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8. Banking Algebra
Problems:
14. Given appropriate information about deposits and reserves, calculate the money multiplier.
15. Given the money multiplier and base money predict the money supply.
16. Given the money multiplier and a change in base money, predict the change in the money supply.
17. Given the growth rate of the money multiplier and the growth rate of base money, predict the growth rate of the money supply.
Money Multiplier
The money supply is equal to the money multiplier multiplied by base money.
Ms = m*B
The money multiplier is equal to one plus the currency deposit ratio divided by the currency deposit ratio plus the reserve deposit ratio.
m = (1+c)/(c+r)
m-money multiplier, c-currency deposit ratio, r-reserve deposit ratio.
The currency deposit ratio is the amount of currency held by a member of the nonbanking public per dollar of checkable deposits and the reserve deposit ratio is equal to the amount of reserves held by banks per dollar of checkable deposits issued.
Base money is equal to currency held by the public plus bank reserves. Base money is directly controlled by the Federal Reserve.
The change in the money supply is equal to the money multiplier multiplied by the change in base money. An increase is a positive change and a decrease is a negative change.
Ms = m*B
Assuming many of the factors influencing the money multiplier can be respresented by a random variable, then a dynamic-stochastic version of the money supply process is that the natural log of the money supply is equal to the natural log of the money multiplier plus the natural log of base money plus a random shock to the money multiplier.
lnMst = lnBt + lnmt + t
The growth rate of the money supply is equal to the growth rate of the money multiplier plus the growth rate of base money.
. . . .
Ms = B + m + t
The growth rate of a macroeconomic variable for a period is equal to the natural log of the variable for the period minus the natural log of its value the previous period.
Examples
The public holds $2 of currency for every $10 of checkable deposits. The banks are required to hold $1 of reserves for every $10 of checkable deposits. The reserve requirements are binding. Base money is $300 billion.
1. Find the reserve deposit ratio and the currency deposit ratio.
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r=1/10 =.1 c=2/10=.2
2. Calculate the money multiplier.
m = (1+c)/(c+r) = (1+.2)/(.2+.1) = 1.2/.3 = 4
3. Predict the money supply.
Ms = m*B = 4*300b=1200b
4. Base money increases by $8 billion and the money multiplier is
2. Predict the change in the money supply.
Ms = m * B = 2 * 8b = 16b 16b increase
5. Base money decreases by $6 billion and the money multiplier is 2. Predict the change in the money supply.
Ms = m * B = 2 * -6b = -12b 12b decrease
6. Base money is Shrinking 2% per year. The money multiplier is constant. Predict the growth rate of the money supply.
. . .
Ms = m + B = 0% + -2% = -2% money supply is shrinking 2% each year
7. Base money is growing 4% per year. The money multiplier is shrinking 1% per year. Predict the growth rate of the money supply.
. . .
Ms = m + B = -1% + 4% = 3% money supply is growing 3% each year
8. Base money is growing 1% per year. The money multiplier is shrinking 3% per year. Predict the growth rate of the money supply.
. . .
Ms = m + B = -3% + 1% = -2% money supply is shrinking 2% each year
The Empirical Money Multiplier
The money multiplier can be calculated from data regarding base money and various measures of the money supply. Simply divide the money supply by base money and you find the proper money multiplier.
Mt = Mst/Bt