Job Assignments under Moral Hazard - School of Economics and ...

if he were instead assigned to job l:
fh(e SB
h (θ SB ), θ SB ) − fl(e SB
h (θ SB )) =
−k α
α + √ α 2 + 4 k
< 0.
Effort eSB h (θ) > eSB l for θ > θSB : At θSB there is an upward jump in effort:
e SB
h (θ SB ) − e SB
l =
The effort gap is increasing in θ: d
d 2
d θ2 eSB h
d θ eSB h
2 k
α + √ α 2 + 4 k .
(θ) = 1
2
(θ) > 0 and d
d θ eSB
h (θ) θ=θ SB > 0, as is easy to check.
k α − α θ2
SB > 0 for θ > θ , because
Expected earnings: For an agent assigned to job h, expected earnings (EE) are higher
than they would be if he remained in job l.
EEh(e SB
h (θ), θ) = fh(e SB
h (θ), θ) × ¯wh(e SB
h (θ), θ) = (α2 θ2 − k) (α2 θ2 + k)
4 (α2 θ2 ,
)
EEl(e SB
l ) = fl(e SB
l ) × ¯wl(e SB
l ) = α2
4 ,
EEh(e SB
h (θ), θ) − EEl(e SB
l ) = α4 θ2 (θ2 − 1) − k2 4 α2 θ2 .
At θ = θ SB there is an upward jump in the expected earnings: 19
EEh(e SB
h (θ SB ), θ SB ) − EEl(e SB
l ) =
Since the difference is increasing in θ,
d
d θ
α k
α + √ α 2 + 4 k
EEh(e SB
h (θ), θ) − EEl(e SB
l ) = α4 θ 4 + k 2
2 α 2 θ 3
> 0.
> 0,
we conclude that the expected earnings difference is positive for all θ > θ SB .
Expected utility: The expected utility (EU) in job j is given by the expected earnings,
calculated above, net of the effort cost c(e SB
j (θ)):
EUh(e SB
h (θ SB ), θ SB ) = (α2 θ2 − 3 k) (α2 θ2 + k)
8 (α2 θ2 , (5)
)
EUl(e SB
l ) = α2
8 .
19 Alternatively, this can be seen also from our earlier result that the success probability at second-
best effort is higher in job h, combined with the fact that the wage after success in job h is also higher:
¯wh(e SB
h (θSB ), θ SB ) − ¯wl(e SB
l ) = 8 k
(1+ √ 1+16 k) 2 > 0.
27