Economics of taxation

Transcription

1 Economcs of axaon Lecure 3: Opmal axaon heores Salane (2003)

2 Opmal axes The opmal ax sysem mnmzes he excess burden wh a gven amoun whch he governmen wans o rase hrough axaon. Opmal axes maxmze socal welfare, gven governmen s revenues.?????? Rgh combnaon of effcency and equy maes axes opmal. No sraghforward. Very conroversal. Wha s he relaon beween effcency and equy?

3 Opmal axes Why s he equy creron conroversal? Vercal equy: axes are mposed subec o he axpayers ncomes and her ables o gan ncome. Horzonal equy: dencal ndvduals should pay he same axes.

4 Opmal axes The general concluson of mahemacal models s ha opmal axes should nclude hgh average ax raes for ndvduals wh hgh ncomes and low average ax raes for low ncomes. A he same me he margnal ax raes should be low for everybody: hose wh low ncomes as well as hose wh hgh.

5 Opmal axes Opmal axaon heory goes round he problem of socal welfare by assumng ha here exs: Bergson-Samuelson funconal W(V,,V n ) where V s he uly ndex of consumer x, y - vecors of feasble socal choces x s socally preferred o y f and only f W(V (x),,v n (x))> W(V (y),,v n (y)) The monooncy of he funcon W reflecs he effcency, whle s concavy reflecs redsrbuve properes,.e. equy. Thus, maxmzng W mples he rade-off beween equy and effcency.

6 A paral equlbrum on a goods mare A smple model, n whch here are only axes on goods and lnear ax on wages. We mpose a small ad valorem ax on good. The deadwegh loss s equal: () pdx D S 2 p x D 2 2 The oal deadwegh loss from he ax sysem s equal : (2) D()=D ( )+ +D n ( n ) Whle ax revenues are gven by: D (3) R()=p x + + p n x n n S

7 mn D(), s.. R()=T (4) where s he Lagrange mulpler assocaed o he governmen s budge consran We can rewre as nverse elasces rule : (5) A paral equlbrum on a goods mare D D S S D S I suggess ha may be beer o nroduce a ax on a good whose demand and supply are less elasc.

8 Opmal axaon n general equlbrum model I consumers-worers wh uly funcons U (,L ) where s consumpon of n goods and L s he labor supply. Assume ha he producon has consan reurns: goods are produced from labor only. Producon of one un of good requres a uns of labor. In equlbrum p =a w Taxes: lnear axes on goods, whch rase consumer prces o (+ ), lnear ax on wages, whch lowers he ne wage o (- τ).

9 Opmal axaon n general equlbrum model The budge consran of consumer, who only owns hs labor, s: (6) Then he ax on wages s equvalen o a unform ax on goods: (7) The budge consran can be ransformed no: (8) ' n n ' L ' L The ax sysem (, τ) s equvalen o he ax sysem (, 0) n whch wages are no axed.

10 The governmen collecs from he consumer : (9) In boh ax sysems he governmen collecs exacly he same revenue. Consumers maxmze her uly: V (q) - he ndrec uly of consumers q =+ - he vecor of consumpon prces (0) n V q L max n U,, L L ' n ' Opmal axaon n general equlbrum model under q L

11 Opmal axaon n general equlbrum model The governmen mus maxmze W(q) n q, subec o s budge consran: () where are he demands of he varous q consumers and q =+. λ - he Lagrange mulpler of he governmen s budge consran Dfferenang Lagrangan n q yelds: (2) I n q W V V I I n q q T ' q

12 Opmal axaon n general equlbrum model By Roy s deny: V where Defne q s he margnal uly of ncome of consumer W V - he socal margnal uly of ncome of consumer. I s he ncrease n he value of he Bergson- Samuelson funconal when s gven one addonal un of ncome.

13 Opmal axaon n general equlbrum model (3) I I n ' q We can use Slusy s equaon: (4) q S R where S q U

14 Opmal axaon n general equlbrum model Usng Slusy s equaon we ge: (5) Tang ou of braces, we can nroduce a new parameer: (6) n b ' I S n I ' b - he ne socal margnal uly of ncome of consumer. The socal margnal uly of ncome of consumer, dvded by he cos of budge resources for governmen. R I I n ' R The ncrease n ax revenue colleced from when hs ncome ncreased by one un.

15 Opmal axaon n general equlbrum model The aggregaed demand for good s equal: Then, rearrangng and usng he symmery of he Slusy marx, we ge: n I I (7) ' S b However, I Le b be he average of all b. The emprcal covarance across consumers s defned as: b I cov, b I

16 Opmal axaon n general equlbrum model Fnally we oban Ramsey s formula: (8) n ' I S b b The LHS of hs formula s called he dscouragemen ndex of good. A ax on good decreases he consumpon of good by consumer by ' S a a fxed uly level. In oher words s he relave change of compensaed demand for good caused by he ax sysem. The RHS of he formula s called he dsrbuve facor of good.

17 Opmal axaon n general equlbrum model Ramsey s formula (smpler verson): (9) p d U where proporonaly coeffcen, ax, p a ne prce (afer ax), s d he elascy of compensaed demand, U s he elascy of supply. Ths formula says ha axes on goods, whch mnmze he excess burden, are proporonal o he sum of he recprocals of elasces of supply and demand.

18 Opmal axaon n general equlbrum model Formula (8) ndcaes ha he governmen should ax less he goods ha are more nensvely consumed by agens wh a hgh ne socal margnal uly of ncome,.e. goods wh a posve covarance. Ths suggess ha he ax sysem should dscourage less he consumpon of he goods ha he poor buy more. Formula (9) suggess ha he governmen should mpose hgher margnal ax raes on goods wh low elascy of demand (or low elascy of supply). Ths means ha we should ax more he goods consumed by he poor. Ths conradcon resuls from gnorng redsrbuve obecves n formula (9). In addon, hs formula s rue only for goods wh no nerdependen demands.

19 Opmal axaon of ncome (Mrrlees, 97) Prevous model was a very smple one. In more complex cases we have o ae more facors no accoun. If we wan o sudy ncome axes, we have o consder e.g. he dscouragng effec of axes on labor supply. Ths problem was solved by Mrrlees (97): - Worers have heerogeneous earnng capaces w. - All ndvduals have he same uly funcon U(C,L), wh one consumpon good C and a labor supply L. - Snce ndvduals have he same preferences we do no need o worry abou horzonal equy.

20 Opmal axaon of ncome (Mrrlees, 97) Governmen collecs axes and uses hem o acheve s redsrbuve obecves, whch maxmze he addve Bergson-Samuelson funconal: (20) W where Ψ U w df U(w) - he afer-ax uly of consumer w, F - he cumulave dsrbuon funcon of w n he populaon, w Ψ - an ncreasng and concave funcon ha weghs he ules of he ndvduals accordng o he governmen s redsrbuve obecves.

21 Opmal axaon of ncome (Mrrlees, 97) Governmen s preferences:. Ularan - he governmen maxmzes he sum of he ndvdual ules. or 2. Rawlsan maxmn - he governmen ams a maxmzng he uly of he leas favored member of socey. The governmen plans o collec from each ndvdual a ax revenue T(w) o fnance publc good expendures R.

22 Opmal axaon of ncome (Mrrlees, 97) In he compeve labor mare each ndvdual s pad hs producvy level wl(w). Everyone chooses he labor supply o maxmze he afer-ax uly: (2) The governmen canno observe he producves of worers. I can only observe ncomes, wha changes he ndvdual maxmzaon problem no: (22) L L w w argmaxu L argmaxu L wl wl, L If he governmen canno observe acual producves, canno mpose axes whch are boh equable and effcen. T T w wl, L

23 Opmal axaon of ncome (Mrrlees, 97) The governmen problem s o choose he ncome ax schedule T(.) o maxmze (23) W where (25) U wl T wl, L all of hs under he governmen s budge consran (26) 0 U w df (24) U w U wl w T wl w, L and L(w) maxmzes over L 0 T wl w df w w R w

24 Crcsm of opmal axaon. Opmal axaon gnores many facors whch are mporan for fscal polcy. - The opmal axaon focuses on he vercal equy: axes should be mposed subec o he axpayers ncomes and her ables o gan ncome. - Opmal axes could be very dffcul and expensve o collec and conrol, no menonng he complance coss for axpayers.

25 Crcsm of opmal axaon 2. Many soluons and conclusons of hs heory can be reached n more nuve way. - Governmens, whle desgnng ax sysems, do no buld models based on Bergson-Samuelson funconal. - Any changes n ax sysems are nroduced slowly and gradually, wh he obecve o mprove suaon under Pareo opmaly. - If we have a nonlnear ncome ax, s always possble o nroduce ax reforms mprovng suaon n Pareo sense.

26 Crcsm of opmal axaon 3. The opmal axes analyss does no gve clear conclusons for he fscal polcy. - Is resuls depend on he economc relaons, whch are dffcul o sudy or measure n pracce, and on nformaon, whch are no accessble. - I s relavely easy o nroduce a small change gvng a Pareo mprovemen, bu very dffcul o run a complex reform of ax sysem. - Ofen we canno ranslae he resuls of opmal axaon models no precse, praccal polcal acons.