A continuous model of income insurance

Abstract

In this paper we treat an individual’s health as a continuous variable, in contrast to the traditional literature on income insurance, where it is assumed that the individual is either able or unable to work. A continuous treatment of an individual’s health sheds new light on the role of income insurance and makes it possible to capture a number of real-world phenomena that are not easily captured in the traditional, dichotomous models. In particular, we show that moral hazard is not necessarily outright fraud, but a gradual adjustment of the willingness to work, depending on preferences and the conditions stated in the insurance contract. Further, the model can easily encompass phenomena such as administrative rejection of claims, and it clarifies the conditions for the desirability of insurance in the first place.

Appendices

Appendix A: Proofs

Proof of Proposition 1

Condition (i) in the proposition is necessary since if $\theta_{\mathrm{upper}} \le\hat{\theta}_{F} = - u'(1 - F(\hat{\theta}_{F}))$, no one will ever work in the presence of insurance and thus, $F(\hat{\theta}_{F}) = 1$. In this case, the inequality can be written θ _upper≤−u′(0) and no insurance can be financed. Thus, θ _upper>−u′(0) is a necessary condition for insurance to be feasible. Condition (ii) in the proposition is also necessary since if $\theta_{\mathrm{lower}} \ge\hat{\theta}_{F} = - u'(1 - F(\hat{\theta}_{F}))$, everyone will always work in the presence of insurance and thus insurance will not be utilized. In that case, $F(\hat{\theta}_{F}) = 0$ and as a result the inequality can be written θ _lower≥−u′(1). Thus, θ _lower<−u′(1) is necessary for insurance to be desirable.

To prove sufficiency, we note that, by definition, an interior solution $\hat{\theta}_{F}$ satisfies $\theta_{\mathrm{lower}} < \hat{\theta}_{F} < \theta_{\mathrm{upper}}$. To prove that a utility function u and a distribution function F satisfying (i) and (ii) must also satisfy this inequality, we define the function φ(θ)≡θ+u′(1−F(θ)). We have φ(θ _lower)=θ _lower+u′(1) which, by (ii), is negative. We also have φ(θ _upper)=θ _upper+u′(0) which, by (i), is positive. The continuous and monotone function φ(θ) must therefore take the value of zero for one (unique) value of θ somewhere in the open interval (θ _lower,θ _upper). By (9), the φ(θ) is zero for $\theta= \hat{\theta}_{F}$; thus $\hat{\theta}_{F}$ is located in the interval (θ _lower,θ _upper). □

Proofs of some properties of optimal insurance in Sect. 5.2

(i) An optimal contract implies less than full insurance

The first-order conditions of the maximization of EU subject to the budget constraint, with respect to p and b, can be written

$$\alpha\cdot u'(1 - p) + (1 - \alpha) \cdot u'( - p) \ge\lambda,\qquad \beta\cdot u'(1 + b) + (1 - \beta) \cdot u'(b) \le\lambda, $$

where

$$\alpha\equiv\frac{\int_{ \tilde{\tilde{\theta}}}^{ \infty} q(\hat{\theta} - \theta)\,dF(\theta)}{\int_{ - \infty}^{ \infty} q(\hat{\theta} - \theta)\,dF(\theta)},\qquad \beta\equiv\frac{\int_{ \tilde{\theta}}^{ \infty} [ 1 - q(\hat{\theta} - \theta) ]\,dF(\theta)}{ \int_{ - \infty}^{ \infty} [ 1 - q(\hat{\theta} - \theta) ]\,dF(\theta )}. $$

With an interior solution, these conditions are satisfied as equalities, implying that the weighted average of u′(1−p) and u′(−p) should be equal to the weighted average of u′(1+b) and u′(b). Since u′(−p)>u′(1+b), the two weighted averages can be equal only if u′(b)>u′(1−p), i.e., only if 1−p>b. □

(ii) The effects on labor supply are ambiguous

The introduction of insurance will induce some individuals with a lucky outcome to choose to stay home from work even though they would have gone to work in the absence of a lottery. These individuals will reduce their labor supply. The number of such individuals is $\int_{ \theta _{0}^{ *}}^{ \tilde{\theta}} dF(\theta)$. Similarly, a number of individuals with a bad outcome in θ and bad luck in the lottery will choose to work, even though they would have stayed home in the absence of insurance; such individuals contribute to an increase in the labor supply. The number of these individuals is $\int_{ \tilde{\tilde{\theta}}}^{\theta_{0}^{ *}} dF(\theta)$. Without adding more structure to the model, it is not possible to determine whether $\int_{ \tilde{\tilde{\theta}}}^{\theta_{0}^{ *}} dF(\theta)$ is smaller or larger than $\int_{ \theta_{0}^{ *}}^{ \tilde{\theta}} dF(\theta)$. □

(iii) Desirability of insurance

A necessary condition for insurance, based only on the signal s, to be desirable is that there is some mass between $\tilde{\tilde{\theta}}$ and $\tilde{\theta}$, where $\tilde{\theta} \equiv u(b) - u(1 + b)$ and $\tilde{\tilde{\theta}} \equiv u( - p) - u(1 - p)$. This condition is satisfied if there is some mass around $\theta_{0}^{ *}$.

To prove this, we first note that $\tilde{\tilde{\theta}} < \theta_{0}^{ *} < \tilde{\theta}$. Assume that all mass of the distribution is to the right of $\tilde{\theta}$. Then everyone will always work, regardless of the outcome of the lottery. The lottery would thus only cause variability in income, and would therefore be undesirable to a risk-averse individual. Thus $\theta_{\mathrm{lower}} < \tilde{\theta}$ is a necessary condition for the lottery to be desirable. Assume now that all mass of the distribution is to the left of $\tilde{\tilde{\theta}}$. Then no one would ever work. In such a case, the lottery would only cause income variability, which is undesirable. Hence $\theta_{\mathrm{upper}} > \tilde{\tilde{\theta}}$ is also a necessary condition for the lottery to be desirable. These two conditions combined imply that some mass between $\tilde{\tilde{\theta}}$ and $\tilde{\theta}$ is necessary for the lottery to be desirable. Since $\tilde{\tilde{\theta}} < \theta_{0}^{ *} < \tilde{\theta}$, some mass around $\theta_{0}^{ *}$ is sufficient for this to occur. □

Appendix B: Effects on labor supply of insurance under full observability

Under what conditions does the introduction of insurance result in a positive (b) or a negative (−p) income change, and hence a fall or an increase in labor supply? It depends on whether $\theta_{0}^{ *} = u(0) - u(1)$ is larger than, or smaller than $\hat{\theta}_{F} = - u'( 1 - F(\hat{\theta}_{F}) )$, which in turn depends on the u(⋅) and F(⋅) functions.

Assume first that $\theta^{*}<\hat{\theta}_{F}$. The individual would earn 0 if not working, and 1 if working, in the absence of insurance. With insurance, and with a realization $\theta^{*}<\theta<\hat{\theta}_{F}$, he would earn b if not working, and 1+b if working (since the benefit is not conditioned on his labor supply decision, but only on the realization θ). Thus, the introduction of insurance provides a positive income increase b regardless of whether the individual works or not. As a result, the individual will demand more pain relief, hence reducing his labor supply.

This reasoning applies for a realization $\theta^{*}<\theta<\hat{\theta}_{F}$. For a realization outside that interval, the introduction of insurance will not have any effect on labor supply; an individual who does not work in the absence of insurance has even stronger reasons not to work when he gets an income transfer b. Correspondingly, an individual who works in the absence of insurance will have even stronger reasons to work when he is exposed to a lump-sum income reduction –p. The configuration $\theta^{*}<\hat{\theta}_{F}$ therefore implies that the introduction of insurance will reduce aggregate labor supply.

Assume instead that $\hat{\theta}_{F}<\theta^{*}$. As always, the individual would earn 0 if not working, and 1 if working, in the absence of insurance. With insurance, and with a realization $\hat{\theta}_{F}<\theta<\theta^{*}$, he would earn –p if not working, and 1−p if working. Thus, the introduction of insurance provides a negative income change, regardless of whether the individual works or not. As a result, he will settle for less pain relief, hence increasing his labor supply.

This reasoning applies for a realization $\theta^{*}<\theta<\hat{\theta}_{F}$. With the same argument as above for realizations outside the interval, we conclude that the configuration $\hat{\theta}_{F}<\theta^{*}$ implies that the introduction of insurance will increase aggregate labor supply.

The analysis shows that under full observability, the effect of insurance on labor supply is driven only by exogenous lump-sum changes in income, and not by any price distortions. These lump-sum changes are positive or negative, depending on the u(⋅) and F(⋅) functions.