Geoscience Reference
In-Depth Information
(
6.14
) holds, (
6.15
) will apply for sufficiently large
ρ
.For
s
¼
0.127 and
T
¼
400,
the minimum
for (
6.15
) to hold is around 1.52.
10. A boundary in Fig.
6.3
is convex to
A
0
a
s long as
x
A
, the distance from
A
0
to the
boundary, is less than or equal to the
x
A
, given by
ρ
1
=
2
ρ
2
ð
Þ
exp 1
1
ρ
x
A
¼
ð :
=
s
ð
6
:
16
Þ
For
x
A
greater than
x
A
the boundary is concave to
A
0
. Notice that (
6.14
) implies
that the e
x
p(1/
s
)of
x
A
ex
ce
eds
T
. For the boundarie
s
displayed in Fig.
6.3
, the
values of
x
A
increase from
x
A
¼
1.3. The
boundaries are the parametric curves implied by increasing
x
A
from a minimum
equal to the
x
*
of Appendix 6.1. In Fig.
6.3
the values of
x
*
increase from
x
*
1, 613 for
ρ
¼
1.05 to
x
A
¼
1, 697 for
ρ
¼
1.05 to
x
*
¼
213 for
ρ
¼
¼
266 for
ρ
¼
1.3.
)
1/(2
ρ
2)
goes to exp(
11. In (
6.16
) the proportion (1/
ρ
0.5)
¼
0.607 as
ρ!
1 + and
increases from 1.
12. It is possible for a boundary to be always concave to
A
0
in Fig.
6.3
. What is
r
equired is a value of
x
*
from Appendix 6.1 which is greater than or equal to the
x
A
of (
6.16
). An example is
T
increases as
ρ
0.18, where
x
*
¼
400,
ρ
¼
1.1 and
s
¼
¼
202 and
160 are implied.
In Fig.
6.3
, where
s
x
A
¼
2, 628 and, given the information
in paragraph (11), the
x
A
of (
6.16
) must be at least 1,595. It follows
th
at for a
¼
0.127 we have exp(1/
s
)
¼
ρ
where
x
A
condition (
6.15
) does not hold, it is necessarily the case that
x
A
>
and the
convexity of paragraph (10) applies.
13. When
n
s
(ln
N
), as in Eq. (
6.6
), we have
¼
n
s
ln
Mx
ðÞ
¼
ln
N
n
ln
x
ðÞ
¼
½
1
s
ln
x
ðÞ
for Eq. (
6.2
). It follows that
x
A
¼
exp(1/
s
) always satisfies Eq. (
6.10
) and so
defines a point common to each of the family of boundaries defined by varying
x
B
¼
ρ
for
a given
s
. Note that the equality
x
A
¼
x
B
¼
exp(1/
s
) implies that
M
(
x
)
¼
1, and
defines a point on the perpendicular bisector of
A
0
B
0
, i.e., a point on the
ρ
¼
1
p
4exp 2
T
2
boundary. The vertical distance to the point is given by 0
:
5
ðÞ
=
s
. With
s
400 this is distance is 2,620, a calculation which is of theoretical
rather than practical consequence.
¼
0.127 and
T
¼
References
Alonso W (1964) Location and land use: toward a general theory of land rent. Harvard University
Press, Cambridge, MA
Barkley DL, Henry MS, Bao S (1996) Identifying 'spread' versus 'backwash' effects in regional
economic areas: a density function approach. Land Econ 72:336-357
