Equilibrium and Profit Levels in Cournot Duopoly

This article discusses the equilibrium and profit levels in a Cournot duopoly scenario. It explains the response function, equilibrium quantity, output levels, and profit calculations. The analysis is based on the assumption of equal marginal costs for both firms. The article also explores the scenario of a leader-follower duopoly and the Nash-Bertrand equilibrium. Additionally, it touches upon the market demand for boxes and the concept of monopoly pricing. Lastly, it mentions the calculation of aggregate industry profit.

1a) Note: We will assume the subscripts for typing cases.
Response function= (a-c-bQ2)/2b
Q1 and Q2 rep the quantities of either output and a b c are the intercepts of the demand
function K= a-bQ
C= marginal cost
For cournot firms, we get our Q1 and Q2 as follows
P= 12-Q
MC1=3
MC2=3
Q1= (12-3-3Q2)/3X3
= (9-3Q2)/4
=9/4-3Q2/4
Since MC1 and MC2 are equal, then Q1=Q2
Thus the response function is
12-Q= 9/4-3Q2/4
9.75=1/4Q
Hence Q= 39
b)

The equilibrium is at b1(0)
c) From a) it is clear that the output levels of the equilibrium cluster at Q=39. Since the MCI
and MC2 are on same response function
d) Equilibrium:12-Q=Q
Assumption, Q1=Q2=Q
Thus P=12
e) Profit level
The two firms are operating in the same market structure. In my case, I assume the duopoly
situation
Profit =TR-TC
12-Q =3
Q=9
2a) MCA=1, and

MCB= 2
Q1= (12-Q-Q2)/1X1
= (9-Q2)
Q2= (12-Q-3Q2)/4
=3-Q2
b) For the leader firm A, we use: Q1= (9-Q2)
And for follower=3-Q2,
For A and B, it follows that:
(9-Q2)+ (3-Q2) =12-Q, But Q=9, Thus,
(9-Q2)+ (3-Q2) =12-9
(9-Q2)+ (3-Q2) =3
C) Profit level:
Profit =TR-TC
For firm A: 12-Q =1
Q=13
For firm B: 12-Q =2
Q=14

d)ATC = qs + 3 + 36/qs = 2qs + 3 = MC
36/qs = qs
36 = qs, on square root, 6 = qs
3a) let aggregate market demand for boxes p = 12 − Q/2, p=price per box and Q=number of
boxes sold.
Case1: A undercuts B on setting pA = cB −= 8 −, is a small number.
Firm B sets pB = cB = $8.
Consumers purchase brandA only,
Thus, solving 8 = 12 − qA/2 givesqA = 8 and qB = 0.
ProfitsπA = (8 − 6)8 = $16,πB = 0.
Case2: A sets a monopoly price.
MR = 12 − qA = cA = 6 yields qA = 6.
p = 12 − 6/2 = $8 > $8 = pB.
Firm A would not charge its monopoly price.
Hence Nash-Bertrand equilibrium: p b A = $8 − and p b B = $8.
Equilibrium profits: π b A = (8 − 6)8 = $16, π b B = 0.
b) We check where A undercuts B on setting pA = cB − = 8 −, is small number. B sets pB =
cB = $8. Consumers, buy brand A only, solving

8 = 12 − qA/2 gives qA = 8 and qB = 0.
Profits πA = (8 − 2)8 = $48 and πB = 0.
Case 2 to be checked when A has monopoly price.
MR = 12 − qA = cA = 2 results to qA = 10.
p = 12 − 10/2 = $7 < $8 = pB.
πA = (7 − 2)10 = $50 > $48.
c) The Nash-Bertrand equilibrium is p b A = $7 and p b B = $8,
Equilibrium profit π b A = (7 − 2)10 = $50 and π b B = 0.
4a) q b A = 100 − 2 · 56 = 44 = 112 and
q bB = 180 − 2 · 44 + 56 = 88.
Thus, π b A = 56 · 112 = $6272 and
π b B = 44 · 88 = $3872. Finally, aggregate industry profit is: Πb = π b A + π b B = $10, 144.
b) 100-4t=Q, so Q=20, P=20,
1-t, P=MC, so 20=4+4q, so q=4
T is a variable to be depended on since it marks the VHS extinct