US city size distribution revisited: Theory and empirical evidence

Ramos, Arturo and Sanz-Gracia, Fernando (2015): US city size distribution revisited: Theory and empirical evidence.

Abstract

We develop a urban economic model in which agents locate in cities of different size so as to maximize the net output of the whole system of cities in a country. From this model two new city size distributions are exactly derived. We call these functions “threshold double Pareto Generalized Beta of the second kind” and “double mixture Pareto Generalized Beta of the second kind”. In order to test empirically the theory, we analyze the US urban system and consider three types of data (incorporated places from 1900 to 2000, all places in 2000 and 2010 and City Cluster Algorithm nuclei in 1991 and 2000). The results are encouraging because the new distributions clearly outperform the lognormal and the double Pareto lognormal for all data samples. We consider a number of different tests and statistical criteria and the results are robust to all of them. Thus, the new distributions describe accurately the US city size distribution and, therefore, support the validity of the theoretical model.

• Item Type: MPRA Paper
• Language: English
• Keywords: human capital; congestion costs; lower tail, body, and upper tail; Pareto and Generalized Beta of the second kind distributions
• Subjects: C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics; D - Microeconomics > D3 - Distribution > D39 - Other; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R11 - Regional Economic Activity: Growth, Development, Environmental Issues, and Changes; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R12 - Size and Spatial Distributions of Regional Economic Activity

References

Abdel-Rahman, H. M. and Anas, A. (2004). Theories of systems of cities. In Henderson, V. and Thisse, J. F., editors, Handbook of Regional and Urban Economics, volume 4, chapter 52, pages 2293–2339. Elsevier.

Anderson, G. and Ge, Y. (2005). The size distribution of Chinese cities. Regional Science and Urban Economics, 35(6):756–776.

Arcaute, E., Hatna, E., Ferguson, P., Youn, H., Johansson, A., and Batty, M. (2014). Constructing cities, deconstructing scaling laws. Journal of the Royal Society Interface, 12.

Batty, M. (2006). Rank clocks. Nature, 444(7119):592–596.

Batty, M. (2008). The size, scale, and shape of cities. Science, 319:769–771.

Batty, M. (2013). The new science of cities. The MIT Press.

Bee, M., Riccaboni, M., and Schiavo, S. (2013). The size distribution of US cities: not Pareto, even in the tail. Economics Letters, 120:232–237.

Beeson, P. E., DeJong, D. N., and Troesken, W. (2001). Population growth in US counties, 1840–1990. Regional Science and Urban Economics, 31(6):669–699.

Behrens, K., Mion, G., Murata, Y., and S¨udekum, J. (2014). Spatial frictions. DICE Discussion Paper No. 160.

Berliant, M. and Watanabe, H. (2015). Explaining the size distribution of cities: Extreme economies. Quantitative Economics, 6:153–187.

Berry, B. J. L. and Okulicz-Kozaryn, A. (2012). The city size distribution debate: resolution for US urban regions and megalopolitan areas. Cities, 29:517–523.

Bettencourt, L. and West, G. (2010). A unified theory of urban living. Nature, 4667:912–913.

Black, D. and Henderson, V. (1999). Spatial evolution of population and industry in the United States. American Economic Review, 89(2):321–327.

Black, D. and Henderson, V. (2003). Urban evolution in the USA. Journal of Economic Geography, 3(4):343–372.

Blank, A. and Solomon, S. (2000). Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components). Physica A, 289:279–288.

Bosker,M., Brakman, S., Garretsen, H., and Schramm,M. (2008). A century of shocks: the evolution of the German city size distribution 1925-1999. Regional Science and Urban Economics, 38(4):330–347.

Brzezinski, M. (2015). Robust estimation of the Pareto tail index: a Monte Carlo analysis. Empirical Economics. DOI 10.1007/s00181-015-0989-9.

Budd, E. C. (1970). Distribution issues: Trends and policies. Postwar changes in the size distribution of income in the U. S. . American Economic Review, 60(2):247– 260.

Burnham,K. P. and Anderson,D. R. (2002).Model selection andmultimodel inference: A practical information-theoretic approach. New York: Springer-Verlag.

Burnham, K. P. and Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods and Research, 33:261–304.

Chiang, A. C. (1992). Elements of dynamic optimization. McGraw-Hill.

Cirillo, P. (2013). Are your data really Pareto distributed? Physica A, 392:5947–5962.

Combes, P. P., Duranton, G., and Gobillon, L. (2008). Spatial wage disparities: sorting matters! Journal of Urban Economics, 63(2):723–742.

Combes, P. P., Duranton, G., Gobillon, L., Puga, D., and Roux, S. (2012). The productivity advantages of large cities: Distinguishing agglomeration from firm selection. Econometrica, 80:2543–2594.

Combes, P. P. and Gobillon, L. (2015). The empirics of agglomeration economies. In Duranton, G., Henderson, V., and Strange,W. C., editors, Handbook of Regional and Urban Economics, forthcoming, volume 5. Elsevier.

Córdoba, J. C. (2008). On the distribution of city sizes. Journal of Urban Economics, 63(1):177–197.

Dagum, C. (1979). Generating systems and properties of income distribution models. In Conference of Canadian Incomes, May 10-12.

Desmet, K. and Rappaport, J. (2015). The settlement of the United States, 1800-2000: The long transition towards Gibrat’s Law. Journal of Urban Economics. DOI 10.1016/j.jue.2015.03.004.

Duranton, G. (2006). Some foundations for Zipf’s law: product proliferation and local spillovers. Regional Science and Urban Economics, 36:542–563.

Duranton, G. (2007). Urban evolutions: The fast, the slow, and the still. American Economic Review, 97(1):197–221.

Duranton, G. and Puga, D. (2004). Micro-foundations of urban agglomeration economies. In Henderson, V. and Thisse, J. F., editors, Handbook of Regional and Urban Economics, volume 4, chapter 48, pages 2063–2117. Elsevier.

Eeckhout, J. (2004). Gibrat’s law for (all) cities. American Economic Review, 94(5):1429–1451.

Eeckhout, J. (2009). Gibrat’s law for (all) cities: Reply. American Economic Review, 99:1676–1683.

Efron, B. and Hinkley,D. V. (1978). Assessing the accuracy of themaximum likelihood estimator: Observed versus expected Fisher information. Biometrika, 65(3):457– 482.

Favaro, J. M. and Pumain, D. (2011). Gibrat revisited: an urban growth model incorporating spatial interactions and innovation cycles. Geographical Analysis, 43:261–286.

Fazio, G. and Modica, M. (2015). Pareto or log-normal? Best fit and truncation in the distribution of all cities. Journal of Regional Science. DOI 10.1111/jors.12205.

Gabaix, X. (1999). Zipf’s law for cities: An explanation. Quarterly Journal of Economics, 114:739–767.

Gabaix, X. and Ioannides, Y. (2004). The evolution of city size distributions. In Henderson, V. and Thisse, J. F., editors, Handbook of Regional and Urban Economics, volume 4, chapter 53, pages 2341–2378. Elsevier.

Giesen, K. and Suedekum, J. (2012). The French overall city size distribution. Región et Développement, 36:107–126.

Giesen, K. and Suedekum, J. (2014). City age and city size. European Economic Review, 71:193–208.

Giesen, K., Zimmermann, A., and Suedekum, J. (2010). The size distribution across all cities-double Pareto lognormal strikes. Journal of Urban Economics, 68(2):129–137.

Glaeser, E. L. (1998). Are cities dying? Journal of Economic Perspectives, 12(2):139–160.

Glaeser, E. L. (2011). Triumph of the city. The Penguin Press.

Glaeser, E. L., Gottlieb, J. D., and Tobio, K. (2012). Housing booms and city centers. NBER Working Paper No. 17914.

Glaeser, E. L. and Kahn, M. E. (2010). The greenness of cities: carbon dioxide emissions and urban development. Journal of Urban Economics, 67(3):404–418.

Glaeser, E. L. and Resseger, M. G. (2010). The complementarity between cities and skills. Journal of Regional Science, 50(1):221–244.

González-Val, R. (2010). The evolution of US city size distribution from a long term perspective (1900–2000). Journal of Regional Science, 50:952–972.

González-Val, R., Ramos, A., Sanz-Gracia, F., and Vera-Cabello, M. (2015). Size distribution for all cities: Which one is best? Papers in Regional Science, (1):177–197.

Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5):1163–1174.

Holmes, T. J. and Lee, S. (2010). Agglomeration Economics, chapter Cities as six-bysix-mile squares: Zipf’s Law? The University of Chicago Press.

Hsu, W.-T. (2012). Central place theory and city size distribution. The Economic Journal, 122:903–932.

Hsu, W.-T., Mori, T., and Smith, T. (2014). Spatial patterns and size distribution of cities. Kier Discussion Paper No. 882.

Ioannides, Y. M. and Overman, H. G. (2003). Zipf’s law for cities: An empirical examination. Regional Science and Urban Economics, 33(2):127–137.

Ioannides, Y.M. and Overman, H. G. (2004). Spatial evolution of the US urban system. Journal of Economic Geography, 4(2):131–156.

Ioannides, Y. M. and Skouras, S. (2013). US city size distribution: Robustly Pareto, but only in the tail. Journal of Urban Economics, 73:18–29.

Kamien, M. I. and Schwartz, N. L. (1991). Dynamic optimization. The calculus of variations and optimal control in Economics and Management. Elsevier.

Kim, S. (2000). Urban development in the United States, 1690-1990. Southern Economic Journal, 66(4):855–880.

Kleiber, C. and Kotz, S. (2003). Statistical size distributions in Economics and actuarial sciences. Wiley-Interscience.

Krugman, P. R. (1996). The self-organizing economy. Blackwell Publishers. Cambridge,MA.

Lee, S. and Li, Q. (2013). Uneven landscapes and city size distributions. Journal of Urban Economics, 78:19–29.

Levy, M. (2009). Gibrat’s law for (all) cities: Comment. American Economic Review, 99:1672–1675.

Lucas Jr., R. E. (1988). On the mechanics of economic development. Journal of Monetary Economics, 22:3–42.

Luckstead, J. and Devadoss, S. (2014a). A comparison of city size distributions for China and India from 1950 to 2010. Economics Letters, 124:290–295.

Luckstead, J. and Devadoss, S. (2014b). Do the world’s largest cities follow Zipf’s and Gibrat’s laws? Economics Letters, 125:182–186.

Luckstead, J. and Devadoss, S. (2015). Pareto tails and lognormal body of US cities size distribution. Working Paper, unpublished.

Luckstead, J., Devadoss, S., and Danforth, D. (2015). Size distributions of all indian cities. Working Paper, unpublished.

Macaulay, F. R. (1922). Income in the United States, its amount and distribution, 1909-1919, Volume II: Detailed Report, chapter The personal distribution of income in the United States. National Bureau of Economic Research.

Mandelbrot, B. (1960). The Pareto-L´evy law and the distribution of income. International Economic Review, I:79–106.

McCullough, B. D. and Vinod, H. D. (2003). Verifying the solution from a nonlinear solver: A case study. American Economic Review, 93(3):873–892.

McDonald, J. B. (1984). Generalized functions for the size distribution of income. Econometrica, 52(3):647–665.

McDonald, J. B. and Xu, Y. J. (1995). A generalization of the beta distribution with applications. Journal of Econometrics, 66:133–152.

Melo, P., Graham, D., and Noland, R. (2009). A meta-analysis of estimates of urban agglomeration economies. Regional Science and Urban Economics, 39(3):332–342.

Moretti, E. (2004). Human capital externalities in cities. In Henderson, V. and Thisse, J. F., editors, Handbook of Regional and Urban Economics, volume 4, chapter 51, pages 2243–2291. Elsevier.

Overman, H. G. and Ioannides, Y. M. (2001). Cross-Sectional evolution of the US city size distribution. Journal of Urban Economics, 49(3):54–566.

Parker, S. C. (1999). The generalised beta as a model for the distribution of earnings. Economics Letters, 62:197–200.

Parr, J. B. (1985). A note on the size distribution of cities over time. Journal of Urban Economics, 18:199–212.

Parr, J. B. (2015). The city and the region as contrasts in spatial organization. The Annals of Regional Science. DOI 10.1007/s00168-015-0686-9.

Parr, J. B. and Suzuki, K. (1973). Settlement populations and the lognormal distribution. Urban Studies, 10(3):335–352.

Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E. F. (1962). The mathematical theory of optimal processes. Wiley Interscience Publishers.

Puente-Ajovín,M. and Ramos, A. (2015). On the parametric description of the French, German, Italian and Spanish city size distributions. The Annals of Regional Science, 54(2):489–509.

Puga, D. (2010). The magnitude and causes of agglomeration economies. Journal of Regional Science, 50(1):203–219.

Rauch, F. (2014). Cities as spatial clusters. Journal of Economic Geography, 14(4):759–773.

Rauch, J. E. (1993). Productivity gains from geographic concentration of human capital: evidence from the cities. Journal of Urban Economics, 34(3):380–400.

Razali, N. M. and Wah, Y. B. (2011). Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. Journal of Statistical Modeling and Analytics, 2:21–33.

Reed, W. J. (2002). On the rank-size distribution for human settlements. Journal of Regional Science, 42:1–17.

Reed, W. J. (2003). The Pareto law of incomes–an explanation and an extension. Physica A, 319:469–486.

Reed, W. J. and Jorgensen, M. (2004). The double Pareto-lognormal distribution–a new parametric model for size distributions. Communications in Statistics-Theory and Methods, 33(8):1733–1753.

Reggiani, A. and Nijkamp, P. (2015). Did Zipf anticipate spatial connectivity structures? Environment and Planning B: Planning and Design. DOI 10.1068/b120012p.

Rosenthal, S. S. and Strange, W. C. (2004). Evidence on the nature and sources of agglomeration economies. In Henderson, V. and Thisse, J. F., editors, Handbook of Regional and Urban Economics, volume 4, chapter 49, pages 2119–2171. Elsevier.

Rossi-Hansberg, E. and Wright, M. L. J. (2007). Urban structure and growth. Review of Economic Studies, 74(2):597–624.

Rozenfeld, H. D., Rybski, D., Andrade, J. S., Batty, M., Stanley, H. E., and Makse, H. A. (2008). Laws of population growth. Proceedings of the National Academy of Sciences, 105(48):18702–18707.

Rozenfeld, H. D., Rybski, D., Gabaix, X., and Makse, H. A. (2011). The area and population of cities: new insights from a different perspective on cities. American Economic Review, 101:2205–2225.

Rybski, D., García-Cantú-Ros, A., and Kropp, J. P. (2013). Distance-weighted city growth. Physical Review E, 87:042114.

Saiz, A. (2010). The geographic determinants of housing supply. The Quarterly Journal of Economics, 125(3):1253–1296.

Schluter, C. and Trede, M. (2013). Gibrat, Zipf, Fisher and Tippet: City size and growth distributions reconsidered. Working Paper 27/2013 Center for Quantitative Economics WWU M¨unster.

Schmidheiny, K. and Suedekum, J. (2015). The pan-European population distribution across consistently defined functional urban areas. Economics Letters, 133:10–13.

Soo, K. T. (2005). Zipf’s Law for cities: A cross-country investigation. Regional Science and Urban Economics, 35(3):239–263.

Soo, K. T. (2014). Zipf, Gibrat and geography: evidence from China, India and Brazil. Papers in Regional Science, 93(1):159–182.

Yankow, J. J. (2006). Why do cities pay more? an empirical examination of some competing theories of the urban wage premium. Journal of Urban Economics, 60(2):139–161.

Zanette, D. H. (2007). The Dynamics of Complex Urban Systems. An Interdisciplinary Approach, chapter Multiplicative processes and city sizes. Springer, Berlin.