Cream-skimming, incentives for efficiency and payment system

Journal of Health Economics 22 (2003) 419–443

Cream-skimming, incentives for efficiencyand payment system

Pedro Pita Barros∗Faculdade de Economia, Universidade Nova de Lisboa, Travessa Estevão Pinto,

P-1099-032 Lisboa, Portugal

Received 2 November 2000; received in revised form 17 July 2002; accepted 24 October 2002

Abstract

Reform proposals of health care systems in several countries have advocated variations of a riskadjustment/capitation system. These proposals face a serious objection: incentives to risk selectionare prevalent in the system. By now, considerable literature has been devoted to finding waysof mitigating, if not eliminating, this problem, while at the same time preserving incentives toefficiency. We contribute to this debate presenting a transfer system that, under some circumstances,attains both provider efficiency and no risk selection. The transfer system extends typical linearpayment systems. It can be interpreted as a fixed transfer in the beginning of the period plus an ex-postfund at the end of the period. The novelty rests in the way contributions to this fund are defined.© 2003 Elsevier Science B.V. All rights reserved.

JEL classification:I11; I18

Keywords:Payment system; Health care financing

1. Introduction

The increasing relevance of the health care sector in modern economies, and the numerousreform attempts around the world, are well documented by now. One reform proposal inseveral instances is the development of sophisticated financing schemes involving capitationtransfers from a funds’ collector (the Government, for example) to health care purchasersand/or providers (the purchaser can coincide with the provider, as in HMOs and fund-holdingGPs). Such transfers call for adequate risk adjustment. FollowingKeenan et al. (2001),formal risk adjustment is defined as the adjustment of premiums paid to health plans (or

∗ Tel.: +351-213-886-73; fax:+351-138-860-73.E-mail address:[email protected] (P.P. Barros).

0167-6296/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved.PII: S0167-6296(02)00119-4

420 P.P. Barros / Journal of Health Economics 22 (2003) 419–443

to insurance companies) based on a formula employing individual level diagnostic and/ordemographic information. Capitation systems and risk adjustment have been used in TheNetherlands, Israel, and the United Kingdom, among other countries. A similar systemappears in proposals for the US, characterized by a single payer contracting with competinghealth plans (Newhouse, 1994). The issue of risk adjustment is also important in the UScontext, as clear incentives for risk selection in health plans have been empirically identified(Newhouse et al., 1997).1

Two main problems with this approach have been exposed in the literature, and severalremedies to mitigate them have been put forward. The problems are risk selection, on theone hand, and providing incentives to efficiency of health care delivery, on the other hand.

Cream-skimming means selection by providers (or entities responsible for health careprovision) of those consumers expected to be profitable, given the system of risk-adjustedcapitation payments. A central element of health systems in some countries is a capitationsystem against which providers can play strategically.

Avoidance of cream-skimming has been discussed along two main lines: adequate riskadjustment and pro-competitive regulation. The latter typically involves open enrollmentrules and definition of standardized benefits. The former encompasses two aspects exten-sively investigated: the refining of the risk-adjusted capitation, and imposition of some sortof high-risk pooling.2 The risk selection issue is far from being settled. In a recent accountby Newhouse (1998)a pessimistic picture of future prospects is drawn, as the followingquote illustrates:

The physician treating the patient will have more information about the patient’s likelyfuture spending than the risk-adjustment formula will incorporate. As a result, the incen-tives to cream and dump will remain.

The recent paper byvan de Ven et al. (1998)discusses the current difficulties in improvingcapitation formulas by estimation of average risks. Marginal improvements in the capi-tation formula are obtained at a considerable research cost. Moreover, it is argued thatavoidance of cream-skimming requires strong regulations and possibly some sort of manda-tory high-risk pooling. In the papers byShmueli et al. (1998)andSmith (1998), the sametype of analysis is carried out, in the sense that both attempt to econometrically approxi-mate risk-adjusted capitations.3 Newhouse (1996b)states that current adjustments to capi-tated payments do leave substantial “between-person variance in expected health care costsunexplained.” Nonetheless, in a recent work,van de Ven et al. (2000), based on simulationsof risk-adjusted premiums, conclude that such risk-adjustment is the appropriate strategyto avoid risk-selection and dumping of patients.

Another way to proceed is to recognize the difficulties in estimating risk-adjusted cap-itations, which must not be vulnerable to superior information by recipients. This leads

1 For a detailed analysis of risk adjustment mechanisms in several countries, seevan de Ven and Ellis (2000),Tables 5 and 6. Another recent account of the importance of defining appropriately the payment system can befound in the special issue ofHealth Care Management Sciencein 2000, devoted to risk adjustment and capitation.SeeRice and Smith (2000).

2 The literature on risk adjustment is too vast to be fully reviewed here. See the recent overview byvan de Venand Ellis (2000).

3 See alsovan Vliet and van de Ven (1992).

P.P. Barros / Journal of Health Economics 22 (2003) 419–443 421

to the economic analysis of risk adjustment. In this vein,Glazer and McGuire (2000)propose a different adjustment rule. This rule essentially establishes that above averagepayments must be associated with higher risk individuals. In a similar spirit, but in a dif-ferent direction,Encinosa (1999)addresses the economic fundamentals behind risk ad-justment in capitation rates. The major point raised by Encinosa is the importance of costsavings resulting from treating more than one type of patient. A different way to proceedis to offer a menu of reimbursement policies to suppliers of health care, as explored bySappington and Lewis (1999). They find a common result in the literature: both capitation(prospective) payment and cost-sharing (reimbursement) are present in the optimal transfersystem.

An alternative to a simple capitation transfer has been considered: each insurance com-pany contributes with a given percentage of its insureds to a common pool (a 5% valuewas used in the simulations conducted byvan Barneveld et al. (1996)). Typically, eachfirm contributes with the worst risks in the portfolio to this fund. The existence of the poolreduces the incentive to risk selection. The theoretical foundations of this approach seemto rest on two assumptions. First, risk selection is increasingly costly as we move from theright-end tail of risk characteristics to the left. This means that cream-skimming with allrisks is less costly than cream-skimming after the worst 5% risks of each company havebeen transferred to a compulsory pool. Second, the incentive for making a cost-reducingeffort is not much affected by this 5% contribution to the pool.

The argument has not been formalized, to our knowledge, although the reasoning behindit seems compelling. The merits of the approach have been discussed in the context ofsimulations.4 Besides these options, another one was explored by the Swiss federal gov-ernment, which introduced a risk-equalization fund.5 The Swiss risk equalization schemeis described inBeck (2000). The main feature to highlight here is its definition: “The riskadjustment formula is based on the difference between capitation and average health careexpenditures of all insured.” (p. 111). There is no discussion of the theoretical propertiesof this fund.

The idea of an ex-post clearing fund is another avenue to counteract cream-skimming. Inthis paper, we explore its merits. The paper makes precise the desired characteristics of thefund. We focus on how transfers to health plans can be set such that provider-level efficiencyis achieved, without giving rise to cream-skimming. The discussion is not only applicableto private health insurance plans, but also to sickness funds (Israel), Health MaintenanceOrganizations (US) and general practitioner fundholders (UK).

We propose a payment rule which, within the class of linear payments, enlarges the usualset of instruments. The payment rule, acting on routine information available, avoids thecosts of very fine risk adjustment and does not promote an “information race” between thepayer and the provider.Schokkaert et al. (1998)discuss the properties of risk-adjustmentschemes: there should be no incentives to selection, insurers should have incentives to be

4 For recent advances in this approach, seevan Barneveld et al. (2000).5 “Insurers having an age and sex structure favorable contribute, while others with unfavorable structure receive

it.” ( Beck and Zweifel, 1998). See also the 2000 Switzerland report of the European Observatory on HealthSystems. The state of New York also makes use of a pool for risk sharing purposes. See New York InsuranceDepartment Regulation No. 146 and its amendments.


efficient and a budget constraint must be satisfied. The payment rule proposed here extendstheir linear payment system in a natural way, and satisfies these requirements.

The reason that the system works is simple. We start, in the usual context, with threeobjectives: patients should pay an equal amount, whatever their initial condition is (equityconsiderations), moral hazard and risk selection problems should be minimized. However,only two instruments are available in the typical capitation/prospective payment schemes:a fixed amount and a proportion of actual costs. The payment rule we propose adds onemore instrument—a component based on the average population risk. The transfer systemis defined by: (i) at the beginning of the year, the central fund pays to the insurance com-pany a per capita value, independent of risk characteristics; the insurance company sets anamount to be collected directly from consumers. No price discrimination is allowed andopen enrollment is enforced; (ii) at the end of the year, insurance companies participate inan ex-post clearing fund, where the contribution (positive or negative) of each company isdefined as the difference of the company’s risk mix to the population’s average risk mix,multiplied by the cost per patient treated in the company. The proposed ex-post fund differsconsiderably from the “voluntary reinsurance pool” discussed, for example, invan de Venet al. (2000).6 The disadvantage of that pool was clearly identified with reduced incentivesfor efficiency. We will show this to be a feature of the particular form of that pool, not ageneral characterization of ex-post funds.

Another close example is the allocation formula proposed bySchokkaert and van deVoorde (2000), based on the previous theoretical work ofSchokkaert et al. (1998). Again,the proposed allocation formula introduces a term reflecting the difference between the setof individuals associated with the insurer and the average value for the population. We offera payment scheme that improves on this.

Moreover, this seemingly complex rule turns out to have a fairly simple implementableform. It can be applied as a direct payment by patients (equal for each and every patient)plus a payment from a central fund (again, based only on counts of enrollees), and anend-of-the-year adjustment fund. The contribution to the fund will be determined by bothrisk selection and moral hazard effects. A provider that selects good risks will be a netcontributor, thus reducing incentives to cream skim the market. A provider with high costsper patient will have an incentive to be more efficient. If the company has a lot of bad risks,higher inefficiency means higher transfer. However, the actual costs of delivering healthcare to patients counteracts the incentive and keeps the provider interested in achievingefficiency.

The paper unfolds in the following way. First, inSection 2, the basic model is presented.The selection issue is highlighted in the model, as well as the incentive problem in thechoice of cost-reducing effort. Next, inSection 3, the imposition of open-enrollment rulesis addressed. Then, inSection 4, we introduce risk-selection on the basis of an informationaladvantage by insurers, and show that an extended linear payment system is able to solveboth risk selection and cost-reduction incentives problems. This extended linear system hasan interpretation in terms of a per capita payment, plus an ex-post fund. In principle, itscharacteristics make it simple to implement this financing structure.Section 5considers a

6 “To finance the pool, each insurer pays to the pool, for each of its pooled clients, the average expenditures ofall pooled individuals in the pool year.”van de Ven et al. (2000, p. 327).


different formulation for the efficiency-effort cost and shows how the transfer system hasto be adjusted.Section 6extends the model to the case of multiple health risks. Finally,Section 7concludes.

2. The model

For the sake of concreteness, we consider the organizational architecture described invan de Ven and van Vliet (1992, 1994)andvan Vliet (1992), among others. As is obvious,the same ideas apply to many other settings where capitation transfers are an issue (such asfund-holding GPs and health plans).

A paper close to ours isMa (1994), in the sense that it looks at the cost-reduction andquality-enhancing incentive effects from different payment systems. The main result of hispaper is that a mixed payment system (combining prospective payment and reimbursement)can give the appropriate incentives to achieving the socially optimal choices of quality andcost-reducing effort, even if there is the possibility of dumping the worst cases. Our settingdiffers fromMa (1994)in several respects, although the main essential elements are present:discussion of payment systems and the issue of giving incentives to cost-reducing efforts.

The main differences are that we extend the possible payment system in one direction,different from the optimal rule derived inMa (1994)(a piecewise linear rule), and no qualitychoices are considered. We introduce the cost-reducing effort as a moral hazard variable,as the transfer recipient is not the provider of health care, as inMa (1994).

The on-going debate seems to indicate that competition coupled with risk adjustmentis a promising strategy for containing rising health care costs.Newhouse (1996a)arguesin this direction, and supports his view with a model that explicitly considers a sunk costof developing new contracts. In the presence of sunk costs, a pooling equilibrium may besustainable. The essence of the argument is that sunk costs make it unprofitable to fullydiscriminate across consumers. We contribute to this debate by presenting a model that both(i) formalizes the arguments of risk selection and incentives to efficiency; and (ii) allowsthe study of optimal transfer system.

The essential features of the model are the following. There is a Government endowedwith compulsory power to collect contributions from the population. These contributionsare a function of income alone and denoted byT (y), wherey is the individual income.In tax-financed systems, the functionT (y) is the tax system; in health systems financedby compulsory social insurance, this function denotes the way contributions are linked toearned income. The functionT (y) incorporates all the distributive (equity) concerns in thefinancing of health care systems.

The Government makes a payment to purchasers of health care (insurance firms, in ourcontext), denoted byS. This payment system, made on a per capita basis, may not cover thefull insurance premium, in which case, an extra payment is made by individuals directly toinsurers. The exact value of this payment is determined by health care purchasers. Staying asclose as possible to real-life facts, we impose that this direct payment to insurance companiesbe the same for all consumers that choose a particular health insurance company. That is,insurance companies are not allowed to price discriminate between consumers accordingto their risk characteristics. This type of constraint on the behavior of insurance companies


has been recognized as inducing opportunities for risk selection.7 The constraint of uniformprices across individuals is widespread. Not do only several European countries have thisfeature in their national health systems; health plans in the US have it too. Thus, this is aconstraint to be included in the model. It is at the root of incentives to select risks, as well.

The extensive form of the game is the following. In the first stage, the Government definesthe elements/parameters of the financing system, that is, the contribution function and thepayment rule. Next, in the second stage, insurance companies set a flat rate contribution tobe paid directly by consumers. Then, consumers choose the insurance company to contractwith. In the third stage, insurance companies set their effort level. Finally, Nature deter-mines whether each consumer falls ill, according to his probability of illness. Health careis demanded by sick consumers, and the contract terms are applied.

The insurance system provides full insurance. The assumption is intended to simplify theexposition. The introduction of contracts with less than full insurance has been advocatedas a way to discipline demand (moral hazard on the consumption decision taken by sickindividuals facing zero price at this point), and no extreme departure from relatively smallcopayments has been seriously proposed. These copayments are not regarded as mainfinancing instruments. We choose not to cloud our analysis of the financing system, and weleave out of the model these well-known arguments for positive copayments.8

Only two states of nature are considered: healthy and ill. The ‘ill’ state of nature occurswith probabilityp. Individuals in the population differ in the probability of being ill. Thepopulation has unit size and the probabilityp is distributed in the population according toa density functiong(p) and a distribution functionG(p). Consumers are risk averse. Weuse a strictly concave functionu(·) to represent the preferences over consumption in eachstate of the nature. Expected utility of a consumerj endowed with illness probabilitypjand incomeyj is:

Vj = U(yj − T (yj )− F)− pj B (1)

where B is the utility cost of being ill net of the benefits from consumption of healthcare (treatment). We take it to be zero, without loss of generality. Individuals take thecontribution to the central fund as exogenous to their decisions, and choose the insurancecompany which announces a lower value for the insurance premiumF paid directly byconsumers to insurance companies.

For expositional purposes, we consider a homogeneous population with respect to healthneeds: conditional on falling ill, the amount of health care provided is the same to allindividuals. Note that expected health spending differs across the population according tothe probability of being sick. This assumption centers our attention on the essential featuresof the transfer system.

The insurance company has to contract the supply of health services with health careproviders. In case of illness, an individual must receive some given units of care (fixedacross the population), which entail a monetary cost ofX.9

7 SeeNewhouse (1984)andPauly (1984).8 There is also the well-known argument ofRothschild and Stiglitz (1976)for offering contracts with less than

full insurance. This does not appear here, as no discrimination between consumers is allowed.9 We assume that no services are provided to healthy people.


This monetary cost is dependent on the effort of insurance companies to get the bestpossible conditions (e.g. value for money). Denote bye the effort exerted by insurancecompanies. ThenX = X(e), with X′(e) < 0, X′′(e) > 0: more effort decreases totalexpenses per patient but does so at a decreasing rate.

The effort is our efficiency variable and it is, by assumption, not observable (more rig-orously, it is not contractible). The assumption of unobservable effort by insurance firmsto achieve better provider performance is reasonable (and even more when the purchaser isintegrated with the provider). It seems to be quite hard to specify in a contract, and monitorfor compliance, the exact stance of insurance firms in negotiation processes with providers,for example, or in the specification, implementation and monitoring of medical protocolsor guidelines. On the other hand, we assume that medical expensesX(e) are observableand contractible.

The linear payment system corresponds to what we observe in reality, and simplicity oflinear rules suggests that applied theory should use this type of assumption if informativepolicy conclusions are to be drawn from the analysis. The same approach has been followedin previous literature (Ma, 1994, for example).10 Constraining the set of possible paymentsis a way of incorporating policy-implementation constraints into the analysis. As is wellknown, focusing on linear rules comes at a generality cost. However, non-linear sharing rulesare not observed often due to the informational requirements needed for implementation.

Exerting effort for cost containment implies a costC(e) per insured. The marginal costof effort is positive and increasing, that is,C′(e) > 0 andC′′(e) > 0. Examples of thiscost are monitoring the care provided, checking the costs incurred by the provider, etc. Theincentive for insurance companies to perform check-ups and early screening can be seen asincluded in the efforte. A higheremeans pressure of insurance companies to get physiciansto perform such screening activities or to act as gatekeepers to differentiated care.

The insurance company maximizes expected profits, which are defined as:

Πi =∫p∈χi

[Si(p,X(ei), χi)+ Fi − C(ei)− pX(ei) ]g(p|χi)dp (2)

whereχi is the set of probabilities of consumers that choose insurance companyi andg(p|χi) is the distribution ofp restricted to the setχi . We also useχi to denote the setof consumers that choose firmi. There is some abuse of notation here:g(p|χi) is notthe conditional distribution ofp on χi . An example illustrates the notation. Suppose thatfor somep, two firms share equally the density of consumers,g(p). Then g(p|χ1) =g(p|χ2) = 0.5g(p). The first term inEq. (2)is the expected transfer payment (sum of percapita transfer over all clients of companyi, possibly risk adjusted). The second term is thevalue of insurance premiums paid directly by consumers to the insurance company. Thethird term denotes the cost of efficiency effort exerted by the firm. For the moment, weassume constant returns to scale of effort cost with respect to clients.11 Finally, the fourthterm denotes expected payments of health care costs to treat consumers that become ill ingroupχi .

10 For a recent use of this assumption in health care settings, seeSappington and Lewis (1999). SeeHolmstrom(1979)for the seminal contribution of this type of model.11 This assumption is relaxed below.


The assumption of competition in the insurance market and the rule that precludes thepractice of insurance premium discrimination according to risk characteristics of the individ-ual means that the only way to make some price discrimination is through cream-skimming(choice of best risks), which in our model is reflected in a selective choice ofχi .

One way to act against this type of behavior is to impose rules designed so that insurancecompanies cannot refuse to contract with a consumer-open-enrollment rules. Another wayto avoid risk selection is to design the transfer ruleS in an appropriate way. The followingsections investigate the properties of each of these alternatives.

3. The traditional argument

Under a system of open enrollment, each insurance company has to announce a directcontribution from individuals,Fi , and cannot refuse a consumer that accepts to pay thepublicly announced value.

According to the timing of the game, we first characterize the optimal efficiency effort byinsurance companies. The effort choice by the insurance company is made after insurancecontracts are signed. The insurance company chooses the effort level that maximizes itsexpected profits. This choice is characterized by the following first-order condition, obtainedby maximizing expected profit(2) with respect to the effort,e:12

∂Πi

∂ei=∫p∈χi

(∂Si

∂XX′(ei)− C′(ei)− pX′(ei)

)g(p|χi)dp = 0 (3)

The assumption of a competitive insurance market implies that, in equilibrium, insurancecompanies must have zero economic profits. The premium set by the company in a com-petitive insurance market must satisfy:

Fi = C(ei)+∫p∈χi pX(ei)− Si(p,X(ei)) g(p|χi)dp∫

p∈χi g(p|χi)dp(4)

Take now the system whereS is a proportion of expected health care costs. This is, of course,a strong restriction on the transfer system. While a linear system can be advocated on thebasis of its simplicity, the assumption of simple proportionality may be too restrictive, evenwithin the set of linear transfer systems. In the next section, we will argue in favor of theuse of a more general linear transfer system.

Since open enrollment rules are usually discussed in the context of proportional paymentsystems, we maintain this assumption in order to evaluate the conventional wisdom on thecharacteristics of this organization of the health care sector. The open enrollment require-ment is not free of problems. For example, incentives to provide poor quality of care tohigh-risk individuals may appear. See further discussion invan de Ven et al. (2000, p. 319).Accordingly,13

Si = αpX(ei), α ≤ 1 (5)

12 Standard regularity assumptions ensure that second-order conditions are met.13 The analysis can be easily generalized toSi = α(p)X(ei), thus being non-linear on the probability.


The constraint onα ensures that no transfer exceeds expected health expenditures. Theequilibrium value for the insurance premium charged by insurance firms to consumers is:

Fi = C(ei)+X(ei)(1 − α)

∫p∈χi p g(p|χi)dp∫


The first question is whether in equilibrium firms must have the same risk-mix of patientsor not. A second issue is to design the optimal transfer system in the model, given theconstraints imposed upon the transfer system.

Before addressing these two questions, it is useful to establish the following result.14

Lemma 1. Under open enrollment and in a competitive insurance market, insurance firmswill choose the same effort level for cost reduction.

Proof. The optimal effort choice by insurance companies is governed by the condition (forα < 1)

−C′(ei) = X′(ei)(1 − α)

∫p∈χi p g(p|χi)dp∫


Definepi as the average probability of illness of the set of consumers who choose to contractwith companyi:

pi =∫p∈χi pg(p|χi)dp∫p∈χi g(p|χi)dp

(8)

The zero-profit condition for each insurance company implies:

Fi = C(ei)+X(ei)(1 − α)pi (9)

Consumers select the insurance company with the lowest valueFi . Thus, in equilibrium,it must be the case thatFi = F,∀i. Otherwise, firms withFj > minFi will not haveconsumers and will exit the market.

From the two equilibrium conditions (zero-profit condition plus the first-order conditionfor the choice ofei), we obtain:

Fi = C(ei)− X(ei)C′(ei)

X′(ei)(10)

As Fi = F , it must be the case thatei = e,∀i (andpi = p,∀i), as the right-hand side ismonotone onei . The equilibrium value ofei is given byEq. (7). The adjustment leading toequilibrium is made through the set of consumersχi that a firm has. �

The risk mix of consumers in a company’s portfolio can, thus, differ across companies. Theconstraint implied by a competitive market is the same value ofpi prevails across firms.

14 As pointed out by a referee, it depends on the assumptions of non-heterogeneity of consumers, and theassumption that only one dimension of effort is allowed. Of course, if either assumption is relaxed, then it may bepossible for different types of firms to be supported in equilibria.


Of course, this has some implications for the admissible configurations of consumers ineach firm’s risk distribution. The most important one is that an equilibrium with companiespicking up only consumers at one tail of the distribution ofp is not possible. Nonethe-less, this is quite less restrictive than requiring an equal distribution of individuals acrossfirms.

This simple model reproduces some conventional-wisdom results. In particular,for apure cost-reimbursement system, there is no incentive for insurance companies to pursuecost-reducing effort. For incentives to cost-saving effort to emerge, a system with a prospec-tive component is necessary.15

This can be readily seen by makingα = 1 (full prospective payment), and recalling thefirst-order condition for the optimal effort choice:

∂Πi

∂ei

∣∣∣∣α=1

= −C′(ei)∫p∈χi

g(p|χi)dp < 0 (11)

Thus,ei is set at the lowest possible value.This is a result from the general structure of moral hazard problems. The corollary is the

need to setα < 1 for a positive effort level result, which can be seen by straightforwardcomputation of the effect of decreasingα in the level of effort,ei :16

∂ei

∂α= piX

′(ei)(1 − α)piX′(ei)+ C′′(ei)

< 0 (12)

It means that the payment does not cover the full expected health care expenses if incentivesfor cost-reducing effort are to be provided. The remaining amount must be obtained throughthe chargeF , in a competitive environment.

Open enrollment rules are seldom fully enforceable. Insurance companies still haveavailable some ways to select risks which are quite difficult to monitor. The emergenceof cream-skimming even under open enrollment rules is leading to proposals of changesaimed at reducing risk-selection activities. The proposals, so far, have had mainly an em-pirical support and are based on the idea of “high-risk pooling”.17

To investigate the optimal transfer, we must first define the objective function of the Gov-ernment. We take it to be the welfare of consumers, aggregated according to a utilitarianview. The maximization of this social welfare function is constrained by three sets of con-ditions: (i) the behavior of insurance companies with respect to the choice of cost-reducingeffort; (ii) the competitive insurance market equilibrium condition; and (iii) the budgetconstraint on the financing of the system.18

The third constraint requires that contributions raised are enough to pay for the expectedhealth care costs. This modeling option makes an extra assumption: higher efficiency inthe system is passed through to consumers by the lowering of contributions. Another alter-native use of savings associated with efficiency gains would be to maintain contributions

15 Other papers that have proposed a partial capitation areEllis (1998), Ellis and McGuire (1986), Newhouse(1996a,b, 1998)andSelden (1990), among others. See alsoSappington and Lewis (1999).16 The denominator has a negative sign as∂2Πi/∂e

2i ≤ 0 to satisfy second-order condition of the maximization

problem.17 Seevan Barneveld et al. (1996).18 We retain the assumption of full insurance.


and increase services in other areas of Government activity. This would, however, divertus from the main features of the health system. The other assumption is that the contri-bution system does not give rise to inefficiencies in the economy. It is widely recognizedthat tax systems usually have some distortion cost. Since this cost is general to the de-sign of tax systems, not specific to the health care sector, we opt to set it aside in theanalysis.

At the present stage, the set of admissible transfers includes only those proportional tothe expected health spending per capita. The problem of a social planner can be formallystated as:

max{α,ei ,F }∫Yh(y)U(y − T (y)− F)dy

s.t. −X′(ei)pi(1 − α) = C′(ei)Fi = F = C(ei)+X(ei)pi(1 − α),∀iT (y) = αpX(ei)

Program A

whereh(y) is the distribution of consumers by income level, independently distributed fromhealth risks, with support inY ; andp = ∫ 1

0 pg(p)dp is the average probability of illness inthe whole population. DefineΘA = {(α, ei, F ) : solvesProgram A}. This set containsall of the triples that can implement the socially optimal allocation.

Proposition 1. Consider(α, e, F ) ∈ ΘA. Thenα < 1.

Proof. Substituting the last two constraints into the objective function, we can rewrite theprogram as:

max{α,ei } Ey [U(y − αpX(ei)− C(ei)−X(ei)(1 − α)pi ]

s.t. −X′(ei)pi(1 − α) = C′(ei)

The solution to this problem yields the result. To show this, it suffices to write down theconstraint to the problem as

α = 1 − C′(e)−piX′(e)

< 1 � (13)

That is, the optimal transfer system in a proportional rule framework implies a transfer thatdoes not fully cover expenditures. Firms breakeven by chargingF > 0 to consumers, due tothe need to provide incentives to efficiency effort developed by insurance firms. Again, theessential feature is that insurance companies anticipate the effect of own decisions of efforton the transfer received. A full transfer would provide no incentives for the developmentof efficiency effort. Thus, the usual proportional transfer cannot induce both the efficientprovision of effort and absence of cream-skimming, when the open enrollment rule is notfully enforceable.

This completes our digression on the conventional wisdom about health care systems.We have replicated in a formal way the standard arguments for partial prospective transfersto insurance companies, as an incentive device to cost-reducing efforts. We are now in aposition to present our main result.


4. An improved transfer system

The rationalization provided in the previous section is still vulnerable to issues of risk-selection that cannot be ruled out by open enrollment rules. We now explicitly recognize thecream-skimming problem. The problem was referred to above in general terms, as selectivechoice of consumers. We now introduce it in a more specific way. However, by enlarging ina simple and feasible manner the set of linear transfers, we are able to propose a new transfersystem that solves both the cream-skimming and the incentives-to-efficiency problems.

One feature that characterizes real world health insurance markets is an informationaladvantage of insurance companies (or providers) over the Government (or central fund).The advantage of a better knowledge of risk characteristics of individuals can translate intoa risk selection activity by insurance companies. This, of course, stems from the impositionof the constraint of no price discrimination across consumers.

In terms of economic modeling, one way of introducing this informational advantage isto assume that the central fund cannot identify the risk characteristicp, while the insurancecompany is able to observe it.

Maintaining the assumption of a transfer system proportional to health care expenses,under the assumed informational asymmetry between insurance companies and the Gov-ernment, the value to be transferred must be the same for all individuals, as the Governmentis not able to differentiate. This implies that only the individuals with the lowest valueof p would be chosen by insurance companies, which can mean that consumers endowedwith a high probability of illness will not be able to get private insurance or do so at rather“unfavorable” terms, from a society’s point of view. In the absence of an open enrollmentrule, or if there is no way of monitoring compliance with it, the transfer system must providethe incentives for firms not to engage in cream-skimming activities.

An obvious way to eliminate the negative effect of cream-skimming that some people(the worst risks) are not able to get health insurance is to set a transfer per individual highenough, such that all individuals are profitable to insurance companies.19 The distributiveissues involved (high profits for insurance companies) as well as efficiency considerations(distortion costs arising from the taxes/contributions levied to get the necessary contribu-tions) make it an undesirable solution.

We are then left with the alternative of determining the optimal transfer system. The nat-ural option of refining the payment formula, in such a way that no significant informationalasymmetry exists between the Government and insurance companies, has obvious difficul-ties. It may lead to a cost escalation in the monitoring and management of the system, withno significant effect in decreasing cream-skimming opportunities.

The search for the optimal transfer system must be done in a careful way. With theassumption of informational asymmetry, the transfer must be a fixed payment (or, at least,based on a coarser risk discrimination than the one available to firms). And any fixedpayment is not able to counteract the incentive to cream-skim the market. However, throughenlargement of the set of instruments available, we may consider the option of creating anex-post clearing fund, which uses information available on a routine basis. The existenceof financial flows with an ex-post clearing fund makes the transfer relevant to the insurance

19 This does not avoid competition for the better risks. However, it does eliminate the incentive to reject consumers.


company to include not only the fixed capitation payment but also the ex-post adjustment,as the insurance company will anticipate and internalize the effect of risk selection on theex-post contribution to this clearing fund. We now formally present this argument.

Due to its simplicity, we restrict our analysis to linear schedules. The linear transfer percapita we propose is of the form:

Si = α0 + α1piX(ei)+ α2pX(ei) (14)

The transfer is defined by a fixed amountα0, plus a proportionα1 of expected healthexpenditures, where the expectation is based on the risk mix in the setχi , and a proportionof expected health expenses, where the expectation is based on average population risk.The definition of the optimal transfer rule consists in the specification of the parameters(α0, α1, α2), which fully characterizes the transferSi . The main (linear) payment structuresanalyzed in previous literature are particular cases ofEq. (14). For α1 = α2 = 0, onehas the pure prospective system, while forα0 = α2 = 0, α1 = 1, we have the purefull-cost-reimbursement model. The partial cost-sharing model emerges forα0 > 0, α1 <

1 andα2 = 0.The profit of the insurance companyi is:

Πi =∫p∈χi

(Fi − C(ei)+ Si − pX(ei))g(p|χi)dp (15)

Substituting for the transfer system in the profit of the insurance company:

Πi = Gi (Fi − C(ei)+ α0 + α1piX(ei)+ α2pX(ei)− piX(ei)) (16)

where

Gi =∫p∈χi

g(p|χi)dp and p =∫ 1

0pg(p)dp

The transfer system must comply with two different sets of constraints: absence of cream-skimming and provision of incentives to cost-reducing effort. Cream-skimming usuallymeans, in health insurance contexts, selection of good risks. That is, consumers who haveexpected health expenses far below the (risk-adjusted) capitation payment. We formalize thisintuitive notion in terms of a constraint of “no cream-skimming” in the following way. Takea partitionχi of consumers from the population. There is no incentive for cream-skimmingprovided that any other partitionχj gives the same expected profit per capita to the insurancecompany. That is,Πi(χi)/Gi = Πi(χj )/Gj , whereGi (respectivelyGj ) is the number ofconsumers in partitionχi (respectivelyχj ). This definition is somewhat different from theintuitive notion, as it does not concentrate only on the idea of “good risks”. This is so becausewe allow for configurations where an insurance company has both very good and very badrisks, but the risk-mix is, on average, similar to that of other insurance companies that takeonly intermediate risks (recall that the competitive equilibrium does not restrict consumers’distribution to be equal across companies). It also accommodates the possibility that a biggerpartition (more consumers) may imply more good-risk consumers (in absolute terms), andalso more bad risks. It is, therefore, important to define the notion of “no cream-skimming”in terms of expected per capita profit to the insurance company. We assume that selection


of patients can be done at no cost. This assumption is the least favorable one, thus being auseful benchmark.20

The decision problem of the Government (or the health plan) is given by Problem B,defined by:

max{α0,α1,α2,F,e}∫YU(y − T (y)− F)h(y)dy

s.t.∫YT (y)h(y)dy = ∑

i

∫p∈χi Si g(p|χi)dp

Si = α0 + α1piX(ei)+ α2pX(ei)

Πi(χi)/Gi = Πi(χk)/Gk, ∀i, kΠi(χi) = 0, ∀i−C′(ei)+ (α1pi − pi + α2p)X

′(ei) = 0, ∀i

Program B

Define the solution toProgram B asΘB = {(α0 · α1, α2, ei, F ) : solvesProgram B}.

Proposition 2. Takeθ ∈ ΘB . Then, α1 = 1 andα2 = −1. That is, the optimal transfer isof the form:

Si = α0 +X(ei)(pi − p) (17)

We proceed now to a sketch of the proof (a formal proof is provided inAppendix A). Forcream-skimming to be unprofitable,α1 = 1 has to be the case. Then, there is no incentivefor the insurance company to engage in risk selection. The individual profit earned on eachconsumer is given by the term in parentheses inEq. (15)). This is independent ofpi , theonly variable determined by the set of consumers. There is no gain in making a selectivechoice of consumers. Withα1 = 1, the private optimum for the choice of cost-reductioneffort is given by:

−C′(ei)+ α2pX′(ei) = 0 (18)

To find the optimal value ofα2, which has the role of aligning the private incentives withthe social incentives in the choice of effort, we need to look at the social planner’s problem.Note that the proportional rule restrictsα2 = 0, which means that effort is set at its lowestpossible value, unlessα1 < 1, as shown previously.

Taking the social welfare function, the problem to be solved by the social planner is:

maxeW =

∫Y

h(y)U(y − T (y)− F)dy s.t.

∫Y

(T (y)+ F) h(y)dy =∫ 1

0(C(e)+ pX(e)) g(p)dp

Substituting the budget constraint into the objective function, the problem can be written as:

maxeW =

∫Y

U

(y − T (y)+

∫Y

T (y)h(y)dy − C(e)− pX(e)

)h(y)dy (19)

20 Of course, if firms have to use resources to select consumers, as long as such cost is not very high, the selectionissue remains. The profit definition would have to be changed accordingly, but the results of the model would beessentially the same.


The first-order condition of this problem is

Ey(U′(·))(−C′(e)− pX′(e)) = 0 (20)

or simply

C′(e) = −pX′(e) (21)

as marginal utility is always positive. Going back to the problem of defining the opti-mal transfer system, to have the social optimum choice implemented, it suffices to makeα2 = −1.

This implies a transfer formula given by:

Si = α0 +X(ei) (pi − p) (22)

We can interpret this transfer system as a lump-sum payment,α0, plus an ex-post adjustmentfund,X(pi − p). The proposed transfer formula allows for a socially optimal choice ofeffort and at the same time it gives no incentive to cream-skim. The valueα0 is determined,jointly with F , by the zero-profit equilibrium condition for the firm. In particular, eithera valueα0 = 0 or a valueα0 such thatF = 0, are possible. The perfect substitutabilitybetweenF andα0 in the optimal solution is easily observed in the zero-profit condition forinsurance companies.

The choice of a particular value forα0 depends on the social planner’s preferences aboutpayment to insurance companies in a direct way and through the central fund. That is, theprecise valueα0 hinges upon the social preferences associated with fund transfers and directpayments. The former may be associated with both income redistribution (positive effect)and heavier use of compulsory power to set contributions according to income, giving riseto other distortions in the economy (a negative effect). The latter has a negative equity effectbut induces smaller distortions in the economy. We do not formalize this choice as it is notessential to our point.

The ex-post adjustment fund has two desirable properties. First, it is a financially balancedscheme (except for the operating costs of the system).21 Second, at the equilibrium, thesystem induces a socially optimal level of cost-reducing effort.

The first term in the adjustment formula,piX(ei), is simply the value of health careper capita costs of the insurance company consumers. The second term,pX(ei) can beapproximated, for implementation purposes, by the ratio of “sick” people in the population,multiplied by the average per capita cost of individuals that have been sick in the company’sset of consumers. The information and computations required are of the same order ofcomplexity as for other ex-post funds. Of course, problems with information gathering andaccurate reporting by insurance companies may still exist and be of importance.22 In thisrespect, the proposed payment system does not differ from other ex-post funds.

This payment system implies that gains from cost reduction stay with the firm, providedit does not engage in cream-skimming. As the incentive to select consumers is eliminated,

21 A proof of this claim is provided inAppendix A.22 For example, assigning costs to diseases is not always straightforward. The interested reader can find an

example of computation of this fund and how it compares with the more standard one in a technical appendixavailable athttp://ppbarros.fe.unl.pt/papers.html, or upon request.

http://ppbarros.fe.unl.pt/papers.html


because the net profit per consumer does not depend on individual risk characteristics,the payment system provides incentives for efficiency. This constitutes the crucial differ-ence to other adjustment rules based on average cost per capita in the whole health caresector.

Some comments are in order here. First, if there is some categorization of risks, by sex, ageor other risk adjuster, the same idea of transfer system can be applied within each type of riskthat is identified by the Government. The issue is relevant whenever there is an associationbetween the observed characteristics and health costs. In such cases, risk selection mightstill be possible. On the other hand, for the ex-post fund to be applied to specific typesof risk requires assignment of spending to each one. This is far from straightforward, andconstitutes a general argument against fine risk classification for the purpose of computingex-post adjustments.

There is a recurrent issue in health care financing discussions: that lower quality of caremay be induced by reforms. The explicit introduction of quality in provision raises severalissues. To our knowledge, in a context similar to ours, onlyEncinosa (1999)states its role.In his model, the problem of choice of quality is solved in a very simple way, as it is set atthe maximum value to attract consumers. Of course, other papers have treated quality as an(un)observable element in the definition of the payment system. These discussions are com-plementary to our main point, which is related to incentives for efficiency by intermediateagents, such as insurance companies and Health Maintenance Organizations.

Another feature of the extended transfer system we propose is administrative feasibility.The information needed to implement it is available on a routine basis, with a reasonabledegree of accuracy and a (presumably) small time lag.

5. Increasing returns to scale in effort cost

A fair criticism of the model in the previous section is the assumption that efficiency-effortcosts are determined on a per capita basis. One can readily find examples of efforts thatresult in cost savings and have a fixed-cost nature. A typical case is the definition of aservice protocol with a general practitioner. It will be applied to every consumer but thecost incurred (the protocol definition process) is independent of the number of individualsthat will be treated. Therefore, it is interesting to assess the optimal transfer structure underthe assumption that efficiency-effort costs are independent of the number of enrollees ofeach insurance company. It will be shown below that a straightforward generalization ofthe transfer mechanism will adjust for the fixed cost nature of efficiency-effort costs.

We first characterize the constrained optimal allocation, where the relevant constraint isthe number of active firms (insurers or providers) in the market. The social welfare functionis: ∫

Y

U

(Y −

∑i

C(ei)−∑i

GipiX(ei)

)h(y)dy (23)

The optimal choice of effort for each firm is given by:

−C′(ei)−GipiX′(ei) = 0,∀i (24)


Although the number of firms, denoted byn, is taken as exogenous by the social planner, forthe latter to be indifferent to the number of firms the following must be true. Every individualis indifferent, in utility terms, between insurance companies. Given that social welfare wasdefined as the sum of individual welfare (across the population), if the above condition isnot met, then social welfare can increase by an appropriate change of one individual fromone firm to another.

We now address the issue of whether a social planner would like to change the riskcomposition of an insurer. Defineqi ≡ ∫

χipg(p)dp = Gipi . A social planner does not

want to change the risk mix of consumers if (taking two insurers, say firm 1 and firm 2):

−X(e1)dq1 −X(e2)dq2 = 0 (25)

This requiresX(e1) = X(e2) or e1 = e2. To have equalei,∀i, one needsqi = q,∀i. Bydefinition,

∑i qi = p. From which it followsq = p/n. Therefore, the optimal effort choice

is given by:

−C′(ei)− pX′(ei)n

= 0 (26)

We turn now to the definition of the transfer system. The problem to be solved is:



i



Πi(χi)/Gi = Πi(χk)/Gk, ∀i, kΠi(χi) = SiGi − C(ei)+GiFi −GipiX(ei) = 0, ∀i−C′(ei)+Gi(α1pi − pi + α2p)X

′(ei) = 0, ∀i

Program C

Program C is equal toProgram B, with the last two constraints, zero-profit conditionfor each firm and optimal choice of effort by firmi, adjusted to the context of increasingreturns to scale in efficiency effort.

The no-selection constraint is again satisfied byα1 = 1. No-selection requires that,for the same number of enrollees, the insurance company does not profit from selecting adifferent risk mix. The valueα1 = 1 makes profits independent of the risk-mix of enrollees,as described previously.

Next, to induce the optimal efficiency effort, one needs to have

Giα2p = − pn

(27)

From this, it results that

α2 = − 1

nGi(28)

This means that the payment system is firm-specific, and adjusts for the number of enrolleesand the number of operating firms, due to the existence of scale economies.


Define the solution toProgram CasΘC = {(α0, α1, α2, ei, Fi) : solvesProgram C}.The following proposition follows.

Proposition 3. Takeθ ∈ ΘC . Thenα1 = 1 and α2 = −1/(nGi ). That is, the optimaltransfer is of the form:

Si = α0 +X(ei)

(pi − p

nGi

)(29)

This proposition can be more easily interpreted if we take the total transfer to an insurancecompany:

GiSi = α0Gi +Gi(pi − p)X(ei)+ p

(Gi − 1

n

)X(ei) (30)

The second term on the right-hand side is similar to the adjustment under constant returnsto scale in the efficiency effort with relation to the number of enrollees. It corresponds tothe selection component. An insurance company with a better risk mix will be a contributor,according to this term.

The last term is a new one, and it results from the fixed cost nature of the effort. To giveincentives for taking efficiency effort, under increasing returns to scale the payment systemmust also correct for the activity level. The valueGi is the number of enrollees in firmi (orits market share, as total population has been normalized to one), while 1/n is the marketshare if the market is equally divided among the existingn firms. Firms larger than 1/nreceive a higher transfer, as it pays to provide them with stronger incentives for efficiency.

As before, the payment system can be interpreted as an initial payment, followed by anend-of-the-year compensation fund. It is straightforward to check that this transfer systemis balanced, as in equilibrium the sum over all firms of contributions to the ex-post fundcancel out.

6. Multiple health risks

A natural extension of the model is to consider multiple health risks. Consider thatan individual facesτ different health risks. We maintain the assumption of one variableof effort by insurance companies. This focuses on broad organization effort rather thanillness-specific effort. Multiple efforts have been considered inGlazer and McGuire (2000,in press). In their context, multiple efforts lead to undesirable service level distortions. Theexpected health expenditures of this individual are:∑

τ

pτ xτ (ei) (31)

wherexτ is the cost of treating illnessτ andpτ is the associated probability of occurrence.The expected payments of an insurance company with a setχi of consumers are given by∫

p∈χi

∑τ

pτ xτ (ei)g(p|χi)dp (32)


wherep denotes the vector of illness probabilities andg(p|χi) is the joint density of prob-abilities in setχi (defined in a similar way to the single-illness case). We again take theinitial assumption of constant returns to scale in the efficiency-effort technology. Accord-ingly, profits of the insurance company are:

Πi =∫p∈χi

(Si(p, x)+ Fi − C(ei)−

∑τ

pτ xτ (ei)

)g(p|χi)dp (33)

The transferSi is defined by

Si = α0 + α1

∫p∈χi

∑τ pτ xτ (ei)g(p|χi)dp∫p∈χi g(p|χi)dp

+ α2

∫ 1

0g(p)

∑τ

pτ xτ (ei)dp (34)

These expressions are the natural generalizations of the single-illness case. The constraintof “no cream-skimming” is, in a similar way,

(α1 − 1)

(∫p∈χi

∑τ

pτ xτ (ei)g(p|χi)dp −∫p∈χj

∑τ

pτ xτ (ei)g(p|χj )dp

)= 0

As before, this constraint is satisfied forα1 = 1. Turning to the optimal effort level, theinsurance company chooses the efficiency level according to the following first-order con-dition:

α2

∫ 1

0

∑τ

pτ∂xτ

∂eig(p|χi)dp − C′(ei) = 0 (35)

It is relatively straightforward to establish the socially optimal effort level as the solution to

−C′(e)−∫ 1

0

∑τ

pτ∂xτ

∂eig(p|χi)dp = 0 (36)

Direct comparison of the two first-order conditions shows thatα2 = −1 leads again tothe optimal effort level from the social point of view. Hence, the results of the previoussection are generalized to multiple health risks in a natural way. Implementation of themultiple-risk version is more demanding in information. In particular, if it is not possibleto assign spending to specific diseases, this will undermine the implementation of themodel.

7. Concluding remarks

In the analysis of financing systems of the health care sector that rely, at least partially,on private health insurance markets, it is widely recognized that design of health financingstructures that use risk adjustment must pay attention to two problems: moral hazard in thechoice of effort to achieve/enhance efficiency and risk selection (cream-skimming).

We present a model that addresses the two issues. It is first shown that, within a simplemodel of linear transfers, for incentives for effort to exist, capitation transfers must be


partial ones. This result is, in itself, a particular case of the general result of incentive theoryapplied to health care financing issues. It shows that the proposed model is able to replicatethe conventional wisdom arguments.

The existence of risk selection adds more constraints to the design of the payment sys-tem. Nonetheless, we are able to propose an expanded transfer rule that has two desirablefeatures: it does not induce cream-skimming and maintains the incentives for insurancefirms to exert cost-reducing effort. The proposal has the additional advantages of rely-ing only on information that is usually available, it is financially balanced in equilibriumand is of relatively easy computation. For the diseases to be explicitly covered by thepayment transfer, the information needed, at the company level, amounts of total expendi-ture per disease, the number of people treated of each disease and the number of insuredconsumers.

The transfer system is defined by: (i) at the beginning of the year, the central fundpays to the insurance company a valueα0 per capita, independent of risk characteris-tics; the insurance company sets some value to be collected directly from consumers. Noprice discrimination is allowed and open enrollment is enforced; (ii) at the end of theyear, insurance companies participate in an ex-post clearing fund, where the contribu-tion (positive or negative) of each company is defined as the difference of the company’srisk mix to the population (average) risk mix, times the cost per patient treated in thecompany. At the equilibrium, as firms are symmetric in our model, average risk and ef-ficiency effort are equal across companies and no adjustment is necessary in expectedvalue. The existence of the ex-post fund is necessary to guarantee the right incentives tofirms. Its primary role is to work as a discipline device of deviations to the equilibriumvalues.

The proposed ex-post fund differs considerably from the “voluntary reinsurance pool”discussed, for example, invan de Ven et al. (2000).23 The disadvantage of that pool wasclearly identified with reduced incentives for efficiency. We showed this to be a feature ofthe particular form of that pool, not a general characterization of ex-post funds.

The use of ex-post funds has both advantages and disadvantages, when compared toalternative payment systems: administrative simplicity, desirable incentives, accuracy andresistance to manipulation. The use of simple rules defining pool funds is an attractivealternative to efforts aimed at refining risk-adjusted formulas. The latter typically requireincreasing information available, and there are serious doubts on how far it can go. Thesimplicity of rules and the mild information requirements of funds (when compared withrisk-adjustment regressions, for example) are on the positive side.

On the second criterion, compared with risk-adjusted capitations, the proposed paymentsystem maintains the incentives for efficiency without creating incentives for risk-selection.This also means that it enjoys better incentive properties than a pool based on total-systemaverage values.

In terms of accuracy and gaming possibilities, the proposed payment system shares thesame features of other ex-post funds. The relative position vis-à-vis formal risk adjustmentis more difficult to establish. The latter typically have low predictive power of individual

23 “To finance the pool, each insurer pays to the pool, for each of its pooled clients, the average expenditures ofall pooled individuals in the pool year.”van de Ven et al. (2000, p. 327).


expenditures, but this is a natural outcome to a considerable extent, as medical needs aretypically unpredictable in their timing and amount (the exception being chronic conditions,of course).

The main disadvantage, in implementation terms, of the proposed payment system isthat, unlike other alternatives, contributions and claims to the fund are only balanced inequilibrium. According to the model, losses and surpluses are absorbed by the Government.Whether this is a feasible characteristic in current health policy decisions is, naturally, achallenge to the model.

Of course, there are some caveats to the model. One of the issues not treated here wasprevention effort. If prevention effects can, to some extent, be seen as smaller spending inthe event of illness (possibly, even reducing it to zero), then prevention effort is included ina broad interpretation of the effort variable. The implications of considering prevention asa direct change in the risk distribution are left for future research.

Moral hazard on patients’ demand is a well-known phenomenon in health care. Thearguments for not providing full insurance are well understood. We opted to leave themout of the model. Moreover, other rationing efforts, namely acting on the supply side,and typically associated with managed care, such as utilization review and implementationof protocols, can be included in the effort variable. Similarly, it is widely recognized thatphysicians possess superior information on patients’ characteristics and on patients’ needs ineach specific illness episode. The efficiency problems raised by this information asymmetryand possible measures taken to mitigate distortions are included, once again, in our broaddefinition of “effort”.

Another caveat to the model is the introduction of further dimensions, be they in qualityof health care or the efficiency efforts made by insurers.

An aspect not explicitly mentioned was quality of care. In health care administration,quality has a definition closely related to the “efficiency” ideas from the economics literature(e.g. reduced number of post-surgery complications). If we see quality in this sense, it iscaptured by our efficiency effort variable. Quality as a service (or good) that provides highervalue to the consumer in the event of illness is a more complex issue, and it is left for futureresearch.

Recent literature on formal risk-adjustment addresses the issues of perverse incentivesregarding quality choices. Quality distortions at the service level may be used to indirectlyselect consumers (Glazer and McGuire, 2000, in press; Frank et al., 2000). Thus, qualityissues should receive further attention in future research.

Overall, we believe the approach in this paper to be more relevant, from an implemen-tation viewpoint, to the issues at hand (payment systems, risk selection and incentives forefficiency) than the introduction of non-linear payment systems.

Acknowledgements

I benefited from the comments of Fernando Branco, Bill Encinosa, Miguel Gouveia, JohnHassler, Izabela Jelovac, Diogo de Lucena, several seminar participants and two anonymousreferees. Financial support from PRAXIS XXI/CEG/15/96 is gratefully acknowledged. Theusual disclaimer applies.


Appendix A

A.1. A formal solution to the choice of the transfer system

Formally, the social planner’s problem can be written as:



i



Πi(χi)/Gi = Πi(χk)/Gk,∀i, kΠi(χi) = 0,∀i−C′(ei)+ (α1pi − pi + α2p)X

′(ei) = 0, ∀iThe first constraint establishes that the funding system must be balanced (a more generoussystem of transfers requires heavier contributions). The second expression is nothing morethan the definition of the allowable transfers.

The third expression reproduces the set of conditions to be satisfied in order to eliminateincentives to cream-skimming (the profit of the insurance company is independent of theset of consumers selected by the insurer).

The fourth condition is a free-entry condition for a competitive health insurance market.Finally, the last constraint is the first-order condition that characterizes the optimal choiceof effort. We assume that the regularity conditions for this approach to be valid hold.

The “no cream-skimming” condition can be re-written as:∫p∈χi

(Si − pX(ei)) g(p|χi)dp =∫p∈χk

(Si − pX(ei)) g(p|χk)dp (A.1)

or ∫p∈χi

(α1 − 1)pX(ei)g(p|χi)dp =∫p∈χk

(α1 − 1)pX(ei) g(p|χk)dp (A.2)

It is straightforward to see that this condition holds for any two sets of consumers only ifα1 = 1. This result can thus be substituted into the original program.

Take now the zero profit condition in a competitive health insurance market. Solving itin order toF yields:

Fi = C(Ei)+ piX(ei)− Si (A.3)

The first constraint can also be written as :

T =∫Y

T (y)h(y)dy = α0 + α1

∑i

piX(ei)Gi + α2

∑i

pX(ei)Gi (A.4)

FromLemma 1, ei = e, and it is possible to write, usingα1 = 1,

T = α0 + (1 + α2)pX(e) (A.5)

From the imposition of open enrollment rules and from the existence of competition amonginsurance firms on the insurance premium directly charged to consumers, it must be the


case that, in equilibrium, all firms will charge the same,Fi = F,∀i. Moreover, asGi > 0and

∑i Gi = 1 by definition, it follows thatF = ∑

i FiGi . This characteristic of the modelallows us to express social welfare in terms of a representative consumer, for each incomelevel. Hence, the social welfare function can be written as:

W =∫Y

U(y − T (y)+∫Y

T (y)h(y)dy −∑i

GiC(ei)−∑i

GipiX(ei))h(y)dy

(A.6)

Substitution forα0 in the problem to be solved simplifies it to:

maxα2

∫Y

U(y − T (y)− C(e)+ T − pX(e)

)h(y)dy s.t.

−C′(e)+ α2pX′(e) = 0

From this, it is clear thatα2 can be freely set to induce the optimal choice of effort level,which occurs forα2 = −1 (in which case, the Lagrange multiplier associated with theincentive constraint has a zero value).

Therefore, a net transfer

Si = α0 +X(ei) (pi − p) (A.7)

allows the system to achieve the optimal level of cost-reducing effort, without creatingincentives to risk selection.

The valueα0 is determined in order to ensure, together withF , that the participationconstraint of the insurance firm is satisfied:

F + α0 = C(e)+ piX(e) (A.8)

A.2. The proposed ex-post fund is financially balanced

We show now the claim made in the text that the ex-post adjustment process proposed isfinancially balanced.

The contribution of each firm to the fund is

Ai = (pi − p) X(ei)Gi (A.9)

Summing over all firms,∑i

Ai =∑i

(pi − p) X(ei)Gi (A.10)

As ei = e,∀i,∑i

Ai = X(e)

(∑i

(∫i∈χi

pg(p)dp

)− p

)= 0 (A.11)

The last equality sign follows from the fact that the set of consumers satisfies

χi ∩ χj = ∅ and ∪i χi = 1


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Cream-skimming, incentives for efficiency and payment system

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