Economics 0281

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Homework #2: Answers

 

1. †††††† Assume that all banks in the banking system have ER = 0 and that one bank sells $1,000 of GS to the Fed. If r = 10%, fill in the rest of the table below:

 

†† Round††† ΔTR†††† †††††ΔRR†††ΔER†† †††††ΔLL†††††ΔDD ††††††SΔDD

 

†††† 1†† ††$1,000.00 †††$0.00 $1,000.00 $1,000.00 $1,000.00$1,000.00††

 

†††† 2††$1,000.00$100.00††$900.00 ††$900.00$900.00$1,900.00

 

†††† 3†††† †††$900.00 ††$90.00 ††$810.00††$810.00 $810.00$2,710.00

 

†††† 4†† †††††$810.00†† $81.00†††$729.00$729.00 †††$729.00$3,439.00††

 

†††† 5†††† †††$729.00†† $72.90†††$656.10 $656.10†††$656.10$4,095.10

 

†† All

Other

Rounds††††††††††††††††††††††††† ††††††††††††††††††††$5,904.90$5,904.90

 

Totals††††††† †††††††††††††††††††††††††††††††††††††$10,000.00 $10,000.00††††

 

NOTE: (1) All Other Rounds and Totals categories should be computed only for ΔLL and ΔDD.

 

(2) Use the multiplier for computing the Totals. Assume c = C/DD = 0 and er = ER/DD = 0.

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The multiplier is simply m = 1/r = 1/0.1 = 10.

 

Note that about 41% of the increase in the money supply is completed by the end of the 5th round.

 

 

 

 

 

 

 

 

 

 

 


 

 

2. ††††††† Assume that all banks in the banking system have ER = 0 and that one bank sells $1,000 of GS to the Fed. If r = 10%, the excess reserve ratio er = ER/DD = 10% (banks desire to hold positive ER), and the currency-deposit ratio c = C/DD = 10%, fill in the rest of the table below:

††††††††††††††††††††††††††††

Round†††††ΔTR††† †††††ΔRR††† ΔER(1) ††ΔER(2)†† ††ΔLL††††ΔDD†††††ΔC††††SΔDD

 

††††† 1 †††$1,000.00 ††$0.00 $1,000.00 ††$909.09 ††$909.09 $826.45 $82.64 ††††$826.45

 

††††† 2†† ††††$826.45 $82.64††$743.81 ††$676.19 ††$676.19 $614.72 $61.47 $1,441.17

 

††††† 3†† ††††$614.72 $61.47†† $553.25 ††$502.95 ††$502.95 $457.23 $45.72 $1,898.40

 

††††† 4†† ††††$457.23 $45.72†† $411.51 ††$374.10 ††$374.10 $340.09 $34.01 $2,238.49

 

††††† 5†† ††††$340.09 $34.01†† $306.08 ††$278.25 ††$278.25 $252.95 $25.30 $2,491.44

 

All

Other

Rounds††††††††††††††††††††††††††††† ††††††††††††††††††††††††††††††††††$926.09 $841.89 $84.19

 

Totals†††††††††††††††††††††††††† ††††††††††††††††††††††††††††††††††††$3,666.67 $3,333.33 $333.33

 

NOTE: (1) ΔTR = ΔRR + ΔER(1)

 

†††††††††† (2) ΔER(2) = ΔLL

 

(3) ΔLL = ΔER(1)/(1 + er)

 

(4) ΔDD = ΔLL/(1 + c) and ΔC = ΔLL Ė ΔDD

 

††† ††††††† For example: ΔER(1) = $1,000

 

††††††††††† ΔLL = ΔER(1)/(1 + er) = $1,000/1.1 = $909.09

 

ΔDD = ΔLL/(1 + c) = $909.09/1.1 = $826.45

†††††††††††††††††

††††††††††† ††ΔC = $909.09 - $826.45 = $82.64

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The multiplier is m = (1 + c)/(r + er + c) = 1.1/0.3 = 3.667

 

The increase in the MS = ΔDD + ΔC = $3,666.67. Note also that about 75% of the increase in the money supply is completed by the end of the 5th round.

 

 

 

3. ††††††† Suppose that the following data are available:

 

†††††††††††† r = 12%, c = C/DD = 35%, and er = ER/DD = 3%

 

†††††††††††† TR = $400B, C = $100B

 

a. Compute the MS using the multiplier formula.

 

m = (1.35)/(0.5) = 2.70

 

MS = m(MB) = m(TR + C) = 2.7($400B + $100B) = $1,350B

 

b. Suppose that the inflation rate has been increasing and the Fed wants to reduce it by contracting the money supply by 1%. What kind of open market operation must the Fed use and how large must this open market operation be to accomplish its goal? Calculate and explain.

 

0.01(-MS) = -$13.5B

 

Fed must sell GS to the banking system. This increases GS in the banking system and decreases TRs by the same amount, causing ER < 0, a contraction of loans and demand deposits, and ultimately a decrease in the money supply. The magnitude of this open market sale must be:

 

††††††† †††††††††††††††-$13.5B/2.7 = - ΔMB = -$5B

 

††††††††††††††††††††††† Or an alternative calculation:

 

$1,336.5B = 2.7(X + $100B) = 2.7X + $270B

 

$1,066.5B = 2.7X†† X = $395B†† ΔX = -$5B

 

This means that the Fed must sell $5B in GS to the Banking system (the minus means sell, + means buy)

 


 

 

 

 

 

 

 

 

 

 

 

 

c. Given the situation in part b., suppose that the following also occurs: c decreases to 30% and FDL rise by $5B. Now what kind of open market operation(s) must the Fed use and how large must it be to accomplish the Fed's goal? Calculate and explain.

 

0.01(-MS) = -$13.5B

 

The Fed must still contract the money supply by $13.5B. But now it must attempt to offset an increase in the MB of $5B and an increase in the money multiplier.

 

††††††††† ††††††††††††† $1,336.5B = m(MB) = [(1.3)/(0.45)]($505B + X)

 

$1,336.5B = 2.89($505B + X)

 

$1,336.5B = $1,459.45B + 2.89X

 

-2.89X = $122.95B

 

†††† X = -$42.54B†††††††

 

[Check: 2.89($462.46B) = $1,336.51B]

 

The Fed must now sell $42.54B worth of GS to the banking system.

 


d. Compare the Fed's policy actions in parts b. and c. Explain what you observe. Will the Fed always have the information it needs to make the correct decisions? Explain.

 

The Fed's actions in part b. assumed all other factors were constant and so its response was relatively mild. The Fedís actions in part c. occurred because the Fed wanted to decrease the money supply while other factors were increasing the money supply. These factors had to be offset in order for the Fed to accomplish its goal. It is highly unlikely that the Fed will have exact information about the shifts in currency preferences so it will have to guess now and get exact data later. That means that the Fed will probably have to make some adjustments later on as it becomes more aware of what has happened. In contrast, information on borrowed reserves should be available to the Fed immediately. The main point here is the emphasis on the Fed's lack of complete control over the money supply in the short run (due to information lags) when it most needs to counter short-run cyclical factors of the business cycle.

 


4. ††††††† Use the following data to do the following problems assuming r = 6.25%:

††††††††††††† ††The K9 Bank ††††††††††††††The Banking System

†††††††††††††† DD = $100,000††††††††††††††† DD = $100.00M

†††††††††††††† ER =†† $2,500†††††††††††† ††††ER =$4.00M

††††††††††††† FDL =$10,000††††††††††††† FDL =†† $3.00M

†††††††††††††† NW =$25,000††††††††††† ††NW =†† $7.00M

†††††††††††††† GS =$50,000†††††††††††††††† GS =$75.00M

 

†††††††††††††† LL =$76,250†††††††††††††††† LL =$24.75M

†††††††††† ††††TR =$8,750††††††††††††††††† TR =$10.25M

††††††††††† †† RR = $6,250††††††††††††††††††† RR = $6.25M

 

a. Construct T-Accounts for The K9 Bank and The Banking System.

 

 

 

 

 

 

 

 

 

 

 

 

 

b. Show how The K9 Bank and The Banking System restore lending equilibrium with ER = 0.

††††††††††††††††††††††††††††††††††††††††††††††††††

†††††††††††

 

 

 

 

 

Note the following:

 

(1) The individual bank can only create a maximum number of New LLs and New DDs equal to its ER. Why? Because it will lose these created DDs to another bank. As a result of this loss the individual bank will also have to transfer TR to another bank through the clearing mechanism and debit the original account when it receives the check from some other bank.

 

(2) The banking system can create New LLs and New DDs that are a multiple of its ER. Why? Because the banking system does not lose the created DDs but merely has the created DDs shuffled around from bank to bank. The reserve ratio in this case is 6.25% and the multiplier is 1/r = 1/0.0625 = 16. Therefore New LLs and New DDs are 16 times ER.

 

5. ††††††† Using the data from problem #5, do the following:

 

a. Using the T-Accounts for The K9 Bank and The Banking System from 5.b. (when ER = 0), show what happens when the Fed sells $5,000 in GS to The K9 Bank. Briefly explain your result.

 

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†††††††††††

GS increased for both the individual bank and the banking system, causing TR to decline by an equal amount. As a result, ERs become negative on each T-Account. Since the contraction of deposits cannot be done for an individual bank, the remaining part of the individual bank's T-Account does not have to be shown. But the banking system adjustment can be shown as a multiple (m = 1/0.0625 = 16) contraction of loans and demand deposits as shown in the last T-Account.

 

b Using the T-Account for The Banking System from 5.b. (when ER = 0), show what happens when the Fed purchases $5M GS from The Banking System. Briefly explain your result.

 

†††††††††††

 

†††††††††††††††††††† ††††††††††††

 

The Fed purchase of GS from the banking system has caused reserves to increase by the same amount as the decline in the banking system's GS. ER > 0 and loans and DDs can be expanded by a multiple of ER for the banking system only.

 

6. ††††††† Use data below from the Great Depression for the following questions:

 

a. What was the reserve ratio at the start of the first banking crisis? (HINT: Calculate this figure by using the data below)

 

er = 0.010

c = 0.167

MB = $6.5B

MS = $24.8B

 

MS = [(1 + c)/(r + er + c)]MB

 

$24.8B = [(1.167)/(r + 0.177)]$6.5B

 

1.167/(r + 0.177) = $24.8B/$6.5B = 3.815

 

1.167 = 3.815r + 0.675

 

3.815r = 0.492

 

r = 0.492/3.815 = 12.9%

 

 

b. Use the data below to calculate the money supply at the end of the final banking crisis in 1933. Explain your results.

 

r = 0.129

c = 0.340

†† †††††††††††††††††††† er = 0.125

††† ††††††† MB = $7B

 

MS = (1.34/0.594)$7B = (2.256)$7B = $15.8B

 

Both the currency-deposit ratio and the excess reserve ratio rose sharply in about 29 months time. This caused the money multiplier to decline sharply (by about 40% from 3.815 to 2.256). With a much smaller money multiplier and monetary base that was only 7.5% higher, the money supply fell by about 36% (Friedman has estimated that the money supply fell by about a third).

 

c. The Fed, noticing that banks ERs were growing very large in 1937, feared that these ERs would be used to create loans and cause an outburst of inflation. So the Fed raised the reserve requirement to wipe out all of the banks ERs. Use the multiplier analysis to explain why the recession of 1937-38 occurred. (Hint: If r = RR/DD rises by the same amount as er = ER/DD falls, then what happens to the multiplier? What must happen for the increase in r to cause a sharp recession?)

 

If r rises by the same amount as er falls then excess reserves have merely been converted into required reserves with no effect on the multiplier. This is the effect for which the Fed was aiming.

 

But when er falls by less than r rises (or remains constant), the multiplier decreases and causes the money supply to fall. Banks apparently reduced their loans in order to partially restore their ER positions because they were worried about liquidity (Remember: Banks can use their ERs to satisfy depositors' desires to convert their DDs into C; RRs cannot be used for this purpose.)

Suppose

 

m = (1 + c)/(r + er +c) with r = 12.5%, c = 35% and er = 12.5%

 

††††††††††††††††††††††† m = 1.35/0.6 = 2.25

 

If r = 25% and er = 0% then the multiplier does not change.

 

But if r = 25% and er = 10% while c = 35% then m = 1.35/.70 = 1.929