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How do Chinese cities grow? A distribution dynamics approach

This paper examines the dynamic behavior of city size using a distribution dynamics approach with Chinese city data for the period 1984-2010. Instead of convergence, divergence or paralleled growth, multimodality and persistence are the dominant characteristics in the distribution dynamics of Chinese prefectural cities. Moreover, initial city size matters, initially small and medium-sized cities exhibit strong tendency of convergence, while large cities show significant persistence and multimodality in the sample period. Examination on the regional city groups shows that locational fundamentals have important impact on the distribution dynamics of city size.


Introduction
The issue of city size distribution has received considerable attention from urban economist.
Most of early works focused on testing and explaining the Zipf 's law or the rank-size rule [1]. The Zipf 's law states that the city size distribution follows a Pareto distribution with exponent one [2].
However, Zipf 's Law can only explain the rank of large cities, but has very limited explanatory power for the distribution of small-and medium-sized cities. Therefore, recent studies tried to test whether city size distribution follows a Pareto distribution, q-exponential distribution, double Pareto lognormal distribution, or other distributions without restrictions on city size.
In relation with the empirical literature on the rank-size rule, a considerable number of theoretical models have been developed to shed light on the dynamics of urban growth process. The random growth model, the parallel growth models and the sequential growth models are among the most important models of them. However, both theoretic and empirical literature shows that city growth patterns may be complicated. As indicated by the sequential growth theories , the growth rate of cities may differ from city size or time. City growth patterns may be random, convergent, divergent, parallel or even hybrid [3,4]. In the hybrid case, for example, cities may exhibit parallel growth in the long-run, but random growth in the short-run due to exogenous shocks [3,4].
However, there are some limitations in existing literature on the distribution of city size. First, the existing literature on the evolution of city size distribution has mainly focused on explaining rank-size rule, paying little attention to other interesting aspects of the growth process. In particular, not much research has been devoted to the transitional dynamics of a system of cities towards its steady state. The rank-size rule is an abstract of real city size; however, this simplification may 3 conceal some important information carried in the real city size. Second, most exiting studies in current literature use parametric approaches in their analysis. As indicated by Quah [5], parametric approaches only provide a statistical mean coefficient for a representative sample. This may neglect the entire shape of city size distribution and changes in the shape. Moreover, the distribution should include all sized cities, but not just the cities in the upper tail. A more useful way to explain the dynamic behavior of city size is to use a continuous distribution dynamics approach. Third, most studies use data from developed countries; few of them focus on the size distribution dynamics of cities in developing countries. This paper focuses on the distribution dynamics of city size in China. We make the following contributions in this paper: first, instead of rank size of upper tail cities, this paper examines the cities of all sizes, providing more dynamic information about the city size, and in particular, about the entire shape of city size distribution and changes in the shape over time.
Second, this paper adopts a continuous distribution dynamics approach which can provide the dynamic law of the entire shape of city size. The distribution dynamics approach can provide full information for the transitional dynamics and long-run steady distribution of all sized cities. Several studies, such as Eaton and Eckstein [6], Anderson and Ge [7], and Xu and Zhu [8], employ discrete Markov chain approaches in their examination on the transitional dynamics of city size distribution.
Compared with Markov chain approach, our continuous distribution dynamics approach requires no arbitrary discretization on the sample. Moreover, this approach can provide the full information of the transition dynamics of the entire shape of city size distribution.
Third, this paper uses Chinese city data to examine the distribution dynamics of city size.
China is experiencing rapid urbanization since 1980's. The urbanization rate has increased from 4 23.1 percent in 1984 to 49.7 percent by the end of 2010 in China. Moreover, Chinese cities experienced quite different growth trajectories during the past three decades due to governmental policy interventions. For example, eastern coastal cities experienced far more rapid growth in population than central and western cities did. Some of the northeast and western cities even loss their population in recent years. Unbalanced economic growth may have strong impact on the growth of city size. Therefore, the dynamics of city growth in China may be significantly different from that of the stable city growth in most developed countries.
We have no intention to examine which theory can predict the distribution dynamics of Chinese city size. Instead, this paper tries to shed light on the dynamic behavior of Chinese cities during the economic transition process. We found significant persistence and multimodality in the size distribution dynamics of cities in China. Initial city size has important influence on the following transitional dynamics. Initially small-and medium-sized cities converge to their own steady states, while large cities exhibit significant persistence and multimodality. Focusing only on upper tail large cities may neglect important patterns of city growth. Moreover, locational fundamentals also matter in the distribution dynamics of city size.
Examination on the distribution dynamics of city size has important policy implications.
Urbanization is not just about the increasing population in urban areas, but also about the spatial distribution of population and economic activities across different cities. This spatial distribution of population and economic activities is closely related with economic efficiency. During the rapid urbanization process in recent decades, urban investment accounts for largest share of the total investment in China. Most of these investments are extremely durable, knowing the distribution dynamics may help to make reasonable urban planning. 5 The remainder of the paper is structured as follows: Section 2 reviews some of the related empirical and theoretical literatures. Section 3 introduces the methodological issues and data.
Section 4 provides some preliminary analyses on the current city size distribution in China. The main empirical findings are provided in Section 5. The final section concludes the paper.

Literature review
The city size distribution has received much attention in economic research. In general, there are two kinds of literature on the distribution of city size. The first concerns about the shape of the city size distribution from a static perspective. The second examines how cities growth relative to each other from the dynamic perspective, i.e., the dynamic evolution of city size. In this section, we provide a brief review of these two strands of literature.
Empirically, most of the existing literature on the city size distribution has mainly focused on testing and explaining Zipf 's law or the rank-size rule. However, the Zipf 's Law is the special case of Pareto law when the exponent equals one [2]. Therefore, a considerable number of studies try to test if city size follows a Pareto law [7,[9][10][11][12][13][14][15]. Beside the traditional Pareto distribution, there are also studies on other distributions, such as the q-exponential distribution [16,17], double Pareto lognormal distribution [18][19]2], and the combination of lognormal and a Pareto distribution [20]. Due to differences in samples and econometric models, the empirical results are quite mixed. More empirical researches are expected to shed light on the distribution of city size.
Three strands of literature are closely related with the dynamic evolution of city size distribution. The first is the theory of random growth, which assumes that city growth process have the common expected growth rate and common standard deviation. The random growth 6 theory predicts that the city size distribution follows Gibrat's law. Gabaix [21] establishes the relationship between Gibrat's laws and Zipf 's Law by formally proving that Gibrat's law implies Zipf 's Law. Córdoba [22] derives restrictions on some factors, such as preferences, technologies, and stochastic properties of the exogenous driving forces, that urban models must satisfy to explain why city size distribution follows Pareto distribution. Rossi-Hansberg and Wright [3] develop a general equilibrium model of economic growth in an urban environment. Their model produces a city size distribution that is well approximated by Zipf 's law.
The second strand of literature is the theory of the parallel growth. The theory of the paralleled growth can well explain the Zipf 's law. Eaton and Eckstein [6] examine the evolution of the transition matrices of France's and Japan's largest metropolitan areas. Their results show that the size-distributions in steady states similar to what they have been in the history. Instead of convergence to an optimal city size or the divergent growth of the largest cities, they conclude that city size takes the form of parallel growth pattern. Black and Henderson [11] develop an endogenous urban growth model to describe how human capital accumulation, and knowledge spillover foster the parallel growth of cities. Sharma [23] uses Indian city data and time-series approach, and finds that the growth of cities may be parallel in the long-run. However, there are may be deviations from the long-run parallel growth of cities due to exogenous shocks in the shortrun. Wang and Zhu [24] find a parallel growth pattern of Chinese cities in the long run. Chen et al. [4] use Chinese city data and time-series econometric techniques to examine the parallel growth.
Their results show that overall Chinese city growth does not follow parallel growth. The parallel growth only exists in some city groups with certain characteristics, such as similar location advantages or policy regime. 7 The third strand of literature examines the Sequential growth of city size. Henderson and Venables [25] and Cuberes [26] develop models of city formation in which urban agglomerations grow sequentially. In these models, the initially largest cities are the first to grow until they reach a critical size, at which point they are followed by the second-largest cities, then the third-largest ones, and so on. Empirically, using two comprehensive data sets on populations of cities and metropolitan areas for a large set of countries, Cuberes [27] finds strong evidence for the sequential growth of city size. Sánchez-Vidal et al. [28] use the US new-born cities in the twentieth century to examine the distribution dynamics of city size. Their findings support the sequential growth of city size. Using the US city size data, Sheng et al. [29] also find that cities grow in sequential patterns.
Their results also show that city size in the upper tail exhibits significant conditional convergence characteristics.

Methodology and data
Traditional econometric techniques only provide information on the growth behavior of an average or representative economy [5]. They provide no information how cities of different sizes evolve relative to each other. They shed little light on some interesting questions on the long run distribution of city sizes. For example, is the city sizes growth parallel, diverge or converge? If they converge, is the long run steady state unimodal, bimodal or multimodal? To address these questions, one needs to know the entire cross-sectional distribution dynamics of city size in the sample.
The distribution dynamics approach developed by Quah [5,30,31] is widely used to examine the dynamic behavior of per capita income distribution. There are two types of distribution dynamics approaches: the discrete approach and the continuous approach. Early studies use a 8 discrete approach to estimate transition probability matrix and the ergodic distribution. This arbitrary discretization of state space into limited state have several limitations：first, it cannot provide any mobility information within state; second, it may change the probabilistic properties of variable in question; third, the estimated results of this approach are sensitive to this discretization approach [31][32][33]. To overcome these limitations, Quah [31] develops a stochastic kernel approach to estimate the transition probability and ergodic distribution. This approach requires no discretisation for the variables. Therefore, it can be considered as the upgraded version of the discrete one. This paper use the continuous distribution dynamics approach to examine the dynamic behavior of city size in China.
To estimate the long run (ergodic) distributions of city size, we need to calculate the transition probability distribution first. Let " ( ) denote the distribution of variable x at time t, and "&' denote the distribution of variable y at time t+τ, where τ>0. Assuming that the evolution of the distribution is time-invariant and first-order, that is, the future distribution of variable y in t+τ depends only on the distribution of variable x in time t. The relationship between the two distributions, namely " ( ) and "&' , can be written as follows: where the conditional density term ' ( | ) is a transition probability operator, which is a stochastic kernel mapping the transition process of the distribution from time t to time + . For [33] for further calculation details).
In this paper, we use kernel density method to estimate the transition probability distribution and its ergodic distribution. The joint natural kernel estimator of ","&' , x can be estimated as follows: (2) where ℎ : and ℎ = denote the bandwidth of x and y respectively. > and > are the population size values of the cities at time t and time + , respectively. Following most studies [31,33,34], the bandwidths in this paper are calculated as the optimal method described in Silverman [35].
Similarly, we can further define the marginal kernel of x as " x = With the joint and marginal kernel density function defined above, the conditional density can be calculated as follows: Given that the transition probability function ( ' ) remains unchanged, this transition dynamics will evolve into a long run equilibrium state, i.e. the ergodic distribution. Therefore, the ergodic distribution (denoted by ∞ ), which corresponds to a given ' , given that it exists, can be estimated through: In the continuous dynamic distribution approach, three-dimension plots and contour plots are the most often used tools to present the transition probability, while the ergodic distribution is displayed with a kernel density plot. To get more information on the transition dynamics of the distribution, this paper adopts a net transition probability approach proposed by Cheong and Wu [36]. This approach provides the precise values of net upward probability at each point. Intuitively Following common practice in distribution dynamics approach, this paper uses relative city size (RCS) which is the population size of each city divided by its yearly average in the analysis.
Chinese city data sets are collected at two levels: county level and prefectural level. From the perspective of agglomeration and administration, prefectural city tends to be more like an urban system than county level city. Due to data availability, this paper uses 237 prefectural cities for the Anderson and Ge [7], which shows that the distribution of city size in China remains stable before the Reform but show significant convergence during the post-reform period.

Distribution dynamics in different size groups
Many studies [25,26] have indicated that initial city size have significant impact on the following growth of cities, implying the existence of path dependence effect. To further understand how initial city size affects the following distribution dynamics, we divided all cities into three equal groups according to their population size in 1984, namely small cities (79), mediumsized cities (79) and large cities (78). Therefore, this sub-section will focus on the distribution dynamics in various size groups.
(1) Large cities The medium-sized cities in the initial year of 1984 with RCS values smaller than 0.8 have a net positive transitional probability, implying that these cities have a strong tendency to increase their relative sizes. However, the medium-sized cities with RCS greater than 0.8 have net negative transition probabilities, indicating that those cities have strong tendency to reduce their relative sizes. Correspondingly, the ergodic distribution in panel (d) of figure 7 exhibit significant unimodality, with the peak situates around the RCS value of 0.8. This implies that, keeping this transitional dynamics unchanged, the medium-sized cities in 1984 will converge to 0.8 times average city size in the long run steady state.

Distribution dynamics in regional groups
Locational fundamentals may play crucial role in the formation and distribution dynamics of city size [38]. China is a large country with many heterogeneous regions, which differ greatly in terms of natural resource endowment, weather conditions, transport accessibility and government policy intervention. These factors may have strong impact on the distribution and growth of cities [4]. Therefore, in this subsection we will examine the distribution dynamics of city size in different regional groups. Following most studies, we divide the full sample of Chinese cities into three sub- (1) The eastern cities

Conclusion and discussion
The issues whether city size growth parallel, converge or diverge have attracted many attentions in urban studies. This paper examines the size distribution dynamics of Chinese cities during the period 1984-2010. It differs from existing studies in using a new framework named distribution dynamics approach, which can provide dynamic behavior of the entire distribution shape of city size. In particular, this approach has advantages in analyzing persistence and convergence clubs. The results in this paper show that persistence and multimodality are the dominant characteristics in the distribution dynamics of the Chinese cities. Some studies use persistence as the evidence of the paralleled growth of city size. However, the multimodality found in long run steady distribution for all cities and regional city groups implies that this may be misleading. The distribution dynamics of city size may be more complicated than one expected.
New models should be developed to shed light on this new stylized fact. Our results show that Chinese cities exhibit significant convergence in the sample period, 61.8 percent of cities have tendency to move down in the intra-distribution dynamics.
Similar to that predicted by the theory of sequential growth, our results show that initial city size may have important impact on the following distribution dynamics of city size. Initially small 25 and medium-sized cities exhibit strong tendency of convergence, while large cities show significant persistence and multimodality in our sample period. This result does not support the sequential growth theory that large cities have more mobility than small-and medium-sized cities. The path dependence effect, if any, would be stronger in the large cities than in small-and medium-sized Our findings suggest that locational fundamentals may have impact on the distribution dynamics of city size. Future research should focus on how location-specific factors, such as weather conditions, natural resources, and governmental regional development programs, affect distribution dynamics of city size.
The findings in this paper have important policy implication. China is experiencing rapid urbanization. The Chinese government play important role in this process through investment policy. Most of the urban investment is extremely durable. Knowing the distribution dynamics of city size in the long run may help the government to make reasonable urban planning and improve economic efficiency in the urbanization process.