Geography Reference
In-Depth Information
Appendix
Proof of Proposition 1 and Corollary 1
Average proi ts follow in a straightforward way by computing p 5
x
p
1
(
x
)
1
(
1 2
x
)
p
2
(
x
)
with p
1
(
x
)
and p
2
(
x
)
as in (24.12) and in (24.17). Total costs in each location are equal
because (24.14) implies:
a
j
5
N
a
2
a
i
5
a
i
51,
n
1
a
j
51,
n
2
Proof of Proposition 2
First, the unique distribution of i rms where proi ts are equal is
x
= 0.5. Indeed taking
proi ts as in (24.17), setting p
1
(
x
)
equal to p
2
(
x
)
, gives a i rst order equation in
x
whose
unique solution is
x
= 0.5.
Geographical equilibria are pure strategy Nash equilibria (PSNE) of the one stage
game where each i rm in a group of
N
(
N
even) has to choose to be located in
l
=1 or
l
=
2 and proi ts are given by (24.17). Denote the i rm
i
= 1, . . .,
N
strategy as
s
i
. Firm
i
can
choose to locate either in 1,
s
i
= 1, or in 2,
s
i
= 0. The strategy space has thus 2
N
elements.
A strategy proi le will be denoted as
s
while
s
−
i
will denote the strategy proi le of the
N
− 1
i rms apart from
i
. Dei ne also:
x
(
s
)
5
a
i
5
1,. . .,
N
s
i
N
To complete the formalization of the game we have to specify each i rm payof for any
strategy proi le
s
. When
s
i
= 1, i rm
i
payof p
i
is given by:
p
i
(
1,
s
2
i
)
p
1
(
x
(
s
))
;
where p
1
(
x
)
is as in (24.17) and
x
(
s
) is dei ned above. When
s
i
= 0, i rm
i
payof is:
p
i
(
0,
s
2
i
)
p
2
(
x
(
s
))
;
where p
2
(
x
)
is again from (24.17). To give an example, if all i rms choose location
l
=
1, so that
x
= 1, one has p
i
5 p
1
(
1
)
for all
i
= 1,. . .,
N
. If, instead, half of the i rms are
located in
l
= 1 and the other half in
l
= 2, so that
x
=0.5, one has p
i
5 p
1
(
0.5
)
if
s
i
= 1
and p
i
5 p
2
(
0.5
)
otherwise. A strategy proi le
s
* is a PSNE if and only if:
p
i
(
s
i
,
s
2
i
)
$ p
i
(
s
i
,
s
2
i
)
for all
s
i
= 0, 1,
i
= 1,. . .,
N
(24A.1)
The only candidates to be PSNE are those strategy proi les
s
for which
x
(
s
)
[
{
0, 1, 0.5
}
.
We start by showing that every
s
* such that
x
(
s
*
)
= 1 or
x
(
s
*
)
= 0 is a PSNE proi le.
From (24.17) it holds that p
1
(
1
)
. p
2
(
x
)
for all
x
[
(
0, 1
)
and p
2
(
0
)
. p
1
(
x
)
for all
x
[
(
0,1
)
, so that (24A.1) is satisi ed. Then consider a strategy proi le
s
* such that
x
(
s
*
)
5
x
* 5 0.5. Given (24A.1), it is a PSNE if and only if:
H
1
(0.5) H
2
;
0.5
1
0.5 2
1
p
1
(
0.5
)
$ p
2
a
N
<
N
b4
i
5 1, . . .,
N
(24A.2)