The Basic New Keynesian Model, the
Labor Market and Sticky Wages
Lawrence J. Christiano
August 25, 2013
• Baseline NK model with no capital and with a competitive
labor market.
— private sector equilibrium conditions
— Details: http://faculty.wcas.northwestern.edu/~lchrist/course/Korea_2012/intro_NK.pdf
• Standard Labor Market Friction: Erceg-Henderson-Levin sticky
wages.
New Keynesian Model with Competitive
Labor Market: Households
• Problem:
•
max E0 Â bt
t=0
1+ j
N
log Ct  exp (t t ) t
1+j
!
, t t = lt t1 + #tt
s.t. Pt Ct + Bt+1  Wt Nt + Rt1 Bt + Profits net of taxest
• First order conditions:
1
1 Rt
= bEt
(5)
Ct
Ct+1 p̄ t+1
Wt
j
exp (t t ) Ct Nt =
.
Pt
New Keynesian Model with Competitive
Labor Market: Goods
• Final good firms:
— maximize profits:
Pt Yt 
subject to:
Yt =
Z
Z 1
0
1
0
Pi,t Yi,t dj,
# 1
#
Yi,t dj
 ## 1
.
— Foncs:
Yi,t = Yt

Pt
Pi,t
#
z
"cross price restrictions"
! Pt =
Z
1
0
}|
(1 # )
Pi,t
di
{
 11 #
New Keynesian Model with Competitive
Labor Market: Goods
• Demand curve for ith monopolist:
Yi,t = Yt
• Production function:

Pt
Pi,t
#
.
Yi,t = exp (at ) Ni,t , at = rat1 + #at
• Calvo Price-Setting Friction:
Pi,t =

• Real marginal cost:
st =
dCost
dworker
doutput
dworker
=
P̃t
Pi,t1
with probability 1  q .
with probability q
minimize monopoly distortion by setting = ## 1
z }| {
(1  n )
exp (at )
Wt
Pt
Brief Digression
• With Dixit-Stiglitz final good production function, there is an
optimal allocation of resources to all the intermediate activities,
Yi,t
— It is optimal to run them all at the same rate, i.e., Yi,t = Yj,t
for all i, j 2 [0, 1] .
• For given Nt , it is optimal to set Ni,t = Nj,t for all i, j 2 [0, 1]
• In this case, final output is given by
Yt = eat Nt .
• Best way to see this is to suppose that labor is not allocated
equally to all activities.
— But, this can happen in a million di§erent ways when there is a
continuum of inputs!
— Explore one simple deviation from Ni,t = Nj,t for all i, j 2 [0, 1] .
Suppose Labor Not Allocated Equally
• Example:
i
0,
1
2
Nt i
1
2
,1
2 Nt
N it
21
,0
1.
• Note that this is a particular distribution of
labor across activities:
1
0
N it di
1 2 Nt
2
12 1
2
Nt
Nt
Labor Not Allocated Equally, cnt’d
1
Yt
0
1
1
2
1
Yi,t di
0
1
2
eat
0
1
2
eat
0
eat Nt
eat N
1
Yi,t di
t
eat Nt f
1
1
2
1
1
N i,t di
1
2
1
2 Nt
1
2
0
1
2
1 2
2
1
1
1
Yi,t di
di
di
1
N i,t di
1
1
2
1
1
2
21
1
1
21
1 21
2
1
Nt
1
di
1
1
di
1
1 2
2
f
1 21
2
1
1
1
Efficient Resource Allocation Means Equal Labor Across All Sectors
1
10
0.98
0.96
f
0.94
0.92
6
0.9
0.88
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Optimal Price Setting
• Let
p̃t 
• Optimal price setting:
where
P̃t
Pt
, p̄ t 
.
Pt
Pt  1
p̃t =
Kt
,
Ft
#
st + bqEt p̄ t#+1 Kt+1 (1)
#1
1
= 1 + bqEt p̄ t#+
1 Ft+1 .(2)
Kt =
Ft
• Note:
Kt =
#
#
st + bqEt p̄ t#+1
st+1
#1
#1
#
+ ( bq )2 Et p̄ t#+2
st+2 + ...
#1
Goods and Price Equilibrium Conditions
• Cross-price restrictions imply, given the Calvo price-stickiness:
h
i 1
(1 # )
(1 # ) 1 #
Pt = (1  q ) P̃t
+ qPt1
.
• Dividing latter by Pt and solving:
"
1  q p̄ t#1
p̃t =
1q
#
1
1 #
"
# 1
1  q p̄ t#1 1#
Kt
!
=
(3)
Ft
1q
• Relationship between aggregate output and aggregate inputs:
Ct = pt At Nt , (6)
2
3 1
! #
( # 1) 1 #
#
1  q p̄ t
p̄
where pt = 4(1  q )
+ q  t 5 (4)
1q
pt  1
Linearizing around E¢cient Steady State
• In steady state (assuming p̄ = 1, 1  n = ## 1 )
p = 1, K = F =
1
#1
, s=
, Da = t = 0, N = 1
1  bq
#
• Linearizing the Tack Yun distortion, (4), about steady state:
p̂t = q p̂t1 !
p̂t = 0, t large
• Denote the output gap in ratio form by Xt :
Ct

 = pt Nt exp
Xt 
tt
At exp  1+ j
so that (using xt  X̂t ):
xt = N̂t +
dt t
1+ j

tt
1+j

,
NK IS Curve, Baseline Model
• The intertemporal Euler equation, (5), after substituting for Ct
in terms of Xt :
1
1
R

 = bEt

 t
t t+1 p̄ t+1
tt
Xt At exp  1+
Xt+1 At+1 exp  1+
j
j
1
1
Rt
= Et
,

Xt
Xt+1 Rt+1 p̄ t+1
where
Rt+1
then, use


1
t t+1  t t
 exp at+1  at 
b
1+j


\
ut
zd
= ût  ẑt
t ut = ẑt + ût ,
zt
NK IS Curve, Baseline Model
• The intertemporal Euler equation, (5), after substituting for Ct
in terms of Xt :
1
1
R

 = bEt

 t
t t+1 p̄ t+1
tt
Xt At exp  1+
Xt+1 At+1 exp  1+
j
j
1
1
Rt
= Et
,

Xt
Xt+1 Rt+1 p̄ t+1
where
Rt+1
then, use
to obtain:


1
t t+1  t t
 exp at+1  at 
b
1+j


\
ut
zd
= ût  ẑt
t ut = ẑt + ût ,
zt



b̄ t+1  R̂t+1
X̂t = Et X̂t+1  R̂t  p
NK IS Curve, Baseline Model
• Let:
Rt
• Also,
 exp (rt )
d exp (rt )
exp (r) drt
! R̂t =
=
= drt  rt  r.
exp (r)
exp (r)
Rt+1


= exp log Rt+1



d
exp
log
R
t+1
! R̂t+1 =
= d log Rt+1
exp (log R )
=r in e¢cient steady state, with p̄ =1
=
log Rt+1

• So, (letting rt  Et log Rt+1 )
z }| {
log R


Et R̂t  R̂t+1 = drt  Et d log Rt+1 = rt  rt .
NK IS Curve, Baseline Model
• Substituting
into


Et R̂t  R̂t+1 = rt  rt



b̄ t+1  R̂t+1 , xt  X̂t ,
X̂t = Et X̂t+1  R̂t  p
and using
b̄ t+1 = p t+1 , when p̄ = 1,
p
we obtain NK IS curve:
xt = Et xt+1  Et [rt  p t+1  rt ]
• Also,
rt
=  log ( b) + Et


t t+1  t t
at+1  at 
.
1+j
Linearized Marginal Cost in Baseline Model
• Marginal cost (using dat = at , dt t = t t because a = t = 0):
st
w̄t
j
, w̄t = exp (t t ) Nt Ct
At
b̄ t = t t + at + (1 + j) N̂t
! w
= (1  n )
• Then,
b̄ t  at = ( j + 1)
ŝt = w
h
tt
j +1
i
+ N̂t = ( j + 1) xt
Linearized Phillips Curve in Baseline Model
• Log-linearize equilibrium conditions, (1)-(3), around steady
state:


b̄ t+1 + K̂t+1 (1)
K̂t = (1  bq ) ŝt + bq #p
b̄ t+1 + bq F̂t+1 (2)
F̂t = bq (#  1) p
q b̄
K̂t = F̂t + 1q p t (3)
• Substitute (3) into (1)


q
q
b̄ t+1 + F̂t+1 +
F̂t +
p̂ t = (1  bq ) ŝt + bq #p
p̂ t+1
1q
1q
• Simplify the latter using (2), to obtain the NK Phillips curve:
pt =
(1q )(1 bq )
ŝt
q
+ bp t+1
The Linearized Private Sector Equilibrium
Conditions of the Competitive Labor
Market Model
xt = xt+1  [rt  p t+1  rt ]
(1q )(1 bq )
pt =
ŝt + bp t+1
q
ŝt = ( j + 1) xt
h
rt =  log ( b) + Et at+1  at 
t t+1  t t
1+ j
i
Equations of Actual Equilibrium
Closed by Adding Policy Rule
Et
rt
rt
1
1
rt
Et
t 1
t
at
rt
Etxt
1
1
xt
t 1
1
t
0 (Phillips curve)
1
xt
0 (IS equation)
x xt
rt
0 (policy rule)
1
t
0 (definition of natural rate)
Solving the Model
at
st
0
0
t
st
Ps t
t 1
xt
1
0 0 0 0
rt
1
0 0 0 0
rt
1
0 0 0 0
0
1
0
1
0 0 0
0
rt
1
0 0 0
0 0 0 0
rt
1
0 0 0
Et
0 zt 1
1 zt
2 zt 1
0
t
1
1
xt
1
0
rt
0
1
rr t
x
0
0 0 0
t 1
0
1
1
xt
0 0
t
1
t 1
1 0 0
0 0 0 0
t
1
t
1
0 0 0
1
at
st
1
0
0
0
0
0
0
1
0 st 1
1 st
0
1
st
Solving the Model
Et
0 zt 1
1 zt
2 zt 1
0st 1
1 st
Ps t
st
0
t
1
0.
• Solution:
zt
Az t
1
Bs t
1A
2I
• As before:
0A
F
0
2
0B
P
1
0,
0A
1
B
0
0,
x
1. 5,
0. 99,
1,
0. 2,
0. 75,
0,
0. 2,
0. 5.
Dynamic Response to a Technology Shock
inflation
output gap
nominal rate
0.2
0.15
0.03
0.15
0.02
0.1
0.1
0.01
0.05
0.05
0
2
4
natural real rate
6
0
2
4
log, technology
6
natural nominal rate
actual nominal rate
0
2
4
output
6
0.2
1.2
1
0.15
0.1
1.1
0.5
0.05
1.05
0
2
4
employment
0
6
0
2
4
0.2
0.15
0.1
natural employment
actual employment
0.05
0
natural output
actual output
1.15
0
5
10
6
1
0
2
4
6
Dynamic Response to a Preference Shock
inflation
output gap
nominal rate
0.25
0.1
0.25
0.08
0.2
0.2
0.06
0.15
0.04
0.1
0.1
0.02
0.05
0.05
0
2
4
natural real rate
6
0.25
0
2
4
6
preference shock
0
2
4
output
6
1
-0.1
0.2
0.15
-0.2
0.5
0.1
-0.3
natural output
actual output
-0.4
0.05
0
2
4
employment
6
0
0
2
4
6
-0.1
natural employment
actual employment
-0.2
-0.3
-0.4
-0.5
natural real rate
actual nominal rate
0.15
0
2
4
6
-0.5
0
2
4
6
Reasons to consider frictions in the labor
market:
• Play an essential role in accounting for response to a monetary
policy shock.
— With flexible wages, wage costs rise too fast in the wake of
expansionary monetary policy shock.
— High costs limit firms’ incentive to expand employment.
— High costs imply sharp rise in inflation.
— But, the data suggest that after an expansionary monetary
policy shock inflation hardly rises and output rises a lot!
— Wage frictions play essential role in making possible an
account of monetary non-neutrality (see CEE, 2005JPE).
• Important for understanding employment response to other
shocks too.
• Introducing sticky wages and monopoly power in labor market
provides a theory of unemployment.
Sticky Wages
• Basic model is due to Erceg-Henderson-Levin.
— We will follow the interpretation of EHL suggested by Gali, so
that we have a theory of unemployment (see also
Gali-Smets-Wouters).
• We will not go into the details here
— see Christiano-Trabant-Walentin Handbook chapter.
— also, detailed online lecture notes (links in course syllabus).
Outline
• Provide a broad sketch of the model.
• Discuss some linearized equations of the model.
• Provide a critical assessment of the model.
Ba
Model
There are many identical households.
The representative household
contains all workers:
Many workers of each type.
Differentiated by utility cost of working
Every worker enjoys the same level of
consumption.
Household A
plumbers
painters
etc.
Monopoly
plumber union
Household B
plumbers
Monopoly
painter union
painters
etc.
Type j Monopoly Union
Wage
Labor supply
unemployment
monopoly
wage
Labor demand
markup
#of people employed
Labor force
Quantity of labor
Does(the(Degree(of(Union(Power(
Affect(the(Unemployment(Rate?(
•  OECD(Employment(Outlook((2006,(chap(7)(
•  Norway(and(Denmark(have(unioniza8on(rates(
near(80(percent.(Before(the(current(crisis(their(
unemployment(rate(was(under(3.0(percent.(
Union(Density(Rates(
•  Jelle(Visser,(2006(Monthly(Labor(Review(
–  Union(density(rates,(1970,(1980(and(1990–2003,(
adjusted(for(comparability.(
–  Defini8on:(union(membership(as(a(propor8on(of(
wage(and(salary(earners(in(employment.(
Unemployment(Rates:(Sources(
•  BLS,( Interna8onal(Comparisons(of(Annual(
Labor(Force(Sta8s8cs, (Adjusted(to(U.S.(
Concepts,(10(Countries,(1970-2010,(Table(1-2.(
•  Finland,(Norway(and(Spain(taken(from(ILO,(
Comparable(annual(employment(and(
unemployment(es8mates,(adjusted(averages (
Monopoly(Power(Hypothesis(
•  If(union(density(in(country(A(grows(faster(than(
union(density(in(US,(then(
–  Expect(unemployment(in(country(A(to(rise(more(
than(unemployment(in(US.((
•  Test(is(based(on(low(frequency(part(of(the(
data,(not(on(the(levels.(
Wage Setting with Frictions
• Suppose, for simplicity, that there are no shocks to labor
preferences.
• The solution to union problem gives rise to a ‘wage Phillips
curve’:
0
1
=C[
tN
j
p̂ w,t
C
z }| t {
kw B
B
C
b̄
=
BĈ + jN̂t  wt C + bp̂ w,t+1
1 + j#w @
A
kw =
Wt
Wt
(1  q w ) (1  bq w )
, p w,t 
, w̄t 
.
qw
Wt  1
Pt
Wage Setting with Frictions
• Suppose, for simplicity, that there are no shocks to labor
preferences.
• The solution to union problem gives rise to a ‘wage Phillips
curve’:
0
1
=C[
tN
j
p̂ w,t
C
z }| t {
kw B
B
C
b̄
=
BĈ + jN̂t  wt C + bp̂ w,t+1
1 + j#w @
A
kw =
Wt
Wt
(1  q w ) (1  bq w )
, p w,t 
, w̄t 
.
qw
Wt  1
Pt
• Wage inflation rises (falls) when cost of working is greater
(less) than the real wage.
Collecting the Equations
• Variables to be determined:
b̄ t , p̂ w,t
xt , rt , p t , r̂t , ŝt , w
• Equations:
xt = xt+1  [rt  p t+1  rt ]
(1q )(1 bq )
pt =
ŝt + bp t+1
q
b̄ t  at
ŝt = w


w
b̄ t + bp̂ w,t+1
p̂ w,t = 1+kj#
Ĉt + jN̂t  w
w
rt =  log ( b) + Et [at+1  at ]
• Need more equations: relate Ĉt , N̂t to xt and shocks, and
b̄ t to p̂ w,t and p t .
connect w
• Recall definition of level of output gap: Xt :
Xt 
Ct
= pt Nt .
At
• Then,
in the linear approximation about undistored steady state, p
Ĉt
= xt + at , N̂t
! Ĉ + jN̂t = (1 + j) xt + at .
z}|{
=
• Also,
p w,t

Wt
=
Wt1
Wt
Pt
Wt1
Pt1
Pt
w̄t
=
p̄ t
Pt  1
w̄t1
b̄ t  w
b̄ t1 + p
b̄ t
! p̂ w,t = w
Equations of the Sticky Wage Model
• Six private sector equations in seven variables:
xt = xt+1  [rt  p t+1  rt ]
(1q )(1 bq )
pt =
ŝt + bp t+1
q
b̄ t  at
ŝt = w


w
b̄ t + bp̂ w,t+1
p̂ w,t = 1+kj#
(1 + j) xt + at  w
w
b̄ t  w
b̄ t1 + p̂ t
p̂ w,t = w
rt = Dat+1
where rt and rt now stand for their deviations from steady
state, r.
• Monetary policy rule:
rt = art1 + (1  a) [fp p t + fx xt ] + ut ,
where ut denotes a monetary policy shock.
Conclusion
• Sticky wages e§ective in getting wage not to rise much after a
monetary policy shock, limiting the rise in inflation and
amplifying the rise in output.
• But,
— Underlying monopoly power theory of unemployment does not
receive strong support in the data.
— Important recent policy question: what will be the e§ect of
extending the duration of unemployment benefits?
• some say, more unemployment benefits will make an already
weak economy even weaker.
• others say, the number of available jobs is low because of
weakness in aggregate demand.
• “if workers search less in response to better unemployment
benefits, that won’t change the (low) number of existing jobs
because it won’t a§ect aggregate demand (see Christiano,
Eichenbaum and Trabandt (in process))”.
• We will discuss alternative approaches to labor markets next
(see Christiano, Eichenbaum and Trabandt (2013)).