Consider two types of household (A and B) that face a risky prospect. Their initial prospects are (e,4, e,4; 8) = (100, 200: 0.2) and (e,5, e,B; 0) = (50,120: 0.5). Both household types rank prospects according to the expected utility of the gamble defined using the cardinal utility function u(x) = In(x). An insurance provider wishes to offer contracts to the two types of household. Assume to start that the firm knows that the probability of loss (state 1) for the types are respectively 8 = 0.2 (type A) and 0 = 0.5 (type B). (4 marks) Find the households' optimal choices (x,4,x,4) and (x,, x,) if each household type is able to purchase actuarily fair insurance that reflects their true probability of loss (state 1). (2 marks) What is the size of the insurance benefit (b), and the premium (p) for each household type implied by your answer in i)? i) ii) For the rest of the question, assume the insurance firm is unable to distinguish between type A and B households. ii) (2 marks) If the low risk households make up 80% of the population, what terms (relationship between premium and benefit) would a pooling insurance contract have to have if it was to yield zero expected economic profit for the firm? (3 marks) Find the pooling contract in iii), (b", p') when the low-risk households make up 80% of the population. (2 marks) What state-contingent consumption pair (x,", x,) can household B obtain using the pooling contract in iv)? (4 marks) Does the high-risk group prefer the pooling contract (in iv) and v) to fully insuring as found in i)? Explain your reasoning. Use a graph in (x,. x2) space to illustrate your answer. iv) v) vi) vii) (4 marks) Now consider a scenario where the share of the population that is low-risk (type A) is considerably lower. Find the population share (call it A, 0