4721: Econ Money and Banking, Fall 2008 Homework 4 Answer Key
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a competitive bank oers demand deposit contracts to all consumers, specifying interest rates r1 and r2 for deposits withdrawn in periods 2 and 3, respectively.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 runs, but 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?
Answer
(a) Consumer invests half of 20, or 10 goods in each asset, so 10 goes to money and 10 goes to capital. In the second period, consumer nds out her type. An impatient person would want to consume everything now, since she gets no utility from consumption in period 3. Return on money is 1, so 10 goods from investment into money turn into 10 goods now, and one-period return on capital is 0.9 so 10 goods from investment into capital turn into 9 goods now. Total consumption of an impatient person will hence be: c1 = 10 + 9 = 19 2 where the superscript 1 denotes persons type. A patient person would want to defer consumption until next period, since for her consumption in period 2 and in period 3 are perfect substitutes, and in period 3 she will get higher return on capital. Return on money is 1, so 10 goods from investment into money turn into 10 goods in period 3, and two-period return on capital is 1.5 so 10 goods from investment into capital turn into 15 goods in period 3. Total consumption of a patient person will hence be: c2 = 10 + 15 = 25 3 where again the superscript 2 denotes persons type. (b) Assuming any banking panics away, a bank would expect half of its customers to turn impatient and the other half to turn patient. Hence the bank would need to keep half the deposits in cash and half in capital. Total deposits will be 200 (20) = 4000, so 2000 must be invested into cash and 2000 into capital. This will allow to give 20 to each of 100 impatient people and 30 to each 100 of the patient. (c) A patient person would come to the bank in period 3, seeking to get 30 (since she was lured in by the rate of r2 = 1.5) and an impatient person would withdraw her deposit in period 2 and will get 20 (since she was promised to have a rate of return r1 = 1).
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(d) First of all, the bank will be able to pay 20 to the rst 100 people who come (there may be a mix of patient and impatient between them, we cannot be sure). After this, the bank will have to start withdrawing capital prematurely. Its 2000 units in capital will yield only 1800 units in period 2, which will be enough to satisfy the needs of 90 people (by providing them the return of 1). In the end, there will be a situation where 10 people havent got anything yet and the bank is not able to give them anything, lacking some 200 units of consumption. The number of people the bank will be able to service would thus be 190. (e) Yes, if you are a patient person and you believe other patient people will rush to the bank, you would want to do this as well. If you wait, have you a chance of becoming one of the unlucky 10 rm part (d), and in any case, the bank wont have assets to pay you back at period 3 if you wait. So it would be better to rush with the mob and try to get at least something out of the bank back. (f) The mutually benecial agreement would work as follows. The bank of city B will see in period 2 that only 25% of people bothered to show up for withdrawals (assuming panics away). It therefore has some 50 (20) = 1000 units of consumption that were invested in cash. On the other hand, bank in city A faces more early withdrawals that he could accommodate without tapping the capital investments. So it would be ideal for bank B to lend the extra 1000 to bank A charging r2 = 1.5 in gross interest. As period 3 comes, bank A will have 4500 units of consumption from capital investments, of which only 3000 will be demanded, so it would be exactly in a position to pay back 1.5 (1000) = 1500 to bank B.
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 40 goods when young and nothing when old. In order to consume when old, agents can save in terms of at money or government bonds. 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?
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(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.)
Answer
(a) Constant population and constant money supply together imply that the rate of return on at money is 1: vt+1 n 1 = = =1 vt z 1 and since r = 1.03, people would prefer to save using bonds, and not cash. The government clearly cannot borrow more than the total amount of resources in the economy, which is constant and equal to Nt y = 40 (100) = 4000 (for all t, population does not grow). The initial debt is 500, and it grows at a rate of 1.03: B1 B2 = rB0 = rB1 = r2 B0 . . . = = r t B0 (1) (2) (3) (4)
Bt
As long as Bt Nt y, rolling over the debt is possible, but since debt is growing, there will be some t = T , after which it will no longer be possible for the government to continue the roll over. To nd this period, we solve the following equation: BT r T B0 1.03T (500) 1.03T = NT y = Ny = = 4000 8 (5) (6) (7) (8)
From here there are several ways to proceed. The rst is just to use the denition of a logarithm with arbitrary base: T = log1.03 8 70.35 (9)
and since T should be an integer, we conclude that from period 71 onward rolling the debt over will not be possible.
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The second way is to take the natural logs of both sides: T ln [1.03] T = = ln [8] ln 8 70.35 ln [1.03] (10) (11)
which, not surprisingly, gives precisely the same answer (b) I will be rather brief here now, since the solution idea stays the same. The way debt grows does not change: Bt = rt B0 , but we now have population growth (and hence GDP growth). Population growth rate of 1.02 and constant money supply together imply that the rate of return on at money is 1.02: vt+1 n 1.02 = = = 1.02 vt z 1 and since r = 1.03, people would still prefer to save using bonds, and not cash. Now, the GDP grows at a rate of 1.02: N1 y N2 y = nN0 y = nN1 y = n2 N0 y . . . = = nt N0 y (12) (13) (14) (15)
Nt y
So now we just have to solve a similar equation again: BT r T B0 (1.03)
T
=
NT y
(16) (17) (18)
= nT N0 y = (1.02) 8
T
and here it is probably easier to use the second approach: T ln [1.03] T = T ln [1.02] + ln 8 ln [8] 213.14 = ln [1.03] ln [1.02] (19) (20)
so from period 214 onward rolling the debt over wont be possible. (c) Population growth rate of 1.05 and constant money supply together imply that the rate of return on at money is 1.05: vt+1 n 1.05 = = = 1.05 vt z 1 and since r = 1.03, people would now prefer to hold cash instead of bonds, so the government wont be able to roll over the debt starting from period 1.
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(d) Population growth rate of 1.05 and money supply growth rate of 1.05 together imply that the rate of return on at money is 1: vt+1 n 1.05 = = =1 vt z 1.05 and since r = 1.03, people again prefer to save using bonds, and not cash. Notice however, that now the rate of GDP growth, n, is higher that the rate of debt accumulation, r, so the government may roll the debt over forever here, and hence there is no such T .
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