OPTIMAL INFLATION TARGETING UNDER ALTERNATIVE FISCAL REGIMES

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1 OPTIMAL INFLATION TARGETING UNDER ALTERNATIVE FISCAL REGIMES Pierpaolo Benigno New York Universiy Michael Woodford Columbia Universiy Inflaion argeing has become an increasingly popular approach o he conduc of moneary policy worldwide since he early 1990s. Mos of he counries ha have adoped inflaion argeing judge he experimen favorably, a leas hus far. In many counries, he adopion of inflaion argeing has been associaed wih reducions in boh he average level and he volailiy of inflaion. Inflaion argeing has been especially successful in sabilizing inflaion expecaions. 1 This sems from inflaion-argeing cenral banks emphasis on a clear medium-erm commimen for inflaion (while emporary deparures from he inflaion arge are allowed) and heir increased communicaion wih regard o he oulook for inflaion over he nex few years. This approach o moneary policy, however, may no be equally suiable for all counries, regardless of heir exising insiuions, he disurbances o which heir economy is subjec, and he oher policies pursued by he governmen. One quesion worhy of discussion is how a counry s fiscal policies migh affec he suiabiliy of inflaion argeing as an approach o he conduc of moneary policy. The heoreical lieraure ha develops he case for inflaion argeing largely neglecs he fiscal consequences of a commimen o We hank Rómulo Chumacero, Norman Loayza, Eduardo Loyo, and Klaus Schmid- Hebbel for useful commens on an earlier draf, Vasco Curdia and Mauro Roca for research assisance, and he Naional Science Foundaion for research suppor. 1. Levin, Naalucci, and Piger (2004) compare inflaion expecaions in inflaionargeing and non-inflaion-argeing counries. Moneary Policy under Inflaion Targeing, edied by Frederic Mishkin and Klaus Schmid-Hebbel, Saniago, Chile Cenral Bank of Chile. 37

2 38 Pierpaolo Benigno and Michael Woodford an inflaion arge. 2 The models used o analyze moneary sabilizaion policy usually absrac from he governmen s budge and he dynamics of he public deb, so ha any fiscal effecs of moneary policy decisions are acily assumed o be irrelevan. This may be an accepable simplificaion if one is choosing a policy for an economy wih sound governmen finances, by which we mean one wih relaively nondisoring sources of revenue and an unquesionable poliical will o mainain governmen solvency. The degree o which such an idealizaion of he circumsances of fiscal policy is realisic varies across counries. As inflaion argeing becomes popular in developing counries ha have recenly had serious problems wih inflaion precisely because of heir precarious governmen finances, one may wonder how safe i is o ignore he inerrelaion beween moneary and fiscal policy choices. A number of auhors sugges ha he appropriaeness of inflaion argeing as a policy recommendaion may depend criically on he naure of fiscal policy. For example, Fraga, Goldfajn, and Minella (2004), in heir discussion of inflaion argeing for developing counries, remark ha...he success of inflaion argeing (...) requires he absence of fiscal dominance (p. 383). They go on o sress ha no only mus fiscal policy be sound in his respec, bu is coninued soundness mus be credible. Their inen is no o sugges developing counries ough no adop inflaion argeing, bu raher o emphasize he imporance of enacing credible fiscal reforms, as well. Neverheless, heir insisence on he need for fiscal commimens ha are no obviously presen in many developing counries raises he quesion of wheher inflaion argeing is illadvised in such counries. Sims (2005) enunciaes exacly his view. He argues ha some counries fiscal policies may make he achievemen of a arge rae of inflaion by he cenral bank impossible, in he sense ha here is no possible raional-expecaions equilibrium in which he arge is fulfilled, regardless of he conduc of moneary policy. He furher assers ha in such a case, aemping o arge inflaion may no only be doomed o frusraion, bu could even be harmful, by leading o less sabiliy (even less sabiliy of he inflaion rae) han migh 2. See, for example, King (1997), Svensson (1997, 1999, 2003), Woodford (2003, chaps. 7 8), Walsh (2003, chap. 11), or Svensson and Woodford (2005) for he heoreical case for some version of inflaion argeing as an opimal policy.

3 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 39 have been achieved hrough oher policies. His essenial argumen is ha if he fiscal regime ensures ha primary budge surpluses are no (sufficienly) increased in response o a moneary ighening, hen a policy inended o conain inflaion namely, raising nominal ineres raes sharply when inflaion rises above he inflaion arge may cause an explosion of he public deb, which ulimaely requires even larger price increases han would have been necessary had he deb no grown. Loyo (1999) and Blanchard (2005) provide examples of models in which orhodox moneary policies of his kind lead o explosive deb dynamics. Our goal in his paper is o analyze he characer of an opimal moneary policy commimen under alernaive assumpions abou he characer of fiscal policy, in order o deermine he condiions under which an opimal policy will be similar o inflaion argeing and he exen o which he form of an opimal moneary policy rule depends on he naure of fiscal policy. To address hese issues, we exend he framework used o analyze opimal moneary sabilizaion policy in Benigno and Woodford (2005a), which allows us o explicily model deb dynamics and he condiions required for ineremporal governmen solvency and also o rea he effecs of ax disorions. We consider a variey of assumpions regarding he characer of fiscal policy, including he kind of fiscal regime under which he real primary budge surplus is no adjused o preven explosion of he public deb as a resul of an ineres rae hike ha is a he hear of he Loyo (1999) and Blanchard (2005) examples of possible perverse effecs of igh-money policies. 1. A MODEL WITH NONTRIVIAL MONETARY AND FISCAL POLICY CHOICES We use a sandard new Keynesian model of he rade-offs involved in moneary sabilizaion policy, augmened o ake accoun of ax disorions The Model The goal of policy is assumed o be he maximizaion of he level of expeced uiliy of a represenaive household. In our model, each household seeks o maximize 3. Furher deails of he derivaion of he srucural equaions of our model of nominal price rigidiy can be found in Woodford (2003, chap. 3).

4 40 Pierpaolo Benigno and Michael Woodford 1 o U E u( C ) v( H ( j) ) dj 0 0 β ; ξ ; ξ 0, (1) = 0 where C is a Dixi-Sigliz aggregae of consumpion of each of a coninuum of differeniaed goods, 1 θ ( θ 1) C c ( i) di 0 ( θ 1) θ, (2) wih an elasiciy of subsiuion equal o θ > 1, and H (j) is he quaniy supplied of labor of ype j. Each differeniaed good is supplied by a single monopolisically compeiive producer. We assume ha here are many goods in each of an infinie number of indusries; he goods in each indusry, j, are produced using a ype of labor ha is specific o ha indusry, and heir prices are also changed a he same ime. The represenaive household supplies all ypes of labor and consumes all ypes of goods. To simplify he algebraic form of our resuls, we resric aenion in his paper o he case of isoelasic funcional forms, 1 σ σ C C u ( C;ξ) 1 1 σ 1 1 and λ + ν v ( H;ξ) H H 1 + ν 1 ν, where σ, ν > 0 and { C, H } are bounded exogenous disurbance processes. (We use he noaion ξ o refer o he complee vecor of exogenous disurbances, includingc and H.) We assume a common echnology for he producion of all goods, in which (indusry-specific) labor is he only variable inpu, y ( i)= A f ( h ( i) )= A h ( i) 1 φ, where A is an exogenously varying echnology facor, and φ > 1. We inver he producion funcion o wrie he demand for each ype of labor as a funcion of he quaniies produced of he various differeniaed goods, and we use he ideniy Y = C + G

5 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 41 o subsiue for C, where G is exogenous governmen demand for he composie good. We can hen wrie he uiliy of he represenaive household as a funcion of he expeced producion plan, {y (i)}. 4 The producers in each indusry fix he prices of heir goods in moneary unis for a random inerval of ime, as in he model of saggered pricing inroduced by Calvo (1983). We le 0 α < 1 be he fracion of prices ha remain unchanged in any period. A supplier ha changes is price in period chooses is new price p (i) o maximize E T j α Q T ( p() i pt PT YT τt ), Π,, ;,, ξ T, (3) T = where Q,T is he sochasic discoun facor by which financial markes discoun random nominal income in period T o deermine he nominal value of a claim o such income in period, and α T is he probabiliy ha a price chosen in period will no have been revised by period T. In equilibrium, his discoun facor is given by Q, T = β u ( C ; ξ ) P T c T T u ( C ; ξ ) P The funcion, c T. (4) j Π( p, p, P; Y, τ, ξ) ( 1 τ) py ( p P) µ w θ θ I 1 v h f Y ( p / P) / A ; ξ 1 θ p f ( ) ( ; ) Y p / P / A u Y G ξ c indicaes he afer-ax nominal profis of a supplier wih price p, in an indusry wih common price p j, when he aggregae price index is equal o P, aggregae demand is equal o Y, and sales revenues are axed a rae τ. Profis are equal o afer-ax sales revenues ne of he wage bill. The real wage demanded for labor of ype j is assumed o be given by an exogenous markup facor µ w (which is allowed o vary over ime, bu is assumed common o all labor markes) imes he marginal rae 4. We assume ha he governmen needs o obain an exogenously given quaniy of he Dixi-Sigliz aggregae in each period, in a cos-minimizing fashion. The governmen hus allocaes is purchases across he suppliers of differeniaed goods in he same proporion as do households, and he index of aggregae demand, Y, is he same funcion of he individual quaniies, {y (i)}, as C is of he individual quaniies consumed, {c (i)}, defined in equaion (2).

6 42 Pierpaolo Benigno and Michael Woodford of subsiuion beween work of ype j and consumpion, and firms are assumed o be wage akers. We allow for wage markup variaions in order o include he possibiliy of a pure cos-push shock ha affecs equilibrium pricing behavior while implying no change in he efficien allocaion of resources. Variaion in he ax rae, τ, has a similar effec on his pricing problem (and hence on supply behavior), so when he evoluion of he ax rae is reaed as an exogenous poliical consrain, variaions in he ax rae also ac as pure cos-push shocks. We absrac here from any moneary fricions ha would accoun for a demand for cenral bank liabiliies ha earn a subsandard rae of reurn; we noneheless assume ha he cenral bank can conrol he riskless shor-erm nominal ineres rae i. 5 This, in urn, is relaed o oher financial asse prices hrough he arbirage relaion, i =( EQ + 1),. (5) We assume ha he zero lower bound on nominal ineres raes never binds under he opimal policies considered below. We herefore do no need o inroduce any addiional consrains on he possible pahs of oupu and prices associaed wih he need for he chosen evoluion of prices o be consisen wih a nonnegaive nominal ineres rae. Our absracion from moneary fricions, and hence from he exisence of seignorage revenues, does no mean ha moneary policy has no fiscal consequences, since ineres rae policy and he equilibrium inflaion ha resuls from i have implicaions for he real burden of governmen deb. In our baseline analysis, we assume ha all public deb consiss of riskless nominal one-period bonds. 6 The nominal value, B, of end-of-period public deb hen evolves according o a law of moion, B = ( 1 + i 1) B 1 P s, (6) where he real primary budge surplus is given by s τ Y G ζ, (7) where ζ represens he real value of (lump-sum) governmen ransfers. Raional-expecaions equilibrium requires ha he expeced pah 5. For a discussion of how his is possible even in a cashless economy of he kind assumed here, see Woodford (2003, chap. 2). 6. The consequences of longer-mauriy public deb are discussed in secion 3.3.

7 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 43 of governmen surpluses mus saisfy an ineremporal solvency condiion, b 1 P P 1 = E R, T st, (8) T = in each sae of he world ha may be realized a dae, where R,T Q,T P T /P is he sochasic discoun facor for a real income sream. We consider alernaive assumpions abou he degree of endogeneiy of he various conribuions o he governmen budge in equaion (7). In he convenional lieraure on opimal moneary sabilizaion policy, boh G and τ are exogenous processes (among he real disurbances o which moneary policy may respond), bu ζ can be adjused endogenously o ensure ineremporal solvency in a way ha creaes no deadweigh loss, so ha he fiscal consequences of moneary policy are no significan for welfare. We also consider a more realisic case in which G and ζ are exogenous disurbances, and addiional governmen revenue has a posiive shadow value, bu τ can be varied endogenously o minimize deadweigh loss. In he mos consrained case, where he concerns sressed by Sims (2005) arise, G, τ, and ζ are all exogenous processes deermined by poliical consrains. 1.2 An Associaed Linear-Quadraic Policy Problem We approximae he soluion o our opimal policy problem by he soluion o an associaed linear-quadraic (LQ) problem; he derivaion of he approximaions is presened in deail in Benigno and Woodford (2004). We show ha we can define an LQ problem wih he propery ha he soluion o he LQ problem is a linear approximaion o opimal policy in he exac model when he exogenous disurbances are small enough. Firs, we show ha maximizaion of expeced uiliy is (locally) equivalen o minimizaion of a discouned loss funcion of he form E β q y Y Y q π *, (9) 2 π = 0 where he arge oupu level, Y *, is a funcion of exogenous disurbances. If seady-sae ax disorions are no oo exreme, hen q y, q π > 0 and he loss funcion is convex, as assumed in convenional accouns of he goals of moneary sabilizaion policy.

8 44 Pierpaolo Benigno and Michael Woodford The consrains on possible equilibrium oucomes are given by log-linear approximaions o he srucural equaions of he model described above. Here we omi derivaions and proceed direcly o he log-linear forms. Firs, here is an aggregae supply relaion beween curren inflaion and real aciviy, π = κ Yˆ + ψτ + c βeπ ˆ ξ +, (10) ξ + 1 where κ, ψ > 0. This is he familiar new Keynesian Phillips curve, augmened o include he cos-push effecs of variaions in he sales ax. We can wrie he consrain in erms of he welfare-relevan oupu gap, y Y Y *, in which case equaion (10) becomes π = κ( y + ψτ + u)+ βeπ, + 1 where u is a composie cos-push erm associaed wih exogenous disurbances oher han variaions in he ax rae. 7 In oher words, π = κ y + ψ τ τ * βeπ +, (11) + 1 where τ * is a funcion of exogenous disurbances ha indicaes he ax change needed o offse he oher cos-push erms. Anoher consrain on he possible equilibrium pahs of inflaion, oupu, and ax raes is he condiion for ineremporal governmen solvency (equaion 8). 8 A log-linear approximaion o equaion (8) akes he form 7. An obvious source of such disurbances would be variaions in he wage markup, µ w. This is he only source of variaions in u when he seady-sae involves no disorions. In he case of a disored seady sae, however, mos oher kinds of real disurbances also have cos-push effecs (as shown in Benigno and Woodford, 2004), since hey do no move he flexible-price equilibrium level of oupu o precisely he same exen (in percenage erms) as hey move he efficien level of oupu. The laer sources of cos-push erms become more imporan as he magniude of he seady-sae disorions increases. 8. This does no amoun o requiring ha fiscal policy be Ricardian; we consider below he consequences of non-ricardian fiscal policies of he kind assumed in he warnings of Sims (2005). Insead, equaion (8) is a condiion ha mus hold in equilibrium under any policy, and i consrains he possible oucomes ha can be achieved in deermining he bes equilibrium under cerain consrains on possible policies.

9 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 45 * b 1 T y f E by yt b = + ( ) + 1 π σ 1 β β τ τt τ T, (12) T = where f is a composie of he various exogenous disurbances ha we refer o as fiscal sress. We have wrien he consrain in erms of he oupu gap and he ax gap, τ τ T * T, which indicaes deparures of he ax rae from he level consisen wih complee sabilizaion of boh inflaion and he oupu gap. Therefore, he erm f (or, more precisely, he sum b 1 + f ) measures he exen o which ineremporal solvency prevens complee achievemen of he sabilizaion goals represened in equaion (9). Here we have subsiued equaion (4) for he sochasic discoun facor (and replaced C by Y G ) o obain a relaion ha involves only he iniial public deb and he pahs of inflaion, oupu, axes, and he various exogenous variables. The effecs of ineres rae policy on deb dynamics are he key o he scenarios of Loyo, 1999, and Blanchard, 2005, under which igh money can be inflaionary. We ake hese effecs ino accoun hrough he presence of he sochasic discoun facor in equaion (8), which is linked o he ineres rae conrolled by he cenral bank hrough equaion (5). Ineres raes do no appear in equaion (12) because we have already subsiued for hem using he connecion beween ineres raes and he pahs of oupu and inflaion ha mus hold in equilibrium, bu he effec of igh money on he burden of he public deb is noneheless aken ino accoun in his equaion. In wriing equaion (12) in he form given, we have reaed ζ (real ne ransfers) as one of he exogenous disurbances ha affecs he fiscal sress erm. For he case in which ne ransfers are endogenous and can be varied o ensure solvency, we need o separae ou he ζ erm from he oher (exogenous) deerminans of f. The solvency consrain ceases o bind, however, given ha he level of ransfers affecs neiher he aggregae supply rade-off (equaion 11) nor he loss funcion (equaion 9), so ha policymakers are free o vary ζ as necessary o saisfy equaion (12). Thus we do no need o wrie he solvency consrain, excep for he case in which ζ is exogenous. 2. OPTIMAL INFLATION TARGETING: THE CONVENTIONAL ANALYSIS We begin by using he framework skeched in he previous secion o recapiulae well-known argumens for a form of flexible inflaion

10 46 Pierpaolo Benigno and Michael Woodford argeing as a way of implemening an opimal sae-coningen moneary policy, highlighing he role of (ofen aci) assumpions abou fiscal policy in deriving hese familiar resuls. 9 The convenional analysis of opimal moneary sabilizaion policy in a new Keynesian model corresponds o he case of he above model in which he processes {G, τ } are boh exogenously given as poliical consrains on wha policy can achieve, while he level of ne lump-sum ransfers, ζ, is an endogenous policy variable (along wih he shor-erm nominal ineres rae). When lump-sum ransfers can be chosen o faciliae sabilizaion policy, he ineremporal solvency consrain ceases o bind, and i can be omied from our descripion of he policy problem. We can similarly omi any reference o he pah of he public deb. Moreover, when he level of disoring axes is given exogenously, we can rea he τ erm in equaion (10) he same as he oher cos-push erms. The problem of opimal sabilizaion policy is hen simply o find pahs {π, y } o minimize equaion (9) subjec o he single consrain, π = κ( y + u )+ βe π, (13) +1 where he definiion of u is now modified o include he cos-push effecs of variaions in τ (if hese are presen). This is he opimal policy problem reaed, for example, in Clarida, Galí, and Gerler (1999). Here we emphasize how his concepion of he goals of moneary sabilizaion policy provides an argumen for inflaion argeing. A firs, simple conclusion abou opimal policy under hese assumpions is ha in he absence of cos-push disurbances, opimal policy would involve adjusing ineres raes as necessary o mainain zero inflaion a all imes. This is easily seen from he fac ha if u = 0 a all imes, equaion (13) is consisen wih mainaining boh a zero inflaion rae and a zero oupu gap a all imes, and such an oucome obviously minimizes he loss funcion (equaion 9). This provides one argumen for inflaion argeing: if cos-push shocks are unimporan (because disorions from marke power and axes are boh small, on average, and fairly sable over ime), hen a low, sable inflaion rae is opimal, regardless of he degree of variabiliy in real aciviy ha his may enail (owing o he effecs 9. See, for example, Clarida, Galí, and Gerler (1999), Svensson (2003), Woodford (2003, chaps. 7 8; 2004), or Svensson and Woodford (2005) for more deailed presenaions of he argumens summarized here.

11 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 47 of preference and echnology disurbances on Y *). I also implies somehing of more general validiy: even when random cos-push shocks of subsanial magniude do occur, opimal policy should involve zero inflaion, on average. (This follows from he previous resul using he cerainy-equivalence propery of linear- quadraic opimizaion problems.) 10 The opimal long-run inflaion arge is hus quie low (zero, in our simple model), regardless of he degree of disorions in he economy or he degree o which he opimal oupu level may exceed he level associaed wih sable prices. Given ha any deparures from his consan long-run average inflaion rae semming from cos-push shocks should be ransiory, expeced inflaion in he medium erm should always be near zero. Therefore, our resul jusifies a policy ha seeks o mainain low and sable medium-erm inflaion expecaions, as a leas one crierion ha an opimal policy should saisfy. The concepion of opimal sabilizaion policy jus proposed provides an imporan reason for a cenral bank o commi iself o an explici arge for inflaion, raher han for oher variables (such as real aciviy), even when cos-push shocks are expeced o be nonrivial. In he opimal conrol of a forward-looking sysem he kind of problem jus posed above he advance commimen of policy generally offers advanages by influencing expecaions a earlier daes in a way ha improves he available sabilizaion oucomes a hose daes. Bu wha aspec of fuure expecaions maer? When he only consrain on wha policy can achieve is he aggregae supply relaion (equaion 13), he only aspec of fuure expecaions ha affecs he inflaion and oupu gap ha can be achieved in some period is expecaions regarding fuure inflaion, E π +1. Hence, his ype of commimen is direcly relevan: commiing o achieve a paricular inflaion rae in he fuure, which migh be differen from wha would oherwise be chosen laer o bes achieve one s sabilizaion goals. Given ha he role of a policy commimen should be o anchor he public s inflaion expecaions, a commimen regarding fuure inflaion and he cenral bank s communicaion of he oulook for inflaion are sraighforward ways o achieve he benefis associaed wih an opimal policy commimen. 10. See Svensson and Woodford (2003) for a discussion of cerainy equivalence in he conex of policy problems wih forward-looking consrains, like he one considered here.

12 48 Pierpaolo Benigno and Michael Woodford Beyond hese general consideraions, one can easily characerize he opimal sae-coningen evoluion of prices and quaniies under a paricular assumpion abou he characer of he disurbances affecing he economy (hough his aspec of our conclusions depends srongly on he precise deails of our assumed model of he ransmission mechanism of moneary policy). The following firs-order condiions are associaed wih he policy problem saed above: 1 q π π = κ ϕ ϕ 1 ( ) and (14) q y y = ϕ, (15) each of which mus hold for each 0. Here, ϕ is he Lagrange muliplier associaed wih he aggregae supply consrain (equaion 13). We can solve condiions (14) and (15), ogeher wih he aggregae supply relaion (equaion 13), for he opimal evoluion of {π, y } given he disurbances {u }. The opimal sae-coningen responses can be implemened hrough commimen o a consan arge for he oupu-gap-adjused price level: p qy p q y +, (16) κ π where p denoes log P, as discussed in Woodford (2003, chap. 7). A argeing rule of his form deermines he opimal rade-off beween price increase and oupu decline ha should be seleced when he shock occurs; he policy sance should be neiher so igh as o cause p o decline (as would be required for here o be no increase in prices) nor so loose as o allow p o rise (as would be required for here o be no reducion in oupu relaive o arge oupu). A he same ime, commimen o adhere o such a rule in he fuure auomaically implies invariance of he expeced long-run price level and oupu gap, and i deermines he opimal rae of reurn of boh variables o hose long-run levels. The oupu gap should no reurn o zero oo quickly (which would allow prices o remain high and so involve an increase in he gap-adjused price level) or oo slowly (which would cause he gap-adjused price level o fall once he cos-push disurbance had dissipaed). Figure 1 provides an example of he opimal impulse responses of inflaion and he oupu gap o a purely ransiory posiive cos-push shock (ha is, he soluion o he firs-order condiions lised

13 Figure 1. Impulse Responses o a Transiory Cos-Push Shock under Discreionary Policy and an Opimal Commimen. Inflaion Oupu Price level Source: Woodford (2003, chap. 7, fig. 7.3).

14 50 Pierpaolo Benigno and Michael Woodford above in he case of such a disurbance). 11 The dynamic pahs of he log price level and he oupu gap are perfec mirror images of one anoher, up o scale, so ha p is no allowed o vary. This is an example of a robusly opimal policy rule in he sense of Giannoni and Woodford (2002): commimen o he same arge crierion is opimal, regardless of he saisical properies of he disurbance process. (The opimal dynamic responses shown in figure 1 will be differen in he case of a shock ha is no compleely ransiory or no wholly unexpeced when i occurs, bu he opimal responses of p and y will always mirror one anoher in he way shown in he figure.) The firs-order condiions of equaions (14) and (15) can be used direcly o show ha p mus no change over ime under an opimal policy, wihou making any assumpions abou he naure of he disurbance. Such a policy prescripion can be viewed as a form of flexible inflaion argeing, since he requiremen ha p = 0 can equivalenly be wrien as π q + y = 0. y κq π In his form, he rule saes ha he accepable rae of inflaion a any poin in ime should vary depending on he rae of change of he oupu gap. Svensson and Woodford (2005) discuss a more realisic version of his prescripion, which incorporaes delays in he effecs of moneary policy on spending and prices. Here, we are ineresed in he ways in which his familiar analysis mus be complicaed under alernaive assumpions abou fiscal policy. 3. OPTIMAL POLICY WHEN ONLY DISTORTING TAXES ARE AVAILABLE: THE CASE OF OPTIMAL TAX SMOOTHING I is more realisic o assume ha lump-sum axes are no available o offse he fiscal consequences of moneary policy decisions. When we assume he process {ζ } o be exogenously given, he ineremporal solvency condiion represens an addiional binding consrain on he se of possible equilibrium pahs for inflaion and oupu. In Benigno 11. This calculaion is furher explained in Woodford (2003, chap. 7). The parameer values assumed are β = 0.99, κ = 0.024, and q y /q π = The figure also shows, for purposes of comparison, he equilibrium responses ha would occur under discreionary opimizaion. In his case, he gap-adjused price level does no change in he period of he shock, bu i is expeced o be allowed o rise subsequenly. This expecaion resuls in a less favorable inflaion-oupu rade-off for he cenral bank in he period of he shock.

15 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 51 and Woodford (2004), we consider opimal moneary policy in such an environmen, under he assumpion ha he pah of he disoring ax rae, {τ }, is chosen opimally in response o he various ypes of real disurbances considered in he model. Here we recapiulae he main conclusions of ha analysis, before urning o cases in which fiscal policy is assumed o be less flexible or no opimally deermined. In his case, we view moneary and fiscal policy decisions as being joinly deermined in a coordinaed fashion, so as o solve a single social welfare problem. The planning problem is o find sae-coningen pahs {π, y, τ } o minimize equaion (9) subjec o he wo consrains of equaions (11) and (12). An especially simple version of his problem is he limiing case in which prices are perfecly flexible. This case clearly illusraes why he absence of lump-sum axes can make i opimal for he inflaion rae o be highly responsive o fiscal developmens, conrary o wha inflaion argeing is generally assumed o imply. Some auhors argue ha his kind of analysis is relevan o he choice of moneary insiuions in Lain America (Sims, 2002). 3.1 Opimal Policy If Prices Are Flexible In he flexible-price limi of he above model, he coefficien q π in equaion (9) is equal o zero, and κ in equaion (11) is also zero (ha is, he aggregae supply relaion is compleely verical). The policy problem reduces o he minimizaion of 1 q E 0 2 y y 0 β, (17) 2 = 0 subjec o he consrains ψ τ τ * = 0 (18) y + and equaion (12). Using equaion (18) o subsiue for y in equaion (17) allows us o equivalenly wrie he sabilizaion objecive as 2 0 E 0 β τ τ *, = 0 in which case he policy objecive can be hough of as ax smoohing, as in Barro s (1979) classic analysis Thus our sabilizaion objecive (equaion 9) does no omi he concerns of he lieraure on opimal ax smoohing; he welfare losses associaed wih a failure o opimally ime he collecion of axes are implici in he oupu-gap sabilizaion objecive.

16 52 Pierpaolo Benigno and Michael Woodford The soluion will involve y = 0 a all imes, since i is feasible o achieve his if he moneary and fiscal auhoriies cooperae. The fiscal auhoriy mus choose τ τ = * a all imes o ensure his, while he moneary auhoriy mus vary he inflaion rae, π, o ensure governmen solvency. Equaion (12) requires ha in such an equilibrium, π = b 1 + f. Thus unexpeced changes in he fiscal sress erm mus be accommodaed enirely by surprise variaions in he inflaion rae, as in Chari and Kehoe (1999). The ax rae should flucuae only o he exen ha τ * flucuaes; ha is, only o he exen ha variaions in he ax rae are useful as a supply-side policy, o offse inefficien supply disurbances. 13 This conclusion implies ha an opimal policy will involve highly volaile inflaion and exreme sensiiviy of inflaion o fiscal shocks. This is he basis of Sims (2002) criique of dollarizaion as a policy prescripion for Mexico; a leas a sric form of inflaion argeing would presumably be rejeced on he same grounds. This analysis, however, neglecs he welfare coss of volaile inflaion, which are sressed in he lieraure on inflaion argeing. Here we consider he imporance of he Chari-Kehoe argumen in he presence of a realisic degree of price sickiness. 3.2 Opimal Policy If Prices Are Sicky In he more general case of our model (wih some degree of price sickiness), he firs-order condiions for he opimal policy problem saed above are 1 q π π = κ ( ϕ1 ϕ1 1 ) ϕ2 ϕ2 1 q y ϕ,, (,, ), (19) = ( ) b ϕ1 1 β σ ϕ + σ ϕ, (20), 2, 2, 1 y y = E ϕ, and (21) + 2, 2, As shown in Benigno and Woodford (2004), a wide range of inefficien supply disurbances may require such an offse, if he seady sae is sufficienly disored as a resul of eiher marke power or a large public deb.

17 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 53 ψϕ = ( 1 β) b ϕ, (22) 1, τ 2, where now ϕ 1, is he Lagrange muliplier associaed wih he aggregae supply relaion and ϕ 2, is he muliplier associaed wih he ineremporal solvency condiion. Condiions (19) (22), ogeher wih he wo srucural equaions (11) and (12), are o be solved for he pahs of he endogenous variables, {π, y, τ, b, ϕ1,, ϕ 2, }, given an exogenous process for {f }. The ype of response o shocks implied by hese equaions can be illusraed using a numerical example. We adop he same numerical parameer values as in Benigno and Woodford (2004), implying ha β = 0.99, σ 1 = 0.157, κ = , ψ = 0.397, b τ = 8.33, and ha he relaive weigh on oupu-gap sabilizaion is q y /q π = As in ha paper, we examine he effecs of an exogenous increase in ransfer programs, ζ, equal o one percen of seady-sae GDP. Here, however, we consider he consequences of alernaive degrees of persisence of such a disurbance; we assume ha he value of ζ following he shock is expeced o decay a he rae ρ, where he coefficien of serial correlaion, ρ, is allowed o ake values beween zero (he case shown in he earlier paper) and 0.7. Figure 2 shows he impulse response of he shock, ζ, for he differen values of ρ considered. Figure 3 hen shows he impulse response of he public deb, b, in response o a pure fiscal shock of his kind under he opimal policy, for each of he alernaive values of ρ. Figure 4 shows he corresponding responses of he ax rae, τ, under he opimal policy, and figure 5 he associaed responses of he inflaion rae. In conras o he opimal policy in he case of flexible prices (discussed furher in Benigno and Woodford, 2004), i is opimal o respond o a pure fiscal shock of his kind by permanenly increasing he level of real public deb and planning a corresponding permanen 14. Here he ineres-sensiiviy of expendiure and he slope of he Phillips curve are calibraed o agree wih he economeric esimaes of Roemberg and Woodford (1997) for he US economy, and he fiscal parameers are calibraed o imply ha seady-sae ax revenues are 20 percen of GDP and ha he seady-sae public deb is 60 percen of annual GDP. The assumed weighs on he wo sabilizaion objecives in he loss funcion (9) are he ones ha correspond o maximizaion of expeced uiliy, given he parameers of he model, as explained in Benigno and Woodford (2004). Noe ha in our presen noaion, π is a quarerly inflaion rae; if we insead wrie he loss funcion in erms of an annualized inflaion rae, he relaive weigh on oupu-gap sabilizaion would insead be This is slighly smaller han he value quoed in Roemberg and Woodford (1997), mainly as a consequence of he ax disorions assumed here, bu absraced from in ha paper.

18 54 Pierpaolo Benigno and Michael Woodford increase in he ax rae. (The increase in he level of real public deb under he opimal policy is more gradual he more persisen he fiscal shock, whereas i was immediae in he case of he purely ransiory shock considered in our previous paper.) Opimal policy does involve some unanicipaed inflaion a he ime of he shock, as in he Chari-Kehoe analysis, bu i is no nearly large enough o compleely offse he fiscal sress, which is why fuure axes are also increased. Figure 2: Alernaive Fiscal Shocks Source: Auhors' compuaions. Figure 3: Impulse Response of he Public Deb o a Pure Fiscal Shock, for Alernaive Degrees of Persisence Source: Auhors' compuaions.

19 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 55 Figure 4: Impulse Response of he Tax Rae o a Pure Fiscal Shock, for Alernaive Degrees of Persisence Source: Auhors' compuaions. Figure 5: Impulse Response of he Inflaion Rae o a Pure Fiscal Shock, for Alernaive Degrees of Persisence Source: Auhors' compuaions. As shown in figure 5, he inflaionary impac of a fiscal shock under he opimal policy regime is quie small. In he case of a purely ransiory (one-quarer) increase in he size of ransfer programs by an amoun equal o one percen of GDP, opimal policy allows an increase in he inflaion rae ha quarer of only wo basis poins (a an annualized rae). 15 Moreover, he increase in inflaion is 15. The log price level is hus allowed o increase ha quarer by only half a basis poin.

20 56 Pierpaolo Benigno and Michael Woodford limied o he quarer of he shock. This compares wih an increase in he inflaion rae of nearly wo percenage poins under he opimal policy in he case of flexible prices. This conclusion ha he opimal inflaion response is small does no depend on an exreme calibraion of he degree of price sickiness. In Benigno and Woodford (2004), we show ha he opimal response o a purely ransiory fiscal shock is similarly small even if prices are assumed o be much less sicky han under he calibraion used here; here is a dramaic difference beween opimal policy under fully flexible prices and under even slighly sicky prices (ha is, he shor-run aggregae supply rade-off is no compleely verical). The opimal inflaion response is larger if he shock is more persisen, since in his case he cumulaive cos of he increased ransfers and hus he oal increase in fiscal sress is several imes as large. Even when ρ = 0.7, however, he opimal increase in he inflaion rae is only abou seven basis poins. Finally, he effec on inflaion is purely ransiory under opimal policy, regardless of he degree of persisence of he fiscal shock iself. This las conclusion ha variaions in inflaion should be purely ransiory under he opimal policy, so ha he expeced rae of inflaion never varies a all is quie robus o he ype of shock considered. The conclusion follows direcly from he firs-order condiions ha characerize opimal policy. Condiion (19) implies ha forecasable variaions in he inflaion rae should be allowed only o he exen ha here are forecasable variaions in one or he oher of he Lagrange mulipliers. Condiion (21) implies ha here are no forecasable variaions in he muliplier associaed wih he solvency consrain, while condiion (22) implies ha he wo mulipliers should covary perfecly wih one anoher, so ha here are no forecasable variaions in he muliplier associaed wih he aggregae supply consrain eiher, under an opimal policy. The fiscal consequences of moneary policy hus maer if all sources of governmen revenue are disoring. This creaes addiional reasons for deparures from sric price sabiliy o be opimal. I is now opimal for he inflaion rae o vary, a leas o some exen, in response o disurbances (such as a change in he size of governmen ransfer programs) ha are irrelevan in he classic analysis reviewed in he previous secion. Even so, opimal policy coninues o possess imporan feaures of an inflaion argeing regime. The rae of inflaion ha is forecasable for he

21 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 57 fuure should never vary, regardless of he kind of disurbances hiing he economy, and he unforecasable variaions in inflaion ha should be allowed are quie small. I is no longer opimal o arge a consan value for he oupugap-adjused price level, p. In fac, he opimal policy now involves some degree of base drif in he price level, since he ransiory inflaion shown in figure 5 permanenly shifs he price level. Noneheless, opimal moneary policy can be characerized by commimen o a arge crierion ha is only a sligh generalizaion of he one presened above for he case of lump-sum axes. We reurn o his opic in secion 6 below. 3.3 Consequences of Addiional Fiscal Insrumens The analysis of Benigno and Woodford (2004) assumes ha a small and quie specific se of policy insrumens are available o he fiscal auhoriy: he only source of governmen revenue is a proporional sales ax, and he only kind of governmen deb ha may be issued is a very shor-erm (one-period) riskless nominal bond. Here we briefly discuss he consequences of allowing for addiional insrumens and, hence, a broader range of possible fiscal policies. No surprisingly, addiional fiscal insrumens, if used skillfully enough, can allow a beer equilibrium o be achieved. This can make i simpler o characerize opimal moneary policy, since we no longer have o rely on a limied se of insrumens o simulaneously serve muliple sabilizaion objecives. Suppose, for example, ha i is possible o independenly vary he level of several differen ypes of disoring axes. Wih wo disinc ax raes, he cos-push erm, ψτ, in equaion (10) becomes ψ τ 1 1 ψ2 τ, + 2,, while he erm b τ τ in equaion (12) becomes b b - 1 τ 1 2 τ, + 2,. In general, no only will here be differen elasiciies in he case of differen axes, bu he raios of he elasiciies will no be he same in he wo equaions; he fac ha a given percenage increase in one ax rae resuls in a 20 percen larger increase in revenues han ha resuling from a similar increase in a second ax rae does no imply ha i also resuls in a 20 percen larger cos-push effec. The exisence of muliple axes ha can be independenly varied (and are no a some boundary value under an opimal policy) hus allows he fiscal auhoriy o independenly shif he aggregae supply relaion and affec he governmen s budge.

22 58 Pierpaolo Benigno and Michael Woodford If his is possible, hen a lump-sum ax is essenially possible, as some combinaion of ax increases and decreases will be able o increase ax revenues wihou any ne effec on he aggregae supply relaion. 16 We are no, however, reurning o he classic siuaion analyzed in secion 2. This seup acually simplifies maers, for ax policy can now be used o offse he cos-push effecs of oher disurbances, wihou any consequences for governmen solvency. Consrain (12) herefore ceases o bind, as in secion 2, bu ax policy can be used o shif he aggregae supply relaion, as in secions 3.1 and 3.2. Opimal policy hen involves using axes o offse he cos-push erm, u, enirely and hen applying moneary policy o compleely sabilize boh inflaion and he oupu gap. (Taxes are also used o ensure ha his equilibrium is consisen wih ineremporal governmen solvency.) In such a case, he opimal moneary policy will be a sric inflaion arge ha mainains π = 0 a all imes, regardless of he shocks o which he economy may be subjec. 17 The case for inflaion argeing is hus quie srong indeed when ax policy can be varied in any of a range of direcions and he fiscal auhoriy can be expeced o exercise is power skillfully. This may no be of he greaes pracical ineres, however. For insance, if he ax raes are each required o be nonnegaive, hen i may be opimal o raise all revenue using only one ax namely, he one wih he lowes raio of ψ j o b j (and hus wih he leas disorion creaed per dollar of revenue raised). The opimal policy problem would hen end up being similar o he one reaed above, in which here is assumed o be only a single ype of disoring ax. Allowing for he possibiliy of issuing oher forms of governmen deb would also increase he flexibiliy of fiscal policy and reduce he consrains on wha moneary policy can achieve. For example, if i were possible o issue arbirary kinds of sae-coningen deb, hen in principle i would be possible o arrange for b 1 o vary wih he sae ha is realized a dae in such a way ha b 1 + f never varies, regardless of he exogenous disurbances. Complee sabilizaion of boh inflaion and he oupu gap would again be possible, and he opimal moneary policy would be a sric inflaion arge of zero. 16. Here we assume ha he various axes in quesion affec all secors of he economy idenically, as in he presence of boh a sales ax and a wage income ax. Under his assumpion, axes creae no disorions oher han he effec indicaed by he cos-push erm in he aggregae supply relaion. 17. Our abiliy o achieve he firs-bes oucome wih a sufficien number of axes is reminiscen of he conclusion of Correia, Nicolini, and Teles (2003) in he conex of a model wih a differen kind of price sickiness.

23 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 59 However, he supposiion ha sae-coningen payoffs on governmen deb can be arranged wih such sophisicaion is hardly realisic. One possibiliy is for counries o use mauriy o vary he kind of deb ha hey issue. If governmen deb does no all maure in one period, hen b 1 is no longer a predeermined sae variable; insead, i depends on he marke valuaion of bonds in period, which generally depends on he shocks ha occur a ha dae. Since he prices of bonds wih differen mauriies respond disincly o shocks occurring a dae, differen mauriy srucures of he public deb will have varying effecs on he sae coningency of b 1. Wih a sufficien number of mauriies available, i may well be possible once again o bring abou he kind of sae coningency ha makes b 1 + f independen of shocks, hereby eliminaing he need for sae-coningen deb, as proposed by Angeleos (2001). Boh inflaion and he oupu gap can hus be fully sabilized, and a sric inflaion arge would again be he opimal moneary policy. To develop hese poins in more deail, we exend ou analysis o allow for he exisence of longer-mauriy nominal governmen deb. In he mos general case, he ineremporal budge consrain (equaion 8) akes he form E R T st E R b = T =,, T 1, T T = P 1 P, T where for any T, b 1,T denoes he real value a ime 1 of he deb ha maures a ime T. A log-linear approximaion can be compued as before, yielding b E d σ y + T T s T = s= f + ( 1 β) E β T by yt + b T T T = τ τ * τ. Here we have defined T π = (23) b b b T 1, T T = β, T = b where b i is he seady-sae real value of i-period deb, and b is he seady-sae real value of all ousanding governmen liabiliies, given by

24 60 Pierpaolo Benigno and Michael Woodford i b = β 1 bi. i= 1 The weighs, d i, are defined as d i = b b i β 1 i for each i 1. Finally, he composie fiscal sress erm, f, is now defined by 1 T f = E dt gt YT E T ( + ) 1 σ ˆ * 1 β β byyˆ * T + b T + b T τ * ˆ ξ, τ ξ = T = which can be wrien more compacly as T f = E d h ξ + ( β) E β f ξ, (24) T + 1 ξ T 1 T = T = again using he noaion defined in Benigno and Woodford (2004). The planning problem is o find sae-coningen pahs {π, y, τ } o minimize equaion (9) subjec o consrains (11) and (23). As before, he composie disurbance, f, compleely summarizes he informaion a dae abou he exogenous disurbances ha inerfere wih complee sabilizaion of inflaion and he oupu gap. In conras o he case of one-period deb, oupu and inflaion can now be sabilized a heir opimal level even when prices are sicky by appropriaely choosing he seady-sae srucure of mauriy. This is because he sochasic properies of he fiscal sress erm now depend on he mauriy srucure. Wih an appropriae choice of he mauriy srucure, one can even ensure ha f is idenically equal o zero a all imes, in which case complee achievemen of boh sabilizaion objecives will be possible. Le governmen deb have a maximum mauriy of N periods and le J be he number of sochasic disurbances of he model. Le us furher suppose (purely for illusraive purposes, for our argumen could easily be generalized) ha he disurbances are all firs-order auoregressive, or AR(1), processes, j j ξ = ρ ξ + ε j j, 1 ξ T

25 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 61 where ε j is a whie-noise process and ρ j < 1 for each disurbance j. In his case, equaion (24) akes he form i j f = d ρ h ξ + ( 1 β) ( 1 ρ β) f ξ N J i i= 1 j= 1 where hξ j j J 1 j ξ j j ξ j, j= 1 and f are he jh componens of he vecors h ξ j ξ and f ξ, respecively. I now follows (generically) ha for f o be zero a all imes, i is necessary and sufficien ha N i ρ = j i i= 1 d z j, (25) where z j is defined by z = ( )( ) β 1 ρ β hξ f, ξ j j j j for each j. Then he se of J equaions (25) ogeher wih he ideniy N d = i i= 1 1, (26) forms a se of J + 1 equaions in he N unknowns, {d i }. We can wrie his sysem of linear equaions using marix noaion. To his end, we define he marix N 1 ρ1 ρ1 N 1 ρ2 ρ2 A N 1 ρj ρj , and le z be he vecor whose firs J elemens are he z j, and whose final elemen is 1. We can hen wrie he sysem of linear equaions in he compac form, Ad = z, (27)

26 62 Pierpaolo Benigno and Michael Woodford where d is he vecor of coefficiens d i. Sandard resuls ensure ha here is a soluion of equaion (27) as long as A is of full rank. In his case, here is a leas one vecor d ha is, a leas one seady-sae mauriy srucure such ha f = 0, so ha complee sabilizaion of boh inflaion and he oupu gap can be achieved. In paricular, if N = J + 1, here is exacly one soluion for any given z, when A is of full rank. For example, in he case of a single sochasic disurbance (J = 1), he marix A is always of full rank, and he firs-bes oucome can be achieved simply by issuing nominal deb wih one- and wo-period mauriies. The opimal mauriy srucure in his case depends on he persisence of he shock, as well as on is conribuion o movemens in he fiscal sress measure, f. If J > 1, A is of full rank if and only if ρ i ρ j for each i and j. (Oherwise here generally is no soluion.) Angeleos (2001) shows in a flexible-price model ha o complee he markes, i is necessary and sufficien o issue nominal deb ha has a leas N-period mauriy, where N is he number of saes of naure in he model. Here we esablish ha in a log-linear model, wha maers is no he number of disinc saes of naure, bu only he number of sochasic disurbances, as Angeleos conjecured on he basis of his numerical resuls. As long as deb can be issued in moderaely long mauriies, i will generally be possible, a leas in principle, o choose a mauriy srucure ha achieves he firs-bes oucome. The opimal moneary policy will simply aim a complee price sabiliy, while he disoring ax rae will be used o offse cos-push disurbances, so ha zero inflaion is compaible wih a zero oupu gap. As Buera and Nicolini (2004) noe in a relaed conex, however, he mauriy srucure required for such an oucome may be implausible, involving very large long and shor posiions in differen mauriies. They also show ha he opimal mauriy srucure may be exremely sensiive o small changes in model parameers, such as small changes in he serial correlaion of disurbance processes. 18 Here again, while in principle he opporuniy o increase he flexibiliy of fiscal policy in his way can grealy faciliae moneary sabilizaion policy, he pracical relevance of his case is open o quesion. We accordingly resric he remainder of our analysis o he case of a single mauriy of governmen deb, specifically, one very shor-erm (single-period) deb. In fac, mos 18. This can be seen from our analysis above, since a small change in hese parameers can cause he rank condiion o fail.

27 Opimal Inflaion Targeing under Alernaive Fiscal Regimes 63 counries wih serious fiscal imbalances issue almos exclusively shormauriy deb, so our assumpion seems likely o represen he mos relevan case for he counries facing he concerns addressed in his paper. This emphasis is also consisen wih our desire o consider he cases in which possible consrains on fiscal policy are mos likely o creae problems for inflaion argeing. The presence of a larger number of fiscal insrumens, or fewer consrains on how hey are used, will generally srenghen he case for inflaion argeing. Our ineres, however, is in he exen o which a form of inflaion argeing coninues o be desirable even when fiscal policy is much less helpful. 4. OPTIMAL MONETARY POLICY WHEN FISCAL POLICY IS EXOGENOUS This secion explores a sill more consrained case, in which {G, ζ, τ } are all assumed o be exogenous processes, deermined by poliical facors ha he cenral bank canno influence. This is he ype of fiscal policy assumed by Loyo (1999), which Sims (2005) uses in his criique of inflaion argeing. In a flexible-price model such as Loyo s, his policy implies a purely exogenous evoluion of he real primary governmen budge surplus, {s }. The cenral bank mus beware ha a igh-money policy does no cause explosive growh of he public deb, for i is assumed ha neiher axes nor governmen spending will be adjused o preven such dynamics. In his case, he ineremporal solvency condiion (equaion 12) consrains he possible pahs for inflaion and oupu ha can be achieved by any moneary policy, and no endogenous fiscal insrumens are available o adjus his consrain. A he same ime, he possible pahs for inflaion and oupu are consrained by he aggregae supply radeoff (equaion 11), and here is no endogenous fiscal insrumen ha can shif his relaion eiher, in conras wih he assumpion in he previous secion. The cenral bank s abiliy o achieve is inflaion and oupu-gap sabilizaion objecives is accordingly more ighly consrained. As Sims (2005) noes, full price sabiliy (or even complee sabilizaion of he inflaion rae a some nonzero value) will ypically be infeasible under hese assumpions unlike he siuaion considered in he previous secion, where his is a possible, hough no quie opimal, moneary policy. Condiion (11) allows us o easily derive he unique oupu-gap process consisen wih complee sabilizaion of he inflaion rae. However, he process {y } obained in his way (ogeher wih he assumed consan inflaion rae and he exogenously given