Computation of the Compensating Variation within a Random Utility Model Using GAUSS Software

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1 Modern Economy, 211, 2, do:1.4236/me Publshed Onlne July 211 ( Computaton of the Compensatng Varaton wthn a Random Utlty Model Usng GAUSS Software Abstract Marlena Locatell, Stenar Strøm Unversty of Turn and Chld Centre, Turn, Italy, Frsch Centre, Oslo, Norway E-mal: marlena.locatell@unto.t, stenar.strom@econ.uo.no Receved February 23, 211; revsed Aprl 1, 211; accepted Aprl 22, 211 To evaluate a tax reform n terms of change n household welfare one possblty s to estmate the compensatng varaton usng a sutable model to assess the change n the household utlty. When a random utlty model s used, the computaton of compensatng varaton s not straghtforward, partcularly when utlty s not lnear n household ncome. It can be carred out usng a methodology recently proposed n the lterature. In ths paper we descrbe a software nstrument, mplemented usng GAUSS programmng language for computng the compensatng varaton to evaluate the 1991 tax reform ntroduced n Norway. The program s flexble and adaptable to dfferent tax systems and dfferent reference years. JEL Classfcaton: C63, B21 Keywords: Compensatng Varaton, Gauss Applcaton, Tax System Evaluaton 1. Introducton In 1992 the Norwegan tax system was reformed towards lower and less progressve tax rates, wth a reducton n the total tax revenue. In the next followng years the tax structure was kept nearly unchanged. To evaluate a tax reform n terms of change n household welfare one possblty s to estmate the compensatng varaton (CV) usng a sutable model to assess the utlty of the households. The theoretcal framework and the emprcal results of the evaluaton of the above tax reform are reported n [1] n whch there s also an extensve lterature revew on ths subject. Novelty of the paper s the use of a random utlty labor supply model, takng nto accountng also sectoral choces (publc and prvate), to assess the mpact of tax reforms on household welfare. Some partal detals on the algorthm mplementaton were frst gven n [2]. The computaton of CV s not straghtforward n a random utlty model, n partcular when utlty s not lnear n household ncome. A random utlty functon mples that the expendture functon s also random. Untl recently, no analytc formulas have been avalable for calculatng the dstrbuton of CV. However, [3] have developed analytc formulas for ths purpose, and we apply ther methodology to calculate the dstrbuton of CV and the mean and varance of ths dstrbuton. What we thus do s to calculate the expected value of CV for each household and ts dstrbuton n the populaton. The purpose of ths note s to descrbe the software nstrument, mplemented usng Gauss software package [4], to evaluate the above tax reform n terms of change n household welfare wthn a random utlty model. The program s flexble and adaptable to dfferent tax systems and dfferent reference years. The paper s organzed as follows. In the next Secton, the data used n estmatons are descrbed. The model s explaned n Secton 3. In Secton 4 we analyze the man steps of the procedure and n Secton 5 the program structure s outlned. The results are reported n Secton 6. Secton 7 draws some conclusons. The Table upon the estmaton s reported n Appendx A. The nterpolaton method followed n the computaton of CV s descrbed n Appendx B. In Appendx C we report the GAUSS program flow. The complete program code can be downloaded [2]. 2. The Data Data on the labor supply of marred women n Norway used n ths note consst of a merged sample from Survey of Income and Wealth, 1994, Statstcs Norway Copyrght 211 ScRes.

2 384 M. LOCATELLI ET AL. (1994) and Level of Lvng Condtons, 1995, Statstcs Norway (1995). Data cover marred couples as well as cohabtng couples wth common chldren. The age of the spouses ranges from 25 to 64. None of the spouses are self-employed and none of them are on dsablty or other type of benefts. All taxes pad are observed and n the assessment of dsposable ncome, all detals of the tax system are accounted for. The sze of the sample used n estmatng the labor supply model s 81 (referred to as n_record n the program). 3. The Model To evaluate the tax reform of 1992 we calculate the change n household welfare. One way s to apply the measure of CV. The calculaton of CV s not straghtforward n a random utlty model when utlty s not lnear n household ncome. A random utlty functon mples that the expendture functon s also random. A general treatment of ths ssue was undertaken by [3] whle n [1] the method was adapted to the change n household welfare wth labor supply random utlty model. What we do s to calculate the expected value of CV,, for each ndvdual and thereafter the dstrbuton of ths value n the populaton from whch we can derve mean, medan etc. We wll assume that the utlty functon has the structure UC, h, z =vc,h z, for z,1, 2,3, (1) where C s the household dsposable ncome, whch equals the sum of the after-tax labor ncome of husband and wfe plus the after tax captal ncome and publc transfer lke chld allowances; h are hours of work. In detals: (not workng), 315, 78, 14, 156, 1976, 234, 26 (both for publc and prvate sector); v(.) s a determnstc functon and ε(z) s a postve random taste shfter. The taste shfter accounts for unobserved ndvdual characterstcs and unobserved job-specfc attrbutes z. {ε(z)}, are ndependently dstrbuted wth c.d.f. exp( x 1 ), x >. From Equaton (1) we get the followng mplct defnton of the CV when the tax regme of 1991 (pror to the tax reform) s compared to the tax regme of 1994 (after the tax reform): z U C, h, Tax regme 1991 U C CV, h, z Tax regme1994 In Equaton (1) we have suppressed the subscrpt of the ndvdual and we should also keep n mnd that the choce of each ndvdual s to choose to work or not, and gven work, to choose sector and hours of work, gven (2) the job opportunty sets and the budget constrants under the dfferent tax regmes and CV. In the calculaton of the expected value of CV we take ths choce structure nto account. If an ndvdual benefts from the tax reform, the expected value of CV for ths ndvdual s postve, meanng that ths amount has to be subtracted from household ncome under the 1994 tax regme n order to make the ndvdual ndfferent between the two tax regmes. To proceed wth the calculaton we need some notaton. Note frst that the determnstc part of the utlty functon can be wrtten v v Ch, h X (v referred to as consumpton functon n the followng), where denotes the dfferent job alternatves. In the examned case the choce alternatves (na) are 15: = 1 the ndvdual s not workng, and 2,3,,8 denote hours of job n the publc sector, whle 9,,15 denote hours of job n the prvate sector. X s a vector of all exogenous characterstcs. Now let v =vch Tax regme 1991,h X (3) and let * v y v C* h, y, h X where C* h, y Fh y, F h wh Twh (4) and T(.) s the tax functon for 1994, w s the wage rate. Let be the expected value of the compensatng varaton, whch can be calculated for each ndvdual as follows [3]: ECV I* 15 v gb y 15 1 dy * max vgb, v y gb (5) where, accordng to estmates gven n [5] and reported n Appendx A, g 1 for all except for 4, 6,11,13 g4 exp(.68) g6 exp(1.58) g11 exp(.8) g13 exp(1.6) b1 1 b exp X4 ; for 2, 3,,8 b exp X ; for 9,1,,15. and y s gven by the followng equaton: 4 * v v y (6) Copyrght 211 ScRes.

3 M. LOCATELLI ET AL. 385 I* (see Equaton (5)) equals the sum of the after tax ncome of husband earnngs and captal ncome, plus chld allowances. The tax reform of 1992 s a combnaton of a change of the tax structure and reducton n tax revenues. 4. Steps of the Procedure In 1992 the Norwegan tax system was reformed towards lower and less progressve tax rates wth a reducton n the total tax revenue. We have thus organzed the program to allow also the computaton of the expected between the 1994 tax system and the flat tax system (n our case 29%) that equals tax revenue of year The Algorthm To calculate the followng steps are requred: 1) Load the matrx fle, prevously prepared 1, wth the values v (the household s determnstc utlty (.e. consumpton functon)) for the reference year 1991 and workng hours h. v v C h Tax functon1991, h X (7) where v s a n_record na matrx, beng n_record the number of records of the matrx fle, and na the number of alternatves of the choce set (.e. 15 alternatves). We also calculate wth reference to a flat tax system (n our case the revenue neutral tax rate smulated on the choce model s 29%). The model under the flat tax system gves the reference values when the 1994 tax regme s evaluated aganst the flat tax system. Now, load the matrx fle wth the averaged values v (the household determnstc utlty) under the flat tax system. 2) Load the data sets ncludng: a) the varables (dsposable ncome, etc.) for the tax system 1991; b) the varables (dsposable ncome, etc.) for the tax system 1994; 3) Load the matrx fles that allow us to dentfy the decles assocate wth poor (frst decle), mddle (from second to nnth decle), and rch (tenth decle) of the dstrbuton of dsposable ncome computed accordng to 1994 tax system. 4) For each observaton and each alternatve compute a matrx Fh (n_record na) whose elements are: Fh = wh -Twh, 1,2,,15 (8) 1 Of course these data sets must be prevously prepared wth the specfc program that consders the dfferent tax systems to be estmated. where w h s the hourly wage multpled by the hours assocated to each of the 15 alternatves, T(.) s the tax functon for The choce alternatves are not workng ( = ), workng n the publc sector at dfferent hours ( = 2,3,4,5,6,7,8) and n the prvate sector ( = 9,1,11,12,13,14,15) 5) Compute a matrx C*, (n_record na), whose elements are C* h,y = F h +y (9) where y s a matrx (n_record na) determned by an teratve procedure so that for y = y, at each observaton the followng equalty holds v =v F h +y, = 1,2,,15 (1) The value of y s determned usng an teratve procedure, descrbed n Appendx B Computaton of the Integral The ntegral s computed numercally dvdng the ntegraton nterval n small steps (a length of NOK 1 was found suffcent) and then summng up the partal contrbuton of the ntegrand functon related to each step. The fnal result for each observaton s obtaned by summng the sngle ntegrals, evaluated for each alternatve, over the total number of alternatves Fnally, s computed subtractng from I* (Equaton (5)) the ntegral evaluated n the prevous step. 5. Program Structure The program, whose flow-chart s shown n Appendx C, conssts of a man part whch resorts to several procedures to accomplsh dfferent tasks. They are brefly descrbed below. Man program: The man program ncludes the computaton of: 1) consumpton functon usng the procedure V and V_SCALAR 2) dsposable ncome computed wth: a) woman wage before tax usng the procedure W_WAGE b) woman net wage usng the procedure NETWAGE_W_94 c) men net wage when woman s not workng usng the procedure NETWAGE_M_94 d) men net wage when woman s workng usng the procedure NETWAGE_MW_94 3) y : monetary value to be added or subtracted to the Copyrght 211 ScRes.

4 386 M. LOCATELLI ET AL. woman net wage of 1994 (Fh 94) for whch the utlty of 1994 equals that of It s evaluated resortng to the procedure INCR and the command lnes lsted n the man program for the refnement of the nterval were the soluton les. 4) the ntegral usng the procedure INTEGRAND. 5) and statstcs for the total sample and for decles of dsposable ncome dstrbuton (poor (frst decle), mddle (from second to nnth decle), and rch (tenth decle). Procedures (lsted n alphabetc order): HALF: soluton refnement through half nterval search INCR: terate untl an nterval s found wth functon values of dfferent sgn INTEGRAND: computaton of the functon to be ntegrated and the ntegral (Equaton (5)), for gven y, observaton, and alternatve j INTERP: search a soluton va lnear nterpolaton NETWAGE_M_94: compute man net wage when woman s not workng (1994 tax system). NETWAGE_MW_94: compute man net wage, when woman works (1994 tax system). NETWAGE_W_94: calculaton of woman net wage (.e. women workng) usng tax functon 1994 V: computaton of consumpton functon The procedure returns the matrx v (of dmenson n_record na) of the consumpton functon for a gven matrx of dsposable ncome dsp (of dmenson n_record na), accordng to the followng equaton: v, j exp X 9.2X C, j hj X2.97X C, j hj (11) where C,j = dsposable ncome of record and alternatve j, passed as a parameter to the procedure. The other parameters are passed as global varables and have the followng meanngs: X 1 s the logarthm of age of the woman, X 2 s the number of chldren aged - 6, and X 3 s the number of chldren aged V_SCALAR: computaton of the value of consumpton functon v for a sngle value of dsposable ncome, a gven sample and alternatve j. Ths s the scalar verson of procedure V. The procedure returns the value (scalar) of the consumpton functon v for a gven dsposable ncome dsp, sample and alternatve j accordng to the equaton reported for procedure V above. W_WAGE: Woman wage ncome before tax. 6. Results The expected CV obtaned from the above procedure s reported n the followng Table 1. From Table 1, we observe that the mean household n the sample ganed NOK 2778 from the 1992 tax reform. The rchest household ganed almost 1 tmes more than the poorest or 4 tmes more n relatve ncome terms. The dstrbuton of expected gan across households s gven n Fgure 1, and we observe that most of the households wll beneft from the 1992 tax reform. Thus, such a reform would have attaned support from a clear majorty of households wth marred and cohabtng women at an electon. We have also calculated the expected value of compensatng varaton of a flat tax reform. In the calculatons, the tax-revenue-neutral flat tax reform of 29% s used as a reference. Negatve values mean that the numercal values have to be subtracted from household ncomes under the flat tax regme n order to make the households ndfferent n welfare terms between the 1994 regme and the flat tax regme. The expected CV s reported n the followng Table 2. Ths Table then says that, on average, the households wll gan NOK f there s a shft from the 1994 tax regme to a flat tax regme. The rchest households gan around 8 tmes more than the poorest. Thus, n a dstrbutonal sense, the rchest household benefted more from havng the 1991 regme replaced wth the 1994 tax regme than they would have n the case of a shft from the 1994 tax regme to a flat tax regme. In Fgure 2, we show the populaton densty of the ndvdual mean CV. We observe that a vast majorty wll beneft from the replacement of the 1994 tax regme wth a flat tax regme. More detals of the results are reported n [1]. Copyrght 211 ScRes.

5 Table 1. Expected value of compensatng varaton (n NOK 1994) for the 1992 tax reform. The 1991 tax system s used as a reference aganst the 1994 tax system. M. LOCATELLI ET AL. 387 All 27, n percent of observed dsposable ncome* Decles n the dstrbuton of household dsposable ncome*: 1 (poor) 6, (mddle) 24, (rch) 64, *Decle(s) refers to the decles n the dstrbuton of dsposable ncome, Fgure 2. Populaton densty of expected Compensatng Varaton. Dstrbuton of, wth the flat tax system of 29% used as a reference aganst the 1994 tax regme. Fgure 1. Populaton densty of expected Compensatng Varaton. Dstrbuton of, comparng the 1991 tax regme aganst the 1994 tax regme. Table 2. Expected value of compensatng varaton (n NOK 1994) for a flat tax reform. A flat tax regme s used as a reference aganst the 1994 tax system. All 51,437 Decles n the dstrbuton of household dsposable ncome, flat tax: 1 (poor) 17, (mddle) 53,93 1 (rch) 146, Conclusons In ths note we have descrbed a GAUSS software nstrument, to compute the value of expected compen- satng varaton wthn the dscrete choce settng suggested n [1]. The program refers to the followng tax reforms: a) tax systems n force n 1991 and n 1994 usng 1991 as reference year, b) 1994 tax system and a flat tax system takng 1994 as reference year. The program s flexble and can be easly modfed to take nto account dfferent tax systems and dfferent reference years. The results show the dfferent mpact of the tax reform: 1) from the 1992 tax reform we observe that most of the households wll beneft from the 1992 tax reform; the rchest household ganed almost 1 tmes more than the poorest or 4 tmes more n relatve ncome terms. Thus, such a reform would have attaned support from a clear majorty of households wth marred and cohabtng women at an electon. 2) from the 1994 tax system and a flat tax system takng 1994 as reference year we observe that the rchest households gan around 8 tmes more than the poorest. We observe that a vast majorty would beneft from the replacement of the 1994 tax regme wth a flat tax regme. 8. Acknowledgements M. Locatell gratefully acknowledges fnancal support by Frsch Centre, Oslo, Norway. 9. References [1] J. K. Dagsvk, M. Locatell and S. Strøm, Tax Reform, Sector-Specfc Labor Supply and Welfare Effect, Scandnavan Journal of Economcs, Vol. 111, No. 2, 29, pp do:1.1111/j x Copyrght 211 ScRes.

6 388 M. LOCATELLI ET AL. [2] M. Locatell and S. Strøm, Computaton of the Compensatng Varaton wthn a Random Utlty Model Usng GAUSS Software, Chld Workng Paper No. 2/26. [3] J. K. Dagsvk and A. Karlstrøm, Compensatng Varaton and Hcksan Choce Probabltes n Random Utlty Models That are Non-Lnear n Income, Revew of Economc Studes, Vol. 72, No. 1, 25, pp do:1.1111/ [4] Aptech System, Inc., GAUSS Software (Verson 1), Black Damond, Rdgefeld, 29. [5] J. K. Dagsvk and S. Strøm, Sectoral Labor Supply, Choce Restrctons and Functonal Form, Journal of Appled Econometrcs, Vol. 21, No. 6, 25, pp do:1.12/jae.866 [6] S. S. Kuo, Computer Applcatons of Numercal Methods, Addson-Wesley, Boston, Appendx A Table A1. Estmaton results for the parameters of the labor supply probabltes. Unformly dstrbuted offered hours wth part-tme and fulltme peaks Varables Parameters Estmate t-values Preferences: Consumpton: Exponent α Scale 1 4 α Subsstence level C n NOK per year 6 Lesure: Exponent α Constant α Log age α (log age) 2 α # chldren - 6 α # chldren 7-17 α Consumpton and Lesure, nteracton α Subsstence level of lesure n hours per year 512 The parameters b 1 and b 2 ; log b = f + f S j j1 j2 Constant, publc sector (sector 1) f Constant, prvate sector (sector 2) f Educaton, publc sector (sector 1) f Educaton, prvate sector (sector 2) f Opportunty densty of Offered hours, g k2 (h), k = 1,2 Full-tme peak, publc sector (sector 1) * log 1 Full 1 Full-tme peak, prvate sector (sector 2) log 2 Full 2 Part-tme peak, publc Sector log 1 Part 1 Part-tme peak, prvate Sector log 2 Part 2 g h g h g h g h g h g h g h g h # observatons 81 Log lkelhood Copyrght 211 ScRes.

7 M. LOCATELLI ET AL. 389 Appendx B To determne the value y we must solve the followng equaton (for each record and each alternatve) v = v(fh94 + y ). That means that we must fnd the zero of the functon f x defned as: f x v v x, wth x Fh94 y, where y s a generc amount of ncome to be added to the 1994 woman net wage. Callng x * the value for whch f x, e have y x* Fh94 To determne the value of x * the followng steps are done: 1) terate untl an nterval s found where the soluton les, usng procedure INCR; 2) refne the nterval teratng untl an approxmate soluton s found. For the very frst teratons (NITER 5), the new value s searched usng lnear nterpolaton (see Fgure 1), mplemented by the procedure INTERP. Then (NITER > 5) the soluton s refned usng a half-nterval search method, mplemented by the procedure HALF. Fore more detals on these methods see, for example, Ch. 6 of [6]. The ext test s performed only after the soluton has been refned usng HALF (.e. only f NITER > 5) and s f(x 1 ) based both on an absolute and relatve tolerance. Consderng the last nterval x 1, x 2 where the soluton s sought, the functon value f x m at the mean value xm.5 x1 x2 s computed. Furthermore the relatve error on x, rel_err = x2 x1 x1, s evaluated. The soluton s accepted f f x m abs_tol or rel_err rel_tol. A satsfactory trade-off between speed and accuracy has been found assumng abs_tol = 1 and rel_tol = 1e-8. * Then the procedure exts assumng x = x m. Appendx C: the GAUSS Program Flow Load V ( V V _91 or V V 29) Set dy =1 Compute Fh94 (dsposable ncome, year 1994) Call: w_wage, netwage_w_94, netwage_m_94, netwage_mw_94 Intal values: fx V V(x ) Call V: computaton of consumpton functon 1 1 x Fh94 1 fx? 1 YES x2 Fh94 y where y s the male dsposable ncome for 1994 fx V V(x ) 2 2 NO fx fx 1 2? YES NO Gve an ncrement to y to fnd an nterval where fx 1 and fx 2 have dfferent sgns: y y dy Call: INCR Refne the soluton and call t x * Call: INTERP, HALF, and V_scalar x Fh94 Soluton NO Are tolerance lmts satsfed? YES x app x 1 x 2 x s the soluton found. Set Compute the ntegral Call: INTEGRAND y x* Fh94 f(x 2 ) Fgure 1. The lnear nterpolaton method: xapp = x f x x x f x f x 1 2 * I Integral END Compute statstcs Copyrght 211 ScRes.