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 profitmaximizing quantity, take the derivative of the profit function with
respect to q
1
and set this firstorder 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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Arnold
 Game Theory, Substitute good, fi rm, best response, Cournot

Click to edit the document details