Two firms compete as a Stackelberg duopoly. The demand they face is P = 40 - Q. The cost function for each firm is C(Q) = 4Q. The profits of the two firms are: A) π_L = $162; π_F = $81. B) π_L = $81; π_F = $40.5. C) π_L = $81; π_F = -$40.5. D) π_L = $162; π_F = $40.5.

Question

Quizplus · Accepted Answer

To find the profits of each firm, we need to first determine the quantity produced by both firms. In a Stackelberg duopoly, one firm (the leader) determines its output level first, and then the other firm (the follower) determines its output level in response. So, let's first determine the output level of the leader. The leader's profits can be expressed as: πS1 = (P - MC)Q where MC is the marginal cost, which is equal to the derivative of the cost function, MC(Q) = 4. The leader's profit-maximizing output level can be found by setting the derivative of the profit function equal to zero: dπS1/dQ = (40 - 2Q - 4Q) - 4 = 32 - 6Q = 0 Q1 = 5.33, which we round down to 5 (since we can only produce in discrete quantities). Now, we can determine the output of the follower. The follower's profits can be expressed as: πF = (P - MC)Q - FC where FC is the fixed cost, which is equal to zero. Since the leader is producing 5 units, the demand facing the follower is: Q = 40 - Q1 - QF QF = 40 - Q1 - Q = 40 - 5 - QF 2QF = 35 QF = 17.5, which we round down to 17 (since we can only produce in discrete quantities). Now we can calculate the profits of both firms: πS1 = (40 - 4(5))5 = $160 πF = (40 - 4(17))17 = -$306 Total profits: π_L = πS1 + πF = $160 - $306 = -$146 π_F = 2πS1 = $320 Therefore, the best choice is A.

Two Firms Compete as a Stackelberg Duopoly