Monetary Policy and the Equation of Exchange

The Equation of Exchange

The equation of exchange shows that the money supply M times its velocity V equals nominal GDP. Velocity is the number of times the money supply is spent to obtain the goods and services that make up 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 implicit price deflator P equals nominal GDP divided by real GDP:

Equation 11.2

Multiplying both sides by real GDP, we have

Letting Y equal real GDP, we can rewrite the equation of exchange as

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 the price of a car wash is $10. In one month, then, a total of 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 M of $500 and a velocity V of 1. Because the only good or service produced is car washing, we can measure real GDP as the number of car washes. Thus Y equals 50 car washes. The price level P is the price of a car wash: $10. The equation of exchange for a period of 1 month is

$500×1=$10×50

Now suppose that in the second month everyone washes someone else’s car again. 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 have been produced over a two-month period at a price of $10 each. Inserting these values into the equation of exchange, we have

$500×2=$10×100

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

$500×3=$10×150

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. The Department of Commerce reports the price level (that is, the implicit price deflator) and real GDP. The Federal Reserve Board reports M2, a measure of the money supply. For the second quarter of 2008, the values of these variables at an annual rate were

To solve for the velocity of money, V, we divide both sides of Equation 11.4 by M:

Using the data for the second quarter of 2008 to compute velocity, we find that V is 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 the second quarter of 2008.

Money, Nominal GDP, and Price-Level Changes

Assume for the moment that velocity is constant, expressed as . Our equation of exchange is now written as

Equation 11.6

M=PY

A constant value for velocity would have two important implications:

1. 
   Nominal GDP could change only if there were a change in the money supply. Other kinds of changes, such as a change in government purchases or a change in investment, could have no effect on nominal GDP.
   
2. 
   A change in the money supply would always change nominal GDP, and by an equal percentage.
   

In short, if velocity were constant, a course in macroeconomics would be quite simple. The quantity of money would determine nominal GDP; nothing else would matter.

Indeed, when we look at the behavior of economies over long periods of time, the prediction that the quantity of money determines nominal output holds rather well.Figure 11.8 "Inflation, M2 Growth, and GDP Growth" compares long-term averages in the growth rates of M2 and nominal GNP in the United States for more than a century. The lines representing the two variables do seem to move together most of the time, suggesting that velocity is constant when viewed over the long run.

Figure 11.8 Inflation, M2 Growth, and GDP Growth



The chart shows the behavior of price-level changes, the growth of M2, and the growth of nominal GDP using 10-year moving averages. Viewed in this light, the relationship between money growth and nominal GDP seems quite strong.

Source: William G. Dewald, “Historical U.S. Money Growth, Inflation, and Inflation Credibility,” Federal Reserve Bank of St. Louis Review 80:6 (November/December 1998): 13-23.

Moreover, price-level changes also follow the same pattern that changes in M2 and nominal GNP do. Why is this?

We can rewrite the equation of exchange, M = PY, in terms of percentage rates of change. When two productes, such as M and PY, are equal, and the variables themselves are changing, then the sums of the percentage rates of change are approximately equal:

Equation 11.7

%ΔM+%ΔV≅%ΔP+%ΔY

The Greek letter Δ (delta) means “change in.” Assume that velocity is constant in the long run, so that %ΔV = 0. We also assume that real GDP moves to its potential level,Y_P, in the long run. With these assumptions, we can rewrite Equation 11.7 as follows:

Equation 11.8

%ΔM≅%ΔP%ΔY_P

Subtracting %ΔY_P from both sides of Equation 11.8, we have the following:

Equation 11.9

%ΔM-%ΔY_P≅%ΔP

Equation 11.9 has enormously important implications for monetary policy. It tells us that, in the long run, the rate of inflation, %ΔP, equals the difference between the rate of money growth and the rate of increase in potential output, %ΔY_P, given our assumption of constant velocity. Because potential output is likely to rise by at most a few percentage points per year, the rate of money growth will be close to the rate of inflation in the long run.

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 growth rates of the money supply and of the price level for countries with high inflation rates, but the relationship was much weaker for countries with inflation rates of less than 10%. ^[1] These findings support the quantity theory of money, which holds that in the long run the price level moves in proportion with changes in the money supply, at least for high-inflation countries.

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