Problem Set 3 EC2450A. Fall ) Write the maximization problem of the individual under this tax system and derive the first-order conditions.

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1 Problem Se 3 EC450A Fall 06 Problem There are wo ypes of ndvduals, =, wh dfferen ables w. Le be ype s onsumpon, l be hs hours worked and nome y = w l. Uly s nreasng n onsumpon and dereasng n hours worked. u = u, l Suppose here s a ax T y on nome. Wre he budge onsran of ype under hs ax sysem. The budge onsran s: y T y Wre he maxmzaon problem of he ndvdual under hs ax sysem and derve he frs-order ondons. max,l u, l y T y The FOC s: u / l u = w T y / 3 Express he margnal ax rae as a funon of he prmves margnal ules and ably levels. Ths gves you a haraerzaon of he margnal ax rae as a funon of he alloaons. We an wre he margnal ax rae as a funon of he alloaons as: T y = + w u / l u /

2 4 Defne V, y ; w = u, y w o be he uly expressed as a funon of observable y. V Express he dervaves:, V y, V w as funons of he dervaves of u. Express he margnal ax rae T as a funon of he dervaves of V. V = u ; V y = u l w ; V w = u l y w = V y y w V / y V / = T y 5 Wha frs-order ondon would haraerze he frs-bes alloaon n he frs bes alloaon, here are lump-sum, ype-spef axes avalable sne here s no prvae nformaon. In he frs-bes here s lump-sum axaon only: V / y V / = 6 Suppose ha w > w. Draw one ndfferene urve for eah ype n he y, spae. Whh one s flaer? Explan nuvely whh ype experenes a hgher reduon n uly from he same nrease n nome dy. Type experenes a smaller uly hange for he same nrease dy. The hgh ype s wllng o rade a smaller amoun of onsumpon for a reduon n nome sne she needs o forgo a lower amoun of effor o reah he same nome. 7 Types w are unobserved. Explan why hoosng opmal axes n hs seng s equvalen o hoosng a menu of, y pars. Is here a more general mehansm ha we ould ome up wh ha would do beer han seng hs menu please jusfy your answer. We know we an desgn a mehansm ha hooses alloaons for every ype beause of he revelaon prnple see also PS. 8 Suppose ha here are n ndvduals of ype and ha here s a revenue requremen R. Wre he revenue onsran n erms of alloaons y,. The revenue onsran s: R = y n + y n R 9 Se up he Pareo problem: he Pareo problem maxmzes he uly of ype subje o ype reahng some arge uly ū and subje o onsrans. onsrans are don forge ha ypes are unobservable. The Pareo problem s: max V, y,,y,y Thnk arefully wha he

3 s.. V, y Ū V, y V, y V, y V, y R = y n + y n R 0 Wre he Lagrangan for hs onsraned maxmzaon and provde he frs-order ondons. Show ha here are only hree possble regmes, dependng on whh onsrans bnd. L = V, y + µv, y + λ V, y V, y + λ V, y V, y + γ [ y n + y n R ] The FOCs are: = µ V V V λ + λ γn = 0 = µ V V V λ + λ γn = 0 y y y y = µ V V V + λ λ γn = 0 = µ V V V + λ λ γn = 0 y y y y Three regmes arse: λ = 0, λ = 0 λ = 0, λ > 0 λ > 0, λ = 0 3

4 Is here a ase n whh he frs-bes soluon would apply? Charaerze n erms of he values of he mulplers and he alloaon. When λ = 0 and λ = 0 he equlbrum s fully revealng and he frs-bes soluon apples. Every agen srly prefers her bundle and here s no rsk of maon. Suppose ha he nenve onsran on ype s bndng. Usng your answer o 3 and 0, show ha he margnal ax rae faed by ype s zero and explan hs nuvely. Also show ha he margnal ax rae faed by ype s posve and explan why. When s hs regme lkely o our as a funon of he prmves of he problem? Usng he FOCs wr and y we derve: V / y V / = u / y u / w = Ths equaon s equal o he FOC for ype. Therefore, a he opmum agen faes a zero margnal ax rae. From he FOCs for ype we derve: V / y V = λ V / y /n γ / + λ V / /n γ < To see hs defne: α = V / y V / Then we an rewre he expresson above as: Sne by assumpon α > α, follows ha: v = λ V / N γ α = + vα + v = α + α + v α < α < Problem Consder an overlappng generaons model. Suppose ha here are wo ypes of onsumers, and n eah generaon. In perod here are N = N +n young of ype whose produvy s v < and N = N + n young of ype wh produvy. Everyone earns a wage equal o her produvy sne he labor marke s perfely ompeve. There s an aggregae produon funon F suh ha: 4

5 Q = F K, E wh: E = vn l + N l E are effen uns of labor. Denoe by w he wage per un of effen labor. Type s uly s u, l where = y, o,+ are hs onsumpons when he s young and when he s old. Suppose ha he governmen an freely hoose K and K + effeve savng every perod and he neres rae s r = 0 so a un saved jus gves you one un of onsumpon laer. The governmen olles axes o fnane revenue requremens R. Wre down he governmen budge onsran a me as a funon of onsumpon, apal, and revenue requremens no axes. Wre oal effen uns of labor as E = vn l + N l. The governmen budge onsran s: F K, E + K = N y, + N y, + N o, + N o, + K + + R Argue why, f here s prvae nformaon on he ypes, he governmen an resr self o hoosng a menu of alloaons for eah ype. Wha do hese alloaons spefy for eah ype? By he revelaon prnple, he governmen mus hoose a dre revealng mehansm C, Y, C, Y under a self-seleon onsran ha saes ha he sklled do no preend o be unsklled so as o pay lower axes. 3 Argue ha n general, ype s nenve onsran wll be bndng and wre. You an assume for now ha he nenve onsran on ype s slak. Type s nenve onsran wll be bndng when he governmen has redsrbuonary preferenes. If was no, he governmen ould lump-sum ax ype n order o ransfer resoures o ype, whose soal margnal welfare wegh s hgher. We an wre: U, Y w = U, Y w 4 The governmen objeve s: W = W = µ N u + µ N u = = Wre he Lagrangan of hs problem. The Lagrangan s: L = W +λ U, Y U, Y w w +γ F K, E +K N y, N y, N o, N o, K + R 5 Suppose ha he governmen an freely hoose K as a onrol varable. Explan wha hs 5

6 means and wha would jusfy hs assumpon. Ths amouns o assumng ha he governmen has he means o fx he apal sok a s opmal level by ssung publ deb. Suppose we are n an eonomy where deb s onraed for one perod and pays he same neres rae as apal - whh mus be rue n equlbrum sne here s no rsk. Then prvae savngs fnane boh nvesmen and publ deb and he governmen an ssue deb o fx he oal apal sok, whh s he aggregae level of nergeneraonal ransfers. Ths allows o separae he problem of opmzng he aggregae apal sok ranferred aross perods from he opmal apal alloaon whn eah perod. 6 Take he frs-order ondons of he governmen s maxmzaon problem. Hn: If you need o ake he FOC wh respe o y, may be easer o do a hange of varables and o maxmze wh respe o l = y w v and l = y w. FOCs for onsumpon are: y = N µ + λ U, l N γ = 0 y 0,+ = N µ + λ U, l N γ + = 0 o,+ FOCs for labor are: l = µ N U l, l λ U l, l + γ vn F l K, E = 0 l = µ N U l, l + λ U l, l + γ N F l K, E = 0 The FOC for apal s: = γ + + F K K, E γ = 0 K + 7 Usng he FOC wh respe o y and 0,+ and K +, show ha here should no be a margnal ax on he neres nome pad o ype you need o defne wha neres nome means here n erms of prmves,.e., margnal uly of onsumpon n dfferen perods. Wha does hs mean and why? Combnng he equaons above we ge: 6

7 U, l y U o,+, l = γ = + F K K+, E + γ + The lef-hand sde s jus he margnal rae of subsuon of ype beween urren and fuure onsumpon, and hs s equal o one plus he afer-ax neres rae a he opmum of he onsumers opmzaon program. On he oher hand, he margnal produvy of apal on he rgh-hand sde equals he before-ax neres rae a he opmum of he produers program. I follows mmedaely ha a he soal opmum, he governmen should no ax he neres nome pad o ype. A hs sage hs resul brngs o mnd he fa ha he margnal ax rae on he labor nome of he mos produve agen s zero, whh s no very surprsng. I s sll possble for he neres nome pad o ype o be axed. Ths s where he weak separably of uly funons omes no play. 8 Take he FOCs wh respe o y and o,+. Impose a weak separably assumpon,.e., for some funon h. u, l = ũ h, l Show ha we an rewre he FOC wh respe o y so as o make he FOC for o,+ appear n. Ths wll help you argue ha he ax on ype s apal nome should also be zero. The frs-order ondons n y and o,+ are: y = N µ U y, l λ U, l v N γ = 0 y 0,+ = N µ U, l λ U, l v N γ + = 0 o,+ o,+ If we mpose weak separably U, l = Ũ h, l hen for all l and for eah =, U, l y U o,+, l = h / y h / o,+ 7

8 The equaon y = 0 an hen be rewren as h / y h N / µ o,+ U, l λ U o,+ o,+, l v = N γ and we ge by subsung n he oher frs-order ondon h / j h = γ = + F / K K+, E + v,+ γ + Ths las equaon means ha he margnal rae of subsuon of he onsumer and s equvalen for he frm are equal. Thus a he soal opmum, ype should no be axed on hs neres nome, no more han ype. Therefore he opmal axaon of neres nome s zero. 9 Wha s drvng he resul n 8? Thnk of several hngs and ry o reason hrough hem arefully. Hns: you know wha made he resuls go hrough n he FOCs mehanally. Sare a hose FOCs and hnk wha happens f you relaxed some assumpons. Queson all FOCs you used! The resul s due o Aknson Sglz preferenes. As long as he governmen an hoose he opmal level of apal, he level of onsumpon n he wo perods s no a good ag for he ype and herefore here s no reason o ax. 8