Assume there is one insurance plan. Consumers decide whether to enroll in this insurance plan or remain uninsured. Consumers are heterogeneous in how much they value insurance and how costly they are to insure. Assume that a consumer of type θ gets utility equal to VI (θ) = θ from insurance.
The costs of providing this consumer with insurance are CI(θ)= α + 2βθ where -0.5 < β < 0.5 and -β < α < 1-2β . The utility and costs of remaining uninsured are VU (θ) = CU (θ) = 0. θ is distributed uniformly between zero and one.

a) How do the allocations that emerge in competive equilibrium compare to the social planners allocation when β = 0, β < 0, and β > 0? Why might β be less than zero (θ can represent consumer characteristics other than unobserved health)? Explain why when β = 0, the competitive equilibrium is efficient.

b) Assume now that β > 0. The government wants to subsidize the insurance plan to induce the efficient allocation. At what level must the government set this subsidy?