10.7 The Equation of Exchange

We can relate the money supply to the aggregate economy by using the equation of exchange:

[latex]MV=\text{Nominal GDP}[/latex]

The equation of exchange shows that the money supply [latex]M[/latex] times its velocity [latex]V[/latex] equals nominal GDP. Velocity is the number of times the money supply is spent to obtain the goods and services that makeup GDP during a particular time period.

To see that nominal GDP is the price level multiplied by real GDP, recall from an earlier chapter that the GDP deflator equals nominal GDP divided by real GDP: We denote GDP deflator as [latex]P[/latex] here:

[latex]\begin{align*}P=\frac{\text{Nominal GDP}}{\text{Real GDP}}\end{align*}[/latex]

Therefore,

[latex]\text{Nominal GDP}=P\times\text{Real GDP}[/latex]

Letting [latex]Y[/latex] equal real GDP, we can rewrite the equation of exchange as,

[latex]MV=PY[/latex]

We shall use the equation of exchange to see how it represents spending in a hypothetical economy that consists of 50 people, each of whom has a car. Each person has $10 in cash and no other money. The money supply of this economy is thus $500. Now, suppose that the sole economic activity in this economy is car washing. Each person in the economy washes one other person’s car once a month, and a car wash costs $10. In one month, then, 50 car washes are produced at a price of $10 each. During that month, the money supply is spent once.

Applying the equation of exchange to this economy, we have a money supply [latex]M[/latex] of $500 and a velocity [latex]V[/latex] of 1. Because the only good or service produced is car washing, we can measure real GDP as the number of car washes. Thus, [latex]Y[/latex] equals 50 car washes. The price level [latex]P[/latex] is the price of a car wash: $10. The equation of exchange for a period of 1 month is,

[latex]\$500\times 1= \$10\times 50[/latex]

Suppose that in the second month, everyone rewashes someone else’s car. Over the full two-month period, the money supply has been spent twice—the velocity over a period of two months is 2. The total output in the economy is $1,000 — 100 car washes produced over two months at a price of $10 each. Inserting these values into the equation of exchange, we have

[latex]\$500\times 2=\$10\times 100[/latex]

Suppose this process continues for one more month. For the three-month period, the money supply of $500 has been spent three times, for a velocity of 3. We have,

[latex]\$500\times 3=\$10\times 150[/latex]

The essential thing to note about the equation of exchange is that it always holds. That should come as no surprise. The left side, [latex]MV[/latex], gives the money supply times the number of times that money is spent on goods and services during a period. It thus measures total spending. The right side is nominal GDP. But that is also a measure of total spending on goods and services. Nominal GDP is the value of all final goods and services produced during a particular period. Those goods and services are either sold or added to inventory. If they are sold, then they must be part of total spending. If they are added to inventory, some firms must have either purchased them or paid for their production; thus, they represent a portion of total spending. In effect, the equation of exchange says simply that total spending on goods and services, measured as [latex]MV[/latex], equals total spending on goods and services, measured as [latex]PY[/latex] (or nominal GDP). The equation of exchange is thus an identity, a mathematical expression that is true by definition.

To apply the equation of exchange to a real economy, we need measures of each of the variables in it. Three of these variables are readily available. Suppose we have the following data for an economy in a certain year:

[latex]\begin{align*}M&=\$7,635.4\;\text{billion}\\[2ex]P&=1.22\\[2ex]Y&=11,727.4\;\text{billion}\end{align*}[/latex]

To solve for the velocity of money, [latex]V[/latex], we divide both sides of the exchange equation, [latex]MV=PY[/latex] by [latex]M[/latex]:

[latex]\begin{align*}V=\frac{PY}{M}\end{align*}[/latex]

Using the data to compute velocity, we find that [latex]V[/latex] was equal to 1.87. A velocity of 1.87 means that the money supply was spent 1.87 times in the purchase of goods and services in a particular year.

10.7.1 Money, Nominal GDP, and Price-Level Changes

We can rewrite the equation of exchange, [latex]MV=PY[/latex], in terms of percentage rates of change.

[latex]\%\Delta M+\%\Delta V\cong\%\Delta P+\%\Delta Y[/latex]

Assume that velocity is constant, expressed as [latex]V[/latex], then the above equation becomes,

[latex]\%\Delta M\cong\%\Delta P+\%\Delta Y_P[/latex]

Rewriting this as,

[latex]\%\Delta P\cong\%\Delta M-\%\Delta Y_P[/latex]

That is,

[latex]\text{The rate of inflation}=\text{the growth in money supply}–\text{growth rate of real GDP}[/latex]

The equation above helps us make the following predictions:

1. If money supply changes faster than that of GDP, the economy experiences inflation
2. If the growth rate of GDP changes faster than the money supply, the economy experiences deflation
3. When both money supply and GDP change in the same proportion, inflation remains unchanged.

Several recent studies that looked at all the countries on which they could get data on inflation and money growth over long periods found a very high correlation between the money supply growth rates and the price level for countries with high inflation rates. Still, the relationship was much weaker for countries with inflation rates of less than 10%. These findings support the quantity theory of money, which holds that in the long run, the price level moves in proportion to changes in the money supply, at least for high-inflation countries.