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246
O ’ D O N O G H U E A N D R A B I N
amount. Because welfare analyses are often the main contribution economists can
make, distinguishing between these two hypotheses is crucial. To further empha-
size this point, consider the more policy-relevant example of an economic analy-
sis of cigarette taxation that a priori assumes away self-control problems. This
analysis may (or may not) yield a very accurate prediction of how cigarette taxes
will affect consumption. But by ignoring self-control and related problems, it is
likely to be either useless or very misleading as a guide to optimal cigarette-tax
policy.
There are clearly many reasons to be cautious about welfare analyses that abandon
rational-choice assumptions, and research ought to employ the most sophisticated
methods available to carefully discern whether behaviors truly reflect harmful
self-control problems. But the existence of present-biased preferences is over-
whelmingly supported by psychological evidence, and strongly accords to com-
mon sense and conventional wisdom. And recall that our analysis in section 4
suggests that even relatively mild self-control problems can lead to significant
welfare losses. Hence, even if the psychological evidence, common sense, and
conventional wisdom are just a little right, and economists’ habitual assumption
of time consistency is just a little wrong, welfare economics ought be attentive to
the role of self-control problems.
By analyzing the implications of present-biased preferences in a simple model,
and positing some general lessons that will likely carry over to other contexts, we
hope that our paper will add to other research in developing a tractable means for
economists to investigate both the behavioral and welfare implications of present-
biased preferences.
Appendix
Proof of Proposition 1
(1) We show that when costs are immediate, for any period if naïfs do it then TCs
do it. Consider period
t
, and let
t
9
;
max
t
.
t
(
y
t
2
c
t
). Naïfs do it in period
t
only
if
by
t
2
c
t
$
b
(
y
t
9
2
c
t
9
), or
y
t
2
(1/
b
)
c
t
$
y
t
9
2
c
t
9
; TCs do it in period
t
if
y
t
2
c
t
$
y
t
9
2
c
t
9
; and
y
t
2
c
t
$
y
t
2
(1/
b
)
c
t
for any
b
#
1. The result follows.
(2) We show that when rewards are immediate, for any period if TCs do it then
naïfs do it. Consider period
t
, and let
t
9
;
max
t
.
t
(
y
t
2
c
t
). TCs do it in period
t
only if
y
t
2
c
t
$
y
t
9
2
c
t
9
; naifs do it in period
t
if
y
t
2
b
c
t
$
b
(
y
t
9
2
c
t
9
), or (1/
b
)
y
t
2
c
t
$
y
t
9
2
c
t
9
; and (1/
b
)
y
t
2
c
t
$
y
t
2
c
t
for any
b
#
1. The result follows.
Proof of Proposition 2
We show that for any period, if naïfs do it then sophisticates do it. Recall naïfs and
sophisticates have identical preferences. The result follows directly because naïfs
do it in period
t
only if
U
t
(
t
)
$
U
t
(
t
) for all
t
.
t
, while sophisticates do it in pe-
riod
t
if
U
t
(
t
)
$
U
t
(
t
9
) for
.
′ =
=
>
τ
τ
τ
τ
min
{ |
}
t
s
s
Y
Proof of Proposition 3
(1) We first argue that when costs are immediate, for any
t
,
t
9
such that
s
t
s
5
s
t
9
s
5
Y
,
U
0
(
t
)
$
U
0
(
t
9
). This follows because for any
t
and
t
9
;
min
t
.
t
{
t
|
s
t
s
5
Y
},
s
t
s
5
Y
only if
by
t
2
c
t
$
b
(
y
t
9
2
c
t
9
)
, which implies
y
t
2
c
t
$
y
t
9
2
c
t
9
.
Now let
;
min
t
.
t
tc
{
t
|
s
t
s
5
Y
}, so is when sophisticates would do it if they
waited in all
t
#
t
tc
. If
U
0
(
t
s
)
,
U
0
(
t
tc
) then
, so either
t
s
5
or
t
s
,
t
tc
. But using the result above, in either case
U
0
(
t
s
)
$
U
0
( ), which implies
U
0
(
t
tc
)
2
U
0
(
t
s
)
#
U
0
(
t
tc
)
2
U
0
( ). Given the definition of ,
only if
by
t
tc
2
c
t
tc
2
b
U
0
( ) or
2
((1
2
b
)/
b
)
c
t
tc
1
U
0
(
t
tc
)
,
U
0
( ). Given the upper
bound on costs , we must have
U
0
(
t
tc
)
2
U
0
(
t
s
)
,
((1
2
b
)/
b
) . It is straight-
forward to show we can get arbitrarily close to this bound, so sup
(
v
,
c
)
[
U
0
(
t
tc
)
2
U
0
(
t
s
)]
5
((1
2
b
)/
b
) . Hence, lim
b
→
1
(sup
(
v
,
c
)
[
U
0
(
t
tc
)
2
U
0
(
t
s
)])
5
0.
(2) Fix
b
,
1. We will show that for any
«
P
(0,
) there exist reward/cost
schedule combinations such that
U
0
(
t
tc
)
2
U
0
(
t
n
)
5
2
2
«
, from which the re-
sult follows. Choose
g
.
0 such that
b
1
g
,
1. Let
i
be the integer satisfying
(
«
)/(
b
1
g
)
i
,
#
(
«
)/(
b
1
g
)
i
1
1
, and let
j
be the integer satisfying
2
j
((1
2
b
)/(
b
1
g
))
.
0
$
2
(
j
1
1)((1
2
b
)/(
b
1
g
))
. Consider the
following reward and cost schedules where
T
5
i
1
j
1
3 is finite:
Under
v
and
c
,
t
tc
5
1 so
U
0
(
t
tc
)
5
2
«
, and
t
n
5
T
so
U
0
(
t
n
)
5 2
.
Hence, we have
U
0
(
t
tc
)
2
U
0
(
t
n
)
5
2
2
«
.
Proof of Proposition 4
(1) When rewards are immediate, by proposition 1
t
n
#
t
tc
. For any
t
,
t
tc
, naïfs
believe they will do it in period
t
tc
if they wait. Hence,
υ
τ
n
−
β
c
τ
n
≥
β
U
0
(
t
tc
),
which we can rewrite as ((1
2
b
)/(
b
)
y
t
n
1
U
0
(
t
n
)
$
U
0
(
t
tc
). Given the upper
bound on rewards , we have
U
0
(
t
tc
)
2
U
0
(
t
n
)
#
((1
2
b
)/
b
) . Since the bound
is easily achieved, sup
(
v
,
c
)
[
U
0
(
t
tc
)
2
U
0
(
t
n
)]
5
((1
2
b
)/
b
) , and lim
b
→
1
(sup
(
v
,
c
)
[
U
0
(
t
tc
)
2
U
0
(
t
n
)])
5
0.
(2) Fix
b
,
1. We will show that for any
«
P
(0,
) there exist reward/cost
schedule combinations such that
U
0
(
t
tc
)
2
U
0
(
t
s
)
5
2
2
«
, from which the result
follows. Let
i
be the integer satisfying (
«
)
/
(
b
9
)
,
#
(
«
)/(
b
i
1
1
), and let
j
be the
integer satisfying
2
j
((1
2
b
)/
b
)
.
0
$
2
(
j
1
1)((1
2
b
)/
b
) . Con-
sider the following reward and cost schedules where
T
5
i
1
j
1
3 is finite:
v
c
=
=
−
−
−
−
−
−
( , / ( ), / (
), . . . , / (
), , , . . . , )
( , , . . . , ,
((
) / ) ,
((
) / ) , . . . ,
((
) / ) , ).
ε ε β ε β
ε β
β β
β β
β β
2
1
2 1
1
0
i
X X
X
X X
X X
X X
X
X
j
X
X
X
X
X
X
X
X
X
X
X
X
c
X
v
c
=
−
−
+
−
−
+
−
−
+
=
+
+
+
( , , . . . ,
,
((
) / (
)) ,
((
) / (
))
, . . . ,
((
) / (
)) , )
( , / (
), / (
) , . . . ,
/ (
) , , , . . . ,
).
X X
X X
X X
X
X
j
X
X X
X
i
1
2 1
1
0
2
β
β γ
β
β γ
β
β γ
ε ε β γ ε β γ
ε β γ
X
X
X
X
X
X
X
X
X
X
τ
τ
s
N
tc
s
τ
=
τ
τ
τ
τ
s
N
tc
s
τ
=
τ
τ
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