Consider a simple version of the Diamond-Dybvig model. The model has three periods (7= 0, 1, 2) and a single perfectly divisible good. There is a unit mass of agents [0, 1]. Every agent is endowed with one unit of the good at T' = 0. If the good is invested at 7 = 0, it yields a return of 2 units of the good if it is held until 7" = 2, but only 1 unit if the project is stopped in T' = 1. Agents can privately store their good at no cost. Ex ante, all agents are the same. Only at T' = 1, every agent learns his type. Type 1 is impatient and cares only about consumption Question continued... Turn Over Page 3 of 4 at T' = 1, while type 2 is patient and wants to consume only at T' = 2. Every agent has the same utility function u(x) = 1 - 1 for all x > 0. With probability 0.4 agent is type 1 and with probability 0.6 agent is type 2. Agents of type 2 have a discount factor p = 0.8. Firstly, calculate the optimal allocation of and of that a competitive bank can offer to agents who deposit at T' = 0 but consume at T = 1 or 7" = 2. Show that 1