Equation of exchange states:
$ MV=PY $
$ M $ is Quantity of Money
$ V $ is Velocity of Money
$ P $ is Price Level
$ Y $ is Real GDP
Could someone please share how the above common form of equation is converted to the growth rate form:
$ \%\Delta M + \%\Delta V = \%\Delta P + \%\Delta Y $

Edit: Sorry the latex doesn't seem to be working. Unable to figure that out.


1 Answer 1


Taking logs of the equation $MV=PY$ gives: $ln(M) + ln(V) = ln(P) + ln(Y)$.

Next, differentiating with respect to time yields $\frac{dln(M)}{dt} + \frac{dln(V)}{dt} = \frac{dln(P)}{dt} + \frac{dln(Y)}{dt}$.

Or, using $\frac{dln(X)}{dt} = \frac{\dot{X}}{X}$, with $\dot{X} = \frac{dX}{dt}$, the derivative of a variable $X$ with respect to time, you can immediately derive $\frac{\dot{M}}{M} + \frac{\dot{V}}{V} = \frac{\dot{P}}{P} + \frac{\dot{Y}}{Y}$, which equals $\%\Delta M + \%\Delta V = \%\Delta P + \%\Delta Y$.


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