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I wonder know the process of this equation What does the first step mean? Thanks! enter image description here

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    $\begingroup$ 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? $\endgroup$
    – Athaeneus
    Oct 23, 2022 at 19:50
  • $\begingroup$ Yes Sir I do not the meaning of the first step and how it comes from $\endgroup$
    – Gavin
    Oct 23, 2022 at 20:06
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    $\begingroup$ 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). $\endgroup$
    – Alex
    Oct 23, 2022 at 20:45

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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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