4721: Econ Money and Banking, Fall 2008 Homework 4 Due Monday, December 8, at the beginning of class.
Problem 1. Bank Runs
Consider an economy in which consumers live for three periods (called period 1, 2, and 3). Each consumer has an endowment of 20 goods when young, and nothing when middle-aged and when old. Utility from consumption depends on what type a consumer is, as follows: log [c2 ] for type 1 impatient u (c1 , c2 , c3 ) = log [c + c ] for type 2 patient
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There are 200 consumers, and half will become each type. However, consumers only learn their type in period 2. Assume no one else can observe the types (an agent knows her type only). There are two assets available in the economy, money and capital, oering the following rates of return: One-Period Return Fiat Money Capital 1 0.9 Two-Periods Return 1 1.5
(a) Suppose a consumer, on her own, saves half her goods in each asset in period 1. After nding out her type, she consumes all the savings in the appropriate period. Calculate the consumption the consumer would receive in either period. (b) Suppose a competitive bank oers demand deposit contracts to all consumers, specifying interest rates r1 and r2 for deposits withdrawn in periods 2 and 3, respectively. Which assets, and in what amounts, would the bank need to hold in order to oer r1 = 1 and r2 = 1.5? (c) Calculate the consumption a consumer would receive in each period from depositing with a bank. (d) Suppose a bank run occurs in this economy (all type 2 people pretend to be type 1 people and withdraw early). How many people, in total, would the bank be able to pay in period 2 at the promised rate of return before the bank runs out of assets? How many additional goods would the bank need in order to pay back all its depositors? (e) Suppose that in the period after you made your deposit at the bank you turn out to be a type 2 person and you learn that all the other type 2 people are about to pretend to be type 1 people so that they can withdraw early. Is it in your self-interest to also try to withdraw early?
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(f) Suppose that there are two cities, A and B, each with consumers and a bank as described above. There are no bank but runs, in each city the true distribution of consumer types turns out to be dierent from what people initially expected. Specically, in city A only 25% of the people learn they are type 2 and in city B 75% of the people learn they are type 2 (instead of 50% in each city). The banks in each city, however, expected that half of their customers would be of each type, so each bank kept half of their deposits in reserves. Explain how these banks could make a mutually benecial lending agreement with each other. Which bank would borrow and which will lend? What would be the interest rate on this inter-bank loan?
Problem 2. Rolling Over Debts
This problem considers the ability of the government to roll over its debt in every period. Rolling over works as follows: in period 1, the government borrows some amount, and in each consecutive period, it keeps borrowing just enough so that it could repay the debt due from the previous period. As we know, if debt grows faster than the economy does, it may not be possible, and here you are to explore this question further a bit. Consider an overlapping generations economy with 2-period lives and constant money supply. Consumers born in each generation are endowed with 20 goods when young and nothing when old. In order to consume when old, agents can save in terms of at money or government bonds. Suppose that each young consumers wants to save half of the endowments. Government bonds pay net real interest rate of 3%. In the initial period, the population is 100 people (N0 = 100) and the government issues bonds valued 500 units of consumption goods (B0 = 500). The government attempts to roll over the debt in the subsequent periods. (a) In what period would it be impossible to roll over the debts if population is constant? (b) In what period would it be impossible to roll over the debt if population is growing at 2% each period? (c) In what period would it be impossible to roll over the debt if population is growing at 5% each period? (d) In what period would it be impossible to roll over the debt if population is growing at 5% each period and money stock grows at a rate of 5% as well? (Hint: be careful here.)
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