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The quantity theory of money
Why printing money ends up, in the long run, raising prices — not output.
M·V = P·Y → π ≈ g(M) − g(Y)Problem / motivation
A central bank doubles the quantity of money. The factories, the workers, the machines have not changed. What becomes of that extra money?
To answer, you first need an image. Follow a €20 note for a year: it pays for a haircut, the barber settles a supplier's bill, the supplier pays a wage… The SAME note finances several purchases in the year. That is the VELOCITY of circulation, written V: the number of times each euro is used, on average, to pay for a year's output. Hence an equation of simple counting, the equation of exchange: M·V = P·Y — the money in circulation (M, the money supply measured by the aggregates of the course “The functions and aggregates of money”), multiplied by the number of times it is used (V), equals the value of what is produced and paid for in the year: NOMINAL GDP, P·Y (the quantities Y at prices P — course “Nominal vs real GDP”).
As it stands, that equation is an IDENTITY: it is true by construction, like “revenue = price × quantity”. It becomes a THEORY of inflation — that of Milton Friedman's formula, “inflation is always and everywhere a monetary phenomenon” — only by betting on its terms: V stable, Y determined by the real economy. The course “The causes of inflation” already had you derive its version in rates, gently; here we keep its promise: doing it EXACTLY, then discussing step by step the conditions that make the theory hold — or break.
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
Take the equation of exchange again: M·V = P·Y. An honest remark — and saying it is a strength, not a weakness: since we MEASURE V by dividing nominal GDP by the money supply (V = P·Y ÷ M), the equation is true by construction. So its content cannot be the equation itself: the whole quantity-theory bet lies in assumption 1 — that ratio V is STABLE, anchored in payment habits. Stable, V passes money through to prices; unstable, it absorbs everything — money created that SLEEPS instead of circulating pushes no price up (you will see 2008 in limit 1). That is what has to be judged.
Inflation is a story of GROWTH RATES, not of levels. A year goes by: M becomes M × (1 + g(M)), V becomes V × (1 + g(V)), P becomes P × (1 + π), Y becomes Y × (1 + g(Y)). The identity holds at both dates, so the coefficients match: (1 + g(M)) × (1 + g(V)) = (1 + π) × (1 + g(Y)). Hence the EXACT version, checkable on a calculator: 1 + π = (1 + g(M)) × (1 + g(V)) ÷ (1 + g(Y)). With 8% money growth, V stable and 2% output: 1.08 ÷ 1.02 = 1.0588 → exact π = 5.88%. (Textbooks often write this step in logarithms; their simplified version is nothing other than the shortcut of the next milestone.)
For small rates, multiplying (1 + small) terms is almost the same as adding the small ones: the shortcut π ≈ g(M) + g(V) − g(Y) — the one the course “The causes of inflation” had you build — gives 8 + 0 − 2 = 6%, against 5.88% exactly. The gap (0.12 point) is the cross term the addition neglects; tiny here, it swells with the rates — the simulator will let you watch the shortcut come adrift in hyperinflation territory. Hence the “≈”: honest, never decorative.
What remains is to make the quantity-theory bet: V stable, so g(V) ≈ 0 — and out comes π ≈ g(M) − g(Y), the formula in the heading. It reads like a sentence: long-run inflation is the excess of money growth over output growth. And it can be turned round: aiming at 2% inflation means making M grow by about 2% + g(Y) — Friedman's famous money growth “rule” (exercise 4 will have you compute it). Click each term:
Solving / calculation
Let us do the two computations side by side — the exact one, then the shortcut — with g(M) = 7%, g(Y) = 2%, V stable (g(V) = 0).
- The exact version: the coefficients match (2nd milestone)
1 + π = 1.07 × 1 ÷ 1.02 - We compute
1.07 ÷ 1.02 = 1.049exact π = 4.9% - The shortcut in sums (3rd milestone)
π ≈ 7 + 0 − 2π ≈ 5% - The gap: the price of the shortcut
5 − 4.90.1 point — negligible here - Economic reading
7 points of money − 2 absorbed by output≈ 5 points into prices
Out of 7 points of money growth, real output absorbs 2; most of the rest ends up in prices — 4.9% exactly, 5% by the shortcut. The two readings tell the same story as long as the rates are small; the simulator takes you to where they diverge: push g(M) towards 20% and watch the shortcut come adrift.
1 + π = (1 + g(M)) × (1 + g(V)) ÷ (1 + g(Y))Set the growth of money, of real GDP and of velocity: inflation is computed EXACTLY and by the shortcut, with the gap between them. Go after the large rates — that is where the shortcut cracks.
Economic interpretation
An identity, a bet on V, a formula: here is what it lights up — and what it demands in order to stay true.
Friedman's formula is about LASTING inflation: no great entrenched inflation has ever occurred without money growth durably exceeding output growth. Hyperinflations are the proof by the extreme — Germany in 1923 (course “The causes of inflation”: the printing press, prices doubling within days), Zimbabwe, Venezuela: when M grows by several thousand per cent and Y by nothing, the exact version of the formula does the rest.
Turned round, the formula becomes a recipe: for 2% inflation with output growing at g(Y), make money grow by about 2% + g(Y). That is the money growth rule advocated by Friedman, and the spirit of the aggregate-targeting policies of the 1980s. Today's central banks have changed tool — they steer the policy rate, not the quantity (course “The policy rate and transmission”) — and limit 2 explains why that choice is not innocuous for the theory.
The whole bridge from M to P runs through the stability of V. Let a shock to confidence push everyone to HOLD ON to their money — hoarding — and V collapses: money created sleeps instead of circulating, and the link g(M) → π breaks. That is exactly the story of the years after 2008 (limit 1): central bank money in abundance, velocity falling — and no immediate inflation. The theory was not “false”: its assumption 1 was suspended.
Faced with a figure, always ask which coat the equation is wearing. M·V = P·Y as an identity is irrefutable — and predicts NOTHING: V absorbs everything, by definition. It predicts only as a theory, with the assumptions in place: V stable, Y real, M steered. That is the grid of the course “The causes of inflation”: the formula measures, the assumptions qualify — and each of the three has its Limits station.
Limits / critiques
Exercises
Money supply +10% a year, real GDP +2.5%, velocity stable. What inflation does the shortcut predict (in %)?
True or false: the equation M·V = P·Y can turn out to be false in some years, when velocity varies a great deal.
g(M) = 20%, g(Y) = 2%, V stable. What is EXACT inflation — (1 + g(M)) ÷ (1 + g(Y)) − 1 (in %)?
2008-2014: the Fed triples central bank money (the monetary base). The strict version of the theory would announce a surge in prices — which did not happen. Why?
g(M) = 5%, g(Y) = 2%, but velocity FALLS by 4% (g(V) = −4%). What inflation does the shortcut π ≈ g(M) + g(V) − g(Y) give (in %)?
Friedman's rule: the central bank aims at 2% inflation and output grows at 1.5% a year (V stable). What growth of M should it aim for (in %)?
Germany 1923: the printing press makes M grow by several thousand per cent a year, while output stagnates. What does the quantity theory predict?
True or false: according to the quantity theory, durably doubling the money supply raises real output Y in the long run.