ANALYSES Progressvty of Taxes, Skeweness of Income Dstrbuton and Volatons of the Progressve Prncple n Income Tax Systems Edyta Mazurek Wrocław Unversty of Economcs, Wrocław, Poland Abstract Kakwan and Lambert state the three axoms, whch should be respected by an equtable tax system. They also proposed a measurement system to evaluate the volatons of the axoms. One of the axoms, axom 2, formulates the progresson prncple n tax systems. Vernzz and Pellegrno mproved the alternatve ndex to evaluate volatons concernng the progressve command n a tax system. The man am of ths paper s to compare the two ndexes n order to evaluate volatons of progressve prncple n tax system usng the real data. We also check how the progressvty of taxes and skewness of dstrbuton affect the measurement of the progressve prncple volaton. Keywords JEL code Personal tax, progressve prncple, redstrbutve effect, progressve of taxes, equtable tax system C8, H23, H24 INTRODUCTION Many authors defne equty n taxaton by horzontal and vertcal equty [Urban, Lambert 2008]. In ths paper the equty n taxaton s defned by means of three axoms, ntroduced by Kakwan and Lambert n 998. Tax system s equtable f all axoms are satsfed. Volaton of them by a personal tax system produces negatve nfluence on the redstrbutve effect of the tax. Ths negatve nfluence provdes the means to characterze the type of nequty present n a tax system. The three general rules requrement for the personal tax system are named axoms by Kakwan and Lambert. As an axom s defned as a mathematcal statement that s accepted as beng true wthout a mathematcal proof (t s a logcal statement that s assumed to be true), we propose to name Department of Statstcs, Wrocław Unversty of Economcs, Komandorska 8/20, 53-345 Wrocław, Poland. E-mal: edyta.mazurek@ue.wroc.pl. 54
STATISTIKA 204 94 (2) these postulates as rules. Despte ther arbtrary character, tax systems that volate them ntentonally are very rare. Practcal solutons n personal tax systems are not, however, so clear. Tax deducton and exemptons commonly used tax nstruments often cause volaton of these rules. Let x, x 2,..., x n mean pre-tax of n unts, who are payng t, t 2,..., t n n tax. We can wrte X as a vector of x, x 2,..., x n and T as a vector of t, t 2,..., t n. In our analyss, household s set as an unt, so: x wll denote pre-tax of household and t tax payment of household. In ths notaton y = x t denotes post-tax of household and a = t x - tax rate for household. The frst rule Rule says that tax duty should ncrease monotoncally wth respect to taxpayers ablty to pay. Ths rule they wrtten as: x x t t. () Because the nequaltes are weak, postulate of equal treatment of equals could be treated as a specal case of ths rule. It also enables government to exempt taxpayers wth the lowest s from havng to pay tax. Ths rule s named mnmal progresson prncple. Accordng to Rule 2, the rcher people must pay taxes at hgher rates. Of course, a volaton of mnmal progresson automatcally entals a volaton of ths prncple. The weak nequaltes n rule 2 mean that proportonal taxaton s permtted. Ths second rule progresson prncple s defned n the followng way: x x and t t a a. (2) If tax system s ruled out by prncples and 2 taken together then t means exstence of regresson n the tax system. The last rule Rule 3 says that a tax, whch satsfes the other two rules, should cause no rerankng n taxpayers post-tax. Ths rule s called no-rerankng crteron and can be wrtten as: x x and t t and a a x t x t. (3) The Rule 3 can be seen as a vertcal restrcton, rulng out too much progresson. VIOLATION OF THE PROGSSIVE PRINCIPLE The most mportant for ths paper s the second rule, the progresson prncple [Lambert 200]. Volatons of progresson prncple (also the others of rules) produces negatve nfluence on the redstrbutve effect of the tax. In ths context we should to be able to assess when the progresson prncple s not upheld and how much lost the redstrbutve effect produces. The redstrbutve effect s defned as dfference between the Gn ndex for pre-tax and the Gn ndex for post-tax [Lambert 200] could be decomposed nto followng way [Kakwan and Lambert 998]: = G X G X T = V S S 2 S 3 (4) where: S measures loss n redstrbutve effect, caused by a volaton of rule, S 2 loss n redstrbutve effect, caused by a volaton of rule 2, S 3 loss n redstrbutve effect, caused by a volaton of rule 3, V value of redstrbutve effect that mght be acheved f all rules are upheld. The measures n decomposton (4) are defned by Gn and concentraton ndex. 55
ANALYSES Let C Z,X denotes concentraton ndex for attrbute Z. Ths measure s calculated n the same way as Gn ndex, but vector values of Z s ordered by s before taxaton (X). If both orderngs are dentcal (attrbute Z causes no rerankng of ), Gn and concentraton ndexes calculated for the same vector of s take the same value. Whereas: and ( CT, X GX ) + ( G ) T C V T, X X X = τ, ( ) S G T C T, X S = τ, ( G C ) ( GT CT X ) 2 =τ T T, X, X X S 3 = G X T CX T, X t τ. = = n ( x t ) = n, S 2 always takes non-negatve values and accordng Kakwan-Lambert methodology: The progresson prncple s volated S 2 > 0 If S 2 s zero, the progresson prncple s upheld. Volaton of rule about mnmal progresson automatcally entals a volaton of the progressve prncple (rule 2). It means that unt pars (, ) for whch rule fals cannot provde volatons of the progressve prncple. In next secton we check how S 2 measures volaton of progresson prncple for real data. 2 EMPIRICAL ANALYSIS The values of measure S 2 as a measure of volaton of progresson prncple was analyzed on the bass of Polsh data from Wrocław-Fabryczna tax offce for fscal year 2007. Ths set of data contans nformaton on and tax pad for taxpayers that fle ther tax return n the Muncpalty of Wrocław, tax offce (dstrct dentfcaton) Fabryczna. In ths analyss households are equated wth couples of taxpayers who take advantage of ont taxaton and flled up the formulate PIT 37. The analyses were performed by author own programmes, wrtten n the R language. Populaton of 9 487 households was dvded nto subpopulatons wth respect to the number of dependent chldren. We created 4 sets: famly wthout chldren, famly wth one chld, famly wth two chldren, famly wth three or more chldren. Table presents measures S 2 for each type of famly and for pars of unts whch satsfed or not rule. When rule s volated and rule 2 s upheld, the measure of loss n due to volaton rule to should be equal 0. It s not true for S 2. 56
STATISTIKA 204 94 (2) Table Values of the measure S 2 for fourth type of famly Parameter Rule volated Rule upheld Total famly wthout chldren S 2 0.00379 0.0077 0.003096 famly wth one chld S 2 0.0023 0.000729 0.00942 famly wth two chldren S 2 0.0074 0.000443 0.0067 famly wth three or more chldren S 2 0.00582 0.0002 0.00792 Source: Own calculatons For each group of taxpayers we observe that the measures S 2 are greater than 0 for pars of unts (,) for whch rule s volated. Accordng to Kakwan and Lambert methodology t means that, progresson prncple s volated. If we look at the mathematcal record of the rule 2 (see formula (2)) t could be observed that for pars of unts (,) for whch rule s volated the rule 2 s not volated and the measure S 2 could be zero. If we want to use ths measure S 2 we should frstly elmnate from set of data the pars of unts (,) for whch rule s volated and next calculate measure S 2. Elmnaton from data the pars of unts (,) for whch rule s volated s not a smple the task. Pellegrno and Vernzz (203) ntroduced the correcton of the measure of loss n redstrbutve effect, caused by a volaton of rule 2 S * 2 whch can be used for full set of data. The measures s defned as follows: S k k * τ a a 2 = 2 2n μ A = = A A/ X T T / X [( I I ) ( I I )], (5) where: n s a sample sze, a = x t k, p, p are weghts assocated to a and a, p = n, μ A s the average of a, =, k. I, Z Z / X = I are ndcator functon for attrbute Z: I Z = z z z < z Table 2 presents values of the measure S * 2 for analyzed sets of data. p p Z / X I = Z I f f f x > x x < x x = x. Table 2 The measure S 2 for each type of famly Parameter Rule volated Rule upheld Total famly wthout chldren S * 2 0 0.0077 0.0077 famly wth one chld S * 2 0 0.000729 0.000729 famly wth two chldren S * 2 0 0.000443 0.000443 famly wth three or more chldren S * 2 0 0.0002 0.0002 Source: Own calculatons 57
ANALYSES We can observe that f S * 2 = 0, t not necessarly true that S 2. The measure S * 2 s demonstratng approprate behavors for each of analysed data sets. In every case of pars of unts (,) for whch rule s volated the value measure S * 2 s zero correctly. It proves that S * 2 could be better measure for lost of redstrbutve effect due to volaton of progressve prncple. Table 3 presents the results of decomposton of accordng formula (4) for four Polsh data sets. For famles wthout chldren, the personal tax system reduces the nequalty of by.6 percentage ponts. Losses n ths redstrbutve effect due to volaton of rule, 2 and 3 are 0.4 percentage ponts accordng to KL methodology or 0.3 percentage ponts accordng to VP methodology. The dfference appears as a result of dfference between estmaton of the loss of redstrbutve effect due to volaton of Rule 2 accordng to KL and VP. The nequty, resultng from volaton of Rule 2, reduces overall redstrbutve effect by 0.3 (accordng to KL) percentage ponts whch s 9.3 % of or 0.7 (accordng to VP) percentage ponts whch s only 0.80 % of. Table 3 decomposton for taxpayers dvded nto subpopulatons wth respect to the number of dependent chldren Kakwan and Lambert Vernzz and Pellegrno Kakwan and Lambert Vernzz and Pellegrno Kakwan and Lambert Vernzz and Pellegrno Kakwan and Lambert Vernzz and Pellegrno pre-tax post-tax famly wthout chldren Potental equty Rule Rule 2 Rule 3 Total Rules 0.37782 0.355399 0.06382 0.020589 0.00095 0.00334 0.0002 0.004206 percentage of (%): 00.00 25.68 5.80 9.3 0.74 25.68 0.37782 0.355399 0.06382 0.0995 0.00095 0.00740 0.0002 0.00282 percentage of (%): 00.00 9.9 5.92 0.80 0.75 7.46 famly wth chld pre-tax post-tax Potental equty Rule Rule 2 Rule 3 Total Rules 0.346474 0.323702 0.022772 0.025682 0.000854 0.00942 0.0005 0.0029 percentage of (%): 00.00 2.78 3.75 8.53 0.50 2.78 0.346474 0.323702 0.022772 0.02447 0.000854 0.000729 0.0005 0.00698 percentage of (%): 00.00 07.45 3.75 3.20 0.50 7.45 famly wth two chldren pre-tax post-tax Potental equty Rule Rule 2 Rule 3 Total Rules 0.346507 0.387 0.027796 0.030329 0.000797 0.0067 0.0009 0.002533 percentage of (%): 00.00 09. 2.87 5.82 0.43 9. 0.346507 0.387 0.027796 0.02955 0.000797 0.000443 0.0009 0.00359 percentage of (%): 00.00 04.89 2.87.59 0.43 4.89 pre-tax post-tax famly wth three or more chldren Potental equty Rule Rule 2 Rule 3 Total Rules 0.387007 0.35328 0.033726 0.036399 0.000782 0.00792 9.95E-05 0.002673 percentage of (%): 00.00 07.93 2.32 5.3 0.29 7.93 0.387007 0.35328 0.033726 0.03487 0.000782 0.0002 9.95E-05 0.0009 percentage of (%): 00.00 03.23 2.32 0.62 0.29 3.23 Source: Own calculatons 58
STATISTIKA 204 94 (2) It s almost twofold ncrease for KL methodology n comparson wth VP methodology. The dfference s so bg that t s worth dong an nvestgaton of why S 2 S * 2 > 0and condtons when S 2 can be a reasonable approxmaton of S * 2. The second aspect of ths problem s the fact that value of S 2 nfluences on the potental redstrbutve effect. It s mportant because potental redstrbutve effect nforms us, what s worth mentonng, how removal of nequtes due to volaton of rules could potentally mprove the redstrbutve effect of taxaton wthout ncreasng the margnal tax rates for the taxpayers groups. The total nequty n the Polsh tax system reduces the redstrbutve effect of taxaton for group of famly wthout chldren by 0.42 percentage ponts (accordng to KL) or by 0.28 percentage ponts (accordng to VP). These results suggest that the absence of all mentoned nequtes could reduce the nequalty of by 2.06 percentage ponts or by.9 percentage ponts (nstead of.64 percentage ponts). For the group of taxpayers wth one, two or three or chldren we observe smlar connecton between S 2.and S * 2. Table 4 presents dfferences between ths two values for each type of famly, whch are always greater than 0. The dfferences are from 4.22 to 8.5 ponts of percentage of for dfferent type of famles. Table 4 The dfferences between S 2 and S * 2 for each type of famly Type of famly S 2 S * 2 S 2 S * 2 S 2 S * 2 as percentage of (%) 0 chldren 0.00334 0.0074 0.00394 8.5 chld 0.00942 0.000729 0.0023 5.33 2 chldren 0.0067 0.000443 0.0074 4.22 3 or more chldren 0.00792 0.0002 0.00582 4.69 Source: Own calculatons Where do the dfferences come from? We are lookng for condtons when we can use S 2 as good approxmaton of S * 2. We can gve some thought f dfference S 2 S * 2 depends on tax progressvty or on the skweness of dstrbuton. Below table presents S 2 S * 2 and the measure of tax progressvty defned by Kakwan (977) as a dfference between the concentraton ndex of taxes and the Gn ndex of the pre-tax. Π K = D T G X. (6) Values of ths measure are ncluded n a range: Π K ϵ [ G X, G X]. Postve values, Π K > 0, mean the progressve tax system. For the proportonal system we receve: Π K = 0. The negatve values Π K < 0 are descrbng the regressve tax system. The measure Π K could be nterpreted as the percent of total fscal charges whch remaned changed from worse earnng to the better earnng for the effect of the progressons of tax system. Table 5 The name and way of create data sets for dfferent types of skwenesses name of set descrpton 0 chldren 80% contans 80% taxpayers wth the lowest from set 0 chldren 0 chldren 90% contans 90% taxpayers wth the lowest from set 0 chldren 0 chldren 95% contans 95% taxpayers wth the lowest from set 0 chldren 0 chldren 97% contans 97% taxpayers wth the lowest from set 0 chldren 0 chldren 99% contans 99% taxpayers wth the lowest from set 0 chldren 0 chldren 00% taxpayers wth 0 dependent chldren Source: Own presentaton 59
ANALYSES We can gve also some thought f the dfference S 2 S * 2 depends on skweness of dstrbuton. In order to do that we created data sets by cuttng down the orgn sets. In ths way we created the followng sets presented n Table 5. In the same way we created the sets of taxpayers wth, 2 or 3 dependent chldren. Table 6 presents results of analyss from each data sets. We calculated apart from dfferences S 2 S * 2, nfluence the dfferences on redstrbutve effect S2 S* 2 as well as the skewness of dstrbuton and progressvty ndex. Table 6 Results of analyss for created data sets Data set S 2 S * 2 S 2 S * 2 Skweness Π K 0 chldren 80% 0.002 0.00693 0.68 0.4 0.089936 0 chldren 90% 0.0009 0.00700 0.562 0.44 0.086222 0 chldren 95% 0.00 0.00842 0.32 0.74 0.097939 0 chldren 97% 0.004 0.00966 0.83 0.96 0.08245 0 chldren 99% 0.0022 0.0202 0.06.48 0.26605 0 chldren 00% 0.0039 0.0638 0.085 3.82 0.57335 chld 80% 0.000853 0.009299 0.098 0.06 0.63209 chld 90% 0.00087 0.0029 0.0790 0.35 0.7039 chld 95% 0.00090 0.03396 0.0679 0.66 0.8808 chld 97% 0.000950 0.04993 0.0634 0.90 0.98983 chld 99% 0.0004 0.08037 0.0577.44 0.2937 chld 00% 0.0022 0.022772 0.0532 3.77 0.247409 2 chldren 80% 0.00073 0.00824 0.0659 0.03 0.240979 2 chldren 90% 0.000763 0.03927 0.0548 0.38 0.252356 2 chldren 95% 0.00085 0.06688 0.0488 0.69 0.26456 2 chldren 97% 0.000856 0.08205 0.0470 0.89 0.27002 2 chldren 99% 0.000960 0.02927 0.0438.46 0.289748 2 chldren 00% 0.0074 0.027796 0.0422 8.23 0.3756 3 chldren 80% 0.000580 0.008092 0.076 0.2 0.36255 3 chldren 90% 0.000738 0.02809 0.0576 0.6 0.37738 3 chldren 95% 0.000839 0.06909 0.0496 0.9 0.383747 3 chldren 97% 0.000924 0.09023 0.0486.23 0.384923 3 chldren 99% 0.0022 0.024289 0.0462.89 0.399589 3 chldren 00% 0.00582 0.033726 0.0469 6.64 0.49609 Source: Own calculatons We can observe that for the lowest tax progressvty we have the bggest value of dfference S 2 S * 2 for each group. Consstently for the hghest tax progressvty we have the smallest value of dfference S 2 S * 2. Generally, the hgher the skewness s the hgher the dfference between S 2 and S * 2. Only for 0 chldren 90% set we have lowest dfference S 2 S * 2 and hghest skweness n compare wth 0 chldren 80%. On the other sde f we are lookng for condtons when we can use S 2 as good approxmaton of S * 2 we should analyze nfluence the S 2 S * 2 on redstrbuton effect. 4 th column n Table 6 presents ths nfluence. We observe that the hgher the skewness s, the lower the nfluence of the S 2 S * 2 dfference on redstrbuton effect. Ths relaton we observe also for 0 chldren 90% set. 60
STATISTIKA 204 94 (2) CONCLUSIONS We presented and compared two measures of volatons of progressvty prncple: S 2 and S * 2. We carred out an nvestgaton for dfferent dstrbuton and one tax system. We tred to understand the dfference between these ndexes and condtons when S 2 could be a reasonable approxmaton of S * 2. If we want to only check f progresson prncple s upheld or not we can use both methods: orgnal Kakwan and Lambert or modfed by Vernzz and Pellegrno. If we want to assess the loss n the redstrbutve effect, caused by a volaton of progresson prncple we should use recast ndex S * 2. We observe that the hgher the skewness of dstrbuton s, the lower nfluence dfference S 2 S * 2 on the redstrbutve effect. We observe smlar smple correlaton between the tax progressvty ndex and nfluence dfference on the redstrbuton effect. References KAKWANI, N. C. Measurement of tax progressvty: an nternatonal comparson. Economc Journal of Poltcal Economy, 977, 87, pp. 7 80. KAKWANI, N. C., LAMBERT, P. J. On measurng nequty n taxaton: A new approach. European Journal of Poltcal Economy, 998, Vol. 43, pp. 369 380. LAMBERT, P. J. The Dstrbuton and Redstrbuton of Income. Manchester and New York: Manchester Unversty Press, 200. LAMBERT, P. J., ARONSON, J. R. Decomposton Analyss and the Gn Coeffcent Revsted. The Economc Journal, 993, Vol. 03, No. 420, pp. 22 227. PELLEGRINO, S., VERNIZZI, A. On Measurng Volatons of the Progressve Prncple n Income Tax Systems. Emprcal Economcs, 203, 45, pp. 239 245. URBAN, I., LAMBERT, P. J. Redstrbuton, horzontal nequty and rerankng: how to measure them properly. Publc Fnance Revew, 2008, 36(5), pp. 563 587. 6