PERSONAL INCOME TAX REFORMS: A GENETIC ALGORITHM APPROACH

Workng Paper 47/4 PERSONAL INCOME TAX REFORMS: A GENETIC ALGORITHM APPROACH Matteo Morn Smone Pellegrno

Personal Income Tax Reforms: a Genetc Algorthm Approach November 9 th, 04 Matteo Morn ENS Lyon, Insttut Rhône-Alpn des Systèmes Complexes (IXXI), Lyon (FR) Unversty of Torno, Department of Economcs and Statstcs, Torno (IT) Smone Pellegrno Unversty of Torno, Department of Economcs and Statstcs, Torno (IT) Abstract Gven a settled reducton n the present level of tax revenue, and by explorng a very large combnatoral space of tax structures, n ths paper we employ a genetc algorthm n order to determne the best structure of a real world personal ncome tax that allows for the maxmzaton of the redstrbutve effect of the tax, whle preventng all taxpayers beng worse off than wth the present tax structure. We take Italy as a case study. JEL Codes: C63, C8, H3, H4 Keywords: Personal Income Taxaton, Genetc Algorthms, Mcro-smulaton Models, Reynolds-Smolensky Index, Tax Reforms

. Introducton Personal ncome tax (hereafter, PIT) around the world s characterzed by several parameters that defne ts structure: margnal tax rates, upper lmts of the thresholds, allowances and deductons, as well as tax credts. Appled to the dstrbuton of ncome observed n a specfc country, the PIT structure of that country determnes a gven tax revenue and a gven redstrbutve effect. Startng from ths poston, a Government may want to cut down PIT revenue n order to ncrease the purchasng power of taxpayers; conversely, t may want to ncrease the redstrbutve effect of the tax leavng the tax revenue unchanged; fnally, t may want to ncrease tax revenue by lettng the rchest taxpayers face all the tax ncrease. In order to acheve one of these specfc targets, how should the whole PIT structure change? For several reasons, t s not the case that polcy makers consder ths queston when thnkng of a PIT reform. The PIT structure observed n a country s ndeed the result of several and partal adjustments that have occurred over the past years and, gven a revenue constrant, whether those tax reforms were amed at achevng the best way to obtan the specfc target s debatable. Wth respect to these arguments, Italy s perfect case study: the Italan PIT s very complcated and ts structure ncorporates more than thrty parameters. Moreover, n order to ncrease the purchasng power of poor PIT taxpayers, as well as taxpayers belongng to the mddle class (a proxy of the redstrbutve effect maxmzaton), the Italan Government recently reduced the PIT revenue by about 9.3 bllon euros by ntroducng a cash transfer of 80 euros per month, only for employees wth a PIT gross ncome n the range of 8-6 thousand euros (about 0.9 mllon taxpayers). Then two questons arse. Is ths tax cut allocaton the best one the Government could have consdered? Or, gven ths settled amount of the tax cut, whch s the best way to reform the whole PIT structure n order to acheve the hghest redstrbutve effect, whlst havng no taxpayers beng worse off, wth respect to the present tax structure? Ths s a revsed verson of the workng paper Personal Income Tax Reforms: a Genetc Algorthm Approach by Matteo Morn and Smone Pellegrno. Some of the results have been updated to reflect better results provded by the algorthm. An earler verson (September 04) of ths workng paper can be found at https://deas.repec.org/p/tur/wpapnw/06.html. The paper was presented at the XXVI Annual Conference of the Italan Assocaton of Publc Economcs, Pava, September 04. We would lke to thank Tto Boer, Paola Profeta, Federco Revell and Ivca Urban for ther useful comments that helped us mprove the paper. Accordng to the offcal statstcs made avalable by the Government, the tax cut amounts to 9.5 bllon euros. Here we consder the tax cut resultng from the mcro-smulaton model employed n ths work.

The soluton of ths problem can face an equty-effcency trade-off: n order for the redstrbutve effect to be the hghest, the effcency of the tax, (.e. the level of the effectve margnal tax rates), can worsen. As an example, n ths paper we manly focus on the equty sde of the problem. Ths does not mply that we forget about the effcency sde; we suggest a few constrants to the allowable parameters of tax structure, n order not to arrve at both trval and neffcent solutons. To answer the above questons, we rely on a statc mcro-smulaton model wrtten n STATA (techncal detals are avalable n Pellegrno et al. (0)) that employs, as nput data, those provded by the Bank of Italy n ts Survey on Household Income and Wealth, publshed n 0 wth regard to the 00 fscal year. The results of ths mcrosmulaton model are very close to the offcal statstcs made avalable by the Italan Mnstry of Economy and Fnance for the 00 fscal year; as a consequence, ths nstrument s sutable for the type of emprcal analyss we propose. The statc mcro-smulaton model had to be re-mplemented n a more versatle way, n order for the knd of analyss we are nterested n to be feasble. We rewrote the modules of the mcro-smulaton model evaluatng the Italan PIT n Python, a very manstream language that convenently allows for the use of parallel computng technques, dstrbuted across multple nodes. Python also offers an excellent compromse between aglty n programmng provdng the developers wth several lbrares optmzed for numercal calculatons and computatonal performance. We then employed a genetc algorthm; that s, a search heurstc nspred by natural selecton, well suted to the dentfcaton of the most promsng soluton to the problem under consderaton. We were nterested n comng up wth a reasonable tax structure that nhbts both trval and neffcent solutons. The genetc algorthm then had to be provded wth a few specfc constrants that had to be obeyed, n terms of some parameters of the tax structure. If ths were not the case, problematc solutons would appear. For example, havng to fnd the hghest redstrbutve effect wth no constrants at all, the genetc algorthm would certanly mpose excessvely hgh margnal tax rates on hgher ncome earners and a zero margnal tax rate on too many of the poorest taxpayers; as a result, a polarzaton of tax rates and bandwdths of thresholds would appear and the tax revenue would consequently be too hgh. Or the genetc algorthm would dsproportonately favour hgh levels of some pecular tax credts, smply

because they are enjoyed by a small group of taxpayers, resultng n a neglgble mpact on the tax revenue, but n awkward preferental treatment for some ncome groups. In order to avod these unpleasant outcomes, we mposed two constrants. Frst, we mpose a condton that no taxpayers should be worse off as a result of the tax reform; that s, all taxpayers must pay a lower (or, at most, equal) amount of taxes than the present one. Therefore, snce the Italan personal ncome tax does not allow for negatve ncome taxaton, and as we are lookng for a tax reform wth no losng taxpayers, we let the no tax area be greater, or at least equal, to the present one. We also requre the hghest margnal tax rate, as well as the lower lmt of the top threshold, to be lower, or at most equal, to the present values. Second, we keep the rank appled by the present tax structure to certan knds of tax credts unchanged: for example, the present tax credt appled to employees s greater than the one appled to pensoners, and the one appled to pensoners s greater than that whch s appled to self-employed taxpayers; smlarly, the tax credt for tenants s greater wth regard to younger ones. In dong so, we do not allow the genetc algorthm to run free wth too magnatve solutons. Then, we have to defne our target: we are nterested n obtanng the hghest possble redstrbutve effect of the tax. To measure t, we refer to the Reynolds-Smolensky ndex, gven by the dfference between the Gn coeffcent for the pre-tax ncome dstrbuton and the correspondng concentraton coeffcent for the post-tax dstrbuton. Instead of the equvalent household gross and net ncome dstrbutons, we refer to the taxpayers ones. The man reasons are twofold: the cash transfer ntroduced by the Italan Government favours taxpayers; ths s the frst exercse that employs a genetc algorthm for a tax system optmzaton, so that by observng the composton of the tax cut by ncome classes, we can ensure that our result s among the best. By referrng to the equvalent household ncome dstrbuton, ths check could be much harder to assess. We then let the genetc algorthm set up a populaton of 00 dfferent tax structures, each of them composed of 33 dfferent parameters defnng the present structure of the Italan personal ncome tax, and we let t evolve for 0 thousands of generatons untl the desred result was acheved. 3

Even f by employng the best tax structure no taxpayer s worse off, ts actual applcablty could face poltcal resstance, snce all parameters of the tax change and, consequently, taxpayers could hardly beleve that no one s worse off. We do not dscuss these poltcal economy nconvenences. Fnally, t has to be noted that here we also do not consder taxpayers responses to the new parameters of the tax structures: t s a short run soluton that can help polcy makers when they thnk of a PIT reform. In order to consder taxpayers responses, agent-based models could be employed. Ths s the baselne of our further research. The structure of the paper s as follows. Secton descrbes n greater detal the 00 structure of the Italan personal ncome tax, the baselne for our analyss. Secton 3 brefly presents how tax progressvty, and the redstrbutve effect exerted by the tax, can be measured. Secton 4 shows the data and peculartes of the statc mcrosmulaton model employed for smulatons. Secton 5 frst descrbes how genetc algorthms work and then presents the mplementaton used n ths work. Secton 6 shows the results, whlst secton 7 offers a concluson.. The Personal Income Tax n the 00 Fscal Year: Techncal Detals Let x be the personal gross ncome of taxpayer,,..., n. The 00 Italan tax law consders two dfferent knds of deductons: d s deducton for the man resdence cadastral ncome; d s the sum of deductons for socal securty contrbutons and almones as well as donatons. The taxable ncome y s evaluated as: y x d 0 d f f d d d d x x From 007 onwards the rate schedule S y contemplates 5 thresholds as reported n Table. () TABLE AROUND HERE 4

j The upper lmts j LL UL,,3,4 j of thresholds are 5, 8, 55, 75 thousand euros, beng the frst lower lmt LL 0 ; tax rates t j range between 3 and 43 percent. By applyng the rate schedule to the tax base the gross tax lablty GT s obtaned. In order to determne the net tax lablty T, tax law admts three dstnct knds of effectve tax credts. They are: tax credts for earned ncome c x dependent ndvduals wthn the household c x, tax credts for, and tax credts for tems of expendture 3 c, where x x d. The net tax lablty T s then evaluated as: 3 x c x c f GT c x c x f GT c x c x 3 GT c c T () 3 0 c In what follows we do not consder regonal and muncpal surtaxes and then we evaluate taxpayer s net ncome as z x T. Focusng on tax credts for employees and pensoners as well as self-employed, tmr f x mr LL x ( t mr ar ) ar f mr x LL LL m r c ( x ) (3) LL 4 x ( t mr ar ) b f LL x LL4 LL4 LL 0 f x LL4 where t s the lowest margnal tax rate (3 percent); m wth r (,,3,4 ) (the level of x below whch taxpayer has a nl net tax lablty) s equal to 8,000 euros for employees than 75 m, 7,500 for pensoners younger than 75 m m, 4,800 for the self-employed m 3 4 r, 7,750 for pensoners older, and zero for non-workng taxpayers; a r s equal to 50 euros for employees a, 470 for pensoners younger than 75 486 for pensoners older than 75 a 3, zero for self-employed a 4 a, ; b, that ranges from 0 to 40 euros n the bandwdth 3-8 thousand euros, s appled only to employees (as dscussed later, we always set b 0 n smulatons). Non-workng taxpayers have no 5

tax credt for earned ncomes. Fnally, ths tax credt decreases from zero to m 4, and from m 4 to LL 4 only for self-employed taxpayers. Four dfferent tax credts for type of relatonshp are allowed: tax credt for dependent H chldren c x S tax credt for dependent spouse c x O components c x. The overall value for x H HF S O c x c x c c x c x. In partcular,, further tax credt for households wth more than three chldren c, HF, and tax credt for other household c s then c H x c 0 Hp q ( f ) e x q ( f ) e f f 0 x x q ( f ) e q ( f ) e (4) where 4 f l l f s the overall number of dependent chldren; f s the number of dependent chldren older than 3 years f the dependent chldren wthn the household are 3 or less; f s the number of dependent chldren younger than 3 years f the dependent chldren wthn the household are 3 or less; f 3 s the number of dependent chldren older than 3 years f the dependent chldren wthn the household are more than 3; f 4 s the number of dependent chldren younger than 3 years f the dependent chldren wthn the household are more than 3; e s equal to 5,000 euros; q s equal to 95,000; 4 Hp Hpl Hp c f c ; the present values for the potental tax credts are: c 800, l l Hp Hp3 Hp4 c 900, c, 000, c, 00 euros. H Moreover, whenever 0 c x and the dependent chldren wthn the households are more than 3 the tax law admts a further tax credt HF c equal to,00 euros for all benefcares. The tax credts for dependent chldren have to be splt between spouses whenever both of them have a postve gross ncome. Fnally, 6

c S x c c c 0 Sp Sp Sp x u LL u u k x k w f f f f LL w x x x LL x k k w (5) and c O x c 0 Op k x k f f x x k (6) where u s equal to 0 euros, w s equal to 40 thousand euros, k s equal to 80 thousand k euros, Sp c s equal to 800 euros and Op c s equal to 750 euros. The present tax code consders hgher values than c Sp u n the ncome range 9,000-35,00 euros. Instead of 690 euros, n ths ncome range values rangng from 700 to 70 euros are appled. We do not consder these dfferences n smulatons, always lettng a same value. Tax credts for tems of expendtures c Sp u be equal to 3 c can be classfed n two groups accordng to the percentage of the expense the tax law admts as a tax credt. There are expenses that allow a tax credt of 9 percent and 36 percent, respectvely. 3 The 9 percent tax credts (we label ths varable expendture) are very large, 9 dfferent cases, such as expenses for health care, mortgage nterests, etc.; 36 percent tax credts (expendture) are allowed for home restructurng-related expenses. All together, tax law admts 30 dfferent tax credts for tems of expendture. Fnally, tax law admts a tax credt for tenants; t s 300 euros f x 5, 494 (we label ths varable tenants); 50 f 5,494 x 30,987 (tenants); 99 euros f x 5, 494 and f the taxpayers are younger than 30 (tenants3). 3 The tax code consders also a 55 percent tax credt for nterventons for energy savng and a 0 percent tax credt for purchasng of a washng machne. Because of the low number of taxpayers nterested n these two knds of tax credts, we dd not consdered them n the mcro-smulaton model. 7

3. Dstrbuton of Income and Personal Income Tax Progressvty Let x...,, x, xn be the pre-tax ncome levels assocated to n ncome unts. The correspondng post-tax ncome levels and tax levels are z...,, z, zn and T, T,..., Tn, respectvely. We denote the pre-tax and the post-tax ncome dstrbuton as well as the tax dstrbuton by X, Z and T, respectvely. As s well known, nequalty among pre- and post-tax ncome levels as well as tax levels can be evaluated by the Gn coeffcent. Let G X, G Z and G T be the correspondng Gn coeffcent for pre-tax ncome, post-tax ncomes and taxes, respectvely. Then, cov, F( ) G (7) where X, Z, T, s the average value for pre-tax and post-tax ncomes and taxes, cov represents the covarance and F s the cumulatve dstrbuton functon. After the tax, t s not guaranteed that post-tax orderng be equal to the pre-tax ncome one. Indeed, t s most lkely that these two orderngs dffer because of the re-rankng due to the tax. Therefore, the nequalty of Z and T can be evaluated once these dstrbutons are ordered accordng to the correspondng pre-tax ncomes, ranked n a non-decreasng order. For what concerns post-tax ncomes and taxes, the correspondng concentraton coeffcent can then be evaluated as follows: cov, F( X ) C (8) Progressve taxaton produces two dfferent effects on the dstrbuton of pre-tax ncomes: post-tax ncome nequalty s lower than that measured on pre-tax ncome dstrbuton, whlst tax nequalty s greater. The frst effect s known as the redstrbutve effect of the tax and the second one as departure from proportonalty of the progressve taxaton (Lambert, 00). The overall redstrbutve effect of the tax RE can be evaluated as X Z APK G C G C RS R RE G G (9) where RS X Z Z Z G C s the APK G X CZ s the Reynolds-Smolensky ndex, whlst Z Z Atknson-Plotnck-Kakwan ndex. The more the tax s progressve, the greater RE and RS; the more the tax causes re-rankng, the greater the negatve contrbuton of re- R 8

rankng to the overall redstrbutve effect. Note that f the tax does not cause re-rankng APK R 0, then RE RS. The departure from proportonalty of the progressve taxaton can nstead be evaluated by the Kakwan ndex K C T G ndexes are lnked by the overall average tax rate, namely n n T x As a consequence, X. The Kakwan and the Reynolds-Smolensky (0) RS K. Ths formula tells us that the Reynolds-Smolensky ndex has two determnants: the overall average tax rate and the Kakwan ndex. In what follows we focus only on the Reynolds-Smolensky ndex and then we are nterested n fndng the best tax structure able to determne a gven tax revenue (smaller than the present one) and to yeld to the greatest RS whle gettng no loser taxpayers. Snce we mpose a reducton of the tax revenue, note that the value of wll be smaller than the present one. Note also that the smulated K wll be greater than the present one n order for RS to be the hghest. 4. The Data and the Statc Mcro-smulaton Model The mcro-smulaton model used n ths work estmated the most mportant taxes and contrbutons whch characterse the Italan fscal system. Here we employ the mcrosmulaton model module concernng the PIT. It consders as nput data those provded by the Bank of Italy n ts 0 Survey on Household Income and Wealth (hereafter, BI-SHIW). The survey contans nformaton on household ncome and wealth n the year 00, coverng 7,95 households and 9,836 ndvduals (Bank of Italy, 0). The sample s representatve of the Italan populaton, composed of about 4 mllon households and 60 mllon ndvduals. 9

The BI-SHIW provdes nformaton only on each ndvdual s dsposable ncome, whch consders tems of ncome that are taxed wthn the PIT or that can be exempt from the tax, as well as can be taxed under a separate regme. Therefore, the mcro-smulaton model frst dstngushes all ncomes ncluded n the PIT taxable ncome defnton, ncomes exempt from any taxes and ncomes taxed under a separate regme. Then the PIT gross ncome dstrbuton s evaluated, startng from the PIT net ncome dstrbuton. The transton from the post- to the pre-tax personal ncome of each ndvdual has been computed by applyng the algorthm proposed by Immervoll and O Donoghue (00). Usng orgnal sample weghts, the grossng-up procedure smply proportons the sum of ndvduals sample weghts to the dmenson of the populaton as estmated by the Natonal Statstcal Offce (ISTAT). Then the grossed-up number of PIT taxpayers has been obtaned by consderng ndvduals wth a postve gross ncome wthn the sample (3,79 taxpayers, correspondng to about 40 mllon n the populaton. Consderng the ncome unts, results concernng the PIT gross ncome dstrbuton are very close to the Mnstry of Fnance s (0) offcal statstcs, both consderng the gross ncome dstrbuton by ncome classes and the composton of PIT ncome unts by work status, as well as by ther mean gross ncome. In addton, the overall tax revenue resultng from the mcro-smulaton model (48.75 bllon euros) s very close to that showed n the offcal statstcs. Consderng all ndvdual taxpayers, Fgure compares the frequency densty functon obtaned wth the mcro-smulaton model and the one obtaned usng the Mnstry of Fnance s offcal data by ncome classes. Smlar pctures emerge consderng the frequency densty functon for pensoners and employees, as well as the self-employed. FIGURE AROUND HERE The column Present value of Table shows the nequalty ndces for ndvdual taxpayers n the 00 fscal year, whch s our reference stuaton for the Reynolds- Smolensky ndex maxmzaton. The Gn coeffcent for the gross ncome dstrbuton s 0.44338, whlst that for the net ncome dstrbuton s 0.3938. The overall redstrbutve effect RE s 0.0500. The concentraton coeffcent for the net ncome 0

dstrbuton s 0.39076, whlst that on the net tax lablty dstrbuton s 0.675; therefore, the Reynolds-Smolensky RS ndex s equal to 0.056 and the Kakwan ndex K s 0.877. The overall average tax rate s 0.8699, whlst the Atknson-Plotnck- Kakwan APK R ndex s equal to 0.0006. TABLE AROUND HERE Table (column Wth cash transfer ) also shows the overall redstrbutve mpact of PIT joned wth the cash transfer. favours only employees wth a PIT gross ncome n the range of 8-6 thousand euros (about 0.9 mllon taxpayers), as follows: 960 6,000 x 960,000 0 f f f m x 4,000 & 4,000 x x 6,000 & 6,000 & GT c ( x GT c ( x GT c ( x ) 0 It does not modfy the PIT structure at all. Benefcares obtan the same amount of PIT net tax lablty T. The net effect ) 0 ) 0, and contnue to pay z * y T s an ncrease n these taxpayers dsposable ncome by 960 euros per year, n the ncome range 8-4 thousands euros and a decreasng amount up to 6 thousand euros. 4 Note also that f T benefcares obtan a subsdy. Taxpayers other than employees, on the contrary, do not gan from ths cash transfer. The cost of ths measure s 9.3 bllon euros. Even f only one taxpayer out of four obtans the cash transfer, the redstrbutve effect of the PIT joned wth the transfer consderably mproves: the Reynolds-Smolensky RS ndex ncreases by 8.5 per cent; on the contrary, the overall redstrbutve effect RE ncreases only by 7.9 per cent, snce the Atknson-Plotnk-Kakwan R APK ndex worsens by about 53 per cent. On the effcency sde, the effectve margnal tax rate resultng n the ncome range of 4-6 thousand euros ncreases by up to 80 per cent; 4 There s a poltcal explanaton behnd ths choce. The Government announced that t would reduce PIT tax lablty by 80 euros per month for all taxpayers. Of course, such an announcement proved to be too expensve; moreover, t would have been very dffcult to reach through a PIT structure reform, snce t s hard to reform such a complex tax structure and, n the meantme, ensure an equal tax reducton for all taxpayers. In order to at least partally meet ts commtment, the Government decded to apply the 80 euros pledge only to a group of taxpayers, gven the revenue constrant of 9.3 bllon euros. In order not to modfy the PIT structure, t chose the cash transfer nstrument.

for all other ncome levels, on the contrary, the effectve margnal tax rates do not change wth respect to the present tax structure. By employng a genetc algorthm, n the next sectons, we show that a more equtyorented and more effcent reform s possble. From the methodologcal pont of vew, the specfc measure employed by the Italan Government s not partcularly nterestng; t refers only to employees and then, a few parameters of the tax could be smultaneously changed n a genetc algorthm framework,.e. parameters defnng the structure of the tax credt for employees. We dscuss an overall reform, whch consders all the parameters of the tax are able to be trmmed. Snce we splt the tax cut between all taxpayers, both the average reducton of the net tax lablty for all taxpayers and the value of transfer. APK R are smaller than that guaranteed to employees by the Government s cash 5. Genetc Algorthm 5.. General Overvew Genetc algorthms (henceforth, GAs) are a heurstc search whch belong to the feld of evolutonary algorthms, a subfeld of artfcal ntellgence. Snce ther ncepton (Holland, 975), GAs found a wealth of applcatons n the most vared research dscplnes, beyond computatonal scence, mathematcs, physcs, bonformatcs, etc. Applcatons n economcs also exst, broadly ncludng game theory, fnance related works, schedule optmzaton and whenever some sort of learnng mechansm s needed; hstorcally, the frst attempt at employng GAs n economcs s due to Mller s (986) research on adaptve behavour. To our knowledge, no prevous attempts at employng GAs for tax system optmzaton exst, to date. The huge solutons search space, whch s the aftermath of the combnatoral effect of very many parameters, posng a serous challenge to tradtonal optmzaton technques; brute force methods are out of the queston, just lke teratve methods (cfr. Newton s); GAs appear as an obvously approprate choce.

Canddate solutons, whch n the GA are nternally represented as ndvduals (each of them s characterzed by her own genome, a vector of chromosomes ) 5, are generated as an ntal populaton at random. Evolutonary operators teratvely select, crossbreed and mutate the best (most ft, accordng to an objectve functon called ftness functon ) ndvduals, n order to produce an offsprng of ndvduals the subsequent generaton that wll enter a new reproducton step. The ndvduals average ftness ncreases after every generaton, untl a satsfactory soluton s found. The stoppng crteron normally employed s related to the populaton homogenety: as the search process becomes closer to an optmum, the ndvduals become more and more smlar among them. The GA mplementaton employed n ths work s based on Python s open-source Pyevolve lbrary (Perone, 009); the populaton selecton mechansm across generatons s the standard roulette wheel (ftness proportonal) wth eltsm selecton (the very best ndvduals n each generaton are kept unaltered and carred over across successve generatons), whle the evolutonary crossover operator s a standard onepont. A low mutaton rate value and a hgh crossover rate have also been utlzed, n order to let the search process converge reasonably quckly on solutons, whlst mantanng the ablty to escape local maxma. The populaton sze vs. number of generatons trade-off has been tackled and solved, favourng a small populaton vs. numerous evolutonary steps. Detalng the trmmng of the GAs techncal parameters s out of the scope of ths work; suffce t to say, as agreed upon by a vast lterature, t s an ad-hoc process, to be performed mostly by tral and error, on every specfc search doman. 5.. The Structure of the Genetc Algorthm We Employ As the startng pont, we let the GA set up a populaton of 00 dfferent tax structures ( ndvduals from the GA pont of vew) and then we let t evolve them for 0,000 generatons. Consequently, the GA has to evaluate as much as mllon canddate solutons, applyng all these dfferent tax structures to the same pre-tax ncome 5 In the early GA mplementatons (BCGA, bnary-coded genetc algorthms), the solutons space had to be coded n bnary numbers; RCGA, real-coded genetc algorthms, allow workng wth varables n contnuous domans; cfr. Herrera et al. (998). 3

dstrbuton, composed of 3,79 taxpayers. We set the crossover rate as equal to 0.75, as well as the mutaton rate as equal to 0.05 and the eltsm selecton equal to 7. For each tax structure and each taxpayer, the GA computes all the relevant tax varables n the transton from the pre- to the post-tax ncome. For each tax structure, t then computes the overall tax revenue, the share of loser taxpayers by consderng each taxpayer s actual net tax lablty, the average loss for the loser taxpayers, as well as the Reynolds-Smolensky RS ndex (the four parts of the objectve ftness functon to maxmze, see below); t then saves all these resultng values, n addton to all the parameters of the tax, on a dump fle. We employ a computer powerful enough to evaluate mllon runs n about 4 days; the duraton of our average run s then 0.8 seconds. The GA has to maxmze a ftness functon. We employed RS e 0 ftness () where s the percentage devaton of the computed tax revenue of each run from the target one (39.44 bllon euros, 9.3 bllon euros less than the present tax revenue), whlst s the share of taxpayers losng wth the smulated tax structure, s the average loss (n euros) for the loser taxpayers,,, and are all postve parameters. We fx 575, 53, 0 and 50. We made several attempts, wth dfferent parameters and dfferent functonal forms; ths ftness functon has proved to be the most effectve. The exponental form of the ftness functon helps the GA to always converge to the best soluton, snce generaton after generaton, more-than-lnearly hgh scores are assgned to the most promsng canddate solutons. The frst term of the exponent shows the part of the ftness functon dependng on RS. We hghly favour RS wth respect to,, and, snce we are nterested n obtanng the hghest Reynolds-Smolensky ndex. The second and thrd, as well as the fourth terms of the exponent of the ftness functon, show the penaltes we mpose on the ftness value when, and became too large. As a result, the smaller, and, the more the ftness value ncreases. The parameter s crucal for the convergence of the GA: there are combnatoral spaces where RS ncreases even f ncreases; a low value for allows the GA to attrbute low scores to those canddate solutons. 4

When the GA has evolved for a reasonable number of generatons, we obtan 0, 0, the smallest, and the hghest RS. The parameters of the best structure of the tax can then be observed. Fgure shows the maxmum value of the ftness functon appled n ths work for each generaton. As can be noted, the maxmum value tends to ncrease generaton after generaton; top per thousand hghest values are reached startng from generaton,500. The hghest ftness value s 5.47, and t s reached n generaton 8,94. FIGURE AROUND HERE Each canddate tax structure s characterzed by 33 dfferent parameters (see Table 3), each of them related to a specfc parameter of the Italan PIT structure descrbed n Secton. In order to evaluate each of the 33 parameters, the GA couples 36 chromosomes that are values rangng from zero to. We now turn to descrbng how we let the GA trm each chromosome. Frst of all, we let the GA choose fve margnal tax rates as n the actual tax code. Gven the constrants we mpose, we know that the top margnal tax rate cannot be too much hgher than the present one (snce we mpose a no loser taxpayers constrant); moreover, we do not want t to be hgher than the present value. Conversely, we do not know the mnmum allowable value of the bottom margnal tax rate; we set the lowest margnal tax rate at not lower than 0 per cent (beng the present value equal to 3 per cent). In partcular, the GA randomly sets a group of sx chromosomes servng for the defnton of the fve margnal tax rates. It then adds them up n order to obtan a normalzaton value as follows: norm t 0.43 0.0 6. chromosome The GA fnally chooses the fve tax rates t wth (,,3, 4,5) as follows: t 0.0 chromosome * norm t. 5

We then set a second group of 5 chromosomes (7-) defnng the four upper lmts j LL j j UL of the thresholds, beng LL 0 by defnton and UL j UL. We appled an emprcal strategy smlar to that employed for the defnton of the margnal tax rates and we mpose UL 5, 00 (see on for the choce of ths value) and we let the hghest value of UL 4 be 75 thousand euros (as n the present tax structure). As a consequence, norm UL 75,000 5,00 7 chromosome and, for j,,3, 4, j ULj 5,00 chromosome * norm UL. Afterwards, we defne 5 chromosomes related to the tax credts structure. Startng at the tax credt for employees (Equaton (3)), we let the GA choose the no tax area m appled to employees, that s the lmt of pre-tax ncome below whch these taxpayers face a zero net tax lablty, between 8 thousand euros (the present value) and *UL : m,000chromosome * * UL 8,000. We set 8 8 5, equal to the present value. As a consequence, UL has to be set greater than or, at most equal, to 5 8 thousand euros beng *5,000 8,000 m. 5 Note that the choce of the parameter nfluences the equty-effcency trade-off. The hgher the value chosen by the GA, the more lkely the slope of the effectve margnal tax credt n the ncome level m r LL s hgh; therefore, the hgher the effectve margnal tax rate for ths ncome bandwdth. 6 The constrant we mpose on the share of loser taxpayers lets the GA chose the hghest admssble value for m and at the same tme keep under control the level of the effectve margnal tax rates (see on). The GA then choose m, m3, m4 as follows: 6 If we let the GA to run free wth respect to ths parameter, for example by lettng the GA choose t up to, a lower than value would be chosen (gven the constrants we mpose), a hgher redstrbutve effect would be obtaned, as well as a confscatory effectve tax rate for ncomes just above the no tax area. The reason s clear: ceters parbus, havng to maxmze the Reynolds- Smolensky ndex gven a revenue constrant, the larger the share of taxpayers wth a nl net tax lablty, the hgher the Reynolds- Smolensky ndex (and the hgher the tax rates and narrower the upper lmt of each threshold n order to obtan the target tax revenue). As a consequence, the larger the no tax area, the narrower the ncome bandwdth between the no tax area and the upper lmt of the bottom threshold, and then the more sharply the subsequent reducton of the effectve tax credt. 6

m m m 3 4 0.9375* m chromosome3 * 0.* 0.9375* m m chromosome * 0.96875* m m 0.6* m 4 chromosome 5 * 0.6m where 0. s arbtrarly chosen, whlst 0.9375 m m, m 0.96875 m 3, and 0.6 m m 4 are the correspondng values accordng to the present tax structure. In so dong, we let the GA choose rank of m r. m wth r (,,3,4 ) n a large combnatoral space preservng the present r Havng the GA chosen t and m r, note that the potental tax credts t mr are automatcally defned. We then let the GA choose also the parameters a r wth r (,,3) n the range 0 t m r by defnng chromosomes 6, 7, and 8 (as descrbed n Secton, parameter a 4 s equal to zero and we keep t unchanged). In so dong, a very large combnaton of tax credts for earned ncomes s allowable. Fnally, we always set the parameter b equal to zero. At present, t s appled only to employees and ts values range from 0 to 40 euros for levels of x belongng to the threshold 3-8 thousand euros. We prefer ths parameter to be fxed at zero snce, f t were postve, t would not let the tax credt under dscusson be a contnuous functon for all levels of x. Fnally, note that we do not let the effectve tax credts for earned ncome be pecewse decreasng wth respect to lmts, others than those observed n the rate schedule. If c x were pecewse decreasng wth respect to other thresholds, the number and the level of effectve margnal tax rates would not be under control, leadng to unpleasant and neffcent outcomes. We contnue by defnng specfc chromosomes, n order to set the combnatory space H for the three tax credts for dependent ndvduals wthn the household: x S c O x and c x. H Startng wth the tax credts for dependent chldren c x c,, we let the GA choose the potental level of the tax credts Hpl c n the range 600-3,000 euros (beng the present values rangng between 800 and,00 euros). Smlar to the choce of the tax rates and the 7

upper lmts of the thresholds, we then defne 5 dfferent chromosomes (from 9 to 3) to set the 4 knds of tax credts for dependent chldren. In so dong, we set specfc constrants, n order to let the potental tax credt be hgher for households wth more than 3 chldren and lower for those wth fewer than 3 chldren, as well as hgher for chldren aged 3 or less and lower for a chld aged more than 3. Then we ntroduce chromosomes 4 and 5 for the choce of parameters q and e. We let the GA choose q between 50 and 30 thousand euros (beng the present value equal to 95 thousand), and the parameter e between zero and 40 thousand euros (beng the present value equal to 5 thousand). Note that these are very large ranges, so that the GA can choose an extremely large set of combnatons. Fnally, the GA chooses chromosome 6 n order to set and,500 euros (beng the actual value,00 euros). S Turnng to the effectve tax credt for the spouse x n order for Sp c HF c between zero c, we generate chromosome 7 to range between 500 and thousand euros (beng the present value equal to 800 euros), and a further chromosome 8 n order for the parameter u to range between zero and Sp c. Chromosomes 7 and 8 let the effectve tax credt for the spouse be a non-ncreasng functon wth respect to x and let the GA choose among a very large combnaton of structures for ths tax credt. S Lookng at Equaton (5), ths effectve tax credt c x s pecewse lnearly decreasng wth respect to three thresholds: from zero to LL, from LL to LL3 w LL 4, and from w to k LL5. In order to defne w and k, we ntroduced chromosomes 9 and 30 as follows: w LL chromosome9 * LL4 LL k w chromosome 30 *(80,000 w) Fnally, concernng the tax credt for other dependent ndvduals wthn the household, we ntroduce chromosome 3 n order for 0.95* O c, and we mpose c x Hp to be zero f x k. c Op to range between 0.75* c and Hp to be lnearly decreasng between zero and k, and Afterwards, we let the GA choose chromosomes 3, 33 and 34 n order to set tax credts for tenants: 8

tenants chromosome 3 tenants *,000 chromosome33 *tenants tenants3 tenants chromosome34 *( tenants *) Accordng to the present tax code, these tax credts are appled to two ncome thresholds: 5,494 and 30,987 euros; we consder ths aspect by lettng them change accordng to UL and UL. Fnally, the GA chooses chromosomes 35 and 36 n order to set the percentage of the expenses the tax law admts as further tax credts for tems of expendture. We let t chose expendture between 0 and 40 per cent (at present equal to 9), and expendture between 0 and 70 per cent (at present equal to 36). 6. Results Table 3 shows all the parameters of the best tax structure able to maxmze the Reynolds-Smolensky RS ndex, gven that the tax revenue s 9.3 bllon euros lower than the present one and no taxpayers have to be worse off due to the tax reform. As can be noted, the bottom margnal tax rate t sgnfcantly decreases from 3 to 9.4 per cent; ths reducton lowers the gross tax lablty, not only for the poorest taxpayers but also for all the other taxpayers. t 4 decreases from 4 to 39.95 per cent, whlst t 5 remans unchanged as expected. The other two margnal tax rates ncrease: t from 7 to 7.96 per cent, and t 3 from 38 to 39.95 per cent. Note that t3 t4 : accordng to the GA, the hghest redstrbutve effect can be obtaned wth 4 nstead of 5 thresholds. In terms of the bandwdth of the thresholds, the frst one broadens from 0-5,000 to 0-,87 euros, whlst the second narrows from 5,000-8,000 to,87-6,454 euros. Then the orgnal thrd and fourth thresholds unte and the after-reform thrd threshold goes from 6,454 to 57,58 euros. Fnally, the top margnal tax rate s appled to ncomes above 57,58 euros nstead of 75 thousand euros. TABLE 3 AROUND HERE 9

The no tax area enlarges for all the four knds of taxpayer: m ncreases from 8,000 to,08 euros, m from 7,500 to 0,339, m 3 from 7,750 to 0,684, whlst m 4 rses from m 4,800 to 5,6. Note that 0. 480 s not set to ts maxmum value. UL The parameters defnng the shape of the effectve tax credts for earned ncome consderably ncrease: a from 50 to,4 euros, a from 470 to,008, and a 3 from 486 to,075 euros. Note that n all cases the GA sets ar tmr. Ths means that the shape of the tax credts for earned ncomes changes wth respect to those orgnally observed. Snce we mpose the tax credt c ( ) pecewse decreasng wth respect to LL and x LL 4, note also that after the tax reform, t s postve up to LL. As an example, Fgure 3 compares the effectve tax credts for employees before and after the tax reform. After the reform, the slope of the effectve tax credt s hgher (n absolute value) n the ncome range m LL. Ths shape also affects the level of the effectve margnal tax rates (EMTR) n ths ncome bandwdth, whch ncrease wth respect to the ones observed before the tax reform (Fgure 4). Therefore, an equty-effcency trade-off emerges: n order for the Reynolds-Smolensky ndex to be the hghest, we have to agree to hgher effectve margnal tax rates. FIGURE 3 AROUND HERE FIGURE 4 AROUND HERE S The shape of the tax credt for a spouse c x s very smlar to the present one (Fgure 5). The potental tax credt s a lttle bt hgher than before, and the effectve one s hgher n the ncome range 0-w; t becomes zero above 80 thousand euros. In terms of the tax credts for dependent chldren, the two tax credts for chldren f the dependent chldren wthn the household are 3 or less are very smlar to those observed before the tax reform: Hp c decreases from 800 to 788 euros, whlst Hp c ncreases from 0

900 to 907. The other two tax credts are sgnfcantly hgher: to,687, whlst Hp4 c from,00 to,65 euros. Hp3 c ncreases from,000 FIGURE 5 AROUND HERE Note also that snce Hp3 c and Hp4 c are so much hgher than before, HF c can be set equal to zero. The ncome lmts above whch ths tax credt becomes zero, also change: q s equal to 86,803, whlst e s equal to 5,55. A smlar pcture emerges when consderng the tax credt for other dependent ndvduals wthn a household: the potental tax credt s a lttle lower than the present value (748 euros), and s postve for ncome below 80 thousand euros. Focusng on the remanng parameters of the tax, the tax credt for tenants wth gross ncome below UL s approxmately the same (30 euros aganst 300); the tax credt for tenants wth ncome n the range UL UL s set to 09 euros nstead of 50, whlst the tax credt for younger tenants s lower (589 euros nstead of 99). Fnally, the percentages of expenses the tax law admts as a tax credt also reman relatvely unchanged: 8.54 per cent nstead of 9 per cent and 4.7 per cent nstead of 36 per cent. Very few taxpayers are worse off as a result of ths tax reform (. per cent and, on average, they lose 43 euros per year), whlst 4.5 per cent are unaffected (we consder as unaffected the taxpayers for whom the absolute value of the computed net tax lablty dffers from the present one by, at most, one euro). The remanng 74.4 per cent of taxpayers gan from the reform. Lookng at the composton of the tax cut n terms of ncome classes (Table 4), 85.6 per cent of the tax cut favours taxpayers n the ncome range 8-8 thousands euro, whlst.5 per cent favours taxpayers wth lower ncomes. Ths s due to the fact that the Italan personal ncome tax system does not admt negatve ncome taxaton; therefore, taxpayers wth a nl net tax lablty (almost all taxpayers wth ncome lower than 8 thousand euros) are not affected by the tax reform. If the Italan PIT allowed negatve ncome taxaton, the tax reform would show a dfferent dstrbuton n terms of the tax cut among ncome classes: n partcular, there would be lower gans for the top ncome earners and hgher gans for the bottom ones.

TABLE 4 AROUND HERE Only 9 per cent of the tax cut favours taxpayers wth ncomes n the range 8-55 thousand euros, whlst the remanng 3. per cent favours rcher taxpayers. It can be observed that the RS could be hgher, were the (low) gans of the rcher taxpayers transferred to the poorest ones. Gven the structure of chromosomes descrbed n subsecton 5., ths s not possble, or at most, not lkely, snce the GA has to balance the effects on RS,, and due to 33 parameters. Fnally, Table 5 compares the nequalty ndexes for taxpayers accordng to the present tax structure, and those obtaned by applyng the new structure of the tax to the same pre-tax ncome dstrbuton. As can be noted, RS s 8.3 per cent hgher than the present value: snce the overall average tax rate decreases from 8.70 per cent to 7.53 per cent, the Kakwan ndex ncreases by 7. per cent. Note also that ths tax reform postvely affects APK R. TABLE 5 AROUND HERE 7. Concludng Remarks In ths paper, we propose a new methodology to mplement a personal ncome tax reform. In partcular, gven a settled tax cut decded upon by the Government, (note that a smlar strategy can be appled f the tax revenue ncreases), we show how a genetc algorthm can be employed, n order to fnd out the values of all parameters defnng the structure of the personal ncome tax able to satsfy a specfc target. Our methodology can be appled to any other specfc target; as an example, n ths work our target s the maxmzaton of the redstrbutve effect of the tax, whle preventng all taxpayers beng worse off wth respect to the present tax structure. We apply ths methodology to the Italan personal ncome taxaton system for two reasons: the tax structure s qute complcated, and recently the Government decded to reduce tax revenue by about 9.3 bllon euro startng from 05. The am of ths tax cut s to ncrease the purchasng

power of poor taxpayers and taxpayers belongng to the mddle class, and the nstrument s the ntroducton of a cash transfer (not related to the structure of the personal ncome tax) only for employees wth gross ncomes n the range 8-6 thousand euros (n order for the yearly gan to be about one thousand euros), whlst all other knds of taxpayer are not affected by ths money transfer. Here we show that a better and more equty-orented reform s possble. Ths methodology allows a short run reform, and can help polcy makers when they thnk of a tax reform. 3

References Bank of Italy (0). Household Income and Wealth n 00, Supplements to the Statstcal Bulletn, XXII (New Seres), No. 6. Holland, J. H. (99). Adaptaton n Natural and Artfcal Systems: an Introductory Analyss wth Applcatons to Bology, Control, and Artfcal Intellgence, MIT Press, Cambrdge, MA; frst edton (975). Unversty of Mchgan Press, Ann Arbor, MI. Herrera F., Lozano M. and Verdegay J.L. (998). Tacklng Real-Coded Genetc Algorthms: Operators and Tools for Behavoural Analyss, Artfcal Intellgence Revew, Vol. (4), pp. 65-39. Immervoll, H. and O Donoghue, C. (00). Imputaton of Gross Amounts from Net Incomes n Households Surveys. An applcaton usng EUROMOD, Workng paper EM, EUROMOD. Lambert, P. J. (00). The Dstrbuton and Redstrbuton of Income, Thrd edton, Manchester Unversty Press, Manchester and New York. Mller, J. H. (986). A Genetc Model of Adaptve Economc Behavor, Unversty of Mchgan Workng Paper, Ann Arbor, MI. Mnstry of Fnance, Department of Fnance (0). Statstcal Reports. Pellegrno, S., Pacenza, M. and Turat G. (0). Developng a Statc Mcrosmulaton Model for the Analyss of Housng Taxaton n Italy, The Internatonal Journal of Mcrosmulaton, Vol. 4(), pp. 73-85. Perone, C. S. (009). Pyevolve: a Python Open-Source Framework for Genetc Algorthms, ACM SIGEVOluton Newsletter, vol. 4, n., pp. -0, ACM, New York. 4

Fgure : Frequency densty functon for all ndvdual taxpayers Frequency densty functon 0 8 6 4 0 8 6 4 0 0 0000 40000 60000 80000 00000 PIT gross ncome (euro) Mnstry of Fnance Mcrosmulaton model Fgure : Maxmum values of the ftness functon 8 6 Maxmum ftness value 4 0 8 6 4 0 500 000 500 000 500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 0000 Generaton 5

Fgure 3: The effectve tax credt for an employee Tax credt (euros) 400 00 000 800 600 400 00 000 800 600 400 00 0 0 5000 0000 5000 0000 5000 30000 35000 40000 45000 50000 55000 60000 Gross ncome (euro) 00 Tax credt GA Tax credt Fgure 4: The effectve margnal tax rates for a celbate employee wthout chldren 50 45 40 35 30 5 0 5 0 5 0 0 0000 0000 30000 40000 50000 EMTR 60000 70000 80000 90000 00000 Gross ncome (euros) 00 EMTR GA EMTR 6

Fgure 5: The effectve tax credt for a spouse 000 900 800 700 600 500 400 300 00 00 0 0 0000 0000 30000 Tax credt (euros) 40000 50000 60000 70000 80000 90000 Gross ncome (euros) 00 Tax credt GA Tax credt 7

Threshold (j) Source: Italan Tax Code. Table : Rate schedule Lower lmt (LL) Taxable ncome (euros) Upper lmt (UL) Tax rate (%) (t) 0 5,000 3 5,000 8,000 7 3 8,000 55,000 38 4 55,000 75,000 4 5 75,000-43 Index Table : Inequalty ndexes for taxpayers Present value Wth cash transfer Absolute dfference Percentage dfference Gn coeffcent for the gross ncome 0.44338 0.44338 0.00000 0.0 Gn coeffcent for the net ncome 0.3938 0.3877-0.004 -. Concentraton coeffcent for the net ncome 0.39076 0.3863-0.00445 -. Gn coeffcent for the net tax lablty 0.6850 0.7355 0.0504 7.5 Concentraton coeffcent for the net tax lablty 0.675 0.787 0.0397 5.9 Redstrbutve effect 0.0500 0.056 0.004 7.9 Reynolds-Smolensky ndex 0.056 0.05707 0.00445 8.5 Kakwan ndex 0.877 0.6849 0.0397 7.4 Atknson-Plotnk-Kakwan ndex 0.0006 0.00095 0.00033 53.3 Average tax rate (%) 8.70 7.53 -.7-6.3 Source: Own elaboratons based on BI-SHIW. 8

Table 3: Present and computed parameters of the tax Present Best Parameters value value t 0.3 0.94 t 0.7 0.7955 t 3 0.38 0.3995 t 4 0.4 0.3995 t 5 0.43 0.43000 UL 5,000,87.0 UL 8,000 6,453.94 UL 3 55,000 44,96.30 UL 4 75,000 57,57.63 m 8,000,08.4 m 7,500 0,339.3 m 3 7,750 0,683.77 m 4 4,800 5,5.9 a 50,4.85 a 470,007.98 a 3 486,074.9 Sp c 800 868.8 u 0 53.46 w 40,000 9,870. k 80,000 80,000.00 Op c 750 748.8 Hp c 800 787.56 Hp c 900 907. Hp3 c,000,687.3 Hp4 c,00,65.06 q 95,000 86,80.79 e 5,000 5,54.69 HF c,00 0.00 tenants 300 30.47 tenants 50 08.55 tenants 3 99 588.65 expendtures 0.9 0.8538 expendtures 0.36 0.47 Source: Own elaboratons based on BI-SHIW. 9

Income class (thousand euros) Composton of the tax cut (%) Table 4: The composton of the tax cut by ncome classes Wnner (%) Indfferent (%) Loser (%) Total (%) Average wn (euros) Average loss (euros) 0-8.5 6. 9.3 0.0 5.3 9 0 8-5 40.5 7.3 4.7 0.0.9 54 0 5-8 45. 34.4 0.4 0.5 35.3 94 5 8-55 9.0 3.0 0. 0.5 3.5 58 39 55-75.8. 0.0 0.0. 90 4 above 75 0.3.5 0.0 0..7 4 4 Total 00.0 74.4 4.5. 00.0 307 43 Source: Own elaboratons based on BI-SHIW. 30

Index Table 5: Inequalty ndexes for taxpayers Present value Best tax structure Absolute dfference Percentage dfference Gn coeffcent for the gross ncome 0.44338 0.44338 0.00000 0.0 Gn coeffcent for the net ncome 0.3938 0.38696-0.0044 -. Concentraton coeffcent for the net ncome 0.39076 0.38639-0.00437 -. Gn coeffcent for the net tax lablty 0.6850 0.700 0.03870 5.7 Concentraton coeffcent for the net tax lablty 0.675 0.750 0.03935 5.9 Redstrbutve effect 0.0500 0.0564 0.0044 8.5 Reynolds-Smolensky ndex 0.056 0.05699 0.00437 8.3 Kakwan ndex 0.877 0.68 0.03935 7. Atknson-Plotnk-Kakwan ndex 0.0006 0.00057-0.00005-8.3 Average tax rate (%) 8.70 7.53 -.7-6.3 Source: Own elaboratons based on BI-SHIW. 3

Our papers can be downloaded at: http://cerp.carloalberto.org/en/publcatons N 47/4 Matteo Morn Smone Pellegrno N 46/4 Maracrstna Ross Eva Sermnska N 45/4 Johannes G. Hoogeveen Maracrstna Ross Daro Sansone CeRP Workng Paper Seres Personal Income Tax Reforms: a Genetc Algorthm Approach Sngle agan? Asset and portfolo changes due to wdowhood shock Drvers of performance n prmary educaton n Togo N 44/4 Elsa Fornero Economc-fnancal lteracy and (sustanable) penson reforms: why the former s a key ngredent for the latter N 43/4 Kees de Vaan Danele Fano Heralt Mens Govanna Ncodano N 4/4 Elsabetta Cagna Gulo Casucco N 4/4 Massmo Baldn Costanza Torrcell Mara Cesra Urzì Brancat N 40/4 Cecla Boggo Elsa Fornero Henrette Prast Jose Sanders N 39/4 Laura Banchn Margherta Borella A Reportng Standard for Defned Contrbuton Penson Plans Equally-weghted Rsk Contrbuton Portfolos: an emprcal study usng expected shortfall Famly tes: occupatonal responses to cope wth a household ncome shock Seven Ways to Knt Your Portfolo: Is Investor Communcaton Neutral? Cogntve Functonng and Retrement n Europe N 38/3 Claudo Morana Insghts on the global macro-fnance nterface: Structural sources of rsk factors fluctuatons and the cross-secton of expected stock returns N 37/3 Claudo Morana New Insghts on the US OIS Spreads Term Structure Durng the Recent Fnancal Turmol N 36/3 Anna Lo Prete Inequalty and the fnance you know: does economc lteracy matter? N 35/3 Rk Dllngh Henrette Prast Maracrstna Ross Cesra Urzì Brancat N 34/3 Annamara Lusard Olva S. Mtchell N 33/3 Annamara Lusard Perre-Carl Mchaud Olva S. Mtchell N 3/3 Rccardo Calcagno Sona Falconer N 3/3 Rccardo Calcagno Mara Cesra Urzì Brancat The psychology and economcs of reverse mortgage atttudes: evdence from the Netherlands The Economc Importance of Fnancal Lteracy: Theory and Evdence Optmal Fnancal Knowledge and Wealth Inequalty Competton and dynamcs of takeover contests Do more fnancally lterate households nvest less n housng? Evdence from Italy

N 30/ Maela Gofré Fnancal Educaton, Investor Protecton and Internatonal Portfolo Dversfcaton N 9/ Mchele Bellon Rob Alesse Adraan Kalwj Chara Marnacc N 8/ Fabo Cesare Baglano Claudo Morana N 7/ Maracrstna Ross Serena Trucch N 6/ Margherta Borella Flava Coda Moscarola Maracrstna Ross Lfetme Income and Old Age Mortalty Rsk n Italy over Two Decades Determnants of US Fnancal Fraglty Condtons Lqudty Constrants and Labor Supply (Un)expected retrement and the consumpton puzzle N 5/ Carolna Fugazza Trackng the Italan employees TFR over ther workng lfe careers N 4/ Agnese Romt Maracrstna Ross N 3/ Elsa Fornero Mara Crstna Ross Mara Cesra Urzì Brancat Should we Retre Earler n order to Look After our Parents? The Role of mmgrants Explanng why, rght or wrong, (Italan) households do not lke reverse mortgages N / Serena Trucch How credt markets affect homeownershp: an explanaton based on dfferences between Italan regons N / Elsa Fornero Chara Montcone Serena Trucch The effect of fnancal lteracy on mortgage choces N 0/ Govann Mastrobuon Flppo Tadde N 9/ Maarten van Rooj Annamara Lusard Rob Alesse N 8/ Luca Beltramett Matteo Della Valle N 7/ Rccardo Calcagno Chara Montcone N 6/ Annamara Lusard Danel Schneder Peter Tufano N 5/ Adele Atknson Flore-Anne Messy N 4/ Leora Klapper Georgos A. Panos N 3/ Dana Crossan Davd Fesler Roger Hurnard N / Johan Almenberg Jenny Säve-Söderbergh N / Elsa Fornero Chara Montcone Age Before Beauty? Productvty and Work vs. Senorty and Early Retrement Fnancal Lteracy, Retrement Plannng, and Household Wealth Does the mplct penson debt mean anythng after all? Fnancal Lteracy and the Demand for Fnancal Advce Fnancally Fragle Households: Evdence and Implcatons Assessng fnancal lteracy n countres: an OECD Plot Exercse Fnancal Lteracy and Retrement Plannng n Vew of a Growng Youth Demographc: The Russan Case Fnancal Lteracy and Retrement Plannng n New Zealand Fnancal Lteracy and Retrement Plannng n Sweden Fnancal Lteracy and Penson Plan Partcpaton n Italy

N 0/ Rob Alesse Maarten Van Rooj Annamara Lusard Fnancal Lteracy, Retrement Preparaton and Penson Expectatons n the Netherlands N 09/ Tabea Bucher-Koenen Fnancal Lteracy and Retrement Plannng n Germany Annamara Lusard N 08/ Shzuka Sekta Fnancal Lteracy and Retrement Plannng n Japan N 07/ Annamara Lusard Fnancal Lteracy and Retrement Plannng n the Unted States Olva S. Mtchell N 06/ Annamara Lusard Fnancal Lteracy Around the World: An Overvew Olva S. Mtchell N 05/ Agnese Romt Immgrants-Natves Complementartes n Producton: Evdence from Italy N 04/ Ambrogo Rnald Penson awareness and naton-wde auto-enrolment: the Italan experence N 03/0 Fabo Baglano Claudo Morana N 0/0 Nuno Cassola Claudo Morana The Great Recesson: US dynamcs and spllovers to the world economy The 007-? fnancal crss: a money market perspectve N 0/0 Tetyana Dubovyk Macroeconomc Aspects of Italan Penson Reforms of 990s N 00/0 Laura Patt Guseppe Rocco N 99/0 Fabo Baglano Claudo Morana N 98/0 Annamara Lusard Danel Schneder Peter Tufano N 97/0 Carlo Maccheron Tzana Barugola N 96/0 Rccardo Calcagno Maracrstna Ross N 95/0 Flava Coda Moscarola Elsa Fornero Maracrstna Ross N 94/0 John A. Lst Sally Sadoff Maths Wagner L educazone e la comuncazone prevdenzale - Il caso talano The effects of US economc and fnancal crses on euro area convergence The Economc Crss and Medcal Care Usage E se l aspettatva d vta contnuasse la sua crescta? Alcune potes per le generazon talane 950-005 Portfolo Choce and Precautonary Savngs Parents/chldren deals : Inter-Vvos Transfers and Lvng Proxmty So you want to run an experment, now what? Some Smple Rules of Thumb for Optmal Expermental Desgn N 93/0 Maths Wagner The Heterogeneous Labor Market Effects of Immgraton N 9/0 Rob Alesse Mchele Bellon N 9/09 Annamara Lusard Olva S. Mtchell Vlsa Curto N 90/09 Annamara Lusard Olva S. Mtchell Retrement choces n Italy: what an opton value model tells us Fnancal Lteracy among the Young: Evdence and Implcatons for Consumer Polcy How Ordnary Consumers Make Complex Economc Decsons: Fnancal Lteracy and Retrement Readness N 89/09 Elena Vgna Mean-varance neffcency of CRRA and CARA utlty functons for portfolo selecton n defned contrbuton penson schemes

N 88/09 Maela Gofré Convergence of EMU Equty Portfolos N 87/09 Elsa Fornero Annamara Lusard Chara Montcone N 86/09 Margherta Borella Flava Coda Moscarola N 85/09 Cathal O Donoghue John Lennon Adequacy of Savng for Old Age n Europe Mcrosmulaton of Penson Reforms: Behavoural versus Nonbehavoural Approach The Lfe-Cycle Income Analyss Model (LIAM): A Study of a Flexble Dynamc Mcrosmulaton Modellng Computng Framework Stephen Hynes N 84/09 Luca Spataro Il sstema prevdenzale talano dallo shock petrolfero del 973 al Trattato d Maastrcht del 993 N 83/09 Annamara Lusard Peter Tufano N 8/09 Carolna Fugazza Massmo Gudoln Govanna Ncodano N 8/09 Fabo Baglano Claudo Morana Debt Lteracy, Fnancal Experences, and Overndebtedness Tme and Rsk Dversfcaton n Real Estate Investments: Assessng the Ex Post Economc Value Permanent and Transtory Dynamcs n House Prces and Consumpton: Cross-Country Evdence N 80/08 Claudo Campanale Learnng, Ambguty and Lfe-Cycle Portfolo Allocaton N 79/08 Annamara Lusard Increasng the Effectveness of Fnancal Educaton n the Workplace N 78/08 Margherta Borella Govanna Segre N 77/08 Govann Guazzarott Petro Tommasno N 76/08 Rccardo Calcagno Elsa Fornero Maracrstna Ross N 75/08 Harold Alderman Johannes Hoogeveen Maracrstna Ross Le penson de lavorator parasubordnat: prospettve dopo un decenno d gestone separata The Annuty Market n an Evolvng Penson System: Lessons from Italy The Effect of House Prces on Household Savng: The Case of Italy Preschool Nutrton and Subsequent Schoolng Attanment: Longtudnal Evdence from Tanzana N 74/08 Maela Gofré Informaton Asymmetres and Foregn Equty Portfolos: Households versus Fnancal Investors N 73/08 Mchele Bellon Rob Alesse N 7/08 Annamara Lusard Olva Mtchell The Importance of Fnancal Incentves on Retrement Choces: New Evdence for Italy Plannng and Fnancal Lteracy: How Do Women Fare? N 7/07 Flava Coda Moscarola Women partcpaton and carng decsons: do dfferent nsttutonal frameworks matter? A comparson between Italy and The Netherlands N 70/07 Radha Iyengar Govann Mastrobuon N 69/07 Carolna Fugazza Massmo Gudoln Govanna Ncodano N 68/07 Massmo Gudoln Govanna Ncodano The Poltcal Economy of the Dsablty Insurance. Theory and Evdence of Gubernatoral Learnng from Socal Securty Admnstraton Montorng Investng n Mxed Asset Portfolos: the Ex-Post Performance Small Caps n Internatonal Dversfed Portfolos

N 67/07 Carolna Fugazza Maela Gofré Govanna Ncodano Internatonal Dversfcaton and Labor Income Rsk N 66/07 Maarten van Rooj Annamara Lusard Rob Alesse Fnancal Lteracy and Stock Market Partcpaton N 65/07 Annamara Lusard Household Savng Behavor: The Role of Lteracy, Informaton and Fnancal Educaton Programs (Updated verson June 08: Fnancal Lteracy: An Essental Tool for Informed Consumer Choce? ) N 64/07 Carlo Casarosa Luca Spataro Rate of Growth of Populaton, Savng and Wealth n the Basc Lfe-cycle Model when the Household s the Decson Unt N 63/07 Claudo Campanale Lfe-Cycle Portfolo Choce: The Role of Heterogeneous Under- Dversfcaton N 6/07 Margherta Borella Elsa Fornero Maracrstna Ross Does Consumpton Respond to Predcted Increases n Cash-onhand Avalablty? Evdence from the Italan Severance Pay N 6/07 Irna Kovrova Effects of the Introducton of a Funded Pllar on the Russan Household Savngs: Evdence from the 00 Penson Reform N 60/07 Rccardo Cesar Guseppe Grande Fabo Panetta N 59/07 Rccardo Calcagno Roman Kraeussl Chara Montcone N 58/07 Elsa Lucano Jaap Spreeuw Elena Vgna N 57/07 Govann Mastrobuon Matthew Wenberg N 56/07 John A. Turner Satyendra Verma La Prevdenza Complementare n Itala: Caratterstche, Svluppo e Opportuntà per Lavorator An Analyss of the Effects of the Severance Pay Reform on Credt to Italan SMEs Modellng Stochastc Mortalty for Dependent Lves Heterogenety n Intra-Monthly Consumpton. Patterns, Self- Control, and Savngs at Retrement Why Some Workers Don t Take 40(k) Plan Offers: Inerta versus Economcs N 55/06 Antono Abatemarco On the Measurement of Intra-Generatonal Lfetme Redstrbuton n Penson Systems N 54/06 Annamara Lusard Olva S. Mtchell Baby Boomer Retrement Securty: The Roles of Plannng, Fnancal Lteracy, and Housng Wealth N 53/06 Govann Mastrobuon Labor Supply Effects of the Recent Socal Securty Beneft Cuts: Emprcal Estmates Usng Cohort Dscontnutes N 5/06 Lug Guso Informaton Acquston and Portfolo Performance Tullo Jappell N 5/06 Govann Mastrobuon The Socal Securty Earnngs Test Removal. Money Saved or Money Spent by the Trust Fund? N 50/06 Andrea Buffa Chara Montcone Do European Penson Reforms Improve the Adequacy of Savng? N 49/06 Maracrstna Ross Examnng the Interacton between Savng and Contrbutons to Personal Penson Plans. Evdence from the BHPS N 48/06 Onorato Castellno Elsa Fornero Publc Polcy and the Transton to Prvate Penson Provson n the Unted States and Europe

N 47/06 Mchele Bellon Carlo Maccheron N 46/05 Annamara Lusard Olva S. Mtchell Actuaral Neutralty when Longevty Increases: An Applcaton to the Italan Penson System Fnancal Lteracy and Plannng: Implcatons for Retrement Wellbeng N 45/05 Claudo Campanale Increasng Returns to Savngs and Wealth Inequalty N 44/05 Henrk Cronqvst Advertsng and Portfolo Choce N 43/05 John Beshears James J. Cho Davd Labson Brgtte C. Madran N 4/05 Margherta Borella Flava Coda Moscarola N 4/05 Massmo Gudoln Govanna Ncodano The Importance of Default Optons for Retrement Savng Outcomes: Evdence from the Unted States Dstrbutve Propertes of Pensons Systems: a Smulaton of the Italan Transton from Defned Beneft to Defned Contrbuton Small Caps n Internatonal Equty Portfolos: The Effects of Varance Rsk. N 40/05 Carolna Fugazza Massmo Gudoln Govanna Ncodano Investng for the Long-Run n European Real Estate. Does Predctablty Matter? N 39/05 Anna Rta Bacnello Modellng the Surrender Condtons n Equty-Lnked Lfe Insurance N 38/05 Carolna Fugazza Federca Teppa An Emprcal Assessment of the Italan Severance Payment (TFR) N 37/04 Jay Gnn Actuaral Farness or Socal Justce? A Gender Perspectve on Redstrbuton n Penson Systems N 36/04 Laurence J. Kotlkoff Pensons Systems and the Intergeneratonal Dstrbuton of Resources N 35/04 Monka Bütler Olva Huguenn Federca Teppa What Trggers Early Retrement. Results from Swss Penson Funds N 34/04 Chourouk Houss Le Vellssement Démographque : Problématque des Régmes de Penson en Tunse N 33/04 Elsa Fornero Carolna Fugazza Gacomo Ponzetto N 3/04 Angelo Marano Paolo Sestto A Comparatve Analyss of the Costs of Italan Indvdual Penson Plans Older Workers and Pensoners: the Challenge of Ageng on the Italan Publc Penson System and Labour Market N 3/03 Gacomo Ponzetto Rsk Averson and the Utlty of Annutes N 30/03 Bas Arts A Swtch Crteron for Defned Contrbuton Penson Schemes Elena Vgna N 9/0 Marco Taboga The Realzed Equty Premum has been Hgher than Expected: Further Evdence N 8/0 Luca Spataro New Tools n Mcromodelng Retrement Decsons: Overvew and Applcatons to the Italan Case N 7/0 Renhold Schnabel Annutes n Germany before and after the Penson Reform of 00 N 6/0 E. Phlp Davs Issues n the Regulaton of Annutes Markets N 5/0 Edmund Cannon Ian Tonks The Behavour of UK Annuty Prces from 97 to the Present

N 4/0 Laura Ballotta Valuaton of Guaranteed Annuty Converson Optons Steven Haberman N 3/0 Ermanno Ptacco Longevty Rsk n Lvng Benefts N /0 Chrs Soares Mark Warshawsky N /0 Olva S. Mtchell Davd McCarthy Annuty Rsk: Volatlty and Inflaton Exposure n Payments from Immedate Lfe Annutes Annutes for an Ageng World N 0/0 Mauro Mastrogacomo Dual Retrement n Italy and Expectatons N 9/0 Paolo Battoccho Francesco Menoncn Optmal Portfolo Strateges wth Stochastc Wage Income and Inflaton: The Case of a Defned Contrbuton Penson Plan N 8/0 Francesco Daver Labor Taxes and Unemployment: a Survey of the Aggregate Evdence N 7/0 Rchard Dsney and Sarah Smth N 6/0 Estelle James and Xue Song The Labour Supply Effect of the Abolton of the Earnngs Rule for Older Workers n the Unted Kngdom Annutes Markets Around the World: Money s Worth and Rsk Intermedaton N 5/0 Estelle James How Can Chna Solve st Old Age Securty Problem? The Interacton Between Penson, SOE and Fnancal Market Reform N 4/0 Thomas H. Noe Investor Actvsm and Fnancal Market Structure N 3/0 Mchela Scatgna Insttutonal Investors, Corporate Governance and Penson Funds N /0 Roberta Romano Less s More: Makng Shareholder Actvsm a Valuable Mechansm of Corporate Governance N /0 Mara Facco and Amezane Lasfer N 0/0 Vncenzo Andrett and Vncent Hldebrand Insttutonal Shareholders and Corporate Governance: The Case of UK Penson Funds Penson Portablty and Labour Moblty n the Unted States. New Evdence from the SIPP Data N 9/0 Hans Blommesten Ageng, Penson Reform, and Fnancal Market Implcatons n the OECD Area N 8/0 Margherta Borella Socal Securty Systems and the Dstrbuton of Income: an Applcaton to the Italan Case N 7/0 Margherta Borella The Error Structure of Earnngs: an Analyss on Italan Longtudnal Data N 6/0 Flava Coda Moscarola The Effects of Immgraton Inflows on the Sustanablty of the Italan Welfare State N 5/0 Vncenzo Andrett Occupatonal Pensons and Interfrm Job Moblty n the European Unon. Evdence from the ECHP Survey N 4/0 Peter Damond Towards an Optmal Socal Securty Desgn N 3/00 Emanuele Baldacc Luca Inglese N /00 Per Marco Ferrares Elsa Fornero Le caratterstche soco economche de pensonat n Itala. Anals della dstrbuzone de reddt da pensone (only avalable n the Italan verson) Socal Securty Transton n Italy: Costs, Dstorsons and (some) Possble Correcton N /00 Gudo Menzo Optng Out of Socal Securty over the Lfe Cycle