Christopher W. Foster
Does MV really Equal PT?
Introduction:
Irving Fisher first developed the idea that MV=PT (the quantity theory of money); M being the money supply, V being the Velocity of money, P being price, and T being quantity of output. Fisher used this model to explain that an increase in the money supply would lead to an increase in the level of prices. To do this he assumed that you could hold both the velocity of money and the quantity of output constant. Thus, the equation is transformed:
MV=PT
%∆M+%∆V=%∆P+%∆T
Holding %∆V and %∆T constant, then,
%∆M=%∆P
With this equation in mind Milton Friedman developed the idea that if one can control the money supply one could also control prices. It sounds simple given the above equation, but Freidman was a realist and he knew that V and T were not held constant in the real world. In his book, A Monetary History of the United States from 1867 to 1960, Freidman, and Anna Schwartz, did their best to track changes in the money supply and GDP (Gross Domestic Product), and inflation, as well as other interesting measures. Repeatedly Freidman explained in the book that changes that were happening to inflation were caused by different monetary changes. He always held to the idea that MV=PT and so every time that the %∆M did not equal the %∆P+%∆T, he contended that Velocity had changed to account for the difference. Throughout the book Freidman has charts of the velocity of money and its changes over time, but at no place does he explain how the velocity of money is measured, and since I have been unable to discover anyone else that has managed to measure velocity I began to wonder if MV=PT was really true. Understandably most resources say that you measure velocity:
V=PT/M
But that analysis is not satisfactory for the following reason. If I measure the money supply and GDP (GDP=PT), I could calculate Velocity via the above formula. But if I then tried to use the calculated Velocity and the measured GDP, and money supply to prove that MV=PT, I would have a circular argument, and no credibility.
Research Question: Does MV=PT?
Purpose:
The purpose of this paper is to test the theory that MV=PT.
Definitions:
M—The money supply, denoted by M1 or M2
V—Velocity of money, denoted by V
P—Price level.
T—Quantity of goods
PT—Measured by GDP
Expectations:
I would expect that the formula MV=PT is in fact true. However, because of the assumption that velocity is held constant I would also expect that the data collected would show variances that could be explained with an accurate measure of velocity.
Resources:
Quarterly GDP and Monthly M1, M2, and CPI data were all gathered from the St. Louis Federal Reserve. All of the data was seasonally adjusted.
Analysis:
The first step in testing the theory that MV=PT is to gather the data necessary to measure M, V, P, and T. The measures I used for M and PT were collected from the St. Louis Federal Reserve database. The only problem is that GDP was measured quarterly and M1and M2 were all measured monthly. So I converted the monthly measures into quarterly ones by adding every 3 months in a quarter together and then dividing by 3. I then took the data ranges of all 3 measures and selected data from the first quarter of 1959 to the second quarter of 2006. I chose 1959 because Milton Freidman’s study ended in 1960.
Measuring Velocity posed a much bigger problem. Currently Velocity is calculated by V=PT/M. Velocity is not something that is simply measured in the economy as are CPI, GDP, and the money supply. So I calculated Velocity using the current GDP and M1/M2 data that I had.
However, using this method of calculating Velocity would be to assume that the equation (MV=PT) that I am trying to test is in fact already true. So in order to better test the equation I cannot simply calculate Velocity. I next thought that if I could find a data set that tracked the dollar value of everything purchased in the US in a particular year, then I could divide that dollar amount by the money supply and therefore measure velocity. However, such data is currently unavailable, and would probably only be accurate for the last 15 years when computers have tracked most purchases. Also, unreported cash transactions could greatly skew the numbers. That leaves only calculating velocity as an option.
The next step was deciding how to compare the data. I cannot simply measure every quarter of GDP to see if it equals MV because I have no measured values for V. Instead I decided on comparing the percentage changes in the money supply and velocity to the percentage changes in GDP. Thus I took the formula:
MV=PT
And converted it into
M1 x V=GDP, or M2 x V=GDP
%∆M1+%∆V=%∆GDP, or %∆M2+%∆V=%∆GDP
So my next task was to convert the data that I had into %∆ data. Doing so gave me the following data sets from 1959 to 2006: %∆M1, %∆M2, %∆GDP. From my GDP and M1/M2 data sets I calculated the %∆V for M1 and M2. I then calculated the following correlations (I threw in measures of CPI for interesting, but unrelated, comparisons):
CORRELATIONS:
GDP&M1=0.9804254
GDP&M2=0.9959567
GDP&CPI=0.9812048
M1&CPI=0.9876655
M2&CPI=0.9791334
%∆(GDP&CPI)=0.3532553
%∆(M1&CPI)=0.0945259
%∆(M2&CPI)=0.0575103
%∆M1+%∆V(M1)&%∆GDP=1.0
%∆M2+%∆V(M2)&%∆GDP=1.0
Testing the correlation between %∆M1+%∆V or %∆M2+%∆V and %∆GDP came out to be a perfect 1.0 as was expected after having used the formula to calculate velocity. Therefore that correlation cannot prove that MV=PT is in fact true.
At this point I decided that I needed to remove velocity from the equation and test just the money supply against GDP. If my data showed that there was a reasonable correlation between the %∆M(1 or 2) and %∆GDP, then that would be acceptable as at least evidence that MV=PT might be true. So I held velocity constant and ran the correlations:
%∆M1&%∆GDP=0.2096311
%∆M2&%∆GDP=0.2843379
Those 2 correlations were the most important for this study. While the correlations appear to be low, the important part is that they are positive. A correlation of 0 or negative between either measures of the %∆ money supply and %∆GDP would have been enough to reject the formula MV=PT because the probability that velocity would change so much as to cause the correlation between M and %∆PT to be 0 or negative is very low. Because the correlations show a moderate positive relation they seem to suggest that if we could accurately measure Velocity with out simply calculating it, then the formula MV=PT would hold true. Small variances would then be due to accuracy of the measures. The regression plots for comparing M1 and M2 to %∆GDP are as follows:
Conclusion:
We can see from these plots that except for some outliers, the majority of the points are located in a positive correlation. While this evidence is not enough to fully prove that MV=PT, it is enough to say that the probability of MV=PT being true is pretty good. Therefore I find it acceptable to use the formula MV=PT in making calculations.
Further Study:
If an accurate measure of the Velocity of money is found, then the theory that MV=PT can be accurately tested by finding the correlation between %∆M+%∆V and %∆P+%∆T.
Also, if more accurate measures of the money supply and GDP are found, the correlations can be rerun.
