Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Economics of Financial Markets (ECON0001) - Problem Set 2

Reading Assignment: Allen and Gale, “Understanding Financial Crises,” Chapter 3.1-3.4.

(1) In class, we have shown that in our three-date economy, the financial market (at time 1) is not efficient while the banking system is efficient. Explain why this is the case.

(2) Suppose R < 1: one unit of long asset produces nothing at t = 1 and produces R < 1 at t = 2.

a) Suppose there is a market for the long asset at time 1. Can we find the equilibrium price for the long asset?

b) Find the competitive bank’s solution, assuming that all early consumers liquidate the bank deposit at date 1 and all late consumers liquidate it at date 2.

(3) (Past Exam 2013. Question B.1) Consider an economy with three dates t = 0, 1, 2 and a single, all-purpose good at each date.  There is a continuum of ex-ante identical agents of measure 1. Each agent has an endowment of one unit of the good at date t = 0 and nothing at dates t = 1, 2.  In order to provide for future consumption, each agent can invest in two assets, a short asset and a long asset. The short asset produces one unit of the good at date t + 1 for every unit invested at date t = 0, 1.  The long asset produces R = 1.5 units of the good at date 2 for every unit invested at date 0 (and produces nothing at date 1).

At date 0 each agent is uncertain about his preferences over the timing of consumption. With probability 1/2 he expects to be an early consumer, who only values consumption at date 1.  With the complementary probability he expects to be a late consumer, who only values consumption at date 2.  Note that he never values consumption at date 0.  Each agent has

preferences represented by a Von Neumann-Morgenstern utility function, with u(c) = ln(c).

Suppose that the agents can invest in the two assets and can trade the long asset at date 1, i.e., after they learn whether they are early or late consumers. At date 0 each agent invests the amount x in the long asset and the amount y in the short asset. The portfolio (x,y) must satisfy the budget constraint x + y ≤ 1. Let c1  denote the amount consumed at date 1 by an early consumer and c2  denote the amount consumed by a late consumer at date 2. Note that he will consume either c1  or c2  but not both.

a) Denote the price of the long asset at date 1 by P .  What is the feasible consumption set for each agent?

b) Find the equilibrium price for the long asset at date 1 (make sure you explain carefully why the price you indicate is the equilibrium price). Show that at this price the date 0 portfolio choice of each agent is irrelevant (i.e., show that his consumption plan is independent of y for any y ∈ [0, 1]).  Write down the consumption plan and compute the expected utility in this equilibrium.

c) Write down the equilibrium in this economy, i.e., the price and the investments in the short and the long assets. Explain carefully why this is an equilibrium.

d) Find out the feasibility constraints in the case of a benevolent social planner who maximizes the agents’expected utility. Illustrate such constraints in a graph. In the same graph, show the equilibrium market allocation. Find the efficient (i.e., social planner) solution for the consumer’s portfolio (x,y) and consumption plan.

e) Suppose, now, the agents are risk neutral. What would their utility function be? What is the efficient solution in this case?

(4) The three-date model concludes that the asset market equilibrium is generally not efficient and does not achieve the optimal liquidity insurance, unless σ  = 1.  What can we do to improve the efficiency of asset market equilibrium within our theoretical framework? What provision or mechanism would be helpful for market efficiency in the real-life financial market?