TwoSec_RQ

# TwoSec_RQ - Review Questions Two Sector Models Econ720 Fall...

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Review Questions: Two Sector Models Econ720. Fall 2009. Prof. Lutz Hendricks 1 A Planning Problem The economy is populated by a unit mass of in°nitely lived households with prefer- ences given by 1 X t =0 ° t u ( c Mt ; c Ht ) where c jt denotes consumption of good j . The household has a unit time endow- ment in each period. There are two goods in the economy, indexed by j = M; H . The production function for good M is F ( k Mt ; h Mt ) ; it is used for investment and consumption ( c Mt ). The production function for good H is G ( k Ht ; h Ht ) ; it is consumed as c Ht . k jt denotes capital input in sector j and h jt denotes labor input. Capital goods depreciate at the common rate ± . (a) Assume that capital cannot be moved between sectors. Once installed in sector j it stays there forever. Formulate the Dynamic Programming problem solved by a central planner. (b) For the remainder of the question assume that capital can be moved freely between sectors. Formulate the planner±s Dynamic Program. (c) De°ne a solution to the Planner±s problem. 1.1 Answer Sketch: Planning Problem (a) The planner solves (in sequence language): max 1 X t =0 ° t u ( c Mt ; c Ht ) subject to c Ht = G ( k Ht ; h Ht ) k jt +1 = (1 ° ± ) k jt + i jt i jt ± 0 c Mt + i Mt + i Ht = F ( k Mt ; h Mt ) 1

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There are other ways of writing this. The state variables are both capital stocks. The Dynamic Program is therefore: V ( k M ; k H ) = max u ( F ( k M ; h M ) ° i M ° i H ; G ( k H ; h H ))+ °V ((1 ° ± ) k M + i M ; (1 ° ± ) k H + i H ) subject to i j ± 0 . (b) The constraint set changes if capital can be moved between sectors. E/ec- tively, the non-negativity constraints on investment are dropped. But it is then more convenient to write the constraints as c Ht = G ( k Ht ; h Ht ) k t +1 = (1 ° ± ) k t + F ( k t ° k Ht ; 1 ° h Ht ) ° c Mt The Dynamic Programming problem is now V ( k ) = max u [(1 ° ± ) k + F ( k ° k H ; 1 ° h H ) ° k 0 ; G ( k H ; h H )] + ° V ( k 0 ) (c) The °rst order conditions are u M F k = u H G K (1) u M F H = u H G H (2) u M = ° V 0 ( k 0 ) The envelope condition is V 0 ( k ) = u M [(1 ° ± ) + F K ] Combining the last 2 equations yields the standard Euler equation u M = ° u M ( : 0 ) [(1 ° ± ) + F K ( : 0 )] (3) A solution to the planner±s problem (in sequence language) consists of sequences f k t ; k Ht ; c Mt ; c Ht g which solve the °rst-order conditions (1) through (3) and the con- straint c Ht = G ( k Ht ; h Ht ) . 2 Consumption Taxes in a Growth Model Consider the following version of the growth model. There is a single representative agent with preferences given by: 1 X t =0 ° t log c t 2
where c t is consumption in period t , and 0 < ° < 1 . The worker is endowed with one unit of time in each period but does not value leisure. There are two production sectors. One sector produces the consumption good using a Cobb-Douglas technology: c t = k ° ct n 1 ° ° ct where k ct and n ct are capital and labor inputs to this sector at time t respectively.

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