Macroeconomics For Dummies®, uk edition Published by: John Wiley & Sons, Ltd

Converting the quantity equation from levels to growth rates

Here’s the full derivation of how to convert the quantity equation from levels to growth rates.

You need basic calculus to follow it.

The quantity equation links the money supply (M), the velocity of circulation (V), the price level (P) and real output (Y):

The subscript t is added to make clear that these refer to the values of the variables at time t.

Take the natural logarithm of both sides:

Now, differentiate both sides with respect to time (t):

Using the chain rule:

But these are just the growth rates of the respective variables:

Think about it this way: the price level tells you the rate at which you can convert money into goods. If the price level is high, you need a lot of money in order to buy goods and vice versa:


If the amount of money in the economy (M) increases and the amount of goods (Y) stays unchanged, goods become relatively scarce and money is relatively abundant. This situation causes the prices of things to increase, because the rate at which you can turn money into goods worsens.


If the amount of goods increases but the amount of money is unchanged, money becomes relatively scarce and goods are relatively abundant. This causes the price of things to fall because you’re able to turn money into goods at a better rate.


Identifying the quantity equation as always being true

The quantity equation is a bit of a misnomer; really it should be called the quantity identity.

Here’s the difference between an equation and an identity:

An equation: Is true for some values of the variables (for example, x + 5 = 7 is an equation because it’s only true for x = 2).

An identity: Is always true whatever values the variables take (for example, the relationship between the temperature in Fahrenheit (F) and in Celsius (C): F = (9/5)C + 32 is always true by definition).

The quantity equation MV = PY is an identity because it’s true by definition – the left-hand side is total expenditure in the economy, and the right-hand side is the value of everything produced. These two things must always be equal no matter what values the variables take.

The relationship we derived from the quantity equation between

inflation, growth in the money supply and output growth ( ) must, because it’s derived from an identity, also always be true (assuming constant velocity).