Geography Reference
In-Depth Information
whereas the right panel plots the corresponding long-run distributions as characterized
by Proposition 4.
With low transportation costs (t = 0.95) the long-run distribution is clustered around
the two extreme values,
x
= 0 and
x
= 1, coni rming the prediction of the static analysis.
However, the simulation of the entry-exit process (left panel of Figure 24.2) shows that
agglomeration is only a meta-stable state. One location can become much larger than
the other for several time steps, like location 2 which, in the simulation shown, attracts
almost all i rms in the periods between 2000 and 3500, but at some point the cluster
abruptly disappears and the other location can take over. This behavior is well in accord-
ance with the bimodal nature of the equilibrium distribution (see right panel of Figure
24.2). In fact, the equilibrium distribution represents the unconditional probability of
i nding the system in a give state. This probability can thus be very dif erent from the
frequency with which this particular state is observed over a i nite time window.
Conversely, for higher transportation costs (t =0.7), agglomeration is 'almost' never
observed: i rms spatial distribution is now l uctuating around
x
* =0.5 (see left panel of
Figure 24.2). Even if the static analysis predicts agglomeration, the equilibrium distribu-
tion of the stochastic system, reported in the right panel of Figure 24.2, shows that the
most likely geographical distribution has an equal number of i rms per location, irre-
spective of the fact that the point
x
* = 0.5 is never a static geographical equilibrium. In
general one has the following:
Proposition 5
Consider the entry and exit process described in Proposition 4
.
When the
marginal proi t is bigger than the intrinsic proi t, b > a, the stationary dis-
tribution (24
.
22) is bimodal with modes in x =
0
and x =
1
; when b < a the
stationary distribution is unimodal with mode in x =
0.5
; and when a = b
the stationary distribution is uniform
.
Proof
. See Appendix.
Given our dynamic locational decision process, the previous proposition clarii es that
the shape of the geographical equilibrium distribution does ultimately depend on the
relative size of the marginal proi t
b
and the intrinsic proi t
a
. When marginal proi ts are
bigger than intrinsic proi ts the distribution has mass on the borders of the [0, 1] interval.
When marginal proi ts are lower than intrinsic proi ts the distribution has higher mass
in the middle of this interval, and when they are equal every value of the geographical
distribution is as likely.
Rewriting the relation
b
_
a
in Proposition 5 using the dei nitions of
a
and
b
in (24.21),
it is straightforward to derive the conditions for the unimodality or bimodality of (24.22)
in terms of the values of t,
I
, μ, and a:
_
(
1 1 t
s21
)
2
t
s21
The left panels of Figures 24.3 and 24.4 have been obtained using the latter inequality:
they show which distributional shape is observed in the dif erent regions of the plane
(
I
, t) and (a, t) respectively. In the white area agglomeration is most likely (bimodal
distribution), whereas in the dark area equidistribution is most likely (unimodal distri-
bution). In the right panels the stationary distributions computed at the corresponding
1 1
as
4a
a
I
m
b
