Inference
Jesús Fernández-Villaverde
University of Pennsylvania
1
A Model with Sticky Price and Sticky Wage
• Household j ∈ [0, 1] maximizes utility function:


´1− 1
³
´1+γ


j
j
σ
j 1−ξ
∞
G
C
N
X
 t

η  Mt 
t
t
t


E0
β 
−
+

1
1
+
γ
1
−
ξ
P
1
−
t
t=0
σ
³
0 < β < 1 is the discount factor, σ > 0 the elasticity of intertemporal
substitution, ξ > 1 the elasticity of money holdings, and γ > 0 the
inverse of the elasticity of labor supply with respect to real wages.
2
• Subject the budget constrain given by:
j
j
j
PtCt + Mt − Mt−1 +
X
j
j
Qt(ht+τ )Dt (ht+τ ) +
Bt+1
Rt
ht+τ
j j
j
j
j
j
= Wt Nt + Πt + Tt + Dt + Bt ,
j
j
j
where Πt are firms’ profits, Tt are nominal transfers, Dt (ht+τ ) dej
notes holdings of contingent bonds, Bt+1 denotes holdings of an unj
contingent bonds, and Wt is the hourly nominal wage.
3
Technology
• Intermediate Goods producer i ∈ [0, 1] use the following production
function:

"

 Z 1³
δ
Yti = AtKsr


ij
Nt
0
´ φ−1
φ
dj
#

1−δ
φ

φ−1 


Atis a technology factor, which is common to the whole economy.
ij
Nt is the amount of hours of type j labor used by intermediate good
producer i. φ > 1 is the elasticity of substitution between different
types of labor, and 0 > δ > 1 is the capital share of output. The
production function is concave in the labor aggregate, and we assume
that capital is fixed in the short run at a level Ksr .
4
• Final good:
Yt =
"Z
1³
0
´ εt−1
i εt
Yt
#
di
εt
εt −1
εt > 1 the elasticity of substitution between intermediate goods,
Λt = εt/ (εt − 1) price markup. There is a shock to the elasticity
of substitution, εt.
5
Final Goods Price Setting
• Final good producers are competitive and maximize profits.
• The input demand functions associated with this problem are
Yti =
"
#−εt
i
Pt
Yt
Pt
∀i,
• The zero profit condition ⇒ the price of the final good
Pt =
"Z
1
0
#
Pti1−εt di
6
1
1−εt
Intermediate Goods Producers Problem
• Operate in a monopolistic competition environment. They maximize
profits taking as given all prices and wages but their own price.
• The profit maximization problem of the intermediate good producers is
divided
n
o into two stages: In the first stage, given all wages, firms choose
ij
Nt
to obtain the optimal labor mix. Hence, the demand of
j∈[0,1]
producer i for type of labor j is


1
j −φ " i # 1−δ
W
Yt
ij
Nt =  t 
Wt
At
7
∀j,
• Where the aggregate wage Wt can be expressed as
Wt =
"Z
1³
0
´
j 1−φ
Wt
dj
8
#
1
1−φ
.
• In the second stage, they set prices. They can reset their price only
when they receive a stochastic signal to do so. This signal is received
with probability 1 − θp.
• If they can change the price, they choose the price that maximizes:
Et
∞
X
τ =0
subject to

 i i
θτp Qt+τ
Pt Yt+τ − Wt+τ
t
i
=
Yt+τ
"
Pti
Pt+τ
#−εt+τ
9
Yt+τ
Ã
i
Yt+τ
At+τ
!
∀i, τ
1
1−δ



• The solution is:



i,∗


t+τ  Pt
τ
i
i
Et
θp Qt
− ΛtM Ct  Yt = 0,
 Pt+τ

τ =0
∞
X
• The evolution of the aggregate price level is:
h
1−εt
Pt = θp (Pt−1)
i 1
1−εt 1−εt
∗
+ (1 − θp) (Pt )
10
Consumers problem
• Intertemporal susbstitution Equation
GtCt
− σ1
Pt
− σ1
= βEt{Gt+1Ct+1Rt
}
Pt+1
• Demand for money, at a given interest rate, always satisfied.
11
Wage Setting Problem
• Consumers operate in a monopolistic competition environment. They
maximize utility given all wages, but their own. They reset wages if
signal to do so. They receive the signal with probability (1 − θw ). As
before, the signal is independent across intermediate good producers
and past history of signals.
j,∗
• If they can change their wage, they choose the wage, Wt , that
maximizes:



j,∗
∞


³
´γ
X
− σ1 Wt
∗j
∗j
τ
Et
(βθw ) Gt+τ Ct+τ
− ϑ Nt+τ  Nt+τ = 0


Pt+τ
τ =0
12
subject to


1
j,∗ −φ Z 1 Ã i ! 1−δ
Yt+τ
W
∗j
di
Nt+τ =  t 
Wt+τ
At+τ
0
• The evolution of the aggregate wage level is:
h
∀j, τ
i 1
1−φ
1−φ
1−φ
Wt = θw Wt−1 + (1 − θw ) (Wt∗)
.
13
Fiscal and Monetary Policy
• On the fiscal side, the government cannot run deficits or surpluses, so
its budget constraint is
Z 1
0
T (ht, j)dj = M(ht) − M(ht−1),
• On the Monetary side, as suggested by Taylor (1993), we assume that
the monetary authority conducts monetary policy using the nominal
interest rate, through a Taylor rule.
µ
r
rt
= t−1
r
r
¶ρ µ ¶γ Ã !γ y
r πt
π yt
π
14
y
emst
Dynamics
at + (1 − δ)nt − yt = 0
mct − (wt − pt) + yt − nt = 0
1
ct + γnt − gt − mrst = 0
σ
h
i
ρr rt−1 + (1 − ρr ) γ π π t + γ y yt + mst − rt = 0
−yt + ct = 0
wt − pt − (wt−1 − pt−1 − ∆wt + π t) = 0
15
Et [−σrt + σπ t+1 − σgt+1 + σgt − ct + ct+1] = 0
Et [κpmct + κpµt − π t + βπ t+1] = 0
Et [κw mrst − κw (wt − pt) − ∆wt + β∆wt+1] = 0
16
where
at = ρaat−1 + εat
µt = εµt
mst = εmst
gt = ρg gt−1 + εgt
and
κp = (1 − δ)(1 − θpβ)(1 − θp)/(θp(1 + δ(ε̄ − 1)))
κw = (1 − θw )(1 − βθw )/ [θw (1 + φγ)]
17
Solve the Model (Uhlig Algorithm)
• Derive the system:
0 = Ast + Bst−1 + Cet + Dzt
0 = Et[F st+1 + Gst + Hst−1 + Jet+1 + Ket + Lzt+1 + M zt]
zt+1 = N zt + εt+1
εt+1 ∼ N (0, Σ)
• st = (wt − pt, rt, π t, ∆wt, yt)0 is the endogenous state, et = (nt, mct, mrst, ct)
are endogenous variables, and zt = (at, mst, µt, gt)0 is the exogenous
state.
18



• N =

ρa
0
0
0
0
0
0
0
0 0
0 0
0 0
0 ρg





• Solution
st = P P st−1 + QQzt
et = RRst−1 + SSzt
19
Writing the Solution in State Space Form
• Transition equation
Ã
st
zt
!
=
Ã
P P QQ ∗ N
0
N
Ã
st
zt
!
st =
Ã
Ã
st−1
zt−1
!Ã
!
=F
• Measurement equation
I 0
0 0
!Ã
st
zt
st−1
zt−1
!
!
+
Ã
Q
I
!
εt =
+ Gεt
=H
Ã
st
zt
!
• Evaluate the Likelihood function using the Kalman Filter.
20
• Apply Metropolis-Hastings Algorigthm to get a draw from the posterior.
• Compute moments and marginal likelihood.
21
1
1−θp
1
1−θw
Prior Distribution
gamma(2, 1) + 1
gamma(3, 1) + 1
γπ
normal(1.5, 0.25)
γy
normal(0.125, 0.125)
ρr
uniform[0, 1)
σ −1
gamma(2, 1.25)
γ
normal(1, 0.5)
ρa
uniform[0, 1)
ρg
uniform[0, 1)
σ a(%)
uniform[0, 1)
σ m(%) uniform[0, 1)
22
σ λ(%)
uniform[0, 1)
σ g (%)
uniform[0, 1)
Mean/std
3.00
(1.42)
4.00
(1.71)
1.5
(0.25)
0.125
(0.125)
0.5
(0.28)
2.5
(1.76)
1.0
(0.5)
0.5
(0.28)
0.5
(0.28)
50.0
(28.0)
50.0
(28.0)
50.0
(28.0)
50.0
(28.0)
Algorithm (gencoeffsehl.m)
Step 0 Read data (usadefl1d.txt)
Step 1 Intial value for θ0, N and set j = 1.
Step 2 Evaluate f (Y T |θ0) and π(θ0) and make sure f (Y T |θ0), π(θ0) > 0
(a) Given θ0 evaluate prior π(θ0) (priorehl.m)
(b) Given θ0: Uligh algorithm to solve the model (modelehl.m and
solve2.m)
(c) Kalman Filter to evaluate f (Y T |θ0) (likeliehl.m)
23
Step 3 θ∗j = θj−1 + ε ∼ N (0, Σε) and u from Unif orm[0, 1]
(a) Given θ∗j evaluate prior π(θ∗j ) (priorehl.m)
(b) Given θ∗j : Uligh algorithm to solve the model (modelehl.m and
solve2.m)
(c) Kalman Filter to evaluate f (Y T |θ∗j ) (likeliehl.m)
³
Step 4 If u ≤ α θj−1, θ∗j
θj−1 otherwise.
´
= min
³ ´

 f (Y T |θ∗)π θ∗
j
j
 f (Y T |θ
j−1 )π (θ j−1 )
Step 5 If j ≤ N then j à j + 1 and got to 3.
24


then θj = θ∗j , θj =

,1
Prior
Mean
(Std)
1
1−θp
1
1−θw
gamma(2, 1) + 1
γπ
normal(1.5, 0.25)
γy
normal(0.125, 0.125)
ρr
uniform[0, 1)
σ −1
gamma(2, 1.25)
γ
normal(1, 0.5)
ρa
uniform[0, 1)
ρg
uniform[0, 1)
σ a(%)
uniform[0, 1)
3.00
(1.42)
gamma(3, 1) + 1
σ m(%) uniform[0, 1)
σ λ(%)
uniform[0, 1)
σ g (%)
uniform[0, 1)
4.00
(Std)
4.37
(0.35)
2.72
(1.71)
(0.27)
1.5
1.08
(0.25)
(0.09)
0.125
0.26
(0.125)
0.5
(0.28)
2.5
(0.06)
0.74
(0.02)
8.33
(1.76)
(2.50)
1.0
1.74
(0.5)
(0.29)
0.5
0.74
(0.28)
0.5
(0.28)
50.0
25
Mean of Posterior
(0.05)
0.82
(0.03)
3.88
(28.0)
(1.09)
50.0
0.33
(28.0)
(0.02)
50.0
31.67
50.0
11.88
(28.0)
(28.0)
(5.32)
(3.28)