The Equation of Exchange and Neutrality

In micro, money barely appeared; barter ran the markets. In macro, money moves to the center.

Double the money and prices double too. The pile of real goods does not grow. As the equation below says, money only rescales the ruler. Inflation is money growth minus real growth.

For example, ten 🧑 classmates share ten 🍬 candies. If the 💰 paper money with candy pictures doubles, each candy just costs twice as much.

MV=PYMV = PY
π=gM+gV−gY\pi = g_M + g_V - g_Y

Part of these learning paths

Examples & Exercises

The central bank grows money at 10%10\% a year, real output grows at 3%3\% a year, and velocity is constant.

What is inflation, and where is the price level (starting at 100100) after 1010 years?

Money growth 10%10\%, real growth 3%3\%, velocity constant.
Show proof

  1. Inflation is what is left of money growth after real growth absorbs its share, as in the formula below. Output growing at 3%3\% soaks up that much money growth without pushing prices up.

    π=gM+gV−gY=10+0−3=7%\pi = g_M + g_V - g_Y = 10 + 0 - 3 = 7\%

  2. The price path multiplies by 1.071.07 each year: P(10)=100⋅1.0710≈196.7P(10) = 100 \cdot 1.07^{10} \approx 196.7. Seven percent a year sounds small, but over a decade prices nearly double.

Overnight, the money supply exactly doubles (M=10000→20000M = 10000 \rightarrow 20000). Velocity (V=1V = 1) and real output (Y=100Y = 100) stay put in the long run.

What happens to the price level and to real money balances?

Flat paths at zero growth — the doubling experiment is read off these levels.
Show proof

  1. Solving the exchange equation for prices gives the formula below.

    Before, P=10000⋅1/100=100P = 10000 \cdot 1 / 100 = 100; with twice the money, P=20000⋅1/100=200P = 20000 \cdot 1 / 100 = 200 — the price level exactly doubles.

    With every other string held fixed, prices alone restore the equation's balance.

    P=MV/YP = MV / Y

  2. Real money balances are M/PM / P: 10000/100=10010000 / 100 = 100 before, 20000/200=10020000 / 200 = 100 after.

    The purchasing power people actually hold is unchanged.

    Real output, Y=100Y = 100, never moved either.

  3. Double the money and the real world is yesterday's world with the nominal ruler rescaled. In the long run money is neutral — though this picture is the one after prices have finished adjusting.

4. Money and Prices