain profit π = 0.0042. The out-of-pocket price i k p = 6.593 < c, which i till higher than the optimal level τ = 5.996. From the lower two diagram, we ee that the uage rate with inurance i cloer to the ocial optimal level τ = 5.053 than that without inurance. That mean the inurance improve ocial welfare. Figure 1: The health care provider profit and ocial welfare 18
5. Concluding Remark It i often uggeted that health inurance not only create extra aymmetric information problem, but alo exaggerate conumer demand for health care. It ditort the reource allocation by inducing conumer purchae too much inurance coverage. However, Wigger and Anlauf (2007) how that from the ocial welfare criterion, conumer hould purchae more inurance coverage than they chooe to do in the market equilibrium. The current paper generalize their model with two health tate to a model with a continuum of health tate. It reaffirm the finding that health inurance typically hurt conumer but benefit health care provider. The market equilibrium with inurance actually entail an inefficiently low inurance coverage. The reult alo ugget that health inurance increae ocial welfare, at leat in the abence of aymmetric information. Hence the inurance bring in a 19
econd-bet outcome. Reference Arnott, Richard and Joeph Stiglitz, 1991. "Equilibrium in Competitive Inurance Market with Moral Hazard," NBER Working Paper 3588. Arrow, Kenneth J., 1963. Uncertainty and the Welfare Economic of Medical Care, American Economic Review, 53(5), pp. 941-973. Arrow, Kenneth J., 1968. The Economic of Moral Hazard: Further Comment, American Economic Review, 58(3), pp.537-539. Chiu, W. Henry, 1997. Health inurance and the welfare of health care conumer. Journal of Public Economic, 64, pp. 125-133. Crew, Michael, 1969. Coinurance and the Welfare Economic of Medical Care, American Economic Review, 59(5), pp. 906-908. Feldman, Roger and Bryan Dowd, 1991. A new etimate of the welfare lo of exce health inurance. American Economic Review,81, 297 301. Feldtein, Martin. S, 1973. The welfare lo of exce health inurance. Journal of Political Economy, 81, 251 280. Gaynor, Martin, Deborah Haa-Wilon, and William B. Vogt, 2000. Are Inviible Hand Good Hand? Moral Hazard, Competition, and the Second-Bet in Health care Market, Journal of Political Economy, 108(5), pp. 992-1005. Pauly, Mark V., 1968. The Economic of Moral Hazard: Comment, American Economic Review, 58(3), pp. 531-537. Pauly, Mark V., 1974. Overinurance and Public Proviion of Inurance: The Role of Moral Hazard and Advere Selection, Quarterly Journal of Economic, 88(1). Pp. 44-62. Png, Ivan and Hao Wang, 2010, Buyer Uncertainty and Two-Part Pricing: Theory and Application, Management Science, 56(2), pp. 334-342. Ma-Colell, Andreu, Michael D. Whinton and Jerry R. Green, 1995. Microeconomic Theory, Oxford. Wang, Hao, 2010, Health Care, Health Inurance and Health Management. (In Chinee) Journal of World Economy, 33(1), pp. 34-48 Wigger, Berthold U and Marku Anlauf, 2007. Do conumer purchae too much health inurance? The role of market power in health-care market. Journal of Public Economic Theory, 9(3), pp. 547-561. Vaithianathan, Rhema, 2006. Health inurance and imperfect competition in the health care market, Journal of Health Economic, 25, pp. 1193-1202. Appendix Proof of Lemma 1: The proof i imilar to the Propoition 1 of Png and Wang (2010). Since 20
the inurer are perfectly competitive, Condition (14) mut be binding. Given health care price p > 0, the inurer problem i: 0 k 1 ( + ) max U g( q( kp, ), ) I ( p kp ) q( kp, ) dg( ) kp q( kp, ) dg( ). (A1) Denote the objective function of the problem a U ˆ ( p, k), that i: ( ) U ˆ ( p, k ) = U g ( q ( kp, ), ) + I ( p kp ) q ( kp, ) dg ( ) kp q ( kp, ) dg ( ). (A2) With expreion (11) we have: Uˆ k q = U '. () p q( kp, ) dg() ( p kp) pdg() p q( kp, ) dg() kp. (A3) When k = 1, it become: Uˆ = U'. () p q( p, ) dg() p q( p, ) dg() k k = 1 ( '( )) ( ) ( '( ) ) ( '(.), ) = p E U E q E U q = p Cov U q. (A4) Differentiating both ide of Equation (4) with repect to give g qq q + gq = 0, or equivalently (A5) q gq = > 0. (A6) g qq Thu q i increaing in. Differentiating Equation (12) with repect to, and then with (11) we have: dx g q g q g kp = + = < 0. (A7) d q which implie that x i decreaing in. Since U i concave, U ' mut be increaing in. So both U ' and q are increaing in, we have Uˆ k k = 1 < 0. Thu reducing k from k = 1 increae conumer utility, which mean the competitive equilibrium coinurance rate k mut be le than 1. Q.E.D. 21
Proof of Lemma 2: For given health care price p > 0, an inurer problem i: 0 k 1 ( + + π ) max U gqkp ( (, ), ) I ( p kpqkpdg ) (, ) ( ) kpqkp (, ) dg ( ). (A8) Denote the target function a U ˆ ( p, k). The optimal k i determined by the firt-order condition: q U () pqkpdg (, ) ( ) p ( p kp ) dg ( ) pqkp (, ) dg ( ) = 0 kp. (A9) Suppoe the optimal coinurance rate et by the inurer i k 1 when the price of medical care i p 1. Conider the price of health care riing from p 1 to p 1 + ε, whereε > 0. If the inurer chooe the coinurance rate that leave conumer out-of-pocket price unaffected, i.e., chooe k = k1 ξ, ubject to ( k ξ )( p + ε ) = k p. 1 1 1 1 Now we check the effect of k marginal change on ˆ U. Subtituting (16) and (22) into (35) we have: 1 1 1 1 1 1 1 1 1 1 x( ) = gqkp ( (, ), ) + I F+ ( kp c) qkp (, dg ) ( ) kp qkp (, ). (A10) Thu after the price change the welfare level tay the ame, a a reult the ditribution of U () remain unchanged. In thi cae, Uˆ ( p, k ) q = U ()( p + ε)[ q( k p,) dg() ( p + ε k p ) ε dg() k ξ p = p 1 + ε 1 1 1 1 1 1 p = p 1 + k= k kp 1 ξ k= k1 q q( k1p 1,)] dg() = U ()( p 1+ ε) ε p= p dg() dg() 0 1+ ε >. (A11) kp k= k1 ξ So the conumer welfare will be improved if the inurer raie the coinurance rate. Thu when the health care price goe up, the out-of-pocket price alo goe up. For given an inurance policy ( Tk,, ) the health care provider olve problem: 22
The firt-order condition i: The correponding econd-order condition i: max ( p c) q( kp, ) dg( ) F. (A12) p qkp (, ) q( kp, ) dg( ) + ( p c) kdg( ) = 0. (A13) kp 2 qkp (, ) 2 qkp (, ) k 2* + ( p c) k dg( ) < 0. (A14) 2 kp ( kp ) Differentiating Equation (A13) with repect to k, we have: dp k = p + dk qkp (, ) c dg() kp. (A15) 2 qkp (, ) qkp (, ) 2* + ( p ck ) dg ( ) 2 kp ( kp ) One can infer the derivative of conumer out-of-pocket price kp with repect to k i: qkp (, ) c dg() dkp dp kp = p + k =. (A16) 2 dk dk qkp (, ) qkp (, ) 2* + ( p ck ) dg ( ) 2 kp ( kp ) Conider the lat term of (A16), it i eay to ee that the numerator i negative when c > 0. dkp By (A14) the denominator i alway negative. Thu 0 dk, where the equality hold only when c = 0. Q.E.D. Proof of Lemma 3: We prove the lemma in two tep. (i) When p = c, the health care provider profit i F. The inurer olve problem: 0 k 1 ( + ). (A17) max U g( q( kc, ), ) I ( c kc) q( kc, ) dg( ) kc q( kc, ) F dg( ) Denote the optimal coinurance rate a k objective function of the problem a Uk, ˆ ( ) i.e., = k, which i le than 1 by Lemma 1. Denote the 23