Question

The market (inverse) demand function for a homogenous good is P(Q) =
10 – Q. There are three firms: firm 1 and 2 each have a total cost of Ci(qi) = 4qi for i ∈ {1.2}. and firm 3 has a total cost of C3(q3) = 2q3. The three firms compete by setting their quantities of production, and the price of the good is determined by a market demand function given the total quantity. Calculate the Nash equilibrium in this game and the corresponding market price.

Answer 1

P = 10 - Q1 - Q2 - Q3

MC1 = dC1/dQ1 = 4

MC2 = dC2/dQ2 = 4

MC3 = dC3/dQ3 = 2

For Firm 1:

TR1 = P x Q1 = 10Q1 - Q12 - Q1Q2 - Q1Q3

MR1 = TR1/Q1 = 10 - 2Q1 - Q2 - Q3

Setting MR1 = MC1,

10 - 2Q1 - Q2 - Q3 = 4

2Q1 + Q2 + Q3 = 6.......(1) [Best response, firm 1]

For Firm 2:

TR2 = P x Q2 = 10Q2 - Q1Q2 - Q22 - Q2Q3

MR2 = TR2/Q2 = 10 - Q1 - 2Q2 - Q3

Setting MR2 = MC2,

10 - Q1 - 2Q2 - Q3 = 4

Q1 + 2Q2 + Q3 = 6.......(2) [Best response, firm 2]

For Firm 3:

TR3 = P x Q3 = 10Q3 - Q1Q3 - Q2Q3 - Q32

MR3 = TR3/Q3 = 10 - Q1 - Q2 - 2Q3

Setting MR1 = MC1,

10 - Q1 - Q2 - 2Q3 = 2

Q1 + Q2 + 2Q3 = 8.......(3) [Best response, firm 3]

Cournot equilibrium is obtained by solving (1), (2) and (3).

(2) x 2 yields:

2Q1 + 4Q2 + 2Q3 = 12.............(4)

2Q1 + Q2 + Q3 = 6...............(1)

(4) - (1) yields:

3Q2 + Q3 = 6.............(5)

(3) x 2 yields:

2Q1 + 2Q2 + 6Q3 = 16.............(6)

2Q1 + Q2 + Q3 = 6...............(1)

(5) - (1) yields:

Q2 + 5Q3 = 10.............(7)

(7) x 3 yields:

3Q2 + 15Q3 = 30..........(7)

3Q2 + Q3 = 6............(4)

(7) - (4) yields:

14Q3 = 24

Q3 = 1.71

Q2 = 10 - 5Q3 [from (6)] = 10 - (5 x 1.71) = 10 - 8.55 = 1.45

Q1 = 6 - 2Q2 - Q3 [from (2)] = 6 - (2 x 1.45) - 1.71 = 4.29 - 2.9 = 1.39

Q = 1.39 + 1.45 + 1.71 = 4.55

P = 10 - 4.55 = 5.45