Paternalism against Veblen: Optimal Taxation and Non-Respected Preferences for Social Comparisons **

Transcription

1 atenalsm Aganst Veblen atenalsm aganst Veblen: Optmal Taaton and Non-Respected efeences fo Socal Compasons ** Thomas Aonsson * and Olof Johansson-Stenman Januay 07 Abstact Ths pape compaes optmal nonlnea ncome ta polces of welfast and patenalst govenments, whee the latte does not espect ndvdual pefeences egadng elatve consumpton. Consstent wth pevous fndngs, elatve consumpton concens unde welfasm typcally mply hghe magnal ncome ta ates. Remakably, the optmal magnal ta ules ae vey smla n the patenalst case. Fo eample, f elatve consumpton concens ae based on mean value compasons and all consumes ae equally postonal, then the fst-best ta ules ae dentcal between the govenments. Etensve numecal smulatons supplement the theoetcal esults, and make t possble to compae also ta levels and oveall edstbuton. Keywods: atenalsm; nonlnea taaton; edstbuton; status; postonal goods JEL Classfcaton: D6, H, H3 ** The authos would lke to thank Dan Slveman, Lay Samuelson and fou anonymous efeees fo vey helpful comments and suggestons. Reseach gants fom the Bank of Sweden Tecentenay Foundaton, the Swedsh Councl fo okng Lfe and Socal Reseach, and the Swedsh Ta Agency (all of them though poject numbe RS0-39:) ae also gatefully acknowledged. * Addess: Depatment of Economcs, Umeå School of Busness and Economcs, Umeå Unvesty, SE Umeå, Sweden. E-mal: Thomas.Aonsson@econ.umu.se Addess: Depatment of Economcs, School of Busness, Economcs and Law, Unvesty of Gothenbug, SE Gothenbug, Sweden. E-mal: Olof.Johansson@economcs.gu.se

2 atenalsm Aganst Veblen Eve snce the wtngs of Adam Smth n the 8 th centuy, t has been well-known n economcs that people cae about status and socal compasons and that elatve consumpton mattes to most people. Ta and othe polcy mplcatons of such compasons have moe ecently been eploed fom dffeent ponts of depatue n a numbe of studes, ncludng Boskn and Sheshnsk (978), Oswald (983), Tuomala (990), esson (995), Coneo and Jeanne (997), Ljungqvst and Uhlg (000), Dupo and Lu (003), Abel (005), Fank (008), Aonsson and Johansson-Stenman (008, 00, 05), and Kanbu and Tuomala (03). A typcal fndng n ths lteatue s that the etenaltes geneated by elatve consumpton concens motvate consdeably hghe magnal ta ates than n the conventonal model of optmal taaton wthout socal compasons. Howeve, as s always the case, the theoetcal esults depend on the undelyng assumptons. A common assumpton n all of these studes s that the ta polcy s decded by a welfast govenment,.e., a govenment that fully espects all aspects of consume pefeences, ncludng concens about elatve consumpton. hle the welfast assumpton n nomatve economc analyss s standad, and often seen as uncontovesal, one may ague that ths assumpton s less obvous when t comes to socal compasons. Indeed, seveal authos, ncludng Sen (979), Hasany (98), and Goodn (986), ague that the govenment should not espect ant-socal pefeences. Hasany (98, p. 56) specfcally mentons envy as an eample of ant-socal pefeences and states that such pefeences must altogethe be ecluded fom ou socal-utlty functon. Goodn (986) smlaly dscusses how to launde pvate pefeences n ode to make them sutable as aguments n the govenmental objectve functon. Snce postonal concens mply that an ndvdual s utlty depends negatvely on othe people s consumpton, one can ntepet such concens n tems of envy. Followng Hasany and Goodn, one could then ague that the govenment should not espect such pefeences and hence not nclude the effects Ths agument also fnds suppot n ecent eseach on happness and questonnae-based epements showng that elatve consumpton s an mpotant detemnant of ndvdual well-beng (e.g., Easteln 995, 00; Johansson-Stenman et al. 00; Blanchflowe and Oswald 004; Fee-- Cabonell 005; Solnck and Hemenway 005; Calsson et al. 007; Clak and Senk 00), whle Cad et al. (0) found a causal effect of pee salay on ndvdual well-beng based on a natual epement.

3 atenalsm Aganst Veblen of elatve consumpton n the socal objectve functon. Yet othe authos, such as Blackoby et al. (005), have eplctly agued aganst the vew of Hasany and Goodn and nstead poposed that the govenment should espect pefeences that may be peceved as ant-socal. Fnally, n a compehensve suvey pape on optmal ncome taaton, ketty and Saez (03) ae geneally postve to nclude elatve concens n the optmal taaton famewok, but seem to hestate egadng what the govenment should eally mamze: hethe such etenaltes should be factoed n the socal welfae functon s a deep and dffcult queston. Suely, hutng somebody wth hghe taes fo the sole satsfacton of envy seems moally wong. Hence, socal welfae weghts should not be allowed to be negatve fo anybody no matte how stong the envy effects. At the same, t seems to us that elatve ncome concens ae a much moe poweful and ealstc way to justfy socal welfae weghts deceasng wth ncome than standad utltaansm wth concave utlty of consumpton. (p. 453) As eseaches on nomatve polcy ssues, we do not see t as ou man ole to judge whch socal objectves ae appopate and whch ae not. Rathe, we beleve t s mpotant to analyze the mplcatons of dffeent nomatve o ethcal ponts of depatue that govenments may have, egadless of whethe we shae these values o not. Theefoe, n the pesent pape we do not take a stand on the appopateness of dffeent assumptons egadng the socal objectve. Instead, we smply analyze the polcy mplcatons of a patenalst appoach, whee the welfae effects of elatve consumpton ae emoved fom the socal objectve functon, and compae them wth those of the conventonal welfast appoach. As ou fundamental wokhose, we wll utlze the dscete self-selecton appoach to the Mleesan optmal nonlnea ncome ta model wth two poductvty types developed by Sten (98) and Stgltz (98), whee nfomaton asymmetes typcally pevent the govenment fom mplementng a fst-best esouce allocaton. Ths appoach s etended to accommodate consume pefeences fo elatve consumpton and povdes a useful analytcal famewok based on a easonably smple stuctue fo undestandng the polcy ncentves assocated wth coecton Accodng to Fank (005), ths s also one lkely eason many economsts have been eluctant to base polcy analyses on models whee the consumes ae postonal. Yet, as also agued by Fank, postonal concens need not necessaly eflect ant-socal pefeences. Instead they mght eflect nstumental easons such as the need fo famles to keep up wth communty spendng to be able to lve n aeas whee the chlden may attend schools of easonable qualty. 3

4 atenalsm Aganst Veblen and edstbuton. Snce much eale lteatue on the self-selecton appoach to optmal taaton s based on the two-type model, t allows fo staghtfowad compasons wth eale studes. In addton, a fst-best ta polcy follows natually fom the specal case whee the self-selecton constant does not bnd, whch smplfes the pesentaton consdeably. Ou study paallels much eale eseach on the self-selecton appoach to optmal taaton by chaactezng aeto-effcent magnal ta polces. It also dstngushes between a welfast polcy based on the ndvduals own pefeences and a polcy based on the laundeed pefeences mposed on them by a patenalst govenment. An mpotant stength of ths appoach s that the polcy ules fo magnal ncome taaton deved below apply fo any such aeto-effcent allocaton and, thus, also fo any undelyng aetan socal welfae functon based on the ndvduals tue and laundeed pefeences, espectvely. Snce ou pmay am s to compae the coectve motve fo taaton faced by welfast and patenalst govenments, and how the self-selecton constant modfes ths motve fo coectve taaton n a second-best settng, the focus on polcy ules fo magnal ta polcy s natual. Howeve, we also povde a functonal fom eample allowng fo some compason between levels of magnal ncome ta ates as well. One may pehaps conjectue that the nduced hghe magnal ncome taes due to socal compasons based on the welfast appoach wll vansh f the analyss s nstead based on a patenalst appoach whee pefeences fo socal compasons ae not espected. It tuns out, howeve, that such a conjectue s mpotantly wong. In fact, a patenalst govenment may espond n a way smla, o even dentcal, to a welfast govenment, although fo a dffeent eason. The ntuton s that the etenalty mposed by each ndvdual on othe people (whch s of mpotance to the welfast govenment) concdes wth the ndvdual s own behavoal falue as peceved by the patenalst govenment. Moeove, these nsghts ae staghtfowad to genealze to a case wth moe than two poductvty types. Consequently, t seems that elatve consumpton concens ae geneally mpotant fo the polcy outcome, espectve of whethe the govenment ams at coectng fo postonal etenaltes o tes to make the consumes behave as f they wee not concened wth the elatve consumpton. Ths majo fndng s also confmed based on numecal smulatons wth a utltaan socal welfae functon, togethe wth specfc functonal foms of the utlty 4

5 atenalsm Aganst Veblen functons and dffeent efeence consumpton levels. In these smulatons, we ae able to compae the two govenments n tems of how the optmal magnal and aveage ta ates, as well as optmal labo supply and oveall edstbuton, vay wth the degee of postonalty,.e., the etent to whch people s utlty gan fom nceased consumpton s dven by the pefeence fo elatve consumpton. Although these smulaton esults, not supsngly, show lage heteogenety, e.g., elated to the assumptons undelyng the socal compasons, we can conclude that elatve consumpton concens wll n geneal substantally affect the optmal ncome ta polcy, not only fo the welfast but also fo the patenalst govenment. The outlne of the pape s as follows. Secton I pesents elated lteatue and the contbuton of the pesent pape n elaton to ths lteatue. Secton II pesents a benchmak model whee each ndvdual compaes hs/he consumpton wth the aveage consumpton n the oveall economy. The mplcatons fo fst-best and second-best taaton ae analyzed n Sectons III and IV, espectvely. Secton V concens the ta polcy mplcatons of two altenatve compason foms: wthn-type and upwad compasons, espectvely. Secton VI pesents the esults of etensve numecal smulatons, whle Secton VII povdes a summay and a dscusson. oofs ae pesented n the Append. I. Relaton to the Lteatue As ndcated n the ntoducton, a numbe of eale studes have eamned the ta polcy mplcatons of elatve consumpton concens based on welfast objectves. 3 Fst-best polcy ules to coect fo the assocated etenaltes often efeed to as postonal etenaltes have been deved n a vaety of contets; see, e.g., Layad (980), esson (995), Ljungqvst and Uhlg (000), and Dupo and Lu (003). The most mpotant ealy contbutons to the study of optmal second-best taaton unde 3 Fo a long tme n the 0 th centuy, thee was lttle dscusson on nomatve mplcatons of elatve consumpton concens, yet thee wee of couse eceptons. Moeove, such ssues wee often taken moe seously by classcal economsts. Fo eample, Mll (848) agued that consume choce qute often s not ncued fo the sake of the pleasue affoded by the thngs on whch the money s spent, but fom egad to opnon, and an dea that cetan epenses ae epected fom them, as an appendage of staton. He concluded that: I cannot but thnk that ependtue of ths sot s a most desable subject of taaton (ncples of oltcal Economy, Book 5, Chapte 6). 5

6 atenalsm Aganst Veblen elatve consumpton concens ae Boskn and Sheshnsk (978), Oswald (983), and Tuomala (990), all of whch eamned ta polcy poblems whee etenalty coecton and edstbuton ae caed out smultaneously. heeas Boskn and Sheshnsk focused on a model wth a lnea negatve ncome ta, the studes by Oswald and Tuomala ae based on Mleesan models of optmal nonlnea taaton. Aonsson and Johansson-Stenman (008) wee the fst to analyze ths poblem by usng the Sten (98) and Stgltz (98) two-type veson of the Mleesan model. The appoach allows fo moe pecse esults and cleae ntuton as to why the selfselecton constant modfes the ncentve to coect fo postonal etenaltes. In addton, they epessed the polcy ules fo effcent magnal taaton n tems of degees of postonalty,.e., the etent to whch nceased consumpton contbutes to hghe utlty though nceased elatve, athe than absolute, consumpton. Ths smplfes the pesentaton, and makes t possble to ntepet the theoetcal esults n lght of avalable empcal estmates of such degees, e.g., based on questonnaeepemental eseach. 4 The pesent pape genealzes the model n Aonsson and Johansson-Stenman (008) to allow fo a patenalst govenment. Ou contbuton s to systematcally compae the magnal ta polcy mplemented by welfast and patenalst govenments n economes whee the consumes ae concened about the elatve consumpton. 5 Thee ae a few pevous studes on patenalst appoaches to optmal taaton n such economes. Dodds (0) and Kanbu and Tuomala (00) 6 compae the optmal magnal ncome ta polcy of welfast and patenalst govenments n the contet of numecal models. A lnea ncome ta s consdeed n the fome pape, wheeas the latte deals wth optmal nonlnea ncome taaton. The numecal esults show that elatve consumpton concens among consumes may motvate much hghe magnal ta ates than n the absence of such concens, even f the consume pefeence fo elatve consumpton does not affect the polcy objectve (povded that the 4 See, e.g., Johansson-Stenman et al. (00), Alpza et al. (005), and Calsson et al. (007). 5 Othe poposed easons fo devatng fom welfasm nclude the pesence of met goods (Sandmo 983), allevaton of povety (Kanbu et al. 994), self-contol poblems (Gube and Köszeg 00), and based sk peceptons (Johansson-Stenman 008). Yet, and not supsngly, thee s no scentfc consensus n any of these cases egadng the appopateness of patenalsm. 6 Ths s the wokng pape veson, whch was subsequently publshed as Kanbu and Tuomala (03). Howeve, n the jounal veson the secton based on a patenalst govenment was dopped. 6

7 atenalsm Aganst Veblen govenment ecognzes the assocated behavoal effects). Eckestofe and endne (03) nstead eamne the optmal stuctue of commodty taaton n a theoetcal model and allow the consumpton etenalty caused by elatve consumpton compasons to be non-atmosphec (such that ndvduals dffe n the magnal contbuton to ths etenalty) and asymmetc (meanng that people use dffeent efeence ponts). They show that both a welfast and a patenalst govenment may mplement ts fst-best esouce allocaton though pesonalzed commodty taaton, and that the pncple of tagetng does not geneally apply f the (welfast o patenalst) govenment s estcted to usng unfom commodty taes. The pape closest to ous s Mcheletto (0), who analyzes optmal ncome taaton n a second-best settng whee he also consdes the case of patenalsm. He uses a qute specfc model, whee each poductvty type compaes hs/he consumpton wth that of the adjacent type wth hghe poductvty (meanng that the hghest poductvty type s not concened about elatve consumpton). e wll etun to hs esults below. Ou study s moe geneal and dffes fom hs n seveal mpotant ways. Fst, we consde a boade spectum of possbltes by analyzng the ta polcy mplcatons of () the mean value compason (whch s the conventonal assumpton n eale compaable studes based on the welfast appoach), () wthn-type compasons, and () upwad compasons. 7 Second, we consde the ncentves undelyng both fst-best and second-best taaton, meanng that we ae able to compae ou esults wth a faly lage body of lteatue on ta polcy and elatve consumpton based on welfast models. Thd, we pesent the optmal magnal ta polcy n tems of degees of postonalty, whch makes t possble to ntepet the esults n lght of such estmates fom the empcal lteatue on socal compasons. 7 The empcal evdence hee s scace. Some evdence suggests that people compae the own consumpton wth that of smla othes (e.g., Runcman 966; McBde 00; Clak and Senk 00), whch n ou settng may justfy compasons wthn poductvty goups, whle othe evdence s moe n accodance wth upwad compasons (e.g., Bowles and ak 005). e also ntepet Veblen (899) n tems of upwad compasons, as he agued that people n othe socal classes ae nfluenced by the behavo of, and ty to emulate, the wealthy lesue class. 7

8 atenalsm Aganst Veblen II. A Two-Type Economy wth Relatve Consumpton and Nonlnea Taaton Consde an economy wth two types of consumes, a low-poductvty type (type ) and a hgh-poductvty type (type ), whee poductvty s measued by the befoe-ta wage ate. Thee ae n ndvduals of the low-poductvty type and n ndvduals of the hgh-poductvty type; N n n denotes total populaton. Output n ths economy s poduced by a lnea technology such that the befoe-ta wage ates ae fed. 8 A. Consume Behavo and efeences fo Relatve Consumpton Each consume deves utlty fom hs/he absolute consumpton and use of lesue, espectvely, as well as fom hs/he consumpton elatve to that of efeent othes. The utlty functon faced by a consume of poductvty type (=,) s gven by () U u (, z, ), whee denotes consumpton, z lesue, and the ndvdual s elatve consumpton. Fo techncal convenence, the elatve consumpton s defned as the dffeence between the ndvdual s own consumpton and a measue of efeence consumpton,, such that (as n, e.g., Akelof 997; Coneo and Jeanne 997; Ljungqvst and Uhlg 000; Bowles and ak 005; and Calsson et al. 007). 9 To begn wth, we consde the conventonal mean value compason, whee the efeence consumpton s gven by the aveage consumpton n the economy as a whole,.e., 0 8 Ths assumpton smplfes the calculatons; t s of no sgnfcance fo how elatve consumpton concens affect the optmal ta polcy. 9 An obvous altenatve would be to assume that the ndvdual s elatve consumpton s detemned by the ato between the ndvdual s own consumpton and the elevant efeence measue (e.g., as n Boskn and Sheshnsk 978; Layad 980; Abel 005; and endne and Goulde 008). It s not mpotant fo the qualtatve esults whch opton s chosen. 0 Eale studes on optmal ncome taaton and elatve consumpton typcally assume that ndvduals compae the own consumpton wth the aveage consumpton n the economy as a whole. Eceptons nclude Aonsson and Johansson-Stenman (00), who also analyze the polcy mplcatons 8

9 atenalsm Aganst Veblen n n. N e assume that the functon u () s nceasng n ts aguments and stctly quasconcave. Note also that equaton () allows fo dffeences n pefeences between types. Altenatve compason foms and measues of efeence consumpton wll be addessed n Secton V. e show below that the stengths of the elatve consumpton concens ae mpotant detemnants of the optmal ta polcy, espectve of whethe the govenment has a patenalst o welfast objectve. Based on Johansson-Stenman et al. (00), the stength of the consume pefeence fo elatve consumpton wll be measued by the degee of postonalty, whch s ntepetable as the facton of an ndvdual s oveall utlty gan fom an addtonal dolla spent on consumpton that s due to nceased elatve consumpton. Ths means that f the degee of postonalty equals zeo, then only absolute consumpton mattes, as n the conventonal model, wheeas a value equal to one means that only elatve consumpton mattes on the magn. An altenatve ntepetaton s that the degee of postonalty eflects the welfae cost to the ndvdual, measued pe unt of consumpton, of an ncease n the level of efeence consumpton. Fo an ndvdual of poductvty type, the degee of postonalty s gven by () u u u. Thoughout the pape, subscpts attached to the utlty functon denote patal devatves such that u u / and u u /. The assumptons made eale mply that (0,), wheeas would be equal to zeo n the absence of any pefeence fo elatve consumpton. The aveage degee of postonalty measued ove all consumes n ths economy can then be wtten as (3) n n. N The aveage degee of postonalty gves an ndcaton of how mpotant elatve consumpton concens ae on aveage n the economy as a whole. th mean-value compasons, t s also a measue of the magnal postonal etenalty pe unt of of wthn-geneaton and upwad compasons, espectvely, faced by a welfast polcy make, and Mcheletto (0), who consdes a vaant of upwad compasons. 9

10 atenalsm Aganst Veblen consumpton (snce all ndvduals contbute to ths etenalty to the same etent unde such compasons). Empcal estmates of the aveage degee of postonalty suggest that elatve consumpton s an mpotant detemnant of ndvdual wellbeng; endne and Goulde (008) ague that ths numbe s typcally found n the nteval , wheeas Alpza et al. (005) and Calsson et al. (007) fnd that the aveage degee of postonalty (measued fo ncome) s aound 0.5. Some estmates fom happness studes suggest even hghe values. These numbes ae clealy consstent wth Fank s (005) agument that postonal etenaltes cause lage welfae losses. The ndvdual budget constant can be wtten as (4) w l T( w l ) 0, whee w denotes the befoe-ta wage ate and l the hous of wok, measued by a tme endowment less the tme spent on lesue. The functon T() epesents the ncome ta. e assume that each consume s small elatve to the economy as a whole and behaves as an atomstc agent by teatng fst-ode condton fo wok hous can then be wtten as (5) u u w ( T '( w l )) u 0. In equaton (5), T '( ) s the magnal ncome ta ate. z w and as eogenous. The B. Constants Facng the Govenment The govenment s assumed to be able to obseve ncome (the poduct of the befoe-ta wage ate and the hous of wok), wheeas ndvdual poductvty (and consequently the hous of wok) s pvate nfomaton. Smla to a geat deal of othe lteatue on optmal taaton, we assume to begn wth that the govenment wants to edstbute ncome fom hgh-poductvty to low-poductvty ndvduals, whch s efeed to as the nomal case by Stgltz (98). By makng the conventonal assumpton that the hgh-poductvty type has flatte ndffeence cuves n the goss e can also thnk of ths assumpton as the case whee the govenment n a fst-best optmum edstbutes ncome fom the hgh-poductvty type to the low-poductvty one, athe than the othe way aound; cf. Stgltz (98). 0

11 atenalsm Aganst Veblen ncome consumpton space than the low-poductvty type, the followng selfselecton constant s mposed to pevent hgh-poductvty ndvduals fom becomng mmckes: 3 (6) ˆ U u (, z, ) u (, l, ) U. The weak nequalty (6) constans the edstbuton polcy: It mples that ths polcy must not be such that a hgh-poductvty ndvdual wll pefe the allocaton ntended fo the low-poductvty type (whch the hgh-poductvty ndvdual can each by educng hs/he hous of wok and selectng the ncome consumpton pont ntended fo the low-poductvty type). and w / w the elatve wage ate. Theefoe, chosen by the mmcke and ˆ U denotes the utlty of a hgh-poductvty mmcke l zˆ l epesents the labo supply s ntepetable as the lesue used by the mmcke (wth the tme-endowment nomalzed to unty). At the end of Secton IV below, we eamne the consequences of eplacng equaton (6) wth a self-selecton constant mposed on the low-poductvty type, based on the assumpton that hghpoductvty ndvduals ae net ecpents fom the edstbuton system. By usng n T ( w l ) 0 togethe wth the pvate budget constants gven n equaton (4), we can wte the publc budget constant as (7) n w l n. The publc decson poblem s to desgn a aeto-effcent ta polcy satsfyng the self-selecton and budget constants gven n equaton (6) and (7), espectvely. e follow conventon n wtng the publc decson poblem as a dect decson poblem, whee consumpton and wok hous seve as dect decson vaables. e can then nfe the magnal ncome ta ates mplct n the socally optmal esouce allocaton smply by compang the fst-ode condtons of the publc decson poblem wth the pvate fst-ode condtons fo wok hous. Note that we efe to ndvdual ndffeence cuves hee, whee utlty-effects of elatve consumpton ae thus taken nto account by the ndvduals. 3 As ponted out by Boadway and Keen (993) n the contet of a model wth poductvty typespecfc utlty functons (as n ou famewok), ths assumpton ules out the possblty that the othe self-selecton constant bnds at the same tme,.e., low-poductvty ndvduals wll stctly pefe the allocaton ntended fo them ove the allocaton ntended fo the hgh-poductvty type. Ths assumpton coesponds to the agent-monotoncty condton of Seade (98) n the specal case whee all ndvduals shae a common utlty functon.

12 atenalsm Aganst Veblen Note n patcula that not only the publc budget constant but also the selfselecton constant s ndependent of whethe the govenment s patenalst o welfast. The eason s, of couse, that the govenment cannot dectly foce the ndvduals not to take elatve consumpton concens nto account, even f t would lke to. C. The atenalst Govenment s oblem The patenalst govenment does not shae the consume pefeence fo elatve consumpton. Instead, t wants each consume to behave, at the magn, as f he/she s not concened wth elatve consumpton compasons, and thus desgns a ta polcy that elmnates ths peceved behavoal falue. To be able to focus on patenalsm n tems of elatve consumpton, we also assume that the govenment espects all othe aspects of consume pefeences. 4 That s, we assume that the govenment would lke to mamze utltes fo all ndvduals based on the actual utlty functons as gven by equaton (), wth the only dffeence that the elatve consumpton s held constant, such that coespondng welfae effects due to changes n elatve consumpton ae not taken nto account by the govenment. In tun, ths means that the patenalst govenment would lke each ndvdual of poductvty type to mamze u (, z, K ), whee the elatve consumpton K s teated as eogenous n equlbum, nstead of mamzng u (, z, ). Howeve, the ndvdual wll of couse not follow ths wsh of the govenment. The ndvdual wll stll mamze u (, z, ), whle the govenment n ts socal optmzaton bases the welfae evaluatons on u (, z, K ), such that s teated as fed n equlbum. As eplaned below, ths means that the magnal utlty of consumpton s gven by u fom the pespectve of the patenalst govenment, whle t s gven by fom each ndvdual s own pespectve. An llustatve specal case ases when the ndvdual pefeences ae addtvely sepaable n, such that equaton () can be wtten as U v (, z ) ( ) u u.ths 4 See Blomqust and Mcheletto (006) fo a two-type model of optmal ncome taaton wthout elatve consumpton compasons, whee the govenmental objectve functon s fomulated ndependently of the ndvdual utlty functons.

13 atenalsm Aganst Veblen means that the govenment would base ts objectve on the fst pat, v (, z ), wheeas the second pat, ( ), would be teated as eogenous and hence dsegaded. Nevetheless, thoughout the pape we consde the geneal fomulaton n equaton () nstead of estctng the analyss to the case of addtve sepaablty. The publc decson poblem then becomes (-Gov) Ma u (, z, K ) s.t. u (, z, K ) U 0, (6) and (7). l,, l, In poblem (-Gov), U 0 s a fed mnmum utlty level that the govenment mposes on the hgh-poductvty type. Note that although the govenment does not deve utlty fom the consumes pefeences fo elatve consumpton, these pefeences wll, nevetheless, affect the self-selecton constant gven n equaton (6), snce ths constant seves to make each hgh-poductvty ndvdual choose the combnaton of wok hous and consumpton ntended fo hs/he poductvty type. Indvduals wll of couse base the choces of consumpton and wok hous on the own pefeences (whee elatve consumpton mattes) and not on those of the govenment. Also, the govenment s assumed to ecognze that the efeence consumpton s endogenous and gven by ( n n ) / ( n n ). The Lagangean coespondng to ths decson poblem can be wtten as (8) u (, z, K ) u (, z, K ) U 0, u (, z, ) uˆ (, l, ) n ( w l ) whee K and K thus ae teated as eogenous. Subscpt efes to patenalst, whle,, and ae Lagange multples. The fst-ode condtons fo l, and (9a) (9b) become u uˆ n w 0, z z n u ( ˆ ˆ u u) n 0, N u n w, (9c) 0 (9d) z n u u u n 0. N 3, l, Two thngs ae woth notng. Fst, snce the patenalst govenment wants the consumes to behave as f they ae not concened wth the elatve consumpton (although t accepts all othe aspects of consume pefeences), the socal magnal

14 atenalsm Aganst Veblen utlty of pvate consumpton s gven by u fo =,,.e., the magnal utlty of absolute consumpton, whch s seen fom equatons (9b) and (9d). In tun, u equals the total pvate magnal utlty of consumpton (the measue of elevance fo the ndvdual consume), U u u, tmes the degee of non-postonalty,. Second, the patal devatve of the Lagangean wth espect to, /, measues the change n socal welfae (fom the pespectve of the patenalst govenment) of nceased efeence consumpton, cetes pabus, and wll be analyzed moe thooughly below. D. The elfast Govenment s oblem Fo puposes of compason, we also addess the optmal ta polcy decded by a welfast govenment, whch ncopoates the consume pefeences fo elatve consumpton n ts own objectve. Ths decson poblem was pevously eamned by Aonsson and Johansson-Stenman (008) and s gven by (-Gov) Ma u (, z, ) s.t. u (, z, ) U 0, (6) and (7). l,, l, The coespondng Lagangean becomes (0) The fst-ode condtons fo u (, z, ) u (, z, ) U 0. u (, z, ) uˆ (, l, ) n ( w l ) l and espectvely, wheeas the fst-ode condtons fo (9b ) n u ( ˆ ˆ u u u) n 0, N (9d ) n N u u n l concde wth equaton (9a) and (9c), 0. and change to ead In equatons (9b ) and (9d ), / measues the patal welfae effect of nceased efeence consumpton fom the pespectve of the welfast govenment. In the followng two sectons, we wll addess the mplcatons of equatons (9) fo optmal ncome taaton. 4

15 atenalsm Aganst Veblen III. Fst-Best Magnal Ta Rates In the economy set out above, the govenment s unable to obseve ndvdual poductvty and must, theefoe, edstbute subject to the self-selecton constant. As a consequence, the govenment cannot ely on poductvty type-specfc lumpsum taes fo puposes of edstbuton. Howeve, f ndvdual poductvty wee obsevable, the self-selecton constant would be edundant, meanng that nothng would pevent the govenment fom edstbutng though poductvty type-specfc lump-sum taes. In that case, the sole pupose of magnal ncome taaton would be to coect fo maket falues (unde a welfast govenment) o behavoal falues (unde a patenalst govenment). Ths case povdes a natual statng pont. e stat by analyzng fst-best taaton befoe tunng to the second-best ta polcy n Secton IV. The fst best (.e., full nfomaton) esouce allocaton follows as the specal case of ou model whee the self-selecton constant does not bnd, n whch 0. It s mpotant to emphasze that the concept of fst best just means the best that each govenment can accomplsh unde full nfomaton about ndvdual poductvty, gven ts objectve and esouce constant. Thus, snce the patenalst and welfast govenments have dffeent objectve functons, t follows that the fst-best allocaton based on a patenalst objectve typcally dffes fom the fst-best based on a welfast objectve. Ou pupose hee s to compae the magnal ta polcy used by a patenalst govenment to mplement ts fst-best allocaton wth the coespondng magnal ta polcy used by a welfast govenment. If 0, t s staghtfowad to deve (see the Append) (a) 0, (b) N 0. ( ) Theefoe, whle nceased efeence consumpton s of no concen to the patenalst govenment as long as ndvdual poductvty s obsevable, nceased efeence consumpton leads to a welfae loss fom the pont of vew of the welfast govenment though nceased postonal etenaltes. Despte ths, the coectve ta polcy mplemented by a patenalst govenment need not necessaly dffe fom that 5

16 atenalsm Aganst Veblen of ts welfast countepat. To see ths, let T '( w l ) and T '( w l ) denote the magnal ncome ta ate mplemented fo poductvty-type by the patenalst and welfast govenment, espectvely, and consde oposton. 5 oposton. Suppose that ndvdual poductvty s obsevable to the govenment and that the elatve consumpton concens ae based on mean value compasons. The optmal magnal ncome ta ates mplemented by the patenalst govenment can then be wtten as T '( w l ) fo all, whle the welfast govenment mplements the followng ates: T '( w l ) fo all. oof: See the Append. oposton elates the optmal ta polcy to the degees of postonalty,.e., the etent to whch the utlty gan of nceased consumpton s dven by the pefeences fo elatve consumpton. Recall that the welfast govenment espects the consumes pefeences fo elatve consumpton and tes to ntenalze the etenaltes that the consumes mpose on one anothe though these concens. th mean value compasons, each consume contbutes to the postonal etenaltes to the same etent at the magn. The aveage degee of postonalty,, thus epesents the value of the magnal etenalty pe unt of consumpton, whch eplans the second fomula n the poposton. Ths welfast ta fomula s analogous to esults deved n the contet of epesentatve agent models by, e.g., Ljungqvst and Uhlg (000) and Dupo and Lu (003), and of couse also to the two-type model n Aonsson and Johansson-Stenman (008). A patenalst govenment, on the othe hand, s not concened wth etenalty coecton, as t gves no weght to elatve consumpton concens n the socal objectve functon, whch can also be seen fom equaton (a). In lght of ths obsevaton, the optmal ta polcy of the patenalst govenment may seem both hghly supsng and unntutve. Yet, the undelyng ntuton s actually 5 It s staghtfowad to show that all fst-best esults pesented n ths pape, ncludng the fomulas n oposton, take the same fom espectve of whethe thee ae two o moe poductvty types. 6

17 atenalsm Aganst Veblen staghtfowad to eplan, as follows: Snce the patenalst govenment does not nclude elatve consumpton concens n ts objectve functon, t wants the consumes to behave as f they wee not concened wth the elatve consumpton. Hence, the govenment desgns the magnal ta polcy accodngly and taes away people s utlty gans fom nceased elatve consumpton. The sze of ths elatve utlty gan s, n tun, obvously measued by the ndvdual s own degee of postonalty,. Theefoe, the magnal ncome ta ate mposed by the patenalst govenment depends on the ndvdual s own degee of postonalty. The followng coollay s an mmedate consequence of oposton : Coollay. Suppose that (a) ndvdual poductvty s obsevable to the govenment, (b) the elatve consumpton concens ae based on mean value compasons, and (c) the type-specfc degees of postonalty, paametes. and, ae fed () A patenalst govenment mposes a lowe magnal ncome ta ate than the welfast govenment on the less postonal type and a hghe magnal ncome ta on the moe postonal type. () If both consume types shae the same degee of postonalty such that, then T '( w l ) T '( w l ) fo all. Unde the condtons of Coollay, the common magnal ncome ta ate decded by the welfast govenment would equal the economy-wde aveage of the two ates (one fo each poductvty type) ntoduced by the patenalst govenment. The second pat of the coollay s a vey stong esult as t mples that, gven a common degee of postonalty, the patenalst govenment would mplement eactly the same magnal ta polcy as ts welfast countepat, although fo a dffeent eason. Thus, gven the edstbuton between types, t does not matte at all whethe o not the govenment espects the consume pefeences fo envy o jealousy the magnal ta polcy mplcatons would be the same n both cases. To take ths dscusson a bt futhe, consde the followng utlty functon:. () U f, z f ( ) ( ), z Fom the epesson afte the second equalty sgn, t s obvous that the paamete s ntepetable as the common degee of postonalty. Ths means that esult () of 7

18 atenalsm Aganst Veblen Coollay holds fo the set of utlty functons satsfyng equaton (). A specfc functonal fom consstent wth equaton () s gven by ( ) ( ( ) ( )) U z z fo =,, whee,, and ae fed paametes (wth ntepetable as the degee of postonalty). Such a utlty functon, but based on models wth only one consume type, has been analyzed by Ljungqvst and Uhlg (000) and late dscussed by Dupo and Lu (003). Ths econcles the patenalst appoach wth esults n eale studes on optmal magnal taaton based on epesentatve agent models wth a welfast govenment. IV. Second-Best Magnal Ta Rates Let us now tun to the moe geneal second-best settng whee asymmetc nfomaton pevents the govenment fom edstbutng though poductvty typespecfc lump-sum taes. Hee the magnal ta stuctue wll eflect both the selfselecton constant and a motve fo coecton (fo maket falues n the welfast case and behavoal falues n the patenalst case). e stat by analyzng the model set out n Secton II above and then contnue wth a modfed veson of the model wth a bndng self-selecton constant on the low-poductvty type. A. The Nomal Case The welfae effects of nceased efeence consumpton n the patenalst and welfast cases,.e., equatons (a) and (b), wll then change to ead (3a) u ˆ ( ) ˆ ( ˆ ˆ u u u u u) (3b) d N, ( ) d whee ( uˆ uˆ )( ˆ ) / ( N) s an ndcato of the dffeence n the degee of postonalty between the mmcke and the low-poductvty type. If the mmcke s moe (less) postonal than the low-poductvty type, so that ˆ ( ), then d 0 ( 0). 8,

19 atenalsm Aganst Veblen Equaton (3b) was ognally deved by Aonsson and Johansson-Stenman (008) and shows that a welfast govenment has two dffeent motves fo adjustng though ta polcy: to ntenalze the postonal etenalty (captued by ) and ela the self-selecton constant by eplotng that the elatve consumpton concens may d dffe between the mmcke and the low-poductvty type (captued by ). The latte effect povdes an ncentve fo the welfast govenment to ela the selfselecton constant though an ncease n f the mmcke s moe postonal than the low-poductvty type ( ˆ ), and though a decease n f the low- poductvty type s moe postonal than the mmcke ( ˆ ). In contast, the patenalst govenment s not concened wth the postonal etenalty pe se, whch eplans why does not appea n equaton (3a). Thus, the patal welfae effect of an ncease n faced by the patenalst govenment s due solely to the self-selecton constant. Futhemoe, fo a patenalst govenment, t s not an ssue whethe a mmcke s moe o less postonal than the low-poductvty type, snce has no dect effect on the objectve that the govenment mposes on the low-poductvty type. Instead, what mattes s just that dectly affects the self-selecton constant though U and U ˆ, whch n tun eplans equaton (3a). In what follows, we dstngush between ndvdual magnal ates of substtuton between lesue and pvate consumpton as evaluated by a patenalst and welfast govenment, espectvely. Fom the pont of vew of a patenalst govenment, the magnal ate of substtuton between lesue and pvate consumpton fo poductvty type and the mmcke s gven by (MRS-) MRS u fo =,, and, z z, 0 u 9 ˆ uˆ MRS 0,, z z, uˆ espectvely, wheeas the coespondng magnal ates of substtuton fo a welfast govenment become (MRS-) MRS u, z z, 0 u u fo =,, and ˆ uˆ MRS 0., z z, uˆ ˆ u Now, to be able to shoten the notaton n the subsequent analyses, note that the optmal magnal ta polcy mplct n the ognal Stgltz (98) model (the veson wth fed befoe-ta wage ates) follows as the specal case of ou model whee thee s (a) no coectve motve fo taaton and (b) no motve to ela the self-selecton constant va polcy-nduced changes n the level of efeence consumpton. If based

20 atenalsm Aganst Veblen on the MRS- functons, the optmal magnal ncome ta ates n the ognal Stgltz (98) model can be wtten as (4a) uˆ nw, ˆ, MRS z, MRSz, and 0, and f based on the MRS- functons, they can be wtten as (4b) ( uˆ uˆ ) ˆ nw and 0.,, MRS z, MRSz, The mplcatons of equatons (4a) and (4b) ae well known fom pevous studes: In the ognal two-type model wth fed befoe-ta wage ates, thee s an ncentve to ela the self-selecton constant though magnal ncome taaton of the lowpoductvty type. In dong ths, one utlzes the dffeence n the magnal value attached to lesue between the mmcke and the low-poductvty type, whle thee s no coespondng ncentve to dstot the behavo of the hgh-poductvty type though magnal taaton. The eason fo pesentng these fomulas hee s that the vaables and ae pat of the patenalst polcy chaactezed below, wheeas the vaables and play a coespondng ole fo a welfast polcy. Consde oposton. oposton. Suppose that the elatve consumpton concens ae based on mean value compasons. The second-best optmal magnal ncome ta ates mplemented by a patenalst govenment can then be wtten as whee T T n u n uˆ w n N '( w l ) ( ) ( ) n u n uˆ, w n N '( w l ) ( ) MRS fo =,, whle a welfast govenment mplements the, z, c / 0 followng second-best optmal magnal ncome ta ates: T T oof: See the Append. d '( w l ) ( ) ( )( ) d d '( w l ) ( ) d 0

21 atenalsm Aganst Veblen Note fst that the ta fomulas pesented n oposton follow as the specal case whee 0, n whch also d 0. The welfast fomulas n oposton can also be found n Aonsson and Johansson- Stenman (008) and eflect thee basc ncentves fo ta polcy: () elaaton of the self-selecton constant by eplotng that the low-poductvty type and the mmcke attach dffeent magnal values to lesue,.e., though, () ntenalzaton of postonal etenaltes as eflected n the aveage degee of postonalty,, and () elaaton of the self-selecton constant by eplotng that a mmcke may ethe be d moe o less postonal than the low-poductvty type as measued by. Snce 0 by ou eale assumptons and 0, t follows that the coectve component n the fomula fo the low-poductvty type,.e., the second tem on the ght-hand sde, s scaled down by the facto ( ). The eason s that the facton of the magnal ncome that s aleady taed away fo othe easons does not gve se to any postonal etenaltes. Note also that the welfast govenment mplements lowe (hghe) magnal ncome ta ates fo both poductvty types than t would othewse have done f the mmcke s moe (less) postonal than the low-poductvty type, cetes pabus,.e., d f 0 ( 0), n whch case an ncease (decease) n the efeence consumpton contbutes to ela the self-selecton constant. If the utlty functons ae gven by (), t follows that the degee of postonalty s the same fo all, ncludng the d mmcke, such that 0, mplyng T '( w l ) ( ) fo =,. The patenalst fomulas ae novel and wtten n a fomat compaable to the coespondng welfast fomulas. Thus, thee ae thee basc polcy ncentves hee as well: () elaaton of the self-selecton constant by eplotng that the lowpoductvty type and the mmcke attach dffeent magnal values to lesue, as eflected n, () coecton fo behavoal falues, and () elaaton of the selfselecton constant though polcy-nduced changes n the efeence consumpton. The fst two aspects ae emnscent to the countepats n the welfast case n tems of qualtatve mplcatons fo ta polcy, wheeas the thd aspect s dffeent. The fst tem on the ght-hand sde of the epesson fo T '( w l s agan the )

22 atenalsm Aganst Veblen standad ncentve fo magnal ncome taaton of low-poductvty ndvduals found n the ognal Stgltz (98) model, although n ths case t s based on the MRS- nstead of MRS- functons. th ths modfcaton, the component n the patenalst ta fomula fo the low-poductvty type s ntepetable n the same geneal way as n the coespondng welfast fomula. Thee s no smla component n the epesson fo magnal ncome taaton of the hgh-poductvty type, snce 0. The motve to coect fo behavoal falues s captued by the second tem n the fomula fo T '( w l and the fst tem n the fomula fo ) T '( w l ). As eplaned n the contet of oposton, ths behavoal falue s captued by the ndvdual s own degee of postonalty. By analogy to the welfast case, the coectve ta component mposed on the hgh-poductvty type s the same as unde fst-best taaton,.e.,, wheeas the coectve component s scaled down fo the low- poductvty type f 0. The ntuton behnd the scale facto s, n ths case, that magnal ncome taes mposed fo othe easons than coecton wll, nevetheless, elmnate pat of the behavoal falue that the govenment wants to coect fo. Thus, f the facton of an addtonal dolla s aleady taed away, only the facton may be used fo pvate consumpton. The fnal component of each patenalst ta fomula s connected to the welfae effect of nceased efeence consumpton n equaton (3a),.e., /, as well as to dect effects of on the self-selecton constant. As such, t eflects an ncentve to ela the self-selecton constant though polcy-nduced changes n the consumpton, and dffes n a fundamental way fom ts countepat n the welfast case. heeas the coespondng effect unde a welfast ta polcy takes the same fom and sgn fo both poductvty types (whee the sgn depends on whethe the mmcke s moe o less postonal than the low-poductvty type), t dffes n sgn between the poductvty types unde a patenalst ta polcy. Moe specfcally, and although / cannot be sgned unambguously, the fnal tem n the ta fomula fo the low-poductvty type s postve, whle t s negatve n the fomula fo the hgh-poductvty type. Ths esult follows because affects the self-selecton constant though two channels,.e., a dect effect and an ndect effect va.

23 atenalsm Aganst Veblen These two effects patly cancel out, leavng a postve net effect n the fomula fo the low-poductvty type and a negatve net effect n the fomula fo the hghpoductvty type (see the Append fo techncal detal). Theefoe, wth mean value compasons, an ncease n elaes, the selfselecton constant (ecognzng that tghtens, and an ncease n and affect ). The govenment may thus ela the self-selecton constant by tang the hgh-poductvty type at a lowe magnal ate than motvated by pue coecton and coespondngly ta the lowpoductvty type at a hghe magnal ate. Thus, and f we assume that 0 n accodance wth the Stgltz (98) model, the followng esult s an mmedate consequence of oposton : Coollay : th a patenalst govenment and unde mean-value compasons, the optmal second-best polcy satsfes T '( w l T '( w l ) ). Can we go futhe and compae the levels of magnal ncome taaton mplemented by the two govenments? In geneal, ths s not possble snce the degees of postonalty ae endogenous vaables. Howeve, fo the set of utlty functons that can be wtten as equaton (), whee the degee of postonalty s a fed paamete and takes the same value fo all ndvduals (ncludng potental mmckes), some such compasons can be made. Ths scenao means that the patenalst govenment mplements a lowe magnal ncome ta ate fo the hghpoductvty type than the welfast govenment. e can see ths esult dectly fom the magnal ncome ta fomulas n oposton, snce 3 and d 0 f all ndvduals shae a common utlty functon gven by equaton (). It s less staghtfowad to compae the magnal ncome ta ates that the two govenments mplement fo the low-poductvty type. Ths s because the vaables and ae lkely to dffe n equlbum, although thee s no a po eason to beleve any of these tems to be lage than the othe. Stll, f we agan consde utlty functons consstent wth equaton (), and f we smply assume that, t follows that the second-best effcent magnal ncome ta fo the low-poductvty

24 atenalsm Aganst Veblen type s hghe unde the patenalst than unde the welfast govenment,.e., the othe way aound compaed wth the hgh-poductvty type. B. The Non-Nomal Case: Redstbuton towads the Hgh-oductvty Type In lne wth all eale studes on optmal edstbutve taaton unde elatve consumpton concens that we ae awae of, we have so fa assumed that the govenment wants to edstbute fom the hgh-poductvty to the low-poductvty type, mplyng unde ou assumptons that equaton (6) consttutes the elevant selfselecton constant. Yet, one cannot a po ule out othe edstbuton pofles, suggestng that the case wth a bndng self-selecton constant on the low- 6, 7 poductvty type should also be addessed. th edstbuton fom the low-poductvty to the hgh-poductvty type, and by etanng the assumpton that the hgh-poductvty type has flatte ndffeence cuves n the goss ncome-consumpton space than the low-poductvty type, the selfselecton constant that may bnd can be wtten as follows: (5) ˆ U u (, z, ) u (, l, ) U, meanng that equaton (5) eplaces equaton (6); othewse, the model s the same as befoe. Let denote the Lagange multple attached to equaton (5), whle and denote the Lagange multples of the mnmum utlty estcton on the hghpoductvty type and esouce constant, espectvely, as befoe. e can then poceed n the same way as n subsecton IV.A. Based on the MRS- and MRS- 6 e ae gateful to an anonymous efeee fo suggestng ths etenson. 7 e can thnk of ths assumpton as the case whee the govenment n a fst-best optmum edstbutes ncome fom the low-poductvty type to the hgh-poductvty one, athe than the othe way aound; cf. Stgltz (98). One motvaton, followng, e.g., Gossman (99) and Acemoglu and Robnson (006, Chapte 5), fo analyzng such a case s that non-democatc govenments may not cae at all about the well-beng of a lage pat of the populaton but must stll keep these people fom becomng too dsapponted and hence make an upoa wth bad consequences fo the ulng elte that the govenment epesents. Yet, such a case typcally does not ft the model pefectly, snce, n nondemocatc countes both today and hstocally, govenments have been able to use also othe measues than ncome o consumpton to sepaate dffeent goups fom each othe, such as heedtay class belongng o ethncty. 4

25 atenalsm Aganst Veblen functons used above, we can once agan shoten the notaton by fst chaactezng the optmal magnal ta polcy mplct n the ognal Stgltz (98) model whee thee s no coectve motve fo taaton and no motve to ela the self-selecton constant va polcy-nduced changes n the level of efeence consumpton. Fo a patenalst govenment and assocated MRS- functons, and based on the selfselecton constant gven n equaton (5), we obtan (6a) 0 and uˆ ˆ nw,,, MRS z, MRS z, whle the coespondng fomulas fo the welfast govenment and assocated MRS- functons become: (6b) 0 and ( uˆ uˆ ) ˆ nw.,, MRS z, MRS z, In a way smla to equatons (4a) and (4b), equatons (6a) and (6b) just epoduce the magnal ta polcy mplct n the Stgltz (98) model; albet n ths case wth a self-selecton constant mposed on the low-poductvty type (nstead of the hgh-poductvty type). Ths means that the magnal ta s zeo fo the lowpoductvty type, whle the magnal ta fo the hgh-poductvty type s typcally negatve (at least when based on the MRS- functons). Equatons (6a) and (6b) wll thus play the same ole hee as equatons (4a) and (4b) dd n the nomal case eamned n subsecton IV.A. o Now, defne ( uˆ uˆ )( ˆ ) / ( N) to eflect the dffeence n postonal concens between the low-poductvty mmcke and the (mmcked) hgh- d poductvty type. Ths vaable s analogous to n subsecton IV.A. The magnal ta polcy mplemented by the patenalst govenment can then be chaactezed as (7a) (7b) whee n u n uˆ, w n N '( ) ( ) T w l n u n uˆ, w n N '( ) ( ) ( ) T w l MRS, whle the welfast govenment mplements the, z, c / 0 followng magnal ta polcy: (8a) T w l, o '( ) ( ) o 5

26 atenalsm Aganst Veblen (8b) T w l. o '( ) ( ) ( )( ) o Equatons (7) and (8) ae analogous to the countepats n the case of edstbuton towad the low-poductvty type (.e., Stgltz nomal case) pesented n oposton. Statng wth the patenalst polcy, note that the fnal tem on the ght-hand sde of equatons (7a) and (7b), espectvely, woks to educe the magnal ta ate mplemented fo the low-poductvty type and ncease the magnal ta ate mplemented fo the hgh-poductvty type,.e., these components thus have the opposte qualtatve effects compaed wth n subsecton IV.A. Consequently, equaton (7a) mples T '( w l ). The ntuton s that the patenalst govenment may ela the self-selecton constant though a polcy-nduced ncease n the low- poductvty type s consumpton. Yet, f s negatve, o fall shot of T '( w l may ethe eceed ) despte that a polcy-nduced decease n the hgh-poductvty type s consumpton n tself woks to ela the self-selecton constant. Tunng to the welfast polcy, we see that o T w l f 0. In ths case, '( ) 0 the mmcked hgh-poductvty type s at least as postonal as the mmcke (.e., ˆ ), whch means that a decease n the level of efeence consumpton educes the postonal etenalty wthout tghtenng the self-selecton constant. Howeve, f the mmcke s moe postonal than the hgh-poductvty type ( ˆ ), a decease n the efeence consumpton wll contbute to tghten the self-selecton constant. Thus, T '( w l can be ethe postve o negatve, although the latte does not seem a ) vey plausble outcome. Fo the hgh-poductvty type, the optmal magnal ta ate can also be ethe postve o negatve, snce 0 by ou eale assumptons and 0. Note also that the coectve ta element mplemented fo the hghpoductvty type eceeds the aveage degee of postonalty,.e., ( ), snce the subsdy component desgned to ela the self-selecton constant ( 6 ) wll contbute to ncease the postonal etenalty that each hgh-poductvty ndvdual mposes on othe people. As ndcated above, these esults ae pefectly analogous to those pesented n subsecton IV.A: the only dffeence s that equaton (5) s desgned to dete the lowpoductvty type fom mmckng the hgh-poductvty type (and not the othe way aound). th a patenalst govenment, ths means that the ncentve to ela the self-