From a Reserve Change to a Deposit Change
When reserves available to banks increase, banks may be able to support additional deposits through lending and the redepositing of loan proceeds. The simple deposit multiplier gives a compact estimate of the maximum deposit change associated with a change in reserves. It is a model result, not a guarantee that deposits will change by that amount in practice.
The key calculation is direct: divide the reserve change by the required reserve ratio. The direction matters. An increase in reserves implies a potential increase in deposits; a decrease implies a potential contraction. Before calculating, identify whether the question gives a change in reserves or asks for the reserves needed to support a target change in deposits.
Conditions for Using the Calculation
Use the calculation as a system-wide estimate under a simplified model. The required reserve ratio is fixed, and banks hold required reserves but no excess reserves. Newly available funds remain in the banking system: the public does not withdraw loan proceeds as currency, and loan proceeds are redeposited. Banks are willing and able to make the loans that use their available lending capacity. There are no offsetting reserve or deposit changes elsewhere in the system.
These conditions matter because the calculation tracks successive rounds of deposit expansion supported by the original reserve change. If funds leave the banking system, banks hold additional reserves voluntarily, or lending is limited for another reason, the observed deposit change may be smaller. Therefore, describe the result as a maximum under the model, not as a prediction of actual lending or deposits.
The ratio \(r\) must be written as a decimal. For example, a required reserve ratio of 12% is \(r=0.12\), so the multiplier is \(1/0.12\), not \(1/12\). A lower reserve ratio gives a larger multiplier because each dollar of reserves can support more dollars of deposits.
Why the Multiplier Works
The calculation can be justified by considering successive deposit rounds. Suppose an initial reserve-supported deposit increase is \(\Delta R\). The bank retains the fraction \(r\) as required reserves and can lend the remaining fraction \(1-r\). If all loan proceeds are redeposited, each subsequent round is a fraction \(1-r\) of the previous round.
The total deposit increase is therefore a geometric series. Since \(0 \leq 1-r < 1\), the series converges, and its sum is the first round divided by one minus the common ratio:
Thus the multiplier is \(1/r\). The result includes the initial deposit round as well as later rounds; it is not just the amount of new lending after the first bank receives reserves. When the reserve change is negative, the same relationship gives a negative deposit change, representing the maximum contraction under the same assumptions.
Worked Example: A Reserve Injection at a 20% Requirement
A hypothetical banking system receives a net reserve increase of $80,000. The required reserve ratio is 20%. Under the simple multiplier assumptions, estimate the maximum increase in deposits and check the result against the required reserves.
Method. Convert the percentage to a decimal, calculate the multiplier, and apply it to the reserve increase.
The ratio is \(r=20\%=0.20\). The multiplier is:
The maximum deposit increase is:
Check. Required reserves against the additional deposits are \(0.20 \times \$400{,}000=\$80{,}000\), equal to the reserve increase. Equivalently, the rounds are $80,000, $64,000, $51,200, and so on; the geometric series sums to $400,000.
Common mistake avoided. Multiplying by \(0.20\) would give $16,000, which is not the deposit increase. The reserve increase is the fraction \(r\) of the deposit increase, so solve \(\Delta R=r\Delta D\) by dividing by \(r\).
Applying the Multiplier to an Increase or Decrease
For an increase, the result is a maximum expansion. For a decrease, use the signed change or calculate the magnitude and then label it as a decrease. Keeping the sign explicit prevents an answer that gives the right size but the wrong direction.
Worked Example: A System-Wide Reserve Loss
A hypothetical banking system initially has $25,000,000 in deposits and $2,000,000 in reserves. Its required reserve ratio is 8%. A net reserve loss of $320,000 occurs. Under the simple multiplier assumptions, calculate the maximum deposit contraction and the resulting deposits.
Method. First confirm that the starting balances are consistent with the stated ratio. Then use the multiplier on the reserve loss and subtract the resulting contraction from starting deposits.
The starting required reserves are \(0.08 \times \$25{,}000{,}000=\$2{,}000{,}000\), matching the stated reserves. The multiplier is:
The reserve change is \(\Delta R=-\$320{,}000\). Thus:
The maximum deposit contraction is $4,000,000, so resulting deposits are \(\$25{,}000{,}000-\$4{,}000{,}000=\$21{,}000{,}000\). The resulting reserves are \(\$2{,}000{,}000-\$320{,}000=\$1{,}680{,}000\). The check is \(0.08 \times \$21{,}000{,}000=\$1{,}680{,}000\), exactly the resulting reserves.
Common mistake avoided. Reporting a positive $4,000,000 change ignores the direction of the reserve movement. A reserve loss produces a deposit contraction in this model, so the signed change is negative.
Working Backward from a Deposit Target
Sometimes the desired deposit change is given and the unknown is the reserve change needed to support it. Rearrange the multiplier relationship: \(\Delta R=r\Delta D\). If the problem gives target deposits rather than a target increase, first subtract current deposits to find \(\Delta D\). Applying the reserve ratio to the entire target balance when only an incremental change is asked for is a frequent source of error.
Worked Example: Finding Reserves Needed for a Deposit Goal
A hypothetical banking system has $9,000,000 in deposits and a required reserve ratio of 12%. It aims to increase deposits to $10,200,000. Assuming the simple multiplier conditions, find the additional reserves required and verify the final reserve balance.
Method. Calculate the deposit increase from the target, then multiply that increase by the reserve ratio.
The target deposit increase is:
The reserves required for that increase are:
Initially, required reserves are \(0.12 \times \$9{,}000{,}000=\$1{,}080{,}000\). After the additional $144,000, reserves are \(\$1{,}080{,}000+\$144{,}000=\$1{,}224{,}000\). The final check is \(0.12 \times \$10{,}200{,}000=\$1{,}224{,}000\), as required.
Common mistake avoided. Multiplying the full target balance of $10,200,000 by 12% gives $1,224,000, the final required reserve balance, not the additional reserves needed. The question asks for the change in reserves, which is $144,000.
A Reliable Calculation Sequence
A short sequence helps keep the model, the units, and the direction of change clear. In particular, distinguish a reserve balance from a change in reserves, and a deposit target from a change in deposits.
Identify the required reserve ratio, the reserve change or deposit target, and whether the problem asks for an increase, decrease, or final balance.
Write the percentage as a decimal, such as 8% as \(0.08\).
For a reserve change, use \(\Delta D=\Delta R/r\). For a required reserve change, use \(\Delta R=r\Delta D\).
State the direction and whether the result is a maximum under the model. Check that \(r\Delta D=\Delta R\), then add to or subtract from the initial balance if a final balance is requested.
Common Calculation Traps
- Multiplying instead of dividing. When reserve change is given, deposits are \(\Delta R/r\), not \(r\Delta R\). Multiplication is appropriate when finding the reserves required for a given deposit change.
- Using a whole-number percentage. Use \(0.08\) for 8%, not 8. This changes the result by a factor of 100.
- Dropping a negative sign. A reserve decrease gives a deposit contraction under this model. Show the sign or explicitly call the amount a decrease.
- Confusing a level with a change. To move from current deposits to target deposits, subtract the current level first. The multiplier applies to the change, not automatically to the full target.
- Overstating what the estimate means. A multiplier result is a maximum conditional on the stated model. It does not establish that actual banks will lend or that actual deposits will expand by that amount.
Check Your Understanding
For each question, use the simple multiplier assumptions stated or implied: a fixed required reserve ratio, no excess reserves, no currency withdrawals, full redeposit of loan proceeds, and no offsetting changes. Give the direction as well as the amount.
- A hypothetical system has a 10% required reserve ratio and receives an additional $75,000 in reserves. What is the maximum deposit change? (Answer: deposits increase by $750,000.)
- At a 25% required reserve ratio, reserves fall by $90,000. What is the maximum deposit change? (Answer: deposits decrease by $360,000.)
- A system has $5,000,000 in deposits and $600,000 in reserves, with a 12% required reserve ratio. Reserves increase by $36,000. What is the maximum deposit increase and the resulting deposit balance? (Answer: deposits increase by $300,000, to $5,300,000.)
- A hypothetical system targets an additional $2,400,000 in deposits. Its required reserve ratio is 15%. How much additional reserves are needed? (Answer: $360,000.)
- With a 7.5% required reserve ratio, a net reserve injection of $240,000 occurs. Calculate the multiplier and the maximum deposit increase, then check the required reserves for that increase. (Answer: multiplier \(13.333\ldots\); deposits increase by $3,200,000; required reserves on the increase are $240,000.)