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lecture4 - Lecture 4 Log-Linearizing the Consumption RBC...

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Lecture 4 Log-Linearizing the Consumption RBC Model

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ECN/APEC 7240 IV - 2 Our Approach There is no exact analytic solution to the Consumption RBC Model. We can solve for the balanced growth path the model converges to in the absence of technology shocks. We can also solve for the linear response to small deviations from this balanced growth path caused by technology shocks.
ECN/APEC 7240 IV - 3 Decomposition of Log Variables For each variable X t , we will assume x t 0 is the balanced growth behavior. For extrinsic variables this will be of the form x t 0 = x 0 + gt . Intrinsic variables will be constant. x t 1 is the response to technology shocks. . 1 0 t t t x x x + =

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ECN/APEC 7240 IV - 4 Model Parameters We assume Cobb-Douglas technology. The parameters are α , β , δ , θ , and g . We assume that ( K / Y ) 0 , ( C / Y ) 0 , α , and g are observable. As in the CKR model, the preference parameters β and θ are not separately identified to zeroth order. They are separately identified when we consider the response to technology shocks.
ECN/APEC 7240 IV - 5 Factor Prices The wage is The gross interest rate is Income-product identity: α α α - - = 1 ) 1 ( ) , ( A K A K W δ α α α - + = - - 1 1 1 ) , ( A K A K R K Y K A K K A K R A K W ) 1 ( ) 1 ( ) , ( ) , ( 1 δ δ α α - + = - + = + -

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ECN/APEC 7240 IV - 6 Zeroth-Order Equations Budget constraint Factor prices Euler equation ( 29 θ β / 1 0 0 1 ) exp( R C C g t t = = + ( 29 1 1 ) exp( 0 0 = - + + Y K g Y C δ δ α - + = 1 0 0 K Y R δ 0 R θ β ,
ECN/APEC 7240 IV - 7 Calibrating the Model Let us set α = 1/3, ( K / Y ) 0 = 3, ( C / Y ) 0 = 0.8, g = 0.015. This implies Depreciation is δ = 0.0516.

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