Pontiggia, Dario (2009) Essays on optimal monetary policy under rule-ofthumb behaviour by price setters. PhD thesis.

Transcription

1 Poniggia, Dario (2009) Essays on opimal moneary policy under rule-ofhumb behaviour by price seers. PhD hesis. hp://heses.gla.ac.uk/399/ Copyrigh and moral righs for his hesis are reained by he auhor A copy can be downloaded for personal non-commercial research or sudy, wihou prior permission or charge This hesis canno be reproduced or quoed exensively from wihou firs obaining permission in wriing from he Auhor The conen mus no be changed in any way or sold commercially in any forma or medium wihou he formal permission of he Auhor When referring o his work, full bibliographic deails including he auhor, ile, awarding insiuion and dae of he hesis mus be given Glasgow Theses Service hp://heses.gla.ac.uk/ heses@gla.ac.uk

2 Essays on Opimal Moneary Policy under Rule-of-humb Behaviour by Price Seers Dario Poniggia Submied in ful lmen of he requiremens for he Degree of Docor of Philosophy Deparmen of Economics Faculy of Law, Business and Social Sciences Universiy of Glasgow December 2009

3 Absrac The aim of his hesis is o sudy he e ecs of in aion persisence due o rule-ofhumb behaviour by price seers on opimal moneary policy. We sar wih a canonical log-linearised New Keynesian model, which we exend by allowing a fracion of price seers o follow a rule-of-humb when seing a new price. We consider di eren speci caions for he rule-of-humb. In all models, seady-sae disorions are assumed o be small so o guaranee he feasibiliy of opimal moneary policy analysis wihin a linear-quadraic framework. We derive uiliy-based objecive funcions for he moneary auhoriy and analyse a range of opimal commimen policies. We perform welfare analysis in order o rank he range of opimal commimen policies. We analyically derive he opimal seadysae in aion raes associaed wih each commimen policy. We show ha rule-of-humb behaviour by price seers generaes an incenive for posiive seady-sae in aion. A ype of imeless perspecive commimen policy is also capable of delivering posiive seady-sae in aion, even in he absence of rule-of-humb behaviour by price seers. The opimal seady-sae in aion raes are direcly proporional o he gap measuring he seady-sae disorions and urn ou o be small in magniude. We depar from he assumpion of small seady-sae disorions and consider he case of a largely disored seady sae wihin a nonlinear medium-scale model, which adds boh nominal rigidiies and real rigidiies o he basic New Keynesian model. We exend he model by allowing a fracion of price seers o follow a rule-of-humb when posing a new price. We numerically characerise he opimal rae of in aion in he Ramsey seady sae. We nd ha rule-of-humb behaviour implies opimal posiive in aion only in he absence of ransacional fricions. We nd ha he gap re ecing seady-sae disorions is only slighly larger han in he case of small seady-sae disorions. Finally, we sudy Ramsey dynamics and he implemenaion of opimal moneary policy via simple ineresrae rules, which we expand o explore he imporance of welfare-relevan oupu gaps. 2

4 Conens Absrac 2 Lis of Tables 4 Lis of Figures 5 Acknowledgemens 6 Declaraion 7 Chaper. Inroducion 3 Chaper 2. Basic New Keynesian Model Households and Marke Srucure Firms Marke Clearing Price Seing Behaviour Flexible Price Equilibrium and E cien Equilibrium Log-linearised Model Hybrid Phillips Curves 38 Chaper 3. Policy Objecive Funcions Theoreical Background Uiliy-based Welfare Crierion Uiliy-based Objecive Funcions 52 Chaper 4. Opimal Moneary Policy: Opimal Long-run In aion Raes Di eren Perspecives on Opimal Moneary Policy The Policy Problem 62 3

5 4.2. Basic New Keynesian Model Rule-of-humb Behaviour Rule-of-humb Behaviour à la Galì and Gerler Rule-of-humb Behaviour à la Seinsson Discussion 79 Chaper 5. Quaniaive Analysis and Welfare Analysis Calibraion Opimal Long-run in aion raes Welfare Analysis Welfare on he basis of he Deerminisic Equilibrium Basic New Keynesian Model Rule-of-humb Behaviour à la Galì and Gerler Rule-of-humb Behaviour à la Seinsson Welfare on he basis of he Sochasic Equilibrium Basic New Keynesian Model Rule-of-humb Behaviour à la Galì and Gerler Rule-of-humb Behaviour à la Seinsson 06 Chaper 6. Opimal in aion rae in a Medium-Scale Macroeconomic Model wih Rule-of-humb Price Seers The model Households and Marke Srucure The Governmen Firms Marke Clearing Solving he Model Calibraion Preferences 38 4

6 Technology and Shocks Marke Srucure The Ramsey Seady Sae The Cashless Model The Social Planner Allocaion 50 Chaper 7. Ramsey Dynamics and Opimal Operaional Ineres-rae Rules Ramsey Dynamics Opimal Operaional Ineres-rae Rules 69 Chaper 8. Conclusions 75 Chaper 9. Appendices Appendix A. Deailed Derivaion of he Hybrid Phillips Curve under Rule-of-humb Behaviour à la Seinsson (2003) Appendix B. Deailed Derivaion of he Uiliy-based Objecive Funcion under Rule-of-humb Behaviour à la Seinsson (2003) Appendix C. Funcional Forms and Equilibrium Condiions in Saionary Variables Funcional Forms Saionary Variables Equilibrium Condiions in erms of Saionary Variables Cashless model Appendix D. The Social Planner Allocaion: Equilibrium Condiions in Saionary Variables Equilibrium Condiions in erms of Saionary Variables 200 References 202 5

7 Lis of Tables 4. Opimal Long-run In aion Raes Benchmark Calibraion Comparaive Saics Benchmark Calibraion Model wih Money: Sandard Deviaions under he Ramsey Opimal Sabilisaion Policy Cashless Model: Sandard Deviaions under he Ramsey Opimal Sabilisaion Policy 65 6

8 Lis of Figures 5. Purely forward-looking model and model wih rule-of-humb behaviour à la Galì and Gerler (999). Annualised percenage opimal seady-sae in aion implied by he alernaive imeless perspecive policy Model wih rule-of-humb behaviour à la Galì and Gerler (999). Annualised percenage opimal seady-sae in aion implied by he sandard imeless perspecive policy Model wih rule-of-humb behaviour à la Seinsson (2003). Annualised percenage opimal seady-sae in aion implied by he sandard imeless perspecive policy Model wih rule-of-humb behaviour à la Seinsson (2003). Annualised percenage opimal seady-sae in aion implied by he alernaive imeless perspecive policy Purely forward-looking model. Welfare on he basis of he deerminisic equilibrium Model wih rule-of-humb behaviour à la Galì and Gerler (999). Welfare on he basis of he deerminisic equilibrium for alernaive values of he degree of price sickiness Model wih rule-of-humb behaviour à la Galì and Gerler (999). Welfare on he basis of he deerminisic equilibrium for alernaive values of he degree of rule-of-humb behaviour Model wih rule-of-humb behaviour à la Seinsson (2003). Welfare on he basis of he deerminisic equilibrium for alernaive values of he degree of price sickiness.00 7

9 5.9 Model wih rule-of-humb behaviour à la Seinsson (2003). Welfare on he basis of he deerminisic equilibrium for alernaive values of he degree of rule-of-humb behaviour Purely forward-looking model. Uncondiional welfare on he basis of he sochasic equilibrium Purely forward-looking model. Condiional welfare on he basis of he sochasic equilibrium Model wih rule-of-humb behaviour à la Galì and Gerler (999). Uncondiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of price sickiness Model wih rule-of-humb behaviour à la Galì and Gerler (999). Condiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of price sickiness Model wih rule-of-humb behaviour à la Galì and Gerler (999). Condiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of rule-of-humb behaviour Model wih rule-of-humb behaviour à la Galì and Gerler (999). Condiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of rule-of-humb behaviour Model wih rule-of-humb behaviour à la Seinsson (999). Uncondiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of price sickiness Model wih rule-of-humb behaviour à la Seinsson (999). Condiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of price sickiness. 0 8

10 5.8 Model wih rule-of-humb behaviour à la Seinsson (2003). Uncondiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of rule-of-humb behaviour Model wih rule-of-humb behaviour à la Seinsson (2003). Condiional welfare on he basis of he sochasic equilibrium for alernaive values of he degree of rule-of-humb behaviour. 6. Model wih money: degree of price sickiness and he Ramsey seady-sae rae of in aion Model wih money: degree of rule-of-humb behaviour and he Ramsey seady-sae rae of in aion Model wih money: ransacion cos parameers and he Ramsey seady-sae rae of in aion Cashless model: degree of price sickiness and he Ramsey seady-sae rae of in aion Cashless model: degree of rule-of-humb behaviour and he Ramsey seady-sae rae of in aion Model wih money: he seady-sae oupu gap for alernaive values of he degree of price sickiness Model wih money: he seady-sae oupu gap for alernaive values of he degree of rule-of-humb behaviour Cashless model: he seady-sae oupu gap for alernaive values of he degree of price sickiness Cashless model: he seady sae oupu gap for alernaive values of he degree of rule-of-humb behaviour Model wih money: Ramsey response o a neural produciviy shock Cashless model: Ramsey response o a neural produciviy shock. 67 9

11 7.3 Model wih money: Ramsey response o a governmen expendiure shock Cashless model: Ramsey response o a governmen expendiure shock Model wih money: Ramsey response o an invesmen-speci c produciviy shock Cashless model: Ramsey response o an invesmen-speci c produciviy shock. 70 0

12 Acknowledgemens I would like o express my graiude o my supervisors, Professor Campbell Leih and Docor Ioana Moldovan, for heir guidance and advice. No word can express my graiude o Eleni and my family.

13 Declaraion The maerial conained in his hesis has no been previously submied for a degree in his or any oher universiy. The copyrigh of his hesis ress wih he auhor. No quoaion from i should be published in any forma, including elecronic and Inerne, wihou he auhor s prior wrien consen. All informaion derived from his hesis mus be acknowledged appropriaely. 2

14 CHAPTER Inroducion The purpose of his hesis is wofold. Firs, i aims o sudy how rule-of-humb behaviour by price seers a ecs opimal moneary policy in an oherwise basic small-scale log-linearised New Keynesian model. In doing so, we follow Woodford (2003) as we derive a uiliy-based policy objecive funcion and subsequenly analyse he policy problem in a linear-quadraic framework. Second, mainaining he presence of rule-of-humb price seers, i aims o he exend he analysis of opimal moneary policy o he medium-scale economy developed in Alig e al. (2005). We characerise Ramsey-opimal moneary policy using he mehodology and algorihms developed in Schmi-Grohé and Uribe (2004b, 2007). New Keynesian economics embeds nominal rigidiies and imperfec compeiion ino he dynamic general equilibrium framework of he Real Business Cycle paradigm. In is basic formulaion, a New Keynesian model feaures one nominal rigidiy modelled in erms of a consrain on he frequency of price seing. Three main models of nominal price rigidiy are used in he lieraure: he quadraic price adjusmen cos model of Roemberg (982), he random price adjusmen signal model of Calvo (983), and he model of saggered conracs of Taylor (980). The Calvo model is he mos widely used in he lieraure for reasons of racabiliy and we mainain he assumpion of sicky prices à la Calvo (983) hroughou his hesis. However, he analysis carried ou in his hesis applies equally well o he oher models of nominal price rigidiy since, as shown in Roemberg (987) and Robers (995), hey all lead, up o a log-linear approximaion, o he same form of aggregae-supply relaion. 3

15 4 In is basic formulaion, a New Keynesian model derived from a discree-ime version of he Calvo price seing model is purely forward-looking. In aion dynamics are described by wha Robers (995) labels he New Keynesian Phillips curve (NKPC henceforh) 2. The NKPC relaes in aion oday o a measure of excess demand and expeced fuure in aion. On he one hand, he appealing feaures of he NKPC are well known. Firs, i is microfounded in he idea ha moneary non neuraliy is due o nominal price rigidiies. Second, i recognises he imporance of in aionary expecaions in he deerminaion of in aion oday as rsly sressed by Friedman (968) and Phelps (968). Third, i is simple enough o be useful for heoreical moneary policy analysis. As a resul, he New Keynesian model derived from a discree-ime version of he Calvo price seing model has become he workhorse for much research on moneary policy and i has been described by McCallum (997) as "he closes hing here is o a sandard speci caion" 3. On he oher hand, he failures of he NKPC are equally well-known as discussed in Mankiw (200). Firs, as iniially poined ou by Ball (994), i implies cosless disin aion, namely no shor-run rade-o beween oupu and in aion. Second, as poined ou by Fuhrer and Moore (995) i fails o capure he empirical fac ha in aion is highly persisen. These wo problems imply ha a disin aion of any size could be achieved coslessly and immediaely by a cenral bank ha could commi o se he pah of fuure oupu gaps equal o zero. Third, evidence from VAR sudies also show ha he response of in aion o shocks is hump-shaped raher han fron loaded as prescribed by he NKPC. Sudies which seek o esimae he NKPC nd ha i s he daa poorly (e.g. Fuhrer and Moore (995), Fuhrer (997), Nelson (998), Galì and Gerler (999), Robers (2005), and Sbordone (2005)). These sudies frequenly rejec he NKPC in favour of a The rs use of a discree-ime version of Calvo s model of price seing, in he conex of dynamic general equilibrium model, is in he work of Yun (996). Oher early applicaions of he same device inlcude King and Wason (996), King and Wolman (996), and Goodfriend and King (997). 2 Aggregae-supply relaion and Phillips curve are used inerchangeably in his hesis. 3 The survey aricle by Clarida e al. (999) and he landmark work by Woodford (2003) are only wo examples among many ohers.

16 5 hybrid Phillips curve, which enails ha curren in aion depends on boh in aionary expecaions and pas in aion, alhough he esimaed relaive values of he forward-looking componen and he backward-looking componen vary grealy beween sudies. To accommodae he persisence in in aion daa, wo main varians of he basic New Keynesian model have been pu forward in he lieraure. Boh varians generae he dependence of curren in aion no only on expeced fuure in aion bu also on lagged in aion by making an addiional assumpion abou he price seing mechanism. The rs varian is due o Galì and Gerler (999). They assume ha a proporion of he rms randomly assigned o reopimise heir prices in he Calvo model do no behave raionally bu follow a rule-of-humb 4. Speci cally, he rule-of-humb prices are a weighed average of he opimal forward-looking prices se in he previous period plus an adjusmen based on lagged in aion. The second varian is due o Chrisiano e al. (2005). They assume ha he rms no assigned o reopimise heir prices will insead index heir prices o lagged in aion. In his hesis we consider backward-looking rule-of-humb behaviour by price seers. Rule-of-humb behaviour by price seers is appealing for a leas ve reasons. Firs, i involves virually no compuaional burden: all ha is needed is for rule-of-humb price seers o observe las period s oupu and/or price seing decisions. Second, i involves passive learning of he behaviour of forward-looking opimising price seers. Third, i implies convergence among individual choices once he e ecs of all shocks are eliminaed from he economy. Fourh, Galì and Gerler (999) provide empirical evidence of he presence of rule-of-humb behaviour in price seing 5. Fifh, using backward-looking price indexaion as a modeling sraegy is less appealing as i implies ha all prices are revised a every poin in ime, which no only conradics empirical evidence ha some 4 An earlier example of he uilizaion of his ype of assumpion in order o beer explain he deviaions of acual behavior from he predicions of models which assume fully raional agens is Campbell and Mankiw (989). They use his ype of assumpion o explain he relaion beween consumpion and income. 5 Once we ake he heoreical economies o he daa, we carefully review he empirical evidence provided by Galì and Gerler (999).

17 6 prices are xed for a cerain amoun of ime in nominal erms (e.g. Bils and Klenow (2004) and Nakamura and Seinsson (2008)) bu also clashes wih he raionale as o why models wih nominal price rigidiy were developed. Moreover, while he implicaions of backward-looking indexaion are horoughly analysed in Woodford (2003), he consequences of rule-of-humb behaviour on he opimal long-run in aion rae have, o he bes of our knowledge, no been analysed previously in he lieraure 6. In characerising opimal moneary policy under rule-of-humb behaviour by price seers, we depar from he widespread pracice in New Keynesian economics of resricing he aenion o models in which he deerminisic seady sae is e cien. The Pareo ef- ciency of he deerminisic seady sae is achieved by assuming he exisence of subsidies which eliminae he seady-sae disorions originaing from monopolisic compeiion 7. This widespread pracice has wo poenial shorcomings. Firs, he insrumen necessary o eliminae seady-sae disorions (i.e. subsidies nanced by lump-sum axaion) is empirically uncompelling. Second, i is ex ane no clear wheher a policy ha is opimal for an economy wih an e cien seady sae remains opimal for an economy where he seady sae is disored. For hese reasons, we do no make he e cien-seady-sae assumpion bu insead work wih models whose seady sae is disored. This implies ha hree equilibrium levels of oupu coexis in he model: ) he acual level oupu, which obains in he presence of boh nominal rigidiies and monopolisic compeiion; 2) he naural level of oupu, which obains in he presence of monopolisic compeiion and in he absence of nominal rigidiies; and 3) he e cien level of oupu, which obains in he absence of boh nominal rigidiies and monopolisic compeiion. 6 Rule-of-humb behaviour has been exensively used o invesigae various issues: responses o supply shocks (Seinsson (2003)), moneary policy rules (Amao and Laubach (2003)), uncerainy (Kimura and Kurozumi (2007)), and open economy (Kirsanova e al. (2007)). 7 Seady-sae and long-run are used inerchangeably in his hesis.

18 7 For he purpose of he rs aim of his hesis, we consider he case of small seady-sae disorions as discussed in Woodford (2003). The degree of ine ciency of he deerminisic seady sae is assumed o be minimal so ha i can be reaed as an expansion parameer. This in urn guaranees ha i su ces o approximae he equilibrium of he model o rs order o obain a second-order accurae measure of welfare. Seady-sae disorions inroduce a gap beween he he naural level of oupu and he e cien level of oupu. This wedge is consan and invarian o shocks so ha he "divine coincidence" in Blanchard and Galì (2007) holds 8. However, allowing for seady-sae disorions maers for he opimal average levels of in aion and oupu, namely for he deerminisic descripion of opimal moneary policy. This is because, from a welfare poin of view, he consan-over-ime gap beween he naural level of oupu and he e cien level of oupu appears in he cenral bank s uiliy-based loss funcion. In oher words, he combinaion of general equilibrium foundaions and seady-sae disorions provides a microfoundaion for argeing a level of oupu above he naural level of oupu. Following he heoreical lieraure on opimal moneary policy, we assume ha he cenral bank s policy insrumen is he shor-erm nominal ineres rae. The combinaion of a cashless economy and he cenral bank s conrol of he nominal ineres rae implies ha he economy is fully described by he aggregae-supply relaion and he cenral bank s objecive funcion. This model of cenral bank behaviour hen allows deermining analyically he long-run in aion rae associaed wih a given policy. In paricular, as sressed by Woodford (2008) "The fac ha he equaions are log-linearized does no mean ha one simply assumes an average in aion rae; he equaions allow one o derive he average in aion rae corresponding o a given policy". We consider hree heoreical economies: ) he purely forward-looking New Keynesian model; 2) he model wih rule-of-humb behaviour à la Galì and Gerler (999); and 3) he model wih rule-of-humb behaviour à la Seinsson (2003). All models are small-scale New Keynesian models as hey feaure 8 Tha is, percenage changes in he naural level of oupu correspond in he log-linear approximaion o percenage changes in he e cien level of oupu.

19 8 only one nominal rigidiy (i.e. price sickiness) and one real rigidiy (i.e. monopolisic compeiion in produc markes). We consider di eren ypes of opimal commimen policy ha have been proposed in he lieraure: he zero-opimal policy, he imeless perspecive commimen policy in Woodford (999), and he alernaive imeless perspecive commimen policy pu forward by Blake (200), Jensen and McCallum (2002), and Damjanovic e al. (2008). Our preference for commimen is based on our focus on analysing he opimal long-run in aion rae when he deerminisic seady sae is disored 9. For he purpose of he rs aim of his hesis, hree resuls sand ou from he lieraure concerning opimal moneary policy in log-linearised New Keynesian models. Firs, ruleof-humb behaviour by price seers, speci ed eiher à la Galì and Gerler (999) or à la Seinsson (2003), breaks he opimaliy of zero long-run in aion found in New Keynesian models. Indeed, wihin New Keynesian economics, he opimaliy of a moneary policy ha aims a zero in aion is surprisingly robus. Full price sabiliy is opimal despie he ine ciency of he nonsochasic seady sae and he exisence of a posiively sloped long-run Phillips-curve rade-o. Moreover, as shown in Woodford (2003), zero longrun in aion is also robus o he presence backward-looking price indexaion. Ruleof-humb behaviour, regardless of is speci caion, implies ha he simulaive e ec of higher in aion is greaer han he oupu cos of higher in aion hus generaing a longrun incenive for posiive in aion under an opimal commimen. Second, a ype of imeless perspecive commimen policy is also capable of delivering posiive seady-sae in aion, even in he purely forward-looking New Keynesian model. Third, all he opimal long-run in aion raes are direcly proporional o he gap beween he naural level of oupu and he e cien level of oupu. Hence, wha we show here is ha he widespread assumpion of an e cien seady sae is no innocuous: a policy ha is opimal for an economy wih an e cien seady sae does no remain opimal in an economy where he 9 Of course, discreionary conduc of moneary policy would resul in he well-known in aion bias sressed by Kydland and Presco (977) and Barro and Gordon (983).

20 9 seady sae is disored. Moreover, he posiive long-run in aion raes urn ou o be small in magniude for empirically realisic values of he models srucural parameers. For he purpose of he second aim of his hesis, we consider he medium-scale model developed in Alig e al. (2005). This model emphasises he imporance of combining nominal as well as real rigidiies in explaining business-cycle ucuaions. Speci cally, he model feaures four nominal rigidiies, sicky prices, sicky wages, a ransacional demand for money by households, and a cash-in-advance consrain on he wage bill of rms, and four real rigidiies, invesmen adjusmen coss, variable capaciy uilisaion, habi formaion, and imperfec compeiion in produc and labour markes. We exend he model by allowing a fracion of price seers o behave in a rule-of-humb manner à la Galì and Gerler (999). We depar from he assumpion of small seady-sae disorions and consider he case of a largely disored seady sae. We characerise boh he Ramsey seady sae and Ramsey dynamics and address he quesion of implemenaion of opimal moneary policy by characerising opimal, simple, and implemenable ineres-rae rules. In doing so, we use he algorihms developed in Schmi-Grohé and Uribe (2004b, 2007). Speci cally, large seady-sae disorions imply ha o obain a second-order accurae measure of welfare i does no su ce o approximae he model s equilibrium condiions up o rs order. In characerising ineres-rae rules, we use he mehodology and he algorihm developed in Schmi-Grohé and Uribe (2004b) for second-order approximaions o policy funcions of dynamic and sochasic models. As for he Ramsey seady sae, he key policy problem faced by he cenral bank is he rade-o beween he sabilisaion of he degree of price dispersion and he sabilisaion of ransacional fricions, which calls for he Friedman rule, namely a de aion which is consisen wih a zero nominal ineres rae. We nd ha he resuls in Schmi-Grohé and Uribe (2007), who consider he possibiliy of backward-looking price indexaion, generally hold. Rule-of-humb behaviour by price seers does no aler he high sensiiviy of he long-run in aion rae wih respec o he degree of price sickiness: he opimal long-run in aion is always negaive and i varies beween he level implied by he Friedman rule

21 20 and a level close o price sabiliy. We depar from he analysis in Schmi-Grohé and Uribe (2007) and consider he case of a cashless medium-scale macroeconomic model 0. Indeed, we seek o esablish a link beween he analysis of opimal moneary policy in a basic log-linearised New Keynesian model and is counerpar in a much richer nonlinear heoreical economy. We nd ha, in he absence of ransacional fricions, rule-of-humb behaviour by price seers enails ha he opimal long-run in aion in he seady sae of he Ramsey equilibrium is posiive. We hus obain he same resul ha we analyically derive in he linear-quadraic framework. Indeed, he opimal long-run in aion rae is no only posiive bu also small in magniude. In he linear quadraic framework he in aion rae is direcly proporional o he gap beween he naural level of oupu and he e cien level of oupu. Hence, we solve he social planner problem in he medium-scale economy so o compare he seady-sae gap beween he social planner level of oupu and he Ramsey level of oupu wih he seady-sae e ciency gap in he log-linearised smallscale economies. We nd ha he di erence beween he wo is in fac raher small. We sudy Ramsey dynamics. In doing so, we are ineresed in addressing wo issues. Firs, we wan o assess wheher he zero lower bound on he nominal ineres rae consiues an impedimen o opimal moneary policy. Indeed, one argumen agains seing a negaive in aion rae, as recommended by he model in he presence of money demand by households and rms, or a near-zero in aion rae, as recommended by he cashless model, is ha a negaive or near-zero raes of in aion he risk of incurring in he zero lower bound on nominal ineres rae would resric he cenral bank s abiliy o sabilise he economy. We nd ha his argumen is of no relevance in he conex of boh he model wih money and is cashless counerpar. The reason for his is ha under he Ramsey-opimal policy, he zero lower bound poses an impedimen o moneary policy only in he case of an adverse shock ha forces he ineres rae o be roughly 8 sandard 0 Schmi-Grohé and Uribe (2007) go on o analyse he opimal long-run rae of in aion by aking ino accoun he scal side of he opimal policy problem. They do so by replacing he assumpion of lumpsum axes wih he assumpion of disorionary income axes. The opimal long-run in aion, alhough remaining always negaive, is hen found o be much closer o price sabiliy.

22 2 deviaions below is mean. The probabiliy of his happening is so small ha he zero lower bound on he nominal ineres rae does no impose an economically imporan consrain on he conduc of opimal moneary policy. Second, we characerise he Ramsey-opimal impulse responses o he hree shocks ha drive aggregae ucuaions. Speci cally, we presen he responses of key macroeconomic variables and we focus on how he Ramsey planner uses moneary policy o respond o each of he hree shocks. We show how he Ramsey-opimal sabilisaion policy is robus o he presence/absence of money in he model. Finally, we consider he implemenaion of opimal moneary policy by characerising opimal, simple, and implemenable ineres-rae rules, using he mehodology and he algorihm developed in Schmi-Grohé and Uribe (2004b) for second-order approximaions o policy funcions of dynamic and sochasic models. Iniially, we show how he implemenaion of opimal moneary policy is virually una eced by he presence-absence of money. We characerise he operaional ineres-rae rule, which is de ned exacly as in Schmi-Grohé and Uribe (2007), in boh he medium-scale model wih money and is cashless counerpar. In boh cases, he opimal operaional ineres-rae rule is con rmed o be acive in price and wage in aion, mue in oupu growh and moderaely inerial. We also consider a modi caion of he operaional ineres-rae rule, which prescribes a concern no for oupu growh per se bu for sabilisaion of oupu around a welfarerelevan measure of oupu, namely he gap beween he Ramsey level of oupu and he e cien level of oupu. We nd ha he opimal operaional ineres-rae rule remains acive in price and wage in aion and moderaely inerial, bu also implies a posiive coef- cien on oupu sabilisaion. Regardless of he presence/absence of money in he model, i is opimal for a cenral bank o sabilise oupu gap, namely he log-di erence beween he level of oupu and he e cien level of oupu. The remainder of he hesis is organised as follows.

23 22 Chaper 2 lays ou a basic New Keynesian model ha we exend by allowing a fracion of price seers o behave in a backward-looking rule-of-humb manner, speci ed eiher à la Galì and Gerler (999) or à la Seinsson (2003). Chaper 3 presens uiliy-based objecive funcions for he cenral bank. We exend he analysis in Woodford (2003) by allowing a fracion of price seers o behave in a backward-looking rule-of-humb manner. The backward-looking behaviour is speci ed in wo ways: in he manner of Galì and Gerler (999) and in he manner of Seinsson (2003). Chaper 4 sudies wha consiues opimal moneary policy in he hree heoreical economies. We consider di eren ypes of opimal commimen policy ha have been proposed in he lieraure: he zero-opimal policy and wo ypes of imeless-perspecive policy. Our preference for commimen is based on our focus on analysing he opimal long-run in aion rae when he seady-sae is disored. Chaper 5 discusses he calibraion of he models srucural parameers, evaluaes he opimal long-run in aion raes, and sudies welfare under he alernaive commimen policies. Chaper 6 characerises he opimal seady-sae in aion rae of he Ramsey planner in he medium-scale macroeconomic model developed in Alig e al. (2005), which we exend by allowing a fracion of price seers o behave in a backward-looking rule-ofhumb manner à la Galì and Gerler (999). Chaper 7 sudies Ramsey dynamics and address he quesion of implemenaion of opimal moneary policy by characerising opimal, simple, and implemenable ineresrae rules. Chaper 8 provides concluding remarks.

24 CHAPTER 2 Basic New Keynesian Model In his chaper, we lay ou a basic New Keynesian model ha we exend by allowing a fracion of price seers o behave in a backward-looking rule-of-humb manner. This resuls in a Phillips curve where curren in aion depends on boh expeced fuure in aion and on lagged in aion, namely a hybrid Phillips curve. Backward-looking rule-of-humb behaviour is speci ed in wo ways. Firs, following Galì and Gerler (999) we allow he rule-of-humb price seers o index heir prices o lagged in aion. Second, following Seinsson (2003) we allow he rule-of-humb price seers o index heir prices o boh lagged oupu gap and lagged in aion. The case of rule-of-humb behaviour à la Seinsson (2003) conains an original conribuion o he lieraure. Speci cally, we correc he hybrid Phillips curve repored in he original paper. The misake in Seinsson (2003) relaes o he coe cien on he erm in curren oupu gap in he hybrid Phillips curve. I has been acknowledged by he Seinsson in he Erraum o Opimal Moneary Policy in an Economy wih In aion Persisence, available on he auhor s webpage. The heoreical economy is assumed closed and here is no capial accumulaion. The model consiss of households ha supply labour and purchase goods for consumpion and rms ha hire labour and produce and sell di ereniaed goods in monopolisically compeiive markes. Households and rms behave opimally: households maximise uiliy and rms maximise pro s. The model is basic in ha i feaures only one real rigidiy and one nominal rigidiy. The real rigidiy sems from monopolisic compeiion in he goods markes, which is The hybrid Phillips curve in he case of backward-looking rule-of-humb behaviour à la Galì and Gerler (999) coincides wih he one repored in Amao and Laubach (2003). 23

25 24 modelled as in Dixi and Sigliz (977). The nominal rigidiy is given by saggered price adjusmen as in Calvo (983). Speci cally, he New Keynesian model laid ou here is he basic neo-wicksellian model in Woodford (2003). Woodford (2003) calls models of his kind neo-wicksellian in order o sress he imporance of a moneary policy ransmission mechanism in which ineres raes a ec ineremporal spending decisions. Ye, following Clarida e al. (999) among ohers, he erminology "New Keynesian" has become common place. We share he basic neo- Wicksellian model s assumpions and general formalism. Appendix A repors a deailed derivaion of he hybrid Phillips curve ha obains in he presence of rule-of-humb à la Seinsson (2003). 2.. Households and Marke Srucure There is a coninuum of households of size one. The represenaive household seeks o maximise a discouned sum of uiliy of he form X (2.) E 0 U = E 0 X =0 =0 2 4u (C ; ) Z 0 3 v(h (i); )di5 where 0 < is he discoun facor, C is an aggregae of he household s consumpion of a coninuum of individual goods which are indexed by i over he uni inerval, is a vecor of exogenous real shocks, namely exogenous shocks o household s impaience o consume and o he household s willingness o supply labour, and h (i) is he supply of ype i labour. Following Dixi and Sigliz (977), he consumpion aggregae is de ned as (2.2) C = 4 2 Z c (i) ( )= di 3 5 =( ) 0 where c (i) is he consumpion of good i and > is he consan elasiciy of subsiuion beween goods. For any given realisaion of, he period uiliy funcion, u (C ; ), is assumed o be concave and sricly increasing in C whereas he period disuiliy of

26 25 supplying labour of ype i, v(h (i); ), is assumed o be convex and increasing in h (i). We assume speci c labour markes in he sense ha ype i labour is only used in he producion of good i. Moreover, he represenaive household is assumed o simulaneously supply all ypes of labour. Considering di ereniaed labour inpus, as we shall see below, has he advanage of delivering a model wih labour markes ha is equivalen o he frequenly used yeoman farmers model, in which households are assumed o supply goods direcly. Moreover, if one were o replace speci c labour markes wih a single homogenous labour marke, our resuls would no change qualiaively bu only quaniaively. On he one hand, he assumpion on he srucure of he labour marke a ecs he way in which he oupu gap eners he Phillips curve. On he oher hand, as we shall see below, wha maers for our resuls is ha he aggregae-supply relaion implies he exisence of a posiively sloped long-run rade-o beween in aion and he oupu gap. As shown in Woodford (2003, Ch. 3), a posiively sloped Phillips-curve long-run rade-o obains under boh assumpions abou he srucure of he labour marke. Financial markes are assumed o be complee, such ha, hrough risk sharing, households face he same budge consrain, which is given by (2.3) Z Z Z p (i)c (i)di + E [Q ;+ B + ] B + W (i)h (i)di + (i)di T where p (i) is he price of good i, B is he nominal value of nancial wealh brough ino he period, Q ;+ is he sochasic discoun facor for one period ahead payo s, T is ne nominal ax collecion by he governmen, W (i) is he nominal wage for labour of ype i, and (i) is he nominal pro s from sales of good i. The governmen is no modelled endogenously and i is assumed o run a balanced budge a all imes. G is he exogenous process ha describes governmen purchases of he aggregae good. Speci cally, denoing wih g (i) he governmen s consumpion of good i, G akes he Dixi and Sigliz (977) aggregae form, namely

27 26 (2.4) G = 4 2 Z 0 g (i) ( )= di 3 5 =( ) The budge consrain saes ha, in any period, nancial wealh carried ino he subsequen period plus consumpion canno be worh more han he value of nancial wealh brough ino he period plus afer-ax non nancial income earned during he period. Noe ha we assume ha every household owns an equal share of all he rms operaing in he economy. The assumpion of complee nancial markes implies ha he assumed rms ownership and he assumpion ha he represenaive household supplies all ypes of labour direcly are innocuous; dropping hese assumpions would no change he condiions ha deermine equilibrium prices and quaniies. The household s opimal behaviour is described by hree ses of condiions. Firs, households face a decision in each period abou how much o consume of each individual good. They adjus he share of a paricular good in heir consumpion bundle so o exploi any di erences in relaive prices. Minimising he level of oal expendiure, given he consumpion aggregae in (2.2), yields he demand for each individual good p (i) (2.5) c (i) = C where he aggregae price level, P, is given by P (2.6) P = 4 2 Z p (i) 3 di5 = 0 This speci caion of he price index has by consrucion he propery ha P C gives he minimum price for which an amoun C of he aggregae consumpion can be purchased. Marke clearing implies ha he oal non nancial income, ha is he economy-wide sales revenues, can be wrien as P Y. Here, Y is he economy s oal oupu, which is he Dixi and Sigliz (977) aggregae of he quaniies supplied of he various di ereniaed

28 27 goods, denoed wih y (i), namely (2.7) Y = 4 2 Z y (i) ( )= di 3 5 =( ) The household s budge consrain can hus be rewrien as 0 (2.8) P C + E [Q ;+ B + ] B + P Y T The absence of arbirage opporuniies implies ha here exiss a unique sochasic discoun facor, Q ;+. The riskless shor-erm nominal ineres rae, i, has a simple represenaion in erms of he sochasic discoun facor, namely (2.9) + i = E [Q ;+ ] A complee descripion of he household s budge consrain requires ruling ou Ponzi schemes. The implied consrain for nancial wealh carried ino he subsequen period, B +, is given by (2.0) B + X T =+ E + [Q +;T (P Y T )] < wih cerainy, ha is, in each sae of he world ha may be reached in he subsequen period. Here Q ;T discouns nominal income received in period T back o period, Q ;T = TQ Q s ;s. Equaion (2.0) implies ha a household s deb in any sae of he world is s=+ bounded by he presen value of fuure afer-ax non nancial income, which is assumed o be nie. Furhermore, prevening unlimied consumpion also requires ha he nominal ineres rae sais es he zero lower bound, i 0, a all imes: a negaive nominal ineres rae would in fac allow households o nance unbounded consumpion by selling enough bonds. The enire in nie series of ow budge consrains and borrowing consrains in

29 28 urn de nes he lifeime budge consrain for he household X (2.) E 0 Q 0; [P C ] B 0 + E 0 =0 X =0 Q 0; [(P Y T )] We can now complee he descripion of opimal household behaviour. Maximising uiliy (2.) subjec o he ineremporal budge consrain (2.) yields he opimal allocaion of consumpion across ime (2.2) u c (C ; ) = u c C + ; + Q ;+ P P + and he opimal supply of labour of ype i (2.3) v h (h (i); ) u c (C ; ) = W (i) P where u c and v h denoe respecively he parial derivaive of u wih respec o he level of consumpion and he parial derivaive of v wih respec o he supply of labour. Subsiuing for he riskless shor-erm nominal ineres rae, as given by (2.9), in he opimal ineremporal allocaion of consumpion delivers he familiar Euler equaion for consumpion (2.4) E " uc C + ; + u c (C ; ) P P + # = + i Allowing for iling due o ineres raes di ering from he household s discoun facor, raional consumers are hus aemping o smooh consumpion over ime such ha he marginal uiliy of consumpion is equal across periods Firms We assume ha each good i has he linearly homogeneous producion funcion (2.5) y (i) = A h (i)

30 29 where A is a ime-varying exogenous echnology facor. I follows ha he nominal marginal cos of supplying a quaniy y (i) of good i is given by (2.6) MC (i) = W (i)a Noe ha he assumpion of speci c labour markes does no imply ha each price seer is a monopsonis in heir labour marke. The possibiliy of rms having any marke power in heir labour marke is ruled ou by assuming ha price seers ha change heir prices a he same ime also hire labour from he same marke 2. Speci cally, his is achieved by assuming a double coninuum of di ereniaed goods, indexed by (I, j) wih an elasiciy of subsiuion of beween any wo goods. Goods belonging o he same indusry I are hen assumed o change heir prices a he same ime and o be produced using he same ype of labour, namely ype I labour. The fac ha now a coninuum of price seers demand ype I labour eliminaes he possibiliy of marke power in heir labour marke wihou any change for he degree of marke power of each price seer in heir produc marke. Combining he opimal supply of labour of ype i (2.3) and he nominal marginal cos speci caion (2.6), he real marginal cos is given by (2.7) mc(y (i); C ; e ) = v h(y (i)=a ; ) u c (C ; ) A where labour is expressed in erms of oupu and e denoes he vecor of exogenous disurbances, which includes exogenous real shocks o echnology, o household s impaience o consume, and o he household s willingness o supply labour Marke Clearing Marke clearing requires, for each good i and a all imes (2.8) y (i) = c (i) + g (i) 2 The Calvo loery is over indusries prices raher han goods prices.

31 30 equivalenly, in aggregae erms (2.9) Y = C + G Subsiuing he marke clearing condiion ino he Euler equaion for consumpion (2.4) yields " eu c (Y + ; (2.20) E e + ) eu c (Y ; e ) wih P P + # = + i (2.2) eu(y ; e ) = u(y G ; ) Equivalenly, subsiuing he marke clearing condiion ino he real marginal cos speci- caion (2.7) yields (2.22) mc(y (i); Y ; e ) = ev y(y (i); e ) eu c (Y ; e ) wih (2.23) ev(y (i); e ) = v(y (i)=a ; ) Equaions (2.2) and (2.23) are he indirec uiliy funcions 3. The former, which is increasing and concave in Y for each possible realisaion of vecor e, indicaes he uiliy ow o he represenaive household as a funcion of is aggregae demand for resources, where aggregae demand adds he household s share of governmen purchases o he household s privae consumpion. Under he assumpion of G being exogenously deermined, variaions in he level of governmen expendiure are simply anoher source of exogenous 3 Noe ha we use he same noaion as in Woodford (2003). Subscrip c denoes parial derivaives of he indirec uiliy funcion eu wih respec o he level of producion, as hese derivaives are idenical o he parial derivaives of he direc uiliy funcion wih respec o he level of aggregae consumpion.

32 3 variaion in he Euler equaion for consumpion 4. The laer, which is increasing and convex in y (i) for each possible realisaion of vecor e, convers he household s disuiliy of supplying labour used for he producion of good i ino he household s disuiliy of direcly supplying good i. Accordingly, he model laid ou here is idenical o he one ha obains under he assumpion of a single yeoman farmer (i.e. coninuum of yeoman farmers), which is used in boh Seinsson (2003) and Amao and Laubach (2003). The represenaive household s discouned sum of uiliy can in fac be rewrien solely as a funcion of all y (i) X (2.24) E 0 U = E 0 =0 X =0 2 4eu(Y ; e ) Z 0 3 ev(y (i); e )di5 wih he period uiliy U being concave in he level of producion of each of he di ereniaed goods Price Seing Behaviour We now urn o he descripion of price seing behaviour. Following Calvo (983), we assume ha only a fracion of indusries prices are rese in each period. The probabiliy of no reseing he price in each period, 0 < <, is independen of boh he ime ha has gone by since he las price revision and he misalignmen beween he acual price and he price ha would be opimal o charge, namely pricing decisions in any period are independen of pas pricing decisions. Furhermore, we assume ha pro s are discouned using a sochasic discoun facor ha equals on average. We now depar from full raionaliy by inroducing backward-looking rule-of-humb behaviour by price seers. Following Galì and Gerler (999), we assume ha only a fracion! of indusries behave opimally (i.e. in a forward-looking manner) when seing he price, he remaining fracion of indusries use he same backward-looking ruleof-humb when revising heir prices. The degree of rule-of-humb behaviour, 0! <, 4 Henceforh, he vecor e includes exogenous real shocks o echnology, o Governmen purchases, o household s impaience o consume, and o he household s willingness o supply labour.

33 32 is hus consan over ime and price seers canno swich beween backward-looking and forward-looking behaviour. If follows ha in each period all forward-looking price seers will se he same price, which we denoe wih p f, and all backward-looking price seers will as well charge a common price, which we denoe wih p b. The aggregae price level in (2.6) hence evolves according o (2.25) P = ( )(p ) + P where (2.26) p = (!)p f +!p b denoes he overall rese price. The rms allowed o change heir price a ime choose p f so o maximise expeced fuure pro s subjec o he demand hey face. The price seer s objecive is given by X (2.27) E () s (p (i); p I +s; P +s ; Y +s; e +s ) s=0 The price seer s nominal pro funcion,, is linearly homogeneous in is rs hree argumens (i.e. good s price, indusry s price, p I, and aggregae price level) and, for any value of he indusry price and he aggregae price level, single-peaked for some posiive value of he good s price 5. The common forward-looking rese price, p f, is implicily de ned by he relaion X (2.28) E () s (p f ; p f ; P +s ; Y +s; e +s ) = 0 s=0 5 Under y (i) = A h (i), he nominal pro funcion is given by (p (i); p I ; P ; Y ; e p (i) v h ( p I =P Y =A ); ) = p (i) Y ) p (i) Y P P u c (C ; ) P A

34 33 Here (p f ; p f ; P +s ; Y +s; e +s ) = 0 implicily de nes wha Woodford (2003, Ch. 3) labels he noional shor-run aggregae supply curve, which indicaes how relaive prices would be se if here was no consrain on he frequency of price seing. (2003) The common rule-of-humb backward-looking rese price, p b, is speci ed as in Seinsson (2.29) p b = p P P 2 Y where Y n denoes he naural level of oupu, which we precisely de ne below, and 0 is he degree of indexaion o pas demand condiions. Rule-of-humb price seers hus index he previous period overall rese price o pas in aion, fully, and pas demand condiions, according o. In he case of zero indexaion o pas demand condiions, he Y n rule-of-humb collapses o he speci caion in Galì and Gerler (999), p b = p P P Flexible Price Equilibrium and E cien Equilibrium We now consider an environmen in which all rms can adjus prices opimally each period, aking he pah of aggregae variables as given. Pro -maximising behaviour under perfecly exible prices hen implies ha rms will operae a he poin a which he relaive price is a mark-up over he real marginal cos (2.30) where = =( p (i) P = mc(y (i); Y ; e ) ) > is he desired consan mark-up, which is common o all rms. The relaive supply of good i hus sais es (2.3) = y (i) = mc(y (i); Y ; e ) Y Given idenical prices and demand condiions he equilibrium under perfecly exible prices is symmeric: each good is produced and consumed in he same quaniy. Wihin New Keynesian economics, he naural level of oupu in Friedman (968), which we

35 34 denoe wih Y n ( e ), has a precise meaning: i is simply he equilibrium level of oupu under perfecly exible prices. I follows from (2.3) ha he naural level of oupu is implicily de ned by (2.32) mc(y n ; Y n ; e ) = In he case of fully exible prices, equilibrium oupu hus equals he naural level of oupu a all imes, Y = Y n ( e ). The naural level of oupu in urn depends only on he exogenous real shocks, which enails ha he equilibrium oupu under perfecly exible prices is compleely independen of moneary policy. In he case of sicky prices, equilibrium oupu can insead di er from he naural level of oupu. The concep of oupu gap in fac plays a major role in New Keynesian economics, boh as a force bringing abou ucuaions in in aion as well as a arge for moneary policy. Speci cally, he oupu gap, which we denoe wih x, has a precise meaning: i is he deviaion of acual oupu from he naural level of oupu, namely x = b Y b Y n = log(y =Y n ). The naural seady-sae level of oupu is he equilibrium level of oupu ha obains in he presence of fully exible prices and in he absence of exogenous real shocks, ha is e = 0 a all imes. We denoe a variable s value a seady sae wih a bar. The naural seady-sae level of oupu, Y, is implicily de ned by (2.33) mc(y ; Y ; 0) = Noe ha we do no use he superscrip n in denoing he naural seady-sae level of oupu. This is because, if e = 0 and Y = Y a all imes, (2.25) has a soluion wih zero in aion a all imes. In oher words, in he presence of zero seady-sae in aion, he seady sae of he economy wih sicky prices coincides wih he naural seady-sae level of oupu. Indeed, we laer log-linearise he model around a seady-sae wih zero in aion and he naural seady-sae level of oupu, Y.

36 35 We mus sress ha he naural level of oupu is no Pareo e cien. The e cien level of oupu is in fac he equilibrium level of oupu under boh perfecly exible prices and perfec compeiion. The e cien level of oupu, Y ( e ), is hus implicily de ned by (2.34) mc(y ; Y ; e ) = Accordingly, he e cien seady-sae level of oupu, Y, is implicily de ned by (2.35) mc(y ; Y ; 0) = 2.6. Log-linearised Model Log-linearising requires choosing he seady sae around which he log-linear approximaion is aken. We log-linearise he model around a seady-sae wih zero in aion and he naural seady-sae level of oupu. We denoe a variable s log-deviaion from is seady-sae value, which is denoed wih a bar, wih a ha. Taking a rs-order Taylor series expansion o eu c (Y ; ) yields (2.36) b euc Y ; e = eu ccy eu c by + eu c e eu cc Y where b Y = log(y =Y ) and all parial derivaives are evaluaed a seady sae 6. Similarly, aking a rs-order Taylor series expansion o ev y (y (i); e ) gives (2.37) b evy y (i); e = ev yyy ev y where by (i) = log(y (i)=y ). by (i) + ev y e ev yy Y Log-linearising he Euler equaion for consumpion (2.20) yields (2.38) bi = br + E + 6 In wha follows we use he noaion in Woodford (2003). subscrip denoes parial derivaives of he indirec uiliy funcions wih respec o all exogenous real disurbances in vecor e.