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Question
Hello sir,

I have a problem to solve which appears hard.
given demand and cost functions: Q=300-2P and TC=150+10Q; show that a monopolist will produce half the output under perfect competition.

Thanks for your help.

Answer
Output under Monopoly and Output under Perfect Competition
Output under Monopoly  

Monopoly Output/Perfect Competition Output
Monopoly Output/Perfec  
Dear Yana,

It appears you have some difficulty in understanding what answer I posted to your question, probably because you are not quite familiar with how mathematically I have arrived at your answer. It is likely for a question like this which appears simple but has some technicality hidden. To make it clearer to you, I am sending you the reply slightly changing the way it is written in order that you may have facility to understand it without difficulty. Thank you anyway for posing a very interesting question on output determination under (a) monopoly and (b) perfect competition.

This is actually a very simple question only if you know a little bit of elementary calculus. You have not mentioned what your mathematical background is. However, looking at the question, I presume you are at least at intermediate level which requires minimum of mathematics. Let me walk you along to the answer at five parts:

PART I – BASICS OF OUTPUT DETERMINATION THEORY;
PART II – OUTPUT DETERMINATION UNDER MONOPOLY;
PART III – OUTPUT DETERMINATION IN INDUSTRY UNDER PERFECT COMPETITION;
PART IV –CONCLUSION

PART I – BASICS OF OUTPUT DETERMINATION THEORY
Even if you do not know much calculus but at least know the following two things, you do not have to worry:

(1)   POSTULATE 1:  TOTAL REVENUE = OUTPUT x PRICE  or TR = QP
(2)   POSTULATE 2:  MARGINAL REVENUE = CHANGE IN TOTAL REVENUE DUE TO CHANGE IN OUTPUT (mathematicians write it as d(TR)/dQ, where “d” means “a change in”)
(3)   POSTULATE 3:  OUTPUT IS PRODUCED –BOTH UNDER MONOPOLY AND UNDER PERFECT COMPETITION –WHEN  “MR = MC”
(4)   POSTULATE 4: Monopolist’s MR falls “at twice the rate” of his AR (=price). [Check any introductory book on monopoly.] This implies monopolist’s price is above his MR [please study the curves in the attached diagram carefully].
(5)   POSTULATE 5: Under perfect competition a firm’s Price=MR [remember firm’s demand curve (=AR = Price) is a horizontal straight line with infinite elasticity of demand and AR (=P) =MR=MC=AC in long-run equilibrium.]. However, the demand curve of the industry under perfect competition –all myriad facing the total demand of all the consumers for the output Q –is “downward sloping.” So, while the firm has a horizontal demand curve, the perfect-competition industry has a downward-sloping demand curve. In that case, we also have
(6)   POSTULATE 6: The industry demand for the output Q is the output Q demanded by all the consumers and is produced by “one produce” or the monopolist. The same industry demand for the output Q demanded by all the consumers is produced by “many producers” under perfect competition. The difference is between “who” produces –monopolist (the only firm in the industry or perfect competition (all the firms in the industry). In both the cases, whoever produces the output, the demand curve is determined by the willingness of the consumers to buy different quantities at different prices.
(7)   POSTULATE 7: Under the same demand function –such as Q = 300 – 2P—there are two sub-postulates:
(i)   Monopolist’s price or P is greater than his marginal cost or MC, i.e., P > MC [MR=MC]
(ii)   The industry under perfect competition has P=MC (=MR)


PART II – OUTPUT DETERMINATION UNDER MONOPOLY
On the basis of the POSTULATE 3, let us determine the monopolist’s output Q equating MR with MC. Our demand function  

Q = 300 – 2P    [1]

shows Q as a function of P. However, MR and TR according to POSTULATE 2 are “functions of Q,” [not functions of P]. In order to define these two concepts as “functions of Q,” we transform the demand function in [1] into the “inverse demand function” adding 2P to both sides of [1], subtracting Q from both sides, and finally dividing both sides by 2. So [1] yields the inverse demand function [2]

         P = 150 −½Q   [2]

From [2] we get total revenue TR = PQ and differentiating that w.r.t. to Q, we get MR (see attachment 2)

MR = 150 – Q     [3]

Differentiating the cost function Q = 150+10Q w.r.t. Q yields MC as under

        MC = d(150+10Q)/dQ = 10  [4]

Since output Q is produced when MR=MC, we have
150 – Q = 10      [5]

From [5], we get          Q=140   [6]

Monopolist’s P>MC. How much is P? We get it from demand function [1] and from [6]

300 – 2P = Q or 300 – 2P= 140 or 160 – 2P= 0, i.e., 160 = 2P => P = 80 (see attachment 2)


PART III – OUTPUT DETERMINATION IN INDUSTYRY UNDER PERFECT COMPETITION
You may well know from your introductory microeconomics course that, by our aforesaid POSTULATE 5, a firm under perfect competition must have the following equalities:

P=MC or equilibrium level of output occurs at MC=MR=PRICE.

Since under perfect competition the industry is composed of many, many firms, which the firm under perfect competition facing a horizontal demand curve has no control over price but produces when MR=MC and P=MC, the industry demand curve under perfect completion is downward-sloping like the monopolist’s demand curve. Here we consider all the consumers’ willingness to buy different quantities at different prices under the industry.

Assume under perfect competition P=10, as calculated above. Now substitute the value of P=10 in Equation [1] to arrive at the solution in Equation [1.B]:

Q = 300 – 2P    [1]
Q = 300 – 2x10    = 300 −20     [1.B]
Q = 280          [1.B]


PART 5 –CONCLUSION
Let us now put in juxtaposition Equation [6] and Equation [1.B]
Q = 140      [8]    when MONOPOLIST produces at MC=MR but P>MR
Q = 280      [1.B]   PERFECT COMPETITION produces at P=MC =10

Thus, since Equation [8] gives an output level 140 which is half the output level 280 given by Equation [1.B], we prove: A MONOPOLIST WILL PRODUCE HALF THE OUTPUT UNDER PERFECT COMPETITION.


I have tried as best I could to explain this, taking your level of education, and am glad to re-send this rewritten answer. I hope, Yana, this solves your problem. If you have any problem, please write back. Best of luck.  

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Eklimur Raza

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It appears some students in this website are confused about elasticity of demand and the slope of the demand curve when they are trying to figure out why rectangular hyperbola comes up in case of unitary demand curve. First, they don't know that RH can be depicted in a positive quadrant of price,quantity plane. Secondly, they make the mistake that the slope of RH is constant at -1. Two points could help them: first, e=1 at each and every point of the RH, because the tangent at any point shows lower segment=upper segment (another geometric definition of e); yet slopes at different points,dQ/dP, are different; second, e is not slope but [(Slope)(P/Q)]in absolute terms. Caveat: only if we measure (log P) along the horizontal axis and (log Q) up the vertical axis, can we then say slope equals elasticity --in which case RH on P,Q plane is transformed into a straight-line demand curve [with slope= -tan 45 deg] on (log Q),(logP) plane, and e= -d(log Q)/d(log P). [By the way, logs are not used in college textbooks --although that is helpful in econometric estimation of elasticity viewed as an exponent of P, when demand equation is transformed into log-linear form.] I have not found the geometrical explanation I have given in any textbook followed in undergraduate and college classes in Canada (including the book followed in a university where I taught for a short time and in the book followed in George Brown College, Toronto, where I teach.

Experience

About 11 years' teaching economics and business studies, and also English, history and elementary French.Practical experience in a development bank, working with international donor agencies like the World Bank and the ADB. Experience in free-lance journalism, including Canada's "National Post."

Organizations
I teach micro- and macroeconomics at George Brown College (continuing education), Toronto, ON, Canada.

Publications
Many articles and editorials, on different subjects, in English newspapers. Recently an applied Major Research Paper, based on a synthesis of the Solow growth model and the Lewis two-sector model, has be accepted by Ryerson University, Toronto. Professors Thomas Barbiero and Eric Cam, Ryerson University, accepted the paper.

Education/Credentials
Master degree in Interantional Economics and Finance and diploma with honours in Business Administration from Canada.

Awards and Honors
Received First Prize in an inter-university Literary Contest.

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