Technology and the two margins of labor adjustment: a New Keynesian perspective

Francesco Furlanetto; Tommy Sveen; Lutz Weinke

doi:10.1515/bejm-2018-0217

Published by De Gruyter August 5, 2019

Technology and the two margins of labor adjustment: a New Keynesian perspective

Francesco Furlanetto , Tommy Sveen and Lutz Weinke

From the journal The B.E. Journal of Macroeconomics

https://doi.org/10.1515/bejm-2018-0217

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Abstract

Canova et al. [Canova, F., J. D. López-Salido, and C. Michelacci. 2010. “The Effects of Technology Shocks on Hours and Output: A Robustness Analysis.” Journal of Applied Econometrics 25: 755–773; Canova, F., J. D. López-Salido, and C. Michelacci. 2012. “The Ins and Outs of Unemployment: An Analysis Conditional on Technology Shocks.” The Economic Journal 123: 515–539] estimate the dynamic response of labor market variables to technological shocks. They show that investment-specific shocks imply predominantly an adjustment along the intensive margin (i.e., hours per worker), whereas for neutral shocks the largest share of the adjustment takes place along the extensive margin (i.e., employment). In this paper we develop a New Keynesian model featuring capital accumulation, two margins of labor adjustment and a hiring cost. The model is used to analyze a novel economic mechanism to explain that evidence.

Keywords: labor market; sticky prices; technological shocks

JEL Classification: E22; E24; E32

Funding source: Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)

Award Identifier / Grant number: 402884221

Funding statement: This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Funder Id: http://dx.doi.org/10.13039/501100001659, 402884221. Their financial support is gratefully acknowledged. The usual disclaimer applies.

Acknowledgement

Thanks to seminar participants at the 68th European Meeting of the Econometric Society, as well as to managing editor Gueorgui Kambourov and two anonymous referees. Patrick Brock, Clara Hoffmann and Dennis Zander provided excellent research assistance.

Appendix A

Linearized Equilibrium Conditions

In what follows we consider a log-linear approximation to the equilibrium dynamics around a zero inflation steady state. Unless stated otherwise lower case letters denote the log-deviation of the original variable from its steady state value. The consumption Euler equation reads

(25)ct=Et{ct+1}−(rrt−ρ),

where parameter ρ denotes the household’s time preference rate and rrt≡rt−Etπt+1 is the real interest rate. Up to the first order aggregate production is given by

(26)yt=(1−α)(zt+nt+ht)+αkt.

Linearizing and aggregating the law of motion of capital gives

(27)kt+1=(1−δ)kt+δit+δzI,t,

and the first-order conditions associated with investment and capital can be log-linearized as

(28)qt=βEt{qt+1}+(1−β(1−δ))Et{rt+1K}+βδρIzI,t−(rrt−ρ),

(29)it−kt=ϵψ(qt+zI,t),

where the following relationship holds true

(30)rtK=mct+(yt−kt).

Aggregating the linearized law of motion of firm-level employment results in

(31)nt=(1−s)nt−1+Φ(V/U)VN[(1−γ)vt+γut],

where we have used the notation that a variable without a time subscript denotes the steady state value of that variable. Linearized search unemployment reads

(32)ut=−(1−s)NUnt−1.

Period unemployment is given by

(33)utm=−NUMnt.

Aggregating and linearizing the first-order condition for firm-level employment implies

(34)Ξξt+ϵnΔnt+WH(wt+ht)=1μYN(mct+yt−nt)+ϵnβEt{Δnt+1}+(1−s)βΞEt{(rrt−ρ)+ξt+1},

where Δ is the difference operator and

ξt=γ(vt−ut)+zt+α1−αzI,t.

The following relationships holds true

(35)ft=(1−γ)(vt−ut).

The real wage is given by

(36)wt=χCH1+η1+ηWH(ct+(1+η)ht)−ht+ΥWHυt,

and

(37)υt=zt+α1−αzI,t+(1−ϕ)ϕΞΥ{ξt+β(1−s)[(1−F)(rrt−ρ)+Et{Fft+1−(1−F)ξt+1}]}.

The real marginal cost reads

(38)mct=ct+ηht−(yt−nt−ht).

The following inflation equation is derived

(39)πt=βEtπt+1+κ mct,

where parameter κ is computed numerically using the method outlined in Woodford (2005). Market clearing implies

(40)yt=CYct+IYit+cVY(vt+zt+α1−αzI,t),

and value added reads

gdpt=CGDPct+IGDPit.

Last, monetary policy is given by

(41)rt=ρRrt−1+(1−ρR)[ρ+ϕππt]+ert.

Appendix B

Computational Algorithm

We posit rules for price-setting and for employment

(42)p^t∗(i)=p^t∗+κ1n^t−1(i),

(43)n^t(i)=ξ1p^t(i)+ξ2n^t−1(i).

where N^t(i)≡Nt(i)Nt, P^t(i)≡Pt(i)Pt denote, respectively, firm i’s relative price and its relative to average employment. We have also used the definitions P^t∗(i)≡Pt∗(i)Pt and P^t∗≡Pt∗Pt, where Pt∗ is the average newly set price.

Let us first impose stability. Invoking the pricing and employment rules, as well as the definition of the price index we obtain

(44)[Etp^t+1(i)Etn^t+1(i)]=A[p^t(i)n^t(i)],

where A≡[10−ξ11]−1[θ(1−θ)κ10ξ2]=[θ(1−θ)κ1θξ1κ1ξ1(1−θ)+ξ2]. Stability requires that all roots of matrix A are inside the unit circle. Our goal is to find conditions for the unknown coefficients in the rules. To this end we first express key firm level variables (production, hours worked, capital and the real marginal cost) as a function of the two variables in the rules. We have

(45)[y^t(i)h^t(i)k^t(i)mc^t(i)]=B[p^t(i)n^t(i)],

where

B≡[10001−(1−α)−α010−111−(1+η)01]−1[−ϵ001−α0001]=11+αη[−ϵ(1+αη)0−ϵ−1−ϵ(1+η)−(1−α)η−ϵη(1−α)−η(1−α)]

With those preparations at hand, we next consider the linearized equation for the relative to average employment at the firm level.

(46)Δn^t(i)=βEt{Δn^t+1(i)}+1ζnh^t(i),

where ζn≡μNϵn(1−α)Y1η. We therefore have

(47)(1+β−β(κ1ξ1(1−θ)+ξ2)−b22ζn)n^t(i)=(βθξ1+b21ζn)p^t(i)+n^t−1(i),

which imposes the following two constraints on the undetermined coefficients ξ₁ and ξ₂ in the employment rule

ξ1=ξ2(βθξ1+b21ζn),ξ2=11+β−β(κ1ξ1(1−θ)+ξ2)−b22ζn.

Last, we consider price-setting. We can write the newly set price chosen by firm i as follows

p^t∗(i)=∑j=1∞(βθ)jEtπt+j+(1−βθ)∑j=0∞(βθ)jEtmct+j+(1−βθ)∑j=0∞(βθ)jEtmc^t+j(i).

Using equation (45) we have

∑j=0∞(βθ)jEtmc^t+j(i)=b41 Et∑j=0∞(βθ)j(p^t∗(i)−πt,t+j)+b42Et∑j=0∞(βθ)jn^t+j(i).

Using the above rules as well as the Calvo assumption we find

n^t+j(i)=ξ1n^t+j−1(i)+ξ2(p^t∗(i)−πt,t+j)=ξ1[ξ1n^t+j−2(i)+ξ2(p^t∗(i)−πt,t+j−1)]+ξ2(p^t∗(i)−πt,t+j).

We therefore have

∑j=0∞(βθ)jEtn^t+j(i)=ξ11−ξ1βθn^t−1(i)+ξ2(1−βθ)(1−ξ1βθ)p^t∗(i)−ξ2(1−βθ)(1−ξ1βθ)∑j=1∞(βθ)jEtπt+j.

Combining the last equations and invoking the Calvo assumption, i.e. noting that the average value of n^t−1(i) is zero in the group of time t price setters we have

(48)p^t∗(i)=p^t∗+11−b41−b42 ξ21−ξ1βθb42 ξ1 (1−βθ)1−ξ1βθn^t−1(i).

We can therefore impose the following condition on the unknown parameter in the pricing rule

κ1=11−b41−b42 ξ21−ξ1βθb42 ξ1 (1−βθ)1−ξ1βθ.

The average newly set price reads

(49)p^t∗=∑j=1∞(βθ)jEtπt+j+1−βθω∑j=0∞(βθ)kmct+j,

where

ω≡(1+ϵη)(1−ξ2βθ)+ηξ1(1−ξ2βθ).

Solving the last equation forward and invoking the linearized price index gives

(50)πt=βEt{πt+1}+κ mct,

where

κ≡(1−βθ)(1−θ)θ1ω.

For candidate parameter values which satisfy the stability requirement we therefore solve the following system

κ1(ξ1,ξ2)=ξ2(1−βθ)η(ξ2βθ−1)(1+ϵη)−ξ1η,ξ1=ϵζξ2ξ2βθ−1,0=1−(1+β)ξ2−ξ2ζ+βξ22+βξ1ξ2(1−θ)κ1.

This pins down the coefficients (ξ1,ξ2,κ1).

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Published Online: 2019-08-05

Technology and the two margins of labor adjustment: a New Keynesian perspective

Abstract

Acknowledgement

Appendix A

Linearized Equilibrium Conditions

Appendix B

Computational Algorithm

References

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