Intuitive Sums: Making the Equation of Exchange Easy to Interpret

Today I’m going to work some fast math magic. Yesterday I reviewed the equation of exchange. Working with the products can be somewhat confusing. So, briefly, I will show the reader how to transform the equation of exchange into (approximate) sums of percent changes. The process is straight forward.

1. Begin with the equation in its basic form

MV = Py

2. Log both sides

ln(MV) = ln(Py)

3. Convert into sums of logs

ln(M) + ln(V) = ln(P) + ln(y)

4. Take the first derivative of both sides

(dM/M) + (dV/V) = (dP/P) + (dy/y)

5. The result is approximately the sum of percent changes

%∆M + %∆V = %∆P + %∆y

6. Rearrange so that desired variable is on the left hand side of the equation. I’ll leave this last step to you.

Notice that this is useful for setting up a regression. If you have data for the 3 variables, you can estimate the fourth. It should be kept it mind that measures of velocity tend to be measures of this sort. If I were to attempt to measure changes in velocity, the form of the equation would be,

%∆V = %∆P + %∆y - %∆M.

Notice that this is similar to the log version

ln(V) = ln(P) + ln(y) – ln(M).

A regression with the log measures will estimate the log of velocity while the equation using the derivative of the logs will estimate percent changes.

One last trick: P and y can be merged, and rightly so. P is a variable that is estimated. Better sometimes to use the aggregate, Y instead.

MV = Py 

MV = Y 

V = Y/M 

ln(V) = ln(Y) – ln(M) 

(dV/V) = (dY/Y) - (dm/M) 

%∆V = %∆Y - %∆M

Enjoy!