Online Appendix for A Simpler Theory of Capital Taxation

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1 Onlne Appendx for A Smpler Theory of Captal Taxaton Emmanuel Saez, UC Berkeley Stefane Stantcheva, Harvard July 2, Proofs of Propostons n the Text 1.1 Proofs for Secton 2 Proof of Proposton 2. We derve the optmal captal tax. The optmal labor tax s derved exactly n the same way. Consder a small reform δt (rk) n whch the margnal tax rate s ncreased by δτ n a small band from captal ncome rk to rk +d(rk), but left unchanged anywhere else. Ths reform has a mechancal revenue effect, a behavoral effect, and a welfare effect. The mechancal revenue effect above captal ncome rk s d(rk)δτ [1 H (rk)] The behavoral effect comes only from taxpayers wth captal ncome n the range [rk, rk+d(rk)]. Thanks to the lnear utlty (.e., no ncome effects), tax payers above rk do not respond to the Stantcheva: Socety of Fellows, Harvard Unversty, Cambrdge, MA 2138 (e-mal: sstantcheva@fas.harvard.edu); Saez: Unversty of Calforna, 53 Evans Hall #388, Berkeley, CA 9472 (e-mal: saez@econ.berkeley.edu). We thank Emmanuel Farh and Thomas Pketty for useful dscussons and comments. We acknowledge fnancal support from the MacArthur Foundaton, and the Center for Equtable Growth at UC Berkeley. 1

2 tax rates snce they do not face a change n ther margnal tax rate. Taxpayers n the small band have a behavoral response to the hgher margnal tax rate. They each reduce ther captal ncome by δ(rk) = e δτ /(1 T (rk)) where e s the elastcty of captal ncome rk wth respect to the net-of-tax return r(1 T (rk). As there are h (rk)d(rk) taxpayers affected by the change n margnal tax rates, the resultng loss n tax revenue s equal to: T d(rk)δτ h (rk)e (rk)rk (rk) (1 T (rk)) wth e (rk), as defned n the text, the average elastcty of captal ncome n the small band. The change n tax revenue s rebated lump-sum to all taxpayers. The value of ths lump-sum transfer to socety s g = 1 due to the absence of ncome effects (the lumpsum rebate also does not change any behavor wth lnear utlty). The welfare effect on the tax payers above rk who pay more tax δτ d(rk) s: g δτ d(rk) :rk rk Recall the defnton of the average socal margnal welfare weght above rk, Ḡ(rk), n (11) and note that: (1 H (rk))ḡ(rk) = g :rk rk g At the optmum, the sum of the mechancal revenue effect, the behavoral effect, and the welfare effect needs to be zero, whch requres that: d(rk)δτ g d [ 1 H (rk) h (rk) e (rk) rk T (rk) ] 1 T (rk) d(rk)δτ g d (1 H (rk)) Ḡ(rk) = We can dvde everythng by d(rk)δτ g d and re-arrange to obtan: T (rk) 1 T (rk) = 1 e (rk) 1 H (rk) (1 Ḡ(rk)) rk h (rk) 2

3 Usng the defnton of the local Pareto parameter α (rk) = rkh (rk)/(1 H (rk)), we obtan the captal tax formula n the proposton. The optmal margnal labor tax formula s derved n the same way, replacng captal ncome rk wth labor ncome z. Dervaton of the optmal top tax rate. Suppose we change the lnear tax rate above threshold k top. Only those wth ncomes above k top wll adjust ther captal ncome decsons. The followng three effects occur. Frst, there s a mechancal revenue collecton effect dτ [k m,top k top ] (1 H (rk top )) from every agent above k top where k m,top = E (k k k top ). Then there s a welfare effect on those wth captal ncome above rk top : :rk rk top g r(k k top ). Fnally, there s a behavoral tax revenue effect rebated lump-sum to everyone and equal to: r d(km (1 H (rk top ) d(1 τ ) dτ. d(km,top (1 H (rk top )) r d r (k m,top (1 H (rk top )) e top s defned as the elastcty of the total captal ncome of top captal ncome earners (those wth captal ncome above k top to the net of tax return. The total change n socal welfare dsw F dτ wth g = ω and g = 1. [ r g (k m,top k top )(1 H (rk top )) rτ e top ] (k m,top (1 H (rk top )) r g (k k top ) r :rk rk top s, dsw F 1 τ e top k m,top dτ 1 τ (k m,top k top ) g (k k top ) :rk rk (k top m,top k top )(1 H (rk top )) where the last term s also equal to: Note that: and ω (k k top ) :rk rk ω (k top m,top k top )(1 H (rk top )) = k m,top (k m,top k top ) = atop. Proof of Proposton 3. 3 :rk rk top ω (k k top ) :rk rk top (k k top )d

4 Let G be government revenue. The change n revenue from a change n the captal ncome tax dτ s: [ dg = rk m 1 τ e τ ] L z m e L,(1 τ ) dτ 1 τ 1 τ rk m Hence the change n socal welfare s: ( dsw F = g rk + dg ) = g ( g rk dτ dτ g + dg ) dτ Settng ths to zero and usng the defnton of ḡ = g k g k m, yelds: τ = 1 ḡ τ L e L,(1 τ ) zm rk m 1 ḡ + e whch s the optmal captal tax formula wth jont preferences and cross-elastctes. The optmal labor tax formula wth cross elastctes can be derved exactly symmetrcally. Proof of Proposton 4. The dervaton of the optmal tax on comprehensve ncome follows exactly the proof of Proposton 2 above, replacng captal ncome rk wth total ncome y. Proof of Proposton 5. The government maxmzes: SW F = ω U (c, k, z, x) wth U (c, k, z, x) = rk + z(1 τ L ) + x(τ L τ ) + τ L (z m x m ) +τ (rk m + x m ) + a (k) h (z) d (x) + δ (k nt k) The frst order condtons wth respect to τ L and τ are: ω (z m x m dz m (z x )) τ L d(1 τ L ) (τ L τ ) dxm = dτ L 4

5 ω (rk m + x m dk m (rk + x )) τ r d(1 τ ) (τ L τ ) dxm = dτ Snce x only depends on τ L τ, we have that: dx m dτ L The FOCs can be rewrtten as: z m x m ω (z x ) τ dz m d(1 τ L ) = dxm dτ = dxm d(τ L τ ). Let τ τ L τ. dx m d(τ L τ ) dz m d(1 τ L ) = τ L rk m + x m ω (rk + x ) + τ r dkm r d(1 τ ) Let us smplfy notaton a bt and denote: dx m d(τ L τ ) dkm d(1 τ ) = τ z dzm d(1 τ L ) k dk m d(1 τ ) x dx m d(τ L τ ) Takng the dfference of those two equatons, we can express τ as τ ( ( x z + 1 )) = zm x m ω (z x ) rk z rkm + x m ω (rk + x ) (1) rk Snce ( 1 + x ( 1 z + 1 rk )) >, the sgn of τ s that of the rght-hand sde of the above expresson. τ > zm x m ω (z x ) z > rkm + x m ω (rk + x ) rk Defne the dstrbutonal factor of shfted ncome, by analogy to the dstrbutonal factors ḡ and ḡ L for captal and labor ncome. ḡ X = The rght-hand sde of (1) can be rewrtten as: ω x z m RHS = 1 xm ḡ z m L + ḡ X e L 1 τ L 1 + xm z ḡ rk m ḡ m X e 1 τ rk m 5

6 Hence: τ > 1 xm ḡ z m L + ḡ X e L 1 τ L > 1 + xm z ḡ rk m ḡ m X e 1 τ Suppose that ḡ X s small enough otherwse, encouragng shftng may be good for dstrbutonal reasons. Formally, suppose that for x m >, rk m x m rk m ḡ X z m xm > and rkm z ḡ m X > We can then wrte: ( 1 τ 1 + x m τ > e > e ḡ rkm ḡ zm L ( X ) 1 τ L 1 x m ḡ z m L + ḡ X rk m ) And note that: If τ = : e = e L (1 ḡ ) (1 ḡ L ) If τ >, then x m > and e > e L (1 ḡ ) (1 ḡ L ). Conversely, f τ <, then x m < and e < e L (1 ḡ ) (1 ḡ L ). Thus: We can now rewrte the FOCs as: τ e e L (1 ḡ ) (1 ḡ L ). z m (1 xm z ḡ m L + ḡ X ) τx = z m e L 1 τ L τ L We dstngush three cases: rk m (1 + xm rk ḡ z m m ḡ X rk ) + m τx = rk m e (1 ḡ If e > e ) L (1 ḡ L, then τ > and ) τ 1 τ e L τ L < 1 xm 1 τ L z ḡ m L + ḡ X < 1 ḡ L 6

7 and n ths case: e τ > (1 + xm 1 τ rk ḡ z m m ḡ X rk ) > 1 ḡ m So that the optmal tax rates wth shftng are bracketed by ther revenue maxmzng rates. If there s no shftng, x then revenue maxmzng rates apply. If x s very large (very senstve shftng to any tax dfferental), then from equaton (1), we have that τ and hence τ L τ. Summng the FOCs and usng ths equalty yelds τ L = τ = τ Y where τ Y s the optmal lnear tax rate on comprehensve ncome derved n Proposton 4. Proof of Proposton 6. Let us compare the followng two regmes consdered n the text: Regme 1 Consumpton tax regme: ( r, T L, τ C ), wth an ntal lump-sum transfer τ C k nt /(1 τ C ) to wealth holders wth ntal wealth k nt. Regme 2 No consumpton tax regme: ( r, ˆT L, τ C = ) wth (z ˆT L (z)) = (z T L (z)) (1 τ C ). Let k denote the steady state wealth choce under ths regme. We wll show that these regmes are equvalent n the steady state, n the consumer s dynamc optmzaton problem, and n the government s revenue rased, as clamed n the text. Steady-state equvalence: The budget constrant n regme 1 s: k = [ rk + z TL (z)] c/(1 τ C ) + G, where G = τ L z m + τ rk m + t C c m s the lump-sum transfer rebate of tax revenue. The budget constrant can be rewrtten n terms of real wealth as: kr = rk r + (z T L (z)) (1 τ C ) + G (1 τ C ) c. Utlty s: u = c + a (k r ) h (z ) 7

8 The frst-order condtons of the agent are: (1 T L(z )) (1 τ C ) = h (z ), a (k r ) = δ r Gven that (1 ˆT L (z )) = (1 T L (z )) (1 τ C ) for all z, the steady-state choces of labor ncome and real captal of the agent are unaffected. Usng the steady state budget constrant, real consumpton c s also not affected as long as the real lump-sum transfer G (1 τ C ) s not affected, whch we prove rght below. The lnk between the two captal levels s: k = (1 τ C ) k (snce real steady state wealth s unaffected). Equvalence of the dynamc consumer optmzaton problem. The law of moton n real-wealth equvalent, k r = rk r +(z T L (z)) (1 τ C )+G (1 τ C ) c, s the same n regme 1 and regme 2 as long as the real lump-sum transfer (1 τ C ) G s the same, whch we show below. The ntal wealth after the lump-sum transfer τ C k nt/(1 τ C ) from the government becomes k nt + τ C k nt /(1 τ C ) = k nt /(1 τ C ), so that real wealth after the transfer s k nt, the same t was n the tax regme wthout a consumpton tax. Equvalence of government revenue. In regme 1, there s frst the ntal cost of provdng the lump-sum τ C knt /(1 τ C ) to all ntal wealth holders. At the same tme, the ntal consumpton change s taxed, whch yelds: τ C (knt k )/(1 τ C ). In real terms, ths s worth: A = τ C The nomnal tax flow per perod under ths regme s (whch s also equal to the lump-sum transfer per-perod n nomnal terms s G: k G = τ C 1 τ C c + T L (z ) + τ rk 8

9 We can express consumpton under ths regme as: c = (z T L (z ))(1 τ C ) + r(1 τ C )k + G(1 τ C ) and aggregate consumpton as: c = (1 τ C ) (z T L (z )) + r(1 τ C ) k + G(1 τ C ) Solvng for G usng the defnton of G and the expresson for aggregate consumpton yelds: G = T L (z ) + τ C 1 τ C ( z + r k ) + 1 τ rk 1 τ C In real terms, revenue s: (1 τ C ) G = (1 τ C ) T L (z ) + τ C z + τ C r k + τ rk In Regme 2, the (real) revenue s: ˆT L (z ) + τ r k Usng the map between the labor ncome taxes: (z ˆT L (z)) = (z T L (z)) (1 τ C ), we obtan that the real revenue n Regme 2 s: (τ C z + T L (z) (1 τ C )) + τ r k The dfference between the per-perod real revenue n regme 1 and that n regme 2 s hence: τ C rk. Recall that the ntal change n revenue n regme 1 was A = τ C k, whch, converted nto a per-perod equvalent s exactly A r = τ C rk and cancels out perfectly the change n per-perod revenue between the two regmes. 9

10 1.2 Proofs for Secton Generalzed Model Proof of Proposton 8 The steady state s characterzed by: u k /u c = δ r(1 T ), u c (1 T L ) = u z and c = rk + z T (z, rk ) Wth lnear taxes, ths smplfes to: u k /u c = δ r, u c (1 τ L ) = u z and c = rk + z (1 τ L ). Frst, consder the case wth exogenous labor ncome. Let us assume that the economy has converged to steady state wth τ and we consder a small reform dτ that takes place at tme and s unantcpated. Let us denote by e (t) the elastcty of aggregate k m (t) wth respect to 1 τ. e (t) converges to e from the orgnal analyss (the steady state elastcty). Usng the envelope theorem (.e., behavoral responses dk t can be gnored when computng dv ), the effect on the welfare of agent s V s: dv = dτ δ [ u c (c (t), k (t))rk m (t) e δ t τ 1 τ In the steady state, k m (t) and c (t), k (t) are tme-constant so that: dv = dτ rk m [u c (c, k ) u c (c, k ) k k τ δ m u c (c, k ) 1 τ The change n socal welfare s hence: dsw F = ω dv = u c (c (t), k (t))rk (t) e δ t u c (c (t), k (t))rk m (t)e t e δ t dt] e t e δ t dt] dτ rk m ω [u c (c, k ) u c (c, k ) k k τ δ m u c (c, k ) 1 τ Recall the normalzaton of socal welfare weghts: ω u c = 1 and g = ω u c. dsw F 1 g k k τ m 1 τ δ g e t e δt dt e t e δ t dt] 1

11 Wth endogenous labor supply, the change n agent s welfare, dv : dv = dτ δ [ u c (c (t), k (t), z (t))rk m (t) e δ t τ 1 τ τ L 1 τ u c (c (t), k (t), z (t))rk (t) e δ t u c (c (t), k (t), z (t))rk m (t)e (t) e δ t dt u c (c (t), k (t), z (t))e L,1 τ (t)z m (t)e δ t dt] In the steady state, k m (t), c (t), z (t), and k (t) are tme-constant and labor ncome adjusts mmedately so that e L,1 τ = e L,1 τ so that the change n agent s utlty s hence: τ δ u c (c, k, z ) 1 τ and the change n socal welfare s: τ dv = dτ rk m [u c (c, k.z ) u c (c, k, z ) k e (t) e δ t dt k m τ L 1 τ u c (c, k, z ) zm rk m e L,1 τ ] dsw F = ω dv = dτ rk m ω [u c (c, k, z ) u c (c, k, z ) k δ u c (c, k, z ) 1 τ e (t) e δ t dt k m τ L 1 τ u c (c, k, z ) zm rk m e L,1 τ ] Usng the normalzaton of socal welfare weghts: ω u c = 1 and g = ω u c. dsw F 1 g k k τ m 1 τ whch yelds the formula n the text. δ g e (t) e δt dt τ L 1 τ z m rk m e L,1 τ Proof of Proposton 9 We consder the top tax rate τ on captal above threshold k top. Suppose we change the top tax rate on captal by dτ. Let N denote the mass of ndvduals above k top,.e., N = P rob(rk (t) rk top ). We agan 11

12 use the notaton k m,top to denote the average wealth above the top threshold,.e.: k m,top = :rk (t) rk top rk P rob(rk (t) rk top ) 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 ),.e.,: (t) = d(nkm,top ) r d r (Nk m,top ) e top For all agents 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 δ t τ 1 τ Startng from the steady state, captal levels are constant so that: [ dv = u c r(k m,top k top )Ndτ 1 (k k top ) (k m,top k top )N τ a top 1 τ where a top = k m,top (k m,top k top ). Let For agents below the cutoff, the change n utlty s: [ dv = u c r(k m,top k top )Ndτ 1 τ a top 1 τ The change n socal welfare s such that: 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) e δ t dt] (k k top ) dsw F 1 g :k k (k top m,top k top )N τ α g δ 1 τ ] δ e top (t) e δ t dt ] δ e top (t) e δ t dt e top (t) e δ t dt ḡ top :k k top g (k k top ) (k m,top k top )N and ētop g δ e top (t) e δ t dt 12

13 τ = 1 ḡ top 1 ḡ top + α ē top Wth endogenous labor, let e L,(1 τ )(t) = dzm (t) (1 τ ) d(1 τ ) z m (t) = dzm (t) d r r z m (t) be the elastcty of aggregate (average) labor ncome z m wth respect to the top captal tax rate, n the two bracket tax system. Not everyone s affected by ths tax change, dependng on the jont dstrbuton of captal and labor ncome. e L,(1 τ ), s constant over tme startng from the steady state. For all agents wth captal ncome above the cutoff: dv = dτ δ [ τ L 1 τ u c (c (t), k (t), z (t))r(k (t) k top ) e δt 1 τ τ u c (c (t), k (t), z (t))nr(k m,top (t) k top ) e δ t u c (c (t), k (t), z (t))z m (t)e L,(1 τ )(t) e δ t u c (c (t), k (t), z (t))nrk m,top (t)e top (t) e δ t dt] Startng from the steady state, captal ncome, labor ncome, as well as the elastcty of labor ncome to the top captal tax rate are constant over tme: dv = u c Nr(k m,top k top )dτ [1 (k k top ) (k m,top k top )N τ L z m 1 τ r(k m,top k top )N e L,(1 τ ) The change n socal welfare s: τ a top 1 τ (k k top ) dsw F = ω dv 1 g :rk rk (k top m,top k top )N τ L z m e L,(1 τ ) 1 τ r(k m,top k top )N τ a top g δ e top 1 τ (t) e δ t dt] δ e top (t) e δ t dt] 13

14 ē top Defne: g δ e top (t) e δ t dt ḡ top = :rk rk top g (k k top ) (k m,top k top )N = :rk rk top g (k k top ) :rk rk top (k k top ) The optmal tax rate takes the form: τ top = 1 ḡtop τ L 1 ḡ top z m e r(k m,top k top )N L,(1 τ ) + αtop ētop = 1 ḡtop τ L z m e r(k m,top k top )(1 H (rk top )) L,(1 τ ) 1 ḡ top + atop ētop Ayagar (1995) Model wth and wthout antcpaton effects 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: dsw F = t<t ( 1 ) t τ = dτ ( (1 τ ) t<t + ( ) t 1 ( [ τ = dτ (1 τ ) ω u t (τ rdkt m ) + [ ( ) t 1 rkt m e t ω u t r(kt m k t )) t ( ) t 1 ω u t (rdτ (kt m k t ) + τ rdkt m ) ω u t + ( ) t ] 1 rkt m e t ω u t ( ) t ] 1 rkt m e t ω u t ( ) t ) 1 ω u t r(kt m k t ) If varable 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: 14

15 dsw F τ (1 τ ) ( δ t<t ( ) t T 1 e t + δ ( ) t T 1 e t) 1 + g k t g kt m Defne the antcpaton responses e ante, the post-reform response epost, and the total response ē to be: e ante = δ t<t ( ) t T 1 e t, e post = δ and the dstrbutonal factor ḡ = g k t g k m t tax n the Ayagar (1995) model s gven by: ( ) t T 1 e t and ē = e ante + e post. Then we have as n the text that the optmal captal τ = 1 ḡ 1 ḡ + ē For the unantcpated reform at tme T = that s studed n the text, assume that the economy s already n the steady state as of tme, and set e ante ē = δ t ( ) t 1 e t = so that: If varable 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 ( τ (1 τ ) [ t 1 ( ) t ] 1 rkt m e t ω u t ( ) t ) 1 ω u t r(kt m k t ) 15

16 Dvdng by dsw F ( 1 ) t ω 1+δ u t kt m τ (1 τ ) [ t 1 yelds: ( ) t 1 k m 1 + ( 1 t e t ( 1 1+δ ) t Now we have to redefne the average welfare weght as: and the total elastcty as: ḡ ē = t 1 ( 1 ( ) t 1 k m ) t t e t u t k t ) t ( 1 1+δ Wth these redefned varables, the same formula holds. ( 1 1+δ ω u t ) t ω u t km t ω u t k t ) t ( 1 1+δ u t km t u t ) t u t km t ω u t km t ] Judd (1985) Model In the Judd (1985) 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 [ ( 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 1 τ rk m (t)e (t)]] 16

17 In the steady state dv = dτ r[ ( u c e δ (c )t + δ (c )u e δ (c )t ) e δ (c )(s t) ds [k m (t) k (t) τ k m (t)e (t)]] t 1 τ ) = dτ r[ (u c e δ (c )t + δ (c )u e δ (c )t 1 [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)] 1 τ = dτ rk m 1 ( ) [ u c + δ (c ) δ (c ) δ (c ) u 1 k k τ ] δ m (c ) e δ (c )t e (t) 1 τ We can hence see that the formulas from our model apply but that g = ( ) 1 ω δ (c u ) c + δ (c ) u δ (c ) ( ) and ē = g δ (c ) 1 δ (c u ) c + δ (c ) u δ (c ) ω e δ (c )t e (t) 2 Antcpated Reforms n the Generalzed Model In ths secton, we extend our analyss from the man text Secton 4 to consder antcpated reforms. Suppose that an antcpated reform to the captal ncome tax dτ happens at tme T >. Captal and labor already start adjustng n antcpaton of the reform before tme T. The change n the utlty of ndvdual s: dv = dτ δ [ T u c (c (t), k (t))rk m (t) e δ t τ 1 τ T u c (c (t), k (t))rk (t) e δ t u c (c (t), k (t))rk m (t)e (t) e δ t dt] In the steady state, k m (t) and c (t), k (t) are tme-constant. Assume that T s large enough so that the antcpatory responses only start when the economy s already n the steady state. In ths case, all terms n e (t) are zero before the steady state, so we can wrte: 17

18 dv = dτ rk m e δ T [u c (c, k ) u c (c, k ) k k τ δ m u c (c, k ) 1 τ τ δ u c (c, k ) 1 τ t<t e (t) e δ (t T ) dt e (t) e δ (t T ) dt] We also assume here that the dscount rates are the same across all agents. g = u cω = 1. k 1 g k τ m 1 τ k = 1 g k m g δ t<t τ 1 τ δ e (t) e δ(t T ) dt e (t) e δ(t T ) dt t<t Defne the dstrbutonal factor ḡ = g k k m elastcty e post e ante Then: and the total elastcty ē to be: τ δ 1 τ g τ 1 τ δ Use that dsw F = ω dv e (t) e δ(t T ) dt] e (t) e δ(t T ) dt and the antcpaton elastcty e ante, the post = δ e (t) e δ(t T ) dt, e post = δ e (t) e δ(t T ) dt and ē = e ante + e post t<t τ = 1 ḡ 1 ḡ + ē Our formulas hence apply exactly (can menton t n the text), but the total elastcty ē now contans antcpaton effects as well. Ths formula s derved under the assumptons that T s large, the antcpaton responses start happenng only after the economy has already converged to ts steady state, and dscount rates are homogeneous across agents. Endogenous Labor Supply wth Antcpaton Effects The antcpaton effects through the cross-elastctes can also start before the reform. The assumpton needed s agan that those antcpaton effects only start once the economy has 18

19 already converged to ts steady state path. In ths case, the formula looks as n the text wth cross-elastctes. 2.1 Steady State and Antcpaton Elastctes We now prove two further results. Steady state elastctes are fnte wth wealth n the utlty. Wth a general utlty and wealth n the utlty, the frst-order condton for agent n the steady state s: u k = (δ r)u c In the steady state, the budget constrant s: c = rk + z hence the steady state can be rewrtten as: (δ r)u c ( rk + z, k ) = u k ( rk + z, k ) whch s a smooth functon of k, as long as the functon u (c, k ) s smooth and concave n consumpton and captal. Hence, the responses of consumpton and captal to the net-of-tax return r are smooth and non-degenerate. The same argument holds wth endogenous labor supply, whch s chosen smoothly. Antcpaton elastctes are nfnte wth wealth n the utlty and certanty, but fnte wth uncertanty (wth or wthout wealth n the utlty). We can also show that the antcpaton elastctes 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 (213) for the Chamley-Judd model (wthout wealth n the utlty). Wth full certanty, the frst-order condton of the agent wth respect to captal always holds: u c,t = (1 + r)/( )u c,t+1 + 1/( )u k,t+1 Suppose we start from a stuaton n a well-defned steady state: (δ r)u c = u k where we 19

20 have perfect consumpton smoothng. The ntertemporal budget constrant s: t Consumpton smoothng mples: ( ) t 1 c t + lm 1 + r k t = ( ) t 1 z t + k t 1 + 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 = 1 1 ) ( ( ) ) t 1 z t + k k 1 + r 1 + r t (2) 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=1 ( r ) t z t + ( ) t 1 z t 1 + r The change n resources evaluated at τ = s: dy = ( ) T 1 ( ) t T +1 ( ) T 1 1 t z tdτ dτ (1 + r) (1 + r) (1 + r) ( T 1 Hence, consumpton pre-reform wll shft down by a factor proportonal to (1+r)) dτ. From the aggregated budget constrant we have that: k m t = (1 + r) t k m c m (1 + (1 + r) + (1 + r) (1 + r) t 1 ) + (z m t (1 + r) t 1 z m ) 2

21 Therefore, the change n the aggregate captal stock s: ( ) (1 + r) t 1 1 dk m t = dc m r ( T 1 Recall that the change n consumpton (from (2)) s proportonal to (1+r)) dτ. Hence: dk m t ( 1 (1 + r) ) T ( (1 + r) t 1 1 r ) ( ) (1 + r) dτ = (1 + r) T t 1 1 dτ r Hence: ( ) (1 + r) e t kt m (1 + r) T t 1 1 dτ r Recall that the antcpaton elastcty e ante e ante = δ t<t ( ) t T 1 e t δ s defned as: t<t ( ) t T ( ) 1 (1 + r) kt m (1 + r) T t 1 1 dτ r Snce we have δ > r, lm T ( 1+δ 1+r) T =, whch makes the sum above (to whch the antcpaton elastcty s proportonal) converge to nfnty when T goes to nfnty. 3 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 2.3.4, the optmal revenue-maxmzng rates are: τ R L = 1 1+e L and τ R = 1 1+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 (16): τ Y = 1 ḡ Y 1 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 21

22 (labor ncome only). Generalzed socal welfare weghts that capture horzontal equty concerns are such that: () If τ L = τ, then g are standard, for nstance 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 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 1. 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. 22

23 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. 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 only labor ncome have a non-zero weght. snce only people wth 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 23

24 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 (216) 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. 3.1 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 2.3.4, 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 whch does not dscrmnate between captal and labor ncome condtonal on total ncome. We say that a tax system unambgously favors captal (respectvely, labor) 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 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 24

25 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 (rk, z) = T (rk + z) + δt (rk, z) s such that: T Y (rk (T ) + z (T ))d > T Y (rk ( T ) + z ( T ))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. 25

26 4 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 = y T C (y),.e., y y T C (y) s the nverse functon of c c + t c (c) and hence 1 + t C = 1/(1 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 to see why 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 = 1 T C,.e., and extra dollar of nomnal wealth s worth 1 T C k r () =. In that case, we have n steady-state n real terms, and c = rk + z T C ( rk + z) = rk r + z T C (z) 26

27 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 consumpton tax depends on the labor choce and the frst-order condton for labor ncome s h (z) = 1 T C ( rk + z) + a (k r )[T C ( rk + z) T C (z)]/ r. 27

28 References Ayagar, Rao, Optmal Captal Income Taxaton wth Incomplete Markets, Borrowng Constrants, and Constant Dscountng, Journal of Poltcal Economy, 1995, 13 (6), Judd, enneth L, Redstrbutve taxaton n a smple perfect foresght model, Journal of publc Economcs, 1985, 28 (1), Pketty, Thomas and Emmanuel Saez, A Theory of Optmal Inhertance Taxaton, Econometrca, 213, 82 (4), Saez, Emmanuel, Usng elastctes to derve optmal ncome tax rates, The revew of economc studes, 21, 68 (1), and Stefane Stantcheva, Generalzed Socal Margnal Welfare Weghts for Optmal Tax Theory, Amercan Economc Revew, 216, 16 (1),