On the best functions to describe city size distributions

González-Val, Rafael and Ramos, Arturo and Sanz-Gracia, Fernando (2010): On the best functions to describe city size distributions.

Abstract

This paper analyses in detail the features offered by a function which is practically new to Urban Economics, the q-exponential, in describing city size distributions. We highlight two contributions. First, we propose a new and simple procedure for estimating their parameters. Second, and more importantly, we explain the characteristics associated with two traditional graphic methods (Zipf plots and cumulative density functions) for discriminating between functions. We apply them to the lognormal and q-exponential, justifying them as the best functions for explaining the entire distribution, and that the relationship between them is of complementarity.

• Item Type: MPRA Paper
• Language: English
• Keywords: city size distribution; q-exponential; lognormal
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R0 - General > R00 - General

References

Anderson, G. and Y. Ge (2005). “The size distribution of Chinese cities”, Regional Science and Urban Economics 35, 756-776.

Bates, D. M. and D. G. Watts (1988). “Nonlinear regression analysis and its applications”, New York: Wiley.

Black, D. and J. V. Henderson (2003). “Urban evolution in the USA”, Journal of Economic Geography 3, 343-372.

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

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, 1339-1373.

Cheshire, P. C. and S. Magrini (2006). “Population growth in European cities: weather matters-but only nationally”, Regional Studies 40(1), 23-37.

Choulakian, V. and M. A. Stephens (2001). “Goodness-of-fit tests for the generalized Pareto distribution”, Technometrics 43, 478-484.

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.

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?” Mimeo.

González-Val, R. (2010). “The evolution of US city size distribution from a long term perspective (1900-2000)”, Journal of Regional Science, forthcoming.

González-Val, R., L. Lanaspa and F. Sanz (2010). “Gibrat’s Law for cities revisited”, Mimeo, Universidad de Zaragoza.

Grimshaw, S. D. (1993). “Computing maximum likelihood estimates for the generalized Pareto distribution”, Technometrics 35, 185-191.

Hosking, J. R. M. and J. R. Wallis (1987). “Parameter and quantile estimation for the generalized Pareto distribution”, Technometrics 29, 339-349.

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”, Economics Department Working Paper, Tufts University.

Kalecki, M. (1945). “On the Gibrat distribution”, Econometrica 13(2), 161-170.

Krugman, P. R. (1996a). “Confronting the mystery of urban hierarchy”, Journal of the Japanese and the International Economies 10, 399-418.

Krugman, P. R. (1996b). “The self-organizing economy”, Oxford: Blackwell.

Levy, M. (2009). “Gibrat’s Law for (all) cities: a comment”, American Economic Review 99(4), 1672-1675.

Malacarne, L. C., R. S. Mendes and E. K. Lenzi (2001). “q-exponential distribution in urban agglomeration”, Physical Review E 65, (017106) 1-3.

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

Rosen, K. and M. Resnick (1980). “The size distribution of cities: an examination of the Pareto law and primacy”, Journal of Urban Economics 8, 165-186.

Sharma, S. (2003). “Persistence and stability in city growth”, Journal of Urban Economics 53, 300-320.

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 44(1), 1-14.

Tsallis, C. (1988). “Possible generalization of Boltzmann-Gibbs statistics”, Journal of Statistical Physics 52, 479-487.

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