Download A DSGE Model for an Emerging Open Economy Oil

A DSGE Model for an Emerging Open Economy
Oil-Producer: Foreign Exchange Interventions as a
Policy Instrument∗
Fred Iklaga
University of Surrey
June 2, 2016
Preliminary and not to be Quoted
Abstract
This paper develops an open-economy DSGE model of an emerging oilproducing economy and incorporates a number of features important for emerging economies in general and the Nigerian economy in particular: creditconstrained consumers, incomplete exchange rate pass-through, oil revenue and
foreign exchange (FX) interventions a second monetary instrument. FX interventions has been extensively used as a policy instrument to manage the
impact of capital flows on the exchange rate in emerging economies with questions raised regarding its effectiveness. Simulations are undertaken to assess
the effects of FX interventions on macroeconomic management in this study.
JEL Classification: E52, E37, E58
Keywords: Nigerian economy, open economy, DSGE model, simulations, foreign exchange interventions, monetary interest rate rules, oil exporter.
∗
To be presented at the 1st Eurasian Conference Baku, Azerbaijan, August 28 -31, 2016.
Contents
1 Introduction
1
2 Open Economy Model
2.1 The Central Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 An Oil Sector and Oil Price Changes . . . . . . . . . . . . . . . . . .
2
4
4
3 Calibration of Parameters and Results
3.1 Parameter Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
4 Conclusions
6
A Summary of the Full Open Economy Model
A.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Zero-Growth Steady State . . . . . . . . . . . . . . . . . . . . . . . .
7
7
12
B A Balanced-Non-Zero-Growth Steady State
15
1
Introduction
Developments in commodity markets have stimulated renewed interest in the study
of the impact of large external shocks on oil exporters. The Nigerian economy is
heavily dependent on revenues from oil exports and is therefore considerably vulnerable to oil price fluctuations. Over the past four decades, oil and gas revenues
have accounted for about 80.0 per cent of total government financing and represents
around 90.0 per cent of its total exports. With proven crude oil reserves estimated at
37 billion barrels and a production output of approximately only 2.5 million barrels
of crude daily (OPEC Annual Statistical Bulletin 2015), Nigeria remains a ‘pricetaker’ subject to the interplay of market forces which are determined by the dictates
of dominant producers and the global demand for oil and gas. Consequently, the
mono-product nature of exports and fiscal revenue in the economy makes macroeconomic outcomes susceptible to the vagaries of oil prices. Against this perspective and
inherent uncertainties, it is evident that the determination of crude oil prices exhibit
a volatile process which poses considerable challenges to macroeconomic stability and
management particularly in heavily dependent oil exporting economies. Indeed, this
scenario shows a fundamental feature of oil exporting economies which is significant
exposure to fluctuations in foreign markets and business cycle dynamics (Bergholt,
2014).
As highlighted by the conclusions in the literature on SSA economies, the effects
of terms of trade shocks on oil producers could be substantial and is of special interest
particularly where the shock is strongly positively correlated with domestic output
and capital inflows. Capital flows may witness a sharp reversal or even a dramatic
“sudden stop” during a negative oil price shock as emphasized by Calvo (1998) and
evident for some oil exporters as a result of the ‘second round effects’ of the 2008
crises, output developments, on the other hand may vary depending on the structure
of the sectoral contributions to overall output. This implies that the debilitating effects of such reversals on the financial system may sometimes be more amplified than
the somewhat transient impact of oil price shocks on output. Therefore, macroeconomic management requires intervention markets including the FX markets while
maintaining the broad objectives of price stability and output growth.
This paper develops an open-economy DSGE model of an emerging oil-producing
economy and incorporates a number of features important for emerging economies
in general and the Nigerian economy in particular: a large proportion of creditconstrained consumers, incomplete exchange rate pass-through, oil revenue and foreign exchange (FX) interventions a second monetary instrument. FX interventions
has been extensively used as a policy instrument to manage the impact of capital flows
1
on the exchange rate in emerging economies. First we will introduce Foreign reserves
intervention as a second monetary instrument in this paper. Widely acknowledged
as a monetary policy instrument, the use of foreign exchange (FX) interventions by
central banks to dampen currency appreciation or depreciation has become common
place in not just emerging economies but also in some developed economies post the
financial crisis of 2008/09. Formalising such interventions as an additional monetary
policy instrument in macroeconomic models should further epitomize the actions of
central banks when faced with the two-pronged effects of volatile developments in international financial markets and oil price shocks. In particular, monetary authorities
may undertake regular interventions in FX market to moderate supply or demand
pressures by sterilizing the surplus proceeds from oil exports to control liquidity in
the system or conversely intervene by providing FX from its international reserves
in order to maintain exchange rate stability when there is a deficit of inflows to the
market as it is the case for oil producers. I follow research by Benes et al. (2015)
to introduce sterilized interventions in the DSGE model for Nigeria. This involves a
study of various hybrid monetary policy rules and managed exchange rate regimes to
analyse their implications for inflation, output and the exchange rate in the presence
of various domestic and external shocks. I present details of the model in this paper.
The DSGE model for an oil exporting economy in this paper follow the prototypical new-Keynesian framework of the open economy (Gali, 2008). The rest of the
paper is organized as follows. Section 2 presents the oil exporting emerging economy
model. Section 3 concludes.
2
Open Economy Model
A standard New Keynesian (NK) DSGE model with features associated with emerging economy oil exporters is developed in this section. The model description is
summarised, while the details of the central bank actions are presented.
Households
A single-period utility for the representative agent in terms of consumption, Ct , external habit χCt−1 and leisure, Lt , as
%(1−σ)
UC,t = (1 − %)(Ct − χCt−1 )(1−%)(1−σ)−1 Lt
%(1−σ)−1
UL,t = %(Ct − χCt−1 )(1−%)(1−σ) Lt
2
(1)
(2)
In this utility function σ ≥ 1 is a risk-aversion parameter which is also the inverse of
the intertemporal rate of substitution. The parameter % ∈ (0, 1) defines the relative
weight households place on consumption.
The value function at time t of the representative household is given by
"
Ωt = Et
∞
X
#
β s U (Ct+s , Lt+s )
(3)
s=0
where β is the discount factor. In a stochastic environment, the household’s problem
at time t is to choose state-contingent plans for consumption {Ct }, leisure, {Lt } and
holdings of financial savings to maximize Ωt given its budget constraint in period t
Bt+1 = Bt Rt−1 + Wt ht − Ct + Γt − T Lt + T CBt
(4)
where Bt is the net stock of real financial riskless assets at the beginning of period
t, Wt is the real wage rate, Γt are dividends from ownership of firms, T Lt are lumpsum taxes, T CBt central bank transfer to households and Rt is the real interest rate
paid on assets held at the end of period t. Hours worked are ht = 1 − Lt ∈ (0, 1)
and the total amount of time available for work or leisure is normalized at unity.
Government spending is financed by lump-sum non-distortionary taxes throughout.
The first-order conditions for this optimization problem are
UC,t = Rt βEt [UC,t+1 ]
UL,t
= Wt
UC,t
(5)
(6)
An equivalent representation of the Euler consumption equation (5) is
1 = Rt Et [Λt,t+1 ]
(7)
U
where Λt,t+1 ≡ β UC,t+1
is the real stochastic discount factor over the interval [t, t + 1].
C,t
Equation (5) (or (7)) is the Euler consumption function.
Firms
For production I assume that output Yt is produced using hours, ht and beginningof-period capital Kt with a Cobb-Douglas production function
1−α
Yt = F (At , ht , Kt−1 ) = (At ht )α Kt−1
3
(8)
where At is a technology parameter and Yt , ht and end-of-period capital, Kt , are all in
per-capita (household) units. Assume first that capital can adjust instantly without
investment costs. Then equating the marginal product of labour with the real wage
and the marginal product of capital with the cost of capital (given by the real interest
rate plus the depreciation rate, Rt + δ), we have
Yt
= Wt
ht
Yt
= (1 − α)
= Rt + δ
Kt−1
Fh,t = α
FK,t
(9)
(10)
Let investment in period t be It . Then capital accumulates according to
Kt+1 = (1 − δ)Kt + It
(11)
In setting up the NK features of the model a differentiated goods market and
nominal rigidities that constraint firms price adjustment frequency are introduced.
For investment adjustment costs in the functional form
S(X) = φX (Xt − X))2
2.1
(12)
The Central Bank
The gross nominal and ex post real interest rates are related by
Rt =
Rn,t−1
Πt
(13)
where the nominal interest rate is a policy variable, typically given in the literature
by a standard Taylor-type rule:
log
2.2
Rn,t
Rn
= ρ log
Rn,t−1
Rn
+ θπ log
Πt
Π
+ θy log
Yt
Y
+ M P S,t
(14)
An Oil Sector and Oil Price Changes
We introduce an oil sector treating output as an exogenous endowment Y O . Revenues
∗
is then driven only by the price of oil PO,t
denominated in foreign currency which is
a exogenous process as for the other shock processes in the model. The only change
to the model is in the trade balance which now becomes
∗
T Bt = St PO,t
Y O + PH,t Yt − PC,t Ct − PI,t It − PH,t Gt
4
(15)
The details of the model is in the Appendix.
3
3.1
Calibration of Parameters and Results
Parameter Calibrations
We simulate the model based on the theoretical underpinnings of the model and findings from previous studies on oil exporters and open economies. The parameters are
calibrated as follows:
Calibrated parameter
Symbol
Discount factor
β
Depreciation rate
δ
Risk premium - scaling
kB
FA risk premium
θ
Labour share
α
Risk premium elasticity
χB
Implied steady state relationship
hours worked/time available
h
Preference parameter
%
Imported investment share
Imported consumption share
Exported investment share
Exported consumption share
oil revenues/GDP
isimport
csimport
isexport
csexport
oil
Value for Nigeria
0.99
0.025
1.00
1.00
1.00
0.50
0.35
calibrated to hit
proportion of hours worked
0.15
0.10
0.02
0.23
0.20
Table 1: Calibrated Parameters
5
4
Conclusions
Developments in New-Keynesian DSGE models have improved how optimising agents
are modelled with the ultimate goal to provide a better informed framework for policy decision making. Incorporating the peculiar features of oil exporting economies
further supports this conclusion. FX intervention can be incorporated as an additional policy instrument into an emerging oil exporting economy DSGE model with
an inflation objective.
The model confirms some recent conclusions in the literature on the benefits of
FX interventions by central banks that respond to significant external shocks which
may drive sudden and huge capital flows. Foreign exchange interventions can be
useful depending on agents perception on overall policy objectives. Therefore, FX
interventions may be used to support the achievement of inflation stability and output
goals of a central bank.
References
Benes, J., Berg, A., Portillo, R., and Vavra, D. (2015). Modeling sterilized interventions and balance sheet effects of monetary policy in a new-keynesian framework.
Open Economies Review, 26(1), 81–108.
Bergholt, D. (2014). Monetary policy in oil exporting economies. Working paper,
CAMP Working Paper Series No 5.
Calvo, G. A. (1998). Capital Flows and Capital-Markets Crises. Journal of Applied
Econometrics, 1(1), 35–54.
6
A
Summary of the Full Open Economy Model
We now bring together all the features in the context of an open economy - creditconstrained consumers, international financial frictions, incomplete exchange rate
pass-through and oil revenue.
A.1
Dynamic Model
C1,t
1
Rn,t
Ct
UL1 ,t
UC1 ,t
ht
= Wt h1,t
UC2 ,t+1
= βEt
UC2 ,t Πt+1
= λC1,t + (1 − λ)C2,t
UL2 ,t
=
= Wt
UC2 ,t
= λh1,t + (1 − λ)h2,t
UC,t =
UL,t =
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(1−%)(1−σ)−1
(1 − %)Ct
(1 − ht )%(1−σ) ;
(1−%)(1−σ)
−%Ct
(1 − ht )%(1−σ)−1 ; for
"
1 =
wC
PH,t
PC,t
for Ct = C1,t , C2,t ; ht = h1,t , h2,t
(A.6)
Ct = C1,t , C2,t ; ht = h1,t , h2,t (A.7)
1−µC
+ (1 − wC )
PF,t
PC,t
1−µC # 1−µ1 C
(A.8)
PH,t
1
=
1
PC,t
[wC + (1 − wC )Tt1−µC ] 1−µC
PF,t
where Tt ≡
PH,t
−µC
PH,t
CH,t = wC
Ct
PC,t
−µC
PF,t
CF,t = (1 − wC )
Ct
PC,t
CH,t
∗
= (1 −
ωC∗ )
PH,t
PC,t RERC,t
7
∗`
St PH,t
θ + (1 − θ)
PH,t
(A.9)
(A.10)
(A.11)
!!−µ∗C
Ct ∗ (A.12)
PCP Price Setters in Home Currency
Ht − ξH βEt [Πζ−1
H,t+1 Ht+1 ] = Yt UC,t
1
Jt − ξH βEt [ΠζH,t+1 Jt+1 ] =
M St Yt UC,t M Ct
1 − ζ1
(A.13)
(A.14)
Wt
W
W
h
PH,t
PH,t
/PC,t
PC,t t
M Ct =
=
=
P
PH,t
PH,t /PC,t
αYt PH,t
C,t
1−ζ
Jt
1 = ξH Πζ−1
H,t + (1 − ξH )
Ht
(A.15)
(A.16)
LCP Price Setters in Foreign Currency
`
`
Ht` − ξH βEt [(Π∗H,t+1
)ζ−1 Ht+1
] = Yt∗ St UC,t
1
`
`
Jt` − ξH βEt [(Π∗H,t+1
)ζ Jt+1
] =
M St Yt∗ St UC,t M Ct`
1 − ζ1
M Ct
M Ct` = S P ∗ `
(A.17)
(A.18)
(A.19)
t H,t
PH,t
` ζ−1
ξH (Π∗H,t
)
1 =
+ (1 − ξH )
Jt`
Ht`
1−ζ
(A.20)
Define the LoP gap for LCPers as
Then
∗`
St PH,t
Ψt ≡
PH,t
(A.21)
`
Ψt
St Π∗H,t
=
Ψt−1
St−1 ΠH,t
(A.22)
Aggregate export inflation in foreign currency is then
Π∗H,t = θ
ΠH,t
`
+ (1 − θ)Π∗H,t
St
but this is not an input into the rest of the model.
8
(A.23)
YtW = (At ht )α Kt1−α
(1 − c)YtW
Yt =
∆t
∆t =
ξΠζt ∆t−1
(A.24)
(A.25)
+ (1 − ξ)
Jt
Ht
−ζ
(A.26)
W
PH,t
PH,t
= M Ct
PC,t
PC,t
Et Λt,t+1
Et [Λt,t+1 Rt+1 ] =
hPW
H,t+1
Pt+1
(1 − α)
W
Yt+1
Kt
Qt
Rn,t−1
Πt
= (1 − δ)Kt−1 + (1 − S(Xt ))It
Rt =
Kt
(A.27)
+ (1 − δ)Qt+1
i
(A.28)
(A.29)
S 0 , S 00 ≥ 0 ; S(1 + g) = S 0 (1 + g) = 0
(A.30)
It
Xt =
(A.31)
It−1
φI
S(Xt ) =
(Xt − (1 + g))2
(A.32)
2
2
It+1
PI,t
0
0
= Qt (1 − S(Xt ) − Xt S (Xt )) + Et Λt,t+1 Qt+1 S (Xt+1 ) 2
PC,t
It
(A.33)
9
−µI
PH,t /PC,t
= wI
It
PI,t /PC,t
−µI
PF,t /PC,t
= (1 − wI )
It
PI,t /PC,t
IH,t
IF,t
∗
IH,t
= (1 − ωI∗ )
PI,t
=
PC,t
"
wI
PH,t
PC,t
∗`
St PH,t
θ + (1 − θ)
PH,t
1
1−µI # 1−µ
I
PF,t
+ (1 − wI )
PC,t
PH,t /PC,t
PI,t /PC,t RERI,t
1−µI
(A.34)
(A.35)
!!−ρ∗I
It∗ (A.36)
(A.37)
∗
∗
Yt = CH,t + IH,t + CH,t
+ IH,t
+ Gt
St
RERC,t Πt
=
St−1
RERC,t−1 Π∗t
Tt
ΠF,t
=
Tt−1
ΠH,t
1
RERC,t = 1
1 − wC + wC TtµC −1 1−µC
(A.38)
(A.39)
(A.40)
(A.41)
1
RERI,t = 1
1 − wI + wI TtµI −1 1−µI
(A.42)
1
Πt = [wC (ΠH,t )1−µC + (1 − wC )(ΠF,t )1−µC ] 1−µC
(A.43)
log Rn,t /Rn = ρr log Rn,t−1 /Rn + (1 − ρr )(θπ Et [log Πt+1 ]/Π
+ θs log St /S) + r,t+1
1
St B ∗
∗
Rn,t
φ( PH,tF,t
)
Yt
∗
UC,t
UC,t
∗
Rn,t−1
=
Π∗t
(A.44)
RERtr =
(A.45)
Rt∗
(A.46)
∗
∗
St BF,t
= St BF,t−1
+ T Bt
(A.47)
∗
∗ St BF,t
φB St BF,t
φ(
) = exp
; φB < 0
(A.48)
PH,t Yt
PH,t Yt
∗
T Bt = St PO,t
Y O + PH,t Yt − PC,t Ct − PI,t It − PH,t Gt (A.49)
10
Then the real exchange rate is given by
RERC,t = RERtd RERtr
(A.50)



r
d
RERt+1 
UC,t+1 RERt+1 1 
1
0 =
Et 
−
∗
r
∗
St BF,t
UC,t RERt Πt+1 φ(
RERtd
) exp(U IP,t+1 )
PH,t Yt
(A.51)
∗
∗
An alternative to (A.51) which uses an exogenous process for Rn,t
rather than UC,t
(which drives RERtr ) is
1
φ
S B∗ t
F,t
PH,t Yt
∗
exp(U IP,t )Rn,t
= Et
UC,t+1 St+1
UC,t Πt+1 St
(A.52)
Shocks There are 7 domestic shocks:
At+1
A
Gt+1
log
G
M St+1
log
MS
M P St+1
log
MP S
U IPt+1
log
U IP
r,t+1
ΨC,t+1
log
ΨC
ΨI,t+1
log
ΨI
log
At
+ a,t+1
A
Gt
ρg log
+ g,t+1
G
M St
ρms log
+ ms,t+1
MS
M P St
ρmps log
+ mps,t+1
MP S
U IPt
ρU IP log
+ uip,t+1
U IP
ρe r,t + νr,t+1
ΨC,t
ρψc log
+ ψc ,t+1
ΨC
ΨI,t
ρψi log
+ ψi ,t+1
ΨI
= ρa log
(A.53)
=
(A.54)
=
=
=
=
=
=
(A.55)
(A.56)
(A.57)
(A.58)
(A.59)
(A.60)
If the ROW is not modelled explicitly we close the model with 4 exogenous AR1
shocks
∗
log Rn,t
/Rn∗
Π∗
log t+1
Π∗
C∗
log t+1
C∗
I∗
log t+1
I∗
∗
= ρ∗r log(Rn,t−1
/Rn∗ + ∗r,t+1
Π∗
= ρ∗π log t∗ + ∗π,t+1
Π
∗
C
= ρ∗c log t∗ + ∗c,t+1
C
I∗
= ρ∗i log t∗ + ∗i,t+1
I
11
(A.61)
(A.62)
(A.63)
(A.64)
and
∗
∗
UC,t
= (1 − %∗ )(Ct∗ − χ∗ Ct−1
)(1−%
A.2
∗ )(1−σ ∗ )−1
∗ (1−σ ∗ )
(1 − h∗t )%
(A.65)
h∗t = h
(A.66)
Y ∗ = C∗ + I∗
(A.67)
Zero-Growth Steady State
First assume zero growth in the steady state: g = g ∗ = 0 and non-negative inflation
Π = Π∗ = Π∗H = Π∗F > 0. Then we have
Rn =
C1
1
Rn
C
UL1
UC1
h
Rn∗ φ
SB
P
= W h1
= β
= λC1 + (1 − λ)C2
UL2
=
=W
UC2
= λh1 + (1 − λ)h2
UC = (1 − %)C (1−%)(1−σ)−1 (1 − h)%(1−σ) ; for C = C1 , C2 ; h = h1 , h2
UL = −%C (1−%)(1−σ) (1 − h)%(1−σ)−1 ; for C = C1 , C2 ; h = h1 , h2
12
"
1 =
PH
PC
CH
CF
CH ∗
wC
PH
PC
1−µC
+ (1 − wC )
1
=
1
[wC + (1 − wC )T 1−µC ] 1−µC
−µC
PH
= wC
C
PC
−µC
PF
= (1 − wC )
C
PC
−µ∗C
PH
∗
= (1 − ωC )
C∗
PC RERC
ζ
1
∆ =
J
H
=
PW
=
P
H(1 − ξH βΠζ−1 ) =
MC =
J(1 − ξH βΠζ ) =
(1 − ξ) 1−ζ (1 − ξΠζ−1 ) 1−ζ
1 − ξΠζ
1
1 − ξΠζ−1 1−ζ
1−ξ
1
J(1 − βξΠζ )
1−
ζ H(1 − βξΠζ−1 )
Y UC
1
Y UC M C
1 − ζ1
H` = H
J` = J
M C` = M C
Ψ=1
13
PF
PC
1−µC # 1−µ1 C
PHW
PC
PH
PC
(1 − α)M C PPHC Y
= MC
K =
(R + δ)Q
(A.68)
(A.69)
(A.70)
Y W = (AH)α K 1−α
(1 − c)Y W
Y =
∆
Rn
R =
Π
I = (g + δ)K
X = 1
S(X) = S 0 (X) = 0
PI
Q =
PC
−µI
PH /PC
wI
I
PI /PC
−µI
PF /PC
(1 − wI )
I
PI /PC
−µ∗I
PH
∗
(1 − ωI )
I∗
P RER
1
" 1−µI
1−µI # 1−µ
I
PH
PF
wI
+ (1 − wI )
PC
PC
(A.71)
(A.72)
(A.73)
(A.74)
(A.75)
(A.76)
(A.77)
IH =
IF =
∗
IH
=
PI
PC
=
Y
= CH + IH + EXC + EXI + Gt
−µ∗C
PH
∗
∗
EXC = CH,t = (1 − ωC,t )
C∗
PC RERC
−µ∗I
PH
∗
∗
EXI = IH,t = (1 − ωI,t )
I∗
PI RERI
1
RERC =
1
[1 − wC + wC T µC −1 ] 1−µC
1
RERI =
1
[1 − wI + wI T µI −1 ] 1−µI
1 = βRn∗
Rn∗
∗
R =
Π∗
14
(A.78)
(A.79)
(A.80)
(A.81)
(A.82)
(A.83)
(A.84)
(A.85)
(A.86)
(A.87)
(A.88)
The problem now there are 31 variables but only 30 SS equations! The model is
only complete if we pin down the steady state of the foreign assets or equivalently
the trade balance. In other words there is a unique model associated with any choice
of the long-run assets of our SOE.1 .
Our missing equation is therefore the trade balance
T B = PO∗ SY O +PH Y −PC C−PI I−PH G = PH EXC − (PC C − PH CH ) + PH EXI − (PI I − PH IH )
|
{z
} |
{z
}
Net Exports of C-goods
Net Exports of I-goods
(A.89)
using Y = CH + IH + EXC + EXI + Gt , for some choice of T B, say zero.
B
A Balanced-Non-Zero-Growth Steady State
We can easily set up the model with a balanced-exogenous-growth steady state. Now
the process for At is replaced with
At = Āt Act
Āt = (1 + g)Āt−1 exp(A,t ) ⇒ log Āt = µ + log Āt−1 + trend,t
log Act − log Ac = ρA (log Act−1 − log Ac ) + A,t
where µ = log(1+g) ≈ g and At is a labour-augmenting technical progress parameter
which we decompose into a cyclical component, Act , modelled as a temporary AR1
process, a stochastic trend, a random walk with drift, Āt . Thus the balanced growth
deterministic steady state path (bgp) is driven by labour-augmenting technical change
growing at a net rate g. If we put g = trend,t = 0 and Āt = 1, we arrive at our previous
formulation with Act = At .
1
The same point applies to government debt when we introduce fiscal policy
15
Now stationarize variables by defining cyclical and stationary components:
α
Ytc ≡
Ktc ≡
Ctc ≡
Itc ≡
Wtc ≡
Kt−1
Āt
(at Ht )
Yt
=
∆t
Āt
Kt
Āt
Ct
Āt
It
Āt
Wt
Āt
1−α
(at Ht )
=
α
c
Kt−1
(1+gt )
1−α
∆t
for all non-stationary variables. Then dynamic equation in cyclical components involving a lead or lag needs modifying as follows:
Utc =
c
UC,t
≡
Λt,t+1 =
Ktc =
Xtc =
S(Xtc ) =
c
((Ctc − χCt−1
/(1 + gt ))(1−%) (1 − ht )% )1−σc − 1
1 − σc
UC,t
c
= (1 − %)(Ctc − χCt−1
/(1 + gt ))(1−%)(1−σc )−1 (1 − ht )%(1−σc )
(1−%)(1−σc )−1
Āt
c
c
UC,t+1
UC,t+1
UC,t+1
β
= β(1 + gt+1 )(1−%)(1−σc )−1 c ≡ βg,t+1 c
UC,t
UC,t
UC,t
Kc
(1 − δ) t−1 + (1 − S(Xtc ))Itc
1 + gt
Ic
(1 + gt ) c t
I t−1
φX (Xtc − 1 − g)2
S 0 (Xtc ) = 2φX (Xtc − 1 − g)
where the growth-adjusted discount rate is defined as
βg,t ≡ β(1 + gt )(1−%)(1−σc )−1 ,
(B.1)
Et [Λt,t+1 Rt+1 ]
(B.2)
the Euler equation is still
and
c
c c
Htc − ξEt [Π̃ζ−1
t+1 Ht+1 βg,t+1 (1 + gt+1 )] = Yt UC,t
c
c
Jtc − ξEt [Π̃ζt+1 Jt+1
βg,t+1 (1 + gt+1 )] = Ytc UC,t
M Ct M St
16
where
gt ≡
(Āt − Āt−1 )
= (1 + g) exp(A,t ) − 1
Āt
is the stochastic steady state growth rate.
The steady state for the rest of the system is the same as the zero-growth one
except for the following relationships:
R =
Rn =
Λ =
Ic =
Xc =
Jc
=
Hc
MC =
(1 + g)1+(σc −1)(1−%)
1
Rc
≡
= n
β
βg
Π
ΠR
1
R
(δ + g)K c
1+g
1+g
1
1 − ξΠζ−1 1−ζ
1−ξ
1
J(1 − βg (1 + g)ξΠζ )
1−
ζ H(1 − βg (1 + g)ξΠζ−1 )
(B.3)
(B.4)
where R and Rn are the real and nominal steady state interest rates and Π is inflation.
17