Stackelberg and Bertrand Games

Stackelberg and Bertrand Games - S tackelberg Game For the...

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Stackelberg Game For the Stackelberg problem P = 100 – Q and marginal cost equals 10. Firm 1 (the leader) determines what output firm 2 (follower) will produce by deriving firm 2’s best response function. This is the same best response function that the firm would have if in a Cournot game: q 2 = (90 – q 1 )/2. Firm 1 substitutes this best response function for q 2 in its own profit function as follows: Profit = Pq 1 – 10q 1 = (100 – q 1 – q 2 )q 1 - 10q 1 = (100 – q 1 - (90 – q 1 )/2)q 1 - 10q 1 = (45 – q 1 )q 1 . To determine the profit-maximizing quantity, take the derivative of the profit function with respect to q 1 and set this first-order condition equal to 0 => 45 – q 1 = 0 => q 1 = 45. Note that the follower firm will set its output based on its best response function => q 2 = (90 – q 1 )/2 = (90 – 45)/2 = 22.5. Total output is the sum of the two firms’ outputs => 45 + 22.5 = 67.5. You will find a similar numerical example on page 465. Note in that example that the first
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Stackelberg and Bertrand Games - S tackelberg Game For the...

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