NotesECN741-page56

NotesECN741-page56 - ECN 741 Public Economics Fall 2008...

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Unformatted text preview: ECN 741: Public Economics Fall 2008 Deﬁnition 11 An allocation is incentive compatible if T β t−1 T π z (z ) t=1 z T ∈Z T πθ (θT |z T ) u(ct (θT , z T )) − v θT ∈ΘT T T πz (z ) β t−1 t=1 z T ∈Z T T T T T πθ (θ |z ) u(ct (αt (θ , z ))) − v θ T ∈ ΘT yt (θT , z T ) θt ≥ (54) yt (αt (θT , z T )) θt for all α : Θ −→ Θ (αt is (θt , z t ) − measurable). The planing problem T β t−1 T πz (z ) max z T ∈Z T t=1 πθ (θT |z T ) u(ct (θT , z T )) − v θT ∈ΘT yt (θ T , z T ) θt sub. to (53) and (54). The inverse Euler equation Suppose (c∗ , y ∗ , K ∗ ) is the solution to the above problem. Fix a public history z t . We perturb ¯ ∗ the solution to (c , y , K ) such that u(ct (θt , z t )) = u(c∗ (θt , z t )) + β ¯ ¯ t πz (¯t , zt+1 )δt+1 (θt , z t , zt+1 ) + γ z ¯ ∀θt (55) zt+1 u(ct+1 (θ t+1 t , z , zt+1 )) = ¯ u(c∗+1 (θt+1 , z t , zt+1 )) ¯ t θt (56) π (θt |z t )c∗ (θt , z t ) + Kt∗+1 (¯t ) ¯t ¯ z π (θt |z t )ct (θt , z t ) + Kt+1 (¯t ) ≤ ¯ ¯ z − δt+1 (θt , z t , zt+1 ) ¯ (57) θt πθ (θt+1 |z t , zt+1 )ct+1 (θt+1 , z t , zt+1 ) − Kt+1 (¯t )(1 − δ ) − F (Kt+1 (¯t ), Yt∗ (¯t , zt+1 ), z t , zt+1 )(58) ¯ ¯ z z z ¯ θt+1 ≤ πθ (θt+1 |z t , zt+1 )c∗+1 (θt+1 , z t , zt+1 ) − Kt∗+1 (¯t )(1 − δ ) − F (Kt∗+1 (¯t ), Yt∗ (¯t , zt+1 ), z t , zt+1 ) ¯ ¯ z z z ¯ t θt+1 The idea is that (c∗ , y ∗ , K ∗ ) must be the solution to the following maximization problem (with γ = 0) 0= max γ δt+1 ,ct ,ct+1 ,γ,Kt+1 56 ...
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