Using the Diamond-Dybvig Model: Now suppose there exists a social planner that individual's entrust all their endowment at time 0. The social planner will pay individuals who want to withdraw from the bank early at C₁* and those who wait till the second period C₂*. Suppose the social planner's problem is:

Maximize: U = (1/2) u(C₁) + (1/2) u(C₂)

subject to: (1/2) C₁ + (1/8) C₂ = 1

Where u(C₁) = -1/C₁ and u(C₂) = -1/C₂

Question: What is the optimal level of period 1 consumption, C₁*, the optimal level of period 2 consumption, C₂*, and the optimal level of expected utility, U* ?