TwoSec_SL

# TwoSec_SL - Two Sector Models Prof Lutz Hendricks August 7...

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Two Sector Models Prof. Lutz Hendricks August 7, 2009 L. Hendricks () Two Sector Models August 7, 2009 1 / 28

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Two Sector Models We relax the assumption that there is only one good at each date. There are no major changes in methods. Multi-sector models are used to study issues such as: technical change that is °embodied± in capital goods, human capital, international trade. L. Hendricks () Two Sector Models August 7, 2009 2 / 28
Planning Problem There is a unit mass of households who live forever. Preferences are t = 0 β t u ( c t , 1 ° v t ) ν is work; 1 ° ν is leisure. L. Hendricks () Two Sector Models August 7, 2009 3 / 28

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Planning Problem Consumption goods are produced according to Y 1 = F ( K 1 , L 1 ) and capital goods according to Y 2 = G ( K 2 , L 2 ) The resource constraints are L 1 t + L 2 t = ν t K 1 t + K 2 t = K t Y 1 t = c t Y 2 t = K t + 1 ° ( 1 ° δ ) K t L. Hendricks () Two Sector Models August 7, 2009 4 / 28
Planning Problem The planner maximizes t = 0 β t u ( c t , 1 ° v t ) subject to the resource constraints. The planner chooses c t , L 1 t , L 2 t , and ϕ t . ϕ t is the fraction of capital employed in sector 1: K 1 t = ϕ t K t K 2 t = ( 1 ° ϕ t ) K t The planner²s state variable is K t . We would need 2 states ( K 1 t , K 2 t ) if there were a cost of reallocating capital. L. Hendricks () Two Sector Models August 7, 2009 5 / 28

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Planning Problem The Bellman equation is V ( K ) = max u ( F ( ϕ K , L 1 ) , 1 ° L 1 ° L 2 ) + β V ( K ( 1 ° δ ) + G ([ 1 ° ϕ ] K , L 2 )) where the choice variables are L 1 , L 2 , and ϕ . L. Hendricks () Two Sector Models August 7, 2009 6 / 28
Planning Problem FOCs: u l = β V 0 ( K 0 ) G L = u c F L u c F K = β V 0 ( K 0 ) G K Envelope: V 0 ( K ) = ϕ F K u c + β V 0 ( K 0 ) f 1 ° δ + ( 1 ° ϕ ) G K g L. Hendricks () Two Sector Models August 7, 2009 7 / 28

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Planning Problem Euler equation u c F K G K = β u c ( . 0 ) F K ( . 0 ) G K ( . 0 ) f 1 ° δ + G K ( . 0 ) g Static condition F K / F L = G K / G L L. Hendricks () Two Sector Models August 7, 2009 8 / 28
Planning Problem Solution Sequences f c t , ν t , K t + 1 , ϕ t , L 1 t , L 2 t g that satisfy: 2 FOCs; 4 feasibility conditions; TVC: lim t !

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