ARTICLE IN PRESS. Journal of Health Economics xxx (2011) xxx xxx. Contents lists available at SciVerse ScienceDirect. Journal of Health Economics

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1 Journal of Health Eonomis xxx (20) xxx xxx Contents lists available at SiVerse SieneDiret Journal of Health Eonomis j ourna l ho me page: Optimal publi rationing and prie response Simona Grassi a,, Ching-to Albert Ma b a Faulty of Business and Eonomis, Department of Eonomis and Eonometris and Institut d Eonomie et de Management de la Santé, University of Lausanne, Building Internef, CH-05 Lausanne, Switzerland b Department of Eonomis, Boston University, United States a r t i l e i n f o Artile history: Reeived 3 January 200 Reeived in revised form 2 May 20 Aepted 22 August 20 Available online xxx JEL lassifiation: D6 H42 H44 I Keywords: Rationing Prie response Means-testing Cost effetiveness a b s t r a t We study optimal publi health are rationing and private setor prie responses. Consumers differ in their wealth and illness severity (defined as treatment ost). Due to a limited budget, some onsumers must be rationed. Rationed onsumers may purhase from a monopolisti private market. We onsider two information regimes. In the first, the publi supplier rations onsumers aording to their wealth information (means testing). In equilibrium, the publi supplier must ration both rih and poor onsumers. Rationing some poor onsumers implements prie redution in the private market. In the seond information regime, the publi supplier rations onsumers aording to onsumers wealth and ost information. In equilibrium, onsumers are alloated the good if and only if their osts are below a threshold (ost effetiveness). Rationing based on ost results in higher equilibrium onsumer surplus than rationing based on wealth. 20 Elsevier B.V. All rights reserved.. Introdution Publi supply of health are servies is very ommon. Beause of limited budgets, free health are for all is infeasible. The limited publi supply is usually distributed by nonprie rationing. Rationed onsumers often an turn to the private market and purhase at their own expense. In this paper we study optimal publi rationing poliies and prie responses in the private market. The design of rationing poliies should take into aount private market reations; otherwise, unintended onsequenes may arise. For example, expansions in Mediaid and similar programs for the indigent may atually redue onsumers purhases in the private market, a phenomenon alled rowd out (see Cutler and Gruber, 996; Gruber and Simon, 2008). The literature has not investigated the mehanism behind it. By expliitly onsidering private market responses, we exhibit a mehanism for rowd out. Two mehanisms are often used for distributing publi health servies. The first is means testing, supply based on wealth or inome. For example, Mediaid in the United States and many state programs target the indigent. The seond is ost effetiveness, supply based on a ratio of benefit to ost. For example, in most Corresponding author. Tel.: +4 (0) addresses: Simona.Grassi@unil.h (S. Grassi), ma@bu.edu (C.-t.A. Ma). European ountries and Canada, a medial servie is overed by national insurane only if its benefit ost ratio is higher than a threshold. Cost-effetiveness rationing and means-testing rationing yield different prie responses in the private market. Crowd out an be avoided under ost-effetiveness rationing. Furthermore, we show that optimal ost-effetiveness rationing results in higher equilibrium onsumer utility than means testing. In our model, onsumers are heterogenous in two dimensions: they have different wealth levels, and they have different illness severities. Wealth heterogeneity is a natural assumption, and it means that rih onsumers are more willing to pay for servies than poor onsumers. Illness severity heterogeneity is also natural. Eah illness severity is assoiated with a treatment ost and a benefit. For onveniene, we simply let severity be the treatment ost. Consumers treatment benefits are inreasing in severity, but at a dereasing rate. Our assumption on ost and benefit is similar to ommon ones in the health eonomis literature (see for example, Ellis, 998). We onsider rationing in two information regimes. In the first, rationing is based on onsumers wealth; means-testing rationing poliies belong to this regime. In the seond, rationing is based on onsumers wealth and ost; ost-effetiveness rationing poliies belong to this regime. In eah regime, we study equilibria of the following extensive form. First, the publi supplier hooses a rationing sheme. Seond, the private firm, unable to observe /$ see front matter 20 Elsevier B.V. All rights reserved. doi:0.06/j.jhealeo Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

2 2 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx onsumers wealth levels, sets its pries aording to onsumers osts of provision. Third, onsumers who are rationed by the publi supplier may purhase from the private firm. The publi supplier aims to maximize aggregate onsumer utility, while the private market onsists of a profit-maximizing monopolist. Rationing determines whom among onsumers are entitled to publi provision. In the first regime with wealth-based rationing, in equilibrium the publi supplier must ration both poor and rih onsumers, and implement prie redution in the private setor. What is the intuition behind this result? If poor onsumers are supplied, then only rih onsumers will be in the private market. The private firm ream-skims rih onsumers by setting a high prie. The publi supplier an mitigate ream-skimming by rationing some poor onsumers, making them available to the private market. The firm may then find it attrative to set a low prie when osts are low. Rationing some poor onsumers always yields a first-order gain in the form of prie redutions. In the seond information regime, rationing an be based on both wealth and ost information. Clearly, the publi supplier s equilibrium payoff must be higher ompared to rationing based only on wealth. Surprisingly, in equilibrium the publi supplier rations onsumers aording to ost information alone, ignoring wealth information altogether. The most effiient use of the publi budget is to serve those onsumers with the highest benefit ost ratio. Using rationing to implement prie redution is suboptimal beause ost effetiveness is already ahieved. The private market is an option for higher-ost onsumers who are willing to pay for the good, and remains so even if it sets a high prie. Clearly, if the publi supplier an pik one piee of information for rationing, it will hoose ost rather than wealth information. One ost information is available, wealth information does not improve the design of optimal rationing. Crowd out higher pries in the private setor is not a onern when the publi supply an be based on osts. In equilibrium, poor and rih onsumers are treated equally beause publi supply is only based on osts. Our information assumptions are plausible. The publi setor has aess to wealth information through tax returns. It may well have aess to ost information beause of servie provisions. The firm has aess to ost information. Dumping and ream-skimming are ommon problems in the health market. These problems are based on the premise that firms get to selet less ostly patients, so we follow a well reognized assumption in the literature. In Grassi and Ma (forthoming), we study a similar model, but the publi rationing and private prie shemes are hosen simultaneously. That model offers a longer term perspetive on the interation, beause publi rationing and private prie shemes must be mutual best responses. In Grassi and Ma (forthoming), ost effetiveness is an equilibrium when rationing is based on wealth and ost. If rationing is based on wealth, the game has a ontinuum of equilibria, all of whih differ from the equilibrium here. In the equilibrium with the highest welfare, all poor onsumers are supplied in the publi setor while all rih are rationed and available in the market. Prie redution is never implemented there. A ommon result in the literature of publi provision of private goods is that the publi setor serves poor onsumers while the private setor serves rih onsumers. This is the theme in Besley and Coate (99) and Epple and Romano (996). In our model, when rationing is based on wealth, the private setor will serve some poor onsumers. Contrary to the standard result, a omplete separation of the poor and rih does not obtain. In both Besley and Coate (99) and Epple and Romano (996), taxes and inome redistributions are a onern, while we studynonprie rationing under a fixed budget. Also, while both assume a perfetly ompetitive private market, we onsider a monopolisti private market. A ompetitive private market is a ommon assumption in the literature. Barros and Olivella (2005) onsider dotors working in the publi setor who self-refer patients to their private praties. Pries paid by patients in the private setor are fixed, while dotors only refer low-ost patients. Iversen (997) studies waiting-time rationing when there is a private market. Hoel and Sæther (2003) onsider the effet of ompetitive supplementary insurane on a national health insurane system. Also the extensive literature on rationing by waiting times either assume away the private setor, or use a perfetly ompetitive private market (see for example, Gravelle and Siiliani, 2008, 2009). In fat, when the private market priing rule is fixed, one only an study how it influenes publi poliies. By ontrast, we study how publi poliies influene private market responses. Cost effetiveness as a riterion to alloate sare resoures has been advoated for a long time (see for example, Weinstein and Zekhauser, 973, or Garber and Phelps, 997). Hoel (2007) disusses how ost effetiveness should be modified when treatments are also available in a ompetitive market. Following Hoel (2007), we study ost effetiveness when a private market exists, but we believe we are the first to derive ost effetiveness as the optimal rationing poliy given a monopolisti private market. Setion 2 lays out the model. Setion 3 and its subsetions desribe the firm s hoie of the profit-maximizing pries and the equilibrium rationing when the publi supplier observes only onsumers wealth level. Setion 4 and its subsetions fous on the information regime where wealth and ost levels are observed by the publi supplier. The last setion ontains some onluding remarks. Appendix A ontains proofs. 2. The model 2.. Consumer utility and benefit There is a set of onsumers. Eah onsumer s wealth is either w or w 2, with 0 < w < w 2. Let m i > 0 be the mass of onsumers with wealth w i, i =, 2. We all onsumers with wealth w poor onsumers, and onsumers with wealth w 2 rih onsumers. Eah onsumer may onsume, at most, one unit of a health are good or treatment. Consumers differ in illness severity, and the ost of providing the good inreases with severity. We use treatment ost to measure severity. Aordingly, we let the monetary ost of providing the good vary on the positive interval [, ], with a distribution funtion G : [, ] [0, ] and an assoiated density g. Let be the expeted value of. We identify a onsumer by his wealth and provision ost, and all him either a rih or poor type- onsumer. The lower support an be interpreted as the minimum severity level above whih treatment may be warranted. A type- onsumer reeives a health benefit from the treatment. This benefit varies aording to severity. Let the funtion H : [, ] R + denote the utility benefits, so a type- onsumer reeives a utility H() from treatment. We let the funtion H be stritly inreasing and onave. A siker onsumer reeives more benefit from treatment, but this benefit inreases at a noninreasing rate. If a type- onsumer with wealth w i pays a prie p for the good, his utility is U(w i p) + H(), while if he does not onsume the good (and pays nothing), his utility is U(w i ). The funtion U is stritly inreasing and stritly onave. We an use a general utility funtion where the utilities from onsuming the good at prie p, and from not onsuming the good, are U(w p, H()) and U(w, 0), In Grassi and Ma (2009), the benefit H() is onstant and normalized to. All the results presented here remain valid under the assumption of a onstant benefit. Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

3 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx 3 respetively. A separable utility funtion simplifies the analysis, but it does mean that rih and poor onsumers reeive the same utility from treatment. A rih or poor type- onsumer s willingness to pay is denoted by i (), and defined impliitly by U(w i i ) + H() = U(w i ), i =, 2, () so i () is the maximum prie a type- onsumer with wealth w i is willing to pay Publi supplier and rationing poliies A publi supplier has a budget B whih is insuffiient to provide the good for free to all onsumers, so we assume B < (m + m 2 ). We onsider two information regimes. In the first, the publi supplier an use a nonprie rationing mehanism based on wealth. In the seond, the publi supplier uses a nonprie rationing mehanism based on both wealth and ost. The first regime orresponds to a means-test poliy regime. For example, in the U.S., indigent onsumers qualify for health insurane provided by Mediaid. The seond regime inludes a ost-effetiveness riterion that is ommonly used in European ountries. For example, all onsumers are overed under a national insurane or health servie, but servies are only provided when they satisfy ost-effetiveness riteria. When rationing is based on onsumers wealth, a rationing poliy is a pair of frations (, 2 ), 0 i, i =, 2. For eah wealth lass w i, the publi supplier rations i m i onsumers, and supplies ( i )m i onsumers. When rationing is based on onsumers wealth and osts, a rationing poliy is a pair of funtions (, 2 ), i : [, ] [0, ], i =, 2. The value i ()g() is the density of onsumers with wealth w i and ost who are rationed. For eah wealth lass, the mass of rationed onsumers with ost less than is m i i(x)g(x)dx, while the mass of supplied onsumers is m i [ i (x)]g(x)dx. The publi supplier s payoff is the sum of onsumer utilities. We fous on the optimal publi supply, not the optimal regulation of the entire market. Therefore, it is natural to assume that the publi supplier is onerned with onsumer surplus. We onsider an unweighted sum of onsumer surplus, but will disuss how our results will hange when the poor s utility is given a higher weight than the rih s. We now write down the benhmark rationing poliies when there is no private supply. First, the aggregate onsumer utility when rationing is based on wealth is } m i { i U(w i ) + ( i ) [U(w i ) + H()]g()d, i= where, for eah wealth lass, the rationed onsumers have utility U(w i ), while supplied onsumers have utility U(w i ) + H(). The budget onstraint is m i [ i ]g()d = i= m i ( i ) = B, i= whih says that the expeted ost of supplying onsumers is equal to the budget. Any rationing poliy (, 2 ) that exhausts the budget is optimal. When rationing is based on wealth alone, supplying a poor onsumer yields the same expeted benefit as supplying a rih onsumer. Seond, the aggregate onsumer utility, when rationing is based on wealth and ost, is { } m i i ()U(w i )g()d + [ i ()][U(w i ) + H()]g()d. i= The aggregate onsumer utility simplifies to m i U(w i ) + i= m i [ i ()]H()g()d. i= The budget onstraint is m i [ i ()]g()d = B. i= An optimal rationing poliy based on wealth and ost is a pair (, 2 ) that maximizes the aggregate onsumer utility subjet to the budget onstraint. The optimal poliy is the familiar ost effetiveness priniple. Consider the benefit less the ost adjusted by the multiplier of the budget onstraint: H(). It is optimal to supply a type- onsumer if and only if this is positive. 2 We have interpreted as the severity threshold for warranted treatment, so we let H() be suffiiently high. From this and the onavity of H, we have H() > 0 if and only if < B where B exhausts the budget if onsumers with ost lower than B are supplied: (m + m 2 ) B g()d = B. As severity inreases, the health benefit inreases but at a noninreasing rate, so it is not ost effetive to treat very severe ases. Also, the ost effetiveness priniple gives equal treatment to the rih and poor onsumers beause they reeive the same benefit. This implies that wealth information is not required for optimal rationing Private market and onsumers willingness to pay There is a private market whih we model as a monopoly. The firm observes a onsumer s ost, but not his wealth w i. To maximize profits, and given the publi supplier s rationing poliy, the private firm hooses pries as a funtion of osts. Beause a onsumer buys, at most, one unit of the indivisible good, prie disrimination in the form of quantity disount is infeasible. 3 In this subsetion, we present properties of the onsumer s willingnessto-pay funtions, as well as the monopolist s priing strategy in a benhmark ase of zero publi supply. Reall that the willingness to pay, i, in () is impliitly defined by U(w i i ) + H() = U(w i ), i =, 2. Beause U is stritly onave, () < 2 () for eah ; a rih type- onsumer is willing to pay more for the good than a poor type- onsumer. From H and U stritly inreasing, the willingness to pay, i (), is stritly inreasing. Indeed, for i =, 2 we have i () = H () U (w i i ()) > 0. (2) 2 The Lagrangean is L = m [ ()]H() + m 2[ 2()]H() + {B m [ ()] m 2[ 2()]}, and the first-order derivative with respet to i is m i (H() ). It is optimal to set i = 0 if and only if H() >. 3 In our model, if the firm managed to observe onsumers wealth and osts, it would extrat all onsumer surplus. In this ase, the existene of the private market would be irrelevant to strategi onsideration of rationing. Also, it is implausible to assume that the firm has no information about ost. A firm must eventually learn about ost, and it may renege on provision if the ost turns out to be higher than prie. Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

4 4 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx prie-ost margin. When there is no publi supply, the profits from these two pries are ( (); ) (m + m 2 )[ () ] (3) τ () o 45 τ 2 () 2 Fig.. Willingness to pay and 2. ( 2 (); ) m 2 [ 2 () ]. (4) By the strit onavity of i, the profit funtions in (3) and (4) are stritly onave in. The ream-skimming literature typially hypothesizes that firms prefer to treat less severe patients. In most priing models, a firm s profit is dereasing in ost. To rule out profits inreasing in osts, we assume that both () and 2 () are smaller than. In this ase, the derivatives of (3) and (4) with respetive to are negative, so that profit does derease with severity. From the expression for i () in (2), if H is smaller than U, the assumption that i () < is valid. Next onsider the differene between the profits from setting a low prie and a high prie, namely the differene between (3) and (4). After simplifiation, this differene is m [ () ] m 2 [ 2 () ()], and its derivative is m [ () ] m 2 [ 2 () ()] < 0, beause 2 () > () and () <. Hene, the profit funtions (3) and (4) ross, at most, one. We will analyze situations in whih the firm will find it optimal to redue the prie from 2 () to () at some ost. Our interest is how rationing implements a prie redution. This issue would be moot if the prie always stayed high at 2 (). We therefore assume that Also, the willingness to pay, i (), is stritly onave. 4 Furthermore, at eah, we have () < 2 (): the rih onsumer s willingness to pay funtion is both higher and inreasing faster than the poor onsumer. 5 Next we define a ost threshold. From our assumption that H() is suffiiently high, we also have () > : the firm is able to sell the good to onsumers with low severities. We assume that () is suffiiently onave and that is suffiiently large so that at some <, we have ( ) =. In sum, we assume that at low severity levels, a poor onsumer s willingness to pay is higher than the treatment ost, but there will be a ost suffiiently high (at ) at whih the benefit H() is not worthwhile to him. We an also analogously define 2 by 2 ( 2 ) = 2 (if there is suh a 2 ). Fig. illustrates the properties of the two willingness-to-pay funtions. There, the two inreasing and onave funtions graph the and 2 for the poor and rih onsumers. We assume that before reahes, the onave funtion must ut the 45-degree ost line from above. At any, the two willingness to pay, () and 2 (), are the firm s andidate profit-maximizing pries. Clearly, if, the firm annot sell to poor onsumers, beause their willingness to pay is lower than ost. Therefore, at any, the firm sets the prie at 2 (), selling only to rih onsumers. At ost <, there are two andidate pries, () and 2 (). If the firm sells to both rih and poor onsumers, it harges the lower prie (), but if it sells only to rih onsumers, it harges the higher prie 2 (). There is the usual trade-off between selling to less onsumers at a higher prie-ost margin and selling to more onsumers at a lower 4 From (2), we have i () = ((U (w i i )H () + H ()U (w i i ) i ())/(U (w i i ()) 2 )) < 0. 5 By definition, U(w ()) + H() = U(w ) < U(w 2) = U(w 2 2()) + H(), so U(w ()) < U(w 2 2()), and U (w ()) > U (w 2 2()). From (2), It follows that () < 2 (). (m + m 2 )[ () ] > m 2 [ 2 () ], (5) whih says that at the lowest ost, the firm s optimal prie is () to sell to both poor and rih onsumers. Beause at =, ( ) =, so 0 = (m + m 2 )[ ( ) ] < m 2 [ 2 ( ) ], the firm s optimal prie is the high prie 2 ( ) at. Our assumption (5) implies that there must exist a unique m between and suh that (m + m 2 )[ ( m ) m ] = m 2 [ 2 ( m ) m ], whih simplifies to m [ ( m ) m ] = m 2 [ 2 ( m ) ( m )]. (6) The ost level m is where prie redution ours. At ost > m, the firm will harge the high prie 2 (), but at < m, it will harge the low prie (). Fig. 2 illustrates the determination of m. The two downward sloping, onave graphs are the profit funtions (3) and (4), and their intersetion defines m Extensive forms We onsider the following extensive-form games: Stage 0: For eah onsumer who has either wealth w or w 2, Nature draws a ost realization aording to the distribution G. The private firm observes a onsumer s ost realization, but not his wealth. Under rationing based on wealth, the publi supplier observes a onsumer s wealth, but not the ost realization. Under rationing based on wealth and ost, the publi supplier observes a onsumer s wealth and ost. Stage : Under rationing based on wealth, the publi supplier sets a rationing poliy (, 2 ), 0 i, supplying ( i )m i of onsumers with wealth w i, i =, 2. Under rationing based on wealth and ost, the publi supplier sets a rationing poliy (, 2 ), i : [, ] [0, ], supplying [ i ()]m i of onsumers with wealth w i and ost. Stage 2: The firm sets a prie for eah ost realization. Stage 3: Consumers who are rationed by the publi supplier may purhase from the firm at pries set at Stage 2. Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

5 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx 5 π () (m + m )[ τ () ] (m + m )[ τ () ] m [ τ () ] m [ τ () ] m Fig. 2. Profits from low and high pries. We study subgame-perfet equilibria. In Stage the publi supplier sets the rationing poliies. A subgame in Stage 2 is a ontinuation game given the rationing poliy in Stage. An equilibrium in Stage 2 refers to the equilibrium of the ontinuation subgame defined by a rationing poliy in Stage. 3. Equilibrium rationing and pries in wealth-based rationing than for a prie redution to our. In an extreme, if only the rih onsumers are rationed and all the poor are supplied, the firm will never redue the prie. We summarize the firm s equilibrium pries in Stage 2 by the following (the proof omitted): Lemma. Given a rationing poliy (, 2 ), if r in (9) is greater than, in equilibrium the firm sets the high prie 2 () if > r, and the low prie () if < r. Otherwise, in equilibrium the firm always sets the high prie 2 (). 3.. Equilibrium pries 3.2. Equilibrium rationing In this subsetion, we derive the equilibrium in Stage 2. Given a rationing poliy (, 2 ), only m of poor onsumers and 2 m 2 Given the equilibrium pries in Stage 2, the aggregate onsumer of rih onsumers are available to the firm. The firm may set a low utility is: { } { r }] m [( ) U(w ) + H()dG + [U(w ()) + H()]dG + U(w )dg + m 2 [( 2 ) { U(w 2 ) + H()dG } { r + 2 [U(w 2 ()) + H()]dG + r r [U(w 2 2 ()) + H()]dG }]. prie (), selling to both rih and poor onsumers, or a high prie 2 (), selling only to rih onsumers. These strategies yield profits: ( (); ) (m + m 2 2 )[ () ] (7) ( 2 (); ) m 2 2 [ 2 () ]. (8) These profit funtions are both dereasing and onave, as in the ase when the firm has aess to all onsumers (ompare with (3) and (4)). Reall that m is the ost threshold at whih the equilibrium prie swithes from 2 () to () when the firm has aess to the entire market of onsumers. Analogously, we an haraterize the equilibrium in Stage 2 by the ost level r at whih the prie swithes from 2 () to () under the rationing poliy (, 2 ). If there is suh a ost level r between and, it is given by (m + m 2 2 )[ ( r ) r ] = m 2 2 [ 2 ( r ) r ], whih simplifies to m [ ( r ) r ] = m 2 2 [ 2 ( r ) ( r )]; (9) otherwise we set r at. As the ost drops below, a prie redution is worthwhile only if there are enough poor onsumers relative to rih ones. If there are few poor onsumers in the market, the ost has to be muh lower In this expression, terms involving ( i ) are onsumers utilities when they reeive the publi supply at no harge. Terms involving i are the market outomes. For poor onsumers, if their osts are below r, they purhase at (), whih atually leaves them no surplus (see definition of i () in ()). Similarly, for rih onsumers, if their osts are above r, they purhase at prie 2 (), earning no surplus. However, if rih onsumers osts are below r, they earn a surplus U(w 2 ()) + H() U(w 2 ) () > 0 sine the prie () is lower than their willingness to pay, 2 (). Using the definitions of the willingness to pay, i, i =, 2, we simplify the aggregate onsumer utility to [m U(w ) + m 2 U(w 2 )] + [m ( ) + m 2 ( 2 )] H()dG r + m 2 2 ()dg, (0) where r haraterizes the firm s equilibrium prie strategy. The first term is the onsumers utility from wealth. The middle term is the total expeted benefit from publi supply, while the last term is the sum of rih onsumers inremental surplus () when they purhase at prie (). Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

6 6 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx We introdue a new notation ˇ B/. Beause B denotes the available budget, and the expeted ost, ˇ is the number of supplied onsumers. In equilibrium the budget B must be exhausted. Hene, we replae m ( ) + m 2 ( 2 ) by ˇ, and simplify (0) to [ ] r m U(w ) + m 2 U(w 2 ) + ˇ H()dG + m 2 2 ()dg. () An equilibrium is a rationing poliy (, 2 ) and the equilibrium prie-redution ost threshold in (9) that maximize (), subjet to the budget onstraint m ( ) + m 2 ( 2 ) = ˇ B (< m + m 2 ), (2) and the boundary onditions r, and 0 i, i =, 2. Proposition. In equilibrium, the publi supplier rations onsumers in eah wealth lass: > 0 and 2 > 0, while the firm harges the low prie () when the onsumer s ost is below a threshold r, where < r <. Proposition (whose proof is in Appendix A) says that for any budget, the publi supplier must ration some poor onsumers and some rih onsumers, and prie redution must our. By assumption, if the firm has aess to all onsumers, it will redue the prie from 2 () to () at < m. The publi supplier an always implement prie redution by setting = 2 > 0, whih maintains the same ratio of rih to poor onsumers as in the full market (ompare (6) and (9)). Some surplus in the private market must be available to onsumers. If = 0, then all poor onsumers are supplied, and the prie must remain high at all osts. If 2 = 0, all rih onsumers are supplied, so they do not partiipate in the private market. In either ase, trade surplus in the private market annot be realized, but this annot happen in equilibrium. Therefore, we must have > 0 and 2 > 0, and ost redution. How does the publi supplier set the rationing poliy? What sort of trade-off is involved? The publi supplier s objetive is to maximize the onsumer surplus in (). Without the onstant terms, the objetive funtion is r m 2 2 ()dg. (3) This is the inremental surplus enjoyed by rationed rih onsumers buying at prie (); all of them have osts below the prieredution ost threshold r. Obviously, the publi supplier would like threshold r to be high, and would like 2 to be high. In that ase, more rih onsumes an realize more surplus from the market. But these two goals, raising the prie-redution ost threshold and rationing more rih onsumers, are inompatible. Consider rationing more rih onsumers. This inreases 2. Some of the budget is now available to supply poor onsumers, so dereases. In other words, there are more rih onsumers and less poor onsumers in the market. The firm finds it less profitable to redue prie, so ost must fall lower before prie redution happens in equilibrium. The value of the ost threshold r dereases as 2 inreases. Raising both 2 and r is impossible. 6 The basi trade-off is between a bigger range of ost redution for fewer rih onsumers and a smaller range of ost redution for more rih onsumers. 6 If 2 inreases, then must derease due to the budget onstraint (2). From (9), when 2 inreases and dereases, r must derease. This is beause for all, 2() () is inreasing in whereas () is dereasing in. Changes in 2 and r are onstrained by the budget as well as the equilibrium in Stage 2. We use (9) and (2) to eliminate and obtain m 2 2 = K ( r ) r, (4) 2 ( r ) r where K m + m 2 ˇ > 0. Substituting (4) into (3), we now an haraterize the equilibrium by the hoie of r that maximizes [ ] r ( r ) r K ()dg (5) 2 ( r ) r subjet to the boundary onditions. The objetive funtion in (3) is a produt of r [ ( r ) r ]/[ 2 ( r ) r ] and ()dg. They are, respetively, the ratio of prie-ost margins at low and high pries, and the inremental onsumer surplus. The total effet on (5) as r hanges depends on the proportional hanges in the produt omponents as r hanges. The first is dereasing in r while the seond is inreasing. We now present the haraterization of the equilibrium in the following proposition (whose proof is in Appendix A): Proposition 2. If the budget B is suffiiently large, the equilibrium prie-redution ost threshold r is the unique solution of [ ] 2 ( r) ( r) + ( r)g( r ) 2 ( r ) r ( r ) r r ()dg = 0 (6) and the equilibrium rationing poliy (, 2 ) an be reovered from (2) and (4): ] < and = m [ + m 2 ˇ 2 (r ) (r ) m 2 (r ) r 2 = m + m 2 ˇ m 2 [ ( r ) r 2 ( r ) r ] <. (7) If the budget is small, either or 2 may be equal to, and the publi supplier may ration an entire wealth lass. If i =, then j = (ˇ/m j ), i, j =, 2, and i /= j, and the value of r then is obtained from (9) with i =. Eq. (6) in Proposition 2 is the first-order ondition for the maximization of (5) when the boundary onditions for i do not bind. If a boundary ondition on i binds, then the onstraint set uniquely determines the optimum. The trade-off is between rationing rih onsumers so they enjoy the inremental surplus in the private market and rationing poor onsumers to implement more prie redution. The optimal tradeoff is ahieved by differentially supplying rih and poor onsumers. With a large budget, the manipulation of this ratio is easier. This orresponds to the first part of Proposition 2 when the boundary onditions i are slak. With a small budget, the manipulation is to withhold supply to a whole lass of onsumers. This orresponds to a binding boundary ondition. In Fig. 3, we graph the downward-sloping budget line (2), and the dotted lines for the boundary onditions for m i i. The feasible set is the triangle formed by the boundary onditions and the budget line. A bigger budget means more onsumers an be supplied (a higher ˇ), so the budget line shifts downward. The upward-sloping line graphs the ombinations of m and m 2 2 that implement a prie redution at ost threshold r. In Fig. 3, the boundary onditions i do not bind. The prieredution ost threshold r (in (6)) is implemented by the poliy in () (the intersetion between the two solid lines in the figure). Here, there is enough budget to implement r. The ost threshold r is independent of the budget, as is the ratio between and 2. Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

7 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx 7 m 2 θ 2 mθ m θ = m m2 β * * τ (r ) m θ 2 2 = mθ r * * τ 2(r ) τ(r ) It sets the high prie if (8) is violated, and it may randomize between () and 2 () if (8) holds as an equality. These are the equilibrium pries in Stage 2. We now define an indiator funtion for equilibria when <. Let p : [, ] [0, ]. Given a poliy (, 2 ), we set p() = if (8) holds as a strit inequality, p() = 0 if (8) is violated, and p() to a number between 0 and if (8) holds as an equality. When p() takes the value 0, the firm hooses the high prie, so there is no prie redution. When p() takes the value, the firm hooses the low prie, so there is a prie redution. Lemma 2. For between and, any equilibrium in Stage 2 is given by a funtion p : [, ] [0, ] satisfying the following two inequalities: m θ p(){m ()[ () ] m 2 2 ()[ 2 () ()]} 0 (9) Fig. 3. Budget onstraint, ost threshold, and boundary onditions. The seond part of Proposition 2 is about the equilibrium when the budget is small. Suppose that the ratio of rih to poor onsumers in the market should derease to favor prie redution, so this requires supplying more rih onsumers than poor ones. With a small budget, this may mean rationing all poor onsumers so all of them are in the private market. A boundary ondition binds. In general, the equilibrium ost threshold r may be higher or lower than m. Nevertheless, if 2 =, we must have <, and r < m. Rationing all rih onsumers means that the budget must be spent on poor onsumers. With less poor onsumers in the market, prie redution is less often. Then publi supply redues transations in the private market. This explains rowd out. The publi supplier s objetive is to maximize the sum of poor and rih onsumers utilities. If there is any equity onern, more weight will be given to poor onsumers. In this ase, rationing will favor the poor, so fewer poor onsumers will be in the market. The equilibrium prie-redution ost threshold will fall, so pries tend to be higher. Equity onern tends to redue the likelihood of prie redution, and generates a larger extent of rowd out. 4. Equilibrium rationing and pries in wealth-ost based rationing 4.. Equilibrium pries We begin with the equilibrium pries given a rationing poliy (, 2 ), i : [, ] [0, ]. Again, there are only two possible equilibrium pries in the private market, the low prie () and the high prie 2 (). For any >, the firm s unique best response is 2 (). For any between and, the firm hooses between the low prie, (), and the high prie, 2 (). The firm s profit from the low prie is [m () + m 2 2 ()][ () ]; the profit is m 2 2 ()[ 2 () ] from the high prie. The firm sets the low prie if [m () + m 2 2 ()][ () ] m 2 2 ()[ 2 () ], or m ()[ () ] m 2 2 ()[ 2 () ()]. (8) [ p()]{m ()[ () ] m 2 2 ()[ 2 () ()]} 0 (20) Lemma 2 (whose proof is in Appendix A) defines a prieredution funtion p() to indiate the equilibrium in Stage 2. The term inside the urly brakets of (9) and (20) is the profit differene between harging the low prie and the high prie (see (8)). The inequalities (9) and (20) are omplementary onditions for prie redution. When the firm harges the low prie, p() must be equal to for (9) and (20) to hold simultaneously; onversely, when the firm harges the high prie, p() must be equal to 0. For ease of exposition, we extend the funtion p from the domain [, ] to [, ], and set p() = 0 for >. This simply says that there is no prie redution for >. This extensions allows us to write payoffs in a simpler way Equilibrium rationing We begin with the publi supplier s payoff given the equilibrium pries: {m [ ()][U(w ) + H()] + m 2 [ 2 ()][U(w 2 ) + H()]}dG() + m (){[ p()]u(w ) + p()[u(w ()) + H()]}dG() + m 2 2 (){[ p()][u(w 2 2 ()) + H()] + p()[u(w 2 ()) + H()]}dG(). In this expression, the first integral is the sum of utilities of supplied onsumers; eah onsumer gets the benefit H() without inurring any ost. The seond integral is the sum of utilities of rationed poor onsumers. A poor type- onsumer will enounter a prie redution with probability p(). If there is no prie redution, the poor onsumer does not buy, so his payoff is U(w ). If there is a prie redution, the poor onsumer buys at prie (), hene the term U(w ()) + H(). The last integral is the sum of utilities of rationed rih onsumers. If there is no prie redution, the rih onsumer buys at 2 (), hene the term U(w 2 2 ()) + H(). If there is a prie redution, he buys at (), hene the term U(w 2 ()) + H(). The gain in utility when onsumers partiipate in the market is due to the rih onsumers purhasing at the low prie (). Poor onsumers either do not buy or buy at their reservation prie (), gaining no surplus from the private market. We use the definitions of () and 2 () to simplify the payoff to: m U(w )+m 2 U(w 2 ) + {m [ ()] + m 2 [ 2 ()]}H()dG() + m 2 2 ()p() ()dg(). (2) Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

8 8 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx CASE p() = 0 φ =φ2 = 0 Consumers supplied for free Prie redution never ours B φ =φ2 = Consumers rationed CASE 2 p() = 0 φ = φ2 = m 0 B φ =φ2 = CASE 3 φ =φ2 = 0 φ =φ2 = B m p() = 0 p() = p() = 0 Prie redution Fig. 4. Equilibrium rationing and prie redution. In (2), the first integral is onsumers utility gain from the publi supply, and the seond integral is the inremental gain of rationed rih onsumers who purhase in the private market at the low prie (). (Reall () U(w 2 ()) + H() U(w 2 ).) The optimal rationing poliy is one that maximizes (2) subjet to the budget onstraint, and the equilibrium pries in the private market. By Lemma 2, the equilibrium prie in Stage 2 is haraterized by the prie-redution funtion p(). Ignoring the onstant terms in (2), we write down the maximization program for the publi supplier s equilibrium poliy: hoose a poliy (, 2 ) and a funtion p to maximize {m [ ()] + m 2 [ 2 ()]}H()dG() subjet to B + m 2 2 ()p() ()dg() (22) {m [ ()] + m 2 [ 2 ()]}dg() 0 (23) p(){m ()[ () ] m 2 2 ()[ 2 () ()]} 0 (24) [ p()]{m ()[ () ] m 2 2 ()[ 2 () ()]} 0, (25) and the boundary onditions 0 i (), i =, 2, 0 p(), eah in [, ], and p() = 0 for >. Inequality (23) is the budget onstraint. For ompleteness, we have rewritten the two inequalities in Lemma 2 as (24) and (25). Proposition 3. In the optimal rationing poliy based on wealth and ost, the publi supplier rations onsumers if and only if their osts are above a threshold. That is, in an equilibrium, () = 2 () = 0 for < B () = 2 () = for > B, where the ost threshold B is defined by B (m + m 2 )dg() = B. Proposition 3 (whose proof is in Appendix A) says that equilibrium rationing oinides with ost effetiveness when the private market is absent. Rih and poor onsumers are treated equally, and a type- onsumer is given publi supply if and only if the net benefit is high: H() >, where is the multiplier of the budget onstraint (23). Beause the benefit from onsumption H() is onave, the benefit ost ratio, H()/, is higher at low osts and dereases with, so low-ost onsumers get the publi supply. The ost level B in the proposition refers to one at whih supplying the good to onsumers with osts below B will exhaust the budget. The presene of the firm does not hange the ost-effetiveness priniple. What is behind this result? Unlike the regime when rationing is based only on wealth, implementing ost effetiveness is possible when rationing is based on wealth and ost. The firm sets the high prie 2 () when there are many rih onsumers, but the low prie () if there are few rih onsumers. How does the firm s best response interat with ost effetiveness? If the publi supplier provides for the rih, prie redution is irrelevant. If the publi supplier provides for the poor, prie redution annot be an equilibrium: without poor onsumers, the firm will set the high prie. Cost effetiveness, however, alls for equal treatment to the rih and the poor. At eah ost level, the publi supplier either provides for both rih and poor onsumers, or none at all. The ratio between rih and poor onsumers in the private market is the same as if the firm had aess to all onsumers. If a prie redution ours, it follows the same fashion as if there was not any publi supply. Crowd out does not happen in equilibrium. Fig. 4 shows the three ases that make up the proof of Proposition 3. Prie redution happens if and only if ost falls below m. In Case, the budget is large so that it is ost effetive to supply all onsumers with osts up to a threshold above. In Case 2, the budget is medium sized, and may over some onsumers with ost above m. There is still no prie redution at ost between m and beause m is the minimum ost level at whih prie redution begins to be profitable. In Case 3, the budget is small. Here, prie redution ours at < m. Clearly, the publi supplier s equilibrium payoff aggregate onsumer utility under rationing based on ost and wealth annot be lower than rationing based on wealth alone. In Proposition 3 the optimal rationing rule is based only on ost. One ost information is available, wealth information does not improve the publi supplier s payoff. We summarize by the following: Corollary. Equilibrium aggregate onsumer utility is higher under rationing based on ost than wealth. If the publi supplier must pik between ost and wealth information to administer rationing, it optimally will hoose ost information. Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

9 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx 9 Finally, we omment on equity onern. Atually, under rationing based on wealth and ost, the publi supplier rations rih and poor onsumers equally. If there is an expliit onstraint on supplying poor onsumers more, then the ost effetiveness priniple annot be applied diretly. When publi supply favors the poor, fewer of them will be in the market, and the rih will be less likely to experiene a prie redution. 5. Conluding remarks We have presented a model to study the effet of rationing on pries in the private market. Publi poliies should take into aount market responses. We show that if rationing is based on wealth information, the optimal poliy must implement a prie redution in the private market. This is ahieved by leaving some poor onsumers in the private market. If the publi supplier observes onsumers wealth and ost, optimal rationing is based on ost effetiveness; wealth information is not neessary. Our model sheds light on rowd out, and the design of publi programs when private market responses are important. We assume two wealth lasses to make the model tratable. Extending the model and deriving the equilibrium rationing sheme for many, or a ontinuum of wealth lasses involve omplex omputation. Many possible prie redution onfigurations must be onsidered. We believe that our basi result is robust. In other words, some onsumers with lower wealth will be rationed to implement more prie redutions. We have used a separable utility assumption. In Grassi and Ma (forthoming), we list some fators that may influene the results when utility funtions are not separable in money and benefits. A seondary effet from onsumption on the marginal utility of inome will have to be onsidered. If inome effets are small, our results extend to the general utility funtion. We have assumed a fixed budget. Extending the model to onsider an optimal budget is fairly straightforward. We have obtained the optimal poliies in the two information regimes, so that the optimized aggregate onsumer utility is available. One the ost of publi funds is speified, the usual optimization steps an be taken to haraterize the optimal budget. The analysis here is limited to free publi supply, but obviously a fixed user fee an be inluded. Due to risk aversion, publily provided health insurane usually does not impose signifiant opayments. Nevertheless, a general analysis of optimal monetary subsidy may be fruitful. Aknowledgement We thank seminar partiipants at Universitat Autonoma de Barelona, University of Bern, Universidad Carlos III de Madrid, Universit de Lausanne, Mihigan State University, and Universit degli Studi di Milano; and onferene partiipants at XXIV Jornadas de Eonomia Industrial in Vigo, ECARES-CEPR in Brussels, and Seminars in Health Eonomis and Poliy in Crans Montana; and espeially Pedro Barros for omments and suggestions. We also thank Editor Tony Cuyler, and two referees for their advie. Appendix A. Proof of Proposition. Beause all terms in square brakets in the objetive funtion () are onstant, we alternatively an write r the objetive funtion as m 2 2 ()dg. The boundary onditions r and 0 2 do not bind. If either r = or 2 = 0 at a solution, then the optimized value r is m 2 2 ()dg = 0. We show that a rationing poliy with = 2 = k > 0 does stritly better. This poliy satisfies the budget onstraint (2) for some 0 < k <. Moreover, from (6) and (9), we have r = m > by assumption. Therefore, the rationing poliy m = 2 = k is feasible, and yields a payoff m 2 k ()dg > 0. This implies that at a solution r > and 2 > 0. Beause r >, it follows from (9) that must be bounded away from 0. Proof of Proposition 2. The steps for simplifying the objetive funtion into (5) are already laid out before the Proposition. We differentiate the logarithm of (5) to get the first-order ondition (6). We now show that the solution to this first-order ondition is unique. The objetive funtion (5) is the produt of [ ( r ) r ]/[ 2 ( r ) r ] and ()dg. We show that the r derivative of the square-braketed term is negative. The derivative of its logarithm is the term in square brakets in (6), and that is negative. To see this, note that 2 ( r ) r > ( r ) r, and 2 ( r) < ( r), and all these terms are positive. So we have {[ 2 ( r)]/[ 2 ( r ) r ]} < {[ ( r)]/[ ( r ) r ]}. Obviously the integral in (5) is inreasing in r, and its derivative is the other term in (6). We onlude that (6) has a unique solution. The equilibrium rationing poliy in (7) is obtained by solving (2) and (4) simultaneously at r = r. If ˇ is suffiiently large, the right-hand side values in (7) will be less than, and the omitted boundary onditions i are satisfied. Otherwise, a boundary ondition binds. Proof of Lemma 2. Consider any equilibrium pries in Stage 2. In this equilibrium, at ost the firm will harge either () or 2 () depending on whether (8) is satisfied. If we have defined p by the method just before the statement of the Lemma, inequalities (9) and (20) are satisfied. Conversely, let a funtion p : [, ] [0, ] satisfy inequalities (9) and (20). We show that it haraterizes a best response priing strategy to any poliy (, 2 ). Suppose that p() =. Inequality (20) is satisfied by any () and 2 (). Inequality (9) requires the term inside the urly brakets to be positive, and this means that (8) is satisfied. Next, suppose that p() = 0. Inequality (9) is always satisfied. Inequality (20) requires the term inside the urly brakets in (20) to be negative, and this means that (8) is violated. Last, if p() is a number stritly between 0 and, both (9) and (20) must hold as equalities, so that (8) must be an equality. Eah value of p() satisfying (9) and (20) orresponds to an equilibrium prie. Proof of Proposition 3. We use pointwise optimization to solve for the optimal rationing poliy. We onsider a relaxed program in whih onstraint (25) is omitted; we will show that in the solution of the relaxed program onstraint (25) is satisfied. To simplify notation, we multiply (24) by g(), so that g() an be ignored for pointwise optimization. Let denote the multiplier for the budget onstraint (23), and () the multiplier for (24) at. The Lagrangean is L = m [ ()]H() + m 2 [ 2 ()]H() + m 2 2 ()p() () + {B m [ ()] m 2 [ 2 ()]} + ()p(){m ()[ () ] m 2 2 ()[ 2 () ()]}, where we have omitted the boundary onditions on i and p. For >, p() = 0, so there is no need to optimize over p, and the first-order derivatives are = m H() + m (26) 2 = m 2 H() + m 2. (27) Please ite this artile in press as: Grassi, S., Ma, C.-t.A., Optimal publi rationing and prie response. J. Health Eon. (20), doi:0.06/j.jhealeo

10 0 S. Grassi, C.-t.A. Ma / Journal of Health Eonomis xxx (20) xxx xxx For <, the first-order derivatives are = m H() + m + ()p()m [ () ] (28) = m 2 H() + m 2 ()p()m 2 [ 2 () ()] 2 + m 2 p() () (29) p = m 2 2 () () + (){m ()[ () ] m 2 2 ()[ 2 () ()]}. (30) We onsider three ases, aording to the size of the budget. Case is when the budget is large: B > ; that is, the budget is suffiient to over osts up to a level where poor onsumers willingness to pay equals ost ( ) =. To prove the proposition, we set = H( B )/ B. Now onsider > B. Beause H()/ is dereasing with, the first-order derivatives (26) and (27) beome (after dividing eah terms by ), m (H()/) + m (H( B )/ B ), and m 2 (H()/) + m 2 (H( B )/ B ), respetively. Both are stritly positive. Hene it is optimal to set i () =. Next, onsider < < B. Then the first-order derivatives (26) and (27) beome stritly negative, and it is optimal to set i () = 0. Now onsider < <. We laim that i () = p() = 0. At these values, the derivatives (28), (29), and (30) are negative. At i () = 0, the derivative (30) is zero; hene it is optimal to set p() = 0. At p() = 0, (28) and (29) redue to m (H()/) + m (H( B )/ B ), and m 2 (H()/) + m 2 (H( B )/ B ), respetively, and both are stritly negative. It is optimal to set i () = 0. Finally, the omitted onstraint (25) is satisfied sine i () = 0. Case 2 is when the budget B is lower, between m and, m < B <. Reall that m is the ost level at whih the firm will set the low prie () if it has aess to all onsumers (m [ ( m ) m ] = m 2 [ 2 ( m ) ( m )], see also (6)). Again, we set = H( B )/ B ). For >, the first-order derivatives (26) and (27) are m (H()/) + m (H( B )/ B ) and m 2 (H()/) + m 2 (H( B )/ B ), respetively. Both are stritly positive. Hene it is optimal to set i () =. Next, onsider B < <. We set () to satisfy m 2 () + (){m [ () ] m 2 [ 2 () ()]} = 0. (3) Beause > B > m, we have m [ () ] < m 2 [ 2 () ()]. Therefore, () > 0. We laim that p() = 0, i () =. Given p() = 0, first-order derivatives (28) and (29) are m (H()/) + m (H( B )/ B ) and m 2 (H()/) + m 2 (H( B )/ B ), respetively. Both are stritly positive. Hene. it is optimal to set i () =. Given i () =, by the hoie of () satisfying (3), the derivative (30) is zero. Hene, setting p() = 0 is optimal. Obviously, the omitted onstraint (25) is satisfied sine i () =. Next, onsider < < B. We laim that i () = p() = 0. Given p() = 0, the first-order derivatives (28) and (29) are both negative when < B. Hene it is optimal to set i () = 0. Next, given that i () = 0, the derivative (30) is zero. Hene it is optimal to set p() = 0. Again, the omitted onstraint (25) is satisfied sine i () = 0. Case 3 is when the budget is small, B < m. We set = H( B )/ B. For >, we use the same argument as in Case and Case 2, and i () =. For m < <, we laim that i () = and p() = 0. We show this by the same argument in Case 2. When () is set to be suffiiently large, the first-order derivative (30) is zero, so that p() = 0 is optimal when i () =. When p() = 0, setting i () = is optimal. The omitted onstraint (25) is satisfied beause i () = and > m. Next, for B < < m, we laim that p() = and i () =. We set () = 0. When i () =, first-order derivative (30) beomes p = m 2 () > 0, and it is optimal to set p() =. Given p() = and () = 0, firstorder derivatives (28) and (29) are stritly positive sine B <. Hene, it is optimal to set i () =. The omitted onstraint (25) is satisfied beause p() =. Finally, for < < B, we laim that i () = p() = 0. Given p() = 0, the first-order derivatives (28) and (29) are stritly negative beause < B. Hene it is optimal to set i () = 0. Given i () = 0, the first-order derivative (30) is zero. It is optimal to set p() = 0. The omitted onstraint (25) is satisfied beause i () = 0. Referenes Barros, P.P., Olivella, P., Waiting lists and patient seletion. 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