ps7macro2sp08anspart1 - Econ 387L: Macro II Spring 2008,...

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Econ 387L: Macro II Spring 2008, University of Texas Instructor: Dean Corbae Answers Problem Set # 7 Part 1: (1) V 1 ( k 1 ,z 1 )=max k 2 u ( z 1 f ( k 1 )+(1 δ ) k 1 k 2 ) Note that, since u 0 > 0 , k 2 ( s 1 )=0 .Thu s ,w ehav e V 1 ( k 1 ,z 1 )= u ( z 1 f ( k 1 )+(1 δ ) k 1 ) (1) Since u is continuous, we ger that V 1 is continuous. Also, for increasing in s 1 , ∂V 1 ( k 1 ,z 1 ) ∂k 1 =( z 1 f 0 ( k 1 )+(1 δ )) u 0 ( z 1 f ( k 1 )+(1 δ ) k 1 ) 0 (2) ∂V 1 ( k 1 ,z 1 ) ∂z 1 = f ( k 1 ) u 0 ( z 1 f ( k 1 )+(1 δ ) k 1 ) 0 (3) Concavity is demonstrated by 2 V 1 ( k 1 ,z 1 ) ∂k 2 1 = z 1 f 00 ( k 1 ) u 0 ( c 1 )+( z 1 f 0 ( k 1 )+(1 δ )) 2 u 00 ( c 1 ) 0 (4) (2) V 0 ( k 0 ,z 0 )=max k 1 © u ( z 0 f ( k 0 )+(1 δ ) k 0 k 1 )+ βE z 1 | z 0 [ V 1 ( k 1 ,z 1 )] ª (5) The FOC for this problem and (2) yield: H 0 = u 0 ( z 0 f ( k 0 )+(1 δ ) k 0 k 1 )+ βE z 1 | z 0 [( z 1 f 0 ( k 1 )+(1 δ )) u 0 ( z 1 f
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ps7macro2sp08anspart1 - Econ 387L: Macro II Spring 2008,...

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