3. Suppose the two firms cannot collude and instead compete in the Cournot Model in the...

P = 18 - Q = 18 - Q1 - Q2 [since Q = Q1 + Q2]

C1 = 0.5Q1²

C2 = 0.5Q2²

(a)

For firm 1,

TR = P x Q1 = 18Q1 - Q1² - Q1Q2

Profit (Z1) = TR1 - TC1 = 18Q1 - Q1² - Q1Q2 - 0.5Q1² = 18Q1 - 1.5Q1² - Q1Q2

Firm 1's optimization problem is:

Maximize Z1 = 18Q1 - 1.5Q1² - Q1Q2

Subject to Q1 >= 0, Q2 >= 0.

(b)

Firm 1 maximizes profit when Z1/Q1 = 0

18 - 3Q1 - Q2 = 0

3Q1 + Q2 = 18...........(1) [Best response, firm 1]

(c)

For firm 2,

TR2 = P x Q2 = 18Q2 - Q1Q2 - Q2²

Profit (Z2) = TR2 - TC2 = 18Q2 - Q1Q2 - Q2² - 0.5Q2² = 18Q2 - Q1Q2 - 1.5Q2²

Firm 2's optimization problem is:

Maximize Z2 = 18Q2 - Q1Q2 - 1.5Q2²

Subject to Q1 >= 0, Q2 >= 0.

Firm 2 maximizes profit when Z2/Q2 = 0

18 - Q1 - 3Q2 = 0

Q1 + 3Q2 = 18...........(2) [Best response, firm 2]

(2) x 3 yields:

3Q1 + 9Q2 = 54.........(3)

3Q1 + Q2 = 18.........(1)

(3) - (1) yields:

8Q2 = 36

Q2 = 4.5

Q1 = 18 - 3Q2 [from (2)] = 18 - (3 x 4.5) = 18 - 13.5 = 4.5

Q = 4.5 + 4.5 = 9

P = 18 - 9 = 9

(d)

The firms do not have an incentive to deviate from their respetive output at the equilibrium price, since both are maximizing profit at this price-output combination. So this is a Nash equilibrium.

NOTE: As HOMEWORKLIB's Policy, only 1st 4 parts can be answered. You need to post (e) and (f) in separate question.