Optimal taxation, credit constraints and the timing of income-tested transfers

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1 Opmal axaon, cred consrans and he mng of ncome-esed ransfers Robn Boadway, Jean-Dens Garon 2, and Lous Perraul 3 Queen s Unversy 2 Unversé du Québec à Monréal 3 Georga Sae Unversy November 2, 207 EARLY DRAFT. PLEASE DO NOT CITE. Absrac We sudy opmal ncome and commody ax polcy wh cred-consraned lowncome households. Workers are assumed o receve an even flow of ncome durng he ax year, bu make ax paymens or receve ransfers a he end of he year. They use her dsposable ncome o purchase mulple commodes over he year. We show ha dfferenaed subsdes on commodes can be welfare-mprovng even f he Aknson- Sglz Theorem condons apply. The exen of he dfferenaon s shown o be a funcon of he cos of ransferrng ressources beween perods and he rao of coss comng from ncome effecs of he subsdes hrough dfferen levels of consumpon made by consraned ndvduals of each commody.

2 Inroducon Many governmen ransfer programs are ncome-esed and delvered hrough he ax sysem. Examples nclude refundable ax creds ha declne n ncome, such as employmen ax creds he Earned Income Tax Cred n he U.S., ax creds for chldren and chldcare he Canada Chld Benef, value-added ax creds he Goods and Servces Tax Cred n Canada, dsably ax creds, and penson ax creds. A key feaure of hese ransfer programs s ha enlemens canno be fully deermned unl he axpayer s ncome ax form has been fled and approved by he ax auhores. In he above examples, ransfer paymens are pad perodcally n a gven year based on axable ncome or famly ncome of he prevous year. In some cases, adjusmens can occur whle he ransfers are beng receved f he axpayer s crcumsances change n a way ha can be verfed by he governmen, such as chldbrh or change n employmen or dsably saus. The consequence s ha ransfer recpens ncome flow s lumpy. Those wh low enough ncome o be elgble for a ransfer from he governmen wll have low possbly zero ncome durng he year and a large ransfer sarng afer he year ends. Indvduals who ancpaes a ransfer would lke o smooh her consumpon sream over he year by borrowng. However, hey may be precluded from dong so by a cred consran. Fnancal nsuons may be unwllng o lend o hem excep a exorban neres raes, especally f hey do no have a cred rang or f he fnancal nsuon canno verfy he expeced ransfer. We adop an opmal ncome and commody ax perspecve o sudy polcy responses o hs ssue. The nformaonal assumpons of opmal axaon accord well wh he problem. The model we use s sylzed and mean o capure he essenal feaures of he nformaon consran faced by he governmen and he cred consran faced by ransfer recpens. Unlke n he sandard opmal ncome ax seng, we assume ha ndvduals receve an even flow of ncome durng he ax year, bu make ax paymens or receve ransfers a he end of 2

3 he year. Indvduals use her dsposable ncome o purchase a flow of mulple commodes over he ax year. The governmen knows only he workers labor ncomes a he end of he year. However, followng Guesnere 995, we assume ha he governmen observes all anonymous ransacons on commody markes and can mpose a se of lnear commody axes or subsdes a he me he purchases occur. Therefore, f he governmen wans o underake some redsrbuon durng he fscal year, mplc ransfers can be made hrough commody subsdes and could be argeed o he nended ndvduals by a dfferenal rae srucure. Our man focus s on he case where ndvduals are cred consraned whch can preven hem from smoohng her consumpon over he fscal year. The cred consran becomes especally relevan when he governmen s redsrbuon scheme mples payng ransfers a he end of he year. Wh perfecly funconng cred markes, hose ancpang ransfers would borrow hroughou he year o smooh he consumpon fnanced by her fuure ransfer. Then, he sandard resuls of opmal ax heory would hold, ncludng he well-known Aknson & Sglz 976 heorem when labor and consumpon are separable. However, when ransfer recpens face a bndng cred consran ha precludes hem from smoohng her consumpon, gvng ransfers a he end of he year does no acheve he governmen s redsrbuve objecves earler n he year. We show ha dfferenaed subsdes on commodes can be welfare-mprovng even f he Aknson-Sglz Theorem condons apply. The dea ha consumpon racks ncome due o cred consrans s well esablshed. For example, n he buffer-sock model of Deaon 99, consumers nably o borrow and mpaence predcs ha consumpon wll rack ncome and ha cred consrans can be bndng. Usng evdence on calorc nake of food samp recpens, Shapro 2005 fnds ha he shor-erm dscoun rae of hese ndvduals s very hgh and hardly reconclable wh geomerc dscounng. Sudyng he effec of smulus paymens from he 200 ax cu In pracce, ax remance are ofen made hroughou he year by employers hrough payroll deducons, bu hs only apples for axpayers and no ransfer recpens. Ignorng hese remances wll have no effec on our analyss snce hose who pay axes face no cred consran. 3

4 epsode o explan he phenomenon of wealhy hand-o-mouh who mosly own llqud asses, Gruber 997 fnds evdence ha unemploymen nsurance, whch s pad on a frequen bass, sgnfcanly smoohs household consumpon. Parker 999 fnds ha consumers do no perfecly smooh her demand for goods when hey expec a change n her ncome alhough, n her case, he complexy n he ax code may be a sake. More recenly, Agula e al. 207 found n a naural expermen ha smoohng cash-ransfers over he year faclaes consumpon smoohng. In parcular, hey fnd ha more frequen cashransfer programs are assocaed wh more conssen spendng on basc needs, such as food and docor apponmens. Anoher source of evdence comes from household behavor durng he monhs when he Earned Income Tax Cred EITC s receved. McGranahan & Schanzenbach 203 fnd ha households who are elgble for he EITC spend relavely more on healhy ems durng he monhs when mos refunds are pad. Among hese healhy ems one fnds vegeables, mea, poulry and dary producs. In a recen survey paper, Nchols & Rohsen 206 sress ha [households] are ofen unable o borrow a reasonable neres raes as evdenced by he hgh ake-up of exremely hgh neres refund ancpaon loans. If cred consrans are bndng, a lump-sum paymen has a smaller effec on he household s uly han would a seres of smaller paymens hroughou he year. They also noe ha unl 200, EITC recpens could apply for a paral advance paymen hroughou he year. Alhough a small proporon of ndvduals oped-n, he mos plausble explanaon for akng up he cred would be ha ndvduals are severely cred consraned. In a recen work, Baker 207 fnds ha he ncome elascy of consumpon s sgnfcanly hgher for hghly ndebed households afer conrollng for ne asses. He concludes ha cred consrans play a domnan role n drvng dfferenal household consumpon responses across households wh varyng levels of deb. Also, usng daa from households who experenced a emporary ncome reducon durng he U.S. federal governmen shudown n 203, Baker & Yannels 205 fnd ndcaons ha households who have a beer 4

5 access o cred or who have more accumulaed savngs exhb sgnfcanly smaller spendng reducons durng he ransonal shock. In he followng secons, we sudy opmal ncome and commody ax polcy wh credconsraned low-ncome households n a mul-ype nonlnear ncome axaon seng. The model feaures several skll-ypes of households who supply labor and consume wo commodes. To smplfy maers, we assume ha ransacons can occur a wo dscree pons: n he mddle of he perod and a he end. Preferences are weakly separable so n he absence of cred consrans opmal commody axes wll be unform a ndeermnae raes gven ha proporonal commody axaon s equvalen o proporonal ncome axaon. The wo commodes are no consumed n he same proporons by dfferen skll-ypes, and hs wll lead o dfferenal commody subsdzaon n he presence of cred consrans. The cred consran wll ake he smples of forms. As well, for reasons o be explaned, we assume ha s cosly for he governmen o make budgeary expendures durng he perod. Dong so requres o borrow agans s end-of-perod ax revenues. In our analyss, we make he smplfyng assumpon ha he governmen canno use unversal lump-sum paymens o all ndvduals as frequenly as ransacons on consumpon goods occur. To fully exhb he polcy consequences of hs n a wo-perod model, we smply assume away lump-sum ransfers. Laely, here have been debaes n several counres abou he nroducon of a unversal basc ncome. However, nohng guaranees ha such ransfer paymen wll be pad n a mely enough manner so ha he mos vulnerable agens n socey can fully benef from hem. From a more echncal sandpon, nroducng he possbly of subsdzng consumpon goods and he paymen of lump-sum ransfers wh he exac same mng would be, no our model, smlar o allowng he governmen o run a dsnc lnearprogressve ax scheme n every perod a hgh frequency. We brefly explore he consequence of hs laer. 5

6 2 Model Consder N ypes of ndvduals who are ndexed by {,..., N}. There are n ype-s, each of whom has exogenous producvy w. The whole populaon s normalzed o one so ha N n =. We dvde he year no wo sub-perods =, 2, and assume ha each ndvdual = works wh he same nensy n boh sub-perods and earns a gross ncome Y /2 n each. A he end of = 2, a ype- agen pays an ncome ax T or receves a ransfer f akes a negave value. When agens choose her labor supples ex-ane, hey know her end-of perod ncome ax lably and herefore her dsposable ncome over boh sub-perods. We use he mehodology of Chrsansen 984 o nroduce consumpon of commodes n he model. In each sub-perod, agens choose a consumpon bundle conssng of wo goods c, d. The producer prces of goods c and d are se o uny, and he consumer prces can nclude a commody ax, whch can equvalenly be eher per un or ad valorem: q c + c and q d + d. Commody axes per un raes c and d are he same for boh sub-perods and for all households snce oherwse arbrage opporunes would exs. The agen s uly funcon s U c, d, Y = Y uc c, d h w where Y /w s oal labor supply over he wo sub-perods and h s a srcly convex cos or dsuly funcon. The funcon u, s he per-perod uly of consumng he bundle of goods. To ensure ha commody ax dfferenaon s no a by-produc of nonlnear Engel curves, we somemes assume ha u s quas-homohec by nroducng a basc need c on good c. Ths would sand for a mnmal quany of food, or dwellng. Indvduals do no dscoun her uly across perods, whch does no resrc our resuls and seems reasonable snce we consder raonal expecaons over a shor perod of me a fscal year. Noe ha alhough he agen supples labor n boh sub-perods, he dsuly of labor supply s defned 6

7 over oal annual labor supply. Snce commodes are separable from labor or lesure n he uly funcon, he Aknson-Sglz Theorem would apply n hs model n he absence of a cred consran. 2 We nroduce mperfecons n he cred marke n he form of a cred consran. The cred consran applyng n he frs sub-perod s q c c + q d d Y 2 + τ + φ, 2 where φ s exogenously gven. Indvduals have access o a compeve cred marke f hey wan o borrow or save. Those who save do so a rae r and hose who borrow do so a rae r, wh r r. Ths reflecs he cos of fnancal nermedaon. For an ndvdual, we denoe by r {r, r} dependng on wheher, n he opmum, he s respecvely a ne saver or borrower a =. If he governmen borrows, can do so a rae r g > r, meanng ha borrows a a hgher rae han he rsk-free rae a whch ndvduals can nves her shor-erm savngs. 3 Under hese assumpons, we shall see ha he wo sub-perods seng gves he same soluon as a sandard Mrrlees problem when here s no cred spread, ha s, when r = r = r g. Ths s our benchmark case. Then, we nroduce a borrowng consran ha prevens ndvduals from usng more han φ dollars of her end-of-year ransfers as a collaeral when applyng for a loan. The smples case s when φ = 0, whch mmcs he corner soluon one would oban f borrowers faced an neres rae ha s prohbvely hgh. Gven ha our model absracs from solvency ssues and fnancal rsks relaed o lendng o ndvduals, hs s a smple way o nroduce cred marke frcons whou explcly modelng solvency rsks. 4 2 The model assumes ha ndvduals comm o her labor supply and ha labor supply s he same across perods. Ths does no drve he resuls and smplfes he analyss. 3 In parcular, hs prevens he fscal polcy from beng a Ponz scheme and evacuae arbrage opporunes. 4 A more complex model would nvolve rsk. Then, would be cosler o banks o lend o ndvduals and he neres rae for borrowers would be hgh. Ths would gve us he same nuon, bu would sgnfcanly 7

8 To be precse, n he case where here would be no cred consran, yearly budge consrans are from an end-of-year sandpon q c c + q d d + r + q c c 2 + q d d 2 Y 2 + r + Y 2 T. Snce ndvduals earn Y /2 every sub-perod and only pay her axes ge her ransfers T a he end of he year and ha ndvduals can make ransacons n he fnancal markes, 2 + he nonlnear ax problem amouns o choosng I r Y T. Therefore, one can 2 rewre he consran as q c c + q d d + r + q c c 2 + q d d 2 I Borrowng consrans and he role of commody axes: nuon Now ha he man pars of he model are se up, le us provde he nuon behnd our resuls. Snce sub-uly funcons uc c, d are he same across perods and here s no whn-perod dscounng, whou a bndng cred consran and f r = r = 0, ndvduals wll perfecly smooh her consumpon over me and consume half of her ne yearly ncome n each perod. Then, f he governmen mposes undfferenaed commody axes or subsdes c = d, can reach he same allocaon by axng everyone s yearly ncome a rae Y = c / + c = d / + d. In hs case, we can normalze one consumpon ax o zero and le he fla revenue-collecon componen be capured by he proporonal ax on ncome lesure. Recall, however, ha ncome axes are colleced a he end of he perod, whle commody axes apply n each sub-perod. Thus, o replcae he effec of a unform consumpon ax ha apples each sub-perod on he budge sream of axpayers, he governmen would have o mpose a equal lump-sum ax or subsdy o all ndvduals n complcae he problem. 8

9 boh sub-perods. However, when an ndvdual s borrowng consran bnds, hs equvalence does no hold. Snce ndvduals wll consume a dfferen mx of he wo goods n he wo sub-perod, he revenue-rasng subsdy componen n c and d canno be maed by a proporonal ax subsdy on ncome. The ncome ax s no pad n he frs perod, so he bndng cred consran s, wh φ = 0, + c c + + d d = Y 2. 4 Noe ha does no conan a ax on s rgh-hand sde. Therefore, f he governmen wan o ax ncome specfcally n he frs perod, has o do hrough he axaon of goods. Smlarly, f he wans o redsrbue n he frs perod, has o do eher hrough a subsdy on goods or hrough a unform lump-sum subsdy o all ndvdual n he frs sub-perod snce canno denfy ndvduals by ype hen. Suppose we dvde boh sdes of he cred consran by + c : c + αd = + c Y 2, 5 where α q d /q c s he prce rao chosen by he governmen, and / + c s he change n purchasng power of frs-perod dsposable ncome nduced by axaon or subsdzaon of c. Alhough he opmal prce rao, whch ells us wheher axes should be dfferenaed n he opmum, may be unque, he opmal ner-emporal ransfer of purchasng power on he rgh-hand sde can be acheved hrough axaon or subsdzaon of eher good. Le us denoe by α he opmal prce rao chosen by he governmen. Suppose nsead we dvde he cred consran by + d : βc + d = Y + d 2 + φ, 6 where he opmal prce rao s now β q c /q d. The opmal prce rao ha wll be chosen 9

10 by he governmen wll be β = /α, bu he ransfer of purchasng power durng he year wll be performed by a ax or a subsdy on good d, d. 2.2 Governmen s budge consran The governmen s budge consran n absolue erms s 2 + r Y I + + r g q c 2 c + q c c r g q d d +q d d = R. Normalzng by q c, seng α q d /q/c, we ge n erms of relave prces: 2 + r Y 2 q c I ++r g q c q c c + q c c 2++r g α q c d + α q c d = R q c The fac ha r g > r makes socally cosly o ransfer ressources n =. 2.3 Opmal ax mx We derve he governmens opmal ax srucure usng a sandard mechansm desgn problem for ncome axes augmened by a choce of commody ax raes. The governmen offers bundles of ncome Y, I nended for ypes. Then, axes pad a he end of he perod are resdually gven by T = Y I, where T can be negave for he low-producvy ype. The governmen also chooses c, or equvalenly q c, and he prce rao α = q d /q c. When an ndvdual s cred consraned n he opmum, α and q c wll generally dffer from uny. In 0

11 he absence of a bndng cred consran, α = q c =, snce he governmen reles enrely on ncome axaon. We solve he ndvdual s problem n wo seps n reverse order. In he second sep, knowng Y, I, q c and q d, he chooses bundles c, d for =, 2. In he frs sep and ancpang he oucomes of he second sep, he ndvdual chooses from he bundles of ncome and dsposable ncome Y, I offered by he governmen Sep 2: Choce of commody bundles Gven Y, I, q c, α, ndvduals of he wo ypes choose bundles c, d o maxmze uly subjec o he annual budge consran 3 and he cred consran 2. The value funcon for hs problem s: ψ Y, I, q c, α, τ = max c,d uc c, d +θ [ I q c + + r τ q c + r c + αd [ µ c + αd ] Y q c 2 + τ + φ, 7 where φ {0, }, dependng on he specfc case ha s under sudy. ] The soluon o he problem of a ype- ndvdual s characerzed by he followng frsorder condons for consumpon n he wo sub-perods: u c c, d θ + r µ = 0, u d c, d θ + r α µ α 0 8 By he envelope heorem, u c c 2, d 2 θ = 0, u d c 2, d 2 θ α 0 9 ψ I = θ q c, ψ Y = µ 2q c, ψ α = θ [ + r d + d 2] µ d. 0 5 For a smlar approach, see Edwards e al. 994

12 Noe also ha dψ /dφ = µ. Snce consumer uly s non-decreasng n he sze of he cred consran φ, ha mples µ 0 wh he nequaly applyng when he consran s bndng. Noe also ha, by defnon, µ = 0 when φ Sep : Choce of ncome and ne ncome bundles Gven commody ax raes c, α and ancpang sep 2 above, he governmen on behalf of ndvduals of he wo ypes offers consumpon bundles Y, I. Ths yelds oal uly for a ype- person of: Y V Y, I, q c, α, τ = ψ Y, I, q c, α, τ h. Usng he envelope resuls 0 on ψ, V sasfes he followng properes: w V Y = µ 2q c w h Y w, V I = ψ I, V τ = ψ τ, V q c = ψ q c, V α = ψ α. 2 Preferences of an ndvdual of ype n Y, I-space have a slope of: di dy = V Y V I = q [ c θ w h ] Y µ w 2q c Fnally, denoe V as he oal ndrec uly of a ype who mmcs a ype. The mmcker wll have he same ncome sream so wll face he same cred consran as he ndvdual beng mmcked. Analogously o V n, ndrec uly s gven by: Y V q c, α, Y, I = ψ h. 3 Smlar envelope properes o 2 apply, and he slope of he mmcker s ndfference curves w 2

13 wll be: dî dŷ = V Y V I = q c θ [ w h Ŷ w ] µ 2q c Compensaed demands For furher use, we presen he expressons used for compensaed demands. We compensae demands by varyng I bu akng Y as gven. When an ndvdual s unconsraned, hey sasfy c α = c α q c c + r I 2 d, d α = d α q d c + r I 2 d. If an ndvdual s consraned, hen n he frs perod c / I = d / I = 0. However, gven me-separably we can make use of he fac ha he frs-perod expendure sasfes q c c + αq d d = Y /2 + φ, evaluaed locally a φ = 0, o derve he compensaed demands. A = hey are, c α = c α q cd c φ, d α = d α q cd d φ. Therefore, followng a margnal change n he prce rao, keepng labor effor consan, ncome compensaon can be acheved hrough a margnal ncrease n gross labor ncome or, equvalenly, by allowng he consraned ndvdual o borrow margnally more. In he second perod, c 2 c 2 α = c 2 α q c I d 2, d 2 d 2 α = d 2 α q c I d 2. 3

14 2.4 Tax mplemenaon Le us fnd he formulas for ax mplemenaon. The governmen mplemens a nonlnear ax funcon T Y. Le us denoe ndrec uly n he followng way: UY, I = ψ I hy /w. Snce hs s mplemened by makng ndvdual free choose hs ncome Y, we rewre explcly as 2 + U = ψ r Y T Y h 2 Usng he envelope condons, hs mply ha he frs-order condon for hs ndvdual s Y w. ψ I I Y + ψ Y h Y Y w = θ 2 + r T Y + µ q c 2 2 q c w h Y = 0. w Isolang he margnal ax rae gves T Y = 2 + r 2 q c θ w h Y w + µ 2 θ. 2.5 Governmen s problem In our problem, he governmen redsrbue from more producve o less producve ndvduals. We use he mehodology developed by Hellwg 2007 also appled by Basan 205 o derve opmal ax schedules wh a fnely large number of ypes. The governmen maxmzes socal welfare: W = n ΦV 4

15 subjec o N ncenve compably IC consrans ha ake he form of downward adjacen consrans, V Y, I, q c, α, τ; w V Y, I, q c, τ, α; w, γ and o he budge consran: N 2 + r Y n I + N n +r g 2 c 2 q c q c q + α N n +r g 2 d c q c = = = R q c = 0 λ where ΦV s a concave socal uly funcon, wh Φ V > 0 and Φ V 0, and R s he exogenous revenue requremen of he governmen. The equaon ndcaors γ and λ represen he Lagrangan mulplers of hese consrans n he governmen s problem. Noe ha for R small enough, a leas one ype he lowes receves a ransfer. The funcon V Y, I, q c, α, τ; w n he IC consrans s he ndrec uly obaned by f he mmcs he adjacen lower ype, so Y V = ψ h. w Gven our assumpon abou preferences, all ndvduals would perfecly smooh her consumpon across sub-perods and 2 n he absence of cred consrans. If cred s consraned, can only be bndng for hose expecng a ransfer a he end of he perod snce hen hey wll wan o borrow n perod. Those who pay posve axes wll save a = o spread her ax lables across sub-perods. We consder he governmen problem n hree successve seng of ncreasng complexy. We begn wh he benchmark case n whch no one s cred-consraned. We hen assume a cred consran s bndng on a leas one ype he lowes, bu resrc he governmen o 5

16 usng a nonlnear ncome ax. Fnally, we le he governmen choose dfferenaed commody axes or subsdes alongsde he nonlnear ncome ax. We denoe by L he Lagrangan funcon of he governmen. For clary, we explcly rean all mulplers on he frs-order condons of he governmen s problem. The frs-order condons for he governmen s problem n he hrd, mos general, seng where he cred consran s bndng for a leas one ype and he governmen chooses commody ax raes are lsed n he Appendx Benchmark: unconsraned ndvduals and no cred spread Ths case corresponds o he sandard opmal nonlnear ncome ax problem wh lnear commody axes. All ndvduals and he governmen can borrow a he unque neres rae r. Frs, we can esablsh wheher he governmen wll use commody axaon a all. I can drecly opmze on relave commody prces α = q d /q c and ses q c = snce we can normalze one ax o zero. Therefore, choosng α s equvalen o choosng q d = + d. Snce ndvduals are no cred-consraned, µ = 0,, n he ndvdual s value funcon 7. Usng he envelope properes for he ndvduals n 0 and 2, he governmen s frsorder condons shown n he Appendx lead o he sandard Aknson-Sglz heorem: Proposon. When =,..., N are unconsraned and here s no cred spread, hen he Aknson-Sglz Theorem holds and commody axes are undfferenaed. Proof: See appendx. The Aknson-Sglz Theorem spulaes only ha commody axes should be unform f used, bu redundan and hus unnecessary. Incenve-compably mples ha, n hs case, he opmal ax schedule, usng 2, sasfes 2 + r T Y = + V Y 2 VI 2 + r = 2 q c θ w h Y w. 4 6

17 The frs-order condons gve us he followng margnal ax formulas: T Y = θ γ + λn h Y /w h Y /w +. 5 θ w θ w + As clarfed by Hellwg 2007, he rghmos erm n parenheses s always posve when he sngle-crossng condon s sasfed and when lesure s an normal good. Therefore, margnal ax raes are everywhere posve excep for w N, for whch here s no dsoron and T w N = 0. We now urn o cases wh eher or boh bndng cred consrans or cred spreads Bndng cred consrans; no commody axes Suppose ha ransfers o he lowes ype s suffcenly large ha he cred consran on a leas one ype s bndng, so µ > 0 for a leas some. Those whose cred consran does no bnd pay axes a he end of he perod and save for, whereas he poores ones would have lked o borrow usng fuure ransfers as collaeral bu hey canno. There are no commody axes n hs specfc case, so q c = α =. As a consequence, he governmen has no revenues and no expenses n he frs sub-perod, and r g does no need o be specfed. Those who save do a rae r = r. The frs-order condons of he governmen s problem are presened n he appendx. Frs, noce ha he frs-order condons of a consraned ndvdual gve he followng wedge beween dsuly of labor and labor ncome: 2 + r T Y = q c Y 2 θ w h w + µ 2 θ. Thus, because of he bndng cred consran, an ndvdual wll be more nclned o work more o generae ncome n he frs-perod. No lookng a he ax formula, whch gves us 7

18 he opmal wedge, we ge he sandard expresson T Y = θ γ + λn h Y /w h Y /w +, 6 θ w θ w + whch wll end o gve lower margnal ax raes for he cred consraned for wo man reasons. Frs, bndng cred consrans lower uly, and herefore reduces he ncenve of hgher ypes o mmck, whch has an effec on γ +. Also, he governmen has an neres n makng consraned ndvduals work more, whch s he only way o ncrease her oherwse subopmal consumpon n he frs perod Cred consrans no bndng n he opmum; use of commody axes We now dscuss he case n whch ndvduals face cred consrans, and where he governmen can smulaneously use commody and ncome axaon. Here, we need o descrbe wo ypes of opmal polcy. In he frs one, whch happens when r g r s small, he governmen can use commody axes o unconsran all ndvduals. In hs case, boh goods are subsdzed a he same rae. Snce he frs-perod of fundng subsdes on commodes s low enough, he governmen subsdze hem enough, hereby ncreasng frs-perod purchasng power, so ha everyone s fnally unconsraned. In hs case, he commody ax subsdy sysem acs as a proporonal subsdy on ncome. Snce cred consrans are no bndng, he equvalen of a second-bes alhough wh borrowng neres raes hgher han savng reurn raes s recovered. To ensure ha he whole ax sysem maxmzes socal welfare n an ncenvecompable way, ncome ax raes boh average and margnal are adjused, so ha he effecve margnal ax rae of ndvdual s dencal o he one we would have obaned whou cred consrans and whou commody subsdes. Proposon 2: If n he opmum µ = 0,, meanng ha he polcy can relax all cred 8

19 consrans n he economy, hen c = d < 0. Proof: See Appendx. In hs parcular case, he margnal ax rae faced by ndvdual s T Y = θ γ + q c λn where z h Y /w h Y /w + θ w θ w + c + r g 2 z T Y I + d, I = 2 + r /2. The ax formula, analogous o ha derved by Edwards e al. 994, shows ha when he governmen subsdzes consumpon proporonally, hs creaes purchasng power ha s dencal o an ncrease n ne ncome. Therefore, a share z T Y of he ncome value of he subsdy has o be lef n he ndvdual s pockes, adjused for he fundng cos of he subsdes r g. The ax formula reflecs he fac ha he wedge beween labor and consumpon mus encompass he margnal ncenves and dsncenves generaed by all ax nsrumens. 7 In parcular, one can use 7 and rewre n erms of margnal effecve ax rae: T Y + c +r g 2 z T Y I + d = θ γ + q c h Y /w I λn h Y /w + θ w θ w + whch shows ha subsdes on consumpon goods mus be clawed back by ncreases n margnal ncome ax raes. I also shows ha he sandard properes of he opmal ax sysems are no volaed, ncludng posve margnal ax raes a all ncome levels and a zero effecve margnal ax rae a he op., 8 Fnally, one should noe ha he mos exreme case of such an opmum would be when subsdes can be funded a no opporuny cos for he governmen, or when r g = r. Then, ransferrng purchasng power from he second o he frs perod s done a no cos for he governmen, who can always lower he prces of commodes so as o slack all cred consrans n he economy. In hs unrealsc example, he mng of ncome-esed paymens 9

20 has no effecve consequence on he overall ax polcy and socal opmum. As soon as subsdes become cosly, for nsance when hey have o be a leas parally funded wh deb, here s a hreshold level of he cos from whch he governmen wll leave some ndvduals cred consraned Cred consrans sll bndng n he opmum; use of commody axes When he cos of fundng commody subsdes n he frs perod are hgh enough, he governmen may decde o no slack some ndvdual s cred consrans. When hs happens, may be opmal o eher dfferenae he subsdes, or o fund one subsdy wh a posve ax on anoher good. The nuon s he followng. Commody axaon generaes boh ncome and subsuon effecs. When here were no bndng consrans, he nonlnear ax sysem could adjus o offse ncome effecs, and ensued ha was opmal o no creae subsuon effecs. Wh bndng cred consrans, for hose who are consraned n he opmum some ncome effecs canno be clawed back by proporonal adjusmens n he ncome ax schedule. These ncome effecs are cosly for he governmen because hey nduce an ncrease n consumpon n he frs perod whch s subsdzed and mus be fnanced hrough shor-erm deb. On he oher hand, subsdzng a leas one commody s benefcal because helps reducng he pressure of cred consrans. Thus, he governmen needs o compromse and spend s resources on he good ha s proporonaely more consumed by he consraned. As urns ou, hs may happen even when uly funcons u feaure lnear Engel curves. Our smple case n whch here s a basc need c on one good jusfes eher subsdzng good c a a hgher rae, or smply subsdzng c and axng d f r g r s very hgh. These resuls are llusraed n he numercal secon, and proposon 3 gves he general condon under whch dfferenaon happens under lnear Engel curves. 20

21 Proposon 3: Assumng lnear Engel curves, when he opmal ax sysem nvolves µ > 0 for a leas some, dfferenaon happens when, n he opmum, [ q E d α = qc E c c φ + d ] φ µ > 0 ]/ π c π d [ d c φ + d φ µ > 0 9 or c < d when [ ] E c c + d φ φ µ > 0 [ ] E d c + d > µ > 0 φ φ π c, 20 π d where π = µ /λ [n Φ V + γ γ + ] n r g r s a wegh ha s srcly negave for unconsraned ndvduals and whch s ncreasng wh frs-perod oal expense. We nerpre he frs erm n he rgh-hand sde of 9 as he cos rao of remanng ncome effecs hrough good c over hose of good d, for all consraned ndvduals. We nerpre he second erm n he rgh-hand sde of 9 as a rao of ne benefs of slackng consrans hrough a reducon n axes or an ncrease n subsdes on good c versus on good d. When he whole erm n he rgh a cos-benef rao of dfferenang c downwards hand sde of 9 s greaer han one, we have eher c < d < 0 or c < 0 < d, dependng on r g r and on he varous parameers, gvng 20. Here agan, he governmen mus adjus he margnal and average ncome ax raes o nclude he effecs of ndrec axes. The margnal ncome ax raes are T Y = θ γ + q c λn [ c h Y /w h Y /w + θ w θ w + + r g 2 c I z T Y + c Y ] d [ ] d + r g 2 I z T Y + d, Y 2 where z = 2 + r /2. In hs parcular case, unconsraned ndvduals consumpon of commodes does no reac o a change n Y snce he governmen may adjus he ncome 2

22 ax schedule followng a change n labor effor. For ndvduals who are sll consraned, frs-perod consumpons do no reac o I, and second perod consumpons do no reac o Y. Example of dfferenaon wh Cobb-Douglas and basc need Le us gve an example wh a per-perod uly funcon uc, d = c c α d α. Sar wh he case where c = 0. Usng he well-known marshallan demands assocaed wh hese preferences, one can verfy ha here s no dfferenaon snce q d /q c = α αα α. Wh he basc need, denong R = Y /2 for he consraned and R = I /2 for he unconsraned, c < d < 0 f α α π [R /q c + α c] π [R /q d cq c /q d ] > α α c n z q c φ + d φ c n z q d φ + d φ [π R /q c + α c] [R /q d cq c /q d ] Calculang funconal forms for he dervaves of demands wh respec o ncome, evaluaed a q c = q d, s welfare-mprovng o dfferenae q c when π [R + αq c c] π [R q c c] > c n z φ + d φ c n z φ + d φ [R + αq c c] [R cq c ] Defnng N C he number of consraned, and ER C her average frs-perod ncome, hs s only verfed f π [R E[R C ]] < 0. Ths resul depends on he parameers of he model, bu for havng no dfferenaon, we should have o observe no correlaon beween he weghs π and frs-perod expense, whch. 22

23 s no a feaure of he second-bes ax sysem. 3 Numercal examples To llusrae our resuls we underake a seres of numercal smulaons. We use he followng quas-homohec per perod consumpon uly funcon uc, d = κ c c ρ ρ + κ d ρ ρ, n addon o he followng labor dsuly funcon hl = l+σ + σ. We se ρ = 0.9 and σ = 2. The basc level of consumpon of good, c, s se o 5 whch roughly represens a $5.75 a day spendng or $200 annually n our model. 6 κ s se o /4 ρ. 7 The consan For he number of workers a each wage level ype n, we use a lognormal dsrbuon wh parameers µ, σ = 2.757, 0.56 aken from Mankw e al The auhors esmae hese parameers from he 2007 March wave of he Curren Populaon Survey CPS. We hen dscreze he dsrbuon o oban 00 wage levels wh he dsance beween each wage beng fxed. The probably mass funcon s rescaled so ha n =. To creae an neres rae spread large enough o hghlgh our man resuls, we use r = 0 and r g = Ths level of mnmum consumpon s farly conservave. The average Supplemenal Nuron Asssance Program SNAP benef s abou $4 a day and f we consder c o nclude oher basc necesses n addon of food, $5.75 s relavely low. In addon he 2007 Federal povery lne s se a $0,20 $28 a day for a sngle person household, and hus our level of c s roughly 20% of ha hreshold. 7 Ths s chosen so he soluon of he opmal ax problem would gve he same uly level and opmal ax funcon as n he case where c = 0 wh r = 0 and he yearly uly of an ndvdual would be. uc, l = c ρ ρ l+σ + σ 23

24 In our smulaons, due o he nroducon of commody axes and subsdes, we also calculae he effecve ax burden pad a a gven level of ncome and he effecve margnal ax raes as n Edwards e al The oal ax burden of a worker a labor ncome Y s ] ] T E Y = T Y + c [ + r 2 c + d [ + r 2 d and he margnal effecve ax rae a Y s T E Y = T + c [ ] [ c + r 2 ] I z T + c d + Y d + r 2 I z T + d, Y where T T Y and z 2 + r /2 for noaonal convenence. To sar, we consder hree scenaros. The frs scenaro s he one called No cred consran, hs scenaro encompasses many oher possble cases. The frs one s he case where here are no cred consrans and he ndvduals can boh save and borrow a he same rae. The no cred consran scenaro wll also gve he same allocaon and same effecve margnal ax raes as he case where he planner can coslessly ransfer money n he frs perod hrough subsdes or any oher scheme. The second scenaro ha we consder s he Cred consran where he ndvdual s borrowng lm,.e. φ, s se o 0. The planner n hs case only has access o nonlnear axaon on labor ncome o acheve hs goals. Due o he fac ha we have se r = 0, ndvduals are consraned whenever T Y < 0. The hrd scenaro, Cred consran wh subsdes s smlar o he case where we mpose a cred consran bu now he planner s able o use axaon on commodes n addon o nonlnear labor ncome axaon o acheve hs goals. Under all hree scenaros he planner 24

25 Fgure : Opmal Labor Income Tax Raes under Dfferen Scenaros 0.5 Margnal ncome ax raes Effecve Margnal ncome ax raes T'y Effecve T'y No cred. consran Cred cons. Cred cons. wh subsdes Annual Income Average Tax Rae 0. No cred. consran Cred cons. Cred cons. wh subsdes Annual Income Effecve Average Tax Rae Ty/y No cred. consran 0.8 Cred cons. Cred cons. wh subsdes Annual Income Effecve Ty/y No cred. consran 0.8 Cred cons. Cred cons. wh subsdes Annual Income has ularan socal preferences,.e. he socal welfare funcon s W = n V. The characerscs of he opmal soluons o all hree cases are shown n Fgure and Table. From Fgure, s possble o see ha he scenaro wh he cred consran and subsdes feaures he hghes margnal ax raes excep a he very boom of he ncome dsrbuon and he hghes average ax raes. As argued above, he opmal soluon o hs scenaro where he governmen has access o commody axaon wll feaure subsdes o 25

26 Table : Characerscs of Opmal Allocaon under Dfferen Scenaros Scenaro q c q d q d /q c % Consraned n V % LF No Cred Consran 0% % Cred Consran 62.23% % Cred. Cons. and Subsdes % % Source: Auhors calculaons. Numbers rounded. consumpon goods. To fnance hese subsdes he planner mus rase revenues hrough he use of labor ncome axes. Lookng a he opmal effecve margnal ax raes s possble o see ha he cred consran and subsdes scenaro s n fac an n-beween case of he no cred consran case and he cred consran case wh only nonlnear labor ncome axaon avalable o he planner. The no cred consran scenaro feaures he hghes effecve margnal ax raes because s able o ransfer money o he ndvdual eher hrough s own savngs or oher schemes, hs mples ha he planner s able o dsor labor more a he boom ensurng a hgher level of redsrbuon. Whereas he cred consran case whou commody axes wll feaure he lowes effecve margnal ax raes. In hs scenaro, due o he basc need c, low skll workers mus supply much more labor o mee her needs and snce he planner s unable o ransfer ressources o hem n he frs perod, he only hng he can do s o no dscourage work all ogeher. In addon, because of he cred consran and he nably of low sklled workers o smooh consumpon, ransferrng uly o hem hrough ransfers n he second perod s more dffcul and hus lowers he value of redsrbung o hem. Ths reduces he need for he planner o ax workers o rase revenue for redsrbuon. The scenaro wh cred consran and commody axaon perms he planner o ransfer some ressources, albe a a cos, o he frs perod hrough subsdes on goods. The same paern from he cred consran case emerges a he boom of he ncome dsrbuon snce he planner does no wan o dscourage work. Bu now labor ncome s 26

27 slghly more valuable snce he subsdes on goods ncrease he purchasng power of workers and hus allows for more redsrbuon and lower number of consraned workers. Table shows ha, hrough subsdes, less workers are consraned n he case where commody axaon s avalable o he planner. In he same Table, he ularan socal welfare of all hree cases s repored. Usng he socal welfare numbers we see ha he case wh cred consran and commody axes s preferred o he case wh he cred consran bu whou he addonal axaon ool. To gve anoher measure of welfare o compare scenaros, we measure welfare gans of he opmal allocaons from he lassezfare allocaon. Ths gan s calculaed by he consan percenage ncrease n consumpon % LF offered o boh goods n boh perods o all ypes o he lassez-fare economy o reach he same level of ularan socal welfare n each scenaro. 8 Agan, we are able o see ha he case wh commody axes s preferred o he case whou. The small dfferences are mos lkely due o he ularan socal welfare preference and he lmed curvaure of he consumpon uly funcon whch pus less wegh on he lower sklled workers. Therefore any socal welfare ha would pu more wegh on he lower sklled workers would also pu more value on polces ha reduces he effec of he cred consran on he workers. We fnsh hs secon by consderng wo ses of smulaons ha explore changes n he cos of provdng subsdes and changes o he level of basc needs. The frs se of smulaons ha we consder changes he spread of neres raes, r g r. To focus on he changes n he coss and no on ncreases n he amoun of ressources avalable o he economy due o hgher neres raes on savngs, we only change he level of r g. The resuls of hese smulaons are shown n Fgure 2 and Fgure 3. Fgure 2 presens he same nformaon n hree dfferen manners. The man pon s ha he cheaper s for he governmen o borrow he more subsdzes good and he less dfferenae commody axes. In he case where coss are hgh enough, s possble o oban a ax on good d nsead of a subsdy. The goal s for he planner o gve every cosly ressource avalable o subsdze he good wh a basc level 8 Ths s he same as expeced lfeme uly of a worker from behnd he vel of gnorance. 27

28 Fgure 2: Prces and Opmal Commody Taxes under Dfferen Ineres Rae Spreads Consumer Prces Consumpon Taxes qc qd 0.2 c d Prce 0.6 Tax Spread rg r Prce of good C and Prce Rao qd/qc qc Spread rg r Rao qd/qc qc Spread rg r 28

29 Fgure 3: Opmal Labor Income Tax Raes under Dfferen Ineres Rae Spreads T'y Margnal ncome ax raes Effecve T'y Effecve Margnal ncome ax raes rg r= rg r=0.5 rg r= Annual Income Average Income Tax Rae rg r= rg r=0.5 rg r= Annual Income Effecve Average Income Tax Rae Ty/y rg r=0.05 rg r=0.5 rg r= Annual Income Effecve Ty/y rg r=0.05 rg r=0.5 rg r= Annual Income 29

30 of consumpon ha mus be me. Fgure 3 shows he effec of dfferen coss on he opmal labor ncome ax schedule. Whenever coss o subsdze are low, he opmal ax sysems feaures large subsdes and requres hgher labor ncome axaon as can been seen by boh he margnal ax raes and average ax raes. As above, when he planner s able o ransfer more money n he frs perod, he s also able o dsor he labor decson of lower sklled workers a b more and aemp more redsrbuon. Ths can be seen by lookng a he effecve margnal ax raes. The case wh he hghes spread, and hus he hghes coss, also feaures he lowes effecve margnal ncome ax raes snce he planner mus encourage work especally a he boom of he dsrbuon. Smulaons found n Fgure 4 and Fgure 5 consder changes n he level of basc need c. From Fgure 4, we see ha he hgher he level of basc need he more he planner wll wsh o subsdze consumpon bu also he more he wll wan o dfferenae he commody axes. We can see ha wh c = 0 he planner wll sll wsh o subsdze consumpon. Ths s due o he cred consran and he cos of ransferrng ressources o he frs perod. The greaer he basc need he harder and more cosly s n erms of uly o acheve hs level of consumpon hrough work effor, hence he governmen wans o use as much ressources as can o help low wage workers by ncreasng he value of her labor earnngs. Fgure 5 llusraes he effec on labor ncome ax raes of an ncrease n he basc need. To fnance he ncrease n subsdes he planner requres hgher levels of labor ncome axaon. Ths s evden from he plos showng margnal ncome ax raes and average ax raes. The hgher level of basc need also appears o ncrease he level of redsrbuon whch s ndcaed by hgher effecve margnal ax raes a he boom of he ncome dsrbuon bu also for all workers. Average effecve ax raes are also hgher for hgher levels of basc need. A bgger basc need leads o hgher margnal uly of consumpon of he necessy good for a gven level of consumpon, snce reduces he dsance beween ha level of consumpon and he basc need. Ths would ncrease he wllngness of a ularan planner, who seeks 30

31 Fgure 4: Prces and Opmal Commody Taxes under Dfferen levels of Basc Need Consumer Prces qc qd Consumpon Taxes c d Prce 0.7 Tax Need $/Per day.06 Prce of good C and Prce Rao Need $/Per day Rao qd/qc qd/qc qc 0.8 qc Need $/Per day 0.5 3

32 Fgure 5: Opmal Labor Income Tax Raes under Dfferen levels of Basc Need Margnal ncome ax raes Effecve Margnal ncome ax raes T'y Need=$0/Day Need=$2.30/Day Need=$5.75/Day 0. Need=$8.05/Day Need=$.50/Day Annual Income Average Income Tax Rae Effecve T'y Need=$0/Day Need=$2.30/Day Need=$5.75/Day 0. Need=$8.05/Day Need=$.50/Day Annual Income Effecve Average Income Tax Rae Ty/y Need=$0/Day Need=$2.30/Day Need=$5.75/Day Need=$8.05/Day Need=$.50/Day Annual Income Effecve Ty/y Need=$0/Day Need=$2.30/Day Need=$5.75/Day Need=$8.05/Day Need=$.50/Day Annual Income o equalze levels of margnal uly across ypes, o aemp more redsrbuon. 4 Implcaons of nroducng a lump-sum ransfer n he frs sub-perod As menoned n he nroducon, he possbly ha he planner could use unversal lumpsum paymens o address drecly he ssue ackled n hs paper would requre hose pay- 32

33 mens o be made a a smlar frequency as ransacons on consumpon goods. If we suppose ha hs was a possbly, hen he nroducon of lump-sum paymens opens he door o neresng neracons beween commody axaon and hese paymens. If he lump-sum ransfers could be made coslessly, e.g. whenever r = r, he planner wll wsh o fron-load redsrbuon n he frs perod by offerng a unversal lump-sum paymen o all ndvduals and hen adjusng he end of perod labor ncome ax schedule o ensure ha no one wll wsh o borrow and be cred consraned. If hs scheme s avalable o he planner, hen he allocaon would be dencal o he one whou cred consrans. There would be no need o use commody subsdes and he Aknson-Sglz Theorem would hold. In he case where r g > r, he planner faces a cos o ransfer money from he second sub-perod o he frs sub-perod. In hs scenaro, he planner s no longer able o fron-load compleely redsrbuon n he frs sub-perod and adjus hs labor ncome ax schedule o leave all workers unconsraned. Ths suaon s smlar o he one where he planner can only use commody axes and subsdes whch offers only a paral relef o he cred consran problem. Ths leaves room for he combnaon of boh commody axes and lump-sum ransfer o ncrease welfare. When he planner as access o boh lump-sum paymens and lnear axes on goods, under he assumpon ha ndvdual preferences gve lnear Engel curves, he planner wll wsh o creae s own nra-perod lnear-progressve ax sysem. Snce ransferrng ressources across perods s cosly, he planner wll se hgh unform commody axes on boh goods and use he revenue o fnance a large lump-sum ransfer n he frs sub-perod. In hs suaon, he planner neher saves nor borrows, budges are balanced n each sub-perod. The planner decdes o reduce nequaly nra-perod nsead of ryng o smooh consumpon across sub-perods by decreasng he number of cred consraned workers. In fac, he opmal allocaon appears o push owards more cred consraned workers. Ths s due o he hgh level of commody axes affecng boh perods whch leads he planner o adjus he labor ncome ax schedule o make he allocaon ncenve compable. To do hs, he ax burden 33

34 of workers s drascally reduced o he pon of a majory of workers facng ne-ransfers from he labor ncome ax. Ths leads many workers o wan o borrow bu are unable o do so due o he cred consran. The unform commody axes, wh cred consraned workers, ac as a lnear ax on labor ncome n he frs sub-perod wh he revenues gong o fnance he large lump-sum paymen. We hen recover he resul of Deaon 979 ha demonsraes ha an opmal lnear progressve ax sysem wll no use dfferenaed commody axaon when Engel curves are lnear. We conjecure ha he unformy resul obaned wll break down f Engel curves are nonlnear. Furhermore, even n he case f lnear Engel curves, we conjecure ha dvorcng he mng of he collecon of commody ax revenues from he paymen of he lump-sum ransfer wll also lead o dfferenaed commody axes. In fac, we should be able o recover many of he resuls found n hs paper. Provng hese conjecures s lef o furher research. 5 Concluson Ths paper sudes an opmal ax sysem when ransacons on goods happen more frequenly han he paymen of ncome-esed ransfers. The cred consrans arse because ndvduals canno fully use fuure ransfers as collaeral. Our resuls show ha when he opmal polcy s able o unconsran all ndvduals, nvolves proporonal subsdes on goods. When he cos of dong so s oo hgh, dfferenaon may happen when consraned ndvduals spend a hgher proporon of her dsposable ncome on a good for nsance a necessy han he general populaon. Then, he governmen can eher subsdze all goods, wh a hgher subsdy on commodes, or ax some goods o fund he subsdes on commodes. 34

35 References Agula, E., Kapeyn, A., & Perez-Arse, F Consumpon smoohng and frequency of benef paymen of cash ransfer programs. Amercan Economc Revew Papers and Proceedngs, 07, Aknson, A. B. & Sglz, J The desgn of ax srucure: drec vs. ndrec axaon. Journal of Publc Economcs, 6, Baker, S Deb and he consumpon response o household ncome shocks. Journal of Polcal Economy, Forhcomng. Baker, S. & Yannels, C Income changes and consumpon: Evdence from he 203 federal governmen shudown. Revew of Economc Dynamcs. Basan, S Usng he dscree choce model o derve opmal ncome ax raes. FnanzArchv, 7, Chrsansen, V Whch commody ax should supplemen he ncome ax? Journal of Publc Economcs, 24, Deaon, A. 99. Savng and lqudy consrans. Economerca. Edwards, J., Keen, M., & and, M. T Income ax, commody axes and publc good provson: A bref gude. Fnanzarchv, 5, Gruber, J The consumpon smoohng benefs of unemploymen nsurance. Amercan Economc Revew, 87, Guesnere, R A conrbuon o he pure heory of axaon. Cambrdge Unversy Press. Hellwg, M A conrbuon o he heory of opmal ularan ncome axaon. Journal of Publc Economcs, 9,

36 McGranahan, L. & Schanzenbach, W The earned ncome ax cred and food consumpon paerns. Federal Reserve Bank of Chcago Workng Papers, WP Nchols, A. & Rohsen, J The earned ncome ax cred ec. In R. A. Moff Ed., Economcs of Means-Tesed Transfer Programs n he Uned Saes, Volume I chaper 2. Unversy of Chcago Press. Parker, J. A The reacon of household consumpon o predcable changes n socal secury axes. Amercan Economc Revew, 89, Shapro, J. M Is here a daly dscoun rae? evdence from he food samp nuron cycle. Journal of Publc Economcs, 89,

37 A Frs-order condons of he general problem The Lagrangan of he governmen s L = N n ΦV + = N γ [V Y, I, q c, α; w V Y, I, q c, α; w ] =2 [ N 2 + r Y +λ n I + N n + r g 2 c + α N ] n + r g 2 d. 2 q c q c q c q c = = = We presen he frs-order condons n her mos general form o keep he noaon as compac as possble. Noe ha, by he defnon of he problem, γ 0 snce he lowes ype canno mmc any lower adjacen ype. Noe, also, ha all ypes ha are no consraned n he opmum have µ = 0. Those who are consraned have c / I = d / I = 0 and hose who are no consraned have c / Y = d / Y = 0. The frs-order condons are: L I = n Φ V θ + γ θ + θ γ λ n + λn q c q c q c q c q c + λn α q c c I + r g 2 d I + r g 2 = 0, ; 22 L µ Y = n Φ V 2q c w h Y µ +γ w 2q c w h Y µ γ + w h 2q c w+ Y w r n +λ +λn 2 q c q c c Y +r g 2 +λn α q c d Y +r g 2 = 0,, 23 37

38 L α = +λ q c N n Φ V θ + r 2 d µ d + γ γ + θ n c α +r g 2 +λ α q c = n d α +r g 2 +λ + r 2 d µ d n d +r g 2 = B Proofs Proposon : When =,..., N are unconsraned and here s no cred spread, hen he Aknson-Sglz Theorem holds and commody axes are undfferenaed. Proof: Take he frs-order condon wh respec o I n 22, mulply by q c + r 2 d, he se q c = and oban n Φ V θ + r 2 d + γ γ + θ + r 2 d λn + r 2 d +λn α q c d I + r2 + r 2 d = 0, ; Reorganzng he las erm, one ges n Φ V θ + r 2 d + γ γ + θ +λn α + r 2 d λn + r 2 d d q c + r 2 d I + r 2 = 0,. 38

39 Subsung he compensaed demands no hs equaon and summng over all gves n Φ V θ + r 2 d + +λ n α γ γ + θ + r 2 d λ d α d + r 2 = 0, α n + r 2 d whch, afer subsung for 24 gves us λ n α d α + r2 = 0. whch requres ha α = 0. Proposon 2: If n he opmum µ = 0,, meanng ha he polcy can relax all cred consrans n he economy, hen c = d < 0. Proof: When nonlnear ncome axes are opmal, he opmal commody ax sysem s characerzed by c c n + r g 2 + d n d + r g 2 q c q c π c = 0. c c n + r g 2 + d n d + r g 2 q d q d Resang he sysem n marces gves n n c q c + r g 2 n c q d + r g 2 n d q c + r g 2 d q d + r g 2 d c = π d = 0. π c π d where he lefmos marx s a lnear combnaon of per-perod Sluzky marces. Wh me-separable uly, hs marx s necessarly negave sem-defne. We denoe by S 39