Consider the model of insurance markets discussed in class. Suppose the consumer's expected utility function is...

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Consider the model of insurance markets discussed in class. Suppose the consumer's expected utility function is U(1, 2) = (1 - p) Inx + pln2, where p is the probability of loss, is consumption in the event of no loss, and r is consumption in the event of loss. The consumer's initial wealth is w = 100, and the amount of loss is = 50. The consumer considers buying an insurance contract (a, I), which costs I upfront and pays out a + I in the event of a loss. (a) Suppose both the consumer and the insurance company know that p = 0.1. Find the contract (a,I) that maximizes the consumer's expected utility subject to the zero-profit constraint: (1 - p)r-pa = 0. For parts (b),(c) and (d), suppose there are two types of consumer: a high-risk type and a low-risk type. The probability of loss is 0.4 for the high-risk type and 0.1 for the low-risk type. The proportion of high-risk types in the population is 0.5. The consumer knows their type, but the insurance firm doesn't. (b) Prove that there is no pooling equilibrium where the insurance firm offers only one contract (a,1) = (30,30). (c) Prove that there is no separating equilibrium where the insurance firm offers two contracts (H,H)=(3,2) and (, IL) = (9,1), the high-risk type buys (H, TH) and the low-risk type buys (aL, IL). (d) Prove that there is no separating equilibrium where the low-risk type buys the contract (a,T) = (45, 5), which gives this type full insurance. Consider the model of insurance markets discussed in class. Suppose the consumer's expected utility function is U(1, 2) = (1 - p) Inx + pln2, where p is the probability of loss, is consumption in the event of no loss, and r is consumption in the event of loss. The consumer's initial wealth is w = 100, and the amount of loss is = 50. The consumer considers buying an insurance contract (a, I), which costs I upfront and pays out a + I in the event of a loss. (a) Suppose both the consumer and the insurance company know that p = 0.1. Find the contract (a,I) that maximizes the consumer's expected utility subject to the zero-profit constraint: (1 - p)r-pa = 0. For parts (b),(c) and (d), suppose there are two types of consumer: a high-risk type and a low-risk type. The probability of loss is 0.4 for the high-risk type and 0.1 for the low-risk type. The proportion of high-risk types in the population is 0.5. The consumer knows their type, but the insurance firm doesn't. (b) Prove that there is no pooling equilibrium where the insurance firm offers only one contract (a,1) = (30,30). (c) Prove that there is no separating equilibrium where the insurance firm offers two contracts (H,H)=(3,2) and (, IL) = (9,1), the high-risk type buys (H, TH) and the low-risk type buys (aL, IL). (d) Prove that there is no separating equilibrium where the low-risk type buys the contract (a,T) = (45, 5), which gives this type full insurance.