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solpr7ec70204 - Solutions to Problem Set 7 Econ 702 Spring...

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Solutions to Problem Set 7 Econ 702, Spring 2004 Prepared by Ahu Gemici March 24, 2004 Problem 1 (Lucas Tree Model) Solution: From the First Order Conditions we get: q i = β j u c ( c ) u c ( c ) Γ ij [ q j + z j ] i (1) In equilibrium, c=z, q i = β j u c ( z j ) u c ( z i ) Γ ij [ q j + z j ] i (2) The above system of equations characterizes the prices of the tree at each state { q i } I i =1 . Now we will write this system of equations in the matrix form: q 1 .. .. q I = β Γ 11 u ( z 1 ) u ( z 1 ) β Γ 12 u ( z 2 ) u ( z 1 ) .. β Γ 1 J u ( z J ) u ( z 1 ) .. β Γ 22 u ( z 2 ) u ( z 2 ) .. .. .. .. .. .. β Γ I 1 u ( z 1 ) u ( z I ) .. .. β Γ IJ u ( z J ) u ( z I ) q 1 .. .. q I + z 1 .. .. z I Let q= q 1 .. .. q I and d= z 1 .. .. z I and A= β Γ 11 u ( z 1 ) u ( z 1 ) β Γ 12 u ( z 2 ) u ( z 1 ) .. β Γ 1 J u ( z J ) u ( z 1 ) .. β Γ 22 u ( z 2 ) u ( z 2 ) .. ..
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