# Famous formulas in economics (II): The equation of exchange

J Brauer | © Stone Garden Economics

In October 2011, I extolled the virtues of Mr. Lerner’s Index, a formula that describes a firm’s market power, that is, the degree to which it can mark-up its prices relative to its costs. This stimulates me to ask what other famous formulas there are in economics. Here is one from the realm of monetary theory. Called the equation of exchange, one of its mathematical variations states that the (year-over-year) percentage change in the amount of money, %M, sloshing around in an economy plus the percentage change in money’s turn-over rate or velocity, %V, must equal the percentage change in prices (inflation, %P) plus the percentage change in the inflation-adjusted sum of all goods and services produced (%Y).

That’s a mouthful. The more readable short-hand notation is %M+%V=%P+%Y.

Before going on, it is probably a good idea to explain why this equation must be true. Put in absolute rather than percentage terms, the dollar-size of the U.S. economy in 2011 was about US\$15 trillion. But there was only about US\$9.3 trillion worth of money available, in the form of cash and of holdings in people’s checking and savings accounts. But cash and money in checking accounts, especially, is used more often than just once a year. That is, each dollar “turns over” several times, say from you to the grocery store (an expenditure) and from the store to its employee (an income) who then pays his cable-TV bill (an expenditure), and so on. For the United States, in 2011, the turn-over rate of money, or its velocity, was about 1.63, so that the average dollar was used 1.63 times to make purchases of goods and services. (Cash turns over more often then do checking account balances and these, in turn, turn over more often than saving account balances.)

The significance of the equation of exchange stems from the contention that the year-over-year percentage change of the turn-over rate, %V, is zero. The reason for this contention is that each society sports certain ingrained financial customs, such as the steady mixture of bi-weekly and monthly paychecks. (In the U.S., salaried workers tend to be paid monthly, but waged employees are paid bi-weekly. In Europe, almost everyone is paid monthly.) Similarly, expenditure patterns are fairly constant, as we see in the payment of monthly cable-TV and utility bills for example. Thus, one might expect that a specific economy’s velocity is constant from year to year. If so, then the year-over-year percentage change must be zero. (The percentage change from 1.63 in one year to 1.63 in the next would be zero: %V=0.)

If this were true then it must also be true that any growth in the money supply (%M) on the left-hand side of the %M+%V=%P+%Y equation must show up, on the right-hand side, either as inflation (%P), or as economic growth (%Y), or as a combination of the two. Thus, if money grows, say, by 6.9 percent (as it did in 2011), then one could have 2.1 percent inflation and yet 4.8 percent economic growth as well.

Figure 1: Money and velocity (year-over-year percentage changes), 1960-2011.

As it turns out, however, economic growth was only 1.7 percent. So we have 6.9%+? = 2.1%+1.7%, and it follows that the missing item—velocity—was not zero at all. Instead, it was -3.1 percent, or else the equation would not work. This is not an isolated aberration. In fact, the 52-year record depicted in Figure 1 shows that velocity is rather volatile. It also shows that money (measured here as M2; there are different measures of money) and its velocity tend to move countercyclically: When money grows, velocity shrinks, and vice versa.

Put differently, when the Federal Reserve Bank steps on the economic gas pedal in an attempt to speed up the economy, people react by slowing down the rate at which the money is circulated within the economy, and vice versa. The effect of the 2008/9 recession can clearly be seen. It induced folks to spend a whole lot less money, drastically slowing down the turn-over rate of cash and checking and savings accounts, and even though the Federal Reserve increased the money supply by a hefty 7.6 percent in 2009, this was not enough to compensate for the 10.1 percent drop in velocity. It follows that the left-hand side of our equation equals +7.6%-10.1% = -2.5%. And since inflation rose by about one percent (+1%), economic growth must have dropped by 3.5 percent (-3.5%) from 2008 to 2009, as indeed it did. (To check: +7.6%-10.1% = +1.0%-3.5%.)

Of course, the governors of the Federal Reserve Bank are well aware that stepping on the monetary gas pedal does not do much good when the economic engine is leaking. In the end, it’s the engine itself and not the amount of gas that counts. And to fix the still sputtering engine is an entirely different task, and one that the Federal Reserve is not equipped to fix.

J Brauer is Professor of Economics, James M. Hull College of Business, Augusta State University, Augusta, Georgia, USA.