Adverse selection, moral hazard and propitious selection

Abstract

We propose a simple model with preference-based adverse selection and moral hazard that formalizes the cherry picking/propitious selection argument. This argument assumes that individuals differ in risk aversion, potentially resulting in more risk averse agents buying more insurance while being less risky. The propitious selection argument is summarized by two properties: regularity (more risk averse agents exert more caution) and single-crossing (more risk averse agents have a higher willingness to pay for insurance). We show that these assumptions are incompatible with a pooling equilibrium, and that they do not imply a negative correlation between risk and insurance coverage at equilibrium.

Appendix

1.1 Example of propitious selection with the dual theory of choice under risk

We model individuals’ risk preferences using Yaari’s (1987) theory, which is dual to the expected utility theory in the sense that it is linear in wealth but non linear in probabilities. This formulation allows us to separate attitude towards risk from attitude towards wealth: with Yaari’s approach, risk aversion is entirely driven by a transformation of probabilities whereby bad outcomes are given more weight while good outcomes are given less weight.

Following De Donder and Hindriks (2003), the utility function of an individual who exerts an effort e with cost c(e), faces a loss probability p(e) and who buys the insurance contract (π,δ) is

$$\begin{array}{lll} u(\pi ,\delta ,p,e;\alpha ) &=&(1+\alpha )p(e)\left( w-\pi -(1-\delta )L\right) \\ &&+\left( 1-(1+\alpha )p(e)\right) \left( w-\pi \right) -c(e) \\ &=&w-\pi -(1+\alpha )p(e)(1-\delta )L-c(e) \end{array}$$

(3)

where w denotes his exogenous income and α ≥ 0 is his risk aversion parameter. In words, the individual overestimates by a fraction α his expected financial damage. The utility without insurance is

$$u(0,0,p,e;\alpha )=w-(1+\alpha )p(e)L-c(e)$$

and the reservation premium is

$$r(\delta ,p;\alpha )=(1+\alpha )p(e)\delta L.$$

(4)

Therefore risk aversion α in our model takes the form of a relative markup over the actuarially fair price.

We first solve for the optimal precaution choice given the insurance contract (δ,π) bought:

$$\max_{e}w-\pi -(1+\alpha )p(e)(1-\delta )L-c(e).$$

We assume that c(e) = e ² , L = 1, $p(e)=\frac{(1-e)}{2}\in \lbrack 0, \frac{1}{2}]$ for $e\in \lbrack 0,1]$ and define θ = (1 + α)/2 (with $\theta \in \lbrack \frac{1}{2},1]$). The optimal precaution choice for type θ is

$$e_{\theta }(\delta )=\frac{(1-\delta )\theta }{2}\in \left[ 0,\frac{1}{2}\right]$$

and the corresponding risk for type θ is

$$p_{\theta }(\delta )=\frac{1}{2}-\frac{(1-\delta )\theta }{4}\in \left[\frac{1}{4},\frac{1}{2}\right].$$

Property 1 (regularity) is then satisfied, and the induced utility function for type θ from insurance contract (δ,π) is

$$\begin{array}{lll}V_{\theta }(\delta ,\pi ) &=&w-\pi -2\theta (1-\delta )p_{\theta }(\delta )-c(e_{\theta }(\delta )) \\ &=&w-\pi -\theta (1-\delta )+\frac{(1-\delta )^{2}}{4}\theta ^{2}. \end{array}$$

With our functional forms, marginal willingness to pay for insurance (using an envelope argument with respect to precaution choice) is

$$\left. \frac{d\pi }{d\delta }\right| _{V_{\theta }}=\theta -\frac{(1-\delta ) }{2}\theta ^{2}\equiv s(\delta ,\theta ).$$

We thus obtain naturally the single-crossing condition (Property 2).

We turn to the insurance contracts offered to the individuals. The contract curve composed of all fair contracts proposed to an individual of type θ is given by

$$\pi \left( \delta ,\theta \right) =\left( \frac{1}{2}-\frac{\left( 1-\delta \right) }{4}\theta \right) \delta$$

where $\delta \in \lbrack 0,1]$.

The slope of this contract curve in the space (π,δ) is given by

$$\frac{\partial \pi (\delta ,\theta )}{\partial \delta }=\frac{1}{2}-\frac{ \theta (1-2\delta )}{4}\equiv \sigma (\delta ,\theta ).$$

It is readily verified that the contract curve is convex, due to moral hazard. Moreover, we have $\pi \left( 0,\theta \right) =0,$ $\pi \left( 1,\theta \right) =1/2$ (since no precaution is exerted and everybody shares the same risk) and $\pi \left( \delta ,L\right) >\pi \left( \delta ,H\right)$ for all 0 < δ < 1 and all L < H, reflecting the higher risk level of less risk averse agents.

The reader can check that the optimal choice of an insurance contract along the actuarially fair contract curve of each individual implies that more risk averse agents buy more coverage. On the other hand, moral hazard results in both indifference curves and contract curves being convex in the (π,δ)-space, with indifference curves less convex than contract curves (because utility functions incorporate the cost of precaution while contract curves do not). Using the dual theory of risk preferences, we then end up with the same situation as the one described in the text: although Properties 1 and 2 are satisfied, propitious selection generates a positive correlation between risk and insurance purchase.