We are in an Insurance Adverse Selection. Assume that consumers differ in their own risk $\pi_i$ distributed on the interval $[\underline \pi, \bar\pi ]=[0, 0.5] $. CDF is as follows $F(\pi)= 2\pi^2 + \pi.$ where $\pi$ is private information and Loss (denoted as L) is equal to $1$ for all consumers. Consider the risk-neutral, competitive insurance companies which offer full insurance at price $p$. The consumers’ utility function is $U(w)=\sqrt{w}$ and initial wealth $w_0 = 5$ are equal for all consumers.
The problem asks for $h(p)$. With which price $p$ only the consumer with $\pi_i = 0.5$ will buy the insurance? And then to find equilibrium price $p^*.$
Now to find $h(p)=\frac{u(w)-u(w-p)}{u(w)-u(w-L)}$ and $p^*=\frac{\int_{h(p^*)}^{\pi}\pi dF(\pi)}{1 − F(h(p^∗)) }L$
For the first part: $h(p)=\frac{u(w)-u(w-p)}{u(w)-u(w-L)}=\frac{\sqrt{5}-\sqrt{5-p}}{\sqrt{5}-2}$. The consumer with $\pi_i=0.5$ will buy the insurance if $\frac{\sqrt{5}-\sqrt{5-p}}{\sqrt{5}-2}\leq 0.5$ (after some math) we find $p=0.5$, only consumer with $\pi_i=0.5$ buys.
For the equilibrium part:
$p^*=\frac{\int_{h(p^*)}^{0.5}\pi dF(\pi)}{1 − F(h(p^∗)) }=\frac{\frac{7}{9}-1/2(h(p^*))^4-1/3(h(p^*))^3}{1-2(h(p^*))^2-h(p^*)}$
Combining both: $h(p^*)\frac{\sqrt{5}-\sqrt{5-p^*}}{\sqrt{5}-2}$ and considering $p^*=0.5$
$h(p^*)=0.5 \rightarrow$ $p^*=E[\pi_i|\pi_i\geq h(p^*)]$ holds when $p^*=0.5$ thus it is an equilibrium. Only those with $\pi_i=0.5$ buys
