Willy owns a small chocolate factory, located close to a river that occasionally floods in the spring, with disastrous consequences. Next summer, Willy plans to sell the factory and retire. The only income he will have is the proceeds of the sale of his factory. If there is no flood, the factory will be worth $500,000. If there is a flood, then what is left of the factory will be worth only $50,000. Willy can buy flood insurance at a cost of $.10 for each $1 worth of coverage. Willy thinks that the probability that there will be a flood this spring is 1/10. Let cF denote the contingent commodity dollars if there is a flood and cNF denote dollars if there is no flood. Willy’s von Neumann-Morgenstern utility function is U (cF , cNF) = .1 √ cF + .9 √ cNF.
(a) If he buys no insurance, then in each contingency, Willy’s consumption will equal the value of his factory, so Willy’s contingent commodity bundle will be (cF , cNF) = ______.
(b) To buy insurance that pays him $x in case of a flood, Willy must pay an insurance premium of .1x. (The insurance premium must be paid whether or not there is a flood.) If Willy insures for $x, then if there is a flood, he gets $x in insurance benefits. Suppose that Willy has contracted for insurance that pays him $x in the event of a flood. Then after paying his insurance premium, he will be able to consume cF = _________. If Willy has this amount of insurance and there is no flood, then he will be able to consume cNF = ___________.
(c) You can eliminate x from the two equations for cF and cNF that you found above. This gives you a budget equation for Willy. Of course there are many equivalent ways of writing the same budget equation, since multiplying both sides of a budget equation by a positive constant yields an equivalent budget equation. The form of the budget equation in which the “price” of cNF is 1 can be written as .9cNF+ ______ cF = _________.
(d) Willy’s marginal rate of substitution between the two contingent commodities, dollars if there is no flood and dollars if there is a flood, is MRS(cF , cNF) = −.1 √ cNF / .9 √ cF . To find his optimal bundle of contingent commodities, you must set this marginal rate of substitution equal to the number = _____. Solving this equation, you find that Willy will choose to consume the two contingent commodities in the ratio cNF/cF = 1.
(e) Since you know the ratio in which he will consume cF and cNF, and you know his budget equation, you can solve for his optimal consumption bundle, which is (cF , cNF)= ____________. Willy will buy an insurance policy that will pay him ______ if there is a flood. The amount of insurance premium that he will have to pay is _________.