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

an interesting direction for future research.
Appendix
A Job assignments given inefficient effort levels
This section builds intuition for additional effects that inefficient second-best effort levels
can have on the job assignment rule. To illustrate these effects, we assume that job h
is locally more informative about effort than job l (Condition 2). We discuss here the
case where the principal implements the inefficient low effort in job l (e SB
l
Condition 1 is violated for job l). The driving forces for the other case (e SB
l
= e, i.e.,
= ē) are
similar to those in Section 2, and we leave the details for the proof of Proposition 5.
Suppose first that in job h Condition 1 is also violated for all ability levels, i.e., e SB
l
e SB
h
(θ) = e. Then no incentives are needed because the principal implements low effort
in either job. Hence, the job assignment problem reduces to maximizing the expected
revenue fj(e, θ) over j, resulting in threshold θ SB = ˆ θ(e, e) > θ F B = ˆ θ(ē, ē). In other
words, the downward distortion of effort in both jobs pushes the job assignment threshold
above the first-best level. As we saw in Section 2, if second-best effort levels in both
jobs are efficient (i.e., Condition 1 holds for all ability levels in both jobs), the principal
distorts the job assignment threshold in the opposite direction: assignment to job h leads
to an implementation cost saving effect that pulls the threshold below the first-best level:
θ SB < ˆ θ(ē, ē).
We expect both forces to be at work whenever Condition 1 holds for job h, but not
for job l. To illustrate that this is indeed the case, and to see how these two forces
interact, suppose that for job h Condition 1 holds for all θ. 16 Because no incentives are
needed in job l, placing the agent into job h cannot lead to cost savings. But it may be
cheaper to implement high effort in job h than it would be in job l. For that reason,
assigning the agent to job h and then implementing high effort might be worthwhile for
some θ < ˆ θ(e, e). Indeed, it is even possible that θ SB is lower than θ F B . When does this
occur? Assigning the agent to the higher level job and implementing high effort increases
output by fh(ē, θ) − fl(e). This has to outweigh the increase in implementation costs of
Ch (ē, θ). Hence, if there is a strict gain in profit from assigning an agent with ability
θ = θ F B to job h we know that θ SB < θ F B (by continuity). The following proposition,
which we prove formally in Appendix B, summarizes our results.
16 Dropping the qualifier “for all θ”, implies that multiple thresholds are possible. Then θ SB in the
explanations and results below refers to the lowest threshold for assigning the agent to job h (see the
proof of Proposition 5).
22
=