# Differential form of the quantity theory of money

The quantity theory of money is $$MV = PY$$.

Why is the differential form $$\delta M + \delta V = \delta P + \delta Y$$?

Shouldn't it be $$V\delta M + M\delta V = Y\delta P + P\delta Y$$?

Well you do not provide any source for the $$dM+dV = dP+dY$$ so it is hard to say why someone written that. That is certainly not normal total derivative of QTM.

You are right that the correct total derivative of QTM will be given by:

$$VdM+MdV=YdP+PdY$$

Possible explanation why you saw it somewhere written differently:

1. Note the above can be divided by $$MV=PY$$ to get:

$$(M dV)/(MV) + (V dM)/(MV) = (P dY)/(PY) + (Y dP)/(PY)$$

which after simplification yields:

$$\frac{dV}{V} + \frac{dM}{M} = \frac{dY}{Y} + \frac{dP}{P}$$

the above expression is quite common in the literature and it is possible someone just decided to substitute $$dx = \frac{dX}{X}$$ to clean up the expression: $$dv + dm = dy +dp$$,

1. Alternatively it is possible that the 'unnamed source' wanted to express the equation in percentage changes via log linearization and just used the same symbol for change as differential symbol. Log linearizing QTM yields:

$$\ln M + \ln V = \ln P + \ln Y$$

which is equivalent to:

$$\Delta \% M + \Delta \% V = \Delta \% P + \Delta \% Y$$

Some people will use $$d$$ or $$\delta$$ instead of $$\Delta$$ and omit $$%$$ sign.

1. There are some missing assumptions transformations that would make it right.

2. Source you got it from made a mistake.