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Stackelberg Model

Stackelberg Model. Like Cournot, there are 2 firms that set quantity However, game is not simultaneous, it is sequential What does this mean? One firm, the leader, chooses quantity first The other firm, the follower, observes the leader’s quantity and then chooses quantity

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Stackelberg Model

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  1. Stackelberg Model • Like Cournot, there are 2 firms that set quantity • However, game is not simultaneous, it is sequential • What does this mean? • One firm, the leader, chooses quantity first • The other firm, the follower, observes the leader’s quantity and then chooses quantity • Once the two quantities are chosen, price is set to clear the market

  2. How do we solve this game? • Work backwards -- use backward induction • Start at the last step -- setting price to clear the market • P = a - b(qL + qF ) • Next step before that -- follower chooses quantity to maximize profit given leader’s choice. • F = (a - b(qL + qF ) - c) qF • Take derivative and set = 0 to get BR • a - bqL - 2bqF - c = 0 • qF* = (a - bqL - c)/2b

  3. Now go the first step -- leader chooses quantity to maximize profit • L = (a - b(qL + qF ) - c) qL • However, leader knows how follower will respond -- leader can figure out follower’s BR, so: • L = (a - b(qL + (a - bqL - c)/2b) - c) qL • Simplify to get L = (a - bqL - c)/2 qL • Take derivative and set equal to 0 to get BR: (a - 2bqL - c)/2 = 0  qL = (a - c)/2b • And qF*=(a-bqL-c)/2b=(a-b(a-c)/2b-c)/2b=(a-c)/4b

  4. Leader has the advantage -- he sets higher quantity and gets a higher profit than the follower • Often called the “first-mover” advantage • Total output = (a-c)/2b + (a-c)/4b = 3(a-c)/4b • Greater than total Cournot output of 2(a-c)/3b • So Stackelberg competition not as “bad” as Cournot, but firms still make profit.

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