<p>I suppose that <span class="math-container">$MR_1$</span> is Marginal Revenue for firm <span class="math-container">$1$</span>.</p>
<p>Revenue <span class="math-container">$R$</span> for firm <span class="math-container">$1$</span> is:</p>
<p><span class="math-container">$$R=p(Q)\cdot q_1,$$</span></p>
<p>where <span class="math-container">$p(Q)$</span> is the (inverse) demand function.</p>
<p>Remembering that <span class="math-container">$Q=q_1+q_2+...q_n$</span>, we can write <span class="math-container">$Q$</span> as a function of <span class="math-container">$q_1$</span>, <span class="math-container">$Q(q_1)$</span>:</p>
<p><span class="math-container">$$R=p[Q(q_1)]\cdot q_1.$$</span></p>
<p>Marginal revenue is the derivative of the revenue <span class="math-container">$R$</span> with respect to <span class="math-container">$q_1$</span>. Remembering the rule for the derivative of a product and for composition of functions, and assuming <span class="math-container">$q_2,..,q_n $</span> fixed, we have:</p>
<p><span class="math-container">$$MR_1=dR/dq_1= p+ q_1 dP(Q)/dQ\cdot dQ/dq_1.$$</span></p>
<p>But <span class="math-container">$dQ/dq_1=1$</span>, so we have</p>
<p><span class="math-container">$$MR_1= p+ q_1 dP(Q)/dQ.$$</span></p>

CC BY-SA 4.0
2022-10-24T05:46:54.240Z
2022-10-24T05:46:54.240Z
BakerStreet
<p>I wonder know the process of this equation
What does the first step mean?
Thanks!
<a href="https://i.sstatic.net/Gwhd9.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Gwhd9.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-10-23T19:50:42.460Z
Could you please clarify what is the problem you are asking about? Are you not sure about the reasoning behind the first equation? Or do you find the problem to be somewhere in the procedure of the expression modification?
Athaeneus
CC BY-SA 4.0
2022-10-23T20:06:53.503Z
Yes Sir I do not the meaning of the first step and how it comes from
Gavin
CC BY-SA 4.0
2022-10-23T20:45:28.087Z
https://www.mit.edu/~rpindyck/Courses/Pricing_10.pdf - where **c** refers to the right-hand side of your first equation (and equals marginal costs - hence MR = MC).
Alex
<microeconomics>
Cournot equilibrium
Gavin
# Cournot equilibrium

(Asked by: username_0 on 2022-10-23)

I wonder know the process of this equation
What does the first step mean?
Thanks!
[![enter image description here](https://i.sstatic.net/Gwhd9.jpg)](https://i.sstatic.net/Gwhd9.jpg)

**Tags:** microeconomics

> **Comments:**
> **username_1**: Could you please clarify what is the problem you are asking about? Are you not sure about the reasoning behind the first equation? Or do you find the problem to be somewhere in the procedure of the expression modification?  
> **username_0**: Yes Sir I do not the meaning of the first step and how it comes from  
> **username_2**: https://www.mit.edu/~rpindyck/Courses/Pricing_10.pdf - where **c** refers to the right-hand side of your first equation (and equals marginal costs - hence MR = MC).  

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## 1 answers

### Answer by username_3 on 2022-10-24. Score: 4 | Accepted answer

I suppose that $MR\_1$ is Marginal Revenue for firm $1$.

Revenue $R$ for firm $1$ is:

$$R=p(Q)\cdot q\_1,$$

where $p(Q)$ is the (inverse) demand function.

Remembering that $Q=q\_1+q\_2+...q\_n$, we can write $Q$ as a function of $q\_1$, $Q(q\_1)$:

$$R=p[Q(q\_1)]\cdot q\_1.$$

Marginal revenue is the derivative of the revenue $R$ with respect to $q\_1$. Remembering the rule for the derivative of a product and for composition of functions, and assuming $q\_2,..,q\_n $ fixed, we have:

$$MR\_1=dR/dq\_1= p+ q\_1 dP(Q)/dQ\cdot dQ/dq\_1.$$

But $dQ/dq\_1=1$, so we have

$$MR\_1= p+ q\_1 dP(Q)/dQ.$$


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