A grocery store owner’s utility is given by the function u(y) = ln y

A grocery store owner’s utility is given by the function u(y) = ln y. The grocer earns $50,000 a year, but there is a 10% chance of his store catching fire, in which case he stands to lose $10,000 in terms of merchandise destroyed.

(a) (20 Points) If an insurance company offers to insure the grocer fully against his loss, how much is he willing to pay as a premium?

(b) (20 Points) In light of the above, do you think the grocer and a risk-neutral insurance company charging a fair premium will actually sign a contract? (Fair premium is the premium that equates the expected utility of the insurance company to zero.)