### Some algebra on adverse selection in insurance markets

12/05/2014 02:17:00 PM
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Here's an extremely simple model of adverse selection in an insurance market. I have in mind health insurance, but nothing about this example is specific to health.

There are two types of individuals in this health insurance market, indexed by $i\in\left\{H,L\right\}$ where $H$ denotes those with a high probability $p_H$ of incurring a hospitalization cost $h \gt 0$ and $L$ denotes those with a low probability $p_L$ of hospitalization. We'll assume there are equal numbers of each, and represent them as two representative individuals. Let $C_i$ denote individual $i$'s consumption of all non-hospital goods and services, and $y_i$ is income. Each individual seeks to maximize expected utility from non-medical consumption according to the risk-averse utility function $$U_i=E_i\left[ln\left(C_i\right)\right]$$ subject to the budget constraint, which is $C_i\leq y_i-h$ if they get sick without insurance, or $C_i\leq y_i$ if they do not buy insurance and don't get sick, or $C_i\leq y_i-m_i$ if they buy insurance at premium $m_i$ whether or not they get sick. This means expected utility without insurance is $$\left(1-p_i\right) ln\left(y\right)+p_i ln\left(y-h\right)$$ versus $$ln\left(y-m_i\right)$$ if he does buy insurance. The insurer is perfectly competitive with no loading costs, so the premium is equal to the insurers' expected costs for each plan.

It is Pareto Efficient for both individuals to buy insurance. To see that result, set prices $m_H=p_hh$, and $m_L=pLh$, and note that \begin{align*} \left(1-p_H\right) ln\left(y\right)+p_H ln\left(y-h\right)&\leq ln\left(y-m_H\right) \\ \left(1-p_L\right) ln\left(y\right)+p_L ln\left(y-h\right)&\leq ln\left(y-m_L\right) \end{align*} With strict inequality guaranteed by the risk-aversion of each individuals (ie, ln is a concave function). Thus, there exists prices at which insurer's profit-maximization conditions are satisfied and both individuals are better off with insurance than without. Not only is this Pareto Efficient, but this is also the competitive equilibrium whenever the risk types $p_L,p_H$ of every individual are publicly known and insurers are allowed to discriminate on the basis of that information (Aka risk-rating).

Suppose that the individuals know their own risk types but that the insurer either doesn't know which is which (asymmetric information), or is prohibited from discriminating on the basis of risk type (community-rating). Either way, the insurer is ultimately forced to offer the same premium to both individuals, and the same profit-maximization that the premium equals the expected cost of the plan still applies. In this case, there are two possible types of equilibrium.

### Pooling Equilibrium

Both individuals will buy insurance at the price $m\equiv m_H=m_L=\frac{p_L+p_H}{2}h$ if \begin{align} \left(1-p_H\right) ln\left(y\right)+p_H ln\left(y-h\right)&\leq ln\left(y-\frac{p_L+p_H}{2}h\right), and \label{htype}\\ \left(1-p_L\right) ln\left(y\right)+p_L ln\left(y-h\right)&\leq ln\left(y-\frac{p_L+p_H}{2}h\right) \end{align} and you can verify that the insurer's profit conditions are also satisfied. Note that the right hand sides are the same, and the left hand side is bigger for the $L$ type, so this says we end up at a pooling equilibrium if $p_H$ is not too much larger than $p_L$, thus achieving an equilibrium that is also Pareto Efficient.

### Separating Equilibrium

Equation \eqref{htype} is always true because the pooled premium is actually less than the $H$ type's expected cost, so this result follows directly from the definition of risk-aversion. However, for $p_H$ that is too much larger than $p_L$ we could have $$\left(1-p_L\right) ln\left(y\right)+p_L ln\left(y-h\right)\gt ln\left(y-\frac{p_L+p_H}{2}h\right)$$ because the pooled premium is higher than the $L$ type's expected cost, and there is a limit to how large a risk premium individuals are willing to pay above their expected costs. If this happens, then we end up in an equilibrium where $m=p_Hh$ and only the $H$ type buys insurance. This equilibrium is not Pareto Efficient because the social costs of giving the $L$ are less than what he's willing to pay, and yet the equilibrium fails to give him insurance.

### Numerical Example

To prove that the above is non-vacuous, consider a numerical example: let $y=100$, $h=50$, and $p_L=0.1$. The graph below shows how the utilities for each with and without insurance changes as $p_H$ (plotted on x-axis) changes: The horizontal axis gives values of $p_H$ while the y-axis plots the utility as given by the functions above. If both individuals buy insurance, then their utilities will be the same, given by the red line in the graph. The $H$ type's utility if he doesn't buy insurance is the blue line, which is horizontal because with him out of the market, his utility does not depend on the $H$ type's risk. Notice that for $p_H \gt 0.17$ the blue line is higher than the red line, implying that the $L$ type is better off not having insurance than paying the high pooled risk-premium. However, the $H$ type's utility without insurance, denoted by the green line, is everywhere less than his utility in the pooling equilibrium.

Thus, if $p_H \gt 0.17$ this market has adverse selection resulting in inefficiently low coverage rates. One solution for this is to subsidize. Suppose $p_L=0.1$ and $p_H=0.2$, then without any intervention we have adverse selection as the low risk type drops out. Importantly, notice that the insurance premium for those who continue to buy insurance, $m=10$ is higher than if the low risk type had not dropped out, which would have been $m=7.5$. It does not take much of a subsidy here to induce the low type to buy: a subsidy of just 0.81 is more than enough, and in fact is also low enough that the $H$ type would still be better off even if he has to pay the taxes to fund the subsidies--a cost to him of 8.31 instead of 10. Hence, a tax-and-subsidy scheme to induce everyone to buy insurance does make every person individually better off in the presence of adverse selection. With the subsidies, the market is once again Pareto Efficient.