1. (The Market for Health Insurance) Consider a market for health insurance, in which there is a continuum of consumer types, differentiated by their risk of becoming ill, indicated by 0. The risk type e is private information to the consumer, and is distributed according to a distribution function F(0), which has positive density f(0) on the interval [e, #]. Consumers can be offered health plans that differ in the generosity of coverage, or quality of the plan, indicated by q 2 0. Let v(q, #) denote the utility that a consumer of type o derives from having a plan of quality q, and let c(q, 0) denote the cost of serving a customer of type with plan q. We will assume that more generous plans are preferred by every risk type, but are more expensive to provide. Thus we assume that v. >> 0 and cq >> 0. Furthermore, we assume that riskier types derive more vale from any given plan, but are more costly to serve, i.e. ve >0 and c > 0. Let S(q, 0) denote the surplus created by offering a plan of quality q to a consumer of risk type 8. Thus we have S(q, # ) = v(q, e) - c(q, #). (a) Suppose that S,(0, 0) 2 0, S,(1, ) 1, Saq 0. Interpret these conditions, and characterize the first best optimum quality q*(@). Is the first best coverage increasing in the risk type 0? We now wish to investigate whether the first-best outcome is implementable, i.e. whether there exists a transfer scheme *(0) such that the pair (q*(0), t*(@)) is incentive compatible and individually rational. The next few (sub)questions address this issue