Deviations from Zipf’s Law for American cities: an empirical examination

Rafael, González-Val (2010): Deviations from Zipf’s Law for American cities: an empirical examination.

Abstract

This paper presents a simple method for calculating deviations between actual city size and the size which would correspond to it with a Pareto exponent equal to one (Zipf’s Law). The results show two differentiated behaviours: most cities (80.25%) present a greater size than that which would fulfil Zipf’s Law, while small cities (19.75%) tend to be too small. Our aim is to analyse the distribution element by element, using data about city characteristics from all American cities in 2000, and to explain the deviation between the size predicted by Zipf’s Law and the actual size of each city. To do this a Multinomial Logit Model is used. The most important variables affecting the probability of a city presenting a negative or positive deviation are per capita income, human capital levels and the percentage of the population employed in some sectors.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cities; Zipf’s Law; deviations; Pareto distribution; Multinomial logit
• Subjects: R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R0 - General > R00 - General; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R1 - General Regional Economics > R12 - Size and Spatial Distributions of Regional Economic Activity

References

Ades, A. F., and E. L. Glaeser, (1995). Trade and circuses: explaining urban giants. The Quarterly Journal of Economics, 110: 195–227.

Batty, M., and L. March, (1976). The method of residues in urban modelling. Environment and Planning A, 8: 189–214.

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

Benguigui, L., and E. Blumenfeld-Lieberthal, (2007). A dynamic model for city size distribution beyond Zipf’s law. Physica A: Statistical Mechanics and its Applications, 384(2): 613–627.

Black, D., and V. Henderson, (1998). Urban evolution in the USA. Brown University Working Paper No. 98-21.

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

Cheshire, P., (1999). Trends in sizes and structure of urban areas. In: Handbook of Regional and Urban Economics, Vol. 3, P. Cheshire and E. S. Mills, eds. Amsterdam: Elsevier Science, Chapter 35, pp.1339–1373.

Coleman, J. S., (1964). Introduction to Mathematical Sociology. Free Press of Glencoe, New York.

Córdoba, J. C., (2008). A Generalized Gibrat’s Law. International Economic Review, 49(4): 1463–1468.

Duranton, G., (2007). Urban Evolutions: The Fast, the Slow, and the Still. American Economic Review, 97(1): 197–221.

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(4): 1676–1683.

Gabaix, X., (1999). Zipf’s Law for cities: An explanation. Quarterly Journal of Economics, 114(3): 739–767.

Gabaix, X., and Y. M. Ioannides, (2004). The evolution of city size distributions. In: Handbook of urban and regional economics, Vol. 4, J. V. Henderson and J. F. Thisse, eds. Amsterdam: Elsevier Science, North-Holland, pp.2341–2378.

Garmestani, A. S., C. R. Allen, and C. M. Gallagher, (2008). Power laws, discontinuities and regional city size distributions. Journal of Economic Behavior & Organization, 68: 209–216.

Garmestani, A. S., C. R. Allen, C. M. Gallagher, and J. D. Mittelstaedt, (2007). Departures from Gibrat's Law, Discontinuities and City Size Distributions. Urban Studies, 44(10): 1997–2007.

Glaeser, E. L., J. Gyourko, and R. E. Saks, (2006). Urban growth and housing supply. Journal of Economic Geography, 6: 71–89.

Glaeser, E. L., and M. E. Kahn, (2004). Sprawl and Urban Growth. In: Handbook of urban and regional economics, Vol. 4, J. V. Henderson and J. F. Thisse, eds. Amsterdam: Elsevier Science, North-Holland, Chapter 56, pp.2482–2527.

Glaeser, E. L., and A. Saiz, (2003). The Rise of the Skilled City. Harvard Institute of Economic Research, Discussion Paper number 2025.

Glaeser, E. L., J. A. Scheinkman, and A. Shleifer, (1995). Economic growth in a cross-section of cities. Journal of Monetary Economics, 36: 117–143.

Glaeser, E. L., and J. Shapiro, (2003). Urban Growth in the 1990s: Is city living back? Journal of Regional Science, 43(1): 139–165.

Gibrat, R., (1931). Les inégalités économiques. Paris: Librairie du Recueil Sirey.

Giesen, K., A. Zimmermann, and J. Suedekum, (2009). The size distribution across all cities - Zipf, lognormal, double Pareto, or what? Paper presented at: 56th Annual North American Meetings of the Regional Science Association International, San Francisco (USA).

Goldstein, M. L., S. A. Morris and G. G. Yen, (2004). Problems with fitting to the Power-law distribution. The European Physical Journal B - Condensed Matter, 41(2): 255–258.

González-Val, R., (2010). The Evolution of US City Size Distribution from a Long Term Perspective (1900–2000). Journal of Regional Science, forthcoming.

Guérin-Pace, F., (1995). Rank-Size Distribution and the Process of Urban Growth. Urban Studies, 32(3): 551–562.

Helpman, E., (1998). The size of regions. In: Topics in Public Economics, D. Pines, E. Sadka, and I. Zilcha, eds. Cambridge University Press.

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

Ioannides, Y. M., and S. Skouras, (2009). Gibrat's Law for (All) Cities: A Rejoinder. Discussion Papers Series, Department of Economics, Tufts University 0740, Department of Economics, Tufts University.

Krugman, P., (1991). Increasing returns and economic geography. Journal of Political Economy, 99: 483–499.

Krugman, P., (1996). The Self-organizing economy. Cambridge: Blackwell.

Long, J. S., and J. Freese, (2006). Regression Models for Categorical Dependent Variables Using Stata. Stata Press Publication, 2nd ed.

Malacarne, L. C., R. S. Mendes and E. K. Lenzi, (2001). q-exponential distribution in urban agglomeration. Physical Review E, 65, 017106.

Malevergne, Y., V. Pisarenko, and D. Sornette, (2009). Gibrat's law for cities: uniformly most powerful unbiased test of the Pareto against the lognormal. arXiv:0909.1281v1 [physics.data-an].

Michaels, G., F. Rauch, and S. Redding, (2009). Rethinking urbanisation. VoxEU Jan 3, 2009.

Newman, M. E. J., (2006). Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46: 323–351.

Nishiyama, Y., S. Osada and Y. Sato, (2008). OLS estimation and the t test revisited in rank-size rule regression. Journal of Regional Science, 48(4): 691–715.

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

Petrakos, G., P. Mardakis, and H. Caraveli, (2000). Recent Developments in the Greek System of Urban Centres. Environment and Planning B: Planning and Design, 27(2): 169–181.

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: 165–186.

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

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

Semboloni, F., (2008). Hierarchy, cities size distribution and Zipf’s law. The European Physical Journal B, 63: 295–301.

Simon, C. J., and C. Nardinelli, (2002). Human capital and the rise of American cities, 1900–1990. Regional Science and Urban Economics, 32: 59–96.

Small, K.A., and C. Hsiao, (1985). Multinomial logit specification tests. International Economic Review, 26: 619–627.

Song, S., and K. H. Zhang, (2002). Urbanisation and City Size Distribution in China. Urban Studies, 39(12): 2317–2327.

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

Soo, K. T., (2007). Zipf's Law and Urban Growth in Malaysia. Urban Studies, Vol. 44(1): 1–14.

Tabuchi, T., (1998). Urban Agglomeration and Dispersion: A Synthesis of Alonso and Krugman. Journal of Urban Economics, 44: 333–351.

Wheaton, W. C., and H. Shishado, (1981). Urban concentration, agglomeration economies and the level of economic development. Economic Development and Cultural Change, 30:17–30.

Zipf, G., (1949). Human Behaviour and the Principle of Least Effort. Cambridge, MA: Addison-Wesley.