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Analysing monopolist's demand and cost functions

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A monopolist produces a single homogeneous good, which he sells in two markets between which discrimination is possible. His total cost function is:
TC = Q3 /3 - 40Q2 + 1800Q + 5000
where annual total cost is in dollars and annual output in tons. The demand curves in the two markets are given by the equations
q1 = 320 - 0.4 p1 and
p2 = A - Bq2
The monopolist achieves a profit-maximizing equilibrium at which his total output (Q = q1 + q2) is 60 tons per annum and his annual pure profit is $5,000.
(a) What is the marginal cost at the profit maximizing output?
(b) What are the values of q1 and p1 at this output?
(c) What are the values for total cost and total revenue at this output?
(d) What are the individual values for total revenue in each of the markets?
(e) Having calculated all of the above, it is now easy to calculate the values for A and B in equation p2 .
What are these values?

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Solution:

(a) What is the marginal cost at the profit maximizing output?
TC= Q^3/3-40Q^2+1800Q+5000
Marginal Cost=MC=d(TC)/dQ=Q^2-40*2Q+1800
=Q^2-80Q+1800
Profit maximization output level is 60 tons (i.e. Q=60)
MC=60^2-80*60+1800=600 dollars

(b) What are the values of q1 and p1 at this output?
q1= 320-0.4p1
p1=(320-q1)/0.4=800-2.5q1 ...

Solution Summary

Solution describes the steps for determining marginal cost and profit maximizing output for a monopolist where price discrimination is possible.

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Monopolist and Price discrimination

A monopolist produces a single homogeneous good, which she sells in two distinct markets
between which price discrimination is possible. Her total cost function is:
TC = 1/3 Q3 - 7.5Q2 + 370Q + 100
The demand curves in the two markets are given by:
q1 = 80 - 0.2p1 and q2= Ap2-5
The monopolist achieves a profit-maximizing equilibrium at which her total output (Q = q1 + q2) is 10 and she charges a price of $360 in market 1.

(a)What is the profit-maximizing output in markets 1 and 2?
(b)Calculate marginal revenue in either market.
(c)What is the cost of producing an extra unit at the profit maximizing output?
(d)What price is charged in market 2?

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