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Will Upvote!-Cournot duopoly: Consider a Cournot duopoly, where each firm has marginal cost MC = 40, and the market demand is Q = 200 − ( 1 /2 ) p
Will Upvote!-Cournot duopoly:
Consider a Cournot duopoly, where each firm has marginal cost MC = 40, and the market demand is Q = 200 − ( 1 /2 ) p.
a) What are the best-response functions of each firm? (2 marks)
b) What is the equilibrium output level for each firm? (1 mark)
c) How does the total output level compare to the cartel (collusive) output level?
Expert Solution
There are two firms in cournot dupoly.
Let named them, firm 1 and firm 2.
Q1 is the output produce by firm 1
Q2 is the output produced by firm 2
Q is the total output.
=> Q = Q1 + Q2
---------------------------------------------
Q = 200 - (1/2)P
=> P = 2*(200 -Q)
=> P = 400 - 2Q
=> P = 400 - 2(Q1 + Q2)
=> P = 400 - 2Q1 - 2Q2
-------------------------------------------------------------------------------------------
(a)
Best response function of firm 1.
Firm 1 maximizes profir at MR1 = MC
TR1 = P* Q1
=> TR1 = (400 - 2Q1 - 2Q2) Q1
=> TR1 = 400Q1 - 2Q12 - 2Q2Q1
--
MR1 = dTR1/ dQ1
=> MR1 = 400 - 4Q1 - 2Q2
and
MC = 40,
Set MR1 = MC
=> 400 - 4Q1 - 2Q2 = 40
=> 400 - 2Q2 - 40 = 4Q1
=> 360 - 2Q2 = 4Q1
=> Q1 = (360 - 2Q2)/4
=> Q1 = 90 - 0.5Q2
Hence, the best response function of firm 1 is Q1 = 90 - 0.5Q2
-----------------------
Best response function of firm 2.
Firm 2 maximizes profit at MR2 = MC
TR2 = P * Q2
=> TR2 = (400 - 2Q1 - 2Q2) Q2
=> TR2 = 400Q2 - 2Q1Q2 - 2Q22
---
MR2 = dTR2 / dQ2
=> MR2 = 400 - 2Q1 - 4Q2
and
MC = 40
Set MR2 = MC
=> 400 - 2Q1 - 4Q2 = 40
=> 400 - 2Q1 - 40 = 4Q2
=> Q2 = (360 - 2Q1) /4
=> Q2 = 90 - 0.5Q1
Hence, best response function of firm 2 is Q2 = 90 - 0.5Q1
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(b) Best response function of firm 1: Q1 = 90 - 0.5Q2 --------------eq(1)
Best response function of firm 2: Q2 = 90 - 0.5Q1 ------eq(2)
Put eq(2) in eq(1)
=> Q1 = 90 - 0.5(90 - 0.5Q1)
=> Q1 = 90 - 45 + 0.25Q1
=> Q1 - 0.25Q1 = 45
=> 0.75Q1 = 45
=> Q1 = (45 / 0.75)
=> Q1 = 60
and
put Q1 = 60 in eq(2)
=> Q2 = 90 - 0.5Q1
=> Q2 = 90 - 0.5(60)
=> Q2 = 60
In case of cournot dupoly each firm will produce 60 units in equilibrium.
----------------------------------------------------------------------------------------------------------
(c)
Total output level: Q = Q1 + Q2
=> Q = 60 + 60
=> Q = 120
Total output produced in cornout dupoly is 120 units;
----------------
Both firms join together and form a cartel. And act as a monopoly.
Profir maximizes at MR = MC
=> Q = 200 - (1/2)P
=> (1/2)P = 200 -Q
=> P = 2(200 -Q)
=> P = 400 - 2Q
------
TR = PQ
=>TR = (400 -2Q)Q
=> TR = 400Q - 2Q2
---
MR = dTR / dQ
=> MR = 400 - 4Q
set MR = MC
=> 400 - 4Q = 40
=> 400 -40 = 4Q
=> Q = 360 / 4
=> Q = 90
In case of cartel, total production would be 90 units.
--------
Hence, the total output in cournot (i.e. 120) is higher than total output of cartel (i.e., 90)
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