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

Π SB
l (ē) (these are not the maximized second-best profit levels, as implementing high
effort is not optimal in job l, and may not be in job h either). From the proof of
Proposition 1 we know that ˜ θ < ˆ θ(ē, ē). Condition 2 implies that ˆ θ(ē, ē) < ˆ θ(e, e).
Denote by θ SB the lowest threshold for assigning the agent to the high ability job. We
now show that θ SB ∈ ( ˜ θ, ˆ θ(e, e)], for which we will divide the ability interval [θ, ¯ θ] into
subintervals over which Condition 1 may hold or fail for job h. Note that whether or
not Condition 1 holds for θ ≥ ˆ θ(e, e) or θ ≤ ˜ θ is irrelevant: all types above ˆ θ(e, e) are
assigned to job h, all below ˜ θ to job l.
Suppose first that Condition 1 also does not hold for any θ ≤ ˆ θ(e, e) in job h. Then
because ΠSB h (e, θ) = fh(e, θ) ≤ fl(e) = ΠSB l (e) for θ ≤ ˆ θ(e, e) and maxe ΠSB h (e, θ) ≥
fh(e, θ) > fl(e) = ΠSB l (e) for θ > ˆ θ(e, e) it follows that θSB = ˆ θ(e, e).
Suppose now that Condition 1 holds for θ ≥ ˜ θ in job h. Then, for these ability levels we
have Π SB
h
(ē, θ) ≥ ΠSB
h
(e, θ) and ΠSB h θ (ē, θ) > 0 (Lemma 1). Because Condition 1 does not
hold for job l, ΠSB l (e) > ΠSB l (ē) = ΠSB h (ē, ˜ θ), as ΠSB l (e) = fl(e) > ΠSB l (ē) = fl(ē)−Cl(ē).
Hence, it follows that θSB > ˜ θ. Moreover, ΠSB h (ē, ˆ θ(e, e)) > ΠSB h (e, ˆ θ(e, e)) = ΠSB l (e), so
θ SB < ˆ θ(e, e).
Finally, if Condition 1 holds only for some θ ∈
( ˜ θ, ˆ θ(e, e)].
˜θ, θ(e, ˆ e) , then a fortiori θSB ∈
Part (1b): To prove the if part suppose that fh(ē, ˆ θ(ē, ē)) − fl(e) > Ch(ē, ˆ θ(ē, ē)).
Together with Condition 2 this implies that fh(ē, ˆ θ(ē, ē)) − fh(e, ˆ θ(ē, ē)) > Ch(ē, ˆ θ(ē, ē)),
so Condition 1 holds at θ = ˆ θ(ē, ē). Thus, the maximized second-best profit in job h is
Π SB
h (ē, ˆ θ(ē, ē)) > Π SB
l (e). By continuity, θ SB < θ F B = ˆ θ(ē, ē).
To prove the only if part, suppose θ SB < θ F B = ˆ θ(ē, ē). By definition of the second-
best threshold Π SB
h (ē, θSB ) = fl(e). The next steps show that Π SB
h (ē, ˆ θ(ē, ē)) > fl(e)
follows from (Condition 3). Condition 2 implies that ˆ θ(ē, ē) < ˆ θ(e, e) so fl(e) > fh(e, θ)
for θ ∈ [θ SB , ˆ θ(ē, ē)]. From the previous step and Π SB
h (ē, θSB ) = fl(e) it follows that
Π SB
h (ē, θSB ) > fh(e, θ SB ), which is equivalent to Condition 1 holding at θ SB . By Lemma
1, Π SB
h θ (ē, θSB ) > 0. We now show that the profit never falls back to or below fl(e)
over the relevant range. By way of contradiction, suppose that there exist values
θn ∈ (θSB , ˆ θ(ē, ē)] for which ΠSB h (ē, θn) = fl(e), and let θ1 denote the smallest of these.
Because ΠSB h (ē, θSB ) = fl(e) and ΠSB h θ (ē, θSB ) > 0 it follows that ΠSB h θ (ē, θ1) < 0. This,
however, implies that Condition 1 is violated at θ1, i.e., ΠSB h (ē, θ1) < fh(e, θ1), which
leads to a contradiction because fl(e) > fh(e, θ1).
Part (2): Denote by ˜ θ the ability level where expected profits in jobs h and l would be
equalized if high effort was implemented in both jobs, as in the proof of Part (1a). We now
show that if Condition 1 holds for job l, then Condition 1 holds also for job h at θ = ˜ θ:
24