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The Labor Market in the NewKeynesian Model:
Incorporating a Simple DMP Version ofthe Labor Market and Rediscovering the
Shimer Puzzle
Lawrence J. Christiano
April 1, 2013
Outline• We present baseline NK model with aDiamond-Mortensen-Pissarides type labor market.— First paper to combine NK and DMP is Walsh (2003). Nowthis is a huge literature.
• The New Keynesian DMP model.— Describe equilibrium conditions of household and goodsproducers
— These coincide with those in the simple, standard NK model(‘Clarida-Gali-Gertler model’)• Details: http://faculty.wcas.northwestern.edu/~lchrist/course/Korea_2012/intro_NK.pdf
— Describe the DMP labor market.• Compare the DMP model with a version in which specificationin which labor market is competitive.— discover ‘Shimer puzzle for monetary models’:employment/unemployment respond very little to anexpansionary monetary policy.
— replace hiring with search costs.
Households• Many identical households, each having a unit measure ofworkers.
• All workers are sent to the labor market, where they are eitheremployed or unemployed.
• Household problem:
max E
0
•
Ât=0
bt
log C
t
,
subject to: P
t
C
t
+ B
t+1
W
t
number of employed householdsz}|{
l
t
+
number of unemployed households, times unemployment benefitsz }| {(1 l
t
)D
+R
t1
B
t
+ Profits net of taxest
First order condition:1
C
t
= bE
t
1
C
t+1
R
t
¯pt+1
Obtaining the NK IS Curve• Let X
t
C
t
exp (a
t
) and substitute into household Fonc:
1
X
t
exp (at
)= bE
t
1
X
t+1
exp (at+1
)
R
t
¯pt+1
or,
1
X
t
= E
t
1
X
t+1
R
t+1
R
t
¯pt+1
, R
t+1
1
bexp (a
t+1
a
t
) .
• log-linearizing:
ˆ
X
t
= E
t
ˆ
X
t+1
ˆ
R
t
b̄pt+1
ˆ
R
t+1
.
• linearizing about a zero inflation steady state:
x
t
= E
t
x
t+1
E
t
[rt
pt+1
r
t
]r
t
= log (b) + E
t
[at+1
a
t
] .
Goods Production• Final good firms solve
max P
t
Y
t
Z
1
0
P
i,t
Y
i,t
dj, s.t. Y
t
=
Z1
0
Y
#1
#i,t
dj
##1
.
• Demand curve for i
th monopolist:
Y
i,t
= Y
t
P
t
P
i,t
#
.
• Production function:
Y
i,t
= exp (at
) h
i,t
, a
t
= ra
a
t1
+ #a
t
,
where h
i,t
is a homogeneous input (not labor!) withafter-subsidy nominal price, (1 n) J
t
P
t
.• Calvo Price-Setting Friction:
P
i,t
=
˜
P
t
with probability 1 qP
i,t1
with probability q .
Optimal Price Setting• As in baseline NK model:
˜
p
t
=K
t
F
t
,
˜
p
t
˜
P
t
P
t
, F
t
= 1+ bqE
t
¯p#1
t+1
F
t+1
(a)
K
t
=#
# 1
s
t
+ bqE
t
¯p#t+1
K
t+1
(b)
˜
p
t
=
"1 q ¯p#1
t
1 q
# 1
1#
!K
t
F
t
=
"1 q ¯p#1
t
1 q
# 1
1#
(c)
s
t
=(1 n) J
t
exp (at
)
• Log-linearizing (a)-(c) around zero inflation steady state (and,assuming 1 n = (# 1) /#)
pt
= (1q)(1bq)q ˆ
s
t
+ bpt+1
pt
¯pt
1.
Linearized Equilibrium Conditions forHouseholds and Goods Producing Firms• Equations
pt
= (1q)(1bq)q ˆ
s
t
+ bpt+1
x
t
= E
t
x
t+1
E
t
[rt
pt+1
r
t
]r
t
= log (b) + E
t
[at+1
a
t
] .ˆ
s
t
= ˆJt
a
t
.
• If we also have a monetary policy rule, would have 5 equationsin 6 variables:
pt
,
ˆ
s
t
, x
t
, r
t
, r
t
,
ˆJt
• Need on net, one more equation.— In simple model with competitive labor markets, have
Jt
= C
t
l
jt
.
• Here, j is a utility parameter and this equation is not availablenow.
— Equations provided by search/matching framework of DMP.
Labor Market• Large number of identical and competitive firms; producehomogeneous output using only labor, l
t
, for real price, Jt
.• At the start of period t, a firm may post a single vacancy forfixed cost k, in which case it meets a worker with probability,Q
t
.— Immediately after they meet, worker and firm agree on a wageand begin employment.
— Match survives into period t+ 1 with fixed, exogenousprobability r.
• Number of workers searching for work at the end of periodt 1 :
— 1 l
t1
unemployed workers plus the (1 r) l
t1
employedworkers that separate.
— total searching workers
1 l
t1
+ (1 r) l
t1
= 1 rl
t1
.
Notation and Identities• Probability a searching worker finds a period t job denoted f
t
.— Law of motion of employment is:
l
t
=
workers that remain attached to their firmz}|{rl
t1
+
flow from unempl to empl plus flow from empl to emplz }| {f
t
(1 rl
t1
)
— Alternative law of motion of employment:
l
t
= (r+ x
t
) l
t1
,
where x
t
denotes the hiring rate, as a proportion of l
t1
.
• For the two laws of motion to be compatible, require:
f
t
=x
t
l
t1
1 rl
t1
=net new jobs
number of workers searching.
Labor Market Tightness• Probability a firm meets a worker is denoted Q
t
:
Q
t
=net new jobs
number of firms posting vacancies=
x
t
l
t1
v
t
l
t1
=x
t
v
t
,
where v
t
denotes the vacancy rate, as a proportion of l
t1
.
• Matching function
# new hiresz }| {x
t
l
t1
= sm
0
@# people searchingz }| {
1 rl
t1
1
As0
@# vacanciesz }| {
l
t1
v
t
1
A1s
.
— divide by 1 rl
t1
:
f
t
=x
t
l
t1
1 rl
t1
= sm
l
t1
v
t
1 rl
t1
1s
= sm
Q1st
,
where Qt
denotes labor market tightness.— note, f
t
= Q
t
Qt
, soQ
t
= sm
Qst
Value Functions•
J
t
is the value to a firm of an employed worker:
J
t
= Jt
¯
w
t
+ rE
t
m
t+1
J
t+1
,
¯
w
t
W
t
P
t
.
• Jt
and m
t+1
are given to firm and worker.• Value of employment to a worker:
V
t
= ¯
w
t
+E
t
m
t+1
[rV
t+1
+ (1 r) (ft+1
V
t+1
+ (1 f
t+1
)U
t+1
)]
• Value of unemployment to a worker:
U
t
= D+ E
t
m
t+1
[ft+1
V
t+1
+ (1 f
t+1
)U
t+1
]
• Free entry and zero profits dictate: k = Q
t
J
t
.
Nash Bargaining• Workers and firms choose a wage that solves a jointoptimization problem, taking as given future wages.
• Surplus of worker:
S
t
¯
w
t
+ ˜
V
t
U
t
,
˜
V
t
E
t
m
t+1
[rV
t+1
+ (1 r) (ft+1
V
t+1
+ (1 f
t+1
)U
t+1
)]
• Surplus of firm:
˜
J
t
¯
w
t
,
˜
J
t
Jt
+ rE
t
m
t+1
J
t+1
• Agreed-upon wage rate solves
max
¯
w
t
¯
w
t
+ ˜
V
t
U
t
h ˜
J
t
¯
w
t
1h
,
taking ˜
V
t
and ˜
J
t
as given. FONC (Nash sharing rule):
J
t
=1 h
h(V
t
U
t
) .
Labor and Goods Market Clearing• Total purchases of homogeneous inputs by intermediate goodproducer:
h
t
=Z
1
0
h
i,t
di.
• Labor market clearingh
t
= l
t
.
• Goods market clearing
C
t
+ v
t
l
t1
k = p
t
exp (at
) l
t
,
where
p
t
=
2
4(1 q)
1 q ¯p
(#1)t
1 q
! #1#
+ q¯p#
t
p
t1
3
51
.
Equilibrium Conditions from Labor Market13 equilibrium conditions associated with 12 new variables:
l
t
, x
t
, Q
t
, J
t
, V
t
, U
t
, m
t
, f
t
, Qt
, Y
t
, v
t
,
¯
w
t
(1) l
t
= (r+ x
t
) l
t1
, (2) k = Q
t
J
t
(3) J
t
= Jt
¯
w
t
+ rE
t
m
t+1
J
t+1
(4) V
t
= ¯
w
t
+E
t
m
t+1
[rV
t+1
+ (1 r) (ft+1
V
t+1
+ (1 f
t+1
)U
t+1
)]
(5) U
t
= D+ E
t
m
t+1
[ft+1
V
t+1
+ (1 f
t+1
)U
t+1
]
(6) J
t
=1 h
h(V
t
U
t
) , (7) Q
t
= sm
Qst
, (8) Qt
=l
t1
v
t
1 rl
t1
(9) m
t+1
= bu
0 (Ct+1
)
u
0 (Ct
), (10) C
t
+ kv
t
l
t1
= Y
t
,
(11) f
t
=x
t
l
t1
1 rl
t1
(12) Y
t
= p
t
exp (at
) l
t
, (13) Q
t
=x
t
v
t
with p
t
= 1 bec we linearly approximate around zero inflation ss.
Solving the Model and Assigning Values toParameters
• Shortage of one equation in goods and household sector.• The labor market added, on net, one equation (i.e., 12 extravariables and 13 extra equations).
• The model has the following 10 parametershousehold and goods productionz }| {
b, ra
, sa
, q, # ,
labor marketz }| {D, k, h, s
m
, s
• Interestingly, # plays no role in the solution to the model, to afirst order approximation— This reflects, in part, how we set the subsidy to monopolists.— Also reflects constant marginal costs of intermediate goodproducers (not obvious).
• Parameter values of goods and household sector easy tochoose, since there is a lot of experience with them— b = 0.99, r
a
= 0.95, sa
= 0.01, q = 0.75.
Monetary Policy
• Taylor rule:
R
t
R
=
R
t1
R
a
exp {(1 a) [fp ( ¯pt
1) + fx
log (Ct
/C)] u
t
} ,
where u
t
is an iid (expansionary) policy shock.• Parameters were set to the following (fairly conventional)values:
a = 0.80, fp = 1.5, fx
= 0.05.
Calibration of Labor Market ParametersGenerally, follow Ravenna-Walsh (2008) and Walsh (2003), but alsoShimer (2012) and Hall-Milgrom (2008).
Variable name symbol value end/exunemployment rate 1 l 0.05 endogreplacement ratio D
¯
w
0.4 endogsearch cost/gross output of homog good kvl/Y 0.01 endogvacancy filling rate Q 0.7 endogelasticity of matching w.r.t. unemp s 0.5 exogdiscount factor b 0.99 exogmatch survival rate r 0.9 exog
Since we exogenously set the values of four endogenous variables,we must allow four exogenous parameters to be endogenous:
D, k, h, sm
.
• Steady state computations
— first, go down first column, then go down second.— In each case, which equilibrium condition that is used isindicated.
R = 1/b (intertemporal Euler)(9) m = b k = (kvl/Y) Y/(lv)(1) x = 1 r (2) J = k
Q
(8) Q = xl
Q(1rl)(3) ¯
w = J+ rbJ J
(11) f = xl
1rl
D = D
¯
w
¯
w
(7) sm
= QQs (10) C = l kvl
J = 1 (pricing equations) (4)-(5) VU = ¯
wD
1br(1f )
(12) Y = l (5) U = D+bf (VU)1b
(13) v = x/Q (6) h = VU
VU+J
.
Comparing the DMP Labor Market withCompetitive Labor Markets
• The DMP version of the model was described in detail above.• The competitive labor market version of the model is obtainedby deleting equations 1-13 above and:
— Interpret Jt
as the real wage.— Allow households to set employment, l
t
, on the intensivemargin and equate
Jt
= C
t
l
jt
,
where 1/j denotes the Frisch labor supply elasticity (see thefirst set of lectures on the labor market in the New Keynesianmodel for details).
Comparing...results
• Run the Dynare file, DMP.mod, to see responses of the systemto an expansionary monetary policy shock.
• "Shimer puzzle" for monetary economics:
— note that the response of output is weaker in the DMP modelthan it is in the (already very weak) version of the model inwhich the labor market is competitive.
— The intuition is that in an expansion,• the labor market tightens (i.e., Q
t
rises) and the probabilitythat a vacancy draws a worker, Q
t
, decreases,• with costs fixed in terms of vacancies, the implied cost ofmeeting a worker rises,
• this negative feedback loop limits the expansion itself.
Hiring versus Search Costs• One response to this ‘Shimer puzzle’ is to replace the searchcosts of meeting a worker in the DMP model with hiring costs.— Approach taken in New Keynesian model versions of DMP byGT and GST.
• Hiring cost specification: firm pays k to meet a worker.— vacancies must be posted too, but they are costless.— the only actual cost of meeting a worker is k.
• Replace equations (2) and (10) above by
k = J
t
(2)0
C
t
+ kx
t
l
t1
= Y
t
(10)0
— do same replacement in the steady state equations, alsoreplace the calibration equation for k :
k = (kxl/Y) Y/(lx)
The E§ect of Switching to Hiring Costsfrom Search
• The switch cancels the negative feedback loop mentionedearlier.
• With a monetary expansion, the rise in employment is greaterand inflation, smaller.— This can be confirmed by running the Dynare software,DMP.mod.
• Unfortunately, this change is not su¢cient to allow the modelto display the response to shocks observed in the data.— in practice, exogenous frictions in the adjustment of wages arestill required (see GT, GST and others).
— Christiano-Eichenbaum-Trabandt (2013) stress this point too,and suggest an alternative to exogenous stickiness in stickywages.• this is the subject of the next handout.
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