~ ~ c N ~ l ~ ~ l * ANALYSIS,
l l l ~ : + l ~Vcd.
IX (July 1977). Submitted 5/75. Acwpted 4/76.
Roger W. White*
Dynamic Central Place Theory: Results of a
Simulation Approach
A dynamic central place theory is formulated as a simulation model
in which retail activities, described by cost equations, and consumers,
described by spatial interaction equations, interact to generate a central
place system. The behavior of the model is then examined. Simulation
results show that the basic character of the system-whether it is
agglomerated or dispersed-depends primarily on a single parameter
in the interaction equation, and only secondarily on the specification
of the cost function. The results are highly robust in that they are largely
independent of the initial sizes and locations of centers, and even
independent of the type of interaction equation used. The patterns
generated appear plausible.
INTRODUCTION
Classical central place theory has an honorable position in the field
of geography; but like many venerable institutions, although it is highly
honored, it is little used. It is in fact largely an unusable theory. The
reasons are well known and have to do essentially with the theory’s
being based on rigid and unrealistic assumptions. There is thus a clear
need for a theory that is more realistic and workable. There have been
several advances towards such a theory: the simulations of central place
patterns by Morrill [ 5 ] ,and the well-known work of Berry and Garrison
[Z, 3, 41 and Berry and Barnum [ I ] , constitute two early and notable
examples; but the situation is still not satisfactory.
An earlier paper by the present author [81, proposed a dynamic theory
of central places. The basic premise of the theory is that any central
place pattern is the result of the differential growth (or decline) of the
various centers making up the system. In addition, it is assumed that
‘This research was supported by grants from the Macrogeo raphic Systems Research
Workshop and the Dean’s Research Fund of the University of 8estern Ontario.
Roger W . White i s assistant professor o f geography, the University of Western
Ontario at London.
Roger W. White
/ 227
the growth (or decline) of each center depends on what may be called
its profitability: when the revenue attracted by a centralplace significantly
exceeds the cost of providing the goods and services, the center will
grow, whereas if costs exceed revenue, the center must in the long run
decline. Since the revenue received by a retail center depends on the
spatial behavior of consumers, and the costs incurred depend on the
cost structures of the firms or retail sectors involved, the theory is
essentially a fusion, within a dynamic framework, of spatial interaction
theory and the theory of the firm.
Specifically, the central place system is represented by a system of
nrn first-order difference equations, one equation for each sector (or
activity) in each center:
t+l
S , = tSij
+g,
['Rij- ' C i j l ,
for all places i = 1,.. . n, sectors j = 1,. . . m, and times t = 0, 1 , . . . ,
where
'Sij > 0 = size at center i of sector j at time t
g , = growth function (gi, > 0 by assumption)
' R , = revenue function for sector j in center i at time t
tCij= cost function for sector j in center i at time t.
Taken at face value, however, these equations represent only an elaborate
description of central place dynamics. The content of the theory lies
in the implications of the equations.
In the original paper, an attempt was made to examine their implications
for hierarchical structures and locational patterns but because of the
impossibility of achieving solutions to large sets of simultaneous nonlinear difference equations, only very simple systems could be examined.
In order to get around this difficulty, it was decided to use simulation
techniques to investigate the implications of the theory for complex
systems. A brief description of this approach has previously been
presented [9].The present paper is a more systematic and thorough,
though still far from comprehensive, investigation of the implications
of the theory.
The use of simulation techniques for problems of this type requires
the specification of the particular models whose behavior is to be
observed. But since each model is precisely specified, it represents only
a particular situation. Thus, in order to build up a reasonably comprehensive picture of the implications of the theory in its general form; a
large number of simulation models must be specified and run, each
model in effect representing a sample point in the domain of applicability
of the theory. Because of the size of the undertaking, it has not so
far been possible to complete a comprehensive examination of the theory.
Instead, certain aspects were chosen for relatively detailed investigation
and others were given only cursory treatment or passed over altogether.
First, since each equation is constructed from a revenue equation
(based on an interaction equation) and a cost equation, it is clear that
228
/ Geographical Analysis
the implications of the theory will depend largely on the specification
of these constituent equations.
Second, because the theory is essentially one of competition in space,
the present investigation is designed to reveal the basic features of that
competition. Thus, the simplest possible background conditions are
chosen. For example, it is assumed that the population distribution is
uniform (except for the population of the centers themselves), and that
there are no exogenous trends in population or mobility. Again, for
a complete understanding of the theory it will be necessary to relax
these assumptions. In empirical applications, of course, the specifications
must be chosen to reflect actual conditions in the system being studied.
Finally, the theory can, in principle, be applied at any scale. It can
be used to describe the evolution of a system consisting of only two
or three centers, or it can be used to describe the behavior of an entire
regional or national system of centers. In the first case, interest would
normally be focused on the behavior of individual centers; in the second
case, however, it is the behavior of the system as a whole that is of
concern. In the latter case questions are asked about the aggregate
properties of the system rather than about the location or size of individual
centers. For example, we may want to know whether the retail hierarchy
will be nested or not, or characterized by primacy, or whether the aggregate
locational pattern will be clustered, random, or dispersed. At this scale,
then, the theory does function as a location theory, even though it says
nothing about the location of specific centers; and it is at this level
that it constitutes the analog of existing central place theories. The present
paper is concerned exclusively with this aspect of the theory.
THESIMULATION
MODEL
In the present model for simulating the growth of a central place
system, the initial size of each sector of each center is given, and center
locations are fixed. Growth (or decline) of each sector of each center
is assumed to be proportional to profit (or loss) in the previous time
period, and profit is the difference of revenue, given by an interaction
equation, and costs, assumed to be a linear function of sector size.
Specifically, the model is as follows:
1. The locations of a number of centers, the number of sectors at
each center and the initial size of each sector at each center are
all specified as input data.
2. Population is assumed to be distributed equally among all grid
cells on a rectangular map (this assumption is a matter of convenience
rather than necessity), except that those cells containing a center
have populations that are functions of the amount of retail activity
at the center:
Roger W. White /
229
where '+'Pi is the population of center i at time t + 1; a j are the
population parameters specified in the input data; and tSi, is the
size of sector j in center i at time t.
In the present paper the population parameters are set equal
to zero; that is, it is assumed that the centers themselves have
no population. Sector size is measured in some appropriate unit
indicating scale of operation, such as number of establishments
or aggregate retail floor space.
3. The cost of offering the goods of each sector at each center is
a function of sector size:
tCi,= b,
+ cj t S T ,
(2)
where ' C is the cost incurred by sector j at center i at time t;
b, is fixecfcost; and c j and m, are marginal cost parameters. The
parameters for fixed and marginal cost are read in as input data.
4. Revenue is calculated either according to a gravity equation,
or according to an exponential interaction equation:
where
'PXu= population of grid cell x, y; 'Px, = 1 unless the cell
contains a center, in which case it is set equal to ' P i
p , = per capita expenditures on goods of sector j ; in the present
paper, for simplicity, p , = 1 for all j
tRi, = revenue of sector j in center i at time t
Dixu
= distance from center i to grid cell (x,y)
n, = the interaction parameter for sector j (read in as input
data)
x, y = grid cell coordinates.
In other words, for each sector, the proportion of revenue from
a given grid cell going to a given center is calculated, and the
total revenue at that center is then the sum of the revenue from
all grid cells.
5. Change in sector size is proportional to profit
t+l
Sij = t S ,
+g,
('Rij - 'Ci,).
(4)
Here g, is the growth factor; these parameters are read in as input
data.
230
/ Geographical Analysis
6. Calculations are repeated for successive time periods until an
equilibrium position is approached.
Although the model is conceptually quite simple, it is evident that
in terms of the variety of situations that can be simulated, and thus
in terms of the range of results that can be obtained, it is 'quite complex.
The question is, then, what sort of central place systems are generated,
and under what circumstances, by the model? This question is a very
general one, and a number of difficulties lie in the way of giving a
complete and satisfactory answer to it.
The major practical problem involved in determining the behavior
of the model is simply the very large number of situations that must
be investigated. The growth equation for each center involves five
different kinds of parameters, and ideally a reasonable range of sample
values for each parameter should be tried. The number of combinations
of trial values for the parameters, then, is obviously very large. I n order
to distinguish the effects of the parameters from the effects of the initial
conditions, it is necessary to run each set of parameters several times,
each time with different starting conditions-different
configurations
of centers, for example, or different starting sizes.
In view of the magnitude of the problem, it is necessary to restrict
the investigation of the behavior of the model to certain aspects and
ignore others. Thus, the present paper explores the behavior of the model
primarily as it depends on the interaction equations and secondarily
as it depends on the cost equations. A subsequent paper will examine
the role of the population parameters, Before examining the simulation
results, it will be necessary to di:scuss in some detail (1)the parameters
that determine the behavior of the model, (2) the initial conditions relevant
to the operation of the model, and (3) the techniques for characterizing
the results.
OF THE MODEL
PARAMETERS
GOVERNING
BEHAVIOR
The Interaction Equation and the Parameter n
Two kinds of interaction equations were tested. Emphasis was on
the gravity equation (3a), with the exponential equation (3b) being
introduced mainly to give some indication as to whether the results
obtained using the gravity equation are entirely specific to the use of
that equation, or whether they are of a more general or robust nature.
In both cases, a number of values for the parameter n (the distance
exponent in the gravity equation, the distance coefficient in the exponential equation) were tried. For the gravity equation, only values of n
from 0.5 to 3.0 were tried, since this range includes almost all values
which, in empirical studies of consumer behavior, it has been found
that n assumes (see, for example [6]).
Also, indications are that values
outside this range affect mainly the convergence rate and not the results
themselves. As for the exponential equation, values of n ranging from
Roger W. White /
231
0.01 to 0.02 were used, because values in this range are equivalent
to the values used in the gravity equation, and because values of the
parameter derived empirically from interaction data typically fall within
this range. Values much greater than n = 0.2make very little difference
in the behavior of the model.
The Cost Equation and the Fixed and Marginal Cost Parameters
Cost equations with zero fixed cost and constant returns to scale were
used extensively in the present paper, both in order to simplify study
of the effects of other parameters, primarily n, and in order to provide
base data against which to judge the effects of nonzero levels of fixed
cost. Fixed cost is of particular interest in connection with central place
studies because it is so closely related to threshold size; thus a large
number of runs were made with fixed costs set at 30. This level was
chosen because it is usually in reasonable proportion to the average
revenue in the system. Some trials were made with other levels of fixed
cost.
A range of values for the marginal cost parameters corresponding
to both economies and diseconomies of scale were tried, in order to
examine the effects of a nonlinear cost structure on the central place
systems generated.
Population Parameters. Population parameters were set equal to zero,
both to provide a standard against which to judge the effects of nonzero
levels, and to facilitate observation of the effects of other parameters.
Results for systems with nonzero population parameters are available,
and will be reported in a future paper.
Growth Factors. Growth factors, which also represent response time
of the system, were in every case set equal to 0.5.Their effect on the
behavior of the model was thus not examined.
THEINITIAL
CONDITIONS
The initial conditions consist of specifications for the number of
centers, the location of each center, the number of sectors, and the
size of each sector at each center. Since the purpose of the study is
to investigate general system behavior rather than the behavior of
individual centers within specific contexts, it is desirable to work with
the largest possible systems. The constraints imposed by computing
time were such, however, that the largest system with which it was
feasible to work regularly was one of 20 centers and two sectors on
a map with 2,500 grid cells. A system of 25 centers and 3,600 map
grid cells was used in a very limited number of cases.
Initial Sizes. Three sets of starting sizes were used: same (all centers
are equal in size), Zage, and small. In the case of large and small sizes,
sizes were assigned to centers on a purely arbitrary basis. The three
sets of sizes were constructed with reference to mean equilibrium sizes
when fixed cost and population parameters are equal to zero. Thus for
232 /
Geographical Analysis
both the same and the small starting sizes, the average size is less than
the average equilibrium size, whereas the average initial size of the
large sizes is greater than average equilibrium size. The same sizes are
used as the standard starting sizes. The other two sets are used primarily
to check results based on same sizes for sensitivity to changes in initial
sizes.
Maps. Seven maps were used (Fig. 1).
1. The standard map consists of 20 centers located irregularly on
a 50 x 50 grid. The first near neighbor statistic characterizes the
configuration as nearly random (R = 1.11)but the map has been constructed so that several spatial situations appear: the lower half has
,
0
250
-
- 200
0
-
-
-
150
-
; :ln
0
:
-
0
-
0
-
0
0
0
100
-
-
- so
0
-
0
-
0
0
-
0
0
-
1
5
FIG. 1. Maps Showing Configurations of Centers Specified as Initial Conditions
Roger W. White /
233
a generally low density of centers; the upper half, a high density; and
the highest density of centers occurs in the small cluster at the center.
The coefficient of correlation ( r , ) between aggregate distance to three
nearest neighbors and distance to map edge for the centers of the standard
map is -0.29. (The significance of r , is explained in the following
section.) This is the map on which most of the simulations discussed
in the present paper are based.
2. The irregular map is another 50 x 50, 20 center map with centers
located irregularly. For this map r , = 0.00.
3. The strip map has dimensions of 10 x 250 (it thus has the same
area as the first two maps) and, again, 20 centers. It is a linear counterpart
of the standard map: the west half has a high density of centers and
the east half, a low density, and there are two noticeable clusters-one
near the west end and one near the center. For this map, r = 0.20.
4. The regular map is 60 x 60 and has 25 centers arrangea in a regular
rectangular grid pattern. On this map, the aggregate distance to the
three nearest neighbors is the same for all centers; thus r , is not calculated.
5-7. Another three maps were created by scale transformations on
the standard map in which all distances (except distance from a center
to itself) were multiplied in turn by 0.1, 0.5, and 10.
MEASURES
OF THE CENTRAL
PLACE
PATTERN
Once a central place pattern is generated by the simulation model,
the problem is to find useful ways of characterizing it. What are the
relevant features to use to describe it? Several kinds of solutions are
possible: measures can be designed that most clearly and systematically
distinguish one generated pattern from another, or result in the most
distinct set of categories, without regard to considerations external to
the model itself; measures facilitating comparison with the results of
other models of urban systems, such as the classical central place models,
can be developed; or we can choose measures that are useful in comparing
the generated pattern with real urban systems (as, for example, in rank-size
relationships). Clearly, however, the three approaches are not mutually
exclusive. In fact, one test of the model would consist of showing that
measures developed in accord with criteria internal to the model itself
were also reasonable descriptions of real central place systems.
The two measures relied on in the present paper are ones developed
specifically to yield simple, clear, and unequivocal characterizations
of the simulation results. The first measure is the degree of correlation
between size of a given sector at a center and aggregate distance from
that center to the three nearest neighboring centers, where each map
edge is treated as a center (except on the strip map, where only the
ends are so treated). In every case where there is a strong relationship
between size and aggregate distance, it is linear. The second measure
is the degree of correlation between size of a given sector at a center
and distance from the center to the nearest edge of the map (or nearest
end, in the case of the strip map). This relationship, too, is usually
234 /
Geographical Analysis
linear. Clearly, the first measure is an indicator of the importance of
the local situation of a center, whereas the second is an indicator of
the importance to a center of centrality in the region. Since both measures
are used on the same central place pattern, the question of multicollinearity arises. For the maps used in the present study, the problem is
not serious. Coefficients of correlation between aggregate distance and
distance to edge range from r , = 0.00 to r , = -0.29. Neither measure
has yet been tested for suitability in describing data from real central
place systems, for a number of definitional and methodological problems
would have to be solved first.
THESIMULATION
RESULTS
What kinds of central place systems are generated, and under what
circumstances, by the model2 To answer this question we first discuss
simple one-sector systems with zero fixed cost and zero population
parameters, designed to reveal both the effect of the interaction parameter
and sensitivity to initial conditions; then systems with various positive
levels of fixed cost will be examined, and finally systems characterized
b y diseconomies and economies of scale are investigated.
Effect of the Interaction Parameter, n
The type of central place system generated by the model depends
fundamentally on the specification of the interaction equation. For both
the gravity formula and the equivalent exponential interaction equation,
there is a relatively small transition zone of values for the distance
exponent n, such that the kind of central place system generated depends
on whether n lies above or below the transition zone. This effect can
be seen clearly in Figure 2, which displays the simulation results for
three values of n, with three sets of starting sizes each. Graphs in the
first row, for which n = 1, show a poor correlation between size and
aggregate distance to the three nearest neighbors. At the same time,
there is a noticeable association between size and distance to the edge
of the map, as indicated by the four categories of symbols plotted. Graphs
in the second and third rows, on the other hand, for which n = 2 and
n = 3, respectively, show a good correlation between size and aggregate
distance, but no relation between size and distance to edge. This
distinction is also evident in the regression equations, shown in Table
1 (lines 1-3 and 13-18). Furthermore, in examining that table, it can
be seen that there is no single critical value of n that divides the
low-exponent patterns from the high-exponent patterns. Rather, there
is a gradual transition from one kind of pattern to the other as n increases
from 1 to 1.9 (lines 4-12), though the major change occurs between
n = 1.2 (line 5) and n = 1.7 (line 10).
In general terms, then, in the case of the higher exponents, the local
relative location determines the size of a center, whereas in the case
of low exponents, the location with respect to the center of the entire
Roger W. White /
300
"-3
SAME
235
t "-'
LARQE
zoo
AQQREQAR DISTANCE TO 3 NEAREST NEIBHBOURS
FIG.2. Relationship between Center Size and Aggregate Distance to Three Nearest
Neighbors for Nine Single-Sector Systems. Fixed cost = 0, population parameter = 0,
standard map; n and initial sizes as indicated. Distance to edge is indicated as follows:
5, circle; 10, triangle; 15, square; 20-22.5, cross.
region is the major determinant of size. This result regarding the role
of the distance exponent n is a very strong one, one which is largely
independent of the initial sizes of centers, the configuration of the centers,
or the form of interaction equations.
A comparison of the graphs in each row of Figure 3, or an examination
of the corresponding regression equations in Table 1, shows that the
results, as they depend on n, are virtually unaffected by the initial sizes
of the centers.
A change in the configuration of the centers apparently disturbs the
results even less than a change in initial sizes: compare the equations
for the standard map (lines 1,3,16, 17) with the corresponding equations
TABLE 1
COEFFICIENTS
OF THE REGRESSION
EQUATION
Exponent
and
Initial
Sizes
VARIABLE
Intercept
Aggregate
Distance
Distance
to Edge
Mult.
R
Part A: Standard M a p and Graoity Equation
Standard map
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
n = l
Same
Large
Small
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
n=2
Same
Large
Small
I1 = 3
Same
Large
Small
n=
n=
n=
n=
n=
n=
n=
n=
n=
-222.16
-209.68
-253.42'
-225.11
-220.80
-209.17
- 194.76
- 177.46
- 162.29
- 147.69
- 143.35
- 123.72
7.86
7.01
10.34'
8.92
9.72
10.23
10.49
10.52
10.53
10.44
10.47
10.18
14.01
14.67
11.74
12.16
10.18
8.19
6.40
4.86
3.52
2.46"
1.90°5
(0.92)
0.93
0.92
0.77
0.93
0.94
0.94
0.95
0.95
0.95
0.95
0.94
0.95
- 135.78
-91.93
- 115.23
10.85
8.93
10.15
(0.82)
(0.25)
(0.24)
0.95
0.95
0.94
-70.40''
-70.43"
-70.26"
9.29
9.29
9.31
(-1.89)
(-1.89)
(- 1.96)
0.91
0.91
0.91
Part B: Other Maps and Equations
Irregular map
n = l
19. Same
20. Small
n=3
21. Same
22. Large
- 195.46
-212.98'
7.93
9.28"
13.66
12.97
0.97
0.79
(-38.99)
(-39.42)
10.48
10.50
(-3.12)
(-3.11)
0.78
0.78
(- 19.64)
(- 1.76)
2.84
2.92
(0.14)
0.87
0.94
n.a.
n.a.
16.50
-5.66
0.95
-0.69
(-0.52)
(3.72)
9.43
12.39
22.90
21.36
15.87
3.67'"
0.94
0.91
0.90
0.91
9.81
10.90
6.67
(1.51)
0.91
0.94
Strip map
23. n = 1
24. n = 3
(0.4)
Regular map
25. n = 1
-5.49
26. n = 3
193.91
Scale: x 0.1, standard map
27. n = 1
-130.21"
-210.88'
28. n = 1.5
-280.09
29. n = 2
-207.36
30. n = 3
Scale: x 0.5, standard map
31. n = 1.5
- 187.69
- 147.50
32. n = 2
Scale: x lo, standard map
33. n = 1.5
- 148.56
34. n = 2
-92.13'
10.35
9.47
2.71'
(-0.38)
0.95
0.94
Roger W. White /
237
TABLE 1 (continued)
Exponent
and
Initial
Sizes
Exponential equation
35. n = 0.01
36. n = 0.05
37. n = 0.1
38. n = 0.2
39. n = 3
VARIABLE
Intercept
-55.78"
(- 122.22)
-266.46
- 127.55"
(-50.57)
Agmgate
Distance
Distance
to Edge
(0.42)
(-1.37)
10.15
11.51
8.80
14.63
23.93
13.24
(- 1.41)
-2.62"
Mult.
R
0.97
0.93
0.93
0.85
0.90
Note: L k r d e n t variable is sector size. Fixed cost = 0. Po ulation parameter = 0. All values are significant at
the O.(MM)5 eve1 unless otherwise indicated; figures in parentEeses are not significant at the .05 level. Where no
size is shown. initial sizes are same.
'Significant at the 0.005 level; **significant at the 0.05level. The significance tests should be interpreted liberally
sincw in the cases of n = 1, 2, and 3. we have three samples each, and thus more information than was used id.
the tests.
for the irregular map (lines 19-22). For the regular map, it is not possible
to make a direct comparison, since one characteristic of this map is
that the aggregate distance to the three nearest neighbors is the same
for every center. Nevertheless, the change in size of the regression
coefficient' for distance to edge is consistent with the results using
the standard and irregular maps. The strip map is an exception: here
the configuration does affect the role of the exponent. The equation
for n = 1 (line 23) is very similar to that for n = 3 (line 24) and is
of the high exponent type: distance to edge is not a significant determinant
of center size. There may, however, be values for n sufficiently small
that the low exponent type of equation (in which distance to edge is
a significant variable) appears. Since in the case of the two-dimensional
maps the upper end of the transition zone for n is at n < 2, it seems
reasonable to expect that for the one-dimensional map, if a transition
zone exists, its upper end will fall at n < 1.
Changes of scale on the standard map, as might be expected, do have
some effect on the role of n. In general, shrinking distances is equivalent
to lowering n, and increasing distances is equivalent to raising n. Changing
the scale by multiplying all distances on the map (except distance from
a center to itself) by 0.1 is equivalent to lowering the exponent from
n = 2 to somewhere in the range n = 1.0 to n = 1.2. Similarly, for n = 3,
the scale change is equivalent to a lowering of the exponent to about
n = 1.6. When distances are multiplied by 0.5, the effect is much smaller.
Multiplying distance by 10 has the opposite effect, equivalent to raising
n, but the effect is very slight.
'The equations for the regular map are shown only in order to permit some sort of
comparison. In fact, both relationships are curvilinear, and polynomial equations would
appear to be the more appropriate.
238
/ Geographical Analysis
50
n = l
n = l
n = l
0
FIG.3. Equilibrium Pattern of Centers for Nine Single-Sector Systems. Fixed cost = 30,
population parameter = 0, initial sizes as indicated. Area of circle is proportional to size
of center.
An exponential interaction equation (equation 3b) was substituted for
the gravity equation, and a number of values of n were tried. As can
be seen in Table 1, Part B (lines 35-39), there was again a transition
zone for n values, such that for larger n, aggregate distance is the primary
determinant of center size, whereas for smaller n, distance to edge is
the major determinant. In the case of the exponential equation, however,
the transition occurs between n = 0.1 and n = 0.2.' Comparing these
results to those obtained using the gravity equation, the equation for
n = 0.1 (line 37) is seen to correspond closely to that for n = 1.1 (line
4), whereas the equation for n = 0.2 (line 38) is very similar to that
for n = 3 (line 16).Values for n in the exponential equation much larger
than n = 0.2 make very little difference in the results (cf. lines 38 and
39, Table 1, Part B).
2The most important factor determining the position of the transition zone seems to
be the distance at which interaction values fall to insignificant levels.
Roger W. White /
239
Effect of the Fixed Cost Parameter: Threshold Size
A glance at Figures 3 and 4 shows the most obvious effect of introducing
a nonzero level of fixed cost: some centers are eliminated. This represents
the threshold effect. A center must remain large enough to attract enough
revenue to cover fixed costs; if it falls below this critical size, it disappears
rapidly. The higher the level of fixed cost, the greater the threshold
size, and the fewer the number of centers that can survive (cf. Fig.
3, bottom row, and Fig. 4).
The exponent n is also an important determinant of the number of
centers to survive (Fig. 3). Furthermore, n is effective in determining
not only how many centers survive, but which ones; thus it has a direct
effect on the locational pattern. In particular, the findings for the case
of zero fixed costs regarding the relative importance of aggregate distance
and distance to edge for different values of n remain relevant. But with
a nonzero level of fixed cost, they must be interpreted to refer more
to center survival than to center size. For n = 1 there appears to be
a tendency for the surviving centers to be in the central part of the
map (Fig. 3, first row). This result is to be expected on the basis of
the results with zero fixed cost, since in that case distance from edge
was the most important determinant of center size for n = 1.
In the case of n = 2, the surviving centers are relatively evenly spaced
on the map. Since for n = 2 it was found that center size depends only
on aggregate distance to the three nearest neighbors-in other words,
that spatial competition is largely with local centers-it is not surprising
that with some centers falling below their threshold size, the surviving
centers tend to be ones maximally removed from each other. Maximal
distance, of course, implies regular spacing. If for n = 2 size is determined
by aggregate distance, it is to be expected that relatively regularly spaced
centers will be similar in size. This expectation is confirmed. For the
three cases, same (Fig. 3, second row; left), large (center) and small
(right), the ratio of size of largest center to size of smallest is, respectively,
n=3
n = 3
n = 3
SAME
LARQE
SMALL
50
t
0
lo
a
1
3
FIG.4. Equilibrium Pattern of Centers for Three Single-Sector Systems. Fixed cost =
100, n = 3, population parameter = 0, initial sizes as indicated. Area of circle is proportional
to center size.
240
/ Geographical Analysis
2.8:1, 3.2:1, and 2.3:1.3 For the corresponding cases with zero fixed
cost, the ratios are on the order of 1 O : l (See Fig. 2).
The cases for n = 3 (Fig. 3, bottom row) are generally like those for
n = 2, but since fewer centers were eliminated, the final pattern still
resembles the initial map of 20 centers. Nevertheless, the patterns are
more regular than the original ones: in terms of the first near neighbor
statistic, R = 1.46 for same (Fig. 1, bottom left) and R = 1.47 for large
and small (bottom center and right); by comparison, for the original
map, R = 1.11.
In the three cases with fixed cost equal to 100 and n = 3 (Fig. 4),
the centers are again widely and fairly regularly spaced. But there is
considerable difference in detail among the three maps in terms of which
particular centers are present. By contrast, the corresponding maps for
fixed cost = 30 are identical (Fig. 3, bottom row). This variability reflects
the difference in the relative starting sizes of the various centers in
the three cases. However, in each case the expected relationship between
size and aggregate distance continues to hold.
Effect of the Marginal Cost Parameters: Economies and Diseconomies
of Scale
In the cost equation (2) the two parameters c j and m . together determine marginal cost, since MCij = d C i j / d S i j= c m Srn;-'. The role of
each parameter was, however, examined separateiy .j ii
In the first case, m was set equal to unity, so that M C , = ci; that
is, there are constant returns to scale. The results are not surprising.
Decreasing the rate (c,) at which costs increase as size increases results
in a greater range of sizes at equilibrium, and conversely (Table 1,
lines 8 and 13, and Table 2, lines 1-4).
In the second case, ci was set equal to unity so that M C , = m i S T - ' .
Clearly, the effect of the cost exponent m depends on whether its value
is greater or less than one -that is, on whether diseconomies or economies
of scale (or urbanization) are being modeled. In the case of diseconomies
of scale ( m > l), a comparison of Table 2 (lines 3-5) with Table 1 (line
8) shows that the major effect on the central place pattern is again
that the range of sizes decreases as m increases. It might also be expected
that higher values of the cost exponent would affect the pattern of results
as it depends on the distance exponent n. In particular, it would seem
reasonable to find that increasing the diseconomies of scale (increasing
m ) would have an effect much like that of lowering the order of good
(increasing n ) , at least in the critical intermediate range of values for
n. However, there is little indication of such an effect (cf. lines 6, 8,
and 10 of Table 2 with lines 1, 8, and 10 of Table 1).
In the case of economies of scale ( m < 1) the results cannot be
3This ratio was calculated excluding the small center at (45, 15) which is about to
disappear.
Roger W. White /
241
TABLE 2
COEFFICIEXTS
OF THE REGRESSION
EOUATION
FOR SYSTEMS
WITH ECONOMIES
AND DISECONOMIES
OF SCALE
Cmt Coefficient c
and Enpcrnents of
Distnncv n and
Cost Ill
Linear cost
function ( m = 1)
n = 1.5
1. c = 0.8
2. c = 1.2
n=2
3. c = 0.8
4. c = 1.2
Nonlinear cost
fiinction (c = 1)
n= 1
5. m = 1.1
6. m = 1.2
n = 1.5
7. m = 1.1
8. m = 1.2
9. m = 1.5
n = 1.7
10. m = 1.2
VARIABLE
Distance
to Edge
Mult.
Intercept
Aggregate
Distance
-252.27
- 174.84
14.41
9.88
6.17
4.13
0.94
0.93
- 147.68
-99.11
12.76
8.53
(0.48)
(0.33)
0.94
0.94
-52.19
-6.23
3.11
1.49
5.08
2.29
0.97
0.97
-47.43
- 10.84
7.33
4.34
2.22
0.57
2.23
1.23
0.34
0.97
0.96
0.94
- 10.53
2.43
0.78'
0.95
R
Note: Dependent variable is sector size. Fixed cost = 0. Population parameter = 0. All values are significant at
the 0.oWS level unless otherwise indicated; figures in parentheses are not significant at the 0.05 level. Initial sizes
are sanie.
'Significant at the 0.005 level.
characterized by a regression equation, because some centers are effectively eliminated. Furthermore, with low values for the cost exponent,
convergence is extremely slow, so it is not as a rule feasible to continue
calculations until something approaching equilibrium is attained. In
general it is not possible to say how many centers are eliminated in
various situations. However, two systems were followed until equilibrium
was approached. In the first case (n = 1.7, m = 0.8), after some 600
iterations the system consisted of two centers of unequal size, both
near the center of the map, a result consistent with the low-n pattern.
Apparently it is possible for competing centers to survive under a regime
of economies of scale or urbanization. Perhaps more surprising was
the complex behavior of this system before equilibrium was attained.
One center maintained a nearly constant size for 150 iterations, whereas
neighboring centers grew and declined. I n the second case (n = 2,
m = 0.8), eleven centers were ultimately eliminated, but the remaining
nine were evenly distributed, a result consistent with the high-n pattern.
Perhaps the most important result regarding the effect of a nonlinear
cost function is simply the indication that the presence of diseconomies
or economies of scale does not fundamentally alter the behavior of the
system as it depends on the interaction parameter n.
242 /
Geographical Analysis
CONCLUSIONS
The simulation results presented in the last section, taken together,
constitute a guide to the behavior of the model under various circumstances. The most striking feature of the behavior of the model is its
essential simplicity. In spite of the complexity of the model itself, and
the number of parameters that might be expected to affect the results,
the nature of the central place systems generated depends fundamentally
only on the interaction parameter n. For low values of n (roughly n I1)
the centrality of a center with respect to the entire map area is the
primary determinant of center size, with distance to neighboring centers
a secondary factor. For high values of n (roughly n 2 2), distance to
neighboring centers is the only determinant of center size. When fixed
costs are set at levels appreciably greater than zero, or when economies
of scale are introduced, some centers are eliminated, and so the center
size effects are translated into loetion pattern effects: for low values
of n, centers tend to be few and concentrated in the middle of the
map area, whereas for high values of n, centers are dispersed, relatively
evenly spaced, and of roughly similar size. These results are largely
independent of the initial sizes and locations of centers. Thus the basic
premise of the dynamic central place theory, that the economic characteristics of tertiary activities and the spatial behavior of consumers
together determine the form and structure of the central place system,
is supported.
The results are highly gratifying, for in most empirical investigations
of spatial interaction using a gravity equation, the distance exponent
n has been found to lie within the range 1 <n < 3, with low values
of n generally associated with high-order goods or activities and high
values of n with low-order goods. Furthermore, there is a generally
observed tendency for establishments offering high-order goods (low
n ) to cluster, and frequently to cluster in the center of the region (the
CBD is usually centrally located in the city), and for establishments
offering low-order (high n ) goods to be widely dispersed and relatively
sensitive to local competition. Thus there appears-to be a very general
correspondence between the simulation results and the real phenomena.
In conclusion, judged on internal grounds, the dynamic central place
theory, formulated as a simulation model, appears quite promising. A
future paper will present a first empirical test, one based on rank-size
and other distributions generated in systems characterized by centers
with two tertiary sectors (high and low order) and populations dependent
on sector size.
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BARNUM.“A gregate Relations and Elemental
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2. BERRY,
BRIAN . L., and WILLIAM
L. GARRISON.
“The Functional Bases of the Central
Place HierarcI! y.” Economic Geography, 34 (1958), 145-54.
Roger W. White
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9.
-. “A
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Note on Central Place Theory and the Range of a Good.” Economic
Geography, 34 (1958), 304-11.
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MORRILL,
RICHARD
L. “Simulation of Central Place Patterns over Time.” In Proceedings
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OLSSON,
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Distance and Human Interaction. Philadelphia: Regional Science
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RUSHTON~,GERARD.
“Postulates of Central Place Theory and Properties of Central Place
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