[SOLVED] Markets and Friction Problem Set 3 Matlab

Markets and Friction

Problem Set 3

Problem Set 3 is worth 15% of your grade.  Please upload your assignment to Moodle by 4pm of November 8th. 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. Do Question 4 of Problem Set 2. Note that we are assuming R > r1+σ .

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. There are two possible states of aggregate liquidity demand:

µ ∈ {µH, µL }, with 1 > µH  > µL  > 0.

The probability of the aggregate state being H is π ∈ (0, 1) . We will let the subscript. i  ∈  {H, L} denote the aggregate state of the economy.  With probability µi, 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 the aggregate state and whether they are early or late agents in t  =  1. The liquidity shock is independent across agents.  The discount factor is β = 1. The agents’ utility is

u (c) = ln (c) .

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 pi.

a. Setup the self-insuring agent’s optimization problem where the secondary mar- ket is open.

b. What are the supply and demand curves for the long asset in the secondary market?

c. Let s*  denote the equilibrium short asset allocation in t  =  0. Write out the equilibrium prices (pH(*), pL(*)) and prove that pL(*) > pH(*) .

d. Let

µH  = 2ϵ,    µL  = ϵ,

where ϵ  >  0. Take the limit of ϵ to 0 and show that the price difference is disproportionately larger than the difference in µ when both prices are cash-in- the-market.

e. Consider the case when only one of the prices is cash-in-the-market. For a fixed ϵ > 0, as r increases, does the volatility in prices increase or decrease? Explain. Hint: Draw a supply and demand graph and remember that s* would increase (no need to prove this).

3. Consider an extension of the environment in Question 2 with competitive banks without default. Suppose all banks adopt the same asset portfolio and offer depos- itors a return of b for withdrawing at t = 1, regardless of the liquidity state, and its residual assets to those who withdraw at t = 2. This question considers an environ- ment without sunspot-driven bank runs.

a. Write down the condition that prevents late agents from withdrawing early.

b. Suppose the condition in Part a.  is violated, what is the consumption of the early and late agents?

c. Setup the optimization problem for the banks.

d. Prove that the equilibrium price of the long asset in the secondary marketsat- isfies PL  = r/R and PH  < r.

e. Explain how, if defaults are allowed, asset prices can be even more volatile.

4. Warren Buffett, a risk-neutral investor with deep pockets, is seeking investment opportunities.  Two ambitious young women reached out—Aiko and Leila—each proposing a new project.  Both are asking for the same investment amount of $7 million. Given the small scale of these investments, Warren wishes to act charitably and aims only to break even on his investment in expectation.

Both projects have a 60% chance of success and will yield $3 per dollar of invest- ment, which is unverifiable, if successful but nothing if they fail.  Aiko and Leila, both cash-strapped entrepreneurs, offer a fraction of their patents as collateral.

The value of these patents will only be known once they hit the market.  A good patent is worth $15 million, while a bad one has no value.  Verifying the patent’s value requires Warren to spend $0.35 million on market research. Warren estimates that Aiko’s patent has an 80% chance of being good, while Leila’shas a 98% chance. Warren proposes to give each a contract specifying the loan amount (i), the face value of the debt (R), and the share of the collateralized patent (x). Aiko and Leila can commit to handing over the promised share of their patents if they default. War- ren must decide whether to conduct the market research and determine the contract terms to offer Aiko and Leila. He seeks your guidance in making these decisions.

a. If Warren conducts market research, what is the optimal information-sensitive debt offered to Aiko? What about Leila?

b. If Warren does not conduct market research, what is the optimal information- insensitive debt offered to Aiko? What about Leila?

c. What are the expected profits for Aiko and Leila under both the information- sensitive and information-insensitive debt? Which type of debt does Aiko pre- fer? Which does Leila prefer?

d. Aiko and Leila, who are classmatesatUNSW, discover that both are seeking in- vestments from Warren. Their instructors—Han, Nick, and PC—suggest they pool their patents and offer Warren two identical new patents to support each project. Each of these pooled patents succeeds with probability 0.89, calculated as (0.8 + 0.98)/2,and yields $15 million if accepted by the market. The success of the new patents is independent. Assuming Warren does not conduct market research, how much can Aiko and Leila each borrow?

e. Can you create a plot to demonstrate why Aiko and Leila can borrow more from Warren after pooling? To do this, (i) use MATLAB (or any other graphing software) to replicate Figure 1 from Lecture 6 using the parameter values given in this question; (ii) clearly label all elements in the figure, including the scales of the x- andy-axes and the meaning of each curve; (iii) identify the location of optimal profits for Aiko and Leila before and after pooling.