handoutrbc06 - Solving RBC Models 1 Planner's Problem The...

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Solving RBC Models 1 Planner’s Problem The allocation we look for solves the planner’s problem: max { C t ,K t +1 } t =0 E 0 " X t =0 β t ln( C t ) # subject to C t + K t +1 = Z t K θ t ¡ (1 + γ ) t ¢ 1 θ +(1 δ ) K t (1) Z t =(1 ρ )+ ρZ t 1 + ε t (2) where ε t is a mean zero, iid fundamental shock. Necessary and su cient conditions for solution of this problem are: 1 C t = βE t " £ θZ t +1 (1 + γ ) ( t +1)(1 θ ) K θ 1 t +1 +(1 δ ) ¤ C t +1 # (3) Transversality condition: lim T →∞ E 0 β T (1 α ) K T +1 C T =0 . (4) 2C a l i b r a t i o n Along a balanced growth path ( Z t =1 , t ), Y t +1 Y t , K t +1 K t , C t +1 C t all grow at a common constant rate γ , consistent with the stylized facts. γ measured from average annual growth rate of per capita output in data Y t +1 Y t =1 . 016 . θ measured by average capital’s share in output ( r + δ ) K Y =0 . 36 . Dividing the de f nition of net investment K t +1 =(1 δ ) K t + I t by Y t we get K t +1 Y t +1 Y t +1 Y t =(1 δ ) K t Y t + I t Y t . Given average I K =0 . 076 in the data, then δ = I K γ =0 . 06 annually. From (3) we have (1+ γ ) β = θ Y K +(1 δ )or given average K Y =3 . 32 in the data then β = (1+ γ ) θ Y K +(1 δ ) =0 . 9691 annually. Solow’s approach to estimating technology shock processs: From produc- tion function ln( Z t )=ln ( Y t ) θ ln( K t ) . Run a regression of ln( Z t )on ln( Z t 1 ) . Estimates arearound ρ =0 . 95 and σ =0 . 007 .
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handoutrbc06 - Solving RBC Models 1 Planner's Problem The...

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