Commodity Taxation an Social Welfare: The Generalised Ramsey Rule

Transcription

1 Commodty Taxaton an Socal Welfare: Te Generalsed Ramsey Rule by Davd Coady Internatonal Food Polcy Researc Unt, Wasngton, DC and Jean Drèze Del Scool of Economcs Te Suntory Centre Suntory and Toyota Internatonal Centres for Economcs and Related Dscplnes London Scool of Economcs and Poltcal Scence Hougton Street DEDPS 27 London WC2A 2AE September 2000 Tel: (020) * We are most grateful to Avnas Dxt and Agnar Sandmo for llumnatng comments.

2 Abstract Commodty taxes ave tree dstnct roles: (1) revenue collecton, (2) nterpersonal redstrbuton, and (3) resource allocaton. Te paper presents an ntegrated treatment of tese tree concerns n a second-best general equlbrum framework, wc leads to te "generalsed Ramsey rule" for optmum taxaton. We sow ow many standard results on optmum taxaton and tax reform ave a stragtforward counterpart n ts general framework. Usng ts framework, we also try to clarfy te noton of "deadwegt loss" as well as te relaton between alternatve dstrbutonal assumptons and te structure of optmum taxes. Keywords: Commodty taxes, effcency, redstrbuton, sadow prces. JEL Nos. : H21, H23. Te autors. All rgts reserved. Sort sectons of text, not to exceed two paragraps, may be quoted wtout explct permsson provded tat full credt, ncludng notce, s gven to te source. Contact addresses: Dr Davd Coady, Food Consumpton Nutrton Dvson, Internatonal Food Polcy Researc Unt, 2033 K St NW, Wasngton, DC 2006, USA. Emal: d.coady@cgar.org Dr Jean Drèze, Centre for Development Economcs, Del Scool of Economcs, Del , Inda. Emal: dreze@econdse.org

3 COMMODITY TAXATION AND SOCIAL WELFARE: THE GENERALISED RAMSEY RULE 1. Introducton Ts paper s concerned wt te welfare economcs of commodty taxaton. It focuses bot on te "tax reform" problem (.e. ow a gven system of commodty taxes can be mproved) and on te "optmum taxaton" problem (.e. te coce of optmum taxes). As s already clear from te lterature, tere s a stragtforward connecton between te two problems, snce an optmum s essentally a pont were no welfare-mprovng reform s possble. In tat sense, muc of te teory of optmum taxaton can be seen as a specal applcaton of te teory of tax reform. 1 Te paper as tree specfc objectves. Frst, we present a syntess of te classcal Ramsey tax problem (Ramsey, 1927) wc attempts to clarfy te relaton between dfferent approaces to ts problem. Second, we sow ow many of te standard results can be generalsed n a second-best general equlbrum framework usng te noton of "sadow taxes", defned as te dfference between consumer prces and sadow prces. Trd, we try to clarfy te relaton between alternatve dstrbutonal assumptons and te structure of optmum taxes. Our analyss makes consderable use of te "generalsed Ramsey rule" for optmum taxaton, derved n Guesnere (1979) and Drèze and Stern (1987). Ts s a general-equlbrum extenson of te classcal "Ramsey rule", due to Ramsey (1927). Te generalsed Ramsey rule allows for a wde range of dstortons n te economy, wc are captured by a vector of "sadow prces". Many of te standard results contnue to apply, after replacng actual commodty taxes by "sadow taxes". Condtonal on te knowledge of sadow prces, te generalsed Ramsey rule s a powerful tool for 1 Most of te teory of tax reform s concerned wt margnal reforms. A pont were no welfare-mprovng reform of ts type exsts s only a local optmum. Te optmum taxaton lterature, owever, rarely dstnguses between local and global optma (and makes lberal use of ad oc assumptons to avod undesrable local optma).

4 te analyss of commodty taxaton n dstorted economes. Te ssue of ow sadow prces are computed s outsde te scope of ts paper, and belongs to te teory of cost-beneft analyss. Te relaton between cost-beneft analyss, tax reform and optmum taxaton s explored n Drèze and Stern (1987); ts paper bulds on te general approac presented tere. 2 In general terms, ndrect taxes fulfll tree dstnct roles. Frst, tey generate government revenue. Second, ndrect taxes can be used for dstrbutonal purposes. Trd, tey ave allocatve effects. Wle te classcal Ramsey rule focuses on te revenue-rasng objectve, te generalsed Ramsey rule ntegrates tese tree concerns: (1) revenue collecton, (2) nterpersonal redstrbuton, and (3) resource allocaton. Ts nsgt provdes an mportant brdge between te lterature generated by Ramsey s orgnal paper, on te one and, and a somewat ndependent lterature wc focuses on taxes as a means of acevng a better allocaton of resources (e.g. te focus on Pgovan taxes n envronmental economcs belongs to te latter tradton). Te outlne of te paper s as follows. In te next secton, we explore varous aspects of te classcal Ramsey tax problem. Secton 3 sows ow sadow prces can be used to extend te classcal results n a general equlbrum framework, based on te generalsed Ramsey rule. Secton 4 examnes te relaton between ncome dstrbuton and te structure of optmum taxes. Te last secton concludes. 2. Te Classcal Ramsey Tax Problem For expostonal clarty, we begn wt a bref overvew of te classcal Ramsey tax problem. 2 On te calculaton of sadow prces and ter applcaton to tax reform and publc polcy, see also Amad, Coady and Stern (1988); Amad and Stern (1991); and Coady (1997). 3

5 Te results presented n ts secton are not new, but we ope to clarfy te relaton between dfferent 3 ways of dervng and nterpretng tem. We also take our frst step towards generalsaton, by lookng at te problem from te pont of vew of "tax reform" rater tan "optmum taxaton". 2.1 Te model Te problem ntally posed by Ramsey (1927) s as follows: ow sould commodty taxes be set so as to mnmse te loss of utlty to te consumer subject to rasng a gven amount of 4 revenue? Formally, ts problem may be wrtten as: Max. V(p+t) s.t. t.x(p+t) = T (1) t were p / (p,...,p ) s a vector of producer prces, t / (t,...,t ) a vector of commodty taxes, V(.) te 1 n 1 n ndrect utlty functon, x(.) te n-dmensonal vector of net consumer demands and T te amount 5 of revenue to be rased. Te frst-order condtons of ts optmsaton problem lead drectly to te "Ramsey rule", on wc more below. Two aspects of ts formulaton of te problem are wort notng: () tere s a sngle consumer, and () producer prces are assumed to be fxed. Te problem 3 Important contrbutons to te analyss of te Ramsey tax problem nclude Ramsey (1927), Dxt (1970, 1975), Damond and Mrrlees (1971), Damond (1975), Atknson and Stern (1974), Sandmo (1974), Atknson and Stgltz (1976), and Deaton and Stern (1986). Useful presentatons of te man results can be found n Sandmo (1976), Atknson and Stgltz (1980), Auerbac (1985), Dxt (1987), and Myles (1995), among oters. 4 As Ramsey mself put t: Te problem I propose to tackle s ts: a gven revenue s to be rased by proportonate taxes on some of or all uses of ncome, te taxes on dfferent uses beng possbly at dfferent rates; ow sould tese rates be adjusted n order tat te decrement of utlty may be a mnmum? (Ramsey, 1927, p.47). 5 Most of te notaton n ts paper follows Drèze and Stern (1987), partly to ensure tat te relaton between te two papers s as transparent as possble. In partcular, commodtes are ndexed by subscrpts =1,...n, and consumers (wen tere are several) by superscrpts =1,...,H; oter subscrpts ndcate partal dervatves. Also, we use bold type to ndcate vectors. 4

6 also appears to ave a "partal equlbrum" format, but n secton 3 below we sow tat t does ave general equlbrum foundatons n a specfc class of cases. Our own startng pont s a varant of te problem ntally posed by Ramsey. Followng te same notaton as n (1), consder an economy wt a sngle consumer wo as utlty V(p+t,m), 6 were m s a lump-sum transfer from te government. Let T / t.x - m denote government revenue. Socal welfare n ts economy may ten be wrtten as: * V / V(p+t,m) + ë(t.x - m) (2) 7 were ë s a sutable (postve) "sadow prce" for government revenue. We start from an arbtrary 0 0 vector of commodty taxes, say t, and consder a small cange dt from t (a "tax reform"). We begn 8 wt te smple case were te tax reform conssts of cangng a sngle tax, say t. Gven a small cange dt n t (wt dq/dt ) te cange n socal welfare (or te margnal socal value of t ) s:, MV ( Mt ' &â x % ë (x % t. Mx Mt ) (3) were â/v m s te margnal utlty of ncome. As (3)ndcates, a small cange dt nduces: (1) a cange n utlty equvalent to a lump-sum transfer of -x (usng Roy's dentty), and (2) a cange n 6 Textbook presentatons of te Ramsey tax problem often rule out lump-sum transfers (.e. m/0); we return to ts ssue furter on. Our formulaton of te problem elps to clarfy te underlyng dstrbutonal assumptons. 7 * Note tat V reflects a partcular (arbtrary) cardnalzaton of V, wc also affects ë. Any cardnalzaton (e.g. money-metrc utlty) wll do. 8 Te generalzaton to smultaneous margnal canges n all taxes s stragtforward and s dscussed later on. 5

7 revenue of (x + t.mx/mq ). Te two components ave to be added, wt sutable wegts, to derve te total cange n socal welfare. Te two components dffer by te term t.mx/mq, wc may be termed te "margnal deadwegt loss" (MDL) from te proposed tax cange. 2.2 Te deadwegt loss Equaton (3) may also be wrtten as MV ( Mt / (ë&â) x % ë ä (4) were ä / t. Mx Mq (5) s te margnal deadwegt loss and can be nterpreted as te ndrect cange n revenue arsng from te tax cange. For convenence we also defne te "normalsed" MDL as: ó / t. Mx Mq x (6) wc, muc lke an elastcty, s ndependent of te coce of unts. Te frst term n (4) s zero f and only f ë and â are te same,.e. government revenue and prvate ncome are deemed to ave te same margnal socal value. In tat case, te MDL s a correct frst-order measure of te cange n socal welfare nduced by a small cange n t. If ë and â are not te same, ten MDL can stll be 6

8 tougt of as a useful notonal quantty wc measures some knd of (usually negatve) effcency effect of extra taxes. Te nterpretaton of ä as an effcency effect can be clarfed by relatng te margnal deadwegt loss to a more general crteron of cost-beneft analyss, te "aggregate benefts crteron" 9 (ABC). Te ABC crteron for evaluatng a partcular polcy cange (e.g. a publc-sector project, a tax reform, or a modfcaton of quantty controls) conssts of askng every agent n te economy ow muc te cange s "wort" to m or er n terms of some pre-specfed numérare, and takng te unwegted sum of all tese net benefts as a measure of te socal wort of te proposed cange. By analogy, t s temptng to suggest usng te MDL as a measure of te welfare effect of a small tax cange. Te dea, muc as wt te ABC crteron, s tat f te MDL s postve, ten te loser (te consumer) can notonally compensate te ganer (te government) for renouncng te proposed cange: for a small cange dt, x s te net loss to te consumer, (x + t.mx/mq ) s te net beneft to te government, and te MDL s te dfference. Ts suggeston, owever, capszes not only on te usual ptfalls of "compensaton crtera" but also because te noton tat prvate ncome and government revenue do not ave te same margnal socal value (.e. â ë) s a central motvaton for commodty taxaton n te frst place. Deadwegt loss measures, owever, do ave a role n dentfyng desrable tax reforms tat leave revenue uncanged. To see ts, let us consder te trade-off between consumer welfare and government revenue wen t vares (locally). More precsely, let us ask ow muc revenue s rased wen te tax on commodty s ncreased to te extent tat te consumer suffers a declne n utlty 9 On ts, see Drèze (1998), and te lterature cted tere. 7

9 10 equvalent to te loss of one unt of ncome. Te latter conventon mples dt = 1/x, and te ncrease n revenue from suc a tax cange (say Ä) s ten: Ä / x % t. Mx Mq x / 1 % ó (7) Here, te numerator measures te responsveness of tax revenue (T) to a small cange n t and te denomnator captures te responsveness of consumer utlty. Te rato between te two s te tradeoff between revenue and utlty. From (7), t mmedately follows tat, f ó > ó j, ten a (margnal) swtc from t to t s desrable. Tat s, a small cange (dt,dt ) wc leaves T uncanged, wt dt j j 11 > 0, mproves consumer utlty and terefore (gven tat T s uncanged) socal welfare. Tus, tax reform sould am at swtcng away from commodtes wt a g normalzed MDL towards tose wt a low normalzed MDL. In sort, te MDL as two possble nterpretatons. Frst, t s a measure of cange n socal welfare under specfc (and strong) dstrbutonal assumptons. Second, t can be nterpreted n terms of a trade-off between revenue and utlty,.e. t s a correct measure of welfare cange for tax reforms tat leave revenue uncanged. 2.3 An alternatve formula Gong back to (3) and (4), te same denttes may be wrtten as follows, usng te Slutsky 10 Equvalently, one could consder ow muc consumer welfare declnes wen te tax on commodty s rased just enoug to rase one unt of extra revenue. In bot cases, wat matters s te trade-off between utlty and revenue. 11 Alternatvely, a small cange (dt,dt ) wc leaves utlty uncanged wt dt >0, ncreases te tax revenue. j 8

10 equaton: MV ( Mt ' & b x % ë t. Mx c Mq (8) were b / MV( Mm Mx ' â & ë % ë t. Mm (9) c and x (.) denotes compensated demand functons. Followng Drèze and Stern (1987), we refer to b as te "margnal socal value of transfer". Indeed, b measures te cange n socal welfare nduced by a small ncrease n te lump-sum transfer m. Note tat, muc as n (3) and (4), te last term n (9) (te "tax propensty" t.mx/mm, multpled by ë) may be tougt of as a margnal deadwegt loss from a lump-sum transfer. It may seem surprsng tat lump-sum transfers nvolve a deadwegt loss. However, n a second-best envronment, actual lump-sum transfers (as opposed to te "notonal" transfers nvolved n compensaton crtera) ave non-neutral allocatve effects tat need to be taken nto account. To put t anoter way, followng muc te same reasonng as n (7) we may nterpret (1+ t.mx/mm) as te trade-off between consumer welfare and government revenue wen m vares. By analogy wt (4), (8) may also be wrtten as: MV ( Mt ' & b x % ë ä c (10) 9

11 were ä c / t. Mx c Mq (11) c s an alternatve measure of te MDL. To dstngus ä from ä, we sall refer to te former as te "margnal excess burden" (MEB), even toug te terms "excess burden" and "deadwegt loss" tend to be used ntercangeably n te lterature. Muc as before, we defne te normalsed MEB as ó c c / ä /x. Followng te same reasonng as n te precedng sub-secton, t s ten easy to sow from c c (10) tat, f ó > ó, ten a swtc from tax j to tax s desrable. Ts result also follows from j c notng tat ó and ó dffer only by a constant. Indeed, usng te Slutsky equaton agan, we ave: ó / 1 x t. Mx Mq ' 1 x t.( Mx c Mq & x Mx Mm ) ' óc! c (12) were c / t.mx/mm s te "tax propensty" (more precsely, te margnal propensty to pay taxes out c of lump-sum ncome). Gven tat ó and ó dffer only by a constant, t does not really matter weter te analyss focuses on te margnal deadwegt loss (5) or on te margnal excess burden 12 (11). Wen we extend te analyss to a many-consumer economy, owever, ts dstncton wll matter. For future reference, we note tat, usng te symmetry of te Slutsky matrx, te normalsed margnal excess burden can also be wrtten as: 12 See Hoff (1994) for a wder dscusson of ts ssue. 10

12 ó c / t. Mx c Mq (13) x In te optmum-taxaton lterature, te rgt-and sde s known as te "ndex of relatve dscouragement" of commodty (Mrrlees, 1976). For small taxes, te ndex of relatve dscouragement measures te proportonate reducton n te consumpton of commodty nduced by te tax system. For non-margnal taxes, an alternatve nterpretaton s also possble: te ndex of relatve dscouragement s a frst-order approxmaton to te proportonate cange of compensated demand tat would be nduced by a small ntensfcaton of te tax system (.e. a cange dt suc tat dt=k.t for some k>0). 2.4 Vector tax canges Te precedng analyss extends wtout dffculty to te evaluaton of a "tax reform" dt wc nvolves smultaneous canges n many taxes. In ts case, te cange n socal welfare s: dv ( ' (ë & â) x.dt % ë t Mx.dt (14) Mq were te last term contnues to be nterpreted as te MDL. Smlarly, te extenson of (10) s smply: dv ( ' & b x.dt % ë t. Mx c.dt (15) Mq were te last term s te MEB. Also as before, a tax reform dt wc leaves revenue uncanged 11

13 (.e. suc tat dt/dt / (x + t.mx/mq).dt = 0) s socally desrable f and only f t reduces te excess burden. 2.5 Optmum taxes and te Ramsey rule * Usng (4), te frst-order condton for optmalty of t, MV /Mt = 0, may be wrtten as: ó / t. Mx Mq x ' â&ë ë (16) Let us say tat t s "unconstraned" f t can be cosen optmally. Recallng (6), (16) states tat te normalzed MDL sould be te same for all unconstraned taxes. Ts statement can also be seen as a corollary of our earler result tat swtces from g-mdl to low-mdl taxes are welfaremprovng. Usng (10) and (11) and te symmetry of te Slutsky matrx, te frst-order condton (16) may also be restated as: t. Mx c Mq x ' b ë (17) Tat s, te ndex of relatve dscouragement sould be te same for all unconstraned taxes. Ts s known as te Ramsey rule. If te amount of revenue to be rased s small, (17) mples tat te proportonate reducton n compensated demand nduced by te tax system sould be te same for all taxable commodtes - te formula proposed by Ramsey (1927) Ramsey's (1927) semnal analyss focuses on te case were t =0 and nfntesmal taxes are ntroduced. 0 Tree ponts are wort notng about ts specal case. Frst, wen nfntesmal taxes are ntroduced at t =0, bot MDL and MEB are zero for any dt (.e. all tax canges tat yeld a unt of revenue lead to te same frst-order cange n 12

14 Varous nterpretatons and specal cases of te Ramsey rule ave been dscussed n te lterature. For nstance, several autors ave dscussed condtons under wc te Ramsey rule bols down to proportonal taxes (varously defned). Te nterested reader s referred to te papers mentoned n footnote 4. It s nstructve to rewrte te rgt-and sde of (17) as [â+ët.(mx/mm)-ë]/ë, were te numerator can be nterpreted as te dfference between te actual margnal socal cost of rasng 14 revenue troug taxes (.e. ë) and te cost f t were rased va lump-sum taxes. As dscussed below, under reasonable assumptons t can be sown tat (b/ë) s negatve. 2.6 Optmum lump-sum transfer 15 So far, we ave assumed tat te consumer's lump-sum ncome m s exogenous. Ts s n te sprt of Ramsey's orgnal formulaton, wc focuses on te use of commodty taxaton as a (second-best) substtute for lump-sum taxaton. Indeed, f m s a coce varable, ten commodty taxes are redundant. To see ts, note tat te frst-order condton for optmum coce of m s smply (from (9)): utlty). Second, as Ramsey demonstrated, examnng second-order terms leads to s orgnal "rule": "... te taxes sould be suc as to reduce n te same proporton te producton of eac taxed commodty (Ramsey, 1927, p.52). Trd, snce ncome effects are second order n ts specal case, t does not matter weter te proportonate reducton s measured along compensated and uncompensated demands. In general, owever, te dstncton does matter. Wle Ramsey stated s orgnal rule n terms of uncompensated demands, te "Ramsey rule" (equaton 17) s best expressed n terms of compensated demands. Indeed, outsde of ts specal case, te prncple tat commodty taxes sould lead to an equproportonate reducton n uncompensated demands makes no ntutve sense. 14 On ts nterpretaton, see Auerbac (1985, p. 88). 15 Some readers may fnd t easer to tnk n terms of a lump-sum tax L / (-m). Te two approaces are, of course, equvalent. 13

15 b = 0 (18) * Wt b = 0, MV /Mt evaluated at t 0 = 0 s zero for eac (see (8)). Tat s, tere s no welfare gan from a small move away from zero commodty taxaton. Alternatvely, we may note tat, f m s a coce varable, ten ë = â and t = 0 solve (16) and (18). Now consder te case analysed by Ramsey, were m s not a coce varable and all taxes 16 are eter unconstraned or zero. Multplyng bot sdes of (17) by t x and summng over, we get: b = (ë/t)(t.s.t) (19) were T / t.x s total tax revenue and S s te Slutsky matrx. Rememberng tat te Slutsky matrx 17 s negatve sem-defnte, te second-term on te r..s. s negatve. Wt ë > 0 by assumpton, (19) mples tat b and T ave opposte sgns. In most sensble applcatons of ts problem, T s postve, mplyng b < 0. Tus, a swtc towards lump-sum taxaton s desrable. 18 If tere are constraned non-zero taxes, b need not be negatve. For nstance, f te consumer s already burdened wt large taxes on some commodtes, a small ncrease n lump-sum ncome (dm>0) may be socally desrable (b>0). Note also tat n te event were some taxes are constraned and non-zero, an optmum lump-sum tax (b=0) does not ental equalzng â and ë (te 16 If all taxes are unconstraned, te assumpton m=0 s needed to make te problem non-trval (see next secton) We rule out te trval case were b=0. See Atknson and Stern (1974) for furter dscusson. 14

16 margnal socal values of prvate ncome and government revenue, respectvely). 2.7 Prce normalzaton and te untaxed good One ssue tat as caused muc confuson n te lterature s weter t s permssble to assume tat a partcular commodty s untaxed (and, f so, weter te coce of untaxed commodty matters). In oter words, wen does ts assumpton amount to notng more tan an nnocuous prce normalzaton rule? To begn wt, consder te standard textbook case were m / 0 and all taxes are "unconstraned". Remember tat producer prces are fxed; by coce of unts, we can set tem equal to one, wtout loss of generalty. Furter, ts model s omogenous (of degree 0) n consumer prces: f q and q' are collnear, notng canges. It follows tat te consumer prce of one commodty can be gven an arbtrary value, wtout loss of generalty. In partcular, we can set q/p for some pre-specfed, tat s, assume tat one commodty s untaxed. To put t anoter way, let t* be a soluton of te Ramsey tax problem. Ten t s easy to verfy tat t' / k.t* + (k-1).p s also a soluton, for any postve k (ndeed, t* and t' correspond to te same consumer prces). For any * gven commodty, we can opt for ts commodty beng untaxed by settng k / p /(p +t ). Next, consder te extenson were m s not dentcal to zero. As before, and n keepng wt te sprt of Ramsey's model, we assume tat m s not a coce varable; nstead, t takes an 19 exogenous value. Note tat te model s no longer omogeneous n q (toug t s omogeneous n (m,q)). Hence, an a pror restrcton on q, e.g. q / 1 or q / p for some, would (n general) be 1 19 Alternatve assumptons are possble, e.g. m / M/I(q) were M s an exogenous parameter and I(q) s an ndex of consumer prces. Te coce of assumpton s not (n general) "neutral", and as to reflect te actual bass on wc lump-sum transfers are determned. 15

17 a real restrcton, not just a normalzaton rule. However, tere s a complcaton: some restrcton s needed to make te problem nterestng. Indeed, f all taxes are "unconstraned", te problem becomes trval. Ts s because, f m s non-zero, proportonal taxes (t = k.p for some k>0) are a form of mplct lump-sum taxaton and present an obvous "soluton", toug te problem tey are 20 meant to solve as effectvely vansed. Ts feature of te model, owever, s not a matter of concern: n te real world t s not dffcult to dentfy non-taxable commodtes (e.g. ome-grown 21 food), and ts s enoug to make te problem non-trval. Te crucal pont s tat, n ts case, te coce of non-taxable commodty s not just a normalzaton rule: t s a matter of emprcal enqury and affects te structure of optmum taxes. 2.8 Te "nverse elastcty rule" A popular rule of tumb n te feld of commodty taxaton s tat te tax system sould weg eavly on commodtes wt relatvely nelastc demands. Te basc dea s tat (1) n Ramsey s model, commodty taxes are a second-best substtute for lump-sum taxaton, and (2) a tax on a commodty wt nelastc demand works muc lke a lump-sum tax, and s terefore a natural substtute for t. Te ntuton can also be developed as follows: te Ramsey rule requres equal dscouragement of eac taxable commodty, and commodtes wt nelastc demands requre ger taxes to aceve te same dscouragement. Cross-prce effects, of course, potentally nvaldate ts reasonng. 20 See Dxt (1970), and te rejonder by Sandmo (1974). 21 Lesure s often cosen as te untaxed good. In some models ts s an nnocuous normalzaton rule, n oters t s a substantve restrcton. 16

18 Formally, suppose tat t s unconstraned and tat Mx /Mq s zero for all j suc tat j and j t 0. Equaton (16) ten mples: j (t /q ) = µ/å (20) were å s te own-prce elastcty of demand for good and µ/(â-ë)/ë s ndependent of. Tus, 22 t sould be nversely proportonal to å. An analogous result olds n terms of compensated demand elastctes. Ts follows mmedately from (17), under smlar assumptons. 2.9 Te many-person Ramsey rule Te results derved so far extend qute easly, but wt some mportant qualfcatons, to te case were tere are many consumers. As far as te structure of te problem s concerned, we smply ave to replace te utlty functon V(.) n (2) wt a sutable socal welfare functon. Assumng, as n muc of te lterature, tat te latter takes te Bergson-Samuelson form, we rewrte (2) as: * 1 1 H H V / W(V (p+t,m ),...,V (p+t,m )) + ë(t.x - Óm ) (21) were m s a lump-sum transfer to consumer (=1...,H). In order to extend te earler results, we frst ntroduce te followng notaton: 22 Note tat tere s no guarantee (n general) of µ/å beng postve. For nstance, f te ntal poston s suc tat some commodtes bear eavy postve taxes tat cannot be removed (for poltcal or oter reasons), te planner may ws to compensate for ts by subsdsng te commodtes wt unconstraned taxes. Te nverse elastcty rule ten suggests tat subsdes sould be concentrated on commodtes wt a low elastcty of demand. 17

19 â / MW MV MV Mm (22) â / 3 â x x (23) b / â & ë % ë t Mx Mm (24) b / 3 b x x (25) Te term â can be nterpreted as te socal valuaton of te margnal utlty of ncome of ndvdual ; we sall also use te more standard (but less precse) term socal margnal utlty of 23 ncome. We sall refer to b, a stragtforward generalzaton of b (see (9)), as te margnal socal value of a transfer to person. Te quanttes â and b are smply commodty-specfc wegted averages of (â ) and (b ), respectvely, were te sares of dfferent persons n total consumpton of commodty are used as wegts. Te term b s known as te net dstrbutonal 23 Anoter standard term s welfare wegt. On te nterpretaton and measurement of welfare wegts, see Stern (1977). 18

20 caracterstc of commodty. Wt tese notatonal conventons, most of te earler equatons go troug after replacng âx everywere wt â x and bx wt b x. In partcular, te Ramsey rule becomes: t. Mx c Mq x ' b ë (26) Equaton (26) s known as te many-person Ramsey rule (Damond, 1975). Note tat te rgtand sde s no longer ndependent of. Instead of beng te same for eac commodty, te ndex of relatve dscouragement sould now be proportonal to te net dstrbutonal caracterstc. In te many-person case, we can stll defne te margnal deadwegt loss and te margnal excess burden as n (5) and (11), respectvely. Note, owever, tat te dfference between (normalsed) MDL and MEB s no longer constant across commodtes. Furter, t s no longer desrable to equate (normalsed) MDL or MEB across commodtes, unless one s prepared to make te furter assumpton tat te dstrbuton of ncome s n some sense optmal. More precsely, under te assumpton tat â s dentcal for eac, te normalsed MDL sould be equalsed across commodtes; on te oter and, f b s dentcal for eac, te normalsed MEB sould be 24 equalsed across commodtes. Te dstncton between tese two dstrbutonal assumptons s dscussed n secton Furter observatons 24 Tese results are easly derved from (5), (11) and (26). 19

21 We end ts secton by notng, for future reference, tat wen lump-sum transfers are feasble t s possble to dentfy comparatvely smple drectons of welfare-mprovng tax reform (Dxt, ). To llustrate, let us go back to equaton (15), wt b=0. Suppose tat dt = -á.t for some small á > 0. Tus, we are consderng a small proportonate reducton of all taxes. Substtutng nto (15), and bearng n mnd tat te Slutsky matrx s negatve sem-defnte, we fnd tat te rgtand sde s postve: a small proportonate reducton of all taxes s welfare-mprovng. Note tat n te case beng consdered, optmum commodty taxes are zero. Tus, n ts case a small move n te drecton of te optmum from an arbtrary ntal pont t s always welfare-mprovng. o Te above llustraton descrbes wat s know as a "radal reform" of commodty taxes. Anoter example s te "concertna reform", wc conssts of reducng te gest taxes only. For furter dscusson of tese and oter welfare-mprovng tax reforms under optmum lump-sum transfers, te reader s referred to Dxt (1975). 3. Te Generalsed Ramsey Rule 3.1 Te model We now consder a generalsed verson of te problem dscussed n te precedng secton. 26 To start wt, we rewrte te vector of net demands of consumer as x (q, xg, m ), were xg s an n-dmensonal vector of quantty constrants and x (.) solves: 25 In te many-person case, te frst term on te rgt-and sde of (15) as to be approprately re-wrtten, but t s stll zero f tere are optmum lump-sum transfers (b = 0 for eac ). 26 For a more detaled exposton, see Drèze and Stern (1987). 20

22 Max. U (x ) s.t. q.x = m x x # xg for eac Note tat, subject to regularty condtons, te constraned demand functons xg (.) ave te standard propertes wt respect to q and m. For nstance, Roy s dentty and te Slutsky equaton contnue to apply. Smlarly, te Slutsky matrx S s symmetrc and negatve sem-defnte and q.s = 0. Te man dfference s tat te matrx S as columns of zero entres for commodtes suc tat te quantty constrant s bndng (because a small cange n te prce of suc a commodty works lke a cange n lump-sum ncome). Te fxed ncome m of ndvdual conssts of te sum of s or er sare n prvate profts and a lump-sum transfer (r ) from te government: m / r % jg è g. Ð g (27) g g g were Ð / p.y s te proft of frm g (see below) and è s te sare of ndvdual n frm g's profts. g g g g Next, we wrte te net supply vector of frm g (g = 1,...,G) as y = y (p, ), were s a g vector of quantty constrants and y (.) solves: g g g Max. p.y s.t. y 0 Y y g g g y # for eac g 27 wt Y denotng te producton set of frm g wc s assumed to be convex. Here agan, te 27 Strctly speakng we only need convexty n te space of commodtes for wc te quantty constrants are not bndng. 21

23 g constraned supply functons y (.) ave te standard propertes wt respect to p, te vector of producer prces. Note also tat te quantty constrants enable us to treat te net supply vectors of frms operatng under constant returns to scale as functons rater tan correspondences. g g Let (p, t, {xg }, { }, {r }, {è }) be te vector of "sgnals" to wc ndvduals and frms respond (bearng n mnd (27) and te dentty q / p + t). Te sgnals are of two types: exogenous sgnals or "parameters", and control varables. Te dea s tat te "planner", wo conducts te optmzaton exercse below, cooses te varables wtn s or er control, takng oter varables 28 as gven. Denotng te vectors of parameters and control varables as ù and s respectvely, te planner cooses s to solve te followng maxmzaton problem: g Max. W(...,V (s;ù),...) s.t. 3x (s;ù)-3gy (s;ù)-z=0 (P) s were z s te vector of net supples from te publc sector. As before (see (21)), V s s ndrect utlty functon and W s a Bergson-Samuelson socal welfare functon. Te model s restrctve n some respects (e.g. te dcotomous parttonng of sgnals between exogenous varables and control varables), but t does encompass a wde range of models n te second-best tradton. Te classcal Ramsey tax problem, for nstance, s a specal case of ts model. So s Damond-Mrrlees's classc model of "optmum taxaton and publc producton" (Damond and Mrrlees, 1971). Smlarly, te general equlbrum model underlyng te project evaluaton manual by Lttle and Mrrlees (1974) can be expressed n te present format. Note also 28 A more sopstcated model would take nto account te possblty of endogenous polcy responses by government agences not under te control of te planner. 22

24 tat te basc model can be readly extended n varous drectons, e.g. to take nto account Boteuxtype budget constrants n te publc sector. 3.2 Polcy reform * Let V (ù,z) denote te maxmum value functon of (P). From te envelope teorem, we know tat te gradent of V* s te same as te gradent of te Lagrangean. Te latter may be wrtten as: = W(...,V (s;ù),...) - í [x(s;ù) - y(s;ù) - z] g were x/ó x and y/ó y denote aggregate (net) consumer demands and aggregate (net) producer g supples, respectvely, and í s a vector of Lagrangean multplers or sadow prces. If ù s a k partcular component of te vector ù of parameters (e.g. a tax, a lump-sum transfer, or a quantty constrant), te margnal socal value of ù k, say MSV k, s: MSV k / MV( Mù k / M Mù k ' j MW MV MV Mù k & í. M(x&y) Mù k (28) Te frst term on te r..s. of (28) s te drect effect on socal welfare, and te second term s te socal value of te addtonal excess demands generated by te proposed reform. Applyng te same reasonng to a small cange dz n te publc producton plan, we fnd tat te project dz s welfare mprovng f í.dz>0,.e. f t makes a proft at sadow prces. Of course, te sadow prces depend on te specfcaton of "coce varables" For an early dscusson of ts pont, see Sen (1972). 23

25 As sown n Drèze and Stern (1987), usng Walras law, te Lagrangean may be rewrtten as: (s;ù) / W(...,V (s;ù),...) % ë R (29) were R s te "sadow revenue" of te government. Te latter s defned as: R / í (.z % ô.x % ô p.y % 3 g è 0g.Ð g & 3 r (30) were í*/í/ë s a vector of normalzed sadow prces, ô/(q-í*) s a vector of sadow consumer p 0g taxes (or sadow taxes for sort), ô /(í*-p) s a vector of sadow producer taxes, and è /(1-3 g è ) s te government s sare n te profts of frm g. Note tat ë s bascally a normalzaton parameter: a dfferent cardnalzaton of te socal welfare functon W(.) leads to a dfferent ë, but leaves í*/í/ë uncanged. Equaton (29) s extremely useful n so far as t converts ts complex general-equlbrum model nto te standard format of a trade-off between consumer welfare and (sadow) government revenue, as n secton 2. To llustrate, let us consder a specfc "parameter", say r (te lump-sum transfer gong to ndvdual ). From (28), we may wrte te "margnal socal value" of r (say b ) as: b / MV( Mr ' â & í.( Mx (31) Mm ) were, as before, â denotes te socal margnal utlty of 's ncome (see secton 2.9). Tus, te margnal socal value of a lump-sum transfer to consumer s smply te dfference between s or er socal margnal utlty and te sadow value of te addtonal consumer demands generated by 24

26 a transfer. Equvalently, we can derve b from (29) and (30) as: b / MV( Mr ' â & ë (1 & ô. Mx (32) Mm ) Te latter expresson also follows from (31), te defnton of í and te addng-up property. Note tat (32) s exactly te same as (24), wt sadow taxes replacng actual taxes. 3.3 Commodty taxaton Turnng to te specfc ssues of tax reform and optmum taxaton n ts general model, te key pont to note s tat te gradent of (29) wt respect to t s muc te same as n te classcal * Ramsey tax problem (see (3)). Indeed, substtutng q/p+t and ô/(q-í ) n (29) and usng te standard propertes of x (.) and V (.), t s easy to derve: M Mt / &â x % ë (x % ô. Mx Mt ) (33) Comparng (33) wt (3), tere are two dfferences. Frst, te term âx s replaced wt â x, exactly as n te extenson from te one-person to te many-person Ramsey rule (see secton 2.9) - notng new ere. Second, n te last term on te rgt-and sde, te vector ô of sadow taxes replaces te vector t of actual taxes. Hence, te earler results are readly extended, by smply replacng actual taxes wt sadow taxes were approprate. For nstance, settng te rgt-and-sde of (33) equal to zero and followng te usual steps (as wt te dervaton of (17) and (26)), te frst-order condtons for optmum taxaton lead to te generalsed Ramsey rule: 25

27 ô. Mx c Mq x ' b ë (34) were, as before, b/3 (x /x )b and b s te margnal socal value of a transfer to ndvdual (dscussed n te prevous secton). By analogy wt (13) n secton 2, we can refer to te l..s. of (34) as te "generalsed ndex of relatve dscouragement". Te generalsed ndex of dscouragement s a useful tool to restate te general prncples of optmum taxaton, bearng n mnd te tree core objectves stated n te ntroducton: (1) resource allocaton, (2) revenue collecton, and (3) nterpersonal dstrbuton. Te mplcatons of te frst objectve can be seen by assumng optmum lump-sum transfers (.e. b = 0 for eac ). In tat case, te generalsed Ramsey rule states tat te generalsed ndex of dscouragement sould be zero for eac commodty. If all taxes are unconstraned, ts s aceved by brngng consumer prces n lne wt sadow prces (ô=á.q for some á 0, so tat te left-and 30 sde of (34) s zero for eac ). To llustrate te mplcatons, n a labour-surplus economy ts would typcally requre subsdsng labour-ntensve commodtes, snce te latter would tend to ave a low sadow prce (Drèze and Stern, 1987). Smlarly, n a foregn-excange-scarce economy, t may requre taxng mported goods. Note tat ts dervaton s based on dfferentatng te Lagrangean wt respect to t as f producer prces (p) were constant. Ts, owever, does not requre te assumpton tat prces are 30 Te stuaton were consumer prces and sadow prces are collnear s known as "C-C effcency"; for furter dscusson, see Guesnere (1979). 26

28 actually constant. If producer prces are among te "control varables", te dervaton remans vald, by te envelope teorem. Furter, t must be remembered tat te "control varables" formally nclude te market-clearng varables,.e. te varables tat mplctly adjust to clear te scarcty 31 constrants. Seen n tat lgt, te devce of oldng producer prces constant s muc more general tan t appears at frst sgt. More precsely, te dervaton s vald f any of te followng olds: (1) producer prces are fxed (as n Damond and Mrrlees, 1975); (2) producer prces adjust endogenously to clear te scarcty constrants; (3) producer prces are drectly controlled by te 32 planner. If none of tese assumptons apply, te mecansm troug wc producer prces are determned as to be specfed n te form of addtonal constrants n te optmsaton problem (e.g. te requrement tat margnal revenue equals margnal cost, n te case of a commodty suppled by a monopoly). If te vector t enters n tese addtonal constrants, te frst-order condtons for optmum taxaton ave to be correspondngly modfed. Te requrements of revenue collecton can be seen by contnung to assume tat te nterpersonal dstrbuton of ncome s optmal (b = b for eac, for some b), and also assumng tat a swtc from ndrect taxaton to lump-sum taxaton would be welfare-mprovng (b > 0). In tat case, te generalsed Ramsey rule states tat consumer prces sould move away from sadow prces n suc a way tat te generalsed ndex of relatve dscouragement remans te same for eac commodty. In te specal case were one commodty can be assumed to ave zero sadow tax (troug sutable normalzaton) and compensated cross-prce demand elastctes among oter commodtes are zero, we obtan te generalsed nverse elastcty rule: 31 For furter dscusson on ts pont and related ssues, see Drèze and Stern (1987, 1990). 32 Combnatons of tese dfferent assumptons for dfferent commodtes are also possble (e.g. fxed prces for traded commodtes and market-clearng prces for domestc commodtes). 27

29 ô ' b e c were e/(mx /Mq )(q /x ) s te compensated prce elastcty of demand for good. Fnally, to see te mplcatons of concern for nterpersonal equty, we relax te assumpton tat b s te same for eac. In tat case, tere as to be a furter adjustment of consumer prces to brng te vector of generalsed ndces of dscouragement n lne wt te vector of net dstrbutonal caracterstcs. Typcally, ts would mean sftng taxes away from commodtes eavly consumed by transfer-deservng ndvduals. All tese prncples make good ntutve sense. Te results llustrate te power of sadow prces n actng as "suffcent statstcs" for a general equlbrum model: dfferent applcatons of te general model outlned n secton 3.1 lead to dfferent sadow prces but, subject to te calculaton 33 of sadow prces, te basc rules of tax reform and optmum taxaton reman te same. Sadow prces also provde a useful brdge between te teory of taxaton, te teory of cost-beneft analyss, and te teory of te second-best. In te specal case were sadow prces are proportonal (wtout loss of generalty, equal) to producer prces, sadow taxes are te same as actual taxes and te results derved n secton 2 apply as stated tere. Tus, te classcal Ramsey tax problem, wc appears to ave a partalequlbrum format, does ave a general-equlbrum foundaton, as long as sadow prces and 33 Sadow prces, of course, are not (generally) ndependent of te tax system. Strctly speakng, all te relevant control varables and sadow prces are smultaneously determned. Neverteless, te sadow-prce approac as muc to commend n terms of conceptual clarty. Furter, n te context of "tax reform", sadow prces can be taken as gven even toug tey may depend on te ntal vector of taxes. 28

30 producer prces concde. 34 Te precedng dscusson focuses on optmum taxaton, but smlar extensons of te standard results apply n te context of tax reform. For example, te condtons for radal and concertna reforms to be welfare mprovng stll go troug but wt tax reductons sutably defned n te space of sadow taxes. Te detals are left to te reader. 4. Optmum Dstrbuton: Interpretaton and mplcatons Polcy reforms are often evaluated from te pont of vew of "effcency" by lookng at ter 35 mplcatons as f dstrbuton dd not matter. But wat s meant by te latter assumpton? At least two nterpretatons are possble: (A1) Equal margnal socal utltes: â = â for all (for some â). (A2) Equal margnal socal values of transfer: b = b for all (for some b). Assumpton (A1) states tat te socal valuaton of te margnal utlty of ncome s te same for all ndvduals. (A2) states tat a small ncome transfer between any two ndvduals would leave socal welfare uncanged. Te two assumptons ave often been used ntercangeably n te lterature. However, tey are not dentcal, bearng n mnd tat actual transfers ave allocatve mplcatons 34 Te condtons under wc sadow prces are proportonal to producer prces are dscussed n Drèze and Stern (1987, 1990). In te openng paragrap of s paper, Ramsey (1927) sows some awareness of ts feature of s model: "...I sall suppose tat, n Professor Pgou's termnology, prvate and socal net products are always equal or ave been made so by State nterference not ncluded n te taxaton we are consderng" (p.47). 35 For one example (among many), see Dxt (1975). 29

31 (captured n b, but not n â - see sectons 2.9 and 3.2). To llustrate, t s sometmes argued tat oterwse welfare-mprovng transfers from te rc to te poor may not be desrable n a savngsconstraned economy, f te rc ave a ger margnal propensty to save tan te poor (see Drèze and Stern, 1987, p.966). In oter words, equalzng margnal socal values of transfer need not ental equalzng margnal socal utltes (and vce-versa). It s also useful to ntroduce a stronger verson of (A2): (A2') Optmum transfers: b = 0 for all. Note tat (A2') nvolves te furter assumpton of optmum dstrbuton between consumers and te government: a small transfer from te government to any ndvdual leaves socal welfare uncanged. ave: Wen are (A1) and (A2) equvalent? Recallng te defnton of b (see (31) and(32)), we â & b / í Mx Mm Mx / ë (1 & ô (35) Mm ) Tus, â and b essentally dffer by a constant mnus te "sadow tax-propensty" (te last term n (35)), so tat (A1) and (A2) are equvalent f and only f te latter s te same for all ndvduals. 36 It s easy to see tat a suffcent condton for ts to old s te followng: 36 Te sadow tax-propensty s essentally a wegted average of sadow tax rates, wt margnal propenstes to consume as wegts. Wt sutable normalzaton, te sadow tax-propensty may also be nterpreted as te covarance between te vector of sadow tax rates and te vector of margnal propenstes to consume. 30

32 (B1) Proportonal sadow taxes: í = ã.q for some ã > 0 (.e. ô/q s te same for all ). Condton (B1) s also necessary for (A1) and (A2) to be equvalent f te (nxh) matrx R wt elements R / (Mx /Mm ) as rank n. To see ts, frst wrte (35) n vector form as follows: â & b/ í R (36) were â / {â } and b / {b }. Note also tat q.r = 1, were 1 ndcates a vector of ones. If bot (A1) and (A2) old, ten tere s a constant c suc tat b = â - c.1. Tus í.r / â - b = c.1 = c.q.r (37) If R as rank n, (38) mples tat í and q are collnear,.e. sadow taxes are proportonal. For te rank condton to be satsfed, tere ave to be at least as many consumers as tere are commodtes, and te H vectors of margnal propenstes to consume sould span a space of dmenson no less tan n. In oter words, tere as to be adequate dversty of (margnal) spendng patterns. Wen te rank condton olds, any dvergence between í and q makes t mpossble for bot (A1) and (A2) to be satsfed. Te followng s a corollary of te reasonng so far: Proposton 1: If R as rank n, ten (B1) s necessary and suffcent for (A1) and (A2) to be equvalent; furter, any two of (A1), (A2) and (B1) mply te trd. 31

33 Wen te rank condton s not satsfed, assumptons oter tan (B1) can be made to ensure tat sadow tax-propenstes are dentcal (so tat (A1) and (A2) are equvalent). One suc assumpton s tat everyone as te same margnal propensty to consume, for eac commodty (te extreme opposte of te assumpton of "adequate dversty"). Anoter one s tat, for eac ndvdual, margnal propenstes to consume are uncorrelated wt sadow taxes. Obvously, all tese assumptons are restrctve. To see wy te dvergence of sadow tax propenstes s an ssue, consder any feasble "reform" suc tat te ganers gan more tan te losers lose (were any person's net gan or loss s 37 measured n terms of ncome equvalent). If (A1) olds, ten te reform s welfare-mprovng. Sometmes t s argued tat even f (A1) does not old, te reform can be justfed on effcency grounds, n te sense tat "te ganers could compensate te losers". However, unless sadow taxpropenstes are dentcal, transfers from te ganers to te losers ave effcency mplcatons tat need furter examnaton, so tat te status of te compensaton argument s unclear. At best, t s a "notonal compensaton" argument of dubous etcal relevance. To see te problem from anoter angle, we ave noted tat under assumpton (A2'), smple drectons of welfare-mprovng tax reform can be dentfed. Furter results along tose lnes (wt stragtforward generalsatons n te space of sadow taxes) are presented n Dxt (1975). As te autor puts t, te results old "assumng tat tere are no dstrbutonal problems". However, n general (A2') conflcts wt (A1), n wc case tere s "a dstrbutonal problem" even f (A2') olds. 37 Tere s no need to dstngus ere between "compensatng" and "equvalent" varatons, snce we are lookng at margnal canges. 32

34 We now turn to te respectve mplcatons of (A1), (A2) and (A2') for te structure of 38 optmum taxes. Under (A2), te one-person Ramsey rule apples: for optmum taxes, te generalsed ndex of relatve dscouragement as to be te same for all commodtes. Ts follows from (34), wt b = b for all. A corollary s tat, under te stronger assumpton of optmum transfers (A2'), optmum taxes are suc tat sadow taxes are proportonal (.e. C-C effcency); ts as already been dscussed n te precedng secton. Fnally, returnng to (A1), te correct optmum tax rule n ts case s tat te normalsed MDL (evaluated usng sadow taxes) sould be te same for all commodtes,.e.: ô. Mx Mq x ' c for some non-zero constant c (ts follows from settng te rgt-and sde of (33) equal to zero). In sort, n a second-best world tere s no smple way of avodng dstrbutonal ssues, even by assumpton (.e. tere s no unque way of statng te assumpton tat "dstrbutonal ssues ave been resolved"). Dstrbutonal assumptons ave to be made explct, bearng n mnd tat tey lead to dfferent tax rules. 5. Concludng Remarks In a wde class of models, te problem of optmum taxaton s usefully splt nto two parts: te calculaton of "sadow prces" and te applcaton of te generalsed Ramsey rule. Te relevant 38 Te argument focuses on frst-order condtons; as noted earler, second-order condtons ave rarely been examned n te lterature. 33

35 sadow prces depend on te structure of te economy, te range of coce varables and te socal welfare functon, but te generalsed Ramsey rule tself s qute robust. Ts way of approacng te problem s more transparent and productve tan te complaton of a long catalogue of tax rules for dfferent types of dstorted economes, as as been done n muc of te lterature on te second-best. In models were te generalsed Ramsey rule apples, te man ssue s to dentfy te relevant sadow prces. In oter models, te basc task s to dentfy te reasons wy te rule does not apply (e.g. nterdependence between consumer demands and producer prces, ndependently of consumer prces), and to derve sutable extensons of te general rule. Smlar prncples apply n te feld of tax reform. Here agan, te problem s usefully splt between calculaton of sadow prces and applcaton of general prncples for te dentfcaton of welfare-mprovng reforms n te space of "sadow taxes". One of tese prncples s tat, under specfc dstrbutonal assumptons, "radal" reforms n te space of sadow taxes (.e. small tax canges tat brng consumer prces closer to sadow prces) are welfare-mprovng. Care must be taken, owever, n dentfyng and assessng te relevant dstrbutonal assumptons. In partcular, we ave noted a possble tenson between te assumptons of "equal margnal socal utlty" and "equal margnal socal values of transfer", wc are often used ntercangeably n te lterature. References Amad, E., Coady, D., Stern, N., A Complete Set of Sadow Prces for Pakstan, Pakstan Development Revew, 27, Amad, E., Stern, N., Te Teory and Practce of Tax Reform n Developng Countres, Cambrdge Unversty Press, Cambrdge. Atknson, A., Stern, N., Pgou, Taxaton and Publc Goods, Revew of Economc Studes, 41, 34

36 Atknson, A., Stgltz, J., Te Desgn of Tax Structure: Drect Versus Indrect Taxaton, Journal of Publc Economcs, 6, Atknson, A., Stgltz, J., Lectures on Publc Economcs, McGraw-Hll: New York. Auerbac, A., Te Teory of Excess Burden and Optmal Taxaton, n Auerbac, A., Feldsten, M. (Eds), Handbook of Publc Economcs, vol. 1, Nort-Holland:Amsterdam. Coady, D., Agrcultural Prcng Polces n Developng Countres: An Applcaton to Pakstan, Internatonal Tax and Publc Fnance, 4, Deaton, A., Stern, N., Optmally Unform Commodty Taxes, Taste Dfferences, and Lumpsum Grants, Economcs Letters, 20, Damond, P., A Many-Person Ramsey Rule, Journal of Publc Economcs, 4, Damond, P., Mrrlees, J., Optmal Taxaton and Publc Producton I: Producton Effcency, and II: Tax Rules, Amercan Economc Revew, 61, 8-27 and Damond, P., Mrrlees, J., Prvate Constant Returns and Publc Sadow Prces, Revew of Economc Studes, 43, Dxt, A., On te Optmum Structure of Commodty Taxes, Amercan Economc Revew, 60(2), Dxt, A., Welfare Effects of Tax and Prce Canges, Journal of Publc Economcs, 4, Dxt, A., Tax Polcy n Open Economes. In Auerbac, A., Feldsten, M. (Eds), Handbook of Publc Economcs, vol. 2, Nort-Holland: Amsterdam. Drèze, J., Dstrbuton Matters n Cost-Beneft Analyss, Journal of Publc Economcs, 70(3), 35