NotesECN741-page45

# NotesECN741-page45 - ECN 741: Public Economics Fall 2008...

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ECN 741: Public Economics Fall 2008 Immiseration result For this part assume βR = 1 . Consider the following planning problem w 0 = max T X t =1 β t - 1 X θ T D π ( θ t ) ± u ( c t ( θ t )) - v ² y t ( θ t ) θ t ³´ (46) sub. to. T X t =1 X θ t π ( θ t ) ± u ( c t ( θ t )) - v ² y t ( θ t ) θ t ³´ T X t =1 X θ t π ( θ t ) ± u ( c t ( α 0 t ( θ t ))) - v ² y t ( α 0 t ( θ t )) θ t ³´ for all α 0 : D -→ D ( α 0 t is θ t - measurable). X θ t T X t =1 π ( θ t ) µ c t ( θ t ) - y t ( θ t ) /R t - 1 0 The solution to this problem must also be the solution to the following dual problem K ( w 0 ) = min X θ t T X t =1 π ( θ t ) µ c t ( θ t ) - y t ( θ t ) /R t - 1 (47) sub. to T X t =1 X θ t π ( θ t ) ± u ( c t ( θ t )) - v ² y t ( θ t ) θ t ³´ T X t =1 X θ t π ( θ t ) ± u ( c t ( α 0 t ( θ t ))) - v ² y t ( α 0 t ( θ t )) θ t ³´ T X t =1 β t - 1 X θ T D π ( θ t ) ± u ( c t ( θ t )) - v ² y t ( θ t ) θ t ³´ w 0 In which K ( U 0 ) is the cost of delivering ex-ante utility U 0 to everyone. We want to write this recursively. Consider an allocation sequence ( c t ( θ t ) ,y t ( θ t )) . Consider a history ¯ θ t . Then deﬁne the ex ante utility of an agent with history
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## This note was uploaded on 12/10/2011 for the course MAT 121 taught by Professor Wong during the Fall '10 term at SUNY Stony Brook.

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