代写Markets and Friction Problem Set 2代写Java编程

Markets and Friction

Problem Set 2

Problem Set 2 is worth 10% of your grade.  Please upload your assignment to Moodle by 4pm of October 25th. You may type or write your answers by hand. Handwritten assignments must be legible or they will be dismissed. 5% will be deducted from the mark of late submissions for each day (before solutions are posted). Submissions made after the solutions are posted will not be marked. Each question is worth 25 points, and each part of the question is worth 5 points.

1. We extend the corporate liquidity demand model to include the possibility of liqui- dation. Consider a three-period model: t = 0, 1, 2. In t = 0, the entrepreneur with net worth A chooses the scale of the project I. In t = 1, an exogenous liquidity shock ρ is realized. The shock can take one of two values: ρ ∈ {ρL, ρH }, with ρH  > ρL . The probability of ρH occurring is q ∈ (0, 1). The entrepreneur can choose to continue at a scale i  ≤ I at an additional cost of ρi. In t  = 2, the project’s return z is realized. Specifically, the value of the project is zi. The pledgeable income is p  < z, so the competitive investors can receive at most pi. The following relationships hold:

0 < ρL  < p < ρH  < z,

1 > (1 − q) (p − ρL ) ,

(1 − q) (z − ρL ) > 1.

At t = 1, after learning the liquidity shock, the entrepreneur can liquidate the as- set that is not used for a return of δ . Specifically, if the entrepreneur continues at scale i < I, then I − i can be liquidated in exchange for δ (I − i) . The entrepreneur appropriates the whole return of the liquidated asset.  Assume that z − ρH   ≥ δ . The financial contract specifies the continuation scale for each shock:  (iL, iH ) with iL, iH  ≤ I.

a. Explain the significance of the assumption z − ρH  ≥ δ .

b. Write down the participation constraint.

c. Write down the optimization problem and explain why at the optimum iL  = I?

d. Let x = I/iH . Compare your answer with the setting without the option of liqui-

dation (δ = 0) and show that liquidation makes continuing at full scale under shock ρH more difficult.

e. Explain the result in Part d.

2. Consider a three-period model with time indexed by t = 0, 1, 2. At t = 0, all agents are endowed with wealth 1. There is a unit mass of identical agents. For each agent, consumption either takes place in t  = 1 or t  = 2. However, at t  = 0, agents are unsure which dates they would want to consume. With probability µ ∈ (0, 1), they would want to consume at t  = 1, which we refer to as the early agents.  With the remaining probability, they would want to consume in t = 2, which we refer to as the late agents.  Agents learn whether they are early or late agentsin t  =  1, and the liquidity shock is independent across agents. The discount factor is β = 1. The agents’ utility is

Agents can buy short and long assets.  The short asset transforms x units of date t consumption to rx units of date t + 1 consumption.  The long asset transforms x units of date t consumption to Rx units of date t + 2 consumption.  Assume that R > r2  > 1. The agents transfer their wealth across periods using these two assets. Furthermore, a secondary market for long assets existsint = 1, where agents can buy or sell their long asset positions at price p.

a. Explain the significance of the assumption R > r2 .

b. Suppose the secondary market for long assets is closed at t = 1. What is the optimal portfolio? Hint: Do not forget about the corner solution!

c. What is the equilibrium price p of long assets in t = 1?

d. What is the optimal portfolio when the secondary market is open?

e. Compare the utility in Part b.  with Part d.  Is the utility gain/loss from the secondary market increasing or decreasing in σ?Explain.

3. Continuing with the environment in Question 2, consider an environment where

the agents’ liquidity risk can be pooled. Furthermore, we assume that R > r1+σ

a. Setup the planner’s problem and derive the social optimal consumption (c1(*), c2(*)) .

b. Explain how the ratio c*1/c*2 changes as σ increases from 0.5 to 1.5.

c. Consider the existence of competitive banks with agents depositing all of their wealth in the banks.  Let bt  denote the returns that the banks pay to the de- positors for withdrawing at date t. Setup the banks’ optimization problem and

derive the optimal returns (b1(*), b2(*)) .

d. Explain why late agents would not want to withdraw at t = 1 if only the early agents withdraw at t = 1.

e. Assume that the long asset can be liquidated at t = 1 for a return of 1. Calculate the tipping point m* such that if the mass of agents trying to withdraw at t = 1 exceeds m*, then all agents would want to withdraw at t  = 1. Hint:  Do not forget the possibility of a corner solution.

4. Consider an extension of the environment in Question 3 with sunspots. The sunspot is a signal ξ ∈  {Red, Green} that is independent of the economic fundamentals. With probability λ, ξ = Red and agents panic and believe that everyone else will withdraw early. With probability 1 − λ, ξ = Green and agents are calm.

a. Given your answer in Part e.  of Question 3, find a condition on σ such that bank runs may occur.

b. Find the values for the following:

i. The early and late consumers’ consumption in a bank run. Hint:  They are not the same!

ii. Maximum b1 such that a bank run will never occur.

c. Suppose the condition you found in Part a. holds. What are the optimal returns (b(ˆ)1, b(ˆ)2 ) such that bank runs never occur.

d. Write down the condition such that banks are willing to risk a bank run.

e. Let the returns from the assets be (r, R)  =  (1.5, 4) and σ = 2 with µ = 0.5.

Find the threshold λ(¯) such that banks use  (b(ˆ)1, b(ˆ)2 ) to avoid runs when λ ≥ λ(¯) .

Hint: Banks may want to choose a different asset portfolio if it chooses to risk a run. Therefore, you may need to first solve for s (λ) before solving for λ .