We now examine some applications of our theory of choice under uncertainty. Continue with he utility function U(x1; x2) = 1 p x1+2 p x2, and let x1 = 36, x2 = 64 and 1 = 2 = 1=2. (a) Find the expected utility of this lottery. (b) Suppose now there are two agents. Each faces this lottery, but the outcomes are independent. Hence, with probability 1 4 each receives 36, with probability 1 4 , each receives 64, and with probability 1 2 , one receives 36 and the other 64. The two agents make an agreement that if one receives 36 and the other 64, the fortunate agent transfers 14 to the unfortunate agent. Calculate their expected utility under this arrangement. Such an arrangement is the essence of an insurance policyó people facing uncorrelated risks can use the fact that they are unlikely to both experience losses to eliminate some of the risk they face. (c) Suppose again that there are two agents, each facing this lottery, but with perfectly correlated lotteries. Hence, either both receive 36 or both receive 64. Can an arrangement like that of [b] give both agents a higher expected utility than that calculated in [a]? (d) In light of your answers to [a]ñ[c], explain why it may be easier to insure houses against Öres than áoods. What you have just discovered is that if insurance is to be e§ective, it must involve uncorrelated rather than correlated risks. An insurance policy cannot help with risks that tend to either impose a loss on everyone, or on no one. What risks are likely to fall into this category?
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