# Mathematical explanation of transformations of Marginal Revenue

MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity

This is the same as this : MR = P(Q) + dP/dQ

I did it with an example and it amazed me how both are equal but I still dont get how. They seem to be different formulas, maybe somebody can get from one to other step by step and then I can connect it.

Also, the second formulas makes that : MR = P(Q) * (1+ 1/elasticity(Q)) And then it is converted to: MR = P(Q) * (1- absolute value of (1/elasticity(Q))

This last step of converting the parenthesis in negative I guess its because usually the demand function looks like this : Q = 100 - P; Therefore the inverse demand : P = 100 - Q

That is what I'm guessing but I would appreciate a deeper explanation.

• This is typically covered in an intermediate micro course. The trick is to note that in an inverse demand curve, price is a function of quantity, i.e. $P(Q)$. Because of this, you'd have to apply product rule when differentiating total revenue, and the result is your second expression of MR (which you missed a $Q$ at the end). Mar 26, 2022 at 2:42

You have two questions. The first one is (I broke lines in the quotes where I tought this improved legibility):

MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity
This is the same as this:
MR = P(Q) + dP/dQ

As Herr K. points out, an important thing to note is that $$P$$ and $$Q$$ are not independent of each other; in this context $$P$$ is a function of $$Q$$. After this observation, the answer is purely mathematical: the product rule. From Wikipedia:

In calculus, the product rule (or Leibniz rule1 or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as

$$(u\cdot v)' = u'\cdot v+u\cdot v'$$

Your second question seems to be

MR = P(Q) * (1+ 1/elasticity(Q))
And then it is converted to:
MR = P(Q) * (1- absolute value of (1/elasticity(Q))
[...] converting the parenthesis in negative I guess its because...
...I would appreciate a deeper explanation...

It is assumed that $$P(Q)$$ is decreasing in $$Q$$, and that both $$P$$ and $$Q$$ are positive. The (point) elasticity formula is $$\epsilon_P(Q) = \frac{\text{d}P(Q)}{\text{d}Q} \frac{Q}{P}.$$ The first fraction is negative (because $$P(Q)$$ is decreasing in $$Q$$), and the second fraction is positive (both $$P$$ and $$Q$$ are positive). Thus the quantity elasticity of price is indeed negative, and hence $$\epsilon_P(Q) = - | \epsilon_P(Q) |.$$

• Thanks!!! Really helpful everybody Mar 27, 2022 at 12:49