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Suppose an insurance provider wishes to offer contracts to two types of household that differ by their risk of exposure to a loss. The rm knows that the probability of loss (state 1} for the types are respectively 6 = 0.2 (type A) and 6' = 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 110:) = ln(x). The two household types' initial prospects are (91A,32A; 6) = (100, 200; 0.2) and (913,623; 6) = (50, 100; 0.5). i} _Express each household's budget constraint in term of prices p1\" and p13 expressed in terms of units of x2 per unit of xi , if each household type is able to purchase actuarily fair insurance that reflects their true probability of loss (state 1}. ii} - Find the household's optimal choices (xlA,x2'\"') and (x13,x23) if each household type are able to purchase actuarily fair insurance that reflects their true probability of loss (state 1). iii} aWhat 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) N Use your answer in ii) to nd the limit on the low-risk type insurance contracts so those contracts are not more attractive to the high-risk households (B'sl than their fully-insu red prospect. That is, identify the self-separating contracts for the two household types. v) \"Use a graph in (x1,x2) space to explain your reasoning in iv)