Microeconomics, monopoly pricing, game theory
mags1020
ps4answers.docSection I: Monopoly pricing (50 points)
Milwaukee Utilities has a complete monopoly over the generation and transmission of energy. The following information on this company is given as follows:
Demand = 500 - 6Q
Average cost = 250 - Q
Where Q is measured in megawatts and prices and costs are measured in dollars.
How much energy would be sold and at what price if
a.) The firm sets price as a profit maximizing monopolist? Note: The marginal cost curve is twice as steep as the average cost curve.
b.) What is the firm’s profits at the monopoly price determined in part a?
c.) Now, suppose the firm adopts a two-part tariff pricing scheme for its customers such that the access fee is equal to the profit-maximizing marginal cost and the user fee is the difference between the profit maximizing monopoly price and marginal cost. Please calculate the user and access fees based on this information.
d.) Now suppose the firm practices 3rd degree price discrimination and charges the profit-maximizing price to the high reservation price customers and charges a 10 percent discount on the monopoly price to low reservation price customers. Note, low reservation price customers are those who would never pay the monopoly price. What is the price charged to the low reservation price customers? What is the profit generated by charging these profits? Are the profits greater than the profits in part ‘b’? Please explain.
e.) Now suppose the state public utility commission requires this firm to charge the competitive price, how much energy would be sold and at what price? What is the firm’s profits?
f.) Based on the profits obtained when forcing this monopoly to charge a competitive price, the regulator now requires this monopoly to set price equal to average cost (this is called second-best pricing). What is the firm’s profits when charging second-best prices?
Please show all work to receive full credit.
Section II: Game theoretic approach toward analyzing output behavior of rivals (50 points)
Firms X and Y are duopolists facing the same two strategy choices. They can either tacitly collude or they can compete in a Cournot fashion. The market demand for their product, as well as their respective cost curves are as follows:
C(qx) = C(qy) =50qi (firm X and Y’s total cost curves), where i=x or y
MC(qy) =MC(qy) = 50 (firm X and Y’s marginal cost curves)
P=500-5Q, (market demand), where Q = qx + qy .
C(q) and have the same cost structure: marginal cost and average cost both=50
a.) Calculate the respective output levels of each firm if they collude to set monopoly prices.
b.) Calculate the respective output levels of each firm if they adhere to the Cournot model.
c.) What four possible output combinations are available in this game?
d.) Derive the for possible profit outcomes for each firm that arise from producing the four possible output combinations available in this game.
e.) Use these profit outcomes to construct a 2×2 normal representative matrix for this game.
f.) Does either firm have a dominant strategy? If so, what is it?
g.) Is there a Nash equilibrium for this game? If so, what is it?
h.) Is the outcome of this game a prisoner’s dilemma? Please Explain?
Please show all work to receive full credit.
Section I: Monopoly pricing
Milwaukee Utilities has a complete monopoly over the generation and transmission of energy. The following information on this company is given as follows:
Demand = 500 - 6Q
Average cost = 250 - Q
Where Q is measured in megawatts and prices and costs are measured in dollars.
How much energy would be sold and at what price if
Approach used to answer succeeding questions: Graph that depicts monopoly and 3rd degree price discrimination and the competitive price set by the regulator for this example:
$
PH =PM
PL AC
MC
PC MR D
QM Q* QC Q
Since, Average revenue (Demand) = 500 -6Q
And Average cost = 250 - Q
Then
Marginal revenue = 500 -12Q
Marginal cost = 250 -2Q
So for monopoly pricing (MR=MC) So for Competitive Pricing (P=MC)
500-12Q=250-2Q 500-6Q=250-2Q
250=20Q 250=4Q
QM =25 QC =62.5
So, for the low reservation price customer
PL = 350- (.10×350)=315
g.) the firm sets price as a monopoly?
Ans. So for monopoly pricing (MR=MC)
500-12Q=250-2Q
QM =25
P=500-6Q, so for the monopoly output level P=500-(6×25) =$350= PM
h.) What is the firm’s profits at the monopoly price determine in part a?
Π= (P×Q) – TC; note TC=AC×Q
= $3125.
i.) Now, suppose the firm adopts a two-part tariff pricing scheme for its customers such that the access fee is equal to the profit-maximizing marginal cost and the user fee is the difference between the profit maximizing monopoly price and marginal cost. Please calculate the user and access fees based on this information.
Ans. User fee=350-200 = 150
Access fee =250-(2(25) =200
j.) Now suppose the firm practices 3rd degree price discrimination and charges the profit-maximizing price to the high reservation price customers and charges a 10 percent discount on the monopoly price to low reservation price customers. Note, low reservation price customers are those who would never pay the monopoly price. What is the price charged to the low reservation price customers? What is the profit generated by charging these profits? Are the profits greater than the profits in part ‘b’? Please explain.
Answer: The monopoly price is the price charged to the high reservation price consumers so PH = 350. The price charged to the low reservation price consumers is 10% less that the monopoly price so PL = 315. Remember the monopoly output is 25=QH and the number of customers at the low discount rate is 5.8333=QL. Hence, the firm’s profit when engaging in the described 3rd degree price discrimination is 3125+413.192=3538.192. The firm generates higher profits compared to single monopoly pricing because it obtains an additional amount of profit equaling 413.192 due to charging the lower price to low reservation price consumers.
k.) Now suppose the state public utility commission requires this firm to charge the competitive price, how much energy would be sold and at what price? What is the firm’s profits?
Answer:
(P=MC)
500-6Q=250-2Q
QC =62.5
Π =(125×62.5) – (187.5×62.5) = -3906.25
Note that the firm experiences a loss because its average cost decreases as output increases, hence marginal cost is always less than average cost for a given output level.
l.) Based on the profits obtained when forcing this monopoly to charge a competitive price, the regulator now requires this monopoly to set price equal to average cost (this is called second-best pricing). What is the firm’s profits when charging second-best prices?
Answer:
(P=AC)
500-6Q = 250-Q. Thus, Q=50 and P=200 and π=0
Section III: Game theoretic approach toward analyzing output behavior of rivals—25 points
i.) If the two firms collude and play as monopoly, they will produce the same quantity and set the same price.
Ans. Here’s the approach that uses calculus:
P=500-5Q
Where 1/2Q= qx =qy
Total revenue is
TR=P×Q = (500-5Q)Q =-5Q2 + 500Q
The profit maximizing condition for monopoly is
MRMC
=
, which is-10Q +500 = 50
So, Q=45/10
Thus the respective output of each firm is 22.5 if they collude and set monopoly prices.
Here’s the approach that does not use calculus:
P=500-5Q
Since the MR curve is twice as steep as the demand curve MR=500-10Q
Setting MR=MC give the following:
500-10Q= 50, hence Q=45
Thus, the respective output of each firm is 22.5 if they collude and set monopoly prices.
j.) Ans. Here’s the approach that uses calculus
In the Cournot model, each firm chooses price and output to maximize its own profit.
For firm X;
(1)
FOC:
(2)
For firm Y, it will produce
(3)
Solve (2) and (3) simultaneously,
qx = qx =30 (4)
Here’s the approach the doesn’t use calculus
P= 500 – 5qx -5qy
Since the slope of the MR curve is twice as steep as the slope of the demand curve
MR1 =500 - 10qx -5qy
Setting MR1 =MC gives 500 - 10qx -5qy = 50
Thus, 450-5qy= 10qx So the reaction function for firm X is qx= 45-(1/2)qy
The same approach gives qy= 45-(1/2)qx for firm Y.
Solving simultaneously gives
Thus, qx and qy equal 30 if they adhere to the Cournot model
Here are the answers for the rest of this question.
c.) qx =22.5 and qy=22.5
qx =22.5 and qy=30
qx =30 and qy=22.5
qx =30 and qy=30
d.) The potential profit levels for Firm X are as follows:
(x=5062.5 if both firms collude
(x=4500 if both firms adhere to the Cournot model
(x=4668.75 if Firm X produces at the collusion output level and Firm Y produces at the Cournot output level
(x=6225 if Firm Y produces at the collusion output level and Firm X produces at the Cournot output level
The potential profit levels for Firm Y are as follows:
(y=5062.5 if both firms collude
(y=4500 if both firms adhere to the Cournot model
(y=4668.25 if Firm Y produces at the collusion output level and Firm X produces at the Cournot output level
(y=6225 if Firm X produces at the collusion output level and Firm Y produces at the Cournot output level
e.)
Firm X's strategy
Collude Cournot
qx =22.5 qx = 30
(y (x (y (x
Collude
qy = 22.5 $5062.5 $5062.5 $4668.75 $6225
Firm Y’s
Strategy
Cournot
qy= 30 $6225 $4668.75 $4500 $4500
f.) Yes. Cournot for both firms.
g.) Yes. Cournot for both firms.
h.) Yes, since collusion would maximize both firms’ pay-off.
Section III: Game theoretic approach toward analyzing output behavior of rivals in a repeated game—25 points
Here are the answers
a.) Yes. Both firms compete.
b.) Yes. Both firms compete.
c.) (2=20. Yes, because 20>5.
d.) (2=65
e.) (2=100
f.) No. 65<100.
g.) The payoff is 110 for firm-2.
h.)
Firm-1
Collude Collude 1st nine periods and
entire 10 periods cheat in 10th period
(2 (1 (2 (1
Collude 100 100 90 110
entire 10 periods
Firm-2
Collude 1st nine 110 90 95 95
periods and cheat in
10th period
i.) Yes. Both firms cooperate and collude for the first 9 periods then compete (cheat) the last period.
j.) Yes, since both firms would be better off colluding throughout the 10 period game.
k.) No. Firm 2’s profit falls by five for each period. (i.e. (2=100 if it cheats in period 10 and(2=95 if it cheats in period 9…and last, (2=65 if it cheats in period 1.)
