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Economics of Financial Markets (ECON0001) - Problem Set 3

Reading Assignment: Allen and Gale, “Understanding Financial Crises,” Chapter 3.4-3.6.

(1)  (Past Exam 2018.  Question B.1) In our economy there are 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 has an endowment of one unit of the good at time t = 0 and nothing at dates t = 1, 2. There are 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. If instead it is liquidated at date 1, it produces r = 0.5.

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 there is a bank operating in a perfectly competitive sector (e.g., because of free entry).  At date 0 the agents deposit their endowments in the bank.  The bank allocates all agents’endowments in a portfolio of x units of the long asset and y units of 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) What is the banking solution?  That is, what portfolio and consumption levels will the bank choose?

b) Explain why with this solution a bank run is possible.

c) Suppose the government wants to introduce a deposit insurance. What levels of consump- tion (c1 , c2 ) should it offer to the consumers to avoid a bank run?

d) Finally, suppose agents’preferences are represented by u(c) = −  . Is a bank run still possible?

(2) Prove that the investment in the short asset is higher when there is a positive probability (π > 0) of a bank run than when there is not (π = 0), provided that r < 1. Would the answer change if r ≥ 1?

(3) Banking crises are frequent, even in advanced economies.  Discuss why this is the case and give some historical examples of measures that regulators and policy makers have adopted to make the banking system less fragile.

∗ Reference: List of Bank Runs - Wikipedia; Reinhart and Rogoff, “This Time is Different: Eight Centuries of Financial Folly,”Chapter 10.