Capitation contracts: access and quality

Journal of Health Economics 18 Ž1999. 315–340

Capitation contracts: access and quality Hugh Gravelle

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National Primary Care Research and DeÕelopment Centre, Centre for Health Economics, UniÕersity of York, Heslington, YO1 5DD, UK Received 4 October 1996; revised 24 June 1998; accepted 10 July 1998

Abstract The implications of competition amongst providers in both private and public systems for the quality of service and the number of care providers are analysed. Strong conditions must be imposed on preferences and cost conditions for quality to be efficiently supplied. In a median voter model of a public system the capitation fee and quality are lower than under competitive market equilibria and the number of practices inefficiently small. Entry control by a union which maximises gross provider income reduces the number of practices until the market is only just covered. q 1999 Elsevier Science B.V. All rights reserved. JEL classification: I1; L13 Keywords: Capitation; Quality; Access; General practice; Product differentiation

1. Introduction This paper examines capitation contracts in which health care providers receive an initial payment for each patient who registers with them and in return agree to provide services in the future under conditions and terms laid down by the contract. The HMO is one example: the patient pays an initial insurance premium to the HMO and the terms of the insurance contract include the type and price

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Tel.: q44-1904-433663; Fax: q44-1904-433644; E-mail: [email protected]

0167-6296r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 2 9 6 Ž 9 8 . 0 0 0 4 6 - 0

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Žcopayment. of care to be provided when the patient is ill. A second example is the British NHS: patients register with a general practitioner ŽGP. who receives a capitation fee for each patient on their list. The capitation fee is paid by the state rather than the patient and care provided by the GP to patients has a zero price. In both systems competition between providers takes place when patients are deciding which HMO or GP list to join, rather than when they require care. HMOs can compete via the capitation fee or insurance premium, though NHS GPs cannot. Both compete for patients via quality. Some changes in the characteristics of services delivered by providers are regarded as improvements by all patients: lower copayment rates, wider range of conditions for which care will be provided, better diagnostic facilities, shorter waiting times for appointments, longer surgery opening hours. We refer to these aspects as the quality of the service provided. Other changes in service characteristics may make some patients better off and others worse off, for example, replacement of an afternoon with an evening surgery. Perhaps the most obvious type of horizontal differentiation is the location of the GP or HMO premises. Distance has a marked impact on choice ŽMartin and Williams, 1992; Dixon et al., 1997. and use of GP ŽCarr-Hill et al., 1996.. The paper examines the welfare properties of market equilibria where providers compete for patients via quality and price. The market equilibria with free entry also determine the number and location of providers and hence the accessibility of services to patients. Section 2 sets out the basic model of a private HMO system and the possible types of market equilibria and Section 3 examines the implications of provider collusion. Section 4 considers the welfare properties of the market equilibria. Section 5 discusses the optimal regulation of a public capitation system in which the capitation fee is paid by the state and financed by taxation. Section 6 outlines a positive, public choice, model of the NHS type system and examines the choice of the capitation fee when voters consider its tax implications and may face an entry controlling profession. Section 7 summarises the results. The analysis is complementary to the existing literature on capitation which is primarily concerned with the design of insurance and reimbursement contracts in the face of asymmetrical information. For example, Selden Ž1990. shows that when insurers are distinct from providers, the efficient contract can have full cover insurance for the patient with a mixed capitation and fee per item contract to reimburse the provider. Both Rogerson Ž1994. and Ma Ž1994. derive optimal reimbursement schemes for insurers facing a monopoly provider who can influence the demand from fully insured patients by varying quality. Rogerson assumes that the provider is non-profit and cannot select patients, whilst in Ma Ž1994. the provider is profit maximising and can turn away patients who are expensive relative to the revenue they generate. Ellis Ž1998. examines the implications for patient selection and quality of capitation payment by an insurer to profit maximising duopoly providers when patients are heterogeneous with respect to location and severity of condition.

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The current paper differs from this literature in a number of ways. Its concern is not the detailed structure of the contract between provider and insurer but with the way in which competition between providers affects the quality of service and, via the number of providers, patient access to services. These issues have also been examined by Economides Ž1993. using a different specification in which firm cost functions are separable in quality and quantity. Quality then has many of the characteristics of a public good for the customers of a firm in that the cost of increases in quality do not vary with the number of consumers. For example, in the GP context, a GP may spend more money on better software for organising patient records. In the model used in the current paper the cost function is not separable: increasing quality increases the average and marginal cost to the practice of each patient. One example would be the GP spending longer with each patient. Our focus also differs in that the main welfare analysis examines restrictions on preferences and cost functions which lead to efficient or welfare maximising quality, rather than assuming a simple set of preferences and showing that quality is efficient for a given number of firms. The conditions for quality to be welfare maximising are quite stringent. The institutional context of the current paper also differs from the literature to date. We focus on the implications of competition amongst providers in both private-HMO type systems where providers set the capitation fee to be paid by patients and in a public-NHS GP type system where the capitation fee is chosen and paid by the state. In the private system we assume that providers are also the insurers of patients, as in some staff-model HMO systems. In such systems the issue of the optimal insurer–provider contract considered in the literature referenced above does not arise. In the analysis of the public NHS type system where providers are paid by the state rather than patients, we take the terms of the GP contract with the state, except the capitation fee, as given in order to consider whether a regulator who cannot observe the quality of individual GPs can nevertheless induce them to supply it efficiently by varying the capitation fee. We also contribute to the literature by considering the implications of provider collusion and of having the capitation fee in a public system chosen by voters.

2. Model For definiteness the analysis is set in the context of a market for GPs paid by capitation contracts. We talk in the paper about patients joining practice lists and getting service of a particular quality. The model of the private system can be reinterpreted to cover patients buying an insurance contract from a provider with a particular set of terms and conditions, including the prices at which care will be supplied. The model is an extension of Salop Ž1979. to allow for endogenous product quality, more general preferences and cost functions, and equilibria in which the

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market may not be completely covered. We use the Salop framework because it is perhaps the simplest specification which allows examination of equilibria where both quality and distance affect demand and where there is free entry to the industry. The Salop model is relatively straightforward and well known, so that the implications of the extensions we adopt can be readily assessed. 2.1. Preferences and technology In a private system the contract offered by the ith provider is defined by the capitation fee pi paid by the patient to the provider and by the other terms of the contract qi . Patients also care about their distance from their service provider and utility is decreasing in distance. Patients have identical preferences and differ only in their location. Utility from choosing the contract offered by provider i is y Ž pi ,qi , d . where d is the patient’s distance from provider i. Utility if no contract is chosen is normalised to zero. The R patients are distributed uniformly around a circle of circumference K with density r s RrK. Quality q would in general be a vector of contract characteristics including the range of services to be provided, the circumstances under which they are provided, the fees charged for the services, and the hours when services are provided. By suitable definition of the variables we can suppose that patient utility is increasing in q. To reduce notational clutter q is a scalar measure of ‘quality’: a characteristic of the service which increases the patient’s utility. Although the model context is taken as competition amongst GPs for patients, it can apply to more general health care contracts whose design may reflect asymmetries of information. For example, let

yˆ s 1 y b Ž z . yˆ 0 Ž y y z y p, d . q b Ž z . yˆ 1 Ž y y z y p y fx , g Ž x . , d . s yˆ Ž p, f , x , z , d . be expected utility, where b is the probability of ill health, y is income, f is the fee per item, x the amount of care consumed when ill Žstate 1. and g the benefit from care when ill. The patient chooses a costly preventive activity z which reduces the probability of ill health. Changes in the contract alter preventive activity and the care received when ill, both of which affect the provider–insurer’s expected costs. If neither z nor x are controllable via the contract the patient’s choice of prevention and care when ill are z Ž f, p, d . and x Ž f, p, d .. Defining q s yf, the expected utility from the contract can be written y s yˆ Ž p, f, z Ž f, p, d ., x Ž f, p, d ., d . s y Ž p,q, d . and yq s byˆ 1 x ) 0. If x was directly controllable by the contract then z s z Ž f, x o , p, d . and we could define quality as a vector with q1 s yf, q2 s x o. Both cases would enable us to write the patient’s

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preferences as an indirect utility function uŽ p,q, d . which is increasing in quality and decreasing in the contract price p and distance d . GPs are profit maximizers. A full account of quality determination would include more complex doctor objectives and information asymmetry ŽEllis and McGuire, 1985; Selden, 1990.. We neglect these aspects to make tractable the analysis of those aspects of provider decision making which are our main concern. The N GP firms are located around the circle and have identical cost functions. Total cost of firm i is Ci s C Ž qi , Di . q Fi , where Fi is fixed cost and C ŽP. is a convex variable cost function. We interpret Fi as a reservation wage for the GP and assume that the distribution of reservation wages across potential GPs is such that there is a positively sloped inverse supply curve FsFŽ N . ,

FX Ž N . ) 0

Ž 1.

A practice’s quality is the same for all its patients. As a practice reduces its price, starting from some initial choke price at which it attracts no patients, it initially attracts patients who would not have joined the list of any other firm because they were too far away. In this monopoly range the practice does not attract patients from other lists. As the price is reduced further the practice’s additional patients are those who switch from neighbouring practices. In this competitive range of prices the demand curve for the practice is steeper since the alternative for the new patients is the list of the other practice, rather than receiving no care. The demand curve for the practice has a kink at the point where it starts to attract patients from neighbouring practices and at this point its marginal revenue curve is discontinuous. Depending on the parameters of the model, there are three possible equilibria: on the monopoly segment of the practice demand curves, at the kink, and on the competitive segment where practices are affected by the prices and qualities of neighbours. For each equilibrium we first examine the symmetric Nash equilibrium price and quantities with a given number of equally spaced practices. The equilibrium prices and quantities produce a profit for each practice which is non-increasing in the number of firms. We then close the model by assuming free entry and an increasing supply price for GP firms. 1 The general specification of preferences and costs is used to investigate the welfare properties of the market equilibrium quality. We also use a simple case with cost function: Ci s a qi2 Di and linear preferences u i s pi y qi y t d i , where t is a distance cost parameter, to derive explicit solutions and to facilitate comparison with the standard Salop model of product differentiation. The simple case is

1

See Economides Ž1989, 1993. for demonstrations that in a model with preferences and technology similar to those of our simple case, the Nash equilibrium is symmetric and firms equally spaced.

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also the basic framework for the welfare analysis of the market equilibrium number of firms and the examination of the public system. 2.2. CompetitiÕe equilibrium In the competitive case all consumers join the list of some firm and are strictly better off than with no firm. Since utility from its contract declines with patient distance, practice i will get patients to its right up to the distance d iiq1 defined by

y Ž pi ,qi , d iiq1 . s y Ž piq1 ,qiq1 , Diiq1 y d iiq1 . ) 0 where Diiq1 is the distance between i and its rival i q 1 to its right. ŽFirms are labelled so that i q 1 is the next firm to right and i y 1 the next firm to the left... Its left market area is defined analogously by

y Ž pi ,qi , d iiy1 . s y Ž piy1 ,qiy1 , Diiy1 y d iiy1 . and its total demand is Di s rd iq1 q rd iy1 s Di Ž pi ,qi , piq1 , piy1 ,qiq1 ,qiy1 .

Ž 2.

Increases in pi and reductions in quality reduce the firm’s market areas. For example,

Ed iiq1 E pi

sy

y p Ž pi ,qi d iiq1 . y d Ž pi qi , d iiq1 . q yd Ž piq1 ,qiq1 , Diiq1 y d iiq1 .

In a symmetric Nash equilibrium where all firms choose the same price and quality and are evenly spaced, each firm has a market segment of KrN s 2 l and the effects of price and quality on demand are Di p i s yr

yp Ž p,q,l . yd Ž p,q,l .

, Di q i s yr

yq Ž p,q,l . yd Ž p,q,l .

Ž 3.

Gross profit is

p i s pi Di Ž pi ,qi ; piq1 , piy1 ,qiq1 ,qiy1 . y C Ž qi , Di Ž pi ,qi ; piq1 , piq1 , piy1 ,qiq1 ,qiy1 . .

Ž 4.

There is a three-stage game: firms choose whether to enter; given that they enter they choose a location Žhorizontal differentiation.; having chosen a location they compete in vertical quality and capitation fee. We assume there is a symmetric Nash equilibrium in which firms locate equidistantly around the circle and choose the same quality and capitation fee.

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Table 1 Market equilibria in simple case: y s q y py t d , C s a q 2 D. Marginal surplus calculated with l s1 Private market

Monopoly

Kink

Competition

Price

p

3r8 a

Quality List size Gross profit

q D p

1r2 a R r4a tK R r32 a 2 tK

Ž1r4a . qŽ tK r N . 1r2 a RrN tKR r N 2

Marginal surplus Range of N Public system

d Srd N

3 R rŽ64a 2 tK . w1,4a tK x pr3qwŽ pa . 2 q3 pa x1r 2 4r3 a

Ž1r2 a . yŽ tK r2 N . 1r2 a RrN Ž R r4a N . yŽ tKR r2 N 2 . tKR r4 N 2 w4a tK,6 a tK x pqŽ tK r2 N .

q Ž p, N .

tKR r4 N 2 Ž6 a tK,`. wŽ tK r N . 2 qŽ pr a .x1r2 yŽ tK r N .

Given the number of firms and their location, firm i chooses its capitation fee and quality to maximise gross profit:

p i p i s D i q w p i y C i D x Di p i s 0

Ž 5.

p i q i s w pi y C i D x Di q i y C i q i s 0

Ž 6.

taking the fees and qualities of its rivals as given. With identical firms equally distributed we get the Nash equilibrium for a given number of firms by solving Eqs. Ž5. and Ž6. to yield the price, quality and profit p c Ž N . as a function of the number of firms. Table 1 summarises the fee, quality, list size, and gross profit for the simple case. 2 2.3. Monopoly equilibrium In the monopoly case some potential patients are so far from practices that they choose not to join any practice and firms have market areas l Ž pi ,qi . to their right and left defined by

y Ž pi ,qi ,l . s 0 Substituting the monopoly demand function Di s 2 rl Ž pi ,qi . into the profit function ŽEq. Ž4.. and maximising yields the monopoly fee, quality, and maximised profit p m Ž N .. Price, quality and profit are unaffected by the number of firms since there are gaps in the coverage of the market and each firm’s profit depends only on its decisions. 2

Gravelle Ž1996. has a fuller discussion of the market equilibria in the simple case, including their comparative static properties.

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2.4. Kink equilibrium The slopes of the monopoly demand curve with respect to price and quality, which are Dp s 2 rl p s y2 r

yp Ž p,q,l . yd Ž p,q,l .

, Dq s 2 rl q s y2 r

yq Ž p,q,l . yd Ž p,q,l .

,

Ž 7.

differ from the slopes of the firm’s demand curves under competition Žsee Eq. Ž3... In the kink case the equilibrium is at the kink in the demand curve where the market is just covered. If the price is raised the marginal consumers at distance l will not join the practice, whilst if price is reduced some patients will be attracted from neighbouring practices. Each practice’s marginal revenue curve is discontinuous. At a kink equilibrium the price and quantity must be such that the monopoly demand is just equal to the competitive demand which is, since the market is covered, RrN. Hence Di s 2 rl Ž p,q . s RrN. Maximizing profit by choice of price and quality subject to this constraint gives the kink price, quality and profit p k Ž N .. 2.5. Prices, profit and number of practices Practice profit varies with the number of firms. With free entry of firms the equilibrium number of firms is determined by the condition that the marginal GP earns a gross profit just equal to her reservation wage:

p Ž N . sFŽ N .

Ž 8.

where p Ž N . is the Nash equilibrium gross profit. The type of equilibria depends on the shape of the gross profit function and the inverse GP supply function. 3 Fig. 1 plots gross practice profit p Ž N . against the number of practices. With less than N m k practices we have the monopoly solution where practices do not compete. Over the range N m k to N k c the number of practices supports the kink equilibrium in which practices compete but the marginal consumer has no surplus. With more than N k c practices competition amongst practices leaves even the marginal consumer with a positive surplus. As the p Ž N . curve indicates, increases in the number of practices reduce profit except in the monopoly equilibrium where profit is unaffected. The equilibrium number of practices is determined by the intersection of the inverse supply curve with the gross profit curve. In Fig. 1 the 3 Other papers developing the model of Salop Ž1979. retain its assumption that there is a perfectly elastic supply of firms. See, for example, Economides Ž1989, 1993. and Norman Ž1989.. The assumption that F does not vary with the number of firms simplifies the analysis in the competitive and kink cases but does not permit analysis of the monopoly case since the number of firms is indeterminate.

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Fig. 1. Equilibrium with different supply curves: competitive Ž F 1 ., kink Ž F 2 , F 3 . and monopoly Ž F 4 ..

equilibrium is competitive when the supply curve is at F 1 , kinked with supply curves F 2 , F 3 and monopoly with F 4 . 4

3. A self-regulating profession The medical profession is often able to persuade policy makers that it should be permitted to regulate entry, capitation fees or quality. The self-regulatory equilibrium depends on what the profession can control and on its objectives. 5 Using the simple case specifications of preferences and costs, we examine two models in both of which the profession seeks to maximise total gross profits Np . 6 3.1. Entry control Suppose the profession can only control entry and the free entry unregulated equilibrium would be competitive. In terms of Fig. 1 the profession seeks a point 4 The other curves are explained below. The boundaries of the ranges of N for the three types of equilibrium are found by noting that the profit Žor price. functions are continuous in N. Thus, for example, at the boundary N mk between the monopoly and kink equilibria ranges, p m Ž N mk . s p k Ž N mk .. Solving gives N mk s 4a tK. 5 See the literature on the objectives of trade unions, for example, Booth Ž1995.. 6 One possible justification for such an objective is a union run by officials who seek to maximise their income from subscriptions which are proportional to members’ gross income.

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on the p Ž N . curve where p Ž N . N is maximised. The profession’s indifference curves in Žp , N . space are rectangular hyperbola, such as Iu , Iuo . Clearly the solution will not lie on the monopoly segment of the profit function since over this range additional entrants increase total profit at a constant rate. The solution must lie on the downward sloping portion of the profit curve. By comparing the slope of the profit curve over the kink and competitive segments with the slope of the indifference curve Žsee Appendix A., we have: 7 Proposition 1. The total profit maximising number of practices is at the boundary between the ranges of firms for which there are kink and monopolistic competition equilibria. If the supply curves are such that the number of practices with free entry is in the kink or monopoly range, as in Fig. 1 with supply curves F 4 , F 3, or F 2 , the profession cannot achieve the total profit maximising solution and its control of entry has no effect. If the free entry solution is in the competitive range, as in the Figure with supply curve F 1, the self-regulating profession restricts the number of practices, driving up the capitation fee and practice profit until the market is just covered. The marginal patient gets a zero surplus and is indifferent between joining a list and not joining. 3.2. Collusion oÕer fee and quality Assume now that the members of the profession can collude, rather than compete, on quality and capitation fee but take the number of GPs as given. For any given N they aim to maximise practice profit. They do so by choosing a combination of fee and quality which leaves the marginal patient with no surplus. 8 Suppose not, so the marginal patient has a positive surplus. If all practices increase their fees no individual practice will lose market share because the difference between practice fees is maintained. Since demand is unaltered and the fee has increased all practices are better off. The collusive profit function p coll coincides with the non-collusive monopoly profit and kink profit function but lies above the non-collusive competitive profit function for levels of N which would lead to competition in the absence of collusion. Fig. 1 illustrates. 7 If the union maximised Np Ž N . minus practices reservation wage costs HF Ž N .d N, the solution X would be where the curve p Ž N .q Np Ž N . cuts the supply curve F Ž N .. The marginal gross income kc curve is discontinuous at N and lies above the horizontal axis to the left of N kc and below it to the right. Hence the solution could be at N kc or in the kink or monopoly range, even if the free entry equilibrium would be competitive. 8 The kink equilibrium is also achieved if firms do not collude but have Loschian conjectures that all ¨ other firms will follow their price and quality variations ŽNorman, 1989..

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Since Žsee Appendix A. total gross profit is increasing in N along p coll the collusive solution is at the point where the collusive profit function cuts the reservation wage function. This is true whether the profession can control entry Žin which case it chooses the largest number of practices consistent with the marginal practice not making a profit less than its reservation wage. or whether it has no control of entry. Proposition 2. If the non-collusiÕe equilibrium is competitiÕe (a) collusion oÕer capitation fee and quality will yield more practices and smaller list sizes than if the union can only control entry; (b) if the union can only control entry the number of practices is reduced compared with the competitiÕe equilibrium.

4. Welfare and market equilibrium We first consider the welfare properties of the capitation market equilibrium price and quality for a given number of firms. Section 4.3 examines the welfare properties of the market equilibria in respect of the number of firms. We address two issues. First, is the equilibrium Pareto efficient? Second, given that allocations which are Pareto efficient may have objectionable distributional properties and the focus of the paper is on quality, does the equilibrium have a welfare maximising level of quality for all welfare functions which are an arbitrarily weighted sum of consumers’ utilities and firms’ profits? 4.1. Pareto efficiency of price and quality The unregulated private capitation regime with a given number of firms has the usual Pareto efficiency properties: Proposition 3. CompetitiÕe and kink equilibria haÕe Pareto efficient price and quality for a giÕen number of firms. Monopoly equilibria haÕe inefficient price and quality. The inefficiency of the monopoly solution is standard: for a given quality level each monopoly firm sets too high a price and has too few patients. The marginal patient values joining the list at p m whereas the cost to the firm of the marginal consumer is CD - p m . When the market is covered a Pareto improving change in price and quality must be one which does not change the demand of any firm. If a proper subset of firms change prices and qualities and the demand for some other firm falls, that firm must be worse off. Conversely, if the demand for some other firm increases the consumers who switch to it must be worse off or they would have chosen it at the original set of prices and qualities. Restricting attention to changes in price and

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quality which leave demand unchanged, a firm’s marginal rate of substitution between quality and price is C qrD. Rearranging the firm’s first order conditions for profit maximization and using the demand derivatives ŽEq. Ž3.. gives Cq D

sy

Dq

sy

yq Ž p,q,l . yp Ž p,q,l .

Dp

so that the indifference curves of the firm and the consumers at l are tangent. All changes in price and quality must either make the firm or the consumers at l worse off and cannot be Pareto improvements. Since firms have market power in that each faces a downward sloping demand curve, the conclusion that there are market equilibria with Pareto efficient price and quality is at first sight surprising, given the well known inefficiency of quality choice under monopoly ŽSpence, 1975.. The explanation is that when the market is covered the firm’s choice of price and quality is governed by its effect on the marginal consumer with demand held constant, and so the firm and the marginal consumer’s marginal rates of substitution between price and quality are equated. Firms efficiently exploit the marginal consumer when the market is covered. 4.2. Welfare: quality and prices Pareto efficiency is a weak criterion for judging allocations. In the cases where the market is covered it places a disproportionate weight on the consumers who are at the edges of firms’ market areas and neglects the implications of changes in price and quality on the infra-marginal consumers. We adopt a welfare function which weights all consumers equally and assumes that all firms have the same price and qualities and are equally spaced: W Ž p,q,n, l .

½H

sN 2

l

5

H0NF Ž N˜ .d N˜

y Ž p,q, d . rd d q lw pDy c Ž q, D . x y l

0

H0N F Ž N˜ .d N˜

s N w V Ž p,q . q lp Ž p,q . x y l

Ž 9.

H0N F Ž N˜ .d N˜

s S Ž p,q, N, l . y l

where 2 l is the segment of the market served by one firm. l g Ž0,`. is the welfare weight on practice profits. Welfare maximising price and quality satisfy Vp q lpp s 2

l

H0 y Ž p,q, d . rd d q l Ž p y C

Vq q lpq s 2

p

D

. Dp q D s 0

Ž 10 .

D

. Dq y C q s 0

Ž 11 .

l

H0 y Ž p,q, d . rd d q l Ž p y C q

and in general vary with the welfare weight.

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Proposition 4. No monopoly equilibria is welfare maximising for any l g (0,`). All equilibria in which the market is coÕered haÕe welfare maximizing p,q for some l if and only if consumer preferences are weakly separable in distance: y s y (u(p,q),d ). Since Vq ) 0, Vp - 0 and the monopoly equilibria have pp s 0, pq s 0 they do not have optimal p, q. When the market is covered changes in the prices and qualities of all firms have no effect on demand and the marginal rate of substitution between price and quality for firms is C qrD. With weakly separable preferences all consumers have the same marginal rate of substitution between price and quality, so that, using the profit maximization conditions ŽEqs. Ž5. and Ž6.. and the demand derivatives ŽEq. Ž3.. C q Ž q e , D Ž p e ,qe . . D Ž p e ,q e .

sy

Dq Ž p e ,q e . Dp Ž p e ,q e .

sy

u q Ž p e ,q e . u p Ž p e ,q e

sy

Vq Ž p e ,q e . Vp Ž p e ,q e .

Ž 12 . Defining le s yVp Ž p e ,q e .rDŽ p e ,q e ., the market equilibrium p e , q e maximises V q lep . The result is not equivalent to Proposition 3. An allocation is Pareto efficient if it maximises a welfare function which is an increasing function of agents’ utilities. To ensure that the Pareto efficient allocation maximises a particular example of such a welfare function requires restrictions on preferences or technology. The fact that a market equilibrium is welfare maximising for some welfare weight l may be of limited interest if we cannot agree on what the weight should be. We therefore consider whether market equilibria can provide a level of quality which is welfare maximising for all l g Ž0,`.. The market equilibria depend on preferences and technology and not on the welfare weight, so we investigate whether there are circumstances in which the welfare maximising quality q ) does not vary with l and q e s q ) . Proposition 5. Market equilibria in which the market is coÕered haÕe welfare maximising quality for all Õalues of the welfare weight l if and only if consumer preferences are weakly separable in distance and consumers haÕe zero income elasticity of demand for quality: y s y (u(p,q),d ), E (u q r u p ) r E p s 0. Weak separability ensures that the social marginal value of changes in quality, which depends on the marginal rates of substitution of all consumers, equals its value to the firm, which depends on the marginal rate of substitution of the marginal consumer. With zero income elasticity of demand, changes in l which change the price have no effect on consumers marginal value of quality, so that the welfare maximising quality is the same for all l.

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No restrictions on technology are required when the market is covered because changes in price and quality have no effect on the number of patients for each firm. However, when there are gaps in the market and some consumers do not join any list, changes in price and quality change the number of consumers per firm. Even if consumers’ marginal valuation of quality is the same at all prices, optimal quality will independent of l only if the firms’ marginal rates of substitution between quality and price C q Ž q, D .rDŽ p,q . are independent of the number of patients. Hence we have Proposition 6. The market equilibrium has welfare maximising quality for all Õalues of the welfare weight for any giÕen number of firms if and only (a) consumer preferences are weakly separable in distance and consumers haÕe zero income elasticity of demand for quality: y s y (u(p,q),d ), E (u q r u p ) r E p s 0 and (b) the cost function is linear in quantity: C s c(q)D. The conditions on preferences in Propositions 5 and 6 for market equilibria to yield optimal quality for a wide range in of welfare judgements appear to be quite stringent in the case of health care. First, patients who expect to be heavy users of GP services may choose to live near the practice. This suggests that the marginal valuation of quality is unlikely to be independent of distance to practice, in contradiction of our assumption that patients are homogeneous except with respect to location. Second, as the example in Section 2.1 suggests, taking account of the dependence of preferences on health status and the risk aversion of consumers means that there are likely to income effects on their valuation of contract terms. Third, if there is patient heterogeneity, patients at any given distance d who place a low valuation q may choose not to join the list of any firm, even though other patients at d do so. In these circumstances changes in p and q affect demand for the firms and the situation is similar to case in which there are identical consumers but gaps in the market. The equilibria will be suboptimal because firms neglect the effect of price and quality on the number of consumers. Our assumption that patients at d are homogenous can be relaxed somewhat, to let y Ž p,q, d . be the expected utility over all consumers located at delta. If we restrict attention to competitive equilibria where all types of consumers at all locations join a list, as in Economides Ž1993., the results about the welfare properties of competitive equilibria are essentially unchanged. We can also note that the cost condition in Proposition 6 rules out cases in which quality has public good aspects. For example the case considered in Economides Ž1993., where total cost depends on quality and not on quantity would imply that no monopoly equilibrium could yield welfare maximising quality for any welfare weight. 9 9

Economides Ž1993. restricts analysis to cases where there are no gaps.

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The simple case, illustrated in Fig. 1 and Table 1, satisfies both conditions. Setting l s 1 10 the welfare function is tl N W s N 2 lr q y y a q 2 y F Ž N˜ . d N˜ Ž 13 . 2 0 where l s Kr2 N when the market is covered and l s Ž q y p .rt when there are gaps. The optimal quality in both cases is 1r2 a , which, see Table 1, is the quality in all three types of market equilibrium. When the consumer utility function is q y p y t d firms can appropriate all the gains to consumers from increasing quality by increasing the price. They therefore internalise the social gains from quality improvements and equate them to their marginal cost. Consumers are exploited but efficiently. Since the price does not affect W when l s 1 and the market is covered, the kink and competitive equilibria also have optimal prices. When there are gaps the price does affect W and the optimal capitation fee, which is the marginal cost of an additional patient, is less than the monopoly price: p ) s CD s a Ž q m . 2 s 1r4a - 3r8 a s p m .

ž

/

H

4.3. Welfare: numbers of firms Remembering that the number of firms in equilibrium is determined by the condition that the profit of the marginal firm is equal to the reservation wage F Ž N ., the marginal effect of an increase in the number of firms at the market equilibrium is dW Ž p e ,q e , N e . dN

s Sp s Sp s Sp

dp dN dp dN dp dN

q Sq q Sq q Sq

dq dN dq dN dq dN

q SN y l F Ž N e . qV q lp q N w Vl l N q lp N x y l F Ž N e . qV q N w Vl l N q lp N x

For monopoly equilibria there is a very general result which does not require any restrictions on preferences or technology. Proposition 7. At a monopoly equilibrium there are too few firms. The price and quality chosen by firms in the monopoly equilibrium do not depend on the number of firms so that dW Ž p m ,q m , N m . dN 10

s SN y l F Ž N m . s V q l p Ž N m . y F Ž N m . s V ) 0

Since Wp s Ž l y1. R r N there are uninteresting corner solutions when the market is covered if l /1.

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Because the monopoly equilibrium leaves gaps in the market, an additional firm enables extra consumers to be served. The marginal firm however only considers its profit and not the gain to the extra consumers and so there are too few firms in equilibrium. Access is inefficiently low. In the competitive and kink equilibria, price and quality vary with the number of firms and we require stronger assumptions to get firm welfare conclusions. In the simple case and with l s 1, we know that both equilibrium quality and price are welfare maximising. However, the gross value of an additional practice at competitive or the kink equilibria is Žsee Table 1. dW Ž p e ,q e , N e . dN

s SN y F Ž N e . sV qp Ž N e . q N w Vl l N q lp N x y F Ž N e . tKR s 4Ž N e .

y FŽ Ne.s 2

tKR 4Ž N e .

Ž 14 . yp Ž N e . - 0, 2

es c,k

and we have Proposition 8. In the simple case, with l s 1, the competitiÕe and kink equilibria haÕe too many firms. Practice profit exceeds the marginal social value of an additional practice and so conveys the wrong entry signal. Note that the social value of an additional practice is less than in the monopoly case. Once the market is covered additional practices do not supply additional patients with care. Fig. 1 illustrates the relationship between the equilibria Ždetermined by the intersection of p Ž N . with the inverse supply function F Ž N .. and social optima Ždetermined by the intersection of the marginal social value curves SN with F Ž N ... 11 A variety of supply functions are used to illustrate the possible relationships between the optimal and equilibrium number of practices in the competitive Ž F 1 ., kink Ž F 2 , F 3 . and monopoly Ž F 4 . cases. Note that it is possible that entry restriction by a self regulating profession could be welfare improving with some supply functions. When the market equilibrium has too many GPs policy is apparently very simple: restrict the number of firm either by direct entry control or by a tax on each practice. The resulting increase in the capitation fee increases gross practice profits. Patients are worse off because they pay a higher capitation fee but this is exactly offset, in welfare terms, by the increased gross profits of the firms. Since quality does not vary with the number of firms the equilibrium quality continues to be socially optimal. 11 Although S is continuous in N, its derivative is not. With incomplete cover SN reflects both the gains from reduced distance costs to patients and the gains from additional patients. Only the former occur when the market is covered.

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5. A public capitation system In a public system, like the NHS, GPs are paid a tax financed capitation fee p per patient on their list. Patients pay nothing when joining a practice list. The proportion of GP income from capitation payments in the NHS was increased in 1990 as part of the introduction of the internal market and regulations were amended to make it easier for patients to move from one practice to another. The aim of the changes was to increase the incentives for GPs to attract and retain patients by maintaining and improving the quality of their services. We investigate the public system using the simple specification of preferences and technology in order to derive some explicit solutions and clear results. We also focus on the competitive case, where the marginal consumer has positive utility, since the vast majority of NHS patients are registered with some GP. The practice list is found from Eq. Ž2. but with the price paid by patients set to zero: Di s Di Ž0,qi ; 0,0,qiq1 ,qiy1 .. Practices can compete only via quality. Substituting the demand function into the first order condition on quality ŽEq. Ž6.. and assuming symmetry, we can solve for the Nash equilibrium quality q Ž p, N .. The explicit solutions for the simple case are shown in Table 1. We show in Appendix A that competitive equilibrium quality is increasing in the number of practices under public capitation. Increasing the number of practices increases competition for patients and since practices cannot compete on price they have greater incentives to increase quality to attract additional patients. Even when quality is not observable by the regulator she has an effective instrument for controlling it since quality is increasing in the capitation fee in a public system. Raising the fee makes marginal patients more valuable and induces practices to compete for them by raising quality. The result confirms those found in somewhat different settings ŽRogerson, 1994; Ma, 1994; Ellis, 1998.. By setting the public capitation fee equal to the private market equilibrium fee the regulator can induce firms to supply the optimal quality. The regulator mimics the outcome of an unregulated private market to induce optimal quality and does not need to monitor the quality level of individual practices. Unfortunately, with free entry the equilibrium number of firms would be too large. The planner has two targets Žoptimal quality and number of firms. and requires two instruments to achieve a first best. The first best optimal capitation remuneration scheme has a negative lump sum component to control the number of practices and a capitation component to control quality. An alternative to such a two part non-linear scheme is a capitation fee coupled with direct control of the number of GPs. 6. Public choice models Instead of comparing the welfare properties of the private capitation outcome with a public capitation scheme run by a welfare maximising regulator it may be

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more appropriate to adopt a comparative institutional approach based on a positive model of a public capitation system ŽDemsetz, 1969.. Suppose that in the public system the capitation fee is decided by majority voting, rather than by a welfare maximising regulator. Assume for simplicity that the public system will be in equilibrium in the competitive range. We again use the simple case with a welfare weight l s 1. Each voter has utility q y t d y p where p is paid as a poll tax rather than directly as a fee for joining the list of a particular practice. Given our assumption of full coverage this is equivalent to a model in which the patient pays the fee directly to her GP and the fee is determined by voting of voter-patients. We assume that there are so few GPs that their votes have no effect on the outcome. Suppose that voters realise that firms respond to higher fees with higher quality via q Ž p, N . and that increases in fees raise profit and therefore the supply of firms. The capitation fee will be chosen to maximise the voter objective function U s U Ž p, N . s q Ž p, N . y p y

tK 4N

.

Ž 15 .

subject to the constraint that p Ž p, N . G F Ž N .. We can interpret Eq. Ž15. by supposing that voters do not know where additional practices will be located. They therefore evaluate the effect of the increases in the number of practices by their effect on the expected distance cost. If voters can control the number of practices, directly or indirectly via a lump sum tax or subsidy, as well as the capitation fee, the voting equilibrium will be first best efficient. If voters control only the fee but take account of its effects on both quality and numbers the outcome will be second best efficient: they will trade off the effects of increases in the fee on quality and access subject to the firm supply curve. 6.1. Voter equilibrium In some circumstances it may be plausible to suppose that voters ignore the effect of the fee on firm profit and the number of firms. They may be myopic and not realise that higher fees have implications in addition to an increase in quality. More plausibly, though somewhat more loosely, suppose that decisions by GPs to enter are investment decisions made in the light of their expectations about future capitation fees over their working lives. Once they have entered GPs choose quality in each period to maximize their profit in the period. Voters make decisions on the fee in each period and are unable to commit themselves, or their successors, to the future level of fees, even if they realise the implications for the supply of GPs. In these circumstances GPs will anticipate that the capitation fee will be chosen in each period to maximize the utility of voters in that period, neglecting the longer term effects on GP supply.

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With voters concerned only about the effect of the fee on the quality of care, the fee chosen for given N will satisfy E q Ž p, N . y1s0 Ž 16 . Ep Using Table 1, the fee is o

p Ž N.s

1

a Ž Kt .

2

-

1

tK

Ž 17 . 4a 4a N N and the resulting quality level is 1 tK 1 qoŽ N . s y Ž 18 . 2a N 2a With free entry equilibrium will be established by adjustments in the number of practices until p Ž p o Ž N ., N . s F Ž N .. 12 y

2

q

Proposition 9. The Õoting equilibrium has a lower price, lower quality and fewer practices than the unregulated competitiÕe equilibrium. Quality is less than the first best welfare maximising leÕel. The marginal social Õalue of additional GPs at the Õoting equilibrium is positiÕe. When voters are concerned only with the effects of the capitation fee on quality and their tax bill they increase their utility, given the number of firms, by reducing price and quality. Firms are worse off and so few of them enter that their marginal social value in terms of improved access and lower distance costs is positive, unlike the unregulated competitive private market equilibrium where there are too many firms. 6.2. Stackelberg equilibrium: Õoters and a self-regulating profession Suppose entry is controlled by a professional union which seeks to maximise total practice profit Np . The capitation fee chosen by the voters each period implies a particular profit level for each practice: p o Ž N . s w p o y a Ž q o . 2 x RrN. Substituting in for the voting equilibrium fee ŽEq. Ž17.. and quality ŽEq. Ž18.., we derive the union’s choice of N by maximising Np o Ž N .. Proposition 10. At the Stackelberg equilibrium with a union controlling entry and the Õoters controlling the capitation fee the market is only just coÕered and the marginal social Õalue of additional GPs is positiÕe. With the voter equilibrium under free entry in the competitive range, where the marginal patients have a positive surplus, the union will restrict entry, driving up 12 An equilibrium in which there is full coverage and practices compete via quality, at a given fee, can only occur for combinations of N and p satisfying 2 M w q Ž p, N .y p xr tK ) Mr N. In what follows we assume that supply function of practices lies, in part, in this region.

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the fee and reducing quality until the marginal patient has a zero surplus. 13 Since we know that the number of firms is inefficiently low at the free entry voting equilibrium, the number of firms is even further from the optimum when the union control entry.

7. Conclusions In private markets a major policy issue is whether the equilibrium quality and numbers of providers is welfare maximising and if it is not what policies are required. We have shown that the conditions for quality to be welfare maximizing are fairly stringent, even when all patients are contracted with some provider. In general the market will also have too many or too few providers. Even if quality cannot be directly controlled by the policy maker, it can be influenced by regulating the capitation fee or by suitable taxes or subsidies to patients for purchase of capitation contracts. Because firm profits give the wrong entry signal an additional policy instrument is required, such as taxes or direct entry control in the case of excessive entry or subsidies in the case of gaps in the market. Regulation of the capitation fee would also affect provider profit and therefore numbers but would involve a trade off between access and quality objectives. Given that competition amongst providers does not yield optimal equilibria, the welfare effects of provider collusion over entry, price or quality are ambiguous. An entry limiting cartel, for example, might increase welfare compared with a free entry equilibrium which has an excessive number of suppliers. Very similar conclusions follow for regulation of a public system in which the tax financed capitation fee is paid by the state, rather than the patient: the optimal capitation scheme requires a lump sum component to control provider numbers and an appropriately fee per patient to provide the right incentives for quality. Comparison of public and private systems requires a positive model of the determination of the capitation fee in the public system. It also requires a very detailed specification of preferences and technology. Using a simple specification as an example, and restricting attention to situations where all patients join the list of some provider, we calculated the voting equilibria when the fee is chosen by voter-patients who take account of its effect on their tax bill as well as quality. In these circumstances the private system has welfare maximizing quality but too many providers. If voters neglect the effect of the fee the number of providers, perhaps because of their inability to bind themselves to future fee levels, the public system will have too low a quality level and too few providers. 13 Strictly, for combinations of p and q where the marginal patient has zero surplus, so that the kink case q Ž p, N . applies, the voting equilibrium is indeterminate since voters are indifferent over all such p,q. We assume that at N s 4a tK voters choose the limit from above of p o Ž N ..

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8. Notation p q d t y Ž p,q, d . R r s RrK 2 l Ž p,q . DŽ p,q . s r 2 l C s C Ž q, D . F p s pD y C N l

capitation fee quality of practice Žcontract terms. patient distance from practice patient cost per unit of distance patient utility function number of patients uniform density of patients around circle of circumference K market segment of firm demand Žnumber of patients; practice list. variable cost function fixed cost gross profit number of firms welfare weight on profit

Acknowledgements Support from the Department of Health to the NPCRDC is acknowledged. The views expressed are those of the author and not necessarily those of the Department of Health. I am grateful to participants in the Health Economics Study Group, Brunel University, June 1996, Gianni de Fraja, and two referees for comments and suggestions.

Appendix A. Proofs Proposition 1. At N s 6 a tK the left and right derivatives of p Ž N ., and the union objective function are dp k

tKR sy

dN

2N

3

)y

dp

tKR N

3

p

s

)

sy dN

pN

N

dp c

2 tKR sy

dN

N3

.

Proposition 2. The collusive kink profit function p coll s Rr4a N y tKRr2 N 2 has slope tKRrN 3 y Rr4a N 2 and is flatter Žabsolutely. than the contour of p N which has slope yprNs tKRr2 N 3 y Rr4a N 2 . Proposition 4. The competitive allocation is welfare maximizing only if

yq Ž p c ,q c ,l c . yp Ž p c ,q c ,l c .

s

Vq Ž p c ,q c . Vp Ž p c ,q c .

Ž 19 .

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336

where l c s Kr2 N. Define VˆŽ p,q,l . s V Ž p,q . s H0l y Ž p,q, d . rd d so that the competitive equilibrium is welfare maximizing only if

yq Ž p c ,q c ,l c . yp Ž p c ,q c ,l c .

s

Vˆl q Ž p c ,q c ,l c . Vˆl p Ž p c ,q c ,l c .

s

Vˆq Ž p c ,q c ,l c . Vˆp Ž p c ,q c ,l c .

Ž 20 .

which implies that VˆŽ p,q,l . is weakly separable in l: Vˆ s VˆŽ uŽ p,q .,l .. But then Vˆl qrVˆl p s Vˆl u u qrVˆl u u p s u qru p s yqryp and y must also be weakly separable in d. Proposition 5. We seek conditions on preferences and technology under which equilibria in which the market is covered have welfare maximising quality, for all welfare weights. The first order conditions ŽEqs. Ž10. and Ž11.. on p and q imply Vq Ž p ) ,q ) . Vp Ž p ) ,q ) .

y

pq Ž p ) ,q ) ,k . pp Ž p ) ,q ) ,k .

s a Ž p ) ,q ) . yb Ž p ) ,q ) ,k . s h Ž p ) ,q ) ,k . s 0

Wp s Vp Ž p ) ,q ) . q lpp Ž p ) ,q ) ,k . s 0

Ž 21 .

The comparative static properties of the welfare maximising solution are determined in the usual way from hp

hq

Wp p

Wp q

yhl pl) ) s W ql pl

Since the market equilibrium is independent of l we must have ql) Ž l,k . s 0, and, using Cramer’s rule, this is true if and only if h p Ž p ) Ž l . ,q ) Ž l . . s a p Ž p ) Ž l . ,q ) Ž l . . y bp Ž p ) Ž l . ,q ) Ž l . . s 0

Ž 22 . We consider the competitive case first. Since the market is covered D s RrN and b s yC q Ž q, RrN . NrR and bp s 0. Hence ql) Ž l. s 0 if and only if a p Ž p ) Ž l.,q ) Ž l.. s 0. At the competitive equilibrium C q Ž q c , RrN . NrR s

Dq Ž p c ,q c . Dp Ž p c ,q c .

s

and for welfare maximization )

C q Ž q , RrN . NrR s

Vq Ž p ) ,q ) . Vp Ž p ) ,q ) .

yq Ž p c ,q c ,l . yp Ž p c ,q c ,l .

Ž 23 .

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337

so the competitive equilibrium has q c s q ) for all l if and only if

yq Ž p c ,q ) ,l . yp Ž p c ,q ) ,l .

s

Vq Ž p ) ,q ) . Vp Ž p ) ,q ) .

for all p ) which implies, given that we also require a p s 0, Vq Ž p c ,q ) . Vp Ž p c ,q ) .

s

yq Ž p c ,q ) ,l . yp Ž p c ,q ) ,l .

which implies, see the proof of Proposition 4, that y s y Ž uŽ p,q ., d . is weakly separable. With y weakly separable as

Vq Ž p ) ,q ) . Vp Ž p ) ,q ) .

s

yq Ž p ) ,q ) . yp Ž p ) ,q ) .

and a p s 0 is equivalent to a zero income effect on demand for quality. Sufficiency is easily established. The proof for kink equilibria is very similar and makes use of the fact that at a kink equilibrium a firm will not be maximising profit unless it chooses a combination of price and quantity which maximise profit at the kink level of demand. The slope of the demand curve is discontinuous with respect to price and quantity at the kink but, the ratio of DqrDp is the same for increases or reductions in p and q. ŽSee Eqs. Ž3. and Ž7... Firms will therefore satisfy C qrD s DqrDp . We can now follow the same argument as for the competitive case. We have implicitly assumed in the proof that the welfare maximising, q ) , p ) leave the market covered if parameters are such that the equilibrium has no gaps. To show that this is indeed the case note that if the equilibrium is competitive then since l - `, p ) must be less than the collusive price which maximises firms’ profits and so the marginal consumer must have a positive surplus. If the equilibrium is at the kink firms’ profits are already maximised and the welfare maximising price will be less than the equilibrium price. Proposition 6. We must show that the stated conditions are necessary and sufficient for monopoly to yield welfare maximising quality, since by proposition 5 they are sufficient for the cases in which the market is covered. We will establish necessity, since sufficiency is obvious. The proof follows the initial stages of the proof of Proposition 5. Suppose that the welfare maximising q ) , p ) are such that the market is covered, even though the equilibrium has gaps. Then we can just apply the argument for the competitive case to show that the preference conditions are necessary and sufficient. Suppose instead that q ) , p ) leave gaps in the market. It is no longer immediately the case that bp s 0. We require that the proposition must hold for all

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convex cost functions. Let k be some parameter which enters the cost function of all firms. Since we require ql) s 0 for all l and k we must also have d h p Ž p ) Ž l ,k . ,q ) Ž l ,k . ,k .

s h p p pl) q h p q ql) s h p p pl) s 0 dl so that h p p Ž p ) Ž l,k .,q ) Ž l,k .,k . s 0 and so d h p Ž p ) Ž l ,k . ,q ) Ž l ,k . ,k . dk

Ž 24 .

s h p p p k) q h p q qk) q h p k s h p q qk) q h p k s 0

Ž 25 . using the fact that h p s 0 implies qk) s yh krh q . With suitably continuous functions the cross partials are equal, and so hq p hk p

s

hq

Ž 26 .

hk

Žexamination of h k s ybk shows that h k p / 0.. Eq. Ž26. is equivalent to weak separability: h s hŽ p, z Ž q,k .., which implies that h s aŽ q . y bŽ q,k . and so a is independent of p and since by assumption q ) does not vary with l, a does not vary with l. When q m s q ) welfare maximization implies yC q Ž q m , D Ž p ) ,q m . . s aD Ž p ) ,q m . q Ž p y CD . aDp Ž p ) ,q m . yDq Ž p ) ,q m . Žwe do not write k explicitly in the cost function since we no longer need it.. In the limit as l ™ ` the welfare maximization solution tends to the monopoly solution. a does not depend on l and so is well defined in the limit. ŽThe limit of b s pq Ž p ) ,q ) .rpp Ž p ) ,q ) . is not well defined in general since numerator and dominator have zero limits and we cannot apply L’Hopital’s rule since the limits ˆ of the derivatives of p ) and q ) with respect to l are also zero. However, because we require ql) s 0 at all l we get lim b s lim pq p pl)rlim pp p pl) s lim pq prlim pp p s pq p Ž p m ,q m .rpp p Ž p m ,q m . which is well defined.. Hence we must have aD Ž p m ,q m . q Ž p m y CD . aDp Ž p m ,q m . y Dq Ž p m ,q m . s D Ž p m ,q m .

Dq Ž p m ,q m . Dp Ž p m ,q m .

which implies, since p m ) CD , as

Dq Ž p m ,q m . Dp Ž p m ,q m .

m

Vq Ž p m ,q m . Vp Ž p m ,q m .

s

yq Ž p m ,q m ,l . yp Ž p m ,q m ,l .

H. GraÕeller Journal of Health Economics 18 (1999) 315–340

339

which is equivalent to the weak separability and zero income effect conditions on preferences. Using a s DqrDp in Eq. Ž21. and rearranging gives )

)

)

)

f Ž p ,q . s D Ž p ,q .

Dq Ž p ) ,q ) . Dp Ž p ) ,q ) .

q c q Ž q ) , D Ž p ) ,q ) . . s 0

Ž 27 .

But, from the firm’s profit maximising first order conditions, the firm will set q m s q ) if and only if f Ž p m ,q ) . s D Ž p m ,q ) .

Dq Ž p m ,q ) . Dp Ž p m ,q ) .

q C q Ž q ) , D Ž p m ,q ) . . s 0

Ž 28 .

The same quality level satisfies both Eqs. Ž27. and Ž28. if and only if fp

Dq p Dp y Dq Dp p

sD

Dp2 Dq p Dp y Dq Dp p

sD

Dp2

q

ž

Dq Dp

/

q C q D Dp

q Ž Cq D D y Cq .

Dp D

s0

The first square bracketed term is zero given zero income elasticity of demand for quality. The second term is zero generically if and only if the cost function has the form C Ž q, D . s DcŽ q .. Effects of N and p on quality under public capitation ŽSection 2.1.. When the equilibrium is competitive we see from Table 1 that qN s y qp s

Ž tK . N

1 2 ab

2

by1 q

3

tK N

2

tK s N

2

ž

tK y N

2

/

by1 q 1 s

tK N2

by1 ) 0.

)0

where b s wŽ tKrN . 2 q Ž pra .x1r2 . Proposition 9. Substitute Eqs. Ž17. and Ž17. in the welfare function ŽEq. Ž13.. to get W

s R qo y po y sR

1

tK

q 4a

tK 4N

ž / N

y Np o Ž N . y

2

ay

5tK 4N

yN

N

H0 F Ž N . d N

tK N

y2a

tK

ž / N

2

R y N

N

H0 F Ž N . Ž 29 .

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340

Differentiating with respect to N and remembering that at the equilibrium p o Ž N o . s F Ž N o . gives dW Ž N o . dN

sR

sR

5tK

N2

2

2a

4a tK

2

ž / ž / ž / ž /ž /

4N tK

tK

2

y

5

4

N

N

y 1 q R2 a

qR tK

N

2

N

2

1

y

N

n

N

tK y

N2

Ž 30 . )0

Proposition 10. The union chooses N to maximise Np o Ž N . s Rw tKrN y 2 a Ž tKrN . 2 x which is maximised at N u s 4a tK. Substituting N u in q o and p o gives marginal consumers at Kr2 N u a utility of zero.

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