City Size Distribution and Economic Growth: The Case of China

Transcription

1 City Size Distribution and Economic Growth: The Case of China Ting JIANG, Ryo OKUI and Danyang XIE Department of Economics, School of Business and Management The Hong Kong University of Science and Technology December 27, 2008 Abstract This paper empirically explores the relationship between city size distribution and economic growth, based on a panel analysis using China provincial data from 1984 to We first estimate Zipf s coefficient, a measure of city size distribution, for each province in each sample year using Gabaix and Ibragimov s (2006) method. Second, we propose and verify a nonlinear relationship between Zipf s coefficient and GDP growth rate capturing the idea that government intervention on labor migration distorts city size distribution and entails productivity loss, so that a deviation of Zipf s coefficient from its equilibrium value has a negative impact on economic growth. Third, we construct a VAR model with which the co-evolution of city size distribution and economic growth is illustrated. Simulation results based on the estimated VAR model uncover the effect of an exogenous shock to city size distribution on economic growth and provide policy implications. Keywords: City size distribution; Economic growth; Zipf s law; Migration JEL Classifications: O18; R11; R12; R23 We have benefited from Kazuhiro Yamamoto and seminar participants at the Renmin University of China and the Hong Kong University of Science and Technology for helpful comments and suggestions. This research is fully supported by a grant from the Research Grants Council of the Hong Kong SAR (Project No. HKUST6470/06H). The usual disclaimer applies. 1

2 1 Introduction Urbanization is a ubiquitous phenomenon accompanying economic development because in cities most of industrial production and innovation take place. 1 This is especially true for developing countries when they experience transformation from agricultural to industrial economies. China is a vivid example. Since the economic reform and open-door policy was implemented in 1978, China has witnessed explosive economic growth for thirty years, with annual real GDP growth rate exceeding 8% in most of the years. At the same time, consistently rapid urban expansion has totally changed the geographical composition of the population. In 1978 when the reform began, only 17.92% of the total population lived in cities. In 2005, urban population represented 42.99% of the total population. This number is close to the world average (48.7%), although still much lower than that for developed countries (74.1%). 2 An important consequence of urbanization is a change in city size distribution. Here, the size of a city is measured by the number of people residing in that city. City size distribution is described by the number of cities and individual city sizes. In Figure 1, we compare the difference between city size distributions of China in 1984 and 2005 by dividing all the cities into six size categories. Two basic observations can be made: First, there is a large increase in the total number of cities (from 295 in 1984 to 656 in 2005, more than doubled), due to creation of new cities; Second, there is a large increase in the size of individual cities, which results in a boom of medium-sized cities and emergence of mega-cities. These results are striking, even when we take into account the fact that the total population of China increases by about 25% during this period. Despite that the coexistence of accelerated urbanization and economic growth seems to be self-explanatory, the evolution of city size distribution in the process of economic development is far from straightforward. A natural question is whether a faster growth leads to a more equal distribution of cities as we see more and more medium-sized cities are formed, or to a more unequal city size distribution as people might concentrate in large cities and smaller cities might decay and die? Yet it is more apposite to ask in the opposite way: Is more equal city size distribution conducive or harmful to economic growth? This paper aims to investigate the interaction between 1 The causal relationship between urbanization per se and economic growth is not the theme of this paper. See Gallup et al. (1999) which implies, and Bertinelli and Black (2004) which explicitly models such causality that urbanization promotes economic growth. 2 Source: World Urbanization Prospects: The 2005 Revision, available at the United Nations web-site, 2

3 city size distribution and economic growth in the context of China, and provide insights into the above questions via empirical analysis. We measure city size distribution by a single coefficient obtained from an auxiliary regression following the existing literature. For a group of cities, we first order them by size and record their ranks, and then regress the ranks on corresponding sizes using ordinary least squares (OLS) after taking natural logs on both sides. The estimated coefficient associated with the log of size (i.e., α 1 in the following equation) is a statistic that represents the size distribution of this group of cities: ln (Rank) = α 0 α 1 ln (Size) + Error. (1) A larger α 1 means city sizes are more evenly distributed, 3 while smaller α 1 means that there is a wider dispersion in city sizes. Remarkably, when researchers apply this simple exercise to various sets of real world data, they find in most cases α 1 is very close to unity (see Gabaix and Ioannides (2004) for a comprehensive survey). It has now become a well-accepted empirical regularity bearing the name Zipf s law. We call α 1 Zipf s coefficient henceforth throughout the text. Like many other studies, this paper acknowledges Zipf s law as being a good approximation to city size distribution in the real world. Moreover, many theoretical works predict that in a standard decentralized economy the city size distribution will evolve to its steady state with Zipf s coefficient approaching unity. We thus conjecture that city size distribution with Zipf s coefficient equal to one is actually the optimal distribution that would be observed in the absence of a friction or an obstacle to the functioning of the economy. A deviation of Zipf s coefficient from one could be viewed as a direct consequence of some distortion of individual decisionmaking (e.g., their migration and residence choices), which may be due to exogenous shocks or government intervention to the economic environment. We may reasonably assume that those distorted decisions are detrimental to economic growth. This hypothesis is the starting point of our attempt to build a relationship between city size distribution and economic growth. We argue that internal migration restriction in China is an important reason why Zipf s coefficient fails to be one in Chinese data, and Zipf s coefficient has a nonlinear effect on economic growth when it serves as a proxy of the enforcement of the migration restriction policy. This paper examines the effect of variation of Zipf s coefficient on aggregate economic performance, the speed of economic growth in particular. As far as we are aware, this is new in the literature. We estimate dynamic panel data models for the relationship between Zipf s coefficient and economic growth using Chinese data. 3 Imagine the extreme case when all the cities in an economy were of the same size, then if they were plotted in a rank-size diagram, we would have a vertical line and hence an infinitely large α 1. 3

4 Province is our definition of economy where a system of cities exists and evolves. By using provincial data, we are able to construct a rich panel dataset. The application of panel data analysis enables us to study the dynamics of city size distribution and economic growth and to avoid the bias caused by unobserved time-invariant province specific factors. We note that cross-sectional data do not allow us to address these issues. We estimate the models by GMM estimators for panel data. The results show that a deviation of Zipf s coefficient from its optimal value has a negative impact on economic growth. Besides, we also estimate a VAR (vector autoregressive) model which describes the co-evolution of city size distribution and economic growth. We then examine how the economy behaves in the short and long run when it faces exogenous shocks to initial Zipf s coefficient or growth rate. The rest of the paper is organized as follows: Section 2 summarizes the existing studies on the link between Zipf s law and economic development; Section 3 describes the dataset and discusses the choice of control variables and the theories behind them; Section 4 discusses how to construct a panel of Zipf s coefficients; Section 5 demonstrates the nonlinear effect of Zipf s coefficient on economic growth; Section 6 examines an economic theory in which economic growth affects Zipf s coefficient; In Section 7, we estimate a VAR model and carry out some simple simulations; Section 8 concludes the paper. 2 Zipf s Law and Economic Development The discovery of Zipf s law has inspired a considerable amount of subsequent research. Numerous researchers try to model an economy that generates a city size distribution which satisfies Zipf s law at its steady state. These studies not only justify our adoption of Zipf s coefficient as a proper measure of city size distribution, but also motivate our design of empirical strategies. Gabaix (1999) introduces exogenous idiosyncratic amenity shocks to explain the growth and decline of cities. Both wage and amenity enter workers utility function multiplicatively. A positive amenity shock leads to an inflow of workers but their utility increase is partly offset by wage decrease. The resulting population shock to a city is proportional to its current size, therefore in the steady state there must be a hierarchy of cities with different sizes. Rossi-Hansberg and Wright (2007) study a system of mono-centric cities each of which specializes in only one industry. Industryspecific exogenous technology shock affects city population through its effect on the average product of labor in the city, as residents seek to get highest earnings net of commuting costs associated with city size growth. Their theory also produces a city size distribution that is well approximated by Zipf s law. Duranton (2006) develops a 4

5 Romer-type endogenous growth model of product proliferation. Urban population is proportional to the number of varieties of industries (products) located in a city, and the growth of city size depends solely on the creation of new industries through R&D. The existence of local externality in R&D activities leads to a similar mechanism as in Gabaix (1999) in which large cities are more likely to expand, which in turn gives rise to Zipf s law. Another paper by Duranton (2007) examines the same issue using a Grossman- Helpman-type endogenous growth model of quality ladders. The channel through which economic development occurs in this model is not the enrichment of product varieties, but the improvement of product qualities. City sizes fluctuate as the locations of industries switch from one city to another. This is so far the only model in which the steady-state economic growth and the steady-state city size distribution are simultaneously solved. Using this model, we exploit an explicit link between economic growth and city size distribution. We empirically examine the validity of this theory in Section 6. Empirical studies related to our paper try to answer the question what factors drive the variation of Zipf s coefficient (i.e., treating Zipf s coefficient as the dependent variable in a regression). 4 Most of them are cross-sectional studies with country level data. Rosen and Resnick (1980) is an early comprehensive study on this topic. They estimate Zipf s coefficients for 44 countries in 1970 (or around that year), and find that variation in Zipf s coefficients is found to be positively associated with population level and GNP per capita, and negatively associated with land area and railway density. Alperovich (1993) reproduces Rosen and Resnick s results by employing a richer set of explanatory variables and also reaches some new findings. In particular, he finds that high degree of government involvement in the economy and high proportion of GDP produced by industrial sectors promote urban concentration (lower Zipf s coefficient). In a recent paper, Soo (2005) deals with a dataset of 75 countries with the latest record in He also uses OLS in the first stage to estimate Zipf s coefficient. In the second stage by OLS with panel corrected standard errors, he finds that political variables have more explanatory power of the variation than economic variables. All the four variables in Rosen and Resnick (1980) plus the size of non-agricultural sectors, the size of international trade, and the degree of scale economy either are insignificant or enter with opposite sign to what theoretical models would predict. Interestingly, he finds that the size of government expenditure is positively related to Zipf s coefficient, which contradicts Alperovich s (1993) 4 The validity of Zipf s law is also a serious concern in empirical works. Dobkins and Ioannides (2001), Ioannides and Overman (2003), and Black and Henderson (2003) are a few examples of recent attempt in this direction. 5

6 finding. Alternative approaches have been taken in closer examination of Zipf s coefficient. First, different techniques other than simple OLS could be applied. Alperovich (1992) carries out a pooled OLS analysis on panel data of Israel cities for various years between 1922 and He simultaneously estimates Zipf s coefficient and investigates their variation. Moomaw and Alwosabi (2004) is a panel analysis based on fixed effects models. The dataset consists of 30 countries from Asia and America for 7 time periods between 1960 and They use urban primacy 5 rather than Zipf s coefficient as the measure of city size distribution and find economic and political variables both to be important. Second, different empirical questions have been raised and tested. Alperovich (1992) finds out that Zipf s coefficient first decreases and then increases with per capita GNP when the country goes through different phases of development. Henderson (2003) investigates how city size distribution would affect economic growth. He demonstrates a nonlinear relationship between urban primacy and productivity growth across countries. This result implies that there might be an optimal degree of urban concentration, and over- or under-concentration may be very costly. Third, the study of developing countries is of independent interest, since they are experiencing more or less dramatic transition and restructuring of the socioeconomic system in which human activities are being organized and their city size distributions tend to exhibit larger fluctuation. The case of China is considered by Anderson and Ge (2005). They estimate Zipf s coefficients for the nation from 1961 to 1999 and check from a pure statistical perspective that there is a structural change of city size distribution due to economic reform and one child policy. 3 Data We make use of the data on 23 provinces in China from year 1984 to Four municipality cities (Beijing, Tianjin, Shanghai, and Chongqing) and four provinces (Hainan, Tibet, Qinghai, and Ningxia) which have few cities are excluded. The choice of time span is constrained by data availability. On the other hand, restricting the data in post-reform period (after 1978) may avoid the problem of structural change pointed out by Anderson and Ge (2005). There are two key variables. One is annual real GDP per capita growth rate which is our measure of the speed of economic growth. The other is Zipf s coefficient, which has to be estimated from city population data for each province in each year. Besides, we have a candidate set 5 The concept of urban primacy will be reviewed in Section 4. 6

7 of control variables which are discussed in Section 3.2. The population data come from China Population Statistics Yearbook and China Urban Statistics for various years. The data for GDP and control variables come from China Compendium of Statistics, and China Statistical Yearbook for various years. 3.1 Issues on population data In China, there are two kinds of cities, one is prefecture-level city (some of them are further defined as sub-provincial-level city), controlled by the province, the other is county-level city, most of which are controlled by the prefecture-level cities while some are controlled directly by the province. We refer the notion of city to both prefecture-level cities and county-level cities. Regarding city population, there are four categories. They are population in the whole region, population in town, nonagricultural population in the whole region, and non-agricultural population in town. Since the data for the last category is the most complete, we choose it as our measure of city size. The National Bureau of Statistics of China also uses this data to construct a size ranking of all cities in China every year. 6 We eliminate two tiny cities from the data. 7 We make these adjustments to remove sudden and sizable rise and fall in the evolution of Zipf s coefficient. We believe that city size distribution results from the aggregation of numerous uncoordinated individual actions, so that it should evolve only smoothly. The existence of these tiny cities can only be explained by administrative reasons rather than representing the effects of economic forces, therefore they are eliminated from the dataset. 8 6 Shen (2006) points out that urban population data for China may not be reliable and consistent. We are fully aware of this possibility. There are two concerns. One is the employment of rural labor in TVEs (township and village enterprises). The other is temporary population of cities mostly composed of rural migrants. The former is also related to another concern that economically meaningful definition of city agglomeration may be different from administratively defined cities. The latter casts more doubt on the credibility of existing data, since temporary population unquestionably contribute to city production and also place burden on city amenity and infrastructure. For some cities, the volume of economic activities by temporary population takes up a non-negligible share of the total number. Potential drawbacks notwithstanding, no systematic adjustment has been made in the present study simply because there is no proper and accepted way of doing so, especially with respect to city level data. 7 Wuda Lianchi is deleted from Heilongjiang from 1985 to It formed in 1985, remained at a low population level of less than 10 thousand before its population explosively increased to more than 110 thousand in Second, Wanting is deleted from Yunnan for all the sample years. It formed in 1985 and deceased in 1999, with its population never exceeding 6 thousand, far less than other cities in the same province. 8 Gabaix and Ioannides (2004) also mention the choice of cutoff size to remove tiny cities (see footnote 4 of their paper). This criterion is not applicable here because the sample size for a 7

8 3.2 Choice of control variables In previous empirical studies seeking for the determinants of variation in city size distributions, besides the variables indicating GDP level or growth, control variables often include population level, land area, size of industrial (or non-agricultural) sectors, size of external (or international) trade, road (or railway) density, and political variables such as size of government, bureaucratic efficiency, political freedom. In addition, growth accounting exercises typically use (physical and human) capital investment rates, inflation rate, black market exchange rate premium, and geographical dummies as independent variables. In the current context, some of the variables are impossible to obtain (such as measures of provincial bureaucratic efficiency or political freedom), some are invariant over time so that they are already captured by fixed effects in panel analysis (such as land area or the dummy indicating whether a province is along the coastal line), and some are unlikely to affect or be affected by city size distribution (such as capital investment rates, inflation rate, or black market exchange rate premium). The remaining candidates are population level, size of industrial sectors, size of external trade, road density, and size of government. It is helpful to first address very briefly two distinct features of cities before we continue to discuss the hypothesized effects of these variables on the direction of change in Zipf s coefficient. The benefit of urbanization comes from agglomeration economies. A static view is that in a city where labor is affluent and trade among firms is costless, production can be organized cheaply. The production network also enables each firm to specialize in the industry where it has comparative advantage, and economy-wide specialization leads to higher efficiency. A dynamic view is that when many firms and industries are gathered around, increased stock of knowledge will greatly facilitate R&D and make technological innovations much easier to take place. The cost of urbanization is due to congestion and hence decreased individual utility from amenities. Looking at the production side, employees will ask firms to compensate for the decrease of living standard, which results in higher production cost and less output. Cities of different sizes will typically face different scales of trade-off between these benefit and cost. When the population level is high, congestion cost may outweigh agglomeration benefit in large cities, so that people are motivated to move to smaller cities. When the size of industrial (non-agricultural) sectors is small and that of agricultural sectors is large, industrial cores have to be scattered to serve many agricultural peripheries, so that many cities are formed. The size of industrial sectors can also be viewed as an indicator of the degree of scale economy. The more prevalent modprovince is fairly small, so we have to keep as many data points as possible. 8

9 ern industries are, the more profitable to form large cities to extract agglomeration externalities. When an economy is involved in a greater extent of external trade (enjoying higher degree of openness), it is less pressing that the demand of its people for different varieties of goods has to be satisfied by internal trade, and that suppliers have to locate themselves near local consumers to sell more of their goods. Thus there is less demand-driven force to form large cities. The effect of road density on city formation is straightforward. More roads mean less transportation cost between the cores and the peripheries and more firms can be located together to form larger cities. More roads also mean less commuting cost, and make it much easier for people to work in mega-cities and live in suburb areas. The effect of the size of government on city distribution is ambiguous. There are two competing hypotheses. One emphasizes the distributive role of governments. The stronger the government is, the more able it is to reduce regional inequality and provide infrastructure and other public service to small cities to foster their growth. This point of view predicts a province with large government size to have equal city size distribution. The other mechanism relies on the fact that larger government size is usually associated with more political rents, and rent-seeking activities are more possible in large cities (usually provincial capital cities). This perspective would predict the opposite effect. Because of the fairly long time span of our dataset, we should avoid using any level variables that would exhibit certain trend over time. Another limitation is that there is no appropriate measure of the size of external trade for a province, which is the counterpart of international trade for a country. We do have the data on total imports and exports for each province, but the data on cross-province trade volume, probably more relevant in our case, are unavailable. Finally, the control variables (all of which are yearly data on the provincial level) we choose are: population natural growth rate (P opu); change in the size of industrial sectors (Ind) as a share of the provincial total GDP; 9 the size of government (Gov), measured by provincial government expenditure as a share of provincial GDP; growth rate of road mileage (Road); 10 and the size of international trade (T rade), measured by total imports and exports as a share of total GDP. Summary statistics for the five control variables are provided in Table 1. Their predicted effects on the direction of change of Zipf s coefficient are summarized in the following table. To reiterate, an increase in Zipf s 9 We also construct the relative size of industrial sectors, measured by first calculating the size of industrial sectors for each province (and the nation) and then taking the difference between the number for each province and that for the nation. But it does not affect the qualitative results and is not reported. 10 We also construct the growth rate of railway mileage. Since the data is incomplete and for many years there is no variation, it does a much worse fit in all regressions and is not reported. 9

10 coefficient means more equal size distribution, or more small and medium-sized cities are formed. On the other hand, a decrease in Zipf s coefficient means more unequal size distribution, or people are more concentrated in large cities. Control variable Predicted effect on Zipf s coefficient Population natural growth rate (P opu) + Change in size of industrial sectors (Ind) Size of government (Gov)? Growth rate of road mileage (Road) Size of international trade (T rade) + 4 Estimation of Zipf s coefficient Although the entire city size distribution may be drawn for any economy of interest, a representative yet operational measure that characterizes the property of the distribution should be made available before direct empirical testing. Several candidate measures have been proposed in the literature. One is spatial Gini index, constructed following exactly the same idea which has been extensively applied to measure income inequality. A perfectly equal distribution of city sizes implies a zero spatial Gini index and more unequal distribution corresponds to higher index. A similar analogy can be made to construct a spatial Hirschman-Herfindahl index, using the sum of the squares of the population share of all cities in an economy. The magnitude of the index is also positively associated with the degree of inequality of the city sizes distribution. A third measure is called urban primacy. It has a couple of variants. The most commonly used is the ratio of the size of the largest city to that of the whole economy. Others include the ratio of the size of the largest city to that of the second largest city, the ratio of the size of the largest city to the next three largest cities, and the ratio of the size of the two largest cities to the next two largest cities, etc.. Again, an economy with more unequal city size distribution is likely to have a larger urban primacy. However, the measure that attracts economists and geographers interest most is Zipf s coefficient. As was first discovered by Auerbach (1913) and later propagated by Zipf (1949), the size distribution of a group of cities is such that the number of cities whose sizes exceed a certain level S is proportional to the inverse of S. Concisely, we have the following probabilistic version of Zipf s law: P rob (Size > S) = α 0 S α 1 with α 1 1, (2) 10

11 and refer to α 1 as Zipf s coefficient. 11 If we take natural logs on both sides it becomes: ln (Rank) = α 0 α 1 ln (Size). (3) Statistically we should observe α 1 1 if Zipf s law is satisfied. Equation (1) is the empirical counterpart of this rank-size rule. However, Gabaix and Ibragimov (2006) show that the OLS estimator based on equation (1) or (3) is heavily biased and should be modified in the following way to correct the bias: 12 ( ln Rank 1 ) = α 0 α 1 ln (Size) + Error. (4) 2 Zipf s law implies that the null hypothesis α 1 = 1 should not be rejected. The interpretation on the magnitude of α 1 is that α 1 gets larger as the city size distribution becomes more even. We illustrate the idea in Figure 2, where we plot the samples of all cities in China in 1984 and 2005 respectively, and the fitted lines are depicted according to the regression equation (4). We can see that the slope of the fitted line for 2005 is steeper than that for 1984, suggesting the city size distribution of China in 2005 has become more equal as opposed to that in We calculate the four aforementioned measures for the entire period and the results are displayed in Figure 3. The evolution of the four measures over time generates similar patterns. For example, before year 2000, all the first three measures show a downward trend while Zipf s coefficient is going upward. The spatial Gini index curve has a U-shape, hitting the bottom in 1999, which matches the inverted- U-shape observed in the Zipf s coefficient curve. Actually, Zipf s coefficient has a strong pairwise correlation with the other three measures. 13 Although we cannot prefer one measure over another on pure theoretical ground as we do not know the underlying data-generating process, we shall expect that the results does not rely heavily on which measure to be used. Since Zipf s coefficient is generally favored in both theoretical and empirical literature, we choose it as the measure of city size distribution in carrying out our empirical study. Henderson (2005) asserts that spatial Gini index might be more reliable than Zipf s coefficient if we want to make any inference in a time-series framework, because 11 It is sometimes labeled Pareto exponent, because equation (2) is the defining property of Pareto distribution. Zipf s law is a special case of Pareto law with shape parameter α 1 = 1. (See Gabaix (2008) for an introduction on Pareto law.) 12 The formula for the standard error of α 1 is S.E. (α 1 ) = 2/nα The correlations between Zipf s coefficient and spatial Gini index, spatial HHI, and urban primacy are.904,.937, and.910, respectively. 11

12 the rank-size rule may not approximate the real-world distributions in all cases. His argument does not necessarily challenge the validity of our approach because we use a different estimation procedure for Zipf s coefficient from former empirical studies. 14 A panel of Zipf s coefficients for all 23 provinces during the years is estimated using the new rank-size rule (equation (4)). Summary statistics for the panel are shown in the first row of Table 1. Note that the mean value is larger than one, implying that provinces in China on average have a more even distribution of city sizes than in the ideal case where Zipf s law holds perfectly. Figure 4 further provides an overview of the evolution of Zipf s coefficient for each province from 1984 to A general conclusion is that provinces exhibit quite distinct time paths and no uniform pattern of evolution can be observed. For instance, cities in Fujian and Sichuan are consistently very evenly distributed (high Zipf s coefficient), while Hebei and Inner Mongolia have a wide dispersion of city sizes (low Zipf s coefficient) which also persists over time. Guangdong and Heilongjiang have once reached the highest and lowest values of Zipf s coefficient among the panel, respectively, although the deviation (from one) seems to be decreasing in recent years for both cases. There are also a number of provinces (e.g., Guizhou and Shanxi) whose city size distributions are nearly perfectly described by Zipf s law. For all the Zipf s coefficients we estimate, the null hypothesis α 1 = 1 is never rejected due to large standard errors. We now turn to formal examination of the interrelationship between city size distribution and economic growth, using the estimated panel as input for secondstage regressions. 5 Migration restriction, city size distribution and economic growth China has implemented strong migration restriction through household registration (hukou) system for many years. The system was set up in 1958, prescribing each individual a place of legal residence. Differential treatments are given to urban and rural residents, in favor of the former, regarding various civil rights pertaining to employment, housing, education, social security, and health care, etc., although the original intention of the system was only to find a way to ration citizens when supplies in necessities of life were scarce. Au and Henderson (2006a, 2006b) argue that migration restriction results in substantial loss in GDP. This argument can be justified in three aspects. First, it would impede large number of implicit rural unemployment 14 Henderson and Wang (2007) is an empirical work using spatial Gini index as the measure of city size distribution. 12

13 to be absorbed by urban industries, retard the formation and prosperity of labor market and market for human capital, and distort the allocative role of market economy for economic factors to achieve efficiency. Second, it reduces aggregate demand, since numerous illegal or temporary laborers are not expected to have growing consumption profiles before legitimate identity is obtained, basic needs for survival are met, and stable working condition is secured. The low life quality of a substantial number of de facto but not de jure urban population results in a vicious circle of low consumption and human capital investment, low productivity, and low earnings, which eventually jeopardizes the health of national economy. Third, migration restriction violates the idea of social justice based on equal opportunity and brings on urban-rural stratification, which would exacerbate social tensions and possibly may cause massive social conflicts that generate adverse economic effects. If people were allowed to move freely across cities, ceteris paribus, the economy would operate on the efficiency frontier. At the same time, the size distribution of cities will be shaped according to Zipf s law. This idea is drawn from the theoretical literature addressing the steady-state city size distribution in an endogenously growing economy, especially Rossi-Hansberg and Wright (2007) and Duranton (2006). In Rossi-Hansberg and Wright s (2007) formulation, land is owned by property developers who extract rents from their land. To achieve this goal, they offer a rich set of subsidies to attract firms and workers to migrate into the city. Developers maximize total rents net of subsidy costs, and by doing so they can fully internalize productivity externalities originated from industrial agglomeration. Therefore the equilibrium allocation is Pareto optimal, and government intervention which blocks free mobility across cities will inevitably reduce the efficiency and growth of the economy. Duranton (2006) does not make explicitly clear whether and how government policy can play any efficiency-enhancing role. Since the growth engine of the economy is R&D activities of competitive research firms aiming to invent new varieties of products based on existing general stock of knowledge, we might think that the government could boost the economy by coordinating independent research efforts in order to avoid duplication of experiments. However, it is hard to believe that the migration restriction mimics this sort of coordination policy. 15 We cannot measure the enforcement of migration restriction policy in different provinces, but we do know that the policy will have a direct effect on city size distribution. A large deviation of Zipf s coefficient from one could be viewed as a large 15 The only positive effect of migration restriction on such an economy we can barely think of is that by prohibiting high-skilled labors from changing their working places, firms have right incentive to invest in human capital accumulation. Not only is this possibility difficult to exploit in a general case, but in China, high-skilled labors actually find themselves easier to move. 13

14 distortion of government policy on the economic activities of individual citizens, and will entail great efficiency loss on the economy. Therefore we should observe a nonlinear relationship between GDP growth rate and Zipf s coefficient, in which Zipf s coefficient serves as a proxy for the intensity of the migration restriction policy s impact on economic performance. We model the nonlinear relationship between city size distribution and economic growth using the quadratic function of α: g it = β 1 g i,t 1 + β 2 α i,t 1 + β 3 α 2 i,t 1 + X itγ + D tδ + η i + u it, (5) where g is real GDP per capita growth rate, α is Zipf s coefficient, X it is a vector of control variables, D t is a vector of year dummies. η i denotes time-invariant provincespecific factors. Error term u it is independently and identically distributed. i is province index and t is time index. The marginal effect of α on g is: g it α i,t 1 = β 2 + 2β 3 α i,t 1. It varies with different values of α, and the sign of the effect and the direction of change depend on the signs of β 2 and β 3. Since Zipf s coefficient is obtained using end-of-year population data, we expect it to have potential influence on the next year s economic growth, so both α and α 2 enter the equation with lagged rather than contemporaneous terms. Fixed-effects estimation results of equation (5) are reported in columns (1) to (3) of Table 2. We can see that output growth is moderately persistent, which is captured by a significantly positive β 1. The parameter associated with α 1 is positive and the parameter associated with α 1 2 is negative, both of which are significant no matter whether control variables and year dummies are added or not. This result implies that the impact of an increase in lagged Zipf s coefficient on GDP growth is positive when it is small (too unequal size distribution), and the impact is negative when Zipf s coefficient is already large enough (too equal size distribution). One concern with fixed-effects estimator is that after within-group transformation, we have E [( g i,t 1 1 T 1 ) ( T 1 g i,j j=1 u i,t 1 T 1 )] T u i,j 0, so that the regressor and the error term after the transformation are correlated. 16 It is now well-known that the fixed-effects estimator has a downward bias of order 16 u it is correlated with g i,k 1 for k t + 1. j=2 14

15 O (1/T ), i.e., it will retain consistency only when T tends to infinity. In particular, Judson and Owen (1999) show that the bias is not negligible even when T = 20. To overcome this problem, Holtz-Eakin et al. (1988) and Arellano and Bond (1991) proposed a first-differenced GMM estimator. We first take the first difference of equation (5): g it g i,t 1 = ( ) β 1 β 2 β 3 γ δ g i,t 1 g i,t 2 α i,t 1 α i,t 2 α 2 i,t 1 α 2 i,t 2 X it X i,t 1 D t D t 1 + (u it u i,t 1 ). Given that u is serially uncorrelated, g i,t k (k 2) can serve as valid instruments for (g i,t 1 g i,t 2 ). We assume that α it (α 2 it) and X it are assumed to be endogenous, i.e., they are uncorrelated with future realizations of u but correlated with current and lagged values of u: E(α it u is ) = E(α 2 itu is ) = 0, E(X it u is ) = 0 for s > t, E(α it u is ) 0, E(α 2 itu is ) 0, E(X it u is ) 0 for s t. This assumption allows the simultaneity of g and α (α 2, X), as well as the potential feedback of α (α 2, X) on g. The difference-gmm estimator is the GMM estimator based on the following moment conditions: E [Φ i,t s (u it u i,t 1 )] = 0 for s 2; t = 3,... T ; i where Φ = { g, α, α 2, X }. (6) For the GMM estimation, we use the weighting matrix that is optimal in homoskedastic cases, but we report the cluster robust standard errors (Arellano (1987)). The results for difference-gmm estimation are shown in columns (7) and (8) of Table 2. Because T is large in our sample, we have to use a truncated set of instruments to ensure the number of instruments does not exceed the sample size. In column (7), we report the result using a maximum of two lagged variables as instruments. In column (8), we also report the result using only one lag of all pertinent variables as instruments, for the purpose of robustness check. The parameters associated with α 1 and α 2 1 remain significant and the estimate of growth rate persistence is adjusted slightly upward, as expected. The Arellano-Bond tests at the last row indicates that there is no evidence of serial correlation in the error term (thus, there is no evidence against instrument validity). We also consider a restricted specification derived from our hypothesis g it = β 1 g i,t 1 + β 2 (1 α i,t 1 ) 2 + X itγ + D tδ + η i + u it, (7) 15

16 where β 2 is expected to be negative. This specification is based on the idea that it is the deviation of Zipf s coefficient from unity that is associated with distortion to the economy. Fixed-effects estimation results of equation (7) are reported in columns (4) to (6) of Table 2, and difference-gmm estimation is similarly implemented and reported in columns (9) and (10). Note that the instrument set, Φ, is now { g, (1 α) 2, X }. The coefficient β 2 appears significantly negative in all the specifications. In particular, our preferred estimates, given by GMM, are.115 and.142. However, the effects of control variables vary upon the inclusion of year dummies and different choices of instrument sets. We now state our main result formally: Other things being equal, any deviation from optimal city size distribution will result in lower economic growth. Specifically, when Zipf s coefficient deviates from one by x, real GDP per capita growth rate will decrease by about 11.5x 2 % to 14.2x 2 %. 6 City size distribution in an innovation-driven economy The previous section identifies the influence that city size distribution exerts on economic growth. However, economic development may also affect the evolution of city size distribution. As mentioned in the Introduction, Duranton (2007) provides a plausible argument. We test the theory of Duranton (2007) using Chinese data. In Duranton (2007), city size is determined by the number of industries located in that city. An industry means a line of same goods with different qualities. In equilibrium only the good with highest quality is produced and each industry is located in only one city. Research is undertaken to improve the quality of current goods, but it would also accomplish unexpected cross-industry innovations with some probability. When research in city A leads to an innovation of some other industry located in city B, the production of that industry will be transferred to city A. Therefore the size of city A increases by one unit while the size of city B decreases by the same unit. A steady state requires that the expected number of cities of each size remains constant. Here, output growth has no direct effect on city size distribution, but it serves as a legitimate proxy of the likelihood of technological innovation. The latter is the driving force for the city size distribution to converge to its steady state. Consider an increase in the probability that a cross-industry technological innovation takes place. It will increase output growth rate for sure, and will also speed up the convergence of city size distribution to the steady state Strictly speaking, Duranton s model does not produce a steady-state city size distribution with 16

17 If we initially have a Zipf s coefficient larger than one, then it should drop more drastically when output growth rate is higher. If we instead have an initial Zipf s coefficient smaller than one, then it should have a larger increase along with higher output growth rate. We start from the following first-order autoregressive structure: α it = b 0 + b 1 α i,t 1. (8) The speed of convergence b 1 depends on the probability of cross-industry technological innovation, for which we use output growth rate as a proxy. The larger b 1 is, the slower it converges to its steady state. We assume a linear structure upon this relationship: b 1 = c 0 + c 1 g i,t, where c 1 < 0. Moreover, Zipf s law predicts the limit of α it to be one, that is Rewriting (8), we get lim α it = b 0 = 1. t 1 b 1 α it = 1 b 1 + b 1 α i,t 1 = 1 c 0 + c 0 α i,t 1 c 1 (1 α i,t 1 ) g i,t. Therefore, an econometric specification we propose is: α it = β 1 α i,t 1 + β 2 (1 α i,t 1 ) g i,t + X itγ + D tδ + η i + u it, where all the terms are defined in the same way as in equation (5) and (7). We are supposed to observe β 2 > 0 in light of Duranton s model. However, the fixed-effects estimates in columns (1) to (3) of Table 3 reveal that the theory is at odds with our data. The coefficient β 2 is found to be insignificantly negative in all the specifications. 18 Of course, this result may be caused by the bias of the fixed effects estimator. As in the previous subsection, we employ GMM estimators to overcome this problem. However, Arellano and Bover (1995) and Blundell and Bond (1998) argue that when the individual series are highly persistent, instruments used in the standard first-differenced estimator are likely to be weak and may underlying Zipf s coefficient equal to one, rather, it produces a complete distribution profile of city sizes, which could be well approximated by Zipf s law. 18 It is possible that GDP growth affects the convergence of city size distribution gradually, but replacing GDP growth rate in the interaction term by its lagged value does not change the result. 17

18 induce serious bias in finite samples. This argument is relevant in the current context as we notice that the estimated β 1 is around 0.8. To cure this problem, in addition to the instruments described in (6), they propose a new set of moment conditions in levels where lagged difference of endogenous (or predetermined) explanatory variables serve as corresponding instruments: E [(α i,t 1 α i,t 2 ) (η i + u it )] = 0 for t = 3,... T ; i, E [((1 α i,t 2 ) g i,t 1 (1 α i,t 3 ) g i,t 2 ) (η i + u it )] = 0 for t = 4,... T ; i. Following the existing empirical studies, we assume all control variables to be strictly exogenous, so they do not serve as GMM-type instruments. 19 The results of this system-gmm estimation are shown in columns (6) and (7) of Table 3, where the maximum numbers of lagged variables used as instruments for differenced equations are again fixed at two and one, respectively. We find that β 2 is still negative. Of course it would be hasty to say that the direction of causality goes solely from city size distribution to economic growth, not the other way round, based on the above result. What may be concluded is that even if output growth does influence city size distribution, the effect is not likely to take place through the channel described in Duranton (2007). This result casts some doubt on the applicability of Duranton s story to China and is a potential topic for future research. 7 A vector autoregressive framework In this section, we illustrate how well our model fits the data and then examine how city size distribution and economic growth evolve in face of exogenous shocks. We first set up a VAR structure in the form of simultaneous equations system: { git = β 11 g i,t 1 + β 12 (1 α i,t 1 ) 2 + X itγ 1 + D tδ 1 + η g,i + u g,it α it = β 21 α i,t 1 + β 22 g i,t 1 + X itγ 2 + D tδ 2 + η α,i + u α,it (9) The first equation is identical to equation (7), which we have discussed in Section 5. As Section 6 shows, the theory we borrow from Duranton (2007) predicting the effect of GDP growth on city size distribution is not supported by the data, so we simply assume a reduced-form linear relationship of g on α, as is presented in the second equation of the above VAR system. 19 If X it is indeed endogenous with city size distribution, OLS and fixed-effects panel estimation will no longer be valid since they both lead to inconsistent estimates. However, in our GMM case results are not affected qualitatively if X it are treated as endogenous and thus not reported. 18

19 Fixed-effects estimation results of the second equation in equations (9) are shown in columns (4) and (5) of Table 3. The corresponding results of System-GMM estimation are reported in columns (8) and (9) of the same table. Note that in a VAR system such as equations (9), control variables should be treated in the same way across equations. Because the control variables are clearly endogenous in the first equation, we assume they are also endogenous in the second one. 20 The estimated effect of GDP growth rate is insignificant in all specifications. The size of government is only significant when assuming no time-specific effect, indicating that the rent-seeking effect of having large government more than offsets the distribution effect. The growth of industrial sectors is significant except when we use a smaller set of instruments in the system-gmm estimation. However, the sign of the effect is opposite to what we have argued in Section 3.2. These results show potential limitation of the existing cross-sectional studies and panel studies using pooled OLS estimates. The results obtained in those studies explaining the variation of city size distribution may simply be due to their failure of taking account of time-specific factors and explicitly modeling the dynamic nature of city size distribution. Next we use our estimated model to interpolate GDP growth and city size distribution during the sample period Since significance of estimates is not a concern here, the parameter values of the above system are taken from column (9) of Table 2 and column (8) of Table 3. The fitness of the VAR system can be illustrated by comparing the real data and the fitted values of g and α. In Figure 5 we plot two examples. Province Guangdong has the largest variation of Zipf s coefficient during the sample period, and Shanxi has the largest variation of real GDP per capita growth rate. For the fitted lines, g 1984 and α 1984 from real data are set as initial values, then g and α for subsequent years are computed in succession according to equations (9). Because of the endogeneity of control variables, we cannot simply insert the real data year by year. Instead, we substitute the sample mean into the equations and keep them constant over the years. We can see from Figure 5 that the model fits the path of GDP growth fairly well, and generates less variation than the real data of Zipf s coefficient, but still matches the trend. The model can also be used for simulation exercises to predict the pattern of co-evolution of economic growth and city size distribution when an exogenous shock is introduced. The long run values of real GDP per capita growth rate and Zipf s coefficient predicted by this model are g = 9.61% and α = The finding α > 1 implies that migration restriction will inhibit the growth of mega-cities in the long run. This finding is in line with Henderson s (2005) observation for developing 20 Again, in column (8), the first two lags of all possible instruments are used for differenced equations, and in column (9) instruments are restricted to include only the first lag. 19

20 countries. Simulation results are shown in Figure 6. Suppose initially the provincial GDP growth and city size distribution are at their respective steady states. The magnitude of shocks are assumed to be one standard deviation of Zipf s coefficient and real GDP per capita growth rate in the whole sample. If the economy is hit by a positive growth shock, the Zipf s coefficient will jump up in the first three years and then decrease to the steady state level. During the transition period, we will observe more equal city size distribution. An interesting observation is that if the economy experiences some sort of negative distribution shock (driving down Zipf s coefficient) that leads to a more unequal city size distribution (we call it polarization shock ), the economy will grow faster in the following years than if there were no such shock. On the contrary, a positive distribution shock (driving up Zipf s coefficient) which has an impact on the equalization of city sizes, will result in a slowdown of GDP growth, and the decrease is disproportionately high as compared to the increase associated with a negative shock of the same magnitude. Imagine that the source of such distribution shock could be government policies. Our simplistic model would generate strong and unambiguous policy implications. One example is the policy that promotes increased enrollment of higher education, which has been intensely debated during recent years in China. Since most universities in China are located in large cities, this policy induces many students to flow into large cities. Most of them will get employed in the cities after graduation. They will be entitled citizenship (hukou) and become permanent residents of the cities. Now that more people are concentrated in large cities than before, the city size distribution will be more unequal, which, according to our model, is conducive to future economic growth. An opposite example is available in the history of Cultural Revolution, during which China has witnessed a swelling tide of educated urban youth going and working in the countryside and mountain areas. This migration choice was, to a large extent, also forced by government policy or Chairman Mao s personal will. The analysis is a bit more complicated than in the previous example, since the outflow of young population happened not only in large cities, but also in smaller cities. Moreover, this forced migration policy aimed not only at college students, but high school students as well. If large cities and small cities have roughly the same proportion of educated youth to total population, reducing the sizes of all cities proportionally will not change the size distribution. However, part of the educated youth in small cities who would have moved to large cities to pursue higher education were actually driven to countryside. Therefore large cities lost more population than smaller cities, and as a consequence, city size distribution became more equal. Again, our model implies 20

21 that the policy during the Cultural Revolution was harmful to economic growth. 8 Conclusion In urban economics literature, a vast quantity of research has emerged in answering how and why city size distributions differ across regions and evolve over time, but few study has been conducted to investigate the effect of city size distribution on economic growth. This paper is an attempt in this direction. Rather than directly modeling the mechanism through which city size distribution would possibly influence the level or pace of economic growth, this paper takes a different strategy. Our results are based on two crucial assumptions. First, in acknowledging that Zipf s law is a widely accepted empirical regularity, especially in developed countries where there is no contrived restriction distorting free internal labor migration, we assume that the optimal city size distribution in the absence of any exogenous intervention is such that Zipf s coefficient of the distribution equals to one. The optimal city size distribution can be viewed as a necessary demographic condition to ensure an optimal growth path. Second, internal migration restriction across cities in China introduces efficiency losses in aggregate production, as well as distorts the convergence of city size distribution. We then argue that there is an inverted-u-shape relationship between Zipf s coefficient and economic growth rate, peaked at the point where Zipf s coefficient equals to one. The argument is not rejected in the context of China. A VAR model characterizing the co-evolution of city size distribution and economic growth is further constructed and estimated. The simulation results based on the model reveal that, given migration restriction as a binding institutional constraint in the long run, any exogenous distribution shock that brings the cities toward a more uneven distribution will be accompanied by faster economic growth. This result has a clear policy implication. References [1] Alperovich, G. (1992). Economic Development and Population Concentration. Economic Development and Cultural Change 41 (1): [2] Alperovich, G. (1993). An Explanatory Model of City-Size Distribution: Evidence From Cross-Country Data. Urban Studies 30 (9): [3] Anderson, G. and Y. Ge (2005). The Size Distribution of Chinese Cities. Regional Science and Urban Economics 35 (6):

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23 [17] Gabaix, X. (2008). Power Laws, entry in The New Palgrave Dictionary of Economics, 2nd Edition, Steven N. Durlauf and Lawrence E. Blume (eds.), MacMillan. [18] Gabaix, X. and R. Ibragimov (2006). Rank -1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents. Working paper. [19] Gabaix, X. and Y. M. Ioannides (2004). The Evolution of City Size Distributions. Handbook of Regional and Urban Economics Volume 4: [20] Gallup, J. L., Sachs, J. D. and A. D. Mellinger (1999). Geography and Economic Development. International Regional Science Review 22 (2): [21] Henderson, J. V. (2003). The Urbanization Process and Economic Growth: The So-What Question. Journal of Economic Growth 8 (1): [22] Henderson, J. V. (2005). Urbanization and Growth. Handbook of Economic Growth Volume 1: [23] Henderson, J. V. and H. G.Wang (2007). Urbanization and City Growth: The Role of Institutions. Regional Science and Urban Economics 37 (3): [24] Holtz-Eakin, D., W. Newey and H. S. Rosen (1988). Estimating Vector Autoregressions with Panel Data. Econometrica 56 (6): [25] Ioannides, Y. M. and H. G. Overman (2003). Zipf s Law for Cities: An Empirical Examination. Regional Science and Urban Economics 33 (1): [26] Judson, R. A. and A. L. Owen (1999). Estimating Dynamic Panel Data Models: A Guide for Macroeconomists. Economics Letters 65 (1): [27] Moomaw, R. L. and M. A. Alwosabi (2004). An Empirical Analysis of Competing Explanations of Urban Primacy Evidence From Asia and the Americas. Annals of Regional Science 38 (1): [28] Rosen, K. T. and M. Resnick (1980). The Size Distribution of Cities: An Examination of the Pareto Law and Primacy. Journal of Urban Economics 8 (2): [29] Rossi-Hansberg, E. S. and M. A. Wright (2007). Urban Structure and Growth. Review of Economic Studies 74 (2):

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25 Figure 1: The composition of cities in China in 1984 and 2005 by size. Note: City size is measured by non-agricultural population in town. 25

26 Figure 2: Zipf s law in China in 1984 and

27 Figure 3: Comparison of different measures of city size distribution. Notes: (a) Spatial Gini index is computed by first drawing a Lorenz curve of the distribution of city sizes and then taking the size ratio of the area enclosed by the Lorenz curve and the line of perfect equality to the lower-right triangle. (b) Spatial HHI (spatial Hirschman-Herfindahl index) = n i=1 (s i/ S 100) 2, where s i is the population of city i, S is the total population, n is the number of cities. (c) Urban primacy is computed as the ratio of the population of the largest city to the total population. (d) Zipf s coefficient is equal to the absolute value of the slope of the fitted line as in Figure 2. 27

28 Figure 4: Evolution of Zipf s coefficient from 1984 to 2005 by province. 28

29 Figure 5: Fitness of the model: two examples. 29

30 Figure 6: Simulation of city size distribution and economic growth. Notes: The figure above illustrates the evolution of Zipf s coefficient, deviating from the steady state, after a positive growth shock and a negative growth shock, respectively. The figure below illustrates the evolution of real GDP per capita growth rate, deviating from the steady state, after a positive (equalization) shock and a negative (polarization) shock on city size distribution, respectively. 30