Question 1. (16 marks) Suppose an insurance provider wishes to offer contracts to two types of household that differ by their risk of exposure to a loss. The firm knows that the probability of loss (state 1) for the types are respectively 6 = 0.25 (type A) and 0 = 0.5 (type B). The firm also knows that the two household types rank prospects according to the expected utility of the gamble defined using the cardinal utility function u(x) = In(x). The two household types' initial prospects are (e1 4, e24; 6) = (100, 200; 0.25) and (e1 , ez; 0) = (50, 100; 0.5) i) (4 marks) Find the household's optimal choices (x1 4, x24) and (x15, x2B) if each household type is able to purchase actuarily fair insurance that reflects there true probability of loss (state 1). ii) (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)? For the rest of the question, we will assume insurance firms are unable to distinguish between type A and B households. You will examine two situations differentiated by the population proportion of low-risk households. The aim will be to show how a change in the population proportion will change the attractiveness of a pooling contract for the low-risk group. iii) (3 marks) Suppose the population proportion of low-risk (A) type is relatively high, y = 0.6. Find the pooling contract chosen by the low-risk households if it exists. iv ) (3 marks) Suppose the population proportions of low-risk (A) type is relatively low, y = 0.25. Find the pooling contract chosen by the low-risk households if it exists. V ) (4 marks) Use a pair of graphs in (X1, X2) space to compare the choices of the low-risk households in iii) and iv) above