Debt management and optimal fiscal policy with long bonds 1

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1 Deb managemen and opimal fiscal policy wih long bonds Elisa Faraglia 2 Alber Marce 3 and Andrew Sco 4 Absrac We sudy Ramsey opimal fiscal policy under incomplee markes in he case where he governmen issues only long bonds of mauriy. We find ha many feaures of opimal policy are sensiive o he inroducion of long bonds in paricular ax variabiliy and he long-run behaviour of deb. When governmen is indebed i is opimal o respond o an adverse shock by promising o reduce axes in he disan fuure as his achieves a cu in he cos of deb. Hence deb managemen concerns override ypical fiscal policy concerns such as ax-smoohing. In he case where he governmen leaves bonds in he marke unil mauriy we find wo addiional reasons why axes are volaile due o deb managemen concerns: deb has o be brough o zero in he long run and here are -period cycles. We formulae our equilibrium recursively applying he Lagrangean approach for recursive conracs. However even wih his approach he dimension of he sae vecor is very large. To overcome his issue we propose a flexible numerical mehod he condensed PEA which subsanially reduces he required sae space. This echnique has a wide range of applicaions. To explore issues of policy coordinaion and commimen we propose an alernaive model where moneary and fiscal auhoriies are independen. Keywords: Compuaional mehods deb managemen fiscal policy governmen deb mauriy srucure ax-smoohing yield curve JEL classificaion: C63 E43 E62 H Marce is graeful for suppor from DGES Monfispol and Excellence Program of Banco de España. Faraglia and Sco graefully acknowledge funding from he ESRC s World Economy and Finance programme. We are graeful o Gerno Müller for commens and seminar paricipans a EEA (Oslo) IAE UPF Monfispol. All errors are our own. Universiy of Cambridge and CEPR. Insiu d Anàlisi Econòmica CSIC ICREA UAB BGSE and CEPR. London Business School and CEPR. BIS Papers o 65 77

2 . Inroducion Table shows he average mauriy of ousanding governmen deb for a variey of counries and displays clear differences across naions. Any heory of deb managemen needs o explain he coss and benefis for fiscal policy of varying he average mauriy in his manner. As he curren European sovereign deb crisis emphasises he mauriy srucure of governmen deb is a key variable. Deciding fiscal policy independenly of funding condiions in he marke is a doomed concep: axes public spending and fiscal deficis should all ake ino accoun he funding condiions in he marke for bonds. Therefore deb managemen should no be subservien o fiscal policy and should no focus simply on minimising coss. Raher fiscal policy and deb managemen should be sudied joinly. Table Average mauriy governmen deb 200 Counry Average mauriy (years) Unied Kingdom 3.7 Denmark 7.9 Greece 7.7 Ialy 7.2 Ausria 7 France 6.9 Ireland 6.8 Spain 6.7 Swizerland 6.7 Porugal 6.5 Czech Republic 6.4 Sweden 6.4 Germany 5.8 Belgium 5.6 Japan 5.4 eherlands 5.4 Canada 5.2 Poland 5.2 Ausralia 5 orway 4.9 Unied Saes 4.8 Finland 4.3 Hungary 3.3 Sources: OECD; The Economis. A number of recen conribuions have sudied his ineracion beween deb managemen and axaion policy in a Ramsey equilibrium seing. Angeleos (2002) Barro (2003) Buera 78 BIS Papers o 65

3 and icolini (2004) use models of complee markes. osbusch (2008) explores a simplified model of incomplee markes and Lusig Slee and Yelekin (2009) examine an incomplee marke model wih muliple mauriies and nominal bonds. In his paper we build on our recommendaions in Faraglia Marce and Sco (200) and exend he seup of Aiyagari Marce Sargen and Seppälä (2002) who sudied opimal fiscal policy wih incomplee markes and shor bonds o he case when bonds maure periods afer having been issued. We describe he behaviour of opimal policy wih long bonds and we show how o navigae compuaional problems. The equilibrium in our model shows some well known feaures of opimal fiscal policy under incomplee markes: he governmen ries o smooh axes axes follow a near-maringale behaviour and deb is used as a buffer sock o spread ax increases over all periods afer an unexpeced adverse shock. We also find ha if he governmen is indebed and an adverse shock occurs he governmen should promise o cu axes in fuure periods when he newly issued long bonds generae a payoff. These fuure ax cus wis curren long ineres raes so as o reduce he burden of pas deb. This means ha a ypical deb managemen concern ie reducing he coss of deb overrides a ypical concern of fiscal policy namely ax-smoohing. This promise o cu axes is he reason ha opimal policy is ime-inconsisen: if he governmen could i would renege on he promise o cu axes. A furher problem ha arises only when dealing wih long bonds is wha decision o make abou ousanding deb a he end of each period. Mos of he lieraure assumes ha he governmen buys back each period all previously issued deb and hen reissues new bonds. This assumpion is innocuous in models of complee markes bu maers under incomplee markes. Furhermore as shown in Marchesi (2004) governmens rarely buy back ousanding deb before redempion. To quoe he UK Deb Managemen Office (2003) he UK s deb managemen approach is ha deb once issued will no be redeemed before mauriy. For his reason we also sudy opimal policy when he governmen leaves long bonds in circulaion unil he ime of mauriy. We call his he hold o redempion case. In his case a any momen in ime he governmen has a full specrum of ousanding deb wih mauriy unil redempion of hrough o one year even hough he governmen only ever issues period deb. The mauriy profile of governmen deb is herefore much more complex wih long bonds and hold o redempion and his will poenially impac deb managemen and fiscal policy. We find ha opimal ax policy is even more volaile in his case: he governmen promises o cu axes permanenly and here are -period cycles in ax policy. Obaining numerical simulaions is no sraighforward. A firs difficuly lies in obaining a recursive formulaion of he model. To do so we exend he recursive conracs reamen of Aiyagari e al (2002). A second difficuly arises because he vecor of sae variables is ypically of dimension 2. Hence i grows rapidly wih mauriy: many OECD counries issue 30-year bonds and boh France and he UK issue 50-year bonds. Solving a non-linear dynamic model wih hese many sae variables is no feasible. 5 To reduce his compuaional complexiy we propose a new mehod he condensed PEA ha reduces he dimensionaliy of he sae vecor while allowing in principle for arbirary precision. We show how in he case of a 20-year bond he sae space is effecively only four variables. We believe his compuaional mehod has wide applicabiliy o oher cases. The fac ha he fiscal auhoriy finds i opimal o wis ineres raes o minimise funding coss raises issues of commimen and policy coordinaion. To assess his we inroduce a model where he fiscal auhoriy is separae from he moneary auhoriy seing ineres raes. In his 5 Linearisaion of he policy funcion is undesirable. Firs because i urns ou ha non-linear erms in he policy funcion play a crucial role even near he seady sae mean. Second because of he presence of deb limis. BIS Papers o 65 79

4 way he wising of ineres raes is no possible since he fiscal auhoriy akes ineres raes as given. This seup provides a framework o undersand he role of commimen in he Ramsey policy and in he case wih buyback i reduces he dimensionaliy of he sae vecor as he usual co-sae variables of opimal Ramsey policy are no longer presen. We find ha he second momens of he model are no highly dependen on mauriy. In a calibraed example allocaions ineres raes and persisence of deb are similar across mauriies and across he hree models of policy considered. The main difference is he long-run level of deb as longer mauriies are associaed wih more deb. The srucure of he paper is as follows. Secion 2 oulines our main model a Ramsey model wih incomplee markes and long bonds when he governmen buys back all ousanding deb each period. Secion 2 shows some properies of he model using analyic resuls. Secion 3 sudies numerical issues inroducing he condensed PEA and describing he behaviour of he model numerically. Secion 4 sudies he model of independen powers whils Secion 5 considers he case of hold o redempion. A final secion concludes. 2. The model: analyic resuls Our benchmark model is of a Ramsey policy equilibrium wih perfec commimen and coordinaion of policy auhoriies in which he governmen buys back all exising deb each period. In Secions 4 and 5 we relax hese assumpions. The economy produces a single non-sorable good wih a echnology c g x () for all where x c and g represen leisure privae consumpion and governmen expendiure respecively. The exogenous sochasic process g is he only source of uncerainy. The represenaive consumer has uiliy funcion: E0 u c v x 0 (2) and is endowed wih one uni of ime ha i allocaes beween leisure and labour and faces a proporional ax rae on labour income. The represenaive firm maximises profis boh consumers and firms ac compeiively by aking prices and axes as given. Consumers firms and governmen have full informaion ie hey observe all shocks up o he curren period and all variables daed are chosen coningen on hisories g g... g0. All agens have raional expecaions. Agens can only borrow and lend in he form of a zero-coupon risk-free -period bond so ha he governmen budge consrain is: g p b x p b (3) where b denoes he number of bonds he governmen issues a ime each bond pays one uni of consumpion good in -periods of ime wih complee cerainy. The price of an i - period bond a ime is p i. In his secion we assume ha a he end of each period he governmen buys back he exising sock of deb and hen reissues new deb of mauriy ; hese repurchases are refleced in he lef side of he budge consrain (3). In addiion governmen deb has o remain wihin upper and lower limis M and M so ruling ou Ponzi schemes eg M b M (4). 80 BIS Papers o 65

5 The erm in his consrain reflecs he value of he long bond a seady sae so ha he limis M M appropriaely refer o he value of deb and hey are comparable across mauriies. 6 We assume ha afer purchasing a long bond he household has only wo opions: one is o resell he governmen bond in he secondary marke in he period immediaely afer having purchased i he oher possibiliy is o hold he bond unil mauriy. 7 Leing s be he sales in he secondary marke he household s problem is o choose sochasic processes c x s b o maximise (2) subjec o he sequence of budge consrains: 0 c p b x p s b s wih prices and axes p p aken as given. The household also faces deb limis analogous o (4). We assume for simpliciy ha hese limis are less sringen han hose faced by he governmen so ha in equilibrium he household s problem always has an inerior soluion. The consumer s firs-order condiions of opimaliy are given by v u p p x c (5) c E uc (6) u E uc c (7) u 2. The Ramsey problem We assume he governmen has full commimen o implemening he bes sequence of (possibly ime-inconsisen) axes and governmen deb knowing equilibrium relaionships beween prices and allocaions. Using (5) (6) and (7) o subsiue for axes and consumpion he Ramsey equilibrium can be found by solving max E0 u c v x c b s.. 0 E uc b S E uc b and (4) and x implicily defined by (). To simplify he algebra we define he governmen and se up he Lagrangian S u v c g u g as he discouned surplus of c x c (8) 6 7 Obviously he acual value of deb is p b we subsiue nohing much changes if he limis are in erms of p b. p by is seady sae value for simpliciy We need o inroduce secondary marke sales s in order o price he repurchase price of he bond. BIS Papers o 65 8

6 0 c c 0 M bn 2 b M L E u c v x S u b u b where is he Lagrange muliplier associaed wih he governmen budge consrain and and 2 are he mulipliers associaed wih he deb limis. The firs-order condiions for he planner s problem wih respec o c and b are u v u c u v c g v c x cc c xx x ucc b 0 c c 2 E u E u (0) wih These FOC help characerise some feaures of opimal fiscal policy wih long bonds. Following he discussion in Aiyagari e al (2002) we see ha in he case where deb limis are non-binding (0) says ha is a risk-adjused maringale wih risk-adjusmen measure u c indicaing ha in his model he presence of he sae variable λ in he policy E u c funcion impars persisence in he variables of he model. The erm D b (9) in (9) indicaes ha a feaure of opimal fiscal policy will be ha wha happened in period has a special impac on oday s axes. Since we have u v 0 and zero axes in he firs bes a high D pulls he model away from he firs bes and zero axes. If D 0 i c x g can be hough of as inroducing a higher disorion in a given period. In periods when is very high we have ha he cos of he budge consrain is high so is high and if he governmen is in deb D 0 so ha axes should go down a. Of course his is no a igh argumen as also responds o he shocks ha have happened beween and and also plays a role in (9) bu his argumen is a he core of he ineres rae wising policy we idenify below. In order o build up inuiion for he role of commimen and o provide a igher argumen we now show wo examples ha can be solved analyically. 2.2 A model wihou uncerainy Assume now ha governmen spending is consan g he governmen is hen g. The only budge consrain of u S b p c 0 0 uc0 S b uc 0 where S or is he non-discouned surplus of he governmen. This shows ha for a uc S given se of surpluses he funding coss of iniial deb b 0 can be reduced by manipulaing consumpion such ha c c for all. As long as he elasiciy of () 82 BIS Papers o 65

7 consumpion wih respec o wages is posiive as occurs wih mos uiliy funcions his can be achieved by seing for all This achieves a reducion of uc reducing he cos of ousanding deb. In oher words he long end of he yield curve needs o be wised up. 8 Ineresingly even hough here are no flucuaions in he economy (2) shows ha opimal policy implies ha he governmen desires o inroduce variabiliy in axes. In oher words opimal policy violaes ax-smoohing. This policy is clearly ime-inconsisen: if he governmen is able o reopimise by surprise a some period 0 i will hen promise insead a cu in axes in period. (2) 2.3 A model wih uncerainy a The previous subsecion absraced from uncerainy. We now inroduce uncerainy ino our model. In he ineres of obaining analyic resuls we assume uncerainy occurs only in he firs period ie g is given by: 9 g g g ~ F g for 0 and 2 for some non-degenerae disribuion F g. Since fuure consumpion and s are known he maringale condiion implies uc uc and I is clear ha in he case of shor bonds equilibrium implies c and consan for 2 reflecing he fac ha even hough markes are incomplee he governmen smoohs axes afer he shock is realised. For he case of long bonds when he FOC wih respec o consumpion (9) is saisfied D b for D 0 for 0and (3) D b D b (4) Hence equilibrium saisfies c c * g for 2 and (5) for a cerain funcion c *. ie consumpion is he same in all periods 2 and alhough his level of consan consumpion depends on he realisaion of he shock g. Clearly c c also depend on he realisaion of g. 8 9 This is of course a manifesaion of he sandard ineres rae manipulaion already noed by Lucas and Sokey (983) excep ha in our case he wising occurs in periods. Formally his economy is very similar o ha of osbusch (2008). BIS Papers o 65 83

8 Therefore here is more ax volailiy han in he case of shor bonds: axes vary in periods and even hough by he ime he economy arrives a hese periods no more shocks have occurred for a long ime An analyic example To make his argumen precise consider he uiliy funcion c x c B c for B 0. c (6) Resul Assume uiliy (6) and b 0. For a sufficienly high realisaion of g we have for all The inequaliies are reversed if b 0 or if he realisaion of g is sufficienly low. Proof Since he FOC of opimaliy yield u v B c F x c B where F ucc b. B c 0 for Consider. For any long mauriy we have ha 0 when so ha u v c B B x c Therefore we can wrie u v c c x x u F0 for (8) v Tha for all and follows from (5). ow we show ha F 0 for. Since B 0 we have ha B u v clearly (7) implies ha 0. Since c x 0 c B 0. Since we consider he case of iniial governmen deb b 0 his leads o b 0 0 and since ucc 0 we have F 0 for. Since for we have 0 0 i follows u v c c x x u for all. v (7) 84 BIS Papers o 65

9 Also i is clear from (7) ha high g implies a high. Since he maringale condiion implies E uc 0E0 uc for slighly high g we have 0. Therefore for and if g was high enough we have 0 0 so ha (8) implies uc uc uc v v v x x x Inuiively in period here is a ax cu for he same reasons as in Secion 2.2. ew in his secion is he ax cu (for high g ) a. The inuiion for his is clear: when an adverse shock o spending occurs a he governmen uses deb as a buffer sock so ha b b0 as his allows ax-smoohing by financing par of he adverse shock wih higher fuure axes. Bu since fuure surpluses are higher han expeced as he higher ineres has o be serviced he governmen can lower he cos of exising deb by announcing a ax cu in period since his will reduce he price p 0 of period ousanding bonds b 0. The ax cu a is a sochasic analog of he ax cu described in Secion Conradicing ax-smoohing The above resul shows ha in his model ax policy is subordinae o deb managemen. In models of opimal policy he governmen usually desires o smooh axes. Taxes would be consan in he above model if he governmen had access o complee markes. Bu we find ha he governmen increases ax volailiy in period long afer he economy has received any shock. Therefore governmen forfeis ax-smoohing in order o enhance a ypical deb managemen concern namely reducing he average cos of deb. Obviously his policy is ime-inconsisen: if he governmen could unexpecedly reopimise in period 2 given is deb b i would renege on he promise o cu axes in period. Insead i would promise o lower axes in period. I is clear from his discussion ha wha will maer for he policy funcion is he erm D 0 b0. Therefore i is he ineracion beween pas s and pas b s ha deermines he size and he sign of oday s ax cu. A linear approximaion o he policy funcion would fail o capure his feaure of he model and i would be quie inaccurae. To summarise we have proved ha in he presence of an adverse shock o spending he governmen has o ake hree acions: (i) increase axes permanenly (ii) increase deb and (iii) announce a ax cu when he ousanding deb maures. Effecs (i) and (ii) are well known in he lieraure of opimal axaion under incomplee markes effec (iii) is clearly seen in his model wih long bonds since he promise is made periods ahead. Obviously in he case of shor mauriy of Aiyagari e al he effec of D would be fel in deciding opimally bu his effec would be confounded wih he fac ha g is sochasic so effec (iii) is harder o see in a model wih shor bonds. 3. Opimal policy: simulaion resuls We now urn o he case where g is sochasic in all periods. As is well known analyic soluions for his ype of model are infeasible so we uilise numerical resuls. The objecive is o compue a sochasic process c b ha solves he FOC of he Ramsey planner namely (8) (9) and (0). Firs we obain a recursive formulaion ha makes compuaion BIS Papers o 65 85

10 possible hen we describe a mehod for reducing he dimensionaliy of he sae space and finally we discuss he behaviour of he economy. 3. Recursive formulaion Using he recursive conrac approach of Marce and Marimon (20) he Lagrangean can be rewrien as: 0 c 0 L E u c v x S u b M b b M for Assuming g is a Markov process as suggesed by he form of his Lagrangean Corollary 3. in Marce and Marimon (20) implies he soluion saisfies he recursive srucure 0 b Fg... b... b c... 0 given b for a ime-invarian policy funcion F. This allows for a simpler recursive formulaion han he promised uiliy approach as he co-sae variables do no have o be resriced o belong o he se of feasible coninuaion variables. The sae vecor in his recursive formulaion has dimension 2. I is unlikely ha furher reducions in his dimension can be achieved purely by heoreical resuls. In order o overcome he problem of dimensionaliy some auhors model long bonds as perpeuiies wih decaying coupon paymens where he raes of decay mimic differences in mauriy (Woodford (200) Broner Lorenzoni and Schmulker (2007) Arellano and Ramanarayanan (2008)). One jusificaion for assuming a decaying payoff is ha i mimics a bond porfolio wih fixed shares ha decay wih mauriy. However since our goal is o build a model of deb managemen where he objec is precisely o sudy he appropriae porfolio weighs he assumpion of fixed porfolio weighs would be inappropriae. Furher alhough modelling bond payoffs in his way would yield smaller sae space vecors i is conrary o he srucure of mos governmen porfolios where mos of he payoff occurs a he ime of mauriy as in his model. (9) 3.2 The condensed PEA We wish o find non-linear soluions firs because he deb limis are likely o be occasionally binding if we wan o keep deb a levels similar o hose observed in he real world second because per our discussion a he end of Secion 2.3 a linear approximaion of he policy funcion F will miss key aspecs of opimal policy. Since bonds of mauriy 0 30 or 50 years are no uncommon a non-linear approach rapidly becomes inracable for a sae 0 In his model i is possible o reduce he sae space even furher by recognising ha he only relevan sae variables are lags of s b. We do no exploi his feaure of he model as i is very specific o his version of he model. For example he no buyback case of Secion 5 needs all sae variables. 86 BIS Papers o 65

11 vecor of dimension 2. To overcome his difficuly we inroduce a soluion mehod based on he parameerised expecaion algorihm of den Haan and Marce (990). This allows us o reduce he dimensionaliy of he policy funcion acually solved for while keeping an accurae soluion. Using PEA is useful because i does capure he relevan non-lineariies described in Secion 2.3 even if he expecaions are parameerised as linear funcions and because i allows for a naural space reducion mehod ha we call condensed PEA. This mehod goes as follows. Denoe he sae vecor as X g... b... b. The idea is ha even hough heoreically all elemens of X are necessary in deermining decision variables a i is unlikely ha in he seady sae disribuion each and every one of hese variables plays a subsanial role in deermining he soluion. For us mos likely some funcion of hese lags will be sufficien o summarise he feaures from he pas ha need o be remembered by he governmen in order o ake an opimal decision. In he conex of PEA his can be expressed in he following way. One of he expecaions requiring approximaion is E u c+ (20) appearing in (0). This expecaion is a funcion in principle of all elemens in X bu i is likely ha in pracice a few linear combinaions of X may be sufficien o predic uc. There are wo reasons for his. Firs he very srucure of he model suggess ha elemens of X are very highly correlaed wih each oher suggesing ha a few linear combinaions of X have as much predicive power as he whole vecor. Anoher way of saying his is ha i is enough o projec any variable on he principal componens of X. Oher mehods available for reducing he dimensionaliy of sae vecors have emphasised his aspec. The second reason is ha some principal componens of X may be irrelevan in predicing u c in equilibrium and herefore hey can be lef ou of he approximaed condiional expecaion. So he goal is o include only linear combinaions of X ha have some predicive power for uc he remaining linear combinaions can be undersood as appearing in he condiional expecaion wih a coefficien of zero. More precisely we pariion he sae vecor ino wo pars: a subse of n sae core X X where n 2 n is small and an omied subse of sae variables variables ou core X X X of dimension 2 n in he parameerised expecaions. If he error uc Euc. We firs solve he model including only core X found using jus hese ou core variables is unpredicable wih X we would claim he soluion wih core variables is ou ou he correc one. If X can predic his error we hen find he linear combinaion of X ha has he highes predicive power for. We add his linear combinaion o he se of sae ou variables solve he model again wih his sole addiional sae variable check if X can predic and so on. Formally given he se of core variables we define he condensed PEA as follows. 2 This definiion assumes we are ineresed in he seady sae disribuion. Of course i could be modified in he usual way o ake ransiions ino accoun. BIS Papers o 65 87

12 Sep Parameerise he expecaion as core E uc X (2) Find values for n R denoed f ha saisfy he usual PEA fixed poin ie core f X causes his o be he bes where he series generaed by parameerised expecaion. This soluion is of course based on a resriced se of ou sae variables. I is herefore necessary o check if he omission of X affecs he ou approximae soluion. The nex sep orhogonalises he informaion in X which will be helpful in arriving a a well condiioned fixed poin problem in Sep 4. Sep 2 Using a long-run simulaion run a regression of each elemen of variables. ou Leing X i be he i h elemen we run he regression core X X b u ou i i i b R and calculae he residuals 2 2 n i res ou core i i i ou X on he core X X X b. (22) res I is clear ha X adds he same informaion o core advanage ha i is orhogonal o X. n Sep 3 Using a long-run simulaion find R such ha core X as ou X bu res X has he core res u 2 c X X T argmin (23) If is close o zero he soluion wih only sop here. Oherwise go o Sep 4 Apply PEA adding expecaion as X core res res 2 E uc X X 2 n2 where R. Find a fixed poin f is a fixed poin since combinaion for he ieraions of he fixed poin 2 f n f 2 core X is sufficienly accurae and we can as a sae variable ie parameerising he condiional core X and 2f for his parameerised expecaion. Since res X are orhogonal and since he linear has high predicive power i makes sense o use as iniial condiion 2 For convenience we describe hese seps wih reference only o he expecaion c expecaions E uc and c E u. In pracice he E u appearing in he FOC also need o be parameerised concurrenly and he seps need o be applied joinly o all condiional expecaions. 88 BIS Papers o 65

13 Go o Sep 2 wih X ec. core res X in he role of core X find a new linear combinaion Two remarks end his subsecion. In he presence of many sae variables i has been cusomary in dynamic economic models o ry each sae variable in order. The idea is o add sae variables one by one unil he nex variable does no much change he soluion found. For example if many lags are needed we add he firs lag hen he second lag and so on. If a some sep he soluion changes very lile i is claimed ha he soluion is sufficienly accurae. Bu i is easy o find reasons why his argumen may fail. For insance perhaps he variables furher down he lis are more relevan. 3 This is he case by he way in our model where sae variable and b play a key role in deermining he soluion a. Or i can be ha all he remaining variables ogeher make a difference bu hey do no make a difference one by one. Our mehod gives a chance o all hese variables o make a difference in he soluion. I is herefore more efficien in finding relevan sae variables as Sep 3 indicaes auomaically if hey are needed and which of hem are o be inroduced. The whole argumen in his secion is made for linear condiional expecaions as in (2). Of course he same idea works for higher-order erms. In order o check he accuracy for higherorder erms one can use he condensed PEA wih he higher-order polynomial erms ie one can check if linear combinaions of say quadraic and cubic erms of X have predicive ou power in Sep 2 include hese in X and go hrough Seps 2 o 4 above. core The variables included in X are no he only ones influencing he soluion. Due o he naure of PEA pas variables can have an effec even if hey are excluded from he core parameerised expecaion. For example even if we find a soluion X b g ha excludes and b from he parameerised expecaion hese sae variables will influence he soluion a hrough heir presence in (9). 3.3 Solving he model wih condensed PEA The uiliy funcion (6) was convenien for obaining he analyic resuls of Secion 2.3. In his secion we use a uiliy funcion more commonly used in DSGE models: 2 c x 2 We choose 0.98 and 2 2. The choice of discoun facor implies ha we hink of a period as one year. We se such ha if he governmen s defici equals zero in he nonsochasic seady sae agens work a fracion of leisure of 30% of he ime endowmen. For he sochasic shock g we assume he following runcaed AR() process: g if g * g g g g if g * g g g * g oherwise 3 For anoher example incomplee marke models wih a large number of agens need as sae variable all he momens of he disribuion of agens which is an infinie number of sae variables. Usually hese models are solved firs by using he firs momen as a sae variable and checking ha if he second momen is added nohing much changes. Bu i could be of course ha he hird or fourh momen are he relevan ones especially since he acual disribuion of wealh is so skewed. BIS Papers o 65 89

14 We assume ~ 0.44 g* 25 2 wih an upper bound g equal o 35% and a lower bound g 5% of average GDP and M is se equal o 90% of average GDP and M M. core ou We choose X b g hence X b 2... b To es if sufficien variables are included for an accurae soluion in Sep 3 we use as our olerance saisic: R dis R R 2 2 aug 2 2 where R and R 2 aug denoe he goodness of fi of he original regression based on he condensed PEA and augmened wih he linear combinaion of residuals respecively. We use for olerance crierion dis Table 2 summarises he number of linear combinaions needed for each mauriy whils Table 3 gives deails and shows he number of 2 linear combinaions needed for each approximaions and he R and dis. Table 2 Holding-o-redempion model: linear combinaions inroduced wih condensed PEA 2 umber of linear combinaions uc uc oe: recall ha denoes mauriy and 2 is he dimension of he sae vecor. In all cases hree variables. # of linear comb refers o how many linear combinaions of he accuracy crierion. We denoe each expecaion o be approximaed by uc E u c and E u. uc c core X has ou X had o be added o saisfy E u c The advanages of he condensed PEA are readily apparen. In nearly half he cases he core variables are sufficien o solve he model and a mos only one linear combinaion of omied variables is required o improve accuracy. Clearly he condensed PEA can be used o solve models wih large sae spaces wih relaively small compuaional cos since he sae vecor is in principle of dimension 4 bu uilising a dimension of 4 is sufficien. Whils we have focused on a case of opimal fiscal policy and deb managemen his mehodology clearly has much broader applicabiliy. 90 BIS Papers o 65

15 Table 3 Benchmark model: accuracy measures in condensed PEA Adding linear comb uc uc Adding linear comb uc uc 2 # lin comb in R aug dis # lin comb in R aug dis # lin comb in R aug dis # lin comb in R aug dis # lin comb in R aug dis oe: see oe of previous Table. 2 R aug and dis are defined in Secion Opimal policy: he impac of mauriy 3.4. Ineres rae wising We compue he policy funcions 4 and display he implied response funcions of key variables o an unexpeced shock in g in Figures and 2. The verical axis is in unis of each of he variables and expresses deviaions from he value ha would occur for he given ss iniial condiion if g g. Figure is for he case when he governmen has zero deb on impac. I shows minor differences beween long and shor bonds. As usual in models of incomplee markes i is opimal o use deb as a buffer sock so ha deb displays considerable persisence. 4 Since deb is very persisen o ensure we visi all possible realisaions in he long-run simulaions of PEA we iniialise he model a nine differen iniial condiions simulae i for 5000 periods for each iniial condiion doing his 000 imes per iniial condiion and compue condiional expecaions discarding he firs 500 observaions for each simulaion. BIS Papers o 65 9

16 Figure Responses o a shock in g benchmark model mauriies and 0: b 0 Figure 2 shows he same impulse response funcions when we assume he governmen is indebed on impac more precisely b 0.5 y * where y * is seady sae oupu. 92 BIS Papers o 65

17 Figure 2 Responses o a posiive shock in g benchmark model mauriies and 0: b 0.5 y * We see ha wih long bonds of mauriy 0 here is a blip in axes a he ime of mauriy of he ousanding bonds. This is a reflecion of he promise o cu axes wih he aim o wis ineres raes as discussed in Secion 2.3 only now he ineres rae wising occurs each period here is an adverse shock. BIS Papers o 65 93

18 3.4.2 Opimal policy wih shor bonds This discussion helps o elucidae he role of commimen in he model of shor-erm bonds as in Aiyagari e al (2002). Consider he case where he governmen is indebed when an adverse shock occurs as in Figure 2. As we explained in Secion 2.3 opimal policy is o increase curren axes bu promise a ax cu in periods. In he case of long bonds he promised ax cu is clearly disinc from he curren increase in axes. Bu in he case of shor bonds he wo effecs are confounded as hey happen in he same period. Figure 3 Responses o a posiive shock in g benchmark model mauriies 5 0 and 20: Taxes: b 0.5 y * 0.5 y * Taxes: b This is clearly seen in he response of axes depiced in Figure 3 for mauriies Given our previous discussion i is clear why he blip in axes keeps moving o he lef as we decrease he mauriy unil he blip simply reduces he reacion of axes on impac a. 94 BIS Papers o 65

19 Therefore opimal policy for shor bonds is o increase axes on impac bu less han would be done if consideraions of ineres rae wising were absen. In he case where he governmen has asses he blip in axes goes upwards as he governmen desires o increase he value of asses. This is shown in he response of axes for he case of asses shown in Figure 3. So comparing he dashed lines in he response of axes in Figures 3 i is clear ha for shor bonds he increase in axes on impac if he governmen iniially has asses is much larger han if he governmen is indebed The level of deb persisence Table 4 shows second momens for he economy a seady sae disribuion for differen mauriies. Mos of he momens differ only o he second or hird decimal place across mauriies. The main excepions are he levels of deb and defici: he governmen on average holds asses bu less under longer mauriy. The value of asses when bonds are of 20 years halves he average deb for shor bonds. Table 4 Second momens seady sae Model: Benchmark model c y Defici R MV pb mean sd oe: o provide a more inerpreable quaniy we repor annualised ineres raes insead of bond prices namely R p 00. The inuiion for he lower level of asses as mauriy grows is as follows. I is well known ha in models of opimal policy wih incomplee markes if he governmen has he same discoun facor as agens he governmen accumulaes asses in he long run. More precisely i is easy o exend he resuls in Aiyagari e (2002) Secion III for he case of a linear uiliy of consumpion uc c o prove ha governmen asses go o a very high level. Therefore i is no surprising ha all seady saes for deb have a negaive mean. On he oher hand i is also well known ha wih long bonds fiscal insurance recommends ha he governmen issues long bonds. As argued in Angeleos (2002) and Buera and icolini (2004) governmens should issue long bonds in a model wihou capial accumulaion because long ineres raes are higher when he governmen runs deficis so ha issuing long bonds provides fiscal insurance. osbusch (2008) argues ha he same endency for issuing long deb is presen in an incomplee markes model. For he same reason if a BIS Papers o 65 95

20 governmen accumulaes deb in long bonds he implied volailiy of axes will be higher. I is herefore no surprising ha long-run deb is lower for longer mauriies as holding long bonds causes axes o be more volaile. In oher words accumulaing asses of long mauriy is derimenal o fiscal insurance. This is no he case wih shor bonds as hey provide fiscal insurance when issued. Therefore he level of asses is lower for longer mauriies. Given ha average asse holdings are lower i is naural ha average primary deficis are lower for higher since he value of asses is equal o he expeced presen value of primary deficis also under incomplee markes. For his reason also axes are higher in seady sae for higher. Anoher way of examining he impac of varying he average mauriy of deb is o see wheher his influences how close o he complee marke oucome hese incomplee marke models can ge. Marce and Sco (2009) show ha measures of relaive persisence are a good way of assessing he exen of marke incompleeness and so Figure 4 shows for various variables he measure: P k y Var y y kvar y y k. Figure 4 k-variances benchmark model mauriies and 20: 96 BIS Papers o 65

21 The closer o 0 his measure he less persisence he variable shows whereas he closer o he measure he more he variable shows uni roo persisence. Alhough he long bond model shows less persisence suggesing ha in he case of persisen governmen expendiure shocks he issuance of longer bonds helps provide more fiscal insurance he difference beween he wo cases are minor. Given ha axes are disorionary we are no in a Modigliani-Miller world and how he governmen finances is expendiure can affec he real economy. However he fac ha he differences across mauriies are so small is perhaps no surprising. Wih he governmen only issuing one ype of bond in each case and he yield curve showing broadly similar behaviour a differen mauriies he ax-smoohing properies of deb issuance are achieved mainly hrough he role of deb as a buffer raher han hrough fiscal insurance. Furher we are a his poin following he res of he lieraure in assuming ha every period he governmen buys back all exising deb and hen reissues. So alhough he governmen is issuing 0-period bonds i always buys hem back afer a year. Thus i is effecively always borrowing hrough one-period deb reducing he disincion beween one-period and 0-period bonds. We shall reurn o his issue in a laer secion. 4. Independen powers In Secions 2 and 3 we found ha full commimen implied a igh connecion beween ineres rae policy deb managemen and ax policy: when governmen is in deb and spending is high he governmen promises a ax cu in periods knowing ha his will increase fuure consumpion and hus increase long ineres raes in he curren period. The reader may hink ha his opimal policy is no relevan for he real world for a leas wo reasons. Firs as differen auhoriies influence ineres raes and fiscal policy i is unlikely ha hey will coordinae in he way described above and second i is unlikely ha governmens can commi o a ax cu in he disan fuure and acually carry hrough wih he promise. Some papers in he lieraure reac o his ype of criicism by wriing down models where governmen policy is discreionary. Bu he assumpion ha he governmen has no possibiliy of commiing is also problemaic as governmens frequenly do hings for he very reason hey have previously commied o do so. For hese reasons we change he way policy is decided in his model. We relax he assumpion of perfec coordinaion and assume he presence of a hird agen a moneary auhoriy ha fixes ineres raes in every period. The fiscal auhoriy now akes ineres raes as given and implemens opimal policy given hese ineres raes. We examine an equilibrium where he wo policy powers play a dynamic Markov ash equilibrium wih respec o he sraegy of he oher policy power and hey boh play Sackelberg leaders wih respec o he consumer. More precisely he fiscal auhoriy chooses axes and deb given a sequence for ineres raes while he moneary auhoriy simply chooses ineres raes ha clear he marke and he fiscal auhoriy maximises he uiliy of agens. This assumpion sideseps he issues of commimen because here is now no room for ineres rae wising on he par of he fiscal auhoriy. I is easy o hink of models where even if he moneary auhoriy is independen i can no deviae oo much from he equilibrium ineres raes of he flexible price model. Therefore we ake a limi case and assume ha he moneary auhoriy simply ses in equilibrium ineres raes as: p c E uc (24) u E uc p. u c BIS Papers o 65 97

22 given agens consumpion. ow he fiscal auhoriy will no be able o manipulae ineres raes so i will lose any ineres in making promises o cu fuure axes. To solve his model 2 2 we are looking for an ineres rae policy funcion : R R such ha if long ineres raes a are given by p p g b (25) hen (24) holds and wih he fiscal auhoriy maximising consumer uiliy in he knowledge of all marke equilibrium condiions bu aking he sochasic process for ineres raes as given i chooses a bond policy such ha (25) holds. For he fiscal auhoriy he problem now is a sandard dynamic programming one and as a resul he sae space now only consiss of he variables b and g. An advanage of his model is ha here is no reason now for longer lags o ener his sae vecor as pas Lagrange mulipliers do no play a role. Therefore his separaion of powers approach is an alernaive way o reducing he sae space and simplifying he model s soluion. In his case of independen powers he Lagrangian of he Ramsey planner becomes 0 0 L E u c v x S p b p b M b b M 2 The firs-order condiion wih respec o consumpion is uc v x (uccc uc v xx(c g ) v x ) u cc(pb p b ) 0 and using he governmen s budge consrain gives (26) v x uc vx ucc c uc vxx c gvx ucc g x 0 u c To see he impac of independen powers we calibrae he model as in Secion 3 and consider he case 0. Figure 5 compares he impulse responses o a one sandard deviaion shock o he innovaion in he level of governmen spending when he governmen has deb beween independen powers and he benchmark model of Secion 3. As can be seen he model of independen powers does no show he blip in axes a mauriy. In his case deb managemen is subservien o ax-smoohing and is aimed a lowering he variance of deficis. To beer undersand he magniude of he ineres wising channel we can compare our independen powers model wih our earlier benchmark model. We simulaed he model a differen ime horizons T 40 T 200 and T 5000 discarding he firs 500 periods. We calculaed he sandard deviaion of axes for each realisaion and we averaged i across simulaions. We repea he same exercise for Figure 6 shows he resuls. In shorer sample periods he effec of wising ineres raes in connecion wih iniial period deb is significan and provides a higher level of ax volailiy in he benchmark model. aurally as we increase he sample size he iniial period effec diminishes. The second momens of he model in his secion are shown in Table 5. They are exremely similar o hose of he benchmark model in Table 4. We have essenially a very similar amoun of bond issuance deb persisence ax-smoohing ec he only difference being ha he ineres rae wising adds some ax volailiy bu his volailiy shows up only in second momens wih shor samples as shown in Figure 6. We conclude ha he model of (27) 98 BIS Papers o 65

23 independen powers may be a good model o have in he oolki as i reains many of he ineresing feaures of he Ramsey models i has he same seady sae momens i avoids he echnicaliies arising from he very large sae vecor and i avoids discussion on he role o commimen a very long horizons. There are however issues of ax volailiy showing up in small samples where he wo models differ. Figure 5 Responses o a posiive shock in g benchmark and independen power model Mauriy 0: b 0.5 y * BIS Papers o 65 99

24 Figure 6 Tax volailiy a differen horizons: benchmark and independen powers model 200 BIS Papers o 65

25 Table 5 Second momens seady sae Model: Independen powers c y defici R MV pb mean sd Hold o redempion Wih long bonds he governmen has a choice o make a he end of every period. I can buy back he period bonds issued las period as assumed in Secions 2 and 3. Alernaively i can leave some or all of he ousanding bonds in circulaion unil hey maure a heir specified redempion dae. In models of complee markes wheher or no here is buyback in each period is immaerial: all prices and allocaions remain unchanged. Bu in his paper here are wo reasons why he oucome is differen. The firs reason is ha he sream of payoffs generaed by each policy is quie differen from he poin of view of he governmen: wih buyback he bond pays he random payoff p nex period; if he bond is lef in circulaion unil mauriy he bond pays wih cerainy a. As is well known under incomplee markes no only he presen value of payoffs of an asse are relevan; he iming of payoffs also maers. A second reason for he differences is ha he possibiliies for governmens o wis ineres raes are differen. In Secion 2 we made he exreme assumpion ha he governmen each period buys back he whole sock of ousanding bonds issued las period. As shown in Marchesi (2004) i is normal pracice for governmens no o buy back deb deb is issued and i is paid off a mauriy. In his secion we assume ha bonds are lef o maure o heir redempion dae. In he case of buyback here are only -period bonds ousanding. In he case of holding o redempion here exis bonds a all mauriies beween and even hough he governmen only issues period bonds. Alhough we model he implicaions of holding o redempion an explanaion for why no buyback is sandard pracice 5 is considered beyond he scope of his paper. In his secion we se up a model where deb managers do no buy back deb a he end of each period show how full commimen gives rise o a differen kind of ineres rae wising 5 Conversaions wih deb managers sugges some combinaion of ransacion coss a desire o creae liquid secondary markes a mos mauriies or worries over refinancing risk. For simpliciy we rule ou a hird possibiliy ha governmens choose o buy back only a cerain proporion of ousanding deb. BIS Papers o 65 20

26 ouline how o use condensed PEA o solve for opimal fiscal policy and we show he behaviour of he model. Since we follow closely he analysis of Secions 2 and 3 we omi some deails and focus on he differences. The economy is as before excep ha he governmen budge consrain is now b x g p b (28) so ha he paymen obligaions of he governmen a are he amoun of bonds issued a. We include he deb limis i M b M i (29) Again his limi mimics he value of he newly issued deb a seady sae prices: if he governmen issued b bonds a all periods i would have b unis of bonds of mauriies 2... ousanding so he oal value of deb a seady sae would be budge consrain of he household s problem changes in a parallel way. i i b. The 5. Opimal policy wih mauring deb Subsiuing in equilibrium bond prices and wages ne of axes (28) becomes. c c s.. b u s E u b (30) The Ramsey problem is now o maximise uiliy (2) over choices of c b subjec o his consrain and he deb limis (29) for all. The Lagrangian becomes M b b M 0 c c 0 L E u c v x S u b u where 2 is he Lagrange muliplier associaed wih (30) and 2 are he ones associaed wih he deb limis and i i i i. M M M M The firs-order condiions wih respec o c and b are u v u c u v c g v u b (3) c x cc c xx x cc 0 E u E u (32) c c 2 Wih... M BIS Papers o 65

27 In shor hese FOC have wo differences relaive o he buyback case: in equaion (3) we now have ( ) insead of ( ) and we now have insead of in he maringale condiion (32) o uncerainy and hold o redempion Le us now consider he no uncerainy case when g g. Proceeding in an analogous way o he case of Secion 2.2 we could wrie he implemenabiliy consrain as uc S b i p i0 0 uc0 i or (33) i S b i uc i 0 i (34) for p0. Bonds issued in periods i 2... appropriaely appear in he righ side of he above consrain as wha maers now is he oal value of deb iniially. Le us consider he problem of maximising uiliy when (34) is he sole implemenabiliy consrain. If he governmen is in deb wih b i 0 for all i... i is clear ha in his case ineres rae wising will involve changing ineres raes in he firs periods hence he governmen will promise o cu axes in all periods beween The FOC for consumpion indicaes ha he ax cu will be larger for periods 0... where he mauring deb b is larger. Therefore ax cus now las periods. For consumpion and axes are consan. Bu assuming ha (34) is he sole implemenabiliy consrain as we did in he previous paragraph is no correc for our model. I would be correc in a slighly differen model where he deb limis would be in erms of he oal value of deb for example if deb limis would be MV MV i i i M b p M (35) Take for simpliciy he case 2. I is clear ha he opimal allocaion described in he previous paragraph can be implemened for bond issuances saisfying u c j b 2 b Sj for all Given iniial condiions his provides a uc j 0 difference equaion on b ha saisfies he period- budge consrain (30) and he value of HTM HTM deb limis if M and M were sufficienly large in absolue value. Bu for our model (34) is no sufficien for an equilibrium. This is perhaps surprising as we hink ha wihou uncerainy and one asse one can always complee he markes for sufficienly high deb limis. To see his poin noice ha for he opimal allocaion described above he surplus is consan equal o a level say S for all. The bonds ha would S saisfy he period budge consrain saisfy b 2 b for all.... This pah for bonds would saisfy he difference equaion 6 In he case of hold o redempion he assumpion of independen powers would no simplify he analysis in erms of reducing he sae space. One would sill need lags of b as sae variables. BIS Papers o