An identity first, a theory second
Written in levels, MV = PY cannot be wrong. It says that total spending counted from the money side (the money stock M times how often each dollar turns over, V) equals total spending counted from the goods side (the price level P times real output Y) — both are just nominal GDP. Because velocity is measured residually as nominal GDP divided by the money supply, V is by construction whatever number balances the equation. The identity becomes the quantity theory only when you add assumptions: that velocity is stable, and that real output is pinned down by capacity rather than by money. Grant those two, and any growth in M beyond the growth in Y has nowhere to go but P.
For inflation intuition, the useful version is the growth-rate form. Taking growth rates of both sides turns multiplication into (roughly) addition: money growth plus velocity change on one side, inflation plus output growth on the other. Rearranged for inflation, it says prices rise by whatever nominal spending growth is left over after real output has absorbed its share.
The equation of exchange
M × V = P × Y
%ΔM + %ΔV ≈ %ΔP + %ΔY
implied inflation: %ΔP ≈ %ΔM + %ΔV − %ΔY
where M is the money supply, V its velocity,P the price level, and Y real output. This calculator takes P as an index (base year = 100), so in levels mode it solves M × V = (P ÷ 100) × Y — that way (P ÷ 100) × Y is nominal GDP in the same dollars as M. The growth form is a linear approximation; the exact version is (1 + m)(1 + v) ÷ (1 + y) − 1, and the calculator shows both.
Worked example
Suppose the money supply grows 7.00% this year, velocity holds steady, and real output grows 3.00%. What inflation does the quantity theory imply?
| Step | Amount |
|---|---|
| Money supply growth (%ΔM)the money stock grows this much over the year | 7.00% |
| + Velocity change (%ΔV)the classic stable-velocity assumption — each dollar turns over as often as before | 0.00% |
| − Real output growth (%ΔY)extra goods and services absorb some of the new money | 3.00% |
| = Implied inflation (approximation)%ΔM + %ΔV − %ΔY | 4.00% |
| Exact, compounded(1.07 × 1.00) ÷ 1.03 − 1 — the linear form drops small cross-terms and runs 0.1165 pp hot | 3.88% |
Computed with this calculator's default settings (growth mode) — open the tool above and you'll see the same numbers, then swap in your own growth rates.
The honest caveat: velocity is not stable
Irving Fisher, who formalized the equation in The Purchasing Power of Money (1911), treated velocity as set by slow-moving payment habits and institutions, and Milton Friedman's 1956 restatement rebuilt the theory on the claim that money demand — and hence velocity — is a stable function of a few variables. The data have not been kind to the strong version of that claim. US M2 velocity drifted for decades, then fell sharply after the 2008 financial crisis and collapsed outright in 2020, when households parked stimulus payments in bank accounts while spending shut down (see measured velocity in the FRED series linked below). When V can swing like that, money growth alone does not map one-for-one into inflation: the same %ΔM can coexist with deflationary pressure if velocity falls, or amplify inflation if it rebounds. That is why this calculator asks for %ΔV explicitly instead of assuming it away — setting it to zero is a modeling choice, and often the wrong one over short horizons.
The implied inflation here is a supply-side story about prices — compare it with what inflation does to your own purchasing power in theinflation calculator, see how the banking system turns base money into the broader M used here with themoney multiplier calculator, or measure the price level P the way national accounts actually do with theGDP deflator calculator.
Frequently asked questions
What is the quantity theory of money?
The quantity theory of money links the money supply to the price level through the equation of exchange, MV = PY: the money stock times how often each dollar is spent equals the price level times real output. Written in levels it is an accounting identity — true by definition. It becomes a theory when you add two assumptions: velocity is roughly stable, and real output is set by capacity rather than by money. Under those assumptions, growth in the money supply beyond output growth must show up as inflation, which is the intuition behind Milton Friedman’s line that inflation is "always and everywhere a monetary phenomenon."
What is velocity, and how is it measured?
Velocity is the number of times an average dollar is spent on final goods and services in a year. Nobody tracks individual dollars — velocity is measured residually, as nominal GDP divided by the money supply. If nominal GDP is $24,200B and M2 is $21,000B, velocity is about 1.15. That residual construction is exactly why MV = PY always holds in the data: V is defined as whatever number makes the equation balance. The interesting empirical question is not whether the identity holds but whether V moves predictably.
Does printing money always cause inflation?
Not one-for-one, and not on a fixed timetable. The growth-rate form says inflation ≈ money growth + velocity change − output growth, so two escape valves sit between the printing press and prices. If velocity falls — people and banks hold the new money rather than spend it — money growth is absorbed without price pressure, which is what happened after 2008. And if the economy has slack, some new spending shows up as more real output rather than higher prices. Over long horizons and in extreme cases (hyperinflations), the link is strong; over a year or two it is loose.
What is the difference between the Fisher and Cambridge versions?
Irving Fisher’s 1911 formulation is the transactions version used here: MV = PT (with output Y standing in for transactions T), focused on how fast money circulates. The Cambridge economists (Marshall, Pigou) flipped the same relationship into a demand for money: M = kPY, where k is the fraction of nominal income people choose to hold as cash. Algebraically k is just 1/V, so the two are equivalent — but the Cambridge form asks why people hold money, which led to Keynes’s liquidity preference and Friedman’s 1956 restatement of the quantity theory as a theory of money demand.
Why didn’t the huge money growth of 2020 raise prices one-for-one?
US M2 grew roughly 25% in 2020, yet CPI inflation that year stayed near 1.4% — because velocity collapsed at the same time. Households banked stimulus checks, spending on services shut down, and each dollar turned over far less often, so %ΔM + %ΔV was much smaller than %ΔM alone. Inflation did arrive in 2021–2022 as reopening spending, supply constraints, and fiscal transfers collided — but the episode is a textbook illustration that the quantity theory needs its stable-velocity assumption, and that assumption fails exactly when behavior shifts sharply.
Sources
- OpenStax, Principles of Economics 3e — Pitfalls for Monetary Policy (the quantity equation and unpredictable velocity)
- Federal Reserve Bank of St. Louis (FRED) — Velocity of M2 Money Stock (see measured velocity)
The official figures this page quotes are drawn from the primary sources above — check them (or a qualified professional) before relying on a result.
Disclaimer: This calculator is foreducation and illustration only. The equation of exchange is an accounting identity, and its inflation prediction rests on assumptions — stable velocity, output at capacity — that regularly fail over short horizons. Results are a textbook calculation, not an inflation forecast, and nothing here is investment, financial, or policy advice.