Consider a market with demand P (Q) = 422Q in which two firms compete. Firm 1 fa
ID: 1109888 • Letter: C
Question
Consider a market with demand P (Q) = 422Q in which two firms compete. Firm 1 faces TC1(Q) = 9Q and firm 2 faces TC2(Q) = 4Q. Suppose that firm 1 chooses their quantity first, then firm 2 sees that quantity, and then firm 2 sets their quantity (Stackelberg duopoly).
(a) Find the reaction function for firm 2.
(b) What is the marginal revenue for firm 1 (given that they know that firm 2 will best respond)?
(c) Find the Stackelberg equilibrium (Q1,Q2,P).
(d) Determine profits for each firm and consumer sur-
plus.
60 s. Consider a market with demand P(Q) = 42-2Q in which two firms compete. Firm 1 faces TC1 (Q) 9Q and firm 2 faces TC2(Q) 4Q. Suppose that firm 1 chooses their quantity first, then firm 2 sees that quantity, and then firm 2 sets their quantity (Stackelberg duopoly). ity (a) Find the reaction function for firm 2. (b) What is the marginal revenue for firm 1 (given that they know that firm 2 will best respond)? (c) Find the Stackelberg equilibrium (Qi, Q2, P*) (d) Determine profits for each firm and consumer sur- et. ty? ach plus. (e) Which firm is better off? mer 6. Suppose that Pepsi and Coke are competing in a horizon- tally differentiated Bertrand market and setting prices. The demand curves are as follows: 2 inExplanation / Answer
In Stackelberg model where firm 1 is a first mover, it must take the reaction function of firm 2 in its computation of marginal revenue.
Derivation of firm 2’s reaction function
Total revenue of firm 2 = P*(q2) = (42 – 2(q1 + q2))q2 = 42q2 – 2q22 – 2q1q2
Marginal revenue = 42 – 4q2 – 2q1
Marginal cost = 4
Solve for the reaction function
42 – 4q2 – 2q1 = 4
4q2 = 38 – 2q1
q2 = (9.5 – 0.5q1)
Incorporate this in the reaction function of firm 2
Total revenue for firm 2 = P*(q1) = (42 – 2(q1 + q2))q1
= (42 – 2q1 – 2(9.5 – 0.5q1))q1
= (23 – q1)q1
MR = 23 – 2q1
Equate MR = MC
23 – 2q1 = 9
2q1 = 14
q1 = 7
q2 = 9.5 – 0.5*7 = 6
Price = 42 – 2(6 + 7) = $16
Profit for firm 1 = 16*7 – 9*7 = $49
Profit for firm 2 = 6*16 – 4*6 = $60
Firm 2 is better off as its profits are higher.
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