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)
 
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