A note on deriving the New Keynesian Phillips Curve under positive steady state inflation

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1 A noe on deriving he New Keynesian Phillips Curve under posiive seady sae inflaion Hashma Khan Carleon Univerisiy Barbara Rudolf Swiss Naional Bank Firs version: February 2005 This version: July 2005 Absrac Recen sudies have considered he New Keynesian Phillips Curve under posiive seady sae inflaion-he NKPC-PI. This noe presens an explici derivaion of he NKPC- PI. This specificaion derived reveals how posiive seady sae inflaion affecs he coefficiens and srucure of he Phillips curve relaive o he sandard NKPC which assumes zero seady sae inflaion. s: hashma khan@carleon.ca, barbara.rudolf@snb.ch

2 This noe presens an explici derivaion of he New Keynesian Phillips Curve under posiive seady sae inflaion NKPC-PI. Implicaions of NKPC-PI have been recenly examined by Bakhshi, Burriel, Khan, and Rudolf 2005 and Ascari The model The benchmark model for a closed economy is based on Woodford o be 1 UC, H C1 σ 1 1 σ 1 0 H i 1+φ di φ where C is aggregae household consumpion, σ 1 is he ineremporal elasiciy of subsiuion for privae expendiure by he households, φ is capures he disuiliy of work effor, H i represens he supply of labour of ype i. Households maximise 1.1 subjec o a budge consrain. The firs-order condiion for he opimal labour-leisure choice is W i H i φ C σ 1 :Specific facor markes P :Common facor markes H φ Cσ Noe ha for his common facor markes case he specificaion 1.1 will be in erms of H. 1.1 Firms Consider an economy wih monopolisically compeiive firms each producing a differeniaed good, Y i. Each firm faces a consan-elasiciy downward sloping demand curve for is produc P i θ Y i Y 1.3 where P i is he price of firm i s good, P and Y are he aggregae price and oupu indices, and θ is he elasiciy of subsiuion beween he differeniaed goods. I also deermines he desired markup,, ha a firm would like o charge. θ Each firm uses a Cobb-Douglas producion echnology P Y i A H i a 1.4 where H i is he labour inpu and A is he echnology shock. The parameer 0 < a < 1 is he elasiciy of oupu wih respec o labour. Firms face consrains on adjusing nominal price of heir respecive producs. 1

3 Under he Calvo 1983 se up, each firm receives a random signal 1 α o re-opimize is price and wih probabiliy α i keeps is price unchanged in a given period. Given 1.3 and 1.4, a firm chooses is nominal price P i o maximise he discouned sum of expeced profis E Q,+j α j P i P i θ Y +j W P +ji i θ Y +j P +j P +j P +j P +j A +j aking as given {W +j i, P +j, Y +j, A +j }. The firs-order-condiion is P E Q,+j α j θ 1 θ Y +j W +ji 1 P P +j P +j P +j a P +j θ Y +j A +j 1 a a 1.5 P +j θ Y +j θp A +j where P is he opimal nominal price same for all firms who re-opimize upon receiving he Calvo price adjusmen signal. Muliplying 1.6 by P and dividing by 1 θ we ge E Q,+j α P j 1 θ Y +j W +ji θ 1 P P +j P +j θ 1 a P +j 1 θ a Y 1 +j P θ Y +j 0 A +j P +j A +j Combining 1.2 wih 1.4 and 1.3, and Y C aggregae marke clearing in equilibrium, we obain 1.7 W i P P θφ a P A φ a Y φ a +σ A φ a Y φ a +σ 1 :Specific facor markes :Common facor markes We consider he specific facor marke case. Therefore, we subsiue 1.8 W +j i P θ φ φ a 1 a φ Y a +σ 1 +j 1.9 P +j P +j A +j in 1.7 and ge E P Q,+j α j 1 θ Y +j P +j θ aθ 1 E P Q,+j α j θ1+ω 1 1+ω Y+j 1+ω+σ P +j A +j 2

4 Subsiuing P P X and P P +j 1 Π j Π +k in 1.10 we ge E Q,+j α j X Π j Π +k θ aθ 1 E 1 θ Y +j Q,+j α j X Π j Π +k θ+ωθ 1 From 1.11 we can solve for he opimal relaive price X and ge A +j 1+ω Y 1+ω+σ 1 +j 1.11 θ 1 X 1+ωθ θ 1 a E Q,+jα j 1 Π j Π +k E Q,+jα j θ+ωθ 1+ω 1 A +j Y 1+ω+σ 1 +j 1 Π j Π +k 1 θ Y +j 1.12 Noe ha under he common marke case he erm ωθ will no appear in The aggregae price level evolves as P [ 1 αp 1 θ 1 + αp 1 1 θ 1 θ 1.13 Dividing boh sides of 1.13 by P 1 θ we can re-wrie as 1 1 αx 1 θ 1.2 Log-linearizing he aggregae price level + απ 1.14 From 1.14, in seady sae noe: he gross seady sae inflaion rae Π π for noaional convenience [ X 1 α 1 Log-linearizing 1.14 around he seady sae we ge Log-linearizing he opimal relaive price Wihou loss of generaliy, we assume A +j 1 j. π x 1.16 From 1.12, no ha in he seady sae, he following expression would hold 3

5 X 1+ωθ µay ω+σ 1 1 αβπ 1 αβπ θ+ωθ, µ θ θ where π 1 is he gross seady sae inflaion rae. The expression 1.17 is he specific-facor marke version of average markup expression derived in King and Wolman Consider he lef-hand-side, L, of Log-linearizing 1.18 gives L E Q,+j α j X Π j Π +k l Y X 1 θ E αβπ [q j,+j + 1 θx + θ 1 1 θ Y +j 1.18 π +k + y +j 1.19 where all lower-case variables denoe log-deviaions from seady sae: z lnz /Z. Noe π lnπ /π. Consider he righ-hand-side, R, of R Log-linearizing 1.20 gives θ aθ 1 E Q,+j α j X Π j Π +k r µ a Y 1+ω+σ 1 X θ+ωθ E αβπ θ+ωθ [q j,+j θ + ωθx + θ + ωθ Equaing 1.19 and 1.21, hen using 1.17 we can solve for x o ge θ+ωθ Y 1+ω+σ 1 +j 1.20 π +k ω + σ 1 y +j αβπ θ+ωθ x E αβπ θ+ωθ [q j,+j + θ + ωθ π +k ω + σ 1 y +j 1 αβπ E αβπ [q j,+j + θ 1 π +k + y +j See Bakhshi e al for a discussion and comparison of how he average markup varies wih seady sae inflaion under he wo cases. 4

6 1.4 The NKPC under posiive seady sae inflaion Using 1.16 and 1.22 we can eliminae x o ge an expression for curren inflaion π as 1 αβπ θ+ωθ π E αβπ θ+ωθ [q j,+j + θ + ωθ 1 αβπ E αβπ [q j,+j + θ Furher simplificaions π +k ω + σ 1 y +j π +k + y +j 1.23 We can simplify 1.23 furher and derive a compac expression for he NKPC under posiive seady sae inflaion. 2 A. Isolaing E π +1 erms from 1.23 To isolae and collec wo E π +1 erms from he wo double summaions of 1.23 we make use of he following mah fac for 0 < γ < 1. 1 γ γ j π +i γe π +1 + γ1 γ i1 i1 B. Isolaing y erms from he summaions in Afer going hrough seps A and B we ge γ j π +1+i 1.24 [ θ + ωθ θ 1 π αβπ θ+ωθ αβπ E π +1 [ 1 + ω + σ αβπ θ+ωθ 1 αβπ y + αβπθ+ωθ 1 αβπ θ+ωθ E αβπ θ+ωθ [θ j + ωθ π +1+i ω + σ 1 y +1+j i1 αβπ 1 αβπ E αβπ [θ j 1 π +1+j + y +1+j 1.25 C. Simplifying he erm in {.} in Lead 1.23 by one period and muliply he resuling equaion by αβπ θ+ωθ o ge 2 In doing so we ignore variaions in he sochasic discoun facor, Q,+j β j Y +j /Y σ 1. See Khan and Rudolf 2005 where hese addiional variaions are incorporaed in he expression for curren inflaion. i1 5

7 αβπ θ+ωθ 1 αβπ θ+ωθ E π +1 αβπ θ+ωθ E αβπ θ+ωθ j [ θ + ωθ π +1+k ω + σ 1 y +1+j αβπ θ+ωθ 1 αβπ E αβπ [θ j 1 π +1+k + y +1+j 1.26 Re-wriing 1.26 as αβπ θ+ωθ E π +1 +αβπ θ+ωθ 1 αβπ E αβπ [θ j 1 1 αβπ θ+ωθ αβπ θ+ωθ E αβπ θ+ωθ [θ j + ωθ π +1+k + y +1+j π +1+k ω + σ 1 y +1+j Noe ha he righ hand side of 1.27 is he same erm as in he {.} in So replacing ha erm wih he lef hand side of 1.27 and collecing erms we ge 1.27 [ θ + ωθ θ 1 π αβπ θ+ωθ αβπ + αβπ θ+ωθ E π +1 [ 1 + ω + σ αβπ θ+ωθ 1 αβπ y [ + αβπ θ+ωθ 1 αβπ αβπ 1 αβπ E αβπ θ j 1 π +1+i + y +1+i 1.28 D. Simplifying he coefficiens in i1 1. Coefficien on E π +1. [ θ + ωθ θ 1 β P I αβπ θ+ωθ αβπ + αβπ θ+ωθ [ θ + ωθβπ 1+ωθ θ 1β + αβπ θ+ωθ [ θ + ωθ β 1 π 1+ωθ θ 1 + απ θ+ωθ

8 Noe ha when π 1, β P I β which is he coefficien on E π +1 in he sandard NKPC derived under he assumpion of zero seady sae inflaion. 2. Coefficien on y : he slope of he NKPC-PI. κ P I [ 1 + ω + σ 1 1 αβπ θ+ωθ 1 αβπ [ β βπ1+ωθ + 1 αβπθ+ωθ ω + σ 1 1 αβπ θ+ωθ ω + σ 1 + β 1 π 1+ωθ Noe ha when π 1, κ P I 1 α1 αβ α ω+σ 1 1+ωθ 1.30 κ which is he slope of he sandard NKPC derived under he specific facor markes assumpion see Woodford Coefficien on he hird erm in 1.28 δ P I [ αβπ θ+ωθ 1 αβπ π1+ωθ 11 1 αβπ αβπ π1+ωθ 1β1 1 αβπ αβπ 1 αβπ 1.31 Noe ha when π 1, δ P I 0 herefore he exra forward-looking srucure in he NKPC-PC due o increased fron-loading in firms opimal pricing decision does no arise. Using β P I, κ P I and δ P I we can re-wrie 1.28 as π β P I E π +1 + κ P I y + δ P I E αβπ θ j 1 E. Quasi-differencing Lead 1.32 by one period, ake E [., and muliply i by αβπ o ge i1 π +1+i + y +1+i 1.32 αβπ E π +1 αβπ β P I E π +2 + αβπ κ P I E y +1 + αβπ δ P I E αβπ θ j 1 i1 π +2+i + y +2+i

9 Subrac 1.33 from 1.32 o ge π αβπ E π +1 β P I E π +1 αβπ β P I E π +2 + κ P I y αβπ κ P I E y +1 E αβπ θ j 1 π +1+i + y +1+i δ P I Consider he las wo erms of 1.34: i1 αβπ δ P I E αβπ θ j 1 i1 π +2+i + y +2+i 1.34 θ 1δ P I E [αβπ αβπ π +2 + αβπ 2 π +2 + π αβπ αβπ αβπ π +3 + αβπ 2 π +3 + π δ P I E [αβπ 0 y +1 + αβπ 2 y αβπ αβπ 0 y +2 + αβπ y Inspecion of 1.35 reveals ha all he higher order erms ge canceled excep for he following: 1.35 θ 1δ P I αβπ 1 αβπ E π +2 + δ P I E y Replacing he las wo erms of 1.34 wih 1.36 we ge π β P I + αβπ αβπ E π +1 + θ 1δ P I 1 αβπ αβπ β P I E π +2 + κ P I y + δ P I αβπ κ P I E y F. Simplifying he coefficien on E π +2 in Using he expressions for β P I and δ P I, and afer some furher algebra, i can be shown ha αβπ β 2 θ 1δ P I 1 αβπ αβπ β P I αβ 2 π θ+ωθ

10 1.6 The compac version of NKPC-PI We can hus wrie he compac version of NKPC-PI as π β 1 E π +1 + β 2 E π +2 + κ P I y + δ 1 E y where κ P I is he slope of he NKPC-PI as in 1.30 and β 1 β P I + αβπ β 2 αβ 2 π θ+ωθ δ 1 δ P I αβπ κ P I A comparison of he compac form of NKPC-PI in 1.39 wih he sandard NKPC under zero seady sae inflaion π βe π +1 + κy 1.40 reveals ha posiive seady sae inflaion affecs boh he coefficiens and he srucure of he Phillips curve. 2 Relaed work Calvo conracs: Khan and Moessner 2004 consider he NKPC-PI and examine responses of oupu and inflaion o moneary policy shocks in a general equilibrium model. Ascari and Ropele 2004 examines he implicaions for opimal moneary policy. Roemberg 2003 considers ime variaion in he Calvo non-adjusmen signal α in a posiive seady sae environmen. Sahuc 2004 consider dynamic parial indexaion and posiive seady sae inflaion. The specificaion derived, however, ignores an addiional forward-looking srucure implied by model. Cogley and Sbordone 2004 implemen a similar parial indexaion formulaion on U.S. daa. 9

11 Khan and Rudolf 2005 derive a closed-form specificaion under parial indexaion and posiive seady sae inflaion and implemen i in an esimaed DSGE model. Taylor conracs: Kiley 2004 considers 2-period Taylor conracs and posiive seady sae inflaion. 10

12 References Ascari, G.: 2004, Saggered price and rend inflaion: some nuisances, Review of Economic Dynamics 7, Ascari, G. and Ropele, T.: 2004, The opimal sabiliy-oriened moneary policy: opimal moneary policy under low rend inflaion, Manuscrip, Universiy of Pavia. Bakhshi, H., Burriel, P., Khan, H. and Rudolf, B.: 2005, The New Keynesian Phillips Curve under sraegic complemenariy and rend inflaion, Forhcoming Journal of Macroeconomics. Calvo, G.: 1983, Saggered prices in a uiliy-maximizing framework, Journal of Moneary Economics 12, Cogley, T. and Sbordone, A.: 2004, A search for a srucural Phillips curve, Manuscrip, Federal Reserve Bank of New York. Khan, H. and Moessner, R.: 2004, Compeiiveness, inflaion, and moneary policy, Bank of England working paper no Khan, H. and Rudolf, B.: 2005, Esimaed closed and open economy models of he UK, Manuscrip, Bank of England and Swiss Naional Bank. Kiley, M.: 2004, Is moderae-o-high inflaion inherenly unsable?, Manuscrip, Federal Reserve Board. King, R. and Wolman, A.: 1996, Inflaion argeing in a S.Louis model of he 21s cenury, Federal Reserve Bank of S. Louis Review 78, Roemberg, J.: 2003, Cusomer anger a price increases, ime variaion in he frequency of price changes and moneary policy, NBER working paper no Sahuc, J.-G.: 2004, Parial indexaion, rend inflaion, and he hybrid Phillips curve, Working Paper 04-05, Universie D Evry. Woodford, M.: 2003, Ineres and prices: foundaions of a heory of moneary policy, Princeon Universiy Press, Princeon. 11