Geography Reference
In-Depth Information
Proposition 4
Consider an economy with two locations, l =
1, 2
, where Assumptions 1-4
are valid
.
The economy is populated by N i rms, distributed according to n =
(
n
1
, n
2
).
At the beginning of each period of time a i rm is randomly selected,
with equal probability over the entire population, to exit the economy
.
Let
m
[{1, 2}
be the location af ected by this exit
.
After exit takes place, a
new i rm enters the economy and, conditional on the exit that occurred in
m, has a probability:
Prob
l
a
b
(
n
<
l
,
m
)
2
a
bN
Prob
l
5
a
1
b
(
n
1
2 d
l
,
m
)
2
a
1
bN
of choosing location l, where a and b are given by (24
.
1)
.
This process
admits a unique stationary distribution:
l
p
(
n
)
5
N
!
C
(
N
,
a
,
b
)
2
1
n
l
!
n
l
(
a
,
b
)
(24.22)
Z
(
N
,
a
,
b
)
q
l
51
where:
1 2
1
C
(
N
,
a
,
b
)
5 2
a
1
a
N
b
bN
(24.23)
n
h
51
[
a
1
b
(
h
2 1
)]
n
. 0
e
q
n
(
a
,
b
)
5
n
5 0
(24.24)
1
and Z
(
N
,
a
,
b
)
is a normalization factor which depends only on the total
number of i rms N, and the coei cients a and b
.
Proof
. See Propositions 3.1-3.4 of Bottazzi and Secchi (2007).
Figure 24.2 shows results from a simulation of the entry-exit process for two dif erent
values of the transportation cost t. The left panel shows 50,000 iterations of the process,
1
6
= 0.95
5
= 0.7
0.8
4
0.6
3
0.4
2
0.2
1
= 0.95
= 0.7
0
0
0
10000
20000
30000
40000
50000
0
0.2
0.4
0.6
0.8
1
x
Time
Figure 24
.
2
Entry-exit process for dif erent values of the transportation costs
.
Left
panel: 50,000 simulations of the entry-exit process for dif erent values of the
transportation cost
t.
Right panel: Long-run stationary distribution of the
entry-exit process simulated in the left panel
.
In both panels the parameters
are
s
= 4,
a
= 1, μ =0
.
5 and I = 800
