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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
CAPITOL INCOME TAXES WITH
HETEROGENOUS DISCOUNT RATES
Peter A.
Diamond
Johannes Spinnewijn
Working Paper 09-22
July 14,2009
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50 Memorial Drive
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Capital Income Taxes with Heterogeneous Discount Rates^
Peter Diamond*
Johannes Spinnewijn*
MIT
MIT
July 14, 2009
Abstract
With heterogeneity
in
Stiglitz result that savings
both
skills
and preferences
for the future,
the Atkinson-
should not be taxed with optimal taxation of earnings does
not hold. Empirical evidence shows that on average people with higher
higher rates.
skills
save at
Saez (2002) suggests that with such positive correlation taxing savings
can increase welfare. This paper analyzes this issue in a model with
correlation between ability
and preference
the same earnings
number
to the analysis
is
level,
the
that types
switch to a lower earning job.
who
for the future.
of types of jobs in the
less
than perfect
To have multiple types
economy
is
restricted.
at
Key
value future consumption less are more tempted to
We show that introducing both a small savings tax on
the
high earners and a small savings subsidy on the low earners increase welfare, regardless
of the correlation between ability
and preferences
However, a uniform
for the future.
savings tax, as in the Nordic dual income tax, increases welfare only
is
sufficiently high.
results
*We
There are also some
results
if
that correlation
on optimal taxes that
parallel the
on introducing small taxes.
Keywords: Optimal
Taxation, Capital Income, Discount Rates
JEL
Number: H21
Classification
Emmanuel
Werning and seminar parand the NBER for
valuable comments, and the National Science Foundation for financial support (Award 0648741). The research reported herein was supported by the Center for Retirement Research at Boston College pursuant to
are grateful to Jesse Edgerton, Louis Kaplow,
ticipants at
MIT,
Saez, Ivan
Berkeley, Stockholm University, the Stockholm School of Economics
a grant from the U.S. Social Security Administration funded as part of the Retirement Research Consortium.
The findings and conclusions are solely those of the authors and should not be construed as representing the
opinions or policy of the Social Security Administration or any agency of the Federal Government; or the
Center
for
Retirement Research at Boston College.
of Economics, E52-344, 50 Memorial Drive, Cambridge,
department
'Department
of
Economics, E51-090, 50 Memorial Drive, Cambridge,
MA
MA
02142, pdiamond@mit.edu.
02142, jspinnew@mit.edu.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
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Libraries
http://www.archive.org/details/capitalincometaxOOdiam
Introduction
1
The
Atkinson-Stiglitz (1976) theorem shows that
linear earnings taxes, optimal taxation
assumptions are
tween consumption and labor and
the available tax tools include non-
inconsistent with taxing savings
when two key
consumers have preferences that are separable be-
(1) that all
satisfied:
is
when
(2) that all
consumers have the same sub-utility function
of consumption. Empirical evidence suggests that on average those with higher
(Dynan, Skinner and Zeldes, 2004, Banks and Diamond, 2008).
at higher rates
relax the second condition
skill
skills
We
therefore
and analyze the taxation of savings with heterogeneity
We
and savings propensity.
save
both
in
consider both uniform and earnings-varying taxation of
savings.
This paper uses a simple model
restricted.
This sheds
light
on the
role of the positive correlation
argument
we
skill
and the
is
be taxed, whereas the savings of the low
independent of the correlation between ability
optimum has
factors, provided that the
A
is
of savings progressive in earnings. In a two-skills model,
earners should be subsidized. This result
productive job.
economy
and savings propensity. The paper provides an
find that the savings of the high earners should
and discount
in the
desirability of earnings-dependent savings taxes
between
making the taxation
for
which the number of types of jobs
in
the high skilled workers on the more
uniform savings tax, however, only increases welfare
if
that correlation
is
sufficiently high.
Our paper
builds
on the analysis
Saez (2002). He derives conditions on endogenous
in
variables to sign the effect on social welfare of introducing a uniform
subsidy,
when consumers have heterogeneous
an optimal non-linear earnings
net marginal social value
is
restricted to the
same
on savings increases welfare
if
or a
With
sub-utility functions of consumption.
tax, a small tax
either the
negatively correlated with savings, conditional on earnings, or
on average those who choose to earn
if
commodity tax
earnings.
less
By
save less than those
restricting the
the importance of the (exogenous) correlation between
number
skills
who choose
to earn more,
of types of jobs,
we analyze
and savings preferences
for the
taxation of savings.
Primary attention
and two
skill levels.
screening problem.
is
focused on a model with four worker types
Thus we
The model assumes
model where each worker can
setting
are examining a particular
with two discount factors
example of a multidimensional
the existence of two jobs, rather than the standard
select the
where workers with the same
-
skill
number
of hours to be worked.
1
This results in a
but different discount factors choose the same job
and so have the same earnings. With the introduction of earnings-related savings tax
:
A
limited
number
of jobs
was assumed
in
Diamond
(2006).
rates,
We
they are subject to the same tax rates.
assume that
optimum both
at the
high-skill
types work at the high-skill job and that redistribution from high earners to low earners
is
Given these assumptions
the important redistribution.
social welfare increases
with the
introduction of a tax on the savings of high earners and with the introduction of a subsidy
on the savings of low earners. The
relative frequencies of the four types in the population
plays no role in the derivation of this result, conditional on the assumed structure of the
optimum.
The underlying assumption
is
that those valuing the future
work than those valuing the future
less,
more are more
conditional on the disutility of work. This
that an incentive compatibility (IC) constraint just binding on a high
value for the future
not binding on a high
is
willing to
skill
worker with low
skill
worker with high value
Earnings-dependent taxes and subsidies on savings allow an increase
for the future.
tempted to switch
jobs.
all
are
high earners) eases the binding IC constraint
transfers resources from the high saver to the low saver for
it
who
In particular, introducing taxation of savings of high earners
(and transferring the revenue back equally to
since
by
in redistribution
targeting types in a given job with saving preferences different than those of types
just
means
whom
the IC constraint
binding. Introducing a subsidy on savings for low earners (financed by equal taxation on
low earners) also eases the binding IC constraint by making switching to the lower job
is
all
less
attractive to the high earner with low savings. In extensions, the case for taxing the savings
of high earners appears to be
earners.
more robust than the case
While the focus of the paper
is
for subsidizing the savings of
the introduction of small taxes,
we
low
also consider
optimal taxes under stronger assumptions. 2
The assumption
is
in line
that those with less discounting of the future are
with standard modeling, representing preferences by u (x)
alternative specification j-u (x)
types with higher 5
+
u
(c)
—
v (~/n
;
)
prefer to save more, but to
l
+
more
S,u (c)
willing to
—
v
work
would imply the exact opposite. That
work
less.
We
on education and
is,
examine some empirical
support for our assumption, using data from the Survey of Consumer Finances (SCF).
find that conditional
An
(z/rii).
We
age, people with higher discount factors tend to earn
more. To proxy for the discount factor, we use reported savings and the time horizon people
report having in
mind when making spending and savings
to revisit the positive correlation
between
skills
decision.
We also use these proxies
and savings propensities.
This paper contributes to the literature on the optimal choice of the tax base and the
joint taxation of labor
and capital incomes
in particular.
Banks and Diamond (2008) review
the literature on the inclusion of capital income in the tax base.
The
analysis assumes rational savings by
relevant for retirement savings policies.
all
Gordon (2004) and Gordon
workers. Concern about too
little
individual savings
is
also
and Kopczuk (2008) argue that capital income reveals information about earnings
thus should be included in the tax base.
people with different
an income
tax,
skills
may be
and
ability
and
Blomquist and Christiansen (2008) analyze how
different preferences for leisure
separated with a commodity tax.
who cannot be
The
separated with
four-types model with hours
chosen by workers has been studied by Tenhunen and Tuomala (2008), which calculates a set
of examples, but explores the analytics only in two- and three-type models.
We
both welfarist and paternalist objective functions.
We
examples to some of our results below.
in a two-types
They
consider
relate the results in their calculated
focus on the four-types model since the result
model, while striking, does not seem relevant for policy inferences. 3 While
the focus of this paper
is
on capital taxation, the intuition generalizes to the taxation of
other commodities for which the preferences are heterogeneous, since this heterogeneity
may
impact the labor choice as well (Kaplow, 2008a).
The paper
two
jobs.
is
Section 2 sets up the model with four types and
organized as follows.
Section 3 characterizes respectively the
referring to
best and the restricted
first
no taxation of savings and an 'equal job, equal pay'
restriction.
first
best,
Section 4
introduces incentive compatibility constraints and characterizes the second best including
Optimal savings tax rates are
the introduction of earnings-varying savings tax rates.
considered.
also
Section 5 considers a uniform savings tax, rather than one varying with the
level of earnings.
For comparison, Section 6 reviews a two-types model. Section 7 discusses
empirical support for the assumptions and Section 8 has concluding remarks.
Model
2
We
consider a model with two periods. Agents consume in both periods, but work only in
the
first
period. Preferences are
and work. Denoting
earnings by
z,
first
assumed to be separable over time and between consumption
period consumption by x, second period consumption by
> 0,u" <
u'
producing output
factor
We
and
preferences satisfy
U(x,
with
c,
and
z.
An
v'
>
c,
z)
0,v"
=
>
u (x)
0.
+
An
5u
(c)
—
v (z/n)
agent's ability
,
n determines the
disutility of
agent's preference for future consumption depends on the discount
5.
consider heterogeneity in both ability n and preference for future consumption
5.
''Kocherlakota (2005) provides an argument for regressive earnings-varying wealth taxation. He analyzes
skills, which is not present in this model.
a model with asymmetric information about stochastically evolving
Although robust insights
for
optimal taxation have been derived in models with two types,
considering heterogeneity in two parameters in a model with two types implies perfect cor-
between the two parameters. The inference based on a simple two-types economy,
relation
although simple,
we consider a
fij
may
therefore be misleading.
and welfare- weights
factors 6
and
1
We
four-types model.
and
with Sh
Sh,
The
rj^.
>
Si.
In order to allow for imperfect correlation,
denote the four types by ll,lh,hl,hh with frequencies
first
two types have low
ability ni, but differ in discount
The second two types have high
also differ in discount factors
Si
and
rih
factor
hh
hi
lh
11
low ability ni
in
the economy, h and
The
/.
The output from a job
high-ability types can hold either job.
redistribution to the low-skilled types
balanced that
all
is
>
n-i,
Si
of the worker's type, while the disutility of holding a job varies with ability.
types can only hold the low job.
n/,
low discount
factor Sh
There are only two jobs
,
Sh.
high discount
high ability
n h with
ability
sufficiently
is
independent
The
We
low-ability
assume that
important and the type mix sufficiently
high-skilled workers hold high-skilled jobs at the various
optima analyzed.
This requires a restriction on the weights in the social welfare functions and the population
distribution,
We
which we do not explore.
begin with the
produced on a job
The
first
is
first best,
the
same
which
for
differs
everyone holding the job.
best has the property that there
consider a restricted first best (the term
constraints, the
term
'restricted'
from the usual treatment
'first
is
We
in that the
assume a
output
linear technology.
no marginal taxation of savings.
Then we
best' refers to a lack of incentive compatibility
means limited tax
tools,
but not limited by IC constraints)
with zero taxation of savings and the requirement that everyone holding a job receives the
same pay (no taxes based on
social welfare
We
identity, only
on potential earnings).
We
calculate whether
can be improved by taxing or subsidizing savings.
then turn to the second best, with taxes based on earnings, not potential earnings,
so that there
is
an incentive compatibility constraint.
restriction, thus preserving the condition of equal
potential gains from taxing or subsidizing savings.
pay
We
assume a zero taxation of savings
for equal
work. Again
we ask about
—
First Best
3
In the
is
first
worker
best, each
is
assigned to the matching job and the social welfare function
maximized with respect to the
consumption
type-specific
With the
periods and the job-specific output levels, subject to a resource constraint.
weight of type
of
ij
i]
,
t
the
first
and second
levels in the first
welfare
best solves:
£ /y^j (u
Maximize I|C ,,
+
[xtj]
6jU
-
[cy]
v
{z l /n,})
(1)
subject
E+
to:
^2
fij
i
x ij
+#
Forming a Lagrangian with A the Lagrange multiplier
£ = Yl
f*rt*i
u
(
^+
~
Si u fad
v
\
Zl l
_1
type
ij
~~
<
z%)
the resource constraint,
for
n^ ~ A
5Z fa
(
Xii
+R
' lc
H ~
we have
z i)
i,j
hi
We
cy
define the net marginal social value of
first
period consumption for an individual of
[xij]
-
as
9ij
Along the relevant portion of the
gij
=
+
VtjU
A.
social welfare optima,
and u
v'
ifulu
=
[zi/nA
L
fihVih)
[x^]
=
=
(/«
5jRu'
+
fih)
[cy]
A
=
we have the
for all i,j,
fuVuu'
and
+
[xu]
following properties:
fihVih.u' [x ih ]
,
rii
for
both the high-skilled and the low-skilled jobs. The net marginal social value of first period
consumption
for
each type equals
the required output for a given job
and the saving of each type
is
is
undistorted. Given that
independent of an individual's type, the earnings are
marginally distorted upward for one discount- factor type and downward for the other type,
since v! [xu]
^
u' [x^], unless
the welfare weights satisfy
T]
d
=
>] lh
.
The output
is
undistorted
'on average' though.
3.1
Equal Pay
Restricted First Best:
for
Equal Work and
No
Taxation of Savings
If
the (after-tax) earnings on a job, y
; ,
is
restricted to be type- independent
and savings can
not be taxed, there are further constraints, which we approach using the indirect
utility-of-
consumption function,
This function
R\.
iuj [y,
u>j
[j/j
=
R]
subject
satisfies
max u
x
to:
[x]
+ R~
1
c
+
5jU
—
y.
[c]
For later use, we note that
dwj
u
\x]
dy
j£
where
Sj [y,
We
all
R]
is
= R- 2 cu'[x} = R- (y-x)u'[x} = R-h J
1
iy,R}u'[x}
the savings function of someone with discount factor
5j.
continue to assume that the welfare weights and population fractions are such that
high skilled are on the more productive job at the optimum.
The
restricted first best
solves the following problem,
Maximize^
]T fljrjij ( Wj
R]
[y n
-
v [zjn,])
(2)
subject
E+
to:
J2
fij (Vi
~
zi)
<
Forming a Lagrangian, we have
hi
hi
The
first
order conditions
(FOC)
are
5Z fctfijU' [Xij] =
A^/y
3
Yl fitful
3
\
zil ni\ l n
i
=
A
3
for
i
—
h,
I.
5Z^'
3
Recalling the definition of the net marginal social
utility,
g tJ
=
rj^u' [xij]
—
A,
the population-weighted values add to zero at each job,
}] fij9ij =
for
i
=
h,l.
3
Thus, the welfare weights determine the direction of desired redistribution (given the equal
pay condition) between workers on each job. Also,
vl [xij]
The
FOC
for
job outputs,
z,,
are the
=
5jRu'
same
absence of savings taxation,
in the
for all i,j.
[cij]
as given above.
Restricted First Best with Small Earnings-Dependent Savings
3.2
Taxes
Given the observability of earnings, small
linear taxes
on savings (collected
in the first period)
could be set differently for high and low earners. This can for instance be implemented by
IRA and
the rules on retirement savings accounts, like the
desire to redistribute can be
met by a small
401 (k) in the US.
them by
(local)
on savings by workers on
linear tax or subsidy
a given job with the revenues returned equally to
The
raising net-of-tax earnings
on the
job.
Differentiating the Lagrangian with respect to a savings tax rate r,
level
i/i,
evaluated at a zero tax
=
A
(
d7~
The impact
H
of a savings tax
level:
fa
(tt
?/i,
the derivative can be written
=
-fr
~ xv)
H fvW'
)
on the Lagrangian
revenue constraint and the impact on
by
on those with earnings
11 fa
utilities.
is
made up
Using the
\
x ij\
of
FOC
(Vi
two
~ x u)
•
pieces: the
with respect to
impact on the
j/j,
multiplied
as:
(%'"'
(^
~
A ) X 'J
=
11 fa9U x H-
3
3
Recall that
Hfij9ij
=
for
i
=
h J-
3
This implies that a tax on the savings by the two types on a given job increases welfare
the savings of the one type towards which redistribution
compared
The
is
desirable saves sufficiently
if
little
to the other type.
welfare weights imply the desired direction of redistribution within productivity
types and so the signs of g }J
With equal incomes and
.
>
Cih
Thus,
if first
period
utilities get
Cu
the same weights for both types, n il
implying a desire to redistribute to the high saver. In contrast,
the
same weights
for
both types,
redistribute to the low saver.
we have
rj
hh u' [xhh]
= Vm u
given discount factors,
i]
'
ht
\
If
x m}
=
r)
we have
different discount factors,
if
=
r)
lh
,
<
gu
second period
<
g,h,
utilities get
u Si
=
n lh 5h, the signs are reversed, implying a desire to
there
is
no desire to redistribute
for high (low) skill types
m general,
with uniform weights for
i]
{Vih u
'
l
—
x ih}
Va u> x u})\
u we do not satisfy both conditions.
,
Second Best
4
We
draw a distinction between
restricted first-best analyses
and second-best ones based on
the absence or presence of IC constraints involving taking a job with lower productivity
(the reverse having been ruled out by assumption).
observability of productivity.
IC constraints are binding.
The prime
We
is,
the distinction depends on the
issue in second-best analyses
is
determining which
start with the further restriction, as above, that savings not
be taxed. With no taxation of savings and equal pay
imitating the other discount rate type
if
That
who
is
for
equal work, the IC constraint of not
holding the same job does not bind. Similarly,
a high productivity worker were to take the low productivity job, the person imitated
would be the one with the same discount
need not be such a worker
for
a high
skill
factor.
Imitation
is
a misnomer here since there
worker to optimize savings while taking a low
skill
job given the assumed policy tools and information.
We add the critical
assumption that earnings distribution issues are
that at the second-best
period consumption g tJ
optimum
=
sufficiently
important
(with IC constraints) the net marginal social value of
r/^u' [x l j]
— A
is
first
negative for both of the worker types holding
the high-skill job and positive for both of the types holding the low-skilled job. Without a
binding IC constraint, this condition could not hold at the optimum as noted above.
Assumption
1
The net marginal
9hj
social values of first period
<
®,9ij
>
0,
for j -h,l.
consumption
satisfy
We
No
Second best with
4.1
Taxation of Savings
assume that the Pareto-weights and population fractions are such that
workers work at the high-skilled job and the desired
is
sufficient that at least
one IC constraint
Maximize,,,,
subject
J2 f^q^
E+
to:
Wh
wi
Forming a Lagrangian with
and assuming that
C =
Y fan
(
WJ
at the
t
y"
-
v
R]
~
v
[yh,
-v
R]
~ ~0 <
fij {Vi
[yh,R]
°
]
(
wh
/n h > W[
[z h
\
[yh
R]
R]
[yu
-v
-
3)
[zi/n h ]
v [zi/n h
]
.
the Lagrange multiplier for the corresponding IC constraint,
fj,j
v
l
]
[zjn,])
/n h >
[z h
optimum each worker
R ~
level of redistribution to lower earners
binding.
is
(vjj [yi}
J2
high-skilled
all
Zi /
n^ - A
Y
i,j
assigned to the matching job,
is
fa ( y
>
we have
~ *)
hi
+ 22 ^j wj
(
[
y/"
R ~
_ w
i
v z >J n h\
]
[
\y<->
R +
]
v z i/ n h])
[
3
Since the first-period consumption of type hj
period consumption of type
the
Ij,
Y fWhj u
Y
'
[
FOC
switching to the low job equals the
if
with respect to earnings are
YW
-Y
x hj] +
[xhj]
~
X
j
;
hsiij 11 '
[
x ij]
Y Ai
=
°'
=
°-
j
vj u> x u\
[
- x Yf<j
3
Given the definition of the net
Y^
Y
i)
y ii'
—
[x,j]
Y^ ^ <
Y M>
= -
9ho
u>
A, this implies
°>
3
3
=
fato
3
The population-weighted
=
social utility g,j
^'"'
°-
3
values add to a positive expression
Y
i,j
f'J 9 'J
=
Y
Pi ("' Xl i\
l
3
LO
&
~ u Xh
'
\-
>
°'
first-
That
is,
transfers which
would be worth doing without an IC constraint are
restricted, raising
the social marginal utilities of consumption, on average, above the value of resources in the
hands of the government. Since the IC constraints are on the high
more
redistribution from the high earners to the low earners
IC constraints
straints
is
Given the equal pay constraint,
binding, and
it is
it
is
skilled types,
on average
desirable.
follows that only one of the
the one on the low discount factor type.
To
IC con-
see this consider
the difference in consumption utility from different incomes,
A [yh,Vu £j. R] = w
This difference in consumption
utility is increasing in
dA[yh ,yi,Sj,R]
u
35
The
difference in labor disutility does not
constraint
is
[c kj ]
-
The low discount
more tempted
[yi,
R]
the discount factor,
u
[cij]
>
0.
depend on the discount
binding on the low discount factor type,
factor type.
therefore
- w3
R]
[Vh,
j
it is
factor.
Thus
if
the IC
not binding on the high discount
factor type values earnings in the first period less
and
is
to switch to the less productive job.
Second Best with Small Earnings-Dependent Taxes on Savings
4.2
As above, the
sign of the welfare impact of introducing a small linear savings tax or subsidy
depends on the welfare weights.
Given observability of earnings, the small linear tax on
savings could be different for high and low earners.
tax on savings (collected in the
first
=
is,
f
]P fhj
(y h
- x hj
-
)
J
^7 =
That
A
A
(
5Z
/« ( y
'
" x «)
)
n
on
utilities,
Y^ fhjVhj u>
" Yl fljlll J u
the impact on the Lagrangian
constraint, the impact
welfare impacts of introducing a
period) are obtained by differentiating the Lagrangian
(with savings taxation included and the tax rates
O^r
The
is
'
i
x hj]
M
made up
set at zero):
(Vh
- x hj) -
fa ~ x 'j)
+
W
Vi u '
1
r "l
[-
[x h i] (y h
(w
-
-
r »)
,
•
of three pieces: the impact on the revenue
and the impact on the binding IC constraint.
11
- xu )
The
FOC
for earnings are
53 fhJ9hj + V-iu'
=
[xhi\
0,
3
j
Multiplying these by the earnings level at the job,
dC
and substituting, we have
j/*,
= 53 fhjQhjXhj +
u x hl] x hl,
'
l-h
[
J
= 53 fa9U x U ~
E<9r
;
Substituting for n
t
u' [xu]
FOC
from the
^
we
for earnings,
— = fhh9hh (xhh -
t^
x "] x "'
t
find
>
Xhi)
and
=
KOTi
The
JihQih [xih
)
0.
signs follow from the assumption on the net social marginal utilities
in savings
behavior by types ih and
il
for
i
plays no role in signing these expressions.
Proposition
skill
- xu <
1
jobs and g^j
that falls
At the second
<
0,gij
on high earners
>
is
0,
best
for j
h,
is
I.
The
optimum, assuming that
=
differences
correlation between skill and discount
The Proposition immediately
all
h,l, then introduction of a
welfare improving;
savings that falls on low earners
One can
—
and the
high
follows.
skill
workers hold high
small linear tax on savings
and introduction of a small
linear subsidy
on
welfare improving.
increase the redistribution from high earners/high savers by taxing savings, but
increasing net-of-tax earnings just enough that the high earners/low savers remain indifferent
to job
change and thus the binding IC constraint
is
unchanged.
One can
also increase
the redistribution towards the low earners/high savers by subsidizing their savings, but
decreasing net-of-tax earnings such that the low earners/low savers remain indifferent so
that
it
does not become more attractive for the high earners/low savers to take the low job.
12
Second Best with Optimal Linear Earnings-Dependent Taxes
4.3
on Savings
We
have considered the introduction of small savings taxes on high and low earners. Part of
the interest in this analysis comes from the possible link to the signs of the optimal taxes.
FOC
Derivation of the
it
for the
optimal linear savings taxes
straightforward;
is
we show
that
matches the signs of the small improvements given the additional condition that workers
save more
One
first
if
4
the after-tax return to savings are higher.
difference in analysis
that changes in both the earnings and savings taxes have a
is
order effect on tax revenues through the behavioral change in savings. In
n
revenue from a linear savings tax
units, the tax
discount factor
5j
and earnings y equals T
l
denote optimal savings
dependence on the
Sj [y,,
R(l —
A
second difference
sv u
depends on the
level of the savings tax.
l^\° K
M F«J
i_ Ti
{
Forming a Lagrangian, and assuming that
at the
is
no
that the relative
is
marginal increase in the savings tax compared to the
of a marginal increase in earnings
dr
For notational convenience,
r,)].
by s^. (Given preference separability, there
r,)]
effort to achieve gross earnings.)
size of the utility loss of a
—
period
on the savings of workers with
levied
[yi,R(l
sd
x
first
That
utility gain
is,
i-ndVi'
optimum each worker
is
assigned to the
matching job, we now have
C =
Y
The
htfij
FOC
H
for
[W. (!
~
+
fi t
T i)
R \
v zil n i])
\
(wi [y h (1
-
,
rh )
-
x
Y
Ai
tt yi
~ z^ ~
R]-v [z h /n h - w
]
t
[y u (1
TiS v [W>
-
r
(
)
R)
i
l
-
T i)
R }}
+ v [zi/n h
])
.
earnings are
Y
Y
fhjVhj u ' x hj]
\
+
htfljU*' [Xij]
fru'
-
[xhi]
fliU
-
[Xu]
Y ^
X
-
Yl Mlj
"
2
(
(
1
Th
l
~FT
dy
=
)
°'
- Ti-
dy
'Consideration of earnings-dependent nonlinear savings taxation would raise the issue of the degree of
complexity that
is
interesting for policy purposes.
L3
FOC
The
Y
for savings
tax rates are
ftJ
IhjVhjU
+
[Xhj]
L
i
ftu' [xta]
Y
-r^
h
'h
•!•
J
X fhJ
s
{
^+
^
1
Th
=
o,
OTh
I
3
SU
Denote by
=
/?/,
respectively the high
= R(l — n)
R(l - r^) and Ri
and low
skill
find that the optimal linear savings tax
fhhQhh {Xhh
~ xm) =
rh
is
r^}
0.
the after-tax returns to savings for
Combining the
types.
+
first
order conditions as before,
we
such that
Y
X fhj
s hj
\
- -Q^Shi + ^y!r Rh
(
\
4)
and
fihQlh (xih
The
~
Xll)
=
T
iY X
f'J
Sl J
I
~
~d^
S"
+
TR
Rl
^
'
|
left-hand sides in equations (4) and (5) correspond to the welfare changes of introducing
earnings-dependent taxes on the high earners and low earners respectively. Thus,
of the terms in brackets
if
on the right-hand
the introduction of a small tax
additive, -^-
<
1,
and so
—
s tj
is
>
—
dRi
n
u
is
-p^-Su
>
is
for
i
=
and vice
h,l.
versa. Since preferences are
Hence, a sufficient condition
for
that savings are increasing in the after-tax return,
the second best
optimum, assuming that savings are increasing in the
after-tax returns, all high skill workers hold high skill jobs,
and g hj <
the optimal linear savings tax
and negative for
is
positive for the high earners
0,gij
>
for j
with hours chosen by workers.
CES
They
in their
Figure
derive the
1.
h,l,
(2008)
(2008) consider two-, three- and four-types models
mechanism design optimal
preferences with varying correlations between discount and
marginal taxes shown
—
the low earners.
Mechanism Design Optimum in Tenhunen and Tuomala
As noted above, Tenhunen and Tuomala
suming
sum
-
Proposition 2 At
4.4
the
positive, the optimal linear tax is positive
welfare- improving
the right-hand side term to be positive
^ii
side
if
For
allocations as-
skill,
with implicit
but very high correlation, they find that sav-
all
ings are implicitly marginally taxed for the high skill worker with low discount factor (type
3),
savings are implicitly subsidized for the low
2),
and there are no other marginal savings
skill
worker with high discount factor (type
distortions.
1
I
With very high
correlation, the low
05
——
i
t*
1
type
1
1
1
....... type 2
1
1
1
1
t
/
/
025
/
/
Wffffnm>tllM»t«»tltl»l«»tlHllllltM»JH»»l
1
i
/
•025
0
0:
5
correlation
FIGURE
1
Marginal tnx on savings
worker with low discount factor (type
skill
m
the welfarisf cnse
1) is implicitly
taxed, which also happens in the
two type model, which has perfect correlation. The potential relevance of the pattern we find
is
enhanced by
their findings.
mechanism design optimal
In contrast with our model, the
allocation allows distortion of the savings of each type separately.
is
not too high, on average the savings tax
low
skill
positive for the high skill
we
levels in
main
allow different ability levels for the two high earner types. Second,
all
four types.
Third,
High Earners
assume that the two types with high
high
skill
is
skill
allow different
the analysis extends to three
sufficiently
more
1
continues to hold.
skilled,
skill
if
skill
which IC constraint
is
\r>
As long
skill
as the
type with low
may be more tempted
required. For given skill of type hh,
binding holds
higher than h M (> n hh ), where the cut-off level n M
model above, we
the type with low discount
the type with high discount factor
is
skill.
than the high
However,
to switch to the low earner job for which less output
this reversal of
In the four-types
have exactly the same
type with high discount factor has higher
discount factor, Proposition
factor
we show how
we
propositions.
workers and jobs, preserving the various assumptions.
Different Ability Levels of the
,
for the
types.
discount factors for
is
and negative
consider three extensions to highlight the extent of robustness of the
First,
n hh
as the correlation
Robustness
4.5
We
is
As long
is
when
the ability level n M of type hi
such that the IC constraint
is
just
binding on both types,
{wi
[y h
,
R]
-
v {zh /h M )}
~
W'i
[Vi,
R]-v {zi/hhi)} =
{wh
With
is
[yh,
R]-v (zh /nhh )} - {w h
[y t
,
R]-v (zi/nhh )}
v [z/n] convex, the difference in labor disutilities between jobs, {v (zi/n)
n^
decreasing in n. Hence, for values of
for the
n«
higher than
the IC constraint
is
—
.
v (zh/n)},
more
stringent
high discount saver. In this case, a savings subsidy on the high earners and a savings
tax on the low earners are welfare improving. This
among
Different Discount Factors
the opposite of Proposition
is
the High and
earnings and no taxation of savings, a high
Low
With
Savers
1.
job-specific
worker considering switching to the low job
skill
chooses optimal savings without needing to match any particular worker holding the low
job. Thus, with the
is
same
skill
always higher for the high
the discount rates
among
among
skill
worker
the low
introducing taxation of savings
high earners, the gain from switching to the low job
skill
among
who
has lower preference for savings, regardless of
We
workers.
continue to have a welfare gain from
high earners as in Proposition
1.
Subsidization of savings of low earners will continue to generate a welfare gain as long
as the discount factor of the high-skill low-saver
of discount factors
among
holders of the low
consumption of the high-skilled low saver
on that job
if
small enough relative to the distribution
is
skill job.
Denoting by
taking the low
skill job,
x^i the first-period
the
FOC
for earnings
is:
3
The impact
of a savings tax on low earners
7^7
= Yl
is
f'j9ijXij
~
hu
'
[x'hi]
xM
.
3
Comparing the consumption
among low
in the
earners, this derivative
IC constraint with a weighted average of consumptions
is
savings of low earners in Proposition
negative (and the gain from the subsidization of the
1
_
continues to hold)
__
and only
if
xM >
xi,
where
Ej fij9ijXij
Yl j
With the
if
fijSij
net marginal social values assumption, gy
16
>
0, for j
—
h,
/, x~i is
a proper weighted
—
—
average of the Xy. Since the discount rates for the marginal high
high to meet this condition,
we
type
skill
may
well be too
consider the tax of the savings of higher earners to be a more
robust policy conclusion than the subsidization of the savings of low earners.
We
are exploring two extensions to the basic model, one with an education choice
and
one with a continuum of worker types. In both cases, preliminary work suggests the same
pattern of greater robustness of the taxation of higher earners than of the subsidization of
lower earners. 5
Three Ability Levels, Three Jobs
We extend
optimum
skilled
We introduce an intermediate skill level in the model.
the assumption that welfare weights and population fractions are such that at the
the high skilled are on the most productive job to also have
all
on the intermediate job.
We
all
the intermediate
again consider the case in which agents
may be tempted
Only two downward constraints are relevant
to switch to jobs designed for less skilled people.
though.
First, as above, for
constraint
two agents with the same
but different discount factors, the IC
skill,
slack for the type with the higher discount factor
is
with the lower discount factor. The reason
A
both
[2/1,2/2, 5j,
— —
dA[ym ,yi,6j,R]
— -
>
-
=
R]
- w3
Wj [yi,R]
and
binding for the type
that, with
is
=
if it is
—
[y 2
—
R]
,
d^{yh ,y
— ,8J- ,R]
do
l
,
.
>
-
for
i
—
m,
I.
ad
Second, with v [z/n] convex, we have a similar condition for the difference in the' disutility
of labor between jobs.
That
is,
A'
with
[z h
dA'[z h ,zi,n]
Thus,
for
,
zi,n]
= {-v
,
=
v
[z h
[z h /n\
two agents with the same discount
low-skilled job
is
/n]
zh
+
-
v [z,/n\
,
,
v [z[/n\ z
factor, the
L
)
2
jn <
0.
IC constraint of switching to the
slack for the type with the highest ability
if it is
satisfied for the
type with
the intermediate ability switching to the low-skilled job and for the type with highest ability
switching to the intermediate job.
That
is,
the local IC constraints imply the global IC
constraint.
In a similar
way
'With heterogeneity
in education.
If
as for the four-types model,
in
discount factors, people
we can
who discount the
only education determines a worker's
factors than low-skilled workers.
17
skill level,
set
up the Lagrangian
future less
may
for the
choose to invest more
high-skilled workers have higher discount
,
.
The two
constrained maximization problem.
Wi
wi
The impact
[y
R]-v [z h /n h
[y h
,
m
R]
,
-
v
[z
m /n m
relevant
]
> w
t
[y
}
> w
t
[y h
IC constraints are
m
,
-
R]
R]
-
m /nh
v
[z
v [zi/n m
]
]
of the introduction of earnings-dependent savings taxes on the Lagrangian equals
respectively
—
=
a
OT
h
fhh9hh {xhh
- xM >
)
0,
DC
JmhQmh \%mh
~7\
DC
=
77-
or
This implies that Proposition
The
-
xu )
<
<
^i
0.
1
continues to hold for the high earners and the low earners.
1
following Proposition applies for the intermediate earners.
Proposition 3 In a model with
redistribution
three ability levels
on savings that
linear tax (subsidy)
if
fihQih (xih
%ml)
from
(to) the
falls
and
three jobs, the introduction of a small
on the intermediate earners
is
welfare improving
intermediate earners to (from) general revenues
is
welfare
improving.
Proposition 3 implies that there
is
a single sign change
in the response of welfare to taxing
savings as a function of earnings. This result generalizes for more than three jobs as well,
the welfare weights are non-increasing in
The
skill.
if
savings of workers with earnings below
a given level are subsidized, the savings of workers with earning above that level are taxed.
The
result
depends on the assumption that types with the same
skill
are at the
same
job,
which becomes increasingly strained with many jobs.
The
single sign
change of the welfare impact of introducing a savings tax as a function of
earnings also holds for the optimal linear earnings-dependent savings taxes
[x]
=
optimal savings tax rate
is
CRRA preferences,
u
distributed across jobs, f }]
-t^-
=
0,
jz~i an d 7
fij
1-
With logarithmic
—
f3
for Vz, j.
fj
for \/i,j
and
~
xu)
first-period
workers
if
u
=
T,
i
=
log
[x],
the
s,j
=
-j^yi
and
satisfies
^ \fij-~Xu.
consumption x io
18
[x]
they are uniformly
Since for logarithmic preferences
the optimal tax on the savings of earners at job
—
preferences,
strictly increasing in the earnings of
fih9ih (Xih
With
<
when workers have
—
j^-yi,
we
find the following
,
,
expression for the optimal savings tax,
-
{Si
fh
A
S h ) (r\ ih
7"i
EiVi&;\»
Since
<$/,
>
5i,
with the welfare weights non-increasing
linear savings tax
is
+
l
5
in skill, this implies that the
optimal
increasing in earnings,
dTi
>0
7T
oyi
-
Second Best with Uniform Taxes on Savings
5
Proposition
1
what
leaves the natural question of
the tax rate on savings
oc
- = dc
a
Ot
dr h
oc
,
^r- =
for
both earnings
levels.
~
JhhQhh {Xhh
.
.
Xhl)
+
.
~
Jlh9lh {%ih
Xii)
Oti
In contrast with the earnings-varying tax
on savings, the correlation between
count factor plays a role here.
If
there
is
no desire to redistribute within a job,
r
hh9hh
r
Jih9ih
Jhh
=
y^
=
^
ST~^ r
y
,
Thj9hj
,
Y~^
>
Jlh
,
c
=
gu, for
— - vJhh
=
Jij9ij
##,
,
f' h
^
I
l-H
r
h,l,
r
then
-i
[x
fJ-iU
I
,
=
i
M
\
1
u F«J
•
Thus, the welfare impact of a change in the uniform tax on savings equals
It is
Adding the
we have
responses to the two separate tax changes,
"H
do with a Nordic dual income tax where
same
required to be the
is
to
9£
(
»T
\L,-/m
fhh
,
r
n
/
flh
N
A
n]/
Lj/y
/
convenient to write this as
9£
_
. „__„ w
flh
,
r
„
!
,
{x „
_
^J^^n
'
fhhTLjfh]
with
_
u'\x hl (x hl
]
U' [x a ] {Xu
19
- x hh
- Xih)
)
-
\
i
j
skill
and
dis-
Since Xu
>
xih,
S ,g n
_
(_j
{-jj^jrO ~
S , 9n
1
j
The
sign of this expression depends on the distribution of types and the ratio of the weights,
Q.
That
1h
>
12 j fhj
Q >
is,
y/"
Hj
,
1
is
a sufficient condition for a positive correlation between aspects,
to imply that introducing a savings tax increases social welfare.
1
,
flj
Assuming homothetic
u
Chl
\f
This expression
.
{
preferences, so that
y—
=
we
[x]
Thus,
the relative risk aversion 7
if
between
is
larger than
is
factor
smaller than
(i.e.
y/
hf
t
1,
>
if
7
is
Proposition 4
with
CRRA
greater than
is
no desire
// there is
-preferences,
A uniform
1
then
y/" ,
0,
>
1
and a
positive correlation
implies that |^
1
)
when
the correlation
to redistribute within a job,
positive.
is
one and the correlation between
ability
small subsidy on savings increases welfare
If there is
—
g,f,
gu, for
no desire
if the
if
and discount factor
the correlation between ability
As with the
some
tax returns to savings and by
the savings of type
and the after-tax
we
Sij
interest rate.
interesting cases.
CAR A
When
h,l,
relative
is
the relative risk aversion
g^ —
negative.
is
gu, for
1
=
h,l, with
if
and
positive (negative).
ij
Denote by
R T = R(l—r)
the after-
as a function of after-tax earnings
Setting the derivative of the Lagrangian with respect to r
- xM ) +
fihQih {xih
~
xu)
=
\J d
t }
U
''For
is
find the following condition for the optimal linear tax,
fhh9hh (xhh
negative.
7
earnings- varying taxes, the sign result for introducing a uniform tax matches
that for optimal linear taxation in
equal to zero,
is
—
and discount factor
and discount factor
within a job,
to redistribute
if
i
logarithmic preferences, a uniform small tax (subsidy) on savings increases welfare
only
If
negative, |^
is
a uniform small tax on savings increases welfare
greater than one and the correlation between ability
Corollary
prefer-
This implies the following proposition.
1.
risk aversion is smaller than
positive.
CRRA
For
the sign of |^ depends on the size of the correlation and the magnitude
1,
of the earnings difference between jobs. Conversely,
negative
becomes
f2
find
and discount
ability
the expression for
-,
equal to one for the log utility function.
is
ences, u
,
= £M
^M
preferences,
§y
the correlation
is
is
Xf
i:j
\
s^
(-/
O
ij
- ~-Su
+ w^-Rt
dyi
1
-i
dRT
negative when the correlation between ability and discount factor
dc
is positive if the absolute risk aversion is sufficiently sma
dr
positive,
^
20
is
If
sum
the
positive
of the terms in brackets
Proposition 5
If there is
logarithmic preferences or
is
we have
the introduction of a small uniform tax
if
logarithmic preferences and
savings
positive,
is
CRRA
welfare improving. This
within a job, g,h
to redistribute
preferences with 7
positive if the correlation between ability
<
1,
=
g,i,
for
the optimal linear
and discount factor
is
is
7 <
preferences with relative risk aversion
no desire
CRRA
is
that the optimal uniform tax
is
the case for
1.
i
=
h,l, with
uniform tax on
positive.
Two Types
6
Using a model with two types of workers, a high-skilled worker with high discount factor and
a low-skilled worker with low discount factor, the empirical finding of a positive correlation
between
skill
and savings rates
is
treated as a perfect correlation. In this model,
positive (negative) marginal earnings taxation then there
(negative) marginal savings taxation.
The
corollary
is
if
design
optimum has the same
property.
to realistic diversity in the economy.
The source
that introducing savings taxation
is
If
The
full
is
is
mechanism
seem robust
of this inference does not
With two-dimensional
earners with both high and low savings rates.
there
a gain from introducing positive
redistribution goes from the high earner to the low earner.
a gain
if
heterogeneity, there are low
a high earner can imitate the savings of a
low earner with the same savings propensities, a savings tax on the low earner does not work
to discourage the high earner from imitating.
Thus
to
model less-than-perfect
correlation,
we use the
four-types model with high and low earners with both high and low concern for
the future.
We
The proof parallels
model.
is
in
to as
Diamond
1
report the results for two types here to
and
(2003).
same
result for the
contrast with the four types
mechanism design optimum, which
consider the second-best Pareto frontier with the types referred
2.
Proposition 6 In
sign (nj
We
that of the
mark the
— n 2 ),
level is welfare
a two-types model without taxes on savings
the introduction of a small linear tax (subsidy)
improving
if
and only
if
and with sign{5\ —
on savings
=
at a given earnings
earnings at that level are marginally taxed (subsidized).
Corollary 2 In a two-types model without taxes on savings and with sign
the introduction of a savings tax
62)
on the lower earner
is
(5i
—
62)
=
sign (nj
welfare improving if redistribution
goes from the higher earner to the lower earner.
The
proposition combines the properties of the mechanism design optima in the two
separate two-types models with heterogeneity in one dimension.
21
When
both types have the
— n 2 ),
same discount
but different
factor,
are marginally taxed or subsidized
(Mirrlees, 1971).
the earnings of the potentially imitated type
abilities,
the ability of that type
if
when both types have the same
Similarly,
same discount
Stiglitz, 1976).
lower or higher respectively.
is
factor, distorting savings does not help separate the
Similarly,
both types have the same
if
marginal taxation.' However, when the two types
both the marginal taxation
subsidization) of savings
is
but different discount
ability,
type are marginally taxed or subsidized
factors, the savings of the potentially imitated
the discount factor for that type
lower or higher respectively
is
both
(or subsidization) of earnings
both types have the
If
two types (Atkinson and
earnings are not subject to
ability,
differ in
and discount
ability
factor,
and the marginal taxation
(or
used to separate types.
Preferences and IC Constraints
7
Above we used the
utility functions
u
[x]
+
8jU
[c]
—
v
[z/rij].
This family of
has the property that those with higher savings rates (larger values of
work
to increase
for
a given amount of additional pay.
But that
utility functions {u [x]
reversed
pay.
If
-
+
5jU
[c])
—
/5j
v
=
[z/rij]
u
[x]
/8j
+
u
—
[c]
utility functions
5j) are
more
not the only
is
which the savings and labor supply decisions can be connected.
is
if
willing
way
in
For example, with the
v
[z/rij],
the relationship
those with higher savings rates are less willing to increase work for additional
we had assumed
this class of functions, then
of desirable savings taxes in Proposition
saver would imply that
it
is
1
we would have reversed the pattern
having the IC constraint bind for the high
-
not binding for the low saver, implying, in turn, that there
should be a subsidy of savings for high earners and a tax on savings for low earners. More
generally, a one-dimensional family of separable utility functions,
any pattern between the variation
in the interaction
in the subutility function of
between consumption and labor. This
empirical basis for distinguishing which case
is
more
to write utility in the form employed does not, by
U
[x, c, j]
[<p
is
different. Generally,
(but not borrowing against
role for discounting
it).
So modeling
on both aspects
-
in
can have
consumption and the variation
raises the question of identifying
relevant.
itself,
That
it
shed light on
work precedes
pay,
is
its
While the formal model has consumption and work simultaneous
experience in real time
,z,j],
an
standard practice
empirical reality.
in the first period,
which precedes spending
it
continuous time would naturally have a similar
saving and willingness to work. But that does not rule
out the possibility that the preferences of high and low savers differ in other ways than just
'This can be considered an implication of the Atkinson-Stiglitz theorem, since there
everyone has the same subutility function over
tax
is
first
not allowed, the two types can be usefully separated by an earnings tax.
earnings of the potentially imitated type
is
is
separability
period earnings and consumption. Notice that
positive
if
and only
22
if
if
The marginal tax on
that type saves less for the
same
and
a savings
the
earnings.
a discount rate on otherwise identical
and
utility
There are reasons based on casual empiricism
for
disutility functions
(assuming additivity).
supporting the appropriateness of using the
Modeling savings with rationality and discounting combines
formulation employed above.
underlying preferences and issues of self-control. As discussed in Banks and
Diamond
(2008),
psychological analyses suggest these are mixed together.
We
does not apply to working as well as to consuming
whether that
is
working
for later
consumption or working to influence future work opportunities. That
is,
working
(at
with disutility) involves self-control
which would be captured
human
may
in the
a future payoff.
for
those with less difficulty in self-control
-
may show
standard
see
And
no reason to think that
this
a job
saving involves self-control. So
greater willingness to both work and save,
utility function expression.
capital investment involves discounting in a similar
way
In a richer model,
to savings decisions
and so
generate the pattern in the standard model structure, although formal modeling would
distinguish between
It
financial capital accumulations.
not easy to find data applying directly to this issue.
is
answer
human and
is
whether, for a given level of
A
greater labor supply functions.
in circumstances
The
we want
question
to
those with higher savings rates tend to have
skill,
complication in looking at data comes from the differences
with age, which we address by considering separate age
We
cells.
report a
few correlations supportive of a positive correlation among savings propensities, discounting
and earnings
abilities
using the Survey of
on time horizon and savings
Before turning to the data,
we have been
levels.
Education choices
rates,
which matter
for
both
them.
education
skill
A
and discount
model
also report
is
skill
some
is
correlations with
of the complications in interpreting the
also a complication in interpreting the
both ex ante
"skill"
Presumably, the
and discount
factor,
level of
and can not be used
affect
wage
completed education
is
on average. In addition to affecting ex
affect one's discount rate thereafter.
rate
data across education
and discount rate and then
in
Thus education
is
a proxy
a simple way to distinguish between
further difficulty in interpreting the correlations
variable while skill
is
that education
is
a discrete
continuous and varying within education classes.
Three-period Model
7.1
We
briefly consider a three-period version of the two-period
for later taxation.
may
some
effort.
reflect
increasing in both ex ante
skill,
We
considering. This will bring out
data at different ages. There
post
Consumer Finances, which includes some questions
work
practice.
we
8
set
up a three-period model with the same preference structure as the two-period model
analyzed, assuming the
same discounting
for the utility of
consumption and the
disutility of
For discussion of correlations with experimentally measured discount rates, see Chabris et
23
al,
2008.
,
work and allowing
for different skills in the
u
+
[xi]
8u
\x 2
]
+
5
2
u
two working periods.
[c]
-
-
v [zi/m]
5v
[z 2 /?i 2
]
Considering the special case with
u
we have the time
[x]
=
series of
log
and v
[x]
=
[z/n]
k (z/nf
+1
/
{(3
+
1)
consumption and earnings behavior, assuming that hours are
a control variable and the marginal tax rate
is
constant over time (with derivation in the
Appendix):
X2
=
—
= 5R
%2
--
(5R)-
~2
z 2 /n 2
--
1/0 (
1+1//3
n2
\
(5Ry 1/0
1/73
( n2
\
zi/n-i
Thus, those with higher discount factors have more rapidly growing consumption but
rapidly growing earnings (for given
earnings and work effort
The
may
skills).
This suggests that the cross section pattern of
be different at different ages.
cross-section pattern of time series behavior
may be more
illuminating than that of
single-period behavior since the single-period cross section patterns are dependent on the
pattern over time in
skills.
be concerned about income
If
we added uncertain
effects in
rates of return to the model,
we would
also
skills.
But
these issues, just reporting simple correlations.
Data Analysis
7.2
We
full
both consumption and earnings choices. Consideration
of wealth or wealth/earnings ratios are also affected by the time series pattern of
we do not explore
less
first
SCF
consider the relations
among
discounting, saving, education and age.
We
use the
panels in 1998, 2001 and 2004, containing information on 13,266 households in total.
For savings rates we consider two proxies.
worth to earnings
regularly or not.
9
for households.
The sample
is
The
first
proxy
The second proxy
is
is
the logarithm of the ratio of net
whether people report that they save
divided into age-education cells (5-year age groupings from
among different statements. We use for this second proxy whether subjects confirm
"Save regularly by putting money aside each month." The results are similar (with sign
''Subjects can choose
the statement:
reversal) with the statement "Don't save
-
usually spend about as
24
much
as income."