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Step-by-step mathematical derivation with typeset equations and expandable detail.
Sections
Base money, deposits, and reserves
The money multiplier describes how the central bank's monetary base (also called high-powered money) is leveraged into a larger broad money supply through the fractional-reserve banking system. The monetary base consists of currency held by the public and reserves held by banks : . Broad money is the sum of currency in circulation and demand deposits : .
Banks are required to hold a fraction of their deposits as reserves (the reserve ratio), but they lend out the remainder. Each loan creates a new deposit at another bank, which in turn holds reserves against it and lends the rest. This cascade of lending and redepositing is the mechanism that multiplies the base into a larger stock of money. The currency-deposit ratio captures the public's preference for holding cash versus deposits.
The monetary base: currency in circulation plus bank reserves.
Broad money supply: currency plus demand deposits.
The reserve ratio: fraction of deposits held as reserves by banks.
The currency-deposit ratio: the public's cash preference relative to deposits.
Deriving the money multiplier
The multiplier is defined as . To derive it, divide both numerator and denominator by deposits . The numerator becomes after division by . The denominator becomes . The money multiplier is therefore .
When the public holds no currency (), the multiplier simplifies to the textbook inverse of the reserve ratio: . With a 10% reserve requirement, each dollar of base money supports ten dollars of deposits. The geometric-series interpretation makes this transparent: a dollar deposited generates in loans, which returns as a new deposit, of which is lent again, and so on. The sum . Introducing currency leakage () reduces the multiplier because each round of redepositing loses some funds to cash holdings that do not re-enter the banking system.
The money multiplier: broad money per dollar of base money, expressed in terms of the currency-deposit and reserve ratios.
With no currency drain, the multiplier is the inverse of the reserve ratio.
The deposit-expansion process as a convergent geometric series.
Policy instruments and the multiplier
Central banks influence the money supply through two channels: changing the monetary base and changing the multiplier . Open market operations are the primary tool for adjusting . When the central bank buys government bonds on the open market, it pays with newly created reserves, expanding . The banking system then multiplies this injection by , so . Selling bonds reverses the process, draining reserves and contracting the money supply.
Reserve requirements directly affect and hence the multiplier itself. Lowering the required reserve ratio raises , amplifying the money supply for a given base. However, most modern central banks rarely change reserve requirements because the effect is blunt and disruptive to bank balance-sheet management. Instead they rely on interest-rate targeting and open market operations. The money multiplier framework remains valuable as a conceptual tool for understanding how policy actions at the central bank propagate through the banking system to determine the total money supply available to the economy.
The money supply is the product of the multiplier and the monetary base.
An open market operation that changes the base by changes the money supply by the multiplier times that change.