## Macro: Equation of the exchange; Money Velocity (compilation)

**To do: clean this up and properly cite sources**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. **

P = Nominal GDP / Real GDP

Multiplying both sides by real GDP, we have

**Equation 11.3. **

Nominal GDP = P × real GDP

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

**Equation 11.4. **

*MV = PY*

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

The essential thing to note about the equation of exchange is that it always holds. That should come as no surprise. The left side, *MV*, 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 a measure of total spending on goods and services as well. 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, then some firm must have either purchased them or paid for their production; they thus represent a portion of total spending. In effect, the equation of exchange says simply that total spending on goods and services, measured as *MV*, equals total spending on goods and services, measured as *PY* (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. 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

*M* = $7,635.4 billion

*P* = 1.22

*Y* = 11,727.4 billion

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

**Equation 11.5. **

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. **

A constant value for velocity would have two important implications:

- 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. - 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.