Question

3. Two firms in the market, 1 and 2, face an inverse demand function given by P(Q1 +Q2) = 400 – 2Q1 – 202 where Q1 is the level of production of firm 1 and Q2 is the level of production of firm 2. The cost function of firm 1 is C1 (Q1) = (Q1) and the cost function of firm 2 is C2 (Q2) = (Q1). The two firms compete in quantities (i.e., Cournot competition). (a) Set up the maximization problem of firms 1 and 2. (10 Points) (b) Find the equilibrium quantities of firms 1 and 2. (10 Points) (c) Find the equilibrium profits of firms 1 and 2. (10 Points)

Answer 1

(a)

P = 400 - 2Q1 - 2Q2

For firm 1,

TR1 = P x Q1 = 400Q1 - 2Q12 - 2Q1Q2

C1 = Q12

Profit (Z1) = TR1 - C1 = 400Q1 - 2Q12 - 2Q1Q2 - Q12 = 400Q1 - 3Q12 - 2Q1Q2

For Firm 2,

TR2 = P x Q2 = 400Q2 - 2Q1Q2 - 2Q22

C2 = Q12

Profit (Z2) = TR2 - C2 = 400Q2 - 2Q1Q2 - 2Q22 - Q12

(II)

For firm 1, profit is maximized when $1584457081205_blob.png$Z1/$1584457081245_blob.png$Q1 = 0.

$1584457081228_blob.png$Z1/$1584457081271_blob.png$Q1 = 400 - 6Q1 - 2Q2 = 0

6Q1 + 2Q2 = 400.........(1) [best response, firm 1]

For firm 2, profit is maximized when $1584457081236_blob.png$Z2/$1584457081259_blob.png$Q2 = 0.

$1584457081253_blob.png$Z2/$1584457081262_blob.png$Q2 = 400 - 2Q1 - 4Q2 = 0

2Q1 + 4Q2 = 400.........(2) [best response, firm 2]

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

6Q1 + 12Q2 = 1200........(3)

6Q1 + 2Q2 = 400.........(1)

(3) - (1) gives us:

10Q2 = 800

Q2 = 80

Q1 = (400 - 4Q2)/2 [from (2)] = [400 - (4 x 80)]/2 = (400 - 320)/2 = 80/2 = 40

(c)

Q = 40 + 80 = 120

P = 400 - 2 x (Q1 + Q2) = 400 - 2Q = 400 - (2 x 120) = 400 - 240 = 160

Profit, Firm 1 = (400 x 40) - (3 x 40 x 40) - (2 x 40 x 80) = 16,000 - 4,800 - 6,400 = 4,800

Profit, Firm 2 = (400 x 80) - (2 x 40 x 80) - (2 x 80 x 80) - (40 x 40) = 32,000 - 6,400 - 12,800 - 1,600 = 11,200

NOTE: You can just check once if C2(Q2) = (Q1)2, since in general, cost function of one firm does not depend on output of another firm. I've solved this only on basis of information given by you.