Comparing Alternative Reimbursement Methods
in a Model of Public Health Insurance
¤
Francesca Barigozzi
yFirst version: October 1998
This 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 has a negative impact on labor productivity. Consumers are heterogeneous with respect to intensity of preferences for treatment which is their private infor-mation. Then the social planner has a choice of two kinds of reimbursement structure: 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 welfare point 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 Economic Theorists (ASSET), Bologna 1998, and at the CNR workshop: ”Equity, e¢ciency and the public provision of private goods”, Bologna, January 1999.
yGREMAQ, University of Toulouse and University of Bologna. E-mail:
1. Introduction
Risk averse consumers demand health insurance. They insure against the …nancial risk associated with buying medical care. I will study alternative health insurance reimbursement 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. If illness occurs consumers would receive a cash payment related to the severity of disease, so that consumer’s sovereignty would be completely preserved. In reality we 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 medical services. 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 reimbursetreat-ment is on treattreat-ment cost, insurance payment depends on consumers’ expenditures upon health care. In this case 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 determined by the insurer. Imposing a ceiling on treatment available to consumers, insurance is able to prevent high demand for care. This implies that in-kind reimbursement allows 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 consumption constraint 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 the insurer.
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 least
GP services are provided in-kind.
2In some cases, in France, third party payer principle may take place also for outpatient care.
Conversely, when reimbursement is on treatment cost (TC) I assume that consumers are free to choose the quantity of treatment they desire. A consequence is that, not internalizing the entire health care cost, they demand an excessive quantity 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 insurance claim, receive a partial reimbursement from the insurer. As an example this reimbursement 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 the population. This suggests that it could exist a threshold value in consumers’ het-erogeneity such that when hethet-erogeneity is not too high in-kind reimbursement is better, 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 reimbursed on 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 consumers share a part of treatment costs, but they maintain an important level of freedom in choosing treatment quantity. On the other side, for inpatient care, access to care 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 consumers directly pay for the services and the treatment prescribed. Later they ask for reimbursement to the Social Insurance Administration and they are paid back approximately from the 60 to the 80% 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. The aim 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 had joined the scheme) because of the fear that they may be more controlled by the health insurance system. In fact in this scheme they are obliged to keep detailed patient records.
• • Freedom in choosing quantity In-kind reimbursement Reimbursement insurance 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 the more general reimbursement schedule of his model. Second, concerning in-kind reimbursement, I relate to the literature on in-kind transfers and optimal taxation (among others Cremer and Gahvari (1997)). In that literature the self-selecting property of in-kind transfers in second-best economies has been analyzed. Third, more generally I refer to the literature on income taxation with uncertainty in which taxation is used to insure consumers against various types of wage and health 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 public health insurance4. I assume that consumers are heterogeneous with respect both to their state of health and to their preferences for treatment consumption. The public 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 of falling ill, then a public health insurance is an e¢cient tool to redistribute welfare when income taxation 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.
consumers’ state of health. In the …rst part of the work I constrain the insurance plan to be uniform in the sense that tastes heterogeneity is not taken into account. Analyzing pooling allocations I …nd that TC dominates IK reimbursement. This means that, contrary to intuition, there is no trao¤ between TC and IK de-pending on consumers’ heterogeneity. In the second part, I consider self-selecting allocations, i.e. allocations where consumers can choose insurance plans which takes into account their preference for treatment. In this case I show that the two reimbursement 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, concon-sumer needs inpatient care. More precisely, with probability p1; consumer is in good health and has a full earning ability, his marginal labor productivity is w1 (ability is normalized to equal the wage rate). With probability p2; consumer is ill and, as a consequence, he partially looses his earning ability; his marginal labor productivity falls to w2 < w1: Finally, with probability p3 = 1 ¡ p1 ¡ p2; consumer is seriously ill and looses all his earning ability (he is hospitalized); in this case marginal labor productivity fall to w3 = 0:5
Consumer’s preferences are state-dependent and twice separable: Uij(C; X; L) = u (Ci) ¡ v (Li) ¡ Hi + µjiÁi(Xi)
i = 1; 2; 3 indicates health status as stated above, C is an aggregated consumption good taken as numeraire, X is health care consumption and L is labor supply. Hi is a …xed, state dependent, utility loss which can be partially recovered through health care consumption. The term µjiÁi(Xi) indicates utility from health care consumption. In particular Ái(Xi) is health improvement from treatment con-sumption, while the parameter µji (0 < µ
l i · µ
h
i; 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 (he is low-type), while, with probability ¹h = 1 ¡ ¹l he has high preference for health care 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 to
Standard hypothesis on utility functions hold: u0(C
i) > 0; u00(Ci) < 0; v0(L
i) > 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 is always 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 of health too. As it is shown in …gure 2, the social planner decides interim, while consumer 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 described here …ts both the case of a public provider (vertically integrated with the public insurer) and of a private one in a competitive market. In both cases, assuming a linear technology, the health care unitary cost is constant. This allows us to say that 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 the marginal labor productivity wi) and his type µji (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), collusion between patient and physician is impossible: physician acts as a perfect agent for insurance. As a consequence reimbursement can be contingent to the health status. Conversely, I assume that, in each state of health, preference for treatment
is not observable and public insurance can reimburse consumers according to a pooling 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 Government implements pooling allocations. In particular the considered plans are: cash, in-kind, treatment costs reimbursement and, …nally, a mix of the two previous methods. In the second part I will analyze the same reimbursement methods when the social planner implements self-selecting allocations. To summarize, the considered 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, patient is 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-uniform monetary transfers (P; Rl2; Rh2; Rl3; Rh3): Consumption in the three states of health is: C1 = w1L1¡ P Cl 2 = w2Ll2+ Rl2¡ qX2l C2h = w2Lh2 + Rh2 ¡ qX2h Cl 3 = R3l ¡ qX3l Ch 3 = Rh3 ¡ qX3h
where P is premium payed by healthy consumer, R2j (j = l; h) is net from premium reimbursement for outpatient care and Rj3 (j = l; h) is reimbursement
for inpatient care. With cash reimbursement consumer decides to purchase the quantity of treatment he prefers. Note that in state of nature 3 cash transfer Rj3 must be enough to let consumer purchase both health care and aggregated consumption.
The social planner maximizes the utilitarian6 social welfare function SW = ¹lEU (µ
l
) + ¹hEU (µ h
), where EU (µl) is low-type consumer’s expected utility and EU (µh) is high-type consumer’s expected utility. Expected utility of low-type and high-type individuals are respectively multiplied for the proportion of low-type and high-low-type consumers in the population7: low and high type consumers have the same weight for the social planner. Note that, when healthy, the two consumer’ types are identical. The social planner solves:
8 > > > > > > > > > < > > > > > > > > > : M ax P;Rij;Lji;Xij p1[u(w1L1¡ P ) ¡ v(L1)] + +p2 X j=l;h ¹j £ u¡w2Lj2+ R j 2¡ qX j 2 ¢ ¡ H2+ µj2Á2 ¡ X2j ¢ ¡ v(Lj2) ¤ + +p3 X j=l;h ¹j £ u¡R3j¡ qX j 3 ¢ ¡ H3+ µ j 3Á3 ¡ X3j ¢¤ s:t: : p1P = p2(¹lRl2+ ¹hR2h) + p3(¹lRl3+ ¹hRh3) (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 welfare is increasing with respect to heterogeneity. This is equivalent to say that high-type consumers have the highest weight in this economy. As a consequence an utilitarian 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 maximin
principle of Rawls is less applicable to cases which deals with health and the allocation of health care. In fact while the need for Rawls’ primary goods (e.g. food and clothing) are more or less the same for all, there is a much more unequal distribution of the need for health care re‡ecting the ”natural lottery”. There are consequently much wider variations in the resources required to meet such unequal distribution of needs. In particular the crucial issue for the maximin criterion is the severity of the worst o¤. As long as it is feasible to improve the health of this individual, resources would be directed to him irrespective of the forgone improvement for the others.
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, ¹
j is equivalent, ex-post, to the
proportion of the j-type.
8For this reason, in the social welfare function, giving a higher (than ¹
>From FOCs we …nd the full-insurance result9: C1 = C2l = C h 2 = C l 3 = C h 3 (= C) (3.2) Moreover it is: L¤1 : w1u0(C) = v0(L1) (3.3) L¤2 : w2u0(C) = v0(L2) (3.4) X2¤j : µ j 2Á02 ¡ X2j ¢ = qu0(C) ; j = l; h: (3.5) X3¤j : µj3Á03¡X3j¢= qu0(C) ; j = l; h: (3.6) As we expected, in every state of health labor supply and treatment quantity are determined such that marginal bene…t equals marginal cost, as a consequence M RSL2;X2 =
q w2
. Moreover in state of health 2 it is: L¤l
2 = L¤h2 and X2¤h > X2¤l; and in state of health 3 it is X¤h
3 > X3¤l: Concerning the monetary transfers, not surprisingly one …nds: Rhi > Rli; i = 2; 3:
Note that the choice of Xij and Li can be decentralized because consumers face prices wi and q. As a consequence the social planner can obtain …rst-best o¤ering the …rst-best contract and letting consumers choose (ex-post) labor supply and treatment quantity.
Here I brie‡y introduce the structure of in-kind and treatment cost reimburse-ment with full information on consumers’ preferences. The two reimbursereimburse-ment plans will be treated in detail in the case of asymmetric information in section 4.2 and 4.3, 7.3 and 7.4.
In-kind reimbursement
Recall that, when reimbursement is in-kind, access to care is free and con-sumers receive the package of care ¹Xij which is determined by insurance. I as-sume that the transfer ¹Xij has to be entirely consumed: no intermediate levels of consumption are possible. This interpretation of in-kind reimbursement, which represents a good approximation of reality, will become important analyzing self-selecting allocations (section 7.3). Individuals’ consumption in the three states of health is:
consumers would be more equitable. In fact it would allow to redress the relative importance of the two consumers types. I leave it for future research.
9Here utilities are state-dependent and separable. Then, for any given income level, ill
health does not alter the marginal utility of income. As a consequence full insurance is optimal. Moreover the full insurance condition concerns only aggregate consumption.
C1 = w1L1¡ PIK C2j = w2Lj2+ R IKj 2 ; ¡ X2j = ¹X2j¢ C3j = RIKj3 ; ¡X3j = ¹X3j¢
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 a double constraint on seriously ill consumers: aggregate consumption and treat-ment quantity. In section 4.2 it will be clear that this double constraint concerns also not seriously ill consumers.
Reimbursement on treatment cost
With 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 it was said before, the moral-hazard problem due to the subsidization of health care corresponds 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 treattreat-ment consumption prices. Consumption in the three states of health is:
C1 = w1L1¡ PT C C2j = w2Lj2+ R T Cj 2 ¡ ® j 2qX j 2 C3j = RT Cj3 ¡ ®j3qX3j
It is evident that under perfect information in-kind and treatment cost reim-bursement are both equivalent to …rst-best. In fact, with both reimreim-bursements methods the social planner can use four additional “instruments” (respectively
¹
Xij with in-kind and ® j
i with treatment cost reimbursement, i = 2; 3; j = l; h) such that it can do at least as well: Actually when monetary transfers contingent to preference for treatment are available, these additional instruments are useless. Concerning treatment cost reimbursement, obviously with full information the social 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 cost reimbursement 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 ®j
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 dCi dXi = ¡µ j i Á0 i(X j i) u0(Cj i) : Cij Uih Uil Cil = Cih Xij Xih Xil
Figure 3: ill consumers’ …rst-best allocation.
When µj is not observable; …rst-best allocations cannot be implemented. In fact, in state of health 2, …rst-best payment implies: Cl
2 = C2h; X2h > X2l and Lh2 = Ll211. And in state of health 3: C3l = C3h; X3h > X3l: This means that low-type consumers would mimic high-type ones.
4. Uniform plans
Dealing with low-types incentive constraints the public insurance has a choice of two kinds of reimbursement structure. Those in which the insurer is unable to distinguish (ex-post or ex-ante) among individuals: this corresponds to a pooling allocation. And those in which the high-type and the low-type can (ex-post)
11In particular it is: RIKl
i = RIKhi = R j i ¡ qX j i = R j i ¡ q ¹X j i:
be identi…ed as a result of the action undertaken by the di¤erent groups: this corresponds to a self-selecting allocation.
The reimbursement plans that we observe in the European public health in-surance systems are essentially uniform: public inin-surance does not o¤er contracts which di¤er according to their recipients’ taste. In fact the same plan is proposed to consumers with the same illness without caring about their (di¤erent) pref-erences for treatment. A possible reason for implementing pooling allocations is just 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 due to the presence of political constraints and/or administrative costs. Anyway, to investigate 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¡ PC C2j = w2Lj2+ RC2 ¡ qX j 2 C3j = RC 3 ¡ qX j 3
Note that the only di¤erence with respect to …rst-best is that here we add the uniformity constraint Rh
i = Rli = RCi :
Recalling that consumer maximizes ex-post, with cash reimbursement con-sumers’ programs are as follow. Good health:
max L1
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: max Lj2;Xj2 u¡w2Lj2+ R C 2 ¡ qX j 2 ¢ ¡ H2+ µj2Á2 ¡ X2j¢¡ v(Lj2)
Then in state of health 2 labor supply and purchased treatment quantity are respectively de…ned according to equations:
L¤j2 : w2u0(C2j) = v0(L j
X2¤j : µj2Á02¡X2j¢ = qu0¡C2j¢; j = l; h: (4.3) Inpatient care: max X3j u¡RC3 ¡ qX j 3 ¢ ¡ H3+ µj3Á3 ¡ X3j ¢
Then in state of health 3 purchased treatment quantity is chosen according to equation: X3¤j : µ j 3Á03 ¡ X3j ¢ = qu0¡C3j ¢ ; j = l; h: (4.4)
The social planner solves the following program: 8 > > > > > > > > > < > > > > > > > > > : M ax PC;RC 2;RC3 p1 £ u(w1L¤1¡ PC) ¡ v(L¤1) ¤ + +p2 X j=l;h ¹j£u¡w2L¤j2 + RC2 ¡ qX2¤j ¢ ¡ H2+ µj2Á2 ¡ X2¤j¢¡ v(L¤j2 )¤+ +p3 X j=l;h ¹j£u¡RC 3 ¡ qX3¤j ¢ ¡ H3 + µj3Á3 ¡ X3¤j¢¤ s:t: : p1PC = p2RC2 + p3RC3 where labor supplies L¤
i and treatment quantities Xi¤j 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 aggregated consumptions. Full insurance is preserved only in average. This distortion from the full insurance represents the cost the uniformity constraint imposes.
Totally di¤erentiating equations (4.2) and (4.3) one …nds that dL2 dX2
> 0 and dX2
dµ2
> 0. Then, not surprisingly, it is: X2h > X2l and Lh2 > Ll2: As a consequence we are not able to say neither if Ch
2 is higher or lower than C2l 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 that dX3
dµ3 > 0 so that 12E [u0(Ci)] = ¹ lu0 ¡ Cl i ¢ + ¹hu0¡Ch i ¢ : 13In fact it is: Uh 2 ¡ U2l = u ¡ Ch 2 ¢ ¡ u¡Cl 2 ¢ + µh2Á2¡Xh 2 ¢ ¡ µl2Á2 ¡ Xl 2 ¢ + v¡Ll 2 ¢ ¡ v¡Lh 2¢ 7 0; where Uh
2 and U2l respectively are h-type and l-type not seriously ill consumer’s utility and
Xh
Xh
3 > X3l: Then C3l > C3h: 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 of care: (PIK; RIK
2 ; RIK3 ; ¹X2; ¹X3):
Individuals’ consumption in the three states of health is: C1 = w1L1¡ PIK C2 = w2L2+ RIK2 ; ¡ X2j = ¹X2 ¢ C3 = R3IK; ¡ X3j = ¹X3 ¢
With respect to …rst-best I added both the uniformity constraint Rh
i = Rli = RIKi and the consumption constraint ¹X
j
i = ¹Xi: That is one more constraint with respect to (uniform) cash reimbursement. This allows us to say that, with in-kind reimbursement, consumers cannot be better o¤. In session 5 it will be clear that consumers are always worse o¤ (proposition 2).
Healthy consumers’ program is the same I showed in the previous case and equation (4.1), where PIK substitute P; still holds.
Outpatient care: max L2 u¡w2L2 + RIK2 ¢ ¡ H2+ µj2Á2 ¡¹ X2 ¢ ¡ 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 C2h = C2l = 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 RIK
3 and
¹
X3 and their utility is: u(RIK3 ) ¡ H3 + µj3Á3( ¹X3): In other words in-kind reim-bursement imposes the following two constraints to ill consumers: Ch
i = Cil = Ci and ¹Xh i = ¹Xil = ¹Xi: 14In fact it is: Uh 3 ¡ U3l = u ¡ Ch 3 ¢ ¡ u¡Cl 3 ¢ + µh3Á3¡Xh 3 ¢ ¡ µl3Á3 ¡ Xl 3¢ 7 0; where U3h and U3l
respectively are h-type and l-type seriously ill consumer’s utility and C3h< C3l; X3h> X3l. 15In fact, being the health status observable, the social planner is always able to use the
monetary transfers Ri: Note that the transfer RIK2 can be, and presumably is, negative (the
With in-kind reimbursement, in both states of illness it is: Uih¡ Uil= Ái¡X¹i
¢ ¡
µhi ¡ µli¢> 0 i = 2; 3
where Uih and Uil respectively are high-type and low-type utility. This inequality means that high-type utility is always larger than low-type utility: with in-kind reimbursement low-type consumers are always worse o¤16. The di¤erence between the two utility levels is proportional to heterogeneity.
The public insurance program is: 8 > > > > > > > < > > > > > > > : M ax PIK;RIK i ; ¹Xi p1 £ u(w1L¤1¡ PIK) ¡ v(L¤1) ¤ + +p2 h u(w2L¤2+ RIK2 ) ¡ H2+ eµ2Á2 ¡¹ X2 ¢ ¡ v(L¤ 2) i + +p3 h u¡RIK 3 ¢ ¡ H3+ eµ3Á3 ¡¹ X3 ¢i s:t: : p1PIK = p2 ¡ RIK 2 + q ¹X2 ¢ + p3 ¡ RIK 3 + 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 PIK; RIK
2 and RIK3 we …nd the
full-insurance condition:
C1 = C2 = C3 = ¹C (4.7)
Moreover treatment packages are determined according to: ¹ X2 : eµ2Á02 ¡¹ X2 ¢ = qu0( ¹C) (4.8) ¹ X3 : eµ3Á03 ¡¹ X3 ¢ = 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 care and 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 h 2; µ l 3 = µ h 3 ¢ ; we are back to …rst-best.
16While with cash reimbursement it was Uh
4.3. Reimbursement on treatment cost
As I said before ®i represents the coinsurance parameter, it is di¤erent for out-patient and inout-patient care. The uniform plan is characterized by three monetary transfers17 and by the two coinsurance parameters: ¡PT C; RT C
2 ; RT C3 ; ®2; ®3 ¢
. In-dividuals’ consumption in the three states of health is:
C1 = w1L1¡ PT C
C2j = w2Lj2+ RT C2 ¡ ®2qX2j C3j = RT C3 ¡ ®3qX
j 3
With respect to …rst-best the uniformity constraints on the monetary transfers (Rh
i = Rli = RT Ci ) and on the coinsurance parameters (® j
i = ®i) has been added. Consumers’ programs are the following. Healthy consumers’ decision is the same I showed in the previous cases and equation (4.1) still holds.
Outpatient care: max Lj2;X2j u¡w2L j 2 + R T C 2 ¡ ®2qX j 2 ¢ ¡ H2+ µ j 2Á2 ¡ X2j ¢ ¡ v(Lj2)
As a consequence labor supply is determined, again, according to equation (4.2), moreover treatment quantity is:
X2¤j : µj2Á20 ¡X2j¢ = ®2qu0 ¡ C2j¢; j = l; h (4.10) Inpatient care: max X3j u¡RT C3 ¡ ®3qX3j ¢ ¡ H3+ µj3Á3 ¡ X3j¢ As a consequence treatment quantity is:
X3¤j : µj3Á30 ¡X3j¢ = ®3qu0 ¡
C3j¢; j = l; h (4.11)
17As in the case of in-kind reimbursement, the transfer RT C
The public insurance program is:18 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : M ax PT C;RT C i ;®i p1 £ u(w1L¤1¡ PT C) ¡ v(L¤1) ¤ + +p2 X j=l;h ¹j£u¡w2L2¤j+ RT C2 ¡ ®2qX2¤j ¢ ¡ H2 + µj2Á2 ¡ X2¤j¢¡ v(L¤j2 )¤+ +p3 X j=l;h ¹j£u¡RT C 3 ¡ ®3qX3¤j ¢ ¡ H3+ µj3Á3 ¡ X3¤j¢¤ s:t: : p1PT C = p2(1 ¡ ®2) q P j=l;h ¹jX2¤j+ p2RT C2 + +p3(1 ¡ ®3) q P j=l;h ¹jX3¤j + p3RT C3 >From FOCs with respect to PT C; RT C
2 and RT C3 one …nds the following equation:
E [u0(Ci)] = u0(C1) · 1 + (1 ¡ ®i)qE · @Xi @RT C i ¸¸ 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 reimbursement policy19. 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) E µ @Xi @®i ¶ + E (Xi) ¸ u0(C1) = E [Xiu0(Ci)] (4.13) The interpretation of equation (4.13) is as follows: the left hand side represents consumers’ marginal cost and the right hand side consumers’ marginal bene…t
18As 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 than
unity.
19In fact, from equation (4.14) below, which describes the optimal coinsurance parameter, we
know that ®i = 1 implies
u0(C1) =
E [Xiu0(Ci)]
E (Xi)
, i = 2; 3; and from equations (4.12) that ®i = 1 implies u0(C1) =
E [u0(C
i)] ; i = 2; 3. This means that it must be E [u0(Ci)] E (Xi) = E [Xiu0(Ci)] ; which is
from a negative variation of ®i (a fall in treatment price). When ®i decreases, consumers out of pocket expenses decrease as well, while insurance reimbursement expenses 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 fact premium is paid by healthy consumers. In the right hand side the positive income e¤ect from a negative variation of ®i is measured by the product of treatment quantity and consumption marginal utility in the illness status. Mean values appear because a uniform plan is implemented. To …nd the optimal coinsurance parameters, equation (4.13) can be rewritten as:
®i = 1 ¡ u0(C 1)E(Xi) ¡ E [Xiu0(Ci)] u0(C1)E · @Xi @®i ¸ (4.14)
The coinsurance parameters are positively correlated to treatment demand mean derivatives with respect to ®i; these terms are a measure of moral hazard. More-over @Xi
@®i is related to price elasticity of demand for treatment so that equation
(4.14) reminds us the inverse elasticity rule in Ramsey taxation: the commodity whose 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 ¡ Cil ¢ + ¹hu0 ¡ Cih ¢ 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 healthy individuals 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 care and 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 expect
that ®2> ®3:
21>From (4.14) the following yields:
is characterized by the following instruments: (PM T; RM T
2 ; RM T3 ; ®2; ¹X3): Con-sumption levels are:
C1 = w1L1¡ PM T C2j = w2Lj2+ RM T2 ¡ ®2qX2j C3 = R3M T; ¡ X3j = ¹X3 ¢
As before, healthy consumers choose their labor supply according to FOC (4.1). Not seriously ill consumers choose simultaneously their labor supply and treatment quantity such that they respectively verify FOCs (4.2) and (4.10). As with in-kind reimbursement, seriously ill consumers’ utility is u(RM T3 )+µ
j 3Á3 ¡¹ X3 ¢ : The public insurance program is:
8 > > > > > > > > < > > > > > > > > : M ax PM T;RM T i ;®2; ¹X3 p1 £ u(w1L¤1¡ PM T) ¡ v(L¤1) ¤ + +p2 X j=l;h ¹j£u¡w2L2¤j+ R2MT ¡ ®2qX2¤j ¢ ¡ H2+ µj2Á2 ¡ X2¤j¢¡ v(L¤j2 )¤+ +p3 h u¡RM T 3 ¢ ¡ H3+ eµ3Á3 ¡¹ X3 ¢i s:t: : p1PM T = p2(1 ¡ ®2) q P j=l;h¹jX2¤j + p2RM T2 + p3 ¡ RM T 3 + q ¹X3 ¢ >From FOCs with respect to PMT and RM T3 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 RM T
2 ; ®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 and equivalent to uniform cash payment.
In fact when consumers are homogeneous the uniformity constraint has no consequence 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 symbols
Proposition 2 In the case of pooling allocations a …rst reimbursement meth-ods ranking is the following: treatment cost reimbursement º cash reimbursement º in-kind reimbursement.
Proof. (i) Uniform cash weakly dominates uniform in-kind reimbursement. In fact, recalling the discussion in session 4.2, cash reimbursement is characterized only by the uniformity constraint while in-kind reimbursement has one more con-straint on treatment consumption. Once the monetary transfers R2 and R3 have been …xed it is always better to let consumers choose treatment, being consump-tion prices not distorted. (ii) Uniform treatment cost weakly dominates uniform cash reimbursement. In fact cash reimbursement is characterized by three mone-tary transfers (PC; R2C; RC3) while treatment cost is characterized by three mone-tary transfers and by two tax/subsidies on treatment price23 (PT C; RT C
2 ; RT C3 ; ®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 treatment cost 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 a cost. In fact the distortion they impose on treatment price has a positive e¤ect on social welfare. The reason is that ®2 and ®3 are used to smooth consumption between 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 to indirectly and partially avoid the consequences of the uniformity constraint24.
Comparing uniform mix of types with uniform in-kind and uniform treatment cost reimbursement one …nds:
Proposition 3 In the case of pooling allocations a second reimbursement
methods ranking is the following: treatment cost reimbursement º a mix of types º in-kind reimbursement.
Proof. A mix of types pays on treatment cost for outpatient care and in-kind for 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 = Cl 3¡ C3h = ®3q ¡ Xh 3 ¡ X3l ¢
be lower with respect to ¢C3(cash) = q
¡ Xh
3 ¡ X3l
¢
: (The same argument holds for ¢C2) Note that for ®3 = 0 it would be C3l = C3h, but in this case
moral-hazard would be too costly. This is the standard trade-o¤ between moral-moral-hazard and optimal risk spreading.
in-kind. Then, considering MT with respect to TC, seriously ill consumers are worse o¤, while not seriously ill consumers are indi¤erent. As a consequence TC weakly dominates MT. On the other side, considering MT with respect to IK, not seriously ill consumers are better o¤, while seriously ill consumers are indi¤erent. As a consequence MT weakly dominates IK. One can conclude that treatment cost 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 serious illness, the price elasticity of demand for treatments is small25. A reasonable interpretation is that patients, for such an illness, have the sentiment that there is only one appropriate treatment. Moreover, this allows us to say that, in the case of inpatient care, heterogeneity is small and, as a consequences, a uniform consumption constraint will have a low impact on social welfare. Given these considerations I set µh2 ¡ µ l 2 ¸ µ h 3 ¡ µ l
3 ¸ 0; so that heterogeneity is lower in the case 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 are homogeneous, a mix of types is equivalent to treatment cost reimbursement.
In fact, if seriously ill consumers are homogeneous, from corollary 1 we know that all the reimbursement methods are equivalent to cash. This implies that, concerning inpatient care, treatment cost and a mix of types are equivalent. While concerning outpatient care, treatment cost and a mix of types are the same by de…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 elasticities
belong 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 of inpatient care services.” (Manning and others (1987), page 258)
Proposition 4 In the case of pooling allocations if seriously ill consumers are homogeneous (unidimensional heterogeneity), the complete reimbursement meth-ods ranking is as follows: treatment cost reimbursement ´ mix of types º cash reimbursement 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 be either -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 better o¤ with cash reimbursement. At the same time treatment cost weakly dominates cash, as a consequence not seriously ill consumers are better o¤ with the mix of reimbursement 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 weakly dominates uniform cash reimbursement. In fact, …rst of all, in reality inpatient care are less frequent than outpatient ones: p3 < p2. Second, I assumed that heterogeneity is lower in the case of serious illness: A simple way to represent this situation it is to normalize µl2 = µl3 = 1 such that µhi ¡ 1 measures heterogene-ity. Then it is µh2 ¸ µ
h
3 ¸ 1. In this way, the heterogeneity structure gives even more 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 of social 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-kind reimbursement. Notice that this is not a general result because it depends on a more assumption (p3 < p2) and on a speci…c normalization (µl2 = µ
l 3 = 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 of heterogeneity (see …gure 1). Proposition 2 shows that this is not the case. The reason is that treatment cost makes use of two more instruments and imposes no constraints on consumption. This result is strictly related to a crucial assumption of the model: the health status is observable. This assumption implies that the social planner can always use monetary transfers contingent on the health status.
6. Self-selecting allocations
Dealing with self-selecting allocations, the previous ranking of reimbursement methods may be substantially a¤ected. With this respect, a …rst important fact is that now the constraint on treatment quantity imposed by in-kind reimburse-ment becomes a useful instrureimburse-ment. Directly providing (indivisible) in-kind health services, 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 is always 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 on health care consumption arises.27 Obviously, if treatment consumption Xj
i is not observable, then consumer’s aggregate consumption (Cij = wiLji+R
jT C i ¡(® j i)qX j i) is not observable too. With in-kind reimbursement, on the contrary, treatment quantity is observable. This follows from the interpretation of in-kind transfer ¹Xij as an indivisible package of care (see section 3).
The social planner’s programs addressed in this section are standard cases of mechanism design under adverse selection. Looking for the optimal mechanism of each reimbursement scheme, I will then employ the well known Revelation Principle28. 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 the relevant 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. This means 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 cost reimbursement as well. I will come back to these considerations later (see note (31)).
not necessarily allow to obtain the second-best optimum. The reason is obviously that 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 the mimickers. Standard mechanism design techniques with discrete types (see Fun-denberg and Tirole (1991), pages 246-250) show that it is optimal to make the mimikers’ incentive compatibility constraints binding thus implying that all the other 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: one for the low-type not seriously ill consumers and another for the low-type seriously ill 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 insurance o¤ers the following contracts: (P; L1) for healthy consumers, the couple of con-tracts ¡Ll 2; Rl2 ¢ and ¡Lh 2; Rh2 ¢
respectively for low and high-type not seriously ill consumers and the uniform transfer R3 for both seriously ill consumers’ types as, in this case, the only variable the social planner can control is the monetary transfers 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, only low-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):
X2¤j : µj2Á02(X2) ¡ qu0 ¡ w2Lj2+ R j 2¡ qX2 ¢ = 0 (6.1) X3¤j : µj3Á03(X3) ¡ qu0 ¡ Rj3¡ qX3 ¢ = 0 (6.2)
while the mimicker will choose the preferred quantity according to the following equation: X2¤lh: µ l 2Á02(X2) ¡ qu0 ¡ w2Lh2 + R h 2 ¡ qX2 ¢ = 0 (6.3)
29A formal proof of this result is standard and then omitted. A complete proof is available
The social planner program then is: 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : M ax P;Rj2;R3;Lji p1[u(w1L1¡ P ) ¡ v(L1)] + +p2 X j=l;h ¹j£u¡w2Lj2+ R j 2¡ qX2¤j ¡ µj2; Lj2; R2j¢¢¡ H2+ µj2Á2 ¡ X2¤j¡µj2; Lj2; Rj2¢¢+ ¡v¡Lj2 ¢¤ + p3 X j=l;h ¹j £ u¡R3¡ qX3¤j ¢ ¡ H3+ µj3Á3 ¡ X3¤j ¢¤ s:t: : p1P = p2 ¡ ¹lRl2+ ¹hRh2 ¢ + p3R3 (°) u¡w2Ll2+ R2l ¡ qX2¤l ¡ µl2; Ll2; Rl2 ¢¢ + µl2Á2 ¡ X2¤l ¡ µl2; Ll2; Rl2 ¢¢ ¡ v(Ll2) ¸ u¡w2Lh2 + Rh2 ¡ qX2¤lh ¡ µl2; Lh 2; Rh2 ¢¢ + µl2Á2¡X¤lh 2 ¡ µl2; Lh 2; R2h ¢¢ ¡ v(Lh2) (¸) where ° 6= 0 and ¸ ¸ 0 are respectively the budget constraint Lagrange multiplier and 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 is such that marginal bene…t equals marginal cost of labor supply.
>From FOC with respect to Rl
2 one …nds: p2¹l+ ¸ p2¹l u0¡C2l ¢ ¡ ° = 0 (6.6)
such that, using equation (6.4), C2l > C1. >From FOC with respect to Ll
2 it follows: w2u0 ¡ C2l ¢ ¡ v¡Ll2 ¢ = 0 (6.7)
and there is no distortion for the low-type not seriously ill consumer. >From FOC with respect to Rh
2 one …nds: u0¡C2h ¢ ¡ p¸ 2¹h u0¡C2lh ¢ ¡ ° = 0 (6.8) where Clh
2 is the mimicker’s aggregate consumption. From equations (6.4) and
(6.8) it follows that Ch 2 < C1.
>From FOC with respect to Lh 2 one …nds: w2u0 ¡ C2h ¢ ¡ v¡Lh2 ¢ w2u0 ¡ Clh 2 ¢ ¡ v¡Lh 2 ¢ ¡ p¸ 2¹h = 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) shows that Ll
2 < Lh2: Concerning high-type distortion, for the (6.3) the mimicker will choose Xlh
2 < X2h; then C2lh > C2h: As a consequence, (6.9) does not contradict that p2¹h ¡ ¸ > 0 only if w2u0 ¡ Clh 2 ¢ ¡ v¡Lh 2 ¢ < w2u0 ¡ Ch 2 ¢ ¡ v¡Lh 2 ¢ < 0, and then w2u0 ¡ Ch 2 ¢ < v¡Lh 2 ¢
: This means that high-type not seriously ill consumer is forced to supply too much labor and to under-consume (with respect to aggregate consumption).
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 higher treatment 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 X3h
Note that the two point in …gure 4 are incentive compatible (no type prefers the consumption bundle of the other type) but, obviously, no one of the two incentive constraints is binding (recall that in the seriously ill state no separation can be obtained because a unique instrument can be used, the monetary transfer). This anticipates 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; Rl2 ¢ and ¡Lh2; Rh2 ¢ verify Ll
2 < Lh2 and C2l > C2h: There is no distortion for low-type consumer. On the contrary high type consumer is forced to supply too much labor and to consume too less aggregate consumption ( w2u0(C2h) < v0(Lh2)). Both seriously ill types receive a monetary transfer R3 and choose treatment and aggregate consumption such that Xl
3 < X3h and C3l > C3h:
6.2. In-kind reimbursement
In-kind reimbursement is characterized by …ve monetary transfers (P; R2j; Rj3); by consumers’ labor supplies (Lji) and by the transfers ¹Xij; i = 2; 3; j = l; h. As I an-ticipated in section 3, it is reasonable to assume that the in-kind transfer ¹Xij is an indivisible package of care30 such that the social planner can observe treatment consumption. As a consequence, with in-kind reimbursement, aggregate con-sumption, treatment quantity and labor supply are all observable. The contracts proposed in the three states then are (C1; L1) ; (C2l; X2l; Ll2); (C2h; X2h; Lh2); (C3l; X3l) and (C3h; X3h). It is interesting to notice that in-kind represents the unconstrained direct mechanism in the sense that, given the agent’s type announcement, all the relevant variables are chosen by the social planner. As a consequence we can an-ticipate that the in-kind optimal allocation corresponds to the allocation which weakly dominates the others.
30Letting consumers choose the preferred treatment quantity under the constraints Xj i · ¹X
j 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 Cij is observable.
The social planner’s program then is31 8 > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > : M ax Cij;Lji; ¹Xij p1[u(C1) ¡ v(L1)] + +p2 X j=l;h ¹j£u¡C2j¢¡ H2+ µj2Á2 ¡ ¹ X2j¢¡ v¡Lj2¢¤+ +p3 X j=l;h ¹j£u¡C3j¢¡ H3+ µj3Á3 ¡¹ X3j¢¤ s:t: : p1(w1L1¡ C1) = p2 X j=l;h ¹j¡C2j ¡ w2Lj2 ¢ + p3 X j=l;h ¹jC3j+ +p2 X j=l;h ¹jq ¹X2j + p3 X j=l;h ¹jq ¹X3j (°) u¡Cl 2 ¢ + µl2Á2¡X¹l 2 ¢ ¡ v¡Ll 2 ¢ ¸ u¡Ch 2 ¢ + µl2Á2¡X¹h 2 ¢ ¡ v¡Lh 2 ¢ (¸2) u¡C3l ¢ + µl3Á3 ¡¹ X3l ¢ ¸ u¡C3h ¢ + µl3Á3 ¡¹ X3h ¢ (¸3)
where ° 6= 0 and ¸2;¸3 ¸ 0 respectively are the budget constraint Lagrange multiplier 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 Cl
2 it follows: p2¹l+ ¸2 p2¹l u0¡C2l ¢ ¡ ° = 0 (6.11)
and then (6.4) and (6.11) imply Cl
2 > C1: From FOC with respect to Ll2 : p2¹l+ ¸2
p2¹l
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 distorequa-tion for low-type consumer con-cerning labor supply. Finally from FOC with respect to ¹Xl
2 one …nds: p2¹l+ ¸2 p2¹l µl2Á02 ¡¹ X2l ¢ ¡ q° = 0 (6.13)
31The monetary transfers Rj
>From (6.11) and (6.13) it follows that µl2Á02¡X¹l 2 ¢ ¡qu0¡Cl 2 ¢ = 0: As a consequence there is no distortion for low-type consumer concerning treatment quantity.
Regarding high-type consumers, from FOCs with respect to C2h : p2¹h¡ ¸2
p2¹h
u0¡C2h¢¡ ° = 0 (6.14)
Comparing the previous equation to (6.4) one …nds Ch 2 < C1: FOCs with respect to Lh2 and ¹X2h respectively yield:
p2¹h¡ ¸2 p2¹h v0¡Lh2¢¡ w2° = 0 (6.15) p2¹hµ h 2Á02 ¡ ¹ X2h¢¡ ¸2µl2Á02 ¡¹ X2h¢¡ p2¹hq° = 0 (6.16)
Comparing (6.15) to (6.5) it is evident that Lh 2 > L1: While from (6.14) and (6.15) it follows that w2u0
¡ Ch 2 ¢ ¡ v¡Lh 2 ¢ = 0: This im-plies that there is no distortion for high-type not seriously ill consumer concerning labor supply.
I showed that Ch
2 < C2l and Lh2 > Ll2; as a consequence, it must be Rh2 < Rl2: Moreover, from the binding incentive constraint it follows ¹Xh
2 > ¹X2l: Solving (6.14) and (6.16) together yield to µh2Á02¡X¹h
2 ¢ < qu0¡Ch 2 ¢ (see Stiglitz (1987) page 1005). This means that high-type not seriously ill consumer is forced to consume too much treatment and too less aggregate consumption.
Concerning seriously ill consumers, from FOC with respect to Cl
3 one …nds: p3¹l+ ¸3
p3¹l
u0¡C3l¢¡ ° = 0 (6.17)
such that, comparing with (6.4), C3l > C1 holds. From FOC with respect to ¹X3l one …nds:
p3¹l+ ¸3 p3¹l
µl3Á03¡X¹3l¢¡ q° = 0 (6.18)
>From (6.17) and (6.19) it follows that µl3Á03¡X¹l 3 ¢ ¡qu0¡Cl 3 ¢ = 0: As a consequence there is no distortion for low-type consumer concerning treatment quantity.
>From FOC with respect to C3h : p3¹h¡ ¸3
p3¹h
Comparing the previous equation to (6.4) it follows that Ch 3 < C1: As before, recalling that Ch
3 < C3l; from the binding incentive constraint it must be ¹X3h > ¹X3l:
FOC with respect to ¹Xh
3 yields: p3¹hµh3Á03 ¡ ¹ X3h¢¡ ¸3µl3Á03 ¡¹ X3h¢¡ p3¹hq° = 0 (6.20)
Solving (6.19) and (6.20) together yield to µh3Á03 ¡¹ X3h ¢ < qu0¡C3h ¢ (again as in Stiglitz (1987) page 1005). This means that high-type seriously ill consumer is forced to consume too much treatment and too less aggregate consumption. Seri-ously ill consumers’ self-selecting allocation in the case of in-kind reimbursement is represented in the following …gure:
Uh 3 Ul 3 Xl 3 Xh3 Ch 3 Cl 3 Xj3 Cj 3
Figure 5: seriously ill consumers’ self-selecting allocation with in-kind reimbursement.
Note that in-kind self-selecting allocation devotes more resources to high-type than 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 (C2l; ¹X2l; Ll2) and (C2h; ¹X2h; Lh2) verify C2l > C2h; ¹X2l < ¹X2h and Ll2 < Lh2:0 The monetary transfers are such that Rl
2 > R2h: There is no distortion for low-type consumer. High low-type consumer is forced to consume too much treatment and too less aggregate consumption ( µh2Á02( ¹X2h) < qu0(C2h)). Contracts (C3l; X3l) and
(Ch
3; X3h) verify C3l > C3h and ¹X3l < ¹X3h: There is no distortion for low-type con-sumer and high type concon-sumer is forced to consume too much treatment and too less aggregate consumption ( µh3Á03( ¹X3h) < qu0(C3h)).
It is important to notice that in-kind reimbursement exactly corresponds to the direct mechanism in this adverse selection setting. Consumers announce their type and receive the second-best (due to asymmetric information) allocation. All the other relevant decisions are taken by the social planner. Interestingly, I shall show in the next section that treatment cost reimbursement turns out to be an indirect mechanism with which the social planner is able to implement the very same 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, (Ll
2; Rl2; ®l2) and (Lh2; Rh2; ®h2) respectively for low and high-type not seriously ill consumers and …nally (Rl
3; ®l3) and (R3h; ®h3) respectively for low and high-type seriously ill consumers. Contrary to the case of cash reimbursement, here also the low-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):
X2¤j : µ j 2Á02(X2) ¡ ® j 2qu0 ¡ w2L j 2+ R j 2¡ ®jqX2 ¢ = 0 (6.21) X3¤j : µ j 3Á03(X3) ¡ ® j 3qu0 ¡ Rj3¡ ® j 3qX3 ¢ = 0 (6.22)
while the mimickers will choose the preferred treatment quantity according to the following equations32:
32In the Optimal Taxation literature, a standard assumption is that only linear commodity
taxes are implementable because transactions are anonymous. This means that the quantity consumed 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 consequence treatment quantity is ex-post veri…able. Then, with this reimbursement method, the linear taxation (subsidization) used by the insurer (the parameters ®ji) corresponds to an ad hoc restriction of the insurance instruments. Anyway, in the real word, the Health Authorities frequently use linear subsidization of treatment.
X2¤lh : µl2Á02(X2) ¡ ®h2qu0 ¡ w2Lh2 + R h 2 ¡ ® h 2qX2 ¢ = 0 (6.23) X3¤lh: µ l 3Á03(X3) ¡ ®h3qu0 ¡ Rh3 ¡ ® h 3qX3 ¢ = 0 (6.24)
The social planner’s program then is 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : M ax P;Rji;®ji;Lji p1[u(w1L1¡ P ) ¡ v(L1)] + +p2 X j=l;h ¹j £ u¡w2L j 2+ R j 2¡ ® j 2qX2¤j ¡ µj2; L j 2; R j 2; ® j 2 ¢¢ ¡ H2+ +µj2Á2¡X2¤j¡µj2; Lj2; R2j; ®j2¢¢¡ v(Lj2)¤+ +p3 X j=l;h ¹j £ u¡RC3 ¡ qX3¤j ¡ µj3; R j 3; ® j 3 ¢¢ ¡ H3+ µ j 3Á3 ¡ X3¤j ¡ µj3; R j 3; ® j 3 ¢¢¤ s:t: : p1P = p2 £ ¹l ¡ Rl2+ ¡ 1 ¡ ®l2 ¢ qX2¤l ¢ + ¹h ¡ Rh2 + ¡ 1 ¡ ®h2 ¢ qX2¤h ¢¤ + +p3 £ ¹l ¡ R3l + ¡ 1 ¡ ®l3 ¢ qX3¤l ¢ + ¹h ¡ Rh3 + ¡ 1 ¡ ®h3 ¢ qX3¤h ¢¤ (°) u¡w2Ll2+ Rl2¡ ®l2qX2¤l ¡ µl2; Ll2; Rl2; ®l2 ¢¢ + µl2Á2 ¡ X¤l 2 ¡ µl2; Ll2; Rl2; ®l2 ¢¢ ¡ v(Ll 2) ¸ u¡w2Lh2 + Rh2 ¡ ®h2qX2¤lh ¡ µl2; Lh 2; R2h; ®h2 ¢¢ + µl2Á2¡X¤lh 2 ¡ µl2; Lh 2; Rh2; ®h2 ¢¢ ¡ v(Lh 2) (¸2) u¡Rl 3¡ ®l3qX3¤l ¡ µl3; Rl 3; ®l3 ¢¢ + µl3Á3 ¡ X¤l 3 ¡ µl3; Rl 3; ®l3 ¢¢ ¸ u¡Rh 3 ¡ ®h3qX3¤h ¡ µl3; Rh 3; ®h3 ¢¢ + µl3Á3¡X¤lh 3 ¡ µl3; Rh 3; ®h3 ¢¢ (¸3)
Where ° 6= 0 and ¸2;¸3 ¸ 0 respectively are the budget constraint Lagrange multiplier 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), Cl
2 > C1 holds. Moreover, from FOC with respect to Ll2 one …nds again equation (6.7): there is no distortion for low-type not seriously ill consumer concerning labor supply.
>From FOC with respect to ®l
2 it follows: X2l £ p2¹lu0 ¡ C2l ¢ + ¸2u0 ¡ C2l ¢ ¡ °p2¹l ¤ + °p2¹l @Xl 2 @®l 2 ¡ 1 ¡ ®l2 ¢ = 0 (6.25)
such that, substituting (6.6), ®l
2 = 1 holds: low-type not seriously ill consumer’s treatment price is not distorted.
>From FOC with respect to Rh
2 equation (6.8) follows, such that, using the (6.4), Ch
>From FOC with respect to Lh
2 one …nds again equation (6.9). As in session 7.2, it follows that Ll 2 < Lh2: Moreover w2u0 ¡ Ch 2 ¢ < v¡Lh 2 ¢
holds. That is, also with non-uniform treatment cost, high-type not seriously ill consumer supplies too much labor and consumes too less aggregate consumption.
Rearranging together FOCs with respect to ®h
2 and Rh2 it follows: ®h2 = 1 + ¸2u 0¡Clh 2 ¢ ¡ Xh 2 ¡ X2lh ¢ °p2¹h @Xh 2 @®h 2 (6.26) where @X2h @®h
2 < 0; ° > 0 and, from (6.21) and (6.23), X
h
2 ¡ X2lh > 0: (6.26) shows that ®h
2 < 1 : high type not seriously ill consumer’s treatment price is subsidized. Concerning seriously ill consumers, from FOC with respect to Rl
3 one …nds
equation (6.17), such that, again, Cl
3 > C1: Moreover, FOC with respect to ®l3 yields: X3l £ p3¹lu0 ¡ C3l ¢ + ¸3u0 ¡ C3l ¢ ¡ °p3¹l ¤ + °p3¹l @X3l @®l 3 ¡ 1 ¡ ®l3 ¢ = 0 (6.27)
such that, substituting (6.17), ®l
3 = 1 holds: low-type seriously ill consumers’ treatment price is not distorted.
While, from FOC with respect to Rh
3 one …nds:
u0¡C3h¢¡ ¸3 p3¹h
u0¡C3lh¢¡ ° = 0 (6.28)
Comparing the previous equation to (6.4), one …nds C3h < C1. Finally, from FOCs with respect to ®h
3 and Rh3 together it follows:
®h3 = 1 + ¸3u0 ¡ Clh 3 ¢ ¡ Xh 3 ¡ X3lh ¢ °p3¹h @Xh 3 @®h 3 (6.29) where @X3h @®h
3 < 0; ° > 0 and, from (6.22) and (6.24), X
h
3 ¡ X3lh > 0: (6.29) shows that ®h
3 < 1 : high-type seriously ill consumer’s treatment price is subsidized. Totally di¤erentiating (6.22) it is easy to verify that dRj3
d®j3 > 0: As a consequence Rl
3 > Rh3:
Seriously ill consumers’ self-selecting allocation in the case of treatment cost reimbursement is represented in …gure 6 by the points A and B. In the …gure
low-type consumer is indi¤erent between the allocation he can reach with the budget constraint de…ned by ¡Rl
3; ®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 cost reimbursement.
Proposition 7 summarizes the previous results.
Proposition 7 The optimal treatment cost self-selecting allocation is such that: contracts (Ll
2; Rl2; ®l2) and (Lh2; Rh2; ®h2) verify Ll2 < Lh2; C2l > C2hand ®h2 < ®l2 = 1: There is no distortion for low-type consumer. On the contrary treatment consumed by high-type consumer is subsidized and high type consumer is forced to supply too much labor and to consume too less aggregate consumption (w2u0(C2h) < v0(Lh2)): Contracts (Rl
3; ®l3) and (Rh3; ®h3) verify C3l > C3h; Rl3 > 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 the social planner has decided the parameters of the insurance contract. Considering seriously ill consumers and looking at …gures 5 and 6 we see that with treatment cost 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
