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.
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 
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  into the “inverse demand function” adding 2P to both sides of , subtracting Q from both sides, and finally dividing both sides by 2. So  yields the inverse demand function 
P = 150 −½Q 
From  we get total revenue TR = PQ and differentiating that w.r.t. to Q, we get MR (see attachment 2)
MR = 150 – Q 
Differentiating the cost function Q = 150+10Q w.r.t. Q yields MC as under
MC = d(150+10Q)/dQ = 10 
Since output Q is produced when MR=MC, we have
150 – Q = 10 
From , we get Q=140 
Monopolist’s P>MC. How much is P? We get it from demand function  and from 
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  to arrive at the solution in Equation [1.B]:
Q = 300 – 2P 
Q = 300 – 2x10 = 300 −20 [1.B]
Q = 280 [1.B]
PART 5 –CONCLUSION
Let us now put in juxtaposition Equation  and Equation [1.B]
Q = 140  when MONOPOLIST produces at MC=MR but P>MR
Q = 280 [1.B] PERFECT COMPETITION produces at P=MC =10
Thus, since Equation  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.