Online Appendix for A Simpler Theory of Optimal Capital Taxation by Emmanuel Saez and Stefanie Stantcheva

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1 Onlne Appendx for A Smpler Theory of Optmal Captal Taxaton by Emmanuel Saez and Stefane Stantcheva A. Antcpated Reforms Addtonal Results Optmal tax wth antcpated reform and heterogeneous dscount rates Wth heterogeneous dscount rates, the change n socal welfare s: dsw F ω e δ T [ u c (c, k, z ) u c (c, k, z ) k k m Defne the normalzed socal welfare weght g (T ) the reform and rewrte: dsw F τ δ u c (c, k, z ) e a τ (t) e δ (tt ) dt τ L δ u c (c, k, z ) zm e a τ rk m L,τ (t) e δ (tt ) dt ω u c e δ T ω u c e δ T whch depends on the tme of g (T ) k k τ δ m g (T ) e a τ (t) e δ (tt ) dt τ L z m δ τ rk m g (T ) e a L,τ (t) e δ (tt ) dt whch yelds the same optmal tax formula as n the text, but wth ḡ = g (T ) k /k m, e a = δ g (T ) e a (t) e δ (tt ) dt and e a L,τ = δ g (T ) e a L,τ (t) e δ (tt ) dt Relatve to the case wth homogeneous dscount rates, the socal welfare weghts g (T ) now depend on the tme of the reform T, and on each person s dscount rate δ. Agents who are more mpatent (larger δ ) get dscounted more heavly n the antcpated socal welfare objectve. Ths effect s starker the longer n advance the reform s announced (the larger T ). In the lmt, only the most patent agents wth the lowest δ wll be counted n the socal welfare objectve. Fnte and nfnte antcpaton elastctes We now show that antcpaton elastctes are nfnte under full certanty wth wealth n the utlty and certanty. We also show that they are fnte wth uncertanty (wth or wthout wealth

2 n the utlty). Frst, the antcpaton elastcty to a reform dτ for t T s nfnte when there s full certanty, even wth wealth n the utlty. The proof s as n Pketty and Saez (23) for the Chamley-Judd model (wthout wealth n the utlty). To see ths, note that under full certanty the frst-order condton of the agent wth respect to captal always holds: u c,t = ( + r)/( + δ )u c,t+ + /( + δ )u k,t+ Suppose we start from a stuaton n a well-defned steady state: (δ r)u c = u k where we have perfect consumpton smoothng. The ntertemporal budget constrant s: t Consumpton smoothng mples: ( ) t c t + lm + r k t = ( ) t z t + k t + r t u c ( rk + z, k ) = λ for the multpler λ on the budget constrant. Hence, k = lm t k t >. Gven that there s perfect consumpton smoothng, usng the budget constrant to solve for consumpton yelds: ( c = ) ( ( ) ) t z t + k k + r + r t (A) Consder what happens f the captal tax rate ncreases by dτ > for t T. The present dscounted value of all resources, denoted by Y for agent s: Y = k + T t= ( + r ) t z t + ( ) t z t + r The change n resources evaluated at τ = s: dy = ( ) T ( ) tt + ( ) T t z tdτ dτ ( + r) ( + r) ( + r) 2

3 ( T Hence, consumpton pre-reform wll shft down by a factor proportonal to (+r)) dτ. From the aggregated budget constrant we have that: k m t = ( + r) t k m c m ( + ( + r) + ( + r) ( + r) t ) + (z m t ( + r) t z m ) Therefore, the change n the aggregate captal stock s: dk m t = dc m ( ) ( + r) t ( T Recall that the change n consumpton (from (A)) s proportonal to (+r)) dτ. Hence: and: dk m t ( ( + r) The antcpaton elastcty e ante e ante = δ + δ t<t ) T ( ( + r) t r r ) ( ) ( + r) dτ = ( + r) T t dτ r ( ) ( + r) e t kt m ( + r) T t dτ r s defned as: ( ) tt e t δ + δ + δ t<t ( ) tt ( ) ( + r) kt m ( + r) T t dτ + δ r Snce we have δ > r, lm T ( +δ +r) T =, whch makes the sum above (to whch the antcpaton elastcty s proportonal) converge to nfnty when T goes to nfnty. Optmal tax formula startng away from the steady state: If the economy s not n steady state, the sprt of the formula stll holds, but t s no longer possble to treat margnal utltes and aggregate varables as constant. In that case, not just the elastctes, but also the weghtng factors multplyng them and the dstrbutonal characterstc take nto account the full transton path. For concseness, let u c (t) u c (c (t), k (t), z (t)). The 3

4 change n socal welfare s (wth each term of the formula marked n underbraces): u c (t) dsw F ω δ T ω δ u T c(t)rk m (t) e δ t dt rk (t) e δt dt }{{} ḡ τ L ω δ u c (t)e a z m (t) L,τ τ (t) ω δ u T c(t)z m (t) e δ t dt eδ t dt ω δ u T c(t)z m (t) e δt dt }{{} ω δ u T c(t)rk m (t) e δ t dt }{{} e a L,(τ ) z m /(rk m ) τ u c (t) ω δ τ ω δ u T c(t)rk m (t) e δ t dt rkm (t)e a (t) e δt dt }{{} e a For an unantcpated reform startng from an arbtrary tax system and away from the steady state, set T = n the above expresson for dsw F and replace the antcpated elastctes by ther unantcpated counterparts e u L,(τ ) and eu. A.2 Comparson to Other Models Ths secton formally compares our model to earler model descrbed n Secton 5. A.2. Judd (985) Model In the Judd (985) model, ndvdual utlty s: V ({c (t), z (t), k (t)} t ) = u (c (t), k (t), z (t))e t δ (c (s))ds dt. The effect on V from a small change n the captal tax dτ s now: dv = dτ [ t= ( u c (c (t), k (t), z (t))e t δ (c (s))ds + δ (c (t)) ( rk m (t) rk (t) t ) u (s)e s δ (c (m))dm ds ) τ τ rk m (t)e (t) dt. 4

5 In the steady state, we can hence wrte dv as: [ dτ r ( u c e δ (c )t + δ (c )u e δ (c )t ) ( e δ (c )(st) ds k m (t) k (t) τ ) k m (t)e (t) dt t τ [ ) = dτ r (u c e δ (c )t dt + δ (c )u e δ (c )t [k m (t) k (t) δ (c ) ( ) u c e δ (c )t + δ (c )u e δ (c )t e δ (c )t τ ds k m (t)e (t) τ = dτ rk m ( ) [ u c + δ (c ) δ (c ) δ (c ) u k k τ δ m (c ) e δ (c )t e (t). τ We can hence see that the formulas from our model apply but wth g and e as redefned n the text. A.2.2 Ayagar (995) Model Note that all proofs below would be exactly the same as the proofs for wealth-n-the-utlty f we reformulated t n dscrete tme, replacng the standard utlty wthout wealth n the utlty, u t (c t ), by u t (c t, k t ). Ths s done by lettng u t denote u t(c t,k t ) c t nstead of u t(c t ) c t. We apply the envelope theorem, whch states that the changes n the captal tax rate dτ only has a drect mpact on utlty through the drect reducton n consumpton that t causes. Usng ths, and takng the dervatve of the socal welfare SW F wth respect to dτ yelds: ( ) t + δ dsw F = ω u t (τ rdkt m ) + t<t ( [ τ ( ) t = dτ rkt m e t ω u t + τ + δ t<t + ( ) t ) ω u t r(kt m k t ) + δ ( [ τ = dτ τ t ( ) t ω u t (rdτ (kt m k t ) + τ rdkt m ) + δ ( ) t rkt m e t ω u t + δ ( ) t rkt m e t ω u t ( ) t ) ω u t r(kt m k t ). + δ + δ If varables have already converged to ther ergodc paths when the antcpaton responses start: then all terms n e t are zero before the steady state has been reached and hence, we can dvde through by ω u tk m t = g k m t whch s constant across t. Thus: 5

6 dsw F τ ( τ ) ( δ + δ t<t ( ) tt e t + δ + δ + δ ( ) tt e t) + g k t + δ g. kt m Let the dstrbutonal factor ḡ = s gven by: g k t g k m t. The optmal captal tax n the Ayagar (995) model τ = ḡ ḡ + e. ( wth e = δ ) tt +δ t> et. For an unantcpated reform, the formula apples wth T = +δ when the economy s already n the steady state as of tme. If varables have not converged to ther ergodc paths when the antcpaton responses start: we have to take nto account the transton of the margnal utltes and the captal stock across tme. dsw F = dτ ( Dvdng by dsw F τ ( τ ) [ t<t ( ) t ω +δ u t kt m ( ) t rkt m e t ω u t ( ) t ) ω u t r(kt m k t ). + δ + δ yelds: [ τ ( ) t k m ( τ ) + δ t<t + ( + δ t e t ( +δ ) t ω u t ) t ω u t km t ω u t k t ) t ( +δ ω u t km t. Now we have to redefne the average welfare weght as: ḡ ( + δ ) t u t k t ) t ( +δ u t km t, and the total elastcty as: e = t ( ) t k m + δ t e t ( +δ u t ) t u t km t. Wth these redefned varables, the same formula holds. 6

7 A.3 Optmal Nonlnear Taxes n the Generalzed Model Let e top (t) be the average elastcty of total captal ncome of those ndvduals wth captal ncome above threshold rk top. It s measured at tme t followng a small reform of the top bracket tax rate dτ takng place at tme. The elastcty s weghted by captal ncome. Let e L,τ (t) be the elastcty of labor ncome of those ndvduals wth captal ncome above threshold rk top. Proposton A. Optmal top captal tax rate n the steady state. Suppose there s a lnear tax on labor ncome τ L. The optmal top captal tax rate above captal ncome level rk top takes the form: wth ē top τ top g δ t= etop (t) eδ t dt. = ḡtop τ L z m ḡ top ḡ top ē r(k m,top k top ) L,(τ ), + atop ētop = :k k top g (k k top ) :k k top (k k top ) s the average captal ncome weghted welfare weght n the top captal tax bracket, and a top = km,top s the Pareto parameter of the captal ncome dstrbuton. ē L,(τ ) g k m,top k top δ e t= L,(τ )(t) e δt dt. Proof of Proposton A: We consder the top tax rate τ on captal above threshold k top. As r s unform, ths s equvalent to a top tax rate applyng above captal ncome threshold rk top. Let N denote the fracton of ndvduals above k top. We agan use the notaton k m,top to denote the average wealth above the top threshold,.e.: k m,top = :k rk (t) k top, N Suppose we change the top tax rate on captal by dτ. As defned n the text, let e top (t) be the elastcty of captal holdng of top captal earners (the wealth elastcty of total wealth to the tax rate of those wth captal ncome above rk top ). For all ndvduals above the cutoff, the change n utlty s: [ dv = dτ δ u c (c (t), k (t))nr(k m,top (t) k top )e δt τ τ u c (c (t), k (t))r(k (t) k top )e δ t u c (c (t), k (t))nrk m,top (t)e top (t) eδ t dt. 7

8 Startng from the steady state, captal levels are constant so that: [ dv = u c r(k m,top k top )Ndτ (k k top ) (k m,top k top )N τ a top τ where a top = k m,top (k m,top k top ). For ndvduals below the cutoff, the change n utlty s: [ dv = u c r(k m,top k top )Ndτ τ a top τ The change n socal welfare s such that: (k k top ) dsw F g :k k (k top m,top k top )N τ a top g δ τ δ e top (t) eδ t dt. δ e top (t) eδ t dt, e top (t) eδ t dt. Let ḡ top :k k top g (k k top ) (k m,top k top )N and etop g δ e top (t) eδ t dt. Then, we obtan the optmal tax rate τ such that dsw F = : τ = ḡ top ḡ top + atop etop. Wth endogenous labor, let e L,(τ )(t) = dzm (t) ( τ ) d( τ ) z m (t) = dzm (t) d r r Nz m (t). be the elastcty of aggregate (average) labor ncome z m wth respect to the top captal tax rate, normalzed by N, n the two bracket tax system. For all ndvduals wth captal ncome above the cutoff: [ dv = dτ δ u c (c (t), k (t), z (t))nr(k m,top (t) k top ) e δ t τ L τ τ τ u c (c (t), k (t), z (t))z m (t)ne L,(τ )(t) e δ t u c (c (t), k (t), z (t))r(k (t) k top ) e δ t u c (c (t), k (t), z (t))nrk m,top (t)e top (t) eδ t dt. 8

9 Startng from the steady state, captal and labor ncome are constant over tme: τ L z m τ r(k m,top k top ) The change n socal welfare s: τ L z m τ r(k m,top k top ) dv = u c Nr(k m,top k top )dτ δ e L,(τ )(t) e δ t dt τ a top τ [ (k k top ) (k m,top k top )N δ e top (t) eδ t dt. (k k top ) dsw F = ω dv g :rk rk (k top m,top k top )N g δ e L,(τ )(t) e δt dt τ a top g δ e top τ (t) eδ t dt. Defne e top, e L,(τ ), and ḡ top as n the text. The optmal formula n the text s then obtaned by rearrangng the prevous condton. It s straghtforward to generalze ths to the case of an antcpated reform by dscountng the elastctes usng e δ(tt ) and usng the antcpated elastctes rather than the unantcpated ones. A.4 Optmal Taxaton wth Horzontal Equty Concerns. In ths secton, we formally consder optmal captal and labor taxaton under horzontal equty concerns. As derved n Secton 3.2., the optmal revenue-maxmzng rates are: τ R L = +e L and τ R = +e. Wthout loss of generalty, we suppose that captal s more elastc so that τ R < τ R L. The optmal lnear comprehensve tax on ncome s, as derved n (6): τ Y = ḡ Y ḡ Y + e Y wth ḡ Y = g y y Suppose that the dstrbuton of captal and labor ncome s dense enough, so that at every total ncome level y = rk + z, there are agents wth y = rk (captal ncome only) and y = z (labor ncome only). Generalzed socal welfare weghts that capture horzontal equty concerns are such that: () If τ L = τ, then the socal welfare weghts g are standard. For nstance, we can set g = u c for all agents. Any reform that changes taxes should put zero weght on those who after the reform are such that τ L z +τ rk < max j {τ L z j +τ rk j z j +rk j = z +rk },.e., on those who 9

10 pay less taxes at a gven total ncome y = rk + z, or, equvalently, have the hghest dsposable ncome and consumpton at any ncome. Ths means that f labor taxes are ncreased, g = for those wth any postve captal ncome at each total ncome level. Conversely, ncreasng captal taxes wll yeld g = for those ndvduals wth some labor ncome at each total ncome level. () If τ L > τ, then all the socal welfare weghts are concentrated on those wth τ L z + τ rk > max j {τ L z j + τ rk j z j + rk j = z + rk },.e., on those agents wth only labor ncome. Conversely, f τ L < τ, all the socal welfare weghts are on agents wth only captal ncome. Suppose that, startng from a stuaton wth τ L = τ we ntroduce a small tax break on captal ncome, dτ <. Captal ncome earners now get an unfar advantage and all the weght s concentrated on those wth no captal ncome (equvalently, everyone wth k > receves a weght g = ). As a result, a small tax break on captal can only be optmal f t rases tax revenue and, hence, allows to lower the tax rate on labor ncome as well. Ths can only occur f τ Y > τ R,.e., the optmal comprehensve tax rate s above the revenue-maxmzng rate on captal ncome. Proposton A2. Optmal labor and captal taxaton wth horzontal equty concerns. () If τ Y τ R, taxng labor and captal ncome at the same comprehensve rate τ L = τ = τ Y s the unque optmum. () If τ Y > τ R, a dfferental tax system wth the captal tax rate set to the revenue maxmzng rate τ = τ R < τ L (wth both τ and τ L smaller than τ Y ) s the unque optmum. Proof. Let us consder the two cases n turn. () If τ Y τ R. To see why τ L = τ = τ s an equlbrum, suppose that we tred to lower the tax rate on captal ncome. Then, all the weght wll concentrate on people wth only labor ncome, whch wll then n turn make t optmal to ncrease the tax on captal agan. Ths equlbrum s unque. There s no other equlbrum wth equal taxes on captal and labor that can rase more revenue wth a lower tax rate, by defnton of τ Y as the optmal rate on comprehensve ncome. There s also no equlbrum wth non-equal tax rates on captal and labor. Suppose that we tred to set (wthout loss of generalty) τ < τ L. Then to rase enough revenue we would requre that τ < τ Y < τ L. Snce captal owners are now advantaged, all the socal welfare weght concentrates on people wth only labor ncome. Snce then a fortor τ < τ R, ncreasng τ would mean that more revenue would be rased, whch would allow us to lower τ L, whch s good snce all weght s on people wth only labor ncome. () If τ Y > τ R.

11 In ths case, the equlbrum has τ = τ R < τ Y and τ Y > τ L > τ R. Clearly ths s an equlbrum snce we cannot decrease τ L wthout losng revenue and we cannot rase more revenue through τ (snce t s already set at the revenue-maxmzng rate for the captal tax base). In addton, we cannot decrease τ further wthout ncreasng τ L, whch s not desrable snce t would beneft people captal ncome earners, who already receve a weght of zero. Ths equlbrum s also unque. If we set τ L = τ equal, we should set them equal to τ Y whch s the optmal tax rate on comprehensve ncome. But then, snce τ s now above ts revenue maxmzng rate, we could lower both τ and τ L wthout losng revenues, so ths would not be an equlbrum. On the other hand, as long as we set τ < τ L, captal ncome earners get zero weght and the only possblty s to go all the way to τ = τ R snce only people wth only labor ncome have a non-zero weght. As a result, horzontal equty concerns wll be a force pushng towards the comprehensve ncome tax system derved n Secton In the text, we provded an effcency argument n favor of a tax on comprehensve ncome (based on ncome shftng opportuntes) whle the argument here s based on equty consderatons. Wth horzontal equty preferences, devatons from a comprehensve ncome tax system can only be justfed f they rase more revenue and generate a Pareto-mprovement, whch drastcally reduces the scope for them. In Saez and Stantcheva (26) we argue that ths s akn to a generalzed Rawlsan prncple whereby dscrmnaton aganst some groups (e.g., captal owners versus labor provders) s only permssble f t makes the group dscrmnated aganst better off,.e., f t generates a Pareto mprovement. A.4. Horzontal Equty wth Nonlnear Taxaton The same reasonng as for lnear taxaton wth horzontal equty also apples to nonlnear taxes. Startng from a comprehensve tax system T Y (z + rk) as derved n Secton 3.2., lowerng the tax rate on captal ncome, condtonal on a gven total ncome level, wll generate a horzontal nequty and concentrate all socal weght on those wth no captal ncome condtonal on that total ncome level. Such a preferental tax break for captal ncome earners wll only be acceptable f t generates more revenue and allows to lower the tax rate on labor ncome as well. We show ths below. Formally, suppose that we start from the optmal tax on comprehensve ncome, T Y (rk + z), as derved n Secton 3.2. whch does not dscrmnate between captal and labor ncome condtonal on total ncome. We say that a tax system unambgously favors captal at ncome level y = rk + z, f for any (rk, z) such that y = rk + z, and any ε, z, T Y (rk, z) > T (rk + ε, z ε) (havng more captal ncome, condtonal on a gven total ncome leads to lower

12 taxes). (Note that t may be the case that a tax system favors captal only at some y levels or only at some rk, z ranges.) Denote a change n the tax by δt (rk, z). A devaton δt (rk, z) s sad to ntroduce horzontal nequty, f, startng from a comprehensve tax system T Y (rk + z), the resultng tax system T Y (z + rk) + δt (rk, z) cannot be expressed as T Y (rk + z) for some functon T Y. Wth nonlnear taxes, we can agan defne the generalzed socal welfare weghts as follows. ) If there s a comprehensve tax T Y (z + rk), then everybody has standard weghts, such as, for nstance, g = u c. For any devaton δt (rk, z) that ntroduces horzontal nequty, the weghts concentrate on the agents who pay the hghest tax at a gven total ncome level,.e., on those wth T Y (z + rk ) + δt (rk, z ) = max j {T Y (z j + rk j ) + δt (rk j, z j ) z j + rk j = rk + z } (whch s equvalent to puttng all the weght on the agent(s) wth lowest dsposable ncome at any total ncome level). Hence, the weghts also need to depend on δt (z, rk), the drecton of the tax reform. ) If the tax s such that T (rk, z) cannot be expressed as T Y (rk + z) for some functon T Y, then the weghts concentrate on those wth T (z, rk ) = max j {T (z j, rk j ) z j + rk j = rk + z },.e., on the agents whch pay the hghest tax (equvalently, have the lowest dsposable ncome) condtonal on total ncome. Equlbra: Suppose that, at the comprehensve tax rate, no small reform δt (rk, z) that ntroduces horzontal equty and favors captal (accordng to our defntons above) can ncrease total tax revenues,.e., for all δt (rk, z) that favor captal and ntroduce horzontal nequty, the alternatve tax system T Y (rk, z) = T Y (rk + z) + δt (rk, z) s such that: T Y (rk (T Y ) + z (T Y ))d > T Y (rk ( T Y ), z ( T Y ))d where naturally, the choces z (T ) and k (T ) depend on the tax system T. Then the unque equlbrum has the comprehensve tax system n place, as derved n No horzontal nequty can be an equlbrum unless t ntroduces a Pareto mprovement. Suppose on the other hand that f the revenue maxmzng tax rate on captal, T R(rk) were mplemented, and a labor ncome tax T L (z) was used to complement t, more revenue could be rased than wth the tax on comprehensve ncome T Y (rk + z) and the tax burden on all agents would be lower than under the comprehensve ncome tax. Then, the optmum s to set dfferental taxes on captal and labor ncome, wth the captal tax at ts optmal revenue-maxmzng schedule. Horzontal nequty s an equlbrum because t generates a Pareto mprovement. 2

13 A.5 Progressve Consumpton Taxes The progressve consumpton tax s defned on an exclusve bass as t C (.) such that k = rk + z [c + t c (c) Equvalently, we can agan defne the nclusve consumpton tax T C (y) on pre-tax resources y devoted to consumpton such that c+t c (c) = y s equvalent to c = yt C (y),.e., y yt C (y) s the nverse functon of c c + t c (c) and hence + t C = /( T C ). The case of a progressve consumpton tax s most easly explaned wth nelastc labor ncome (possbly heterogenous across ndvduals). Real wealth k r n the presence of the progressve consumpton tax s: k r (k) = k T C( rk + z) T C (z) r Recall that real wealth s defned as nomnal wealth adjusted for the prce of consumpton. There are two ways to see that the above s the rght expresson. Frst, wealth k provdes an ncome stream rk whch translates nto extra permanent consumpton equal to the ncome mnus the tax pad on the extra consumpton rk [T C ( rk + z) T C (z) whch can be captalzed nto wealth k r by dvdng by r. If labor ncome s heterogeneous across agents, then k r (k, z) should also be ndexed by z. Another way to see ths s to ask what the captal k r would be that would yeld the same dsposable ncome as the nomnal captal under the consumpton tax. Dsposable ncome n terms of real captal k r s rk r T C (z). Dsposable ncome expressed n terms of nomnal captal s: rk T C ( rk + z). These two must be equal, whch yelds the expresson for k r above. k r has three natural propertes: wth no consumpton tax, real and nomnal wealth are equal, dk r /dk = T C,.e., and extra dollar of nomnal wealth s worth T C n real terms, and kr () =. In that case, we have n steady-state c = rk + z T C ( rk + z) = rk r + z T C (z) and the frst order condton for utlty maxmzaton s a (k r ) = δ r. Hence, real captal s chosen to satsfy the same condton as nomnal captal when there s no consumpton tax. Put dfferently, any consumpton tax wll be undone by agents n terms of ther savngs and wll have no effect on the real value of ther wealth held (and, hence, by defnton of the real wealth, on ther purchasng power). Hence, the consumpton tax s equvalent to a tax on labor ncome only. The equvalence s not exact wth elastc labor supply, as n that case, the margnal con- 3

14 sumpton tax depends on the labor choce and the frst-order condton for labor ncome s h (z) = T C ( rk + z) + a (k r )[T C ( rk + z) T C (z)/ r. References Ayagar, R. (995). Optmal captal ncome taxaton wth ncomplete markets, borrowng constrants, and constant dscountng. Journal of Poltcal Economy 3 (6), Judd,. L. (985). Redstrbutve taxaton n a smple perfect foresght model. Journal of publc Economcs 28 (), Pketty, T. and E. Saez (23). A theory of optmal nhertance taxaton. Econometrca 82 (4), Saez, E. and S. Stantcheva (26). Generalzed socal margnal welfare weghts for optmal tax theory. Amercan Economc Revew 6 (),