Finnish Economic Papers – Volume 20 – Number 2 – Autumn 2007
Estimating Equilibrium Search Models
from Finnish Data*
Tomi Kyyrä
Government Institute for Economic Research, P.O. Box 1279,
00101 Helsinki, Finland; e-mail: tomi.kyyra@vatt.fi
Empirical equilibrium search models have attracted a growing interest in recent
years. Estimation of such models has not been completely successful, however.
This has led researchers to develop more sophisticated versions of the models in
an attempt to get a better fit to the data. We estimate various proposed specifications of the Burdett-Mortensen model from Finnish panel data. We begin with a
pure search model in which search frictions are the only source of wage dispersion. Then we proceed to more complex specifications by introducing measurement
error in wages and unobserved employer heterogeneity. Our results show how
search frictions vary with education, age and sex, and how these differences are
reflected in wage differentials in the Finnish labour market.
(JEL: C41, E24, J31, J64)
1. Introduction
The equilibrium models of the labour market
provide a useful framework for analysing various labour market issues. In such models a
policy reform or shock that directly affects some
subgroup of firms or workers can change the
behaviour of all agents on both sides of the market, leading to changes in the equilibrium wage
distribution and unemployment rate. When both
sides of the labour market are modelled in a
dynamic environment, a number of simplifying
assumptions must be adopted to keep the model
* This study is based on a chapter of my doctoral thesis.
The paper has benefited from helpful comments received at
the 56th European Meeting of the Econometric Society, the
1st Nordic Econometric Meeting and various seminars held
in Finland. Michael Rosholm provided most of Gauss programs needed in the estimations. I thank three anonymous
referees and (co-editor) Heikki Kauppi for their suggestions
and comments. All the remaining errors and misinterpretations are my responsibility.
tractable. Despite simplifying assumptions, the
equilibrium solutions are typically rather complex, involving highly nonlinear functions of
structural parameters. A consequence is that the
equilibrium models are difficult to estimate, and
thereby such models are usually calibrated, not
estimated.
In calibration, one collects numbers from
various sources for some parameters of the
model and solves the model with respect to the
remaining parameters. This procedure is quite
arbitrary, and therefore the estimation of the
structural parameters should be preferred. Another issue is to which extent stylized equilibrium models are able to describe the real-world
labour market. If the model has not been tested
with the data, one must be gullible to take the
predictions of the model seriously. Estimating
structural equilibrium models from micro-data
139
Finnish Economic Papers 2/2007 – Tomi Kyyrä
produces more credible parameter estimates, allows us to assess the fit of the model, and yields
valuable information for developing a more rigorous basis for equilibrium labour market analysis.
This study considers the estimation of the
equilibrium search model of Burdett and Mor­
tensen (1998).1 The model generates wage differentials between homogenous workers even
when all firms are assumed to be identical. This
main prediction of the model is consistent with
empirical evidence on the existence of considerable wage differentials which cannot be explained by worker and job characteristics (see
Bowlus et al., 1995). Other strong predictions
concerning the relationship between wages, job
durations and firm sizes follow from this simple
model as well. Since many of these predictions
are consistent with the stylized features of the
labour market, the Burdett-Mortensen model
has attracted a growing interest in the empirical
literature. The various specifications of the
model have been applied to data from the Netherlands (Van den Berg and Ridder, 1998), the
US (Bowlus, 1997, and Bowlus et al., 2001),
Denmark (Bunzel et al., 2001) and France (Bontemps et al., 2000). Bunzel et al. (2001) compare different versions of the model using Danish data. This study provides a similar exercise
in the context of the Finnish labour market.
We estimate different variants of the BurdettMortensen model using maximum likelihood
from a sample of workers who entered unemployment in Finland during 1992. In the analysis we distinguish between separate segments of
the labour market by stratifying the data according to education, sex and age. The estimation
results are discussed and compared across the
model specifications as well as across the worker groups.2 We begin with the homogeneous
version of the model. This simplest version of
the model gives a poor fit to the data because
the shape of the theoretical wage distribution
is at odds with the observed distribution. We
1
See Mortensen and Pissarides (1999), Van den Berg
(1999), Rogerson et al. (2005), and Eckstein and Van den
Berg (2007) for the surveys of search models.
2
Unlike Bunzel et al. (2001), we include also a model
specification with continuous productivity dispersion in our
set of the equilibrium search models to be compared.
140
proceed by estimating an augmented specification of the homogenous model by assuming that
the wage data are subject to measurement error
(e.g. Christensen and Kiefer, 1994a). This is
followed by analysis of the theoretical extensions of the model in which firms differ in their
unobserved labour productivity. We consider
specifications with a finite number of firm types
(Bowlus et al., 1995, 2001, and Bowlus, 1997)
as well as continuous productivity heterogeneity (Van den Berg and Van Vuuren, 2000, and
Bontemps et al., 1999, 2000). Although the fundamental structure of the model remains unchanged, the source and interpretation of observed wage differentials depend upon the underlying assumptions on measurement error and
productivity heterogeneity.
The results of the paper are useful in a variety of ways. First, a comparison of the fit of
different extensions is informative on what aspects of the theory and empirical specification
are likely to be important to obtain an acceptable fit to the Finnish data. Second, by investigating variation in the parameter estimates
across the model specifications, we can test how
robust the estimates of the fundamental parameters are with respect to different ways of deviating from the homogenous version of the model. Third, our results are informative about the
relationship between unemployment durations,
job durations and wages in the Finnish labour
market.
The rest of the paper is organized as follows.
In the next section the concepts of the equilibrium search theory are discussed. Section 3 introduces the data and provides some summary
statistics. Estimation results from various empirical specifications are reported in Section 4.
The final section concludes.
2. Equilibrium search theory
In this section we briefly describe equilibrium
search theory along the lines of Burdett and
Mortensen (1998) and Bontemps et al. (2000).
We begin with a pure search model in which all
jobs are equally productive and all workers are
identical. Then we introduce employer heterogeneity in the model but retain the assumption
tion 4. The final section concludes.
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Equilibrium
search theory
of homogeneity
of the worker population than for the employed (λ 0 > λ 1), the reservation
throughout the paper.
wage r exceeds the value of non-market time b.
s section we briefly describe equilibrium search theory along the
lines
of it
Burdett
In that
case
is more rewarding to search while
the worker rejects wage offers
ortensen (1998)
Bontempssearch
et al. (2000).
We begin
with aunemployed
pure searchand
model
2.1 and
Equilibrium
with identical
agents
in
the
interval
(b,
though
thisjob
causes
a the optimal
From (1) one can see how the possibilityr),ofeven
search
on the
affects
ch all jobs are equally productive and all workers are identical.utility
Then loss
we introduce
in the short run. If the offers arrive
2.1.1 Worker behaviour
search strategy of an unemployed
worker.
If wage
offers arrive more frequently for the
a higher
rateofwhen
yer heterogeneity in the model but retain the assumption of at
homogeneity
the employed (λ 0 < λ 1), enterThe supply side of theunemployed
labour market
popuing employment
the subsequent
thanis for
the employed
(λ > λ1 ) , improves
the reservation
wage r exceeds the value
populationlated
throughout
the paper.of ex ante identical work- chances of0 receiving
by a continuum
higher offers. It follows
of non-market
time with
b. In that
is more
search
while
unemployed and
ers. Behaviour of workers
is characterised
thatcase
the itworker
is rewarding
willing to to
accept
also
some
Equilibrium
identical
agents
the search
standardwith
job search
with
search
onoffers
jobsinthat
less (b,
than
When
the arrival
rate a utility loss
themodel
worker
rejects
wage
the pay
interval
r),b.even
though
this causes
the job. In particular workers are assumed to be is independent of employment status (λ 0 = λ 1),
Worker behaviour
themaximising
short run. the
If the
at aishigher
rate when
employed
(λ0 < λ1 ) , entering
risk-neutral agents whoinare
ex-offers
thearrive
worker
indifferent
between
searching
pected present value of
future
income
stream
while
employed
and
while
unemployed.
the subsequent
of receiving higher offers. Any
It follows that the
upply side of the labour market is employment
populated byimproves
a continuum
of ex antechances
identical
with infinite horizon. Workers in the labour job that compensates for the foregone value of
workerwith
isorwilling
to accept
somemodel
jobs
that
pay less than
b. When
the arrival rate is
s. Behaviour
of workers
is characterised
the standard
jobalso
search
with
market
are either
employed
unemployed.
non-market
time
is acceptable
in this
case and
Each worker is facing a known distribution
of thus
r =(λ
b.0 =
status
λ1 )are
, the worker is indifferent between searching
on the job. In particular workers independent
are assumedoftoemployment
be risk-neutral
agents
who
wage offers F with associated jobs, from which
while
employed
while
unemployed.
Any job that compensates for the foregone value
mising the expected
present
value wage
of future
income
stream
infinite horizon.
he randomly
samples
offers
bothand
on
and with
2.1.2 Firm behaviour
off
the
job.
Wage
offers
arrive
at
the
Poisson
of non-market
time is acceptable
in this
case and
rs in the labour market are either employed
or unemployed.
Each worker
is facing
a thus r = b.
rate λ 0 when unemployed and at the Poisson The demand side of the labour market consists
distributionrate
of wage
offersemployed.
F with associated
jobs, from
which he
samples
λ 1 when
Unemployed
workers
of randomly
a continuum
of ex ante identical firms. The
2.1.2 Firm behaviour
search
for
an
acceptable
job
and
employed
firms
are
assumed
offers both on and off the job. Wage offers arrive at the Poisson rate λ0 whento use only labour inputs in
workers for a better job. Jobs are destroyed at production. Each worker generates a flow of revThe demand side of the labour market consists of a continuum of ex ante identical firms.
ployed and at
Poissonrate
ratedλ
employed.
workers
search
for
thethePoisson
, 1inwhen
which
case theUnemployed
worker enue
p to his
employer.
We assume that p is inThe
firms
are
assumed
to
use
only
labour
inputs
in
production.
Eachrefer
worker generates
who
holds
the
job
is
laid
off
and
becomes
undependent
of
the
size
of
the
workforce and
eptable job and employed workers for a better job. Jobs are destroyed at the Poisson
employed.
to p as the (labour) productivity of the firm. The
a flowjob
of isrevenue
pand
to becomes
his employer.
We assume that p is independent of the size of
in which case Jobs
the worker
who holds
laid off
unemployed.
are identical
apartthe
from the
wage
associfirm sets
its wage so as to maximise the expecttheassociated
workforce
and
refer to
p aassteady-state
the (labour)
productivity
of the
The firm sets its
atedapart
with from
them.the
As wage
a result,
employed
workers
profit
flow taking
thefirm.
optimal
bs are identical
with
them.
Ased
result,
employed
are willing to move into higher-paying jobs search behaviour of workers and wages set by
wage jobs
so aswhenever
to maximise
the expected
steady-state
profit flow taking the optimal search
s are willingwhenever
to move into
the opportunity
arises.
The To attract
the higher-paying
opportunity arises.
The current
other firms
as given.
workers the firm
behaviour
of
workers
and
wages
set
by
other
firms
as
given.
To
attractranworkers the firm
wage
thus
serves
as
the
reservation
wage
for
posts
wage
offers,
among
which
workers
t wage thus serves as the reservation wage for employed workers. In the limiting
employed workers. In the limiting case of zero domly search using a uniform sampling scheme.
postsofwage
offers, amongworker
which can
workers
randomly search using a uniform sampling scheme.
zero discounting,
the reservation
wage
the unemployed
betoexpressed
discounting,
the reservation
wage
of the unem- Contrary
the competitive setting, the presence
Contrary to
the presence
searchmarket
frictions
in the labour market
ployed worker can be expressed
as the competitiveofsetting,
search frictions
in theoflabour
generates
dynamic monopsony power for wage-setting
h
generates
dynamic monopsony power for wage-setting firms. As workers cannot find a
1 − F (z)
firms. As workers cannot find a higher-paying
dz,
(1) r = b + (κ0 − κ1) higher-paying job instantaneously,
firms can offer
strictly
smaller
job instantaneously,
firmswages
can offer
wages
strict-than marginal
1
+
κ
[1
−
F
(z)]
1
r
ly smaller than marginal labour productivity.
labour
productivity.
b is the value
of non-market
time,
including
unemployment
of search
where
b is the value
of non-market
time, includ- benefits
The net
expected
profit flow of a firm paying
The
expected
profit
flow
of
a
firm
paying
wage
a steady
ing
unemployment
benefits
net
of
search
costs,
wage
w
in
a
steady
statewisingiven
by state is given by
κ0 = λ0 /δ, κ1 = λ1 /δ and h is the upper bound of the support of F (see Burdett
κ 0 = λ 0 /d, κ 1 = λ 1/d and h is the upper bound of
ortensen, 1998).
This equation
defines
the and
reservation
wage r (2) as a function
the support
of F (see
Burdett
Mortensen,
(2)
π(p, w) =of(pthe
− w) l(w),
1998).
This
equation
defines
the
reservation
ural parameters of the model.
wage r as a function of where
the structural
w) isworkforce
the expected
size of the
workforce
l(w) is parameters
the expected where
size ofl (the
(associated
with
a given F ). The firm
of the model.
(associated with a given F ). The firm would emwould
employ
as manyofworkers
to maximise
its profit
flow as long as p > w.
From (1) one can see
how
the possibility
ploy as
as possible
many workers
as possible
to maximise
4
search on the job affects
searchserves
its profit
as long wage
as p > w.
the current
Sincethe
theoptimal
current wage
as theflow
reservation
for Since
employed
workers, the number
strategy of an unemployed worker. If wage of- wage serves as the reservation wage for emof workers
to the ployed
firm in workers,
equilibrium
with
the wage
offered. In other
fers arrive more frequently
for theavailable
unemployed
theincreases
number of
workers
availa-
words, the labour supply curve the firm is facing with is upward-sloping. The firm takes
141
the function l as given and offers a wage that maximises its expected steady-state profit
5
Finnish Economic Papers 2/2007 – Tomi Kyyrä
ble to the firm in equilibrium increases with the for all w on the common support of F and G.
wage offered. In other words, the labour supply Since workers tend to move up the wage range
curve the firm is facing with is upward-sloping. over time, the earnings distribution lies to the
The firm takes the function l as given and offers right of the wage offer distribution, or more fora wage that maximises its expected steady-state mally, G first-order stochastically dominates F
profit flow. Obviously, a firm never offers a as F (w) – G (w) ≥ 0 for all w and κ 1 ≥ 0. The diswage above p as the profits would be negative, crepancy between the earnings and wage offer
nor it offers a wage less than r as such a wage distributions depends on κ 1 which is equal to
would not attract any workers.
the expected number of wage offers during a
productive
receive
spell of nor
employment (which may consist of sevsly, a firm never Equally
offers a wage
above pfirms
as themust
profits
wouldthe
be negative,
same
expected
profit
flow
in
equilibrium.
This
eral
consecutive
job spells)
flow.than
Obviously,
a firm
never
offers
wage above
p as the profits would be negative,
nor and can be thought
ge less
r as such
a wage
would
nota attract
any workers.
does not mean that wage offers need to be equal, of as a relative measure of competition among
it offersfirms
a wage
lessreceive
than
r the
aspaying
such
wage
would
not
attract
any
workers.
however.
A firm
higher
wage
makes
ain equilibrium.
firms
for workers.
roductive
must
sameaaexpected
profit
flow
lower
profit
per
worker
but
makes
it
up
in
volBy
equating
inflow and outflow of workver offers
a
wage
above
p
as
the
profits
would
be
negative,
nor
productive
firms
receive
the same
expected
flow in the
equilibrium.
t meanEqually
that wage
offers need
to must
be equal,
however.
A firm
payingprofit
a higher
ume as the higher wage attracts more workers ers to the firm offering wage w, we find that the
r asThis
suchdoes
a wage
would
not
attract
any workers.
not
mean
that
wage
to firm
be
equal,
however.
A firmsize
paying
a higher
from
other
firms
and
enables
the
to higher
retain
expected
of the
workforce in such a firm
lower profit
per
worker
but
makes
itoffers
up
inneed
volume
as the
wage
attracts
By equating
the inflowasand outflow of workers to the firm offerin
them
for
a
longer
time.
It
follows
that
some
can
be
expressed
ms wage
mustmakes
receive
the
same
expected
profit
flow
in
equilibrium.
a lower
perthe
worker
makesthem
it up for
in volume
as time.
the higher
from other firms
and profit
enables
firm but
to retain
a longer
It wage attracts
firms choose to offer low wages with a cost
theofexpected size of the workforce in such a firm can be expressed
wage
offers
need
to
be
equal,
however.
A
firm
paying
a
higher
more
workers
from
otherturnover,
firms
andwhile
enables
the pay
firm
to retain
for a longer time. It
high
labour
others
higher
some
firms
choose
to
offer
low
wages
with
a cost
of high
labourthem
turnover,
By equating the inflow and outflow of workers
firm offering
find
λ1 G(w)
(m − u)wage w, we κ
+ κ1 ) m
λ0 uto+the
0 (1that
wages
and
experience
lower
labour
turnover.
perfollows
worker but makes
it up inchoose
volumetoasoffer
the higher
wagewith
attracts
=
(5)
l(w) =
some
low turnover.
wages
a cost
of tradehigh
labour
turnover,
pay higher that
wages
and firms
experience lower
labour
Due
to (5) this
δ +λ
− F (w)] as
(1 + κ0) (1 + κ1 [1 −
the expected
of the offered
workforce in such a firm can
be1 [1
expressed
Due to this trade-off
between size
the wage
firms
andothers
enables
thehigher
firm to
retain
them
for a longer
It turnover. Due to this tradepay
wages
and
experience
lowertime.
labour
and
the
same
profit
he while
wage offered
andlabour
labourturnover,
turnover,
the
sameexpected
expected
profit
level
can
be
that−the
λ1 G(w) (m
u) size of the firm
+ κ1 ) m is normalized to one. O
κ0 (1population
λ0 u +provided
=
(5)
l(w)
= turnover,
can beoffered
attained
bylabour
paying
different
wages.
hoose
to offer level
low
wages
with
aand
cost
of high
labour
2,
off
between
the
wage
turnover,
the
same
expected
profit
level
can
be
δ +inλ1w[1and
− Fcontinuous
(w)]
aying different wages.
(1
+ κ0F
) (1is+continuous.
κ1 [1 − F (w)])
where
Note that no assumpt
ages
and experience
labour
turnover.
to this
tradeattained
by payinglower
different
wages.
provided
that Due
the size
of the
firmbeen
population
normalized
to one.quit
Obviously
increasing
have
made sois far.
Since workers
and arel is
hired
and laid o
2.1.3 Steady-state
outcomes
dy-state outcomes
provided
that
the
size
of
the
firm
population
is
red and labour turnover, the same
expected
profit
level
can
be
in w and continuous wherethe
F is continuous.
Note
that anorandom
assumptions
on the
shape
of F
of the
varying
Denote the fixed
size of the labour force with mworkforce
normalized
tofirm
one.isObviously
lvariable,
is increasing
in around its
2.1.3 Steady-state
outcomes
nt
wages.
xed size of the labour
force
with m
andbeen
the steady-state
of unemployed
have
madeofsounemployed
far.number
Since
workers
quit
and are also
hired
and
laid
off
and the
steady-state
number
continuous
where
Fprofit
is continuous.
Noteintervals,
time.
Asw aand
consequence,
the
flowatisrandom
a random
variable.
workers
with
u.
In
a
steady
state
the
flows
into
that
no
assumptions
on
the
shape
of
F
have
been
the fixed
sizethe
of the
labour
force
with
m
and
the
steady-state
number
of
unemployed
u.Denote
In a steady
state
flows
into
and
out
of
unemployment
are
equal:
the workforce of the firm is a random and
variable, varying
around
its expected
value
(1998)
prove
there
existsl aover
unique
tcomes
and out of unemployment are equal, so thatBurdett
made so Mortensen
far. Since workers
quit that
and are
hired
with
u.
In
a
steady
state
the
flows
into
and
out
of
unemployment
are
equal:
= λworkers
udt
for
a
short
time
interval
dt.
Thus
the
steady-state
unemployment
0
time.
a consequence,
also
the profit
flow
a random
variable.
d (m – u) dt = λ 0 udt
for As
a short
time interval
dt.
and
laid
offis at
random
intervals,
thesuch
workforce
librium
which
consists
of a triple
(r, F, π),
that (i ) r satisfie
labour
with
and
the
number
ofdt.
unemployed
Thus
unemployment
rate
is the steady-state
of
the
firm
is
a
random
variable,
varying
around equiδ (m force
− u) dt
=m
λ0the
udtsteady-state
for steady-state
a short
time
interval
Thus
unemployment
Burdett and Mortenseneach
(1998)
prove
that
there
exists
a
unique
non-cooperative
w on the support of F maximises π(p, w), yielding the expec
adyrate
state
and out of unemployment are equal: its expected value l over time. As a consequence,
is the flows into
a triple
(r,to
F,the
π),3profit
such flow
that is(i )a they
r satisfies
(1) given
F and (ii ) solu
δ librium1which consists of flow
u
also
random
variable.
equal
π.
Furthermore,
show
that
the equilibrium
= Thus the
= steady-state
,
(3) a short time interval
dt.
unemployment
Burdett and Mortensen (1998) prove that
m
δ + λeach
1wu+on
κ0 theδ support 1of F maximises
0
π(p, w), with
yielding
the
expected
steady-state
absolutely
continuous
thenon-cooperative
common
support
[r, h] . profit
=
(3)
=
,
there
exists a unique
equilibm
δ
+
λ
1
+
κ
0
0
3
analogous argument
can derive
the
steady-state
earnings
distribution
G, consists
∞ for
flowargument
equal
to we
π. can
Furthermore,
they
show
that
the
equilibrium
and
G are
Using we
an analogous
derive theSince
rium
which
of a istriple
(r, F,allπ
), F
such
the
expected
profit
flow
π solutions
across
firms
in equilibrium
steady-state
earnings
distribution
G,
the
crossδ
u
1
that
(i)
r
satisfies
(1)
given
F
and
(ii)
each
w
on
Using
argument
we can
derive the
steady-state
distribution
G,
ion wage
distribution
employed
workers,
associated
with=earnings
aπgiven
continuous
with
the π(r)
common
support
[r, h]all. w on
= an analogous
=of currently
, absolutely
= π(w)
support
of F. Taking this t
that
of currently employed
the support
of for
F maximisesthe
π (p,
w), yielding
m
δ section
+ λ0 wage
1 + κdistribution
0
the cross-section
wage
distribution
of
currently
employed
workers,
associated
with
a
given
stribution
F. By
equating
worker
flows
into
and
out
of
jobs
paying
w
or
∞ particular
Sinceathe
expected
profit
flow
is
across
allsteady-state
firms
in equilibrium,
holdstoinπ
workers, associated with
given
wage offer
distheπ expected
profit 4flowitequal
.3
the
equilibrium
wage
offer distribution
gument
we
can
derive
the
steady-state
earnings
distribution
G,
tribution
F. relationship
By
worker
flows
intointo
and and
Furthermore,
they
show
the equilibrium
offer steady-state
distribution
F. equating
By
flows
out
of jobs of
paying
w orthat
thewage
following
wage
offer
earnings
= the
π(w)
for all
w and
on the
support
F. Taking
together
thatequating
π(r)between
= πworker
p −with
w (5) gives
1 +absolutely
κthis
1
out
of
jobs
paying
w
or
less,
we
find
the
followsolutions
for
F
and
G
are
continuous
, w ∈ [r, h]
1
−
(6)
F
(w)
=
ribution
of find
currently
employedsteady-state
workers, associated
withbetween
a given the wage
less, we
thesteady-state
following
relationship
offer and support
earnings
κ1[r, h].
p−r
the
equilibrium
wage offer
distribution
ing
relationship
between
the wage
with 4the common
density
profit flow is π
the
. By
equating offer
worker
into distributions:
and out of jobs payingwith
w orthe associated
∞ across all
andflows
earnings
Since
expected
distributions:
F (w)
λ0 u
F (w) F (w) = 1 + κ1 1 − p − w , w ∈ [r, h] ,
(6)
firms
in
equilibrium,
it
holds
in
particular
that
G(w) = relationship between
·
=
steady-state
the
and
earnings κ1
p−r
1 + κ1
δ + λ1 [1 − F (w)] m −
u wage
1 + offer
κ1 [1λ−
F
(w)]
∞
F (w)
u
F
(w)
π
(r)
=
π
=
π
(w)
for
all
w
on
the
support
of
0
f(w) =
,F. w ∈ [r, h] .
(4)
·
= (7)
(4) G(w) = with the associated
2κ
(p
−
r)
(p
−
w)
density
1
δ
+
λ
[1
−
F
(w)]
m
−
u
1
+
κ
[1
−
F
(w)]
the common support of F and G.1Since workers tend to move 1up the wage
Moreover, by1 substituting
(6) into (1) and recognising that F (h)
+ κ1
Fall
(w)
λ0 u
w on the
of F
andofG.
workers
tend
to move
up are
the
(7)toF (w)
f(w)
= distribution,
, ruled
wwage
∈ out
[r, h]
3
me,for
the
earnings
distribution
lies
the
right
theSince
wage
offer
Trivial
solutions
by .making the natural
· common
= support
2κ
(p
−
r)
(p
−
w)
1 supportthat
bounds assumptions
of
of F
as
δ + λ1 [1 − F (w)] m − u
1 + κ1 [1 − F (w)]
∞ > p > b and ∞ >κ i > 0 for i = 0,1.
range
over time,stochastically
the earnings dominates
distribution
the−right
wage
offer distribution,
ally,
G first-order
F lies
as Fto(w)
G(w)of≥the
0 for
all w
Moreover, by substituting (6) into (1) and recognising that F (h) = 1, we can write the
support
of
F
and
G.
Since
workers
tend
to
move
up
the
wage
(8)F as F (w)
r = αb + (1 − α) p,
more formally,
G first-order
stochastically
− G(w) ≥ 0 for all w
Theordiscrepancy
between
the earnings
and wage dominates
offer distributions
depends
142
bounds of support of F as
ngsand
distribution
lies
to
the
right
of
the
wage
offer
distribution,
The discrepancy
between
the earnings
offer distributions depends
(9)wage
h = βb + (1 − β) p,
1 ≥
s equal κto
the0.expected
number of
wage offers
during aand
spell
of employment
(8)
r
=
αb
+
(1
−
α)
p,
order
stochastically
dominates
F
as
F
(w)
−
G(w)
≥
0
for
all
w
on κof
equal to thejob
expected
number
offers
a spell of employment
1 which
onsist
severalisconsecutive
spells) and
can of
bewage
thought
of during
as a relative
where the weights are given by
ncy(which
between
the
earnings
and
wage
offer
distributions
depends
(9)consecutive job spells) and canhbe =thought
βb + (1
may consist of several
of −
asβ)
a p,
relative
aMortensen
consequence,
also
the
profit
flow isexists
a varying
random
variable.
orce
of the 3firm
is aprove
random
variable,
around
its expected value
l over
(1998)
that
there
a satisfies
unique
non-cooperative
equion the
of(r,
F F,
maximises
π(p,that
w),
the
onsists
π),
such
that
(i ) ryielding
(1)expected
given
F steady-state
andfor
(iiF) andprofit
equal
toofsupport
π.a triple
Furthermore,
they
show
the equilibrium
solutions
G are
ttconsequence,
and Mortensen
prove
that
exists
a unique
non-cooperative
equiaonsists
also(1998)
the
profit
flow
is there
a (irandom
variable.
of
a
triple
(r,
F,
π),
such
that
)
r
satisfies
(1)
given
F
and
(ii
)
3
ual tocontinuous
π.
they
show
that
the
equilibrium
solutions for
F and G are
upport
of FFurthermore,
maximises
π(p,
w),
yielding
the[r,
expected
steady-state
profit
utely
with the
common
support
h] .
hich
of a (1998)
tripleπ(p,
(r,
F,
π),
such
that
(iexpected
) ra satisfies
(1)Finnish
givenprofit
FEconomic
andequi(ii ) Papers 2/2007 – Tomi Kyyrä
tt
andconsists
Mortensen
prove
that
there
exists
unique
non-cooperative
upport
of
F
maximises
w),
yielding
the
steady-state
3tely
continuous
with
theflow
common
support
h] solutions
. in equilibrium,
Furthermore,
they
show
that
equilibrium
for F and
G arein particular
ince
the expected
profit
is πthe
across
all [r,
firms
it holds
n3hich
the
support of
ofthey
maximises
π(p,
w),
yielding
the
expected
consists
aF triple
(r,that
F, π),
such
that
(i ) solutions
r satisfies
givenGFare
andprofit
(ii ) a limiting case when search
Furthermore,
show
the
equilibrium
for(1)Fsteady-state
Taking
this
together
with
gives
equilibsolution
emerges
ce
the=with
expected
profit
flow
across
all
inthe
equilibrium,
itand
holds in
particular
nuous
common
support
[r,support
h] (5)
. firms
π =the
π(w)
for
all
w isonπ the
of F.
Taking
this together
with
(5) as
gives
π(r)
4
3
rium
offer
distribution
frictions
disappear.
l the
to π.support
Furthermore,
they
show
that
the
equilibrium
solutions
for F and
G are As a second extreme, if only
ofcommon
Fwage
maximises
π(p,
w),
yielding
the expected
steady-state
profit
nuous
with
the
support
[r,
h]
.
4
= π(w)
w on all
thefirms
support
of F. Taking
this together
with (5)
gives receive offers (0 < λ < ∞
(r)
= πprofit
pected
flowfor
is πallacross
in equilibrium,
it holds
inunemployed
particular
quilibrium
wage
offer
distribution
workers
0
ypected
continuous
with
the
common
support
h] . [r,
in
to
π.3profit
Furthermore,
they
show
the
equilibrium
solutions
for
F
and
G
are
flow
is
π
across
all
firms
equilibrium,
it
holds
in
particular
4that
and
λ =
0),
the
firms cannot increase their
1
distribution
p − wthis together with (5) gives
1 + κ1 of F. Taking
=uilibrium
π(w) forwage
all woffer
on the
support
, w ∈ [r, it
h]holds
,workforce
−
F (w)
=is π across
(6) by offering higher wages. Thus all
Taking
of 1F.
expected
profit
flow
all
firms
inequilibrium,
particular
continuous
with
the
common
ph]−
=the
π(w)
for all
w on
the
(5)ingives
4 support
p[r,−
w. rthis together with
1 +κ
κ11 support
wage
offer
distribution
firms
offer
the same wage equal to r which in
, wthis
∈ [r,together
h] ,
1 − of F. Taking
F all
(w)w=
=
π(w)
fordensity
the
support
with
(5) gives
= πexpected
4on
κ
p − rin equilibrium,
the
profit
flow
is π
firms
it holds
in particular
the
associated
1across all
wage
offer
distribution
turn
converges
to b. In this case the equilibrium
p−w
1+
κ1associated
the
, w ∈ [r, h] ,
1 − 4 density
Fwage
(w) with
=
earnings
distribution
G limits to a mass point at
brium
offer
distribution
he
associated
density
1r + κof1 F. Taking this together with (5) gives
w on the psupport
= π = π(w) for
κall
−w
1+
1κ1
f(w)1=− , w∈ [r,,h] ,w ∈ [r, h] . b, and the Diamond’s (1971) paradoxical
F (w) =
κdistribution
r−κ
p1r)
− (p
w − w)
1 1 + κ12κ41p −
1(p+
ted density
brium
wage (7) offer
, , ww∈∈[r,[r,h]h]
, . monopsony solution emerges. Moreover, all
F (w)f(w)
= =
1 −
κ
p
−
r−w)
12κ
r)
(precognising
over,density
by substituting1 (6)
(1)−and
that F (h) employment
= 1, we can write
ted
1 (p
wouldthebe uniformly distributed
1+
κ
p
−
w
+
κinto
1
1
f(w)
=
,
w
∈
[r,
h]
.
,
w
∈
[r,
h]
,
1
−
F
(w)
=
across
the
firms
associated
density
ver,
substituting
into
(1)−and
that F (h) = 1, we can write as
thel (w) = m – u by virtue of (5)
+r)
κ1(p
−r
ds ofbysupport
of2κ
F as(6)
(p1κ−
w) precognising
1
f(w)Moreover,
= 1 by substituting
, (6)
w ∈into
[r, h]
. and rec- and (3).
(1)
1+
(pand
− r)recognising
(p
−κw)
1
1 (1)
sbstituting
of support
off(w)
F2κas
associated
density
(6)
into
that
= h]
1, .we can write
r ==
=that
, −Fα)
w(h)
∈
Otherthe
strong predictions follow from the simognising
F (h)
1, we
the
αb−can
+w)
(1write
p,[r,bounds
2κ
(p
−
r)
(p
1
bstituting
(6)
into
(1)
and
recognising
that
F
(h)
=
1,
we
can
write
the
ple
model
outlined above. First, workers with
of
support
of
F
as
1 + κ1
rt of F as
r hand
== αb
++
(1(1
−
α)w
p,∈p,[r,Fh]
f(w)(6)
= into ,−
. = 1, longer
employment
βb
β)
by
substituting
(1)
recognising
that
(h)
we
can
write
the history are predicted to be
2κ1 (p − r) (p − w)
ort of F as
more
likely
to
be located at the upper end of the
αbh+ (1
α)+
p,(1 − β) p,
= −βb
support
of (8) F as r =
into
(1) and
recognising
that F (h) = 1, wage
we can
write the This is because wage growth
distribution.
eby
thesubstituting
weights are (6)
given
by
r = αb + (1 − α) p,
h = βb + (1 − β) p,
in the model results from job-to-job transitions.
2
support
of F
the
weights
areasgiven byr = αb + (1(1
−+
α)κp,
1)
Second, the model implies a positive relation(9) h = αβb=+ (1(1−+β)κ p,)2 + (κ2 − κ ) κ ,
1 + κ1 ) 0
1
1
(1
ts are given by
ship between the size of workforce and the
= αb
βb + (1 − α)
β)1 p,
α rh==
,
2
wage paid by the firm. Firms offering higher
2
(1
+
κ
)
+
(κ
−
κ
)
κ
ts are givenwhere
by
1
0
1
1 .
β =
the weights
(1 +are
κ1 )given
2 by
(1
+
κ
)
+
(κ
−
κ
)
κ
α
=
,
h
=
βb
(1
−
β)
p,
1
0
1
1
wages grow at a larger size because a higher
1
2
weights are given by
+(κ
κ10)2− 2κ1 ) κ1
(1β+ κ=1 )(1
+
.
workers
to a firm from other
α
=
,
(1 + κthe
+ (κ assumptions
− κ1 ) κ1 that ∞wage
rivial solutions are ruled out by making
> p >attracts
b and ∞more
> κi >
0
1 ) natural
+κκ11))κ210
(1 + κ1 )2 +1(κ(1
0−
are βgiven
by
firms
and
reduces
the
quit
rate, λ 1 [1– F (w)].
=weights
0, 1.
=
.
α =
,
(10) 2 natural assumptions that ∞ > p > b and ∞ > κi > 0
ialassociated
solutions are
ruled out
the
1(κ
(1solutions
+byκmaking
)2 +
+
κ(κ
he
equilibrium
forκ
the
earnings
follows directly
from
the steady-state
+
−
κ
1(1
1 )0κ
1distribution
These
results
are
driven
by
on-the-job
search.
1 )0 −
1 ) κ1
2
=
(1 + κ1 ) .
0,
1. outlinedβin (4).
2
onship
αsolutions
(1=
+the
κ1natural
)for
+the
(κassumptions
κ1 ) distribution
κ1 that ∞, >
An
interesting
prediction of the model is that
02 − 1
sassociated
are ruled equilibrium
out by making
p
>
b
and
∞
>
κ
>
0
earnings
follows
directly
from
the
steady-state
i
β = (1 + κ1 ) 2 + (κ0 − κ1 ) κ1 .
change
in the unemployment benefit b does
ship
in (4).
+ κ1 )assumptions
+ (κ0 − κthat
1 ) κ∞
1 > p > b anda∞
s areoutlined
ruled out
by making the (1
natural
> κi > 0
1 7 follows
equilibrium solutions for
the
earnings
distribution
directly
from
the
steady-state
not
affect
equilibrium
unemployment as long as
β =
. that ∞ > p > b and ∞ > κ > 0
(11) 2
solutions
ruled out by making
natural
assumptions
i
d
in (4). aresolutions
(1 +the
κdistribution
κ1 ) κdirectly
1 ) + (κ0 −
1
equilibrium
for the earnings
follows
from
the
steady-state
b
<
p.
For
example,
an
increase in b increases
7
d in (4).
the
reservation
r by virtue of (1). Howociated equilibrium
solutions
for thethe
earnings
follows
from
the ∞
steady-state
solutions
are ruled out
by making
naturaldistribution
assumptions
that directly
∞>p>
b and
> κi > wage
0
7
outlined in (4).
In other words, both
the support and functional ever, to retain a positive workforce, firms offerociated equilibrium
the
distribution follows
directly from
the steady-state
formsolutions
of the for
wage
offer distribution
depends
7 earnings
ing wages
below the new value of r must react
outlined in (4).
only on the structural parameters of the model.
7
The fact that h is a weighed average of b and p
further implies that the7 highest wage offered in
the market is strictly smaller than p.
The main outcome of equilibrium search
theory with on-the-job search is that wages are
dispersed in equilibrium even when all workers
and firms are homogenous. When the arrival
rates of job offers, λ 0 and λ 1, tend to infinity,
the equilibrium earnings distribution G converges to a mass point at p, and both the steadystate unemployment rate and the equilibrium
profit rate tends to zero. Thus the competitive
4
The associated equilibrium solutions for the earnings
distribution follows directly from the steady-state relationship outlined in (4).
by increasing their wage offers which in turn
affects the wage offers of other firms. The net
result is that the exit rate out of unemployment
and thus the unemployment level remain unchanged. Using a similar reasoning one can see
that a decrease in b does not affect unemployment either.
2.2 Employer heterogeneity
The model of the previous section makes several predictions which can be expected to be
consistent with empirical data. However, a closer look at (7) reveals that the density of the
wage offer distribution (and, consequently, that
of the earnings distribution) is strictly increasing and convex on its whole support. This con143
i
i
i
that (pi − w) l(w) = (pi − wi ) l(wi ), where l (w) is as defined in (5), holds for all firms
equilibrium distribution of wage offers:
implies the
with productivity pi offering a wage on the interval [wi , wi ) . This
following
1 + κ1 1 − γ i−1
pi − w
1 + κ1
Finnish Economic
Papers
2/2007
–
Tomi
Kyyrä
1−
, w ∈ [wi
equilibrium distribution of (12)
wage offers:F (w) = κ
1 + κ1
pi − wi
1
tradicts with the shape of wage
the associated
− γ i−1
1 +associated
κ1 1with
pi − w density
1 + κ1distributions
with
density
(12)
F (w)
= as empirical
1 − the
, w ∈ [wi , wi ) ,
usually observed
in the
data
wage
1 + κ1
pi − w i
2.2 Employer heterogeneityκ1
1 + κ1 1 − γ i−1
densities are typically unimodal and skewed
, w ∈ [wi , wi ) ,
(13)
f(w)
=
with
thetail.
associated
density
with a long
right
This calls
to doubt wheth- (13) 2κ
(piexpected
− w) (pi −
wi )
1 be
The model of the previous section makes several predictions
which
can
to
er the simple equilibrium search model1 with
+ κ1 1 − γ i−1
, w ∈that
[wi , w
(13)
f(w)
=
be
consistent
with
empirical
data.
However,
a
closer
look
at
reveals
the
identical
agents
can
provide
an
acceptable
fit
to
i ) ,density
heterogeneity
9
(pi −(7)
2κ1realis(pi − w)
wi )
the wage data. To make the model more
of the wage offer distribution (and, consequently, that of the earnings distribution) is
tic,makes
we extend
thepredictions
basic model
by allowing
for where
previous section
several
which
can be expected
to γ i = F(∞wi), with the convention that γ 0
= 0.9As
shown inwith
Mortensen
(1990)
strictly
increasing
and convexInon
whole
contradicts
the shape
of and in Buremployer
heterogeneity.
theitsfirst
casesupport.
we in- This
empirical data.troduce
However,
a closer look
at (7) reveals
thatdisthe density
dett and Mortensen (1998), an equilibrium is
heterogeneity
assuming
a discrete
wage distributions usually observed in the data as empirical wage densities are typically
∞ 1, …, π
∞ q), where (i) r is
by (r, F, π
tribution
for productivity
firms along
the characterized
distribution (and,
consequently,
that ofacross
the earnings
distribution)
is
the common
wage
satisfying (1), (ii)
unimodal
andMortensen
skewed with
a long
right
tail. and
This calls
to doubtreservation
whether the
simple
lines of
(1990)
and
Burdett
and convex onMortensen
its whole support.
This
contradicts
the shape
F isofthe wage offer distribution given in (12)
(1998). This
is followed
by with
an extenequilibrium search model with identical agents can provide an ∞ acceptable fit to the wage
and (iii) π i = (pi – w) l (w) is the expected steadysioninofthe
Bontemps
al. (2000)
which
allowsare
fortypically
usually observed
data as et
empirical
wage
densities
flow ofbyfirms
with for
productivity pi
dispersion.
data.continuous
To makeproductivity
the model more
realistic, we extendstate
the profit
basic model
allowing
wed with a long right tail. This calls to doubt whether the simple
offering wages on the interval [Kwi, ∞wi), i = 1, 2,
employer heterogeneity. In the first case we introduce heterogeneity assuming a discrete
q. In general, when there are q types of
model with identical
can provide
an acceptable
fit to the …,
wage
2.2.1 Aagents
finite number
of firm
types
the resulting
distribution
of wage offers
distribution for productivity across firms along the linesfirms,
of Mortensen
(1990)
and Burdett
he model moreAssume
realistic,that
wethere
extend
basic ofmodel
allowing
forabsolutely continuous with the support
F is
arethe
q types
firmsby
which
and Mortensen (1998). This is followed by an extension of Bontemps et al. (2000) which
[r, h] and has q –1 “kinks” corresponding to the
differ
labourheterogeneity
productivityassuming
such that
neity. In the first
caseinwetheir
introduce
a discrete
p1 < p
<…< pq. Let
γ i be the fraction
of firms wage cuts (∞w1, ∞w2, …, ∞wq –1).5
allows
for 2continuous
productivity
dispersion.
ductivity across
firms
along the lines
Mortensen
(1990)
and Burdett
Recall from the previous section that the
with
productivity
p orofless.
Keeping
all other
i
equilibrium wage densities are convex over their
aspects
model of
unchanged,
Mortensen
A finite
number
firm types
98). This 2.2.1
is followed
byofanthe
extension
of
Bontemps
et al. (2000) which
(1990) and Burdett and Mortensen (1998) show
whole support when all jobs are equally produc-
us productivitythat
dispersion.
tive. This
property
is at odds
the equilibrium
thiswhich
case results
Assume that
there are q solution
types of in
firms
differ in their
labour
productivity
such with
that the long flat
in a complete segmentation of the wage offer
right tail commonly observed in the wage data.
<range
p2types
< among
... < pqfirm
. Lettypes.
γ i be Optimal
the fraction
firms with
Keeping
i or less.
umber ofp1firm
In productivity
contrast, the pmodel
of this
section with prowageofsetting
ductivity
implies that the wage
implies
that of
thethewage
offered
increases
with (1990)
all other
aspects
model
unchanged,
Mortensen
andheterogeneity
Burdett and Mortensen
are q types of productivity
firms which differ
in their
labourofproductivity
that f is discontinuous at the wage cuts, beand that
all firms
type i offersuchdensity
(1998)
showonthat
the
equilibrium
solution
in bounds
this case results
in a complete segmentation
tween
wages
the
interval
[K
w
, ∞
w
),
where
the
i
i
Let γ i be the fraction of firms with productivity
pi or less. Keeping which it exhibits locally increasing patterns.setting
This theoretical
distribution
of intervals
arerange
such among
that w
∞ i =firm
w
K i +1types.
for i = 1,2, … ,
of the
wage offer
Optimal wage
implies that
the wage can take the
the model unchanged,
Mortensenthe
(1990)
andwage
Burdett
and Mortensen
q –1. In addition,
lowest
offered
is functional form able to mimic the shape oboffered increases
with productivity
and
that
firms ofserved
type iinoffer
on the interval
the wages
data. Moreover,
since more proto the
reservation
wage,
that
w
K 1 all
= r.We
he equilibrium equal
solution
in this
case results
inso
a complete
segmentation
ductive
firms
wages,
also
∞wq = h in
to beare
consistent
[wi , w
where that
the bounds
of order
intervals
such that w
for ioffer
= 1,higher
2, ..., q −
1. In they attract
i ) , define
i = wi+1
ange among firm
Optimalnotation.
wage setting implies that the more
wage workers, face lower quit rates and, consewithtypes.
our previous
addition,
the
lowest wage
offeredofis given
equal to
themust
reservation
wage, so that w1 = r. We also
quently,
In
equilibrium
all
firms
type
ith productivity and that all firms of type i offer wages on the interval are larger on average. High productivity firmsnotation.
make also more profit on average in
have
profit
flow, so
define
thatthe
wq same
= h inexpected
order to be
consistent
withthat
our previous
bounds of intervals
w
=
w
for
i
=
1,
2,
...,
q
−
1.
In
i
equilibrium
than less productive firms.
(pi – w) are
l (w)such
= (pthat
–
w
K
) l
(
w
K
),
where
l (
w)
is
as
dei+1
i i
i
In
equilibrium
all firms
offirms
givenwith
typeproductivmust have the same expected profit flow, so
fined
in
(5),
holds
for
all
t wage offered is equal to the reservation wage, so that w1 = r. We also
wage
on
the interval [Kwi, ∞wis
thatity
(pip−i offering
w) l(w) =a (p
defined
in (5), holds
for all
firms
i). as 2.2.2 i −w
i ) l(wi ), where l (w)
A continuum
of firm
types
in order to beThis
consistent
previous
notation. distribuimplieswith
the our
following
equilibrium
withtion
productivity
pi offering a wage on the interval [w
the(see
following
i , w i ) . This
Bontemps
et implies
al. (2000)
also Bontemps et
of
wage
offers:
all firms of given
type
must
have the same expected profit flow,
so
al., 1999, and Burdett and Mortensen, 1998)
equilibrium distribution of wage offers:
= (pi − wi ) l(wi ), where l (w) is as defined
in (5),
holds for
all firms
1 + κ1 1 − γ i−1
pi −5 wVan den Berg (2003) points out that there may exist
1 + κ1
(w)interval
=
, w ∈characterized
[wi , wi ) , by a different reservation
pi offering (12)
a wage
[w , 1w−) . This implies the following
(12) onFthe
other equilibria
κ1 i i
1 + κ1
pi wage
− wiand a different number
of active firm types. This posution of wage offers:
sibility arises from the two-way relationship between the
with
the
associated
density
reservation wage of the unemployed and minimum produc1 + κ1 1 − γ i−1
pi − w 1 + κ 1 − γ 1 + κ1
tivity level in use. In other words, the reservation wage af.
1
i−1
1(13)
−
, w ∈ [wi , w
i) ,
, fects
w ∈the
[wminimum
f(w)
=wi κ1
1 + κ1
pi −
i , w i ) , level of profitable productivity and vice
2κ1 (pi − w) (pi − wi ) versa.
d density
9
1144
+ κ1 1 − γ i−1
, w ∈ [wi , wi ) ,
f(w) =
2κ1 (pi − w) (pi − wi )
9
n average in equilibrium than less productive firms.
uum of firm types
Finnish Economic Papers 2/2007 – Tomi Kyyrä
an alternative
of Burdett
the BurdettGiven Γ an equilibrium is characterized by
(2000) (seepropose
also Bontemps
et al.,extension
1999, and
and Mortensen,
Mortensen model which allows for continuous
(r, F ) , where (i) the common reservation wage
alternative productivity
extension of the
Burdett-Mortensen
model which allows
for (1) and (ii) the wage offer distribution
dispersion.
For this specification
r satisfies
we suppose
that specification
productivity we
p issuppose
continuously
the workers face while searching F is given by
ctivity dispersion.
For this
that productivity
distributed across active firms according to the
(14) and (16).6 A
s before the equilibrium distri­
only one wage offer can be profit-maximizing
a closed-form expression in general. Despite
distributed across
active function
firms according
to the
distribution
distribution
Γ, with
support
[K p, ∞ p], function
bution oΓ,f wage offers F is absolutely continu
et show
al. (2000)
showone
that
over
K p ≥ r. Bontemps
, where p ≥where
r. Bontemps
et al. (2000)
that only
wageous
offer
canits support [r, h]. Neither K nor F has
ing for each for
firmeach
and the
offer increases
with
firmoptimal
and thewage
optimal
wage offer
in-productivity.
this the model outlined above makes some re­
creases
with
productivity.
It followsdistribution
that there Γ strictions
re exists a direct
map
between
the productivity
and wage on the shape of the wage distribu-
tions, excluding certain shapes for F and G. One
of these restrictions says that for given κ 1,
f (w) (1 + κ 1 [1– F (w)]) decreases over the whole
support of F (i.e. the second-order condition of
F
(K(p))
=
Γ(p),
(14) profit maximization holds everywhere). This
003) points out that there may exist other equilibria characterized by a different
where
K(p)
denotes
the
wage
offered
by
firms
prediction can be used as a specification test in
a different number of active firm types. This possibility arises from the two-way
with
p andand
K minimum
is an increasing
andlevelthe
empirical
analysis of the model. It imposes
the reservation
wageproductivity
of the unemployed
productivity
in use.
In
continuous
function.
differently,
the fracrvation wage affects
the minimum
levelStated
of profitable
productivity
and vicethe
versa.
restriction how steeply f can increase for
e K(p) denotes the wage offered by firms with productivity p and K is an increasing
tion of offers no higher than K(p) equals the given κ 1. Where the density f is decreasing the
continuous
function.
differently,
thep and
fraction
no higheristhan
K(p) met regardless of the
he wage offered
by firms
withwith
productivity
K less.
is of
anoffers
increasing
fraction
ofStated
firms
productivity
p or
condition
obviously
Given
that
K
is
strictly
increasing
in
p,
firms
value
of
κ .
For
large
κ 1 the condition allows f
where
K(p)differently,
denotes
wage
offeredofby
productivity
p and 1K is an increasing
ls
the Stated
fraction
of firmsthe
with
productivity
pfirms
or less.
ion.
the
fraction
offers
nowith
higher
than K(p)
with the lowest productivity K p must offer a to increase quite steeply, but for small κ 1 the
and that
continuous
function.
Stated
ther and
fraction
ofcondition
offers no pis
higher
thaneven
K(p)if f increases only
Given
Kproductivity
is strictly
increasing
in differently,
p, firmswage
with
productivity
must
offer
firms
withoffered
p10
or
less.
wage
equal
to
the
reservation
theincreasing
violated
the
wage
by
firms
with
productivity
p andthe
K lowest
is an
highest
wage
in the
market
is offered
firms slightly.
equals
the
fraction
firms
with
p orby
less.
ge
equal
to the
reservation
wage
rproductivity
and
highest
wage
theoffer
market is offered by
rictly
increasing
in p,offirms
with
the
lowest
productivity
p in
must
tion.
Stated
differently,
the
fraction
ofthe
offers
no the
higher
with
the highest
productivity
∞p. Since
wagethan K(p)
When productivity is dispersed across firms,
Given
that
K
is
strictly
increasing
ininwage
p,
firms
with
the
lowest
productivity
p must offer
with
highest
productivity
Since
offer
for
p inofthe
wage
ris
and
thep for
highest
the
market
isthe
offered
by
offer
unique
any
pwage
inthe
the
interior
ofisΓ,unique
theany
shape
F interior
obviously
depends on the shape
feservation
firmsthe
with
productivity
or p.
less.
optimal
wage
offer wwage
= K(p)
solves
the first-wageofinΓ.the
However,
whereas
κ 1by
> 0 is necessary and
a
wage
equal
to
the
reservation
r
and
the
highest
market
is
offered
the optimal
wage
offer
wwage
=with
K(p)
theproductivity
first-order
∂π (p, w) /∂w = 0.
productivity
p. Since
offersolves
islowest
unique
for
any p incondition
the interior
trictly
increasing
p,the
firms
offer
order in
condition
∂π (p,the
w) / ∂w
= 0. For givenpFmust
sufficient
for wage dispersion, it should be notthe
highest
productivity
p. Since
the wage
offer
is unique
for any p in the interior
given
F with
and
p this
condition
writes
as
p rthis
writes
as
e firms
offer
w
=and
K(p)
solves
the highest
first-order
condition
∂π (p,
w)
0.
ed=that
reservation
wage
andcondition
the
wage
in the market
is /∂w
offered
by productivity dispersion is neither necesnor sufficient
for wage
of
Γ, the optimal
wage offer w = K(p) solves the first-order sary
condition
∂π (p, w) /∂w
= 0. dispersion. In the
condition
writes
as
ts productivity
p. Since
the wage offer is unique for any p in the interior
presence
of
productivity
differentials the degree
(p − w) writes
− (1 +as
κ1 [1 − F (w)]) = 0,
(15) 1 f(w)
For given F
and 2κ
p this
condition
of
wage
dispersion
may
be decomposed into
ge offer w = K(p) solves the first-order condition ∂π (p, w) /∂w = 0.
2κ1 f(w) (p − w) − (1 + κ1 [1 − F (w)]) = 0,
two parts: one which is due to search frictions
ded
that w =
K(p)as≥ r.2κ
The
first-order condition gives an implicit
of the
his
condition
writes
to the
homogeneous case and an(15)
= 0, function
1 f(w) (p − w) − (1 + κ1 [1 − F (w)])analogously
provided
that
w
=
K(p)
≥
r.
The
first-order
conother
offer≥ofr. aThe
firmfirst-order
with productivity
given an
κ1 implicit
and the function
distribution
of which
wages results
offered from
by variation in productiv(p)
conditionp gives
of
the
dition
anκ≥1implicit
function
the wagegivesityanacross
firms.
In particular
2κ1 f(w)that
(p
−w
w)gives
−K(p)
(1 +
[1
−The
F (w)])
= 0, ofcondition
provided
=
r.
first-order
implicit
function
the it can be shown
rith
firms.
As shown
al.distribution
(2002),pthis
the
following
optimal wageofofthe
productivity
andet
the
ofimplies
wages
offered
offer pofingiven
aBontemps
firmκ1with
productivity
given
κ 1 and
thatbythe equilibrium
homogeneous verwage
offer
of
a
firm
with
productivity
p
given
κ
and
the
distribution
of
wages
offered
by
1
the
distribution
of
wages
offered
by
other
firms.
sion
of
the
Burdett-Mortensen
models emerges
forBontemps
a firm
with
productivity
p : implies
ny in
et
al. (2002),
this
optimal of
wage
K(p)
≥
r. The
first-order
condition
givesthe
an following
implicit function
the
As
shown
in
Bontemps
et
al.
(2002),
this
imas
a
limiting
case
when
the
degree
of productivother firms. pAs: shown in Bontemps et al. (2002),
this implies the following optimal wage
p
productivity
dzaoffered
with
productivity
p given
κ1 andoptimal
the distribution
of wages
by
plies the
following
wage
policy
for
ity
dispersion
goes
to
zero
(see
Bontemps
et al.,
2
=
p −productivity
(1 + κ1 [1 − Γ(p)])
2 , for details).
policy for firm
aK(p)
firmwith
with
p
productivity
p: p :
2000,
(1
+
κ
[1
−
Γ(z)])
r
1
n in Bontemps et al. (2002), this implies
dz the following optimal wage
F:
exists a direct map between the productivity
distribution Γ and wage offer distribution F:
) = p − (1 + κ1 [1 − Γ(p)])2
, p
2
dz of the structural
κ1 [1 − Γ(z)])
r (1 +we
r from
(1) =
into
express
2 K(p) in terms
hubstituting
productivity
p : K(p)
(16)
p −(16),
(1 + κ1 [1can
− Γ(p)])
(16) 2,
r (1 + κ1 [1 − Γ(z)])
p
meters
λ0 ,(16),
λ1, δ,we
p, Γ)
. express
om
(1) (b,
into
can
K(p) dz
in terms of
the structural
2
.(1)
p)By
= psubstituting
− (1 + κ1 [1 −
,
r Γ(p)])
from
into (16),bywe(r,can
K(p)
terms of the structural
2where
Given
Γ. an equilibrium
is characterized
F ) , express
(i)
the in
δ, p, Γ)
− Γ(z)])
6common reservation
r (1 + κ1 [1
To be specific, there can be a single equilibrium, mul
tiple equilibria or equilibrium may not exist at all, dependλ0 , λ(ii
p, Γ)
1 , δ,the
rparameters
satisfies
(1)(b,
and
wage
offer
distribution
thethe
workers
face while searching
brium
characterized
F. ) , where
(i) the
common
reservation
om
(1)is into
we )by
can(r,express
K(p)
terms
ing on the values of structural parameters of the model.
By(16),
substituting
r from (1)
intoin(16),
we of
can ex-structural
Given
Γ
an
equilibrium
is
characterized
by
(r,
F
)
,
where
(i)
common
6
Bontemps
et
al.offers
(2000)reservation
and
(16).
As before
the
equilibrium
distribution
ofthe
wage
Fpoint
is out that it is in general hard
nd
(iiΓ)
)by
the
wage
offer
distribution
the
workers
face
while
searching
press
K(p)
in terms
of the
structural
parameters
,given
δ, p,
. (14)
to differentiate between alternative cases. In the text we
(b,
λ 0over
,(1)
λ the
δ,equilibrium
p,
. wage
wage
r6 satisfies
(iiΓ )) the
offer distribution
thea arbitrarily
workers
while
searching
assume
that the
unique equilibrium exists.
1,and
utely
continuous
its
support
h]
. Neither
F has
closed-form
expression
nd
(16).
before
distribution
ofnor
wage
offers
F is face
ibrium
is As
characterized
by
(r, F ) ,[r,
where
(i) theKcommon
reservation
6
F isitsgiven
by this
(14)
and
(16). K
As
the
equilibrium
distribution
of wage
offers F is
neral.
model
outlined
above
makes
someexpression
restrictions on
the shape
over
support
[r,offer
h]the
. Neither
norbefore
F has
a closed-form
and
(ii )Despite
the wage
distribution
the
workers
face while
searching
145
absolutely
continuous
over itsmakes
support
[r, h]
. Neither
Konnor
F shape
hasofathese
closed-form
expression
6
end
wage
distributions,
excluding
certain
shapes
for
F
and
G.
One
restrictions
his
the
model
outlined
above
some
restrictions
the
(16). As before the equilibrium distribution of wage offers F is
in general.
Despite
this(1the
above
makes
some
restrictions
that
for
given
κ1[r,
, f(w)
+ for
κmodel
[1Fnor
−and
Foutlined
(w)])
over
the whole
supportonofthe
F shape
ons,
certain
G.
of these
restrictions
1K
s overexcluding
its support
h] shapes
. Neither
F
hasOne
adecreases
closed-form
expression
the wage
excluding
certain
shapes
forsupport
F and G.
of these
restrictions
the
second-order
profit maximization
holds
everywhere).
This
prediction
,off(w)
(1 + κdistributions,
[1condition
− F (w)])of
decreases
over the
whole
of FOne
1
1
Finnish Economic Papers 2/2007 – Tomi Kyyrä
3. Data
3.1 Sample details
Christensen and Kiefer (1997) discuss data requirements for identification of the structural
parameters of the Burdett-Mortensen model.
They show that the model can be estimated
from data on individual labour market histories
where at least some of the workers are observed
with both unemployment duration and job duration with the associated wage. Empirical analysis of this study is based on a sample of individuals drawn from the worker data of the Integrated Panel of Finnish Companies and Workers
(the IP data).7 Underlying source of information
on workers in the IP data is the Employment
Statistics (ES) database of Statistics Finland.
The ES database is a longitudinal database
which combines information from over 20 administrative registers. Since 1987 the ES database has been updated regularly, and it covers
effectively all people with a permanent residence in Finland.
Each individual in the ES database who holds
a job at the last week of the year is associated
to his or her employer with a company and establishment identifier. The worker panel of the
IP data covers all people from the ES database
with an identifier of the private-sector employer
at least in one of the years between 1988 and
1996. As a result, the underlying worker panel
covers practically all persons who have been
employed in the private sector during the period
1988–1996 (at least at the end of one year). The
total number of persons in the IP data is slightly below two million. For these people a set of
variables, collected by combining the annual
records of the ES database, is available over the
period 1988–1996.
In this paper we focus on a subsample of the
worker panel of the IP data. As a first step we
select all individuals between the ages of 16 and
65 who entered unemployment during 1992. In
choosing a sample of unemployed workers we
follow the practice of Bowlus et al. (2001) and
Bunzel et al. (2001). We exclude workers who
7
A detailed description of the IP data is given in Kor­
keamäki and Kyyrä (2000). More recently, the Finnish Longitudinal Employer-Employee data (FLEED) have replaced
the IP data.
146
have been self-employed as well as those have
been employed by the public sector or nonprofit organization during the period 1990–
1996. These groups are excluded as the underlying model does not describe their labour market
experiences.
For all individuals selected in the sample, we
record the duration of the unemployment spell
(d ) and information on whether unemployment
ended because of finding a job (cd = 0) or for
some other reason (cd = 1).8 Unemployment
spells not followed by a job are treated as rightcensored in the empirical analysis. This may
occur due to a drop out of the labour force, participation in the active labour market programme or the spell continuing beyond the observation period. It should be stressed that we
treat unemployment spells ended in a job replacement programme as right-censored as well.
Thus we make a difference between finding a
job from the open labour market and becoming
employed by labour administrative measures.
For those workers who found a job, we further record the accepted wage (w) and the duration of the subsequent job spell ( j ) along with
the reason for termination. The wage rate is
computed using information on annual earnings
and the days worked. A job spell may end in a
layoff (a = 0, cj = 0), a quit (a = 1, cj = 0) or be
right-censored (cj = 1). Job spells followed by
unemployment are classified to be ended in a
layoff, whereas job spells consecutively followed by another job spell with a new employer are interpreted to be ended in a quit for a
better job. We identify changes in employer by
comparing establishment identifiers attached to
workers on the basis of the employer.9 Job spells
terminated due to a drop out of the labour force
and those continuing beyond the observation
period are treated as right-censored. All durations are measured in months, and wages are
8
If the worker has several unemployment spells started
in 1992, we choose the first one.
9
The establishment identifier is available only at the last
week of each year, and hence we do not observe the employer for jobs that started and ended within the calendar
year. Subsequent job spells result in a quit only if we actually observe the change in the establishment identifier.
Hence, we may be underestimating the number of quits to
some extent.
Finnish Economic Papers 2/2007 – Tomi Kyyrä
converted into monthly rates to match the duration measures.
Recall that our theoretical model is concerned
with the population of homogeneous workers.
While all workers are different in practice, we
cannot allow the parameters to be different for
each individual as the model will be of no use
at all in such a case. Instead we assume that the
labour market consists of a large number of segments, each of which forms a single market of
its own. These segments are assumed to differ
from each others according to observed characteristics of workers. To deal with this kind of
heterogeneity, the model can be applied separately to each group of workers, allowing for all
parameters to vary freely across the groups.
This approach corresponds to controlling observed heterogeneity with a complete set of
discrete­ regressors (Christensen and Kiefer,
1994a). To pursue this approach, we stratify the
data by education, sex and age. We categorize
education as follows: lower vocational education or less (11 years or less), upper vocational
education (12–13 years), lower university (a
Bachelor degree or the lowest level of university education, 13–15 years), and upper university (a Master degree or higher, 16 years or
more). The age groups considered are: 16–21
years, 22–30 years, 31–50 years, and 51–65
years. As only few workers with lower or upper
university education are aged below 22 or above
50, we combine the two lowest age groups as
well as the two highest age groups for these
education groups.
It is worth noting that the segmentation assumption does not preclude firms from employing different types of workers provided that the
production function is additive in different types
of labour inputs (e.g. Van den Berg, 1999). In
this case, there may be several types of jobs
with possibly different productivity levels within a firm but only a particular type of workers
are suitable for each type of job. Since workers
of different type do not compete for the same
jobs, wage offers for workers occupying a particular labour market segment are independent
of labour market conditions in other segments.
This is an unrealistic assumption since heterogeneous workers (say, women and men with the
same education) compete partly for the same
jobs in the real-world labour market. One may
try to relax this restriction by introducing a
more realistic production technology but it will
probably make the model completely intractable. This unappealing implication of the segmentation assumption should be kept in mind
when interpreting our empirical results.
As some of the estimation procedures used
are sensitive to outliers in the wage data, some
concern needs to be taken with our wage measures. Since the monthly wages are computed
from annual earnings without information on
hours worked, the wage data can be expected to
contain some measurement error. To deal with
outliers in the wage data, we first require that
all wages must be at least 80% of the lowest
salary grade of the central government, after
which we trim the lowest and highest 3% of
wage observations in each subgroup.
Finally, the maximum size of the estimation
sample is restricted to 3,000 observations. This
is because the computational burden of the estimation method for the model with a discrete
distribution of productivity increases rapidly
with the sample size and the number of firm
types. Thus, we have drawn random subsamples
of 3,000 workers (after trimming the wage data)
from the underlying groups, defined by education, sex and age, which exceed this size threshold.
3.2 Descriptive statistics
It is worth emphasizing that the period under
investigation is exceptional one. An overheating
period of the Finnish economy in the last years
of the 1980s was followed by a deep recession
in the early 1990s. The annual change in the
GDP was negative during the period 1991–1993,
and in the worst year, 1991, the GDP decreased
by over 7%. According to the Labour Force Survey, the unemployment rate rose from 3.2% in
1990 to over 16% in 1993, remaining at the
level beyond 14.5% until 1996. The labour market experiences of the sampled workers thus
took place in a period of record high and stable
unemployment. We should keep this in mind
when interpreting the results.
Table 1 gives some descriptive statistics for
the worker groups to be analysed. The aggre147
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 1: Summary statistics
N
w
wmin
wmax
d
cd
j
cj
a
Lower vocational and less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
17,051
40,145
73,142
20,129
7,604
13,721
31,212
13,724
7,819
9,526
10,483
10,320
7,071
7,638
7,932
7,871
4,625
4,845
5,067
4,940
4,546
4,607
4,685
4,647
16,713
23,464
26,669
27,002
18,325
18,858
18,172
17,741
7.56
10.49
11.58
16.43
7.07
10.06
11.94
19.17
.78
.58
.57
.73
.80
.71
.66
.79
12.54
12.28
12.52
9.94
12.24
14.63
16.71
13.95
.32
.20
.22
.20
.33
.31
.29
.22
.20
.22
.20
.19
.17
.22
.14
.10
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
9,544
11,233
8,993
1,811
6,769
11,002
6,273
695
7,843
9,298
11,742
13,645
6,919
7,734
9,298
10,284
4,659
4,834
5,278
5,724
4,535
4,680
4,837
5,028
15,324
22,003
28,770
27,963
14,799
16,232
22,809
23,591
5.83
8.81
11.22
18.08
5.41
7.51
10.35
18.98
.78
.57
.60
.79
.73
.60
.65
.82
11.91
14.92
17.82
14.02
11.45
15.72
17.60
13.98
.41
.28
.31
.30
.37
.31
.32
.26
.27
.27
.22
.12
.25
.26
.23
.07
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
5,042
4,495
1,831
1,512
10,531
13,755
8,541
11,241
5,185
5,338
4,725
4,938
23,859
37,047
16,842
28,597
7.36
9.28
6.55
10.83
.59
.54
.56
.61
17.77
18.88
17.31
17.97
.35
.34
.36
.36
.28
.24
.32
.25
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
502
1,321
526
543
12,883
17,874
11,756
12,887
5,636
5,667
5,683
4,831
27,255
53,488
34,397
34,092
7.21
12.12
6.33
10.32
.47
.56
.51
.59
20.87
22.18
18.09
16.64
.35
.38
.40
.34
.55
.32
.38
.32
Notes: N is the number of observations in the underlying population from which the estimation sample was drawn.
w is the average accepted wage (FIM) in the estimation sample. wmin and wmax are the minimum and maximum
of observed wages (FIM) in the estimation sample respectively. d and j are the average durations of unemployment
and job spells respectively. cd is the share of censored unemployment spells, and cj is the share of censored job
spells in the estimation sample. a is the share of uncensored job spells ending in a quit for a better job.
gate number of observations in the underlying and most of subsequent job spells ended in a
group (N) is given in the first column of the layoff.
table. Where N exceeds 3,000, all sample statisUnemployment duration increases with age
tics are computed from a random subsample of and is exceptionally high among low educated
3,000 workers, describing the sample to be used workers aged over 50. There are no clear differin the estimations. It appears that the size of the ences in the average duration of unemployment
underlying worker group is much lower for by sex. The rate of censoring in the unemployhighly educated groups. This does not reflect ment data is found to be very high. It is also
only the education structure of the labour force worth emphasizing that the average duration of
but also a lower incidence of unemployment 36 censored spells is over two times higher than
among more educated workers. The fact that the that of uncensored spells (not shown in the taperiod under investigation is characterized by ble). This is because long-duration spells of
high unemployment levels is reflected to the unemployment are often terminated by labour
figures in the table. Unemployment durations administrative measures. This explains partly
are relatively long with a high rate of censoring, the higher censoring rates for the groups with
148
Finnish Economic Papers 2/2007 – Tomi Kyyrä
the longest unemployment durations. Young job Some technical details on the estimation proceseekers are often regarded as a special target dures are given in the Appendix.
group of the labour administrative measures,
resulting in a relatively highlikelihood
censoringfunction
rate forwithout
specifying
the functional
form for the wage offer distribu
4.1 The
log-likelihood
function
workers aged under 22. There are no large difIn the subsequent sections we report the empirical results. Some technical details on
ferences in job duration across age groups. The structural parameters of interest are (λ 0, λ 1,
, r)given
with in
a scalar
p for the homogeneous modHighly educated workers experience
slightly δare
estimation procedures
the Appendix.
longer job spells and are more likely to quit for el, the set (p1, p2, …, pq) of productivity terms
model with a discrete distribution of proa better job. Compared to unemployment
spells, for the function
4.1 The log-likelihood
job spells are longer on average and the rate of ductivity, and Γ for the model with a continuous
productivity.
correcensoring in the job duration
data
is much parameters
low- distribution
The
structural
of interestofare
(λ0, λ1 , δ, r) With
with athe
scalar
p for the homogen
sponding parameter estimates in hand, we can
er.
(p1the
, p2 , ...,
pq ) ofan
productivity
terms
for the
model
withour
a discrete distribu
10
obtain
estimate for
b using
(1).
Since
Wages increase with age model,
at leastthe
up set
until
were
drawn
from
the inflowdistribution
of unemployinterval 31–50 years of age.ofThere
are no clear
productivity,
and Γ data
for the
model
with
a continuous
of productivity. W
wage differentials between workers at the two ment, we observe a spell of unemployment
the corresponding
estimates
in hand, we can obtain
an estimate for b u
along with
the post-unemployment
destination
lowest levels of education, whereas
higher edu-parameter
10
individual
in the
data.
those whose we observe a
cation yields slightly higher(1).
return.
Given
Since
ouredudata for
wereeach
drawn
from the
inflow
of For
unemployment,
cation and age women receive uniformly lower unemployment ended in a new job we further
unemployment
along
with the
for each individu
observe
thepost-unemployment
wage rate accepted destination
as well as the
wages than men do. Despiteofthe
trimming produration
of
the
subsequent
job
along
with
the
cedure there are still wage the
observations
which
data. For those whose unemployment ended in a new job we further observe
are relatively low compared to minimum re- reason for termination. Since we do not have
rate accepted
as the information
duration of the
job along
complete
on subsequent
all observations,
the with the reaso
quirements, reflecting somewage
measurement
prob-as well
possibility
censored
observations
unemlems in the wage data. Empirical
wage densities
termination.
Since we do
not have of
complete
information
on allonobservations,
the possib
for each worker group are shown in Figures 1 ployment and job durations is explicitly acobservations
on unemployment
and job indicators
durations isc explicitly
accounte
for using censoring
to 3 in the Appendix, where of
thecensored
thick solid
lines counted
d and
cj. cd and cj .
represent the kernel densityusing
estimates
obtained
censoring
indicators
The likelihood contribution from an individusing Gaussian kernels with the bandwidth chocontribution
an individual
is unemployed
for d periods, acc
ual whofrom
is unemployed
forwho
d periods,
accepts
sen by a rule of thumb. Other The
lineslikelihood
depict the
then
a
job
with
an
associated
wage
w,
keeps
that
predicted densities obtainedthen
froma job
the with
different
an associated wage w, keeps that jobs for j periods until he gets lai
specifications of the equilibrium search model job for j periods until he gets laid off (a = 0) or
(a =on.
0) The
or finds
another
job another
(a = 1) has
a general
finds
job (a
= 1) hasform
a general form
and they will be discussed later
empirical wage densities are generally unimodal and
1−c
skewed with a long right tail.
(17)
(17) � = ϕ (d) [f(w)φ (j, a |w )] d ,
ϕ isof the
density function
of unemploywhere ϕ is the densitywhere
function
unemployment
duration,
f is the density of the
4. Econometric analysis
ment duration, f is the density of the wage offer
offer distribution and φdistribution
is the density
job duration
and
condit
andfunction
f is theofdensity
function
of destination
job
We have derived the explicit
solutions for the w. The censoring
and destination
conditional
on the ac- duration c t
on the accepted wage duration
indicator
for unemployment
d
equilibrium wage offer distribution with and cepted wage w. The censoring indicator for una valueFrom
of zero
the unemployment
spell iscdfollowed
by a new
job,ifand a value of
without employer heterogeneity.
the ifasemployment duration
takes a value
of zero
sumptions underlying the theoretical
model
it
is
the
unemployment
spell
is
followed
by
a
new
otherwise.
straightforward to derive distributions for un- job, and a value of one otherwise.
jobKnowloffers arrive Since
at thejob
Poisson
λ0 and
allPoisson
offers are
acceptable
to the u
employment and job durations Since
as well.
offersrate
arrive
at the
rate
λ 0
edge about these distributions
allows
us
to
write
and
all
offers
are
acceptable
to
the
unemployed
ployed in equilibrium, unemployment duration d is exponentially distributed with inte
down the likelihood function for the various
, so that 10 It can be argued that b is the ’deep’ structural paramspecifications of the model.parameter
In the nextλ0section
we derive the general form of the likelihood eter of the model rather than r, which follows implicitly from
strategy1−c
of dunemployed
workers. Howd
function without specifying (18)
the functional form the optimal search
ϕ (d)does
= λnot
e−λ0any
. difference in pracever, this distinction
0 make
for the wage offer distribution. In the subse- tice due to the one-to-one relationship between b and r
10
It can be argued
that outlined
b is the ’deep’
quent sections we report the empirical
results.
in (1). structural parameter of the model rather than r, which fo
implicitly from the optimal search strategy of unemployed workers. However, this distinction doe
make any difference in practice due to the one-to-one relationship between b and r outlined in (1).
149
17
ro if the unemployment spell is followed by a new job, and a value of one
Economic
– Tomi
Kyyrä to the unemoffers arrive at Finnish
the Poisson
rate λ0Papers
and all2/2007
offers are
acceptable
wage
offer distribution generally does not dein equilibrium,
unemployment
duration
d is ex- with
librium, unemployment
duration
d is exponentially
distributed
intensity
pend on λ 0. This observation taken together
with (20) suggests that λ 0 is identified from the
unemployment duration data only. It follows
1−cd −λ0 d
that the estimator of λ 0 is stochastically indeϕ
(d)
=
λ
e
.
(18) 0
pendent of all other parameters of the model,
gued that b is the ’deep’ structural parameter of the model rather than r, which follows
As
workers
search
randomly
among
employers
being robust with respect to different model
the optimal search strategy of unemployed workers. However, this distinction does not
using
a
uniform
sampling
scheme,
the
wage
ofspecifications.
Despite the apparent simplicity
ence in practice due to the one-to-one relationship between b and r outlined in
(1).
fers are random draws from the equilibrium of the log-likelihood function (20), its maximiwage offer distribution F. To derive the condi- zation is a difficult task, and some nonstandard
17 among employers using a uniform sampling scheme, the wage
As workers search randomly
tional distribution of job duration j and destina- procedures are called for. We discuss these difAs can
workers
search
randomly
among
employers
using
a uniform
scheme, the wage
As workers
search
randomly
among
employers
using
a offer
uniform
sampling
scheme,
the wage
offers
are
draws
from
equilibrium
wage
distribution
Tosampling
derive
the
tionrandom
a,
we
useusing
the
standard
competing
risks
ficulties
in
the F.
Appendix.
rch randomly
among
employers
athe
uniform
sampling
scheme,
the
wage
framework
for
exponential
duration
models.
Reoffers
are of
random
draws
from
the
equilibrium
offer
F. To derive the
randomly
among
employers
using
awage
uniform
sampling
scheme,
the
wage
offers
areusing
random
draws
from
the
equilibrium
wage
offer
distribution
F.distribution
To
the
conditional
distribution
job
duration
j the
and
destination
a, wage
we
use
the derive
standard
mong
employers
a uniform
sampling
scheme,
wage
om draws
from
equilibrium
distribution
To
derive
thecan
callthe
that
layoffs occur
at offer
the Poisson
rate dF.
and
4.2 Identical workers and firms
conditional
distribution
duration
j and
destination
a,the
westandard
can use
from
equilibrium
wage
offer
distribution
F.
To
the
conditional
distribution
ofdestination
jobexponential
duration
jjob
and
destination
a,
we can
alternative
offers
arrive
at
the
Poisson
rate
λ derive
competing
risks
framework
for
duration
models.
Recall
thatuse
layoffs
occur
at the standard
mdraws
the equilibrium
wage
offer
distribution
F. To
the
tribution
of the
job
duration
j and
a,of derive
we
can
use
1.the standard
Since
only
wage
offers
exceeding
the
current
We
begin
our
empirical
analysis
with the simcompeting
risks
framework
for
exponential
duration
Recall
that
ution
ofthe
jobPoisson
j framework
destination
a,offers
we
can
use
standard
competing
risks
exponential
duration
models.
Recall
that
layoffs
occur
atlayoffs occur at
rate
δand
and
alternative
arrive
atthe
the
Poisson
rate
λ
only
wage
b
duration
j duration
andexponential
destination
a,
weforcan
use
the
standard
s framework
for
duration
models.
Recall
occur
at models.
1 . Since
wage will be accepted,
the actual
quitthat
ratelayoffs
is plest
specification
of the model with homogenethe
rate
δwill
and
alternative
offers
arrive
at
theisλPoisson
λwage
. Since only
wage
amework
for
exponential
duration
models.
Recall
that
occur
atworkers
the
Poisson
δ Poisson
and
alternative
offers
arrive
at
the
Poisson
rate
.1 Since
only
offers
exceeding
the
current
wage
bereceiving
accepted,
the
actual
quit
rate
λ
(1 firms.
− Frate
(w))
,1the
λ 1 (1–rate
(models.
w)),
the probability
of
ofous
Equilibrium
solutions
rteexponential
duration
Recall
that
layoffs
occur
δ and
alternative
offers
arrive
at
the
Poisson
rate
λlayoffs
.at
Since
only
wage
1and
F 1an
feroffers
times
the exceeding
probability
that
the
received
offer
for
F and
f are given
by
(6)isand
(7) respectively.
offers
the
wage
will
beonly
accepted,
the
quit
rate
and
alternative
arrive
atanthe
Poisson
rate
λ1 . is
Since
wage
offersarrive
exceeding
the
current
wage
will
be
accepted,
quit
rateactual
is λ
(1
−
(w))
,λthe
1 (1 − F (w)) , the
probability
of receiving
offer
the
probability
that
the
received
offer
is F
acceptable
offers
at
the
Poisson
rate
λtimes
.current
Since
only
wage
gative
the current
wage
will
be
accepted,
the
actual
quit
rate
λthe
−
FFor
(w))
, the
1
1 (1actual
is acceptable
given the
current
wage
w. Condithe
reasons1 given in the Appendix, we estiprobability
receiving
an
offer
times
probability
that
the isreceived
is acceptable
current
wage
will
beof
accepted,
the
actual
quit
rate
is
λduration
(1
− Fthe
(w))
,the
the
probability
receiving
anof
offer
times
the
probability
offer
acceptable
given
the
current
wage
w.
Conditional
the
wage
w, received
job
j sample
has offer
an minimum
tional
on
the
current
wage
w,
the
job
jthat
mate
rthe
and
h duration
using
the
and
ge
will be
accepted,
the
actual
quit
rate
isthat
λ
−on
F
(w))
, the
1current
receiving
an
offer
times
the
probability
the
received
offer
is
acceptable
1 (1
has
an
exponential
distribution
with
intensity
maximum
respectively.
These
order
statistics
given
the
current
wage
w.
Conditional
current
w,
thejthis
job
duration
j has an
ving
anexponential
offer
times
the
probability
that
the
received
isδ acceptable
given
the current
wage
Conditional
onacceptable
theoffer
current
wage
w,
job
duration
hasjob
an
distribution
with
intensity
+on
λ
(1j −
Fthe
(w))
.wage
Exit
from
rnttimes
the
that
thew.
received
offer
isparameter
wage
w. probability
Conditional
current
wage
w,
the
job
duration
has
an of
1 the
parameter on
d +the
λ 1 (1–
estimates
(r, h) for each group can be found
F (w)). Exit from this job
exponential
distribution
intensity
δand
+ .λ1,
−
F (w))
. Exit
wage
w.into
Conditional
on
the
current
wage
w,
the
duration
jparameter
an
exponential
intensity
parameter
δ 1+from
λhas
(1
− FTable
(w))
Exit
from
this
job from thistojob
1 (1
into
udistribution
nemployment jpδ/
robability
where
correspond
the
unemployment
occurs
with
[δ
+
(1
−
Fthis
(w))]
exit
into
athey
highernditional
on
the
current
wage
w,with
the
job
has
anλ
stribution
with
intensity
parameter
δoccurs +probability
λduration
−with
Fjob
(w))
. Exit
job
1from
1 (1with
d
/ [d + λ 1 (1– F (w))] and exit into a higher- minimum and maximum accepted wage (i.e.
into
occurs
with
probability
δ/
[δ + Putting
λand
− Fthese
(w))]
and
exit into a higherution
with
intensity
λλ111(1
−
FF(w))
./this
Exit
from
into
unemployment
occurs
with
probability
δ/
[δ
λ(1 (1
−(w))].
Fjob
(w))]
into atogether
higher1 (16 exit
paying
jobprobability
with
[δ +
λ+
−this
F
intensity
parameter
δparameter
+probability
λ unemployment
(1δ/
−δ[δ
F+
(w))
.(1
Exit
from
job
ment
occurs
with
+
−
higherpaying
job 1with
probability
λ 1(w))]
(1– F and
(w)) exit
/ [1d 1+into6ra = w
min and h = wmax). Estimates of other pa(w))
/together
[δ + λ1Putting
(1
F (w))].
these together
thoccurs
with probability
δ/−[δ/Putting
+
λwith
−
Fexit
(w))]
and
exit
into
a−higher1 (1
λ (1w))].
these
together
yields
rameters
of−the
model
are presented
in Table
paying
with
probability
λ(1
(1
−
Finto
(w))
[δ
+−
λ1F(1
F
(w))].
these Putting
together
yields
probability
δ/
[δ
λpaying
(1
Fjob
(w))]
a/λ
higher1λ(1
1 probability
λjob
(1+
−
F
(w))
[δ
+
−
F
(w))].
Putting
these
1 (1–
F 1
1and
2.
yields
λ1 (1
λ1 (1 −
F (w))].these
Putting
these together
λrobability
(w))
/ [δ−+Fλ(w))
−[δF+
(w))].
Putting
together
1 (1 − Fyields
1 (1 /
[1−F (w)])j
It is 1found
that
(19) (19) φ (j, a |w ) = [(1 − a)δ + aλ1 (1 − F (w))]1−cj e−(δ+λ
. λ 0 is uniformly higher than
λ ,
suggesting
that
wage offers arrive more fre1
1−cj −(δ+λ1 [1−F (w)])j
1−c(1
−(δ+λ
1 [1−F (w)])j
1−c
(w)])j
1 [1−F
(j,
a j|we+−(δ+λ
)aλ
= 1[(1
−
+.aλ
F (w))]
e.
.
1 jquently
φ(19)
(j,
|w−
)=
[(1φ −
a)δ
(1
− a)δ
F
(w))]
e−
φ (j, a |w(19)
) = [(1 − a)δ +
aλ1a(1
F (w))]
when unemployed
than when
employed.
Substituting 1−c
(18) and
(19)
into
(17)
and
taking
logarithm
gives
the
individual
contri1−c
−(δ+λ1 [1−F (w)])j
j (w)])j
This
corresponds
to
the
case
where
the
reserva−(δ+λ
[1−F
j
1
a−|w
)=
+(w))]
aλ1 (1 − eF (w))]
e
.
a)δ
+ [(1
aλ1−(1a)δ
−F
.
tion
wage
r
exceeds
the
value
of
non-market
Substituting
(18)
and
(19)
into
(17)
and
taking
logarithm
gives
the
individual
contriSubstituting
(18)
and
(19)
into
(17)
and
taking
Substituting
(18)
and
(19)
into
(17)
and
taking
logarithm
gives
the
individual
contribution
to
the
log-likelihood
function:
g (18) and (19) into (17) and taking logarithm gives the individual contrilogarithm gives the individual contribution to time b. Moreover, as d is uniformly higher than
bution
to the
log-likelihood
function:
8)
and(17)
(19)
into
and
taking
logarithm
gives the
individual contribution
to(17)
the
log-likelihood
function:
into
and
taking
logarithm
gives
the individual
contriog-likelihood
function:
λ , jobs are more likely to end in a layoff than
the
log-likelihood
function:
log � = (1 − cd ) (1 − cj ) ln [(1 − a)δ + aλ1 (11 − F (w))] − λ0 d
in a quit for a better job. Ignoring workers aged
kelihood function:
nction:
log
�
=
(1
−
c
)
(1
−
c
)
ln
[(1
−
+ λ aλ
− λage,
log
�
=
(1
−
c
)
(1
−
c
)
ln
[(1
−
a)δ
+
aλ
(1
− a)δ
F22,
(w))]
λ0. d− F (w))]
d
j
1decreases
0 d being excepbelow
with
og � = (20)
(1 − c(20) )
(1
−
c
)
ln
[(1
−
a)δ
+
aλ
(1
−
F
(w))]
−
λ
d
0−
+ (1 −dcj ) (ln
j)(1
d
j
1 λj0 + ln f(w) −0(δ + 1λ
1 [1 − F (w)])
tionally
low
for
less
educated
workers
aged over
(1c−
clnd(1
)[(1
(1
− c)j(ln
) ln
+(1
aλ
(w))]
−
(1 −
+
aλ
−+F
(w))]
λλ0F
d[1
(20)
−(w)])
(δ + λ
F (w)]) j) .
+
(1F
−
cλj0)d(ln
d )=
j )+
1−(1
(20)
)−
(ln
+
ln
f(w)
−.λ(δ
+ ln
λ
[1 Moreover,
−F
j)1λ .[1 −increases
0+
λ[(1
lna)δ
f(w)
−−1(δc(1
+
(w)])
j)
−
j−
0−
1f(w)
cja)δ
0+
1λ
50.
with
education,
0
Estimations of different specifications of the Burdett-Mortensen model will all
be based on
though not uniformly. In contrast, λ 1 does not
(ln λ0−+(δln+f(w)
λ1 [1
+λ
(10 −
+ cln
λ1 [1−−(δF +
(w)])
j)−
. F (w)]) j) .
cj ) (ln
j )f(w)
Estimations
of different
specifications
of
Burdett-Mortensen
model
will
all be based
on
Estimations
different
specifications
of
the
Burdett-Mortensen
model
will patterns
all
be
based
on
(20),
the onlyofdifference
between
the specifications
being
the
functional
form
assumed
for
Frespect
exhibit
any clear
with
to educadifferent
specifications
of the Burdett-Mortensen
model
will
allthe
be
based
on
of different
specifications
of specifications
thebased
tiononnor
with
respect
to age.
Less educated
11Estimations
(20),
the
difference
between
the
functional
form
rent specifications
of the
Burdett-Mortensen
model
willthe
all
beassumed
(20),
only
difference
between
specifications
being
thedistribution
functional
assumed
F assumed for F
and
f.the
Recall
that
the only
shape
of the
equilibrium
wage
offer
generally
doesfor
not
cations
of
the
Burdett-Mortensen
model
will
all
be based
on
difference
between
the
specifications
being
the
form
forbeing
F form
Burdett-Mortensen
model
willfunctional
all be based
on women
aged
over 50 have the lowest chances of
11
11 specifications
and
f.
Recall
that
the
shape
ofwith
the
generally
does not
being
the
functional
assumed
for
F wage
(20),
the
only
difference
between
theform
specificaand
f.the
that
the
shape
of
the
equilibrium
wage
distribution
does
not
. This
observation
taken
together
(20)offer
suggests
thatoffer
λ0generally
isdistribution
identified
from
depend
on
λthe
en
thebetween
specifications
the
functional
form
assumed
for
Fequilibrium
lence
that
the
shape
ofRecall
equilibrium
wage
offer
distribution
generally
does
not
0being
tions being
the
functional
form
assumed
for
F
This
observation
taken
together
with
(20)
that λfrom
on
λ0 .offer
tofthe
shape
of the
wage
distribution
generally
does
not
. depend
This
observation
taken
together
with
suggests
that
depend
onequilibrium
λwage
0 is identified from
the
unemployment
duration
data
only.
It follows
that
the
estimator
of λλsuggests
is identified
stochastically
theobservation
equilibrium
offer
distribution
generally
does
This
together
suggests
that
λnot
identified
from
00 is
0 is(20)
andtaken
f.011 Recall
thatwith
the (20)
shape
of the equilibrium
quit or layoff, i.e. observations on a are not crucial for iden-
, so that
ponentially distributed with intensity parameter
λ 0, so that
the
duration
data
only.
It
the
of λ0ofisλ stochastically
tification.
Theof
separate
identification
taken
together
with parameters
(20)
suggests
that
λ
isfrom
identified
from that
the
unemployment
duration
data
It
that
thefollows
estimator
λ0estimator
is stochastically
independent
of(20)
allItunemployment
other
of
thefollows
model,
robust
with
respect
to different
1 and d even withnobservation
taken
together
with
suggests
that
λ0only.
isestimator
identified
0 of
ment
duration
data
only.
follows
that
the
λ
is stochastically
0 being
out knowledge of a follows from the fact that the condi-
11
of
all
parameters
of of
the
model,
beingrespect
robust
respect
different
duration
data
only.
It
follows
that
the
ofwith
λ0 isrespect
stochastically
Christensen
and
Kiefer
(1997)
show
that
identification
tional
jobwith
hazard
decreases
with
w.
A higherto
wage
does not
independent
ofindependent
all
other
parameters
the
model,
being
robust
towith
different
model
specifications.
Despite
the
apparent
simplicity
log-likelihood
function
(20),
ata
It
follows
that
the
estimator
ofestimator
λother
isofstochastically
allonly.
other
parameters
of
the
model,
being
tothe
different
0 robust
of all structural parameters of the model does not necessar-
affect the layoff rate but implies a lower quit rate and the
model
specifications.
Despite
the
apparent
simplicity
ofare
the
log-likelihood
other parameters
ofapparent
the
model,
being
robust
with
respect
tofunction
model
specifications.
Despite
the
apparent
simplicity
of extent
the
log-likelihood
function
its
maximization
is
arobust
difficult
task,
and
some
nonstandard
procedures
called
for.
ily
require
information
on
whether
the
job
ends
in
a different
of this effect
depends
on
λ 1(20),
. We function (20),
meters
of
the
model,
being
with
respect
tospell
different
ations.
Despite
the
simplicity
of
the
log-likelihood
(20),
itsof
maximization
is alog-likelihood
difficult
and
some
nonstandard
s. is
Despite
thesimplicity
apparent
simplicity
of task,
the
function
(20),
its maximization
is
a difficult
and
sometask,
nonstandard
procedures
areprocedures
called for. are
We called for. We
these
in the
Appendix.
the
apparent
the
log-likelihood
function
(20),
on
a discuss
difficult
task, difficulties
and
some
nonstandard
procedures
are
called
for.
We
150
11
discuss
these
difficulties
in thefor.
Appendix.
a difficult
task,
and difficulties
some
nonstandard
procedures
are
called
for. We parameters of the model does
Christensen
and
Kiefer
(1997)
show
thatcalled
identification
of all structural
discuss
in the
Appendix.
task,
and
some
nonstandard
procedures
are
We
ifficulties
in
thethese
Appendix.
not necessarily require information on whether the job spell ends in a quit or layoff, i.e. observations on
11
11
Christensen
and
show that
identification
of allwithout
structural
ulties
inathe
Appendix.
and
Kiefer
(1997)of
show
that(1997)
identification
ofofall
structural
parameters
of knowledge
theparameters
model of
does
Appendix.
nd
Kiefer
(1997)
that
allKiefer
structural
parameters
the
areChristensen
notshow
crucial
foridentification
identification.
The
separate
identification
of
λ1model
and δdoes
even
aof the model does
not that
necessarily
require
information
on
whether
the
job
spell
in i.e.
adoes
quit
or affect
layoff,the
i.e.
not necessarily
require
information
on whether
the
spell
inw.
aA
quit
or
layoff,
observations
on observations on
quire information
on the
whether
the
job
ends
in
a quit
or job
layoff,
i.e.ends
observations
onends
follows
from
fact
the spell
conditional
job
hazard
decreases
with
higher
wage
not
Kiefer
(1997)
show
of quit
all
structural
parameters
of
theeffect
model
does on
are
crucial
for
identification.
The
separate
δ even without
howidentification.
that
of aall
structural
parameters
ofthe
theextent
does
aidentification
are
not that
crucial
for
identification.
The
identification
of
λidentification
of a knowledge of a
1 and δofeven
for
The
separate
identification
of
λseparate
δ model
even
without
knowledge
a λof
layoff
rate
but identification
implies
anot
lower
rate
and
of
this
depends
. λ1 andknowledge
1 and
1without
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 2: Estimation results with identical agents
λ0
λ1
δ
p
b
Lower vocational and less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0285
.0399
.0375
.0166
.0282
.0292
.0286
.0109
(.0011)
(.0011)
(.0010)
(.0006)
(.0012)
(.0010)
(.0009)
(.0004)
.0099
.0130
.0117
.0138
.0070
.0082
.0051
.0054
(.0011)
(.0009)
(.0008)
(.0013)
(.0009)
(.0007)
(.0005)
(.0008)
.0470
.0547
.0531
.0701
.0486
.0401
.0384
.0519
(.0024)
(.0018)
(.0018)
(.0029)
(.0025)
(.0017)
(.0015)
(.0024)
42,697
58,533
70,946
78,136
62,871
50,235
65,841
78,050
(3,271)
(2,844)
(3,629)
(5,373)
(6,164)
(3,240)
(5,314)
(9,152)
2,461
754
332
4,532
1,753
1,228
811
3,988
(229)
(279)
(312)
(205)
(257)
(264)
(238)
(114)
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0385
.0494
.0353
.0116
.0505
.0534
.0335
.0095
(.0015)
(.0014)
(.0010)
(.0006)
(.0018)
(.0015)
(.0010)
(.0009)
.0128
.0118
.0080
.0071
.0113
.0106
.0079
.0035
(.0013)
(.0008)
(.0006)
(.0013)
(.0010)
(.0007)
(.0006)
(.0014)
.0405
.0390
.0327
.0454
.0458
.0357
.0324
.0506
(.0022)
(.0014)
(.0012)
(.0029)
(.0022)
(.0013)
(.0013)
(.0055)
29,884
46,673
71,490
94,126
33,238
33,173
55,604
152,083
(1,751)
(1,950)
(3,825)
(13,130)
(1,982)
(1,426)
(3,089)
(55,257)
1,740
—2,358
—3,464
4,687
635
—1,366
—1,489
3,970
(298)
(416)
(535)
(342)
(291)
(346)
(419)
(337)
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0562
.0495
.0675
.0357
(.0016)
(.0013)
(.0025)
(.0015)
.0096
.0075
.0119
.0080
(.0007)
(.0005)
(.0010)
(.0009)
.0296
.0292
.0284
.0293
(.0011)
(.0010)
(.0014)
(.0017)
48,443
91,523
28,814
66,729
(2,054)
(4,474)
(1,272)
(4,988)
—7,453 (697)
—14,893 (1,031)
—5,081 (705)
—4,881 (895)
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0735
.0363
.0770
.0395
(.0047)
(.0016)
(.0050)
(.0027)
.0180
.0078
.0102
.0123
(.0020)
(.0008)
(.0015)
(.0019)
.0184
.0218
.0249
.0310
(.0018)
(.0013)
(.0023)
(.0029)
34,727
110,639
63,666
64,873
(1,350)
(7,135)
(5,355)
(6,069)
—16,266
—20,876
—26,298
—5,850
(2,945)
(2,368)
(3,901)
(1,781)
Notes: Standard errors are in parentheses.
finding a job when unemployed, but women parameter not only a function of unemployment
with university education tend to receive more benefits but also of search costs and perhaps
offers than their male counterparts when unem- even of the disutility of unemployment (Bunzel
ployed. Layoff rate d decreases with education et al., 2001). There is also a tendency for b to
but there is little difference by sex. Workers be lower for women than men among less eduaged below 22 and those aged over 50 are more cated workers aged below 22 and over 50, while
likely to be laid off than other workers.
the reverse is true among workers between 22
Productivity p increases with age and is often and 50 years old.
higher for men than for women with some exEquilibrium unemployment rates, as implied
ceptions. Young workers with upper vocational by the estimates of λ 0 and d, are record high,
education are found to be less productive than being over 50% for some groups (not reported
their less educated co-workers. Otherwise pro- in the table). It is obvious that our sample drawn
ductivity differentials across education groups 37 from the inflow of unemployment is not a repdo not exhibit very clear insight. The value of resentative sample of the labour force but worknon-market time b appears to be positive among ers with poor employment prospects are likely
workers with the lowest level of education. For to be over represented. In particular, the layoff
more educated groups b is typically negative, rates are expected to be much higher in our
being more negative for higher education levels. sample than among the employed population on
Negativity of b reflects the need to interpret this average, which leads to the overwhelmingly
151
Finnish Economic Papers 2/2007 – Tomi Kyyrä
large unemployment rates. Although a flow
sample is a natural choice to explain differences
in post-unemployment wages, it is obvious that
our results do not describe the labour market as
a whole very well.
Overall all structural parameters of the model are estimated accurately and their estimates
allow for a meaningful economic interpretation.
The fit to the wage data is less satisfactory,
however. This is illustrated in Figures 1 to 3 in
the Appendix where empirical wage distributions and predicted wage offer distributions obtained from the different specifications of the
model are shown. The predicted theoretical density for the pure homogeneity model is computed by inserting the parameter estimates into
(7). While the empirical (kernel) densities are
unimodal and skewed with a long right tail, the
equilibrium search model with identical agents
restricts the predicted densities to be increasing
and convex over the whole support. Such a
shape is obviously not supported by the data.
The predicted densities are flat over their whole
support, leading to a poor fit to the wage data in
all worker groups.
Recall that our data do not contain information on working time. This with some other inaccuracies in the available data suggests that
our wage variables are subjected to measurement error. The estimates of the structural parameters of the model can be expected to be
affected by measurement errors. The order statistics estimators of (r, h) are clearly sensitive
to measurement error. The dependence of (r, h)
on other parameters of the model implies that
the maximum likelihood estimates of frictional
parameters (λ 1, d ) are also affected by measurement error in wage data (Van den Berg and Ridder, 1993). Taking the possibility of measurement error in wages explicitly into account may
hence improve the performance of maximum
likelihood estimation.
Following Christensen and Kiefer (1994a)
and Bunzel et al. (2001), we assume that the
wage data are subject to a multiplicative error
term with a Pearson Type V distribution with
unit mean. Estimation results for the homogeneous model with measurement error in wages are
reported in Table 3. While the underlying theoretical model remains unchanged, the estima152
tion procedure deals with a more complex
measurement process now (see the Appendix
for details). In contrast to the previous case, we
now estimate r and h along with other parameters using maximum likelihood.
The results are missing for four groups of
low-educated women as the estimates of r and
h converged to the same value in their cases.
Compared to the previous results, λ 1 is now uniformly much higher and d uniformly slightly
lower, while λ 0 is of course not affect by the
introduction of measurement errors. Among
workers with an upper university degree λ 1 now
exceeds d, while the reverse still holds for other
groups. Moreover p is uniformly lower and b
uniformly higher than previously. The presence
of measurement errors in the wage data suggests that a range of wage offers is narrower
than previously, so less variation in p and b is
required to explain the observations in the data.
The estimates of r and h are generally very
close to each other, resulting in a small difference between p and b. As another implication,
dispersion in the sequence of wages that the
worker can receive over time is predicted to be
quite narrow.
The predicted densities for observed wages
obtained from the measurement error model are
shown in Figures 1 to 3 in the Appendix. It is
evident that the measurement error specification
results in a rather good (statistical) fit to the
wage data. In particular, both tails are captured
quite nicely and the mode point is very close.
As a consequence of the improved fit to the
wage data, the estimates of frictional parameters
are likely to be more appropriate here. On the
other hand, the finding that measurement errors
account for such a large part of the observed
wage variation can be viewed as a failure of the
theoretical model as it implies that the theory is
unable to explain wage dispersion within the
labour market segments. In other words, the results suggest that search frictions do not play an
important role in explaining wage dispersion.
But this is a too hasty conclusion since workers
and jobs are likely to be different within the
narrowly defined labour market segments as
well.
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 3: Estimation results with identical agents and measurement error in wages
λ0
Low voc. & less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0285
.0399
.0375
.0166
λ1
9,278
(920)
11,915
(848)
11,329
(996)
14,336 (1,499)
—
—
8,013 (1,022)
11,457 (1,793)
(.0027)
(.0024)
(.0022)
(.0033)
—
—
.0122 (.0013)
.0124 (.0018)
(.0023)
(.0018)
(.0017)
(.0028)
—
—
.0366 (.0015)
.0502 (.0024)
.0385
.0494
.0353
.0116
.0505
.0300
.0286
.0189
.0133
.0301
.0363
.0353
.0301
.0439
.0412
(.0031)
(.0020)
(.0015)
(.0025)
(.0029)
—
—
.0075 (.0031)
.0437
.0508
.0495
.0658
p
(.0011)
(.0011)
(.0010)
(.0006)
—
—
.0286 (.0009)
.0109 (.0004)
(.0015)
(.0014)
(.0010)
(.0006)
(.0018)
—
—
.0095 (.0009)
.0236
.0317
.0291
.0339
δ
b
7,052
8,121
9,995
8,855
σ
(454)
(475)
(545)
(517)
—
—
7,849
7,026
(420)
(391)
.3012
.3417
.3739
.3743
(.0114)
(.0108)
(.0100)
(.0173)
—
—
.2700 (.0069)
.2883 (.0123)
(.0021)
10,900
(660)
5,611 (409)
.2294
(.0013)
10,682
(635)
8,056 (540)
.3320
(.0012)
16,688 (1,268)
8,244 (889)
.3549
(.0029)
34,816 (4,476)
8,013 (652)
.2657
(.0021)
7,436
(479)
6,464 (364)
.2594
—
—
—
4.3 Discrete
productivity
dispersion
—
—
—
.0496 (.0054)
40,043 (13,510)
5,802 (674)
.2217
(.0183)
(.0091)
(.0170)
(.0327)
(.0076)
—
—
(.0516)
Next we consider the extended model with a discrete distribution of
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
4.3 (.0016)
Discrete
.0562
.0228 productivity
(.0016) we
.0264
15,660
(612)
4,556 error
(607)in wages,
.2506 (.0151)
do (.0011)
notdispersion
allow
for measurement
so comparisons
.0495 (.0013)
.0183 (.0013)
.0265 (.0010)
22,672 (1,127)
4,951 (1,007)
.3550 (.0201)
.0675 (.0025)
.0267 (.0024) version
.0251 (.0014)
9,190 can(570)
7,545
.3018 (.0103)
of
the model
be done
in a (827)
straightforward
manner.Here
The
Next we consider
the extended
model
with
a discrete
distribution
of productivity.
.0357 (.0015)
.0195 (.0022)
.0265 (.0016)
13,554 (1,640)
9,305 (1,332)
.3832 (.0188)
wit
ind
to the log-likelihood
function
is still givenwith
by (20),
the only differen
we do not allow for measurement
error in wages,
so comparisons
the homogeneous
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0735 (.0047)
.0431 (.0047) homogeneous
.0140 (.0016) case
17,131
(528)measurement
2,864 (1,355) errors
.2071 being
(.0210)that F and f
without
version of the.0197
model
can be.0189
done
in a straightforward
manner.
The individual
contribution
.0363 (.0016)
(.0020)
(.0012)
27,896 (1,760)
6,656 (1,901)
.4462 (.0388)
.0770 (.0050)
.0294 (.0044) and
.0205
(.0022)
12,490
(1,350)
9,911
(2,646)
.3409
(.0206)
(13)still
respectively.
Bowlus
etonly
al. (1995)
develop
an estimation
to the
log-likelihood
function
(20),
the
difference
compared
to the
.0395
(.0027)
.0305 (.0050)
.0271 is
(.0028) given
19,978by(1,861)
6,219
(1,642)
.4045 (.0604)
ar
m
to deal
with the errors
ill-behaved
of given
this extension
homogeneous case without
measurement
being likelihood
that F andfunction
f are now
by (12) (se
Notes: Standard errors are in parentheses
details).
the other
parameters
of the method
model are
estimated,
and (13) respectively. Bowlus
et Once
al. (1995)
develop
an estimation
which
is able an e
obtained
using function of this extension (see the Appendix for
to deal with the ill-behaved
likelihood
q
4.3 Discrete productivity
details).dispersion
Once the other parameters of the model
an estimate
of b 2δ
can(1be
λ0 −are
λ1 estimated,
− μi )
(21) (21) b = r −
(pi − wi ) 1 − μ2i −
λ
1
− γ i−
δ
+
λ
1
1
obtained using
i=1
Next we consider the extended model with a
discrete distribution of productivity. Here we
do follows
q
from the substitution
λ0which
− λ1 2δ of
(1 (12)
− μi )into (1).
2
.
not allow for measurement
error
in
wages,
so
,
(21)
b=r−
(pi − wi ) 1 − μi −
γ 1i−1
δ+
, wq ) can be found from T
statistics estimates for
(r,λh)
≡−(w
1 1
comparisons with the homogeneous versionλ1ofOrder
i=1
the model can be done in a straightforward
Other parameter
which follows from the substitution
of (12)estimates
into (1). are shown in Tables 4 to 6. Estimates of
manner. The individual contribution to the log- which follows from the substitution of (12) into
(λ
δ) ≡
are
very close to the estimates obtained fro
1 ,h)
likelihood function isOrder
still given
by estimates
(20),meters
the for(1).
statistics
(r,
(wgenerally
1 , w q ) can be found from Table 1 as previously.
Order
statistics estimates
for (r,Compared
h) = (Kw1, ∞wto
only difference compared to the homogeneous
q) the correspon
model
measurement
wages.
Other parameter estimates
are with
shown
in Tables 4 error
to 6. inEstimates
of the frictional
paracase without measurement errors being that 38
F can be found from Table 1 as previously. Other
the measurement
specification,
there
is a tendency
parameter
estimates
are shown
in Tables
4 tohomogeneous
6. for λ1 to be s
and f are now given
by (12)
andare
(13)
respecmeters
(λ1 , δ)
generally
very
close to error
the estimates
obtained
from
the
Estimates
of
the
frictional
parameters
(
λ ,
d
)
tively. Bowlus et al. (1995) develop an estima1
does not
exhibitCompared
any systematic
Overall
the difference
model with measurementδ error
in wages.
to the differences.
corresponding
estimates
from
tion method which is able to deal with the ill- are generally very close to the estimates obare so moderate
that
can draw
conclusions
tainedthere
from
homogeneous
withsame
measbehaved likelihoodthe
function
of thiserror
extension
measurement
specification,
isthe
aone
tendency
for basically
λ1model
to be the
slightly
smaller
while conc
urement
error
in
wages.
Compared
to
the
cor(see the Appendix for details). Once the other
parametersdifferences.
from this model
andthe
fromdifferences
the homogeneous
with the
δ does not exhibit any systematic
Overall
in thesemodel
estimates
parameters of the model are estimated, an esti- responding estimates from the measurement
error
specification,
there
is a tendency
for λ 1the
to frictional
mate of b can be obtained
using that one can draw
are so moderate
basically
the same
conclusions
concerning
[ Tables
to exhibit
6 aboutany
here ]
be slightly smaller while
d does 4not
parameters from this model and from the homogeneous model with the measurement error.
To get an idea of productivity differences, the average
153 productivit
[ Tables 4 to 6 about here ]
computed and shown in Table 4. Conditional on the education level,
tivity p tends
to increase
with age.productivity
Except for across
workers
over
To get an idea of productivity
differences,
the average
theaged
firms
is 50,
for p to be lower for workers with upper vocational education than f
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 4: Estimation results with discrete productivity dispersion
λ0
λ1
δ
q
p
b
Lower vocational and less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0285
.0399
.0375
.0166
.0282
.0292
.0286
.0109
(.0011)
(.0011)
(.0010)
(.0006)
(.0012)
(.0010)
(.0009)
(.0004)
.0219
.0288
.0234
.0298
.0166
.0193
.0123
.0122
(.0023)
(.0019)
(.0016)
(.0026)
(.0020)
(.0017)
(.0012)
(.0017)
.0444
.0507
.0499
.0662
.0464
.0370
.0365
.0503
(.0023)
(.0018)
(.0017)
(.0028)
(.0025)
(.0017)
(.0015)
(.0024)
5
6
5
6
4
6
6
4
21,539
28,438
36,920
37,507
27,227
22,808
29,556
36,217
4,266
4,049
3,761
5,793
4,069
3,971
3,465
4,715
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0358
.0494
.0353
.0116
.0505
.0534
.0335
.0095
(.0015)
(.0014)
(.0010)
(.0006)
(.0018)
(.0015)
(.0010)
(.0009)
.0254
.0254
.0181
.0133
.0276
.0230
.0174
.0072
(.0024)
(.0016)
(.0013)
(.0023)
(.0024)
(.0015)
(.0013)
(.0029)
.0373
.0353
.0301
.0439
.0420
.0323
.0298
.0498
(.0021)
(.0013)
(.0012)
(.0029)
(.0021)
(.0013)
(.0013)
(.0054)
4
4
5
3
5
6
4
2
17,497
23,945
35,160
53,937
15,697
17,499
27,277
77,089
3,827
2,491
2,320
5,972
3,636
2,547
2,913
4,798
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0562
.0495
.0675
.0357
(.0016)
(.0013)
(.0025)
(.0015)
.0213
.0180
.0245
.0205
(.0014)
(.0012)
(.0019)
(.0021)
.0267
.0266
.0250
.0263
(.0011)
(.0010)
(.0014)
(.0016)
5
6
6
6
25,167
41,202
16,807
29,217
275
—1,758
373
2,460
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0735
.0363
.0770
.0395
(.0047)
(.0015)
(.0050)
(.0027)
.0410
.0208
.0257
.0265
(.0043)
(.0020)
(.0033)
(.0039)
.0138
.0188
.0206
.0275
(.0016)
(.0012)
(.0022)
(.0028)
5
6
5
5
19,919
45,053
26,853
33,265
—863
—584
—3,965
2,109
Notes: Standard errors are in parentheses. q is the number of firm types and p =
average productivity across firms.
systematic differences. Overall the differences
in these estimates are so moderate that one can
draw basically the same conclusions concerning
the frictional parameters from this model and
from the homogeneous model with the measurement error.
To get an idea of productivity differences, the
average productivity across the firms is computed and shown in Table 4. Conditional on the
education level, the average productivity ∞ p tends
to increase with age. Except for workers aged
over 50, there is a tendency for ∞ p to be lower for
workers with upper vocational education than
for those with lower vocational education. Overall these differences across the worker groups
are in line with the findings from the homogeneous model, even though the absolute values
of productivity estimates are quite different.
154
q
i=1
γ i − γ i−1 pi is the
Namely, the average productivity estimates ∞ p
are approximately only half of the corresponding productivity estimates obtained from the
homogeneity model without measurement error
but are clearly higher than the estimates from
the measurement error specification. Furthermore, it turns out that the value of non-market
time b is typically positive, though there are few
groups for which b takes a negative value. Differences in b with respect to education and age
are similar to those observed in the case of the
homogeneous
model.
39
Estimates of productivity parameters, wage
cuts and their weights are shown in Tables 5 and
6. Individual productivity values and wage cuts,
say pi and ∞wi, exhibit increasing patterns with
respect to age among groups with university
education while the picture is less clear for less
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 5: Estimation results with discrete productivity dispersion, continued
p1
p2
p3
p4
p5
Lower vocational and less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
13,643
17,295
22,671
20,879
11,969
12,354
16,398
19,360
30,392
17,190
37,564
26,750
33,784
17,426
17,708
43,418
40,487
19,007
47,012
35,892
114,231
25,163
24,134
123,891
44,072
24,177
56,809
40,360
679,097
42,016
28,632
540,357
108,892
41,443
148,293
62,652
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
13,375
15,146
22,338
38,261
9,757
11,946
16,759
49,328
19,258
21,668
27,545
72,673
15,158
13,236
26,769
152,647
26,360
27,620
30,704
141,549
29,653
16,207
48,668
721,354
62,415
75,768
41,592
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
18,213
25,616
12,812
17,605
20,020
45,985
13,507
25,184
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
16,589
26,719
17,355
22,495
16,960
28,891
18,697
26,773
47,827
45,129
p6
136,864
211,517
125,113
162,682
100,363
39,350
26,410
195,431
87,954
56,156
56,535
24,891
48,888
14,208
30,545
35,263
83,992
17,024
44,957
113,867
156,640
22,032
59,843
446,879
41,729
127,858
19,525
31,640
25,678
34,878
21,083
44,102
50,459
40,743
40,111
58,421
151,310
95,614
168,181
Notes: All pi ’s are statistically significant at the 5 per cent level.
educated workers. Of course, these kinds of
comparisons are complicated by the fact that the
number of firm types q varies across the groups.
Given the shape of the empirical wage distribution, it is not very surprising that the bulk of
firms is found to be low productivity ones.
Firms with the lowest level of productivity represent over half of all firms in each submarket
as g 1 > .5 holds for all worker groups. Some
productivity terms take very high values in some
groups of workers, though their relative weights
are very low (see the associated g i’s). It is worth
noting that all wage differentials that cannot be
explained by differences in the frictional parameters are attributed to productivity differences.
Thus the productivity parameters may capture
also other sources of wage dispersion than pure
productivity differences (Bowlus, 1997).
In Figures 1 to 3 in the Appendix the estimated density functions of the model with discrete productivity dispersion are characterized
by discontinuous jumps at the estimated wage
cuts (∞w1, …, ∞wq –1), between which the densities
exhibit locally increasing patterns. It turns out
that the model with discrete productivity dispersion is able to capture the shape of the wage
distribution quite well but has some difficulties
with the both tails of the distribution. In particular the estimated density has generally the
40 tail which is too fat compared to the obleft
served wage distribution. As a result, the model
has problems also to match the mode point accurately. These failures are not unique to the
Finnish data but appear in the previous empirical applications of the same model as well (see
Bowlus et al., 1995, Bowlus, 1997, Bunzel et
155
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 6: Estimation results with discrete productivity dispersion, continued
w1
w2
w3
w4
w5
Lower voc. & less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
8,700
9,768
12,458
11,330
7,352
7,913
8,270
8,849
10,704
10,008
14,803
11,478
9,303
8,150
8,647
11,177
11,755
10,273
15,684
12,988
14,857
10,445
9,378
13,265
12,511
11,889
16,329
13,132
18,042
11,352
9,629
12,211
12,102
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
9,379
9,563
12,429
16,607
7,118
8,074
10,213
13,736
10,197
10,242
13,539
19,303
8,296
8,349
11,963
17,585
11,728
14,532
14,280
10,438
9,603
18,128
10,646
11,238
11,580
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
12,405
15,407
8,763
10,846
12,811
21,470
8,951
13,638
13,030
21,528
9,742
14,769
17,481
25,153
10,257
15,852
31,672
12,828
21,243
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
12,846
16,284
13,065
13,888
14,301
17,287
13,086
15,327
16,066
19,185
16,651
17,958
17,204
23,020
22,223
21,768
15,874
17,680
29,622
γ1
γ2
γ3
γ4
γ5
.7857
.6143
.7477
.7283
.8027
.7080
.6623
.7947
.8917
.6493
.8620
.7403
.9153
.7357
.7290
.9433
.9297
.6807
.8933
.8183
.9927
.8880
.8063
.9823
.9543
.8053
.9110
.8247
.9467
.9173
.8270
.9410
.9383
.7967
.6310
.6340
.7914
.7297
.6483
.7013
.8653
.8673
.6810
.7100
.8767
.8663
.6957
.8103
.9686
.9450
.9067
.7527
.7490
.7200
.5913
.6153
.5553
.5637
.7102
.6160
.9420
.8757
.9517
.8377
.9770
.9573
.9157
.9213
.7897
.9040
.6199
.7864
.8030
.9057
.7297
.8373
.9563
.9520
.7763
.8635
.9903
.9206
.9533
.7085
.6181
.7122
.6979
.8255
.7062
.8796
.7914
.8872
.8022
.9592
.9006
.9103
al., 2001, and Bowlus et al., 2001). Addition- ductivity parameters pi being in a more reasonally, there are difficulties in explaining wage able range and the associated values of g i being
observations at the upper end of the distribution well below one. Changing the upper value of
which is often thin, covering wide ranges. Add- trimming has of course a direct effect on the
ing more firm types serves as a way of obtain- estimate of h. A brief sensitivity analysis done
ing a more accurate fit to the right tail of the with the different trimming thresholds suggests
distribution. The estimation procedure aims to that the other parameters and conclusions of
attach different firm types to each of these ob- interest are reasonably robust, however.
servations, leading to implausibly high productivity values sometimes. However, these high 4.4 Continuous productivity dispersion
productivity values have only a minor overall
effect as their weights are very low (in terms of Next we turn our attention to the version of the
41model with a continuous productivity distributhe associated values of g i’s).
It should be stressed that the trimming proce- tion. The individual log-likelihood contribution
dure applied to the wage data is related to the has the same general form as previously defined
number of heterogeneity terms needed to match in (20) but the equilibrium distribution of wage
the right hand tail of the wage distribution. In- offers is now given by F(w) = Γ (K–1(p)). This is
deed by trimming a higher fraction of wage val- a highly nonlinear function of unknown paramues from the upper end of the distribution leads eters and does not have a closed-form expresto a smaller choice of q, with the highest pro- sion in general, suggesting that the standard
156
Finnish Economic Papers 2/2007 – Tomi Kyyrä
maximum likelihood estimation could be very
cumbersome. The first point to note is that the
integrals within the expressions for F and f must
be evaluated numerically in each iteration. An
additional difficulty follows from the fact that
the productivity distribution Γ is not generated
by the model but it must be taken as exogenously given. This is not a problem as such but, as
argued by Bontemps et al. (2000), the most
well-known parametric specifications for Γ are
unlikely to generate the wage offer distribution
consistent with the shape usually observed in
the wage data.
To deal with these difficulties Bontemps et al.
(2000) propose a flexible estimation procedure
which does not restrict Γ to belong in any parametric family (see the Appendix for details). In
the first step the wage distribution is estimated
using some nonparametric method, which does
not impose any restrictions on the shape of
equilibrium wage distributions. Conditional on
the nonparametric estimates of F and f, maximum likelihood estimation in the second step
relies only on assumptions about the behaviour
of individual workers who are taking the wage
offer distribution as given. In other words, the
assumptions about the wage-setting strategies
of firms do not affect the estimation of frictional parameters (λ 0, λ 1, d ). Only in the final step
of the estimation procedure, that part of the
model which describes firm behaviour (i.e. the
first-order condition of firm’s problem) is exploited to estimate productivity dispersion.
Bontemps et al. (2000) emphasise that the estimates of the frictional parameters can be expected to be consistent under a wide range of
assumptions on the demand side of the story.
This class of models includes, among others,
the specifications of equilibrium search models
outlined and estimated in the previous sections.
The estimates of the frictional parameters are
given in Table 7, whereas the kernel estimates
of f are shown in Figures 1 to 3 in the Appendix.
It appears that λ 1 is generally very close to the
estimates obtained from the homogeneous model with measurement error and from the model
with discrete productivity dispersion while d is
almost identical. Since the estimates of (λ 0, λ 1,
d ) in this setting can be expected to be robust
with respect to different mechanisms determining the wage distributions, we can conclude that
all specifications of the Burdett-Mortensen
model generating an acceptable fit to the wage
data produce appropriate estimates for the frictional parameters. This suggests that to the extent we are concerned with the estimation of
frictional parameters it is not so important
whether the deviations from the theoretical distribution predicted by the homogeneous model
are attributed to measurement error or to employer heterogeneity as long as the shape of the
wage distribution is captured by the specification. This is essentially the same result as found
by Bunzel et al. (2001) with the Danish data.
The average productivity values in the last
column are computed by taking the averages of
point estimates p over workers who found a job
(see (28) in the Appendix). However, the estimated relationship between the wage offer and
productivity is not consistent with the theory.
The condition that f (w)(1 + k 1 [1–F(w)]) decreases everywhere is violated for small wages
in all worker groups. In other words, the model
fails to capture a steeply increasing wage density observed at the lower end of the wage distribution. Recall that the model with a discrete
distribution of productivity also fails to explain
the shape of the left tail of the wage distribution.
The increasing pattern of f (w)(1 + k 1 [1–
F(w)]) on small wages implies that the relationship K(p) is downward-sloping for small values
of p. This suggests the wages offered by some
less productive firms are not optimal as they
could increase their profits by reducing their
wages. A binding minimum wage is an obvious
candidate to explain this failure of the model, as
it may prevent low paying firms from lowering
their wages further. As pointed out by Van den
Berg and Van Vuuren (2000), omitted worker
heterogeneity may also provide an explanation.
To see this, suppose that workers within a given
labour market segment are heterogeneous with
respect to their value of non-market time (this
heterogeneity may result, for example, from differences in unemployment benefits or in the
value of leisure). In this case workers will apply
different reservation wages when searching
from unemployment. Thus a firm which lowers
157
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Table 7: Estimation results with continuous productivity dispersion
λ0
λ1
δ
p
Lower vocational and less
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0285
.0399
.0375
.0166
.0282
.0292
.0286
.0109
(.0009)
(.0014)
(.0016)
(.0007)
(.0010)
(.0010)
(.0011)
(.0006)
.0234
.0291
.0254
.0298
.0179
.0194
.0118
.0117
(.0029)
(.0021)
(.0022)
(.0030)
(.0025)
(.0019)
(.0011)
(.0016)
.0437
.0508
.0495
.0658
.0454
.0368
.0366
.0502
(.0034)
(.0026)
(.0025)
(.0040)
(.0032)
(.0022)
(.0019)
(.0034)
20,921
28,443
34,973
36,894
25,973
22,520
30,019
38,504
Upper vocational
Men, 16-21
Men, 22-30
Men, 31-50
Men, 51-65
Women, 16-21
Women, 22-30
Women, 31-50
Women, 51-65
.0385
.0494
.0353
.0116
.0505
.0534
.0335
.0095
(.0014)
(.0017)
(.0012)
(.0006)
(.0023)
(.0019)
(.0013)
(.0012)
.0293
.0262
.0183
.0129
.0280
.0232
.0184
.0077
(.0026)
(.0020)
(.0014)
(.0026)
(.0026)
(.0017)
(.0015)
(.0034)
.0362
.0352
.0301
.0440
.0412
.0322
.0296
.0495
(.0029)
(.0019)
(.0018)
(.0043)
(.0029)
(.0017)
(.0016)
(.0075)
16,134
23,465
33,879
56,461
15,990
17,573
26,052
71,444
Lower university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0562
.0495
.0675
.0357
(.0018)
(.0018)
(.0026)
(.0015)
.0224
.0174
.0245
.0185
(.0014)
(.0011)
(.0025)
(.0020)
.0264
.0266
.0251
.0265
(.0014)
(.0013)
(.0018)
(.0020)
24,535
42,881
16,971
31,605
Upper university
Men, 16-30
Men, 31-65
Women, 16-30
Women, 31-65
.0735
.0363
.0770
.0395
(.0067)
(.0018)
(.0051)
(.0031)
.0384
.0189
.0271
.0294
(.0054)
(.0018)
(.0042)
(.0051)
.0139
.0190
.0204
.0272
(.0021)
(.0016)
(.0026)
(.0035)
20,448
48,668
24,157
30,916
Notes: Standard errors are in parentheses. p is the average productivity over workers who found a job.
its wage offer may become unattractive for
some groups of workers. The firms should take
this effect into account when setting wages,
which may explain the failure of the theoretical
model at the lower end of the wage distribution.
5. Conclusion
This paper has provided quite an extensive
structural empirical analysis of various specifications of the Burdett-Mortensen model. We
found that, in the absence of measurement error
in wages, the equilibrium search model with
identical agents does not fit to the wage data.
This failure is due to the prediction of the theory that the shape of the equilibrium wage density is convex which is at odds with the wage
158
data. Introduction of the measurement error in
wages or employer heterogeneity in terms of
labour productivity across firms provides a way
of making the model more flexible. These extended versions of the basic model proved to
give a much better fit to the wage data. The
frictional­ parameters of the model, i.e. the layoff rate and arrival rates of job offers, were
found to be fairly robust across the model specifications which fit to the wage data. Stated differently, it does not make much difference for
the estimates of the frictional parameters whether 42
the wage distribution is matched by allowing
for the measurement error in wages or unobserved productivity differences. However, the
estimates of the other parameters – the value of
non-market time and productivity terms – vary
across the different specifications to a large extent.
Finnish Economic Papers 2/2007 – Tomi Kyyrä
In the case of the homogeneous model with
the measurement error in wages almost all wage
dispersion was attributed to the measurement
error. This indicates that the model without (unobserved) worker or employer heterogeneity
cannot explain the observed wage variation. Although the equilibrium models with employer
heterogeneity match the overall shape of the
wage distributions relatively well, they do have
problems in explaining the shape of the lower
end of the wage distribution, and in particular
the testable theoretical conditions implied by
the model with continuous productivity dispersion were rejected in the empirical analysis. Of
course, one can expect that incorporating employer heterogeneity and measurement error
into the same model provides a way of capturing the shape of the lower end of the wage distribution as well. On the other hand, unobserved
heterogeneity and measurement errors allowed
for in a sufficiently flexible form can be used to
’explain’ any discrepancy between the theory
and data.
One may call into question whether introducing more unobservables in the empirical analysis of equilibrium search models stands for any
progress. As long as the shape of empirical
wage distributions is captured mainly by unobservables, we cannot be very satisfied with our
empirical analysis. Since the equilibrium search
theory describes the joint behaviour of workers
and firms, a natural way to proceed is to exploit
data on both sides of the labour market. Matched
firm-worker data are currently available for
many countries, including Finland. Such data
contain information on firm sizes, labour turnover and productivity, opening up new possibilities for testing existing models and, more importantly, for estimating more general models
(see Postel-Vinay and Robin, 2002, and Christensen et al., 2005, for steps in this direction).
But it may be difficult to make significant
progress without reconsidering the theoretical
structure of the model. In particular, the demand
side of the equilibrium search story raises
doubts. The assumption that the only choice the
firm has to make is to set its wage optimally is
obviously not realistic. A consequence is that
the labour demand side of the BurdettMortensen model is too naive for describing
observed labour turnover in the data. In reality
firms face external shocks which lead to the
creation and destructions of jobs as wages cannot be fully adjusted in response to these shocks.
Equilibrium search models with endogenous job
creation and destruction are necessarily much
more complicated than the Burdett-Mortensen
model. Therefore, a serious challenge for empirical equilibrium analysis of the labour market
is to introduce endogenous labour turnover
without making the resulting model empirically
completely intractable.
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r) . The properties
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1 , δ, p,however.
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andusing
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esti- the the
fromone
(λ , λ , δ,
p, r) tothe
(λ estimation
, 0λ, λ , 1δ,, dh,
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0 simulated
1
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0
the (1993).
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solve
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this route,
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system
(8) and(8)
(9)and
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“Wage
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.
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.
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the
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(
λ , λ , d
)
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1
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the superco
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√
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to and
theirtherefore
trueinvalues
at treat
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in√ n,
the maximum
0 , λ1 , δ),
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This
Van den Berg, G. J., and G. Ridder
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“Estimating
inference
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(λ0of
, λthe
and therefore
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asthey
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exogenou
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ignoring variation
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of (r, h)are
does
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43–56. Eds.ofH.the not
lihood estimation
frictional
parameters,
even
though
they
are
not
exogenous
affect asymptotic inference about (λ 0, λ (λ
1, 0d, ),
and Kiefer,
λ1 , δ, r, h) , w
Bunzel, P. Jensen, and N. C.Christensen
Westergård-Nielsen.
Am- 1994b). Given the consistent estimates of
and therefore
treat (r, h)
as fixed
in 0the
sterdam: North-Holland. Christensen and Kiefer, 1994b).
Given we
the can
consistent
estimates
of (λ
, λ1 , δ, r, h) , w
estimate
p and
b using (22)
and (23)
and compute
their of
standard
errors using the
Van den Berg, G. J., and G. Ridder
(1998).
“An equilibmaximum
likelihood
estimation
the frictionrium search model of the labor
market.”
Econometrica
estimate
p and
b using
(22)
and (23)
and
compute
errors using the
al 2001,
parameters,
even
thoughtheir
they standard
are not exogemethod (see
Bunzel
et al.,
for details).
66, 1183–1221.
nous
(see
Christensen
and
Kiefer,
1994b).
GivVan den Berg, G. J., and A. van
Vuuren
(2000).
“Using
method
(see
Bunzel
et al., 2001, for details).
firm data to assess the performance of equilibrium search en the consistent estimates of (λ 0, λ 1, d, r ,h) , we
and firms with measurement error
models of the labour market.”A.2
AnnalesIdentical
d’Economie etworkers
de
can estimate p and b using (22) and (23) and
Statistique 67–68, 227–256.A.2 Identical workers and firms with measurement error
compute
standard
usingobservation
the delta in the data
To deal with measurement
errors, their
we assume
thaterrors
the wage
method
(see
Bunzel
et
al.,
2001,
for
details).
To deal with measurement errors, we assume that the wage observation in the data
x, is the product of the true unobserved wage w and an error term ε, so that x =
Appendix
x, is the product of the true unobserved wage w and an error term ε, so that x =
The multiplicative
errorworkers
is assumed
to bewith
independently and ident
A.2Identical
and firms
A.1This appendix gives details
of the estima-measurement
measurement
error
The
multiplicative
measurement
error
is
assumed
to
be
independently
and ident
tion procedures. Identical workers and
31
firms without measurement error
To deal with measurement errors, we assume
31
Substituting (6) and (7) into (20) and summing that the wage observation in the data, say x, is
over the observations gives the log-likelihood
function in terms of (λ 0, λ 1, d, p ,r) . The properties of the maximum likelihood estimators are
not standard in this case, however. This is due
to the dependence of the support of F on the
unknown parameters of the model. Kiefer and
Neumann (1993) show that one can simplify the
estimation problem considerably by reparametrizing the model from (λ 0, λ 1, d, p ,r) to
160
the product of the true unobserved wage w and
an error term e, so that x = w · e. The multiplicative measurement error is assumed to be independently and identically distributed across individuals and to be independent of all other
variables in the model. Following Christensen
and Kiefer (1994a) and Bunzel et al. (2001), we
assume that e has a Pearson Type V distribution
with unit mean, variance s 2 and density
tributed across individuals and to be independent of all other variables in the model.
A.3
Discrete productivity dispersion
llowing Christensen and Kiefer (1994a) and Bunzel et al. (2001), we assume that ε has
In
addition
to the
problem
that the
bounds
of the
support
of F depend on
Finnish
Economic
Papers
2/2007
– Tomi
Kyyrä
Pearson
V be
distribution
withofunit
mean,variables
variance
σ 2 the
and
density
ividualsType
and to
independent
all other
in
model.
−2 parameters,
the
model specification
with acan
discrete producti
(∞estimation
w1, …, ∞wq –1of). the
An estimation
method which
−2 2+σ
1 +etσal.
1 +that
σ −2 ε has
n and Kiefer (1994a) and Bunzel
(2001), we assume
4)
gε (ε) =
exp
−
,
deal
with
these
complications
has
been
devel(24) distributed across individuals
and
be εindependent
offurther
allε other
variables in
3+σ −2bution is
complicated
bythe
themodel.
fact that the likelihood function is not di
(2
Γ
+ σto−2
oped by Bowlus et al. (1995). Once again it is
tribution with unit mean, variance
σ 2) ·and
density
Following
Christensen
and
Kiefer(1994a)
and
Bunzel
et al.cut
(2001),
assume
has estimation
denotes
the
wage
points
(w
...,reparametrize
wthat
).ε An
the−2
2+σ
−2 function.
convenient
the modelmethod
in a sim-which can deal
1 ,to
q−1 error
at
here
Γ
gamma
A consequence
of allowing
for
thewe
measurement
1+σ
1 + σ −2
ilar
fashion
as
was
done
in
the
homogeneous
. expwith
gεPearson
(ε) = Type V distribution
−
, variance σ 2 and
density
complications
hasthe
been developed
by Bowlus et al. (1995). Once again it is
−2
that weand
add
integral
each
wage
observation
in
likelihood
viduals
be−2
independent
offor
allunit
other
variables
in the model.
3+σ
need
ε mean,
Γ
(2to
+toσ
) · εan
case.individual
Evaluating
(12) at w = ∞wi and solving for
−2
2+σ
−2
−2
to assume
reparametrize
model
in a similar
fashion asbetween
was done
1(17)
+
gives
the following
relationship
thein the homogen
1 + σεphas
ntribution.
Formally,
weBunzel
replaceet
by forwe
and
Kiefer (1994a)
and
al.σ (2001),
that
ithe
amma
of
allowing
the
measurement
error
24) function. A consequence
gε (ε) =
exp
−
,
−2
Γ
where Γ̃.denotes
the
gamma
function.
A
conseproductivity
terms,
wage
cuts
and
the
fractions
−2
3+σ
x/r
ε
1−c
d = w and solving for p gives the following relationship b
(2 +σσ
) · εdensity
2 and
x 1
(12)
at w
i
i
withquence
unit
mean,
variance
x
dibution
an integral
for each
observation
theEvaluating
individual
likelihood
in
ofϕwage
allowing
for
the
measurement
error
of firm
φ
j,
a
(ε)dε
, types:
f
g
5)
�
=
(d)
ε
−2
where Γ denotes
the
gamma
function.
A
consequence
of
allowing
for
the
measurement
error
2+σ
ε
ε
ε
−2
−2
productivity
wage cuts and the fractions of firm types:
is σ(17)
that
to x/h
add an
for
each
wageterms,
1+
1 +integral
σof
y, we
replace
by
oss
individuals
andwe
to need
be independent
all other
variables
in the model.
(ε) = observation −2
exp
−
,
in
the
individual
likelihood
contri
s
that
we
need
to
add
an
integral
for
each
wage
observation
in
the
individual
−2
3+σ
1 likelihood
ε
1−cdsection, and 1/ε is the
here ϕ,Γφ(2and
in
the previous
Jacobian
of the μ2i
σ f )are
·(1994a)
ε as
x given
and
xwe
Bunzel
(26)
=
wi −
w , i = 1, 2, ..., q,
tensen and+x/r
Kiefer
et (17)
al. (2001),
we assume(26) that εpihas
bution.
Formally,
by
1replace
2
1 − μi
1 − μ2i i
φ consequence
j, a the
, given the error
freplace
g(17)
=
ϕ (d)
ε (ε)dε
ontribution.
Formally,
wetrue
byfor2the
ansformation
between
and
observed
wages
term.
In
this
case
we
mma
function.
A
of
allowing
measurement
error
ε mean,
ε ε
x/h
V distribution
with unit
1−cd
variance σ and density
x x 1
x/r −2h)
not
use
order
statistics
for
(r,
but
estimate
them
simultaneously
Jacobian
an
integral
for
each
wage
observation
in1/ε
the
individual
likelihood
is
2+σ
e25)
as given in
the
previous
section,
and
the
of
the , with (λ0 , λ1 , δ, σ) 32
−2
−2
φ j, a 1 + σf where gε (ε)dε where
(25) 1 �+=σ ϕ (d)
ε
ε
ε
gε (ε) =
− on (25), ,in which the integral is evaluated
x/h exp
maximizing
the
likelihood
wethe
replace
(17)
by
3+σ −2
observed
ε term. In this case we
en
true and
given based
the error
Γ
(2
+ σ −2 )wages
· εfunction
1−c
where ϕ, φx/rand f are
as
given
in
the
previous
section, and 1/ε is the Jacobian
theγ i )
(1 −
1 + κ1 of
d
xthem
The
merically
in h)
each
of measurement
where
∈ (0, 1) , i = 1, 2, ..., q.
xA
1 presence
(27)
μi = the support
for (r,
butiteration.
simultaneously
(λ0 , λerrors
, δ, σ)makes
sistics
the gamma
function.
of allowing
forwith
the measurement
error
estimate
1(27) . fconsequence
φ
j,
a
(ε)dε
,
ϕ
(d)
g
1
−
γ i−1
1
+
κ
ε
1 case we
ransformation
between
the
true
and
observed
wages
given
the
error
term.
In
this
ε
ε wages
ε
x/h
the
distribution
of
observed
independent
ofκ the
parameters, so the
kelihood
function
based
on (25),
which
the in
evaluated
to add
an
integral
for
each
wagein
observation
(1 is−unknown
γ i ) likelihood
1integral
+the
1 individual
simultaneously
i ’s∈out
(0,
,(12)
i=
1,
=
(27)
μ
Substituting
p
ofwith
and
using
do
not
use
order
statistics
for
(r,
h)
but
estimate
them
(λ
λ(13)
, δ,q.
σ) (26) allows us to write the likelih
i
0 , 2,
1...,
as given in the previous
section,
and 1/εinisthis
of the 1)
aximum
estimation
is standard
γ i−1
1the
+case.
κJacobian
1 1−
teration.likelihood
presence
of measurement
the
ormally,
weThe
replace
by
support
where
ϕ(17)
, φ and
f are as givenerrors
in themakes
previous
in
terms
λ0,we
λ1 , δ,integral
r, h, w1 ,is...,evaluated
wq−1 , γ 1 , ..., γ q−1 . Since there are kinks at
by
maximizing
the
likelihood
function
based
on
(25),
in of
which
the
1−c
n With
the
true
and
observed
wages
given
the
error
term.
In
d this case
the
estimates
of
(λ
,
λ
,
δ,
r,
h)
in
hand,
the
estimates
of
(p,
b)
can
be
computed
section,
and
1/
e
is
the
Jacobian
of
the
transforSubstituting
p
’s
out
of
(12)
and
(13)
(26)
x/r
0
1
Substituting
(13) usingso(26)
function
observed wages independent
the
unknown
parameters,
theallows usi to write the likelihoodusing
xout
1 of (12) and
x ofpi ’s
φ
j,
a
(ε)dε
,
f
g
�
=
ϕ
(d)
in
F,
the
density
f
and
hence
the
likelihood
function
is
discontinuous
at the
mation
between
the
true
and
observed
wages
allows
us
to
write
the
likelihood
function
in
numerically
in
each
iteration.
The
presence
of
measurement
errors
makes
the
support
ε
tics
for (r,
h) (23)
butx/h
estimate
them
simultaneously
with (λ0 , λ
1 , δ, σ) for observed wages
ng (22)
and
asinbefore.
predicted
εIn
ε1 , δ,ε 1-3
are kinks at the wage cuts
terms
of
λFigures
r, h,the
w
, γdensities
γ q−1 of. Since
estimation
isgiven
standard
in this
case.
0
1 , ...,
q−1use
1 , ...,
the
error
term.
In, λthis
case
we
do wnot
terms
(λ 0, λ there
1, d, r ,h, ∞w1, …, ∞wq –1, γ 1, …, γ q –1).
Bowlus
et al. (1995)
show that
the maximum likelihood estimates
f the distribution
of observed
wages
independent
of isthe
unknown
parameters,
so the
function
based
onprevious
(25),
insection,
which
the
integral
evaluated
elihood
estimates
into
d
fof are
given
in
and
is likelihood
the
Jacobian
of
theisare
statistics
for
(r, h)
estimate
them
Since
there
kinks at theat
wage
cuts
in F, the
F,the
theparameter
density
fbut
and
hence
the
function
discontinuous
these
points.
sobtained
(λ0 ,as
λ1by
,order
δ,inserting
r, h)
inthe
hand,
the
estimates
of
(p,1/ε
b)
can
besicomputed
(wcase.
wq−1
) density
come from
the hence
set of observed
wages.function
As in theishomogenous ca
multaneously
with
(isλ 0x/r
, λ 1, d, x
s )in
bythis
maximizing
f and
the likelihood
maximum
likelihood
estimation
standard
1 , ...,
ration.
The
presence
of
measurement
errors
makes
the
support
1
between
true
and1-3
observed
wages
the
error
term.the
In this
case welikelihood estimates of wage cuts
Bowlus
et
al.
(1995)
that
maximum
s before. the
Inthe
Figures
the
predicted
densities
for
fgiven
(ε)dε.
gεshow
likelihood
function
based
onε (25),
in observed
which wages
discontinuous
at these points.
ε
der
statistics
to
estimate
(r,be
h).computed
Conditional on these estimates, the likelihood
With
the
estimates
of
(λ
,
λ
,
δ,
r,
h)
in
hand,
the
estimates
of
(p,
b) can
x/h
0
1
observed
wages
independent
of thethem
unknown
parameters,
the
er
statistics
for integral
(r,
h)
simultaneously
with
, λ1 , δ,As
σ)
the
evaluated
numerically
in of
each
it- so (λ
etinal.the
(1995)
show that
thewe
maximum
0Bowlus
(wbut
...,estimate
wq−1
) come
from
the set
observed
wages.
homogenous
case,
use orting
the parameter
estimates
into
1 , is
be
maximized
using
an
iterative
procedure
with
two
steps
inq –each
iteration. In
eration.
The
presence
of
measurement
errors
likelihood
estimates
of
wage
cuts
(∞
w
, …, ∞
w
using
(22)
and
(23)
as
before.
In
Figures
1-3
the
predicted
densities
for
observed
wages
1
1)
x/r
timation
is
standard
in
this
case.
3
Discrete function
productivity
dispersion
the likelihood
based
on
in which
integral isonevaluated
x statistics
1
der
to(25),
estimate
(r, h).the
Conditional
these
estimates,
likelihood
function
makesf the
support
of
the distribution
of ob- come
from
the set ofthe
observed
wages.
As incan
the
gε (ε)dε.
likelihood
function is maximized with respect to (w1 , ..., wq−1 ) holding (
re
obtained
by
inserting
estimates
into
ε εthe
of (λ
, λ1 , δ,served
r,x/h
h)
inwages
hand,
theparameter
estimates
ofthe
(p,unknown
b)the
can
bemakes
computed
0iteration.
each
The
of using
measurement
errors
the
support
independent
of
pahomogenous
case,
weiteration.
use orderInstatistics
esbepresence
maximized
an
iterative
procedure
with
two steps
each
the first to
step
addition to the problem
that the
bounds
of
the
support
of
F depend
oninunknown
x/r
x
1
fixed,
using
simulated
annealing
which
randomly
searches
over
the possible w
rameters,
so
the
maximum
likelihood
estimation
timate
(r, h).
Conditional
on
these
estimates,
the
before.
In Figures
1-3 the
predicted of
densities
observed
wages so the
f theis unknown
gfor
tion
of observed
wages
independent
parameters,
ε (ε)dε.
oductivity
the in
likelihood
function
respect
to (w
holding
(r, h, λusing
1 , ..., wdistriq−1
0 , λ1 , δ)
ε maximized
ε
rameters, the
estimation
of
thecase.
model
with with
a discrete
productivity
isdispersion
standard
this
likelihood
function
can) be
maximized
an
x/h specification
12
binations according an optimal stopping rule. Given the estimates of (w1 ,
ng
theestimation
parameter
into
hood
isestimates
standard
in simulated
this
With
the estimates
of case.
(λ 0, λ annealing
hand,randomly
iterative
procedure
two steps
in each
itera1, d, r ,h) in
fixed,
using
which
over with
the possible
wage
cut com
tion
is
further
complicated
by
the
fact
that
the
likelihood
issearches
not differentiable
roblemDiscrete
thatthe
the
bounds
support
F depend
on function
unknown
x 1 ofof the
x/restimates
corresponding
discontinuity
points
in
the
wage
offer
distribution
γ 1 , ..., γ
(p, b)
can beofcomputed
using
tion.
In
the
first
step
the
likelihood
function
is
A.3
productivity
dispersion
12
timates of (λ0 , λf1 , δ,binations
r, h)gin
hand,
the
estimates
of
(p,
b)
can
be
computed
(ε)dε.
according
an
optimal
stopping
rule.
Given
the
estimates
of
(w
,
...,
w
),
the
ε...,before.
1 w q−1
the
wage
cut
points
(w
,
w
).
An
estimation
method
which
can
deal
with
these
(22)
and
(23)
as
In
Figures
1–3
the
premaximized
with
respect
to
(∞
w
, …, ∞
)
holding
q−1with a discrete productivity distri–1
ε1
ation of thex/h
model εspecification
mated by observed frequencies in the wage1 data. q In
the second step the lik
as before.
Indensities
Figures
1-3
the
predicted
for
observed
wages
dicted
for
observed
wagesdensities
obtained
(r, h, d )on
fixed,
using
simulated
annealing
n(23)
addition
to
the
problem
that
the
bounds
ofare
the
support
of
F λ depend
unknown
0, λ 1, distribution
are esti,
...,
γ
corresponding
discontinuity
points
in
the
wage
offer
γ
1
q−1
mplications
has fact
beenthat
developed
by Bowlus
et al.is (1995).
Once again it is convenient
plicated by by
the
the
likelihood
function
not
differentiable
with
respect tosearches
(λ0, λ1 , δ)over
conditional
on r, h, w1 , ..., wq−1
inserting the parameter estimatestion
intois maximized
which
randomly
the possible
ductivity
dispersion
inserting the
intofrequencies
parameters,
theparameter
estimation
of the model
specification
a discrete
productivity
distrimatedestimates
by
in with
the
wage
data.
In combinations
the second
stepaccording
the likelihood
funcin aobserved
similar
fashion
was
done
inwage
the homogeneous
case.
cut
an optimal
tsreparametrize
(w1, ..., wq−1 ).theAn
estimation
method
which as
can
model
Sincedeal
thiswith
partthese
of the maximization
problem is smooth, standard maximu
x/r 12 x
1
rule.on
Given
bution
is further
complicated
the
fact
the likelihood
is not
differentiable
1, …,
tion
is
maximized
with
respect
to following
(λ
.
λ1 ,function
δ)stopping
conditional
r, h, w
, ...,estimates
wq−1 , γ 1 , of
..., (∞γwq−1
blem
that
theatbounds
thebysupport
ofpthat
F
depend
on , unknown
1the
fand
gε (ε)dε.
aluating
(12)
=
wiof
for
the
relationship
between
the
i gives
en
developed
by w
Bowlus
et εal.solving
(1995).
Once
again
it is0 convenient
ε
algorithms
can
be
applied.
These
two
steps
are
then
iterated
until converge
w
),
the
corresponding
discontinuity
points
in
x/h
q –
1
cut points
t theofwage
(wthis
wwith
estimation
method
which
deal with
thesemaximum likelihood
1 , ..., part
q−1 ).
Since
the
maximization
problem
is can
smooth,
standard
tion
the model
specification
a An
discrete
distrioductivity
wage
cuts
and
theof
fractions
ofproductivity
firm
types:
the
wage
offer distribution
(γ 1, …, γ q –1) are estimodel in aterms,
similar
fashion
as was
done
in the
homogeneous
case.
In addition to the order statistics estimators for (r, h), the maximum li
te
productivity
dispersion
mated
by observed
frequencies
in the wage
data.
omplications
has
developed
etis
al.
Once
it is convenient
algorithms
can by
be Bowlus
applied.
These
two steps
are again
then
iterated
until
convergence
occurs.
icated
by the
fact been
that
the
likelihood
function
not(1995).
differentiable
1the
μ2i relationship
A.3Discrete
productivity
dispersion
=
w
and
solving
for
p
gives
following
between
the
6) i
pii =
wi −
wi , mators
i = 1,of
2,the
..., q,
wage
cuts
in thestep
wagethe
offer
distribution
(w1 , ...,
In
the
second
likelihood
function
iswq−1) also con
2
2
1−
1 which
−fashion
μi can
o(wreparametrize
theestimation
model
inμamethod
asdeal
waswith
done
in the for
homogeneous
case.
i similar
In
addition
to the
order
statistics
estimators
(r,
h), respect
the
maximum
likelihood
esti..., wq−1
).that
An
these
1 ,problem
√
the
the
bounds
of
the
support
of
F
depend
on
unknown
maximized
with
to
(
λ , λ , d
)
conditionIn
addition
to
the
problem
that
the
bounds
of
the
0
1
age cuts and the fractions of firm types:
true value at a rate faster than n. It follows
that
they are asymptotically in
alrelationship
on (r ,h, ∞(w
w11, …, ∞
wq−1
γ also
γ q –1). Since
this
support
of
F
on unknown
parameters,
Evaluating
(12)
at wmators
=2w
and
solving
for
piin
gives
the
following
between
the
q –1, 1, …, i depend
of
the
wage
cuts
the
wage
offer
distribution
,
...,
w
)
converge
to their
developed
by
Bowlus
et
al.
(1995).
Once
again
it
is
convenient
32
estimation
1 of the model
μi specification with a discrete productivity distrithe
maximum
likelihood
estimator
of
(λ
,
λ
,
δ)
and
the
theory
of local cuts b
part
of
the
maximization
problem
is
smooth,
the
estimation
of
the
model
specification
with
a
pi =
w
−
w
,
i
=
1,
2,
...,
q,
√
0
1
i and the fractions of firm types:
2 i
2cuts
productivity
terms,
μdiscrete
1wage
−fact
μvalue
true
atthe
adistribution
rate
faster
n.comfollows
model
in1 a− similar
fashion
as was
done
in the
homogeneous
case.that they are asymptotically independent of
i by the
i that
r complicated
likelihood
function
isItnot
differentiable
productivity
isthan
further
and Kiefer (1994b) justifies conditioning on them in the second step of the pr
1 that
μ2iestimator
12 and the theory of local cuts by Christensen
plicated
by
the
fact
the
likelihood
the
maximum
likelihood
of
(λ2,
, ...,
λ1,the
δ)
Forthese
simulated annealing, see Goffe et al. (1994) or
wpoints
for
p
gives
the
following
relationship
between
0
i and solving
i
p
=
w
−
w
, function
i=
1,
q,
t26)
(w1, ..., w
).
An
estimation
method
which
can
deal
with
i
i
32 1 − μ2 at the1 −
separate
maximization
Bowlus
et al. (1995). can be shown to lead to a joint maximum of
μ2i iiterative
is notq−1
differentiable
wage
cut points
i
and
Kiefer
(1994b)
justifies Once
conditioning
in the second step of the procedure. The
ge cuts
the fractions
of firm
has
beenand
developed
by Bowlus
ettypes:
al. (1995).
again iton
is them
convenient
function on convergence.
2
32
161
1
μi
iterative
separate maximization
can homogeneous
be shown to lead
to a joint maximum of the likelihood
zepi the
case.
= model
win
w fashion
, i = 1,as2,was
..., q,done in the
i −a similar
There is no formal test for choosing a value of q, the number of firm typ
1 − μ2i
1 − μ2i i
function
convergence.
at w = wi and solving
for pon
i gives the following relationship between the
the authors of this estimation technique argue that the likelihood ratio test o
is noof formal
test for choosing a value of q, the number of firm types. However,
32 There
rms, wage cuts and the
fractions
firm types:
q against another based on the standard χ2 -criterion can be expected to wo
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Figure 1. Wage offer densities for workers with lower vocational education or less.
162
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Figure 2. Wage offer densities for workers with upper vocational education.
163
Finnish Economic Papers 2/2007 – Tomi Kyyrä
Figure 3. Wage offer densities for workers with university education.
164
The estimation method of Bontemps et al. (2000) for the model with a continuous
F and
f from
the
wage
using
some
nonparametric
procedure.
In t
F and
f from
the
wage
datadata
using
some
nonparametric
procedure.
ductivity
distribution
involves
three
steps.
The first
step of the procedure
isIn
to the
esti
the
likelihood
function
is
maximized
with
respect
to
(λ
,
λ
,
δ)
condition
0
1
likelihood
function
is maximized
with respect procedure.
to (λ0, λ1 , δ)Inconditional
F and the
f from
the wage
data Economic
using
somePapers
nonparametric
the secondo
Finnish
2/2007 – Tomi Kyyrä−1
parametric
estimates
of
F
and
f.
As
the
final
step
p
=
(w)
−1K
parametric
estimates
of F and
f. As the final step p = K (w) andand
γ(p)γ(p
ar
thealgorithms
likelihood
function
is maximized
14 with respect to (λ0 , λ1 , δ) conditional on the
standard maximum likelihood
can
be
mated
using
14
using
14
applied. These two steps areparametric
then using
iterated
until of F and f. As the final step p = K −1 (w) and γ(p) are estim
estimates
convergence occurs.
F (w)]
1 [1
−1
using14 estimators
1 + 1κ1+[1κ−
F−
(w)]
(28) In addition to the order statistics
K −1K
(w) (w)
= =
w +w +
(28)(28)
, ,
2κ
f(w)
1
2κ1 f(w)
for (r, h), the maximum likelihood estimators of
3
1 + κ1 [1 − F (w)] 2κ1 f(w)
2κ,1 f(w)3
K −1(w) = w
+ =
the wage cuts in the wage(28)
offer distribution
(29)
γ(p)
2
(29)
, ,
2κκ11f(w)
(29) γ(p) =
f(w)
f � (w)
F (w)])
2 − f �−
1 [1
(∞w1, …, ∞wq –1) also converge to their true value at
κ1 f(w)
(w)
(1 +(1κ1+[1κ−
F−
(w)])
3
2κ1 f(w)
a rate faster than √ n. It follows
they are γ(p) � =
(29) thatwhere
,
F,and
f and
replaced
byf �their
estimates
� f are κ
2−
f(w)
(w)
(1nonparametric
+ κ1 [1 − Festimates
(w)])
where
F,
f
f
are
replaced
by
their
nonparametric
fromfrom
the th
fir
1
asymptotically independent of the maximum 1
= λ� 1by
/δ by maximum
its maximum
likelihood
estimate
the
second
step.
κ1 λ
15
likelihood estimator of (λ 0, λ 1, d ) and
theory
=
likelihood
estimate
fromfrom
thefrom
second
κ1 fthe
where F,
and1 /δ
f are its
replaced by their
nonparametric
estimates
the step.
first stepT
of local cuts by Christensen and Kiefer (1994b) where F, f and f ' are replaced by their nonparathe wage
offer
density
f, apply
we apply
the
standard
Gaussian
kernel density
offer
density
f,
we
the the
standard
Gaussian
est
metric estimates
from
first step
and k kernel
= 15density
justifies conditioning on them
step
λthe
/δwage
by its
maximum
likelihood
estimate
from
the
second
To esti
κ1in=the
1second
1 step.
choose
the
bandwidth
by
a
rule
of
thumb
that
minimizes
the
mean
integrate
λ /
d
by
its
maximum
likelihood
estimate
from
of the procedure. The iterative separate
maximichoose
bandwidth
a rule
of standard
thumb that
minimizes
the mean
integrated
the wage
offerthe
density
f,1 we by
apply
the
Gaussian
kernel
density
estimators
the wage offer
zation can be shown to lead to a joint maximum the second step.15 To estimate
�
Corresponding estimates
ofand
F and
f are
obtained
by integration
� are
Corresponding
Fapply
fthe
thenthen
obtained
bykernel
integration
andand
diff
density
we
standard
Gaussian
of the likelihood function on
convergence.
choose
the bandwidth estimates
by
a rulef,ofof
thumb
that
minimizes
the mean
integrated square
e
of
the
kernel
density
estimate.
Standard
errors
are
finally
obtained
by
boo
density
estimator
and
choose
the
bandwidth
by
There is no formal test for choosing
a
value
of the kernel
density
estimate.
Standard
errors are
obtained
bootst
Corresponding
estimates
of F
and f � are
then obtained
byfinally
integration
and by
differentia
of q, the number of firm types. However, the a rule of thumb that minimizes16the mean intewhole
estimation
procedure
outlined
above.
16
whole
estimation
procedure
outlined
above.
grated
square
error.errors
Corresponding
of bootstrapping
authors of this estimation technique
argue
that estimate.
of the kernel
density
Standard
are finally estimates
obtained by
13
A Monte
evidence
ofthen
Bowlus
et al. (2001)
indicates
a tendency
towards
overfi
13 q against
Fevidence
and
f ' ofare
integration
andtowards
the likelihood ratio test of one value of
16 (2001) by
A Monte
CarloCarlo
Bowlus
etobtained
al.
indicates
a tendency
overfittin
whole estimation
procedure
outlined
above.
of
heterogeneity
types
using
this
criterion.
However,
they
further
find
that
choosing
a
differentiation
of
the
kernel
density
estimate.
another based on the standard c 2-criterion
can
of heterogeneity types using this criterion. However, they further find that choosing a valu
than true
the true
has only
a minor effect
on estimates
the estimates
(λ1, ,while
δ) , while
the order
st
13
than
valuevalue
has
a minor
on
the
of (λof
, towards
δ)
the order
statist
1
Awell
Monte
Carlo evidence
of only
Bowlus
et al.effect
(2001)
indicates
a tendency
overfitting
the nu
Standard
errors
are
finally
obtained
by
bootbe expected to work reasonably
inofthe
practice.
(r,and
h) and
the estimator
ML estimator
ofare
λ0 obviously
are obviously
unaffected
by value
the value
q chosen.
of (r, h)14
the
ML
of λ0whole
the
of qa of
chosen.
of heterogeneity
types
using
this criterion.
However,
they unaffected
furtherprocedure
findbythat
choosing
value
−1 of q g
strapping
the
estimation
outThis is so even though the exact
distribution
of
The
first
equation
is
simply
the
first-order
condition
(15)
solved
for
p
=
(w).
Th
14
−1K
estim
firsthas
equation
is simply
the
first-order
condition
for porder
= Kstatistics
(w).
The
sec
than the trueThe
value
only a minor
effect
the estimates
of (λ1(15)
, δ) ,solved
while the
16on
−1
can
be
found
by
differentiating
the
first
equation
with
respect
to
w
and
noting
that
K� −1
lined
above.
the test statistics is not known
due
to
the
non
found
differentiating
theobviously
first equation
with
respect
to
w
and
noting
that
K
(w)
of (r, h) can
andbe
the
ML by
estimator
of λ0 are
unaffected
by
the
value
of
q
chosen.
= Γ−1K −1(w) .
by choose
virtue of relationship
the relationship
(w)
It the
is worth
emphasising
theforestimation
regular estimation procedure.14Thus,
we
virtue
F (w)F=
Γcondition
K (w) (15)
. that
Theby
first
equation
is simply
first-order
solved
p = K −1 (w).
equ
15 of the
The
second
Obviously,
the
final
step
requires
the denominator
of (29)
be positive.
holds
15
the method
finalthe
step
requires
thewith
denominator
ofw(29)
to
betopositive.
This� This
holds
thei
of
Bontemps
et
al.
(2000)
is
very
simple
the number of firm types can
by be
comparing
two
foundObviously,
by differentiating
first
equation
respect
to
and
noting
that K −1
(w)
= if
f (w)
−1is satified
condition
of
the
firm’s
problem
or,
in
other
words,
if
f
(w)
(1
+
κ
1 [1 − F (w)])
condition
of
the
firm’s
problem
is
satified
or,
in
other
words,
if
f
(w)
(1
+
κ
[1
−
F
(w)])
decr
1
by
virtue
of
the
relationship
F
(w)
=
Γ
K
(w)
.
in computational respect. Substitution of the
times the improvement in the
log-likelihood
support
15
support
of final
F.of F.
Obviously,
the
step
requires
the denominator
(29) to be positive.
This
holds if the second16
kernel
estimates
in slightly
theof
likelihood
function
cirfunction with each additional firm
type
to
the
Our
estimation
procedure
differs
that
usedBontemps,
by Bontemps,
al. (200
16
estimation
procedure
differs
fromfrom
that
used
et decreases
al.et (2000)
condition of Our
the firm’s
problem
is satified
or, inslightly
other words,
if f (w)
(1 by
+ κ1 [1 − F (w)])
ovb
different
sampling
scheme.
Contrary
to
the
inflow
sample
of
unemployment,
Bontemps,
cumvents
the
need
for
numerical
integrations,
c 2.05 critical value.13 Once the
other
parameters
sampling scheme. Contrary to the inflow sample of unemployment, Bontemps,
et al
support different
of F.
sample
from
the
French
Labour
Force
Survey
drawn
from
the
stock
of
employed
and
un
16
from procedure
the French
Labour
Forcestep
Survey
drawn
the stock
andbecause
unempl
while
the slightly
third
does
not
require
evenofiteraof the model are estimated, unobserved
heteroOursample
estimation
differs
from
that
usedfrom
by Bontemps,
etemployed
al. (2000)
Consequently,
wage
observations
in their
come
from
G, from
not from
asour
in data.
our da
Consequently,
wagetions.
observations
inclear
their
datadata
come
from
G, not
F asFinet
samplingusing
scheme.
Contrary
to the
inflow
sample
of unemployment,
Bontemps,
al. (2000)
This
a
advantage
compared
to
the
geneity terms (p1, ..., pq) candifferent
be estimated
estimate
G (instead
of
F ) using
a nonparametric
procedure
and
recover
the assoc
estimate
G (instead
of FForce
) using
a nonparametric
procedure
and
thenthen
recover
the associated
sample
from
the
French
Labour
Survey
drawn
from
the
stock
of
employed
and
unemployed
wo
estimation
procedure
ofreplace
Bowlus
al. (29)
(1995)
forcorresponding
(26) and (27).
equilibrium
flow
relationship.
They
also
(28)etand
by corresponding
the
equ
equilibrium
relationship.
They
also
replace
(28)
equatio
Consequently,
wage flow
observations
in their
data
come
from
G,and
not(29)
frombyF the
as in our data. As
such
�
p
and
γ
as
functions
of
G,
κ
the
density
of
the
earnings
distr
�g, g and productivity
1 , where g is dispersion.
the
case
of
discrete
γ as functions
of aG,nonparametric
g, g and κ1 , procedure
where g isand
thethen
density
of the
distributi
estimatepGand
(instead
of F ) using
recover
theearnings
associated
F usin
derivative.
derivative.
flow relationship.14They also replace (28) and (29) by the corresponding equations expr
A.4Continuous productivityequilibrium
dispersion
The
first equation is simply the first-order condition
�
p and γ as functions of G, g, g and κ1 , where–1g is the density of the earnings distribution and
(15) solved for p = K (w). The second equation can be
found by differentiating the first equation with respect to w
and noting that (K–1)' (w) = f (w)/γ (p) by virtue of the relationship F (w) = Γ (K–1 (w)).
34 34
15
Obviously, the final step requires the denominator of
(29) to be positive. This holds
34 if the second-order condition
of the firm’s problem is satified or, in other words, if
f (w) (1+k 1 [1–F (w)]) decreases over the support of F.
16
Our estimation procedure differs slightly from that
used by Bontemps et al. (2000) because of the different
sampling­ scheme. Contrary to the inflow sample of unemployment, Bontemps et al. (2000) use a sample from the
French Labour Force Survey drawn from the stock of employed and unemployed workers. Consequently, wage ob13
A Monte Carlo evidence of Bowlus et al. (2001) indi- servations in their data come from G, not from F as in our
cates a tendency towards overfitting the number of hetero- data. As such they estimate G (instead of F) using a nongeneity types using this criterion. However, they further find parametric procedure and then recover the associated F
that choosing a value of q greater than the true value has using the equilibrium flow relationship. They also replace
only a minor effect on the estimates of (λ 1, d), while the (28) and (29) by the corresponding equations expressing p
order statistics estimators of (r, h) and the ML estimator of and γ as functions of G, g, g' and k 1, where g is the density
of the earnings distribution and g' its derivative.
λ 0 are obviously unaffected by the value of q chosen.
The estimation method ofderivative.
Bontemps et al.
(2000) for the model with a continuous productivity distribution involves three steps. The first
step of the procedure is to estimate F and f from
the wage data using some nonparametric procedure. In the second step the likelihood function
is maximized with respect to (λ 0, λ 1, d ) conditional on the nonparametric estimates of F and
f. As the final step p = K–1(w) and γ (p) are esti-
165