Publi Setor Rationing and Private Setor Seletion Simona Grassi Faulty of Business and Eonomis Institut d Eonomie et de Management de la Santé Université de Lausanne simona.grassi@unil.h Ching-to Albert Ma Department of Eonomis Boston University ma@bu.edu June 2010 Abstrat We study the interation beteen nonprie publi rationing and pries in the private market. Under a limited budget, the publi supplier uses a rationing poliy. A private firm may supply the good to those onsumers ho are rationed by the publi system. Consumers have different amounts of ealth, and osts of providing the good to them vary. We onsider to regimes. First, the publi supplier observes onsumers ealth information; seond, the publi supplier observes both ealth and ost information. The publi supplier hooses a rationing poliy, and, simultaneously, the private firm, observing only ost but not ealth information, hooses a priing poliy. In the first regime, there is a ontinuum of equilibria. The Pareto dominant equilibrium is a means-test equilibrium: poor onsumers are supplied hile rih onsumers are rationed. Pries in the private market inrease ith the budget. In the seond regime, there is a unique equilibrium. This exhibits a ost-effetiveness rationing rule; onsumers are supplied if and only if their ostbenefit ratios are lo. Pries in the private market do not hange ith the budget. Equilibrium onsumer utility is higher in the ost-effetiveness equilibrium than the means-test equilibrium. Aknoledgement: We thank many seminar and onferene partiipants for their omments and suggestions. We also thank the editor John Conley and to referees for their advie. Various parts of the researh here ere done hile the authors ere at the Universidad Carlos III de Madrid; e are grateful to their hospitality. The first author reeived partial finanial support from the Italian Fulbright Foundation.
1 Introdution Many governments and publi organizations provide or subsidize goods and servies suh health are and eduation. Free or subsidized publi provision often oexists ith a private market. In this paper, e study the interation beteen rationing poliies in the publi setor and profit-maximizing pries in the private setor. We derive equilibria of games beteen the publi and private setors, and ompare equilibrium pries and aggregate onsumer utilities. All publi programs operate under limited budgets. Unable to over all intended onsumers, a publi supplier must use a rationing rule. A variety of rationing and subsidy praties exist. Current Mediaid poliies in the United States provide health insurane to indigent individuals. In the pending US health are reform, families up to 400% of the Federal Poverty Level reeive either free health insurane or substantial subsidies. These rationing poliies are a means-test mehanism, hih alloates publi supply to poor individuals. In Canada and many European ountries, publi health systems ration are aording to illnesses, patients medial onditions, and treatment osts. This form of rationing is based on a ost-benefit orost- effetiveness mehanism, hih alloates publi supply to individuals for hom it is orthhile, in some qualified sense. Rationed individuals nevertheless may onsider purhasing from the private market. Here, an important issue is seletion of profitable onsumers by a private firm. A fous of our paper is on ho a private firm s profit-maximizing strategy ill reat to the publi rationing mehanism. For example, hen an individual does not qualify for Mediaid, his inome should not be very lo; neither should be his illingness to pay for health insurane. On the other hand, if a patient does not qualify for a ertain treatment aording to the publi setor ost-effetiveness measure, a private firm may not infer about his illingness to pay. Our model onsists of a set of onsumers, a publi supplier, and a private firm. Eah onsumer ould like at most one unit of an indivisible good (a medial treatment, a ourse of eduation, et.). Consumers have different ealth or inome levels, and the osts of providing the good to them also differ. We use ealth heterogeneity to model differenes in onsumers valuations of the good. Rih onsumers are more illing to pay for the good than poor onsumers. Cost heterogeneity arises beause a onsumer s harateristis 1
may determine ho muh it osts to supply the good to him. For example, the ost of a medial treatment depends on illness severity, and the ost of helping a student to ahieve an aademi standard depends on the student s ability and aptitude. Variations in these harateristis affet provision osts. To fous on rationing and priing issues, e abstrat from finaning. The publi supplier has a limited budget for providing the good at zero ost to onsumers, and aims to maximize total onsumer utility. 1 We onsider to information regimes. In the first, rationing is based on onsumers ealth information. In the seond, rationing is based on onsumers ealth and ost information. The first regime orresponds to the means-test senario above. The seond regime is somehat more general than the ost-effetiveness senario, but e ill exhibit a unique equilibrium in hih the publi supplier ignores ealth information, so ost-effetiveness does obtain. We study the equilibria of the folloing games. In stage 1, nature dras randomly and independently the ealth level and ost for eah onsumer. Consumers ost information is learned by the firm. In the first regime, the publi supplier learns onsumers ealth information, but not the osts. In the seond regime, the publi supplier learns both onsumers ealth and ost information. In stage 2, the publi supplier designs a rationing sheme based on the available information in eah regime. Simultaneously, the private firm sets a prie shedule depending on osts. In stage 3, the publi supplier s rationing sheme is implemented, and rationed onsumers deide hether to purhase from the firm at pries set in the seond stage. We model the private setor by a monopoly firm, and the analysis extends to Cournot ompetition. Our model addresses the folloing issues. First, the private setor may reat to publi supply by seleting or ream-skimming onsumers. Ho does the publi setor reat to ream-skimming? Seond, different onsumers enjoy different surpluses from publi and private supplies. Ho does this affet publi supply and pries in the private setor? Third, publi rationing poliies may be based on different information. Ho do equilibria hange as the information struture hanges? 1 Nonprie rationing is ubiquitous. Many governments set negligible pries for publi eduation and health are. Presumably, this may be due to fairness or politial onsiderations. There are also eonomi reasons for favoring rationing. For the health market, insuring onsumers finanial risks due to illness is a fundamental goal. Under soial insurane, onsumers should not be exposed to too muh finanial risk upon beoming sik. For the eduation market, a government may enourage the investment of human apital, hih may enhane eonomi groth and reate externalities. Again, reduing osts of eduation may be a sensible poliy. 2
In the first regime, hen publi rationing is based on onsumers ealth, there is a ontinuum of equilibria. In the Pareto dominant equilibrium, the publi supplier uses the budget on poor onsumers and rations all rih onsumers. The means-test riterion emerges in equilibrium. Pries set by the private firm rises as the available budget inreases. As poor onsumers are supplied, the private firm realizes that it sells only to onsumers ith higher illingness to pay, so raises pries aordingly. In the seond regime, hen publi rationing is based on onsumers ealth and ost, there is a unique equilibrium. The publi supplier spends the entire budget on onsumers hose ost per unit of benefit is lo, and ignores onsumers ealth information. The ost-effetiveness riterion emerges in equilibrium. The private and publi setors ill appear to be separated, ith the publi setor serving onsumers ith lo ost-benefit ratios, hile the private setor serving those ith high ost-benefit ratios. Equilibrium private market pries for available onsumers remain the same as if the publi setor ere inative, and equilibrium pries are independent of the budget. At any given budget level, equilibrium aggregate onsumer utility is higher hen rationing is based on the ost-effetiveness riterion than the means-test riterion. What explains these results? Riher onsumers have higher illingness to pay than poor onsumers. In the private market, onsumers fae pries that are dependent on their osts, not ealth. Hene, riher onsumers obtain higher inremental surplus from trades in the private setor. This trade-surplus effet is ommon aross both information regimes. When rationing is based on ealth, releasing riher onsumers to the private market allos them to realize more inremental trade surplus there than poorer onsumers. This motivates the publi supplier to ration the rih. Simultaneously, hen publi supply is for poor onsumers, the firm knos that only riher onsumers are in the market. These onsumers have higher illingness to pay, so the firm raises pries. In the seond information regime, ost information is available. Here, absent the private setor, a osteffetiveness priniple applies: publi supply is for onsumers ith a lo ost-benefit ratio. Theost- effetiveness priniple ontinues to apply hen there is a private market. For a given ost level, a onsumer obtains more surplus from free publi supply than the private market. Rationing lo-ost and rih onsumers annot our in equilibrium. If suh onsumers ere rationed, the private firmouldunderstandthatlo-ost 3
onsumers must be rih, and ould raise pries aordingly. The trade-surplus effet annot be implemented. Our goal is to study the effet of a prie-reative private setor, so e have rejeted a perfetly ompetitive market here pries ould alays be marginal osts (but for ompleteness, e have inluded related results). We use the monopoly setup, but all results extend to Cournot ompetition. We have let the monopolist observe osts, but not ealth. If the firm ould also observe ealth, it ould be able to extrat all surplus, and the publi rationing poliy ould not affet onsumers trade surplus in the private market. In any ase, firms seldom possess information on ealth. Given that the good is indivisible and a onsumer buys at most one unit, nonlinear pries annot be implemented. It ould be a less interesting model if the firm did not observe onsumers osts; seletion and ream-skimming issues ould be assumed aay. 2 Most other papers assume that the government is a first mover. Generally, an ability to ommit to a rationing rule is valuable. Hoever, as e sho in a ompanion paper, Grassi and Ma (2009), the publi supplier s Stakelberg rationing rule is time inonsistent. By ommitting to ration some poor onsumers, the publi supplier an implement loer pries beause the firm ill ant to sell to poor onsumers hen osts are lo. Hoever, given loer pries, the publi supplier ould exploit the trade-surplus effet by reneging, supplying poor onsumers and rationing rih onsumers. We have adopted a simultaneous-move game, and the publi supplier and the private firm have symmetri ommitment poer. This is a longer-term perspetive beteen the players beause both players hoose mutual best responses. Finally, our fous is on nonprie rationing. Free publi provision is the main mehanism in publi health and eduation systems, so e have not inluded taxes and subsidies in the publi supplier s strategies. We study interations beteen publi rationing and a nonompetitive private market, hile models in the literature usually assume a ompetitive private market, or exogenous priing rules. Barros and Olivella (2005) fous on publi physiians referring patients to their on private praties. The publi setor uses aitingtime rationing, and physiians refer patients hen the patients osts are lo. Iversen (1997) onsiders the effet of a private setor on aiting time in the publi setor. Hoel and Sæther (2003) onsider supplementary private health are hen publi health are is subjet to aiting time rationing. In the above papers, the 2 Also, hen the prie is based on the expeted ost, the firm may renege one it learns that the ost turns out to be higher than the prie it has harged. 4
prie in the private market is fixed. Hoel (2007) derives the optimal ost-effetiveness rule hen patients have aess to a ompetitive private market, here pries do not respond to the publi setor s alloation rule. By ontrast, in our model, pries in the private setor respond to publi poliies. Rationing is similar to transfers in kind. Blakorby and Donaldson (1988) sho ho transfers in kind may solve asymmetri information problems. The literature has also studied redistribution effets. In Besley and Coate (1991), the government uses a poll tax to provide for free a good at a lo quality. Rih onsumers optimally hoose the good at a high quality from a ompetitive private market, hile poor onsumers do not. The government in effet taxes rih onsumers to subsidize poor ones. Segregation beteen rih and poor onsumers is an equilibrium of our game hen rationing is based on ealth, but this stems from a trade-surplus effet in the private market. In Besley and Coate (1991) the budget is endogenous hile e abstrat from finaning issues, and assume an exogenous budget. Our formal model is like a ommon ageny model. The publi supplier and the private firm are to prinipals hose ations ill affet the onsumer, ho is the agent; see Bernheim and Whinston (1986). In line ith the ommon ageny model, e use a symmetri setup, so that both setors reat against eah other s strategy. We are unaare of a paper that models ho publi setor rationing and private setor priing strategies mutually reat. We also depart from the mehanism design literature on the provision of publi goods (see Norman 2004, and Hellig 2003 on exludable publi goods), here inentive ompatible, individually rational and budget-balaned shemes are derived. In Norman (2004) the publi supplier may set individualized user fees to provide aess to the exludable publi good. Our interest is on nonprie rationing, so e have not alloed publi supplier to use taxes and subsidies. Setion 2 and its subsetions lay out the model. In Setion 3, e first derive a foal equilibrium in the ealth rationing regime. Then e haraterize a ontinuum of other equilibria. Setion 4 ontains the analysis of rationing based on ealth and ost, and ompares equilibrium onsumer utilities aross the to rationing regimes. The last Setion dras some onlusions. Appendix A ontains proofs of all Lemmas, Propositions, and the last Corollary; proofs of other Corollaries are omitted. Appendix B disusses alternative assumptions on onsumer preferenes, information, and ost-benefit strutures. 5
2 The model We begin ith the desription of onsumers. Next, e introdue a publi setor, and derive benhmark optimal rationing poliies. Then e introdue a private firm. We omplete the model by desribing the extensive-form games beteen onsumers, the publi supplier, and the private firm. 2.1 Consumers and their illingness to pay There is a set of onsumers. Eah onsumer may onsume at most one unit of an indivisible good. We let there be a ontinuum of these onsumers, ith a total mass normalized to 1. Eah onsumer is indexed by to parameters, and. Thevariable denotes the onsumer s ealth. The variable denotes the ost of supplying the good to the onsumer. The ost of provision is idential hether the good is supplied by the publi or private setors; e do not onsider any produtive omparative advantage beteen the private and publi setors in order to fous on information and priing problems. We often use the term onsumer (, ) torefertoonehohasealth and ost. Let F :[, ] [0, 1] be the distribution funtion of. WeassumethatF is differentiable, and the orresponding density f stritly positive. Similarly, let G :[, ] [0, 1] be the distribution funtion of. We also assume that G is differentiable, and the orresponding density g stritly positive. Let γ R dg denote the expeted ost. The domains of both distributions are stritly positive and bounded. The variables and are assumed to be independently distributed. In Appendix B e ill disuss the independene assumption. For a general speifiation of preferenes, e an let a onsumer s utility be U(, 0) hen he does not onsume the good, and U( p, 1) hen he onsumes the good at a prie p 0. The utility funtion U is stritly inreasing, and stritly onave in, andu(, 1) > U(, 0). It saves on notation and simplifies the analysis if e let the utility funtion U be separable in the to arguments. The separability assumption says that a unit of the good generates the same utility inrement, independent of the onsumer s ealth. If the onsumer ith ealth pays a prie p to onsume the good, his utility is U( p) +1; his utility is U() if he does not onsume the good. We disuss the nonseparable utility funtion in Appendix B. 6
If a onsumer ith ealth is indifferent beteen paying τ for the good and the status quo, e have: U( τ)+1=u(). (1) This equation impliitly defines a illingness-to-pay funtion τ :[, ] R + for onsumers ith various ealth levels. Beause U is onave, hene almost everyhere differentiable, the illingness-to-pay funtion is differentiable. From total differentiation of (1), e have: dτ d =1 U 0 () U 0 > 0. (2) ( τ) A onsumer s illingness to pay for the good is stritly inreasing in ealth due to the strit onavity of U. We ill assume that the loest illingness to pay τ() is larger than the loest ost. This assumption ensures that there is some sope for any onsumer to benefit from a trade in the private market. We illustrate our desription of onsumer preferenes and osts ith examples in the health market. The good may refer to a surgial proedure (for example, a hip replaement). Patients differ in their illness severity levels (some hip replaements are more diffiult than others). For a fixed amount of improvement in health, interpreted as a unit inrement of utility (for example, the ability to alk about ithout pain), siker patients require more resoures, and riher patients are more illing to pay. In our setup, onsumer preferenes do not diretly depend on the provision ost. Inthehealthare example, this means that patients ith different severity levels obtain the same inremental utility from the good. One interpretation is that the good provides a standardized unit of improvement in ell-being. In other situations, onsumers obtain different inremental utilities depending on their severity levels. Consumer preferenes then may depend on ost, and e ill disuss this alternative assumption in Appendix B. 2.2 The publi setor and rationing The publi setor has a budget B hih is insuffiient to supply the good to all onsumers for free, so 0 <B<γ. We onsider to information regimes. First, only onsumers ealth information is available to the publi supplier, and seond, onsumers ealth and ost information is available. In eah ase, nonprie rationing ill be used to alloate the budget for providing the good to onsumers. 7
In the first regime, the publi supplier s rationing rule is a funtion θ :[, ] [0, 1]. For [, ], the publi supplier provides onsumers ith ealth belo a total of R (1 θ(x))f(x) dx units of the good. The rationing rule θ splits the density f so that at, [1 θ()]f() of onsumers are supplied at zero prie, and θ()f() of onsumers are rationed. Beause ealth and ost are independently distributed, the ost among rationed onsumers remains distributed aording to G. In the seond regime the rationing rule is a funtion φ :[, ] [, ] [0, 1]. It has the same interpretation as in the first regime. For onsumer (, ), the density φ(, )f()g() is available to the private firm. 3 In eah regime, the publi supplier s objetive is the sum of onsumer utilities. The rationing shemes θ and φ orrespond to random rationing, but an be implemented by aiting times. We an add to the onsumer preferene speifiation a ne parameter, say δ, a random variable hose distribution depends on ealth, ost, or both. The utility of a onsumer is no U()+1 δt if he gets the good after a delay of t units of time. The parameter δ desribes the onsumer s marginal aiting ost. An impatient onsumer (one ith a high value of δ) may deide against the publi system if he expets a long delay. By setting the delay t, the publi supplier determines the fration of onsumers ithin a ealth group or a ealth-ost group ho hoose to ait for the good in the publi setor. 2.3 Benhmark optimal rationing poliies ith an inative private market For no suppose that the publi setor is the sole provider. Consider the first information regime here rationing is based on ealth. For a rationing rule θ, total onsumer benefit from the publi supply is R (1 θ())f()d as eah unit of onsumption inreases a onsumer s utility by one unit. The onsumer elfare index, hih the publi supplier maximizes, is V (θ) Z U() df + Z [1 θ()] f() d. (3) 3 We restrit rationing rules to those that leave the funtions θ()f() and φ(, )f() integrable, so that θ(x)f(x) dx φ(x, )f(x) dx at eah are ell defined for [, ]. We an restrit the publi provider to supply to either all and or none of the onsumers ithin a ealth lass or a ealth-ost lass. Rationing shemes are then funtions that map [, ] to {0, 1} and [, ] [, ] to {0, 1}. The general rationing funtions an no be interpreted as mixed strategies. For ease of exposition, e do not use the mixed strategy interpretation. 8
The rationing rule must satisfy the budget onstraint hih says that the expeted ost must not exeed the available budget. γ Z [1 θ()]f() d B, (4) The determination of a rationing rule that maximizes (3) subjet to (4) is rather trivial. Any rationing rule that exhausts the budget is optimal. The publi supplier alloates the good to onsumers ithout olleting any payment. Due to the separable utility funtion, the utility inrement is independent of. Any rationing sheme that exhausts the budget results in the same level of the elfare index, and is optimal. No e onsider the seond information regime, here rationing an be based on ealth and ost. For a rationing rule φ, theelfareindexis V (φ) Z Z The rationing rule must satisfy the budget onstraint Z Z {U()+[1 φ(, )]} f()g() d d. (5) [1 φ(, )] f()g() d d B. (6) By pointise optimization ith respet to φ, the optimal rationing rule is given by 4 φ(, ) =0 for < s and φ(, ) =1 otherise, here R s dg() =B. The supplier has perfet information, and the optimal rationing rule is based on a ost-effetiveness measure. Eah unit of the good yields a fixed inrement of utility. The optimal rationing poliy therefore supplies the good to onsumers if and only if their osts are belo a threshold. 2.4 The private setor and profits There is a monopoly firm in the private setor; Cournot and perfet ompetition in the private setor ill be disussed. For onsumer (, ), the private firm observes the ost of providing a unit of the good to the onsumer, but not his ealth. 4 The Lagrangean is U() +[1 φ] +λ[b (1 φ)], anditsfirst-order derivative ith respet to φ is 1 +λ, hihis stritly positive if and only if is larger than a threshold, say, s. 9
By setting a prie p, the monopolist sells to those onsumers ith illingness to pay higher than p. Obviously, the monopolist ill not set a prie outside the range of illingness to pay τ. Setting a prie p is equivalent to seleting the ealth level of the marginal onsumer, herep = τ(). By the stritly monotoniity of τ, onsumers ith 0 > have τ( 0 ) > τ(), hene are illing to pay p = τ() to purhase the good. The funtion τ is like a demand funtion; e simply restate the ommon priniple that a monopolist may hoose equivalently beteen a prie and a quantity hile respeting the demand funtion. It is more onvenient to let the firm hoose the quantity or the marginal onsumer by setting the prie τ(). A a quantity funtion is denoted by b :[, ] [, ]. We present the profit funtions under the to rationing rules. First, suppose that rationing is based on ealth. At ost the density of onsumers available to the firm is θ()f(). At a prie τ(), onsumers ith ealth higher than ill buy, and the profit is π(;, θ) = Z θ(x)f(x) dx [τ() ]. (7) Here, the integral is the total quantity purhased, and τ() is the prie-ost margin. Seond, suppose that rationing is based on ealth and ost. At ost the density of onsumers available to the firm is φ(, )f(). At a prie τ(), onsumers ith ealth higher than ill buy, and the profit is π(;, φ) = 2.5 Interation beteen the publi and private setors Z φ(x, )f(x) dx [τ() ]. (8) We study the folloing games and look for their subgame-perfet equilibria: Stage 1: Nature dras (, ) aording to the distributions F and G, respetively, for eah onsumer. The private firm observes. The publi supplier observes either, or both and. Stage 2: In eah information regime, the publi supplier hooses a rationing rule, θ or φ, and the private firm hooses a quantity funtion b. Stage 3: Consumers supplied by the publi setor get the good for free, and onsumers not supplied by the publi setor may purhase from the private firm at pries set in Stage 2. 10
3 Equilibrium rationing and pries hen rationing is based on ealth We begin ith the private firm s profit-maximizing pries and onsumers utilities. Then e present an equilibrium in hih the publi supplier uses the entire budget on onsumers ith loer ealth levels. Next, e present a ontinuum of equilibria, and sho that the one e have presented is Pareto dominant. Finally, e disuss some omparative statis on the budget, as ell as Cournot and perfet ompetition in the private market. 3.1 Profit-maximizing pries and onsumer utilities To haraterize an equilibrium, e need to refer to a profit-maximizing quantity funtion hen the firm has aess to all onsumers. Let this funtion be b m (). Suppose that the publi supplier rations all onsumers, so in (7) e set θ() =1for all. The funtion b m :[, ] [, ] is given by b m () argmax Z f(x) dx [τ() ]. (9) We assume that the profit funtion in (9) is onave, and that b m () is single-valued. By the Maximum Theorem b m () is ontinuous. We further assume that as varies over [, ], the marginal onsumers vary over a proper subset of [, ], so < b m () < b m () <. This requires that variation in ealth is suffiiently large relative to variation in osts. The optimal quantity b m () is given by the usual marginalrevenue-equal-marginal-ost ondition. The quantity funtion b m () is stritly inreasing. As marginal ost inreases, the optimal quantity is adjusted to ahieve a higher level of marginal revenue. This means setting a higher prie and selling to less onsumers. 5 By the onavity of the profit funtion for any > b m (), the derivative of profit ith respet to is negative: d d Z f(x) dx [τ() ] < 0 for > b m (), any. (10) The solid line in Figure 1 illustrates suh a quantity funtion. We have assumed that the budget is insuffiient to over all onsumers at zero ost (B <γ). For a very small budget, the interation beteen 5 The sign of the derivative of m () isthesameasthesignoftherosspartialof f(x)dx [τ() ], hihispositive. 11
the publi and private setor may be irrelevant. Suppose that the budget an only over some onsumers ith < b m (). The private firm ill never sell to these onsumers even if they are available beause their illingness to pay is too lo. Any publi supply to them ill not lead to a prie response from the private firm. To rule out equilibria ithout any interation beteen setors, e assume that the budget is suffiient to over some onsumers ho may purhase from the firm. This assumption is B>F( b m ())γ. At the loest ost, thefirm sells to those ith ealth above b m (), so this assumption says that the budget is suffiient to eliminate some onsumers ho otherise may buy from the firm. Given a rationing rule θ, the private firm s profit from selling to onsumers ith ealth higher than is in (7). Let b () be the optimal quantities, and bπ () the maximum profit: b () arg max π(;, θ), (11) bπ () π( 0 ;, θ), 0 b(). (12) For some rationing rules, there may be multiple quantities that maximize profit, so b() is a orrespondene. Aording to the Maximum Theorem, the orrespondene b () is upper semiontinuous. The profitmaximizing prie τ( b ()) may not be stritly inreasing in ost, although the maximum profit isstritly dereasing, as the next Lemma shos. Lemma 1 The maximum profit is stritly dereasing in. Any seletion from the profit-maximizing quantities b () arg max π(;, θ) is inreasing in. Thatis,if 1 < 2,then 1 2,here 1 b( 1 ) and 2 b( 2 ). A best response is a seletion from the profit-maximizing quantity orrespondene, and need not be ontinuous. Nevertheless, beause it must be (eakly) inreasing, any point of disontinuity of b() must be an upard jump (see also Figure 3 belo). Beause there is no risk of onfusion, e also denote suh a (eakly inreasing) seletion by the notation b :[, ] [, ]. When ill an equilibrium quantity funtion fail to be stritly inreasing? Suppose that the rationing rule speifies that θ() =0for < e, and θ() =1for > e. This rationing sheme supplies onsumers (at zero prie) if and only if their ealth is belo a threshold e; onsumersith< e are not in the market. 12
ˆ m () ˆ () ~ Figure 1: Quantity funtions b m () and b(). Let the ost threshold e be defined by b m (e) = e; see Figure 1. For >e, the profit-maximizing quantity remains at b m (). At >e, thefirm ould not sell to those onsumers ith ealth belo e anyay. At <e, the profit-maximizing quantity funtion is onstant at e. Lemma 1 says that the profit-maximizing quantity annot rise as ost falls belo e. Thefirm has no available onsumers ith ealth belo e, so the optimal quantity stays at e. 6 Under the rationing sheme, in Figure 1 the optimal quantity funtion beomes the horizontal, dotted line hen ost falls belo e. When ill an equilibrium quantity funtion fail to be ontinuous? Suppose that θ() =0for [ 1, 2 ] here < 1 < 2 <, andθ() =1otherise. The publi setor supplies only to onsumers ith medium ealth. Figure 2 illustrates the density of onsumers available to the private firm. The profit-maximizing quantity funtion b() is in Figure 3. For < 1 or > 2,theprofit-maximizing quantity is unique. For ( 1, 2 ), the prie remains onstant beause all onsumers ith ealth in [ 1, 2 ] are supplied by the publi setor. At ost 1,thefirm makes equal amounts of profit hether it harges a prie τ( 2 ) selling to 6 Any < yields a profit so setting = is optimal. = θ(x)f(x) dx + f(x) dx [τ() ] < θ(x)f(x) dx [τ() ] f(x) dx [τ( ) ] 13
onsumers ith > 2,orτ( 0 ) selling to onsumers ith beteen 0 and 1 and above 2. Finally, some quantities may never be hosen; in Figure 3, the firm never sets b() to any [ 0, 2 ). f( ) θ( ) f( ) 1 2 Figure 2: Consumer density under rationing sheme θ. We need to rite don aggregate onsumer utility given rationing funtions and quantity funtions that are eakly inreasing and possibly exhibiting upard jumps. Given a quantity funtion, b(), rationed onsumer (, ) buys from the private firm if and only if b(). In Figure 3, this is the set above the graph of b(). Itismoreonvenienttoviethesetofpurhasing onsumers as one indexed by a funtion b :[, ] [, ] that is like an inverse of b. Define b() =sup{ : b()}; if there is no [, ] suh that b(), setb() =. Suh a funtion b is illustrated in Figure 3. While the funtion b gives the ealth of the marginal onsumer in terms of his ost, the funtion b gives the threshold ost level belo hih a onsumer ith ealth ill buy at prie τ( b()). Whenever b is stritly inreasing and ontinuous, the funtion b is its inverse. When b is onstant on an interval, then b exhibits disontinuities at the to ends of the interval. Finally, b() beomes hen the firm does not sell to onsumer (, ). Clearly b() is inreasing henever its value is not. Thesetofonsumersho purhase are those ith (, ) belo the graph of b(), and this differs from those above the graph of b() 14
' ( ) 2 1 ( ˆ ( ) 2 0 ] ' 1 2 1 ' ( ) = 0 1 2 ' Figure 3: The quantity funtion b() and its "inverse" b(). at most for a set of measure zero. Funtions b and b are to equivalent ays of keeping trak of onsumer types ho purhase from the private firm. Given a quantity funtion b (and its equivalent b), and a rationing sheme θ, the elfare index V (θ) is R () Z Z {U( τ( b ())) + 1} g() d [1 θ()]f()[u()+1] d + θ()f() + R d. (13) U()g() d () In this expression, the first term is the utility of onsumers supplied by the publi setor. The seond term is the utility of rationed onsumers. The top integral inside the big square brakets is the utility of onsumer (, ) buying from the firm at prie τ( b ()), hile the bottom integral is the utility of onsumers ho do not buy. The elfare index an be simplified to V (θ) = Z Z [U()+(1 θ())] f() d + (14) " Z # () θ()f() {U( τ( b ())) + 1 U()} g() d d, here the first term is the base utility U() plus the inrease of utility from publi supply, and the seond term is the onsumer inremental surplus from the private market. 15
3.2 An equilibrium: rationing rih onsumers An equilibrium is a pair of rationing and quantity funtions (θ, b) that are mutual best responses. That is, θ maximizes aggregate onsumer utility subjet to the budget onstraint given quantity funtion b, and b maximizes profit for every given θ. Wefirst present an equilibrium in hih onsumers are rationed if and only if they are rih, and in hih the firm hooses the quantity funtion b() in Figure 1. This is a means-test equilibrium: onsumers are supplied if and only if their ealth is lo. Proposition 1 The folloing is an equilibrium. The publi supplier rations all onsumers ith ealth above a threshold E andsuppliesallonsumersithealthbelo E : θ() =1, > E and θ() =0, < E. The threshold E exhausts the budget and is given by F ( E )γ = B. The private firm sets the prie to the monopoly prie hen ost is above a threshold E,defined by b m ( E )= E,andtoafixed prie τ( E ) hen ost is belo E. We explain the intuition for Proposition 1. First, hen disussing the property of b() in Figure 1, e already sho that it is the profit-maximizing quantity funtion hen the rationing poliy is the one in Proposition 1. Hene, the firm s quantity funtion in Proposition 1 is a best response. We no explain hy the rationing poliy is a best response. To ompute aggregate onsumer utility, e apply the quantity funtion in the Proposition to (14). First, for > E e let the inverse of b m be b m ; this takes the role of b in (14): for > E,onsumer(, ) buys at τ( b m ()) if <b m (). Seond, for < E, b m takes the value : rationed onsumers ith ealth belo E never buy beause the firm never sets a prie belo τ( E ). Under rationing poliy θ, aggregate onsumer utility is Z [U()+(1 θ())] f() d (15) Z + θ() E R E U( τ( E )) + 1 U() g() d+ + R m () [U( τ( b m ())) + 1 U()] g() d E f() d. The elfare index is desribed as follos. The integral on the first line of (15) is the sum of the base utility U() plus the utility inrease from the publi supply. The seond line is the private market inremental 16
surplus of rationed onsumers. Only rationed onsumers ith ealth above E ill buy from the private market. For these onsumers, if their osts are belo E, they purhase at prie τ( E ),andobtainthe inremental surplus in the integral ith limits beteen and E ;iftheirostsareabove E, they purhase at prie τ( b m ()) if their osts are belo b m (), and obtain the inremental surplus in the integral ith limits beteen E and b m (). A rationing poliy θ is a best response if it maximizes (15) subjet to the budget onstraint (4). We onsider the trade-off in rationing a onsumer ith ealth. The benefit of rationing a onsumer is the saving of expeted ost γ, a onstant. The ost of rationing a onsumer depends on the onsumer s ealth level. If is belo E, this onsumer does not buy from the private market, so the ost is one unit of utility due to nononsumption. If is above E, this onsumer may gain some inremental surplus from the private market (the seond line in (15)), so the ost of rationing him is less than one unit of utility. Rationing a rih onsumer is less ostly preisely beause the rih onsumer has the opportunity to buy from the private market. Therefore, it is optimal to ration riher onsumers, those ith ealth above E. This equilibrium is similar to many pratial shemes in hih poor onsumers reeive free supplies hile the rih do not, but this means-test equilibrium is not due to an equity onern. The publi supplier selets among onsumers ith different ealth levels to partiipate in the private market. Wealthy onsumers realize larger gains in trade in the private market, so they are rationed. The private market fully antiipates that poor onsumers are unavailable, so even hen ost dereases, the equilibrium prie stops falling. 3.3 Charaterization of a ontinuum of equilibria In this subsetion, e haraterize all equilibria. We ill sho that in equilibrium the firm s quantity funtion is the monopoly quantity funtion for onsumers ith ost higher than a threshold, and a onstant otherise, but this threshold must be higher than the one in Proposition 1. In equilibrium the publi supplier must ration rih onsumers, but may also ration some poor onsumers. To haraterize the equilibrium rationing poliies, e let the publi supplier hoose the net density of rationed onsumers θf, and impose the requirement that 0 θf f. The onsumer elfare index (14) is 17
linear in θf, andforeah its first-order derivative ith respet to θf is V θf = Z () {U( τ( b ())) + 1 U()} g() d 1. (16) This expression measures the hange in aggregate onsumer utility at ealth level. It is the expeted inremental surplus from onsumer ith ealth buying from the firm (the integral) less the unit inremental utility of onsumption at zero ost. We establish a monotoniity in the supplier s preferenes. Lemma 2 The first-order derivative V θf in (16) is inreasing in. It is stritly inreasing in [ 1, 2 ] unless b() = for eah suh. Lemma 2 says that the publi supplier favors rationing the onsumer over supplying as the ealth level inreases. This is a basi priniple in our model. Pries in the private market depend only ost, so onsumer (, ) gets more surplus from a trade at prie τ( b()) as inreases: U( τ( b()))+1 U() is inreasing in. The publi supplier s marginal utility from rationing, (16), is stritly inreasing in for onsumers ho buy from the firm. When onsumers do not buy from the firm, there is no inremental surplus, so the integral in (16) is 0, and the derivative in (16) beomes 1, independent of. Lemma 2 does not take into aount the budget, the onsideration of hih is our next step. Against a quantity funtion b() (and the orresponding b()), the publi supplier hooses θf to maximize (13) subjet to the budget onstraint (4). Using pointise optimization, e onsider the Lagrangean R () {U( τ( b ())) + 1} g() d θ()f() + R +[1 θ()]f()[u()+1] λ[γ(1 θ(x))f(x) B], U()g() d () here λ is the multiplier. The first-order derivative of the Lagrangean ith respet to θf is V θf + λγ = Z () {U( τ( b ())) + 1 U()} g() d 1+λγ. (17) From Lemma 2, the first-order derivative of the Lagrangean is stritly inreasing in henever some onsumers ith ealth less than purhase from the private market. Lemma 3 In any equilibrium, the publi supplier rations onsumers ith ealth above a threshold e. That is,inanequilibriumthereis e < suh that θ() =1for > e. 18
Given the limited budget, some onsumers must be rationed. There is alays some sope for trade for rationed onsumers at the private market beause a onsumer s illingness to pay is higher than the loest ost. No one there is some trade by rationed onsumers, ealthier onsumers get more inremental surplus. If it is optimal to ration a onsumer ((17) positive at ), then it is also optimal to ration onsumers riher than him ((17) stritly positive at 0 >). Lemma 3 is onsistent ith the publi supplier rationing some poor onsumers in equilibrium. If the firm does not sell to very poor onsumers, there is no inremental surplus for them. The first-order derivative (17) beomes 1+λγ, independent of. In fat, hen the value of (17) is 0, the publi supplier is indifferent beteen rationing a onsumer and supplying him. We an use this indifferene to selet various rationing rules to support equilibria. Lemma 4 In any equilibrium, the private firm sets a onstant prie hen ost falls belo a threshold e. That is, in an equilibrium there is e suh that b() is onstant for <e. Lemma 4 says that an equilibrium quantity funtion must beome a onstant hen ost is suffiiently lo. An equilibrium quantity funtion b() is alays inreasing, and there must be some interation beteen the to setors beause the budget satisfies B>F( b m ())γ. If b() is stritly inreasing at lo osts, similar to the solid line in Figure 1, the inremental surplus is higher for ealthy onsumers. The publi setor supplies poor onsumers and rations ealthy onsumers. When the poor onsumers have been taken out of the market, it is no longer profit maximizing for the firm to redue prie hen ost beomes lo. This is inonsistent ith an equilibrium funtion being alays stritly inreasing at lo osts. The last to lemmas establish the form of an equilibrium, but do not pin don the exat strategies. Besides the equilibrium in Proposition 1, e an onstrut many equilibria exhibiting properties in Lemmas 3 and 4. Take the ost and ealth thresholds E and E in Proposition 1, and hange the equilibrium 19
f( ) f( ) θ( ) f( ) + ε E +δ Figure 4: Equilibria in hih some poor onsumers are rationed. rationing poliy there to: θ() = 1 for <+ θ() = 0 for + << E + δ θ() = 1 for E + δ<, here >0 and δ>0 are both small numbers. In this rationing rule the supplier shifts some resoures from those ith ealth just above the loest value to those onsumers ith ealth just above E. Figure 4 shos the density of onsumers available to the private firm in suh an equilibrium. Values of and δ an be so hosen that the ne sheme satisfies the budget: B = γ[f ( E +δ) F (+ )]. Against this rationing sheme, the private firm sets a quantity funtion equal to b m () for > E + η and b m ( E + η) for < E + η, here E is the ost threshold in Proposition 1, and η > 0 satisfies b m ( E + η) = E + δ. In this equilibrium, the publi supplier gives the good to some onsumers ith ealth slightly higher than E, but rations onsumers ith ealth lose to the loest level. These rationed onsumers have suh 20
lo illingness to pay that the firm ill not redue prie in order to sell to them even hen ost is loest. Furthermore, beause onsumers ith ealth slightly higher than E are no supplied by the publi, the private firm s prie ill not fall all the ay to τ( E ). In Appendix A, e provide a formal proof for this equilibrium. Infinitely many equilibria an be onstruted in a similar fashion. As long as the private firm does not find it profit-maximizing to redue prie in order to sell to onsumers ith lo illingness to pay, a quantity funtion like the one in Figure 1 remains a best response. In all these equilibria the publi supplier rations some onsumers ith lo ealth, but must ration all onsumers ith ealth above a threshold. The equilibrium in Proposition 1 is foal. This is the one that ahieves the highest elfare index for the publi supplier. This is beause it has the idest range of prie redution as ost dereases. The equilibrium also allos the private firm to make the highest equilibrium profit. Any equilibrium different from the one in Proposition 1 ould have feer transations in the private market. Proposition 2 The equilibrium in Proposition 1 ahieves the highest equilibrium onsumer utility, and the highest equilibrium profit for the private firm. In any other equilibrium, the publi supplier sets θ() =1, for > e e,here e e > E (defined by F ( E )γ = B in Proposition 1) and the firm sets a prie equal to τ( b m ()) for >e e,andaprieequaltoτ( e e ) for <e e,here b m (e e )= e e and e e > E (defined by E = b m ( E ) in Proposition 1). Ho are onsumers utilities affeted by the publi supply and the prie reation in the private market? In Proposition 1, onsumers ith ealth above E are rationed, here γf( E )=B, andthefirm s equilibrium pries range beteen τ() and τ( E ). Suppose that the budget inreases by B, then E ill inrease by E,hereγF( E + E )=B + B, so the minimum prie in the private market beomes higher. In Figure 1, the ne equilibrium is obtained by shifting the dotted horizontal line upard. An inrease in the budget ill be used in equilibrium to supply onsumers ith ealth just above E so available onsumers in the private market are ealthier, and the firm redues its prie less hen ost falls. Define E by b m ( E + E )= E + E. Consider onsumers ith ealth above E + E, those that remain rationed after the budget inrease. They still are offered the monopoly pries hen their osts are above E + E, 21
so their utilities remain unhanged, but those ith osts belo E + E fae a higher prie τ( E + E ) although all of them still prefer to purhase. Wealthy and lo-ost onsumers are hurt by the budget inrease, hile more poor onsumers benefit. Corollary 1 Under rationing based on ealth, private market equilibrium pries are higher hen the publi supplier s budget inreases. Consumers ho remain rationed after the budget inrease fae a stritly higher prie hen their osts are lo. We have assumed a monopolisti private setor. The extension to an imperfetly ompetitive setor poses no oneptual problem. For our model of a homogeneous good, e onsider a Cournot model. Let there be N firms in the private setor. Given a rationing sheme θ, let eah firm hoose a quantity funtion bq i (), here i =1,..., N. The total supply is q() = P N i=1 q i(). For the market to lear the marginal onsumer is b() here R θ()f() d = q(), and the prie in the private setor is τ( b()). All results derived () above ontinue to hold for any given number of firms in the private setor. Next, e an extend our model to the ase of a perfetly ompetitive private setor. Here, the prie in the private setor is marginal ost: τ() =. Given this priing funtion, the orresponding quantity funtion b() is impliitly defined by U( b )+1 = U( b). Lemma 2 an be applied to this quantity funtion. Beause the perfetly ompetitive quantity funtion is stritly inreasing, the derivative (17) is stritly inreasing for all values of. Lemma 3 ontinues to hold. In sum e have the folloing. Corollary 2 If the private market is perfetly ompetitive so that pries are equal to marginal osts, the publi setor uses the entire budget on onsumers ith lo ealth levels: θ() = 0, for < E, and θ() =1,for> E here F ( E )γ = B. 4 Equilibrium rationing and pries hen rationing is based on ealth and ost In this setion e let the supplier observe both ealth and ost information. A rationing poliy is φ : [, ] [, ] [0, 1]. Suppose that the firm observes that a onsumer s ost is, the density of onsumer 22
available to the private firm is φ(, )f(). If it sets a prie τ(), the total mass of onsumers purhasing is R φ(x, )f(x)dx, andtheprofit is Z We use the same notation and let b() maximize profit (18). φ(x, )f(x)dx [τ() ]. (18) Consider a rationing funtion φ :[, ] [, ] [0, 1] and a quantity funtion b :[, ] [, ]. Consumer (, ) buys from the private firm if and only if U( τ( b())) + 1 U(). Therefore, hen the supplier rations onsumer (, ), that onsumer obtains a utility max [U( τ( b())) + 1, U()] from the private setor. Aggregate onsumer utility from a poliy φ is Z Z {φ(, )max[u( τ( b())) + 1,U()] + [1 φ(, )] [U()+1]} f()g() d d. (19) Our next Proposition reports that in the unique equilibrium the publi supplier s alloation rule depends only on onsumers ost level. Consumers are supplied the good if and only if their osts are loer than a threshold, irrespetive of their ealth levels. The rationing rule based on and, is the same as the optimal alloation ithout a private setor. Proposition 3 If the publi supplier rations onsumers based on ealth and ost information, the equilibrium rationing funtion is idential to the optimal rationing funtion hen the private setor is inative. In the equilibrium, onsumer (, ) is rationed hen his ost is above a threshold s >, and supplied hen his ost is belo s.thatis,φ(, ) =1for > s,any, andφ(, ) =0otherise, here R s dg() =B. The private firm hooses the monopoly quantity b m () for > s,and b() = for < s. When the supplier observes ealth and ost information, a standard ost-effetiveness priniple applies. If the ost is too high relative to the benefit, the onsumer should not be given the good. No the availability of the private supply to high-ost onsumers does not alter this priniple. Next, onsider lo-ost onsumers. The ost-effetiveness priniple points against rationing. Hoever, some ealthy and lo-ost onsumers may obtain a higher surplus from the private setor if the prie is lo. Might the publi supplier ant to ration them, and allo them to get the surplus from the private market? Proposition 3 says that this annot happen in equilibrium. Consider the marginal onsumer (, ); he pays 23
aprieτ() in the private market and obtains a zero inremental surplus. No if his ost is lo, the publi supplier prefers to alloate the good to him, giving him a positive inremental surplus. By ontinuity, the supplier also assigns the good to those ith ealth slightly above, eliminating these onsumers from the private market. Given that rationed onsumers are riher, the best response by the private firm is to raise the prie. This unravelling ontinues until the private firm raises the prie to τ(). In an equilibrium, it is as if lo-ost onsumers did not have a private option, so the supplier provides the good to them. Proposition 3 is onsistent ith pratial ost-effetiveness poliies. We have used a standardized benefit of one unit of utility inrement from the onsumption of the good. Therefore, Proposition 3 should be understood to say that in equilibrium only onsumers ith suffiiently lo osts per unit of benefit ill be supplied. This is indeed hat most publi programs aim to ahieve. In ontrast to the regime hen rationing is based on ealth information only, here onsumers are not hurt by publi supply. An inrease in the budget, say by B, ill raise the value of s,sayby s,somore onsumers benefit from publi supply. The prie shedule ill not hange. Those onsumers ho remain rationed after the budget inrease fae the same monopoly prie shedule. Let b m ( s + s )= s + s. The pries for onsumers ith ost above s + s no range beteen τ( s + s ) to τ(), thesameas before the budget inrease. Corollary 3 Under rationing based on ealth and ost, rationed onsumers fae the same equilibrium pries hen the budget inreases. Clearly, Proposition 3 applies diretly to Cournot ompetition in the private market. Equilibrium rationing hen the private market is perfetly ompetitive is a little different. Here, some lo-ost and ealthy onsumers may be rationed. When the private market is ompetitive, pries are given by marginal osts, τ() =. The onsumer elfare index is Z Z {φ(, )max[u( )+1,U()] + [1 φ(, )] [U()+1]} f()g() d d. (20) Corollary 4 Suppose the private market is ompetitive so that pries are equal to marginal osts. The publi setor rations all onsumers ith ost above a threshold. For those onsumers ith osts belo the threshold, 24