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  denote the contingent commodity dollars if there is a flood and  denote dollars if there is no flood. Willy’s von Neumann-Morgenstern utility function is U(, ) = 0.1√ + 0.9√.
(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 (,) = _________________
(b) To buy insurance that pays him $x in case of a flood, Willy must pay an insurance premium of 0.1x. (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 =_________. If Willy has this amount of insurance and there is no flood, then he will be able to consume =________________
(c) You can eliminate x from the two equations for  and  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  is 1 can be written as 0.9 +____=________.
(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(,) = 0.1√. To find his optimal bundle 0.9√
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 ______________.
     
(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 (,)=___________ 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 ______________.