A Note on Public Debt, Tax-Exempt Bonds, and Ponzi Games

Transcription

1 WP/07/162 A Noe on Public Deb, Tax-Exemp Bonds, and Ponzi Games Berhold U Wigger

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3 2007 Inernaional Moneary Fund WP/07/162 IMF Working Paper Fiscal Affairs Deparmen A Noe on Public Deb, Tax-Exemp Bonds, and Ponzi Games Prepared by Berhold Wigger Auhorized for disribuion by Michael Keen July 2007 Absrac This Working Paper should no be repored as represening he views of he IMF The views expressed in his Working Paper are hose of he auhor(s) and do no necessarily represen hose of he IMF or IMF policy Working Papers describe research in progress by he auhor(s) and are published o elici commens and o furher debae By issuing ax-exemp bonds, he governmen can incur deb and never pay back any principal or ineres, even if he economy wihou public deb evolves on a dynamically efficien growh pah The welfare effecs of such a Ponzi ype borrowing scheme are mixed The curren young will unambiguously benefidepending on preferences and he aggregae echnology, also a finie number of subsequen generaions may benefi The welfare of all generaions hereafer, however, will be lower han in he economy wihou public deb JEL Classificaion Numbers: E60, H62, H63 Keywords: Public deb, ax-exemp bonds, capial axaion, Ponzi game Auhor s Address: berholdwigger@wisouni-erlangende

4 2 Conens Page I Inroducion 3 II The Model 4 III The Feasibiliy of a Ponzi Game 6 IV Welfare Implicaions 10 V Conclusion 13 VI VII Appendix 14 References 18

5 3 I Inroducion The Diamond (1965) overlapping generaions model may generae compeiive equilibria in which he growh rae of he labor force exceeds he long-run reurn on capial In hese cases he economy is said o be dynamically inefficien and he governmen can play a so-called Ponzi game Tha is, he governmen can issue bonds and roll over ineres and principal from period o period by perpeually issuing new bonds o render deb service Such a Ponzi game is beneficial as i removes overaccumulaion of capial associaed wih dynamic inefficiency 1 The presen noe demonsraes ha he governmen can even run a Ponzi game if he economy wihou public deb is dynamically efficien This becomes possible when he reurn on privae bonds or equiy is axed and he governmen issues ax-exemp bonds 2 Unlike a radiional Ponzi game, however, he welfare effecs of a Ponzi game based on he issuance of ax-exemp bonds are raher mixed The curren young will unambiguously benefi Depending on preferences and he aggregae echnology, also a finie number of subsequen generaions may benefi Thereafer, however, welfare is lower han in he economy wihou public deb 1 Many auhors have sudied he properies of Ponzi games (or bubbles) in dynamically inefficien economies See, eg, Tirole (1985), O Connell and Zeldes (1988), Ball e al (1998), Chalk (2000), and Wigger (2005) 2 Some counries, eg, India, exemp ineres income on public bonds from income axaion In he Unied Saes ineres income on bonds issued by lower levels of governmen is exemped from federal income axaion See Norregaard (1997) for a comprehensive survey on he ramificaions of ax-exemping income on public bonds

6 4 II The Model Consider an overlapping generaions economy in which a each ime a new generaion, referred o as generaion, is born and lives for wo periods In he firs period of life individuals supply one uni of labor, consume, and save for old-age by purchasing one period bonds in he capial marke In he second period of life individuals reire and consume ou of he proceedings of heir savings The size of each generaion is an N muliple of is forerunner, ha is N is he economy s exogenous growh facor Uiliy of a generaion individual is assumed o be u = ln c + β ln z, where c and z are young- and old-age consumpion and β > 0 is a discoun facor Young-age consumpion is deermined by c = w s and old-age consumpion by z = R n +1 s, where w and s are he wage rae and individual saving a ime and R n +1 is he ne of ax rae paid on one period (privae or public) bonds a ime + 1 Saving is deermined by uiliy maximizaion and reads s = γ w, (1) where γ β/(1 + β) is he marginal propensiy o save ou of labor income Compeiive firms finance invesmen by issuing one period bonds and hire labor o produce a homogenous good which serves for boh consumpion and invesmen purposes Aggregae producion of all firms a ime is deermined by Y = AK L1, where A > 0 and > 0 are echnological parameers, K is he capial sock a ime, and L is he labor force a ime which equals he size of generaion Each facor of producion is rewarded by is marginal produc Assuming ha capial fully depreciaes wihin one period, he gross rae ha

7 5 compeiive firms pay on one period bonds a ime hus reads R = AK 1 and he wage rae a ime is given by L 1, (2) w =(1 ) AK L (3) The governmen imposes a capial ax a he rae τ so ha he ne rae ha individuals receive on heir savings a ime reads R n =(1 τ)r Moreover, a ime 0 he governmen generaes addiional revenue by issuing bonds in an amoun equal o B 0 Boh, ax revenue and revenue by issuing bonds a ime 0 are used for some (waseful) expendiure which does no affec individual uiliy Raher han paying back he deb, he governmen rolls over ineres and principal from period o period by issuing new bonds a each ime If such a Ponzi game is feasible, he amoun of bonds issued by he governmen a ime + 1 will equal B +1 = R b +1 B, (4) where R b +1 is he rae paid on one period public bonds In conras o privae bonds, public bonds are ax-exemp Therefore, non-arbirage in he bond marke requires R b =(1 τ)r (5) Equilibrium in he capial marke obains, when aggregae savings equal he amoun of public and privae bonds As he laer deermine he sock of capial a ime + 1, he capial marke equilibrium condiion may be

8 6 wrien as L s = B + K +1 (6) The evoluion of he economy can be characerized by wo difference equaions deermining he dynamics of public bonds and he sock of capial I is convenien o express boh magniudes in inensive form Thus, le k = K /L denoe he capial sock per uni of labor and b = B /Y he deb-gdp raio a ime 3 Then, sraighforward manipulaion of eqs (1) o (6) yields k +1 = 1 N [γ(1 ) b ] Ak, (7) b +1 = (1 τ) b γ (1 ) b, for = 0, 1, (8) III The Feasibiliy of a Ponzi Game Consider firs, as a benchmark, an economy wihou public deb Thus, le b = 0 for all = 0, 1, Proposiion 1 esablishes he condiion which guaranees ha he economy wihou public bonds is dynamically efficien 4 Proposiion 1 Le b = 0 for all Then, he economy evolves on a dynamically 3 Expressing public deb in per GDP unis (raher han expressing i in per labor unis) faciliaes he analysis as i allows o separae he bond dynamics from he dynamics of he capial sock per worker 4 Galor and Ryder (1991) discuss a variey of condiions ha guaranee dynamic efficiency in he overlapping generaions model

9 7 efficien growh pah if γ < 1 (9) Proof : The presen economy is dynamically efficien if lim R < N When b = 0 for all, i follows from (7) ha lim k = [A γ (1 )/N] 1/(1 ) Therefore, considering (2), one ges lim R = [A γ (1 )/N] 1 I follows ha lim R < N is equivalen o γ < /(1 ) QED Nex, i will be analyzed under which condiion he governmen can play a Ponzi game even hough γ < /(1 ), ha is, even hough he economy is dynamically efficien To sar wih, consider difference equaion (8) and observe ha i has a unique non-rivial fixed poin a b = γ (1 ) (1 τ) (10) Obviously, b > 0 is equivalen o γ > (1 τ)/(1 ) Assume ha his condiion is saisfied and consider he dynamics of he deb-gdp raio as ploed in Figure 1 b +1 b 45 o b Figure 1: Bond Dynamics

10 8 If he governmen issues bonds a ime 0 in an amoun which saisfies b 0 < b, he deb-gdp raio decreases over ime and evenually becomes zero In his case, he economy converges o he economy wihou public deb and a Ponzi game is feasible 5 If, on he oher hand, he governmen issues bonds a he rae b 0 = b, he deb-gdp raio says consan over ime and, again, a Ponzi game is feasible, bu he economy will no converge o he one wihou public deb Things urn ou o be compleely differen, if he governmen issues bonds a ime 0 a a rae b 0 > b In his case, he deb-gdp raio increases over ime A some ime in he fuure he oal amoun of public bonds necessary o render deb service will become larger han aggregae savings Then, he young generaion a ha ime will no be able o purchase all he issued bonds and, in urn, he governmen will no be able o pay he full amoun of ineres and principal o he hen old As he old will have anicipaed his when young, hey will no have invesed in public bonds By backward inducion i follows ha already he young a ime 0 will no accep bonds As a consequence, he governmen canno play a Ponzi game a a rae ha exceeds b Proposiion 2 summarizes hese resuls Proposiion 2 The governmen can play a Ponzi game, ha is, i can issue bonds a ime 0 up o an amoun b > 0 and roll over principal and ineres perpeually, if and only if γ > (1 τ) 1 (11) Noe ha (11) is equivalen o (1 τ) R < N, 5 Noe ha when he economy wih public deb converges o he economy wihou public deb, he absolu amoun of public deb, B, sill grows indefiniely a he rae lim R n 1

11 9 when b = 0 for all, where R = lim R is he seady sae reurn o capial Tha is, in he presence of ax-exemp public bonds a Ponzi game is feasible if and only if in he absence of public bonds he seady sae ne of ax reurn o capial is smaller han he economy s growh facor In conras, wihou ax exempion a Ponzi game would only be feasible if he gross of ax reurn o capial was smaller han he economy s growh facor Compounding he resuls saed in Proposiions 1 and 2, he following corollary can be esablished Corollary 1 The economy is boh dynamically efficien and he governmen can play a Ponzi game if (1 τ) 1 < γ < 1 (12) Condiion (12) is equivalen o (1 τ) R < N < R, when b = 0 for all Thus, when he economy s growh facor is jus beween he seady sae ne of ax reurn o capial and he seady sae gross of ax reurn o capial, a Ponzi game is feasible and he economy is dynamically efficien There is a similariy beween he resul saed in Corollary 1 and a resul derived by Grossman and Yanagawa (1993) and King and Ferguson (1993) in an endogenous growh model where an exernaliy from invesmen in physical capial susains long-run per capia income growh The exernaliy creaes a wedge beween he privae and he social reurn o capial If he endogenously deermined growh rae lies beween he privae and he social reurn o capial, hen a Ponzi game is feasible alhough

12 10 he economy is dynamically efficien (as he growh rae is below he social reurn o capial) IV Welfare Implicaions In he Diamond (1965) economy a Ponzi game when feasible is beneficial as i removes overaccumulaion of capial associaed wih dynamic inefficiency 6 In conras, under he condiion saed in Corollary 1 here is no dynamic inefficiency despie he fac ha a Ponzi game is feasible In fac, a Ponzi game based on he issuance of ax-exemp bonds canno be generally welfare improving as here is no such hing as overaccumulaion However, as he nex proposiion saes, a Ponzi game will unambiguously benefi generaion 0 and, depending on preferences and he aggregae echnology, a finie number of subsequen generaions Proposiion 3 Le he governmen issue bonds a ime 0 in an amoun saisfying b 0 = b and le i roll over principal and ineres perpeually Then, i he welfare of generaion 0 increases, ii he welfare of generaion, = 1, 2, increases if and only if γ > , iii if he welfare of generaion increases, he welfare of all generaions j = 1,, 1 will also increase 6 See Tirole (1985)

13 11 Proof : See he Appendix Members of generaion 0 benefi because hey are affeced by he launch of he Ponzi game a ime 0 only hrough an increase in he reurn on heir saving a ime 1 This is because he wage rae ha he members of generaion 0 receive when young is predeermined by he sock of capial accumulaed a ime 1 All subsequen generaions are in wo ways affeced by he Ponzi game Since he Ponzi game has a negaive effec on capial accumulaion, i boh lowers he wage rae received when young and increases he reurn on saving received when old For a finie number of generaions he overall resul of hese wo effecs may be posiive In fac, he more generaions will benefi he larger is γ and he smaller is A large γ implies a large saving and, hus, a large benefi from an increase in he reurn on saving A small, on he oher hand, implies ha aggregae producion inelasically responds o a decrease in he capial sock, which dampens he negaive effec of a Ponzi game on capial accumulaion Since he sequence { (1 )/(1 +1 ) } converges o 1, only a =1 finie number of generaions will benefi from he Ponzi game if γ < /(1 ), ha is, if he economy is dynamically efficien (see Proposiion 1) All generaions born hereafer, in conras, will be made worse off This case is illusraed in Figure 2, where v measures indirec uiliy of a member of generaion, and, as a poin of reference, ˆv measures indirec uiliy on a growh pah wihou public bonds (see he Appendix for he exac form of indirec uiliy; in Figure 2 i is assumed ha he iniial capial sock per uni of labor, k 0, is below is long-run level) Proposiion 3 confines aenion o he case b 0 = b Ifb 0 < b, he deb- GDP raio becomes zero over ime so ha he economy converges o he seady sae wihou public bonds In his case, herefore, he Ponzi game can only affec welfare on a ransiory growh pah The nex proposiion explains he welfare consequences of a Ponzi game wih b 0 < b

14 12 v 0 v, ˆv ˆv 0 Figure 2: Welfare for b 0 = b Proposiion 4 Le he governmen issue bonds a ime 0 in an amoun saisfying 0 < b 0 < b and le i roll over principal and ineres perpeually Then, i he welfare of generaion 0 increases, ii here is some 1 so ha he welfare of all generaions decreases if γ < 1 Proof : See he Appendix The resul is illusraed in Figure 3 Like in he case b 0 = b, he Ponzi games benefis he curren young and, possibly, a finie number of subsequen generaions If he economy wihou public deb is dynamically efficien, however, all generaions born afer some ime will be made worse off Since he economy converges o he economy wihou public deb, he welfare of fuure generaions will converge o he level of welfare hey would have obained, if he governmen had no launched a Ponzi game a ime 0

15 13 v, ˆv v 0 ˆv 0 Figure 3: Welfare for 0 < b 0 < b V Conclusion The presen noe has shown ha he governmen may issue bonds and roll over ineres and principal, ha is run a Ponzi game, even if he economy wihou public deb is dynamically efficien This becomes possible when he governmen axes capial income and he ne of ax ineres rae is smaller han he economy s rae of growh, whereas he gross of ax ineres rae is larger Then, if he governmen exemps ineres paymen on public bonds from capial axes, he oal amoun of public deb will grow a a lower rae han aggregae income and, as a consequence, a Ponzi game becomes feasible Such a Ponzi game will benefi he curren young generaion and, depending on he parameers of ase and echnology, a finie (maybe large) number of fuure generaions Thereafer, however, welfare is lower han in an economy wihou public deb Thus, ax-exemp bonds may be employed by governmens ha wan o please curren generaions and he generaions ha live in he no oo disan fuure

16 14 Appendix In order o prove Proposiions 3 and 4, he following Lemma will be esablished Lemma 1 The sysem of difference equaions (7) and (8) has he soluion k = k 0 [ A [ ] ] j γ(1 ) b N j 1, (13) j=1 b = bb 0 ( ) γ(1 ), (14) ( b b (1 0 )+b 0 τ) for = 1, 2, Proof : Eq (8) can be ransformed ino x +1 = γ(1 ) (1 τ) x 1, for = 0, 1,, (1 τ) where x = 1/b Solving his linear difference equaion and hen ransforming back leads o eq (14) Eq (7) can be ransformed ino y +1 = y + m, for = 0, 1,, where y = ln k and m = ln[(γ(1 ) b )a/n] The soluion o his difference equaion is y = y 0 + j m j 1, for = 1, 2, j=1 Transforming back yields eq (13) QED

17 15 Proof of Proposiion 3 Welfare of generaion may be expressed in erms of indirec uiliy as v = [1 γ (1 )] ln k γ (1 ) ln[γ (1 ) b ]+C, (15) where C is a consan given by C =(1 γ) ln(1 γ)+γ ln[(1 τ) γa] +(ln[(1 ) A]+γ (1 ) ln N γ (1 ) ln A Noe ha indirec uiliy a ime is, in fac, a 1 + β muliple of v Since he facor 1 + β is inconsequenial, i has been omied for simpliciy Proof of i Welfare of generaion 0 reads v 0 = [1 γ (1 )] ln k 0 γ (1 ) ln[γ (1 ) b 0 ]+C Obviously, v 0 is sricly increasing in b 0 Proof of ii Now le b 0 = b, which implies b = b for all = 0, 1, 2, and subsiue for k in (15) by employing (13) Indirec uiliy of generaion hen becomes where v = +1 [1 γ (1 )] ln k 0 + [1 γ (1 )] j ln[γ (1 ) b] j=1 γ (1 ) ln[γ (1 ) b]+ C, C = C + [1 γ (1 )] j [ln A ln N] j=1

18 16 If he governmen had no sared o issue bonds a ime 0, indirec uiliy of generaion would be ˆv = +1 [1 γ (1 )] ln k 0 + [1 γ (1 )] j ln[γ (1 )] j=1 γ (1 ) ln[γ (1 )] + C Now subrac ˆv from v o find afer a few manipulaions ha v > ˆv is equivalen o γ > 1 j j=1 1 + j j=1 Considering ha j=1 j =(1 )/(1 ), his reduces o γ > Proof of iii In ligh of ii i holds ha v 1 > ˆv 1 if Since γ > > ,

19 17 i follows ha v 1 > ˆv 1 if γ > By backward inducion one ges v j > ˆv j for all j = 1,, 1 QED Proof of Proposiion 4 Par i has already been shown in he proof of Proposiion 3 To prove par ii subrac ˆv from v o find ha v ˆv = [1 γ (1 )] j [ ln[γ (1 ) b j 1 ] ln[γ (1 )] ] j=1 γ (1 ) [ln[γ (1 ) b ] ln[γ (1 )]] From (14) i follows ha b < b j for all j = 1,, Therefore [ ] v ˆv < [1 γ (1 )] j γ (1 ) j=1 [ln[γ (1 ) b ] ln[γ (1 )]] Since ln[γ (1 ) b ] ln[γ (1 )] < 0, i follows ha a sufficien for v ˆv o be negaive is γ < For γ < /(1 ) here is some 1 so ha γ < QED

20 18 References Ball, L, DW Elmendorf, and NG Mankiw, 1998, The defici gamble, Journal of Money, Credi, and Banking 30, Chalk, NA, 2000, The susainabiliy of bond-financed deficis: an overlapping generaions approach, Journal of Moneary Economics 45, Diamond, P, 1965, Naional deb in a neoclassical growh model, American Economic Review 55, Galor, O and HE Ryder, 1991, Dynamic inefficiency of seady-sae equilibria in an overlapping-generaions model wih producive capial, Economics Leers 35, Grossman, GM and N Yanagawa, 1993, Asse bubbles and endogenous growh, Journal of Moneary Economics 31, 3-19 King, I and D Ferguson, 1993, Dynamic inefficiency, endogenous growh, and Ponzi games, Journal of Moneary Economics 32, Norregaard, J (1997), The ax reamen of governmen bonds, IMF Working Paper 97/25 O Connell, SA and SP Zeldes, 1988, Raional Ponzi games, Inernaional Economic Review 29, Tirole, J, 1985, Asse bubbles and overlapping generaions, Economerica 53, Wigger, BU, (2005), Public deb, human capial formaion, and dynamic inefficiency, Inernaional Tax and Public Finance 12, 47-59