The Spatial Perspective

Abstract

In this chapter, we start showing that development processes are spatially uneven. In advanced regions there are factors that lead to increasing returns: economies of specialization, economies of scale, and external economies. Opening markets might thus bring about regional divergence as a result of a cumulative inflow of mobile resources to the more advanced regions. We then observe that in lagging regions there may be, however, untapped immobile resources, and their valorization justifies place-based policies. Current profitability may be in favor of a certain spatial distribution of activities, but potential profitability may be in favor of a different distribution. The possible movements that may arise will depend on the formation of ex ante expectations. It is therefore reasonable to think that intentional actions such as place-base policies—supporting the best exploitation of untapped, immobile resources where they exist—are justified and may produce significant results. We will lastly examine whether and how realistic it is to assume that these resources are untapped, taking into account a strong objection: if resources are available, they will be spontaneously exploited in a market capitalism system.

Appendix: Size and Growth of European Cities

2.1.1 Introduction

The number of cities in a given territory is always in an inverse geometrical progression of their size (for instance, there are about 23,000 urban agglomerations larger than 10,000 inhabitants in the world, 2,000 are larger than 100,000 inhabitants and about 200 above 1 million, after Moriconi-Ebrard 1993). This persistent scaling behaviour has been questioned for more than one century, giving rise to a large variety of interpretations. (Pumain 2004, 1).

The essay by the German physicist, Felix Auerbach (Das Gesetz der Bevölkerungskonzentration), published a century ago, is always cited. He observed that the probability of finding a city larger than x was inversely proportional to x. This kind of distribution is found in many natural and social processes, but Auerbach was the first to apply it to the urban concentration of population. The American linguist George Kingsley Zipf, who discovered the same distribution in word frequency, evidenced this: «The first person to my knowledge to note the rectilinear distribution of communities in a country was Felix Auerbach in 1913» (Zipf 1949–2012, 374). The term rectilinear refers to the fact that, taking cities in order of decreasing size, the second city tends to have half of the population compared to the first, the third one-third, the fourth one-fourth and so on. In this way, the logarithms of rank as a function of the logarithms of size, follow a straight line with an angular coefficient equal to minus one.

The studies on this subject have never ceased, and thanks to the wider availability of new statistics, they have become numerous in the late 1990s. There are many tests today that show the applicability of this distribution, or its variants. There are also different interpretative proposals concerning the mechanisms that may lie behind it, as shown in the Berry and Okulicz-Kozaryn (2012) review.

As to Zipf’s distribution applicability to different contexts and different samples of cities, a specific problem has always hindered the easiest exercise, which could be the regression of the rank logarithm to the size logarithm, in order to verify whether the estimated coefficient is significantly different from −1. The reason is fairly obvious, underlined a long time ago (Quandt 1964; Rapoport 1978) and mentioned on several occasions (Gabaix and Ioannides 2004; Gan et al. 2006). The variable rank is not the result of independent observations but it is obtained by ordering variable size. It is endogenous. Estimates are therefore biased and inconsistent. This was tested by using Monte Carlo simulations. With generated random numbers and applied econometric rank-size estimates, there are significant results in favor of Zipf-type distribution, while this should not happen for construction (Ganet al. 2006, 259, 262). The problem is serious and the different solutions offered (including the remarkable one Gabaix and Ibragimov (2011) proposed), confirmed Urzua’s judgment (2000, p. 260): «strictly speaking, Zipf’s law cannot hold except for a certain sample size». Generally, after a great deal of theoretical-methodological and empirical work, a substantial consensus emerges (Malevergne et al. 2009) on expected distributions. For large samples of cities, we can expect (even in different contexts like the U.S., Europe, and emerging countries) a lognormal distribution in the range of medium-small cities. A Zipf distribution should prevail among large cities. A parabolic fractal²³ distribution is expected in the middle (see also Giesen and Suedekum 2012; Rybski et al. 2013). To illustrate the distinctive features of these distributions, Figure 2.13 (based on an artificial sample of 2,000 cases) shows the percentage deviations, with respect to the linear Zipf, of the logarithms of the lognormal and parabolic fractal distribution size.

Fig. 2.13

Percentage deviations, respect Zipf’s log of cities’ size, of that corresponding to lognormal (DIF_LN) and parabolic fractal (DIP_PF) distributions. Artificial simulation 2,000 cases

The lognormal distribution presents deviations from Zipf’s, which take the shape of a heightened bell displaced to the left, while the deviations of the parabolic fractal are lower.

A second problematic aspect concerns the definition of a city. It seems now undisputed (Rozenfeld et al. 2011; Veneri 2013) that the results of the analysis concerning the shapes of the distributions also depend on the criteria adopted to identify what a city is. It often emerges that the indicated rank-size rule is best applied to urban realities defined by substantive criteria rather than administrative boundaries. It is therefore appropriate to consider temporal data over not too long periods relying on statistical data based on administrative boundaries. In this way, it is at least possible to conjecture a constant error.

Finally, concerning the mechanisms that can lead to these distributions, Gabaix (1999) and Malevergne et al. (2009) showed that they may be the result of the long term growth rates, independent of size (Gibrat law).

In the following section, we will introduce an exercise on data concerning 352 European cities. We will see that the conjecture concerning distribution forms is verified, indicating that in the past city growth rates were independent of size. We will then see that in recent times the growth rates, by contrast, appear to increase—in a nonlinear manner—in relation to size. This requires a specific interpretation which we will attempt to provide.

2.1.2 Data

The explanations for city growth usually evoke mechanisms that affect productivity (and hence the degree of utilization of resources, in an open economy): economies and diseconomies of scale, human capital, innovation, public goods. It seems therefore appropriate to consider employment, rather than population. Employment size and growth are more immediately linked to productivity and its effects on the use of resources. Eurostat (with additions²⁴) provides, for a total of 352 European cities,²⁵ the average employment levels for the periods 1999–2002, 2003–2006, 2010–2012. The city sample (determined by the availability of data but fairly representative also of medium-small size cities) appears as shown in Table 2.6. From the same source, we have data on area size and population, and we can calculate the population density relative to 2012. Finally, with Google Maps, we have, obtained distances (in minutes of travel by car) between the towns of various countries and their capitals. Size, growth, density, distances are, in fact, considered essential in the recent debate on the analysis of urban area evolution.

Table 2.6 European cities in the sample

2.1.3 The Exercise

Let us consider first (Fig. 2.14) the differences between the logarithms of the actual city size and those that would occur applying a Zipf distribution (taking Gabaix-Ibragimov’s correction).

Fig. 2.14

Percentage deviations, respect Zipf’s log of cities’ size, of actual log sizes: DIF2 = 1999_2002, DIF12 = 2010_2012. Source Figure obtained by processing data as indicated in note 50

As expected, the distribution is lognormal with transition to the parabolic fractal starting at about 100,000 employees, while the tail of large cities drops rapidly to zero indicating that, in this range, size distribution becomes Zipf-type. From 1999–2002 to 2010–2012, the shift of distribution to the right and down is evident. This displacement will be in part a statistical result, owing to the fact that the sample of 352 cities is constant. New cities do not enter in this sample in the period considered, producing an increase in average size. We cannot exclude, however, that this may also depend on a change of the relative weight structure, regardless of this statistic effect. In fact, there is evidence for this when we consider frequency by size groups. The percentage frequency of the cities with less than 40,000 employed people decreases (from 18 to 15 %). However, the percentage frequency of the cities between 40,000 and 250,000 employees does not increase and stays at 69 %, while the share of the largest cities increases from 13 to 16 %. There is confirmation and explanation for this change considering ratios between 2010–2012 and 1999–2002 cumulative employment (Fig. 2.15).

Fig. 2.15

Ratios between cumulative employment 2010_2012 on 1999_2002: rapcum actual figures; R simulation assuming proportional growth (Gibrat’s law). Source Figure obtained by processing data as indicated in note 50

Figure 2.15 shows that cities in the size class 40–250,000 employed (logarithms from 4.6 to 5.4) had the lowest gains comparing the actual data with a linear interpolation. Smaller cities had higher increases and large cities even more so. The interpolating line indicates employment growth of individual cities regularly increasing with the size-related logarithm. If we make a comparison considering constant growth rates independent of size (R line), however, the transition between the upper and lower growth is around the city size of 100,000 employees.

Summing up: (1) The relationship between growth and size in the ten-year period considered deviates from Gibrat’s law, which probably operated in the long run in the past. (2) One part of the cities grows less and one part more than a constant growth rate; the dividing threshold is around the size of 100,000 employed. (3) A recent positive correlation between growth and size is not linear and the cities that are growing relatively less are those between 40 and 250,000 employed.

These results, however, may arise from a spurious correlation between size and growth. Employment could have increased more, independently of city size, in countries where the sample share of small and large cities is greater. While we cannot completely exclude this possible distortion using the available data, we can limit it considering the growth rates of cities net of countries’ growth rate and other possible effects. An equation is therefore estimated and the results are shown in Table 2.7.

Table 2.7 Regression results

Some country dummies were significant, indicating city employment growth rates larger than the overall nationwide growth (in Belgium, France and Italy), and smaller instead in Hungary and Spain. Distance from the capital and population density (in logarithms) are significantly negative, while positive distance divided by the maximum distance is not very significant. The negative effect of population density probably depends on congestion costs, intended in the broadest sense, which detract from the benefits of agglomeration. The negative effects of distance from the capital, while positive in terms of relative distance, are more difficult to interpret. Distance probably captures an adverse effect, related to the physical size of the country, as the cities grow more in small countries. Relative distance, however, captures a positive effect as the net result of conflicting mechanisms, with the prevalence of dispersion over concentration. The non-linear trend of the relative rates of growth of employment in relation to the size of the city seems to be confirmed. The estimated value of dimensional dummy coefficients, in the range of 40–250,000 employed, is the smallest. The coefficient is greater in the range of 30–40,000, and over 250,000. The standard estimate errors, however, are high enough to make these differences not very significant. Nevertheless, the significance of the non-linear trend is clear considering the residuals. We computed, for size ranges, averages of residue growth rates after removing effects of variables not related to the size: country dummies, population density, and distance. We obtained the following residues which measure cities’ annual relative growth rates. These residues depend on the size of the cities, on other unknown variables, and on a random component. They are: 1.6 % (30–40,000 employed), 1.5 % (from 40 to 250,000), 2.0 % (from 250 to 400,000), 2.3 % (over 400,000). Finally, we compared actual data and simulated data by means of these residues as ratios of cumulative employment in 2010–2012 and 1999–2002. The non-linearity in the relation between growth and size is now less strong but not canceled. The growth of large cities is also reduced. An increasing relationship between growth and size of cities remains however (Fig. 2.16).

Fig. 2.16

Ratio 2010_2012 to 1999_2002 cumulative employment: rapcumEF actual data; rapcumSEF simulated data depending on cities’ sizes. Source Figure obtained by processing data as indicated in note 50

2.1.4 Suggested Interpretation

The exercise shows that, during the last 10 years in Europe, medium-sized cities appear to have grown less than in the past, while larger and, to a lesser extent, smaller cities have grown relatively more. It is not easy to give an interpretation for this intervened non-linear relation between size and growth.

The Acemoglu and Zilibotti’s growth model (1997) suggests considering it a possible result of an increase in uncertainty, which in fact appears to have taken place, fueled by intensified international competition. In Fig. 2.17, we represent unemployment rates as moving averages of three four-month terms from 1970 to 2013. Since the beginning of the 1990s unemployment has not only remained high (having progressively increased since the 1970s), but it has also become more unstable.

Fig. 2.17

Moving three terms averages of unemployment percentages on work force (Europe 12). Source Figure obtained by processing data from Drèze e Bean (1990) and from Eurostat data base

The Acemoglu and Zilibotti model seems suitable to explain how this increased instability and uncertainty might have penalized growth in medium-sized cities. The microfounded model contains several assumptions, the most important of which are the following three: (1) Savings are used by agents to make investments, which are classified into two types: risky investments with greater expected earnings, and safe investments with lower returns. (2) Different projects madepossible by risky investments are imperfectly correlated so that agents have the possibility of reducing the risk through portfolio diversification. (3) The allocation problem is not trivial (it would be so if all agents could invest in all projects by diversifying most of the risks) as there is a minimum investment threshold for each project which gives rise to a trade-off between expected returns and risk insurance.

The authors suggest applying the model to national economies. The reason is that the considered economic system must have boundaries. However, the model can also be applied to a city if it is admitted that the city constitutes a system with limits. This does not exclude the city having large external relations. It implies that the considered possibilities for agents to act for potential projects are internal. An external wider system of relations may exist, but we assume that the system is constant while internal activities change.

Figure 2.18 shows the dynamics of accumulation of productive capital in the city under uncertainty conditions (standardized variables).

Fig. 2.18

Economic growth under condition of uncertainty

G(Kt + 1) and B(K + 1) indicate the stock of capital at time t + 1 in the case of “good news” and “bad news”, respectively. At initial accumulation level (small town) the ease of finding small-scale, not too risky projects allows growth in every case, with both good and bad luck. The two curves are both, in fact, to the left of the bisector. However, as the process goes on, in the case of bad news, growth stops (in the figure to the level Kt = Kt + 1 = 0.11). It remains positive in the case of good news. Therefore, the system may remain, for the initial phase of its growth at least, subject to a condition oscillating between stop and go. Such a condition would be overcome and growth could restart without interruption over a certain threshold (in the figure above Kt = Kt +1 = 0.23), when the curve indicates that the effect of bad luck goes to zero and when the G curve is always to the left of the bisector.

This result, for a given risk aversion [which determines the saddle shape of the curve G(Kt +1)] depends on the difference between the returns on risky projects and safe projects. Reducing this parameter by half, we obtain Fig. 2.19, where circumstances are very different.

Fig. 2.19

Economic growth with uncertainty and reduced difference between risky and safe projects’ returns

Now a condition exists (Kt between 0.29 and 0.37) in which there is no growth, as the curve G(Kt + 1) is not on the left of the bisector. Here the city is fairly big and take-off difficulties have already been surmounted. These difficulties of medium-size cities are born because of a low yield spread between safe and risky assets owing to an increase of all risks resulting from the increased instability and uncertainty. In a medium-size city, this lower return differential does not find a balance in risk distribution, while in larger cities it is guaranteed by a broad diversification of assets. In a medium-size city, furthermore, lower return differential is not offset by the lower cost of the projects as in the case of small cities.