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Working Paper Series

Inflation Dynamics with Search
Frictions: A Structural Econometric
Analysis

WP 08-01

Michael U. Krause
Deutsche Bundesbank
Thomas A. Lubik
Federal Reserve Bank of Richmond
David Lopez-Salido
Federal Reserve Board

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Inflation Dynamics with Search Frictions:
A Structural Econometric Analysis∗
Michael U. Krause
Deutsche Bundesbank

David Lopez-Salido
Federal Reserve Board

Thomas A. Lubik†
Federal Reserve Bank of Richmond
March 2008
Working Paper No. 08-01

Abstract
The New Keynesian Phillips curve explains inflation dynamics as being driven by
current and expected future real marginal costs. In competitive labor markets, the
labor share can serve as a proxy for the latter. In this paper, we study the role of
real marginal cost components implied by search frictions in the labor market. We
construct a measure of real marginal costs by using newly available labor market data
on worker finding rates. Over the business cycle, the measure is highly correlated with
the labor share. Estimates of the Phillips curve using GMM reveal that the marginal
cost measure remains significant, and that inflation dynamics are mainly driven by the
forward-looking component. Bayesian estimation of the full New Keynesian model with
search frictions helps us disentangle which shocks are driving the economy to generate
the observed unit labor cost dynamics. We find that mark-up shocks are the dominant
force in labor market fluctuations.
JEL Classification: E24; E32; J64.
Key Words: Phillips curve; Bayesian estimation; marginal costs; labor market frictions.

∗
This paper was prepared for the Carnegie-Rochester Conference on Public Policy, November 9-10, 2007.
We are grateful for comments by our discussant Miguel Casares, as well as Olivier Blanchard, Bill English,
Marvin Goodfriend, Ben McCallum, Per Krusell, Julio Rotemberg, Carlos Thomas, Carl Walsh, and the
participants at the Carnegie-Rochester Conference. We also would like to thank seminar audiences at
the Federal Reserve Bank of Dallas, the University of Montreal, and the Hungarian National Bank. The
views expressed in this paper are those of the authors and should not be interpreted as those of the Deutsche
Bundesbank, the Federal Reserve Bank of Richmond, the Board of Governors, or the Federal Reserve System.
†
Corresponding author: Research Department, P.O. Box 27622, Richmond, VA 23261. Email:
thomas.lubik@rich.frb.org.

Introduction

This paper studies the determinants of real marginal cost fluctuations when there are search
frictions in the labor market. Without such frictions, or any other type of labor adjustment
costs, real marginal costs are identical to unit labor costs. Search frictions are a particular
form of labor adjustment costs that are determined by aggregate labor market conditions,
rather than being internal to the firm. They therefore give rise to long-term employment
relationships since both firm and worker save future search costs by continuing their match.
This dual role of search frictions motivates our interest in how they alter the nature of real
marginal costs, which in turn are the key determinants of inflation dynamics in business
cycle models with monopolistic price setting and price rigidities.
We illustrate the linkages between inflation and real marginal costs in a New Keynesian
model with search and matching frictions in the labor market.1 We first use the model
to derive a (linearized) equation for real marginal costs. Our strategy is then to generate
a synthetic time series for real marginal costs by calibrating the additional parameters in
the equation. We use newly available labor market data for 1960 to 2005 on job finding
rates, vacancies, unemployment and wages to construct the series. This, in turn, forms
the basis of a limited-information (GMM) estimation of the hybrid New Keynesian Phillips
curve (NKPC) from the model. In a third step, we estimate the full general equilibrium
model, using the same variables, to further understand the interaction between labor market
variables and inflation, and the driving forces of their joint dynamics.
We find that search frictions do indeed matter for inflation dynamics, in that they tend
to reduce the role of backward-looking price setting for generating persistence, and that they
change the sensitivity of inflation to real marginal costs. At the same time, the synthetic
measure of real marginal costs is fairly closely related to unit labor costs. We also find
that, among the variables that matter for real marginal costs, the real wage has become
more volatile since the 1980s, even though consumption is less volatile. Furthermore, real
marginal costs have become procyclical from the 1980s, while they are countercyclical for
the whole sample.
This paper is among the first to estimate an aggregate labor market search and matching
model in a full-information setting.2 The estimation allows us to disentangle the determinants of the joint fluctuations of labor market variables and inflation. Our findings confirm
1

We follow a literature that has adopted the labor market model by Pissarides (2000) into dynamic
general equilibrium frameworks, such as Merz (1995), Andolfatto (1996), and Den Haan et al. (2000) in real
models, and Walsh (2005), Trigari (2006), and Krause and Lubik (2007a) for monetary models.
2
Other recent contributions are Christoﬀel et al. (2006), and Gertler et al. (2007).

those from the calibration-based analysis in that search and matching frictions do not dramatically alter inflation dynamics. However, this conclusion hides three aspects of marginal
cost dynamics that are not apparent from a limited-information perspective.
First, the main driving force of labor market variables are mark-up shocks, which substantiates the argument of Rotemberg (2006). Mark-up shocks generate volatile vacancies
and unemployment since they do not lead to wage increases that reduce firms’ incentive
to post vacancies. Second, we find that unit labor costs and real marginal costs can move
together positively or negatively depending on the underlying shock. Whether labor market
frictions are helpful in capturing inflation dynamics therefore depends on the incidence of
specific shocks. We argue that over our sample period mark-up shocks played a smaller role
overall, hence the measured conclusion regarding the importance of the labor market for
inflation dynamics. Finally, we emphasize the importance of a fully structural analysis in
addressing these questions. We extract an implied marginal cost series from the estimation
that diﬀers significantly from the calibrated series. The smoothing algorithm decomposes
the ‘residual’ in the NKPC into an endogenous variable component and an exogenous shock
component. In other words, the persistence and volatility of inflation stems from sources
besides those already captured by the imputed marginal cost series.
The model we employ is standard in most of its components. We deviate from the search
and matching model in that we assume that hiring of workers is instantaneous rather than
with a lag as in most models. In this respect we unify the specifications of Rotemberg
(2006) and Blanchard and Galí (2007). The former author assumes large firms with costs of
job creation that are concave in the number of vacancies posted,3 while the latter authors
make this timing assumption to allow a representation of the New Keynesian Phillips curve
in terms of inflation and unemployment, rather than the output gap. The virtue of this
specification is that real marginal costs can be expressed in terms of observable labor market
variables. In contrast, the standard model implies a real marginal cost expression in terms of
unobservable shadow values of employment. However, we note that the timing assumption
as such does not deliver substantially diﬀerent dynamics.
The paper proceeds as follows. The next section describes the full New Keynesian
DSGE model. In section 3, we derive an equation for real marginal costs from the model’s
job creation condition that explicitly shows the role of the labor market variables implied by
search frictions and discuss the construction of a real marginal cost series using calibrated
parameters and labor market data. Section 4 conducts the GMM estimation of the NKPC
3

The standard model features constant-returns-to-scale job creation costs.

under labor market frictions. In section 5, we take a general equilibrium perspective and
estimate the full model using Bayesian methods. Section 6 concludes, while an Appendix
provides key derivations.

A New Keynesian model with search frictions

Consider an economy that consists of households, firms, a government, and a central bank.
Households choose consumption over time and the allocation of consumption across diﬀerentiated products. They supply labor at both the intensive and extensive margins: workers
search in order to find employment, and when employed, they supply hours and earn wages
determined in bilateral Nash bargaining. Firms simultaneously choose hiring and prices
subject to hiring and price adjustment costs. They hire workers in a frictional labor market
and separate from them at an exogenous rate, and choose the price of their diﬀerentiated
product in a monopolistically competitive product market. Employment is the outcome
of firms’ and workers’ search behavior, while wages and hours worked are the outcome of
bargaining. Wages are fully flexible. The government issues a one period bond and levies a
lump-sum tax. The central bank sets the nominal interest rate in response to inflation and
output.

2.1

Households

Households are distributed along the unit interval and consist of a continuum of workers of
measure one. The welfare of household i is given by
"
#
∞
X
h1+μ
(cit − ςct−1 )1−σ − 1
t
it
− χt nit
Et
β ζt
W(nit ) = max
,
1
−
σ
1
+
μ
{cit }∞
t=0
t=0

(1)

where β ∈ (0, 1) is the discount factor, cit consumption, ct−1 aggregate consumption, where

ςct−1 is an index of external habits, and nit the number of employed workers. The welfare
function already assumes that all members of the household consume the same amount of

goods, and supply the same amount of hours hit when employed. The parameter σ governs
risk aversion, and μ is the inverse Frisch elasticity of labor supply. The labor market will
be considered in more detail later on. Finally, we allow household welfare to be aﬀected by
an intertemporal preference shock ζ t and an intratemporal labor supply shock χt .
All households consume the familiar constant-elasticity-of-substitution bundles of differentiated goods
cit =

µZ

( t −1)/

cit (j)

t /( t −1)

(2)

with

the stochastic elasticity of substitution between goods. The associated minimum-

expenditure price index is
pt =

µZ

1−

pt (j)

¶1/(1−

(3)

where pt (j) are the prices charged by each monopolistic competitor producing variety j.
The households’ flow budget constraint is
cit +

Bit−1
Bit
= Rt−1
+ wt nit hit + (1 − nit )b + dit + Tit ,
pt
pt

(4)

where households enter period t with bonds Bit−1 , that pay a gross interest rate Rt−1 . At
the beginning of the period, they receive lump-sum nominal transfers Tit , labor income
wt nit hit , where wt denotes the real wage, and a nominal dividend dit from firms, which are
owned by the households. The term (1 − nit )b denotes income of unemployed household
members, which can be interpreted as total output of a home production sector where

the technology parameter b > 0. Alternatively, this parameter captures the access to
unemployment benefits by the non-working members of the household.
The first-order conditions with respect to bonds and consumption imply
λit = ζ t (cit − ςct−1 )−σ ,
∙
¸
λit+1 pt
,
1 = βEt Rt
λit pt+1

(5)
(6)

where λit is the marginal utility of consumption. In general equilibrium, this condition
governs the stochastic discount factor used in the firms’ problem. Due to perfect risk
sharing, the sole problem of the household is to determine the consumption path of its
members. There is no explicit household labor supply choice as it is chosen at the firm level
during negotiations.

2.2

The labor market

The labor market is subject to search and matching frictions. In order to form new employment relationships, workers must search, and firms must post vacancies. We assume that
the number of matches Mt depends on the aggregate matching function
Mt = mt uξt vt1−ξ ,

(7)

which gives the number of new employment relationships available at the beginning of
period t. ut represents the number of searching workers, vt is the total number of vacancies
5

posted; mt is stochastic match eﬃciency, and 0 < ξ < 1 is the elasticity of the matching
function with respect to unemployment.
The evolution of aggregate employment is given by
nt = (1 − ρ)nt−1 + Mt ,

(8)

where ρ is an exogenous rate of job destruction, which takes place at the end of a period.
Note that we assume hiring to be contemporaneous, that is, at the beginning of period
t, firms observe the realization of the stochastic variables and post vacancies accordingly.
These vacancies are matched with the pool of searching workers which are given by the
workers not employed at the end of period t − 1, so that ut ≡ 1 − (1 − ρ)nt−1 .4

The matching function is homogeneous of degree one, increasing in each of its arguments,

concave, and continuously diﬀerentiable. Homogeneity implies that a vacancy gets filled
with probability q (θt ) ≡

mt uξt vt1−ξ
vt

= mt θ−ξ
t , which is decreasing in the degree of labor

market tightness θt ≡ vt /ut . Similarly, an unemployed worker finds a job with probability

p(θt ) ≡

mt uξt vt1−ξ
ut

= θt q(θt ), which is increasing in θt . Note that θt is taken as given by both

firms and households.

2.3

Firms

We assume that there is a continuum of firms of measure one. Each firm is a monopolistic
competitor and produces a diﬀerentiated good sold to households. Let pjt and yjt denote
nominal price and output for firm j, and yt aggregate income. A firm’s output is sold in
a monopolistically competitive market with demand, derived from consumer preferences,
given by
yjt =
where yt =

³R
1
0

( −1)/

yjtt

t /( t −1)

pjt
pt

¶−

yt ,

(9)

, consistent with the consumption bundles consumed

by households. A firm produces its diﬀerentiated good using njt workers according to the
following technology
yjt = At (njt hjt ) α ,

(10)

where At is aggregate productivity, and 0 < α < 1.
4

Equivalently, ut is the number of workers not employed in period t − 1, 1 − nt−1 plus those workers,
ρnt−1 , who lost their jobs at the end of the period. This timing convention is exactly analogous to that of
Rotemberg (2006) and Blanchard and Galí (2007), but in the notation familiar from the search and matching
literature.

During period t, a firm sets its nominal price pjt subject to the requirement that demand
be satisfied. Following Rotemberg (1982), the firm faces a quadratic cost of adjusting its
nominal price between periods, measured in terms of aggregate output and given by
µ
¶2
1 pjt
ψ
− 1 yt ,
(11)
Pjt =
2 π
et−1 pjt−1

with ψ > 0 controlling the importance of price adjustment costs, and π
et−1 = π γt−1 π 1−γ .

Inflation is defined as gross inflation π t = pt /pt−1 . The parameter 0 ≤ γ ≤ 1 governs the

degree of backward-looking price setting; and finally, π represents steady-state inflation,
which is equal to the central banks inflation target (Ireland, 2007).
This cost function penalizes deviations of the firms price change from an average between

past aggregate inflation π t−1 and steady-state inflation π. When γ = 0, price setting is
purely forward-looking, in the sense that it is costless for firms to increase their prices in
line with steady-state inflation. When γ = 1, price setting is purely backward-looking,
in the sense that it is costless for firms to increase their prices in line with the previous
period’s actual rate of inflation. This formulation yields a Phillips curve analogous to the
one derived from Calvo-price setting with backward-looking firms, as in Galí and Gertler
(1999), or with backward indexation, as in Christiano et al. (2005).
The evolution of employment at the firm level corresponds to that of aggregate employment. We assume that the new matches at firm j at the beginning of period t are proportional to the ratio of its vacancies to total vacancies posted, vjt /vt , so that vjt Mt /vt =
vjt q (θ) is hiring by firm i. Evolution of employment at firm j can then be written as
njt = (1 − ρ)njt−1 + vjt q(θt ).

(12)

For its posted vacancies vjt , the firm has to pay a flow labor adjustment cost Njt = c(vjt ).

Allowing for c00 6= 0 follows Rotemberg (2006) and departs from the standard search and

matching model where costs of recruiting are assumed to be linear (Pissarides, 2000).5 As

emphasized by Rotemberg (2006), if this is interpreted as the cost of advertising openings
in an information source it can easily be subject to economies of scale at the firm level, so
that c00 < 0.
Firms produce diﬀerentiated goods in a monopolistically competitive product market
and they maximize the present value of discounted flow profits
"µ ¶
#
∞
X
pjt 1− t
t
β λt
yt − wjt njt hjt − Njt − Pjt ,
Jt (njt ) = E0
pt
t=0

(13)

5
Note that in models in which firms consist of only one worker, the assumption of returns to scale in
vacancy posting would be immaterial.

with respect to pjt , njt , and vjt, subject to the constraint that demand (9) equals production
(10), the employment constraint (12), and labor and price adjustment costs, Njt and Pjt ,

respectively. The discount factor β t λt derives from consumer preferences in the presence of
perfect capital markets and is taken as exogenous by firms.
The first-order conditions for prices, employment, and vacancies are
¶
µ
¶
µ
π jt
π jt+1
π jt+1 yt+1
π jt
−1
= Et β t+1 ψ
−1
+
ψ
π
et−1
π
et−1
π
et
π
e t yt
µ ¶− t
µ ¶1− t
pjt
pjt
+ t mcjt
,
+(1 − t )
pt
pt
α
μjt = mcjt αAt nα−1
jt hjt − wjt hjt + (1 − ρ)Et β t+1 μjt+1 ,

c0 (vjt )
q(θt )

= μjt ,

(14)
(15)
(16)

where β t+1 = βλt+1 /λt is the stochastic discount factor and μjt is the Lagrange-multiplier
associated with the employment constraint. It represents the current-period marginal value
of workers for the firm.6 The first equation is the optimal price setting condition, which
in its linearized form reduces to the familiar New Keynesian Phillips curve. The multiplier
mcjt on the constraint that demand equals production is the contribution of an additional
unit of output to revenue and is equal to the firm’s real marginal cost. Combining the latter
two conditions yields the job creation condition
c0 (vjt+1 )
c0 (vjt )
α
= mcjt αAt nα−1
.
h
−
w
h
+
(1
−
ρ)E
β
jt jt
t t+1
jt
jt
q(θt )
q(θt+1 )

(17)

Intuitively, firms expand employment up to the point where the benefit from employing
an additional worker (the right-hand side) is equal to the cost of posting a vacancy (the
left-hand side). In symmetric equilibrium, vjt = vt , for all t, and the indices j disappear.
A few remarks concerning the job creation condition are in order. With symmetry in
vacancy posting, we implicitly assume that all firms have equal employment levels and that
all workers work the same amount of hours. Given the assumed homogeneity of employed
workers and of firms, this turns out to be true in equilibrium.
A second point concerns the implied dynamics of vacancies. As a firm increases vacancies, it immediately increases employment. If the derivative of the vacancy cost function
is suﬃciently small (that is, negative enough), raising vacancies reduces the marginal cost
of a vacancy. At the same time, rising employment lowers the marginal product of labor.
Thus, in partial equilibrium, marginal posting costs must not fall too quickly for incentives
6

This is not to be confused with the Frisch labor supply elasticity μ, which is not subscripted.

to post vacancies to be exhausted at some point. In general equilibrium, however, this eﬀect
may be mitigated if q(θt ) increases fast enough.
The third point to note is that firms take wages as given when choosing employment
(and vacancies). Strictly speaking, large firms’ employment adjustment should take into
account that employment potentially aﬀects wages if they depend on the marginal product
of labor. This will indeed be the case under Nash bargaining. In fact, Rotemberg (2006)
takes this ‘intra-firm bargaining’ eﬀect into account. Here, we deviate for two reasons. One
is merely computational convenience. The other is that intra-firm bargaining is not likely
to significantly aﬀect business cycle dynamics, as shown in Krause and Lubik (2007b).

2.4

Wage determination

In general equilibrium, job creation incentives are aﬀected by wages, and wages in turn
depend on labor market conditions. We assume, as in most of the labor search literature,
that worker and firm bargain at the individual level over the joint surplus of their match,
split according to the Nash bargaining solution. Bargaining takes place both over hours per
worker and the wage, in order to maximize the Nash product
µ
¶ µ
¶
1 ∂Wt (nt ) η ∂Jt (nt ) 1−η
,
λt ∂nt
∂nt

(18)

where the two terms are, respectively, the marginal contribution of a worker to household’s
welfare, and to the present value of profits of the firm.7 The parameter 0 < η < 1 reflects
the bargaining power of the worker. The two resulting optimality conditions are a wage
equation
£
¤
hαt + (1 − ρ)Et β t+1 θt+1 c0 (vt+1 )
wt ht = η mct αAt nα−1
t
"
#
h1+μ
1
t
+ (1 − η) b + χt ζ t
,
λt
1+μ

(19)

and a labor supply equation
ht =

mct α2 At nα−1
t
λt
χt ζ t

1
¶ 1+μ−α

(20)

The first equation is familiar from the equilibrium unemployment literature (e.g. Pissarides,
2000). It expresses the total wage payment to the worker as a weighted average between the
marginal revenue product of the worker plus the cost of replacing the worker, and the outside
option of the worker plus the marginal disutility of labor, at the level of hours worked ht .
7

From now on, we ignore household and firm indices for ease of exposition.

The bargaining weight determines how close the wage is to either the marginal product or
to the outside option of the worker. A new element is the presence of expected labor market
tightness and marginal vacancy posting cost, whereas the standard setup features current
values of these variables. The intuition is that, in case negotiations break down, worker and
firm will have to look for partners in next period’s matching market. This saving of search
costs is incorporated in the wage.8
The second equation determines the amount of hours worked. It is derived from a
condition that equalizes the marginal product of hours worked with the worker’s marginal
rate of substitution between leisure and consumption
mrst =

1
ζ χ hμ = mct α2 At nα−1
hα−1
= mplt .
t
t
λt t t t

(21)

Thus, hours are chosen as in a competitive labor market. However, optimal hours are
independent of the wage. The condition also highlights the driving forces of hours variation
in the search model. Higher marginal utility of wealth λ, and higher labor productivity
increase hours supply, while it declines whenever the disutility of labor or the intertemporal
preference increase.9

2.5

Closing the model

The model is closed by specifying monetary and fiscal policies. The government’s budget
constraint is
Bt−1
Bt
=
+ Tt ,
(22)
pt
pt
where Tt is the sum of transfers, and Bt is the aggregate of bonds held by the public. The
Rt−1

central bank is assumed to set the nominal interest rate Rt according to the Taylor-type
interest rate rule:
Rt
=
R∗

Rt−1
R∗

¶ρr ∙³

π t ´rπ
π∗

yt
y∗

¶rY ¸1−ρr

εR
t ,

(23)

where 0 < ρr < 1 captures interest rate smoothing, and where rπ ≥ 0 and rY ≥ 0. An

asterisk denotes the steady state value of the corresponding variable.

We assume a symmetric equilibrium throughout, which entails identical choices for all
variables. Defining aggregates as the averages of firm specific variables, we have that nt =
R1
R1
njt = 0 njt dj, and vt = vjt = 0 vjt dj. Furthermore, as pjt = pt , yjt = yt , for all t
and j. Thus, all firms produce the same amounts of output, employ equal amounts of
8

The details of the derivation are given in the Appendix.
In models where firms choose the amount of hours (such as in the right-to-manage setup of Trigari, 2006,
and Christoﬀel and Linzert, 2006), hours are determined to equate their marginal product of labor with the
bargained wage.
9

labor, and, in particular, face the same marginal costs mct . Similarly, for all households
R1
Tt = Tit = 0 Tit di. Finally, using the household budget constraint, firms profits, and the
government budget constraint, the resulting aggregate income identity is yt = ct + c(vt ).
Output is used for consumption and for posting vacancies.10

The cyclical behavior of marginal costs

In this first part of our empirical analysis, we use the model’s equilibrium condition on
the posting of vacancies to derive an expression for real marginal costs in the presence
of search frictions. We construct a synthetic measure of real marginal costs, based on
data on the labor share and other labor market variables associated with search frictions.
With this measure at hand, we then proceed to estimate the New Keynesian Phillips curve
using limited-information techniques. The empirical analysis is complemented by Bayesian
estimation of the full model in section 5.

3.1

Real marginal costs and search frictions

We show that a first-order approximation to real marginal cost can be decomposed into real
unit labor costs — as in the standard model without search — and terms that arise in the
presence of search frictions. We use the job creation condition (17)
c0 (vt+1 )
c0 (vt )
yt
= mct α − Wt + (1 − ρ)Et β t+1
,
q(θt )
nt
q(θt+1 )

(24)

where Wt = wt ht denotes the wage per worker, and yt /nt = At nα−1
hαt . Note that, by
t
symmetry, the condition is the same for all jobs.
The key equation for understanding the role of search frictions in the labor market is
obtained by rewriting the equation above
Wt
+
mct =
α(yt /nt )

c0 (vt )
q(θt )

(vt+1 )
− (1 − ρ)Et β t+1 cq(θ
t+1 )

α(yt /nt )

(25)

In the presence of search frictions, a firm’s real marginal cost has two components: unit
labor costs (the labor cost over the marginal product of labor), and a correction for the
current unit hiring cost relative to expected hiring costs next period. This latter term
can be interpreted either as cost savings from not needing to hire the worker again next
period. Equivalently, it equals the future marginal benefit of that worker, which needs to
be subtracted from current costs.
10
See the Appendix for a summary of the model’s equilibrium conditions and constraints, its steady state,
as well as specification of the stochastic variables.

In more compact form, the previous expression can be written as
mct = st (1 + xt ),
where st =

Wt nt
αyt

(26)

is real unit labor costs, which equals the labor income share divided by

the elasticity of output to employment. The term
¸
∙
c0 (vt+1 )
1 c0 (vt )
− (1 − ρ)Et β t+1
,
xt =
Wt q(θt )
q(θt+1 )

(27)

captures the eﬀects of labor adjustment costs relative to the real wage. In the absence
of labor market frictions, mct = st . This is familiar from New Keynesian models with
competitive labor markets: real marginal costs are proportional to the labor share, St = αst .
In the steady state, Eq. (14) implies that real marginal cost is a constant that solely
depends on the demand elasticity . That is, mc = (1 − 1/ ), which in turn is the inverse

of the steady-state markup. This is the standard implication of monopolistic competition.
In addition, it follows that
1
(1 − )M P L = W (1 + x),

(28)

where M P L = α ny is the marginal product of labor. This equation shows that the benefit
of hiring an additional employee — the marginal revenue product of labor — equals the
marginal cost of adjusting labor that include the hiring decisions. Thus, Eq. (28) is our
representation of expression (19) in Rotemberg (2006). In steady state, it follows that
0

(v)
[1 − (1 − ρ)β]. On average, real marginal revenue covers the wage plus
W (1 + x) = W + cq(θ)

the annuity costs of hiring per period.

As mentioned above, Rotemberg (2006) uses the large firm assumption for the purpose
of motivating increasing returns to vacancy posting at the firm level. To see the eﬀects of
this assumption, we assume that vacancy posting costs are specified as
c(vjt ) = cv vjtc ,
where

(29)

= 1 corresponds to the linear case discussed by Pissarides (2000). Together with

the functional form of the matching function, the correction factor xt then becomes
i
1 h c cv ξ c −1
c cv ξ
c −1
θ vt
θ v
.
xt =
− (1 − ρ)Et β t+1
Wt m t
m t+1 t+1

(30)

Noting that total hirings in this model are ht = θt q(θt ) = mt θ1−ξ
. This implies θξt =
t
−ξ/(1−ξ) ξ/(1−ξ)
ht
,

so that
1 h ξ/(1−ξ)
xt =
vt
Bht
Wt

c −1

ξ/(1−ξ)

− (1 − ρ)Et β t+1 Bht
12

i
c −1
vt+1
,

(31)

−1/(1−ξ) .
c cv m

where B =
mc
c t = sbt
+

Linearizing mct = st (1 + xt ) then yields

∙

ξ b
1−φ
e tb
(ht − βE
ht+1 ) + (
e
1
−
ξ
1−β

where φ =

s
mc

1
1+x ,

e = β(1 − ρ).
and β

(32)

b
e t vbt+1 ) − βE
e tβ
e bt ,
− 1)(b
vt − βE
t+1 − (1 − β)w

In a Walrasian labor market, mc = s, so that φ = 1 and hence mc
c t = sbt . This

corresponds to the baseline specification in Rotemberg and Woodford (1999) and Galí and
bt+1 =
Gertler (1999). Marginal costs are aﬀected by the stochastic discount factor, Et β

bt . The variable ht has two alternative interpretations. First, from the workers’
bt+1 − λ
Et λ
perspective, it represents the probability of being hired in period t, that is, the job-finding

rate. Second, it is an index of labor market tightness since it is proportional to the ratio of
aggregate vacancies to the unemployment rate.

Our general marginal cost equation nests two cases. For linear utility, β t is constant, so
that the discount factor is irrelevant for first-order dynamics. This corresponds to the model
by Rotemberg (2006). On the other hand, when

= 1, we obtain the equation implied by

the model of Blanchard and Galí (2007), who include a stochastic discount factor.11

3.2

Calibration

We now study the properties of real marginal costs based on the derivations above. We
calibrate the parameters of the model and use data on labor market variables to generate the
implied marginal cost series. We then describe the statistical properties of this series, and,
in particular, contrast it with proxies that have typically been used in empirical studies.
Each period is assumed to correspond to a quarter. With regard to preference parameters, the benchmark value of the relative risk aversion parameter σ is set equal to 1 (as
in Blanchard and Galí, 2007), although we also consider the case of linear preferences as
1

in Shimer (2005a) and Rotemberg (2006). We set the discount factor β = 1.03− 4 which
implies a 3 percent annual real interest rate. We keep the steady state labor income share
S equal to 0.64 as in Cooley and Prescott (1995).
The (quarterly) steady-state rate of exogenous and endogenous separation ρ = 0.05, a
value consistent with evidence by Yashiv (2006), slightly higher than 0.034 of Den Haan
et al. (2000), but lower than 0.086, the value used by Merz and Yashiv (2007). We set
= 11 as our benchmark value, which implies a steady state mark-up of 10 percent, and
11

The Appendix shows the corresponding derivations.

which is consistent with the evidence presented in Basu and Fernald (1997). We set the
short run elasticity of output to labor α = 0.75. Using this calibration, the steady state
value for marginal recruiting costs over the marginal product is φ = 0.95. This value is
in line with the calibration in Rotemberg (2006) and Blanchard and Galí (2007). Finally,
e = 0.943, so that the contribution of
given the calibration of β and ρ the discount factor β

the labor market variables to the fluctuations of the real marginal costs becomes small, i.e.,
1−φ
h
1−β

= 0.055.

We conduct some sensitivity analysis below that focuses on the calibration of the elas-

ticity of the matching function with respect to vacancies, 1 − ξ, and the concavity of the
hiring costs,

The elasticity of the matching function with respect to vacancies determines

how the job-finding rate responds to changes in its driving forces but it also determines the
sensitivity of marginal costs to the tightness ratio and the finding rate. Thus, the lower
1 − ξ, the higher is the sensitivity of marginal costs variations to the relevant labor market

variables. We set the elasticity of the matching function with respect to vacancies, 1 − ξ,
equal to 0.5 as our benchmark value. This value is in line with the upper bound of the

range reported by Petrongolo and Pissarides (2001) in their review of the literature on
the matching function, and it has recently been used by Blanchard and Galí (2007) and
Mortensen and Nagypal (2007). Nevertheless, we also consider the alternative value 0.3 in
line with the estimates by Shimer (2005a). Regarding the elasticity of vacancy creation

we follow Pissarides (2000) and assume that recruiting costs are linear in vacancies posted,
i.e.,

= 1. As a robustness check, we follow Rotemberg (2006) and also consider concave

recruiting costs, which implies a value of

3.3

= 0.2.

Results

Figure 1 presents a brief summary of some basic stylized facts about unemployment, vacancies and the finding rates.12 As shown in the top panel, the unemployment rate is strongly
countercyclical, and sometimes with large fluctuations. Vacancies (measured as the helpwanted index) are even more strongly procyclical, so that the vacancy-unemployment ratio
(labor market tightness) is procyclical (see the bottom panel of Fig. 1.) Finally, as evident
from the bottom panel the correlation between labor market tightness and the finding rate
is very high (in fact, 0.9). Recessions are periods when there is a substantial fall in the
probability of finding a job, and are periods where the vacancy-unemployment ratio is low
relative to its average level.
12

A complete description of the data used in the paper can be found in the Appendix.

Figure 2 depicts our imputed measure of marginal cost and real unit labor cost st in the
baseline calibration. We also plot the three main components of marginal cost associated
with the presence of search frictions, i.e. Eq. (32): the contribution of the expected
e tb
e
ht+1 ), the contribution of the
changes in the finding rates, (1 − φ) /(1 − β)ξ/(1
− ξ)(b
ht − βE

e β[E
e tλ
bt ], and
bt+1 = (1 − φ) /(1 − β)
bt+1 − λ
e βE
e tβ
stochastic discount factor, (1 − φ) /(1 − β)

the contribution of the cyclical component of the real wage, (1 − φ)w
bt .13

The top left panel compares the typical marginal cost proxy in the New Keynesian

Phillips-curve literature, real unit labor cost, with our synthetic measure. As the figure

shows, the two series are very similar. At first glance, it appears that the influence of search
and matching frictions on inflation dynamics is not very strong. The two series comove
closely, with similar turning points, and exhibit similar persistence and volatility. From the
1980s, though, the new series is somewhat less volatile and smoother. As can be seen from
Figure 3, this result remains qualitatively unchanged if we set the elasticity of the matching
function with respect to vacancies (1 − ξ) equal to 0.7, and use the tightness ratio as an

observable. The stochastic properties of real marginal costs remain unaltered.14 Overall,
this alternative calibration tends to reduce the volatility of marginal costs.
The reason for this result is illustrated in the other three panels of Fig. 2. The contribution of the stochastic discount factor and the cyclical variation of the real wage are
negligible relative to the variability of unit labor costs. There is an interesting pattern,
however. Consistent with the ‘great moderation’ period, after the mid-80s the reduction in
the variability of consumption growth reduces the contribution of the stochastic discount
factor to the variation in the marginal costs. Notwithstanding, the variation in the real
wage is somewhat higher, but its contribution is still very small.
Table 1 reports some basic statistics underlying the visual evidence in Figs. 2 and
3. In particular, we report second moments for quadratically detrended (log) output as
an indicator of the business cycle, real unit labor costs, and two measures of real marginal
costs for the model with the calibration following Rotemberg (mcR ) and Blanchard and Galí

(mcBC ), respectively. Note first that the percent standard deviation of real marginal costs
is larger than the one of detrended output and real unit labor costs.15 In the Rotemberg
calibration, where marginal costs depend explicitly on the variation in the cost of vacancies,
13
14

The trend component was obtained using the HP filter method with a smoothing parameter λ = 100, 000.
This result is not surprising given the high correlation between the finding rate and tightness (see Fig.

1).
15
Furthermore, the role of job separations as a means to smooth hiring is eliminated. See also Krause et
al. (2007) for more details on the computation of the marginal costs in models where the separation rate is
endogenous.

we accordingly obtain a reduction in the variation of the marginal costs.
In addition, the departures of marginal costs from steady state are persistent, but less
than the persistence of output and real unit labor costs. Over the first part of the sample
marginal costs are somewhat negatively correlated with detrended output, while over the
second marginal costs are more procyclical. We conclude that for the second half of the
sample period the presence of search frictions enhances the countercyclical movement in the
price mark-up by making marginal cost more procyclical (e.g., Rotemberg and Woodford,
1999).
In Figure 4 we display the robustness of real marginal costs consistent with the Rotemberg calibration. Under this specification, marginal costs inherit the eﬀects of the expected changes in vacancies. The importance of this eﬀect for marginal cost dynamics
depends on the value of the elasticity of vacancy creation

The bottom panel of Fig. 4

presents the time series of this component, viz. the contribution of vacancies corresponds
e vt − βE
e t vbt+1 ). As can be seen from Fig. 4 and Table 1, both
to (1 − φ)( c − 1)/(1 − β)(b
volatility and persistence remain similar to the specification by Blanchard and Galí, albeit
marginal costs under the Rotemberg specification are slightly less volatile (see Table 1).

To summarize, we find that adding search and matching frictions in the labor market
appears to aﬀect the cyclical behavior of marginal costs only slightly in terms of comovement, persistence and volatility. A typical proxy measure for real marginal costs, such as
unit labor costs, behaves similarly. This does not, however, allow us to conclude that these
measures have no substantial eﬀects on inflation dynamics. We investigate this issue further
along two dimensions. First, we look at the correlation between inflation and marginal cost
using a limited information approach. Second, we analyze empirically how the presence of
search frictions aﬀect inflation dynamics from a general equilibrium perspective.

Inflation and marginal costs: a limited-information approach

In this section we extend the analysis of Galí and Gertler (1999) to the case with search
frictions in the labor market. We present estimates of the New Keynesian Phillips curve
using the measure of real marginal costs constructed in the previous section. We begin by
noting that a log-linear approximation of the price setting condition (14) yields the familiar New Keynesian Phillips curve, which describes inflation as driven by lagged inflation,
expected future inflation, and real marginal cost:
ct −
π̂ t − γ π̂ t−1 = βEt [π̂ t+1 − γ π̂ t ] + κ mc
16

1
bt .
ψ

(33)

π̂ t is price inflation expressed as a log deviation from steady state, mc
c t represents real

marginal cost, and bt represents exogenous variations in the mark-up associated with changes
in the elasticity of demand. Since we allow for partial indexation to lagged inflation, current

inflation is aﬀected by inflation in the previous period. Finally, the pass-through from

marginal costs to inflation, κ, is a function of the elasticity of demand

and the price

adjustment cost parameter ψ. Given a value for the elasticity of demand, the slope coeﬃcient
κ=

−1
ψ

pins down the price adjustment cost parameter. The previous expression can be

rewritten in a more familiar form as a hybrid New Keynesian Phillips curve
bt+1 + γ b π
bt−1 +
π
bt = γ f Et π

κ
1
mc
ct −
bt ,
1 + βγ
ψ(1 + βγ)

(34)

where γ f = β/(1 + βγ), and the parameter on past inflation γ b = γ/(1 + βγ). As in
the original model of Galí and Gertler (1999), Eq. (34) corresponds to the hybrid NKPC.
When γ = 0, the model corresponds to Rotemberg’s (1982) original contribution, so that
the model reduces to the purely forward-looking NKPC.
In this paper, we deviate from the analysis in Krause et al. (2007), where we use Eq.
(34) to define the set of orthogonality conditions for all t: Et {(π t − γ b π t−1 − γ f Et {π t+1 } −
κ
1+βγ

mc
c t ) zt } = 0, which can then be used to estimate the model using generalized method

of moments (GMM). Instead, we rewrite Eq. (33) as a relationship between inflation and
the expected discounted value of the future values of real marginal cost and mark-up variations:16
π
bt = γb
π t−1 + κ

∞
X
k=0

β k Et [mc
c t+k + ϕbt+k , ] .

(35)

To estimate our model using (35), we need to forecast real marginal cost. Define mc
c t = ρmc

mc
c t−1 + ut , where 0 < ρmc < 1, and the innovation ut is an i.i.d. innovation that is uncorrelated with bt+k .17 It is straightforward to compute the forecasts as follows: Et mc
c t+k = ρkmc

mc
c t . Consequently, the equation for inflation that we estimate is:
π̂ t = γ π̂ t−1 +

κ
mc
ct +
1 − βρmc

πt .

(36)

We jointly estimate the inflation equation and the AR(1) process for marginal costs. Since
the exogenous variation in markups may be correlated with our measures of marginal costs,
we use lagged variables as instruments. Our benchmark set of instruments includes one lag
of inflation, one lag of marginal costs, and two lags of the output gap, measured as the
16
17

The methodology closely parallels the present-value approach used in the empirical finance literature.
We used the Box-Jenkins methodology to pin down the best AR model for marginal costs.

deviation of non-farm business sector output from a quadratic trend.18
In Tables 2 and 3 we present the results for the hybrid model over the period 1985:I to
2005:IV, for the specification of the marginal costs under the Blanchard-Galí and Rotembergcalibration, respectively. We distinguish the baseline calibration of marginal costs from the
alternative calibration, as specified in the previous section. Finally, we also present the
robustness of the results to alternative ways of calculating marginal costs, i.e., using information on the finding rates or labor market tightness. The first two columns report the
estimates of the indexation parameter γ, and the slope coeﬃcient κ. The next column gives
the implied backward-looking parameter γ b obtained from the value of γ and the calibrated
discount factor β. The final column shows the JT test of overidentifying restrictions and
below its corresponding p-value in parenthesis.
The degree of indexation is well estimated across all the specifications and it ranges
between 0.6 and 0.7. Hence, even if the forward-looking component is slightly more relevant,
the backward-looking component plays a significant role on inflation dynamics with a value
for the coeﬃcient γ b around 0.4. These estimates are fairly stable across specifications,
and are in line with the results of Galí et al. (2005). Finally, the slope coeﬃcient on the
marginal costs is significant but somewhat less precisely estimated, and implies that the
estimated probability of changing prices, i.e., the duration of prices being fixed, is slightly
larger than the one estimated in the literature.19

Inflation and marginal costs: a Bayesian full-information
approach

We now turn to an analysis of inflation and marginal cost dynamics in a full information
setting and estimate the full model. Our motivation is twofold. First, we are interested
in the plausibility and robustness of the calibration and limited-information analysis. The
18
Since it is possible that our instruments are only weakly correlated with the endogenous variables in
our model, we follow Stock et al. (2002) and Stock and Yogo (2002) and check for the presence of weak
instruments based on the gmin statistic of Cragg and Donald (1993). We compare this statistic against the
critical values compiled by Stock and Yogo (2002), who show how to test for the presence of weak instruments
based on this test statistic.
19
As shown by Sbordone (2002), Rotemberg’s menu cost model of price rigidity due to firms facing convex
adjustment costs is observationally equivalent to a model based on Calvo (1983), where the price rigidity is
determined by a random draw of the firms that are allowed to change prices. The slope coeﬃcient κ = −1
ψ

under the first interpretation is equal to κ = (1−βω)(1−ω)
under the second one, where ω represents the
ω
probability of changing prices. Hence, it is possible to use the slope coeﬃcient κ to pin down the probability
of chaging prices, ω. For the baseline calibration, the Blanchard-Galí model implies a value or ω = 0.85.
As recently shown by Thomas (2007), introducing strategic complementarities in a search setup allows
reinterpretation of the slope coeﬃcient in terms of a lower ω.

model is richly parameterized and includes labor market parameters for which quantitative information is diﬃcult to come by or which are controversial in the literature.20 Full
information estimates might thus give us an indication to what extent the marginal cost
series is correctly imputed. Furthermore, we are interested in the unobservable marginal
cost series that is consistent not only with the dynamics of inflation, but with full aggregate
dynamics. This allows us to ascertain not only the contribution of individual shocks, but
also to decompose the movements of inflation into endogenous components (those arising
from marginal costs and of lagged and future inflation itself) and exogenous driving forces.
It is precisely the latter aspect that is neglected in the limited information setting and that
may oﬀer clues as to the small role of the labor market in explaining inflation.
Our empirical approach is Bayesian. We log-linearize the non-linear model around a
deterministic steady state and write the linearized equilibrium conditions in a state-space
form. After solving the model, we employ the Kalman-filter to evaluate the likelihood function of the observable variables which we then combine with the prior distribution of the
model parameters to obtain the posterior distribution. We evaluate the posterior numerically by employing the random-walk Metropolis-Hastings algorithm. We report posterior
means and 90% coverage intervals as our estimation results. Further details on the computational procedure can be found in Lubik and Schorfheide (2005) or An and Schorfheide
(2006).
We proceed as follows. We first discuss selection of the priors and the data employed
in the estimation. The posterior estimates are reported next, followed by impulse response
functions and variance decompositions which we use to discuss the sources of fluctuations
in our estimated model. We then report and discuss the filtered, model-consistent marginal
cost series and contrast it with the calibrated series from above. We conclude this section
by oﬀering an interpretation of the role of labor market frictions in explaining inflation
dynamics.

5.1

Priors and data

We choose priors for the Bayesian analysis from a variety of sources. We roughly distinguish
between two groups of parameters, those associated with production and preferences, and
the labor market parameters. We choose tight priors for the former, but fairly uninformative
priors for the latter. One aspect of our analysis is a characterization of the information
content of the data with respect to these parameters. Share parameters are assigned a Beta20

Chief examples are the bargaining parameter η and the worker’s outside option b.

distribution with support on the unit interval, while Gamma-distributions are employed for
positive-valued parameters. The priors are reported in Table 4.
The discount factor β is fixed at 0.98. All other parameters are estimated. We choose
fairly wide priors for the intertemporal substitution elasticity σ and the (inverse of) the
labor supply elasticity μ with a mean of one for both parameters. Similarly, the mean
of the habit parameter ς = 0.5. The labor input elasticity α is tightly centered around
0.67, the average labor share in the U.S. economy, while the demand elasticity

is set to

a mean value of 11 which implies a steady state mark-up of 10%, a customary value in
the literature. The prior mean of the firm’s price adjustment cost ψ is set to 20, and the
indexation parameter γ to 0.5.
The priors of the matching function parameters are chosen to be consistent with two
labor market facts, the observed job finding rate of 70% per quarter (Shimer, 2005a) and
the average unemployment rate over the sample of 6.3%. This leads to a prior mean of 0.7
for the match elasticity ξ and of 0.7 for the match eﬃciency parameter m. We set the mean
separation rate at ρ = 0.1, which is the value reported in den Haan et al. (2000). The
elasticity parameter of the vacancy creation cost function

is chosen to have a mean of

one with a wide coverage region. It is centered at the baseline value in the literature, which
typically assumes constant creation costs. This allows us to evaluate the empirical evidence
provided in Yashiv (2006) against the calibration in Rotemberg (2006) who selects

¿ 1.

We choose to be agnostic about the bargaining parameter η and use a uniform prior
over the unit interval. Similarly, the value of the outside option of the worker is crucial
to the debate on whether the search and matching model is consistent with labor market
fluctuations (Hagedorn and Manovskii, 2008). We pick a mean value of b = 0.4 with a wide
standard deviation, which implies a replacement ratio of slightly more than one-half of the
aggregate wage.
Finally, we choose a prior mean of the response of the monetary authority to inflation γ π
to 1.5, and to output γ Y = 0.25. The prior mean of the interest rate smoothing parameter
ρr is 0.7. These values are commonly found in empirical Taylor rules. We consider a
Beta distribution for the autocorrelation parameters of the shocks and an Inverse Gamma
density for the standard deviation of the shocks, which are assumed to have a high degree
of persistence and identical innovation variances with a wide coverage region. The baseline
specification of the model consists of six exogenous shocks: production technology, matching
technology, monetary policy, mark-up, discount factor, and the disutility of working. This
would allow us to estimate the model on at most 6 data series. For our baseline estimation,

however, we use five series only.21
We estimate the model on five data series: output, inflation, the interest rate, unemployment and vacancies. Our data are quarterly and cover a sample period from 1984:1 to
2007:1. Although observations on all variables are available at least from 1964 onward, we
concentrate on a time period which is characterized by a single monetary policy regime, i.e.,
the tenures of Paul Volcker and Alan Greenspan as Federal Reserve Chairmen. The macroeconomic variables were extracted from the Haver database. All series, except interest and
inflation rates are passed through a Hodrick-Prescott filter with smoothing parameter 1600
and are demeaned prior to estimation. The output series is real GDP in chained 2000$,
which we scale by the labor force measured as over-16-year-old employed civilians. The
interest rate is the quarterly eﬀective federal funds rate, while inflation is the percentage
change in the GDP deflator. Unemployment is measured by the unemployment rate of over
16 year olds. The series for vacancies is the index of help-wanted advertisements in the 50
major metropolitan areas.

5.2

Parameter estimates

We report posterior means and 90% coverage intervals in Table 5. Three estimates stand
out. First, the labor supply parameter μ has a posterior mean of 5.06 with a 90% coverage
region ranging from 3.4 to 6.8. This is quite high and considerably shifted away from the
prior, which indicates that the data are informative with respect to this parameter. Since
the model is estimated without data on hours worked, identification of the hours supply
parameter might be regarded as problematic. However, the estimation procedure constructs
an implied hours series to be consistent with the comovement in the observables.
The estimated value implies that hours do not vary much over the business cycle. Workers are unwilling to substitute out of leisure, once employed, in order to incrementally work
longer. This is not to say that the overall labor supply elasticity for total hours worked is
low. It simply implies that labor input adjusts mainly along the extensive margin in line
with the findings of Cho and Cooley (1994).22 Workers do not have direct control over the
level of employment since it is governed by the matching process. However, firms can increase employment by posting more vacancies, subject to two constraints: hiring costs cv vt c
21

Estimation turned out to be unstable, on account of likely identification problems, when the model was
specified with five exogenous disturbances. Adding an additional shock helped disentangle this. Furthermore,
in our robustness exercise, we add additional information in form of an observable to investigate the stability
of the estimates.
22
It is likely that this finding is not robust to changes in the specification of the disutility of work. This
is left to future research, however.

and wage payments, both of which reduce a firm’s incentive to seek employees. Since wages
and hours are jointly chosen in the bargaining process, they represent a ‘benefits package’
that resolves the relative volatilities of the intensive and extensive margin by smoothing
hours.
The posterior estimate of η is 0.67 with a 90% coverage region that lies between 0.38
and 0.97. Recall that the prior on η is uniform. The estimation shifts probability mass
towards stronger worker bargaining power. Wages, therefore, are more closely in line with
the marginal product, which depresses vacancy creation and thus increased movement in
employment. At the same time, the elasticity

is estimated with a posterior mean of 3.35,

which makes vacancy creation very costly and compounds the eﬀect of a fairly high degree
of wage flexibility. In other words, the algorithm largely shuts down the labor market
as a source of persistence and volatility, which echoes the findings in Krause and Lubik
(2007a).23
Estimates of the other labor market parameters are much less dramatic. All of them
show substantial overlap with the priors. The benefit parameter b is estimated at almost
the prior mean of 0.4, but is more concentrated. This indicates that aggregate data, seen
through the prism of this DSGE model, would adjudicate in favor of the Shimer (2005a)
calibration and its implication that the standard search and matching model cannot replicate the dynamics of unemployment and vacancies.24 The posterior means of the matching
function parameters are in line with other values in the literature. The match elasticity ξ
of 0.68 is not far away from the prior, as is the match eﬃciency parameter m. Finally, the
estimate of the level parameter in the vacancy cost function cv simply replicates the prior,
and therefore is not identified in an econometric sense. This is to some extent an artifact
of the specific parameterization of the cost function.
The demand elasticity

and the labor share parameter α come in close to the prior.

Since the specific data set used appears to be uninformative, we fix both parameters at
their prior means of

= 11 and α = 0.67 in the rest of the paper.25 The estimate of σ as

0.92 is not implausible and reasonably tight and diﬀerent from the prior. Since identifying
23

These estimates are substantially diﬀerent from what is typically assumed in the calibration literature.
In most papers, vacancy creation costs are linear, i.e. c = 1. Rotemberg (2006) even assumes values as low
as c = 0.2. In contrast, Yashiv (2006), estimates a convex cost function ( c > 1), albeit using micro-data
and a slightly diﬀerent specification.
24
Hagedorn and Manovskii (2008) argue that values of b as high as 0.9 are more plausible, to which,
however, the posterior distribution assigns zero probability.
25
Estimating the baseline version with either or both parameters fixed shows virtually no diﬀerences in
parameter estimates. Using marginal data densities as measures of fit, we find that the preferred specification
allows for variation in . The diﬀerences in posterior odds are tiny, however, and it is well known that they
are sensitive to minor specification changes.

information on this parameter comes largely from the optimal hours choice we leave this
parameter unrestricted.
The estimates of the habit parameter and the inflation indexation parameter are somewhat surprising. Both are commonly seen as sources of intrinsic inflation persistence in
the NKPC. The habit parameter is eﬀectively zero, while the posterior mean of γ is 0.29.
Both parameter distributions are locationally diﬀerent from their respective priors and quite
concentrated. The model thus relies on extrinsic sources of persistence, which, given our
discussion of the estimates of the labor market parameters, are likely to be the exogenous
shocks. This confirms the findings in Galí and Gertler (1999) who argue for a small, but
significant degree of indexation in price-setting, and, at the same time, confirms the results
above from the calibration and limited information exercise. The model generates enough of
a propagation mechanism to not have to rely on an intrinsic source of inflation persistence.
What the Bayesian estimation shows is that this is almost exclusively coming through the
exogenous shocks, rather than the elements added by the search and matching frictions in
the labor market. This is confirmed by the estimates (not reported) of the autoregressive
coeﬃcients of the shocks which are largely clustered around 0.95.
We conduct one robustness check by adding more information to the model in the form
of an additional observable, namely the labor share. Our main concern is how this helps
with the identification of the labor market parameters. The series is the same as used before
in the limited information analysis; it is the total wage bill in the non-farm business sector
scaled by the respective output. Posterior estimates are reported in Table 6. The most
notable finding is that the labor share series adds persistence to implied marginal cost and,
therefore, inflation. This is evident from the reduction of the intrinsic persistence parameter
from 0.3 to 0.2. As an observable the labor share captures enough of the second moments
inherent in inflation to serve as a good proxy for the unobserved marginal cost (which is
a main reason why this variable has been used from Galí and Gertler, 1999, on forward).
Search and matching frictions add to this, so that only a smaller component of inflation
persistence is explained by the backward-looking term.
However, the estimates also reveal potential identification problems. The posterior for
the benefit parameter b essentially overlaps with the prior, while the vacancy cost elasticity
c

is not markedly diﬀerent from unity. The price stickiness parameter is now estimated an

order of magnitude smaller than the prior mean, which stems from the use of the labor share
as an observable. The labor supply elasticity μ is much smaller than in the baseline, but
still shifted away from the prior. This indicates that an inelastic hours supply is required to

match the joint behavior of the labor market and inflation, unless there are other sources
of persistence.

5.3

Impulse response functions and variance decomposition

We report the impulse responses of the observables to the structural shocks in Figure 5. The
eﬀects of a one-standard-deviation technology shock are standard. Output increases, and
firms respond by posting vacancies and letting current employees work longer. This stands
in contrast to the widely-discussed finding by Galí (1999) that a technology shock has a
contractionary eﬀect on hours worked.26 The diﬀerence lies in the labor market framework.
Persistent technology shocks raise the value of a job to the firm. In order to cover the
required outlays for vacancy postings, firms produce more output through higher hours.
This obtains despite a low estimated hours supply elasticity and convex vacancy costs.
The eﬀects of shocks to the demand elasticity

are virtually identical to those of tech-

nology shocks, except for scaling. This addresses the suggestion by Rotemberg (2006) that
demand shocks can help explain the volatility puzzle in search and matching models. Technology shocks raise the marginal product of labor and thereby put upward pressure on
wages, which in turn reduces the incentive of firms to post vacancies. In contrast, expansionary demand shocks, which amount to a reduction in market power, make firms want to
increase employment, but at the same time reduce marginal revenue and thus wage pressures. Consequently, incentives to post vacancies remain high. Note also that the output
response is of the same order of magnitude as under a technology shock, while the response
of unemployment and vacancies is much larger for this type of demand shock. Both shocks
generate similar endogenous propagation and appear as equally important driving forces
for labor market variables. Finally, innovations to the match eﬃciency mt have a strong
negative eﬀect on unemployment and vacancy creation, but are on balance expansionary
(although not to the same degree as technology shocks).
Impulse response analysis can only give an indication as to the most important driving
forces of the business cycle. We compute the model’s variance decomposition in order to
investigate this issue further. The results are reported in Table 7. In the estimated model,
unemployment and vacancies are exclusively driven by mark-up and matching shocks. This
is not surprising since the latter are identified from the employment equation as the innovation to the rate of transformation of old into new employment. This is reminiscent of
an investment-specific shock in the RBC literature. Similarly, the mark-up shock mainly
26

Note that total hours also increase over the entire adjustment path.

operates through the job creation condition as it aﬀects the expected value of a job. An
important implication of this finding is that search and matching models that do not include
either shock oﬀer an incomplete characterization of business cycle dynamics.
The picture for the other variables is more balanced. 67% of output variations are
explained by technology shocks, and 27% by the mark-up shock, the eﬀect identified by
Rotemberg (2006). The labor supply shock χ has virtually no eﬀect on the model’s estimated
business cycle behavior. The shock to the discount factor ζ, on the other hand, almost
fully explains the behavior of the nominal interest rate. The intertemporal preference
shock functions as the residual in the Euler-equation. Since the model implies consumption
smoothing and the anti-inflationary policy renders inflation less volatile, variations in the
nominal interest rate are picked up by the unobserved component, that is, ζ t+1 /ζ t .
This also helps explain the slightly counterintuitive behavior of inflation, which is driven
to almost 75% by the policy shock. This is due to two factors. First, the model is estimated
over a period during which monetary policy was conducted in an anti-inflationary manner,
as evidenced by the high estimated reaction coeﬃcient. Disturbances are counteracted by
the endogenous policy reaction with the result that inflation does not move much. Whatever dynamics there are, are explained by the exogenous component to policy, the policy
shock. Second, the variance decomposition indicates that independent movements in the
components of marginal cost eﬀectively counteract each other. While mark-up and matching shocks are the main driving forces of unemployment and vacancies, their contribution
to inflation is minimal, which supports the conclusion arrived at before that search and
matching frictions are seemingly negligible for marginal costs dynamics.
When we include the labor share in our set of observables, the variance decomposition
reveals some diﬀerences to the baseline (see Table 6). The labor share is explained to 66%
by the mark-up shock, which is due to its impact on the marginal revenue product. This
also aﬀects the role of mark-up shocks for vacancy fluctuations, whose main driving forces
are now matching shocks. Interestingly, the determinants of movements in unemployment
are roughly unchanged. As in the baseline version, these three shocks do not have any
influence on interest and inflation dynamics.
To summarize, evidence from the baseline estimates confirms the findings from the
calibration-based analyses above. Search and matching frictions in the labor market do not
seem to matter much for the dynamics of inflation. However, the contributions of individual
shocks are not robust to the inclusion of additional information.

5.4

Model-consistent marginal costs

We now extract the implied marginal cost series by using the Kalman-smoother at the posterior mean. This procedure is essentially similar to the approach pursued in the calibration
analysis as we attempt to describe the behavior of an unobservable by using observable variables. The crucial diﬀerence is that the likelihood-based estimation imposes the optimal
instruments and the cross-equation restrictions from the model. More importantly, however, the smoother extracts the pure marginal cost series conditional on the model’s shocks.
The imputed series is reported in Figure 6 along with the calibrated series from the baseline
specifications using the Rotemberg and Blanchard-Galí timing conventions.
Two observations are immediate. The filtered series is more persistent and volatile, and,
second, the turning points do not line up. At first glance, the two types of series appear
to be unrelated. The explanation for this seeming disconnect is that the calibrated series
include the contributions of the exogenous shocks impacting the economy, while the filtered
series is constructed such as to strip them out. In the reduced form of the full model, the
equation associated with the NKPC can be decomposed into endogenous variables, which
make up the implied marginal cost term, and the model’s shocks. The filtered series is
thus conditional on the identified shocks, whereas the calibrated series shows the average,
unconditional behavior of marginal cost over the business cycle.
The flip side of this argument is that search and matching frictions do contribute significantly to marginal cost dynamics. Their contribution, however, is obscured by the fact
that the variables in the model are buﬀeted by shocks that imply specific comovement.
Consequently, the contribution of the labor market is reduced. In that sense, the calibrated
version of the marginal cost series is truly from a partial equilibrium, reduced-form perspective, whereas the filtered series depicts the endogenous contribution of marginal costs
to aggregate dynamics inclusive of labor market frictions. This suggests that our calibration analysis, and therefore the limited information approach, is not well primed to address
this question in the absence of explicitly modeling the interactions of the endogenous and
exogenous variables. This, of course, is akin to conducting a full-information analysis.

5.5

Interpretation

Estimation of the full New Keynesian model using likelihood-based methods allows us to
draw a few conclusions regarding inflation and marginal cost dynamics. The first observation concerns the robustness of the parameter estimates. While the matching function
parameters ξ and m are tightly estimated across specifications, there is scarce information
26

in the data about, notably, the benefit parameter b. The source of intrinsic persistence is
robust over all specifications: habit formation plays virtually no role; on the other hand,
backward-looking price-setting, i.e., the lag coeﬃcient in the NKPC, is significant but small.
For other parameters, the estimates vary widely, across specifications. The labor supply
parameter μ and the bargaining share η are a case in point. Essentially, they are not identified, both in an econometric sense (which judicious use of a Bayesian prior can remedy) and
an economic sense. The use of more information, such as data on hours worked and wages
can potentially alleviate this. However, this leads us to the caveat that a calibration-based
analysis which rests on these parameter values should be approached with caution.
The second conclusion concerns the results from the limited-information analysis. Search
and matching frictions do not appear to add much explanatory power for marginal cost and
inflation dynamics. The two most important determinants in that respect are the labor share
(as a proxy for unit labor cost) and a lagged inflation term. The full-information approach,
however, illuminates subtle qualifications to this conclusion. The calibration-based approach
from above is essentially an unconditional analysis. It captures the average behavior of
inflation, marginal cost and labor market variables over the business cycle. Moreover,
the calibrated marginal cost series, which we use as input for the GMM estimation, may
violate the cross-equation restrictions that are implicit when we estimate the full model as
a data-generating process. When we impute the unobservable marginal cost series, we use
individual data series irrespective of how they were generated. This procedure therefore
conflates the responses of endogenous variables and the exogenous impact of shocks.
Third, we find that mark-up shocks are the main driving forces behind labor market
variables, but they are not as important in explaining output fluctuations. Vice versa,
technology shocks have low explanatory power for the labor market. This result is in line
with the Shimer (2005a) puzzle, which essentially demonstrates a disconnect between the
labor market side and aggregate dynamics (see also the discussion in Krause and Lubik,
2007a). We also substantiate the argument in Rotemberg (2006) who demonstrates the
importance of mark-up shocks. As already evident from the simple sample correlations in
section 3, comovement between marginal cost and inflation has changed both over time and
across the incidence of shocks. We would therefore argue that findings in the empirical
NKPC literature which identify marginal cost as a variably unimportant driving force of
inflation rest on specific historical episodes.
Finally, we also oﬀer a few perspectives on the labor market literature. Based on our
estimates, the aggregate evidence points towards a low replacement ratio. Contrary to

Hagedorn and Manovskii (2008), high values of b appear not to be a solution to the Shimerpuzzle, as the estimation puts very low probability weight on their values. We also identify
mark-up shocks and matching shocks as crucial for explaining labor market dynamics.

Conclusion

In this paper, we develop and estimate a baseline New Keynesian model with labor market
frictions. We find a relatively low impact of search frictions on the cyclical variation of
real marginal costs beyond that of real unit labor costs. From the structural estimation
of the full model we identify the sources of macroeconomic fluctuations that are needed
to explain labor market dynamics and inflation. Exogenous movements in the marginal
revenue product of labor associated with changes in desired mark-ups help explain vacancy
and unemployment fluctuations without any eﬀect on inflation.
We arrive at these results using both limited- and full-information estimation techniques.
The limited-information estimation based on GMM shows that the labor share (or unit
labor costs) and real marginal costs exhibit very similar dynamics, which appear unaﬀected
by the inclusion of labor market variables. The full-information Bayesian estimation of the
model confirms the dichotomy between labor market frictions and inflation, and furthermore
identifies the importance of mark-up shocks as a driving force of business cycles.
Whether mark-up shocks are to be regarded as structural shocks or endogenous responses
of mark-ups through some unmodeled channel is an issue for further research. Mark-ups do
vary over the business cycle, as has been observed before (see, for example, Rotemberg and
Woodford, 1999). We do not take a stand on the ultimate sources of mark-up variations.
But their ability to generate realistic labor market dynamics in the search and matching
framework casts doubt on the pessimistic assessments by Hall (2005) and Shimer (2005a),
who solely focused on real supply shocks as the driving forces of vacancy and unemployment
fluctuations. Allowing for demand shocks potentially changes the picture.

Appendix
1. Derivation of the wage equation under Nash bargaining and endogenous
hours
Firms and workers bargain each period over how the joint surplus of their match is
divided. The solution maximizes the Nash product
¶ µ
¶
µ
1 ∂Wt (nt ) η ∂Jt (nt ) 1−η
,
λt ∂nt
∂nt
the arguments of which are what each party would lose, if the match broke up. The surplus
of the workers is equal to the marginal value of the job to household i
h1+μ
1
1 ∂Wt (nt )
= wit hit − b −
ζ χ it
λit ∂nit
λit t t 1 + μ
i (n
1 ∂Wt+1
it+1 )
[(1 − ρ)(1 − θt+1 q(θt+1 ))] .
+ Et β t+1
λit+1
∂nit+1
This is the derivative of the household’s value function with respect to employment nit .
Division by λit translates the utility units of W in terms of goods. The firm’s surplus is the

marginal value of a worker:

#
"
j
∂Jt+1
(njt+1 )
∂Jtj (njt )
α−1 α
= mct At αnjt hjt − wjt hjt + Et β t+1
(1 − ρ) .
∂njt
∂njt+1

The first-order condition with respect to the wage is (dropping the indices j and i):
(1 − η)

∂Jt (nt )
1 ∂Wt (nt )
=η
.
λt ∂nt
∂nt

This equation can be interpreted as a sharing rule according to which each party obtains a
´
³
∂Jt (nt )
t (nt )
1 ∂Wt (nt )
fraction of the joint surplus. That is: λ1t ∂W∂n
and accordingly
=
η
+
∂nt
λt
∂nt
t
for the firm. To find the wage, insert the value functions into the sharing rule
Ã
∙
¸!
h1+μ
1
1 ∂Wt+1 (nt+1 )
t
(1 − η) wt ht − b − ζ t χt
+ Et β t+1 (1 − ρ)(1 − θt+1 q(θt+1 ))
λt
1+μ
λt+1
∂nt+1
µ
∙
¸¶
∂Jt+1 (nt+1 )
hαt − wt ht + Et β t+1 (1 − ρ)
= η mct At αnα−1
,
t
∂nt+1
and rearrange:

!
1+μ
h
1
hαt + (1 − η) b + χt ζ t t
wt ht = ηmct At αnα−1
t
λt
1+μ
∙
¸
1 ∂Wt+1 (nt+1 )
− (1 − η) Et β t+1 (1 − ρ)(1 − θt+1 q(θt+1 ))
λt+1
∂nt+1
∙
¸
∂Jt+1 (nt+1 )
+ ηEt β t+1 (1 − ρ)
.
∂nt+1
29

1 ∂Wt+1 (nt+1 )
Because of continuous renegotiation, the sharing rule must also hold in the future (1 − η) λt+1
=
∂nt+1

(nt+1 )
. Simplifying, yields:
η ∂Jt+1
∂nt+1

h1+μ
1
t
α
wt ht = ηmct At αnα−1
h
+
(1
−
η)
b
+
χ
ζ
t
t
λt t t 1 + μ
∙
¸
∂Jt+1 (nt+1 )
+ ηEt β t+1 (1 − ρ)θt+1 q(θt+1 )
.
∂nt+1

Use the first-order condition for vacancies posted, noting that the Lagrange multiplier of
the firms optimization problem is μt+1 = ∂Jt+1 (nt+1 )/∂nt+1
c0 (vt+1 )
∂Jt+1 (nt+1 )
=
,
q(θt+1 )
∂nt+1
to arrive at the equation for the wage paid to a worker:
Ã
!
1+μ
h
1
hαt + (1 − η) b + χt ζ t t
wt ht = ηmct At αnα−1
+ η(1 − ρ)Et β t+1 θt+1 c0 (vt+1 ).
t
λt
1+μ
To determine the choice of hours, maximize the Nash product
¶ µ
¶
µ
1 ∂Wt (nt ) η ∂Jt (nt ) 1−η
,
λt ∂nt
∂nt
t (nt )
t (nt )
with respect to hours and obtain (using the previous result (1 − η) λ1t ∂W∂n
= η ∂J∂n
)
t
t

1
χt ζ t hμt = mct α2 At nα−1
hα−1
,
t
t
λt
or:
ht =

mct α2 At nα−1
t
λt
χt ζ t

1
¶ 1+μ−α

Hours are set so that the marginal rate of substitution of labor is equal to the marginal
product of labor, just as in the competitive model.
3. Comparing the Rotemberg and Blanchard-Galí models
Blanchard and Galí (2007) analyze a version of the New Keynesian model with search
and matching frictions that diﬀers from our baseline specification in that hiring is instantaneous. That is, vacancies are assumed to be filled immediately by paying the hiring cost,
which is assumed to be a function of labor market tightness. This implies that optimal
hiring decisions are not determined by an intertemporal condition. Jobs are destroyed at
the fixed rate ρ and employment evolves as:
njt = (1 − ρ)njt−1 + Hjt .
30

Current-period employment depends on last period’s employment that survives the separation shock, and current period hiring. Hiring Hjt is given by vjt q(θt ) as before. Hiring costs
per firm are Hjt Gt where Gt = Bhδt , with ht = Ht /Ut and δ ≥ 0. B is a positive constant
satisfying ρB < 1. Gt and q(θt ) are taken as given by firms. Thus, firm j hiring costs are

Njt = vjt q(θt )Gt . The first-order condition for employment is
Bhδt = mct α

yt
− Wt + (1 − ρ)Et β t+1 Bhδt+1 ,
nt

which can be used to write the marginal costs as in our general expression (26), where now,
xt =

1
[Bhδt − (1 − ρ)Et β t+1 Bhδt+1 ].
Wt

As emphasized by Blanchard and Galí, the first term in xt captures the cost of hiring a
marginal employed worker, while the second relates to the savings in hiring costs resulting
from the reduced hiring needs in period t + 1. To avoid potential confusion, note that our
, so that
ht corresponds to their xt = Ht /Ut . In our setup, ht = Ht /Ut = θt q(θt ) = mθ1−ξ
t
(1−ξ)δ

hδt = mθt

. Rotemberg (2006) uses the large firm assumption to motivate increasing

returns to vacancy posting at the firm level. To see the eﬀects of this assumption, notice that
the cost of posting vjt can be specified by the following, in principle, non-linear function:
c(vjt ) = cv vjtc ,
where

≤ 1, and

= 1 corresponds to the linear case discussed by Pissarides (2000).

Crucially, Rotemberg assumes that the hiring costs are incurred one period later, and that
aggregate conditions in t+1 are observed at the end of period t, before vacancies are chosen.
Essentially, this amounts to hiring taking place contemporaneously, as in Blanchard and
Galí above. Therefore, we write from the outset the following evolution of employment27
njt = (1 − ρ)njt−1 + vjt q(θt ).
Note that in Rotemberg’s specification vacancies and labor market tightness would be timed
t−1. The diﬀerence to our baseline setup is that new jobs are not aﬀected by job destruction.
The first-order condition for this setup is therefore:
c cv vt

c −1

q(θt )

= mct α

c −1
yt
c cv vt+1
.
− Wt + (1 − ρ)Et β t+1
nt
q(θt+1 )

It follows from the previous expressions that the only diﬀerence to Blanchard and Galí
(2007) is the specification of the returns to scale in vacancy posting. This expression can
27

In Rotemberg (2006), vacancies and labor market tightness would be timed t − 1.

be used to write the marginal costs as in our general expression (26) with
xt =
where B =

c cv

1 h ξ
Bθt vt
Wt

c −1

i
c −1
,
− (1 − ρ)Et β t+1 Bθξt+1 Vt+1

. Note that Rotemberg assumes that the households’ utility of consumption

is linear, so that β t = β, for all t. For

ξ/(1−ξ)

= 1, and ht = θ1−ξ
⇔ θξt = ht
t

, with

δ = ξ/(1 − ξ), this expression is equivalent to the formulation above, i.e. Blanchard and

Galí (2007) is identical to Rotemberg (2006). The new element is the negative dependence
of hiring costs on vt , arising from returns to scale in vacancy posting when

< 1. It is worth

mentioning that contrary to our Eq. (27), the two expressions have two distinctive features.
First, the extra-term depends positively on current hiring, vacancies and labor market
tightness and negatively upon expected values. This implies that the cyclical behavior of
the marginal cost is modified in diﬀerent forms depending upon the form and timing of
both firing and hiring costs. Second, the last two expressions require the specification of a
stochastic discount factor, β t+1 .
4. A complete description of the baseline model and the steady state

Euler equation
Production
Resource constraint
Unemployment
Job creation
Wage
Hours
Inflation

Taylor rule

The Stochastic Model
h
i
t
, λt = ζ t (ct − ςCt−1 )−σ
λt = Et βλt+1 Rt PPt+1

Yt = At nt α hαt
Yt = Ct + c(Vt )
ut = 1 − (1 − ρ)nt−1 , nth= (1 − ρ) [nt−1 + Mt−1 ], Mt = mt uξt vt1−ξ
i
c0 (vt )
vt+1
α−1 α
0 (v
mc
v
=
(1
−
ρ)E
β
αA
n
h
−
w
h
+
c
)
t
t
t+1
t
t+1
t+1
t+1
t+1
t+1
t+1
Mt
Mt+1
´
³
h1+μ
α−1 α
1
t
wt ht = ηmct αAt nt ht + (1 − η) b + λt χt ζ t 1+μ + η(1 − ρ)Et β t+1 θt+1 c0 (vt+1 )
1
´ 1+μ−α
³
mct α2 At nα−1
t
ht =
λt
χt ζ t
³
´
−γ γ−1
γ−1
ψ π t π t−1 π
− 1 π t π −γ
t−1 π
³
´
−γ γ−1 Yt+1
γ−1 − 1 π
= Et β t+1 ψ π t+1 π −γ
t+1 π t π
t π
Yt + (1 − t ) + t mct ,
´ρr h¡ ¢ ¡ ¢r i1−ρr
³
r
Rt−1
Rt
π t π Yt Y
εR
t
R∗ =
R∗
π∗
Y∗

Marginal utility
Euler equation
Production
Resource constraint
Employment
Unemployment
Matching
Job creation
Wage
Hours
Inflation

The Steady State
λ = (1 − ς)−σ C −σ
1 = βRπ −1
Y = (nh)α
Y = C + c(v)
n = ρ−1 (1 − ρ)M
u = 1 − (1 − ρ)n
M = uξ v 1−ξ
£
¤
c0 (v)
Y
v
0
M v = (1 − ρ)β mcα n − wh
³+ c (v) M1+μ ´
wh = ηmcαnα−1 hα + (1 − η) b + λ1 χ h1+μ + η(1 − ρ)βθc0 (v)
³
´ 1
2 α−1
1+μ−α
h = mcα χn λ
mc =

−1

4. Data
We take the series for the job separation and the job finding rate from Shimer (2005b).
They are quarterly averages of monthly rates. Shimer calculates two diﬀerent series for
the job separation and job finding rate. The first two are available from 1948 up to 2004.
He uses data from the Bureau of Labor Statistics for employment, unemployment, and
unemployment duration to calculate the instantaneous rate at which workers move from
employment to unemployment and vice versa. The two rates are computed under the assumption that workers do not make labor force participation decisions. Hence, they are
an approximation to the true underlying labor market rates. Starting from 1967:2, Shimer
also uses the monthly Current Population Survey microdata to directly calculate the flow of
workers that move in and out of the three possible labor market states (employment, unemployment, and out of the labor force). With this information he calculates the instantaneous
rates at which workers move in and out each state. This yields an exact instantaneous rate
at which workers move from employment to unemployment and from unemployment to
employment.
We also compare the results by using two data sets of two recent studies that have
modified and extended Shimer’s original calculation. We first use the hazard rates series
from Fujita and Ramey (2006). The series are available at monthly frequency and cover
the period of January 1976 through December 2005. We compute the quarterly averages
of monthly rates. These authors correct by potential margin error —inconsistency in the
stock-flow identities— in the CPS. In addition, they also account for time aggregation problems. Elsby et al. (2007) also propose some refinements of the correction methods used by
Shimer based on publicly available data from the CPS. Interestingly, they also distinguish
employment-to-unemployment flows stemming from job loss and from job leaving, and they
33

show that these two flows have very diﬀerent cyclical properties. Thus, we use their disaggregated analysis of unemployment where we distinguish three categories: job losers, job
leavers, and labor force entrants.
We use the index of help-wanted advertisements released by the Conference Board as an
approximation for vacancies (HW). We also use the stock of unemployed —16 years and over—
from the BLS, and the Unemployment Index equals to

U(t)
U (June87) ,

which is consistent with

HW Index. We construct the quarterly averages of monthly rates that are available starting
at January 1951. Finally, our measure of marginal costs corresponds to the Nonfarm BusiR
ness Sector. The data are drawn from FRED°
II database and the variables correspond to:
real output (OUTNFB), the output deflator (IPDNBS), the aggregate number of hours worked
(HOANBS), and the compensation per hour (COMPNFB), respectively. Real consumption corresponds to the sum of real non-durable (PCNDGC96) and services (PCESVC96), respectively.
Finally, CNP16OV is the civilian non institutional population.

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# 2445.

Table 1. Basic Statistics: Baseline Calibration
Sample P eriod : 1960 − 2005
Sample P eriod : 1985 − 2005
Variable s.d.(%) ρ
Cross Correlation
s.d.(%) ρ
Cross Correlation

y
s
mcBG
mcR

2.95
2.09
2.62
2.39

.95
.92
.69
.63

s
-0.12
1

mcBG
-0.10
0.83
1

mcR
-0.14
0.84
0.97
1

1.93
1.78
2.27
2.08

.95
.91
.53
.45

s
0.30
1

mcBG
0.28
0.76
1

mcR
0.21
0.75
0.97
1

Table 2. GMM Estimates: 1985-2005
Blanchard-Galí Specification
γ
κ
γb
Finding Rates:
Baseline Calibration
Alternative Calibration
Using Tightness:
Baseline Calibration
Alternative Calibration

0.682

(0.124)

0.635

0.013

(0.007)

0.026

(0.119)

(0.013)

0.698

0.0062

0.692

0.0085

(0.124)

(0.0037)

(0.005)

0.407

(0.045)

0.389

(0.045)

0.406

(0.044)

0.410

(0.043)

6.55

(0.044)

6.26

(0.044)

7.17

(0.044)

6.97

(0.044)

Note: In all cases the dependent variable is quarterly inflation measured using the GDP Deflator.
The sample period is 1985:I-2004:IV. Standard errors are shown in brackets. The instrument set
includes two lags of detrended output, and one lag of real marginal costs. The hazard rates are from
Shimer (2005b). The results are unchanged under alternative hazard rates.

Table 3. GMM Estimates: 1985-2005
Rotemberg Specification
γ
κ
γb
Finding Rates:
Baseline Calibration
Alternative Calibration
Using Tightness:
Baseline Calibration
Alternative Calibration

0.682

(0.124)

0.704

(0.129)

0.698

(0.124)

0.692

(0.124)

0.007

(0.0040)

0.007

(0.0039)

0.007

(0.0041)

0.007

(0.0041)

0.416

(0.045)

0.414

(0.044)

0.416

(0.045)

0.417

(0.046)

6.72

(0.044)

6.87

(0.044)

7.68

(0.044)

6.70

(0.044)

Table 4. Prior Distributions
Definition

Parameter

Density

Mean

Std.

σ
ς
μ
α
ξ
m

Gamma
Beta
Gamma
Beta
Beta
Gamma
Gamma
Gamma
U nif orm
Beta
Beta
Beta
Gamma
Gamma
Beta
Gamma
Gamma
Beta
Inv. Gamma

1.00
0.50
1.00
0.67
0.70
0.70
1.00
0.05
0.50
0.40
0.10
0.50
20.00
11.00
0.70
1.50
0.25
0.90
0.01

0.10
0.20
0.50
0.02
0.05
0.10
0.50
0.02
0.25
0.10
0.02
0.20
5.00
1.00
0.02
0.10
0.05
0.05
1.00

Relative Risk Aversion
Habits
Inverse of Labor Supply Elasticity
Elasticity of Output to Labor Input
Elasticity of Matching to Unemployment
Match Eﬃciency
Elasticity of Vacancy Creation
Scaling Factor on Vacancy Creation
Bargaining Power of the Worker
Unemployment Benefit
Separation Rate
Indexation
Price Adjustment Costs
Elasticity of Demand
Interest Rate Smoothing
Interest Rate Response to Inflation
Interest Rate Response to Output
AR-Coeﬃcients of Shocks
Std. Deviation of Shocks

cc
η
b
ρ
γ
ψ
ρr
γπ
γY
ρ0 s
σ0s

Table 5. Posterior Estimates: Baseline Model
Prior
Posterior
Mean Mean 90% Interval
Relative Risk Aversion
σ
1.00
0.92 [0.75, 1.08]
Habits
ς
0.50
0.03 [0.00, 0.05]
Input Elasticity
α
0.67
0.65 [0.61, 0.68]
Labor Supply Elasticity
μ
1.00
5.06 [3.42, 6.75]
Elast. of Matching
ξ
0.70
0.68 [0.62, 0.74]
0.70
0.72 [0.57, 0.86]
Scaling Factor Matching Function m
1.00
3.35 [2.35, 4.35]
Elast. of Vacancy Cost
c
Bargaining Power
η
0.50
0.67 [0.38, 0.97]
Worker Outside Option
b
0.40
0.42 [0.38, 0.46]
Separation Rate
ρ
0.10
0.06 [0.04, 0.08]
Indexation
γ
0.50
0.29 [0.04, 0.55]
Price Adjustment Costs
ψ 20.00
9.06 [5.27, 12.73]
Elasticity of Demand
10.00 10.71 [9.09, 12.41]
0.70
0.61 [0.58, 0.68]
Interest Rate Smoothing
ργ
Inflation Response
γπ
1.50
2.24 [2.05, 2.43]
Output Response
γy
0.25
0.14 [0.10, 0.19]

Table 6. Robustness: Labor Share
Parameter Estimates

σ
ς
μ
ξ
c

Mean
0.93
0.03
1.96
0.72
1.22

90% Interval
[0.75, 1.10]
[0.00, 0.05]
[0.79, 2.83]
[0.64, 0.80]
[0.87, 1.57]

η
b
γ
ψ

Mean
0.29
0.34
0.21
2.08

90% Interval
[0.00, 0.83]
[0.17, 0.53]
[0.03, 0.41]
[1.08, 2.98]

Variance Decomposition

y
u
v
ls

Technology
0.80
0.01
0.11
0.21

Mark-Up
0.03
0.45
0.05
0.66

Matching
0.06
0.53
0.81
0.08

Lab. Supply
0.10
0.00
0.02
0.02

Preferences
0.00
0.01
0.00
0.00

Policy
0.00
0.00
0.00
0.03

y
π
R
u
v

Table 7. Variance Decomposition: Baseline Model
Technology Mark-up Matching Lab.Supply Preferences
0.67
0.27
0.00
0.05
0.00
0.05
0.02
0.00
0.00
0.19
0.07
0.02
0.00
0.00
0.90
0.03
0.60
0.37
0.00
0.00
0.05
0.94
0.01
0.00
0.00

Policy
0.00
0.74
0.00
0.00
0.00

Labor Market Variables
Unemployment and Vacancies
225

(Stoc k) Unemploy
Vacanc ies

200
175
150
125
100
75
50

1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005

Labor Market Tightness and Finding Rate
200

Tightnes s
Finding

175

140
130

150

120

125

110

100

-25

50
1960

1964

1968

1972

1976

1980

1984

1988

1992

1996

2000

2004

Figure 1: Some labor market variables. Top panel: unemployment rate (solid line) and
vacancies (dotted line). Bottom panel: tightness ratio (solid line) and finding rate (dotted
line). The shadowed areas correspond to the NBER recession dates.

Components of the Marginal Costs
Blanchard-Gali Baseline Calib ration (Finding Rates)
10.0

Real Unit Labor Costs v s Marginal Costs

Finding Rate Contribution

5
unit labor cost
marginal cost

7.5

5.0

2
2.5
1
0.0
0
-2.5

-1

-5.0

-2

-7.5

-3
1960 1964

0.36

1968 1972 1976 1980 1984 1988 1992 1996

2000 2004

1960 1964

Discount Factor Contribution

0.06

0.24

1968 1972

1976 1980

1984 1988

1992 1996

2000 2004

Real Wage Contribution

0.04

0.12
0.02
0.00
0.00
-0.12
-0.02

-0.24

-0.36

-0.04
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Figure 2: Real marginal costs and search frictions. In the top panel we plot unit labor costs
(solid line) and real marginal costs (dotted line). The shadowed areas correspond to the
NBER recession dates.

Real Unit Labor Costs and Marginal Costs
Blanchard-Gali Approach
Baseline Calibration
10.0
7.5
5.0
2.5
0.0
-2.5
-5.0

unit labor c ost
marginal c os t

-7.5
1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

1990

1993

1996

1999

2002

2005

1993

1996

1999

2002

2005

Alternativ e Calibration
6
4
2
0
-2
-4
-6

unit labor cos t
marginal c ost

1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

1990

Figure 3: Real marginal costs under alternative calibrations. The solid line represents unit
labor costs and the dotted line real marginal costs. The shadowed areas correspond to the
NBER recession dates.

Components of the Marginal Costs
Rotemberg Specification
Real Unit Labor Costs vs Marginal Costs
7.5
5.0
2.5
0.0
-2.5
-5.0

unit labor cost
marginal c ost

-7.5
1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

1990

1993

1996

1999

2002

2005

1990

1993

1996

1999

2002

2005

1990

1993

1996

1999

2002

2005

Finding Rate Contribution
5
4
3
2
1
0
-1
-2
-3
1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

Vacancies Contribution
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
1960

1963

1966

1969

1972

1975

1978

1981

1984

1987

Figure 4: Real marginal costs in the Rotemberg model. In the top panel the solid line
represents unit labor costs and the dotted line real marginal costs. The shadowed areas
correspond to the NBER recession dates.

Figure 5: Impulse Response Functions: Baseline
50

15
Blanchard-Gali Baseline
Rotemberg Baseline
Filtered
10

-5

-10

-15
1985

1990

1995

Figure 6: Marginal Cost Series

2000

2005

Full text of Working Papers (Federal Reserve Bank of Richmond) : Inflation Dynamics with Search Frictions : A Structural Econometric Analysis, Working Paper 08-01

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