The Basic New Keynesian Model, theLabor Market and Sticky Wages

Lawrence J. Christiano

August 25, 2013

• Baseline NK model with no capital and with a competitivelabor 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 stickywages.

New Keynesian Model with CompetitiveLabor Market: Households

• Problem:

max E

0

•

Ât=0

bt

log C

t

exp (tt

)N

1+jt

1+ j

!, t

t

= ltt1 + #

tt

s.t. Pt

C

t

+ Bt+1 WtNt + Rt1Bt + Profits net of taxest

• First order conditions:

1

C

t

= bEt

1

C

t+1

R

t

¯pt+1

(5)

exp (tt

)Ct

N

jt

=W

t

P

t

.

New Keynesian Model with CompetitiveLabor Market: Goods

• Final good firms:

— maximize profits:

P

t

Y

t

Z

1

0

P

i,t

Y

i,t

dj,

subject to:

Y

t

=

Z1

0

Y

#1#

i,t

dj

##1

.

— Foncs:

Y

i,t

= Yt

P

t

P

i,t

#!

"cross price restrictions"z }| {

P

t

=

Z1

0

P

(1#)i,t

di

11#

New Keynesian Model with CompetitiveLabor Market: Goods

• Demand curve for ith monopolist:

Y

i,t

= Yt

P

t

P

i,t

#.

• Production function:

Y

i,t

= exp (at

)Ni,t

, a

t

= rat1 + #

a

t

• Calvo Price-Setting Friction:

P

i,t

=

˜

P

t

with probability 1 qP

i,t1 with probability q.

• Real marginal cost:

s

t

=dCost

dworkerdoutputdworker

=

minimize monopoly distortion by setting = #1#z }| {(1 n) Wt

P

t

exp (at

)

Brief Digression• With Dixit-Stiglitz final good production function, there is anoptimal allocation of resources to all the intermediate activities,Y

i,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 N

t

, it is optimal to set Ni,t

= Nj,t

for all i, j 2 [0, 1]• In this case, final output is given by

Y

t

= eatNt

.

• Best way to see this is to suppose that labor is not allocatedequally to all activities.— But, this can happen in a million di§erent ways when there is acontinuum of inputs!

— Explore one simple deviation from Ni,t

= Nj,t

for all i, j 2 [0, 1] .

Suppose Labor Not Allocated Equally

• Example:

• Note that this is a particular distribution oflabor across activities:

Nit2 Nt i 0, 12

2 1 Nt i 12 , 1, 0 1.

0

1Nitdi 12 2 Nt

12 2 1 Nt Nt

Labor Not Allocated Equally, cnt’dYt

0

1Yi,t

1di

1

0

12 Yi,t

1di

12

1Yi,t

1di

1

eat0

12 Ni,t

1di

12

1Ni,t

1di

1

eat0

12 2 Nt

1di

12

12 1 Nt

1di

1

eatNt0

12 2

1di

12

12 1

1di

1

eatNt 12 21 1

2 2 11 1

eatNtf

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.88

0.9

0.92

0.94

0.96

0.98

1

f

Efficient Resource Allocation Means Equal Labor Across All Sectors

6

10

f 12 21 1

2 2 11 1

Optimal Price Setting• Let

˜

p

t

˜

P

t

P

t

,

¯pt

P

t

P

t1.

• Optimal price setting:

˜

p

t

=K

t

F

t

,

where

K

t

=#

# 1s

t

+ bqEt

¯p#t+1Kt+1(1)

F

t

= 1+ bqEt

¯p#1t+1Ft+1.(2)

• Note:

K

t

=#

# 1s

t

+ bqEt

¯p#t+1

#

# 1s

t+1

+ (bq)2 Et

¯p#t+2

#

# 1s

t+2 + ...

Goods and Price Equilibrium Conditions• Cross-price restrictions imply, given the Calvo price-stickiness:

P

t

=h(1 q) ˜P(1#)

t

+ qP(1#)t1

i 11#

.

• Dividing latter by Pt

and solving:

˜

p

t

=

"1 q ¯p#1

t

1 q

# 11#

!K

t

F

t

=

"1 q ¯p#1

t

1 q

# 11#

(3)

• Relationship between aggregate output and aggregate inputs:

C

t

= pt

A

t

N

t

, (6)

where pt

=

2

4(1 q)

1 q ¯p(#1)

t

1 q

! #1#

+ q¯p#

t

p

t1

3

51

(4)

Linearizing around E¢cient Steady State• In steady state (assuming ¯p = 1, 1 n = #1# )

p

= 1, K = F =1

1 bq, s =

# 1#

, Da = t = 0, N = 1

• Linearizing the Tack Yun distortion, (4), about steady state:ˆ

p

t

= q ˆpt1 !

ˆ

p

t

= 0, t large

• Denote the output gap in ratio form by Xt

:

X

t

C

t

A

t

exp

tt

1+j

= pt

N

t

exp

t

t

1+ j

,

so that (using xt

ˆXt

):

x

t

= ˆNt

+ dtt1+j

NK IS Curve, Baseline Model• The intertemporal Euler equation, (5), after substituting for C

t

in terms of Xt

:1

X

t

A

t

exp

tt

1+j

= bEt

1

X

t+1At+1 exp

tt+1

1+j

Rt¯p

t+1

1

X

t

= Et

1

X

t+1Rt+1

R

t

¯pt+1

,

where

R

t+1

1

bexp

a

t+1 at t

t+1 tt1+ j

then, use

dz

t

u

t

= ˆzt

+ ˆut

,

\u

t

z

t

= ˆu

t

ˆzt

to obtain:ˆ

X

t

= Et

ˆ

X

t+1

ˆ

R

t

b̄pt+1 ˆR

t+1

NK IS Curve, Baseline Model• The intertemporal Euler equation, (5), after substituting for C

t

in terms of Xt

:1

X

t

A

t

exp

tt

1+j

= bEt

1

X

t+1At+1 exp

tt+1

1+j

Rt¯p

t+1

1

X

t

= Et

1

X

t+1Rt+1

R

t

¯pt+1

,

where

R

t+1

1

bexp

a

t+1 at t

t+1 tt1+ j

then, use

dz

t

u

t

= ˆzt

+ ˆut

,

\u

t

z

t

= ˆu

t

ˆzt

to obtain:ˆ

X

t

= Et

ˆ

X

t+1

ˆ

R

t

b̄pt+1 ˆR

t+1

NK IS Curve, Baseline Model• Let:

R

t

exp (rt

)

! ˆRt

=d exp (r

t

)

exp (r)=

exp (r) drt

exp (r)= dr

t

rt

r.

• Also,

R

t+1 = exp

log R

t+1

! ˆRt+1 =

d exp

log R

t+1

exp (log R)= d log R

t+1

= log Rt+1

=r in e¢cient steady state, with ¯p=1z }| {log R

• So, (letting rt

Et

log R

t+1)

E

t

ˆ

R

t

ˆRt+1

= dr

t

Et

d log R

t+1 = rt r

t

.

NK IS Curve, Baseline Model• Substituting

E

t

ˆ

R

t

ˆRt+1

= r

t

rt

into

ˆ

X

t

= Et

ˆ

X

t+1

ˆ

R

t

b̄pt+1 ˆR

t+1

, x

t

ˆXt

,

and usingb̄p

t+1 = pt+1, when ¯p = 1,

we obtain NK IS curve:

x

t

= Et

x

t+1 Et [rt pt+1 rt

]

• Also,

r

t

= log (b) + Et

a

t+1 at t

t+1 tt1+ j

.

Linearized Marginal Cost in Baseline Model

• Marginal cost (using dat

= at

, dtt

= tt

because a = t = 0):

s

t

= (1 n)¯

w

t

A

t

,

¯

w

t

= exp (tt

)Njt

C

t

! b̄wt

= tt

+ at

+ (1+ j) ˆNt

• Then,

ˆ

s

t

= b̄wt

at

= (j+ 1)h

tt

j+1 +ˆ

N

t

i= (j+ 1) x

t

Linearized Phillips Curve in Baseline Model• Log-linearize equilibrium conditions, (1)-(3), around steadystate:

ˆ

K

t

= (1 bq) ˆst

+ bq# b̄p

t+1 + ˆKt+1(1)

ˆ

F

t

= bq (# 1) b̄pt+1 + bq ˆFt+1 (2)

ˆ

K

t

= ˆFt

+ q1qb̄p

t

(3)

• Substitute (3) into (1)

ˆ

F

t

+q

1 qˆp

t

= (1 bq) ˆst

+ bq

# b̄p

t+1 + ˆFt+1 +q

1 qˆp

t+1

• Simplify the latter using (2), to obtain the NK Phillips curve:

pt

= (1q)(1bq)q ˆst + bpt+1

The Linearized Private Sector EquilibriumConditions of the Competitive Labor

Market Model

x

t

= xt+1 [rt pt+1 r

t

]

pt

= (1q)(1bq)q ˆst + bpt+1ˆ

s

t

= (j+ 1) xt

r

t

= log (b) + Et

ha

t+1 at t

t+1tt1+j

i

Equations of Actual EquilibriumClosed by Adding Policy Rule

Et t 1 xt t 0 (Phillips curve)

rt Et t 1 rt Etxt 1 xt 0 (IS equation)

rt 1 1 t 1 xxt r t 0 (policy rule)

rt at 11 1 t 0 (definition of natural rate)

Solving the Model

stat

t

00

at 1

t 1

t

t

st Pst 1 t

0 0 01 1 0 00 0 0 00 0 0 0

t 1

xt 1rt 1rt 1

1 0 00 1 1 1

1 1 x 1 00 0 0 1

t

xtr trr t

0 0 0 00 0 0 00 0 00 0 0 0

t 1

xt 1rt 1rt 1

0 0 00 0 00 0 00 0 0

st 1

0 00 00 0

1 1

st

Et 0zt 1 1zt 2zt 1 0st 1 1st 0

Solving the Model

• Solution:

• As before:

Et 0zt 1 1zt 2zt 1 0st 1 1st 0

st Pst 1 t 0.

zt Azt 1 Bst

0A2 1A 2I 0,

F 0 0B P 1 0A 1 B 0

x 0, 1.5, 0.99, 1, 0.2, 0.75, 0, 0. 2, 0.5.

0 2 4 6

0.01

0.02

0.03

inflation

0 2 4 6

0.05

0.1

0.15

output gap

0 2 4 6

0.05

0.1

0.15

0.2nominal rate

natural nominal rateactual nominal rate

0 2 4 6

0.05

0.1

0.15

0.2natural real rate

0 2 4 60

0.5

1

log, technology

0 2 4 61

1.05

1.1

1.15

1.2

output

natural outputactual output

natural employmentactual employment

Dynamic Response to a Technology Shock

0 5 100

0.05

0.1

0.15

0.2employment

0 2 4 6

0.02

0.04

0.06

0.08

0.1inflation

0 2 4 6

0.05

0.1

0.15

0.2

0.25

output gap

0 2 4 6

0.05

0.1

0.15

0.2

0.25nominal rate

natural real rateactual nominal rate

0 2 4 6

0.05

0.1

0.15

0.2

0.25natural real rate

0 2 4 60

0.5

1preference shock

0 2 4 6-0.5

-0.4

-0.3

-0.2

-0.1

output

natural outputactual output

natural employmentactual employment

Dynamic Response to a Preference Shock

0 2 4 6-0.5

-0.4

-0.3

-0.2

-0.1

employment

Reasons to consider frictions in the labormarket:

• Play an essential role in accounting for response to a monetarypolicy shock.

— With flexible wages, wage costs rise too fast in the wake ofexpansionary 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 monetarypolicy shock inflation hardly rises and output rises a lot!

— Wage frictions play essential role in making possible anaccount of monetary non-neutrality (see CEE, 2005JPE).

• Important for understanding employment response to othershocks too.

• Introducing sticky wages and monopoly power in labor marketprovides a theory of unemployment.

Sticky Wages

• Basic model is due to Erceg-Henderson-Levin.

— We will follow the interpretation of EHL suggested by Gali, sothat we have a theory of unemployment (see alsoGali-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

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.

Model

plumbers

painters

etc.

Household A

Household B

plumbers

painters

etc.

Monopolyplumber union

Monopolypainter union

Type jMonopoly Union

Labor demand

unemployment

monopolywage

markupQuantity of labor

Wage

#of people employedLabor force

Labor supply

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 laborpreferences.

• The solution to union problem gives rise to a ‘wage Phillipscurve’:

ˆpw,t

=k

w

1+ j#w

0

BBB@

=[Ct

N

jtz }| {

ˆ

C+ j ˆNt

b̄wt

1

CCCA+ b ˆp

w,t+1

kw

=(1 q

w

) (1 bqw

)

qw

, pw,t

W

t

W

t1,

¯

w

t

W

t

P

t

.

• Wage inflation rises (falls) when cost of working is greater(less) than the real wage.

Wage Setting with Frictions• Suppose, for simplicity, that there are no shocks to laborpreferences.

• The solution to union problem gives rise to a ‘wage Phillipscurve’:

ˆpw,t

=k

w

1+ j#w

0

BBB@

=[Ct

N

jtz }| {

ˆ

C+ j ˆNt

b̄wt

1

CCCA+ b ˆp

w,t+1

kw

=(1 q

w

) (1 bqw

)

qw

, pw,t

W

t

W

t1,

¯

w

t

W

t

P

t

.

• Wage inflation rises (falls) when cost of working is greater(less) than the real wage.

Collecting the Equations• Variables to be determined:

x

t

, r

t

, pt

,

ˆ

r

t

,

ˆ

s

t

,

b̄w

t

,

ˆpw,t

• Equations:

x

t

= xt+1 [rt pt+1 r

t

]

pt

= (1q)(1bq)q ˆst + bpt+1ˆ

s

t

= b̄wt

at

ˆpw,t

= kw1+j#

w

ˆ

C

t

+ j ˆNt

b̄wt

+ b ˆp

w,t+1

r

t

= log (b) + Et

[at+1 at]

• Need more equations: relate ˆCt

,

ˆ

N

t

to xt

and shocks, andconnect b̄w

t

to ˆpw,t

and pt

.

• Recall definition of level of output gap: Xt

:

X

t

C

t

A

t

= pt

N

t

.

• Then,

ˆ

C

t

= xt

+ at

,

ˆ

N

t

in the linear approximation about undistored steady state, pt

'1z}|{= x

t

! ˆC+ j ˆNt

= (1+ j) xt

+ at

.

• Also,

pw,t

W

t

W

t1=

W

t

P

t

W

t1P

t1

P

t

P

t1=

¯

w

t

¯

w

t1¯p

t

! ˆpw,t

= b̄wt

b̄wt1 + b̄pt

Equations of the Sticky Wage Model• Six private sector equations in seven variables:

x

t

= xt+1 [rt pt+1 r

t

]

pt

= (1q)(1bq)q ˆst + bpt+1ˆ

s

t

= b̄wt

at

ˆpw,t

= kw1+j#

w

(1+ j) x

t

+ at

b̄wt

+ b ˆp

w,t+1

ˆpw,t

= b̄wt

b̄wt1 + ˆpt

r

t

= Dat+1

where rt

and rt

now stand for their deviations from steadystate, r.

• Monetary policy rule:

r

t

= art1 + (1 a) [fppt + fxxt] + ut,

where ut

denotes a monetary policy shock.

Conclusion• Sticky wages e§ective in getting wage not to rise much after amonetary policy shock, limiting the rise in inflation andamplifying the rise in output.

• But,— Underlying monopoly power theory of unemployment does notreceive strong support in the data.

— Important recent policy question: what will be the e§ect ofextending the duration of unemployment benefits?• some say, more unemployment benefits will make an alreadyweak economy even weaker.

• others say, the number of available jobs is low because ofweakness in aggregate demand.

• “if workers search less in response to better unemploymentbenefits, that won’t change the (low) number of existing jobsbecause 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)).

Outline