Money Multiplier

The money multiplier expresses the relationship between the money supply and the monetary base. It has two distinct uses: an observed ratio between these quantities, and a theoretical coefficient describing how changes in base money might affect money held by the public. The distinction matters because an accounting relationship does not establish a causal mechanism. In particular, the familiar textbook multiplier based on reserve requirements is not a general description of modern money creation. (federalreserve.gov)

Definition and measurement

For a specified monetary aggregate $M$ and monetary base $B$, the observed multiplier is

The monetary base consists of central-bank currency and reserve balances. Monetary aggregates include currency held by the public and selected financial liabilities, principally bank deposits. Different aggregates include different instruments, so an M1 multiplier and an M2 multiplier need not have the same value. In the United States, M2 includes M1, small-denomination time deposits, and retail money market mutual fund shares. (federalreserve.gov)

An observed multiplier is therefore a dimensionless ratio, not necessarily a stable parameter. It can change because deposits grow, currency preferences change, or bank reserves increase independently of deposits. Empirical research has found that the tight reserve–money relationship assumed in elementary multiplier models does not adequately describe U.S. banking. (federalreserve.gov)

The simple deposit-expansion model

The elementary model assumes that banks retain a fixed fraction $r$ of deposits as reserves and expand lending with the remainder. Its essential assumptions are:

• a positive, uniform required reserve ratio;
• no reserves held above that requirement;
• no withdrawal of funds into currency held outside banks;
• redepositing of loan proceeds within the banking system;
• sufficient willingness to borrow and lend.

Under these assumptions, the simple deposit multiplier is

This stylized model describes banking-system expansion, not the ability of an individual bank to expand its deposits without limit. (stlouisfed.org)

For example, suppose a hypothetical banking system receives an additional $1,000 of reserves accompanied by $1,000 of deposits, and $r=0.10$. The first round permits $900 of additional lending; redepositing the proceeds permits another $810, followed by $729, and so on. The resulting deposits form a geometric series:

In this example, the final deposit increase includes the initial $1,000. Additional lending totals $9,000. Thus, “tenfold deposit expansion” does not mean that the initial deposit generates ten times its value in additional loans.

The example also depends on how the initial transaction is specified. Depositing currency already held by the public initially exchanges currency for deposit money, rather than increasing the public’s total money holdings. A central-bank transaction introducing new base money is a different starting point. (bankofengland.co.uk)

Currency and excess reserves

A more general formulation allows the public to hold currency and banks to hold excess reserves. Define:

• $C$: currency held by the nonbank public;
• $D$: deposits included in the simplified monetary aggregate;
• $R$: total bank reserves;
• $c=C/D$: the currency-to-deposit ratio;
• $r$: the required reserve ratio;
• $e$: excess reserves divided by deposits.

With $M=C+D$, $B=C+R$, and $R=(r+e)D$, substitution gives

Setting $c=e=0$ yields $m=1/r$. This formulation is a simplified representation: actual monetary aggregates and reserve rules can cover different instruments and categories of deposits. (federalreserve.gov)

For an illustrative calculation, let $c=0.20$, $r=0.10$, and $e=0.05$. Then

The identity $M=mB$ implies

Consequently, treating the multiplier as a prediction of money growth requires assumptions about its stability. If it changes, multiplying a base-money increase by the previous ratio is insufficient.

Money creation in modern banking

Modern bank lending does not generally begin with a mechanical sequence of receiving reserves and lending them onward. When a commercial bank grants a loan, it normally records a loan asset and a matching deposit liability on its balance sheet. Payment to another bank can subsequently require settlement using reserves. Repayment of loan principal reduces both the loan and the corresponding deposit money. (bankofengland.co.uk)

This does not make lending unconstrained. Banks must consider profitability, credit risk, funding costs, liquidity, and regulatory requirements, including capital requirements. These constraints are distinct from the quantity of reserves already held. (bankofengland.co.uk)

Where a central bank implements monetary policy by targeting an interest rate, reserve provision responds to the operating framework and banks’ settlement needs. The Bank of England’s 2014 account emphasized that lending and deposit creation can generate demand for base money, rather than base money initiating a fixed multiple of lending. (bankofengland.co.uk)

Historical changes and policy interpretation

The reserve-based multiplier has long appeared in introductory accounts of monetary policy. Its relevance weakened as banks developed ways to reduce reservable liabilities and central banks adopted operating systems with abundant reserves. A 2021 Federal Reserve Bank of St. Louis teaching article argued that the simple multiplier should no longer be used to explain the Federal Reserve’s operating framework. (stlouisfed.org)

The Federal Reserve reduced reserve requirement ratios to zero effective March 26, 2020. This makes the textbook expression $1/r$ undefined; it does not imply unlimited lending or make the observed ratio $M/B$ undefined. (federalreserve.gov)

Quantitative easing further illustrates the distinction. Asset purchases can increase reserves substantially without producing a fixed multiple of new loans. Purchases from nonbank sellers can also create deposits directly, while purchases from banks primarily exchange one bank asset for another. Their effects therefore cannot be inferred from the simple reserve multiplier alone. (bankofengland.co.uk)

The observed multiplier remains a descriptive statistic, provided the monetary aggregate and base are clearly defined. It should not be confused with a bank’s leverage ratio or with the fiscal multiplier, which concerns the response of economic output to fiscal changes.

References

1. Money, Reserves, and the Transmission of Monetary Policy: Does the Money Multiplier Exist?federalreserve.gov
2. What is the money supply? Is it important?federalreserve.gov
3. Teaching the Linkage Between Banks and the Fed: R.I.P. Money Multiplierstlouisfed.org
4. Money creation in the modern economybankofengland.co.uk
5. Money creation in the modern economybankofengland.co.uk
6. How is money created?bankofengland.co.uk
7. Reserve Requirementsfederalreserve.gov