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Question
Consider a monopolistic facing the following demand and cost curves
P=50-2Q    
C=25+10Q
Suppose the firm is able to separate its customers in two distinct markets with the following demand functions
P1= 40-2.5Q1
P2=90-10Q2
Please help me calculate the total demand, marginal revenue and marginal cost from the above equation.
Thank You.

Answer
Elasticity of Demand
Elasticity of Demand  
YOUR QUESTION IS REPHRASED MORE PRECISELY AS:
Consider a monopolist [not “monopolistic”] facing the following demand and cost curves for a commodity Q
P=50-2Q    
C=25+10Q
Suppose the firm is able to separate its customers in two distinct markets –the two markets with DIFFERENT price elasticities for a commodity, Q, the resale of which is NOT possible –with the following demand functions
P1= 40-2.5Q1
P2=90-10Q2
Please help me calculate the total demand, marginal revenue and marginal cost from the above equations for the two markets [I believe this is not a homework question].

Dear Indrani:
This is a question on intermediate-level microeconomics, and the student is supposed to have an exposure to elementary differential calculus. This is a very simple question for any student who with college-level mathematics has had a good introduction to advanced demand theory and cost theory. As I don’t know of your mathematical background, I assume you have at least grounding in school-level algebra and an idea about derivative and on that basis I am going to give a detailed answer to your question with as little mathematics as possible. To facilitate your understanding, rather than just giving you the answer, I place my answer under the following rubrics.

THE BACKGROUND AGAINST WHICH THE QUESTION HAS BEEN SET
This is a question on price discrimination practiced by a monopolist. The monopolist can practice price discrimination in respect of the commodity Q when he can sell the same product, Q, in two different markets, Market1 and Market2, at two different prices, P1 and P2, provided the monopolist meets two conditions. The two markets for the same product Q have two different price elasticities, and the monopolist can ensure that the product Q cannot be resold from Market 1 to Market 2, or vice versa.

As a result, since the monopolist produces Q at a certain cost no matter in which market he sells the product, the cost of production of Q is the SAME for both the markets: C=25+10Q.
If the monopolist were to sell the product in only ONE market, without any price discrimination, the price that would be pervasive without discrimination throughout the market would be the ONE price: P=50-2Q.

However, because of price discrimination, even though cost remains the same, prices charged in the two markets are different: P1=40-2.5Q1 and P2=90-10Q2 in the Market1 and Market2, respectively.



PRICE [AVERAGE REVENUE] AND TOTAL REVENUE IN THE TWO MARKETS
Market1
P1=40-2.5Q1
TR1=P1Q1= (40-2.5Q1)Q1
TR1=40Q1-2.5Q^2         [1]

Market2
P2=90-10Q2
TR2=P2Q2=(90-10Q2)Q2
TR2=90Q2-10Q^2.         [2]

MARGINAL REVENUE (MR) AND MARGINAL COST (MC) IN TWO MARKETS
From equation [1], we get the marginal revenue (MR) in Market1 by differentiating TR1 with respect to Q1 (any elementary introduction to differential calculus is enough for this). Similarly, we calculate MR2 for Market2.
MR1= 40-5Q1
MR2=90-20Q2

MARGINAL COST (MC) IS THE SAME FOR BOTH THE MARKETS
We get the marginal cost (MC) which is the same for both Market1 and Market2 by differentiating the total cost function, C=25+10Q, with respect to output Q

MC=10

THE MIONOPOLIST PRODUCES WHERE MR=MC. THERE ARE TWO OUTPUTS IN THE TWO MARKETS.
Market1
MR1=MC
40-5Q1=10
30=5Q1
Q1=6         [3]

Market2
MR2=MC
90-20Q2=10
80=20Q2
Q2=4         [4]

So the monopolist produces Q=10, sells Q1=6 in Market1 and Q2=4 in Market2. Here Q=Q1+Q2=10=6+4. Check: Monopolist’s total demand function and cost function are given by
P=50-2Q          [5]
C=25+10Q      [6]
From equation [5], we get total revenue and marginal revenue as a whole:
TR=(50-2Q)Q =50Q-2Q^2
MR= 50Q-4Q      [7]
MC=10         [8]
Equating [7] with [8],
50Q-4Q =10      [9]
40=4Q
Q=10         [10]
Thus [10]=[3]+[4] gives us Q=Q1+Q2=10=6+4.

THE MONOPOLIST CHARGES TWO DIFFERENT PRICES IN TWO DIFFERENT MARKETS
Market1
P1=40-2.5Q1
P1=40-2.5x6   (since Q1=6 by equation [3])
P1=40-15
P1=25

Market2
P2=90-10Q2
P2=90-10x4   (since Q2=4 by equation [4])
P2=90-40
P2=50

OBSERVATION IN TWO MARKETS
The monopolist charges P1=25 –less than what he would otherwise have charged had there been no price discrimination, with P=30 [P=50-2Q, C=25+10Q, MR=50-4Q, MC=10, Q=10, P=50-2x10= 30]. However, he charges P2=50, much higher than what he would have charged without price discrimination. The total revenue indeed goes up.

TR1 = P1xQ1= 25x6 =150
TR2 = P2xQ2 = 50x4 =200
TR =TR1+TR2= 150+200 =350

TR with price discrimination (350) is indeed higher than TR without price discrimination (30x10=300).  


ELASTICITIES ARE DIFFERENT IN THE TWO MARKETS
Although you haven’t asked the question, I am adding this in case you may stumble into it or most likely the next question could that –different elasticities. Elasticity in Market2 (e=5/4) is less than elasticity in Market1 (5/3). You can reason it this way: The monopolist sells the same product at a low price in Market1 where there are many poor people; and sells the same product at a high price in Market1 where there are few rich people. The rich people have low price elasticity of demand for the product, because they don’t care. Or it could simply be because the monopolist can extract more money from few. Look at Microsoft CD. The same CD is sold at almost half the price in California than the price it is sold in Toronto, Canada. [By the way for a straight-line demand curve ab, with a=price intercept, b=quantity intercept, (a/b)= the slope of dQ/dP, price elasticity works out at e=P/(a-P), or the lower segment of the demand curve over the upper segment of the demand curve. For example, E1=25/( 40-25)=5/3; E2=50/(90-50)=5/4; and E2<E1, allowing the monopolist to charge a higher price in Market2.

SUMMARY
From the equations under the questions, in addition to some other adventitious but nonetheless pertinent results, the following results as have been asked are placed below.

Total Demand=10
Marginal Revenue in Market1= MR1= 40-5Q1
Marginal Revenue in Market2 =MR2=90-20Q2
Marginal Cost=10

I hope, Indrani, I have been able to help you with a solution to, and a full-dress explanation of, your question. I wish you best of luck in your pursuit of knowledge.  

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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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