Assumptions Behind the Simple Multiplier

Why the Multiplier Depends on Assumptions

The simple deposit multiplier turns a reserve change into a potential change in bank deposits. Its arithmetic is compact, but its result depends on what banks, borrowers, and depositors do at every stage. If a bank holds reserves beyond the required amount, a borrower keeps some loan proceeds as cash, or funds do not return as deposits, the sequence assumed by the simple model is interrupted.

The multiplier is therefore best treated as a conditional benchmark: it describes deposit expansion under a specified set of assumptions, not a mechanical rule that every reserve change must produce the same deposit change. The earlier material on reserves, lending, and deposit expansion provides the balance-sheet setting for interpreting those assumptions.

Definition: Let $r$ be the required reserve ratio, so required reserves equal $r$ times deposits subject to that requirement. Under the simple multiplier model, with no excess reserves and no currency leakage, the simple deposit multiplier is $1/r$. A reserve increase $\Delta R$ is associated with a potential deposit increase $\Delta D=\Delta R/r$, provided the model's assumptions hold.

The Assumptions Behind the Simple Model

The simple multiplier requires more than a stated reserve ratio. It assumes a chain of events that repeats in a consistent way across the banking system. In the model, each bank retains the required fraction of each new deposit and lends the rest. Borrowers spend the loans, and those payments become deposits in banks rather than leaving the deposit system as currency. Those banks then repeat the process.

A stable, applicable reserve ratio. The same ratio $r$ applies to the relevant deposits throughout the calculation. The model does not account for changes in the requirement or different effective ratios across deposits.
No excess reserves. Banks hold only the reserves required against deposits. Every additional reserve dollar not needed for the required reserve position can support lending under the model.
No currency leakage. Loan proceeds are not withdrawn and held as cash by the public. Payments are deposited in the banking system.
Repeated lending and redepositing. Banks lend the funds available above required reserves, and recipients redeposit the proceeds so that the next round can occur.
No other changes that interrupt the sequence. The calculation isolates the reserve change and deposit expansion. It assumes no unrelated deposit outflows or reserve losses that alter the modeled process.

These are simplifying assumptions, not definitions of banking. For instance, a bank may receive reserves but choose to hold some as excess reserves rather than lend them. A recipient of a payment may also withdraw cash. Either event changes the sequence that produces the simple multiplier.

Why the Formula Is a Geometric Series

Consider an initial new deposit $D_0$ in a system with a constant required reserve ratio $r$, where $0<r\leq 1$. The bank retains $rD_0$ as required reserves and can lend the remaining $(1-r)D_0$. If the entire loan is spent and redeposited, the next round's deposit is $(1-r)D_0$. Each later round is the preceding round multiplied by $1-r$.

D_{\text{total}}=D_0+(1-r)D_0+(1-r)^2D_0+\cdots

Because $0<r\leq 1$, the common ratio $1-r$ is between zero and one. The infinite geometric series therefore sums to $D_0/[1-(1-r)]$, which is $D_0/r$. In this setup, the initial deposit is the reserve injection that starts the process; the total required reserves against the resulting deposits equal $r(D_0/r)=D_0$. The result depends on every round retaining the same lending-and-redepositing pattern.

D_{\text{total}}=\frac{D_0}{1-(1-r)}=\frac{D_0}{r} \qquad\Longrightarrow\qquad \frac{D_{\text{total}}}{D_0}=\frac{1}{r}

Worked Example: The Idealized Deposit Sequence

A new $8,000 deposit enters a banking system. The required reserve ratio is 20%. Assume banks hold no excess reserves, lend every dollar above required reserves, and borrowers spend and redeposit every dollar of each loan. Find the first three deposit rounds, the total deposit expansion, and the amount retained as required reserves against the total.

Method. Calculate the lending share as $1-r=1-0.20=0.80$. Each round's deposit is 80% of the preceding round.

Round 1: The initial deposit is $8,000. Required reserves are $0.20\times \$8{,}000=\$1{,}600$, leaving a loan of $\$8{,}000-\$1{,}600=\$6{,}400$. After spending and redepositing, the next deposit is $6,400.

Round 2: Required reserves against $6,400 are $0.20\times \$6{,}400=\$1{,}280$. The bank lends $\$6{,}400-\$1{,}280=\$5{,}120$, which becomes the next deposit.

Round 3: Required reserves against $5,120 are $0.20\times \$5{,}120=\$1{,}024$. The bank lends $\$5{,}120-\$1{,}024=\$4{,}096$, which becomes the next deposit.

The full sequence is $8{,}000+6{,}400+5{,}120+\cdots$. The multiplier is $1/0.20=5$, so total deposits are $\$8{,}000\times5=\$40{,}000$. Required reserves against that total are $0.20\times\$40{,}000=\$8{,}000$, matching the initial reserve injection. The check confirms the model's accounting. A common mistake is to report $32,000 by counting only the later deposits and omitting the initial $8,000 deposit; the multiplier's total includes that first round.

Check the Assumptions Before Calculating

A scenario may give a reserve ratio and a reserve change without satisfying the other assumptions. Before applying $1/r$, ask whether excess reserves are being held, whether any proceeds are withdrawn as currency, and whether the lending-and-redepositing sequence is stipulated. If a condition fails, the simple multiplier may be a useful comparison, but it is not the scenario's supported prediction.

Excess reserves have a direct interpretation: they are reserves beyond the amount required. If part of a reserve injection is held as excess reserves, only the remainder is available to support deposits at the required ratio in a simplified calculation. This does not mean that the bank has no ability to lend; it means the simple model's assumption that all available reserves support the deposit sequence is not met.

Worked Example: An Excess-Reserve Holdback

A reserve injection of $10 million enters a system with a required reserve ratio of 8%. Banks collectively choose to hold $2 million of the injection as excess reserves. Assume the remaining $8 million supports deposits under the no-currency-leakage model, and that banks lend and redeposit as specified. Calculate the ideal simple-multiplier benchmark and the deposit expansion supported after the holdback.

Step 1: Calculate the unadjusted benchmark. With $r=0.08$, the simple multiplier is $1/0.08=12.5$. Applying it to the full $10 million gives a benchmark deposit expansion of $\$10\text{ million}\times12.5=\$125\text{ million}$.

Step 2: Identify the reserve amount that supports the sequence. The banks retain $2 million as excess reserves, so the amount used for required-reserve support is $\$10\text{ million}-\$2\text{ million}=\$8\text{ million}$.

Step 3: Calculate the deposit expansion under the stated holdback. Applying the multiplier to $8 million gives $\$8\text{ million}\times12.5=\$100\text{ million}$.

Check: Required reserves against $100 million of deposits are $0.08\times\$100\text{ million}=\$8\text{ million}$, exactly the portion assigned to required-reserve support. The $25 million difference from the unadjusted benchmark is $ \$2\text{ million}\times12.5$. The common mistake is to apply the multiplier to the entire injection after the problem says that some is held as excess reserves. The correction is to apply it only to the portion supporting the deposit sequence.

Currency and Excess Reserves Change the Result

When currency leakage and excess reserves are present, the simple deposit multiplier's assumptions fail. The earlier material on deriving a multiplier with currency and excess reserves gives a more general relationship. To use it, define $c=C/D$ as currency held by the public per dollar of deposits and $e=ER/D$ as excess reserves per dollar of deposits. With a constant required reserve ratio $r$, a constant $c$, and a constant $e$, the money multiplier for money $M=C+D$ relative to the monetary base $B=C+R$ is $(1+c)/(r+c+e)$. This formula is not the simple deposit multiplier $1/r$; it also requires the ratios to remain constant.

The relationship follows from the balance-sheet ratios: $C=cD$, required reserves are $rD$, and excess reserves are $eD$. Therefore, total reserves are $R=(r+e)D$, the monetary base is $B=(r+e+c)D$, and money is $M=(1+c)D$. Dividing $M$ by $B$ gives the stated multiplier. When $c=0$ and $e=0$, it reduces to $1/r$.

Worked Example: Comparing Simple and Generalized Benchmarks

Suppose the required reserve ratio is 10%, currency held by the public equals 20% of deposits, and excess reserves equal 5% of deposits. Treat these ratios as constant. For a $7 million increase in the monetary base, calculate the simple-model benchmark and the change in money implied by the generalized multiplier. Also find the corresponding deposit change under the stated ratios.

Step 1: Calculate the simple benchmark. The simple deposit multiplier is $1/0.10=10$. If all the simple-model assumptions held, a $7 million reserve injection would be associated with a deposit expansion of $\$7\text{ million}\times10=\$70\text{ million}$. This is a benchmark, not the result implied by the stated currency and excess-reserve ratios.

Step 2: Calculate the generalized multiplier. Here $r=0.10$, $c=0.20$, and $e=0.05$. Thus the money multiplier is \[ \frac{1+c}{r+c+e}=\frac{1.20}{0.10+0.20+0.05} =\frac{1.20}{0.35}=\frac{24}{7}\approx3.4286. \]

Step 3: Find the money change. Multiply the $7 million base increase by $24/7$: $\$7\text{ million}\times24/7=\$24\text{ million}$.

Step 4: Find the deposit change. Since the monetary base is $B=(r+c+e)D=0.35D$, the deposit change is $\$7\text{ million}/0.35=\$20\text{ million}$. Currency changes by $c\Delta D=0.20\times\$20\text{ million}=\$4\text{ million}$, so the money change is $\$20\text{ million}+\$4\text{ million}=\$24\text{ million}$, as calculated. The common mistake is to call the simple result, $70 million, the money change. It ignores the given currency and excess-reserve ratios; under the stated generalized model, deposits rise by $20 million and money rises by $24 million.

Common Exam Traps

A quick calculation can be arithmetically correct and still answer the wrong question if its assumptions are overlooked. In particular, distinguish a potential multiplier result from an actual deposit change described by the scenario.

Using the reserve ratio without checking excess reserves. The formula $1/r$ assumes no excess reserves. If a scenario specifies a reserve holdback, account for it or state that the simple prediction does not apply as written.
Assuming redepositing when cash is retained. A loan that is withdrawn and held as currency does not create the next deposit round assumed by the simple sequence.
Confusing deposits with money. Currency held by the public is part of money but is not a bank deposit. A deposit multiplier and a money multiplier do not necessarily measure the same change.
Leaving out the initial deposit. The geometric series includes the first deposit as well as all later redeposits.

Key takeaway: Apply the simple deposit multiplier only when the required reserve ratio is stable, banks do not hold excess reserves, loan proceeds return as deposits, and the lending sequence continues as modeled. If a problem states otherwise, identify the failed assumption before using the multiplier as a prediction.

Check Your Understanding

For each scenario, check which assumptions hold before using a multiplier.

A $6,000 initial deposit enters a system with a 25% required reserve ratio. Banks hold no excess reserves, and all loan proceeds are redeposited. Find the simple multiplier and total deposits. (Answer: Multiplier $=4$; total deposits $=\$24{,}000$.)
A system has a 12% required reserve ratio and receives $5 million in new reserves. Banks hold $0.5 million as excess reserves; assume the rest supports deposits, with no currency leakage. Find the deposit expansion supported by the remaining reserves. (Answer: $\$4.5\text{ million}/0.12=\$37.5\text{ million}$.)
A 10% required reserve ratio is given, but a scenario states that recipients withdraw some loan proceeds and keep them as cash. Can the simple multiplier alone determine the deposit expansion? State why. (Answer: No; currency leakage violates the full-redeposit assumption, and the amount or ratio of cash held is needed to calculate an adjusted result.)
Let $r=0.08$, $c=0.10$, and $e=0.02$, with all ratios constant. Find the generalized money multiplier and the change in money for a $3 million increase in the monetary base. (Answer: Multiplier $=1.10/0.20=5.5$; money increases by $\$16.5\text{ million}$.)
In the setting of question 4, find the associated deposit change and verify the money change using currency. (Answer: Deposits increase by $\$3\text{ million}/0.20=\$15\text{ million}$; currency increases by $0.10\times\$15\text{ million}=\$1.5\text{ million}$, so money increases by $\$16.5\text{ million}$.)

What You'll Learn