Economcs 2450A: Publc Economcs Secton 10: Educaton Polces and Smpler Theory of Captal Taxaton Matteo Parads November 14, 2016 In ths secton we study educaton polces n a smplfed verson of framework analyzed by Stantcheva (2016). We then revew a smpler theory of captal taxaton proposed by Saez and Stantcheva (2016) n a contnuous tme model. 1 Educaton Polces We study a statc model wth human captal nvestments based on Bovenberg and Jacobs (2005, 2011) and Stantcheva (2016). Suppose ndvduals are heterogeneous n ablty dstrbuted accordng to f ( ). Agents can nvest n educaton payng a monetary cost M (e) such that M 0 (e) > 0 and M 00 (e) 0. Wages are a functon of ablty and human captal that we can wrte as w (, e). Denote wth e the Hcksan coe cent of complementarty between ablty and educaton defned as: e w ew w w e Consder two specal cases for the wage functon: () multplcatve w (, e) e mples e 1; () constant elastcty of substtuton w (, e) 1 +(1 ) e 1 1 1 mples e. The elastcty of the wage to ablty s " w, @ log (w) /@ log ( ). Suppose utlty s quas-lnear n consumpton such that: u (c, l) c h (l) The agent consumes everythng that s left after taxes and educaton nvestments such that c ( ) w (, e ( )) l ( ) M (e ( )) T (w (, e ( )) l ( ),e( )). Solvng the ndvdual maxmzaton problem we can defne ncome and educaton wedges as: y ( ) 1 h 0 (l ( )) w (, e ( )) (1) The elastcty of labor to the net of tax wage s: e ( ) w e (, e ( )) l ( )(1 y ( )) M 0 (e ( )) (2) " h0 (l) h 00 (l) l We can wrte the ndrect utlty as u ( ) c ( ) ncentve constrant: h (l ( )). Usng the Envelope we can derve the local 1
@u( ) l ( ) w (, e ( )) @ w (, e ( )) h0 (l ( )) (3) It d ers from the one derved n the standard problem by the term w (, e ( )), that takes nto account the e ect of ablty on the wage. In the stardard problem we normalzed t to 1. The resource constrant s: (w (, e ( )) l ( ) u ( ) h (l ( )) M (e ( ))) f ( ) d E (4) The government assgns welfare weght ( ) to each and solves: ( ) u ( ) f ( ) d max c( ),l( ),e( ),u( ) subject to (3) and (4). The frst order condtons wrt u ( ), l ( ) and e ( ) are respectvely: ( ) f ( ) f ( ) µ 0 ( ) (5) h w µ ( ) w h0 (l)+l w w h00 (l) apple µ ( ) Takng the ntegral over (5) we derve µ ( ) ( ) elastcty of labor supply we get: µ ( ) w apple w h0 (l) 1+l h00 (l) h 0 (l) y ( ) 1 y ( ) + [w h 0 (l)] f ( ) 0 (6) l w 2 w ew h 0 (l)+ l w w eh 0 (l) + [w e l M 0 (e)] f ( ) (7) apple µ ( ) w (1 y ( )) 1+l h00 (l) h 0 (l) ( ) F ( ) w f ( ) w 1+" " F ( ). Usng (6), (1) and the defnton of the w y ( ) f ( ) w y ( ) f ( ) ( ) F ( ) " w, f ( ) 1+" The optmal ncome wedge s smlar to the one we studed n Secton 3. The formula has an extra term proportonal to the elastcty of the wage to ablty. Labor dstortons are hgher at the optmum when ncome s hghly elastc to ablty: the government dstorts labor more when ncome s mostly explaned by ablty and less by e ort or nvestment n educaton. Notce that: " (8) w e l M 0 (e) e + M 0 (e) y 1 y Rearrangng (7) and usng (1) and (2) we get: µ ( ) l w apple ew w 2 h0 (l) 1+ w ew + e + M 0 (e) y f ( ) 0 w w e 1 y e ( )+M 0 (e) y ( ) (1 y ( )) 2 ( ) F ( ) l w w e f ( ) w (1,e) (9) 2
The optmal educaton wedge decreases n the Hcksan coe cent of complementarty. When educaton and ablty are complements (.e. e > 0) the government wants to dscourage human captal nvestments n order to redstrbute ncome more. On the other hand, f the coe cent s negatve or low, educaton benefts low ablty ndvduals more and a government subsdy to educaton helps redstrbuton. Suppose the wage functon s w (, e) e and the monetary cost s lnear n e, the optmal wedge s e ( ) y ( ). Ths s the specal case studed by Bovenberg and Jacobs (2005) whose result s that ncome and educaton taxes are Samese Twns and both margns should be dstorted the same way. They also prove that the optmal lnear educaton subsdy s equal to the optmal lnear ncome tax rate, whch s equvalent to makng human captal expenses fully tax deductble. 2 A Smpler Theory of Captal Taxaton We ntroduce n ths paragraph a contnuous tme model wth wealth n the utlty functon. We study the case where utlty s quas-lnear n consumpton that allows us to transofrm the problem n a statc taxaton problem. Suppose ndvdual has utlty u (c, k, z) c+a (k) h (z) wherea ( ) s ncreasng and concave and h ( ) s the standard dsutlty from labor. Agents have heterogeneous dscount rates. The dscounted utlty s: 1 V ({c (t),k (t),z (t)}) [c (t)+a (k (t)) h (z (t))] e t dt (10) Captal accumulates accordng to: 0 dk (t) rk (t)+z (t) T (z (t),rk (t)) c (t) (11) dt where T (z (t),rk (t)) s the tax pad by ndvdual and s dependent on ncome and captal returns. Wealth accumulaton depends on the heterogeneous ndvdual preferences, as emboded n the taste for wealth a ( ) and n the mpatence. It also depends on the net-of-tax return r r (1 T k ): captal taxes dscourage wealth accumulaton through a substtuton e ect (there are no ncome e ects). The Hamltonan for the ndvdual maxmzaton problem s: H (c (t),k (t),z (t), (t)) [c (t)+a (k (t)) h (z (t))] e t + (t)[rk (t)+z (t) T (z (t),rk (t)) c (t)] Takng frst order condtons we have: @c e t (t) 0 @z h 0 (z (t)) e t + (t)[1 T z (z (t),rk (t))] 0 @k (t) a0 (k (t)) e t + (t) r (1 T k (z (t),rk (t))) 0 (t) Rearrangng: (t) 1, h 0 (z (t)) 1 T z (z (t),rk (t)), a 0 (k (t)) r (1 T k (z (t),rk (t))) Snce utlty s quas-lnear n consumpton, the model converges mmedately to a steady state. Denote (c,z,k ) the steady state allocaton, the problem collapses to a statc optmzaton of the followng objectve functon: 3
V ({c (t),k (t),z (t)}) [c + a (k ) h (z)] + k nt k where k nt s the nherted level of captal and k nt k s the utlty cost of gong from k nt to the steady-state level. The government maxmzes the followng: SWF! U (c,k,z ) d wth! 0 s the Pareto weght on ndvdual. The socal margnal welfare weght s g! U c. Optmal Lnear Taxes Suppose the government sets lnear ncome and captal taxes L and K.The ndvdual chooses labor and captal accordng to a 0 (k ) r and h 0 (z )1 L wth r r (1 K ). The government balances the budget through lump-sum transfers for a total of G K rk m ( r) + L z m (1 L ), where z m (1 L ) z d s the aggregate labor ncome and k m ( r) k s aggregate captal. Total consumpton s c (1 K ) rk +(1 L ) z + K rk m ( r) + L z m (1 L ) and the government maxmzes: SWF! (1 K ) rk +(1 L ) z + K rk m + L z m + a (k ) h (z )+ k nt k d Usng the Envelope theorem we get: dsw F d K apple! rk + rk m + K r @km @ k apple rk m k! 1 k m d d K 1 K e K where e K s the elastcty of aggregate captal wth respect to the net of tax return r. At the optmum dsw F/d K 0 and the optmal lnear tax s: 1 ḡ K k 1 ḡ K + e K where ḡ K g k / k. Ths s the standard formula for optmal lnear taxes that we studed n Secton 2 appled to captal. Notce that whenever captal accumulaton s uncorrelated wth socal margnal welfare weghts (.e. ḡ K 1) the optmal tax s zero. The reason s that f captal has no tag value the government does not fnd optmal to tax captal for redstrbutve purposes. We also know from prevous sectons that the revenue maxmzng tax rate corresponds to the case of ḡ K 0 and t s K 1/1+e K. Optmal Non-Lnear Separable Taxes The ndvdual budget constrant s: Suppose the government optmally sets T K (rk) and T L (z). c rk T K (rk )+z T L (z ) Defne wth ḠK (rk) the average relatve welfare weght on nvduals wth captal ncome hgher than rk. Wehave: {:rk Ḡ K (rk) rk} g d P (rk rk) 4
Let h K (rk) be the dstrbuton of captal ncome so that the Pareto parameter assocated to the captal ncome dstrbuton s: K (rk) rk h K (rk) 1 H K (rk) Denote e K (rk) the elastcty of captal ncome wth respect to the net of tax return r (1 TK 0 (rk)). Suppose the government ntroduces a small reform T K (rk) where the margnal tax rate s ncreased by K n a small nterval of captal ncome from rk to rk + d (rk). The mechancal e ect assocated to the reform s: d (rk) K [1 H K (rk)] The welfare e ects just weghts the mechancal e ect by Ḡ (rk), the socal margnal welfare weght assocated to captal ncomes above rk. Indvduals who face the ncrease n the tax rate change ther captal ncomes by (rk) e K K / (1 TK 0 (rk)). There are h K (rk) d (rk) ndvduals n the wndow a ected by the tax change. Therefore, the total behavoral e ect s: h K (rk) d (rk) rk T K 0 (rk) 1 TK 0 (rk)e K (rk) Summng up the three e ects and rearrangng we fnd: TK 0 (rk) 1 TK 0 (rk) 1 e K (rk) 1 H K (rk) rk h K (rk) 1 Ḡ K (rk) Usng the defnton of the Pareto parameter we derve: T 0 K (rk) 1 Ḡ K (rk) 1 Ḡ K (rk)+ K (rk) e K (rk) whch looks lke the standard optmal non-lnear tax formula. References [1] Bovenberg, L. and B. Jacobs. 2005. ìredstrbuton and Educaton Subsdes are Samese Twns,îJournal of Publc Economcs, 89(11-12), 2005-2035 [2] Bovenberg, L. and B. Jacobs. 2011. Optmal Taxaton of Human Captal and the Earnngs Functon, Journal of Publc Economc Theory, 13, (6), 957-971. [3] Saez, Emmanuel, and Stefane Stantcheva. Workng Paper. A Smpler Theory of Optmal Captal Taxaton. NBER Workng Paper 22664. [4] Stantcheva, Stefane. Optmal Taxaton and Human Captal Polces over the Lfe Cycle, Journal of Poltcal Economy, forthcomng. 5