The Role of Commodity Taxation in Pareto Efficient Tax Structures for Redistribution

Transcription

1 The Role of Commodity Txtion in Preto Efficient Tx Structures for Redistribution John urbidge 15 ugust 2016 bstrct The tkinson-stiglitz theorem (1976) rests on sophisticted nonliner ernings tx system in which lump-sum txes re possible t ech level of ernings. For exmple, in two-type world, the optiml mrginl ernings tx rte on the high erner is zero (the ernings tx on the high erner is lump sum) nd differentil commodity txtion is useless if leisure is wekly seprble from goods. Since for most countries mrginl rtes exceed verge tx rtes t higher levels of ernings it is importnt to know whether differentil commodity txtion would increse efficiency in n environment with lesssophisticted nonliner ernings tx. I build on Deton s (1979,b,1981) use of distnce functions for understnding optiml commodity tx rtes. mong other things I show tht, even in model where the nonliner ernings tx system is more efficient thn one where mrginl rtes exceed verge tx rtes t the top, nonuniform commodity txtion my be efficient even if leisure is wekly seprble from goods. Keywords: Optiml txtion, distnce function, seprbility JEL clsifiction H21 Deprtment of Economics, University of Wterloo, Wterloo, Ontrio, Cnd, N2L 3G1, jburbidg@uwterloo.c. I thnk Lutz-lexnder usch, Christoph Lülfesmnn, John Revesz, Césr Sos-Pdill, Michel Vell, nd seminr prticipnts t McMster University, the University of Wterloo nd the 2014 nd 2016 Cndin Economics ssocition Meetings for helpful comments. 1

2 1 Introduction In the most recent Hndbook of Public Economics, Piketty nd Sez (2013, pp ) summrize the development of optiml tx theory. From Rmsey (1927) until Mirrlees (1971) optiml txtion ws primrily bout the problem of how to structure commodity txes efficiently when some commodities, like leisure, cnnot be txed directly. Mirrlees (1971) explored the problem of optiml nonliner ernings txes for redistribution when the government cn observe only ernings nd not wge rtes. segment of the subsequent literture (e.g., Sdk (1976), Sede (1977), Stiglitz (1982)) focused on the result tht the optiml mrginl ernings tx rte for the highest erner is zero. nd tkinson nd Stiglitz (1976) shifted the optiml tx literture further wy from differentil commodity txtion towrds nonliner ernings txtion. tkinson nd Stiglitz (1976) derived the very importnt nd influentil result tht under seprbility nd homogeneity ssumptions on preferences, differentited commodity txtion is not useful when ernings cn be txed nonlinerly. This fmous result ws influentil both for shping the field of optiml tx theory nd in tx policy debtes. Theoreticlly, it contributed gretly to shift the theoreticl focus towrd optiml nonliner txtion nd wy from the erlier Dimond nd Mirrlees (1971) model of differentited commodity txtion (itself bsed on the originl Rmsey (1927) contribution). Prcticlly, it gve strong rtionle for eliminting preferentil txtion of necessities on redistributive grounds, nd using insted uniform vlue-dded-tx combined with income-bsed trnsfers nd progressive income txtion. (Piketty nd Sez, 2013, p. 402) 1 The nonliner ernings tx in this literture is one where the government hs the bility to set the mrginl ernings tx rte nd the totl ernings txes pid, t ech level of pre-tx ernings. For exmple, tht the optiml mrginl ernings tx rte for the highest erner is zero implies tht the ernings tx on the highest erner is lump-sum tx. nd the modern proof of the tkinson-stiglitz theorem relies on lump-sum chnges in totl ernings txes pid t ech level of ernings. 2 Since most countries use less-sophisticted nonliner ernings-tx systems, e.g. t higher ernings typiclly mrginl tx rtes exceed verge tx rtes, it is importnt to know wht role differentil commodity txtion should ply in the context of these 1 The sentence following this pssge is: Even more importntly, the tkinson nd Stiglitz (1976) result hs been used to rgue ginst the txtion of cpitl income nd in fvor of txing solely ernings or consumption. There is literture on the optimlity of cpitl txtion; e.g. Eros nd Gervis (2002), Cones, Kito nd Krueger (2009) nd urbidge (2015). 2 See Piketty nd Sez, p

3 tx systems. This pper tkes step towrds nswering this question by studying the reltionships between the Rmsey nd the Mirrlees problems under different nonliner ernings tx systems. Intuition suggests, nd the models in this pper comprise exmples, tht the less sophisticted the nonliner ernings tx system the greter the role for non-uniform commodity txtion. ssume prtil-equilibrium model in which the equilibrium with no txes is Preto efficient. The hert of the Rmsey problem is the existence of some element in the consumer s utility function nd budget constrint, like leisure, tht cnnot be txed directly. If every item could be txed directly uniform proportionl tx rte on ll items in the budget constrint would produce first-best outcome. The nswer to the Rmsey problem must be tht it is second-best efficient to levy higher tx rtes on goods tht re more complementry with leisure. The Mirrlees problem is bout designing efficient ernings tx structures for redistribution. With lump-sum txes nd trnsfers, nd full informtion on ech person s wge rte, the government could chieve ny point on the first-best utility-possibility frontier (upf). When the government cnnot observe wge rtes it must condition ernings txes on reported ernings. long the upf, wy from the privte equilibrium, redistribution my proceed to the point where high-bility worker would be better off by working less, reporting the ernings of lower-bility worker to the tx uthority, nd receiving the csh trnsfer given to the lower erner, tht is, the high-bility worker mimics the low bility worker. t this point the key to efficient redistribution is to use the tx system to seprte the types of workers. One wy to do this is to crete n incentive for the low-wge worker to work less tke more leisure, which discourges the high-wge type from mimicking. Juxtposition the Rmsey nd Mirrlees problems: in Rmsey, the objective is to set commodity tx rtes to tx leisure indirectly; in Mirrlees it my be efficient to do the opposite, tht is, to rejig commodity tx rtes to tx work more hevily. The results in this pper re driven by the tension between these models in every model exmined, the optiml tx structure chnges t the point on the upf where mimicking constrints first bind. I begin with the Rmsey problem in the simplest setting possible: two goods nd leisure. bsent proportionl tx rte to tx leisure directly the government must try to tx leisure indirectly by setting higher tx rte on whichever good is more complementry with leisure. It is well known tht the dedweight loss of tx rtes rises solely from substitution effects so the complementrity between goods nd leisure must be mesured with utility held constnt. The distnce function is idelly suited to this tsk. In this context it is the number by which bundle of goods nd leisure must be scled to deliver prticulr level of utility, sy u 0 ; the rgu- 3

4 ments of the distnce function re the sme s those of the ordinry utility function, nd u 0. The observtion tht distnce functions offer the clerest understnding of optiml commodity txes ws mde by Deton in series of ppers tht includes Deton (1979,b, 1981). mong other things, he derived optiml commodity-tx formule tht showed goods more complementry with leisure should be txed t higher rtes nd he proved the formule were equivlent to the better-known inverse elsticity rule. Perhps more importntly, he showed tht most empiricl work imed t estimting the prmeters needed to implement these formule ws bsed on ssumptions (e.g. wek seprbility between leisure nd goods, liner Engel curves) tht implied uniform commodity txtion is efficient. distnce function pproch to solving the Rmsey problem cn be implemented by choosing goods nd leisure to mximize government revenue subject to minimum-utility constrint nd constrint tht rules out lump-sum txes. This pproch my be dpted to ddress the Mirrlees problem: ssume two types nd let the wge rte for the s be greter thn the wge rte for the s, w > w. In Model 1 in this pper, the government chooses goods nd leisure for ech type to mximize totl revenue subject to minimum-utility constrints for ech nd for ech (u 0, u 0 ), constrint tht s nd s py the sme prices for goods, nd constrints tht yield the outcome tht the s not mimic the s. The vlue function for this problem gives mximum revenue s function of u 0, u 0 nd other prmeters. Setting this function to zero implicitly defines u 0 s function of u 0, tht is, the upf. I show tht Model 1 delivers the fmous results listed bove, including zero mrginl ernings tx rte on the high erner, nd the tkinson-stiglitz theorem zero proportionl tx rtes on goods 1 nd 2 if leisure is wekly seprble from goods. Even in Model 1, however, there is strong link to the Rmsey problem. In Rmsey nd, if leisure is not wekly seprble from goods in Model 1, good 1 should be txed t higher rte thn good 2 if good 1 is more complementry with leisure. In the literture discussed bove nd in Model 1, lump-sum txes on the high erners re not ruled out, so, on the utility possibility frontier (upf) between the privte equilibrium nd the point where the no-mimicking constrints bind, Model 1 delivers first-best results: lump-sum txes on the high erners fund csh trnsfer to ech low erner. fter the mimicking constrints bind it is efficient to supplement lump-sum txes on the high erners with mrginl ernings tx rte on the low erners to discourge them from working, together with the possibility of different proportionl tx rtes on goods 1 nd 2 if leisure is not wekly seprble from goods. 3 3 Every point on the upf will be ssocited with prticulr bundle of goods nd leisure for ech nd nother bundle of goods nd leisure for ech. nd for ech point on the upf there will be continuum of equivlent tx systems depending on whtever normliztion of tx rtes is 4

5 Model 2 in this pper is identicl to Model 1 except tht I impose the (Rmseylike) constrint of no lump-sum txes on the high erners. Clerly the ddition of constrint forces the upf for this model inside, or t best to touch, the upf of Model 1 (it will touch t the privte equilibrium). If one dopts the normliztion tht the ernings tx rte on the low erners is zero, I prove tht, long the upf, from the privte equilibrium to the point where mimicking strts, the ernings tx rte on the high erners is the primry source of revenue for csh pyments to the low erners. Even if leisure is wekly seprble from goods (but preferences re not homothetic) 4 the tx rte will be positive on whichever good is more complementry with leisure, nd negtive on the other good. long the upf, fter the mimicking constrints bind, one cn show tht the ernings tx rte on the high erners is reduced to discourge mimicking, nd increses in goods tx rtes grdully replce the ernings tx rte on the high ernerss s the source of revenue for redistribution. The Rmsey rule holds long the entire upf the commodity tx rte is higher on whichever good is more complementry with leisure. The nonliner ernings tx system in Model 2, which rules out the lump-sum ernings tx instrument of Model 1, opens the door to much greter role for nonuniform commodity txtion. Model 3 dds one constrint to those of Model 2. In Model 3 the government is forced to set the ernings tx rte on the two types t the sme rte, or, equivlently to set the ernings tx rte to zero nd to operte only with tx rtes on goods 1 nd 2. In this setting, the government s bility to redistribute is severely limited it hs exctly two instruments for the two objectives of rising revenue nd deling (eventully) with mimicking constrints. If leisure is wekly seprble from goods nd preferences re homothetic then the government hs, in effect, one instrument nd the upf must stop t the point where the mimicking constrints bind. 5 If one modifies preferences or prices to mke one good more complementry with leisure thn the other the good more complementry with leisure should be txed t higher rte long the upf from the privte equilibrium to the point where mimicking constrints bind. ut to del with mimicking the government must discourge the low erners from working the government hs to flip from txing leisure (indirectly) chosen. distinct dvntge of the distnce-function pproch over other pproches is tht its instruments re the vribles tht determine ech point on the upf. When both income tx rtes nd commodity tx rtes my be nonzero it is much less confusing to stte optiml tx results in terms of reltions between mrginl rtes of substitution nd rtios of privte equilibrium prices p 1, p 2, w, w. Then, hving chosen some normliztion of tx rtes, one cn tell stories bout the ernings nd commodity tx rtes tht would implement prticulr lloction. 4 In the ppendix I prove tht in ll the models in this pper wek seprbility between leisure nd goods together with homothetic preferences imply equl tx rtes on goods 1 nd 2. 5 See Sttement 3 below. 5

6 to txing work. For Model 3 I prove tht long the upf fter mimicking begins the higher goods tx rte flls nd the lower goods tx rte increses. 6 Section 2 resttes Deton s distnce-function method for solving the Rmsey problem. Section 3 develops the set of constrints nd presents the results for ech of the three models. Section 4 summrizes nd concludes. 2 Rmsey with two goods nd leisure Denote goods consumption nd leisure by (x 1, x 2, l) nd prices without txes by (p 1, p 2, w). ssuming tht leisure cnnot be txed directly nd tht the government uses proportionl tx rtes, one cn write the individul s budget constrint s (1 + t 1 ) p 1 x 1 + (1 + t 2 ) p 2 x 2 + (1 t e ) wl = (1 t e ) wl, (1) where L is the time endowment. If u(x 1, x 2, l) is the ordinry utility function the distnce function, d(x 1, x 2, l, u 0 ), is defined by ( ) x 1 u d(x 1, x 2, l, u 0 ), x 2 d(x 1, x 2, l, u 0 ), l = u 0, (2) d(x 1, x 2, l, u 0 ) tht is, d(x 1, x 2, l, u 0 ) is the number by which ny vector of goods nd leisure must be scled to deliver utility level u 0, nd u(x 1, x 2, l) u 0 if nd only if d(x 1, x 2, l, u 0 ) 1. (3) Deton (1979) showed the distnce function nd the expenditure function, e (p 1, p 2, w, u 0 ), re relted to ech other in the following wy. e (p 1, p 2, w, u 0 ) = Min x 1, x 2, l p 1 x 1 + p 2 x 2 + wl. (4) d(x 1, x 2, l, u 0 ) d (x 1, x 2, l, u 0 ) = Min p 1, p 2, w p 1 x 1 + p 2 x 2 + wl e (p 1, p 2, w, u 0 ). (5) The derivtives of the expenditure function with respect to prices re Hicksin demnds, h i (p 1, p 2, w, u 0 ), i = 1, 2, 3, which mp from prices to quntities. The derivtives of the distnce function with respect to quntities, denote these by 6 I provide numericl exmple in which there is point on the upf (with mimicking) where the commodity tx rtes re equl. 6

7 i (x 1, x 2, l, u 0 ), i = 1, 2, 3, re like inverse Hicksin demnds tht mp from quntities to prices. pplying the envelope theorem to (5) with tx rtes in plce obtin 1 (x 1, x 2, l, u 0 ) d (x 1, x 2, l, u 0 ) = (1 + t 1)p 1 x 1 (1 t e ) wl 2 (x 1, x 2, l, u 0 ) d (x 1, x 2, l, u 0 ) = (1 + t 2)p 2 x 2 (1 t e ) wl 3 (x 1, x 2, l, u 0 ) d (x 1, x 2, l, u 0 ) l = (1 t e) w (1 t e ) wl = 1 L (6) (7) (8) Deton proved i / j, i j is the mrginl rte of substitution between i nd j, denoted MRS ij, nd it equls the rtio of the price of i to the price of j. Just s the Hessin of the expenditure function, the Slutsky mtrix, must be symmetric nd negtive semi-definite, so must the Hessin of the distnce function, the ntonelli mtrix, be symmetric nd negtive semi-definite. In ddition, pre-multiplying the Slutsky mtrix by prices yields vector of zeros nd pre-multiplying the ntonelli mtrix by [x 1 x 2 l] yields row vector of zeros. 7 The Rmsey problem cn be solved by ssuming the government mximizes revenue subject to minimum utility constrint, inequlity (3), nd constrint tht rules out lump-sum txes. In this model government revenue is whtever the person does not consume, tht is, wl p 1 x 1 p 2 x 2 wl. first-best solution to the problem would be to employ lump-sum tx, T. From (8), lump-sum tx would imply 3 (x 1, x 2, l, u 0 ) = (1 t e ) w (1 t e ) wl T > (1 t e) w (1 t e ) wl = 1 L. Thus in the bsence of lump-sum tx the government must del with the constrint tht 1 L 3 (x 1, x 2, l, u 0 ) 0, (9) nd the Lgrnge multiplier on this constrint must be nonnegtive. The Lgrngin for the government s problem cn be written s 7 I ssume the elements on the min digonl of the ntonelli mtrix re negtive nd the individul consumes positive mount of ech good. See Deton (1981), p nd p

8 ( ) 1 L = wl p 1 x 1 p 2 x 2 wl + λ u (d (x 1, x 2, l, u 0 ) 1) + λ L L 3 (x 1, x 2, l, u 0 ). The first-order conditions re 0 = L = p 1 + λ u 1 λ L 31 x 1 0 = L = p 2 + λ u 2 λ L 32 x 2 0 = L = w + λ u 3 λ L 33 l Using the the symmetry of the ntonelli mtrix λ u = p λ L 13 = p λ L 23 = w 3 + λ L 33, where ij ij / i. From the second equlity p 2 1 ( 1 p ) 1 2 p 2 = λ L ( 13 23). (10) ssuming the constrints bind the sign of 1 / 2 p 1 /p 2 mtches the sign of j3 mesures the degree of complementrity between good j nd leisure holding utility constnt, nd from (6) nd (7), 1 / 2 = (1 + t 1 )p 1 /((1 + t 2 )p 2 ). The lgebr confirms the Corlett-Hgue intuition tht good 1 should be txed t higher rte thn good 2 if good 1 is more complementry with leisure thn is good 2, nd vice vers. Since ln ( 1 / 2 ) l nd 1 nd 2 re positive I cn write = ( 1 / 2 ) t 1 t 2 if nd only if 0. (11) l In prticulr, necessry nd sufficient condition for the efficiency of equl proportionl txtion of commodities 1 nd 2 is tht 1 / 2 be independent of leisure. 8

9 The result in (11) is the sme s eqution (52) in Deton (1979) nd, given the dulity between the expenditure function nd the distnce function described by Deton (1979), (5.1) in esley nd Jewitt (1995). (h 1 /h 2 ) t 1 t 2 if nd only if 0, (12) w where h j = e (p 1, p 2, w, u 0 ) / p j is the Hicksin demnd for commodity j. 8 3 Txtion for redistribution Consider n economy with two types of price-tking gents like the gent discussed bove. s nd s differ only in their wge rtes with w > w. Ernings hve to be spent on goods 1 nd 2, which re txed t proportionl rtes t 1, t 2. ssume the government wishes to redistribute money from the s to the s but it cn observe only n individul s ernings, not her wge rte. This is the Mirrlees (1971) problem with two types. To derive the chrcteristics of efficient lloctions suppose the government mximizes government revenue subject to minimum-utility constrints for nd nd vrious other constrints listed below. The vlue function for this problem gives mximized revenue s function of the nd utility levels nd other prmeters such s the time endowment. If the only purpose of txtion is to rise revenue for redistribution setting this vlue function to zero implicitly defines s utility s function of s utility, tht is, the upf. The government s bility to redistribute efficiently depends on the instruments t its disposl. The equivlent of equtions (6)-(8) in the present context re 1 = (1 + t 1) p 1 E ; 2 = (1 + t 2) p 2 E ; 3 = 1 = (1 + t 1) p 1 E ; 2 = (1 + t 2) p 2 E ; 3 = ( 1 t H ) w (13) ( E ) 1 t L w, (14) E where E j is the totl expenditure of gent j =, nd t j, j = H, L re mrginl 8 In note on the literture tht followed tkinson nd Stiglitz (1976), uerbch (1979) showed tht with the dditively seprble utility function u (x 1, x 2, l) = x 1/2 1 x 1/2 2 + x 1/2 1 + l 1/2, uniform commodity txtion is never efficient in the Rmsey setting the optiml level of t 1 will lwys exceed the optiml level of t 2. See footnote 15. 9

10 ernings tx rtes on high nd low erners. 9 Note tht these equlities build in the ssumption tht nd py the sme prices for goods. Government revenue is n ( w L p 1 x 1 p 2 x 2 w l ) + n ( w L p 1 x 1 p 2 x 2 w l ), n j, j =, is the number of ech type. Let λ j, j =, be the Lgrnge multipliers on the minimum utility constrints for nd : 10 d ( x j 1, x j 2, l j, u j 0) 1 0. j =, (15) Ech low erner will be given csh trnsfer equl to totl revenue divided by n. T L 0 denotes the lump-sum tx for ech low erner. I ssume the government cnnot condition the proportionl commodity tx rtes on person type the s nd s py the sme prices for the two goods. From (13) nd (14) ( ) ( ) (1 + t 1 ) p 1 = 1 x 1, x 2, l, u 0 (1 + t 2 ) p 2 2 (x 1, x 2, l, u 0 ) = 1 x 1, x 2, l, u 0 2 (x 1, x 2, l, u 0 ) or 1 2 ( ) ( ) x 1, x 2, l, u 0 2 x 1, x 2, l, u 0 ( ) ( ) x 1, x 2, l, u 0 1 x 1, x 2, l, u 0 = 0. (16) Denote the Lgrnge multiplier on this constrint by λ p. t this point I cnnot plce restrictions on the sign of this multiplier but more cn be sid in the models presented lter in the pper. 11 I turn now to mimicking constrints. Strting t the privte equilibrium where ll tx rtes re zero there is no incentive for n to mimic. ut s tx rtes re incresed nd the revenue given to the low erners one moves up the utility possibility frontier (drw the upf with u on the horizontl xis) in the direction of lower utility. t some point ech will relize tht her utility would be 9 The rguments of the j i functions, i = 1, 2, 3, j =, re (xj 1, xj 2, lj ) nd u j 0, j i /j k equls the rtio of the price of good i to the price of good k for person j, nd the price of good 1 is (1 + t 1 )p 1 for everyone. Thus E is the cost of (x 1, x 2, l ) t ((1 + t 1 )p 1, (1 + t 2 )p 2, (1 t H )w ) nd E is the cost of (x 1, x 2, l ) t ((1 + t 1 )p 1, (1 + t 2 )p 2, (1 t L )w ). 10 These must be positive; see footnote See the discussion of Model 3 below. 10

11 higher if she pretended to be low erner nd ws eligible for the csh trnsfer. When n mimics, the mimicking chooses leisure, l m, to mke her ernings w (L l m) equl the ernings of, w (L l ). If she does this she receives the csh trnsfer, T L, nd will fce budget constrint for goods 1 nd 2 tht is identicl to tht fced by ech. With mimicking there re three extr constrints nd two new choice vribles, x 1m nd x 2m. One of the extr constrints is tht s utility cting s n be t lest s high s the utility of n mimicking, tht is, u ( x 1, x 2, l ) u ( x 1m, x 2m, l m). This constrint together with the constrint tht u ( x 1, x 2, l ) u 0 cn be written s (15) nd (17). 1 d ( x 1m, x 2m, l m, u 0 ) 0 (17) The Lgrnge multiplier on (17), λ m, must be nonnegtive. second constrint is tht mimicking s py the sme prices for goods s everyone else. Then the other side of the observtion tht to prevent mimicking the government must mke the utility of n t lest s lrge s the utility of n mimicking is tht, to discourge mimicking, the government would like to hve n instrument tht would discourge mimicking by pushing the goods budget of mimicking below the goods budget for. Using (14) nd 1m/ 2m = 1 / 2 this could be written s 1 1 ( ) ( ) x 1, x 2, l, u 0 x x 1, x 2, l, u 0 x 2 > ( ) ( ) x 1, x 2, l, u 0 x 1m + 2 x 1, x 2, l, u 0 x 2m. The bsence of such n instrument mens tht the second nd third constrints rising from mimicking cn be collpsed into ( ) ( ) x 1, x 2, l, u 0 x x 1, x 2, l, u 0 x 2 ( ) ( ) x 1, x 2, l, u 0 x 1m + 2 x 1, x 2, l, u 0 x 2m or ( ) ( ) ( ) ( ) x 1, x 2, l, u 0 x 1 x 1m + 2 x 1, x 2, l, u 0 x 2 x 2m 0. (18) The Lgrnge multiplier on this constrint, λ c, must be non-positive. efore going into the detils of the vrious models it is useful to think bout goods nd leisure choices on the upf bove the point where mimicking constrints bind (recll tht I ssumed u is on horizontl xis). Consider step up the upf. u must increse nd the fll in u must equl the fll in u m. If dl were negtive 11

12 or zero, from w (L lm) = w (L l ), it follows tht dlm = (w /w )dl, tht is, lm would hve to fll by less or the sme mount s l. Therefore, for u to rise, the goods budget for would hve to increse but this goods budget is the sme for mimicking nd, nd the mrginl utility of leisure is lower for mimicking thn. There my be utility functions where, moving up the upf fter mimicking begins, l decreses (in prticulr, leisure would hve to be n inferior good) but I m going to rule them out. I ssume tht l nd lm increse on the upf fter the point where mimicking begins nd the reson u m flls s u increses, even though t ech step the goods budgets chnge by the sme mount, is tht dlm = (w /w )dl < dl. 3.1 Model 1 In Model 1 the government uses x 1, x 2, l, x 1, x 2, l to mximize government revenue subject to (15) nd (16) if mimicking constrints do not bind. If they do the government hs two extr choice vribles, x 1m, x 2m, to del with the three mimicking constrints, (17), (18) nd 1m/ 2m = 1 / In the ppendix I prove the results in the following description of Model 1 s Preto-efficient tx structures. Sttement 1: () From the privte equilibrium on the upf until the mimicking constrints bind, (i) 1 / 2 = 1 / 2 = p 1 /p 2, (ii) 1 / 3 = p 1 /w nd (iii) 1 / 3 = p 1 /w. Further up the upf, fter mimicking constrints bind: (b) if leisure is wekly seprble from goods, (i) nd (ii) re still true nd (iv) i / 3 > p i /w, i = 1, 2; (c) if leisure is not wekly seprble from goods, ltering preferences or prices so tht j 13 j 23, j =, rises bove zero mkes (v) j 1/ j 2 p 1 /p 2 > 0, j =,, (vi) 1 / 3 p 1 /w > 0, (vii) 2 / 3 p 2 /w < 0 nd (iv) still holds; (d) if leisure is not wekly seprble from goods, ltering preferences or prices so tht j 13 j 23, j =, flls below zero reverses the signs in (v), (vi) nd (vii) nd (iv) still holds. 13 Unlike the Rmsey model in section 2, Model 1 llows lump-sum txes in Model 1 there is no constrint tht is the equivlent of (9) in the Rmsey model. Until the mimicking constrints bind the first-best upf is ttinble; T H 0, T L 0 nd ll other tx rtes re set equl to zero. 12 Recll tht lm is chosen to mke the ernings of mimicking equl the ernings of. 13 In the ppendix I prove tht if uniform txtion is efficient with the mimicking constrints binding, tht is (i) is true, then (ii) is true nd x im = x i, i = 1, 2. ut leisure does not hve to be wekly seprble from goods. u(x 1, x 2, l) = x 1/2 1 x 1/2 2 + x 1/2 1 l 1/2 + x 1/2 2 l 1/2, p 1 = p 2 is one exmple where leisure is not wekly seprble but uniform commodity txtion is efficient. 12

13 s noted bove continuum of normliztions is possible in ny model. Since the optiml ernings tx literture hs focused on zero mrginl ernings tx rte for the high erner nturl normliztion to choose for this model is t H = 0. Then, for the section of the upf where the mimicking constrints bind: Sttement 1(b) implies tht with wek seprbility between leisure nd goods, t 1 = t 2 = 0 (the tkinson-stiglitz theorem), nd then (iv) implies t L 0. Wek seprbility dicttes tht nothing useful would follow from differentil commodity txtion nd the government wnts to get the s to work less to discourge the s from mimicking them. In fct, (iv) holds, t L 0, with or without wek seprbility between leisure nd goods; the government hs to get the s to work less, full stop. Wht bout commodity txtion when leisure is not wekly seprble form goods nd the mimicking constrints bind? Here, just s in the Rmsey model, if good 1 is more complementry with leisure thn is good 2, tht is j 13 > j 23, j =,, then mimicking will consume more of good 1 nd less of good 2 thn becuse the hs more leisure. Now it is efficient to rise the price of good 1 reltive to the price of good 2. With t H = 0 (vi) nd (vii) imply t 1 > 0, t 2 < 0. If j 13 < j 23, j =, which is cse (d) then t 1 < 0, t 2 > 0. Note tht, if leisure is not wekly seprble from goods, the Rmsey rule tells us how to structure commodity tx rtes even in Model Model 2 This model is the sme s Model 1 except the government does not hve the bility to levy lump-sum txes. The extr constrint which is the equivlent of (9) is 1 L 3 Sttement 2 lists the implictions of Model ( ) x 1, x 2, l, u 0 0. (19) Sttement 2: () If preferences re homothetic nd leisure is wekly seprble from goods, (i) without or with mimicking, 1 / 2 1 / 2 = 1 / 2 = p 1 /p 2, (ii) without or with mimicking, i / 3 p i /w, (iii) without mimicking, i / 3 = p i /w, nd (iv) i / 3 p i /w increses with mimicking nd it rises more quickly thn i / 3 p i /w. (b) ltering preferences wy from homotheticity or wek seprbility between leisure nd goods (v) without or with mimicking, the sign of 1 / 2 p 1 /p 2 mtches the sign of j 13 j 23, j =,, nd (vi) sttements (ii) nd (iv) re still true. 14 Proofs re in the ppendix. 13

14 Once gin, there is continuum tx normliztions possible for this (or ny other) model. The one tht offers perhps the clerest view of how the model opertes is to set the ernings tx rte on the low erners to zero, t L = 0. Then, with wek seprbility of goods nd leisure, nd homothetic preferences, 2()(iii) implies tht from the privte equilibrium to the point where mimicking begins t 1 = t 2 = t L = 0. Since t H = 0 t the privte equilibrium 2()(ii) implies tht t H increses long the upf from the privte equilibrium to the point where mimicking begins. In this rnge, s one would expect from the Rmsey model, if good 1 is more complementry with leisure thn is good 2, then t 1 t 2 should be positive. This is implemented by setting t 1 > 0 nd t 2 < 0. fter the mimicking constrints bind then Sttement2()(iv) nd 2(b)(vi) imply tht t H becomes the primry instrument for deling with mimicking t H is reduced to encourge s not to mimic s nd commodity txtion becomes the primry instrument for generting the revenue for redistribution. s s one would expect from the Rmsey model, imposing the constrint tht rules out lump-sum txes increses the role for commodity txtion the good most complementry with leisure is txed t the higher rte. 3.3 Model 3 With lump-sum tx on the higher erners ruled out the previous section showed government tht hd the bility to set different proportionl ernings tx rtes would use this bility to mke t H t L 0 from the privte equilibrium to pst the mimicking-constrints point. Wht would hppen if the government did not hve this bility, tht is, if the government were forced to operte with proportionl ernings tx? Using (13) nd (14) for leisure nd good one one could write (1 + t 1 ) p 1 3 w 1 (1 + t 1 ) p 1 3 w 1 = 1 t H = 1 t L or, subtrcting the second eqution from the first, ( ) (1 + t 1 ) p = t L t H. w 1 w 1 Without lump-sum tx on the high erners (T H = 0), the government would like to mke t L t H negtive or 14

15 3 w 1 3 w 1 < 0. Therefore, in the bsence of the bility to mke t H nd t L unequl the government hs to function with the constrint tht which cn be rewritten s 3 w 1 3 w 1 0, 3 w (20) w If the reltive price of leisure nd good two hd been used insted of the reltive price of leisure nd good one the corresponding inequlity would hve been 3 w (21) w Clerly, whether one employs the reltive price of leisure nd good one or the reltive price of leisure nd good two to frme the government s problem my ffect the wy constrints re written nd the signs of the ssocited Lgrnge multipliers, but this choice cnnot lter the conditions for Preto-efficient lloctions when these re described by reltionships between MRSes nd producer prices. Observe tht inequlity (20) implies (21) when (22) nd (21) implies (20) when I use inequlity constrints tht re the equivlent of (20) nd (22). w 1 ( ) ( ) x 1, x 2, l, u 0 3 x 1, x 2, l, u 0 w 1 15 ( ) ( ) x 1, x 2, l, u 0 3 x 1, x 2, l, u 0 0,

16 with the Lgrnge multiplier denoted, λ w 0, nd 1 ( ) ( ) ( ) ( ) x 1, x 2, l, u 0 2 x 1, x 2, l, u 0 2 x 1, x 2, l, u 0 1 x 1, x 2, l, u 0 0, with the Lgrnge multiplier denoted λ p 0. Simply put, these two constrints re one wy of imposing single proportionl ernings tx rte on ll consumers who must py the sme prices for goods 1 nd 2. The implictions of this model re summrized in Sttement 3. Sttement 3: () If preferences re homothetic nd leisure is wekly seprble from goods, (i) without mimicking, 1 / 2 1 / 2 = 1 / 2 = p 1 /p 2, (ii) there re no points on the upf with mimicking binding, tht is, with λ m > 0. (b) lter producer prices or preferences (to step wy from homotheticity or wek seprbility between leisure nd goods) so tht j 13 j 23, j =,, (iii) without mimicking, the sign of 1 / 2 p 1 /p 2 mtches the sign of j 13 j 23, j =, nd (iv) long the upf pst the point t which mimicking begins, 1 / 2 p 1 /p 2 is flling if 1 / 2 > p 1 /p 2 when mimicking strts, nd vice vers. Sttement 3 () (i) nd (b) (iii) re proved in the ppendix. Given the results in section 2 one would expect tht, long the upf before mimicking strts, the sign of t 1 t 2 mtches the sign of j 13 j 23, j =, if good 1 is more complementry with leisure thn is good 2, it is efficient to tx good 1 t higher rte. Wht hppens when mimicking begins? If preferences re homothetic nd leisure is wekly seprble from goods, uniform commodity txtion is efficient nd there is in effect only one instrument to cope with redistribution nd the mimicking constrints. Sttement 3 () (ii) is the logicl consequence of the government being short of tx instruments. The upf ends t the point where mimicking begins. If j 13 j 23, j =, there re two instruments, t 1 nd t 2, to cope with the trgets of incresing the utility of whilst keeping the utility of n mimicking s high s the utility of n. Redistribution with mimicking constrints is possible, but brely so. Consider step up the upf. s noted bove, dlm = (w /w )dl > 0 nd the goods budget for nd mimicking must fll. This could not occur if both tx rtes incresed; one must fll nd the other must increse. Chnging commodity txes to increse the consumption of the good most complementry with leisure encourges the s to work less which discourges the s from mimicking them. In other words, the mimicking constrint switches the government s problem from trying to tx leisure indirectly to trying to tx work; the Corlett-Hgue intuition 16

17 is reversed. For exmple, if good 1 is more complementry with leisure thn is good 2, j 13 > j 23, j =,, then t 1 > t 2 on the upf when the mimicking constrints first bind, nd, s we move further up the upf, t 1 flls nd t 2 rises. t some point t 1 cn equl t 2 which demonstrtes tht homotheticity nd wek seprbility re not necessry for the efficiency of uniform commodity txtion in this model Summry nd conclusions This pper juxtpositions the Rmsey nd Mirrlees problems in the simplest setting possible two types, two goods nd leisure. Model 1 shows tht the fmilir results in the literture, like the tkinson-stiglitz theorem, require lump-sum ernings tx on the high erner. This is more sophisticted nonliner ernings tx thn those observed in most countries where typiclly mrginl ernings tx rtes exceed verge tx rtes t higher ernings. Model 2 shows tht the tkinson-stiglitz theorem does not hold even with nonliner ernings tx where mrginl equls verge ernings tx rte for the high erner. nd the commodity tx rtes follow Rmsey-like rule where the commodity more complementry with leisure is txed t higher rte. The pper highlights the contrst between the functioning of model without nd with mimicking constrints tht bind. The results for Model 1 show tht the first-best solution to deling with mimicking constrints is to crry on with lumpsum ernings txes on the high erner nd to levy mrginl ernings tx rte on the low erner to increse this gent s choice of leisure. In Model 2, where lump-sum ernings txes re ruled out, second-best wy to cope with mimicking constrints is to reduce the proportionl ernings tx rte on the high erner nd to rely on higher 15 Tbles of numericlly-simulted upfs for some utility functions, including the following, u (x 1, x 2, l) = x 1/2 1 x 1/2 2 + x 1/2 1 + l 1/2 u (x 1, x 2, l) = x 1/2 1 x 1/2 2 + x 1/2 2 l 1/2 u (x 1, x 2, l) = x 1/2 1 x 1/2 2 + x 1/2 1 l 1/2 + x 1/2 2 l 1/2 u (x 1, x 2, l) = x α 1 x β 2 l1 α β re t: In the first utility function leisure is dditively seprble nd preferences re not homothetic. In the second leisure is not wekly seprble nd preferences re homothetic. The third utility function with p 1 = p 2 = 1 is used to show tht in ny of the models in this pper t 1 = t 2 cn be efficient even if leisure is not wekly seprble from goods. The fourth utility function is Cobb-Dougls; leisure is wekly seprble nd preferences re homothetic. The R code for the simultion progrms nd further detils re vilble from the uthor. 17

18 commodity tx rtes to generte the revenue required for redistribution. Here, with or without mimicking constrints, the good more complementry with leisure should be txed t higher rte. In Model 3, with liner ernings tx, or no ernings tx nd only commodity tx rtes, coping with mimicking constrints is impossible unless there is difference cross the commodities in their complementrity with leisure. If this difference exists there is switch t the point on the upf where mimicking constrints first bind from setting commodity tx rtes to tx leisure indirectly before mimicking to setting tx rtes to tx work indirectly fter mimicking. While the simplicity of the models studied is n dvntge in showing the tkinson- Stiglitz theorem does not generlize to more plusible nonliner ernings tx systems there is the obvious question of whether the results developed here would still hold in more generl models. The forces identified here would still be t work in more generl models but they might be overwhelmed by other effects. Whether they re or they re not dominted by other effects the techniques used in this pper e.g. using distnce functions to study optiml tx problems should be useful in more generl settings. There is lrge distnce between the model in ny optiml tx pper nd the relity in which ctul governments operte. Still it is remrkble tht ctul governments use nonuniform commodity tx rtes much more thn one would think they should given the literture in public economics (see the pssge quoted in the introduction). Could it be tht higher tx rtes on items like lcohol nd tobcco could be re-interpreted s txes on goods tht re strong complements with leisure nd perhps optiml in world with unsophisticted nonliner ernings txtion? 18

19 ppendix Models 1 nd 2 re specil cses of Model 3. The optimiztion problem for Model 3 with mimicking constrints is Opt x 1, x 2, l x 1, x 2, l x 1m, x 2m λ j, j =,, p, L, w, m, c n ( w L p 1 x 1 p 2 x 2 w l ) + n ( w L p 1 x 1 p 2 x 2 w l ) + λ ( d ( ) ) x 1, x 2, l, u λ ( d ( ) ) x 1, x 2, l, u λ ( ( ) ( ) ( ) ( )) p 1 x 1, x 2, l, u 0 2 x 1, x 2, l, u 0 2 x 1, x 2, l, u 0 1 x 1, x 2, l, u 0 + λ L ( 1 L 3 ( ) ) x 1, x 2, l, u 0 + λ ( ( ) ( ) ( ) ( )) w w 1 x 1, x 2, l, u 0 3 x 1, x 2, l, u 0 w 1 x 1, x 2, l, u 0 3 x 1, x 2, l, u 0 + λ ( m 1 d ( )) x 1m, x 2m, lm, u 0 + λ ( ( ) ( ) ( ) ( c 1 x 1, x 2, l, u 0 x 1 x 1m + 2 x 1, x 2, l, u 0 x 2 x2m)) where w ( L lm) ( = w L l ) (23) The first-order conditions for the eight goods nd leisure vribles re 0 = n p 1 + λ 1 + λ ( p ) λ L 31 + λ ( ) w w 3 11 w = n p 2 + λ 2 + λ ( p ) λ L 32 + λ ( ) w w 3 12 w = n w + λ 3 + λ ( p ) λ L 33 + λ ( ) w w 3 13 w

20 0 = n p 1 + λ 1 + λ ( p ) ( + λ w w 1 31 w 3 11) + λ ( ( ( ) ) c 11 x 1 x1m) + 21 x 2 x 2m = n p 2 + λ 2 + λ ( p ) ( + λ w w 1 32 w 3 12) + λ ( ( ( ) ) c 12 x 1 x1m) + 22 x 2 x 2m = n w + λ 3 + λ ( p ) ( + λ w w 1 33 w 3 13) + λ m w 3m w + ( ( ) ( λc 13 x 1 x 1m + 23 x 2 x2m)) 0 = λ m 1m λ c 1 0 = λ m 2m λ c 2. Note the lst two equtions imply tht 16 λ c 1 = λ m 1m Using these equtions obtin λ c 2 = λ m 2m λ c = λ m E. Em 0 = n p 1 + λ 1 + λ ( p ) λ L 31 + λ ( ) w w 3 11 w = n p 2 + λ 2 + λ ( p ) λ L 32 + λ ( ) w w 3 12 w = n w + λ 3 + λ ( p ) λ L 33 + λ ( ) w w 3 13 w = n p 1 + λ 1 + λ ( p ) ( ) + λ w w 1 31 w 3 11 λ m E ( ( ) ( )) Em 11 x 1 x 1m + 21 x 2 x 2m λ m 1m 0 = n p 2 + λ 2 + λ ( p ) ( ) + λ w w 1 32 w 3 12 λ m E ( ( ) ( )) Em 12 x 1 x 1m + 22 x 2 x 2m λ m 2m 0 = n w + λ 3 + λ ( p ) ( ) + λ w w 1 33 w 3 13 λ m E ( ( ) ( )) Em 13 x 1 x 1m + 23 x 2 x 2m λ m w 3m w. 16 E m is the cost of buying (x 1m, x 2m, l m) t ((1 + t 1 )p 1, (1 + t 2 )p 2, w m). w m is the shdow price of leisure for mimicking nd equls (1 + t 1 )p 1 3m/ 1m. 20

21 Recll ij ij / i nd use (13), (14) nd the symmetry of the ntonelli mtrix to obtin n p 1 = n E 1 = λ + λ ( ) p λ L t 1 λ ( ) w w 3 11 w 1 13 n p 2 = n E 1 = λ + λ ( ) p λ L t 2 λ ( ) w w 3 21 w 1 23 n w = λ + λ ( ) p λ L λ ( w w 3 31 w 1 33 = n E 1 n p 1 1 λ m E E m n p 2 2 λ m E E m n w 3 λ m E E m ) = λ + λ ( p t 1 ( ( ) ( 11 x 1 x 1m + 12 x 2 x 2m )) λ m E = n E 1 = λ + λ ( p t 2 ( ( ) ( 21 x 1 x 1m + 22 x 2 x 2m = λ + λ p ( (24) (25) (26) ) ) ( + λ w w 1 13 w 3 11 E m )) λ m E (27) ) ( + λ w w 1 23 w 3 21 E m ) + λ w ( w 1 33 w 3 31 ) ) (28) ( ( ) ( )) 31 x 1 x 1m + 32 x 2 x 2m λ m 3m w 3 w. (29) Then (25) minus (24), nd (28) minus (27) yield t the privte equilibrium (in ny of the models) ll Lgrnge multipliers re zero except λ nd λ, nd E j = w j L, j =,. Using (24) nd (27), λ j = n j w j L, j =,. 21

22 n E t 1 t 2 (1 + t 1 ) (1 + t 2 ) = ( λp 2 λ L ( ) + λ w ( w 3 n E t 1 t 2 (1 + t 1 ) (1 + t 2 = ( λp 1 λ ( ( ) w w w 3 (( λ m E E m ( ) ( ( ) + w 1 ( ( ) ( ( x 1 x1m) ( ) ( )) + )) 22 + )) )) + (30) ) ( )) x 2 x 2m. (31) Finlly, subtrct n E times (31) from n E times (30), divide by E, nd use (13) nd (14) to write ( ) 0 = n λ L λ { p n [ ( ) ( )] [ 11 + n 1 [ λ w 1 n w ( ) n w ( )] λ w w w [ ( n 1 t H) ( E λ m n 1m 1 ) + n ( 1 t L) ( )] + ( ) ( )]} [( ) ( ( ) ( )] x 1 x1m) x 2 x 2m. (32) Proof tht homotheticity nd wek seprbility of leisure nd goods imply 1 / 2 = p 1 /p 2 (t 1 = t 2 ) 18 If the utility function is homothetic there is strictly incresing trnsformtion of it tht is homogeneous of degree 1. Thus, with wek seprbility of leisure, u (x 1, x 2, l) = f (x 1, x 2 ) g(l) nd for ll dmissible vlues of goods nd leisure, for ll α > 0 f (αx 1, αx 2 ) g (αl) = αf (x 1, x 2 ) g(l) = α γ f (x 1, x 2 ) α 1 γ g(l). Thus f is homogeneous of degree γ nd using Euler s theorem its first derivtives re homogeneous of degree γ 1 nd its second derivtives re homogeneous of degree γ 2. Then for i, j, k = 1, 2 18 The proof shows tht these ssumptions imply j = j 23, j =,.

23 f ij (αx 1, αx 2 ) f k (αx 1, αx 2 ) = αγ 2 f ij (x 1, x 2 ) α γ 1 f k (x 1, x 2 ) = 1 f ij (x 1, x 2 ) α f k (x 1, x 2 ). Letting α = 1/x 2 f ij (x 1, x 2 ) f k (x 1, x 2 ) = 1 x 2 f ij (x 1 /x 2, 1) f k (x 1 /x 2, 1), nd since preferences re homothetic nd nd py the sme prices for goods 1 nd 2, x 1 /x 2 = x 1 /x 2. Thus for q =, f ij (x q 1, x q 2) f k (x q 1, x q 2) = 1 f ij (x 1 /x 2, 1) x q 2 f k (x 1 /x 2, 1), nd the second term on the RHS is independent of type. Turning to the distnce function, u (x 1, x 2, l) homogeneous of degree 1 implies tht from we hve ( u x 1 d (x 1, x 2, l, u 0 ), x 2 d (x 1, x 2, l, u 0 ), ) l = u 0 d (x 1, x 2, l, u 0 ) d (x 1, x 2, l, u 0 ) = u 1 0 u (x 1, x 2, l) = u 1 0 f (x 1, x 2 ) g(l). Remembering tht ij ij / i, for i, j = 1, 2 nd q =, nd q ij = f ij (x q 1, x q 2) f i (x q 1, x q 2) = 1 f ij (x 1 /x 2, 1) x q 2 f i (x 1 /x 2, 1) 1 x q fij, (33) 2 q 13 = g (l) g(l) = q 23. (34) Recll tht mimicking nd hve the sme budget constrint for goods nd therefore with wek seprbility of leisure nd goods their goods consumption levels re the sme nd the lst term in (32) is zero. Further inspection of (32) shows tht in Models 1 nd 2 where λ w = 0 it must be tht λ p = 0 19 nd therefore t 1 = t 2 from (30). To prove t 1 = t 2 in Model 3 is little more work. 19 On my ssumptions the coefficient of λ p is positive; see footnote 7. 23

24 Using (34) nd (14), E times (30) yields n E E t 1 t 2 (1 + t 1 ) (1 + t 2 ) = ( ( ) ( )) λp (1 + t 1 ) p (1 + t2 ) p λ w w w ( ) Then using (33) x 2 E times (30) produces E E n x 2 λ w w w (f 21 f 11) t 1 t 2 (1 + t 1 ) (1 + t 2 ) = λp ((1 + t 1 ) p 1 (f 12 f 22) + (1 + t 2 ) p 2 (f 21 f 11)) + Likewise, x 2 E times (31) produces E E n x 2 λ w w w (f 11 f 21). dding the lst two equtions t 1 t 2 (1 + t 1 ) (1 + t 2 ) = λp ((1 + t 1 ) p 1 (f 22 f 12) + (1 + t 2 ) p 2 (f 11 f 21)) + E E ( n x 2 + n x 2 Proof of Sttement 1 ) t 1 t 2 (1 + t 1 ) (1 + t 2 ) = 0 or t 1 = t 2. On the upf between the privte equilibrium nd the point where mimicking strts, λ L = λ w = λ m = 0. The coefficient of λ p in (32) is positive (see footnote 7) so (32) implies tht λ p = 0. Then Sttement 1() (i) 1 / 2 = 1 / 2 = p 1 /p 2, (ii) 1 / 3 = p 1 /w nd (iii) 1 / 3 = p 1 /w follow immeditely from (24) - (29). Wht hppens long the upf fter the point where mimicking begins? If leisure is wekly seprble from goods then mimicking s spend their money the sme wy s do, x i = x im, i = 1, 2, nd from (32) λ p = 0 nd gin Sttement 1() (i) nd (ii) follow from (24), (25) nd (26); Sttement(iii) chnges nd I will del with it immeditely below. ut before doing this note tht, with mimicking, wek seprbility of leisure nd goods is not necessry for the efficiency of uniform commodity txtion. If uniform commodity txtion is efficient (30) implies λ p = 0. Then (24) to (26) imply i / 3 = p i /w nd (31) implies 24

25 ( ) ( ( ) ( ) x 1 x1m) x 2 x 2m = 0. Since < 0, > 0, nd (x 1 x 1m) positive implies (x 2 x 2m) negtive, nd vice vers, x i = x im, i = 1, 2. With regrd to (b) (iv), use (27), (28), (29) nd (1 + t 1 )p 1 = 1 E = 1mE m to write i 3 w p i = λ 3m w λm 3 w λ λ m 1m 1 So s λ m increses from zero the left-hnd side will rise bove 1 if nd only if. 1m 1 > 3m 3 w w. (35) This inequlity holds. 1 / 3 is the MRS between good 1 nd leisure, nd mimicking s enjoy more leisure nd the sme utility s s, so 1m 3m > 1 3 or 1m 3 3m 1 w w > w w. nd the right side equls unity when mimicking strts. This proves Sttement 1() (iv), i / 3 > p i /w, i = 1, 2. Wht hppens fter mimicking strts on the upf if leisure is not wekly seprble from goods? Strt with j 13 = j 23, j =, nd mke the first terms slightly lrger thn the second, tht is, mke good 1 more complementry with leisure thn is good 2. Since mimicking s hve the sme goods budget s s but more leisure then in (32), x 1m > x 1, x 2m < x 2, the coefficient of λ m is negtive nd λ m itself is positive, so on these ssumptions λ p rises from zero strting t the mimicking point on the upf. The rtio of (25) to (24) is ( ) = λ + λ p p 1 λ + λ p ( ) 1 p 2 2 The coefficient of λ p is positive in the numertor nd negtive in the denomintor so s λ p rises the left-hnd side must increse. Since the MRS between the goods is 25

26 the sme for nd, Sttement 1(c) (v) j 1/ j 2 p 1 /p 2 > 0, j =, holds. The rtio of (26) to (24) is 1 3 w p 1 ( ) = λ + λ p λ + λ p ( ) Observe tht the coefficient of λ p in the numertor cn be rewritten s 1 2 ( 23 )/ 3, so it is smll by ssumption, nd thus the movement in the rtio is determined by the λ p term in the denomintor, which is negtive. Thus, s mimicking strts, the left-hnd side of this eqution moves bove unity nd Sttement 1(c) (vi) is true. Similrly, (vii), 2 / 3 p 2 /w < 0, follows from inspecting the rtio of (26) to (25). Why does (iv), i / 3 > p i /w, i = 1, 2, still hold? Given the current ssumption tht j 13 j 23, j =, re smll positive numbers nd therefore 1 / 2 is incresing fter the point where mimicking strts I focus on the rtio of (29) to (28) w p 2 ( ) ( ( ) ( )) = λ + λp λ m E E m 31 x 1 x 1m + 32 x 2 x 2m λ m 3m w 3 w. λ + λ p ( ) λ m E ( Em 21 (x 1 x 1m) + 22 (x 2 x 2m)) λ m 1m 1 The λ p term nd the first λ m term in the numertor re smll by ssumption. The corresponding terms in the denomintor hve opposite signs nd since the rtio of (25) to (24) must equl the rtio of (28) to (27) nd the corresponding λ p terms on either side of the equl sign hve opposite signs the first λ m term in the denomintor must outweigh the λ p term in the denomintor. ut, s mimicking strts, these terms will be smll reltive to the difference in size in the lst λ m terms in the numertor nd denomintor; see (35). With mimicking, (iv) holds with or without wek seprbility between leisure nd goods. Muttis mutndis, the bove prgrph proves Sttement 1(d), tht is, wht hppens fter mimicking strts on the upf if leisure is not wekly seprble nd j 13 j 23, j =, re smll negtive numbers. Proof of Sttement 2 In this model λ w = 0; ll other Lgrnge multipliers my be nonzero. Erlier in this ppendix I proved Sttement 2()(i): if preferences re homothetic nd leisure is wekly seprble from goods, then without or with mimicking, 1 / 2 = p 1 /p 2. To see why 2()(ii) nd the first prt of 2(b)(vi) re true, tht is, without or with mimicking, i / 3 p i /w, look t the rtio of (26) to either (24) or (25): 26

27 i 3 ( ) w = λ + λ p λ L 33 p i λ + λ p ( 2 i1 1 i2 λl i3 ( ) = λ + λ p / 3 λ L 33 λ + λ p ( 2 1 i2 ) λl i3 i1 (36) Homothetic preferences nd wek seprbility between leisure nd goods imply j 13 = j 23, j =, so from (32) λ p = 0. Then the result follows from i3 > 0 nd 33 < 0. If one moves wy from homotheticity keeping wek seprbility (32) implies tht λ p hs the opposite sign of If one moves wy from wek seprbility in (32) the coefficients of λ L nd λ m hve opposite signs nd consequently the mgnitude of λ p will be lower. ut, t lest initilly, the λ p terms in this eqution will be smll reltive to the λ L terms nd 2()(ii) nd the first prt of 2(b)(vi) will still be true. Now look t the rtio of (29) to either (27) or (28). w p i i 3 = λ + λ ( ) p λ m ( ( ) ( )) 1m 31 x 1 x 1m + 32 x 2 x 2m λ m 3m w 1 3 w λ + λ p ( 1 i2 2 i1 ) λm 1m ( i1 (x 1 x 1m) + i2 (x 2 x 2m)) λ m 1m 1 1 (37) With homothetic preferences nd wek seprbility, nd without mimicking, λ p = 0 nd thus i / 3 = p i /w ; this proves Sttement2()(iii). With mimicking, the second λ m terms in the numertor nd the denomintor kick in nd tht i / 3 p i /w is incresing follows from (35). In ddition, note tht λ m terms enter (37) but not (36), so s mimicking begins it must be tht i / 3 p i /w is incresing fster thn i / 3 p i /w. This proves Sttement 2()(iv). The second prt of Sttement2(b)(vi) sys tht s we move wy from homothetic preferences or wek seprbility, whtever the sign of i / 3 p i /w hppens to be s mimicking begins, s we move up the upf this expression increses. This is true becuse, t lest initilly, the λ p nd new λ m terms in (37) re smll reltive to the gp between the lst λ m terms in the numertor nd denomintor. gin, this is driven by (35). nd the sme rgument supports (iv) still being true. Finlly, to see why Sttement 2(b)(v) holds, 1 / 2 p 1 /p 2 mtches the sign of j 13 j 23, j =,, look t the rtio of (28) to (27). 27