Consider a model of bank run. There are three periods (T = 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 in a bank at T = 0, it yields a return of 2 units of the good if it is held until T = 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 her type. Type 1 is impatient and cares only about consumption at T = 1, while type 2 is patient and wants to consume only at T = 2. Agent types are private information. Every agent has the same utility function u(x) = 1 -e-21 for all x. With probability 0.5 an agent is type 1 and with probability 0.5 an agent is type 2. Agents of type 2 have a discount Page 3 of 4 factor p = 0.6. Let c and ca denote the consumption that an agent can possibly obtain at time T = 1 or 2, respectively. Firstly, calculate the optimal allocation q and that a competitive bank can offer to agents who deposit at T = 0 but consume at T = 1 or T = 2. Prove that 1