THE ECONOMICS OF TAXATION Statc Ramsey Tax School of Economcs, Xamen Unversty Fall 2015
Overvew of Optmal Taxaton Combne lessons on ncdence and effcency costs to analyze optmal desgn of commodty taxes. What s the best way to desgn taxes gven equty and effcency concerns? 1
From an effcency perspectve, would fnance government purely through lump-sum taxaton. Wth redstrbutonal concerns, would deally levy ndvdual-specfc lump sum taxes. Tax hgher-ablty ndvduals a larger lump sum. Problem: cannot observe ndvduals types. Therefore must tax economc outcomes such as ncome or consumpton, whch leads to dstortons. 2
Ramsey vs. Mrrleesan Approaches Two approaches to optmal taxaton: 1. Ramsey: restrct attenton lnear (t x) tax systems 2. Mrrleesan: non-lnear ( tx) ( ) tax systems, wth no restrctons on tx ( ) Ramsey approach: rule out possblty lump sum taxes by assumpton and consder lnear taxes. Mrrleesan approach: permt lump sum taxes, but model ther costs n a model wth heterogenety n agents sklls. 3
Prmal vs. Dual Approaches Regardless of whch approach s used, there are two ways of solvng the optmal taxaton problem. 1. Prmal approach: the government chooses allocatons drectly. The optmal tax formulas are then typcally expressed drectly n terms of the prmtves of the model. 2. Dual approach: the government chooses the taxes drectly. The optmal tax formulas are easly expressed n terms of supply and demand elastctes. 4
Four Central Results n Optmal Tax Theory 1. Ramsey (1927): nverse elastcty rule 2. Chamley (1985), Judd (1986): no captal taxaton n nfnte horzon Ramsey models 3. Damond and Mrrlees (1971): producton effcency 4. Atknson and Stgltz (1976): no consumpton taxaton wth optmal non-lnear ncome taxaton 5
Ramsey Tax Problem Government sets taxes on uses of ncome n order to accomplsh two obectves: 1. Rase total revenue of amount R 2. Mnmze utlty loss for agents n economy Key assumptons: 1. Lump sum taxaton prohbted 2. Cannot tax all commodtes (e.g., lesure untaxed) 3. Producton prces fxed (and normalzed to one): p 1 q 1 6
Ramsey Model: Setup One ndvdual (no redstrbutve concerns) As n effcency analyss, assume that ndvdual does not nternalze effect of on government budget Captures dea that any one ndvdual accounts for a small fracton of economy Indvdual maxmzes utlty subect to budget constrant ux (,, x l) 1 N, q x q x wl Z 1 1 N N Z = non wage ncome, w = wage rate 7
Ramsey Model: Consumer Behavor Lagrangan for ndvdual s maxmzaton problem: L u( x,, x, l) ( wl Z ( q x q x )) Frst order condton: 1 N 1 1 N N u x q where V Z s margnal value of money for the ndvdual Yelds demand functon x ( qz, ) and ndrect utlty functon V( qz, ) where q ( wq, 1,, q N ) 8
Ramsey Model: Government s Problem Government solves ether the maxmzaton problem max V( qz, ) subect to the revenue requrement N x x( qz, ) R 1 Or, equvalently, mnmze excess burden of the tax system mn EBq ( ) eqv (, ( qz, )) e( pv, ( qz, )) E subect to the same revenue requrement 9
For maxmzaton problem, Lagrangan for government s: LG V( qz, ) x( qz, ) E LG V x x q q mechancal effect q prv. welfare behavoral response loss to ndv. V Usng Roy s dentty ( x ): q ( ) x x q 0 Note connecton to margnal excess burden formula, where 1 and 1. 10
Ramsey Optmal Tax Formula Optmal tax rates satsfy system of N equatons and N unknowns: x x ( ) q Same formula can be derved usng a perturbaton argument, whch s more ntutve. 11
Ramsey Formula: Perturbaton Argument Suppose government ncreases by d Effect of tax ncrease on socal welfare s sum of effect on government revenue and prvate surplus. Margnal effect on government revenue: dr x d dx Margnal effect on prvate surplus: V du d x d q Optmum characterzed by balancng the two margnal effects: du dr 0 12
Ramsey Formula: Compensated Elastcty Representaton Rewrte n terms of Hcksan elastctes to obtan further ntuton usng Slutsky equaton: Substtuton nto formula above yelds: x h x x q q Z ( ) x h q x x Z 0 where ( x ). Z h 1 x q 13
s ndependent of and measures the value of the government of ntroducng a $1 lump sum tax ( x ) Z Three effects of ntroducng a $1 lump sum tax: 1. Drect value for the government of 2. Loss n welfare for ndvdual of 3. Behavoral effect loss n tax revenue of ( x ) Z 14
Intuton for Ramsey Formula: Index of Dscouragement h 1 x q Suppose revenue requrement R s small so that all taxes are also small. Then tax on good reduces consumpton of good (holdng utlty constant) by h approxmately dh q Numerator of LHS: total reducton n consumpton of good Dvdng by x yelds % reducton n consumpton of each good = ndex of dscouragement of the tax system on good Ramsey tax formula says that the ndexes of dscouragements must be equal across goods at the optmum. 15
Inverse Elastcty Rule Introducng elastctes, we can wrte Ramsey formula as: N c 1 1 Consder specal case where 0 f Slutsky matrx s dagonal Obtan classc nverse elastcty rule: 1 1 16
Example 1: Two Commodtes If N=2, then we can have: Ths yelds 1 1 1 c 2 c 1 2 1 2 1 11 22 12 2 1 2 c c c c 11 21 c c If the goods are complementary ( 12 21 0), the tax rates 1 and 2 wll tend to be closer to each other. Intutvely, complementary goods look more alke, and there s less need to dfferentate between them. 17
Example 2: Corlett-Hague (1953) We can renterpret the results n the followng way. The compensated elastctes satsfy the addng-up property: So, c c c 0 1 2 0 2 1 2 11 22 20 1 1 1 ( ) ( ) c c c c c c 11 22 10 c What matters for the relatve tax rates s therefore the magntude of 0. If the good zero as lesure, a good s more complementary wth lesure than good. c c f 0 0 Goods that are more complementary wth lesure should be taxed more heavly. 18
Example 3: Unform Commodty Tax Rate If utlty functon s weakly separable: UGx ( ( ), l) UGx ( ( )) vl ( ) and the consumpton component s homothetc,.e. G() s a homothetc functon, then, 1 1 or Therefore, all goods should be taxed at the same rate. 19
Ramsey Formula: Lmtatons Ramsey soluton: tax nelastc goods to mnmze effcency costs But does not take nto account redstrbutve motves Necesstes lkely to be less elastc than luxures Therefore, optmal Ramsey tax system s lkely regressve Damond (1975) extends Ramsey model to take redstrbutve motves nto account: Basc ntuton: replace multpler wth average margnal utlty for consumers of that good 20
Applcaton of Ramsey Approach to Taxaton of Savngs Standard lfecycle model of consumpton max ut( ct) subect to qc t t t W where q (1 ) p and 0 0 t t t Consumpton n each perod somorphc to consumpton of dfferent goods Can apply standard Ramsey formula to calculate t Captal ncome tax s a constant tax on nterest rate: 1 1 (1 ) r q t t 21
Optmal Captal Income Tax Rate For any 0, mpled tax t approaches as t : qt p t 1 r 1t 1 (1 ) r lm t Ramsey formula mples that optmal t t cannot be for any good t Therefore optmal captal ncome tax rate converges to 0 n long run (Judd 1985, Chamley 1986) Best polcy s for gov t to tax captal untl t accumulates suffcent assets to fund publc goods and never tax captal agan 22
Zero Captal Taxaton n Ramsey Models Farly robust result n pure Ramsey framework (Bernhem 2002) But not robust to Allowng for progressve ncome taxaton (Golosov, Kocherlakota, Tsyvnsk 2003) Allowng for credt market mperfectons (Ayagar 1995, Farh and Wernng 2011) Fntely-lved agents wth general utlty functon and age-dependent tax system (Erosa and Gervas 2003) or fntely-lved agents wth fnte bequest elastctes (Pketty and Saez, 2013) 23