Cornell University
Economics 3130
Problem Set 11 Solutions
1. In a given market, there are two ﬁrms
i
=
{
1
,
2
}
that face the demand curve
Q
D
= 16

p
for
p
≤
16 and
Q
D
= 0 for
p >
16. As usual,
Q
D
is the aggregate quantity demanded.
The ﬁrms have no ﬁxed cost. Each has a marginal cost of production of
c
i
>
0. The
ﬁrms compete by simultaneously choosing an amount of (identical) goods
x
i
to produce
(fractions are allowed), and the amount each produces must be less than or equal to
some high number ¯
x
.
For parts (a)(d), assume that
c
1
=
c
2
= 4.
(a) What is the cost function for ﬁrm
i
?
c(w,y) = 4y
(b) What are the possible actions for ﬁrm
i
?
a
i
= [0
,
¯
x
]
(c) What is ﬁrm
i
’s best response function?
BR
i
(
x
j
) = 6

x
j
2
(d) Which actions for ﬁrm
i
are strictly dominated? Can this game be solved through
iterated elimination of strictly dominated strategies/actions. If so, solve it. If not,
explain why not.
Any strategy of
x
1
>
6 is dominated. If ﬁrm 2 chooses to produce zero, then
the best response is
x
i
(0) = 6. By the same logic, any strategy of of
x
2
>
6 is
dominated.
Knowing that
x
1
≤
6, ﬁrm 2 will never choose
x
2
<
3. The same is true for ﬁrm
1.
Knowing that
x
1
∈
[3
,
6], ﬁrm 2 will never choose
x
2
>
4
.
5. The same is true for
ﬁrm 1.
Knowing that
x
1
∈
[3
,
4
.
5], ﬁrm 2 will never choose
x
2
<
3
.
75. The same is true
for
q
2
.
You should recognize a pattern. After
n
eliminations, a range of
12
2
n
remains. In
the limit, the range contains only the single strategy
x
i
= 4.
1
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View Full Document(e) What are the pure strategy Nash equilibria? What are the proﬁts of each ﬁrm in
each equilibrium?
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 Spring '06
 MASSON
 Economics, Microeconomics, Game Theory, Firm, Pareto efficient outcomes, CORNELL UNIVERSITY Economics

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