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