Solutions
1.
Demand Schedule for Pepsi is given as Q
2
(P
1
,P
2
) = 49.52 5.48P
2
+1.40P
1
Marginal Cost is given as C
2
= 3.96
Follow Steps:
I.
Begin with the Profit function of Pepsi
∏
2
= P
2
Q
2
–
C
2
Q
2
= (P
2
–
C
2
) Q
2
Sub in known Q
2
(P
1
,P
2
) and C
2
∏
2
= (P
2
–
3.96) (49.52 5.48P
2
+1.40P
1
)
∏
2
=71.2208 P
2
–
5.48 P
2
2
+1.40 P
1
P
2
5.544 P
1
–
196.0992
II.
Differentiate with respect to P
2
and set the derivative equal to zero.
III.
Solve for P
2
* (reaction function )
P
2
* = 6.49 +0.1277P
1
2.
This is a standard Cournot Model except marginal costs differ.
Firm 1’s profit function: ∏
1
= (aq
1
q
2
) q
1
–
c
1
q
1
∏
1
=aq
1
q
1
2
q
2
q
1
 c
1
q
1
Differentiate with respect to q
1
and set equal to zero:
∂∏
1
/∂
q
1
=a 2 q
1
–
q
2
–
c
1
= 0
Solve for b
1
(q
2
) i.e.
Firm 1’s best response function:
b
1
(q
2
) = ½ (a
–
q
2
–
c
1
)
By symmetry of the process we can get Firm 2’s best response (just change c
1
to c
2
, and change q
1
to
q
2
): b
2
(q
1
) = ½ (a
–
q
1
–
c
2
)
Sub in b
2
(q
1
) for q
2
in the b
1
(q
2
) (N.B. Logic of this step, Firm 1’s best approximation of
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 Spring '11
 JULES
 Economics, Microeconomics, Game Theory

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