# Fréchet Distribution Applied to Salary Incomes in Spain from 1999 to 2014. An Engineering Approach to Changes in Salaries’ Distribution

^{1}

^{2}

^{a}Planta, 28010 Madrid, Spain

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The data from Table 1 and Table 2 was post-processed in order to obtain an estimation of the salaries earned in brackets of half minimum wage from 5 to 7.5 times the minimum wage and from 7.5 to 10 times the minimum wage. To do this, the evolution of the minimum wage in Spain from 1999 to 2014 is required (see Table 3).
- (2)
- The data (salaries paid) was made non-dimensional in order to directly compare the distributions from different years.
- (3)
- A corrected Fréchet distribution was fitted to the data (once post-processed), the evolution from 1999 to 2014 of this distribution being analyzed in order to detect changes in their main parameters and study the data through them.

## 2. Methodology

- the number of salaried people depends linearly on the salary, and
- the average salary corresponding to each new sub-bracket of half minimum wage is centered in relation to the aforementioned bracket.

_{1}, and the reduction in the number of earners from one sub-bracket to the following one, Δ. Therefore, if S is the population (salaried people) within the bracket to be split into five new sub-brackets, M is the total amount of salaries paid in the brackets to be split, Φ is the annual minimum wage, and φ

_{1}is the center of the first sub-bracket (φ

_{1}= 5.25 times the minimum wage in the case of the 5 to 7.5 times the minimum wage salaries bracket and φ

_{1}= 7.75 times the minimum wage in the case of the 7.5 to 10 times the minimum wage salaries bracket, see Figure 3), the following expressions can be derived for Δ and s

_{1}:

_{1}and φ

_{2}minimum salaries:

#### The Fréchet Distribution Fitted to the Data

_{n}is the percentage of the salaries paid at the income bracket corresponding to the salary φ

_{n}and f (φ

_{n}) is the figure from the selected PDF at φ

_{n}. The results show similar values of this error (RMSE = 4.423 × 10

^{−2}[Fréchet]; RMSE = 4.410 × 10

^{−2}[log-normal]; RMSE = 4.443 × 10

^{−2}[gamma]; RMSE = 4.415 × 10

^{−2}[Dagum]; and RMSE = 4.403 × 10

^{−2}[GB2]) for the five distributions.

_{1}and φ

_{2}minimum salaries:

## 3. Results and Discussion

_{30%}and φ

_{70%}:

_{30%}and φ

_{70%}, are calculated, the salaried people in the brackets mentioned in Equation (7) can be estimated:

_{0%–30%}and Δ

_{30%–70%}) are compared. The data shows a very good correlation indicating that each increase (decrease) in the number of earners within the 0%–30% salaries paid bracket is well correlated with a proportional decrease (increase) in the number of earners within the 30%–70% salaries paid bracket (see right graph of Figure 7). As it can be observed in Table 5, the boundary salary φ

_{30%}has a value of approximately φ

_{30%}= 2, which was identified as one of the points where a higher transfer of salaried people from one side to the other of this salary level was produced from 1999 to 2014. The estimated results (based on the Fréchet distribution fitting), however, do not reflect the aforementioned correlation. In addition, it should also be said that, according to the graph in Figure 7, the percentage of earners within the more than 70% salaries paid bracket remains constant throughout the studied period (1999–2014).

_{kw}, of the Fréchet distribution is calculated using the gamma function (De Gusmão and Ortega 2011; Khan et al. 2008):

_{kw}, was calculated in order to compare it with the suggested symmetry parameter, ψ. The results, made non dimensional with the value of the skewness in 1999, are also included in Figure 8. A similar trend between the skewness and parameter ψ can be observed. In addition, it seems that parameter ψ can filter some noise shown by the skewness in those points where the scale parameter, γ, is close to γ = 3, as the skewness of the Fréchet distribution presents a singularity at this point.

_{1}% of households, what percentage p

_{2}% of the total income they have’. In Figure 9, the Lorenz curves corresponding to 1999, 2005, and 2013 are shown. In this graph, the percentage of the salaried people, p, is plotted on the x-axis, whereas the percentage of the wages received by this number of salaried people, L(p), is plotted on the y-axis. The Lorenz curves were calculated, from 1999 to 2014, with the data from Table 1 and Table 2. As can be observed in Figure 9, all curves have the same pattern. Nevertheless, some additional information can be derived in order to establish an evolution pattern followed by the wages distribution. If the difference between the Lorenz curve and the theoretical values that represent the maximum equality (Krause 2014), p − L(p), is plotted as a function of the percentage of the salaried people, p, it is possible to analyze the position of the maximum of this curve (see Figure 10). If this maximum is displaced towards p = 1, the existence of an elite within the salaried group is revealed, whereas if it is displaced towards p = 0 it could be said that a group with extreme low wages exists. The evolution of this maximum position, p

_{max|p}

_{−}

_{L}

_{(p)|}, from 1999 to 2014 is shown in Figure 11. It can be appreciated that this maximum detaches its position from p = 1, which indicates a reduction in the importance of the higher wages (the elite) in relation to the whole group. This result agrees with the pattern shown by the symmetry parameter ψ in Figure 8; that is, the salary distribution in Spain has changed towards a more balanced situation since 2007.

_{kw}. It seems that the value of this coefficient remains quite constant, with a very slight increase starting in 2007, indicating an increase of inequality. This result contrasts with the previous results, based on the asymmetry of the wage distributions, and indicates a margin for further research.

## 4. Conclusions

- The Fréchet distribution has proven to fit the studied salaries’ distributions well, having a similar accuracy in relation to the data when compared to other distributions (Log-Normal, Gamma, Dagum, GB2) that can be considered more complex.
- The analysis of the results showed that changes in the salary distribution as a result of the economy evolution are reflected in the 0%–30% and 30%–70% brackets of the total salaries paid, the top 30% (affecting 9% of the salaried people) being quite resilient to those changes from 2002 to 2014 in Spain.
- Finally, the results based on the distributions’ asymmetry (skewness) indicate an increasingly more balanced salaries distribution (i.e., less skewed) starting in 2007. However, this seems to be in contrast with the evolution of the Gini coefficient.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Abbas, Kamran, and Yincai Tang. 2012. Comparison of Estimation Methods for Frechet Distribution with Known Shape. Caspian Journal of Applied Sciences Research 1: 58–64. [Google Scholar]
- Alonso, Mario. 2016. El sector público en España está poco auditado. El País, January 3. Available online: http://economia.elpais.com/economia/2015/12/29/actualidad/1451383564_744232.html (accessed on 21 April 2017).
- Argimón, Isabel, and Ángel Luis Gómez. 2006. Empleo y salarios en las AAPP : Una perspectiva macroeconómica. Revistas. Presupuesto Y Gasto Público 41: 73–92. [Google Scholar]
- Asplund, Rita, and Erling Barth, eds. 2005. Education and Wage Inequality in Europe. A literature Review. Helsinki: ETLA. [Google Scholar]
- Atkinson, Anthony B., and François Bourguignon. 2015. Handbook of Income Distribution. Amsterdam: Elsevier B.V. [Google Scholar]
- Azad, Abul Kalam, Mohammad Golam Rasul, and Talal Yusaf. 2014. Statistical Diagnosis of the Best Weibull Methods for Wind Power Assessment for Agricultural Applications. Energies 7: 3056–85. [Google Scholar] [CrossRef]
- Banerjee, Anand, Victor M. Yakovenko, and T. Di Matteo. 2006. A study of the personal income distribution in Australia. Physica A: Statistical Mechanics and its Applications 370: 54–59. [Google Scholar] [CrossRef]
- Botella, Marta, Pablo Hernández de Cos, and Javier J. Pérez. 2009. Algunas consideraciones sobre los efectos macroeconómicos de los salarios y del empleo de las Administraciones Públicas. Boletín Económico. Banco de España 9: 57–74. [Google Scholar]
- Burkhauser, Richard V., and Kenneth A. Couch. 2009. Intragenerational inequality and intertemporal mobility. In The Oxford Handbook of Economic Inequality. Edited by Wiemer Salverda, Brian Nolan and Timothy M. Smeeding. New York: Oxford University Press, pp. 522–45. [Google Scholar]
- Carmona, René. 2014. Statistical Analysis of Financial Data in R. New York: Springer. [Google Scholar]
- Carrasco, Raquel, Juan F. Jimeno, and A. Ortega. 2015. Returns to Skills and the Distribution of Wages: Spain 1995–2010. Oxford Bulletin of Economics and Statistics 77: 542–65. [Google Scholar] [CrossRef]
- Chotikapanich, Duangkamon, D. S. Rao, and Kam Ki Tang. 2007. Estimating income inequality in China using grouped data and the generalized beta distribution. Review of Income and Wealth 53: 127–47. [Google Scholar] [CrossRef]
- Cran, G.W. 1988. Moment estimators for the 3-parameter Weibull distribution. IEEE Transactions on Reliability 37: 360–63. [Google Scholar] [CrossRef]
- Cubas, Javier, Santiago Pindado, and Marta Victoria. 2014. On the analytical approach for modeling photovoltaic systems behavior. Journal of Power Sources 247: 467–74. [Google Scholar] [CrossRef]
- Cubas, Javier, Santiago Pindado, and Carlos de Manuel. 2014. Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function. Energies 7: 4098–115. [Google Scholar] [CrossRef]
- De Gusmão, Felipe R. S., and Edwin M. M. Ortega. 2011. The generalized inverse Weibull distribution. Statistical Papers 52: 591–619. [Google Scholar] [CrossRef]
- Dorvlo, Atsu S. S. 2002. Estimating wind speed distribution. Energy Conversion & Management 43: 2311–18. [Google Scholar]
- Espejo, Isabel García, and Marta Ibáñez Pascual. 2007. Los trabajadores pobres y los bajos salarios en España: Un análisis de los factores familiares y laborales asociados a las distintas situaciones de pobreza. Empiria. Revista de Metodología de Ciencias Sociales 14: 41–67. [Google Scholar] [CrossRef]
- Febrer, Antonia, and Juan Mora López. 2004. Wage Distribution in Spain, 1994-1999: An Application of a Flexible Estimator of Conditional Distributions. Alicante: University of Alicante, pp. 1–38. [Google Scholar]
- Fisher, Ronald Aylmer, and Leonard Henry Caleb Tippett. 1928. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society 24: 180–90. [Google Scholar] [CrossRef]
- Galindo, Cristina. 2015. Lo que aprendimos de la crisis. El País, December 6. Available online: http://economia.elpais.com/economia/2015/12/03/actualidad/1449157907_637737.html (accessed on 21 April 2017).
- García, Carmelo, Mercedes Prieto, and Hipólito Simón. 2014. La modelización paramétrica de las distribuciones salariales. Revista de Economía Aplicada 22: 5–38. [Google Scholar]
- Goda, Yoshimi, Masanobu Kudaka, and Hiroyasu Kawai. 2010. Incorporating of Weibull distribution in L-moments method for Regional Frequency Analysis of Peak over Threshold wave heights. Paper presented at the 32nd International Conference on Coastal Engineering (ICCE 2010), Shangai, China, June 30–July 5; pp. 1–11. [Google Scholar]
- Jagielski, Maciej, and Ryszard Kutner. 2013. Modelling of income distribution in the European Union with the Fokker–Planck equation. Physica A Statistical Mechanics & Its Applications 392: 2130–38. [Google Scholar]
- Jiménez, Miguel. 2015. El Sueldo Medio Declarado a Hacienda Cae al Nivel Más Bajo Desde 2007. El País, November 18. Available online: http://economia.elpais.com/economia/2015/11/17/actualidad/1447782510_551814.html (accessed on 21 April 2017).
- Jiménez, Miguel. 2015. Los Hombres Acaparan el 82% de los Sueldos Más Altos, Según Hacienda. El País, November 19. Available online: http://economia.elpais.com/economia/2015/11/18/actualidad/1447872182_528635.html (accessed on 21 April 2017).
- Justus, C.G., W. R. Hargraves, Amir Mikhail, and Densie Graber. 1978. Methods for Estimating Wind Speed Frequency Distributions. Journal of Applied Meteorology 17: 350–53. [Google Scholar] [CrossRef]
- Khan, M. Shuaib, G. R. Pasha, and Ahmed Hesham Pasha. 2008. Theoretical analysis of inverse weibull distribution. Wseas Transactions on Mathematics 7: 30–38. [Google Scholar]
- Kleiber, Christian. 2008. A guide to the Dagum distributions. In Modeling Income Distributions and Lorenz Curves. New York: Springer, pp. 97–117. [Google Scholar]
- Kleiber, Christian, and Samuel Kotz. 2003. Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken: John Wiley and Sons. [Google Scholar]
- Klein, Nadja, Thomas Kneib, Stefan Lang, and Alexander Sohn. 2015. Bayesian structured additive distributional regression with an application to regional income inequality in Germany. The Annals of Applied Statistics 9: 1024–52. [Google Scholar] [CrossRef]
- Krause, Melanie. 2014. Parametric Lorenz Curves and the Modality of the Income Density Function. The Review of Income Wealth 60: 905–29. [Google Scholar] [CrossRef]
- Krugman, Paul. 2016. On Invincible Ignorance. The New York Times, March 21. Available online: https://www.nytimes.com/2016/03/21/opinion/on-invincible-ignorance.html?_r=0 (accessed on 21 April 2017).
- Krugman, Paul. 2016. La irreductible ignorancia. El País, March 26. Available online: http://economia.elpais.com/economia/2016/03/23/actualidad/1458724332_015099.html (accessed on 21 April 2017).
- Lubrano, Michel. 2016. The Econometrics of Inequality and Poverty. In Lecture 4: Lorenz Curves and Parametric Distributions. Marseille: Centre National de la Recherche Scientifique, Groupement de Recherche en Économie Quantitative d’Aix-Marseille (GREQAM). [Google Scholar]
- Lun, Isaac Y. F., and Joseph C. Lam. 2000. A study of Weibull parameters using long-term wind observations. Renewable Energy 20: 145–53. [Google Scholar] [CrossRef]
- Machado, José AF, and José Mata. 2005. Counterfactual decomposition of changes in wage distributions using quantile regression. Journal of applied Econometrics 20: 445–65. [Google Scholar] [CrossRef]
- Mandelbrot, Benoit. 1962. Paretian Distributions and Income Maximization. Quarterly Journal of Economics 76: 57–85. [Google Scholar] [CrossRef]
- Mann, Nancy R. 1984. Statistical estimation of parameters of the Weibull and Frechet distributions. In Statistical Extremes and Applications. Dordrecht: Springer, pp. 81–89. [Google Scholar]
- Marek, Luboš, and Michal Vrabec. 2013. Model wage distribution—Mixture Density Functions. International Journal of Economics and Statistics 1: 113–121. [Google Scholar]
- McDonald, James B., and Michael R. Ransom. 1979. Functional forms, estimation techniques and the distribution of income. Econometrica: Journal of the Econometric Society 47: 1513–25. [Google Scholar] [CrossRef]
- McDonald, James B., and Michael Ransom. 2008. The generalized beta distribution as a model for the distribution of income: Estimation of related measures of inequality. In Modeling Income Distributions and Lorenz Curves. New York: Springer, pp. 147–66. [Google Scholar]
- Mendaña Saavedra, Felipe, and Carlos Pindado Carrión. 2013. Relleno con bicomponente del gap de los anillos de dovelas en los escudos no presurizados. Revista de Obras Públicas 160: 21–35. [Google Scholar]
- Orsini, Kristian. 2014. Wage adjustment in Spain: Slow, inefficient and unfair? ECFIN Country Focus 11: 1–8. [Google Scholar]
- Pijoan-Mas, Josep, and Virginia Sánchez-Marcos. 2010. Spain is different: Falling trends of inequality. Review of Economic Dynamics 13: 154–78. [Google Scholar] [CrossRef]
- Piketty, Thomas. 2014. El capital en el siglo XXI. Mexico: Fondo de Cultura Económica. [Google Scholar]
- Pindado, Santiago, Imanol Pérez, and Maite Aguado. 2013. Fourier analysis of the aerodynamic behavior of cup anemometers. Measurement Science and Technology 24: 065802. [Google Scholar] [CrossRef]
- Pindado, Santiago, Javier Cubas, and Felix Sorribes-Palmer. 2015. On the harmonic analysis of cup anemometer rotation speed: A principle to monitor performance and maintenance status of rotating meteorological sensors. Measurement 73: 401–418. [Google Scholar] [CrossRef]
- Rehman, Shafiqur, T. O. Halawani, and Tahir Husain. 1994. Weibull parameters for wind speed distribution in Saudi Arabia. Solar Energy 53: 473–79. [Google Scholar] [CrossRef]
- Rigby, Robert A., and D. Mikis Stasinopoulos. 2015. The Generalized Additive Models for Location, Scale and Shape. Journal of the Royal Statistical Society: Series C (Applied Statistics) 54: 1–2. [Google Scholar] [CrossRef]
- Seguro, J. V., and T.W. Lambert. 2000. Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis. Journal of Wind Engineering & Industrial Aerodynamics 85: 75–84. [Google Scholar]
- Selezneva, Ekaterina, and Philippe Van Kerm. 2016. A distribution-sensitive examination of the gender wage gap in Germany. Journal of Economic Inequality 14: 21–40. [Google Scholar] [CrossRef]
- Shatnawi, Dina, Ronald L. Oaxaca, and Michael R. Ransom. 2013. Movin’ on up: Hierarchical occupational segmentation and gender wage gaps. Journal of Economic Inequality 12: 315–38. [Google Scholar] [CrossRef]
- Shittu, Olanrewaju I., and K. A. Adepoju. 2014. On the Exponentiated Weibull Distribution for Modeling Wind Speed in South Western Nigeria. Journal of Modern Applied Statistical Methods Jmasm 13: 431–45. [Google Scholar]
- Sohn, Alexander, Nadja Klein, and Thomas Kneib. 2014. A New Semiparametric Approach to Analysing Conditional Income Distributions. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2404335 (accessed on 21 April 2017).
- Sohn, Alexander, Nadja Klein, and Thomas Kneib. 2015. A Semiparametric Analysis of Conditional Income Distributions. Schmollers Jahrbuch 135: 13–22. [Google Scholar] [CrossRef]
- Ulgen, Koray, and Arif Hepbasli. 2002. Determination of Weibull parameters for wind energy analysis of Izmir. Turkey. International Journal of Energy Research 26: 495–506. [Google Scholar] [CrossRef]
- Weibull, Waloddi. 1951. A statistical distribution function of wide applicability. Journal of Applied Mechanics 18: 293–97. [Google Scholar]

1 | Agencia Tributaria de España. Mercado de Trabajo y Pensiones en las Fuentes Tributarias. http://www.agenciatributaria.es/AEAT.internet/datosabiertos/catalogo/hacienda/Mercado_de_Trabajo_y_Pensiones_en_las_Fuentes_Tributarias.shtml. |

2 | Average annual Spanish salary falls to lowest level since 2007 (Jiménez 2015a). |

3 | Men account for 82% of highest salaries in Spain, says new report (Jiménez 2015b). |

4 | According to Kleiber and Kotz (2003), the Weibull distribution is surrounded by some controversy as the “French would argue that this is nothing else but Fréchet distribution”. |

5 | Selezneva and Van Kerm, published another interesting work on gender discrimination in wage distribution in Germany, showing a larger gender gap at the bottom of the distribution (Selezneva and Van Kerm 2016). |

6 | This work was dedicated by B. Mandelbrot to Maurice Fréchet, who proposed in 1927 the distribution selected in the present work to study the wages distribution in Spain. |

7 | This work by Sohn et al. is the 2014 working version of the 2015 paper ‘A Semiparametric Analysis of Conditional Income Distributions’ (Sohn et al. 2015). |

8 | This work, published initially in 2013 in French (and in 2014 in English), was cited more than 4700 times by the end of 2016, according to Google Scholar. |

9 | MacDonald and Ransom claim that the GB2 distribution fits the income distributions better than other simpler distributions such as log-normal or Weibull (McDonald and Ransom 2008). |

10 | Mario Alonso. President of the Institute of Auditors and the Spanish Accounting (Instituto de Censores Jurados de Cuentas) (Alonso 2016). |

11 | The change of the labor market law in 2010 started to be studied in 2008. During two years the government of President Rodríguez Zapatero tried to reach a wide agreement that could include both the employers’ association and the trade unions. |

12 | This fact agrees with what Paul Krugman said, quoting a work by Burkhauser and Couch (2009); ‘The majority of economic mobility occurs over fairly small spans of the distribution’. On Invincible Ignorance (Krugman 2016a). Also: La irreductible ignorancia (Krugman 2016b). |

**Figure 1.**Distribution of salaries paid in Spain (2014) as a function of the minimum wage (m.w.) brackets established by the Spanish Tax Agency (Agencia Tributaria de España).

**Figure 2.**Evolution from 1999 to 2014 of salary earners (left y-axis) and total salaries paid (right y-axis) in Spain.

**Figure 3.**(

**Left**) Estimated number of salary earners as a function of the non-dimensional salary paid, φ (expressed in multiples of the minimum wage), within the 0.5 minimum wage sub-brackets, which divide the official 5–7.5 and 7.5–10 times the minimum wage brackets, in Spain (2014). The variables corresponding to Equations (3) and (4) in relation to the official 5–7.5 bracket are indicated in the graph. (

**Right**) Salary earners in Spain (2014) as a function of the non-dimensional salary paid, φ. (Official Statistics 2014, see footnote 1)

**Figure 4.**Salaries paid (in terms of percentage of the total amount, TSP) in 2014 as a function of the non-dimensional salary paid, φ (expressed in times the minimum wage). The Fréchet distribution has been fitted to the data (Equation (8); λ = 0.479; γ = 8.55; c = 13.6; d = −11.18), together with the Log-Normal (Equation (9); λ = 0.478; a = 1.15; b = 0.658), the Gamma (Equation (10); λ = 0.463; a = 2.856; b = 1.188), the Dagum (Equation (11); λ = 0.490; a = 3.740; b = 4.160; p = 2.287; d = −2.445), and the GB 2 (Equation (12); λ = 0.493; a = 5.218; b = 5.046; ξ = 2.112; η = 0.740; d = −3.532) distributions.

**Figure 6.**(

**Left**) RMSE related to Equation (8) fittings to the official statistics data, see Table 2. (

**Right**) Average values of the aforementioned fittings (dashed lines represent the higher and the lower values of that fittings) in relation to the perceived salary, φ (expressed in times the minimum salary). On the right y-axis: averaged error when comparing the fittings and the official statistics data (the standard deviation bars have also been included; the highest levels of the standard deviation have been indicated by dashed ellipses).

**Figure 7.**(

**Left**) Evolution from 1999 to 2014 of the of the percentage salary earners within the 0%–30%, 30%–70%, and more than 70% of the total salaries paid brackets; (

**Right**) Yearly variations of the aforementioned the percentage salary earners within the 30%–70% bracket in relation to the variations of the the percentage salary earners within the 0%–30% bracket.

**Figure 8.**Evolution from 1999 to 2014 of the Fréchet distribution fittings asymmetry parameter, ψ, (open circles; defined with equation (23)). The calculated skewness of the Fréchet distribution fittings, s

_{kw}, (open squares), and the evolution of the Gini coefficient (open rhombi) have been also included in the graph.

**Figure 9.**Lorenz curves representing the wage distributions in 1999, 2005, and 2013. Percentage of the wages, L(p), perceived by a percentage p of the salaried people in relation to the percentage of the salaried people, p. The dashed straight line corresponds to the theoretical maximum equality level.

**Figure 10.**Difference between the Lorenz curves and the theoretical maximum equality level, p – L(p), in relation to the percentage of salaried people, p, in 2007 and 2012 (left). Location of the maximum point on these curves (right).

**Figure 11.**Evolution from 1999 to 2014 of the salaried people percentage corresponding to the maximum difference between the Lorentz curve applied to the salaries distribution and the theoretical maximum equality level.

**Table 1.**Salary earners per income bracket in Spain from 1999 to 2014 (expressed in million people). Source: Agencia Tributaria de España.

Income | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 to 0.5 min. wage | 2.784 | 2.774 | 2.795 | 2.833 | 2.911 | 2.851 | 3.232 | 3.286 | 2.988 | 3.090 | 3.405 | 3.420 | 3.466 | 3.494 | 3.642 | 3.695 |

0.5 to 1 min. wage | 1.688 | 1.739 | 1.760 | 1.817 | 1.829 | 1.887 | 2.145 | 2.187 | 2.214 | 2.284 | 2.251 | 2.207 | 2.211 | 2.122 | 2.110 | 2.197 |

1 to 1.5 min. wage | 1.743 | 1.819 | 1.839 | 1.870 | 1.872 | 1.970 | 2.211 | 2.359 | 2.492 | 2.481 | 2.284 | 2.215 | 2.175 | 2.051 | 1.982 | 2.047 |

1.5 to 2 min. wage | 2.182 | 2.341 | 2.388 | 2.440 | 2.418 | 2.583 | 2.824 | 3.010 | 3.169 | 2.966 | 2.678 | 2.592 | 2.500 | 2.417 | 2.224 | 2.220 |

2 to 2.5 min. wage | 1.595 | 1.760 | 1.926 | 2.024 | 2.144 | 2.174 | 2.188 | 2.281 | 2.318 | 2.281 | 2.038 | 1.979 | 1.956 | 1.896 | 1.762 | 1.769 |

2.5 to 3 min. wage | 1.033 | 1.114 | 1.215 | 1.282 | 1.374 | 1.399 | 1.411 | 1.467 | 1.528 | 1.548 | 1.430 | 1.389 | 1.373 | 1.344 | 1.251 | 1.244 |

3 to 3.5 min. wage | 0.799 | 0.848 | 0.899 | 0.950 | 0.982 | 1.013 | 1.035 | 1.081 | 1.109 | 1.122 | 1.048 | 1.059 | 1.073 | 1.017 | 0.984 | 0.990 |

3.5 to 4 min. wage | 0.642 | 0.689 | 0.727 | 0.746 | 0.785 | 0.793 | 0.809 | 0.847 | 0.875 | 0.879 | 0.831 | 0.823 | 0.807 | 0.768 | 0.761 | 0.756 |

4 to 4.5 min. wage | 0.524 | 0.548 | 0.579 | 0.605 | 0.641 | 0.643 | 0.639 | 0.657 | 0.677 | 0.689 | 0.654 | 0.641 | 0.630 | 0.553 | 0.567 | 0.569 |

4.5 to 5 min. wage | 0.374 | 0.417 | 0.454 | 0.484 | 0.521 | 0.522 | 0.480 | 0.491 | 0.511 | 0.521 | 0.498 | 0.456 | 0.419 | 0.342 | 0.357 | 0.361 |

5 to 7.5 min. wage | 0.718 | 0.791 | 0.866 | 0.935 | 1.030 | 1.003 | 0.936 | 0.953 | 0.958 | 0.979 | 0.916 | 0.851 | 0.803 | 0.727 | 0.722 | 0.726 |

7.5 to 10 min. wage | 0.205 | 0.224 | 0.247 | 0.265 | 0.290 | 0.282 | 0.263 | 0.267 | 0.276 | 0.277 | 0.250 | 0.236 | 0.225 | 0.201 | 0.195 | 0.197 |

More than 10 min. wage | 0.143 | 0.156 | 0.177 | 0.187 | 0.204 | 0.199 | 0.187 | 0.185 | 0.194 | 0.194 | 0.168 | 0.156 | 0.149 | 0.133 | 0.125 | 0.128 |

Total amount | 14.431 | 15.220 | 15.871 | 16.438 | 17.001 | 17.321 | 18.360 | 19.070 | 19.309 | 19.311 | 18.452 | 18.025 | 17.788 | 17.063 | 16.682 | 16.899 |

**Table 2.**Salaries payed per income bracket in Spain from 1999 to 2014 (expressed in billion euros). Source: Agencia Tributaria de España.

Income | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 to 0.5 min. wage | 3.428 | 3.499 | 3.615 | 3.733 | 3.856 | 4.122 | 4.991 | 5.303 | 5.221 | 5.632 | 6.371 | 6.392 | 6.513 | 6.407 | 6.586 | 6.797 |

0.5 to 1 min. wage | 7.363 | 7.746 | 8.002 | 8.424 | 8.659 | 9.420 | 11.542 | 12.440 | 13.320 | 14.420 | 14.684 | 14.571 | 14.789 | 14.203 | 14.216 | 14.791 |

1 to 1.5 min. wage | 12.778 | 13.601 | 14.031 | 14.534 | 14.831 | 16.453 | 19.914 | 22.471 | 25.085 | 26.241 | 25.107 | 24.716 | 24.544 | 23.128 | 22.464 | 23.172 |

1.5 to 2 min. wage | 22.269 | 24.417 | 25.470 | 26.584 | 26.922 | 30.288 | 35.597 | 39.996 | 44.331 | 43.662 | 40.927 | 40.203 | 39.278 | 37.991 | 35.191 | 35.111 |

2 to 2.5 min. wage | 20.731 | 23.326 | 26.059 | 27.944 | 30.219 | 32.280 | 34.993 | 38.500 | 41.291 | 42.758 | 39.756 | 39.181 | 39.223 | 38.002 | 35.558 | 35.704 |

2.5 to 3 min. wage | 16.480 | 18.110 | 20.168 | 21.698 | 23.731 | 25.460 | 27.674 | 30.349 | 33.338 | 35.536 | 34.118 | 33.635 | 33.707 | 33.073 | 30.925 | 30.757 |

3 to 3.5 min. wage | 15.097 | 16.348 | 17.707 | 19.081 | 20.113 | 21.888 | 24.105 | 26.549 | 28.721 | 30.570 | 29.697 | 30.445 | 31.226 | 29.571 | 28.807 | 28.978 |

3.5 to 4 min. wage | 14.007 | 15.329 | 16.516 | 17.279 | 18.533 | 19.749 | 21.738 | 24.008 | 26.151 | 27.623 | 27.164 | 27.292 | 27.099 | 25.768 | 25.710 | 25.547 |

4 to 4.5 min. wage | 12.931 | 13.828 | 14.899 | 15.871 | 17.152 | 18.140 | 19.458 | 21.106 | 22.943 | 24.528 | 24.203 | 24.110 | 23.975 | 20.996 | 21.671 | 21.755 |

4.5 to 5 min. wage | 10.305 | 11.732 | 13.052 | 14.202 | 15.600 | 16.467 | 16.299 | 17.573 | 19.310 | 20.718 | 20.581 | 19.133 | 17.779 | 14.550 | 15.258 | 15.415 |

5 to 7.5 min. wage | 24.948 | 28.028 | 31.311 | 34.474 | 38.690 | 39.767 | 40.058 | 42.975 | 45.575 | 48.938 | 47.509 | 44.870 | 42.974 | 38.942 | 38.911 | 39.099 |

7.5 to 10 min. wage | 10.173 | 11.350 | 12.779 | 13.986 | 15.623 | 16.000 | 16.109 | 17.227 | 18.786 | 19.842 | 18.658 | 17.795 | 17.218 | 15.364 | 14.955 | 15.148 |

More than 10 min. wage | 13.049 | 14.792 | 17.513 | 18.397 | 20.460 | 21.172 | 21.600 | 22.819 | 25.165 | 26.353 | 23.370 | 22.163 | 21.463 | 19.402 | 18.443 | 19.006 |

Total amount | 183.56 | 202.11 | 221.12 | 236.21 | 254.39 | 271.21 | 294.08 | 321.32 | 349.24 | 366.82 | 352.15 | 344.51 | 339.79 | 317.40 | 308.70 | 311.28 |

Year | Minumum Wage [€] | Year | Minumum Wage [€] |
---|---|---|---|

1998 | 5725.02 | 2007 | 7988.4 |

1999 | 5828.48 | 2008 | 8400 |

2000 | 5947.2 | 2009 | 8736 |

2001 | 6068.3 | 2010 | 8866.2 |

2002 | 6190.8 | 2011 | 8979.6 |

2003 | 6316.8 | 2012 | 8979.6 |

2004 | 6659.1 | 2013 | 9034.2 |

2005 | 7182 | 2014 | 9034.2 |

2006 | 7572.6 |

Year | k_{1} | k_{2} | γ | c | d | λ |
---|---|---|---|---|---|---|

1999 | 3.93 | 1.89 | 3.78 | 6.40 | −3.86 | 0.488 |

2000 | 4.11 | 1.92 | 3.42 | 5.83 | −3.25 | 0.489 |

2001 | 4.21 | 1.95 | 3.37 | 5.78 | −3.15 | 0.486 |

2002 | 4.32 | 1.97 | 3.32 | 5.76 | −3.10 | 0.488 |

2003 | 4.38 | 2.01 | 6.39 | 10.15 | −7.68 | 0.480 |

2004 | 4.55 | 1.98 | 3.25 | 5.62 | −2.95 | 0.487 |

2005 | 5.01 | 1.90 | 3.51 | 5.78 | −3.24 | 0.484 |

2006 | 5.24 | 1.89 | 3.35 | 5.44 | −2.92 | 0.485 |

2007 | 5.27 | 1.90 | 3.13 | 5.06 | −2.54 | 0.486 |

2008 | 5.29 | 1.90 | 3.68 | 6.04 | −3.51 | 0.485 |

2009 | 5.16 | 1.87 | 4.63 | 7.71 | −5.18 | 0.486 |

2010 | 5.10 | 1.84 | 5.41 | 8.78 | −6.29 | 0.483 |

2011 | 5.09 | 1.82 | 6.39 | 10.15 | −7.68 | 0.480 |

2012 | 4.95 | 1.79 | 6.93 | 10.49 | −8.08 | 0.475 |

2013 | 4.87 | 1.78 | 8.62 | 13.61 | −11.18 | 0.479 |

2014 | 4.93 | 1.78 | 8.55 | 13.60 | −11.18 | 0.479 |

**Table 5.**Wage levels, φ

_{30%}and φ

_{70%}, that indicate 30% and 70% of the total amount of salaries paid and the number of salaries paid (salaried people) within the 0%–30%, 30%–70%, and more than 70% salaries paid brackets from 1999 to 2014. Two different figures are included: ‘calc.’ stands for the figures calculated by interpolating on the official statistical data, whereas ‘est.’ stands for the figures estimated with the fittings of Equations (6) and (8).

Year | φ_{30%} | φ_{70%} | S_{0%–30%} [%] | S_{30%–70%} [%] | S_{70%-}_{∞} [%] | |||||
---|---|---|---|---|---|---|---|---|---|---|

calc. | est. | calc. | est. | calc. | est. | calc. | est. | calc. | est. | |

1999 | 2.04 | 2.27 | 4.50 | 4.72 | 63.1 | 72.1 | 27.8 | 22.5 | 9.1 | 8.5 |

2000 | 2.07 | 2.30 | 4.56 | 4.79 | 62.6 | 72.4 | 28.2 | 22.6 | 9.2 | 8.5 |

2001 | 2.12 | 2.36 | 4.67 | 4.90 | 62.4 | 72.7 | 28.4 | 22.5 | 9.2 | 8.4 |

2002 | 2.14 | 2.38 | 4.71 | 4.93 | 62.3 | 72.8 | 28.5 | 22.5 | 9.3 | 8.5 |

2003 | 2.01 | 2.23 | 4.29 | 4.48 | 62.3 | 69.5 | 28.4 | 23.1 | 9.3 | 11.2 |

2004 | 2.15 | 2.40 | 4.72 | 4.95 | 61.8 | 73.1 | 28.8 | 22.5 | 9.4 | 8.6 |

2005 | 2.05 | 2.29 | 4.50 | 4.73 | 62.2 | 72.5 | 28.5 | 22.5 | 9.2 | 8.6 |

2006 | 2.03 | 2.27 | 4.45 | 4.69 | 61.9 | 72.6 | 28.8 | 22.4 | 9.3 | 8.7 |

2007 | 2.02 | 2.27 | 4.45 | 4.69 | 61.2 | 72.3 | 29.4 | 22.6 | 9.5 | 8.8 |

2008 | 2.04 | 2.28 | 4.46 | 4.69 | 61.6 | 72.8 | 29.0 | 22.5 | 9.5 | 8.8 |

2009 | 2.04 | 2.26 | 4.42 | 4.62 | 62.7 | 73.3 | 28.0 | 22.3 | 9.3 | 8.8 |

2010 | 2.02 | 2.25 | 4.35 | 4.54 | 62.8 | 73.5 | 27.9 | 22.0 | 9.3 | 8.9 |

2011 | 2.01 | 2.23 | 4.29 | 4.48 | 62.9 | 73.6 | 27.7 | 21.7 | 9.3 | 9.0 |

2012 | 1.97 | 2.20 | 4.18 | 4.37 | 63.0 | 73.4 | 27.7 | 21.4 | 9.3 | 9.0 |

2013 | 1.99 | 2.20 | 4.24 | 4.40 | 63.9 | 73.8 | 26.9 | 21.5 | 9.2 | 8.9 |

2014 | 1.98 | 2.19 | 4.24 | 4.40 | 64.1 | 73.4 | 26.8 | 21.7 | 9.1 | 8.8 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pindado, S.; Pindado, C.; Cubas, J. Fréchet Distribution Applied to Salary Incomes in Spain from 1999 to 2014. An Engineering Approach to Changes in Salaries’ Distribution. *Economies* **2017**, *5*, 14.
https://doi.org/10.3390/economies5020014

**AMA Style**

Pindado S, Pindado C, Cubas J. Fréchet Distribution Applied to Salary Incomes in Spain from 1999 to 2014. An Engineering Approach to Changes in Salaries’ Distribution. *Economies*. 2017; 5(2):14.
https://doi.org/10.3390/economies5020014

**Chicago/Turabian Style**

Pindado, Santiago, Carlos Pindado, and Javier Cubas. 2017. "Fréchet Distribution Applied to Salary Incomes in Spain from 1999 to 2014. An Engineering Approach to Changes in Salaries’ Distribution" *Economies* 5, no. 2: 14.
https://doi.org/10.3390/economies5020014