Suppose a consumer satisfies the von NeumannMorgenstern axioms and is (strictly) risk averse. She initially has monetary wealth w. There is some probability p that she will lose an amount L. The consumer can purchase insurance that will pay her q dollars in the event that she incurs this loss. The amount she has to pay for this insurance is q; here is the premium per dollar of coverage.

(a) How much insurance will the consumer purchase? That is, what is the choice of q (as a function of the parameters w, p, L and ) that maximizes her expected utility? (Note: It's fine to just give an equation that implicitly defines q.) (Hint: state the first order condition without further expressing/deriving q as a function of the other variables/parameters).

(b) In the special case where the insurance is actually fair (that is, = p), how much insurance will the consumer purchase? What about if > p? Discuss.