Monopolistic Competition and Nominal Stickiness inGeneralized New Keynesian Model
Rui WANG
November 25,2019
Discussion Paper No.1913

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANMonopolistic Competition and Nominal Stickiness inGeneralized New Keynesian Model
Rui WANG∗
Department of Economics, Kanto Gakuen University†
November 25, 2019

Abstract
In this paper, we extend the standard New Keynesian Dynamic Stochastic General Equilibrium(DSGE) model to allow both staggered price setting and staggered wage setting and derive a generalized version of New Keynesian model to study how these distortions affect the steady state anddynamics of model given different annual target inflation rates. The main finding is that the imperfection of labor market has more distortionary power on aggregate output and aggregate welfare given
positive target inflation rate. Sensitivity analysis of structural parameters in the context of static steadystate provides us a macroeconomic structural model-based explanation for the stylized fact that many
central banks set the target inflation rate within a range from 1% to 2%. Also, given positive target
inflation rate, the dynamic responses of macroeconomic variables on exogenous shocks are more sensitive to the change of structural parameters related to labor market. By comparing the dynamics
generated under different sets of calibration, we find that the structure of labor market with high degree of monopolistic competition, low wage stickiness and low wage indexation is more desirable inan economy with positive target inflation rate.
Key Word: Price Stickiness, Wage Stickiness, Generalized New Keynesian Model, Distortion
JEL classification: E43, E44, E52, E58

1

Introduction

Monopolistic competition and nominal stickiness are two basic microeconomic-foundations in the standard New Keynesian model, which is the workhorse in the modern monetary macroeconomics. Monopolistic competition follows the mechanism proposed by Dixit and Stiglitz (1977). Differentiatedintermediate-goods are not perfect substitutes for each other and intermediate-good firms are in monopolistic competition market, which means intermediate-good firms have some monopolistic power
to add a markup on the marginal cost of product. The competition degree of monopolistic market canbe represented by a specific structural parameter, elasticity of substitution. Nominal stickiness can be introduced by the Calvo (1983) staggered pricing mechanism which is the most popular and widely-used
way to introduce nominal stickiness into the New Keynesian model. The Calvo (1983) price setting
mechanism is simple and easy to track and interpret. In each period, an intermediate-good firm hasa random change to optimize its price with a fixed probability. Those intermediate-good firms that
can’t re-optimize price setting just index the price to the last period inflation rate. The role of priceindexation is to fit and generate the high persistence in the dynamic response of inflation on exogenous shock, which is usually observed in empirical analysis. Given these basic features, the standard
New Keynesian model1 has the property of super-neutrality, which means that at steady state with zeroinflation, aggregate output is totally decided by real factors such as endogenous economic structure
represented by structural parameters and exogenous shock process. However, if we allow steady state∗ Assistant professor at Faculty of Economics, Department of Economics, Kanto Gakuen University, Japan. Research fellow at
Graduate School of Economics, Kobe University, Japan. E-mail: kizuna.wr@gmail.com† 200 Fujiagu-cho, Ōta, Gunma Prefecture, 373-0034, Japan.
1 For the standard literature about the New Keynesian model and its application in monetary policy analysis, please refer toWoodford (2003), Galı́ (1999, 2000, 2015), Miao (2014) and Walsh (2017).

1

with non-zero inflation rate, super-neutrality will not hold anymore. New Keynesian model with nonzero inflation steady state, in another terminology, trend inflation, is also known as Generalized New
Keynesian model2 . Note that non-zero steady state inflation or trend inflation in the New Keynesianmodel can also be explained as the target inflation rate anchored by the central bank in the practice ofmonetary policy. This is why we should be concerned with this issue. Most of existing literatures on
generalized New Keynesian model only focus on monopolistic intermediate-good market and assume
that labor market is completely competitive. In this paper, following Erceg, Henderson, and Levin(2000), we introduce monopolistic labor market, wage stickiness and wage indexation into the generalized New Keynesian model proposed by Ascari and Sbordone (2014). Under this setting, we have 6
key structural parameters that dominate these nominal and real distortions in the model.
εp
εw
θp
θw
$p
$w

monopolistic competition in intermediate-good market
monopolistic competition in labor market
nominal stickiness in staggered price setting
nominal stickiness in staggered wage setting
backward-looking indexation in staggered price setting
backward-looking indexation in staggered wage setting
Table 1: Economic Explanation of Structural Parameters

As listed in Table 1, these structural parameters have explicit economic interpretation. For the valueof parameters related to price (wage) setting, θ p (θw ) and $ p ($w ), these parameters can be identified andestimated in the empirical estimation of DSGE models such as Smets and Wouters (2003, 2007) andChristiano et al. (2005). These parameters can also be estimated from micro-level data. For ε p and ε w ,these parameters are generally calibrated to fit the markup rate. As we will show in the later3 , if target
inflation rate is zero, no matter what the values of these structural parameters are, they don’t effect pricedispersion or wage dispersion, super-neutrality does hold in this case. However, if target inflation rate is
not zero, aggregate output and aggregate welfare will be directly affected through the price dispersion
and wage dispersion. Based on this reason, in this paper, we investigate these how these structuralparameters affect the steady state and dynamics of generalized New Keynesian model.
We find that in our generalized version of New Keynesian model, in the context of steady state,increasing of elasticity of substitution in intermediate-good market (ε p ) always leads to increasing of
aggregate output given every target inflation rate from 0% to 8%. But increasing of elasticity of substitution in intermediate-good market (ε p ) only leads to positive aggregate welfare gain given moderatetarget inflation rate from 0% to 4%. Besides, high elasticity of substitution in labor market (ε w ) only
leads to high aggregate output given target inflation from 0% to 2%. Also, increasing of elasticity of
substitution in labor market (ε w ) only leads to positive aggregate welfare gain given strict 0% target
inflation rate. These findings may imply us a macroeconomic structural model-based explanation forthe stylized fact that the most central banks set their target inflation rate within a range from 1% to 2%.
From the sensitivity analysis of structural parameters in the context of dynamic impulse response, wefind that impulse response functions are more sensitive to the change of structural parameters of labormarket than the counterparts of intermediate-good market. Labor market with high degree of monopolistic competition (low ε w ), low wage stickiness (low θw ) and low wage indexation (low $w ) is desirable
in the economy with positive target inflation rate. Labor market with low degree of monopolistic competition, in another word, high degree of competition (high ε w ) is not desirable in this situation.
This remaining paper is organized as follows. In Section 2, we firstly derive out generalized New
Keynesian model. Then we linearize the equilibrium conditions around the steady state associated with
non-zero target inflation rate. In Section 3, we check how these nominal and real distortions dominatedby the key structural parameters affect the deterministic steady state of nonlinear model by changing
the key structural parameters given different positive target inflation rates. Section 4 checks the sensitivity of key parameters in the context of dynamic impulse response. Finally, Section 5 summarizes thispaper, implies policy implications and prospects the future research.
2 Ascari and Sbordone (2014) provides a survey about empirical results of trend inflation. They also show a baseline generalized New Keynesian model.
3 Please refer to Appendix.

2

2

Generalized New Keynesian Model

2.1

Household and Labor Supply
C 1− σN1+ ϕ

ς t h,t
The period utility function U (Ch,t , Nh,t ) = 1h,t−σ − χe 1+ ϕ of the representative household with indexation h ∈ (0, 1) is assumed to be separable in consumption Ch,t and labor Nh,t , where ς t is an
exogenous labor supply shock. σ and ϕ represent the inverse of inter-temporal elasticity of substitution and the inverse of Frisch elasticity of labor supply, respectively. Households supply differentiated kinds of labor to competitive labor ”packer” which combines heterogeneous types of labor intoa homogenous composite labor good that it then leases to intermediate-good producers at prevailingnominal wage rate Wt . The imperfect substitutability between differentiated types of labor gives households some monopolistic power in setting their own wage. The period budget constraint is given byW
B
B= Ph,tNh,t + Dh,t + h,tPt−1 where it is nominal interest rate, Bh,t is one-period risk-free bondCh,t + P (1h,ttt +it )
holdings, Wh,t is the nominal wage rate paid to household h that has specialized kind of labor h, andDh,t is distributed dividends. The representative household maximizes the expected sum of discountedtinter-temporal utility ∑∞t=0 β U (Ch,t , Nh,t ) subject to period budget constraint. β is subjective discountfactor. According to Erceg, Henderson, and Levin (2000), if there exist state contingent claims that insure households against idiosyncratic wage risk, and if preferences are separable in consumption andleisure, all households will be identical in their optimal choice of consumption and bond-holdings, andwill only differ in the wage they charge and labor supply. Based on this reason, we can omit the indexation h in the first-order conditions of consumption Ch,t and bond-holdings Bh,t . In our model, there
doesn’t exist government and capital accumulation, final output Yt is all used as consumption Ct .

Yt = Ct(1)

Euler equation derived from utility maximization can be written as"!#
1 + it
1
1
= βEt
Ytσ
π t +1Ytσ+1where πt+1 =

(2)

Pt+1
Pt is defined as gross inflation rate from period t to period t + 1. Competitive laborpacker uses CES technology Nt =R1

ε w −1εw

0 Nh,t εw

ε w −1dh

to pack differentiated kinds of heterogeneous la-bor into homogenous labor which is used by intermediate-good producers as input. ε w > 1 is the elasticity of substitution among differentiated kindsof labor with indexation
 h. The profit maximization
R1
problem of the competitive labor packer is max Wt Nt − 0 Wh,t Nh,t dh from which we can solve the{ Nh,t }−ε wWdemand of specialized kind of labor h as Nh,t = Wh,tt
Nt . Substituting the demand equation of Nh,tR 1
1− ε w1
1− ε w
into the definition of Nt leads to the aggregate level of nominal wage Wt as Wt =
W
dh
.
0
h,t
∗
Optimal nominal wage set by the representative household that supplies Nh,t is denoted as Wh,t . The
wage setting here follows the assumption of Calvo (1983) and Erceg, Henderson, and Levin (2000).
There is a probability θw that a household will be stuck with the current wage in each period. Thosehouseholds can’t optimize their wage just index the nominal wage to previous period gross inflation$
rate as Wh,t+ j = πt+w j−1 Wh,t+ j−1 . $w ∈ [0, 1] indicates the degree of backward-looking indexation. Realtwage rate wt is defined as wt = W. Wage optimization problem of the representative household h can
P"
t
#

j
be written as max Et ∑∞j=0 ( βθw )
∗ }
{Wh,t

1+ ϕ

N +j
Ct−+σj Nh,t+ j Wh,t+ j− χeς t+ j 1h,t
Pt+ j
+ϕ

which means the representative house-

hold maximizes the expected sum of discounted real labor income

Wh,t+ jPt+ j Nh,t+ j measured by marginal∗ at period t subject to the labor demand
utility of consumption Ct−+σj by choosing optimal real wage wh,t
N

1+ ϕ

+j
Nh,t+ j , given the disutility of labor supply χeς t+ j 1h,t
+ ϕ . The first-order condition of wage optimization is



∗
wh,t

1+ ε w ϕ=3

ε w ψw,t
ε w − 1 ϕw,t(3)

∗ is optimal real wage set by household at period t. ψ
where wh,tw,t and ϕw,t are two auxiliary variables
with recursive structure which are defined to rewrite the first-order condition of wage optimization.
i
h
1+ ϕ
− $ ε (1+ ϕ )ε w (1+ ϕ )ψw,t = χeς t wtε w Nt
(4)
+ βθw πt w w
Et π t +
ψ
w,t
+
1
1
$ (1− ε w )ϕw,t = Ct−σ wtε w Nt + βθw πt w

i
h
w −1
Et πtε+
ϕ
w,t
+
1
1

(5)

According to Law of Large Numbers (LLN), aggregate level of wage Wt can be written as Wt = 
1− ε w
1−ε w  1−1ε w
R 1
1− ε w$w
1
1− ε w∗+1−
θW
from which we can solve the
W
dh
=
θ
π
W
(
)
w
w
t −1 t −1h,t0h,t∗ by writing this equation into real term.
optimal real wage wh,t$

1 = θw

πt−w 1πt!1− ε w 

Define nominal wage dispersion as sw,t =sw,t = (1 − θw )2.2

w t −1wt+ (1 − θ w )R 1  Wh,t −ε w
0

Wt

∗
wh,t

!−ε w

!1− ε w
(6)

wt

dh and write sw,t into recursive formulation.
$

+ θw

wt

∗
wh,t

 1− ε wwt−1 πt−w 1
wt πt

!−ε w
sw,t−1(7)

Intermediate-good Producer and Final-good Producer

Final-good producers aggregate intermediate-goods Yi,t to final-good Yt with CES production technol! εp
ε −1
ε p −1R 1 pε pwhere ε p > 1 indicates the elasticity of substitution among differentiateddiogy Yt =Y
0 i,tintermediate-goods. Final-good producer maximizes the profit Pt Yt −R1

Y di subject to this pro0 Pi,t
 i,t

R1
duction technology. The profit maximization of final-good producer is max Pt Yt − 0 Pi,t Yi,t di from
{Yi,t }
 −ε pP
which we can solve the optimal input of each intermediate-good Yi,t as Yi,t = Pi,tt
Yt . Substi-

tuting this demand equation of Yi,t into production function of Yt leads to definition of general price R 1− ε  11− ε p1
p. The production technology of intermediate-good producer is Yi,t =level Pt =di
0 Pi,t1− αAt Ni,twhere At is total factor productivity that is identical for all intermediate-good producers.
  1
Wt Yi,t 1−αr and real marginal cost MC r are given as TC r = Wt N
Real total cost TCi,t
and
i,t
i,t
Pt i,t = PtAtrr = ∂TCi,t
MCi,t
∂Yi,t
1

r =
MCi,t

1

=

Atα−1 Wt
1−α PtAtα−1 Wt 1−αrespectively.
1−α Pt Yi,t



Pi,t
Pt

α

−ε p

r leads to
Substituting demand equation of Yi,t into MCi,t

 α
Yt

1− α

which means the different intermediate-good producers charging

different prices would produce different levels of output and have different marginal costs if α 6= 0.
If α = 0, the model economy has the property of constant return to scale and all intermediate-good
producers have the same real marginal cost. The pricing mechanism of intermediate-good producer
is based on Calvo (1983). In each period, there is a fixed probability 1 − θ p that an intermediate-good∗ . With probability θ , instead, theproducer can optimize its nominal price and set it to optimal level Pi,tpintermediate-good producer that can’t optimize its price setting just indexes its price to the precious$pperiod gross inflation rate like Pi,t = πt−1 Pi,t−1 where $ p ∈ [0, 1] indicates the degree of backwardlooking indexation. Intermediate-good producer optimizes
its price to maximize
the expected sum of
discounted profit in real term as max Et ∑∞j=0 Dt,t+ j θ p
j∗}{ Pi,tPi,t+ jr
Pt+ j Yi,t+ j − TCi,t+ j

λwhere Dt,t+ j = β j λt+t j is thestochastic discount factor and λt+ j denotes the marginal utility of consumption Ct−+σj . The first-ordercondition of price optimization is∗pi,t1+ αε p

1− α

=

εp
ψ p,t

(1 − α) ε p − 1 ϕ p,t4(8)∗ as p∗ = P∗ /P and define two auxiliary variables ψ
where we rewrite pi,ttp,t and ϕ p,t to simplify thei,ti,tfirst-order condition. ψ p,t and ϕ p,t can be rewritten into recursive form.
1

ψ p,t = wt Yt1−α

−σ

 εp

$p ε p− 1−
At 1−α + βθ p πt 1−α Et πt1+−1α ψt+1
$ p (1− ε p )ϕ p,t = Yt1−σ + βθ p πt(9)ih ε −1p
Et πt+1 ϕ p,t+1(10)

According to Law of Large Numbers (LLN), 1 − θ p proportion of all intermediate-good producers can
optimize price and θ p proportion of all intermediate-good producers just index the price to last periodgross inflation rate. The aggregate price level Pt evolves as following equation.
Pt =

Z 1
0

1− ε pPi,t di 1−1ε

p



= θp



1− ε p$p+πt−1 Pt−1

1 − θp∗ 1− ε p
Pi,t





1
1− ε p∗ .
Rewrite this equation by dividing Pt and solve the optimal relative price pi,t

$ p (1− ε p )ε p −11 − θ p π t −1
πt
∗
pi,t
=
1 − θp

11− ε p(11)

Price dispersion s p,t can be written into recursive form under the assumption of the Calvo (1983).
s p,t = 1 − θ p

2.3



∗
pi,t

εp

− ε p−$p ε p+ θ p πt1−α πt−11−α s p,t−1

1− α

(12)

Aggregate Output and Potential Output

Denote N(h,i),t as the type of labor h packed by labor packer that provides homogenous labor to intermediate
R 1 R 1dhdi.
good producer i. Aggregate labor demand Nt is defined as Nt = 0
N
(
h,i
)
,t
0
Nt =

Z 1
0

Ni,t

Z 1

N(h,i),t

0

Ni,t


dh


di = sw,t s p,t

Yt
At

 1

1− α

(13)

f

We use f to denote the variables under flexible price and wage. Yt can be solved by setting θ p = θw = 0
in equilibrium conditions. When price and wage are both flexible, every household and intermediategood producer will make identical optimal decisions for price and wage setting, so price dispersion andffwage dispersion will disappear, which means sw,t = s p,t = 1.
f∗pi,t = 1 =

f

f

p,t

w,t

(1 − α)(ε p − 1) ψ p,t  f ∗ 1+ε w ϕ
ε w ψw,t, wh,t=fεpεw − 1 ϕ f
ϕ
f∗
f
f
f
f
wh,t = wt , Yt = Ct , Nt =

f

Yt
At

! 1−1 αfCombining these equilibrium conditions leads to Yt .
 1+ ϕ  ϕ+α+1−σ(α1−α)# 1− αϕ + α + σ (1− α )
1− α
(
1
−
α
)
ε
−
1
ε
−
1
(
)
p
w
f
 At Yt =χε p ε w
eς t
"

f

(14)

From this result, we can find that potential output Yt is totally decided by real factors and not affectedby target inflation rate.

5

2.4

Monetary Policy

The central bank follows a standard Taylor rule with weight ϕπ p on deviation of inflation from its targetfπ, weight ϕπw on wage inflation and weight ϕy on output deviation from potential output level Yt . vt
is a monetary policy shock.
! ϕy
  



1 + itπt ϕπ pwt πt ϕπw Yt
(15)
evt
=
f
1+i
π
w t −1 πYt

Output gap can be defined as the difference between output under decentralized equilibrium withsticky price and sticky wage and potential output under decentralized equilibrium with flexible priceand flexible wage.
Yt
xt = f
(16)
Yt

2.5

Aggregate Welfare Function

Aggregate welfare Wt can be defined as the aggregation of each individual household’s welfare Wh,t
 1− σ1+ ϕ 
R1
t Ch,t − χeς t Nh,t
as Wt = 0 Wh,t dh, where Wh,t is given as Wh,t = ∑∞
β
t =0
1− σ
1+ ϕ .
∞

Wt = ∑ β t
t =0

Ct1−σ
1−σ

−

Z 1
χeς t

1+ϕ

0

!
1+ ϕ

Nh,t dhRewrite Wt in recursive form and define
Wt =



∞

=∑β

1− σ
t  Ctt =01−σ

R 1  Wh,t −ε w (1+ ϕ)
0

Wt

1+ ϕ Z 1 Ct1−σχeς t Nt
−
1−σ
1+ϕ

0

Wh,t
Wt

−

Z 1
χeς t

1+ϕ



0

Wh,t
Wt

!1+ ϕ

−ε w
Nt

dh

dh as ∆w,t .

 − ε w (1+ ϕ )dh + βEt Wt+1
(17)

1+ ϕ

C 1− σχeς t Nt
= t
−
1−σ
1+ϕ



∆w,t + βEt Wt+1

Similar to the wage dispersion sw,t , we rewrite ∆w,t in recursive formulation.
∆w,t = (1 − θw )∗wh,t

! − ε w (1+ ϕ )

wt

$

+ θw

wt−1 πt−w 1wt πt! − ε w (1+ ϕ )∆w,t−1(18)

We have 18 equilibrium conditions from equation (1) to equation (18) with 18 endogenous variables
n
o
f
∗
∗
Ct , Yt , it , πt , wt , Nt , pi,t, wi,t, ψ p,t , ϕ p,t , ψw,t .ϕw,t , s p,t , sw,t , Yt , xt , ∆w,t , Wt
and 3 shock process variables { At , ς t , vt }.
2
log At = ρ A log At−1 + ε A,t , ε A,t ∼ N (0, σA)ς t = ρς ς t−1 + ε ς,t , ε ς,t ∼ N (0, σς2 )vt = ρv vt−1 + ε v,t , ε v,t ∼ N (0, σv2 )Linear approximating all nonlinear equilibrium conditions around the steady state associated with target inflation rate leads to linear form of generalized New Keynesian model4 . By setting zero annual
target inflation rate as π = 1, after some algebraic manipulations, we can reduce our model to a morecompact form5 .
4 Please refer to Appendix.
5 Please refer to the New Keynesian model with price stickiness and wage stickiness in Chapter 6 in Galı́ (2015)6

3

Analysis of Deterministic Steady State

Following standard literature of New Keynesian model such as Woodford (2003), Galı́ (2015), Walsh(2017) and Miao (2014), we calibrate the structural parameters given in Table 2 as benchmark case.
σ = 1 means the representative household has the log utility. α = 0 means the model economy has theproperty of constant return to scale. ε p = 6 means the intermediate-good producer add 20% markup
rate6 on marginal cost. Similarly, household adds 28% markup rate7 on marginal rate of substitution. θ pand θw are both calibrated to be 0.75, which means the price and wage can be reset every 4 quarters. $ p
and $w are calibrated to be zero, implying that there doesn’t exist backward-looking indexation in pricesetting and wage setting. Once price or wage has been optimized to its optimal value, it will not change
until next change of re-optimization. We calibrate the aggregate output and potential aggregate output
to be 1 under zero target inflation rate. We can calibrate χ with reverse engineering8 from the steadystate of potential aggregate output Y f . According to Galı́ (2015), ϕπ p + ϕπw > 1 must hold to ensure theexistence of a unique stationary equilibrium. Without loss of generality, we set ϕπ p = ϕπw = 1, whichmeans that the nominal interest rate it responses to price inflation and wage inflation with same weight.
σ

β

ϕ

α

εw

$w

θw

εp

$p

θp

ϕπ p

ϕπw

ϕy

π

χ

ρA

ρς

ρv

1

0.99

1

0

4.5

0

0.75

6

0

0.75

1

1

0.125

1

5.8333

0.9

0.9

0.5

Table 2: Benchmark Calibration of Structural Parameters
For the analysis of deterministic steady state, all exogenous shock processes disappear, we can solve
the steady state of endogenous variables given in Table 3.
C

Y

i

π

w

N

pi∗

wh∗

ψp

ϕp

ψw

ϕw

sp

sw

Yf

x

∆w

W

0.33330.33330.0101

10.83330.33331

0.83333.23623.88350.48781.7096

1

1

1

1

1

−142.2686

Table 3: Benchmark Steady State at Zero Target Inflation Rate
Figure 1 shows the steady state of main model variables given positive annual target inflation rate9
from 0% to 8%. At the steady state associated with zero target inflation rate, each variable attains its
benchmark value given in Table 3. With the increasing of target inflation rate π, optimal relative price pi∗
increases from 1 to 1.075, about 7.5% increasing. Optimal real wage wh∗ shows different behavior compared with optimal price pi∗ . Optimal real wage wh∗ decreases slightly from 0.833 to 0.818, about 1.86%
decreasing. The increasing of target inflation rate doesn’t have much impact on the optimal decisionof household and intermediate-good producer. Price dispersion s p also increases with the increasing oftarget inflation rate, but very slightly, 8% annual target inflation rate corresponds with 2.5% increasing
in price dispersion compared with zero target inflation rate. The situation for wage dispersion sw isquite different. Wage dispersion sw increases from 1 to 1.37 with the increasing of annual target inflation rate from 0% to 8%. 37% increasing of wage dispersion is the main distortion that erodes thesteady state aggregate output Y and aggregate labor input N. Given the benchmark calibration of otherstructural parameters and the 3% annual target inflation rate, Y is 0.294 and N is 0.328, which means
11.9% loss and 1.68% loss compared with their corresponding steady states associated with zero target
inflation rate.
In New Keynesian model with price stickiness and wage stickiness, there are two type, four sourcesof distortions. First, monopolistic competition in intermediate-good market and labor market can generate low employment and low output. This first type of two sources of distortions can be corrected byproviding employment subsidy to firms. This subsidy can be financed by non-distortionary lump-sum
taxes levied from households. The second type of distortion comes from the staggered price setting
6 p∗ = ε p MC riε p −1ϕ
7 w∗ = ε w χNhε w −1 C − σ
8 χ = ( ε p −1 ) ( ε w −1)ε p εw
9 For example, 4% annual target inflation rate means that in the model, steady state of quarterly inflation target rate π is given1

as π = (1.04) 4 ≈ 1%.

7

and staggered wage setting. This second type of two sources of distortions can’t be corrected simultaneously because there exists only one policy instrument in New Keynesian model. We summarize
the structural parameters that dominate these distortions in Table 1. We also check how the degree of
backward-looking indexation affect the model10 . We change the values of these key structural parameters to check how these distortions affect the steady state given positive target inflation rate.

3.1

Steady State of Price Dispersion

Given the non-zero target inflation rate π and the assumption that the model economy has the property εp1−θ p π (ε p −1)(1−$ p ) ε p −1
1− θ pof constant scale to scale (α = 0), price dispersion is given as s p =1− θ pε p (1− $ p )1− θ p π

which means there are 3 structural parameters, ε p , θ p and $ p , can affect the price dispersion s p at steadystate. ε p represents the real distortion of monopolistic competition in intermediate-good market. θ p represents the nominal distortion of price stickiness in staggered price setting. $ p represents the backwardlooking indexation of non-optimizing firms in the Calvo (1983) framework of staggered price setting.
Figure 2 shows how the increasing of these 3 structural parameters change the price dispersion s p at thesteady state. For example, the panel (a) of Figure 2 shows that, given the same elasticity of substitution
ε p , price dispersion s p increases with the increasing of target inflation rate π. Given the same target
inflation rate π, higher elasticity of substitution ε p leads to higher price dispersion s p . Higher elasticity
of substitution means the less monopolistic power of firm. The intuition of this result is that the firmwith high monopolistic power is not willing to change its pricing frequently, hence less price adjustment and less price dispersion. Similar pattern can be confirmed in the panel (c) of Figure 2. Given
the same target inflation rate π, higher price stickiness θ p causes more lack of synchronization in priceadjustment, which means high price dispersion. Also, given the same θ p , higher target inflation rate
leads to larger price dispersion because of the forward-looking nature of firm. Finally, non-optimizing
intermediate-good firms index their prices to the last period gross inflation rate. $ p = 1 means total
indexation and $ p = 0 means no indexation. Given the same positive target inflation rate π, less indexation which means a smaller $ p leads to larger price dispersion. Price dispersion s p changes withthe change in key parameters, but still in very small range. For example, given θ p = 0.8 and 8% target
annual inflation rate, s p = 1.053, only 5.3% increasing from the benchmark case (s p = 1).

3.2

Steady State of Wage Dispersion

1− θ w
Similar to the analysis of price dispersion, wage dispersion is given as sw =which means
1 − θ w π ε w (1− $ w )that there are 3 structural parameters, ε w , θw and $w , can affect the wage dispersion at steady state. ε w
represents the real distortion of monopolistic competition in labor market. θw represents the nominal
distortion of wage stickiness. $w represents the backward-looking indexation of non-optimizing households in the Calvo (1983) framework of staggered wage setting. Figure 2 shows the change of wage
dispersion sw . The change in key structural parameters leads to significant change of wage dispersion,
given the same target inflation rate. At 2% annual target inflation rate, sw is 1.072 given ε w = 4.5 and sw
is 1.118 given ε w = 7. The increasing of ε w from 4.5 to 7 results in almost 4.3% increasing of sw , whichmeans the less monopolistic power of household, the larger wage dispersion. The change of θw and $w
also results in significant change of wage dispersion, which can be confirmed from Figure 2. For example, given 2% annual target inflation rate, sw = 1.031 with θw = 0.5 and sw = 1.137 with θw = 0.8 meanalmost 10.3% increasing of wage dispersion sw . Higher wage stickiness means larger wage dispersion.
Note that in the panel (a) of Figure 2, the range of price dispersion s p is from 1 to 1.033, but in the panel(b) of Figure 2, the range of wage dispersion sw is from 1 to 1.762. These findings imply that the structural parameters of labor market affect the wage dispersion in a more aggressive way compared to theway that the structural parameters of intermediate-good market affect the price dispersion.

3.3

Steady State of Aggregate Output and Aggregate Welfare

Following the way in previous section that we used to check the relationship between structural parameters and price (wage) dispersion given different target inflation rate, we check the steady state of
10 It is generally known that the backward-looking indexation is an ad-hoc assumption to generate the inertia of inflation, but
this assumption has less empirical evidence in microeconomic level of household and wage behavior. In this paper, we don’t
have much discussion about this assumption and just show the numerical results.

8

aggregate output given different combinations of target inflation rate and structural parameters. By
combing the equilibrium conditions at steady state, we can solve the aggregate output Y as the functionof A and structural parameters11 .
! 1− α + ε p ( α + ϕ ) ϕ+α+1−σ(α1−α)1 − θ p π (ε p −1)(1−$ p ) (1−ε p )(1−α)  (1 − α ) ε p − 1 ( ε w − 1)


×


χε p ε w
1 − θp



 

! ϕ
ε p (1− $ p )


1+ ϕ
ε
1
−
$
(
)


w
w
1
−
α
1 − θpπ1 − θw π
Y = A ϕ + α + σ (1− α ) ×1 − θp1 − θw





 

!
ε p (1− $ p )ε w (1+ ϕ)(1−$w )


1− α
1
−
βθ
π
1
−
βθ
π
p
w

×
×
ε
−
1
1
−
$
)
(
)(
w
w
ε
−
1
1
−
$
(
)(
p
p)
1
−
βθ
π
w
1 − βθ p πGiven the benchmark calibration of structural parameters (α, σ, ϕ, χ), we change the value of otherstructural parameters to investigate how monopolistic competition and nominal stickiness in intermediategood market and labor market affect the output at steady state under different target inflation rate.
Aggregate welfare can be checked in a similar way.
 1− σC
1 − θw
χN 1+ ϕ
1
∆w =
− ∆w
, W=
1−β 1−σ1+ϕ1 − θw π ε w (1−$w )(1+ ϕ)
At steady state, only if target inflation rate is zero (π = 1) or wage stickiness disappears (θw = 0), ∆wwill be unit. If either of these two conditions doesn’t hold, ∆w will be larger than unit and generate
the loss of aggregate welfare12 . Figure 3 and Figure 4 show the sensitivity of aggregate output Y andaggregate welfare W to the change of key structural parameters given each target inflation rate. We
find that steady state aggregate output Y is more sensitive to the change of structural parameters that
dominates the imperfection of labor market, especially at the high target inflation rate. Also, given zerotarget inflation rate, when ε w increases from 4 to 7, Y has 6.9% increasing from 0.3273 to 0.3499. W also
increases from -142.9 to -140.7. But given 2% target inflation rate, Y doesn’t have change much but W
decreases from -153.8 to -161.9.

Flexible Price and Flexible Wage
Sticky Price and Sticky Wage
Flexible Price and Sticky Wage
Sticky Price and Flexible Wage

θ p = 0 and θw = 0
θ p = 0.75 and θw = 0.75
θ p = 0 and θw = 0.75
θ p = 0.75 and θw = 0

Table 4: Model Variation
By setting the value of θ p and θw , we can turn-off/-on the stickiness in the model as shown in Table 4.
Figure 5 shows that how the target inflation rate affects the aggregate output Y and aggregate welfare
W given other parameters calibrated as benchmark case. Given θ p = θw = 0, the model economy has
both flexible wage and flexible price, so the target inflation rate doesn’t have real effect on output andwelfare. By comparing other 3 variations of model economy, we can find the distortion caused by wage
stickiness is more severe than the counterpart caused by price stickiness, especially given a high target
inflation rate.

4

Analysis of Dynamic Impulse Response

As shown in Appendix, the coefficients of linearized model contain target inflation rate π. For example,
given zero target inflation rate (π = 1), equation (5) of price dispersion bs p,t will reduce to bs p,t = θ p b
s p,t−1and equation (10) will reduce to b
sw,t = θw b
sw,t−1 . This means bs p,t = b
sw,t = 0 and the price dispersion andwage dispersion will not affect the model dynamics at least at first-order approximation. But if target
11 The steady state of technology A is normalized to be A = 1.
12 Note that in the computation of steady state welfare, σ = 1 reduces C1−σ to log C.
1− σ

9inflation rate is positive, bs p,t and b
sw,t will directly affect the model dynamics. Considering this point,
we run a sensitivity analysis of structural parameters as showed in Table 5.

εp
εw
θp
θw
$p
$w

BenchmarkHigh ε pHigh ε w

Low θ pLow θw

High $ p

High $w

6
4.5
0.75
0.75
0
0

9
4.5
0.75
0.75
0
0

6
7.5
0.75
0.75
0
0

6
4.5
0.5
0.75
0
0

6
4.5
0.75
0.5
0
0

6
4.5
0.75
0.75
0.9
0

6
4.5
0.75
0.75
0
0.9

Table 5: Structural Parameter Sensitivity

ε p and ε w
θ p and θw$ p and $w

Productivity Shock ε A,tMonetary Policy Shock ε v,tLabor Supply Shock ε ς,tFigure 6
Figure 7
Figure 8

Figure 9
Figure 10
Figure 11

Figure 12
Figure 13
Figure 14

Table 6: Sensitivity Analysis of Structural Parameters in Dynamic Impulse Response

We run two sets of sensitivity analysis, given 0% target inflation rate and 2% target inflation rate andcompare the dynamic response of 3 kinds of exogenous shocks. All results are presented from Figure 6
to Figure 14. We pick up some interesting findings from these results.

4.1

Productivity Shock ε A,tbt and theFrom Figure 6, we can find that high ε w can significantly strength the response of output Y
bt at 2% target inflation rate. From Figure 7, we can find that given 0% target
response of inflation π
inflation rate, the change of θ p doesn’t result in significant change in the dynamic impulse response
bt to productivity shock ε A,t , but low wage stickiness (θw = 0.5) can weaken theof aggregate output Y
response. The situation is different given 2% target inflation rate. Low price stickiness (θ p = 0.5) canstrengthen the impulse response. Figure 8 shows the sensitivity analysis of $ p and $w . The impulse
bt and π
bt are less sensitive to the change of target inflation rate. Also, given same
response functions of Y
target inflation rate, high wage indexation ($w = 0.9) can generate the strongest impulse responses
compared to the case of benchmark calibration.

4.2

Monetary Policy Shock ε v,t

The panel (a) and panel (b) of Figure 11 are worth noting. High wage indexation ($w = 0.9) neutralizes the effect of monetary policy shock given 2% target inflation rate. This finding may imply a policy
implication that the flexible and rapid adjustment of wage contract is preferable in an economy with
non-zero target inflation rate. The changes incurred by other four structural parameters are less significant, which can be confirmed from Figure 9. Comparing the panel (c) and panel (d) of Figure 10, we
can find that given the same θw , the change of θ p can generate obvious change of impulse response in
the case of 2% target inflation rate, but no obvious change in the case of 0% target inflation rate.

4.3

Labor Supply Shock ε ς,tComparing the panel (a) and panel (b) of Figure 12, we can find that high ε w can strength the response
bt in the case of 2% target inflation rate. But in the case of 0% target inflation rate, the 3of output Y
sets of impulse response function are almost same given 3 different sets of calibration. The panel (b) of
Figure 14 is also worth noting. High wage indexation ($w = 0.9) neutralizes the effect of labor supply
shock, but not in the case of 0% target inflation rate.

10Summarizing all these findings, we conclude that given 2% target inflation rate, impulse response
functions are more sensitive to the change of structural parameters, especially to those structural parameters of labor market.

5

Conclusion and Policy Implication

Finally, we conclude our paper to summarize the main findings. Standard New Keynesian model withprice stickiness generally assumes the steady state associated with zero target inflation rate. We extend
the standard New Keynesian model to allow both price stickiness and wage stickiness, and allow thesteady state associated with positive target inflation rate. These two modifications change the static
properties of the model. Aggregate output and aggregate welfare responses more aggressively to thechange in the key structural parameters that dominates the imperfection of labor market. From the sensitivity analysis of structural parameters in the context of static steady state, we can find that if we allow
a non-zero target inflation rate, given the reasonable range of key structural parameters, 1% ∼ 2% is thereasonable range of annual target inflation rate, which is consistent with the most of central banks. Fromthe sensitivity analysis of structural parameters in the context of dynamic impulse response, we can findthat the change of structural parameters related to labor market can generate significant changes in the
responses of macroeconomic variables on exogenous shocks compared those counterparts of structuralparameters related intermediate-good market. Given positive target inflation rate, high degree of monopolistic competition (low ε w ), low wage stickiness (low θw ) and low wage indexation (low $w ) aremost desirable compared to other calibrations, at least in the context of this paper.
Some questions are remained for the future research. For example, we need to identify the determinacy and indeterminacy regions of the structural parameters in monetary policy rule. Optimalmonetary policy should be evaluated by a rigorous welfare function which is derived by second-orderapproximating utility function. Previous research only treated the similar question with the generalized New Keynesian model that doesn’t consider the imperfection of labor market. We will discuss thisquestion in the future research.

References[1] Dixit, A. K., & Stiglitz, J. E. (1977). Monopolistic Competition and Optimum Product Diversity. The
American Economic Review, 67(3), 297-308.
[2] Ascari, G., & Sbordone, A. M. (2014). The Macroeconomics of Trend Inflation. Journal of Economic
Literature, 52(3), 679-739.
[3] Erceg, C. J., Henderson, D. W., & Levin, A. T. (2000). Optimal Monetary Policy with Staggered
Wage and Price Contracts. Journal of Monetary Economics, 46(2), 281-313.
[4] Christiano, L. J., Eichenbaum, M., & Evans, C. L. (2005). Nominal Rigidities and the Dynamic
Effects of A Shock to Monetary Policy. Journal of Political Economy, 113(1), 1-45.
[5] Calvo, G. A. (1983). Staggered Prices in A Utility-maximizing Framework. Journal of Monetary
Economics, 12(3), 383-398.
[6] Clarida, R., Gali, J., & Gertler, M. (1999). The Science of Monetary Policy: A New Keynesian Perspective. Journal of Economic Literature, 37(4), 1661-1707.
[7] Galı́, J. (2015). Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New
Keynesian Framework and its Applications. Princeton University Press.
[8] Clarida, R., Galı́, J., & Gertler, M. (2000). Monetary Policy Rules and Macroeconomic Stability:
Evidence and Some Theory. The Quarterly Journal of Economics, 115(1), 147-180.
[9] Walsh, C. E. (2017). Monetary Theory and Policy. MIT Press.
[10] Miao, J. (2014). Economic Dynamics in Discrete Time. MIT Press.
[11] Smets, F., & Wouters, R. (2003). An Estimated Dynamic Stochastic General Equilibrium Model ofthe Euro Area. Journal of the European Economic Association, 1(5), 1123-1175.
11

[12] Smets, F., & Wouters, R. (2007). Shocks and Frictions in US Business Cycles: A Bayesian DSGE
Approach. American Economic Review, 97(3), 586-606.

(a) (π, pi∗ ) plane(b) (π, wh∗ ) plane(c) (π, s p ) plane(d) (π, sw ) plane(e) (π, Y ) plane(f) (π, N ) plane(g) (π, W) plane(h) (π, i ) plane

Figure 1: Steady State under Different Annual Target Inflation Rates with Benchmark Calibration

12

(a) (ε p , π, s p ) heat-map given θ p = 0.75 and $ p = 0(b) (ε w , π, sw ) heat-map given θw = 0.75 and $w = 0(c) (θ p , π, s p ) heat-map given ε p = 6 and $ p = 0(d) (θw , π, sw ) heat-map given ε w = 4.5 and $w = 0(e) ($ p , π, s p ) heat-map given ε p = 6 and θ p = 0.75(f) ($w , π, sw ) heat-map given ε w = 4.5 and θw = 0.75

Figure 2: Steady State of Price Dispersion s p and Wage Dispersion sw

13

(a) (ε p , π, Y ) surface(b) (ew , π, Y ) surface(c) (θ p , π, Y ) surface(d) (θw , π, Y ) surface(e) ($ p , π, Y ) surface(f) ($w , π, Y ) surface

Figure 3: Steady State of Aggregate Output Y

14

(a) (ε p , π, W ) surface(b) (ew , π, W) surface(c) (θ p , π, W) surface(d) (θw , π, W) surface(e) ($ p , π, W) surface(f) ($w , π, W) surface

Figure 4: Steady State of Aggregate Welfare W

15

(a) (π, Y ) plane(b) (π, W) plane

Figure 5: Steady State of Aggregate Output Y and Aggregate Welfare W

16

bt (0%)(a) ε A,t → Y

bt (2%)(b) ε A,t → Y

bt (0%)(c) ε A,t → π

bt (2%)(d) ε A,t → π

Figure 6: Sensitivity Analysis of ε p and ε w in Productivity Shock ε A,tbt (0%)(a) ε A,t → Y

bt (2%)(b) ε A,t → Y

bt (0%)(c) ε A,t → π

bt (2%)(d) ε A,t → π

Figure 7: Sensitivity Analysis of θ p and θw in Productivity Shock ε A,t17

bt (0%)(a) ε A,t → Y

bt (2%)(b) ε A,t → Y

bt (0%)(c) ε A,t → π

bt (2%)(d) ε A,t → π

Figure 8: Sensitivity Analysis of $ p and $w in Productivity Shock ε A,tbt (0%)(a) ε v,t → Y

bt (2%)(b) ε v,t → Y

bt (0%)(c) ε v,t → π

bt (2%)(d) ε v,t → π

Figure 9: Sensitivity Analysis of ε p and ε w in Monetary Policy Shock ε v,t18

bt (0%)(a) ε v,t → Y

bt (2%)(b) ε v,t → Y

bt (0%)(c) ε v,t → π

bt (2%)(d) ε v,t → π

Figure 10: Sensitivity Analysis of θ p and θw in Monetary Policy Shock ε v,tbt (0%)(a) ε v,t → Y

bt (2%)(b) ε v,t → Y

bt (0%)(c) ε v,t → π

bt (2%)(d) ε v,t → π

Figure 11: Sensitivity Analysis of $ p and $w in Monetary Policy Shock ε v,t19

bt (0%)(a) ε ς,t → Y

bt (2%)(b) ε ς,t → Y

bt (0%)(c) ε ς,t → π

bt (2%)(d) ε ς,t → π

Figure 12: Sensitivity Analysis of ε p and ε w in Labor Supply Shock ε ς,tbt (0%)(a) ε ς,t → Y

bt (2%)(b) ε ς,t → Y

bt (0%)(c) ε ς,t → π

bt (2%)(d) ε ς,t → π

Figure 13: Sensitivity Analysis of θ p and θw in Labor Supply Shock ε ς,t20

bt (0%)(a) ε ς,t → Y

bt (2%)(b) ε ς,t → Y

bt (0%)(c) ε ς,t → π

bt (2%)(d) ε ς,t → π

Figure 14: Sensitivity Analysis of $ p and $w in Labor Supply Shock ε ς,tAppendix
Deterministic Steady State
Y=C
π = β (1 + i )sw =1 − θw
1 − θ w π ε w (1− $ w )ψw =χ ( w ε w N )1+ ϕ
1 − βθw π ε w (1+ ϕ)(1−$w )ϕw =C −σ wε w N
1 − βθw π (ε w −1)(1−$w )wh∗ = w =ε w χN ϕ 1 − βθw π (ε w −1)(1−$w )ε w − 1 C −σ 1 − βθw π ε w (1+ ϕ)(1−$w )
"

pi∗ =

1 − θ p π (ε p −1)(1−$ p )1 − θpεpα( pi∗ )1+ 1−α =εpψpε p − 1 (1 − α ) ϕ p
1

ψp =

1

wA− 1−α Y 1−α −σ
1 − θ p βπϕp =# 1−1ε p

ε p (1− $ p )1− α

Y 1− σ1 − θ βπ (ε p −1)(1−$ p )
p

21


N = sw s pYAAMCir = MCr =

 1

1− α

1
α −1

1−α

α

wY 1−αεpsp =1 − θp1 − θpπε p (1− $ p )1− α( pi∗ )

εp
− 1− α1 − θp=ε p (1− $ p )"1 − θ p π (ε p −1)(1−$ p )1 − θp#( ε p −1 ) (1− α )1 − θ p π 1− αε p −1)(1−$ p )
(
αε p
1
−
θ
βπ
ε
α
1
p
p

 wA− 1−α Y 1−α( pi∗ )1+ 1−α =ε p (1− $ p )ε p − 1 (1 − α )1 − θ p βπ 1−α
ε p −1)(1−$ p )
(
αε p
ε
1
−
θ
βπ
p
p
 MCr

⇒ ( pi∗ )1+ 1−α =ε p (1− $ p )εp − 11 − θ p βπ 1−α"# 1− αϕ + α + σ (1− α )
(
1
−
α
)
ε
−
1
ε
−
1
(
)
p
w
Yf =
χε p ε wx=

YYf

The Linearized Modelbt = CbtYbt = Et Y
bt+1 − 1 bit − Et π
bt + 1
Y
σ

1 b
bt
bt = b
Yt − ANs p,t + b
sw,t +1−αfbit = ϕπ π
b
b
b
b
b
b
+
ϕ
w
−
w
+
π
+
ϕ
Y
−
Y
+ vt
(
)
t
π
t
t
y
t
t −1
p
w
t






ε p (1− $ p )ε p (1− $ p )

εp
εp
∗
bt − $ p π
bt − 1 + bbpbi,t+ θ p π 1− αs p,t = −1 − θ p π 1− απs p,t−1
1−α
1−α


αε p
∗
pbi,t
= ψbp,t − ϕbp,t
1+
1−α


(1)
(2)
(3)
(4)
(5)
(6)


 
1 b
1
bt
At +
−σ Y
1−α
1−α


bp,t+1
bt + 1 − $ p π
bt + E t ψ
Et π

(7)



bt
bp,t = 1 − θ p βπ (ε p −1)(1−$ p ) (1 − σ )Y
ϕ




bt + 1 − $ p π
bt + E t ϕ
bp,t+1+ θ p βπ (ε p −1)(1−$ p ) ε p − 1 Et π

(8)bp,t =
ψ1 − θ p βπ

+ θ p βπε p (1− $ p )1− αε p (1− $ p )
1− α

∗
pbi,t
=





εp
1−α

bt −
w


θ p π (ε p −1)(1−$ p )bt − $ p π
bt − 1π
ε p −1)(1−$ p )
(
1−θ π

(9)

p




∗
b
bt − wbh,t
sw,t = ε w 1 − θw π ε w (1−$w ) w
bt − $ w π
bt − 1 ) + θ w π ε w ( 1 − $ w ) b
bt − wb t −1 + π+ θ w ε w π ε w (1− $ w ) ( wsw,t−1
22

(10)

∗
bh,t
(1 + ε w ϕ ) w= ψbw,t − ϕbw,t

(11)

h


i

bt + ς t
bw,t = 1 − βθw π ε w (1+ ϕ)(1−$w ) (1 + ϕ) ε w w
bt + Nψbw,t+1
bt + 1 − $ w πbt ) + E t ψ+ βθw π ε w (1+ ϕ)(1−$w ) ε w (1 + ϕ) (Et π

(12)




bt + ε w w
bt
bt + Nbw,t = 1 − βθw π (ε w −1)(1−$w ) −σC
ϕbt + 1 − $ w πbt ) + E t ϕ
bw,t+1 ]+ βθw π (ε w −1)(1−$w ) [(ε w − 1) (Et π∗
bt − 1 − πbt ) + ( 1 − θ w ) w
bt = θw π (ε w −1)(1−$w ) (wb t −1 + $ w π
bh,t
w

bf =
Y
t

1+ϕ
1−α
bt −
A
ςt
ϕ + α + σ (1 − α )ϕ + α + σ (1 − α )

(13)

(14)
(15)

bt − Ybf
xbt = Yt(16)


1+ ϕ ∆1− σ
χN 1+ ϕ ∆w  b
w b
bt − χNc t +1ct = CC
Nt −W∆w,t + ς t + βEt W
WW(1 + ϕ )W

(17)



∗
b w,t = ε w (1 − θw )(1 + ϕ) w
bt − w
bh,t
∆
∆w
h
i
ε w (1+ ϕ)(1−$w )b w,t−1
bt − w
bt − 1 ) + ∆bt + πb t −1 − $ w π+ θw πε w (1 + ϕ ) ( w(18)18 equilibrium conditions consist of 18 endogenous variablesno∗ bbt , Ybt , bit , π
b f , xbt , ∆b w,t , Wct
b t , pb∗ , w
bw,t , ϕbt , w
bt , Nbp,t , ψbw,t , bCs p,t , bsw,t , Yi,t b h,t , ψ p,t , ϕtand 3 shock process variablesnobt , ς t , vt .
A
bt = ρ A A
bt−1 + ε A,tA
ς t = ρς ς t−1 + ε ς,tvt = ρv vt−1 + ε v,t23