Maximizing city sizes or maximizing spatial interaction between cities. A look into the relative evolution of Romanian city regions from 1948 to 2021

This paper aims to analyse the spatial interaction of Romanian Cities from 1948 to 2021 trying to perceive if, along seven decades of quite different political regimes – from Central Planning to Market Interaction–, there have been changes in the logic of spatial interaction between City Regions. The study looks into the economic-demographic evolution of Romanian city regions and tests whether, throughout time, adding to the role of the economic base, their evolution follows a logic of size maximization, or a logic of maximization of interaction flows subject to the spatial structure of Romanian cities. Results show that the logic of spatial interaction changed during the period. Except for the first periods from 1948-1956-1966, the spatial evolution of Romanian City Regions is better explained by a logic of size maximization, either in the period of Central Planning (1966-1992) or in the period of Market Interaction where the population of the previous period, or the logic of size maximization, has a strong effect in the relative performance of city-regions when compare with the logic of maximization of interaction flows. Furthermore, exports


Introduction
Cities are political entities (Cole & Payre, 2016) which, constrained by the spatial structure and by the economic, political, environmental, and technological context, compete with each other. The spatial structure of City-Regions (Pumain, 2021) is very much influenced by the intrinsic potential of places and locations shown by the resilience of the ranking of each city/region within the city/regions of a country (Pascariu et al., 2022;Banica et al., 2020;Reggiani & Nijkamp, 2015;Zipf, 1949) and synthesising not only the distance between and within cities but also their relative attractiveness implicit in the recurring concept of enduring proximities and resilience (Sandu et al., 2021;Sigler et al., 2021). Notwithstanding this, there are external impacts that also have a role in the profile of the system of city/regions: i) the political interference in the spatial redistribution of public money and the creation of barriers to the natural flow of people, goods and capital (Dentinho, 2017); the collapse of environmental or natural resources (Diamond, 2005); or technological changes related to mobility and communication (Reggiani & Nijkamp, 1998).
When interaction is free, each agent act generates impacts that modify the context for other agents. With many free agents, interactions become complex, leading to chaotic behaviours (Goldberger, 1996), new equilibriums (Gribbin, 2004) or self-organized dynamic equilibriums (Waldrop, 1992).
In ecology, organisms tend to maximize their stock and adapt through evolution (Pearce & Merletti, 2006) into spatially defined ecosystems (Müller-Schloer et al., 2011) eventually explained by network models (Riley et al., 2015). Nevertheless, when people and places interact freely within space they can choose to maximize their stock constrained by the context or, instead, opt to maximize the benefit from their interaction, constrained by available resources (Dentinho & Rodrigues, 2021).
Based on the maximization of land rent, Von Thünen (1826) analysed interactions between rural and urban areas. Assuming the minimization of transport and transaction costs of public goods, Chistaller (1966) understood the spatial structure of interaction between different city sizes. Losch (1954) added the role of rational behaviour and saw the possibility of spatial and economic equilibrium further included in the works of Bos (1965) and Paelinck and Nijkamp (1976). William Alonso (1964) reinforced the idea of spatial equilibrium and the role of land bid rent. Before, Hoteling (1929) perceived that the maximization of location benefits can be explained by using game theory, and the rationality of people and places was present in the works of Krugman (1991) and Sen and Smith (1995), then operationalized by the researchers inspired by the works of Wilson (1970) and Echenique et al. (2013), or econometric approaches to space, or spatial error as it is often called (Anselin et al., 1997). A step further was taken in trying to understand the evolution of the network of cities (Barthelemy, 2021).
The issue is whether places or city regions maximize interaction flows as assumed in regional economic models associated with gravity functions (Sen & Smith, 1995) or whether they maximize stock or size (Dentinho & Rodrigues, 2021). In the paper of Dentinho (2011), the argument is that a tragedy of the commons and urban non-sustainability will occur when cities try to maximize their size instead of managing size to maximize net benefits. According to Simões Lopes and Pontes (2010), size is maximized when the average cost of the city equals the average benefit, whereas net benefits are maximized when the marginal cost of the city equals the marginal benefit as assumed in the bid rent of Alonso urban models (Alonso, 1964).
Using the reasoning explained in Dentinho and Rodrigues (2021), this paper aims to understand whether Romanian Cities have maximized their size or the benefits of their spatial interaction, from 1948 to 2021, during seven decades of quite different political regimesfrom Central Planning to Market Interaction.
More specifically, the analysis looks into the economic-demographic evolution of Romanian City-Regions and tests whether, throughout time, the spatial interaction between City-Regions follows the maximization of their size, or the maximization of the interaction between them, all of this subject to the spatial structure of Romanian City-Regions and their economic base. The novelty derives from the application of the methodological approach proposed by Dentinho and Rodrigues (2021) on models of complex spatial interaction where the authors differentiate between rational spatial interaction that maximizes interaction flows between City-Regions, on the one hand and, on the other hand, organic spatial interaction that maximizes the size of each City-Region, both constrained by their spatial structure and economic base (Sirkin, 1959), suitable for small open economies of City-Regions.
In a previous work (Dentinho & Pantelica, 2020), it is shown that the Romanian integration in the socialist block followed by the integration in the European Union created winners in the North East and on the Coast and losers in the other parts of the country, but did not decisively change the path of increasing concentration in the capital region. Nevertheless, whereas Central Planning favoured City-Regions in the middle of the country, market interaction benefited the more populated City-Regions. This signals different logics of spatial interaction between City-Regions. What we would like to test is if the logic of spatial interaction is more organic, with the maximization of size, or more rational, with the maximization of interaction.
Point 2 presents the models of spatial interaction both for the case of maximization of City-Regions sizes and for the case when the logic is to maximize the interaction between the City-Regions. Section 3 estimates both models for the different periods by using the evolution of the population of Romanian City-Regions and assuming that, with free movement of people between City-Regions and rational fertility rates, the demographic evolution mirrors the economic change. Section 4 discusses the results regarding urban sustainability and Section 5 presents conclusions and recommendations.

Maximizing city sizes or maximizing spatial interaction
Organisms and humans adapt and evolve, adjusting to environmental, institutional, economic and technological contexts (Chadwick, 1978). The difference is that humans maximize interaction flows, whereas organisms tend to maximize the size of their communities. The difference between size maximization, or organic spatial interaction, and maximization of spatial interaction, or rational spatial interaction, is the human perception of scarcity that signals the congestion and underuse of contextual resources expressed in prices and shadow prices (Scott & Pearse, 1992) stimulating technological and institutional evolution in the long-run.
The approach taken in this paper proposes the use of demographic data per City-Region of Romania and tests different models of spatial interaction. The organic spatial interaction, the rational spatial interaction and a combination of the two according to the period of the recent history of Romania: the period of Central Planning and the Period of Market Interaction. The approach proposed takes the Organic Spatial Interaction Model and the Rational Spatial Interaction Model developed by Dentinho and Rodrigues (2021) into account.

Organic spatial interaction model
The Organic Spatial Interaction Model presented in expression (1)

Combined models
Finally, based on the previous results, we test spatial interaction growth models considering organic and rational interaction for different periods. 1

Table 1. Coefficients and variables of organic and rational spatial interaction growth models
Organic Spatial Interaction Urban Growth Model

Source: Authors' representation
Notice that in the Organic Spatial Interaction Growth Model, growth depends on the existing population ( ) in the entropy variable and the export's correction variable. This does not happen in the Rational Spatial Interaction Growth Model, where the existing population does not have a role and growth is only explained by the exports and by the permanent features of the territory ( ). Figure 1 shows the evolution of the population of the Romanian city regions. The fifties and the sixties of the twentieth century marked the relative take-off of Bucharest discernible in black and the surrounding regions, very connected to the establishment and reinforcement of the socialist regime in the country. In the seventies and the eighties' growth, assessed by demography, decelerated, but some city regions were revealed to be more resilient than others.

Data on population
The fall of the Berlin Wall and the Romanian integration into the European Union led to a strong decrease in the population in most of the regions (Holland et al., 2011;Ibanescu et al., 2020), except for the suburbs of Bucharest that increased sharply, revealing the growth of the metropolitan area of the capital. Contrary to what happens in most of the country, Iasi, Timisoara and a few others seemed to find their path along the last decade. Figure 2 represents the quartiles of the distribution of the population of Romanian City-Regions from 1948 to 2020. It is clear that, just immediately the war, Southwestern regions concentrate the population and that most of the population is, on the one hand, more distributed throughout the country and, on the other hand, more concentrated in fewer city regions around the capital. Since we take the basic sector as 20% of the population assuming a population multiplier effect equal to 5 ( = 0,2 ) the distribution shown in Figure 2 is a relevant explanatory proxy variable of the demographic and economic growth of the City-Regions. ) }, where = Population of (i), = Population of (j), = distance between (i) and (j) and = the impedance parameter calibrated to secure that the average interaction within each city-region is 50% of the average interaction with Figure 3 reveals the strong Potential of the capital Bucharest, the relative potential of Ploiesti centrality in the country and the relative importance of some border regions because the distance between Kyiv and Vienna was also considered as edges of the network of Romanian City-Regions (Rozenblat & Neal, 2021). The Permanent Potential of Spatial Interaction is a proxy for the natural vocation of places, following the suggestion of Paul Krugman (1996) that nature or urban ranks flow along the hierarchy of river basins and that cities rooted in the natural capital also have a strong hierarchy.

Figure 3. Permanent Potential of Spatial Interaction
Source: Authors' representation 3. Estimates of the organic, rational and mix spatial interaction models Table 2 presents the results of the Organic Spatial Interaction Growth Model that explains demographic-economic growth as a function of organic entropy, exports, and exports' correction factor. The model with fixed effects presents an overall R 2 =0.3725 with a within R 2 =0.6865 and a between R 2 =0.1913. Apart from the robust intercept, the only robust statistical indicators relate to exports in the periods (1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002) and (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011), organic entropy for (1948-1956, 1956-1966, 2002-2011) and organic correction of exports for (1948-1956, 1966 -1977, 2002-2011).    Figure 3 should eventually be higher for these regions. Spatial Autocorrelation is negative and not strong, but it indicates a high-high correlation in the centre of Bucharest and, around Iasi, spread effects to the West and backwash effects to the South and North.  Table 3 presents the results of the Rational Spatial Interaction Growth Model that explains demographic-economic growth as a function of rational entropy, exports, exports correction factor and autonomous spatial interaction.
Summing up, the interaction variable of the Rational Organic Spatial Interaction Growth Model seems to present a robust explanation for the economic and demographic evolution of the Romanian Regions from 1948 to 2020.  Figure 6 shows for the Rational Model the evolution of the Export (a), Entropy (b) Export Correction (c) and Spatial Interaction Exports Correction Coefficients and respective errors from 1956 to 2011. It is evident that the role of the economic base in regional growth decreased from 1948 until the end of the last century, to raise in the XXI century but, in the rational model, the entropy of the territory complemented the process with the correction of the Exports' Correction Factors. Figure 7 shows the percentile of the estimated Fixed Effects of the Rational Growth Model (a), with the LISA Maps and the spatial autocorrelation coefficients. With the Rational Growth Model, the regions of Iasi, Ilfov (surroundings of Bucharest), Ploiesti and Brasov have higher values indicating that, for the calibration of this Rational Model, the Permanent Potential of Spatial Interaction of Figure 3 should eventually be higher for these regions. Spatial Autocorrelation is positive and not strong, but it indicates a high-high correlation in the centre of Bucharest, Ploiesti and Targoviste.

Figure 6. Evolution of the Coefficients of the Rational Model (Exports (a), Entropy (b) Exports Correction(c) and Exports Spatial Interaction Correction (d))
Source: Authors' representation

Mixed model of spatial interaction growth
Finally, by joining the variables of the rational and organic spatial interaction growth and by selecting the ones that are more robust for each period, it is possible to see if, from the Second World War until 2020, there were periods when the demographic and economic growth of Romanian City-Regions followed a more organic or more rational logic. Table 4 presents the results of the Mixed Model of Spatial Interaction Growth.
The novelty of this approach is that, in the same regression, we can consider different functional theoretically rigorous expressions that, for different periods, better explain the economic and demographic growth of regions. The advantage is that it is possible to identify the logic of spatial interaction for a long period of time. The disadvantage is that selecting the best model requires a lot more time and, for this case, we additionally consider an expression of Interaction included in the rational interaction even when the period was better explained by the organic interaction -a reinforcement of the mixed model, but not a clear shift from rational to organic spatial interaction.
All presented coefficients are robust. The intercept, rational entropy from 1956 to 1966 and organic entropy onwards, exports from 1977 to 2011, organic correction of exports from 1977 to 2011, and Rational Interaction from 1948 to 2011. Summing up, the interaction variable of the Rational Organic Spatial Interaction Growth Model seems to present a robust explanation for the economic and demographic evolution of the Romanian Regions from 1948 to 2020.   Figure 8 shows, for the Mix Model, the evolution of the Exports (a), Entropy (b) Export Correction (c) and Spatial Interaction Exports Correction Coefficients (d) and respective errors from 1956 to 2011. It is evident that the regions of the country evolved uniformly from 1948 to 1992 and, from then on, were supported by the economic base of the regions. Regarding the entropy factor, it is clear that, besides having no strong effect from 1948 until 1956, when the whole country was recovering from the war, they followed a rational pattern from 1956 to 1966balancing the North-eastern part of the country, more affected by World War II (see Figure 2), but then following an organic growth system with the increase of the surroundings of Bucharest and Constanta, with compensating effects from export correction factor and the spatial interaction factors. Figure 9 shows the percentile of the estimated Fixed Effects of the Mixed Growth Model (a), with the LISA Maps and the spatial autocorrelation coefficients. With the Mixed Growth Model, the regions of Bucharest and Ilfov (surroundings of Bucharest), Brasov and Galati have higher values, indicating that, for the calibration of this Mixed Model, the Permanent Potential of Spatial Interaction of Figure 3 could eventually be higher for these regions. Spatial Autocorrelation is positive but not strong, and it indicates a high-high correlation in the centre of Bucharest and Brasov.

Figure 8. Evolution of the coefficients of the mix model (exports (a), entropy (b) exports correction(c) and exports spatial interaction correction (d))
Source: Authors' representation

Discussion
The scale of the Graphs of the model coefficients in Figure 4 is the same as the scale of the Graphs in Figure 6 and Figure 8. So, it is easy to compare and discuss the results of the three models. The mixed model is the best in terms of R2 and the one that shows better results for all the coefficients.
Looking at the export coefficients, it is interesting to notice that they play a crucial role in both the organic and rational models, whereas the only significant export coefficients for the mixed model are for 1992 onwards. And growing as happens for that period for the organic and rational model. It seems that, with the mixed model, spatial interaction features become more relevant. First, the organic entropy, with a residual impact in the organic model, reveals more of its importance in the mixed model. Second, the exports' correction factor that explains how the impact of exports in one area spreads across the territory becomes more relevant and of increasing importance. Finally, the autonomous interaction factor that appears in the theoretical formulation of the rational model (Rodrigues & Dentinho, 2021), appears with robust coefficients, mainly in the period of rational interaction (1948)(1949)(1950)(1951)(1952)(1953)(1954)(1955)(1956)(1957)(1958)(1959)(1960)(1961)(1962)(1963)(1964)(1965)(1966), but with decreasing importance from 1966 to 2011. The best model, the Mix Model, refrains the role of exports, indicating that, beyond pushing exports, it is important to understand the role of each city within the network of city regions.
These changes in the logic of spatial interactionfrom rational to organic from 1948 to 2020 -did have a role in the sustainability and resilience of Romanian cities assessed with reference to the Zipfs Curve (Zipf, 1949) for Romanian City-Regions (Dentinho & Pandelica, 2020): for resilience, by the vertical change of population share within the same rank; and for sustainability, by the horizontal change in the ranking for the same population share.
The analysis shows that space does not need to be threatened as homogenous with a centre as proposed by Von Thünen (1826) and William Alonso (1964), not even as distance throughout networks as assumed in the works of Chistaller (1966), Losch (1954), Paelinck and Nijkamp (1976). Space is, also, more than a game table where people and places compete with each other, as interestingly pointed out by Hoteling (1929) and later on by (Krugman, 1991). Following the works of Wilson (1970), Sen and Smith (1995) and Echenique et al. (2013), space can be operationalized by interaction networks and even more than the evolution of the networks (Barthelemy, 2021). Notwithstanding this, space is not an econometric error (Anselin et al., 1997) to be tentatively neutralized. As we try to show, the logic behind spatial interaction is crucial for understanding spatial interaction and the flows and channels it encompasses.

Conclusions
The question was to know if the evolution of Romanian City-Regions followed a logic of rational optimization of spatial interaction, a logic of organic maximization of spatial interaction or a mixed logic, with rational logic for some periods and organic logic for other periods. The paper shows that the mixed model is the best, indicating that the logic of spatial interaction changed over the period. Actually, except for the first periods, from 1948 to 1966, the evolution of the spatial structure of Romanian City Regions is better explained by the organic formulation, either in the period of Central Planning , or in the period of Market Interaction , where the previous period population has a strong effect indicating a maximization of the population of each city region rather than a maximization of the interaction between city regions.
On the other hand, comparing estimates for the various models, it is clear that exports play a less significant role in regional growth when space, represented by spatial interaction logics, is appropriately integrated. The Permanent Potential of Spatial Interaction, presented as a proxy representation of the permanent features of the territory, is a crucial factor to explain the demographic and economic performance of City Regions, meaning that the stable structure of the territory seems to be, together with exports, the main explanatory factor of city-regions growth. Future work will try to adjust the Permanent Potential of Spatial Interaction and relate it to the permanent features of the territory.