Income Tax Statistics Analysis: A Comparison of Microsimulation Versus Group Simulation

Transcription

1 INTERNAONAL JOURNAL OF MICROSIMULAON (009) (1) 3-48 Income Tax Statstcs Analyss: A Comparson of Mcrosmulaton Versus Group Smulaton Heko Müller 1 and Caren Suret 1 Rur-Unversty of Bocum, Faculty of Economcs, Unverstätsstr 150, Bocum, Germany, and arqus, Quanttatve Tax Researc, wwwarqusnfo; emal: HekoMueller@rur-un-bocumde Unversty of Paderborn, Faculty of Busness Admnstraton and Economcs, Warburger Str 100, Paderborn, Germany, and arqus, Quanttatve Tax Researc, wwwarqusnfo; emal: csuret@notesupbde ABSTRACT: Mcrosmulaton based on ncome tax statstcs may be useful n tax reform dscussons Unfortunately, access to approprate data s stll rater restrcted and expensve for ad-oc analyses, and ndvdual data s often even not avalable at all In ts paper we take Germany and ts data stuaton as a proxy for many countres restrctons n terms of tax data avalablty Analyzng ow muc relablty and robustness of results we lose f we employ group smulaton nstead of mcrosmulaton, we compare bot metods Investgatng effects by te group model leads to very good results Determnng te fnancal effects of modfed tax bases, te devaton from te mcrosmulaton results ncreases, especally f tax base cuts vary between taxpayers In addton, we take account of te class of taxpayers wt a negatve taxable ncome Neglectng ts class we dentfy a systematc underestmaton of te fnancal consequences of a modfed tax base wt te group model assumng a progressve If te group smulaton data s not arranged accordng to te taxable ncome, but rater accordng to te total amount of ncome, we also fnd a tendency towards ger devatons from te mcrosmulaton results Quantfyng te tax revenue effects of alternatve tax settngs te group smulaton model represents a good compromse between te desre to capture te complex realty and te acevable accuracy wen facng lmted resources and data Furtermore, for tose cases n wc group smulaton s te approprate tool, we provde a very smple metod to nterpolate a sutable ncome dstrbuton and tereby te tax dstrbuton wtn te classes Ts nterpolaton makes future estmates of tax revenues a lot easer We conclude tat, altoug mcrosmulaton n general s te superor approac, a group smulaton model remans of nterest, especally for analyses of rater old data and cross-country analyses, wen suffcently detaled data for mcro analyses s mssng Keywords: mcrosmulaton; group smulaton; tax revenue; personal ncome tax; tax statstcs I INTRODUCON Mcrosmulaton models of ncome tax systems are usually employed to analyze te fscal and dstrbutve ssues of taxaton Tese are mportant felds of researc Te results may be useful n tax reform, budget and ncome dstrbuton dscussons and terefore may contrbute substantally to solvng tese tree major economc questons As long as complete mcrodatasets are avalable mcrosmulaton s te preferable tool However, access to approprate data n a number of countres s stll rater restrcted or expensve for ad oc analyses In addton, n te case of analyses based on data from prevous assessment perods ndvdual data are often not avalable at all Even n te ndustralzed countres data collected earler tan 10 to 15 years ago s usually not mcrodata but grouped data Consequently analyses of tmes seres often ave to fall back on group data Ts s true even for ndustralzed countres were mcrosmulaton models ave become a wdespread tool for te analyss of newly collected data Hence, group smulaton models often ave to be appled for specfc countres, for cross-country analyses and for long-term tme seres analyses Aganst ts backdrop t s mportant to fnd out ow robust te results from group smulaton are and tus ow bg s te error arsng from te more aggregate group model n comparson to a mcrosmulaton model Conversely, gven te expense and effort nvolved n settng up a mcrosmulaton model, t s also mportant to consder under wat crcumstances sample-based mcrosmulaton tat uses ncomplete mcrodatasets remans superor to group smulaton After te amendment of te German Act on Fscal Statstcs n 1996 t was for te frst tme possble to consoldate te ndvdual data records from te local statstcal offces centrally and to use tem for auxlary and specal analyses (cf Zwck, 001:640, see furter Dell, 007) Now te data can be prepared more flexbly and used for mcrosmulatons for researc and polcy purposes However, because of te generally lmted access to mcrodata or for reasons of economy t s sometmes recommendable for several types of analyses of tax revenue effects to refer nstead to classfed data from ncome tax statstcs In te followng nvestgaton we take Germany and ts data stuaton as a proxy for many countres restrctons n tax data avalablty Ts analyss enables us to draw some general conclusons about ow to deal wt tese lmtatons n future researc n countres wt a gly developed tax admnstraton and tax statstcs but nsuffcently detaled and publsed tax data A vast body of lterature examnes te mpact of ncome taxaton on ncome dstrbuton and tax revenues referrng to dfferent sources of data usng eter mcro or group models For an overvew see, for example, Atknson and Bourgugnon (000)

2 MÜLLER and SURETH Income tax statstcs analyss 33 and Morrsson (000) Based on te semnal work of Orcutt (1957) te potental of mcrosmulaton as a new analytcal tool emerged Orcutt, Merz and Qunke (1986) and Ctro and Hanusek (1991) provde contrbutons of varous autors and descrbe te opportuntes and lmtatons of researc based on mcrosmulaton models for polcy support purposes Wt specfc relevance to ts paper, Cowell (1984) and Zandvakl (1994) examne mcrodata from ouseold surveys to dentfy redstrbutve effects of taxaton, wlst Merz (000) employs sampled mcrodata from te German ncome tax statstcs to analyze te redstrbutonal mpact of te German tax system Bork and Petersen (000), Wagenals (001) and Haan and Stener (005) smlarly employ mcrosmulaton to analyze German tax reform effects A more detaled overvew of te recent lterature on mcrosmulaton models relyng on German data s provded by Wagenals (004) Furter researc applyng mcrosmulaton taxbeneft models, based on mcrodata from several countres, provdes a deep nsgt nto te tax effects of varyng taxaton systems Suterland (1995) gves an overvew of statc mcrosmulaton models n fve European countres and prepares te feld for a European model Callan and Suterland (1997) explore te prospects and lmtatons of suc models referrng to a case study Tey pont out tat te level of detal nerent n a mcro model based on mcrodata allows researcers to adjust smulatons for transnatonal approaces But stll, ts callenge s very demandng as dfferences n data avalablty, qualty and defntons may ave an mpact on te results of eac country For example, Atknson (007) stresses tat n specfc cases tax data may be superor to data from ouseold surveys employng UK tax data, wlst Pudney and Suterland (1994) dscuss te relablty of mcrosmulaton results and sow tat samplng error n mcro models often can be very sgnfcant On te oter and, Zandvakl (1994) ponts out tat mcrodata s usually superor to aggregated data wt comparable varable defnton In contrast to te mcrosmulaton lterature, Kakwan (1977) focuses on te problem of measurng progressvty n taxaton and publc expendture and conducts an nter-country comparson usng group data from te offcal ncome tax statstcs Kraus (1981) employs suc data as well to nvestgate ncome nequalty Lozdes (1988) also uses group data from te offcal Greek tax statstcs to measure progressvty effects Dfferences between twelve OECD countres are dentfed by Wagstaff, van Doerslaer, van der Burg et al (1999) and Wagstaff and van Doorslaer (001) usng ouseold survey and grouped OECD data Pketty (003) glgts Frenc tax data defcts and estmates ncome nequalty n France on te bass of tax statstcs Pketty and Saez (003, 007) and Saez and Veall (007) look at US and Canadan grouped tax data Dell (007) uses group data from te German tax return statstcs, dentfes several breaks n data over tme, and stresses certan lmts of recent data on tax bases and taxes pad All of tem nvestgate te tax mpact on dstrbuton, especally on top ncomes over te twentet century Wereas several papers pont out tat usng group data lmts te relablty of ter studes n general (cf Kakwan, 1977:75, Orcutt, 198, Caldwell, 1985, McClung, 1986, Wagstaff and van Doorslaer, 001:313), tere s no analyss about te extent of naccuracy arsng from data defcences One am of our paper s to partly fll ts vod As te nstrument of mcrosmulaton cannot always be appled, due to eter lack of data or resource, t s also of nterest to consder ow muc relablty and robustness of results we lose f we use group smulaton nstead A second am of our paper, terefore, s to compare te outcomes of bot metods We apply mcrosmulaton to mcrodata and group smulaton to classfed data, drawn from te same underlyng dataset Te results allow us to draw conclusons about te opportuntes and lmtatons of group smulaton compared to tose of mcrosmulaton models Specfcally, we are able to sow under wc crcumstances mcrosmulaton s undoubtedly te superor approac and wen group models provde reasonable estmates Furtermore, we are able to sow tat n tose cases n wc group smulaton s an approprate tool, a very smple metod to nterpolate a sutable ncome dstrbuton and tereby te tax dstrbuton wtn te classes can be appled Ts result makes future estmates of tax revenues a lot easer Te remander of ts paper begns wt an ntroducton to te tax statstcs of te German Federal Statstcal Offce n secton II In secton III we descrbe te man caracterstcs, advantages and lmtatons of mcro and group smulaton models We present our model n secton IV and te smulaton results n secton V On ts bass we summarze and draw fnal conclusons on te applcablty, relablty and robustness of results obtaned from te alternatve metods n secton VI II TAX STASCS OF THE FEDERAL STASCAL OFFICE One am of ts paper s to compare mcrosmulaton models wt group models and dentfy settngs for wc one or te oter s preferable In contrast to group models mcrosmulaton s gly demandng of bot data and uman resource nputs In tese crcumstances, an alternatve modellng tool tat requres less detaled data and uman resource nput mgt well be more approprate Settng resource nputs to one sde, te problem of data (non-)avalablty can be llustrated f we take a look at te example of te German Federal Income Tax Statstcs Income statstcs are secondary statstcs, e te tax autortes provde summary tax statstcs

3 MÜLLER and SURETH Income tax statstcs analyss 34 based upon data collected durng te tax assessment procedure Tese data are not collected troug questonnares but extracted from personal tax assessments recorded by te fscal admnstraton for statstcal reasons Te ncome tax statstcs, owever, are only assembled every tree years by te German Federal Statstcal Offce, wt a tme-lag of at least of four or fve years A multtude of data from wage tax cards, tax returns and from offcal tax assessment notes are documented n te tax statstcs Marred couples tat are jontly assessed are regarded legally as one tax payer (cf Wagstaff and van Doorslaer, 001:307 on te problems of a tax unt referrng to an ndvdual or a couple) Te 1995 tax statstcs contan approxmately 30 mllon data records coverng 38 mllon persons, wt around 400 attrbutes per record (Zwck, 001:641) Besdes tecncal and socoeconomc nformaton tese attrbutes nclude te data necessary to determne te ndvdual tax base and people's personal tax labltes Te German Federal Statstcal Offce publses part of tese data n tables tat provde grouped nformaton only In tese tables, group-specfc nformaton s gven for sets of taxpayers wtn ntervals of a gven ncome defnton, for example, for classes of total ncome or classes of taxable ncome Researcers do not ave access to te complete mcrodatasets For te years before 199 mcrodata s not avalable at all for researc purposes Hence, analyses on earler years ave to fall back on grouped data In contrast, for te years 1998 and 001 te German Federal Statstcal Offce as provded researcers wt access to scentfc usefles Tese fles contan a stratfed sample of te complete mcrodata base compled for mcrosmulaton purposes In lne wt te offcal calculaton procedure for tax assessment all adjustments to ncome declared taxable, suc as allowances for specal expenses and expenses for extraordnary fnancal burdens, are consdered as legtmate Begnnng wt te ncome from dfferent sources of taxable ncome tese adjustments are conducted and fnally produce te tax base, e te taxable ncome In addton to te tax base te tax lablty s documented n te ncome tax statstcs Applyng te to te taxable ncome leads to te tax scale ncome tax Ten, tax credts, tax prepayments (for example, by wage tax and source taxes), tax refunds and so on ave to be taken nto account to arrve at te assessed tax lablty Te tables publsed by te German Federal Statstcal Offce dstngus between classes of "total ncome" and classes of "taxable ncome" Te total amount of ncome s a knd of prelmnary tax base, e a tax base before specal ndvdual expenses and expenses for extraordnary fnancal burdens Te tables contan te aggregated value of te underlyng attrbutes from te taxpayers mcrodata, for all tax payers or for certan selected groups of taxpayers, suc as tose subject to te basc or splttng Wereas te base ncome s appled to ndvdual taxpayers, marred couples are subject to te splttng tax scale To determne te ncome tax of a couple, te ncomes of bot spouses s summed and ten alved Ts alved ncome s subject to te basc ncome Te resultng ncome tax as to be doubled to calculate te couple's ncome tax Ts procedure s called applyng te splttng tax scale Tus te publsed tables provde group-specfc nformaton about te tax base and te assessed tax Te tables used for group smulaton only provde mean values for eac attrbute and class Wen te underlyng mcrodata ave not been released, tese grouped data ave to be used nstead Overall a substantal nformaton loss arses from aggregatng data n eac tax class n comparson to te correspondng ndvdual tax mcrodata Te remander of ts paper nvestgates weter ts nformaton loss leads to g or rater neglgble smulaton dfferences III MICRO VS GROUP SIMULAON Referrng to te most mportant dstnctve feature - te degree of aggregaton of te appled data - n economcs and te socal scences - we fnd tree basc types of smulaton model: Models tat are essentally based on te aggregates from te natonal accountng system, lke macroeconomc models and general equlbrum models (g aggregaton level), Group models tat refer to selected attrbutes of omogeneous groups of economc unts (medum aggregaton level), and Mcroanalytc models tat focus on ndvdual mcro unts (strong dsaggregaton) Macroeconomc models and equlbrum models are not generally sutable for analyzng ncome tax revenue In general equlbrum models a normally complex formula ncludng a macroeconomc growt rate and oter macro parameters are used to estmate te effects of monetary and fscal polcy on prces or employment or oter macroeconomc varables If suc models are employed to estmate tax revenue effects relatvely g predcton errors occur n comparson to more detaled approaces (group or mcro models), due to te ger degree of aggregaton Attrbutes of te ouseolds, taxpayers and structural factors are nsuffcently consdered bot n te model and n te results In comparson, te more ntensvely dsaggregated group models and mcroanalytc models offer structural advantages Generally, group models ave a relatvely smple and transparent structure compared wt te mcroanalytc models Ts facltates ter mplementaton and modfcaton and makes tem a flexble and low cost nstrument for nvestgatng revenue effects Ts advantage

4 MÜLLER and SURETH Income tax statstcs analyss 35 as to be offset aganst te prevously mentoned nformaton loss caused by usng data aggregated wt respect to a specfc attrbute Hence te feld of applcaton of group models s restrcted by te underlyng aggregaton pattern If mcrodata are not avalable, te ensung analytcal lmtatons of group models ave to be accepted Te queston s weter te adverse mpact of tese lmtatons, n terms of analytcal outcomes, s acceptable In contrast, f sutable data are avalable te ger degree of dsaggregaton tat can be aceved usng mcroanalytc smulaton models s superfcally desrable and necessary, weter analyzng te dstrbutve effects of varous tax and transfer systems or undertakng beavoural smulatons Mcroeconomc models take explct account of taxpayers' ndvdual attrbutes and ence allow us to determne te tax base and tax lablty more precsely It s terefore teoretcally possble to make a more accurate and dfferentated assessment of te revenue effects of, for example, a tax reform In a (pure) mcroanalytc smulaton eac ndvdual mcro unt wt ts attrbutes s referred to drectly Ts can be realsed on te bass of ndvdual cases, a sample or te parent populaton Te advantage of compreensve and detaled structural nformaton can only be exploted f an approprate multplcty of attrbutes of te mcro unts s avalable n te database In order to aceve a smulaton as close to realty as possble nterdependences of tax reform and ndvdual beavour ave to be taken nto account Tus, we ave to refer to te relevant elastctes, utlty functons and so on n te model on eter an emprcal or teoretcal bass Ts ncreases te complexty of te model as well as te number of attrbutes Even f te mcroanalytc models are teoretcally superor to te group models, te requred specfcaton and format of te data and te necessty to update t often lmt or even prevent te applcaton of mcrosmulatons, partcularly wen dealng wt long-term tme seres and crosscountry analyses In partcular for ad oc analyses or analyses of earler tax perods we may ave to fall back to te publsed aggregated data as no oter detaled data s avalable In tese cases only group smulaton models can be employed In any case, mcro models are often de facto group models, as data lmtatons sometmes necesstate te assumpton tat all ndvduals n te same group sare te same attrbute or dstrbuton 1 As a result scenaros can be dentfed for wc group model results ardly dffer from tose of mcrodata analyses A dsadvantage of group tax smulatons s tat tey tend to lead to tax revenues tat are too small Ts s because progressve ncome taxaton s usually not smulated correctly, a result of referrng to aggregate ncome per ncome class and aggregate ncome tax per class nstead of exact ndvdual ncome In te case of mcrosmulaton an emprcal ncome dstrbuton s nerent n te underlyng mcrodatasets In contrast, for group smulaton purposes an emprcal frequency dstrbuton as to be formally estmated from te avalable aggregated data by applyng specfc dstrbuton functons Under tese crcumstances group smulaton potentally becomes an attractve and powerful nstrument and alternatve to mcrosmulaton models Ts estmaton of unknown emprcal dstrbutons can be aceved n prncple by two metodologcal approaces: Applyng analytc dstrbuton functons wose parameters are derved from emprcal materal by approxmaton, or Applyng nterpolaton functons In comparson to mcro models one major drawback of group smulaton models relyng on an analytcal dstrbuton functon s tat te matematcal approxmaton of te analytc dstrbuton functons to te unknown emprcal dstrbuton s very tmeconsumng and complex Furtermore, tere are often substantal devatons, n partcular n te upper and lower ncome classes It sould also be noted tat te advantage of usng an analytc dstrbuton functon s often lmted by te lack of a usable economc nterpretaton of te functon parameters If no acceptable matematcal approxmaton can be aceved we ave to abstan from a teoretcal approac to emprcal ncome dstrbuton and conduct an nterpolaton nstead In te next secton of te paper we descrbe te constructon of a group tax smulaton model and, as part of ts descrpton, put forward one possble approac to approxmatng te emprcal ncome dstrbuton IV THE MODEL In te followng, we ntroduce a dscrete ncome tax smulaton model based on classfed data from German Fscal Statstcs Te am of ts group model s to dentfy te revenue effects of alternatve tax rules or systems, partcularly te fscal consequences of specfc tax regulatons, rapdly and flexbly Te group model s based upon avalable aggregate data from te ncome tax statstcs After presentng te group model we compare te results of mcro and group smulaton calculatons n order to assess te relatve accuracy of te group and mcrosmulaton models and ence fnd out under wc crcumstances group smulaton s or s not an approprate approac, and n partcular under wat crcumstances mcrosmulaton models cannot be substtuted by group models n an acceptable way IV1 Dscrete ncome dstrbuton Tax revenue analyses can normally be conducted wtout an analytc ncome dstrbuton Analytcal teory-based ncome dstrbutons only approxmate real world dstrbutons Wdely used analytc approxmatons nclude te log-normal (eg, Berglas, 1971:534) and Pareto dstrbutons (eg, Pketty and Saez, 003:6; Saez and Veall, 007:

5 MÜLLER and SURETH Income tax statstcs analyss 36 30) It s preferable, we argue, to deduce te ncome dstrbuton drectly from te avalable data As data on te number of tax payers for eac specfc unt of taxable ncome s unavalable we need to nterpolate We derve te results presented n te followng by applyng a group smulaton model and determnng te dstrbuton of ncome by means of a lnear nterpolaton of te group smulaton An artmetcal seres (e a dscrete functon), rater tan a contnuous functon, s cosen to approxmate te dstrbuton of ncome Generatng dscrete ncome dstrbuton functons s approprate for tax revenue analyss snce te doman of te ncome functon contans only natural numbers and tus dscrete arguments 3 Durng te nterpolaton, te aggregate taxable ncome of all taxpayers n eac tax class s also consdered Te dscrete model presented for smulatng personal ncome taxaton based upon grouped data ensures tat n eac class aggregated taxable ncomes and numbers of taxpayers are dentcal to te orgnal mcrodata totals Terefore, a degree of precson n dsaggregaton can be aceved tat leads n eac class to a 100% correct agreement between te aggregated taxable ncomes and te amounts ndcated n te tax statstcs Te absolute frequency of taxpayers wt a specfc taxable ncome s () and yelds from te closed ncome nterval wt te nterval bounds [a,b ], wt a +1 =b +1 of te dscrete densty functon of te taxpayers: b (11) = ( ) = a Only te gest ncome nterval as an open upper bound wt b n = Ts set of numbers (eq 11) s a unque transformaton of a set of natural numbers (taxable ncome) on a set of ntegers (absolute frequency of te taxpayers) Te sum of te taxable ncome of te taxpayers n te nterval s and can be determned as follows from te densty functon: b (1) = ( ) = a Applyng te ncome to te tax base, neglectng prelmnary specal regulatons, we receve ncome tax t 4 () Te sum of te determned ncome tax of all taxpayers of te nterval s T and s gven by: b (13) T = ( ) t( ) = a IV Taxable ncome class As already descrbed, te publsed tables from te ncome tax statstcs separated nto taxpayers underlyng te basc scale and taxpayers underlyng te splttng nclude aggregate data for a varety of tax relevant facts An example s provded n Table 1 For eac band of taxable ncome te number of taxpayers and relevant sum n DM s dsplayed For te purposes of tax revenue analyss t s approprate to run a group smulaton usng data grouped wt respect to classes of taxable ncome, snce te range of values for te tax base of te taxpayers n eac class s explctly gven and, tus, te nterpolaton of te dstrbuton of te taxpayers s lmted to ts nterval We use nformaton relatng to te number of taxpayers wt a taxable ncome, te sum of te taxable ncome of tese taxpayers and te sum of assessed ncome tax from te ncome tax statstcs Ts database can formally be descrbed for taxpayers subject to te basc or splttng as follows: Gven are classes of "taxable ncome" for =1 to n classes wt te class lmts [a, b ], were a 1 =-, b 1 =0, a =1 and b n = For every class we know: te class frequency (number of taxpayers of te class for wom a taxable ncome as been assessed), te sum of te taxable ncome of te taxpayers of class, and te sum of te assessed ncome tax AT of te taxpayers of te class Te assessed ncome tax AT of all taxpayers results from te applcaton of all relevant tax rate regulatons, tax reductons and tax base addtons wtout mputable taxes Unfortunately, te ncome tax statstcs do not nclude te ncome tax but te sum of te assessed ncome tax of eac class In contrast to te assessed ncome tax te ncome tax results from te assessment process at a stage before specal regulatons, tax reductons and tax base addtons are consdered Furtermore, te absolute frequency of te taxpayers wt a specfc taxable ncome, (), te sum of tese taxable ncomes, (), as well as te correspondng ncome tax from (), t (), cannot be found n te aggregate data of te ncome tax statstcs Only te average taxable ncome of eac class,, can be determned by dvdng te sum of te taxable ncomes and te number of taxpayers of te class Furter nformaton tat may be elpful to analyze te dstrbuton of te taxpayers wtn te class s not avalable Snce te total assessed tax, T, s te result of assessment after consderng all ndvdual relevant tax regulatons no addtonal nformaton about te dstrbuton of te taxpayers can be ganed by referrng to sums of assessed ncome tax n te respectve ncome classes (AT ) publsed n te ncome tax statstcs 5 Even f we

6 MÜLLER and SURETH Income tax statstcs analyss 37 Table 1 Example of grouped data provded by te German Statstcal Offce, postve and negatve ncome from dfferent sources and assessed ncome tax (1995 ncome dstrbuton, basc ) Amount assessed negatve ncome postve ncome Assessed usng from dfferent sources from dfferent sources ncome tax 1990 tax taxable ncome scale from to under DM number of taxpayers sum n DM '000 Source: German Statstcal Offce, Wesbaden number of taxpayers sum n DM '000 number of taxpayers sum n DM '000 sum n DM '000 Under 1 66,105-1,415,50 61,453 9,16,448 8, ,670 70, ,317 1,7,1 1,318, ,060 54,34-5,670-5,670 8,154 9,30-315, ,39 9,407,0 375, , ,739 8,154-1,096 4, ,175 1,133,053 18,737, ,91 59,00 965,99 1,096-1,366,634-3,13 65,535 1,7,609 64,97 59,071 83,707 1,366-13,068 6,541-7,11 159,406 3,180, , ,071 19,6 13,068-18,036 48,1-574, ,65,60, ,305 1,795,389 1,837,818 18,036-5,00 77, ,391 1,17,176 36,818,996 1,15,761 4,031,413 4,018,86 5,00-30,03 61, ,561 99,610 33,453,11 99,514 4,411,434 4,408,774 30,03-40,013 13,580-1,389,631,146,19 91,73,03,146,147 14,380,689 14,380,66 40,013-50, ,496-1,334,815 1,659,698 86,684,169 1,659,673 15,650,731 15,660,133 50,004-55,78 53,055-61,01 599,575 36,517, ,574 7,160,71 7,173,713 55,78-58,644 1,470-64,00 31,363 15,15,855 31,360 3,100,941 3,108,599 58,644-60,048 9,7-135,733 98,916 6,747,897 98,916 1,399,886 1,403,31 60,048-66,366 38, ,366 35,999 5,53,735 35,999 5,449,135 5,465,481 66,366-70,038 18,540-8,451 15,869 11,943,497 15,866,644,97,65,440 70,038-75,006 1, ,06 158,898 13,170, ,898 3,001,504 3,013,301 75, ,008 61,81-1,177, ,395 35,597, ,365 8,730,816 8,784, ,008-10,04 0, ,376 94,303 11,909,994 94,301 3,63,494 3,99,181 10,04-40,084 34,145-1,370,69 107,606 19,967, ,577 6,68,57 6,458,174 40, ,168 9,64-669,580 3,53 8,93,469 3,518 3,94,57 3,488, ,168-1,000,06 3,404-40,661 7,793 5,90,376 7,787,337,854,549,395 1,000,06 or more, ,475 4,30 15,3,935 4,99 6,096,409 7,6,559 total 1,146,16-6,018,587 13,545, ,1,441 10,485,77 93,967,818 96,379,068 assume dentcal tax bases for every taxpayer of an ncome class dfferent ncome tax assessments may arse, as specfc tax regulatons may lead to dfferent reductons and addtons A strct functonal relaton between te assessed ncome tax and te assessed tax base "taxable ncome" cannot be assumed In contrast to mcrosmulaton, we ence ave to be aware of te fact tat durng a smulaton based on classfed data te above mentoned problem for progressve ncome wll occur If we determne te ncome tax revenues referrng to average taxable ncome per ncome class by multplyng te ncome tax on te average taxable ncome, t ( ), of te class wt te number of taxpayers of te class,, te deduced tax revenue wll generally be too low Ts s due to te fact tat wtn te segment of te progressve rse of te ncome tax rate, te ncome tax on te average assessed tax base may not map te effect of te progressve structure precsely Furtermore, te effects of a transton between two zones of te tax scedule cannot be reproduced wtn a class because te average taxable ncome of te class can le only n one zone 6 Ts affects partcularly te smulaton of te revenues from reformed tax bases and reformed tax scedules wt dfferent zones In te followng, n order to reduce tese naccuraces wen determnng ncome tax revenues by means of a group smulaton based on classfed data, we develop a dscrete model for te taxpayer dstrbuton wtn a class by applyng lnear nterpolaton (cf Wagstaff and van Doorslaer, 001:307; Atknson, 007:91-9; Saez and Veall, 007:30) Te lnear nterpolaton requres te descrpton of m elements between two numbers z 1 and z wt te dfference z z 1 = d n suc a way tat a fnte artmetc seres of numbers emerges wose frst element s z 1 and wose (m+)t element s z If d denotes te dfference of te wanted artmetcal seres of numbers, ten d = 1 1 e d (m 1) z z + ( m +1)d = z + d, In a frst step we assume tat te taxpayers n te closed nterval (class) wt te nterval bounds

7 MÜLLER and SURETH Income tax statstcs analyss 38 [a,b ] are equally dstrbuted In ts case te average taxable ncome of a class s dentcal to te md-pont of class: ( a = + b ) Te sum of te taxable ncome of all taxpayers of a class s gven accordng to eq (1) by te product of te average taxable ncome and te number of taxpayers of ts class: = Te aggregated ncome tax of te class can easly be determned by eq (13) snce te absolute frequency of te taxpayers for every taxable ncome wtn te nterval s dentcal and can be descrbed by ( b ) ( b ) ( a ) ( a 1) 1 ( b a 1) However, te average taxable ncome of a class s usually not equal to te md-pont of a class, meanng tat te dstrbuton of taxpayers wtn te class s obvously not unform In suc a case, an assumpton about te dstrbuton of te taxpayers wtn te class s necessary Startng wt te unform dstrbuton a dscrete functon (artmetcal sequence of numbers) tat s strctly monotonously ncreasng or fallng as to be assumed for te dstrbuton of te taxpayers n te class Ts functon s condtoned on te poston of te average taxable ncome n te class n relaton to te md-pont of te class We presume tat te number of taxpayers n te md-pont of te class s equal to te quotent of te total number of te taxpayers of ts class and te class breadt, e a b ( b a In ts way, te problem s reduced to redstrbutng a certan number of taxpayers between te lower and upper class alves so tat te sum of te ncome of te class corresponds to te emprcal value Ts redstrbuton s standardzed suc tat te number of taxpayers at te begnnng and end of te class dffer exactly by two taxpayers, e ( a ) ( b ) Tus, te dfference between te number of taxpayers n te md-pont of te class and te number of taxpayers at te class begnnng or te class end s exactly: 1) 1; ( a ) ( b ) a b a b n oter words, one taxpayer 7 Wtn te class te number of taxpayers rses and falls wt /(b a ) wenever te underlyng taxable ncome s amended by one DM 8 Ts standardzaton ensures te requred strct monotony Te degree of redstrbuton wtn a class u, can now be determned by referrng to te emprcal taxable ncome of te class: a b b a a b a (14) u b a Te number of taxpayers wt a specfc taxable ncome under te gven set of assumptons s: (15) ( ) u b a 1 b a Consderng u and te number of taxpayers, (), and tereby () and () t () can be estmated for every taxable ncome Insertng te frequences of te taxpayers from eqn (15) nto te eqns (1) and (13) we fnd for every class tat te sum of taxable ncome equals exactly te emprcal value from te ncome tax statstcs Ts s true snce () s determned va Furtermore, te total ncome tax T of ts class can be estmated Proceedng lke ts wen determnng te aggregate ncome tax of a class we succeed n reducng te systematc underestmaton n group models fundamentally If te aggregate tax of a class s determned by multplyng te ncome tax on te average taxable ncome of te class wt te number of taxpayers of te class under a progressve tax, we receve te mnmum level of te possble total tax of te class If we nstead employ a strctly monotonous dscrete functon tat s defned on te bass of te emprcally determned number of taxpayers and te sum of te taxable ncome of te class, ten te total tax of a class vares between te teoretcal mnmum and maxmum possble total tax of ts class V COMPARING TAX REVENUES EFFECTS OF MICROSIMULAON AND GROUP SIMULAON MODELS V1 Tax scale smulaton based on taxable ncome Ts type of group smulaton allows us to obtan qute exact results nvolvng relatvely low effort, partcularly wen smulatng dfferent s Te qualty of ts smulaton approac can be empaszed n te followng by comparng te results of a mcrosmulaton, carred out by te a b a

8 MÜLLER and SURETH Income tax statstcs analyss 39 German Statstcal Offce, wt tose of te dscrete group smulaton model ntroduced ere Te smulatons of te German Statstcal Offce consulted for comparson purposes were carred out on te base of ndvdual datasets from a 10% sample of te 1995 ncome tax statstcs Te 10% sample s a formally anonymzsed sample taken from te entrety of te recorded ncome tax assessments of te 1995 assessment perod n te ncome tax statstcs Ts sample s a stratfed random sample provded by te German Statstcal Offce In te followng, te smulaton of tax patterns s stylzed, e algned wt te man caracterstcs of te tax code Tus, specfc regulatons, suc as German tax relef for commercal earnngs applcable only n 1995, ave been neglected Te ntal values of te sample and te results of te sample from te smulaton were extrapolated to te parent populaton by te German Statstcal Offce On bass of te aggregated data of te extrapolated ntal values for te number of taxpayers and te aggregated taxable ncomes of te classes, we run smulatons usng te dscrete group model Snce te German Statstcal Offce defnes te lowest ncome class as avng no lower and te upper as avng no upper lmt, tese class borders for group smulaton purposes are eurstcally determned Terefore, assumng a unform dstrbuton, te average taxable ncome of te class s equated wt te md-pont of te class: a b Tus, te upper lmt of te nterval s equvalent to twce te md-pont of te class, e = b Ts also apples to te lower class lmt of te frst class, e = a, because ts class contans all taxpayers wt a taxable ncome of less tan one DM and terefore, te taxable ncome may even be negatve n ts class 9 Te results presented n Table sow tat te dfferences between te results from usng our group smulaton model and te results from te smulaton conducted by te German Statstcal Offce based upon ters sample mcrodata, bot applyng te basc tax rate and te 1990 and 1996 ncome s, are very small Ts result s robust even f we analyze te splttng nstead Te observable devatons, as expected, are muc lower tan te teoretcally derved relatve underestmaton of te tax lablty f we refer to te md-pont of te class It s remarkable tat te g qualty of te group smulaton results arse wen comparng not only te total tax revenues but also n almost every sngle class Te sometmes substantal devatons found by oter models n te lower and upper ncome classes (cf, eg, Pketty and Saez, 003:55, concernng te eterogenety n te top ncome decle) are consderably reduced wen we employ our dscrete group smulaton model Moreover, te qualty of te results of te dscrete group smulaton model s not dependent on te class lmts cosen by te German Statstcal Offce Even for smulatons wt s wose basc tax-exempt amount does not correspond to te class lmts set by te German Statstcal Offce, dfferences of smlar structure and dmenson occur, e agan very small devatons V Tax base deductons smulaton based on taxable ncome It s desrable to fnd out weter te degree of precson of our group model obtaned for smulatons (Secton V1) s acevable for te smulaton of tax revenue effects caused by reforms of fxed (flat) amount tax base deductons as well Unfortunately, no mcrosmulaton was carred out by te German Statstcal Offce for ts scenaro, so comparson wt our group smulaton results s not possble Instead, n te followng we focus on te problem of tax deductons from te tax assessment base (cf O Donogue and Suterland, 1999: ) In order to measure te fscal mpact of tese deductons, ter tax revenue effects are determned by consderng a correspondng ncrease n te tax base wtn te smulaton Our conclusons can n prncple be transferred to tax regulatons tat lead to an ncrease of te tax base and ter tax revenue effects by smulatng an adequate tax base reducton However, n case of suc smulatons te dfferences between mcro and group analyses may ncrease f te underlyng fxed amount s not deductble by all taxpayers and, furter, f te (relatve) dstrbuton of te taxable ncome of te taxpayers wo enjoy ts deducton does not correspond to te (relatve) dstrbuton of te taxable ncome of all taxpayers In order to mprove te qualty of te results of our group model, nformaton about te dstrbuton of te taxpayers enjoyng ts fxed tax prvlege, as far as ts nformaton s avalable, sould be consdered explctly n te smulaton From te publsed ncome tax statstcs, as outlned already, te number of taxpayers and te total amount of fxed amount tax base deductons n tousands of deutscmarks per class s gven Terefore we ave nformaton about te dstrbuton among dfferent ncome classes, but not about te dstrbuton of tese amounts among te taxpayers wtn te classes If te tables n te ncome tax statstcs do not provde data on te taxable ncome of te taxpayers wo beneft from ts deducton, ten for group smulaton purposes we ave to fall back on te sum of te taxable ncomes of all taxpayers and ence, on te dstrbuton of all taxpayers n ts class derved from te group smulaton Ts may nvolve a larger, and possbly unacceptable, devaton from te results of a mcrosmulaton

9 MÜLLER and SURETH Income tax statstcs analyss 40 Table Tax scale based mcro and group smulaton of tax revenue for te basc (1995 ncome dstrbuton) tax scale class no taxable ncome (DM) ncome tax n (DM ' 000) mcrosmulaton (German Statstcal Offce) group smulaton relatve dfference (%) relatve dfference (%) 1 under , ,670-8, , , ,154-1, ,99 965, ,096-1,366 83,707 83, ,366-13,068 19,6 19, ,068-18,036 1,837,818 1,837, ,036-5,00 4,018,86 4,018, ,00-30,03 4,408,774 4,408, ,03-40,013 14,380,66 14,381, ,013-50,004 15,660,133 15,660, ,004-55,78 7,173,713 7,173, ,78-58,644 3,108,599 3,108, ,644-60,048 1,403,31 1,403, ,048-66,366 5,465,481 5,465, ,366-70,038,65,440,65, ,038-75,006 3,013,301 3,013, , ,008 8,784,647 8,78, ,008-10,04 3,99,181 3,98, ,04-40,084 6,458,174 6,458, , ,168 3,488,455 3,488, ,168-1,000,06,549,395,549, ,000,06 or more 7,6,559 7,6, Total (ALL BANDS) 96,379,068 96,377, Source: German Statstcal Offce, Wesbaden; own calculatons Usng te symbols defned n secton IV1 te problem can be presented formally as follows From te aggregated data of te ncome tax statstcs we know for eac class te frequency g of te exstng tax facts (number of taxpayers, wo are affected by ts fact) and te sum of ts value, G, were te average value of a class s gven by G G g In te case of a fxed tax base deducton G s constant for eac class Te fnancal consequences of ts tax rule per class arse from te dfference, ΔT, between te respectve sum of te ncome tax of te class bot ncludng te effects of te deducton (T g ) and excludng ts effect (T ): were T T T, g g g T ( ) t( + G ) = Here g () s te number of taxpayers wt a specfc wo are affected by g Furtermore, t ( + G ) denotes te ncome tax for te tax base wc s ncreased by G Te degree of precson of te smulaton s also nfluenced by weter or not we are nformed about te sum of te taxable ncomes of te taxpayers for te class wo deducted an amount ( g ) due to specal fxed tax regulatons Determnng u and g () usng te equatons (14) and (15) t s mportant weter we refer to te taxable ncome of all taxpayers () or to te taxable ncome ( g ) of tose taxpayers wo enjoy tax prvleges and tus are ncluded n g If g s known, ten g = g Oterwse u as to be determned on bass of and, for reasons of smplcty, we set g ( ) = ( ) Proceedng lke ts, an dentcal dstrbuton of te taxpayers wt a specfc taxable ncome for te respectve class s assumed for all examned tax facts Precson s furter reduced wen applyng a dscrete group smulaton model to determne tax revenue effects caused by tax base deductons tat vary between taxpayers Ts s magnable n te cases of, for example, deprecaton and loss offset allowances g

10 MÜLLER and SURETH Income tax statstcs analyss 41 Snce te actual dstrbuton of taxpayers cannot be determned from te aggregated data we need approprate assumptons on te dstrbuton of te underlyng tax deductons n eac class analogous to tose made wt respect to te dstrbuton of taxable ncome Tese assumptons are necessary even f te dstrbuton of taxpayers, te deductble amount n eac ncome classes and even te sum of te taxable ncomes of te taxpayers n queston can be taken from te tables of te German ncome tax statstcs For our analyss, agan for reasons of smplcty, we assume a unform dstrbuton so tat for every taxpayer of a gven class te average value G G g tat can be deduced from te sum of tax deductons of eac class s taken as a proxy for te ndvdual amount Use of a class-specfc average s s preferable to deductng te same fxed fxed amount (te overall average tax deducton) regardless of class Te results of te mcrosmulaton by te German Statstcal Offce on ncome tax revenue effects n case of lmted loss offset are compared wt tose of our dscrete group model n Table 3, usng te same taxable ncome classes as Table We analyze loss offset restrcton as losses could not be compensated wt postve earnngs from oter sources In lne wt te comparson n Table we apply te basc 1990 to determne te ncome tax Applyng te 1990 rater tan te 1996 allows us to test weter te degree of modellng accuracy s ndependent of te class borders cosen by te German Statstcal Offce for a partcular tax year Te relatve dvergence of te ncome tax calculated on te bass of te group smulaton and te ncome tax calculated on te bass of te mcrosmulaton s presented n Table 3 for eac ncome class as well as for all taxpayers Furtermore, we dstngus between te basc and te splttng Table 3 also ncludes relatve dfferences n smulated fnancal consequences Te fnancal consequences are based upon te sum of te ncome tax of all taxpayers wt negatve earnngs, n te case of eter a complete or lmted loss offset (cf Wagstaff and van Doorslaer, 001:307) Te relatve dfference between te fnancal Table 3 Tax base based mcro and group smulaton of tax revenue and te fnancal effects usng tables n case of vertcal loss offset restrcton (1995 ncome dstrbuton) class no Percentage dfference n te results of mcrosmulaton (German Statstcal Offce) relatve to te dscrete group model g nterpolaton by nterpolaton by basc splttng basc splttng total total wtout class fnancal effects fnancal effects wtout class Source: German Statstcal Offce, Wesbaden; own calculatons

11 MÜLLER and SURETH Income tax statstcs analyss 4 consequences of a refusal to allow vertcal loss offset s sown at te end of te table In addton, te group smulaton was carred out on te bass of two dfferently aggregated data sets Te frst group smulaton s based on tabulated data from te sample projected by te German Statstcal Offce Ts sample contans data for taxpayers wt a negatve ncome, e te sum of te taxable ncome of tese taxpayers per class s known ( g ) Ts group specfc nformaton cannot be found n te publcly avalable model results Rater, t was prepared by te German Statstcal Offce as a specal statstcal evaluaton for ts researc project only In contrast, te second group smulaton used te sum of te taxable ncome of all taxpayers of te class ( ) provded n te tabulated data to smulate te dstrbuton of te tax bases wtn te class Te relevant detals for all taxpayers are ncluded n te publsed statstcs Concentratng on te tax revenue effects of tax base deductons (wc may be dfferent for every taxpayer), a comparson of te results of Tables and 3 sows tat te devatons of group smulaton results from tose of te mcrosmulaton model are substantally greater tan tose found wen smulatng dfferent tax scales Wen nterpolatng usng te class sum of te taxable ncome of te taxpayers wt a negatve ncome, g, we fnd tat te group smulaton results are lower, for all taxable ncome classes, tan te mcrosmulaton model results (negatve relatve dfferences) Te total effect, across all taxable ncome classes, s a devaton - 19% (basc ) and -37% (splttng tax scale) Wen nterpolatng usng, te class sum of te taxable ncome of all taxpayers of te class, te group smulaton results are once agan consstently lower tan for te mcrosmulaton model, wt te notable excepton of very gest and lowest taxable ncome bands Tese apparently mnor dfferences, owever, ave a sgnfcant mpact Te overall net devaton, summed across all classes, becomes postve (rater tan negatve), wt postve devatons of 6% (basc ) and 30% (splttng tax scale) Te dfferences are largest n te lower ncome classes and decrease as te tax base ncreases Te greatest relatve dfference s observed for class 1, wc ncludes taxpayers wt a taxable ncome less tan one DM Snce ts class s not furter subdvded n te ncome tax statstcs but covers a wde range of negatve taxable ncomes, ere te group smulaton model s gly naccurate As a consequence, estmatng te number of taxpayers wt postve ncome greater tan te basc tax-exempt amount due to vertcal loss offset restrcton s rater unrelable Besdes, te results n ts class depend on te lower class boundary wc must be determned eurstcally Includng te class of te taxpayers wt a taxable ncome less tan one DM s reasonable only for mcrosmulaton of tax revenue effects f we want to analyze an ncrease n te tax base - as far as tese taxpayers are affected by t 10 Due to te lack of data, n ts case a group model can only arbtrarly lead to smlar results as a mcrosmulaton If te class of taxpayers wt a taxable ncome less tan one DM s neglected n smulaton, comparng mcro and group models leads to relatve devatons n tax revenues for all taxpayers wt a negatve ncome employng nterpolaton usng g of 19% (basc ) and -18% (splttng ) and furter, usng of -35% (basc ) and -1% (splttng ) We realze tat te tax revenue calculated by mcrosmulaton for te unmodfed tax base (Table ) does not dffer as muc as from te one determned by group smulaton as do te tax revenues assumng a modfed tax base (Table 3) (Te modfed taxable ncome s gven by te taxable ncome ncreased for example by losses tat ave not yet been offset aganst profts) Terefore, te fnancal consequences of te tax base modfcaton nvoke substantally greater relatve devatons between te mcrosmulaton and te group smulaton Te dfferences occurrng n te lower ncome classes partcularly preponderate Te relatve devatons between te mcrosmulaton and te group smulaton for te overall fnancal effects ncludng all ncome classes are -69% (basc ) and -146% (splttng ) usng g and are 147% (basc ) and 131% (splttng ) referrng to If we neglect te lowest ncome class, relatve devatons of about -75 % (basc and splttng ) arse n te context of te nterpolaton of g -77 % (basc ) and - 76% (splttng ) can be found by employng Obvously, a group smulaton excludng te naccurate values of te frst class leads n prncple to an underestmaton of te fnancal effects Ts fndng meets te expectatons snce by relyng on te average amount of tax base deductons per correspondng taxpayer we determne te lower boundary of te possble tax revenue sortfall Furtermore, Table 3 clarfes tat te results of te group smulaton tat are based on te class sum of te taxable ncome of te taxpayers wt a negatve ncome ( g ) nvolve as expected a tendency towards fewer devatons from te mcrosmulaton results tan s te case n a smulaton tat refers to te class sum of te taxable ncome of all taxpayers of te class ( ) From ts, we cannot conclude tat te structure of devaton dentfed ere wll generally be observable because te (unknown) dstrbuton of te taxpayers wtn a class n prncple may dffer by class and by te examned tax facts Ts s clarfed comparng te class specfc results n Table 3

12 MÜLLER and SURETH Income tax statstcs analyss 43 V3 Tax scale smulaton based on total amount of ncome Most of te tables provded by te German Statstcal Offce on ncome tax, n partcular tose relatng to specfc tax rules, are not arranged accordng to sze of te taxable ncome but rater to sze classes of te total amount of ncome Of course, taxable ncome would be a better group attrbute for te underlyng researc queston Terefore, and n general, t would be desrable tat te Statstcal Offce releases tables of ts type Ts would mprove group model results and tus tax effect smulatons On te oter and, te Statstcal Offce already releases tables coverng more tan 1000 dfferent attrbutes, and t s clearly not possble to produce a set of tables suffcent to satsfy all possble researc questons n advance In addton tere are known problems n defnng total taxable ncome (cf O Donogue and Suterland, 1999; Goolsbee 000) In te future te fscal autortes mgt peraps make t easer to obtan specally commssoned tables, n so far as ts s possble wtout compromsng on respondent confdentalty In te meantme, gven te lack of data on taxable ncome, we ave to make use of te data provded on total amount of ncome Once agan analyzng taxpayers tat are subject to eter te basc or splttng s, te database can be descrbed formally as follows Te suppled data provde a categorzaton per total amount of ncome for j=1 to m classes wt class borders [c j, d j ], were c 1 =-, d 1 =-1, c =0 and d m = For classes j>1 te taxpayers ave a taxable ncome greater tan zero DM Te frst class (j=1) contans te so called cases of loss wc occur f te taxpayer as an assessed negatve ncome A negatve value can result wen determnng of te sum of te earnngs from dfferent sources of ncome or, later n te assessment pattern, wen determnng te taxable ncome, for example due to te deducton of extra expendtures and extraordnary expenses For eac class j we know: te frequency j n class j (number of taxpayers n te class for wom a taxable ncome as been assessed), te frequency g j of a tax fact (number of taxpayers wo meet ts fact) and te value G j of ts tax fact (n tousands of DM or ), te sum of te taxable ncomes of all taxpayers n ts class j and te sum of te assessed ncome tax of all taxpayers n ts class T j Applyng te group smulaton model to data from tables tat are arranged accordng to total amount of ncome (TAI) te followng problem arses Te dstrbuton of taxpayers wt a specfc taxable ncome ( () ) s dffcult to estmate due to te fact tat for te taxpayers of a TAI-class j only te average taxable ncome of te class, j j =, j can be determned drectly Te nterval range [a, b ] of te possble taxable ncome of tese taxpayers cannot be deduced from te TAI tables By mappng a taxpayer to a certan TAI class we can only determne te upper lmt of te taxable ncome b as te teoretcal maxmum taxable ncome of te class by reducng te upper lmt of te TAI class d j by te mnmum fscal reductons, for example allowances for specal expenses In contrast, a teoretcal lower lmt for te taxable ncome a cannot be determned because te taxable ncome can adopt any value below te upper bound of te TAI class d j due to varous dscounts on te total amount of ncome, for example specal expenses, loss offset or extraordnary expendtures Consequently, n ts case te lower nterval lmt a (smallest possble taxable ncome) must be estmated rougly, mplyng relatvely g naccuracy of te results of smulaton In order to reduce te devatons n group smulaton caused by ts defct of nformaton cross tables were provded by te German Statstcal Offce for our analyss Tese cross tables allow us to restructure part of te aggregated data of te ncome tax statstcs tat are grouped accordng to total amount of ncome (TAI) and rearrange tem accordng to classes of taxable ncome () In tese cross tables te absolute frequency of te taxpayers,, wt a taxable ncome n class and te sum of te taxable ncome,, are brougt togeter wt te absolute frequency of te taxpayers, j, wt a total amount of ncome n class j and te sum of te taxable ncomes of tese taxpayers, j As a result, we obtan a matrx of te absolute frequences of te taxpayers, j, and te necessary sums of te taxable ncome, j Usng ts matrx t s possble to estmate te dstrbuton of te taxpayers wt a specfc taxable ncome from te aggregated data of te ncome tax statstcs grouped accordng to te class attrbute total amount of ncome, as n eqns (14) and (15): j (18) ( ) uj, ( b a 1) b a were a b a