Deriving a Multiplier with Currency and Excess Reserves

Why Currency and Excess Reserves Change the Multiplier

A simple deposit multiplier assumes that reserves support deposits and that funds do not leave the banking system as currency. In practice, households and businesses may hold some money as currency, while banks may retain reserves above the required minimum. Both choices affect how much deposit money corresponds to a given monetary base. The multiplier that includes these features is derived from the balance-sheet identities and three ratios.

Definition: Let $D$ denote customer deposits, $C$ currency held by the public, $RR$ required reserves, and $ER$ excess reserves. Total reserves are $R=RR+ER$. The monetary base is $B=C+R$, and the money stock considered here is $M=C+D$. Define the required-reserve ratio $r=RR/D$, the excess-reserve ratio $e=ER/D$, and the currency-deposit ratio $c=C/D$, for $D>0$.

Each ratio is measured relative to deposits. For example, $c=0.20$ means that the public holds 20 cents of currency for each dollar of deposits. The money multiplier is the ratio of the money stock to the monetary base, $m=M/B$. It is not the same as the deposit multiplier $D/B$, because the money stock includes currency as well as deposits.

Deriving the Money Multiplier

The derivation uses only the definitions above and the maintained ratios. Required reserves are $rD$, excess reserves are $eD$, and currency held by the public is $cD$. Consequently, the monetary base and money stock can both be written in terms of deposits.

Write reserves in terms of deposits.
Since $RR=rD$ and $ER=eD$, total reserves are $R=(r+e)D$.

Express the monetary base.
The base includes public currency and bank reserves, so $B=C+R=cD+(r+e)D=(c+r+e)D$.

Express the money stock and take the ratio.
Since $M=C+D=(1+c)D$, dividing the expression for $M$ by the expression for $B$ gives the money multiplier.

m=\frac{M}{B} =\frac{(1+c)D}{(r+e+c)D} =\frac{1+c}{r+e+c}

The deposit multiplier is instead

\frac{D}{B}=\frac{1}{r+e+c}.

These two expressions differ in the numerator: the money stock counts both deposits and currency, so $M/D=1+c$. This explains why omitting the numerator $1+c$ gives the deposit multiplier, not the money multiplier.

Key result: When the three ratios are held fixed, a change in the monetary base is associated with a change in the money stock given by $\Delta M=m\Delta B$, where $m=(1+c)/(r+e+c)$. The corresponding change in deposits is $\Delta D=\Delta B/(r+e+c)$. These are related but distinct calculations.

Worked Applications

Worked Example: Currency and Excess Reserves in a Base Expansion

In a hypothetical banking system, the required-reserve ratio is 10%, the excess-reserve ratio is 4%, and the currency-deposit ratio is 20%. The monetary base increases by $17 million. With the ratios unchanged, calculate the change in the money stock and show how it is divided among deposits, currency, required reserves, and excess reserves.

Method. First calculate the money multiplier. Then use the deposit multiplier to find the deposit change and apply each ratio to that change. This avoids confusing the deposit change with the total money-stock change.

The ratios are $r=0.10$, $e=0.04$, and $c=0.20$. Thus the multiplier is

m=\frac{1+0.20}{0.10+0.04+0.20} =\frac{1.20}{0.34} =\frac{60}{17} \approx 3.5294.

The change in the money stock is

\Delta M=3.5294118\times \$17\text{ million} =\$60\text{ million}.

For the deposit change, use the deposit multiplier:

\Delta D=\frac{\$17\text{ million}}{0.34} =\$50\text{ million}.

The associated currency change is $c\Delta D=0.20\times \$50\text{ million}=\$10\text{ million}$. Required reserves increase by $r\Delta D=0.10\times \$50\text{ million}=\$5\text{ million}$, and excess reserves increase by $e\Delta D=0.04\times \$50\text{ million}=\$2\text{ million}$. Total reserves therefore increase by $\$5+\$2=\$7$ million.

Check. The base increase is currency plus reserves: $\$10+\$7=\$17$ million. The money-stock increase is currency plus deposits: $\$10+\$50=\$60$ million, matching the multiplier calculation.

Common mistake avoided: Using $1/0.34$ as the money multiplier would produce $\$50$ million. That is the deposit change, not the money-stock change; it omits the $\$10$ million increase in currency.

Worked Example: Recovering the Multiplier From Deposit Data

A hypothetical bank system has deposits of $140 million. Its required-reserve ratio is 8%, its excess-reserve ratio is 2%, and the public holds currency equal to 25% of deposits. Calculate the amounts of currency and each type of reserve, then find the money stock, monetary base, and money multiplier.

Method. Apply each ratio to deposits, then form the two totals from their definitions. This makes it possible to check the multiplier against the underlying balances.

Currency is $C=0.25\times \$140\text{ million}=\$35\text{ million}$. Required reserves are $RR=0.08\times \$140\text{ million}=\$11.2\text{ million}$, and excess reserves are $ER=0.02\times \$140\text{ million}=\$2.8\text{ million}$. Therefore total reserves are $R=\$11.2+\$2.8=\$14$ million.

The money stock is

M=C+D=\$35\text{ million}+\$140\text{ million} =\$175\text{ million}.

The monetary base is

B=C+R=\$35\text{ million}+\$14\text{ million} =\$49\text{ million}.

Hence $m=M/B=\$175/\$49=25/7\approx 3.5714$. Direct use of the ratio formula gives the same result:

m=\frac{1+0.25}{0.08+0.02+0.25} =\frac{1.25}{0.35} =\frac{25}{7} \approx 3.5714.

Check. The balance-sheet totals give $\$175/\$49\approx3.5714$, which agrees with the ratio formula. The calculation uses the stated deposit base for all three ratios.

Common mistake avoided: Applying the currency ratio to reserves rather than deposits would give the wrong currency amount. By definition, $c=C/D$, so here currency is 25% of $\$140$ million in deposits.

Worked Example: The Effect of a Higher Currency-Deposit Ratio

A hypothetical system has a monetary base of $24 million, a required-reserve ratio of 8%, and an excess-reserve ratio of 2%. Initially, the currency-deposit ratio is 10%. Calculate the money stock. Then recalculate it if the currency-deposit ratio rises to 20%, with the base and the two reserve ratios unchanged.

Method. For each currency ratio, calculate the multiplier and multiply by the fixed base. Then verify the second result using its implied deposits, currency, and reserves.

Initially, $r+e+c=0.08+0.02+0.10=0.20$, and $1+c=1.10$. Thus $m=1.10/0.20=5.5$, and the money stock is $M=5.5\times \$24\text{ million}=\$132\text{ million}$.

After the currency-deposit ratio rises to 20%, the denominator becomes $0.08+0.02+0.20=0.30$ and the numerator becomes $1.20$. The new multiplier is $1.20/0.30=4$, so the money stock is $4\times \$24\text{ million}=\$96\text{ million}$.

Check. Under the new ratios, $B=(0.30)D=\$24$ million, so $D=\$24/0.30=\$80$ million. Currency is $0.20\times \$80=\$16$ million, and reserves are $(0.08+0.02)\times \$80=\$8$ million. The base is $\$16+\$8=\$24$ million, and money is $\$16+\$80=\$96$ million.

Common mistake avoided: Holding deposits fixed while changing $c$ would not respect the fixed monetary base. With the base fixed, the ratio change alters the deposit level consistent with the base.

How the Ratios Affect the Multiplier

The formula shows directly that a higher required-reserve ratio or excess-reserve ratio raises the base needed to support a given amount of deposits. In both cases the denominator $r+e+c$ increases while the numerator is unchanged, so the multiplier falls. A higher currency-deposit ratio also lowers the multiplier under the usual condition $r+e<1$. Although it increases the numerator $1+c$, it increases the denominator by the same amount; the net effect is negative when reserve ratios sum to less than one.

The effect of $c$ can be established by differentiating the multiplier with respect to each ratio, holding the others fixed:

\frac{\partial m}{\partial r} =-\frac{1+c}{(r+e+c)^2}<0,\qquad \frac{\partial m}{\partial e} =-\frac{1+c}{(r+e+c)^2}<0

\frac{\partial m}{\partial c} =\frac{(r+e+c)-(1+c)}{(r+e+c)^2} =\frac{r+e-1}{(r+e+c)^2}.

If $r+e<1$, the last derivative is negative as well. This condition is satisfied when required and excess reserves together are less than deposits. In that case, increasing any of the three ratios reduces the multiplier, though for different balance-sheet reasons: required reserves and excess reserves use more of the base per dollar of deposits, while currency held by the public is part of the money stock but is not a bank deposit.

Ratio rises	Direct effect in the formula	Effect on the multiplier when $r+e<1$
Required-reserve ratio $r$	Raises the denominator	Falls
Excess-reserve ratio $e$	Raises the denominator	Falls
Currency-deposit ratio $c$	Raises numerator and denominator	Falls

Common Exam Trap: Using the Wrong Multiplier

A useful check is to ask what the question wants: deposits, or the broader money stock that includes currency? For deposits, the relevant ratio is $D/B=1/(r+e+c)$. For money, use $M/B=(1+c)/(r+e+c)$. The units provide a second check: a multiplier is dimensionless, while multiplying it by a change in the monetary base gives a dollar change in the relevant total.

Key takeaway: Currency and both types of reserves are measured relative to deposits. Write $B=(r+e+c)D$ and $M=(1+c)D$ before calculating. Their ratio gives the money multiplier, while the reciprocal of the base-to-deposit ratio gives only the deposit multiplier.

Check Your Understanding

For each problem, use the definitions and multiplier appropriate to the quantity requested.

A system has $r=0.12$, $e=0.03$, and $c=0.10$. If the monetary base increases by $12 million and the ratios remain fixed, what is the change in the money stock? (Answer: $52.80 million.)
Deposits are $200 million, with $r=0.10$, $e=0.025$, and $c=0.05$. Find the monetary base, money stock, and money multiplier. (Answer: $B=\$35$ million, $M=\$210$ million, and $m=6$.)
With $r=0.08$, $e=0.04$, and a fixed monetary base of $16 million, compare the money stock when $c=0.10$ with the money stock when $c=0.20$. (Answer: $80 million initially and $60 million after $c$ rises.)
A system has money multiplier $m=3$, required-reserve ratio $r=0.10$, and currency-deposit ratio $c=0.20$. Find its excess-reserve ratio. (Answer: $e=0.10$.)
A system has $r=0.12$, $e=0.08$, and money multiplier $m=2.5$. Find the currency-deposit ratio. (Answer: $c=1/3$, approximately 0.3333.)

Ratio rises	Direct effect in the formula	Effect on the multiplier when \(r+e<1\)
Required-reserve ratio \(r\)	Raises the denominator	Falls
Excess-reserve ratio \(e\)	Raises the denominator	Falls
Currency-deposit ratio \(c\)	Raises numerator and denominator	Falls

What You'll Learn