# Nonlinear reimbursement rules for preventive and curative medical care

Helmuth Cremer, Jean-Marie Lozachmeur

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Helmuth Cremer, Jean-Marie Lozachmeur. Nonlinear reimbursement rules for preventive and curative medical care. Journal of Health Economics, 2025, 103, pp.103049. 10.1016/j.jhealeco.2025.103049. hal-05464902

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Revised version

July 2025

# "Nonlinear reimbursement rules for preventive and curative medical care"

Helmuth Cremer and Jean-Marie Lozachmeur
Nonlinear reimbursement rules for
preventive and curative medical care<fnref n="1" />

Helmuth Cremer<fnref n="2" />
and Jean-Marie Lozachmeur<fnref n="3" />

June 2024, revised July 2025

<footnote n="1">¹We thank the two Reviewers and the Editor, Odd Rune Straume, for their constructive and helpful comments. This paper has been presented at the PET 2024 conference in Lyon; we thank the participants for their comments. Financial support from the Chaire “*Marché des risques et creation de valeur*” of the FdR/SCOR is gratefully acknowledged. We also gratefully acknowledge the fundings received by TSE from ANR under grant ANR-17-EURE-0010 and the funding received by IHU HealthAge from ANR under grant ANR-23-IAHU-0011 (Investissements d'Avenir program).</footnote>

<footnote n="2">²Toulouse School of Economics, University of Toulouse Capitole, France.</footnote>

<footnote n="3">³Corresponding author. Toulouse School of Economics, CNRS, email: jean-marie.lozachmeur@tse-fr.eu.</footnote>
Abstract

This study examines nonlinear reimbursement rules for secondary preventive and ther-
apeutic care. Individuals may be healthy or ill, with illness severity determining their
ex post type. Preventive care is chosen beforehand, while curative care is decided after
health status is known.

In an ideal scenario where health status is observable, optimal insurance provides
lump-sum payments unrelated to expenditures. However, when severity is unobservable
(causing ex post moral hazard), this approach is not incentive-compatible. Instead,
optimal insurance designs benefits that increase with both preventive and curative care,
as higher expenditures reduce informational rents and align incentives.

Preventive care, though chosen before illness occurs, affects incentive constraints
due to two factors: (1) it is more effective for severely ill individuals, and (2) they have
lower marginal utility of income, meaning preventive expenditures impact them less.
These effects shape the optimal reimbursement structure.

Additionally, when individuals misperceive preventive care benefits, the main results
hold, but an extra corrective (Pigouvian) term appears in the reimbursement formula
to adjust for this misperception.

JEL Codes: I11, I13, I18.

Keywords: *ex post* moral hazard, health insurance, secondary prevention.
# 1 Introduction

There are several ways individuals can protect themselves against risks. An obvious way is to transfer the risk to a third party via insurance without modifying the risk itself. Another option is to engage in prevention that is to act directly on the risk by altering either its occurrence or its consequence. Affecting the occurrence means changing in the probability to be in a bad state of nature. Alternatively (or in addition), it may be possible to reduce the utility loss in case of a negative shock. The study of prevention started with the seminal work of Ehrlich and Becker (1972), which has led to a flourishing literature in the field of risk and insurance economics.<fnref n="1" /> Most papers deal with positive questions for instance, how an increase in risk aversion affects both levels of prevention; see Dionne and Eeckhout (1985).

The literature on health insurance design has predominantly concentrated on curative or therapeutic care. This in itself is a complex issue. Even if one abstracts from redistributional considerations and supply side effects associated with imperfect competition, the appropriate insurance coverage is not a trivial problem. Because of asymmetric information there is typically a tradeoff between insurance coverage and *ex post* moral hazard which can be mitigated by an appropriate design of reimbursement schemes and the use of copayments. The earlier literature, concentrates on linear reimbursement rules; see for instance Besley (1988). More recent papers consider general, nonlinear policies; see Blomqvist (1997) or Martinon *et al.* (2018).<fnref n="2" />

The insurance coverage of preventive care has received less attention. In practice it varies from country to country and is influenced by the specific healthcare system, the role of public and private insurance, and societal values and priorities.

One can distinguish two types of preventive care. Primary prevention reduces the probability of illness. That is, it aims at reducing occurrence of diseases and health

<footnote n="1">¹For a review of the literature, see Courbage et al., 2013.</footnote>

<footnote n="2">²For a survey see Zweifel *et al.* (2009), Ch. 6.</footnote>
conditions before they develop. Examples include behavioral patterns like exercise, a balanced diet, not smoking or limiting the alcohol intake. While insurers can try to promote primary prevention via counseling and education, potential insurance coverage is limited by the fact that it is typically not observable or at least not contractible (verifiable).<fnref n="3" />

Secondary prevention refers to measures aimed at detecting and treating diseases and conditions in their early stages to prevent further progression or complications. In other words, it does not affect the probability of illness but is intended to reduce its severity. Examples include checkups and diagnostic screening like mammographies colonoscopies, pap smears, blood tests, and other tests that can detect cancer, diabetes, heart disease, dental care and other health conditions. Secondary prevention is typically observable and verifiable so that it can be covered by an insurance scheme.<fnref n="4" /> Vaccines are another prominent example of prevention, both primary and secondary. They reduce the probability to contract a disease and they reduce the severity of the disease if nevertheless infected. Vaccines are at least potentially observable but raise specific problems which go beyond the scope of this paper. In particular, they create a positive externality and even when they are available for free adherence may be too small. And political considerations often imply that mandates are not a realistic option.

We study the design of reimbursement rules of preventive and curative (therapeutic) care. Our approach is normative and we study a mandatory social insurance scheme that maximizes expected utility. Because most of primary prevention is not verifiable

<footnote n="3">³Some types of behavior (like smoking, lack of exercise) may not just increase the probability of illness but also increase the severity of otherwise unrelated affections. Consequently, they are in part also secondary prevention. In this study we concentrate on secondary prevention which is verifiable.</footnote>

<footnote n="4">⁴The insurance scheme may include copayment rates or a direct provision of secondary prevention. For example, in the US co-payment rates apply to recommended diagnostic follow-up tests (e.g. see Ngo et al, 2023) or blood pressure monitoring (see Desai et al., 2020). This also the case for secondary prevention targeted to frail and old individuals (see Tavassoli, N. et al., 2023). Finally, the insurance mechanism that we study can include a direct provision of prevention. This is just a matter of implementation. We come back to this issue in section 2.3.</footnote>
we concentrate on secondary prevention.<fnref n="5" /> The main contribution of this paper is that it considers nonlinear policies. In other words, we determine the best policy given the information available to the insurer. Most of the existing literature restricts policies to be linear (affine). This is an *ad hoc* assumption which is not justified by the information structure. For curative care, a notable exception is Blomqvist (1997) who studies nonlinear reimbursement rules, albeit for a somewhat restrictive utility function. This approach is formally similar to Mirrlees (1971) who studies redistribution between agents differing in their unobservable productivity. Beyond the technicalities, a major result is the asymmetric information limits possible redistribution. Roughly speaking the problem is that when individuals have a low income, one cannot discern whether this is because they have a low earning ability or because they have a high ability but do not work much. In our model, agents are ex ante identical but differ ex post in their unobservable severity of illness. Consequently, ex post, insurance redistributes across states of nature and thus across individuals.

To our knowledge, few papers consider therapeutic and secondary preventive care. The most noticeable example is Barigozzi (2004) who also considers secondary prevention along with therapeutic care. As will become clear below the generalization to nonlinear rules is not just of methodological interest; it also has a drastic impact on the results. With linear rules, Barigozzi shows that while treatment expenses should always be subsidized this is true for prevention if and only if prevention reduces the cost of treatment, that is in the case where the two activities are substitutes. We will show that this result is an artifact of the linearity assumption. With nonlinear schemes both types of care should be subsided (at the margin) irrespective of the substitutability or complementarity of prevention and treatment. Intuitively, linear copayments mechanically create substitution effects (direct and across types of care) which can be avoided with nonlinear policies. When reimbursement rules are restricted solely by the available

<footnote n="5">⁵Ellis and Manning (2007) also consider prevention and treatment but concentrate on primary prevention.</footnote>
information, they have to be designed according to their impact on informational rents.<fnref n="6" />

Our model considers a large number (or a continuum) of *ex ante* identical individuals who are endowed with a given disposable income. With some probability they are healthy and do not need any curative (therapeutic) health care. Otherwise they become ill and their health status (the severity of their disease) is realized and identifies their *ex post* type. Preventive care is determined *ex ante*, that is before the health status is determined while therapeutic care is chosen *ex post*.

We study the design of a social insurance scheme that maximizes individual's expected utility subject to the resource constraint, which requires that total contributions (payroll taxes or premium payments) equal expected health insurance benefits. Insurance benefits depend on preventive and curative care in a possibly nonlinear way, and marginal benefits can be positive or negative. A reimbursement policy thus consists of premium, the coverage of prevention, and the reimbursement as a function of services. We first study the case where the health status is ex post publicly observable which yields the first best optimum. Then we turn to the case where severity of disease is not observable to the insurer. Throughout the paper we assume that expenditures on preventive care and health status (healthy or sick) are observable at the individual level. We concentrate on reimbursement policies and thus on the demand side. We do not deal with provider payment schemes.<fnref n="7" />

In the first best (under full information), insurance benefits are flat (lump sum payments) and do not depend on expenditures. When the severity of the disease is not observable, this solution cannot be implemented because individuals in good health would mimick the less healthy individuals. The optimal insurance implies benefits that increase with both types of care. This is because health expenditures reduce informational rents and they are upward distorted. For therapeutic care this generalizes

<footnote n="6">⁶Barigozzi et al. (2017) also consider linear reimbursement rule for preventive care but in a positive, game theoretical setting. Insurance coverage is provided by a risk pooling agreement.</footnote>

<footnote n="7">⁷Consequently, the term policy always refers to reimbursement policy.</footnote>
the result of Blomqvist (1997) and the intuition is easily understood. Less healthy
individuals value care more than healthy individuals. Consequently, an increase in
expenditures on (therapeutic) care relaxes the incentive constraint.

The case of preventive care is more complex. One might at first be tempted to think that a solution would leave the choice of preventive care undistorted and just provide a flat payment like under full information. Indeed, preventive care is chosen *ex ante*, at a point where there is uncertainty but no asymmetric information. Individuals and insurers alike do not know the (future) realization of the state of health and the severity in case of disease. Consequently it is not immediately obvious what positive effect a distortion might bring about. Our formal analysis shows that this first intuition is misleading; prevention does have an impact on the incentive constraint and thus on informational rents. Specifically as preventive care increases, utility decreases less fast with the severity of the disease so that the rents enjoyed by healthier individuals decrease. Intuitively this is due to two concurring effects. First, prevention is more effective for the more severely ill. Second, these individuals also have a lower marginal utility of income so that a given level of expenditure on preventive care has less impact on their utility.

We present the model in Section 2. The optimal contract under full information is studied in Section 3, while Section 4 studies the case where the severity of the disease is not publicly observable. Throughout these sections we assume that individuals are rational and well informed so that they correctly assess the effect of prevention on the utility loss in case of illness.

In a final section, we consider an extension in which individuals may misperceive the
effectiveness of prevention. We show that our results remain valid, but that there is now
an extra corrective (Pigouvian) term in the expression for the marginal reimbursement
of preventive care.

Misperception is a widely discussed issue in the medical literature to explain in-
sufficient levels of prevention; see Carman and Koreman (2014) for a discussion and references. In the economic literature, misperception (and myopia) has been studied mainly in the taxation and pension literature, see for instance Allcott et al. (2019), Cremer and Pestieau (2011) or Farhi and Gabaix (2020).<fnref n="8" /> Leroux et al. (2011) do consider misperception of preventive care but concentrate on primary prevention;<fnref n="9" /> in their setting prevention affects the probability of illness and this effect is misperceived.<fnref n="10" />

# 2 The model

## 2.1 Individuals

There is a large number (or a continuum) of ex ante identical individuals who are endowed with a disposable income $\omega$. With probability $\pi$, they are healthy (state of nature $H$) and do not need any curative (therapeutic) health care. Because we do not consider primary prevention, we assume that $\pi$ is exogenous. The utility of healthy individuals is given by $v(c_0)$, where $c_0$ is net consumption; we assume $v'(c_0) > 0$ and $v''(c_0) < 0$.

With probability $(1 - \pi)$ they become ill (state of nature $S$) and their health status (the severity of their disease) is represented by a parameter $\theta$, which is also used to identify their ex post type. The random variable $\theta$ is distributed over $\Theta \equiv [\underline{\theta}, \bar{\theta}] \subset \Re_+$ with a density $f(\theta)$ and a distribution function $F(\theta)$. Note that a larger value of $\theta$ corresponds to a more severe disease thus a worse health status. Individual of type $\theta$

<footnote n="8">⁸Cremer and Roeder (2017) study insurance design under misperception of risk. They do not consider prevention and concentrate on issues of asymmetric information in the (competitive) insurance market.</footnote>

<footnote n="9">⁹They do not consider the severity of illness and concentrate on a two type model. From these perspectives our model is thus more general than theirs. However, unlike the current paper, they also consider ex ante heterogeneity and income tax design.</footnote>

<footnote n="10">¹⁰Our study of misperception is also related to the literature on sin taxes. People consume for instance sugary beverages because they misperceive (or ignore for reasons of self-control) their potentially harmful effect on their health. Taxing these goods in order to reduce consumption is thus a kind of prevention, though mostly primary because it affects the probability of contracting various health issues. Cremer et al. (2012) combine sin taxes with curative care but they concentrate on linear instruments.</footnote>
has preferences

$$u(c, m, e, \theta), \qquad (1)$$

where c denotes consumption of a numeraire good, m medical expenditures (curative care) and e secondary prevention expenditures. We will be more specific on the timing below, but it is important to note from the outset that preventive care is determined ex ante when individuals commit to a level of prevention, that is before the health status is determined while m is chosen ex post.<fnref n="11" /> Consequently, m can be conditioned on θ while e is by definition the same in all states of nature. An individual’s expected utility is thus given by

$$EU = \pi v(c_0) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u[c(\theta), m(\theta), e, \theta] f(\theta) d\theta. \qquad (2)$$

We assume $u_c > 0$, $u_m > 0$, $u_e > 0$ so that consumption as well as both types of medical care increase utility for any level of θ, and $u_{cc} < 0$ which implies that individuals are risk averse.<fnref n="12" /> Furthermore, we have $u_θ < 0$ which reflects the assumption that a larger θ corresponds to a more severe disease. We also assume $u_{mθ} > 0$ and $u_{eθ} > 0$ so that the marginal benefits of medical care and preventive care increase with the severity of the illness. Consequently, absent of any insurance, individuals with a larger θ choose a larger level of m.<fnref n="13" /> Finally, we suppose that $u_{cθ} \le 0$. In words, the marginal utility of net income decreases with the severity of the illness. Empirically, a strict decrease appears to be the most plausible assumption; see Finkelstein et al. (2013). However, some of the literature, including Blomqvist (1997), assumes $u_{cθ} = 0$ for tractability. Consequently, we did not want to rule out this special case.<fnref n="14" />

$$\frac{\partial m}{\partial \theta} = -\frac{u_{m\theta}}{SOC},$$

<footnote n="11">¹¹As in Ellis and Manning (2007) or other papers treating prevention, we implicitly assume that individuals commit ex ante to a level of prevention e (which is paid for ex post).</footnote>

<footnote n="12">¹²Subscripts refer to partial derivatives.</footnote>

<footnote n="13">¹³Absent of any insurance we have</footnote>

<footnote>where SOC < 0 is the second-order condition for an interior solution which is assumed to hold.</footnote>

<footnote n="14">¹⁴Barigozzi (2004) uses a utility function given by $u(c + H(e, m))$; there are just two states of nature</footnote>
## 2.2 Policy design

We study the design of a social insurance scheme that maximizes individual’s expected utility subject to the resource constraint. This constraint requires that total contributions (payroll taxes or premium payments) equal expected health insurance benefits. Social insurance “covers” both preventive and curative care. To be more precise, benefits depend on $e$ and $m$ in a possibly nonlinear way and marginal benefits can be positive or negative. We first study the case where the health status $\theta$ is ex post publicly observable. Then we turn to the case where individuals $\theta$’s are not observable to the insurer. Throughout the paper we assume that $e$ and the health status, $H$ or $S$ are observable at the individual level.

Note that while we focus on social insurance the same equilibrium would emerge in a private insurance market with identical insurers, perfect competition and free entry. In equilibrium, profits are zero; there is no loading factor. Under these assumptions the problem of a private insurer is to maximize the expected utility of the representative individual under a zero profit constraint, which is exactly the same as that of welfare maximizing social insurance.

## 2.3 Timing

Formally a policy consists of a premium $P$ and a benefit function

$$B(e, m(\theta)) = I(e) + R(m(\theta)), \quad (3)$$

where $I(e)$ is the (positive or negative) reimbursement of preventive care while $R(m(\theta))$ is the reimbursement associated with curative care expenditures. Note that splitting $B$ into two part $I$ and $R$ is done only for the ease of exposition and has no impact on the

<footnote>corresponding to our $S$ and $H$ and the severity of illness is not considered. The preferences considered by Besley (1988), Cremer and Lozachmeur (2022) and by Martinon et al. (2018) are also encompassed by (1); they account for the severity $\theta$ but do not have prevention.</footnote>
results. This is because $e$ is chosen ex ante and is thus the same in all states of nature ex post.<fnref n="15" />

The timing is as follows. First, the social insurer announces the policy {$P, I(e), R(m(\theta))$}, specifying the premium $P$, paid ex ante and the benefit rule defined in (3). Second, individuals choose their level of preventive care $e$. Note that because preventive care is by definition determined ex ante the same level is chosen by all individuals and it cannot be conditioned on $\theta$. Third, the state of nature is realized and the variable $\theta$ is drawn for all individual in state $S$ and revealed to them. Finally, individuals in state $S$ choose their level of health care expenses $m(\theta)$ depending on their health status $\theta$. As mentioned in the introduction, non linear policies defined by (3) can also be implemented via a direct provision of prevention $e$ together with a reimbursement scheme $R(m(\theta))$.<fnref n="16" /> Consequently, a marginal subsidy on secondary prevention is equivalent to a direct provision of prevention that is larger than its first-best value.

## 2.4 Individual problem

To determine the optimal reimbursement policy we shall use a mechanism design approach and determine first the allocation that is induced by this policy. To examine how the optimal policy can be implemented by the considered instruments we have to study an individual’s problem. For a given policy {$P, I(e), R(m(\theta))$}, agents choose $e$ and $m(\theta)$ by solving the following problem

$$ \max_{e,m(\theta)} \pi v[\omega - P + I(e) - e] + (1 - \pi) \int_{\theta}^{\bar{\theta}} u[\omega - P - m(\theta) - e + I(e) + R(m(\theta)), m(\theta), e, \theta] f(\theta) d\theta. $$

<footnote n="15">¹⁵The functional form of $R(m)$ does of course depend on the level of $e$. But nothing can be gained by writing it as a separate argument since ex post it is a constant.</footnote>

<footnote n="16">¹⁶This is the case if secondary prevention is observable and not tradable which is assumed throughout the paper. See Cremer and Gahvari (1997) on this issue.</footnote>
Differentiating with respect to e and m(θ) and rearranging yields

$$MRS_{cm} = \frac{u_m}{u_c} = 1 - R'(m(\theta)) \quad \forall \theta \in \Theta, \qquad (4)$$

$$MRS_{ce} = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} u_c f(\theta) d\theta}{\pi v'(c_0) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_c f(\theta) d\theta} = 1 - I'(e). \qquad (5)$$

Condition (4) states that the marginal benefit of m, expressed in monetary terms must equal its marginal cost accounting for the reimbursement. Since m(θ) is chosen ex post no uncertainty is involved. The interpretation of (5) concerning e is similar. Note that since e is chosen ex ante its benefits are uncertain and depend on the realization of H and θ. When R(m(θ)) = I(e) = 0 and R'(m(θ)) = I'(e) = 0 for all levels of m and e we obtain the laissez-faire solution with no insurance.

For future reference also note that

$$\frac{\partial MRS_{cm}}{\partial \theta} = \frac{u_c u_{m\theta} - u_{c\theta} u_m}{(u_c)^2} > 0. \qquad (6)$$

In words at any given point in the (m, c) space, individuals with a larger θ (who are in worse health) have steeper indifference curves and thus a higher willingness to pay for m. This, in turn implies the single crossing property of indifference curves in the (m, c) plane.

## 3 The full information optimum

Our main focus is of course on the policy design when θ is not observable. To understand its properties, the full information optimum provides an interesting benchmark. Define $d_0 = c_0 + e = \omega + I - P$ and

$$d(\theta) = c(\theta) + m(\theta) + e = \omega + R(\theta) + I - P. \qquad (7)$$
Intuitively, $d(\theta)$ denotes the total resources available to an individual in state $S$ and of type $\theta$, including reimbursement of medial care net of the premium. This budget is allocated to consumption and both types of medical care. The variable $d_0$ has a similar interpretation for an individual in state $H$ for whom it is allocated to consumption and preventive care. For consistency with the solution under asymmetric information we use $d_0$, $d(\theta)$, $m(\theta)$ and $e$ as decision variables. The problem of the social planner is

$$ \max_{d_0, d(\theta), m(\theta), e} \pi v(d_0 - e) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u[d(\theta) - m(\theta) - e, m(\theta), e, \theta] f(\theta) d\theta, \quad (8) $$

$$ \text{s.t.} \quad \omega - \pi d_0 - (1 - \pi) \int_{\theta}^{\bar{\theta}} d(\theta) f(\theta) d\theta \geq 0. \quad (9) $$

In words, we maximize expected utility of a representative individual subject to the resource constraint. Note that the reimbursement policy does not explicitly appear in this problem but is implicitly defined by (7) together with (9) and the definition of $d_0$. Note that combining these two equations we obtain

$$ \pi(I - P) + (1 - \pi) \int_{\theta}^{\bar{\theta}} (R(\theta) + I - P) f(\theta) d\theta = I - P + (1 - \pi) \int_{\theta}^{\bar{\theta}} R(\theta) f(\theta) d\theta = 0. $$

As shown in Appendix A, one obtains

$$ v'(c_0) = u_c \quad \forall \theta \in \Theta, $$

so that marginal utilities of income are equalized in all states of nature. In other words, individuals are fully insured. Furthermore, we show that

$$ MRS_{cm} = \frac{u_m}{u_c} = 1, \quad (10) $$

which from (4) requires

$$ R'(m(\theta)) = 0 \quad \forall \theta \in \Theta. \quad (11) $$
Similarly one has:

$$ MRS_{ce} = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} u_{e} f(\theta) d\theta}{\pi v'(c_{0}) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_{c} f(\theta) d\theta} = 1, \quad (12) $$

so that from (5) we have

$$ I'(e) = 0. \quad (13) $$

In words, (11) and (13) mean that the marginal reimbursement of expenditures on medical care e and m(θ) are equal to zero. Consequently medical care levels are not distorted: their marginal benefits, as measured by the marginal rates of substitution in (10) and (12), are equal to their marginal costs (namely 1).

These properties are not surprising. With θ observable and absent any ad hoc restrictions on instrument the solution is of course first-best efficient. This, in turn requires full insurance and undistorted medical expenses.

Note that with purely linear instruments this would of course not be possible. But with nonlinear instruments and given full information we can give each individual the appropriate flat benefit, which does not directly depends on the individual's expenditures on medical care.

The main results of this section are summarized in the following proposition.

**Proposition 1** When there is full information so that the health status and the severity of the illness are observable and nonlinear instruments are available, the optimal solution, $d_0^*(\theta), d^*(\theta), m^*(\theta), e^*$ implies

(i) Full insurance, so that the marginal utility of consumption is equalized across states of nature.

(ii) The reimbursement rules of curative and preventive care are flat; marginal reimbursement rates are zero: $R'(m) = I'(e) = 0$.
(iii) Medical care levels are not distorted: their marginal benefits are equal to their marginal costs.

# 4 Solution under asymmetric information

Providing each individual with the appropriate flat benefit is of course only possible when $\theta$ is publicly observable. When this is not the case the reimbursement policy has to be based on observable variables and specifically $m(\theta)$ and $e$. Since $e$ is chosen ex ante, it is not immediately obvious that the information asymmetry would be relevant. By contrast, as far as $m$ is concerned, this is obvious. When the reimbursement is based on the individual's level of expenditure there is an obvious problem of ex post moral hazard an issue which is well known in the health economics literature. Specifically a positive reimbursement rate will tend to lead to excessive consumption of care.

When $\theta$ is not observable, the insurance policy must be incentive compatible ex post so that all individuals must prefer their own consumption bundle to that available to any other type. This requires that more healthy individuals receive a larger utility than when there was full insurance. Using the terminology of contract theory one can refer to this extra utility as informational rent. To understand this, note that when there is full insurance, there is full redistribution (equal marginal utilities of consumption) ex post, across states of nature and thus across individuals. Specifically, the healthy individuals pay for the less healthy. But when the severity of illness is not observable, this is not feasible because healthy individuals would have an incentive to mimick the less healthy ones to get a larger benefit. These rents conceded to healthier individuals decrease expected utility because they preclude full insurance.

As usual we solve this problem by first deriving the best incentive compatible allocation and then study how it can be implemented by an insurance policy specifying the reimbursement rules of medical care.
## 4.1 The problem

We continue to use the same decision variables as in the previous section namely $d_0$, $d(\theta)$, $m(\theta)$ and $e$. The problem of the social planner is now given by

$$ \max_{d_0, d(\theta), m(\theta), e} \pi v(d_0 - e) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u[d(\theta) - m(\theta) - e, m(\theta), e, \theta] f(\theta) d\theta, \quad (14) $$

$$ \text{s.t.} \quad \omega - \pi d_0 - (1 - \pi) \int_{\theta}^{\bar{\theta}} d(\theta) f(\theta) d\theta \geq 0, \quad (15) $$

$$ \begin{gathered} u(d(\theta) - m(\theta) - e, m(\theta), e, \theta) \geq \\ u(d(\theta') - m(\theta') - e, m(\theta'), e, \theta), \quad \forall \theta, \theta' \in \Theta \end{gathered} \quad (16) $$

which differs from the problem under full information (8)–(9) in that we have added an incentive constraint for each type $\theta$, equation (16).

## 4.2 The local incentive constraint

To solve this problem we use a first-order approach which leads us to consider a relaxed problem. Specifically we consider a direct mechanism consisting of a bundle $\{d_0, d(\theta), m(\theta)\}$ for each $\theta$. Individuals choose their reported type $\theta'$ which maximizes their utility given their true type and the policy $\{d(\theta), m(\theta)\}$. Formally, they solve

$$ \max_{\theta'} \pi v(d_0 - e) + (1 - \pi) u(d(\theta') - m(\theta') - e, m(\theta'), e, \theta). \quad (17) $$

Using the FOC associated with this problem the local incentive constraint can be written as<fnref n="17" />

$$ \dot{U}(\theta) = (1 - \pi) u_{\theta}(d(\theta) - m(\theta) - e, m(\theta), e, \theta), \quad (19) $$

$$ [\dot{d}(\theta') - \dot{m}(\theta')]u_c[d(\theta') - m(\theta') - e, m(\theta'), e, \theta] + \dot{m}(\theta') u_m(d(\theta') - m(\theta') - e, m(\theta'), e, \theta) = 0, \quad (18) $$

and to derive (19), we have used the fact that to ensure truthful revelation it must be satisfied for $\theta' = \theta$. Intuitively this amounts to using the envelope theorem which implies that the total derivative of $U$ with respect to $\theta$ is equal to the partial derivative.

<footnote n="17">¹⁷The FOC is given by</footnote>
where $U(\theta) = \pi v(d_0 - e) + (1 - \pi)[d(\theta) - m(\theta) - e, m(\theta), e, \theta]$ and where $\dot{U}(\theta)$ denotes the total derivative of $U$ with respect to $\theta$.<fnref n="18" />

The local approach is valid if the second-order condition of problem (17) is satisfied for which a sufficient condition is $\dot{m}(\theta) > 0$. For simplicity we assume that this property holds in equilibrium.<fnref n="19" />

Observe that equation (19) implies $\dot{U}(\theta) < 0$ so that utility decreases with $\theta$. In other words, it increases as $\theta$ decreases. Note that the faster $U$ decreases with $\theta$, the larger are the rents enjoyed by more healthy individuals.

## 4.3 The relaxed problem

As usual in contract theory we determine the solution by using an optimal control approach with $U(\theta)$ as state variable and $m(\theta)$ as control, while also optimizing with respect to $e$ which being set ex ante is not contingent on $\theta$. To be consistent with standard optimal control theory we add the control variable $z(\theta)$ and impose the constraint that $z(\theta) = e$.<fnref n="20" />

<footnote n="18">¹⁸While our model treats illness severity as a continuous variable $\theta$ realized after illness occurs, one might loosely interpret a given illness label (e.g., “COVID”) as a fixed state. However, as the referee rightly points out, preventive actions can affect the nature of the illness itself — that is, “COVID with prevention” and “COVID without prevention” are not equivalent. In our model, such differences are captured through $\theta$: preventive care affects the utility loss associated with illness by modifying the distribution of severity. Hence, we are not claiming that prevention reduces treatment costs within a fixed illness state, but rather that it changes the composition of outcomes conditional on being ill. This distinction is important for interpreting our incentive results, and we are grateful to the referee for prompting this clarification.</footnote>

<footnote n="19">¹⁹If it is violated then the solution involves bunching over some interval(s). As in much of the literature on contract theory and in particular optimal taxation, we neglect this complication as it adds little to the understanding of the underlying economic intuition.</footnote>

<footnote n="20">²⁰As a reviewer correctly pointed out we could simply differentiate the Hamiltonian with respect to $e$ yielding the same result. But since $e$ is not strictly speaking a control variable this would no longer be consistent with standard textbook control theory which requires that the derivatives of the Hamiltonian with respect to the control variables is zero. Consequently it would be necessary to prove that the shortcut is valid which we effectively do by introducing the extra control variable.</footnote>
Formally, the problem of the government can then be stated as follows

$$ \max_{U(\theta), m(\theta), z(\theta), e} \int_{\theta}^{\bar{\theta}} U(\theta) f(\theta) d\theta, \qquad (20) $$

$$ \text{s.t.} \quad \omega - \pi d_0 - (1 - \pi) \int_{\theta}^{\bar{\theta}} d(\theta) f(\theta) d\theta \geq 0, \qquad (21) $$

$$ U(\theta) = \pi v(d_0 - e) + (1 - \pi) u(d(\theta) - m(\theta) - z(\theta), m(\theta), e, \theta), \quad (22) $$

$$ z(\theta) - e = 0, \qquad (23) $$

$$ \dot{U}(\theta) = (1 - \pi) u_{\theta}(d(\theta) - m(\theta) - e, m(\theta), e, \theta). \qquad (24) $$

The Hamiltonian associated with this problem is

$$ \begin{aligned} \mathcal{H} = & f(\theta) U(\theta) \\ & - \alpha(\theta) [U(\theta) - \pi v(d_0 - e) - (1 - \pi) u(d(\theta) - m(\theta) - z(\theta), m(\theta), z(\theta), \theta)] \\ & + \mu [\omega - \pi d_0 - (1 - \pi) d(\theta)] f(\theta) \\ & - \eta(\theta)[z(\theta) - e] \\ & + \lambda(\theta) (1 - \pi) u_{\theta}(d(\theta) - m(\theta) - z(\theta), m(\theta), z(\theta), \theta), \end{aligned} \qquad (25) $$

where $\lambda(\theta)$ is the costate variable associated with equation (24) while $\mu$, $\alpha(\theta)$ and $\eta(\theta)$ are the Lagrange multipliers associated respectively with constraints (21), (22) and (23). The transversality conditions are given by:

$$ \lambda(\theta) = \lambda(\bar{\theta}) = 0. \qquad (26) $$

We show in Appendix B that the solution yields

$$ R'(m) = \frac{\lambda(\theta) \left( u_{\theta m} - u_{c\theta} \frac{u_m}{u_c} \right)}{\mu f(\theta)}. \qquad (27) $$

Recall that $u_{\theta m} > 0$ and $u_{\theta c} \le 0$ and based on the usual properties of Lagrangian multipliers we have $\mu > 0$. In Appendix C we also show that $\lambda(\theta) > 0$ for $\theta \in ]\theta, \bar{\theta}]$.
Consequently, equation (27) implies that $R'(m(\theta)) > 0$ for $\theta \in ]\underline{\theta}, \bar{\theta}[$. Together with $\dot{m}(\theta) > 0$ this means that $R'(m) > 0$ for $m \in ]m(\underline{\theta}), m(\bar{\theta})[$. In words the marginal reimbursement rate of curative health care is positive except at the endpoints of the interval. This implies that, compared to the efficient full information outcome, health care is distorted upward. Intuitively, this distortion is explained by the usual rent reduction effect. To understand this, recall that in this setting, mimicking goes from low $\theta$'s (more healthy individuals) to higher $\theta$'s (less healthy individuals). Furthermore, equation (6) implies that individuals with a higher $\theta$ (the mimicked) have a larger willingness to pay than the mimicking individual with a lower $\theta$. Consequently the upward distortion relaxes the otherwise binding incentive constraint so that the rents of healthy individuals are reduced.

This result is rather intuitive and in line with standard properties obtained in optimal tax models.<fnref n="21" /> A similar result was already obtained by Blomqvist (1997) and our contribution regarding $m$ is mainly that we generalize Blomqvist’s analysis. For practical purposes this means that $R'(m) > 0$ is a very robust result and does not rely on the specific assumptions imposed by Blomqvist (like the separability).

Anyway, the main focus of our paper is preventive care to which we now turn. One might at first be tempted to think that a solution would leave the choice of $e$ undistorted and just provide a flat payment like under full information. Indeed, preventive care is chosen *ex ante*, at a point where there is uncertainty but no asymmetric information. Individuals and insurers alike do not know the (future) realization of the state of health and in state $S$, the severity $\theta$. Consequently it is not immediately obvious what positive effect a distortion might bring about.

Our formal analysis show that this first intuition is misleading; prevention does have an impact on the incentive constraint and thus on informational rents. We first establish

<footnote n="21">²¹Except that all signs are reversed because a high $\theta$ refers to the “bad” type so that the incentive constraint binds upwards. In optimal tax models by contrast a large $w$ corresponds to the “good” type and the downward incentive constraint is binding.</footnote>
this result formally and then further discuss the intuition. We show in Appendix D that

$$I'(e) = \frac{(1 - \pi) v'(c_0) \int_{\theta}^{\bar{\theta}} \lambda(\theta) (u_{\theta e} - u_{\theta c}) d\theta}{\mu EU_c} > 0. \quad (28)$$

Consequently, preventive care must be subsidized at the margin which in turn implies an upward distortion compared to the full information solution. The numerator on the right-hand-side of this expression measures the welfare benefit of an increase in e via its impact on the incentive constraint. To understand this note that $(u_{\theta e} - u_{\theta c}) > 0$ is the derivative of (24) with respect to e (which affects the first and the third argument of $u_{\theta}$). Since it is positive it means that $\dot{U}$ increases, which since $\dot{U} < 0$ implies that the absolute value of $\dot{U}$ decreases. Consequently, utility decreases less fast as $\theta$ increases so that the rents enjoyed by healthier individuals decrease. Intuitively, preventive care is more beneficial for unhealthy individuals than for more healthy ones. Consequently, distorting it upward relaxes the incentive constraint because it hurts the (healthy) mimicker more than the (less healthy) mimicked. In other words it reduces informational rents conceded to more healthy individuals, which in turn allows for more ex post redistribution and thus better insurance. Note that in practice an upward distortion can be achieved by subsidizing e or by providing it publicly at a level which is larger than otherwise optimal. To sum up, even though e is chosen ex ante, it affects the rents enjoyed by healthier individuals ex post.<fnref n="22" /> Intuitively this is due to two concurring effects. First, prevention is more effective for the more severely ill and second, these individuals also have a lower marginal utility of income so that a given level of e has less impact on their utility.

A striking feature of this result is that it does not depend on the sign of $u_{me}$ which can be interpreted as the degree of complementarity between preventive and curative care. More precisely, when $u_{me} > 0$ the two types of care can be consider as complements (prevention makes treatment more effective) while they are substitutes when $u_{me} < 0$ (so

<footnote n="22">²²The denominator of (28) is simply the expected marginal utility of consumption which normalizes the welfare impact to express it in monetary terms. This makes it comparable to $I'$ which is also in monetary terms.</footnote>
that the marginal benefit of curative care decreases with the level of prevention). This is in sharp contrast to the results obtained in linear models and particularly to Barigozzi (2004) who shows that the sign of $I'$ crucially depends on the degree of complementarity. In her setting it is possible that preventive care is taxed.<fnref n="23" /> Our analysis shows that these results are merely an artifact of the linearity assumption which in turn is *ad hoc* and in no way justified by informational considerations. When instruments are restricted solely by the information structure and can be nonlinear, the results are clear and simple in the sense that both types of care should be subsidized at the margin.

Finally let us return to the extent of insurance coverage. We have shown that, as expected, the full information solution involves full insurance in the sense that the marginal utility of income is equalized across states of nature. This implies that both the risk of illness and of its severity are fully insured. In that context the result was easily obtained and followed directly from equations (A1) and (A2). Under asymmetric information, the counterparts to these conditions are (A5) and (A6). Because (A6) depends on the incentive constraint (via the third term) a simple inspection of the expressions shows that they no longer (directly) imply $v' = u_{c(\theta)}$ for all $\theta \in \Theta$ and there is no reason to believe that this property would hold in general. And indeed, we show in Appendix B that

$$[v'(c_0) - u_{c(\theta)}] = \lambda(\theta) u_{c\theta}/\alpha(\theta) < 0 \text{ for all } \theta, \quad (29)$$

so that $v'(c_0) < u_{c(\theta)}$. Consequently, there is underinsurance for the risk of being ill that is for the state $S$.

Turning to the severity of illness, the extent of insurance coverage is less obvious. Indeed, it is not clear whether $u_c$ is increasing or decreasing in $\theta$. Observe that an increasing $u_c$ would reflect underinsurance while a decreasing profile would involve overinsurance. The first-order condition (18) in footnote 17 implies that $c(\theta)$ is decreasing.

<footnote n="23">²³More precisely both types of care can be taxed or subsidized.</footnote>
ing which everything else equal would imply that $u_c$ is increasing.<fnref n="24" /> However, there are other effects and in particular the one associated with assumption that $u_{c\theta} < 0$, which tends to make $u_c$ decreasing. This effect disappears if, as in Blomqvist (1997), one assumes that $u_{c\theta} = 0$, but as discussed above (and acknowledged by the author) this is not a realistic assumption.<fnref n="25" /> However, it follows by continuity that when the absolute value of $u_{c\theta}$ is sufficiently small, there is also underinsurance for the severity of illness. By contrast when this cross derivative is large (in absolute value) the possibility of overinsurance cannot be ruled out. To sum up, while it is clear that the solution does not in general involve full insurance or the severity of disease, it does not appear to be possible to determined whether it involves over- or underinsurance without making further assumptions on the utility function.

The main results of this section are summarized in the following proposition.

**Proposition 2** *When the severity of illness is not publicly observable and nonlinear instruments are available the optimal solution*

(i) Does not imply full insurance; marginal utility of income is not in general equalized across states of nature. The risk of illness (being in state S) is underinsured. Regarding the severity of the disease (the realization of $\theta$), when $u_{c\theta}$ is zero or sufficiently small (in absolute value), there is underinsurance. But when the absolute value of $u_{c\theta}$ is large, the result is ambiguous and no general conclusion can be reached without further restrictions on the utility function.

(ii) Implies a marginal subsidy on both types of care so that $R'(m(\theta)) > 0$ (except at the endpoints of the support of $\theta$ when $R'(m) = 0$), and $I'(e) > 0$. In other words $m$ and $e$ are distorted upwards, irrespective of the degree of complementarity between

$$(\dot{d} - \dot{m})u_c + \dot{m}u_m = 0,$$

so that $(\dot{d} - \dot{m}) < 0$, which implies that $c(\theta) = d(\theta) - m(\theta) - e$ is decreasing.

<footnote n="24">²⁴To see this observe that simplifying notation, equation (18) can be written as</footnote>

<footnote n="25">²⁵And even with this assumption one cannot make a definitive conclusion because $m$ increases with $\theta$ which in turn might affect $u_c$, and we haven't made any assumption regarding $u_{cm}$.</footnote>
preventive and curative care.

(iii) Implies levels of both types of care to be distorted to mitigate rents (relax the incentive constraint).

(iii)a For m this is intuitively explained by the relative slopes of the mimicked and the mimicker's indifference curves in the (m, c) space, exactly like in an optimal income tax model.

(iii)b Since e is chosen ex ante and the same for all the effects at work are more complex. A larger e provides benefits that increase with the severity of the illness. Consequently utility decreases less fast as θ increases so that the rents enjoyed by healthier individuals decrease.

# 5 Misperception

We now assume that individuals misperceive ex ante the impact of prevention on their health. Ex post, that is when requiring therapeutic care, the individuals correctly perceive how their state of health (and thus their utility) has been affected by prevention. Their actual expected utility thus continues to be given by (2), while their perceived (subjective) expected utility is now given by

$$EU^s = \pi v(c_0) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u^s[c(\theta), m(\theta), e, \theta] f(\theta) d\theta. \quad (30)$$

We assume that the government is paternalistic in the sense that it continues to maximize individuals' actual utility. This is in line with typical approach in the literature on myopia and/or misperception; see for instance Allcott et al. (2019), Cremer and Pestieau (2011) or Farhi and Gabaix (2020).

Misperception is a widely acknowledged problem in the medical literature; see Kenkel (2000) and Carman and Kooreman (2014) for a discussion and references. Since we study secondary prevention, we concentrate on the misperception of the effectiveness of
prevention. This may be due to a lack of information (Kenkel, 2000, Section 4.2). We continue to assume that probabilities are perceived correctly.<fnref n="26" /> In reality probabilities of various affections and the effects of primary prevention are often underestimated.

Given the policy {$P, I(e), R(m(\theta))$} an individual ex ante now solves

$$ \max_{e^s, m^s(\theta)} \pi v[\omega - P + I(e^s) - e^s] + (1 - \pi) \int_{\theta}^{\bar{\theta}} u^s[c^s(\theta), m^s(\theta), e^s, \theta] f(\theta) d\theta, \quad (31) $$

where $c^s(\theta) = \omega - P - m^s(\theta) - e^s + I(e^s) + R(m^s(\theta))$.

Ex post individuals in state $H$ do not make any decision, while those is state $S$ choose $m(\theta)$, given $\theta$ and given $e$ which was chosen ex ante. Individuals now correctly perceive the impact of $e$ so that they solve

$$ \max_{m(\theta)} u[\omega - P - m(\theta) - e + I(e) + R(m(\theta)), m(\theta), e, \theta], \quad (32) $$

which the same as their problem absent of misperception.

The formal analysis of the full information and asymmetric information solutions is presented in Appendix E where we establish the following proposition.

**Proposition 3** Assume that individuals misperceive ex ante the impact of prevention on their health but are rational ex post, when choosing therapeutic care. The optimal contract is determined by maximizing individuals' true expected utility. We have:

(i) The optimal solution both under full and asymmetric information is the same as absent of misperception.

(ii) The benefit function $R(m)$ is not affected by misperception. Consequently we continue to have $R'(m^*(\theta)) = 0$ under full information and $R'(m^a(\theta)) > 0$ for $\theta \in [\underline{\theta}, \bar{\theta}]$ and given by expression (A27) which is equivalent to its full information counterpart.

(iii) The implementing benefit rule for $e$, $I(e)$ is affected by misperception and given by (A26) under full information and (A30) when $\theta$ is not publicly observable. In both

<footnote n="26">²⁶In reality probabilities of various diseases and the effects of primary prevention are often underestimated. Considering this type of misperception would be a natural extension of our model.</footnote>
cases, it now includes a Pigouvian (corrective) term which is given by the difference
in willingness to pay for e between a rational and a misperceiving individual. When
misperception reduces the demand for e the Pigouvian term is positive.

Point (i) of this proposition states that neither the full information nor the asym-
metric solutions are affected by misperception. Observe that this result is about the
allocation achieved and not necessarily about its implementation. Intuitively, this is
because the government is paternalistic and maximizes individuals true utility. Conse-
quently, the result is obvious under full information. Under asymmetric information one
has to show that, in addition, the incentive constraint are not affected. This in turn is
also intuitive because asymmetric information is only relevant *ex post*, when the health
state is realized but at that point misperception is no longer pertinent.

The remaining results concern the implementation which *may* be affected because
individuals make their decisions based on the misperceived preferences. Point (ii) fol-
lows because *m* is chosen *ex post* when misperception is no longer relevant so that its
implementation via the benefit rule *R*(*m*) is not affected. The level of prevention, *e*, on
the other hand is chosen *ex ante* and individuals base their decision on a (misperceived)
utility which differs from social (true) preferences. Consequently, a corrective term is
needed irrespective of the information structure. We refer to it as Pigouvian because it
arises since *ex ante* social and individual preferences differ exactly like in presence of an
externality. Given that an individual creates and is affected by misperception it does
not strictly speaking create an externality but what is often called in the literature an
internality; see for instance Allcott et al. (2019).

6 Conclusion

We have studied the design of nonlinear reimbursement rules of preventive and curative
(therapeutic) care. We have concentrated on secondary prevention which is typically
verifiable. Most of the existing literature restricts policies to be linear (affine). By contrast, we determine the best policy given the information available to the insurer without imposing such an *ad hoc* assumption. This has a drastic impact on the results. With linear rules, prevention should be subsidized if and only it reduces the cost of treatment, that is when the two types of care are substitutes. With nonlinear schemes both types of care should be subsided (at the margin) irrespective of the substitutability or complementarity of prevention and treatment.

We have shown that in the first best (when the severity of illness is observable) insurance benefits are flat (lump sum payments) and do not depend on expenditures. When the severity of the disease is not observable, there is *ex post* moral hazard and this solution is not incentive compatible. The optimal insurance implies benefits that increase with both types of care. This is because health expenditures reduce informational rents and they are upward distorted. For therapeutic care this generalizes the result of Blomqvist (1997) and the intuition is easily understood. Less healthy individuals value care more than healthy individuals. Consequently, an increase in expenditures on (therapeutic) care relaxes the incentive constraint.

The case of preventive care is more complex because preventive care is chosen *ex ante*, at a point where there is uncertainty but no asymmetric information. We have shown that prevention nevertheless does have an impact on the incentive constraint and thus on informational rents. Specifically as preventive care increases, utility decreases less fast with the severity of the disease so that the rents enjoyed by healthier individuals decrease. Intuitively this is due to two concurring effects. First, prevention is more effective for the more severely ill. Second, these individuals also have a lower marginal utility of income so that a given level of expenditure on preventive care has less impact on their utility.

Concerning insurance coverage, we have shown that while the first best implies full insurance, the second best does not. The risk of disease is underinsured while no general
conclusion regarding insurance coverage can be reached without further restrictions on
utility. In particular when the marginal utility of income decreases sufficiently fast with
the severity, overinsurance cannot be ruled out.

Finally, we have considered the possibility that individuals misperceive (underestimate) the benefits of preventive care. The government is paternalistic and continues to maximize individuals actual utility. Consequently, the optimal solution both under full and asymmetric information is not affected by misperception. The implementing benefit rule $R(m)$ is not affected because individuals are rational ex post. However, the reimbursement rule for $e$, $I(e)$ is affected by misperception and includes a Pigouvian (corrective) term, which (roughly speaking) is positive when misperception reduces the demand for preventive care.<fnref n="27" />

We have ignored a number of potentially relevant issues that might affect insurance
design. In particular, we have not considered *ex ante* income heterogeneity.<fnref n="28" /> Clearly,
the insurance coverage of health care involves many redistributive issues. In particular,
subsidizing preventive care can also help promote health equity by making these services
more accessible to a broader range of individuals, regardless of their income or financial
situation. The redistributive role of health insurance (to supplement taxation) has
been studied by Rochet (1991) and Cremer and Pestieau (1996). These papers have
shown the complexity of the underlying problem because it involves multidimensional
heterogeneity. Either way neither of these papers considers preventive care.

Insurance may also take into account age and risk factors when determining coverage
for secondary prevention. For example, certain screenings may be recommended at
specific ages or for individuals with known risk factors, and insurance should cover

<footnote n="27">²⁷For the sake of interpretation we have concentrated on this case which intuitively means that individuals underestimate the effectiveness of prevention. Our formal analysis is also valid in the opposite case which is, however, empirically not very appealing.</footnote>

<footnote n="28">²⁸*Ex post* the health states is also likely to induce differences in income. While this is not explicitly considered, it is effectively included in our analysis. With our general utility, one can think of the income loss explaining part of the utility cost of disease.</footnote>
these as appropriate. While conditioning coverage of curative care on observable risk
factors is problematic from and equity perspective and indeed typically ruled out by
anti-discrimination laws, encouraging screening test for specific risk groups is common
practice. Formally this would amount to introducing tagging into our setting.

Finally, we have abstracted from administrative costs and insurer profits. As empha-
sized by Arrow (1963), their presence implies that full insurance is no longer optimal,
and result (10) no longer holds. Including such costs would be a relevant extension and
might affect the structure of the optimal reimbursement rules.
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# Appendix

## A The first-best solution

Denoting by $\mu$ the Lagrange multiplier associated with the government budget constraint and differentiating, the Lagrangian associated with the first-best problem is:

$$ \mathcal{L} = \pi v(d_0 - e) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u[d(\theta) - m(\theta) - e, m(\theta), e, \theta] f(\theta) d\theta \\ + \mu \left( \omega - \pi d_0 - (1 - \pi) \int_{\theta}^{\bar{\theta}} d(\theta) f(\theta) d\theta \right), $$

where $\mu$ is the Lagrange multiplier associated to the resource constraint. The first order conditions (FOCs) with respect to $d_0, d(\theta), m(\theta)$ and $e$ are respectively given by:

$$ \frac{\partial \mathcal{L}}{\partial d_0} = \pi[v'(c_0) - \mu] = 0, \qquad (A1) $$

$$ \frac{\partial \mathcal{L}}{\partial d(\theta)} = (1 - \pi)[u_c - \mu]f(\theta) = 0, \quad \forall \theta \in \Theta \qquad (A2) $$

$$ \frac{\partial \mathcal{L}}{\partial m(\theta)} = (1 - \pi)[u_m - u_c]f(\theta) = 0, \quad \forall \theta \in \Theta \qquad (A3) $$

$$ \frac{\partial \mathcal{L}}{\partial e} = (1 - \pi) \int_{\theta}^{\bar{\theta}} u_e f(\theta) d\theta - [\pi v'(c_0) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_c f(\theta) d\theta] = 0. \qquad (A4) $$

Combining (A1) and (A2) yields

$$ v'(c_0) = u_c \quad \forall \theta \in \Theta, $$

Furthermore, (A3) and (A2) imply

$$ MRS_{cm} = \frac{u_m}{u_c} = 1, $$
Similarly (A4) and (A2) imply

$$ MRS_{ce} = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} u_{ef}(\theta) d\theta}{\pi v'(c_0) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_{ef}(\theta) d\theta} = 1. $$

## B The second best solution

Differentiating $\mathcal{H}$ with respect to $d_0, d(\theta), m(\theta), z(\theta), e$ and applying Pontryagin's maximum principle yields the following necessary conditions defining the solution<fnref n="29" />

$$ \alpha(\theta)\pi v'(d_0 - e) - \mu\pi f(\theta) = 0, \qquad (A5) $$

$$ \alpha(\theta)(1 - \pi)u_c - (1 - \pi)\mu f(\theta) + \lambda(\theta)(1 - \pi)u_c\theta = 0, \qquad (A6) $$

$$ \alpha(\theta)(1 - \pi)(u_m - u_c) + \lambda(\theta)(1 - \pi)(u_{\theta m} - u_{c\theta}) = 0, \qquad (A7) $$

$$ -\eta(\theta) + \alpha(\theta)(1 - \pi)(u_e - u_c) + \lambda(\theta)(1 - \pi)(u_{\theta e} - u_{c\theta}) = 0, \qquad (A8) $$

$$ -\alpha(\theta)\pi v'(d_0 - e) + \eta(\theta) = 0, \qquad (A9) $$

$$ \dot{\lambda}(\theta) = -\frac{\partial \mathcal{H}}{\partial U(\theta, e)} = \alpha(\theta) - f(\theta). \qquad (A10) $$

and the transversality conditions are given by:

$$ \lambda(\theta) = \lambda(\bar{\theta}) = 0. \qquad (A11) $$

Conditions (A5)-(A11) along with the resource constraint (21) define the optimal allocation. From (A6), one has

$$ \alpha(\theta) = \frac{\mu f(\theta)}{u_c} - \lambda(\theta) \frac{u_{c\theta}}{u_c}. \qquad (A12) $$

<footnote n="29">²⁹See for instance Takayama (1985), pages 602–603.</footnote>
Substituting in (A7) and rearranging successively yields

$$ 
\begin{gathered}
\left( \frac{\mu f(\theta)}{u_c} - \lambda(\theta) \frac{u_{c\theta}}{u_c} \right) (u_m - u_c) + \lambda(\theta) (u_{\theta m} - u_{c\theta}) = 0, \\
\mu f(\theta) \frac{u_m}{u_c} - \mu f(\theta) - \lambda(\theta) u_{c\theta} \frac{u_m}{u_c} + \lambda(\theta) u_{c\theta} + \lambda(\theta) (u_{\theta m} - u_{c\theta}) = 0, \\
\mu f(\theta) \left( \frac{u_m}{u_c} - 1 \right) + \lambda(\theta) \left( u_{\theta m} - u_{c\theta} \frac{u_m}{u_c} \right) = 0.
\end{gathered}
\tag{A13}
$$

Combining equation (A13) with the individual's first-order condition under the implementing benefit rule (4) yields (27). Moreover, using equations (A5) and (A6) directly yields (29).

### C Proof that $\lambda(\theta) \ge 0$

The approach we use is inspired by Werning (2000). Substituting (A6) in (A10) yields

$$ f(\theta) - \dot{\lambda}(\theta) - \lambda(\theta) \frac{u_{c\theta}}{u_c} - \frac{\mu f(\theta)}{u_c} = 0. \tag{A14} $$

Assume that $\lambda(\theta) < 0$ on some interval $[\theta_a, \theta_b]$. We thus have:

$$ \lambda(\theta_a) = \lambda(\theta_b) = 0, \tag{A15} $$

$$ \lambda'(\theta_a) \le 0 \text{ and } \lambda'(\theta_b) \ge 0. \tag{A16} $$

Now consider the dual problem associated to the expenditure minimization in state $\theta$:

$$ 
\begin{array}{ll}
\min_{c(\theta), m(\theta)} & \mathcal{E} = c + m, \\
\text{s.t.} & u(c, m, \theta) - \tilde{u} \ge 0.
\end{array}
$$

Denoting by $\sigma$ the Lagrange multiplier associated to the utility constraint, the first order conditions are

$$ 1 - \sigma u_c = 0, $$

$$ 1 - \sigma u_m = 0, $$
and $u(c, m, \theta) - v(\tilde{u}) = 0$. The solution to this problem yields $c(\tilde{u})$ and $m(\tilde{u})$ and $\mathcal{E}(\tilde{u})$.

Differentiation of the utility constraint yields:

$$u_c \frac{\partial c}{\partial \tilde{u}} + u_m \frac{\partial m}{\partial \tilde{u}} - 1 = 0$$

so that using the envelope theorem

$$\frac{\partial \mathcal{E}(\theta, \tilde{u})}{\partial \tilde{u}} = \frac{\partial c(\theta, \tilde{u})}{\partial \tilde{u}} + \frac{\partial m(\theta, \tilde{u})}{\partial \tilde{u}} \\ = \frac{1}{u_c} > 0. \qquad (A17)$$

Now combining (A14), (A17) and (A15) yields

$$\frac{\partial \mathcal{E}(\theta, \tilde{u})}{\partial \tilde{u}} = \frac{1}{\mu} - \frac{\dot{\lambda}(\theta)}{\mu f(\theta)} \quad \text{for } \theta = \{\theta_a, \theta_b\},$$

which using (A16) implies

$$\frac{\partial \mathcal{E}(\theta_a, \tilde{u}(\theta_a))}{\partial \tilde{u}} \geq \frac{\partial \mathcal{E}(\theta_b, \tilde{u}(\theta_b))}{\partial \tilde{u}}. \qquad (A18)$$

We now show that this inequality implies $\tilde{u}(\theta_b) \leq \tilde{u}(\theta_a)$. Assume instead that $\tilde{u}(\theta_a) < \tilde{u}(\theta_b)$. One has

$$\frac{\partial^2 \mathcal{E}(\theta, \tilde{u})}{\partial \tilde{u}^2} = - \frac{\frac{\partial c(\theta, \tilde{u})}{\partial \tilde{u}} u_{cc} + \frac{\partial m(\theta, \tilde{u})}{\partial \tilde{u}} u_{mm}}{u_{c2}} \\ = - \frac{\frac{\partial c(\theta, \tilde{u})}{\partial \tilde{u}} u_{cc} + \left( \frac{1-u_c}{u_m} \frac{\partial c}{\partial \tilde{u}} \right) u_{mm}}{u_{c2}} \\ = - \frac{\frac{\partial c(\theta, \tilde{u})}{\partial \tilde{u}} (u_{cc}-1) + u_{mm}}{u_c^2} > 0.$$

and

$$\frac{\partial^2 \mathcal{E}(\theta, \tilde{u})}{\partial \tilde{u} \partial \theta} = \frac{-u_{c\theta} + u_{m\theta}}{u_c^2} > 0,$$

so that $\tilde{u}(\theta_a) < \tilde{u}(\theta_b)$ implies $\partial \mathcal{E}(\theta_a, \tilde{u}(\theta_a)) / \partial \tilde{u} < \partial \mathcal{E}(\theta_b, \tilde{u}(\theta_b)) / \partial \tilde{u}$, which contradicts (A18). Consequently, we must have $\tilde{u}(\theta_b) \geq \tilde{u}(\theta_a)$ but this clearly violates the incentive constraint. Indeed, we have $\theta_b > \theta_a$ and the incentive constraint implies that $\dot{U} < 0$.

Summing up, we have shown that if $\lambda(\theta) < 0$ on some interval $[\theta_a, \theta_b]$, both $\tilde{u}(\theta_a) < \tilde{u}(\theta_b)$ and $\tilde{u}(\theta_b) \geq \tilde{u}(\theta_a)$ are impossible. Consequently, $\lambda(\theta) < 0$ is not possible.
## D Derivation of (28)

Substituting (A6) in (A8) yields:

$$-\alpha(\theta)\pi v'(d_0 - e) + \alpha(\theta)(1 - \pi)(u_e - u_c) + \lambda(\theta)(1 - \pi)(u_{\theta e} - u_{\theta c}) = 0.$$

Dividing by $\alpha(\theta)$ and multiplying by $f(\theta)$ yields:

$$-\pi v'(d_0 - e) f(\theta) + (1 - \pi)(u_e - u_c)f(\theta) + \frac{\lambda(\theta)}{\alpha(\theta)}(1 - \pi)(u_{\theta e} - u_{\theta c})f(\theta) = 0.$$

Recall that $u_{\theta c} < 0$ while $u_{\theta e} > 0$ so that $(u_{\theta e} - u_{\theta c}) > 0$. Integrating over $\theta$ and rearranging yields

$$\pi v'(d_0 - e) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_c f(\theta) d\theta = (1 - \pi) \int_{\theta}^{\bar{\theta}} u_e f(\theta) d\theta + \int_{\theta}^{\bar{\theta}} \frac{\lambda(\theta)}{\alpha(\theta)} (1 - \pi) (u_{\theta e} - u_{\theta c}) f(\theta) d\theta. \quad (A19)$$

Dividing by $EU_c = [\pi v'(d_0 - e) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_c f(\theta) d\theta]$ and using (5) implies:

$$I'(e) = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} \frac{\lambda(\theta)}{\alpha(\theta)} (u_{\theta e} - u_{\theta c}) f(\theta) d\theta}{EU_c}, \quad (A20)$$

which using (A6) to substitute for $f(\theta)/\alpha(\theta)$ implies (28).

## E Misperception: formal analysis

Differentiating (31) with respect to $e^s$ and $m^s(\theta)$ and rearranging yields

$$MRS_{cm}^{s} = \frac{u_{m}^{s}(\cdot)}{u_{c}^{s}(\cdot)} = 1 - R'(m^{s}(\theta)) \quad \forall \theta \in \Theta, \quad (A21)$$

$$MRS_{ce}^{s}[c^{s}(\theta), m^{s}(\theta), e^{s}] = 1 - I'(e^{s}), \quad (A22)$$

where

$$MRS_{ce}^{s}[c^{s}(\theta), m^{s}(\theta), e^{s}] = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} u_{e}^{s}[c^{s}(\theta), m^{s}(\theta), e^{s}, \theta] f(\theta) d\theta}{\pi v'(c_{0}^{s}) + (1 - \pi) \int_{\theta}^{\bar{\theta}} u_{c}^{s}[c^{s}(\theta), m^{s}(\theta), e^{s}, \theta] f(\theta) d\theta}. \quad (A23)$$
These conditions are the counterparts to equations (4) and (5) which are relevant absent of misperception. They have a similar structure but differ in two ways. First, marginal rates of substitution are calculated for $u^s$ rather than for $u$. Second all expressions are evaluated at the perceived (hypothetical) levels $[c^s(\theta), m^s(\theta), e^s]$ obtained from Problem (31).

Turning to the *ex post* problem, equation (32) brings us back to the problem absent of misperception with a first-order condition

$$MRS_{cm} = \frac{u_m}{u_c} = 1 - R'(m(\theta)), \qquad (A24)$$

which is the same as (4).

## E.1 Full information

First observe that the government maximizes the individuals' actual utility. Consequently its problem continues to be given by (8)–(9) and the solution is the same as in Section 3, that is $[c^*(\theta), m^*(\theta), e^*]$ satisfying conditions (10) and (12). However misperception may affect the properties of the implementing policy $\{P, I(e), R(m(\theta))\}$. While the equilibrium levels $I(e^*)$ and $R(m^*(\theta))$ must be the same as absent of misperception (because the solution to be implemented is not affected) the marginal reimbursements $I'(e)$ and $R'(m)$ may differ.

Starting with $R(m)$ implementing the full information solution (A24) together with (10) thus require $R'(m(\theta)) = 0$ like in the absence of misperception.

Turning to preventive care, note that to achieve $e^*$ a simple solution is to set $I(e)$ such that

$$\left\{ \begin{array}{l} I(e^*) = I^*(e^*), \\ I(e) = -C \quad \text{if } e \neq e^*, \end{array} \right. \qquad (A25)$$

where $C$ is a sufficiently large constant to make deviations unattractive. To achieve a “smoother” decentralization (with differentiable functions) we can combine (A22) and (12) recalling that the latter implies $MRS_{ce}[c^*(\theta), m^*(\theta), e^*] = 1$. Consequently,
implementing $e^*$ requires

$$I'(e^*) = MRS_{ce}[c^*(\theta), m^*(\theta), e^*] - MRS_{ce}^s[c^s(\theta), m^s(\theta), e^*], \quad (A26)$$

where $c^s(\theta) = \omega - P - m^s(\theta) - e^* + I(e^*) + R(m^s(\theta))$.

Consequently, for $e$ there is a Pigouvian terms to correct for the misperception. Using (A4) the Pigouvian term is positive i.e.:

$$MRS_{ce}[c^s(\theta), m^s(\theta), e^*] < MRS_{ce}[c^*(\theta), m^*(\theta), e^*],$$

which using (5) means that the FOC of a myopic individual taking $I$ as given at $e^*$ is negative. In other words myopic persons choose a level of $e$ that is too small. It is then normal that it should be subsidized.

## E.2 Asymmetric information

We now drop the assumption that the severity of an individual's illness, $\theta$, is publicly observable. We thus revisit Section 4 to accommodate the possibility of misperception.

Observe that the asymmetry of information does not affect the individual's problem given the policy $\{P, I(e), R(m(\theta))\}$, which continues to be given by (31) so that expressions (A21) and (A22) continue to apply.

Turning to the optimal solution, first note that since individuals correctly perceive the impact of $e$ ex post the IC constraint does not change so that the optimal IC solution (maximizing actual expected utility) is the same as in the absence of misperception. Consequently, the optimal $e$ continues to be determined by (A19).

Let $c^a(\theta)$, $m^a(\theta)$ and $e^a$ denote the solution under asymmetric information determined in Section 4. To implement this solution we can use the same function $R(m)$ as described in that section. Recall that $m$ is chosen ex post when individuals are rational and correctly perceive the impact of $e$. We thus have

$$R'(m^a(\theta)) = \frac{\lambda(\theta) \left( u_{\theta m} - u_{c\theta} \frac{u_m}{u_c} \right)}{\mu f(\theta)}, \quad (A27)$$
which is the same as expression (27). In words, we continue to have a distortion due to rents but the misperception does not imply any extra term.

Turning to e, using an expression similar to (A25) also continues to be possible. The remaining question is to find an implementation of e^a via a differentiable function I(e).

To achieve this (A22) must be satisfied at e^a thus requiring

$$MRS_{ce}^{s}[c^{s}(\theta), m^{s}(\theta), e^{a}] = 1 - I'(e^{a}). \qquad (A28)$$

To simplify notation define

$$A = \frac{(1 - \pi) \int_{\theta}^{\bar{\theta}} \frac{\lambda(\theta)}{\alpha(\theta)} (u_{\theta e} - u_{\theta c}) f(\theta) d\theta}{EU_c},$$

which, as shown by (A20), is the rent term for e under asymmetric information. We can then rewrite (A19) as

$$1 - A = MRS_{ce}[c^a(\theta), m^a(\theta), e^a]. \qquad (A29)$$

Combining (A28) and (A29) setting e^s = e^a, we obtain

$$MRS_{ce}^{s}[c^{s}(\theta), m^{s}(\theta), e^{a}] = MRS_{ce}[c^{a}(\theta), m^{a}(\theta), e^{a}] + A - I'(e^{a}),$$

or

$$I'(e^a) = \{MRS_{ce}[c^a(\theta), m^a(\theta), e^a] - MRS_{ce}^s[c^s(\theta), m^s(\theta), e^a]\} + A. \qquad (A30)$$

The first term in brackets is the Pigouvian term with the same interpretation as under full information. The second term is the rent term like absent of misperception. Note that this term is evaluated for a rational individual. Recall that the asymmetric information is relevant ex post when individuals correctly perceive the impact of preventive care.