A note on health insurance under ex post moral hazard

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Introduction
The linear coinsurance problem, originally examined by Mossin (1968), plays an important role in the analysis of economic and …nancial decisions under risk, and this is for at least two reasons.Firstly, this model is suitable for tractable comparative statics analysis, in order to study wealth and income e¤ects on insurance demand in various settings (e.g., with or without background risk, in a static or dynamic setting, etc...).Secondly, its conclusions can be straighforwardly adapted to the analysis of static portfolio choices when agents can invest in one risk-free asset and in one risky asset.An important property of this model states that the individual's degree of risk aversion with respect to wealth in the sense of Arrow-Pratt goes hand in hand with a higher optimal coinsurance rate: more risk averse individuals choose a higher coinsurance rate.
In this note, we will show that this property does not hold for health insurance under ex post moral hazard.There is ex post moral hazard in medical insurance when insurers do not observe the severity of illness and policyholders may exaggerate their health care expenses -Arrow (1963), Pauly (1968) and Zeckhauser (1970).This should be distinguished from ex ante moral hazard that occurs when the insurance contract distorts the policyholder's incentives to make precautionary e¤orts.Linear coinsurance under ex post moral hazard (i.e., when insurers pay the same fraction of the health care cost whatever the individuals' expenses) has been considered by many authors, including Zeckhauser (1970), Feldstein (1973), Arrow (1976), and Feldman and Dowd (1991) to analyze the trade-o¤ between two con ‡icting objectives: providing risk coverage on one side, and incentivizing policyholders to moderate their health expenses on the other side.
In order to show that ex post moral hazard breaks the link between the degree of risk aversion and the optimal coinsurance rate, we will proceed through a simple model, in which utility depends on wealth and health in an additive way and where the utility derived from health is linear.Furthermore, the only private information of individuals is about the severity of their illness.All other preference parameters, including health risk exposure and risk aversion are either observed by insurers, or rather recovered from observable variables such as age, education, occupation, marital status or from past loss experience.These very crude assumptions are obviously not chosen for the sake of realism, but because they allow us to focus on the ex post moral hazard problem in a simple way, without interfering with adverse or advantageous selection issues.
It will turn out that, in this model, the positive e¤ect of absolute risk aversion on the optimal coinsurance rate may vanish.In particular, an increase in initial wealth does not a¤ect the optimal coinsurance rate even if the policyholder displays decreasing or increasing absolute risk aversion.We will also consider a computable example with constant absolute risk version w.r.t.wealth.In that case, the optimal coinsurance rate does not depend on the degree of absolute risk aversion: it is fully determined by the probability distribution of health states, independently of the policyholder's risk aversion.
The intuition for these results goes through two e¤ects of an increase in the coinsurance rate.On one hand, for a given pattern of health care expenses, a larger coinsurance rate o¤ers a better risk protection to risk averse individuals: thus, the larger the degree of risk aversion, the larger the bene…t drawn for this more complete risk coverage.This is the standard channel that links together the intensity of risk aversion and the optimal insurance coverage.On the other hand, an increase in coverage exacerbates health care overexpenses, and ultimately this leads to an increase in the cost borne by the policyholder.From this standpoint, coinsurance works as a self-discipline device, and this incentive mechanism will be more bene…cial to more risk averse people.When the index of absolute risk aversion increases, the risk protection e¤ect and the incentive e¤ect push the optimal coinsurance rate upwards and downwards, respectively.This mechanism will be illustrated in two di¤erent ways.In Section 2, we introduce our model of linear coinsurance under ex post moral hazard.We show how risk protection and incentives interact in the determination of the optimal coinsurance rate, and how both e¤ects are a¤ected by the degree of absolute risk aversion.More speci…cally, we also show that changes in the policyholder's wealth do not a¤ect the optimal coinsurance rate, although they may make the policyholder more or less risk averse when absolute risk aversion is not constant.In Section 3, we consider the case where the individual displays constant risk aversion w.r.t.wealth: the optimal coinsurance rate can then be explicitly calculated.It is shown that this rate is independent from the index of absolute risk aversion: it only depends on the probability distribution of health states.In other words, in that case, the variations in the risk protection e¤ect and in the incentive e¤ect exactly balance each other when risk aversion change, and ultimately the coinsurance rate remains unchanged.

Optimal coinsurance under ex post moral hazard
Let us consider an individual whose welfare depends both on monetary wealth R and health level H, with a bi-variate von Neumann-Morgenstern utility function U (R; H).Some preliminary comments have to be made at this stage.Many studies on economic decision-making have focused attention on the e¤ect of risk aversion on optimal choices under risk, including insurance coverage, …nancial choices, prevention behavior and numerous other topics.
In the case of health care, the problem is further complicated by the de…nition of risk aversion itself, because of the bi-variate nature of utility.Firstly, we should distinguish the usual risk aversion for gambles on wealth alone from multivariate risk aversion.Multivariate risk aversion has been considered by several authors, following seminal papers by Keeney (1972) and Richard (1975).Roughly speaking, a decision maker who faces multivariate lotteries with "good" and "bad" outcomes for two attributes is considered multivariate risk-averse if she prefers getting some of the "good" and some of the "bad" to taking a chance on all of the "good" or all of the "bad".She is multivariate risk neutral if she is indi¤erent between these two prospects and she is risk seeking when her preferences are reversed.Multivariate risk preferences do not depend on risk preferences for gambles on any attribute alone.In particular, in our setting, an individual may display risk aversion with respect to her wealth R and be multivariate risk averse or risk seeking w.r.t.R; H.
Secondly, in an expected utility setting, the optimal choice of a decision maker who may substitute an attribute to another one is simultaneously a¤ected by her risk preference for gambles on each attribute alone, and by her multivariate risk aversion.For example, when the two attributes correspond to time-dating of wealth, multivariate risk aversion is often referred to as correlation aversion.In that case, the intertemporal elasticity of substitution depends at the same time on atemporal risk aversion and on correlation aversion. 2e will analyze health insurance choices in a model with a separable utility function with u 0 > 0; u 00 < 0. Thus, the individual is assumed to be risk averse w.r.t.wealth and bivariate risk neutral.The assumption of bivariate risk-neutrality is not made for its realism, but -in addition to technical simplicity -because it allows us to separate the e¤ects of risk aversion w.r.t.wealth from those of bivariate risk averse or risk seeking preferences. 3For the sake of technical simplicity, we also assume v(H) = H; > 0. Hence, the marginal utility of health is constant and equal to , which means that the individual is risk-neutral w.r.t.health.Her Marginal Willingness to Pay for a health improvement is and, for R given, the larger , the larger this marginal willingness to pay. 4ealth may be negatively a¤ected by illness, but it increases with the health care expenses.This is written as where h 0 is the initial health endowment, x is the severity of illness and m is the health care expense level.Illness severity is distributed as a random variable X over an intervall [a; b];with a > 0 and b < h 0 , and the parameters of the problem are such that m 2 [0; 1].Thus, the health level H increases linearly from h 0 x to h 0 when m increases from 0 to 1.
The individual's insurance contract speci…es that a fraction of the monetary expenses are reimbursed and that the insurance premium P is actuarial.In what follows, is called the co-insurance rate.5 Thus, the individual's wealth is R = w (1 )m P: It is assumed that insurers observe all the characteristics of insurance seekers, including their risk exposure and risk aversion.In more concrete terms, insurers are supposed to be able to recover these information through observable characteristics, such as age, gender or level of education. 6et e m(x; ; w; P ) denote the health expenses in state x when the individual owns initial wealth w and she has an insurance contract with coinsurance rate and premium P .She chooses the medical expenses that maximize her utility.Thus, we have e m(x; ; w; P ) 2 arg max Let us consider an interior solution where e m(x; ; w; P ) 2 (0; 1) for all x, is characterized by the …rst-order optimality condition more ‡exibility in order to characterize the attitude toward …nancial risk, independently from the marginal willingness to pay for a health improvement.See Abdellaoui et al. ( 2007) for an experimental approach and Bleichrodt et al. (2000) for an application to medical decision analysis.
for all x.Di¤erentiating (1) yields the partial derivatives of function e m: is the absolute risk aversion index.We assume that there is no transaction cost, and thus the insurer charges actuarial premiums.Hence, we have P = E[ e m(x; ; w; P )]; which gives an equilibrium insurance premium P = e P ( ; w), with be the health care expense, after taking into account the endogenous determination of the insurance premium.We may rewrite the policyholder's expected utility as which is a function of .The optimal coinsurance rate maximizes EU in [0; 1].Using (1) allows us to write the …rst-order optimality condition for an interior optimum 2 (0; 1) as 7 ; w) and e P 0 e P 0 ( ; w).Using ( 1) and ( 7) yields which de…nes the optimal coinsurance rate8 .Risk aversion a¤ects both terms in equation ( 12).The …rst term is and it corresponds to the positive e¤ect of an increase in due to the correlation between health care expenses and the marginal utility of wealth.We may intuitively understand the drivers of this correlation by calculating the derivative of Hence, for a given trajectory x !b m(X; ; w), the larger b A, the larger and thus a psoitive e¤ect on @EU=@ .This is the standard risk protection e¤ect of insurance: wealth is redistributed toward lower income states, and the index of absolute risk aversion measures the gain from such a redistribution of wealth across states.However, in the present case, risk aversion also a¤ects the trajectory x !b m(X; ; w) through an incentive e¤ect.Using (2) allows us to rewrite (13) as which reverses the sign of the relationship between b The second term in equation ( 12) is another component of the incentive e¤ect of coinsurance.It corresponds to the additional insurance premium induced by the change in the policyholder's behavior caused by a unit increase in : for a marginal increase d , this additional expected net payment is the di¤erence between the premium increase e P 0 d and the increase in the insurer's expected cost for unchanged behavior E( b m)d .( 7) shows that this di¤erence is equal to e 1).Multiplying by expected marginal utility of wealth E(u 0 ( b R)) = E(X)=( 1) provides the second term of equation ( 12).The larger E(1= b A), the larger this net expected cost.Put di¤erently if the policyholder is very risk averse, she will react to an increase in the coinsurance rate by moderately increasing her health care expenses, and the net expected cost of this adaptation will be small.
All in all, since risk aversion is a determinant of function !b m(x; ; w), i.e., of the incentive e¤ects of insurance coverage, we cannot easily predict whether more risk aversion leads to a larger or lower optimal coinsurance rate.Going further in this direction requires making additional assumptions, as we will do in Section 3. Here, with in mind the fact that wealth may be an important driver of risk aversion, we may focus on the relationship between the initial wealth w and the optimal coinsurance coe¢ cient .From the implicit theorem, the e¤ect on of a change in w is given by d dw = @ 2 EU=@ @w @ 2 EU=@ 2 ; with @ 2 EU=@ 2 < 0 at a maximum of EU .Hence, d =dw and @ 2 EU=@ @w have the same sign.Using (4),( 5) and ( 8) gives b m 0 w = e m 0 w + e m 0 P b P 0 w = 1 for all x; ; w, and thus @[ b m E( b m)]=@w = 0. Furthermore, (6) gives @E( b T )=@w = 0. We deduce that @ 2 EU=@ @w = 0, which implies d =dw = 0. Hence, an increase initial wealth w -which, for instance, would make the individual less risk averse under DARA preferences -does not a¤ect the optimal coinsurance rate. 9

A computable example
Let us specify preferences furthermore, by assuming that the individual displays CARA preferences w.r.t.wealth.We write u(R) = expf Rg; > 0; where is the index of absolute risk aversion, and we still assume v(H) = H; > 0. In that case, we obtain b R(x; ) = u 0 1 ( x=( 1)) = 1 ln 9 The conclusions of Section 2 have been reached for a given value of parameter , and the optimal coinsurance rate may depend on as well as on function u(R).Since M W P = =u 0 (R), we may consider an exogenously given wealth level R 0 as a reference point, and de…ne M W P 0 = =u 0 (R 0 ) as the reference M W P of the individual.With this de…nition, an individual is fully characterized by function u(R) that represents her preferences among …nancial gambles, by M W P 0 that measures her willingness to pay for a better health and by her initial wealth w.Our conclusion about the invariance of the optimal coinsurance rate w.r.t.initial wealth hold for u(R) and M W P 0 unchanged.By disregarding the constant term h 0 E[X], (14) and ( 18) allow us to write the individual's expected utility as We have V 00 (z) = E[X] z 2 < 0:

Conclusion
Risk aversion may depend on several parameters, including wealth, age, marital status and occupation, among others.Consider the case of a background risk, such as business interruption, assumed to be uninsurable and in force for self-employed people, but not for employees.Under risk vulnerability, such a background risk makes the individual more averse to other independent risks, including health care expenditures.If insurance expenses were perfectly monitored by the insurer, then this background risk would increase the coinsurance rate for health care.In other words, everything else given, self-employed people should choose a more complete health insurance than employees.This may not be the case under ex post moral hazard.
7 (11) is obtained …rst by subtituting b R(x; ) = w (1 ) b m(x; ; w) e P ( ; w) in EU and then by observing that, for all x, the derivative of U with respect to m vanishes when m = b m(x; ; w) because of equation (1).The pointwise derivative of U with respect to is thus written as u 0 ( e R(x; ))[ b m(x; ; w) e P 0 ( ; w)].The optimal coinsurance rate cancels the expected value of this pointwise derivative, which gives (11).