Optimal mixed taxation, credit constraints and the timing of income tax reporting

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1 Opmal mxed axaon, cred consrans and he mng of ncome ax reporng Robn Boadway, Jean-Dens Garon 2, and Lous Perraul 3 Queen s Unversy 2 Unversé du Québec à Monréal 3 Georga Sae Unversy January 2, 208 DRAFT. 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 depends on he cos We hank Kaherne Cuff, Phlppe De Donder and Chrsan Moser for her commens on an earler draf. We also hank parcpans a he 207 IIPF Congress (Tokyo and he 207 Naonal Tax Assocaon Conference (Phladelpha.

2 of ransferrng resources 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

3 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 he Earned Income Tax Cred, he Addonal Chld Tax Cred and he Healh Coverage Tax Cred n he U.S.; he Workng Income Tax Benef, he Canada Chld Benef and he Goods and Servces Tax Cred n Canada; and he Workng Tax Cred, he Chld Tax Cred, and he Unversal Cred n he U.K. 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 3

4 end of 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 before he end of 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 weakly 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. And, he governmen canno provde opmal ransfers before he end of he ax year snce does no have he requred nformaon o deermne who s enled o hem. 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 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. 4

5 be bndng. Varous sudes usng U.S. daa confrm hs. 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 epsode o explan he phenomenon of wealhy hand-o-mouh who own mosly 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 cash-ransfer 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 5

6 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 beer access o cred or who have accumulaed more 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 sandard 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, wll be cosly for he governmen o make budgeary expendures before he end of he perod. Dong so requres o borrow agans s end-of-perod ax revenues. In prncple, he governmen could make a unform lump-sum paymen o all persons a he begnnng of he perod. Combnng a lump-sum ransfer wh non-dfferenaed commody axes would be equvalen o a lnear progressve ax sysem and would allow he governmen o redsrbue a he begnnng of he perod even f had no nformaon on ndvduals ncomes. If preferences were weakly separable n goods and labor and quashomohec n good he Deaon (979 condons non-dfferenaed commody axes would be opmal, and hs would have mplcaons for our analyss. In our analyss, we assume ha he governmen does no use a unform lump-sum ransfer a he begnnng of he perod. In parcular, we assume ha all componens of he drec ax-ransfer sysem are mplemened a he end of he perod. We reurn o a dscusson of begnnng-of-perod 6

7 lump-sum axes a he end. 2 Model There are N ypes of ndvduals who are ndexed by {,..., N}. The number of ype ndvduals s n, each of whom has exogenous producvy w. The whole populaon s normalzed o one so ha N = n =. The economy lass for one perod, whch we can hnk of as a ax year. We dvde he perod 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 ndvdual pays an ncome ax T (or receves a ransfer f akes a negave value. When ndvduals 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 no he model. In each sub-perod, ype ndvduals 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 c and d are he same for boh sub-perods and for all ndvduals snce oherwse arbrage opporunes would exs. An ndvdual s uly funcon s assumed for smplcy o ake he followng addve form: U (c, d, Y = ( Y u(c c, d h w ( where Y /w s labor supply n each of 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 n c and d by nroducng a 7

8 basc need c on good c and leng u(, be homohec n c c and d. The quany c could sand for a mnmal quany of food or sheler. For smplcy, we assume ha ndvduals do no dscoun her uly across perods, whch does no resrc our resuls. Noe ha alhough ndvduals supply labor n boh sub-perods, he dsuly of labor supply s defned 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, as we confrm below. 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 + φ, (2 where φ s exogenously gven. In wha follows, we assume φ = 0 so ndvduals are precluded from borrowng. Indvduals have access o a compeve cred marke f hey wan o save or are able o borrow. 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-perod 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 whch we sudy frs. 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. As menoned, a smple case s when 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 elmnaes arbrage opporunes. 8

9 φ = 0, whch mmcs he corner soluon one would oban f borrowers faced an neres rae r 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 To be precse, n he case where here s no cred consran, he annual budge consran for a ype ndvdual s (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 hey can make ransacons n he fnancal markes, he nonlnear ax problem amouns o choosng annual dsposable ncome defned as ( 2 + I r Y T. (3 2 Therefore, one can rewre he ndvdual s annual budge consran as (q c c + q d d ( + r + q c c 2 + q d d 2 I. (4 2. Tax normalzaons In he sandard sac opmal ncome and commody ax analyss, unform commody axes are equvalen o a proporonal ncome ax. Ths mples ha he absolue level of commody axes s ndeermnae: reducng commody ax raes proporonaely and ncreasng he ncome ax rae by he same amoun wll have no effec on equlbrum oucomes. Commody axes can hen be normalzed by, for example, seng one commody ax rae 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 complcae he problem. 9

10 o zero. In our seng, hs s no possble f cred consrans are bndng. Tha s because whle commody axes are pad on purchases n boh sub-perods, ncome axes apply only a he end of he perods. To llusrae, suppose he governmen mposes undfferenaed commody axes c = d. In he absence of bndng cred consrans and assumng no neres rae spread beween borrowng and lendng, can reach he same allocaon by axng everyone s yearly ncome a he proporonal 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, he me sream of ax lables wll dffer under he wo sysems. A unform commody ax sysem wll generae ax lables n boh sub-perods whle ncome ax revenues wll be pad a he end of he perod. Ths dfference n mng has no real effec n he absence of cred consrans and neres rae spreads. The analogous resul apples n he case where a unform commody subsdy s appled. However, when an ndvdual s borrowng consran bnds, hs equvalence does no hold. The ncome ax s no pad n he frs perod, so wh φ = 0 he bndng cred consran (2 becomes ( + c c + ( + d d = Y 2. (5 A proporonal ncrease n commody ax raes wll ghen he cred consran n (5, whle a correspondng proporonal decrease n he ncome ax rae wll no undo hs ghenng. Therefore, proporonal commody axes or subsdes are no equvalen o proporonal ncome axes or subsdes. The absolue level of commody ax raes maers so we canno normalze one rae o zero. Noe furher ha (5 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 0

11 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. In wha follows, we rea he absolue levels of commody ax raes as governmen polcy varables along wh he nonlnear ncome ax sysem. Unlke n he sandard models of opmal ncome and commody axaon, our analyss yelds a well-defned ax mx. 2.2 Governmen s budge consran The governmen s budge consran n absolue erms n end-of-perod values s ( (2 + r Y I + ( + r g (q c 2 c + (q c c 2 + ( + r g (q d d + (q d d = R. (6 where R s an exogenous revenue requremen. Noe ha he dscoun facor r g s used o oban he end-of-perod values of commody ax receps n he frs sub-perod. Tha s because we are assumng ha he governmen s a ne borrower. If subsdzes commodes n he frs sub-perod, mus borrow a he rae r g o fnance hose subsdes. Some of he benef of he subsdes accrues o hgh-ncome ndvduals who are savers and oban a reurn r on her savngs. The fac ha r g > r makes socally cosly o ransfer resources o hem n =. By he same oken, f he governmen axes commodes n he frs subperod, reduces s borrowng and he savng of hgh-ncome ndvduals also decreases, whch agan saves resources snce r g > r. However, he cred consran s ghened for low-ncome ndvduals for whom bnds.

12 3 Opmal ax mx We derve he governmen s 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 and dsposable ncome (Y, I nended for ypes, where ncome s earned equally over he wo sub-perods. Then, usng (3 axes pad a he end of he perod are resdually gven by T = (2 + r Y /2 I, where T can be negave for low-producvy ypes. The governmen also chooses c and d, or equvalenly q c and q d. As we shall see, when an ndvdual s cred consraned n he opmum, he opmal prce rao q c /q d wll generally dffer from uny. We begn by characerzng ndvdual behavor, and hen urn o he governmen s problem. 3. Indvdual behavor We solve he ype- ndvdual s problem n wo seps n reverse order. In he second sep, knowng Y, I, q c and q d, he ndvdual 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, q d, ndvduals of ype choose commody bundles (c, d o maxmze uly ( subjec o he annual budge consran (4 and he cred consran (2. The value funcon for hs problem s: ψ (Y, I, q c, q d = max c,d [ u(c c, d + θ I ] ( + r (q c c + q d d =,2 =,2 5 For a smlar approach, see Edwards e al. (994 2

13 ( ] Y µ [q c c + q d d 2 + φ, (7 where he cred consran akes he values φ {0, }, dependng on he specfc case under sudy. Applyng he envelope heorem o he value funcon ψ (, ψi = θ, ψy = µ 2, ψ q c = θ ( + r c µ c, ψq d = θ ( + r d µ d. =,2 Noe 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. =,2 ( Sep : Choce of ncome and ne ncome bundles Gven commody ax raes ( c, d and ancpang sep 2 above, he governmen on behalf of ndvduals of he wo ypes offers ncome-consumpon bundles (Y, I. In an opmum, ndvduals choose he bundles nended for hem. Ths yelds oal uly for a ype- person: ( Y V (Y, I, q c, q d = ψ (Y, I, q c, q d h. (9 Usng he envelope resuls (8 on ψ, V ( sasfes he followng properes: w V Y = µ 2 w h ( Y w, V I = ψ I, V q c = ψ q c, V q d = ψ q d. (0 Preferences of an ndvdual of ype n (Y, I-space have a slope: di dy = V Y V I = θ [ w h ( Y w ] µ 2 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 3

14 ndvdual beng mmcked. Analogously o V n (9, ndrec uly s gven by: ( Y V (q c, q d, Y, I = ψ (Y, I, q c, q d h. ( Smlar envelope properes o (0 apply, and he slope of he mmcker s ndfference curves wll be: dî dŷ = V Y V I [ = θ w h (Ŷ w ] µ 2 w 3.2 Tax mplemenaon Tax mplemenaon nvolves fndng margnal ax raes ha mplemen he opmaly condons derved usng mechansm desgn analyss. Dong so nvolves relang margnal ax raes o ndvdual behavor as follows. The governmen mplemens a nonlnear ax funcon T (Y. Usng (3, we can rewre he expresson for ndrec uly n (9 as ( ( 2 + V ( = ψ (Y r Y, Y T (Y, q c, q d h. 2 w The ndvdual chooses ncome Y o maxmze V (. Usng he envelope condons (8, he frs-order condon can be wren ψy + ψi I Y h Y ( ( Y = µ 2 + w 2 + r θ T (Y w ( Y 2 h = 0. w Isolang he margnal ax rae gves T (Y = 2 + r 2 θ w h ( Y w + µ 2 θ. (2 Ths expresson for T (Y s he margnal ax wedge facng a ype- ndvdual. Below we use (2 o characerze he margnal ax raes ha mplemen he soluon o he governmen s opmal ax problem. 4

15 3.3 Governmen s problem In our problem, he governmen redsrbues 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 Φ(V subjec o he budge consran (6 and o N ncenve compably (IC consrans ha ake he form of downward adjacen consrans, V (Y, I, q c, q d ; w V (Y, I, q c, q d ; w. (γ where Φ(V s a concave socal uly funcon, wh Φ (V > 0 and Φ (V 0. The funcon V (Y, I, q c, q d ; w n he IC consrans s he ndrec uly obaned by a ype who mmcs he adjacen lower ype, so s gven by (. The equaon ndcaors γ represen he Lagrangan mulplers of he ncenve consrans n he governmen s problem, and λ s he Lagrangan mulpler for he budge consran (6. Noe ha for R small enough, a leas one ype (he lowes receves a ransfer. Gven our assumpon abou preferences, all ndvduals would smooh her consumpon across sub-perods and 2 n he absence of cred consrans, albe mperfecly. Cred consrans wll be bndng only for hose expecng a ransfer a he end of he perod snce hen hey wll wan o borrow n sub-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 sengs of ncreasng complexy. We begn wh he benchmark case n whch no one s cred-consraned and here s no cred spread. We hen assume a cred consran wh φ = 0 ha s bndng on a leas one ype (he lowes, bu resrc he governmen o usng a nonlnear ncome ax. The 5

16 cred spread s rrelevan n hs case snce no ndvduals wll borrow, and he governmen ges all s revenues a he end of he perod. In he fnal case, boh cred consrans and a cred spread apply, and 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. 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 Appendx A Benchmark case: 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 and lend a he common neres rae r. Frs, we esablsh ha he governmen need no use commody axaon a all, and hen we characerze he opmal ncome ax sysem. The governmen can normalze commody axes by seng q c =, and hen opmze on relave commody prces whch we denoe by α = q d /q c. 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 (8 and (0, he governmen s frs-order condons shown n Appendx A 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 B. Thus, he Aknson-Sglz heorem connues o apply even hough consumpon and labor supply occur sequenally over he ax year. The heorem spulaes only ha commody axes should be unform f used, bu snce unform commody axes are equvalen o 6

17 a proporonal ncome ax n hs benchmark case, hey are redundan and hus unnecessary. Snce cred consrans are no bndng n hs case, µ = 0, so he margnal ax wedge n (2 smplfes o ( 2 + r T (Y = 2 θ w h ( Y w. (3 Usng he frs-order condons n Appendx A o rewre he rgh-hand sde of (3, we oban he followng margnal ax formulas n an opmum: T (Y = θ γ + λn ( h (Y /w h (Y /w +. (4 θ w θ w + As shown by Hellwg (2007, he erm n parenheses s always posve when he snglecrossng condon s sasfed and when lesure s an normal good. Moreover, γ N+ = 0 snce here s no downward ncenve consran a he op. Therefore, margnal ax raes are everywhere posve excep a Y N, for whch T (w N = 0 so here s no dsoron. These are he sandard opmal ncome ax resuls Case wh bndng cred consrans and no commody axes Suppose now ha ransfers o he lowes ypes are 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 = q d =. 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 so a rae r = r. To characerze he opmal ncome ax sysem, noe frs ha he ax wedge (2 can 7

18 now be wren ( 2 + r T (Y = 2 θ w h ( Y w + µ 2 θ. For hose who are no cred-consraned, µ = 0 as above. For cred-consraned lowerncome ndvduals, µ > 0. They wll be more nclned o work more o generae ncome n he frs-perod. As above, we can use he governmen s frs-order condons from Appendx A o oban opmal margnal ncome ax raes. They ake exacly he same form as he sandard case above gven by (4. However, compared wh he benchmark case, hs wll resul n lower margnal ax raes for he cred-consraned ndvduals for wo man reasons. Frs, bndng cred consrans lower uly, and hs reduces he ncenve of hgher ypes o mmc, whch has an effec on γ +. consumpon n he frs perod Case wh bndng cred consrans and ncome and commody axes When he governmen has access o commody axes or subsdes, can use hem o relax he bndng ncenve consrans by subsdzng consumpon n sub-perod. Bu, hs comes a a cos snce mus borrow a he rae r g o fnance he subsdes. Gven ha commody subsdzaon also benefs he unconsraned ndvduals, savng of he laer s ncreased. Snce r g > r governmen savng accompaned by prvae dssavng resuls n a resource cos. In hs case, wo oucomes are possble. Frs, subsdzaon of commodes may be suffcen o elmnae he cred consran of all low-ncome ndvduals. Alernavely, he cos of commody subsdzaon may be suffcenly large ha n an opmum, some low-ncome ndvduals reman cred-consraned. Snce he qualave resuls dffer n he wo cases, we consder hem separaely. Case A. Cred consrans elmnaed n he opmum In hs case, whch happens when r g r s small enough, he governmen can use commody 8

19 axes o undo he cred consran of all ndvduals. The followng proposon s proved n Appendx B. Proposon 2: If n he opmum µ = 0,, so polcy relaxes all cred consrans n he economy, hen c = d < 0. Thus, boh goods are subsdzed a he same rae. The commody subsdy sysem acs as a proporonal subsdy on ncome. Snce cred consrans are no bndng, he equvalen of a second-bes s recovered, alhough wh neres raes hgher for borrowers (he governmen han for savers. To ensure ha he enre ax sysem maxmzes socal welfare n an ncenve-compable way, ncome ax raes are adjused o reflec he fac ha unform commody subsdes are equvalen o a subsdy on ncome. In he opmum, he effecve margnal ncome ax rae of ndvdual akng accoun of boh ncome ax and commody subsdy dsorons s dencal o he one obaned n he benchmark case whou cred consrans and commody subsdes. In parcular, he margnal ax rae faced by ndvdual s now T (Y = θ γ + λn ( h (Y /w h (Y /w + θ w θ w + ( + r g 2 ( z T (Y ( c I + d, I where c = d and z = (2 + r /2. Ths 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 ndvduals pockes, adjused for he fundng cos of he subsdes r g. (5 The ax formula n (5 reflecs he fac ha he wedge beween labor and consumpon mus encompass he margnal ncenves and dsncenves generaed by all ax nsrumens. 9

20 Ths can be seen by rewrng (5 as a margnal effecve ax rae: T (Y + ( ( c ( + r g 2 (z T (Y I + d = θ γ + h (Y /w I λn h (Y /w + θ w θ w +, (6 where he rghhand sde s he opmal labor wedge n (4. Ths 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. Fnally, 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, and can always lower he prces of commodes so as o make all cred consrans n he economy slack. In hs unrealsc example, he mng of ncome-esed paymens has no effecve consequence on he overall ax polcy and socal opmum. As soon as subsdes become cosly, here s a hreshold level of he cos beyond whch he governmen wll leave some ndvduals cred consraned. Case B. Some cred consrans bndng n he opmum When he cos of fundng commody subsdes n he frs perod are hgh enough, may be opmal for he governmen o leave some ndvduals cred consrans bndng. When hs happens, dfferenal subsdy or ax raes should apply, bu may be opmal eher o subsdze boh goods, or o subsdze one and ax he oher. The nuon s as follows. Commody axaon generaes boh ncome and subsuon effecs. Wh no bndng cred consrans, he nonlnear ax sysem can adjus o offse ncome effecs, and hs ensures ha s opmal no o creae subsuon effecs. Wh 20

21 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-of-consumpon funcons u( feaure lnear Engel curves. Our smple case n whch here s a basc need for good c, c, 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 below. Proposon 3, whose proof s gven n Appendx B, gves he general condon under whch dfferenaon happens under lnear Engel curves. Proposon 3: Le us denoe B (µ /λ(φ (V +γ /n γ + /n (r g r. Also denoe by C he subse of ypes whose cred consran bnd n he opmum: C µ > 0. Then, he opmal polcy has d > c f and only f ( n B c C n c n B d C n > 0. d To undersand he nuon conaned n he proposon, le us consder he followng expresson: n B (µ /λ(n Φ (V + γ γ + n (r g r. (7 Ths s he margnal socal valuaon, accouned for n dollars, assocaed wh gvng one exra lump-sum dollar o all ype- ndvduals n he frs perod, whle he second-perod ncome ax schedule adjuss o reman opmal. I ncludes he margnal socal welfare wegh of ndvdual and he margnal effecs on mmckng (adjacen, below and above f he s 2

22 consraned, and he mechancal cos on he budge consraned assocaed wh fundng he dollar hrough deb. The erm B s necessarly negave for / C snce he ncome effecs generaed by commody subsdzaon are repad for by an adjusmen n T (Y, excep for he ne cos n (r g r relaed wh fundng n dollars of subsdes. However, can ake eher sgn f C, and s presumably declnng when w ncreases, snce he mulpler on he cred consran, µ, declnes wh and because he socal welfare funcon s concave. 6 The erms C n c and C n d are he coss for he governmen ha are assocaed wh ncome effecs ha reman for consraned ndvduals. When a dollar of purchasng power s gven n he frs perod o consraned ndvduals hrough ndrec axaon, hese ndvduals ncrease her consumpon of boh goods n he frs perod due o he ncome effec. The more one good s consumed, he cosler hs ncome effec s for he governmen. Ths ncome effec does no show up for unconsraned ndvduals, snce followng an ncrease n a subsdy on a good, he governmen can adjus her second-perod ncome ax schedule o repay he subsdy and no ncome effecve ncome effec s hen experences. Overall, Proposon 3 saes ha he opmal polcy nvolves dfferenang commody axes, wh c < d, f he opmal polcy nvolves a hgher benef/cos rao of subsdzng good c han good d. Corrolary, whch s proven below, saes ha f Engel s curves are lnear and no basc need, hen s opmal o no dfferenae commody ax raes. Ths resul s ndependen of he shape of he socal welfare funcon Φ(. The nuon s ha he governmen wans o subsdze more a good ha s proporonaely consumed more by consraned ndvduals. When c = 0, all ndvduals consume goods c and d n same proporons, and he governmen decdes no o generae subsuon effecs. Ineresngly, he Corrolary mples ha cred consrans alone, or he presence of a basc need alone, are no suffcen o jusfy dfferenaon. 6 A suffcen condon for n Φ (V +γ γ + > 0 s ha max{ c, d } < (+r /(+r g when boh goods are normal. Ths condon, whch rgh-hand sde s larger han, s obaned by decomposng equaon (22 and applyng Walras s Law. 22

23 Corrolary : If c = 0, hen here s no dfferenaon and c = d < 0. Ths resul s ndependen of he specfc socal welfare funcon Φ( used by he socal planner. Proof: Denoe by α c /d n he opmum. Snce Engel curves are lnear and affne hrough he orgn, α = α,. Thus, ( n B c n C n c ( B d C n = α n B d d n α C n d B d C n = 0. d Ths resul s ndependen of he specfc form of Φ(, whch only appears n B,. 4 Numercal examples We llusrae our resuls wh some numercal smulaons. funcon and dsuly of labor are respecvely The quas-homohec uly u(c, d = κ (c c ρ ρ + κ d ρ ρ ; (8 h(l = l+σ, l Y/w. (9 + σ We se he values of he parameers o ρ = 0.9 and σ = 2. The consan κ s chosen so he soluon of he opmal ax problem gves he same uly level and opmal ax funcon as n he fcve case where c = 0 wh r = 0, and where he yearly uly of an ndvdual would be u(c, l = c ρ ρ l+σ + σ. Is value s herefore equal o (/4ρ. The number of workers a each wage level, n, s obaned usng a lognormal dsrbuon wh parameers (µ, σ = (2.757, I s explcly aken from Mankw e al. (2009. who 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 a fxed dsance 23

24 beween any wo wage levels. The probably mass funcon s rescaled so ha n =. We nally se he neres rae spread o 5 percenage pons (whch s a b lower han borrowng raes on cred cards, wh r = 0 and r g = 0.5, alhough sensvy analyses are are also compued. We use he basc need c = 5, whch approxmaely represens a $5.75 daly, or $2,00 annually n our model. Ths amoun appears o us as beng farly conservave. For nsance, he average Supplemenal Nuron Asssance Program (SNAP, commonly referred o as he food samp program, s abou $4 a day. Moreover, he 2007 Federal povery lne s se a $0,20 ($ 28 per day for a sngle-person household. The year 2007 s used o calbrae our smulaons, n order o make hem comparable wh Mankw e al. (2009. Due o he nroducon of commody axes and subsdes, we repor he effecve ax burden pad a a gven level of ncome, and he effecve margnal ax raes as n Edwards e al. (994. The oal ax burden of a worker a labor ncome Y, n he eyes of he governmen, s [ ] [ ] T E (Y = T (Y + c ( + r g 2 c + d ( + r g 2 d. (20 We also repor margnal effecve ax raes as derved n equaon (6. Baselne smulaons We sar consderng hree scenaros. The frs one s he No cred consran scenaro. Indvduals can save and borrow a he same rae. Ths frs scenaro s a useful benchmark as gves he same opmal allocaon and effecve margnal ax raes as when he planner can subsdze frs-perod consumpon a no cos. In hs sense, hs s analogous o a sandard second-bes ax regme. In he second scenaro (Cred consran ndvdual s borrowng lm s φ = 0. The planner resors o nonlnear ncome axaon o acheve hs redsrbu- 24

25 ve objecve. Wh r = 0, ndvduals are consraned whenever hey expec recevng a ransfer a he end of he fscal year (T (Y < 0. Commody axes and subsdes are assumed away. The hrd scenaro ( Cred consran wh subsdes nroduces commody axaon/subsdzaon. As a benchmark, we sar usng an ularan socal welfare funcon (W = n V, followed by oher examples feaurng Pareo weghs. [Fgure abou here ] The characerscs of he opmal ax sysems for all hree scenaros are shown n Fgure and n Table. From Fgure, s possble o see ha he Cred consran and subsdes scenaro has he hghes margnal ax raes, excep a he very boom of he ncome dsrbuon, and also he hghes average ax raes. The opmal soluon o hs scenaro where he governmen has access o commody axaon wll feaure subsdes o consumpon goods. To fnance hese subsdes he planner mus rase revenues usng of labor ncome axes. Lookng a he opmal effecve margnal ax raes, one sees ha he Cred consran and subsdes scenaro s an nermedae case, where only he nonlnear labor ncome ax avalable as a polcy nsrumen. The No cred consran scenaro feaures he hghes effecve margnal ax raes. As s ypcal wh second-bes opmal ncome ax models, n he absence of cred consran he governmen heavly dsors labor supply a he boom of he dsrbuon, and he margnal effecve (ncome ax raes declne when we move owards he op of he sklls dsrbuon. Conrasngly, he Cred consran case, whch precludes he use of subsdes, has he lowes margnal effecve ax raes for all he sklls dsrbuon. Snce here s a basc need c > 0, low-skll workers mus supply more labor n order o mee her basc need, he planner beng unable o ransfer resources o hem n he frs perod. Moreover, he nably of low sklled workers o smooh consumpon reduces he redsrbuve poenal of he ncome ax schedule, hereby reducng he need o collec axes for redsrbuve purposes. 25

26 The Cred consran wh subsdes scenaro allows he planner o ransfer resources n he frs perod usng ndrec axaon, albe a a cos, o he frs perod hrough subsdes on goods. The same qualave paern as wh he scenaro precludng ndrec axaon emerges a he boom of he ncome dsrbuon because he planner does no wan o dscourage work. When goods are subsdzed, labor ncome gves a hgher margnal uly snce subsdes ncrease he purchasng power of each addonal dollar earned early n he fscal year. [Table abou here ] Table shows ha, hrough subsdzaon, fewer workers are consraned f commody axaon s a polcy ool. The ularan socal welfare levels are repored for all hree scenaros. Tha wh cred consrans and commody axes/subsdes domnaes he scenaro wh cred consrans bu no commody axaon. The able also presens an alernae measure of socal welfare, whch s, a calculaon of he welfare gans sarng from he lasser-fare allocaon. Ths gan nerpreed as he mnmal percenage ncrease n consumpon from he lasser-fare, requred o aan he same welfare levels as n he relevan scenaros (% LF. Ths percenage s fxed across ndvduals and me-perods. 7. Once more, we observe ha he case wh commody axes s preferred o ha whou. The small dfferences are 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. Pung more wegh on he poor We now we use a weghed ularan socal welfare funcon wh weghs ω w η n w η, η 0; n ω = (2 7 Ths way o accoun for welfare varaons s documened n Farh & Wernng (203 and? 26

27 where he socal welfare s gven by W n ω V. Ths funcon pus more wegh on workers wh low wages as η ncreases. Then, allowng he governmen o subsdze goods wll capure mos of he gans from he No cred consran scenaro, measured agan by he percenage ncrease n consumpon from he lasser-fare allocaon. Table 2 repors he welfare gans of he Cred consran and he Cred consran wh subsdes scenaros. 8 The more he governmen cares abou he low-wage ndvduals, he less palaable becomes o leave hem cred consraned. Ths means ha relaxng her cred consran usng ndrec axaon becomes a prory for he governmen. [Table 2 abou here ] Changes n basc need We consder changes n he level of basc need c. Fgure 4 shows ha an ncrease n he basc need nduces he socal planner o subsdze consumpon more. I also nduces hm o dfferenae dfferenae more he ax raes. We can see ha wh c = 0 he planner wll sll wsh o subsdze consumpon, bu ha relave prces reman undsored. The greaer he basc need he harder and cosler s n o acheve ha basc level of consumpon hrough work effor. Hence, he governmen wans o use ncreasngly more resources a helpng low wage workers by ncreasng he value of her labor earnngs. Ths nvolves subsdzng more heavly he goods ha are proporonaely consumed more by he cred consraned ndvduals. [Fgure 4 abou here ] Fgure 5 llusraes he effec of an ncrease n he basc need on labor ncome axes. The planner requres hgher ncome ax levels o fnance he subsdes. Ths s vsble from he 8 For every dfferen levels of η, new smulaons are compued where he opmal polces were chosen wh he new socal welfare as he governmen s objecve. 27

28 plos depcng margnal and average ax raes. The hgher basc need also ncreases he level of redsrbuon, whch resuls n hgher effecve margnal ax raes for all workers, bu especally a he boom of he ncome dsrbuon. Average effecve ax raes also ncrease wh c, snce a hgher basc need ncreases margnal uly of consumpon of he necessy good a a gven level of consumpon. Ths renforces he redsrbuve move of he governmen. [Fgure 5 abou here ] Changes n neres raes We fnsh hs secon by explorng changes n he cred spread, r g r. To focus on he changes n he coss and no on ncreases n he amoun of resources avalable o he economy due o hgher neres raes on savngs, we only change he level of r g, keepng r = 0. Fgure 2 presens he same nformaon n hree dfferen manners. The man pon s ha reducng he governmen s cos of borrowng bolsers s ncenve o subsdze commodes, and reduce s wllngness o dfferenae he raes of hese subsdes. If he cos s hgh enough, we evenually oban a posve ax rae he non-necesscy good d and a ax on c. The planner wans o use every cosly resource avalable o ensure ha all ndvduals consume a leas c. 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 28

29 margnal ncome ax raes snce he planner mus encourage work especally a he boom of he dsrbuon. [Fgure 3 abou here ] [Fgure 2 abou here ] 5 Conclusons and an exenson o lump-sum ransfers In hs paper, we suded an opmal ax sysem when ransacons on goods happen more frequenly han he paymen of ncome-esed ransfers. Cred consrans arse because ndvduals canno fully use fuure ransfers as collaeral. Our resuls show ha when he opmal polcy s able o relax he consran on all ndvduals, nvolves proporonal subsdes on all goods. When he cos of provdng he subsdes 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 a dfferenal raes, or subsdze some commodes whle axng ohers. As menoned n he Inroducon, f he governmen could make lump-sum ransfers n sub-perod separaely from s ncome ax-ransfer sysem a he end of he perod, could use hem o amelorae bndng cred consrans. Ths opens he door o neresng neracons beween commody axaon and hese paymens. Consder frs he exreme case where r g = r = r, so he lump-sum ransfers could be made coslessly by he governmen borrowng agans fuure ax revenues. The governmen wll offer a unversal lump-sum ransfer a = suffcenly large o relax all cred consrans and reduce all ax lables a he end of he perod by an equvalen amoun. If hs scheme 29

30 s avalable o he planner, hen he allocaon would be dencal o our benchmark case whou cred consrans. There would be no need o use commody subsdes and he Aknson-Sglz Theorem would hold. Suppose nsead ha r g > r so he governmen faces a cos of ransferrng money from he second o he frs sub-perod. In hs scenaro, he governmen may no longer be able o fron-load compleely redsrbuon n he frs sub-perod and adjus he ncome ax schedule o leave all workers unconsraned. If he governmen uses only nonlnear ncome axaon along wh he unversal lump-sum ransfer, he suaon s smlar o our second case above. Some ndvduals reman cred-consraned, and he sandard form of he opmal margnal ncome ax raes bu snce some ndvduals reman cred-consraned socal welfare s less han n he benchmark case. When he governmen can use commody axes along wh perod- lump-sum ransfers, hngs change poenally dramacally. Unform commody axes combned wh a lumpsum ransfer s equvalen o a lnear progressve ax sysem whch ransfers ncome from hgh- o low-ncome ndvduals. Moreover, he commody axes generae ax revenue for he governmen o fnance he lump-sum ransfers hereby offseng he need o borrow. If he governmen ses he commody ax rae hgh enough, would seem could fnance an amoun of lump-sum ransfers suffcen o relax all cred consrans of low-ncome ndvduals whle a he same me avodng he need o borrow. And, f he uly-ofconsumpon funcon u( sasfes he Deaon condons, would seem o be opmal o use undfferenaed commody axes. In fac, he opmal allocaon acheved n hs manner appears lkely o lead o more cred-consraned workers. Ths s because he hgh level of commody axes mus apply n boh perods. Ths nduces he governmen o adjus he labor ncome ax schedule o make he allocaon ncenve compable. To do hs, he ax burden of workers s drascally reduced o he pon where a majory of workers face 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 30

31 consran. We conjecure ha he unform commody ax resul obaned wll break down f Engel curves are nonlnear. Furhermore, he ably of he governmen o use commody ax revenues obaned from perod- commody purchases o fnance he lump-sum ransfer requres ha he governmen acually receve hose revenues n perod. In pracce, he frms collecng commody axes wll no rem hem o he governmen unl he end of he ax year and hs wll cause he above mechansm o break down. 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 earler n hs paper. Provng hese conjecures s lef o furher research. 3

32 References Agula, E., Kapeyn, A., & Perez-Arse, F. (207. Consumpon smoohng and frequency of benef paymen of cash ransfer programs. Amercan Economc Revew Papers and Proceedngs, 07, Aknson, A. B. & Sglz, J. (976. The desgn of ax srucure: drec vs. ndrec axaon. Journal of Publc Economcs, 6, Baker, S. (207. Deb and he consumpon response o household ncome shocks. Journal of Polcal Economy, Forhcomng. Baker, S. & Yannels, C. (205. Income changes and consumpon: Evdence from he 203 federal governmen shudown. Revew of Economc Dynamcs. Basan, S. (205. Usng he dscree choce model o derve opmal ncome ax raes. FnanzArchv, 7, Chrsansen, V. (984. 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. (994. Income ax, commody axes and publc good provson: A bref gude. Fnanzarchv, 5, Farh, E. & Wernng, I. (203. Insurance and axaon over he lfe cycle. Revew of Economc Sudes, 80, Gruber, J. (997. The consumpon smoohng benefs of unemploymen nsurance. Amercan Economc Revew, 87, Guesnere, R. (995. A conrbuon o he pure heory of axaon. Cambrdge Unversy Press. 32

33 Hellwg, M. (2007. A conrbuon o he heory of opmal ularan ncome axaon. Journal of Publc Economcs, 9, Mankw, N. G., Wenzerl, M., & Yagan, D. (2009. Opmal axaon n heory and pracce. Journal of Economc Perspecves, 23, McGranahan, L. & Schanzenbach, W. (203. The earned ncome ax cred and food consumpon paerns. Federal Reserve Bank of Chcago Workng Papers, WP Nchols, A. & Rohsen, J. (206. 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. (999. The reacon of household consumpon o predcable changes n socal secury axes. Amercan Economc Revew, 89, Shapro, J. M. (2005. Is here a daly dscoun rae? evdence from he food samp nuron cycle. Journal of Publc Economcs, 89,

34 A Frs-order condons of he general problem The Lagrangan of he governmen s L = N n Φ(V + = N γ [V (Y, I, q c, q d ; w V (Y, I, q c, q d ; w ] =2 +λ [ N = (( 2 + r n Y I + (q c 2 + (q d N n ( + r g 2 c = ] N n ( + r g 2 d. = 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 γ + θ λn + λn (q c c I ( + r g 2 + λn (q d d I ( + r g 2 = 0, (22 ( L µ Y = n Φ (V 2 w h γ + ( µ 2 h w+ ( ( Y µ + γ w ( Y w + 2 w h ( 2 + r + λ n 2 ( Y w + λn (q c c Y ( + r g 2 + λn (q d d Y ( + r g 2 = 0, (23 34

35 +λ (q c +λ (q c L q c = N + ( n Φ (V θ ( + r 2 c µ c ( (γ γ + θ = ( + r 2 c µ c c n ( + r g 2 + λ (q d q c + λ n L q d = N + n d q c ( + r g 2 c ( + r g 2 = 0. (24 ( n Φ (V θ ( + r 2 d µ d ( (γ γ + θ = ( + r 2 d µ d c n ( + r g 2 + λ (q d q d + λ n n d q d ( + r g 2 d ( + r g 2 = 0. (25 Effecve margnal ax raes By (22, oban ( n Φ (V + γ γ + θ c = λn ( c I ( + r g 2 d d I ( + r g 2. (26 Usng z (2 + r /2, z T = h (Y /w µ, and addng and subracng w θ θ 2 ( µ γ + θ θ 2 h (Y /w w θ 35

36 n (23, oban ( n Φ (V + γ γ + θ ( ( h T (Y z γ + θ (Y /w h (Y /w + w θ w + θ ( + λn z + c c Y ( + r g 2 + d Subsung (26 no (27 and reorganzng, one obans d Y ( + r g 2 = 0. (27 ( c T (Y + c ( + r g 2 I (z T + c Y ( d + d ( + r g 2 I (z T + d Y ( = γ+ θ h (Y /w q c h (Y /w +. (28 λn w θ w + θ B Proofs For furher use, we presen he expressons used for compensaed demands. We compensae demands by varyng dsposable ncome I bu akng annual earnngs Y as gven. For an unconsraned ndvdual and usng a lde o denoe compensaed demands, he Slusky equaons can be wren c = c c ( + r q c q c I 2 c, =,2 d = d q c q c d ( + r I 2 c, (29 =,2 d = d d ( + r q d q d I 2 d, =,2 c = c c ( + r q d q d I 2 d. (30 =,2 If an ndvdual s consraned, hen n he frs perod c / I = d / I = 0. Gven meseparably we can make use of he fac ha frs-perod expendures sasfes q c c + q d d = 36