Consider an insurance setting where all losses are strictly financial. All consumers have initial wealth > 0 and incur loss (0, ) with probability > 0, where is the consumers type, which may be R (risky) or S (safe). Assume > and that fraction (0,1) of consumers are of type R.

There is one risk-neutral insurer, while consumers are risk-averse with utility over wealth . Let be type s willingness to pay for full insurance.

a) Write down the equation that determines and use it to prove that > .

b) Suppose the insurer cannot observe the consumers type. Write down the necessary and sufficient condition under which, due to adverse selection, it is not possible to achieve a Pareto efficient outcome that is also a Pareto improvement (considering both types of consumers and the insurer) from a complete lack of insurance. Show your reasoning. Be sure to eliminate redundant conditions.