1. Begin with the equation in its basic form

MV = Py2. 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 + %∆y6. 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!

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