The primal versus the dual approach to the optimal Ramsey tax problem

Transcription

1 The primal versus he dual approah o he opimal Ramsey ax prolem y George Eonomides a, Aposolis Philippopoulos,, and Vangelis Vassilaos a Deparmen of Inernaional and European Eonomi Sudies, Ahens Universiy of Eonomis and Business, 76 Paission sree, Ahens 434, Greee. Deparmen of Eonomis, Ahens Universiy of Eonomis and Business, 76 Paission sree, Ahens 434, Greee. ESifo, Munih, Germany. April, 28 Asra: There are wo soluion approahes o he dynami opimal puli finane (also nown as Ramsey prolem: he primal and he dual. Mos papers ha have fully solved a Ramsey prolem (y full soluion, we also mean a quaniaive soluion of poliies and ouomes aross differen ax regimes have used he primal approah; his is eause i is simpler han he dual. This paper fully solves a Ramsey poliy prolem y applying oh approahes, and ompares hem. JEL: H2; O4. eywords: Dynami opimal axaion; Endogenous growh. orresponding auhor. Tel: Fax: aphil@aue.gr Anowledgemens: We han Marios Angeleos, evin Lansing, Theodore Palivos and Hyun Par for ommens. Any errors are ours.

2 . Inroduion The dynami opimal puli finane prolem, referred o as Ramsey prolem, is one of he mos fundamenal and influenial eonomi prolems. In his prolem, he governmen hooses is axde poliy mix o maximize he household s welfare y aing ino aoun he equilirium reaion of privae agens o he ax poliy. A eleraed resul is ha he ax rae on (apial inome should e high in he iniial periods and hen roughly zero (see hamley, 986, and Judd, 985. Two approahes have een used o solve his prolem: he primal and he dual. In he primal, we eliminae axes and pries, so ha he governmen an e hough as direly using he quaniies as onrols. In he dual, he governmen uses he ax raes or pries as onrols. Mos papers ha have fully solved he Ramsey prolem (y full soluion, we mean no only he aove eleraed qualiaive resul, u also a quaniaive soluion of poliies and ouomes aross differen ax regimes 2 have used he primal approah. We are no aware of a full soluion o a dual prolem. This is eause, as is widely reognized (see e.g. Jones e al., 997, p. 99, he primal is onsideraly simpler han he dual. 3 This paper fully solves a Ramsey ax poliy prolem y applying oh approahes, and ompares hem. To mae he analysis lear and raale, we hoose a simple seup. The model is as in he lieraure exep ha we use a single inome ax and a linear A produion ehnology. The former is no imporan eause an inome ax inheris he feaures of a apial inome ax. The A ehnology has he advanage ha here are no ransiion dynamis wihin eah ax poliy regime; his redues he omplexiy of he Ramsey prolem wihou affeing he ey poins. As is nown, here are hree ax poliy regimes: an iniial period during whih he ax rae is exogenously given; he period(s of relaively heavy hosen ax raes; and he laer period(s of hosen zero ax raes. We will herefore presen hree suses of equaions assoiaed wih hese hree regimes and show how he hree suses are ineronneed o give a sysem of equaions ha haraerizes he full Ramsey prolem. Beyond his, we will solve he Ramsey sysem numerially. We will do so under oh he primal and he dual approah. See e.g. Ljungqvis and Sargen (2, haper 2 for a reen review of he lieraure. The opimal long-run ax rae an e differen from zero if here are imperfeions. Bu he ey logi remains: i is opimal o heavily ax inelasially supplied inpus. 2 As Ljungqvis and Sargen (2, p. 349 poin ou, a qualiaive analysis provides imporan insighs u anno yield definie resuls. Wha is he value of he non-zero ax rae(s in he iniial period(s? Wha is for onsumpion, growh, e, aross differen ax regimes? These quesions have o e sudied numerially. 3 In he Ramsey prolem, irrespeively of he soluion approah used, one anno solve firs for he long run and in urn sudy he ransiion period, as is ypially he ase in growh models. Insead, one has o solve simulaneously for he long run and he whole opimal pah. hari e al. (994 were he firs ones who oained numerial resuls y applying he primal approah. Bu, as far as we now, here are no analogous resuls y applying he dual approah (hamley, 986, p. 68, has provided a seh of a full dual soluion.

3 The main resuls are as follows. The dual prolem is indeed more omplex o solve (in erms of equaions and unnowns. I is hene harder o find ranges of parameer values, iniial ondiions and exogenous governmen spending ha yield a dual soluion. The primal soluion is easier o oain. This an parly explain he la of full, numerial soluions o he dual prolem. Neverheless, given a soluion, he wo approahes give idenial resuls along he whole opimal pah as expeed. 4 The res of he paper is as follows. Seion 2 presens he eonomy and solves for is ompeiive equilirium. Seion 3 solves he Ramsey poliy prolem y using he primal approah, while Seion 4 uses he dual approah. Seion 5 onludes. 2. ompeiive equilirium 2. Desripion of he eonomy and how we are going o wor onsider a losed eonomy wih an individual agen and a governmen. The individual onsumes, saves in he form of apial and governmen onds, and produes a single good aording o a linear A produion ehnology. The governmen imposes an inome ax and issues onds o finane puli servies, where he laer ener he individual s uiliy funion. We assume disree ime, infinie ime-horizons and perfe foresigh. The governmen is enevolen and hooses he pah of he ax rae one-and-for-all a ime y aing ino aoun he ompeiive equilirium. 5 Afer we presen he ompeiive equilirium, we solve he primal prolem. The dual will follow nex. This is for onveniene. 2.2 Individuals Using for simpliiy a log-linear uiliy funion, he individual maximizes: β [ ν ln ( ν ln H ] ( where and H are respeively privae onsumpion and puli onsumpion a, < β < is he disoun rae and <ν < is he weigh given o privae relaive o puli onsumpion. The wihin-period udge onsrain is: 4 Solving he prolem also under he dual approah is no only for inelleual uriosiy. In addiion o he ig numer of papers ha have used he dual approah o ge qualiaive resuls, here are ases (in riher seups where i is no possile o redue he onsrains o he Ramsey prolem ino a simple implemenailiy (udge onsrain and a resoure equaion. 5 Following mos of he lieraure, we assume ha governmen expendiure is exogenous. Our main resuls do no hange if governmen expendiure is also hosen opimally (resuls are availale upon reques. 2

4 B B ( τ A( B (2 where is end-of-period apial, B is end-of-period onds, τ < is he ax rae a and A > is a parameer. We assume for simpliiy ha apial and onds pay he same gross reurn, A, and are axed a he same rae, τ. We also assume zero apial depreiaion. The iniial sos, and B, are given. The household hooses {, } B, < o maximize ( suje o ( Governmen udge onsrain To finane puli expendiure, H, he governmen axes all ypes of inome a a rae τ < < and issues onds. The wihin-period governmen udge onsrain is: B B H AB τ A( B (3 2.4 ompeiive equilirium (E Given he pahs of he independen poliy insrumens {, H } τ and iniial ondiions for and B, a E is an alloaion {, } B suh ha he individual s prolem is solved,, mares lear and udge onsrains are saisfied. There are wo equivalen ways of presening he E. In he firs, he E is summarized y he resoure onsrain holding in eah period and a single implemenailiy (udge onsrain in period (see Appendix A ha also ompares our implemenailiy onsrain (4 o he lieraure: 6 H A (4a [ ( τ β A ]( B (4 where, in (4a-(4, we have eliminaed pries and axes apar from τ, whih is exogenously given o mae he poliy prolem nonrivial (see elow for deails. Seondly, and equivalenly, he E an e summarized y he individual Euler equaion, he resoure onsrain and governmen udge onsrain in eah period (see Appendix A: 6 and B are exogenously given iniial sos, while period- onsumpion,, is endogenous. Thus, in our seup, follows from (4; see also elow. 3

5 ( β r (5a H A (5 B B H r B r A (5 where r ( τ A is he ne (afer ax reurn o asses. onerning he exogenous poliy insrumens, we assume ha he governmen ses is expendiure as an exogenous fraion of he eginning-of-period apial so, H h (his saisfies saionariy. For simpliiy, in he ompuaions elow, we will assume ha h h is onsan over ime. Also, he period- ax rae, τ <, will e aen as given; oherwise he governmen would use i as a lump-sum ax whih maes he poliy prolem firs-es and hene rivial (see e.g. Ljungqvis and Sargen, 2, pp The primal approah o he Ramsey prolem, In he primal approah, he governmen hooses he pahs of { } o maximize ( suje o (4a-(4. To mae our resuls easily omparale, we follow hari e al. (994 and Ljungqvis and Sargen (2, pp The Lagrangean is: β { ν ln ( νln H [ A H ] } ( B [ ( τ A] ξ (6 β where ξ is an aemporal muliplier assoiaed wih (4 and is a dynami muliplier assoiaed wih (4a. As is nown (see e.g. hari e. al, 994, p. 625, and Ljungqvis and Sargen, 2, p. 322, he period- alloaions differ from he same rules governing ehavior from period onward. This is eause he period- firs-order ondiions inlude erms relaed o he iniial so of asses, and B. Speifially, a he firs-order ondiions for and are: 7 Well-nown papers ha use he primal approah inlude Luas (99, hari e al. (994 and Jones e al. (997. 4

6 ν ξ ( B [ ( τ A] (7a 2 β ( A (7 while, a he firs-order ondiions for and are: ν (8a β ( A (8 In addiion, in all periods, he firs-order ondiions inlude he onsrains o he governmen s prolem, namely (4a-. 3. Qualiaive feaures In he asene of exogenous upper ounds on he ax rae, here an e one period only wih nonzero axaion, and his is a. 8 Aually, i is sraighforward o show y woring as in hari e al. (994, pp ha, in his lass of uiliy funions, he opimal ax rae is zero a 2 onward. Thus, here are hree ax poliy regimes ha orrespond o, and 2, where τ is exogenously given and τ for 2. All his is onfirmed elow. 3.2 The full Ramsey sysem We now presen he full sysem. We wor in wo seps. Firs, we omine he firs-order ondiions - equaions (7a-, (8a- and he onsrains (4a- - so as o saisfy oninuiy aross poliy regimes. Reall ha here are hree disin poliy regimes, whih orrespond o periods, and 2. Seond, sine he model allows for long-erm growh, we ransform he variales o mae hem saionary. In pariular, we define a all, whih is a jump variale. Thus, as in he asi A model, afer period 2 here are no ransiional dynamis; as soon as he zero ax rae regime sars a 2 wih given values of apial and onds, 2 and B 2, all saionary 8 If here are exogenous upper ounds on he ax rae, i is opimal o se he ax rae a is upper ound for as many periods as neessary and zero hereafer. See he disussion in hari e al. (994, p

7 variales jump o heir long-run values where all quaniies grow a he same onsan rae. 9 Therefore, we have (see Appendix B for deails: Γ A h ] βν ( Firs poliy regime, (9a [ A β ( A A h β ( A A h Seond poliy regime, (9 Third poliy regime, 2 (9 ν where Γ ξ [ ( τ A] 2. Throughou, numers in susrips denoe ime periods, while variales wihou ime susrips denoe long-run values (here he long run is reahed a 2. We also have he implemenailiy onsrain (4 rewrien in saionary form as: ( β [ ( τ A] (9d are (9a-d summarize he Ramsey prolem. We have four equaions in four unnowns whih B,,, ξ. This is given he pah of h, he iniially given and he period- ax rae, τ. In urn, he implied opimal ax rae ( τ for an follow from he individual s opimaliy ondiions (see Appendix B. This is a simple sysem ha an e solved even analyially. 3.3 Numerial soluion We solve (9a-d numerially o mae our resuls omparale o hose in seion 4 elow. As a aseline ase, we se he following values for parameers, ν. 85, A, β. 9 ; B iniial ondiions,. 25 ; and exogenous variales, τ. 4 and h H.667. Soluions are in Tale ha also repors he implied values of he ax rae, τ. 9 See hari e al. (994, pp for a riher model wih ransiion dynamis wihin poliy regimes. The values of and h are he means of he US eonomy. 6

8 Tale : Soluion of he primal prolem endogenous variales ax poliy regimes τ ξ (se Noe: We use Mala 7.. All values mae sense. For insane, he ax rae in he firs period is posiive, τ.7, while i is zero in he long run (a 2. The value of he aemporal muliplier assoiaed wih he implemenailiy onsrain, ξ, is posiive. I is worh poining ou ha, exep for he ax rae, all oher variales jump o heir long run values in he firs period (his is proaly a propery of he A model. We repor ha omparaive sai exerises give inuiive resuls. For insane, a higher τ leads o a fall in τ and an inrease in he growh rae of onsumpion and apial in period onward. A higher leads o opposie effes. We finally repor ha our resuls are rous o hanges in he values of parameers, iniial ondiions and exogenous variales. 4. The dual approah o he Ramsey prolem τ,,, B o maximize ( suje o (5a- (5. We now follow hamley (986 and Ljungqvis and Sargen (2, p. 36. Sine τ < is In he dual approah, he governmen hooses { } aen as given o mae he poliy prolem nonrivial, he governmen an hoose,, B only a. Afer his period,, he governmen hooses τ,,, B, or equivalenly r,,, B, where r ( τ A. The Lagrangean is: { β ν ln ( ν ln H A H ] [ [ H ( r B B r A ] β ( r ]} [ ( 7

9 where, and are dynami mulipliers assoiaed wih (5a, (5 and (5 respeively. The period- firs-order ondiions again differ from he same rules governing ehavior from period onward. Speifially, a he firs-order ondiions for,, B are respeively: ν β r ( (a β A β ( r ( ( A β ( r ( while, a he firs-order ondiions for r,,, B are respeively: B (2a ( ν β ( r β (2 β ( A β ( r A (2 β ( r (2d In addiion, in all periods, he firs-order ondiions inlude he onsrains o he governmen s prolem, namely (5a-. 4. Qualiaive feaures In addiion o he feaures disussed in he primal approah (see suseion 3., noe ha equaion (5a is linearly dependen wih equaions ( and (2d. In pariular, a any ime,..., whih means ha φ is onsan over ime as in hamley 2 2 (986, equaion 34. eep in mind ha φ is a new endogenous variale. Following hamley (986, φ p. 68, in wha follows, we use o susiue ou (he sign of φ is he sign of, whih is expeed o e negaive in a seond-es prolem and omi ( and (2d from he sysem. All 8

10 his also onfirms ha one has o solve simulaneously for he long run and he ransiion pah inluding period, as was oviously he ase in he primal approah (see elow. 4.2 The full Ramsey sysem We now presen he full sysem woring as in he primal approah. Thus, we firs omine he firs-order ondiions - equaions (a-, (2a-d and he onsrains (5a- - so as o saisfy oninuiy aross he hree poliy regimes. In urn, we ransform he variales o mae hem saionary; in pariular, we define, m Λ, Λ a all, whih are, B all jump variales. 2 Therefore, we have (see Appendix for deails: Firs poliy regime, β ( r A h m β ( r m mh m m r A βφ Λ ( A h β Λ( A ( r (3 A (3a (3 ν Λ Λ β ( r (3d Seond poliy regime, β ( r A h m β ( r m mh m m r A (4a (4 βφ Λ [ A h ] β Λ ( A ( r A (4 φ φ Λ (4d m While he need o solve simulaneously for he long run and he ransiion pah is, y onsruion, he ase in he primal approah, in he dual approah we ould solve for he long run independenly if here were no puli de (i.e. he governmen udge is alaned; see e.g. Par and Philippopoulos (24 and Eonomides and Philippopoulos (28. 2 We repor ha he soluion is rous o he ransformaions used. 9

11 ν Λ Λβ ( r Λ [ A β h ] (4e Third poliy regime, 2 3 β ( r A h β ( r mh m m r A (5a (5 βφ Λ [ A h ] β Λ ( A ( r A (5 φ φ Λ (5d m ν Λ Λ β ( r Λ[ A β h ] (5e Equaions (3a-d, (4a-e, (5a-e and m summarize he Ramsey prolem. B B We have fifeen equaions in fifeen unnowns whih are,,, r, r,,, Λ Λ Λ,,, Λ Λ Λ, m, m, m,φ. This is given he pah of h, iniial ondiions for and B, and he period- ax rae, τ. In erms of he numer of equaions and unnowns, his is a more omplex sysem o solve han he one in he primal approah (see (9a-d aove. 4.3 Numerial soluion To solve he aove sysem, we use he same parameer values used in seion 3. Aually, i was harder o find ranges of parameer values, iniial ondiions and exogenous variales ha yield a dual soluion; a soluion o he primal prolem was muh easier o oain. We hus sared our searh for a soluion from he dual. Numerial resuls for all endogenous variales are presened in Tale 2. We also repor he implied values of B m and he ax raes, τ. 3 As in he primal soluion, sine all ransformed variales are jump, he eonomy is a is long run a 2.

12 Tale 2: Soluion of he dual prolem endogenous variales ax poliy regimes m Λ Λ r τ φ (se Noe: We use Mala 7.. The soluions of and τ are he same as in Tale. In addiion, here we have soluions for Λ, Λ and he onsan value of φ, whih are presen in he dual prolem only. All endogenous variales have he righ sign. Regarding he soluion for, he idea is ha i is opimal for he governmen o raise all ax revenue hrough a ime- apial levy, lend he proeeds o he privae seor and finane governmen expendiure y using he ineres from he loan (hene he negaive value of m, 5. onlusions We solved he dynami opimal puli finane (Ramsey prolem y applying he dual and primal approah. As expeed, hey yield he same resuls for poliies and alloaions.

13 APPENDIX Appendix A: ompeiive equilirium The firs-order ondiions of he individual s prolem inlude (2 and he Euler equaion (5a. Using he governmen s udge onsrain (3 ino he individual s udge onsrain (2, we ge he resoure onsrain, (4a or (5. Equaion (5 is equaion (3 rewrien as in hamley (986, equaion 2. To ge he implemenailiy onsrain (4, one an wor as in Ljungqvis and Sargen (2, haper 2. Aually, our model wih a log-linear uiliy funion and an A produion ehnology is a speial ase of equaions (2.3 and 2.32 in Ljungqvis and Sargen if: (i we se u( (ii we ignore heir laor supply erms (iii sine here apial and onds are axed a he same rae and earn he same reurn, we use [ ( τ A ]( B on he righ-hand side of (2.32. This gives [ ( τ A]( β B, whih is (4. Appendix B: Primal soluion o he Ramsey prolem onsider ondiions a. If we use (7a for and (8a for ino (7, we have ( A ν βν ξ [ ] τ 2 B ( A Γ, where H A from (4a. This gives (9a. onsider ondiions a. Equaions (8a-(8 imply β ( A or ( A H β ( A β, where A from (4a. Thus, a,. A h When we are in he seond poliy regime a, whih also means ha a 2 he eonomy will e on is alaned growh pah where variales remain onsan and are denoed wihou a ime susrip, his is wrien as β ( A A h β ( A growh pah, so his is wrien as, whih is (9. A h Equaion (9d is equaion (4 rewrien in saionary form., whih is (9. A 2, we are on he alaned 2

14 Finally, given he (primal soluion, he gross growh rae of onsumpion is ( A β a, while a we have ( A βν. Also, a any, he gross Γ growh rae of apial follows from he resoure onsrain, A h. Having solved for, he ax rae an follow from β [ ( τ ] A. Appendix : Dual soluion o he Ramsey prolem ν onsider (a whih holds a only. This is wrien as β ( r, whih is (3d. onsider (2a and (2 whih hold a only. Equaion (2a is rewrien as B or, y using B φ φb. This, as B gives (4d a and (5d a 2. Equaion (2 is rewrien as ν β( r, where β A H from (5. This gives (4e a and (5e a 2. onsider now hose ondiions ha hold a any ime,. The Euler ondiion, (5a, and he resoure onsrain, (5, imply β ( r A h. This gives (3a a, (4a a and (5a a 2. The governmen udge onsrain, (5, joinly wih he Euler ondiion, m B β ( r (5a, imply. This gives (3 a, (4 a m B mh m m r A and (5 a 2. Finally, he Euler ondiions for apial, ( or (2, imply ha a any β ( A β ( r A, where This gives (3 a, (4 a and (5 a 2. φ and A H. 3

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