= There are three dates t = 0, 1, 2 and a continuum of consumers with...

 

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= 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 a 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/c₂), where where c₂ 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 investmen 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 tota welfare of its depositors when it chooses c², c2, 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 c²,c2. Does the allocation cỉ, c3 always constitute an equilibrium? If not, derive the necessary condition for c²,c² to be an and carefully explain your finding. II. III. equilibrium IV. Now, suppose the condition that you derived in iii) above does not hold. Derive the feasible (incentive compatible) allocation cí, c 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 pM and the consumption levels in the two statesc, c. 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. = 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 a 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/c₂), where where c₂ 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 investmen 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 tota welfare of its depositors when it chooses c², c2, 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 c²,c2. Does the allocation cỉ, c3 always constitute an equilibrium? If not, derive the necessary condition for c²,c² to be an and carefully explain your finding. II. III. equilibrium IV. Now, suppose the condition that you derived in iii) above does not hold. Derive the feasible (incentive compatible) allocation cí, c 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 pM and the consumption levels in the two statesc, c. 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.

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a To set up the banks optimization problem we need to maximize the total welfare of its depositors which consists of both impatient and patient consumers Let cc be the promised payouts for early t 1 a... View the full answer