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

As derived in the appendix, the corresponding first order conditions are
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
�
Yt+1Pi,t+1
− 1
ΠssP 2 i,t
⎫⎤
⎬
⎦
⎭
Y 1
ν
t+1
1
ν
X− i,t+1
where Qi,k,t, Qi,n,t, and Λi,t are the shadow values of physical capital, R&D capital and price of intermediate
goods, respectively. 14
Central Bank The central bank follows a modified Taylor rule specification that depends on the lagged
interest rate and output and inflation deviations:
ln
� Rt+1
Rss
�
� Y 1
ν
Li,t
t X
1 − ν
i,t
� � � � � �
Rt
Πt
�Yt
= ρr ln + ρπ ln + ρy ln + σξξt
Rss
Πss
�Yss
where Rt+1 is the gross nominal short rate, � Yt ≡ Yt
Nt is detrended output, and ξt ∼ N(0, 1) is a monetary
policy shock. Variables with a ss-subscript denote steady-state values. Given this rule, the central bank
chooses ρr, ρπ, ρy, and Πss.
Symmetric Equilibrium In the symmetric equilibrium, all intermediate firms make identical decisions:
Pi,t = Pt, Xi,t = Xt, Ki,t = Kt, Li,t = Lt, Ni,t = Nt, Ii,t = It, Si,t = St, Di,t = Dt, Vi,t = Vt. Also, Bt = 0.
14 Φi,k,t ≡ Φk
� Ii,t
tional convenience.
K i,t
�
, Φi,n,t ≡ Φn
� Si,t
N i,t
�
, Φ ′ � �− 1
Ii,t ζk ′
i,k,t ≡ α1,k
, Φ K i,n,t ≡ α1,n
i,t
11
� Si,t
N i,t
⎫⎤
⎬
⎦
⎭
�
� − 1
ζn are defined for nota