Taxing bequests and consumption in the steady state. Johann K. BRUNNER and Susanne PECH. Working Paper No October 2017

Transcription

1 DEPARTMENT OF ECONOMICS JOHANNES KEPLER UNIVERSITY OF LINZ Taxng bequests and consumpton n the steady state by Johann K. BRUNNER and Susanne PECH Workng Paper No October 2017 Johannes Kepler Unversty of Lnz Department of Economcs Altenberger Strasse 69 A-4040 Lnz - Auhof, Austra johann.brunner@jku.at, susanne.pech@jku.at

2 Taxng bequests and consumpton n the steady state Johann K. Brunner and Susanne Pech Unversty of Lnz October 13, 2017 Revsed Verson Abstract We study the optmal tax system n a dynamc model where dfferences n wages nduce dfferences n nhertances, and the transton from parent ablty to chld ablty s descrbed by a Markov chan. In accordance wth emprcal evdence, we assume that n any generaton more able ndvduals are lkely to have a more able parent, whch mples that n the steady state they also tend to receve larger nhertances than less able ndvduals. We show that the Atknson-Stgltz result on the redundancy of ndrect taxes does not hold n ths framework. In partcular, gven an optmal ncome tax, a bequest tax as well as a consumpton tax are potental nstruments for addtonal redstrbuton. For the bequest tax the sgn of the overall welfare effect depends on the reacton of bequests and on nequalty averson, whle for the consumpton tax the sgn s always postve because the dstortng effect s outweghed by the nduced ncrease n wealth accumulaton. Keywords: Optmal taxaton, estate tax, consumpton tax, wealth transmsson. Emal: johann.brunner@jku.at. Emal: susanne.pech@jku.at. both: Department of Economcs, Altenberger Straße 69, 4040 Lnz, Austra.

3 1 Introducton In developed economes each year a substantal amount of wealth s transmtted from one generaton to the next (see Pketty 2011, Wolff 2002). Hence, nhertances represent an mportant source of ndvdual wealth, but ther varaton across ndvduals also creates addtonal nequalty n a socety, on top of dfferences n ncome. In ths paper we study the mplcatons of ths fact for the desgn of the tax system. In partcular, we ask whether the exstence of nhertances requres specfc tax nstruments such as an estate tax to mtgate nequalty. As s well-know, nhertance taxaton s one of the most controversal ssues n tax polcy. To provde an answer to ths queston we formulate a dynamc optmal-taxaton model of an economy wth ndvduals who dffer n ther labor productvty and who leave bequests to ther descendants. We ask how a shft from labor-ncome taxaton to bequest or consumpton taxaton affects welfare n the steady state of ths economy. The startng pont s the wellknown result by Atknson and Stgltz (1976) derved n a model wthout bequests: f labor ncome s taxed optmally then any other - ndrect - tax s redundant, gven weak separablty of preferences n consumpton and labor. 1 As a consequence, f leavng bequests s just seen as one form of consumpton, there s no role for a tax on estates; 2 nether s any knd of consumpton or captal taxaton useful. In partcular, an estate tax does not allow more redstrbuton than what s possble through an ncome tax alone. 3 The stuaton changes f one takes nto account that, as the very consequence of wealth transmsson from parents to chldren, ndvduals of some generaton dffer not only n abltes but also n nherted wealth. Brunner and Pech (2012ab) have shown that wth exogenously gven nherted wealth a tax on wealth transfers ndeed has a redstrbutve effect, gven that more productve ndvduals receve more nhertances. 4 In contrast, n the present paper we analyze the welfare consequences of taxes n the steady- 1 Ths holds n case of an optmal nonlnear ncome tax. If t s restrcted to be lnear the condton that preferences over consumpton are homothetc s needed (see Deaton 1979). 2 Unless bequests are a complement to lesure, see e.g., Gale and Slemrod (2001), Kaplow (2001). 3 On the contrary, one can also take the vew that even a subsdy for bequests ncreases welfare. The reason s the postve external effect assocated wth bequests, because n addton to the value they provde for the donor, they also contrbute to utlty of the recpent (see Farh and Wernng (2010), among others). Recently, Kopczuk (2013a) has presented a formula for the optmal nonlnear tax on bequests whch accounts for the postve externalty as well as for the negatve ncome effect of receved nhertances on labor supply of chldren. 4 In a related approach, Boadway et al. (2000) and Cremer et al. (2001, 2003) show that ndrect taxes as well as a tax on captal ncome make sense f ndvduals dffer n nherted wealth. They assume, however, that bequests and nhertances are unobservable and, thus, cannot be taxed. In Saez (2002) exogenous dfferences n the tastes for consumpton goods make ndrect taxes useful. 1

4 state equlbrum of an economy, when nhertances are completely endogenous. More precsely, we formulate a dynamc model where n each perod a new generaton consstng of n dfferent ablty groups of ndvduals exsts. Each ndvdual s the parent of one chld, who agan belongs to one of the n groups n the next generaton. The transton probabltes from parent ablty to chld ablty are constant over tme; the stochastc process can be descrbed by a Markov chan and we consder the steady-state dstrbuton of abltes. Each ndvdual uses her labor ncome and her receved nhertances for own consumpton and for leavng bequests to her sngle chld of the next generaton, and ths process goes on n the followng perods. Under the assumptons of a joy-of-gvng motve for bequeathng and of aff ne-lnear Engel curves for consumpton and bequests n each perod, we are able to characterze the nhertances of each ndvdual n the steady state. Wth a gven budget share devoted to bequests n each ablty group, the transton probabltes determne how the dstrbuton of nhertances develops over tme. Eventually, the process leads to a steadystate dstrbuton of nhertances, wthn and across ablty groups. Ths dstrbuton s rather dverse, as t depends on all bequest decsons of all pror generatons. As a result, we arrve at a Mrrlees-type model wth dfferng labor productvtes, extended by the exstence of endogenously determned nhertances. We then apply ths model to study the optmal desgn of a tax system, determned by maxmzaton of expected socal welfare of a generaton. For ths analyss, t s crucal how the dstrbuton of labor ncomes s related to the dstrbuton of nhertances, whch n turn depends on the ndvduals probabltes of havng ancestors of a certan ablty. We make the assumpton that n each generaton the probablty dstrbuton of a more able ndvdual over her parent s abltes frst-order stochastcally domnates the probablty dstrbuton of a less able ndvdual. We are able to show that ths property ndeed mples that n the steady state the dstrbuton of nhertances receved by a more able ndvdual frst-order stochastcally domnates the dstrbuton of nhertances receved by a less able ndvdual. To put t dfferently, the property that n any generaton a more able ndvdual s lkely to have a more able parent than an ndvdual of lower ablty entals that n the steady state the more able ndvdual also tends to receve larger nhertances. That ths stochastc-domnance assumpton conforms to realty can be concluded from a number of emprcal studes on the correlaton between parent and chld ncome. The poneerng work by Solon (1992) and Zmmerman (1992) found an ntergeneratonal correlaton n the 2

5 long-run ncome (between fathers and sons) of around 0.4 for the Unted States (suggestng less socal moblty than prevously beleved). By now, there s a large set of estmates for a range of dfferent countres (for a recent overvew see Black and Devereux 2011 and Legh 2007). Although these studes used dfferent estmaton methods, varable defntons and sample selectons (lmtng ther comparablty), all of them estmated a postve ntergeneratonal correlaton n earnngs, rangng from 0.1 up 0.4, whch corroborates the above condton. 5 Frst, we analyze the optmal lnear ncome tax n the extended model wth endogenous nequalty of nherted wealth. We fnd that the redstrbutve effect of ths tax s more pronounced, compared to the purely statc Mrrlees-model, because t affects not only labor ncome as such, but also ndrectly the steady-state nhertances fnanced by labor ncome of the ancestors. On the other hand, the dstorton of labor supply causes an ndrect negatve effect on the amount of what s left to the chldren, that s, on nhertances. Altogether, the desrablty of a postve margnal tax rate s guaranteed f the labor supply elastcty s not too large. 6 Next, we address the man queston, namely whether the ntroducton of ndrect taxes on bequests or consumpton ncreases steady-state socal welfare, gven an optmal lnear ncome tax. The answer to ths queston s unclear a pror, because on the one hand we know for the bequest tax that t s desrable f ndvduals dffer n a second exogenously gven characterstc, namely nherted wealth (n addton to labor productvtes). On the other hand, however, n the steady state the dfferences n nhertances are n fact the consequence of the dfferences n labor productvtes. Therefore, as the latter represent the only (exogenous) source of heterogenety, one mght suppose that agan the Atknson-Stgltz result apples and no other (ndrect) tax than the optmal labor ncome tax s desrable. Our analyss shows that ths conjecture s wrong; we fnd a specfc role of ndrect taxes n the steady state of the economy. To determne the sgn of ther welfare effect requres a deep-gong analyss of how total nhertances n the steady state are affected by the mposton 5 In partcular, these studes suggest that the correlaton n the Nordc countres (Sweden, Fnland, Denmark, Norway), Canada and Australa s about and thus lower than n the U.S. and U.K. Interestngly, Jäntt et al. (2006) fnd that the greatest cross-country dfferences arse at the tals of dstrbuton. Moreover, these studes estmate hgher correlatons between son and father than between daughter and father. Moreover, there s a growng emprcal lterature that studes multgeneratonal persstence of socoeconomc status (that s, over three or more generatons). It provdes some evdence that nequalty s more persstent than estmates from parent-chld correlatons would suggest (for an overvew see Solon 2015 and Braun and Stuhler 2017). 6 Ths s n contrast to the statc, one-perod model where the optmal margnal tax rate s unambguously postve (see e.g., Hellwg 1986, Brunner 1989). 3

6 of ndrect taxes n each sngle perod when bequest decsons are made. It turns out that an addtonal property must hold n order to derve a clear-cut result. Ths new condton, whch s closely related but not dentcal to the frst-order stochastc domnance mentoned above, requres that ndvduals who receve hgher nhertances from some specfc prevous perod, also tend to receve hgher nhertances from ther mmedate parents. We argue by the use of extensve smulatons that ths condton can typcally be taken as beng satsfed n our framework. Gven these assumptons, the bequest tax has a redstrbutve effect, on top of what s acheved by the optmal lnear ncome tax. However, t also dstorts the bequest decson and nduces, thus, lower accumulated wealth n the steady state. The overall welfare effect s postve n case that the substtuton elastcty between bequests and consumpton s small or that the degree of nequalty averson n the socal welfare functon s hgh. Further, we fnd for the consumpton tax, ntroduced n addton to an optmally chosen tax on labor ncome, that under the same assumptons t has a redstrbutve potental n our model, smlar to the bequest tax. A closer analyss shows that both taxes, though beng targeted on some form of spendng of the ndvduals budgets (for bequests versus own consumpton), n fact are nstruments for an equalzaton of receved (nherted) wealth. The dfference between them s, however, that the dstorton caused by the consumpton tax works n favor of bequests and nduces, thus, a hgher accumulaton of wealth. For small tax rates, ths accumulaton effect s large enough to outwegh the deadweght loss so that the overall welfare effect of a consumpton tax s postve. Ths result can be seen as a justfcaton of a specfc tax on consumpton whch n fact exsts n many countres, lke the VAT n the European Unon. It should be stressed that n standard optmal-taxaton models such a tax has no partcular role: f the ndvduals budget conssts only of labor ncome whch s spent for the consumpton of dfferent commodtes, a unform tax on the latter s equvalent to an ncome tax. In our model, the analogous nstrument s a unform tax on bequests and consumpton whch we call an expendture tax. We show that such a tax ncreases socal welfare, because t combnes the redstrbutve effect of both separate taxes and causes no dstorton. Our study establshes a role for ndrect taxaton as an nstrument for ncreasng equalty of opportunty. In a related work, Pketty and Saez (2013) also consder the steady state of a dynamc model. They assume that utlty functons as well as abltes follow two stochastc 4

7 processes and study both, the case of joy-of-gvng preferences and the case of altrustc preferences for bequests. Ther central result s a formula for the optmal lnear bequest tax n terms of estmable parameters such as the elastcty of bequests wth respect to the tax. For the dervaton, they take the amount of government transfers as gven, and the formula states to whch extent, dependng on the parameter constellaton, these transfers should be fnanced by a tax on bequests as opposed to a tax on labor ncome. Calbraton of the model wth data from France and the US, leads to the fndng that a tax rate on bequests up to 50 percent may be optmal. Farh and Wernng (2010) study a model wth dynastc preferences and nonlnear taxes on ncome and bequests n a two-perod economy, whch later s extended to an nfnte-horzon settng. Ther man dscusson refers to the case that the socal planner values the future generaton separately, n addton to the value gven to t through the altrustc preferences of the present generaton, whose utlty s also part of the socal welfare functon (ths s sometmes called double-countng). As already mentoned, they fnd that the optmal tax rate on bequests s negatve (to correct for the postve externalty assocated wth leavng bequests) and progressve (the margnal tax rate decreases wth ncome n order to reduce nequalty over generatons). If for some reason margnal tax rates are restrcted to be nonnegatve, a rate of zero up to a certan threshold ncome s optmal. Kopczuk (2001) consders the steady-state of a dynamc economy where nequalty n nhertances arses as a result of dfferences n abltes, smlar to our model. However, he makes the smplfyng assumpton that the ablty levels wthn a dynasty are perfectly correlated (complete mmoblty of abltes over generatons) and derves general formulae whch characterze optmal nonlnear taxes on labour ncome and bequests and whose components reflect the role of these taxes for the reducton of nequalty and for the correcton of externaltes. In contrast to these studes, an mportant feature of our contrbuton s that we provde a precse model of ntergeneratonal transfers, by formulatng the parent-chld relaton of abltes as a Markov process. Ths model allows us to consder the jont dstrbuton of ncome and nherted wealth n the steady-state, and to analyze n detal the long-run relatonshp between drect and ndrect taxes. To our knowledge, ths has not been done before. In partcular, we are able to characterze under whch condtons taxes on ncome and bequests ndeed have a redstrbutve effect n the long run. Moreover, we also study the role of the consumpton tax. 5

8 As to the ssue of double countng, we should stress that our dscusson of the steady-state effects means that double countng s present n our analyss and, thus, the above-mentoned postve external effect s fully taken nto account. 7 That s, f some tax change nfluences bequests left by a generaton, we measure the drect welfare effect on the donor (who has a joy-of-gvng motvaton for net bequests), but also the ndrect effect because smultaneously we nclude the change of the receved nhertances of ths (steady-state) generaton as well. As steady-state nhertances result as the sum of all pror bequests, we thus ncorporate how a tax modfes the postve external effect caused by all pror bequests. In the followng Secton 2 we present the basc model of ndvdual decsons on labor supply, consumpton and bequests. Secton 3 provdes an analyss of the dynamcs of wealth transmsson over generatons, whose steady state s studed n Secton 4. Secton 5 descrbes the maxmzaton problem of the government and ntroduces the stochastc-domnance property mentoned above. In Secton 6 the optmal lnear tax on labor ncome s characterzed and n Secton 7 the results on the desrablty of addtonal taxes on bequests and consumpton, respectvely, are derved. Secton 8 provdes a dscusson of these results. 2 The Model In each perod t a populaton of mass one exsts. It s splt nto n groups wth dfferent abltes w 1 < w 2 <. < w n, whose shares are f 1,, f n. That s, f 1,, f n are the probabltes of an ndvdual to belong to the respectve ablty group. Indvduals of a generaton lve for one perod; they have common preferences over consumpton c, labor tme l, and leavng bequests b, = 1,, n. Preferences are descrbed by the common utlty functon u(c, b, l ) = ϕ(c, b ) g(l ), where ϕ s a concave functon, ncreasng n both arguments, and g s strctly convex and ncreasng. Ths formulaton ndcates that we assume bequests to be motvated by joy of gvng. Pre-tax labor ncome s w l and the ndvdual receves nhertances e. There exsts a lnear tax on labor ncome, gven by α + σw l wth α as a unform negatve tax and σ as the margnal tax rate on gross labor ncome. Moreover, the tax system conssts of a proportonal tax τ b on bequests and of a proportonal tax τ c on consumpton. For 7 In the lterature there s no unanmty whether double countng represents the correct representaton of socal welfare or not (see Cremer 2003, among others). 6

9 gven taxes the maxmzaton problem of an ndvdual s max ϕ(c, b ) g(l ), (1) s.t. (1 + τ c )c + (1 + τ b )b e + α + (1 σ)w l, (2) c, b, l 0. (3) The frst-order condtons for an nteror soluton wth λ as the Lagrange varable assocated wth (2) read as ϕ c λ(1 + τ c ) = 0, (4) ϕ b λ(1 + τ b ) = 0, (5) g (l ) + λ(1 σ)w = 0. (6) We assume that ndvduals need some fxed mnmum consumpton c a 0 and that the ncomeconsumpton curve s lnear, orgnatng n c a. As a consequence, expendtures (1 + τ b )b and (1+τ c )(c c a ) can be wrtten as shares γ and 1 γ, wth γ (0, 1), of the avalable budget, after deductng the expendtures (1 + τ c )c a for mnmum consumpton. Hence, wth γ b γ/(1 + τ b ) and γ c (1 γ)/(1 + τ c ) the demand functons for c and b can be wrtten as c = γ c (e + x (1 + τ c )c a ) + c a, (7) b = γ b (e + x (1 + τ c )c a ). (8) Here x denotes net ncome of the household, x α + (1 σ)w l. We generally assume that the expendtures for c a are not larger than net ncome, that s, x c a (1 + τ c ) 0. Such a type of demand functons, as descrbed by (7) and (8), arses f ϕ can be wrtten as a functon ϕ(c c a, b ), homogeneous n c c a and b. In addton, we assume that ϕ s homogeneous of degree 1. Note that n general γ tself depends on τ b and τ c, as n case of a properly adjusted CES utlty functon wth the functonal form ϕ(c c a, b ) = ((1 δ) 1 η (c c a ) η + δ 1 η b η ) 1 η where the parameters δ and (1 δ), respectvely, express the 7

10 weghts of b and c c a, whle η determnes the constant elastcty of substtuton 1/(1 η). 8 Indrect utlty of an ndvdual s gven by evaluatng ϕ(c, b ) g(l ) usng (7) and (8), and wth l determned by (6). Lnear homogenety of ϕ mples that ϕ(γ c (e +x (1+τ c )c a ), γ b (e + x (1 + τ c )c a ) = λ(e + x (1 + τ c )c a ), where λ = ϕ(γ c, γ b ) s the margnal utlty of ncome whch s ndependent of w, e and x but depends on τ c and τ b. Hence, ndrect utlty V s gven by V (e ; α, σ, τ c, τ b ) = λ(e + x c a (1 + τ c )) g(l ) (9) where l s determned by (6) as l = (g ) 1 (λ(1 σ)w ), dependng on the after-tax wage rate (1 σ)w and on the margnal utlty of ncome λ. Note that from (6) the relaton x 1 < x 2 < < x n follows, because λ s ndependent of w. 3 Dynamcs of wealth transmsson and abltes Havng specfed the model of the economy n any gven perod, ncludng the ndvdual decson to bequeath part of the avalable budget to the descendants, we now turn to the dynamcs. We consder the transmsson of wealth n detal and study the dstrbuton of nhertances n the steady-state of ths process. We assume that potental ablty levels w 1 < w 2 <. < w n reman constant over tme (that s, over generatons). Each ndvdual has a sngle descendant to whom she leaves all her bequests. There s a constant transton probablty p j that an ndvdual wth ablty w has a descendant wth ablty w j, where n j=1 p j = 1 for = 1,, n. Wth constant transton probabltes the dynamcs of the shares of the ablty groups over tme represents a Markov chan. We assume the Markov chan to be ergodc and to converge to a steady-state dstrbuton π = (π 1,, π n ). π s unque and ndependent of the ntal dstrbuton f (f 1,, f n ); t s determned by the equaton π = πp together wth the normalzaton n =1 π = 1, where P s the n n transton matrx wth entres p j. As s well-known, π can be found by consderng the lmt W lm t P t. All n rows of W are dentcal and equal to π. We assume n the followng that the dstrbuton of abltes s already n the steady-state, that s f = π, = 1,, n. Thus, f has the property f = fp. (10) 8 Only n case of a Stone-Geary utlty functon (c c a) 1 δ b δ (η converges to zero), γ = δ s a constant. 8

11 Obvously, then also f = fp s for any s = 1, 2,, because fp s = fp P s 1 = fp s 1 = = fp. Next we study the transmsson of bequests from generaton to generaton. In the followng we ntroduce a tme ndex s to characterze a generaton that lved s perods before some partcular generaton ndexed by 0. Moreover, to ndcate the ablty level of a member of generaton s we use the ndex r s = 1,, n. Hence e r s and b r s denote bequests receved and left, respectvely, by an ndvdual wth ablty w r s n perod s. Smlarly, x r s denotes net ncome of ndvdual of type r s n perod s. Now consder the generaton n perod 0. Each ndvdual = 1,, n of ths generaton receves potental nhertances b r 1 = γ b (e r 1 + x r 1 (1 + τ c )c a ), where r 1 = 1,, n can be any type n the prevous perod 1. Note that wth homothetc preferences, bequests left by an ndvdual r 1 n generaton 1 can be separated nto those out of receved nhertances, γ b e r 1, and those out of own labor ncome, γ b (x r 1 (1 + τ c )c a ). In a next step, we consder nhertances e r 1 of ndvdual r 1 n perod 1, whch arse as some b r 2 = γ b (e r 2 + x r 2 (1 + τ c )c a ) and can, thus, agan be splt nto bequests out of receved nhertances n perod 2 and those out of own labor ncome n perod 2. Pluggng n b r 2 for e r 1 n the formula for bequests left by ndvdual r 1 n perod 1, one gets for nhertances e (= b r 1 ) of some ndvdual n perod 0 e = γ 2 b e r 2 + γ 2 b (x r 2 (1 + τ c )c a ) + γ b (x r 1 (1 + τ c )c a ). (11) Contnung ths separaton further nto the past untl some perod t, we fnd the general formula e = γ t b e r t + γ t b (x r t (1 + τ c )c a ) + γ t 1 b (x r t+1 (1 + τ c )c a ) (12) + + γ 2 b (x r 2 (1 + τ c )c a ) + γ b (x r 1 (1 + τ c )c a ). Here r t, r t+1,, r 2, r 1 denote some ndces from the set {1,, n}. Neglectng for the moment the frst term γ t b e r t, we fnd that nhertances of an ndvdual n perod 0 result from avalable labor ncome n the prevous generatons up to t. It s mportant to note that what an ndvdual of generaton 0 nherts from some partcular generaton s, 1 s t, depends only on the type r s {1,, n} of the ndvdual who orgnated the bequests, but not on the 9

12 types n the n-between generatons s + 1, s + 2,, 1: these generatons just pass forward a share γ b of ther nhertance. In addton, each of these n-between generatons agan orgnates bequests out of own avalable labor ncome. As a consequence, stll leavng asde the term γ t b e r t n (12), all possble nhertances n perod 0, from the perods 1, 2,, t can be wrtten as the sum t γ s b (x r s (1 + τ c )c a ), r s {1,, n} for all s = 1, 2,, t. (13) Note that these possble nhertances are the same for all ablty groups (all potental nhertors) n perod 0. However, the probabltes of the varous realzatons of the nhertances dffer accordng to the type of the recevng nhertor n perod 0. They are determned by the probabltes that has a specfc type r s as her great-great-grandparent n perod s, s = 1,, t. For a closer characterzaton, let µ j be the probablty that an ndvdual of type has a parent of type j. To determne µ j we observe that the probablty of a descendant to belong to ablty group s n j=1 f jp j, whch s equal to f n the steady-state populaton. Thus, the condtonal probablty of a descendant n group to have a parent of group j s µ j = Prob(parent = j descendant = ) = f jp j f. (14) Obvously, µ j s ndependent of t, gven the steady-state dstrbuton of abltes. In matrx notaton, where M denotes the n n matrx wth elements µ j, we have M = p 11 f 2 p 21 f 1 f 1 p 12 f 2 p 22 f np n1 f 1 f np n2 f 2 f 1 p 1n f n f 2 p 2n f n p nn. (15) By defnton, M s also a stochastc matrx, that s, all elements are nonnegatve and n j=1 µ j = 1, for all = 1,, n. Gong back one step further, µ r 1 µ r 1 r 2 s the probablty for ndvdual n perod 0 of havng a parent of type r 1 {1,, n} n perod 1 together wth a grandparent 10

13 of type r 2 {1,, n} n perod 2. Generally, we defne µ (r 1, r 2,, r s ) as the probablty of ndvdual of havng skll types r 1, r 2,, r s as ther ancestors up to generaton s and get µ (r 1, r 2,, r s ) = µ r 1 µ r 1 r 2 µ r s+1 r s. (16) Moreover, n r 1 =1 µ r 1 µ r 1 r 2 s the probablty of ndvdual havng a grandparent r 2 n perod 2, allowng for all possble types as the parent. One observes mmedately that ths probablty s just the element (, r 2 ) of the Matrx M M = M 2. Clearly, M 2 s agan a stochastc matrx, that s, n n r 1 =1 r 2 =1 µ r 1 µ r 1 r 2 = 1, for each. The same holds for M s wth any number of perods s, thus n n r 1 =1 r n 2=1 r µ (r 1, r 2,, r s ) = 1. The elements (, r s ) of the matrx M s are denoted by µ s;r s, that s, µ s;r s n Each µ s;r s r 1 =1 r 2 =1. r s+1 =1 µ r 1 µ r 1 r 2 µ r s+1 r s, = 1,, n, r s = 1,, n. (17) descrbes the probablty that an ndvdual of type n perod 0 has an ancestor of type r s n perod s. Then n r 1 =1 n r s =1 µ (r 1,, r s ) = n r s =1 µ s;r s = 1. Let p s;j be defned n a completely analogous way to µ s;j. Then µ 2;j = n k=1 µ kµ kj = n k=1 f kp k f j p jk /(f f k ) = ( n k=1 p jkp k )f j /f = p 2;j f j /f, and repeated applcaton of ths argument shows that generally µ s;j = f jp s;j f, for any s = 1, 2, (18) As a consequence, lm s M s = W. Ths follows from lm s P s = W, that s, for s, p s;j f whch mples µ s;j f j, j = 1,, n, dentcal for all = 1,, n. We note for later use that wth respect to the constant dstrbuton f of the ablty types, M has the same property as P : f = fm s, for any s = 1, 2, (19) To proof ths, consder that fm = (f 1 (p 11 + p p 1n ),, f n (p 11 + p p 1n )) = f. Moreover, fm 2 = fmm = f, and the same logc apples for any s. 11

14 4 The steady-state We already assumed wthout mentonng that the tax rates τ c and τ b on consumpton and bequests, respectvely, are constant over tme. Extendng ths assumpton to the ncome tax parameters α and σ mples that labor supply and net ncome of each type are also constant over tme. Remember that we have assumed from the begnnng that the populaton s constant and the dstrbuton of abltes s n the steady-state, that s, the shares f of all groups are dentcal n each perod. We defne a steady state of ths economy as a stuaton where the realzatons of nhertances for the ndvduals of a generaton reman constant over tme. Thus, also all possble bequests left by the ndvduals do not change. In vew of the constancy of each type s net ncome over tme, as a consequence of the fxed tax parameters, t follows mmedately from (12) that all possble nhertances e of an ndvdual of a partcular steady-state generaton 0 are gven by all possble nfnte sums γ s b (x r s (1 + τ c )c a ), x r s {x 1,, x n } for all s = 1,. (20) Obvously, all these nfnte sums ndeed have a fnte value as γ b < 1 and all x r s x n whch s a fnte number. 5 The Government Problem Formula (20) characterzes steady-state nhertances n terms of net ncomes x r s (resultng from labor supply l r s ), r s = 1,, n, n all pror generatons s, and of the share γ b of bequests whch n turn depend on the tax parameters α, σ, τ b, τ c, fxed by the government. In the followng we study how the government should choose the tax parameters n order to maxmze socal welfare n the steady-state. In partcular, we want to clarfy the queston of whether taxes on bequests or consumpton (or on both) are approprate supplements to an optmal lnear ncome tax. In our model the ndvduals dffer n labor productvtes and n receved nhertances, but the latter arse endogenously as a result of the exogenous dfferences n labor productvtes. Therefore, t s a pror unclear whether taxes on bequests or consumpton represent desrable addtonal nstruments or are redundant as n the Atknson-Stgltz case 12

15 (wth productvty as the only dmenson of heterogenety). The fact that ndvduals now dffer wth respect to an nfnte number of potental nhertances, n addton to ther skll types, makes the analyss of the long-run consequences of the taxes a rather complex task. In order to keep t tractable, we take nto account only a fnte, though arbtrarly large, number of prevous generatons, from whch a steady-state generaton actually receves nhertances. In other words, n each perod those bequests whch are ntally left by a generaton more than t perods before are dscarded. Ths smplfcaton s certanly justfed by the fact that n (20) the value of nhertances declnes wth the dstance s of earler generatons and approaches zero for s gong to nfnty. Moreover, as all nvolved functons are contnuous, the sgn of the welfare effect of a tax can never swtch to the opposte when the number of prevous generatons taken nto account goes to nfnty. Denote n the followng by the ndex 0 an arbtrary perod of our economy n the steady state. Let t N be arbtrarly large. Consder (20) for a fnte number t of prevous generatons, then each realzaton of nhertances n perod 0 s wrtten as some e(r 1,, r t ) t γ s b (x r s (1 + τ c )c a ) wth r s {1, 2,, n} for all s = 1, 2,, t. (21) Remember that the potental realzatons e( ) are the same for all skll-types, but the assocated probabltes µ (r 1,, r t ) dffer accordng to the type, wth µ ( ) gven by (16). Let E [e] = n r 1 =1 n µ (r 1,, r t )e(r 1,, r t ) denote expected nhertances of a type- ndvdual of some steady-state generaton 0, thus, f E [e] represents actual aggregate nhertances of group. Let further V (r 1,, r t ) denote ndrect utlty of an ndvdual of ablty type, as gven by (9), f ths ndvdual receves an nhertance e = e(r 1,, r t ). 9 We formulate the government s problem of maxmzng steady-state socal welfare n the followng way: let ρ > 0 be the parameter of nequalty averson; the government maxmzes the sum (weghted by the populaton shares f ) of the groups expected socal utlty, each wrtten as E [V 1 ρ /(1 ρ)] = n r 1 =1 n µ (r 1,, r t )V (r 1,, r t ) 1 ρ /(1 ρ), gven stochastc nhertances e(r 1,, r t ). It has to observe the budget constrant that the revenues from the ncome tax σ and the taxes on bequests (τ b ) and consumpton (τ c ) cover the unform subsdy 9 Clearly, V also depends on the tax parameters α, σ, τ c, τ b, but for smplcty of notaton we do not ndcate ths explctly. 13

16 α. Thus, the government s problem reads as max n =1 f E [V 1 ρ /(1 ρ)], (22) s.t. σ n f w l α + τ b =1 τ c =1 f γ b (E [e] + x (1 + τ c )c a )+ =1 f {γ c (E [e] + x (1 + τ c )c a ) + c a } 0, (23) where the publc budget constrant s formulated by use of the demand functons (8) and (7). A crucal ssue for the analyss of optmal taxes wthn ths model s how the dstrbuton of nhertances, over whch the expectaton s computed, s related to the dstrbuton of abltes n each steady-state generaton. Clearly, ths n turn depends on the probabltes of wealth transmsson across ablty types from one generaton to the next. From a socal-welfare pont of vew, redstrbuton of ncome from hgh to low skll types only appears acceptable f low-able ndvduals do not typcally receve larger estates than hgh-able ndvduals. As mentoned n the ntroducton, there s ample emprcal evdence that ths condton s fulflled n realty, that s, a postve correlaton holds between the sze of receved nhertances and own abltes. A natural way to account for such a postve correlaton n our model s to assume that each subsequent row of the matrx M frst-order stochastcally domnates the prevous row, that s, n each generaton an ndvdual of type + 1 s lkely to have a more able parent than an ndvdual of type. Defnton 1 (Row-FOSD) An n-dmensonal stochastc matrx M wth elements µ j fulflls frst-order stochastc domnance of subsequent rows f for each = 1,, n 1, the row vector (µ +1,1,, µ +1,n ) frst-order stochastcally domnates the row vector (µ 1,, µ n ), that s, j k=1 µ k j k=1 µ +1,k, for all j = 1,, n. As an mportant result, we fnd that ths property of ablty transton from one perod to the next ndeed mples for the steady state that overall receved nhertances of a more-able ndvdual tend to be larger than those of a less-able ndvdual. Proposton 1 If M fulflls Row-FOSD then n the steady state the dstrbuton of receved nhertances of a type-( + 1) ndvdual frst-order stochastcally domnates the dstrbuton of nhertances of a type- ndvdual. 14

17 Proof. See Appendx B. An mmedate consequence of ths fundamental characterzaton s that expected nhertances E [e] and utlty levels E [V ] ncrease wth the abltes = 1,, n. On the other hand, the welfare weght of an ndvdual, that s, her expected socal margnal utlty of ncome E [V 1 ρ /(1 ρ)]/ x = λe [V ρ ] = λ n r 1 =1 n µ (r 1,, r t )V (r 1,, r t ) ρ s decreasng wth abltes (remember that λ = V / x s the ndvdual margnal utlty of ncome, ndependent of an ndvdual s wage w ). These propertes whch are essental for the analyss of optmal taxes n the next sectons, are summarzed n the followng. Corollary 1 If M fulflls Row-FOSD then for any σ < 1 the followng nequaltes apply: E [e] < E +1 [e], E [V ] < E +1 [V +1 ] and E [V ρ ] > E +1 [V ρ ], = 1,, n Proof. See Appendx B. The assumpton of Row-FOSD s made throughout the upcomng analyss. As an mportant property we note Lemma 1 If M fulflls Row-FOSD, then M s fulflls Row-FOSD as well. Proof. See Appendx B. 6 The lnear ncome tax As a frst step towards an analyss of whether ndrect taxes have a role n the steady state of ths model, we need to study the welfare effect of the (lnear) ncome tax. That s, we ask f and under whch condtons a margnal ncrease of σ from zero, wth a smultaneous ncrease of α to satsfy the resource constrant, s desrable accordng to the socal welfare functon, settng τ b = τ c = 0. Let wl n =1 f w l denote the average gross ncome and let, for any gven s = 1,, t and r s = 1,, n, E [V ρ r s ] = n r 1 =1 n n r s+1 =1 r n s 1=1 µ (r 1,, r t ) V (r 1,, r t ) ρ /µ s;r s havng an ancestor of type r s n perod s. be the expected socal welfare weght of ndvdual, condtonal on 15

18 Proposton 2 Let τ b = τ c = 0. The welfare effect of an ntroducton of a lnear ncome tax s λ n =1 λ n f E [V ρ ](wl w l )+ t f =1 r s =1 f =1 r s =1 λ n t µ s;r s E [V ρ r s ]γ s (wl w r s l r s )+ µ s;r s E [V ρ r s ]γ s w r s l r s σ. (24) The frst two terms are postve and descrbe the mmedate and the long-run welfare eff ect, respectvely, of ncome redstrbuton. The thrd term s negatve and represents the long-run dstortng eff ect. The overall welfare eff ect s postve f the labor-supply reacton s not too strong. Proof. See Appendx C. The frst lne of (24) captures the mmedate welfare effect, for gven nhertances, thus gnorng the mplcatons of ncome taxaton n pror perods. It corresponds to the standard welfare effect of a lnear ncome tax n a purely statc Mrrlees-model wthout nhertances. A margnal ncrease of the tax rate σ (from zero) hts ndvduals accordng to ther gross ncome w l, whle an dentcal amount α = wl (overall revenue) s returned to everyone, thus wl w l ndcates the net ncome change for a specfc ndvdual wth wage w. Ths net ncome change s clearly postve for low-able and negatve for hgh-able ndvduals; the assocated welfare effect depends on the socal evaluaton of ths redstrbuton, whch s expressed by the socal welfare weghts of the groups, that s, ther expected socal margnal utlty of ncome λe [V ρ ]. It was stated n Corollary 1 that these weghts decrease wth the ablty level f M has the property of Row-FOSD. Gven ncreasng w l, ths mples, together wth n =1 f (wl w l ) = 0, that the frst lne of (24) s unambguously postve. (Note that the mmedate dstortng effect of the tax s of second order, thus zero at σ = 0.) Clearly, the magntude of ths postve welfare effect ncreases wth the spread of ndvdual ncomes and wth the degree of nequalty averson ρ. The second lne of (24) descrbes the long-run effect of ncome redstrbuton. As wth the statc part dscussed above, the mmedate ncome change for group r s n perod s, of ncome 16

19 redstrbuton through the lnear ncome tax, s gven by the dfference wl w r s l r s. However, the socal weght of ths redstrbuton s now more complex: t conssts of the consequences of ths ncome change n perod s (that s, of the change γ s (wl w r s l r s ) of receved nhertances n the generaton 0), for welfare of all ablty groups n the steady-state generaton 0. Clearly, the relevant welfare evaluaton of each group s µ s;r s E [V ρ r s ],.e., ts probablty of havng type r s as an ancestor multpled by the assocated condtonal expected socal welfare. Summng up over all = 1,, n and r s = 1,, n gves us the welfare consequences of ncome redstrbuton n a sngle prevous perod s, and the sum over s then descrbes the total long-run welfare effect. It turns out that redstrbuton of ncome n prevous perods n effect means redstrbuton of nhertances n the steady-state generaton 0. Ths can formally be seen by wrtng the second lne of (24) n an equvalent way as (see (52) n Appendx C) λ n =1 f E [V ρ (e e( ))], (25) where e n =1 f n r n 1=1 µ (r 1,.., r t )e(r 1,.., r t ) = n =1 f E [e] denotes average nhertances receved by generaton 0. For each group, the expectaton term n (25) can be nterpreted as descrbng the socal-welfare evaluaton f receved nhertances n the steadystate are redstrbuted va a lnear tax. Ths relates the analyss of the ncome tax to that of the bequest tax, as wll be dscussed more extensvely n the next secton. As we show formally n Appendx C, the long-run welfare effect of ncome redstrbuton represented by the second lne of (24) s postve. The essental dea s to splt the effect of ncome redstrbuton n prevous perods nto one of nhertance redstrbuton across ablty groups n generaton 0, and an effect wthn groups. Intutvely, Row-FOSD of M mples that a more able ndvdual receves larger expected nhertances, thus ncome redstrbuton n prevous perods produces a desrable equalzaton of nhertances across ablty groups n the steady state. In addton, also wthn each group, for ndvduals wth the same sklls, a desred equalzaton from those who receve large to those who receve small nhertances takes place. Altogether, the long-run redstrbutve effect of the ntroducton of an ncome tax s postve. Fnally, turnng to the thrd lne of (24), ths effect s clearly negatve because l r s / σ < 0 for all s = 1,, t. Its structure s smlar to that of the second lne; t represents the long-run 17

20 welfare loss caused by the dsncentve effect of the ncome tax on labour supply n each perod s, whch transforms nto lower steady-state nhertances. Note that ths effect drops out f redstrbuton wthn a generaton alone s consdered (as n the statc Mrrlees model), because for each ndvdual the negatve mpact on labor supply and ncome s just compensated by the welfare gan through the assocated ncrease n lesure (the deadweght loss s zero at σ = 0); nevertheless ths dsncentve effect occurs n the steady-state of our dynamc model, because the tax affects the sze of bequests left by prevous generatons. In the followng we take as gven that a postve margnal tax rate σ s optmal, whch means that some redstrbuton of ncome s desrable. In other words, we assume that labour supply s not too elastc, such that for an ntal ncrease of σ the dsncentve effect (lne three of (24)) s domnated by the postve effects. Obvously, wth larger σ, the dstortng effect on labor supply of the margnal tax rate becomes ncreasngly mportant, and σ < 1 must hold n the optmum; otherwse labor supply would be zero, as follows from (6) wth g (l ) > 0 for l > 0. 7 Optmal ndrect taxes For a precse analyss of whether an ndrect tax can contrbute to a desrable redstrbuton of wealth n the steady state, on top of what s acheved by an optmal ncome tax, we need to ntroduce a new property, n addton to Row-FOSD. Let s N, s t 1. Remember that µ s;jk denotes the element (j, k) of M s (thus, µ 1;jk = µ jk ), then clearly µ s+1;k = n j=1 µ jµ s;jk s the element (, k) of MM s = M s+1. Defnton 2 An n-dmensonal stochastc matrx M wth elements µ j fulflls the condton of Path Domnance (abbrevated by PD) of order t, f for any s = 1,, t 1, = 1,, n and k = 1,, n 1, the probablty vector wth elements µ j µ s;j,k+1 /µ s+1;,k+1, j = 1,, n, frstorder stochastcally domnates the probablty vector wth elements µ j µ s;jk /µ s+1;k, j = 1,, n. and k. By defnton, n j=1 µ jµ s;j,k+1 /µ s+1;,k+1 = 1 = n j=1 µ jµ s;jk /( n j=1 µ jµ s;jk ), for any As dscussed above, the elements µ s+1;k of M s+1 descrbe the probablty that an ndvdual of skll type has a type-k great-, great-, grandparent, s + 1 perods before. Thus, the µ j µ s;jk /µ s+1;k, j = 1,, n descrbe the probablty that an ndvdual of skll type and 18

21 wth a type-k great-, great-, grandparent, s + 1 perods before, has an mmedate parent of type-j, n perod 1. Intutvely then, condton PD requres that f we consder two ndvduals of the same type n the steady-state generaton 0 wth dfferent great-, great-, grandparents of type k and type k + 1, respectvely, s + 1 perods before, the one wth a type-k + 1 great-, great-, grandparent tends to have a more-able mmedate parent than the other. For a two-type economy we fnd: Lemma 2 If n = 2, then PD s mpled by Row-FOSD of M. Proof. We have to check that µ 1 µ s;11 /(µ 1 µ s;11 + µ 2 µ s;21 ) µ 1 µ s;12 /(µ 1 µ s;12 + µ 2 µ s;22 ) whch reduces to µ s;11 µ s;22 µ s;12 µ s;21. Ths nequalty holds because Row-FOSD of M s (see Lemma 1) for n = 2 mples µ s;11 µ s;21 as well as µ s;12 µ s;22. Though PD s related to Row-FOSD, t represents a separate condton. Indeed, for n > 3 one can fnd matrces for whch PD s not mpled by Row-FOSD. On the other hand, Path Domnance s automatcally fulflled for large s (as s Row-FOSD), whch follows mmedately from the fact that lm s M s = W (mentoned n Secton 3), together wth the property that all rows of W are dentcal and equal to the steady-state dstrbuton f. The relaton between Row-FOSD and Path Domnance of M s dscussed n more detal below and n Appendx D. 7.1 The bequest tax We begn wth the analyss of the welfare effect of a proportonal bequest tax and ask whether the latter can play a role as a complement to an optmally chosen labor ncome tax. Let for any s = 1,, t 1, e s (r s 1,, r t ) t ŝ=s+1 γŝ s b (x r ŝ (1+τ c )c a ) denote what generaton s receves as nhertances from some seres r s 1,, r t of ancestors up to perod t, wth average e s n r s =1 f r s n r s 1 =1 n µ r s (r s 1,.., r t )e s (r s 1,.., r t ). Wth ε as the elastcty of substtuton between bequests and consumpton, Ω as defned n (74) n Appendx C and S(τ b, τ c ) as the optmal value functon of the maxmzaton problem (22) - (23), we fnd for the bequest tax: Proposton 3 Let τ c = 0. The welfare effect of an ntroducton of a tax τ b on bequests, gven 19

22 that the lnear ncome tax s chosen optmally, s S τ b τ b =0 =λ n =1 + n =1 n =1 f E [V ρ ( )γ(e e( ))] t 1 f f E [V ρ E [V ρ ( )γ s γ(e s e s ( ))] + γω ( )ε(1 γ)(e( ) + t 1 γ s e s ( ))]. (26) The frst lne descrbes the mmedate redstrbutve eff ect; t s postve. The second lne descrbes the long-run redstrbutve eff ect; t s postve, gven Path Domnance. The thrd lne captures the long-run dstortve eff ect and s negatve (or zero). The overall effect of τ b on socal welfare s postve () f Path Domnance holds and the elastcty of substtuton between bequests and consumpton s small, or () f nequalty averson s large and nhertances at the lower end of the nhertance dstrbuton are small. Proof. See Appendx C. As wth the lnear ncome tax, two separate effects can be dstngushed as a consequence of the ntroducton of the bequest tax: the welfare effect when nhertances are held fxed, whch s the term n the frst lne of the above formula, and the welfare effect due to changes n steady-state nhertances (as a result of changes n bequests left by prevous generatons), whch s descrbed by the second and the thrd lnes of (26). The mmedate redstrbutve effect. Ths effect represents the man role of the bequest tax: t allows addtonal redstrbuton of γe( ),. e., of that partcular part of bequests left by the steady-state generaton 0, whch s fnanced out of receved nhertances. The mportant pont s that n fact the bequest tax s an nstrument for equalzaton of receved nhertances, because these ndeed represent a second characterstc of the ndvduals, n addton to abltes. Ths redstrbuton of receved nhertances s performed ndrectly, va the share γ whch s used for bequests left to own hers. Note that γ(e e( )) s clearly postve for ndvduals who nhert lttle whle t s negatve for those who nhert much. Ths effect also occurs n the statc model wth exogenous nhertances (Brunner and Pech 2012b). Clearly, the bequest tax s also mposed on the other part of bequests, γx, whch s fnanced out of net labor ncome. But for ths part no effect shows up n the frst lne of (26) because - as the Atknson-Stgltz result tells 20

23 us - no further redstrbuton s possble n addton to that performed by the optmal ncome tax. One observes that the mmedate redstrbutve effect n lne one of (26) s very smlar to the long-run redstrbutve effect of the ncome tax, as already mentoned above (see (25)); t dffers only by the factor γ. Ths relates to the common vew that nhertance taxaton may represent an undesrable double-taxaton, because t affects the bequeathed part of ncome, and the later was already subjected to the ncome tax. Note, however, that ths s clearly true for any ndrect tax, thus also for a consumpton tax (see below). More mportantly, n the present context the essental pont s that the bequest tax allows addtonal redstrbuton, above what s performed by the optmal ncome tax. The ntutve reason s that, compared to the latter, t causes a smaller dstorton of the labor supply decson, as can mmedately be seen from the relaton l / τ b = γ(1 σ) l / σ (see (59) n Appendx C), where γ(1 σ) < 1. On the other hand, n the longer run the bequest tax has a dstortng effect on captal accumulaton, as s dscussed n the followng. Redstrbuton n prevous perods. The frst term n the lne two of (26) comes from the fact that n each perod s the bequest tax redstrbutes that partcular part γe s (r s 1,, r t ) of bequests whch s fnanced out of receved nhertances from ancestors up to t. 10 Ths s completely analogous to the mechansm dscussed above for the redstrbuton of bequests fnanced out of nhertances n the steady state (descrbed by the frst lne of (26)). Of ths redstrbuton n s, fnally γ s γ(e s e s (r s 1,, r t )) arrves at the generaton 0. What matters for the socal evaluaton s how aggregate welfare of all groups = 1,, n n ths steady-state generaton 0 s affected by the total of ths redstrbuton takng place n all prevous perods s, 1 s > t, and ths s descrbed by the frst term n the second lne of (26). To determne the sgn of ths term turns out to be a complex task. For a two-type economy, the Row-FOSD condton mples a clear postve effect, whle for n > 2 the Path-Domnance condton on M s needed to guarantee that redstrbuton of bequests n any prevous perod s mproves perod-0 welfare. PD requres, essentally, that for an ndvdual who receves 10 Remember our assumpton that only bequests from generatons up to t arrve at generaton 0. Therefore nhertances of generaton s orgnatng from ancestors pror to t do not occur n the formula. 21

24 hgher nhertances from a specfc generaton s, total nhertances from all prevous perods together are hgher as well. It must be stressed, however, that ths condton s by no means necessary; the frst term n the second lne of (26) s qute lkely to be postve, even f the frst-order stochastc domnance, requred n PD, s volated n some components. In fact, a number of smulaton results whch we present n Appendx D, corroborate ths clam. These smulatons show that for randomly chosen matrces M whch fulfll Row-FOSD and whose rows are unmodal wth the maxmum beng the dagonal element, 90 percent of them ether fulfll PD or volate PD n less than 8 percent of all relevant comparsons. Moreover, the extent by whch the requred nequaltes are not fulflled when the comparsons fal, s small. As a consequence, these results ndeed allow the concluson that redstrbuton of bequests n prevous perods has a postve mpact on steady-state welfare. As to the second term γω n the second lne of (26), we show n Appendx C that t s postve and goes to zero for t gong to nfnty, that s, the larger the number of generatons whose bequests we actually take nto account. The dstorton of the bequest decson and the overall effect. The thrd lne of (26) descrbes the socal evaluaton of the substtuton effect assocated wth the ntroducton of the bequest tax n prevous perods. It s negatve or zero because of ε 0. Gven that redstrbuton n prevous perods has a postve effect (lne two of (26)), the overall effect s clearly postve for ε = 0. Intutvely, a zero substtuton elastcty means that the rato b /(c c a ) does not change wth the tax τ b ; the latter only redstrbutes the avalable budget but does not dstort the accumulaton process. In other words, t only causes an ncome effect but no substtuton effect. By contnuty, we can extend ths concluson and state that a bequest tax ncreases socal welfare f the share of bequests does not react too much, that s f ε s small. For a larger value of the elastcty of substtuton a postve welfare effect of the bequest tax arses f ρ, the parameter of nequalty averson, s large and nhertances at the lower end of the dstrbuton are small. The reason s that for large ρ only the ndvduals wth lowest ncomes and lowest nhertances count n the socal welfare functon. If ther nhertances are close to zero (all ther ancestors n the perods s = 1,, t had net ncomes just suff cent to cover necessary consumpton c a ) then obvously the nhertances of ther ancestors are also close to zero and, thus, the thrd lne of (26) s close to zero for any ε. As an llustraton consder the 22