Notes on Inverse Euler equation and Savings Distortions
• Rogerson (1985)
• moral hazard model
• two periods t =
0, 1
– effort in first period e
0
– consumption in both periods c
0
, c
1
– stochastic output in second period y
1
= θ
1 with denisty f ( θ
1
| e
0
)
• separable utility
U ( c
0
ˆ
) − h ( e ) + β U ( c
1
( θ
1
)) f ( θ
1
| e
0
)
• incentive compatible { c
0
, e
0
, c
1
( θ
1
) } requires:
U ( c
0
ˆ
) − h ( e ) + β U ( c
1
( θ
1
)) f ( θ
1
| e
0
) ≥ U ( c
0
) − h ( e '
ˆ
) + β U ( c
1
( θ
1
)) f ( θ
1
| e ' )
• rewrite in terms of utility assignements: u t
= U ( c t
) ....
u
0
ˆ
− h ( e ) + β u
1
( θ
1
) f ( θ
1
| e ) ≥ u
0
− h ( e '
ˆ
) + β u
1
( θ
1
) f ( θ
1
| e ' )
• planning problem
� min C ( u
0
ˆ
) + q [ C ( u
1
( θ
1
)) − y
1
( θ
1
)] f ( θ
1
| e
0
)
� u
0
ˆ
− h ( e ) + β u
1
( θ
1
) f ( θ
1
| e ) = v
0 u
0
ˆ
− h ( e ) + β u
1
( θ
1
) f ( θ
1
| e ) ≥ u
0
− h ( e '
ˆ
) + β u
1
( θ
1
) f ( θ
1
| e ' )
• here q = R − 1
1

• if agent could save at safe rate of return R = q − 1 then we would have
U ' ( c
ˆ
0
) = β R U ' ( c
1
( θ
1
)) f ( θ
1
| e
0
) standard Euler equation.
• we will show this does not hold: there is a distortion in savings.
• fix e
0 and consider variations in consumption/utility: u
0
= u
0
− β Δ u
1
( θ
1
) = u
1
( θ
1
) + Δ
• no effect on utility or incentive constraint since: u
0
ˆ
− h ( e ' ) + β u
1
( θ
1
) f ( θ
1
| e ' ) = u
0
ˆ
− h ( e ' ) + β u
1
( θ
1
) f ( θ
1
| e ' ) for all e '
• we need to solve
� u
0 min C
, u
1
( · ) , Δ
( u ˆ
ˆ
0
) + q C ( u
1
( θ
1
)) f ( θ
1
| e
0
)
� u
0
= u
0
− β Δ u
1
( θ
1
) = u
1
( θ
1
) + Δ
• substituting � min
Δ
C ( u
0
ˆ
− β Δ ) + q C ( u
1
( θ
1
) + Δ ) f ( θ
1
| e
0
)
�
• note: similarity with savings problem ( Δ looks like assets; − C ( − x ) looks like the utility function)
• FOC is
C ' ( u
0
ˆ
− β Δ ) β = q C ' ( u
1
( θ
1
) + Δ ) f ( θ
1
| e
0
)
2

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
U ' (
1 c
0
1
ˆ
)
=
β R U ' ( c
1
1
( θ
1
)) f ( θ
1
| e
0
)
Inverse Euler equation
• since
1 x is convex we can apply Jensen’s inequality
• if Var [ c
1
( θ
1
)] > 0 then
U ' ( c
ˆ
0
) < β R U ' ( c
1
( θ
1
)) f ( θ
1
| e
0
) agents are “savings constrained”
• wedge
U ' ( c
0
ˆ
) = β ( 1 − τ ) R U ' ( c
1
( θ
1
)) f ( θ
1
| e
0
) is positive
τ ≥ 0
• Mirrlees model [Golosov et al]
• work time at t =
1 u ( c
0
ˆ
) + β [ u ( c
1
( θ
1
)) − h ( y
1
, θ
1
)] f ( θ
1
)
• 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

• Utility
E t
∞
∑
= 0
β t [ U ( c t
) − h ( y t , θ t
)]
• { θ t
} general stochastic process and private information
• again: rewrite in terms of u t
• incentive constraint
E t
∞
∑
=
0
β t [ u ( θ t ) − h ( y ( θ t ) , θ t
)] ≥ E t
∞
∑
=
0
β t [ u ( σ t ( θ t )) − h ( y ( σ t ( θ t )) , θ t
)]
• Planner’s net cost:
E t
∞
∑
= 0 q t [ C ( u ( θ t )) − y ( θ t )]
• variations as before: [Farhi-Werning] u ˆ ( θ t ) = u ( θ t ) + Δ ( θ t − 1 ) − β Δ ( θ t ) and a “No Ponzi” condition lim β t E Δ ( σ t ( θ t )) =
0
• preserve utility and incentive compatibility since
E t
∞
∑
= 0
β t u ( σ t ( θ t )) = E
∞
∑ t = 0
β t ( σ t ( θ t ))
• hence, must minimize cost
E t
∞
∑
= 0 q t [ C ( u ( θ t ) + Δ ( θ t −
1 ) − β Δ ( θ t ))
• looks like savings problem:
Farhi-Werning exploit this to solve this partial reform
• implementation: see slides
4

MIT OpenCourseWare http://ocw.mit.edu
14.471 Public Economics I
Fall 20 12
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