Research in Economics

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1 Reearch in Economic 64 (2010) Content lit available at ScienceDirect Reearch in Economic journal homepage: Health inurance: Medical treatment v diability payment Geir B. Aheim a, Anne Wenche Emblem b,c, Tore Nilen a, a Univerity of Olo, Norway b Sørlandet Hopital, Norway c Univerity of Agder, Norway a r t i c l e i n f o a b t r a c t Article hitory: Received 30 September 2009 Accepted 23 February 2010 Keyword: Health inurance Health demand Diability inurance We preent argument for treating health inurance and diability inurance in an integrated manner in economic analyi, baed on a model where each individual utility depend on both conumption and health and her income depend on her earning ability. When purchaing inurance, he may chooe a contract that offer le than full medical treatment. We find that high-ability individual demand full recovery and equalize utility acro tate, while low-ability individual demand partial treatment and cah compenation and uffer a lo in utility if ill. Our reult carry over to the cae where health tate are not obervable Univerity of Venice. Publihed by Elevier Ltd. All right reerved. 1. Introduction Random change in health, e.g. due to illne, affect a peron well-being in everal way. One way i the direct effect of health on well-being. But there are alo ome indirect effect. Firt, reduced health may affect the peron utility from conumption. Secondly, medical expene neceary to recover from illne reduce affordable conumption. Finally, illne may affect the ability to generate income. In thi paper, we argue that a proper treatment of health rik and health inurance hould take all thee effect of an illne into account, and we offer a theoretical model in which to do thi. In thi model, an individual utility depend on both conumption and health and her income depend on her earning ability. When purchaing health inurance, he may chooe to receive le than full medical treatment when ill, and upport conumption from a combination of cah compenation and her remaining earning ability. While health inurance and diability inurance are in fact integrated in a number of European countrie with public taxfinanced (ocial) inurance ytem, the economic literature ha treated the two rik a eparate problem, with the rik of medical expenditure to be covered by health inurance and the rik of loe in labour market productivity to be covered by diability inurance. The preent work i an attempt at correcting thi, putting a coordinating perpective on health and diability inurance. In particular, we expand the concept of health inurance to include not only coverage againt medical cot but alo againt permanent lo in earning ability, arguing that thee are two type of conequence of the ame rik, i.e. the rik of uffering a lo in health. Wherea health and diability inurance in mot European countrie i heavily ubidized in effect, a cro-ubidization take place from individual with high earning abilitie to thoe with low abilitie our analyi addree the quetion of We are grateful to Agnar Sandmo and Sören Blomquit for helpful comment on earlier verion. Part of Aheim and Nilen reearch i financed by the Reearch Council of Norway through HERO The Health Economic Reearch Programme at the Univerity of Olo. They are both aociated with the ESOP centre at the Univerity of Olo. The ESOP centre i alo financed by the Reearch Council of Norway. Emblem gratefully acknowledge financial upport from the SiS Programme at the Univerity of Agder. Correponding addre: Department of Economic, Univerity of Olo, P.O.Box 1095, Blindern, NO-0317 Olo, Norway. Fax: addree: [email protected] (G.B. Aheim), [email protected] (A.W. Emblem), [email protected] (T. Nilen) /$ ee front matter 2010 Univerity of Venice. Publihed by Elevier Ltd. All right reerved. doi: /j.rie

2 138 G.B. Aheim et al. / Reearch in Economic 64 (2010) what the outcome would be without any tranfer. We find that individual with low earning abilitie indeed trade off health for conumption. Letting inurance contract offer individual variou combination of medical treatment and cah compenation of income lo if they become ill, we find, in particular, that high-productivity individual chooe contract providing full medical treatment while low-productivity individual chooe contract offering partial medical treatment and partial compenation for lo in earning. Low-productivity individual conequently chooe not to fully recuperate from an illne but rather receive cah payment that partly offet the income lo due to partial impairment. 1 Moreover, even in thi etting where there i ymmetric information about health rik and health tate, uch low-productivity individual end up being le than fully inured, in the ene that they have lower utility when ill than when healthy. Thee reult have intereting policy implication. Wherea there i much focu among many policy maker on the iue of providing health inurance that cover all individual in a fair and uniform manner, our analyi point to reaon for offering a menu of health inurance contract, where cah compenation may ubtitute for the right to full recovery. 2 In the cae where health inurance cheme are alo ued a a mean of reditributing income, our analyi indicate that thi iue i related to reditribution in cah, i.e., through the tax ytem: a low-productivity individual may actually prefer to receive upport from government in the form of cah rather than in the form of improved health, ince better health ha a maller effect on conumption for low-killed people than for high-killed one. 3 Like the traditional literature on health inurance, we focu on illnee for which a treatment i available that fully retore pre-illne health and ability. However, in other work, individual deire to retore health i taken for granted. 4 In Marchand and Schroyen (2005), for example, there i no lo of health for an individual who fall ill, only a lo of time caued by illne. In their model, high-productivity individual get well immediately at a private practice, while low-productivity one uffer a time lo while waiting in the public health-care ytem. Thi time lo contitute an inefficiency, wherea the outcome in our model i efficient, with low-productivity individual getting compenation in cah intead of full treatment. In other work where non-monetary conequence of illne are taken into account, it i aumed that utility i tate dependent and that health i either irreplaceable or not retorable. 5 In thi paper, we allow for both monetary and nonmonetary conequence of illne without impoing aumption either that health will alway be fully retored or that it i irreplaceable. Thu, we provide a bridge between the health-inurance literature, which typically take only monetary conequence of illne into account, and the diability-inurance literature potulating that health i irreplaceable or nonretorable. Our model ha the following crucial feature. Firt, we make the reaonable aumption that an individual productivity i affected by her health: If he uffer an illne and health i not fully retored, then her productivity will be negatively affected by the illne. The effect i that individual with low full productivity have lower incentive for retoring health and therefore will tend to prefer contract with cah compenation for illne. 6 Second, we ue a bivariate formulation of utility that allow for interaction between conumption and health. In particular, we make the aumption that the two are complement, i.e., that an individual marginal utility of conumption i increaing in health. 7 Our main analyi take place in a world of ymmetric information about health rik and health tate, while the individual ability may be private information. However, we how that our finding hold alo in a ituation where an individual health i non-verifiable, i.e., when inurer face problem of ex-pot moral hazard. In fact, if ex-pot moral hazard i a problem, then integrating medical inurance (with in-kind proviion of medical treatment) and diability inurance (with cah compenation) reduce the inured individual incentive to falely claim to be ill when in good health; in other word, integration induce elf-election. 8 The outline of thi paper i a follow. The model i preented in Section 2 while our main finding are derived in Section 3. We dicu the cae of ex-pot moral hazard in Section 4. Our reult are dicued in a concluding Section 5. Proof are relegated to an Appendix. 1 Our finding are in accordance with the empirical obervation that individual with le chooling are more likely to be diabled than thoe with more chooling [e.g., Haveman and Wolfe (2000)] and may provide an explanation for thi correlation in addition to thoe traditionally put forward. 2 Of coure, there are normative argument for giving all individual the right to full recovery whenever feaible; uch iue are not dicued here. 3 Thi theme ha been picked up by Fleurbaey (2006), who note it conequence for dicuion of health and equity: the poor being le healthy than the rich may in part be a reult of individual preference. 4 One exception i Byrne and Thompon (2000), who argue that, when the probability of ucceful treatment i mall, the inured may be better off with cah compenation if ill, rather than going through the treatment. We hall aume that full treatment, if choen, provide full retoration of health with certainty. 5 Analye baed on health being non-retorable include Zeckhauer (1970), Arrow (1974), Vicui and Evan (1990), Evan and Vicui (1991), and Frech (1994). Health i irreplaceable if individual value retored health lower than pre-illne health; ee Cook and Graham (1977) and Schleinger (1984). 6 In related work by Jack and Sheiner (1997) and Koç (2004), the demand for health inurance i dicued in a ituation where, like in our model, the conumption of health care i endogenou. An important difference, however, i that thee author diregard the effect of illne on an individual earning ability. 7 Such interaction i tandard in analye of diability inurance where it i called tate-dependent utility; ee reference in footnote 5. Our aumption of complementarity between conumption and health i in line with the reult from the empirical tudy by Finkeltein et al. (2008). 8 Aheim et al. (2003) analyze a verion of our model with aymmetric information about an individual probability of illne, i.e., a ituation where inurer face problem of advere election.

3 G.B. Aheim et al. / Reearch in Economic 64 (2010) The model Conider an individual who ha preference over conumption, c, and health, h. The individual face exogenou uncertainty with repect to her tate of health. She may either be healthy, which correpond to tate 1, or he may fall ill, which correpond to tate 2. The two tate are mutually excluive, jointly exhautive, and verifiable. In tate 1, the level of health i normalized to 1: h 1 = 1. In tate 2, the individual i ill and, without any medical treatment, uffer a complete lo in health: h 2 = 0. Health if ill may, however, be partially or fully retored (intantly and with certainty) if the individual receive medical treatment: t [0, 1], i.e., treatment i aumed to be a continuou variable. Medical treatment leading to full recovery i available at cot C, while treatment at cot tc lead to partial recovery. Health in tate 2 i henceforth meaured by the fraction of C pent on treatment, that i, h 2 = t. Conumption in the two tate i denoted c 1 and c 2, repectively. The probability of falling ill i known to the individual and given by π (0, 1). The individual eek to maximize the von Neumann Morgentern expected utility (1 π)u(c 1, 1) + πu(c 2, t), where u(c, h) i a Bernoulli utility function. We aume that u : R 2 + R i twice continuouly differentiable, trictly concave, and atifie: (c, h) R 2 ++, u c > 0 and u h > 0, where partial derivative are denoted by ubcript. In particular, a trictly concave u implie that the individual i rik avere. Moreover, health and conumption are aumed to be complement in utility: u ch > 0. Thi aumption i in accordance with the empirical reult of Finkeltein et al. (2008) and implie that individual take more pleaure in conumption when health i good than when health i poor. We alo aume that u c (c, h) a c 0 whenever h > 0, u h (c, h) a h 0 whenever c > 0, and u c (c, h) or u h (c, h) a c 0 and h 0. Note that our aumption on u imply normality. Strict concavity implie that marginal utility from treatment i higher at a low treatment ratio than at a high one, and that an intermediate level of treatment i preferred to an uncertain propect of either complete or zero treatment with the ame expected cot. Since leiure i not included in the utility function, we implicitly aume that leiure i contant (and thu, labour upply i fixed) acro tate. There exit a competitive inurance market in which profit maximizing inurer offer inurance at an actuarially fair premium. Information about the individual probability π of falling ill, which dieae he i uffering from and, conequently, the aociated cot of treatment, i ymmetrically ditributed among the market participant. Moreover, health tate i verifiable, o that inurance can be offered contingent on it. (A ituation where health tate i non-verifiable i dicued in Section 4.) The individual ability, i.e., her inherent capacity to generate earning, i denoted A (>0). By normalization, A alo denote the individual labour earning when well, i.e., in tate 1. Labour earning in tate 2 are proportional to the amount pent on medical treatment: By pending tc, the individual obtain an ability equal to ta in tate 2. The analyi doe not require inurance companie to know the individual ability; hence A may be private information. Illne entail two type of lo: financial and non-financial. The financial lo include reduced earning due to lower ability (productivity) and medical expenditure. The non-financial lo i in the form of reduced utility due to poorer health. Health if ill i however endogenou, and o the ize of the non-financial lo i alo endogenou. Indeed, if the individual chooe treatment leading to full recovery (t = 1), then he uffer a financial lo only, viz., the cot of treatment, while if he chooe partial treatment (0 < t < 1), then he uffer both a financial and a non-financial lo. The individual inurance deciion take place prior to her knowing which tate ha occurred. Her budget contraint in tate 1 and 2 are repectively given by: c 1 + πi = A, and c 2 + πi + tc = ta + I, where πi i the inurance premium to be paid in both tate of the world in order to receive compenation equal to I if ill. It follow that I = tc + c 2 ta + A c 1, that i, the inurance provide for both medical expenditure, tc, and a cah compenation, c 2 ta + A c 1. 9 By eliminating I from the two budget contraint, it follow that the individual i contrained by: A c 1 = π[tc + (c 2 ta + A c 1 )] (1) when ex ante making her choice of c 1, c 2, and t. In the following, we characterize the individual demand for inurance with repect to both level and type of coverage. In particular, we analyze how the individual ability A influence her choice of compenation: whether to be compenated in the form of health retoration, i.e., medical treatment, and/or in the form of cah, i.e., compenation for lo in income due to incomplete recovery. 9 Since the premium πi = A c1 mut be paid in both tate, dipoable income net of the premium equal ta (A c 1 ) if no cah compenation i received. Hence, the cah compenation i c 2 [ta (A c 1 )].

4 140 G.B. Aheim et al. / Reearch in Economic 64 (2010) P high A low A t= 1 t Fig. 1. The ingle-croing property. 3. Analyi Treatment leading to a health level t i available at a cot tc when ill. For the purpoe of our analyi, however, we ak more generally what i the maximum utility achievable if the individual ha to pay P( 0) for treatment t: U(t, P; A) := max {(1 π)u(c 1, 1) + πu(c 2, t)} (c 1,c 2 ).t. (1 π)c 1 + π(c 2 + P) = (1 π)a + πta, where, for the purpoe of defining and analyzing the U function, we allow t > 1, o that U : R ++ [0, (1/π (1 t))a) R ++ R. Solving thi problem, we find the conumption function in each tate: atifying (c 1 (t, P; A), c 2 (t, P; A)) R 2 ++, u c (c 1 (t, P; A), 1) = u c (c 2 (t, P; A), t) and the budget contraint in (1). Conumption in each tate i a function of treatment t (i.e., the degree of recovery in tate 2), the price of treatment P, and ability A. In optimum, the individual marginal utility of conumption i equal acro tate. We write the utility function U a: U(t, P; A) = (1 π)u(c 1 (t, P; A), 1) + πu(c 2 (t, P; A), t), and have that U i trictly increaing in t, trictly decreaing in P, and trictly increaing in A. The marginal rate of ubtitution between t and P, given A, i: U t MRS(t, P; A) = U P = u h(c 2, t) u c (c 2, t) + A, where the econd equality follow from the envelope theorem and Eq. (2). A hown in the Appendix, the utility function U ha the ingle-croing property, i.e., MRS i increaing in A, and feature diminihing willingne to pay for treatment, i.e., it i quai-concave. Thee two propertie of U are illutrated in Fig. 1: diminihing willingne to pay for treatment implie that indifference curve are trictly concave, and the ingle-croing property implie that the lope of indifference curve are teeper, the higher i A. Due to the diminihing willingne to pay for treatment, an individual being faced with the poibility of purchaing treatment t at cot P = tc, contrained by t 1, will have a unique level of treatment t(a) maximizing U(t, tc, A). Furthermore, due to the ingle-croing property, t(a) i (weakly) increaing in A. In fact, whenever 0 < t(a) < 1, t(a) i determined by MRS(t, P; A) = C, i.e., marginal willingne to pay for treatment equal marginal cot of treatment. It follow that t(a) i trictly increaing in A when 0 < t(a) < 1. We have that t(a) = 1 for all A A, where A i that level of ability, unique by the ingle-croing property, for which the indifference curve through (1, C) ha lope C, o that uncontrained maximization of U(t, tc, A ) lead to t = 1. We define A by MRS(1, C; A ) = C. (2) (3) (4) (5)

5 G.B. Aheim et al. / Reearch in Economic 64 (2010) P C higha low A t = 1 t Fig. 2. The optimal level of treatment. We have that A < C, ince MRS(1, C; A) > A. Moreover, it follow from (1) and (2) that c 1 = c 2 = A πc when t = 1 and P = C, implying that t = 1 i not feaible when A < πc. Finally, ince u c (c, h) a c 0 whenever h > 0, it follow that MRS(1, C; A)/ t 0 a A πc. Now, at A = πc, full treatment will not be choen. It follow that A > πc. The optimum level of treatment i illutrated in Fig. 2 for two different value of ability: A l < A h = A. In that figure, a high-ability individual indifference curve in (t, P)-pace i tangent to the marginal-cot line for t = 1, while that of a low-ability individual i tangent to the marginal-cot line for ome t (0, 1). In the propoition below, we make ue of the following aumption on u: u h (c, 1) u c (c, 1) u h(c, t) u c (c, t), if 0 < c < tc and 0 < t < 1. (6) Any concave homothetic function of c and h atifie thi, but the aumption i alo atified by other demand ytem. 10 The propoition tate that the individual utility i contant acro tate if he chooe full treatment. Her utility if ill i lower than that if well if he chooe le than full treatment. Moreover, with full treatment, he will not receive any cah payment in addition to what i required to pay for treatment, while in the cae of partial treatment, her compenation will exceed the amount pent on medical treatment. The obervation in the text above partially prove the propoition; the proof i completed in the Appendix. Propoition 1. Aume that the condition in (6) hold. Then there exit a level of ability, A, where πc < A < C, uch that: 1. If the individual ability A i high, in particular, if A A, then her optimum level of treatment i the maximum one and doe not vary with A: t(a) = 1. Moreover, her level of conumption i identical in the two tate: c 1 (1, C, A) = c 2 (1, C, A) = A πc, a i her utility: u(c 1, h 1 ) = u(c 2, h 2 ) = u(a πc, 1). Her inurance coverage i in the form of medical treatment only. 2. If, however, the individual ability A i low, in particular, if 0 < A < A, then her optimal level of treatment i poitive but le than one, 0 < t(a) < 1, and increaing in A: t(a)/ A > 0. Moreover, her level of conumption if ill i lower than if healthy: c 2 (t(a), t(a)c, A) < c 1 (t(a), t(a)c, A), and her utility if ill i lower than if healthy: u(c 2, h 2 ) < u(c 1, h 1 ). Her inurance coverage i partly in the form of medical treatment and partly in the form of cah. Propoition 1 can be illutrated by the following Bernoulli utility function: u(c, h) = c r h, with r > 0, > 0 and r + < 1, atifying all our aumption; ee the Appendix for detailed calculation. In thi cae, A = r + π r + C, and if A < A, then the following expreion obtain for the cah compenation: c 2 t(a)a + A c 1 = r + (1 π) t(a)( A ) A. (7) 10 The multiplicatively eparable pecification u(c, h) = v(c)w(h), where v, w > 0 and v, w < 0, which i a one-period verion of the utility function propoed by Bleichrodt and Quiggin (1999), atifie all our aumption. The pecification u(c, h) = f (c + ah) + bh, where f, a, b > 0 and f < 0, ued by Ma and Riordan (2002), atifie all our aumption (including (6)), except that it ha u ch < 0. See Rey and Rochet (2004) for a dicuion of variou bivariate utility function ued in health economic.

6 142 G.B. Aheim et al. / Reearch in Economic 64 (2010) Thi mean that the critical level A increae with both a greater probability π of falling ill and a greater cot of treatment, C. Furthermore, for given value of π and C (and hence, for a given A ), the individual level of ability, A, ha an ambiguou effect on the ize of the cah compenation. On the one hand, provided that A < A, a maller A lead to a greater relative importance of the cah compenation through the term A A. On the other hand, an individual with a maller A ha a maller initial endowment and will chooe an inurance contract with a lower level of total compenation. Thi moderating effect on the ize of the cah compenation i reflected by the term t(a), which decreae with a reduction in A when A < A. 4. Ex-pot moral hazard We have, in Propoition 1, hown that an ill individual with ability A lower than the critical level A receive partial treatment, t(a) < 1, and, in addition, a poitive cah compenation: c 2 (A) t(a)a + A c 1 (A), where we from now on write c 1 (A) = c 1 (t(a), t(a)c, A) and c 2 (A) = c 2 (t(a), t(a)c, A). If, contrary to what we aumed above, the two tate (healthy/ill) are not obervable, the availability of uch a diability payment may tempt the individual to claim that he ha fallen ill, although he i in fact in good health. In thi ection, we how that our analyi goe through even if we allow for uch ex-pot moral hazard, 11 provided that (i) the cah compenation i paid only in combination with treatment; and (ii) the diutility of receiving treatment while healthy i ufficiently great. Conequently, in order to prevent the individual from falely claiming to be ill, he hould uffer a lo in expected utility from undergoing redundant medical treatment. Moreover, the diutility hould at leat balance the gain in expected utility from maquerading a ill. To enure that a healthy individual with ability A doe not falely claim to be ill, we mut conider the poibility that he not only mirepreent her health tate but alo her ability, in order to receive the higher cah compenation deigned to be paid to an individual with a different ability A. Uing the obervation that an individual ha no incentive to lie about her ability unle he intend alo to mirepreent her health tate, we provide, in the Appendix, a proof of the following reult. Propoition 2. The ex-pot moral hazard of a healthy individual maquerading a ill in order to obtain cah compenation doe not contitute an incentive problem if, for any true ability A and claimed ability A, the additional utility that A obtain from the cah compenation, c 2 (A ) t(a )A + A c 1 (A ), to be paid to A in cae of illne, doe not exceed the diutility that A uffer from undergoing, when healthy, redundant treatment at the level t(a ) that A i entitled to. It i clear from thi reult that a healthy individual falely claim to be ill only after having purchaed the optimal inurance contract of ome ability A maller than A. The reaon i that, according to Propoition 1, for any ability A A, the optimal inurance contract include no cah compenation, implying that the diutility from undergoing redundant treatment will dominate. On the other hand, if A < A, then there i a poitive cah compenation. What can be aid about how thi cah compenation varie with A for 0 < A < A? To invetigate thi quetion, it might be intructive to look at the pecial Cobb Dougla cae conidered at the end of Section 3. It follow from the expreion for the cah compenation in (7) that the level of claimed ability, A, ha an ambiguou effect on the ize of the cah compenation. While, with A < A, a maller A lead to a greater relative importance of the cah compenation through the term A A, there i a moderating effect through the term t(a ), which reflect that a maller A lead to an inurance contract with a lower level of total compenation. Hence, provided that the diutility of receiving treatment while healthy doe not decreae ignificantly with a maller t, and thu with a maller A, thi diutility exceed, for any true ability A and claimed ability A, the additional utility obtained from the cah compenation paid to A in cae of illne. When health tate i not verifiable, i.e., when ex-pot moral hazard i a problem, the individual will have an incentive to maquerade a ill in order to acquire a cah compenation. However, when cah compenation i made conditional on medical treatment, we have hown that the individual incentive to maquerade i reduced ince he will uffer a diutility from receiving redundant treatment. The ex-pot moral hazard problem aociated with cah compenation i hence olved in our model through the integration of treatment for illne and payment for diability. The lack of uch integration can help explain why private market for diability inurance are of little empirical ignificance. Naturally, one may argue that even if treatment and cah compenation were not integrated, then information about whether an individual i ill could be obtained if the inurer offering diability inurance could require information from the inurer offering medical inurance. In thi cae, information on the (contractually) adequate level of treatment a well a the level of treatment actually undertaken i required. It follow that the informational cot would be higher relative to a ituation in which the two type of inurance are integrated. 11 Ex-pot moral hazard refer to the effect of inurance on the inured individual incentive to reveal her true health tate (i.e., the inured individual know the tate of the world, while the inurer doe not, or verification of health tate i too cotly for the inurer). The analyi of ex-pot moral hazard wa pioneered by Spence and Zeckhauer (1971). The idea that in-kind tranfer, uch a medical treatment, can alleviate ex-pot moral hazard i due to Nichol and Zeckhauer (1982).

7 G.B. Aheim et al. / Reearch in Economic 64 (2010) Dicuion Our focu ha been on how an individual inherent ability at full functionality (i.e., when healthy) influence her exante choice of inurance contract. Inurance allow the individual to allocate income between the two health tate prior to knowing which tate occur and, if falling ill, between conumption and health. It i of no importance, in a world of ymmetric information, whether the coverage for medical cot i paid in cah intended to cover medical bill, or directly in the form of medical treatment. The individual ex-ante deciion concerning what level of treatment to chooe i unaffected by the way he i compenated; the fundamental deciion concern to what extent health i to be retored. 12 However, a dicued in Section 4, if health tate i not eaily verifiable, then it become eential whether medical expenditure are compenated in cah or in kind. When information about health tate i aymmetric, integration of a cah compenation of income lo and an in-kind compenation of medical expenditure reduce the individual incentive to falely claim to be ill. Our finding are driven by the fact that the potential lo in income i larger, the higher the ability. Thi implie that the price of the two type of contract differ depending on the individual ability. The higher the potential income lo due to reduced ability, the cheaper i the contract offering indemnity in kind (i.e., treatment), compared to a contract offering cah compenation of income lo. Thu, the cot benefit ratio on medical treatment i lower the higher the level of ability at full functionality. Appendix A.1. Propertie of U The ingle-croing property: By differentiation in (3), we find that MRS i increaing in A: MRS A = [ ] uh (c 2 ( t, P, A), t) + A > 1, A u c (c 2 ( t, P, A), t) ince u cc < 0, u ch > 0, and c 2 / A > 0. Thi i the ingle-croing property. Diminihing willingne to pay for treatment: We need to how that U i trictly quai-concave a a function of t and P. Thi i done by demontrating that, if (t, P ) and (t, P ) are different combination yielding the ame utility level given A, then any interior convex combination (t, P) = (αt + (1 α)t, αp + (1 α)p ), 0 < α < 1, will yield a trictly higher utility level. Accordingly, aume U(t, P, A) = U(t, P, A), and introduce ome notation: c = 1 c 1(t, P, A) c 1 c 1(t, P, A) c = 2 c 2(t, P, A) c 2 c 2(t, P, A). Alo, let (c 1, c 2 ) = (αc 1 + (1 α)c, 1 αc 2 + (1 α)c 2 ). Since (c, 1 c ) 2 atifie the ex-ante contraint (1) given (t, P, A) and (c, 1 c ) 2 atifie contraint (1) given (t, P, A), it follow that alo (c 1, c 2 ) atifie contraint (1) given (t, P, A), implying that (c 1, c 2 ) i feaible. Hence, U(t, P, A) (1 π)u(c 1, 1) + πu(c 2, t) > (1 π)[αu(c 1, 1) + (1 α)u(c, 1)] + 1 π[αu(c, 2 t ) + (1 α)u(c, 2 t )] = αu(t, P, A) + (1 α)u(t, P, A) where the firt inequality follow ince (c 1, c 2 ) i feaible, and the econd inequality follow ince u i trictly concave. A.2. Proof of Propoition 1 Part (1). Given the obervation in the text prior to the propoition, it remain to how that the individual utility i contant acro tate, and that he ha inurance coverage in the form of medical treatment only. 12 Arrow (1963) mention three different way in which cot of medical care can be covered in an inurance contract: payment directly in medical ervice, a fixed cah payment, and a cah payment that cover the actual cot involved in providing the neceary medical treatment. In a perfect market, individual wihing to receive medical treatment would be indifferent between a payment directly in the form of medical treatment and it cah equivalent.

8 144 G.B. Aheim et al. / Reearch in Economic 64 (2010) Contant utility acro tate follow ince c 1 = c 2 = A πc and h 1 = h 2 = 1. Since cah payment equal c 2 ta+a c 1 (ee footnote 9), it follow that cah payment i zero. Part (2). Given the obervation in the text prior to the propoition, it remain to how that the individual utility if ill i lower than if healthy, and that he receive a poitive cah compenation if ill. By the definition of A, 0 t(a) < 1 whenever 0 < A < A. Moreover, ince u h (c, h) a h 0 whenever c > 0, and u c (c, h) or u h (c, h) a c 0 and h 0, it follow from A > 0 and Eq. (3) that MRS > C if t i ufficiently mall; hence, t(a) > 0. Now, the ingle-croing property implie that dt(a)/da > 0. From Eq. (2) and the propertie of u, it follow that c 1 > c 2, ince h 1 = 1, and h 2 = t(a) < 1. Thi in turn implie that u(c 1, 1) > u(c 2, h 2 ). To how that cah payment i poitive, i.e., that c 2 ta + A c 1 > 0, we tart out with the obervation that t(a) i determined by MRS = C whenever 0 < t(a) < 1. Uing the expreion for MRS in (3), we have that the marginal willingne to pay for treatment equal the marginal cot of treatment: u h (c 2, t)/u c (c 2, t) + A = C. In the hypothetical cae where treatment were available alo if healthy, or inverely, where health could be old at price C A, the acce to actuarially fair inurance would imply the ame level of health in both tate. Since thi i not the cae, it i a binding contraint that healthy individual cannot ell health at price C A, implying that: u h (c 1, 1)/u c (c 1, 1) < C A = u h (c 2, t)/u c (c 2, t). Hence, effectively, the relative price of health in term of conumption i lower if healthy than if ill. It follow by aumption (6) that tc 1 c 2. Moreover, contraint (1) entail that c 1 A πtc if and only if c 2 ta πtc. Therefore, c 1 A πtc lead to the following contradiction: tc 1 t (A πtc) > ta πtc c 2. Thu, we have that c 1 < A πtc and c 2 > ta πtc. Thi in turn mean that c 1 A < c 2 ta, or c 2 ta + A c 1 > 0. A.3. Calculation for the Cobb Dougla cae The following Cobb Dougla function i a Bernoulli utility function that atifie all aumption lited in Section 2, a well a the condition in (6): u(c, h) = c r h, with r > 0, > 0 and r + < 1. With thi function, it i poible explicitly to calculate A. We have that MRS (1, C; A) = u h(c 2, 1) u c (c 2, 1) + A = u h(a πc, 1) u c (A πc, 1) + A = (A πc) + A, r where the firt equality follow from (3), the econd equality follow ince c 2 = A πc when t = 1 and P = C, and the third equality follow ince u h (c, h) u c (c, h) = r c h when u i given by the Cobb Dougla function above. Since A i defined by MRS (1, C; A ) = C, we can find A by olving r (A πc) + A = C, which implie that A = r + π r + C. Since, for A < A, t i determined by MRS = C, we get, by invoking Eq. (3) and (A.1), that r c2 + A = C. t Moreover, by letting the cah compenation, c 2 ta + A c 1, be denoted by x, it follow from Eq. (1) that x c 2 + ta = π[tc + x]. We now have that (1 π)x = πtc + c 2 ta = πtc + r t(c A) ta = 1 [ ] (r + π)c (r + )A t = r + ( A ) A t, where the firt equality follow from (A.4), the econd equality follow from (A.3), and the fourth equality follow from (A.2). We have thereby hown that the cah compenation can be expreed a in (7). (A.1) (A.2) (A.3) (A.4)

9 G.B. Aheim et al. / Reearch in Economic 64 (2010) A.4. Proof of Propoition 2 Denote by v(c, t) the (direct) diutility of receiving treatment t while healthy and conuming c; to be precie, v(c, t) i the difference between the utility derived from conumption c when healthy and not receiving unneceary treatment and the utility derived from conumption c when healthy and receiving unneceary treatment t. Aume that v atifie, (c, t) R 2 ++, v(c, t) > 0 and v t 0. To prevent problem caued by ex-pot moral hazard, the following inequality mut hold, for any true ability A and claimed ability A : (1 π)u(c 1 (A), 1) + πu(c 2 (A), t(a)) (1 π) [ u(c 2, 1) v(c 2, t(a )) ] + πu(c 2 (A ) + t(a )(A A ), t(a )). (A.5) Here, c = 2 c 2(A ) + (A t(a )A ) repreent the conumption that a healthy individual with ability A receive having purchaed the optimal inurance contract of an individual with ability A and maquerading a ill, while c 2 (A )+t(a )(A A ) i the conumption that an ill individual with ability A receive having purchaed the optimal inurance contract of ability A and truly claiming to be ill. An individual with ability A > (<) A generate higher (lower) earning and can, therefore, utain a higher (lower) level of conumption than can an individual with ability A. In condition (A.5), we allow the claimed ability A to take any value, including the real ability A. Hence, we do not require that an individual alo lie about her ability when he mirepreent her health tate, but we allow for thi poibility. Of coure, a miinterpretation of ability mut, if it occur, take place before the individual know whether he ha fallen ill or not; thi i reflected by the lat term on the right-hand ide of condition (A.5). To find a ufficient condition for (A.5) to hold for any true ability A and any claimed ability A, the following obervation i ueful: (c 1 (A), c 2 (A), t(a)) maximize expected utility over all triple (c 1, c 2, t) atifying (1); in particular, (1 π)u(c 1 (A), 1) + πu(c 2 (A), t(a)) (1 π)u(c 1, 1) + πu(c 2(A ) + t(a )(A A ), t(a )), (A.6) where c = 1 c 1(A ) + (A A ) i the conumption that a healthy individual with ability A receive having purchaed the optimal inurance contract of ability A and not maquerading a ill. Thi mean that an individual with ability A doe not lie about her ability unle he intend alo to mirepreent her health tate. The increae in conumption that an individual with ability A, having purchaed the inurance contract of ability A, obtain by maquerading a ill, c 2 c 1, i equal to the cah compenation deigned to be paid to an individual with ability A : c 2 c 1 = c 2(A ) t(a )A + A c 1 (A ). Moreover, by (A.6), it i a ufficient condition for (A.5) to be atified, for any true ability A and claimed ability A, that u(c 2, 1) u(c 1, 1) v(c 2, t(a )) hold for any true ability A and any claimed ability A. Thi complete the proof of Propoition 2. Reference Arrow, K.J., Uncertainty and the welfare economic of medical care. American Economic Review 53, Arrow, K.J., Optimal inurance and generalized deductible. Scandinavian Actuarial Journal Reprinted in: Collected Paper of Kenneth J. Arrow, vol. 3: Individual Choice under Certainty and Uncertainty, Blackwell (1984), pp Aheim, G.B., Emblem, A.W., Nilen, T., Deductible in health inurance: pay or pain? 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