Floran Scheuer 2/23/2016 Mxed Taxaton and Producton Effcency 1 Overvew 1. Unform commodty taxaton under non-lnear ncome taxaton Atknson-Stgltz (JPubE 1976) Theorem Applcaton to captal taxaton 2. Unform commodty taxaton under lnear ncome taxaton Prmal approach to commodty taxaton Useful for thnkng about captal taxaton Applcaton: producton effcency theorem (Damond-Mrrlees AER 1971) 2 Mxed Taxaton Atknson-Stgltz (1976) consumpton goods x R m, labor Y R preferences U (x, Y), arbtrary heterogenety B budget set: (x, Y) B (x, Y) s affordable may be non-lnear set If we had B Frst-best achevable Here: B anonymous (B s the government s choce varable) Assume U [G(x), Y]. I.e. weak separablty and no heterogenety n G Result: unform commodty taxaton Proof: Gven B, consumers choose (x, Y ) arg max U (x, Y) (x,y) B (1) 1
(not necessarly unque max; f they are ndfferent, just pck one) Technology: lnear (can be relaxed, see below) j p j x j Y (2) {x, Y } I, B feasble ff (1) and (2) hold Start from B 0 {x 0, Y 0 } feasble. B 0 can be arbtrary, e.g. wth commodty taxes. Look for reform (B and feasble {x, Y }) that leaves utltes unchanged and saves resources. If U = U (G(x), Y), have 2-stage optmzaton: 1. for a gven G, decde on optmal x 2. then choose between G and Y optmally (1) (G(x 0 ), Y 0 ) arg max (g,y) b 0 U (g, Y) wth b 0 = {(g, Y) g = G(x) (x, Y) B 0 } (so b 0 s fully pnned down by B 0 ) Look for reform for B that has the same mpled budget set b 0 Indvduals choose the same g, Y and hence get same utlty But now let them choose the x s effcently Cost functon e G (g, p) = mn x p x s.t. g = G(x) Then B AS = {(x, Y) p x e G (g, p) and (g, Y) b 0 } For example ˆb = {(y, Y) y = e G (g, p) (g, Y) b 0 }, wth Y denotng before-tax ncome and y after-tax ncome. Hence, ˆb corresponds to a nonlnear ncome tax schedule. B AS = {(x, Y) p x y and (y, Y) ˆb} 2
ndvduals choose same g, Y as before x s potentally changed, but by defnton of e, t mnmzed p x, so LHS of resource constrant s decreased! (RHS unchanged) (2) s now satsfed wth nequalty (strct f x 0 = x AS ) More general technology: F ( x π, Y π ) 0 set p j = F x j at the new allocaton Applcaton to captal taxaton: see next class 3 Prmal Approach to Commodty Taxaton Return to lnear tax framework from the begnnng of the quarter Introduce an alternatve approach (called the prmal approach) that wll be useful to characterze optmal captal taxaton next class 3.1 Setup representatve consumer, no heterogenety lnear taxaton, no lump-sum tax to fnance exogenous government expendture (the Ramsey problem ) numerare good labor l, untaxed n consumpton goods c 1,..., c n, prces p, taxed at lnear rate t consumers: max U(c 1,..., c n, l) s.t. c 1,...,c n p (1 + t )c l. (3),l frms: CRS technology F(x 1,..., x n, l) 0, e.g. p x l 0, where 1/p s the productvty of labor n good 3
proft maxmzaton max x 1,...,x n,l p x l s.t. F(x 1,..., x n, l) 0 (4) government: exogenous government expendtures {g }, budget constrant p g p t c (5) note: snce {g } s fxed, we could easly allow for preferences U(c 1,..., c n, l; g 1,..., g n ) wthout changng results 3.2 Compettve Equlbrum A compettve equlbrum wth taxes {t } and government expendtures {g } s an allocaton {c, x, l} and prces {p } such that 1. {c, l} solves the consumers problem (3) gven prces {p } and taxes {t } 2. {x, l} solves (4) gven {p } and frms make zero profts 3. {c, g } and {p, t } satsfy the government budget constrant (5) 4. all markets clear,.e. c + g = x = 1,..., n (6) Lemma 1. {c, l} and {p } s part of a compettve equlbrum wth {t } and {g } f and only f and {c, l} solves (3) gven {p, t }. F(c 1 + g 1,..., c n + g n, l) = 0, (7) p = F (c 1 + g 1,..., c n + g n, l) F l (c 1 + g 1,..., c n + g n, l) Proof. only f: clear set x = c + g, so 4. s satsfed 4
necessary condtons for (4) are p = γf for some γ and 1 = γf l. f p = F /F l, then these condtons are satsfed and profts are p x l = F F l x l = 1 F l by CRS and Euler s theorem, so 2. s satsfed as for 3., note ( ) F x + F l l = 0 ( ) ( ) ( ) p g = t p c p g = p (1 + t )c l p c l = p c l by the consumers budget constrant, p (g + c ) l = p x l = 0, snce profts are zero as shown above. Thus, 3. s satsfed. 3.3 Ramsey Problem s.t. and max U(c 1,..., c n, l) c,l,t,p F(c 1 + g 1,..., c n + g n, l) = 0 {c 1,..., c n, l} arg max U(c 1,..., c n, l) s.t. c 1,...,c n c (1 + t )p = l,l we optmze over quanttes {c, l} and prces/taxes {p, t }, but the two are related through the last condton 5
two approaches: 1. dual: solve quanttes as a functon of prces and optmze over prces (as we dd at the begnnng of the quarter) 2. prmal: solve prces as a functon of quanttes and optmze over quanttes we pursue the second approach now as t wll be very useful for dynamc taxaton later 3.4 Prmal Approach by convexty of the consumers problem, FOCs are necessary and suffcent: U = λ(1 + t )p = λ solve for prces (1 + t )p = U substtute n budget constrant U c + l = 0 (8) mplementablty constrant, no prces left Proposton 1. Consder any allocaton {c, l } that satsfes the mplementablty constrant (8) and the feasblty constrant (7). Then there exst prces and taxes {p, t } such that {c, l } and {t, p } s part of a compettve equlbrum wth taxes. note: many solutons snce 2 constrants, but n + 1 varables Proof. set so 2. s satsfed p = F (c 1 + g 1,..., c n + g n, l ) F l (c 1 + g 1,..., c n + g n, l ) gven ths, FOC for consumers p (1 + t ) = U (c 1,..., c n, l ) (c 1,..., c n, l ), 6
thus set 1 + t = U U l F l F (9) moreover, consumers budget constrant s satsfed snce p (1 + t )c = l s equvalent to (substtutng {p, t } from above) U c = Ul l, whch s the mposed mplementablty constrant (8). Hence, 1. s satsfed. market clearng was guaranteed when dervng p government budget constrant s satsfed by Walras law: consumers budget constrant zero profts and subtractng p (1 + t )c = l p (c + g ) = l p t c = p g 3.5 Optmal Tax Rules Ramsey problem max U(c 1,..., c n, l) c 1,...,c n,l s.t. (7) and (8) Lagrangan L = U(c 1,..., c n, l) + µ ( ) U c + l γf(c 1 + g 1,..., c n + g n, l) 7
FOCs for good c j and l or we know from (9) that (1 + µ)u j + µ (1 + µ) + µ U j ( ( ) U j c + j l ) c + l l 1 + µ + µ U j c + j l U j 1 + µ + µ = c + l l = γf j = γf l F j F l 1 + t j = U j F l F j = 1 + µ + µ c + l l 1 + µ + µ U j c + j l U j 1 + µ µh l 1 + µ µh j (10) 3.5.1 Unform Taxaton Rule suppose U s separable such that U(c 1,..., c n, l) U(G(c 1,..., c n ), l), where U Gl = 0 and G(.) s homogeneous of degree one f double total spendng on all consumptons goods, then double demand for each ndvdual good, and demand for consumpton goods does not depend on l then U j = U G G j and U j = U GG G G j + U G G j and j = G G j = 0 8
thus U j c + j l U j = ( ) UGG G G j c + U G G j c U G G j now use and U GG G G j c = U GG G j G c = U GG G j G U G G j c = U G G j c = 0 by Euler s theorem and snce G s homogeneous of degree one and G j s homogeneous of degree zero hence U j c + j l U j = U GGG U G therefore ndependent of j 1 + t j = 1 + µ + µ ll 1 + µ + µ U GGG U G exercse: show that the same result more generally goes through when G(.) s homothetc,.e. G(c 1,..., c n ) k(k(c 1,..., c n )), where k (K) = 0 and K(c 1,..., c n ) s homogeneous of degree ρ. can also be generalzed to heterogeneous agents wth preferences U k (G(c 1,..., c n ), l), where k s the household ndex. Then the unform commodty taxaton rule apples n any Pareto optmum f () U k (.) s separable for all k, () the sub-utlty G(.) s the same for all k and () G(.) s homothetc. Note the dfference to the Atknson-Stgltz (1976) theorem: Non-lnear taxaton of labor, no homothetcty of G(.) (nor U Gl = 0) requred. 3.5.2 Inverse Income Elastctes Rule rearrange (10) to t j 1 + t j = µ H j H l 1 + µ µh l 9
thus t j > t f H j > H suppose now that U s separable across all arguments, so that H j = U jjc j U j and H l = ll to consder ncome effects, suppose consumers are endowed wth some exogenous non-labor ncome I, so that the FOC for ther utlty maxmzaton problem s U (c (q, I)) = λ(q, I)q, where dfferentate w.r.t. I so that H becomes q = [p 1 (1 + t 1 ),..., p n (1 + t n )] c U I = q λ I = U λ λ I H = U c U = c λ/ I λ c / I defne the ncome elastcty to get η c I H = I c λ I I λ η numerator s postve, so H s negatvely related to ncome elastcty η hence goods wth a hgher ncome elastcty are taxed at a lower rate: tax necesstes at a hgher rate than luxury goods 4 Producton Effcency How should goods that consumers do not consume drectly be taxed, such as ntermedate goods? A general result (Damond and Mrrlees, 1971) s that the economy should always be on the producton possblty fronter wth optmal taxes. Ths mples that ntermedate goods should not be taxed. 10
We wll use the prmal approach from the precedng secton to demonstrate a smple verson of ths result. Consder an economy wth two sectors. The fnal goods sector has technology f (x, z, l 1 ) = 0, where x s the fnal good, z s an ntermedate good and l 1 s labor used n the fnal goods sector. The ntermedate goods sector has technology h(z, l 2 ) = 0, where l 2 s labor used n the ntermedate goods sector. Let us frst characterze a compettve equlbrum n ths economy. Consumers maxmze ther utlty subject to ther budget constrant takng prces as gven. max U(c, l 1 + l 2 ) c,l 1,l 2 s.t. p(1 + τ)c w(l 1 + l 2 ), where τ s the consumpton tax and we normalzed the tax on labor to be zero. p s the prce of consumpton and w s the wage. The FOCs are U c = p(1 + τ)λ and = wλ. Substtutng p(1 + τ) and w from ths n the budget constrant, we obtan the mplementablty constrant U c c + (l 1 + l 2 ) = 0. The fnal goods sector maxmzes profts subject to the feasblty constrant, takng prces as gven max px wl 1 q(1 + τ z )z x,l 1,z s.t. f (x, z, l 1 ) = 0, where τ z s the tax on the ntermedate good and q s the prce of the ntermedate good. The FOCs are w = γ f l 11
q(1 + τ z ) = γ f z so that f l f z = w q(1 + τ z ). (11) The ntermedate goods sector also maxmzes profts subject to the feasblty constrant and takng prces as gven wth the FOCs max z,l 2 qz wl 2 s.t. h(z, l 2 ) = 0 q = γh z w = γh l so that and hence usng (11) h l h z = w q h l h z = (1 + τ z ) f l f z. (12) Fnally, the government s budget constrant s τpc + τ z qz = pg and market clearng requres c + g = x. The socal planner s problem can be wrtten as max U(c, l 1 + l 2 ) c,l 1,l 2,z s.t. U c c + (l 1 + l 2 ) = 0 f (c + g, z, l 1 ) = 0 h(z, l 2 ) = 0, where we used the prmal approach by substtutng out prces from the consumers 12
and w.r.t. l 2 = h l γ h, FOCs and budget constrant as before. The FOC of the plannng problem w.r.t z s f z γ f + h z γ h = 0 or f z h z = γ h γ f. The FOC w.r.t. l 1 s = f l γ f whch mples that or f l h l = γ h γ f f l f z = h l h z. Ths shows that when taxes are set optmally, the margnal rate of transformaton s undstorted across goods. Comparng wth the condton for a compettve equlbrum n (12), we see that n the optmum τ z = 0. Hence, no tax on ntermedate goods should be mposed. 13