RDP 2015-11 – Unprecedented Changes in the Terms of Trade:
Online Appendix
Mariano Kulish∗and Daniel M. Rees†
August 2015
∗ School
of Economics, UNSW, m.kulish@unsw.edu.au
Research, Reserve Bank of Australia reesd@rba.gov.au
† Economic
1
1
The Model
This section outlines the non-linear model.
1.1
Households
The representative household maximises its expected lifetime utility:
E0
∞
X
β t ζt U (Ct , LH,t , LN,t , LX,t )
t=0
where ζt is an intertemporal preference shock:
ln ζt = ρζ ln ζt−1 + uζt
(1)
The household’s utility function is given by:
h
i 1+ν
L
t
1+ω
1+ω
1+ω 1+ω
ξH LH,t + ξN LN,t + ξX LX,t
U (Ct , LH,t , LN,t , LX,t ) = ln (Ct − hCt−1 ) −
1+ν
where L
t is a shock to labour supply:
L
ln L
t = ρ ln t−1 + ut
L
(2)
Households maximise subject to the budget constraint:
∗
F
Pt Ct + PI,t It + Bt+1 + St Bt+1
≤ (1 + Rt−1 ) Bt + 1 + Rt−1
St Bt∗
K
K
K
+WH,t LH,t + WN,t LN,t + WX,t LX,t + RN,t
KN,t + RH,t
KH,t + RX,t
KX,t
+ΓN,t + ΓH,t + ΓX,t − Tt
and the capital accumulation equations:
Ih,t
JH,t
KH,t+1 = (1 − δ) KH,t + Vt 1 − Υ
Ih,t−1
In,t
KN,t+1 = (1 − δ) KN,t + Vt 1 − Υ
JN,t
In,t−1
Ix,t
JX,t
KX,t+1 = (1 − δ) KX,t + Vt 1 − Υ
Ix,t−1
(3)
(4)
(5)
where:
Vt =
v
Ṽt
(1 + zI )t
ln Ṽt = ρV ln Ṽt−1 + uVt
(6)
(7)
Note that JN,t refers to investment that produces capital for use in the non-tradeable sector. It is distinct from IN,t which
refers to non-tradeable goods used to produce investment goods via the investment goods aggregator.
The interest rate that the household receives on foreign bonds follows the process:
St Bt∗
− b∗ + ψ̃b,t
RtF = Rt∗ exp −ψb
Pt Yt
2
(8)
where b∗ is the steady-state net foreign asset-to-GDP ratio, ψ˜b,t is a risk-premium shock that follows the process:
ψ̃b,t = ρψ ψ̃b,t + uψ,t
(9)
and Rt∗ follows the process
∗
ln Rt∗ = (1 − ρR∗ ) ln R∗ + ρR∗ ln Rt−1
+ uR∗ ,t
(10)
The household’s consumption bundle is a CES aggregate of traded and non-traded goods, while the traded goods is itself a
CES aggregate of home- and foreign-produced traded goods:
η
η−1
1 η−1
η−1
1
η
η
η
η
Ct = γT,t CT,t + γN,t CN,t
CT,t =
γH γF
CH,t
CF,t
H
γ
γFγF
γH
Note that the Cobb-Douglas specification allows us to assume that γH,t = γH and so on.
The non-traded, home-produced traded and imported consumption goods are themselves bundles of imperfectly substitutable
goods:
1
ˆ
Cj,t ≡
Cj,t (i)
θj −1
θj
j
θjθ−1
di
0
The price indices corresponding to the consumption goods aggregates are:
h
i 1
1−η
1−η 1−η
Pt = γT,t PT,t
+ γN,t PN,t
γH γF
PT,t = PH,t
PF,t
(11)
(12)
The investment good is similarly a CES aggregate of non-traded and traded goods:
γI
It = (1 + zv )t
γI
IT,t =
γI
T
N
IT,t
IN,t
γTI
I
γT
I
γN
γN
γI
H
F
IH,t
IF,t
I
γH
γH
γFI
γF
The price indices corresponding to the investment goods aggregates are:
I
PtI = (1 + zv )−t PT,t
I
γH
I
γT
γI
N
PN,t
I
γF
I
PT,t
= PH,t PF,t
(13)
(14)
The non-traded, home-produced traded and imported investment goods are themselves bundles of imperfectly substitutable
goods:
ˆ
Ij,t ≡
1
Ij,t (i)
0
3
θj −1
θj
j
θjθ−1
di
1.2
1.2.1
Production
Commodity Firms
Commodity firms produce using the Cobb-Douglas production function:
1−αX
αX
YX,t = At ZX,t KX,t
(Zt LH,t )
(15)
where ZX,t is a stationary sector-specific TFP shock:
ZX,t = (1 + zX )t Z̃X,t
ln Z̃X,t = ρX ln Z̃X,t−1 + uX,t
(16)
At is a stationary technology shock that is common across sectors:
ln (At ) = ρa ln (At−1 ) + uA,t
(17)
and Zt is a labour-augmenting technology shock that is common across sectors whose growth rate 1 + zt =
Zt
Zt−1
follows:
ln zt = (1 − ρz ) ln z + ρz ln zt−1 + uz,t
(18)
∗
follows
We assume that the foreign price of commodities PX,t
∗
PX,t
= κt Pt∗
(19)
where Pt∗ is the foreign price level and κt follows the process:
κt = exp (κ̃t )
1 + z∗
∗
1 + zX
t
(20)
κ̃t = (1 − ρκ )κ + ρκ κ̃t−1 + εκ,t
(21)
The law of one price holds and so the domestic price of commodities is:
∗
PX,t = St PX,t
(22)
and the firm takes this as given when making its production decisions.
1.2.2
Non-tradeable Firms
Non-tradeable firms sell differentiated products, which they produce using the Cobb-Douglas production function:
1−αN (i)
αN
YN,t (i) = At ZN,t KN,t
(i) (Zt LN,t (i))
(23)
ZN,t is a stationary sector-specific TFP shock:
ZN,t = (1 + zN )t Z̃N,t
ln Z̃N,t = ρN ln Z̃N,t−1 + uN,t
4
(24)
and Zt is a labour-augmenting technology shock defined above.
Firms can only change prices at some cost, following a Rotemberg (1982) pricing mechanism:
ψN
2
2
PN,t (i)
− 1 PN,t YN,t
Π̄N PN,t−1 (i)
The output of the non-traded sector is an aggregate of the output of each of the non-traded firms
ˆ
1
YN,t ≡
YN,t (i)
θN −1
θN
θ θN−1
N
di
0
1.2.3
Tradeable Firms
Tradeable firms produce using the Cobb-Douglas production function:
αH
YH,t (i) = At ZH,t KH,t
(i) (Zt LH,t (i))
1−αH
(25)
ZH,t is a stationary sector-specific TFP shock:
ZH,t = (1 + zH )t Z̃H,t
ln Z̃H,t = ρH ln Z̃H,t−1 + uH,t
(26)
and Zt is a labour-augmenting technology shock defined above.
Firms can only change prices at some cost, following a Rotemberg (1982) pricing mechanism:
ψH
2
2
PH,t (i)
− 1 PH,t YH,t
Π̄H PH,t−1 (i)
The output of the non-traded sector is an aggregate of the output of each of the non-traded firms
ˆ
YH,t ≡
1
YH,t (i)
θH −1
θH
θ θH−1
H
di
0
1.3
Importing Firms
Importing firms purchase foreign good varieties at the price ςSt Pt∗ (i) and sell them in the domestic market at price PF,t (i).
The parameter ς represents a subsidy to imported firms, funded by lump-sum taxation. We set the subsidy equal to
ς = (θf − 1)/θf , thereby ensuring that markups in this sector are zero in equilibrium.
Importing firms can only change prices at some cost, following a Rotemberg [1982] pricing mechanism:
ψF
2
1.4
2
PF,t (i)
−
1
PF,t YF,t
Π̄F PF,t−1 (i)
(27)
Foreign Sector
The rate of foreign goods price inflation is Π∗t , which follows the process,
ln Π∗t = (1 − ρΠ∗ ) ln Π∗ + ρΠ∗ ln Π∗t−1 + uΠ∗ ,t
5
(28)
∗
We also assume that foreign demand for the domestically produced tradable, CH,t
, follows the process below;
∗
CH,t
PH,t
St Pt∗
−η∗
Yt∗
(29)
Yt∗ = Zt (1 + z ∗ )t Ỹt∗
ln Ỹt∗ = ρ∗Y ln Ỹt∗ + u∗t
(30)
=
∗
γH,t
where
∗
and γH,t
follows the process:
∗
∗
γH,t
= γH
1.5
1 + zH
1 + z∗
t(1−η∗ )
Relative Prices and Current Account
In what follows it will be convenient to define a number of relative prices:
TN,t =
TT,t =
I
TT,t
=
TH,t =
TF,t =
TI,t =
TX,t =
TF ∗ ,t =
PN,t
Pt
PT,t
Pt
I
PT,t
Pt
PH,t
Pt
PF,t
Pt
PI,t
Pt
PX,t
Pt
St Pt∗
Pt
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
Nominal net exports are given by:
∗
N Xt = PH,t CH,t
+ PX,t YX,t − St Pt∗ YF,t
(39)
And the current account equation is given by:
∗
∗
St Bt∗ + N Xt
St Bt+1
− Bt∗ = Rt−1
1.6
(40)
Monetary Policy
Rt
=
R
Rt−1
R
ρR "
Pt /Pt−1
Π̄
φπ 6
Yt /Yt−1
¯ t−1
Yt /Y
φy #(1−ρR )
exp(uR,t )
(41)
1.7
Market Clearing
Investment goods:
Labour market:
It = JH,t + JN,t + JX,t
(42)
h
i 1
1+ω
1+ω 1+ω
Lt = L1+ω
+
L
+
L
H,t
N,t
X,t
(43)
Non-tradeable goods:
YN,t
ψN
= CN,t + IN,t +
2
2
ΠN,t
− 1 YN,t
Π̄N
Tradeables:
∗
YH,t = CH,t + CH,t
+ IH,t +
ψH
2
2
ΠH,t
YH,t
−
1
Π̄H
(44)
(45)
Imports:
2
ΠF,t
−
1
YF,t
Π̄F
(46)
N GDPt = PH,t YH,t + PN,t YN,t + PX,t YX,t
(47)
Yt = TH YH,t + TN YN,t + TX YX,t
(48)
YF,t = CF,t + IF,t +
ψF
2
Nominal GDP:
and Real GDP:
2
2.1
Derivation of equilibrium conditions
Households
∗
The household’s problem is to choose a sequence for Ct , LH,t , LN,t , LX,t , Ih,t , IN,t , IX,t , KH,t+1 , KN,t+1 , KX,t+1 , Bt+1 , Bt+1
to maximise expected lifetime utility subject to the budget constraint and capital accumulation equation. The Lagrangian
for this problem is:
L
=
E0
∞
X
(
χt−1 ζt ln (Ct − hCt−1 ) −
t=0
h
i 1+ν
ξL
t
1+ω
1+ω 1+ω
L1+ω
+
L
+
L
H,t
N,t
X,t
1+ν
∗
+Λ̃t (1 + Rt−1 ) Bt + 1 + Rt−1
St Bt∗ + WH,t LH,t + WN,t LN,t + WX,t LX,t
K
K
K
+RH,t
KH,t + RN,t
KN,t + RX,t
KX,t + ΓN,t + ΓH,t + ΓX,t
∗
−Pt Ct − PI,t It − Bt+1 − St Bt+1
JH,t
+ΦH,t (1 − δ) KH,t + Vt 1 − Υ
JH,t − KH,t+1
JH,t−1
JN,t
+ΦN,t (1 − δ) KN,t + Vt 1 − Υ
JN,t − KN,t+1
JN,t−1
)
JX,t
+ΦX,t (1 − δ) KX,t + Vt 1 − Υ
JX,t − KX,t+1
JX,t−1
The first order conditions for the household’s problem are:
7
ζt
ζt+1
0=
− hEt β
− Λt
Ct − hCt−1
Ct+1 − hCt
Λt+1 (1 + Rt ) Pt
0 = −Λt + Et β
Pt+1
Λt+1 (1 + Rt∗ ) Pt St+1
0 = −Λt + Et β
Pt+1 St
"
(
#)
K
Λt+1 RH,t+1
0 = −QH,t + Et β
+ (1 − δ) QH,t+1
Λt
Pt+1
"
#)
(
K
Λt+1 RN,t+1
+ (1 − δ) QN,t+1
0 = −QN,t + Et β
Λt
Pt+1
"
#)
(
K
Λt+1 RX,t+1
+ (1 − δ) QX,t+1
0 = −QX,t + Et β
Λt
Pt+1
JH,t
JH,t
JH,t
− Υ0
0 = −TtI + QH,t Vt 1 − Υ
JH,t−1
JH,t−1 JH,t−1
)
(
2
Λt+1 QH,t+1 Vt+1 0 JH,t+1 JH,t+1
+ Et β
Υ
2
Λt
JH,t
JH,t
JN,t
JN,t
JN,t
I
0
0 = −Tt + QN,t Vt 1 − Υ
−Υ
JN,t−1
JN,t−1 JN,t−1
)
(
2
Λt+1 QN,t+1 Vt+1 0 JN,t+1 JN,t+1
+ Et β
Υ
2
Λt
JN,t
JN,t
Ix,t
Ix,t
JX,t
0 = −TtI + QX,t Vt 1 − Υ
− Υ0
JX,t−1
JX,t−1 JX,t−1
)
(
2
Λt+1 QX,t+1 Vt+1 0 JX,t+1 JX,t+1
+ Et β
Υ
2
Λt
JX,t
JX,t
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
and
WH,t
Pt
WN,t
ν−ω ω
L
0 = −ζt ξt (Lt )
LN,t + Λt
Pt
W
X,t
ν−ω ω
0 = −ζt ξL
LX,t + Λt
t (Lt )
Pt
ν−ω
0 = −ζt ξL
t (Lt )
Lω
H,t + Λt
(58)
(59)
(60)
where Λt ≡ Λ̃t Pt is the discount factor on nominal claims and Qj,t ≡ Φt /Λt is Tobin’s Q for sector j.
Demand for each type of consumption good:
CN,t = γN,t (TN,t )
−η
Ct
(61)
Ct
(62)
CT,t
(63)
CT,t
(64)
−η
CT,t = γT,t (TT,t )
CH,t = γH (TH,t )
−1
−1
CF,t = γF (TF,t )
8
and for investment goods:
IN,t
IT,t
IH,t
IF,t
2.2
2.2.1
−1
TN,t
=
It
TI,t
−1
TT,t
I
It
= γT,t
TI,t
−1
TH,t
I
= γH
IT,t
TI,t
−1
TF,t
I
= γF
IT,t
TI,t
I
γN,t
(65)
(66)
(67)
(68)
Firms
Commodity Firms
Given prices, commodity firms choose LX,t and KX,t to maximise profits, given by:
ΓX,t = PX,t YX,t − WX,t LX,t − RtKX KX,t
The resulting quantities are:
PX,t YX,t
K
= RX,t
KX,t
PX,t YX,t
(1 − αX )
= WX,t
LX,t
αX
(69)
(70)
Note that this implies that ΓX,t = 0.
2.2.2
Tradeable Firms
Profits for tradeable firms are given by:
K
ΓH,t (i) = PH,t (i)YH,t (i) − WH,t LH,t (i) + RH,t
KH,t (i)
2
ψH
PH,t (i)
−
− 1 PH,t YH,t
2
Π̄H PH,t−1 (i)
The optimality conditions of the cost minimisation problem are:
αH WH,t
LH,t
K
1 − αH RH,t
1−αH αH
1−αH K αH
WH,t
RH,t
1
1
=
1−α
1 − αH
α
ZH,t Zt H PH,t
KH,t =
M CH,t
where M Ct denotes real marginal cost.
When setting prices, the problem of firm i is to choose PH,t (i) to maximise:
∞
X
χt−1 Λt
t=0
Pt
9
(ΓH,t (i))
(71)
(72)
subject to the demand constraint that: YH,t (i) =
ΠH,t
0= H
Π
2.2.3
PH,t (i)
PH,t
−θH
YH,t . Optimal price setting implies that:
(
)
ΠH,t
θH − 1 θH
Λt+1 YH,t+1 TH,t+1 ΠH,t+1 ΠH,t+1
−1 +
−1
−
M CH,t − Et β
ΠH
ψ
ψ
Λt YH,t TH,t
ΠH
ΠH
(73)
Non-tradeable Firms
Profits for non-tradeable firms are given by:
K
ΓN,t (i) = PN,t YN,t (i) − WN,t LN,t (i) + RN,t
KN,t (i)
2
ψN
PN,t (i)
−
−
1
PN,t YN,t
2
Π̄N PN,t−1 (i)
The optimality conditions of the cost minimisation problem are:
αN WN,t
LN,t
K
1 − αN RN,t
1−αN αN
1−αN K αN
WN,t
RN,t
1
1
=
1−α
1 − αN
α
ZN,t Zt N PN,t
KN,t =
M CN,t
(74)
(75)
where M CN,t denotes real marginal cost.
The firm’s pricing problem is to choose PN,t (i) to maximise:
∞
X
χt−1 Λt
Pt
t=0
"
ψ
PN,t (i)YN,t (i) − M CN,t Pt YN,t (i) −
2
PN,t (i)
−1
N
Π PN,t−1 (i)
#
2
PN,t YN,t
Optimal price setting implies that the inflation rate of non-tradeable goods is given by:
ΠN,t
0= N
Π
2.3
(
)
θN − 1 θN
Λt+1 TN,t+1 YN,t+1 ΠN,t+1
ΠN,t+1
ΠN,t
−1 +
−
M CN,t − Et β
−1
ΠN
ψ
ψ
Λt TN,t YN,t
ΠN
ΠN
(76)
Importing Firms
Profits for importing firms are given by
ΓF,t = (PF,t −
ςSt Pt∗ ) YF,t (i)
ψF
−
2
PF,t (i)
−1
F
Π̄ PF,t−1 (i)
2
PF,t YF,t
(77)
Optimal price setting implies that the inflation rate of imported goods is given by:
ΠF,t
0= F
Π
3
(
)
ΠF,t
θF − 1 θF
Λ
T
Y
Π
Π
t+1
F,t+1
F,t+1
F,t+1
F,t+1
−1 +
−
ςSt Pt∗ − Et β
−1
ΠF
ψ
ψ
Λt TF,t YF,t
ΠF
ΠF
Normalised Equilibrium Conditions
We normalise the variables as follows:
10
(78)
Non-tradeable Firms
Households
1. ct =
Ct
Zt
20. yN,t =
YN,t
Zt (1+zN )t
38. yX,t =
YX,t
Zt (1+zX )t
39. kX,t =
KX,t
Zt
K
40. rX,t
=
K
RX,t
Pt
2. cN,t =
CN,t
Zt (1+zN )t
21. iN,t =
IN,t
Zt (1+zN )t
3. cT,t =
Ct
Zt (1+zT )t
22. kN,t =
KN,t
Zt
4. cH,t =
CH,t
Zt (1+zH )t
K
23. rN,t
=
K
RN,t
Pt
CF,t
Zt (1+zF )t
24. wN,t =
5. cF,t =
6. λt = Λt Zt
26. jN,t =
8. Yt =
N GDPt
Pt Zt
27. qN,t = QN,t
10. nxt =
11. πt =
N Xt
Pt Zt
Pt
Pt−1
42. jX,t =
Relative Prices
44. τI,t = TI,t (1 + zI )t
Home-tradeable Firms
28. yH,t =
YH,t
Zt (1+zH )t
29. c∗H,t =
∗
CH,t
Zt (1+zH )t
PN,t
PN,t−1
30. iH,t =
IH,t
Zt (1+zH )t
13. πT,t =
PT ,t
PT ,t−1
31. kH,t =
KH,t
Zt
14. πH,t =
PH,t
PH,t−1
K
=
32. rH,t
K
RH,t
Pt
15. πF,t =
PF,t
PF,t−1
33. wH,t =
16. πI,t =
PI,t
PI,t−1
34. mcH,t = M CH,t
I
=
17. πT,t
PTI ,t
PTI ,t−1
35. πH,t =
PH,t
PH,t−1
It
Zt (1+zI )t
36. jH,t =
JH,t
(1+zI )t Zt
19. vt = Vt (1 + zI )t
JX,t
(1+zI )t Zt
43. qX,t = QX,t
JN,t
(1+zI )t Zt
12. πN,t =
18. it =
WX,t
Pt Zt
25. mcN,t = M CN,t
7. kt =
St Bt+1
Pt GDPt
41. wX,t =
WN,t
Pt Zt
Kt
Zt
9. b∗t+1 =
Commodity Firms
45. τN,t = TN,t (1 + zN )t
46. τT,t = TT,t (1 + zT )t
47. τH,t = TH,t (1 + zH )t
48. τF,t = TF,t (1 + z ∗ )t
49. τF ∗ ,t = TF ∗ ,t (1 + z ∗ )t
I
I
(1 + z ∗ )t
= TT,t
50. τT,t
WH,t
Pt Zt
Foreign Economy
51. πt∗ =
Pt∗
∗
Pt−1
Miscellaneous Variables
52. ∆st =
37. qH,t = QH,t
St
St−1
The Cobb-Douglas assumption for the tradeable consumption bundle implies that the growth rate of the tradeable consumption bundle is 1 + zT = (1 + zH )γH (1 + z ∗ )γF . A similar relationship holds for the tradeable investment good.
where it is implied by the bundles that:
1 + zT = (1 + zH )γH (1 + z ∗ )γF
and
I
I
1 + zTI = (1 + zH )γH (1 + z ∗ )γF
and
I
I
1 + z I = (1 + zv )(1 + zTI )γT (1 + zN )γN
UIP implies that that s staisfies
1 + z∗ =
11
π
sπ ∗
and if there is a steady state for τI,t then it follows that
1 + zI =
π
πI
Also, note that zt = Zt /Zt−1 . With these normalisations, the equilibrium conditions are as follows.
Household optimisation:
ζt+1
ζt zt
− hEt β
− λt
ct zt − hct−1
ct+1 zt+1 − hct
λt+1 (1 + rt )
0 = −λt + Et β
zt+1 πt+1
λt+1 (1 + rt∗ ) st+1
0 = −λt + Et β
zt+1 πt+1
λt+1 K
0 = −qH,t + Et β
rH,t+1 + (1 − δ)qH,t+1
λt zt+1
λt+1 K
rN,t+1 + (1 − δ)qN,t+1
0 = −qN,t + Et β
λt zt+1
λt+1 K
rX,t+1 + (1 − δ)qX,t+1
0 = −qX,t + Et β
λt zt+1
jH,t zt (1 + zI ) jH,t zt (1 + zI )
jH,t zt (1 + zI )
0 = −τtI + qH,t vt 1 − Υ
− Υ0
jH,t−1
jH,t−1
jH,t−1
(
2 )
λt+1 qH,t+1 vt+1 0 jH,t+1 zt+1 (1 + zI )
jH,t+1 zt+1 (1 + zI )
Υ
+ Et β
λt zt+1 (1 + zI )
jH,t
jH,t
jN,t zt (1 + zI )
jN,t zt (1 + zI ) jN,t zt (1 + zI )
I
0
0 = −τt + qN,t vt 1 − Υ
−Υ
jN,t−1
jN,t−1
jN,t−1
(
2 )
jN,t+1 zt+1 (1 + zI )
λt+1 qN,t+1 vt+1 0 jN,t+1 zt+1 (1 + zI )
Υ
+ Et β
λt zt+1 (1 + zI )
jN,t
jN,t
jX,t zt (1 + zI )
jX,t zt (1 + zI ) jX,t zt (1 + zI )
0 = −τtI + qX,t vt 1 − Υ
− Υ0
jX,t−1
jX,t−1
jX,t−1
(
2 )
λt+1 qX,t+1 vt+1 0 jX,t+1 zt+1 (1 + zI )
jX,t+1 zt+1 (1 + zI )
Υ
+ Et β
λt zt+1 (1 + zI )
jX,t
jX,t
0=
(79)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(87)
and
ν−ω
Lω
H,t + λt wH,t
(88)
ν−ω
Lω
N,t
+ λt wN,t
(89)
ν−ω
Lω
X,t + λt wX,t
(90)
0 = −ζt ξL
t (Lt )
0=
−ζt ξL
t
(Lt )
0 = −ζt ξL
t (Lt )
12
Capital accumulation:
jH,t
0 = kH,t+1 Et {zt+1 } − (1 − δ) kH,t − vt 1 − Υ
zt (1 + zI ) jH,t
jH,t−1
jN,t
0 = kN,t+1 Et {zt+1 } − (1 − δ) kN,t − vt 1 − Υ
zt (1 + zI ) jN,t
jN,t−1
jX,t
0 = kX,t+1 Et {zt+1 } − (1 − δ) kX,t − vt 1 − Υ
zt (1 + zI ) jX,t
jX,t−1
(91)
(92)
(93)
Price and inflation indices:
h
i 1
1−η
1−η 1−η
πt = γN (πN,t τN,t−1 (1 + zN ))
+ γT (πT,t τT,t−1 (1 + zT ))
γH γF
πF,t
πT,t = πH,t
I
πtI = πT,t
I
γH
I
γT
(94)
(95)
I
γN
πN,t
(96)
I
γF
I
πT,t
= πH,t πF,t
(97)
Consumer demand:
−η
cN,t = γN (τN,t )
−η
cT,t = γT (τT,t )
ct
(98)
ct
(99)
−1
cH,t = γH γT (τH,t )
−1
cF,t = γF γT (τF,t )
1−η
(τT,t )
(τT,t )
1−η
ct
(100)
ct
(101)
Investment demand:
iN,t
iT,t
iH,t
iF,t
−1
τN,t
=
it
τI,t
!−1
I
τT,t
I
it
= γT
τI,t
!−1
!−1
I
τT,t
τH,t
I I
= γH γT
it
I
τI,t
τT,t
!−1
!−1
I
τT,t
τF,t
I I
= γF γT
it
I
τI,t
τT,t
I
γN
(102)
(103)
(104)
(105)
Production:
αX 1−αX
yX,t = at Z̃X,t kX,t
lX,t
(106)
αX 1−αH
yH,t = at Z̃H,t kH,t
lH,t
(107)
αN 1−αN
at Z̃N,t kN,t
lN,t
(108)
yN,t =
13
Tradeable firms:
αH wH,t lH,t
K
1 − αH rH,t
1−αH αH 1−αH K αH
wH,t rH,t
1
1
0 = mcH,t −
1 − αH
αH
τH,t Z̃H,t
πH,t πH,t
θH − 1
θH
λt+1 yH,t+1 τH,t+1 πH,t+1 πH,t+1
0=
−1 +
−
mcH,t − Et β
−1
πH
πH
ψH
ψH
λt yH,t τH,t
πH
πH
0 = kH,t −
(109)
(110)
(111)
Non-tradeable firms:
αN wN,t lN,t
K
1 − αN rN,t
1−αN αN 1−αN K αN
wN,t rN,t
1
1
0 = mcN,t −
1 − αN
αN
τN,t Z̃N,t
πN,t πN,t
θN − 1
θN
λt+1 yN,t+1 τN,t+1 πN,t+1 πN,t+1
0=
−1 +
−
mcN,t − Et β
−1
πN
πN
ψN
ψN
λt yN,t τN,t
πN
πN
0 = kN,t −
(112)
(113)
(114)
Commodity firms:
τX,t yX,t
K
− rX,t
kX,t
τX,t yX,t
− wX,t
0 = (1 − αX )
lX,t
0 = αX
(115)
(116)
Importing firms
πF,t πF,t
θF − 1
θF
λt+1 yF,t+1 τF,t+1 πF,t+1 πF,t+1
0=
−1 +
−
mcF,t − Et β
−1
πF
πF
ψF
ψF
λt yF,t τF,t
πF
πF
τF ∗ ,t
mcF,t = ς
τF,t
(117)
(118)
Law of one price:
0 = τX,t − κt τF ∗ ,t
14
(119)
Relative Prices:
τN,t
πN,t (1 + zN )
τN,t−1
πt
τT,t
πT,t (1 + zT )
0=
−
τT,t−1
πt
0=
0=
I
τT,t
I
τT,t−1
−
−
I
πT,t
(1 + zTI )
πt
τH,t
πH,t (1 + zH )
−
τH,t−1
πt
τF,t
πF,t (1 + z ∗ )
0=
−
τF,t−1
πt
τF ∗ ,t
∆st πt∗ (1 + z ∗ )
0=
−
τF ∗ ,t−1
πt
τI,t
πI,t (1 + zI )
−
0=
τI,t−1
πt
0=
Foreign sector:
c∗H,t
=
∗
γH
τH,t
τF ∗ ,t
−η∗
Ỹt∗
(120)
(121)
(122)
(123)
(124)
(125)
(126)
(127)
Market clearing:
2
ψN πN,t
yN,t
−
1
2
π̄ N
2
ψH πH,t
0 = yH,t − cH,t − c∗H,t − iH,t −
−
1
yH,t
2
π̄ H
2
ψF πF,t
0 = yF,t − cF,t − iF,t −
− 1 yF,t
F
2
π
0 = it − jn,t − jh,t − jx,t
h
i 1
1+ω
1+ω
1+ω 1+ω
0 = lt − lH,t
+ lN,t
+ lX,t
0 = yN,t − cN,t − iN,t −
(128)
(129)
(130)
(131)
(132)
0 = gdpt − τH,t yH,t − τN,t yN,t − τX,t yX,t
c∗H,t
yX,t
cF,t − iF,t
0 = nxt − τH,t
+ τF,t
+ τX,t
gdpt
gdpt
gdpt
∗ ∗
b
r
gdp
t−1
− nxt
0 = b∗t+1 − t t
πt gdpt zt
(133)
1 + rtF = (1 + rt∗ ) exp(−ψb (b∗t − b∗ ) + ψ̃t )
ρ "
φy #(1−ρR )
1 + rt
1 + rt−1 R πt φπ yt zt
=
exp(uR,t )
1+r
1+r
Π
yt−1 z̄
(136)
(134)
(135)
Interest rates and monetary policy:
4
Steady State
Household optimisation:
15
(137)
z − hβ
− cλ
(z − h)
πz
0=1+r−
β
1+r
0=s−
1 + r∗
β K
0 = −qH +
r + (1 − δ)qH
z H
β K
0 = −qN +
rN + (1 − δ)qN
z
β K
rX + (1 − δ)qX
0 = −qX +
z
0 = −τ I + qH v
0=
(138)
(139)
(140)
(141)
(142)
(143)
(144)
0 = −τ I + qN v
(145)
I
(146)
0 = −τ + qX v
and
ω
0 = −ξlν−ω lH
+ λwH
0=
ω
−ξlν−ω lN
(147)
+ λwN
(148)
ω
0 = −ξlν−ω lX
+ λwX
(149)
0 = kH (z − 1 + δ) − vjH
(150)
0 = kN (z − 1 + δ) − vjN
(151)
0 = kX (z − 1 + δ) − vjX
(152)
Capital accumulation:
(153)
Inflation indicies:
h
i 1
1−η
1−η 1−η
π = γN (πN τN (1 + zN ))
+ γT (πT τT (1 + zT ))
γ
γ
I
γH
I
γF
πT τT = (πH τH ) H (πF τF ) F
γ I
γI
π I τI = πTI τTI H (πN τN ) N
πTI τTI = (πH τH )
(154)
(155)
(156)
(πF τF )
(157)
16
Consumer demand:
−η
cN = γN (τN )
cT = γT (τT )
−η
c
(158)
c
(159)
−1
cH = γH γT (τH )
cF = γF γT (τF )
−1
1−η
c
(160)
1−η
c
(161)
(τT )
(τT )
Investment demand:
−1
τN
i
iN =
τI
I −1
τT
iT = γTI
i
τI
−1
τH
I
iT
iH = γH
τTI
−1
τF
iF = γFI
iT
τTI
I
γN
(162)
(163)
(164)
(165)
Production:
α 1−α
yX = kX
lX
(166)
α 1−α
kH
lH
(167)
α 1−α
yN = kN
lN
(168)
yH =
Tradeable firms:
αH wH lH
K
1 − αH rH
1−αH αH 1−αH K αH
wH
rH
1
1
0 = mcH −
1 − αH
αH
τH
θH − 1
0=
− mcH,t
θH
0 = kH −
(169)
(170)
(171)
Non-tradeable firms:
αN wN lN
K
1 − αN rN
1−αN αN 1−αN K αN
wN
rN
1
1
0 = mcN −
1 − αN
αN
τN
θN − 1
0=
− mcN
θN
0 = kN −
(172)
(173)
(174)
Commodity firms:
τX yX
K
− rX
kX
τX yX
0 = (1 − αX )
− wX
lX
0 = αX
17
(175)
(176)
Importing firms
θF − 1
− mcF
θF
τF ∗
0 = mcF − ς
τF
0=
(177)
(178)
Law of one price:
0 = τX − κτF ∗
(179)
Relative Prices:
πN (1 + zN )
π
πT (1 + zT )
π
πTI (1 + zTI )
π
πH (1 + zH )
π
∆sπ ∗ (1 + z ∗ )
π
πF (1 + z ∗ )
π
πI (1 + zI )
π
0=1−
0=1−
0=1−
0=1−
0=1−
0=1−
0=1−
(180)
(181)
(182)
(183)
(184)
(185)
(186)
Market clearing:
0 = yN − cN − iN
(187)
0 = yH − cH − c∗H − iH
(188)
0 = y F − cF − i F
(189)
0 = i − jh − jn − jx
1+ω
1
1+ω
1+ω 1+ω
0 = l − lH
+ lN
+ lX
(190)
0 = gdp − τH yH − τN yN − τX yX
c∗
yF
yX
0 = nx − τH H − τF
+ τX
gdp
gdp
gdp
b∗ r∗
0 = b∗ −
− nx
πz
(192)
(191)
(193)
(194)
Miscellaneous equations:
b = b∗
(195)
π=Π
(196)
18
5
The Posterior Sampler
To simulate from the joint posterior of the structural parameters and the date breaks, p(ϑ, T|Y ), we use the MetropolisHastings algorithm following a strategy similar to Kulish et al. [2014]. As we have continuous and discrete parameters we
modify the standard setup for Bayesian estimation of DSGE models. We separate the parameters into two blocks: date
breaks and structural parameters. To be clear, though, the sampler delivers draws from the joint posterior of both sets of
parameters.
The first block of the sampler is for the date breaks, T. As is common in the literature on structural breaks (Bai and Perron
[1998]), we set the trimming parameter to 25 per cent of the sample size so that the minimum length of a segment has 20
observation. Within the feasible range we draw from a uniform proposal density and randomize which particular date break
in T to update. This approach is motivated by the randomized blocking scheme developed for DSGE models in Chib and
Ramamurthy [2010].
The algorithm for drawing for the date breaks block is as follows: Initial values of the date breaks, T0 , and the structural
parameters, ϑ0 , are set. Then, for the j th iteration, we proceed as follows:
1. randomly sample which date break to update from a discrete uniform distribution with support ranging from one to
the total number of breaks, in our case two.
0
2. randomly sample the corresponding elements of the proposed date breaks, Tj , from a discrete uniform distribution
[Tmin , Tmax ] and set the remaining elements to their values in Tj−1
0
3. calculate the acceptance ratio αjT ≡
p(ϑj−1 ,Tj |Y)
p(ϑj−1 ,Tj−1 |Y)
0
4. accept the proposal with probability min{αjT , 1}, setting Tj = Tj , or Tj−1 otherwise.
The second block of the sampler is for the nϑ structural parameters.1 It follows a similar strategy to the date-breaks-block
described above - we randomize over the number and which parameters to possibly update at each iteration. The proposal
density is a multivariate Student’s t− distribution.2 Once again, for the j th iteration we proceed as follows:
1. randomly sample the number of parameters to update from a discrete uniform distribution [1, nϑ ]
2. randomly sample without replacement which parameters to update from a discrete uniform distribution [1, nϑ ]
0
3. construct the proposed ϑj by drawing the parameters to update from a multivariate Student’s t− distribution with 10
degrees of freedom and with location set at the corresponding elements of ϑj−1 , scale matrix based on the corresponding
elements of the negative inverse Hessian at the posterior mode multiplied by a tuning parameter ι = 0.15 .
0
4. calculate the acceptance ratio αjϑ ≡
p(ϑj ,Tj |Y)
p(θj−1 ,Tj |Y)
0
or set αjϑ = 0 if the proposed ϑj includes inadmissible values (e.g. a
proposed negative value for the standard deviation of a shock or autoregressive parameters above unity) preventing
0
calculation of p(ϑj , Tj |Y )
0
5. accept the proposal with probability min{αjϑ , 1}, setting ϑj = ϑj , or ϑj−1 otherwise.
We use this multi-block algorithm to construct a chain of 575,000 draws from the joint posterior, p(ϑ, T|Y ), throwing out
the first 25 per cent as burn-in. Trace plots show that the sampler mixes well.
1 In
our application nϑ = 37.
hessian of the proposal density is computed at the mode of the structural parameters.
2 The
19
References
Jushan Bai and Pierre Perron. Estimating and Testing Linear Models with Multiple Structural Changes. Econometrica, 66:
47–78, 1998.
Siddhartha Chib and Srikanth Ramamurthy. Tailored Randomized Block MCMC Methods With Application to DSGE
Models. Journal of Econometrics, 155:19–38, 2010.
Mariano Kulish, James Morley, and Tim Robinson. Estimating the Expected Duration of the Zero Lower Bound in DSGE
Models with Forward Guidance. UNSW Business School Research Paper 2014-32, UNSW, 2014.
Julio J. Rotemberg. Monopolistic Price Adjustment and Aggregate Output. Review of Economic Studies, 49:517–531, 1982.
20