solpr11ec70204

# solpr11ec70204 - Problem Set 11 Econ 702 Spring 2004...

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Problem Set 11 Econ 702, Spring 2004 Prepared by Ahu Gemici May 3, 2004 Problem 1 (Romer Endogenous Growth) Deﬁne the decentralized problem in Romer’s endogenous growth model that we went over in class. Get a formula for the balanced growth path in equilibrium and compare to the social planner’s problem (Look at Problem 5 in Homework 10 for the social planner’s problem). Solution : From the FOC of the ﬁrm in the ﬁnal goods sector, we have: q t ( i ) = (1 - α ) L α 1 t x t ( i ) - α (1) w t = αL 1 t α - 1 A t x t ( i ) 1 - α (2) and since x t = K t ηA t , (2) becomes, w t = αL 1 t α - 1 A t ( K t ηA t ) 1 - α (3) From the FOC of the ﬁrm in the intermediate good sector: (1 - α ) 2 L α 1 t ( K t ηA t ) - α = R t η (4) p P t = w t ζA t (5) Also, in equilibrium, total proﬁt a patent generates will be equal to the price of it so that the zero proﬁt condition for the intermediate goods sector is satisﬁed. p P t = X τ = t π t ( i ) ( R ) τ - t (6) Now write down the consumer’s problem ( δ = 1): max c t ,L 1 t ,L 2 t ,k t +1 X t =0 β t ± c 1 - σ t - 1 1 - σ ² (7) 1

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s.t. c t + k t +1 = R t k t + w t ( L 1 t + L 2 t ) From the FOC to the consumer’s problem, we get: ( c t +1 c t ) σ = βR t (8) On the balanced growth path: K t +1 = γK t A t +1 = γA t c t +1 = γc t L 1 t = L 1 L 2 t = L 2 w t +1 = γw t x t = x = K t ηA t R t = R π t ( i ) = π p P t = p P q t ( i ) = q Use these above BGP conditions to write, From (8), γ σ = βR (9) From (4), (1 - α ) 2 L α 1 ( K t ηA t ) - α = (10) From (5) and (3), p P = αL α - 1 1 ( K t ηA t ) 1 -
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solpr11ec70204 - Problem Set 11 Econ 702 Spring 2004...

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