Motivational Goal Bracketing - School of Economics and ...

Similarly, the incentive constraint of self 2 after he observes that self 1 shirked is:
ab β (µ − 1) [Pr(loss|e2, e1) − Pr(loss|ē2, e
1)] ≥ c − β p ¯y. (15)
1−(1−p)=p
Again this coincides with the incentive constraint for self 1, if self 1 expects self
2 to shirk. Note also that the incentive constraint (15) coincides with the single-
task/narrow-goals incentive constraint (5), so it binds at ab = ân. As ân < âb, self
t ∈ {1, 2} will work hard under âb, even if the self responsible for the other task does
not.
ii) Part (i) of Lemma 2.
The upper bound on an “easy” broad goal, ab ≤ ¯y, implies that âb is feasible if and
only if
iii) Part (ii) of Lemma 2.
c ≤ β p ¯y [(1 − p) µ + p] ⇔
1
(1 − p) µ + p βsc ≤ β.
The proof of Part (ii) is analogous to that of Lemma 1 and therefore omitted. Again,
the precise formula for the cutoff value is of little importance, but we state it here for
completeness:
βb =
A.4 Proof of Lemma 3
See Section 4.2.
A.5 Proof of Lemma 4
Comparing the narrow goal ân =
forward that âb = ân/(1 − p).
A.6 Proof of Proposition 1
[µ (1 − p) 2 + p (2 − p)] c
p [µ ((1 − p 2 ) µ + p 2 ) ¯y − 2 (µ − 1)(1 − p) c] .
c−β p ¯y
β p (µ−1) with the broad goal âb =
For narrow goals the (normalized) expected utility of self 0 is:
For the broad goal it is:
U0(ân, ân; ē1, ē2)/β = 2 (p ¯y − c) − ân [2 p + 2 (1 − p) µ].
U0(âb; ē1, ē2)/β = 2 (p ¯y − c) − âb [2 p − p 2 + (1 − p) 2 µ].
22
c−β p ¯y
, it is straight-
β p (1−p) (µ−1)