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Optimal social health insurance with supplementary private insurance
Journal of Health Economics 18 Ž1999. 727–745 www.elsevier.nlrlocatereconbase
Optimal social health insurance with supplementary private insurance Alessandro Petretto
)
Department of Economics, UniÕersity of Florence, UniÕersity ‘‘L. Bocconi’’, Milan and Public Expenditure Commission, the Treasury, Rome, Italy Received 13 July 1998; received in revised form 6 January 1999; accepted 22 March 1999
Abstract This paper investigates the structure of a National Health Service in which there is compulsory social insurance covering a package of essentials, a given part of individuals’ health expenditure, and supplementary private policy topping up the remaining services. The latter insurance contract provides for a co-payment by patients, limiting the so-called ‘‘third-party payer’’ effect. Thus, an individual’s health expenditure is divided into three parts: the first covered by social insurance, the second by a private policy and the third out-of-pocket. Such mixed system design has received increasing attention in recent years and has been adopted by several industrialized countries. The conditions for optimal rates of social insurance coverage and of private coinsurance are analysed and discussed. The optimality requirements refer to efficiency as well as equity concerns. q 1999 Elsevier Science B.V. All rights reserved. JEL classification: I18; G22; H21 Keywords: Social insurance; Ex post moral hazard; Risks sharing and optimal redistribution
1. Introduction If we consider the principal trends in the reform of health care systems in industrialised countries we observe a shift from publicly financed systems to )
Corresponding author. Tel.: q00-39-5527-10415; fax: q00-39-5527-10424; E-mail: [email protected] 0167-6296r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 2 9 6 Ž 9 9 . 0 0 0 1 7 - X
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A. Petrettor Journal of Health Economics 18 (1999) 727–745
mixed ones, with significant patient co-payments for services ŽBesley and Gouveia, 1994; Zweifel and Breyer, 1997.. The problem investigated in this paper refers to the case of compulsory social insurance covering a package of essentials Ža given part of health expenditures. and a supplementary private policy topping up the remaining services, with co-payments by patients. In this way an individual’s health expenditure is divided into three parts: the first covered by social insurance, the second by a private policy and the third out-of-pocket. Examples often considered in literature include the U.S. Medicare plan, in which private insurers offer supplementary ‘‘Medigap’’ plans, and the Australian Medicare, in which supplementary private insurance contracts are offered by Registered Private Health Funds. Similar funding schemes have been applied during the recent reform of National Health System in Italy, where the task of organising a system of supplementary Private Health Funds has been delegated to regions and municipalities. In many institutional contexts individual contributions to social insurance are income-dependent Ža payroll tax or an income tax. and the integrative private policy applies a differentiated premium plus a coinsurance rate in order to limit the so-called ‘‘third party payer’’ effect. In this paper we propose to merge two streams of literature to facilitate study of such a mixed system from a normative perspective: the first focuses on the classical problem of trade-off between gain from risk-sharing and the deadweight loss of ex post moral hazard ŽSpence and Zeckauser, 1970; Pauly, 1974; Besley, 1988; Blomqvist, 1997.; 1 the second deals with combining the redistributive roles of income taxation and social insurance ŽBlomqvist and Horn, 1984; Rochet, 1991; Cremer and Pestieau, 1996.. Recently, Blomqvist and Johansson Ž1997. have also considered a mixed publicrprivate health insurance system, but they have not modelled an income-dependent contribution to social insurance with a variable supply of labour. They have raised, following the example of Besley Ž1989. and Selden Ž1993., an interesting debate on the efficiency of a mixed system with respect to a purely private competitive one Žsee also Selden, 1997.. In our case, preference for an institutional system with respect to mixed insurance is taken for granted, thus we aim to investigate its optimal structure in terms of efficiency and equity. The social insurance system we refer to is essentially compulsory and universal, although it provides only for partial coverage, and does not allow low-risks or rich people to opt-out of the NHS. 2 However, since flexibility may lead to greater efficiency, we outline some feasible extensions of the model in order to present some alternative solutions. 1 On empirical aspects of the trade-off between the gain from risk-sharing and the deadweight loss from ex post moral hazard see Manning and Marquis Ž1996.. For an analysis of taxation benefit and private health insurance see Jack and Sheiner Ž1997. and for the combination of optimal insurance and optimal provider contracts see Ma and McGuire Ž1997.. 2 Regarding the characteristics of such a social insurance system, see Barr Ž1998, p. 125..
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As to the ‘‘third-party payer’’ effect, we recall that we are speaking of ex post moral hazard Ži.e., once nature has determined the state of the world. because the insurer Žpublic as well private. is unable to determine the health status of the insured with certainty. Health care expenditure itself serves as an indicator of illness, causing benefits to depend on expenditure by means of a specific function. For the insured there is no possibility of manipulating the risk of illness, as in the ex ante moral hazard case, but there is a choice between various alternative treatments once an illness has occurred. 3 We may furthermore assign a redistributive role to social health insurance given that compulsory membership offers mandatory coverage for all, along with a given fraction of potential losses, thus acting as a redistribution in kind. In order to develop these concepts, in Section 2 we construct a two-state-of-the world model of health insurance. In Section 3 we describe the solution to the model according to a three-stage maximization process. In Section 4 we study the individual’s equilibrium co-payment formula for the supplementary private policy; in Sections 5 and 6 we propose and discuss the optimal social insurance coverage conditions. Section 7 contains some concluding comments.
2. The model We may generally distinguish two states of the world Žhealthy and sick. and surmise that n individuals are identified by a pair of parameters: wages Žskills. and probability of sickness: 4 wi , pi 4 , i s 1, . . . n. This approach is quite general: it could for example represent a situation with two risk classes Žwith one type needier than the other, as far as the consumption of health services is concerned. and many productivity parameter values. If this is the case, no hypothesis is made about the relative level of skill Žwage rate., so it may be that high skill parameters are associated with higher risk and vice versa. Moreover, while the probabilities pi ) 0, i s 1, . . . n, are known Žno adverse selection in this respect. by the insurance companies and by the government, skill parameters wi are not. 5 Therefore, the government is not able to apply a first best optimal taxation linked to this specific characteristic. As in the familiar optimal 3
In reality, treatments are only indirectly chosen by patients since the true choice is made by the physician acting as the best-informed agent in a principal–agent relationship ŽMooney and Ryan, 1993; Zweifel and Breyer, 1997, Chap. 7.. 4 In order to characterise the type of individual i we use subscripts: e.g., x i . When using the familiar short form for denoting partial derivatives with a subscript, we make use of two letters; hence x i q denotes the partial derivative of x with respect to q for individual i. 5 This asymmetry in the insurer’s information is clearly assumed for simplicity. In this respect we are following Žand generalising. the framework by Rochet Ž1991. and Cremer and Pestieau Ž1996.. For a clear discussion of government information on individuals’ risk and skill see Blomqvist and Horn Ž1984..
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income taxation context, only individual labour income Žthe expected value in our framework. is taken into account by the government for modelling the taxation function ŽAtkinson and Stiglitz, 1980.. Furthermore, we implicitly surmise that the government is also unable to differentiate the income taxes imposed on individuals according to their risk of illness. 6 We may now imagine a state-contingent concave utility function for individual i as follows: u ih s u ih Ž c ih ,0, Lhi . ,
Ž 1a .
when healthy, and u is s u is Ž c is , Hi , Lsi . ,
Ž 1b .
when sick. Hi represents the health service expenditure and L ij, c ij, j s h, s, represent labour and consumption in the two states. Hi is observable only ex post, i.e., only after the payment has been made Žex post moral hazard.. We consider this expenditure to differ among individuals due to the different coinsurance rates chosen by each consumer-type. Furthermore, we have two budget constraints c ih s wi Lhi y T Ž wi Lhi . y Pi
Ž 2a .
c is s wi Lsi y T
Ž 2b .
Ž
wi Lsi
. y Pi q A i q Bi y Hi
where T Ž.. is the payroll tax function, Pi is the premium of the private health insurance; A i and Bi are the benefits of social and private insurance, respectively. If we denote the coverage Žreimbursement. rate of social insurance with a and the private coinsurance rate with k i , we obtain the following expressions for benefits Žassumed to be linear for simplicity. and premiums A i s a Hi
Ž 3.
Bi s Ž 1 y a . Ž 1 y k i . Hi
Ž 4.
Pi s pi Ž 1 q d . Bi s Ž 1 y a . Ž 1 y k i . Ž 1 q d . pi Hi s p i Hi .
Ž 5.
The function of public benefits ŽEq. Ž3.. contains an implicit constraint, i.e., a i s a all i. If instead a type-specific social insurance coverage were available, this might mean that, at the optimum, a j s 0, for some j, and a i ) 0, for the others. Thus, this would present the possibility of excluding some share of the population from NHS coverage — that is, those individuals able to privately insure all risks. However, the uniformity constraint a i s a all i is not imposed in 6 In fact this opportunity should be considered since we are assuming that probabilities are known and also since in a Second Best framework it is efficient to use all available information to design the optimal taxation structure. For a discussion of this opportunity see Blomqvist and Horn Ž1984..
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Eq. Ž3. exclusively to simplify analysis, given that it may involve an objective of universality defined, as in some European countries, at the Constitutional level ŽBarr, 1998.. The private insurance premium ŽEq. Ž5.. is proportional to individual benefit and subsequently to health expenditure according to the ‘‘price’’ p i ; d is a loading factor, which, in case of fairness, is d s 0 ŽZweifel and Breyer, 1997, p. 163.. Furthermore, at aggregate level, we observe the following government budget constraint:
aÝ pi Hi s ÝE T Ž wi L i . 4 i
Ž 6.
i
which shows that the expected public expenditure for the NHS is financed by the expected revenues from pay-roll taxation. 7 If, for simplicity, we assume a linear payroll tax T Ž wi L ij . s twi L ij y M ,
j s h, s
Ž 7.
the government budget constraint ŽEq. Ž6.. becomes
aÝ pi Hi s tÝwi L i y nM i
Ž 8.
i
which may be, finally, reduced to the more compact one:
a H s tY y M where we use the notations: H s Ž1rn.Ý i Hi , Hi ' pi Hi q Ž1 y pi .0, L i s pi Lsi q Ž1 y pi . Lhi , Yi ' wi L i , Y ' Ž1rn.Ý i Yi . The lump sum transfer M, which may be positive as well negative Ža poll-tax., is meant to represent the usual non-proportional structure of payroll-tax Žoften regressive.. 7 If we pursued the idea of allowing differentiated rates of social coverage we would have the following budget constraint:
Ýa i pi Hi sÝE T Ž wi Li . 4 i
i
This means that all individuals Žalso those excluded from the NHS. contribute by paying income tax. If, instead, we want to consider reducing the tax liability of excluded persons when they obtain approved substitute private insurance, we have to add a further constraint such as: Tj sT8 Žwith T8 possibly 0. as a j s 0. We must note that this is not a flexible ‘‘opt-out system’’ because the choice to be ‘‘in’’ or ‘‘out’’ is not made individually but decided by the government on the basis of personal incomes; a social insurance system like this one is now applied is Germany. For an analysis of general models of redistribution in-kind in which the provision of public and private services are mutually-exclusive, see Blomquist and Christiansen Ž1998. and Boadway et al. Ž1998..
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By substituting Eqs. Ž3. – Ž5. into Eqs. Ž2a. and Ž2b., and taking account of Eq. Ž7., we easily obtain the following expressions for the individual’s budget constraints: c ih s wi Ž 1 y t . Lhi q M y p i Hi c is s wi
Ž1yt .
Lsi q M y p i Hi y qi Hi
Ž 9a . Ž 9b .
where qi s Ž1 y a . k i is the relevant ‘‘price’’ for the choice of H by the individual i when ill Žtaking the private insurance premium as given.; it is the individual unit cost of health service out-of-pocket consumption. 8 Four cases of special interest may be considered. 1. If a s 0 Žno social insurance. we have qi s k i , p i s Ž1 y k i .Ž1 q d . pi . If, moreover, we put k i s 0 Žfull private insurance, i.e., no coinsurance. and d s 0 Ž‘‘fair premium’’., we obtain the canonical result qi s 0, p i s pi . 2. If a s 1 Žfull social insurance coverage, and no supplementary private insurance. we have qi s 0 and p i s 0. This is the case of a purely public NHS. 3. If 0 - a - 1 and k i s 0 Žprivate ‘‘full coverage’’ insurance, i.e., no patient co-payment. we have qi s 0 and p i s Ž1 y a .Ž1 q d . pi . 4. If 0 - a - 1 and k i s 1 Žpartial social insurance coverage, without a supplementary private policy. we have qi s Ž1 y a . and p i s 0. This case may arise if loading is included in the premium formula, since individuals may consider zero private coverage to be convenient. In general, as we shall see, the mixed system is characterised by 0 - a - 1 and 0 - k i - 1, i s 1, . . . n. It is ultimately clear that the model applies to a system in which social insurance covers all risks, with private insurance paying for the part of health expenditure not covered by social insurance. In order to consider the interesting case of social and private insurance covering different risks, it would have been necessary to construct a multiple-risks model ŽVon der Schulenburg, 1995. which, however, would have much complicated the analysis.
3. A three-stage maximization problem for mixed publicr r private insurance decision-making In our model the decision process runs in three steps: Ži. in step 1, the government chooses social insurance coverage a and the structure of taxation Ž t and M .; Žii. in step 2, each consumer chooses a private insurance contract and subsequently a coinsurance rate k i ; Žiii. in step 3, the consumer chooses, in the 8
The cost of treatment proves to be in some way subsidised by social and private insurance because of the two payment systems Žtaxes vs. premiums.. This determines a loss of efficiency, as it prevents the total insurance premium from being risk specific and therefore causes moral hazard. However, this loss is a somehow unavoidable consequences of the chosen Second Best framework.
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event of illness, the labour supply and health expenditure and, in the event of healthy status, only the labour supply Žconsumption in both conditions takes place as permitted by contingent budget constraints.. Thus, the decisions regarding structures of social and private insurance contracts are ex ante decisions following an expected utility maximization framework, while decisions concerning commodity bundles by the two consumers are instead ex post decisions which consider as given the terms of the insurance contracts. The model is solved, using game-theory terminology, by ‘‘backward induction’’, requiring passage through the following three-stage maximization process. 3.1. First stage Consumer i chooses Lhi if healthy and the bundle Ž Lsi , Hi . if ill, given a , k i , t and M; the choices in sick status are made while taking private insurance premiums as given. Consumption levels in the two states arise from Eqs. Ž9a. and Ž9b.. This maximisation process gives the following ex post decisions for i: Hi Ž qi ,wi Ž 1 y t . , M . s Arg max H u i Ž wi Ž 1 y t . Lsi q M y p i Hi yqi Hi , Hi , Lsi .
Ž 10 .
with F.O.C., ŽEu is .rŽEHi . s qi g is, where g is s ŽEu is .rŽEc is . is the marginal utility of income ŽMUI. when ill. Lsi Ž qi ,wi Ž 1 y t . , M . s Arg max L u i Ž wi Ž 1 y t . Lsi q M y p i Hi yqi Hi , Hi , Lsi . with F.O.C.
Ž 11 .
yŽEu is .rŽELsi . s wi Ž1 y t .g is.
Lhi Ž wi Ž 1 y t . , M . s Arg max L u i Ž wi Ž 1 y t . Lhi q M y p i Hi ,0, Lhi .
Ž 12 .
with F.O.C. yŽEu ih .rŽELhi . s wi Ž1 y t .g ih , where g ih s ŽEu ih .rŽEc ih . is the MUI when healthy. 3.2. Second stage Consumer i chooses his or her individual k i taking into account the supply and demand functions for the equilibrium values Ž10., Ž11., Ž12.: Hi Ž.., Lsi Ž.. and Lhi Ž... In other words, the second stage ex ante decision for k i derives from the maximization of the following indirect utility function
n i Ž a ,k i ,t , M . s pi u i wi Ž 1 y t . Lsi Ž . . q M y p i Hi Ž . . yqi Hi Ž . . , Hi Ž . . , Lsi Ž . . q Ž 1 y pi . u i wi Ž 1 y t . Lhi Ž . . qM y p i Hi Ž . . ,0, Lhi Ž . .
Ž 13 .
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3.3. Third stage The government chooses t, M and a , given each individual decision for k i , and the functions Hi Ž.., Lsi Ž.. and Lhi Ž.., i s 1, . . . n. Therefore, the third stage ex ante decisions on a , t and M arise from the maximization of a welfare function, which we assume of the utilitarian type, 9 W s Ýn i Ž a ,kUi ,t , M .
Ž 14 .
i
subject to the government budget constraint ŽEq. Ž8.. and to the first-order condition of the previous stage-maximization for kUi .
4. The individual’s equilibrium private coinsurance rate It is useful, in order to solve the second-stage maximization problem, to state some preliminary comparative statistical results with the following Proposition 1. Eqi Ži. s Ž1 y a .; Ek i Ep i Žii. s yŽ1 y a .Ž1 q d . pi ; Ek i EHi Žiii. ' Hi k s ŽEHirEqi .ŽEqirEk i . s Hi q Ž1 y a .; Ek i En i Ž iv . ' n i k s y Ž g i s y g i Ž 1 q d .. H i Ž 1 y a . y g i p i H i k , Eki where g i ' pi g is q Ž1 y pi .g ih. Proof. Ži. and Žii. derive from straightforward direct computation, while Žiii. arises from Eq. Ž10.. Furthermore, by differentiating Eq. Ž13. with respect to k i and then applying the Envelope theorem, we obtain this modified Roy identity:
y i k s ypi g is Ž Ž EqirEk i . q Ž Ep irEk i . . Hi y pi g is Ž qi q p i . Hi k q pi g isqi Hi k y Ž 1 y pi . g ih Ž Ep irEk i . Hi y Ž 1 y pi . g ihp i Hi k Thus, by substituting and rearranging the terms, we reach Živ..
B
9 We refer to this type of welfare function only for simplicity; any concave Bergson–Samuelson welfare function works equally well. Of course we are following a strictly normative approach with its usual shortcomings and hitches stressed by Public Choice literature. In particular, the implicit hypothesis of ‘‘benevolent’’ and ‘‘welfarist’’ social decision-making, given the complexity of the actual context, seems to be quite strong.
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Discussion With 0 - a - 1, Ži. and Žii. of Proposition 1 are positive and negative, respectively. By assuming health service to be a normal good Žwith a decreasing demand schedule., the partial derivative in Žiii. is negative; that is, the standard result by which an increase of the coinsurance rate Žthe ‘‘price’’. reduces health expenditure. According to Živ., the net welfare loss for individual i due to an increase of the coinsurance rate has two conflicting components. The first is related to the greater expected cost of risk deriving from the health expenditure not covered by social insurance; the term is negative if the insurance premium may be considered ‘‘fair’’ Ži.e., the loading factor d is negligible., as g is ) g i Žthe MUI in illness is greater than expected MUI.. The second component represents the expected gain from the reduction of private insurance premiums due to the health expenditure decrease; the term, given the normality of H, is positive. We now solve the private insurance problem in the second step, when each consumer i makes herrhis own choices. Proposition 2. In the described mixed insurance system, the optimal priÕate coinsurance rate kUi , i s 1, . . . n, is giÕen by the condition
Ž i.
Ž1yki .
where qi '
ki
s qire i
g is y g i Ž 1 q d . gi Ž 1 q d .
Ž 15 . , e i ' yŽEHirEqi .Ž qirHi . ) 0, or by the condition
Ž ii . Covp Ž g i , Hi . s yg i Ž 1 y k i . Ž 1 q d . Hi k qdg i Hi
Ž 16 .
where Covp Žg i , Hi . ' pi g is Hi y g i Hi ) 0. Proof. n i k s 0 is the condition of equilibrium coinsurance rate: k i s kUi ; so from Ži. – Živ. of Proposition 1, by rearranging the terms, we have y Ž g is y g i Ž 1 q d . . pi Hi Ž 1 y a . s g i Ž 1 y a . Ž 1 y k i . Ž 1 q d . pi Hi k As a result, taking into account the definition of health care demand elasticity e i and the definition of covariance Covp , we immediately reach conditions Ž15. and Ž16.. B Discussion Condition Ž15. may be interpreted according to the one-consumer optimal Ramsey tax rule. The left hand side ŽL.H.S.. of Eq. Ž15. may be considered as an optimal ad Õalorem subsidy rate which is inversely correlated to the elasticity of demand wBesley, 1988; Blomqvist, 1997; Zweifel and Breyer, 1997, p. 192x. qi ,
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the coefficient of proportionality of R.H.S. of Eq. Ž15., is represented by the relative difference between MUI in illness and the expected MUI, the latter weighted with the loading factor. If this is negligible Ž d 0., qi ) 0 and, with moral hazard, 0 - kUi - 1. 10 Recall that the First Best solution requires g is s g ih and Hi k Žno moral hazard., thus p i s pi and kUi s a U s 0: the risk is entirely insured on the competitive insurance market. The L.H.S. of Eq. Ž16. is the gain from risk-sharing ŽRSG. due to a marginal reduction of the coinsurance rate Ž risk-pooling . and is measured by the covariance between marginal utility of income g i and health expenditure Hi . The term is positive, as explained in literature on optimal insurance. 11 The R.H.S. has two components: the marginal cost of moral hazard ŽMHC. associated with a reduction of the coinsurance rate Ža deadweight loss. and the loss associated with the expected loading and administrative expenditures. We see, in Eqs. Ž15. and Ž16., that the degree of social insurance coverage does not appear explicitly in the formula for the optimal choice of kUi . However, this does not mean that in equilibrium kUi is, in this mixed health insurance, independent of a , but simply that the choice of the optimal individual value of coinsurance rate follows the same rule as in a purely private health insurance policy Ž a s 0.. In other words, it reaches a marginal condition of standard equilibrium in an insurance market with ex post moral hazard and two states of the world — RSG s MHC — which is the equilibrium of the contract signed in the second step. Actually, conditions Ž15. and Ž16. implicitly define a function of the type
™
k i s k i Ž a ,t , M . ,
i s 1, . . . n
Ž 17 .
which gives rise, in a certain sense, to the ‘‘reaction function’’ of private insurance contracts with respect to social policy instruments. Particularly crucial in our analysis is the partial derivative ŽEk irEa . ' k i a , which specifies a sort of substitution relationship between private and social insurance coverage. We may reasonably assume that k i a ) 0, because it is easy to realise that an increase of the social insurance coverage rate Žan extension of health services paid by the NHS. should increase the co-payment rate of private insurance. With sai ' k i a Ž ark i . we denote the elasticity of k i with respect to a .
10 By observing Eq. Ž15. it is straightforward to see that the optimal rate of coinsurance is ceteris paribus increasing along with health risk Žindividuals with higher probability of illness agree to stipulate an insurance contract with a higher coinsurance rate. as well as with the elasticity of demand for medical care ŽZweifel and Breyer, 1997, p. 194.. 11 Generally, it is assumed that sicker people consume more medical services Žas it is, of course, in our case., so that a sufficient condition for a positive covariance is that MUI increase in moving from health to illness. See for instance Besley Ž1988. and more recently Jack and Sheiner Ž1997..
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5. The government’s choice of social insurance rate The solution, with regard to social insurance coverage, is obtained by applying the following maximization Max a W s Ýn i Ž a ,k i ,t , M . i
s.t.
aÝ pi Hi s tÝwi L i y nM i
i
k i s k i Ž a ,t , M . ,
i s 1, . . . n.
It is also relevant to determine the conditions for the optimal tax rate tU and optimal transfer M U , since the mix between social and private insurance and the distributive effects of the former are bound to change in line with the parameters of linear income taxation; we shall discuss this further on. As in Cremer and Pestieau Ž1996. paper, for the solution of government maximization, we require that 0 F a F 1 Žand also 0 F t F 1.. Later we shall also consider the conditions by which 0 - a U - 1; for the moment, we will simply assume that this the case. Before solving the government maximisation problem for an interior positiÕe solution of a U we need to establish some useful comparative static results. Proposition 3
Ž i.
Eqi Ea
s yk i q Ž 1 y a . k i a s ybi k i
Ž 18 .
where bi ' w1 y sai Ž1 y a .ra x
Ž ii .
Ep i
s y Ž 1 y k i . pi y Ž 1 y a . pi k i a Ž 1 q d .
Ea
s ypi w 1 y bi k i x Ž 1 q d .
Ž iii . Ž iv .
EHi Ea Ey i Ea
' Hi a s Ž EHirEqi . Ž EqirEa . s yHi q bi k i
' y i a s g i Hi Ž 1 q d . q Ž g is y g i Ž 1 q d . . bi k i Hi y g ip i Hi a
where Õi (a ,t,M) ' n i (a ,k i (a ,t,M),t,M). Proof. Eq. Ž18.i and Eq. Ž18.ii derive from direct computation, after having inserted Eq. Ž17. in the relevant functions; Eq. Ž18.iii comes from Eq. Ž10. and Eq. Ž18.i.
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Furthermore, by differentiating Eq. Ž13. with respect to a and then applying the Envelope theorem to this particular context, we obtain the following modified Roy identity:
yi a s n i a q n i k k i a s ypi g is Ž EqirEa . q Ž Ep irEa . Hi y pi g is Ž qi q p i . Hi a q pi g isqi Hi a y Ž 1 y pi . g ih Ž Ep irEa . Hi y Ž 1 y pi . g ihp i Hi a Therefore, by substituting Eq. Ž18.i and Eq. Ž18.ii and some further standard manipulations, we reach Eq. Ž18.iv. B Discussion Eq. Ž18.i is the change in the price of out-of-pocket health expenditure following an increase in the social insurance rate. The term is negative if bi ) 0, or if sai - Ž arŽ1 y a ..; so, for a sufficiently inelastic reaction of private insurance to its social counterpart, an increase in the social insurance rate decreases the out-of-pocket unit cost of health expenditure. Eq. Ž18.ii illustrates the response of the private premium coefficient to the social insurance rate and it is negative as we have assumed k i a ) 0. Note that, from Eq. Ž18.i and Eq. Ž18.ii we have 0 - bi k i - 1, consequently, by means of Eq. Ž18.iii and the normality hypothesis, we conclude that the health expenditure increases with social insurance. According to Eq. Ž18.iv, the net welfare gain for individual i of an increase of the social insurance coverage is determined by three elements: the first one Žpositive. is the expected gain by the increase of the ‘‘subsidy’’ on H; the second one Žpositive if d is negligible. represents the expected gains of risk-pooling for the health expenditure not covered by private insurance; the third one Žnegative, giving the sign of Eq. Ž18.iii represents the welfare loss from the increase in the private insurance premium due to the health expenditure increase. In order to lend more intuition to the F.O.C.s of government optimization we are going to calculate, we may consider the special case of partial equilibrium analysis, i.e., no income effects on the demand for health care and independent demand schedules for both health care and leisure. In the optimal taxation literature this case is usually referred as the simplified Feldstein framework ŽAtkinson and Stiglitz, 1980, p. 388. and is, despite its strict implied restrictions on preferences, also considered in several papers on health insurance Žsee, e.g., Besley, 1988; Cremer and Pestieau, 1996.. Before stating the main Proposition we shall summarise our list of assumptions. Assumptions. A1. Hi is a normal good A2. With 0 - a - 1, k i a ) 0 and sai - Ž arŽ1 y a ..;
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A3. There is a partial equilibrium analysis (Feldstein) framework; A4. Loading factor d is negligible. The following Proposition offers the solution of the model, as far as social insurance coverage is concerned. Proposition 4. GiÕen the preÕious assumptions A1–A4, in the described mixed insurance system the optimal social insurance coÕerage rate a U is represented by the following condition:
Ý
Covp Ž bi , bi k i Hi . q n bj H y 1 H s Ýt i Hi a
i
Ž 19 .
i
where bi is the expected social MUI of indiÕidual i, i.e., bi ' pi bis q Ž1 y pi . bih ' Žg irm .; bis ' Žg isrm .; bih ' Žg ihrm .; b ' Ž1rn.Ý i bi , m is the Lagrange multiplier of the goÕernment budget constraint (the marginal social cost of taxation), j H ' (1 r n)Ý i (bi r b)(Hi r H) s CoÕ(bi ,Hi ) q 1 is the distributional characteristic of H and t i ' [bi (1 y a )(1 y k i ) q a ] is a social term attached to the effect of a on i expected health expenditure. Proof. The Lagrangean of the problem is
C s Ýy i y m aÝ pi Hi y tÝwi L i q nM . i
i
i
As the necessary condition for a s a U is ŽEC .rŽEa . s 0, using assumption A3 Žby which, L ija s 0, j s h, s . we obtain
Ýy i a y m Ý pi Hi q aÝ pi Hi a i
i
s0
i
Now, by substituting Eq. Ž18.i, Eq. Ž18.ii, Eq. Ž18.iii and Eq. Ž18.iv, and taking into account assumption A4, the definitions of expected social MUI bi , its average value b, and of the social weight t i , we obtain the following equation:
Ýbi Hi y Ýbi bi k i Hi q Ý pi bis bi k i Hi y Ý pi Hi i
i
s
žÝ
i
i
pi bis bi k i Hi y Ýbi bi k i Hi q
i
i
/ žÝ i
bi Hi y nH
/
s Ýt i Hi a i
The first term of the L.H.S. in parenthesis is equal to Ý i Covp Ž bi , bi k i Hi . and B the second to nw bj H y 1x H, consequently we reach condition Ž19..
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6. Discussion and interpretation of the optimality condition for the social insurance rate 6.1. On the optimal social insurance rate a U The first component of the L.H.S. of Eq. Ž19. may be called the social risk-sharing gain ŽSRSG. from a marginal increase of social insurance coverage, since it arises from the sum of individuals’ covariance terms between bi and bi k i Hi , i.e., between the social Ždivided by m and then normalised in terms of government revenue. MUI and the health expenditure not covered by private insurance. 12 Recall that, as in the traditional optimal taxation framework, the expected social MUI may be thought of as declining in line with the skill parameter w across individuals. The term SRSG gives a measure of the gain for society as a whole from ‘‘buying’’ a further insurance policy with a uniform reimbursement rate for all members; the term is positive, given Proposition 3 and assumption A2, by which 0 - bi k i - 1. Note that the optimal level of social coverage depends on the level of equilibrium of each consumer’s private coinsurance rate, given by Proposition 2, and on this rate’s response to the social insurance level, as from Eq. Ž17.. Therefore, in practice, in order to compute the optimal a U , the government must take into account the contract structure chosen by insured types. The second component of the L.H.S. of Eq. Ž19. is the social redistribution gain ŽSRG., stressing the role of a for vertical equity aims. It is clearly closely linked to the distributional characteristic of goods subject to the standard many-consumers optimal commodities taxation rule ŽAtkinson and Stiglitz, 1980, p. 388.. In this case, Ž j H y 1. is the ‘‘normalised covariance’’ between expected social MUI and expected health expenditure across individuals. As we have stated that the prior is declining with w, it is j H ) 1 Ž- 1. if health expenditure is also decreasing Žincreasing. with w. 13 The sign of SRG also depends on the level of b and this, as we shall see, is dependent on tax parameters; at the moment we may say state that SRG tends to be positive Žnegative. if health expenditure is a necessity Žluxury. and consequently a U tends to increase Ždecrease.. In this three-commodity model social insurance works like a subsidy on a good Žhealth expenditure., in the presence of linear income tax and with an untaxed good, the numeraire Žthe consumption of the private good.. Then the choice of tU , M U and a U follows rules in some sense similar to those specified in the classical problem of optimal indirect taxation and linear direct taxation ŽAtkinson and 12
A similar expression has been obtained by Besley Ž1988. for designing the optimal reimbursement rule of a given health good. 13 As clearly specified by Atkinson and Stiglitz Ž1980, p. 432., this assumptions does not come from the normality property of that ‘‘good’’.
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Stiglitz, 1980, p. 428–435.. Also in our application, as far as distributional objectives are concerned, social insurance plays two roles: first, if health expenditure is a necessity Žluxury. an increase Ždecrease. of a U may increase the progressiveness of the system; second, a reduction of a U may allow the progressive lump-sum payment M to be increased. Therefore, the final rate of social insurance has to be set to balance the two types of considerations. The R.H.S. of Eq. Ž19. is the social moral hazard cost ŽSMHC., as a weighted sum of individual moral hazard effects. It is positive given Eq. Ž18.iii. Note that, by substituting the term t i Žthe social value of an extra unit of health expenditure by i ., we may state SMHCs Ž 1 y a . Ýbi Ž 1 y k i . Hi a q a nHa ,
Ž 20 .
i
so that SMHC is the sum of two types of moral hazard effects: the first comes from private budget constraints and is linked to the sum of individual effects in terms of expected health expenditure increase; the second is determined by government budget constraint and is related to the effect on public expenditure due to social health insurance. Until now we have analysed questions concerning the optimal level of a ‘‘positive’’ and less than one rate of social insurance coverage a U ; nothing has been said of the possibility of obtaining corner solutions. Actually, the level of a U comes from a balance of two opposite effects: the social gain ŽSRSGq SRG. vs. the social cost ŽSMHC. of an increase in the rate. When these terms are finite and positive in absolute terms it should follow that 0 - a U - 1 as result of a social trade-off. In particular, we see that a U ) 0 if, at a s 0, it is SRSGq SRG ) Ý i bi Ž1 y k i . Hi a and a U - 1 if, at a s 1, it is SRSGq SRG - nHa . However, we cannot exclude corner solutions, represented by full social insurance or no social insurance. In the special case where k i s 0 Žno private coinsurance rate., the first term of the L.H.S. of Eq. Ž19. — SRSG — vanishes, so a has only a redistributive effect, SRG, to be equated, at the margin, with the deadweight loss, SMHC, as in the standard general many-consumers Ramsey rule. So, as in that context, if we impose the separability and homotheticity restrictions set by Deaton Ž1979; 1981., it turns out that a U s 0, since the optimal linear income tax is the only instrument to be used for both efficiency and redistribution aims. If, instead, we imagine a lack of moral hazard effect, Eq. Ž19. becomes simply CovŽg i , pi . s 0: a U should be increased as long as this is beneficial for redistribution, given that it has no deadweight loss. As in the Cremer and Pestieau Ž1996. theoretical context, if that covariance is always positive it is a U s 1. 6.2. On interdependence with optimal tax parameters tU and M U Integration between the optimal social insurance rate and optimal linear taxation parameters as redistributive instruments is more evident when observing the
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conditions for M U and tU . These, in a Feldstein framework and with ‘‘fair’’ premiums Žassumption A3 and A4., are, respectively 14 b y Ž 1rn . ÝCovp Ž bi ,k i M Ž 1 y a . Hi . s 1
Ž 21 .
i
ÝCovp Ž bi ,k i t Ž 1 y a . Hi . q n
t sy 1yt
i
b Cov Ž bi ,Yi . q 1 y 1 Y
ž
/
Ý´ Li Yi i
Ž 22 . ´ iL
where is the elasticity of the expected labour supply at the marginal net wage and CovŽ bi , Yi . s Ž1rn.Ý i Ž birb .Ž YirY . y 1 is the ‘‘normalised covariance’’ between expected social MUI and expected gross labour income across individuals. Thus, according to Eq. Ž21. b ) 1 if the response of the private coinsurance rate is positive with respect to an increase of the lump sum payment, i.e., if, for all i, k i M ) 0, then Covp Ž bi , k i M Ž1 y a . Hi . ) 0, the latter being the risk-sharing term referring to the ‘‘new’’ out-of-pocket health expenditure. The condition Ž21. says that the lump-sum payment should be adjusted so that the social marginal valuation of the transfer of a unit of numeraire, net of this covariance term, should on aÕerage be equal to the cost Ž1 pound.. With b ) 1 the role of optimal M U is to reinforce the progressiveness of the system to be pursued through a U because SRG in Eq. Ž19. tends to increase, given the higher weight attached to j H . As far as the optimal marginal rate is concerned, it may be noted that in Eq. Ž22. Covp Ž bi , k i t Ž1 y a . Hi . - 0 if the response of the coinsurance rate is negative with respect to an increase in the marginal tax rate: k i t - 0. Therefore, given that, as reasonably assumed in optimal taxation literature, it is ´ Li ) 0 and CovŽ bi , Yi . - 0, we have 0 - tU - 1 if b - Ž1rŽCovŽ bi ,Yi . q 1... Hence, we cannot, given the integration with social insurance, exclude a corner solution, i.e., a zero optimal marginal tax rate.
7. Summarising and concluding remarks In the previous sections we modelled a National Health Service in which compulsory social insurance, covering a package of essentials, is integrated by a 14
These derive easily from the F.O.C.s CM s 0, Ct s 0 and by taking into account Eq. Ž17. for computing, by means of the Envelope theorem, respectively, y i M and y i t . Both conditions Ž20. and Ž21. extend to this case the standard formula designing the parameters of the optimal linear income tax ŽAtkinson and Stiglitz, 1980, p. 407.. For a relevant reference to the integration between social insurance and optimal linear income taxation in a somewhat different model see Blomqvist and Horn Ž1984..
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private policy topping up the remaining services, with co-payments by patients. Thus, an individual’s health expenditure is divided in three parts: the first is covered by social insurance, the second by a private insurance policy and the third is out-of-pocket expenditure. The social insurance is financed by a linear income Žpay-roll. taxation whose structure is optimally chosen, together with the social insurance coverage, by the government. This considers n types of individuals distinguished by two parameters, the probability of illness and skills Žwage in efficiency units.. There is ex post moral hazard with regard to health expenditure and adverse selection in the sense that the individual’s skill parameter is not observable by the government. Moreover, although the probabilities of illness are however known, the government cannot differentiate its tax policy between high-risk and low-risk individuals, but only according to their personal incomes. In this framework we have obtained the following health care policy conclusions. The individual’s equilibrium private coinsurance rate follows the standard condition of optimal insurance with ex post moral hazard. This requires equalising, at the margin, the gain of risk-sharing with the deadweight loss due to moral hazard effect for each individual. In any case, the private insurance contract signed by each individual includes a coinsurance rate which is a function of social policy instruments Žpayroll-tax and social insurance rate.. Such a function has a meaning similar to that of a following ‘‘reaction function’’ in a Stackelberg framework. On the subject of optimal social insurance coverage, we have reached some formal results which almost duplicate mere common intuition. First, the higher the distributional characteristic of health services, the higher the optimal rate, as in the standard many-consumers optimal commodity taxation model. Indeed, social insurance coverage works like a subsidy applied to health expenditure at a uniform rate. It is also higher the higher is the sum of individuals’ gains of risk-sharing, referring to out-of-pocket health expenditure. The gain from risk-pooling is in some sense ‘‘socialised’’ in two ways: first, by following a utilitarian welfare function rule, i.e., summing up the individual covariances between marginal utility of income and out-of-pocket health expenditure; second, by considering, as in the optimal taxation framework, each individual’s ‘‘social’’ Ži.e., measured in terms of government revenue. marginal utility of income. It is noteworthy that the optimal level of social coverage indeed depends, marginally, on the equilibrium level of each consumer private coinsurance rate and on the response of each reaction function Ža sort of strategic substitution elasticity term.. Therefore, from the information requirements point of view, the government, in order to compute the optimal social insurance rate, must infer the private contract typology Žspecifically the co-insurance rate and its degree of response to social insurance coverage. signed by the different types of insured individuals. Both effects of pushing up the coverage rate, the social gain of risk-sharing and
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the social redistributive effect, are balanced by the aggregate Žsocial. moral hazard effect. The analytical results of this paper might suggest some implications for modelling a mixed insurance system in industrialized countries. First, we have seen that a positive, but less than one, social insurance rate may be justified on the basis of efficiency and equity considerations. However, as long as health expenditure is treated as a luxury good, whose consumption is largely concentrated on the wealthy Žas is often bound to happen in industrialized countries., the latter consideration is of reduced relevance. Therefore, because the moral hazard ‘‘third-payer’’ effect proves to be a strong deterrent, the principal implication is the need to limit social insurance coverage. This recalls the attempt recently made in some institutional frameworks 15 to introduce forms of rationing and selection of services and treatments to be covered by social insurance according to their cost-effective performance. We must ultimately admit that our analysis is strictly normative Žreferring to a pure ‘‘welfarist’’ point of view as well. and unable to consider some relevant flexible alternatives in social insurance systems — such as the possibility of placing restrictions on individuals’ rights to acquire supplementary private insurance on one hand and, of allowing opt-out solutions on the other — which may improve the efficiency of a mixed system. In the course of the paper we have suggested some possible extensions of our model in these directions which may serve as interesting starting points for future research.
Acknowledgements I would like to thank Alessandro Balestrino, Giuseppe Pisauro and two anonymous referees for helpful comments and suggestions.
References Atkinson, A.B., Stiglitz, J.E., 1980. Lectures on Public Economics. MacGraw Hill, New York. Barr, N., 1998. The Economics of the Welfare State, 3rd edn. Oxford Univ. Press, Oxford. Besley, T., 1988. Optimal reimbursement health insurance and the theory of Ramsey taxation. Journal of Health Economics 7, 321–336. Besley, T., 1989. Publicly provided disaster insurance for health and the control of moral hazard. Journal of Public Economics 39, 141–156. Besley, T., Gouveia, B., 1994. Alternative systems of health care provision. Economic Policy, October, 203–258. Blomquist, S., Christiansen, V., 1998. Topping-up or opting-out? The optimal design of public provision schemes. International Economic Review 39, 399–411.
15
See in this respect the Dutch health care reform proposal ŽVan de Ven, 1995..
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Blomqvist, A., 1997. Optimal non-linear health insurance. Journal of Health Economics 16, 303–321. Blomqvist, A., Horn, H., 1984. Public health insurance and optimal income taxation. Journal of Public Economics 24, 353–371. Blomqvist, A., Johansson, P.-O., 1997. Economic efficiency and mixed publicrprivate insurance. Journal of Public Economics 66, 505–516. Boadway, R., Marchand, M., Sato, M., 1998. Subsidies versus public provision of private goods as instruments for redistribution. Scandinavian Journal of Economics 3, 545–564. Cremer, H., Pestieau, P., 1996. Redistributive taxation and social insurance. International Tax and Public Finance 3, 281–295. Deaton, A., 1979. Optimally uniform commodity taxes. Economic Letters 2, 357–361. Deaton, A., 1981. Optimal taxes and the structure of preferences. Econometrica 49, 1245–1260. Jack, W., Sheiner, L., 1997. Welfare-improving health expenditure subsidies. American Economic Review 87 Ž1., 206–221. Ma, C.A., McGuire, T.G., 1997. Optimal health insurance and provider payment. American Economic Review 87 Ž4., 685–704. Manning, W.G., Marquis, M.S., 1996. Health insurance: the trade-off between risk pooling and moral hazard. Journal of Health Economics 15, 609–639. Mooney, G., Ryan, M., 1993. Agency in health care: getting beyond first principles. Journal of Health Economics 44, 125–135. Pauly, M.V., 1974. Overinsurance and public provision of insurance: the roles of moral hazard and adverse selection. Quarterly Journal of Economics 88, 44–62. Rochet, J.C., 1991. Incentives, redistribution and social insurance. The Geneva Papers on Risk and Insurance 16, 143–165. Selden, T.M., 1993. Should the government provide catastrophic insurance?. Journal of Public Economics 51, 241–247. Selden, T.M., 1997. More on the economic efficiency of mixed publicrprivate insurance. Journal of Public Economics 66, 517–523. Spence, M., Zeckauser, R., 1970. Insurance, information and individual action. American Economic Review 61, 380–387. Van de Ven, W.P.M.M., 1995. Regulated competition in health care: with or without a global budget?. European Economic Review 39, 786–794. Von der Schulenburg, J., 1995. The relationship between basic and complementary health system: theoretical considerations. In: Schneider, M. ŽEd.., Complementary Health Schemes in the European Union. Augsburg Basys. Zweifel, P., Breyer, F., 1997. Health Economics. Oxford Univ. Press, Oxford.
Optimal social health insurance with supplementary private insurance Alessandro Petretto
)
Department of Economics, UniÕersity of Florence, UniÕersity ‘‘L. Bocconi’’, Milan and Public Expenditure Commission, the Treasury, Rome, Italy Received 13 July 1998; received in revised form 6 January 1999; accepted 22 March 1999
Abstract This paper investigates the structure of a National Health Service in which there is compulsory social insurance covering a package of essentials, a given part of individuals’ health expenditure, and supplementary private policy topping up the remaining services. The latter insurance contract provides for a co-payment by patients, limiting the so-called ‘‘third-party payer’’ effect. Thus, an individual’s health expenditure is divided into three parts: the first covered by social insurance, the second by a private policy and the third out-of-pocket. Such mixed system design has received increasing attention in recent years and has been adopted by several industrialized countries. The conditions for optimal rates of social insurance coverage and of private coinsurance are analysed and discussed. The optimality requirements refer to efficiency as well as equity concerns. q 1999 Elsevier Science B.V. All rights reserved. JEL classification: I18; G22; H21 Keywords: Social insurance; Ex post moral hazard; Risks sharing and optimal redistribution
1. Introduction If we consider the principal trends in the reform of health care systems in industrialised countries we observe a shift from publicly financed systems to )
Corresponding author. Tel.: q00-39-5527-10415; fax: q00-39-5527-10424; E-mail: [email protected] 0167-6296r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 2 9 6 Ž 9 9 . 0 0 0 1 7 - X
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mixed ones, with significant patient co-payments for services ŽBesley and Gouveia, 1994; Zweifel and Breyer, 1997.. The problem investigated in this paper refers to the case of compulsory social insurance covering a package of essentials Ža given part of health expenditures. and a supplementary private policy topping up the remaining services, with co-payments by patients. In this way an individual’s health expenditure is divided into three parts: the first covered by social insurance, the second by a private policy and the third out-of-pocket. Examples often considered in literature include the U.S. Medicare plan, in which private insurers offer supplementary ‘‘Medigap’’ plans, and the Australian Medicare, in which supplementary private insurance contracts are offered by Registered Private Health Funds. Similar funding schemes have been applied during the recent reform of National Health System in Italy, where the task of organising a system of supplementary Private Health Funds has been delegated to regions and municipalities. In many institutional contexts individual contributions to social insurance are income-dependent Ža payroll tax or an income tax. and the integrative private policy applies a differentiated premium plus a coinsurance rate in order to limit the so-called ‘‘third party payer’’ effect. In this paper we propose to merge two streams of literature to facilitate study of such a mixed system from a normative perspective: the first focuses on the classical problem of trade-off between gain from risk-sharing and the deadweight loss of ex post moral hazard ŽSpence and Zeckauser, 1970; Pauly, 1974; Besley, 1988; Blomqvist, 1997.; 1 the second deals with combining the redistributive roles of income taxation and social insurance ŽBlomqvist and Horn, 1984; Rochet, 1991; Cremer and Pestieau, 1996.. Recently, Blomqvist and Johansson Ž1997. have also considered a mixed publicrprivate health insurance system, but they have not modelled an income-dependent contribution to social insurance with a variable supply of labour. They have raised, following the example of Besley Ž1989. and Selden Ž1993., an interesting debate on the efficiency of a mixed system with respect to a purely private competitive one Žsee also Selden, 1997.. In our case, preference for an institutional system with respect to mixed insurance is taken for granted, thus we aim to investigate its optimal structure in terms of efficiency and equity. The social insurance system we refer to is essentially compulsory and universal, although it provides only for partial coverage, and does not allow low-risks or rich people to opt-out of the NHS. 2 However, since flexibility may lead to greater efficiency, we outline some feasible extensions of the model in order to present some alternative solutions. 1 On empirical aspects of the trade-off between the gain from risk-sharing and the deadweight loss from ex post moral hazard see Manning and Marquis Ž1996.. For an analysis of taxation benefit and private health insurance see Jack and Sheiner Ž1997. and for the combination of optimal insurance and optimal provider contracts see Ma and McGuire Ž1997.. 2 Regarding the characteristics of such a social insurance system, see Barr Ž1998, p. 125..
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As to the ‘‘third-party payer’’ effect, we recall that we are speaking of ex post moral hazard Ži.e., once nature has determined the state of the world. because the insurer Žpublic as well private. is unable to determine the health status of the insured with certainty. Health care expenditure itself serves as an indicator of illness, causing benefits to depend on expenditure by means of a specific function. For the insured there is no possibility of manipulating the risk of illness, as in the ex ante moral hazard case, but there is a choice between various alternative treatments once an illness has occurred. 3 We may furthermore assign a redistributive role to social health insurance given that compulsory membership offers mandatory coverage for all, along with a given fraction of potential losses, thus acting as a redistribution in kind. In order to develop these concepts, in Section 2 we construct a two-state-of-the world model of health insurance. In Section 3 we describe the solution to the model according to a three-stage maximization process. In Section 4 we study the individual’s equilibrium co-payment formula for the supplementary private policy; in Sections 5 and 6 we propose and discuss the optimal social insurance coverage conditions. Section 7 contains some concluding comments.
2. The model We may generally distinguish two states of the world Žhealthy and sick. and surmise that n individuals are identified by a pair of parameters: wages Žskills. and probability of sickness: 4 wi , pi 4 , i s 1, . . . n. This approach is quite general: it could for example represent a situation with two risk classes Žwith one type needier than the other, as far as the consumption of health services is concerned. and many productivity parameter values. If this is the case, no hypothesis is made about the relative level of skill Žwage rate., so it may be that high skill parameters are associated with higher risk and vice versa. Moreover, while the probabilities pi ) 0, i s 1, . . . n, are known Žno adverse selection in this respect. by the insurance companies and by the government, skill parameters wi are not. 5 Therefore, the government is not able to apply a first best optimal taxation linked to this specific characteristic. As in the familiar optimal 3
In reality, treatments are only indirectly chosen by patients since the true choice is made by the physician acting as the best-informed agent in a principal–agent relationship ŽMooney and Ryan, 1993; Zweifel and Breyer, 1997, Chap. 7.. 4 In order to characterise the type of individual i we use subscripts: e.g., x i . When using the familiar short form for denoting partial derivatives with a subscript, we make use of two letters; hence x i q denotes the partial derivative of x with respect to q for individual i. 5 This asymmetry in the insurer’s information is clearly assumed for simplicity. In this respect we are following Žand generalising. the framework by Rochet Ž1991. and Cremer and Pestieau Ž1996.. For a clear discussion of government information on individuals’ risk and skill see Blomqvist and Horn Ž1984..
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income taxation context, only individual labour income Žthe expected value in our framework. is taken into account by the government for modelling the taxation function ŽAtkinson and Stiglitz, 1980.. Furthermore, we implicitly surmise that the government is also unable to differentiate the income taxes imposed on individuals according to their risk of illness. 6 We may now imagine a state-contingent concave utility function for individual i as follows: u ih s u ih Ž c ih ,0, Lhi . ,
Ž 1a .
when healthy, and u is s u is Ž c is , Hi , Lsi . ,
Ž 1b .
when sick. Hi represents the health service expenditure and L ij, c ij, j s h, s, represent labour and consumption in the two states. Hi is observable only ex post, i.e., only after the payment has been made Žex post moral hazard.. We consider this expenditure to differ among individuals due to the different coinsurance rates chosen by each consumer-type. Furthermore, we have two budget constraints c ih s wi Lhi y T Ž wi Lhi . y Pi
Ž 2a .
c is s wi Lsi y T
Ž 2b .
Ž
wi Lsi
. y Pi q A i q Bi y Hi
where T Ž.. is the payroll tax function, Pi is the premium of the private health insurance; A i and Bi are the benefits of social and private insurance, respectively. If we denote the coverage Žreimbursement. rate of social insurance with a and the private coinsurance rate with k i , we obtain the following expressions for benefits Žassumed to be linear for simplicity. and premiums A i s a Hi
Ž 3.
Bi s Ž 1 y a . Ž 1 y k i . Hi
Ž 4.
Pi s pi Ž 1 q d . Bi s Ž 1 y a . Ž 1 y k i . Ž 1 q d . pi Hi s p i Hi .
Ž 5.
The function of public benefits ŽEq. Ž3.. contains an implicit constraint, i.e., a i s a all i. If instead a type-specific social insurance coverage were available, this might mean that, at the optimum, a j s 0, for some j, and a i ) 0, for the others. Thus, this would present the possibility of excluding some share of the population from NHS coverage — that is, those individuals able to privately insure all risks. However, the uniformity constraint a i s a all i is not imposed in 6 In fact this opportunity should be considered since we are assuming that probabilities are known and also since in a Second Best framework it is efficient to use all available information to design the optimal taxation structure. For a discussion of this opportunity see Blomqvist and Horn Ž1984..
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Eq. Ž3. exclusively to simplify analysis, given that it may involve an objective of universality defined, as in some European countries, at the Constitutional level ŽBarr, 1998.. The private insurance premium ŽEq. Ž5.. is proportional to individual benefit and subsequently to health expenditure according to the ‘‘price’’ p i ; d is a loading factor, which, in case of fairness, is d s 0 ŽZweifel and Breyer, 1997, p. 163.. Furthermore, at aggregate level, we observe the following government budget constraint:
aÝ pi Hi s ÝE T Ž wi L i . 4 i
Ž 6.
i
which shows that the expected public expenditure for the NHS is financed by the expected revenues from pay-roll taxation. 7 If, for simplicity, we assume a linear payroll tax T Ž wi L ij . s twi L ij y M ,
j s h, s
Ž 7.
the government budget constraint ŽEq. Ž6.. becomes
aÝ pi Hi s tÝwi L i y nM i
Ž 8.
i
which may be, finally, reduced to the more compact one:
a H s tY y M where we use the notations: H s Ž1rn.Ý i Hi , Hi ' pi Hi q Ž1 y pi .0, L i s pi Lsi q Ž1 y pi . Lhi , Yi ' wi L i , Y ' Ž1rn.Ý i Yi . The lump sum transfer M, which may be positive as well negative Ža poll-tax., is meant to represent the usual non-proportional structure of payroll-tax Žoften regressive.. 7 If we pursued the idea of allowing differentiated rates of social coverage we would have the following budget constraint:
Ýa i pi Hi sÝE T Ž wi Li . 4 i
i
This means that all individuals Žalso those excluded from the NHS. contribute by paying income tax. If, instead, we want to consider reducing the tax liability of excluded persons when they obtain approved substitute private insurance, we have to add a further constraint such as: Tj sT8 Žwith T8 possibly 0. as a j s 0. We must note that this is not a flexible ‘‘opt-out system’’ because the choice to be ‘‘in’’ or ‘‘out’’ is not made individually but decided by the government on the basis of personal incomes; a social insurance system like this one is now applied is Germany. For an analysis of general models of redistribution in-kind in which the provision of public and private services are mutually-exclusive, see Blomquist and Christiansen Ž1998. and Boadway et al. Ž1998..
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By substituting Eqs. Ž3. – Ž5. into Eqs. Ž2a. and Ž2b., and taking account of Eq. Ž7., we easily obtain the following expressions for the individual’s budget constraints: c ih s wi Ž 1 y t . Lhi q M y p i Hi c is s wi
Ž1yt .
Lsi q M y p i Hi y qi Hi
Ž 9a . Ž 9b .
where qi s Ž1 y a . k i is the relevant ‘‘price’’ for the choice of H by the individual i when ill Žtaking the private insurance premium as given.; it is the individual unit cost of health service out-of-pocket consumption. 8 Four cases of special interest may be considered. 1. If a s 0 Žno social insurance. we have qi s k i , p i s Ž1 y k i .Ž1 q d . pi . If, moreover, we put k i s 0 Žfull private insurance, i.e., no coinsurance. and d s 0 Ž‘‘fair premium’’., we obtain the canonical result qi s 0, p i s pi . 2. If a s 1 Žfull social insurance coverage, and no supplementary private insurance. we have qi s 0 and p i s 0. This is the case of a purely public NHS. 3. If 0 - a - 1 and k i s 0 Žprivate ‘‘full coverage’’ insurance, i.e., no patient co-payment. we have qi s 0 and p i s Ž1 y a .Ž1 q d . pi . 4. If 0 - a - 1 and k i s 1 Žpartial social insurance coverage, without a supplementary private policy. we have qi s Ž1 y a . and p i s 0. This case may arise if loading is included in the premium formula, since individuals may consider zero private coverage to be convenient. In general, as we shall see, the mixed system is characterised by 0 - a - 1 and 0 - k i - 1, i s 1, . . . n. It is ultimately clear that the model applies to a system in which social insurance covers all risks, with private insurance paying for the part of health expenditure not covered by social insurance. In order to consider the interesting case of social and private insurance covering different risks, it would have been necessary to construct a multiple-risks model ŽVon der Schulenburg, 1995. which, however, would have much complicated the analysis.
3. A three-stage maximization problem for mixed publicr r private insurance decision-making In our model the decision process runs in three steps: Ži. in step 1, the government chooses social insurance coverage a and the structure of taxation Ž t and M .; Žii. in step 2, each consumer chooses a private insurance contract and subsequently a coinsurance rate k i ; Žiii. in step 3, the consumer chooses, in the 8
The cost of treatment proves to be in some way subsidised by social and private insurance because of the two payment systems Žtaxes vs. premiums.. This determines a loss of efficiency, as it prevents the total insurance premium from being risk specific and therefore causes moral hazard. However, this loss is a somehow unavoidable consequences of the chosen Second Best framework.
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event of illness, the labour supply and health expenditure and, in the event of healthy status, only the labour supply Žconsumption in both conditions takes place as permitted by contingent budget constraints.. Thus, the decisions regarding structures of social and private insurance contracts are ex ante decisions following an expected utility maximization framework, while decisions concerning commodity bundles by the two consumers are instead ex post decisions which consider as given the terms of the insurance contracts. The model is solved, using game-theory terminology, by ‘‘backward induction’’, requiring passage through the following three-stage maximization process. 3.1. First stage Consumer i chooses Lhi if healthy and the bundle Ž Lsi , Hi . if ill, given a , k i , t and M; the choices in sick status are made while taking private insurance premiums as given. Consumption levels in the two states arise from Eqs. Ž9a. and Ž9b.. This maximisation process gives the following ex post decisions for i: Hi Ž qi ,wi Ž 1 y t . , M . s Arg max H u i Ž wi Ž 1 y t . Lsi q M y p i Hi yqi Hi , Hi , Lsi .
Ž 10 .
with F.O.C., ŽEu is .rŽEHi . s qi g is, where g is s ŽEu is .rŽEc is . is the marginal utility of income ŽMUI. when ill. Lsi Ž qi ,wi Ž 1 y t . , M . s Arg max L u i Ž wi Ž 1 y t . Lsi q M y p i Hi yqi Hi , Hi , Lsi . with F.O.C.
Ž 11 .
yŽEu is .rŽELsi . s wi Ž1 y t .g is.
Lhi Ž wi Ž 1 y t . , M . s Arg max L u i Ž wi Ž 1 y t . Lhi q M y p i Hi ,0, Lhi .
Ž 12 .
with F.O.C. yŽEu ih .rŽELhi . s wi Ž1 y t .g ih , where g ih s ŽEu ih .rŽEc ih . is the MUI when healthy. 3.2. Second stage Consumer i chooses his or her individual k i taking into account the supply and demand functions for the equilibrium values Ž10., Ž11., Ž12.: Hi Ž.., Lsi Ž.. and Lhi Ž... In other words, the second stage ex ante decision for k i derives from the maximization of the following indirect utility function
n i Ž a ,k i ,t , M . s pi u i wi Ž 1 y t . Lsi Ž . . q M y p i Hi Ž . . yqi Hi Ž . . , Hi Ž . . , Lsi Ž . . q Ž 1 y pi . u i wi Ž 1 y t . Lhi Ž . . qM y p i Hi Ž . . ,0, Lhi Ž . .
Ž 13 .
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3.3. Third stage The government chooses t, M and a , given each individual decision for k i , and the functions Hi Ž.., Lsi Ž.. and Lhi Ž.., i s 1, . . . n. Therefore, the third stage ex ante decisions on a , t and M arise from the maximization of a welfare function, which we assume of the utilitarian type, 9 W s Ýn i Ž a ,kUi ,t , M .
Ž 14 .
i
subject to the government budget constraint ŽEq. Ž8.. and to the first-order condition of the previous stage-maximization for kUi .
4. The individual’s equilibrium private coinsurance rate It is useful, in order to solve the second-stage maximization problem, to state some preliminary comparative statistical results with the following Proposition 1. Eqi Ži. s Ž1 y a .; Ek i Ep i Žii. s yŽ1 y a .Ž1 q d . pi ; Ek i EHi Žiii. ' Hi k s ŽEHirEqi .ŽEqirEk i . s Hi q Ž1 y a .; Ek i En i Ž iv . ' n i k s y Ž g i s y g i Ž 1 q d .. H i Ž 1 y a . y g i p i H i k , Eki where g i ' pi g is q Ž1 y pi .g ih. Proof. Ži. and Žii. derive from straightforward direct computation, while Žiii. arises from Eq. Ž10.. Furthermore, by differentiating Eq. Ž13. with respect to k i and then applying the Envelope theorem, we obtain this modified Roy identity:
y i k s ypi g is Ž Ž EqirEk i . q Ž Ep irEk i . . Hi y pi g is Ž qi q p i . Hi k q pi g isqi Hi k y Ž 1 y pi . g ih Ž Ep irEk i . Hi y Ž 1 y pi . g ihp i Hi k Thus, by substituting and rearranging the terms, we reach Živ..
B
9 We refer to this type of welfare function only for simplicity; any concave Bergson–Samuelson welfare function works equally well. Of course we are following a strictly normative approach with its usual shortcomings and hitches stressed by Public Choice literature. In particular, the implicit hypothesis of ‘‘benevolent’’ and ‘‘welfarist’’ social decision-making, given the complexity of the actual context, seems to be quite strong.
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Discussion With 0 - a - 1, Ži. and Žii. of Proposition 1 are positive and negative, respectively. By assuming health service to be a normal good Žwith a decreasing demand schedule., the partial derivative in Žiii. is negative; that is, the standard result by which an increase of the coinsurance rate Žthe ‘‘price’’. reduces health expenditure. According to Živ., the net welfare loss for individual i due to an increase of the coinsurance rate has two conflicting components. The first is related to the greater expected cost of risk deriving from the health expenditure not covered by social insurance; the term is negative if the insurance premium may be considered ‘‘fair’’ Ži.e., the loading factor d is negligible., as g is ) g i Žthe MUI in illness is greater than expected MUI.. The second component represents the expected gain from the reduction of private insurance premiums due to the health expenditure decrease; the term, given the normality of H, is positive. We now solve the private insurance problem in the second step, when each consumer i makes herrhis own choices. Proposition 2. In the described mixed insurance system, the optimal priÕate coinsurance rate kUi , i s 1, . . . n, is giÕen by the condition
Ž i.
Ž1yki .
where qi '
ki
s qire i
g is y g i Ž 1 q d . gi Ž 1 q d .
Ž 15 . , e i ' yŽEHirEqi .Ž qirHi . ) 0, or by the condition
Ž ii . Covp Ž g i , Hi . s yg i Ž 1 y k i . Ž 1 q d . Hi k qdg i Hi
Ž 16 .
where Covp Žg i , Hi . ' pi g is Hi y g i Hi ) 0. Proof. n i k s 0 is the condition of equilibrium coinsurance rate: k i s kUi ; so from Ži. – Živ. of Proposition 1, by rearranging the terms, we have y Ž g is y g i Ž 1 q d . . pi Hi Ž 1 y a . s g i Ž 1 y a . Ž 1 y k i . Ž 1 q d . pi Hi k As a result, taking into account the definition of health care demand elasticity e i and the definition of covariance Covp , we immediately reach conditions Ž15. and Ž16.. B Discussion Condition Ž15. may be interpreted according to the one-consumer optimal Ramsey tax rule. The left hand side ŽL.H.S.. of Eq. Ž15. may be considered as an optimal ad Õalorem subsidy rate which is inversely correlated to the elasticity of demand wBesley, 1988; Blomqvist, 1997; Zweifel and Breyer, 1997, p. 192x. qi ,
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the coefficient of proportionality of R.H.S. of Eq. Ž15., is represented by the relative difference between MUI in illness and the expected MUI, the latter weighted with the loading factor. If this is negligible Ž d 0., qi ) 0 and, with moral hazard, 0 - kUi - 1. 10 Recall that the First Best solution requires g is s g ih and Hi k Žno moral hazard., thus p i s pi and kUi s a U s 0: the risk is entirely insured on the competitive insurance market. The L.H.S. of Eq. Ž16. is the gain from risk-sharing ŽRSG. due to a marginal reduction of the coinsurance rate Ž risk-pooling . and is measured by the covariance between marginal utility of income g i and health expenditure Hi . The term is positive, as explained in literature on optimal insurance. 11 The R.H.S. has two components: the marginal cost of moral hazard ŽMHC. associated with a reduction of the coinsurance rate Ža deadweight loss. and the loss associated with the expected loading and administrative expenditures. We see, in Eqs. Ž15. and Ž16., that the degree of social insurance coverage does not appear explicitly in the formula for the optimal choice of kUi . However, this does not mean that in equilibrium kUi is, in this mixed health insurance, independent of a , but simply that the choice of the optimal individual value of coinsurance rate follows the same rule as in a purely private health insurance policy Ž a s 0.. In other words, it reaches a marginal condition of standard equilibrium in an insurance market with ex post moral hazard and two states of the world — RSG s MHC — which is the equilibrium of the contract signed in the second step. Actually, conditions Ž15. and Ž16. implicitly define a function of the type
™
k i s k i Ž a ,t , M . ,
i s 1, . . . n
Ž 17 .
which gives rise, in a certain sense, to the ‘‘reaction function’’ of private insurance contracts with respect to social policy instruments. Particularly crucial in our analysis is the partial derivative ŽEk irEa . ' k i a , which specifies a sort of substitution relationship between private and social insurance coverage. We may reasonably assume that k i a ) 0, because it is easy to realise that an increase of the social insurance coverage rate Žan extension of health services paid by the NHS. should increase the co-payment rate of private insurance. With sai ' k i a Ž ark i . we denote the elasticity of k i with respect to a .
10 By observing Eq. Ž15. it is straightforward to see that the optimal rate of coinsurance is ceteris paribus increasing along with health risk Žindividuals with higher probability of illness agree to stipulate an insurance contract with a higher coinsurance rate. as well as with the elasticity of demand for medical care ŽZweifel and Breyer, 1997, p. 194.. 11 Generally, it is assumed that sicker people consume more medical services Žas it is, of course, in our case., so that a sufficient condition for a positive covariance is that MUI increase in moving from health to illness. See for instance Besley Ž1988. and more recently Jack and Sheiner Ž1997..
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5. The government’s choice of social insurance rate The solution, with regard to social insurance coverage, is obtained by applying the following maximization Max a W s Ýn i Ž a ,k i ,t , M . i
s.t.
aÝ pi Hi s tÝwi L i y nM i
i
k i s k i Ž a ,t , M . ,
i s 1, . . . n.
It is also relevant to determine the conditions for the optimal tax rate tU and optimal transfer M U , since the mix between social and private insurance and the distributive effects of the former are bound to change in line with the parameters of linear income taxation; we shall discuss this further on. As in Cremer and Pestieau Ž1996. paper, for the solution of government maximization, we require that 0 F a F 1 Žand also 0 F t F 1.. Later we shall also consider the conditions by which 0 - a U - 1; for the moment, we will simply assume that this the case. Before solving the government maximisation problem for an interior positiÕe solution of a U we need to establish some useful comparative static results. Proposition 3
Ž i.
Eqi Ea
s yk i q Ž 1 y a . k i a s ybi k i
Ž 18 .
where bi ' w1 y sai Ž1 y a .ra x
Ž ii .
Ep i
s y Ž 1 y k i . pi y Ž 1 y a . pi k i a Ž 1 q d .
Ea
s ypi w 1 y bi k i x Ž 1 q d .
Ž iii . Ž iv .
EHi Ea Ey i Ea
' Hi a s Ž EHirEqi . Ž EqirEa . s yHi q bi k i
' y i a s g i Hi Ž 1 q d . q Ž g is y g i Ž 1 q d . . bi k i Hi y g ip i Hi a
where Õi (a ,t,M) ' n i (a ,k i (a ,t,M),t,M). Proof. Eq. Ž18.i and Eq. Ž18.ii derive from direct computation, after having inserted Eq. Ž17. in the relevant functions; Eq. Ž18.iii comes from Eq. Ž10. and Eq. Ž18.i.
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Furthermore, by differentiating Eq. Ž13. with respect to a and then applying the Envelope theorem to this particular context, we obtain the following modified Roy identity:
yi a s n i a q n i k k i a s ypi g is Ž EqirEa . q Ž Ep irEa . Hi y pi g is Ž qi q p i . Hi a q pi g isqi Hi a y Ž 1 y pi . g ih Ž Ep irEa . Hi y Ž 1 y pi . g ihp i Hi a Therefore, by substituting Eq. Ž18.i and Eq. Ž18.ii and some further standard manipulations, we reach Eq. Ž18.iv. B Discussion Eq. Ž18.i is the change in the price of out-of-pocket health expenditure following an increase in the social insurance rate. The term is negative if bi ) 0, or if sai - Ž arŽ1 y a ..; so, for a sufficiently inelastic reaction of private insurance to its social counterpart, an increase in the social insurance rate decreases the out-of-pocket unit cost of health expenditure. Eq. Ž18.ii illustrates the response of the private premium coefficient to the social insurance rate and it is negative as we have assumed k i a ) 0. Note that, from Eq. Ž18.i and Eq. Ž18.ii we have 0 - bi k i - 1, consequently, by means of Eq. Ž18.iii and the normality hypothesis, we conclude that the health expenditure increases with social insurance. According to Eq. Ž18.iv, the net welfare gain for individual i of an increase of the social insurance coverage is determined by three elements: the first one Žpositive. is the expected gain by the increase of the ‘‘subsidy’’ on H; the second one Žpositive if d is negligible. represents the expected gains of risk-pooling for the health expenditure not covered by private insurance; the third one Žnegative, giving the sign of Eq. Ž18.iii represents the welfare loss from the increase in the private insurance premium due to the health expenditure increase. In order to lend more intuition to the F.O.C.s of government optimization we are going to calculate, we may consider the special case of partial equilibrium analysis, i.e., no income effects on the demand for health care and independent demand schedules for both health care and leisure. In the optimal taxation literature this case is usually referred as the simplified Feldstein framework ŽAtkinson and Stiglitz, 1980, p. 388. and is, despite its strict implied restrictions on preferences, also considered in several papers on health insurance Žsee, e.g., Besley, 1988; Cremer and Pestieau, 1996.. Before stating the main Proposition we shall summarise our list of assumptions. Assumptions. A1. Hi is a normal good A2. With 0 - a - 1, k i a ) 0 and sai - Ž arŽ1 y a ..;
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A3. There is a partial equilibrium analysis (Feldstein) framework; A4. Loading factor d is negligible. The following Proposition offers the solution of the model, as far as social insurance coverage is concerned. Proposition 4. GiÕen the preÕious assumptions A1–A4, in the described mixed insurance system the optimal social insurance coÕerage rate a U is represented by the following condition:
Ý
Covp Ž bi , bi k i Hi . q n bj H y 1 H s Ýt i Hi a
i
Ž 19 .
i
where bi is the expected social MUI of indiÕidual i, i.e., bi ' pi bis q Ž1 y pi . bih ' Žg irm .; bis ' Žg isrm .; bih ' Žg ihrm .; b ' Ž1rn.Ý i bi , m is the Lagrange multiplier of the goÕernment budget constraint (the marginal social cost of taxation), j H ' (1 r n)Ý i (bi r b)(Hi r H) s CoÕ(bi ,Hi ) q 1 is the distributional characteristic of H and t i ' [bi (1 y a )(1 y k i ) q a ] is a social term attached to the effect of a on i expected health expenditure. Proof. The Lagrangean of the problem is
C s Ýy i y m aÝ pi Hi y tÝwi L i q nM . i
i
i
As the necessary condition for a s a U is ŽEC .rŽEa . s 0, using assumption A3 Žby which, L ija s 0, j s h, s . we obtain
Ýy i a y m Ý pi Hi q aÝ pi Hi a i
i
s0
i
Now, by substituting Eq. Ž18.i, Eq. Ž18.ii, Eq. Ž18.iii and Eq. Ž18.iv, and taking into account assumption A4, the definitions of expected social MUI bi , its average value b, and of the social weight t i , we obtain the following equation:
Ýbi Hi y Ýbi bi k i Hi q Ý pi bis bi k i Hi y Ý pi Hi i
i
s
žÝ
i
i
pi bis bi k i Hi y Ýbi bi k i Hi q
i
i
/ žÝ i
bi Hi y nH
/
s Ýt i Hi a i
The first term of the L.H.S. in parenthesis is equal to Ý i Covp Ž bi , bi k i Hi . and B the second to nw bj H y 1x H, consequently we reach condition Ž19..
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6. Discussion and interpretation of the optimality condition for the social insurance rate 6.1. On the optimal social insurance rate a U The first component of the L.H.S. of Eq. Ž19. may be called the social risk-sharing gain ŽSRSG. from a marginal increase of social insurance coverage, since it arises from the sum of individuals’ covariance terms between bi and bi k i Hi , i.e., between the social Ždivided by m and then normalised in terms of government revenue. MUI and the health expenditure not covered by private insurance. 12 Recall that, as in the traditional optimal taxation framework, the expected social MUI may be thought of as declining in line with the skill parameter w across individuals. The term SRSG gives a measure of the gain for society as a whole from ‘‘buying’’ a further insurance policy with a uniform reimbursement rate for all members; the term is positive, given Proposition 3 and assumption A2, by which 0 - bi k i - 1. Note that the optimal level of social coverage depends on the level of equilibrium of each consumer’s private coinsurance rate, given by Proposition 2, and on this rate’s response to the social insurance level, as from Eq. Ž17.. Therefore, in practice, in order to compute the optimal a U , the government must take into account the contract structure chosen by insured types. The second component of the L.H.S. of Eq. Ž19. is the social redistribution gain ŽSRG., stressing the role of a for vertical equity aims. It is clearly closely linked to the distributional characteristic of goods subject to the standard many-consumers optimal commodities taxation rule ŽAtkinson and Stiglitz, 1980, p. 388.. In this case, Ž j H y 1. is the ‘‘normalised covariance’’ between expected social MUI and expected health expenditure across individuals. As we have stated that the prior is declining with w, it is j H ) 1 Ž- 1. if health expenditure is also decreasing Žincreasing. with w. 13 The sign of SRG also depends on the level of b and this, as we shall see, is dependent on tax parameters; at the moment we may say state that SRG tends to be positive Žnegative. if health expenditure is a necessity Žluxury. and consequently a U tends to increase Ždecrease.. In this three-commodity model social insurance works like a subsidy on a good Žhealth expenditure., in the presence of linear income tax and with an untaxed good, the numeraire Žthe consumption of the private good.. Then the choice of tU , M U and a U follows rules in some sense similar to those specified in the classical problem of optimal indirect taxation and linear direct taxation ŽAtkinson and 12
A similar expression has been obtained by Besley Ž1988. for designing the optimal reimbursement rule of a given health good. 13 As clearly specified by Atkinson and Stiglitz Ž1980, p. 432., this assumptions does not come from the normality property of that ‘‘good’’.
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Stiglitz, 1980, p. 428–435.. Also in our application, as far as distributional objectives are concerned, social insurance plays two roles: first, if health expenditure is a necessity Žluxury. an increase Ždecrease. of a U may increase the progressiveness of the system; second, a reduction of a U may allow the progressive lump-sum payment M to be increased. Therefore, the final rate of social insurance has to be set to balance the two types of considerations. The R.H.S. of Eq. Ž19. is the social moral hazard cost ŽSMHC., as a weighted sum of individual moral hazard effects. It is positive given Eq. Ž18.iii. Note that, by substituting the term t i Žthe social value of an extra unit of health expenditure by i ., we may state SMHCs Ž 1 y a . Ýbi Ž 1 y k i . Hi a q a nHa ,
Ž 20 .
i
so that SMHC is the sum of two types of moral hazard effects: the first comes from private budget constraints and is linked to the sum of individual effects in terms of expected health expenditure increase; the second is determined by government budget constraint and is related to the effect on public expenditure due to social health insurance. Until now we have analysed questions concerning the optimal level of a ‘‘positive’’ and less than one rate of social insurance coverage a U ; nothing has been said of the possibility of obtaining corner solutions. Actually, the level of a U comes from a balance of two opposite effects: the social gain ŽSRSGq SRG. vs. the social cost ŽSMHC. of an increase in the rate. When these terms are finite and positive in absolute terms it should follow that 0 - a U - 1 as result of a social trade-off. In particular, we see that a U ) 0 if, at a s 0, it is SRSGq SRG ) Ý i bi Ž1 y k i . Hi a and a U - 1 if, at a s 1, it is SRSGq SRG - nHa . However, we cannot exclude corner solutions, represented by full social insurance or no social insurance. In the special case where k i s 0 Žno private coinsurance rate., the first term of the L.H.S. of Eq. Ž19. — SRSG — vanishes, so a has only a redistributive effect, SRG, to be equated, at the margin, with the deadweight loss, SMHC, as in the standard general many-consumers Ramsey rule. So, as in that context, if we impose the separability and homotheticity restrictions set by Deaton Ž1979; 1981., it turns out that a U s 0, since the optimal linear income tax is the only instrument to be used for both efficiency and redistribution aims. If, instead, we imagine a lack of moral hazard effect, Eq. Ž19. becomes simply CovŽg i , pi . s 0: a U should be increased as long as this is beneficial for redistribution, given that it has no deadweight loss. As in the Cremer and Pestieau Ž1996. theoretical context, if that covariance is always positive it is a U s 1. 6.2. On interdependence with optimal tax parameters tU and M U Integration between the optimal social insurance rate and optimal linear taxation parameters as redistributive instruments is more evident when observing the
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conditions for M U and tU . These, in a Feldstein framework and with ‘‘fair’’ premiums Žassumption A3 and A4., are, respectively 14 b y Ž 1rn . ÝCovp Ž bi ,k i M Ž 1 y a . Hi . s 1
Ž 21 .
i
ÝCovp Ž bi ,k i t Ž 1 y a . Hi . q n
t sy 1yt
i
b Cov Ž bi ,Yi . q 1 y 1 Y
ž
/
Ý´ Li Yi i
Ž 22 . ´ iL
where is the elasticity of the expected labour supply at the marginal net wage and CovŽ bi , Yi . s Ž1rn.Ý i Ž birb .Ž YirY . y 1 is the ‘‘normalised covariance’’ between expected social MUI and expected gross labour income across individuals. Thus, according to Eq. Ž21. b ) 1 if the response of the private coinsurance rate is positive with respect to an increase of the lump sum payment, i.e., if, for all i, k i M ) 0, then Covp Ž bi , k i M Ž1 y a . Hi . ) 0, the latter being the risk-sharing term referring to the ‘‘new’’ out-of-pocket health expenditure. The condition Ž21. says that the lump-sum payment should be adjusted so that the social marginal valuation of the transfer of a unit of numeraire, net of this covariance term, should on aÕerage be equal to the cost Ž1 pound.. With b ) 1 the role of optimal M U is to reinforce the progressiveness of the system to be pursued through a U because SRG in Eq. Ž19. tends to increase, given the higher weight attached to j H . As far as the optimal marginal rate is concerned, it may be noted that in Eq. Ž22. Covp Ž bi , k i t Ž1 y a . Hi . - 0 if the response of the coinsurance rate is negative with respect to an increase in the marginal tax rate: k i t - 0. Therefore, given that, as reasonably assumed in optimal taxation literature, it is ´ Li ) 0 and CovŽ bi , Yi . - 0, we have 0 - tU - 1 if b - Ž1rŽCovŽ bi ,Yi . q 1... Hence, we cannot, given the integration with social insurance, exclude a corner solution, i.e., a zero optimal marginal tax rate.
7. Summarising and concluding remarks In the previous sections we modelled a National Health Service in which compulsory social insurance, covering a package of essentials, is integrated by a 14
These derive easily from the F.O.C.s CM s 0, Ct s 0 and by taking into account Eq. Ž17. for computing, by means of the Envelope theorem, respectively, y i M and y i t . Both conditions Ž20. and Ž21. extend to this case the standard formula designing the parameters of the optimal linear income tax ŽAtkinson and Stiglitz, 1980, p. 407.. For a relevant reference to the integration between social insurance and optimal linear income taxation in a somewhat different model see Blomqvist and Horn Ž1984..
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private policy topping up the remaining services, with co-payments by patients. Thus, an individual’s health expenditure is divided in three parts: the first is covered by social insurance, the second by a private insurance policy and the third is out-of-pocket expenditure. The social insurance is financed by a linear income Žpay-roll. taxation whose structure is optimally chosen, together with the social insurance coverage, by the government. This considers n types of individuals distinguished by two parameters, the probability of illness and skills Žwage in efficiency units.. There is ex post moral hazard with regard to health expenditure and adverse selection in the sense that the individual’s skill parameter is not observable by the government. Moreover, although the probabilities of illness are however known, the government cannot differentiate its tax policy between high-risk and low-risk individuals, but only according to their personal incomes. In this framework we have obtained the following health care policy conclusions. The individual’s equilibrium private coinsurance rate follows the standard condition of optimal insurance with ex post moral hazard. This requires equalising, at the margin, the gain of risk-sharing with the deadweight loss due to moral hazard effect for each individual. In any case, the private insurance contract signed by each individual includes a coinsurance rate which is a function of social policy instruments Žpayroll-tax and social insurance rate.. Such a function has a meaning similar to that of a following ‘‘reaction function’’ in a Stackelberg framework. On the subject of optimal social insurance coverage, we have reached some formal results which almost duplicate mere common intuition. First, the higher the distributional characteristic of health services, the higher the optimal rate, as in the standard many-consumers optimal commodity taxation model. Indeed, social insurance coverage works like a subsidy applied to health expenditure at a uniform rate. It is also higher the higher is the sum of individuals’ gains of risk-sharing, referring to out-of-pocket health expenditure. The gain from risk-pooling is in some sense ‘‘socialised’’ in two ways: first, by following a utilitarian welfare function rule, i.e., summing up the individual covariances between marginal utility of income and out-of-pocket health expenditure; second, by considering, as in the optimal taxation framework, each individual’s ‘‘social’’ Ži.e., measured in terms of government revenue. marginal utility of income. It is noteworthy that the optimal level of social coverage indeed depends, marginally, on the equilibrium level of each consumer private coinsurance rate and on the response of each reaction function Ža sort of strategic substitution elasticity term.. Therefore, from the information requirements point of view, the government, in order to compute the optimal social insurance rate, must infer the private contract typology Žspecifically the co-insurance rate and its degree of response to social insurance coverage. signed by the different types of insured individuals. Both effects of pushing up the coverage rate, the social gain of risk-sharing and
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the social redistributive effect, are balanced by the aggregate Žsocial. moral hazard effect. The analytical results of this paper might suggest some implications for modelling a mixed insurance system in industrialized countries. First, we have seen that a positive, but less than one, social insurance rate may be justified on the basis of efficiency and equity considerations. However, as long as health expenditure is treated as a luxury good, whose consumption is largely concentrated on the wealthy Žas is often bound to happen in industrialized countries., the latter consideration is of reduced relevance. Therefore, because the moral hazard ‘‘third-payer’’ effect proves to be a strong deterrent, the principal implication is the need to limit social insurance coverage. This recalls the attempt recently made in some institutional frameworks 15 to introduce forms of rationing and selection of services and treatments to be covered by social insurance according to their cost-effective performance. We must ultimately admit that our analysis is strictly normative Žreferring to a pure ‘‘welfarist’’ point of view as well. and unable to consider some relevant flexible alternatives in social insurance systems — such as the possibility of placing restrictions on individuals’ rights to acquire supplementary private insurance on one hand and, of allowing opt-out solutions on the other — which may improve the efficiency of a mixed system. In the course of the paper we have suggested some possible extensions of our model in these directions which may serve as interesting starting points for future research.
Acknowledgements I would like to thank Alessandro Balestrino, Giuseppe Pisauro and two anonymous referees for helpful comments and suggestions.
References Atkinson, A.B., Stiglitz, J.E., 1980. Lectures on Public Economics. MacGraw Hill, New York. Barr, N., 1998. The Economics of the Welfare State, 3rd edn. Oxford Univ. Press, Oxford. Besley, T., 1988. Optimal reimbursement health insurance and the theory of Ramsey taxation. Journal of Health Economics 7, 321–336. Besley, T., 1989. Publicly provided disaster insurance for health and the control of moral hazard. Journal of Public Economics 39, 141–156. Besley, T., Gouveia, B., 1994. Alternative systems of health care provision. Economic Policy, October, 203–258. Blomquist, S., Christiansen, V., 1998. Topping-up or opting-out? The optimal design of public provision schemes. International Economic Review 39, 399–411.
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See in this respect the Dutch health care reform proposal ŽVan de Ven, 1995..
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Blomqvist, A., 1997. Optimal non-linear health insurance. Journal of Health Economics 16, 303–321. Blomqvist, A., Horn, H., 1984. Public health insurance and optimal income taxation. Journal of Public Economics 24, 353–371. Blomqvist, A., Johansson, P.-O., 1997. Economic efficiency and mixed publicrprivate insurance. Journal of Public Economics 66, 505–516. Boadway, R., Marchand, M., Sato, M., 1998. Subsidies versus public provision of private goods as instruments for redistribution. Scandinavian Journal of Economics 3, 545–564. Cremer, H., Pestieau, P., 1996. Redistributive taxation and social insurance. International Tax and Public Finance 3, 281–295. Deaton, A., 1979. Optimally uniform commodity taxes. Economic Letters 2, 357–361. Deaton, A., 1981. Optimal taxes and the structure of preferences. Econometrica 49, 1245–1260. Jack, W., Sheiner, L., 1997. Welfare-improving health expenditure subsidies. American Economic Review 87 Ž1., 206–221. Ma, C.A., McGuire, T.G., 1997. Optimal health insurance and provider payment. American Economic Review 87 Ž4., 685–704. Manning, W.G., Marquis, M.S., 1996. Health insurance: the trade-off between risk pooling and moral hazard. Journal of Health Economics 15, 609–639. Mooney, G., Ryan, M., 1993. Agency in health care: getting beyond first principles. Journal of Health Economics 44, 125–135. Pauly, M.V., 1974. Overinsurance and public provision of insurance: the roles of moral hazard and adverse selection. Quarterly Journal of Economics 88, 44–62. Rochet, J.C., 1991. Incentives, redistribution and social insurance. The Geneva Papers on Risk and Insurance 16, 143–165. Selden, T.M., 1993. Should the government provide catastrophic insurance?. Journal of Public Economics 51, 241–247. Selden, T.M., 1997. More on the economic efficiency of mixed publicrprivate insurance. Journal of Public Economics 66, 517–523. Spence, M., Zeckauser, R., 1970. Insurance, information and individual action. American Economic Review 61, 380–387. Van de Ven, W.P.M.M., 1995. Regulated competition in health care: with or without a global budget?. European Economic Review 39, 786–794. Von der Schulenburg, J., 1995. The relationship between basic and complementary health system: theoretical considerations. In: Schneider, M. ŽEd.., Complementary Health Schemes in the European Union. Augsburg Basys. Zweifel, P., Breyer, F., 1997. Health Economics. Oxford Univ. Press, Oxford.