The insured is a representative agent with an income $y$. Health insurance is assumed to be compulsory, so there are no uninsured individuals. This allows the insured to pay only $t\left(x\right)$: the total cost of health care $x$, ( $x>0$, single care). The policyholder’s net income is therefore equivalent to his income after health care costs: $y-t\left(x\right)$. The policyholder’s utility function depends on his income $y$, the monetary loss $t\left(x\right)$ and the disutility $w\left(e\right)$ associated with the effort $e$ made to prevent illness. This utility function is separable. It is decomposed into a function of the policyholder’s net income $v\left(y-t\left(x\right)\right)$ where $v^{\prime }>0$, $v^{″}<0$ (the policyholder is risk averse) and an effort disutility function $w\left(e\right)$ with $w^{\prime }>0$; $w^{″}<0$. Formally, we have:

$V=V\left(y-t\left(x\right);e\right)=v\left(y-t\left(x\right)\right)-w\left(e\right)$ (1)

The social insurer’s objective is considered to be to minimise the costs of providing a given state of health. Although it has a duty to cover all or part of the insured person’s healthcare costs, it has limited resources at its disposal. It therefore increases its utility by reducing the social share of healthcare expenditure for each insured person, i.e. $x-t\left(x\right)$.

As this social share of health expenditure is not known with certainty, we associate a probability distribution with it and define an expected utility function for the social insurer, of the Von Neumann-Morgenstern type $U(.)$ with $U^{\prime }>0$. This last condition requires taking as the argument of $U(.)$ the term $\left[-x+t\left(x\right)\right]$ and not $\left[x-t\left(x\right)\right]$. Thus, the insurer’s social utility function is a negative function of the expenditure $x$ (monetary loss) and positive of the amount of financial participation $t\left(x\right)$ (the health expenditure) borne by the insured:

$U=U\left(-x+t\left(x\right)\right)$ (2)

The aim of the social insurer is to minimise costs (reimbursement expenditure). To do this, it puts in place an optimal incentive contract that takes into account the level of effort made by the insured to prevent illness, as well as the proportion of healthcare expenditure to be borne by the patient, in order to make the insured more responsible. In other words, the aim is to build an incentive contract that minimises the cost of social cover while minimising healthcare expenditure for insured individuals. The health insurance contract is drawn up by the State and offered to the insured without any possibility of negotiation or renegotiation. This is tantamount to seeking to maximise the utility of the social insurer under the constraints of policyholder participation and incentives.

The policyholder participation constraint assumes that the expected gain is greater than the opportunity cost: the utility derived should be greater than the disutility generated by participation. Furthermore, an individual’s interest in participating in health insurance is reinforced by the assumption that $y>t\left(x\right)$. This reflects the idea that the absence of insurance leads to the individual becoming poorer with $t\left(x\right)\ge y$. Thus, an individual who has no insurance is considered as an insured who does not pay his contributions. This compulsory nature of health insurance, combined with the fact that it guarantees the individual’s income by opening the door to a wide range of care, then induces that the individual’s participation constraint is always satisfied. Put another way, the compulsory nature of health insurance means that rationing by quantity, i.e., a reduction in the number of claimants, must be ruled out. There is no ex-ante exclusion from the health insurance contract. We should not therefore speak of adverse selection. The policyholder participation constraint is written as follows:

$Ev\left(y-t\left(x\right)\right)-w\left(e\right)\ge 0$ (3)

The incentive constraint assumes that there is a potential conflict between the principal (the social insurer) and the agent (the insured). The principal wants to reduce the social share of individual healthcare expenditure $\left[x-t\left(x\right)\right]$ in order to increase its utility $U(.)$ via the increase in $\left[-x+t\left(x\right)\right]$. The agent, for his part, wants to reduce his financial contribution $t\left(x\right)$ to the total cost of his healthcare. This increases his utility $V$ but reduces that of the principal. Since rationing by quantity is impossible given the compulsory nature of public health insurance, what remains is rationing by cost, involving the effort of the agent (the insured). The agent is self-motivated to make the greatest possible preventive effort in order to benefit from an expected utility that is at least equivalent to the disutility caused by this effort. The principal, who is assumed to be infallible and sufficiently credible, provides an incentive contract which functions as a mechanism that ensures the equivalence between the level of effort chosen by the agent and the level of effort desired by the principal and the reward provided by the latter.

$Ev\left(y-t\left(x\right)\right)-w\left(e\right)\ge Ev\left(y-t\left(\tilde{x}\right)\right)-w\left(\tilde{e}\right)$ (4)

$\tilde{x}$ and $\tilde{e}$ represent respectively the level of the total cost of care and the level of prevention effort desired by the principal.

For the agent (the insured), satisfying the incentive constraint means maximising his utility function as a function of his disease prevention effort. This consists of obtaining the optimal level of effort that provides maximum utility. At this optimal level of effort, the marginal utility linked to net income is equal to the marginal disutility of the effort. See Equation (7).

$e=\underset{e}{\text{Argmax}}Ev\left(y-t\left(x\right)\right)-w\left(e\right)$ (5)

More formally, let us note the level of preventive effort made by the insured $e\in \left[\underset{\_}{e};\bar{e}\right]$, with $\underset{\_}{e}$: low effort and $\bar{e}$: high effort. This level of unobservable effort by the social insurer induces an observable outcome $x$ (which is the total cost of healthcare) according to a probability density distribution $f\left(x;e\right)$ and distribution function $F\left(x;e\right)$. Assume that $F\left(x;e\right)$ and $f\left(x;e\right)$ are differentiable and continuous for all $x$ and $e$.

The first-order approach defines conditions known as the Monotone Likelihood Ratio Condition (MLRC) and the Concavity of Distribution Function Condition (CDFC). These conditions are necessary and sufficient to satisfy the policyholder incentive constraint. This approach has been developed by many authors: [12] - [16] .

The MLRC condition, proved by [17] , implies stochastic dominance; that is, $F_e\left(x;e\right)\le 0$, and states that a higher signal gives more information about the insured’s effort. Formally $f\left(x;e\right)$ must satisfy the MLRC condition if:

$\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\text{is}\text{non-decreasing}\forall x,e$

The CDFC condition is a decreasing stochastic return condition. It applies to the production of information on the insured’s effort. $\left(x;e\right)$ must also satisfy the CDFC condition:

$F_{ee}\left(x;e\right)\ge 0$ (6)

According to the first-order approach, if the probability density $f\left(x;e\right)$ satisfies the MLRC and CDFC conditions, then the incentive constraint can be written as:

$\frac{\partial }{\partial e}\left[Ev\left(y-t\left(x\right)\right)-w\left(e\right)\right]=0$ (7)

Let’s assume that:

The insured reduces the probability of becoming ill by choosing a high level of effort:

$\frac{\partial f\left(0;e\right)}{\partial e}>0$ (8)

In the event of illness, the insured reduces the likelihood of incurring a large monetary loss by making a high level of effort:

$\frac{\partial f\left(x;e\right)}{\partial e}<0$ (9)

${\displaystyle \int \left(x;e\right)}=1-f\left(0;e\right)$: probability of the insured being ill. $f\left(x;e\right)$ is therefore discontinuous at $x=0$.

The insured person derives a positive utility from public health insurance as long as his financial participation is lower than his income ( $t\left(x\right)<y$) and the

share of health expenditure ( $\frac{t\left(x\right)}{x}$) decreases in relation to his total health expenditure ( $x$): $\frac{\partial Q\left(x\right)}{\partial x}<0$ with $Q\left(x\right)=\frac{t\left(x\right)}{x}$.

The programme for determining the conditions of the optimal incentive contract between the social insurer and the insured is formally as follows:

$\begin{array}{l}\{\begin{array}{l}\underset{e;t(.)}{\underset{︸}{\max }}EU\left(-x+t\left(x\right)\right)\\ s/c;Ev\left(y-t\left(x\right)\right)-w\left(e\right)\ge 0\\ \frac{\partial }{\partial e}\left[Ev\left(y-t\left(x\right)\right)-w\left(e\right)\right]=0\end{array}\\ \Rightarrow \{\begin{array}{l}\underset{e;t(.)}{\underset{︸}{\max }}{\displaystyle \int U}\left(-x+t\left(x\right)\right)f\left(x;e\right)\text{d}x\\ s/c:{\displaystyle \int v}\left(y-t\left(x\right)\right)\text{d}x-w\left(e\right)\ge 0\left(\lambda \right)\\ {\displaystyle \int v}\left(y-t\left(x\right)\right)f_e\left(x;e\right)\text{d}x-w^{\prime }\left(e\right)=0\left(\mu \right)\end{array}\end{array}$

Noting λ: Lagrange multiplier associated with the individual’s participation constraint; μ: Lagrange multiplier associated with the insured’s choice of effort level, the optimal incentive contract leads to the following equations:

$\{\begin{array}{ll}\frac{U^{\prime }\left(-x+t\left(x\right)\right)}{v^{\prime }\left(y-t\left(x\right)\right)}=\lambda +\mu \frac{f_e\left(x;e\right)}{f\left(x;e\right)}\hfill & (10)\hfill \\ t^{\prime }\left(x\right)=\frac{U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)-{}^1\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]*{\left[v^{\prime }\left(y-t\left(x\right)\right)\right]}^2}{\left[U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)+v^{″}\left(y-t\left(x\right)\right)U^{\prime }\left(x+t\left(x\right)\right)\right]}\hfill & (11)\hfill \end{array}$

There are two cases depending on whether the social insurer (the principal) is risk averse in relation to the agent’s prevention effort ( ${U^{″}}_e<0$) or risk neutral ( ${U^{″}}_e=0$).

When the social insurer is risk averse, and with $v^{″}<0$, the result is:

$0<t^{\prime }\left(x\right)<1$ (12)

The optimal incentive contract is such that, when the total cost of health care increases by one unit, the insured’s financial participation increases, but proportionately less than the cost of care.

When the social insurer is risk-neutral, and with $v^{″}<0$, we show that:

$t^{\prime }\left(x\right)=\frac{-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]\ast {\left[v^{\prime }\left(y_0-t\left(x\right)\right)\right]}^2}{\left[v^{″}\left(y_0-t\left(x\right)\right)U^{\prime }\left(x+t\left(x\right)\right)\right]}\Rightarrow t^{\prime }\left(x\right)>0$

The optimal contract is such that, as the total cost of healthcare increases, the insured’s financial participation increases.

These results can be summarised in Proposition 1 below:

Proposition 1: The contract between the social insurer and the insured induces, at the optimum, a sharing of risks between them, relating to the assumption of health care costs.

Let’s find the functional form of the optimal financial participation $t\left(x\right)$ of the insured in order to obtain the optimal levels of the compulsory contribution, the co-payment and the excess.

Starting from Equation (11), we obtain a primitive of $t^{\prime }\left(x\right)$:

$t\left(x\right)={\displaystyle \int \frac{U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]*{\left[v^{\prime }\left(y-t\left(x\right)\right)\right]}^2}{\left[U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)+v^{″}\left(y-t\left(x\right)\right)U^{\prime }\left(x+t\left(x\right)\right)\right]}\text{d}x}+b$ (13)

with $b\in ℝ$. The expression for $b$ is obtained from Equation (13) by posing $t\left(0\right)=K=b$. More precisely:

$b=t\left(0\right)-{\displaystyle \int \frac{U^{″}\left(t\left(0\right)\right)v^{\prime }\left(y-t\left(0\right)\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]\ast {\left[v^{\prime }\left(y-t\left(0\right)\right)\right]}^2}{\left[U^{″}\left(t\left(0\right)\right)v^{\prime }\left(y-t\left(0\right)\right)+v^{″}\left(y-t\left(0\right)\right)U^{\prime }\left(t\left(0\right)\right)\right]}\text{d}x}$ (14)

Note:

$A\equiv -\frac{U^{″}\left(t\left(0\right)\right)v^{\prime }\left(y-t\left(0\right)\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]\ast {\left[v^{\prime }\left(y-t\left(0\right)\right)\right]}^2}{\left[U^{″}\left(t\left(0\right)\right)v^{\prime }\left(y-t\left(0\right)\right)+v^{″}\left(y-t\left(0\right)\right)U^{\prime }\left(t\left(0\right)\right)\right]}$ (15)

Note that $A$ is a constant. By substituting Equation (14) into Equation (13), the expression for $b$ reduces to:

$b=K+{\displaystyle \int A\text{d}x}=t\left(0\right)+{\left[Ax\right]}_0^x=K+Ax$ (16)

Noting:

$B\left(x\right)\equiv {\displaystyle \int \frac{U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]\ast {\left[v^{\prime }\left(y-t\left(x\right)\right)\right]}^2}{\left[U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)+v^{″}\left(y-t\left(x\right)\right)U^{\prime }\left(x+t\left(x\right)\right)\right]}\text{d}x}$ (17)

Ultimately, the functional form of the optimal financial contribution is written as:

$t\left(x\right)=K+Ax+B\left(x\right)$ (18)

$K$ is the optimal compulsory contribution paid by the insured; $Ax$ is the optimal co-payment amount; ( $x$) the optimal excess.

It is also possible to determine the fixed amount of the excess that triggers coverage. To do this, we need to return to the non-negativity constraint on the insured’s net income, which guarantees that his financial contribution to health insurance, remaining lower than his income, does not make him poorer:

$y-t\left(x\right)>0\Rightarrow y-K-Ax-B\left(x\right)>0$ (19)

The principal (the State) must ensure that when the total cost of healthcare $x$ exceeds a certain threshold, so that the insured person’s financial participation impoverishes him or her, the latter is exempt from paying the excess. This health care cost threshold is the level of $x$ that cancels out the insured person’s net income. Let’s note $\hat{x}$. We have:

$y-K-A\hat{x}-B\left(\hat{x}\right)=0$ (20)

We simply show that when the healthcare cost exceeds the threshold, $x\ge \hat{x}$, the insured’s net income becomes negative:

$x\ge \hat{x}\Rightarrow y-K-Ax-B\left(x\right)\ge y-K-A\hat{x}-B\left(\hat{x}\right)\text{puisque}B\left(x\right)\ge B\left(\hat{x}\right)\text{et}Ax\ge A\hat{x}$.

So:

$y-K-Ax-B\left(x\right)\ge 0$

In this case, the State (the social insurer) must, as far as possible, set the excess at 0. The insured is exempt from the excess. When the insured’s expenditure on care is below the threshold, $\hat{x}$, his excess is $B\left(x\right)<B\left(\hat{x}\right)$ and his net income becomes positive: $y-K-Ax-B\left(x\right)>0$.

Formally, the social insurer can, at optimum, maintain the excess $B\left(x\right)$. The insured is not exempt from the excess. Noting $B\left(\hat{x}\right)=\hat{B}$, we formally obtain:

$\{\begin{array}{l}B\left(x\right)>0\text{si}B\left(x\right)<\hat{B}:\text{<msup> l <mo> ′ </mo> </msup> insured is not exempt}\text{from}\text{deductible}\\ B\left(x\right)=0\text{si}B\left(x\right)\ge \hat{B}:\text{<msup> l <mo> ′ </mo> </msup> insured is exempt from}\text{the}\text{excess}\end{array}$

Let’s summarise this result in Proposition 2 below:

Proposition 2: The optimal incentive contract proposed by the social insurer to minimise its social expenditure and make the insured more responsible, consists of a combination of compulsory contribution, co-payment, deductible and a fixed threshold of cost in care B, which triggers the assumption of responsibility (exemption from the deductible), such that:

$\{\begin{array}{l}t\left(x\right)=K+Ax+B\left(x\right)\text{si}B\left(x\right)<\hat{B}:\text{<msup> l <mo> ′ </mo> </msup> insured}\text{is}\text{not}\text{exempt}\text{from}\text{deductible}\\ t\left(x\right)=K+Ax\text{si}B\left(x\right)\ge \hat{B}:\text{<msup> l <mo> ′ </mo> </msup> insured}\text{is}\text{exempt}\text{from}\text{the}\text{excess}\end{array}$

With:

$A\equiv -\frac{U^{″}\left(K\right)v^{\prime }\left(y-K\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]\ast {\left[v^{\prime }\left(y-K\right)\right]}^2}{\left[U^{″}\left(K\right)v^{\prime }\left(y-K\right)+v^{″}\left(y-K\right)U^{\prime }\left(K\right)\right]}$ (21)

$B\left(x\right)\equiv {\displaystyle \int \frac{U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)-\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]\ast {\left[v^{\prime }\left(y-t\left(x\right)\right)\right]}^2}{\left[U^{″}\left(-x+t\left(x\right)\right)v^{\prime }\left(y-t\left(x\right)\right)+v^{″}\left(y-t\left(x\right)\right)U^{\prime }\left(x+t\left(x\right)\right)\right]}\text{d}x}$ (22)

We show that the co-payment is an increasing linear function of the total cost of care $x$, for a particular type of care. This is because the co-payment parameter $A$ is a positive constant. In fact, $A>0$ implies that:

$\frac{U^{″}\left(K\right)}{v^{\prime }\left(y-K\right)}<\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]$ (23)

This relation is always verified since $U^{″}\left(K\right)/v^{\prime }\left(y-K\right)<0$ and by virtue of the MLRC condition, we have:

$\forall x,e\ge 0;\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]\ge 0$ (24)

This property of the co-payment increasing linearly with the total cost of health care is consistent with characteristic 1 of the public health insurance system in developing countries mentioned earlier.

In particular, the co-payment parameter $A$ increases with income $y$, for the same type of care. Indeed:

$\frac{\partial A}{\partial y}>0\text{if}\text{and}\text{only}\text{if}{U}^{″}\left(K\right)v^{\prime }\left(y-K\right)<2\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]v^{\prime }\left(y-K\right)$

This condition is always satisfied since:

$\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(0;e\right)}{f\left(0;e\right)}\right]v^{\prime }\left(y-K\right)>0\text{et}{U}^{″}\left(K\right)v^{\prime }\left(y-K\right)<0$

This growth in co-payments with income is in line with characteristic 2 of the public health insurance system in developing countries, as described above.

Finally, to say that the co-payment increases with income does not mean that the co-payment is differentiated according to the income level of the insured. In fact, the income taken into account in our theoretical model is a smoothed income that does not distinguish between high and low incomes. This income smoothes out income inequalities between high and low incomes. The optimal co-payment increases with policyholders’ smoothed income but is independent of policyholders’ income categories. This result is compatible with characteristic 3 of the public health insurance system in developing countries.

The optimal deductible is positive ( $B\left(x\right)>0$) and grows with the cost of care ( $B^{\prime }\left(x\right)>0$). See the Appendix for the proof.

According to identity (21), the optimal deductible grows with smoothed income y. Indeed, we show that:

$\frac{\partial B\left(x\right)}{\partial y}>0\text{si}\text{et}\text{seulement}\text{si}{U}^{″}\left(t\left(x\right)\right)v^{″}\left(y-t\left(x\right)\right)<2\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]v^{\prime }\left(y-t\left(x\right)\right)$

This is always the case since:

$\frac{\text{d}}{\text{d}x}\left[\frac{f_e\left(x;e\right)}{f\left(x;e\right)}\right]v^{\prime }\left(y-t\left(x\right)\right)>U^{″}\left(t\left(x\right)\right)v^{″}\left(y-t\left(x\right)\right)<0$ (25)

The optimal level of Bx grows with the smoothed income of policyholders but is independent of policyholders’ income categories. Consequently, this deductible does not take into account income inequalities between policyholders. These results concerning the optimal level of the deductible are compatible with characteristic 4 of the public health insurance system in DCs.

The ex-ante moral hazard model developed in this article assumes the homogeneity of the insured. As a result, the policyholder taken into account in the model is an “average individual” in the statistical sense of the term. The optimal level of financial participation tx does not take into account income inequalities between policyholders. Are the results of this model in line with characteristic 5 concerning the exclusion of low-income policyholders?

We first show that low income is underestimated while high income is overestimated in the determination of the optimal health insurance contract. To do this, consider that the economy is composed of high-income policyholders $y$ in proportion $\theta$ (avec $\theta \in \left]0,1\right]$) and low-income policyholders $y$ in proportion $1-\theta$. It then follows that the insured’s income is the sum of the low income $y$ and a positive differential reflecting income inequality $\theta \Delta y$. It is also the difference between high income $y$ and $\theta \Delta y$. Indeed, on the one hand:

$y=\theta \bar{y}+\left(1-\theta \right)\underset{\_}{y}=\theta \left(\bar{y}-\underset{\_}{y}\right)+\underset{\_}{y}=\underset{\_}{y}+\theta \Delta y$;

and on the other hand: $y=\bar{y}-\theta \Delta y$, où $\underset{\_}{y}<y<\bar{y}$ et $y\equiv \bar{y}-\underset{\_}{y}>0$.

Secondly, we show how the deductible used in the optimal contract may not be sufficient to guarantee the participation of low-income agents in the health insurance contract. Remember that the rationale behind the deductible was to prevent the insured’s net income from becoming negative (impoverishment). As we have seen, when the total cost of healthcare exceeds the x threshold, the policyholder is exempt from paying the excess. The deductible is triggered in relation to the policyholder’s income and the total cost of care, but not in relation to the policyholder’s expected utility. In fact, it is possible that the level of the total cost of healthcare exempts the policyholder from paying the excess, thus reducing his net income to a positive level, while his expected utility becomes negative. To see this, let’s return to the policyholder participation constraint:

$Ev\left(y-t\left(x\right)\right)-w\left(e\right)\ge 0$ (26)

Introducing the expression ( $\theta \Delta y$) into this participation constraint gives;

$Ev\left(\underset{\_}{y}-\theta \Delta y-t\left(x\right)\right)-w\left(e\right)\ge 0$ (27)

$Ev\left(\bar{y}+\theta \Delta y-t\left(x\right)\right)-w\left(e\right)\ge 0$ (28)

Let $x$ be a level of healthcare costs that saturates the policyholder participation constraint:

$Ev\left(\underset{\_}{y}-\theta \Delta y-t\left(x\right)\right)-w\left(e\right)=0$ (29)

$Ev\left(\bar{y}+\theta \Delta y-t\left(x\right)\right)-w\left(e\right)=0$ (30)

From the fact that $v^{\prime }>0$, it comes that $\forall$ $\theta >0$ and $\Delta y>0$, we have:

$Ev\left(\underset{\_}{y}+\theta \Delta y-t\left(\tilde{x}\right)\right)-w\left(e\right)>Ev\left(\underset{\_}{y}-t\left(\tilde{x}\right)\right)-w\left(e\right)$ (31)

$Ev\left(\bar{y}-\theta \Delta y-t\left(\tilde{x}\right)\right)-w\left(e\right)<Ev\left(\bar{y}-t\left(\tilde{x}\right)\right)-w\left(e\right)$ (32)

Thus, at the level of care $\tilde{x}$, the participation constraint of the low-income insured becomes negative and that of the high-income insured becomes positive:

$Ev\left(\underset{\_}{y}-t\left(\tilde{x}\right)\right)-w\left(e\right)<0$ (33)

$Ev\left(\bar{y}-t\left(\tilde{x}\right)\right)-w\left(e\right)>0$ (34)

With $t\left(\tilde{x}\right)=K+A\tilde{x}+B\left(\tilde{x}\right)$, these inequalities become:

$Ev\left(\underset{\_}{y}-K-A\tilde{x}-B\left(\tilde{x}\right)\right)-w\left(e\right)<0$ (35)

$Ev\left(\bar{y}-K-A\tilde{x}-B\left(\tilde{x}\right)\right)-w\left(e\right)>0$ (36)

For a total cost of care $\tilde{x}$ ( $\tilde{x}\ge \hat{x}$), the low-income insured is exempt from the deductible and his expected utility becomes:

$Ev\left(\underset{\_}{y}-K-A\tilde{x}\right)-w\left(e\right)$.

This utility remains negative as long as $\tilde{x}>\left[\left(\underset{\_}{y}-K-E{v}^{-1}\left(w\left(e\right)\right)\right)/A\right]=\hat{x}$, with $E{v}^{-1}\left(w\left(e\right)\right)=e$.

In this case, the low-income insured, even with an exemption from the deductible, will forgo these types of care ( $\tilde{x}>\hat{x}$).

Under these conditions, even the deductible present in the optimal incentive contract is not sufficient to ensure that low-income agents have access to the types of care for which $\tilde{x}\ge \hat{x}$.

We summarise this result in the following proposition 3:

By guaranteeing the participation of the insured, the optimal public health insurance contract excludes the low-income insured from access to certain types of care.

This result from the ex-ante moral hazard model reproduces feature 5 of the public health insurance system in DCs. We summarise the types of care according to cost and accessibility for the low-income insured as follows:

$\{\begin{array}{l}\tilde{x}<\hat{x}\text{acces}\text{to the care}\\ \tilde{x}=\hat{x}\text{acces},\text{if}B\left(x\right)\ge \hat{B}\\ \tilde{x}=\hat{x}\text{exclusion},\text{if}B\left(x\right)<\hat{B}\\ \tilde{x}>\hat{x}\text{exclusion}\end{array}$

The optimal level of financial participation does not satisfy the criterion of equity, which is consubstantial with any public health insurance system.