This document sets out the details for extending the wage Phillips curve of the New Area-Wide Model (NAWM; cf. Christoffel, Coenen and Warne, 2008) with a partial indexation mechanism linking wages to trend productivity developments. The document first outlines the labour-market setting in which households are offering their labour services. It then derives the first-order condition characterising the optimal wage-setting decision of an individual household as well as the law of motion for the aggregate wage index. Finally, the document derives the implied log-linear wage Phillips curve. An appendix provides additional technical details of the derivations

Coenen, Gunter

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Munich Personal RePEc ArchiveExtending the NAWM with a partialindexation mechanism linking wages andtrend productivitiyCoenen, GunterEuropean Central Bank2 October 2009Online at https://mpra.ub.uni-muenchen.de/86153/MPRA Paper No. 86153, posted 12 Apr 2018 00:39 UTC2 October 2009Extending the NAWM with a partial indexation mechanismlinking wages and trend productivityGünter Coenen∗European Central BankAbstractThis document sets out the details for extending the wage Phillips curve of the NewArea-Wide Model (NAWM; cf. Christoffel, Coenen and Warne, 2008) with a partial in-dexation mechanism linking wages to trend productivity developments. The documentfirst outlines the labour-market setting in which households are offering their labour ser-vices. It then derives the first-order condition characterising the optimal wage-settingdecision of an individual household as well as the law of motion for the aggregate wageindex. Finally, the document derives the implied log-linear wage Phillips curve. Anappendix provides additional technical details of the derivations.JEL Classification System: C11, C32, E32, E37Keywords: DSGE modelling, Bayesian inference, forecasting, policy analysis, euroarea.∗ Correspondence: Günter Coenen: Directorate General Research, European Central Bank, Kaiser-strasse 29, 60311 Frankfurt am Main, Germany, e-mail: gunter.coenen@ecb.int. The views expressed inthis documnet are those of the author and do not necessarily reflect those of the ECB or the Eurosystem.Any remaining errors are the sole responsibility of the author.Labour-Market SetupConsider a continuum of households indexed by h ∈ [ 0, 1 ], each of which supplies differen-tiated labour services Nh,t and acts as wage setter in monopolistically competitive markets.As a consequence, each household is committed to supply sufficient labour services to satisfylabour demand.Further, there is a continuum of firms indexed by f ∈ [ 0, 1 ], all of which take the wagerates set by the households as given and aggregate the differentiated labour services into anhomogenous bundle using a Dixit-Stiglitz technology,Nf,t =(∫ 10(Nhf,t) 1ϕt dh)ϕt, (1)where the possibly time-varying parameter ϕt > 1 is inversely related to the intratem-poral elasticity of substitution between the differentiated labour services supplied by thehouseholds, ηt = ϕt/(ϕt − 1) > 1.1Under these assumptions, each household h faces the following demand for its differen-tiated labour services from any given firm f as a function of its wage rate Wh,t relative tothe aggregate wage index Wt:Nhf,t =(Wh,tWt)− ϕtϕt−1Nf,t (2)with −ϕt/(ϕt − 1) representing the wage elasticity of labour demand.Aggregating over the continuum of firms f , we obtain the following aggregate demandfor the labour services of a given household h:Nht =∫ 10Nhf,t df =(Wh,tWt)− ϕtϕt−1Nt, (3)which, in equilibrium, is met by the household’s commitment to supply sufficient labourservices for any wage rate set; that is,Nh,t = Nht . (4)The wage index Wt can be obtained by substituting the labour index (1) into the labourdemand schedule (2) and then integrating over the unit interval of households:Wt =(∫ 10W11−ϕth,t dh)1−ϕt. (5)1As shown below, the parameter ϕt has a natural interpretation as a markup in the household-specificlabour market.1The Optimal Wage-Setting DecisionEach household h supplies its differentiated labour services Nh,t in monopolistically com-petitive markets. There is sluggish wage adjustment due to staggered wage contracts à laCalvo (1983). Accordingly, household h receives permission to optimally reset its nominalwage contract Wh,t in a given period t with probability 1− ξ.All households that receive permission to reset their wage contracts in a given period tchoose the same wage rate W̃t = W̃h,t. Those households which do not receive permissionare allowed to adjust their wage contracts according to the following scheme:Wh,t = g†z,tΠ†C,tWh,t−1, (6)where g†z,t = gχ̃z,t g1−χ̃z and Π†C,t = ΠχC,t−1Π̄1−χt . That is, the nominal wage contractsare indexed to a geometric average of the current (gross) rate of productivity growth,gz,t = zt/zt−1, and the steady-state (gross) rate of productivity growth, gz, and to a ge-ometric average of past (gross) consumer price inflation, ΠC,t−1 = PC,t−1/PC,t−2, and themonetary authority’s possibly time-varying (gross) inflation objective, Π̄t, with χ̃ and χbeing indexation parameters.Each household h receiving permission to reset its wage contract in period t maximisesits lifetime utility function subject to its budget constraint, the demand for its differentiatedlabour services (3) and the wage-indexation scheme (6).Hence, we obtain the following first-order condition characterising the households’ op-timal wage-setting decision:2Et∞∑k=0(ξβ)kΛt+k (1− τNt+k − τWht+k) g†z;t,t+kΠ†C;t,t+kΠC;t,t+kW̃tPC,t(7)−ϕt+k ǫNt+k (Nh,t+k)ζ)Nh,t+k]= 0,where Λt+k denotes the marginal utility out of income (equal across all individual house-holds in the economy), g†z;t,t+k = g(1−χ̃)kz∏ks=1 gχ̃z,t+s, Π†C;t,t+k =∏ks=1ΠχC,t+s−1Π̄1−χt+s andΠC;t,t+k =∏ks=1ΠC,t+s−1.This expression states that in those labour markets in which wage contracts are re-optimised, the latter are set so as to equate the households’ discounted sum of expected2See the Appendix for details.2after-tax marginal revenues, expressed in consumption-based utility terms, λt+k, to the dis-counted sum of expected marginal cost, expressed in terms of marginal disutility of labour,∆h,t+k = −Nζh,t+k. In the absence of wage staggering (ξ = 0), the factor ϕt representsa possibly time-varying markup of the real after-tax wage charged over the households’marginal rate of substitution between consumption and leisure,(1− τNt − τWht )W̃tPC,t= −ϕt ǫNt∆tΛt, (8)reflecting the existence of monopoly power on the part of the households.3The Aggregate Wage IndexWith households setting the wage contracts for their differentiated labour services accord-ing to equation (6) and equation (7), respectively, the aggregate wage index Wt evolvesaccording toWt =(ξ(gz,tΠ†C,tWt−1) 11−ϕt + (1− ξ)(W̃t) 11−ϕt)1−ϕt. (9)The Log-Linear Wage Phillips CurveWe use π̂C,t = log(ΠC,t/Π̄) to denote the logarithmic deviation of the current consumerprice inflation rate from the monetary authority’s long-run inflation objective, while ̂̄πt =log(Π̄t/Π̄) represents the logarithmic deviation of the current possibly time-varying inflationobjective from its long-run value. Moreover, because those firms which do not receivepermission to reset their prices are allowed to index them to a geometric average of pastinflation and the current inflation objective, it is natural to define the “quasi inflation gap”̂̇πC,t = π̂C,t − π̂†t , where π̂†C,t = log(Π†C,t/Π̄) = log(ΠχC,t−1Π̄1−χt /Π̄). Similarly, we define the“quasi productivity gap” ̂̇gz,t = ĝz,t − ĝ†z,t, where ĝ†z,t = log(g†z,t/gz) = log(gχ̃z,t/gχ̃z ).Finally, we use ŵt, m̂rst and ϕ̂t to denote, respectively, the productivity-adjusted realwage (deflated by the consumer price index and equal across all households), the marginalrate of substitution between consumption and leisure and the wage markup (all variablesexpressed as logarithmic deviations from their respective steady-state values).3Note that, in this case, also the marginal disutility is equal across households; that is ∆t = ∆h,t.3Then, combining the log-linearised first-order condition characterising the households’optimal wage-setting decision (7) and the log-linearised aggregate wage index (5) yields thelog-linear wage Phillips curve4ŵt =β1 + βEt [ŵt+1] +11 + βŵt−1 +β1 + βEt[̂̇πC,t+1]−11 + β̂̇πC,t (10)+β1 + βEt[̂̇gz,t+1]−11 + β̂̇gz,t −(1− βξ) (1− ξ)(1 + β) ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t)withΨ(ϕ, ζ) = 1 +ϕϕ− 1ζand where ŵτt represents the productivity-adjusted after-tax real wage (expressed in loga-rithmic deviation from its steady-state value) or, more compactly,∆ ŵt = β Et [∆ ŵt+1] + β Et[̂̇πC,t+1]− ̂̇πC,t (11)+β Et[̂̇gz,t+1]− ̂̇gz,t −(1− βξ) (1 − ξ)ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t) ,which shows that it relates changes in the productivity-adjusted real wage to current andfuture deviations of the productivity-adjusted after-tax real wage from the markup over themarginal rate of substitution, accounting for developments in inflation and productivity.Alternatively, noting that ̂̇πC,t = π̂C,t − π̂†C,t = π̂C,t − χ π̂C,t−1 − (1 − χ) ̂̄πt and ̂̇gz,t =ĝz,t− ĝ†z,t = (1− χ̃) ĝz,t, the wage Phillips curve can be written in terms of the productivity-adjusted (after-tax) real wage, actual consumer price inflation, the inflation objective andproductivity developments asŵt =β1 + βEt [ŵt+1] +11 + βŵt−1 +β1 + βEt [π̂C,t+1] (12)−1 + βχ1 + βπ̂C,t +χ1 + βπ̂C,t−1 −β(1− χ)1 + βEt[̂̄πC,t+1]+1− χ1 + β̂̄πC,t+β(1− χ̃)1 + βEt [ĝz,t+1]−1− χ̃1 + βĝz,t −(1− βξ) (1 − ξ)(1 + β) ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t) .AppendixIn this appendix, we provide details on the derivation of the first-order condition charac-terising the households’ optimal wage-setting decision (7). We then log-linearise the latter4For details see the Appendix.4as well as the aggregate wage index (9) around the deterministic steady state. We finallyderive the log-linear wage Phillips curve (A.15).The Optimal Wage-Setting DecisionTo derive the first-order condition characterising the households’ optimal wage-setting de-cision, we form the associated Lagrangean, substituting the labour demand schedule (3),the market-clearing condition (4), and the wage-indexation scheme (6), while neglecting allterms that are not relevant for the optimisation,Et∞∑k=0(ξβ)k(−ǫNt+k1 + ζ((g†z;t,t+k Π†C;t,t+kWh,tWt+k)− ϕt+kϕt+k−1Nt+k)1+ζ(A.1)+Λt+k (1− τNt+k − τWht+k) g†z;t,t+k Π†C;t,t+kWh,tPC,t+k(g†z;t,t+k Π†C;t,t+kWh,tWt+k)− ϕt+kϕt+k−1Nt+k)or, more compactly,Et[∞∑k=0(ξβ)k(Λt+k (1− τNt+k − τWht+k)Wt+kPC,t+k(g†z;t,t+k Π†C;t,t+kWh,tWt+k)− 1ϕt+k−1Nt+k (A.2)−ǫNt+k1 + ζ((g†z;t,t+k Π†C;t,t+kWh,tWt+k)− ϕt+kϕt+k−1Nt+k)1+ζ ,where Λt+k denotes the marginal utility out of income (equal across all households),g†z;t,t+k = g(1−χ̃)kz∏ks=1 gχ̃z,t+s, and Π†C;t,t+k =∏ks=1ΠχC,t+s−1Π̄1−χt+s .Differentiating this expression with respect to Wh,t then yields, after some algebra, thefirst-order condition characterising the households’ optimal wage-setting decision,Et∞∑k=0(ξβ)kΛt+k (1− τNt+k − τWht+k) g†z;t,t+kΠ†C;t,t+kΠC;t,t+kW̃tPC,t(A.3)−ϕt+k ǫNt+k((g†z;t,t+k Π†C;t,t+kW̃tWt+k)− ϕt+kϕt+k−1Nt+k)ζ×(g†z;t,t+k Π†C;t,t+kW̃tWt+k)− ϕt+kϕt+k−1Nt+k = 0,where W̃t denotes the optimal wage rate in period t that is chosen by all households thathave received permission to reset their wage contracts and ΠC;t,t+k =∏ks=1ΠC,t+s−1.Invoking again the demand schedule (3) in combination with the market-clearing con-5dition (4) and the indexation scheme (6), the first-order condition (A.3) simplifies toEt∞∑k=0(ξβ)kΛt+k (1− τNt+k − τWht+k) g†z;t,t+kΠ†C;t,t+kΠC;t,t+kW̃tPC,t(A.4)−ϕt+k ǫNt+k (Nh,t+k)ζ)Nh,t+k]= 0,(cf. equation (7) in the main text).Log-Linearisation of the Optimal Wage-Setting DecisionIn order to log-linearise the households’ optimal wage-setting decision, it is convenient todefine the auxiliary variables xt = W̃t/Wt, wt = Wt/(ztPC,t) and λt = zt Λt. Substitutingthese variables, the first-order condition (A.3) can be re-written asEt∞∑k=0(ξβ)kλt+k (1− τNt+k − τWht+k)wt+kg†z;t,t+kgz;t,t+kΠ†C;t,t+kΠC;t,t+kxtwtwt+k− 1ϕt+k−1Nt+k (A.5)−ϕt+k ǫNt+kg†z;t,t+kgz;t,t+kΠ†C;t,t+kΠC;t,t+kxtwtwt+k−ϕt+kϕt+k−1Nt+k1+ζ = 0,or, after re-arranging,Et∞∑k=0(ξβ)k λt+k (1− τNt+k − τWht+k)wt+kg†z;t,t+kgz;t,t+kΠ†C;t,t+kΠC;t,t+kxtwtwt+k− 1ϕt+k−1Nt+k (A.6)= Et∞∑k=0(ξβ)k ϕt+k ǫNt+kg†z;t,t+kgz;t,t+kΠ†C;t,t+kΠC;t,t+kxtwtwt+k−ϕt+kϕt+k−1Nt+k1+ζwhere gz;t,t+k =∏ks=1 gz,t+s−1.Indicating the logarithmic deviation of a variable from its steady-state value by ahat (‘̂’), defining steady-state values implicitly by dropping the time subscripts, and notingthat Π†C = Π, x = ǫN = 1, we obtain the following log-linearised expression:Et[∞∑k=0(βξ)k λ (1− τN − τWh)wN(λ̂t+k −τ̂N + τ̂Wh1− τN − τWh+ ŵt+k (A.7)−1ϕ− 1(k∑s=1(π̂†C,t+s − π̂C,t+s + ĝ†z,t+s − ĝz,t+s)+ x̂t + ŵt − ŵt+k)+ N̂t+k)]= Et[∞∑k=0(βξ)k ϕN1+ζ(ϕ̂t+k + ǫ̂Nt+k −(1 + ζ)ϕϕ− 1(k∑s=1(π̂†C,t+s − π̂C,t+s6+ ĝ†z,t+s − ĝz,t+s)+ x̂t + ŵt − ŵt+k)+ (1 + ζ) N̂t+k)]or, noting that (1 − τN − τWh)w = ϕN ζ/λ (cf. equation (8) in the main text) and re-arranging,Et[∞∑k=0(βξ)k(ŵτt+k − m̂rst+k − ϕ̂t+k (A.8)+Ψ(ϕ, ζ)(k∑s=1(π̂†C,t+s − π̂C,t+s + ĝ†z,t+s − ĝz,t+s)+ x̂t + ŵt − ŵt+k))]= 0,where the termsŵτt+k = −τ̂N + τ̂Wh1− τN − τWh+ ŵt+k,m̂rst+k = ǫ̂Nt+k + ζ N̂t+k − λ̂t+kdenote the log-linearised productivity-adjusted after-tax real wage and the log-linearisedmarginal rate of substitution, respectively, andΨ(ϕ, ζ) = 1 +ϕϕ− 1ζ.Solving for x̂t + ŵt, we obtainx̂t + ŵt = − (1− βξ) Et[∞∑k=0(βξ)k(Ψ(ϕ, ζ)−1(ŵτt+k − m̂rst+k − ϕ̂t+k)(A.9)−k∑s=1(π̂C,t+s − π̂†C,t+s + ĝ†z,t+s − ĝz,t+s)− ŵt+k)]= − (1− βξ) Et[∞∑k=0(βξ)k(Ψ(ϕ, ζ)−1(ŵτt+k − m̂rst+k − ϕ̂t+k)− ŵt+k) ]+(1− βξ)Et[∞∑k=0(βξ)kk−1∑s=0(π̂C,t+s+1 − π̂†C,t+s+1 + ĝ†z,t+s − ĝz,t+s) ]= − (1− βξ) Et[∞∑k=0(βξ)k(Ψ(ϕ, ζ)−1(ŵτt+k − m̂rst+k − ϕ̂t+k)− ŵt+k) ]+βξ Et[∞∑k=0(βξ)k(π̂C,t+k+1 − π̂†C,t+k+1 + ĝ†z,t+s − ĝz,t+s) ].Forming the quasi-difference Zt − βξ Et[Zt+1] then yieldsx̂t + ŵt − βξ Et [x̂t+1 + ŵt+1] = − (1− βξ)Ψ(ϕ, ζ)−1 (ŵτt − m̂rst − ϕ̂t) (A.10)+ (1− βξ) ŵt + βξ Et[̂̇πC,t+1]+ βξ Et[̂̇gz,t+1],7where ̂̇πC,t = π̂C,t − π̂†C,t and ̂̇gz,t = ĝz,t − ĝ†z,t.Log-Linearisation of the Aggregate Wage IndexTo log-linearise the aggregate wage index (9), we first re-write the identity as1 =ξg†z,tgz,tΠ†C,tΠC,twt−1wt11−ϕt+ (1− ξ) (xt)11−ϕt1−ϕt. (A.11)Log-linearisation then yieldsx̂t =ξ1− ξ(ŵt − ŵt−1 + ̂̇πC,t + ̂̇gz,t). (A.12)The Log-Linear Wage Phillips CurveCombining equations (A.10) and (A.12) yields the log-linear wage Phillips-curveŵt =β1 + βEt [ŵt+1] +11 + βŵt−1 +β1 + βEt[̂̇πC,t+1]−11 + β̂̇πC,t (A.13)+β1 + βEt[̂̇gz,t+1]−11 + β̂̇gz,t −(1− βξ) (1 − ξ)(1 + β) ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t) ,or, more compactly,∆ ŵt = β Et [∆ ŵt+1] + β Et[̂̇πC,t+1]− ̂̇πC,t (A.14)+β Et[̂̇gz,t+1]− ̂̇gz,t −(1− βξ) (1 − ξ)ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t) ,Alternatively, recalling that ̂̇πC,t = π̂C,t − π̂†C,t = π̂C,t − χ π̂C,t−1 − (1− χ) ̂̄πt and ̂̇gz,t =ĝz,t− ĝ†z,t = (1− χ̃) ĝz,t, the wage Phillips curve can be written in terms of the productivity-adjusted (after-tax) real wage, actual consumer price inflation, the inflation objective andproductivity developments asŵt =β1 + βEt [ŵt+1] +11 + βŵt−1 +β1 + βEt [π̂C,t+1] (A.15)−1 + βχ1 + βπ̂C,t +χ1 + βπ̂C,t−1 −β(1 − χ)1 + βEt[̂̄πC,t+1]+1− χ1 + β̂̄πC,t+β(1 − χ̃)1 + βEt [ĝz,t+1]−1− χ̃1 + βĝz,t −(1− βξ) (1 − ξ)(1 + β) ξΨ(ϕ, ζ)(ŵτt − m̂rst − ϕ̂t)(cf. equation (A.15) in the main text).8ReferencesChristoffel, Kai, Günter Coenen, and Anders Warne, 2008, “The New Area-Wide Model ofthe Euro Area: A Micro-Founded Open-Economy Model for Forecasting and PolicyAnalysis”, European Central Bank Working Paper No. 944.9

Extending the NAWM with a partial indexation mechanism linking wages and trend productivitiy

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Extending the NAWM with a partial indexation mechanism linking wages and trend productivitiy

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