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An Estimated DSGE Model with Intergenerational
Transfers for the Chinese Economy
Robert Dixona
a.
Zhichao Zhanga
Chuanjie Zhang a,b
Durham University Business School
b.
Corresponding Author. Contact address: Durham University Business School, Green Lane,
Durham, UK, DH1 3LA. Tel: 00-44-07926675321; Email: [email protected]
Abstract
In this paper, a DSGE model developed and estimated for the Chinese economy. Based on the overlapping
generation models, we explicitly consider influences of intergenerational transfers and a new method is
deployed to include intergenerational transfers behaviour in DSGE constitution. The framework of this
model is New Keynesian DSGE through which we explore the impluse response functions of key variables
to tehonology and monetray policy shocks and discuss the model’s calibration. Bayesian estimation and
evaluation techniques are applied to analyse relevant parameters and the performance of modelling.
Keywords: DSGE models; China; Intergenerational transfers; Monetary policy; Net foreign assets,
Jel Classification: E42, E52, E60
1
I Introduction
One of the salient features of the Chinese economy is the large savings by Chinese
households. According to Chinese tradition and economic environment, intergenerational
transfers are a significant reason to explain the saving and consumption behaviour of
Chinese households. Compared to a representative Western household, a representative
Chinese household would like to save and leave much more money to maximize his/her
and his/her children’s utility. Hence intergenerational transfers are an especial and
outstanding character of Chinese households. However, this key feature of the Chinese
economy has been ignored in the exiting research of DSGE models for China. Therefore,
it is desirable and necessary to blend this element into the DSGE framework.
By combining the approach of overlapping generations (OG) model ,Lockwood’s
(2010) life cycle model and the recent new Keynesian DSGE models for China, we set
out a DSGE model which includes intergenerational transfers and the other key
macroeconomic elements in China’s economic development . The framework of the
follows the approaches as put forwards by Smets and Wouters (2003), Zhang (2009), Di
Giorgio and Nistico (2013), among others.
The remainder of the paper is organized as follows. The second section presents
the framework of the model. Section 3 shows the consequential first order conditions
engendered by households’ and firms’ optimization behaviours . We also present
linearized functions of the general equilibrium. We explore the reaction of inflation and
net foreign assets to two different shocks, which are monetary policy and the technology
shock. The third section also undertakes Bayesian estimation to evaluate parameters and
2
the performance of the model , and section four concludes the paper.
2.
Related literature
Interest in DSGE modelling of the Chinese economy has been heightened in recent
years along with the rising economic significance of the country in the world economy.
Several branches have emerged in last few years from this growing interest. the family of
DSGE modelling for China. Vogel (2010) presented an extended multi-country version of
the QUEST III model to analyse the influence of global integration and export expansion
on Chinese economy.
These extensions capture two important elements of macroeconomic policy in China,
which are the restriction of cross-border capital and active exchange rate management. A
series of shocks, such as labour supply, investment incentives, household savings and
foreign savings demand are applied to characterise key features of China’s recent
economic development : High GDP growth; declining consumption and increasing
investment shares in GDP; large current account surpluses and large aggregate NFA
holdings. The main purpose of this research is to investigate individual or combined
impact of the shocks on the whole economy. However, this type of modelignores some
key macroeconomic elements of New Keynesianism, such as monopolistically
competitive firms and real price rigidity.
New Keynesian models for China constitute a second series of researches. Before
the intensive study of DSGE model for China, two leading small-open-economic DSGE
models that have been widely employed by many central banks in the western world are
ECB’s New Area-Wide Model(NAWM) and the Riksbank’s RAMES.(see Chiristoffel et
al. (2008) and Adolfson et al. (2007), respectively). These models specified the structure
of the modelled economy from both the real and nominal sides. On the real side, these
models determine labour input and the main components of the national accounts data,
i.e. GDP, consumption, investment, exports and imports. On the nominal side, these
models specify the dynamics of prices (the CPI and various deflators), wage, the
exchange rate and the interest rate set by the central bank. They contain real rigidities,
3
such as habits in consumption and adjustment costs in investment and exports. They also
include nominal rigidities such as Calvo (1983) style price and wage stickiness and
indexation. These rigidities contribute significantly to the models’ fit and, as mentioned
above, the nominal rigidities establish a link between the nominal and real sides of the
economy.
Straub and Thimann (2010 ) developed a multi-country version of the NAWM for
China’s economy and take account some key macroeconomic elements in China’s
economic development . Improved education, adaptation of new technology, mobilisation
of the labour force in industry, export-led growth strategy, high investment and savings
rates are considered as the foundation of China’s current economic success.
Following the work of He and Wang (2011) and Chen et al. (2011), who investigate a
partial equilibrium model for the banking sector when credits and interest rates are
regulated by the central bank, Chen, Funke and Paetz (2012) implement their
partial-equilibrium modelling work in a fully specified New Keynesian DSGE framework
to analyse the effect on inflation and output and analyse the monetary policy of Chinese
government. They present three distinguish features of China’s central bank (the People’s
Bank of China) : regulating the retail lending rate and the deposit rate; employing
non-market tools; manipulating required reserve ratio. There are two features of this
mode. First, this is a simple two-household closed economy model in the spirit of
Iacoviello (2005). Following the seminal work of Kiyotaki and Moore (1997) on credit
cycles, households are divided into patient and impatient groups. Second, this model
considered a private banking sector restricted by the guidelines of the central bank.
Private banks can also trade financial assets with institutional investors in the interbank
market. The entrepreneurs in the impatient household group produce intermediate goods
to sell in the retail sector. This sector has some monopoly power and sets prices in a
staggered manner. Moreover, Chang, Liu and Spiegel (2012) also built an estimated
DSGE model for China to examine optimal monetary policy responses to the sudden
changes in world interest rates after the 2008-2009 global financial crises with a few key
characteristics unique to the Chinese economy.
4
Moreover, Zhang (2009) built a closed-economy to explore two significant monetary
policy instruments in China, quantity and price, and study their impacts. One feature is
this model is the representation of the growth of nominal money supply. Following the
work of Zhang (2009),
Mehrotra , Nuutilainen and Pääkkönen (2011) build a
closed-economy estimated DSGE model of China. It features price rigidities, habit
formation in consumption and costs in capital adjustment, and calibrate the model with
data for the Chinese economy. The economy of this model consists of four types of
agents: an infinitely-lived representative household, a representative final-good producer,
a continuum of intermediate good producers, and a monetary authority. All agents
maximize their utility (with firms maximizing profits), subject to an intertemporal budget
constraint. Liu and Zhang (2010) also use a small New Keynesian model with behaviour
equations to evaluate the effectiveness of the current monetary policy framework .
All in all, it is probably fair to say that there has been considerably less work done
for one crucial feature of China’s economy, intergenerational transfers, in DSGE
framework. According to Chinese tradition and current situation, we could notice that a
representative Chinese household would like to transfers much more assets to maximize
his/her and his/her children’s utility. However, this feature has not been taken into
consideration in previous research.
A few studies has focused on the modelling method about bequest behaviour and
intergenerational transfers. Alonso-Carrera, Jaime., Jordi Caballe and Xavier Raurich,
(2007) build a overlapping generations (OG) model to discuss the importance of parental
altruism and represent how the process of preference formation affects the bequest
motive. They represent two kind of preference formation: habits and aspirations. Habits
are based on one’s past consumption, while aspirations are determined by the willingness
and standard of parents. They explicate that willingness of parents to leave bequests is
reduced by the influence of their habits. However, their aspirations are making
contribution to the existence of positive bequest. Therefore, the quantity of bequest is
affect by both habits and aspirations.
On the other hand, Lockwood (2010) reveals a life cycle model including the
bequests factor. In this model, bequests are considered as an element of household’s
5
consumption path which is applied to maximize expected utility. He also discusses the
method to calculate bequests according to consumption and personal wealth.
On the basis of previous models of intergenerational transfers and DSGE model with
Chinese features, we integrate the methodology of these two types of models to build an
estimated DSGE model with intergenerational transfers and the other key features of
China’s economy.
II. The Model
2.1. Households
Following the seminal work of Kiyotaki and Moore (1997) on credit cycles,
households are divided into patient and impatient groups. Further, inpatient households
can be modeled as entrepreneurs (Chen, Funke and Paetz, 2012),. Entrepreneurs (denoted
by
E
) borrow money from private banks that, in turn, buy and sell bonds from the central
bank and borrow deposits from patient households (denoted by
P
). Patient households
are general consumers who provide labour and buy final goods from retailers.
Intergenerational transfers impact the utility of Chinese households. This is because,
with intergenerational transfers, the parental generation provides money and other forms
of wealth to their offspring, which raises the offspring’s utility. As a result, the total
utility of Chinese households are improved because the parents consider their offspring’
utility to be at least as significant as theirs.
There are two generations of consumers in our model. The first generation, or the
parental generation, households are in adult ages at the beginning of the period. They
Ct1
decide their consumption
t
in period . The second generation households are the
6
offspring of the previous one. We assume that both generations receive money and assets
from their parents, while both of them will also transfer wealth including money and
Dt
property to their children for which purpose they save. Consequntly, we denote
for
deposits and capital holdings by households. The superscripts 1 and 2 represent the
Bt1
generations. So,
for example represents intergenerational transfers from the first
Bt2
generation
from the second generation, etc.
Then intergenerational transfers at time
t
t
i =0
i =0
t
is
Bt = ∑ β i ×a (Ci − hCi −1 ) + ∑ β i ×aDt
(1)
β
where
hCt −1
is the discount factor,
Ct
denotes the external habit stock with
being
Ct1 − hgCt1−1
aggregate consumption,
is the effective consumption of households.
a
represents the rate of effective consumption and deposits that parents would like to leave
to their children. This rate denotes the effect of aspiration. This equation conforms to
Alonso-Carrera, et al. (2007)’s theory that households’ habits decreases the amount of
their transfers while their aspirations makes contribution to the existence of positive
intergenerational transfers.
The utility function of a representative first generation patient household is:
7
1−γ
∞
 ( (1 − a )(C 1 − hgC1 ) ) 1−σ

N t1+ϕ  
1  M tP 
t
t −1


U = Et ∑ β P
+


÷ −
1−σ
1 − γ  Pt 
1+ ϕ 
 t =0 


1
t
Mt
where
,
(2)
Pt
is money supply and hence with
t denoting he price level we have the
Mt
= mt
Pt
current real money balances:
defined as:
. The equilibrium of real money balances can be
Mt
= mt = Bt = BtC + BtD
Pt
,
t
(3)
where
D
t
i
i =0
t
B = ∑ β i ×aDt
B = ∑ β ×a(Ci − hCi −1 )
C
t
,
i =0
Then we can reason out
BtC = β BtC+1 + a (Ct − hCt −1 )
(4)
BtD = β BtD+1 + a ( Dt )
(5)
1 + rt c
Dt +1 = (
) Dt
1+ πt
(6)
Parameters
σ
ϕ
and
represent the inverse intertemporal elasticities of substitution
with respect to concumption and labour supply, respectively. Assets earn a certain, real
βp
rt c Wt
after-tax return, .
is nominal wage. is the discount factor of the patient
Et
households.
t
is the expectation operator at time .
8
Ct1
t Nt
is comsumption of first generation housholds in period .
is labour supply in
γ
terms of the hours of work.
denotes the inverse of the elasticity of asset holdings by
households given the interest rate. The households contribute a part of their effective
a (Ci − hCi −1 )
consumption, i.e.
, to their children as intergenerational transfers. The rest of
(1 − a)(Ci − hCi −1 )
the effective consumption, which equates to
, is still applied to improve
household’s own utility. The concumption inflation rate is represented
Πt ≡
as
Pt
Pt −1 ⇒ Π t = 1 + π t π t
.
. is the inflation rate.
In this model, each household is assumed to receive equal wages, expend same
consumption and save equivalent deposits. Consequently, the households of the first and
second generations make the same decision about their consumption and saving.
Wt1 = Wt 2
Moreover,
t
is the average wage at time period . This implies:
Ct1 = Ct2 , Dt1 = Dt2 ,Wt1 = Wt 2
,
Bt1 = Bt2 = BtP
we then can reason out that
according to equation (1).
Bt1 = Bt2 = BtP
In the light of equation (3) and that
, equation (2) can be rewritten as
∞
 (1 − a)(C P − h gC P ) 1−σ P

N t1+ϕ  
1
t
t −1 


C
D 1−γ

 Bt + Bt  −
U = Et ∑ β P
+

1−σ P
1− γ 
1+ ϕ 
 t =0 


P
t
(7)
9
The dynamic budge constraint of the first generation patient households is:
CtP +
M t BOt  M t −1  BOt −1 Wt
+
=
+
Nt + Ft R + Ft B + Tt
÷+
Pt Pt Rt  Pt 
Pt
Pt
BOt
where
Rt
is coupon bond,
Wt
denotes the gross returns on bond.
denotes the nominal
Ft R
Pt
wage rate,
(8)
is the consumption price index (CPI).
are the real lump-sum
Ft R
profits received from the retail firms, respectively.
is
Tt
t
the dividends.
denotes the real net transfers from government in period . The main
parameters and variables are summarized in Tables 1 and 2.
Table 1
Parameters in the Model
Parameter
Description
Discount factor
σ
Elasticity of substitution
with respect to concumption
β
γ
Parameter
Elasticity of current nominal
money balance
a
Description
Rate of effective consumption
and intergenerational transfers
ϕ
Elasticity of labour supply
α
Production elasticity with respect
to labour
κ
Technology parameter
δ
Depreciation rate
h
Habits parameter
µ
elasticity of substitution between
any pair of differentiated goods
θ
fraction of firms which
adjust
their price according to
last-period
ξ
fraction of households
setting their wages
l
10
elasticity of substitution among
differentiated labour services
according to last-period
wage inflation
Table 2
Variable Definitions
Variable
Description
Consupmtion
Variable
Nt
Labour supply
Wt
Nominal real wage
wt
Real wage with respect to
price index
Pt
Aggregate price index
Mt
Real money balances
mt
Real money balances with
respect to price index
ηt
Real marginal cost
Kt
Capital servies
Zt
Technology shock
πt
Inflation rate
rt c
Rental rate on capital
BtD
Intergenerational transfers
for deposits
Ct
It
Bt
dt
Investment
Intergenerational transfers
for consumption
Description
Gross intergenerational transfers
Households’ deposits
Gross return of bond
C
t
B
Rt
BOt
Coupon bond
Ft B
Profit for banking sector
Ft R
Profit for retail firms
Yt
Composite final good (output)
2.2 Firms (Entrepreneurs)
11
Thanks to the work of Dixit-Stiglitz (1977) and the further development of
Blanchard and Kiyotaki (1987), monopolistic competition can be appropriately
incorporated in modern macroeconomic models. Moreover, the Dixit-Stiglitz framework
can also be embodied in a general equilibrium setting. In this section, we apply the
Dixit-Stiglitz framework to describe the behaviour of firms and retailers.
Yt
In what follows,
is a composite final good that is produced by a representative firm.
Yt (i )
The final-good producer chooses a continuum of intermediate goods
as production
i ∈ [0,1]
input which is indexed by
, each produced by a unique monopolistically
competitive firm. Furthermore, firms and retailers set prices in a Calvo-staggered
manner. Aggregation of the production is summarized as follows:
µ
A production aggregator
µ −1
 1
 µ −1
Yt =  ∫ Yt (i ) µ di 
0


,
(9)
1< µ < ∞ µ
where
,
is the constant elasticity of substitution between any pair of
Y (i )
differentiated goods.
can be represented by a Cobb-Douglas production function.
Yt (i ) = Z t K t (i )1−α N t (i )α
,
(10)
i ∈ [0,1]
denotes each differentiated good produced by a unique producer.
12
α
represent the
N t (i )
elasticity of output with respect to labor supply
K t (i )
in the production process.
t
denotes capital at time . All firms are assumed to use the same technology and the
technology in turn is subject to the following path:
_
ln( Z t ) = (1 − κ ) ln( Z ) + κ ln( Z t −1 ) + ε tZ
,
_
Z
where
ε tZ
Zt
is the mean of
and
denotes the technological shock,
0 < κ <1
.
The final good producers are competitive and produce the goods according
pi (t )
to the production function (9). They choose intermediate goods whose price is
and
Pt Pt
set the price of the final goods to be
.
is the index for the final goods price.
As a consequence, their maximization problem is:
1
max[ PY
t t − ∫ Yt (i ) pt (i ) di ]
0
Yt ( i )
(11)
The input demand function associated with the intermediate goods is:
−µ
 p (i) 
Yt (i ) =  t ÷ Yt
 Pt 
(12)
Pt
The price of final goods
Pt =
( ∫ p (i) di )
1
0
1− µ
t
1
1− µ
is determined by:
.
(13)
13
For consumption, its aggregation is made by the following process:
µ
µ −1
 1
 µ −1
Ct =  ∫ Ct (i ) µ di 
0


A consumption aggregator
(14)
µ
As in equation (9),
is the substitution elastic parameter。
The behaviour of firms and retailers is in line with conventional Calvo staggered
pricing mechanism (Calvo,1983), as followed in Christiano et al. (2005) and Smets
(1 − θ )
and Wouters (2003). It means a randomly selected fraction
prices while the remaining fraction
θ
of firms adjusts
Pt ( i ) = ( 1 + π t −1 ) Pt −1 ( i )
keep their prices as
.
Pt* ( i )
Defining the index for the “reset” price in period t,
firm
i
, as the optimal price chosen by
in order to maximize the present value of real profits, the aggregate price index
(CPI) is given by:
Pt = [θ ( 1 + π t −1 ) P
1− µ
t −1
1
* 1− µ 1− µ
+ (1 − θ )( Pt )
]
(20)
Pt * ( i )
Pt
The relationship between the aggregate price index
−σ
µ
and the optimal price
C 
P 
Et ∑ j =0 θ β  t + j ÷ ηt + j  t + j ÷ Yt + j G j − µ
*
Pt ( i )  µ 
 Ct 
 Pt 
=
÷
−σ
µ −1
Pt
C
 µ −1 
 Pt + j 
∞
j
j  t+ j 
1− µ
Et ∑ j =0 θ β 
÷ ηt + j 
÷ Yt + j G j
 Ct 
 Pt 
∞
j
j
14
is
Gt (1 + π 1 )(1 + π 2 )....(1 + π t )
t = 0 Gt
t ≥1
when
, =1.
=
, for
.
2.3 Monetary Policy
The Chinese government determines national monetary policy which follows a Taylortype rule (Chen, Funke and Paetz , 2012):
Rt = Φ1 ( Eπ t +1 − π t ) + Φ 2π t + Φ yYt + Φ r Rt −1
,
Rt
where
(21)
Φ1
t
is the nominal interest rate for period ; the policy reaction parameters
,
Φy
Φ2
and
determine the preference of the central bank with respect to inflaction and
Φr
output gap stablilization from steady state.
Rt
indicates the relation between
Rt −1
and
,
and reflects the smoothing of interest rate dynamics. This rule suggests that inflation and
output gap are the main factors which influence the central bank’s decision on the
Φr
nominal interest rate. Moreover, the interest rate is a rate depends on the coefficient
on the previous interest rate. Zhang (2009) argues that following an interest rate rule is
indeed superior to a money supply rule for China, as it leads to less fluctuations in the
economy.
The baseline model which we set up so far is still a closed economy framework. We
will modify this model according to the spirt of Obstfeld and Rogoff (1995) and Lindé,,
Nessen and Soderstrom (2009) in what follows in Section 4.
15
III Estimation and Policy Analysis
3.1. First order conditions (FOCs)
We apply Uhlig’s method (2006) to calculate the first-order conditions (FOCs) of
the model. First, we derive the Lagrangian of equation (2):
1−γ
 (1 − a)(C P − hgC P ) 1−σ P

P


M
N t1+ϕ
1
t
t −1 

t


+

÷ −
∞


1− σ P
1 − γ  Pt 
1+ ϕ
L = max E ∑ β P 

P
P
t =0


M
BO
BO
M
W
P
R
B
t
t
t −1
t −1
t
 −λt (Ct + P + P R − P − P − P N t − Ft − Ft − Tt ) 
t
t t
t
t
t


(22)
Ct P
The first-order condition (FOCs) with respect to
(1 − a )(CtP − hgCtP−1 ) 
−σ P
(1 − a )(CtP − hgCtP−1 ) 
−σ P
is
− λt = 0
= λt
(23)
M tP
The first-order condition (FOCs) with respect to
is
−γ

1  M tP 
1
1 
) = 0

÷ − λt − β P E λt +1 (−
Pt  Pt 
Pt
Pt +1 

−γ
1  M tP 
1
βP
E [ λt +1 ]

÷ = λt −
Pt  Pt 
Pt
Pt +1
(24)
BOt
The first-order condition (FOCs) with respect to
16
is
−λt
 1

1
− β E −
λt +1  = 0
Pt Rt
 Pt +1

 PR

λt = β p E  t t λt +1 
 Pt +1

⇒
(25)
According to equations (24) and (25), we obtain the equilibrium equations:
 P R  C P − hgC P −σ P 
t
 =1
β p E  t t  t +P1
P ÷
P
C
−
h
g
C
 t +1  t

t −1 
(26)
−γ
1  M tP 
1
1
λt

÷ = λt −
Pt  Pt 
Pt
Pt Rt
−γ
 M tP 

1
1

÷ = λt − λt = 1 −
Rt
 Rt
⇒  Pt 
−σ P

P
P
÷(1 − a)(Ct − hgCt −1 ) 

(27)
M t / Pt = mt
Given that
[ mt ]
−γ

1
= 1 −
 Rt
, equation (27) can be rewritten as:
−σ P

P
P
÷(1 − a )(Ct − hgCt −1 ) 

(28)
mt
This is the conventional optimality conditions for current money balances
equilibrium. From equations (23) and (25), we can get:
(1 − a)(CtP − hgCtP−1 ) 
−σ P
−σ P 
 PR
= β E  t t (1 − a )(CtP+1 − hgCtP )  
 Pt +1

17
in
 P R  C P − hgC P  −σ P 
t
1 = β E  t t  t +P1
÷ 
Pt +1  Ct − hgCtP−1  



⇒
(29)
Ct
This is the Euler equation with respect to consupmtion
.
Nt
The first-order condition (FOCs) with respect to
− N t ϕ − λt (−
⇒
Wt
W
)=0
N t ϕ = λt ( t )
Pt
Pt
⇒
N t ϕ = (1 − a)(CtP − hgCtP−1 ) 
−σ P
(
,
(labor) is
λt = (1 − a)(CtP − hgCtP−1 ) 
−σ P
Wt
)
Pt
(30)
Equation (30) represents the optimal labour-leisure decision in equilibrium.
It
Further, we can find the following first order conditions with respect to investment
Ct
and capital stocks
according to the capital stock accumulation equation (19):
σ
 C − hCt −1 
c
β Et  t
÷ ( 1 − δ + rt +1 ) = 1
C
−
hC
t 
 t +1
(31)
Zt
We have mentioned the path subject to the technology shocks
and Cobb-Douglas
production function equation (10):
_
ln( Z t ) = (1 − κ ) ln( Z ) + κ ln( Z t −1 ) + ε tZ
(32)
18
On the basis of equations (31) and (10), we could have the first order condition with
N t (i)
respect to
K t (i )
and
are:
c
Wt  α rt   Kt (i) 
=
÷
÷
Pt  1 − α   N t (i ) 
(33)
This formula could be rewritten as:
α
1−α
 Wt   rt c 
Wt
=α
÷ 
÷
Pt
 α Pt   1 − α 
1−α
 K t (i ) 

÷
 N t (i ) 
(34)
ηt
We set the real marginal cost
1
ηt =
Zt
α
as:
1−α
 Wt   rt c 

÷ 
÷
 α Pt   1 − α 
(35)
In the light of equations (34) and (35), we can reason out
1−α
 K t (i) 
Wt
= α Z tηt 
÷
Pt
 N t (i ) 
(36)
The aggregate wage equation is:
1−l
1−l
Wt = (1 − µ ) ( W * ) + µ ( Wt −1 ) 


1/(1−l )
,
(37)
Wt *
where
t
denotes the wage one can replace in the current time period .
Then we can summarize relevant equations discussed before to obtain the steady state in
equilibrium.
19
Capital accumulation fucntion:
K t = (1 − δ ) K t −1 + I t
Pt = Π t Pt −1 ⇒ Pt = (1 + π t ) Pt −1
πt
, where
represents the inflation rate.
The function of monetary policy is:
Rt = Φ1 ( Eπ t +1 − π t ) + Φ 2π t + Φ yYt + Φ r Rt −1
3.2. Steady states
At the steady state, all variables are constant. We calculate the steady state of equations
which we have discussed in the previous sub- section. The steady state of optimality
mt
conditions for the current money balances
−
 m 
−γ
is:
−σ P
− P
− P 
 1 
= 1 − − ÷ (1 − a )(C − hgC ) ÷

÷

 R 
(38)
The steady state of Euler equations (29) is:
20
− −
P R
1= βE  −
 P






−σ P
− P 
 −P
C
−
h
g
C

÷
− P ÷
 −P
 C − hgC ÷


−
⇒1= β R
⇒
−
R = 1/ β
(39)
The steady state of optimal labour-leisure decision (30) is:
− P
− P 

N =  (1 − a )(C − hgC ) 


− ϕ
−σ P
−
− P 

(−)
N = (1 − a)(1 − h)(C ) 


P ⇒
− ϕ
W
−
−σ P
(
W
−
P
)
(40)
Zt
The steady states of technology shock
are:
_
_
_
and the Cobb-Douglas production function (10)
_
ln( Z ) = (1 − κ ) ln( Z ) + κ ln( Z ) = ln( Z )
(41)
1−α
_
Y = ZK÷
 
_
_
α
_
N÷
 
(42)
The steady states of aggregate price, wage, capital and inflation are:
1
1
1− µ 1− µ
  _ 1− µ
 _ 1− µ  1− µ _
_ 
P = θ  P ÷ + (1 − θ )  P ÷  =  P ÷  = P
  
  
  
_
1−l
1−l

 _ 
 _  
W = (1 − µ )  W ÷ + µ  W ÷ 
 
  

_
−
−
−
K = (1 − δ ) K + I
−
1/(1−l )
1/(1−l )
 _ 1−l 
=  W ÷ 
  
(43)
_
=W
(44)
−
⇒ δK =I
(45)
21
−
Π≡
P
−
=1
_
_
⇒ 1+ π = 1 ⇒ π = 0
P
(46)
The steadt state of equation (31) is:
σ
−
−

C
−
h
C

÷
β Et −
 C − h C− ÷


−
−
−


1
c
c
c
1
−
δ
+
r
=
1
⇒
β
1
−
δ
+
r
=
1
⇒
r
= + δ −1

÷

÷
β




(47)
3.3. Linearized equations
According to Uhlig (2006), the principle of loglineariaztion is :
For
x≈0
ex ≈ 1 + x
,
.
−
^
xt = log( xt / x)
Therefore, we set
xt
to be the log-deviation of
^
−
^
−
xt
xt ×100%
Thus,
from its steady state.
is (approximately) the percent deviation of
−
from its steady state
^
xt = x e xt ≈ x(1 + x t )
So,
According to this method, linearized functions of the equilibrium equations (28),
(29),(30) are：
^ P 
 ^P
 ^ 
 1   Rt ÷  σ P   C t − hgC t −1 ÷
M t − Pt = mt =  ÷
+
÷
_
 γ   1 − R ÷  γ   1 − h ÷
÷




^
^
^
22
(47)
x
.
_
R = 1/ β
where
. Equation (47) is the linearized equation of optimality condition of
^
mt
current money balances
. This condition indicates that current real money balances
depend on the efective consumption and the gross returns on bond:
^ P
^ P
^
^ 
C t +1 + h C t −1  h − 1   ^
Ct =
−
P
−
P
−
R
t
+
1
t
t ÷
÷
1+ h
 (1 + h)σ  

^ P
(48)
^ P
Equation (48) is the linearized equation of Euler equation of consumption
Ct
. This
condition implys that current consumption is determined by prior consumption, expected
h
future consupmtion, the inflation rate and the gross returns on bond. represents habit
factor.
 −σ
Nt =  p
 ϕ
^
wt =
We let
^ P 
 ^P
  C t − hgC t −1 ÷ 1 ^ ^
+ (Wt − Pt )
÷
 1 − h ÷
÷ ϕ


Wt
Pt
^
^
^
Wt − Pt
wt
then
(49)
=
i
. The relationship between labour supply of firm
N t (i )
Nt
and the whole labour supply
is:
−l
W * (i )Gt 
N t (i ) =  t
 Nt
 Wt 
(50)
l
t=0
where is elasticity of substitution among differentiated labour services. When
,
23
Gt
(1 + π 1 )(1 + π 2 )....(1 + π t )
Gt
=1.
=
, for
t ≥1
.
On the basis of equations (49) and (50), we follow Zhang (2009) to rewrite (49), to
obtain the linearized equation of real wage:
^
^
wt
= (1 + ϕ l )ξ ( wt −1 + π t −1 ) − (1 + β )(1 + ϕ l )ξπ t
(1 + β )(1 + ϕ l )ξ + (1 − βξ )(1 − ξ )
^
σ ^ ^ 
 ^
+(1 + ϕ l )ξβ Et ( wt +1 + π t +1 ) + (1 − βξ )(1 − ξ )gϕ N t +
(C t − C t −1 ) 
1− h


(51)
ξ
denotes the fraction of households that set their wages according to the last-period
wage inflation or a combination of last-period price and wage inflation rates. The rest
1− ξ
portion of the households would adjust their wage optimally. This condition
indicates that current nominal wage is a function of inflation, effective consumption,
labour and prior and expected wages.
Linearized equation of equations (31) is:
^
^
^
(1 + h) C t = h C t −1 + C t +1 +
^
(h − 1)
(1 + βδ − β ) r c t +1
σ
(52)
This equation is pretty similar to equation (48). We could derive from this equation that
rental capital returns depend on the current, prior and expected levels of
consumption.
We additionally have:
24
^
^
Z t = (1 − κ ) Z t −1 + ε t
(53)
This is the linearized equations equation (32), which expresses the path of the
technology shock.
Next, we have:
^
^
^
^
Y t = Z t + (1 − α ) K t + α N t
(54)
It is the linearized form of equation (10), i.e. the Cobb-Douglas production function.
Equation (54) shows the linear relation between the three inputs in production, i.e.
technology, capital and labour.
^
Linearized equations of the price index
^
Pt
^
and capital accumulation
K t +1
are:
^
π t = P t − P t −1
(55)
This expression indicates that curent inflation is the difference between the current and
prior levels of price index.
We also have the equation of capital stocks:
^
^
^
K t +1 = (1 − δ ) Kt + δ I t
(56)
Equation (56) implys that future stocks of capital are a function of current capital and
investment. The depreciation rate
δ
affects the present value of changes in investment.
^
ηt
Linearied equation of real marginal cost of labour
25
(equation 35) is:
^
^
c
t
^
^
η t = α wt + ( 1 − α ) r − Z t
(57)
According to equation (57), the real marginal cost of labour comes from three sources:
wage, rental rate on capital and technology shocks. The elasticity of labour and capital
service are the parameters in this equation.
Linearized form of equation (33) is the equilibrium:
^
^
^
^
wt = rt c + K t − N t
^
^
^
^
⇒ N t + wt = rt c + K t
(58)
The equilibrium indicates that the sum of real wage and real unit labour costs equal to the
sum of capital accumulation and rental rate of capital. Underlying this equilibrium is the
assumption that firms finance their wage bills and decide their demand for labor on the
basis of their capital and rental capital income.
Pt * ( i )
Pt
The relationship between the aggregate price index
−σ
and optimal price
is:
µ
C 
P 
Et ∑ j =0 θ β  t + j ÷ ηt + j  t + j ÷ Yt + j G j − µ
*
Pt ( i )  µ 
 Ct 
 Pt 
=
÷
−σ
µ −1
Pt
C
 µ −1 
 Pt + j 
∞
j
j  t+ j 
1− µ
Et ∑ j =0 θ β 
÷ ηt + j 
÷ Yt + j G j
C
P
 t 
 t 
∞
We already know when
⇒
Gj =
j
j
t=0
Gt
,
(1 + π 1 )(1 + π 2 )....(1 + π t )
Gt
=1.
=
, for
Pt + j
Pt
Therefore, we have
26
t ≥1
.
−σ
C 
Et ∑ j =0 θ β  t + j ÷ ηt + jYt + j
*
Pt ( i )  µ 
 Ct 
=
÷
−σ
Pt
 µ −1 
C 
∞
Et ∑ j =0 θ j β j  t + j ÷ ηt + jYt + j
 Ct 
∞
j
j
=
 µ 

÷
 µ −1 
(59)
We can derive the following linearized Phillips curve from equations (20) and (59):
^
2 − ( 1 + β ) θ + βθ
θ
πt =
π
+
E
π
+
1
−
θ
1
−
βθ
η
(
)
(
)
t
t −1
t t +1
1 + βθ 2
1 + βθ 2
2
(60)
The inflation rate is a function of the real marignal cost, prior and expected levels of
β
inflation. As before,
is the discount rate and
θ
denots the fraction of firms that will
adjust their prices according to the last-period price inflation and marginal costs.
The lineraized function of composite production is in the following form:
^
Yt =
1 + α (ε − 1) ^ (1 − α )(ε − 1) ^
Ct +
It
ε
ε
(61)
Equation (61) is derived from equations (59) and (12) . This function indicates that output
also depends on consupmtion demand of households and investment. The elasticity of
substitution between any pair of differentiate goods, i.e.
ε
, plays a key role in this
equilibrium.
BtC
Linearied equations of intergenational transfers
( 4)
equations
( 5)
and
, are:
27
BtD
and
, which are related to
^
^
^
 1− β  ^
BtC = β BtC+1 + 
÷(Ct − h Ct −1 )
 1− h 
^
^
(62)
^
BtD = β BtD+1 + ( 1 − β ) Dt
(63)
These linearised intergenerational transfers functions imply that effective consupmtion
and deposits have a positive influence on the transfers. Currenet amount of
intergenerational transfers is also related to its value in the last period.
Dt
Linearied equation of the deposits of households
^
^
,according to equation (6), is
^
Dt +1 = D t + rt c − π t
(64)
In the log-linearized equilibrium, expected future deposits equal to the sum of current
deposits and the rental rate on capital minus inflation. In the case of monetary policy rule,
we can replace equation (47) with equation (21).
3.4. Parameter Estimates
Following Walsh (2003) and He et al. (2007), we set
α = 0.4
σ = 2 α = 0.4
δ = 0.04
,
and
. If
, one can get the elasticity of the substitution among intermediate goods and final
µ = 4.61
goods
.The average (annual) nominal interest rate is 0.08 from1978 to 2005,
from which one may derive a quarterly rate equaling to 0.02. Thus, The growth rate of
28
Rd = 1 + 0.02 = 1.02
deposits is
. Following Zhang (2009), we consider habit parameter of
Chinese households to be
h = 0.61
β = 0.98
. The discount rate
.
−
R = 1/ β = 1/ 0.98 = 1.0204
In the steady state, the nominal interest rate
. The GMM
estimation of equation (60) with data of 1995 to 2005 shows that the fraction of firms that
adjust their prices according to the last period prices is
θ = 0.84
. We obtain the elasticity
ϕ = 6.16
of labour supply
, the fraction of households who set their wages depending on
ξ = 0.6
wage inflation
, elasticity of substitution among labour services
l=2
, on the
basis of Liu (2007). The estimation of equation (47) with data of 1992-2006 shows that
γ
the elasticity of real money balances
is equal to 3.13. The summary about the values of
parameters are given in Table 3.
Table 3
Parameters in Estimation
Parameter
β
Value
0.98
Parameter
a
Value
0.5
σ
2
ϕ
6.16
γ
3.13
α
0.4
κ
0.5
δ
0.04
h
0.61
µ
4.61
θ
0.84
l
ξ
0.6
29
2
3.5. Simulation and Impluse Response Functions
In this section, we explore the reactions of monetary policy and foreign net assets to
the variables in the model, particularly the technology shocks,. First, we present the reulst
of implues reponse functions of every variables to technology shocks. Impluse response
fucntions relate to the reaction of a dynamic system in some exogenous change. In this
case, the stucture of dynamic system are the linearized models we have produced before.
Following Di Giorgio and Nistico (2013), we consider net foreign asset (NFA) as a new
key variable in our system. NFA could be described by following fucntion:
NFAt =
( Rt −1 )
( 1+ πt )
NFAt −1 + NDt
(65)
NDt
where
is the external trade balance.
The steady state of equation (65) is:
 _
R
NFA = 
_

 1+ π
_
_
 _

_
R
÷NFA+ ND
1 −
_
÷

1
+
π


⇒
,
−
−
R = 1/ β π = 0
We already know
,
, so
 _
_
÷NFA = ND
÷

(66)
_
_
 1
NFA×1 − ÷ = ND
 β
The dynamic linearized function of equation.(65) is:
^
NFAt =
^
^
1 ^
  1 

 NFAt −1 + Rt − π t ÷+ 1 − ÷1 + NDt ÷
β
  β 

30
(67)
As the external trade balance is held constand, equation (67) could be rewritten as:
^
NFAt =
^
1 ^
  1
 NFAt −1 + Rt − π t ÷+  1 − ÷
β
  β
(68)
According to the dynamic general equilibrium that we have showed above, the model’s
responses to a technology shock are given in Figures 1 and 2.
_____________________________________________________________
Figure 1
Estimated Model Responses to a Technology shock (1)
_____________________________________________________________
From Figure1 we can see that the reactions of consumption and inflation to a
technology shock consist of a standard level response and a negative change in
31
adjustment speed of responses from consumption and inflation. Although these estimated
results might not be very precise, they could be applied to explain and analyze the trend
of reaction to differentiate shocks.
An estimated negative response of the change in consumption suggest that a technology
shock would cause decrement of consumption in the short run, because households need
some time to decide their new consumption demand. Thanks to the influence of
technology shocks, households and firms determine to invest more money and capital in
new production. The reactions of real money balances and capital are negative to
technology shocks. Moreover, technology shocks lead to a negative response in labour
supply and in the rental capital rate in the first period but bring out positive responses in
the next four periods. This implies that both demand for labour and for capitaldeclines at
the beginning of industry transformation. However, firms require more labour and
capital with the development of new production industry. The rental rate of capital
increases because of the agumentation in demand for capital. This is the reason why the
reactions of real money balances and the rental rate on capital switch to positive after the
first period. The impluse responses of techonology is positive because the success of
invention and innovation will encourage the technological progress.
Comparing Figure1 with Figure 2, we find that reactions of serval variables are
related. As noted above, the reactions of consumption are negative to a positive
technology shock. Thus, reactions of intergenerational transfers is also negative to
technology shocks. The impluse responses of deposits are postive in the initial period but
are negative in the next five periods. This pattern of responses can also be found in
eactions of intergenerational transfers to deposits, in the face of a technology shock,
households resolve to reduce their deposits to make investments in new production
industry. These responses also conform to reactions of investment to technology shocks,
which are negative in the initial period but positive in the next five period. The estimates
show deposits and investment have negative corelation. The impluse responses of output
are positive after the initial period.
The findings conrespond to macroeconomic theory that most techonology shocks
32
increase the productivity in an economy. Net foreign assets have a positve reaction to
a technology shock, suggesting that demand for and supply of foreign assets
are augmenting with the process of industry transformation. On the other hand, foreign
investors are willing to commit money and assets to emerging economies, which have
significant potential of technological developments. Because they consider these
investments are good opportunity to bring about financial success.
________________________________________________________________________
Figure 2
Estimated and Model Responses to a Technology shock (2)
________________________________________________________________________
In the next step, we consider the impulse responses of variables of interest to monetary
33
policy shocks. As noted above, we replace equation (47) with (21) in dynamic general
equilibrium. Now the monetary policy is regarded as the only shock in this system and so
the stochastic effects in the equation of technology progress are kept constant in this case.
Estimates of impulse responses functions are given in Figures 3 and 4.
________________________________________________________________________
Figure 3
Estimated and Model Responses to Monetary policy (1)
________________________________________________________________________
34
______________________________________________________
Figure 4
Estimated and Model Responses to Monetary policy (2)
On the basis of Figures 3 and 4, we notice that the reactions of inflation to the
35
monetary policy is negative in the first two periods but remain positive later. This result
implies that policy makers have lagged inflation rather than current inflation in their rule
in the initial period. Moreover, a change in the interest rate stimulates potential output
and investment and helps inflation stabilization in the face of upward pressure on
aggregate demand because the sizable positive response of output and investment.
The estimates also indicates that a monetary policy shock leads to a large investment
adjustment cost temporarily. Reactions of investment are significantly negative at
first but turn to positive and smoother in the subsequent periods. Reactions of
consumption and capital to the monetary shocks imply that households will be more
precautious to consumption and make financial decisions in the face of monetary policy
shocks.
The response of deposits to monetary policy shocks is also negative at first several
periods but change to positive after 15th period. The impulse responses of
intergenerational transfers is similar to that of deposits. This estimation conforms to
reactions of households’ consumption. The decrement of consumption brings about the
depression of relevant part in intergenerational transfers. Intergenerational transfers of
deposits are increasing, because households decided to leave more deposits to their
offspring in the situation of high volatility. The reaction of net foreign assets suggests that
foreign investor would reduce their investment temporarily because of the alteration of
monetary policy. In the wake of a monetary policy shock, foreign investors are more
willing to transfers assets to the other economies.
3.6 Bayesian Estimation
The Bayesian estimation and evaluation techniques – as forcefully claimed by An and
Schorfheide (2007) and Fern´andez-Villaverde (2009) – are now the standard tool for
analysis of DSGE models. Estimation with the Bayesian methods is based on the
likelihood generated by the DSGE system. As opposed to GMM estimation which is
36
based on particular equilibrium relationships, Bayesian estimation fits the complete,
solved DSGE model. The Bayesian estimation is also a bridge between calibration and
maximum likelihood. By using the Bayes theorem, one can combine the prior density
with the likelihood function to get the posterior density. In this situation, the posterior
distribution avoids peaking at strange points where the likelihood peaks. Moreover, the
distribution of priors is also very helpful to identify structure parameters and shocks.
We attain the posterior distributions of all estimated parameters in two steps. First,
the
posterior mode and an approximate covariance matrix are obtained by numerical
optimisation on the log posterior density. Second, we apply the
Metropolis-Hastings and
Markov-Chain Monte Carlo (MCMC) algorithm with 3.000 draws to
obtain a sequence
from the unknown posterior distribution. Dynare preprocessor for
Matlab is employed in
computation.
Prior and posterior distributions of estimated parameters
The parameters in Table 1 governing the dynamics of the model are estimated. Most
of them pertain to the nominal frictions and elasticity in the model, while the rest denote
the exogenous shock processes. Detailed descriptions of the prior
distributions for the
structural DSGE parameters and the shock parameters are given in
Table 4.
Table 4
Prior and posterior distributions of structural parameters
37
Prior Distributions
Parameters
Type
Posterior Distributions
Mean
St.Dev.
Mode St.Dev.
0.3999 0.0018
Conf. I
nt.
[0.3814,
0.4245]
mean
α
beta
0.400
0.002
β
beta
0.980
0.0002
0.9802
0.0003
[0.9778,
0.9829]
0.9805
δ
beta
0.040
0.003
0.0388
0.0009
[0.0355,
0.0423]
0.0388
σ
gamma
2.000
0.020
2.0000 0.0026
[1.9751,
2.0256]
1.9974
κ
beta
0.500
0.05
0.4916
0.0090
[0.4586,
0.5559]
0.5120
γ
gamma
3.100
0.003
3.1000
0.0006
[3.0959,
3.1041]
3.1000
ϕ
gamma
6.100
0.003
6.1000
0.0007
6.0993
normal
0.500
0.005
0.5000
0.0006
normal
0.600
0.005
0.6000
0.0007
gamma
0.800
0.003
0.7999
0.0005
gamma
4.600
0.003
4.6000
0.0003
[6.0948,
6.1031]
[0.4934,
0.5076]
[0.5906,
0.6119]
[0.7976,
0.8052]
[4.5969,
4.6050]
gamma
2.000
0.003
2.0000
0.0004
gamma
0.600
0.003
0.6000
0.0003
Inv_gamma
0.010
0.0003
0.0078
0.0003
a
h
θ
0.4047
0.5009
0.6012
0.8017
4.6012
µ
l
ξ
e
38
[1.9948
,
2.0022]
[0.5949,
0.6026]
[0.0071,
0.0095]
1.9985
0.5985
0.0084
We use the inverse gamma (IG) distribution for the standard deviation of the shocks
and set a loose prior with two degrees of freedom. We use the beta distribution for all
parameters bounded between 0 and 1, except
elasticities, such as
γ σ ϕ θ
,
,
,
a
and
h
. For parameters measuring
ξ
and
, we use the gamma
distribution. Normal distribution is applied to estimate
a
and
h
.The elasticity of
µ
substitution between any pair of differentiated goods
is estimated to be higher than the
priors, suggesting a slower response of substitution between goods. The elasticity of
substitution with respect to capital consumption is lower, suggesting a faster response of
consumption to the shocks.
The mean of the reaction coefficient to effective consumption and intergenerational
transfers is estimated to be higher than its prior distribution. Finally, turning to the
exogenous shock variables, the posterior mean of technology shock parameter
κ
is
higher than its prior, with a coefficient of 0.512, pointing a higher persistence than we
considered.
Figure 5 displays prior and posterior distributions of the parameters
and shocks.
Overall, all parameters seem to be well identified, as shown by the fact
that the posterior
Distributions are not centered on the prior or it is centered but with a
smaller dispersion
indicating high significance of the estimates. On the price stickiness
side, 80.17 percent
of firms don’t adjust prices within one quarter. This implies that prices
are re-optimized
39
once every three quarters. The fraction of households setting their wages
according to
ξ
the last-period wage
is estimated to be 0.5985, implying a big share of backward-
looking firms. The structural parameter
a
is estimated to be in the range of 0.4934-
0.5076, highlighting the importance of intergenerational transfers.
40
Figure 5 Prior vs. Posterior Distributions in Metropolis-Hastings Procedure
41
Figure 6 Markov-Chain Monte Carlo (MCMC) Univariate Diagnostics for Parameters
42
Figure 6 presents the Markov-Chain Monte Carlo (MCMC) univariate diagnostics for
the parameters. This is the main source of feedback to gain confidence with results. In
order to diagnose the sensibility and accuracy of the results, two key points of the results
should be paid attention to. First, we should notice whether the results within any of
however many iterations of Metropolis-Hastings simulation are similar. And second, we
should observe the distance between various chains. More specifically, the red and
blue lines on the charts represent measures of parameter vectors within and between
chains respectively. From Figure 6, we notice that the distance between the two chains
are very close for most parameters. Moreover, the plotted moments of structural
α β δ
σ
parameters , , and
are relatively stable and converge. This suggests that the
identification of priors is reasonable and the results from chains are sensible.
Multivariate diagnostics present an aggregate measure based on the eigenvalues of the
variance-covariance matrix of individual parameters. The analysis method of multivariate
diagnostics is very similar to the MCMC univariate diagnostic, because the results of
these two diagnostics are based on the same theory. The plotted moments of multivariate
diagnostic are relatively constant and converge, implying sensibility of the results (see
Figure 7).
43
Figure 7 Multivariate Diagnostics for Parameters
IV CONCLUSIONS
In this paper, we develop a DSGE model for the Chinese economy that includes key
characteristics of New Keynesian tradition and incorporate intergenerational factor. We
estimated the parameters of model by using Bayesian techniques. The
estimation suggests that intergenerational transfers play an important
role in the economy and foreign reserves. The fit of the model is quite
satisfactory, because the plotted moments of structural parameters are relatively
stable and converge. This suggests that the identification of priors is rational and the
results from chains are sensible. According to the impulse response function,
monetary policy shocks lead to an increase in intergenerational
transfers, both in consumption and deposits portion. Monetary policy
44
shock seems to explain the reason of generous intergenerational
transfers partly.
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