### Velocity

While the relationship between money supply, money demand, the price level, and the value of money presented above is accurate, it is a bit simplistic. In the real world economy, these factors are not connected as neatly as the quantity theory of money and the basic money market diagram present. Rather, a number of variables mediate the effects of changes in the money supply and money demand on the value of money and the price level.

The most important variable that mediates the effects of changes in the money supply is the velocity of money. Imagine that you purchase a hamburger. The waiter then takes the money that you spent and uses it to pay for his dry cleaning. The dry cleaner then takes that money and pays to have his car washed. This process continues until the bill is eventually taken out of circulation. In many cases, bills are not removed from circulation until many decades of service. In the end, a single bill will have facilitated many times its face value in purchases.

Velocity of money is defined simply as the rate at which money changes hands. If velocity is high, money is changing hands quickly, and a relatively small money supply can fund a relatively large amount of purchases. On the other hand, if velocity is low, then money is changing hands slowly, and it takes a much larger money supply to fund the same number of purchases.

As you might expect, the velocity of money is not constant. Instead, velocity changes as consumers' preferences change. It also changes as the value of money and the price level change. If the value of money is low, then the price level is high, and a larger number of bills must be used to fund purchases. Given a constant money supply, the velocity of money must increase to fund all of these purchases. Similarly, when the money supply shifts due to Fed policy, velocity can change. This change makes the value of money and the price level remain constant.

The relationship between velocity, the money supply, the price level, and output is represented by the equation M * V = P * Y where M is the money supply, V is the velocity, P is the price level, and Y is the quantity of output. P * Y, the price level multiplied by the quantity of output, gives the nominal GDP. This equation can thus be rearranged as V = (nominal GDP) / M. Conceptually, this equation means that for a given level of nominal GDP, a smaller money supply will result in money needing to change hands more quickly to facilitate the total purchases, which causes increased velocity.

The equation for the velocity of money, while useful in its original form, can be converted to a percentage change formula for easier calculations. In this case, the equation becomes (percent change in the money supply) + (percent change in velocity) = (percent change in the price level) + (percent change in output). The percentage change formula aids calculations that involve this equation by ensuring that all variables are in common units.