Answer ALL parts of this question.
Consider a Diamond-Dybvig economy with a single consumption good and three dates (t = 0, 1, and 2). There is a large number of ex ante identical consumers. The size of the population is N > 0. Each consumer receives one unit of good as an initial endowment at t = 0. This unit of good can be either consumed or invested.
At t = 1, each consumer finds out whether he/she is a patient consumer or an impatient consumer. The probability of being an impatient consumer is 1 ∈ (0,1) and the probability of being a patient one is 2 = 1 − 1. Impatient consumers only value consumption at t = 1. Their utility function is (1), where 1 denotes consumption at t = 1. Patient consumers only value consumption at t = 2. Their utility function is given by (2), where 2 denotes consumption at t = 2 and ∈ (0,1) is the subjective discount factor. The function () is strictly increasing and strictly concave, i.e., ′ () > 0 and ′′() < 0.
Consumers can buy or sell a single risk-free bond after knowing their type (patient or impatient) at t = 1. The price of the bond is p at t = 1 and it promises to pay one unit of good at t = 2. There is a simple storage technology. Each unit of good stored today will return one unit of good in the next time period. Finally, there is an illiquid asset. Each unit of illiquid investment will return > 1 units of good at t = 2, but only ∈ (0,1) units if terminated prematurely at t = 1.
(a) Let be the optimal level of illiquid investment for an individual consumer. Derive the first-order condition for an interior solution of . Show your work and explain your answers. [10 marks]
(b) Explain why the bond market is in equilibrium only when p = 1. Derive the optimal level of illiquid investment in the bond market equilibrium. [10 marks]
(c) Let (1 , 2 ) be the allocation of consumption when the bond market is in equilibrium.
Suppose the utility function is given by () = 1− /1− , with > 0. Derive the condition(s) under which this allocation is Pareto optimal. [10 marks]
