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

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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.
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern