Monetary Conservatism and Fiscal Policy 1

Transcription

1 issn

2 Moneary Conservaism and Fiscal Policy Klaus Adam 2 Robero Billi 3 Firs version: February 2007 This version: July 2008 RWP 07-0 Absrac: Does an inflaion conservaive cenral bank à la Rogoff (985) remain desirable in a seing wih endogenous fiscal policy? To provide an answer we sudy moneary and fiscal policy games wihou commimen in a dynamic, sochasic sickyprice economy wih monopolisic disorions. Moneary policy deermines nominal ineres raes and fiscal policy provides public goods generaing privae uiliy. We find ha lack of fiscal commimen gives rise o excessive public spending. The opimal inflaion rae inernalizing his disorion is posiive, bu lack of moneary commimen generaes oo much inflaion. A conservaive moneary auhoriy hus remains desirable. When fiscal policy is deermined before moneary policy each period, he moneary auhoriy should focus exclusively on sabilizing inflaion. Moneary conservaism hen eliminaes he seady sae biases associaed wih lack of moneary and fiscal commimen and leads o sabilizaion policy ha is close o opimal. Keywords: sequenial non-cooperaive policy games, discreionary policy, ime consisen policy, conservaive moneary policy JEL Classificaion: E52, E62, E63 We hank Marina Azzimoni, Helge Berger, V.V. Chari, Gaui Eggersson, Jordi Galí, Alber Marce, Ramon Marimon, Chris Waller, seminar paricipans a IGIER/Bocconi Universiy, paricipans a he Third Conference of he Inernaional Research Forum on Moneary Policy and he h Inernaional Conference on Compuing in Economics and Finance for helpful commens and discussions. Errors remain ours.. The views expressed herein are solely hose of he auhor and do no necessarily reflec he views of he Federal Reserve Bank of Kansas Ciy or he Federal Reserve Sysem. 2 Mannheim Universiy, Deparmen of Economics, L7, 3-5, 683 Mannheim, Germany; and CEPR, London; adam@mail.uni-mannheim.de 3 Federal Reserve Bank of Kansas Ciy, One Memorial Drive, Kansas Ciy, MO 6498, Unied Saes, Robero.Billi@kc.frb.org

3 Inroducion Which moneary insiuions can overcome he problems associaed wih lack of moneary commimen? A prominen early answer is Rogo s (985) proposal o appoin an in aion conservaive cenral banker. This and oher well-known proposals assume, however, ha scal policy can be reaed as exogenous when sudying he design of moneary insiuions. While his assumpion is a useful saring poin, i is an unsaisfacory aspec of previous sudies. In his paper we ask wheher insalling a conservaive cenral banker remains desirable when scal policy is endogenous and equally subjec o a commimen problem. We analyze a non-cooperaive moneary and scal policy game wihin a sandard sochasic general equilibrium model wihou capial, along he lines of Roemberg (982) and Woodford (2003). This economy feaures hree sources of disorions: () rms operae under monopolisic compeiion, which causes oupu o be ine cienly low; (2) prices are rigid in he shor-erm, which gives rise o real e ecs of moneary policy; (3) policymakers canno credibly commi o a pah for fuure policy, bu insead deermine policy sequenially. In line wih recen moneary policy models, he moneary auhoriy deermines he shorerm nominal ineres rae. We add o his seing a scal auhoriy which decides he level of public goods provision. Public goods generae uiliy for privae agens and are nanced by lump sum axes, so as o balance he governmen s ineremporal budge. 2 While he moneary and scal auhoriy are benevolen, i.e., maximize he uiliy of he represenaive agen, lack of commimen gives rise o subopimal policy oucomes. Since oupu is ine cienly low, boh policymakers are emped o increase oupu, eiher via lowering real ineres raes (moneary auhoriy) or via increasing public spending ( scal auhoriy). See for example Svensson (997) and Walsh (995). 2 Our resuls exend o a seing wih disorionary labor axes, as shown in a companion noe. See Adam and Billi (2008).

4 This resuls in an in aionary bias and in overspending on public goods, compared o a siuaion wih policy commimen, because wih sequenial decision making boh policymakers fail o fully inernalize he welfare cos of generaing in aion oday. In our seing moneary and scal policy inerac in ineresing ways. Speci cally, aking he lack of scal commimen as given makes i opimal for moneary policy o aim a posiive in aion raes. We show ha posiive in aion raes reduce he scal spending bias and hereby increase welfare. This suggess ha, unlike in he sandard case wih exogenous scal policy, a conservaive cenral bank may no always be desirable. A quaniaive exercise suggess, however, ha he opimal deviaions from price sabiliy end o be small. Moreover, in he non-cooperaive Markov-perfec Nash equilibrium wih sequenial moneary and scal policy, he in aion rae lies signi canly above he opimal in aion rae for a wide range of model parameerizaions. 3 This resul suggess ha insalling an in aion conservaive cenral banker is desirable also wih endogenous scal policy. We hen formally inroduce a conservaive moneary auhoriy ha maximizes a weighed sum of an in aion loss erm and he represenaive agen s uiliy. And we characerize he resuling Markov-perfec equilibria. When policies are deermined simulaneously or when moneary policy is deermined before scal policy each period, moneary conservaism alone canno eliminae enirely he seady sae disorions from sequenial policymaking. There is posiive in aion or scal overspending, or boh. Neverheless, we nd ha a su cienly high degree of moneary conservaism eliminaes mos of he seady sae welfare loss arising from he lack of moneary and scal commimen. Moneary conservaism urns ou o be even more desirable if scal policy is deermined before moneary policy each period (arguably he mos relevan iming proocol). In such 3 Markov-perfec Nash equilibria, as de ned in Maskin and Tirole (200), are a sandard re nemen used in he applied dynamic games lieraure, e.g., Klein e al. (2008). 2

5 a seing moneary conservaism is inernalized by scal policy, which makes i possible o eliminae enirely he seady sae biases semming from lack of moneary and scal commimen, provided he moneary auhoriy cares exclusively abou in aion. Overall, he case for a conservaive moneary auhoriy hus remains sronger in a seing wih endogenous scal policy. We also address he issue of how he conduc of sabilizaion policy is a eced by moneary conservaism. We show ha scal leadership in combinaion wih a fully conservaive cenral banker can achieve he exible-price Ramsey allocaion following echnology and mark-up shocks. This resul suggess ha moneary conservaism is desirable also from he viewpoin of cyclical sabilizaion policy. Following he work of Kydland and Presco (977) and Barro and Gordon (983), he moneary policy lieraure has exensively sudied ime-inconsisency problems in dynamic seings and is poenial soluions, see Rogo (985), Svensson (997) and Walsh (995). In his lieraure, scal policy is ypically absen or exogenous o he model. A he same ime, a number of conribuions analyze he ime-consisency of opimal scal plans in dynamic general equilibrium models, e.g., Lucas and Sokey (983), Chari and Kehoe (990) or Klein, Krusell, and Ríos-Rull (2008). Bu his lieraure ypically sudies models wihou money. The nex secion inroduces he model and he implemenabiliy consrains characerizing privae-secor behavior. Secion 3 derives moneary and scal policy wih and wihou commimen and some analyical resuls abou he policy biases. In secion 4 we quanify he biases and heir welfare implicaions. Secion 5 sudies moneary conservaism and secion 6 explains he robusness of he resuls o disorionary axaion. The echnical maerial is available in he web appendix o his paper. 3

6 2 The Model The seing is a sicky-price economy wih monopolisic compeiion, similar o he one sudied in Schmi-Grohé and Uribe (2004). 2. Privae Secor There is a coninuum of idenical households wih preferences given by E 0 X =0 u(c ; h ; g ) () where c is consumpion of an aggregae consumpion good, h 2 [0; ] labor e or, g public goods provision by he governmen in he form of aggregae consumpion goods, and 2 (0; ) he discoun facor. Uiliy is separable in c; h; g and u c > 0, u cc < 0, u h < 0, u hh 0, u g > 0,, hu u gg < 0, and hh are bounded. cucc u c u h Each household produces a di ereniaed inermediae good. Demand for his good is y d( P e =P ), where y is (privae and public) demand for he aggregae good and P e =P he relaive price of he inermediae good compared o he aggregae good. The demand funcion d() sais es d() = and d 0 () =, where 2 ( ; ) is he price elasiciy of demand for he di eren goods. This elasiciy is ime-varying and induces ucuaions in he monopolisic mark-up charged by rms. The demand funcion is consisen wih opimizing individual behavior when privae and public consumpion goods are a Dixi-Sigliz aggregae of he goods produced by di eren households. The household chooses e P, hen hires he necessary amoun of labor e or e h o saisfy he resuling produc demand, i.e.,! h e ep = y d P (2) where is an aggregae echnology shock. The mark-up shock and he echnology shock follow independen AR() sochasic processes wih auocorrelaion coe ciens and z and seady sae values z = and <. 4

7 Following Roemberg (982), sluggish nominal price adjusmen by rms is described by quadraic resource coss for adjusing prices according o 2 P ep ep where > 0 indexes he degree of price sickiness. 4 The households budge consrain is 2 P c + B = R B + P 4 P e!! 3 2 P e y d w e ep h 5 + P w h P l (3) P 2 ep where R is he gross nominal ineres rae, B are nominal bonds ha pay R B in period +, w is he real wage paid in a compeiive labor marke, and l are lump sum axes. 5 Alhough we consider lump sum axes, secion 6 shows ha he main resuls exend o he case wih disorionary axes. Lump sum axes allow us o derive many resuls analyically. Finally, he no-ponzi scheme consrain on household behavior is:! 2 +j lim E Y B +j 0 (4) j! R i=0 i The household s problem consiss of choosing {c ; h ; e h ; e P ; B } =0 o maximize () subjec o (2), (3) and (4) aking as given {y ; P ; w ; R ; g ; l } =0. The rs order condiions of he household s problem are hen equaions (2), (3) and (4) holding wih equaliy and also u h =w R =E = y d(r ) + r y d 0 (r ) w + E + r+ r + y d 0 (r ) r+ r 2 + r r where r e P P is he relaive price and P P he gross in aion rae. Furhermore, he ransversaliy condiion lim j! E ( +j +j B +j =P +j ) = 0 has o hold a all coningencies. 4 Using insead he Calvo approach o nominal rigidiies would considerably complicae maers because price dispersion hen becomes an endogenous sae variable. 5 We absrac from money holdings and hus seignorage by considering a cashless limi economy à la Woodford (998); money only imposes a lower bound on nominal ineres raes (R ). r 5

8 2.2 Governmen The governmen consiss of a moneary auhoriy seing nominal ineres raes R and a scal auhoriy deermining he level of public good provision g. The budge consrain is B = R B + P (g l ) (5) Wih lump sum axes, ax versus deb nancing decisions do no maer for equilibrium deerminaion as long as he implied pahs for he deb level saisfy he no-ponzi scheme consrain (4) and he ransversaliy condiion. For sake of simpliciy, axes are se such ha he level of real deb B P remains bounded from below and asympoically grows a a rae less han. Consrain (4) and he ransversaliy condiion are hen always sais ed and can be ignored from now on. Fiscal policy is hus passive in he sense of Leeper (99). 2.3 Privae Secor Equilibrium In a symmeric price seing equilibrium he relaive price is given by r = for all. From he assumpions made, i follows ha he rs order condiions of households behavior can be condensed ino a price seing equaion ( ) = h + + u h ofen referred o as a Phillips curve, and a consumpion Euler equaion + E + ( + ) + (6) R = E + + (7) A privae-secor raional expecaions equilibrium consiss of a plan fc ; h ; B ; P g saisfying equaions (6) and (7), he governmen budge (5), and he marke-clearing condiion h = c + 2 ( ) 2 + g (8) given he policies fg ; l ; R g, he exogenous processes f ; g, and he iniial condiions (R B ; P ). 6

9 2.4 Time Inconsisency Problems Under commimen policymakers deermine sae-coningen fuure policies a he beginning of ime. Policymakers ha canno commi decide abou policies a he ime of implemenaion, i.e., period by period. We refer o such behavior as sequenial decision making. Sequenial policy is subopimal because i fails o fully inernalize he welfare cos of generaing in aion oday. Since pas prices can be aken as given a he ime curren policy is deermined, sequenial decision making ignores he link beween curren policy decisions and pas pricing decisions. Such a link does exis because he privae secor raionally anicipaes curren in aion and is forward-looking in is pricing decision, see equaion (6). As a resul, sequenially deciding policymakers underesimae he welfare coss of generaing in aion oday and are emped o move oupu closer o is rs-bes level. 3 Moneary and Fiscal Policy Regimes In his secion we sudy he seady sae oucomes associaed wih lack of commimen. The implicaions for sabilizaion policy will be discussed in secion 5. We sar by analyzing he rs-bes allocaion, which absracs from monopoly disorions and nominal rigidiies. Then we deermine he Ramsey allocaion, which akes ino accoun boh disorions, bu sill allows for policy commimen. In a nal sep we relax he commimen assumpion. 3. Firs-Bes and Ramsey Allocaion The rs-bes allocaion solves max fc ;h g g E 0 X u(c ; h ; g ) s.. h = c + g (9) =0 where equaion (9) is he resource consrain. The seady sae rs-order condiions deliver u c = u g = u h 7

10 showing ha, given he echnological consrains, i is opimal o equae he marginal uiliy of privae and public consumpion o he marginal disuiliy of labor e or. This simple resul ceases o be opimal in he Ramsey allocaion, which akes ino accoun he presence of price seing and monopoly disorions, as summarized by he implemenabiliy consrains (6) and (7). Speci cally, he Ramsey allocaions solves 6 max E 0 fc ;h ; ;R ;g g X =0 u(c ; h ; g ) (0) s.. Equaions (6); (7); (8) for all Noe ha he Ramsey allocaion sill allows for commimen o policies a ime zero and full cooperaion beween moneary an scal policymakers. Deriving he rs order condiions of (0) and imposing seady sae condiions delivers 7 = and R = as well as he marginal condiions u c = + u h () u g = + q + u h + q (2) where q c hu hh u c h u h cu cc 0. Equaion () shows ha monopolisic compeiion creaes a wedge beween he marginal uiliy of privae consumpion and he marginal disuiliy of work. This wedge re ecs he fac ha labor fails o receive is marginal produc when rms have monopoly power, which causes housholds o reduce consumpion of produced goods and o increase consumpion of leisure. 8 6 Since Ricardian equivalence holds we ignore he nancing decisions of he scal auhoriy and he iniial deb level R B, which do no maer for equilibrium deerminaion of he oher variables. Since he iniial condiion P simply normalizes he implied price level pah, i can equally be ignored. 7 Deails of he derivaion are in web appendix A.. 8 Seady sae real wages fall shor of heir marginal produc because w = ( + ) = <. 8

11 For u hh < 0, one has q > 0 and equaion (2) implies ha he opimal level of public spending falls shor of equaing he marginal uiliy of public consumpion o he marginal disuiliy of work, unlike in he rs bes allocaion. One migh hink ha he opimal provision of public goods should be una eced by he presence of a monopolisic mark-up, since lump sum axes allow o nance any price mark-up wihou generaing addiional disorions. However, public spending (and axes) below he rs-bes level reduce he marginal disuiliy of work, which increases he ine cienly low level of privae consumpion. 3.2 Sequenial Policymaking We now consider separae moneary and scal auhoriies ha canno commi o fuure policy plans and decide abou policies a he ime of implemenaion. To faciliae he exposiion, each policymaker akes as given he curren policy choice of he oher policymaker, as well as all fuure policies and fuure privae-secor choices. We verify he raionaliy of hese assumpions a he end of his secion Sequenial Fiscal Policy: Spending Bias Given he assumpions made above, he scal auhoriy s problem in period is max fc +j ;h +j ; +j ;g +j g E X j u(c +j ; h +j ; g +j ) (3) j=0 s:: Equaions (6); (7); (8) for all fc +j ; h +j ; +j ; R +j ; g +j g given for j Eliminaing Lagrange mulipliers from he rs order condiions delivers 9 u g = u h 2 2 ( ) ( + + u h u + h hh ) (FRF) 9 Deails of he derivaion are in web appendix A.2. 9

12 The scal auhoriy ses he level of public goods provision g such ha his scal reacion funcion (FRF) is sais ed, each period. Consider a seady sae in which he in aion rae is equal o he one chosen by he Ramsey planner ( = ). The scal reacion funcion hen simpli es o u g = u h (4) showing ha scal policy equaes he marginal uiliy of public consumpion o he marginal disuiliy of labor e or. While such behavior is consisen wih he rs-bes allocaion, i is generally subopimal in he presence of monopolisic disorions, as argued in secion 3.. Sequenial scal policy hus gives rise o a subopimally high level of public spending. This scal spending bias causes he Ramsey allocaion o be unaainable in he presence of sequenial scal policy, because in aion or scal spending, or boh mus deviae from heir Ramsey values. This resul is summarized in he following proposiion. Proposiion For u hh < 0, sequenial scal policy implies excessive scal spending in he presence of price sabiliy. The economic inuiion underlying his resul is as follows. By aking fuure decisions and he curren moneary policy choice R as given, he scal auhoriy considers privae consumpion c o be deermined by he Euler equaion (7). The scal auhoriy hus perceives labor inpu h o move one-for-one wih governmen spending g. In a siuaion wih price sabiliy, he in aion coss of public spending are zero (a he margin) and can be ignored, so ha he sequenial spending rule (4) appears opimal. When 6= he marginal coss of in aion fail o be zero, leading o he opimaliy condiion FRF. 0

13 3.2.2 Sequenial Moneary Policy: In aion Bias Given he previous assumpions, he moneary auhoriy s problem in period is max E fc +j ;h +j ; +j ;R +j g X j=0 j u(c +j ; h +j ; g +j ) (5) s.. Equaions (6),(7),(8) for all fc +j ; h +j ; +j ; R +j ; g +j g given for j Eliminaing Lagrange mulipliers from he rs order condiions of his problem delivers he moneary reacion funcion 0 u hh ( ( ) ) ( ) + h u h u h + 2 u cc ( ) (( ) h ( + )) = 0 (MRF) where he moneary auhoriy ses he nominal ineres rae R such ha MRF is sais ed, each period. In a seady sae wih = his policy reacion funcion simpli es o u c u h = where he expression on he l.h.s. corresponds o he real wage. Monopoly power implies ha real wages fall shor of he marginal produc of labor. As a resul, he moneary reacion funcion MRF is inconsisen wih price sabiliy. We appendix A.4 shows ha he seady sae in aion rae mus be posiive. Proposiion 2 For su cienly close o, sequenial moneary policy implies a sricly posiive rae of in aion in seady sae. Sequenial moneary policy generaes an in aion bias as in he sandard case wih exogenous scal policy, see for example Svensson (997). Inuiively, he moneary auhoriy 0 Deails of he derivaion can be found in web appendix A.3.

14 is emped o simulae demand by lowering nominal ineres raes. Since price adjusmens are cosly, he price level will no fully adjus, and real ineres raes fall, which simulaes demand. The real wage increase required o saisfy his addiional demand generaes in aion Sequenial Moneary and Fiscal Policy We now de ne a Markov-perfec Nash equilibrium wih sequenial moneary and scal policy. We also verify he raionaliy of our iniial assumpions ha sequenially deciding policymakers can ake as given he curren policy choice of he oher policymaker, as well as all fuure policies and fuure privae-secor decisions. The privae secor s opimaliy condiions (6) and (7), he feasibiliy consrain (8), as well as he policy reacions funcions FRF and MRF, all depend on curren and fuure variables only. This observaion suggess he exisence of an equilibrium where curren play is a funcion of he curren exogenous variables and only. Fuure play hen depends on fuure exogenous variables only, which jusi es he assumpion ha fuure equilibrium play (and o -equilibrium play) is independen of curren play. If each period, in addiion, moneary and scal policy are deermined simulaneously, Nash equilibrium requires aking he oher player s curren decision as given. This jusi es he assumpions made in deriving FRF and MRF and moivaes he following de niion. De niion 3 (SP) A saionary Markov-perfec Nash equilibrium wih sequenial moneary and scal policy consiss of ime-invarian policy funcions c( ; ); h( ; ); ( ; ); R( ; ); g( ; ) solving equaions (6),(7),(8), FRF and MRF. We now show ha Sackelberg leadership by one of he policy auhoriies (wih regard o he wihin-period moves) is consisen wih he same equilibrium oucome. While he policy problem of he Sackelberg follower remains unchanged, he Sackelberg leader akes ino accoun he reacion funcion of he follower. Ye, he Lagrange mulipliers associaed 2

15 wih addiionally imposing MRF in he sequenial scal problem (3) or wih imposing FRF in he sequenial moneary problem (5) are zero. These reacion funcions can be derived from he rs order condiions of he leader s policy problem even when he follower s reacion funcion is no being imposed. Inuiively, he leadership srucure does no maer for he equilibrium oucome because boh auhoriies are pursuing he same policy objecive. Any deparure from he Ramsey soluion is hus enirely due o sequenial decision making. The presence of disinc policymakers and he sequence of moves will maer in secion 5 when we consider a moneary auhoriy ha is more in aion averse han he scal auhoriy. 4 How Much In aion is Opimal? We have shown ha a seady sae wih sequenial moneary and scal policy (SP) involves posiive raes of in aion (proposiion 2). We now show ha price sabiliy ( = ) is no opimal whenever scal policy faces a commimen problem, i.e., is described by FRF. Moneary policy should hen aim a posiive raes of in aion because his increases he perceived coss of public spending for he scal auhoriy and reduces he scal spending bias. This leads o a rs order welfare gain, while he coss of locally deviaing from price sabiliy are of second order only. This is formally shown in web appendix A.5: Proposiion 4 Assume u hh < 0. In a seady sae wih sequenial scal policy, welfare increases and scal spending decreases wih he seady sae in aion rae, locally a =. This is so because wih sequenial scal policy u c > u g = u h when =, see FRF. 3

16 Wha is hen he welfare maximizing in aion rae ha correcly inernalizes he sequenial scal policy disorion? This in aion rae can be obained as he soluion o he problem max E 0 fc ;h ; ;R ;g g X =0 u(c ; h ; g ) (OI) s.. Equaions (6),(7),(8),(FRF) for all where moneary policy can commi, while scal behavior is described by FRF. We will refer o his siuaion as he opimal in aion (OI) regime. If he opimal seady sae in aion rae is below (above) he rae emerging in he seady sae of he Markov-perfec Nash equilibrium, hen an in aion conservaive (liberal) cenral banker would appear desirable Quanifying he Policy Biases In his secion we quanify he seady sae under sequenial moneary and scal policy (SP) and compare i o he Ramsey allocaion and also o he allocaion achieved under he opimal in aion (OI) regime. We consider he following preference speci caion, which is consisen wih balanced growh, u(c ; h ; g ) = log (c ) h +'! h + ' +! g log (g ) (6) wih! h > 0,! g 0 and ' 0. As a baseline calibraion, we se = 0:993 implying an annual real ineres rae of 3:5%. The seady sae price elasiciy of demand is se a 6, so ha here is a 20% mark-up over marginal coss. The degree of price sickiness is chosen o be 7:5, such ha he log-linearized version of he Phillips curve (6) is consisen wih he one in Schmi-Grohé and Uribe (2004). Labor supply elasiciy is se o ' = and he uiliy weighs! h and! g are chosen such ha in he Ramsey seady sae agens work 20% 2 Since seady sae in aion depends on seady sae nominal ineres raes only, see equaion (7), he opimal in aion rae implicily deermines opimal moneary policy. 4

17 of heir ime (h = 0:2) and spend 20% of oupu on public goods (g = 0:04). 3 We esed he robusness of our numerical resuls by considering a wide range of alernaive model parameerizaions and by using di eren saring values. 4 The baseline resuls are robus, excep for he exible price limi (! 0) and for inelasic labor supply (large '). In boh cases he ime-inconsisency problems of policy disappear and he real allocaions under SP approach he Ramsey seady sae. The rs row of able presens informaion on he seady sae in he SP regime. All variables are expressed as percenage deviaions from heir corresponding Ramsey seady sae values. 5 The las column of he able repors he seady sae welfare loss, expressed in erms of he permanen reducion in privae consumpion making he Ramsey seady sae welfare equivalen o he considered policy regime. 6 In line wih proposiion 2, he sequenial policy oucome is characerized by an in aion bias, which urns ou o be sizable. In addiion, here is a small scal spending bias. Overall, he welfare losses generaed by he sequenial conduc of policy are fairly large, in he order of % of seady sae consumpion per period. The second row of able shows he oucome under he OI regime. The opimal in aion rae is no only lower han he in aion rae in he SP regime bu also very close o price sabiliy. As suggesed by proposiion 4, reducing in aion from he level of he SP regime o he opimal level increases he scal spending bias. While he scal spending increase is fairly large, he opimal in aion rae neverheless eliminaes large par of he welfare losses associaed wih he SP regime. This suggess ha he scal spending bias, despie being 3 This requires! h = 26:042 and! g = 0:227, see equaions (44) and (45), in web appendix A.6. 4 To keep he Ramsey seady sae invarian o he parmeerizaion, we adjused he weighs! h and! g. We used di eren saring values o es for mulipliciies in he seady sae, as occurred in King and Wolman (2004), and did no ecouner any, possibly due o he fac ha in our seing he privae secor akes all of is decisions afer policy has been deermined. 5 In he Ramsey seady sae c = 0:6, h = 0:2, g = 0:04 and =. 6 Web appendix A.8 explains he compuaion of he welfare losses. 5

18 sizable in absolue value, is no very derimenal in welfare erms. Clearly, his resul hinges parly on he availabiliy of lump sum axes. The resuls from able show ha he opimal in aion rae is well below he one emerging in he SP regime. This holds for a wide range of alernaive model parameerizaions, which suggess ha insalling a conservaive moneary auhoriy should be desirable on welfare grounds whenever sequenial scal policy is described by FRF. 5 Conservaive Moneary Auhoriy This secion analyzes wheher he disorions semming from sequenial moneary and scal policy decisions can be reduced by insalling a moneary auhoriy ha is more in aion averse han sociey. Rogo (985) and Svensson (997) have shown his o be he case if scal policy is reaed as exogenous. Following Rogo (985), we consider a weigh conservaive moneary auhoriy wih period uiliy funcion ( ) u(c +j ; h +j ; g +j ) ( +j ) 2 2 where 2 [0; ] is a measure of moneary conservaism. For > 0 he moneary auhoriy dislikes in aion (and de aion) more han sociey; if = he policymaker cares abou in aion only. The preferences of he scal auhoriy remain unchanged. Wih moneary and scal auhoriies now pursuing di eren policy objecives, he equilibrium oucome will depend on he iming of policy moves, i.e., on wheher scal policy is deermined before, afer, or simulaneously wih moneary policy each period. Casual observaion suggess ha i akes longer o enac scal decisions, which would imply ha scal policy is deermined before moneary policy. A he same ime, he ime lag beween a moneary policy decision and is e ecs on he economy may be subsanial oo. I hus remains o be ascerained which of hese iming srucures is he mos relevan for acual economies. 6

19 For hese reasons, we consider Nash as well as leadership equilibria. 5. Nash and Leadership Equilibria This secion de nes he various Markov-perfec equilibria in he presence of a conservaive moneary auhoriy. As i urns ou, he equilibrium oucomes wih simulaneous moneary and scal decisions each period (Nash case) and wih moneary policy deermined before scal policy each period (moneary leadership) are very similar. This similariy emerges because in boh cases scal policy akes curren moneary decisions as given, so ha scal policy coninues o be described by FRF, i.e., by he reacion funcion in he absence of a conservaive auhoriy. For space consrains we herefore only discuss he Nash case. The siuaion in which scal policy is deermined before moneary policy ( scal leadership) di ers considerably. The scal auhoriy has o ake ino accoun he conservaive moneary auhoriy s wihin-period reacion funcion. Moneary policy can hen use o -equilibrium behavior o discipline he behavior of he scal auhoriy along he equilibrium pah. Fiscal leadership hus opens he possibiliy for oucomes ha are welfare superior o hose achieved in he OI regime. Firs, consider he case wih simulaneous decisions. While he policy problem of he scal auhoriy remains unchanged, he moneary auhoriy now solves max fc +j ;h +j ; +j ;R +j g E X ( j ) u(c +j ; h +j ; g +j ) j=0 s.. Equaions (6),(7),(8) for all 2 ( +j ) 2 (7) fc +j ; h +j ; +j ; R +j ; g +j g given for j Taking rs order condiions of problem (7) and eliminaing Lagrange mulipliers delivers he conservaive moneary auhoriy s reacion funcion ha we denoe by CMRF. 7 For 7 Web appendix A.9 provides he echnical deails. As before, CMRF implies ha curren ineres raes 7

20 = 0, CMRF reduces o he moneary reacion funcion wihou conservaism (MRF). This moivaes he following de niion. De niion 5 (CSP-Nash) A saionary Markov-perfec Nash equilibrium wih sequenial and conservaive moneary policy, sequenial scal policy, and simulaneous policy decisions consiss of policy funcions c( ; ); h( ; ); ( ; ); R( ; ); g( ; ) solving equaions (6), (7), (8), FRF and CMRF. Nex, consider he case of scal leadership (FL). The scal auhoriy mus now ake ino accoun he conservaive moneary reacion funcion (CMRF). The scal auhoriy s policy problem a ime is hus given by max E fc +j ;h +j ; +j ;R +j ;g +j g X j=0 j u(c +j ; h +j ; g +j ) (8) s.. Equaions (6),(7),(8), CMRF for all fc +j ; h +j ; +j ; R +j ; g +j g given for j The rs order condiions associaed wih problem (8) deliver he corresponding scal reacion funcion ha we denoe by CFRF-FL. We propose he following de niion. De niion 6 (CSP-FL) A saionary Markov-perfec equilibrium wih sequenial and conservaive moneary policy, sequenial scal policy, and scal policy deciding before moneary policy, consiss of policy funcions c( ; ); h( ; ); ( ; ); R( ; ); g( ; ) solving equaions (6), (7), (8), CFRF-FL and CMRF. depend on curren economic condiions only, validaing he conjecure in (7) ha in a Markov-perfec equilibrium fuure policy choices can be aken as given. 8

21 5.2 Seady Sae Implicaions We now deermine he seady sae properies for he various iming arrangemens wih an in aion conservaive cenral banker. Our rs resul is ha scal leadership in combinaion wih a fully conservaive moneary auhoriy achieves he Ramsey seady sae. Proposiion 7 The Ramsey seady sae is consisen wih sequenial policymaking in a regime wih scal leadership, if he moneary auhoriy is fully conservaive ( = ). Inuiively, wih scal leadership he scal auhoriy anicipaes he wihin-period reacion of he moneary auhoriy. In paricular, for = he moneary auhoriy is deermined o achieve price sabiliy a all coss. A scal expansion above he Ramsey spending level generaes in aionary pressures and hus riggers an increase in ineres raes o resrain privae consumpion. The scal auhoriy herefore inernalizes ha scal spending crowds ou privae consumpion one-for-one. This e ec disciplines scal spending and allows he achievemen of he Ramsey seady sae despie sequenial policymaking by boh auhoriies. A formal proof is provided in web appendix A.0. The nex proposiion shows ha his fails o be possible wih oher iming arrangemens: Proposiion 8 For u hh < 0, he Ramsey seady sae canno be achieved wih sequenial policymaking in a regime wih moneary leadership or simulaneous decisions, for any degree of moneary conservaism. Wih moneary leadership or simulaneous decisions, he behavior of he scal auhoriy coninues o be described by he scal reacion funcion (FRF). I hen follows from proposiion ha in aion or scal spending, or boh, mus deviae from heir Ramsey seady sae values. Neverheless, he quaniaive ndings from secion 4. sugges ha a conservaive moneary auhoriy should remain desirable. 9

22 Figure illusraes his poin using he baseline calibraion of secion 4.. The gure displays he consumpion equivalen seady sae welfare losses associaed wih inermediae degrees of moneary conservaism 2 [0; ] relaive o he Ramsey seady sae. The upper horizonal line indicaes he welfare losses of he OI regime. Noe ha in he Nash case a fully conservaive moneary auhoriy ( = ) approximaely achieves he seady sae welfare level of he OI regime. 8 Thus, even in he absence of scal leadership large par of he welfare losses associaed wih sequenial moneary and scal policy can be recovered hrough a su cien degree of moneary conservaism. Using again he baseline calibraion, gure 2 illusraes how he seady sae values of privae consumpion, labor e or, in aion and public spending depend on he degree of moneary conservaism. While an increase in moneary conservaism reduces he in aion bias for all iming proocols, is e ec on he scal spending bias depends on wheher scal policy anicipaes he moneary policy reacion. If scal policy akes moneary decisions as given, moneary conservaism resuls in an increased scal spending bias, as suggesed by proposiion 4. Neverheless, an in aion conservaive cenral banker remains desirable, as a value of slighly below recovers he OI oucome. 5.3 Implicaions for Sabilizaion Policy This secion exends he analysis o a sochasic economy, considering sabilizaion policy in response o echnology and mark-up shocks. We resric aenion o he sequenial policy regime ha achieves he Ramsey seady sae, i.e., scal leadership and full moneary conservaism ( = ). Full moneary conservaism implies ha he cenral bank will achieve sable prices a all imes. Thus, a necessary condiion for he opimaliy of he impulse response under his policy 8 The welfare level of he OI regime is achieved by a value of very close bu slighly below. 20

23 arrangemen is ha he Ramsey allocaion can be achieved wih a sable price pah. The nex proposiion saes ha his is also a su cien condiion. 9 Proposiion 9 If he Ramsey response o shocks can be achieved wih a sable pah for prices, hen i is consisen wih sequenial policymaking in a regime wih scal leadership and fully conservaive moneary policy ( = ). The following proposiion provides su cien condiions under which he Ramsey impulse response o shocks involves a sable price pah. Proposiion 0 Assume preferences over c ; h ; and g are of he consan relaive risk class. If privae and public consumpion have he same relaive risk aversion, hen he Ramsey response o a echnology shock involves no deviaion from price sabiliy. The proof is given in web appendix A.2 and shows ha price sabiliy under Ramsey policy requires a sable privae consumpion o oupu raio, as well as a sable public consumpion o oupu raio. Mainaining boh raios consan is no possible if preferences are no homogeneous in (c ; g ). Thus, he Ramsey response o echnology shocks will generally involve deviaions from price sabiliy. The Ramsey response o mark-up shocks will equally involve deviaions from price sabiliy. This is he case even when he assumpions of proposiion 0 are sais ed. We illusrae his poin in gure 3 for he baseline parameerizaion of secion 4. and a posiive mark-up shock of hree sandard deviaions. 20 The Ramsey response involves an iniial rise in in aion followed by a small bu persisen amoun of de aion, while he sequenial policy implemens 9 The proof is given in web appendix A. and shows ha he rs order condiions of he Ramsey problem wih sable prices are idenical o he equilibrium condiions implied by scal leadership and full conservaism. 20 Following Ireland (2004) we se he quarerly auocorrelaion of mark-up shocks o n = 0:96. We choose Ireland s esimae of he sandard deviaions of innovaions, which requires muliplying he value repored in his able by he price adjusmen parameer. 2

24 sable prices a all imes. Overall, he deviaions from price sabiliy in he Ramsey regime seem small (in he order of less han 0.% per quarer) and he responses di er across regimes only for he early periods following a shock. Alhough he sabilizaion policy associaed wih scal leadership and full moneary conservaism is no fully opimal, he following proposiion suggess ha such a policy arrangemen remains close o fully opimal: Proposiion Sequenial policymaking in a regime wih scal leadership and fully conservaive moneary policy ( = ) is consisen wih he Ramsey response o shocks under exible prices. The proof is given in web appendix A.3. Thus, scal leadership and full conservaism eliminae all gaps o he Ramsey equilibrium wih exible prices. The presence of sicky prices allows, however, o improve somewha upon he exible price allocaion, see Adao e al. (2003). Since hese gains are likely o be small, full moneary conservaism achieves desirable sabilizaion oucomes in a seing wih scal leadership. 6 Disorionary Taxaion The discussion so far relied on he availabiliy of lump sum axes. While his assumpion allows o derive he main resuls analyically, i also implies ha he governmen could poenially eliminae he monopolisic disorion, and hereby he commimen problem, via an oupu subsidy o rms or a wage subsidy o workers. In a companion noe, Adam and Billi (2008), we assume ha scal spending mus be nanced by disorionary labor income axes. We nd again ha sequenial moneary policy gives rise o a seady sae in aion bias and ha sequenial scal policy resuls in overspending on public goods whenever prices are sable. Numerical exercises sugges ha hese policy 22

25 biases are considerably larger han in a seing wih lump sum axes: wih he sequenial scal auhoriy no fully inernalizing he cos of public spending and axaion, more overspending implies higher labor axes, hereby lower oupu, and hus even sronger incenives o increase public spending and in aion. When numerically evaluaing he e ecs of moneary conservaism, we nd again ha scal leadership wih fully conservaive moneary policy achieves he Ramsey seady sae, and close o full moneary conservaism is opimal for he oher iming proocols. Moneary conservaism hus remains desirable in a seing wih disorionary axes. 7 Conclusions This paper analyzes he policy biases semming from sequenial moneary and scal policymaking, asking wheher an in aion conservaive cenral banker remains desirable in a seing wih endogenous scal policy. While he lack of scal commimen can make i opimal o aim for posiive in aion raes, he opimal deviaions from price sabiliy urn ou o be quaniaively small. Since lack of moneary commimen generaes oo much in aion, insalling a conservaive cenral banker remains welfare improving when scal policy is endogenous. In a seing wih scal leadership, arguably he mos relevan case, insalling a fully conservaive cenral bank which focuses exclusively on sabilizing in aion eliminaes no only he in aion bias bu also he scal spending bias in seady sae. The case for moneary conservaism may hus be even sronger in a seing wih endogenous scal policy. 23

26 References Adam, K., and R. Billi (2008): Moneary Conservaism and Fiscal Policy: The Case of Disorionary Taxes, Universiy of Mannheim Mimeo. Adão, B., I. Correia, and P. Teles (2003): Gaps and Triangles, Review of Economic Sudies, 70, Barro, R., and D. B. Gordon (983): A Posiive Theory of Moneary Policy in a Naural Rae Model, Journal of Poliical Economy, 9, Chari, V. V., and P. J. Kehoe (990): Susainable Plans, Journal of Poliical Economy, 98, Ireland, P. (2004): Technology Shocks in he New Keynesian Model, Review of Economics and Saisics, 86(4), King, R. G., and A. L. Wolman (2004): Moneary Discreion, Pricing Complemenariy and Dynamic Muliple Equilibria, Quarerly Journal of Economics, 9(4), Klein, P., P. Krusell, and J.-V. Ríos-Rull (2008): Time Consisen Public Policy, Review of Economic Sudies, 75, Kydland, F. E., and E. C. Presco (977): Rules Raher Than Discreion: The Inconsisency of Opimal Plans, Journal of Poliical Economy, 85, Leeper, E. M. (99): Equilibria under Acive and Passive Moneary and Fiscal Policies, Journal of Moneary Economics, 27, Lucas, R. E., and N. L. Sokey (983): Opimal Fiscal and Moneary Policy in an Economy Wihou Capial, Journal of Moneary Economics, 2,

27 Maskin, E., and J. Tirole (200): Markov Perfec Equilibrium: I. Observable Acions, Journal of Economic Theory, 00, Rogoff, K. (985): The Opimal Degree of Commimen o an Inermediae Moneary Targe, Quarerly Journal of Economics, 00(4), Roemberg, J. J. (982): Sicky Prices in he Unied Saes, Journal of Poliical Economy, 90, Schmi-Grohé, S., and M. Uribe (2004): Opimal Fiscal and Moneary Policy under Sicky Prices, Journal of Economic Theory, 4(2), Svensson, L. E. O. (997): Opimal In aion Targes, Conservaive Cenral Banks, and Linear In aion Conracs, American Economic Review, 87, Walsh, C. E. (995): Opimal Conracs for Cenral Bankers, American Economic Review, 85, Woodford, M. (998): Doing Wihou Money: Conrolling In aion in a Pos-Moneary World, Review of Economic Dynamics,, (2003): Ineres and Prices. Princeon Universiy Press, Princeon. 25

28 Policy c h g Consumpion Losses Regime (Deviaions from Ramsey) Relaive o Ramsey SS SP 0:44% 0:67% :46% 0:48% :03% OI 0:83% 0:85% 0:09% 7:5% 0:07% Table : Seady Sae E ecs 26

29 0 Consumpion Losses Relaive o Ramsey SS 0.2 Percenage Poins OI CSP FL CSP Nash Degree of Conservaism (Alpha) Figure : Welfare Gains From Moneary Conservaism 27

30 Deviaion from Ramsey % OI Privae Consumpion CSP FL CSP Nash Degree of Conservaism (Alpha) Deviaion from Ramsey % Labor Effor 0.2 OI 0. CSP FL CSP Nash Degree of Conservaism (Alpha) Gross Inflaion Public Goods Deviaion from Ramsey % OI CSP FL CSP Nash Deviaion from Ramsey % OI CSP FL CSP Nash Degree of Conservaism (Alpha) Degree of Conservaism (Alpha) Figure 2: Seady Sae E ecs of Moneary Conservaism 28

31 Privae Cons. Mark Up Shock Ramsey 0.4 CSP FL (Alpha=) Labor Effor Gross Inflaion Public Goods Quarers Figure 3: Responses o Mark-Up Shocks and Moneary Conservaism 29

32 A Web Appendix - NOT FOR PUBLICATION A. Ramsey Seady Sae The Lagrangian of he Ramsey problem (0) is X max E 0 nu(c ; h ; g ) fc ;h ; ;R ;g g =0 + h ( ) + 2 uc u c+ R h c 2 ( ) 2 g + + u h + ( + ) + The rs-order condiions w.r.. (c ; h ; ; R ; g ), respecively, are given by + u cc h u cc ( ) ( + ) u h u cc 2 u cc 3 = 0 (9) R + + u h u hh + h + 3 = 0 (20) u cc ( ) + 2 uc (2 ) ( ) = 0 (2) 2 R 2 = 0 (22) u g 3 = 0 (23) where j = 0 for j = ; 2. We denoe he Ramsey seady sae by dropping ime subscrips. Equaion (22), > 0 and R imply 2 = 0 Equaions (23) delivers 3 = u g > 0 30

33 This and (2) gives = From (7) i hen follows R = Then (6) delivers + + u h u c = 0 (24) This delivers () shown in he main ex. Using he previous resuls, (20) simpli es o u h h u hh + u g = 0 (25) From (9) one obains h = u c u g u cc ( + ) (26) Subsiuing (26) ino (25) delivers Using (24) o subsiue for u c one ges Using (24) again o subsiue u h u c u g u cc ( + ) u hh + u g = u g = u h + 2 uhh + u hh + u cc u cc delivers (2) shown in he main ex. 3

34 A.2 Sequenial Fiscal Reacion Funcion The scal problem (3) is max E fc +j ;h +j ; +j ;g +j g + +j + 2 +j + 3 +j X j=0 j nu(c +j ; h +j ; g +j ) +j +j h +j +j ( +j ) +j i ) +j+ +j+ ( +j+ uc+j +j+ R +j +j+ +j h +j c +j 2 ( +j ) 2 g +j + +j + u h+j +j +j +j aking as given R +j and oher variables daed + j for j. The rs order condiions w.r.. (c ; h ; ; g ), respecively, are given by + u cc h u cc ( ) u h ( + ) + 2 u cc + + u h + h u hh R 3 = 0 (27) + 3 = 0 (28) (2 ) 3 ( ) = 0 (29) u g 3 = 0 (30) From (29) and (30) one ges = u g( ) (2 ) Using he previous resul and (30) o subsiue he Lagrange mulipliers in (28) delivers FRF shown in he main ex. 32

35 A.3 Sequenial Moneary Reacion Funcion The moneary problem (5) is max E fc +j ;h +j ; +j ;R +j g + +j + 2 +j + 3 +j X j=0 j nu(c +j ; h +j ; g +j ) +j +j h +j +j ( +j ) +j i ) +j+ +j+ ( +j+ uc+j +j+ R +j +j+ +j h +j c +j 2 ( +j ) 2 g +j + +j + u h+j +j +j +j aking as given g +j and oher variables daed + j for j. The rs order condiions w.r.. (c ; h ; ; R ), respecively, are given by + u cc h u cc ( ) u h ( + ) + 2 u cc + + u h + h u hh R 3 = 0 (3) + 3 = 0 (32) (2 ) 3 ( ) = 0 (33) 2 R 2 = 0 (34) Equaion (34), > 0 and R imply 2 = 0 Then solving (3), (32) and (33) for 3 3 = + 3 = 3 = u h delivers, respecively, u cc h u cc ( ) + (2 ) ( ) ( + ) + + u h + h u hh (35) (36) (37) 33

36 Equaions (35) and (37) imply = 2 u cc (( ) h ( + )) (38) While (36) and (37) give = u h u h + h u hh (39) From (38) and (39) one obains MRF shown in he main ex. A.4 Proof of Proposiion 2 We rs show ha MRF canno hold in he neighborhood of =. In seady sae one can rewrie MRF as + u c + O( ) = 0 (40) u h where O( ) summarizes erms ha converge o zero as ( )! 0. In a seady sae wih = equaion (6) delivers uc u h < + <. Since he implici funcion uc u h () de ned by (6) exiss, his implies ha uc u h is bounded away from also in a su cienly small neighborhood around =. Thus, (40) canno hold in he neighborhood of =. Moreover, from R and (7) we have in seady sae. For su cienly close o, i hen follows ha MRF can only hold if >. A.5 Proof of Proposiion 4 The e ec of in aion on seady sae uiliy is given by du d = (4) where c(); h(); g() denoe he seady sae levels emerging under sequenial scal policy when moneary policy implemens in aion rae, and he derivaives u j (j = c; h; g) are 34

37 evaluaed a his seady sae. We rs evaluae (4) a =. Equaion (6) delivers u c = + u h (42) Toally di ereniaing (8) and evaluaing a @ Using his resul and (4), hen (4) can be rewrien as du d = (u c u Equaions (4) and (42) imply u c > u g, hus sign = sign (43) To deermine he sign we oally di ereniae FRF, (6), and (8) and evaluae a =, his delivers 0 0 u hh u gg h u hu cc u 2 c h u hh u c 0 0 C @ 0 = C B h u hu hh u c 0 0 C A @ = u c u hh u c u hh u gg u cc u h u gg u cc u h u hh h u hu hh u c u c u hh + u cc u h u c u hh u gg u cc u h u gg u cc u h u hh h u hu hh u c When u hh < 0, signing hese inequaliy and (43) imply du d > 0, locally a =. > < 0, as claimed. The former 35

38 A.6 Uiliy Weighs For he period uiliy speci caion (6), he Ramsey policy marginal condiions () and (2), respecively, deliver! h = + ch '! g =! h gh ' + + c ' h (44) + c h ' (45) A.7 Solving for he Equilibrium wih Sequenial Moneary and Fiscal Policy We show how o solve for he sochasic Markov-perfec Nash equilibrium wih sequenial moneary and scal policy. We illusrae he mehod for he SP case, i.e., wihou a conservaive cenral banker, bu he mehod readily exends o he case wih a conservaive moneary auhoriy. The Markov perfec Nash equilibrium solves he following problem max E fc +j ;h +j ; +j ;R +j ;g +j g s.. X j=0 Equaions (6),(7),(8) for all E (c +j ; h +j ; +j ; R +j ; g +j ) given for j j u(c +j ; h +j ; g +j ) (46) One should noe ha FRF and MRF need no be imposed, since hey can already be derived from he rs order condiions of his problem, see secions 3.2. and 3.2.2, respecively. The soluion of problem (46) will always saisfy FRF and MRF. 36

39 Then, he recursive formulaion of he Lagrangian of problem (46) is W ( ; ) = s.. min max ( ;2 ;3 )(c ;h ; ;R ;g ) ff () + E W (+ ; + )g (47) + = ( + = ( z ) + z + " z+ ) + + " + where he one-period reurn is f () = u(c ; h ; g ) + h ( ) + 2 uc E IS + 3 R h c 2 ( ) 2 g + + u h E AS wih he expecaions funcions E AS E + ( + ) + (48) E IS E + + (49) aken as given. The addiional conrol variables, 2, 3 are he Lagrange mulipliers of he implemenabiliy consrains (6) and (7), and he feasibiliy consrain (8), respecively. We hen solve for he seady sae using he rs order condiions of he recursive formulaion (47). Thereafer, we compue a quadraic approximaion of he one-period reurn f() around his seady sae. This sep involves quadraically approximaing he implemenabiliy and feasibiliy consrains. Insead, he expecaion funcions E AS approximaed as and E IS are linearly E AS a 0 + a ( ) + a 2 ( ) (50) E IS a a 2 ( ) + a 2 2 ( ) (5) 37

40 Imporanly, posulaing linear expecaion funcions is su cien o obain a rs order approximaion o he equilibrium dynamics and policy funcions. The policymaker akes expecaions funcions as given, herefore, hey do no show up in di ereniaed form in he rs order condiions. Moreover, linear expecaions funcions are su cien o evaluae he Lagrangian, i.e., uiliy, up o second order. This is he case because eiher he implemenabiliy consrains or he Lagrange mulipliers are zero in a su cienly small neighborhood around he seady sae. As a resul, no rs order erms appear when evaluaing he quadraic approximaion of f() a he soluion. Obviously, his is jus a resaemen of he fac ha (47) is an unconsrained opimizaion problem. We now explain how we compue he expecaion funcions (50) and (5). We sar wih an iniial guess for a j i (j = ; 2; i = 0; ; 2), hen we solve (47) wih f() replaced by is quadraic approximaion. We updae j i, as explained below, and coninue ieraing unil he maximum absolue change of he policy funcions drops below he square roo of machine precision, i.e., : Le he soluion for he policy funcions c () and () be given by c + c = cz (+ ) + c ( + ) (52) + = z (+ ) + ( + ) (53) where variables wihou ime subscrip denoe seady sae values. A rs-order approximaion of he expecaion funcions (48) and (49) hen delivers E AS E + E (c + ss ss E (c + ss E AS ss E E ( + ) ss E ( + ) ss where j ss indicaes expressions evaluaed a seady sae. These condiions ogeher wih (52), 38