Economics 111
Winter 2002
Professor Flavin
Sketch of solutions to problem set 3
1. a)
Bank A
Assets
Reserves
Securities
Loans
Liabilities
30 Deposits
20
170
Total Liabilities
Net worth
220 Total Liabilities and Net worth
Total Assets
200
200
20
220
Bank B
Assets
Liabilities
Reserves
Securities
Loans
b)
30 Deposits
40
250
Total Liabilities
Net worth
Total Assets
320 Total Liabilities and Net worth
Required Reserves  200 M .10  20 M
Bank A:
Excess Reserves  30 M  20 M  10 M
Required Reserves  300 M .10  30 M
Bank B:
Excess Reserves  30 M  30 M  0
300
300
20
320
c)
Bank A
Assets
Reserves
Securities
Loans
Total Assets
Liabilities
20 Deposits
30
170
Total Liabilities
Net worth
220 Total Liabilities and Net worth
200
200
20
220
Bank B
Assets
Reserves
Securities
Loans
Liabilities
40 Deposits
30
250
300
2
Total Liabilities
Net worth
320 Total Liabilities and Net worth
Total Assets
300
20
320
d)
Bank A
Assets
Reserves
Securities
Loans
Liabilities
19 Deposits
30
170
Total Liabilities
Net worth
219 Total Liabilities and Net worth
Total Assets
199
199
20
219
Bank B
Assets
Reserves
Securities
Loans
Total Assets
Liabilities
41 Deposits
30
250
Total Liabilities
Net worth
321 Total Liabilities and Net worth
301
301
20
321
e)
Required Reserves  199 M .10  19.9 M
Excess Reserves  19 M  19.9 M  .9 M
Required Reserves  301 M .10  301
. M
Bank B:
Excess Reserves = 41M-30.1M=10.9M
Bank A has a reserve deficiency because its excess reserves are less than zero. Bank A can satisfy its
reserve requirements in several ways. It can:
 Borrow .9 M on the federal funds market, either from bank B or another bank that has excess
reserves of at least .9 M to lend;
 Obtain a discount loan of .9 M from the Federal Reserve System;
 Sell .9 M of its securities;
 Make a .9 M repurchase agreement (REPO or RP). That is, bank A sells its securities and agrees to
repurchase them at a specified date.
Bank A:
2. a) We want to find an expression for m , the money multiplier, that relates the change in the money
supply to a given change in the monetary base. In symbols, M  mB .
 Banking Sector
Let EX  Excess Reserves and RR  Required Reserves .
Reserves are defined as: R  RR  EX , thus
R  RR  EX .
(1)
Required reserves are defined as: RR  rD , thus
RR  rD .
(2)
R  RR EX

The banking sector is in equilibrium when: e 
, thus
D
D
3
EX  eD .
(3)
Substituting equations (2) and (3) into (1), we have:
R  rD  eD  ( r  e )D .
(4)
 Nonbank Private Sector
By definition, money supply is equal to the sum of currency in circulation and deposits:
M  C  D , thus
M  C  D .
(5)
C
The nonbank private sector is in equilibrium when c  , thus
D
C  cD .
(6)
Substituting (6) into (5), we have:
M  cD  D  (1  c )D .
(7)
 By definition B  C  R , thus
B  C  R
(8)
In equilibrium equations (4) and (6) hold, substituting them into, (8), we have:
B  cD  ( r  e )R  ( r  e  c )D .
(9)
Thus, rearranging:
1
(10)
D 
B .
(r  e  c)
Finally, substituting (10) into (7), we have:
1 c
M 
B
(11)
rec
1 c
Therefore, m 
.
rec
1 c
1.15

 411
.
b) m 
r  e  c .10.03.15
1 c
1 1
 
 10
c) m 
r  e  c r .10
d) When e  0 and c  0 the value of the multiplier is lower because they reduce the reserves
available for lending and deposit expansion. In other words, excess reserves and currency holdings
represent leakages in the multiple expansion of deposits.
3.
Fed sells $2 million in Treasury bonds to Wells Fargo:
Federal Reserve
Assets
Securities
-$2M Reserves
Liabilities
-$2M
Wells Fargo
Assets
Reserves
Securities
Liabilities
-$2M
+$2M
Change in Monetary Base = B  C  R  0  $2M  $2M. That is, the monetary base
falls by $2 million.
4. False.
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If r  0 , the money multiplier is infinity only if e and c are also zero:
1 c
1
  .
If r  0 , e  0 , and c  0 , then m 
rec 0
If the required reserve ratio were zero, is it plausible that banks would hold zero excess
reserves? No; even in the absence of reserve requirements, banks would hold some nonzero
level of reserves so that they can redeem deposits on demand. If the required reserve ratio were
zero, these reserves (voluntarily held for liquidity purposes) would count as excess reserves, so
that e would be nonzero. Thus even if the currency ratio were zero, the money multiplier
would still be finite.
5. Consider the expression for the money multiplier which you derived in question 1. The
phenomenon that the public came to distrust bank deposits and therefore hold more of their money
in the form of currency can be modeled as an increase in the currency ratio, c. Similarly, the
response of banks in the form of holding higher levels of excess reserves can be modeled as an
increase in the excess reserve ratio. By inspection, one can see that an increase in e reduces the
multiplier. By taking the derivative, we can also show that an increase in c reduces the money
multiplier. In fact, the money multiplier did fall substantially during the depression, due to the
increases in currency holding by the public and to the increases in excess reserve holdings by the
banking sector. The fact that the Fed did not react sufficiently strongly by increasing the base
caused the money supply to fall, prolonging the depression.
6. a) Whether the Fed is purchasing Treasury securities, gold, or Godiva chocolates, an open market
purchase has exactly the same effect of increasing the monetary base. Since the increase in the
monetary base comes about because the Fed pays for the asset (securities, gold, chocolate) with a
check written on itself and therefore injects into the banking system additional reserves, the
particular nature of the asset which the Fed buys makes no difference in terms of the effect of the
purchase on the monetary base.
b) For practical reasons, conducting open market operations by buying or selling chocolate is not a
good idea for two reasons. First, chocolate is bulky to store (if you’re buying $100 million worth
of it) and does not keep indefinitely. Within a year or so, the value of the chocolate would fall due
to spoilage. Second, if the Fed attempted to conduct open market operations in the chocolate
market, purchases or sales of the dollar amount typically involved in control of the monetary base
would severely distort the price of chocolate relative to other goods in the market. That is, the
market is too narrow to withstand fluctuations in demand of the magnitude that would be required.
Besides, why should my holiday plans to buy chocolate be frustrated when the Fed decides to buy
5 million pounds in order to increase the monetary base? Besides the fact that Treasury securities
are not vulnerable to physical deterioration, and are essentially costless to store, a second
advantage is that the market for Treasury securities is very broad (broad enough to withstand large
purchases and sales by the Fed). Further, when the Fed buys Treasury securities, the relative price
that is affected is the price of bonds (or the interest rate). Thus conducting open market operations
by buying and selling Treasury securities will have an impact on bond prices and interest rates
(financial variables), instead of distorting the relative price of some innocent commodity like
chocolate.
7. a) To avoid having a reserve deficiency, Wells Fargo can borrow reserves from another bank
through the Federal Funds market, borrow reserves from the Fed via a discount loan, sell some of
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its marketable securities, or engage in a repurchase agreement in which it sells a Treasury bond to
another bank or firm with an agreement to repurchase the bond subsequently.
b) 1) The level of excess reserves will fall.
2) The Federal Funds rate will rise.
3) The volume of discount lending will rise.
8. When the Fed sells foreign assets, its international reserves fall. Since its liabilities fall, the
monetary base falls, and the US money supply falls. This increases the domestic (US) nominal
interest rate (causing the vertical line representing the expected return to domestic deposits to shift
right), and also increases the expected nominal exchange rate at t+1 (because the reduction in the
money supply has reduced expectations of US inflation) which causes the line representing the
expected return to foreign deposits to shift up. The dollar appreciates. In the long run, the
domestic nominal interest rate return to its original level, but nevertheless, the exchange rate
remains higher than it was before the intervention.
9. As before, the sale of foreign assets results in a decline in the international reserves of the Fed.
However, because the intervention is offset, or sterilized, by an open market operation, there is no
effect on the monetary base or the money supply, and therefore no effect on the exchange rate.
10. With 3 francs you could buy 3/60 th ounce of gold and exchange the gold for one dollar, so the
exchange rate is 3 francs per dollar.
11. The central bank would need to buy foreign assets, thus increasing its international reserves and the
monetary base. The increase in the money supply causes the domestic interest rate to fall, so that the
line representing the expected return to domestic deposits shifts left. Since Bretton Woods was a fixed
exchange rate system, the expected future exchange rate does not change (i.e., remains as the par
value) so that the expected return to the foreign deposit line does not change. The market exchange
rate declines to the par value.