On the Desirability of Nominal GDP Targeting *

Transcription

1 On he Desirabiliy of Nominal GDP Targeing * Julio Garín Universiy of Georgia Rober Leser Colby College July 21, 2015 Eric Sims Universiy of Nore Dame & NBER Absrac This paper evaluaes he welfare properies of nominal GDP argeing in he conex of a New Keynesian model wih boh price and wage rigidiy. In paricular, we compare nominal GDP argeing o inflaion and oupu gap argeing as well as o a convenional Taylor rule. These comparisons are made on he basis of welfare losses relaive o a hypoheical equilibrium wih flexible prices and wages. Oupu gap argeing is he mos desirable of he rules under consideraion, bu nominal GDP argeing performs almos as well. Nominal GDP argeing is associaed wih smaller welfare losses han a Taylor rule and significanly ouperforms inflaion argeing. Relaive o inflaion argeing and a Taylor rule, nominal GDP argeing performs bes condiional on supply shocks and when wages are sicky relaive o prices. Nominal GDP argeing may ouperform oupu gap argeing if he gap is observed wih noise, and has more desirable properies relaed o equilibrium deerminacy han does gap argeing. JEL Classificaion: E31; E47; E52; E58. Keywords: Opimal Policy; Nominal Targeing; Moneary Policy. * We are graeful o Tim Fuers and seminar paricipans a he Universiy of Nore Dame, he 2015 Midwes Macroeconomics Conference, he Banco Cenral del Uruguay, he Universidad ORT Uruguay, he NGDP Targeing Conference, and he CEF2015 Conference for helpful commens and suggesions. address: jgarin@uga.edu. address: rbleser@colby.edu. address: esims1@nd.edu.

2 1 Inroducion Wha rule should a cenral bank follow in he formaion of moneary policy? Despie exensive research abou his opic, i remains an unseled quesion. Alhough nominal oupu argeing has recenly received aenion in he popular press and wihin policy circles, i has no been scruinized wihin he conex of he quaniaive frameworks commonly used by cenral banks. The objecive of his paper is o sudy he desirabiliy of nominal GDP argeing wihin he conex of a New Keynesian model wih boh price and wage rigidiy. In paricular, we compare he welfare properies of nominal GDP argeing o wo oher popular argeing rules inflaion and oupu gap argeing as well as o a convenional Taylor rule. Alhough here is some disagreemen on which ype of rule cenral banks should follow, economiss agree on several principles in he design of moneary policy. Firs, rules are preferred o discreion. Rules allow agens o beer anchor expecaions which improves he inflaion-oupu gap radeoff. This is rue in models of eiher ad hoc Phillips curves as in Barro and Gordon (1983) or in micro-founded Phillips curves as in Woodford (2003). Second, he policy objecives of cenral banks should be undersandable o he public. As argued by Bernanke and Mishkin (1997), his requires he cenral bank o be more accounable. Even if moneary policy follows some sric rule, forming expecaions is difficul if households do no undersand he rule. Third, he cenral bank faces informaion consrains which should be aken ino accoun in he formaion of moneary policy. Responding o precisely measured variables is superior o responding o imprecisely measured variables or variables ha are hypoheical consrucs of a model. Finally, a desirable moneary policy rule ough o suppor a deerminae equilibrium. In he micro-founded welfare loss funcion from he simples version of he New Keynesian model, a cenral bank ough o care abou sabilizing boh price inflaion and he oupu gap (defined as he gap beween he equilibrium level of oupu and he hypoheical equilibrium level of oupu which would obain if prices were flexible). Inflaion and oupu gap argeing are herefore easily moivaed as poenially desirable policy rules. In a version of he model where only prices are sicky, he so-called Divine Coincidence (Blanchard and Gali (2007)) holds, and eiher argeing rule fully implemens he flexible price allocaion. In oher words, sabilizing inflaion also sabilizes he oupu gap, and vice-versa. Given ha he oupu gap is a hypoheical model-based consruc, whereas inflaion is easily observed a high frequencies, inflaion argeing is herefore ofen oued as a highly desirable and easily implemened policy rule. Inflaion argeing may be less desirable in versions of he model in which he Divine Coincidence does no hold. Erceg e al. (2000) consider a version of he model in which 1

3 boh prices and nominal wages are sicky, he combinaion of which breaks he equivalence beween inflaion and gap sabilizaion and renders i impossible for he cenral bank o fully implemen he flexible price and wage allocaion. For plausible parameerizaions of he parameers governing price and wage sickiness, hey find ha inflaion argeing ends o perform poorly from a welfare perspecive. Oupu gap argeing, in conras, does very well, coming close o implemening he flexible price and wage allocaion. Alhough nominal GDP argeing has recenly gained aenion as poenial policy rule in he popular press and wihin policy circles, i has received relaively lile aenion wihin he conex of he ype of models in widespread use a cenral banks and among academics. Whereas inflaion argeing focuses on nominal variables and gap argeing on real variables, nominal GDP argeing simulaneously arges boh nominal and real variables. And unlike oupu gap argeing, i does no require he cenral bank o observe he hypoheical flexible price and wage level of oupu. In Secion 2 we sudy he meris of nominal GDP argeing relaive o oher policy rules using a sandard parameerizaion of he sicky price and wage New Keynesian model developed by Erceg e al. (2000). We evaluae differen rules on he basis of he average welfare loss relaive o a hypoheical flexible price and wage allocaion. Like Erceg e al. (2000), we find ha oupu gap argeing does well, producing very small welfare losses ha come close o implemening he flexible price and wage allocaion. Nominal GDP argeing does almos as well as gap argeing. I is associaed wih smaller welfare losses han a convenionally parameerized Taylor rule and significanly ouperforms inflaion argeing. Nominal GDP argeing performs bes in a relaive sense when wages are sicky relaive o prices and condiional on supply shocks. These resuls are consisen wih he inuiion laid ou by Sumner (2014) in a exbook aggregae supply / aggregae demand model. We consider a more elaborae medium scale version of he model in Secions 3 and 4. In addiion o price and wage sickiness, he model feaures invesmen and capial accumulaion, several sources of real ineria, and a number of differen shocks. In Secion 4, he parameers of he model are esimaed using Bayesian mehods o fi recen US daa. The resuls of he medium scale model echo hose from he small scale model. Oupu gap argeing produces he lowes welfare losses relaive o he hypoheical flexible price and wage allocaion. Nominal GDP argeing does almos as well, significanly beering boh an esimaed Taylor rule as well inflaion argeing. As in he small scale model, nominal GDP argeing performs bes when wages are sicky relaive o prices and condiional on supply shocks, alhough he relaive desirabiliy of nominal GDP argeing condiional on supply and demand shocks is no as sark as in he small scale model. Though we find oupu gap argeing o be he bes performing of he differen policy 2

4 rules under consideraion, here are some reasons o be weary of acually implemening gap argeing as a policy rule. These reasons relae back o some of he basic principles of desirable moneary policy which are highlighed a he beginning of he Inroducion. Firs, as a hypoheical model consruc, gap argeing may be difficul o successfully communicae o he public, even if he cenral bank can observe he flexible price and wage level of oupu wih precision. Second, i may be difficul for he cenral bank o observe he flexible price and wage level of oupu, paricularly in real ime. This poin has been made by Orphanides (2001), Orphanides and van Norden (2002), and Orphanides (2003). Third, gap argeing may resul in equilibrium indeerminacy. This poin has been made in he conex of sicky price New Keynesian models wih posiive rend inflaion in Hornsein and Wolman (2005), Ascari and Ropele (2009), and Coibion and Gorodnichenko (2011). We consider in Secion 4 a couple of differen exercises o address some of hese poins in he conex of he esimaed medium scale model. In one, we assume ha he cenral bank observes he flexible price and wage level of oupu wih noise, and compue he amoun of noise which would equae he welfare losses associaed wih argeing he mis-measured gap and argeing nominal GDP. We find ha if he noise in he observed flexible price and wage level of oupu is more han abou one-hird he volailiy of he acual flexible price and wage level of oupu, hen nominal GDP argeing produces a lower welfare loss han does gap argeing. In a second exercise, we suppose ha he cenral bank adjuss is perceived flexible price and wage level of oupu o he acual flexible price and wage level of oupu slowly. If he cenral bank adjuss slowly enough, hen gap argeing can resul in significan welfare losses relaive o nominal GDP argeing. Third, we invesigae he deerminacy properies of gap argeing when rend inflaion is posiive. We find ha if rend inflaion is greaer han abou 0.2 percen annually, hen gap argeing resuls in equilibrium indeerminacy. Because in he long run he level of oupu is independen of moneary policy, nominal GDP argeing is equivalen o a price level arge in he long run, and herefore suppors a deerminae equilibrium for any level of rend inflaion. Our paper is relaed o a large lieraure on opimal moneary policy. The relaive meris of nominal GDP argeing versus inflaion argeing have been recenly revived by Billi (2014) and Woodford (2012), who discuss he rules wihin he conex of he zero lower bound. Cecchei (1995) and Hall and Mankiw (1994) show in counerfacual simulaions ha nominal GDP argeing would lower he volailiy of boh real and nominal variables. Neiher of hese laer wo papers have a srucural model o conduc welfare analysis. Mira (2003) sudies nominal GDP argeing in a small scale New Keynesian model wih adapive learning. Jensen (2002) compares inflaion and nominal GDP argeing in a linearized New Keynesian model wih price sickiness and Kim and Henderson (2005) compare he wo rules in a model 3

5 of wage and price sickiness. While boh of hese papers include srucural models similar o our own, Jensen (2002) does no include wage sickiness and Kim and Henderson (2005) uses one period ahead price and wage conracs raher han he more common saggered price and wage conracs. Moreover, Mira (2003), Jensen (2002), and Kim and Henderson (2005) do no analyze nominal GDP argeing in a medium scale model like Smes and Wouers (2007) or Jusiniano e al. (2010) which are exensively used in cenral banks. Schmi-Grohe and Uribe (2007) sudy opimal policy rule coefficiens for a Taylor rule in a sicky price New Keynesian model wih capial. They find ha i is opimal o respond srongly o inflaion and no a all o oupu, hough hey do no allow he cenral bank o arge he oupu gap and heir model does no feaure wage rigidiy. Sims (2013) sudies he relaive meris of a Taylor rule responding o he oupu gap or oupu growh. 2 The Basic New Keynesian Model We begin by sudying a exbook New Keynesian model feauring boh wage and price rigidiy along he lines of Erceg e al. (2000). This model is a special case of he medium scale model we sudy in he nex secion. In he subsecions below we briefly describe he problems of each ype of agen in he model and discuss equilibrium and aggregaion. We hen use he model o develop some inuiion for he relaive benefis of differen moneary policy rules. The full se of condiions characerizing he equilibrium are presened in Appendix A. 2.1 Households There exis a coninuum of households indexed by h [0, 1]. These households are monopoly suppliers of differeniaed labor, N (h). There exiss a labor aggregaing firm which bundles differeniaed labor inpu ino an aggregae labor inpu, N, which is sold o firms a real wage w ; w (h) denoes he real wage paid o household h. The echnology which bundles labor inpu is given by: N = ( 0 1 N (h) ɛw 1 ɛw ) ɛw ɛw 1, ɛw > 1. (1) The parameer ɛ w measures he degree of subsiuabiliy among differen ypes of labor. Profi maximizaion by he labor aggregaing firm gives rise o a downward sloping demand curve for each variey of labor and an aggregae real wage index: N (h) = ( w ɛ w (h) ) N (2) w 4

6 w 1 ɛw = 0 1 w (h) 1 ɛw dh. (3) Households are no freely able o adjus heir wage each period. In paricular, each period here is a 1 θ w probabiliy, θ w [0, 1), ha a household can adjus is wage. This probabiliy is independen of when a household las adjused is wage. As in Erceg e al. (2000), we assume here exiss sae-coningen securiies which insure households agains idiosyncraic wage risk. We also assume ha preferences are separable in consumpion and labor. Combined wih perfec insurance, his means ha households will make idenical non-labor marke choices. We herefore suppress formal dependence on h in wriing he problem of a paricular household wih he excepion of labor marke variables. The problem of a paricular household can be wrien: max C,B,w (h),n (h) E 0 =0 β ν {ln C ψ N (h) 1+η } 1 + η subjec o C + B P w (h)n (h) + Π + (1 + i 1 ) B 1 P (4) N (h) ( w ɛ w (h) ) N (5) w w (h) = w # (h) (1 + π ) 1 w 1 (h) oherwise if w (h) chosen opimally. (6) The discoun facor is given by β (0, 1), ψ is a scaling parameer on he disuiliy from labor, and η represens he inverse of he Frisch elasiciy of labor supply. The exogenous variable ν is a preference shock common o all households. Consrain (4) is a sandard flow budge consrain. A household eners he period wih a sock of nominal bonds, B 1, and can choose a new sock of bonds, B, which pay ou nominal ineres rae i in period + 1. Consumpion is denoed by C and he price of goods is P. Real profi disribued from firms is Π. Consrain (5) requires ha household labor supply mees demand. Wage-seing is described by (6). Wih probabiliy 1 θ w he household chooses a new real wage, denoed by w # (h). Wih probabiliy θ w he household is unable o adjus is nominal wage, so he real wage i charges in period is is period 1 real wage divided by he gross inflaion rae beween 1 and, where 1 + π = P /P 1. Opimizaion gives rise o a sandard Euler equaion for bonds ha is he same across all households. I is sraighforward o show ha all households given he opporuniy will adjus o a common wage, w #. 5

7 2.2 Producion Producion akes place in wo phases. There exis a coninuum of producers of differeniaed oupu indexed by j [0, 1], Y (j). Differeniaed oupu is ransformed ino final oupu, Y, by a compeiive firm using he following echnology: Y = ( 0 1 ɛp Y (j) ɛp 1 ɛp 1 ɛp dj), ɛp > 1. (7) The parameer ɛ p measures he degree of subsiuabiliy among differeniaed oupu. The price of he final oupu is P and he prices of differeniaed oupu are denoed by P (j). Profi maximizaion by he compeiive firm gives rise o demand for each differeniaed oupu and an aggregae price index: Y (j) = ( P ɛ p (j) ) Y (8) P P 1 ɛp = The producion funcion for producer j is given by: 0 1 P (j) 1 ɛp. (9) Y (j) = A N (j). (10) The exogenous variable A is a produciviy shock common across all producers of differeniaed oupu. These firms are no freely able o adjus heir price in a given period in an analogous way o household wage-seing. In paricular, each period here is a 1 θ p probabiliy, θ p [0, 1), ha a firm can adjus is price, which we denoe by P # (j). Oherwise i charges is mos recenly chosen price: P (j) = P # (j) if P (j) chosen opimally P 1 (j) oherwise. (11) Regardless of wheher a firm can adjus is price, i will choose labor inpu o minimize cos, subjec o he consrain of producing enough o mee demand, given by (8). Solving he cos minimizaion problem reveals ha all firms face he same real marginal cos, given by mc = w /A. Firms given he opporuniy o adjus heir price will do so o maximize he expeced presened discouned value of profi reurned o households. I is sraighforward o show ha all updaing firms will choose a common rese price, P #. 6

8 2.3 Exogenous Processes There are wo exogenous variables in he model, he produciviy shock, A, and he preference shock, ν. We assume ha hese boh follow saionary AR(1) processes wih non-sochasic means normalized o uniy: ln A = ρ A ln A 1 + σ A ε A, (12) ln ν = ρ ν ln ν 1 + σ ν ε ν,. (13) The auoregressive parameers, ρ A and ρ ν, lie beween zero and one. The innovaions, ε A, and ε ν,, are drawn from sandard normal disribuions. These innovaions are scaled by σ A and σ ν, which measure he sandard deviaions of he innovaions. 2.4 Marke-Clearing and Aggregaion Marke-clearing requires ha bond-holding is zero a all imes, B = 0, and ha he sum of labor demanded by producers of differeniaed oupu equals oal labor supplied by he labor aggregaing firm. Inegraing he flow budge consrains of households along wih he definiion of firm profis gives he aggregae resource consrain Y = C. The price and real wage indexes can be wrien wihou reference o household or firm subscrips as: P 1 ɛp w 1 ɛw = (1 θ p )P #,1 ɛp = (1 θ w )w #,1 ɛw + θ p P 1 ɛp 1 (14) + θ w (1 + π ) ɛw 1 w 1 ɛw 1. (15) Inegraing over he demand curves for differeniaed oupu, (8), gives rise o an aggregae producion funcion: Y = A N v p (16) where he variable v p is a measure of price dispersion given by: v p = 0 1 ( P ɛ p (j) ) dj. (17) P We define he hypoheical consruc of he flexible price level of oupu (or someimes naural rae ) as he level of oupu which would obain in he absence of price and wage sickiness. We denoe his by Y f, and can solve for i as he equilibrium level of oupu when θ p = θ w = 0. We hen define he oupu gap, X, as he raio of he level of oupu o he 7

9 flexible price level, X = Y /Y f. Aggregae welfare is defined as he sum of he presened discouned value of flow uiliy across all households. This can be wrien recursively in erms of aggregae variables alone as: W = ν [ln C ψv w N 1+η 1 + η ] + βe W +1. (18) Recall ha N is aggregae labor supplied by he labor aggregaing firm. This is only equal o aggregae labor supply in he special case ha all households charge idenical wages. In he more general case, here is a wedge beween aggregae labor supply and labor used in producion due o wage dispersion. This is capured by he variable v w, which can be wrien: 2.5 Moneary Policy v w = 0 1 ( w ɛ w(1+η) (h) ) dh. (19) w Before fully characerizing he equilibrium i remains o specify he conduc of moneary policy. In addiion o a nominal GDP argeing rule, we also consider inflaion and oupu gap argeing rules. Each of hese hree argeing rules can be considered resriced cases of a generalized Taylor (1993) ype insrumen rule, which is ofen used in quaniaive work o model he behavior of moneary policy. The primary objecive of he paper is o evaluae he performance of a nominal GDP argeing rule vis-á-vis oher popular argeing rules. A nominal GDP argeing rule can be wrien as P Y = (P Y ). (20) In his rule he nominal ineres rae adjuss so ha nominal GDP, P Y, equals some exogenous and consan arge, (P Y ). 1 In he long run his rule is equivalen o a price level arge since he long run level of real GDP is independen of he moneary policy rule. In he shor run, commimen o his rule implies ha he growh rae of nominal GDP is zero, or ha (1 + π ) Y Y 1 = 1 each period. The second rule we consider is a sric inflaion argeing rule: π = 0. (21) Under an inflaion argeing rule he cenral bank adjuss he nominal ineres such ha 1 Here we follow convenion in assuming no rend growh in oupu or rend inflaion. I is sraighforward o modify he nominal GDP argeing rule o accoun for hese feaures. 8

10 inflaion his is arge each period (implicily we have normalized he arge inflaion rae o zero, bu could easily exend his o non-zero arges). The final argeing rule we consider is an oupu gap argeing rule: X = 1. (22) In his rule he nominal ineres rae is adjused in such a way ha he equilibrium level of oupu always equals is flexible price level. In a version of he model wih flexible wages he so-called Divine Coincidence would hold (Blanchard and Gali (2007)), and in equilibrium inflaion and gap argeing would be equivalen o one anoher, and would boh implemen he flexible price equilibrium. When prices and wages are simulaneously sicky his equivalence does no hold. Gap argeing requires ha he cenral bank know he flexible price level of oupu, which is no direcly observable, neiher in real ime nor ex-pos. A poenial advanage of inflaion argeing is ha inflaion is observed a high frequencies and does no require a cenral a bank o know anyhing abou he underlying model. Whereas gap argeing focuses on a real variable and inflaion argeing focuses solely on prices, nominal GDP argeing implicily arges boh nominal and real variables. I also only requires a cenral bank o observe endogenous variables, no hypoheical model consrucs like he flexible price level of oupu. Each of he hree argeing rules can be undersood o be special cases of a generalized Taylor rule of he following form: ln(1 + i ) = (1 ρ i ) ln(1 + i ) + ρ i ln(1 + i 1 ) + φ π ln (1 + π ) + φ x ln X + φ y ln ( Y Y 1 ). (23) Here ρ i [0, 1) is a smoohing parameer and φ π, φ x, and φ y are non-negaive coefficiens on inflaion, he oupu gap, and oupu growh, respecively. In wriing his rule we have mainained an implici assumpion of zero rend inflaion and no rend growh. Whereas he hree rules discussed above are argeing rules, argeing he values of endogenous variables and adjusing he nominal rae o hi hose arges, he Taylor rule is an insrumen rule, direcly specifying a process for he nominal ineres rae as funcion of endogenous variables. Taylor rules of his form seem o provide an accurae accoun of observed moneary policy in he las several decades and end o have good normaive properies. Each of he hree argeing rules can be undersood o be special cases of he Taylor rule. The Taylor rule would be isomorphic o an inflaion argeing rule when φ π and ρ i = φ x = φ y and equivalen o a gap argeing rule when φ x and ρ i = φ π = φ y. Because commimen o a nominal GDP argeing rule implies argeing he growh rae of nominal GDP each period, i is equivalen 9

11 o he Taylor rule when φ π = φ y and ρ i = φ x = Quaniaive Analysis We adop a sandard parameerizaion of he model s parameers; in he nex secion wih a more empirically plausible model wih capial accumulaion we esimae hem. The values of he parameers are lised in Table 1. Table 1: Value of Parameers Parameer Descripion Value β Discoun rae η Inverse Frisch elasiciy 1 ψ Labor disuiliy N = 1/3 ɛ w Elasiciy of subsiuion Labor 10 ɛ p Elasiciy of subsiuion Goods 10 θ w Wage sickiness θ p Price sickiness σ A Sandard deviaion Produciviy σ ν Sandard deviaion Preference ρ A Persisence Produciviy 0.97 ρ ν Persisence Preference 0.70 Noes: The able shows he values of he parameers uses in he quaniaive analysis of he small scale model laid ou in his secion. The discoun facor is se o β = 0.995, which implies an annualized risk-free ineres rae of wo percen. The Frisch labor supply elasiciy is se o uniy, implying η = 1. The elasiciies of subsiuion for goods and labor, ɛ p and ɛ w, are boh se o 10, implying seady sae price and wage markups of a lile more han en percen. The scaling parameer on he disuiliy from labor, ψ, is se so ha seady sae labor hours equal 1/3. Our baseline parameerizaion of he price and wage sickiness parameers is θ w = θ p = 0.75, which implies ha wages and prices boh adjus once a year on average. We consider various differen values of hese parameers in he quaniaive work below. The auoregressive parameer in he produciviy process is se o ρ A = 0.97 and he auoregressive parameer for he preference shock is ρ ν = The sandard deviaions of he innovaions o produciviy and 2 One could also map he Taylor rule ino argeing rules ha are no sric in he sense of feauring large bu neverheless finie coefficiens on he arge variables in he Taylor rule. In paricular, an inflaion argeing Taylor rule migh feaure a large value of φ π (say, 10) and values of 0 for he oher parameers, and similarly for a gap arge and he coefficien on φ x and a nominal GDP arge wih large and equal coefficiens on φ π and φ y. We have experimened wih hese specificaions and obain similar resuls o he sric argeing rules considered. 10

12 preferences are σ A = and σ ν = 0.02, respecively. When moneary policy is governed by a convenionally parameerized version of he Taylor rule wih ρ i = 0.7, φ π = 0.45, φ y = , and φ x = 0, 3 his parameerizaion generaes a sandard deviaion of log oupu of 0.036, which is very close o he observed volailiy of linearly derended log GDP in poswar US daa. Furhermore, he produciviy and preference shocks each accoun for 50 percen of he uncondiional variance of log oupu. In oher words, our parameerizaion implies ha supply and demand shocks are equally imporan in driving flucuaions in oupu. We evaluae differen policy rules by compuing he uncondiional mean of welfare for a paricular policy rule and comparing ha o he uncondiional mean of welfare in a hypoheical economy where prices and wages are boh flexible, e.g. θ p = θ w = 0. We compue hese uncondiional means by solving he model using a second order approximaion of he equilibrium condiions abou he non-sochasic seady sae. We hen calculae a compensaing variaion, compuing he percen of consumpion each period which would make a household indifferen beween he flexible price and wage economy and he sicky price and wage economy. Formally, he compensaing variaion is given by: λ = 100 [exp(e W flex E W 0 ) 1] where E W flex is he expeced presen discouned value of uiliy under flexible prices and wages and E W 0 is he expeced discouned value of uiliy in he sicky price and wage economy wih a given moneary policy regime. We can inerpre his compensaing variaion as a welfare loss from price and wage rigidiy. More desirable moneary policy regimes herefore coincide wih lower values of he compensaing variaion. Table 2 presen compensaing variaions for he hree differen argeing rules for differen combinaions of price and wage sickiness parameers. For poin of comparison we also show he compensaing variaion for he Taylor rule parameerized as described above. Focus firs on he case where he price and wage rigidiy parameers are boh The mos desirable policy regime is he oupu gap argeing rule, which produces a compensaing variaion of only 0.02 percen of consumpion. Nominal GDP argeing does almos as well, generaing a welfare loss of only 0.03 percen of consumpion. Inflaion argeing performs very poorly, wih a welfare loss of nearly 20 percen of consumpion. The Taylor rule performs fairly well, albei subsanially worse han eiher nominal GDP or oupu gap argeing, wih a compensaing variaion of 0.3 percen. We now urn aenion o differen combinaions of price and wage sickiness parameers. The oupu gap argeing rule a leas weakly dominaes he oher wo argeing rules as well 3 Noe ha our rule is no wrien as one of parial adjusmen. The long run response of he ineres rae o inflaion and oupu growh is φ π /(1 ρ i ) and φ y /(1 ρ i ), or 1.5 and 0.125, respecively. 11

13 as he Taylor rule for all combinaions of price and wage rigidiy. The oupu gap argeing rule implemens he flexible price allocaion (i.e. he compensaing variaion is zero) if eiher prices or wages are flexible. Inflaion argeing ends o perform very poorly excep in he exreme case where wages are flexible, when sric inflaion argeing also fully implemens he flexible price allocaion. The compensaing variaion associaed wih he Taylor rule is roughly he same a all combinaions of price and wage rigidiy under consideraion. Nominal GDP argeing ends o perform very well, yielding compensaing variaions less han 0.04 percen of consumpion for all combinaions of price and wage sickiness parameers under consideraion. Table 2: Consumpion Equivalen Welfare Losses from Differen Policy Rules θ p = 0.75 θ p = 0.00 θ p = 0.50 θ p = 0.75 θ p = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.50 θ w = 0.00 NGDP Inflaion Oupu Gap Taylor Rule Noes: The able conains he compensaing variaions of hree differen argeing rules as well as a Taylor rule. The parameerizaion of he model is as described in Table 1. The firs hree rows consider sric argeing rules. The row labeled Taylor rule considers he Taylor rule, (23), parameerized wih ρ i = 0.7, φ π = 0.45, φ x = 0, and φ y = Tha inflaion argeing performs poorly, and oupu gap argeing does very well, when boh prices and wages are rigid has been well-known since Erceg e al. (2000). The novel resul here is ha nominal GDP argeing seems o have desirable properies. Nominal GDP argeing is associaed wih subsanially lower compensaing variaions han he Taylor rule, and significanly ouperforms inflaion argeing excep in he case where wages are flexible. While nominal GDP argeing is always weakly worse han oupu gap argeing, he differences in he compensaing variaions associaed wih hese wo argeing rules are very small. From he Table i appears as hough nominal GDP argeing is relaively beer he sickier are wages relaive o prices when wages are rigid and prices are flexible, nominal GDP argeing is equivalen o gap argeing, while when wages are flexible and prices rigid, inflaion argeing ouperforms nominal GDP argeing. Figure 1 shows he loci of wage (horizonal axis) and price (verical axis) rigidiy parameers where nominal GDP argeing and inflaion argeing generae he same compensaing variaion. The area shaded green shows parameer combinaions where nominal GDP argeing sricly dominaes inflaion argeing. For wage sickiness parameers in excess 12

14 of 0.3 nominal GDP argeing sricly dominaes inflaion argeing for any value of he parameer governing price rigidiy. Furhermore, a low levels of wage sickiness inflaion argeing only dominaes nominal GDP argeing if prices are very rigid θp θ w Figure 1: Nominal GDP vs. Inflaion Targeing Noes: This figure races ou he loci of wage sickiness parameers, (θ w, θ p) for which nominal GDP argeing and inflaion argeing yield he same compensaing variaion. The area shaded green depics combinaions of (θ w, θ p) for which nominal GDP argeing sricly dominaes inflaion argeing. We nex consider he role of he wo differen shocks in he model in driving he relaive performance of he differen moneary policy regimes. Tables 3 and 4 repea he exercises in Table 2 condiioning on only he produciviy or he preference shock, respecively. For his exercise, we se he sandard deviaion of one of he shocks o zero, and re-parameerize he sandard deviaion of he remaining shock o generae he same volailiy of log oupu as in he baseline model when policy is characerized by a Taylor rule. 4 The compensaing variaions for he differen policy regimes when here are only produciviy shocks are qualiaively similar o he compensaing variaions when boh shocks are included in he model. Oupu gap argeing always weakly dominaes he oher policies and inflaion argeing does poorly unless wages are flexible. Nominal GDP argeing is significanly more desirable han inflaion argeing unless wages are flexible, and performs beer han he Taylor rule for mos parameer configuraions. Inflaion argeing performs worse condiional on produciviy shocks relaive o he case where boh shocks are in he 4 When condiioning on he produciviy shock, his implies a value of σ A = and σ ν = 0. When condiioning on he preference shock, σ A = 0 and σ ν =

15 model. When here are only preference shocks, each of he hree argeing regimes implemens he flexible price allocaion for any combinaion of price and wage rigidiy parameers, while he Taylor rule performs similarly o when boh shocks are in he model. This equivalence of he hree argeing regimes obains because of he absence of capial in he model and our assumpions abou preferences. Given ha preferences are separable in consumpion and labor and ha consumpion mus equal oupu, in he flexible price allocaion neiher oupu nor real wages change in response o a preference shock. 5 This has he implicaion ha sabilizing prices (inflaion argeing), real aciviy (gap argeing), or a mix of boh (nominal GDP argeing) are equivalen in equilibrium. Table 3: Consumpion Equivalen Welfare Losses from Differen Policy Rules, Only Produciviy Shocks θ p = 0.75 θ p = 0.00 θ p = 0.50 θ p = 0.75 θ p = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.50 θ w = 0.00 NGDP Inflaion Oupu gap Taylor rule Noes: The able conains he compensaing variaions of he policies described in he ex in a specificaion of he model in which here are only produciviy shocks. In paricular, he sandard deviaion of he produciviy shock is chosen o produce he same volailiy under a Taylor rule as in he version of he model wih boh shocks, while he sandard deviaion of he preference shock is se o zero. The sandard deviaion of he produciviy shock is σ A = The resuls in Tables 3 and 4 sugges ha he choice of moneary regime is far more relevan for welfare condiional on supply shocks (i.e. he produciviy shock) han demand shocks (i.e. he preference shock). This confirms he simple aggregae demand - supply inuiion in Sumner (2014). Our baseline parameerizaion assigns he produciviy and preference shocks equal weigh in accouning for oupu flucuaions, and herefore akes a relaively agnosic sand on he relaive imporance of supply and demand shocks. In he nex secion we consider an empirically realisic medium scale model and esimae he relaive imporance of several differen kinds of shocks. In he simple model of his secion, he demarcaion beween supply and demand shocks is clear, because he preference shock 5 A simple way o see his is o noe ha he price and wage markups are fixed when boh prices and wages are flexible. Given our assumpions on preferences, his means ha consumpion and labor mus co-move negaively absen a change in A. Bu since C = Y = A N, consumpion and labor canno co-move negaively absen a change in A. Hence, in he flexible price allocaion consumpion, hours, oupu, and he real wage do no reac o a preference shock. 14

16 does no impac he flexible price level of oupu. Bu when capial and oher feaures are added o he model, he disincion beween supply and demand can become blurry. While he basic inuiion abou he choice of policy regime condiional on supply and demand shocks will indeed carry over largely inac, some cauion is in order when exrapolaing from he simple model o he more realisic one in he nex secion. Table 4: Consumpion Equivalen Welfare Losses from Differen Policy Rules, Only Preference Shocks θ p = 0.75 θ p = 0.00 θ p = 0.50 θ p = 0.75 θ p = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.75 θ w = 0.50 θ w = 0.00 NGDP Inflaion Oupu gap Taylor rule Noes: The able conains he compensaing variaions of he policies described in he ex in a version of he model in which here are only preference shocks. In paricular, he sandard deviaion of he preference shock is chosen o produce he same volailiy under a Taylor rule as in he version of he model wih boh shocks, while he sandard deviaion of he produciviy shock is se o zero. The sandard deviaion of he preference shock is σ ν = Medium Scale Model While he previous secion allows one o undersand some of he inuiion for he welfare properies of nominal GDP argeing relaive o inflaion argeing and a Taylor rule, i did so wihin he conex of a very simplified framework. In his secion, we consider a medium scale version of he model. The simple economy of he previous secion is a special case of he more realisic model in his secion. In addiion o price and wage rigidiy, he model allows for capial accumulaion, habi formaion in consumpion, variable capial uilizaion, wage and price indexaion o lagged inflaion, and several more sochasic shocks. Such a model has been shown o capure he dynamic effecs of moneary policy and he mos salien business cycle facs. 6 Where differen from he simpler model of Secion 2, in he subsecions below we lay ou he deails of he medium scale model. The full se of condiions characerizing he equilibrium is presened in Appendix B. 6 See Chrisiano e al. (2005) as an example of he former and Smes and Wouers (2007) for he laer. 15

17 3.1 Households The basic srucure of he household side of he model is very similar o he simpler model. There again exis a coninuum of households indexed by h [0, 1]. The labor aggregaing firm is he same, giving rise o he same downward-sloping demand for each household s labor, (2), and he same aggregae real wage index, (3). The medium scale model depars from he simpler model in allowing for inernal habi formaion in consumpion, capial accumulaion and capial uilizaion, and indexaion of wages o lagged inflaion. The problem of a paricular household is: subjec o max C,B,u,I,K +1,w (h),n (h) E 0 β ν {ln (C bc 1 ) ψ N (h) 1+η } 1 + η =0 C +I + B P +[γ 1 (u 1) + γ 2 2 (u 1) 2 ] K w (h)n (h)+r u K +Π +T +(1+i 1 ) B 1 P (24) K +1 = Z [1 τ 2 ( I 2 1) ] I + (1 δ)k (25) I 1 N (h) ( w ɛ w (h) ) N (26) w w (h) = w # (h) (1 + π ) 1 (1 + π 1 ) ζw w 1 (h) oherwise if w (h) chosen opimally. (27) Relaive o he simpler model, he preference specificaion is idenical wih he excepion of he inclusion of inernal habi formaion, which is governed by he parameer b [0, 1). In he flow budge consrain, (24), I denoes invesmen, K physical capial, u capial uilizaion, R he renal rae on capial services (where capial services is undersood o represen he produc of uilizaion and physical capial), and T is a lump sum ax paid o a governmen. There is a convex resource cos of capial uilizaion given by γ 1 (u 1) + γ 2 2 (u 1) 2. This cos is measured in unis of physical capial. The law of moion for physical capial is given by (25). Z is an exogenous shock o he marginal efficiency of invesmen along he lines of Jusiniano e al. (2010) and τ 0 is an invesmen adjusmen cos in he form proposed by Chrisiano e al. (2005). The depreciaion rae on physical capial is given by δ (0, 1). A household is required o supply as much labor as is demanded a is wage, jus as in he simpler model. The wage-seing process is virually he same as in he simpler model. Wih probabiliy 1 θ w a household can adjus o a new opimal wage, which we denoe in real erms as w # (h). Oherwise a household mus charge is mos recenly chosen nominal 16

18 wage. Differenly han he simple model, here we permi parial indexaion of nominal wages o lagged inflaion as measured by he parameer ζ w [0, 1]. I is again he case ha all updaing households will adjus o he same rese real wage, which we denoe by w #. 3.2 Producion The producion process again akes place in wo phases. The final oupu aggregaor, demand curve for each differeniaed oupu, and expression for he aggregae price level are he same as in he simpler model, given by equaions (7), (8), and (9), respecively. The producion funcion of firms who produce differeniaed oupu is given by: Y (j) = max {A K (j) α N (j) 1 α F, 0}. (28) Here K is capial services leased from households. The parameer α lies beween 0 and 1. There is also a fixed cos of producion given by F 0. Cos-minimizaion by firms reveals ha all firms face he same real marginal cos and hire capial services and labor in he same raio. As in he simpler model, firms face a probabiliy of 1 θ p ha hey can adjus heir price in a given period. Differenly han he simpler model, we permi indexaion of non-updaed prices o lagged inflaion, governed by he parameer ζ p [0, 1]. A firm s price in any period herefore saisfies: P (j) = P # (j) if P (j) chosen opimally (1 + π 1 ) ζp P 1 (j) oherwise. (29) Updaing firms will choose heir prices o maximize he presen discouned value of flow profis, where discouning is by he sochasic discoun facor of households. 7 I is again sraighforward o show ha all updaing firms choose an idenical price, denoed by P #. 3.3 Fiscal Policy In he medium scale model we allow for governmen spending. Governmen spending is assumed o be exogenous and obeys he following saionary sochasic process: ln G = (1 ρ G ) ln G + ρ G ln G 1 + σ G ε G, (30) 7 Though here is heerogeneiy among households, because of separabiliy and perfec insurance he marginal uiliy of income is idenical across households, so i is safe o alk abou one sochasic discoun facor in spie of he heerogeneiy among households. 17

19 Here G is he seady sae level of governmen spending, he parameer ρ G lies beween zero and one, and he innovaion ε G, is drawn from a sandard normal disribuion. The innovaion is scaled by he sandard deviaion parameer σ G. We assume ha he only source of governmen revenue is lump sum axes, T. This means ha i is innocuous o assume ha he governmen balances is budge each period, wih G = T. 3.4 Exogenous Processes The exogenous processes for produciviy, A, and he preference shock, ν, are he same as in Secion 2. We assume ha he invesmen shock, Z, also obeys a saionary sochasic process wih seady sae value normalized o uniy: ln Z = ρ Z ln Z 1 + σ Z ε Z, (31) The parameer ρ Z lies beween zero and one, ε Z, is drawn from a sandard normal disribuion, and σ Z is he sandard deviaion of he innovaion. 3.5 Moneary Policy Our ulimae objecive is o again consider and compare he hree differen argeing rules discussed in Secion 2. For he purposes of esimaion, however, we assume ha he cenral bank ses policy according o he same Taylor rule described above, bu augmen his o include a policy innovaion, ε i,, which is drawn from a sandard normal disribuion wih sandard deviaion σ i : ln(1+i ) = (1 ρ i ) ln(1+i )+ρ i ln(1+i 1 )+φ π ln (1 + π )+φ x ln X +φ y ln ( Y Y 1 )+σ i ε i,. (32) 3.6 Aggregaion Since we assume ha he governmen issues no deb, in equilibrium bond-holding is always zero, B = 0. Combining his wih he definiion for firm profi and he governmen s budge consrain gives rise o he aggregae resource consrain: Y = C + I + G + [γ 1 (u 1) + γ 2 2 (u 1) 2 ] K. (33) The expressions for he aggregae price and wage indexes are similar o Secion 2, bu accoun for indexaion: 18

20 P 1 ɛp = (1 θ p )P #,1 ɛp + θ p (1 + π 1 ) ζp(1 ɛp) P 1 ɛp 1 (34) w 1 ɛw = (1 θ w )w #,1 ɛw The aggregae producion funcion is given by: + θ w (1 + π ) ɛw 1 (1 + π 1 ) ζw(1 ɛw) w 1 ɛw 1. (35) Y = A (u K ) α N 1 α v p F. (36) The variable v p is he same measure of price dispersion as in he simpler model, defined by (17). Aggregae welfare is defined similarly as in Secion 2, bu akes ino accoun inernal habi formaion: W = ν [ln (C bc 1 ) ψv w N 1+η 1 + η ] + βe W +1. (37) The variable v w is again a measure of wage dispersion, defined above by (19). 4 Quaniaive Analysis In his secion we analyze he welfare properies of nominal GDP argeing, inflaion argeing, oupu gap argeing, and he Taylor rule in he medium scale model. Quaniaively evaluaing differen policy regimes requires selecing values for he parameers of he model. We calibrae some parameers o mach long run momens of he daa and esimae he remaining parameers so as o ensure ha he model provides an empirically realisic fi o observed daa. The calibraed parameers and heir values are lised in Table 5. As in he simpler model of Secion 2, he discoun facor is se o β = 0.995, and he elasiciies of subsiuion for goods and labor, ɛ p and ɛ w, are se o 10. The seady sae level of governmen spending is chosen so ha he seady sae raio of governmen spending o oupu is G /Y = The scaling parameer on he disuiliy of labor is se o ψ = 6, which implies seady sae labor hours of beween one-hird and one-half for reasonable values of he Frisch elasiciy and habi persisence parameer. To firs order, his scaling parameer is irrelevan for equilibrium dynamics. The parameer on he linear erm in he uilizaion adjusmen cos funcion, γ 1, is chosen o be consisen wih a seady sae normalizaion of uilizaion o one. Capial s share is se o α = 1/3, and he depreciaion rae on physical capial is δ = The fixed cos, F, is se so ha profis are zero in seady sae. 19

21 The remaining parameers are esimaed via Bayesian maximum likelihood. The observable variables in he esimaion are he log firs differences of real oupu, real consumpion, real invesmen, he inflaion rae, and he nominal ineres. To faciliae comparison wih he model, we define nominal GDP as he sum of consumpion (non-durables and services consumpion), invesmen (he sum of durables consumpion and privae fixed invesmen), and governmen spending (governmen consumpion expendiures and gross invesmen). These series are all aken from he NIPA ables. We deflae nominal oupu, as well as he individual componens, by he GDP implici price deflaor. We hen divide by he civilian non-insiuionalized populaion, ake logs, and firs difference. Inflaion is defined as he log firs difference of he GDP deflaor. Our measure of he nominal ineres rae is he effecive Federal Funds Rae, aggregaed o a quarerly frequency by averaging monhly observaions. We use daa from he firs quarer of 1984 hrough he hird quarer of The sar dae is chosen o accoun for he large break in volailiy associaed wih he Grea Moderaion, while he end dae is chosen so as o exclude he recen zero lower bound period. Table 5: Calibraed Parameers, Medium Scale Model Parameer Descripion Value β Discoun rae ψ Labor disuiliy 6 ɛ w Elasiciy of subsiuion Labor 10 ɛ p Elasiciy of subsiuion Goods 10 G SS Governmen Spending G /Y = 0.2 α Capial Share 1/3 F Fixed cos Π = 0 γ 1 Uilizaion linear cos u = 1 δ Depreciaion rae Noes: The able shows he values of he calibraed parameers in he medium scale model. Table 6 presens he prior and poserior disribuions of he esimaed parameers. These resuls are broadly consisen wih Smes and Wouers (2007) and Jusiniano e al. (2010). There is significan rigidiy in boh prices and wages. The average duraion beween prices changes is roughly hree quarers, while he average duraion beween wage changes is abou a year. The laer is consisen wih he micro evidence presened in Baraieri e al. (2014). We find moderae levels of wage and price indexaion. The invesmen adjusmen cos parameer, τ, and he parameer governing inernal habi formaion, b, imply significan real ineria. The esimae of he parameer η implies a Frisch labor supply elasiciy of abou wo-hirds. The moneary policy rule feaures significan ineria and a large response o 20

22 inflaion (he implied long run response of he ineres rae o inflaion is abou four). The policy rule feaures a posiive response o oupu growh and no response o he oupu gap, he laer of which is esimaed precisely a zero. This is consisen wih he evidence in Coibion and Gorodnichenko (2011) ha he Fed moved away from responding o he oupu gap and oward a much sronger reacion o oupu growh in he pos-volcker period. Table 6: Esimaed Parameers Prior Poserior Parameer Dis. Mean SD Mode Mean SD 90% Probabiliy Inerval b Bea [0.6634, ] τ Normal [3.6661, ] η Normal [1.3075, ] γ 2 Bea [0.2372, ] θ p Bea [0.6008, ] θ w Bea [0.6517, ] ζ p Bea [0.1982, ] ζ w Bea [0.3610, ] ρ i Bea [0.8470, ] φ π Normal [0.3747, ] φ x Normal [0.0000, ] φ y Normal [0.1036, ] ρ A Bea [0.7506, ] ρ ν Bea [0.4885, ] ρ Z Bea [0.5840, ] ρ G Bea [0.9267, ] σ A Inv. Gamma [0.0046, ] σ ν Inv. Gamma [0.0128, ] σ Z Inv. Gamma [0.0330, ] σ G Inv. Gamma [0.0087, ] σ i Inv. Gamma [0.0011, ] Noes: The variables used in he esimaion are he growh raes of oupu, consumpion, and invesmen, and he levels of inflaion and he nominal ineres rae. Consrucion of hese series is as described in he ex. The poserior is generaed wih 100,000 Meropolis-Hasings draws. The esimaed model produces second momens which align closely wih heir counerpars in he daa. Consumpion growh is less volaile han oupu growh, while invesmen growh is abou hree imes more volaile han oupu growh. Oupu, consumpion, and invesmen are srongly posiively correlaed, while he nominal ineres rae is roughly uncorrelaed wih 21

23 oupu growh and he inflaion rae is mildly negaively correlaed wih oupu growh. The firs order auocorrelaions of oupu, consumpion, and invesmen growh are all significanly posiive. In erms of conribuions o business cycle dynamics, he invesmen shock is he mos imporan disurbance. I accouns for abou 50 percen of he uncondiional variance of oupu growh. The produciviy shock explains abou 30 percen of he variance of oupu growh. The preference and governmen spending shocks each conribue a lile less han 10 percen, while he moneary shock accouns for abou 5 percen of he uncondiional variance of oupu growh. This variance decomposiion is very much in line wih he resuls in Jusiniano e al. (2010). 4.1 Evaluaing Moneary Policy Rules We solve he medium scale model using he calibraed or esimaed values lised in Tables 5 and 6 using a second order approximaion abou he non-sochasic seady sae. Compensaing variaions are consruced comparing average welfare under differen policy regimes o he hypoheical flexible price and wage equilibrium. These exercises are idenical o he ones carried ou in Secion 2. When evaluaing he welfare performance of he esimaed Taylor rule, we se he sandard deviaion of he policy innovaion o zero so as o faciliae comparison wih he oher argeing rules, which feaure no shocks. The compensaing variaions for he differen moneary policy rules in he esimaed model are summarized in he firs main column of Table 7. The relaive performance of he differen argeing rules is he same as in he small scale model from Secion 2. The welfare loss associaed wih nominal GDP argeing is 0.11 percen of consumpion. This is no as good as oupu gap argeing, which has a welfare loss of 0.03 percen of consumpion, bu ouperforms he esimaed Taylor rule (welfare loss of 0.24 percen of consumpion) and does significanly beer han inflaion argeing, which produces a compensaing variaion of 0.86 percen of consumpion. The remaining columns of Table 7 consider differen values of he parameers governing price and wage sickiness. For hese exercises all bu he lised parameer are fixed a heir esimaed or calibraed values. The paerns in he Table again echo hose from he small scale model. If eiher prices or wages are compleely flexible, he oupu gap argeing rule implemens he flexible price and wage allocaion wih a compensaing variaion of zero. If wages are perfecly flexible hen inflaion argeing also implemens he flexible price allocaion. Nominal GDP argeing performs very well, producing a lower welfare loss han he Taylor rule in all specificaions and ouperforming inflaion argeing by a wide margin in all bu he case where wages are flexible. As in he small scale model, he relaive performance 22