Velocity of money calculator

Solve the classroom quantity theory identity for velocity, for nominal GDP, or for money. The identity is MV equals PY. The same statement is MV equals PQ when Q stands for real output. Every step stays visible. This is classroom arithmetic, not a forecast of inflation, and not a recommendation about how fast money should grow. The arithmetic runs entirely in your browser.

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Mode

Solve for velocity. Nominal GDP equals the price level times real output. Velocity equals that nominal GDP divided by money.

Money stock in classroom units you choose. The EXAMPLE default is 100. This is not a live M1 or M2 figure.

A price level, or a GDP deflator index, in the units you choose. The EXAMPLE default is 1. If you type an index that equals 100 in a base year, keep output and money in units that match that choice. This page does not look up a deflator. For the nominal and real split on its own, use the GDP deflator calculator.

Real output in classroom units you choose. The EXAMPLE default is 500. With a price level of 1, nominal GDP is 500. With money of 100, velocity is 5.

Velocity

Identity

How to read this

Arithmetic

How to read this. The result is classroom arithmetic from the numbers you typed. Velocity is nominal spending per unit of money in this identity. A higher velocity means the same money stock lines up with a larger nominal GDP. A lower velocity means the opposite. Velocity is not constant. In the short run, a change in money need not show up one for one in the price level, because velocity and real output can move too. The definition of money changes the velocity you would compute from official data. M1 is narrower than M2. UK broad money is not the same object as US M2. For the aggregates, read money supply M1 and M2. For prices, read the inflation topic. For nominal GDP, read the GDP topic.
EXAMPLE

Solve for velocity uses money of 100, a price level of 1, and real output of 500. Nominal GDP is 500, and velocity is 5. Solve for nominal GDP uses money of 100 and velocity of 5, so nominal GDP is 500. Solve for money uses velocity of 5 and nominal GDP of 500, so money is 100. Those figures are classroom arithmetic. They are not a measured velocity for the UK or the United States, and they are not an official money stock.

Educational only

Not investment advice and not a prediction of inflation. Nothing here says a central bank should expand the money stock, shrink it, or hold it fixed. A velocity of 5 is not a forecast. The calculator does not fetch live data and does not invent an official velocity series.

The classroom formula

This page uses three classroom sketches of one identity. You pick the mode.

Velocity ≈ (P × Y) / M
Nominal GDP ≈ M × V
Money ≈ (P × Y) / V

P is a price level or a GDP deflator index. Y is real output. Their product is nominal GDP. M is a money stock. The identity defines velocity as nominal GDP divided by money. The quantity theory adds a further claim, often met in textbooks: if velocity is stable and real output is set by real forces, then money growth and inflation move together over long horizons. This page keeps the identity. It does not assume velocity is stable, and it does not estimate that long run claim from data.

Rearranged in growth rates, money growth plus velocity growth approximately lines up with inflation plus real output growth. That growth reading is only a guide. This calculator solves the levels. For a separate sketch of how a fiscal impulse can move output, see the fiscal multiplier calculator. For a price path over years, see the inflation adjuster. For a separate sketch of a nominal interest rate and inflation, see the real interest rate calculator. Wider context sits on the monetary and fiscal policy topic.

Where to find the inputs

Type your own numbers. This page does not download official series. For US money, read the Federal Reserve H.6 release and the FRED series M1SL and M2SL, then type a figure you choose. For UK money, read Bank of England money and credit releases. FRED also publishes velocity series such as M1V and M2V. If you use one, read the notes on which money aggregate it uses, then type the number yourself. This page does not look up those series and does not treat them as defaults. The IMF Institute teaches readers to handle macroeconomic statistics with the definitions attached, and IMF Data is a public entry point for many series.

UK and US framing

Do not paste a US M2 velocity onto UK broad money and call the result a fact. The aggregates differ. A definition change can move a narrow aggregate on its own. Around May 2020, the United States reclassified many savings deposits into M1, so M1 jumped for statistical reasons. Velocity computed from M1 and velocity computed from M2 answer different questions. Short run links from money growth to inflation are loose. Credit conditions, supply shocks, and expectations can move as well. Large asset purchases are a separate topic. Read quantitative easing for that mechanism. This calculator does not model bank reserves or a central bank balance sheet.

For the aggregates themselves, read money supply M1 and M2. For prices, read the inflation topic. For the P times Y idea, read the GDP topic.

Sources

All calculators · GDP deflator calculator · Money supply M1 and M2 · Quantitative easing · Inflation topic · GDP topic · Monetary and fiscal policy · Inflation adjuster · Real interest rate calculator · Fiscal multiplier calculator · FAQ