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 ?= 0.1 (type A and ? = 0.4 (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) = ln(x).

The two household types' initial prospects are ii) (81A, 82.4; 3) = (100, 200; 0.1) and (13,223; 9) = (50, 120; 0.4-). [2 marks) Express each household's budget constraint in term of prices p15 and p13 expressed in terms of units of 1:2 per unit of 2:1 , if each household type is able to purchase actuarily fair insurance that reflects their true probability of loss [state 1}. [4 marks) Find the household's optimal choices (x1A,x2A) and (1:13;?) if each household type are able to purchase actuarily fair insurance that reflects their true probability of loss [state 1]. (2 marks] 1What is the size of the insurance benefit lb}, 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} [2 marks) If the low risk households make up 30% ofthe 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? v} [3 marks) Find the pooling contract in iv), {b*,p*] when the low risk households make up 80% of the population