Comparing Alternative Reimbursement Methodsin a Model of Public Health Insurance¤

Francesca Barigozziy

First version: October 1998This version: June 2000

Abstract

I compare in-kind reimbursement and reimbursement insurance. I ex-plicitly consider outpatient and inpatient care in a model where illness hasa negative impact on labor productivity. Consumers are heterogeneous withrespect to intensity of preferences for treatment which is their private infor-mation. Then the social planner has a choice of two kinds of reimbursementstructure: pooling (uniform) and self-selecting allocations.Analyzing pooling allocations I show that reimbursement insurance weakly

dominates in-kind reimbursement. While considering self-selecting alloca-tions I show that the two reimbursement methods are, from a social welfarepoint of view, equivalent.

Keywords: health insurance, in-kind transfers, reimbursement insurance,adverse selection.

¤I thank Alberto Bennardo, Giacomo Calzolari, GianLuca Fiorentini, Umberto Galmarini,Alessandro Lizzeri, Eric Malin, Alessandro Pavan, François Salanié and especially Helmuth Cre-mer for helpful comments. Paper presented at the Association of Southern European EconomicTheorists (ASSET), Bologna 1998, and at the CNR workshop: ”Equity, e¢ciency and the publicprovision of private goods”, Bologna, January 1999.

yGREMAQ, University of Toulouse and University of Bologna. E-mail:barigozz@spbo.unibo.it

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1. Introduction

Risk averse consumers demand health insurance. They insure against the …nancialrisk associated with buying medical care. I will study alternative health insurancereimbursement methods to …nd out which kind of payment is socially preferable.In an ideal world, the optimal insurance contract from the social planner’s

point of view would pay lump-sum transfers contingent on the health status. Ifillness occurs consumers would receive a cash payment related to the severity ofdisease, so that consumer’s sovereignty would be completely preserved. In realitywe normally do not observe this type of reimbursement.Instead of a cash payment, we generally observe either in-kind reimbursement

or reimbursement insurance (later on reimbursement on treatment cost). Gen-erally, when reimbursement is in-kind, consumers are payed directly in medicalservices. Payment is contingent on disease as it would be for cash reimbursement,but, in the case of in-kind, consumers are not free to choose the quantity of treat-ment they prefer. On the contrary, when reimbursement is on treatment cost,insurance payment depends on consumers’ expenditures upon health care. In thiscase instruments as coinsurance and deductible are used to limit overconsumption.The representation of these reimbursement plans that I choose in the model is

not able to capture all their complex features but provides a treatable framework.To be as simple as possible I assume that when reimbursement is in-kind (IK),access to care is free and consumers receive a quantity of treatment determinedby the insurer. Imposing a ceiling on treatment available to consumers, insuranceis able to prevent high demand for care. This implies that in-kind reimbursementallows cost-containment. At the same time an evident disadvantage of IK reim-bursement is the cost on social welfare due to the imposition of a consumptionconstraint to the insured people.Concerning physician’s fee, an important consequence of free access to care

is that, with IK reimbursement, health care providers are payed directly by theinsurer.Considering public health insurance systems which use to reimburse in-kind,

we generally refer to National Health Service type organizations. Great Britain,Germany, Italy1 and, only for inpatient care, also France2 are an example.

1In Italy outpatient care reimbursement is rather complicate, but we can say that at leastGP services are provided in-kind.

2In some cases, in France, third party payer principle may take place also for outpatient care.However it occurs only for chronic or very serious diseases or, more often, as the consequence of

2

Conversely, when reimbursement is on treatment cost (TC) I assume thatconsumers are free to choose the quantity of treatment they desire. A consequenceis that, not internalizing the entire health care cost, they demand an excessivequantity of it (overconsumption). This is an ex-post moral-hazard problem.Concerning again physician’s fee, with treatment cost reimbursement health

care providers generally are payed by consumers. The latter, after the insuranceclaim, receive a partial reimbursement from the insurer. As an example thisreimbursement is used, only for outpatient care, in France3.Figure 1 summarizes the trade-o¤, characterizing the two reimbursement meth-

ods, between consumers’ freedom in choosing treatment quantity and consumers’incentive to not overconsume.The importance of a uniform consumption constraint (in a sense which will

be clari…ed later) directly depends on the level of heterogeneity characterizing thepopulation. This suggests that it could exist a threshold value in consumers’ het-erogeneity such that when heterogeneity is not too high in-kind reimbursement isbetter, while, when heterogeneity is su¢ciently high, treatment cost reimburse-ment is preferable.To my knowledge this institutional comparison between alternative reimburse-

ment methods is still an unexplored issue.Moreover this work provides a treatable framework for systems which use a

mix of in-kind and treatment cost reimbursement: outpatient care are reimbursedon treatment cost and inpatient care are reimbursed in-kind. This is just the reim-bursement plan used in France. In fact, in the case of outpatient care consumersshare a part of treatment costs, but they maintain an important level of freedomin choosing treatment quantity. On the other side, for inpatient care, access tocare is free and treatment quantity is normally decided by the public insurance.

complementary private insurance purchase.3The French system for outpatient care leave complete freedom to consumers: they choose the

provider (both generalist and specialist) and the number of examinations. Moreover consumersdirectly pay for the services and the treatment prescribed. Later they ask for reimbursement tothe Social Insurance Administration and they are paid back approximately from the 60 to the80% of their expenses.Concerning French consumers’ freedom, the o¢ce based doctors convention introduced a vol-

untary scheme in 1987 which o¤ered the possibility to doctors of becoming “médecin référants”.Patients who join this scheme have a moral commitment not to visit a specialist directly. Theaim of this scheme was essentially to check the e¢cacy of a possible cost containment measure.But most doctors where reluctant (up to the end of 1997, only the 12,5 per cent of them hadjoined the scheme) because of the fear that they may be more controlled by the health insurancesystem. In fact in this scheme they are obliged to keep detailed patient records.

3

•

•

Freedom inchoosingquantity

In-kindreimbursement

Reimbursementinsurance

Cost-containment

Figure 1.1: trade-o¤ between consumers’ freedom and cost-containment.

Concerning the related literature, …rst, I have to mention the models on moral-hazard in health insurance, one of the seminal papers being Zeckhauser (1970).The way I treat treatment cost reimbursement represents a particular case of themore general reimbursement schedule of his model. Second, concerning in-kindreimbursement, I relate to the literature on in-kind transfers and optimal taxation(among others Cremer and Gahvari (1997)). In that literature the self-selectingproperty of in-kind transfers in second-best economies has been analyzed. Third,more generally I refer to the literature on income taxation with uncertainty inwhich taxation is used to insure consumers against various types of wage andhealth risks (as an example, Varian (1980) and Cremer and Gahvari (1995)).

In the next pages I compare IK and TC reimbursement in a model of publichealth insurance4. I assume that consumers are heterogeneous with respect bothto their state of health and to their preferences for treatment consumption. Thepublic insurer plays the role of the social planner and he is fully informed on

4A public health insurance has been analyzed for the …rst time in Blonqvist and Horn (1984).The authors show that, if individuals di¤er in their earning ability and also in the probability offalling ill, then a public health insurance is an e¢cient tool to redistribute welfare when incometaxation is linear.Together with their focus on public health insurance, Blonqvist and Horn (1984) presents

another similarity with respect to this paper: in both the models consumers’ utilities are state-dependent.

4

consumers’ state of health. In the …rst part of the work I constrain the insuranceplan to be uniform in the sense that tastes heterogeneity is not taken into account.Analyzing pooling allocations I …nd that TC dominates IK reimbursement. Thismeans that, contrary to intuition, there is no trade-o¤ between TC and IK de-pending on consumers’ heterogeneity. In the second part, I consider self-selectingallocations, i.e. allocations where consumers can choose insurance plans whichtakes into account their preference for treatment. In this case I show that the tworeimbursement methods are, from a social welfare point of view, equivalent.

2. The model

Let us consider a representative consumer and three possible states of health. Con-sumer can be healthy, not seriously ill and seriously ill. When not seriously ill, con-sumer needs outpatient care, while when seriously ill, consumer needs inpatientcare. More precisely, with probability p1; consumer is in good health and has afull earning ability, his marginal labor productivity is w1 (ability is normalized toequal the wage rate). With probability p2; consumer is ill and, as a consequence,he partially looses his earning ability; his marginal labor productivity falls tow2 < w1: Finally, with probability p3 = 1 ¡ p1 ¡ p2; consumer is seriously illand looses all his earning ability (he is hospitalized); in this case marginal laborproductivity fall to w3 = 0:5

Consumer’s preferences are state-dependent and twice separable:

U ji (C;X;L) = u (Ci)¡ v (Li)¡Hi + µjiÁi (Xi)i = 1; 2; 3 indicates health status as stated above, C is an aggregated consumptiongood taken as numeraire, X is health care consumption and L is labor supply. Hiis a …xed, state dependent, utility loss which can be partially recovered throughhealth care consumption. The term µjiÁi (Xi) indicates utility from health careconsumption. In particular Ái (Xi) is health improvement from treatment con-sumption, while the parameter µji (0 < µ

li · µhi ; i = 2; 3; j = l; h) represents

intensity of preferences for treatment, i.e. heterogeneity in consumers’ tastes.With probability ¹l consumer has low preference for health care consumption (heis low-type), while, with probability ¹h = 1¡¹l he has high preference for healthcare consumption (he is high-type), and this for both states of illness (bidimen-sional heterogeneity).

5As I will show in section 5.1, illness severity plays an important role also with respect toconsumers’ tastes on consumption levels.

5

Standard hypothesis on utility functions hold: u0 (Ci) > 0; u00 (Ci) < 0;v0 (Li) > 0; v

00(Li) > 0: H1 = 0 < H2 < H3. The function Á (X) is such

that: Á1 (X) = 0 8X; Ái (0) = 0; Á0i (X) > 0; Á00i (X) < 0; i = 2; 3: Moreover,Hi > µjiÁi (Xi) ; 8i = 2; 3; 8j = l; h and 8X; such that consumer’s utility isalways greater when in good health than when ill.The social planner will be concerned with making comparisons of utility levels

across consumers’ types. Thus, I assume full comparability of consumers’ utilities.The timing of the model is as follows: at t1 (interim) consumer learns his type

and at t2 (ex-post) the health-risk is realized and consumer learns his state ofhealth too. As it is shown in …gure 2, the social planner decides interim, whileconsumer decides ex-post.

t2(state of nature)

t1(types)

t

Social Planner Consumer

Figure 2: timing.

In this model I focus on the relationship between consumer and public insur-ance, the health care provider is not explicitly considered. The situation describedhere …ts both the case of a public provider (vertically integrated with the publicinsurer) and of a private one in a competitive market. In both cases, assuming alinear technology, the health care unitary cost is constant. This allows us to saythat consumer and the public insurer face the same treatment price (q) :Moreover,I assume that the provider behaves as a perfect agent for his patient.Concerning the informational structure of public insurance, in the model con-

sumer has potentially two private informations: his health status (captured by themarginal labor productivity wi) and his type µ

ji (high/low taste for treatment).

I assume that consumers’ health status is observable. This means that, concern-ing this aspect of the examination (as opposed to treatment purchase), collusionbetween patient and physician is impossible: physician acts as a perfect agentfor insurance. As a consequence reimbursement can be contingent to the healthstatus. Conversely, I assume that, in each state of health, preference for treatment

6

is not observable and public insurance can reimburse consumers according to apooling allocation or a self-selecting one.The structure of the work is as follow. In the …rst part I will show the …rst-

best, then I will compare the alternative insurance plans when the Governmentimplements pooling allocations. In particular the considered plans are: cash,in-kind, treatment costs reimbursement and, …nally, a mix of the two previousmethods. In the second part I will analyze the same reimbursement methodswhen the social planner implements self-selecting allocations. To summarize, theconsidered cases are:

Pooling allocations:- …rst-best- uniform plans:

- cash reimbursement- in-kind reimbursement- treatment cost reimbursement- mix of reimbursement types

Self-selecting allocations:- cash reimbursement- in-kind reimbursement- treatment cost reimbursement- mix of reimbursement types

3. First-best

I assume that, exhibiting an illness certi…cation provided by a physician, patientis entitled to receive reimbursement.First-best is represented by a contract contingent both to the health status

and to preference for treatment, that is a plan characterized by …ve non-uniformmonetary transfers (P;Rl2; R

h2 ; R

l3; R

h3): Consumption in the three states of health

is:C1 = w1L1 ¡ PC l2 = w2L

l2 +R

l2 ¡ qX l

2

Ch2 = w2Lh2 +R

h2 ¡ qXh

2

C l3 = Rl3 ¡ qX l

3

Ch3 = Rh3 ¡ qXh

3

where P is premium payed by healthy consumer, Rj2 (j = l; h) is net frompremium reimbursement for outpatient care and Rj3 (j = l; h) is reimbursement

7

for inpatient care. With cash reimbursement consumer decides to purchase thequantity of treatment he prefers. Note that in state of nature 3 cash transferRj3 must be enough to let consumer purchase both health care and aggregatedconsumption.The social planner maximizes the utilitarian6 social welfare function SW =

¹lEU(µl) + ¹hEU(µ

h), where EU(µl) is low-type consumer’s expected utility andEU(µh) is high-type consumer’s expected utility. Expected utility of low-typeand high-type individuals are respectively multiplied for the proportion of low-type and high-type consumers in the population7: low and high type consumershave the same weight for the social planner. Note that, when healthy, the twoconsumer’ types are identical. The social planner solves:8>>>>>>>>><>>>>>>>>>:

MaxP;Rji ;L

ji ;X

ji

p1 [u(w1L1 ¡ P )¡ v(L1)]+

+p2Xj=l;h

¹j£u¡w2L

j2 +R

j2 ¡ qXj

2

¢¡H2 + µj2Á2 ¡Xj2

¢¡ v(Lj2)¤++p3

Xj=l;h

¹j£u¡Rj3 ¡ qXj

3

¢¡H3 + µj3Á3 ¡Xj3

¢¤s:t: : p1P = p2(¹lR

l2 + ¹hR

h2) + p3(¹lR

l3 + ¹hR

h3)

(3.1)Two remarks can be useful. First, premium is fair. Second, because of the

way the heterogeneity parameter µji enters the utility functions, social welfareis increasing with respect to heterogeneity. This is equivalent to say that high-type consumers have the highest weight in this economy. As a consequence anutilitarian social welfare function redistributes from low to high-type individuals8.

6Concerning the choice of the social welfare function it is interesting to say that the maximinprinciple of Rawls is less applicable to cases which deals with health and the allocation of healthcare. In fact while the need for Rawls’ primary goods (e.g. food and clothing) are more or lessthe same for all, there is a much more unequal distribution of the need for health care re‡ectingthe ”natural lottery”. There are consequently much wider variations in the resources requiredto meet such unequal distribution of needs. In particular the crucial issue for the maximincriterion is the severity of the worst o¤. As long as it is feasible to improve the health of thisindividual, resources would be directed to him irrespective of the forgone improvement for theothers.For an interesting discussion on this subject see J.A.Olsen: ”Theory of justice and their

implication for priority setting in health care”, Journal of Health Economics 16, 1997, 625-639.7Considering a large number of representative consumers, ¹j is equivalent, ex-post, to the

proportion of the j-type.8For this reason, in the social welfare function, giving a higher (than ¹l) weight to low-type

8

>From FOCs we …nd the full-insurance result9:

C1 = Cl2 = C

h2 = C

l3 = C

h3 (= C) (3.2)

Moreover it is:L¤1 : w1u

0(C) = v0(L1) (3.3)

L¤2 : w2u0(C) = v0(L2) (3.4)

X¤j2 : µj2Á

02

¡Xj2

¢= qu0 (C) ; j = l; h: (3.5)

X¤j3 : µj3Á

03

¡Xj3

¢= qu0 (C) ; j = l; h: (3.6)

As we expected, in every state of health labor supply and treatment quantityare determined such that marginal bene…t equals marginal cost, as a consequenceMRSL2;X2 =

q

w2. Moreover in state of health 2 it is: L¤l2 = L

¤h2 and X¤h

2 > X¤l2 ;

and in state of health 3 it is X¤h3 > X¤l

3 : Concerning the monetary transfers, notsurprisingly one …nds: Rhi > R

li; i = 2; 3:

Note that the choice of Xji and Li can be decentralized because consumers face

prices wi and q. As a consequence the social planner can obtain …rst-best o¤eringthe …rst-best contract and letting consumers choose (ex-post) labor supply andtreatment quantity.

Here I brie‡y introduce the structure of in-kind and treatment cost reimburse-ment with full information on consumers’ preferences. The two reimbursementplans will be treated in detail in the case of asymmetric information in section 4.2and 4.3, 7.3 and 7.4.In-kind reimbursementRecall that, when reimbursement is in-kind, access to care is free and con-

sumers receive the package of care ¹Xji which is determined by insurance. I as-

sume that the transfer ¹Xji has to be entirely consumed: no intermediate levels of

consumption are possible. This interpretation of in-kind reimbursement, whichrepresents a good approximation of reality, will become important analyzing self-selecting allocations (section 7.3). Individuals’ consumption in the three states ofhealth is:

consumers would be more equitable. In fact it would allow to redress the relative importance ofthe two consumers types. I leave it for future research.

9Here utilities are state-dependent and separable. Then, for any given income level, illhealth does not alter the marginal utility of income. As a consequence full insurance is optimal.Moreover the full insurance condition concerns only aggregate consumption.

9

C1 = w1L1 ¡ P IKCj2 = w2L

j2 +R

IKj2 ;

¡Xj2 = ¹Xj

2

¢Cj3 = R

IKj3 ;

¡Xj3 =

¹Xj3

¢where j = l; h. Note that seriously ill consumers are obliged to consume ac-

cording to the transfer RIKj3 . This means that in-kind reimbursement imposes adouble constraint on seriously ill consumers: aggregate consumption and treat-ment quantity. In section 4.2 it will be clear that this double constraint concernsalso not seriously ill consumers.Reimbursement on treatment costWith reimbursement insurance the social planner uses a cost-sharing param-

eter ®ji · 1 (i = 2; 3; j = l; h) 10 to reduce health care overconsumption. As itwas said before, the moral-hazard problem due to the subsidization of health carecorresponds to the main disadvantage of reimbursement on treatment cost.As in the case of cash reimbursement, consumers choose their preferred treat-

ment quantity, the di¤erence is that here the social planner modi…es treatmentconsumption prices. Consumption in the three states of health is:C1 = w1L1 ¡ P TCCj2 = w2L

j2 +R

TCj2 ¡ ®j2qXj

2

Cj3 = RTCj3 ¡ ®j3qXj

3

It is evident that under perfect information in-kind and treatment cost reim-bursement are both equivalent to …rst-best. In fact, with both reimbursementsmethods the social planner can use four additional “instruments” (respectively¹Xji with in-kind and ®

ji with treatment cost reimbursement, i = 2; 3; j = l; h)

such that it can do at least as well: Actually when monetary transfers contingentto preference for treatment are available, these additional instruments are useless.Concerning treatment cost reimbursement, obviously with full information thesocial planner sets ®ji = 1 such that prices are not distorted.This result is stated in the following proposition:

Proposition 1 Under perfect information, cash, in-kind and treatment costreimbursement are equivalent.

This result is in line with Arrow’s intuition. In fact, in his seminal paper(Arrow (1963), page 962), he says that, in an hypothetically perfect market, the

10With full information considering either ®ji or simply ®i is equivalent:

10

existing di¤erent methods of treatment costs coverage should be equivalent.

In …gure 3 ill consumers’ …rst-best allocation is shown. As the reader can see,the slope of low-type utility function is higher than high-type one, this happens

because dCidXi

= ¡µjiÁ0i(X

ji )

u0(Cji ):

Cij

Uih

Uil

Cil = Ci

h

Xij

XihXi

l

Figure 3: ill consumers’ …rst-best allocation.

When µj is not observable; …rst-best allocations cannot be implemented. Infact, in state of health 2, …rst-best payment implies: C l2 = Ch2 ; X

h2 > X l

2 andLh2 = Ll2

11. And in state of health 3: C l3 = Ch3 ; Xh3 > X l

3: This means thatlow-type consumers would mimic high-type ones.

4. Uniform plans

Dealing with low-types incentive constraints the public insurance has a choice oftwo kinds of reimbursement structure. Those in which the insurer is unable todistinguish (ex-post or ex-ante) among individuals: this corresponds to a poolingallocation. And those in which the high-type and the low-type can (ex-post)

11In particular it is: RIKli = RIKhi = Rji ¡ qXji = R

ji ¡ q ¹Xj

i :

11

be identi…ed as a result of the action undertaken by the di¤erent groups: thiscorresponds to a self-selecting allocation.The reimbursement plans that we observe in the European public health in-

surance systems are essentially uniform: public insurance does not o¤er contractswhich di¤er according to their recipients’ taste. In fact the same plan is proposedto consumers with the same illness without caring about their (di¤erent) pref-erences for treatment. A possible reason for implementing pooling allocations isjust the presence of an hidden incentive constraint as the one treated in this work:facing o¤ the lack of information on consumer’s type, in the real world public in-surance chooses to provide uniform reimbursements. Maybe this choice is dueto the presence of political constraints and/or administrative costs. Anyway, toinvestigate on this interesting issue is not in the aim of this work.

4.1. Cash reimbursement

This plan is de…ned by three monetary transfers: (PC ; RC2 ; RC3 ): Ill consumers

receive reimbursement and choose in the market the preferred treatment quantity.Consumption in the three states of health is:C1 = w1L1 ¡ PCCj2 = w2L

j2 +R

C2 ¡ qXj

2

Cj3 = RC3 ¡ qXj

3

Note that the only di¤erence with respect to …rst-best is that here we add theuniformity constraint Rhi = R

li = R

Ci :

Recalling that consumer maximizes ex-post, with cash reimbursement con-sumers’ programs are as follow. Good health:

maxL1

u(w1L1 ¡ PC)¡ v(L1)

Then in state of health 1 labor supply is de…ned according to the following equa-tion:

L¤1 : w1u0(C1) = v0(L1) (4.1)

Outpatient care:

maxLj2;X

j2

u¡w2L

j2 +R

C2 ¡ qXj

2

¢¡H2 + µj2Á2 ¡Xj2

¢¡ v(Lj2)Then in state of health 2 labor supply and purchased treatment quantity arerespectively de…ned according to equations:

L¤j2 : w2u0(Cj2) = v

0(Lj2); j = l; h; (4.2)

12

X¤j2 : µj2Á

02

¡Xj2

¢= qu0

¡Cj2¢; j = l; h: (4.3)

Inpatient care:maxXj3

u¡RC3 ¡ qXj

3

¢¡H3 + µj3Á3 ¡Xj3

¢Then in state of health 3 purchased treatment quantity is chosen according toequation:

X¤j3 : µj3Á

03

¡Xj3

¢= qu0

¡Cj3¢; j = l; h: (4.4)

The social planner solves the following program:8>>>>>>>>><>>>>>>>>>:

MaxPC ;RC2 ;R

C3

p1£u(w1L

¤1 ¡ PC)¡ v(L¤1)

¤+

+p2Xj=l;h

¹j£u¡w2L

¤j2 +R

C2 ¡ qX¤j

2

¢¡H2 + µj2Á2 ¡X¤j2

¢¡ v(L¤j2 )¤++p3

Xj=l;h

¹j£u¡RC3 ¡ qX¤j

3

¢¡H3 + µj3Á3 ¡X¤j3

¢¤s:t: : p1P

C = p2RC2 + p3R

C3

where labor supplies L¤i and treatment quantities X¤ji verify consumer’s FOCs

(4.1), (4.2), (4.3) and (4.4).>From FOCs we …nd:

u0(C1) = E [u0 (C2)] = E [u0 (C3)] 12 (4.5)

As equation (4.5) shows, low and high-type consumers choose di¤erent aggregatedconsumptions. Full insurance is preserved only in average. This distortion fromthe full insurance represents the cost the uniformity constraint imposes.

Totally di¤erentiating equations (4.2) and (4.3) one …nds thatdL2dX2

> 0 and

dX2dµ2

> 0. Then, not surprisingly, it is: Xh2 > X

l2 and L

h2 > L

l2: As a consequence

we are not able to say neither if Ch2 is higher or lower than Cl2 nor which not

seriously ill consumers’ type is characterized by the higher utility level13. In the

same way, totally di¤erentiating equation (4.4), one …nds thatdX3dµ3

> 0 so that

12E [u0 (Ci)] = ¹l u0¡Cli¢+ ¹h u

0 ¡Chi ¢ :13In fact it is: Uh2 ¡ U l2 = u

¡Ch2¢¡ u ¡Cl2¢+ µh2Á2 ¡Xh

2

¢ ¡ µl2Á2 ¡Xl2

¢+ v

¡Ll2¢¡ v ¡Lh2¢ 7 0;

where Uh2 and U l2 respectively are h-type and l-type not seriously ill consumer’s utility andXh2 > X

l2; L

h2 > L

l2; C

h2 7 Cl2.

13

Xh3 > X

l3: Then C

l3 > C

h3 : But, again, we cannot a priori say which seriously ill

consumers’ type is characterized by the higher utility level14.

4.2. In-kind reimbursement

This plan is characterized by three monetary transfers15 and by two packages ofcare: (P IK; RIK2 ; R

IK3 ; ¹X2; ¹X3):

Individuals’ consumption in the three states of health is:C1 = w1L1 ¡ P IKC2 = w2L2 +R

IK2 ;

¡Xj2 = ¹X2

¢C3 = R

IK3 ;

¡Xj3 = ¹X3

¢With respect to …rst-best I added both the uniformity constraint Rhi = R

li =

RIKi and the consumption constraint ¹Xji =

¹Xi: That is one more constraint withrespect to (uniform) cash reimbursement. This allows us to say that, with in-kindreimbursement, consumers cannot be better o¤. In session 5 it will be clear thatconsumers are always worse o¤ (proposition 2).Healthy consumers’ program is the same I showed in the previous case and

equation (4.1), where P IK substitute P; still holds.Outpatient care:

maxL2

u¡w2L2 +R

IK2

¢¡H2 + µj2Á2 ¡ ¹X2¢¡ v(L2)Then, in state of health 2, labor supply is:

L¤2 : w2u0(C2) = v0(L2) (4.6)

Note that here, because of the separability of preferences, both types of not seri-ously ill consumers have the same labor supply L¤2. As a consequence C

h2 = C

l2 =

C2: This means that imposing the constraint on treatment quantity, in-kind reim-bursement yields to the same aggregate consumption for both consumers’ types.The same holds for inpatient care. In fact consumers are constrained to RIK3 and¹X3 and their utility is: u(RIK3 ) ¡ H3 + µj3Á3( ¹X3): In other words in-kind reim-bursement imposes the following two constraints to ill consumers: Chi = C

li = Ci

and ¹Xhi = ¹X l

i = ¹Xi:

14In fact it is: Uh3 ¡ U l3 = u¡Ch3¢¡ u ¡Cl3¢+ µh3Á3 ¡Xh

3

¢¡ µl3Á3 ¡Xl3

¢7 0; where Uh3 and U l3

respectively are h-type and l-type seriously ill consumer’s utility and Ch3 < Cl3; X

h3 > X

l3.

15In fact, being the health status observable, the social planner is always able to use themonetary transfers Ri: Note that the transfer RIK2 can be, and presumably is, negative (thesocial planner can collect resources also from not seriously ill consumer).

14

With in-kind reimbursement, in both states of illness it is:

Uhi ¡ U li = Ái¡¹Xi¢ ¡µhi ¡ µli

¢> 0 i = 2; 3

where Uhi and Uli respectively are high-type and low-type utility. This inequality

means that high-type utility is always larger than low-type utility: with in-kindreimbursement low-type consumers are always worse o¤16. The di¤erence betweenthe two utility levels is proportional to heterogeneity.The public insurance program is:8>>>>>>><>>>>>>>:

MaxP IK ;RIKi ; ¹Xi

p1£u(w1L

¤1 ¡ P IK)¡ v(L¤1)

¤+

+p2

hu(w2L

¤2 +R

IK2 )¡H2 + eµ2Á2 ¡ ¹X2¢¡ v(L¤2)i+

+p3

hu¡RIK3

¢¡H3 + eµ3Á3 ¡ ¹X3¢is:t: : p1P

IK = p2¡RIK2 + q ¹X2

¢+ p3

¡RIK3 + q ¹X3

¢where eµi = P

j=l;h

¹jµji : In fact, to implement the pooling allocation, the Government

maximizes the utility of a representative consumer: the mean µ-type consumer.Not surprisingly, from FOCs with respect to P IK; RIK2 and RIK3 we …nd thefull-insurance condition:

C1 = C2 = C3 = ¹C (4.7)

Moreover treatment packages are determined according to:

¹X2 : eµ2Á02 ¡ ¹X2¢ = qu0( ¹C) (4.8)

¹X3 : eµ3Á03 ¡ ¹X3¢ = qu0( ¹C) (4.9)

Obviously neither type of ill and seriously ill consumers receive the optimal quan-tity of treatment (determined respectively by equation (3.5) for outpatient careand by equation (3.6) for inpatient care) because ¹X2 and ¹X3 are determined ac-cording to the mean-µ-type consumer.It is evident that, when there is no heterogeneity

¡µl2 = µ

h2 ; µ

l3 = µ

h3

¢; we are

back to …rst-best.16While with cash reimbursement it was Uhi ¡ U li 7 0.

15

4.3. Reimbursement on treatment cost

As I said before ®i represents the coinsurance parameter, it is di¤erent for out-patient and inpatient care. The uniform plan is characterized by three monetarytransfers17 and by the two coinsurance parameters:

¡P TC ; RTC2 ; RTC3 ; ®2; ®3

¢. In-

dividuals’ consumption in the three states of health is:C1 = w1L1 ¡ P TCCj2 = w2L

j2 +R

TC2 ¡ ®2qXj

2

Cj3 = RTC3 ¡ ®3qXj

3

With respect to …rst-best the uniformity constraints on the monetary transfers(Rhi = R

li = R

TCi ) and on the coinsurance parameters (®ji = ®i) has been added.

Consumers’ programs are the following. Healthy consumers’ decision is thesame I showed in the previous cases and equation (4.1) still holds.Outpatient care:

maxLj2;X

j2

u¡w2L

j2 +R

TC2 ¡ ®2qXj

2

¢¡H2 + µj2Á2 ¡Xj2

¢¡ v(Lj2)As a consequence labor supply is determined, again, according to equation (4.2),moreover treatment quantity is:

X¤j2 : µj2Á

02

¡Xj2

¢= ®2qu

0 ¡Cj2¢ ; j = l; h (4.10)

Inpatient care:maxXj3

u¡RTC3 ¡ ®3qXj

3

¢¡H3 + µj3Á3 ¡Xj3

¢As a consequence treatment quantity is:

X¤j3 : µj3Á

03

¡Xj3

¢= ®3qu

0 ¡Cj3¢ ; j = l; h (4.11)

17As in the case of in-kind reimbursement, the transfer RTC2 can be negative.

16

The public insurance program is:188>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

MaxPTC ;RTCi ;®i

p1£u(w1L

¤1 ¡ P TC)¡ v(L¤1)

¤+

+p2Xj=l;h

¹j£u¡w2L

¤j2 +R

TC2 ¡ ®2qX¤j

2

¢¡H2 + µj2Á2 ¡X¤j2

¢¡ v(L¤j2 )¤++p3

Xj=l;h

¹j£u¡RTC3 ¡ ®3qX¤j

3

¢¡H3 + µj3Á3 ¡X¤j3

¢¤s:t: : p1P

TC = p2 (1¡ ®2) qPj=l;h

¹jX¤j2 + p2R

TC2 +

+p3 (1¡ ®3) qPj=l;h

¹jX¤j3 + p3R

TC3

>From FOCs with respect to P TC ; RTC2 and RTC3 one …nds the following equation:

E [u0(Ci)] = u0(C1)·1 + (1¡ ®i)qE

·@Xi@RTCi

¸¸i = 2; 3 (4.12)

It is interesting to remark that equation (4.12) would be equivalent to (4.5) for®i = 1; i = 2; 3: However it is easy to show that, because of consumers’ hetero-geneity, this will never be the case at the optimal treatment cost reimbursementpolicy19. Then ®i; i = 2; 3 are always di¤erent from 1 for positive level of hetero-geneity.Obviously when there is no heterogeneity it is optimal to impose ®i = 1;

i = 2; 3 and …rst-best is obtained.>From FOCs with respect to the coinsurance parameters one …nds:·

¡ (1¡ ®i)Eµ@Xi@®i

¶+ E (Xi)

¸u0 (C1) = E [Xiu0 (Ci)] (4.13)

The interpretation of equation (4.13) is as follows: the left hand side representsconsumers’ marginal cost and the right hand side consumers’ marginal bene…t18As it is normally the case in health insurance models, I do not impose any constraint on

the coinsurance parameters ®i; i = 2; 3, and I will verify ex-post if they are less or higher thanunity.19In fact, from equation (4.14) below, which describes the optimal coinsurance parameter, we

know that ®i = 1 implies

u0(C1) =E [Xiu

0(Ci)]E (Xi)

, i = 2; 3; and from equations (4.12) that ®i = 1 implies u0(C1) =

E [u0(Ci)] ; i = 2; 3. This means that it must be E [u0(Ci)]E (Xi) = E [Xiu0(Ci)] ; which isimpossible because Ci depends also on Xi.

17

from a negative variation of ®i (a fall in treatment price). When ®i decreases,consumers out of pocket expenses decrease as well, while insurance reimbursementexpenses increases. As a consequence insurance premium must increase as well.Marginal cost is measured by marginal variation of insurance premium (in bracket)multiplied for marginal utility of consumption in state ”good health”. In factpremium is paid by healthy consumers. In the right hand side the positive incomee¤ect from a negative variation of ®i is measured by the product of treatmentquantity and consumption marginal utility in the illness status. Mean valuesappear because a uniform plan is implemented. To …nd the optimal coinsuranceparameters, equation (4.13) can be rewritten as:

®i = 1¡ u0(C1)E(Xi)¡E [Xiu0(Ci)]

u0(C1)E·@Xi@®i

¸ (4.14)

The coinsurance parameters are positively correlated to treatment demand meanderivatives with respect to ®i; these terms are a measure of moral hazard. More-over @Xi

@®iis related to price elasticity of demand for treatment so that equation

(4.14) reminds us the inverse elasticity rule in Ramsey taxation: the commoditywhose demand is more inelastic is subsidized more.20

Verifying ex-post if it is optimal to impose a subsidy (®i < 1) or a tax (®i > 1) ;we …nd21 that a su¢cient condition to subsidize treatment is:

u0 (C1) < ¹lu0 ¡C li¢+ ¹hu0 ¡Chi ¢ i = 2; 3 (4.15)

while a necessary condition to tax treatment is the opposite of (4.15).The right hand side of the previous inequality is average marginal utility of

consumption in the illness status. Marginal utility being decreasing, the interpre-tation of (4.15) is the following: it is optimal to subsidize treatment if healthyindividuals consumption is larger than a particular mean of the ill individuals one.This means that the social planner may impose a tax on treatment.

4.4. Mix of reimbursement types

Consider now a reimbursement which pays on treatment cost for outpatient careand which pays in-kind for inpatient care. The mix of reimbursement types (MT)

20As it will be clari…ed in section 5.1, elasticity is higher for outpatient care. Then we expectthat ®2 > ®3:21>From (4.14) the following yields:®i < 1, cov(Xi; u

0(Ci))+E (Xi) [E (u0 (Ci))¡ u0 (Ci)] > 0; where cov(Xi; u0(Ci)) is positive:

18

is characterized by the following instruments: (PMT ; RMT2 ; RMT3 ; ®2; ¹X3): Con-sumption levels are:C1 = w1L1 ¡ PMTCj2 = w2L

j2 +R

MT2 ¡ ®2qXj

2

C3 = RMT3 ;

¡Xj3 = ¹X3

¢As before, healthy consumers choose their labor supply according to FOC

(4.1). Not seriously ill consumers choose simultaneously their labor supply andtreatment quantity such that they respectively verify FOCs (4.2) and (4.10). Aswith in-kind reimbursement, seriously ill consumers’ utility is u(RMT3 )+µj3Á3

¡¹X3¢:

The public insurance program is:8>>>>>>>><>>>>>>>>:

MaxPMT ;RMT

i ;®2; ¹X3p1£u(w1L

¤1 ¡ PMT )¡ v(L¤1)

¤+

+p2Xj=l;h

¹j£u¡w2L

¤j2 +R

MT2 ¡ ®2qX¤j

2

¢¡H2 + µj2Á2 ¡X¤j2

¢¡ v(L¤j2 )¤++p3

hu¡RMT3

¢¡H3 + eµ3Á3 ¡ ¹X3¢is:t: : p1P

MT = p2 (1¡ ®2) qP

j=l;h ¹jX¤j2 + p2R

MT2 + p3

¡RMT3 + q ¹X3

¢>From FOCs with respect to PMT and RMT3 one …nds that: C1 = C3 = RMT3 :As we expected, full-insurance concerns only healthy and seriously ill consumers’aggregated consumption. Moreover FOCs with respect to RMT2 ; ®2 and ¹X3 deter-mine respectively equations (4.12), (4.14) and (4.9).

5. Comparing the alternative uniform reimbursement plans

>From proposition 1 the following corollary holds.

Corollary 1 In the case of pooling allocations if consumers are homoge-neous, in-kind, treatment cost and a mix of types reimbursement are identical andequivalent to uniform cash payment.

In fact when consumers are homogeneous the uniformity constraint has noconsequence on social welfare and we are back to …rst-best.Reintroducing heterogeneity, the following result holds.22

22In the following the ranking among reimbursement schemes will be stated with the symbolsº and ¼ respectively for weak dominance and equivalence.

19

Proposition 2 In the case of pooling allocations a …rst reimbursement meth-ods ranking is the following: treatment cost reimbursement º cash reimbursementº in-kind reimbursement.Proof. (i) Uniform cash weakly dominates uniform in-kind reimbursement. Infact, recalling the discussion in session 4.2, cash reimbursement is characterizedonly by the uniformity constraint while in-kind reimbursement has one more con-straint on treatment consumption. Once the monetary transfers R2 and R3 havebeen …xed it is always better to let consumers choose treatment, being consump-tion prices not distorted. (ii) Uniform treatment cost weakly dominates uniformcash reimbursement. In fact cash reimbursement is characterized by three mone-tary transfers (PC ; RC2 ; R

C3 ) while treatment cost is characterized by three mone-

tary transfers and by two tax/subsidies on treatment price23 (P TC ; RTC2 ; RTC3 ; ®2;®3), i.e. treatment cost has two more instruments. Moreover, for ®2 = ®3 = 1;treatment cost is equivalent to cash reimbursement. As a consequence treatmentcost is at least as well as cash reimbursement.Remark that the parameters ®2 and ®3 that we have introduced as the source

of moral-hazard in treatment cost reimbursement, actually do not represent acost. In fact the distortion they impose on treatment price has a positive e¤ecton social welfare. The reason is that ®2 and ®3 are used to smooth consumptionbetween di¤erent consumers’ types in the same health status, such that TC op-timal allocation can approach full insurance. In other words ®2 and ®3 allow toindirectly and partially avoid the consequences of the uniformity constraint24.

Comparing uniform mix of types with uniform in-kind and uniform treatmentcost reimbursement one …nds:

Proposition 3 In the case of pooling allocations a second reimbursementmethods ranking is the following: treatment cost reimbursement º a mix of typesº in-kind reimbursement.Proof. A mix of types pays on treatment cost for outpatient care and in-kindfor inpatient care. From proposition 2 we saw that treatment cost dominates

23See the discussion at the end of section 4.3.24In fact, loosely speaking, if we consider for example inpatient care, ®3 (< 1) lets ¢C3 =

Cl3 ¡ Ch3 = ®3q¡Xh3 ¡Xl

3

¢be lower with respect to ¢C3 (cash) = q

¡Xh3 ¡Xl

3

¢: (The same

argument holds for ¢C2) Note that for ®3 = 0 it would be Cl3 = Ch3 , but in this case moral-hazard would be too costly. This is the standard trade-o¤ between moral-hazard and optimalrisk spreading.

20

in-kind. Then, considering MT with respect to TC, seriously ill consumers areworse o¤, while not seriously ill consumers are indi¤erent. As a consequence TCweakly dominates MT. On the other side, considering MT with respect to IK, notseriously ill consumers are better o¤, while seriously ill consumers are indi¤erent.As a consequence MT weakly dominates IK. One can conclude that treatmentcost weakly dominates a mix of types which weakly dominates in-kind.To …nd a more general result, that is a ranking of cash and MT, I will introduce

in the next section an assumption on the structure of consumers’ heterogeneity.

5.1. Unidimensional heterogeneity

Regarding heterogeneity, empirical evidence shows that, in the case of seriousillness, the price elasticity of demand for treatments is small25. A reasonableinterpretation is that patients, for such an illness, have the sentiment that thereis only one appropriate treatment. Moreover, this allows us to say that, in thecase of inpatient care, heterogeneity is small and, as a consequences, a uniformconsumption constraint will have a low impact on social welfare. Given theseconsiderations I set µh2 ¡ µl2 ¸ µh3 ¡ µl3 ¸ 0; so that heterogeneity is lower in thecase of serious illness.The particular case of unidimensional heterogeneity (µh2 ¡ µl2 > µh3 ¡ µl3 = 0) is

interesting because it can represent a good approximation of reality.Considering the uniform mix of reimbursement types in the particular case

with no heterogeneity on serious illness the following remark holds:

Remark 1 In the case of pooling allocations if seriously ill consumers arehomogeneous, a mix of types is equivalent to treatment cost reimbursement.

In fact, if seriously ill consumers are homogeneous, from corollary 1 we knowthat all the reimbursement methods are equivalent to cash. This implies that,concerning inpatient care, treatment cost and a mix of types are equivalent. Whileconcerning outpatient care, treatment cost and a mix of types are the same byde…nition.As a result it is possible to de…ne a complete ranking of the four reimbursement

methods when seriously ill consumers are homogeneous:

25Results from the RAND Health Insurance Experiment show that health care price elasticitiesbelong to the range [-0.1 , -0.2]. In particular, concerning serious illness treatment consumption,results show that ”there are no signi…cant di¤erences among the coinsurance plans in the use ofinpatient care services.” (Manning and others (1987), page 258)

21

Proposition 4 In the case of pooling allocations if seriously ill consumersare homogeneous (unidimensional heterogeneity), the complete reimbursement meth-ods ranking is as follows: treatment cost reimbursement ´ mix of types º cashreimbursement º in-kind reimbursement.

5.2. Bidimensional heterogeneity

With bidimensional heterogeneity (µh2 ¡ µl2 ¸ µh3 ¡ µl3 ¸ 0) it exists a trade-o¤between cash reimbursement and a mix of reimbursement types such that it can beeither -cash dominates a mix of types- or the opposite. In fact, from proposition 2,cash weakly dominates in-kind, as a consequence seriously ill consumers are bettero¤ with cash reimbursement. At the same time treatment cost weakly dominatescash, as a consequence not seriously ill consumers are better o¤ with the mix ofreimbursement types.Indeed, concerning this problem, one can make the following remark: the so-

cial welfare function used in the model suggests that uniform mix of types weaklydominates uniform cash reimbursement. In fact, …rst of all, in reality inpatientcare are less frequent than outpatient ones: p3 < p2. Second, I assumed thatheterogeneity is lower in the case of serious illness: A simple way to represent thissituation it is to normalize µl2 = µl3 = 1 such that µhi ¡ 1 measures heterogene-ity. Then it is µh2 ¸ µh3 ¸ 1. In this way, the heterogeneity structure gives evenmore weight to not seriously ill consumers. As a consequence, a mix of types,giving more utility to not seriously ill consumers, should reach the higher level ofsocial welfare. After the previous considerations we expect that, when heterogene-ity is bidimensional, the complete reimbursement methods ranking is as follows:treatment cost reimbursement º a mix of types º cash reimbursement º in-kindreimbursement. Notice that this is not a general result because it depends on amore assumption (p3 < p2) and on a speci…c normalization (µ

l2 = µ

l3 = 1):

As we said in the introduction, it seems natural to expect that the trade-o¤between in-kind and treatment cost reimbursement is a¤ected by the degree ofheterogeneity (see …gure 1). Proposition 2 shows that this is not the case. Thereason is that treatment cost makes use of two more instruments and imposes noconstraints on consumption. This result is strictly related to a crucial assumptionof the model: the health status is observable. This assumption implies that thesocial planner can always use monetary transfers contingent on the health status.

22

6. Self-selecting allocations

Dealing with self-selecting allocations, the previous ranking of reimbursementmethods may be substantially a¤ected. With this respect, a …rst important factis that now the constraint on treatment quantity imposed by in-kind reimburse-ment becomes a useful instrument. Directly providing (indivisible) in-kind healthservices, the social planner can observe the treatment consumed by ill individuals,as we shall see.As both the health status (captured by consumers’ marginal labor productivity

wi) and the pre-tax revenue (wiLi) are observable, labor supply is known and isalways part of the contracts proposed by the social planner to consumers.26

Consider now treatment quantity. Except when reimbursement is in-kind,treatment quantity is not observable by the social planner and mimicking onhealth care consumption arises.27 Obviously, if treatment consumption Xj

i is notobservable, then consumer’s aggregate consumption (Cji = wiL

ji+R

jTCi ¡(®ji )qXj

i )is not observable too. With in-kind reimbursement, on the contrary, treatmentquantity is observable. This follows from the interpretation of in-kind transfer ¹Xj

i

as an indivisible package of care (see section 3).

The social planner’s programs addressed in this section are standard cases ofmechanism design under adverse selection. Looking for the optimal mechanismof each reimbursement scheme, I will then employ the well known RevelationPrinciple28. Hence, I will study direct mechanisms in which consumers (truthfully)announce their type µ and the insurer o¤ers an allocation which speci…es all therelevant variables in the contractual relationship with consumers.Notice that for each reimbursement method we shall look for the social plan-

ner’s optimal allocations attainable within each reimbursement scheme. Thismeans that, as we shall make clearer, the available reimbursement plans will

26See the conclusions for an extension of the model, in line with the Optimal Taxation liter-ature (Stiglitz (1987), among others), with asymmetric information (also) on the health status.27In the case of cash reimbursement it is evident that insurance has no way to control con-

sumers’ treatment purchase. On the contrary, in the case of treatment cost reimbursement,consumers present the physician’s invoice and then receive reimbursement from the insurance.As a consequence, in this case we can say that treatment quantity is ex-post observable. How-ever, in the real world this information is generally not used by the insurance (which, in fact,implements linear commodity tax on treatment, represented in this model by the parameter®ji ). For this reason in the model treatment quantity is not observable with treatment costreimbursement as well. I will come back to these considerations later (see note (31)).28Myerson (1979), among others.

23

not necessarily allow to obtain the second-best optimum. The reason is obviouslythat some of them are instrument-constrained.

To have consumers truthfully report their type, the social planner has to max-imize his objective function under (also) the incentive compatibility constraints.As it has been shown at the end of section 3, the low-type consumers are themimickers. Standard mechanism design techniques with discrete types (see Fun-denberg and Tirole (1991), pages 246-250) show that it is optimal to make themimikers’ incentive compatibility constraints binding thus implying that all theother constraints are satis…ed.29 As a consequence, to recover the separating allo-cations I will add two incentive constraints to the social planner’s program: onefor the low-type not seriously ill consumers and another for the low-type seriouslyill consumers.

6.1. Cash reimbursement

Separating cash reimbursement is characterized by four monetary transfers¡P;Rj2; R3

¢and by consumers’ labor supplies (Lji ); j = l; h; i = 1; 2. In particular insuranceo¤ers the following contracts: (P; L1) for healthy consumers, the couple of con-tracts

¡Ll2; R

l2

¢and

¡Lh2 ; R

h2

¢respectively for low and high-type not seriously ill

consumers and the uniform transfer R3 for both seriously ill consumers’ typesas, in this case, the only variable the social planner can control is the monetarytransfers and then no separation can be obtained in the seriously ill state.Notice that this means the social planner cannot discriminate between the two

seriously ill consumers’ type and is obliged to o¤er a pooling contract. Thus, onlylow-type not seriously ill incentive constraint appears in the insurance program.With respect to treatment, not seriously and seriously ill consumers will re-

spectively choose treatment quantity according to equations (6.1) and (6.2):

X¤j2 : µj2Á

02 (X2)¡ qu0

¡w2L

j2 +R

j2 ¡ qX2

¢= 0 (6.1)

X¤j3 : µj3Á

03 (X3)¡ qu0

¡Rj3 ¡ qX3

¢= 0 (6.2)

while the mimicker will choose the preferred quantity according to the followingequation:

X¤lh2 : µl2Á

02 (X2)¡ qu0

¡w2L

h2 +R

h2 ¡ qX2

¢= 0 (6.3)

29A formal proof of this result is standard and then omitted. A complete proof is availablefrom the author.

24

The social planner program then is:8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

MaxP;Rj2;R3;L

ji

p1 [u(w1L1 ¡ P )¡ v(L1)] +

+p2Xj=l;h

¹j£u¡w2L

j2 +R

j2 ¡ qX¤j

2

¡µj2; L

j2; R

j2

¢¢¡H2 + µj2Á2 ¡X¤j2

¡µj2; L

j2; R

j2

¢¢+

¡v ¡Lj2¢¤+ p3 Xj=l;h

¹j£u¡R3 ¡ qX¤j

3

¢¡H3 + µj3Á3 ¡X¤j3

¢¤s:t: : p1P = p2

¡¹lR

l2 + ¹hR

h2

¢+ p3R3 (°)

u¡w2L

l2 +R

l2 ¡ qX¤l

2

¡µl2; L

l2; R

l2

¢¢+ µl2Á2

¡X¤l2

¡µl2; L

l2; R

l2

¢¢¡ v(Ll2) ¸u¡w2L

h2 +R

h2 ¡ qX¤lh

2

¡µl2; L

h2 ; R

h2

¢¢+ µl2Á2

¡X¤lh2

¡µl2; L

h2 ; R

h2

¢¢¡ v(Lh2) (¸)

where ° 6= 0 and ¸ ¸ 0 are respectively the budget constraint Lagrange multiplierand the incentive constraint Khun Tucker multiplier.>From FOCs with respect to P and L1 one respectively …nds:

u0(C1)¡ ° = 0 (6.4)

w1u0(C1)¡ v0(L1) = 0 (6.5)

which imply ° > 0. Moreover, as we expected, healthy consumer’s allocation issuch that marginal bene…t equals marginal cost of labor supply.>From FOC with respect to Rl2 one …nds:

p2¹l + ¸

p2¹lu0¡C l2¢¡ ° = 0 (6.6)

such that, using equation (6.4), C l2 > C1.>From FOC with respect to Ll2 it follows:

w2u0 ¡C l2¢¡ v ¡Ll2¢ = 0 (6.7)

and there is no distortion for the low-type not seriously ill consumer.>From FOC with respect to Rh2 one …nds:

u0¡Ch2¢¡ ¸

p2¹hu0¡C lh2¢¡ ° = 0 (6.8)

where C lh2 is the mimicker’s aggregate consumption. From equations (6.4) and(6.8) it follows that Ch2 < C1.

25

>From FOC with respect to Lh2 one …nds:

w2u0 ¡Ch2 ¢¡ v ¡Lh2¢

w2u0¡C lh2¢¡ v ¡Lh2¢ ¡ ¸

p2¹h= 0 (6.9)

Substituting (6.8) in (6.9) and rearranging one …nds:

p2¹h ¡ ¸p2¹h

v0¡Lh2¢¡ °w2 = 0 (6.10)

Such that it must be p2¹h¡ ¸ > 0. Moreover, using (6.6) and (6.7), (6.10) showsthat Ll2 < Lh2 : Concerning high-type distortion, for the (6.3) the mimicker willchoose X lh

2 < Xh2 ; then C

lh2 > Ch2 : As a consequence, (6.9) does not contradict

that p2¹h ¡ ¸ > 0 only if w2u0¡C lh2¢ ¡ v ¡Lh2¢ < w2u

0 ¡Ch2 ¢ ¡ v ¡Lh2¢ < 0, andthen w2u0

¡Ch2¢< v

¡Lh2¢: This means that high-type not seriously ill consumer is

forced to supply too much labor and to under-consume (with respect to aggregateconsumption).Finally, from the FOC with respect to R3 one …nds the same result obtained

for uniform cash reimbursement:

E [u0 (C3)] = u0 (C1)

High-type consumers, having a higher preference for treatment, choose an highertreatment quantity with respect to low-type consumers. Seriously ill consumers’pooling allocation in the case of cash reimbursement is represented in …gure 4.

U3l

U3h

X3j

C3j

C3l

C3h

X3l X3

h

Figure 4: seriously ill consumers’ allocation with cash reimbursement.

26

Note that the two point in …gure 4 are incentive compatible (no type prefers theconsumption bundle of the other type) but, obviously, no one of the two incentiveconstraints is binding (recall that in the seriously ill state no separation can beobtained because a unique instrument can be used, the monetary transfer). Thisanticipates that cash will be easily dominated by other reimbursement schemes.The following proposition summarizes the previous results concerning the sep-

arating cash allocation:

Proposition 5 The optimal cash self-selecting allocation is such that: contracts¡Ll2; R

l2

¢and

¡Lh2 ; R

h2

¢verify Ll2 < L

h2 and C

l2 > C

h2 : There is no distortion for

low-type consumer. On the contrary high type consumer is forced to supply toomuch labor and to consume too less aggregate consumption (w2u0(Ch2 ) < v

0(Lh2)).Both seriously ill types receive a monetary transfer R3 and choose treatment andaggregate consumption such that X l

3 < Xh3 and C

l3 > C

h3 :

6.2. In-kind reimbursement

In-kind reimbursement is characterized by …ve monetary transfers (P;Rj2; Rj3); by

consumers’ labor supplies (Lji ) and by the transfers ¹Xji ; i = 2; 3; j = l; h. As I an-

ticipated in section 3, it is reasonable to assume that the in-kind transfer ¹Xji is an

indivisible package of care30 such that the social planner can observe treatmentconsumption. As a consequence, with in-kind reimbursement, aggregate con-sumption, treatment quantity and labor supply are all observable. The contractsproposed in the three states then are (C1; L1) ; (C l2; X

l2; L

l2); (C

h2 ;X

h2 ; L

h2); (C

l3;X

l3)

and (Ch3 ; Xh3 ). It is interesting to notice that in-kind represents the unconstrained

direct mechanism in the sense that, given the agent’s type announcement, all therelevant variables are chosen by the social planner. As a consequence we can an-ticipate that the in-kind optimal allocation corresponds to the allocation whichweakly dominates the others.

30Letting consumers choose the preferred treatment quantity under the constraints Xji · ¹Xj

i ;the in-kind reimbursement self-selecting allocation would be very similar to the cash reimburse-ment one. The only di¤erence would be that with IK Cji is observable.

27

The social planner’s program then is318>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

MaxCji ;L

ji ;¹Xji

p1 [u(C1)¡ v(L1)]+

+p2Xj=l;h

¹j£u¡Cj2¢¡H2 + µj2Á2 ¡ ¹Xj

2

¢¡ v ¡Lj2¢¤++p3

Xj=l;h

¹j£u¡Cj3¢¡H3 + µj3Á3 ¡ ¹Xj

3

¢¤s:t: : p1 (w1L1 ¡ C1) = p2

Xj=l;h

¹j¡Cj2 ¡ w2Lj2

¢+ p3

Xj=l;h

¹jCj3+

+p2Xj=l;h

¹jq ¹Xj2 + p3

Xj=l;h

¹jq ¹Xj3 (°)

u¡C l2¢+ µl2Á2

¡¹X l2

¢¡ v ¡Ll2¢ ¸ u ¡Ch2 ¢+ µl2Á2 ¡ ¹Xh2

¢¡ v ¡Lh2¢ (¸2)u¡C l3¢+ µl3Á3

¡¹X l3

¢ ¸ u ¡Ch3 ¢+ µl3Á3 ¡ ¹Xh3

¢(¸3)

where ° 6= 0 and ¸2;¸3 ¸ 0 respectively are the budget constraint Lagrangemultiplier and the incentive constraints Khun Tucker multipliers.>From FOCs with respect to C1 and L1 one …nds respectively equations (6.4)

and (6.5).>From FOC with respect to C l2 it follows:

p2¹l + ¸2p2¹l

u0¡C l2¢¡ ° = 0 (6.11)

and then (6.4) and (6.11) imply C l2 > C1: From FOC with respect to Ll2 :

p2¹l + ¸2p2¹l

v0¡Ll2¢¡ w2° = 0 (6.12)

(6.5) and (6.12) together yield Ll2 > L1. Moreover (6.11) and (6.12) imply equa-tion (6.7). As a consequence there is no distortion for low-type consumer con-cerning labor supply. Finally from FOC with respect to ¹X l

2 one …nds:

p2¹l + ¸2p2¹l

µl2Á02

¡¹X l2

¢¡ q° = 0 (6.13)

31The monetary transfers Rji will be derived later from the optimal allocation.

28

>From (6.11) and (6.13) it follows that µl2Á02

¡¹X l2

¢¡qu0 ¡C l2¢ = 0: As a consequencethere is no distortion for low-type consumer concerning treatment quantity.Regarding high-type consumers, from FOCs with respect to Ch2 :

p2¹h ¡ ¸2p2¹h

u0¡Ch2¢¡ ° = 0 (6.14)

Comparing the previous equation to (6.4) one …nds Ch2 < C1:FOCs with respect to Lh2 and ¹Xh

2 respectively yield:

p2¹h ¡ ¸2p2¹h

v0¡Lh2¢¡ w2° = 0 (6.15)

p2¹hµh2Á

02

¡¹Xh2

¢¡ ¸2µl2Á02 ¡ ¹Xh2

¢¡ p2¹hq° = 0 (6.16)

Comparing (6.15) to (6.5) it is evident that Lh2 > L1:While from (6.14) and (6.15) it follows that w2u0

¡Ch2¢¡ v ¡Lh2¢ = 0: This im-

plies that there is no distortion for high-type not seriously ill consumer concerninglabor supply.I showed that Ch2 < C

l2 and L

h2 > L

l2; as a consequence, it must be R

h2 < R

l2:

Moreover, from the binding incentive constraint it follows ¹Xh2 > ¹X l

2:Solving (6.14) and (6.16) together yield to µh2Á

02

¡¹Xh2

¢< qu0

¡Ch2¢(see Stiglitz

(1987) page 1005). This means that high-type not seriously ill consumer is forcedto consume too much treatment and too less aggregate consumption.Concerning seriously ill consumers, from FOC with respect to C l3 one …nds:

p3¹l + ¸3p3¹l

u0¡C l3¢¡ ° = 0 (6.17)

such that, comparing with (6.4), C l3 > C1 holds. From FOC with respect to ¹X l3

one …nds:p3¹l + ¸3p3¹l

µl3Á03

¡¹X l3

¢¡ q° = 0 (6.18)

>From (6.17) and (6.19) it follows that µl3Á03

¡¹X l3

¢¡qu0 ¡C l3¢ = 0: As a consequencethere is no distortion for low-type consumer concerning treatment quantity.>From FOC with respect to Ch3 :

p3¹h ¡ ¸3p3¹h

u0¡Ch3¢¡ ° = 0 (6.19)

29

Comparing the previous equation to (6.4) it follows that Ch3 < C1:As before, recalling that Ch3 < C l3; from the binding incentive constraint it

must be ¹Xh3 >

¹X l3:

FOC with respect to ¹Xh3 yields:

p3¹hµh3Á

03

¡¹Xh3

¢¡ ¸3µl3Á03 ¡ ¹Xh3

¢¡ p3¹hq° = 0 (6.20)

Solving (6.19) and (6.20) together yield to µh3Á03

¡¹Xh3

¢< qu0

¡Ch3¢(again as in

Stiglitz (1987) page 1005). This means that high-type seriously ill consumer isforced to consume too much treatment and too less aggregate consumption. Seri-ously ill consumers’ self-selecting allocation in the case of in-kind reimbursementis represented in the following …gure:

Uh3

Ul3

Xl3 Xh

3

Ch3

Cl3

Xj3

Cj3

Figure 5: seriously ill consumers’ self-selecting allocation with in-kindreimbursement.

Note that in-kind self-selecting allocation devotes more resources to high-typethan to low-type consumers just as the …rst-best allocation does (see …gure 3).Proposition 6 summarizes the previous results.

Proposition 6 The optimal in-kind self-selecting allocation is such that: con-tracts (C l2; ¹X

l2; L

l2) and (C

h2 ; ¹X

h2 ; L

h2) verify C

l2 > Ch2 ; ¹X

l2 < ¹Xh

2 and Ll2 < Lh2 :0

The monetary transfers are such that Rl2 > Rh2 : There is no distortion for low-type consumer. High type consumer is forced to consume too much treatment andtoo less aggregate consumption ( µh2Á

02( ¹X

h2 ) < qu0(Ch2 )). Contracts (C

l3;X

l3) and

30

(Ch3 ; Xh3 ) verify C

l3 > C

h3 and ¹X l

3 < ¹Xh3 : There is no distortion for low-type con-

sumer and high type consumer is forced to consume too much treatment and tooless aggregate consumption ( µh3Á

03( ¹X

h3 ) < qu

0(Ch3 )).

It is important to notice that in-kind reimbursement exactly corresponds tothe direct mechanism in this adverse selection setting. Consumers announce theirtype and receive the second-best (due to asymmetric information) allocation. Allthe other relevant decisions are taken by the social planner. Interestingly, I shallshow in the next section that treatment cost reimbursement turns out to be anindirect mechanism with which the social planner is able to implement the verysame in-kind allocation.

6.3. Reimbursement on treatment cost

Treatment cost reimbursement is characterized by …ve monetary transfers (P;Rj2; Rj3),

by consumers’ labor supplies (Lji ) and by four cost-sharing parameters (®ji ); i =

2; 3 j = l; h. In particular insurance contracts are: (P;L1) for healthy consumers,(Ll2; R

l2; ®

l2) and (L

h2 ; R

h2 ; ®

h2) respectively for low and high-type not seriously ill

consumers and …nally (Rl3; ®l3) and (R

h3 ; ®

h3) respectively for low and high-type

seriously ill consumers. Contrary to the case of cash reimbursement, here also thelow-type seriously ill consumers are mimickers.Concerning treatment, not seriously and seriously ill consumers will respec-

tively choose treatment quantity according to equations (6.21) and (6.22):

X¤j2 : µj2Á

02 (X2)¡ ®j2qu0

¡w2L

j2 +R

j2 ¡ ®jqX2

¢= 0 (6.21)

X¤j3 : µj3Á

03 (X3)¡ ®j3qu0

¡Rj3 ¡ ®j3qX3

¢= 0 (6.22)

while the mimickers will choose the preferred treatment quantity according to thefollowing equations32:

32In the Optimal Taxation literature, a standard assumption is that only linear commoditytaxes are implementable because transactions are anonymous. This means that the quantityconsumed by every consumer is not observable.In the setting I analyze here with treatment cost reimbursement, the situation is di¤erent.

In fact consumers ask for reimbursement after their purchase has been done. As a consequencetreatment quantity is ex-post veri…able. Then, with this reimbursement method, the lineartaxation (subsidization) used by the insurer (the parameters ®ji ) corresponds to an ad hocrestriction of the insurance instruments. Anyway, in the real word, the Health Authoritiesfrequently use linear subsidization of treatment.

31

X¤lh2 : µl2Á

02 (X2)¡ ®h2qu0

¡w2L

h2 +R

h2 ¡ ®h2qX2

¢= 0 (6.23)

X¤lh3 : µl3Á

03 (X3)¡ ®h3qu0

¡Rh3 ¡ ®h3qX3

¢= 0 (6.24)

The social planner’s program then is8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

MaxP;Rji ;®

ji ;L

ji

p1 [u(w1L1 ¡ P )¡ v(L1)]+

+p2Xj=l;h

¹j£u¡w2L

j2 +R

j2 ¡ ®j2qX¤j

2

¡µj2; L

j2; R

j2; ®

j2

¢¢¡H2++µj2Á2

¡X¤j2

¡µj2; L

j2; R

j2; ®

j2

¢¢¡ v(Lj2)¤++p3

Xj=l;h

¹j£u¡RC3 ¡ qX¤j

3

¡µj3; R

j3; ®

j3

¢¢¡H3 + µj3Á3 ¡X¤j3

¡µj3; R

j3; ®

j3

¢¢¤s:t: : p1P = p2

£¹l¡Rl2 +

¡1¡ ®l2

¢qX¤l

2

¢+ ¹h

¡Rh2 +

¡1¡ ®h2

¢qX¤h

2

¢¤+

+p3£¹l¡Rl3 +

¡1¡ ®l3

¢qX¤l

3

¢+ ¹h

¡Rh3 +

¡1¡ ®h3

¢qX¤h

3

¢¤(°)

u¡w2L

l2 +R

l2 ¡ ®l2qX¤l

2

¡µl2; L

l2; R

l2; ®

l2

¢¢+ µl2Á2

¡X¤l2

¡µl2; L

l2; R

l2; ®

l2

¢¢¡ v(Ll2) ¸u¡w2L

h2 +R

h2 ¡ ®h2qX¤lh

2

¡µl2; L

h2 ; R

h2 ; ®

h2

¢¢+ µl2Á2

¡X¤lh2

¡µl2; L

h2 ; R

h2 ; ®

h2

¢¢¡ v(Lh2) (¸2)u¡Rl3 ¡ ®l3qX¤l

3

¡µl3; R

l3; ®

l3

¢¢+ µl3Á3

¡X¤l3

¡µl3; R

l3; ®

l3

¢¢ ¸u¡Rh3 ¡ ®h3qX¤h

3

¡µl3; R

h3 ; ®

h3

¢¢+ µl3Á3

¡X¤lh3

¡µl3; R

h3 ; ®

h3

¢¢(¸3)

Where ° 6= 0 and ¸2;¸3 ¸ 0 respectively are the budget constraint Lagrangemultiplier and the incentive constraints Khun Tucker multipliers.>From FOCs with respect to P and L1 one respectively …nds equations (6.4)

and (6.5).>From FOC with respect to Rl2 one …nds equation (6.6), such that, using

the (6.4), C l2 > C1 holds. Moreover, from FOC with respect to Ll2 one …ndsagain equation (6.7): there is no distortion for low-type not seriously ill consumerconcerning labor supply.>From FOC with respect to ®l2 it follows:

X l2

£p2¹lu

0 ¡C l2¢+ ¸2u0 ¡C l2¢¡ °p2¹l¤+ °p2¹l@X l2

@®l2

¡1¡ ®l2

¢= 0 (6.25)

such that, substituting (6.6), ®l2 = 1 holds: low-type not seriously ill consumer’streatment price is not distorted.>From FOC with respect to Rh2 equation (6.8) follows, such that, using the

(6.4), Ch2 < C1 holds.

32

>From FOC with respect to Lh2 one …nds again equation (6.9). As in session7.2, it follows that Ll2 < Lh2 : Moreover w2u

0 ¡Ch2 ¢ < v¡Lh2¢holds. That is, also

with non-uniform treatment cost, high-type not seriously ill consumer suppliestoo much labor and consumes too less aggregate consumption.Rearranging together FOCs with respect to ®h2 and R

h2 it follows:

®h2 = 1 +¸2u

0 ¡C lh2 ¢ ¡Xh2 ¡X lh

2

¢°p2¹h

@Xh2

@®h2

(6.26)

where @Xh2

@®h2< 0; ° > 0 and, from (6.21) and (6.23), Xh

2 ¡ X lh2 > 0: (6.26) shows

that ®h2 < 1 : high type not seriously ill consumer’s treatment price is subsidized.Concerning seriously ill consumers, from FOC with respect to Rl3 one …nds

equation (6.17), such that, again, C l3 > C1: Moreover, FOC with respect to ®l3yields:

X l3

£p3¹lu

0 ¡C l3¢+ ¸3u0 ¡C l3¢¡ °p3¹l¤+ °p3¹l@X l3

@®l3

¡1¡ ®l3

¢= 0 (6.27)

such that, substituting (6.17), ®l3 = 1 holds: low-type seriously ill consumers’treatment price is not distorted.While, from FOC with respect to Rh3 one …nds:

u0¡Ch3¢¡ ¸3

p3¹hu0¡C lh3¢¡ ° = 0 (6.28)

Comparing the previous equation to (6.4), one …nds Ch3 < C1.Finally, from FOCs with respect to ®h3 and R

h3 together it follows:

®h3 = 1 +¸3u

0 ¡C lh3 ¢ ¡Xh3 ¡X lh

3

¢°p3¹h

@Xh3

@®h3

(6.29)

where @Xh3

@®h3< 0; ° > 0 and, from (6.22) and (6.24), Xh

3 ¡ X lh3 > 0: (6.29) shows

that ®h3 < 1 : high-type seriously ill consumer’s treatment price is subsidized.

Totally di¤erentiating (6.22) it is easy to verify that dRj3d®j3

> 0: As a consequence

Rl3 > Rh3 :

Seriously ill consumers’ self-selecting allocation in the case of treatment costreimbursement is represented in …gure 6 by the points A and B. In the …gure

33

low-type consumer is indi¤erent between the allocation he can reach with thebudget constraint de…ned by

¡Rl3; ®

l3 = 1

¢and the allocation he can reach with

the budget constraint de…ned by¡Rh3 ; ®

h3 < 1

¢:

U3l

U3h

X3j

C3j

R3l

R3h

A

B

Figure 6 : seriously ill consumers’ self-selecting allocation with treatment costreimbursement.

Proposition 7 summarizes the previous results.

Proposition 7 The optimal treatment cost self-selecting allocation is such that:contracts (Ll2; R

l2; ®

l2) and (L

h2 ; R

h2 ; ®

h2) verify L

l2 < L

h2 ; C

l2 > C

h2 and ®

h2 < ®

l2 = 1:

There is no distortion for low-type consumer. On the contrary treatment consumedby high-type consumer is subsidized and high type consumer is forced to supply toomuch labor and to consume too less aggregate consumption (w2u0(Ch2 ) < v

0(Lh2)):Contracts (Rl3; ®

l3) and (R

h3 ; ®

h3) verify C

l3 > Ch3 ; R

l3 > Rh3 and ®

h3 < ®l3 = 1:

There is no distortion for low-type consumer and treatment consumed by high-type consumer is subsidized.

Treatment cost represents an indirect mechanism in this adverse selection prob-lem. In fact consumers choose aggregate consumption and treatment after thesocial planner has decided the parameters of the insurance contract. Consideringseriously ill consumers and looking at …gures 5 and 6 we see that with treatmentcost the in-kind allocation cannot be implemented because it is not incentive com-patible. In other words, using treatment cost, the social planner could obtain an

34

allocation with the same characteristics of the second-best one, but this wouldimply an additional cost. Actually this is not a problem and the second-best allo-cation can be obtained with treatment cost too. In fact, referring to the TaxationPrinciple in the mechanism design literature, we know that the social planner cano¤er a non linear schedule C3(X) which corresponds to the optimal non-linear tar-i¤33. In particular this non-linear tari¤ allows to eliminate the ”undesired” partsfrom the consumers’ budget constraints such that only the points correspondingto the second-best allocation will be chosen at the equilibrium. Figure 7 showsan example of optimal non-linear tari¤ which implements the second-best.

C3j

X3j

U3l

U3h

C3(X)

Figure 7: implementing the second-best with treatment cost reimbursement.

Concerning not seriously ill consumers, as before, the social planner imple-ments the optimal non-linear tari¤ C2(X) such that only the second-best alloca-tion is chosen at the equilibrium. Such non-linear tari¤ also depends on laborsupply (Lj2).

34

33Note that this schedule is the non-linear equivalent of the pooling schedule C3(X) =RTC3 ¡ ®3X which has been analyzed in section 4.3 when I treated uniform treatment costreimbursement.34There is another possible interpretation of the self-selecting treatment cost allocation.

Consider seriously ill consumers and the mechanism fR(µ); ®(µ)g where consumers announceboth their type and the quantity of treatment they want to buy. The monetary transfer isR(µ) = C + ®(µ)qX: Notice that here there is no possibility of misrepresenting the quantityof treatment to purchase. In fact in equilibrium only two quantities of treatment are possible:Xli and X

hi : This means that low-type incentive constraints are similar to the in-kind program

35

7. Comparing the alternative separating reimbursement plans

Sections 6.2 and 6.3 show that in-kind and treatment cost reimbursement are,from a social welfare point of view, equivalent. The …rst corresponds to thedirect mechanism while the second corresponds to a payo¤ equivalent indirectmechanism. Both allow to implement the second-best allocation.Consider now the other reimbursement plans. Let start from a mix of reim-

bursement types and recall that it pays on treatment cost for outpatient care andin-kind for inpatient care. From the previous consideration it follows that a mix oftypes allows to implement the second-best allocation too. As a consequence a mixof reimbursement types is, again from a social welfare point of view, equivalentto in-kind and treatment cost reimbursement.On the contrary, concerning cash reimbursement, as it has been said analyz-

ing uniform plans (Proposition 2) this reimbursement method uses one instrumentless with respect to treatment cost. As a consequence cash is weakly dominatedby treatment cost reimbursement. The following proposition establishes the re-imbursement methods ranking in the case of self-selecting allocations.

Proposition 8 In the case of self-selecting allocations the reimbursementmethods ranking is as follows: in-kind reimbursement ¼ treatment cost reim-bursement ¼ mix of types º cash reimbursement.

8. Conclusion

This work presents an institutional comparison of alternative health insurance re-imbursement methods. In the model I compare in-kind reimbursement (IK) andreimbursement insurance (in the paper treatment cost reimbursement (TC)) in amodel of public health insurance. Moreover the model provides a treatable frame-work for systems which use a mix of in-kind and treatment cost reimbursement(as an example the French system): outpatient care are reimbursed on treatmentcost and inpatient care are reimbursed in-kind.The model explicitly considers serious and not serious illness which both have a

negative (but di¤erent) impact on labor productivity. Not seriously ill consumersneed outpatient care, while seriously ill ones need inpatient care. A key feature ofthe model is consumers’ heterogeneity with respect to intensity of preferences for

ones (see the previous section) and the second-best allocation is implemented. As it is evident,following this interpretation treatment cost becomes a direct mechanism too.

36

treatment. Public insurance is fully informed on consumers’ health status but itcannot observe preference for treatment. In this setting low-type consumers wantto mimic high type consumers. Facing low-type consumers incentive constraints,public insurance can choose to implement a pooling allocation or a separatingone.In the …rst part of the work I analyze pooling allocations. The main result is

that TCweakly dominates the mix of IK and TC payment which weakly dominatesIK reimbursement.In the second part of the paper I analyze self-selecting allocations. Intuitively,

with these allocations the rationale for in-kind reimbursement should be stronger:the self-selecting property of in-kind transfers should partially prevents from mim-icking. The result con…rms this intuition: in-kind reimbursement corresponds tothe direct mechanism and then it is not dominated by any other reimbursementmethod. Treatment cost corresponds to an indirect mechanism which is able toimplement the second-best allocation too and then, from a social welfare pointof view, it is equivalent to in-kind reimbursement. Not surprisingly also a mix oftypes reimbursement turns out to be equivalent to in-kind and treatment cost.Finally, the structure of the model may allow to consider also a setting with

asymmetric information with respect to the health status along the lines of theOptimal Taxation literature (Stiglitz (1987)). In that case consumers can mimica worst state of health in order to work less and to get a better reimbursement.Di¤erent health status corresponds to di¤erent earning ability so that there wouldexist three groups of individuals: healthy, not seriously ill and seriously ill con-sumers (w1 > w2 > w3 = 0): As usual, wiLi is observable but earning ability andlabor supply separately are not. Note that this means that seriously ill individ-uals are not able to mimic. If provider behaves as a perfect agent for consumer,when consumer wants to mimic, physicians certi…es a false state of health allow-ing consumers to ask for a better reimbursement. It is reasonable to think thatconsumers are interested in mimicking a worse state of health such that they areable to work less and to obtain a larger reimbursement. In particular this meansthat the sense of mimicking goes from healthy to ill individuals: an healthy con-sumer can mimic a not seriously ill or a seriously ill one and a not seriously illconsumer can mimic a seriously ill one. Consumers’ possibility to mimic dependson the insurance plan structure. Thus, consequences of mimicking will be di¤erentfor di¤erent reimbursement types. In particular we expect that, weakening theincentive constraint, in-kind reimbursement will partially prevent from mimick-ing. To be more realistic, this setting should also explicitly take into account the

37

health care provider, eventually considering collusive behavior between patientand physician. These topics are left for future research.

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