Consider a Diamond-Dybvig economy with a single consumption good and three dates (*t* = 0*,* 1 and 2). There is a simple storage technology and an illiquid investment opportunity. For each unit of good stored, the simple storage technology will return one unit of good in the next period. For each unit of good invested at time *t* = 0*,* the illiquid investment will yield a risk-free return of *R* *>* 1 units of goods at time *t* = 2*,* but only *l* ∈ (0*,* 1) units if terminated prematurely at time *t* = 1*.*

The economy has a large number of *ex* *ante* identical consumers. The size of the population is *N* *>* 0*.* Each consumer receives one unit of good as initial endowment at *t* = 0*.* 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 *π*_{1} ∈ (0*,* 1) and the probability of being a patient one is *π*_{2} = 1 — *π*_{1}*.* An impatient consumer only cares about consumption at *t* = 1*,* denoted by *C*_{1}*.* The consumer’s utility is given by *u* (*C*_{1}) *.* A patient consumer only cares about consumption at *t* = 2*,* denoted by *C*_{2}*.* The consumer’s utility is given by *βu* (*C*_{2}) *,* where *β* ∈ (0*,* 1) is a parameter known as the subjective discount factor that captures the consumer’s time preferences.

- Suppose the consumer’s utility function is given by

*C*1—*u* (*C*) =1 —, with *>* 0*.*

Let (*C*1∗*,* *C*2∗) denote the first-best allocation. Derive an interior solution of (*C*1∗*,* *C*2∗) *.* Show that the corner solutions (*C*_{1}*,* *C*_{2}) = (0*,* *R/π*_{2}) and (*C*_{1}*,* *C*_{2}) = (1*/π*_{1}*,* 0) cannot be a first-best allocation under this type of utility function.

- Suppose now there is a financial intermediary which offers the consumers a deposit contract. Under this contract, the consumers will give their initial endowment at
*t*= 0 to the bank. In return, the bank will offer*C*1∗ units of good to anyone who withdraws at*t*= 1 and*C*2∗ units of good at*t*= 2*.*Based on your answer in part (a), show that if*βR**<*1 then all patient consumers will choose to withdraw early. - Suppose now the consumer’s utility function is given by
*u*(*C*) = √*C.*State the conditions under which a unique interior solution of the autarky problem exists. Derive the interior solution under these conditions.