Why Cash Withdrawals Matter
When a bank makes a loan, it typically credits a deposit to the borrower. If the borrower spends that deposit and the recipient redeposits the payment, funds can support further lending across banks. But if someone withdraws cash and keeps it outside the banking system, that portion is no longer available to be redeposited as a bank deposit. The withdrawal also reduces the bank’s reserves and the customer’s deposit.
The key is to track two different things separately: the payment of a deposit and the withdrawal of currency. A payment between customers at different banks generally transfers reserves between banks while leaving the deposit in the banking system. A withdrawal for cash reduces deposits held at banks and removes reserves from the banking system when the cash leaves. The distinction matters when following lending capacity and deposit creation.
For the calculations below, a reserve requirement is a fixed fraction \(r\) of deposits, and required reserves are \(rD\), where \(D\) is deposits. Unless a problem says otherwise, assume the listed reserves are reserves that count toward the requirement. A bank’s reserve headroom is reserves minus required reserves. The simplified lending-chain examples assume banks lend available reserves above the requirement and that loan recipients make the stated choices about cash and redepositing. They omit other constraints such as capital, loan demand, and banks’ desired excess reserves.
What a Withdrawal Does to a Bank
Suppose a customer takes \(W\) dollars out of a bank and keeps the currency. On the bank’s balance sheet, reserves fall by \(W\), because cash paid to the customer is no longer held as bank reserves. The deposit liability to the customer also falls by \(W\). The two balance-sheet changes are equal: the bank pays out an asset and extinguishes an equal liability.
The withdrawal also changes the amount of required reserves. Deposits fall by \(W\), so required reserves fall by \(rW\). But reserves fall by the full \(W\), not just by \(rW\). Thus, although the reserve requirement becomes smaller, the bank’s headroom generally falls.
Here \(R\) is the bank’s initial reserves. For \(0<r<1\), a cash withdrawal reduces headroom by \((1-r)W\). This result assumes the withdrawal is not followed by a redeposit into the banking system and that no other transaction changes reserves or deposits. It does not mean every withdrawal immediately makes a bank unable to meet the requirement: the bank may begin with enough headroom to absorb it. If the withdrawal takes reserves below required reserves, the bank must respond, for example by obtaining reserves or adjusting its balance sheet.
Worked Example: Measuring the Reserve Effect of a Withdrawal
A hypothetical bank has deposits of $800,000 and reserves of $110,000. Its reserve requirement is 10% of deposits. A customer withdraws $20,000 in cash and keeps it. Find the bank’s deposits, reserves, required reserves, and headroom after the withdrawal. The bank has no other transactions during this calculation.
Method. First calculate the reserve requirement and headroom before the withdrawal. Then reduce both deposits and reserves by the full withdrawal amount.
Before the withdrawal: required reserves are \(0.10 \times \$800{,}000=\$80{,}000\). Headroom is \(\$110{,}000-\$80{,}000=\$30{,}000\).
After the withdrawal: deposits are \(\$800{,}000-\$20{,}000=\$780{,}000\). Reserves are \(\$110{,}000-\$20{,}000=\$90{,}000\). Required reserves are \(0.10 \times \$780{,}000=\$78{,}000\), so headroom is \(\$90{,}000-\$78{,}000=\$12{,}000\).
Check. The withdrawal reduces headroom by \(\$30{,}000-\$12{,}000=\$18{,}000\). The formula gives \((1-0.10)\times \$20{,}000=\$18{,}000\). The bank still has positive headroom, so this withdrawal alone does not put it below its requirement. The common mistake is to subtract only the $2,000 reduction in required reserves from headroom; reserves fell by $20,000, so the net headroom reduction is $18,000.
Payments, Withdrawals, and Redeployment
A purchase paid for by bank transfer is not itself a withdrawal from the banking system. If a customer at Bank A pays a customer at Bank B, the payer’s deposit falls, the recipient’s deposit rises, and reserves move from Bank A to Bank B through settlement. Across the banking system, that payment changes which customers and banks hold the deposits and reserves, but it does not by itself reduce their totals.
A cash withdrawal is different. If the recipient withdraws the payment and keeps the currency, deposits across banks fall and reserves leave the banking system. If some of the cash is later deposited at another bank, that part returns to the banking system as a deposit and as reserves at the receiving bank. The portion still held as currency is the leakage.
This is why a deposit chain cannot be followed by assuming that every dollar is redeposited. At each stage, recipients may keep some funds as cash. The next bank can lend only from the reserves it receives that remain available after meeting its reserve requirement, under the simplified assumptions used here.
Worked Example: Following Cash Through Two Lending Rounds
A hypothetical bank receives $100,000 in reserves and makes a $100,000 loan. The borrower spends the proceeds. The recipient keeps 20% of the payment as currency and deposits the other 80% at another bank. The reserve requirement is 10%. At each subsequent bank, assume the bank lends all reserves above its requirement, and each borrower’s recipient keeps 20% of the loan proceeds as currency and redeposits 80%. Trace the first two rounds of redepositing and lending.
Method. Split each loan into the portion kept as cash and the portion redeposited. The receiving bank holds required reserves against that deposit and can lend the remainder.
First redeposit. The recipient deposits \(0.80 \times \$100{,}000=\$80{,}000\). The recipient keeps \(0.20 \times \$100{,}000=\$20{,}000\) as currency. The receiving bank’s required reserves are \(0.10 \times \$80{,}000=\$8{,}000\), leaving \(\$80{,}000-\$8{,}000=\$72{,}000\) available to lend.
Second lending round. The bank lends $72,000. Its borrower spends that amount; the recipient keeps \(0.20 \times \$72{,}000=\$14{,}400\) as currency and deposits \(0.80 \times \$72{,}000=\$57{,}600\) at the next bank. That bank’s required reserves are \(0.10 \times \$57{,}600=\$5{,}760\), leaving \(\$57{,}600-\$5{,}760=\$51{,}840\) available to lend.
Check. Across these two rounds, the recipients hold \(\$20{,}000+\$14{,}400=\$34{,}400\) in cash. The successive redeposits are $80,000 and $57,600; the second is the redeposited part of the $72,000 loan, not an additional $57,600 loan. The common mistake is to treat all loan proceeds as new deposits at the next bank, ignoring the cash kept outside the system.
Calculating the Net Effect of Partial Redepositing
Sometimes a problem states that some withdrawn cash is redeposited elsewhere. In that case, do not treat the entire original withdrawal as permanent leakage. Calculate the net amount that remains outside banks. If \(W\) is withdrawn and \(Q\) is redeposited, then the net currency leakage is \(W-Q\), provided those are the only relevant flows.
Across the banking system, a cash withdrawal followed by a redeposit has the following net effect when the cash is withdrawn from one bank and redeposited at another: deposits fall by the net leakage, and reserves fall by the same net leakage. The redeposit shifts reserves to the receiving bank; it does not undo the portion retained as currency. Individual banks can have larger inflows or outflows than the system-wide net change.
Worked Example: A Withdrawal Partly Redeposited Elsewhere
A hypothetical banking system has $300,000 in deposits and $40,000 in reserves. The reserve requirement is 12%. A customer withdraws $15,000 in cash, spends it, and the recipient deposits $9,000 at a different bank. The recipient keeps the remaining $6,000 as cash. Calculate the system’s new deposits, reserves, required reserves, and reserve headroom.
Method. Separate the gross cash withdrawal from the amount that returns as a deposit. Only $6,000 remains outside the banking system.
Net leakage. The redeposit is $9,000, so net currency leakage is \(\$15{,}000-\$9{,}000=\$6{,}000\).
New system totals. Deposits are \(\$300{,}000-\$6{,}000=\$294{,}000\). Reserves are \(\$40{,}000-\$6{,}000=\$34{,}000\). Required reserves are \(0.12 \times \$294{,}000=\$35{,}280\). Headroom is \(\$34{,}000-\$35{,}280=-\$1{,}280\), meaning the system’s reserves are $1,280 below the stated requirement.
Check. The $9,000 redeposit is a deposit and reserve inflow at the receiving bank, while the original $15,000 withdrawal is an outflow. The net system change is therefore \(-\$6{,}000\) in both deposits and reserves. The initial required reserves were \(0.12 \times \$300{,}000=\$36{,}000\), and initial headroom was \(\$40{,}000-\$36{,}000=\$4{,}000\). The net leakage reduces headroom by \(0.88 \times \$6{,}000=\$5{,}280\), leaving \(-\$1{,}280\), as calculated. The common mistake is to count the full $15,000 as permanent leakage or to add the $9,000 redeposit to system deposits without accounting for the original deposit withdrawal.
Why Leakage Interrupts the Simple Chain
In a fully redeposited chain, a payment that leaves one bank becomes a deposit at another. The receiving bank gets reserves through settlement and can use reserves above its requirement to support further lending. When some proceeds are kept as currency, fewer reserves return to banks in the form of deposits. That leaves less reserve support for later rounds of lending than in a chain where every payment is redeposited.
The simple deposit multiplier discussed in earlier material assumes that funds return to banks in a consistent way and that banks lend available reserves. Currency leakage violates the full-redepositing part of that simplified picture. It is therefore not correct to apply the simple multiplier mechanically when a problem says that households or businesses hold some of their funds as currency. The precise adjusted multiplier is a separate calculation; here the essential skill is to identify and track the leakage correctly.
Currency held by the public is not the same as an excess reserve held by a bank. Both are outside the amount available for further lending in the simplified chain, but they are held by different parties and have different balance-sheet effects. A withdrawal converts a deposit liability into currency held by the public while reducing bank reserves. Excess reserves remain as bank assets and can be retained rather than lent.
Common Mistakes
- Reducing reserves only by the reserve requirement. If a customer withdraws \(W\) dollars in cash, reserves fall by \(W\); required reserves fall by \(rW\).
- Calling every bank-to-bank payment a withdrawal. A transfer between bank customers moves deposits and reserves between banks but does not necessarily remove either from the banking system.
- Counting a redeposit as new lending. A redeposit is a transfer of funds into a bank account. It is not itself a new loan.
- Ignoring the recipient’s cash choice. If a recipient keeps part of a payment as currency, only the remainder returns as a bank deposit.
- Confusing bank headroom with system-wide reserves. A reserve transfer can improve one bank’s position while reducing another’s. A transfer alone does not add reserves to the banking system.
Check Your Understanding
For each problem, keep deposits, reserves, and currency outside banks distinct. Assume the stated reserve requirement applies to deposits and ignore other transactions unless specified.
- A hypothetical bank has deposits of $500,000 and reserves of $58,000. Its reserve requirement is 10%. A customer withdraws $20,000 and keeps the cash. Find post-withdrawal deposits, reserves, required reserves, and reserve headroom. (Answer: deposits $480,000; reserves $38,000; required reserves $48,000; headroom \(-\$10,000\), a $10,000 shortfall.)
- A hypothetical loan of $60,000 is spent. The recipient keeps 25% as currency and deposits the rest at another bank. With a 10% reserve requirement, how much is deposited, and how much can the receiving bank lend if it lends all reserves above the requirement? If that next loan is spent with the same 25% cash choice, how much is redeposited at the following bank? (Answer: first redeposit $45,000; first receiving bank can lend $40,500; next redeposit $30,375.)
- A banking system begins with deposits of $400,000 and reserves of $50,000; the reserve requirement is 10%. Customers withdraw $12,000 in cash, and 40% of that cash is subsequently deposited at other banks. Find the net currency leakage and the system’s new deposits, reserves, required reserves, and headroom. (Answer: leakage $7,200; deposits $392,800; reserves $42,800; required reserves $39,280; headroom $3,520.)
- A bank receives a $50,000 deposit of reserves and makes a $50,000 loan. The recipient of the loan keeps 10% of the proceeds as cash and redeposits the rest. The reserve requirement is 20%. A second bank lends all reserves above its requirement, and the next recipient also keeps 10% of that loan as cash. Find the second bank’s loan and the total currency kept by the two recipients. (Answer: second bank loan $36,000; total currency $8,600.)
- A hypothetical banking system has deposits of $1,000,000 and reserves of $125,000, with a 10% reserve requirement. Customers withdraw $80,000, and half of the withdrawn currency is redeposited at other banks. Find net leakage and the new system deposits, reserves, required reserves, and reserve headroom. (Answer: leakage $40,000; deposits $960,000; reserves $85,000; required reserves $96,000; headroom \(-\$11,000\), an $11,000 shortfall.)