NBER WORKING PAPER SERIES CROSS-BORDER TAX EXTERNALITIES: ARE BUDGET DEFICITS TOO SMALL? Willem H. Buiter Anne C. Sibert

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1 NBER WORKING PAPER SERIES CROSS-BORDER TAX EXTERNALITIES: ARE BUDGET DEFICITS TOO SMALL? Willem H. Buier Anne C. Siber Working Paper hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA November 2003 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 for helpful commens. The views expressed herein are hose of he auhors and no necessarily hose of he Naional Bureau of Economic Research by Willem H. Buier and Anne C. Siber. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Cross-Border Tax Exernaliies: Are Budge Deficis Too Small? Willem H. Buier and Anne C. Siber NBER Working Paper No November 2003 JEL No. E62, F2, H21 ABSTRACT 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. Willem H. Buier European Bank for Reconsrucion and Developmen One Exchange Square London, EC2A 2JN, UK and NBER buierw@ebrd.com Anne C. Siber Deparmen of Economics Birkbeck College Universiy of London Male Sree Bloomsbury, London WC1E 7AX, UK 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? The issue figures prominenly in he debae abou he meris of he European Union s Sabiliy and Growh Pac and is par of he debae on he meris of G3 policy coordinaion. Two broad classes of cross-border public deb exernaliies are recognised in he lieraure. The firs are exernaliies associaed wih eiher he occurrence of sovereign deb defaul or wih acions underaken by he debor counry or by ohers o preven sovereign defauls. The second are cross-border exernaliies associaed wih he ransmission of naional public deb policies hrough heir effec on he global risk-free real ineres rae; i is his second ype of exernaliy ha is he focus of his paper. We provide an explicily ineremporal equilibrium model wih opimising households and governmens, in which public deb and he ineremporal budge consrain of he governmen provide an explici link beween ax decisions oday and ax decisions omorrow. Such ineremporal models are analyically difficul, especially if hey do no exhibi firs-order Ricardian equivalence or deb neuraliy. The combinaion of budge consrains (where changes in asse socks ener addiively) and equilibrium deerminaion of ineremporal relaive prices (which ener muliplicaively wih asse socks) means ha non-lineariies are inrinsic. This has led us o specify he simples possible supply side for he naional economies (a perishable endowmen echnology), simple household preferences, a represenaive infinie-lived consumer wih log-linear preferences and a simple source of Ricardian non-equivalence, 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. The focus of his paper is on real ineres rae cross-border spillovers ha occur in he 1

4 absence of sovereign defaul risk and wihou sraegic ineracions beween a naional fiscal auhoriy and a naional or supranaional moneary auhoriy. Our formal model is ha of a nonmoneary 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 he axes of he oher governmens as given. Thus, here is commimen bu no inernaional cooperaion. We show his non-cooperaive behaviour resuls in inefficien global equilibria. Households are infinie-lived and here are no overlapping generaions feaures ha could cause problems of dynamic inefficiency. Each counry s supply side is a simple endowmen economy wih a single perishable good. Resources are always fully uilised. There is perfec inernaional mobiliy of financial capial. Inernaional ransmission of naional fiscal policy is only hrough ineres raes. The assumpion ha prevens our model from exhibiing Ricardian equivalence, is he presence of increasing and sricly convex ax adminisraion and collecion coss. 1 Taxes are lump-sum; heir incidence can no be alered hrough changes in privae behaviour, bu because of he sric convexiy of he ax adminisraion and collecion 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. In he formal model, hese adminisraion and collecion coss are all locaed in he public secor. When axes are negaive (governmen subsidies or ransfers) real resource coss resul from privae ren-seeking behaviour. Exending he model o include compliance coss 1 Tax sysems have boh adminisraive and enforcemen coss borne by he public secor and compliance, avoidance and evasion coss born by he privae secor. To keep he noaion simple, we have chosen o model he fiscal ransfer cos here as an adminisraive cos, borne solely by he public secor. 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 he compliance coss o be abou 10 cens per dollar colleced. 2

5 borne by he privae ax payers, eiher o comply wih or o avoid or evade axes, would add noaional complexiy wihou changing our qualiaive conclusions. Wihou hese ax adminisraion and collecion 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. If he represenaive agen assumpion were replaced by ha of overlapping generaions wihou a beques moive, alernaive rules for financing a given public spending programme would give rise o pure pecuniary exernaliies if here were no ax adminisraion and collecion coss. Tha is, even wih symmeric counries, here could be disribuional effecs beween generaions, bu, as long as dynamic inefficiency does no occur, any feasible sequences of lumpsum axes and deb suppor equilibria ha are Pareo efficien. 2 In he single counry special case of our model, he presence of ax adminisraion and collecion coss does no give rise o an inefficiency, as long as one assumes ha, in he counerfacual command economy, resource ransfers beween he privae and public secor would be subjec o he same real fiscal ransfer coss as in our marke economy. Inefficiency comes when here is more han one counry and each counry influences he choice se of he oher counries in a way ha is no adequaely refleced in marke prices. Wihou cosly ax adminisraion and collecion (or convenional disorionary axes), 2 See Buier and Klezer (1991). If here is dynamic inefficiency (he real ineres rae is almos always below he real growh rae), hen fiscal policy ha causes redisribuion from he young can lead o a Pareo improvemens. Wih asymmeric counries, alernaive defici financing policies would, in general, have inernaional as well as inergeneraional disribuional implicaions. 3

6 alernaive governmen financing rules eiher have no exernaliies associaed wih hem (in models such as ours, which would exhibi deb neuraliy) or only purely pecuniary exernaliies ( in OLG models). These are exernal effecs ha, firs, are ransmied only hrough a marke price - he global real ineres rae and, second, do no have efficiency implicaions. 3 Obviously, hese pecuniary exernaliies will have disribuional consequences if some counries are ne lenders while ohers are ne borrowers. Disribuional effecs from policies ha change he global ineres rae need no have efficiency implicaions. I is an implicaion of he firs welfare heorem, ha all compeiive equilibria suppored by differen lump-sum ax-ransfer and borrowing schemes are Pareo efficien. This is rue even if a counry is large in he world capial marke and explois is monopoly power. All ha is required is ha axes and ransfers be lump-sum. Because our model has a represenaive agen and axes are lump-sum, here would be deb neuraliy wihou fiscal ransfer coss. However, wih sricly convex fiscal ransfer coss, here will be ineres rae spillovers. Our naional economies are symmeric and, even when here is cosly ax adminisraion and collecion causing alernaive governmen financing policies o affec he global rae of ineres, here are no disribuional consequences. Changes in he global ineres rae brough abou by domesic ax policy have efficiency effecs because hey affec he ineres bill faced by governmens wih ousanding deb. All governmens mus mee heir ineremporal budge consrain and changes in he ineres bill require changes in axes, now and/or in he fuure. We assume ha naional governmens maximise he welfare of heir represenaive 3 Again, in he case of OLG models, we rule ou dynamically inefficien equilibria. Wih opimising governmens, dynamic inefficiency does no occur in equilibrium (Buier and Klezer (1991)). See he previous foonoe.

7 naional consumer and do no inernalise any fiscal ransfer coss hey may impose on foreign naions. In he presence of fiscal ransfer coss, a naional governmen s financing decision ha raises he world ineres rae inflics a negaive exernaliy on he res of he world if oher governmens have posiive socks of deb ousanding. Thus far our model suppors he convenional wisdom. Where our model depars radically from convenional wisdom is hrough he mechanism by which differen governmen financing choices affec ineres raes. The convenional wisdom associaes policies ha resul in larger governmen deficis wih financial crowding ou. Tha is, given public spending, larger deficis raise ineres raes. In our horoughly neo-classical ineremporal model he opposie is rue. Lower axes and larger deficis early on resul in a lower global rae of ineres. We show ha if governmens are oo small o affec he global ineres rae, hey minimise he coss of collecing axes by smoohing he axes over ime. If hey are able o influence he ineres rae and counries have a posiive iniial sock of deb, hen hey se a lower ax in he iniial period han in subsequen periods. This is because lower ax disorions in he iniial period han in laer periods imply ha aggregae consumpion is higher in he iniial period han in laer periods. Thus, he ineres rae a which he counry can borrow in he iniial period is lower han wih perfecly smooh axes and his lowers he deb service on is ousanding deb and, hence, fuure ax collecion coss. Relaive o he global (cooperaive) opimum, non-cooperaive 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 he issuance of 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 non-cooperaive equilibrium, counries do no ake 5

8 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 raher small fish in he global financial pond. Our analysis is more relevan o ineracion beween he Unied Saes, he European Union as a whole and, possibly, Japan and China. There are few papers analysing he welfare economics of inernaional ineres-rae spillovers from naional ax and borrowing sraegies of naional governmens using opimising sequenial general equilibrium models. Hamada (1986) and Buier and Klezer (1991) sae he problem bu do no develop he excessive deficis bias issue. Kehoe (1987,1989) considers he welfare economics of inernaional fiscal policy cooperaion, bu in a model where governmen budges are always balanced. In secion 2 we presen he model. In secion 3 we exend he model o consider small l variaions 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 consan elasiciy of ineremporal subsiuion 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 6

9 privae radeable, non-sorable consumpion good and each governmen purchases an exogenous amoun of he privae good o produce a public good. The governmens finance heir purchases by issuing deb or by axing heir residen households. We assume ha he ax sysem is cosly o adminiser; he governmen uses up real resources collecing axes. All savings is 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 The households The counry-i household, i = 1,...,N, has preferences over is consumpion pah given by u i ' j '0 M lnc i, (1) where c i is is period- consumpion and M 0 (0,1) is is discoun facor. The household s period-, = 0,1,..., budge consrain is c i % a i %1 ' W & Ni % 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, is (one plus) he ineres rae beween period - 1 and period, W is he household s per-period endowmen of he good and N i is i 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 is 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, R 7

10 lim 6 a i %1 / P 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 & N i )/P R s ' j s'0 '0 c i / P s'0 R s, () The household chooses is pah of asse holdings and is consumpion sream o maximise is uiliy funcion (1) subjec o is ineremporal budge consrain (). The soluion o is problem saisfies equaion () and he Euler equaion %1 ' MR %1 c i, ' 0,1,.... (5) c i 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 ' M P 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 0 ' (1 & M) R 0 a 0 % W & Ni 0 % j c i '1 (W & N i )/ P s'0 R s. (7) Subsiuing equaion (6) ino equaion (1) yields he household s indirec uiliy as a 8

11 funcion of iniial consumpion and he ineres facors u i ' lnc i 0 % (1 & M) j '1 M ln P 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 N i & (Q/2)N i 2 % b i %1 ' G % R b i, (9) where b i is he governmen s ousanding deb a he sar of period, G > 0 is is per-period purchase of he consumpion good. The amoun of resources used up in collecing a ax of N (or adminisering a surplus of - N) is given by and (Q/2)N 2, where Q > 0. The governmen s iniial deb (or credi, if negaive), b 0, is given. We resric he parameers of he model so ha saisfying equaion (9) is feasible; hese 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 / P 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: 9

12 N i 0 & 2 (Q/2)Ni 0 & g 0 % j s'0 where g :' [N i & (Q/2)N i 2 G % R 0 b i 0, ' 0 G, >0. & g ]/PR 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. The global resource consrain requires ha oal household consumpion plus oal governmen purchases of he good plus oal resources used up paying he ax equals oal endowmens. Thus, c ' W & G & Q N 2N j N 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 gives he period- real ineres facor as a funcion of aggregae consumpion in periods 0 and period. P R s ' s'1 c M c 0, ' 1,2,... (1) Equaions (13) and (1) imply ha in equilibrium, he ime- ineres facor is solely a funcion of ime-0 and ime- axes. 10

13 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 & Mlnc 0 % (1 & M) j '1 M lnc. (15) Subsiuing equaions (12) and (1) ino equaion (7) gives he household s iniial consumpion as a funcion of he pahs of axes. The predeermined value of iniial governmen deb eners as a parameer. c i 0 ' (1 & M)c 0j M (w & N i )/c, '0 where w :' W % R 0 b 0, ' 0 W, >0. (16) Subsiuing equaion (16) ino he indirec uiliy funcion (equaion (15)) yields u i ' ln j '0 M (w & N i )/c % (1 & M) j '0 M lnc. (17) Subsiuing he global resource consrain (equaion (13)) ino equaion (17) would allow he household s indirec uiliy o be expressed solely as a funcion of he pahs of axes in he N counries. Subsiuing equaion (1) ino equaion (11) yields he counry-i governmen s budge consrain as a funcion of is own axes and aggregae consumpion 11

14 s i j M s i '0 ' 0, where ' Ni & (Q/2)N i 2 & g, ' 0,1,.... c (18) Subsiuing equaion (13) ino equaion (17) would allow he budge consrain o be expressed solely as a funcion of he pahs of he axes in he N counries. 2. Taxes and revenues Wih ax revenues (ne of collecion coss) bounded in our model and wih exogenous real public consumpion spending and endowmens, we mus impose furher resricions on he parameer space o ensure he exisence of an equilibrium. Thus, we assume max{g,g 0 } # 1/(2Q), w 0 $ 1/Q, W & G >2/Q, g 0 >0. (19) The ne ax revenue funcion, N & (Q/2)N 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 and subsidies. I is maximised a N = 1/Q and (ne) revenue equals 1/(2Q) a his poin. The firs inequaliy in assumpion (19) ensures ha i is possible o finance expendiures of G and g 0. This implies ha an equilibrium wih no governmen borrowing from ime-1 on is feasible. The ime-0 budge is balanced wih a ax of N & 0 := (1-1 & 2Qg 0 )/Q or a ax of N % 0 := (1 + 1 & 2Qg 0 )/Q. Likewise, he ime-, > 0, budge is balanced wih axes of N & := (1-1 & 2QG)/Q or N % := (1 + 1 & 2QG)/Q. The axes N & 0 and N - are on he righ, or upward-sloping par of he ime-0 and ime-, > 0, ne ax revenue curves, respecively. he axes N % 0 and N + are on he wrong or downward-sloping pars of he ime-0 and ime-, > 0, ne ax revenue curves, respecively. There is a convenional governmen budge surplus in counry i in period 0 if and only if N i 0 0 [ N& 0, N % 0 ] and here is a primary (ha is, ne of ineres paymens) surplus in period 12

15 if and only if N i 0 [ N &, N % ]. By he resource consrain, equaion (13), he upper bound on feasible axes in a symmeric equilibrium is YN := (2/Q)(W & G). Assumpion (19) implies ha he axes N % 0 andn % are sricly less han YN, and hence are feasible. By equaions (13) and (18), a symmeric equilibrium wih consan axes has axes of (1 ± 1 & 2Q[G % (1 & M)R 0 b 0 ])/Q. By assumpion (19), his is feasible. We allow for negaive axes, or subsidies. In his case he collecion cos is viewed as he cos of adminisering 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, ' 0,1,2,..., in equaion (18) can be inerpreed as he period-0 value of he governmen s ime- budge surplus (or defici, if negaive), divided by c 0. For period 0, his surplus is he oal surplus and for periods > 0 i is he primary surplus. We will refer o 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 1/Q, he ax ha maximises ne revenue. To see his suppose ha N i ' 1/Q. Marke power gives a counry he abiliy o influence he global ineres facor. Holding oher axes posiive, a marginal increase in N i causes aggregae period- consumpion o fall. As ne revenues are insensiive o axes a 1/Q, hey are unaffeced by a marginal increase in he ax above his poin. Thus, a marginal increase in o rise. N i above 1/Q causes he discouned ime- surplus We have he following resul. Proofs of his and all oher proposiions are in he 13

16 Appendix. Proposiion 1. Given N 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 N i 1/Q. If > 0, he maximising ax is an elemen of [1/Q,N % ] and is increasing (decreasing) below (above) he maximising ax on [N &,N % ]. If = 0, he s i s i 0 maximising ax is an elemen of [1/Q,N % 0 ] and is increasing (decreasing) below (above) he maximising ax on [N & 0,N% 0 ]. Denoe he ax ha maximises s i when axes are symmeric and here are N counries by N (N 0 if = 0 and by N (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 & YN 2 W & W 2 & YN 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 {N 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 1

17 Mu i MN i ' M m i, where m i :' & 1 % QNi N w & N i c c j M w s s & Ns s'0 c s & (1 & M)QNi Nc, ' 0,1,.... (20) Le axes be consan across counries. Then equaion (16) implies [ '0 M (w & N )/c = 1/( 1 - M). Equaion (13) and he definiion of s (in equaion (18)) implies (w & N )/c = 1 & s. These resuls and equaion (20) imply ha wih symmeric axes m i ' m '& 1 & M c 1 % QN 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, an increase in ime- axes has he direc effec of lowering consumer income in period and his ends o lower welfare. Suppose N > 0. Then a ax increase lowers available global resources in period because of he higher collecion coss. Thus, i raises he relaive price of consumpion in period. This ineres rae effec has a posiive effec on welfare if consumers lend o he governmen in period (s < 0). The direc dominaes he ineres rae effec; an increase in he ime- ax lowers wihin-period welfare a ime. Suppose N < 0. Then a ax increase increases available resources in period because of he lower adminisraion coss. Thus, he relaive price of consumpion in period falls. As consumers mus be lending o he governmen in his case, he ineres rae effec also ends o lower welfare and marginal uiliy mus be negaive. By equaions (13) and (18), he marginal increase in he discouned value of he budge 15

18 surplus (he lef-hand side of equaion (18)), ha resuls from a marginal increase in he ime- ax when axes are idenical across counries is M n, where n :' (1/c )(1 & QN % QN 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. 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 N i ' N i, c i ' c i, and s i ' s, > 0. Then equaions (17) and (18) imply ha he governmen maximises ln[(1 & M)(w 0 & N i 0 )/c 0 % M(W & Ni )/c] % (1 & M)lnc 0 % Mlnc (23) subjec o B i :' (1 & M)s i 0 % Ms i ' 0. (2) By equaions (13) and (23), he marginal uiliies associaed wih N i 0 and N i are m i 0 '& 1 & M c 0 1 & QNi 0 w 0 & N i 0 N c 0 C % QN 0 N, m i 1 '& M c 1 & QNi w & N i N c C % QN N, (25) where C :' (1 & M)(w 0 & N i 0 )/c 0 % M(W & Ni )/c, respecively. By equaions (13) and (2), he marginal increases in B i associaed wih increases in N i 0 16

19 and N i are n i 0 ' 1 & M c 0 1 & QN i 0 % QN i 0 s i N, n i 1 ' M c 1 & QNi % QNi s i N, (26) respecively. If he counries ac symmerically, hen equaion (16) implies ha C = 1. Equaion (13) implies ha (W - N 0 )/c 0 = 1 & s 0 and (W & N)/N ' 1 & s. These resuls and equaions (25) and (26) imply m 0 '& 1 & M c 0 1 % QN 0 s 0 N n 0 ' 1 & M 1 & QN c 0 % QN 0 s 0 0 N, m 1 '& M c 1 % QNs N, n 1 ' M c 1 & QN % QNs N. (27) The firs-order condiions for an equilibrium imply m 1 /m 0 = n 1 /n 0. This and equaions (27) imply 5 N 0 /(1 % QN 0 s 0 /N) ' N/(1 % QNs/N). (28) Symmery and equaion (2) imply (1 & M)s 0 % Ms ' 0. (29) The second-order condiion requires ha he bordered Hessian marix associaed wih he opimisaion problem has a sricly posiive deerminan. This requires 5 In deriving equaion (28), boh sides were divided by Q. If Q = 0, he iming of axes is irrelevan as long as he governmen saisfies is ineremporal budge consrain. 17

20 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 N i 0 and N i, respecively, when axes are symmeric and where n 0 and n 1 are he derivaives of n i (as given by equaion (26)) wih respec o N i 0 and N i, respecively, when axes are symmeric. I is sraighforward, bu exceedingly edious o demonsrae ha symmeric axes which saisfy equaions (28) and (29) also saisfy equaion (30). 6 Definiion 1. A symmeric equilibrium is a pair of axes {N 0,N} such ha (i) he feasibiliy condiion (29) is saisfied (ii) he opimaliy condiion (28) is saisfied. We firs esablish ha axes are always posiive. Proposiion 6. An equilibrium canno have subsidies. ( N 0 <0or N <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 are downward sloping; an increase in he fuure ax allows he governmen o reduce he curren ax and sill balance is budge. 7 The curve represening an iniial deb of zero, F 0, goes hrough he poin ( N &,N & ), labelled A. The curve corresponding o a sricly posiive sock of iniial deb, F +, lies above he curve wih zero deb and he curve corresponding o a negaive sock of iniial deb, F -, 6 Deails available on reques. 7 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 lies below i. Boh curves pass hrough he poin poin lies above A; wih a negaive iniial sock of deb i lies below A. 19 (N & 0,N& ). Wih a posiive iniial sock of deb, his 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 O NN % O NO % labelled and represen he case sricly posiive iniial deb and N = NN and N = 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 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; O N) % and O N O % lie above he 5 o line. (iv) O N) lies above % when N > N - and N ; lies below when N < N - 0 < N & 0 O N) O NO & and N 0 > N & 0. % O N O 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 axes, and hence ax disorions, 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 ax collecion coss hrough is influence on he global rae of ineres. If i has an iniial sock of deb, 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 on governmen deb beween periods zero and one. Thus, is required ax revenue falls. Likewise, if he governmen 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 on governmen savings beween periods zero and one. & & &

22 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. 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, N > N 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, N > N 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 he value of he iniial deb is zero, hen here is complee ax smoohing. If counries have some marke power (N < ), hen he 20

23 iniial ax is sricly less (greaer) han he subsequen ax if here is a sricly posiive (negaive) sock of iniial deb. When counries have no marke power, we derive he same resul as Barro (1979). Taxes resul in resource losses, here because hey are cosly o collec and in Barro (1979) because hey are disorionary. 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 an 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. 8 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.. CES Preferences The log-linear preference specificaion of he previous secion is a special case of CES 8 If he governmen begins wih a sricly posiive (negaive) iniial sock of deb, hen Proposiion 9 says ha he iniial ax is lower (higher) han subsequen axes. This implies ha he governmen eners period one wih a sricly posiive (negaive) sock of iniial deb. Thus, if he governmen could re-opimise, beginning 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 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. 21

24 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 1 effec he resuls of he las secion. Le u i ' 1 1 & ] j '0 M (c i 1&7 & 1),0<M <1,] >0, (31) where ] is he reciprocal of he elasiciy of ineremporal subsiuion. As ] 6 1, he above preferences become he logarihmic specificaion of he previous secions. We assume ha ] is arbirarily close o one. 9 Given he above preferences, he Euler equaion of he consumer s opimisaion problem becomes %1 ' (MR %1 )1/] c i, ' 0,1,.... (32) c i Solving he difference equaion (32) yields he household s ime- consumpion as a funcion of is iniial consumpion and he ineres rae c i ' M P R s s'1 1/] c i 0, ' 1,2,.... (33) Averaging boh sides of equaion (33) across counries yields 9 I is easy o generalise he resuls of he las secion o ] < 1, and by coninuiy argumens, o ] wihin a righ-hand side neighbourhood of one. I appears analyically inracable o exend hem o ] sufficienly greaer han one. In his secion we are concerned wih marginal changes a ] = 1. 22

25 P R s ' 1 s'1 M c c 0 ], ' 1,2,.... (3) Subsiuing equaions (33) and (3) ino he household s budge consrain yields c i 0 c 0 ' j '0 M w & N i c ] / j '0 M c 1&], >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 & ] j '0 M w & N i c ] 1&] j '0 M c 1&] ]. (36) Subsiuing equaion (3) ino he governmen s budge consrain (equaion (11)) yields j M ŝ i s'0 ' 0, where ŝ i ' Ni & (Q/2)N i 2 c ] & g, ' 1,2,..... (37) In he previous secion we demonsraed ha a symmeric equilibrium mus have consan axes afer period zero. Subsiue N = N ino equaions (36) and (37). Then he opimisaion problem of he governmen is o choose N 0 and N o maximise [(1 & M)(w 0 & N i 0 )/c ] 0 % M(W & Ni )/c ] ] 1&] [(1 & M)c 1&] 0 % Mc 1&] ] ] /(1 & ]) (38) subjec o (1 & M)ŝ i 0 % Mŝ i ' 0. (39) 23

26 The firs-order condiions evaluaed a a symmeric equilibrium imply N 0 /(1 % Q]N 0 s 0 /N) ' N/(1 % Q]Ns/N). (0) By equaion (39) and symmery (1 & M)ŝ 0 % Mŝ ' 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 ] = ] k, k = 0,1, where summarised in he following proposiion. ] 0 < ] 1. The properies of he curves are Proposiion 11. The curves F 0 and F 1 are downward sloping and inersec a (N & 0,N& ) and on he 5 o line. F 0 lies above F 1 o he lef of (N & 0,N& ) and below he 5 o line; i lies below F 1 o he righ of (N & 0,N& ) 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 ] would make he governmen s radeoff over feasible curren and fuure axes more favourable. To see he inuiion suppose ha he governmen is running a ime-zero defici. I finances his defici by borrowing. 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 in period zero. This higher is ], he lower is he ineremporal elasiciy of subsiuion and he greaer is heir desire o smooh heir consumpion. Thus, he higher is ], he less cosly is i for he governmen o rade off fuure ax increases for ax cus in period zero. The inuiion is similar for he case of an iniial surplus. When R 0 b 0 > 0, he governmen s radeoff is more favourable wih a higher value of ] 2

27 han wih a lower value of ] if consumpion is higher 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 be running 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 curren axes requires higher fuure axes and he higher is ] he higher are hese fuure axes. Consumpion is made less smooh by he governmen s borrowing and he higher is ], 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 ], he less consumers wan o smooh heir consumpion and he less is he ineres rae effec. Thus, he bigger is ] 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 7 causes he socially opimal value of he iniial ax o fall. 10 As well as considering marginal changes in ] around one, we can consider he polar cases 10 As noed, his heorem is for marginal changes a ]'1. I is sraighforward o generalise i o large changes for ] < 1, bu i is no analyically racable o consider large changes above one. 25

28 where ] goes o zero and o infiniy. In he limi as ] 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 ] 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 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 & (Q/2)N &2.. Producion and capial accumulaion An imporan simplifying feaure of he model is ha varying he paern of axes over ime (and hereby changing he paern of real fiscal ransfer coss over ime) is he only way o ransfer real resources beween periods. Households and governmens make saving decisions, bu in equilibrium, ne global saving (privae plus public) is always zero and invesmen is always zero for each individual counry because real goods are perishable. Reducing axes in any given period will increases he real resources available ha period. In equilibrium, privae consumpion in ha period will herefore increase. In our benchmark model, he ineres rae on curren savings falls as a resul of he curren ax cu. In his secion, we exend he model o allow for producion. When capial formaion is added o he model, real resources can be ransferred beween periods no only by shifing he paern of axaion (and of fiscal ransfer coss) bu also by capial formaion. Formally, we assume ha he household has he CES preferences of he las secion and ha he single good in he model is boh a capial and a privae and public consumpion good. 26

29 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 wih a capial share of oupu of _ 0 (0,1). Then if k is he capial-labour raio, he oupu per uni of labour is f(k) = Ak 8, where A > 0. We suppose ha labour is immobile across counries, physical 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 & _)/R ] 1/_. 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 & (Q/2)N 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. The inuiion is ha 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 27

30 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 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, co-operaive 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, which feaures 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 mus be in [N -,N + ] and i mus saisfy s i 28

31 ds i dn i ' 1 & QNi c % QN i s i ' 0 (3) Nc d 2 s i /dni 2 i ' 2QN ds Nc dn i & Q c % Qs i <0. () Nc When N i 0 [N -,N + ], hen N s i $ 0; hence a soluion of (3) mus have N i $ 1/Q. When N i = 1/Q, ds i /dni > 0; when N i = N +, ds i /dni < 0; hence (3) has a soluion on [1/Q,N + ]. By (2) and (3), d 2 s i 2 /dni = 1/ (N i c i ) < 0 a his soluion; hence, he soluion is unique and i is a maximum. For N i 0 [N -,1/Q], ds i /dni > 0; hence, s i is increasing below he maximising ax on [N -,N + ] and decreasing above he maximising ax. I is obvious from (3) and () ha he maximising ax is decreasing in N and goes o 1/Q as N goes o. Proof of Proposiion 2. Suppose o he conrary ha > 0 such ha a leas one of he counries ses is ax on he wrong side of he ne ax revenue curve. Wihou loss of generaliy, suppose he counry wih he highes ax is counry i and le N i = N W. We suppose > 0; he argumen for = 0 is similar. Le ˆN, he average ax in he oher counries in period, be given. Le s W be he value of a N W. Le N * be he ax ha maximises, le he value of a his ax be s *, le he s i posiive value of N i for which c = 0 be YN (. Then N W > N ( and s W 0 (&,s ( ). We have ha s i, maps (&N (,N ( ) ono (&,s ( ); hence, N R 0 (- YN (,N ( ) such ha s i = s W a N R. By equaion (18), a swich from N W o N R has no effec on he governmen s ineremporal budge consrain and, hence, does no require a change in any oher ax. Thus, he governmen prefers N R o N W if indirec uiliy (given by (17) is higher when N i ' N R han when N i ' N W. 29 s i s i

32 We have N R 2 < N W 2 ; hence, c R > c W, where c k is c when N i ' N k, k ' R,W Thus, by (17), indirec uiliy is higher when N i ' N R han when N i ' N W if (W & N R )/c R > (W & N W )/c W. By (13), W & N k c k ' 1 & s W % N &1 N Q ˆN 2 & N k 2. 2 c (5) k Thus, we need o show ha (ˆN 2 & N R 2 )/c R >(ˆN 2 & N W 2 )/c W. By (13), his is rue iff ˆN < YN which mus be he case 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, hen neiher can any oher counry. Proof of Proposiion 3. Suppose > 0. By (21), m canno be posiive unless N s < 0. By Proposiion 1, N 0 ]- YN,N (N ]; hence, m canno be posiive unless N 0 ]0, N & ]. In his case, m < 0 if L(N ):' W & G % (Q/2)N 2 & (Q 2 /2)N 3 & QGN > 0. The funcion L has an inerior minimum a N iff N such ha LN(N) = N - (3Q/2)N 2 - G = 0 and LO(N) = 1-3QN > 0. If 1-6QG < 0, hen no such N exis and L(0) = L(N - ) = W - G > 0 ensures he proposiion holds. If 1-6QG $ 0, hen an inerior minimum exiss. I is sufficien o show ha L is sricly posiive a his poin. Using (3Q/2)N 2 = N - G, we have L(N) = W - G + (Q/2)N(N - QN 2-2G) = (1/3)[3(W - G) + (Q/2)N(N - G)] = (1/9)[9W - 10G + (1-6QG)N]. 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 Proposiion 1 and 2, N u < N & (or N 0 < N & 0 if u = 0). Thus, s i u < 0 and here mus be some period v where s i v >0. By proposiions 1 and 2 N v 0 (N &,N (N ] (or N 0 0 (N & 0,N(N 0 ] if v = 0) and n v >0. Suppose ha he governmen of counry i were o lower N i u marginally and o change N i v o saisfy is ineremporal budge consrain. Then dn i v = & M u&v (n u /n v )dn i u < 0. By marginally lowering axes in boh periods, 30

33 Proposiion 3 ensures ha uiliy rises. Thus conradicion. N u canno be par of an equilibrium. This is a Proof of Proposiion 5. Suppose o he conrary ha N u < N v, u,v $ 1. If he governmen of counry i marginally increases N u, changing N v o saisfy is budge consrain, hen du i =M u m u dn i u + M v m, where d =. Then du i v dn i v N i v & M u&v (n u /n v )dn u = M u (m u - m v n u /n v )dn i u. By Proposiion 1, n u > 0; hence, counry i would defec from he equilibrium if m u /n u > m v /n v. This is rue if m /n is sricly decreasing in N. By (21) and (22), his is rue if N /(1 + QN s /N) is increasing in N. This is rue if 1 > (Q 2 /N) ds /dn. Proposiion and (18) ensure ha ds /dn > 0 in an equilibrium. Thus, if N < 0 he resul mus hold. Suppose N > 0. We show he resul holds when > 0; he proof for = 0 is similar. The resul is rue if 1 > (QN 2 /N) (W - G - QN w + QN2 /2)/c 2 on [0,N (1 ] g [0,N (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(N) :=(YN 2 & N 2 ) 2 & 2N 2 (YN 2 & 2WN % N 2 ) > 0 œn 0 [0, N (1 ]. For G o have an inerior minimum on [0, N (1 ] requires (3W - 9W 2 & 8YN 2 )/2 # N (1. By he definiion of N (1, his is impossible. G(0) = YN > 0 and G(N (1 ) ' (YN 2 & N (12 ) 2 > 0; hence G > 0 œn 0 [0, N (1 ]. Proof of Proposiion 6. Assumpion (19) rules ou negaive axes in boh periods. Rearranging (28) yields N(N 0 & N) ' QNN 0 (s 0 & s). Subsiuing in s 0 /s = -M/(1 - M) (from (29)) yields N(1 & M)(N & N 0 ) ' QNN 0 s. (6) Suppose. Then (19) and (29)imply s > 0. The lef-hand side of (6) is N >0>N 0 posiive, he righ-hand side is negaive. This is a conradicion. Suppose N 0 >0>N. Then s < 0, he lef-hand side of (6) is negaive and he righ-hand side is posiive. This is a conradicion. Proof of Proposiion 7. Le he righ-hand side of (28) be represened by h(n;n) := N(1 + QNs/N) > 0 for N 0 [0,N *1 ]; he lef-hand side by h 0 (N 0 ;N) := N(1 + QN 0 s 0 /N) > 0 for N 0 0 [0,.] These N (1 0 31