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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
The aggregate resource constraint is where Πt ≡ Pt Pt−1 is the gross inflation rate. Yt = Ct + St + It + φR 2 � Πt Πss Nominal Yields The price of a n-period nominal bond P (n) t where P (0) t P (n) t = Et � 1 Mt+1 Πt+1 �2 − 1 Yt P (n−1) � t+1 can be written recursively as: = 1 and P (1) t = 1 . The yield-to-maturity on the n-period nominal bond is defined as: Rt+1 2.2 Exogenous Growth y (n) t ≡ − 1 n log � This setup also nests a fairly standard New Keynesian setup with exogenous growth when the technological appropriability parameter η is set to 0 and the aggregate stock of R&D Nt is exogenously specified. Under P (n) t these conditions the production function of the intermediate firm can be expressed as � Xi,t = K α i,t(ZtLi,t) 1−α Zt = AtNt where Nt follows a stochastic process. Note that TFP is now exogenous and comprised of a stationary component At and a trend component Nt. I consider two versions of the exogenous growth model, one with a deterministic trend and the other with a stochastic trend in productivity, to compare with the benchmark endogenous growth model. Deterministic Trend The law of motion for Nt is Nt = e µt where µ is parameter governing the average growth rate of the economy. Equivalently, this expression can be rewritten in log first differences as ∆nt = µ 12
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Page 39 and 40: Table 2: Macroeconomic Moments Data
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