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NotesECN741-page40

# NotesECN741-page40 - 1 R t-1 ω θ 1 = 1 βR E ³ 1 u c t 1...

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ECN 741: Public Economics Fall 2008 Note that this is the FOC with respect to a c t ( θ T ) at a particular draw θ T . But we know that c t ( θ T ) is θ t - measurable, therefore we don’t need to sum over all θ T D , but only those that contain the particular history θ t X θ T | θ t π ( θ T ) ω ( θ 1 ) u 0 ( c t ( θ T )) β t - 1 = λ/R t - 1 X θ T | θ t π ( θ T ) by measurability of c t ( θ T ) . u 0 ( c t ( θ T )) β t - 1 ω ( θ 1 ) = λ/R t - 1 Note: This implies that optimal c t ( θ T ) is actually θ 1 -measurable, i.e., it is independent from θ t for t > 1 . In other words there is full insurance. Note: The following Euler equation hold u 0 ( c t ( θ T )) = βR E ± u 0 ( c t +1 ( θ T )) | θ t ² planner is happy to allow access to outside trade. Note: Another Euler equation also holds. 1 u 0 ( c t ( θ T )) = λ - 1 β t -
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Unformatted text preview: 1 R t-1 ω ( θ 1 ) = 1 βR E ³ 1 u ( c t +1 ( θ T )) | θ t ´ 3.1.1 Inverse Euler Equation Consider again the original planning problem ( 43 )(with incentive constraints). Let ( c * ,y * ) be the solution to this problem. Now consider the following perturbation around ( c * ,y * ) y = y * c s = c * s for all s 6 = t,t + 1 (for ﬁxed t ) for all histories θ t u ( c t ( θ T )) + βu ( c t +1 ( θ T )) = k + u ( c * t ( θ T )) + βu ( c * t +1 ( θ T )) for all θ t +1 such that π ( θ t +1 | θ t ) > 40...
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