Taxation and Economic Efficiency

Transcription

1 Taxaion and Economic Efficiency Alan J. Auerbach Universiy of California, Berkeley and NBER James R. Hines Jr. Universiy of Michigan and NBER February 2001 This paper has been prepared for a forhcoming volume of he Handbook of Public Economics, edied by Alan Auerbach and Marin Feldsein. We hank Charles Blackorby, Peer Diamond, Kenneh Judd, Louis Kaplow, Gareh Myles, Michel Srawczynski and Ronald Wendner for helpful commens on a previous draf.

2 Taxaion and Economic Efficiency ABSTRACT This paper analyzes he disorions creaed by axaion and he feaures of ax sysems ha minimize such disorions (subec o achieving oher governmen obecives). I sars wih a review of he heory and pracice of deadweigh loss measuremen, followed by characerizaions of opimal commodiy axaion and opimal linear and nonlinear income axaion. The framework is hen exended o a variey of seings, iniially consising of opimal axaion in he presence of exernaliies or public goods. The opimal ax analysis is subsequenly applied o siuaions in which produc markes are imperfecly compeiive. This is followed by consideraion of he feaures of opimal ineremporal axaion. The purpose of he paper is no only o provide an up-o-dae review and analysis of he opimal axaion lieraure, bu also o idenify imporan cross-cuing hemes wihin ha lieraure. JEL Classificaion: H21. Alan J. Auerbach James R. Hines Jr. Deparmen of Economics Office of Tax Policy Research 549 Evans Hall Universiy of Michigan Business School Universiy of California 701 Tappan Sree Berkeley, CA Ann Arbor, MI auerbach@econ.berkeley.edu rhines@umich.edu

3 Table of Conens 1. Inroducion Ouline of he chaper The heory of excess burden Basic definiions Variaions in producer prices Empirical issues in he measuremen of excess burden The design of opimal axes The Ramsey ax problem Changing producer prices The srucure of opimal axes An example The producion efficiency heorem Disribuional consideraions Income axaion Linear income axaion Nonlinear income axaion: inroducion Nonlinear income axaion: graphical exposiion Nonlinear income axaion: mahemaical derivaion Exernaliies, public goods, and he marginal cos of funds The provision of public goods and he marginal cos of public funds Exernaliies and he double-dividend hypohesis Disribuional consideraions and he MCPF Opimal axaion and imperfec compeiion Opimal commodiy axaion wih Courno compeiion Specific and ad valorem axaion Free enry Differeniaed producs Ineremporal axaion Basic capial income axaion: inroducion The seady sae Inerpreing he soluion Human capial accumulaion and endogenous growh Resuls from life-cycle models Conclusions... 94

4 1. Inroducion This chaper considers a subec a he very cener of public finance analysis, he disorions inroduced (and correced) by axaion. Tax-induced reducions in economic efficiency are known as deadweigh losses or he excess burdens of axaion, he laer signifying he added cos o axpayers and sociey of raising revenue hrough axes ha disor economic decisions. Taxes almos invariably have excess burdens because ax obligaions are funcions of individual behavior. The alernaive, pure lump-sum axes, are aracive from an efficiency perspecive, bu are of limied usefulness precisely because hey do no vary wih indicaors of abiliy o pay, such as income or consumpion, ha are funcions of axpayer decisions. Thus, even hough ax analysis ofen sars wih he simple case of a represenaive household, i is household heerogeneiy and he inabiliy fully o observe individual differences ha usify he resricions commonly imposed on he se of ax insrumens. Designing an opimal ax sysem means keeping ax disorions o a minimum, subec o resricions inroduced by he need o raise revenue and mainain an equiable ax burden. The following secions discuss he heory and measuremen of excess burden and he design of opimal ax sysems. The analysis draws heavily on he chapers by Auerbach (1985) and Sigliz (1987) in he original volumes of his Handbook, inerweaving he mos imporan resuls conained in hese wo chapers wih he addiional insighs and areas of inquiry ha have appeared since heir publicaion. For more deailed analysis and a reamen of many oher opics in his lieraure, he reader is referred o hese original essays.

5 1.1. Ouline of he chaper The chaper begins wih he basics and hen urns o seleced opics. Secions 2, 3, and 4 lay ou he heory of excess burden, opimal commodiy axaion, and opimal income axaion. Secion 5 considers he provision of public goods and he correcion of exernaliies, and how hese problems inerac wih he manner in which revenues are raised. Secion 6 discusses he impac on ax design of deviaions from perfec compeiion, and Secion 7 exends he heory of ax design o address issues ha arise in ineremporal seings. Secion 8 offers some brief conclusions regarding he evoluion of he lieraure and promising direcions for fuure research. 2. The heory of excess burden 2.1. Basic definiions Excess burden (or deadweigh loss) is well defined only in he conex of a specific comparison, or concepual experimen. If one simply seeks he excess burden of a paricular ax policy, here are many equally plausible answers, so in order o obain a unique meaning, i is necessary o be more specific. For example, he excess burden of a 10 percen ax on reail sales varies no only wih he iniial condiions of he ax sysem, bu also wih he direcion of change, i.e., wheher he ax is being added or removed. To illusrae his ambiguiy and is resoluion, consider he simple case in which here are wo goods, an unaxed numeraire good and a second good wih a consan relaive producer price of p 0. In he absence of axaion, a populaion of idenical consumers 1 demands quaniy x 0 of he second good, as depiced by poin 0 in Figure 2.1. The imposiion of a ax per uni of p 1 p 0 raises he consumer price of he axed good o p 1, wih he producer price remaining a p 0. Thus, 1 We limi our discussion of excess burden o he case of idenical consumers, hereby sidesepping issues of aggregaion ha arise in he case of heerogeneous consumers. See Auerbach (1985) for furher discussion. 2

6 he quaniy purchased falls o x 1, and he governmen collecs revenue equal o (p 1 p 0 )x 1, as represened in he figure by he shaded area labeled A. Wha is he excess burden of his ax? If one were o use he Marshallian measure of he consumers surplus generaed by consumpion in his marke he area under he demand curve, D, beween x=0 and x=x 0 i would appear ha consumers lose an area equal o ha of regions A+B, or B in excess of he revenue acually colleced. By his approach, he roughly riangular area B commonly known as a Harberger riangle in recogniion of Arnold Harberger s influenial empirical conribuions measures he excess burden of he ax. Unforunaely (see Auerbach 1985), his paricular measure of excess burden is no uniquely defined in a seing wih more han one ax, due o he well-known problem of pah dependence of consumers surplus: he measure of excess burden is affeced by he order in which one envisions he axes being imposed. Pah dependence is disconcering, bu more imporanly reflecs he imprecision of consumers surplus-based measures of excess burden. There is no well-defined economic quesion o which he difference beween he change in consumers surplus and ax revenue is he answer. Thus, economiss have sough alernaive measures of excess burden ha are no pah-dependen and ha answer meaningful quesions. Pah dependence does no arise if excess burden is measured by Hicksian consumers surplus, based on schedules ha hold uiliy, raher han income, consan as prices vary. Because acual ax policy changes ypically do no hold uiliy consan, i is herefore necessary o consruc a measure based on a concepual experimen in which uiliy is held consan. One inuiive experimen is o imagine ha, as a ax is imposed, uiliy is held consan a is pre-ax level. Graphically, in Figure 2.2, his measure is based on he compensaed demand curve D(u 0 ), which by definiion passes hrough he original, no-ax equilibrium poin 0. If he ax is 3

7 imposed, and consumers are compensaed o remain a original uiliy levels, hen demand follows his schedule and he ax reduces consumpion o poin 1. A his poin, revenue raised is he sum of areas A and C, raher han he acual level of revenue represened by area A, because compensaion induces consumers o purchase more of he axed good (if, as is assumed here, he good is normal) and hence pay more axes. Excess burden is defined as he amoun, in excess of his revenue, ha he governmen mus compensae consumers o mainain iniial uiliy in he face of a ax-induced price change. The amoun of compensaion, which corresponds o he Hicksian measure of he compensaing variaion of he price change, may be calculaed using he expendiure funcion as p 1dE( p, U (2.1) = 0) E p1, U0) E( p0, U0) dp = p 0 dp p1 c ( x ( p, U0 p 0 ) dp which is well-defined even for a vecor of changing prices p he Hicksian variaions are singlevalued, regardless of he order of inegraion of he differen price changes in (2.1). For each marke, his measure equals he area beween prices p 0 and p 1 o he lef of he compensaed demand curve D c (U 0 ). Thus, he deadweigh loss equals area D in he figure sill approximaely a Harberger riangle, bu differen han ha defined by he ordinary demand curve in Figure An alernaive concepual experimen is o begin wih he ax already in place and hen remove i, exracing from consumers in lump-sum fashion an amoun ha prevens hem from changing heir uiliy levels while he ax is removed. Because he iniial ax is disorionary, i is 2 Noe ha his definiion is equally well-defined for he case of negaive revenue, in which we would race a pah down he compensaed demand curve from poin 0. There, oo, he ax sysem generaes excess burden, in ha he revenue los exceeds he absolue value of he associaed compensaing variaion. This serves as an imporan reminder ha deadweigh loss is he resul of disorion, no of raising revenue per se. 4

8 necessary o exrac more from consumers han he ax revenue, he difference represening he excess burden of he iniial ax. Saring from poin 1 in Figure 2.2, his experimen follows he compensaed demand curve D c (U 1 ) down o poin 0, where he price reaches is no-ax level bu uiliy remains unchanged. Again using he expendiure funcion o calculae he amoun he governmen exracs in his case he Hicksian equivalen variaion, based on he formula in (2.1) wih U 1 in place of U 0 he amoun equals he area o he lef of demand curve D c (U 1 ) beween prices p 0 and p 1. This exceeds he forgone revenue in his case he acual revenue defined by area A and again does so by a riangle. Alhough hese wo measures are he mos inuiive, hey are acually us examples drawn from a class of measures based on arbirary levels of uiliy, say U i : (2.2) E p, U ) E( p, U ) R( p, p, U ) ( 1 i 0 i 0 1 i c where R p, p, U ) ( p p ) x ( p, U ) is he level of revenue colleced wih axes in place ( 0 1 i i and uiliy fixed a level U i. As Figure 2.3 shows, i is also possible o represen excess burden in a graph in commodiy space. In he figure, he consumer s indifference curve is angen o he original budge line a poin 0, which corresponds o poin 0 in Figure 2.2. The ax roaes he consumer budge line as shown, leading o consumpion a poin 1 (corresponding o poin 1 in Figure 2.2), a which ax revenue, measured in erms of he numeraire commodiy, equals R(p 0,p 1,U 1 ). The consumer could mainain uiliy level U 1 in he absence of axes by consuming a poin 0 (again, as labeled in Figure 2.2), where only E(p 0,U 1 ) of expendiure would be required, which is less (as measured by he numeraire commodiy) han he expendiure necessary o generae uiliy level 5

9 U 1 when consumpion is disored by axes (as i is a poin D). The difference is he equivalen variaion measure of excess burden, based on expression (2.2) for uiliy level U 1. I is sraighforward o generalize his class of measures o siuaions in which iniial equilibria are no Pareo-opimal due o pre-exising axes. The marginal excess burden of a ax change is he difference beween he Hicksian variaion associaed wih he price change and he change in ax revenue (which, in he absence of preexising axes, is simply ax revenue), a he chosen level of uiliy: (2.3) E p, U ) E( p, U ) [ R( p, p, U ) R( p, p, U )] ( 2 i 1 i 0 2 i 0 1 i in which p 2 is he price vecor afer he ax change. For a given reference uiliy level U i,, his definiion has he imporan propery ha he marginal excess burden in moving from poin 1 o poin 2 equals he difference beween he excess burden a poin 2 and he excess burden a poin 1, as defined in expression (2.2). Figure 2.4 illusraes his measure for he case in which an iniial ax in a single marke ha changed he consumer price from p 0 o p 1 is hen increased, raising he price o p 2. The figure illusraes he marginal excess burden of his ax increase, aking he reference uiliy level o be ha obained a poin 1, he consumpion poin wih he iniial ax in place. The Hicksian variaion of he addiional price change equals he sum of areas A and B. The change in ax revenue (wih uiliy held consan) equals he difference beween final ax revenue (areas A+C) and ax revenue prior o he imposiion of he second ax, (C+D), or a difference of A D. Tha is, wih a preexising ax, i is necessary o ne he revenue los on forgone purchases agains he revenue gained from a higher ax on remaining purchases. Thus, he marginal excess burden consiss no only of he riangle B, bu also he recangle D. Marginal excess burden is no 6

10 longer us a second-order phenomenon (he riangle) ha vanishes wih a small ax increase, bu insead is of firs-order significance. The oal excess burden (calculaed a uiliy level U 1 ) of boh axes equals his marginal excess burden plus he excess burden of he iniial ax, equal o area E Variaions in producer prices The analysis hus far adops he simplifying assumpion of fixed relaive producer prices, bu i is possible o exend he various measures of excess burden o he more general case in which producer prices vary. I is helpful o begin wih a graphical exposiion. Figure 2.5 repeas he experimen of Figure 2.3, bu does so in a case in which he relaive producer price of he axed good he inverse slope of he producion possibiliies fronier (PPF), shown in bold varies wih he oupu mix. Saring again a an equilibrium in which a disorionary ax is used o raise revenue from he represenaive household, he household s consumpion bundle is shown a poin 1, which corresponds o poin 1 in Figure 2.3. Producion occurs a poin 1 p in he figure, and he governmen raises revenue in he numeraire commodiy equal o he horizonal disance beween poins 1 and 1 p. The consumer price p 1 exceeds he producer price q 1 by he ax per uni of oupu. The household s income (in unis of he numeraire commodiy) is y 1, and is indifference curve is angen o he consumer price line a poin 1. Also passing hrough poin 1 (bu having a slope 1/q 1 and no angen o he indifference curve) is a privae producion possibiliies fronier he original PPF, displaced o he lef by he amoun of he numeraire commodiy corresponding o governmen consumpion. Because he governmen is assumed o absorb only he numeraire commodiy, his displacemen is horizonal; oherwise, poin 1 would no lie direcly o he lef of poin 1 p. If, insead, he governmen devoed all ax revenues o purchases 7

11 of he axed commodiy, hen poin 1 would lie direcly below poin 1 p. I should be clear ha (unlike in he experimen wih fixed producer prices) he equilibrium is affeced by how he governmen uses is revenue, since governmen purchases influence relaive demand and hence relaive producer prices of he wo commodiies. Excess burden is he amoun of addiional revenue he governmen could collec wihou harming he consumer, were lump-sum axes used insead of disorionary axes. I is necessary o specify he form ha his exra revenue akes. Here, all revenue akes he form of he numeraire commodiy, shifing he privae PPF horizonally o he lef unil angen (a poin 0) wih he indifference curve passing hrough poin 1. Corresponding o consumpion poin 0 is he producion poin 0 p. Excess burden is measured as he horizonal disance beween his undisored poin 0 and he corresponding poin on he privae PPF passing hrough poin 1. Excess burden can be defined algebraically by noing ha he horizonal disance beween poins 0 and 0 p equals he sum of excess burden and ax revenue (he same revenue as ha raised in he iniial equilibrium, R(q 1,p 1,U 1 ). Thus, leing y 0 be he value of he household s income from producion a poin 0, excess burden equals (2.4) y E p, U ) R( q, p, U ) = E( p, U ) E( p, U ) + y y R( q, p, ) 0 ( U1 wih he las sep in (2.4) following from he ideniy ha E( p 1, U1) = E( p1, U( p1, y1)) y1. As in he case wih fixed producer prices, he measure defined in (2.4) may be consruced for differen reference uiliy levels. 3 Also, differences in excess burden as measured by (2.4) correspond o changes in excess burdens due o addiional axes. 3 The expression for excess burden, and is graphical inerpreaion, becomes somewha more complicaed if he governmen absorbs boh axed and unaxed commodiies. See Auerbach (1985) for furher discussion. 8

12 Expression (2.4) collapses o (2.2) when producer prices do no change, for hen income y is fixed and he ne of ax price vecor in he ax-disored equilibrium, q 1, and he price vecor in he undisored equilibrium, p 0, boh are idenical o he original price vecor p 0. The exra erm, y y, is he change in income along he producion possibiliies fronier when moving 0 1 from poin 1 p o poin 0 p. By he envelope heorem, he change in income equals 0 x( q) d q, where x(q) is he quaniy vecor of goods produced a price vecor q. I is hen possible o represen excess burden in a single marke in price-quaniy space, as does he diagram in Figure 2.6, in his case wih an upward sloping supply curve for he axed good, x(q). The excess burden, according o expression (2.4), equals he sum of Hicksian consumers surplus, areas A+B, plus he change in income, areas C+D (someimes known as producers surplus ) minus ax revenue, A+C, for a ne excess burden of areas B+D. For fuure reference, i is useful o presen a very simple expression for he marginal excess burden of axaion. Toally differeniaing he righ side of (2.4) yields p q1 (2.5) de d EB = dp dp 1 dy dq dq 1 ( p 1 c dx q1 ) dp dp x ( dp 1 dq 1 ) = x c ( p 1, U 1 ) dp 1 x( q 1 ) dq 1 ( p 1 c dx q1) dp dp 1 x ( dp 1 c dx dq1 ) = dp dp 1 where he las sep follows from he fac ha x c p, U ) = x( ). Tha is, he change in excess ( 1 1 q1 burden equals he sum of he producs of exising ax raes and changes in oupu. This resul is exremely useful in searching for axes ha impose minimal excess burden. I is someimes expressed as a firs-order Taylor approximaion for discree changes, x, or a second-order approximaion ( x + ½ x). The second-order approximaion aken around he 9

13 undisored poin (=0), wih se equal o he ax vecor iself, approximaes a measure of he oal excess burden of he ax sysem (e.g. Harberger 1964a). From his approximaion comes he common inuiion ha excess burden increases wih he square of a ax. If one considers he second-order approximaion for a single ax i and producer prices fixed, excess burden is ) 1 c 2 i ( dxi di. i 2.3. Empirical issues in he measuremen of excess burden While he heory of deadweigh loss measuremen has a long and colorful hisory ha daes back o he nineeenh cenury conribuions of Jules Dupui (1844) and Fleeming Jenkin (1871/72), economiss seldom measured acual deadweigh losses prior o he pioneering work of Arnold Harberger in he 1950s and 1960s. In wo influenial papers published in 1964, Harberger (1964a) derived he approximaion (2.5) used o measure deadweigh loss and (1964b) applied he mehod o esimae deadweigh losses due o income axes in he Unied Saes. Harberger shorly hereafer (1966) produced esimaes of he welfare cos of U.S. capial axes. A generaion of empirical sudies by oher scholars followed he publicaion of Harberger s subsequen survey aricle (1971). 4 The empirical work ha followed Harberger s effors focussed on he use of simple deadweigh loss formulas o esimae he welfare impac of a wide array of ax-induced disorions, including hose o labor supply (Browning, 1975; Hausman, 1981a), saving (Feldsein, 1978), corporae axaion (Shoven, 1976), and he consumpion of goods, such as housing and non-housing consumpion iems, ha are axed o differing degrees (King, 1983). 5 In addiion, some aenion was devoed o refining he approximaions used in applying 4 See Hines (1999) for an inerpreive survey of his lieraure. 5 See he discussion in Auerbach (1985) and he more recen survey by Slesnick (1998). 10

14 esimaed behavioral parameers o calculae deadweigh losses. The varian of (2.5) used by Harberger, in which a form of uncompensaed demand is used in place of compensaed demand, approximaes a compensaed measure of welfare change (2.4). One quesion of ineres o subsequen invesigaors is he pracical difference beween resuls obained using Harbergersyle approximaions and hose available from more exac measures. As Mohring (1971) and subsequen auhors noe, i is ofen he case ha he same demand informaion necessary o calculae approximaions o (2.5) can, if properly modified, be used o calculae Hicksian deadweigh loss measures of he form (2.4). The exen o which hese wo mehods generae differen answers is, of course, an empirical quesion. Rosen (1978) finds ha (2.4) and approximaions o (2.5) rack each oher raher closely, bu Hausman (1981b) offers some examples in which hey differ considerably. The generaion of empirical work following Harberger calls aenion o he imporance of linking he sraegy used o esimae demand and he ulimae goal of using he esimaes o perform welfare analysis. Specifically, his enails esimaing models ha can be inegraed o obain expendiure funcions from which expressions such as (2.4) can be derived. 6 In he course of performing such esimaion, i is of course desirable o make he model sufficienly flexible ha is funcional form imposes as few answers as possible. For his purpose i can be useful o employ algorihms ha esimae expendiure funcions numerically based on demand parameer esimaes (Varia, 1983). A maor pracical difficuly in measuring he excess burden of a single ax, or of a sysem of axes, is ha excess burden is a funcion of demand ineracions ha are poenially very difficul o measure. For example, a ax on labor income is expeced o affec hours worked, bu 6 Examples of such esimaion sraegies include Deaon and Muellbauer (1980), Gallan (1981), and Jorgenson, Lau and Soker (1982). Hausman and Newey (1995) offer a nonparameric alernaive. 11

15 may also affec he accumulaion of human capial, he inensiy wih which people work, he iming of reiremen, and he exen o which compensaion akes ax-favored (e.g., pensions, healh insurance, and workplace ameniies) in place of ax-disfavored (e.g., wage) form. In order o esimae he excess burden of a labor income ax, i is in principle necessary o esimae he effec of he ax on hese and oher decision margins. Analogous complicaions are associaed wih esimaing he excess burdens of mos oher axes. In pracice, i can be very difficul o obain reliable esimaes of he impac of axaion on us one of hese variables. I is in reacion o he complicaed naure of he problem of separaely esimaing he effec of axaion on all of a axpayer s decision margins ha a number of recen papers esimae varians of (2.5) in which he dependen variable is axable income. The usefulness of his formulaion is eviden from considering he consumer s problem in maximizing (2.6) U x, x, x, ), ( l in which x 1, x 2, and x 3 are commodiies axed o differing degrees, and l is leisure. In order o illusrae he issues involved, we consider he case in which good 1 is an ordinary commodiy ha consumers purchase ou of afer-ax income, purchases of good 2 are fully deduced from axable income, and purchases of good 3 are parially deducible for ax purposes. Given a labor endowmen of L ~, a wage of w, and facing a (fla-rae, for purposes of simpliciy) labor income ax rae of J, he consumer s budge consrain is ~ +, (2.7) p x p x ( τ) + p x ( 1 ατ ) + w( 1 τ) l w( 1 )L τ in which " denoes he degree o which purchases of x 3 are deducible for ax purposes. Feldsein (1999) noes ha he budge consrain (here, 2.7) can be ransformed o yield a varian of 12

16 (2.8) p x 1 1 ( 1 α) + 1 τ p 3 x 3 ~ w( L l) p x 2 2 αp x. 3 3 ~ The righ side of (2.8) equals axable income, since labor effor is given by ( L l), purchases of commodiy 2 are deducible from income, and a fracion " of purchases of commodiy 3 is also deducible. In his environmen, higher labor income ax raes creae deadweigh loss by discouraging consumpion of good 1, and parially discouraging consumpion of good 3, relaive o consumpion of leisure and of good 2. I is herefore possible o esimae deadweigh loss by esimaing he responsiveness of axable income o changes in ax raes, since doing so races he effec of changes in J on he numeraor of he lef side of (2.8). Several empirical sudies, including Lindsey (1987), Feldsein (1995), Auen and Carroll (1999), Goolsbee (2000), and Moffi and Wilhelm (2000), consider he responsiveness of axable income o ax raes, relying on maor U.S. ax changes o provide variaion in ax raes. The American ax reforms of 1981 and 1986 significanly reduced marginal ax raes, paricularly hose of high-income axpayers, while ax reforms enaced in 1990 and 1993 had he opposie effec of raising ax raes on high-income axpayers. The evidence indicaes ha axable income is generally very responsive o ax changes, wih esimaed response elasiciies ha significanly exceed he ypically very modes esimaed effecs of axaion on numbers of hours worked. Lindsey and Feldsein repor elasiciies of axable income in excess of uniy, while Auen and Carroll, Goolsbee, and Moffi and Wilhelm provide a range of somewha more modes esimaes. All of hese sudies repor ha he axable incomes of high-income axpayers are far more responsive o ax rae changes han are he axable incomes of he res of he populaion. 13

17 There are wo imporan consideraions in inerpreing his evidence. The firs is ha, in order o use he framework described by (2.7) as he basis of analysis, i is imporan o esimae he responsiveness o axaion of he presen value of axable income. Tax avoidance ofen akes he form of deferring a ax obligaion from one period ino anoher in order o reduce is presen value. Consequenly, he reacion of shor-erm axable income o a ax change may exceed he reacion of he presen value of axable income, which Goolsbee (2000) finds occurred wih execuive compensaion in response o he 1993 U.S. ax change. In addiion o he difficuly of disinguishing empirically shor-erm from long-erm reacions, here is he added complicaion ha iming behavior depends on anicipaed fuure ax policies ha may no be known o he analys. The second consideraion is ha ax changes ha reduce one ype of axable income may have offseing or reinforcing effecs on oher sources of axable income. For example, increasing he personal income ax rae may encourage some high-income axpayers o incorporae heir personal businesses, hereby reducing oal income earned by individuals hrough proprieorships while increasing corporae income. A simple calculaion of he responsiveness of personal income o changes in personal income ax raes would hen oversae he rue effec of ax changes on oal axable income. Furhermore, individuals purchase commodiies ha are axed o differing degrees, and ax collecions from hese sources are appropriaely included in reacions o ax changes. 7 Properly accouning for all of hese reacions when performing welfare analysis is a dauning ask, bu one ha is more likely han many of he available alernaives o provide useful answers. 7 Noe ha (2.7) would be unchanged if expendiures on commodiy 3 were nondeducible, bu purchases of commodiy 3 were subec o an ad valorem ax a rae (-"J). As a general maer, however, preexising disorions due o axes, imperfec compeiion, and oher sources of divergence beween price and marginal cos should be incorporaed in measuring deadweigh loss. 14

18 3. The design of opimal axes Taxes (oher han lump-sum axes) disor behavior, ye sociey needs o collec revenue o pursue various social obecives. The opimal axaion lieraure idenifies ax sysems ha minimize he excess burden of axaion, subec o various resricions on ax insrumens and informaion available o he governmen, and under differen assumpions abou populaion heerogeneiy and he funcioning of privae markes. Hisorically, here are hree srands in he developmen of he opimal axaion lieraure. One, iniiaed by he seminal work of Ramsey (1927) and carried on, perhaps mos noably, by Diamond and Mirrlees (1971), concenraes on he design of commodiy axes. A second se of conribuions, beginning wih Mirrlees (1971), considers more general nonlinear income axes and focuses on he role of such axes in addressing disribuional concerns. Finally, he work of Pigou (1947) and ohers analyzes he use of axes o address wo ypes of marke failures: financing public goods no provided by he privae secor, and correcing exernaliies associaed wih incomplee privae secor markes. 8 Alhough hese hree srands in he lieraure have converged, i is sill useful o consider hem separaely in urn before discussing heir inerrelaionship The Ramsey ax problem The simples version of he Ramsey ax problem absracs from populaion heerogeneiy and posis ha he governmen mus raise a fixed sum of ax revenue wih proporional 8 One poenially imporan marke failure no considered by his chaper is he incompleeness of markes in saeconingen claims ha migh oherwise be used o diversify risks. In such a seing, i is possible for axaion o improve welfare simply by reducing (afer-ax) privae reurns since he governmen can pool risks hrough is ax and spending acions. Diamond, Helms and Mirrlees (1980), Varian (1980), and Eaon and Rosen (1980) analyze he properies of opimal disorionary axaion in sochasic seings wih missing sae-coningen markes, while Sandmo (1985) provides a more general survey of he impac of axaion in seings characerized by risk. 15

19 commodiy axes, leaving o he side how such revenue is o be spen. Wih a populaion of idenical individuals, ypically analyzed as a single represenaive individual, he goal of opimal ax design is o minimize he excess burden associaed wih raising he needed revenue. We ypically raionalize governmen s inabiliy o use lump-sum axes by saying ha such axes are inequiable, alhough his may seem a bi forced in a seing wih idenical individuals. I may help o hink of his simple problem as a necessary building block, raher han as one ha adequaely models a realisic siuaion. The represenaive consumer maximizes uiliy, U(x), over a vecor of commodiies x i (i = 0,1,..., N), subec o he budge consrain p x y, where p is he corresponding vecor of consumer prices and y is lump-sum income. To raise he required level of revenue, R, he governmen imposes a vecor of axes on he commodiies,, driving a wedge beween consumer prices and producer prices, q. I is useful o assume iniially ha his vecor of producer prices is fixed (perhaps by world prices), bu as will be seen laer, his is no a resricive assumpion in characerizing he opimum. Wih given producer prices, he governmen in seing ax raes is effecively choosing he consumer price vecor, since p=q+. Thus, he governmen s opimal ax problem can be modeled as (3.1) max V ( p, y), subec o ( p q) x R p where V( ) is he household s indirec uiliy funcion. To see he relaionship beween he opimal ax problem and he problem of excess burden, noe ha he problem in (3.1) is equivalen o (3.2) min y E( q, V ( p, y)) R, subec o ( p q) x R p 16

20 because y and R are consans and E(q,V(p,y)) is monoonically increasing in V(p,y). Bu, as y E(p,V(p,y)), expression (3.2) amouns o minimizing he excess burden of axaion subec o he revenue consrain, in which excess burden is evaluaed a he uiliy level V(p,y) ha holds in he presence of axaion (ha based on he Hicksian equivalen variaion 9 ). Wihou furher resricions, he opimal ax problem is acually quie rivial, since excess burden can be avoided enirely simply by raising all prices by a uniform muliple. Tha is, le p=φq, wih φ>1 chosen so ha (φ 1)qx=R. Then excess burden is E( φq, V ( φq, y)) E( q, V ( φq, y)) ( φ 1) q x( φq, y) (3.3) = φe ( q, V ( φq, y)) E( q, V ( φq, y)) ( φ 1) q x( φq, y) = ( φ 1) q x( φq, y) ( φ 1) q x( φq, y) = 0 where he second sep follows from he fac ha he expendiure funcion is homogeneous of degree 1 wih respec o prices, and he hird sep from he ideniy E(q,V(φq,y)) qx c (φq,v(φq,y)) = qx(φq,y). Raising revenue in his way enails no excess burden because i is equivalen o imposing a lump-sum ax; he household s budge consrain in he presence of uniform axaion is (3.4) φ q x = y q x = y ( φ 1) y / φ 9 This measure of excess burden based on he equivalen variaion may be used more generally o compare any wo ax sysems, neiher of which is necessarily opimal. This propery has led some (e.g. Kay 1980) o prefer is use over measures based on oher reference-uiliy levels. 17

21 Thus, i is necessary o impose axes ha creae excess burden only if i is impossible o adus he ax raes freely on all N+1 commodiies, or else if exogenous income y=0, in which case uniform axes raise no revenue. 10 Wha does i mean for consumers o have no exogenous income? The inerpreaion of he condiion ha y=0 depends on he definiion of commodiies x. Consider, for example, he simple case of hree commodiies, including wo ha he household purchases, x 1 and x 2, and a hird, labor, ha he household supplies as a facor o he producion process. I is cusomary o wrie he budge consrain for his problem as (3.5) p1x1 + p2 x2 + wl = wl ~ where l is leisure consumed and ~ L is he household s ime endowmen. Households divide heir ime beween leisure and working a a wage of w per uni of working ime. Wih he budge consrain wrien his way, i is clear ha a uniform ax on consumpion and leisure is equivalen o a lump-sum ax on he household s ime endowmen. I is sandard o rule his ou by ~ specifying ha leisure canno be axed, ha he governmen resriced o axing labor, L = L l Wih such a resricion, if leisure is axed, he governmen mus offer a maching subsidy o he ime endowmen, a requiremen ha eliminaes he possibiliy of lump-sum axaion. Tha is (3.5) can be rewrien as. ~ (3.6) p x + p x + w( l L ) = p x + p x wl = in which i is clear ha uniform axes on x 1, x 2, and L raise no revenue. This resul may seem counerinuiive because he ax on he household s leisure purchases raises he price of labor, 10 Noe ha if y<0, i is possible o raise revenue wih uniform axaion by choosing φ<1. 18

22 corresponding o wha we normally hink of as a wage subsidy. I is possible o raise revenue by lowering he wage while raising prices p 1 and p 2, bu his no longer leaves relaive prices undisored i lowers he real wage in erms of each consumpion good. Indeed, a labor income ax and a uniform ax on he wo consumpion goods are equivalen ax policies. Wih he budge consrain expressed as (3.7) p1x1 + p2x2 = wl, i is clear ha raising commodiy prices is he same policy as reducing wages. Thus, he need o use disorionary axes resuls eiher from a resricion on he use of ax insrumens (e.g., i is no possible o ax leisure, or he consumpion of any oher endowed commodiy, separaely from is endowmen) or on he absence of exogenous income (if labor, raher han leisure, is he relevan commodiy). Because i is sandard o assume ha he governmen canno impose separae axes on endowmens in labor or oher commodiies, 11 i is easier o adop he second inerpreaion, expressing commodiies as flows beween he household and producion secors and leaving only pure economic ren poenially on he righ side of he budge consrain. Wih no lump-sum income, wo ax sysems are equivalen if hey differ by proporional axes on all commodiies. Wihou lump-sum income one is herefore free o normalize one of he axes, say on good 0, o zero, and for convenience choose he same good as numeraire, i.e., q 0 = p 0 =1. The maximizaion problem in (3.1), wih he muliplier µ associaed wih he budge consrain, yields N firs-order condiions: 11 I is cusomary simply o assume ha he governmen canno ax an individual s labor endowmen because his endowmen is no observable; equivalenly, we assume ha we can observe an individual s labor income, bu no he effor expended or leisure forgone in earning ha income. Alhough here has been some work considering modificaions of his assumpion (e.g., Sern 1982), his issue has received relaively lile aenion in he lieraure. 19

23 (3.8) λ x + µ x + i i dx dp = 0 i = 1,..., N i in which λ V ( p, y) y is he marginal uiliy of income. Making use of he Slusky decomposiion, (3.8) implies ( µ α) (3.9) S = x i = 1,..., N i i µ where S i is he i h elemen of he Slusky marix S d x c dp and α = λ + µ dx dy is he social marginal uiliy of income ha includes he value of he addiional ax revenue raised when he household receives anoher uni of income. 12 Alhough here is no independen condiion for good 0, i may be shown (see Auerbach 1985) ha he N firs-order condiions in (3.9) imply a comparable condiion for good 0, a resul ha should no be oo surprising given ha he choice of he good o bear he zero ax is arbirary. Sacking hese N+1 condiions yields µ α (3.10) S = x µ Premuliplying boh sides of (3.10) by he ax vecor, we obain an equaion in which he lef side is a negaive semi-definie quadraic form and he righ side equals he produc of he 12 Samuelson (1951) uses he symmery of he Slusky marix (S i = S i ) o inerpre (3.9) as implying ha opimal axes enail equiproporionae compensaed reducions in demands for all commodiies. While valid locally, his inerpreaion relies on consancy of he elemens of he Slusky marix as ax raes change, a feaure hey do no generally exhibi. 20

24 consan erm (µ α)/µ and ax revenue x. 13 Thus, if revenue is posiive, µ α he marginal social cos of raising addiional revenue, µ, is a leas as large as he cos of raising revenue in lump-sum fashion, α, i.e., marginal excess burden is nonnegaive. This condiion does no hold for arbirary ax schedules, bu saring from an opimal ax sysem for any given level of revenue means ha here is no opporuniy o reduce excess burden while raising axes, for example by bringing up he ax raes on goods ha iniially are underaxed. 14 Noe ha his inequaliy relaes µ o α, no o λ, he privae marginal uiliy of income. By he definiion of α, µ α µ λ only if revenue is nondecreasing in income, i.e., if he ax base is a normal composie good. This disincion is imporan o keep in mind when considering he lieraure ha seeks o idenify he marginal cos of funds. Before inerpreing expression (3.10) furher, i is useful o consider he more general case of variable producer prices Changing producer prices Since he excess burden of a ax is a funcion of he exen o which he ax changes producer prices, i follows inuiively ha allowing producer prices o vary alers he firs-order condiions for he opimal ax schedule. Le he general producion be characerized by (3.11) f(z) 0 where z is he producion vecor and perfec compeiion insures ha q i /q = f i /f i,. Wihou loss of generaliy, he unis of he producion funcion can be chosen such ha q i = f i. If here are 13 Because he firs elemen of he ax vecor is zero, he relevan par of he Slusky marix is he submarix formed by sriking he firs row and column of S. This submarix and he associaed quadraic form will generally be negaive definie, as long as some of he omied subsiuion erms are nonzero. 21

25 consan reurns o scale, hen f( ) is homogeneous of degree zero in z. Oherwise, here may be pure profis, y= qz> 0. Wih changing producer prices, i is no appropriae o specify he consrain in he opimal ax problem as a scalar value of ax revenue o be colleced, so i is necessary o posi ha he governmen absorbs a vecor R of commodiies. This implies ha he consumpion vecor x saisfies f(x+r) 0, hereby incorporaing boh revenue and producion consrains. The opimal ax problem, hen, is o maximize he indirec uiliy funcion V(p,y) subec o his consrain, and no ha given in (3.2). The associaed Lagrangean expression is (3.12) V(p,y) µf(x+r) and he governmen s problem is sill ha of choosing he consumer price vecor p, raher han he ax vecor, even hough he relaionship beween changes in he wo vecors is more complicaed han when producer prices are fixed. 15 The resuling firs-order condiions are (using he normalized form of producion funcion) dy (3.13) λx + λ + µ i dp i q dx dp = 0 i = 1,..., N i Differeniaing he household s budge consrain px= y wih respec o p i yields (3.14) x p dx dy + i N i = 0 = 1,..., dp dp i i 14 Noe ha marginal excess burden is nonposiive when revenue is iniially negaive, because raising revenue means reducing he level of disorions caused by subsidies. 22

26 and adding he lef side of his equaion o he expression inside he brackes in (3.13) yields dy dx dy (3.15) λx + λ + µ x + dp dp dp = 0 i = 1,..., N i i i i i Since producer prices, and hence profis, change wih p, he derivaive dx /dp i in (3.15) includes he indirec effec of p i on profis hrough changes in producion: (3.16) dx dp i x dx dy = + p dy dp i i Using his and he Slusky decomposiion, (3.15) can be rewrien as ( µ α) dy (3.17) S i = xi i = 1,..., N dp µ i which differs from expression (3.9), he firs-order condiion in he case of fixed producer prices, by he erm dy/dp i on he righ side. Thus, if here are consan reurns o scale (y 0), he firsorder condiions are idenical (Diamond and Mirrlees 1971). The same is rue if he governmen imposes a pure profis ax, so ha he afer-ax value of y accruing o households is uniformly zero (Sigliz and Dasgupa 1971). From expression (2.5), he lef side of (3.17) equals he marginal excess burden associaed wih an increase in p i. The second erm on he righ side of (3.17) is he ne 15 As discussed in Auerbach (1985), dp/d=[i-hs] -1, where H is he Hessian of f( ), so here is a one-o-one relaionship beween changes in and changes in p as long as [I-HS] is of full rank. 23

27 compensaion required o mainain he individual s uiliy as p i rises 16 which, by definiion, exceeds he marginal revenue raised by he marginal excess burden induced by he price change. Thus, (3.17) says ha he excess burden of a marginal increase in any ax mus be proporional o he sum of marginal revenue plus marginal excess burden, or: (3.18) d EB = dp ( µ α) dr d EB + µ dp dp i i i i = 1,..., N I follows ha he marginal excess burden per dollar of revenue raised, (µ α)/α, is also consan, (3.19) d EB = dp i ( µ α) dr α dp i i = 1,..., N which is an inuiive condiion for minimizing he oal excess burden induced by raising a given amoun of revenue from alernaive sources The srucure of opimal axes The opimal ax rules us derived generally do no imply ha he governmen should impose axes a uniform raes, even in he simple case in which producer prices are fixed. For example, consider he hree-good case, in which he wo firs-order condiions (3.9) yield (3.20) 1 2 S x S x = + S x + S x dv ( p, y) / dp 16 This erm equals i ; according o Roy s ideniy, his equals he ne increase in income required o dv ( p, y) / dy mainain he household s uiliy level as p i increases. 24

28 which, using he fac ha Σ p i S i = 0, and defining θ i i /p i as he ax rae on good i, may be rewrien as (3.21) θ1 θ 2 ε + ε + ε = ε + ε + ε where ε i is he compensaed cross-price elasiciy of demand for good i wih respec o he price of good. This expression indicaes ha wo goods should be axed a equal raes (i.e., θ 1 = θ 2 ) if and only if he goods are equally complemenary wih respec o he unaxed good 0. The inuiion someimes offered for his resul comes from he case in which he unaxed good 0 is labor, making i desirable o ax more heavily he good ha is more complemenary wih leisure because i is impossible o ax leisure direcly. Bu since expression (3.20) would also apply if a consumpion good were chosen o bear he zero ax, i may be more accurae o say ha complemens o unaxed goods are axed more heavily o achieve reducions in he unaxed goods wihou axing hem direcly. In he special case of zero cross-elasiciies among all axed goods, he firs-order condiions (3.9) yield he inverse elasiciy rule ha θ i 1/ε i, since in his case each good s demand responds only o is own ax, so achieving a reducion of equal proporion means keeping θ i ε i consan An example Suppose ha household preferences over goods and leisure are described by he Sone- Geary uiliy funcion, 25

29 (3.22) U ( x, x, l) = ( x a ) ( x a ) l β 1 β 2 1 β 1 β For his uiliy funcion, he cross elasiciy ε i0 equals (1 β 1 β 2 )(1 a i /x i ), so opimal axes fall more heavily on he consumpion good whose basic need a i represens a larger porion of oal consumpion x i. In erms of underlying preferences, i can be shown ha his is equivalen o axing more heavily he good wih he higher value of p i a i /β i, he good for which expendiures on basic needs are a greaer fracion of he good s discreionary budge share, β i. In he special case where a 1 = a 2 = 0, he Sone-Geary uiliy funcion collapses o he Cobb-Douglas funcion, and uniform axes are opimal. The Cobb-Douglas uiliy funcion is separable ino goods and leisure (or, o be more exac, ino he axed and unaxed commodiies) and homogenous in goods i can be wrien in he form U(φ(x),l), where φ( ) is a homogeneous funcion. This homoheic separabiliy is a sufficien condiion for uniform axaion (Akinson and Sigliz 1972). Separabiliy alone does no suffice as he general Sone-Geary example illusraes The producion efficiency heorem All of he ax insrumens considered so far are proporional axes on ransacions beween he household secor and he producion secor. Producion iself is assumed o face no disorions, and perfec compeiion ensures ha he economy achieves a poin on he producion fronier. However, he governmen has access o policies ha disor producion while raising revenue, eiher hrough explici axes or hrough governmen producion schemes ha allocae inpus and oupus on he basis of crieria possibly differen han hose used by he privae secor. One migh hink ha such policy insrumens would favorably augmen he governmen s opions, bu his may well no be so. 26

30 Consider he case in which here is a second producion secor, say conrolled direcly by he governmen, wih producion funcion g( ) and producion vecor s, wih he producion se defined by g(s) 0. Disorions beween he wo secors occur implicily hrough he governmen s choice of he vecor s, wih each secor, bu no necessarily he wo secors in combinaion, assumed o be on is own producion fronier. Furher assume ha producion in boh secors is subec o consan reurns o scale. Because privae producion now equals he difference beween purchases x+r and governmen producion s, he governmen s problem is o maximize V(p,y) subec o f(x+r s) 0 and g(s) 0. Forming he Lagrangean as before, wih he muliplier ζ associaed wih he second secor s producion, we obain he same firs-order condiions as before wih respec o p, and he condiions ha µf i ζg i =0 i wih respec o he vecor s. This implies ha all marginal raes of subsiuion in producion should be equal, f i /f = g i /g, i.e., producion should no be disored. This resul does no hold if here are pure profis received by he household, and his helps provide insigh ino why i does hold when no such profis are received. In his special case, all household decisions are based on he relaive price vecor p. I is possible o bring abou any configuraion of his vecor ha is consisen wih he revenue consrain, wihou resoring o producion disorions. Thus, producion disorions can serve only o reproduce wha can already be achieved, bu wih he addiional social cos of los producion. Of course, if he governmen is no free o adus all relaive prices direcly, i may find producion disorions useful, and poliical realiies may ofen dicae such an indirec policy Disribuional consideraions The rules derived hus far apply o he case of idenical individuals, bu heerogeneiy wih respec o ase and abiliy is an imporan consideraion. Taking accoun of individual 27

31 differences in a populaion of H individuals means replacing he indirec uiliy funcion of he represenaive individual, V(p,y), wih a social welfare funcion, W(V 1 (p,y 1 ),...V H (p,y H )). Wih eiher fixed producer prices or consan reurns o scale, here is no lump-sum income y h and social welfare is sill simply a funcion of he price vecor p. This has he immediae implicaion ha he producion efficiency heorem us derived sill holds, because here is no scope for improving social welfare once he price vecor is esablished hrough he opimal ax vecor. However, he shape of he social welfare funcion influences he choice of iself. The firs-order condiions corresponding o maximizing his social welfare funcion subec o he revenue consrain in (3.1) are analogous o hose in (3.8): h dx h h (3.23) W x + x + dp = i = hλ i µ i 0 1,..., N h h i where W h is he parial derivaive of W wih respec o he uiliy of individual h, λ h is individual h s marginal uiliy of income, and x i h is individual h s consumpion of good i. Again defining h h h h α W λ + µ dx dy as individual h s social marginal uiliy of income, (3.23) can be h expressed in more compac form (Diamond 1975) as ( µ α ~ i ) (3.24) Si = xi i = 1,..., N µ where S i h = S is an aggregaion of comparable erms from individual Slusky marices and h i h (3.25) α ~ x i i α h x i h 28

32 is he social marginal uiliy of income aken from households via a ax on good i. I is higher, he greaer he share of he ax burden borne by individuals wih a high social marginal uiliy of income, which is ypically hough o be hose of lower income. Equaion (3.24) is easy o undersand by reference o (3.18), which sill holds in his case, for α ~ i in place of α. Now, he marginal excess burden, raher han being equal for each source of funds, should be reduced for hose commodiies for which he associaed loss in real income is cosly ( α ~ i is high). Because he ulimae obecive is o equalize µ across sources of revenue, hose wih higher disribuional coss should have lower efficiency coss. To illusrae his rade-off beween equiy and efficiency in he choice of ax srucure, consider again he hree-good case in which wo consumpion goods are axed. Now, he raio of he ax raes on he wo goods should saisfy (3.26) θ1 θ 2 πε + πε + π ε = π ε + πε + π ε where π i (µ α ~ i )/µ. Here, θ 1 >θ 2 if and only if ε 10 /ε 20 < π 1 /π 2. If he good mos complemenary wih leisure is also he good wih he greaer social valuaion α ~ i, i is no clear which good will be axed more heavily he answer depends in par on he srengh of disribuional preferences. If preferences saisfy he resricion of homoheic separabiliy menioned above in secion 3.4, i will sill be rue ha commodiy axes should be uniform (as long as preferences over consumpion are he same across individuals). When preferences ake his form, Engel curves (relaing consumpion o income) are linear and pass hrough he origin. Thus, here will be no variaion in he relaive budge shares of differen goods among individuals of differen 29