Duopoly again

1. Duopoly again Here we look at the Stackelberg leader/follower model

2. The Cournot model was a simultaneous game in the sense that each player did not know what the other did before their own action. The Stackelberg duopoly model is sequential in that one player will act, the other will see the action and then act. As an example, let’s use the same demand and cost conditions as we used in the Cournot example: P = 100 – 2Q, or = 100 - 2(q1 + q2), and MC = 10 for both. Firm 1 is the leader in the example. There are industries where we have leaders. In cars it is still probably GM. What about in software? Computer Chips?

3. Firm 1 being the leader thinks that with the market demand firm 2 will take what ever part of the market firm 1 leaves behind. In the sense of Cournot, firm 1 understands that firm 2 has a best response function indicating what firm 2 should make. As before in the Cournot case, the best response function firm 2 has is found by: MR = MC for firm 2, giving 100 - 2q1 - 4q2 = 10, or q2 = (90/4) - (2/4)q1, Firm 1 then thinks that since firm 2 will follow in this way I (firm 1) will put this into the demand and see what is my best option where MR = MC.

4. P = 100 - 2q2 - 2q1, is market demand P = 100 – 2[ (90/4) - (2/4)q1 ]- 2q1, by sub. of q2 from firm 2, = 55 - q1. Firm 1 MR = MC is then MR = 55 – 2q1 = 10 = MC, or q1 = 22.5. Then the follower will have q2 = (90/4) - (2/4)22.5 = 11.25. Total output in the market is 33.75 and thus the market price is (from the demand curve) 100 – 2(33.75) = 32.5.

5. Summary: P Q profit Monopoly 55 22.5 1012.5 P. Comp 10 45 0 for each firm Duopoly - Cournot 40 30 450 for each firm Doupoly – Stack. 32.5 33.75 506.25 for leader 253.13 for foller So, the Stackelberg Duopoly leads the industry closer to the competitive solution than did the Cournot solution. This seems ironic because usually, when we have one leader firm, the leader gets criticized. But here it leads to better result, short of competition.