There are three dates t = 0, 1, 2 and a continuum of consumers with measure one, each endowed with wealth one. There are two types of

consumers: a fraction & is impatient, i.e. their utility is 1 – (1/c₁), where c₁ is consumption at t = 1, and a fraction 1 - a are patient whose

utility is 1 – (1/₂), where where ₂ is consumption at t = 2. At t= 0 consumers don't know their type, i.e. it is unknown whether the

consumer wishes to consume at date t = 1 or t= 2. At t = 1 all consumers observe their own type, but cannot observe others' types (i.e.

type is private information). There are two technologies to manage liquidity. Consumers can invest for one period only (short term

technology), in which case the investment will yield r≥ 1. This technology is available at both t = 0 and t = 1. Alternatively, wealth can be

invested for two periods (long term technology) at date t = 0, which yields 0 at t = 1, but R > r² at t = 2. The investment in the long-term

technology can be liquidated at t = 1, which then yields L < 1. Wealth can also be stored across periods without cost.

a)

Suppose there is a bank to manage liquidity risk, all consumers deposit their wealth in the bank and the bank maximizes total

welfare of its depositors when it chooses , c3, the promised payouts for early (t = 1) and late (t = 2) withdrawals, respectively.

I

Set up the bank's optimisation problem and derive the first order condition. What does the first order condition tell you about

the optimal allocation?

Calculate the optimal investment in the long-term technology IB and the optimal allocation c, c

III.

Does the allocation , always constitute an equilibrium? If not, derive the necessary condition for ² to be an

equilibrium and carefully explain your finding.

II.

IV.

Now, suppose the condition that you derived in iii) above does not hold. Derive the feasible (incentive compatible) allocation

that maximizes total welfare and can always be supported in an equilibrium.

b)

Now, suppose there is no financial intermediary to handle liquidity shocks. However, at t = 1 a financial market for bonds

opens up and agents trade their wealth at t = 1 for wealth at t = 2 . Each bond pays 1 at t = 2 and its price is pM. Calculate the

consumer's optimal investment decision IM at t=0, an expression for the price of the bond M and the consumption levels in the two

states ,

c)

Is the bond market allocation efficient? Calculate the price of the bond that would ensure that the market delivers the

optimal allocation and discuss what would be needed for this price to clear the market.