Markets and Friction
Problem Set 1
Problem Set 1 is worth 10% of your grade. Please upload your assignment to Moodle by 4pm of September 27th. 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. Consider a three-period model with a continuum of ex-ante homogeneous agents of measure 1. Time is indexed by t = 0, 1, 2. At t = 0, all agents are endowed with wealth W > 0. 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 agents in t = 1. The discount factor is β = 1. The agents’ utility can be summarized as
Agents can buy (but cannot sell) short and long assets. The short asset transforms x units of date t consumption to x units of date t + 1 consumption. The long asset transforms x units of date t consumption to Rx units of date t + 2 consumption, where R > 1. The agents transfer their wealth across periods using these two assets.
a. Let s denote the fraction of wealth spent on buying short assets. Setup the agents’ problem in t = 0.
b. Derive the optimal fraction of short assets s* .
c. Please answer the following comparative statics questions:
i. In your answer for Part b., how does s* depend on wealth W? Explain.
ii. In your answer for Part b., how does s* depend on γ?Explain.
d. Derive threshold μ(¯) ∈ (0, 1) such that for any μ ≥μ(¯), s* = 1. e. How does your answer in Part d. vary with R? Explain.
2. Continuing with the environment (agents and assets) from Question 1, now con- sider a complete market,i.e., agents can also purchase state-contingent bonds.
At t = 0, agents may buy and sell “early” bonds and “late” bonds. A unit of early (late) bond delivers one unit of consumption at t = 1 (t = 2) if and only if the agent is an early (late) agent. Each unit of early bond costs qE and each unit of late bond costs qL . The equilibrium prices of both bonds must be such that the market clears, i.e. agents are indifferent between holding the state-contingent bonds and other assets, so there are always buyers and sellers of bonds.
a. Prove that when qE = μ in equilibrium, then agents are indifferent between buying the short asset and the early bond.
b. Derive the equilibrium price qL .
c. Given these equilibrium prices, setup the agents’ problem at t = 0 in terms of c1 and c2 .
d. Solve for the optimal consumption c1(∗) and c2(∗) .
e. What happens to c2(∗) − c1(∗) as γ → ∞?Explain.
3. Consider a setting with two identical fixed-scale projects. Each project requires I/2 investment, and pays back R/2 if it is successful and 0 otherwise. For each project, the risk-neutral entrepreneur can either choose to exert high effort or low effort. The entrepreneur’s private benefit from exerting low effort in a project is B/2. The projects are stochastically independent, and the entrepreneur chooses the effort level for each project independently but simultaneously. The entrepreneur holds A ∈ (0, I) initial assets and seeks the investment from competitive outside risk-neutral investors. The investors’ outside option is normalized to 0. Figure 1 illustrates the payoff structure of one project.
Figure 1: Description of the model
Suppose the incentive payment is contingent on the number of successful outcomes, i.e., i = {0, 1, 2} denotes the number of successful outcomes. Denote the wealth of entrepreneurs under incentive payment as
X ≡ {Xi}i∈{0,1,2} ≡ {X0, X1, X2},
where Xi ≥ 0 due to the entrepreneur’s limited liability. Assume that pHR − I > 0 > pLR − I + B, i.e., it is socially beneficial to invest in the project only if the entrepreneur exerts high effort.
a. Write down the investors’ participation constraint for investing in both projects.
b. Write down the incentive compatibility constraints for the entrepreneur to exert high effort on both projects.
c. Consider the following payment schedule
Are the constraints in Part b. satisfied under the above payment schedule?
d. What is the expected rent to the entrepreneur and the pledgeable income to the investors under the payment schedule in Part c.?
e. Which one is larger, the pledgeable income with two projects or the pledgeable income with only one project? Explain.
f. Bonus (10 points): Prove that the payment schedule in Part c. delivers the maximum pledgeable income for the investors.
4. Consider a two-period investment model: t = 0, 1. A risk-neutral entrepreneur has a project with expected returns z ≥ 1 in t = 1. Investment takes place in t = 0. The pledgeable income per unit of investment is 0.9. The entrepreneur has existing net worth of 100 at t = 0.
a. What is the maximum investment scale of the project?
b. What is the entrepreneur’s net worth at t = 1?
c. Suppose z ∈ {1.05, 0.95} with Pr (z = 1.05) = Pr (z = 0.95) = 0.5. What is the growth rate of the entrepreneur’s net worth under each return realization?
d. Explain how a 5% change in the returns can generate the large gains or losses to the entrepreneur’s net worth that you found in Part c.
e. The entrepreneur wishes to reinvest in t = 1. The pledgeable income p (z) in t = 1 is a function of the realized return z : p (1.05) = 0.9 + x and p (0.95) = 0.9 − x with x ∈ [0, 0.1) . Graph maximum investment scales I1 (x; 1.05) when z = 1.05 and I1 (x; 0.95) when z = 0.95 in t = 1 on the domain [0, 0.1) .