Choice under uncertainty

Question

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<c

max 𝑝𝑢(𝑤 − 𝐿 − ca + n) + (1 − 𝑝) 𝑢(𝑤 − 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?

Answer 1

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 $L = n$.