Optimal Income Tax Schedules under Action Revelation

Transcription

1 Optmal Income Tax Schedules under Acton Revelaton Jonathan Hamlton and Steven Slutsky Department of Economcs Warrngton College of Busness Unversty of Florda Ganesvlle FL USA Aprl 03 Earler versons of ths paper were presented at the Econometrc Socety, the Southern Economc Assocaton, and the Assocaton for Publc Economc Theory meetngs and semnars at Academa Snca, CORE, the Delh School of Economcs, ISI-Delh, Sungkyunkwan Unversty, the Unversty of Brtsh Columba, the Unversty of Copenhagen, and the Unversty of Vrgna. We thank Mchael Blake, Rchard Romano, Davd Sappngton, Arunava Sen, and Art Snow for comments. We thank the Warrngton College of Busness for fnancal support for ths research. All errors are solely our responsblty.

2 ABSTRACT Most ncome tax systems rely on reportng after ndvduals earn ncome. Most theoretcal tax models utlze the revelaton prncple and have ndvduals report types ex ante, before effort. Wth an nfnte populaton, acton revelaton and announcement revelaton do not dffer sgnfcantly. However, wth a fnte number of ndvduals, a sngle ndvdual s msrevelaton s detectable, and the government can revse tax schedules after detectng msrevelaton. Pketty showed that any full-nformaton outcome s attanable under acton revelaton. However, the off-equlbrum allocatons used to sustan full-nformaton outcomes on the equlbrum path may not be optmal for the government to actually carry out the government must commt to suboptmal allocatons. We study outcomes for dfferent levels of commtment under acton revelaton. Wth full commtment, dstorted outcomes may arse when budget mbalance would have arsen off the equlbrum path. The dstorton pattern can dffer from the standard Mrrlees-Stgltz pattern. Both types may be subsdzed (or taxed) on the margn. The outcome under acton revelaton s weakly domnated by that under announcement revelaton. Somewhat surprsngly, wthout full commtment, the government can sometmes do better usng acton revelaton. Acton revelaton may ncrease the government s ablty to commt to off-equlbrum polces and thus may allow t to redstrbute more n equlbrum.

3 . Introducton Most ncome tax systems use ex post ncome reportng where ndvduals fle tax returns after choosng labor effort. In contrast, most theoretcal models of ncome taxaton utlze the revelaton prncple and hence have ndvduals reportng types ex ante, before workng. We call the former approach acton revelaton and the latter announcement revelaton. Wth an nfnte populaton whether wth a contnuum of types (Mrrlees [97]) or a fnte set of types (Stgltz [98]) the government cannot detect msrevelaton by a sngle ndvdual. It announces tax schedules ex ante and does not revse them f anyone msreveals. Whether revelaton s by acton or announcement does not affect the optmal tax schedules. Instead, suppose that the number of ndvduals s fnte and the government knows the true pattern of types but not any partcular ndvdual s type. If even one ndvdual msreveals, the planner at some stage wll detect that the revealed pattern of types does not match what t knows for certan to be the actual pattern. If dfferent types pay dfferent taxes, then any msrevelaton causes ether a surplus or a defct that the government would notce. Knowng that someone msrevealed (even though t does not know who), the government could condton the bundles t offers each ndvdual on others revelatons as well as her own. Indeed, when msrevelaton causes a defct, to mantan budget balance, the government must have the taxes of at least some ndvduals depend on what others reveal. In an economy wth a fnte populaton usng acton revelaton, Pketty [993] consdered the use of generalzed tax schedules where each ndvdual s tax depends on others ncomes. He showed that the government not only could do better than the standard Mrrlees-Stgltz second best, but could attan any full-nformaton outcome. However, Hamlton and Slutsky [007] showed that, to sustan many full-nformaton outcomes n equlbrum, the government must

4 announce tax schedules that do not balance the budget off the equlbrum path. Such announcements would not be feasble when a defct occurs and hence would not be credble. For announcement revelaton, they show that the result that any full-nformaton outcome s attanable remans vald even f budget balance s requred on and off the equlbrum path. Not havng a defct n any crcumstance may be necessary for a system of generalzed taxes to be credble but t s not suffcent. Indvduals wll not beleve feasble threats f the government cannot commt to them and t s not n the government s nterest to carry them out. If the government announces off-equlbrum allocatons that run a surplus to sustan a fullnformaton outcome n equlbrum, t wll not be optmal for the government to actually enact them. Attanng full-nformaton outcomes would then requre that the government be able to commt tself to makng those suboptmal allocatons. When the government cannot fully commt, such tax schedules would be revsed, and the approprate soluton concept ncorporates renegotaton-proofness. In that case, the government no longer has the role of a mechansm desgner standng outsde the game settng the rules under whch ndvduals wll play. It s now another player n the game, able to make certan moves but subject to the restrctons of optmzng at each of ts nformaton sets. 3 We study the outcomes for dfferent budget balance requrements off the equlbrum path under acton revelaton. 4 Results dffer sgnfcantly from those under announcement revelaton. Dstorted outcomes may arse, n partcular when budget mbalance would be needed off the equlbrum path. The pattern of dstortons can dffer from the Mrrlees-Stgltz pattern n whch the hgher ablty type faces no margnal dstorton and the lower ablty type s taxed on the margn. In some crcumstances, t s optmal to have both types face negatve margnal taxes (or

5 postve margnal taxes). We also compare outcomes under announcement revelaton and acton revelaton and show that nether approach domnates the other. Secton specfes the model and secton 3 consders some benchmarks for comparson. 5 Sectons 4 and 5 present results for full commtment and renegotaton proofness. Fnally, secton 6 offers some conclusons.. The Model (A) Structure The economy has a fnte number n of ndvduals of two dfferent types. Let or ) denote the number of each type. Each ndvdual consumes a net trade bundle n (for = wth two components, the frst beng labor ncome and the second consumpton. A lnear producton technology determnes feasble pars of net trades, p(n n) 0. Wthout loss of generalty, we can specfy the unts of measurement so that p (,). If trade bundle nvolves no redstrbuton across types. p 0, then that net Each ndvdual has a utlty functon U ( ) over the net trade vector. The utlty functons can dffer across types because of taste, productvty, or endowment dfferences. We make the followng assumptons about preferences: () U ( ) s quas-concave and contnuously dfferentable. Snce labor s a bad, and U/ <0. U/ 0 U () lm 0. Thus, for any bundle wth p < 0, there s an ˆ 0 such that U()=U(0, ˆ ). 3

6 () Indvduals have a maxmum possble labor supply L and thus have maxmum possble pre-tax ncomes and. 6 Furthermore, lm U. From these assumptons, we can characterze the ndfference curves. Indfference curves slope U/ up ( MRS () 0 ). The ndfference curve through any bundle U/ * wth * p < 0 must cross the lne p 0twce at a pont where and at a pont where *. * Next, we assume that nether consumpton nor lesure s nferor. Consder lnes p K and let (K) be the bundle that solves Max U (), s.t. p K. Then, (v) If KK, then (K) (K) and (K) (K). In addton, we make a standard sngle-crossng assumpton across types: (v) At any bundle, MRS () MRS () and hence. Assumpton (v) holds n the standard case n whch ndvduals dffer only n ablty wth type more able than type. (B) Tmng of Actons Nature moves frst and determnes each ndvdual s type. The government learns the preferences and the exact number of each type, but t does not learn the type of any partcular ndvdual. All ndvduals know ther own types and the total numbers of each type. 7 The government wll choose polces n the second and fourth stages to seek to mplement a partcular outcome. We assume that the equal treatment property s satsfed n any allocaton that the government wshes to mplement so that every ndvdual earnng the same ncome receves the same allocaton. Then any outcome that the government wshes to mplement s a par of bundles,,,. In the second stage, the government announces a menu of 4

7 provsonal tax polces condtonal on the vector of labor ncomes that ndvduals report n the n next stage. That s, for each possble vector of ncomes,,,... announces the provsonal consumpton assgned to each ndvdual, Y n, the government Y(),Y(),...Y (). These provsonal consumpton levels vary wth the entre vector of ncomes and not just each ndvdual s own ncome. The government may modfy these bundles later n the game subject to some restrctons. In order for these provsonal tax polces to mplement the par n ndvduals choosng must assgn,,,, any vector that has n ndvduals choosng to the ndvduals choosng and to the and ndvduals choosng. 8 Any bundle to be mplemented that s feasble and effcent must exactly balance the budget. That s: n + n = n + n () In addton, there are restrctons related to budget balance off the equlbrum path as well. These depend on what revsons the government can make to ts provsonal announcements and are presented below. In the thrd stage, after observng the government s provsonal tax polces, ndvduals smultaneously select ther ncomes whch the government observes and are fxed for the remander of the game. In makng these choces, ndvduals are aware of what devatons, f any, the government wll make n the fourth stage. The government wants the equlbrum of ths game of ndvdual choces to be unque. To ensure ths, there must be restrctons on the Y ( ) provsonal polces announced by the government n stage 3. These restrctons are n effect ncentve compatblty constrants. Wth a fnte number of ndvduals, the Mrrlees-Stgltz self-selecton constrants are not approprate snce the government detects devatons by even one 5

8 person. Followng Pketty [993], we assume that desred behavor s nduced by domnance solvablty rather than the stronger noton that t s a domnant strategy. Our problem s more complcated snce the government moves both before and after ndvdual revelaton. Indvduals observe the provsonal polces but also react to ther belefs of what adjustments the government wll make after revelaton. Snce the fnal stage adjustments occur after ndvduals ncome decsons, they cannot be drectly constraned to yeld domnance solvablty. To specfy ths formally, assume for now that the government can prevent any ndvdual from selectng an ncome other than or. We consder below how to do ths under dfferent possbltes of the modfcatons the government can make n stage 4. Hence, the crucal concern s whether ndvduals choose the ncome ntended for ther type or that for the other type. The government s problem then reduces to assgnng consumptons to people who earn these ncomes condtonal on the number who choose each level. Let N j be the total number who choose and let N k = n N j, for j k, be the number who choose j N and k j j, and let k j N be the consumptons assgned to the two ncome levels as functons of the number who choose each one. From the perspectve of any partcular ndvdual, let j ˆN be the number of other ndvduals who choose ncome j j. Let N,Y be the consumptons that someone choosng each type s ncome expects to receve after stage 4 as a functon of how many chose each ncome level and the provsonal polces from stage. Then domnance solvablty holds f the followng condtons are satsfed for j equal to ether or : U (, (N ˆ +, Y)) U (, (N ˆ,Y)), 0 j j j j j k k j U(,(N,Y)) ˆ U(,(N+,Y)) ˆ, n j k k k j k j j j j ˆN n - () j ˆN n (3) 6

9 Condton () mposes that choosng others have chosen j, t s better for type j to choose t s then a domnant strategy for type k to choose j s a domnant strategy for type j: no matter how many j than k. From condton (3), k gven that every type j has chosen In the fourth stage, havng learned the vector that ndvduals chose, the government specfes the actual taxes t wll levy. We specfy these as bundles ( ). We consder two possbltes for what devatons the government can make from ts second stage provsonal announcements. If no devatons are allowed, the government can fully commt to feasble announcements. Formally, ths mposes the restrcton: j. ( ) = Y ( ), (4a) A second possblty s renegotaton proofness where changes can be made only f no one objects. 9 Snce the ncomes are now set, the only changes that can be made are n the consumptons. The no-objecton restrcton s then: () Y(), for all (4b) Anyone can block a change from the provsonal announcement f t lowers hs or her consumpton. (C) Socal Preferences Ths game has n + strategc players: n ndvduals and the government. Indvduals utlty functons defne ther payoffs. The government s preferences derve from a weghted utltaran welfare functon. Gven the vector of ndvdual ncomes, the government forms belefs about the types (or mx of types, f poolng occurs) of ndvduals who choose a partcular ncome. Let ncome be the type that the government beleves an ndvdual who has chosen to be when the complete vector s (for = or ). Let α and α be the 7

10 socal welfare weghts gven to ndvduals beleved to be types and respectvely. 0 Then we assume that socal welfare s n W,Y N U,Y. We assume that ndvduals know ths functon and can accurately predct what polces the government wll mplement n stage 4 condtonal on the chosen vector. (D) Soluton Concept Snce, n stage 3, ndvduals observe the provsonal government polces before revealng ther type and, n stage 4, the government observes ndvduals revelatons before mplementng a fnal polcy, the approprate soluton concept s sequental equlbrum (Kreps and Wlson [98]). The player who moves at any nformaton set maxmzes gven her belefs, and those belefs must be consstent wth equlbrum strateges, where possble. In stage 4, each vector defnes a dfferent nformaton set for the government wth, =,..., n, ts belefs about ndvduals types. In formng these belefs, the government presumes that ths vector arses from mnmum devatons from the ncomes whch t ntended ndvduals to choose. If the vector matches exactly what was ntended, then the government beleves each ndvdual acted as desred and does not assume that offsettng devatons occurred. If all ndvduals chose ntended ncomes but the numbers do not match the dstrbuton, then the government beleves that everyone acted as ntended except for the mnmum number who must have devated. 3. Benchmarks It s useful to compare our equlbrum outcomes to some standard benchmarks. Frst, f the government has full nformaton about ndvduals types, t mplements the soluton to: (FI) Max, αn U() ( α)nu() s.t. p (n n ) 0. 8

11 Choce of the parameter allows ths problem to descrbe any undstorted allocaton. Let (n,α) denote the soluton to FI for any n and α. There exsts an α o such that the soluton to o o 0 FI entals no redstrbuton (p (n,α ) p (n,α ) 0). For α > α, transfers from type s to type s occur (n p (n,α) n p (n,α) 0) wth the reverse for 0 α < α. As shown by Pketty [993], under acton revelaton, any soluton to FI can be sustaned wth generalzed tax schedules when the government does not know types of specfc ndvduals. Pketty placed no feasblty or budget balance restrctons on off-equlbrum bundles. As Hamlton and Slutsky [007] show, sustanng some full nformaton allocatons mght only be possble f the budget does not balance n some off-equlbrum stuatons. To see ths, consder an economy wth just two ndvduals, one of each type. Let,,,, denote the government s desred allocaton whch s assumed to be a constraned effcent allocaton. Assume for now that nether ndvdual chooses an ncome level other than or. Domnance solvablty over these two ncomes can be acheved n two ways. One s for ndvdual to prefer to earn no matter whch ncome ndvdual chooses and then for ndvdual to prefer to earn gven that ndvdual earns. The other s for to prefer earnng regardless of s decson and then for to choose when earns. Note that, for ether way, three condtons are needed. Two come from the requrement that the desred allocaton should be a Nash equlbrum so that each must choose the approprate ncome when the other does. The thrd comes from the requrement that the desred allocaton must be the unque Nash equlbrum. Hence, at least one ndvdual must choose the approprate ncome even when the other does not. When off-equlbrum feasblty or budget balance s mposed, 9

12 there are crcumstances n whch ether of these yelds a superor outcome. Wthout offequlbrum restrctons, only one of the ways s needed; Pketty only needed domnance solvablty wth the lower ablty type havng a domnant strategy for hs results. The followng Lemmas specfy dfferent possbltes for domnance solvablty dependng upon the drecton of the ncome transfer and the type wth the domnant strategy. Denote the bundle the government assgns when both choose by (0), and when both choose by (). Lemma : If type j receves a transfer from type k, then ˆ where U k k k k j j, ˆ U, U, max U,,U, j j j j k k j k k. Ths lemma has mplcatons for when the budget can always be balanced f the taxed ndvdual has the domnant strategy. Snce U j j j j k k, U,, havng a balanced budget f both choose the taxed ndvdual s ntended ncome always ensures that the recpent (regardless of whch drecton redstrbuton takes) would strctly prefer to earn hs ntended ncome than that of the taxed ndvdual. If ˆ, then havng a balanced budget when both k k choose the taxed ndvdual s ntended ncome would also ensure that the taxed ndvdual prefers hs own ntended ncome to that of the recpent ndvdual when the recpent chooses the wrong ncome. If, however, ˆ >, then a defct would be needed when both choose the taxed k k ndvdual s ntended ncome to ensure that the taxed ndvdual would not prefer the recpent s ntended ncome. Snce U j j j j k k, ˆ U, path would bnd wth respect to the taxed ndvdual. Lemma : Assume that the transfer s from type to type., the no-defct constrant off the equlbrum 0

13 (a) Ths allocaton s sustanable by domnance solvablty wth havng a domnant strategy to choose wth (0) satsfyng budget balance and () wth () satsfyng budget balance f U, U, U, U, ; () wth a defct requred at () f U, U, and () wth a surplus requred at () f U, U, (b) to choose. ; and Ths allocaton s sustanable by domnance solvablty wth havng a domnant strategy and wth budget balance at (0) and () ff U, U, U, U, solvablty of ths type.. If, then an off-equlbrum defct s requred at (0) to have domnance Thus, from (a), when taxes are desgned so that the lower ablty person has the domnant strategy, budget balance can be acheved f, at the ncome level ntended for that type, the no-tax 45 lne goes through the gap between the two types ndfference curves through the bundle ntended for the hgher ablty type. A defct s requred f that ndfference curve for type s above the 45 lne at that ncome, and a surplus s requred f type s ndfference curve through that bundle s below the 45 lne at that ncome level. On the other hand, f taxes gve the hgher ablty ndvdual the domnant strategy, then there s off-equlbrum budget balance f and only f type s ndfference curve through the bundle ntended for type s below the no-tax 45 lne at the ncome level ntended for the type. If ths does not hold, then a defct s requred. For ths drecton of domnance solvablty, a surplus s never necessary.

14 When transfers are from to, to show that there exst crcumstances n whch budget balance must be volated off the equlbrum path to sustan an FI allocaton, we need to show that budget balance s nconsstent wth ether way of dong domnance solvablty. From Lemma, ths can happen n two ways. One s that defcts are requred at both (0) and (). The one at (0) s needed for to have a domnant strategy, and the one at () s needed for to have a domnant strategy. The other s that a surplus s needed at () to nsure that chooses when chooses. The frst way occurs f the condtons n (a()) ( U, U, ) and (b) ( U, U, ) both hold. See Fgure for an example of when ths happens. The second way occurs f the condtons n (a()) ( U, U, ( U, U, ) both hold. 3 ) and (b) When s such that the transfer s from the less able to the more able, the possbltes for domnance solvablty wth budget balance are smpler. Lemma 3: Assume that the transfer s from type to type. If (0) and () satsfy budget balance, then has a domnant strategy to choose. Domnance solvablty then holds ff U, U,. If ths does not hold, then a defct at (0) s requred. When transfers go from the lower ablty type to the hgher ablty one, n a domnance solvable allocaton wth off-equlbrum budget balance, both types have a domnant strategy to choose ther ntended ncomes. 4 However, f domnance solvablty cannot be acheved wth budget balance, then t s the requrement that type chooses the bndng constrant. when type chooses that s

15 That off-equlbrum budget mbalances may be needed to sustan some full nformaton outcomes s sgnfcant snce defcts are not feasble even off the equlbrum path, so such stage 3 announcements cannot be mplemented n stage 4 and wll not be beleved. Thus, feasblty condtons must be mposed on the provsonal announcements and on the fnal taxes: n Y for all (5) n n (6) n Under full commtment, (6) follows from (5) and under renegotaton proofness, (5) s needed even though there can be later revsons because of the restrcton of (4b). A second benchmark s the Mrrlees-Stgltz model where the government knows the dstrbuton of types but not any ndvdual s type, and t commts to taxes whch depend only on each ndvdual s own actons. To nduce truthful reportng, the government offers everyone a choce from the same par of net trade bundles, where the bundles satsfy self-selecton constrants. The planner s optmzaton problem s: (MS) Max αnu() ( α)nu(), s.t. U() U() U() U() p(n n) 0. Our next result establshes that generalzed tax schedules wth budget balance everywhere can sustan any allocaton that satsfes the Mrrlees-Stgltz self selecton constrants. 3

16 Lemma 4: Consder an economy wth one ndvdual of each type. If the outcome satsfes the MS constrants, then ths allocaton can be sustaned under generalzed tax schedules wth budget balance off the equlbrum path wth the taxed type havng a domnant strategy. It then follows that, n a fnte economy even wth off-equlbrum budget balance, the government can do at least as well and often better than n the Mrrlees-Stgltz model. Thus, whle Pketty s strong result that all full nformaton outcomes can be acheved wth generalzed taxes s not true when off-equlbrum budget balance s mposed, an mportant kernel remans. Usng generalzed taxes even wth off equlbrum budget balance allows the government to do better than n the standard model whenever dstortons must arse as shown n Theorem. Theorem : If an allocaton solves MS for some α, then there exsts an allocaton that weakly Pareto domnates t and s a domnance solvable equlbrum usng generalzed tax schedules that satsfy budget balance everywhere. If the MS allocaton les off the FI fronter, then the Pareto domnance s strct. There are several mportant ponts about ths result. Frst, as shown n the proof, the taxed ndvdual has a domnant strategy, not the recpent as n Pketty. If transfers are very large so that the MS self-selecton constrants do not hold at the FI allocaton, then the recpent may have the domnant strategy. Second, snce the domnance solvablty condtons all hold wth strct nequalty, they contnue to hold for slghtly larger redstrbutons that volate the MS self-selecton constrants. Hence, FI outcomes can be sustaned wth budget balance and generalzed tax schedules when dstortons are requred n the Mrrlees-Stgltz framework. Thrd, wth more than one ndvdual of each type, there s more flexblty n choosng consumptons off the equlbrum path, so the result contnues to hold. Fnally, note that the MS 4

17 self-selecton constrants are satsfed for near α o so that full nformaton allocatons wth lttle redstrbuton can be sustaned wth generalzed taxes and budget balance. The next results specfy suffcent condtons on preferences and the degree of redstrbuton so that sustanng the FI outcome when redstrbuton s from to requres ether a defct or a surplus at (0) or (). Theorem : (A) For suffcently large, a defct s requred at (0) to ensure domnance solvablty wth havng the domnant strategy. (B) If there exsts an ˆ 0 wth (,α) ˆ, all, then, for suffcently large α, an offequlbrum surplus at () s requred to sustan (,α),, under domnance solvablty. When such a surplus s requred, then a defct s needed at (0) for domnance solvablty wth havng the domnant strategy. However, f budget balance were requred, a no-surplus constrant at () would be the bndng constrant to ensure ether type of domnance solvablty. (C) If (,α) 0, for some ˆα, then a defct at () s requred to sustan the allocatons ˆ (,α),, for ˆα α wth havng the domnant strategy to get domnance solvablty. Several ponts deserve menton. Frst, whch type has the domnant strategy can vary wth the amount of redstrbuton. When lttle redstrbuton s done ( α s near α o ), type has the domnant strategy from Lemma 4. If, after sgnfcant redstrbuton, ether the condton n part (B) holds or the condton n part (C) holds and the restrctons at () are bndng, then type would have the domnant strategy. Second, the condton n (B) that a surplus s requred when type chooses the ncome ntended for type s satsfed f type s demand for lesure s unaffected by the sze of the transfer ((,α)/ α 0). Thrd, these condtons are suffcent 5

18 but not necessary. For example, the suffcent condton n (C) for nfeasblty s only one of many possbltes. If type s demand for consumpton s unaffected by the transfer ((,α)/ α 0), then the off-equlbrum path allocatons must be nfeasble for large. Fnally, when (,α) 0 for all α but s not bounded away from zero, t may be possble to sustan any full-nformaton allocaton wth budget balance. If U((,α), (,α)) U((,α)) and U((,α)) U ( (,α), (,α)) for all α, the achevable utlty possblty fronter wth budget balance s dentcal to the full-nformaton utlty possblty fronter. 4. Equlbra under Full Commtment Under full commtment, the government can commt to run a surplus and dscard the excess revenue. Therefore, gven assumpton (), t s straghtforward for the government to make t strctly domnated for any ndvdual to reveal an ncome other than or by assgnng a consumpton of 0 to any ncome other than those two but assgnng postve ncome to anyone who reveals or. Domnance solvablty then reduces to just condtons () and (3) beng satsfed. In addton, the government cannot commt to nfeasble polces so that (5) must be satsfed. That condton smplfes to: N (N ) + N (N ) N (N ) + N (N ), all N (7) The equlbrum of the game then reduces to beng the soluton to a smple optmzaton problem where, (N ),, and (N ) for all N are chosen to maxmze αn U ( (n )) ( α) n U ( (n )) subject to (), (3), and (7). Note that, whle the government chooses allocatons for every N, only the bundles where N = n (the true number) enter the objectve functon. We call ths the full commtment problem (FC). 5 6

19 Under full commtment, the government can acheve any full-nformaton outcome that has a balanced budget or a surplus off-equlbrum. If an off-equlbrum defct s requred, then the full-nformaton outcome cannot be attaned, and dstortons must be mposed. Theorem 3 characterzes the equlbrum bundles n that case. Theorem 3: Consder an economy wth one ndvdual of each type and assume that the nodefct constrant bnds off the equlbrum path. () If redstrbuton s toward type, then, on the equlbrum path, both types consume bundles wth dstortons n the same drecton where ether (a) type has a domnant strategy; MRS, and transfer exceeds the full-nformaton transfer at these ncome levels; or MRS, ; and the (b) type has a domnant strategy; MRS, and transfer s smaller than full-nformaton transfer at these ncome levels. MRS, ; and the MRS, () If redstrbuton s toward type, then type has a domnant strategy: MRS, ; and the transfer exceeds the full nformaton transfer at these ncome and levels. Whle the dstorton pattern n () s the same as n the Mrrlees-Stgltz model, that s not true n (). Frst, no matter whch type has the domnant strategy, the type payng a postve total tax faces a dstorton. Second, when the optmal tax system gves type the domnant strategy, then both types are subsdzed on the margn nstead of beng taxed. An ncrease n earnngs for both types shfts s ndfference curve through down and restores earnng to be a domnant strategy for type. Whch dstorton pattern arses depends on what type of constrant 7

20 bnds. If a Nash equlbrum constrant bnds, the Mrrlees-Stgltz pattern arses. If a unqueness constrant bnds, the double dstorton arses. The utlty possblty fronters under full commtment and n the MS model dverge from the full nformaton fronter n dfferent ways. Brto et al. [990] show that, n the MS model wth fnte types, the achevable utlty possblty fronter concdes wth the full nformaton fronter over a connected range around the no-redstrbuton utltes. For extensve redstrbuton n ether drecton, dstortons must be mposed and the achevable fronter les below the full nformaton fronter. There are two connected ntervals of dvergence, one for each drecton of redstrbuton. In our model, the nterval around the no-redstrbuton utltes where the FI and FC fronters concde s larger than n the MS model, possbly even coverng the entre full nformaton fronter. When redstrbuton s toward the more able type, f the fronter dverges from the full nformaton one, as n the MS model, t wll do so n a connected nterval. However when redstrbuton s toward the less able type, there can be a segment where the fronter dverges from the full nformaton one but then wth more redstrbuton, the two fronters agan concde. Wth suffcent redstrbuton, exact off- equlbrum budget balance may not sustan a full nformaton allocaton, but sometmes a surplus s requred and sometmes a defct. When a defct s requred, the fronters dverge, but a surplus s sustanable under full commtment, and the outcome s agan effcent. For redstrbuton from s to s, when there s one ndvdual of each type, an undstorted allocaton cannot be sustaned under full commtment when defcts would be needed at (0) and (). Theorem 4 shows that ncreasng the number of ndvduals to two of each type may relax or tghten the government s no-defct constrants and thereby may ncrease or decrease the set of undstorted allocatons sustanable under full commtment. 8

21 Theorem 4: If larger or smaller than when n n, the set of undstorted allocatons whch can be sustaned can be n n. Hence, the falure to sustan all undstorted allocatons s not an artfact of havng just two ndvduals. Wth more ndvduals, t s even possble that fewer undstorted allocatons can be sustaned. Wth more ndvduals, there are more off-equlbrum budget balance restrctons. However, ncentve constrants are relaxed. Havng more ndvduals adds bundles between (n,α)and(n,α) ; these choces gve the government addtonal flexblty. Ths can help overcome the preference of type s for type s full-nformaton bundle wth sgnfcant redstrbuton toward type s. Whch effect domnates depends upon preferences as shown n the proof for the case wth four ndvduals. Whether type s ndfference curve through the noredstrbuton bundle (4) ntersects the vertcal lne through (,α) ˆ above or below the p 0 lne depends upon how much curvature the ndfference curves have and how close (,α) ˆ and (,α) ˆ are to each other. The more they dffer, the more lkely t s that more effcent redstrbuton can be done when there are more ndvduals. 5. Renegotaton-Proof Equlbra To analyze ths case, we must fnd condtons for renegotaton-proofness and then, gven that renegotaton-proofness s satsfed, specfy condtons for domnance solvablty so that the renegotaton-proof equlbrum s unque. Condtons for renegotaton-proofness are more complcated than under full commtment snce the government can change consumptons after ndvduals have chosen ther ncomes. In stage 4, for each, the government wll adjust taxes so that the budget exactly balances. A defct s nfeasble and a surplus could be dstrbuted among ndvduals rasng the utltes of both types and hence rasng socal welfare. Knowng the government s belefs, ndvduals can predct what the government wll do n the fnal stage. 9

22 They wll thus not beleve that polces such as gvng zero consumpton to ndvduals who choose a level of ncome other than or wll be carred out. Furthermore, any outcome that arses after readjustments could have been announced ntally n stage. It thus would have been feasble under wth full commtment and exact budget balance, n contrast to only rulng out defcts. Conversely, any soluton under full commtment and exact budget balance s also possble wth renegotaton proof re-optmzaton. If stage announcements exactly balance the budget, only the consumpton components can change n stage 4 snce ncomes are fxed. Budget balance mples that rasng one type s consumpton at any gven requres lowerng the other type s consumpton whch volates condton (4b) of renegotaton proofness. It mmedately follows from these arguments that exact budget balance n all stuatons s equvalent to renegotaton proofness. 6 Lemma 5: A menu of provsonal tax polces announced n stage s renegotaton proof f and only f condton (5) holds wth equalty. Thus, renegotaton proofness dffers from full commtment only because exact budget balance must hold. The government cannot run ether surpluses or defcts off the equlbrum path. In part, ths result depends upon the assumpton that there are only two components to the bundles. Once ncome s fxed, only consumpton can vary, and the two types have dentcal preferences over changes n consumpton alone. In ths case, t s relatvely easy to make any reoptmzaton nfeasble. 7 Next, consder condtons for domnance solvablty. It s no longer trval to ensure that no ndvdual wll choose an ncome other than one of the equlbrum values. Unlke the full commtment case, budget balance must hold n all stuatons. The government cannot smply threaten to punsh anyone who chooses an ncome level that no one should select by gvng them 0

23 low consumpton wthout volatng budget balance ncreasng ther consumpton ex post would be a Pareto mprovement. It s necessary to specfy what s done wth the consumpton taken from such ndvduals. In Appendx B (avalable on-lne) for the case of one ndvdual of each type, we present generalzed tax schedules defned for all ncome dstrbutons that balance the budget n all crcumstances and that yeld strong ncentves to avod non-equlbrum ncomes. Gven these tax schedules, we then present a suffcent condton on the allocatons that the government seeks to mplement so that domnance solvablty rules out anyone choosng an ncome level other than those n allocatons on the equlbrum path. After rulng out non-equlbrum ncomes, renegotaton proofness requres selectng among equlbrum ncomes n a domnance solvable way just as under full commtment. Therefore, renegotaton proof outcomes can be found as the soluton to the same optmzaton problem determnng full commtment outcomes, except condton (5) must hold wth equalty. It s now straghtforward to fnd the renegotaton proof equlbra and to compare them to those under full commtment. For redstrbutons allowed by the condton n Appendx B, they are dentcal when the full nformaton outcome can be sustaned by off-equlbrum balanced budgets or defcts. The full commtment equlbrum would mpose off-equlbrum budget balance n these cases, so t would also be renegotaton proof. When an off-equlbrum surplus s needed to sustan a full nformaton optmum, then outcomes under renegotaton proofness are nferor to those under full commtment, snce the government can commt to a surplus but must balance the budget f t cannot commt. For one ndvdual of each type, as shown n Theorem B and the dscusson followng, there exst preferences for whch, at α near, a surplus off the equlbrum path s requred to sustan the full-nformaton outcome. These allocatons are attanable under full commtment but not under renegotaton proofness. However, for some

24 preferences, a surplus s never needed to sustan full-nformaton allocatons then outcomes under full commtment and renegotaton proofness are the same for all α. When the no-surplus constrant bnds under renegotaton proofness, Theorem 5 descrbes the resultng pattern of dstortons. Unlke the stuaton when the no-defct constrant bnds, the pattern of dstortons s the same as n the standard Mrrlees-Stgltz model. Theorem 5: Consder an economy wth one ndvdual of each type wth redstrbuton from type s to type s. In a unque renegotaton proof equlbrum, f the no-surplus constrant bnds, then, on the equlbrum path, the type bundle s undstorted, but for type s, MRS. In addton, the government s dong nsuffcent redstrbuton n the sense that, f the ncentve constrant were removed but ncomes were fxed, the government would gan by ncreasng the consumpton transfer from s to s. Fnally, t s worthwhle to compare outcomes under announcement and acton revelaton. Wth full commtment, the optmzaton problem for acton revelaton s FC. Announcement revelaton s essentally the same problem wthout the constrant that an ndvdual s ncome be equal across nformaton sets. Hamlton and Slutsky [007] show that, under announcement revelaton and full commtment, the government can sustan any soluton to FI whle Theorems 3 and 4 show that ths s not true under acton revelaton. Hence, wth full commtment, the government does at least as well (and sometmes strctly better) under announcement revelaton than under acton revelaton. Wth renegotaton proofness, t s no longer true that outcomes under announcement revelaton are as least as good as those under acton revelaton. Under acton revelaton, the only mplcaton of renegotaton proofness s that budget balance must hold n all stuatons. Under announcement revelaton, renegotaton proofness has more mplcatons, and the equlbrum

25 cannot be found as the soluton to an optmzaton problem. Wth one ndvdual of each type, ndvduals compare solutons to three separate stage 4 optmzatons for ther stage 3 decsons. The added flexblty under announcement revelaton may beneft the government n each of these optmzatons ndvdually but harm t n the overall problem. Achevng a better result n one off-equlbrum optmzaton tghtens the pror stage revelaton constrant, and ths sometmes hurts the government. In effect, the addtonal constrants of acton revelaton act as a form of addtonal commtment, lmtng re-optmzaton. For the two-argument utlty functons consdered here, ths can be powerful. Once ncome s fxed, especally off the equlbrum path, the planner has lttle flexblty n re-optmzng snce only consumpton, constraned by budget balance, can be vared. For both types of revelaton, there exst 0 α near α for whch full-nformaton allocatons are sustaned. The set of such α can be larger or smaller under acton revelaton. Under announcement revelaton, the set of sustanable undstorted allocatons s always a proper subset of the full-nformaton Pareto fronter. As dscussed after Theorem, there exst preferences for whch the entre full-nformaton fronter s sustanable under acton revelaton, and thus, for some α, acton revelaton yelds greater socal welfare. In other crcumstances, announcement revelaton does better. Assume one ndvdual of each type, and let β denote the hghest value of α such that the soluton under announcement revelaton s on the full-nformaton fronter. When ndvduals pool, re-optmzaton under renegotaton proofness wll assgn both types bundles on the p 0lne between the optmal ponts for the two types on that lne. The most redstrbuton occurs when type s best bundle s assgned. Consder type s ndfference curve through (,β), the undstorted bundle for the type gvenβ. Preferences can be such that the type ndfference curve les above the 3

26 p 0 lne at (,β). Acton revelaton then sustans a smaller set of full-nformaton allocatons snce the undstorted bundle for α = β cannot be sustaned under t. See Fgure. 5. Conclusons If there s a fnte set of ndvduals and the government knows precsely the dstrbuton of types, then optmal taxes depend sgnfcantly on several factors that are not relevant n the standard Mrrlees-Stgltz framework. These nclude whether ndvduals reveal ther type by actons or announcements, whether the government can run a surplus off the equlbrum path or must exactly balance the budget n all crcumstances, and whether the government can commt to polces specfed before ndvdual revelaton or can adjust those polces after revelaton as long as no ndvdual objects. These factors nteract. Under announcement revelaton wth full commtment, whch off-equlbrum budget restrcton s mposed s rrelevant. Even wth the strong constrant that the budget must exactly balance n all crcumstances, the government can sustan any full nformaton Pareto optmal allocaton. If nstead the government can make adjustments subject to restrctons of renegotaton proofness, then not all full nformaton Pareto optma can be acheved but no ndvdual ever faces margnal dstortons. Under acton revelaton, there s complete algnment between the dfferent levels of commtment and the restrctons on budget balance; mposng a no-defct constrant s equvalent to full commtment and mposng exact budget balance s equvalent to renegotaton proofness. If the government needs to run a surplus off the equlbrum path to sustan some full nformaton optmum, then n the renegotaton proof equlbrum, the same pattern of dstortons arse as n the standard model: the net taxed ndvdual faces no margnal dstorton and the net subsdzed ndvdual s taxed on the margn. When the government needs an off-equlbrum defct to sustan a full nformaton optmum, then n both full commtment and renegotaton proof equlbra, the 4

27 pattern of dstortons s qute dfferent from n the standard model. Both types are dstorted wth a subsdy (tax) on the margn to nduce them to work more (or less) than wth lump sum taxaton. Under acton revelaton, off-equlbrum budget restrctons reduce the set of mplementable undstorted allocatons below what Pketty found wthout those restrctons where every full nformaton optmum was mplementable. However, the set of mplementable undstorted allocatons s stll larger than n the Mrrlees-Stgltz model. Surprsngly, there s ambguty as to whether more full nformaton allocatons can be mplemented wth announcement or acton revelaton. Although acton revelaton mposes extra constrants on the government, only when there s full commtment wth just a no-defct constrant off the equlbrum path do the equlbra under announcement revelaton weakly domnate those under acton revelaton. When re-optmzaton s allowed and exact budget balance s mposed, then, dependng on both ndvdual and socal preferences, ether method mght allow more undstorted redstrbuton than the other. That acton revelaton can be superor to announcement revelaton n some sense volates the revelaton prncple. The nature of the volaton dffers from that of Bester and Strausz [000] n a prncpal-agent model wth multple agents and mperfect commtment by the prncpal. They consder the revelaton prncple to be volated f the sze of the message space s greater than the number of types, whch occurs wth partal poolng. In our case, the sze of the message space under both acton and announcement revelaton equals the number of types. 8 The dfference between the two revelaton methods s whether consstency s requred across onand off-equlbrum allocatons, a factor not consdered by Bester and Strausz. Fnally, although we have focused on the optmal ncome taxaton problem, there are mportant lessons from ths analyss for any mechansm desgn problem wth a fnte number of 5

28 agents. If only to avod mposng nfeasble actons off the equlbrum path, polces affectng one agent cannot be ndependent of what other agents do. Wth such dependence, t becomes sgnfcant how agents reveal ther attrbutes, by announcement or by acton. The prncpal may also be able to alter polces after revelaton. If so, the restrctons of renegotaton proofness we consdered here may be an approprate and realstc possblty lyng between complete commtment and no commtment. References Abreu, D. and A. Sen, 99, Vrtual Implementaton n Nash Equlbrum, Econometrca 59: Bester, H. and R. Strausz, 000, Imperfect Commtment and the Revelaton Prncple: The Mult-Agent Case, Economcs Letters 69: Brto, D., J. Hamlton, S. Slutsky, and J. Stgltz, Pareto Effcent Tax Structures, Oxford Economc Papers 4: 6-77 Farrell, J. and E. Maskn, 989, Renegotaton n Repeated Games, Games and Economc Behavor : Fudenberg, D. and J. Trole, 990, Moral Hazard and Renegotaton n Agency Contracts, Econometrca 58: Gaube, T., 0, Taxaton of Annual Income as a Commtment Devce, workng paper, Department of Economcs, Unversty of Osnabruck. Hamlton, J. and S. Slutsky, 007, Optmal Nonlnear Income Taxaton wth a Fnte Populaton, Journal of Economc Theory 3: Hamlton, J. and S. Slutsky, 0, Judcal Revew and the Power of the Executve and Legslatve Branches, workng paper, Department of Economcs, Unversty of Florda. Kreps, D. and R. Wlson, 98, Sequental Equlbra, Econometrca 50: Mrrlees, J., 97, An Exploraton n the Theory of Optmal Income Taxaton, Revew of Economc Studes 38: Pketty, T., 993, Implementaton of Frst-Best Allocatons va Generalzed Tax Schedules, Journal of Economc Theory 6:

29 Smon, J., 0, Fnancal Markets as a Commtment Devce for the Government, workng paper, European Unversty Insttute. Stgltz, J., 98, Self-Selecton and Pareto Effcent Taxaton, Journal of Publc Economcs 7: One possble dfference s that acton revelaton prevents randomzng both consumpton and earned ncome. Of course, when there s nose n the structure of the problem (for example, f the government does not know the precse dstrbuton of types), then any outcome may be on the equlbrum path wth postve probablty and detectng msrevelaton becomes much more complcated. 3 Gaube [0] and Smon [0] consder optmal taxaton n ntertemporal settngs wth dfferent levels of government commtment. Wthout full commtment, the government becomes a player nstead of a mechansm desgner. 4 Hamlton and Slutsky [0] study how dfferent levels of commtment and budget balance requrements correspond to dfferences n the legal system regardng separaton of powers and judcal revew. 5 Proofs of the results n the text are n Appendx A (avalable on-lne). 6 Ths assumpton means that any redstrbuton problem wll be bounded snce the maxmum amount of redstrbuton s lmted. The government could transfer no more than from a type and and no more than from a type. 7 Except for an ndvdual s knowledge of her own type, ndvduals and the government have the same nformaton about the aggregate dstrbuton. Ths s not a model such as Abreu and Sen [99] where ndvduals have complete nformaton about the type of every ndvdual. 8 For acton revelaton to work, must hold for the government to dstngush between types. As long as lesure s strctly ncreasng n lump-sum ncome when 0, ths occurs for at most one full-nformaton allocaton. Snce the planner could sustan full-nformaton allocatons that are arbtrarly close to that one, we gnore ths case n our formal analyss. Note that ths ssue only arses for transfers toward type, gven normalty of lesure. 9 See Farrell and Maskn [989] and Fudenberg and Trole [990] for dscussons and applcatons of renegotaton proofness. 0 Under acton revelaton as opposed to announcement revelaton, snce no ndvdual can be sad to have drectly led, we do not allow these weghts to vary wth whether the government beleves someone has msrevealed as n Hamlton and Slutsky [0]. 7

30 The government mght gan by actng as f ts preferences dffer from ts true ones n order to affect what ncome ndvduals wll choose n stage 3. We assume that the government cannot commt to a false welfare functon and thus must act n stage 4 accordng to ts true preferences. Assumng that any nformaton set off the equlbrum path arses from the mnmal number of devatons satsfes the lmt restrcton of Kreps and Wlson [98, p. 875]. In Appendx B, we consder belefs after ncome choces other than or. 3 The condton n (b) that U, 4 The condton U, U, U, must hold from normalty. must hold for the desred allocatons to be a Nash equlbrum. When ths does hold, then U ( ) > U (, ), so that type also has a domnant strategy (see footnote n Appendx A). 5 We wrte condtons () and (3) wth weak nequaltes. Typcally, domnance solvablty condtons nvolve terated elmnaton of strctly domnated strateges. When a game s domnance solvable n that sense, t has a unque Nash equlbrum. If one elmnates weakly domnated strateges nstead, the process may elmnate some Nash equlbra when multple equlbra exst (see Gretlen [983] for an explanaton of the dffcultes that arse from elmnatng weakly domnated strateges). Here, we want the mechansm to have a unque equlbrum and ntend to elmnate only strctly domnated strateges. However, snce these condtons serve as constrants n an optmzaton problem, f we specfy strct nequaltes, the feasble set would not be closed, potentally leadng to dffcultes. As developed n the proofs, most of these condtons hold wth strct nequalty (see, for example the analyss just pror to Theorem wth one ndvdual of each type). At least one constrant holds wth equalty. Ths s smlar n effect to havng standard self-selecton constrants hold wth weak nequaltes. When ndvduals are ndfferent between actng as the government ntends or devatng, we assume they act as ntended. A bundle wthn an arbtrarly small of the soluton exsts whch would make the condton hold wth strct nequalty. 6 Whle the lmted degree of commtment gves the government the opportunty to revse consumpton levels after ndvduals choose ther ncomes, revsons may never be observed. A government that forsees ts future decsons could smply announce them at stage and then have no desre to make revsons. The lmted commtment acts as a constrant on stage announcements so that the government does not make revsons. 7 However, f the government cannot dfferentally tax dfferent consumpton goods and can only tax ncome, then a composte commodty theorem would reduce the many consumpton goods to a sngle composte good, makng ths two-good model approprate. 8 Ths s clearly true for announcement revelaton. For acton revelaton, the message space mght seem to be any feasble ncome level for each type whch s a contnuum. However, as consdered n Appendx B, the government can specfy tax schedules so that only one of two levels of ncome wll ever be selected by any ndvdual and thus the sze of the message space reduces to the number of types. 8