Student Number:
QUIZ 5
Section D
Question 1. [6 marks] A multi-plant monopolist faces demand P = 460 − 6Q . The first
plant (call it Plant 1) has total cost, TC1 = 2.5Q 2 and the second plant (call it Plant 2) has
total cost, TC 2 = 5Q 2 .
a) [2.5 marks] If the monopolist owned only plant 2, what is the optimal price this
monopolist should charge, the total number of units the monopolist should supply, and
the profit the monopolist would earn.(Assume that the demand curve faced by the
monopolist is the same as given above).
Answer:
To find the profit-maximizing price, set MR = MC . In this case we have
460 − 12Q = 10Q ⇒ Q = 20.91
At this quantity, the monopolist will charge a price
⇒ P = 460 − 6(20.91) = 334.54
The profit of monopolist is
Profit = 334.54(20.91) − [5( 20.91)^ 2] = 6995.23 − 2186.14 = 4809.09
b) [2.5 marks] Suppose that monopolist owns both, plants 1 and 2. Determine the optimal
price this monopolist should charge, the total number of units the monopolist will supply,
and the number of units the monopolist should produce at each plant. Also calculate the
profit of the multi-plant monopolist
Answer:
The monopolist will produce so that the marginal costs are equal at both plants. To
begin, find the multi-plant marginal cost curve, MCT . This is found by horizontally
summing the marginal cost curves for the individual plants. This implies
QT = Q1 + Q2
MCT MCT
+
5
10
3MCT
QT =
10
10QT
MCT =
3
QT =
The second equation above is obtained from the marginal cost curves for the individual
plants. Because the marginal costs will be equal at each plant, we call this MCT .
To find the optimal price and quantity, set marginal revenue equal to marginal cost. In
this case we have
10Q
3
1, 380 − 36Q = 10Q
Q = 30
At this (total) quantity, the firm will charge a price equal to P = 460 − 6(30) = $280 .
Finally, to determine the amounts to produce at each plant plug the marginal cost,
MCT = 100 , into each plant’s marginal cost curve and solve for Q . For Plant 1 we have
460 − 12Q =
100 = 5Q1
Q1 = 20
and at Plant 2 we have, 10Q2=100, implies Q2=10.
The profit =280(30)-[2.5(20)^2+5(10)^2]=6900.
c) [1 mark] Is the profit calculated in part b) more or less than the profit calculated in part
a), and why?
Answer:
Part b) profit – Part a) profit = 6900-4809.09 = 2090.91.
Thus, the profit calculated in part b) is more and the reason is that the additional firm
owned by the monopolist is a lower marginal cost firm than firm 2. And, in the absence
of any fixed costs, the profit had to rise. Note that the quantity supplied is more in part b).
Question 2. [4 marks] A homogeneous products duopoly faces a market demand function
given by P = 500 − 10Q . Both firms have a constant marginal cost of MC = 200 .
a) [1 mark] What would the equilibrium price in this market be if it were perfectly
competitive?
Answer:
If this market were perfectly competitive, then equilibrium would occur at the point
where P = MC . Assuming two firms, this will occur where
500 − 10Q1 − 10Q 2 = 200 .
Since in equilibrium Q1 = Q2 ,
500 − 20Q1 = 200
300 = 20Q1
Q1 = 15
Since both firms will produce the same level of output in equilibrium, both firms will
produce 15 units. At this level of output, price will be
P = 500 − 10(15) − 10(15) = 200
b) [1 mark] What would the equilibrium price in this market be if the two firms colluded
to set the monopoly price?
Answer:
If the firms colluded to set the monopoly price, then
500 − 20Q = 200
300 = 20Q
Q = 15
P = 500 − 10Q
P = 350
c) [2 marks] What is the Bertrand equilibrium price in this market?
Answer:
If the firms acted as Bertrand oligopolists, the equilibrium would coincide with the
perfectly competitive equilibrium of P = 200 and Q = 30 , with each firm producing onehalf of the market output of 15 units each. If either firm tried to raise its price, it would
lose its entire market share.