Editing Marginal revenue (section)

== Definition ==
Marginal revenue is equal to the ratio of the change in revenue for some change in quantity sold to that change in quantity sold. This can be formulated as:<ref name=":6">{{Cite book|last=Pindyck, Robert S.|url=https://www.worldcat.org/oclc/908406121|title=Microeconomics|others=Rubinfeld, Daniel L.|date=3 December 2014|isbn=978-1-292-08197-7|edition=Global edition, Eighth|location=Boston [Massachusetts]|oclc=908406121}}</ref>

<math>MR = \frac{\Delta TR}{\Delta Q}</math>

This can also be represented as a derivative when the change in quantity sold becomes arbitrarily small. Define the revenue function to be<ref>{{Cite web|date=2020-02-27|title=3.2: Monopoly Profit-Maximizing Solution|url=https://socialsci.libretexts.org/Bookshelves/Economics/Book%3A_The_Economics_of_Food_and_Agricultural_Markets_(Barkley)/03%3A_Monopoly_and_Market_Power/3.02%3A_Monopoly_Profit-Maximizing_Solution|access-date=2020-10-26|website=Social Sci LibreTexts|language=en}}</ref>
:<math>R(Q)=P(Q)\cdot Q ,</math>

where ''Q'' is output and ''P''(''Q'') is the inverse [[demand function]] of customers. By the [[product rule]], marginal revenue is then given by
:<math>R'(Q)=P(Q) + P'(Q)\cdot Q,</math>

where the prime sign indicates a derivative. For a firm facing perfect competition, price does not change with quantity sold {{nowrap|(<math>P'(Q)=0</math>),}} so marginal revenue is equal to price. For a [[monopoly]], the price decreases with quantity sold {{nowrap|(<math>P'(Q)<0</math>),}} so marginal revenue is less than price for positive <math>Q</math> (see '''Example 1''').<ref name=":0" />

'''Example 1:''' If a firm sells 20 units of books (quantity) for $50 each (price), this earns <u>total revenue</u>: P*Q = $50*20 = $1000

Then if the firm increases quantity sold to 21 units of books at $49 each, this earns <u>total revenue</u>: P*Q = $49*21 = $1029

Therefore, using the marginal revenue formula (MR)<ref name=":6" /> = <math>\frac{\Delta TR}{\Delta Q} = \left ( \frac{\$1029 - \$1000}{21 - 20} \right ) = \$29</math>

'''Example 2:''' If a firm's total revenue function is written as <math>R(Q)=P(Q)\cdot Q ,</math><ref>{{Cite book |author=Goldstein, Larry Joel |author2=Lay, David C. |author3=Schneider, David I. |title=Brief calculus & its applications |date=2004 |publisher=Pearson Education  |isbn=0-13-046618-2 |edition=10th |location=Upper Saddle River, NJ |oclc=50235091}}</ref>

<math> R(Q)=(Q)\cdot (200 - Q) </math>

<math> R(Q)=200Q - Q^2 </math>

Then, by first order derivation, marginal revenue would be expressed as

<math> MR = R'(Q)=200- 2Q</math>

Therefore, if Q = 40,

MR = 200 − 2(40) = $120