Measuring Urban Agglomeration: A Refoundation of the Mean City-Population Size Index

Abstract

In this paper, we put forth the view that the potential for urbanization economies increases with interaction opportunities. From that premise follow three fundamental properties that an agglomeration index should possess: (1) to increase with the concentration of population and conform to the Pigou–Dalton transfer principle; (2) to increase with the absolute size of constituent population interaction zones; and (3) to be consistent in aggregation. Limiting our attention to pairwise interactions, and invoking the space-analytic foundations of local labor market area (LLMA) delineation, we develop an index of agglomeration based on the number of interaction opportunities per capita in a geographical area. This leads to Arriaga’s mean city-population size, which is the mathematical expectation of the size of the LLMA in which a randomly chosen individual lives. The index has other important properties. It does not require an arbitrary population threshold to separate urban from non-urban areas. It is easily adapted to situations where an LLMA lies partly outside the geographical area for which agglomeration is measured. Finally, it can be satisfactorily approximated when data is truncated or aggregated into size-classes. We apply the index to the Spanish NUTS III regions, and evaluate its performance by examining its correlation with the location quotients of several knowledge intensive business services known to be highly sensitive to urbanization economies. The Arriaga index’s correlations are clearly stronger than those of either the classical degree of urbanization or the Hirshman–Herfindahl concentration index.

Appendices

Appendix 1: Geometric Interpretation of the Agglomeration Index

Define n ₀ = 0 and let K be the number of LLMAs in geographical area under consideration. Then

$$f_{i} = \frac{{n_{i} }}{{\sum\limits_{j = 1}^{K} {n_{j} } }} = \frac{{n_{i} }}{{\sum\limits_{j = 0}^{K} {n_{j} } }}$$

(15)

is the fraction of population residing in the ith LLMA (with f ₀ = 0), and $F_{i} = \sum\nolimits_{j = 1}^{i} {f_{j} } = \sum\nolimits_{j = 0}^{i} {f_{j} }$ is the cumulative distribution (with F ₀ = 0).¹⁹ LLMAs are assumed to be ordered from smallest to largest. The area above the curve is computed as:

$$I = \sum\limits_{i = 1}^{K} {\left( {1 - F_{i - 1} } \right)\;\left( {n_{i} - n_{i - 1} } \right)}$$

(16)

In our example, this is equal to 0.0242. Note that the first term of formula (14) is the area above the curve to the left of the first LLMA in Fig. 1. This reflects the fact that LLMAs cover the whole territory, so that the threshold between urban and non-urban is irrelevant. The first term in (15) is equal to the size of the smallest LLMA:

$$\left( {1 - F_{0} } \right)\;\left( {n_{1} - n_{0} } \right) = \left( {1 - 0} \right)\;\left( {n_{1} - 0} \right) = n_{1}$$

(17)

Equation (15) can be written as:

$$I = \sum\limits_{i = 1}^{K} {\left( {1 - \sum\limits_{j = 0}^{i - 1} {f_{j} } } \right)\;\left( {n_{i} - n_{i - 1} } \right)}$$

(18)

$$I = \sum\limits_{i = 1}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;\left( {n_{i} - n_{i - 1} } \right)}$$

(19)

$$I = \sum\limits_{i = 1}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i} } - \sum\limits_{i = 1}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i - 1} }$$

(20)

Remembering that n ₀ = 0,

$$I = \sum\limits_{i = 1}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i} } - \sum\limits_{i = 2}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i - 1} }$$

(21)

$$I = \sum\limits_{i = 1}^{K} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i} } - \sum\limits_{i = 1}^{K - 1} {\left( {\sum\limits_{j = i + 1}^{K} {f_{j} } } \right)\;n_{i} }$$

(22)

$$I = f_{K} n_{K} + \sum\limits_{i = 1}^{K - 1} {\left( {\sum\limits_{j = i}^{K} {f_{j} } } \right)\;n_{i} } - \sum\limits_{i = 1}^{K - 1} {\left( {\sum\limits_{j = i + 1}^{K} {f_{j} } } \right)\;n_{i} }$$

(23)

$$I = f_{K} n_{K} + \sum\limits_{i = 1}^{K - 1} {\left( {\sum\limits_{j = i}^{K} {f_{j} } - \sum\limits_{j = i + 1}^{K} {f_{j} } } \right)\;n_{i} }$$

(24)

$$I = f_{K} n_{K} + \sum\limits_{i = 1}^{K - 1} {f_{i} n_{i} } = \sum\limits_{i = 1}^{K} {f_{i} n_{i} }$$

(25)

which is exactly Eq. (3).

Appendix 2: Transfer Principle

A key property of the index is that it correctly reflects the change in the potential for interactions and urbanization economies of any reallocation of population. This property is close to the Pigou–Dalton transfer principle for measures of inequality, which states that any change in the distribution that unambiguously reduces inequality must be reflected in a decrease in its measure.

Let Δn _i represent the change in the relative population size of the ith LLMA. A reallocation of population is restricted by the condition that $\sum\limits_{i = 1}^{k} {\Delta n_{i} = 0}$. Any reallocation can be represented as a series of reallocations between two LLMAs, and any reallocation between two LLMAs can be represented as a series of reallocations between an LLMA and the following or preceding one when LLMAs are ordered according to size. Therefore, we need only to consider a reallocation of population from the (s–1)th LLMA to the sth (from a LLMA to the next higher ranking one in terms of size):

$$\Delta ns = \, {-}\Delta ns{-} 1 { } > \, 0,{\text{ and}}\;\Delta ni = \, 0{\text{ for}}i \ne s,s{-} 1$$

(26)

According to our theoretical a priori, such a reallocation raises the potential for interactions. What effect does it have on the index?

Following Eq. (2), define:²⁰

$$\Delta f_{i} = \frac{{\Delta n_{i} }}{{\sum\limits_{j = 1}^{K} {n_{j} } }} = \frac{{\Delta n_{i} }}{{\sum\limits_{j = 0}^{K} {n_{j} } }}$$

(27)

where, in view of (26),

$$\sum\limits_{j = 1}^{K} {n_{j} } - \Delta n_{s - 1} + \Delta n_{s} = \sum\limits_{j = 1}^{K} {n_{j} }$$

(28)

and

$$\Delta f_{s} = - \Delta f_{s - 1} > 0$$

(29)

The value of the index after the reallocation is:

$$I^{\prime} = \sum\limits_{i \ne s - 1,s}^{{}} {f_{i} n_{i} } + \left( {f_{s - 1} - \Delta f_{s} } \right)\left( {n_{s - 1} - \Delta n_{s} } \right) + \left( {f_{s} + \Delta f_{s} } \right)\left( {n_{s} + \Delta n_{s} } \right)$$

(30)

$$I^{\prime} = \sum\limits_{i \ne s - 1,s}^{{}} {f_{i} n_{i} } + f_{s - 1} n_{s - 1} + f_{s} n_{s} + \left( {f_{s} - f_{s - 1} } \right)\Delta n_{s} + \Delta f_{s} \left( {n_{s} - n_{s - 1} } \right) + 2\Delta f_{s} \Delta n_{s}$$

(31)

$$I^{\prime} = \sum\limits_{i = 1}^{K} {f_{i} n_{i} } + \left( {f_{s} - f_{s - 1} } \right)\Delta n_{s} + \Delta f_{s} \left( {n_{s} - n_{s - 1} } \right) + 2\Delta f_{s} \Delta n_{s}$$

(32)

$$I^{\prime} = I + \left( {f_{s} - f_{s - 1} } \right)\Delta n_{s} + \Delta f_{s} \left( {n_{s} - n_{s - 1} } \right) + 2\Delta f_{s} \Delta n_{s}$$

(33)

Given that the LLMAs are ordered from the smallest to the largest, ns > ns–1, and fs > fs–1, so that I’ > I.

Appendix 3: Relationship With the Pareto Distribution

The empirically estimated exponent of the Pareto city-size distribution (a generalization of Zipf’s rank-size rule) has been used as a measure of the concentration of an urban system (Rosen and Resnick 1980). Following the notation established above, the (discrete) Pareto distribution can be written as:

$$K + 1 - i = An_{i}^{ - a}$$

(34)

where K is the number of cities (ranked from the smallest to the largest), n _i is the size of city i,²¹ and A and a are parameters. Parameter A can be calibrated from the size of the largest city:

$$K + 1 - K = 1 = An_{K}^{ - a}$$

(35)

$$A = n_{K}^{a}$$

(36)

Inverting (34), we obtain:

$$n_{i}^{a} = \frac{A}{K + 1 - i}$$

(37)

$$n_{i} = \left( {\frac{A}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}}$$

(38)

Total urban population is:

$$N = \sum\limits_{i = 1}^{K} {n_{i} } = \sum\limits_{i = 1}^{K} {\left( {\frac{A}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} }$$

(39)

And so it is quite straightforward to construct a cumulative distribution similar to the one in Fig. 1 reflecting a theoretical Pareto distribution. It is then possible to apply our proposed index to a theoretical Pareto distribution using formula (3). There results

$$I = \frac{{\sum\limits_{i = 1}^{K} {\left( {\frac{A}{K + 1 - i}} \right)^{{{2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} } }}{{\sum\limits_{i = 1}^{K} {\left( {\frac{A}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} } }}$$

(40)

where we exploit the identity in Eq. (3).²² If we assume that the number of cities K and the size of the largest city n _K are fixed, then, using (36), (40) can be written as:²³

$$I = \frac{{\sum\limits_{i = 1}^{K} {\left( {\frac{{n_{K}^{a} }}{K + 1 - i}} \right)^{{{2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} } }}{{\sum\limits_{i = 1}^{K} {\left( {\frac{{n_{K}^{a} }}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} } }}$$

(41)

$$I = \frac{{\sum\limits_{i = 1}^{K} {n_{K}^{2} \left( {\frac{1}{K + 1 - i}} \right)^{{{2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} } }}{{\sum\limits_{i = 1}^{K} {n_{K} \left( {\frac{1}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} } }}$$

(42)

$$I = n_{K} \frac{{\sum\limits_{i = 1}^{K} {\left( {\frac{1}{K + 1 - i}} \right)^{{{2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} } }}{{\sum\limits_{i = 1}^{K} {\left( {\frac{1}{K + 1 - i}} \right)^{{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} } }}$$

(43)

The derivative of the index relative to the Pareto parameter is ²⁴

$$\frac{\partial I}{\partial a} = n_{K} \left( \frac{1}{a} \right)^{2}\; \frac{{2\sum\limits_{j = 1}^{K} {\;\sum\limits_{i = 1}^{K} {\left[ {j^{{^{{ - {1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} }} i^{{^{{ - {2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} }} \ln i} \right]} } - \sum\limits_{j = 1}^{K} {\;\sum\limits_{i = 1}^{K} {\left[ {j^{{^{{ - {2 \mathord{\left/ {\vphantom {2 a}} \right. \kern-0pt} a}}} }} i^{{^{{ - {1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} }} \ln i} \right]} } }}{{\left( {\sum\limits_{i = 1}^{K} {i^{{^{{ - {1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} }} } } \right)^{2} }}$$

(44)

The sign of that derivative is the sign of its numerator, but we could not determine that sign analytically. Using numerical simulations,²⁵ we obtain that the derivative is negative for low values of a, and positive for high values. The sign reversal of the derivative is explained by the fact that, for a given number of cities, the size of the smallest city under the rank-size rule, $n_{1} = n_{K} K^{{^{{ - {1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-0pt} a}}} }}$, increases with a, leaving a larger gap to the left of the first point on the cumulative distribution (see Fig. 1). Referring to index computation formula (7), it is easily verified that its first term is equal to n ₁. Indeed, our numerical simulations confirm that, if that first term is omitted, our index is a monotonically decreasing function of parameter a. This is illustrated in Fig. 7).

Fig. 7

Relationship of the proposed index to the Pareto elasticity parameter