Optimal Monetary Policy with Labor Market Frictions:
The Role of the Wage Channel∗

Takeki Sunakawa†

Abstract
This paper introduces right-to-manage bargaining into a labor search model withsticky prices instead of standard ecient bargaining and examines the Ramsey-optimalmonetary policy. Without real wage rigidity, even when the steady state is inecient, price stability is nearly optimal in response to technology or government shocks.
Right-to-manage bargaining creates the wage channel to ination, as there is a direct
relationship between real wages and real marginal cost. In the presence of the wage
channel, price markups consist of only real marginal cost, and real wages and hours per
worker are determined such as in the Walrasian labor market.
Keywords : Ramsey-optimal monetary policy; Right-to-manage bargaining; Wage channel; Real wage rigidity; Wage markups
JEL codes : E52; E24

This paper is originally a part of the dissertation by the author at The Ohio State University. The
author is especially grateful to Bill Dupor and Paul Evans for valuable guidance and support. The author
also appreciates the comments and discussions from anonymous referees, Hiroki Arato, Ippei Fujiwara,
Naohisa Hirakata, Ryoichi Imai, Masaru Inaba, Aubhik Khan, Kengo Nutahara, Pau Rabanal, Paulina
Restrepo-Echavarria, Murat Tasci, Julia Thomas, Kenneth West (the editor), Alexander Wolman, Willem
Van Zandweghe, and Francesco Zanetti, as well as seminar participants at The Ohio State University, theCanon Institute for Global Studies, Kansai University, Econometric Society Meetings 2011, the Bank ofEngland, Shinshu University, and University of Tsukuba. Any remaining errors are the author's.
†
Assistant Professor at Graduate School of Public Policy, the University of Tokyo. E-mail address:
sunakawa@pp.u-tokyo.ac.jp.
∗

1

1

Introduction

In the study of the Ramsey-optimal monetary policy in models with sticky prices and search
frictions in the labor market, it is usually assumed that real wages and hours per worker are1

determined by ecient bargaining (EB) between rms and workers.

Under EB, when thesteady state is ecient (i.e., the Hosios (1990) condition is met and there is a subsidy tooset the distortion stemming from monopolistic competition), price stability to technology
shocks isexactly optimal (Thomas, 2008). The further we deviate from the ecient steadystate, the more deviation from price stability becomes optimal, as endogenous price markups
stemming from the net hiring cost lead to a tradeo between ination and unemployment(Faia, 2009).
However, in actual practice, rms have the right to manage workers; typically, hours
per worker are unilaterally picked by rms rather than determined by bargaining between
rms and workers. Then, real wages are determined by bargaining given the labor demandschedule of rms. Nickell and Andrews (1983) and Trigari (2006) introduced right-to-manage(RTM) bargaining into labor search models instead of EB, based on empirical backgrounds.

2This paper studies the Ramsey-optimal monetary policy in a labor search model withsticky prices and compares the two bargaining schemes, EB and RTM. The results under
RTM contrast starkly with the ones under EB. Even when the steady state is inecient,
price stability isnearly optimal under RTM, as opposed to Faia (2009). This is in line withprevious studies of optimal monetary policy in models with sticky prices and the Walrasianlabor market.

3

Also, when the steady state is ecient, the price stability result does notexactly hold under RTM, as opposed to Thomas (2008).
Under RTM, there is a direct transmission from real wages to real marginal cost, hence

the wage channel to ination (Christoel and Kuester,

leading to ination. This is called2008). In the presence of the wage channel, price markups consist of only real marginal cost,and real wages and hours per worker are determined such as in the Walrasian labor market
with variable wage markups.
Some papers have argued that real wage rigidity is important in explaining empirical4

regularities in labor search models.

With real wage rigidity, deviation from price stability

becomes optimal under both EB and RTM, although the reasons are quite dierent. Whenreal wages are sluggish and cannot respond immediately to a positive technology shock,rms want to increase labor input.

Under RTM, as rms can adjust labor input at theintensive margin (hours per worker) and hours per worker are more volatile than real wages,
wage markups become countercyclical and volatile; therefore, deviation from price stability is
optimal. Under EB, rms mainly adjust labor input at the extensive margin (employment),
which generates uctuations in the net hiring cost and price markups.

2

Recently, a few papers have studied the Ramsey-optimal monetary policy in labor searchmodels with sticky prices under EB. Faia (2009) used a similar setting to the one presented
in this paper in terms of the timing of events and quadratic price adjustment, but with only
the extensive margin. She found that deviation from price stability was optimal with theinecient steady state.

In contrast, Thomas (2008) introduced both of the intensive andextensive margins and showed that price stability is optimal with the ecient steady state.
When nominal wage bargaining is staggered, the case against price stability arises. Blanchard and Galí (2010) presented a simpler framework and demonstrated the price stability
result. Real wage rigidity à la Hall (2005) creates the case against price stability. Ravenna
and Walsh (2011) derived the objective function of monetary policy by using second-orderapproximation of the household's utility and gap terms of unemployment, as well as ination and consumption. They explicitly showed how gap terms are related to welfare costs ofsearch and matching frictions.
This paper examines both cases with an inecient and an ecient steady state, andcompares dierent methodologies to compute the Ramsey-optimal monetary policies: theLagrange and linear-quadratic (LQ) methods. The Lagrange method can be applied evenwhen the steady state is inecient, whereas the LQ method, with rst-order approximationof the equilibrium conditions and second-order approximation of the household's utility, isusually used only when the steady state is ecient.

5

Faia (2009) used the Lagrange method,whereas Thomas (2008); Blanchard and Galí (2010); Ravenna and Walsh (2011) used the
LQ method. This paper is consistent with the previous studies; under EB, when the steadystate is ecient, price stability is exactly optimal, and the Lagrange and LQ methods yieldexactly the same result up to rst order. The further we deviate from the ecient steadystate, the more volatile optimal ination is, and the larger the numerical error between the
Lagrange and LQ methods. Under RTM, however, the optimal volatility of ination is notzero nor minimized even when the steady state is ecient.
In the next section, the labor search model and the two types of bargaining, EB and RTM,
are introduced. The relationship between the labor market allocation and the wage channel
is also examined.

The Ramsey-optimal monetary policies are computed by the Lagrangeand LQ methods, and calibration is discussed in Section 3. Section 4 shows the results. The
case of the inecient steady state is rst considered in Section 4.1, whereas the case of theecient steady state is examined in Section 4.2. Section 5 oers a conclusion. Calculationsof the steady state and proofs of propositions are shown in the appendix.

3

2

Model

In this section, the labor search model used for monetary policy analysis is presented.
Quadratic price adjustment costs, as in Rotemberg (1982), two dierent types of bargaining(EB and RTM), as in Trigari (2006), and a simple form of real wage rigidity, as in Hall
(2005), are introduced. The relationship between labor market allocation and the dierent
types of wage bargaining is also discussed.

2.1 Timing of Events
Time is discrete and innite: t = 0, 1, 2, ..., ∞. The timing of events is as follows:1.

1 − nt−1 is the measure of unemployed workers at the end of the last period. A fractionof the employed workers ρnt−1 are exogenously separated as

ut = 1 − nt−1 + ρnt−1 ,(1)where ut workers seek jobs in the current period.
2. Firms post vacancies

vt to match with the unemployed workers ut by a matching

function m(ut , vt ), which exhibits constant returns to scale. The labor market tightness
is given byθt = vt /ut .

(2)

Each worker and rm are small, in the sense that they take as given the labor market tightness at the aggregate levelθt .

The job nding rate for workers,p(θt ) =m(ut , vt )/ut = m(1, θt ), and the job lling rate for rms, q(θt ) = m(ut , vt )/vt =m(θt−1 , 1), are also given for them.
3. The number of the newly employed workers is given by

nt = (1 − ρ)nt−1 + m(ut , vt ).
Note that the newly employed workers can work immediately.

(3)

6

4. The employed workers enter into production:yt = at f (ht )nt ,(4)

where at is the technology shock. Each employed worker works ht hours and produces

at f (ht ), which is the decreasing returns to scale in terms of hours per worker, so that4

rms earn a positive prot period by period. Total production is linear in nt .

2.2 Households
A family of households chooses consumption ct , one-period nominal bonds Bt that pay grossinterest rate Rt , and the number of the employed workers nt so as to maximize the life-time
utility:

Vt =

c1−σ
t
− g(ht )nt + βEt Vt+1 ,1−σsubject to

Rt−1 Bt−1 Tt
Bt≤ wt ht nt + b(1 − nt ) +
+ ,
Pt
Pt
Pt
nt = (1 − ρ)nt−1 + (1 − nt−1 + ρnt−1 )p(θt ).
ct +A fraction of households nt are employed and earn wage bill wt ht , and 1 − nt are unemployedat the end of the period and receive unemployment benets b.

wt is real wage, and ht ishours per worker, which are determined by bargaining between workers and rms. Tt is thesum of transfers from rms and the government. Pt is the aggregate price index. β ∈ (0, 1) is
a discount factor. σ > 0 is a parameter for the degree of risk aversion. g(ht ) is an increasing
function that measures labor disutility. The job nding rate p(θt ) is taken as given by eachworker. Consumption is pooled among households to insure against the unemployment risk;
therefore, households consume the same amount regardless of their employment status (Merz,
1995; Andolfatto, 1996).
By the rst order necessary conditions (FONCs) of ct and Bt , the standard consumption
Euler equation is derived as(
1 = βRt Et

ct+1
ct

−σ

Pt
Pt+1

)
.

(5)

The FONC of nt and the envelope theorem yields

St = wt ht − g(ht )cσt − b + β(1 − ρ)Et

(

ct+1
ct

)

−σ
(1 − p(θt+1 ))St+1,(6)

where St is the ratio of Lagrange multipliers of the budget constraint and the law of motion
for nt . St is also the matching value of a marginal worker for the family of households.

52.3 Firms
There are two types of rms: a nal-good producing rm and dierentiated intermediategood producing rms. The nal-good producing rm purchases each intermediate good yit
R1
Pit yit di subject to theat price Pit for i ∈ [0, 1] and minimizes its expenditure Pt yt =0Rεp /(εp −1)1 (εp −1)/εp
Dixit-Stiglitz (1977) aggregator yt =ydi
. It yields the demand curve
0 it−εpwith the elasticity parameter εp > 1, yit = (Pit /Pt )yt .
Each intermediate-good producing rm i produces yit in monopolistic competition andsells it to the nal-good producing rm.

The intermediate-good producing rm i chooses

prices and allocations Pit , nit , and vit so as to maximize the discounted sum of future prots

E0

∞
X

βt



t=0

ct
c0

−σ (

(1 + τ )Pitψyit − wit hit nit − cv vit −
Pt
2



2 )
Pit
−1
,
Pit−1

subject toyit = (Pit /Pt )−εp yt = at f (hit )nit ,nit = (1 − ρ)nit−1 + vit q(θt ),t−σwhere β (ct /c0 )is a stochastic discount factor for the rm. The rm i sets the intermediategood price Pit subject to the downward-sloping demand curve and quadratic price adjustment
costs with the parameter ψ > 0. τ ≥ 0 is a subsidy to rms to oset the distortion stemmingfrom monopolistic competion.

7

The rmi also chooses the amount of employment andvacancies nit and vit and pays the total wage bill wit hit nit and vacancy costs cv vit with theparameter cv > 0. The job lling rate q(θt ) is taken as given by each rm. The FONCs are

(
)


−σ 
Pt
ct+1
Pt+1
Pt+1
Pt
−1
= βEt
ψ
−1
∂Pit : ψ
Pt−1
Pt−1
ct
Pt
Pt


εp
+yt (1 − εp ) (1 + τ ) −ϕt ,εp − 1
(
)
−σ
ct+1
∂nit : Jt = ϕt at f (ht ) − wt ht + β(1 − ρ)Et
Jt+1 ,
ct
cv∂vit : Jt =.
q(θt )
(7)(8)

In equilibrium, all of the rms choose the same prices and allocations, so index i is dropped
in the resulting conditions. The FONC of Pit is a nonlinear version of the Phillips curve. ϕtis the Lagrange multiplier on the production function and measures real marginal cost.
is also the price of intermediate goods relative to nal goods.

ϕt8

In the FONC of nit , Jt is the Lagrange multiplier on the law of motion for employment

6and measures the matching value of a marginal job for each rm. The rm's prot period by
period is given by ϕt at f (ht ) − wt ht > 0, and the rm continues to match in the next periodwith the probability (1 − ρ). Note that the value of a vacancy is zero under the assumptionof free entry, and Jt is equal to the expected cost of vacancy in the FONC of vt . The last
two equations are summarized into the hiring condition:

cv= ϕt at f (ht ) − wt ht + β(1 − ρ)Et
q(θt )

(

ct+1
ct

−σ

cv
q(θt+1 )).

(9)

The expected cost of vacancy is equal to the discounted sum of future prots per worker.
The net hiring costcvxt ≡− β(1 − ρ)Et
q(θt )

(

ct+1
ct

−σ

cv
q(θt+1 ))(10)

is also dened; employment is a stock, and the net hiring cost includes the benet of stock
hoarding in the next period.

2.4 Wage Contracts
As shown in Sections 2.2 and 2.3, matching values of workers and rms are given by (6)and (8). A wage bill set [ω t , ω t ], which is compatible with their incentive to continue thematching, is derived from the conditions St ≥ 0 and Jt ≥ 0.

ω t is the workers' reservation

wage bill, and ω t is the rms' reservation wage bill. As long as the wage bill wt ht is in the set[ω t , ω t ], workers and rms continue their matching. To pin down the equilibrium real wages
wt and hours per worker ht within the set, two dierent types of wage contracts between
workers and rms are introduced.

2.4.1 Ecient Bargaining
Under EB, real wages and hours per worker (wt , ht ) are determined by Nash bargaining

η 1−η
between worker and rm to maximize the joint surplus St Jt
subject to the equations (6)and (8), where η ∈ (0, 1) is a parameter for workers' actual bargaining power. The FONCs
are∂ht : ϕt at f 0 (ht ) = g 0 (ht )cσt ,(11)∂wt : ηJt = (1 − η)St .

(12)

7

Equation (11) shows that the marginal rate of substitution is equal to the real marginal
cost times the marginal product of labor. Equation (12) shows that worker's surplus frombargaining is simply a fraction η of the total surplus, St = η(St + Jt ). Plugging the matchingvalues for workers and rms, St and Jt given by (6) and (8) into (12), we have

wt ht = ηϕt at f (ht ) + (1 − η) (g(ht )cσt + b)
(
)
−σ
ct+1
+ ηβ(1 − ρ)Et
cv θt+1 .
ct(13)

The wage bill is equal to a weighted average of the workers' reservation wage and the rms'
reservation wage, wt ht = ηω t + (1 − η)ω t with a constant weight η .

2.4.2 Right-to-Manage Bargaining
Under RTM, there are two stages in the bargaining in each period.

First, real wages wt

are determined by Nash bargaining between workers and rms so as to maximize the joint

η 1−η
surplus St Jt. Second, rms choose the level of hours per worker ht so as to solely maximize
the matching value for rms Jt . In the second stage, given wt , the FONC of ht is

wt = ϕt at f 0 (ht ).

(14)Real wages are equal to the real marginal cost times the marginal product of labor. Equation(14) implies that rms have labor demand schedule on real wages, ht = f

0−1(wt /(ϕt at )) ≡h(wt ). In the rst stage, taking the wage schedule into account, workers and rms bargain
over real wages wt so as to maximize the joint surplus. The FONC of wt isδt = [(1 − η)/η]St /Jt ,(15)

where δt ≡ −(∂St /∂wt )/(∂Jt /∂wt ) is the marginal gain for workers (relative to marginal loss
for rms) by incremental real wages. By plugging the matching values for workers and rms

St and Jt into (15), we have
wt ht = χt ϕt at f (ht ) + (1 − χt ) (g(ht )cσt + b)
(
)
−σ
cv
ct+1
+ χt β(1 − ρ)Et
(1 − (δt+1 /δt )[1 − s(θt+1 )]),ct
q(θt+1 )where χt= ηδt /(1 − η(1 − δt )) is workers' eective bargaining power.

(16)

The wage bill isequal to a weighted average of the workers' reservation wage and the rms' reservation wage,

wt ht = χt ω t + (1 − χt )ω t with a time-varying weight χt .
8

2.4.3 Real Wage Rigidity
Under both EB and RTM, workers and rms bargain over real wages period by period. Realwage rigidity is introduced into the model in the form of Hall's (2005) real wage norm

wt = (1 − γw )wt∗ + γw wt−1 ,∗where wt is the actual wage and wt is the target wage determined by bargaining.

(17)

γw is

a parameter for the degree of real wage rigidity. Although the wage norm itself is an adhoc assumption and not considered when the target wage is determined, the frameworkencompasses various sources of wage rigidity at a more primitive level, such as in Hall and
Milgrom (2008) and Gertler and Trigari (2009). This type of wage rigidity is also robust to
the Barro's (1977) critique; as long as the actual wage bill is inside the wage bill set [ω t , ω t ],both the workers and rms have an incentive to continue the matching.

2.5 Closing the Model
The resource constraint, including posting vacancies and price adjustment costs and unemployment benet, is given by

ct + gt + cv vt +ψ 2
π = yt + (1 − nt )b,
2 t

(18)

where gt is the government expenditure. Note that all rms set the same price so that2
R 1 ψ  Pit−
1
di = (ψ/2)πt2 , where πt = Pt /Pt−1 − 1 is the ination rate. To close the
Pit−1
0 2
model, the Ramsey-optimal policy is given in Section 3.
Competitive equilibrium is dened by the equilibrium conditions (1)-(9), (11), and (13)under EB or (14) and (16) under RTM, (17) and (18) under the Ramsey-optimal policy suchthat (i) households maximize their utility, (ii) rms maximize their prots, (iii) wages andhours per worker are determined via bargaining between rms and workers, and (iv) themarkets clear.

2.6 Labor Market Allocation and the Wage Channel
Labor market allocation is characterized by equations (11) and (13) under EB or (14) and(16) under RTM. In (14), real wages have a direct link to real marginal cost and henceination.

As rms choose the level of hours per worker, real wages are equal to the real

marginal cost times the marginal product of labor.
this transmission mechanism

Christoel and Kuester (2008) called

the wage channel to ination. Real wage rigidity is irrelevant
9to ination persistence under EB (Krause and Lubik, 2007), whereas it is relevant under
RTM (Christoel and Linzert, 2010).
In this section, it is assumed that elasticity of output to hours per worker is givenby α= f 0 (h)h/f (h) ∈ (0, 1), and elasticity of match to unemployment is given by ξ =(∂m(v, u)/∂u)(u/m) ∈ (0, 1).

Under RTM, the marginal gain for workers (relative tomarginal loss for rms) by incremental real wages can be written as∂St /∂wt,∂Jt /∂wt= 1 − εw + εw g 0 (ht )cσt /wt ,δt = −

0where εw = −h (w)w/h(w) = 1/(1 − α) > 1 is elasticity of demand for hours per worker toreal wages. This equation can be solved for

wt = µwt g 0 (ht )cσt ,(19)where µwt = εw /(εw − 1 + δt ) is wage markup over the marginal rate of substitution. Realwages and hours per worker are determined by demand (14) and supply (19), as if they arein the Walrasian labor market with variable wage markups.

0
If the marginal product of labor mplt ≡ at f (ht ) is equal to the marginal rate of substi0
σtution mrst ≡ g (ht )ct , hours per worker is at the ecient level. Galí, Gertler, and LopezSalido (2007) called the wedge between the marginal product of labor and the marginal rate

9of substitution the ineciency gap.

The ineciency gap is decomposed into price and wage

markups. In the present paper, even with search and matching frictions and labor adjustment
at the extensive margin, the ineciency gap at the intensive margin consists of price andwage markups. Price and wage markups are dened by µpt ≡ mplt /wt and µwt ≡ wt /mrst ,and it is assumed that there is no real wage rigidity for the sake of exposition. It can thenbe proven that

Proposition 1. The ineciency gap is given bymplt≡ µpt µwt
mrst


ϕ−1 ,
t
=
ϕ−1
t

10

εw
,
εw −1+δt

under EB,
under RTM,

where price and wage markups are
µpt =

µwt



ϕ−1 α 1 +
t

xt
wt ht



, under EB,

ϕ−1 ,
under RTM,
t


−1
α−1 1 + xt
, under EB,wt ht=
 εw ,
under RTM.

(20)

(21)

εw −1+δt

Proof. See Appendix A.2.1.
Under EB, rms adjust labor input only at the extensive margin, as hours per worker

EB

is determined by bargaining, and µpt



xt
= ϕ−1
α
1
+
holds. This includes endogenoust
wt ht

price markups related to the net hiring cost of labor adjustment at the extensive margin(Faia, 2009). Under RTM, rms adjust labor input not only at the extensive margin but also
at the intensive margin; rms choose the level of hours per worker and smooth out the net
hiring cost; α

1 + wxt ht t



= ϕ−1
= 1 holds under RTM. This implies µRTM
t ; price markups
pt

consist of only real marginal cost.

EB

Under EB, wage markups just oset price markups related to the net hiring cost, µwt =−1
α−1 1 + wxt ht t
so that the ineciency gap consists of only real marginal cost measuringineciency in the goods market. Under RTM, the ineciency gap also includes endogenous

RTM

wage markups µwt= εw /(εw − 1 + δt ) regarding ineciency in the labor market.

Under RTM, the allocation in the decentralized economy is not ecient, even when thestandard Hosios condition is satised (Sunakawa, 2012). It can be proven that

Proposition 2. The allocation in the decentralized economy is ecient in the steady state
if and only if ϕ = 1 andunder EB,
under RTM.

η = ξ,
η = ξ = χ,Also, the allocation in the ecient steady state is common between EB and RTM.
Proof. See Appendix A.2.2.
The eciency condition under RTM imposes more parameter restriction than the standard Hosios condition under EB, as the eective workers' bargaining power, χ = ηδ/(1 −η(1 − δ)), depends on δ and hence other variables.

Therefore, from the viewpoint of its

empirical relevance, the case of the inecient steady state is rst considered in Section 4.1,
whereas the case of the ecient steady state is also examined in Section 4.2.

11

3

Optimal policy and calibration

The equilibrium conditions in the previous section are summarized into (c.f., Schmitt-Grohéand Uribe (2004)):

Et {fi (yt+1 , yt , xt+1 , xt )} = 0,
where fi is the (possibly nonlinear) equilibrium conditions for i = 1, ..., n − 1, xt is a nx × 1
vector of the state variable, yt is a ny ×1 vector of the jump variable, and n = nx +ny . There
are n − 1 equilibrium conditions and n variables. The Ramsey-optimal policy pins down the∗∗equilibrium (if it exists and is unique) yt = g (xt ; σε ) and xt+1 = h (xt ; σε ) + ηε σε εt+1 .

3.1 Ramsey-Optimal Policies
The Ramsey-optimal policy yields the constrained ecient equilibrium, which can be obtained by maximizing the representative household's utility subject to the equilibrium conditions of the economy. There are two dierent methodologies to compute the Ramsey-optimal
policies: the Lagrange and LQ methods.
In the Lagrange method, the Ramsey-optimal policy is computed by using the Lagrangean

L0 = minmax
E0∞∞{Λt }t=0 {Ξt }t=0(∞X(
β t EtU (xt , yt ) +

t=0

n−1
X

))
λit fi (yt+1 , yt , xt+1 , xt ),i=1

where Λt is the set of Lagrange multipliers, Ξt is the set of endogenous variables, and U
is the household's utility period by period. The Lagrangean yields n rst-order necessary
conditions and n − 1 Lagrange multipliers; therefore, there are 2n − 1 equilibrium conditions
and unknowns.
In the LQ method, the household's utility U is approximated up to second order, andthe equilibrium conditions fi s are approximated up to rst order in the Lagrangean shown
above. Benigno and Woodford (2012, Proposition 1) showed that, if the household's utility
is correctly approximated, the Lagrange and LQ methods are exactly the same up to rstorder.

However, computing the correct LQ approximation can be very tedious.

All thelinear terms in the approximated household's utility are substituted out by using secondorder approximation of the equilibrium conditions so that the household's utility is purely
quadratic.

When the steady state is ecient, such a computation is relatively easier asfollows:

Proposition 3. When the steady state is ecient, under both EB and RTM, the discounted

12sum of the household's utility is approximated as
V0 − V∞yc1−σ X t ψ 2E0
βπt + σĉ2t − (ŷt2 − n̂2t )
≈ −
2
c
c
 t=0 h
i2 
αy/c
2
−
n̂t − n̂t + (1 + φ)ĥt
1+φ

cv v/c222,+
ξ ût + (1 − ξ)v̂t − m̂t
1−ξwhere x̂t = ln(xt /x) is the log deviation from the steady state.
Proof. See Appendix A.2.3.
Note that the ecient steady state is common between EB and RTM, and so is the correct
LQ approximation. The correct LQ approximation is used to compute the Ramsey-optimal
policies in the case of the ecient steady state in Section 4.2.

3.2 Calibration
Functional forms are assumed as follows.

The production function is f (ht )= hαt , whereα ∈ (0, 1) is elasticity of production to hours per worker. The labor disutility is g(ht ) =κh1+φ/(1 + φ), where φ > 0 is the inverse of Frisch elasticity and κ > 0 is a normalizationtξ 1−ξparameter. The matching function is m(ut , vt ) = mut vt, where ξ ∈ (0, 1) is elasticity ofmatches to unemployment and m > 0 is a normalization parameter. Unemployment benets
are given by b = bw wh, where bw ∈ (0, 1) is the ratio of unemployment benet to the wage
bill.
Parameter values are calibrated following previous studies based on the postwar U.S. data

10

and summarized in Table 1.

Steady state unemployment after matching 1 − n = 0.4 and11lling vacancy rates for rms q = 0.7 are given as the calibration targets.

β = 0.99 implies

that the annual interest rate is 4%. The risk aversion parameter is set to σ = 2.

εp = 6

implies that the steady state price markup is 20%. The ratio of government expenditure to
output is 25%. These values are standard and in line with the postwar U.S. data. The priceadjustment cost parameter ψ = 20 implies that the elasticity of ination to real marginal
cost in the log-linearized Phillips curve is (εp − 1)y/ψ= 0.15.

The job destruction rate

ρ = 0.1 and elasticity of matches to unemployment ξ = 0.4 are also standard. The ratio of
unemployment benet to wage bill is bw = 0.6. Elasticity of production to hours per worker
is α = 0.9, which also implies elasticity of labor demand to real wages at the intensive margin

εw = 1/(1 − α) = 10. The inverse of Frisch elasticity is φ = 2. The shock parameters are

13

assumed ρa = 0.95 and σa = 0.008, following the evidence on the Solow residual from thepostwar U.S. data.

[Table 1 is inserted here]

In the case of the inecient steady state in Section 4.1, η > ξ= 0.4, and the Hosios

condition is not met under EB. Given the values of η and ξ , the marginal gain for workers
by incremental real wages is δ = 0.26 < 1 and the wage markup is µw = w/mrs = 1.08 > 1
in the steady state under RTM, as shown in Table 2, which implies that there is a positiveineciency gap in the labor market (Galí, Gertler, and Lopez-Salido, 2007). The ineciency
gap also implies that the workers' eective bargaining power χ = δη/(1 − η + δη) = 0.28 isless than the workers' actual bargaining power η = 0.6. When η = ξ = 0.225, η = ξ = χholds and the steady state is ecient under both EB and RTM, as shown in Proposition 2,
which is the case examined in Section 4.2.

[Table 2 is inserted here]

4

Results

In this section, the Ramsey-optimal monetary policies are numericaly investigated.

First,

the case of the inecient steady state (Faia, 2009) without and with real wage rigidity
is examined in Section 4.1.

The case of the ecient steady state (Thomas, 2008) and acomparison between the Lagrange and LQ methods are shown in Section 4.2.

4.1 The inecient steady state
In the case of the inecient steady state, it is shown that price stability in response totechnology shocks isnearly optimal in the model with RTM without real wage rigidity. This

result starkly contrasts with the Faia's (2009) result in the model with EB, and is in line withthe result in a model with the Walrasian labor market. With real wage rigidity, deviationfrom price stability becomes optimal. Under RTM, the optimal volatility of ination is highwhen the workers' bargaining power is low, which also oers an interesting contrast with
Faia's result. Similarly, a larger unemployment benet to amplify the labor market volatility(Hagedorn and Manovskii, 2008) does not increase the optimal volatility of ination under
RTM, whereas it does under EB.

14

4.1.1 Without real wage rigidity
Figure 1 shows the impulse responses to a technology shock (0.8 = 100σa %) without real
wage rigidity under both EB and RTM. First of all, price stability is nearly optimal under
RTM; namely, the optimal response of ination to a positive technology shock is very small.
In response to a positive technology shock, under RTM, real wages respond immediately,
and rms post more vacancies to produce more. A larger labor demand makes the aggregate
labor market condition tighter, and employment gradually adjusts so that the labor supply
is increased. Under EB, real wages are sluggish even without real wage rigidity, which makes
the labor market condition even tighter. Ination deviates from zero and responds negatively.

[Figure 1 is inserted here]

What is the intuition behind the dierent results under EB and RTM? Under EB, as inFaia (2009), endogenous price markups stemming from the net hiring cost and a related tradeo between ination and unemployment undo price stability, even with both the intensive
and extensive margins, compared to the case with the extensive margin only in Faia (2009).
Under RTM, real wages and hours per worker replicate the allocation in the Walrasian labormarket with variable wage markups.

The wage channel to ination leads to stable pricemarkups, as they consists of only real marginal cost.
The left column of Figure 2 breaks down the ineciency gap into the price and wage
markups, as shown in Proposition 1.

Under EB, the price markup responds positively,

as the net hiring cost is higher with a tighter labor market condition.

The wage markup

responds negatively to oset the labor market ineciency measured by the ineciency gap.
The gap responds positively, as the former eect dominates the latter. Under RTM, the wage
markup responds negatively, but it stems from δt in Equation (19), as the marginal gain forworkers (relative to the marginal loss for rms) by incremental real wages gets larger. Pricemarkups are stable in response to the shock, and wage markups are the main driving forceof the ineciency gap.
The allocation in a model with the Walrasian labor market is also computed in Figure
3. Real wages and hours per worker are determined by wt = mrst = ϕt mplt . The divine
coincidence in Blanchard and Galí (2007) holds and uctuations in the price and wage
markups are exactly zero.

The price stability result under RTM is a quantitative one, as

opposed to the case with the Walrasian labor market.

12[Figure 2 is inserted here]154.1.2 With real wage rigidity
Figure 3 shows the impulse responses with real wage rigidity.

It is shown that deviationfrom price stability is optimal both under EB and RTM. Real wages are sluggish and cannot
respond immediately to a positive technology shock. Firms post even more vacancies, andthe aggregate labor market condition becomes even tighter to take advantage of lower real
wages, especially under EB, because rms mainly adjust labor input at the extensive margin,13

as hours per worker is determined by bargaining.

Under RTM, sluggish real wages make wage markups countercyclical and volatile because
the marginal rate of substitution is related to real wages, as shown in the labor supply
equation (19). Hours per worker increase, as rms can adjust labor input at the intensive
margin. Under EB, rms adjust labor input mainly at the extensive margin and hours per
worker decreases instead because there is no direct link between real wages and hours per
worker. Price markups respond strongly, reecting aggressive adjustments at the extensive
margin, whereas wage markups are relatively stable compared to the ones under RTM.

[Figure 3 is inserted here]

The right column of Figure 2 also breaks down the ineciency gap into price and wage
markups in the case with real wage rigitity.

As in the case without real wage rigidity,

price markups are volatile and wage markups just oset the gap under EB, whereas wage
markups mainly explain uctuations in the gap under RTM. The responses under RTM
closely resemble the ones in the model with the Warlasian labor market.
Under RTM, the allocational role of real wages is the key to explain the equilibrium
dynamics. Firms can adjust labor input at the intensive margin and take advantage of low
and sluggish real wages by increasing hours per worker via the wage channel. The adjustment
at the intensive margin, however, yields wage markup uctuations and makes the optimalination volatile, depending on the degree of real wage rigidity, as in the model with theWalrasian labor market. Under EB, the allocational role of real wages is limited, as there isno direct link between real wages and hours per worker. Firms mainly adjust labor input at
the extensive margin, which generates uctuations in the net hiring cost and price markups.

4.1.3 Robustness checks
The price stability result under RTM without real wage rigidity is a quantitative one. Somerobustness checks are done for dierent shocks and parameter values. The analysis herein
focuses on the case without real wage rigidity.

16Government expenditure shocks

Figure 4 shows the response to a positive government

expenditure shock in the case without real wage rigidity. Faia (2009) showed that in responseto the government expenditure shock, deviation from price stability arises, although the response is rather small. Faia's result depends on the assumption that rms adjust labor input

4
at the extensive margin only. When φ = 10 so that adjusting at the intensive margin is too
costly, the model under EB in this paper replicates the case with the extensive margin only
in Faia (2009). Employment decreases through a lower tightness of labor market. However,when φ = 2 so that rms also adjust labor input at the intensive margin, employment increases as well as the labor market tightness and hours per worker, which leads to more pricemarkup uctuation and deviation from price stability. Under RTM, employment decreases
insteadly, whereas hours per worker increase. The response of price markups is muted, andprice stability is as nearly optimal as in the case of technology shocks.

[Figure 4 is inserted here]Dierent values of η and bw

Faia (2009) showed that the optimal volatility of ination

is increasing in η because a higher bargaining power η results in more external congestionand unemployment uctuations under EB. The left window of Figure 5 shows the optimal
volatility of ination for dierent values of15

replicates Faia's result.

η under both EB and RTM.14 Under EB, it

Under RTM, the optimal volatility of ination is high when η islow. The external congestion is irrelevant to the optimal ination because rms can smooththe net hiring cost at the extensive margin by adjusting labor input at the intensive margin.
When η is low (i.e., when rms have more bargaining power), rms adjust hours per worker

16

in the presence of the wage channel, and wage markups become more volatile.

Hagedorn and Manovskii (2008, hereafter HM) showed that a larger value of unemployment benet, b = bw wh, also yields a larger employment uctuation under EB.

17

The right

window of Figure 5 shows the optimal volatility of ination for dierent values of bw . Under
EB, the larger bw , the more volatile ination is. The labor market condition becomes morevolatile, as pointed out by HM. Under RTM, however, the wage channel kills the labor market volatility. The optimal volatility of ination is high when bw is low because workers want
to work more with a lower outside option, which makes hours per worker and wage markups
volatile.

[Figure 5 is inserted here]174.2 The ecient steady state
The case of the ecient steady state is examined to compare it with the price stability result
under EB in Thomas (2008). If and only if ϕ = 1 and ξ = η = χ, the steady state is ecientand common between EB and RTM, as shown in Proposition 2. The condition for eciency
in the steady state is satised when ξ = η∗= 0.225, with the calibration in Section 3.2. The

correct LQ approximation is also common between EB and RTM, as shown in Proposition
3, and is used to compute the Ramsey-optimal monetary policies.
Figure 6 shows the impulse responses to a technology shock (0.8= 100σa %) without

real wage rigidity under both EB and RTM. When the steady state is ecient, the optimal
response of ination is exactly zero as in Thomas (2008). The reason is understood by examining Figure 7, which shows the decomposition of the ineciency gap. Without real wage
rigidity, under EB, the price and wage markups cancel each other out, and the ineciency
gap is exactly equal to zero. In other words, there is no external congestion. Thus, the monetary policymaker can focus on stabilizing ination. Under RTM, even though the (logged)ineciency gap is zero in the ecient steady state, it temporally deviates from zero due towage markup uctuations. With real wage rigidity, deviation from price stability is optimalunder both EB and RTM. The responses of price and wage markups under RTM resemble
the ones in the model with the Walrasian labor market, as in the case of the inecient steadystate.
Figure 8 shows the optimal volatility of ination for dierent values of η . It is shown that∗the optimal volatility of ination is zero under EB when ξ = η , as in Thomas (2008). On∗
the contrary, under RTM, even when ξ = η , the optimal volatility of ination is not zeronor minimized over η . The optimal volatility of ination computed by the LQ method are∗shown with gray lines in Figure 8. When ξ = η , the optimal volatilities computed by the
Lagrange and LQ methods are exactly the same under both EB and RTM.
The results of the paper are consistent with the ones in Thomas (2008) [price stability
when the steady state is ecient] and Faia (2009) [deviation from price stability when thesteady state is inecient].

It is found that, when the steady state is ecient, the pricestability result holds under EB, and the LQ and Lagrangean methods yield exactly the same
results up to rst order. When the steady state is inecient, deviation from price stability
arises, and the further we deviate from the ecient steady state, the larger the numericalerror between the Lagrange and LQ methods.

[Figure 6-8 is inserted here]18

5

Concluding Remarks

This paper analyzed the Ramsey-optimal monetary policy in labor search models with sticky
prices, focusing on the role of the wage channel to ination (i.e., a relationship between real
wages and real marginal cost), based on empirical evidence.

It is found that the natureof the Ramsey-optimal monetary policy under RTM is totally dierent from the one under
EB. Under RTM, even when the steady state is inecient, price stability is nearly optimal,
whereas deviation from price stability is optimal under EB. Real wage rigidity creates thecase against price stability, as hours per worker are more volatile than real wages, and wage
markups are countercyclical and volatile, as in models with the Walrasian labor market.
This paper also studied both cases with the inecient and the ecient steady state. UnderEB, price stability is optimal with the ecient steady state, and the more we deviate fromthe ecient steady state, the more volatilie optimal ination is. Under RTM, however, evenwhen the steady state is ecient, the optimal volatility of ination is not zero nor minimized.
In the present paper, it is found that wage markups are countercyclical and volatile withthe wage channel and real wage rigidity. In the U.S. economy, the ineciency gap between
the marginal product of labor and the marginal rate of substitution is mainly explainedby countercyclical wage markup uctuations, as shown in Galí, Gertler, and Lopez-Salido(2007); therefore, one of the interesting topics for future research will be to investigate thisrelationship more. Within the search and matching framework, the wage channel and real
wage rigidity are potentially able to explain the labor wedge (Chari, Kehoe, and McGrattan,
2007), which is one of the main drivers for business cycles (Cheremukhin and RestrepoEchavarria, 2014; Pescatori and Tasci, 2011).

A

Appendix

A.1 Steady Stater = 1/β and ϕ = (1 + τ )(εp − 1)/εp are immediately obtained.

Given the steady state

unemployment ũ = 1 − n and job lling rate q = m/v , the other steady state values of thelabor market aren = 1 − ũ, v = ρn/q, m̄ = ρn/(uξ v 1−ξ ),u = 1 − n + ρn,θ = v/u,p = ρn/u,

which are common between the two bargaining schemes.

19A.1.1 Ecient Bargaining
We normalize a = 1. Steady state conditions of matching values S and J are given by

ϕαhα−1 = κhφ cσ ,[1 − β(1 − ρ)(1 − p)]S = (1 − bw )wh − κh1+φ cσ /(1 + φ),[1 − β(1 − ρ)]J = ϕhα − wh,
1 = [(1 − η)/η]S/J,These equations can be solved for

wh =S , and J .

Given

η[1 − β(1 − ρ)(1 − p)] + (1 − η)[1 − β(1 − ρ)]α/(1 + φ) αxh ,
η[1 − β(1 − ρ)(1 − p)] + (1 − η)[1 − β(1 − ρ)](1 − bw )q and J , vacancy cost is cv = qJ and steady state consumption c =hα n − g − cv v − (1 − n)b gets pinned down. With the calibration given in Section 3.2, steadystate vacancy cost is about 2.18% of output and steady state consumption is about 72.82%
of output. Steady state reservation wage bills for workers and rms are given by ω = wh − Sand ω = wh + J . Using the calibrated parameters, [ω, ω] = [0.5952, 1.2699]wh is obtained.

A.1.2 Right-to-Manage Bargaining
Steady state conditions of matching values are given by

wh = ϕαhα ,[1 − β(1 − ρ)(1 − p)]S = (1 − bw )wh − κh1+φ cσ /(1 + φ),[1 − β(1 − ρ)]J = ϕhα − wh,δ = [(1 − η)/η]S/J,These equations can be solved forδ=(1 − η)[(1 + φ)(1 − bw ) − α][1 − β(1 − ρ)]α,η[1 − β(1 − ρ)(1 − p)](1 + φ)(1 − α) + (1 − η)[1 − β(1 − ρ)]α(1 − α)(22)

S , and J . Steady state vacancy cost is about 7.65% of output and steady state consumption
is about 67.35% of output. [ω, ω] = [0.5952, 2.0194]wh is obtained.

20

A.2 Proofs
A.2.1 Proposition 1
For price markups, under both EB and RTM, the hiring condition (9) at the extensive margin
becomes

ϕtwhere mplt

xt
wt ht nt
=
1+
,
wt ht
yt


xt
αwt
=
1+
,
wt ht mplt



−1
xt
=
ϕ
α
1
+
.
≡ at f 0 (ht ) = αyt /(ht nt ); therefore, µEB
t
pt
wt ht

Under RTM, in

addition to the hiring condition, the wage channel (15) becomes

wt = ϕt at f 0 (ht ),= ϕt mplt .
RTM

Price markups are equal to the inverse of real marginal cost, i.e., µpt= ϕ−1t , which also

implies α(1 + xt /(wt ht )) = 1; the net hiring cost is smoothed.
For wage markups, under EB, the bargaining between rms and workers over hours per
worker yields the eciency condition (11) at the intensive margin;

mrst = ϕt mplt ,which implies that the ineciency gap is equal to the inverse of real marginal cost. Wage

−1
xt
−1
EB
EB
1 + wt ht
markups are given by µwt = wt /(ϕt mplt ) = 1/(ϕt µpt ) = α. Under RTM,
equation (11) does not hold, and the wage equation (16) becomes

wt = µRTM
wt mrst ,RTM

where µwt

εw
εw
RTM RTM= εw −1+δ
. The eciency gap is given by µptµwt = ϕ−1t εw −1+δt .
tFigure 9 graphically shows how real wages and hours per worker (w, h) are determined.
Under EB, by equation (11), hours per worker are determined at h
wage contracts.

Real wages are determined independently at w = w

18= hEB regardless ofEBby equation (13).

(wEB , hEB ) is determined at the point E1 in Figure 9. If the Hosios (1990) condition is satisedand there is a subsidy to oset the distortion stemming from monopolistic competition,
real wages and hours per worker are at the ecient level at the point∗∗

E0 in Figure 9,(w , h ). Under RTM, the wage channel yields the labor demand schedule equation (14),21whereas the bargaining over real wages between workers and rms yields the labor supply

RTM

schedule equation (19), which shifts with variable wage markups µwt. The equilibrium isthe intersection of the demand and supply at the point E2 in Figure 9, (w

RTM, hRTM ).

[Figure 9 is inserted here]

A.2.2 Proposition 2
The ecient equilibrium is computed by solving the social planner's problem, in absence of
price stickiness and monopolistic competition. The social planner chooses {ct , ht , nt , vt } soas to maximizec1−σ
Vt = t− g(ht )nt + βEt Vt+1 ,1−σsubject to

ct + gt + cv vt = at f (ht )nt + b(1 − nt ),nt = (1 − ρ)nt−1 + muξt vt1−ξ ,where ut = 1 − nt−1 + ρnt−1 and θt = vt /ut . From the FONCs, the equilibrium conditions
for the ecient equilibrium are obtained ascv= (1 − ξ) (at f (ht ) − b − g(ht )cσt )q(θt )
(
−σ 
)
ct+1
cv
+β(1 − ρ)Et
− ξcv θt+1,
ct
q(θt+1 )at f 0 (ht ) = g 0 (ht )cσt .
In the steady state,[1 − β(1 − ρ)(1 − ξp)] =(1 − ξ)q(af (h) − g(h)cσ − b) ,cv

af 0 (h) = g 0 (h)cσ .

(23)
(24)

Under EB, the usual Hosios (1990) condition applies; when ϕ = 1 and ξ = η , the allocationin the decentralized economy is ecient. Under RTM, from equations (9), (19), and (16),22the equilibrium conditions in the decentralized economy are obtained ascv= (1 − χt ) (ϕt at f (ht ) − b − g(ht )cσt )q(θt )
(
)
−σ
ct+1
cv
+β(1 − ρ)Et
(1 − χt + χt (δt /δt−1 )[1 − p(θt+1 )]),ct
q(θt+1 )ϕt at f 0 (ht ) = µwt g 0 (ht )cσt ,where χt = ηδt /(1 − η(1 − δt )) and µwt = εw /(εw + 1 − δt ). In steady state,[1 − β(1 − ρ)(1 − χp)]cv
(1 − χ)q=(ϕf (h) − b − g(h)cσ ) ,qcv

ϕf 0 (h) = µw g 0 (h)cσ ,

(25)
(26)

By comparing equations (23)-(24) and (25)-(26), when ϕ = 1 and ξ = η = χ, the equations
coincide, and the allocation in the decentralized economy is ecient. Note that the ecient
steady state is common between EB and RTM, as it satises (23)-(24) under both EB and
RTM.
As δ can be viewed as a function of η and χ = δ(η)η/(1 − η + δ(η)η) is a monotonefunction of η , there is the unique η = η∗∗satisfying η = χ(η) such that δ(η ) = 1 holds andthe steady state is ecient (Sunakawa, 2012).

19

A.2.3 Proposition 3
Available upon request from the author.

23∗
Equation (22) can be solved for such η .

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26

Notes1 In the present paper, Nash bargaining over both real wages and hours per worker is called ecientbargaining (Christoel and Linzert, 2010). Under ecient bargaining (and perfect competition), hours per
worker is determined at the ecient level such that the marginal rate of substitution is equal to the marginal
product of labor.
2 However, RTM yields a larger ineciency than EB does. Firms with some monopolistic power, instead,may oer a take-it-or-leave contract to workers that species both real wages and hours per worker (Leontief,
1946).
3 Price stability to technology shocks is one of the central results of previous studies (Woodford, 2010).
Erceg, Henderson, and Levin (2000) showed that deviation from price stability is optimal due to wage markup
uctuations in a model with sticky prices and wages and the Walrasian labor market.
4 Shimer (2005; 2010) introduced real wage rigidity as a real wage norm à la Hall (2005) to amplify labormarket volatility to bring the model to the data. Christoel and Linzert (2010) argued that real wage
rigidity under RTM, in contrast with under EB (Krause and Lubik, 2007), is important to explain ination
persistence.
5 Benigno and Woodford (2012) showed that, if the correct LQ approximation is used, the Lagrange and
LQ methods yield exactly the same result up to rst order. They also derived the correct LQ approximation
to the Ramsey-optimal policies when the steady state is inecient.
6 This timing assumption is used in Faia (2009), and dierent from Trigari (2006) and Christoel and
Linzert (2010).
7 If prices are exible in equation (7), ψ = 0, then (1 + τ )P = [ε /(ε − 1)]ϕ̃ holds, where ϕ̃ = P ϕ is
t
p
p
t
t
t t
nominal marginal cost and εp /(εp − 1) is a steady-state price markup. When 1 + τ = εp /(εp − 1), ϕt = 1
holds, namely, there is no markup and hence distortion.
8 Trigari (2006) split rms' price setting and hiring processes into two steps. Here, the two steps are
merged into one step by following Faia (2009).
9 Galí, Gertler, and Lopez-Salido (2007) originally dened the gap as the log dierence between themarginal rate of substitution and the marginal product of labor; their gap is gapt ≡ log mrst − log mplt =−(log µpt + log µwt ), opposed to mplt /mrst in the present paper. When µpt > 1 and µwt > 1, their gap isnegative, whereas this gap (after logged) is positive.
10 All values except (α, φ) are the same as in Faia (2009); (α, φ) are missing in Faia because there is only
the extensive margin in the model. Details of the steady state calculation are found in Appendix A.1.
11 Steady state unemployment seems high, but this value may include discouraged and occasionally participating workers. Andolfatto (1996) used n = 0.57 (before the match), which implies u = 0.43 by including a
non-labor force population.
12 The impulse response in Figure 1 shows an initial rise of ination, although it is admittedly small. Also,
Figure 5 shows that the optimal volatility of ination under RTM is positive.
13 This result is in line with Shimer (2005), who argued that real wage rigidity has an amplifying eect onlabor market variables.
14 For displayed values of η and b in Figure 5, the actual wage bill w h lies in the bargaining set [ω , ω̄ ].
w
t t
t
t
15 Note that, even when the Hosios condition is met at η = ξ under EB, the steady state is not ecientbecause of the distortion stemming from monopolistic competition. See also Figure 3 in Faia (2009).
16 It is analytically shown that as η → 0, the marginal gain for workers by incremental real wages δ =t−(∂St /∂wt )/(∂Jt /∂wt ) increases, and wage markups µwt are more elastic to changes in δt .
27

17 With both intensive and extensive margins, there is an upper bound of b

1w ≤ b̄w = 1 − µw (1+φ) ∈ (0, 1)

with which S ≥ 0 holds; i.e., workers want to continue the matching with rms. The upper bound isb̄w = 0.6912, a much lower value than the one used in HM, with the calibration in the present paper. As
φ → ∞, the model replicates the case with extensive margin only (Faia, 2009), and b̄w approaches one.
18 All variables are logged. A similar gure is found in Galí, Gertler, and Lopez-Salido (2007).
19 δ(η) has the following property: If (1 + φ)(1 − b ) > 1, δ(0) = [(1 + φ)(1 − b ) − α]/(1 − α) > 1, δ(1) = 0,ww
and δ 0 (η) < 0 hold.

28

Table 1: Parameter values.
Parameter

Value

Description

β
σ
εp
ψ
ξ
ρ
η
bw
α
φ
1−n
q
gy
ρa
σa

0.99

Discount factor

2

Risk aversion

6

Elasticity of demand20

Price adjustment cost

0.4

Elasticity of matches to unemp.

0.1

Job separation rate

0.6

Workers' actual bargaining power

0.6

Unemployment benets

0.9

Elasticity of production to hours per worker

2

Inverse of Frisch elasticity

0.4

Steady state unemployment after matching

0.7

Steady state job lling rate for rms

0.25

Steady state govt. expenditure

0.95

Auto corr. of technology

0.008

Std. dev. of technology

29

Table 2: Steady state values.
Variable

n
v
θ
m̄
p
w
b
c/y
cv v/yω/(wh)
ω/(wh)
κ
δ
µw
χ

Value

Description

RTM

EB

0.60000.6000

Employment after matching

0.08570.0857

Vacancies

0.18630.1863

Labor market tightness

0.35750.3575

Scaling parameter in matching function

0.13040.1304

Job nding rate for workers

0.75000.8095

Real wages

0.45000.4857

Unemployment benets

0.67350.7282

Consumption-output ratio

0.07650.0218

Vacancy cost-output ratio

0.59520.5952

Reservation wage bill for workers

2.0194

1.2699Reservation wage bill for rms

4.2546

3.9293

Scaling parameter in labor disutility

0.2648-Marginal gain for workers

1.0794-

Wage markup

0.2843-

Workers' eective bargaining power

Notes: Steady state values are computed by the model's steady state conditions, which are shownin Appendix A.1.

30

Figure 1: Impulse responses to a positive technology shock without real wage rigidity.

Real wages

Labor market tightness

1

0
0

10
20
Employment

0
0

30

1

1

0.5

0

0
0

10
20
Price markup

−1
0

30

0.2

0.2

0

0

−0.2
0

10
20
Hours worked

−0.2
0

30

0.05

0

0
10

20

10
20
Hours per worker

30

10
20
Wage markup

30

10

20

30

20

30

Inflation

1

−1
0

RTM
EB

5

0.5

30

−0.05
0

31

10

Figure 2: Composition of the ineciency gap.

EB, γ =0EB, γ =0.6

w

w

0.2

1

0
−0.2
0

Gap
Price markup
Wage markup

0

10

20

−1
0

30

RTM, γ =0

10

20

30

RTM, γ =0.9

w

w

0.2

1
0

0

−1
−2

−0.2
0

10

20

−3
0

30

Warlasian, γw=0

10

20

30

Warlasian, γw=0.9

0.2

1
0

0

−1
−2

−0.2
0

10

20

−3
0

30

32

10

20

30

Figure 3: Impulse responses to a positive technology shock with real wage rigidity.

Real wages

Labor market tightness

1

0
0

10
20
Employment

0
0

30

1

1

0.5

0

0
0

10
20
Price markup

−1
0

30

1

2

0

0

−1
0

10
20
Hours worked

−2
0

30

0.2

0

0
10

20

10
20
Hours per worker

30

10
20
Wage markup

30

10

20

30

20

30

Inflation

1

−1
0

RTM
EB

5

0.5

30

−0.2
0

10

Notes: The degree of real wage rigidity is γw = 0.6 for EB and γw = 0.9 for RTM.

33

Figure 4: Impulse responses to a positive government expenditure shock without real wage
rigidity.

Consumption

Employment

0

0.1

−0.1

0

−0.2
0

10
20
Hours per worker

−0.1
0

30

0.01

0.05

0
10
20
Price markup

30

−0.01
0

0.05

0.05

0

0

−0.05
0

10
20
Real wages

30

−0.05
0

0

0.5

−0.02

0

−0.04
0

10

20

10

20

30

10
20
Wage markup

30

10
20
Labor market tightness

30

Inflation

0.1

0
0

RTM
EB
4
EB (φ=10 )

−0.5
0

30

10

20

30

Notes: the government spending shock follows ln(gt /g) = ρg ln(gt−1 /g) + εgt , where εgt ∼ N (0, σg2 ),ρg = 0.9, and σg = 0.0074, following to Faia (2009).

34

Figure 5: Optimal volatility of ination.

0.12

0.12

σ (π)

RTM
EB
0.1

0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0.2

0.4

η

0.6

0

0.8

35

0.2

0.4
bw

0.6

Figure 6: Impulse responses to a positive technology shock without real wage rigidity (withthe ecient steady state).

Real wages

Labor market tightness

1

5
RTM
EB

0.5
0
0

10
20
Employment

0
0

30

2

0

1

−0.2

0
0

10
20
Price markup

−0.4
0

30

0.5

0.5

0

0

−0.5
0

10
20
Hours worked

−0.5
0

30

0.02

0.5

0
10

20

30

10
20
Wage markup

30

10

20

30

20

30

Inflation

1

0
0

10
20
Hours per worker

30

−0.02
0

36

10

Figure 7: Composition of the ineciency gap (with the ecient steady state).

EB, γ =0EB, γ =0.6

w

w

1

1

0
−1
0

Gap
Price markup
Wage markup

0

10

20

−1
0

30

RTM, γ =0

10

20

30

RTM, γ =0.9

w

w

1

2
0

0
−2
−1
0

10

20

−4
0

30

Warlasian, γw=0

10

20

30

Warlasian, γw=0.9

0.2

2
0

0
−2
−0.2
0

10

20

−4
0

30

37

10

20

30

Figure 8: Optimal volatility of ination (with the ecient steady state).

0.08
RTM
EB
0.07

0.06

σ (π)

0.05

0.04

0.03

0.02

0.01

0

0.1

0.2

0.3

0.4

0.5
η

0.6

0.7

0.8

0.9

Notes: the vertical line shows ξ = η ∗ = 0.225 at which the steady state is ecient under both EB
and RTM. Dark lines are the ones computed by the Lagrange method, whereas light lines are theones computed by the LQ method.

38

Figure 9: Labor market allocation.

Notes: EB denotes allocation in ecient bargaining. RTM denotes allocation in right-to-manage
bargaining. * denotes allocation at the social optimum.

39