<p>Your derivation looks ok, but maybe a bit too complicated. Notice that if there is perfect competition in the insurance market and if firms only sell to the <span class="math-container">$\pi_i = 0.5$</span> type then expected profits (per contract) is zero and equals:
<span class="math-container">$$
p - 0.5 L = 0
$$</span>
This immediately gives <span class="math-container">$p = 0.5$</span> if <span class="math-container">$L = 1$</span>.</p>
<p>Also, your incentive constraint is somewhat off as:
<span class="math-container">$$
\frac{\sqrt{5} - \sqrt{5 - 0.5}}{\sqrt{5} - 2} = 0.486079
$$</span>
which is not exactly equal to 0.5</p>

CC BY-SA 4.0
2021-06-05T20:01:37.997Z
Thank you for your imput. Yeah, I redid it again and got something similar.
Maybeline Lee
CC BY-SA 4.0
2021-06-04T18:30:14.807Z
2021-06-04T18:30:14.807Z
tdm
<p>We are in an Insurance Adverse Selection. Assume that consumers differ in their own risk <span class="math-container">$\pi_i$</span> distributed on the interval <span class="math-container">$[\underline \pi, \bar\pi ]=[0, 0.5] $</span>. CDF is as follows <span class="math-container">$F(\pi)= 2\pi^2 + \pi.$</span> where <span class="math-container">$\pi$</span> is private information and Loss (denoted as L) is equal to <span class="math-container">$1$</span> for all consumers.  Consider the risk-neutral, competitive insurance companies which offer full insurance at price <span class="math-container">$p$</span>. The consumers’ utility function is <span class="math-container">$U(w)=\sqrt{w}$</span> and initial wealth <span class="math-container">$w_0 = 5$</span> are equal for all consumers.</p>
<p>The problem asks for <span class="math-container">$h(p)$</span>. With which price <span class="math-container">$p$</span> only the consumer with <span class="math-container">$\pi_i = 0.5$</span> will buy the insurance? And then to find equilibrium price <span class="math-container">$p^*.$</span></p>
<p>Now to find <span class="math-container">$h(p)=\frac{u(w)-u(w-p)}{u(w)-u(w-L)}$</span> and <span class="math-container">$p^*=\frac{\int_{h(p^*)}^{\pi}\pi dF(\pi)}{1 − F(h(p^∗)) }L$</span></p>
<p>For the first part: <span class="math-container">$h(p)=\frac{u(w)-u(w-p)}{u(w)-u(w-L)}=\frac{\sqrt{5}-\sqrt{5-p}}{\sqrt{5}-2}$</span>. The consumer with <span class="math-container">$\pi_i=0.5$</span> will buy the insurance if <span class="math-container">$\frac{\sqrt{5}-\sqrt{5-p}}{\sqrt{5}-2}\leq 0.5$</span> (after some math) we find <span class="math-container">$p=0.5$</span>, only consumer with <span class="math-container">$\pi_i=0.5$</span> buys.</p>
<p>For the equilibrium part:</p>
<p><span class="math-container">$p^*=\frac{\int_{h(p^*)}^{0.5}\pi dF(\pi)}{1 − F(h(p^∗)) }=\frac{\frac{7}{9}-1/2(h(p^*))^4-1/3(h(p^*))^3}{1-2(h(p^*))^2-h(p^*)}$</span></p>
<p>Combining both: <span class="math-container">$h(p^*)\frac{\sqrt{5}-\sqrt{5-p^*}}{\sqrt{5}-2}$</span> and considering <span class="math-container">$p^*=0.5$</span></p>
<p><span class="math-container">$h(p^*)=0.5 \rightarrow$</span> <span class="math-container">$p^*=E[\pi_i|\pi_i\geq h(p^*)]$</span> holds when <span class="math-container">$p^*=0.5$</span> thus it is an equilibrium. Only those with <span class="math-container">$\pi_i=0.5$</span> buys</p>

<microeconomics><adverse-selection>
Information economics
Maybeline Lee
# Information economics

(Asked by: username_0 on 2021-06-02)

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

**Tags:** microeconomics, adverse-selection


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## 1 answers

### Answer by username_1 on 2021-06-04. Score: 2 | Accepted answer

Your derivation looks ok, but maybe a bit too complicated. Notice that if there is perfect competition in the insurance market and if firms only sell to the $\pi\_i = 0.5$ type then expected profits (per contract) is zero and equals:
$$
p - 0.5 L = 0
$$
This immediately gives $p = 0.5$ if $L = 1$.

Also, your incentive constraint is somewhat off as:
$$
\frac{\sqrt{5} - \sqrt{5 - 0.5}}{\sqrt{5} - 2} = 0.486079
$$
which is not exactly equal to 0.5

> **Comments:**
> **username_0**: Thank you for your imput. Yeah, I redid it again and got something similar.  

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