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 8 = 0.1 (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 (e, 4, e24; 6) = (50, 100; 0.1) and (e, B, ezB; 0) = (50, 100; 0.5). i) (2 marks) Express each household's budget constraint in term of prices p, " and p1" expressed in terms of units of x2 per unit of x1 , if each household type is able to purchase actuarily fair insurance that reflects their true probability of loss (state 1). ii) (4 marks) Find the household's optimal choices (x, ",x24) and (x, #,x2#) if each household type are able to purchase actuarily fair insurance that reflects there true probability of loss (state 1). iii) (2 marks) What is the size of the insurance benefit (b), and the premium (p) for each household type implied by your answer in ii)? For the rest of the question we will assume insurance firms are unable to distinguish between type A and B households. iv) (3 marks) Use your answer in ii) to find the limit on the low-risk type insurance contracts so those contracts are not more attractive to the high-risk households (B's) than their fully-insured prospect. That is, identify the self-separating contracts for the two household types. V ) (3 marks) Use a graph in (X1, X2) space to explain your reasoning in iv)