Q3. Alter the Diamond-Dybvig model by adding a liquid asset. Suppose that there are two assets, an illiquid asset that returns 1+r units of consumption goods in period 2 for each unit invested in period 0, and a liquid asset that returns one unit of consumption goods in period 1 for each unit invested in period 0. The illiquid asset production technology cannot be interrupted in period 1 (L = 0). The model is otherwise the same as the slides.

(a) Determine a consumer life-time income when there is no bank, show this in a diagram and determine the consumer's optimal consumption for an impatient consumer and patient consumer.

(b) Determine a bank's life time budget, show this in your diagram and determine the optimal deposit contract for the bank in the diagram. Are consumers who deposit in the bank better off than in (a)? Explain it.

(c) Is there a bank run equilibrium? Explain when it will happen.