A Note on the Average Density Function in Urban Analysis

Abstract

It is argued that the density function, commonly used in the study of urban spatial structure, is more appropriately described as the ‘marginal density function’. From such a marginal density function, it is possible to derive two types of average density function, each being concerned with a particular aspect of the spatial structure of population. The first type is consistent with the standard approach to the ‘average’ in economic analysis, while the second more completely takes account of the urban context. The two types of average density function are examined for different underlying forms of the marginal density function. Of the two types, the second has a greater applicability than the first in the analysis of urban spatial structure.

1. Introduction

In the extensive literature on urban density functions, attention is drawn to the manner in which the density of population at a given distance is systematically related to distance from the centre of the city. In this context, density refers to the population per unit area and the city is defined by its wider agglomeration. Since density is only concerned with conditions at a given distance x, it is more accurately regarded as marginal density, representing one point on a marginal density function $M (x)$ . The fact that a marginal function implies (via integration) the existence of a total function is well known. The existence of an average function is less familiar, and this will be central to the subsequent discussion. To complicate matters, there are two types of average density function. The first of these is comparable to the one employed in standard economic analysis, while the second is more relevant to the study of urban spatial structure. In section 2, we explore these two average density functions, when the underlying marginal density function is negative exponential, as popularised by Clark (1951). This approach is extended in section 3, where the underlying marginal density function conforms to the more realistic log-normal form. In the final section, it is argued that the second type of average density function has a relevance in urban analysis with respect to such aspects as the description of urban spatial structure, its application to particular areas of public policy and the estimation of city populations.

2. The Initial Case

The negative exponential form of the marginal density function has figured prominently in the characterisation of urban density patterns, starting with the seminal work of Clark (1951). The micro-economic foundations for the negative exponential function of urban population densities have been considered by a number of authors, including Alonso (1964), Muth (1969), Amson (1972), Mills (1972), Evans (1973) and Anas et al. (1998).¹ This function is generally expressed as

M (x) = C \exp (- c x)

for

C > 0; c > 0; 0 \leq x \leq u

(1)

where, $M (x)$ represents the marginal density (or population per unit area) within a thin annular ring at distance x from the centre; $C = M (0)$ is the density at the centre; and c is the slope (the lower the value of c, the higher the level of decentralisation or suburbanisation within the city). The term u is the boundary of the city—i.e. the distance at which the minimum urban density $M (u)$ is encountered. The graph of equation (1) is indicated by curve M in Figure 1, where the parameters are assumed to be C = 6474 and c = 0.1505.² Based on the marginal density function, two versions of the average density function are now considered.

Figure 1.

Marginal and average densities (negative exponential case).

2.1 The Unweighted Average Density Function

The first type of average density function is described as ‘unweighted’, since, in its derivation from the marginal density function, there is no modification of the value of $M (x)$ . To determine average density at x, we first take the integral of the marginal density function of equation (1) over the interval 0 to x, to obtain $T (x)$ , the total density at distance x, which is expressed as³

T (x) = \int_{0}^{x} M (r) d r = \frac{C [1 - \exp (- cx)]}{c}

The function $T (x)$ can be viewed as the sum of the marginal densities from 0 to x. By dividing this by the distance x, we obtain

A (x) = \frac{C [1 - \exp (- cx)]}{cx}

where, $A (x)$ is the first type of average density function. The graph of this function, which is based on the parameter values given earlier for C and c, is shown in Figure 1 as curve A. The word ‘average’ in the definition of $A (x)$ corresponds to ‘average’ in standard economic theory. Thus, for example, average density is the counterpart of average product. Similarly, marginal density and total density are the respective counterparts of marginal product and total product, although in the case of marginal density there can be no negative range.

The $A (x)$ function lies above the $M (x)$ function for all $x > 0$ . However, the two functions are linked by $r_{A, M}$ , where

r_{A, M} (x) = \frac{A (x)}{M (x)} = \frac{[\exp (cx) - 1]}{cx}

(2)

where

r_{A, M} (x) > 1

The value of $r_{A, M} (x)$ is independent of C, but dependent on c and x. For a given value of c, the value of $r_{A, M} (x)$ increases with x, and with x given, $r_{A, M} (x)$ increases with c.

2.2 The Weighted Average Density Function

In the second type of average density function, the marginal density at distance x is weighted by the ‘area’ $2 π x$ , the circumference of the ring at distance x. Obviously, the weighting necessarily increases with x. Since the product of density and area is equal to population, this process of weighting yields $p (x)$ , the population at distance x, which is expressed as

p (x) = 2 π xM (x) = 2 π xC \exp (- cx)

(3)

Equation (3) may be described as the marginal population function, indicating the additional increment of population as x increases. If equation (3) is integrated over the interval 0 to x, we obtain

\begin{array}{l} P (x) = \int_{0}^{x} p (r) d r \\ = \frac{2 π C [1 - \exp (- cx) (1 + cx)]}{c^{2}} \end{array}

(4)

where, $P (x)$ is total population within an area of radius x, i.e. the sum of area-weighted marginal density values.

The value $V (x)$ , average density (the population per unit area) at x, is obtained by dividing equation (4) by $π x^{2}$ (the area of a circle of radius x), so that

V (x) = \frac{2 C [1 - \exp (- cx) (1 + cx)]}{{(cx)}^{2}}

(5)

where, $V (x)$ indicates the average density of the city out to a perimeter of radius x, this being what is conventionally understood by the term ‘average density’. The particular version of equation (5) appearing in Figure 1 as curve V is based on the parameter values given earlier. The $V (x)$ function lies above the $M (x)$ function for all $x > 0$ , but can be linked to the original marginal density function of equation (1) by $r_{V, M} (x)$ , where

r_{V, M} (x) = \frac{V (x)}{M (x)} = \frac{2 [\exp (cx) - (1 + cx)]}{{(cx)}^{2}}

(6)

where

r_{V, M} (x) > 1

As with $r_{A, M} (x)$ , the value of $r_{V, M} (x)$ is independent of C, increasing in c and increasing in x.

2.3 The Two Average Density Functions Compared

Summarising to this point, there exist two distinct types of average density function. In Figure 1, for $x > 0$ , the $A (x)$ function (curve A) lies above the $V (x)$ function (curve V) and both functions lie above the $M (x)$ function (curve M). The two average density functions are related in such a manner that

r_{A, V} (x) = \frac{A (x)}{V (x)} = \frac{cx [\exp (cx) - 1]}{2 [\exp (cx) - (1 + cx)]}

(7)

where

r_{A, V} (x) > 1

From equations (2), (6) and (7), it can be determined that $r_{A, M} (x)$ , $r_{V, M} (x)$ and $r_{A, V} (x)$ are interrelated, and that

r_{A, M} (x) = r_{V, M} (x) r_{A, V} (x)

The difference between the two functions turns on the question of area. This becomes apparent if we depart from the two-dimensional city and consider the one-dimensional city—i.e. a city extending from the ‘centre’ along a line of zero width. Here, the $A (x)$ function is derived from the $M (x)$ function in the same manner as before (except that instead of x referring to radial distance from the centre, it now refers to distance). In such a setting, the $V (x)$ function does not exist as a separate function. Because of the one-dimensional nature of such a city, it is not possible to speak of ‘area’ at distance x. As a result, $p (x)$ and $M (x)$ are indistinguishable, both referring to the incremental population at distance x, so that the $V (x)$ and $A (x)$ functions also coincide. This contrasts with the situation in the conventional city considered earlier, where area is taken into account, so that the $V (x)$ function exists separately from the $A (x)$ function. This contrast holds, whatever the form of the marginal density function.

2.4 Additional Discussion

In general, where $M (r)$ is unspecified, we have, using integration by parts

\begin{array}{l} \int_{0}^{x} M (r) d r = {[rM (r)]}_{0}^{x} - \int_{0}^{x} rM' (r) d r \\ = xM (x) - \int_{0}^{x} rM' (r) d r \end{array}

where, $M' (r)$ is the derivative of $M (r)$ , so that $A (x)$ can now be given as

A (x) = M (x) - \frac{1}{x} \int_{0}^{x} rM' (r) d r

We also have, for an unspecified $M (r)$

\begin{array}{l} \int_{0}^{x} rM (r) d r = {[\frac{r^{2}}{2} M (r)]}_{0}^{x} - \int_{0}^{x} \frac{r^{2}}{2} M' (r) d r \\ = \frac{x^{2}}{2} M (x) - \int_{0}^{x} \frac{r^{2}}{2} M' (r) d r \end{array}

implying

V (x) = M (x) - \frac{1}{x^{2}} \int_{0}^{x} r^{2} M' (r) d r

(9)

On the basis of equations (8) and (9), we also have

V (x) - A (x) = \frac{1}{x^{2}} \int_{0}^{x} (xr - r^{2}) M' (r) d r

(10)

It follows from equations (8), (9) and (10) that $A (x) > V (x) > M (x)$ holds for all positive x whenever $M' (r)$ is always negative. This includes the negative exponential function but also the inverse power function (Vining, 1955). The latter function is defined by $M (x) = N x^{- n}$ , where, N and n are positive. The existence of $\int_{0}^{x} M (r) d r$ , and therefore the $A (x)$ function, requires that $n < 1$ . Moreover, the existence of $\int_{0}^{x} rM (r) d r$ , and therefore the $V (x)$ function, requires that $n < 2$ .⁴

3. An Alternative Case

We now consider a situation where the underlying marginal density function is of a log-normal form, this being more realistic than the negative exponential. Not only does it reflect the density crest near the centre of the city, but beyond the crest as far as the boundary of the city, it conforms closely to the negative exponential function.⁵ The presence of the density crest raises a number of additional issues, associated with the A(x) and V(x) functions. Again, the A(x) function is based on unweighted marginal densities and the V(x) function involves weighted marginal densities.

3.1 Features of the Log-normal Function

With a density crest present, the log-normal function has the form

\ln M (x) = \ln K + a (\ln x) + g {(\ln x)}^{2}

(11)

for

g < 0; 0 < x \leq u

where, $M (x)$ is the marginal density at distance x, and $K = M (1)$ is a constant representing the marginal density at distance 1, the terms a and g referring to the other constants. The graph of equation (11) is shown as curve M in Figure 2, where the parameters are K = 3771, a = 0.9945 and g = −0.6111.⁶ The function in equation (11) has a maximum (a density crest) at $x^{*}$ , where

Figure 2.

Marginal and average densities (log-normal case).

x^{*} = \exp (\frac{a}{2 {\hat{g}}^{2}})

(12)

with $\hat{g} = \sqrt{- g}$

We proceed as in section 2 and derive the $T (x)$ and the $P (x)$ functions. These are then divided by x and $π x^{2}$ respectively to give the two average density functions

A (x) = \frac{\sqrt{π} K \exp [{\frac{(a + 1)}{4 {\hat{g}}^{2}}}^{2}] Φ_{A}}{\hat{g} x}

where

Φ_{A} = Φ [\sqrt{2} \hat{g} (\ln x - \frac{a + 1}{2 {\hat{g}}^{2}})]

and

V (x) = \frac{2 \sqrt{π} K \exp [{\frac{(a + 2)}{4 {\hat{g}}^{2}}}^{2}] Φ_{V}}{\hat{g} x^{2}}

where

Φ_{V} = Φ [\sqrt{2} \hat{g} (\ln x - \frac{a + 2}{2 {\hat{g}}^{2}})]

Here, $Φ (v)$ is the cumulative distribution function of the standard normal. These are shown in Figure 2 as curves A and V.

For $x < x^{*}$ , the marginal density function is increasing and it follows from equations (8), (9) and (10) that $M (x) > V (x) > A (x)$ , as illustrated in Figure 2. When $x > x^{*}$ , the situation is less clear. The fact that the $A (x)$ function reaches its maximum where $A (x)$ and $M (x)$ are equal, parallels a fundamental result of economic theory. The x value at which this maximum occurs, $x^{* * *}$ , is defined implicitly as the solution to

λ [\sqrt{2} \hat{g} (\ln x - \frac{a + 1}{2 {\hat{g}}^{2}})] = \frac{1}{\sqrt{2} \hat{g}}

(13)

where $φ (v)$ , the probability density function of the standard normal, is required to obtain $λ (v) = \frac{ϕ (v)}{Φ (v)}$ often called the ‘inverse Mill’s ratio’.

In a similar fashion, the $V (x)$ function reaches its maximum where $V (x)$ and $M (x)$ are equal, this being a consequence of

\frac{d V (x)}{d x} = \frac{2 [M (x) - V (x)]}{x}

which follows from equation (9). The x value at which the maximum of $V (x)$ occurs, $x^{* *}$ , is given implicitly by

λ [\sqrt{2} \hat{g} (\ln x - \frac{a + 2}{2 {\hat{g}}^{2}})] = \frac{\sqrt{2}}{\hat{g}}

(14)

The parameter values used in constructing Figure 2 give $x^{*} = 2.26$ , $x^{* *} = 3.43$ and $x^{* * *} = 4.41$ , all in kilometres. The functions $A (x)$ and $V (x)$ are equal at distance $\tilde{x}$ and for $x > \tilde{x}$ the vertical positioning of the $A (x)$ , $V (x)$ and $M (x)$ functions is the same as that in the negative exponential function for $x > 0$ .

3.2 Implications of Parameter Change

By inspection of equations (12), (13) and (14), we find that an increase $Δ a$ in a requires that $\ln x^{*}$ , $\ln x^{* *}$ and $\ln x^{* * *}$ each increases by $\frac{Δ a}{2 {\hat{g}}^{2}}$ . An equivalent result is obtained if $x^{*}$ , $x^{* *}$ and $x^{* * *}$ are each multiplied by $\exp (\frac{Δ a}{2 {\hat{g}}^{2}})$ .

Implicit differentiation and equation (13) give

\frac{\partial x^{* * *}}{\partial \hat{g}} = - \frac{\ln x^{* * *}}{\hat{g}} - \frac{1}{2 {\hat{g}}^{3}} [\frac{1}{λ'} + 1 + a]

(15)

where, $λ' (v) = - λ (v) [v + λ (v)]$ is the derivative of $λ (v)$ . Similarly we can obtain

\frac{\partial x^{* *}}{\partial \hat{g}} = - \frac{\ln x^{* *}}{\hat{g}} - \frac{1}{{\hat{g}}^{3}} [\frac{1}{λ'} + \frac{a + 2}{2}]

(16)

from equation (14). Both equations (15) and (16) appear to be of ambiguous sign. In Figure 2, however, both are negative: −2.43 and −2.21 respectively. Thus an increase in $\hat{g}$ will lead to a decrease in $x^{* *}$ and $x^{* * *}$ in that figure. That an increase in $\hat{g}$ always decreases $x^{*}$ is a consequence of equation (12).

4. Further Comments

The preceding discussion has been concerned with the average density function. Of the two distinct types of average density function identified, the $A (x)$ function refers to the average level of marginal density over the range 0 to x. This function can be regarded as exactly equivalent to average functions used in micro-economic theory, although it has only limited relevance in the analysis of population density. By contrast, the $V (x)$ function considers average density in more familiar terms and refers to the density of population within a given area having a radius of x. The two average density functions are related and the exact nature of this relationship depends on the underlying marginal density function. While this has been restricted to two forms (the negative exponential and the log-normal), the analysis has a generality that extends beyond these.

The $V (x)$ function can be employed in urban analysis in a number of ways and, for the purpose of illustration, attention is confined to the situation where the underlying marginal density function is of the negative exponential form. One such case is the use of this function to construct a measure of decentralisation or suburbanisation. Attention has customarily been focused on the parameter c in equation (1). A drawback of this is that c remains constant for all x, although this has been regarded as convenient for the purposes of description. An alternative measure of decentralisation $D (x)$ may be expressed, using equation (6), as

D (x) = \frac{V (x) - M (x)}{V (x)} = 1 - \frac{1}{r_{V, M} (x)}

where

< D (x) < 1

so that the greater the value of $D (x)$ , the greater will be the level of decentralisation. Thus, within a city, $D (x)$ increases with x (the distance from the city centre). Having derived the value $D (x)$ , the next stage would be to relate this to various land-market and housing-market attributes at distance x.

Another use of the $V (x)$ function in urban analysis is where the concern is with issues of public policy. Within a given radius, the incidence of traffic congestion, crime and environmental pollution, for example, may exhibit patterns which can be correlated with the average density of population inside this radius, although these relations are likely to be complex. Average density may thus provide an important variable in the modelling of these phenomena. Moreover, were such relationships known to be causally connected, it would be possible to determine whether direct attempts at ameliorating a particular urban problem within a given radius from the centre were more or less effective (in terms of net benefits) than indirect efforts to reduce average density of population by means of housing regulation and zoning, for example.

A further application of the $V (x)$ function involves the estimation and forecasting of city populations. For many nations, particularly in the developing world, the census is becoming a costly (and not always reliable) means of determining city populations. Serious consideration has therefore been given to alternatives, these involving remote sensing and satellite imagery (Orgrosky, 1975). The use of such technology is able to specify the areal extent of a city with considerable accuracy. If this is multiplied by an estimate of the average density of the entire city (the special case where x = u), a value for the city population is obtained. This estimate of average density for the city can be derived by determining the values of C and c in the marginal density function of equation (1) and applying these in equation (5). Alternatively, the level of average density can be determined from data on the relationship between area and average density across the cities of a nation. This kind of relationship is similar to the one demonstrated for regions (Stephan, 1972) and is related to regularities among cities identified by Stewart and Warntz (1958). The obvious advantage of such approaches is that these permit estimates of the population of a city to be undertaken at the time required and to be updated as necessary.

It becomes apparent that the $V (x)$ function offers perspectives on the nature of urban spatial structure that are beyond the scope of the $M (x)$ function, the workhorse of urban economics over the past six decades. In drawing attention to the $V (x)$ function, there is no suggestion that this is a replacement for the $M (x)$ function, since the two functions form an important complementarity. In this sense, the $V (x)$ function can be viewed as a missing link in the analysis of urban spatial structure.

Footnotes

Acknowledgements

The authors wish to thank the journal Editors and three anonymous referees for their helpful comments on an earlier version of this paper.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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