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homework5 2009 solutions

homework5 2009 solutions - ECN/APEC 6000/7230 Fall 2009...

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Unformatted text preview: ECN/APEC 6000/7230 Fall 2009 Homework 5 Solutions 1) Consider the Cass-Koopmans model but with the quadratic period utility function 2 ) ( 2 1 ) ( C M C u-- = , where the constant M > 0 is known as the bliss point. a) The Hamiltonian is )] ( ) ( ) ( ) ( ) ( ) ( )[ ( )) ( ( 2 ) ( 2 t C t L t K t r t L t W t t C M t L e t- + +-- =- H . b) Hamiltons equations are ) ( ) ( )) ( )( ( ) ( =-- = - t L t t C M t L e t C t H ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t C t L t K t r t L t W t dt t dK- + = = H ). ( ) ( ) ( ) ( t r t t K dt t d = =- H c) Let us define . ' ) ' ( exp ) ( = t dt t r t R Then ). ( ) ( ' ) ' ( exp ) ( ) ( t R t r dt t r t r dt t dR t = = A solution to the equation is , ) ( ) ( t R t = where is an integration constant: ECN/APEC 6000/7230 Fall 2009 ( 29 . ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 t t r t R t r t R t r t R dt t d - =- =- = The C equation simplifies to...
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homework5 2009 solutions - ECN/APEC 6000/7230 Fall 2009...

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