A Comparative Analysis of Optimal and Systematic Monetary Responses to Exogenous Oil Price Shocks: Extended Version with Technical Appendix

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1 A Comparaive Analysis of Opimal and Sysemaic Moneary Responses o Exogenous Oil Price Shocks: Exended Version wih Technical Appendix Michael Plane Sepember 10, 2008 Absrac This paper compares and conrass opimal and sysemaic moneary responses o an exogenous real oil price shock across a wide variey of modeling assumpions. For he calibraions considered I show ha he opimal policy focuses on sabilizing some combinaion of PPI/Core CPI and nominal wage inaion and never sabilizes eiher value added or headline inaion. The welfare losses generaed by dieren sysemaic policies depend upon how well he policy sabilizes he welfare relevan inaion variables and cerain classes of policies perform well across a number of models. Cerain exensions, such as capial accumulaion, or calibraions, such as how sicky wages or prices are, have imporan quaniaive impacs on he sizes of he welfare losses. Dae: Sepember Any commens would be grealy appreciaed. I would like o hank my advisors Edward Bue, Eric Leeper, Brian Peerson, and Todd Walker as well as James Murray and Hess Chung for helpful commens. Par of his paper was compleed while a he Sveriges Riksbank as a summer inern in heir research deparmen. While here I received very helpful commens from Lars Svensson and ohers. I assume full responsibiliy for any and all errors in he paper. Deparmen of Economics, Indiana Universiy, 100 S. Woodlawn, Bloomingon, IN 47405, or by a mplane@indiana.edu 1

2 1 Inroducion The rise in oil prices over he las half decade has creaed a new ineres in he old quesion of how moneary policy should respond o hese evens. While here is some consensus in he empirical lieraure abou he real and nominal eecs of an oil price shock here remains a healhy debae abou he relaive imporance of he specicaion of moneary policy. Some papers, such as Bernanke, Gerler, and Wason (1997), argue ha how moneary policy responds o hese ypes of shocks is of overriding imporance while ohers, such as Hamilon and Herrera (2004) and Carlsrom and Fuers (2006), disagree abou he srengh of heir ndings. The uncerainy surrounding hese resuls has spurred researchers o use heoreical moneary models o examine he implicaions of using dieren specicaions of moneary policy in response o an oil price shock. The recen heoreical lieraure includes a handful of papers geared owards he Unied Saes. Some of hem, such as Leduc and Sill (2004) and Dhawan and Jeske (2007), have as heir sole focus he examinaion of how various specicaions of moneary policy drive he behavior of he model. Ohers, such as Kormilisina (2008) and Blanchard and Gali (2007), perform oher exercises in addiion o looking a he imporance of moneary policy. These papers have produced a number of useful resuls. Several issues, however, have no been adequaely addressed. Firs, he works so far have examined sysemaic policy and opimal policy in isolaion from each oher. This obscures any connecions ha migh exis beween he wo specicaions of policy and makes comparing resuls beween papers dicul o do. Second, he inclusion of oil ino a New Keynesian model creaes dierences beween he inaion raes of he price of he nal good, he value added price index, and he consumer price index, all of which would be equivalen in models wihou oil. While Dhawan and Jeske (2007) has examined he use of headline inaion in sysemaic rules here has been lile work done on his issue in he conex of opimal policy. Finally, here has been lile discussion regarding he generaliy of he resuls found so far. I would be useful o know if he relaive araciveness of paricular policies are due o cerain modeling feaures or if 1

3 hey hold across a number of models. The goal of his paper is o shed ligh on hese issues and ohers by comparing opimal and sysemaic moneary policy across a variey of modeling assumpions. This is done in he hopes of being able o derive a se of general resuls ha can be used o guide how moneary policy should deal wih exogenous oil price shocks. A baseline model is inroduced which exends he sandard New Keynesian model wih nominal price rigidiies by incorporaing oil as an inermediae inpu and as a good ha provides uiliy for a represenaive agen. The robusness of he resuls derived from ha model are hen checked by considering more complicaed exensions such as capial accumulaion, more general producion funcions and nominal wage rigidiies. Three specicaions of moneary policy are examined. As in he recen works of Schmi- Grohe and Uribe ((2004b), (2004c), (2006a), (2006b), (2006c),) opimal moneary policy is dened as he welfare maximizing Ramsey opimal soluion under commimen. 1 The second class of policies consiss of a se of Taylor rules ha adjus he nominal rae in response o movemens in GDP and various measures of inaion. This paper considers boh opimal and simple Taylor rules where he coeciens in he simple rules are se o values sandard in he lieraure while in he opimal rules hey are chosen o minimize he welfare losses of ha paricular rule. The hird class of policies considered are polices ha perfecly sabilize he inaion rae of a specic price index. I summarize he opimal policy by describing wha i implies for he nominal ineres rae and he various inaion variables and explain hese resuls by building o he inuiion presened in earlier opimal policy works such as Woodford (2003) and Erceg, Henderson, and Levin (2000). To help characerize he sysemaic policies I calculae he welfare losses generaed by using hese policies as opposed o he Ramsey opimal policy. This provides useful informaion regarding which policies perform well, in a welfare maximizing sense, boh wihin and across models. 1 This paper absracs from scal policy so ha opimal policy is solely concerned wih opimally choosing he pah of he nominal ineres rae. 2

4 These exercises produce several ineresing resuls. Firs, I show he only model exensions ha creae signican dierences in welfare losses are capial accumulaion and nominal wage rigidiies. The eecs of capial accumulaion are quaniaive in ha his exensions generally raises he welfare losses of a given policy by 2-3 imes compared o a model wihou capial. Adding nominal wage rigidiies changes he naure of he opimal policy problem and for mos policies leads o much larger welfare losses han he case where only prices are sicky. Under he models considered here are four disinc inaion variables and i is possible o perfecly sabilize only one of hem. The second major resul is ha for all of he models considered i is never opimal o sabilize eiher headline or value added inaion. This occurs because an exogenous oil price shock does no add a disorion o he model so opimal policy sabilizes some combinaion of nominal wage inaion and PPI inaion. When only prices are sicky opimal policy almos compleely sabilizes PPI inaion. When wages are also sicky opimal policy parially sabilizes he inaion rae of he nominal wage and allows for some inaion in he PPI. The hird imporan resul is ha several ypes of policies appear o produce fairly small welfare losses across a number of models. These include policies ha fully sabilize eiher nominal wage, PPI, or value added inaion and Taylor rules which adjus he nominal rae o movemens in one of hose hree inaion variables, condiional on he coecien on inaion being se away from he indeerminacy region. On he oher hand, Taylor rules which respond posiively o GDP movemens, weakly o an inaion variable, or adjus he rae in response o movemens in headline inaion generally produce signicanly larger welfare losses. When wages are sicky policies ha use headline inaion are paricularly unaracive when wages are sicky. The res of he paper is organized as follows. In secion 2 I briey review he relaed lieraure and highligh he similariies and dierences beween his paper and he previous works in he eld. In he hird secion I inroduce a very general model ha conains as special cases a number of simpler models ha are examined in he paper. The fourh secion 3

5 presens resuls for he case where he only disorions presen are nominal price rigidiies and monopolisic compeiion in produc markes. The fh secion adds nominal wage rigidiies and monopolisic compeiion in labor markes. The sixh secion summarizes he key ndings and concludes. 2 Connecions o he Lieraure This paper is an ougrowh of wo lieraures, he opimal moneary policy lieraure and he lieraure ha examines he relaionships beween oil prices and he macroeconomy. Boh of hese lieraures are large so he focus in his secion is no o provide a comprehensive survey bu o insead briey inroduce he works mos closely relaed o his paper. 2 In general seup and mehodology his paper builds o he works of Schmi-Grohe and Uribe ((2004b), (2004c), (2006a), (2006b), (2006c)) by working wih he same deniion of opimal policy. This mehod conrass wih he linear-quadraic approach of Woodford (2003) or he quadraic loss funcion approach used by ohers. A second similariy is ha his paper also uses welfare losses o characerize and rank he various sysemaic policies under consideraion. In opic his paper is a member of he works which focus on he quesion of how moneary policy should respond o exogenous oil price shocks. Paricularly close o his paper are he works of Leduc and Sill (2004), Dhawan and Jeske (2007), and Kormilisina (2008). All of hese papers focus on he Unied Saes and work wih a closed economy New Keynesian model, alhough here are imporan dierences in cerain specics such as how o model oil prices and how o rank various policies. For example, in Leduc and Sill (2004) prices and wages are sicky, oil is an inermediae inpu and moneary policy is modeled wih a variey of simple policy rules. Policies are ranked informally by comparing he sizes of he deviaions of inaion and oupu from heir 2 A more general survey abou oil prices and he economy can be found in Brown and Yucel (2002), amongs ohers. For readings on opimal moneary policy some references can be found in Woodford (2003) or he works of Schmi-Grohe and Uribe previously referenced. 4

6 seady sae values produced by each policy. They nd ha Taylor rules ha have a large coecien on he inaion rae of he price of he nal good produce he bes resuls. Rules which respond posiively o oupu movemens or have low coeciens on inaion produce much worse resuls. Dhawan and Jeske (2007) exend he model in Leduc and Sill (2004) by adding consumer durables. This no only adds a second channel for oil o direcly aec he economy bu also creaes a dierence beween PPI/Core CPI and headline inaion. 3 Policies are ranked along he same lines as Leduc and Sill (2004) and for rules of he same form his paper nds similar resuls. Imporanly, hey also nd ha rules which use headline inaion insead of PPI inaion perform signicanly worse. Kormilisina (2008) exends he medium scale macroeconomic model of Schmi-Grohe and Uribe (2006b) by incorporaing oil as an inermediae inpu. She solves for he welfare maximizing opimal policy, esimaes he model using US daa, and hen compares he opimal response wih he response generaed by a policy rule from he esimaed model. She nds ha he response of he nominal rae is oo small in he esimaed rule compared o he opimal rule. Unlike Leduc and Sill (2004), Dhawan and Jeske (2007), or his paper, she does no aemp o rank sysemaic policy rules. I have recenly discovered a paper by Erceg (2008) ha has been wrien concurrenly wih his paper. In ha paper hey examine opimal policy... This paper diers from he previously menioned works boh in overall focus and in some specic deails. Firs and foremos, his paper examines opimal policy and sysemaic policy rules joinly whereas he previous papers have worked wih one or he oher bu no boh. Second, his paper derives he welfare losses associaed wih various sysemaic policy rules and uses hese losses o rank he policies. This is in conras o Leduc and Sill (2004) and Dhawan and Jeske (2007) where policies are ranked informally by heir eecs on oupu and inaion. I will show laer ha in some cases, bu no all, here is a connecion beween 3 In boh papers here is a dierence beween he PPI and he GDP deaor bu neiher paper examines rules which use GDP deaor inaion. 5

7 he wo ranking mehods. Finally, an explici goal of his paper is o explain he imporance of various exensions wih regards o how hey aec boh opimal and sysemaic policy. The oher papers have generally sared wih fairly complicaed models and while hey have conduced some sensiiviy analysis i is no o he exen considered here. 3 The Model The model is sochasic and ime is discree wih each period represening a quarer. There is a monopolisically compeiive inermediae goods secor ha uses capial, an aggregaed labor inpu, and oil o produce rm specic goods. Prices in his secor are sicky according o Calvo (1983). The aggregae labor inpu is produced using a variey of imperfecly subsiuable labor sub-ypes. The nominal wages of hese labor sub-ypes are also sicky according o Calvo (1983). The oupu of he inermediae goods secor is used by a represenaive rm in he nal goods secor o produce a nal good ha can be ransformed one for one ino a consumpion good, oil, or invesmen spending. Modeling of oil is done in a simple and concise way as in Leduc and Sill (2004) and Dhawan and Jeske (2007). Real oil prices are exogenous in he model and he economy is closed in he sense ha rade decis are no allowed. On he household side a represenaive agen derives uiliy from a consumpion good and from oil producs and derives dis-uiliy from providing labor. The agen can inves in a sock of capial as well as a risk free nominal bond. This paper follows Woodford (2003) and absracs from money demand. 3.1 Producion Producion akes place in wo secors, he rs being a perfecly compeiive nal goods secor and he second being a monopolisically compeiive inermediae goods secor. There is a coninuum of rms of measure 1 indexed by i in he inermediae goods secor. Oupu for each rm is denoed by y i and is sold a a price p i wih he elasiciy of subsiuion beween he various goods given by θ > 1. 6

8 The nal good, Y, is produced by a represenaive rm using he inermediae goods. This rm's problem is max y i,y Γ f = Y 1 0 p i P y i, (1) s.. [ 1 Y = 0 ] y θ 1 θ θ 1 θ i di, (2) where P is he price of he nal good and Γ f is real pros of he nal good rm. The zero pro condiion implies ha he demand funcion for y i is y i = ( pi P ) θ Y, (3) and ha [ 1 P = 0 ] 1 p 1 θ 1 θ i di. (4) P is a price index ha measures he prices received by he rms for heir produc so i will be referred o as he Producer Price Index (PPI). In he models considered here he PPI is also equivalen o he core CPI as P is he price of he consumpion good. The nal good is he numeraire so ha all nominal variables are deaed by he PPI. All inaion variables are gross so ha PPI inaion is Π = P P 1. (5) In he inermediae goods secor each rm i demands capial, k i, he aggregaed labor inpu, h i, and oil, o i o produce y i. The real prices of capial, labor, and oil are denoed as R, W, and P o respecively. As in Yun (1996) inpus are no rm specic bu insead are purchased in compeiive facor markes so ha hese prices are aken as given by each rm. Solving he rm's problem is done in hree seps. In he rs sep he rm produces h i using he labor sub-ypes. In he second sep he rm solves a cos minimizaion problem which gives rs order condiions for k i, h i, and o i. In he nal sep he rm solves a pro maximizaion problem which produces a rs order condiion for p i. Following Schmi-Grohe and Uribe (2006b) he labor inpu h i is produced using a conin- 7

9 uum of labor sub-ypes indexed by j. These ypes are imperfec subsiues for each oher wih he elasiciy of subsiuion given by θ n > 1. Demand by rm i for labor ype j is denoed as h j i. The seup for producing h i from h j i is essenially he same as he one used o produce Y using y i. Each rm produces h i by solving max h i,h j i h i 1 0 w j W h j idj, (6) s.. [ 1 h i = 0 ] h j θn 1 θn θn 1 θn i dj, (7) where w j is he real wage of labor ype j. The demand for labor ype j by rm i is given by and he aggregae wage index is h j i = [ 1 W = 0 ( w j ) θn h i, (8) W ] 1 w j 1 θn 1 θn dj. (9) A deailed explanaion of he wage seing mechanism is deferred unil he agen's problem is described. The cos minimizaion problem for rm i is min T C i = W h i + R k i + P o o i, (10) h i,o i,k i s.. Ah α h i o αo i k α k i = y i, (11) where T C i is oal cos for rm i. Technology is Cobb-Douglas where A is a scaling parameer. Dene φ as he muliplier from he cos minimizaion problem. Afer some basic algebraic 8

10 manipulaion he rs order condiions can be wrien as h i = α h φ y i W, (12) φ y i o i = α o, (13) P o k i = α k φ y i R. (14) As he producion funcion is homogenous degree φ is he uni cos funcion so φ = 1 A ( ) αh ( W R α h α k ) αk ( P o ) αo, (15) α o and T C i = y i φ. (16) Prices in his secor are sicky according o Calvo (1983) so ha rms only change heir prices when hey receive a sochasic signal o do so. The probabiliy of no being able o change is price each period is given by ω. Firms ha receive a signal choose heir price, p i, by solving he pro maximizaion problem max p i Γ i = E j=0 ω j βj λ +j λ ( pi P +j ) 1 θ Y +j ( pi P +j ) θ φ +jy +j. (17) Condiional expecaions a ime are denoed by E and Γ i is he real pro of rm i. The rm discouns expeced fuure pros no only by ω bu also by he sochasic discoun facor given by he βj λ +j λ erm, where β is he discoun facor of he agen and λ is he muliplier on he agen's budge consrain. The rs order condiion for his problem is (θ 1)E j=0 ω j βj λ +j λ ( pi P +j ) 1 θ Y +j = θe ω j βj λ +j j=0 λ ( pi P +j ) θ φ +jy +j. (18) The sandard procedure is o linearize his rs order condiion around a seady sae where Π = 1 which gives he usual New Keynesian Phillips curve. This procedure is decien 9

11 because i may be possible ha he Ramsey opimal seady sae rae of inaion is no 1. This problem can be avoided by re-wriing he rs order condiion as (θ 1)x 2 = θx 1, (19) where i can be shown ha x 1 and x 2 have he following recursive forms: x 1 = ( pi P ) θ Y φ + ωβ λ +j λ ( pi p i,+1 ) θ x 1 +1, (20) x 2 = ( pi P ) 1 θ Y + ωβ λ +j λ ( pi p i,+1 ) 1 θ x (21) These equaions allow for a seady sae rae of inaion dieren from 1 and are also racable analyically and compuaionally. In he equaions given so far a number of he variables are rm specic variables which makes he model inracable. This is deal wih in wo seps. The rs sep is o assume a symmeric equilibrium so ha any rm ha has he opporuniy o change is price in he curren period will chose he same opimal price p so ha p i = p for all i ha receive he signal. This variable is no saionary if seady sae inaion is no exacly 1 bu can be made so by dening he variable p = p P. Equaions (4), (20), and (21) can hen be wrien in saionary form as 1 = (1 ω) p 1 θ x 1 = p θ Y φ + ωβ x 2 = p 1 θ Y + ωβ + ωπ θ 1 (22) ( ) θ p Π θ λ x 1 p +1 λ +1, (23) ( ) 1 θ p Π θ 1 λ x 2 λ +1. (24) p +1 The second sep is o aggregae linearly across rms. Toal demand for labor, oil, and capial 10

12 are herefore given by 1 0 h i di = h d, o i di = O f, k i di = K d. The superscrip f is used on he oil erm o denoe his as oil demanded by rms as opposed o households, which will be given by O h. Inegraing equaion (8) gives he oal demand for labor sub-ype j, h j = w h d. (25) The variable w as is a sae variable measuring real wage dispersion in he model and is dened w = 1 0 ( w j ) θn dj. (26) W Linear aggregaion of rm level oupu using (3) gives 1 0 y i di = Y, (27) where is a measure of price dispersion equal o = 1 ( pi 0 P ) θ di. (28) I can be shown afer imposing a symmeric equilibrium ha he law of moion of is given by = (1 ω) p θ + ω 1 Π θ. (29) Finally, aggregaion of he rs order condiions for he facor inpus, equaions (12), (13), and (14), gives h d = α h φ Y W, (30) 11

13 O f φ Y = α o, (31) P o K d = α k φ Y R. (32) 3.2 The Agen's Opimizaion Problem The represenaive agen derives uiliy from a consumpion good, C, and household demand for oil producs, O h. The agen also derives dis-uiliy from aggregae labor supplied o rms, N s. In erms of uiliy, he agen is concerned only abou aggregae hours worked, given by N s = 1 n j 0, and no abou wha labor ype is provided nor which rm ges he labor. In reurn for supplying labor ype j he agen receives he real wage w j. Nominal wages for each ype of labor are sicky according o Calvo (1983) wih he probabiliy of no being able o change a wage in any given period being ω n. I will discuss shorly he exac naure of he wage seing problem. A his poin, i is only necessary o sae ha for each labor ype j he agen ses he wage when given he opporuniy and hen agrees o provide however much labor is demanded by he rms a ha wage rae. 4 This assumpion implies ha n j = h j = w h d. The agen also supplies a sock of capial o each rm, denoed k s i, which earns a rae of reurn R. As he rae of reurn is equal across rms he agen cares only abou he aggregae 4 Noe ha his seup for sicky wages diers from ha used in Erceg, Henderson, and Levin (2000), for example. In ha paper households provide a specic labor ype which induces wage income heerogeneiy in he model. They hen assume access o complee coningen claims so ha consumpion is idenical across households. I have chosen o work wih a dieren seup since I am no ineresed in heerogeneiy and can avoid coningen claims all ogeher wih his model. 12

14 amoun of capial supplied, denoed as 1 0 k s idi = K s. A hird source of income for he agen is real re-numeraed pros from rms in he inermediae goods secor. These are denoed as Γ i and aggregaed according o 1 0 Γ i di = Γ. The sum of all wages, renal income, and pros is equal o Gross Domesic Income (GDI). In addiion o capial he agen has access o anoher asse in he form of a risk-free nominal bond. The bond, in real erms, is denoed as B. In his paper I follow Woodford (2003) and absrac from money demand. I assume ha C and O h are aggregaed using a CES aggregaor, (C ν 1 ν ) + κ 2 O h ν 1 ν ν 1 ν, where ν is he elasiciy of subsiuion beween non-oil and oil consumpion and κ 2 is a disribuion parameer. 5 aggregaor so ha per period uiliy is given by For convenience I work wih he indirec uiliy version of his ( ) 1 1 Ex τ P CP I 1 1 τ (N s ) 1+ 1 µ κ , µ where τ is he ineremporal elasiciy of subsiuion, µ is he wage elasiciy of labor supply, and κ 1 is a disribuion parameer. The Ex erm is real aggregae consumpion expendiure, 5 A more realisic seing would insead posi ha oil is used in conjuncion wih consumer durables. I urns ou ha in some basic cases he same general resuls abou CPI-H inaion ha I inroduce laer in his paper carry over o a model wih durables. There are, however, some good reasons for underaking a separae reamen of he opic. See Plane (2008b) for a discussion of his and for furher resuls regarding a model wih consumer durables. 13

15 in erms of he nal good, and is given by Ex = C + P o O h. (33) The P CP I erm is he nominal CPI deaed by he price of he nal good. I can be derived from he expendiure minimizaion problem and is P CP I = ( ) 1 + κ ν 2P o1 ν 1 1 ν. (34) Nominal CPI inaion, or headline inaion, is given by Π CP I = Π [ 1 + κ ν 2 (P o ) 1 ν ] 1 1 ν. (35) 1 + κ ν 2(P 1) o 1 ν Headline inaion consiss of a PPI inaion componen as well as a erm ha represens, in a slighly complicaed manner, real oil price inaion. 6 O h can be derived using Roy's ideniy and are The demand funcions for C and 1 C = Ex 1 + κ ν 2P o 1 ν, (36) O h = Ex κ ν 2P o 1 ν 1 + κ ν 2P o 1 ν The appendix provides deails abou he derivaions involved here.. (37) The agen's problem, ignoring he nominal wage choice for now, can be wrien as max Ex,N s,ks,b,ik E β =0 ( ) 1 1 Ex τ P CP I 1 1 τ (N s ) 1+ 1 µ κ , (38) µ 6 This is clearer when he aggregaor is Cobb-Douglas in which case headline inaion is simply Π CP I = Π (Π o ) γo, wih γ o being he weigh of oil producs in he Cobb-Douglas funcion. 14

16 s.. each period Ex + B + I k = h d 1 w j 0 ( w j ) θn dj + R K s + Γ + I 1 B 1, (39) W Π w h d = N s, (40) K s +1 K s = I k δ k K s. (41) Noe ha I k is simply gross invesmen in capial and δ k he depreciaion rae of capial. I denoe he mulipliers on he budge consrain, he labor equaion, and he law of moion for capial as, respecively, λ, λ n,, and λ k,. The rs order condiions for his problem are Ex 1 τ P CP I 1 1 τ = λ, (42) κ 1 (N s ) 1 µ = λ n,, (43) λ = βe I Π +1 λ +1, (44) λ k, = λ, (45) λ k, = βe [λ +1 R +1 + λ k,+1 (1 δ k )]. (46) A rs order condiion mus also be derived for he wage of labor ype j. As saed before, nominal wages are sicky according o Calvo and he wage for each labor ype can be changed wih probabiliy 1 ω n each period. Given ha he job ypes are ex-ane idenical o one anoher he nominal wages for each labor ype j ha are rese in he same period will be idenical in equilibrium. I denoe his real opimal wage choice as w so ha any wage ha can be rese will be given by w j = w, while wages ha are unse are given by w j = wj 1 Π. 15

17 Noe ha he unchanged wages are deaed by inaion because hey are in real erms, no nominal, and over ime he real wage may rise or fall depending on wha happens wih inaion. I is now possible o se up a Lagrangian for w. The wage seing problem for he agen is quie similar o he price seing problem for he inermediae rm. Afer inegraing across he wo saes of receiving and no-receiving a signal, he agen maximizes E k=0 (ω n β) [λ k +k h d +kw θn +k w1 θn k ( 1 s=1 Π +s ) 1 θn λ n,+kh d +kw θn +k w θn k ( 1 s=1 Π +s ) θn ]. (47) The lef hand erm in he bracke represens he bene gained from adjusing he nominal wage while he righ hand side in he bracke represens he marginal coss of doing so. The rs order condiion is θ n 1 θ n E k=0 (ω n β) k λ +k h d +kw θn 1 θn +kw ( k 1 s=1 Π +s ) 1 θn ( = E (ω n β) k λ n,+k h d +kw θn θn 1 +kw k=0 (48) The usual procedure would be o linearize his equaion around a seady sae where Π = 1 which would give a Phillips curve equaion for real wage inaion. This will be inadequae if seady sae inaion is no 1. Insead I dene wo new variables f 1 and f 2 and re-wrie he rs order condiion as where f 1 and f 2 have he following recursive form f 1 = θ n 1 λ h d W θn w 1 θn + ω n β θ n f 2 = λ n, h d W θn w θn f 1 = f 2, (49) + ω n β ( ) ( θn 1 w+1 1 w ( ) ( w+1 1 w Π +1 Π +1 ) f 1 +1 (50) ) θn f 2 +1 (51) Π +s ) θn Previously I dened an equaion for he real wage dispersion variable, w, bu did no give he law of moion for i. Under he assumpions made here regarding he wage seing 16

18 mechanism he law of moion is given by ( ) θn ( ) θn w w W = (1 ω n ) + ωn Π θn 1. (52) W W 1 The aggregae wage index given in (9) can be re-wrien as W 1 θn = (1 ω n ) w 1 θn + ω n Π θn 1. (53) Finally, nominal wage inaion is given by Π w = W W 1 Π. (54) 3.3 Marke Clearing I is now necessary o impose marke clearing in he labor marke, he capial renal marke, he bond marke, and he goods marke. In he labor marke oal labor demand does no exacly equal oal labor supplied because of wage dispersion. Insead, oal labor supply is given by N s = N = w h d. (55) Tha is, any wage dispersion means ha he agen mus spend more ime working o produce a uni of he aggregaed labor inpu because of inecien use of he various sub-ypes of labor. I herefore re-wrie (43) as κ 1 (N ) 1 µ = λn,. (56) Marke clearing in he capial renal marke and he bond marke imply, respecively, K d = K s = K, (57) and B = 0. (58) 17

19 The law of moion for capial in equilibrium is given by I k = K +1 + (1 δ k )K (59) The resource consrain for he economy can be derived by aggregaing he real pro funcion across rms, subsiuing his ino he agen's budge consrain, and hen imposing marke clearing. Doing his gives Ex + I k + P o O h + P o O f = Y. (60) 3.4 Dening Value Added and Value Added Inaion In his model here is a divergence beween gross oupu and value added (Gross Domesic Produc (GDP) or Gross Domesic Income (GDI)). I is no necessary, per se, o dene value added o solve he model. In erms of a measure of oupu, however, policy makers generally use GDP and no gross oupu because GDP is a more naural measure of he oal income of a counry. By deniion value added is equal o gross oupu minus expendiure on inermediae inpus. Dening Y g as real GDP his implies Y g = Y P o O f. (61) Laer in he paper when I specify sysemaic policy rules I will assume ha if hey respond o movemens in an oupu variable ha hey respond o movemens in GDP no o movemens in gross oupu. This emphasis on GDP diers from Leduc and Sill (2004) and Dhawan and Jeske (2007) boh of which work wih policy rules ha respond o movemens of gross oupu. There is also a dierence in he price indices of he nal good and value added and hence in heir inaion raes. I show in he appendix how o use he uni cos funcion o derive formulas for value added inaion when echnology is boh Cobb-Douglas and CES. When 18

20 echnology is Cobb-Douglas value added inaion is given by Π v = Π (Π o ) αo 1 αo, (62) where Π o is real (no nominal) oil price inaion. In he Cobb-Douglas case value added inaion consiss of a componen due o PPI inaion and a second componen due o real oil price inaion. 3.5 Oil Prices Three issues need o be addressed when modeling oil prices. Firs, are we concerned wih nominal or real oil prices? Second, how do we model he sochasic process? Third, are prices exogenous or endogenous? Unforunaely here is no consensus on he bes way o answer any of hese quesions. Leduc and Sill model he log of he real oil price as an AR(1) process, Dhawan and Jeske model he log of he real oil price an ARMA(1,1) process, and Kormilisina assumes ha he nominal price of oil is non-saionary. All of hese papers assume he price o be sricly exogenous o he model. All of hese assumpions and ohers may be reasonable depending on he ime frame being looked a and he goals one has in mind. For example, one could make a fairly good argumen for considering he shocks ha occurred during he 1970s permanen one ime jumps in he price of oil. On he oher hand, from log real oil prices appear saionary and an AR(1) or ARMA(1,1) process appears o reasonably well. Oher more complicaed papers endogenize he price of oil by using muli-counry models. This has is meris and drawbacks. Making he price exogenous grealy simplies he exposiion of he model. I also signicanly reduces he complexiy of he opimal policy problem because having more han one counry inroduces some game heoreic aspecs o he opimal policy problem. Wih ha being said i is probably beer o be pragmaic insead of dogmaic. I assume ha he real price of oil is exogenous and ha all price changes are emporary in naure. 19

21 The exac sochasic process of he log real price of oil is a simple AR(1) process, ln(p o ) = (1 ρ p ) ln(p o ) + ρ p ln(p o 1) + ɛ p, (63) where ɛ p is a mean zeros, i.i.d. shock wih sandard deviaion σ p. Using daa on he log real price of oil from I found an esimae of he sandard deviaion as.08 for a pre-86 sample,.12 for a pos-86 sample, and.10 for he full sample. I calibrae i o a value of.10. The persisence parameer ρ p is se o.90 bu I also examine he case where ρ p is se o.99 and discuss he implicaions of lower values. 3.6 Policy Relevan Inaion Measures The model conains four policy relevan inaion variables: PPI, nominal wage, value added, and headline inaion. PPI and nominal wage inaion are relevan from a heoreical sandpoin because in models wih sicky prices and wages changes in hese raes bring abou price and wage dispersion which leads o welfare losses. Hence any policy maker ineresed in maximizing welfare mus be ineresed in hese variables. Boh value added and headline inaion are no welfare relevan bu are quie relevan from an applied poin of view because hey are ofen explicily used by cenral banks o measure inaion. Consequenly hey form an inegral par of he analysis ha follows. Some useful informaion abou he behavior of value added and headline inaion can be had by simply examining he equaions governing hem under he special assumpion ha uiliy (echnology) is Cobb-Douglas. In his case headline (value added) inaion is given by Π CP I = Π (Π o ) γo, Π v = Π (Π o ) αo 1 αo. For boh variables here is a componen due o PPI inaion and a componen due o real oil price inaion. The direc impac of real oil price inaion on hese wo variables is quie 20

22 dieren hough. Real oil price inaion direcly raises headline inaion as i direcly raises he cos of purchasing oil producs since γ o 0. On he oher hand, he direc eec of real oil price inaion on value added inaion is o lower i as αo 1 α o 0. This resul occurs because he value added price index measures he price one would be willing o pay for a uni of value added and given he assumpions made on he producion funcion a rise in he price of oil brings ou a drop in he price of value added. These wo equaions show anoher paricularly imporan poin: i will never be possible o sabilize more han one of he inaion variables. Any policy which holds consan one of hem necessarily forces he ohers o vary as a funcion of he inaion rae of real oil prices. Sabilizing PPI inaion auomaically means ha he moneary auhoriy allows headline inaion o rise and value added inaion o fall if real oil prices are rising Policies which hold headline inaion consan force PPI inaion o drop and value added inaion o drop even more as Π v = Π CP I (Π o ) αo 1 αo γo. On he oher hand, holding value added inaion consan forces he PPI o rise and creaes an even larger spike in headline inaion because Π CP I = Π v (Π o ) αo 1 αo +γo. 3.7 Moneary Policy To close he model ou i is necessary o specify he behavior of moneary policy. This is done by working wih one of hree specicaions of moneary policy: he opimal policy, a Taylor rule, or a policy which fully sabilizes one of he inaion variables. The opimal policy is dened as he Ramsey opimal soluion under commimen. I assume ha he model begins from he Ramsey opimal seady sae. The Ramsey equilibrium is given by saionary processes for φ, x 1, x 2, p, h d, λ, K, N, Ex, O f, W, Π, R, Y,, 21

23 w, P CP I, I k, λ k,, λ n,, f 1, f 2, w, and I which for 0 maximize ( ) 1 1 Ex τ E β P CP I =0 1 1 τ (N s ) 1+ 1 µ κ µ, given an exogenous process for P o and subjec o equaions (42), (46), (44), (56), (15), (19), (23), (24), (29), (30), (31), (32), and (60), (22), (34), (50), (49), (51), (52), (53), (??), (59), and (45) as well as saring condiions for he relevan variables and for all of he Lagrange mulipliers associaed wih he consrains jus lised. When he model is closed o wih a Taylor rule one of he following four rules are used Rule 1: ln( I I ) = α π ln( Π ), Π (64) Rule 2: ln( I I ) = α π ln( Π Π ) + α g ln( Y g ), Y g (65) Rule 3: ln( I I ) = α π ln( Π Π ) + α po ln( Πo Π Rule 4: ln( I I ) = α π ln( Π Π ) + α g ln( Y g o ), (66) Y ) + α g po ln( Πo ), (67) Πo where he parameers I, Π, Π o, and Y g are he seady sae values of he nominal ineres rae, PPI inaion, real oil price inaion (equal o 1), and real GDP, respecively. 7 rs wo rules are sandard in he lieraure excep ha rule 2 adjuss he nominal rae o movemens in GDP as opposed o oupu. If α po diers from 0 rule 3 and 4 creae an explici adjusmen of he ineres raes in response o inaion in real oil prices above and beyond is eecs on PPI inaion or real GDP. I also consider rules which respond o oher inaion variables insead of PPI inaion. These replace Π wih Π w, Π v, or Π CP I. The rules ha use value added or headline inaion can be re-wrien in erms of PPI inaion. For example, if echnology is Cobb-Douglas he value added version of rule 1 can 7 In an earlier version of his paper I also worked wih implemenable rules which adjused he nominal rae o lagged values of inaion and GDP. These produced implausible impulse response funcions for a wide range of calibraions and hence are omied from his version of he paper. The 22

24 be re-wrien as ln( I [ I ) = α π ln( Π Π ) α ] o ln( Πo 1 α o Π ). o This rule is herefore jus a rule ha uses PPI inaion wih specic resricions on he coeciens. In paricular, he nominal ineres rae is adjused inversely o real oil price inaion. The same general principal holds for headline inaion wih he dierence ha he coecien on oil price inaion mus be posiive. This will have imporan implicaions for some of he laer resuls. The nal class of policies are hose ha fully sabilize a specic inaion variable. This is implemened by adding an addiional equaion of he form Π = Π for he specic inaion variable being sabilized. 3.8 Calibraion and he Ramsey Seady Sae I calibrae he model o an iniial seady sae wih a calibraion ha is represenaive of he works in he moneary lieraure ha focus on he Unied Saes. The ineremporal elasiciy of subsiuion τ is calibraed o.50 and he wage elasiciy of labor supply µ o 1. The discoun facor β is se o.99 o mach he real rae of reurn. Following Erceg, Henderson, and Levin (2000) I se boh he elasiciy of subsiuion beween inermediae goods θ and he elasiciy of subsiuion beween various labor ypes θ n o 4. Boh he probabiliy of a price and a wage being xed in any given period, ω and ω n, are se o.60. The depreciaion rae of capial, δ k, is se o.025. The disribuion parameer κ 1 is se so ha he seady sae value of labor is abou one hird of he agen's oal ime allomen. The elasiciy of subsiuion beween C and O h is se o.25 which ensures ha he price elasiciy of demand for oil producs is fairly low. Real GDP is calibraed o uniy and I calibrae seady sae real aggregae consumpion expendiure, invesmen spending, household demand for oil, and rm demand for oil as percenage shares of GDP. Real consumpion expendiure is se o.80 so gross invesmen spending is.20. Boh household and rm demand for oil producs are se o.05 percen 23

25 of GDP. Boh of hese have ucuaed over ime depending on he period examined due o dierences in boh oal demand for oil and he price of oil. I have chosen slighly high end value for hese shares in order o magnify he poenial for oil o inuence he economy. The values for he parameers from he producion funcion, A, α k, α h, and α o are chosen implicily from hese calibraions. I assume ha he iniial value for PPI inaion is he Ramsey opimal seady sae, he value of which depends crucially upon assumpions embedded in he model. For example, including explici money demand pushes he opimal inaion rae lower because his reduces he disorion from holding money. Sicky prices, on he oher hand, creaes a srong incenive o se Π o 1 in he seady sae because price dispersion creaes dead weigh losses. As his paper absracs from money demand he he opimal seady sae gross inaion rae is 1. Calibraion of he Taylor rules is done in he following way. This paper works wih wo ypes of rules: opimal and simple rules. When working wih opimal rules I use numerical mehods o choose he values of he coeciens α π, α g, and α po so as o minimize he uncondiional welfare loss beween he rule and he Ramsey opimal soluion. In addiion, he coeciens mus generae a unique soluion and α π is resriced from being larger han For he simple rules α π, α g, and α po are calibraed for purposes oher han maximizing he welfare ha he rule could generae. For rule 1 I examine he cases where α π is eiher 1.50 or The former case is a ypical calibraion found in he lieraure while he laer is he borderline indeerminacy case. In rule 2 I calibrae α π o 1.50 and α g o.50, anoher sandard calibraion in he lieraure. In rule 3 I examine he case where α π = 1.50 and he parameer α po is se o a small value of.02. I have se he value of α po so ha large jumps in he price of oil lead o modes bu no overly large movemens in he nominal rae. For example, a doubling of oil prices leads o an increase of abou 1.5 percenage poins in he nominal rae, no including any oher movemens due o he eecs of oil prices on inaion. This seems reasonable. Rule 4 subsumes rules 1-3 and uses α π = 1.50, α po =.02, and α g = In some cases he opimal value of α π was over 1,000,000. I appears ha in hese cases small bu non-zero welfare gains can be had by increasing α π o ever larger values. 24

26 3.9 Calculaion of he Welfare Losses Welfare losses are calculaed as he amoun of aggregae real consumpion required o produce he same welfare in a model wih sub-opimal policy as he Ramsey opimal policy. Dene per period uiliy in he Ramsey soluion as U( Exr as U( Exn P CP I P CP I, N n ). Then he condiional welfare losses, λ c, are given by E =0 β [ U( Exn P CP I, N n ) while he uncondiional losses, λ u, are given by ] = E =0, N r ) and in a non-opimal soluion β [ U(λ c Exr P CP I, N r ) ], E =0 β [ U( Exn P CP I, N n ) ] = E =0 β [ U(λ u Exr P CP I, N r ) ]. I is possible o analyically derive a formula for he welfare losses ha is accurae o second order. Ineresed readers should refer o Schmi-Grohe and Uribe (2006b) for an explanaion of how o do his Equilibrium and Solving he Model The deniion of equilibrium is sandard bu lenghy and is conained in he appendix. The complexiy of he model requires solving i numerically. Furhermore, in order o be able o disinguish he welfare implicaions of dieren policies i is necessary o use a second order approximaion. I use he mehod of Schmi-Grohe and Uribe (2004a) and he code wrien by he auhors in conjuncion wih ha paper. 25

27 4 A Model wih Sicky Prices and Disored Produc Markes One of he goals of his paper is o examine he quaniaive and qualiaive imporance of various modeling exensions. Consequenly I begin he invesigaion from he simples case possible and exend he model in seps. In he rs model considered I se θ n = and ω n = 0 so ha wages are fully exible and he labor sub-ypes are perfecly subsiuable wih each oher. Therefore he only disorions in he model are sicky prices and monopolisic compeiion. Capial accumulaion is also iniially absraced from. Afer presening resuls from hese models I hen examine he implicaions of allowing echnology o be CES or a nesed CES producion funcion ha allows for ne complimenariy beween oil and capial. I always allow for household purchases of oil relaed producs. 9 The resuls are presened in he following manner. I begin by presening he opimal responses of he nominal rae and he dieren inaion variables. I hen presen ables which conain he welfare losses for all of he policies considered. Every able is spli ino secions based upon he inaion variable being considered and hen furher broken down ino opimal and calibraed rules. The rs column of each able idenies he specic rule along wih he inaion variable used in he rule. Columns wo hrough four give he value of he coeciens α π, α g, and α po respecively. The las wo columns presen he condiional and uncondiional welfare losses for a given policy. These welfare losses are convered o percenages so ha an enry of 1 is 1 percenage poin of consumpion, no 100 percen, i.e. each enry is 100 λ c or 100 λ u. Numbers are accurae up o hree decimal places and enries irrelevan for a paricular case are lled in wih '-'. 9 I have examined he case where his is no allowed bu i does no lead o signicanly dieren resuls, exceping for he fac ha here is no such hing as headline inaion when household demand for oil is no allowed. 26

28 4.1 Cobb Douglas Producion Consider rs he case when echnology is Cobb-Douglas and here is no capial accumulaion. This model is jus he usual New Keynesian model exended o include household demand of oil producs and oil as an inermediae inpu. I use his model as a baseline case o derive some basic resuls and o examine he implicaions of adding capial o he model. I rs begin by examining he opimal responses of he four inaion variables and he nominal ineres rae in response o a one sandard deviaion shock o he log price of oil. Figure (1) shows he response of PPI and nominal wage inaion. These variables, and all oher raes, are annualized. I clear from he graph ha opimal policy fully sabilizes PPI inaion while allowing for signican movemen in nominal wage inaion. The nominal wage iniially drops and hen rises monoonically over ime which shows up as a one ime downward spike in nominal wage inaion followed by inaion. As opimal policy fully sabilizes PPI inaion his auomaically means ha value added and headline inaion are no sabilized, which can be seen in gure (2). The opimal policy allows he CPI o spike upwards and hen monoonically decline over ime unil headline inaion reurns o is seady sae value of 1. The value added index iniially spikes downward and hen rises over ime. Wha drives he opimal responses of he inaion variables? The answer o his quesion comes sraigh from resuls already derived in he opimal policy lieraure. In order o maximize welfare opimal policy seeks o minimize he disorions ha creae ineciencies in he economy. In his case he only disorions are sicky nominal prices and monopolisic compeiion. Monopolisic compeiion creaes an incenive for moving around Π o adjus oupu while sicky prices creaes an incenive o sabilize Π so ha price dispersion is kep o a minimum. Given he lack of an analyical soluion he only way o nd ou which disorion dominaes is o examine he resuls. In his case even wih θ se o he low value of 4 he disorion from sicky prices clearly dominaes given ha PPI inaion is fully sabilized. 27

29 Figure 1: Opimal PPI and Nominal Wage Inaion wih Cobb-Douglas Technology Figure 2: Opimal Value Added and Headline Inaion wih Cobb-Douglas Technology 28

30 Figure 3: Opimal Nominal Rae wih Cobb-Douglas Technology Wha does opimal policy imply abou he behavior of he nominal rae? The usual line of hough is o view an oil price shock as a ypical supply shock which reduces oupu and raises inaion so ha he cenral banks should ighen raes if hey wan o gh he inaionary pressures generaed by he shock. Figure (3) shows his o be rue for his model. In response o a one sandard deviaion shock o he price of oil, roughly a 10 percen increase, he nominal rae is raised by abou 10 basis poins and is above is seady sae value on he enire ransiion pah. How do he oher specicaions of policy hold up o he opimal one? Using he welfare losses as guidance I answer he following four quesions. Wha versions of he opimal Taylor rules, if any, reduce he welfare losses o 0 and why? Do any of he simple rules perform relaively well? Wha are he consequences of sabilizing an inaion variable besides he welfare relevan PPI inaion? Under wha circumsances does he choice of he inaion variable maer? The welfare losses from he various Taylor rules are presened in Table 1. The opimal simple rules which respond o PPI inaion, all characerized by large coeciens on PPI inaion, 29

31 always reduce he welfare losses o 0. High coeciens on he inaion variable in a Taylor rule sabilize ha paricular variable and since sabilizing PPI inaion is he rue opimal policy he opimal rules his explains he large coeciens. The welfare losses of he opimal rules which respond exclusively o value added, nominal wage, or headline inaion are no zero bu are very small, usually below a percenage poin of consumpion. Noe he small coeciens on value added and headline inaion. This occurs because having a large coecien on any of hose variables will sabilize he wrong inaion variable a he expense of causing volailiy in he welfare relevan PPI inaion variable. Anoher resul ha sands ou is ha opimal rules which respond o value added inaion, nominal wage inaion, or headline inaion in conjuncion wih real oil price inaion are able o reduce he welfare losses o 0. The reason why his occurs for value added and headline inaion is ha hese rules can be re-wrien as rules which respond o PPI inaion and real oil price inaion. By having he appropriae coecien on real oil price inaion he opimal rule eliminaes he real oil price inaion componen from value added or headline inaion so ha only PPI inaion is lef in he rule. Consequenly he rule behaves very close o he opimal version of Rule 1 ha uses PPI inaion. The resul for nominal wage inaion occurs because nominal wage inaion is a large componen of value added inaion and hence heir behavior is very similar. The welfare losses from he calibraed versions of he rules show ha a rule which moderaely sabilizes an inaion variable by seing α π = 1.50 is able o reduce he welfare losses o negligible amouns. Responding weakly o real oil price inaion, as in rule 3, is of minor imporance. On he oher hand, rules which respond posiively o GDP movemens or weakly sabilize an inaion variable by having α π = 1.01 produce much larger welfare losses. In paricular, a low coecien on PPI or headline inaion produces very large welfare losses in relaion o he oher rules. Assuming ha per capia aggregae consumpion is $30,000 hese losses ranslae ino abou $65 per capia and $142 per capia respecively. The quesion naurally arises as o why hese wo polices seem o perform so horribly. Given 30

32 Figure 4: PPI Inaion when α π = 1.01 in a Taylor Rule ha welfare is maximized by sabilizing PPI inaion i is bes o examine wha happens o PPI inaion under hese policies. Figure (4) shows clearly ha he low coeciens on inaion bring abou large spikes in PPI inaion wih he resuls going from wors o bes for rules ha use headline, PPI, and value added inaion. In he case of headline inaion no only is here a low coecien on PPI inaion bu here is a posiive coecien on real oil inaion. As shown in gure (5) in his case he nominal rae is well above wha happens wih a rule ha simply responds o PPI inaion. The opposie occurs wih value added inaion because he coecien on real oil price inaion is negaive in ha case. This reduces inaion compared o he rule ha responds only o PPI inaion and hence reduces he welfare losses generaed by ha rule. Wha are he implicaions of fully sabilizing a paricular inaion variable? Obviously sabilizing PPI is opimal bu wha occurs when he wrong inaion variable is sabilized? Consider he case of sabilizing value added inaion. Figure (6) shows he resuls ha sabilizing value added inaion has on PPI, value added, and headline inaion. Sabilizing value added inaion forces he PPI o adjus upward in response o he real oil price shock. 31

33 Figure 5: The Nominal Rae when α π = 1.01 in a Taylor Rule The upward jump in he PPI brings abou an even larger jump in he CPI han occurs when he opimal policy is in place. Policies which sabilize headline inaion force he PPI o adjus downwards which brings abou an even larger downward adjusmen in he value added price index. Despie he unpleasan spikes in PPI inaion ha occur when he wrong inaion variable is sabilized hese policies all seem o perform reasonably well in welfare erms. The losses from sabilizing nominal wage inaion are incredibly small, less han 1 percen of a percen of consumpion. Sabilizing value added or headline inaion produce losses ha are relaively large bu sill fairly small in absolue erms. Sabilizing value added inaion creaes losses of abou 5 percen of a percen while sabilizing headline inaion creaes a loss a lile under 3 percen of a percen. These ranslae ino losses of roughly $15 and $9 per capia in aggregae consumpion respecively. How many of he previous resuls are robus o incorporaing capial accumulaion ino he model? I begin answering his quesion by looking a he opimal policy resuls. Figures (7) and (8) replicae he graphs shown in gures (1) and (2) for he case wih capial accumula- 32