<p>Well you do not provide any source for the <span class="math-container">$dM+dV = dP+dY$</span> so it is hard to say why someone written that. That is certainly not normal total derivative of QTM.</p>
<p>You are right that the correct total derivative of QTM will be given by:</p>
<p><span class="math-container">$$VdM+MdV=YdP+PdY $$</span></p>
<p>Possible explanation why you saw it somewhere written differently:</p>
<ol>
<li>Note the above can be divided by <span class="math-container">$MV=PY$</span> to get:</li>
</ol>
<p><span class="math-container">$$(M dV)/(MV) + (V dM)/(MV) = (P dY)/(PY) + (Y dP)/(PY)$$</span></p>
<p>which after simplification yields:</p>
<p><span class="math-container">$$\frac{dV}{V} + \frac{dM}{M} = \frac{dY}{Y} + \frac{dP}{P}$$</span></p>
<p>the above expression is quite common in the literature and it is possible someone just decided to substitute <span class="math-container">$dx = \frac{dX}{X}$</span> to clean up the expression: <span class="math-container">$dv + dm = dy +dp$</span>,</p>
<ol start="2">
<li>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:</li>
</ol>
<p><span class="math-container">$$\ln M + \ln V = \ln P + \ln Y$$</span></p>
<p>which is equivalent to:</p>
<p><span class="math-container">$$\Delta \% M + \Delta \% V = \Delta \% P + \Delta \% Y$$</span></p>
<p>Some people will use <span class="math-container">$d$</span> or <span class="math-container">$\delta$</span> instead of <span class="math-container">$\Delta$</span> and omit <span class="math-container">$%$</span> sign.</p>
<ol start="3">
<li><p>There are some missing assumptions transformations that would make it right.</p>
</li>
<li><p>Source you got it from made a mistake.</p>
</li>
</ol>

CC BY-SA 4.0
2020-12-22T00:51:31.710Z
2020-12-22T17:52:24.800Z
2020-12-22T17:52:24.800Z
1muflon1
<p>The quantity theory of money is <span class="math-container">$MV = PY$</span>.</p>
<p>Why is the differential form <span class="math-container">$\delta M + \delta V = \delta P + \delta Y$</span>?</p>
<p>Shouldn't it be <span class="math-container">$V\delta M + M\delta V = Y\delta P + P\delta Y$</span>?</p>

<quantity-theory-of-money>
Differential form of the quantity theory of money
curiousgeorge
# Differential form of the quantity theory of money

(Asked by: username_0 on 2020-12-22)

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 $V\delta M + M\delta V = Y\delta P + P\delta Y$?

**Tags:** quantity-theory-of-money


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## 1 answers

### Answer by username_1 on 2020-12-22. Score: 3 | Accepted answer

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: $dv + dm = dy +dp$,

2. 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.

3. There are some missing assumptions transformations that would make it right.
4. Source you got it from made a mistake.


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