Skip to main content
Top
Published in:

Open Access 16-10-2024 | Special Issue Paper

Structural change in city systems evolution

Authors: Martin Andersson, Börje Johansson, Thomas Niedomysl

Published in: The Annals of Regional Science | Issue 4/2024

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper analyzes city system dynamics, based on a theoretical framework relating interaction potentials to agglomeration economies and density externalities. It employs new historical time series data on population size of cities in Sweden over two centuries (1810–2010) and introduces two schematic growth factors: (i) the intra-city potential and (ii) the extra-city potential located in in rings encircling each city. The first factor is measured by each city’s population size, while the second is a vector of distance-discounted population size for each of a city’s urban rings. In this way, we can explain a city’s growth as a function of its interaction potential inside the city, as well as inside the first, second hand third ring. A robust finding is that cities with large ring potentials follow different development paths than those with small ring potentials. We also find clear evidence of structural change between the two centuries 1810–1910 and 1910–2010. In the first period, city growth is positively impacted by the size of the intra-city potential, whereas the same potential dampens or reduces the growth in the second period. Moreover, the ring potentials outside the city tend to switch from having negative growth stimulation in the first period to having positive stimulation in the second period.
Notes

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1 Introduction

The process in which cities grow and decline over time affects the size distribution as well as the geographical configuration of population in other locations. The population change of a city is at the same time influenced not only by its own size, but also by its distance-discounted access to the population in its surrounding cities. Bosker et al (2008) report on three regularities that characterize population dynamics in a city system. First, the city size distribution displays a slowly changing pattern over time. Second, the hierarchy of cities displays a high degree of invariance, and this indicates proportionate growth (Black and Henderson 2003). Third, the size distribution can be approximated by a power law in the upper tail of observed distributions, frequently referred to as Zipf’s law (Soo 2005; Nitsch 2005). It has been shown that these results can be expected when each single city grows according to Gibrat’s law of proportional effect (Gabaix 1999).
The present paper studies the long-term population growth of individual cities, where each city is identified as a municipality 2010 and where the city boundaries for 2010 are applied backwards for each city all the way to 1810. For each city, we have information about the intra-city population as well as population size in cities that belong to Ring I (less than 50 km away, Ring II (50–100 km away, and Ring III (more than 100 km away). In addition, we know the distance between the centroids of each pair of cities. Based on this information, we can calculate every city’s interaction potential with regard to the population (i) inside the city, (ii) in Ring 1 of the city, (iii) in Ring II of the city, and (iv) in Ring III of the city. Given this, we investigate how population is changing in response to the size of the four potentials.
Population growth of a city is measured in two alternative ways: absolute growth and relative growth (change in percent) in a framework where city growth is depicted as a response to the size of the city’s internal potential as well as to its Ring I, Ring II, and Ring III potential. These potentials reflect the size (volume and diversity) of a city’s probability-based opportunities to interact with actors inside the city itself and inside each of the three rings. In the model, the individual city responds to the value of each of the four potentials in year t with positive or negative growth between t and \(t + \tau\), where \(\tau = 10\). A fixed effects approach is designed to take care of the heterogeneity of cities that may spring from slowly changing resource endowments and amenities (Irwin et al. 2010).
The intra-city potential reflects how the city offers options for economic and social interaction which may stimulate population growth, but competition for space in a growing city may also repel in-migration flows and thereby reduce population growth. Such a negative impact on relative growth (growth rate) would then be a natural thing for cities that have been growing in a sequence of decades. This phenomenon is investigated for both absolute and relative growth, and we show that the two growth indicators complement each other.
The Ring III potential for each individual city is heavily distance discounted and does not vary so much from city to city. This observation should imply that the Ring III potential has limited impact on city growth, with statistically insignificant influence on growth.
A city’s Ring I and Ring II potentials reflect the presence of agglomeration phenomena with magnified opportunities for labor market matching, supply of diversified household services, and many other forms of localization and urbanization economies (Fujita and Thisse 2002). Established theory suggests that large-potential values imply large agglomeration phenomena, and we assume that the latter associate with population growth in the pertinent city. The growth analysis that is carried out in the present paper searches for explanations of how a city’s opportunities to interact with surrounding cities augments its likelihood to grow fast and steady. Dobkins and Ioannides (2001) study spatial interactions among US cities and show for example that growth rates among neighboring cities are interdependent. Their study focuses on systems phenomena, including presence of neighboring cities, influences from each location’s region, and distance between cities. Our model shares the systems perspective of their study, which relates to Christaller (1933) and Pred (1966), among others. An overview of alternative contributors and predecessors is presented in Fujita and Thisse (2002).
The study examines and ascertains the presence of dynamic regularities in the 200-year (1810–2010) development of the Swedish city system. In the first period (1810–1910), the intra-city potential has a positive impact on city growth. In the second period (1910–2010), the pertinent response parameter switches to have a negative impact. Moreover, the Ring II potential has a negative growth impact in the first period and switches to have a positive impact on growth in the second period. This change represents a shift from a weakly connected to a more integrated city system.
The paper is outlined as follows. Section 2 presents a theoretical framework for guiding the formulation of alternative econometric models of city growth. Section 3 informs about data sources and descriptive statistics. In Sect. 4, we assess regressions that depict city growth in a 200 years’ perspective. The following section compares growth in the two periods 1810–1910 and 1910–2010, and it suggests the presence of structural changes. In Sect. 6, we carry out the same analysis for the subset of cities with a small external potential, followed by a similar investigation of cities with a large external potential. In this way, we can show to what extent “size matters.”

2 Theoretical framework

2.1 Interaction potentials and urban rings

This paper lines up with Dobkins and Ioannides (2001) by emphasizing the interdependence between cities in their population evolution and by applying ideas from central place systems models (CPS models) in combination with agglomeration advantages. Moreover, we recognize that CPS models provide a highly ordered picture of the mutual influence that can develop in a city system. At the same time, we emphasize that interdependencies may change over time as infrastructure technology develops and as households adopt new means of transportation. Denser interaction networks and reduced interaction friction imply that individual cities can benefit from enlarged interaction potentials, stimulating further growth.
The CPS model depicts how cities of different sizes are located in a pattern which reflects a trade-off between inter-city transport costs and increasing returns. Such a pattern takes into account the size of each city and the distance between each pair of cities. Moreover, authors like Beckmann (1996) and Tinbergen (1967) consider that larger cities have a greater multiplicity of services and goods varieties. This implies that factors from CPS models can suggest how to explain city growth. Some of these factors can also be derived from agglomeration economics models and New Economic Geography (Fujita and Thisse 2002). Recent research has developed such enlarged econometric models, designed to enrich our understanding of city system evolution (Partridge and Rickman 2008; Partridge et al. 2008; Irwin et al. 2010). Our study covers 200 years, and the farther back in time we move, the more limited the number available observables are. Thus, we offer an analysis of long-term regularities in Sweden’s city growth, where the long-term issue is stressed at the expense of a reduced set of city characteristics that might influence the city dynamics. At the same time, we employ growth factors that are compatible with explanatory variables in New Economic Geography and CPS models.
We employ new and unique Swedish panel data for municipalities for a period of 200 years (1810–2010), where boundaries remain unchanged for the entire period. Each municipality is defined as a city and is characterized by (i) its centroid and its centroid distances to all other city centroids, (ii) its own population size, (iii) and its access to the population size in Ring I, Ring II, and Ring III. The four latter variables are labeled intra-city potential and extra-city ring potentials. We set up two alternative models of how the population changes in 10-year cycles in response to each city’s own size and to its access to the population size in Ring I, Ring II, and Ring III.
The growth of a city may be measured as absolute change, yielding an AC-model, or as relative change, resulting in an RC-model. Both models employ the same set of explanatory variables. The central observable is \(P_{m} (t)\), which denotes the size of city m’s population at date t. The time series starts 1810 and continuous up to 2010, and in a panel with 10-year intervals we have that t = 1810, 1820, …, 2010. In this setting, the analysis attempts to explain the population change process across cities by identifying regularities in interval changes, defined as absolute change in and as relative change in Eq. (1) and as relative change in Eq. (2):
$$\Delta P_{m} \left( t \right) = P_{m} \left( {t + \tau } \right) - P_{m} \left( t \right),\tau = 10$$
(1)
$$\Delta P_{m}^{R} \left( t \right) = \ln \left[ {\frac{{P_{m} \left( {t + \tau } \right)}}{{P_{m} \left( t \right)}}} \right],\tau = 10$$
(2)
The purpose is to examine regularities that govern this kind of long-run change process. The framework that we have presented assumes that each city, m, changes in response to interaction opportunities, as these rely on (i) population size,\(P_{m} (t)\), (ii) size of the interaction potential in Ring I, signified by \(A{}_{m}^{1} (t)\), (iii) size of the interaction potential, in Ring II, signified by \(A{}_{m}^{2} (t)\), and (iv) size of the interaction potential in Ring III, signified by \(A{}_{m}^{3} (t)\).
In order to calculate the values of the three potentials, we have to collect information about accessibility components like \(P_{j} \exp \left\{ { - \lambda d_{mj} } \right\}\), where \(d_{mj}\) is distance between city m and j, \(\lambda\) denotes the distance discount parameter and \(P_{j}\) denotes the size of city j’s population.1 We can sum these components over all \(j \in R_{m}^{1}\), where \(R_{m}^{1}\) is the set of cities that belong to city m’s Ring I. Then we obtain city m’s Ring I potential as described in formula (3). Applying the same summation over \(j \in R_{m}^{2}\) and \(j \in R_{m}^{3}\), we get measures of city m’s Ring II potential and Ring III potential. Formally, these potentials are defined in formulas (4) and (5).
For each city m, we identify \(R_{m}^{1}\), \(R_{m}^{2}\) and \(R_{m}^{3}\) as the sets of cities belonging to Rings I, II, and III of city m. Given this structure, the three Ring potentials are defined as follows:
$$A_{m}^{1} (t) = \sum\nolimits_{{j \in R_{m}^{t} }} {P_{j} \exp \left\{ { - \lambda d_{mj} } \right\}}$$
(3)
Reflecting m’s potential associated with Ring I
$$A_{m}^{2} (t) = \sum\nolimits_{{j \in R_{m}^{2} }} {P_{j} \exp \left\{ { - \lambda d_{mj} } \right\}}$$
(4)
Reflecting m’s potential associated with Ring II
$$A_{m}^{3} (t) = \sum\nolimits_{{j \in R_{m}^{3} }} {P_{j} \exp \left\{ { - \lambda d_{mj} } \right\}}$$
(5)
Reflecting m’s potential associated with Ring III.
Inspection of formulas (3)-(5) reveals that they qualify as accessibility measures for a location in city m (Weibull 1976 and (1980). As such, they inform about the expected number of contacts or interaction events associated with a location in city m. In this context, we also recognize that our measures can be derived from random choice theory (Johansson et al. 2002). We also observe that \(A_{m} (t) = A_{m}^{1} (t) + A_{m}^{2} (t) + A_{m}^{3} (t)\), where \(A_{m} (t)\) represents m’s overall or total interaction potential. In the regression analysis, we don’t use the total potential as an explanatory variable. Instead, we consider the separate effect of the Ring I, Ring II, and Ring III potential. However, we assess our results by additional regressions for cities with a large and a small total potential, respectively.
The ring distinctions in (3)–(5) can be applied to find out if a nearby interaction potential has greater or smaller growth impact than a more distant interaction potential. We may also conjecture that the ring accessibilities shift in importance as observations shift from the pre-automobile to the automobile society, where the year 1910 separates the two regimes.

2.2 City growth as a response to the city’s potentials

The change processes in (1) and (2) are assumed to be driven by the city and extra-city interaction potentials which reflect the size of opportunities in goods, service and labor markets, as well as public sector opportunities. Formula (1) depicts absolute change as dependent variable in the AC-model, and formula (2) depicts relative change in the RC-model. The basic equation of the AC-model has the following form:
$$\Delta P_{m} \left( t \right) = \propto_{m} + \beta_{1} P_{m} \left( t \right) + \beta_{21} A_{m}^{1} \left( t \right) + \beta_{22} A_{m}^{2} \left( t \right) + \beta_{23} A_{m}^{3} \left( t \right) + \varepsilon_{m} \left( t \right)$$
(6)
where \({\propto }_{m}\) is a city-specific fixed effects which capture any time-invariant city-specific heterogeneity. Equation (6) captures the change process as viewed in the perspective of the AC-model, and the dependent variable is simply the absolute change in population between periods of 10 years.
We specify the RC-model as shown in Eq. (7):
$$\Delta P_{m}^{R} \left( t \right) = \propto_{m} + \beta_{1} {\text{ln}}P_{m} \left( t \right) + \beta_{21} {\text{ln}}A_{m}^{1} \left( t \right) + \beta_{22} {\text{ln}}A_{m}^{2} \left( t \right) + \beta_{23} {\text{ln}}A_{m}^{3} \left( t \right) + \varepsilon_{m} \left( t \right)$$
(7)
Equations (6) and (7) both refer to basic geographic principles. First, they recognize the von Thunen (1826) modeling of how urban location is influenced by distance sensitivity, where—in modern versions—service interactions are considered to be more sensitive to time distances than other interactions. Second, the three potentials reflect urban structures, where the growth of a typical city is influenced in specific ways by its own intra-city potential but also by its three Ring potentials—located nearby, at intermediate and far away distances. Such spatial configurations include Christaller (1933) patterns as a possible urban structure. In this framework, it is natural to contemplate how the evolution of new transportation systems affects distance sensitivity along the sequence of 10-year observations.
Long-term regularities reveal themselves in the form of parameters in Eqs. (6) and (7) that remain significant over the 200-year period 1810–2010. This leads to the supplementary questions: which potentials remain basic growth factors, and are such factors the same during the first and the second 100-year period?

3 Descriptive statistics and data sources

3.1 Data sources

The population data employed in this paper have been derived from the FOLKNET 1810–1990 database, available online from Umeå University (see Demographic Database 2015). The FOLKNET database contains decadal information on the number of inhabitants for parishes, municipalities, and counties for entire Sweden since 1810. We use data for the municipality level, referred to as cities in the paper, and include the 272 municipalities with a population > 0 for all years during 1810–2010.2
The data are to a large extent built from the parish level. For only a few parishes, data are reported as missing, more specifically 42 posts during the nineteenth century. In those cases, an estimate of the population has been employed using the decade before and after the missing decade to approximate the population. For the purposes of this paper, where we use data aggregated to cities (municipalities instead of parishes), small-potential mismeasurements at the parish level are too insignificant to influence the results. For example, Håkansson (2000) has assessed the quality of the data by way of comparing data of the same parishes from different sources and by random samples. These quality assessments confirmed a high accuracy of the data even at the parish level.

3.2 Model variables

With the help of city centroids, the city environment is divided into rings encircling the city. In a basic setting, a city’s Ring I consists of other cities less than 50 km from the city, while Ring II contains cities 50–100 km away, and Ring III contains the remaining set of cities. Having information about distances between all cities, we can calculate a city’s distance-discounted population in each ring.
Table 1 presents descriptive statistics for all variables that enter different model varieties. The mean absolute population change over a 10-year interval is around 1,250 individuals, and the mean population size is just over 20,000. However, as in all data on cities, the data are skewed with large standard deviations compared to the mean values. For example, the standard deviation for population change is almost four times the mean. Likewise, the standard deviation of city size is about twice the mean of city size. This is a natural outcome given what we know of the structure of city size distributions.
Table 1
Descriptive statistics for all cities 1810–2010
 
Obs
Mean
Std. dev
Min
Max
All cities
Population change
5440
1251.897
5209.542
− 93,272
141,328
Relative population change*
5440
.0569486
.1202431
− .5401025
1.652266
City size (internal potential)
5712
20,458.29
38,859.88
360
847,073
Ring I potential
5712
18,790.06
52,148.88
0
770,599.4
Ring II potential
5712
458.9876
451.7954
0
7294.201
Ring III potential
5712
4.267941
4.108314
.0017775
46.58308
Cities with a large potential
Population change
2720
1643.615
6974.423
− 93,272
141,328
Relative population change*
2720
.07285
.1361869
− .1909954
1.652266
City size (internal potential)
2856
20,653.33
51,123.86
360
847,073
Ring I potential
2856
33,792.74
70,487.48
214.386
770,599.4
Ring II potential
2856
575.1186
510.5488
7.340092
7294.201
Ring III potential
2856
4.733501
4.253734
.2436342
46.58308
Cities with a small potential
Population change
2720
860.179
2310.642
− 16,845
25,781
Relative population change*
2720
.0410471
.0993445
− .5401025
1.081939
City size (internal potential)
2856
20,263.24
20,173.85
368
197,787
Ring I potential
2856
3787.379
4600.06
0
46,135.33
Ring II potential
2856
342.8567
347.372
0
4433.294
Ring III potential
2856
3.802381
3.90314
.0017775
45.66363
The table reports descriptive statistics for 272 cities in Sweden over the period 1810–2010. All figures are based on mean values over cities and time. Population change refers to the absolute change in population in intervals of ten years. * relative population change is approximated by log of population in year t divided by population in year t-10
The data on external potentials show that there is a large variation across cities in terms of their hinterland and thus the potential for inter-city interaction effects. For example, the index for total external potential ranges from 0.24 to over 770,000. Looking at the decomposition of the total environment into different distance bands (Rings I–III), it becomes clear that some cities have zero accessibility in Ring I and Ring II. This is a manifestation of a few small and remote cities in the north of Sweden with very limited hinterland, hosting few inhabitants.

3.3 Rising importance of ring potentials

Figure 1a and 1b presents the relationship between the level of overall accessibility of cities and their subsequent population change in each respective period, i.e., 1810–1910 and 1910–2010. It is evident that the association between initial overall accessibility is negative for the period 1810–1910 but turns to positive in the period 1910–2010. In both periods, the relationship is statistically significant.
Figure 2 a and b presents the relationship between the level of overall accessibility of cities and their subsequent relative population change in each respective period, i.e., 1810–1910 and 1910–2010. Relative population change is measured as the log of the ratio between the employment in t and t-100. It is evident that the association between initial overall accessibility is negative for the period 1810–1910, but turns to positive in the period 1910–2010. In both periods, the relationship is statistically significant. Looking at relative population change, it is weakly positive in 1810–1910 but rather clearly positive between 1910 and 2010.
The models that we apply have net population change in cities as a basic observation, reflecting absolute change (AC) and relative change (RC). This implies that net emigration flows are indirectly included in the time series. During the period 1850–1940, net out-migration fluctuated with a large positive mean such that the population growth ended up 1–2 million smaller than otherwise (Hägerstrand et al. 1974).
This reinforces the hypothesis that the reduced association between internal city size and population change is substituted and replaced by a growing role of external size. The more recent pattern which emphasizes a growing role of overall accessibility supports the city systems framework adopted in this paper, which associates with Dobkins and Ioannides (2001), going back to Christaller (1933) and Lösch (1954).
Figure 3 further qualifies the patterns emerging in Fig. 1ab and 2ab. It shows the sum of population in cities with small and large overall accessibility, respectively, throughout the two centuries.3 It is clear that there is a shift in the trend in the beginning of the 1950s where cities with large overall accessibility start to grow at a much higher pace than cities with small overall accessibility. Around 1950, the total population in cities with large overall accessibility also become larger than those with a small accessibility, and throughout the period the gap between the two groups in terms of population widens.
Figure 3 also clarifies that the widening gap between cities with small and large overall accessibility coincides with the growth of modern infrastructure and adoption of personal motor vehicles. The gap begins to widen in the period when the registered number of passenger cars in Sweden grow significantly as well as when passenger kilometers associated with railways grow. This illustrates that the growing role of cities’ overall accessibility was enabled by the growth of modern infrastructure and means of transportation. For example, railways and passenger cars enabled commuting and enlarged the spatial extent of labor markets, for example, enabling rural-to-urban commuting (Partridge et al. 2010). One may argue that the economic potential of having large overall accessibility could be realized once modern transportation developed. Cities with a large hinterland thus got a significant advantage, and this implied a growing role of spatial interdependencies and network relationships between cities in explaining city growth. In short, this suggest that transportation paved the way for structural change in city system evolution.
In the subsequent sections, we will further analyze these structural patterns in the growth of cities and test whether the same patterns emerge in econometric analysis of the long-term population change using panel estimations for the full set of cities. All this is revealed by stronger growth impact of potentials in Ring I and Ring II.

4 Absolute and relative city growth 1810–2010

4.1 Long-term absolute growth

Table 2 presents the results from three separate regressions for the 200-year period, where absolute growth is the dependent variable. The first regression is made for the set of all cities, while the second and third regressions employ cities with a small and a large overall (total) potential, respectively. When we compare these three regressions, we can see that in all three cases the response to the Ring III potential is negative and insignificant. This implies that a given city does not respond positively to a large potential far away. Observe also that we consider that regularities are present when the same set of significant parameters depict (or predict) the process of city change over a 200-year sequence of 10-year intervals.
Table 2
City growth 1810–2010. Panel estimates in a fixed effects AC-model
Variables and statistics
All cities
Cities with a small potential
Cities with a large potential
Intra-city potential,\(\beta_{1}\)
0.0151***
0.0437***
0.0117***
(0.00281)
(0.00385)
(0.00398)
Potential in Ring I,\(\beta_{21}\)
0.0204***
0.0101
0.0179***
(0.00202)
(0.0182)
(0.00293)
Potential in Ring II,\(\beta_{22}\)
0.991***
1.877***
0.196
(0.270)
(0.228)
(0.442)
Potential in Ring III,\(\beta_{23}\)
− 12.13
− 7.729
− 55.69
(31.62)
(20.28)
(57.65)
Number of observations
5,440
2,720
2,720
*** p < 0.01 ** p < 0.05 * p < 0.1
Table 2 presents model estimates of a 200-year model of AC-type. For that model.
applied to all cities, each city expands its population in response to the size of its intra-
city potential, its potential in Ring I and in Ring II. Consider that the data on cities are split into cities with a small and with a large potential. When carrying out our econometric exercises on these two sets, we find that small-potential cities are influenced by their potentials in Ring II, whereas large-potential cities are impacted by the size of their potentials in Ring I.
The importance of having large interaction potentials is evident from the table which shows that city growth responds to (i) the size of its intra-city potential, (ii) the size of the Ring I potential, and (iii) the size of the Ring II potential. This suggests that city growth is favored by agglomeration phenomena that enhance interaction opportunities.

4.2 Comparing long-term absolute and relative growth

Relative growth is measured on another response scale than what applies to absolute growth. We can see this by describing how absolute change, \(\Delta P(t)\), relates to relative change or growth rate, defined as \(\Delta P(t)/P(t - \tau )\). This measure increases when the denominator is reduced and becomes smaller as the denominator is enlarged.
The message from Table 3 is sharp in the sense that typically the growth rate responds negatively to an increase in the intra-city potential (population size), and this holds in the very long-term perspective—as an essential feature of all cities, of cities with a small overall potential, and of cities with a large overall potential. Our second sharp question is as follows: what dynamics drive the growth rate upwards? For the set of all cities, there is a distinct and positive impact of the Ring I potential on the individual city’s growth rate. The same observation can be made for the sub set of cities with a large overall potential.
Table 3
Relative city growth 1810–2010. Panel estimates in a fixed effects RC-model
Variables and statistics
All cities
Cities with a small potential
Cities with a large potential
ln Intra-city potential,\(\beta_{1}\)
− 0.0147***
− 0.0374***
− 0.0412***
(0.00424)
(0.00611)
(0.00607)
ln Potential in Ring I,\(\beta_{21}\)
0.0766***
− 0.0266***
0.140***
(0.00556)
(0.00595)
(0.00927)
ln Potential in Ring II,\(\beta_{22}\)
− 0.0137*
0.00414
− 0.0264**
(0.00773)
(0.00786)
(0.0133)
ln Potential in Ring III,\(\beta_{23}\)
− 0.0255***
− 0.000898
− 0.00771
(0.00801)
(0.00802)
(0.0167)
Number of observations
5,440
2,720
2,720
*** p < 0.01 ** p < 0.05 * p < 0.1
We make the following observation: A negative value of \(\beta_{1}\) applies to all three regression equations. For cities with a large potential, this is compensated by a positive value of \(\beta_{21}\). Thus, for these cities the growth rate is negatively impacted by each city’s own population size. At the same time, the growth rate is positively impacted by the potential in the set of the city’s closest neighbors.
Table 3 reveals a switch point property, such that \(\beta_{21}\) is negative for small-potential cities and positive for large-potential cities. This means that there is a structural change in the dynamic process such that the response behavior is different for cities with a small and a large overall potential. In the sequel, we will identify several switch points of this kind.

5 City growth 1810–1910 and 1910–2010

Section 5 splits the data into two different sets, where the first set is used to examine city growth during the first period (1810–1910) and the second set is used to analyze city growth during the second period (1910–2010). With these two separate sets, we apply the AC-model and the RC-model in regression analyses that characterize the city growth process. In this way, we intend to show (i) that there is a structural difference between the first and second period with the AC-model and (ii) that the AC- and RC-models have the same structure during the second period.

5.1 Absolute growth in the first and the second period

We start the two-period analysis by examining the AC-model. There is an essential feature that remains the same in the long-term model and in the period 1 and period 2 models: The Ring III potential does not impact city growth in a significant way. In other words: Parameter \(\beta_{23}\) is not significant in the 200-year period and the 100-year period interval 1910–2010. This statement is valid for both absolute and relative changes.
As shown in Table 4, the intra-city potential has a positive and significant impact on absolute growth during the first period \(\left( {\beta_{{_{1} }} = 0.233} \right)\). During the second period, the same parameter has shifted to indicate a negative impact \(\left( {\beta_{{_{1} }} = - 0.152} \right)\), reflecting a negative response to city size, caused by congestion, capacity tensions, and other forms of friction. This depicts a switch in the growth mechanism that operates in the first and in the second period.
Table 4
Absolute city growth in two periods. Panel estimates in an AC fixed effects model with all cities
Variables and statistics
1810–2010
1810–1910
1910–2010
Intra-city potential,\(\beta_{1}\)
0.0151***
0.233***
− 0.152***
(0.00281)
(0.00491)
(0.00847)
Potential in Ring I,\(\beta_{21}\)
0.0204***
0.0448***
0.0422***
(0.00202)
(0.00500)
(0.00532)
Potential in Ring II,\(\beta_{22}\)
0.991***
− 2.622***
2.832***
(0.270)
(0.555)
(0.648)
Potential in Ring III,\(\beta_{23}\)
− 12.13
− 150.4**
− 42.91
(31.62)
(63.60)
(76.30)
Number of observations
5,440
2,720
2,720
*** p < 0.01 ** p < 0.05 * p < 0.1
The RHS-column of Table 4 reflects the change process during the second period. As we can see, an individual city reacts with positive (absolute) growth in response to the Ring I potential \(\left( {\beta_{21} = 0.04} \right)\) and in response to the Ring II potential \(\left( {\beta_{22} = 2.83} \right)\). Thus, this regression captures complex dynamics, where city growth is stimulated by the size of Ring I and Ring II potentials, whereas the generated growth is reduced or dampened by the size of the intra-city potential.
The results described in Table 4 include a switching phenomenon, where the Ring II potential has a negative parameter for the first period \(\left( {\beta_{22} = - 2.622} \right)\) and a positive for the second period \(\left( {\beta_{22} = 2.832} \right)\). The result implies that the city growth regime changes in structure as we study city growth in the first and in the second period. In the second period, the Ring II potential has become more important with stronger growth effects. One may argue that the described switch implies stronger and geographically wider integration of urban activities. For the second period, the \(\beta_{22}\) parameter is shifted upwards from a significantly negative to a significantly positive value, suggesting an increasing growth effect of the size of the Ring II potential. Thus, in the second period the growth is spurred by Ring II accessibility. In the preceding 100-year period, the same potential is negative.

5.2 Relative growth in the first and the second period

The second perspective on city growth is represented by the RC-model. If we apply this model to describe city growth during the second period, we will find that it reflects the same structure as the AC-model in Table 4. To verify this, we compare the RHS-column in Table 4 with the corresponding column in Table 5
Table 5
Relative city growth in two periods. Panel estimates in an RC fixed effects model with all cities
Variables and statistics
1810–2010
1810–1910
1910–2010
ln Intra-city potential,\(\beta_{1}\)
− 0.0147***
0.0453***
− 0.108***
(0.00424)
(0.0134)
(0.00798)
ln Potential in Ring I,\(\beta_{21}\)
0.0766***
0.0453***
0.143***
(0.00556)
(0.0118)
(0.0147)
ln Potential in Ring II,\(\beta_{22}\)
− 0.0137*
− 0.0684***
0.0446**
(0.00773)
(0.0160)
(0.0183)
ln Potential in Ring III,\(\beta_{23}\)
− 0.0255***
0.0454***
0.00656
(0.00801)
(0.0157)
(0.0196)
Number of observations
5,440
2,720
2,720
*** p < 0.01 ** p < 0.05 * p < 0.1
Comparing the first and second period in the RC formulation shows that \(\beta_{1} > 0\) in the first period, whereas \(\beta_{1} < 0\) in the second period. The negative parameter value implies that the growth is dampened when city size increases. We recognize this phenomenon as a parameter switch such that the first period’s \(\beta_{1} -\) value is positive and that the second period’s \(\beta_{1} -\) value is negative.
Just as we found in Table 4, we also find in Table 5 that the \(\beta_{22}\)-parameter switches from negative in the first period and positive in the second. Hence, the AC- and RC-models have congruent structure. What exactly do we mean when claiming that the AC- and RC-model have a similar structure for the second period? First, the same parameters are significant and have the same sign, and one parameter is insignificant in both models:
i.
Significant parameters that are positive: \(\beta_{21} > 0, \, \beta_{22} > 0\) (Rings I–II)
 
ii.
Insignificant parameter: \(\beta_{23}\)(Ring III)
 
iii.
Significant and negative parameter: \(\beta_{1} < 0\) (Intra-city)
 
These structural observations imply that during the second period we reach a situation where absolute and relative growth coincide into a similar dynamic structure, with two potentials attracting inflow of people and with city size reducing this inflow. We observe that when a city increases in size, then it holds back the growth with increasing force. This may function as a size-controlling feedback mechanism.

6 Development of cities with a small overall potential

In previous sections, we have been able to present evidence which supports the idea that absolute city growth is associated with the size of the potentials in Ring I and Ring II. This is partly true for relative growth too. In order to assess this regularity further, the set of cities have been divided into the following two groups: cities with a small overall potential and cities with a large overall potential.

6.1 Absolute growth of cities with a small overall potential in two periods

Table 6 presents the estimated parameters for cities with a small overall potential. For these cities with a small overall potential, the corresponding Ring I and Ring II potentials attract population away from the pertinent cities as given by \(\beta_{21} = - 0.{122}\) and \(\beta_{22} = - {3}.{435}\). For the first period, we find a positive growth response to the size of the intra-city potential and negative responses to the size of the potentials in Rings I–III. This observation has a clear conclusion: the transport conditions during the first period were not attractive enough to integrate a city with its rings. Alternatively expressed, cities with a small overall potential in the first period were not stimulated to grow as a response to the size of their surrounding potentials. Instead, they expanded in response to their intra-city potential.
Table 6
Absolute city growth in two periods for cities with a small overall potential. Panel estimates in an AC fixed effects model
Variables and statistics
1810–2010
1810–1910
1910–2010
Intra-city potential,\(\beta_{1}\)
0.0437***
0.0901***
0.00157
(0.00385)
(0.01000)
(0.00801)
Potential in Ring I,\(\beta_{21}\)
0.0101
− 0.122***
0.0904**
(0.0182)
(0.0464)
(0.0423)
Potential in Ring II,\(\beta_{22}\)
1.877***
− 3.435***
3.276***
(0.228)
(0.891)
(0.448)
Potential in Ring III,\(\beta_{23}\)
− 7.729
− 258.2***
− 17.91
(20.28)
(72.51)
(43.75)
Number of observations
2,720
1,360
1,360
*** p < 0.01 ** p < 0.05 * p < 0.1
The second conclusion from Table 6 concerns the structural change that happens in the second period. This change is drastic: (i) the intra-city parameter switches from significant to insignificant, (ii) the parameters associated with the potentials in Ring I and Ring II “bifurcate” from negative to positive, which indicates new transport conditions that are interaction-friendly and therefor capable of integrating the city with its first and second rings. This is stressed by the following observations:
i.
\(\beta_{21} < 0\) in the first period and \(\beta_{21} > 0\) in the second
 
ii.
\(\beta_{22} < 0\) in the first period and \(\beta_{22} > 0\) in the second
 

6.2 Relative growth of cities with a small overall potential in two periods

In this subsection, we present regression results with an econometric model, for which the growth rate is dependent variable and potentials are independent variables. The regression exercises are carried out for the first and second period, with the objective to identify structural differences between the two periods.
Table 7 illuminates differences between the model structure in the first and second period. Our focus is on response parameters, such that coefficients switch sign when estimations are made separately for the first and second period. The following shifts can be detected in the table:
i.
The intra-city potential parameter shifts from insignificant to significantly negative
 
ii.
The potential in Ring I goes from having a negative to having a positive parameter value
 
iii.
The potential in Ring II goes from insignificant to positive and significant
 
iv.
The potential in Ring III goes from positive and significant to insignificant
 
Table 7
Relative growth in two periods. Panel estimates in an RC fixed effects model for cities with a small overall potential
Variables and statistics
1810–2010
1810–1910
1910–2010
ln Intra-city potential,\(\beta_{1}\)
− 0.0374***
− 0.0256
− 0.0746***
(0.00611)
(0.0170)
(0.0122)
ln Potential in Ring I,\(\beta_{21}\)
− 0.0266***
− 0.0381***
0.0347**
(0.00595)
(0.0124)
(0.0137)
ln Potential in Ring II,\(\beta_{22}\)
0.00414
0.0168
0.0383**
(0.00786)
(0.0158)
(0.0165)
ln Potential in Ring III,\(\beta_{23}\)
− 0.000898
0.0755***
0.0192
(0.00802)
(0.0153)
(0.0166)
Number of observations
2,720
1,360
1,360
*** p < 0.01 ** p < 0.05 * p < 0.1
The changes described in (i)–(iv) associate with results present in Table 6. The conclusion is that both the AC- and the RC-models suggest that in the second period we find that cities and their rings seem to integrate into interactive urban structures. These cities are different from those cities that have a large overall potential and that are examined in Sect. 7.

7 Growth of cities with a large overall potential

A recurrent observation in this presentation is that a city with large ring potentials has a growth advantage when compared with cities that have smaller than average potentials. To illuminate this phenomenon, we have calculated the overall potential of each city and formed a set which contains those cities that have an overall potential that exceeds the value of the average potential. This is the set of cities with a large potential. We characterize this set with respect to absolute and relative growth and find a structural difference between the first and the second period.

7.1 Absolute growth in the first and second period for cities with a large potential

In the second period, 1910–2010, the regression exercises demonstrate that there are two basic growth factors. The first is the intra-city potential, and it reduces city growth, as clarified by the significant parameter \(\beta_{1} = - 0.183\). The second is the Ring I potential, and it increases city growth as demonstrated by the significant parameter \(\beta_{21} = 0.0436\).
As shown in Table 8, there is a clear structural difference between estimated parameters for the first period (1810–1910) and the second period (1910–2010). This difference reflects a movement away from intra-city growth stimulation toward a feedback that implies increased resistance to growth of large cities. Thus, in the second period there is a hurdle that must be overcome before the city will be able to grow. The force to overcome the hurdle is generated by the size of the potential in Ring I, as indicated by the estimated parameter \(\beta_{21} = 0.0436\).
Table 8
Absolute growth of cities with a large overall potential in two periods. Panel estimates in an AC fixed effects model
Variables and statistics
1810–2010
1810–1910
1910–2010
Intra-city potential,\(\beta_{1}\)
0.0117***
0.259***
− 0.183***
(0.00398)
(0.00607)
(0.0124)
Potential in Ring I,\(\beta_{21}\)
0.0179***
0.0423***
0.0436***
(0.00293)
(0.00632)
(0.00765)
Potential in Ring II,\(\beta_{22}\)
0.196
− 1.632**
0.943
(0.442)
(0.780)
(1.091)
Potential in Ring III,\(\beta_{23}\)
− 55.69
10.63
− 161.8
(57.65)
(102.3)
(138.7)
Number of observations
2,720
1,360
1,360
*** p < 0.01 ** p < 0.05 * p < 0.1
With reference to Table 8, the structural shift between the first and second period may be summarized as follows:
i.
The intra-city parameter, \(\beta_{1}\), changes from being positive to become negative
 
ii.
The Ring I parameter, \(\beta_{21}\), remains positive and unchanged over the two periods
 
iii.
The Ring II parameter, \(\beta_{22}\), changes from negative and significant to insignificant
 

7.2 Relative growth in the first and second period for cities with a large potential

Section 7 examines the role of cities with a large overall potential in the growth processes of cities. In particular, we compare this role for absolute and relative change, respectively. To accomplish this, we confront the estimated parameters in Tables 8 and 9, finding elements of congruence between these two sets. In other words, absolute change follows a similar path as does relative change. In this sense, cities with a large overall potential have communal growth process.
Table 9
Relative growth of cities with a large overall potential in two periods. Panel estimates in an RC fixed effects model
Variables and statistics
1810–2010
1810–1910
1910–2010
ln Intra-city potential,\(\beta_{1}\)
− 0.0412***
0.0878***
− 0.141***
(0.00607)
(0.0193)
(0.0113)
ln Potential in Ring I,\(\beta_{21}\)
0.140***
0.229***
0.193***
(0.00927)
(0.0233)
(0.0255)
ln Potential in Ring II,\(\beta_{22}\)
− 0.0264**
− 0.192***
0.0116
(0.0133)
(0.0370)
(0.0323)
ln Potential in Ring III,\(\beta_{23}\)
− 0.00771
− 0.106**
− 0.0118
(0.0167)
(0.0491)
(0.0397)
Number of observations
2,720
1,360
1,360
*** p < 0.01 ** p < 0.05 * p < 0.1
The econometric strategy of this study has been to employ the RC- and AC-models to describe and characterize the city system dynamics in Sweden for two 100-year periods. This means that we estimate and compare the estimated parameters. In these comparisons, we examine the presence of switching phenomena, such that a parameter (i) shifts from being significant to becoming insignificant and vice versa and (ii) such that a parameter changes sign. Table 9 identifies such information about cities with a large overall potential. Comparing parameters from Table 8 and Table 9 yields the following conclusions about switches and sign changes:
i.
The intra-city parameter, \(\beta_{1}\), changes from being significant and positive to becoming negative and significant
 
ii.
The Ring I parameter, \(\beta_{21}\), remains significant, positive and unchanged over the two periods
 
iii.
The Ring II parameter, \(\beta_{22}\), changes from negative and significant to insignificant
 

8 8. Summary and discussion

The present paper asks the question: are there any regularities in the Swedish process of city growth during the long period 1810–2010? To examine this, we formulate several regression equations with absolute change and relative change as dependent variables. In this endeavor, we suggest that regularities are revealed by significant regression parameters that describe and explain city growth and decline. The result is that there exist such parameters that remain invariant for a 200-year period. Regularities of this kind are also found for the periods 1810–1910 and 1910–2010.
Although there are long-term regularities, the econometric analyses in the paper estimate parameter structures for the first and second 100-year period and find that there are structural differences between the periods, such as change of the sign of parameters that determine the city formation dynamics. A similar form of difference is found between cities that have a large overall potential and those that have not. A background may be the introduction of new transport systems.
A major observation is that there is an interplay between a city’s internal potential and its ring potentials. When the latter have large values, they stimulate the city to grow, and when that happens the resistance to grow becomes greater and dampens the effect of the stimuli from the ring potentials. This phenomenon (based on intra-city tensions) represents a sequential feedback mechanism that helps the system to stay away from “explosive” trajectories.
The temporal structure of the econometric model is constrained by the fact that we observe each city’s population every 10th year, which means that we cannot consider shorter time intervals than 10 years if we want to contemplate causation patterns. The change process unfolds as follows: The population in each city adjusts during a 10-year interval as a response to the values of its intra-city potential and its ring potentials at the beginning of the time interval.
The number of inhabitants in every individual city (municipality) changes over time in a process where the independent (explanatory) variables are (i) intra-city potential, (ii) Ring I potential, (iii) Ring II potential, and (iv) Ring III potential, where the last factor plays a subordinate role. Results from analyses with two 100-year periods emphasize that a city’s intra-city potential generates additional growth in the first period and reduced growth in the second period. We note that this property is captured by both the AC- and the RC-model. For the RC formulation, the result is especially sharp.
The basic findings may be summarized as follows. First, the intra-city parameter switches from positive to negative as the estimation is done for the first and second period. Second, the Ring I and Ring II potentials have a different switching profile, where first-period parameters tend to be negative, whereas the second-period parameters have a positive growth effect. During the first period local, intra-city interaction opportunities stimulate city growth. In the second period, the interaction territory has widened to include an urban region.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://​creativecommons.​org/​licenses/​by/​4.​0/​.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Footnotes
1
In the computation of environment accessibilities, the value 0.1 is used as distance friction parameter (). This value corresponds to the estimated distance sensitivity of inter-city commuting flows in Sweden (Johansson et al 2002), and represents the best available information.
 
2
The database was originally created by Christian Swärd who drew upon various official sources, mainly from Tabellverket and Statistics Sweden, to compile the data.
 
3
For simplicity, we divided the set of cities in two groups where one was below the median value of the overall accessibility and one was above.
 
Literature
go back to reference Beckmann, M.J. (1996), In: The Location of Market Oriented Industries in a Growing Economy, Paper presented at the 5th World Congress of the RSAI in Tokyo, May 1996 Beckmann, M.J. (1996), In: The Location of Market Oriented Industries in a Growing Economy, Paper presented at the 5th World Congress of the RSAI in Tokyo, May 1996
go back to reference Black D, Henderson V (2003) Urban evolution in the USA. J Econ Geogr 3:343–372CrossRef Black D, Henderson V (2003) Urban evolution in the USA. J Econ Geogr 3:343–372CrossRef
go back to reference Blanchard OJ, Katz LF (1992) Regional evolutions. Brook Pap Econ Act 1992(1):1–75CrossRef Blanchard OJ, Katz LF (1992) Regional evolutions. Brook Pap Econ Act 1992(1):1–75CrossRef
go back to reference Bosker M, Brakman S, Garretsen H, Schramm M (2008) A century of shocks – the evolution of German city size distribution 1925–1999. Reg Sci Urban Econ 38(4):330–247CrossRef Bosker M, Brakman S, Garretsen H, Schramm M (2008) A century of shocks – the evolution of German city size distribution 1925–1999. Reg Sci Urban Econ 38(4):330–247CrossRef
go back to reference Christaller, W. (1933), In translation (1966), Central Places in Southern Germany. Prentice-Hall, Englewood Cliffs, New Jersey Christaller, W. (1933), In translation (1966), Central Places in Southern Germany. Prentice-Hall, Englewood Cliffs, New Jersey
go back to reference Davis DR, Weinstein DE (2002) Bones, bombs, and break points: the geography of economic activity. Am Econ Rev 92(5):1269–1289CrossRef Davis DR, Weinstein DE (2002) Bones, bombs, and break points: the geography of economic activity. Am Econ Rev 92(5):1269–1289CrossRef
go back to reference Dobkins LH, Ioannides YM (2001) Spatial interaction among U.S. citites: 1900–1990. Reg Sci Urban Econ 31:701–731CrossRef Dobkins LH, Ioannides YM (2001) Spatial interaction among U.S. citites: 1900–1990. Reg Sci Urban Econ 31:701–731CrossRef
go back to reference Fujita M (1989) Urban Economic Theory – Land use and city size. Cambridge University Press, CambridgeCrossRef Fujita M (1989) Urban Economic Theory – Land use and city size. Cambridge University Press, CambridgeCrossRef
go back to reference Fujita M, Thisse J-F (2002) Economics of Agglomeration: Cities, industrial location and regional growth. Cambridge University Press, CambridgeCrossRef Fujita M, Thisse J-F (2002) Economics of Agglomeration: Cities, industrial location and regional growth. Cambridge University Press, CambridgeCrossRef
go back to reference Gabaix X (1999) Zipf’s law for cities: an explanation. Quart J Econ 114:739–767CrossRef Gabaix X (1999) Zipf’s law for cities: an explanation. Quart J Econ 114:739–767CrossRef
go back to reference Ganning JP, Baylis K, Lee B (2013) Spread and backwash effects for nonmetropolitan communities in the U.S. J Reg Sci 53:464–480CrossRef Ganning JP, Baylis K, Lee B (2013) Spread and backwash effects for nonmetropolitan communities in the U.S. J Reg Sci 53:464–480CrossRef
go back to reference Glaeser EL (1994) Cities, information, and economic growth. Cityscape 1:9–47 Glaeser EL (1994) Cities, information, and economic growth. Cityscape 1:9–47
go back to reference Hägerstrand, T, Guteland, G, Karlqvist, K, Holmberg, I, and Rundblad, B (1974), The Biography of a People – Past and Future Population Changes in Sweden. Royal Ministry for Foreign Affairs, Allmänna Förlaget, Stockholm Hägerstrand, T, Guteland, G, Karlqvist, K, Holmberg, I, and Rundblad, B (1974), The Biography of a People – Past and Future Population Changes in Sweden. Royal Ministry for Foreign Affairs, Allmänna Förlaget, Stockholm
go back to reference Håkansson, J. (2000) Changing population distribution in Sweden – Long term trends and contemporary tendencies. GERUM — Kulturgeografi 2000:1 Håkansson, J. (2000) Changing population distribution in Sweden – Long term trends and contemporary tendencies. GERUM — Kulturgeografi 2000:1
go back to reference Irwin EG, Isserman AM, Kilkenny M, Partridge MD (2010) A century of research on rural development and regional issues. Am J Agr Econ 92:522–553CrossRef Irwin EG, Isserman AM, Kilkenny M, Partridge MD (2010) A century of research on rural development and regional issues. Am J Agr Econ 92:522–553CrossRef
go back to reference Jacobs J (1984) Cities and the wealth of nations. Camden Press, New York Jacobs J (1984) Cities and the wealth of nations. Camden Press, New York
go back to reference Johansson B, Klaesson J, Olsson M (2002) Time distances and labor market integration. Pap Reg Sci 81:305–327CrossRef Johansson B, Klaesson J, Olsson M (2002) Time distances and labor market integration. Pap Reg Sci 81:305–327CrossRef
go back to reference Krugman P (1991) Increasing returns and economic geography. J Polit Econ 99:483–499CrossRef Krugman P (1991) Increasing returns and economic geography. J Polit Econ 99:483–499CrossRef
go back to reference Lösch A (1954) The economics of location. Yale University Press, New Haven Lösch A (1954) The economics of location. Yale University Press, New Haven
go back to reference Partridge MD, Rickman DS (2008) Distance from urban agglomeration economies and rural poverty. J Reg Sci 48:285–310CrossRef Partridge MD, Rickman DS (2008) Distance from urban agglomeration economies and rural poverty. J Reg Sci 48:285–310CrossRef
go back to reference Partridge MD, Rickman DS, Ali K, Olfert R (2008) Lost in Space: Population Growth in the American Hinterlands and Small Cities. J Econ Geogr 2008:1–31 Partridge MD, Rickman DS, Ali K, Olfert R (2008) Lost in Space: Population Growth in the American Hinterlands and Small Cities. J Econ Geogr 2008:1–31
go back to reference Partridge M, Ali K, Olfert MR (2010) Rural-to-urban commuting – three degrees of integration. Growth Chang 41(2):303–335CrossRef Partridge M, Ali K, Olfert MR (2010) Rural-to-urban commuting – three degrees of integration. Growth Chang 41(2):303–335CrossRef
go back to reference Pred AR (1966) The Spatial Dynamics of US Urban-industrial Growth, 1800–1914: Interpretive and Theoretical Essays. MIT Press, Cambridge Pred AR (1966) The Spatial Dynamics of US Urban-industrial Growth, 1800–1914: Interpretive and Theoretical Essays. MIT Press, Cambridge
go back to reference Rivera-Batiz FL (1988) Inreasing returns, monopolistic competition and agglomeration economies in consumption and production. Reg Sci Urban Econ 18:125–153CrossRef Rivera-Batiz FL (1988) Inreasing returns, monopolistic competition and agglomeration economies in consumption and production. Reg Sci Urban Econ 18:125–153CrossRef
go back to reference Soo KT (2005) Zipf’s law for cities: a cross-country investigation. Reg Sci Urban Econ 35:239–263CrossRef Soo KT (2005) Zipf’s law for cities: a cross-country investigation. Reg Sci Urban Econ 35:239–263CrossRef
go back to reference Thuenen, J.H. von (1826), Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie, Hamburg Thuenen, J.H. von (1826), Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie, Hamburg
go back to reference Tinbergen J (1967) The hierarchy model of the size distribution of centers. Papers of Reg Sci Assoc 20:65–80 Tinbergen J (1967) The hierarchy model of the size distribution of centers. Papers of Reg Sci Assoc 20:65–80
go back to reference Weibull JW (1976) An axiomatic approach to the measurement of accessibility. Reg Sci Urban Econ 6:357–437CrossRef Weibull JW (1976) An axiomatic approach to the measurement of accessibility. Reg Sci Urban Econ 6:357–437CrossRef
go back to reference Weibull JW (1980) On the numerical measurement of accessibility. Environ Plan A 12:53–67CrossRef Weibull JW (1980) On the numerical measurement of accessibility. Environ Plan A 12:53–67CrossRef
Metadata
Title
Structural change in city systems evolution
Authors
Martin Andersson
Börje Johansson
Thomas Niedomysl
Publication date
16-10-2024
Publisher
Springer Berlin Heidelberg
Published in
The Annals of Regional Science / Issue 4/2024
Print ISSN: 0570-1864
Electronic ISSN: 1432-0592
DOI
https://doi.org/10.1007/s00168-024-01322-w