Common Errors in Money-Creation Problems

Why Money-Creation Calculations Go Wrong

Money-creation problems often involve three quantities that are related but not interchangeable: reserves, deposits, and loans. A reserve is an asset of a bank, such as its balance at the central bank. A deposit is a liability of a bank: the bank owes that balance to its customer. A loan is an asset of the bank because the borrower owes repayment. Keeping those balance-sheet roles straight prevents many errors.

A second source of error is applying a multiplier without checking what its input represents. In the simple model, the multiplier is applied to a change in reserves to find the maximum change in deposits. It is not applied indiscriminately to a loan, an existing deposit balance, or a bank’s total assets. The model also depends on assumptions about reserve holdings and how the public handles payments.

Definition: Let $D$ denote deposits in the banking system, $R$ total bank reserves, $RR$ required reserves, and $ER$ excess reserves. If the required reserve ratio is $r$, then $RR=rD$. Excess reserves are reserves above the required amount: $ER=R-RR$. A bank’s basic balance-sheet identity is assets = liabilities + equity. Reserves and loans are assets; customer deposits are liabilities.

This tutorial focuses on diagnosing errors in those relationships. It uses the reserve-based simple deposit expansion model when its assumptions are explicitly stated, then contrasts it with cases where those assumptions fail.

A Loan Creates a Deposit; Settlement Transfers Reserves

When a bank grants a loan and credits the borrower’s deposit account, the bank records a loan asset and a matching deposit liability. The loan itself does not transfer reserves to the borrower. If the borrower then pays a customer of another bank, the payer’s deposit is debited, the recipient’s deposit is credited, and reserves are transferred between the banks to settle the payment. That interbank transfer changes which bank holds the reserves, not the total reserves in the system.

Worked Example: Tracing a Loan and an Interbank Payment

Bank A initially has reserves of $110,000, deposits of $100,000, and equity of $10,000. Bank B has reserves of $50,000, deposits of $40,000, and equity of $10,000. Bank A approves a $24,000 loan and credits the borrower’s deposit. The borrower immediately pays a vendor who banks at Bank B. Assume the payment settles fully between the banks and there are no other transactions. Find the changes in each bank’s reserves and deposits, and in systemwide reserves, deposits, and loans.

Method. Record the loan origination first, then record the payment and reserve settlement. At origination, Bank A adds a $24,000 loan asset and a $24,000 deposit liability. Its reserves do not change at that instant.

When the borrower pays the vendor, Bank A debits the borrower’s deposit by $24,000. Bank B credits the vendor’s deposit by $24,000. Settlement transfers $24,000 of reserves from Bank A to Bank B.

\begin{aligned} \text{Bank A final reserves} &= 110{,}000-24{,}000=86{,}000,\\ \text{Bank A final deposits} &= 100{,}000+24{,}000-24{,}000=100{,}000,\\ \text{Bank A final loans} &= 24{,}000. \end{aligned}

Bank B’s final reserves are $50{,}000+24{,}000=74{,}000$, and its final deposits are $40{,}000+24{,}000=64{,}000$. Its loans are unchanged. Bank A’s final assets are $86,000 in reserves plus $24,000 in loans, or $110,000, matching $100,000 in deposits plus $10,000 in equity. Bank B’s final reserves of $74,000 match its $64,000 in deposits plus $10,000 in equity.

Across both banks, reserves remain $110{,}000+50{,}000=160{,}000$. Deposits increase from $140,000 to $164,000, and loans increase by $24,000. The systemwide deposit increase is the loan-created deposit; the later payment merely changes which customer holds that deposit. Common mistake avoided: subtracting $24,000 from system reserves because the loan was paid out. The settlement transferred reserves between banks, so the systemwide total did not fall.

Using the Simple Deposit Multiplier Correctly

The simple deposit multiplier is a model of the maximum deposit expansion supported by an initial reserve injection. Assume a fixed required reserve ratio $r>0$, no currency withdrawals, no excess reserves, and that every loan is redeposited into the banking system. Also assume banks lend all funds not required as reserves, and do not face other lending constraints. Under those assumptions, each new deposit creates required reserves and leaves the remainder available to support further lending.

Theorem: Under the simple model’s assumptions, the maximum change in total deposits supported by a new reserve injection $\Delta R$ is $\Delta D=\Delta R/r$. This is a model result, not a universal rule for actual lending.

The formula follows from the reserve requirement. If all newly created deposits are backed by required reserves and banks hold no excess reserves, then the reserve injection supporting those deposits must equal $r$ times the deposit increase. Rearranging gives the multiplier formula.

\Delta R=r\Delta D \quad\Longrightarrow\quad \Delta D=\frac{\Delta R}{r}.

Equivalently, the successive deposits form a geometric series. For an initial deposit supported by the reserve injection, the first round leaves $1-r$ available for lending; each later round repeats that fraction. The sum of the rounds converges because $0<1-r<1$.

\Delta D =\Delta R\left[1+(1-r)+(1-r)^2+\cdots\right] =\frac{\Delta R}{1-(1-r)} =\frac{\Delta R}{r}.

Total deposits and total loans are different results. In this simplified expansion, the initial reserve injection remains in the banking system, while the rest of the deposit expansion is matched by loans. Thus the associated increase in loans is $\Delta D-\Delta R$, or $(1-r)\Delta D$. This distinction is useful whenever a question asks for both deposits and lending.

Worked Example: Applying the Multiplier to New Reserves

Assume the required reserve ratio is $r=0.10$. A new $18,000 reserve injection supports an initial $18,000 deposit, and all deposits are redeposited, banks lend all excess reserves, and the public holds no currency. Find the maximum total increase in deposits and the associated increase in loans.

Method. Use the simple deposit multiplier because the problem specifies the assumptions required by the model. The multiplier is $1/r=1/0.10=10$. Apply it to the $18,000 reserve injection, not to a loan amount.

\Delta D=\frac{18{,}000}{0.10}=180{,}000.

Required reserves against the resulting deposits are $0.10(180{,}000)=\$18{,}000$, exactly the reserve injection. The associated increase in loans is total deposits less those reserves:

\Delta L=\Delta D-\Delta R =180{,}000-18{,}000 =162{,}000.

As a check, the bank system holds $18,000 in additional reserves and has $162,000 in additional loans, for $180,000 in additional assets. Those assets match the $180,000 increase in deposits, assuming equity is unchanged. Common mistake avoided: reporting $180,000 as additional loans. That is the deposit increase; the required reserves account for the remaining $18,000 of the matching assets.

When a Simple Multiplier Is the Wrong Model

The simple multiplier does not apply just because a problem gives a reserve ratio. If banks hold excess reserves, some funds are not lent. If the public holds some funds as currency rather than depositing them, those funds do not support repeated deposit creation. Either behavior breaks the simple sequence of full redepositing and relending. The simple formula is therefore a conditional result: check the stated behavior before using it.

One way to represent stable currency and excess-reserve behavior is to define $c=C/D$, the currency-to-deposit ratio, and $e=ER/D$, the excess-reserve-to-deposit ratio. Let $M=C+D$ be currency plus deposits, and let $H=C+R$ be the monetary base. If these ratios remain constant, the money-to-base ratio is $(1+c)/(r+e+c)$. This ratio is not the simple deposit multiplier $1/r$: it relates a change in the monetary base to a change in currency plus deposits under the specified stable ratios.

Worked Example: Accounting for Currency and Excess Reserves

Suppose a banking system has a required reserve ratio of $r=0.10$, an excess-reserve ratio of $e=0.05$, and a currency-to-deposit ratio of $c=0.25$. A change in the monetary base is $32,000, and all three ratios remain constant. Find the associated changes in deposits, currency, reserves, and money $M=C+D$.

Method. Because the problem specifies currency holdings and excess reserves, do not use $1/r$. Use the stated stable-ratio model. The increase in currency and deposits together is given by the money-to-base ratio:

\Delta M =\frac{1+c}{r+e+c}\Delta H =\frac{1+0.25}{0.10+0.05+0.25}(32{,}000) =\frac{1.25}{0.40}(32{,}000) =100{,}000.

Since $C/D=0.25$, total money is $M=1.25D$. Therefore $\Delta D=100{,}000/1.25=\$80{,}000$, and $\Delta C=0.25(80{,}000)=\$20{,}000$. Total reserves equal required plus excess reserves, so $\Delta R=(r+e)\Delta D=(0.10+0.05)(80{,}000)=\$12{,}000$.

Check the base: $\Delta C+\Delta R=20{,}000+12{,}000=\$32{,}000$, as specified. Check money: $\Delta C+\Delta D=20{,}000+80{,}000=\$100{,}000$. Common mistake avoided: using $1/r=10$ and reporting a $320,000 money increase. That calculation ignores both the currency leakage and the excess reserves specified in the problem.

Common Calculation Traps

Before calculating, identify what the question asks for and which balance-sheet quantity the given number describes. Several tempting shortcuts produce answers with the right units but the wrong meaning.

Treating a loan as a reserve transfer. Loan origination creates a loan asset and a deposit liability. Reserves move between banks when a payment is settled; that movement does not reduce systemwide reserves by the amount of the loan.
Calling deposits reserves. Deposits are liabilities owed to customers. Reserves are assets held by banks. A required reserve calculation uses the deposit base specified by the problem; it does not turn deposits into reserves.
Multiplying the wrong input. Under the simple model, $\Delta D=\Delta R/r$. If a problem gives an initial reserve injection, use that as $\Delta R$. If it gives an initial deposit, first determine whether that deposit is the injection described by the model; do not silently treat any loan or existing deposit balance as new reserves.
Confusing deposit expansion with loan expansion. The simple-model deposit increase includes the reserves retained to satisfy the requirement. Where the question asks for additional loans, subtract the retained reserves from the deposit increase.
Ignoring the assumptions. Currency withdrawals, excess reserves, or other limits on lending can make the simple multiplier an inappropriate prediction. A stated reserve ratio alone does not prove that deposits will expand by $1/r$ times a given quantity.

A reliable check is to return to the balance sheet. For the system as a whole, changes in assets must equal changes in liabilities plus changes in equity. In a simplified calculation with unchanged equity, the increase in reserves plus the increase in loans must match the increase in deposits. If the answer does not balance, a quantity has probably been misclassified or counted twice.

Key takeaway: First classify the given amount as reserves, deposits, or loans. Then trace any payment settlement and check the model assumptions. Use the simple multiplier only for a reserve injection under the no-currency, no-excess-reserve, full-redepositing model; keep deposit growth distinct from loan growth.

Check Your Understanding

For each problem, identify the relevant balance-sheet quantities before calculating. Answers are provided for checking.

A banking system receives a $12,000 reserve injection. The required reserve ratio is $0.08$, with no currency withdrawals or excess reserves and full redepositing. Find the maximum deposit increase and the associated loan increase. (Answer: deposits increase by $150,000; loans increase by $138,000.)
A bank grants a $40,000 loan, credits the borrower’s deposit, and the borrower pays a recipient at another bank. Settlement is complete, and there are no other transactions. What are the changes in systemwide deposits and reserves, and what is the interbank reserve transfer? (Answer: deposits increase by $40,000; systemwide reserves change by $0; $40,000 in reserves transfers from the lending bank to the receiving bank.)
Under the simple model, the required reserve ratio is $0.20$, and a new $9,000 reserve injection supports the deposit expansion. Find total deposits and associated loans. (Answer: deposits increase by $45,000; loans increase by $36,000.)
Stable ratios are $r=0.10$, $e=0.05$, and $c=0.25$. A monetary-base increase is $16,000. Find the associated changes in money, deposits, currency, and reserves. (Answer: money increases by $50,000; deposits by $40,000; currency by $10,000; reserves by $6,000.)
Under the simple model, a banking system is to support a $240,000 increase in deposits. The required reserve ratio is $0.125$; there are no currency withdrawals or excess reserves. Find the reserve injection required and the associated increase in loans. (Answer: reserve injection of $30,000; loans increase by $210,000.)

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