NotesECN741-page41

NotesECN741-page41 - ECN 741: Public Economics Fall 2008 (T...

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ECN 741: Public Economics Fall 2008 X θ T | θ t π ( θ T ) ± c 0 t ( θ T ) + c 0 t +1 ( θ T ) /R ² = X θ T | θ t π ( θ T ) ± c * t ( θ T ) + c * t +1 ( θ T ) /R ² Note: ( c 0 ,y 0 ) is feasible and incentive compatible. Note: What we are doing is perturbing u ( c t ( θ T )) by some amount and then make an appro- priate perturbation in every immediate history following θ t so that incentive compatibility is preserved. If ( c * ,y * ) is the solution to ( 43 ), this perturbation cannot improve welfare. One implication of this is that ( c * ,y * ) solves the following maximization problem and k = 0 at the optimal solution. max k,c 0 t ( θ T ) ,c 0 t +1 ( θ T ) k sub. to u ( c 0 t ( θ T )) + βu ( c 0 t +1 ( θ T )) = k + u ( c * t ( θ T )) + βu ( c * t +1 ( θ T )) for all θ t t +1 such that π ( θ t +1 | θ t ) > 0 X θ T | θ t π ( θ T ) ± c 0 t ( θ T ) + c 0 t +1 ( θ T ) /R ² = X θ T | θ t π ( θ T ) ± c * t ( θ T ) + c * t +1 ( θ T ) /R ² let η ( θ t +1 ) and λ be multipliers. Let’s write the FOC X θ t +1 | θ t η ( θ t +1 ) u 0 ( c 0 t ( θ T )) = λ X θ T | θ t π ( θ T ) = λπ ( θ
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