# Choice under uncertainty

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?

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 $$L = n$$.