Equilibrium Growth, Inflation, and Bond Yields - Duke University's ...

Substituting the envelope conditions and definitions above, the first-order conditions can be expressed as:
Λi,t
Pt
Qi,k,t =
Qi,n,t =
= φR
1
Φ ′ i,k,t
1
Φ ′ i,n,t
� Pi,t
⎡
ΠssPi,t−1
Qi,k,t = Et
⎣Mt+1
+ Et
�
Qi,n,t = Et
⎣Mt+1
Wi,t
Pt
+ Et
= (1 − α)(1 − 1
⎡
�
⎧
⎨
⎩
Mt+1Qi,k,t+1
⎩
�
− 1
Yt
− Et
ΠssPi,t−1
α(1 − 1
1
1
ν
ν
ν )Yt+1X1− i,t+1
Ki,t+1
�
+ Λi,t+1
�
1 − δk − Φ′ i,k,t+1 Ii,t+1
⎧
⎨η(1
− α)(1 − 1
ν
Mt+1Qi,n,t+1
1
ν
ν )Y
Li,t
Ni,t+1
)Y 1
ν
t+1
Ki,t+1
1
ν
X1− i,t+1
Mt+1φR
+
� α
ν
� Pi,t+1
� Y 1
ν
t+1
Ki,t+1
+ Φi,k,t+1
Λi,t+1
ΠssPi,t
1
ν
X− i,t+1
��
� �
η(1−α)
ν
Ni,t+1
�
1 − δn − Φ′ i,n,t+1Si,t+1 ��
+ Φi,n,t+1
Ni,t+1
t X
1 1− ν
i,t
+ Λi,t
� 1−α
ν
35
� Y 1
ν
Li,t
t X
1 − ν
i,t
�
Yt+1Pi,t+1
− 1
ΠssP 2 i,t
⎫⎤
⎬
⎦
⎭
Y 1
ν
t+1
1
ν
X− i,t+1
⎫⎤
⎬
⎦
⎭
�