Welcome to your study space
Understand economics at your own pace: step-by-step guided courses, and exercises to practice.
The money multiplier
Where the textbooks' 1/r comes from — and why it is a ceiling, never a forecast.
ceiling: m = 1/r · observed: m = (1 + c)/(c + e)Problem / motivation
One textbook in two teaches this formula: if banks must keep 10% of deposits in reserve, every euro issued by the central bank supports 10. In the euro area, the required reserve ratio is 1%: the multiplier should therefore be 100. The poll below asks what it actually is — and the gap between those two numbers is the whole course.
Three words before we begin, because the whole course rests on the distinction between two kinds of money. The MONETARY BASE is the money issued by the central bank: notes, and the RESERVES banks hold in their account with it — money that circulates only between banks. The MONEY SUPPLY, by contrast, is what non-bank agents can spend: notes and deposits (course “The functions and aggregates of money”). The multiplier is the ratio between the two: how much money supply per euro of base.
This course does three things, in this order. It BUILDS the 1/r instead of announcing it — almost nobody has shown you where it comes from, and it comes from an infinite addition that closes in three lines. It then shows that this number is a CEILING, obtained by cancelling two very real leaks. Then it puts a figure on the gap between that ceiling and reality, and says under what condition the model genuinely bit. What it does not do: settle the direction of causality — do reserves come before loans, or the reverse? That is the subject of the course “Money creation”, which looks at the operation as it is actually recorded on a bank's balance sheet.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Let's follow €200 billion of monetary base, with r = 10%. WAVE 0: the sum arrives as a deposit in bank A, which keeps 10% — €20bn — and can lend the remaining 180. WAVE 1: the borrower spends those €180bn, which come back as a deposit in bank B (assumption 2). B keeps 18 and lends 162. WAVE 2: 162 come back elsewhere, of which 16.2 kept and 145.80 lent. Look at the mechanics: each wave is the previous one MULTIPLIED by (1 − r) = 0.9. The total of deposits created in the system is therefore 200 × [1 + 0.9 + 0.9² + 0.9³ + …] — an endless addition, which we need to know how to close.
Here is the proof that was missing, and it is exactly the trick of the course “The fiscal multiplier” — one technique for both courses, only the ratio changes (c there, 1 − r here). Call S the total for €1 of base: S = 1 + (1−r) + (1−r)² + (1−r)³ + … Multiply S by (1−r): (1−r)·S = (1−r) + (1−r)² + (1−r)³ + … — which is exactly S, minus its first term. So (1−r)·S = S − 1. Rearrange: S − (1−r)·S = 1, that is S·[1 − (1 − r)] = 1, and since 1 − (1 − r) = r, we are left with **S·r = 1, so S = 1/r**. Three lines, and the multiplier is born: m = 1/r. At r = 10%, m = 10 — the €200bn of base supports €2,000bn of deposits, of which 1,800 created by lending. And why does this endless addition give a FINITE total? Because 1 − r is below 1: the waves shrink, like 1 + ½ + ¼ + … which comes to 2 and not infinity.
Now reread assumptions 2 and 3: every euro lent comes back as a deposit, and nothing leaks — no note kept in a pocket, no excess reserve. So 1/r is not a forecast, it is the attainable MAXIMUM, obtained when no bank keeps one euro too many and nobody holds cash. Saying “the multiplier is 100” is as wrong as saying that a car limited to 200 km/h is doing 200. And the model is slow as well as maximal: at r = 10%, it takes 22 waves to reach 90% of the effect (0.9²² ≈ 0.098) — each one needing time to be lent, spent and redeposited.
Let's restore what assumption 3 had cancelled. Write c for the share the public keeps in NOTES, related to deposits, and e for the share banks actually hold in RESERVES, related to deposits too — r being only the legal minimum. Then everything can be recounted: the monetary base is notes + reserves, that is deposits × (c + e); the money supply is notes + deposits, that is deposits × (1 + c). Their ratio gives **m = (1 + c) / (c + e)**. Two decisive remarks. First, this is NOT a rival model: it is an IDENTITY, the decomposition of the multiplier actually observed — and the textbook model is its special case c = 0 and e = r, which does give back (1 + 0)/(0 + r) = 1/r. Second, look at where the leaks sit: in the DENOMINATOR. Every reserve kept in excess, every hoarded note swells it, and so crushes the multiplier. Click each term:
Solving / calculation
Three passes: the toy cascade carried through to its total, the confrontation with the euro area's real figures, then the exact decomposition of the gap. The toy (base 200, r = 10%) is chosen to come out neatly; the euro area figures are the ECB's, end of May 2026.
- 1. The first four waves (base 200, r = 10%)
200 + 180 + 162 + 145.80€687.80bn — and that is only the beginning - 2. The total, through the narrative's sum
m = 1/r = 1/0.10 = 10; deposits = 200 × 10€2,000bn, of which 1,800 created by lending - 3. What the 4 waves represent of the total
687.80 / 2,00034% — the remaining 66% are in the tail - 4. The time it takes
0.9ⁿ ≤ 0.1, that is n ≥ ln 0.1 / ln 0.922 waves to reach 90% of the effect - ⭐ 5. The ceiling, applied to the euro area (r = 1% since 2012)
m = 1 / 0.01100: every euro of base should support €100 of money supply - ⭐ 6. The multiplier ACTUALLY observed
M1 / monetary base = 11,328 / 3,9972.83 — the ceiling is 35 times above it - 7. The two leaks, in figures from the same accounts
notes/deposits: 1,606/9,722 ‖ reserves/deposits: (3,997 − 1,606)/9,722 = 2,391/9,722c = 16.5% and e = 24.6% — that is 25 times the required minimum - 8. The identity of beat 4 gives back the observed value EXACTLY
(1 + 0.165) / (0.165 + 0.246)2.83 ✓ — an identity, not an approximation - 9. Check by absurdity: what if the leaks were nil?
supply = base × 100 = 3,997 × 100€399,700bn of M1 — thirty-five times the M1 actually observed: it really is the LEAKS that make reality
Three things to take away. (1) The 1/r is no convention: it is the sum of a geometric sequence with ratio (1 − r), closed in three lines by the same trick as the fiscal multiplier. (2) It is a CEILING, and it assumes two nil leaks — which they are not: euro area banks keep 24.6% of deposits in reserves when 1% is required, and the public 16.5% in notes. (3) The formula with leaks is not a competing model but the identity that decomposes the observed multiplier, of which the textbook is the limiting case c = 0, e = r. One question remains that this course does not settle: do reserves command loans, or the reverse? The course “Money creation” takes it up on the balance sheet.
ceiling = 1/r · effective = (1 + c)/(c + r + excess)The sliders start in the textbook world: both leaks at zero. First note that the effective multiplier there touches exactly the ceiling 1/r. Then raise excess reserves alone — which is what happened after 2008 — and watch the multiplier collapse without any legal ratio having changed. Finally, look for the setting that reproduces the euro area (r = 1%, notes 16.5%, excess 23.5%): you will land back on 2.8.
Economic interpretation
A false model does not survive sixty years in the textbooks: this one says three true things, and reverses one. Here is how to keep it useful rather than misleading.
A deposit that is lent becomes a deposit elsewhere — the course “Money creation” shows it on the balance sheet, entry by entry: when you pay a seller at another bank, the deposit disappears at one and reappears at the other, and reserves move across. The cascade from bank to bank is therefore no fiction, and that is what makes the model's intuition valuable. What is false is believing that it unfolds all the way to its arithmetic conclusion.
Before 2008, central banks supplied liquidity in small doses: reserves were SCARCE, banks kept almost nothing beyond the minimum, and the leak e stayed close to r. The observed multiplier then hugged its ceiling far more closely, and the reserve ratio was a genuine lever. What changed is not the arithmetic: it is the regime. Since the massive purchases of securities, reserves have been superabundant — banks hold thousands of billions too many — and the ceiling no longer constrains anything.
At 1% in the euro area and 0% in the United States since March 2020, the reserve ratio no longer steers lending. But where it is set high, it steers in earnest: China long raised and lowered it to brake or revive its banks' lending, exactly as the model predicts. So the multiplier is not “false everywhere”: it is toothless where the ratio is symbolic, and operative where it binds.
The model starts from the base and deduces loans from it. The Bank of England wrote the opposite in black and white in 2014, in its quarterly bulletin “Money creation in the modern economy”: banks do not multiply central bank money; it is the loans granted that create deposits, and the central bank then supplies the necessary reserves — at the PRICE it sets, not in the quantity it rations (course “The policy rate and transmission”). That is why this course presents 1/r as a ceiling and not as a machine for producing money: the debate about which way the arrow points belongs to the course “Money creation”.
Limits / critiques
Exercises
The required reserve ratio is 20%. What is the model's money multiplier?
Monetary base = €100bn, reserve ratio = 5%. How much is WAVE 2 of the cascade worth, that is the second deposit arising from the initial loan (in €bn, two decimals)?
Why is 1/r a CEILING and not a forecast?
True or false: since the euro area's reserve ratio is 1%, the money multiplier there is close to 100.
Monetary base = €300bn, reserve ratio = 20%. What MAXIMUM money supply does the model allow (in €bn)?
A multiplier of 4 is observed. What reserve ratio would the textbook model assume (in %)?
The public keeps 20% of deposits in notes (c = 20%) and banks hold 30% of deposits in reserves (e = 30%). What is the EFFECTIVE multiplier (two decimals)?
True or false: the multiplier model assumes reserves exist BEFORE loans, and that is precisely the assumption the course “Money creation” turns round.