Question

Consider a Cournot duopoly with the following inverse demand function: P = 96 - 2Q1 - 2Q2, where Q1 and Q2 are quantities produced by firms 1 and 2, respectively. The firms' marginal cost are identical and given by MCi(Qi) = 2Qi, where i is either firm 1 or firm 2. Based on this information firm 1 and 2's reaction functions are:_____.

a. MR1(Q1, Q2) = 100-2Q1-Q2 and MR2(Q1, Q2) 100-Q1-2Q2.
b. MR1(Q1, Q2) = 100-4Q1-2Q2 and MR2(Q1, Q2) = 100 - 2Q1 - 402.
c. MR1(Q1, Q2) = 100 - 2Q1 - 4Q2 and MR2(Q1, Q2) = 100 - 4Q1 - 2Q2.
d. MR1(Q1, Q2) = 24.5 - 0.5Q2 and MR2(Q1, Q2) = 24.5 - 0.5Q1.
Two identical firms compete as a Cournot duopoly. The demand they face is P = 90-Q. The cost function for each firm is C(Q_i) = 6Qi. Each firm earns equilibrium profits of:_____.

Answer 1

Answer: See explanation

Step-by-step explanation:

The revenue - firm 1 will be:

= P x Q1

= 100Q1 - 2Q1² -2Q1Q2

Then, marginal revenue MR1 will be:

= dR1/dQ1

= 100 - 4Q1 -2Q2

Similarly, the revenue for firm 2 will be:

= P x Q2

= 100Q2 - 2Q2² -2Q1Q2

Then, MR2 will be:

= 100 - 4Q2 - 2Q1

Therefore, MR1(Q1, Q2) = 100-4Q1-2Q2 and MR2(Q1, Q2) = 100 - 2Q1 - 402.

Option B is the correct answer.

P = 90 - Q

Q = Q1 + Q2

The revenue for firm 1 (R1) will be:

= PQ1

= 90Q1 - Q1² - Q2Q1

The marginal revenue MR1 will be:

= 90 - 2Q1 - Q2

The marginal cost MC is:

= dC/dQ

= 6

Since profit is maximized when MR = MC, this will be:

90 - 2Q1 - Q2 = 6

- 2Q1 - Q2 = 6 - 90

- 2Q1 - Q2 = - 84

2Q1 + Q2 = 84

3Q = 84

Q1 = Q2 = 28

Therefore,

P = 90-Q.

P = 90 - (Q1 + Q2)

P = 90 - (28 + 28)

P = 90 - 56

P = 34

Then, the equilibrium profit will be:

= Total revenue - Total cost

= (34 × 28) - (6 × 28)

= 952 - 168

= 784