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.

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 $\log \ell (\alpha ,\beta ,\sigma )$.