Consider an industry in which in which each firm has the following total cost fu
ID: 1194677 • Letter: C
Question
Consider an industry in which in which each firm has the following total cost function:
TC = 50q.
The market demand curve for the product is given by:
Q = 1000 – 2P.
(a). Suppose there are two firms producing this good and they behave as Cournot competitors. Find the Nash equilibrium price, the output that each firm produces, and the profit that each firm earns.
(b). Now, suppose the two firms enter the market sequentially. Firm 1 is the leader and Firm 2 is the follower. When Firm 1 enters the industry, it knows that Firm 2 will choose its output level on the assumption that Firm 1 will not adjust output. Find the Nash equilibrium price, output per firm, and profit per firm.
(c). Finally, suppose these two firms collude to maximize their combined profit. How much output will they produce? What price will they charge? How much profit will each firm earn if they decide to divide production evenly between them?
Explanation / Answer
The demand function is
Q = 1000 – 2P
2P = 1000 – Q
P = 500 – Q/2
Since Q = Q1+Q2,
P = 500 – (Q1+Q2)/2
P = 500 – Q1/2 – Q2/2
which is the market’s inverse demand curve.
(a) Firm 1’s profit is
1 = revenue of firm 1 – cost of firm 1
= PQ1 – 50Q1
= (500 – Q1/2 – Q2/2)Q1 – 50Q1
= 500Q1 – Q12/2 – (Q1Q2)/2 – 50Q1
= 450Q1 – Q12/2 – (Q1Q2)/2
To maximize profit, differentiate the above function with respect to Q1 and equate to 0.
1/Q1 = 0
450 – Q1 – Q2/2 = 0
Q1 = 450 – Q2 /2 …… (1)
which is the best response function of firm 1.
Similarly, firm 2’s best response function is
Q2 = 450 – Q1/2 …… (2)
Substitute (2) in (1).
Q1 = 450 – (450 – Q1/2)/ 2
Q1 = 450 – (225 – Q1/4)
Q1 = 225 + Q1/4
3Q1/4 = 225
Q1 = 300
And
Q2 = 450 – 300/2 = 300
Therefore, under Cournot equilibrium, each firm will produce 300 units. Total output is 600 units.
Nash equilibrium price is
P = 500 – Q/2
= 500 – 600/2
= 200
Profit of each firm = revenue of each firm – cost of each firm
= PQi – 50Qi
= 300P – 50(300)
= 300(200) – 15000
= 60000 – 15000
= 45000
(b) Given firm 1 produces Q1 units of output, firm 2’s best response function is
Q2 = 450 – Q1/2
Firm 1’s profit function is
1 = revenue of firm 1 – cost of firm 1
= PQ1 – 50Q1
= (500 – Q1/2 – Q2/2)Q1 – 50Q1
= 500Q1 – Q12/2 – (Q1Q2)/2 – 50Q1
= 450Q1 – Q12/2 – (Q1Q2)/2
= 450Q1 – Q12/2 – (Q1/2)Q2
Substitute Firm 2’s best response function in firm 1’s profit function.
1 = 450Q1 – Q12/2 – (Q1/2)Q2
= 450Q1 – Q12/2 – (Q1/2)(450 – Q1/2)
= 450Q1 – Q12/2 – (225Q1 – Q12/4)
= 450Q1 – Q12/2 – 225Q1 + Q12/4
To maximize profit, differentiate the above function with respect to Q1 and equate to 0.
1/Q1 = 0
450 – Q1 – 225 + Q1/2 = 0
225 – Q1/2 = 0
Q1 = 450
Therefore, Firm 1 will produce 450 units.
Firm 2 will produce
Q2 = 450 – 450/2
Q2 = 450 – 225
Q2 = 225
Total output is 675 (=450+225) units.
Equilibrium price is $162.5 (= 500 – 675/2).
Firm 1’s profit is
1 = revenue of firm 1 – cost of firm 1
= PQ1 – 50Q1
= (162.5)(450) – 50(450)
= 73175 – 22500
= 50625
Firm 2’s profit is
2 = revenue of firm 2 – cost of firm 2
= PQ1 – 50Q1
= (162.5)(225) – 50(225)
= 36562.50 – 11250
= 25312.50
(c)
Suppose two firms collude and operate as monopoly does. Find the profit maximizing output.
= revenue – cost
= PQ – 4Q
= (500 – Q/2)Q – 50Q
= 450Q – Q2/2
To maximize profit, differentiate the above function with respect to Q and equate to 0.
/Q = 0
450 – Q = 0
Q = 450
Therefore, total output under collusion will be 450 units. And Each firm will produce 225 (=450/2) units.
Nash equilibrium price is
P = 500 – 450//2
= 500 – 225
= 275
Profit of each firm = revenue of each firm – cost of each firm
= PQi – 50Qi
= 225P – 50(225)
= 225(275) – 11250
= 50625