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 (e,4, ez4; 6) = (100, 200; 0.1) and (e, , ez#; 0) = (50, 120; 0.4). i) (2 marks) Express each household's budget constraint in term of prices p, 4 and p1 expressed in terms of units of x2 per unit of x, , 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, 4, x24) and (x, B, x2) if each household type are able to purchase actuarily fair insurance that reflects their 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) (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? V) (3 marks) Find the pooling contract in iv), (b", p") when the low risk households make up 80% of the population. vi) (2 marks) What state-contingent consumption pair (*1, X2 ) can household B obtain using the pooling contract in v)? vii) (4 marks) Does the high risk group prefer the pooling contract (in v) and vi)) to fully ensuring at a price as found in i)? How do you know? Use a graph in (x1, X2) space to illustrate your answer. vili) (3 marks) Suppose the share of the population that is low-risk (type A) is considerably lower. Find the population share (call it ) , 0