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?