Willy owns a small chocolate factory, located close to a river that occasionally floods in the spring, with disastrous consequences. Next summer, Wily 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 the 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 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 , cN F) = .1 √ cF + .9 √ cNF .
a) What are cF and cNF if Willy buys some amount of insurance x?
b) Plug your answers for part a) into Willy’s von Neumann-Morgenstern utility function.
c) Take the derivative of b) with respect to x. Set it equal to zero (because it’s a firstorder condition) and solve for x.
d) How do you know the solution in c) is the x that maximizes Willy’s utility? Could it be the x that minimizes Willy’s utility?