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Department of Economics
Working Paper Series
A NEW KEYNESIAN MODEL WITH UNEMPLOYMENT
Olivier
Blanchard
Jordi Gali
Working Paper 06-22
July 18,2006
Room
E52-251
50 Memorial Drive
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Social Science Research
http://ssrn.com/abstract=920959
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
LIBRARIES
A New
Keynesian Model with
Uneraployment*
Jordi Gali^
Olivier Blancliard^
July 18, 2006
(first draft:
March 2006)
Abstract
We
develop a utility based model of fluctuations, with nominal
unemployment. In doing
nesian model with
its
so,
its
In developing this model,
first
we proceed
in
specification, productivity shocks have
strained efficient allocation.
wage
We
then introduce nominal
how
We
it is
We
mechanisms
real
firms.
New Key-
focus on labor market frictions and unemployment.
leave nominal rigidities aside.
setting
of research: the
and
focus on nominal rigidities, and the Diamond-Mortensen-
Pissarides model, with
We
we combine two strands
rigidities,
two
steps.
We
show
that, under a standard utility
no
effect
on unemployment in the con-
then focus on the implications of alternative
for fluctuations in
rigidities in the
unemployment.
form of staggered price setting by
derive the relation between inflation and unemployment and discuss
influenced
by the presence
of the tradeoff between inflation
of real
wage
rigidities.
and unemployment
We show the
stabilization,
nature
and we draw
the implications for optimal monetary policy.
JEL
Classification: E32, E50.
frictions, search
Keywords: new Keynesian model, labor market
model, unemployment, sticky prices, real wage
We
rigidities.
thank Ricardo Caballero, Peter Ireland, Julio Rotemberg, and participants in seminars
Wliarton, Boston Fed, and at the UQAM conference on "Frontiers of Macroeconomics" for helpful comments. We also thank Dan Cao for valuable research assistance,
t
MIT and NBER
*
at
t
,
ITAM, MIT,
CREI, UPF,
CEPR
and
NBER
.
Introduction
1
Two
different
paradigms have come to dominate macroeconomics research over
the past decade.
On
New
the one hand, the
emerged as a powerful tool
Keynesian model (the
for
NK
monetary policy analysis
model,
for short)
in the presence of
has
nominal
rigidities. Its
adoption as the backbone of the medium-scale models currently
developed by
many
and policy
central banks
That success may be viewed
its success.
existing versions of the
NK
as
the other hand,
of search
a clear reflection of
surprising given that the
in
hours of work or employment.^
in
and independently, the Diamond-Mortensen-Pissarides model
and matching
framework
somewhat
is
paradigm typically do not generate movements
unemployment, only voluntary movements
On
institutions
(the
DMP
for the analysis of labor
model, henceforth) has become a popular
market
frictions, labor
market dynamics, and
the implications of alternative policy interventions on unemployment.^ However,
its
assumption of linear
utility
and
its
abstraction from nominal rigidities (or from
the existence of money, for that matter), has limited
its
usefulness for
monetary
economists.
Our purpose
of research.
in this
We
paper
do so not
is
to provide a simple integration of these
for the sake of integration,
framework that combines the two
is
needed
two strands
but because we believe a
in order to
accommodate simulta-
neously the following properties, which we hold to be relevant in industrialized
economies:
•
Variations in
unemployment
are an important aspect of fluctuations, both
from a positive and a normative point of view.
•
Labor market
frictions
and the nature
of
wage bargaining are
central to
understanding movements in unemployment.
•
The
effects of
technology and other real shocks are largely determined by
the nature of nominal rigidities and monetary policy, thus calling for an
1.
Paradoxically, this
was viewed as one of the main weaknes.ses of the
ample, Summers (1991)), but was then exported to the NK model.
2.
See Pissarides (2000) for a description of the DMP raodel.
RBC
model
(for ex-
analysis of the consequences of alternative
monetary policy
rules for the
nature of fluctuations and the level of welfare.
In our model, frictions in labor markets are introduced by assuming the presence
We
of hiring costs, which increase with the degree of labor market tightness.
start
by showing
that,
under our assumptions on preferences and technology, the
unemployment.
constrained-efficient allocation implies a constant level of
Hiring costs, together with the fact that
it
takes time for unemplo3''ed workers to
find'a job, lead to a surplus associated with existing
with the wage determining
ers.
We
how
that surplus
split
is
employment
relationships,
between firms and work-
examine the consequences of two alternative wage setting structures
N ash-bargaining we
equihbrium. Under
employment
recover the property of a constant un-
rate which characterizes the constrained efficient allocation, though
in the decentralized
efficient.
That
economy the implied
result of constant
level of
under our
unemployment
is
generally in-
equihbrium unemployment, which stands
among
contrast to Pissarides (2000) and Shimer (2005),
fact that,
for
utility specification, the reservation
others,
in
comes Trom the
wage, rather than being
constant, increases in proportion to consumption and thus with productivity.
proportional increase in real wages and productivity leaves
all
The
labor market fiows
unaffected.
The above
real
wage
finding leads us to consider the scope for and the implications of
rigidity.
As
in Hall (2005), real
wage
but not employment in existing matches.
of technology shocks on
rigidity
unemployment
We
rigidity affects hiring decisions,
characterize the
dynamic
effects
as a function of the degree of real
and other characteristics of the labor market. In
particular,
wage
we show that
the size of unemployment fluctuations increases with the extent of real rigidities,
whereas the persistence of those fluctuations
sclerotic labor markets,
i.e.
is
higher in economies with more
markets with lower average job-finding and separation
rates.
Independently of their
always
inefficient in
size
and persistence, fluctuations of unemployment are
our model.
When we
introduce nominal rigidities, mone-
tary policy can influence those fluctuations, motivating the analysis of the conse-
quences of alternative monetary
policies.
We
first
consider two extreme pohcies.
We
show that
ployment
affect
fully stabilizing inflation leads to persistent
in response to productivity shocks.
unemployment
movements
in
in
unem-
As productivity shocks would not
in the constrained efficient allocation, this implies that these
unemployment, and by implication,
The degree
suboptimal.
movements
of persistence of
strict inflation targeting, are
unemployment depends on the under-
lying parameters of the economy. Interestingly, in light of the difference between
the
US and European
more
We
persistent are
labor markets, the more sclerotic the labor market, the
unemployment movements.
then show that stabilizing unemployment can be achieved, but only at the
cost of variable inflation, with inflation increasing for
an adverse technology shock.
We
finally derive the
,
some time
in response to
,
optimal monetary policy. The latter implies stabilizing a
weighted average of the variances of inflation and of unemployment.
It
implies
therefore that an adverse productivity shocks lead to an increase in inflation
and an
in'crease in
unemployment "for some
time. Interestingly, while the implied
responses of inflation are similar under the U.S. and European calibrations, the
required fluctuations in unemployment are larger under the U.S. calibration.
show how such optimal responses can be approximated reasonably
We
well with a
simple Taylor- type rule which has the central bank adjust the short-term nominal
rate in response to variations in inflation and unemployment, with suitably chosen
response coefficients.
We
are not the
first to
point to the need for and attempt such an integration, with
relevant papers ranging from
Merz (1995)
defer a presentation of the literature
and
Put simply, we see our contribution
as the
to Christoffel
and Linzert (2005).
of our relation to
it
We
to later in the paper.
development of a simple, anal5^ically
tractable model, which can be used to characterize the effects of different shocks,
their relation to the underlying parameters of the economy,
and to characterize
optimal monetary policy. Such a framework seems potentially useful
in guiding
the development of larger and more realistic models.
The paper
is
organized as follows.
Section 2 sets up the basic model, leaving out nominal
rigidities.
We
capture
The
labor market frictions through external hiring costs.
market tightness, defined as the ratio of hires to the unemployment
of labor
We
pool.
then characterize the constrained-efficient allocation, and show that
productivity shocks have no effect on unemployment.
is
latter are a function
The source
of this neutrality
that income effects lead to changes in the wage proportional to changes in
productivity-as would be the case in an economy without labor market frictions.
Section 3 characterizes the decentrahzed equilibrium under alternative wagesetting mechanisms.
We
first
economy
ditions under which the
We
assume Nash bargained wages and derive the conreplicates the constrained efficient allocation.
then show that, under Nash bargaining, the neutrality property continues to
hold.
We
then introduce real wage
of productivity shocks on
rigidities,
and characterize the dynamic
effects
unemployment.
Section 4 introduces nominal rigidities, in the form of staggering of price decisions
by
We
firms.
between
relations
lation
it
show how the model reduces,
between
iafiation,
inflation
to the standard
effect of
both the
NK
level
to a close approximation, to simple
marginal cost, and unemployment.
We
derive the re-
and unemployment implied by the model, and contrast
formulation. Put crudely, the
and the change
in
model implies a
unemployment on
significant
inflation, given ex-
pected inflation.
Section 5 then turns to the implications of our framework for monetary policy.
It
shows how stabilizing
large
and
inflation in response to productivity shocks leads to
inefficient (given that constrained-efficient
movements
in
unemployment.
It
unemployment
is
constant)
shows how the persistence of unemployment
is
higher in more sclerotic markets, markets in which the separation and the hiring
rate are lower. It derives optimal
inflation
monetary
policy,
and the implied movements
in
and unemployment.
Section 6 offers two calibrations of the model, one aimed at capturing the United
States, the other
kets. In
each case,
aimed at capturing Europe, with
it
its
more
sclerotic labor
mar-
presents the imphcations of pursuing either inflation-stabilizing,
unemployment-stabilizing, or optimal monetary policy.
Section 7 indicates
how our model
relates to the existing-and rapidly
growing-
on the
literature
nominal price
market
relative roles of labor
rigidities in
wage
frictions, real
rigidities,
and
shaping fluctuations.
Section 8 concludes.
2
A
2.1
Assumptions
Simple Framework
Preferences
The
representative household
by the unit
interval.
is
made up
The household
continuum of members represented
of a
seeks to
maximize
/
EoE/5'
where Ct
tution
The
e,
is
a
CES
at1+(I>\
(logC7,-^^j
function over a continuum of goods with elasticity of substi-
and Nt denotes the fraction of household members who are employed.
latter
must
satisfy the constraint
<
Note that such a specification
the
(1)
DMP
iVt
<
1
.
of utihty differs
from the one generally used
model, where the marginal rate of substitution
to be constant.
Our
business cycle, given
specification
its
is,
is
(2)
in
generally assumed
instead, one often used in
models of the
consistency with a balanced growth path and the direct
parameterization of the Frisch labor supply elasticity by
</).
Technology
We
assume a continuum of firms indexed by
i
€
[0, 1],
each producing a differen-
tiated good. All firms have access to an identical technology
Yt{i)
-
At Nt{i)
The
variable At represents the state of technology, which
mon
across firms
Employment
and
in firm
(0, 1) is
of workers hired
assumed
to
be com-
to vary exogenously over time.
evolves according to
i
Nt{i) =^ {I
with 5 e
is
-
5) Nt-,{i)
an exogenous separation
by firm
i
in period
rate,
+
Htii)
and Ht{i) represents the measure
t.
Labor Market
Flows and Timing.
At the beginning of period
for hire,
and whose
t
there
is
a pool of jobless individuals
we denote by
size
Ut-
We
refer to
who
are available
the latter variable as
beginning-of-the period unemployment (or just unemployment, for short).
make assumptions below
that guarantee
full
participation,
i.e.
We
at all times all
individuals are either employed or willing to work, given the prevailing labor
market conditions. Accordingly, we have
= l-il-5)Nt-x
Ut
Among those unemployed
at the
beginning of period
are hired and start working in the
same
t,
a measure Ht
=
f^
H{i)
di
period. Aggregate hiring evolves according
to
Ht
where Nt
We
=
Jg
N{i)
di
=
Nt
-
(1
-
(5)
Nt-i
(3)
denotes aggregate employment.
introduce an index of labor market tightness,
of aggregate hires to the
unemployment
xt,
which we define as the
rate
Hf
The
tightness index Xt
is
assumed to
lie
ratio
within the interval
,
[0, 1].
^
Only workers
in the
Note
unemployment pool
that,
at the beginning of the period
can be hired {Ht
from the viewpoint of the unemployed, the index
interpretation: It
the job-finding
the probability of being hired in period
is
rate.
Xt
t,
<
Ut).
has an alternative
or, in
other words,
Below we use the terms labor market tightness and job-
finding rate interchangeably.
Hiring
costs.
Hiring labor
is
costly.
Hiring costs for an individual firm are given by GtHt{i),
CES
expressed in terms of the
which
is
bundle of goods. Gt represents the cost per
hire,
independent of Ht (i) and taken as given by each individual firm.
While Gt
taken as given by each firm,
is
is
it
an increasing function of labor
market tightness. Formally, we assume
Gt
where a
>
and
S
is
=
At
Bxf
a positive constant satisfying
6B <
1.
In our framework, the presence of hiring costs creates a friction in the labor
market similar to the cost of posting a vacancy and the time needed to
the standard
DMP
Ut,
it is
useful to define an alternative measure of unemployment,
and given by the fraction of the population who are
a job after hiring takes place in period
full participation,
in
model.
For future reference,
denoted by
fill it
t.
left
without
Formally, and given our assumption of
we have
Ut
= l-Nt
In our model, vacancies are assumed to be
'
immediately by paying the hiring cost,
model, the hiring cost is uncertain,
with its expected value corresponding to the (per period) cost of posting a vacancy times the
expected time to fill it. This expected time is an increasing function of the ratio of vacancies to
3.
which
is
filled
a function of labor market tightness. In the
unemployment which can be
DMP
expressed- in turn as a function of labor market tightness. Thus,
is different, both approaches
have similar implications for firms' hiring decisions and unemploj'ment dynamics.
while the formalism used to capture the presence of hiring costs
The Constrained
2.2
We
Efficient Allocation
derive the constrained-efficient allocation by solving the problem of a benev-
olent social planner
who
and labor market
faces the technological constraints
frictions that are present in the decentralized
economy, but who internalizes the
market tightness on hiring costs and, hence, on the
effect of variations in labor
resource constraint.
Given symmetry
and technology,
be produced and consumed
will
i
in preferences
e
[0,1].
in the efficient allocation,
Ct{i)
lowers hiring costs, for any given level of
(it
hiring), the social planner will choose
Hence the
i.e.
=
good
Ct for
all
Furthermore, since labor market participation has no individual cost
but some social benefit
and
identical quantities of each
maximizes
social planner
(1)
an allocation with
full
employment
participation.
subject to (2) and the aggregate resource-
constraint
=
Ct
where Ht and
At {Nt
Xt are defined in (3)
The optimality condition
as
and
- Bxt
Ht)
(5)
(4).
for the social planner's
problem can be
written
,
XCtNt
T—
—
^
<
-,
1
-
/-,
(1
+
,
+P{1 -
\Tj a
ajSxj
5)
E,
which must hold with equality
Note that the
left
hand
1^ ^B
if A''t
<
(xJVi
+ axt^.il-
x,+i))}
(6)
1.
side of (6) represents the marginal rate of substitution
between labor and consumption, whereas the right hand side captures the corresponding marginal rate of transformation, both normalized by productivity At-
The
latter
first
term corresponds to the additional output, net of hiring
has two components, captured by the two right-hand side terms. The
costs,
by a marginal employed worker. The second captures the savings
resulting from the reduced hiring needs in period
4.
t
+
generated
in hiring costs
1.'^
Note that hiring costs (normalized by productivity) at time
t
are given by
BxfHt
.
The
9
Consider
=
Ct
the case no hiring costs
first
AtNt
with the efficiency condition
,
B—
(i.e.
(6)
XA^^*<1
0).
In that case
we have
simpHfying to
(7)
,
thus implying a level of employment invariant to technology shocks. This
invariance
is
the result of offsetting income and substitution effects on
labor supply, and
is
standard in this class of models. Absent capital ac-
cumulation, consumption increases in proportion to productivity; given a
specification of preferences consistent with balanced growth, this increase
in
consumption leads to an income
effect that exactly offsets the substitu-
tion effect.
In the presence of hiring costs
to (6)
still
{B >
0), it is
easy to check that the solution
implies a constant level of employment,
'
N* =
where x*
is
/
-
,,
=
-
N(x*)
the efficient level for the tightness indicator, implicitly given
as the solution to
{l-SBx") x/V(x)'+^ < l-(l-/3(l-(5))(l+a) Bx^-p{l-5)a Bx^+''
We
assume an
interior solution
<
x*
<
1
(8)
for the previous equation
(which must hence hold with equahty). This in turn imphes an interior
solution for
The
term Bx"
employment and unemployment.^
levels of
consumption and output are proportional to productivity,
in (6) captures the increase in hiring costs resulting
cost per hire unchanged.
The term aBx'f
reflects tlie effect
from an additional hire, keeping
on hiring costs of the change in
the tightness index Xt induced by an additional hire (given Ht).
The savings
in hiring costs at
have two components, both of which are proportional to 1 — 5, the decline in required
hiring. The first component, Bx"_^.^, captures saving resulting from a lower Ht^i given cost per
hire. The (negative) term aBx" (1 — Xt+i) adjusts the first component to take into account
the lower cost per hire brought about by a smaller Xt^i (the effect of lower required hires ift+i
t
+
also
1
,
more than
A
offsetting the smaller
unemployment pool Ut+i).
(8) is given by x(l
condition will be satisfied for any given values for 5 and B, as long as x
(but below) one.
5.
sufficient condition for
an interior solution to
— SB) >
is
1
—
B. That
sufficiently close to
10
»«
g
03
O
_g
"c
o
o
CD
a
CD
c
"03
o
o
CD
»
'
a g
'
and given by C;
Figure
in
1
= AtN*
(1
-
5Bx*<') and
Y;
shows graphically the determinants of the
=
At
N*
.
employment
efficient level of
our model. The dashed (blue) lines represent the efficiency condition in the
case of no hiring costs, while the solid (red) lines correspond to the
hiring costs.
to the left
They
The upward
hand
model with
sloping schedules starting off the origin correspond
side of (7)
and
(8),
respectively, normalized
by productivity.
represent the (normalized) marginal rate of substitution x{Ct/At)N'^, as a
function of the (constant) level of employment
A''.
Note that the red schedule
is
uniformly below the blue one, due to the downward adjustment of consumption
by a factor
(1
hiring costs.
side of (7)
— 6Bx")
resulting from the diversion of a fraction of output to
The schedules
and
(8),
originating at (0,1) correspond to the right
and capture the marginal rate
employment and consumption
of
employment.
declining
when
It is
(again, normalized
of transformation
by productivity),
hand
between
as a function
constant (and equal to one) in the case of no hiring costs, but
hiring costs are present, capturing the fact that the increase in
employment needed
to produce an additional unit of
consumption
is
increasing,
as a result of the increasing level of hiring costs.
Having characterized the constrained-efficient allocation we next turn our attention to the analysis of equilibrium in the decentralized economy.
the case of flexible prices.
3
We
We
consider
first
•
,
Equilibrium under Flexible Prices
start looking at the
problem facing
The
solution to that problem de-
Then we
characterize the equilibrium
firms.
scribes the equilibrium, given the wage.
under two alternative models of wage determination.
3.1
We
Value Maximization by Firms
first
look at firms' behavior, given the wage. Each firm produces a differen-
tiated good,
whose price
it
sets optimally each period, given
demand. Formally,
11
the firm maximizes
Et
its
value
{Pt+kmt+k{i) ~ Pt+kWt+kNt+k{i)
Y, Qt.t+k
-
Pt+kGt+k Ht+k{i))
'
'
k
subject to the sequence of
for
i
=
(real)
0, 1, 2,.
demand
constraints
..and taking as given the paths for the aggregate price level Pt, the
wage Wt, and unit hiring
costs Gt,
and where
Qt,t+k
=
P''
^^
-p^
is
the
stochastic discount factor for nominal payoffs.
The optimal
price setting rule associated with the solution to the above
problem
takes the form:
P^{i)
for all
t,
where M.
=-^
is
the optimal
MC^ = ^^^ Bxt is
the firm's (real) marginal cost.
by productivity,
= MPtMCt
as well as
/3(1
markup and
-
The
(9)
5)
E,
latter
1^ ^
Bx^+i}
(10)
depends on the wage normalized
on current and expected hiring
costs. In particular,
marginal cost increases with current labor market tightness (due to the induced
higher hiring costs), but
it
decreases with the expected tightness index, since a
higher value for the latter implies larger savings in next period's hiring costs
current employment.^
the firm decides to increase
its
In a symmetric equilibrium
we must have
(9)
if
Pt{i)
=
Pt for
all
i
G
[0, 1],
and hence
imphes
Roteraberg and Woodford (1999) were among the first to point out the implications of
the construction of marginal cost measures. They assumed hiring costs were
internal to the firm and took the form of a quadratic fxmction of the change in employment.
6.
liiring costs for
12
for all
Combining
t.
and
(11)
and rearranging terms we obtain
M - Bxt+P{l -
At
A
(10)
S)
£:,|^
^
S:r,v}
characterization of the equilibrium requires a specification of
tion.
We
start
(12)
wage determina-
by assuming Nash bargaining.
Equilibrium with Nash Bargained Wages
3.2
Let Wf^ and W^^ denote the value to the representative household of having a
marginal
member employed
or
pressed in consumption units.
is
given by
unemployed
The
at the beginning of period
i,
both ex-
value of a (marginal) employment relationship
i
,
Wf = Wt-xCtNr
+l3Et
i.e.
the
sum
{^
[(1
-
S{1
-
of the current payoffs (the
x,+i))
W,^i
+
6{l
-
x,+i)
wage minus the marginal rate
tution) and the discounted expected continuation values, with 5{1
probability of being unemployed in period
at time
Wf+i]|
t
+
1,
—
of substi-
Xt+i)
is
the
conditional on being employed
t.
The corresponding
has taken place
value from a
member who remains unemployed
after hiring
is:
wf = PEt
{^
[x,+i
w!i,
+
(1
- x,+0 Wf+j}
Combining both conditions we obtain the household's surplus from an established
job relationship:
Wf-Wf
- Wt-xCtNr
+/?(i-5)£;,|^(i-x,+i)(w/ti->v,'^o}
(13)
13
On
the other hand the firm's surplus from
an!
established relationship
is
simply-
given by the hiring cost Gt, since a firm can always replace a worker at that cost
(with no search time required).
Letting d denote the relative weight of workers in the Nash bargain, the latter
requires
Imposing the
and rearranging terms, we obtain the Nash
latter condition in (13),
wage schedule:
dBxt -
^(1
-
S,
5)
At
At
1^ ^
(1
Equation (14) can now be used to substitute out Wt in
equilibrium under
xCtNf
At
=
Nash bargaining
-
x,+i)
(12),
d 5xf+i|
with the resulting
being described by
1
—-{l + d)Bx?
M
+/?(l-5)E,|^^B(xr+i + ^xr+i(l-x«+i))}
together with
It is
(14)
(3), (4),
and
(5),
and given an exogenous process
(15)
for technology.
easy to verify that the equilibrium under Nash bargained wages involves a
constant level of employment:
Ar"fe
=
(5
where x"^
is
X„nb
-
+
(1
-^ = /v(x"h
-(5)x^
the (constant) equilibrium job finding rate, implicitly given by the
solution to:
X(l
- 5Bx") N{xy+^ = -^ -
(1
- /3(1 - 6))
(1
+ 19)
Ex''
- /3(1 - 5)d Bx"
(16)
14
and where consumption and output
by
C^''
= At{l-
(5S(x"'')°)A^"''
are proportional to productivity,
and
Fj"''
=
At N^'K
imposing the equilibrium conditions
Finally,
and given
in (14),
we obtain an expression
for
the equilibrium Nash bargained wage:
-^ = — -(1-/3(1 -<5))5(x"^r
tJ/nb
Note
that,
in the
is
1
(17)
under our assumptions, productivity shocks are reflected one-for-one
Nash bargained wage and thus have no
on employment. That
effect
result
independent of the elasticity of hiring costs with respect to Xt or the relative
weight of workers in the Nash bargain. Such a result stands in contrast to Shimer
(2005).
The reason
is
that, in line with the
DMP
framework, Shimer assumes
linear preferences and, hence, a marginal rate of substitution that
an increase
to changes in productivity. In that context,
a
than one-for-one increase
less
and inducing firms
to create
jobs.
invariant
productivity leads to
in
Nash bargained wage, increasing
in the
new
is
Shimer then shows that
,
profits
under plausible
assumptions about the matching function and the relative bargaining strength of
workers and firms, the employment response
Our model
is
likely to
be small.
allows instead for concave preferences and an endogenous marginal
rate of substitution between consumption and labor.
specification, the
Nash bargained wage
Under our-standard-utility
increases one-for-one with the marginal
rate of substitution (which, in turn, increases with productivity), at any given
level of
employment. Thus, changes
in productivity
have no
effect
on (un)employment,
independently of the hiring function, or the relative bargaining of workers and
firms. ^
In a
,
way analogous
to the standard
DMP
model
(see Hosios (1990)),
one can
Nash bargaining
will cor-
derive the conditions under which the equilibrium with
7.
Notice however that the equal increase
in
C and
W
is tlie
result of
both our specification of
preferences and the absence of capital accumulation. In the presence of capital accumulation,
employment would
typically move, as
it
does in conventional real business cycle models.
15
respond to the constrained
efficient allocation.
To
see this,
we
just need to
com-
pare equilibrium condition (15) with condition (6) characterizing the constrained
efficient (interior) allocation,
and which we rewrite here
for convenience:
l-(l + a)5xr+/3(l-<5)£;,|^^i?(x,% + axr^i(l-xe+i))|
At
The two equations
are equivalent
if
and only
if
the following two conditions are
satisfied:
M. =
First,
must
1
goods market
hold. In other
words we must have perfect competition
(or alternatively, a
in the
production subsidy which exactly offsets the
market power distortion).
Second, the workers' relative share of the surplus in the Nash bargain,
coincide with the elasticity of the hiring cost function, a.
analogous to the Hosios condition found in the standard
requires that the share of workers in the
of the
i?,
must
That requirement
DMP framework,
Nash bargain corresponds
is
which
to the elasticity
matching function with respect to unemployment.
Figures 2 and 3 show graphically the factors behind the equilibrium under Nash
bargaining, and contrasts
allocation.
The
latter
is
them with those underlying the constrained
represented by the intersection of the solid (red)
which match those in Figure
The downward
ting,
1,
lines,
already discussed above.
sloping dashed line gives the ratio
equation (10), after imposing
MCt =
employment allocation (constant over
mand
efficient
time).
Wt/At implied by
price set-
1/-M, and evaluated at a constant
It
can be interpreted as a labor de-
schedule, determining the (constant) level of
employment consistent with
value maximization by firms, as a function of the (productivity adjusted) wage.
Its slope
becomes steeper with B, while
that the schedule
is
flatter
than
its
its
curvature increases with a. Note also
counterpart in the social planner problem, re-
flecting the fact that in the decentralized
economy
firms do not take into account
the effects of their hiring decisions on hiring costs.
The upward
sloping dashed (green) line gives
Wt/At implied by the Nash wage
16
05
c
'03
sz
if)
03
3
LU
C
"c
'03
&_
05
CO
03
E
3
3
O"
CD
-I—
c
o
it:
LU
X
schedule (14), evaluated at a constant employment allocation. Its slope
ing in
workers' relative weight in the
!?,
line reflects
workers
'
Nash
is
increas-
bargain. Its position above the red
bargaining power.
Figure 2 represents an equilibrium characterized by inefhciently high unemploy-
ment. This
a consequence of two factors:
is
market (which
shifts the labor
bargaining power by workers
As
we have
a result
A''"^
demand
>
{"d
(i)
a positive
markup
schedule downward), and
in the
goods
too
much
(ii)
a) which makes the wage schedule too steep.
< N* which
,
implies an inefficiently high
unemployment
level.
Figure 3 illustrates graphically an equilibrium which leads to an
ployment
level.
we have M. =
mined by
i9)
Two
I.
assumptions are needed
unem-
for this, as discussed earlier. First,
Second, the slope of the upward sloping wage schedule (deter-
exactly offsets the flatter labor
determined by a), making both intersect
We
efficient
restrict ourselves to equilibria in
rate of substitution
when
the latter
at
demand
schedule (whose slope
is
A'^*.
which the wage remains above the marginal
is
evaluated at
employment:
full
W,
-^^>xil-5B)
for all
rather
t.
participation (as
all
the unemployed would
and that those without a job
in
any given period are
This condition guarantees
work than
not),
(18)
full
involuntarily unemployed. Thus, any inefficiency in the equilibrium level of
ployment cannot be attributed to an
it is
inefficiently
low labor supply.^ Note that
only because of the presence of hiring costs that this
private efficiency of existing
employment
em-
is
consistent with the
relationships.^
RBC or NK models, in which suboptimal levels of eman inefficiently low labor supply.
9.
In the absence of such frictions any unemployed worker would be willing to offer his labor
services at a wage slightly lower than that of an employed worker (from a different household),
which would bid down the wage up to the point where (18) held with equality.
8.
This
is
in contrast
ployment are
with standard
also associa.ted with
17
Wage
Equilibrium under Real
3.3
As shown
do not
tivity
leaving
equihbrium wage under Nash bargaining
in the previous section, the
moves one-for-one with productivity
Rigidities
variations.
As
a result, changes in produc-
change the rate at which workers are hired,
affect firms' incentives to
unemployment unchanged. In our model, that invariance
result holds in-
dependently of the elasticity of the hiring cost function and the Nash weights, and
hence, independently of whether the equilibrium supports the efficient allocation
or not.
Following Hall (2005) and Blanchard and Gali (2005), we examine in this sec-
wage
tion the consequences of real
More
rigidity
on employment and unemployment.
and to keep the analysis as simple as
precisely,
possible,
we assume a wage
schedule of the form
Wt = 6
where 7 G
[0, 1]
an index
is
of real
wage
A]--'
(19)
and
rigidities
6 >
a constant that
is
determines the real wage, and hence employment in the steady state. Clearly, the
above formulation
we
shall
is
meaningful only
for
7
=
(and setting
schedule corresponds to
tlie
of a rigid
A'' (1
-
(1
an assumption
=
1,
hand
for
side of (17)) the
any given
wage
relative share d.
equation (19) corresponds to the canonical ex-
we assume that
(2005). Also notice that
—
allow for the possibility that the efficient
/3(1
In addition to (18)
-
5)) g[x*))
(if
we
in addition
M. —
if
\.)
we assume
.
t,
stationary,
wage analyzed by Hall
steady state can be attained
for all
to the right
Nash bargaining wage,
At the other extreme, when 7
G =
is
maintain here.
Note that
ample
technology
if
M
At
which guarantees that firms
will
^
'
want to employ some workers without
incurring any losses.
Combining
(19) with (12)
we obtain the
following difference equation representing
18
the equilibrium
under real wage
rigidities.
9^r = if-B.? + «l-fla{^^B<«}
The previous equation can be rearranged and
Bxt = f;(^(l -
where
!vt^t+k
flexible (7
>
=
0), variations in
market tightness
solved forward to yield
S))" E, |A,,+fc
{Ct/Ct+k) (^t+fc/A)-
We
(21)
(-^ -
A7.)
see that as long as
}
wages are not
fully-
current or anticipated productivity will affect labor
(or, equivalently,
the job finding rate), with the size of the effect
being a decreasing function of the sensitivity of hiring costs to labor market
condition, as measured by parameter a. In particular, a transitory increase in
productivity raises Xt temporarily, which implies a decrease in unemployment,
followed by an eventual return to
From
the anatysis above
it
its
normal
level.
follows that in the presence of real
wage
rigidities the
equilibrium will be characterized by inefficient fluctuations in (un)employment.
We
defer a full characterization of these fluctuations to later, after
duced sticky prices and a
policy,
namely
role for
inflation targeting, the
outcome
economy without nominal
4
Introducing Sticky Prices
much
intro-
monetary policy (Under a particular monetary
in the
Following
we have
for real variables
is
the same as
rigidities.)
of the recent literature
on monetary business cycle models, we
introduce sticky prices in our model with labor market frictions using the
malism due to Calvo (1983). Each period only a fraction
1
—
randomly, reset prices. The remaining firms, with measure
for-
of flrms, selected
^,
keep their prices
unchanged.
19
Appendix A,
As shown
in
in period
t is
tlie
optimal price setting rule
a firm resetting prices
for
given by
Etlf^e"
{P:
Qt,t+kYt+kit
M
-
Pt+k
MCt+k)
lfc=0
where
period
+
fc
for
desired markup.
a firm resetting
The
MCt =
is
newly
P^ denotes the price
f
real
by
set
its
marginal cost
embody
previous equations
t,
and M. =jzi
now given by
is
t,
is.
The marginal
rigidities
cost
is,
The
first
step
around a zero
is
same form
costs:
it
as in the
leads firms to
the price stickiness parameter.
B
and a) and
real
wage
we
turn.
7).
requires log-linearizing the system, the task to which
Dynamics
to derive the equation characterizing inflation. Log-linearization
inflation steady state of the optimal price setting rule (22)
=
((1
-
9){P^y~''
+
and the
6'(Pi_i)^"')'^, yields the standard
dynamics equation^"
nt
10.
9,
captured by hiring cost parameters
price index equation P(
inflation
(23)
however, influenced by the presence of labor market
Log-linearized Equilibrium
4.1
the gross
is
a weighted average of current and expected marginal
(measured by
To make progress
in
the essence of our integration:
with the weights being a function of
frictions (as
output
t.
price setting equation (22) takes the
choose a price that
•
(22)
Yt+k\t is the level of
standard Calvo model7given the path of marginal
costs,
=
eAr + Bx?-P{l-5)E,!^-^^^Bxf^,'^
The optimal
•
at time
price in period
the firm's real marginal cost in period
The two
\
J
See, e.g., Gali
^P
and Gertler (1999)
for
Et{nt+i}
+
\ fnct
(24)
a derivation.
20
= \og{MCt/MC) = log{M MCt)
where mct
represent the log deviations of real
steady state value and A
=
—
—
marginal cost from
its
The second
to derive the equation characterizing marginal cost. Letting
Xt
—
step
\og{xtjx),
is
Ct
=
log(Cf/C), and g
=
(1
I39){1
we can write the
Bx°'
0)/9.
first-order Taylor
expansion of (23) as
mct
= agM Xt-P{l-5)gM
where $
=
MW/A =
Et{ict
-
-
at)
1-(1 — /3(1 — S))gM. <
=
the steady state productivity to unity {A
- at+i) + a Xt+i}-^-r
{ct+i
1
at
(25)
and where we have normalized
,
1), letting at
=
logA* denote log
deviations of productivity from that steady state.
The
third step, using (4) and (5), and letting nt
linear approximations for labor
5xt
ct
The
=
+
at
\og{Nt/N)
is
until
= nt-{l-5){l~x)nt^i
—
l-6g
now) to
Ct
where
it
=
nt
+
I-
p-
(26)
—
-
nt-i
l-5g
5g
5 Xt
is
consumer (which
Et{ct+,}
-
{it
-
Et{TTt+i}
-
(28)
r*t)
= p—
at
+ Et{at+i},
the interest rate that supports the efficient allocation.
The equilibrium
of the
model with wage
rigidities
is
fully characterized
tions (24) to (28), together with a process for productivity,
monetary
(27)
get:
denotes the short-term nominal interest rate and r^
= — log/3,
to derive the log
market tightness and consumption:
last step is to use the linearized first order conditions of the
we have ignored
p
=
by equa-
and a description of
policy.
21
Approximate
4.2
The
characterization
Log-lineeirized
we have
Dynamics
just given
can however be simpHfied considerably
under two further approximations, which we view as reasonable:
The
first is
that hiring costs are small relative to output, so
consumption by
The second
is
Ct
=
at
-\-
we can approximate
nt-^^
that fluctuations in Xt are large relative to those in
rit,
an approx-
imation which follows from (26) and the assumption of a low separation
Combined with our
its
first
assumption
this allows
one to drop the term
rate.-'^
q — a^ and
lead from the expression for marginal cost (25).
These two approximations imply that we can rewrite the expression
for
marginal
cost, equation (25), as:
mct
- agM
These approximations and
{xt
- P Et{xt+i})- $7
this expression for
at
(29)
marginal cost allow us in turn to
derive the following characterization of the economy:
•
First,
combining equation
(29)
with equation (24) gives a relation be-
tween inEation, current labor market tightness and current and expected
productivity:
TTt
= agMX
Xt
- A$7 ^/3'=
Et{at+k}
(30)
k=0
Note that, despite the
(30), inflation is
More
expected inflation does not appear in
a forward looking variable, through
current and future
and expected
fact that
Ot's,
and current
real marginal costs.
xt,
which
itself
its
dependence on
depends on current
-^^
5 are of the same order of magnitude as fluctuations
terms involving gut or 5nt are of second order.
12. In other words, (26) implies that 5 Xt is of the same order of magnitude as rit and, hence,
it cannot be dropped from our linear approximations.
We assume the same is true for terms
g Xt (i.e., we assume that 5 and g have the same order of magnitude).
13. This can be seen by solvmg (29) forward, to get agM Xt = Yl'kLo 0'' Et{mct+k + $7 at+h}11.
precisely
we assume that g and
In Hi, implying that
m
22
•
Second,
=
let ut
(after hiring)
Ut
from
—u
steady state value, u
its
Then, using equation
denote the deviations of the unemployment rate
—
(5(1
and the approximation
(26)
—
•
=
-
(1
-
5){1
-
x) ut-i
(1
-
+
= — (1 —
Uj
5{1
u)
—
rit,
unemployment
us a relation between labor niarket tightness and the
ut
x))/{x
x)).
gives
rate:
u)5 xt
(31)
Third, using the log-linearized Euler equation of the consumer, together
with the approximation
rit
Ct
=
rit
+ at,
gives a relation between the
and the approximation
Ut
unemployment rate and the
—
—(1
— u)
real interest
rate:
Ut
where, as before,
at
+ Et{at+\},
=
Et{ut+i}
it is
with p
+
[l
-
u)
-
{it
Et{7Tt+i}
rl)
(32)
the short-term nominal interest rate and rl
= — log/?.
Note that
(12) relates the
and anticipated deviations of the
rate to current
-
=
p
—
unemployment
real interest rate
from
its
efRcient counterpart.
These equations, together with a specification of monetary
terize the equilibrium.
We
shall look at the
policies in the next section.
We
imphcations of alternative monetary
hmit ourselves here to a few remarks about the
inflation
and unemployment:
Assume that productivity
follows a stationary
relation
between
parameter p^ 6
[0, 1).
We
where
i&
=
AR(1) process with autoregressive
can then rewrite (30)
TTt
X^/{1 - Ppa) >
policy, fully charac-
= agMX
Xt
-
as:
"^I-y
at
(33)
0.
Using the relation between labor market tightness and unemployment, we can
then derive a "Phillips curve" relation between inflation and unemployment:
.
TTf
= -KUt +
K;(l-(^)(l-a;) Ui_i-\E'7
Oi
(34)
23
where
k,
= agMX/5{l —
TTt
= -k(1 -
(1
u).
-
Or
equivalently
-
5)(1
x)) ut
-
k(1
which highhghts the negative dependence of
change in the unemployment
Given that constrained
stabilize
-
J)(l
-
inflation
on both the
efficient
unemployment
both unemployment and
monetary authority to
simultaneously. There
is,
at
level
and the
rate.
inflation.
is
constant,
it
would be best
to
Note however that, to the extent that
the wage does not adjust fully to productivity changes (7
for the
Aui - *7
i)
fully stabilize
>
0), it is
not possible
both unemployment and
to use the terminology introduced
inflation
by Blanchard and
Gall (2005), no divine coincidence.
The reason
affect the
is
the same as in our earlier paper, the fact that productivity shocks
—the unemployment rate that would
wedge between the natural rate
prevail absent
nominal
rigidities
rate. Stabilizing inflation,
which
—and
is
the constrained efficient unemployment
equivalent to stabilizing
unemployment
at its
natural rate, does not deliver constant unemployment. Symmetrically, stabilizing
unemployment does not
The next
5
deliver constant inflation.
section examines the implications of alternative monetary policy regimes.
Monetary
Policy, Sticky Prices
and Real Wage
Rigidities
5.1
We
Two Extreme
start
inflation
Policies
by discussing two simple, but extreme,
policies
and unemployment, leaving the derivation
and
their
outcomes
for
of the optimal poficy for the
next subsection.-'^ Given the paths of inflation and unemployment implied by
each policy regime, equation (32) can be used to determine the interest rate rule
that would implement the corresponding allocation.
14.
Throughout we
rely
on the log-linear approximations made
in the previous section.
24
Unemployment
stabilization. Recall that in the constrained efficient alloca-
unemployment
tion
unemployment and
all
t,
is
A
constant.
poHcy that seeks to
requires therefore that Ut
its efficient level
and hence
that, given
^f
(which
is
= -*7
Xt
=
for
is
increasing in the degree of
wage
.
unemployment
Ut
=
ttj
=
we can determine the
in (34)
evolu-
rate under strict inflation targeting:
(1
-
(5)(1
Note that the previous equation
under
=
a function of underlying parameters other than 7),
Strict inflation targeting. Setting
tion of the
rit
at
the size of the implied fluctuations in inflation
rigidities 7.-^^
=
gap between
'
TTt
Note
stabilize the
-
x) ut-i
-
i^ij/K) at
(35)
also characterizes the evolution of unemplo3''ment
flexible prices, since the allocation consistent
with price stability replicates
the one associated with the flexible price equilibrium.-^^
Equation (35) shows that under a
ployment rate
will display
some
strict inflation
targeting policy, the unem-
intrinsic persistence,
i.e.
beyond that inherited from productivity. The degree of
pends
critically
on the separation rate
6
some
serial correlation
intrinsic persistence de-'
and the steady state job finding rate
x,
while the amplitude of fluctuations are increasing in 7, the degree of real rigidities.
In a sclerotic labor market, with high average
unemployment and a a low
average job finding rate x, unemployment will display strong persistence, well
beyond that inherited from productivity. More active labor markets, on the other
15. Intuitively, the .stabilization of
unemployment (and hiring
cost vary with productivity, according to fnct
The amplitude
of those fluctuations
is
= —$7
at,
costs)
makes the
real
marginal
generating fluctuations in inflation.
increasing in the degree of
wage
rigidities
7 and
in the
persistence of the shock to productivity po, but decreasing in the degree of nominal rigidities
(which is inversely related to A).
16.
The
full
stabilization of prices requires that firms
markups
are stabilized at their desired
marginal cost remains constant at its steady state level, i.e.
In the presence of less than one-for-one adjustment of wages to changes in
level or, equivalently, that the real
fnct
=
for all
t.
productivity, that requues an appropriate adjustment in the degree of labor market tightness
and, hence, in unemployment.
.
25
hand, will be characterized by a high average job finding rate, and will display
persistent
unemployment
fluctuations under the stric inflation targeting pohcy.^'^
Optimal Monetary Policy
5.2
In this section
To
less
we
analj^ze the nature of the optimal
monetary policy
in our
simplify the analysis and avoid well understood but peripheral issues,
sume the unemployment
model.
we
as-
fluctuates around a steady state value which corresponds
As shown
to that of the constrained efficient allocation.
in
Appendix B, a second
order approximation to the welfare losses of the representative household around
that steady state
is
proportional
to:
EoJ2l3'
where au
=
A(l
+
>
0)x(A^*)'*"Ve
Hence the monetary authority
by
+
auU^t)
(36)
0.
will seek to
of equilibrium constraints given
form of the welfare
(TT^
minimize (36) subject to the sequence
(34), for
loss function the
t
=
0, 1, 2,
...
Clearly,
and given the
optimal pohcy will be somewhere between
the two extreme pohcies discussed above.
The
first
order conditions for the above problem take the following form:
Trt
=
Ct
auUt = K Ct- I3{1-6){1- x)k Et{Ct+i}
for
t
=
straint.
0, 1, 2,
...where
(t is
the Lagrange multiplier associated with period
Combining both conditions
t
con-
to eliminate the multiplier gives the optimal
targeting rule:^^
TTt=P{l-5){l-x)
Et{7rt+i}
+
17. The hypothesi.s that more sclerotic markets might lead
ment was explored empirically by Barro (1988).
18.
Note that when a^ corresponds to
utility
{au/K) ut
to
(37)
more persistence to unemploy-
maximization we have ^^
—
=—
"^
^
26
Equivalently, and solving forward,
= (-) Yl(^{l-5){l-x))'Et{ut^k}
V K
an adverse productivity shock, the central bank should
Hence
in response to
partly
accommodate the
crease
up
to the point
Combining
(37)
and
(38)
where
(34)
unemployment
inflationary pressures, while letting
its
we can
in-
anticipated path satisfies (38).
derive a second order difference equation describ-
ing the optimal evolution of the unemployment rate as a function of productivity:
ut
=
+ Pq
q ut-i
Et{ut+i}
-
s at
-
where
.^-au + K^il + Pil-d^il-xy)^''
and
k(1-/?(.1-(^)(1-x)pJ^P7
a„-FAc2(l
The unique
+ /?(!-
^
ipu
Ut-i
where
=
i'u
by
(39), the
(37),
and
ment and
-x)2)
stationary solution to that difference equation
Ut
Given
5)2(1
it
1- Jl -
-
is
given by:
ipa at
(39)
'
4/3g2
\-^-^e
0,1,
s
^a=
>
behavior of inflation under the optimal policy
is
then determined
can be represented by the following linear function of unemploy-
productivity:
Tft
=
ipuUt-
ipa
Oft
where
^
.'^"~
<^
/c(1-^(1-x)(1-5)V'„)
_
^°~
r.
'
Pjl
-
5){\
-
x)^pu^PaPa
k(1-/?(1-x)(1-5)^0
27
Calibration and Quantitative Analysis
6
In order to illustrate the quantitative properties of our
model we analyze the
responses of inflation and unemployment to a productivity shock imphed by the
We
three monetary policy regimes introduced above.
start
by discussing our
cal-
ibration of the model's parameters.
We
take each period to correspond to a quarter. For the parameters describing
preferences
e
=
We
6 (impljdng a gross steady state
=
set 9
which
we
2/3,
the hterature:
in
markup A1
=
set the degree of real
standard
DMP
wage
rigidities, 7,
of real
a we
exploit a simple
model. In the
latter, the
V
H = Z U^V^^'^
,
we have
are typically close to 1/2
Below we analyze two
and unemployment
US and
We
1,
wage
rigidities,
equal to 0.5, the midpoint in the
mapping between our model and the
expected cost per hire
is
is
proportional to
given by
V/H
denotes the number of vacancies. Assuming a matching function of the
V/H — Z'^ [H/U) 1-1
hiring cost function corresponds to 77/(1
77
=
,0
-^^
the expected duration of a vacancy, which in the steady state
of
0.99
which implies an average duration of prices of three quarters,
In order to calibrate
form
—
1.2).
we do not have any hard evidence on the degree
admissible range.
where
j3
roughly consistent with the micro and macro evidence on price setting. At
is
this point
so
we assume values commonly found
,
— rf)
we assume a
in the
=
1
.
Hence, parameter
DMP model.
a
in our
Since estimates
in our basehne cahbration.
different cahbrations for the steady state job finding rate
rate,
which we think of
as capturing the characteristics of the
the European labor markets respectively.
think of our baseline calibration as capturing the US.
We
rate x equal to 0.7. This corresponds, approximately, to a
set the job finding
monthly rate
of 0.3
Under an (overly) strict interpretation of our model, 7 can be obtained tlirough a regression
wage growth on productivity growtli wliich is exogenous in our model. Such a regression
yields a coefficient between 0.3 and 0.4, so a value for 7 between 0.6 and 0.7. Stepping outside
our model, obvious caveats apply, from the measurement of productivity growth, to the direction
19.
of real
—
of causality.
28
we
consistent with U.S. evidence. ^°. In our baseline calibration
which
is
consistent with the average
u we can determine the separation
unemployment
u to
set
0.05,
rate in the U.S.
Given x and
=
— u){l — x)).
rate using the relation 5
ux/{{l
This yields a value for 5 roughly equal to 0.12.
We
choose our alternative calibration to capture the more sclerotic European
labor market.
Thus we assume x
monthly rate of
0.1)
and u
=
=
0.25 (which
0.1, values in line
is
roughly consistent with a
with evidence
Union over the past two decades. The implied separation
Finally,,
we need
to set a value for B,
rate
which determines the
for the
is
(5
=
0.04.
level of hiring costs.
Notice that, in the steady state, hiring costs represent a fraction 5g
GDP. Lacking any
direct evidence
baseline calibration that fraction
upper bound. This implies
B=
is
on the
latter
we choose
European
B
= SBx"
so that under our
one percent of GDP, which seems a plausible
0.01/(0.12)(0.7)
~
0.11
Note that the above calibrated parameters, combined with the assumption
efficient
steady state, allow us to pin
down
The Dynamic
x —
x- Specifically, (8) implies
under the our baseline/U.S. calibration, and x
6.1
of
—
1-22
of an
1-03
under the European one.^^
Effects of Productivity Shocks
Figures 4 and 5 summarize the effects of a productivity shock under alternative
policies. In
Figure 4 we assume a purely transitory shock {pa
=
0),
which allows
us to isolate the model's intrinsic persistence, whereas in Figure 5
=
Pa
0.9,
a more
realistic
degree of persistence. In both figures
we
we assume
display the
responses for both the U.S. and European labor market calibrations. In
we
all
cases
report responses to a one percent decline in productivity, and expressed in
percent terms.
See Blanchard (2006).
20.
(1
—
in
Europe by assuming a
We
compute the equivalent quarterly rate
as x,n
+
(1
— x„^)xm +
Xni)^im, where Xm. is the monthly job finding rate.
21. Note that our model can only account for a higher efficient steady state unemployment rate
larger disutility of labor. Alternatively,
we could have assumed an
steady state only for the U.S., and impose the implied x to the European calibration
as well. In that case, however, the steady state unemployment would not be efficient. and an
efficient
additional linear term would appear in the loss function, which would render our log-linear
approximation to the equilibrium conditions
insufficient.
,
29
constant
constant u policy: inflation
h3
-^
tt
policy:
unemployment
— us
— eu
Ql
I
^ Q.® ..e..®.©^-®~@ @-Q-@-<3 'S"g" S & ®Q
.
''
.
10
5
'
15
'
h'
20
optimal policy: unemployment
optimal policy: inflation
1—
0.14
0.12
0.1
,^
0.08
0.06
0.04
0.02
'
Figure
4.
Dynamic Responses
3 €> &'G @-€h»'^»-@-e' a-Q •& Q
to Transitory Productivity
Shock
Sr^r&-s^-e^-ei4
constant u policy:
constant n policy: unemployment
inflation
^
8
•
/
'i.
6
^/%
4
''
X\
X
\
\.
\sc
2
_i
[
,
'
^^"Q^^gi
10
optimal policy: unemployment
optimal policy: inflation
\
/-\
0.5
-
\^
%^
0.4
\^
0.3
^^t
0.2
0.1
1
r
,
10
Figure
5.
,
'""'^^^i.
20
Dynamic Responses
to a Persistent Productivity
Shock
20
us
us
caltfctration: inflation
0.5
•e—
optimal
-^
taybr
—
1
0.5
0.8
0.4
\x
0.3
^°
calibration:
unempfoyment
\
-
O.S
^N,
•
0.41-
0.2
^
^.
<?•."
^^^
0.1
%
10
eu
15
20
10
eu
calibration: inflation
"-K
0.5
\\
\i\
0.4
0.6
0.5
^=
0.4
0.3
"^^Si.
.o
"n.
caSibratfon:
15
20
unemploymsiit
n
\
0.3
0.2
0.2
'*^*^^;
0.1
0.1
.
10
Figure
15
6.
20
Taylor-type Rule vs. Optimal Policy
10
15
20
We
begin by discussing the case of a transitory shock.
As shown
in Figure 4,
and
implied by the analysis above, under a pohcy that stabilizes unemployment, a
transitory shock to productivity has a very limited impact on inflation, which rises
less
than ten basis points under both calibrations, and displays no persistence.
The
size of the effect is
Under
inflation stabilization
basis points
one.
its
Most
initial
on impact
interestingly,
on the other hand, unemployment
rises
by about 70
European
in the U.S. calibration, 60 basis points in the
and as discussed above, unemployment remains above
value well after the shock has vanished, with the persistence being
significantly greater
is
almost identical for both calibrations.
under the European calibration. The difference
in persistence
a consequence of both a lower job finding rate and a smaller separation rate in
Europe.
Not
surprisingly, the optimal policy strikes a balance
achieves a highly
muted response
of
both
inflation
between the two, and
and unemployment. The
ferences in the responses across the two calibrations
is
dif-
very small, with only a
slightly smaller inflation response of inflation in the U.S. cahbration being de-
tectable on the graph. Interestingly, the persistence in both variables
negligible
is
(though not zero) under both calibrations. Perhaps the most salient feature of
the exercise
is
the substantial reduction in
unemployment
under the
volatility
optimal policy relative to a constant inflation policy, achieved at a small cost in
terms of additional inflation
volatility.
Figure 5 displays analogous results, under the assumption that pa
realistic
degree of persistence.
0.9,
Now the size of the response of inflation is
substantially under the constant
unemployment
than 70 basis points on impact under both
reflects the
—
policy,
amplified
with an increase of more
calibrations. This amplification effect
forward looking nature of inflation and the persistent
effects
marginal costs generated by the interaction of the shock and real wage
Note
a more
on
real
rigidities.
also that inflation inherits the persistence of the shock.
Given the strong inflationary pressures generated by a persistent adverse productivity shock,
it is
not surprising that fully stabilizing inflation
in the face of
those pressures would require large movements in unemployment. This
is
con-
30
firmed by our results, which point to an increase of 6.5 percentage points
(!)
in
the unemployment rate under the U.S. cahbration. Interestingly, the unemploy-
ment increase under the European
calibration
significantly smaller (though
unemployment
huge): 5.2 percentage points. In both cases
with
is
is
highly persistent,
The degree
response displaying a prominent hump-shaped pattern.
its
persistence
is
still
remarkably larger under the European calibration,
of
same
for the
reasons discussed earlier.
Under the optimal
policy, the increase in
unemployment on impact
Note that the
inflation targeting.
size of
The
such responses
price for having a
is
about
1
under the European
percentage point under the U.S. calibration, half that size
one.
is
several times smaller than under
smoother unemployment path
sistently higher inflation, with the latter variable increasing
is
per-
by about 60 basis
points under both calibrations.
path of inflation implied by the optimal pohcy
Interestingly, while the
the same under both calibrations, the increase in unemployment
smaller under the European calibration.
do with the
fact that
The explanation
larger sacrifice ratio. This
is
in turn a
implied by our calibration (0.12
is
reason
for that result
calibration
,
thus implying a
economy with a high separation
that the sensitivity of marginal cost to changes in the
clear, the
has to
consequence of a higher separation rate
rate depends on the effects of the latter on the tightness index x.
makes
substantially
vs. 0.04)
the Phillips curve flatter in an
is
roughly
K-the coefficient measuring the sensitivity of inflation to
unemployment changes-is smaller under the U.S.
Why
is
is
higher
is
the separation rate the smaller
change in unemployment on
x.
This
is
in turn explained
aggregate hirings with respect to unemployment
an economy with high turnover, a change
in
when
5
is
The
unemployment
As equation
(26)
the effect of a given
by a lower
is
rate?
elasticity of
high: intuitively, in
employment of a given
size
can be
absorbed without much upsetting of the labor market.
Figure 6 illustrates
how
the optimal responses of inflation and unemployment to
a productivity shock can be approximated fairly well by having the central bank
31
follow a simple Taylor-type rule of the form
it
=
+
p
(p^
TTt
-
(pu
Ut
Casual experimentation with alternative calibrations of the rule suggest that
(f).,^
=
1.5
and
(j)u
—
0.2
the U.S. calibration.
good job
work
On
fine at
approximating the optimal responses under
the other hand, a rule with ^^
at approximating the optimal responses
~
1.5
and
4>y,
=
0.6 does a
under the European calibration.
Notice that in the latter case the larger coefficient on unemployment
is
consistent
with the smaller required variation in unemployment.
Relation to the Literature
7
Our model combines
of
(4)
consumption and
four
main elements:
leisure), (2) labor
As a
price staggering.
result,
it
(1)
standard preferences (concave
market
is
frictions, (3) real
wage
utility
rigidities,
related to a large and rapidly growing
literature.
Merz (1995) and Andolfatto (1996) were the
first
to integrate (1)
introducing labor market frictions in an otherwise standard
ticular,
RBC
and
(2),
by
model. In par-
Merz derived the conditions under which Nash bargaining would
or
would
not deliver the constrained-efficient allocation. Both models are richer than ours
in allowing for capital accumulation,
and
in
the case of Andolfatto, for both an
extensive margin (through hiring) and an intensive margin (through adjustment
of hours) for labor. In both cases, the focus
nological shocks,
and
Cheron and Langot
in
both
(2000),
cases, the
was on the dynamic
effects of tech-
model was solved through simulations.
Walsh (2003) and Trigari (2006, with a draft
circu-
lated in 2003), integrate (1), (2) and (4), by allowing for Calvo nominal price
setting by firms. Their models are again
much
richer
than ours. Walsh allows
endogenous separations. Cheron and Langot, as well as Trigari allow
extensive and an intensive margin for labor,. with efficient
for
for
both an
Nash bargaining over
hours and the wage. In addition Trigari considers "right to manage" bargaining,
32
with the firm choosing freely hours ex-post. Those models are too large to be
analytically tractable,
and
Trigari's papers
and are solved through simulations. The focus of Walsh
on the dynamic
is
and Langot study the
ability of the
effects of
nominal shocks, while Cheron
model with both technology and monetary
shocks to generate a Beveridge curve as well as a Phillips curve. More recent
papers, by
(2006)
al.
Walsh
among
(2005), Trigari (2005),
others, introduce a
Moyen and Sahuc
number
of extensions,
in preferences, to capital accumulation, to the
models
complex
in these papers are relatively
and Andres
(2005),
from habit persistence
imphcations of Taylor
DSGE
et
The
rules.
models, which need to be
studied through calibration and simulations.
Shimer (2005) and Hall (2005) were the
gued that,
of
in the
unemployment
DMP
standard
first
to integrate (2)
model, wages were too
was too
to technological shocks
and
flexible,
(3).
Shimer
and the response
small. Hall (2005)
showed
the scope for and then the imphcations of real wage rigidities in that class of
from ours because of their assumption
els.
These
(in
addition to their being real models).
mod.els_differ
of this difference.
But our
their conclusion that real
We
results, using a
wage
have shown
standard
rigidities are
real
wage staggering a
for
nominal
cally,
and
is
rigidities.
la Calvo.
utility specification, reinforce
needed to explain fluctuations.
Being a real model, however,
Their model
is
mod-
earlier the implications
Their model allows for standard preferences, labor market
(3).
first
of linear preferences
Gertler and Trigari (2005) have explored the implications of integrating
and
ar-
it
(1), (2)
frictions,
and
has no room
again too complex to be solved analyti-
studied through simulations. Their focus
is
on the dynamic
effects
of technological shocks.
The paper
closest to ours
(1) to (4),
with standard preferences, labor market
is
by ChristofTel and Linzert (2005). Like
and nominal price staggering by
complex than
and either
ours, allowing for
efficient
frictions, real
firms. Again, their
model
is
it
integrates
wage
rigidities,
us,
substantially
both extensive and intensive margins
Nash bargaining,
through simulations. The main focus
or right to
is
on
manage. The model
more
for labor,
is
solved
inflation persistence in response to
monetary policy shocks.
33
This brief (and surely partial) review makes clear that we cannot
Our aim was
originality of purpose.
enough that
its
to develop
claim
fully
and present a structure simple
solution can be characterized analytically, the
dynamic
effects of
shocks can be related to the underlying parameters, and optimal policy can be
derived. In contrast,
any normative
much
analysis,
of the related literature described
due to the complexity
that our analytical model
above
falls
models involved.
of the
short of
We
think
a needed step in the development of these more
is
complex models.
Conclusions
8
We
have constructed a model with labor market
and staggered price
New
setting.
role of
^<,
rigidities,
integrates the key elements of the
in
all
needed
-.«
-
.
effects of
monetary pohcy
one
if
movements
to explain
is
in
changes in productivity on the economy, and the
shaping those
effects.
have derived the constrained-efficient allocation, and the equilibrium in the
decentralized
economy
The optimal monetary
unemployment and
The
-
three ingredients above are
unemployment, the
We
wage
Keynesian framework and the Diamond-A-Iortensen-Pisarrides search model
of labor market flows.
The
The model thus
frictions, real
extent of real
in the presence of real
policy has been
shown
wage
i.e.
of persistence
is
prices.
inflation fluctuations.
wage
rigidities
determines the amplitude of unemployment
unemployment
fluc-
displays intrinsic
The degree
persistence beyond that inherited from productivity.
decreasing in the job finding rate. Hence, the more sclerotic
the labor market the more persistence
In the presence of real wage
the best monetary pohcy.
is
As
in
targeting does not deliver
Blanchard and Gali (2005), the reason
The
is
unemployment.
rigidities, strict infiation
distortions vary with shocks.
inflation,
and sticky
to minimize a weighted average of
tuations under the optimal pohcy. Furthermore,
persistence,
rigidities
best policy implies
is
that
some accommodation
of
and comes with some persistent fluctuations in unemployment.
.
,
34
Appendix A: Derivation
New
of the
Keynesian Phillips Curve with
Hiring Costs
The Optimal Price Setting Problem
X
Let Xt+h\t denote the value of a variable
reset its price in period
ofc
.Ct_Pt_
jg
j^Q^
P^
t.
- nmx
Et
'
i
+
fc
for a firm that last
denotes the price set by this firm at time
t.
Qt,t+k
stochastic discount factor for nominal payoffs.
^j^g
resetting prices in period
T4(A^,_i|0
in period
t
maximizes
J^
its
value, given
Qt,t+k [Pt^Yt+Ht
O'^
-
A
=
firm
by
Pt+k{Wt+kNt+k\t
+
Gt+kHt+k\t)]
k=0
oo
+
{l-9)EtY,^'Qt,t+k+iVt+k+i{Nt+k\t)
(40)
fc=o
subject to
Yt+k\t ^'^At+kNt+k\k
^m-|t=(;^)
where Ht+k\t
=
-
A^t+fc|t
(1
-
{Ct+k
+ G,+,.Ht+k)
and Pt
^) Nt+k-i\t
(41)
=
(^J^
(42)
PtUY''
djj
'"
is
the
aggregate price index.
Notice that, in contrast with the Calvo model without hiring costs, the payoffs
associated with the current price decision
the period after the end of the
life
of the
now
newly
include a term corresponding to
set price.
That term takes the form
of a weighted average of the value of the firm at the time or resetting the price,
with the weights,
(1
—
6)
at each possible horizon.
9'^
,
corresponding to the probability that this happens
The reason
the payoff's of that period through
is
that P^ will
its effect
still
have some influence on
on inherited employment and, hence,
on hiring requirements.
Even though the
by
its
That
profits of a firm resetting its price at
time
t
will
be be influenced
lagged employment, the price decision will be independent of past variables.
result,
which allows our model to preserve
all
the aggregation properties of
35
the basic Calvo framework, follows from our assumption of linear hiring costs at
the firm
level.
and using constraints
Differentiating objective function (40) with respect to P^
(41)
and
(42), as well as the
must hold
=
envelope property V^{Nt-i\t)
=
(1
—
J)
PtGt (which
we obtain the optimality condition
for all t)
EtY,e''-Qt,t+k
yt+k\t
-
(.1
+
ej
^
e
'n+fc|t
t-rh-iL
p*
-
ft+k'^t+k
t-rA.^t--r'v
r^
JD*
k=0
-6(1
-
j)(i
-
E,
9)
j: 9^
Q.,+,,1 '"'^''"^^r-''^"^ Nt.kit
*
k=0
where
dHt+k\t
_
dP;
=
for
fc
=
1, 2, 3,
..and
Collecting terms
=
^
and
^
1
dYt+k\t_
At+k
dPt*
-^
[^t+k\t
-^
letting
M.
_
_
-
(1
-
dPt*
--
mt+k-i\i]
k^O.
Nt\t for
dYt+k-i\t
1
cN
'At+k-i
'
-
=^ denote the gross desired markup we have
EtJ29''Qt,t+k[Yt+k\t
k=0
-M^ (Wt+k +
Pt
Gt+k
-
(1
- 5)Qt+kMk+i^^Gt+k+i)
Pt+k
\
which can be seen to be independent of
setting their price in period
t
will
Nt-i\t, thus
employment
Nt+k\t]
J
implying that
choose the same price
their price history (and, hence, their
is
/.
P/',
all
firms re-
independently of
in the previous period). This
a consequence of our assumption of hiring costs which are linear in the firm's
hiring level, thus implying that lagged
or marginal revenue (even though
it
employment does not
affect
marginal cost
will influence the level of profits). In turn,
that property allows us to preserve the aggregation properties of the basic Calvo
model.
36
Rearranging terms we have
oo
•
= EtJ2^'Qt,t+kYt+k\t{P:-MPt+kMCt+k)
fc=0
where
MC. 5 + .(.,)-«l-.)S.{l^^ .(.,«)}
37
Appendix B: Derivation
Under our assumed
of the Welfare Loss Function
utility specification
u(Ct)
=
we
have:
=
log Ct
c
+
ct
and
v{Nt)
=
X
l
+
(p
2
l
+
.
where we have made use of the
Hence, the deviation of period
is
'2
"^
cf)
fact that
utility
up
from
to second order
its
^'~^
c:i
rit
+
\
nf
steady state value, denoted by Ut,
given by
'
UtC^Ct~xN'+*rit + ^{l + cP)xN'+^nj
Next we derive an equation that
and
nf.. Market
good
clearing for
Integrating over
i
relates,
i
up
(43)
to a second order approximation,
requires that At{Nt{i)
—
g{xt)Ht{i))
=
Ct
Ct{i).
yields:
At {Nt
-
g{xt) Ht)
=
/
Ct{i) di
Jo
= CtDt
where
A=
/o
(^j
di.
Thus we can write
AM-' = -'^''^
n;
38
Notice that the right-hand side term equals
—5g
our assumption on the order of magnitude of 5 and
relative to
employment
fluctuations.
Thus
already of second order
g, it is
suffices to derive a first-order
it
under
in the steady state. Hence,
Taylor
expansion:
H
9{^t) -Tf
-
Sg
+
-
5g
+ g{l-5 + a)nt-
ag5
xt
+ g{l-5)nt- g{l-6)
where we have made use of equation
-
g{l
+
<5)(1
nt-i
-
q(1
x)) rit-i
(26) as well as the fact that g'x
— ag
.
Hence we have
g{l-5 +
CtD-^
^
g{l
-
6){1
,
l-5g
AtNtil --Sg)
Taking logs on both
ihultiplied
a)
sides,
and
1
realizing that the
+
-
terms
-
ajl
x))
^
5g
and Ut-i are
in nt
pre-
by g and hence are already of second order:
^
l-g(l + a)^
,
I-
gll
-
+ a(l 1-Sg
S)(l
5g
Lemma.: up to a second order approximation,
dt
=
logDt
—
x))
^
,
,
I vari[pt{i)).
Proof; see appendix C.
Using
(43)
and
(44),
we can write the expected discounted sum
of period utilities
as follows:
CO
""
t=0
f=0
t=0
«=0
+t.i.p
Assuming that the economy
fluctuates around the efficient steady state,
we can
39
use (8) to show that the coefficient on
The
rit
equals zero.
following result allows us to express the cross-sectional variance of prices as
a function of inflation:
Lemma: ESo/^*
Proof:
vari{pt{i)) =- { YT=q(^^ ^t
Woodford (2003)
Combining the previous
rate ut,
with our definition of the unemployment
results, together
we can write the welfare
losses
from fluctuations around the
efficient
steady state (ignoring terms independent of policy or lagged as
t=o
where
Q'„
=
(A(l
+
(^)x(l
-
u*)'^~^)/e
>
0.
40
Appendix C
From
the definition of the price index, in a neighborhood of the zero inflation
steady state:
=
exp{{l
/
-
e) {pt{i)
- pt)}
di
Jo
c.
1
+
(1
-
- Pt)
[\ptii)
e)
^i^
+
di
^
Jo
f\p,{i)
- p,f
di
Jo
thus implying
Pt^
f^
Pt{i) di
+
—-—
(1
-
^
Jo
By
e)
f^
(ptii)
/
-ptf
di
Jo
definition,
-
'p.M--
"'
^
=
«
L
I
/
exp{-e
I
*
{pt{i)
-pt)]
di
Jo
-
1
W {pS) -Pt) + —
+ ^^^ \\pi{i) - Pif
-
di
1
=
It follows
that dt
1
+
{pt{i)
dx
+
- Ptf
di
"^
1 / {pt{i)-ptf
:^ (e/2)
/
^ Jo
Jo
vari{pt{i))
^ ^\p,{x)
- p,f
di
di
up to a second order approximation.
41
References
Andolfatto, David (1996): "Business Cycles and Labor Market Search," American
Economic Review,
86-1, 112-132
Andres, Javier, Rafael Domenech, and Javier Ferri (2006): "Price Rigidity and
the Volatility of Vacancies and Unemployment," mimeo, Universidad de Valencia.
Barro, Robert (1988):
Review,
vol. 78-2,
Blanchard, Olivier
"The
Persistence of Unemployment," American
Economic
May, 32-37.
J.,
and Jordi Gali (2006): "Real Wage Rigidities and the
New
Keynesian Model," Journal of Money Credit and Banking, forthcoming.
Gheron, Arnaud, and Frangois Langot (2000): "The Phillips and Beveridge Curves
Revisited,"
Economics Letters
Christoffel, Kai,
69, 371-376.
and Tobias Linzert
Labor Market Frictions
for
(2005):
"The Role of Real Wage
Unemployment and
Inflation
Rigidities
and
Dynamics", Discussion
Paper 556, European Central Bank
Gall, Jordi
and Mark Gertler (1999): "Inflation Dynamics:
metric Analysis," Journal of Monetary Economics,
Gertler,
Structural Econo-
vol. 44, n° 2, 195-222,
Wage
Bargaining," mimeo, November.
Robert (2005): "Employment Fluctuations with Equilibrium
ness,"
American Economic Review
Krause, Michael,
in the
1999
Mark, and Antonella Trigari (2005), "Unemployment Fluctuations with
Staggered Nash
Hall,
A
Thomas Lubik
New Keynesian Model
Wage
Sticki-
vol. 95, no. 1, 50-64.
(2005):
"The
(Ir)relevance of Real
Wage
rigidities
with Search Frictions," mimeo.
Merz, Monica (1995), "Search in the Labor Market and the Real Business Cycle",
Journal of Monetary Economics, 36, 269-300
Moyen, Stephane, and Jean Guillaume Sahuc (2005), "Incorporating labour market frictions into an optimizing-based monetary policy model".
Economic Mod-
elling 22, 159-186
Pissarides, Christopher (2000): Equilibrium
Unemployment Theory,
MIT
Press.
42
Rotemberg, Julio and Michael Woodford (1999): "The CycHcal Behavor of Prices
and Costs,"
in
Handbook
of Macroeconomics,
J.
Taylor and M. Woodford eds.,
Elsevier-North Holland.
Shimer, Robert (2005): "The CycHcal Behavior of Equilibrium unemployment
and Vacancies," American Economic Review
Summers, Lawrence
(1991):
"The
Scientific Illusion in
Scandinavian Journal of Economics
Trigari, Antonella (2005):
Dynamics,"
vol. 95, no. 1, 25-49.
Empirical Macroeconomics,"
vol. 93, no. 2, 129-148.
"Equihbrium Unemployment, Job
flows,
Inflation
ECB WP#304.
Trigari, Antonella (2006):
tion Dynamics",
"The Role of Search Frictions and Bargaining of
Infla-
mimeo.
Walsh, Carl (2003): "Labor Market Search and Monetary Shocks",
of
and
Dynamic Macroeconomic
Walsh, Carl (2005):
Rules", Review of
Analysis, S. Altug,
J.
Chadha and
in
Elements
C. Nolan eds.
"Labor market search. Sticky Prices, and Interest Rate
Economic Dynamics,
8,
829-849
43