NotesECN741-page57

NotesECN741-page57 - ECN 741: Public Economics Fall 2008...

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ECN 741: Public Economics Fall 2008 sub. to ( 55 )-( 58 ). Let η ( θ t , ¯ z t ) ( θ t +1 , ¯ z t ,z t +1 ) , μ z t ) and μ z t ,z t +1 ) be multipliers on ( 55 ), ( 56 ), ( 57 ) and ( 58 ). Write the FOCs u 0 ( c * t ( θ t , ¯ z t )) η ( θ t , ¯ z t ) = μ z t ) π θ ( θ t | ¯ z t ) (59) u 0 ( c * t +1 ( θ t +1 , ¯ z t ,z t +1 )) η ( θ t +1 , ¯ z t ,z t +1 ) = μ z t ,z t +1 ) π θ ( θ t +1 | ¯ z t ,z t +1 ) (60) βπ z z t ,z t +1 ) η ( θ t , ¯ z t ) = X θ t +1 ,z t +1 | θ t , ¯ z t η ( θ t +1 , ¯ z t ,z t +1 ) (61) μ z t ) = X z t +1 μ z t ,z t +1 )(1 - δ + F K ( K * t +1 z t ) ,Y * t z t ,z t +1 ) , ¯ z t ,z t +1 )) (62) substitute η ( θ t ) and η ( θ t +1 , ¯ z t ,z t +1 ) from ( 59 ) and ( 60 ) into ( 61 ) βπ z z t ,z t +1 ) μ z t ) π θ ( θ t | ¯ z t ) u 0 ( c * t ( θ t , ¯ z t )) = X θ t +1 ,z t +1 | θ t , ¯ z t μ z t ,z t +1 ) π θ ( θ t +1 | ¯ z t ,z t +1 ) u 0 ( c * t +1 ( θ t +1 , ¯ z t ,z t +1 )) Take μ z t ,z t +1 ) out of summation and rearrange terms βπ z z t ,z t +1 ) π θ ( θ t | ¯ z t ) u 0 ( c * t ( θ t , ¯ z t )) X θ t +1 ,z t +1 | θ t , ¯ z t π θ ( θ t +1 | ¯ z t ,z t +1 ) u 0 ( c * t +1 ( θ t +1 , ¯ z t ,z t +1 )) - 1 = μ z t ,z t +1 ) μ z t ) Note that by the Independence assumption we had π θ ( θ t +1 | ¯ z t ,z t +1 ) π θ ( θ t | ¯ z t ) = π θ ( θ t +1 | ¯ z t ,z t +1 ) π θ ( θ t | ¯ z t ,z t +1 ) = π θ ( θ t +1 | ¯ z t ,z t +1 t ) Let λ z t ,z t +1 ) μ z t ,z t +1 ) μ z t ) π z z
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