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

3.1.2 The continuous effort model
Our setup links promotion decisions to the cost of providing incentives: The principal
can use a promotion as a tool to reduce implementation costs or to extract more effort
from the agent. In our two-effort model, however, we do not have much flexibility in
varying effort levels in and across jobs. To elaborate on the effects that varying effort
levels across hierarchy levels can have, we extend our model to allow for continuous effort.
Expected outputs in jobs l and h are fl(e, θ) = α e and fh(e, θ) = max {α e θ − k, 0},
respectively, with k ≥ 0, θ ∈ [θ, ¯ θ] ⊆ R+. The cost of effort is c(e) = e 2 /2. Further, we
impose the following technical conditions:
Assumption 2
(i) α ∈ (0, 1) and e ∈ [0, 1].
(ii) k/α < ¯ θ < 1/α.
(iii) θ < √ α 2 +2 k
α
< ¯ θ.
The first two parts guarantee proper probability distribution functions. Specifically, (i)
implies that fl(e, θ) ∈ (0, 1). And the upper bound in (ii) ensures that fh(e, θ) < 1 for all
θ ∈ [θ, ¯ θ]. The lower bound guarantees that there exist interior ability levels θ ∈ (θ, ¯ θ)
for which positive output in job h is possible (i.e., fh(e = 1, θ) > 0). Finally, (iii) ensures
that there exists an interior first-best job assignment threshold.
The ‘warm up’ cost k ≥ 0 for the more complex job h allows us to vary parametrically
the difference in informativeness of output across jobs. As argued in Section 2.4, a job
higher up in the hierarchy (job h) often is more informative about effort. Assuming k > 0
captures this. Analogous to the two-effort case, the continuous version of the likelihood
ratio says how informative output in job j ∈ {l, h} is about effort: rj(e, θ) = ∂
∂ e fj(e,θ)
fj(e,θ) .
In our setup, the inverse likelihood ratios across jobs have a simple relation:
1
rh(e, θ) =
1 k
−
rl(e, θ) α θ .
So at k = 0 both jobs are equally informative about effort. With increasing k, job h
becomes more informative about effort relative to job l. This can be seen at an intuitive
level by comparing what success in a given job tells the principal about the possible
effort levels that could have led to this outcome. In job l this could have been any effort
level e > 0. In contrast, success in job h is more informative about effort, because a
success indicates that the agent put in at least an effort of k/(α θ). Similarly, in job h
output becomes less informative about effort with a higher ability level: the more able
an agent the more likely it is that he succeeds even with little effort.
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