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NotesECN741-page58

NotesECN741-page58 - ECN 741 Public Economics Fall 2008 z z...

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ECN 741: Public Economics Fall 2008 1 = E λ 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 )) | ¯ z t Note that (using Jensen’s inequality) λ z t , z t +1 ) = β u 0 ( c * t ( θ t , ¯ z t )) E 1 u 0 ( c * t +1 ( θ t +1 , ¯ z t , z t +1 )) θ t +1 , ¯ z t , z t +1 - 1 < β E u 0 ( c * t +1 ( θ t +1 , ¯ z t , z t +1 )) θ t +1 , ¯ z t , z t +1 u 0 ( c * t ( θ t , ¯ z t )) and therefore u 0 ( c * t ( θ t , ¯ z t )) < β E u 0 ( c * t +1 ( θ t +1 , ¯ 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 )) θ t +1 , ¯ z t , z t +1 Example 1: suppose Θ is singleton. Then λ z t , z t +1 ) = βu 0 ( c * t +1 z t , z t +1 )) u 0 ( c * t z t )) and therefore 1 = E βu 0 ( c * t +1 z t , z t +1 )) u 0 ( c * t z t )) (1 - δ + F K ( K * t +1 z t ) , Y * t z t , z t +1 ) , ¯ z t , z t +1 )) | ¯ z t Example 2: suppose Z is singleton. Then λ t +1 = β u 0 ( c * t ( θ t )) E 1 u 0 ( c * t +1 ( θ t +1 )) θ t +1 - 1 1 = λ t +1 (1 - δ + F K ( K * t +1 , Y * t )) and therefore β (1 - δ + F K ( K * t +1 , Y * t )) u 0 ( c * t ( θ t )) = E 1 u 0 ( c * t +1 ( θ t +1 )) θ t +1 and therefore (using Jensen’s inequality)
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