Cross-Border Tax Externalities: Are Budget Deficits Too Small? Willem Buiter * European Bank for Reconstruction and Development, NBER and CEPR

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1 Cross-Border Tax Exernaliies: Are Budge Deficis Too Small? Willem Buier * European Bank for Reconsrucion and Developmen, NBER and CEPR Anne Siber Birkbeck College, Universiy of London and CEPR 30 March 200 * Willem H. Buier and Anne C. Siber The views and opinions expressed are hose of he auhors. They do no represen he views and opinions of he European Bank for Reconsrucion and Developmen. The auhors are graeful o Jordi Gali and oher paricipans in he conference The Design of Sabilizing Fiscal Policies a he Universiy of Valencia on June 17-28, 2003 and o seminar paricipans a he Universiy of Cambridge and he Universiy of Souhern California for helpful commens.

2 Absrac In a dynamic opimising model wih cosly ax collecion, a ax cu by one naion creaes posiive exernaliies for he res of he world if iniial public deb socks are posiive. By reducing ax collecion coss, curren ax cus boos he resources available for curren privae consumpion, lowering he global ineres rae. This pecuniary exernaliy benefis oher counries because i reduces he ax collecion coss for foreign governmens of curren and fuure deb service. In he non-cooperaive equilibrium, naionalisic governmens do no allow for he effec of lower domesic axes on deb service coss abroad. Taxes are oo high and governmen budge deficis oo low compared o he global cooperaive equilibrium. Even in he cooperaive equilibrium complee ax smoohing is no opimal: curren axes will be lower han fuure axes. JEL classificaion: E62, F2, H21. Key words: fiscal policy, inernaional policy coordinaion, opimal axaion. Willem H. Buier, European Bank for Reconsrucion and Developmen, One Exchange Square, London, EC2A 2JN, UK. Tel: + (0) ; Fax: + (0) ; buierw@ebrd.com Anne C. Siber, Deparmen of Economics, Birkbeck College, Universiy of London, Male Sree, Bloomsbury, London WC1E 7AX, UK. Tel: + (0) ; Fax: + (0) ; a.siber@econ.bbk.ac.uk

3 1. Inroducion Does one counry s borrowing in an inegraed global financial marke impose exernaliies on oher counries? If so, are hese spillovers welfare-enhancing or welfarereducing? This issue figures prominenly in he debaes abou he European Union s Sabiliy and Growh Pac and he meris of G3 policy coordinaion. In his paper we consider cross-border exernaliies associaed wih he ransmission of naional public deb policies hrough heir effec on he global risk-free real ineres rae. 1 We provide a dynamic equilibrium model wih opimising households and governmens, in which public deb and he governmen s ineremporal budge consrain provide a link beween curren and fuure ax decisions. Such models are analyically difficul, especially if hey do no exhibi Ricardian equivalence or deb neuraliy. This has led us o focus on he simples possible supply side for he naional economies (a perishable endowmen), a represenaive infinie-lived consumer wih log-linear preferences and a simple source of Ricardian nonequivalence, or absence of deb neuraliy: fiscal ransfer coss. We assume ha here are increasing and sricly convex real resource coss of adminisering and collecing axes. 2,3 We hen exend he benchmark model o allow for CES preferences and a producion echnology. We focus on real ineres rae cross-border spillovers ha occur in he absence of 1 Anoher exernaliy is ha associaed wih eiher he occurrence of sovereign deb defaul or wih acions underaken by he debor counry or by ohers o preven sovereign defauls. 2 Slemrod and Yizhaki (2000) repor ha he adminisraive cos of he US ax sysem is 0.6 cens per dollar of revenue raised. Slemrod (1996) esimaes compliance coss o be abou 10 cens per dollar colleced. 3 Barro (1979) pioneered hese sricly convex fiscal ransfer coss in a closed-economy seing. He assumed ha he governmen minimises he discouned sum of hese coss, raher han maximises he discouned welfare of households. 1

4 sovereign defaul risk and wihou sraegic ineracions beween a naional fiscal auhoriy and a naional or supranaional moneary auhoriy. We model a non-moneary economy in which every naional fiscal auhoriy saisfies is ineremporal budge consrain. Governmen spending on goods and services is exogenous. We assume ha each governmen can commi o a pah of axes, aking oher governmens axes as given. Thus, here is commimen, bu no inernaional cooperaion. We show his noncooperaive behaviour resuls in inefficien global equilibria. We assume ha resources are always fully uilised. Financial capial is perfecly mobile across counries. Inernaional ransmission of naional fiscal policy is only hrough ineres raes. Taxes are lump-sum; heir incidence canno be alered by changing privae behaviour, bu because of he sric convexiy of he fiscal ransfer coss, he iming of axes maers in his model, jus as i would wih convenional disorionary axes on labour income or asse income in models wih endogenous labour supply and capial accumulaion. Wih negaive axes, or subsidies, resource coss resul from privae ren-seeking behaviour. We model he ax collecion coss as borne by he public secor. Allowing for compliance coss borne by he privae secor would add noaional complexiy wihou changing our qualiaive conclusions. Wihou fiscal ransfer coss our model, wih is represenaive privae agen, would exhibi Ricardian equivalence: any sequence of lump-sum axes and deb ha saisfies he ineremporal budge consrains would suppor he same equilibrium for any given sequence of public spending on goods and services. There would be no inernaional spillovers. This is rue even if a counry is large in he world capial marke and explois is monopoly power. If we had assumed overlapping generaions, insead of a represenaive agen, hen alernaive rules for financing a given public spending programme would cause pure pecuniary exernaliies if here were no fiscal ransfer coss and axes were lump sum. Even wih symmeric counries, here could be disribuional effecs beween generaions, bu as long as dynamic 2

5 inefficiency does no occur, any feasible sequence of lump-sum axes and deb suppors a Pareo efficien equilibrium. In he one-counry special case of our model, fiscal ransfer coss do no cause inefficiency if one assumes ha resource ransfers beween he privae and public secor in he counerfacual command economy are subjec o he same fiscal ransfer coss as in our marke economy. Inefficiency arises when here are muliple counries and each counry affecs oher counries choice ses in a way ha is no adequaely refleced in marke prices. Wih symmeric counries and represenaive, infinie-lived consumers, symmeric ax policies have no disribuional effecs. However, hey can have welfare consequences if hey change he world ineres rae. If, for example, counries have ousanding socks of deb and a change in policy causes he ineres rae o rise, hen counries mus raise axes, now or in he fuure, and fiscal ransfer coss increase. Naional governmens ha maximise heir own residen s welfare do no inernalise he cos of he higher fiscal ransfer coss o oher counries. Thus, a naional governmen s financing decision ha raises he world ineres rae inflics a negaive exernaliy on he res of he world. This is in line wih convenional wisdom. Where our model depars radically from convenional wisdom is hrough he mechanism by which financing choices affec ineres raes. I is convenional o associae defici financing of public spending wih financial crowding ou. Tha is, for a given public spending programme, larger bond-financed deficis brough abou by lower axes raise ineres raes. In our horoughly neoclassical ineremporal model he opposie is rue. Lower axes and larger deficis early on resul in a lower global rae of ineres. See Buier and Klezer (1991). If here is dynamic inefficiency, hen fiscal policy ha causes redisribuion from he young o he old can lead o a Pareo improvemen. Wih asymmeric counries, alernaive defici-financing policies would have inernaional, as well as inergeneraional, disribuional implicaions. 3

6 We show ha if a governmen is oo small o affec he global ineres rae, i minimises he coss of collecing axes by smoohing hem over ime. If i is able o influence he ineres rae and has posiive iniial deb, hen i ses a lower ax in he iniial period han in fuure periods. This is because lower fiscal ransfer coss in he iniial period han in laer periods imply higher aggregae consumpion in he iniial period han in laer periods. Thus, he real ineres rae in he iniial period is lower han wih perfecly smooh axes and his lowers he ineres paymen on he governmen s ousanding deb and, hence, lowers fuure fiscal ransfer coss. Relaive o he global (cooperaive) opimum, noncooperaive counries ax oo much and issue oo lile deb in he iniial period. Reducing curren axes has a posiive welfare spillover, even hough i requires issuing more deb. Lowering he curren ineres rae by lowering curren axes lowers he cos of servicing all counries deb and hus reduces all counries need o collec cosly axes. In a noncooperaive equilibrium, counries do no ake ino accoun his benefi o oher counries and hey ax oo much in he iniial period. Our conclusion ha lack of inernaional cooperaion leads o axes ha are iniially oo high and public deficis ha are iniially oo small seems o conradic he presumpion refleced in he deb and defici ceilings of he Sabiliy and Growh Pac ha deficis are ap o be oo large. However, we do no wan o make oo much of he size of he exernaliies associaed wih alernaive ax and borrowing policies of naional governmens in EMU; even he larger EMU counries are small fish in he global financial pond. Our analysis is more relevan o ineracion beween he Unied Saes, he European Union as a whole, possibly Japan and soon China. Our paper analysing he welfare economics of inernaional ineres rae spillovers from he ax and borrowing sraegies of naional governmens using a dynamic opimising general equilibrium model is relaed o he vas lieraure on oher inernaional fiscal policy linkages, he more modes lieraure on he poliical economy of he iming of axes in closed economies

7 and he sizable lieraure on he opimal iming of muliple disorionary axes in a closed economy. The lieraure on he inernaional ransmission of fiscal policy has wo main srands. Firs, here is he work on he ransmission of governmen-expendiure shocks wih lump-sum axes and wihou fiscal ransfer coss. Examples are Frenkel and Razin (1985, 1987) and Turnovsky (1988); Turnovsky (1997) provides a survey. The papers in his vein are in sharp conras o ours. We ake governmen expendiure as exogenous and ask how he financing maers when here are fiscal ransfer coss. Second, here are papers on he ransmission of ax shocks in models wih disorionary axes in a balanced-budge seing. There is a sizable lieraure going back o Hamada (1966) on he sraegic axaion of capial income in a world economy. In his lieraure, capial-exporing (imporing) counry can increase is naional income by acing as a monopolis (monopsonis) and resricing capial movemens. The resul is a Nash equilibrium where naions wan o ax capial flows. Oher papers consider issues of he feasibiliy of differen ax regimes in an inegraed world economy, ax harmonisaion and ax compeiion. Examples of such papers are Sinn (1990) and Bovenberg (199). In he closed-economy poliical economy lieraure, excessive public deficis and deb may resul from a poliical pary s desire o ie he hand s of a possible successor (Persson and Svensson (1989) and Alesina and Tabellini (1990)), an incumben governmen s incenive o signal is compeency prior o an elecion (Rogoff and Siber (1988)), or a war-of-ariion game over he allomen of he coss of fiscal adjusmen (Alesina and Drazen (1991)). 5 Drazen (2000) provides a discussion of his lieraure. In his paper, we absrac from poliical economy concerns; governmens are able o commi o policies which maximise naional welfare. 5 Tabellini (1990) shows ha inernaional policy coordinaion worsens he oucome in Alesina and Tabellini (1990). 5

8 Chamley (1981, 1986) pioneered he normaive sudy of dynamic opimal axaion in a closed economy seing, when he governmen can borrow or lend. He focuses on he choice beween disorionary capial and labour income axaion and does no consider fiscal ransfer coss of he kind sudied here. The sae s opimal policy is o impose he maximum possible capial levy on he privae secor s iniial, predeermined socks of capial and public deb and hen o swich permanenly o a zero capial income ax rae (see also Lucas (1988)). Incorporaing fiscal ransfer coss of he kind considered here would render Chamley s highly uneven ime profile of ax receips subopimal. In secion 2 we presen he model. In secion 3 we exend he model o consider small changes in he households ineremporal elasiciy of subsiuion. We show ha as his elasiciy falls, he deviaion beween cooperaive and noncooperaive axes rises. In secion we consider a producion economy and CES preferences. We show ha if he world economy is a a seady sae wih posiive deb, a coordinaed reducion in he curren ax financed by higher fuure axes improves welfare. Secion 5 concludes. 2. The Model The model comprises N $ 1 counries, each inhabied by a represenaive infinie-lived household and a governmen. Each period, each household receives an endowmen of he single privae radeable, non-sorable consumpion good and each governmen purchases an exogenous amoun of he privae good. Governmens finance heir purchases by issuing deb or by axing heir residen households. We assume ha he ax sysem is cosly o adminiser; governmens use up real resources collecing axes. All savings are in he form of privaely or publicly issued real bonds. We assume ha households are symmeric and ha endowmens and governmen purchases are consan over ime. There is perfec inernaional inegraion of he naional financial markes, and hence, a common world ineres rae. 6

9 2.1 The households The counry-i household, i = 1,...,N, has preferences over is consumpion pah given by u i ' j '0 β lnc i, (1) where c i is is period- consumpion and β 0 (0,1) is is discoun facor. The household s period-, = 0,1,..., budge consrain is c i % a i %1 ' W & τi % R a i, (2) where a i is he household s sock of asses in he form of real bonds a he sar of period, R is (one plus) he ineres rae beween period - 1 and period, W is he household s per-period endowmen of he good and τ i is is ime- ax bill. The household s iniial asses, a 0, are given. In addiion o saisfying is wihin-period budge consrain, he household mus saisfy he long-run solvency condiion ha he presen discouned value of is asses be non-negaive as ime goes o infiniy. The ransversaliy condiion associaed wih is opimisaion problem ensures ha he presen discouned value of is asses is no sricly posiive. Thus, lim 6 a i %1 / Π s'0 R s ' 0. (3) Equaions (2) and (3) imply ha he presen discouned value of he household s consumpion equals he presen discouned value of is (afer-ax) income plus is iniial asses: a 0 % j '0 (W & τ i )/Π R s ' j s'0 '0 c i / Π s'0 R s. () 7

10 The household chooses is consumpion pah o maximise is uiliy funcion (1) subjec o is ineremporal budge consrain (). The soluion saisfies he Euler equaion c i %1 ' βr %1 c i, ' 0,1,.... (5) Solving he difference equaion (5) yields he household s ime- consumpion as a funcion of is iniial consumpion and he -period ineres facor c i ' β Π R s s'1 c i 0, ' 1,2,.... (6) Subsiuing equaion (6) ino equaion () yields he household s iniial consumpion as a funcion of is axes and he ineres facors c i 0 ' (1 & β) R 0 a 0 % W & τi 0 % j (W & τ i )/ Π R s. (7) '1 s'1 Subsiuing equaion (6) ino equaion (1) yields he household s indirec uiliy as a funcion of iniial consumpion and he ineres facors u i ' lnc i 0 % (1 & β) j '1 β ln Π R s, (8) s'1 where consans ha do no affec he household s opimisaion problem are ignored. 2.2 The governmen The counry-i, i = 1,...,N, governmen s period-, = 0,1,..., budge consrain is τ i & (φ/2)τ i 2 % b i %1 ' G % R b i, (9) 8

11 where b i is he governmen s ousanding deb a he sar of period and G > 0 is is per-period purchase of he good. The fiscal ransfer cos associaed wih a ax τ or a surplus - τ is (φ/2)τ 2, where φ > 0. The governmen s iniial deb (or credi, if negaive), b 0, is given. We resric he model s parameers so ha saisfying equaion (9) is feasible; he resricions are deailed laer in his secion. In addiion o saisfying is wihin-period budge consrain, he governmen also saisfies lim 6 b i %1 / Π s'0 R s ' 0. (10) As wih he household, his is an implicaion of he long-run solvency consrain and he ransversaliy condiion associaed wih he governmen s opimisaion problem. Equaions (9) and (10) imply ha he presen discouned value of he governmen s purchases, plus is iniial deb, equals he presen discouned value of is ax sream, ne of collecion coss: τ i 0 & 2 (φ/2)τi 0 & g 0 % j [τ i & (φ/2)τ i 2 '0 where g :' G % R 0 b i 0, ' 0 G, >0. & g ]/ΠR s ' 0, s'1 (11) 2.3 Marke clearing Marke clearing requires ha he sum of he N households asse holdings equals he sum of he N governmens deb. Thus, a ' b, (12) where variables wihou a superscrip denoe global averages. 9

12 The global resource consrain requires ha he sum of oal household consumpion, oal governmen purchases and oal fiscal ransfer coss equals oal endowmens. Thus, c ' W & G & φ N 2N j τ j2, ' 0,1,.... (13) j'1 Equaion (13) is, of course, also implied by equaions (2), (9) and (12). Averaging boh sides of he Euler equaion (6) over he N counries yields Π R s ' c /(β c 0 ), ' 1,2,... (1) s'1 Equaions (13) and (1) imply ha in equilibrium, he ime- ineres facor is solely a funcion of ime-0 and ime- axes. Equaions (13) and (1) imply ha lower ime-zero axes financed by higher ime-one axes lower he ineres rae beween period zero and period one. Does he iming of axes affec he global risk-free ineres rae in he manner ha his model predics? There is a large body of empirical work aemping o quanify he relaionship beween ineres raes and budge deficis. 6 However, i is problemaic and he resuls are hard o inerpre for several reasons. Firs, boh axes and ineres raes are endogenous and an apparen relaionship beween hem may be due o he influence of oher variables. For example, auomaic sabilisers cause ax revenue o be lower and deficis o be higher during recessions. A he same ime, an expansionary moneary policy may emporarily lower real ineres raes. Thus, he role of moneary policy over he business cycle may cause deficis and real ineres raes o be negaively correlaed. Second, while lowering axes may lower he global risk-free ineres rae, i may also increase sovereign- 6 A recen example is Laubach (2003). 10

13 defaul risk premia. If he effec on he risk premium is larger han he effec on he risk-free ineres rae, i will cause ax decreases o be associaed wih higher measured marke ineres raes. Third, if ax cus in a counry include lower capial axes, hen he counry s before-ax ineres rae may fall o equae afer-ax ineres raes. This causes lower axes o be associaed wih lower (before-ax) ineres raes. Fourh, he scenario here has lower axes in he curren period accompanied by higher fuure axes and unchanged governmen spending. However, households may inerpre lower axes oday as a signal of a change in he governmen s aiude oward fiscal policy and hey may expec lower axes in he medium-run as well. In empirical sudies, i is difficul o conrol for he public s expecaion of fuure ax policy and is beliefs abou fuure spending. Subsiuing equaion (1) ino equaion (8) gives he counry-i household s indirec uiliy as a funcion of is iniial consumpion and he pah of aggregae consumpion: u i ' lnc i 0 & βlnc 0 % (1 & β) j '1 β lnc. (15) Subsiuing equaions (12) and (1) ino equaion (7) gives he household s iniial consumpion as a funcion of he pah of axes. The predeermined value of iniial governmen deb eners as a parameer. 0 ' (1 & β)c 0j β (w & τ i )/c, c i '0 where w :' W % R 0 b 0, ' 0 W, >0. (16) Subsiuing equaion (16) ino he indirec uiliy funcion (equaion (15)) yields 11

14 u i ' ln j '0 β (w & τ i )/c % (1 & β) j '0 β lnc. (17) Subsiuing equaion (1) ino equaion (11) yields j β s i '0 ' 0, where s i ' τi & (φ/2)τ i 2 & g, ' 0,1,.... (18) c Subsiuing he global resource consrain (equaion (13)) ino equaions (17) and (18) would allow he household s indirec uiliy and he governmen s budge consrain o be expressed solely as funcions of he pahs of he axes in he N counries. 2. Taxes and revenues We impose furher resricions on he parameer space o ensure an equilibrium exiss: max{g,g 0 } # 1/(2φ), w 0 $ 1/φ, W & G >2/φ, g 0 >0. (19) The ne ax revenue funcion, τ & (φ/2)τ 2 looks like a Laffer curve, alhough is shape is he resul of ax collecion coss and no he disorions associaed wih non-lump sum axes. Ne revenue has a maximum of 1/(2φ) a τ = 1/φ. Thus, firs inequaliy in assumpion (19) ensures balanced budges are feasible. The ime-0 budge is balanced a τ 0 = τ & 0 := (1-1 & 2φg 0 )/φ or τ 0 = τ % 0 := (1 + 1 & 2φg 0 )/φ. Likewise, he ime-, > 0, budge is balanced wih axes of τ & := (1-1 & 2φG)/φ or τ % := (1 + 1 & 2φG)/φ. The axes τ & and τ - 0 are on he righ, or upward-sloping par of he ime-0 and ime-, > 0, ne ax revenue curves, respecively. The axes τ % 0 and τ + are on he wrong or downward-sloping pars. There is a convenional governmen budge surplus in counry i in period 0 if and only if τ i 0 0 [ τ& 0, τ % 0 ] and a primary (ha is, ne of ineres paymens) 12

15 surplus in period if and only if τ i 0 [ τ &, τ % ]. By he resource consrain, equaion (13), he upper bound on feasible axes in a symmeric equilibrium is τ := (2/φ)(W & G). Assumpion (19) implies ha he axes τ % 0 andτ % are sricly less han τ, and hence are feasible. By equaions (13) and (18), a symmeric equilibrium wih consan axes has axes of (1 ± 1 & 2φ[G % (1 & β)r 0 b 0 ])/φ. By assumpion (19), his is feasible. We allow for negaive axes, or subsidies. In his case he collecion cos is he cos of adminisering and disbursing he surplus. We rule ou, however, he empirically implausible case of an iniial sock of credi ha is so large ha he governmen can achieve a balanced budge (including ineres paymens) in period zero wih a subsidy. The necessary condiion for his, g 0 > 0, is included in (19). The variable s i surplus (or defici, if negaive), divided by in equaion (18) is he period-0 value of he governmen s ime- budge periods > 0 i is he primary surplus. We will refer o. For = 0, his surplus is he oal surplus and for c 0 s i as counry i s discouned ime- surplus. From he governmen s budge consrain, equaion (18), i appears ha a raional governmen wih marke power may se axes on he wrong side of he ne ax revenue curve, ha is, a a ax higher han he ne revenue maximising ax 1/φ. To see his, suppose τ i ' 1/φ. Wih marke power, a counry can influence he global ineres facor. Holding oher axes consan, a marginal rise in τ i causes aggregae period- consumpion o fall. As ne revenues are insensiive o axes a 1/φ, hey are unaffeced by a marginal increase in he ax a his poin. Thus, a marginal increase in τ i above 1/φ causes he discouned ime- surplus o rise. We have he following resul. All proofs of proposiions are in he Appendix. Proposiion 1. Given τ j, j i, s i has a unique maximum in. The maximising ax is decreasing in he number of counries; as N 6 i goes o τ i 1/φ. If > 0, he maximising ax is an elemen of [1/φ,τ % ] 13

16 and is increasing (decreasing) below (above) he maximising ax on [τ &,τ % ]. If = 0, he s i maximising ax is an elemen of [1/φ,τ % 0 ] and is increasing (decreasing) below (above) he maximising ax on [τ & 0,τ% 0 ]. Denoe he ax ha maximises s i s i 0 when axes are symmeric and here are N counries by τ (N 0 if = 0 and by τ (N if > 0. If N = 1 and axes are symmeric hen he maximising ax is if = 0 and i is w 0 & w 2 0 & τ2 W & W 2 & τ 2 if > 0. The relaionship beween imporan ime- values is shown in Figure 1. The posiion of he corresponding ime-0 values is similar. We now show ha i canno be par of an equilibrium for any governmen o ever se a ax above he one ha maximises is discouned surplus. The sraegy of he proof is o show ha if i did so, i could always pick a ax on he oher side of he ne ax revenue curve ha would provide he same discouned surplus and higher uiliy. Proposiion 2. I canno be par of an equilibrium for governmen i o se is ime- ax higher han he one ha maximises s i. 3. Dynamic Opimal Taxaion We assume ha a ime zero, he governmen in counry i can commi o a ax plan {τ i } '0. I akes he ax plans of he oher governmens as given and maximises he indirec uiliy of is household (equaion (17)) subjec o is budge consrain (equaion (18)). We begin analysing he problem by considering he effecs of ime- axes on welfare and he governmen s fiscal posiion. By equaions (13) and (17), he marginal change in uiliy from a marginal increase in he ime- ax is Mu i Mτ i ' β m i, where m i :' & 1 % φτi N w & τ i c c j β w s s & τs s'0 c s & (1 & β)φτi Nc, ' 0,1,.... (20) 1

17 If axes are equal across counries, equaion (16) implies Σ '0 β (w & τ )/c = 1/( 1 - β). Equaion (13) and he definiion of s (in equaion (18)) implies (w & τ )/c = 1 & s. These resuls and equaion (20) imply ha wih symmeric axes m i ' m '& 1 & β c 1 % φτ s N, ' 0,1,.... (21) We nex show ha he marginal (indirec) uiliy of axes mus be sricly negaive. Proposiion 3. A symmeric equilibrium mus have m < 0, = 0,1,.... Holding oher axes consan, raising ime- axes has he direc effec of lowering disposable income in period, ending o lower welfare. Suppose τ > 0. Then he ax increase raises collecion coss, and hus, lowers aggregae consumpion and raises he cos of borrowing a ime. This ends o increase (decrease) welfare if consumers lend o (borrow from) he governmen in period. Proposiion 3 implies ha he direc effec dominaes he ineres rae effec; he ax increase lowers welfare. Suppose τ < 0. Then a ax rise increases aggregae consumpion and lowers he cos of borrowing in period because of lower collecion coss. As consumers mus be lending o he governmen in his case, he ineres rae effec ends o lower welfare. The marginal change in uiliy associaed wih a ax rise mus be negaive. By equaions (13) and (18), he marginal increase in he discouned value of he budge surplus (he lef-hand side of equaion (18)), resuling from a marginal increase in he ime- ax when axes are idenical across counries is β n, where n :' (1/c )(1 & φτ % φτ s /N), ' 0,1,.... (22) We show ha in equilibrium, a marginal increase in he ime- ax mus increase he discouned value of he governmen s sream of budge surpluses. 15

18 Proposiion. A symmeric equilibrium mus have n > 0, = 0,1.... Afer he firs period, he governmen smoohs axes. Proposiion 5. A symmeric Nash equilibrium has consan axes afer period zero. Le τ i ' τ i, c i ' c i, and s i ' s, > 0. Then equaions (17) and (18) imply ha he governmen maximises ln[(1 & β)(w 0 & τ i 0 )/c 0 % β(w & τi )/c] % (1 & β)lnc 0 % βlnc (23) subjec o B i :' (1 & β)s i 0 % βs i ' 0. (2) By equaions (13) and (23), he marginal uiliies associaed wih τ i and τ i 0 are m i 0 '& 1 & β c 0 1 & φτi 0 w 0 & τ i 0 N c 0 C % φτ 0 N, m i 1 '& β c 1 & φτi w & τ i N c C % φτ N, (25) where C :' (1 & β)(w 0 & τ i 0 )/c 0 % β(w & τi )/c, respecively. and τ i are By equaions (13) and (2), he marginal increases in B i associaed wih increases in τ i 0 n i 0 ' 1 & β c 0 1 & φτ i 0 % φτ i 0 s i N, n i 1 ' β c 1 & φτi % φτi s i N, (26) respecively. If he counries ac symmerically, hen equaion (16) implies C = 1. Equaion (13) 16

19 implies (W - τ 0 )/c 0 = 1 & s 0 and (W & τ)/τ ' 1 & s. These resuls and equaions (25) and (26) imply m 0 '& 1 & β 1 % φτ 0 s 0 c 0 N n 0 ' 1 & β 1 & φτ c 0 % φτ 0 s 0 0 N, m 1 '& β c 1 % φτs N, n 1 ' β c 1 & φτ % φτs N. (27) The firs-order condiions for a maximum imply m 1 /m 0 = n 1 /n 0. This and equaions (27) imply 7 τ 0 /(1 % φτ 0 s 0 /N) ' τ/(1 % φτs/n). (28) Symmery and equaion (2) imply (1 & β)s 0 % βs ' 0. (29) The second-order condiion requires ha he bordered Hessian marix associaed wih he opimisaion problem has a sricly posiive deerminan. This requires n 0 (m 00 m 2 1 & 2m 01 m 0 m 1 % m 11 m 2 0 ) & m 0 (n 00 m 2 1 % n 11 m 2 0 )<0, (30) where m 0 and m 1 are he derivaives of m i (as given by equaion (25)) wih respec o τ i 0 and τ i, respecively, when axes are symmeric and where n 0 and n 1 are he derivaives of (as given by equaion (26)) wih respec o τ i 0 and τ i, respecively, when axes are symmeric. I is sraighforward, bu edious o demonsrae ha symmeric axes which saisfy equaions (28) and n i 7 In deriving equaion (28), boh sides were divided by φ. If φ = 0, he iming of axes is irrelevan as long as he governmen saisfies is ineremporal budge consrain. 17

20 (29) also saisfy equaion (30). 8 Definiion 1. A symmeric equilibrium is a pair of axes {τ 0,τ} such ha he feasibiliy condiion (29) and he opimaliy condiion (28) are saisfied. We firs esablish ha axes are always posiive. Proposiion 6. An equilibrium canno have subsidies. ( τ 0 <0or τ <0). We analyse he equilibrium by graphing equaions (29) and (28) in Figure 2. This figure is drawn for sricly posiive axes ha are less han he ones ha maximise he wihin-period discouned surpluses as we have shown ha no oher axes can be par of an equilibrium. The feasibiliy condiion (29) is represened by he solid curves F -, F, and F + ; he differen curves represening differen iniial socks of deb. By Proposiion, hese curves slope down; an increase in he fuure ax allows he governmen o reduce he curren ax and sill balance is budge. 9 The curve represening he case of zero iniial deb, F 0, goes hrough he poin ( τ &,τ & ), labelled A. The curve represening he case of sricly posiive iniial deb, F +, lies above F 0 and passes hrough he poin (τ & 0,τ& ), which is above A. The curve represening he case of sricly negaive iniial deb, F -, lies below F 0 and passes hrough (τ & 0,τ& ), which is below A. The curves represening equaion (28) in Figure 2 are represened by he dashed lines. The curve O 0 represens he case of no iniial deb or an infinie number of counries. The curves labelled and represen he case sricly posiive iniial deb and N = NN and N = NO, O NN % O NO % respecively, where 1 # NO < NN < ; he curves labelled O NN and O NO represen he case sricly negaive iniial deb when N = NN and N = NO, respecively. Proposiion 7. The curves represening he opimaliy condiions in Figure 2 have he following & & 8 Deails available on reques. 9 The curves are drawn as convex o he origin. This is rue if N is sufficienly large, bu need no be rue oherwise. 18

21 properies: (i) The curve O 0 is he 5 o line. (ii) All of he opimaliy curves are upward sloping and pass hrough he origin. (iii) O N) % and O N O lie below he 5 o line; and lie above he 5 o % O N) & O N O & line. (iv) O N) % lies above O N O when τ > τ - and ; lies below when τ < τ - % τ 0 < τ & 0 O N) & O NO & and τ 0 > τ & 0. The inuiion behind he opimaliy curves in Figure 2 is ha he governmen rades off wo objecives. Firs, i wans o smooh consumpion by smoohing fiscal ransfer coss over ime. If his were is sole objecive, opimaliy would be represened by O 0. Second, i wans o lower he discouned value of he fiscal ransfer coss hrough is influence on he global rae of ineres. If i is an iniial debor, i does his by lowering iniial axes and raising fuure axes. Through he global resource consrain (equaion (13)) his raises iniial consumpion and lowers fuure consumpion, hus lowering he ineres rae beween periods zero and one. Thus, is required ax revenue falls. Likewise, if i is an iniial credior i can lower is required discouned ax revenue, and hus is ax collecion coss, by raising iniial axes and lowering fuure axes, hus raising he ineres rae beween periods zero and one. This second objecive means ha he curve represening he opimaliy condiion in Figure 2 is flaer han O 0 when here is iniial deb and i is seeper han O 0 when here is an iniial surplus. The more marke power a counry has (ha is, he smaller is N) he greaer is is abiliy o affec he global ineres rae and he more imporan his second moive becomes. Thus, as he number of counries falls, he opimaliy curve becomes flaer if he counry is an iniial debor and seeper if he counry is an iniial credior. When N 6 counries have no marke power. Only he firs objecive maers and he opimaliy equaion is represened by O 0. Equilibrium occurs a he inersecion of he relevan feasibiliy and opimaliy curves. We show ha a unique inersecion mus occur. 19

22 Proposiion 8. A unique symmeric equilibrium exiss. Differen equilibria are represened by he poins A - G in Figure 2. Poin A is he equilibrium when here is no iniial sock of deb. In his case here is ax smoohing and he budge is balanced each period. Poins B, C and D represen equilibria when here is a posiive sock of iniial deb. If N =, he equilibrium is represened by poin B and here is ax smoohing. Poins C and D lie below he 5 o line; hence, if N < and here is a posiive iniial sock of deb, τ > τ 0. As N falls, he negaive slope of he curve represening equaion (29) ensures ha he iniial ax declines and he fuure ax rises. Likewise, poins E, F and G represen equilibria when here is an iniial negaive sock of deb. If N = (poin G), here is complee ax smoohing. Poins E and F lie above he 5 o line; hence, if N < and here is a negaive iniial sock of deb, τ > τ 0. As N falls, he iniial ax rises and he fuure ax falls.these resuls are summarised below. Proposiion 9. If counries have no marke power (N = ) or if iniial deb is zero, hen here is complee ax smoohing. If counries have some marke power (N < ), hen he iniial ax is sricly less (greaer) han he subsequen ax if here is a sricly posiive (negaive) iniial deb. When counries have no marke power, we have Barro s (1979) resul. Taxes resul in resource losses because hey are cosly o collec. If hese coss are convex, hen an opimising governmen smoohs hem over ime. If, however, he governmen can affec he ineres rae and i has a non-zero iniial sock of deb, hen i lowers he discouned value of is required ax revenue by reducing iniial axes and raising fuure axes. If i is an iniial credior i raises is reurn o is savings by increasing he iniial ax and lowering fuure axes If he governmen begins wih sricly posiive (negaive) iniial deb, hen Proposiion 9 says ha he iniial ax is lower (higher) han subsequen axes. This implies ha he governmen eners period one wih sricly posiive (negaive) deb. Thus, if he governmen could re-opimise in period one, Proposiion 9 implies ha i would se a lower (higher) ax in period one hen in laer periods. This implies ha he equilibrium, which feaures consan axes 20

23 The case of N = 1 corresponds o he social planner s oucome. Hence, we have he following resul. Proposiion 10. Suppose ha N > 1. If here is a posiive (negaive) iniial sock of deb, hen he iniial ax is oo high (low) relaive o he social opimum. The subsequen ax is oo low (high) relaive o he social opimum. If here is a posiive sock of iniial deb, lowering iniial axes causes a posiive exernaliy by decreasing all counry s borrowing coss. Counries do no ake ino accoun he social benefi and hey do no decrease iniial coss enough. The oucome is furhes from he opimal oucome when he number of counries goes o infiniy and counries lose heir marke power. This is in sark conras o he resul in beggarhy-neighbour policy games where naions aemp o exploi heir marke power o gain a he expense of oher counries. In such papers, as he number of counries goes o infiniy and naions lose heir marke power, he noncooperaive oucome converges o he cooperaive oucome. 11 The resul ha he disorion does no vanish, bu increases when naions lose heir marke power, is similar in spiri o ha in Kehoe (1987), alhough he economic mechanism is quie differen. In his paper governmens balance heir budges and do no fully ake ino accoun he negaive effec on world capial accumulaion of axing workers o provide curren governmen spending. Here, governmens do no ake ino accoun he posiive effec of curren ax reducions and larger budge deficis on he world ineres rae. In boh papers, as he number of counries increases and he effec of any counry on global variables declines, he failure of counries o ake ino accoun he effec of heir acions on he world economy becomes more severe. from period one on, is no ime consisen unless R 0 b 0 = 0. The ime inconsisency arises because he iniial deb is aken as exogenous, and hence, unaffeced by axes. 11 This would occur for, for example, in Hamada (1966). 21

24 . CES Preferences The log-linear preference specificaion of he previous secion is he special case of CES preferences for an elasiciy of ineremporal subsiuion equal o one. In his secion, we look a how small changes in he value of he elasiciy of subsiuion in he neighbourhood of one affec he resuls of he las secion. Le u i ' 1 1 & θ j '0 β (c i 1&θ & 1),0<β <1,θ >0, (31) where θ is he reciprocal of he elasiciy of ineremporal subsiuion. As θ 6 1, he above preferences become he logarihmic specificaion of he previous secions. We assume ha θ is arbirarily close o one. 12 In his case, he Euler equaion of he consumer s opimisaion problem becomes c i %1 ' (βr %1 )1/θ c i, ' 0,1,.... (32) Solving he difference equaion (32) yields he household s ime- consumpion as a funcion of is iniial consumpion and he ineres rae c i ' β Π R s s'1 1/θ c i 0, ' 1,2,.... (33) Averaging boh sides of equaion (33) across counries yields Π R s ' (1/β )(c /c 0 ) θ, ' 1,2,.... (3) s'1 12 I is easy o generalise he resuls of Secion 3 o θ < 1, and by coninuiy argumens, o θ wihin a righ-hand side neighbourhood of one. I appears analyically inracable o exend hem o θ sufficienly greaer han one. Here we are concerned wih marginal changes a θ = 1. 22

25 Subsiuing equaions (33) and (3) ino he household s budge consrain yields c i 0 /c 0 ' j '0 β w & τ i c θ / j '0 β c 1&θ, >0. (35) Subsiuing equaions (33) - (35) ino equaion (31) and ignoring consans ha are unimporan o he opimisaion problem yields he indirec uiliy funcion 1 1 & θ j '0 β w & τ i c θ 1&θ j '0 β c 1&θ θ. (36) Subsiuing equaion (3) ino he governmen s budge consrain (equaion (11)) yields j β ŝ i s'0 ' 0, where ŝ i ' [τ i & (φ/2)τ i 2 & g ]/c θ, ' 1,2,.... (37) In he previous secion we showed ha a symmeric equilibrium has consan axes afer period zero. Subsiue τ = τ ino equaions (36) and (37). Then he opimisaion problem of he governmen is o choose τ 0 and τ o maximise [(1 & β)(w 0 & τ i 0 )/c θ 0 % β(w & τi )/c θ ] 1&θ [(1 & β)c 1&θ 0 % βc 1&θ ] θ /(1 & θ) (38) subjec o (1 & β)ŝ i 0 % βŝ i ' 0. (39) The firs-order condiions evaluaed a a symmeric equilibrium imply τ 0 /(1 % φθτ 0 s 0 /N) ' τ/(1 % φθτs/n). (0) 23

26 By equaion (39) and symmery (1 & β)ŝ 0 % βŝ ' 0. (1) The feasibiliy consrain and he opimaliy condiion are represened graphically in Figure 3. In his figure, F k represens he feasibiliy consrain and O k represens he opimaliy consrain for he case of θ = θ k, k = 0,1, where summarised in he following proposiion. θ 0 < θ 1. The properies of he curves are Proposiion 11. The curves F 0 and F 1 are downward sloping and inersec a (τ & 0,τ& ) and on he 5 o line. F 0 lies above F 1 o he lef of (τ & and below he 5 o line; i lies below F 1 0,τ& ) o he righ of (τ & 0,τ& ) and above he 5 o line. The curves O 0 and O 1 are upward sloping and lie below he 5 o line. The curve O 1 lies below O 0. To see he properies of he feasibiliy curves, firs suppose here is no iniial deb. Then increasing θ improves he governmen s radeoff over feasible curren and fuure axes. To see his, suppose ha a = 0 he governmen runs a defici and borrows. Wih lower period-zero axes han fuure axes, consumpion is higher in period zero han in period one. Consumers smooh heir consumpion by lending o governmens a = 0. The higher is θ, he greaer is heir desire o smooh heir consumpion. Thus, he higher is θ, he less cosly is i for he governmen o rade off fuure ax increases for ax cus in period zero. The inuiion is similar if here is an iniial surplus. When R 0 b 0 > 0, he governmen s radeoff is more favourable wih a higher value of θ han wih a lower value of θ if consumpion is higher in he period in which he governmen runs a defici. Wih an iniial posiive sock of deb, however, i is possible for he counry o run a defici in period zero, even hough consumpion is lower in period zero han in period one. This corresponds o he pars of he curves beween he wo inersecing poins. In his case, reducing 2

27 curren axes requires higher fuure axes and he higher is θ he higher are hese fuure axes. Consumpion is made less smooh by he governmen s borrowing and he higher is θ, he more he governmen mus pay o borrow. To see he shape of he opimaliy curves, suppose here is an iniial sock of governmen deb and ha axes are consan across periods. Then he governmens run a defici in he curren period and mus borrow. If firs-period axes were lowered, his would increase curren consumpion and lower he ineres rae ha he governmen mus pay on is deb. If his ineres rae effec is aken ino accoun, hen axes will be lower in he firs period han if he ineres rae effec is no aken ino accoun. This is he argumen of he previous secion. The lower is θ, he less consumers wan o smooh heir consumpion and he less is he ineres rae effec. Thus, he bigger is θ he greaer is he socially opimal reducion in he firsperiod ax below he feasible consan ax. Given Proposiion 11 we have he following. Proposiion 12. Suppose ha N > 1. If here is a posiive iniial sock of deb, hen he iniial ax is oo high relaive o he social opimum and he subsequen axes are oo low relaive o he social opimum. An increase in θ causes he socially opimal value of he iniial ax o fall. 13 As well as considering marginal changes in θ around one, we can consider he polar cases where θ goes o zero and o infiniy. In he limi as θ falls o zero, here is no ineres rae effec as consumers do no care a all abou smoohing heir income. Thus, he socially opimal and uncoordinaed oucomes coincide and axes are smoohed over ime. In he limi as θ goes o infiniy, indifference curves for curren and fuure consumpion become righ angles and only he minimum consumpion maers. If R 0 b 0 >0, cooperaing governmens should borrow marginally 13 As noed, his heorem is for marginal changes a θ'1. I is easy o generalise i o large changes for θ < 1, bu i is no analyically racable o consider large changes above one. 25

28 less han heir ousanding deb in period one and se he curren ax marginally higher han he one ha balances he fuure primary defici. Curren consumpion is hen marginally lower han fuure consumpion so he required gross ineres rae on he borrowing is zero. The fuure ax is hus he one ha balances he fuure primary defici. Minimum consumpion over he curren and fuure can be made arbirarily close o W & G & (φ/2)τ &2.. Producion and capial accumulaion An imporan simplifying feaure of he model is ha varying he iming of axes, and hus fiscal ransfer coss, over ime is he only way o ransfer resources across periods. In equilibrium, ne global privae and public saving is always zero because he good is perishable. Reducing axes in any given period increases he resources available ha period and increases privae consumpion. In our benchmark model, he ineres rae on curren savings falls as a resul of he curren ax cu. In his secion, we allow for producion. Wih capial formaion, resources can be ransferred across periods no only by changing he pah of axes, bu also by capial formaion. Formally, we assume ha he household has CES preferences and ha he single good in he model is boh a capial and a consumpion good. The represenaive households each supply one uni of labour inelasically each period and save boh bonds and he oupu of he curren good in he form of capial. The savings of capial are loaned o he firms o be used in he nex-period s producion process. The firms ransform capial and labour ino oupu via a Cobb-Douglas producion funcion where oupu per uni of labour is f(k) = Ak α, where k is he capial-labour raio, A > 0 and α 0 (0,1). We suppose ha labour is immobile across counries, capial is perfecly mobile and capial depreciaes compleely. Then perfec mobiliy of capial and perfec compeiion imply ha capial-labour raios and wages are equalised across counries and k ' k(r ) ' [A(1 & α)/r ] 1/α. 26

29 A symmeric equilibrium is characerised by he Euler equaion (32), he governmen budge consrain (37) and he global resource consrain, which is now f(k(r )) & G & (φ/2)τ 2 & c & k(r %1 ) ' 0, ' 0,1,.... (2) The model wih capial is far more difficul o analyse han he one wihou. To obain an analyical resul, we resric ourselves o a simple experimen. Imagine ha he world is a a symmeric seady sae wih consan axes and a posiive iniial sock of deb. Can policy makers raise welfare wih a coordinaed symmeric marginal ax cu? Proposiion 13. Suppose counries are a a symmeric seady sae wih consan axes and sricly posiive deb. Then i is possible o increase welfare wih a coordinaed marginal ax cu in he curren period. We show in he proof ha welfare is improved if he curren ax cu is financed wih fuure ax rises ha leave consumpion consan from period one on. Lowering he curren ax and raising fuure axes raises curren consumpion and lowers fuure consumpion, hus lowering he curren ineres rae as in he previous secions. This lowers he cos of servicing he deb and reduces fuure ax collecion coss. To see ha he ineres rae mus fall, suppose ha i did no. Then nex period s marginal produc of capial will rise so curren capial accumulaion falls. Wih lower ax collecion coss and fixed curren oupu, his implies curren consumpion rises. This is inconsisen wih he ineres rae falling in he curren period unless nex period s, and hence every fuure period s, consumpion rises by more han curren consumpion. However, wih lower curren capial accumulaion and higher fuure ax collecion coss his is impossible. Thus, we have a conradicion. Conclusion. We have demonsraed ha, in our baseline model, opimising governmens will perfecly smooh axes if hey have no marke power or if hey have no iniial deb. If counries are large 27

30 enough o affec he world ineres, hen hey will se lower axes in he curren period han in he fuure if hey have a posiive iniial sock of deb. If hey have an iniial sock of credi hey will se higher axes in he curren period han in he fuure. We show ha, relaive o he firs-bes, cooperaive oucome, wih posiive iniial deb, counries se heir curren axes oo high. Thus, relaive o he opimum, iniial budge deficis are oo low. Similarly, if counries are iniial crediors, iniial budge deficis are oo high. We exend our baseline model wih is log-linear preferences, o he case of CES preferences. We show ha a marginal fall in he ineremporal elasiciy of subsiuion increases he deviaion beween he uncoordinaed oucome and he firs-bes oucome; a marginal rise decreases he deviaion. We also consider he case of producion and capial accumulaion. We show ha if here is a seady sae wih consan axes and sricly posiive deb, hen i is possible o increase welfare wih a coordinaed cu in he curren ax. Appendix Proof of Proposiion 1. We show his for > 0; he proof for = 0 is similar. A ax ha maximises s i mus be in [τ -,τ + ] and i mus saisfy ds i /dτi ' (1 & φτ i )/c % φτ i s i (Nc ) ' 0 (3) d 2 s i 2 /dτi ' 2φτ /(Nc ) ds i /dτi & φ/c % φs i /(Nc )<0. () When τ i 0 [τ -,τ + ], hen τ s i $ 0; hence a soluion of (3) mus have τ i $ 1/φ. When τ i = 1/φ, ds i > 0; when = τ +, < 0; hence (3) has a soluion on [1/φ,τ + /dτi τ i ds i /dτi ]. By (3) and (), d 2 s i 2 /dτi = 1/ (τ i c i ) < 0 a his soluion; hence, he soluion is unique and i is a maximum. For τ i 0 [τ -,1/φ], > 0; hence, is increasing below he maximising ax on [τ -,τ + ds i /dτi s i ] and 28

31 decreasing above he maximising ax. By (3) and (), he maximising ax is decreasing in N and goes o 1/φ as N goes o. Proof of Proposiion 2. Suppose o he conrary ha > 0 such ha a leas one counry ses is ax above 1/φ. Wihou loss of generaliy, le counry i be he counry wih he highes ax and le τ i = τ W. Le > 0; he proof for = 0 is similar. Le ˆτ, he average ax in he oher counries in period, be given. Le s W be he value of a τ W, τ * be he ax ha maximises, s * be he s i value of s i a his ax and be he posiive value of a which c = 0. Then > and s W τ ( τ i τ W τ ( 0 (&,s ( ). We have ha s i, maps ono, hus 0 (- ) such ha = s W (&τ (,τ ( ) (&,s ( ) τ R τ (,τ ( s i a τ R. A swich from τ W o τ R does no affec s i ; hence, by (18), i does no require a change in any oher ax. Thus, τ R is preferred o τ W if indirec uiliy (given by (17)) is higher a τ i ' τ R han a τ i ' τ W. We have τ R 2 < τ W 2 ; hence, c R > c W, where c k is c when τ i ' τ k, k ' R,W. Thus, by (17), indirec uiliy is higher a τ i ' τ R han a τ i ' τ W if (W & τ R )/c R > (W & τ W )/c W. By (13), s i W & τ k c k ' 1 & s W % N &1 N φ ˆτ 2 & τ k 2. 2 c (5) k Thus, we need o show (ˆτ 2 & τ R 2 )/c R >(ˆτ 2 & τ W 2 )/c W. By (13), his is rue iff ˆτ < τ which mus be rue as he oher counries have lower axes han counry i. As he counry wih he highes ax canno se is ax on he wrong side of he ne ax revenue curve, neiher can any oher counry. Proof of Proposiion 3. Suppose > 0. By (21), m # unless τ s < 0. By Proposiion 1, τ 0 ]- τ,τ (N ]; hence, m # unless τ 0 ]0, τ & ]. In his case, m < 0 if L(τ ):' W & G % (φ/2)τ 2 - (φ 2 /2)τ 3 & φgτ > 0. The funcion L has an inerior minimum a τ iff τ such ha LN(τ) = τ - (3φ/2)τ 2 - G = 0 and LO(τ) = 1-3φτ > 0. If 1-6φG < 0, hen no such τ exis and L(0) = L(τ - ) = 29

32 W - G > 0 ensures he proposiion holds. If 1-6φG $ 0, hen an inerior minimum exiss. I is sufficien o show ha L is sricly posiive a his poin. Using (3φ/2)τ 2 = τ - G, we have L(τ) = W - G + (φ/2)τ(τ - φτ 2-2G) = (1/3)[3(W - G) + (φ/2)τ(τ - G)] = (1/9)[9W - 10G + (1-6φG)τ]. Assumpion (19) ensures 9W > 10G; hence his is rue. The proof for = 0 is similar. Proof of Proposiion. Suppose o he conrary ha u $ 0 such ha, n u < 0. By Proposiions 1 and 2, τ u < τ & (or τ 0 < τ & 0 if u = 0). Thus, s i u < 0 and here mus be some ime v where s i v >0. By Proposiions 1 and 2, τ v 0 (τ &,τ (N ] (or τ 0 0 (τ & 0,τ(N 0 ] if v = 0) and n v >0. Suppose he governmen of counry i were o lower τ i u marginally, changing τ i v o saisfy (18). Then dτ i v = & β u&v (n u /n v )dτ i u < 0. By marginally lowering axes in boh periods, Proposiion 3 ensures uiliy rises. Thus τ u canno be par of an equilibrium. This is a conradicion. Proof of Proposiion 5. Suppose o he conrary ha τ u < τ v, u,v $ 1. If he counry-i governmen marginally raises τ u, changing τ v o saisfy is budge consrain, hen du i = β u m u dτ i u + β v m, where d =. Then du i v dτ i v τ i v & β u&v (n u /n v )dτ u = β u (m u - m v n u /n v )dτ i u. By Proposiion 1, n u > 0; hence, counry i defecs from he equilibrium if m u /n u > m v /n v. This is rue if m /n is sricly decreasing in τ. By (21) and (22), his is rue if τ /(1 + φτ s /N) is rising in τ. This is rue if 1 > (φ τ 2 /N) ds /dτ. Proposiion and (18) ensure ds /dτ > 0 in an equilibrium. Thus, if τ < 0 he resul holds. Suppose τ > 0. We show he resul holds when > 0; he proof for = 0 is similar. The resul is rue if 1 > (φτ 2 /N) (W - G - φτ w + φτ2 /2)/c 2 on [0,τ (1 ] g [0,τ (N ]. If he lef-hand side is negaive, his is rue. If i is posiive, i is rue if i is rue for N = 1. This is rue if G(τ) :=( τ 2 & τ 2 ) 2 & 2τ 2 ( τ 2 & 2Wτ % τ 2 ) > 0 œτ 0 [0, τ (1 ]. For G o have an inerior minimum on [0, τ (1 ] requires (3W - 9W 2 & 8 τ 2 )/2 # τ (1. By he definiion of τ (1, his is impossible. G(0) = τ > 0 and G(τ (1 ) ' ( τ 2 & τ (12 ) 2 > 0; hence G > 0 œτ 0 [0, τ (1 ]. Proof of Proposiion 6. Assumpion (19) rules ou negaive axes in boh periods. Rearranging (28) yields N(τ 0 & τ) ' φττ 0 (s 0 & s). Subsiuing in s 0 /s = -β/(1 - β) (from (29)) yields 30