# Finance:Marginal revenue

Linear marginal revenue (MR) and average revenue (AR) curves for a firm that is not in perfect competition

Marginal revenue (or marginal benefit) is a central concept in microeconomics that describes the additional total revenue generated by increasing product sales by 1 unit.[1][2][3][4][5] To derive the value of marginal revenue, it is required to examine the difference between the aggregate benefits a firm received from the quantity of a good and service produced last period and the current period with one extra unit increase in the rate of production.[6] Marginal revenue is a fundamental tool for economic decision making within a firm's setting, together with marginal cost to be considered.[7]

In a perfectly competitive market, the incremental revenue generated by selling an additional unit of a good is equal to the price the firm is able to charge the buyer of the good.[3][8] This is because a firm in a competitive market will always get the same price for every unit it sells regardless of the number of units the firm sells since the firm's sales can never impact the industry's price.[1][3] Therefore, in a perfectly competitive market, firms set the price level equal to their marginal revenue $\displaystyle{ (MR = P) }$.[6]

In imperfect competition, a monopoly firm is a large producer in the market and changes in its output levels impact market prices, determining the whole industry's sales. Therefore, a monopoly firm lowers its price on all units sold in order to increase output (quantity) by 1 unit.[1][3][6] Since a reduction in price leads to a decline in revenue on each good sold by the firm, the marginal revenue generated is always lower than the price level charged $\displaystyle{ (MR \lt P) }$.[1][3][6] The marginal revenue (the increase in total revenue) is the price the firm gets on the additional unit sold, less the revenue lost by reducing the price on all other units that were sold prior to the decrease in price. Marginal revenue is the concept of a firm sacrificing the opportunity to sell the current output at a certain price, in order to sell a higher quantity at a reduced price.[6]

Profit maximization occurs at the point where marginal revenue (MR) equals marginal cost (MC). If $\displaystyle{ MR \gt MC }$ then a profit-maximizing firm will increase output to generate more profit, while if $\displaystyle{ MR \lt MC }$ then the firm will decrease output to gain additional profit. Thus the firm will choose the profit-maximizing level of output for which $\displaystyle{ MR = MC }$.[9]

## 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:[10]

$\displaystyle{ MR = \frac{\Delta TR}{\Delta Q} }$

This can also be represented as a derivative when the change in quantity sold becomes arbitrarily small. Define the revenue function to be[11]

$\displaystyle{ R(Q)=P(Q)\cdot Q , }$

where Q is output and P(Q) is the inverse demand function of customers. By the product rule, marginal revenue is then given by

$\displaystyle{ R'(Q)=P(Q) + P'(Q)\cdot Q, }$

where the prime sign indicates a derivative. For a firm facing perfect competition, price does not change with quantity sold ($\displaystyle{ P'(Q)=0 }$), so marginal revenue is equal to price. For a monopoly, the price decreases with quantity sold ($\displaystyle{ P'(Q)\lt 0 }$), so marginal revenue is less than price for positive $\displaystyle{ Q }$ (see Example 1).[6]

Example 1: If a firm sells 20 units of books (quantity) for $50 each (price), this earns total revenue: P*Q =$50*20 = $1000 Then if the firm increases quantity sold to 21 units of books at$49 each, this earns total revenue: P*Q = $49*21 =$1029

Therefore, using the marginal revenue formula (MR)[10] = $\displaystyle{ \frac{\Delta TR}{\Delta Q} = \left ( \frac{\1029 - \1000}{21 - 20} \right ) = \29 }$

Example 2: If a firm's total revenue function is written as $\displaystyle{ R(Q)=P(Q)\cdot Q , }$[12]

$\displaystyle{ R(Q)=(Q)\cdot (200 - Q) }$

$\displaystyle{ R(Q)=200Q - Q^2 }$

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

$\displaystyle{ MR = R'(Q)=200- 2Q }$

Therefore, if Q = 40,

## Notes

1. Bradley R. chiller, "Essentials of Economics", New York: McGraw-Hill, Inc., 1991.
2. Edwin Mansfield, "Micro-Economics Theory and Applications, 3rd Edition", New York and London:W.W. Norton and Company, 1979.
3. Roger LeRoy Miller, "Intermediate Microeconomics Theory Issues Applications, Third Edition", New York: McGraw-Hill, Inc, 1982.
4. Tirole, Jean, "The Theory of Industrial Organization", Cambridge, Massachusetts: The MIT Press, 1988.
5. John Black, "Oxford Dictionary of Economics", New York: Oxford University Press, 2003.
6. Schiller, Bradley R.; Gebhardt, Karen (2017). Essentials of economics (10th ed.). New York: McGraw-Hill/Irwin. ISBN 978-1-259-23570-2. OCLC 955345952.
7. Mankiw, N. Gregory (2009). Principles of microeconomics (5th ed.). Mason, OH: South-Western Cengage Learning. ISBN 978-0-324-58998-6. OCLC 226358094.
8. O'Sullivan & Sheffrin (2003), p. 112.
9. Fisher, Timothy C. G.; Prentice, David; Waschik, Robert G. (2010). Managerial economics : a strategic approach. Routledge. p. 33. ISBN 9780415495172. OCLC 432989728.
10. Pindyck, Robert S. (3 December 2014). Microeconomics. Rubinfeld, Daniel L. (Global edition, Eighth ed.). Boston [Massachusetts]. ISBN 978-1-292-08197-7. OCLC 908406121.
11. Goldstein, Larry Joel; Lay, David C.; Schneider, David I. (2004). Brief calculus & its applications (10th ed.). Upper Saddle River, NJ: Pearson Education. ISBN 0-13-046618-2. OCLC 50235091.
12. Landsburg, Steven E. (2013). Price theory and applications (Ninth ed.). Stamford, CT. ISBN 978-1-285-42352-4. OCLC 891601555.
13. Landsburg, S Price 2002. p. 137.
14. Kumar, Manoj (2015-05-08). "Revenue Curves under Different Markets (With Diagram)" (in en-US).
15. Russell W. Cooper; Alun Andrew John (2011). Microeconomics : theory through applications. Arlington, Virginia. ISBN 978-1-4533-1328-2. OCLC 953968136.
16. McLean, William J. (William Joseph) (2013). Economics and contemporary issues. Applegate, Michael. (9e ed.). Mason, Ohio: South-Western Cengage Learning. ISBN 978-1-111-82339-9. OCLC 775406167.
17. Tuovila, Alicia. "Marginal Revenue (MR) Definition" (in en).
18. Perloff (2008) p. 364.
19. Rekhi, Samia (2016-05-16). "Marginal Revenue and Price Elasticity of Demand" (in en-US).
20. Paul Krugman; Robin Wells; Iris Au; Jack Parkinson (2013). Microeconomics (3rd ed.). New York: Worth Publishers. ISBN 978-1-4292-4005-5. OCLC 796082268.
21. Pemberton, Malcolm; Rau, Nicholas (2011). Mathematics for economists : an introductory textbook (3rd ed.). Manchester: Manchester University Press. ISBN 978-0-7190-8705-9. OCLC 756276243.
22. Pindyck, R & Rubinfeld, D (2001) p. 334.
23. Perloff (2008) p. 371.

## References

• Landsburg, S 2002 Price Theory & Applications, 5th ed. South-Western.
• Perloff, J., 2008, Microeconomics: Theory & Applications with Calculus, Pearson. ISBN:9780321277947
• Pindyck, R & Rubinfeld, D 2001: Microeconomics 5th ed. Page Prentice-Hall. ISBN:0-13-019673-8
• Samuelson & Marks, 2003 Managerial Economics 4th ed. Wiley
• O'Sullivan, Arthur; Sheffrin, Steven M. (2003). Economics: Principles in Action. Pearson Prentice Hall. ISBN:0-13-063085-3.