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.
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 94720-3880 Ann Arbor, MI 48109-1234 auerbach@econ.berkeley.edu rhines@umich.edu
Table of Conens 1. Inroducion... 1 1.1. Ouline of he chaper... 2 2. The heory of excess burden... 2 2.1. Basic definiions... 2 2.2. Variaions in producer prices... 7 2.3. Empirical issues in he measuremen of excess burden... 10 3. The design of opimal axes... 15 3.1. The Ramsey ax problem... 15 3.2. Changing producer prices... 21 3.3. The srucure of opimal axes... 24 3.4. An example... 25 3.5. The producion efficiency heorem... 26 3.7. Disribuional consideraions... 27 4. Income axaion... 30 4.1. Linear income axaion... 30 4.2. Nonlinear income axaion: inroducion... 34 4.3. Nonlinear income axaion: graphical exposiion... 36 4.4. Nonlinear income axaion: mahemaical derivaion... 39 5. Exernaliies, public goods, and he marginal cos of funds... 46 5.1. The provision of public goods and he marginal cos of public funds... 47 5.2. Exernaliies and he double-dividend hypohesis... 51 5.3. Disribuional consideraions and he MCPF... 54 6. Opimal axaion and imperfec compeiion... 57 6.1. Opimal commodiy axaion wih Courno compeiion.... 57 6.2. Specific and ad valorem axaion.... 63 6. 3. Free enry... 67 6.4. Differeniaed producs... 72 7. Ineremporal axaion... 75 7.1. Basic capial income axaion: inroducion... 76 7.2. The seady sae... 78 7.3. Inerpreing he soluion... 79 7.4. Human capial accumulaion and endogenous growh... 82 7.5. Resuls from life-cycle models... 90 8. Conclusions... 94
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.
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
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
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 2.1. 2 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
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 1 0 1 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
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
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. 2.2. 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
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 ( 0 1 1 1 1 1 1 0 1 0 1 1 1 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
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
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
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
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, ), ( 1 2 3 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 1 1 2 2 1 3 3 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
(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
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
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. 3.1. 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
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
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
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 = 1 1 2 2 1 1 2 2 0 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
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
(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
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. 3.2. 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
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
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
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. 3.3. 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 22 1 12 2 11 2 21 1 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
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 20 21 12 10 21 12 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. 3.4. An example Suppose ha household preferences over goods and leisure are described by he Sone- Geary uiliy funcion, 25
(3.22) U ( x, x, l) = ( x a ) ( x a ) l β 1 β 2 1 β 1 β 12 1 2 1 1 2 2 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. 3.5. 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
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. 3.7. 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
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
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 1 20 1 21 2 12 2 10 1 21 2 12 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
abiliies, and hence nohing o gained from a disribuional perspecive by imposing differenial axaion; his leaves he opimaliy of uniform axaion undisurbed. An insance in which disribuional preferences necessarily work in he opposie direcion of minimizing excess burden is ha in which he social welfare is he sum of individual uiliies and individuals have idenical Sone-Geary uiliy funcions of he ype considered in he example above, differing only wih respec o abiliy (as measured by he wages received per uni of labor supplied). To see his, noe firs ha he ordinary demand funcions x i (p,y) are linear in income. Thus, he change in ax revenue generaed when a household changes is consumpion in response o receiving a dollar of income is consan across households. This implies ha differences in α ~ i arise only from differences in consumpion paerns of households wih differing social marginal uiliies of income (W h λ h =λ h ). Nex, noe ha he derivaive of good-i c consumpion wih respec o household uiliy is dx ( p, U ) / du = ( x a ) / U, so ha he elasiciy of x i wih respec o U is (x i a i )/x i. Thus, he good wih he higher elasiciy of consumpion wih respec o uiliy he good more concenraed among higher-uiliy individuals and hence wih he lower value of α ~ i is he good wih he lower value of a i relaive o x i and herefore has a higher demand cross-elasiciy wih respec o leisure. Thus, he good ha is desirable o ax more heavily for disribuional reasons is also he good ha is desirable o ax less heavily for efficiency reasons. i i i 4. Income axaion 4.1. Linear income axaion In analyzing axes on a represenaive individual, i was convenien o side sep he quesion of why he governmen migh no be able o use lump-sum axes. Wih populaion 30
heerogeneiy now an explici aspec of he analysis, i is appropriae o revisi his quesion. In pracice, governmens include uniform lump-sum axes among heir ax insrumens. Indeed, he use of lump-sum axes permis he inroducion of he mos rudimenary of progressive income axes, he linear income ax. For example, in he hree-good case considered earlier, wih he household s budge consrain given by (3.7) and suiably modified by inroducing a lump-sum ax and choosing one of he consumpion goods (good 1) as he unaxed numeraire commodiy, he household faces he budge consrain: (4.1) q x 1 1 + q2 w x2 T L wl T wl 1 = + = ( + τ ) θ ( 1 θ ) 2 0 where τ = θ 0 /(1 θ 0 ) is he household s marginal income ax rae. As (4.1) shows, he governmen has he opion of using differenial commodiy axaion o supplemen he linear income ax schedule. This leads immediaely o wo quesions. Firs, when will he governmen wish o use he commodiy ax θ 2 or, for he case of several commodiies 1,..., N, he commodiy axes θ 2,..., θ N? Second, under wha condiions will he income ax be progressive, wih average ax raes rising wih income (e.g., wih T < 0)? In answer o he firs quesion, a sufficien condiion for he opimaliy of uniform commodiy axes or, equivalenly, axes only on labor income, is ha preferences are weakly separable ino goods and leisure, and ha commodiies have linear Engel curves wih idenical slopes across households (Deaon 1979). 17 Such preferences include he case of homoheic separabiliy, for which Engel curves pass hrough he origin. I is noeworhy ha his condiion is he same as ha required for exac aggregaion of consumers, and ha for an aggregae 17 An example is he Sone-Geary uiliy funcion considered above. 31
measure of excess burden o be independen of he disribuion of resources across consumers. Noe also ha a weaker condiion suffices wih a nonlinear income ax schedule, he design of which is discussed below. In ha case, i is possible o dispense wih he requiremen ha Engel curves be linear, since weak separabiliy of goods and leisure suffices (Akinson and Sigliz 1976). If he governmen axes only labor income, hen equaion (3.25) implies (because purchases of labor are negaive) ha ( µ α ~ 0 ) (4.2) S = L 0 00 ( ) µ where L and S 00 are aggregae measures, wih labor measured in efficiency unis so ha i is possible o aggregae over individuals of differen abiliies. The availabiliy of lump-sum axes adds a marginal condiion ha µ = α, he unweighed average value of α across individuals: since he governmen can use posiive or negaive lump-sum axes a he margin, he marginal cos of funds mus equal he cos of raising funds wih lump-sum axes. Subsiuing his condiion ino (4.2) and rearranging erms yields (4.3) ( ) p ( S ) ( α ~ α) 0 0 00 0 = p L α 0 which, for a household labor price of p 0 = w(1 τ) and 0 = τw (recall ha in his noaion a posiive value of 0 raises he afer-ax wage rae) may be expressed (Dixi and Sandmo 1977) as (4.4) τ ( 1 τ) ( α ~ 0 α) = αε cov L h L = ε h α, α 32
where ε w( 1 τ)( S00 ) / L is he aggregae compensaed labor supply elasiciy (which mus be posiive), L h is household h s labor supply, and L is he average value of L h across households. Since labor is expressed in efficiency unis (a he common wage w), higher abiliy ranslaes, for a given fracion of ime worked, ino higher labor supply. Expression (4.4) says ha he marginal ax rae on labor income is posiive if and only if he marginal social valuaion of income falls as labor supply (in efficiency unis) rises, a condiion ha is me by uiliarian social welfare funcions ogeher wih labor supply schedules ha are increasing in abiliy. The value of marginal ax rae, and wheher i is sufficienly high o make he linear income ax progressive (T < 0), depends on he weigh of he social welfare funcion s redisribuive componen how fas α h declines as L h rises. Properies of he marginal ax rae also depend on he amoun of ax revenue required. To undersand why, consider he case in which he governmen s revenue requiremen is zero. Then i is possible o obain a Pareo opimum by seing he marginal income ax rae, and he lump-sum ax T, o zero. Since he social marginal uiliy of income differs across individuals, and since here is no firs-order excess burden from he inroducion of a small ax, i mus hen be opimal o inroduce some disorion (i.e., a posiive marginal ax rae) o redisribue income from hose wih high incomes and low social marginal uiliy of income o hose wih lower incomes and higher social marginal uiliy of income. Thus, he linear income ax is progressive a zero ne revenue. As he governmen s revenue requiremen rises, holding T consan, he marginal excess burden of raising revenue also rises, and so oo does he cos of redisribuion. As Sigliz (1987) noes, here exiss a poin a which maximum revenue is colleced via marginal ax raes (i.e., he marginal excess burden per dollar of revenue is infinie), a which poin he governmen mus 33
rely on lump-sum axes for addiional revenue. Greaer reliance on lump-sum axes obviously reduces he progressiviy of he ax schedule. Indeed, simulaions confirm ha he lump-sum ransfer falls as revenue rises (Sern 1976), and ha i becomes negaive for sufficienly high revenue requiremens (Slemrod e al. 1994). 4.2. Nonlinear income axaion: inroducion In pracice, governmens use income ax sysems wih muliple marginal ax raes. Alhough he linear income ax us considered can have progressive average ax burdens, is redisribuive poenial is limied by he fac ha he average ax burden mus approach he marginal ax rae asympoically and can rise no higher. Hisorically, many in governmen have fel ha only a schedule of rising marginal ax raes could deliver he appropriae degree of progressiviy oward he op of he income disribuion, and have implemened income ax sysems wih op marginal ax raes in some insances exceeding 90 percen. 18 Governmens cerainly can impose income ax sysems more complicaed han he linear income ax, bu wha should hese sysems look like? As in he case of he linear income ax, he issue involves balancing efficiency and equiy, wih he surprising conclusion ha high and rising marginal ax raes may well no be appropriae even when he governmen has a srong redisribuive moive. A firs, i migh seem ha he abiliy o choose an arbirary income ax funcion T( ) offers he governmen he opporuniy o impose individual-specific lump-sum axes, for he funcion could be chosen o pass hrough values of ax burdens appropriae o individuals a each 18 For example, us prior o he Kennedy-Johnson ax cu of 1964, he op marginal federal income ax rae in he Unied Saes was 91 percen. 34
level of income. However, as is rapidly apparen, he endogeneiy of income srongly limis he governmen s abiliy o impose differenial lump-sum axaion. To begin, suppose ha here is a single consumpion good, ha labor supply is he only source of income, and ha individuals have common preferences U(c,l) over consumpion and leisure, differing only in heir abiliies, as measured by wage raes w. Imagine ha he governmen needs o raise a cerain amoun of revenue, R, using an income ax, and ha i is desirable o assign a lump-sum income ax burden T i o individual i. Wih he consumpion good as numeraire, he problem may be expressed as 1 1 (4.5) 1 2 2 2 H H H h max W( V ( w, T ), V ( w, T ),..., V ( w, T )) subec o T R T h If µ is he Lagrange muliplier associaed wih he revenue consrain, hen he H firs-order condiions are simply ha W h λ h =µ ha he marginal social uiliy of income is he same across all individuals. Wha does his condiion imply for ax burdens? For he uiliarian social welfare funcion W(U 1,...,U H ) = Σ h U h, i implies ha he marginal uiliy of income λ h is consan across individuals, which (from he firs-order condiions for uiliy maximizaion) implies ha he marginal uiliy of consumpion is consan across households, bu ha he marginal uiliy of leisure is proporional o w h. Equaing he marginal social cos of income across individuals, he governmen in effec forces high-wage individuals o work unil hey reach he poin ha leisure is very valuable o hem. In he process, his ax sysem makes high-wage individuals worse off han low-wage individuals, a paradoxical oucome ha is guaraneed if leisure is a normal good. For example, suppose he common uiliy funcion akes he quasi-linear form U(c,l) = c v(1 l), wih v>0 and v>0. Then, wih opimal household-specific axaion, all households 35
have he same level of consumpion, and leisure declines monoonically wih he wage rae. The lowes wage household obains he highes level of uiliy, which illusraes quie clearly he problem o be faced in aemping o implemen such a ax sysem. Aside from he poliical implausibiliy of he oucome, his scheme could be implemened only if governmen knew each household s abiliy level and assigned axes accordingly. Oherwise, all oher households would have incenives simply o masquerade as he household wih he lowes abiliy by supplying he amoun of labor necessary o produce ha household s income level, hereby leaving hemselves beer off han he lowes-abiliy household (because hey forgo less leisure o reach his level of income), raher han worse off. Bu his, in urn, leaves he governmen wih a uniform lumpsum ax and oo lile revenue. While he governmen could respond by increasing he lump-sum ax, i is clear from he previous discussion of he linear income ax ha his policy alone is no likely o be opimal. Raher, he governmen seeks o impose a ax sysem more progressive han he lump-sum ax, while sill accouning for he absence of informaion abou individual ypes and he endogeneiy of household income. A linear income ax is bu one such ax sysem. 4.3. Nonlinear income axaion: graphical exposiion Much of he inuiion behind he design of he opimal nonlinear income ax emerges from consideraion of an income ax imposed on an economy composed of wo individuals, one (H) of high abiliy and one (L) of low abiliy. 19 Because he governmen observes only income, Y=w(1 l), raher han labor supply and wage raes separaely, i is useful o express each individual s preferences over consumpion and leisure (or labor) in erms of preferences over consumpion and income, as depiced in Figure 4.1. On he lef side of he figure is an 19 We follow he mnemonic noaion in he lieraure in denoing he wo abiliy classes as H and L for he following graphical exposiion, bu remind he reader ha he variable L represens labor supply in all oher pars of he chaper. 36
indifference curve over consumpion and leisure, based on he uiliy funcion U(c,l). On he righ are wo corresponding indifference curves for he same level of uiliy bu differen wage raes, based on he same uiliy funcion, U(c,1 y/w). The curve corresponding o he higher wage rae is flaer because a given change in labor ranslaes ino a greaer change in income. This suggess ha when indifference curves of wo individuals do cross, as a poin A, he indifference curve of he higher abiliy individual is flaer. Figure 4.2 illusraes he oucome of aemping o impose he previously-discussed lumpsum ax soluion, wih consumpion equal o c 0 for boh high- and low-abiliy individuals and he higher abiliy ype on a lower indifference curve, as indicaed by he relaive consumpion a zero income (a which abiliy differences are irrelevan). Raher han accep he bundle (c 0,y H ), he high-abiliy household would prefer o earn income y L and receive he same level of consumpion. The problem wih his plan is ha i violaes he self-selecion consrain ha each household prefer is governmen-designaed bundle among he available opions. In his insance, he high-abiliy household prefers he bundle designaed for he low-abiliy household. I is ypically he self-selecion consrain of he high-abiliy person wih which he governmen mus be concerned. As Figure 4.3 illusraes, he self-selecion consrains limi he scope for redisribuion hrough differenial lump-sum axaion. For he sake of exposiion, assume ha he required level of revenue, R, equals zero. Wih no redisribuion, each household s budge consrain has uni slope (since a dollar of income produces a dollar of consumpion) and passes hrough he origin. The high-abiliy and low-abiliy households choose poins H and L, respecively. Each household sricly prefers is own bundle, so neiher self-selecion consrain is binding. As a resul, i is possible o impose a lump-sum ax on H and provide an equal lump-sum ransfer o L 37
unil reaching he poin ha H s self-selecion consrain binds, which occurs a poins H and L. The governmen canno do more wih lump-sum axaion wihou violaing H s self-selecion consrain, bu i can do more. Slopes of he indifference curves of individuals H and L differ a poin L. Because his poin is an opimum for L (since L s indifference curve is angen o he budge line) bu no for H, a sligh movemen in any direcion along he budge line has no firs-order effec on he uiliy of L, bu does have a firs-order effec on he uiliy of H. Moving oward he origin along he budge line makes H worse off, because H already is working inefficienly oo lile a poin L H s indifference curve is flaer han he budge line. This suggess a way o relax H s selfselecion consrain and achieve more redisribuion, as illusraed in Figure 4.4. By shifing individual L from poin L o poin L, he governmen imposes on L only a second-order excess burden (since L is iniially a an undisored poin) bu raises firs-order ax revenue by being able o shif individual H down o poin H. This ax revenue equals he disance CD in Figure 4.4. The exra revenue exraced from H (ne of he amoun disance AB in Figure 4.4 needed o compensae for he small disorion o L s choice) can hen be allocaed beween L and H, wih H receiving us enough o keep he self-selecion consrain saisfied. The final resul is ha L is beer off han a L and H is worse off han a H. The limis ha govern his redisribuion are he governmen s success in carrying i ou (which reduces dispariies in he social valuaion of marginal incomes received by differen households) and by marginal excess burdens ha rise as one moves furher away from he iniial poin L. L s bundle can be hough of as being implemened via a marginal ax rae on L s income ha produces a budge line wih slope less han one. This offers he insigh ha i is opimal o impose a posiive marginal ax rae on individual L no o raise revenue from L, bu o 38
raise revenue from hose wih incomes higher han L s in his case, individual H. A corollary is ha, as here is no one of higher abiliy han H in his example, i is no opimal o impose a marginal ax rae on H s income. Doing so would disor H s behavior and reduce he revenue he governmen could exrac from H wihou violaing H s self-selecion consrain. These lessons are useful in considering he case in which here is a coninuum of agens. 4.4. Nonlinear income axaion: mahemaical derivaion The mahemaics of opimal income axaion wih a coninuum of agens is no sraighforward, because i is no possible o rule ou such phenomena as nondiffereniabiliy of he ax funcion T( ). These phenomena are no simply anomalies. As discussed in Sigliz (1987), nondiffereniabiliy arises in cases in which i is opimal o pool individuals wih differen skill levels a a single poin in (c,y) space. To undersand why, consider he case in which here are many individuals of ype H (as considered above) and an equal number of individuals of ype L. The opimal ax policy is obviously idenical o ha wih one individual of each ype. Then inroduce an addiional individual a some inermediae wage rae beween L and H. If his individual, say M, is offered an allocaion ha H prefers o L s bundle, hen H s self-selecion consrain is violaed. I is possible o mainain individuals of ype H a heir iniial allocaions only by reducing he araciveness of M s bundle. This, in iself, disors he choice of M s bundle, bu if here are many more individuals of ypes H and L han of ype M, sociey gains from doing so unil M s bundle approaches ha of L. In spie of he imporance of his complicaion, i is useful for inuiion o derive resuls for cases in which such problems do no arise. Our approach closely follows ha in Akinson and Sigliz (1980). For furher discussion of he more general mahemaical issues, see Mirrlees (1976, 1986). 39
Coninuing o assume, for simpliciy, ha overall revenue R = 0, he governmen seeks o maximize some general social welfare funcion of individual uiliies, subec o he consrain ha oal consumpion equal oal before-ax income. Leing f(w) be he fracion of he populaion endowed wih wage rae/skill level w, he governmen s obecive is 0 (4.6) max G( U ( w)) f ( w) dw subec o ( c( w) y( w)) f ( w) dw w w where c(w) and y(w) are he levels of consumpion and income chosen by each individual a wage rae w and U(w) is he uiliy of ha individual based on hese values, U(c(w),1 y(w)/w). The opimizaion problem is furher consrained by he requiremen ha wage-w individuals volunarily choose he bundle (c(w),y(w)) he self-selecion consrain discussed above. The requiremen ha he bundle (c(w),y(w)) is individually raional for people of wage w means ha uiliy U(c(w),1 y(w)/w) achieves a maximum a w=w. This may be expressed in erms of he firs-order condiion, (4.7) U c dc U dy dw + y dw = 0 ha indicaes ha he individual canno increase uiliy hrough a local change in labor supply. This hen implies, for common preferences, ha he change in uiliy as he wage rae rises is simply he derivaive of he uiliy funcion wih respec o w, holding c and y fixed: (4.8) du dw U = = w U y = U L 2 2 2 w w Thus, he opimal ax problem is ha expressed in (4.6), subec o he addiional consrain given in (4.8). While i is expressed as one of choosing he bundle (c,y), i can equally well be viewed 40
as a choice of he uiliy level u and he level of labor supply L, as u=u(c,1 L) and y=wl. To solve he problem expressed his way, i is helpful o form he Hamilonian: (4.9) H = [ G( u) µ ( c( L, u) y( L, u))] f ( w) ηu ( L, u) L 2 w wih conrol variable L, sae variable u, Lagrange muliplier µ and cosae variable η. The firsorder condiions are (4.10) (a) H L = 0 (b) H dη = u dw Condiion (4.10a), as applied o (4.9), implies ha (4.11) µ c η + = L u y L u f w U L u L U 2 2 ( ) 0 w w Noe ha y=wl y L u = w and du dl u = 0 = U1 c Lu U 2 c L u = U2 U1. Furher, individual uiliy maximizaion ensures ha U U = w( 1 T ). Thus, (4.11) can be rewrien as 2 1 (4.12) T = U η ψ 1 1 T µ wf ( w) where ψ U L u L 2 U + 1. This expression says ha he opimal marginal ax rae is increasing 2 in (U 1 η/µ) and ψ and decreasing in wf(w). The las of hese effecs is sraighforward: he more 41
effecive labor supply ha is subec o he marginal ax rae a w, he greaer is he excess burden associaed wih ha ax rae. To inerpre he oher wo erms in (4.12) and heir effecs, consider he special case of quasilinear preferences, U(c,l) = c v(1 l) = c v(l), where v( ) is convex. For his case, i may be shown ha ψ=1+1/ε, where ε is he compensaed labor supply elasiciy a w. Thus, a higher labor supply elasiciy leads o a lower value of ψ, which by (4.12) leads o a lower marginal ax rae. This is sensible, as a higher labor supply elasiciy is also associaed wih greaer excess burden per dollar of revenue raised. A similar effec appears in (4.4) for he case of he linear income ax, bu here i is he labor supply elasiciy a he paricular wage rae w, raher han he aggregae labor supply elasiciy, ha is imporan because he governmen is free o choose differen marginal ax raes for differen levels of income. Finally, consider he remaining erm in (4.12), (U 1 η/µ). From he firs-order condiion (4.10b), dη H c y L c (4.13) = = G µ L L f ( w) ηu 21 L dw u u u w u As y u L = 0 and because du du L = 1 dc du L = 1 U1, (4.13) can be rewrien as (4.14) U 21 1 dη G U1 ηu L = 1 f ( w) + µ dw µ. wµ To inerpre his furher, i is again helpful o impose he simplifying assumpion of quasilinear preferences, hereby implying ha U 1 is consan (here normalized o 1) and U 21 =0. Then, inegraing boh sides of (4.14) and imposing he ransversaliy condiion (η 0 as w ) yields 42
U η G w U G w U 1 1 f w dw F w 1 = ( ~ ) ( = f w dw µ µ ~ ) ( ~ ) ~ µ (4.15) 1 ( ~ ) ~ [ 1 ( )] w w where F( ) is he cumulaive densiy funcion based on f( ). 20 This expression equals he social value, scaled by he marginal cos of funds µ, of raising a dollar hrough marginal axaion a wage level w. This value has wo componens. The firs erm is he amoun of revenue raised, equal o he axes colleced from all hose who pay he exra ax hose wih wages raes a leas as high as w. The second erm is he value, again in revenue unis, of he social welfare los by hese individuals in paying he exra ax. Each of hese erms declines wih w, because we collec less revenue and impose less burden by raising axes on fewer people, bu i is he difference beween he erms ha maers. Wha paern does his difference follow? The difference mus be posiive if marginal ax raes are posiive, and he difference converges o zero as w. If Gdeclines wih w, hen he second erm in (4.15) he social cos of an increase in he marginal ax rae a w converges o zero more rapidly han does he firs erm. Hence, here may be a range of w over which he difference beween he wo erms increases. The inuiion is ha high marginal ax raes a high levels of income are very inefficien because hey produce so lile revenue, while high marginal ax raes a low levels of income are inequiable because hey impose burdens on hose wih very high social marginal uiliies of income G. The bes compromise may be o raise marginal ax raes a middle income levels, where ax obligaions are no imposed on hose for whom he burden of higher axes is mos socially cosly bu where higher ax raes sill raise considerable revenue. 20 In recen work, Saez (2000a) derives an analyical expression exending (4.15) o he case of more general preferences. While he offers an inuiive discussion of his expression, i is necessarily more complicaed han he basic inuiion presened here. 43
As should be clear from his discussion, he exac paern ha he erm in (4.15) follows as w rises depends on he social welfare funcion and he shape of he wage disribuion. Even if his erm does indicae higher marginal ax raes somewhere in he middle of he wage disribuion, his is precisely where one of he oher erms in (4.12), wf(w), is also likely o be greaes, which has he effec of reducing T. Thus, i is possible o say very lile abou he general shape of he opimal marginal ax rae schedule, alhough since he seminal work of Mirrlees (1971) here has been a general endency o find ha opimal marginal ax raes should eiher fall hroughou mos of he income disribuion or else have an invered-u shape, reflecing he effec of he erm in (4.15) (see, e.g., Kanbur and Tuomala 1994). This conclusion is in some sense predeermined by findings ha, under cerain circumsances, he opimal marginal ax rae equals zero a boh he op and boom of he income disribuion. The raionale for a zero op marginal ax rae appeared already, in he graphical presenaion of he wo-person case. For he general case wih a bounded disribuion of wage raes, he resul (see Phelps 1973, Sadka 1976 and Seade 1977) follows direcly from he fac ha he erm in (4.15) approaches zero as he wage w approaches is upper suppor, w. As o why he marginal rae migh be zero a he boom of he wage disribuion (see Seade 1977), consider he value of expression (4.15) a he lower suppor of he wage disribuion, say w. As F(w)=0, he expression indicaes ha T/(1 T) 1 α / µ, where α is he average social marginal uiliy of income over he enire disribuion. 21 Bu, as discussed in he case of he linear income ax, α=µ when here is a uniform lump-sum ax available, so Tmus equal zero. The inuiion for his resul follows he algebra. A he very boom of he income disribuion, an increase in he 44
marginal ax rae has he same revenue and disribuional effecs as a uniform lump-sum ax i raises revenue from he enire populaion. Bu i also disors he behavior of he lowes income individuals, which a lump-sum ax does no. Thus, a lump-sum ax dominaes any posiive marginal ax on lowes-wage individuals. However, neiher of hese resuls is robus o reasonable changes in assumpions. As is derivaion suggess, he resul regarding he marginal ax rae a he boom requires ha he enire populaion works. Oherwise, he marginal ax rae applied o he lowes-wage worker does no collec ax revenue from all individuals, and he logic us given breaks down. 22 A he op of he wage disribuion, opimal marginal ax raes need no approach zero, even in he limi, if he wage disribuion is unbounded, nor is he invered-u shape of he marginal ax rae disribuion robus, as demonsraed by Diamond (1998) for he case of a Pareo disribuion of wages and quasilinear preferences. 23 Even for bounded wage disribuions where opimal marginal ax raes mus evenually decline, marginal ax raes may rise over mos of he income disribuion, alhough numerical simulaions of he more resriced opimal wo-bracke linear ax sysem (Slemrod e al. 1994) find ha he second/op marginal rae is lower han he firs. This has quie ineresing implicaions for he recen debae abou he equiy effecs of he fla ax (Hall and Rabushka 21 As here are no income effecs on labor supply for he quasilinear uiliy funcion, i is possible o ignore he indirec effec of income on revenue. 22 A differen deparure from his logic occurs if individuals a he boom end of he income disribuion make discree choices of wheher or no o work, as analyzed by Saez (2000b). In his case, he opimal marginal ax rae on he lowes income is negaive, since he ax sysem hereby induces greaer labor force paricipaion and higher incomes. 23 Diamond finds he opimal marginal ax rae schedule o be u-shaped in he example he analyzes. As clarified by Dahan and Srawczynski (2000), hough, Diamond s resul of a rising marginal ax rae a he op depends on he oin assumpions of an unbounded abiliy disribuion and quasilinear preferences. The resul need no hold, even for he Pareo disribuion of abiliies, if one adops a more general uiliy funcion. For anoher variaion in assumpions, Sigliz (1982) noes ha if he effor of high-skilled workers is an imperfec subsiue for ha of lowskilled workers, i may be opimal o subsidize income a he op of he wage disribuion o increase skilled labor effor and hereby raise he wages of he less skilled. 45
1995), a close relaive of he linear income ax under which ax liabiliies are consrained o be nonnegaive. Alhough some (e.g. Bradford 1986) have suggesed modifying he fla ax o permi addiional, higher marginal ax rae brackes on higher wage individuals, hese simulaion resuls sugges ha adding an addiional bracke should occasion lower, no higher marginal ax raes a higher wage levels. 5. Exernaliies, public goods, and he marginal cos of funds The analysis o his poin ignores he use o which public funds may be pu, oher han redisribuion o oher axpayers. In realiy, of course, a maor reason for raising revenue is o finance public expendiures, and i is imporan o consider how his affecs he conclusions. In urn, i is ineresing o ask how he use of disorionary axaion influences he opimaliy condiions of Samuelson (1954) regarding he provision of public goods. A he same ime, i is convenien o consider how he disorionary naure of axaion alers he prescripions concerning he use of Pigouvian axaion o correc exernaliies. Basic resuls relaing he provision of public goods and he correcion of exernaliies o he use of disorionary axes may be found, respecively, in Akinson and Sern (1974) and Sandmo (1975). Auerbach (1985) presens and inerpres hese resuls in some deail, so we will offer only a brief derivaion here. Boh models assume ha he governmen is limied o he use of indirec proporional axes, and avoid any discussion of disribuion by assuming ha individuals are idenical, i.e., ha he populaion consiss of H copies of he represenaive individual. In his conex, i is naural o assume ha he governmen seeks o maximize he uiliy of each represenaive individual or, equivalenly, he sum of individual uiliies. 46
5.1. The provision of public goods and he marginal cos of public funds Consider firs he case in which he governmen wishes o provide a public good, G, using all is ax revenue. Individuals choose consumpion x reaing G as given, so heir uiliy funcion may be wrien in semi-indirec form as V(p,y; G), wih V / G = U / G x( p, y; G). For simpliciy, he economy s producion funcion f(x,g) (where X = Hx) is aken o obey consan reurns, so ha here are no pure profis and y = 0. This se-up gives rise o he Lagrangean (compare o 3.12): (5.1) HV(p;G) µf(x,g) wih firs-order condiions wih respec o each price and he level of public goods, G. The firsorder condiions wih respec o price are idenical o hose derived above in secion 3 for he case of y = 0, in (3.15), repeaed here for convenience: dx (5.2) λx i + µ X i + = 0 i =1,..., N dpi excep ha X i is now he sum of individual purchases of good i, equivalenly he produc of H and he purchase of he represenaive consumer. The firs-order condiion wih respec o he public good is V X i (5.3) H µ f G + f i = 0. G i G The uiliy funcion implies ha V / G =, in which h U G h U i is individual h s marginal uiliy of good i. The economy s producion consrain and privae producion efficiency impose he 47
condiion ha f i q i, while he consumer s budge consrain implies ha p ' X G = 0. Taking good 0 o be he unaxed numeraire commodiy, and λ o be he marginal uiliy of income, i follows ha U h = λ p = λ, and (5.3) implies 0 0 f 0 h U G µ f G dr (5.4) = h h U λ f 0 dg 0 where R is ax revenue, X, and he variable µ is he shadow cos of he governmen s revenue consrain (measured in unis of uiliy). The raio (µ/λ), which measures he shadow price of revenue unis of he numeraire, is ofen referred o as he marginal cos of public funds (MCPF), because i measures he cos of each uni of public funds, aking accoun of he deadweigh loss from he addiional axes associaed wih hose funds. Expression (5.4) deviaes in wo respecs from he Samuleson rule of equaing he marginal rae of ransformaion, f G /f 0, and he sum of he marginal raes of subsiuion, h U h U h G 0. Firs, i indicaes ha he implici cos of public goods is reduced o he exen ha public spending increases spending on axed commodiies, i.e., dr/dg > 0 a poin noed by Diamond and Mirrlees (1971). Second, i requires ha one adus he relaive price of public goods, f G /f 0, for he MCPF, consisen wih inuiion provided by Pigou (1947). However, as noed by Akinson and Sern, he MCPF as defined need no exceed 1. Recall from secion 3 ha opimal axes ensure ha µ > α, where α = λ + µ dr/dy is he social marginal uiliy of income he value o sociey of giving an individual an exra uni of income, aking accoun of he revenue provided by induced spending on axed goods. However, if dr/dy is negaive, hen i is possible ha he MCPF is equal o or even less han 1. 48
A simple example illusraing his possibiliy is provided by Ballard and Fulleron (1992). Consider he case in which he uiliy funcion is weakly separable ino privae and public goods, so ha dr/dg = 0. Suppose ha here are us wo privae goods, leisure and consumpion, so ha here is us one independen ax insrumen, and normalize his ax insrumen so ha only he ax on labor income is posiive. The firs-order condiion wih respec o he price of labor he wage rae w is, from (5.2), (5.5) λl + µ(l dl/dw) = 0 where L is he aggregae supply of labor and is he ax per uni of labor supplied. 24 Defining η Lw as he uncompensaed labor supply elasiciy and θ as he ax rae /w, (5.5) may be rewrien: (5.6) µ λ 1 = 1 θη Lw from which i is obvious ha he MCPF exceeds 1 if and only if he uncompensaed labor supply elasiciy is posiive. For he benchmark case of Cobb-Douglas preferences, he uncompensaed labor supply elasiciy is zero, and he MCPF = 1. 25 Given ha a zero uncompensaed labor supply elasiciy lies wihin he range of exising esimaes, his resul is no simply a heoreical curiosiy, and suggess ha we may well err in auomaically assuming ha he exisence of disorionary axaion raises he MCPF significanly. 26 24 The erm dl/dw eners in expression (5.5) wih a minus sign because he ax is subraced from he wage. 25 Ballard and Fulleron argue based on an informal survey ha his oucome was generally a surprise o a group of public finance economiss. 26 More generally, if he uiliy funcion is no separable, one may show ha he Samuelson rule holds whenever he supply of labor is unaffeced by he increase in spending on he public good whenever he combined impac on L of he increase in G and he decrease in w equals zero. In his case, he marginal cos of funds as defined in (5.3) is no equal o 1, bu is deviaion from 1 is offse by he dr/dg erm. 49
The reason ha his assumpion has he poenial o go wrong is ha he deadweigh loss of a ax sysem and he MCPF are wo enirely separae conceps. Deadweigh loss is a measure of he poenial gain from replacing disorionary axes wih an efficien lump-sum alernaive, and marginal deadweigh loss is simply he change in his magniude as ax revenue changes. By conras, he MCPF reflecs he welfare cos, in unis of a numeraire commodiy, of raising ax revenue for exhausive governmen expendiure. While his resul seems simple and sraighforward, much has been wrien on he opic of how he MCPF should be defined. Wihou reviewing his exensive lieraure (see, for example, he survey by Håkonsen 1998), we noe ha he disagreemens relae largely o erminology and quesions of normalizaion. As an illusraion (see Schöb, 1997), consider he same example (one public good, labor, and one oher privae good), bu normalize he proporional axes so ha he ax on labor is zero. The firs-order condiion wih respec o he price, p, of he axed commodiy, insead of (5.5), would be (5.7) λx + µ(x + dx/dp) = 0 where X is he aggregae purchase of he commodiy and is he ax per uni of ha commodiy. Defining η Xp as he uncompensaed own-price demand elasiciy and θ as /p, (5.7) can be rewrien as (5.8) µ λ 1 = 1 + θη Xp which says ha he MCPF should exceed 1 if and only if η Xp < 0 i.e., X is no a Giffen good. Since his is a much weaker condiion han ha η Lw > 0, i is easy o see how one migh become 50
confused, given ha hese condiions supposedly reflec he same underlying experimen. Indeed, when η Lw = 0, η Xp = 1, so µ/λ = 1/(1 θ). This apparen paradox is resolved by noing ha he normalizaion does no affec he underlying oucome, bu does change he unis of (µ/λ). In he firs insance, he MCPF is defined in unis of he commodiy; in he second, i is measured in erms of unis of labor. The impac of his difference may be undersood using he sandard approach of cosbenefi analysis (e.g., Harberger 1972), ha weighs he coss of funds according o sources. When he labor supply elasiciy is zero, an increase in he ax on labor has no impac on he amoun of labor supplied. Thus, he exra axes ha finance addiional spending on he public good are absorbed fully hrough reduced consumpion. Hence, he marginal cos of funds equals he marginal value of a uni of he commodiy. Therefore, if he commodiy is chosen as he numeraire, he marginal cos of funds equals 1. If labor is chosen as he numeraire, he marginal cos of funds sill equals 1 uni of he commodiy, bu his equals 1/(1 θ) unis of labor, due o he ax wedge beween labor and privae consumpion. The equilibrium is he same regardless of normalizaion, bu he MCPF is differen. This discussion also highlighs ha he MCPF reflecs only he presence of a disorion on one paricular margin beween he public good and he numeraire. This disorion can be posiive, negaive or zero, independen of he presence of deadweigh loss due o axaion. 5.2. Exernaliies and he double-dividend hypohesis A similar logic applies o he analysis of exernaliies, as in Sandmo (1975). Suppose ha, raher han here being a public good, here is an exernaliy, E, ha eners ino each person s uiliy funcion and which canno be avoided, so ha he represenaive individual s indirec uiliy funcion may be wrien V(p;E). Suppose also, for simpliciy, ha he exernaliy 51
is he produc of aggregae consumpion of a single good, say he good wih he highes index, N. Then, he Lagrangean, (5.9) HV(p;X N ) µf(x) implies he following N firs-order condiions wih respec o he prices of goods 1,, N (compare o 3.8): dx * (5.10) λx i + µ xi + = 0 dpi i = 1,..., N where = * * HV E HV E λ N = N + = N + µ µ λ N Expression (5.10) is he sandard opimal ax soluion, excep ha i calls for he ax on he exernaliy-producing good, N, o equal he sum of he opimal ax ha ignores he exernaliy, * N, plus a erm ha reflecs he cos of he exernaliy. This second erm equals he correcive Pigouvian ax he social cos per uni of consumpion of he good, measured in erms of he numeraire commodiy divided by he MCPF, µ/λ. Thus, in a resul analogous o ha us presened for he provision of public goods, he presence of disorionary axaion leads o undercorrecion of he exernaliy if and only if he MCPF exceeds 1. As before, hough, one mus exercise care in inerpreing his resul. Suppose, following he previous example, ha he exernaliy eners he uiliy funcion in a separable manner, and ha preferences over direc consumpion of goods and leisure are Cobb-Douglas. 52
Also assume ha here are us wo consumpion goods, a clean good and a diry good ha causes he exernaliy. Absen he exernaliy (and if various regulariy condiions are saisfied), he opimal ax srucure calls for equal axes on he wo consumpion goods, i.e., = * * 1 2. This can be achieved eiher hrough a ax on wages alone or hrough uniform axes on he wo consumpion goods. In he firs case, leing he clean good be numeraire, i is clear ha µ/λ =1, so he Pigouvian ax should be implemened wihou adusmen. In he second case, leing labor be numeraire, µ/λ = 1/(1 θ) >1, so i is necessary o undercorrec for he exernaliy. I is emping o conclude in he laer case ha one undercorrecs because he correcive ax is piled on op of he preexising consumpion ax, while in he former case no iniial preexising consumpion ax exiss. However, he wo equilibria are idenical, wih he same disorions presen on all margins. 27 Thus, he inuiion is misleading. While here is no iniial consumpion ax when only labor is axed, here is sill a disorion of he labor-leisure choice. Taxing he diry consumpion good exacerbaes he disorion beween ha good and labor, us as if he iniial ax were on he wo consumpion goods insead. The fac ha i is overall disorions ha maer, and no he levels of individual axes, also exposes a serious inerpreive difficuly in wha is known as he double-dividend hypohesis. This hypohesis, as discussed in much more deail in he chaper in his Handbook by Bovenberg and Goulder, saes ha correcive axes have an added benefi in he presence of oher disorionary axes he revenue ha allows a reducion in he oher ax raes and heir associaed deadweigh loss. Correcive axes do no merely raise revenue and correc exernaliies, bu also exacerbae 27 For example, le q be he producer price of he diry good, and p he Pigouvian ax based on he sandard formula. When he clean good is he unaxed numeraire and labor is axed, he ne wage rae relaive o he price of he diry good is w(1 θ)/(q+ p ). When labor is unaxed, each consumpion good faces a ax ha raises is price by he facor θ/(1 θ), and he diry good also faces he correcive ax of p /(µ/λ) = p /(1 θ), so he ne wage relaive o he price of he diry good is w/[q/(1 θ)+ p /(1 θ)] = w(1 θ)/(q+ p ). 53
exising disorions. Taxing consumpion and using he proceeds o reduce axes on labor has no ne impac on he consumpion-leisure choice in his insance. 5.3. Disribuional consideraions and he MCPF Wih a heerogeneous populaion, he provision of public goods and he correcion of exernaliies ake on added complicaions. Even in he absence of disorionary axaion, he opimal rules hen reflec he social valuaions of uiliies of differen individuals. In addiion, he coss and benefis of public goods, exernaliies, and he axes ha address hem all have disribuional consequences. For example, he governmen migh wish o expand provision of public goods ha have favorable disribuional consequences; Sandmo (1998) offers a deailed analysis of he general problem. Also see Slemrod and Yizhaki (2001), who illusrae how one can decompose boh he coss and benefis of public expendiure proecs in erms of efficiency and disribuional consequences. However, i is also useful o consider circumsances in which he problem becomes much simpler, which is he case when he governmen has sufficien flexibiliy in is choice of ax insrumens. There is a close analogy here o he sandard opimal income ax problem, under which i may no be necessary o ax luxury goods more heavily for purposes of disribuion if he governmen can use a nonlinear income ax (as in Akinson and Sigliz 1976). Indeed, he analysis yields a parallel resul, namely ha disribuional consideraions should no ener ino he provision of public goods or he correcion of exernaliies when here is a nonlinear income ax and preferences are weakly separable ino goods and leisure. This resul is described by Kaplow (1996), building on previous work of Hylland and Zeckhauser (1979). Kaplow s observaion is ha he Samuelson rule for public goods provision is unaffeced by he presence of disorionary axaion when preferences are separable and he governmen 54
uses a nonlinear income ax. The argumen has wo pieces. Firs, following he inuiion given above for he proporional ax case, here will be no change in labor supply, so ha all of he expendiures on he public good come hrough reducions in he unaxed numeraire commodiy. Hence, here is no ax wedge a he margin beween public and privae goods. Second, because of he availabiliy of he nonlinear income ax, he disribuional consequences of an increase in public goods spending can be offse, so ha disribuional weighs will also be absen from he decision. To expand on he reasoning Kaplow provides for his resul, we presen a deailed proof here. Suppose ha households vary wih respec o wage raes, w, bu ha each household s preferences ake he form U(v(c,g),1 L), where c is privae good consumpion, g is he level of he public good, and L is labor supplied. Public goods are financed using a nonlinear ax on labor income T(wL; g), where T 1 is he household s marginal ax rae. Consider an experimen in which g is increased, wih axes raised on each individual so ha ne uiliy is unchanged. (Coninuing o spend and ax in his way will evenually lead o an opimal level of public goods provision, if he governmen persiss o he poin ha marginal revenue from addiional spending is zero.) The claim is ha his policy resuls in no change in labor supply. The household s iniial opimum labor supply decision implies ha U (5.11) = U1 v1( w T1w) U2 = 0 L and ha (5.11) holds as g changes: du1 dc dl dv dl (5.12) v1 + U1 v11 + v12 w( 1 T1 ) U1v1w T11w + T12 U 21 U 22 = 0 dg dg dg dg dg 55
The claim is ha (5.12) holds wih boh U and L consan. Noe ha if U and L remain consan, so mus v, and hence U 1. Thus, he claim implies ha dc v1 (5.13) v11 + v12 = T12 dg 1 T 1 or, using dv/dg = v 1 dc/dg + v 2 = 0 dc/dg = ( v 2 /v 1 ), ( v2 / v1 ) 1 (5.14) = T12 c 1 T 1 By he assumpion ha L is fixed, dc/dg = dt/dg and dt/dg = T 2. Thus, v 2 /v 1 = T 2. Moreover, his equaliy does no hold simply a a paricular poin, bu raher a all poins in he income disribuion. Tha is, he funcions v 2 /v 1 (c,g) and T 2 (wl;g) are equal for any value of c = wl T(wL;g). Thus, (5.15) ( v2 / v c 1 ) = T 21 dwl g = T dc 21 1 1 T 1 Because T 12 = T 21, (5.14) holds, consisen wih he iniial claim. Jus as in he case previously considered in secion 5.2, a parallel analysis applies o exernaliies, wih he implicaion ha, under he mainained assumpions regarding preferences and he use of he nonlinear income ax, no adusmen o he sandard Pigouvian ax formula is warraned. While hese resuls do depend on wo key assumpions, hose concerning he separabiliy of individual preferences and he flexibiliy of he income ax, hey are sill quie imporan because hey idenify he source of deviaions from he basic rules of Samuleson and Pigou. As discussed 56
in his Handbook s chaper by Kaplow and Shavell, hey also have addiional implicaions regarding he exen o which governmen policies should be influenced by disribuional issues. 6. Opimal axaion and imperfec compeiion. The analysis o his poin concerns he opimal design of ax policies in economies wih perfecly compeiive indusries. Since some economic siuaions are characerized by imperfec compeiion, i is useful o consider he implicaions of differing degrees of marke compeiion for opimal ax design. One of he difficulies of summarizing he implicaions of imperfec compeiion for opimal axaion sems from he mulipliciy of imperfecly compeiive marke srucures. Neverheless, i is possible o idenify common welfare implicaions by considering a range of ax insrumens and marke siuaions. Our analysis follows closely ha of Auerbach and Hines (2001). 6.1. Opimal commodiy axaion wih Courno compeiion. I is useful o sar wih he behavior of a firm ha acs as a Courno compeior in an indusry wih a fixed number (n) of firms. The governmen imposes a specific ax on oupu a rae, so firm i s profi is given by (6.1) Px x C x ), i i ( i in which P is he marke price of he firm s oupu, x i he quaniy i produces, and C(x i ) he cos of producing oupu level x i. In his parial-equilibrium seing, i is appropriae o ake P o be a univariae funcion of indusry oupu, denoed X. The firm s firs-order condiion for profi maximizaion is 57
dp 1, dx (6.2) P + x ( + θ) = C ( ) i x i dx in which 2 is firm i s conecural variaion, corresponding o ( 1). Differing marke dx i srucures correspond o differing values of 2. In a Courno-Nash seing, in which firm i believes ha is quaniy decisions do no affec he quaniies produced by is compeiors, hen 2 = 0. In a perfecly compeiive seing, 2 = 1. Various Sackelberg possibiliies correspond o values of 2 ha can differ from hese, and indeed, need no lie in he [ 1, 0] inerval. I is useful o consider he pricing implicaions of (6.2). Differeniaing boh sides of (6.2) wih respec o, aking 2 o be unaffeced by, and limiing consideraion o symmeric equilibria (so ha X x i =, C( x i ) = C( X / n), and, since n dx dp d = d dp dx, i follows ha dx i dp d = ), hen d ndp dx dp d (6.3) = 1 + ( 1+ η) ( X / n) 1 + θ C n ndp dx 1, in which 2 d P X η is he elasiciy of he inverse demand funcion for X. From (6.3), i is 2 dx dp dx dp clear ha can exceed uniy, a possibiliy ha is consisen wih he firm s second-order d condiion for profi maximizaion and wih oher condiions (discussed by Seade, 1980a, 1980b) ha correspond o indusry sabiliy. Equaions (6.2) and (6.3) idenify he poenial welfare impac of axaion in he presence of imperfec compeiion. From (6.2), he combinaion of imperfec compeiion (2 > 1) and a 58
dp downward-sloping inverse demand funcion ( < 0) implies ha firms choose oupu levels a dx which price exceeds marginal cos. Hence here is deadweigh loss in he absence of axaion, and, in his simple parial equilibrium seing, ax policies ha simulae addiional oupu reduce deadweigh loss, while hose ha reduce oupu make bad siuaions worse. In some circumsances he imposiion of a ax may reduce indusry oupu sufficienly ha afer-ax profis acually rise. Tax policy can be used o reduce or eliminae he allocaive inefficiency due o imperfec compeiion, hough oher policy insrumens (such as anirus enforcemen) are also ypically available and may be more cos-effecive a correcing he problem. 28 Taking alernaive remedies o be unavailable, he opimal policy, if he governmen has access o lump-sum axaion, is o guaranee marginal cos pricing by seing = ( 1+θ) X dp. 29 dp Since < 0, his n dx dx correcive mehod enails subsidizing he oupu of he imperfecly compeiive indusry, so in realisic siuaions in which ax revenue is obained hrough disorionary insrumens, i follows ha he opimal policy may no fully eliminae he problems due o imperfec compeiion. In order o explore his issue furher, consider he seup of secion 3.1, in which all commodiies are produced a consan cos. There are N+1 commodiies, of which he firs M+1, indexed 0,, M, are produced by perfecly compeiive firms, and he remaining commodiies, M+1,, N, are produced in imperfecly compeiive markes, each of whose pricing saisfies 28 One possibiliy, explored by Kaz and Rosen (1985), is ha ax auhoriies design correcive policies on he basis of imperfec undersanding of he exen of compeiion in oligopolisic indusries. 29 Such a correcive subsidy was proposed by Robinson (1933, pp. 163-165), who aribues i o her husband and presens i as an ingenious bu impracical scheme. 59
(6.2). 30 Denoing he (consan) per-uni producion cos of commodiy i by q i, i follows ha p = q +, i = 0, K M. As in secion 3, we assume ha he ax on he numeraire commodiy, i i i, good 0, equals 0. Firms in he imperfecly compeiive indusries generae profis, and someone in he economy receives hese profis as income. 31 Taking consumers in he economy o be idenical, i follows ha he uiliy of he represenaive consumer can be represened by (6.4) V(p, B), in which p is he vecor of N+1 commodiy prices, and B represens profis earned by he imperfecly compeiive firms. Commodiy demands are hen funcions of (p, B), bu o simplify he calculaions ha follow, we consider he case in which firms ignore he indirec impac of heir pricing decisions on demand hrough induced changes in profis. In indusry > M, he represenaive firm s firs-order condiion for profi maximizaion is (6.5) p q X = n ( 1+ θ ) X p, where n and θ are defined for indusry in he usual way. Thus, he price-cos margin imposed by imperfec compeiion is m X = n ( 1+ θ ) ( X p ) in indusry. The opimal axaion problem consiss of maximizing (6.4) wih respec o he specific axes subec o hese mark-up condiions, he revenue consrain, 30 We follow much of he lieraure in assuming ha preferences and echnology suppor a unique sable marke equilibrium, which, as Robers and Sonnenschein (1977) noe, need no exis in he presence of imperfec compeiion. 31 In he compeiive conex, assuming a zero ax rae on one commodiy resrics he governmen effecively from imposing a ax on pure profis hrough a uniform ax on all commodiies. Here, hough, before-ax profis would 60
(6.6) X N = 1 = R and he household s budge consrain, (6.7) ( p q ) X = π N = M + 1. Combining he revenue consrain (6.6) and he budge consrain (6.7), we may recas he problem as one of maximizing (6.4) wih respec o consumer prices p, subec o he consrain, N (6.8) ( p q ) X R + π, = 1 where profis are given by 32 (6.9) ( 1+ θ ) X π = X. N = M + 1 n X p Wih µ defined as he muliplier of he consrain given in (6.8), he firs-order condiions for his problem are: respond o such uniform axaion, leaving he governmen s problem unchanged. We show his below, afer presening an expression for equilibrium profis. 32 Examinaion of expression (6.9) clarifies ha axing all goods uniformly would no reduce real profis. Taxing all goods a he same rae would raise prices by a facor λ, so i is necessary o verify ha (6.9) coninues o hold if profis, π, simulaneously increased by λ (and were herefore unchanged in real erms). Muliplying prices and profis by λ has no effec on X, since consumer demands are homogeneous of degree zero in income and prices. Bu his magnificaion of prices and income muliplies X /p by he facor 1/λ, as a uni change in price represens only 1/λ as large a proporional change as before. Thus, he righ-hand side of (6.9) equals is original value, muliplied by λ. As lef-hand side of (6.9) also equals is original value (π) muliplied by λ, he expression sill holds. 61
62 (6.10) N i dp d y X q p p X q p x dp d X i i i i i 1,..., 0 1 ) ( ) ( = = + + + + π µ π λ λ where, as before, λ is he marginal uiliy of income. Once again defining = + = N X 1 π µ λ α o be he social marginal uiliy of income, we may rewrie (6.10) as (6.11) 0 1 * = + = i N i i i dp d p X X X π µ α µ µ λ in which * = M * = p q > M is he oal wedge in marke, equal o + m in noncompeiive indusries. Equaion (6.11) is analogous o (5.10), and carries precisely he inerpreaion offered by Sandmo for he opimal ax condiions in he presence of exernaliies. Inuiively, he exernaliy in he case of imperfec compeiion is he oucome of he oligopolisic oupu selecion, resuling in he exra mark-up m. The definiion of * akes ino accoun he need o correc his pre-exising disorion. Were his he only erm on he righ side of (6.11), hen i would be opimal fully o correc for he exra disorions in noncompeiive indusries and hen impose he sandard opimal axes. Presumably, he ne resul in indusry would be an incomplee offse of oligopolisic mark-ups, he opimal ax componen normally being posiive. The second erm in brackes in (6.11) accouns for he exisence of profis, aking he form laid
ou in expression (3.17) above and explained in ha conex. 33 In his insance, ax-induced price changes affec he profiabiliy of he imperfecly compeiive indusry, he difference (: ") capuring he welfare effec of increasing indusry profis by one uni. To he exen ha a higher price of a commodiy direcly or indirecly augmens oligopoly profis, his mus be included in compuing he price change s overall welfare effec. Doing so has he effec of making he price increase less aracive as a policy ool. 6.2. Specific and ad valorem axaion. In compeiive markes he disincion beween specific and ad valorem axaion arises only from minor ax enforcemen consideraions. In imperfecly compeiive markes hese wo ax insrumens are no longer equivalen, since he imposiion of an ad valorem ax makes he ax rae per uni of sales a funcion of a good s price, which is parly under he conrol of individual firms. As a resul, ad valorem and specific axes ha raise equal ax revenue will ypically differ in heir implicaions for economic efficiency, ad valorem axaion being associaed wih much less deadweigh loss. 34 Inuiively, ad valorem axaion removes a fracion (equal o he ad valorem ax rae) of a firm s incenive o resric is oupu level in order o raise prices. The welfare superioriy of ad valorem axaion is eviden in he simple parial equilibrium seing considered iniially above. Now, he governmen is assumed o have access boh o an ad valorem ax and o a specific ax, and ax revenues are assumed cosly o obain (for reasons omied from he model). In his seing he firm s profis equal (6.1) ( τ) Px x C( x ) 1 i i i 33 Auerbach and Hines (2001) presen a longer, alernaive derivaion of (6.11) ha includes explici expressions for he erms dπ/dp i. 34 Suis and Musgrave (1953) provide a classic analysis of his comparison; heir reamen is grealy expanded and elaboraed by Deliapalla and Keen (1992). 63
in which J is he ad valorem ax rae. Assuming he n-firm oucome o be symmeric, he firsorder condiion for profi maximizaion becomes X dp X 1, n dx n (6.2) ( τ) P + ( 1+ θ) = C and is pricing implicaions are dp d 1 + θ C n ndp (6.12) = ( 1 τ) 1+ ( 1+ η) ( x / n) dx 1 dp dτ X dp n dx dp 1. d (6.13) = P + ( + θ) Since a uni change in J raises more ax revenue han does a uni change in, i is unsurprising ha dp > dτ dp d. Much more revealing is he effec of hese ax insrumens normalized by dollar of marginal ax revenue. Since oal ax revenue is given by Rev = τpx+x, i follows ha drev d + dp + dτ (6.14a) = X 1 τ ( + τ P) drev dτ (6.14b) PX ( + τp) X P dp d τ dp X dp = 1 + +. P dτ P dτ In his simple parial equilibrium model, he change in deadweigh loss associaed wih one of hese ax changes is equal o he produc of he induced change in X and he difference beween marginal cos and price. Consequenly, 64
d( DWL) d d( DWL) dτ = P C X n ( X P) ( dp d ) dp d = ( X P) ( dp dτ) X dp dτ P C n, which, ogeher wih (6.14a) and (6.14b), implies ha (6.15) d( DWL) d d DWL d ( ) τ drev d drev d τ = P X + τ + dp dτ 1 X + τ + dp d ( + τp ) ( + τp ) X P X P. From (6.13), dp dp < P, so if ax revenue is an increasing funcion of ax raes, hen he righ dτ d side of (6.15) is greaer han uniy. Hence revenue-equal subsiuion of ad valorem for specific axaion reduces deadweigh loss a any (, J) combinaion. 35 Of course, such subsiuion works a he expense of firm profiabiliy, and would, if used excessively, drive profis negaive and supply presumably o zero. Bu assuming he firm profiabiliy consrain no o bind, he opimal ax configuraion enails ad valorem raher han specific axaion. The preceding comparison of ad valorem and specific axaion compares heir effeciveness per dollar of foregone revenue, bu does no address he quesion of he opimal rae of ad valorem axaion when he governmen is unable or unwilling o provide specific subsidies. While his problem is ypically hough (e.g., Myles, 1989) o enail a very differen soluion han 35 Consequenly, if he governmen is able o impose negaive specific axes (specific subsidies), hen i can compleely eliminae he disorion due o imperfec compeiion hrough a udicious combinaion of ad valorem ax and specific subsidy, as noed by Myles (1996). The effeciveness of his correcive mehod is limied by any consrains on ad valorem ax raes, such as a resricion ha hey be nonnegaive. 65
ha for specific axaion, properly framed i becomes clear ha he soluion has he same characer regardless of he ype of available ax insrumen. Following he analysis of specific axes, we seek o maximize he indirec uiliy funcion in (6.4) subec o he revenue consrain, (6.16) τ p X R, N = 1 he definiion of profis, (6.17) ( p 1 τ ) q ) X = π N = M + 1 (, and he characerizaion of producer behavior in noncompeiive indusries, (6.18) ( 1+ θ ) X p ( 1 τi ) q = (1 τ ) > M. n X p As before, we combine household and governmen budge consrains o express economy s resource consrain as (6.19) ( p q ) X R + π, N = 1 and analyze he problem as one of maximizing (6.4) wih respec o p, subec o his consrain, where profis are given by (6.20) = + = N q π φ X, where M 1 p φ φ X = n ( 1+ θ ) X. p 66
Noe ha expression (6.20) differs from (6.9) by he erm muliplying φ X on he righ-hand side of (6.20), which equals (1 τ ). Oherwise, he problem is idenical o ha for specific axes, and he firs-order condiions given in (6.11) sill hold, for τ i insered in place of i /p i. The resuling equilibrium will generally be differen, of course, because profis, and hence he erms dπ/dp i, will be differen. Auerbach and Hines (2001) provide some numerical simulaions confirming ha, in cases for which a noncompeiive indusry s ax is posiive under specific axaion, i should be higher in he case of ad valorem axaion. They also exend he analysis o he case in which he governmen is uncerain abou he degree of noncompeiive behavior, as represened by he parameer θ. This uncerainy ends o reduce he exen of he desired correcive subsidy, for he subsidy ends o be mos effecive precisely when i is leas needed, i.e., when θ is small. 6. 3. Free enry. The sandard Courno model akes as is poin of deparure an indusry wih a fixed number of firms. The abiliy of firms o ener and leave an indusry changes he opimal ax problem, and inroduces some ineresing feaures of he soluion (such as he possibiliy of welfare-improving posiive ax raes even if he governmen has access o nondisorionary sources of revenue). In spie of hese differences, many of he main implicaions of he preceding analysis, including he welfare superioriy of ad valorem o specific axaion, persis in a model wih free enry. Consider an indusry consising of idenical firms ha behave according o (6.2). In his model, enry and exi are free, bu new enrans do no necessarily selec oupu levels ha 67
68 minimize cos, since hey behave in a manner ha is cognizan of he effec of oupu on price. 36 The governmen imposes ad valorem and specific axes, so he zero-profi condiion for indusry enry (assuming, for convenience, ha i is possible o have fracional numbers of firms) is (6.21) ( ) 0 1 = n X C n X P n X τ Assuming ha he governmen has access o lump-sum ax insrumens, he social oal cos (TC) of producing indusry oupu is given simply by is resource cos, or = n X nc TC. For a small change in a ax insrumen, > (eiher an ad valorem or a specific ax), i follows ha (6.22) ξ ξ ξ d dn n X C n X n X C d dx n X C d dtc + =. The value o consumers for which he ax change is responsible is given by dξ dx P. Consequenly, he change in he difference beween consumer value and social cos, say 7, is (6.23) ( ) ξ ξ ξ d dn n X n X C n X n X C d dx n X C P d d = Λ. Equaion (6.24) succincly capures he wo compeing consideraions in changing a ax rae ha applies o imperfecly compeiive indusries. The firs erm is he produc of he induced change in oupu and he difference beween price and marginal cos of producion for 36 New enrans are assumed o exhibi he same oligopolisic behavior (as refleced in 2) as do oher firms in he indusry; see Mankiw and Whinson (1986) for an analysis of he welfare effecs of enry in such a seing.
firms in he indusry. If he number of firms in he indusry were fixed, hen his would be he only expression on he righ side of (6.23), and i would carry he previous implicaion ha, wih he availabiliy of lump-sum ax insrumens, efficien axaion consiss of equaing price and marginal cos. The difficuly, of course, is ha i is no he only erm on he righ side of (6.23). In his model i is necessary o subsidize an indusry in order o equae price and marginal cos, and governmen subsidies encourage inefficien enry of new firms. The welfare effec of ax policy on enry is capured by he second erm on he righ side of (6.23). This erm is he produc of he amoun of oupu produced by new enrans and he difference beween average and marginal coss for each firm in he indusry. Subracing (6.2) from (6.21) implies ha C X X n n X n X dp n dx (6.24) C = ( 1 τ) ( 1+ θ) > 0, which simply follows from he fac ha price exceeds marginal cos. Hence average cos exceeds marginal cos, and new enry is inefficien, since marginal oupu is less expensively produced by exising firms han by new enrans. 37 The effec of inroducing axes can be idenified by differeniaing he ideniy ha X n X n, which yields 37 This equilibrium condiion requires he producion echnology o exhibi decreasing average coss over some range of oupu. 69
70 (6.25) ξ ξ ξ d n X d n d dn n X d dx + =. Togeher, (6.21), (6.23) and (6.25) imply (6.26) [ ] ξ ξ τ ξ d n X d n X XC n X nc X N d dx P d d + + = Λ. Saring from = J = 0, i follows from (6.26) and (6.24) ha 0 > Λ dξ d if 0 > dξ n X d, regardless of he effec of axaion on enry and exi. The inuiion behind his resul is ha, while greaer oupu by exising firms promoes efficiency (since price exceeds marginal cos), in he absence of axaion, price equals average cos and here is no welfare impac of marginal enry. Recall from (6.24) ha average cos exceeds marginal cos in equilibrium, and hence is a declining funcion of a firm s oupu. Therefore, increases in oupu per firm will reduce average cos and increase welfare. From he zero-profi condiion (6.21), average cos is (6.27) ( ) P n X n X C n X AC = = τ 1. Hence oupu per firm rises, and herefore welfare rises, in response o he inroducion of axes ha reduce he righ side of (6.27). Equaion (6.2) describes he firm s firs-order condiion for profi maximizaion. By (6.27), average oupu per firm (X/n) can be expressed as a decreasing funcion of [P(1 J) ], while he marke demand curve allows us o express oal oupu, X, as a funcion of P.
Appropriaely differeniaing boh sides of (6.2) wih respec o, evaluaing he resuling expression a J = = 0, and collecing erms yields (6.28) dp d = d 1+ d d( X n) dp 1+ ( 1+ θ) d( P ) dx ( X n) dp ( 1 + θ) C ( P ) dx C X d n d d( X n) d( P ) ( X n) η( 1+ θ) + ( P ) n X n. where 0 is he elasiciy of he inverse demand funcion, as defined above a (6.3). Since he condiions for indusry sabiliy imply ha boh he numeraor and he denominaor of he expression on he righ side of (6.28) are posiive, 38 i follows ha dp 1 d has he same sign as 0. Hence a posiive value of 0 implies ha he inroducion of a (posiive) specific ax increases he marke price by less han he amoun of he ax, expanding per-firm oupu and hereby improving welfare. 39 The reason is ha he reduced indusry oupu due o a higher ax dp rae reduces, which is a facor in he oligopolisic markup by which price is elevaed above dx marginal cos. While he same consideraion applies in oher seings, he exisence of free enry and exi is criical o he welfare resul due o he induced aenuaion of he effec of axes on price. Ad valorem axaion coninues o be more aracive han specific axaion in indusries wih free enry and exi. Saring from J = = 0, he inroducion of an ad valorem ax reduces 38 Seade (1980a) demonsraes ha sabiliy requires C (X/n) > (1+θ) dp/dx, and since d ( X n) d( P ) < 0 follows ha he numeraor of (6.28) is posiive. Seade (1980b) also adops η+ n ( 1 + θ) > 0, i as a sabiliy condiion, noing (1980a) ha i is a sufficien condiion for a firm s marginal revenue o fall when oher firms expand oupu, and ha his condiion implies ha new enry is associaed wih greaer indusry oupu. Togeher, hese sabiliy condiions guaranee ha he denominaors of (6.28) and (6.29) are posiive. 39 See Besley (1989) and Delipalla and Keen (1992) for addiional resuls and inerpreaion. 71
he righ side of (6.27) if dp < P. Appropriaely differeniaing boh sides of (6.2) wih respec dτ o J yields (6.29) d( X n) dp ( 1+ θ) C [ P( 1 τ) ] dx d( X n) dp + ( 1 + θ) d[ P( 1 τ) ] dx d( X n) + [ P( 1 τ) ] d( X n) d[ P( 1 τ) ] X X dp 1+ dp d n d n dx = P dτ X η1 1 C + n n ( + θ) ( 1+ θ) P. Since he sabiliy condiions imply ha he denominaor of he righ side of (6.29) is posiive, i follows ha dp P dτ has he same sign as dp dx X P η. Hence he inroducion of an ad valorem ax improves welfare no only if 0 is posiive, bu also if 0 is negaive bu smaller in absolue value han he elasiciy of he inverse demand funcion. This condiion for welfare improvemen is weaker han ha for he inroducion of specific axes, hereby reflecing he relaively more poen effec of ad valorem axes in reducing an imperfecly compeiive firm s reurn from resricing oupu in order o elevae price. 6.4. Differeniaed producs. Cerain ypes of oligopolisic siuaions ake he form of compeiion among firms selling producs ha are imperfec subsiues. Firms ake acions ha affec produc aribues as well as oupu levels, and hese acions are poenially affeced by ax policies. Since here are many forms of compeiion beween sellers of differeniaed producs, i can be difficul o draw general welfare conclusions concerning he impac of axaion in such seings; i is, however, possible o idenify he maor consideraions on which he resuls urn. 72
Consider an indusry of n firms selling producs ha differ along a univariae qualiy scale, indexed by <, so ha firm i sells producs of qualiy < i, in which < i represens a profimaximizing choice made by he firm. Firm i produces oupu x i a qualiy level < i, wih idiosyncraic coss given by c i (x i, < i ). The represenaive consumer s preferences are hen responsible for he inverse demand funcion p(x, <), and he governmen imposes an ad valorem ax a a uniform rae on all sales in he indusry. Producion akes place in wo sages. Firs, firms selec values of < i, aking as fixed he elemens of he < vecor oher han < i (ineresing generalizaions are possible by incorporaing sraegic ineracion in he choice of <). Second, firms choose oupu levels x i coningen on < and aking he oupu of oher firms as fixed. Of course, firs sage choices of < are made in anicipaion of induced pricing and oupu effecs in he second sage. Condiional on <, firm i s opimal choice of x i in he second sage mus saisfy (6.30) ( τ) p ( x, ν) p ( x, ν) c ( x, ν ) i i i i 1 i + xi x =. i xi Denoing he vecor of values of x i ha solve (6.30) by x*(<), he firs-order condiion for he opimal choice of < i is (6.31) ( τ) p ( x* ( ν), ν) p ( x * ( ν), ν) x ( ν) c ( x, ν ) i i i i i 1 + xi =. νi i x νi ν i Oligopolisic siuaions offer differing inerpreaions of he conex and welfare inerpreaions of (6.30) and (6.31). From (6.30), i is clear ha, condiional on <, imperfec compeiion leads o oo lile producion, in he sense ha prices exceed marginal coss. From 73
his observaion i is emping o conclude ha (as before) he opimal ax policy is one ha subsidizes he oupu of imperfecly compeiive firms. The endogeneiy of < has he poenial o reverse his reasoning, however, since here is no presumpion, from he general form of (6.31), ha qualiy choices are opimal in he absence of axaion. Qualiy choice may be subopimal for many reasons. The firs is ha firms selec qualiy levels based on heir impac on marginal demand and no on he valuaion of inframarginal oupu by he same firm. A second reason is ha one firm s reurn o qualiy may come a he expense of oher firms, and such pecuniary exernaliies affec welfare in siuaions in which prices differ from marginal coss. And a hird reason is ha qualiy choice in he firs sage affecs he oupu decisions of oher firms in he second sage, a sraegic consideraion ha creaes inefficiencies whenever demand for one commodiy is affeced by he prices of ohers. The examples analyzed in he lieraure generally share he feaure ha he inroducion of (posiive) ad valorem axaion can improve welfare. 40 Equaion (6.31) idenifies he sraegic consideraion responsible for his effec, since, if commodiies i and are subsiues in demand p x i x < 0, and sraegic subsiues in supply < 0, hen, in he absence of axaion, νi qualiy is oversupplied in he sense ha p i ci < ν ν i i. Ad valorem axaion ypically reduces qualiy levels, hereby quie possibly improving welfare even hough i serves furher o disor he oupu level choice refleced in (6.30). This implicaion is very similar o he resul (from he previous secion) ha ad valorem axaion is desirable in a model wih free enry and exi, and indeed, hese cases share many similariies. Firms described by (6.30) and (6.31) selec oupu 40 See, for example, Kay and Keen (1983) and Cremer and Thisse (1994). Besley and Suzumura (1992) analyze a wo-sage game of sraegic invesmen in cos-reducing echnology wih similar feaures. Kay and Keen (1991) consider he naure of preferences ha deermine he effec of axaion on produc qualiy. 74
levels a which prices exceed marginal coss, bu also selec qualiy levels a which marginal coss exceed non-sraegic reurns. One can hink of (6.31) as characerizing excessive enry along he qualiy dimension, and herefore posiive ad valorem axaion as being desirable o he exen ha i simulaes oupu per uni of effecive qualiy. Hence, here is poenially a saluary role of axes in reducing qualiy, paricularly if oligopolisic compeiion is aggressive in nonprice dimensions. 7. Ineremporal axaion This secion considers opimal axaion in ineremporal seings, generally resuming he assumpion of perfec compeiion. Due in par o ineres generaed by he consumpion ax advocacy of Fisher and Fisher (1942), Kaldor (1955), and ohers, one ineremporal issue in paricular has received exensive aenion: he opimal ax rae on capial income. One of he noable developmens of modern opimal ax heory is he finding ha, in a simplified secondbes seing wih idenical individuals and in which he governmen can ax boh capial income and labor income, welfare maximizaion implies zero axes on capial income in he seady sae. This finding reflecs, of course, he highly disorionary naure of capial income axes over long periods of ime, bu is neverheless surprising in view of he sandard Ramsey inuiion ha he deadweigh loss is zero for he firs dollar colleced by any ax and herefore, in he absence of spillovers beween markes, all opimal ax raes are sricly posiive. Where his inuiion fails in he ineremporal conex is ha i does no accoun for us how exremely disorionary capial axaion can be even a very low raes of ax specifically, ha low ax raes correspond o disorionary ineremporal ax wedges ha grow over ime. The main findings concerning opimal capial axaion are repored by Chamley (1986) and Judd (1985). Subsequen research by Jones Manuelli, and Rossi (1993, 1997), Milesi- 75
Ferrei and Roubini (1998), and ohers exends is logic o he ineremporal axaion of facors oher han capial. In paricular, o he exen ha wages represen reurns o he accumulaion of human capial, labor income axes have capial componens and are likewise opimally zero in he seady sae. Indeed, he logic of opimal ineremporal axaion is such ha here are plausible circumsances in which all axes may be zero in he seady sae. Of course, governmens ha aemp o implemen such opimal axes would need o amass considerable unspen ax revenue in years prior o he seady sae in order o mainain ineremporal budge balance. Before considering hese implicaions, however, i is useful o review he source of he basic ineremporal resuls concerning capial axaion alone. 7.1. Basic capial income axaion: inroducion The logic of he resul ha capial is unaxed in he seady sae is apparen from working hrough a simplified version of Chamley s problem. Consider he case of an economy consising of idenical consumers who maximize he presen discouned value of uiliy over infinie horizons: = 0 (7.1) β u( C, L ) in which β is he one-period discoun facor (β = (1+δ) 1, δ being an individual s subecive discoun rae), aken o be consan for all individuals in all periods. u(c,l ) is a consumer s conemporaneous uiliy in year, an increasing funcion of consumpion (C ) and a decreasing funcion of labor supplied (L ). Consumers have iniial wealh of K 0 and earn labor income in period zero equal o w 0 L 0, in which w 0 is he afer-ax wage rae in period zero. Labor income is received a he sar of 76
77 each period, and consumpion also akes place a he sar of each period, so any capial income is earned while a period elapses. A consumer herefore dissaves ( ) 0 0 0 L w C in he iniial period, and has he lifeime budge consrain (7.2) ( ) ( ) ( ) 0 0 0 0 1 1 1 1 1 L w C K r w L C s s + = = in which r is he (afer-ax) reurn earned by capial during period. Assuming ha he consrain (7.2) is binding (and ha he soluion enails inerior opima), he firs-order condiions ha characerize he maximum of (7.1) are (7.3) L u C u w = (7.4) ( )β r C u C u + = + 1 1. Equaion (7.4) in urn implies (7.5) ( ) = + = 1 0 0 1 n i i n n r C u C u β Combining he budge consrain, (7.2), and he firs-order condiions, (7.3) and (7.5), yields (7.6) 0 0 0 C u K L L u C C u = β.
As he economy consiss of idenical individuals, we consider he mos noaionally simple case of one such individual. The period-by-period resource consrain for such an economy is (7.7) C + G + K + F ( K, L ) + K, 1 in which G is governmen consumpion in period, and F(K,L ) is he economy s producion funcion. The pah of governmen consumpion is aken o be exogenous and (for simpliciy) capial is assumed no o depreciae. Inequaliy (7.7) expresses he idea ha he sum of privae and public consumpion, plus ne capial accumulaion, canno exceed he oupu of he economy. 7.2. The seady sae The mos sraighforward way o evaluae he properies of opimal axaion is o consider he firs-order condiions ha correspond o maximizing (7.1) subec o (7.6) and (7.7), aking C, L and K o be conrol variables. (I is noeworhy ha (7.7) acually represens a separae consrain for each period.) The firs-order condiion corresponding o an inerior choice of C is 2 2 u u u u (7.8) β λ C L = µ C + C C L C, 2 in which λ is he Lagrange muliplier corresponding o he consrain (7.6), and µ is he Lagrange muliplier corresponding o condiion (7.7) in period. The firs-order condiion corresponding o an inerior choice of K is F (7.9) µ 1 + = µ 1. K 78
Consider an economy ha ulimaely seles ino a long-run seady sae in which economic variables, specifically C and L, are unchanging. Since he erm in braces on he lef side of equaion (7.8) is unchanging in his seady sae, i follows ha µ = βµ 1. Imposing his equaliy on (7.9) yields F (7.10) β 1 + = 1. K Equaion (7.4), one of he consumer s firs-order condiions, implies ha, if C = C + 1 and L, hen β ( 1 r ) = 1 = L +1 +. Consequenly, (7.10) implies ha r F = in he seady sae. K Recall ha r is he afer-ax reurn received by savers during period. In a compeiive marke, F K is he pre-ax reurn o invesors. The equaliy of r and F K herefore implies ha savings are unaxed. 7.3. Inerpreing he soluion The finding ha capial income should be unaxed in he seady sae conradics he naïve inuiion ha, since axes on labor income disor labor-leisure choices in he seady sae, a minor reducion in labor axes financed by a very small ax on capial income would improve he welfare of he represenaive individual. Where his inuiion fails is ha even very low-rae axes on capial income generae firs-order consumpion disorions over long horizons. The reason is ha a capial income ax a a very low rae creaes a small disorion beween consumpion in periods and (+1), bu a large disorion beween consumpion in period and consumpion in period (+n), for large n. 79
I does no by any means follow from he seady-sae properies of he opimal program ha capial income axes are always zero. Indeed, Chamley (1986) offers an example in which consumers have uiliy funcions ha are addiively separable in consumpion and leisure and iso-elasic in consumpion, for which he opimal dynamic ax configuraion is one in which he governmen imposes a capial income ax a a 100 percen rae for an iniial period and 0 hereafer. 41 Chamley offers he inuiion ha high iniial raes of capial ax serve o ax away he value of iniial capial, hereby acing in par as a lump-sum ax and in par as a very disorionary ax on capial accumulaion during he regime of 100 percen ax raes. This inuiive inerpreaion of he opimal ax paern is correc bu incomplee, since even if he governmen possessed an addiional ax insrumen, permiing i o exrac up o 100 percen of he value of iniial capial from he privae secor, i migh sill wish o use sandard capial income axes o raise addiional revenue in he shor run. The reason is ha capial income axes in early years disor he choice beween presen and fuure consumpion, bu leave he margins among consumpion a differen fuure daes unaffeced; nonzero capial income axes in laer years also disor he paern of fuure consumpion. If one hinks of consumpion a differen daes as separae commodiies, hen he Ramsey analysis suggess ha opimal policy enails equal (revenue-adused) marginal disorions o consumpion in each period. Because consumpion axes are no included in he governmen s insrumen se, his oucome is approximaed by he use of capial income axes in early years bu no in laer years. Analyically, he equaions (7.8) and (7.9) ha characerize he opimal pah would be formally unchanged even if he governmen had access o an addiional insrumen ha exracs he value of iniial capial. Of course, hese condiions would hen imply a differen ax rae pah, bu is 41 Chamley consrains he governmen no o impose capial income axes a greaer han a 100 percen rae in order o rule ou nondisorionary lump-sum iniial capial levies as a mehod of governmen finance. 80
general feaure ha capial income ax raes fall over ime would persis, and herefore no reflec he desire o ax he value of iniial capial. The ime-varying naure of opimal capial axaion makes such a policy imeinconsisen, in ha whaever profile of fuure axes ha is opimal as of year would no be opimal as of year +1, and opimizing governmens migh herefore be emped no o follow hrough on previously announced ax plans. Privae agens, anicipaing such behavior by governmens, could no hen be expeced o respond o announced ax plans in he same way ha hey would if he governmen could commi reliably o he axes ha i announces. This is us one of many examples of he ime inconsisency of opimal plans, a feaure ha akes on special significance in an economy in which privae agens hold capial, he value of which governmens migh find aracive o seize hrough heir ax policies. While here are aemps o idenify opimal ime-consisen capial ax policies by somehow consraining governmen behavior, all such effors confron he fundamenal problem ha he mere exisence of capial, ogeher wih he disorionary naure of income axaion, creaes incenives for benevolen governmens o behave in a ime-inconsisen fashion. 42 The analysis of his secion follows he maoriy of he lieraure in considering governmen policies under he assumpion ha i is possible o make credible commimens. 42 There is an enirely separae, bu relevan, issue ha arises concerning he benevolence of governmens over ime. The opimal ax pah is one ha accumulaes enormous governmen revenues in he early years in order o finance expendiures in laer years (in which capial income ax raes will be zero). Given he implausibiliy of acual governmens besowing upon heir successors such hard-won budge surpluses in order o finance efficien axaion in he fuure, i is worh bearing in mind ha opimal axaion is a useful ideal if no a realiy. In pracice, he opposie paern in which governmens run sizable deficis parly o consrain he fiscal choices of fuure governmens (as in Persson and Svensson, 1989) is much more common. 81
7.4. Human capial accumulaion and endogenous growh The model described by (7.1) (7.10) carries implicaions for he axaion of labor income, bu hese are very difficul o characerize succincly (oher han o say ha labor income axes are posiive and unchanging in he seady sae). The reamen of labor as a facor of producion is somewha sylized, in ha all labor is homogeneous and represens forgone leisure opporuniies (wih which individuals are endowed). The economy described by (7.1) (7.10) grows via capial accumulaion (and shrinks during periods of capial decumulaion). As shown by Lucas (1990), Lainer (1995) and ohers, he qualiaive feaures of opimal axaion are unaffeced by inroducing exogenous echnical progress ha generaes economic growh and causes he economy o sele ino a balanced growh pah in he long run. Judd (1999) obains he similar resul ha he long-run average opimal capial income ax rae is likewise zero for economies ha do no converge o seady saes. Exensions o economies wih producion subec o sochasic shocks, such as hose by Zhu (1992) and Chari, Chrisiano, and Kehoe (1994), produce he resul ha he opimal ax on capial income is generally very low or zero. The impac of fiscal policies in seings in which economies grow endogenously is he subec of a closely relaed lieraure. There is more han one poenial source of endogenous growh, perhaps he mos obvious being he accumulaion of human capial, along wih ohers ha include social increasing reurns o scale due o he produciviy-enhancing effecs of infrasrucure and oher public goods. 43 These models share in common he characerisic ha he endogeneiy of he growh rae arises from some posiive exernaliy. As in radiional public finance analysis, he presence of exernaliies means ha an equilibrium wihou disorionary 43 See, for example, Lucas (1990), King and Rebelo (1990), Rebelo (1991), Trosel (1993), and Sokey and Rebelo (1995). The sources of endogenous growh analyzed by Eaon (1981) and Hamilon (1987) differ from hese in reflecing he saving and porfolio preferences of consumers, and need no enail any producive exernaliies. 82
axes will generally no be Pareo-opimal. Thus, opimal ax design mus ake he presence of such exernaliies ino accoun, as discussed in Secion 5.2 above. In endogenous growh models, he accumulaion of human capial generaes exernaliies hrough inergeneraional ransmission of acquired skills. However, one may consider he accumulaion of human capial and is associaed exernaliy separaely, and i is useful o do so in undersanding he effecs on opimal ax resuls. Human capial accumulaion iself (wihou any inergeneraional ransmission of skills) is easily incorporaed in he model (7.1) - (7.10), as labor income hen represens he reurn o pas forgone consumpion and leisure (assuming ha boh goods and ime conribue o he accumulaion of human capial), as well as conemporaneous forgone leisure. Since labor income axes hen effecively ax ineremporal labor/leisure choices in much he same way ha capial income axes effecively ax ineremporal consumpion choices, i is no surprising ha he opimal dynamic ax pah is one in which labor income axes, as well as capial income axes, are zero in he seady sae (as in Jones, Manuelli, and Rossi (1993, 1997) and Milesi-Ferrei and Roubini (1998)). To show his more formally, consider he case in which consumers have hree uses for heir ime: hey can work, for which hey receive a wage, hey can accumulae human capial, which increases fuure wages, and hey can consume leisure. Denoe by E he amoun of ime ha he consumer devoes o human capial accumulaion in period. In he simple case in which uiliy is a funcion only of consumpion and leisure, so ha he disuiliy of ime working equals he disuiliy of devoing he same amoun of ime o human capial accumulaion, he consumer s maximand becomes = 0 (7.1) β u ( C, L + E ). 83
Le H denoe he consumer s period- sock of human capial; purely for simpliciy assume ha human capial does no depreciae. Accumulaion of human capial occurs by devoing ime and valuable goods and services (e.g., educaional resources) o producing addiional human capial. Le M(E, B) denoe he (ime-invarian) human capial producion funcion, in which B represens he value of goods and services devoed o human capial. The accumulaion of human capial is herefore consrained by he relaionship: (7.11) H M ( E, B ) + H,. + 1 The abiliy of consumers o allocae some of he economy s oupu o he accumulaion of human capial requires a modificaion in he economy s resource consrain, as well as a slighly differen specificaion of he producion funcion, so ha (7.7) becomes (7.7) C B + G + K F ( K, L, H ) + K,. + + 1 The exisence of human capial does no change (7.6), he consumer s ineremporal budge consrain. The inroducion of human capial adds a new sae variable (H ) o he opimal ax problem, as well as wo new choice variables (E and B ), a new consrain (7.11), and requires he modificaion of he obecive funcion and one of he previous consrains. Once again, he mos sraighforward way o describe he properies of he opimal soluion is o maximize (7.1) subec o (7.6), (7.7), and (7.11), aking C, L, K, B, and H o be conrol variables. Equaions (7.8) and (7.9) coninue o hold, and so, herefore, does (7.10) and is implicaion ha he reurn o saving is unaxed in he seady sae. The firs-order condiion corresponding o an inerior choice of H is 84
F (7.12) µ + ψ = ψ 1, H in which R is he Lagrange muliplier on he consrain (7.11) in period. The firs-order condiion corresponding o an inerior choice of B is M (7.13) ψ = µ. B Since (7.8) coninues o characerize he opimal soluion, i follows ha a seady sae in which C, L, E and B are unchanging implies ha µ = βµ 1. From (7.13), i hen follows ha, in he seady sae in which M B is unchanging, i mus be he case ha ψ = βψ 1. Togeher, (7.12), (7.13), and ψ = βψ 1 imply (7.14) M F 1 + = B H 1 β. From he seady sae condiion 1/$ = (1+r) i follows ha M F (7.15) = r. B H Equaion (7.15) characerizes he seady sae economy under opimal axaion, so i is insrucive o compare (7.15) o he consumer s firs-order condiions. An individual who defers consumpion invess eiher in physical capial or in human capial. (7.4) describes he (inerior) firs-order condiion for invesing in physical capial; he analogous firs-order condiion for invesing in human capial is 85
u u M w (7.16) = 1 + β, C C+ 1 B H in which w is he afer-ax wage. The erm w H in (7.16) herefore equals he single-period afer-ax privae reurn from accumulaing an addiional uni of human capial. Equaions (7.16) and (7.4) ogeher imply ha M B w H = r, which, ogeher wih (7.15), implies ha (7.17) w H F = H. The lef side of (7.17) is he amoun of addiional afer-ax income received by a worker who accumulaes one more uni of human capial; he righ side of (7.17) is he marginal produc of his addiional uni of human capial. Assuming ha here are no produciviy spillovers, so ha he produciviy gains from addiional human capial are embodied in he effecive labor supply of workers who possess he human capial, facor marke compeiion guaranees ha he righ side of (7.17) equals he effec of human capial accumulaion on preax wages. Since he lef side of (7.17) is he effec of human capial accumulaion on afer-ax wages, i follows ha labor income mus be unaxed in he seady sae. Noe ha his resul depends on (7.16), which applies only if human capial accumulaion requires inpus of goods forgone consumpion as well as leisure. If his is no he case if 86
human capial is accumulaed simply hrough forgone leisure hen he resuls ha follow will no hold. In paricular, he ax on labor income will no longer disor he accumulaion of human capial, because he enire cos of invesmen will be ax deducible. I follows, hen, ha if goods inpus are deducible, he human capial decision will remain undisored by labor income axes, in which case here is no requiremen ha labor income axes equal zero in he seady sae. As shown by Milesi-Ferrei and Roubini (1998), governmens wih a sufficien number of ax insrumens can effecively decouple he axaion of human capial accumulaion from he axaion of he reurn o forgone leisure. The analysis of human capial accumulaion is really a subse of a broader range of issues in which ax insrumens are resriced in one way or anoher. In oher seings, Jones, Manuelli and Rossi (1993, 1997) observe ha resricions on he range of ax insrumens available o he governmen, or he presence of public goods in he aggregae producion funcion, change he naure of even seady sae axaion in a way ha can make i opimal for he governmen o impose axes on capial income. For example, here migh be wo ypes of labor in he economy, wih properies (such as differing labor supply elasiciies) ha would make i opimal o ax he incomes hey generae a differen raes. If he governmen is consrained o selec a single labor income ax rae, hen he opimal ax rae on capial income migh differ from zero in he seady sae in order o compensae for he governmen s inabiliy o ailor is labor income axes. Judd (1997) analyzes he implicaions of resricions on he abiliy of he governmen o conrol monopolisic and oher noncompeiive marke behavior, in which case ax policy may funcion as a differen kind of second-bes correcive mechanism; his work idenifies circumsances under which he opimal ax on capial income may hen be negaive in he seady sae. Coleman (2000) comes o a similar conclusion in a seing in which he governmen can impose separae 87
consumpion and labor income axes, and here are resricions on he range of available ax insrumens. Aiyagari (1995) considers he implicaions of marke incompleeness ha leaves individuals incapable of diversifying idiosyncraic risks. The resuling demand for precauionary saving leads o a posiive opimal ax rae on capial income, even in he seady sae. Correia (1996) noes ha many of hese consideraions sem from he exisence of an imporan producive facor ha he governmen is unable (for some reason) o ax or o subsidize. Depending on he applicaion, his facor migh represen inframarginal profis from decreasing reurns o scale aciviy, he reurns o monopolisic rens, posiive or negaive produciviy spillovers, labor or capial of specific ypes, or he value of goods devoed o human capial accumulaion. The effec of such a facor on opimal capial axaion is insrucive. Consider he case in which consumers provide an addiional producive service, denoed A, for which hey experience disuiliy and which he governmen is unable o ax. The consumer s uiliy becomes = 0 (7.1) β u( C, L, A ) which he governmen maximizes subec o he condiions: (7.6) u u u u β C L A K0 0 C L A = C0 and (7.7) C + G + K + ) + 1 F ( K, L, A K. 88
89 Greaer levels of aciviy A generae preax reurns of A F. The inabiliy of he governmen o ax he reurn o A herefore imposes he addiional consrain: (7.18) C U A U A F The firs order condiion corresponding o an inerior choice of C is (7.19) C U A U C U C U C A U L C L u C u C C u C u µ θ λ β = + 2 2 2 2 2 2 2, in which θ is he Lagrange muliplier corresponding o he consrain (7.18). The firs-order condiion corresponding o an inerior choice of K is (7.20) 1 2 1 = + + K A F K F µ θ µ Taking he Lagrange muliplier θ o grow a rae β in he seady sae, hese condiions ogeher imply ha, in he seady sae, (7.21) K A F K F r + = 2 θ. Equaion (7.21) is inconsisen wih zero capial axaion whenever wo condiions hold simulaneously: ha consrain (7.18) binds, and ha changes in K affec he marginal produciviy of A.
In he case of ordinary human capial accumulaion, he governmen does no seek o ax A (which can be inerpreed as pas labor effor used o accumulae human capial), so θ = 0 and physical capial is unaxed as well. In he case of economies wih public goods or oher ypes of producive exernaliies, or hose in which heerogeneous inpus mus receive idenical ax reamen, a governmen ha canno use correcive axaion o induce efficien decenralized behavior will change is oher axes o accommodae he missing marke. 44 As a resul, seady sae ax raes on capial will be greaer han, equal o, or less han zero according o he naure of he exernaliy (posiive or negaive) and he complemenariy or subsiuabiliy of he unaxed facor wih capial a sandard implicaion along he lines of Corle and Hague (1953) in a saic seing. 7.5. Resuls from life-cycle models Though undoubedly a powerful and illuminaing resul, he convergence of he opimal capial income ax o zero ress on he implausible assumpion ha agens live forever or behave in an equivalen manner wih respec o heir heirs. Wihou infinie lifeimes, no such resul holds, alhough inuiion suggess ha long bu finie lifeimes sill would place srong bounds on he size of he opimal capial income ax. However, wih finie lifeimes also comes he complicaion of heerogeneiy wih respec o age cohor, which ax policy opimizaion mus ake ino accoun. Thus, here is more o learn from consideraion of finie-lifeime, overlapping generaion (OG) models han ha capial income axes should be low, if no zero, in he long run. 44 Auerbach (1979) offers a similar analysis of he opimal axaion of heerogeneous capial goods in he presence of oher consrains. Coleman s (2000) analysis of opimal consumpion and labor income axes akes he pah of fuure governmen spending o be fixed in nominal erms, which implies ha, in he seady sae, he combinaion of a consumpion ax and a labor subsidy relaxes he governmen s revenue requiremen by reducing real governmen spending. Coleman finds ha, if he labor income ax is consrained o be non-negaive, hen he opimal seady sae labor income ax rae is zero and he ax on income from capial (which is a subsiue for labor) is negaive. 90
The Diamond (1965) model, in which each generaion lives for wo periods, consuming in boh and working in he firs, provided he basis for he iniial research on opimal axaion in OG models. In his model, wihou bequess, he lifeime budge consrain for he represenaive household born in period may be wrien: 1 1 2 (7.22) C + C = wl r + 1 1+ + 1 where C 1 is consumpion when young, C 2 is consumpion when old, L is labor supply when young, and subscrips indicae periods in which aciviy occurs. As is clear from his expression, endowing he governmen wih wo insrumens, proporional axes on labor income (which affec w) and capial income (which affecs r), is equivalen o allowing he governmen o ax firs- and second-period consumpion, a possibly differen raes. A zero-ax on capial income a labor income ax would resul in uniform axaion of consumpion in he wo periods. Using his model, papers by Diamond (1973), Pesieau (1974), Auerbach (1979), and Akinson and Sandmo (1980) characerized opimal seady-sae axes under differen assumpions abou insrumens available o he governmen. Two general resuls from his lieraure are ha (1) wih governmen deb available o redisribue resources across generaions, he marginal produc of capial should converge o he ineremporal discoun rae embodied in he governmen s social welfare funcion; and (2) in his equilibrium, opimal axes on labor and capial facing individual cohors should follow he sandard hree-good analysis of saic opimal ax heory, wih a zero ax on capial income being opimal only for a cerain class of preferences. Resul (1) confirms ha Cass s (1965) modified Golden rule resul holds even in he presence of disorionary axaion. I is analogous o he Chamley-Judd resul discussed 91
above. However, as resul (2) confirms, his does no imply ha capial income axes converge o zero. The marginal produc of capial is being equaed o he governmen s discoun rae (for comparing he consumpion of differen cohors a differen poins in ime), no he discoun rae used by individual households in comparing heir own firs- and second-period consumpion. These resuls, like hose derived for he infinie-lifeime case, ell us lile abou he naure of opimal ax schedules in ransiion; nor are hey useful in deermining how he long-run opimum migh differ if he governmen faced consrains on is shor-run policy. For example, if he opimal pah for capial income axes were one of high axes declining o zero (as in Chamley s analysis), bu he governmen s decision wheher or no o abolish capial income axes had o be made on a once-and-for-all basis, would i sill improve economic efficiency o abolish capial income axes? As ransiion consrains are a maor concern of acual ax policy decisions, undersanding he linkage beween ransiion and long-run policy is imporan. Analyzing he efficiency (and incidence) effecs of ax policies in ransiion has been a maor obecive of he lieraure uilizing dynamic compuable general equilibrium (CGE) models based on more realisic characerizaions of life-cycle behavior. Auerbach, Kolikoff and Skinner (1983) and Auerbach and Kolikoff (1987) developed a 55-generaion OG model wih endogenous labor supply and reiremen, in which agens alive during he ransiion from one seady sae o anoher have perfec foresigh abou fuure facor prices and ax raes. Their cenral simulaions consider he impac of swiching immediaely from a uniform ax on labor and capial income o a ax on labor income or a consumpion ax. While such axes appear equivalen in erms of he lifeime budge consrain represened in (7.22), as well as in he 55- period version of his budge consrain, hey are no he same wih respec o ransiion generaions, who begin he ransiion wih previously accumulaed life-cycle wealh. For hese 92
ransiion generaions, a consumpion ax is equivalen o a ax on labor income plus a ax on exising wealh a capial levy. This can be seen by considering an amended version of (7.22) ha has some measure of exising asses, A, on he righ side. Thus, he ransiion o a consumpion ax is more aracive han a ransiion o a labor income ax from he sandpoin of economic efficiency. Deermining he efficiency differences beween hese wo reforms is complicaed by he fac ha he reforms also have differen inergeneraional incidence, he consumpion ax harming iniial generaions a he expense of fuure generaions, he labor income ax doing he reverse. As a resul, he seady-sae welfare gain oversaes he efficiency gain in he case of he consumpion ax, for i reflecs no only efficiency gains bu also ransfers from ransiion generaions. By he same logic, he seady-sae welfare gain undersaes he efficiency gain in he case of he labor income ax. To separae incidence from efficiency effecs, he auhors consruc a hypoheical lump-sum redisribuion auhoriy ha makes balanced-budge lumpsum axes and ransfers among generaions o ensure ha all ransiion generaions are kep a he pre-reform uiliy level and all pos-ransiion generaions enoy an equal increase in lifeime uiliy, an increase ha can be viewed as a measure of he policy s efficiency gain (or loss, if negaive). Wih his adusmen, and for base case parameer assumpions, he ransiion o a consumpion ax is prediced o increase economic efficiency, while he ransiion o a labor income ax would reduce economic efficiency. The key lesson of hese simulaions is ha ax sysems ha appear o be equivalen from he perspecive of a represenaive individual may differ significanly in an economy wih differen age cohors. A corollary is ha adoping a consumpion ax bu simulaneously providing ransiion relief for hose harmed by he ax in ransiion will offse no only adverse 93
disribuional effecs, bu also he efficiency benefis of he capial levy. Auerbach (1996) illusraes his resul in an analysis of a range of consumpion-ype ax reform proposals ha vary in he exen o which hey provide ransiion relief. The puaive efficiency advanage of he consumpion ax relies, of course, on he abiliy of he governmen o use he implici capial levy us once and raises he quesion of dynamic inconsisency discussed above. Jus as i is possible o exend he represenaive-agen, infinie-horizon model o include human capial accumulaion, his has been one direcion in which dynamic CGE models have been exended in recen years, mos noably by Heckman, Lochner and Taber (1998). 8. Conclusions The analysis of excess burden and opimal axaion is one of he oldes subecs in applied economics, ye research coninues o offer imporan new insighs ha build on he original work of Dupui, Jenkin, Marshall, Pigou, Ramsey, Hoelling, and ohers. Fundamenally, i remains rue ha deparures from marginal cos pricing are associaed wih excess burden, ha he magniude of excess burden is roughly proporional o he square of any such deparure, and ha efficien ax sysems are ones ha minimize excess burden subec o achieving oher obecives. The conribuion of modern analysis is o idenify new and imporan reasons for prices and marginal coss o differ, o assess heir pracical magniudes, and o consider heir implicaions for axaion. One of he maor developmens of he las fify years is he widespread applicaion of rigorous empirical mehods o analyze he efficiency of he ax sysem. Empirical work no only assiss he formaion and analysis of economic policy, bu also plays a criical role in disinguishing imporan from less-imporan heoreical consideraions, hereby conribuing o furher heoreical developmen. Properly execued, empirical analysis is no only consisen 94
wih he welfare heory ha underlies normaive public finance, bu also akes he heory furher by esing is implicaions and offering reliable measuremen of parameers ha are criical o he assessmen of ax sysems. Recogniion of he imporance of populaion heerogeneiy and of he poenial complicaions of evaluaing policy reforms wih pre-exising disorions has moivaed much of he recen normaive work in public finance. The new learning serves generally o highligh he value of Ramsey s insighs by demonsraing heir applicaion o a variey of seings, including hose wih populaion heerogeneiy and a wide range of available ax insrumens. Mirrlees differs from Ramsey in focussing on he role of informaional asymmeries beween governmens and axpayers as a deerminan of he shape of opimal ax schedules; neverheless, Ramsey-like condiions characerize opimal ax policy even in his seing. The efficiency of he ax sysem is a opic of enduring imporance and coninuing invesigaion. Economic analysis has much o offer on he opic of efficiency, and indeed, is occasionally criicized for offering oo much. The oher chapers in his Handbook offer wha is perhaps an illusraion of his proposiion by examining boh posiive and normaive aspecs of axaion in a wide variey of seings. 95
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Figure 2.1. The Measuremen of Excess Burden price, p p 1 1 A B p 0 0 D x 1 x 0 quaniy, x
Figure 2.2. Using Hicksian Variaions o Measure Excess Burden price, p 1 1' p 1 A C D p 0 0' 0 D c (U 0 ) D D c (U 1 ) x 1 x 0 quaniy, x
Figure 2.3. Excess Burden: An Alernaive Graphical Represenaion axed commodiy 0 0' 1 E(p 0,U 1 ) D y = E(p 1,U 1 ) numeraire commodiy R(p 0,p 1,U 1 )
Figure 2.4. Marginal Excess Burden of a Pre-Exising Tax price, p p 2 2 2' p 1 A B 1 C D E p 0 0' 0 D D c (U 1 ) x 1 x 0 quaniy, x
Figure 2.5. Excess Burden wih Varying Producer Prices axed commodiy 0 p R(q 1,p 1,U 1 ) 0 1 1 p producer price line: slope = 1/q 1 consumer price line: slope = 1/p 1 E( p 0,U 1 ) y 1 = E(p 1,U 1 ) y 0 numeraire commodiy excess burden
Figure 2.6. Excess Burden wih an Upward Sloping Supply Curve prices, p,q S p 1 A B p 0 q 1 C D D c (U 1 ) x 1 x 0 quaniy, x
Figure 4.1. Indifference Curves over Consumpion and Income c u 0 c u 0 (w=1) u 1 (w=2) u 0 (w=2) A l=1 l y=0 y
Figure 4.2. Violaion of he Self-Selecion Consrain c u H u L u H c 0 y L y H y
Figure 4.3. The Scope for Lump-Sum Taxaion u H c u H u L u L H H L L 45 o y
Figure 4.4. Using Disorionary Income Taxaion c H H L L A B 45 o C y D