Chinese Business Review, ISSN 537-56 Novemer, Vol., No., 974-984 Application of areto Distriution in Wage Models Diana Bílková University of Economics in rague, rague, Czech Repulic This paper deals with the use of areto distriution in models of wage distriution. areto distriution cannot generally e used as a model of the whole wage distriution, ut only as a model for the distriution of higher or of the highest wages. It is usually aout wages higher than the median. The parameter is called the areto coefficient and it is often used as a characteristic of differentiation of fifty percent of the highest wages. areto distriution is so much the more applicale model of a specific wage distriution, the more specific differentiation of fifty percent of the highest wages will resemle to differentiation that is epected y areto distriution. areto distriution assumes a differentiation of wages, in which the following ratios are the same: ratio of the upper quartile to the median; ratio of the eighth decile to the sith decile; ratio of the ninth decile to the eighth decile. This finding may serve as one of the empirical criterions for assessing, whether areto distriution is a suitale or less suitale model of a particular wage distriution. If we find only small differences etween the ratios of these quantiles in a specific wage distriution, areto distriution is a good model of a specific wage distriution. Approimation of a specific wage distriution y areto distriution will e less suitale or even unsuitale when more epressive differences of mentioned ratios. If we choose areto distriution as a model of a specific wage distriution, we must reckon with the fact that the model is always only an approimation. It will descrie only approimately the actual wage distriution and the relationships in the model will only partially reflect the relationships in a specific wage distriution. Keywords: areto distriution, areto coefficient, estimation methods for parameters, least squares method, wage distriutions areto Distriution The question of income and wage distriutions and their models is quite etensively treated in the statistical literature (Bartošová, ; Bartošová & Bína, 9; Bílková, ; Dutta, Sefton, & Weale, ; Majumder & Chakravarty, 99; McDonald & Snooks, 985; McDonald, 984; McDonald & Butler, 987). Acknowledgment: The paper was supported y grant project IGS 4/ called Analysis of the Development of Income Distriution in the Czech Repulic Since 99 to the Financial Crisis and Comparison of This Development With the Development of the Income Distriution in Times of Financial Crisis, According to Socioical Groups, Gender, Age, Education, rofession Field and Region from the University of Economics in rague. Diana Bílková, Ing./Dr., Department of Statistics and roaility, Faculty of Informatics and Statistics, University of Economics in rague. Correspondence concerning this article should e addressed to Diana Bílková, Department of Statistics and roaility, Faculty of Informatics and Statistics, University of Economics in rague, Sq. W. Churchill 938/4, rague 3, Czech Repulic, post code: 3 67. E-mail: ilkova@vse.cz.
ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS 975 areto distriution is usually used as a model of the distriution of the largest wages, not for the whole wage distriution. In this article, we will consider using the areto distriution to model wages higher than median. The % quantile of the wage distriution will e denoted y, < <. This value represents the upper ound of % lowest wages and also the lower ound of ( ) % highest wages. A particular quantile (denoted as ) which will e the lower ound of some small numer of the highest wages is usually set to e the maimum wage. If the following formula () holds for any quantile, the wage distriution is areto distriution. = The parameter of the areto distriution () is called the areto coefficient. It can e used as a characteristic of differentiation of 5% highest wages. We will now consider a pair of quantiles and, <. It follows from equation () that: and = = From what we can derive for the rate of to that: = The rate is an increasing function of the areto coefficient. If the rate of quantiles increases, the relative differentiation of wages increases too. If only asolute differences etween quantiles increase, only the asolute differentiation of wages increases. It follows from the equation () that once the values and are chosen, we can determine the quantile for any chosen or the other way around for any value we can find the corresponding value of. In the first case, it is advantageous to write the equation () as: = (5) or after arithmic transformation as: () () (3) (4) = [(- ) - ( )] (6) in the second case: = ( ) (7)
976 ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS or after arithmic transformation as: ( ) = ( ) + ( The equations ()-(4) will after arithmic transformation have the following forms: ) = (9) = () It follows from the equation (9) that instead of the areto coefficient we can use any other quantile of the areto distriution and it follows from the equation () that the areto coefficient can e calculated using any known quantiles and. Then we can also determine the value using the formulas:, = () = () The model characterized with the relationship () will e practically applicale if the following is known: The value of the quantile that characterizes the assumed wage maimum and the value of the areto coefficient ; The value of the quantile that characterizes the assumed wage maimum and the value of any other quantile; The values of any two quantiles of the areto distriution. Any two quantiles can e written as and +k, where < k <. Using the equation (4), we can derive for the rate of these two quantiles: (8) + k = k (3) The rate (3) will e equal for such pairs of quantiles for which the following formula holds: = c, (4) k where c is a constant, i.e., the rate will e the same for all pairs of quantiles for which: c k = ( ) (5) c We will use the constant c = in equation (5) and we will choose gradually =.5;.6;.8. Then using the equation (3) we can show the equality of rates of some frequently used quantiles:
ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS 977.75.8.9 = = (6).5.6.8 From the relationship (6) we can conclude that areto distriution assumes such a wage differentiation for which the rate of the upper quartile to median is the same as: The rate of the 8th to the 6th decile; And as the rate of the 9th to the 8th decile. If in a particular case, the oserved differences of the rates of the aove mentioned quantiles are negligile, areto distriution will e an appropriate model of the considered wage distriution. In the case, the differences are quite material, the approimation of the considered wage distriution with areto distriution will e more or less inappropriate. More aout the theory of areto distriution is descried in statistical literature (Fores, Evans, Hastings, & eacock, ; Johnson, Kotz, & Balakrishnan, 994; Kleier & Kotz, ; Krishnamoorthy, ). arameter Estimates If the areto distriution is chosen as a model for a particular distriution we have to keep in mind that this model is only an approimation. The wage distriution will e only approimated and the relations derived from the model will also hold for the true distriution only approimately. Which relations will hold more precisely and for which the precision will e lower will e mostly dependent on the method of parameter estimates. There are many possiilities to choose from. In the following tet the quantiles of areto distriution will e denoted as and the quantiles of the oserved wage distriution will e denoted as y. First we need to decide which quantile to choose as. It this article we will assume that =.99. From the equation () we can see that the considered areto distriution will e defined y the equation:.99 = (7). Then we need to determine the value.99 and the value of the areto coefficient. Because it is necessary to estimate the values of two parameters we need to choose two equations to estimate from. A natural choice is the equation = y ; that is in our case.99 = y.99. As the other equation we set a quantile equal to the corresponding oserved quantile, i.e., = y. In this case, the parameters of the model will e: = y (8) and using equation (9): y y = We can get different modifications using different choice of the maimum wage and the second quantile. If we use equation.99 = y.99 and we use the median in the second equation, i.e.,.5 = y.5 we get a model with (9)
978 ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS parameters:.99 = y.99 () y.99 y.5..5 = () Another possiility is setting any two quantiles of the model equal to the quantiles of the oserved distriution: = y () = y Using the formula (), we get the following parameter estimates: and from equations () and () we get: y y = = y = y (3) (4) (5) With this alternative we can also get numerous modifications depending on the choice of quantiles y and y that are used. The third possiility is ased on the request that = y and that the rate of some other two quantiles of the areto distriution / is equal to the rate y /y of correspoding quantiles of the wage distriution oserved. In this case we will estimate the parameters using equation (): = y (6) y y = In this case, notwithstanding that / = y /y holds, the equality of quantiles itself, y and y, does not hold. In this case, we can also arrive to numerous modifications depending on what maimum wage is chosen and what quantiles y and y are chosen. For all of the aove methods the equality of two characteristics of the model and the oserved distriution was required. There are also different approaches to the parameter estimates. The least squares method is frequently used for the areto distriution parameter estimates as well. We will consider the following quantiles of the oserved wage distriution y, y,, y k and corresponding quantiles of the (7)
ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS 979 areto distriution,,, k. The model distriution will e most precise when the sum of squared differences: k i= ( y i ) i (8) is minimized. In this case closed formula solution does not eist. Therefore sum of squared differences of arithms of quantiles is often considered: k i= i i ( y ) (9) Minimizing the ojective function (9), it is possile to derive the following estimates: k k k k y y i= i i= i i= i i = (3) k k k i= i= i i k k y i i i = = (3) i = k k In the case we use this estimating method, it is needed to keep in mind that the equality of model quantiles and oserved quantiles is not guaranteed for any. Again we can arrive to different results depending on what quantiles y, y,, y k are used for the calculations. Furthermore the parameter estimates also depend on the choice of the maimum wage. Characteristics of the Appropriateness of the areto Distriution For the application of areto distriution as a model of the wage distriution, it is crucial that the model fits the oserved distriution as close as possile. It is important that the oserved relative frequencies in particular wage intervals are as close to the theoretical proailities assigned to these intervals y the model as possile. It is needed to note that the same parameter estimation method does not always lead to the est results. It is of particular importance in what direction is the oserved wage distriution different from areto distriution. areto distriution assumes such wage differentiation that the relations (6) hold. With real data we can encounter many different situations: y.75 y.8 y.9 < < (3) y y y.5.6.8 y.75 y.8 y.9 > > (33) y y y.5.6.8 y.75 y.9 y.8 < < (34) y y y.5.8.6 y.75 y.9 y.8 > > (35) y y y.5.8.6 y.8 y.75 y.9 < < (36) y y y.6.5.8
98 ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS y.8 y.75 y.9 > > (37) y.6 y.5 y.8 It follows from equations (3)-(37) that the oserved distriutions will more or less systematically differ from the areto distriution. In the case of equation (3) the differentiation of the oserved wage distriution is higher; in the case of equation (33) the differentiation will e lower than in the case of areto distriution. Some ias occurs in cases equations (34)-(37) as well (ut cannot e so specified). Systematical ias should e a signal for potential adjustment of the model which could e ased for eample on adding one or more parameters into the model. These adjustments usually lead to more complicated models. Therefore, the aove mentioned ias is often neglected and simple models are preferred even though they lead to some ias. Wage Distriution of Males and Females in the Czech Repulic in - The data used in this article is the gross monthly wage of male and female in CZK in the Czech Repulic in the years -. Data were sorted in the tale of interval distriution with opened lower and upper ound in the lowest and highest interval respectively. The source is the we page of the Czech statistical office. The following quantiles were calculated (see Tale ). Tale Median y.5 (in CZK), 6th Decile y.6 (in CZK), Upper Quartile y.75 (in CZK), 8th Decile y.8 (in CZK), 9th Decile y.9 (in CZK) a 99th ercentile y.99 (in CZK) of Gross Monthly Wages in the Czech Repulic in the Years - (Total and Split up to Male and Female Separated) Year y.5 y.6 y.75 y.8 y.9 y.99 Total Males Females,5 5,545 6,735 7,79 8,597 9,54,987,3 4,5 6,985 8,4 9,344,8,99,933 4,498,77 3,746 4,83 5,64 6,454 7,3 8,39 9,399 4,4 7,5 8,458 9,557,566,564 3,7 4,696 5,78 8,667,6,3,446 3,46 5,366 7,5,87 5,8 6,453 7,33 8, 9,,39,6 6,987,5,4 3,77 4,47 5,675 7,59 9,553 9,37,64 4,45 5,36 6,8 8,9 3,84 3,343 4,655 7,77 9,8,93,46,53 4,4 5,558 8,54,93 3,797 4,849 6,38 7,693 9,9 3,769,697 4,99 6,4 7,86 8,989 3,55 3,663 35,5 5,7 8,93,68,56,84 3,966 5,94 7,5 3,39 7,754 9,59 3,8 33,9 35,3 37,89 4,54 6,64 3, 34,564 34,89 37, 39,38 4,85 46,375 8,94 3,9 4,637 5,776 7,53 9,8 3,338 33,45 44,9 47,7 47,79 56,369 56,85 57,36 66,395 68,88 46,78 48,47 48,47 57,54 57,88 58,4 7,5 7,338 37,56 43,339 44,883 5,776 5,58 54,54 58,649 63,68
ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS 98 From Tale, we can see that, with the eception of male in the year, and, all other wage distriutions have lower differentiation than areto distriution. The systematical error occurred also in the case of male in the year, and. It follows from the empirical criterion (6) and from Tale that in all cases the differences etween the rates of the considered quantiles are negligile and therefore areto distriution can e used as the model of the distriution. The 99th percentile will e considered as a characteristic of the maimum wage. The parameters of the areto distriution are estimated using the aove descried methods. First we consider the conditions = y a = y and we chose median as the second quantile, i.e.,.99 = y.99 and.5 = y.5. We estimate the parameter using the formula (). The summary of the parameter estimates is in Tale 3. Tale The Rates of Quantiles y 75 /y 5, y 8 /y 6 and y 9 /y 8 of the Wage Distriutions in the Years - and Its Relations Year y.75 y.8 y.9 y.5 y.6 y.8 Relations etween quantile rates Total.35885.999.77456 (.).34.95897.56 (.).37994.896.43457 (.).33.768.58 (.).3585.86.6454 (.).35734.843.76 (.).3463.8795.679 (.).34653.8643.768 (.) Males.34548.3556.68936 (.).33847.96386.853 (.).3368.9456.3773 (.5).38.79734.7684 (.).3543.953.8363 (.).3586.335.946 (.).354.87669.38 (.4).33.9467.337 (.5) Females.3673.887.43 (.).8964.4537.363 (.).39.53747.9439 (.).97375.465.9556 (.).389.537.676 (.).3488.4834.347 (.).3636.783.884 (.).3749.59954.7448 (.) Net we apply the conditions = y and = y and we choose 6th and 9th decile for y and y. We use the formulas (4) and (5) to estimate the parameters. The summary of the parameter estimates is in Tale 3. arameters of the areto distriution can also e estimated using the equations = y and / = y /y. We choose the 9th and 6th decile in the rate y /y. In this case we use the relations (6) and (7) to estimate the parameters. The summary of the parameter estimates is also in Tale 3.
98 ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS Tale 3 Estimated arameters of areto Distriution for Different Choices of the Estimation Equations Total Males Females Equations used.99 = y.99,.5 = y.5.6 = y.6,.9 = y.9 y =, y = y.99.99.9.9.6.6 arameter estimates arameter estimates arameter estimates Year 44,9.3695 54,43.365843 44,9 47,7.83758 6,89.34893 47,7 47,79.67846 64,8.3445 47,79 56,369.95969 67,96.3349 56,369 56,85.99456 74,95.347455 56,85 57,36.75468 79,64.35483 57,36 66,395.9445 85,46.35345 66,395 68,88.87978 9,35.35755 68,88 46,78 48,47 48,47 57,54 57,88 58,4 7,5 7,338 37,56 43,339 44,883 5,776 5,58 54,54 58,649 63,68.3564.6584.4954.78536.67749.57739.8753.76777.3987.93539.8355.3989.9665.96.9646.33636 6,7 7,63 84,934 78,63 86,65 93,98,4 3,87 39,96 47,48 48,7 49,97 54,55 57,954 63,977 68,97.367449.36846.39464.353784.364658.373653.3776.3878.36679.38749.97.8755.9744.99456.39955.3456 46,78 48,47 48,47 57,54 57,88 58,4 7,5 7,338 37,56 43,339 44,883 5,776 5,58 54,54 58,649 63,68.365843.34893.3445.3349.347455.35483.35345.35755.367449.36846.39464.353784.364658.373653.3776.3878.36679.38749.97.8755.9744.99456.39955.3456 In the end we also estimate the parameters of the areto distriution using the least squares method. We use the relations (3) and (3). In this method, we choose 5th, 6th, 7th, 8th and 9th deciles of the oserved wage distriution, i.e., k = 5. arameters estimated using the least squares method are summarized in Tale 4. Tale 4 arameters Estimated Using the Least Squares Method Year arameter estimates Total Males Females 56.56 64.6 67.9 69.3 76.3 8.7 88. 94.849.3799.358469.3534.34465.356935.3666.36359.366387 63.774 73.77 85.8 8.3 88.5 95.5 3.45 4.3.3799.3785.3967.36986.37535.38.38383.3993 4.5 49.88 5.5 5.763 57.43 6.97 67.57 7.463.3447.368.3987.33849.386.35.35878.33659 The values of the sum of asolute differences of oserved and theoretical asolute frequencies of all
ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS 983 intervals calculated for all cases considered wage distriutions are in Tale 5. In the case of the theoretical frequencies at first we determined theoretical proailities using the formula (8). From these, we determined theoretical asolute frequencies. Tale 5 Sums of the Asolute Differences of the Oserved and Theoretical Frequencies Year Total Males Females.99 = y.99.5 = y.5 37,459 5,358 73,388 3,65 67,946 57,94 68,74 8,396,63 33,576 47,99 6,4 8,55 96,789 4,965 35,96 4,56 3,697 37,5 45,49 5,793 58,4 38,4 4,955.6 = y.6.9 = y.9 3,55 7,37 36,5 64,4 69,93 68,849 6,786 53,373,89 9,7 3,576 3,457 35,349 37,737 38,678 33,953 3,96 6,76 3,9 4,46 4,65 4,37 8,854 3,5 Conclusions Equations used.99 = y.99 85,795 7,44 4,535 49,348 353,66 46,44 3,437 37,7 56,9,796 96,863 78,858,76 5,764,43 73,6 3,687 4,48 4,37 45,46 5,493 74,3 5,58 55,7 y.9.9 = Least squares method.6 y.6 3,859 3,658 39,7 66,49 68,679 69,4 6,4 57,5 9,959,98 3,747 33,76 36,3 37,653 39,48 35,89,7 8,595 3, 4,957 4,449 4,8 7,33 3,4 The appropriateness of particular modifications of the areto distriution can e evaluated comparing the theoretic and empirical frequencies. It is possile to compare oth the asolute and relative differences etween the theoretic and oserved empirical distriutions. In this article we used the asolute differences. The values sums these differences are in Tale 5. The values seem to e relatively high. The question of appropriateness of a given theoretic wage distriution in the case of large samples was descried in statistical literature (Bílková, ). Some more general conclusions can e made from the values of the asolute differences of oserved and theoretic distriutions. With the eception of the wage distriution of women in, the worst results are achieved using the equation.99 = y.99 and setting the ratio of other two quantiles of the areto distriution.9 /.6 equal to the ratio y.9 /y.6 of the corresponding empirical quantiles. This fact is less ovious for female distriution and most ovious for total distriution. This is also due to the larger sample size of the total sample (in comparison with the
984 ALICATION OF ARETO DISTRIBUTION IN WAGE MODELS sample size of the su-groups of male and female). Again with the eception of the wage distriution of women in the second worst model is the estimate ased on the equations.99 = y.99 and.5 = y.5. This fact is again less ovious for female distriution and most ovious for total distriution. In the case of the wage distriution of women in, the worst estimate is ased on the equations.99 = y.99 and.5 = y.5. In the case of the total group is the third worst (second est) method the least squares method (with the eception of ). The est results are achieved with the method ased on the equations.6 = y.6 and.9 = y.9. In the case of the total wage distriution in is the third worst method ased on the equations.6 = y.6 and.9 = y.9 and the est method is the least squares method. In the case of the wage distriution of male (with the eception of the years and ), the third worst (second est) results are again achieved using the least squares method. The est results are achieved with the method ased on the equations.6 = y.6 and.9 = y.9. In the years and (set of men) is the third worst method the method ased on the equations.6 = y.6 and.9 = y.9 and the est is the least squares method. In the case of the female group (with the eception of the years, and ) is the third worst (second est) method ased on the equations.6 = y.6 and.9 = y.9 and the most precise results are achieved with the least squares method. In the years,, and was for the group of women the most precise the least squares method. The very est method for the group of male in was the least squares method. In this case other methods had much higher values of the aove mentioned sum of asolute differences. From the aove descried comparison, it is ovious that the simplest parameter estimating methods can e in the case of the areto distriution competing with more advanced methods. References Bartošová, J. (). Logarithmic-normal model of income distriution in the Czech Repulic. Austrian Journal of Statistics, 35(3), 5-. Bartošová, J., & Bína, V. (9). Modelling of income distriution of Czech households in years 996-. Acta Oeconomica ragensia, 7(4), 3-8. Bílková, D. (). Modeling of income distriutions using normal distriution. In th International Scientific Conference AMSE Applications of matematics and statistics in Ekonomy. oprad-tatry, Slovakia, August 9-Septemer,, CD. Dutta, J., Sefton, J. A., & Weale, M. R. (). Income distriution and income dynamics in the United Kingdom. Journal of Applied Econometrics, 6(5), 599-67. Fores, C., Evans, M., Hastings, N., & eacock, B. (). Statistical diatriutions (4th ed.) (p. ). New Jersey: John Wiley & Sons. Johnson, N. L., Kotz, S., & Balakrishnan, N. (994). Continuous univariate distriutions (nd ed.) (p. 756). New York: John Wiley & Sons. Kleier, Ch., & Kotz, S. (). Statistical size distriutions in economics and actuarial sciences (p. 33). New Jersey: John Wiley & Sons. Krishnamoorthy, K. (). Handook of statistical distriutions with applications (p. 346). Boca Rato: Chapman & Hall/CRC ress. Majumder, A., & Chakravarty, S. R. (99). Distriution of personal income: Development of a new model and its application to U. S. income data. Journal of Applied Econometrics, 5(), 89-96. McDonald, J. B. (984). Some generalized functions for the size distriution of income. Econometrica, 5(3), 647-665. McDonald, J. B., & Butler, R. J. (987). Some generalized miture distriutions with an application to unemployment duration. The Review of Economics and Statistics, 69(), 3-4. McDonald, J. B., & Snooks, G. D. (985). The determinants of manorial income in domesday England: Evidence from Esse. The Journal of Economic History, 45(3), 54-556.