will satisfy
(3.7)
C^22 C ≤ K ≤ 2 C(1) C
C2 C1
, „ ,, ..... ... . τ. γ2 . ,. , ,
Thus all reasonable population densities will give K = ^---^γ~ to within the same order of
magnitude. This is the value we will use in the sequel.
This modeling of the centripetal forces has implicitly been based on a set of simplifying assumptions.
One such assumption is that distance to the city center is the same, no matter where in the CBD the
relevant shopping destination is located. Hence, distance represents an estimate of average distance
to retailing facilities in the city center, and this can be stated as an assumption that internal distances
downtown are so short that they can be ignored when shopping from more peripheral locations is
considered.
Another simplifying assumption is that the agglomeration tendencies are continuously reduced as the
distance from the city center increases. This continuous reduction is not explained by our modeling
framework. One possible explanation is spatial variation in land prices and economic rents within the
city center. This might explain a high density of stores in the most central parts of the CBD. Still, the
analysis in this paper has been based on the assumption of uniform prices of consumption goods
within the urban center; there is no spatial variation in price reductions. Internal distances can be
thought to be too short to allow for spatial price variations in a market equilibrium. We will not,
however, enter into a discussion of spatial price competition in this paper, as we don't consider this
to be of basic importance for the problem that is focused.
As a last step we find the relative level of local sector activities as the net result of centripetal and
centrifugal forces:
(3.8)
R [ d ] Ragglomeration
[d ] + Rlocal sector [ d]
This, too, is a simplification. Internally in the CBD, the two parts are acting together and the model
does not take the effect of this into account. When D is reasonably small, however, the local sector
effect is small anyway so this do not significantly change the model. The curve in Figure 1 shows a
numerical simulation of the function in (3.8). Here we used the parameter values D=10 (km), a=5%
, E0∕L0=250 (pr 1000 customers), R8=200 (pr 1000 customers), and β=0.03 . The values inside the
CBD may seem surprisingly high. Note, however, that the scaling constant K is calculated from a
hypothesis where the population is uniformly distributed within a two dimensional disc. Only a very
small proportion of the population will then be situated close to the center.
4. Extension to several CBDs
The construction in Section 2 and 3 applies to the situation where there is only one CBD. In this
paragraph we wish to extend this construction to the case where there is more than one CBD. To
this end we will base our construction on a convex combination of functions constructed from the
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