Economics 2450A: Public Economics Section 9: Linear Capital Taxation

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1 Economics 2450A: Public Economics Secion 9: Linear Capial Taxaion Maeo Paradisi November 7, 206 In his secion we inroduce a framework o sudy opimal linear capial axaion. We firs focus on a wo-period model, define he concep of ineremporal wedge and derive opimal capial axes using he Akinson Sigliz resul. We hen move o an infinie horizon model wih aggregae uncerainy and derive opimal axes. Finally, we sudy a model wih capialiss and workers and show ha only under some assumpion abou preferences we can recover a zero capial ax in seady sae. A Two-Period Model We inroduce a wo-period model wih capial accumulaion ha will be useful o sudy he problem of capial axaion. The wo ime periods are = 0 and =. The preferences for individual i are U i (c 0,c,y 0 ). The individual can save period 0 income and earn gross ineres rae R on savings. We sar by consraining he governmen s insrumens and focusing on a linear consumpion ax, while keeping a non-linear ax on income. The budge consrains for he wo periods read: ( + 0 ) c 0 apple y 0 T (y 0 ) k Combining he wo we ge: ( + ) c apple Rk Rearranging we have: ( + ) c apple R (y 0 T (y 0 ) ( + 0 ) c 0 ) () c 0 + R apple c apple y 0 T (y 0 ) + 0 Le us denoe + =(+ ) / ( + 0 ) and T (y) =( 0 y + T (y)) / ( + 0 ). The budge consrain can be wrien: c 0 + R [ + ] c apple y 0 T (y0 ) When he budge consrain holds wih equaliy we can wrie consumpion a ime as: c = R + s 0 where s 0 = y 0 T (y0 ) c 0 is he oal level of savings when here are no disorions in he economy. We can inerpre as a capial income ax. I disors iner-emporal consumpion decisions by changing The second par of hese noes is based on lecure noes by Florian Scheuer.

2 he relaive price of c 0 and c. I can be inerpreed as a wedge on he opimal savings decision. Noice ha whenever = 0 we have no disorion in he iner-emporal choice of he agen. Suppose he agen has separable preferences in consumpion and labor of he form: U (c 0,c,y 0 )=u (c 0 )+u (c ) h (y 0 /w) We can rewrie he preferences as U (g (c 0,c ),y 0 ) so ha he uiliy is weakly separable in g ( ) and y 0. I follows ha Akinson-Sigliz applies: if non-linear income axaion is available he governmen finds opimal o se a fla zero ax on c 0 and c. 2 2 Infinie Horizon Model - Chamley (986) In his secion we inroduce a model where capial reurns and wages are endogenous. We focus on linear capial and labor axes in an infinie horizon economy. There is aggregae uncerainy in he economy and each period a sae s is realized so ha he hisory of aggregae uncerainy is a sequence s =(s 0,s,...,s ). Oupu is produced according o a consan reurn o scale producion funcion: he producive capial a ime is he sock ha was chosen a ime profi maximizaion problem: F K s,l s,s, (2) max F K K,L s,l s,s, w s L s r s K s. The firm solves he following Compeiive labor and capial markes imply ha inpu prices are equal o heir marginal produc: w s = F L K s,l s,s, r s = F K K s,l s,s, The economy is populaed by a represenaive agen whose uiliy is: Pr s u c s,l s s where Pr (s ) is he probabiliy of hisory s. The aggregae resource consrain of he economy is: c s + g s + K s ( ) K s apple F K s,l s,s, (3) The oupu produced is employed o finance consumpion, public spending and invesmens. The resource consrain implicily assumes ha aggregae uncerainy resuls from echnology or governmen spending shocks. The governmen opimally chooses axes on labor income l (s ) and axes on capial income k (s ) and sars wih iniial deb B 0. 2 Here is a shor proof of he Akinson Sigliz resul provided in Kaplow (2006). Suppose all individuals have weakly separable preferences V (g (x),y), where x is a vecor of commodiies. Suppose we sar from a siuaion where here are posiive axes on commodiies and we implemen a policy such ha! 0: zero fla ax on all commodiies. Suppose he governmen o ses he uiliy change of he agen wih non-linear income axes such ha labor supply is unchanged a he opimum and V (g (x 0 ),y)=v (g (x),y). Bydefiniioneveryagenhashesameuiliyasbeforeandnooneiswillingo imiae P anoher individual (if hey were no willing o imiae before he ax change). By revealed preference we know ha k p kx k >y T (y): he agen canno a ord he old P bundle under he curren income axaion. Under he old policy scenario he agen P could a ord he bundle and we had k (p k + k ) x 0 k apple y T (y). Combining he wo inequaliies we find ha T (y) > k kx 0 k + T (y) andheoalrevenueraisedaferheaxchangeissriclyhigherhanoalrevenuesbefore he ax change. Since incenive compaibiliy holds and we have no welfare e ec by consrucion, he new policy is welfare improving since i raises more revenues. 2

3 We assume complee markes where he price of an Arrow-Debreu securiy is p (s ). 3 The governmen budge consrain is: p s g s l s w s L s k s r s K s apple B 0,s Taxes on consumpion and capial are employed o finance governmen layous g (s ). Noice ha he capial ax is levied on he capial gain ne of he capial depreciaion. The household budge consrain reads: p s c s + K s w s l s L s R s K s apple B 0 (4),s where R (s )=+ k (s ) (r (s ) ) is he gross ineres rae ne of axes. We can se up he Lagrangian for he consumer problem: L = s Pr s u c s,l s B 0,s p s The firs order condiions are: c s 3 + K s w s l s L s R s K s 5 Pr s u c c s,l s p s = 0 (5) Pr s u L c s,l s + p s l s w s = 0 (6) p s + s + p s + R s + = 0 (7) On op of he FOCs, a non-arbirage condiion mus hold beween capial and Arrow-Debreu securiies: p s = s + p s + R s + (8) We can define a compeiive equilibrium as follows: Definiion: A compeiive equilibrium is a policy g (s ), k (s ), l (s ),anallocaion{c (s ),K(s ),L(s )} and prices {w (s ),r(s ),p(s )}, such ha households maximizes uiliy s.. budge consrain, firms maximize profis, he governmen budge consrain holds and markes clear Combining (5) and (6) we ge he sandard inraemporal condiion for labor supply: Pr (s ) u c (c (s ),L(s )) p (s = ) Pr (s ) u L (c (s ),L(s )) p (s )( l (s )) w (s ) l s w s = u L (c (s ),L(s )) u c (c (s ),L(s )) (9) 3 An Arrow-Debreu securiy is a financial insrumen ha provides one uni of consumpion in a sae s and zero unis in any oher sae. We alk abou complee markes whenever we can price such an asse in every sae of he world. 3

4 From (5) and (7) we derive he so called Euler equaion ha pins down he slope of he consumpion pah of he agen: u c (c 0,L 0 )= Pr (s ) u L (c (s ),L(s )) p (s ) Saring from (4), we can rewrie i using he opimaliy condiions and he no-arbirage condiion:,s,s,s,s p s Pr (s ) u c (c (s ),L(s )) u c (c 0,L 0 ) c s w s l s L s apple B 0 + R 0 K 0 apple c s + u L (c (s ),L(s )) u c (c (s ),L(s )) L s apple B 0 + R 0 K 0 Pr s u c c s,l s applec s + u L (c (s ),L(s )) u c (c (s ),L(s )) L s apple u c (c 0,L 0 )[B 0 + R 0 K 0 ] Pr s uc c s,l s c s + u L c s,l s L s apple u c (c 0,L 0 )[B 0 + R 0 K 0 ] () We call he consrain in () implemenabiliy consrain since i capures he agen s opimal choices subjec o heir feasibiliy. Opimal Taxes The governmen chooses axes o maximize he welfare of he represenaive agen subjec o he resource consrain and he implemenabiliy consrain. The problem reads: (0) s.. max c(s ),L(s ),K(s ), 0 k s Pr s u c s,l s c s + g s + K s ( ) K s apple F K s,l s,s, Pr s uc c s,l s c s + u L c s,l s L s apple u c (c 0,L 0 )[B 0 + R 0 K 0 ],s We assume 0 k is fixed, we denoe wih µ he muliplier on he implemenabiliy consrain and define: W c s,l s = u c s,l s + µ u c c s,l s c s + u L c s,l s L s The governmen problem becomes: s.. max c(s ),L(s ),K(s ) s Pr s W c s,l s µu c (c 0,L 0 )[B 0 + R 0 K 0 ] c s + g s + K s ( ) K s apple F K s,l s,s, For any period 6= 0 he FOCs are: 4

5 Pr s W c c s,l s s =0 Pr s W L c s,l s + s F L K s,l s,s, =0 s + s + s,s + Fk K s,l s,s, +( ) =0 Combining he FOCs we ge an inraemporal condiion: W L (c (s ),L(s )) W c (c (s ),L(s )) = F L K s,l s,s, Using he household FOC in (6), we can rewrie he opimaliy condiion as a funcion of he ax: l s = u L (c (s ),L(s )) W c (c (s ),L(s )) W L (c (s ),L(s )) u c (c (s ),L(s )) From he governmen FOCs we can also derive an ineremporal condiion: (2) W c c s,l s = s + Pr s + s W L c s +,L s + R s + where R (s ) = + F K K s,l(s ),s, is he unaxed gross reurn on capial ne of depreciaion. Again, exploiing he household s Euler equaion we can derive: R s + = R s + W c c s +,L s + u c (c (s + ),L(s + )) u c (c (s ),L(s )) W c (c (s ),L(s )) (3) Proposiion : Suppose ha (i) here is no uncerainy (ii) here is a seady sae. Then in he seady sae k =0is opimal. I is easy o see ha in a seady sae when here is no uncerainy R (ss) =R (ss), which implies: + k (ss) (F k (K (ss),l(ss)) )=+F K (K (ss),l(ss),ss) and ha k (ss) = 0. Now consider a special case wih separable preferences and consan ineremporal elasiciy of subsiuion: hen we have: u (c, L) = c v (L) W (c, L) = c = v (L)+µ c c v 0 (L) L + µ c [v (L)+µv 0 (L) L] herefore: 5

6 W c = (+µ ( )) c = (+µ ( )) u c W c u c =+µ ( ) Equaion (3) reduces o R s + = R s +. Hence, we esablished ha for separable preferences wih consan ineremporal elasiciy of subsiuion we have zero capial axaion even ou of he seady sae and in a model wih uncerainy. Tax Smoohing Take now he special case where v (L) = L / is isoelasic, we have: W (c, L) = + µ c + µ L i follows ha he opimal linear labor ax becomes: l s = W L u L =+µ +µ ( ) +µ Therefore, labor axes are consan across saes and over ime. The governmen finds opimal o smooh disorions o labor supply. This resul depends on he possibiliy of seing sae-coningen capial axes. If he labor elasiciy is consan and shocks can be o se using capial axes, here is no residual reason o di erenially ax labor. 3 Infinie Horizon - Judd (985) We now inroduce he model by Judd (985) where he famous zero seady sae capial ax resul arise. We hen show ha he resul is no general and depends on he agen s preferences as shown in Werning Sraub (205). Suppose here are wo agens: capialiss and workers. The former save, ge he reurn o capial and do no work; he laer supply one uni of labor inelasically and consume everyhing hey earn. The governmen axes reurn o capial and pays ransfers o workers. Oupu is produced according o a consan reurn o scale echnology wih producion funcion F (k,n ). Aggregae labor is n =so ha we can rewrie f (k )=F (k, ). Capialiss have uiliy U (C ) and workers uiliy is u (c ). The resource consrain of he economy is: c + C + g + k + apple f (k )+( ) k Under he assumpion of perfecly compeiive labor markes, wages are: w = F L (k,n )=f(k ) k f 0 (k ) The afer-ax reurn o capial is: R =+( )(R ) where R = f 0 (k )+. 6

7 Capialiss solve he following maximizaion problem: s.. + max C,a + =0 U (C ) C + a + = R a The opimaliy condiion delivers he sandard Euler equaion U 0 (C )= R + U 0 (C + ). Since oal wealh mus equal oal capial sock in equilibrium, using he Euler equaion: rearranging: C + k + = R k = U 0 (C ) U 0 (C ) k U 0 (C )(C + k + )=U 0 (C ) k (4) Equaion (4) is he implemenabiliy consrain. The governmen maximizes he following objecive funcion, where is he Pareo weigh on capialiss: The Lagrangian of he problem is: max c,c,k + =0 (u (c )+ U (C )) L = (u (c )+ U (C )) =0 + + =0 =0 (f (k )+( ) k c C g k + ) µ ( U 0 (C )(C + k + ) U 0 (C ) k ) wih µ 0 = 0 since here is no implemenabiliy in he firs period ( 0 is aken as given). The firs order condiions wr o c, k + and C are respecively: u 0 (c )= (5) µ + = µ k + + (f 0 (k + )+ )= + U 0 (C ) (µ + µ ) (6) C + k + + U 0 (C ) U 00 (C ) + k + U 0 (C ) U 00 (C ) U 00 (C ) I is sraighforward from equaion (6) ha whenever a seady sae exiss i involves zero capial axes and R (ss) =f 0 (k )+ = R. This resul is exremely powerful since i is independen of he welfare weigh aached o capialiss. However, he resul does no hold for he case where =. Rewrie he FOCs (6) and (7) using he inverse ineremporal elasiciy of subsiuion = U 00 (C ) C /U 0 (C ) and defining he raio v = U 0 (C ) /u 0 (c ): (6) ) u0 (c + ) u 0 (c ) (f 0 (k + )+ )= + v (µ + µ ) (7) 7

8 / (7) ) µ + = µ + + k + k v ( v ) Take he case where = (log preferences) and he allocaion converges o a seady sae, hen: µ + µ = R v µ + µ = v kv ) R = v k As long as here is a low enough weigh on capialiss, capial is axed in seady sae. For a long ime we hough ha his was simply an anomaly for he logarihmic case. However, Werning and Sraub (205) show ha he resul does no hold for any > by noicing ha he seady sae does no necessarily exis. Proposiion 2: If > and =0, hen for any iniial k 0 he soluion o he planning problem does no converge o he zero-ax seady sae, or any oher inerior seady sae. Suppose capial axes are raised in he fuure, capialiss will decrease savings oday for he subsiuion e ec. A capial ax increase will also reduce agen s wealh and lower capialiss consumpion hrough he income e ec. When > he income e ec prevails and capialiss save more. The increase in he capial sock increases wages and is beneficial for workers. For his reason he governmen wans o se posiive capial axes in he long-erm. The opposie is rue when <: he subsiuion e ec is larger han he income e ec and zero axes in he fuure increase savings in he shor erm increasing wages and workers consumpion. References [] Chamley, Chrisophe, Opimal Taxaion of Capial Income in General Equilibrium wih Infinie Lives, Economerica, 986, 54 (3), pp [2] Judd, Kenneh L., Redisribuive axaion in a simple perfec foresigh model, Journal of Public Economics, 985, 28 (), [3] Kaplow, Louis, On he undesirabiliy of commodiy axaion even when income axaion is no opimal, Journal of Public Economics, Elsevier, vol. 90(6-7), pages , Augus. [4] Scheuer, Florian (204), Lecure Noes (find a hp://web.sanford.edu/~scheuer/linear_ capial_axaion.pdf) [5] Sraub, Ludwig and Ivan Werning, Posiive Long Run Capial Taxaion: Chamley-Judd Revisied, Working Paper, MIT