Notes on Inverse Euler Equation and Savings Distortions

Notes on Inverse Euler equation and Savings Distortions 1 A Two Period Moral Hazard Model • Rogerson (1985) • moral hazard model • two periods t = 0, 1 – effort in first period e0 – consumption in both periods c0 , c1 – stochastic output in second period y1 = θ1 with denisty f (θ1 |e0 ) • separable utility ˆ U ( c0 ) − h ( e ) + β U (c1 (θ1 )) f (θ1 |e0 ) • incentive compatible {c0 , e0 , c1 (θ1 )} requires: ˆ U ( c0 ) − h ( e ) + β ˆ ' U (c1 (θ1 )) f (θ1 |e0 ) ≥ U (c0 ) − h(e ) + β U (c1 (θ1 )) f (θ1 |e' ) • rewrite in terms of utility assignements: ut = U (ct ).... ˆ u0 − h ( e ) + β ' ˆ u1 ( θ1 ) f ( θ1 | e ) ≥ u0 − h ( e ) + β u1 ( θ1 ) f ( θ1 | e ' ) • planning problem � � ˆ min C (u0 ) + q [C (u1 (θ1 )) − y1 (θ1 )] f (θ1 |e0 ) ˆ u0 − h ( e ) + β u1 ( θ1 ) f ( θ1 | e ) = v0 ˆ u0 − h ( e ) + β ' u1 ( θ1 ) f ( θ1 | e ) ≥ u0 − h ( e ) + β • here q = R−1 1 ˆ u1 ( θ1 ) f ( θ1 | e ' ) 1.1 Savings Distortions • if agent could save at safe rate of return R = q−1 then we would have ' ˆ U ' (c1 (θ1 )) f (θ1 |e0 ) U (c0 ) = βR standard Euler equation. • we will show this does not hold: there is a distortion in savings. • fix e0 and consider variations in consumption/utility: û0 = u0 − βΔ û1 (θ1 ) = u1 (θ1 ) + Δ • no effect on utility or incentive constraint since: ˆ ' u0 − h ( e ) + β ' ' u1 (θ1 ) f (θ1 |e ) = û0 − h(e ) + β ˆ û1 (θ1 ) f (θ1 |e' ) for all e' • we need to solve   � � ˆ min C (û0 ) + q C (û1 (θ1 )) f (θ1 |e0 ) û0 ,û1 (·),Δ û0 = u0 − βΔ û1 (θ1 ) = u1 (θ1 ) + Δ • substituting  � ˆ min C (u0 − βΔ) + q Δ C ( u 1 ( θ 1 ) + Δ ) f ( θ 1 | e0 ) �  • note: similarity with savings problem (Δ looks like assets; −C (− x ) looks like the utility function) • FOC is ' ˆ C (u0 − βΔ) β = q 2 C ' ( u 1 ( θ 1 ) + Δ ) f ( θ 1 | e0 ) this condition is necessary and sufficient for an interior: we can use this to solve for Δ • if original allocation was optimal then Δ = 0 and using that C = U −1 we obtain 1 1 = ' U ( c0 ) βR ˆ U ' (c 1 f ( θ 1 | e0 ) 1 ( θ1 )) Inverse Euler equation • since 1 x is convex we can apply Jensen’s inequality • if Var[c1 (θ1 )] > 0 then ˆ ' U (c0 ) < βR U ' (c1 (θ1 )) f (θ1 |e0 ) agents are “savings constrained” • wedge ˆ ' U ( c0 ) = β (1 − τ ) R U ' (c1 (θ1 )) f (θ1 |e0 ) is positive τ≥0 1.2 Mirrlees Model • Mirrlees model [Golosov et al] • work time at t = 1 ˆ u ( c0 ) + β [u(c1 (θ1 )) − h(y1 , θ1 )] f (θ1 ) • same perturbations • same optimality conditions: Inverse Euler equation • true also for a mixed model of moral hazard and adverse selection where effort affects distribution of θ 3 2 General Horizon and Welfare • Utility ∞ E ∑ βt [U (ct ) − h(yt , θt )] t =0 • {θt } general stochastic process and private information • again: rewrite in terms of ut • incentive constraint ∞ t t ∞ t E ∑ β [u(θ ) − h(y(θ ), θt )] ≥ E ∑ βt [u(σt (θ t )) − h(y(σt (θ t )), θt )] t =0 t =0 • Planner’s net cost: ∞ E ∑ qt [C (u(θ t )) − y(θ t )] t =0 • variations as before: [Farhi-Werning] û(θ t ) = u(θ t ) + Δ(θ t−1 ) − βΔ(θ t ) and a “No Ponzi” condition lim βt EΔ(σt (θ t )) = 0 • preserve utility and incentive compatibility since ∞ ∞ t =0 t =0 E ∑ βt u(σt (θ t )) = E ∑ βt û(σt (θ t )) • hence, must minimize cost ∞ E ∑ qt [C (u(θ t ) + Δ(θ t−1 ) − βΔ(θ t )) t =0 • looks like savings problem: Farhi-Werning exploit this to solve this partial reform • implementation: see slides 4