Journal of Economic Behavior & Organization

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1 Journa of Economic Behavior & Organization 85 ( Contents ists avaiabe at SciVerse ScienceDirect Journa of Economic Behavior & Organization j ourna ho me pag e: ebo Heath insurance, treatment pan, and deegation to atruistic physician Ting Liu a,, Ching-to Abert Ma b, a Department of Economics, Stony Brook University, Stony Brook, NY 794, USA b Department of Economics, Boston University, 27 Bay State Road, Boston, MA 225, USA a r t i c e i n f o Artice history: Received 4 October 2 Received in revised form 8 November 22 Accepted 9 November 22 Avaiabe onine xxx JEL cassification: D82 I3 Keywords: Optima contract Deegation Atruistic physician Commitment a b s t r a c t We study deegating a consumer s treatment pan decisions to an atruistic physician. The physician s degree of atruism is his private information. The consumer s iness severity wi be earned by the physician, and aso wi become his private information. Treatments are discrete choices, and can be combined to form treatment pans. We distinguish between two commitment regimes. In the first, the physician can commit to treatment decisions at the time a payment contract is accepted. In the second, the physician cannot commit to treatment decisions at that time, and wi wait unti he earns about the patient s iness to do so. In the commitment game, the first best is impemented by a singe payment contract to a types of atruistic physician. In the noncommitment game, the first best is not achieved. A but the most atruistic physician earn positive profits, and treatment decisions are distorted from the first best. Pubished by Esevier B.V.. Introduction Physicians have different practice styes. Patients with simiar medica conditions often get treated differenty. Practicestye variations are present across speciaties such as obstetrics (Epstein and Nichoson (29, cardioogy (Moitor (22, and primary care (Grytten and Sørensen (23. Practice variations can be very costy if physicians deviate from using cost-effective treatments. In fact, Pheps and Parente (99 estimated an annua wefare oss vaued at US $33 biions due to hospitaization rate variations. Current theory expains practice variation by information diffusion and physician earning (Pheps (992, Pheps and Mooney (993. Under this hypothesis, practice variation shoud be smaer within markets than between markets, and shoud diminish over time. However, Epstein and Nichoson (29 find the opposite: for risk-adjusted cesarean-section rates, within-market variation is twice that of between-market variation; amost 3% of the variation is due to time-invariant, physician-specific factors other than experience, gender, race, and where a physician received residency training. This timeinvariant, physician-specific factor ikey refects physicians intrinsic preferences about the appropriate treatments for their patients. In this paper, we mode practice styes by physicians heterogenous preferences towards their patients. Physicians are partiay atruistic, their utiities being weighted sums of profits and patients utiities. Physicians have mutipe treatment options, and patients iness severities differ. Physicians tasks are to match patients with different severities to Corresponding author: Te.: ; fax: E-mai addresses: ting.iu@stonybrook.edu (T. Liu, ma@bu.edu (C.-t.A. Ma. Te: ; fax: /$ see front matter. Pubished by Esevier B.V.

2 8 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( different treatment pans. However, physicians possess private information about patient s iness severity, and their treatment decisions are noncontractibe. We study the foowing questions. What is the efficient treatment pan when there are mutipe treatment options? Under what conditions can payment contracts impement the efficient treatment pan? If the efficient treatment pan is not impemented, what are the distortions? Finay, how are insurance premiums affected? Since Arrow (963 observed the importance of atruistic physicians in the heath market, the atruistic-physician assumption has been widey adopted. 2 Whie most papers in the iterature have assumed that the degree of atruism is given and known, we go beyond the fixed-atruism assumption and aow the physician to be of many different types, this being his private information. 3 An atruistic physician may trade off his own profit against the consumer s utiity. This forma construct does permit an utra atruistic physician to run a financia oss to subsidize treatments. This, however, is unreaistic. Being an economic agent, a physician must face some financia constraints, so we assume that a physician must on average earn a minimum profit. We do aow a physician to sustain some financia oss sometimes, but he must expect to earn a minimum profit on average. We normaize this minimum expected profit to zero. 4 The physician practice-stye issue rests on an environment in which many treatment options for an iness are avaiabe. We mode mutipe treatment options in the simpest way. A ess costy treatment succeeds in eiminating a patient s iness disutiity with a ower probabiity. A second treatment is more costy, but succeeds with a higher probabiity. In contrast to papers in the iterature, we et physicians combine treatments. For exampe, a high-cost treatment may be used after a ow-cost treatment fais to eradicate the iness. The physician decides on sequences of treatments, which we ca treatment pans or protocos. Our main findings are the foowing. First, the first-best treatment pan prescribes a conservative approach under a costconvexity assumption, which says that the higher the success probabiity, the higher is the cost per unit success probabiity. If the severity is ow, then no treatment is used; if it is of medium vaue, a ow-cost treatment wi be used; if it is high, then the ow-cost treatment wi be used, foowed by the high-cost treatment if necessary. In other words, the consumer shoud never take the high-cost treatment before trying the ow-cost treatment. Second, the first best can be impemented by a singe contract when the physician can commit to treatment pans before earning about patients severities. This resut is surprising both because in principa-agent modes, information asymmetry often generates information rent and distortions, and because the first best is impemented without the use of any contract menu. Third, the first best is infeasibe when the physician cannot commit to treatment pans; the physician earns excess profits, and treatment decisions are distorted from the first best. To expain our resuts, we shoud first describe the extensive-form game. In Stage, an insurer offers an insurance contract to the consumer, and a payment contract to the physician, which consists of a capitation payment and the physician s share of treatment cost. In Stage 2, nature determines the physician s degree of atruism, which is privatey known to the physician. In Stage 3, the physician and the consumer decide whether to accept the contract. The physician aso decides on a practice stye which is a rue for prescribing a treatment pan for any iness severity. In Stage 4, nature determines the patient s iness severity. The physician earns the iness severity and foows the treatment pan decided in Stage 3. The commitment power manifests in Stage 3. At that time, the physician has not earned the patient s iness information (he aready has the private information about the degree of atruism, but he does anticipate earning that in Stage 4. What he does in Stage 3 is to formuate a rue for how the patient is to be treated: if the severity turns out to be such and such in Stage 4, then this or that treatment wi be used. Stage 3 is aso the contract acceptance stage, and the physician must simutaneousy assess whether the capitation payment and cost share can generate a minimum expected profit. The first best can be impemented by a contract designed as if the physician were the east atruistic type. Suppose the east atruistic physician puts a % weight on consumer s utiity. The insurer shoud offer a contract with a % cost share and a transfer equa to % of the expected first-best cost. The % atruistic physician wi fuy internaize the socia costs and benefits when bearing % of the cost. A ump-sum transfer equa to % of the expected cost in the first best aows the east atruistic physician to break even. Why can this contract sti impement the first best when the physician puts, say, a 5% weight on the consumer s utiity? If the physician accepts the contract and impements the first best, he aso breaks even. The doctor woud have iked to offer more generous treatments because he was more atruistic. But if he had done so, he woud not break even. The transfer is so ow ony % of the expected first-best cost that more generous treatment pans woud put the 5% physician in the red. The nonnegative expected profit constraint is so binding that the 5% physician must foow the strategy of the east atruistic physician. It foows that the 5% atruistic physician impements the first best. 2 A sampe of papers using the atruism assumption in the heath iterature incudes Chakey and Macomson (998, Choné and Ma (2, Dranove and Spier (23, Dusheiko et a. (26, Eis and McGuire (986, 99, Jack (25, Ma (998, Ma and Riordan (22, Makris and Siciiani (2, Newhouse (97, Rochaix (989, and Rogerson ( A the papers in footnote 2 use the known atruism assumption except Choné and Ma (2 and Jack (25. 4 Our resuts remain the same if the minimum profit is stricty positive. The eve of the premium wi be adjusted accordingy, since any profits wi be passed onto consumers.

3 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Next, we study a game in which the physician does not have commitment power. The first two stages of the game remain the same. But now in Stage 3, the physician ony decides on whether to accept the contract. He does anticipate earning the iness severity in Stage 4, but the treatment decision is postponed unti then. The difference, therefore, is that any capitation payment specified in Stage 3 has been paid, and has no bearing on the physician s treatment decision in Stage 4. Now, the singe contract in the game with commitment fais to impement the first best. The 5% atruistic physician wi reject a % cost-share contract. In Stage 4, bearing ony % of costs, the physician now cannot resist offering treatments that are more generous than the first best. It is time inconsistent for the 5% atruistic physician to stick to the first best. However, the ow transfer in the % cost-share contract woud not aow him to break even. Anticipating the deficit in the continuation, the 5% atruistic physician rejects the contract in Stage 3. If the insurer has to retain a physician with high degrees of atruism, contracts with higher cost shares must be offered. In fact, a menu of incentive-compatibe payment contracts wi be offered, and physicians may earn positive profits. Distortions from first-best treatment pans wi resut, and the insurance premium for the consumer wi be higher. Our resuts confirm the efficiency oss due to practice-stye variations. However, our anaysis aso indicates how this oss can be avoided. If treatment pans can be finaized when the financia constraint is reevant, efficiency can be attained. A sort of bottomine medicine principe is being advocated whereby resources, incuding ump-sum payment, and medica treatments shoud aways be considered together. The poicy impication is that the insurer shoud encourage doctors to formuate their treatment pans at the point of contract acceptance, and give doctors incentives to carry out the pan when seeing patients. For exampe, when offering the singe contract, the insurer aso suggests the efficient treatment pan as a medica guideine. In addition, the insurer announces that he wi ony renew contracts with physicians whose tota treatment cost (say in a year is beow a threshod. In economic modes, it has been shown time and again that commitment is powerfu. Yet, it appears that here, a physician s commitment power is being expoited by the insurer. A physician earns a zero profit when he is abe to commit to a treatment pan, but a positive profit otherwise. However, physicians in our mode are atruistic and their preferences are not based on profits aone. In fact, a physician s tota utiity may be higher when he has commitment power and is very atruistic. Athough we anayze games in which physicians may or may not commit to treatment pans, commitment itsef is taken to be exogenous. In the iterature, many researchers have posited that commitment requires a payer of a game to take a costy action, but others have assumed that a payer may be a commitment type that can stick to a strategy. 5 We are agnostic about whether commitment must require a prior costy action or not. Our interest is to identify circumstances in which efficiency can be achieved. As it turns out, our researh points to the importance of medica practice stye as a commitment that may be used for impementing efficienct treatments. As Arrow (963 has pointed out, physician atruism seems so natura, and important in the heath care market. The economic anaysis foowing such a hypothesis has ony been studied quite recenty. A contribution here is that atruism interacts with profit motives. The impementation of the first best depends on physicians caring about their patients, having to make a minimum expected profit, as we as being abe to commit to treatment protocos. In the iterature, the idea that economic agents have nonmonetary motives has been studied intensivey. Here is sampe of such recent papers: Akerof and Kranton (25, Bénabou and Tiroe (23, Besey and Ghatak (25, Defgaauw and Dur (27, 28, Francois (2, Makris (29, Murdock (22, and Prendergast (27, 28. Our paper differs from these works in that the physician s degree of atruism is unknown (see aso footnotes 2 and 3 above. Unknown atruism generay brings in a second dimension of asymmetric information. Our paper contributes methodoogicay to the muti-dimensiona asymmetric information probem. A few papers in the iterature use a imited iabiity constraint, which is identica to our minimum expected income constraint. Makris and Siciiani (2 consider incentive schemes for atruistic providers who possess private information about production efficiency, but who must be abe to break even. Makris (29 uses a sighty different setup in which an agent must not be asked to use any of his own weath. In these two papers, the degree of atruism is common knowedge. Choné and Ma (2 aso use a minimum income constraint. The requirement of minimum profit for atruistic agents appears to be both natura and necessary. Unknown atruism in the heath market has been considered before by Jack (25 and Choné and Ma (2. Nevertheess, our paper differs in many ways. In Jack (25 and Choné and Ma (2, risk aversion and insurance are not considered. Jack s mode considers noncontractibe quaity choices by a provider, and ets the physician suffer some financia osses. We do not consider quaity, and impose a nonnegative expected profit constraint. Choné and Ma study a more genera agency probem in which the physician s preferences may not be atruistic. In addition, in Choné and Ma, heath care quantities are contractibe, and the physician possesses private information about patient iness severity and his degree of physician agency before accepting a contract, so commitment is irreevant. Moreover, in Jack (25 and Choné and Ma (2, there are no equiibria in which the first best is impemented. The iterature on physician payment is arge. An earier survey is McGuire (2, and a more recent one is Léger (28. Despite the prevaence of mutipe treatment options, most existing works either do not mode treatment pans (Pauy, 5 For exampe, in the industria organization iterature, a firm may make an irreversibe investment in order to commit to agressive pricing strategies against rivas. On the other hand, in the resoution of the chain-store paradox, and in many repeated games, some type of a payer is assumed to aways pay a suboptima strategy.

4 82 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( , Zeckhauser (97, Choné and Ma (2, or aow patients to take ony one treatment (Ma and Riordan (22. Severa more recent papers (Chernew et a. (2, Macomson (25, Siciiani (26 aow the patient to choose one treatment out of many options. However, they do not aow the patient to take a treatment sequence. Different from a these works, our mode has mutipe treatment options and examines optima treatment sequences. The rest of the paper is organized as foows. Section 2 presents the mode and the first best. Section 3 studies the two deegation games. Section 4 discusses reated issues and poicy impications. Section 5 draws concusions. Proofs are in the Appendix. 2. The mode and the first best A risk-averse consumer has income Y and suffers from an iness. The oss due to iness is described by a random variabe on a support, ], with distribution and density functions F( and f( >, respectivey. 6 We assume that the upper support of the iness oss,, is sufficienty arge. 7 We et the consumer s utiity function be separabe in income and the oss from iness, and measure the disutiity of iness by the oss, so the consumer s utiity is U(Y when is the iness oss. The function U is stricty increasing, stricty concave, and the margina utiity at zero income is infinite (U (x as x +. The consumer s oss due to iness can be recovered by medica treatments. We assume that there are two treatments; in Section 4 we wi discuss the case when more treatments are avaiabe. A treatment either recovers the oss or does not, and is defined by the probabiity of success and the cost. Treatment can be taken sequentiay, so if a treatment does not succeed, a second treatment can be used. We assume that when a treatment fais once, it wi fai again. In other words, the effectiveness of a treatment is perfecty correated over trias. Given the binary structure, a treatment wi never be used twice. We ca the two treatments, Treatment and Treatment 2. Treatment succeeds with probabiity and costs c. Treatment 2 succeeds with probabiity 2 and costs c 2. These four parameters are stricty positive. Treatment 2 is more effective than Treatment but aso costs more, so we have < 2 and c < c 2. We make an assumption on the reative effectiveness of the treatments: Assumption (Cost convexity. c < c 2 2. Assumption says that the cost per unit of success probabiity of Treatment 2 is higher than Treatment. This is a convexity assumption on treatment costs; the cost per unit of success probabiity increases with the success probabiity. We wi discuss what wi happen if Assumption is vioated. In this paper we consider Treatment protocos. A treatment protoco describes a sequence of treatments. There are five treatment protocos: Protoco : Do not use any treatment. Protoco : Use Treatment ony. Protoco 2: Use Treatment 2 ony. Protoco 3: Use Treatment, and then Treatment 2 if Treatment fais. Protoco 4: Use Treatment 2, and then Treatment if Treatment 2 fais. Because we have assumed that a treatment outcome is perfecty correated across trias, Treatment Protocos do not incude mutipe trias of the same treatment. The ex ante success probabiities of Protocos 3 and 4 are, respectivey, 3 + ( 2 and ( 2. These ex ante success probabiities are the same because each of Protocos 3 and 4 aows the consumer to try both treatments, and offers a higher success probabiity than either Protoco or Protoco 2. The expected costs of Protocos 3 and 4 are, respectivey, c 3 c + ( c 2 and c 4 c 2 + ( 2 c. By Assumption, Protoco 4 costs more than Protoco 3: c 4 c 3 = c 2 c 2 >. Without any insurance, the consumer wi decide on the treatment protoco after she earns her iness oss. For ow vaues of, she may not get any treatment; for high vaues, she may. The consumer faces fuctuations in income since she has to bear treatment costs. The consumer can insure hersef against income fuctuations due to iness by purchasing an insurance contract in a competitive insurance market. Insurers are risk neutra, and they offer insurance contracts to maximize the consumer s expected utiity subject to a zero expected profit constraint. 8 6 Unike most other modes, we do not set up a probabiity of the consumer faing i, upon which the oss occurs. Our mode is sighty more genera because we aow a arge density around =, so it can approximate modes with a fixed probabiity of faing i. 7 If the upper support is not arge enough, the consumer s benefit from Treatment 2 cannot be justified by the cost. Hence, Treatment 2 shoud never be used. See beow. 8 We can repace the perfecty competitive insurance market by a benevoent reguator who wishes to maximize a weighted average of the consumer s utiity and the physician s payoff. Our resuts wi continue to hod as ong as the weight on the physician s payoff is not too high.

5 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( First best In the first best, iness oss is verifiabe. An insurance contract can be made contingent on the vaue of. Due to risk aversion, the first best shieds the consumer from a risks due to treatment costs. A first-best contract specifies a premium P and four treatment protoco functions i :, ], ], i =, 2, 3, 4. The consumer pays P before the reaization of, and wi not incur any payment after is reaized and when treatment is used. The function i, i =, 2, 3, 4, specifies the probabiity that Protoco i is to be used when the consumer s oss is. We have used the nontreatment Protoco as defaut. If the consumer suffers a oss and is treated by Protoco i, her expected payoff is U(Y P + i. The first-best contract (P,, 2, 3, 4 maximizes the consumer s expected utiity ] U(Y P + i ( i df( ( subject to the breakeven constraint P = i (c i df( (2 and the boundary conditions i ( and i (, (3 for each, ] and i =, 2, 3, 4. The utiity function in ( consists of the utiity from the income ess the premium, the utiity oss, as we as the recovery prospects from the four treatment protocos. The breakeven constraint (2 ensures that any insurance firm offering the contract wi make zero expected profit. The remaining constraints in (3 make sure that the treatment protoco probabiities are consistent. First, we rank the reative cost effectiveness of the treatment protocos: Lemma. Under Assumption, c < c 3 < c 4 < c According to Lemma, in terms of cost per unit of success probabiity, the ranking, in ascending order, is Protoco, Protoco 3, Protoco 4, and Protoco 2. Now, 3 = 4 > 2, so both in terms of success probabiity and cost per unit of success probabiity, Protocos 2 and 4 are dominated by Protoco 3. In other words, Protocos 2 and 4 are ess efficient than Protoco 3. 9 Proposition. In the first best, the consumer pays a premium P, receives no treatment if her oss is ower than, Protoco if her oss is between and, and Protoco 3 if her oss is higher than, where U (Y P c < U (Y P c 2. The 2 premium is given by P = c F( ] + ( c 2 F( ]. Proposition presents two principes in the first best. First, the consumer is risk averse, so financia risks due to iness wi be borne by the insurer. Second, by basic cost-benefit consideration, the consumer shoud receive more treatment when her oss is higher. Basic cost-benefit consideration aso eiminates inefficient treatments, so by Lemma, Protocos 2 and 4 are never used. Consider consumer. His expected utiity benefit from Treatment is, and this is equa to the cost of Treatment measured in utiity, U (Y P c. Whie the benefit from Treatment increases in iness oss, the cost remains constant. Therefore, the consumer shoud receive Treatment if and ony if his iness oss is at east. As the iness oss continues to increase beyond, Treatment 2 shoud aso be given if it is needed. At =, the cost of using Treatment 2 is U (Y P c which is equa to 2. Therefore, a consumer with > shoud receive Treatment 2 if and ony if Treatment has faied. This is Protoco Atruistic physician and deegation Suppose now the consumer s iness oss is not observed by the insurer. Athough treatments prescribed by the physician are verifiabe ex post, they are ex ante noncontractibe. The physician wi observe the iness oss and be deegated to make the treatment decision. In the deegation regime, an insurance company estabishes a payment contract with the physician, 9 We briefy comment on the case when the Cost Convexity assumption is vioated. In that case, we have c > c 2. The ranking of cost per unit of success 2 probabiity becomes c 2 < c 4 < c 3 < c, so that Protocos 3 and wi be inefficient. Proposition wi be modified: Protoco 2 wi be used for intermediate vaues of, whie Protoco 4 wi be used for high vaues.

6 84 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( and an insurance contract with the consumer. The physician s decision on a treatment pan can be interpreted as his practice stye. The insurance contract for the consumer consists of a premium P. We focus on physician payment and deegation, so we assume that the patient does not bear any financia risks ex post. In fact, this is what the first best prescribes. The payment contract for the physician is a two-part tariff, (S, T, where S is the physician s share of the incurred treatment cost, and T is a ump-sum or capitation payment. The physician is risk neutra, and partiay atruistic to the consumer. The physician earns about the consumer s iness oss after the payment contract has been accepted. This is a natura assumption in an insurance mode because at the time the insurer offers contracts, the consumer is not yet sick. When the physician treats the consumer with Protoco i, his expected payoff consists of profit and the consumer s utiity: T Sc i + U(Y P + i ]. Both S and T are nonnegative, but we do not restrict S to being ess than. The profit from using Protoco i is T Sc i ; he receives the transfer T, and bears a cost Sc i, with the baance of the cost paid for by the insurer. The parameter measures the strength of the consumer s utiity in the physician s preferences. The atruism parameter is a random variabe, drawn on a stricty positive support, ], with distribution and density functions, respectivey, G( and g( >. We assume that the hazard rate G( g( is increasing in. The physician knows, and this is his private information. We use the term type- physician for a physician with atruism parameter. We assume that F and G are independent. A higher vaue of indicates a physician who cares more about the patient s wefare. The strength of the physician s trade-off between profit and patient utiity is captured by the atruism parameter. In making a decision based on this trade-off, the physician must respect an ex ante nonnegative profit constraint. In practice, a physician treats many patients, and the ikeihood that he makes a oss ex post out of the entire set of patients is negigibe. Indeed, if we interpret the consumer as the representative in a mass, then ex ante nonnegative profit impies ex post nonnegative profit. As in other agency modes, we incude a reservation utiity constraint. If the atruistic physician does not accept the contract, he does not earn any profit, but does not treat the patient either, so his utiity is U(Y ]df(, which is defined to be his reservation utiity. To better understand the equiibria when the physician s degree of atruism is unknown, we first show, in the next subsection, that the first best can be impemented when the physician s degree of atruism is known. 3.. Known atruism In this subsection, we assume that the atruism parameter is common knowedge. The physician, with known atruism parameter, is paid a ump-sum T(, and bears a cost S( c i when he uses Protoco i for the patient. Subject to the payment scheme, the physician makes treatment decisions for the patient. Suppose that, on observing the iness oss, the physician uses treatment Protoco i with probabiity i (. His expected utiity is { T( S( i (c i + U(Y P + i ( i ] df(. (4 He chooses i to maximize (4 subject to a nonnegative expected profit constraint { T( S( i (c i df(, (5 and a participation constraint: { T( S( i (c i + U(Y P + i ( i ] df( U(Y ]df( Aternativey, we can et the consumer be treated by some other physician, and the consumer obtains a higher expected utiity. See footnote 9.

7 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( which says that the utiity from accepting the contract is higher than from refusing it. Now, the participation constraint never binds. Rewrite it as { T( S( i (c i df( U(Y ]df( { U(Y P + i ( i ] df(. The right-hand side of this inequaity is the patient s oss from the ack of insurance. Due to a competitive insurance market, this oss is never positive, so, in fact, minimum profit impies participation. In the games with asymmetric information, given the minimum profit constraint, the participation constraint remains sack for each type of the atruistic physician, so from now on, we wi ignore it. For each, ], define S( ] T( c df( + c 3 df(, (6 (7 where = U (Y P, and P,, and are the first-best premium and threshod oss eves defined in Proposition. Lemma 2. Given S( and T( defined in (6 and (7, the deegation scheme impements the first best. The cost share S( in Lemma 2 makes the physician internaize the consumer s treatment cost and benefit. The partiay atruistic physician vaues the patient s benefit at i. To aign his preferences with the first best, he shoud be made to bear the cost at c i, where, the margina utiity of income at first best, adjusts for the difference in the measurement between benefits (in utiity and cost (in money. This is exacty what S( does. Under this cost share, the physician s expected utiity in (4 becomes { T( S( i (c i + U(Y P + i ( i ] df( = { ] i ({ i c i + T( + U(Y P ] df(, so the term inside the big square brackets is the consumer s benefit ess cost. The transfer T( ensures that the physician makes a zero expected profit. A more atruistic physician is asked to bear a arger share of the cost ex post because he has a greater incentive to overtreat the patient. The ump-sum transfer T( is proportiona to the cost share S(. Given that a types of physician wi incur the first-best cost, a more atruistic physician shoud receive a arger transfer ex ante; otherwise, he wi not be abe to break even. These findings are consistent with Eis and McGuire (986 who show that the first best can be impemented in a mixed payment system when the physician s degree of atruism is common knowedge. Whie Eis and McGuire focus on a singe treatment, we show the same resuts for mutipe treatments. The physician s behavior for the maximization of (4 subject to (5 assumes that he chooses the treatment protocos at the time of contract acceptance and before he observes the iness severity. This assumption is ony made for convenience. When S( and T( are given by (6 and (7, the physician can aso make the treatment decision after he observes. The treatment decisions wi be exacty the same. Hence, the timing for treatment decisions is unimportant when the physician s degree of atruism is common knowedge. This, however, is not true when the physician s degree of atruism is his private information Unknown atruism In this subsection, we study deegation games with unknown atruism. We show that equiibria depend on the physician s timing of treatment decisions. In the case of commitment, the physician foows a predetermined treatment pan before he earns the patient s iness. In the case of noncommitment, the physicians decides on treatment after he earns the patient s iness. We first present the game with commitment; the game without commitment foows. In each game, the insurer We assume that if the consumer is not served by the physician, the iness remains untreated; this is the utiity on the right-hand side of the participation constraint. We can aternativey assume that the consumer is served by another physician, so the right-hand side term is arger. However, we have assumed that the insurer aims to maximize the consumer s expected utiity, so the participation constraint remains sack.

8 86 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( chooses contracts to maximize the consumer s expected utiity, so we assume that the consumer wi accept the contract. We assume that the consumer does not know the physician s degree of atruism, but sti deegates treatment decisions to the physician. One might wonder if there woud be an incentive for a consumer to seek out a more atruistic physician, and we wi discuss this issue in Section Equiibria in deegation with treatment pan commitment We show the first best can be impemented by a singe contract when the physician can commit to a treatment pan made at the point of contract acceptance. The extensive form of the game has four stages. Stage : An insurer offers an insurance contract to the consumer and a payment contract to the physician. Stage 2: Nature draws from the distribution G. The physician earns. Stage 3: The physician decides whether to accept the payment contract, and the consumer decides whether to accept the insurance contract. The game ends if either party refuses to accept; otherwise, the physician aso decides on how he wi prescribe treatment protocos depending on iness oss. Stage 4: Nature draws from the distribution F. The physician earns, and carries out treatment protocos according to the prescription rue decided in Stage 3. The physician wi be paid according to the payment contract. When atruism is unknown, a type- physician wi mimic another type if the fu menu of contracts defined in the regime of known atruism is offered. From Lemma 2, if a type- physician seects (S(, T(, he wi choose the first-best treatment protocos and break even. However, the type- physician can do better by exaggerating and choosing a contract meant for type-, >. Under (S(, T(, he can sti impement the first best and break even, but wi gain by being sighty ess generous than offering first-best treatments. This deviation wi resut in a second-order oss in the consumer s expected utiity but a first-order gain in the profit because T( > T(. Our next resut shows that, surprisingy, each type of physician can sti be made to impement the first best even when the fu menu of contracts defined in the regime of known atruism fais to do so. This is achieved by a very simpe payment contract, namey (S(, T(, defined in (6 and (7. This contract is designed as if the physician were the east atruistic type. A type- physician s best response against (S(, T( is to seect i ( to maximize { T( S( i (c i + U(Y P + i ( i ] df( (8 subject to { T( S( i (c i df(. (9 A type- physician s choice of treatment decision in Stage 3 is made contingent on possibe iness oss. Anticipating that he wi foow this treatment pan after observing the iness severity in Stage 4, the physician decides whether to accept the payment contract. Lemma 3. When given contract (S(, T( defined in (6 and (7, a type- physician, with >, chooses the first-best treatment threshods and. Lemma 3 reports a surprising resut. Under the payment contract (S(, T(, the best response of the type- physician is the first-best treatment protoco. His incentives have been aigned with the first best. Now consider a more atruistic, type- physician. He cares more about the consumer s utiity than type-, so he woud ike to be more generous, offering Protoco at <, and Protoco 3 at <. Indeed, the first-order derivative of (8 with respect to i is i c i ], which is greater i than i c i ], the corresponding first-order derivative in the first best. i The capitation payment T( ony compensates for the cost share S( when treatments are at the first best. The type- suffers a oss if he foows a treatment pan more generous than the first best. The binding nonnegative profit constraint therefore stops the type- physician from being more generous than a type- physician. Since he is abe to commit to a treatment pan, it is a best response for the type- physician to accept the contract (S(, T(, and to impement the first best. 2 To summarize, we present 2 We have assumed that the atruism parameter is in a stricty positive support, ]. If the support of incudes (so that =, our resut wi be modified sighty. Here, the first best can be approximated. Setting the payment contract at (S(, T(, where > and is arbitrariy cose to, wi impement the first best for a physician types higher than. However, the contract (S(, T( wi not impement the first best for any physician type.

9 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Proposition 2. In the equiibrium under deegation with treatment pan commitment, the insurer offers a singe payment contract (S(, T(. In equiibrium, each physician type accepts the contract and deivers the first-best treatment protocos to the consumer. The key to the first-best resut stems from the requirement that treatment pans are made when the nonnegative expected profit consideration is sti reevant. We coud consider an aternative extensive form where the physician decides on treatment pans after he has accepted a contract, but before he observes (sti fuy anticipating that he wi. This kind of commitment has no bite, and the equiibrium wi be exacty the same as if commitment were impossibe (as in the next subsection. This is because once the contract (S(, T( has been accepted, the treatment pan decision wi be determined ony by the cost share S( whie the transfer T( is aready received. Proposition 2 highights the socia vaue of treatment pan commitment. If the physician determines his treatment pan when accepting the payment contract and sticks to it, the insurer can successfuy induce a types of physicians to carry out the efficient treatment pan. Therefore, the unwarranted cost variation due to physicians heterogeneous preferences can be reduced. The next subsection discusses the scenario when the physician acks the abiity to commit to a predetermined treatment pan Equiibria in deegation without treatment pan commitment The first two stages of the game without treatment pan commitment are the same as game with commitment, except that a payment contract is now a menu. The ast two stages are as foows: Stage 3: The physician decides whether to accept the payment menu, and the consumer decides whether to accept the insurance contract. The game ends if either party refuses to accept; otherwise, the physician picks an item from the menu. Stage 4: Nature draws from the distribution F. The physician earns, and decides on treatment protocos. The physician wi be paid according to the payment contract that he has seected in Stage 3. The key difference between games with and without treatment pan commitment is the timing of treatment decisions. Under deegation without treatment pan commitment, the physician makes his treatment decision after he has accepted the contract. In other words, the physician makes the contract acceptance decision and treatment decisions sequentiay. By contrast, in the game with treatment pan commitment, he makes the two decisions simutaneousy. In both cases, however, the physician fuy anticipates receiving the patient s severity information in Stage 4. Ceary, the singe contract (S(, T( can no onger impement the first best. Anticipating using treatment pans more generous than the first best, physician types more atruistic than wi reject this contract. This is because the transfer T( is so ow that they cannot break even. We derive the menu of optima contracts by examining the physician s treatment protoco decisions in Stage 4. Suppose that a type- physician has accepted a payment contract (S(, T( in Stage 3, and earns that the consumer s iness oss is. His decision is ony affected by the cost-share parameter S(, not the transfer T(. ] Given and S(, his payoff U(Y P + 4 i i from choosing Protoco i with probabiity i is S( 4 ic i +. The first-order derivative with respect to i is i S( c i. As in the earier anaysis, the equiibrium treatment is characterized by two threshods ( ; and ( ;. The physician wi never use the inefficient protocos. A consumer with smaer than ( ; receives no treatment; with between ( ; and ( ;, Protoco ; with arger than ( ;, Protoco 3. The equiibrium in Stage 4 is competey characterized by the threshods ( ; S( c = and ( ; S( c 2 =. ( 2 To save on notation, we write ( ; and ( ; as ( and (, respectivey. In contrast to deegation with treatment pan commitment, the equiibrium treatment decisions are to be made without any reference to the nonnegative profit requirement. In Stage 4, the physician does not have the option of rejecting a payment contract. The requirement of making a nonnegative expected profit has no bite here. Next we study the physician s equiibrium choice of a payment contract in Stage 3. Suppose that the menu {(S(, T( has been offered to the physician in Stage 2. We use a generaized version of the reveation principe (Myerson (982. Define a type- physician s expected payoff from seecting contract (S(, T( and the threshods and by ] V(,, ; T( S( c df( + c 3 df( ] + U(Y P E( + df( + 3 df(.

10 88 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( We consider equiibria in which a type- physician seects contract (S(, T(, and adopts the threshods ( and (. Ceary, for any choice of (S(, T( the threshods that maximize V are (, and ( ;, as in the continuation equiibrium (. A menu of contracts is said to be incentive compatibe if V(, (, ( ; V(,, ; for a and, and a and. Given a menu (S(, T(, define the type- physician s maximum payoff by W( max V(,, ;.,, Lemma 4. A menu of contracts {(S(, T(,, ], is incentive compatibe ony if W is convex, ( W ( = U(Y P E( + df( + 3 df(, ( ( ( and both ( and ( are decreasing in. Incentive compatibiity requires that the physician s equiibrium utiity be convex in the atruism parameter. In other words, the change of the physician s equiibrium payoff, W(, must be increasing. Because U is a utiity function of income, its sign can be positive or negative; hence, W and W can be positive or negative. Indeed, signs of W and W are irreevant for incentive compatibiity. Furthermore, Lemma 4 says that the equiibrium threshods must be decreasing so that a more atruistic physician prescribes more treatments. We write the continuation equiibrium condition ( as ( = S( c and ( = S( c 2, (2 2 so incentive compatibiity requires the cost share to atruism parameter ratio, S(, to be decreasing. This is in contrast with the known case where S( is a constant. To see the intuition, suppose S( increases proportionay to. Because the more atruistic physician wi prescribe more treatments but has to bear a arger share of the cost, the ump-sum transfer must increase more than proportionay. Otherwise, the physician cannot break even. However, a disproportionatey arge ump-sum transfer woud provide the physician a greater incentive to exaggerate his degree of atruism because he can gain a arger profit by withhoding treatments. Hence, with constant S(, the insurer has to give up too much information rent to induce truth teing. The insurer can do better by reducing the cost share borne by the physician to trade off efficiency for information rent. Next, we anayze the physician s nonnegative profit constraint. By seecting (S(, T(, a type- physician s expected profit is ( ( T( S( c df( + c 3 df(. ( ( Substituting this expression into W( = V(, (, ( ;, we have W( = ( + W (, or ( = W( W (. (3 Differentiating both sides of this equation, we have ( = W (. The convexity of W( impies that ( is decreasing. The physician s nonnegative profit constraints are therefore simpified to (. In other words, if the most atruistic physician breaks even, so do a other physician types. Athough the physician s profit is decreasing in, his equiibrium payoff is increasing due to the atruistic benefit. Lemma 5. Incentive compatibiity is equivaent to S( / being decreasing, and hence ( and ( decreasing. Nonnegative expected profit for the physician is equivaent to (. We continue with the derivation of the equiibrium contract menu. Foowing the standard method in the iterature, we repace ( by W( to simpify the maximization probem. The insurer must break even given the continuation equiibrium after Stage. The tota expected expenditure by the insurer equas the expected profit and treatment cost, averaged over a physician types. Hence, the premium P satisfies ( P = ( dg( + c df( + c 3 df( dg(. (4 ( ( From W( W( W (xdx, we can substitute for W in the expression for in (3: ( = W( W (xdx W (.

11 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Then we use ( in Lemma 4 to repace W (x. After integration by parts, we can substitute for ( and rewrite (4 as ( P = c df( + c 3 df( dg( + W( ( ( ( G( (5 ( g( + U(Y P E( + df( + 3 df( ( ( dg(. The premium for the patient incudes treatment costs and the physician s utiity, which consists of the base utiity W( ess the consumer s utiity mutipied by the physician s atruism parameter adjusted by the hazard rate (G( /g( +. From (3, we have ( = W( W ( ( = W( U(Y P E( + df( + 3 df(, ( ( so ( if and ony if ( W( U(Y P E( + df( + 3 df(. (6 ( ( The equiibrium in Stage 4 aso requires (2, which says that ( and ( foow a fixed ratio; this wi be shown to be satisfied, so we wi ignore this requirement for now. The equiibrium aocation impemented by the insurer is the soution to the foowing program: choose P, W(, (, and ( to maximize the consumer s expected utiity ( U(Y P E( + df( + 3 df( dg( ( ( subject to the breakeven constraint (5, the physician nonnegative profit constraint (6, and (, ( both decreasing. Let denote the mutipier for the insurer s breakeven constraint (5. We present the characterization of the soution: Proposition 3. Under treatment pan noncommitment, the equiibrium threshods and premium, (, (, and P are given by ( = c ( G( + g( + ] (7 ( = c ( 2 G( + 2 g( + ] (8 U (Y P =. (9 The type- physician earns zero profit, and W( is given by (6 as an equaity; a other physician types earn stricty positive profits. From the equiibrium threshods in Proposition 3 and equation (2, we can find the cost share and transfer functions for the impementation. The cost share function is ( G( ] S( = + g( + (2 and the transfer function is ( T( = W( W ( + S( c df( + c 3 df(, (2 ( (

12 9 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( where W ( is determined by equation ( and W( is obtained by integrating W (. The physician s impementation of Protoco 3 is time consistent. If > (, ( his ] utiity from continuing with Treatment 2 for the consumer is 2 S( c 2. G( From (2, this is 2 + g( + c2, which is stricty positive by Proposition 3. The determination of the equiibrium threshods incudes the term G( g( +, the key difference from Proposition. The first-best threshods are determined by a straightforward cost-effectiveness principe. This has to be modified due to the missing information about the physician s degree of atruism. The equiibrium cost shares and transfers invove the hazard rate, G( g(, a standard, Myerson virtua adjustment due to private information. Furthermore, treatment benefits are vaued by physicians, so the adjustment aso incudes the term in addition to the virtua component. Because the physician s profit is passed on to consumers, we have the foowing coroary: Coroary. The equiibrium premium P is higher than the first best premium P. The comparison between equiibrium threshods in Proposition 3 and the first best is not straightforward. The first best is independent of the distribution of, but the functions and have ranges that depend on the distribution as we as the support of. We suspect that for ow vaues of, equiibrium threshods wi be higher than first best, whie for high vaues of, they wi be ower. That is, ess atruistic physicians provide treatments ess than the first best, and the opposite for more atruistic physicians. The foowing exampe agrees with our conjecture. Let the utiity function be U(Y = n Y, so U (Y = /Y. Suppose that is uniformy distributed on, ] whie is uniformy distributed on, + ], >. By Proposition, the first-best threshods are c = Y P and c 2 = Y P. (22 2 Equation (9 reduces to = /(Y P. The equiibrium threshods in Proposition 3 are ( = c + 2 ] and ( = c ]. (23 Y P Y P Y P 2 Y P By Coroary, the premium P is arger than the first-best premium P. From (22 and (23, ( and ( are arger than the first-best threshods for < P P +, and are smaer than or equa to the first-best threshods otherwise. If the difference 2 P P is between and + 2, there exists a type- physician deivering first-best treatments. Physicians ess atruistic than type- wi provide ess treatment than the first best, whereas physicians more atruistic than type- wi provide more. Proposition 3 and Coroary can expain the wide variations of medica costs. Differences in physician practice styes are here captured by differences in physician atruism. The same iness wi be treated differenty depending on the attending physician s preferences. Such variations, however, can be avoided if atruistic physicians make treatment decisions when the fu financia consequences are respected, as Proposition 2 shows. 4. Discussions and poicies 4.. Poicy impication Our anaysis suggests that the insurer shoud hep a physician with some commitment mechanism when offering the singe contract (S(, T( to impement the first best. The patient s iness severity is unobservabe to the insurer, but perhaps medica trainings can partiay address this issue. When doctors have been trained to perform treatments according to certain (first-best protocos, these protocos become estabished professiona practices, and physicians may subscribe to them out of habit. An insurer therefore has a vested interest in promoting some protocos. Nevertheess, our mode aso suggests another way. When offering (S(, T(, the insurer can propose the first best as medica guideines, and renew physicians contracts when they can maintain a minimum profit, say, at the end of an accounting period. Contract termination punishes the physician for overtreatment. Contract nonrenewa is an effective threat because in equiibrium physicians earn more than their reservation utiity. This is a kind of efficiency wage mechanism to provide incentives for the first best More than two treatments We have assumed that there are ony two treatments avaiabe. Proposition can be extended to an arbitrary number of treatments under Cost Convexity. Suppose that there is aso Treatment 3. We can construct many treatment protocos by various treatment sequences. However, ony three are efficient. These three are (i use Treatment ony; (ii use Treatment, and if it fais use Treatment 2; and (iii begin with Treatment, if it fais use Treatment 2, and if that aso fais, use Treatment 3. The intuition behind the inefficiency of protocos other than those in (i, (ii, and (iii mimics that in Lemma. For exampe, the protoco of Treatment 2 and then Treatment 3 upon faiure of Treatment 2 is dominated by the protoco in (iii. Adding Treatment before Treatment 2 raises the tota probabiity of success, and reduces the expected cost due to Cost Convexity, because Treatment has the owest cost-success probabiity ratio. Retaining the two-treatment assumption saves on notation, whie reaxing it woud not ead to quaitativey new resuts.

13 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Searching for atruistic physicians In Proposition 3, a physician provides more treatments when he is more atruistic. Therefore, ex post, a consumer prefers to be treated by a more atruistic physician. In Lemma 5, a physician reveas his type by seecting an item from the fu costshare-transfer menu. Typicay, however, consumers may not be aware of the financia arrangement between the insurer and the physician, so a physician s atruism information may not be inferred. In repeated interactions, without treatment pan commitment, consumers incentive to search must exist. In our setup, after an initia treatment episode, if a consumer knows the iness severity, then she can update her beief about the physician s atruism. For exampe, suppose that the severity is moderate, but the physician does not recommend Treatment 2 after Treatment has faied. Then the consumer wi infer that the physician is not very atruistic. Searching for more atruistic physicians is irreevant when treatment pan commitment is possibe. In the first-best equiibrium in Proposition 2, a physician types provide the same treatment. When search is reevant, it is associated with inefficiency and higher premium due to the ack of commitment in Proposition 3. Search exacerbates inefficiency. To attract consumers, physicians may offer more treatments even when their own degree of atruism is ow. Custering of consumers among atruistic physicians may ikey increase the premium, too. A poicy impication is that inefficient search can be avoided if treatment pan commitment is possibe Seecting physicians Physicians earn profits when there is a ack of treatment pan commitment. A way to imit profit is to reject some physician types. We have assumed that a types in, ] must earn nonnegative profits. It is possibe to reax this by aowing the insurer to retain ony those with between and <. This can be impemented by reducing the transfer function T in (2 (say, by a constant. Those physicians with arger than wi not accept any contract. A those who accept wi make ess profits, and the distortion can be reduced. The cost of rejecting highy atruistic physician types comes in the form of rationing. We have considered contracts for one consumer and one physician. We impicity have assumed that the aggregate suppy equas aggregate demand. Rejecting some physician types reduces the physician suppy. Even in a competitive insurance market, the premium may have to increase; otherwise, nonprice rationing resuts. 5. Concuding remarks We study how an insurer can reduce the unnecessary cost due to practice-stye variations by designing payment contracts for heterogenous physicians. Our mode consists of two new eements. Treatments can be combined, and physicians are atruistic, with different degrees of atruism. We deveop new principes from this setup. First, we show that the firstbest treatment pan foows a conservative pattern. Second, we consider deegating treatment decisions to physicians, and show that the first best can be impemented ony when a physician can commit to treatment pans at the time of contract acceptance. We offer various poicy impications. Treatment pans invove a time dimension, and it is natura that commitment pays a roe in the anaysis. The physician committing to using particuar pans may resut in time-inconsistent decisions. But such commitment has socia vaue; it reduces premium and inefficient search. The treatment technoogy is richer than the usua heath care quantity approach. This ets us rue out some treatment combinations as inefficient. However, our main resuts for deegation under treatment pan commitment and noncommitment shoud hod without any modification if the physician is choosing a quantity of services. We acknowedge that our mode abstracts from earning. Two issues naturay arise when earning is important. First, the ikeihood of treatment success may itsef be uncertain. A first treatment is often an experimentation for the physician to earn about treatment efficacy. The faiure of a treatment may then update the ikeihood that other treatments may be successfu. Second, iness severity may be uncertain. A first treatment may revea that the iness is more or ess severe than initiay thought. This new information wi impact subsequent treatments. We have focused on payment contracts based ony on the physician s reported type and on fu insurance contracts for consumers. In genera, the physician s cost shares can depend on the chosen treatments, and consumers may incur copayments. These more genera contracts are unnecessary under treatment pan commitment. We aready can impement the first best with the restricted contracts. More genera contracts can potentiay improve outcomes when treatment pan commitment is invaid. However, the trade-off between efficiency, risk sharing, and incentives is compicated. We have found the characterization under such genera contracts intractabe. Apparenty, separate anayses of demand-side and suppyside incentives are common in the iterature, and we have chosen to study suppy-side incentives. It is cear, however, that adding demand-side incentives woud not permit the impementation of the first best because fu insurance of financia risks cannot be achieved.

14 92 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Acknowedgments We thank David Bardey, Pedro Barros, Chiara Canta, Yuk-fai Fong, Michae Luca, and Henry Mak for comments. We aso thank David Martimort for discussing various issues with us. Many seminar and conference participants gave us comments and suggestions. An Associate Editor and two referees gave us hepfu advice. Part of this research was done whie the first author was affiiated with Michigan State University and the second author was visiting the Universidad Caros III de Madrid and Tufts University; the support and hospitaity at these Economics Departments is gratefuy acknowedged. Appendix A. Proof of Lemma : Let k c and k 2 c 2. By Assumption, k < k 2. From the definitions of 3 and c 3, we substitute c 2 and c 2 by k and k 2 2, respectivey, and obtain c 3 3 = ] k + + ( 2 ] ( 2 k 2, + ( 2 which is a weighted average of k and k 2, so c < c 3 < c Because 3 = 4 and c 3 < c 4 by Assumption, we have c 3 < c 4. It remains to show that c 4 < c 2. By Assumption , c 2 < c 2. To both sides of this inequaity we mutipy by ( 2 and then add c 2 2. This resuts in (c 2 + ( 2 c 2 < c 2 ( 2 + ( 2. Since c 4 = c 2 + ( 2 c and 4 = 2 + ( 2, we have c 4 2 < c 2 4, so c 4 < c Proof of Proposition : Omit the boundary conditions. Use pointwise optimization, and form the Lagrangian for : ( L = U(Y P + i ( i ]df( + P i (c i df( where > is the mutipier of the premium constraint. The first-order derivatives are P = U (Y P + (24 ( ( = f ( i c i = f ( i c i, i =, 2, 3, 4. (25 i i The derivatives in (25 are independent of i, so at each, the Protoco with the highest positive vaue of among i =, 2, i 3, 4 wi be used. If a the derivatives are negative, then no treatment wi be used. First, 2 ( = 4 ( = for a ; the consumer never uses Protocos 2 and 4. Because c 3 < c 2 by Lemma and 3 2 < 3, 2 <,. Therefore, we must have 2 2 ( =,. Because c 3 < c 4 by Lemma and = 4, <,. Therefore, we must have 4 ( =,. By Lemma, when < c, the first-order derivatives are a negative. Define c. From Lemma, when <, i i ( =, i =, 2, 3, 4. Hence, the consumer does not use any treatment when <. Next, from (25, we have 3 = ( 3 (c 3 c ] f ( = ( 2 c 2 ]f ( Now define c 2. (Because we assume that is sufficienty arge, we have <, and it is we-defined. The expression 2 in (26 is positive if and ony if >. Both and are positive when >. Together, we have 3 ( = when <, and 3 ( = when < <. Setting the first-order derivative (24 to, we have = U (Y P, so the vaues of and are those in the Proposition. Finay, the premium P is F( F( ]c + F( ]c 3, which simpifies to P = c F( ] + ( c 2 F( ]. (27 There is a unique soution for P between and Y. Let g(p denote the right-hand side of (27, where and are now regarded as functions of P. Since U (Y P increases in P, and increase ] in P. The function g(p ] is decreasing in P. The function g(p reaches the maximum at P =, and g( = c F( U (Yc + ( c 2 F( U (Yc 2 >. The function g(p 2 reaches the minimum at P = Y, and g(y = because U ( =+. We concude that there is a unique soution for P = g(p. (26

15 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( Proof of Lemma 2: First, given the contract (S(, T(, the type- physician chooses treatment protocos i (, i =, 2, 3, 4, to maximize his expected utiity { T( S( i (c i + U(Y P + i ( i ] df(. (28 The first-order derivative of (28 with respect to i ( is f ( i S( c ] i i = f ( i ( c i i (29 upon substitution S( by. The first-order derivative (29 is the first-order derivative (25 for the first best mutipied by, a constant. We concude that the type- physician s optima treatment decision is first best. Given the contract, the physician s expected profit from his optima, first-best treatment decision is T( S( = c df( + c df( + c 3 df( ] c 3 df( ] c df( + c 3 df( so constraint (5 is satisfied. It remains to show that the insurer breaks even. The insurer receives the first-best premium P from the consumer. He pays the physician the transfer T(, and S( share of the cost to the physician. The insurer s expected profit is therefore ] P T( ( S( c df( + c 3 df( ] { ] = P c df( + c 3 df( T( S( c df( + c 3 df(. ] The insurer breaks even in the first-best contract, so P c df( + c 3dF( =. The term inside the big cury brackets in (3 is the physician s profit and has been shown be to zero. Hence, the insurer makes zero expected profit. Proof of Lemma 3: The Lagrangian for the constraint optimization program maximizing (8 subject to (9 is L = { ( + ϕ T( S( ] ( i (c i + U(Y P + ] =, i ( i df(, where ϕ is the mutipier for the nonnegative expected profit constraint. From pointwise optimization, the first-order derivative with respect to i ( is: = f ( i ( ] + ϕc i. i i after substitution by S( =. Define c (+ϕ - ] and = c 2 (+ϕ - 2 (3 (3 ]. From the proof of Proposition, the physician wi not prescribe any treatment if, wi use treatment Protoco if < < and treatment Protoco 3 for <. Next, we show that the Lagrangian mutipier ϕ must equa for a type- physician. When ϕ =, the oss threshods and are identica to the first-best eves, and, respectivey, so the first best is optima. It remains to show that ϕ =, and we do that by contradiction. Suppose that ϕ <. Then the oss threshods satisfy < and <. The difference between the physician s expected profit from choosing threshods and and that from choosing the first-best threshods and is

16 94 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( { T( - S( - = S( - { c df( + ( c 2 df( + c 3 df( ] c df( { T( - S( - <. c df( + c 3 df( Given that under (S(, T( the expected profit from the first-best treatments (the second term, in cury brackets, on the first ine is, the physician s expected profit from choosing threshods and is negative. This vioates the nonnegative expected profit constraint, and contradicts the assumption that ϕ <. Hence, we concude that ϕ. Next, suppose that ϕ >. Then the oss threshods satisfy > and >. The difference between the physician s expected profit from choosing threshods and and that from choosing the first-best threshods is { ] { ] T( - S( - c df( + c 3 df( T( - S( - c df( + c 3 df( { = S( - ( c 2 df( + c df( >. Again, given that under (S(, T( the expected profit from the first-best treatments is, the physician earns a stricty positive expected profit. Hence, the nonnegative expected profit constraint does not bind, and the mutipier ϕ must be zero. This contradicts the assumption that ϕ > >. Hence we concude that ϕ. In sum, we have ϕ =. Proof of Proposition 2: First, Lemma 2 has shown that a type- physician wi accept (S(, T( and impement the first best. Next, consider a type- physician, with >. According to Lemma 3, he wi impement the first-best treatment threshods and given contract (S(, T(. Because the contract aows the physician to just break even on the first best, the type- physician s payoff is U(Y P E( + d + 3 d]. If the type- physician rejects the contract, he receives the reservation utiity U(Y E(]. Because the insurer maximizes the consumer s expected payoff, U(Y P E( + d + 3 d > U(Y E(. Given >, a type- physician stricty prefers to accept (S(, T(. Proof of Lemma 4: Because W( is the upper bound of affine functions of, it is convex (Rockafear, 972, Theorem 5.5, and therefore amost everywhere differentiabe (Rockafear, 972, Theorem Incentive compatibiity impies V(, (, ( ; = W(. By the enveope theorem, W ( = V ( = U(Y P E( + df( + 3 df(, ( ( with the partia derivative being evauated at =, = (, and = (, and we obtain the expression in the Lemma. Next, rewrite W ( as U(Y P E( + df( + ( 2 df(. (32 ( ( Because d ( S( d( = d c and d ( S( d( = d d c 2, d ( d ( and d 2 d d share S( d( the same sign as that of d. If ( and ( were increasing at some, then from (32 W ( woud be decreasing at. This contradicts incentive compatibiity. We concude that ( and ( must be decreasing. Proof of Lemma 5: We ony need to show that any menu satisfying S( / decreasing and ( impies incentive compatibiity and nonnegative expected profit. We start with a given cost-share rue S( with S( / decreasing. From Lemma 4 and the equiibrium condition for Stage 4 in (2, we have the threshods ( and ( being decreasing. We can construct T( so that (S(, T(,, ], is incentive compatibe. First, we set ( and ( by (2 for the continuation ]

17 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( equiibrium in Stage 4. Second, we use ( to construct a function W (. Setting a vaue for W(, we integrate W ( to obtain W(. Third, we set ( T( = W( W ( + S( c df( + c 3 df(. (33 ( ( It is straightforward to check that S( and the T( in (33 satisfy incentive compatibiity. Finay, we can choose W( so that (. Proof of Proposition 3: From the Lagrangian function L, where and are the mutipiers for constraint (5 and (6, respectivey. ( ( L = U(Y P E( + df( + 3 df( + ( ( P W( c df( c 3 df( ( ( ( G( ( + g( + U(Y P E( + df( + 3 df( ( ( ( + W( U(Y P E( + df( + 3 df( ( (. We use pointwise optimization for (, (, and take the derivatives of the Lagrangian function with respect to them at. To simpify, we drop constant terms in the derivatives and. These (simpified derivatives are in expressions (34 - (37. The derivatives of the Lagrangian function with respect to P and W( are in (38 and (39. ( G( = + c < g( + (34 ( G( = + c = g( + + (35 ( G( = 2 + c 2 g( + 2 (36 < ( G( = 2 + c 2 g( (37 = ] ( G( P = U (Y P + U (Y P g( + dg( + U (Y P (38 W( = + (39 We obtain (7 and (8 in the Proposition by setting (34, (36 to zero. From (39, we have =. We then substitute by in and (38, set it to zero, and then appy integration by parts to obtain (9. The first-order conditions for and at = are ( = c + G( ] g( (4 ( = c G( g( ]. The imit of ( as converges to from beow is c ( + G( g( + ]. Ceary, im (4 ( < (. Because incentive compatibiity requires ( to be decreasing, the monotonicity constraint must bind at (, so ( = im (. By the same argument, we have ( = im (.

18 96 T. Liu, C.-t.A. Ma / Journa of Economic Behavior & Organization 85 ( By assumption, the hazard rate G( g( is increasing, so G( g( + is increasing. Hence, ( and ( are decreasing in. Finay, from (7 and (8, the ratio of ( to ( is a constant, so the equiibrium condition in Stage 4, (2, is satisfied. Proof of Coroary : Suppose P P. Then U (Y P U (Y P = c, where the equaity foows from Proposition. From (9, we have = U (Y P U (Y P = c, so c. (42 By (7, we have c = ( ( G( + c + ] g( + ( G( g( + ] > c, where the weak inequaity is due to (42, and the strict inequaity foows from the term inside the square brackets of (43 being stricty positive. Therefore, ( < for a. Repeating the same argument, we have ( < for a. The consumer receives more treatments and the physician receives profits. This therefore impies that P > P, which is a contradiction. References (43 Akerof, G., Kranton, R., 25. Identity and the economics of organizations. Journa of Economic Perspectives 9, Arrow, K., 963. 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