<p>Let <span class="math-container">$c$</span> be the cost per unit of insurance, so the premium is equal to <span class="math-container">$cn$</span>. Then the agent maximises:
<span class="math-container">$$
p(u(w - L - cn + n) + (1-p)u(w - cn).
$$</span>
The first order condition with respect to <span class="math-container">$n$</span> is given by:
<span class="math-container">$$
(1 - c) p u'(w - L - cn + n) - c (1-p) u'(w - cn) = 0
$$</span>
Rearranging gives:
<span class="math-container">$$
\frac{p}{1 - p} = \frac{c}{1 - c} \frac{u'(w - cn)}{u'(w - L - cn + n)}.
$$</span>
if <span class="math-container">$p = c$</span> this simplifies to:
<span class="math-container">$$
1 = \frac{u'(w - cn)}{u'(w - L - cn + n)}
$$</span>
As <span class="math-container">$u'$</span> is strictly decreasing, this can only happen if <span class="math-container">$w - cn = w - L - cn + n$</span> or equivalently that <span class="math-container">$L = n$</span>.</p>

CC BY-SA 4.0
2021-10-05T13:08:33.957Z
2021-10-05T13:08:33.957Z
tdm
<p>I am practising past micro economics questions from the internet and I am not sure how to proceed with this question:</p>
<p>Imagine a situation where a risk averse agent has positive wealth(w) and may face a loss(L) with probability (p). She can buy an insurance (n&gt;0) at cost(ca): How can I maximise this function to show that the agent buys full insurance if c=p, and that the agent’s insurance coverage decreases with wealth(w) when utility is decreasing and p&lt;c</p>
<p>max ( −  − ca + n) + (1 − ) ( − ca)</p>
<p>I have attempted this but i don't think I'm on the right track, your suggestions will be helpful.</p>
<p>I took the first and second derivatives wrt to p and c, how do I proceed after this. To show that full insurance decreases with wealth do I minimise the function or derivate wrt to w?</p>

<microeconomics><risk><uncertainty>
Choice under uncertainty
Gill
# Choice under uncertainty

(Asked by: username_0 on 2021-10-05)

I am practising past micro economics questions from the internet and I am not sure how to proceed with this question:

Imagine a situation where a risk averse agent has positive wealth(w) and may face a loss(L) with probability (p). She can buy an insurance (n>0) at cost(ca): How can I maximise this function to show that the agent buys full insurance if c=p, and that the agent’s insurance coverage decreases with wealth(w) when utility is decreasing and p

max ( − − ca + n) + (1 − ) ( − ca)

I have attempted this but i don't think I'm on the right track, your suggestions will be helpful.

I took the first and second derivatives wrt to p and c, how do I proceed after this. To show that full insurance decreases with wealth do I minimise the function or derivate wrt to w?

**Tags:** microeconomics, risk, uncertainty

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## 1 answers

### Answer by username_1 on 2021-10-05. Score: 3

Let $c$ be the cost per unit of insurance, so the premium is equal to $cn$. Then the agent maximises:
$$
p(u(w - L - cn + n) + (1-p)u(w - cn).
$$
The first order condition with respect to $n$ is given by:
$$
(1 - c) p u'(w - L - cn + n) - c (1-p) u'(w - cn) = 0
$$
Rearranging gives:
$$
\frac{p}{1 - p} = \frac{c}{1 - c} \frac{u'(w - cn)}{u'(w - L - cn + n)}.
$$
if $p = c$ this simplifies to:
$$
1 = \frac{u'(w - cn)}{u'(w - L - cn + n)}
$$
As $u'$ is strictly decreasing, this can only happen if $w - cn = w - L - cn + n$ or equivalently that $L = n$.


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