Motivational Goal Bracketing - School of Economics and ...

So the expected utility of self 0 under narrow bracketing is larger than the one from broad
bracketing if and only if the expected cost of goal setting with narrow bracketing, ân [2 p +
2 (1 − p) µ], is lower than the one for broad bracketing, âb [2 p − p 2 + (1 − p) 2 µ]. Using
âb = ân
1−p
and rearranging:
1
1 − p
Rearranging again then shows that
≥ 2 p + 2 (1 − p) µ
2 p − p 2 + (1 − p) 2 µ .
U0(ân, ân; ē1, ē2) ≥ U0(âb; ē1, ē2) ⇔ p 2 − (1 − p) 2 µ ≥ 0.
Exploiting the monotonicity of the utility functions in β (which follows from ∂
∂β âb < 0, and
∂
∂β ân < 0, respectively), we obtain another equivalence relation in terms of the respective
minimum level of the present bias parameter for which an “easy” broad goal or narrow goals
are feasible:
U0(ân, ân; ē1, ē2) ≥ U0(âb; ē1, ē2) ⇔ p 2 − (1 − p) 2 µ ≥ 0 ⇔ βb ≥ βn. (16)
A.7 Derivation of â info
b
[Equation (12)]
There is an asymmetric effort response of self 2, depending on the outcome in the first task:
ē2(y 1 ) and e 2(¯y1). So the incentive constraint of self 1 is:
β ab (µ − 1) [Pr(loss|e1, ē2(y ), e
1 2(¯y1)) − Pr(loss|ē1, ē2(y ), e
1 2(¯y1)) ]
=1−p−(1−p) 2 =p (1−p)
≥ c − β p ¯y + [β (p ¯y − c) − β (1 − p) (p ¯y − c)].
The goal that makes this incentive constraint just bind is:
Subtracting âb from â info
b
and using âb = ân
1−p
gives (12).
â info
b
â info
b
A.8 Proof of Lemma 5
See Section 5.2.
c − β p ¯y + β p (p ¯y − c)
= .
β p (1 − p) (µ − 1)
− âb =
p ¯y − c
(1 − p) (µ − 1) ,
23