<p>You have two questions. The first one is (I broke lines in the quotes where I tought this improved legibility):</p>
<blockquote>
<p>MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity <br>
This is the same as this: <br>
MR = P(Q) + dP/dQ</p>
</blockquote>
<p>As <a href="https://economics.stackexchange.com/questions/50878/why-marginal-revenue-equals-this#comment82240_50878">Herr K. points</a> out, an important thing to note is that <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> are not independent of each other; in this context <span class="math-container">$P$</span> is a function of <span class="math-container">$Q$</span>. After this observation, the answer is purely mathematical: the product rule. From <a href="https://en.wikipedia.org/wiki/Product_rule" rel="nofollow noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>In calculus, the product rule (or Leibniz rule<a href="https://en.wikipedia.org/wiki/Product_rule" rel="nofollow noreferrer">1</a> 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</p>
<p><span class="math-container">$ (u\cdot v)' = u'\cdot v+u\cdot v'$</span></p>
</blockquote>
<hr />
<p>Your second question seems to be</p>
<blockquote>
<p>MR = P(Q) * (1+ 1/elasticity(Q)) <br>
And then it is converted to: <br>
MR = P(Q) * (1- absolute value of (1/elasticity(Q)) <br>
[...] converting the parenthesis in negative I guess its because... <br>
...I would appreciate a deeper explanation...</p>
</blockquote>
<p>It is assumed that <span class="math-container">$P(Q)$</span> is decreasing in <span class="math-container">$Q$</span>, and that both <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> are positive. The (point) elasticity formula is
<span class="math-container">$$
\epsilon_P(Q) = \frac{\text{d}P(Q)}{\text{d}Q} \frac{Q}{P}. 
$$</span>
The first fraction is negative (because <span class="math-container">$P(Q)$</span> is decreasing in <span class="math-container">$Q$</span>), and the second fraction is positive (both <span class="math-container">$P$</span> and <span class="math-container">$Q$</span> are positive). Thus the quantity elasticity of price is indeed negative, and hence
<span class="math-container">$$
\epsilon_P(Q) = - | \epsilon_P(Q) |.
$$</span></p>

CC BY-SA 4.0
2022-03-27T12:49:07.330Z
Thanks!!! Really helpful everybody
Teko JR
CC BY-SA 4.0
2022-03-26T10:31:39.010Z
2022-03-28T11:43:16.493Z
2022-03-28T11:43:16.493Z
Giskard
<p>MR = d(Total revenue)/dQuantity = d (Price * Quantity)/dQuanitity</p>
<p>This is the same as this : MR = P(Q) + dP/dQ</p>
<p>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.</p>
<p>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))</p>
<p>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</p>
<p>That is what I'm guessing but I would appreciate a deeper explanation.</p>

CC BY-SA 4.0
2022-03-26T02:42:41.250Z
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).
Herr K.
<microeconomics><elasticity><nonexpert><marginal>
Mathematical explanation of transformations of Marginal Revenue
Teko JR
# Mathematical explanation of transformations of Marginal Revenue

(Asked by: username_0 on 2022-03-26)

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.

**Tags:** microeconomics, elasticity, nonexpert, marginal

> **Comments:**
> **username_1**: 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).  

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## 1 answers

### Answer by username_2 on 2022-03-26. Score: 4

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 [username_1. points](https://economics.stackexchange.com/questions/50878/why-marginal-revenue-equals-this#comment82240_50878) 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](https://en.wikipedia.org/wiki/Product_rule):

> In calculus, the product rule (or Leibniz rule[1](https://en.wikipedia.org/wiki/Product_rule) 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
$$
\epsilon\_P(Q) = - | \epsilon\_P(Q) |.
$$

> **Comments:**
> **username_0**: Thanks!!! Really helpful everybody  

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