Economics/calculating total demand, marginal revenue and marginal cost.
Consider a monopolistic facing the following demand and cost curves
Suppose the firm is able to separate its customers in two distinct markets with the following demand functions
Please help me calculate the total demand, marginal revenue and marginal cost from the above equation.
YOUR QUESTION IS REPHRASED MORE PRECISELY AS:
Consider a monopolist [not “monopolistic”] facing the following demand and cost curves for a commodity Q
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
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].
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
MARGINAL REVENUE (MR) AND MARGINAL COST (MC) IN TWO MARKETS
From equation , 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.
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
THE MIONOPOLIST PRODUCES WHERE MR=MC. THERE ARE TWO OUTPUTS IN THE TWO MARKETS.
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
From equation , we get total revenue and marginal revenue as a whole:
MR= 50Q-4Q 
Equating  with ,
50Q-4Q =10 
Thus =+ gives us Q=Q1+Q2=10=6+4.
THE MONOPOLIST CHARGES TWO DIFFERENT PRICES IN TWO DIFFERENT MARKETS
P1=40-2.5x6 (since Q1=6 by equation )
P2=90-10x4 (since Q2=4 by equation )
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.
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.
Marginal Revenue in Market1= MR1= 40-5Q1
Marginal Revenue in Market2 =MR2=90-20Q2
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.