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The nonlinear distribution of employment across municipalities

Abstract

In this paper, the nonlinear distribution of employment across Spanish municipalities is analyzed. Also, we explore new properties of the family of generalized power law (GPL) distributions and explore its hierarchical structure, then we test its adequacy for modeling employment data. A new subfamily of heavy-tailed GPL distributions that is right tail equivalent to a Pareto (power-law) model is derived. Our findings show on the one hand that the distribution of employment across Spanish municipalities follows a power-law behavior in the upper tail and, on the other hand, the adequacy of GPL models for modeling employment data in the whole range of the distribution.

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Acknowledgements

The authors are grateful for the constructive suggestions provided by the reviewers, which improved the paper. The authors thankfully acknowledges the computer resources, technical expertise and assistance provided by the Advanced Computing and e-Science Team at IFCA. Faustino Prieto also acknowledges the Faculty of Business and Economics and the Centre for Actuarial Studies at the University of Melbourne for their special support, since part of this paper was written while Faustino Prieto was visiting The University of Melbourne during the period July-September 2018.

Funding

Faustino Prieto acknowledges funding by the José Castillejo Program (Grant number CAS17/00461, Ministerio de Educación, Cultura y Deporte, Programa Estatal de Promoción de Talento y su Empleabilidad en I+D+i, Subprograma Estatal de Movilidad, del Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016). José María Sarabia thanks to Ministerio de Economía y Competitividad, project ECO2016-76203-C2-1-P, for partial support of this work. Research partially carried out while Calderin-Ojeda visited University of Cantabria as part of his Special Study Program leave (University of Melbourne).

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Appendix

Appendix

Proof of Theorem 1

Note that any function g, with \(\displaystyle \lim \nolimits _{z \rightarrow \infty }g(z;\varvec{\theta })=\alpha >0\), is slowly varying at infinity: \(\displaystyle \lim _{z \rightarrow \infty }g(tz;\varvec{\theta })/g(z;\varvec{\theta })=1, \forall t>0\); then, we can check that \(\displaystyle \lim _{z \rightarrow \infty }\frac{S_Z(tz;\varvec{\theta })}{S_Z(z;\varvec{\theta })}=\displaystyle \lim \nolimits _{z \rightarrow \infty }\frac{(1+tz)^{-g(tz;\varvec{\theta })}}{(1+z)^{-g(z;\varvec{\theta })}}=t^{-\alpha }, \forall t>0\), where \(S_Z(z;\varvec{\theta })\) is the survival function of the corresponding GPL distribution. \(\square \)

Proof of Remark 1

We can check that (a) \(\displaystyle \lim \nolimits _{z \rightarrow \infty }\frac{S_Z(z;\varvec{\theta })}{G_Z(z)}= \displaystyle \lim \nolimits _{z \rightarrow \infty }\frac{(1+z)^{-g(z;\varvec{\theta })}}{(1+z)^{-\alpha }}=1\), where \(G_Z(z)\) is (in this case) the survival function of the Pareto type II distribution;

(b) \(\displaystyle \lim _{z \rightarrow \infty }\frac{S_Z(z;\varvec{\theta })}{G_Z(z)}= \displaystyle \lim _{z \rightarrow \infty }\frac{(1+z)^{-g(z;\varvec{\theta })}}{e^{-\lambda z}}=\infty \), where \(G_Z(z)\) is (in this case) the survival function of the exponential distribution.

Log-likelihood function, GPL(II) model (Eq. 5) with left-censored data:

$$\begin{aligned} \log \ell (\alpha ,\beta ,\sigma )= & {} \log \left[ [1-S_X(x_0)]^r\prod _{\begin{array}{c} i=r+1 \end{array}}^{N} f(x_i)\right] \\= & {} (N-r)[\log (\alpha )-\log (\sigma )]-\sum _{\begin{array}{c} i=r+1 \end{array}}^{N}\log \left( 1+x_i/\sigma \right) \\+ & {} \sum _{\begin{array}{c} i=r+1 \end{array}}^{N}\log \left[ \beta +1+\log \left( 1+x_i/\sigma \right) \right] +\beta \sum _{\begin{array}{c} i=r+1 \end{array}}^{N}\log \left[ \log \left( 1+x_i/\sigma \right) \right] \\- & {} (\beta +1)\sum _{\begin{array}{c} i=r+1 \end{array}}^{N}\log \left[ 1+\log \left( 1+x_i/\sigma \right) \right] -\alpha \sum _{\begin{array}{c} i=r+1 \end{array}}^{N} \displaystyle \frac{[\log \left( 1+x_i/\sigma \right) ]^{\beta +1}}{[1+\log \left( 1+x_i/\sigma \right) ]^\beta }\\+ & {} r\log \left[ 1-\exp {\left[ -\alpha \displaystyle \frac{[\log \left( 1+x_0/\sigma \right) ]^{\beta +1}}{[1+\log (1+x_0/\sigma )]^\beta }\right] } \right] , \end{aligned}$$

where N is the sample size (included the left censored observations), r is the number of left censored observations, \(x_0=4\) is the censoring value, and the maximum likelihood estimates of the unknown parameter vector (\(\alpha \),\(\beta \),\(\sigma \)) is the one that maximizes the log-likelihood function \(\log \ell (\alpha ,\beta ,\sigma )\).

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Prieto, F., Sarabia, J.M. & Calderín-Ojeda, E. The nonlinear distribution of employment across municipalities. J Econ Interact Coord 16, 287–307 (2021). https://doi.org/10.1007/s11403-020-00294-2

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Keywords

  • Labor market
  • Municipalities
  • Generalized power-law models
  • Complex adaptive systems

JEL Classification

  • E24
  • F16
  • J21
  • C46
  • C55