Derivation of DSGE Model with Financial
Frictions by Mohamad Hasni Shaari
Using the dissertation of Mohamad Hasni Shaari, written by me. Sometimes
I almost copy word by word, cause the original text is concise and hard to
simplify.
1 Definitions
Lets start with definitions. consumption basket:
Ct = (1 − γ)
1
η (CH,t)
η−1
η + γ
1
η
(CF,t)
η−1
η
η
η−1
γ measures openness, H and F denote home and foreign goods, η is the
elasticity of substitution.
Both domestic and foreign goods are given by
CH,t =
1
0
CH,t(i)
ε−1
ε di
ε
ε−1
so we get demands
CH,t(i) =
PH,t(i)
PH,t
−ε
CH,t
Note that we express both PH and PF in domestic currency.
We also get demands
CH,t = (1 − γ)
PH,t
Pt
−η
Ct, CF,t = γ
PF,t
Pt
−η
Ct
and price index
Pt = (1 − γ)P1−η
H,t + γP1−η
F,t
1
1−η
, pt = (1 − γ)pH,t + γpF,t.
Now about inflation:
πt = (1 − γ)πH,t + γπF,t
1
Define terms of trade as
TOTt =
PF,t
PH,t
so that
pt = (1 − γ)pH,t + γpF,t = pH,t + γtott, πt = πH,t + γ∆tott.
We have St the nominal exchange rate, increase = depreciation, and RERt =
St
P∗
t
Pt
. We assume incomplete exchange rate pass-through, so that LOPGt =
StP∗
t
PF,t
. For estimation, LOPG is assumed to follow AR(1) process (strange...).
But anyway, we can express the log of RER as
rert = st+p∗
t −pt = st+p∗
t −pH,t−γtott = st+p∗
t −pF,t+pF,t−pH,t−γtott = lopgt+(1−γ)tott
While law of one price does not hold for imports, it holds for exports, so that
P∗
H,t = PH,t/St.
2 Households
Households maximize discounted future utility given by
U(Ct, Lt) = log(Ct − hCt−1) −
L1+Ψ
H,t
1 + Ψ
where Ct is consumption, LH,t is labor supply by household and h is the
parameter of external habit. Ψ is inverse elasticity of labor supply.
Budge constraint
WH,tLH,t+Rt−1Dt−1+R∗
t−1ΨB
(Zt−1, AUIP
t−1 )StBt−1+Πt+Tt = PtCt+Dt+StBt
means that HH gets income from labor LH,t and nominal wage WH,t. HH
gets profits from retailer Πt and left-over equity from entrepreneurs who die
and leave economy Tt. HH can buy two assets: domestic Dt from domestic
intermediary and foreign (denominated in foreign currency) which gives riskadjusted
return R∗
t ΨB
(Zt, AUIP
t ). Risk-adjustment is there to stationarise
the model, so we specify
ΨB
(Zt, AUIP
t ) = exp −ψB
(Zt + AUIP
t ) .
Here Zt = StBt
YtPt
is the net foreign asset position of the domestic economy and
AUIP
t is a shock assumed to follow AR(1) process.
HH chooses quadruple {Ct, LH,t, Dt, Bt} and we get following FOCs
2
• wrt to LH,t:
WH,t = −
LΨ
H,t
λt
• wrt to Ct:
1
Ct − hCt−1
= −λtPt
• wrt to Bt:
λtSt = βλt+1R∗
t ΨB
(Zt, AUIP
t )St+1
• wrt to Dt:
λt = βλt+1Rt
Now combine first two to get labor supply:
WH,t
Pt
= WH,t = Lψ
H,t(Ct − hCt−1).
Now we combine the second and fourth (HH decides if to consume or to
invest in domestic bonds)
βRt =
Ct+1 − hCt
Ct − hCt−1
Pt+1
Pt
.
Similarly, HH decides if to consume or invest in foreign bonds:
R∗
t ΨB
(Zt, AUIP
t ) =
1
β
St
St+1
(Ct − hCt−1)
(Ct+1 − hCt)
Pt
Pt+1
The latter two equations define optimal choice between foreign and domestic
bonds and therefore imply UIP. Combine to get:
R∗
t ΨB
(Zt, AUIP
t ) =
St
St+1
Rt
R∗
t exp −ψB
(Zt + AUIP
t ) = Rt
RERtPt
P∗
t
P∗
t+1
Pt+1RERt+1
When log linearized, these equations become:
lH,t =
1
Ψ
wH,t −
1
1 − h
(ct − hct−1)
(1 − h)(rt − Etπt+1) = (ct+1 − hct) − (ct − hct−1)
rert+1 − rert = (rt − Etπt+1) − (rt ∗ −Etπ∗
t+1) + ψB
zt + AUIP
t
3
3 Entrepreneurs
We add capital and entrepreneurs. Entrepreneurs produce intermediate
goods and capital goods. They also own all capital. We want them to
face a financing constraint, so we need to prevent them from living infinitely
and accumulating enough new worth so that the constraint wouldn’t matter.
We let a fraction of (1 − ξ) of entrepreneurs die every period.
Entrepreneurs produce intermediate (wholesale) goods YH,t and seel them for
price PW
H,t. They use both HH and own labor
Lt = LΩ
H,tL1−Ω
E,t ,
but we normalize LE,t to one for simplicity. HH is paid WH,t for labor unit,
entrepreneur is paid WE,t. The gross nominal return on capital is RG,t.
They employ production function
YH,t = AY
t Kα
t L
(1−α)Ω
H,t ,
where AY
t is productivity common for all firms assumed to follow AR(1)
process. Log-linearize to get
yH,t = αkt + (Ω(1 − α))lH,t + AY
t .
Entrepreneurs every period minimize costs
RG,tKt + LH,tWH,t + WE,t
s.t. the production function. We get following FOCs:
RG,t = λα
YH,t
Kt
WH,t = λΩ(1 − α)
YH,t
LH,t
WE,t = λ(1 − Ω)(1 − α)
YH,t
LE,t
Realize that λ is the nominal marginal cost of producing one more unit of
output, which is equal to PW
H,t (no profit), and that LE,t is 1, and we get the
4
demand schedules:
RG,t = PW
H,tα
YH,t
Kt
WH,t = PW
H,tΩ(1 − α)
YH,t
LH,t
WE,t = PW
H,t(1 − Ω)(1 − α)YH,t
We will manipulate a bit to show how the LOPG and RER influence this.
Define MCH,t =
PW
H,t
PH,t
to be the real marginal costs expressed in terms of the
domestic goods price level. Next, divide all by Pt to get real variables. We
get (also log-lin)
RG,t
Pt
= RG,t = α
YH,t
Kt
MCH,t
PH,t
Pt
rG,t = yH,t + mcH,t − kt −
γ
1 − γ
(rert − lopgt)
WH,t
Pt
= WH,t = Ω(1 − α)
YH,t
LH,t
MCH,t
PH,t
Pt
wH,t = yH,t + mcH,t − lH,t −
γ
1 − γ
(rert − lopgt)
WE,t
Pt
= WE,t = (1 − Ω)(1 − α)YH,tMCH,t
PH,t
Pt
wE,t = yH,t + mcH,t −
γ
1 − γ
(rert − lopgt)
where we use the fact that
pH,t − pt = γtott
rert = (1 − γ)tott + lopgt
tott =
rert − lopgt
1 − γ
To get the expression for the real marginal costs for domestic prduced goods,
just plug equations for lH,t and kt into prioduction function to get:
mcH,t =
(1 − α)(1 + Ω)
α + (1 − α)Ω
yH,t +
1
α + (1 − α)Ω
[αrG,t + (1 − α)wH,t]
+
1
α + (1 − α)Ω
γ
1 − γ
(rert − lopgt) −
1
α + (1 − α)Ω
AY
t
5
Interestingly, depreciation of RER increases MC, while larger LOPG decreases
it.
4 Investment
Entrepreneurs produce capital and sell it for nominal price Qt. Capital is
produced using old capital and investment INVt, which is produced exactly
as consumption goods
INVt = (1 − γ)
1
η (CH,t)
η−1
η + γ
1
η
(CF,t)
η−1
η
η
η−1
so that demand functions of the entrepreneurs are exactly the same as HH’s.
Also, price of the investment is Pt and investment price index is equal to
CPI. Capital accumulates according to
Kt+1 = Φ
INVt
Kt
Kt + (1 − δ)Kt.
Φ is increasing concave, stands for adjustment costs:
Φ
INVt
Kt
=
INVt
Kt
−
ψI
2
INVt
Kt
− δ
2
.
In steady state, we have Φ(SS) = δ, Φ (SS) = 1. The first condition means
that in steady state capital stock doesn’t change, the second one ensures that
the real price of capital is equal to one is SS. In log-lin:
¯K(1 + kt+1) = ¯K + Φ ¯K
¯I
¯K
invt + (1 − δ) ¯Kkt + Φ ¯Kkt − Φ ¯K
¯I
¯K2
¯Kkt
¯Kkt+1 = ¯Kδinvt + (1 − δ) ¯Kkt + δ ¯Kkt − δ ¯Kkt
kt+1 = δinvt + (1 − δ)kt
When deciding how much capital to produce, the entrepreneur solves
max
INVt
QtΦ
INVt
Kt
Kt − PtINVt
6
which gives FOC:
QtΦ
INVt
Kt
= Pt
Qt =
1
Φ INVt
Kt
Lets log-linearize, cause I don’t find it straightforward at all:
Q(1 + qt) =
1
Φ INV
K
+ (−1)
1
Φ INV
K
2 Φ
INV
K
INV
K
invt +
+(−1)
1
Φ INV
K
2 Φ
INV
K
(−1)
INV
K2
Kkt
Qqt =
1
Φ INV
K
2 Φ
INV
K
INV
K2
Kkt +
(−1)
1
Φ INV
K
2 Φ
INV
K
INV
K
invt : Q =
1
Φ INV
K
qt =
−Φ INV
K
Φ INV
K
INV
K
(invt − kt)
Now RK,t =
{RG,t+(1−δ)Qt}Kt
Qt−1Kt
is the real gross return on capital received by
the entrepreneurs. In log-lin:
RK(1 + rK,t) =
[RG + (1 − δ)Q] K
QK
+
RGK
QK
rG,t + (1 − δ)
QK
QK
qt −
[RG + (1 − δ)Q] K
(QK)2
QKq−1
rK,t + qt−1 =
RG
QRK
rG,t +
1 − δ
RK
qt =
RG
RG + (1 − δ)Q
rG,t +
1 − δ
RK
qt =
= (1 −
(1 − δ)Q
RG + (1 − δ)Q
)rG,t +
1 − δ
RK
qt = 1 −
1 − δ
RK
rG,t +
1 − δ
RK
qt
rK,t + qt−1 = 1 −
1 − δ
RK
rG,t +
1 − δ
RK
qt
So two things determine the return on investment by entrepreneurs. First,
capital is used by intermediate (wholesale firms), which pay rental rate rG,t.
Second, since the entrepreneurs own the capital and rent it, any change in
the price of capital influences the return on investment. It also influences the
entrepreneur’s net worth.
7
4.1 Frictions and Net Worth
Entrepreneurs (E) finance their production operations and owning of capital
using their net worth Nt+1 and financing from intermediary Ft+1. So their
budget constraint is
QtKt+1 = Ft+1 + Nt+1.
When borrowing from financial intermediary, E pays not only the gross real
interest rate Rt
Pt
Pt+1
, but also a premium dependent on the leverage ratio of
the E. BGG explain this using principal-agent problem. The premium is
given by
Premium =
Nt+1
QtKt+1
−χ
,
where χ measures the elasticity of the premium. BGG provide detailed explanation
for why the premium should be in such a relation with leverage
ratio, but I find it natural.
E are risk-neutral and choose Kt+1 to maximize profit. Chosen Kt+1 implies
necessary Ft+1. On the optimal margin, expected marginal return on
investment is equal to marginal financing cost:
Et(RK,t+1) = Et
Nt+1
QtKt+1
−χ
Rt
Pt
Pt+1
which again in log-lin is
EtrK,t+1 = rt − πt+1 − χ(nt+1 − qt − kt+1).
Now we need to determine the evolution of E’s net worth. The new net worth
consists of entrepreneurial equity held by the fraction ξ of Es that survive
this period and E labor income WE,t:
Nt+1 = ξVt + WE,t.
The remaining Es who leave the economy transfer their wealth to HHs Tt =
(1−ξ)Vt. We assume that labor income of Es is small, so that (1−Ω) = 0.01.
This mechanism only ensures that net worth is pinned down in steady state.
Equity is given by
Vt = RK,tQt−1Kt −
Nt
Qt−1Kt
−χ
Rt−1
Pt−1
Pt
Ft.
8
So the equity is the return minus the repayment of loans. Note that an
increase in interest rate lowers E’s net worth, which increases the premium
and further lowers the net worth. Before log-linearizing, realize that
RK =
N
QK
−χ
R
P
P
Q =
1
Φ INV
K
= 1
Nt+1 = ξVt + WE,t
V vt =
Nnt+1 − WEwE,t
ξ
Ft = Qt−1Kt − Nt
ft =
K(qt−1 + kt) − Nnt
F
and log-linearize (I wont bother with writing the first term of Taylor expansion,
I mean the one equal to the equation in the steady state, since it
immediately cancels out on both sides):
Nnt+1 − WEwE,t
ξ
= RKQK(rK,t + qt−1 + kt) + χ
N
QK
−χ−1
N
QK
R(QK − N)nt
−χ
N
QK
−χ−1
N
(QK)2
QKR(QK − N) −
N
QK
−χ
RQK (qt−1 + kt)
−
N
QK
−χ
R(QK − N)(rt − πt) +
N
QK
−χ
RNnt
Now we make use of
RK =
N
QK
−χ
R
and Q = 1 to get
Nnt+1 = WEwE,t + ξRKK(rK,t + qt−1 + kt) + ξχRK(K − N)nt −
−ξ [χRK(K − N) + RKK] (qt−1 + kt) − ξRK(K − N)(rt−1 − πt) + ξRKNnt
Nnt+1 = ξRK [K(rK,t + qt−1 + kt) + χ(K − N)nt − χ(K − N)(qt−1 + kt) − K(qt−1 + kt)] +
+ξRK [−(K − N)(rt−1 − πt) + Nnt] + WEwE,t
9
Denote Γ5 = K
N
− 1 to get:
K − N
N
= Γ5
K
N
= Γ5 + 1
WE
N
=
WE
K
K
N
= (Γ5 + 1)
WE
K
Now collect terms and divide by N:
nt+1 = ξRK [(Γ5 + 1)rK,t − χΓ5(qt−1 + kt) + (χΓ5 + 1)nt − Γ5(rt−1 − πt)] + (Γ5 + 1)
WE
K
wE,t
5 Retailers
We assume Calvo pricing on final goods market. There are importers and
domestic firms, that sell on the domestic market and export to the foreign
economy. Retailers buy the consumption good for price PW
H,t. Each period,
fraction (1−θH) resets its price to new optimal price PNEW
H,t . The rest update
their price according to
PH,t(z) = PH,t−1(z)(πt−1)κ
,
where κ measures the degree of inflation indexation. This implies aggregate
price level
PH,t = (1 − θH) PNEW
H,t
1−ε
+ θH (PH,t−1πt−1)1−ε
1
1−ε
.
Let YH,t(z) be the composite good sold by a retailer z in period t. The
aggregate good sold is given by
YH,t =
1
0
YH,t(z)
ε−1
ε dz
ε
ε−1
Expected future demands by households are given by
YH,t+k(z) =
PNEW
H,t
PH,t+k
(πt−1,t+k)κ
YH,t+k
10
The representative firm maximizes
max
PNEW
H,t
Et
∞
k=0
βk
θk
H YH,t+k(z) PNEW
H,t (πt−1,t+k)κ
− PH,t+k
PW
H,t+k
PH,t+k
where note that
PW
H,t+k
PH,t+k
= MCH,t+k.
FOC is
∞
k=0
(βθH)k
EtYH,t+k PNEW
H,t (πt−1,t+k)κ
−
ε
ε − 1
PH,t+kMCH,t+k = 0
It is good to use this FOC to write the optimal price as
PNEW
H,t = µ
(βθH)k
EtYH,t+k [PH,t+kMCt+k]
(βθH)kEtYH,t+k [(πt+k−1)κ]
.
where µ = ε
ε−1
is the gross desired markup.
Log-linearize the equation above to get
pNEW
H,t = (1 − βθH)(pH,t + mcH,t) + (βθH) Et{pNEW
H,t+1 − κπt−1}
where mcH,t+k = pW
H,t+k − pH,t+k. Log-linearize the price index equation to
get
πH,t = (1 − θH) pNEW
H,t − pH,t−1 + θHκπt−1
Substitute and rearrange to get
πH,t =
1
1 + βκ
βEt{πH,t+1} + κπt−1 + ΛH
mcH,t) , ΛH
=
(1 − βθH)(1 − θH)
θH
.
6 Importers
Importers buy for price PW
F,t = StP∗
t (in local currency). They sell at price
PF,t. We assume PF,t = StP∗
t . Again, importers price a la Calvo, so the price
index for imported goods is
PF,t = (1 − θF )(PNEW
F,t )1−ε
+ θF (PF,t−1(πt−1)κ
)1−ε
1
1−ε
.
11
Again importers solve the problem
max
PNEW
F,t
∞
k=0
(βθF )k
Et YF,t+k(z) PNEW
F,t (πκ
t+k−1) − PF,t+k
PW
F,t
PF,t+k
which results in
PNEW
F,t = µ
∞
k=0(βθF )k
Et(YF,t+k[PF,t+kMCF,t+k])
∞
k=0(βθF )kEt(YF,t+k(πt+k−1)κ
which in log-lin becomes
pNEW
F,t = (1 − βθF )[pF,t − mcF,t] + (βθF ) Et{pNEW
F,t+1 } − κπt−1
where mcF,t = pW
F,t − pF,t. By definition pW
F,t = st + p∗
t , so
mcF,t = st + p∗
t − pF,t = lopgt.
Log-linearize the price index for imported goods and combine to get
πF,t =
1
1 + βκ
βEt{πF,t+1} + κπt−1 + ΛF
lopgt , ΛF
=
(1 − βθF )(1 − θF )
θF
Finally, realize that overall CPI inflation
πt = (1 − γ)πH,t + γπF,t
whi after plugging gives
πt =
1
1 + βκ
β{πt+1} + κπt−1 + (1 − γ)ΛH
mcH,t + γΛF
lopgt
7 Monetary policy
Interest rate is given by
rt = (1 − ρ) [βππt+1 + Θyyt+1] + ρrt−1 + εMP
t .
12
8 Market clearing and equilibrium
Foreing agents are assumed to have identical preferences to the domestic
agents, so the foreign demand for domestic exports is given by
C∗
H,t = γ
P∗
H,t
P∗
t
−η
Y ∗
t
We assume that law of one price holds for exports, so P∗
H,t =
PH,t
St
, we know
that RERt = St
P∗
t
Pt
and we have
C∗
H,t = γ
PH,t
Pt
−η
1
RERt
−η
Y ∗
t .
In every period, final good YH,t is either consumed, exported or invested, so
we get aggregate resource constraint
YH,t =
PH,t
Pt
−η
(1 − γ)(Ct + INVt) + γ
1
RERt
−η
Y ∗
t .
In log-lin
yH,t =
C
YH
(1 − γ)ct +
INV
YH
(1 − γ)invt + γy∗
t + ηγ
2 − γ
1 − γ
rert −
ηγ
1 − γ
lopgt.
Financial intermediary lends to entrepreneurs. To finance itself, it collects
funds from households at cost Rt. We assume zero-profit banking and borrowing
only from domestic HHs. So in equilibrium
Ft = Dt.
9 Net foreign assets
Eveolution of net foreign assets is
Zt = R∗
t−1ΨB
(Zt−1, AUIP
t−1 )Zt−1 + (YH,t − Ct − INVt)
where Zt = StBt
YtPt
is the economy share of net foreign assets on NGDP. The
term (YH,t − Ct − INVt) stands for current account balance. In log-lin, we
get
zt =
1
β
zt−1 + (yH,t − ct − invt) −
γ
1 − γ
(rert − γlopgt)
13
10 Log-linearized equations
The model consists of following set of equations:
lH,t =
1
Ψ
wH,t −
1
1 − h
(ct − hct−1) (1)
(1 − h)(rt − Etπt+1) = (ct+1 − hct) − (ct − hct−1) (2)
rert+1 − rert = (rt − Etπt+1) − (rt ∗ −Etπ∗
t+1) + ψB
zt + AUIP
t (3)
rG,t = yH,t + mcH,t − kt −
γ
1 − γ
(rert − lopgt) (4)
wH,t = yH,t + mcH,t − lH,t −
γ
1 − γ
(rert − lopgt) (5)
wE,t = yH,t + mcH,t −
γ
1 − γ
(rert − lopgt) (6)
yH,t = αkt + (Ω(1 − α))lH,t + AY
t (7)
kt+1 = δinvt + (1 − δ)kt (8)
qt =
−Φ INV
K
Φ INV
K
INV
K
(invt − kt) (9)
rK,t + qt−1 = 1 −
1 − δ
RK
rG,t +
1 − δ
RK
qt (10)
EtrK,t+1 = rt − πt+1 − χ(nt+1 − qt − kt+1) (11)
nt+1 = ξRK [(Γ5 + 1)rK,t − χΓ5(qt−1 + kt)] + (12)
+ξRK [(χΓ5 + 1)nt − Γ5(rt−1 − πt)] + (Γ5 + 1)
WE
K
wE,t (13)
πt =
1
1 + βκ
β{πt+1} + κπt−1 + (1 − γ)ΛH
mcH,t + γΛF
lopgt (14)
rt = (1 − ρ) [βππt+1 + Θyyt+1] + ρrt−1 + εMP
t (15)
yH,t =
C
YH
(1 − γ)ct +
INV
YH
(1 − γ)invt + γy∗
t + ηγ
2 − γ
1 − γ
rert −
ηγ
1 − γ
lopgt(16)
zt =
1
β
zt−1 + (yH,t − ct − invt) −
γ
1 − γ
(rert − γlopgt) (17)
lopgt = ρlopg ∗ lopgt−1 + εlopg
t (18)
14
11 Foreign economy
We can either introduce foreign economy as VAR(1) process or try to do it
structurally. The equations describing the foreign block are
l∗
H,t =
1
Ψ
w∗
H,t −
1
1 − h
(c∗
t − hc∗
t−1) (19)
(1 − h)(r∗
t − Etπ∗
t+1) = (c∗
t+1 − hc∗
t ) − (c∗
t − hc∗
t−1) (20)
r∗
G,t = y∗
t + mc∗
t − k∗
t (21)
w∗
H,t = y∗
t + mc∗
t − l∗
H,t (22)
w∗
E,t = y∗
t + mc∗
t (23)
y∗
H,t = αk∗
t + (Ω(1 − α))l∗
H,t + AY ∗
t (24)
k∗
t+1 = δinv∗
t + (1 − δ)k∗
t (25)
q∗
t =
−Φ INV ∗
K∗
Φ INV ∗
K∗
INV ∗
K∗
(inv∗
t − k∗
t ) (26)
r∗
K,t + q∗
t−1 = 1 −
1 − δ
R∗
K
r∗
G,t +
1 − δ
R∗
K
q∗
t (27)
Etr∗
K,t+1 = r∗
t − π∗
t+1 − χ(n∗
t+1 − q∗
t − k∗
t+1) (28)
n∗
t+1 = ξR∗
K (Γ5 + 1)r∗
K,t − χΓ5(q∗
t−1 + k∗
t ) + (29)
+ξR∗
K (χΓ5 + 1)n∗
t − Γ5(r∗
t−1 − π∗
t ) + (Γ5 + 1)
W∗
E
K∗
w∗
E,t(30)
π∗
t =
1
1 + βκ
β{π∗
t+1} + κπ∗
t−1 + Λ∗
mc∗
t (31)
r∗
t = (1 − ρ) βπ∗ π∗
t+1 + Θyy∗
t+1 + ρr∗
t−1 + εMP∗
t (32)
y∗
t =
C∗
Y ∗
c∗
t +
INV ∗
Y ∗
inv∗
t (33)
12 Model without financial frictions
Lets now remove financial frictions. We will assume that entrepreneurs have
enough new worth to be able to cover the cost of their operations
Nt = Qt−1Kt.
Therefore, E don’t need to borrow from banks. We can also let them live
indefinitely and we can remove the entreprenurial labor from the model.
Therefore: nt, wE,t disappear.
15
lH,t =
1
Ψ
wH,t −
1
1 − h
(ct − hct−1) (34)
(1 − h)(rt − Etπt+1) = (ct+1 − hct) − (ct − hct−1) (35)
rert+1 − rert = (rt − Etπt+1) − (rt ∗ −Etπ∗
t+1) + ψB
zt + AUIP
t (36)
rG,t = yH,t + mcH,t − kt −
γ
1 − γ
(rert − lopgt) (37)
wH,t = yH,t + mcH,t − lH,t −
γ
1 − γ
(rert − lopgt) (38)
yH,t = αkt + (Ω(1 − α))lH,t + AY
t (39)
kt+1 = δinvt + (1 − δ)kt (40)
qt =
−Φ INV
K
Φ INV
K
INV
K
(invt − kt) (41)
rK,t + qt−1 = 1 −
1 − δ
RK
rG,t +
1 − δ
RK
qt (42)
EtrK,t+1 = rt − πt+1 (43)
rt = (1 − ρ) [βππt+1 + Θyyt+1] + ρrt−1 + εMP
t (44)
yH,t =
C
YH
(1 − γ)ct +
INV
YH
(1 − γ)invt + γy∗
t + ηγ
2 − γ
1 − γ
rert −
ηγ
1 − γ
lopgt(45)
zt =
1
β
zt−1 + (yH,t − ct − invt) −
γ
1 − γ
(rert − γlopgt) (46)
lopgt = ρlopg ∗ lopgt−1 + εlopg
t (47)
16