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Unformatted text preview: 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 ¡ c Mt + i Mt + i Ht = F ( k Mt ;h Mt ) 1 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 ¡ . (b) The constraint set changes if capital can be moved between sectors. E/ec tively, the nonnegativity 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 ; G ( k H ;h H )] + & V ( k ) (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 ( k ) The envelope condition is V ( k ) = u M [(1 & ¡ ) + F K ] Combining the last 2 equations yields the standard Euler equation u M = & u M ( : ) [(1 & ¡ ) + F K ( : )] (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 &rstorder 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 < & < 1 . The worker is endowed with one unit of time in each period but does not value leisure....
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This note was uploaded on 10/29/2009 for the course ECON 720 at UNC.
 '09
 LUTZHENDRICKS

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