2 + 2 < 4? Monetary Policy in the Presence of Downward Nominal Wage Rigidity and the Zero Lower Bound on the Nominal Interest Rate

Transcription

1 2 + 2 < 4? Moneary Polcy n he Presence of Downward Nomnal Wage Rgdy and he Zero Lower Bound on he Nomnal Ineres Rae Rober Amano a, Sefano Gnocch b a Bank of Canada b Bank of Canada Ocober 9, 216 Absrac We add downward nomnal wage rgdy o a sandard New Keynesan model wh scky prces and wages, where he zero lower bound of he nomnal neres rae s allowed o bnd. We fnd ha wage rgdy no only reduces he frequency of he ZLB, bu also mgaes he severy of he recesson ha occurs when a ZLB epsode maeralzes. As an mplcaon, prevous sudes mgh have overemphaszed he need of ncreasng he nflaon arge o offse he ncreased rsk of hng he ZLB. In addon, he belef n he vrues of wage flexbly ha he recen crses have revved on boh sdes of he Alanc mgh as well be msplaced. The vews expressed n hs paper are hose of he auhors. No responsbly for hem should be arbued o he Bank of Canada. Emal addresses: bamano@bankofcanada.ca (Rober Amano), sgnocch@bankofcanada.ca (Sefano Gnocch) 1

2 1. Inroducon The zero lower bound of he nomnal neres rae (ZLB) and downward nomnal wage rgdy (DNWR) are among he mos promnen reasons ha movae he adopon of posve nflaon arges. Ths vew s suppored by several conrbuons n he leraure, whch has suded he wo ssues separaely, devong lle aenon o her neracon. In an effor o fll hs gap, hs paper explores he mplcaons of DNWR for he cos of he ZLB, as well as for he benefs of devang from prce sably. The few sudes ha analyzed jonly he ZLB and DNWR offer a mxed pcure. On he one hand, Cobon e al. (212) seem o sugges ha DNWR may allevae he cos of he ZLB by reducng s frequency. On he oher hand, Schm-Grohé and Urbe (213) denfy n he lack of nomnal wage adjusmen a possble drver of he severe recesson wnessed by some European counres durng he deb crss. Along smlar lnes, Daly and Hobjn (214) argue ha DNWR can explan he surge n unemploymen observed durng he Grea Recesson. Takng all hese conrbuons ogeher, one may wonder wheher he lack of wage flexbly makes lqudy raps less recurren bu more severe and wha s overall welfare mplcaons are. We reconsder hese quesons n a unfed framework ha blends dfferen frcons. A sandard New-Keynesan model wh scky prces and wages s modfed o ncorporae wo zero lower bounds, one on he nomnal neres rae and one on nomnal wage growh. Moneary polcy s assumed o follow an neres-rae rule ha arges nflaon and oupu flucuaons, as well as he lagged nomnal neres rae. Condonal on hs rule and a se of demand and supply shocks, he model s smulaed over a range of nflaon arges beween zero and fve by usng he pecewse-lnear perurbaon mehod developed by Guerrer and Iacovello (215). We fnd ha DNWR affecs he cos of he ZLB hrough wo margns. On he one hand, reduces he frequency of ZLB epsodes acng hrough an exensve margn because aenuaes he mpac of demand shocks on he real margnal cos and, as a resul, moderaes he sze of neres rae cus needed o counerac deflaon. On he oher hand, f a ZLB epsode maeralzes, ncreases welfare as compared o an economy where nomnal wages are free o fall, acng hrough an nensve margn. In fac, dampens acual and expeced deflaon so ha he real neres rae remans lower, supporng aggregae demand. Overall, DNWR complemens a posve nflaon arge n prevenng he ZLB from bndng, whle subsues moneary polcy n boosng demand when furher polcy space s unavalable. Therefore, reduces he expeced cos of he ZLB by makng lqudy raps less lkely and less severe. Resuls may seem surprsng n lgh of he ngraned belef ha wage nflexbly causes unemploymen. However, such a belef comes from mplc assumpons on goods markes, as recenly emphaszed by Galí (213). If goods markes are compeve and prces are flexble, employmen nversely relaes o he real wage hrough labor demand and DNWR s harmful because dscourages frms from hrng by keepng real wages hgh. If nsead goods markes are mperfecly compeve and prces are scky, employmen s exclusvely drven by aggregae demand, whle wages only affec frms 2

3 markups and profs. Hence, DNWR s benefcal when he polcy rae s consraned by he lower bound and furher moneary accommodaon s needed, because smulaes aggregae demand. Our fndngs conrbues o he leraure n several respecs. Frs, f DNWR s aken no accoun, he nflaon arge becomes less of an approprae polcy nsrumen o ake care of he ZLB, because changes n he arge affec he nensve and he exensve margn n oppose drecons. Even hough he probably of beng bound by he ZLB falls wh he arge, hgher nflaon exacerbaes he recessons due o lqudy raps by dmnshng he ncdence of DNWR. The coexsence of hs opposng forces makes he cos of he ZLB overall less sensve o changes n he nflaon arge, and more so as he arge ncreases. Second, our analyss reconcles he seemngly opposng vews offered by he leraure on he benefs of wage flexbly. Our model accommodaes boh possbles ha n a lqudy rap DNWR promoes or hampers employmen, dependng on wheher prces are scky or flexble. Analyzng he mpac of wage rgdes on labor marke oucomes requres o ake a sand also on he dsorons ha affec goods markes, especally n he case of parcularly harsh and prolonged recessons. Fnally, our analyss speaks o he relevance of DNWR for explanng pen-up wage deflaon.e., he perssen weakness of nomnal wage growh ypcally observed afer recessons accompaned by wdespread wage freezes. As dscussed by Daly and Hobjn (214), workers prefer o delay wage ncreases f hey ancpae he possbly of beng bound by DNWR n he fuure. Accordng o hs vew, he lack of wage growh ha followed he Grea Recesson was no a sgnal of perssen labor marke slack. Our analyss shows ha he srengh of wage resran due o DNWR depends o a large exen on wheher wages are upward flexble or no. If ndeed some rgdy deans wage rses, for example conracs are revsed nfrequenly, workers become more forward lookng and are more nclned o demand a pay rse f hey expec he economy o recover. As a resul, pen-up wage deflaon becomes less relevan and he lack of wage growh s raher explaned by low producvy or weak aggregae demand. All our fndngs reflec a well-known resul from he heory of he second bes: all dsorons nerac and hey canno be suded ndependenly o assess he mers of a parcular polcy (Lpsey and Lancaser (1956)); wo and wo do no always add o four. 2. The Model Consder a closed producon economy populaed by a connuum of households and frms neracng on goods, labor and asse markes. Households are nfnely lved, have dencal preferences over consumpon and lesure, and supply a dfferenaed labor ype. Frms produce a dfferenaed consumpon good usng as npu labor ypes suppled by all households. Boh labor and produc markes are monopolscally compeve, and prces and wages are scky à la Roemberg (1982). In addon, nomnal 3

4 wage growh s subjec o a zero lower bound, capurng he noon ha households and frms mgh face greaer frcons when negoang wage reducons raher han wage ncreases. 1 Fnancal markes are complee and he moneary auhory decdes on he nomnal neres rae n a cashless economy as he one descrbed by Woodford (23) and Galí (28). The res of hs secon descrbes he model and s man equlbrum condons, whle he dervaons are relegaed o he Appendx Prmves Each household [, 1] has preferences defned by [ ] U = E β (C) 1 σ 1 (N ) 1+ϕ Z, (1) 1 σ 1 + ϕ = where E denoes expecaons condonal on he nformaon avalable a me, C and N denoe consumpon and hours worked, respecvely, β (, 1) s he subjecve dscoun facor, σ s he nverse neremporal elascy of subsuon, ϕ s he nverse elascy of labor supply and Z s an aggregae preference shock ha follows he sochasc process ln Z = ρ z ln Z + v z,, ρ z [, 1). (2) Innovaons o he preference shock, v z,, are normal random varables dencally and ndependenly dsrbued wh zero mean and varance σv,z. 2 The flow budge consran n nomnal erms s { } [ ( )] D + (1 + τ w )W N + Γ T E Q,+1 D+1 P C + Φ w Π w,, Π w, W. W (3) Households ener each perod wh fnancal wealh D, earn nomnal reurns W on labor, whch s subsdzed a a rae τ w, receve he dvdends dsrbued by frms Γ, pay lump-sum axes T, and buy a porfolo of Arrow-Debreu secures wh random nomnal value D+1, where Q,+1 s he one-perod ahead sochasc dscoun facor. 2 The prce of a porfolo payng one un of currency wh cerany, E Q,+1, equalzes he prce of rsk-free cenral-bank balances, R, by a sandard no-arbrage argumen. 1 Adjusmens coss can be specfed n such a way ha f he consran on nomnal wage growh s slack, he model s equvalen o he one lad ou by Erceg e al. (2) who assume Calvo (1983) prcng. More precsely, as s showed below, adjusmen cos funcons can be chosen so ha he model delvers he same equlbrum and welfare, a frs- and order second-order accuracy, respecvely, as n Erceg e al. (2). 2 Q,+1 s he me- prce vecor of sae-conngen asses dvded by he condonal probably ha he sae occurs n + 1 gven nformaon avalable a. The general me- dscoun facor for nomnal payoffs j perods ahead s Q,+j = +j s=+1 Q s,s. 4

5 Income ne of savngs fnances consumpon of a fnal good wh prce P and wageadjusmen coss, expressed n uns of he fnal good. Adjusmen coss are modeled as a non-negave, non-decreasng and convex funcon Φ w ( ) of household s wage nflaon. Fnally, we also assume ha nomnal wages canno decrease: Π w, 1. (4) Each frm j [, 1] produces an nermedae consumpon good wh a decreasngreurn-o-scale echnology, Y,j = X N 1 α,j, α [, 1), N,j = [ 1 ( N,j ) η w η w ] ηw η w d, η w > 1, (5) where N,j aggregaes labor ypes suppled by households and η w represens he elascy of subsuon beween labor ypes. Shocks o echnology, X, are dencal across frms and evolve accordng o ln X = ρ x ln X + v x,, ρ x [, 1). (6) Innovaons o echnology shocks are normal random varables dencally and ndependenly dsrbued wh zero mean and varance σv,x. 2 The presen value of curren and fuure nomnal profs s { } E Q, Γ,j, (7) Γ,j P,j Y,j (1 + τ p ) 1 = W N,jd P Φ p (Π,j ), Π,j P,j P,j. P,j s he prce of nermedae good j. Prce-adjusmen coss are modeled as a nonnegave, non-decreasng and convex funcon Φ p ( ) of frm j s prce nflaon, and hey are measured n uns of he fnal good. The laer s a convenonal compose ndex of nermedae goods: [ 1 Y = (Y,j ) η p η p ] ηp [ η p 1 dj, η p > 1, P ] 1 P 1 ηp,j dj 1 η p, (8) where η p denoes he elascy of subsuon beween nermedae goods. 3 Aggregae profs are fnally defned as Γ 1 Γ,jdj. 3 I s equvalen o assume ha a compeve realer buys nermedae goods a prces P,j, produces he consumpon good wh echnology (8) and sells o households a prce P, or ha households buy each good j a prce P,j o maxmze her uly defned over nermedae goods. 5

6 Opmal allocaon of households expendure across varees and frms cos mnmzaon mply sandard demand funcons for labor: N = ( ) W η w N, W [ 1 W ] ( ) 1 W 1 η w 1 η w 1 d, N N,j dj, (9) and nermedae goods: Y,j = ( P,j P ) η p [ C + 1 Φ w ( Π w, ) d + 1 ] Φ p (Π,j ) dj, C 1 C d. (1) Households choose he se of sae-conngen sequences { } C, N, D+1, W o maxmze uly, (1), subjec o consrans (3), (4) and (9), akng as gven aggregae prces and quanes and nal condons, D and W. Frms choose he se of saeconngen sequences {P,j, Y,j, N,j } o maxmze profs (7) subjec o consrans (5) and (1), akng as gven aggregae prces and quanes and he nal condon P,j Prvae-secor equlbrum We focus on symmerc equlbra where W = W, P,j = P, Π w, = Π w, Π,j = Π and D =, for all, and j. Marke clearng mples he followng feasbly consrans Y = C + Φ w (Π w ) + Φ p (Π ), Y = X N 1 α, (11) consumpon s allocaed ner-emporally accordng o a convenonal Euler equaon, { } Υ+1 R βe = 1, Υ C σ Z, (12) Υ Π +1 and nomnal frcons generae New Keynesan Phllps curves for prce and wage nflaon { } ( ) Φ Υ+1 p (Π ) Π = βe Φ p (Π +1 ) Π +1 + η p Y (M p ) ηp 1 (1 + τ Υ η p p ), (13) { } Φ w (Π w ) Π w Υ+1 = βe Φ w (Π w ) Π w Λ w +1 + Λ w Υ +η w W ( ) N (M w ) ηw 1 P η (1 + τ w), w (14) wh prce and wage mark-ups, M p and M w, defned by M p (1 α)p X, M w W N α W P N ϕ C σ 6, (15)

7 The real wage, W /P, s relaed o prce and wage nflaon hrough deny W P = Πw Π W P, (16) whle Λ w capures he coss of reducng he nomnal wage and sasfes complemenary slackness and non-negavy condons 4 Λ w Π w =, Λ w, Π w 1. (17) A (monopolscally) compeve equlbrum s a se of sae-conngen sequences { N, C, Υ, Y, W }, M p, M w, Π p, Π w, Λ w P ha sasfes equaons (11)-(17) gven he exogenous sae varables, {Z, X }, moneary polcy, {R }, subsdes, τ p and τ w, and he nal condon W /P. Taxes are lump-sum and herefore mmaeral for he equlbrum. For smplcy hey are se o balance he governmen budge consran perod by perod. Boh Phllps curves relae nflaon o fuure expeced nflaon and o devaons of markups from her naural values, as n a convenonal New Keynesan model wh scky prces and wages. 5 However, he wage Phllps curve, (14), dffers from he one for prce nflaon by he presence of Λ w, whch measures he value of relaxng nequaly consran (4). When he consran bnds, Λ w s srcly posve, sgnallng ha households would lke o undercu her wage. Snce workers fal o nernalze he effecs of her wage seng decsons on aggregae demand, elmnang he consran does no necessarly enhance welfare. Wheher hs exernaly makes DNWR socally desrable or no s one of he quesons analyzed below. Two are he effecs of Λ w on wage nflaon. Frs, when he consran bnds wage nflaon s hgher han would be n absence of he consran. Second, f he consran s expeced o bnd n he fuure, workers prefer o resran wage demands. In fac, he benef of ncreasng he curren wage mus be weghed agans he cos of beng consraned n he fuure, whch s mgaed by dampenng curren wage nflaon. Ths feaure of wage seng behavor s conssen wh he phenomenon of pen-up wage deflaon : n economes ha experence a recesson severe enough o make DNWR bnd, nomnal wage growh may reurn o pre-recesson levels well afer he recovery maeralzes., 4 See he Appendx for a formal defnon of Λ w, whch s proporonal o he Lagrange mulpler aached o equaon (4). 5 Naural values of prce and wage markups are η p /((η p 1)(1 + τ p )) and η w /((η w 1)(1 + τ w )), respecvely, and hey are obaned by mposng Φ p ( ) =, Φ w ( ) = and Λ w =. 7

8 2.3. Moneary polcy The model s closed by specfyng polcy. The cenral bank s assumed o follow a smple Taylor-ype rule ha akes no accoun he zero lower bound of he nomnal neres rae, R = max { (R ) γr [ Π T β ( Π p Π T ) γπ ( ) γy ] 1 γr Y, 1}, (18) Y where γ r, γ p and γ y are exogenously gven Taylor-rule coeffcens, Π T s an exogenously gven nflaon arge and Y denoes he level of oupu a he non-sochasc seady sae. Ths rule mples seady-sae nflaon raes Π p = Π w = Π T A Canoncal represenaon A frs useful benchmark s gven by he naural equlbrum, where boh prces and wages are fully flexble, upwards and downwards, whch can be obaned by mposng Φ p = Φ w = Λ w =. Naural levels of oupu, hours worked and he real wage a he non-sochasc seady sae are Y = (1 α) Ψy 1+ϕ, N = (1 α) (1 α)ψy 1+ϕ, ω = (1 α) 1 αψy 1 α, (19) respecvely, whle her seady-sae log devaons read as y n = Ψ y x, n n = (1 σ)ψ y 1 + ϕ x, ω n = (σ + ϕ)ψ y 1 + ϕ, Ψ y 1 + ϕ σ(1 α) + ϕ + α. Naural levels are Pareo effcen because subsdes have been chosen o offse he monopolsc dsoron. 6 As far as he dynamcs are concerned, wo feaures of he naural equlbrum are mporan. The level of employmen only depends on echnology shocks. Hence, equlbrum flucuaons n hours worked and oupu, condonal on preference shocks, sgnal neffcen varaons n markups. In addon, for he lmng case of logarhmc uly n consumpon, he level of employmen s consan and any flucuaon n hours s neffcen. A second useful benchmark s he sandard New Keynesan model wh scky prces and wages as n Erceg e al. (2) or Galí (28), whch can be recovered as a parcular case of our framework under some paramerc assumpons. Frs, assume ha prceand wage-adjusmen coss are neglgble a a frs-order approxmaon and nl a he non-sochasc seady sae.e., Φ p (Π) = Φ w (Π w ) = Φ p (Π) = Φ w (Π w ) =. (2) 6 The requred values are τ p = 1/(η p ) and τ w = 1/(η w ). All varables whou me subscrp denoe a seady sae and ω log(w /P ) log(w /P ). 8

9 Then, a frs-order approxmaon of equaons (13)-(16) yelds 7 π = βe π +1 + αδp + δ p ω, δ p ηp Y (1 α)ŷ Φ pπ, (21) 2 ( ) π w = βe π w +1 δ w λ w +1 + (σ + ϕ ) δ w ŷ δ w ω +δ w λ w, δ w ηw ωn 1 α Φ w (Π w ) 2, (22) ω = ω + π w π ω n, (23) afer defnng he oupu gap, ŷ = y y n, and he real wage gap, ω = ω ω n. When adjusmen cos funcons are calbraed as Φ θ p (1 α + αη p )η p Y p (Π) = (1 α)(1 θ p )(1 θ p β)π, (24) 2 Φ w (Π w θ w (1 + ϕη w )η w ωn ) = (1 θ w )(1 θ w β) (Π w ) 2, and f λ w =, he equlbrum of our model concdes a a frs-order approxmaon wh he one mpled by he sandard New Keynesan model wh scky prces and wages, where θ p and θ w represen he me- probables ha nermedae goods and labor servces, respecvely, canno be reprced. 8 The full-blown verson of he model, where wages are scker downwards han upwards, s obaned by allowng he nequaly consran π w π w o be occasonally bndng and hereby leng λ w. Fnally, a second-order approxmaon of he uly funcon (1) yelds welfare creron 9 W = 1 {( 2 E β σ + ϕ + α ) } ŷ 2 + ηp 1 α δ p π2 + ηw (1 α) (π w δ w ) 2, (25) = whch concdes wh he one obaned by Erceg e al. (2) and Galí (28). Snce he welfare funcon s purely quadrac, a pecewse-lnear approxmaon of he equlbrum s enough o characerze welfare a second-order accuracy as n Cobon e al. (212). If R > 1 and π w > π w for all, he polcy predcons of our model are he same as n Erceg e al. (2) and Galí (28). 7 Lower-case varables denoe log devaons from he non-sochasc seady sae, wh he only excepon of he real wage, ω, and λ w. Argumens of funcons Φ p (Π) and Φ w (Π w ) have been suppressed for noaonal convenence. 8 Expressons (24) exend o he case of scky wages he known smlary beween Calvo (1983) and Roemberg (1982) prcng poned ou by Khan (25) and Ascar e al. (211). The equvalence s broken f assumpons BLA 9 Inequaly consrans nduce non-dfferenably of decsons rules wh respec o he sae varables. Noce however ha a second-order Taylor expanson of uly s well defned. In fac, all expressons needed o derve (25) he uly funcon (1) and feasbly consrans (11) are dfferenable wh respec o he expanson varables, whle neher he nequaly consran, (4), nor he Taylor-ype rule, (18), never need o be used n he approxmaon. 9

10 2.5. Soluon mehod and parameerzaon The economy feaures wo consrans ha bnd occasonally and sandard perurbaon echnques canno be appled. The model s hen solved wh he pecewse lnear perurbaon mehod developed by Guerrer and Iacovello (215). The mehod handles nequaly consrans as dfferen regmes of he same model, where regmes are defned by wheher he consrans are bndng or slack. Whle he dynamcs of each regme are solved lnearly, expecaons are compued by akng no accoun ha he probably of swchng across regmes s endogenous. The neracon beween he sae of he economy and he expeced duraon of regmes capures he non-lnear effecs due o he consrans. However, he approxmaon remans lnear condonal on he probably of swchng across regmes and hus canno accoun for precauonary behavor. 1 Parameer values are prmarly borrowed from he exsng leraure on DNWR and/or he ZLB and are summarzed n Table 1. We follow Km and Ruge-Murca (29) n choosng he dscoun facor, Taylor-rule coeffcens, he echnology parameer α, and he elascy of subsuon beween goods and labor ypes. 11 We resrc o he case of logarhmc uly n consumpon and unary Frsch elascy of labor supply, whch consues a useful benchmark. We follow Fernández-Vllaverde e al. (215) n choosng parameer values of sochasc processes. The seral correlaon of echnology s se o.9, whle s sandard devaon reflecs he lower volaly of labor producvy n he las wo decades. For preference shocks, perssence and sandard devaon are se o mach a 3-quarer half lfe and a.25% sandard devaon of rsk premum shocks, respecvely. Unless oherwse saed, he paper manans he assumpons ha equaon (2) holds and ha subsdes are chosen o offse he monopolsc dsoron. Coeffcens Φ p (Π) and Φ w (Π w ) are deermned o mach a range of prce- and wage-conracs duraons, accordng o expressons (24), whle he nflaon arge s vared beween % and 5%. ZLB sascs are compued by smulang he model wh echnology and preference shocks, whch nerac n deermnng frequency and duraon of ZLB epsodes, makng a condonal analyss hardly nformave. However, he sudy of he effecs of DNWR and he ZLB on macroeconomc volaly and welfare s lmed o he case of demand shocks for a varey of reasons. Frs, s already well known ha DNWR 1 For an applcaon of hs mehod o a ZLB problem see Cobon e al. (212). They also compare pecewse-lnear perurbaon wh hgher-order mehods whou fndng quanavely sgnfcan dfferences n mpulse response funcons. Even hough our model s smlar and much smpler o he one consdered by Cobon e al. (212), he usual dsclamer ha accuracy may be model specfc apples. 11 The dscoun facor s equal o he mean of he nverse ex-pos real neres rae n he Uned Saes beween 1964 and 26. The shor-run response of he nomnal neres rae o nflaon and oupu are (1 γ r )γ π = and (1 γ r )γ y =.68, whch concde wh he esmaes by Km and Ruge-Murca (29). 1

11 s cosly condonal on echnology shocks and, even f he cenral bank opmally addresses busness cycle flucuaons, a moderaely posve nflaon arge reduces s ncdence, mprovng welfare (Km and Ruge-Murca (29)). Second, demand shocks are parcularly neresng n our framework: hey are lkely key drvers of ZLB epsodes and hey canno be fully offse f he ZLB bnds, conrary o wha happens n a model whou ZLB. Fnally, echnology shocks canno accoun for ZLB epsodes hrough he lenses of a sandard New Keynesan model: hey have he counerfacual mplcaon of generang economc booms durng lqudy raps. Whle he cos of DNWR mgh be underesmaed due o he omsson of echnology shocks, he Appendx shows ha her ncluson affecs our resuls only margnally. 3. The ZLB, DNWR and macroeconomc volaly Ths secon analyzes he effecs of DNWR on he frequency of he ZLB and on s severy n erms of macroeconomc oucomes. To hs purpose, we compare four alernave versons of he model. A frs verson, labeled as baselne, assumes away all lower bounds. In hs case, he model bols down o a sandard New Keynesan model wh scky prces and wages. A second and a hrd verson nclude eher he lower bound on nomnal wage growh or on he nomnal neres rae and hey are respecvely labeled as DNWR and ZLB he former concdng wh he case consdered by Km and Ruge-Murca (29). Fnally, he full-blown model ncludes boh lower bounds and s labeled as ZLB-DNWR. All versons are smulaed for 15, perods wh he same random draw of nnovaons, and for a seres of nflaon arges ha range from % o 4.5% DNWR and he probably of hng he ZLB The occurrence of perods when he ZLB bnds s acheved by a combnaon of preference and echnology shocks. To apprecae he role of DNWR o s full exen we sar by focusng on he case of wages ha are upward flexble (φ w = ), whle prce duraon s se o wo quarers. The relave conrbuon of shocks n rggerng lqudy raps can be gauged by comparng he uncondonal dsrbuon of shocks wh her dsrbuon condonal on he ZLB beng bndng (Fgure 1). The former symmercally les whn hree sandard devaons from he uncondonal mean. The laer s shfed o he lef n he case of preference shocks and o he rgh for echnology shocks. 12 A ypcal ZLB quarer s assocaed wh lower han usual aggregae demand and hgher han usual producvy. As n Gavn e al. (215) and Fernández-Vllaverde e al. (215), he economy s forced no a lqudy rap when excess capacy, measured by a negave 12 Three sandard devaons amoun o 1.7% for echnology and o 1% for preference shocks, n accordance wh he parameer values dsplayed n Table 1. By consrucon, uncondonal dsrbuons are he same across models and rrespecve of he nflaon arge. 11

12 oupu gap, s large enough, a fac ha manly occurs because of adverse demand shocks. A comparson of condonal dsrbuons across models reveals ha when wages are downward rgd larger shocks are needed for lqudy raps o occur, suggesng ha DNWR mgh reduce he probably of vsng he ZLB. The op-lef panel of Fgure 2 plos he share of quarers spen a he ZLB as a funcon of he nflaon arge and confrms hs nuon. If he nflaon arge s suffcenly hgh, he wo models delver smlar predcons. For an nflaon arge of 3.5%, whch concdes wh average U.S. nflaon snce he early 5s, he ZLB bnds wh a frequency of abou 7.5%, roughly n lne wh he pos-war hsorcal evdence. As argeed nflaon falls and he ncdence of DNWR rses, he ZLB-DNWR model predcs a sgnfcanly lower frequency. Cobon e al. (212) were frs o noce ha DNWR mgaes he concerns relaed o he ZLB. Ther nuve explanaon s ha DNWR makes he real margnal cos downward rgd, moderang he declne n nflaon due o negave demand shocks. As a consequence, reduces he lkelhood of reachng he zero bound n he aemp of conanng deflaon by cung polcy raes. However, anoher mechansm s conssen wh he same premses: DNWR would keep expeced nflaon hgher han oherwse when lqudy raps maeralze, posvely conrbung o manan low he real neres rae and foser he recovery. The remanng panels of Fgure 2 decompose he fracon of me spen a he ZLB no wo componens: he probably of enerng a ZLB spell (.e. a seres of consecuve quarers spen a he ZLB) and he duraon of spells. The analyss suggess ha he laer channel also plays a subsanal role: boh he probably of enerng ZLB spells, rrespecve of her duraon, and her average duraon are unformly lower when wages are downward rgd. The boom-rgh panel fnally documens how DNWR makes shor spells more lkely, relave o long ones. Overall, DNWR affecs he frequency of he ZLB boh hrough an exensve margn, whch makes ZLB spells less lkely o occur, and hrough an nensve margn ha shorens he spells. One may herefore wonder wheher also reduces her severy and conans, raher han amplfyng, macroeconomc volaly a he ZLB. We urn o hs queson n he followng secon DNWR, he severy of lqudy raps and pen-up wage deflaon I s useful o frs consder he case of wo economes ha absrac from he ZLB: he baselne and he DNWR models. Fgure 3 repors mpulse response funcons of seleced varables o a preference shock ha s large enough o make DNWR bndng. The nflaon arge s se o 2 percen. Wage nflaon, whch s nally a s rend level, decreases by 2 percenage pons on mpac and hs s lower bound n he DNWR model. Snce nomnal wages canno fall furher han n he baselne model, prce nflaon and he nomnal neres rae also end o reac by less. Hence, for a gven moneary polcy rule, DNWR dmnshes he accommodaon provded by moneary polcy and causes employmen o declne by more han would oherwse, depressng 12

13 consumpon. Fgure 4 repeas he exercse for he model versons ha accoun for he ZLB. A comparson of he ZLB model wh he baselne repored n Fgure 3 shows how he ZLB amplfes he adverse consequences of he shock. In fac, consumpon and hours worked drop sgnfcanly more when he nomnal neres rae canno urn negave because he economy falls shor of moneary accommodaon, whch s nsead provded n he baselne model. DNWR and he ZLB are boh undesrable consrans: each of hem consdered n solaon prevens he economy from effcenly respondng o he shock. However, a comparson of he models dsplayed n Fgure 4 shows ha DNWR becomes desrable when he ZLB can bnd. Consumpon and hours worked are ndeed less volale n he ZLB-DNWR model and he effec of he shock s mlder on mpac. On he one hand, DNWR has he convenonal effec of placng upward pressure on real wages whch fall o a lesser exen and very gradually, as compared o he model wh flexble wages. On he oher hand, DNWR smulaes aggregae demand because perssenly dampens he response of nflaon and, due o he ZLB, keeps he real neres rae lower han oherwse, mgang he adverse effec of he shock. A frs blush, mgh seem surprsng ha wage rgdy conans raher han amplfyng neffcen employmen flucuaons. I s n fac an ngraned belef ha f wages le above her compeve level, employmen mus be neffcenly low. However, as emphaszed by Galí (213), such a belef comes from he assumpon ha goods prces are fully flexble so ha aggregae demand s rrelevan for employmen. Ths logc s reversed when goods prces are scky, because employmen becomes demanddeermned. Fgure 5 compares he behavor of employmen n he ZLB and n he ZLB-DNWR models for dfferen combnaons of prce and wage rgdy, when he economy s h by a negave preference shock of wo sandard devaons. If prces and wages are fully flexble (op-lef quadran) he ZLB s rrelevan for quanes and employmen neffcenly falls only when DNWR s nroduced. The paern however nvers when oher nomnal rgdes are presen. Fgure 5 also documens ha an ncrease n wage adjusmen coss que obvously makes DNWR less relevan, as compared o he sandard New Keynesan model ha already ncorporaes symmerc adjusmen coss. Our resuls sand n sark conras wh recen conrbuons ha analyze he effecs of DNWR, wh or whou he ZLB, such as Schm-Grohé and Urbe (213) and Daly and Hobjn (214). The belef n he vrues of wage flexbly ha he recen crses revved on boh sdes of he Alanc seems o rely on he assumpon of full prce flexbly. The behavor of nomnal wage nflaon n Fgure 4 shows ha he model also capures pen-up wage deflaon. Under he assumpon of flexble wages, wage nflaon massvely falls on mpac and rses as soon as he demand shock sars dsspang. In he model wh DNWR nsead, nomnal wages fall by less nally, bu hey also recover wh a delay. Ths s because of wo forces: on he one hand, DNWR leaves real 13

14 wages hgher han hey would normally be, elmnang he need of re-adjusng hem upwards durng he recovery; on he oher hand, as he Phllps curve for wage nflaon makes clear, posve expeced values of λ +1 dscourage households from negoang hgher wages o conan he fuure cos of beng consraned. As n Daly and Hobjn (214), DNWR also empers wage ncreases. Faced wh such resul, s naural o ask wheher he reverse also holds.e., wheher upward nomnal wage rgdy affecs he lkelhood of wage cus. Fgure 6 dsplays mpulse responses o a negave preference shock of wo sandard devaons n he ZLB-DNWR model for dfferen degrees of upward wage rgdy, whle prce duraon s kep consan and equal o wo quarers. In all cases wage nflaon hs on mpac s lower bound, bu he larger s he wage adjusmen cos, he faser does nomnal wage growh recover. A frs counernuve, he fndng naurally follows from he fac ha f wages are scky, boh he curren and he fuure value of relaxng he consran on nomnal wage growh fall. Hence, households become less relucan n negoang hgher wage ncreases and pen-up wage deflaon looses relevance. Noce ha he resul s enrely drven by ancpaon: a he me of he shock, he value of relaxng he consran s always large enough o keep wage nflaon a s lower bound and s rrelevan wheher s magnude falls n he wage adjusmen cos. Wha maers s ha he expeced value of λ w +1 s lower, makng households lf wages sooner. Taken ogeher, our fndngs show ha, raher han DNWR, sckness upwards s welfare dermenal because parally offses he benefcal effecs of rgdy downwards. A recenly emphaszed moneary polcy mplcaon of pen-up wage deflaon s ha he cenral bank mgh be mslead and whdraw moneary accommodaon oo lae f he lack of wage nflaon was nerpreed as sgnallng labor marke slack. Ths concern s however empered f wages are upward rgd because, by makng DNWR less relevan, hey would call for more raher han for less moneary accommodaon. Snce he ncdence of DNWR, and hus of pen-up wage deflaon, also depends on upward wage rgdy, we conclude ha sudyng he wo frcons separaely mgh be msleadng. Buldng on hese resuls, one mgh advance he educaed guess ha he benefs of devang from prce sably o ake care of he ZLB are over-esmaed f DNWR s negleced. We hus conclude he paper by urnng o welfare analyss. 4. DNWR and he cos of he ZLB Snce all varans of he model share he same seady sae rrespecve of he nflaon arge, equaon (25) can be used o rank alernave regmes and polces as well as o compue he welfare cos of he ZLB. The expeced cos of he ZLB, boh n he ZLB and n he ZLB-DNWR models, s compued as he fracon of seady-sae consumpon ha households would be wllng 14

15 o gve up n order o swch o he Baselne and o DNWR model, respecvely. 13 The oal expeced cos of he ZLB reflecs boh dfferences n he frequency of ZLB epsodes as well as dfferences n he welfare cos of an average ZLB quarer. Snce he frequency of he ZLB vares across models, may be nformave o compue he cos of he ZLB per quarer spen a he zero bound. Fgure 7 repors hose numbers. In he ZLB model, as expeced, boh coss fall n he nflaon arge. As average nflaon rses, he lkelhood of hng he zero bound dmnshes. In addon, even f he shock s large enough o force he economy no a lqudy rap, nflaon gves he cenral bank enough laude o cu he polcy rae and make he consequen recesson mlder. Therefore, he per-quarer cos also falls. Inflaon reduces he oal cos of he ZLB also n he ZLB-DNWR model. However, he cos s unformly smaller, because DNWR alone mgaes he lkelhood of hng he zero bound. In addon, ncreasng he nflaon arge becomes less effecve because also dmnshes he ncdence of DNWR, whch s n urn useful o preven he occurrence of ZLB evens. Hence, he cos s flaer wh respec o he nflaon arge, whch does no maer much for he cos of hng he zero bound. In addon, DNWR acs hrough an nensve margn, makng lqudy raps less severe should hey maeralze. For hs reason, he cos per ZLB quarer becomes U- shaped. Sarng from he case of zero nflaon, nally falls as n he case whou downward nomnal wage rgdy. However, as he nflaon arge ncreases DNWR becomes less relevan and, gven ha empers he adverse effecs of he ZLB, he cos per quarer sars rsng. Overall, ncreasng he nflaon arge n presence of DNWR seems o make lqudy raps less recurren bu more panful, leavng her oal expeced cos roughly unaffeced. Fgure 8 summarzes he gans of DNWR n reducng he cos of he ZLB for alernave calbraons of prce and wage duraon and decomposes he gans n he share due o changes n he volaly of he oupu gap, prce nflaon and wage nflaon. Gans are szeable: for a 2% percen nflaon arge, whch already elmnaes he ncdence of DNWR o a large exen, gans sll range from.5% o.25% of seady-sae consumpon, dependng on he duraon of prce and wage conracs. Evdenly, mos of he gans are drven by a fall n he volaly of prce and wage nflaon. One mgh hen wonder wheher hese gans would survve he use of an ad-hoc, and more convenonal, welfare funcon ha absracs from wage nflaon. Hence, we repea he exercse wh he followng ad-hoc welfare funcon W = 1 2 E β { } ξ y ŷ 2 + π 2, (26) = and consder hree possble values for he oupu wegh:,.25 and.5. Fgure 9 repors he resuls, whch sll pon owards a sgnfcan fall n he cos of he ZLB. 13 Equaon (25) measures welfare drecly n erms of consumpon equvalens because he approxmaon has been re-scaled by ( U/ C)C. 15

16 5. Concludng Remarks Ths paper sudes how DNWR neracs wh he ZLB and he mplcaons of such an neracon for he opmal nflaon arge. We conrbue o he recenly revved leraure on DNWR n hree respecs. Frs, he concluson ha DNWR s responsble for neffcen employmen flucuaons when moneary polcy s consraned by he ZLB heavly reles on he assumpon ha he economy s no subjec o any oher dsoron. Hence, s no robus o he nroducon of nomnal prce rgdes, nor of upwardly scky wages. Second, DNWR mgh be an mporan par of he puzzlng mssng dsnflaon wnessed durng he Grea Recesson. Fnally, he presence of DNWR alone s no suffcen o argue n favor of a hgher nflaon arge as long as he curren arge adequaely addresses concerns regardng he ZLB. 16

17 References Ascar, Gudo, Efrem Caselnuovo, and Lorenza Ross, Calvo vs. Roemberg n a rend nflaon world: An emprcal nvesgaon, Journal of Economc Dynamcs and Conrol, 211, 35 (11), Calvo, Gullermo, Saggered Prces n a Uly Maxmzng Framework, Journal of Moneary Economcs, 1983, 12 (3), Cobon, Olver, Yury Gorodnchenko, and Johannes Weland, The Opmal Inflaon Rae n New Keynesan Models: Should Cenral Banks Rase Ther Inflaon Targes n Lgh of he Zero Lower Bound?, Revew of Economc Sudes, 212, 79 (4), Daly, Mary C. and Bar Hobjn, Downward Nomnal Wage Rgdes Bend he Phllps Curve, Journal of Money, Cred and Bankng, Ocober 214, 46 (S2), Erceg, Chrsopher J., Dale W. Henderson, and Andrew T. Levn, Opmal moneary polcy wh saggered wage and prce conracs, Journal of Moneary Economcs, Ocober 2, 46 (2), Fernández-Vllaverde, Jesús, Grey Gordon, Pablo Guerrón-Qunana, and Juan F. Rubo-Ramírez, Nonlnear advenures a he zero lower bound, Journal of Economc Dynamcs and Conrol, 215, 57 (C), Galí, Jord, Moneary Polcy, Inflaon and The Busness Cycle, Prnceon Unversy Press, 28., Noes for a New Gude To Keynes: Wages, Aggregae Demand and Employmen, Journal of he European Economc Assocaon, 213, 11 (5), Gavn, Wllam T., Benjamn D. Keen, Alexander W. Rcher, and Nahanel A. Throckmoron, The zero lower bound, he dual mandae, and unconvenonal dynamcs, Journal of Economc Dynamcs and Conrol, 215, 55 (C), Guerrer, Luca and Maeo Iacovello, OccBn: A oolk for solvng dynamc models wh occasonally bndng consrans easly, Journal of Moneary Economcs, 215, 7 (C), Khan, Hashma, Prce-seng behavour, compeon, and markup shocks n he new Keynesan model, Economcs Leers, June 25, 87 (3), Km, Jnll and Francsco J. Ruge-Murca, How much nflaon s necessary o grease he wheels?, Journal of Moneary Economcs, Aprl 29, 56 (3),

18 Lpsey, Rchard G. and K. Lancaser, The General Theory of Second Bes, Revew of Economc Sudes, Roemberg, Julo J., Scky Prces n he Uned Saes, Journal of Polcal Economy, 1982, 9 (6), Schm-Grohé, Sephane and Marn Urbe, Downward Nomnal Wage Rgdy and he Case for Temporary Inflaon n he Eurozone, Journal of Economc Perspecves, 213, 27 (3), Woodford, Mchael, Ineres and Prces, Prnceon Unversy Press,

19 Table 1: Benchmark parameerzaon Descrpon Parameer Value Dscoun facor β.997 Rsk averson σ 1 Labor supply elascy ϕ 1 Technology parameer α 1/3 Elascy of subsuon of goods η p 11 Elascy of subsuon of labor ypes η w 3.5 Ineres rae smooher γ r.381 Taylor coeffcen on nflaon γ π 1.89 Taylor coeffcen on oupu γ y.11 Seral correlaon of echnology shocks ρ x.9 Sandard devaon of nnovaons o echnology shocks σ v,x.25 % Seral correlaon of demand shocks ρ u.8 Sandard devaon of nnovaons o demand shocks σ v,u 2 % 19

20 3 Dsrbuon of echnology shocks Frequency (%) Uncondonal ZLB ZLB DNWR Dsrbuon of preference shocks 3 25 Frequency (%) Fgure 1: Uncondonal dsrbuons of echnology and preference shocks and dsrbuons condonal on he ZLB beng bndng. The op panel refers o echnology shocks and he boom panel refers demand shocks. The nflaon arge s se o and he value of shocks, repored on he horzonal axs, s measured n percenage devaons from he non-sochasc seady sae. 2

21 Frequency (%) ZLB frequency Inflaon arge (%) ZLB mean duraon ZLB DNWR ZLB Frequency (%) Probably of enerng a ZLB spell Inflaon arge (%) Duraon dsrbuon ZLB DNWR ZLB Quarers Frequency (%) Inflaon arge (%) Quarers Fgure 2: ZLB sascs n he ZLB-DNWR and ZLB models. The ZLB frequency (op-lef panel) s defned as he share of quarers spen a he ZLB as a funcon of he nflaon arge. The probably of enerng a ZLB spell (op-rgh panel) s compued as he share of quarers ha a ZLB spell sars, rrespecve of s duraon. The ZLB mean duraon (boom-lef panel) s he average duraon of ZLB spells measured n quarers. The frequency of spells by duraon (boom-rgh panel) plos he share of spells of a gven lengh as a fracon of he oal number of spells for he case of a zero nflaon arge. 21

22 Consumpon Hours % devaon.5.5 % devaon % devaon 3 Prce Inflaon % devaon 5 Wage Inflaon Levels % devaon Real Wage Quarers Real Ineres Rae Baselne DNWR Quarers Sd. Devaons Levels 2 Nomnal Ineres Rae Quarers Preference shock Quarers Fgure 3: Impulse response funcons of seleced varables o a wo-sandard-devaon negave preference shock. Nomnal and real neres raes are expressed n percenage and annualzed. Prce and wage nflaon are expressed n percenage, annualzed and repored n devaon from her seadysae value. All oher varables are repored n percenage devaons from he seady sae. The nflaon arge s se o 2%. Dashed (sold) lnes refer o he model wh DNWR bu no ZLB (wh neher DNWR nor he ZLB). 22

23 Consumpon Hours % devaon 3 % devaon Prce Inflaon Wage Inflaon % devaon % devaon % devaon Real Wage Quarers Real Ineres Rae Levels Nomnal Ineres Rae Quarers Preference shock Levels 2 1 ZLB ZLB DNWR Quarers Sd. Devaons Quarers Fgure 4: Impulse response funcons of seleced varables o a wo-sandard-devaon negave preference shock. Nomnal and real neres raes are expressed n percenage and annualzed. Prce and wage nflaon are expressed n percenage, annualzed and repored n devaon from her seadysae value. All oher varables are repored n percenage devaons from he seady sae. The nflaon arge s se o 2%. Dashed (sold) lnes refer o he ZLB-DNWR (ZLB) model. 23

24 Prce duraon=1 Wage duraon=1 Prce duraon=1 Wage duraon=2 Oupu (% devaons) Prce duraon=2 Wage duraon=1 Prce duraon=2 Wage duraon=2 Oupu (% devaons) ZLB ZLB DNWR Quarers Quarers Fgure 5: Response of oupu for dfferen combnaons of prce and wage rgdy when he economy s h by a negave preference shock of wo-sandard-devaons. The nflaon arge s se o 2%. Dashed (sold) lnes refer o he ZLB-DNWR (ZLB) model. 24

25 % devaon Consumpon Prce Inflaon % devaon 3 Hours Wage Inflaon % devaon % devaon.5.5 1q 2q 3q % devaon Real Wage Quarers Real Ineres Rae Levels Nomnal Ineres Rae Quarers Preference shock Levels Quarers Sd. Devaons Quarers Fgure 6: Impulse response funcons of seleced varables o a wo-sandard-devaon negave preference shock n he ZLB-DNWR model. Nomnal and real neres raes are expressed n percenage and annualzed. Prce and wage nflaon are expressed n percenage, annualzed and repored n devaon from her seady-sae value. All oher varables are repored n percenage devaons from he seady sae. The nflaon arge s se o 2%. 1q, 2q and 3q sand for one-quarer, wo-quarer and hree-quarer duraon of wage conracs. 25

26 Consumpon equvalens (%) Expeced cos of he ZLB ZLB model Toal Per ZLB quarer Inflaon arge.2 Expeced cos of he ZLB ZLB DNWR model Consumpon equvalens (%) Inflaon arge Fgure 7: Expeced cos of he ZLB as a funcon of he nflaon arge n he ZLB (op panel) and n he ZLB-DNWR models (boom panel). Sold (dashed) lnes refer o he (oal cos) cos per ZLB quarer. 26

27 Fall n ZLB cos due o DNWR Conrbuon of he Oupu Gap 6 14 Consumpon Equvalens Flex wage 2Q wage duraon Flex prce Share of gans Inflaon arge Inflaon arge Conrbuon of Prce Inflaon Conrbuon of Wage Inflaon Share of gans Share of gans Inflaon arge Inflaon arge Fgure 8: Fall n he ZLB cos due o DNWR and s componens. The welfare funcon and smulaed volales are used o npu welfare gans o he varables enerng he welfare funcon. The flex wage calbraon assumes flexble wages and a wo-quarer prce duraon. The 2Q wage duraon calbraon assumes a wo-quarer duraon for boh prces and wages. The flex prce calbraon assumes flexble prces and a wo-quarer wage duraon. 27

28 Consumpon Equvalens Fall n ZLB cos due o DNWR Flex wage ξ y = ξ y =.25 ξ y =.5 Consumpon Equvalens Fall n ZLB cos due o DNWR 2Q wage duraon ξ y = ξ y =.25 ξ y =.5 Consumpon Equvalens Fall n ZLB cos due o DNWR Flex prce Inflaon arge ξ y = ξ y =.25 ξ y =.5 Fgure 9: Fall n he ZLB cos due o DNWR wh an ad-hoc welfare funcon. ξ denoes he wegh of he oupu gap, relave o nflaon. The flex wage calbraon assumes flexble wages and a woquarer prce duraon. The 2Q wage duraon calbraon assumes a wo-quarer duraon for boh prces and wages. The flex prce calbraon assumes flexble prces and a wo-quarer wage duraon. 28

29 Appendx Appendx A. Prvae-secor equlbrum Ths secon saes he households and frms problems, her respecve frs-order condons and derves all he equaons ha defne he prvae-secor equlbrum. Appendx A.1. Households Each household faces he followng labor demand funcon of frm j: N,j = ( ) W η w N,j, W (A.1) whch follows from frms j s cos mnmzaon, and, ogeher wh defnon N 1 N,j dj, mples ha he oal demand for labor ype s (A.2) = N = ( W W ) η w N. (A.3) Households choose { } C, N, D+1, W o maxmze [ ] U = E β (C) 1 σ 1 (N ) 1+ϕ Z, [, 1], (A.4) 1 σ 1 + ϕ subjec o { } [ ( )] D + (1 + τ w )W N + Γ T E Q,+1 D+1 P C + Φ w Π w, (A.5) and equaon (A.3), gven aggregae prces and quanes, nal condons D and W, dvdends, Γ, and polcy. Afer usng equaon (A.3) o subsue for N n equaons (A.4) and (A.5), he Lagrangan can be wren as {[ L = E β (C) 1 σ 1 (N ) 1+ϕ 1 σ 1 + ϕ = [ [ ( )] { λ P C + Φ w Π w, + E Q,+1 D+1 D (1 + τ w )W [ + Λ w, Π w, 1 ]}, ( W W ( W W } + ) η ] w N Γ + T 29 ) η w (1+ϕ) ] Z (A.6)

30 and he correspondng frs-order condons are { C } : Υ (C ) σ Z = Λ P, (A.7) { W } : Λ P Φ w W ( ) Π w, Λ (1 + τ w )(1 η w )N + ηw (N ) 1+ϕ Z W { ( Λ + βe +1 P +1 W+1Φ w Π w, +1 + Λ w, W (W ) 2 { Λw, βe +1W+1 (W ) 2 )} } + =, (A.8) { D +1 } : Λ Q,+1 + βλ +1 =, (A.9) {KT - condons} : Λw,, Π w, 1, Λw, (Π w, 1) =. (A.1) The Euler equaon dsplayed n he man ex s obaned by usng equaon (A.7) o subsue for Λ n equaon (A.9), he fac ha E Q,+1 = R and by mposng symmery. To oban he wage Phllps curve, use equaon (A.7) o subsue for Λ n equaon (A.8), whch can be rearranged, η w W [ (η w 1)(1 + τ w ) N P N ϕ ] C σ + Φ P η w w (Π w ) Π w W { } { } Υ+1 ( ) βe Φ w Π w Υ +1 Π w +1 Λ w Υ+1 + βe Λ w +1 =, (A.11) Υ by mposng symmery, dvdng and mulplyng boh sdes of he equaon by Υ and W, respecvely, and applyng he followng defnon Λ w Λ w Π w Υ. (A.12) Equaon (A.11) mples he wage Phllps curve repored n he ex. Snce Λ w and only f Λ w =, equaons (A.1) can be equvalenly rewren as = f Λ w, Π w 1, Λ w (Π w 1) =. (A.13) I s fnally sraghforward o show ha he opmal allocaon of household s expendure mples he followng demand for good j: Y,j = ( ) η p P,j P [ ( )] C + Φ w Π w,. (A.14) 3

31 Appendx A.2. Frms We sar by dervng demand and cos funcons faced by each frm. The opmal allocaon of prce-adjusmen expendure across nermedae goods mples ha demand of frm z, z [, 1], for good j s Y,j,z = ( ) η p P,j P Φ p (Π,z ). Hence, as saed n he ex, oal demand for good j s Y,j = = 1 ( ) η p P,j P Y,j,z dz + Y d ; 1 Y,j d = Y d C + 1 Φ w ( Π w, P X ) 1 d + Φ p (Π,j ) dj. (A.15) (A.16) Cos mnmzaon mples real oal and margnal cos funcons, ne of adjusmen coss, 1 T C,j = W N,jd = W ( ) 1 Y,j 1 α, (A.17) MC,j = W P ( P,j where he aggregae real margnal cos s defned by = P MC (M p ) so ha frm j s prof funcon can be wren as ( ) η p ( P,j E Q, P,j Y d P,j (1 + τ p ) W P P ) αη p 1 α MC, W N α P (1 α)x, ) η p 1 α ( Y,j X ) 1 (A.18) (A.19) 1 α P Φ p (Π,j ), by subsung for 1 W N,jd from equaon (A.17) no he expresson for profs. Is maxmzaon wh respec o P,j yelds he followng necessary condon: (1 η p )(1 + τ p )Y,j + η p P Y,j MC,j + (A.2) P,j Φ P p (Π,j ) + E Q,+1 Φ p (Π +1,j ) P +1P +1,j =. P,j P,j 2 Afer usng equaons (A.9) and (A.7) o subsue for Q,+1 and Λ, respecvely, and mposng symmery, he necessary condon becomes (1 η p )(1 + τ p )Y + η p Y (M p ) Φ p (Π ) Π + (A.21) { } Υ+1 βe Φ p (Π +1 ) Π +1 =, Υ whch, afer rearrangng, gves he prce Phllps curve saed n he ex. 31

32 Appendx A.3. Marke clearng The clearng of markes for all goods j, (A.16), and he aggregaon funcon mply [ 1 Y = (Y,j ) η p η p ] ηp η p dj (A.22) Y = Y d = C + 1 Φ w ( Π w, ) 1 d + Φ p (Π,j ) dj. (A.23) By applyng symmery o equaon (A.23) and o he producon funcon, one mmedaely obans he feasbly consrans saed n he ex. Appendx B. Canoncal represenaon Ths secon derves he naural equlbrum, a frs-order approxmaon of he Phllps curves and a second-order approxmaon of he uly funcon abou he non-sochasc seady sae. Appendx B.1. Naural equlbrum and seady sae The naural equlbrum s easly derved by mposng Φ p = Φ w = Λ w = n equaons (A.11) and (A.21): P N ϕ C σ M w = 1, W M w = W N α M p = 1, (1 α)p X M p = η w (η w 1)(1 + τ w ), whch, ogeher wh he feasbly consrans, mply ha [ 1 α N n = M p M w [ W n = P n ] 1 X 1 σ σ(1 α)+ϕ+α, (M w ) α ( 1 α M p Y n = ) σ(1 α)+ϕ X σ+ϕ η p (η p 1)(1 + τ p ), (B.1) [ ( 1 α ) 1 α X 1+ϕ ] 1 σ(1 α)+ϕ+α, M p M w ] 1 σ(1 α)+ϕ+α, (B.2) where superscrps are used o denoe naural equlbrum values. As saed n he ex, he naural equlbrum n log-devaons from he non-sochasc seady sae s n n = (1 σ)ψ y 1 + ϕ x, y n = Ψ y x, ω n = (σ + ϕ)ψ y 1 + ϕ, (µ p ) n =, (µ w ) n =, Ψ y ϕ σ(1 α) + ϕ + α, (B.3)

33 afer defnng ω n = log(w n /P n ) log(w/p ), and seady-sae values become 1 1 α N = (1 α) σ(1 α)+ϕ+α, Y = (1 α) σ(1 α)+ϕ+α, ω = (1 α) σ(1 α)+ϕ σ(1 α)+ϕ+α M p = 1, M w = 1, (B.4) afer subsung for τ p and τ w from τ p = 1 η p 1, τ w = 1 η w 1, (B.5) n equaon (B.2). The assumpon ha Φ p (Π) = Φ w (Π w ) = also mples, ogeher wh equaons (B.4) and (B.1), ha W N P Y = C, (B.6) = (1 α)y. (B.7) Appendx B.2. Phllps curves A frs-order approxmaon of equaon (A.11) abou he non-sochasc seady sae yelds and η w W N P M w µw + Φ w (Π w ) 2 π w βφ w (Π w ) 2 π w +1 λ w + βλ w +1 = (B.8) µ w = ω ( σ + ϕ ) y + 1 α ϕ 1 α x, (B.9) where lower case varables denoe log-devaons from he non-sochasc seady sae, wh he excepon of λ w = Λ w Λ w. Equaons (B.2) mply ha ( ) (µ w ) n = ω n σ + y n + so ha ϕ 1 α µ w = µ w (µ w ) n = ω ϕ 1 α x =, (B.1) ( σ + ϕ ) ŷ, 1 α (B.11) afer defnng he oupu gap, ŷ y y n, and he real wage gap ω ω ω n. Usng equaon (B.11) o subsue for µ w n equaon (B.8) mmedaely gves he log-lnearzed wage Phllps curve dsplayed n he ex. A frs-order approxmaon of equaon (A.21) abou he non-sochasc seady sae yelds ( ) η p Y 1 (ηp 1)(1 + τ p ) y M p η p ηp Y M p µp Φ p (Π) 2 π + βφ p (Π) 2 π +1 = η p Y µ p Φ p (Π) 2 π + βφ p (Π) 2 π +1 = 33 (B.12)

34 where he second lne follows from equaons (B.4) and Equaons (B.2) mply ha so ha µ p = ω + α 1 α y 1 1 α x. (B.13) (µ p ) n = ω n + α 1 α yn 1 1 α x =, (B.14) µ p = µ p + (µ p ) n = ω + α 1 αŷ. (B.15) Usng equaon (B.15) o subsue for µ p n equaon (B.12) mmedaely gves he log-lnearzed prce Phllps curve dsplayed n he ex. Appendx B.3. Welfare funcon Le U be he nsananeous uly funcon. Is second-order approxmaon abou he non-sochasc seady sae reads as U = C (c 1 σ + 1 ) 2 c2 12 ( σc1 σ c 2 N 1+ϕ n + 1 ) 2 n2 1 2 ϕn 1+ϕ n 2 (B.16) + 1 [ ] C 1 σ c + N 1+ϕ n z +..p. = 2 = 1 [ ] ( ) C 1 σ (1 σ)c 2 N 1+ϕ (1 + ϕ)n 2 + C 1 σ c N 1+ϕ n ( C 1 σ c N 1+ϕ n ) z +..p. We now prove ha he las lne of equaon (B.16) s a hrd-order erm and hereby s zero a a second-order approxmaon. Frs, he producon funcon n s log-lnear form, y = x + (1 α)n, (B.17) holds exacly. In addon, a second-order approxmaon of he resource consran yelds ( Y y + 1 ) ( 2 y2 = C c + 1 ) 2 c2 + Φ pππ + Φ wπ w π w (B.18) Φ pπ 2 π Φ w (Π w ) 2 (π w ) 2 ; y y2 = c c2 + 1 η p 2 y = c + 1 η p 2 δ p π η w W N P δ w (πw ) 2 ; (B.19) δ p π η w (1 α) (π w δ w ) 2 ; (B.2) 34