Econ 485
Fahad Khalil
Answer key 2
1. Unique NE is (T,L), but it is Pareto Dominated by (B,R)
L
C
R
1
, 1
0, 0
0, 0
0, 0
0, 1
3, 0
T
M
B
0, 0
1, 0
2, 2
2. To obtain an understanding of what F does, let us look at the problem of firm 1 given
q
2
.
Firm 2’s problem will be similar.
π
1
(q
1
, q
2
) = (1-q
1
-q
2
)q
1
– F
∂π
1
/∂
q
1
= 1 – 2q
1
– q
2
= 0, which implies that BR
1
(q
2
) is given by
(1-q
2
)/2
if
π
1
(q
1
(q
2
), q
2
)
≥
0,
q
1
(q
2
)
=
0
if
π
1
(q
1
(q
2
), q
2
) < 0.
Note that
π
1
(q
1
(q
2
), q
2
) = (1-q
1
(q
2
)-q
2
)q
1
(q
2
) – F, therefore, we can replace (1-q
2
)/2 for
q
1
(q
2
) to show that
π
1
(q
1
(q
2
), q
2
)
≥
0
⇔
[1 - (1-q
2
)/2 – q
2
] (1-q
2
)/2
≥
F
⇔
1- 2
√
F
≥
q
2
.
This means the BR
1
(q
2
) will look as follows depending on the value of F.
q
2
q
2
1- 2
√
F
q
1
q
1
F
=
0
F
>
0
If F = 0, we get the usual BR with no discontinuity.
But if F > 0, the optimal q
1
is zero
for q
2
≥