Financial Markets & Money

The Money Multiplier

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The money multiplier links central-bank money (monetary base H = currency + reserves) to the broader money supply M = currency + deposits. With reserve ratio θ and currency ratio c, M = mm · H where mm = 1 / [c + θ(1−c)]. If c=0 (no currency), mm = 1/θ.

Derivation

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Multiplier mm

5.00

Money supply M

$1,000

Deposits D

$1,000

Currency CU

Lending rounds (first 6)

#	Deposit	Reserves (θ)	Loan out (1−θ)
1	$200	$40	$160
2	$160	$32	$128
3	$128	$25.6	$102.4
4	$102.4	$20.48	$81.92
5	$81.92	$16.38	$65.54
6	$65.54	$13.11	$52.43

Formula: M = H / [c + θ(1−c)] = $200 / 0.200 = $1,000

Inputs

Base money H$200

Reserve ratio θ0.20

Currency ratio c0.00

Mechanics: each $1 of H → $1/θ of M when c = 0.

Breakdown: high c (cash hoarding) or high θ (tight reserves) shrinks mm.

Modern LM: CB targets i, so M is endogenous — this widget shows the mechanics, not the policy instrument.

Definitions

| Symbol | Meaning | |--------|---------| | $M$ | Money supply (currency + deposits) | | $H$ | Monetary base / CB money (currency + reserves) | | $CU$ | Currency held by public | | $D$ | Bank deposits | | $R$ | Reserves held at the central bank | | $c = CU/M$ | Currency ratio (behavioural) | | $\theta = R/D$ | Reserve ratio (regulatory + prudential) |

The Formula

M = \frac{1}{c + \theta(1-c)} \cdot H

In the no-currency case ( $c = 0$ ):

M = \frac{1}{\theta} \cdot H

The Lending Process (c = 0, θ = 0.2)

CB buys $200 of bonds →$ 200 injected into the banking system → geometric lending expansion: the sum of the series is ΔM = ΔH / θ = $1000.

| Round | Deposit | Reserves | Loan out | |-------|---------|----------|----------| | 1 | $200 |$ 40 | $160 | | 2 |$ 160 | $32 |$ 128 | | 3 | $128 |$ 25.60 | $102.40 | | … | … | … | … | | Total | **$ 1,000** | $200 | — |

Why the Multiplier Can Break Down

Excess reserves: Banks hold more than required θ when confidence is low (2008–15 QE era). Effective multiplier collapses.
Currency hoarding: c rises in crises → mm falls.
ZLB: At zero interest, the opportunity cost of holding reserves is zero → banks prefer reserves over lending.

Policy Implications

Open-market purchase (CB buys bonds): $\uparrow H \Rightarrow \uparrow M$ by $mm \cdot \Delta H$ .
Reserve requirement cut (lower $\theta$ ): $mm$ rises → same $H$ supports more $M$ .
Modern flat-LM view: CB targets the interest rate directly, so $M$ is endogenous. Multiplier describes the mechanics, but the instrument is $i^T$ , not $H$ .

Worked Example

θ = 0.2, c = 0. CB purchases $200 of bonds from an individual.

Round 1: Seller deposits $200. Bank keeps $40, lends $160.
Round 2: Borrower deposits $160. Bank keeps $32, lends $128.
Round 3: Deposit $128. Bank keeps $25.60, lends $102.40.
Geometric series: ΔM = $200 × [1 + 0.8 + 0.8² + …] = $200 × (1/0.2) = $1,000.

ΔM = $1,000. Multiplier = 1/0.2 = 5. $200 of base money → $1,000 of money supply.

Common Mistakes

—Forgetting currency: with c > 0, mm = 1/[c + θ(1−c)], not 1/θ.
—Confusing M (money supply) with H (monetary base): CB controls H, not M.
—Thinking mm is fixed — it changes with c (payment habits) and θ (regulation/behaviour).
—Ignoring excess reserves: banks may hold > θ when confidence is low, shrinking effective mm.

Exam Cues

→No-currency case: mm = 1/θ. Learn by heart.
→Full formula: mm = 1/[c + θ(1−c)]. Verify with c=0 → 1/θ.
→Geometric series: 1 + (1−θ) + (1−θ)² + … = 1/θ.
→Policy: ΔM = mm × ΔH. Tightening: ↓H → ↓M by mm × ΔH.