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Institute of
EDUCATIONAL POLICY AND THE
ECONOMICS OF THE FAMILY
Abhijit V. Banerjee
Working Paper 03-09
November 2002
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MAY
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2003
LIBRARIES
I
Educational Policy and the Economics of the Family
Abhijit V. Banerjee*
November, 2002
Abstract
This paper analyzes the implications of alternative ways to model decision-
making by
families for educational policy.
We
show that many of the policy
impli-
cations associated with credit constraints cannot be distinguished from the implications of models of the family that differ from the conventional Barro-Becker model.
We
then argue that
it is
the combination of credit constraints and non-conventional
preferences that provides a robust basis for government intervention to promote
educational investment.
Keywords: Family; Intergenerational Transactions; Education. JEL Codes: 015;
016: D13.
*I
am
grateful to Esther Duflo. Paras Mehta. Antonio Rangel. Debraj
Angrist and T.N. Srinivasan for their comments.
1
Ray and
particularly Josh
Introduction
1
The
family plays a crucial role in decisions about investment in education, at least in
developing countries.
The
reasons are simple: Education
is
expensive and there
very limited scope for borrowing in order to invest in education. Education
because free and functioning public schools do not exist
they do they are often free only
cannot borrow to invest
in
in
name (The Probe
education because
human
in
many
places,
is
is
only a
expensive
and even when
Report, 1999, Kochar, 2000).
One
capital provides very poor collateral,
and, in any case, credit markets function very poorly in developing countries. Most of the
financial investment in education therefore has to
As a
result,
be funded by the family.
most theoretical and empirical studies of investment
take a stand on
how they think
the family makes
its
decisions.
This
in education
is
have to
an issue especially
with respect to decisions about education that are taken when the children are young and
parents are, at least nominally, in charge of financial decisions for the family. There
however, very
little
is,
agreement about how families make these decisions. Even a cursory
glance at the literature on investment in education reveals a wide variety of views ranging
from the Barro-Becker model (Becker, 1981; Loury, 1981; Mookherjee and Ray, 2000),
where parents and children share a single unified
generation model, where the family
their children
is
who repay them with
utility function, to the
purely a nexus for transactions
old age care (Kotlikoff
pure overlapping
— the old "lend"
and Spivak, 1981; Cox, 1987;
Cox, 1990; Cremer, Kessler and Pestieau, 1992; Cox and Jakubson, 1995, Barham et
1995).
There
is
also a range of formulations that are in
between these two
1
for
example because they
like
al,
—formulations
where parents put some weight on the income, consumption or human capital of
children,
to
their
the "warm-glow" they get from what they have
given the child. 2
However, to the best of our knowledge, there have been no systematic attempts to
'See Behrman. Pollak and
Taubman
(1982).
and Ermisch and Francesconi (2000)
for
examples of
formulations of this class.
2
See
Glomm
and Ravikumar (1992). Galor and Zeira (1993). Banerjee and
and Moav (1999).
Newman
(1994).
and Galor
understand whether and in what way these things matter
making each
the positive and normative implications of
The
goal of this paper
is
to
fill
—
in other words,
what are
of these alternative assumptions.
We
a part of this gap in the literature.
present a simple
one-good, two-sector growth model where goods are produced using skilled and unskilled
labor.
Everyone
by investing
are perfect
in
endowed with a unit of unskilled
human
capital.
We
either financed
is
human
capital,
production of
credit markets,
more
we make the
intensive in
and that there are no human capital
is
skill
has to be accumulated
and investment
human
capital
externalities.
They
potentially controversial.
is
in
human
getting the
plausible and standard 4 assumption that the
we assume that the production function
assumptions
but
by the state 3 or by the parents of the child who
In addition,
skills is
labor,
consider both the case where the credit markets
and the case where there are no
capital
Finally,
is
are
than the production of goods.
for skills exhibits
We
decreasing returns
recognize that each of these last
made mainly
for strategic reasons.
The
assumption of non-convexities in the educational production function has been shown by
Galor and Zeira (1993) to have a number of strong implications when combined with the
—
assumption of imperfect credit markets
in particular, there are
poverty traps and public
Moreover, they argue
investment in education that can increase steady state output.
that their results are independent of the
the non-convexity
is
large enough. Since
way we model parental
we
preferences as long as
are interested in the differences between the
implications of different types of parental preferences, avoiding non-convexities clearly
helps.
in
We
do, however, take comfort in the fact that the evidence for non-convexities
education
is
rather mixed.
Psacharapoulus (1994) argues on the basis of evidence
from a number of countries that the returns are highly concave.
On
the other hand, a
couple of detailed studies from the U.S. find evidence of small non-convexities (Angrist
and Krueger, 1999; Card and Krueger, 1992).
human
3
capital externalities
Where
was driven
in part
Likewise, the decision not to allow for
by the
fact that the
consequences of such
the state could be the local government.
'This assumption goes back at least to
Uzawa
(1965).
Rebelo (1991),
education policy in growth models also makes this assumption.
in his
well-known study of
been widely studied and
externalities have
in part
by the
seems to be very
fact that there
micro-evidence for the presence of large externalities of this type. 5
little
We
study this model under different assumptions about family decision-making.
distinguish between the models of the family along three dimensions:
altruism
is
private utilities of future generations
sum
of the
— the formulation associated with the work of Barro
Second, the nature of the inter-generational contract within the family:
(1974).
complete
many
whether
First,
perfect in the sense that each generation maximizes a discounted
in the sense of
generations does
binding
it
all
We
present and future generations;
span? Third,
is
if
Is it
incomplete,
how
there symbolic consumption in the sense of
people caring directly about consumption or investment outcomes, and not only because
they are a source of utility to someone
their
son going to Harvard because
— the archetypal example
it is
is
someone caring about
Harvard, and not because of he
will
be
richer
because of having been to Harvard.
We
take as benchmark the case where there
no symbolic consumption,
credit markets.
point of time
contrasts with
In this
is
i.e.,
model
is
perfect altruism but
the so-called Barro-Becker model, combined with perfect
(see section 3.1) the
investment
in
human
capital at any
independent of current family wealth and parental preferences.
what we
find
when we shut down the
credit markets in section 4.1
both parental preferences and family wealth matter
this case
no contracts and
This
—
in
for educational investment.
However, when we allow for more general preferences (section 4.2) this simple contrast
between the perfect and imperfect credit market cases no longer holds.
when
there are credit constraints, there
effect.
only
if)
However, both of these
there
is
a family wealth
can
arise
effect
true that
and a parental preference
even with perfect credit markets
if
(and
symbolic consumption of educational investment. This raises questions
about using these
effects as
The steady state
5
effects
is
It is still
a way of identifying credit constrained households.
implications of the
benchmark model, given
See Acemoglu and Angrist (1999) and Duflo (2000).
educational externalities using cross-country data.
Bils
in section 3.2, are
perhaps
and Klenow (1998) make a case against
even more striking. 6
irrespective of
where they started,
the steady state
no inequality
First, there is
will
end up with the same
up with the same
starts with
an abundance of them. 8
run
while a proportional tax on
even
increase the rate of return on
Third,
.
we have either
human
human
capital reduces
7
Second,
an economy which
subsidies have no long-
human
capital investment
capital necessarily raise investment.
when we shut down the
perfect altruism, no symbolic consumption
altruism and perfect contracting. 9
some
level of skills as
lump-sum taxes and
Section 5.1 shows that these results also hold
that
level of education.
then redistributed as a lump-sum educational subsidy. Finally, policies that
if it is
long as
Every dynasty,
unique: Economies that start with very few highly skilled people (and
is
therefore very few teachers) will end
effects,
in the steady state:
To put
things in perspective,
credit markets, as
and no contracting, or no
it is
worth underscoring
of these results are very strong indeed. In particular the idea that an increase
in the rate of return
on education should always
raise the level of
investment
is
not at
all
obvious, since raising the rate of return also raises the price of teachers, which discourages
investment.
Section 5.2 turns to the case of symbolic consumption of educational expenditures. In
this case
none of the
credit markets
five properties listed
above necessarily hold with or without perfect
and irrespective of whether there
is
no altruism or perfect altruism: Long-
run inequality and multiplicity of steady states are now
taxes and subsidies have long-run
6
it is
7
8
The steady
state, as
well-known,
is
is
effects.
Proportional taxes on
an inadequate proxy
for
and lump-sum
real possibilities
what happens
human
capital
may
after the short run,
but
the only one we have.
Remember
there
no randomness
is
This says that even
AIDS seems
if
there
is
in our
model.
a shock which wipes out a huge part of a country's
human
capital (as
to be in the process of doing in sub-Saharan Africa), the country will recover (actually
it
says less than that because even a unique steady state need not be a global attractor).
9
The
fact that the
non-convexities,
is
in
absence of credit markets does not make any difference
in the
long run, absent
Loury (1981). The similarity of the two polar cases of no altruisms and complete
contracts and perfect altruism and no contracts, arises from the fact that both lead to a perfect alignment
of the incentives of parents
and
children. In this sense, these results are related to the intuitions behind
Becker's famous "Rotten Kid Theorem" (Becker, 1981).
sometimes
raise
average investment and increases in the returns to
reduce investment in education by
human
capital
making teachers expensive.
Finally section 5.3 looks at the case of imperfect altruism: While this case
to solve because
we have
may
is
harder
to deal with time consistency issues, the results appear to be
similar to those in section 5.2.
Taken together these
handicapped by the
fact that
elementary questions
capital (as
results suggest that the educational policy debate
like
recommended,
we have no
whether
for
it
model of the
would be good to
family.
substantially
Even
raise the rate of return
relatively
on human
example, by Foster and Rosenzweig, 2000), seem to depend
On
on what we believe about preferences.
of credit constraints
settled
is
the other hand,
by themselves may be
it
suggests that the importance
slightly exaggerated, at least if
we mainly
care
about the long run.
2
The Basic Model
Production Technology: Consider a world where
types of
human
inputs- -skill
and unskilled
labor.
The
there
final
is
only one final good but two
good
is
produced by combining
the two types of inputs using a technology:
y
where
=
f(H,L,a)
H and L are, respectively, the amount of skill and the amount of unskilled labor used
in production,
and a
is
a parameter describing the nature of the technology.
We
assume
that the production function exhibits constant returns with respect to the two inputs
together but diminishing returns with respect to each individual input. This formulation
also imposes the restriction that
all
levels of
of the insights of the paper by Mookherjee
human
capital are perfect substitutes
and Ray,
cited above,
is
—one
that relaxing this
assumption introduces an additional source of persistent inequality.
Labor Supply: Each person
labor. In addition, they
own a
in the
certain
economy
number
is
assumed to own one unit of unskilled
of units of
skills,
which correspond to the
amounts they invested
The
no consumption
is
work
or
—
all
it.
consumption
they do
We
when people work and earn an income.
is
associated with
for
capital.
Life Cycle: Agents in this economy live for three periods. In the
of their lives, there
period
human
in
acquire
is
in the
next period, when they no longer have wage income. In the second
and each cohort
is
the level of
=
S(h)
capital
We
The
labor.
human
capital.
Human
Capital:
of
and
(1
human
capital
is
is
assumed to be
The population
human
>
0,
The second and
gf >
0,
-§^=
< 0,J^ >
third assumptions
capture the idea that the family atmosphere matters
—though at a diminishing
easier to acquire skills
therefore find
it
markets for labor and
skill
wf
Except
in the
this
paper we
skills,
will
make
next section we
Policy Instruments:
The
^
>
human
where
0,
is
last
skilled parents find
it
assumption imposes the
not outweighed by the reduction
now have more education and
We
the assumption of perfectly compet-
will
make
assume that there
is
financed partly or entirely by a tax on the earning
(1
—
-y)w^)s(h t+ i, h
lump-sum and partly a function
t
)
is
w^ and
an educational subsidy of
This
+
being
that of
capital.
E=
("fivj^
t
the extreme assumption that there are
where
sum
and h~
1
The next two assumptions
with the price of labor at time
no capital markets and no assets other than human
is
< 7 <
easier to educate their children.
itive
.
is
js(h,h~) units
acquiring the
and
0,
is
us that the cost function
tell
fact that the next generation will
Markets: Throughout
1
is
—children of
rate.
condition that the increased cost from increasing h
coming from the
capital
who
capital of the parent of the person
increasing and convex, therefore ruling out non-convexities.
in cost
size 1.
^y)s(h,h~) units of unskilled labor, where
§
therefore
produced using a combination of human capital
cost of acquiring h units of
—
assume that
s(h,h).
disutility
This income can be spent on current consumption or saved and invested
stays unchanged over time
and unskilled
The second
assume that labor has no
period of their lives they also give birth to exactly one child.
Human
skills.
period
first
eo
+
&\E,
the amount the family spends on education.
of the taxpayer's
human
taxes are of course standard while the tax on
6
members
capital:
human
of society that
T=
is
1
To+Tihtiuf
.
partly
Lump-
capital earnings will turn out to
.
be a simple way to introduce redistributive taxation. In order to get sharper
limit the
number
we
of cases,
will, for
results
the most part, focus on the case where
we
and
start
from a situation where the family was already spending some amount on education and
then look at the
effect of
very small changes in the taxes and subsidies. This allows us to
avoid the issue of corner solutions. 10
Preferences:
level)
outcomes.
We
allow people to get utility both from private and collective (family-
Private outcomes includes both material consumption and symbolic
consumption. Material consumption
is
middle-aged and when one
to a person of generation
consumption
Pt+i his
"warm glow"
like the
who
is
old:
Uf(c
t, is
t
is
the consumption of the one final good
when one
the utility from material consumption accruing
Uf,
,Pt+i),
where
in the last period of his
ct
is
his
consumption
in middle-age
and
Symbolic consumption covers things
life.
of giving to one's children (Andreoni, 1989), pride in having children
are well-educated or rich
and the pleasure of being able
to say that one's children go
to an expensive school. In other words, the utility from private outcomes
may
take the
form:
Uf =
where h
t
is
the
human
son has a Ph.D."),
The
utility
,
{
1W H +
consumption
{^w H
+
(1
—
y)w L )S(h t+ i)
r
from collective outcomes,
is
While
U
restrictions. In particular,
10
is
the
amount spent on the education
therefore
mU M + sl/f.
t
s=l
utility, i.e.
relatively general,
we have assumed
it
U = Uf +
still
t
additive separability at a
educational subsidies and general income subsidies since even
it
clff
imposes a number of stringent
This formulation does, however, come with the cost that there
the family could always cut back on what
Uf =
£«£.+£ w&-
represent an individual's total
this formulation
that of the next ("my
the next generation ("my daughter
is
is
is
given by
s=l
let
- 7)» l )5(V0),
level of
to Exeter"). Total private utility
&? =
Finally
(1
capital level of current generation, ht+i
c (+1 is the
drives a Porsche" and
("my son goes
Uf{ht, ht +1 ,ct+1
if
is
number
of different
no way to distinguish between
the subsidy were earmarked for education,
was already spending on education.
levels
and we have ruled out children caring about the welfare
A special case of this is
=
case where c
=
1, S\
<5i
altruism" to distinguish
6 S y^ 0, for s
>
1, which
the standard Barro-Becker formulation of altruism: This
=
=
6 and 6 S
=
<5 S
for s
>
from the case where either
it
we
"imperfect altruism".
call
where symbolic consumption
this yields the
of parents.
ruled out,
is
i.e.,
s
=
We
1.
6S
^
An
and
standard Barro-Becker preferences, which
will call this case "perfect
Ss or at least one of 8 S and
,
even more special case
m—
is
the
is
a
checked that
It is easily
1.
utility
one
is
function of the form:
U = J2^U M (c t+s ,p t+s+ i).
(1)
t
s=0
Inter-generational Contracting: Since each generation may have to
rely
on edu-
cational investment decisions taken by previous generations, inter-generational contracts
can potentially play an important role in this analysis. Suppose, for example, generation
t
has to rely on generation
t
—
human
for its
1
does not care about the well-being of the next generation,
possible solution
is
that generation
with old-age care (p t ) in return
for
t
But
capital investment.
if
why would
generation
it
t
invest?
the
human
capital investment (h t ). This
depend on how much he gets to consume,
will
how much he
has a budget constraint, clearly depends on
man
capital of the next generation (h t+ i). In other words,
between generation
between generations
The same
t
and generation
t
+1
and
issues arise even
t
+
One
contracts with the previous generation to provide
would
t
+ 1,
will
it
c( ,
it
clearly
provide some incentives to invest. However, the amount of old-age care generation
happy to provide
—\
t
is
which, given that he
have to invest in the hu-
will
depend on the contract
which, in turn, will depend on the contract
2, etc.
when there
is
some
altruism.
The
basic point
is
that the
preferences of the current generation over future outcomes can be very different from that
who
has control over that decision. Of course, there could be contracts
(implicit or otherwise)
between generations which would minimize these issues but there
of the generation
are obvious difficulties with contracts that span
The
many
generations.
fact that there are externalities across contracts suggests that the ideal contract
may be one
that covers
all
the generations. Under such a contract, {p t c
,
t
,
/i (
+i}t=o°
w^
<
be simultaneously chosen to maximize
as a real contract:
It is
is
^
X
t
U
t
Obviously, one should not think of this
.
perhaps best thought of as a norm that binds
particular dynasty. This
This
T,
is
what we
will call
a
all
generations of
complete contract.
One
obviously an extreme assumption.
alternative would be to go to the
i.e.,
no
contracting. Then each generation would have to be assigned control rights over a
set
other extreme: Simply assume that inter-generational contracts are impossible,
We
of decisions.
impose the sensible
restriction that this set should not include decisions
about actions taken when they are not
of generation
t
D
We
living.
define the potential control set
={pt-i,Pt,Pt+i,Ct-i,<hiCt-uht,ht+i,ht+2}-
t
would be an allocation of a subset of
D
to generation
t
gets to take the exact corresponding set of decisions.
11
A third alternative,
contracting
a subset of
—we assume that generation
D U D t+i
t
average of their
1
where A
.
in
utilities,
Note that
=
in
this
t
and
t
+
1
actual control rights
such a way that each generation
which allows some scope for contracting,
to
is
assume bi-generational
are jointly allocated control over
such a way that each pair of generations get to take the exact
corresponding set of decisions.
and t —
t
The
They then choose the contract
U + XU +\,
t
t
to
maximize a weighted
taking as given the contract between generations
makes no contracting a
t
special case of bi-generational contracting
0.
The Benchmark: The Beckerian Model
3
3.1
Properties of the Short
Run
Equilibrium
Here we consider the case of our model defined by standard preferences,
i.e.,
no symbolic
consumption and perfect altruism combined with perfect credit markets and no contracting.
1.
The
first
two assumptions imply that the preferences of each generation are given by
Perfect credit markets
"In other words,
if
amount
generation
get to decide on a vector
Xt+S
.
t
to
assuming that anyone can borrow and lend as much
gets to decide on a vector
X =
t
{pt- c t
.
h
t
}
then generation
t
+
s will
as they
want at the market
Under
interest rate r.
this assumption, each
dynasty faces
the inter-temporal budget constraint:
uJt+i
=r
t
(uj t
-c -pt+Wt+ht{l-T )w" -(>yw" + (l-~f)w{')s(h t+u h ){l-e ) + e -T
t
t
l
),Vt.
1
(2)
where w t
is
the starting wealth of the
investment
ej) represents the
in
t
th
The term (710^+ (1 — j)wf')s(ht+ i, h
generation.
the next generation's
human
capital
measured
t
){\-
in units
of the good.
Since there
is
no contracting, each generation
consumption when middle-aged
Maximizing the
utility
dUm {c
=
7
<+
t
(1
is
its
and the education
(pt+i),
1
6r t dUm{ct+i,Pt+2)/dc t+i
,Pt+i)/dpt+i
=
6rtdUm (c t+1 ,p t+2 )/dp t+ 2
-
tKMVi, ht)(l - (7<! +
(1
future expected level,
its
own
of its child (h t+ i).
under Equation 2 gives
=
us:
(3)
ei )
(4)
- 7)^+1)^2(^+2, ht+i)(l -
completely determined by the
endowment h t and
assigned control rights over
,p t+ i)/dc t
t
-[tu&iCl -Ti)
Note that h t +i
and old
function given in Equation
dUm {c
(
(c ( )
is
last equation,
ei)].
taking as given the current
h t+ 2- Remarkably no
utility
term enters
this
equation. This has two implications: First, there are no income effects, since the choice of
h t+ i
is
unaffected by whether the marginal utility of consumption
parental preferences do not affect the decision to invest in
results follow
is
human
high or low. Second,
capital.
Both these
from the fact that under perfect capital markets, the decision to invest
human
capital can be analyzed entirely in terms of its net present value of
person
who
invests in his son's education today can immediately
10
income
in
—the
borrow back the implied
increase in his son's future income and
Observation
consume
12
it.
In a model with no symbolic consumption, perfect altruism, no
1:
contracting and perfect credit markets, investment in
how much money
A
I
depends on
the parents have nor on their preferences.
say in the introduction, the main focus here
steady state
w^
=
w(*
capital neither
Properties of the steady state
3.2
As
human
outcome
is
— w H ,io L = w^+1 = w L and h = h t+ =
+ 6s 2 (h,h) =
a position to ask a
in
ht+2
=
=
rt
h.
in
the steady state.
r i+1
=
r, c t
—
c (+1
,
Assuming that such an
above conditions reduce to one key condition:
1
now
i
t
(
exists, the
are
on what happens
economy where
defined to be a state of the
s 1 (h h)
We
is
!~ Tl)/( 1 ei)
7 + (1 — j)w L /w „
(
6
,
number
of questions
L
(5)
-
Ji
about the nature of the steady
state.
Dependence on Parental Preferences:
Preferences enter equation 5 only because
they help determine the steady state interest rate, which turns out to be 1/6. Preferences
of individual parents
Inequality:
Is
do not enter anywhere
into this equation.
the steady state perfectly egalitarian or
in the sense that different dynasties within the
state levels of
fixed value of
12
human
capital?
Or
which
same economy converge
in the
data
may
effects in this
look like a wealth
therefore raises h t+ i. In fact, the
\
to different steady
steady states for a
w L /w H ? 13
amount
of
economy, there can be a parental human capital
effect:
keeping h t+ 2 fixed, lowers the marginal cost of investing
h t+
possible inequality persists
in other words, are there multiple
However, while there are no income
effect,
is it
human
in
For example,
if
5j 2
<
0.
an increase
in h t
,
education (without affecting the benefits) and
capital in all generations will go up.
The
increase in
causes h t +2 to go up, which causes S2{h t+2 ht+i) to go down, which encourages further increases in
,
ht+i, etc.
13
We
also have to check that the value
capital that
we would have
in the
o{w L /xv H
so generated
is
consistent with the
amount
of
human
steady state, but we can always choose the production function to
11
The argument, made
education,
is
tells
above, suggesting that the children of educated parents get more
small enough and the ratio S12/S11
of this paper
where
S12
=
may be
us that such multiplicity
we focus on
is
and
positive
this
this choice: First
to
add to
when the dynasties
only exists
Second,
their results.
it
consequence that S2(0, h) can be
human
capital.
Under the assumption that
—
for 6 close to 1,
less
tells
level of
S12
=
human
Uniqueness: Do otherwise
levels of
human capital
we need
to
0, all
dynasties converge to the
To
In
h t+ 2 enters Equations 3 and
level of
h t+ i. Therefore, everyone
up with same average
level?
To answer
+
6s 2 (K h)
-
(1
8
7
~
+
Tl)/( 1
~
€l
,
1
(l
\
-7)p(/i,Q)
first.
this question
=
^ denotes
=
(6)
the steady state relative
human
1
S(h)(l
-
human
supply of unskilled labor and
capital, the net
S(ti)(l
-
7), a)
-
7)
and
and h -
wL =
sense, this
is
just
fL (h
-
S(/i) 7 1
,
-
an interpretation of what zero human capital means.
12
capital to the
S(h)"f, respectively. Therefore
fit.
some
To
derive the function g(h, a) note that in a steady state where everyone has
Wh = fn(h - S(%, 1 14
level of h.
show that the steady state equation:
production sector are given by
make that
same
same
identical economies that start with very different average
has a unique solution where the function g(h, a)
h units of
independent
capital in every period other than the very
eventually end
Sl (h, h)
to wages.
the
14
us that every dynasty chooses the
must have the same
is
we have
educated parent even when both
see this, observe first that under this condition, neither h t nor
which
and
implying that a more educated parent
strictly negative,
would have a lower educational expenditure than a
4,
when s^ <
However, making this separability assumption has the unfortunate
of initial conditions.
are providing zero
5
namely the case
us that the steady state
tells
if
can be shown that the multiplicity
are sufficiently impatient
equivalent of a Turnpike Theorem, which
the case
Galor and Tsiddon
(1997) have already developed the possibility of multiple steady states
we have nothing
is
large enough. However, in the rest
the case where such multiplicity cannot arise,
There are two reasons why we make
0.
and indeed
possible
S(/i)(l
-
7), a).
—
What happens
when h goes up? Since /()
to g(h, a)
depends only on the ratio
1
denominator of
numerator
_Z h UZ
Since,
homogeneous
is
by assumption S(h)
is
of degree one, g(-)
increasing in h, the
)
this expression clearly goes
down when h goes
however, potentially ambiguous. However,
is,
•
h
if
The
up.
neighborhood of a
in the
is
on the
effect
steady state,
-
—
d(h
S(/i) 7 )
dh
.
=
-
- si7-
1
s 27
>1-
Si7
,
- ^27
=
expression on the right
is
too large there
of
human
number
more than one
of units of
human
the relative price of skills will go
up. For the rest of the paper
into this case
and therefore we
of h, the per capita
is
capital to the
up when the
we
will
will
The
all
no
effect
fall
in the
+
=
,1-7
<5
first
lump-sum tax r
on the long-run
e{) is
not too much
if
the subsidy
will lead to
human
level of
capital in the
is
0,
When
this
a reduction
As a
result,
economy goes
never so large that we get
that g(h,a) is always increasing as a function
capital in the economy.
map
(Equation 6) with
to get the expression
—
(l-r )/(l-e
—gi{h,a).
(7 + (i -i)g\h,a)y
1
and Subsidies: We
policy experiments. Consider
or a
—
Basically,
economy
is
1
are
)
always positive since s u
positive. Therefore, the steady state has to
Effects of Taxes
e
(1
capital to produce.
differentiate the steady state
,
22
level of
In the neighborhood of a steady state this expression
are
large.
assume that subsidy
assume
7
51 (h)
— Tj)/
capital available to the production sector.
respect to h and use the fact that s i2
*
<5s
human
unit of
human
To complete the argument, we
+L
increasing in h as long as the
is
true as long as (1
endowment of human
s„
+ ^g(h, a)
over-investment in education to the point where an extra unit
happens, adding an extra unit of
in the
)
.,
(1
words the subsidy cannot be too
may be
capital costs
This
positive.
is
in other
1;
1
i
Therefore, in the neighborhood of a steady state, g(h)
larger than
6(l-r,)/(l—
—ei—
_,)
-
1
,
S22
and
be unique.
now
in
a position to look at some
a small increase in the lump-sum educational subsidy
-
Since neither enters Equation
6,
they clearly have
investment in education, as long as the non-negativity
13
constraint on investment in education
Changes
and
in e\
T\ clearly
15
not binding.
is
do have an impact on educational investment.
checked that the first-best level of investment will be achieved when Tj
collection
costly,
is
the
to subsidize education
optimum
is
is
presumably T\
= e\ =
0.
=
It is easily
e\.
Since tax
Consequently, the only reason
to counteract the effects of pre-existing taxes
on human
capital.
Moreover, an increase in eo financed by an increase in T\ clearly reduces educational
We
investment.
interpret this as saying that
than taxes on human
if
education subsidies are harder to target
capital, then the net effect of the subsidy
may be
to discourage
education. 16
Raising Returns to Education:
rate of return to education goes up.
g(h, a) for
all
each level of
values of
h,
h:
if
g(-)
We
we can
look at what happens
capture this by assuming that raising
This amounts to an increase
and can be interpreted as the
promoting the adoption of a new
that
Finally,
result, for
when the
a
lowers
in the relative price of skills, for
example, of a government policy
skill-intensive technology.
1
'
It is clear
from Equation 6
goes down, h will have to up. Therefore, an increase in the rate of return on
education does increase investment in education in this model.
Observation
has the same
and
In the
2:
human
their preferences.
benchmark model the steady
state
is
unique. Every dynasty
capital level in the steady state irrespective of
Lump sum
taxes and subsidies have no
taxes and subsidies tend to distort investment in
human
capital.
effect,
where they started
while proportional
Higher returns to
human
capital are associated with higher investment.
15
rate
In the short run, however, changes in eo
and through
it
—
t
,
keeping
all
other policies fixed, do affect the interest
investment in education.
16
Once
17
Foster and Rosenzweig (2000) suggest that the green revolution in India was an example of such a
again, this
is
only true
if
the non-negativity constraint does not bind.
policy.
14
Beyond the Beckerian Model: Short Run Proper-
4
ties
4.1
The Role
Let us
now
credits
markets are too
of Credit Constraints
consider a world where there are no credit markets, or at least the existing
inefficient to
be of relevance to most people.
At
this point
we
impose no restrictions on the class of preferences beyond what has already been imposed
In this world, people invest in their children's schooling using their family
in section 2.
resources. Moreover, investing in education
to the family.
There
Therefore,
budget constraint
its
w\ + htw?{l -
are, however, also taxes
Ti)
-
(yui?
+
This constraint holds for every
negative in order to
make the
in
and
any period
(1
t.
the only investment opportunity available
is
t
subsidies, just as in the previous section.
can be written
- l)w^)s{h t+1 ,h
t
){l
-
Moreover, consumption
ei
is
)
as:
-p +
t
e
-
of investment
Marginal
Q.
(7)
credit constraint meaningful.
its effect
must necessarily
across people.
=
assumed to be always non-
This assumption rules out the possibility that the decision to invest in
could be evaluated in terms of
r
on the present value of income
alter the distribution of
utilities of
human
capital
—changing the
level
consumption over time and/or
consumption must therefore enter the calculation
that determines investment in education, and since changes in income have an effect on
the marginal utility of consumption, they will affect investment in
specifically,
L
u> t
from Equation 7
it
is
marginal
Then
utilities
at least one of c t
t
+\
unchanged and
Therefore h t+1 must go up.
in
.
eo or
Suppose h t+i did
t
would have to go down. Yet, because of our separability assumptions,
new outcome with h
investment
t
More
and p would have to go up and the corresponding
the marginal gain from additional investment
in
capital.
income (say
in current
has gone up) must either go into a higher h t +\ or a higher c t or p
not go up.
the
any increase
clear that
human
human
An
capital.
c<
in
or
human
capital
unchanged and therefore
p having gone up, cannot be an optimum.
t
increase in income
must be associated with an increase
Moreover the extent to which
15
is
it
would go up
will clearly
depend on the
and therefore
This logic
makes
relative weight given to these various
will typically
depend on parental
collective preferences
preferences.
quite easy to formalize, but the wide range of preferences permitted here
is
notationally cumbersome.
it
outcomes in the
We
therefore limit ourselves to stating the implied
result informally.
Observation
3:
As long
as credit markets are imperfect,
and/or different preferences
income
levels
human
capital.
two dynasties with
will typically invest different
This remains true even
if
there
perfect altruism
is
different
amounts
in the
and no symbolic
consumption.
The Role
4.2
we go back
In this section
pand the
This
is
of Non-standard Preferences
to the assumption of perfect credit markets, but
not entirely straightforward, in particular because even small departures from
the Barro-Becker assumption about preferences
is
^2 s=0 6
s
U M (c +s,Pt+s+\))
t
+\.
Then generations
"^X!s=i ^
S_1
UM
(
t
Let generations
and
c t+s, Pt+s+i)
,
+
t
)
utility function builds in
1 will
t
and
see this, combine
the utility function of generation
(A
+
t
+
1
have control over
want to maximize
which gives them an
U M {ctlPt+l +
This
(i.e.,
To
with the assumption of bi-generational contracting
t
stead of no contracting).
h
ex-
symbolic consumption and imperfect altruism.
set of preferences to allow for
the Beckerian formulation can lead to unexpected complexities.
t
now
6)
^
8
s
^
s
s=0 5 [/
c t ,Pt+i
(in-
and
M (c t+s ,pf+s+ i) +
effective utility function:
U M (c t+1+S:Pt+s+2
a form of hyperbolic discounting. As
).
is
well-known from the
literature (Harris-Lai bson, 2001) this generates decision rules that are not time-consistent
and therefore the current generations,
in taking their decisions,
The Beckerian model
the reaction function of the next set of decision-makers.
issue
by making the very
specific
assumption that A
=
have to take account of
avoids this
0.
We now consider a model where there is symbolic consumption of educational spending
16
and potentially imperfect altruism:
M
Uf = U (c
t
,
Pt+l )
+
C/
s
(( 7
<+
(1
s=l
s=l
It is
assumed here that there are no taxes and
implicitly
having control over Ct,p +i and h t+ \.
As noted above,
t
U + \U +i
contracting. Maximizing
"*"(«»«>
dc
t
t
maximum
us that at an interior
Note that
s
((
and
t
+
1
subsumes the case of no
7), tells
that:
x
(8)
t
+
7
(1
+
(1
-
7)^+1 )s 2
- 7)^)5! (ftj+i, /»») (1 -
<+
(1
in writing
consistency issues. This
If S12
this
t
subject to the budget constraint (Equation
must be the case
it
[w?+l {\ - n) - (jw?+1
r t (7«;f
To complete the
subsidies.
assume bi-generational contracting with generation
description of the model,
-L/'
- lM)s(ht+1 ,ht)).
)(l
-
ei
-
)
ej]
- 7 )^L )s(^+i A))(7^f +
down
is
(/7 (+2 ,/?, +1
this condition
(1
-7
K )»i(^i.M(l L
we have not had
e
to deal with the time
because we assume both perfect credit markets and
were not equal to zero, the marginal cost of educational investment
S12
=
0.
in the future
would depend on how much the current generation invested and therefore the current
generation's investment decision would have to take account of the responsiveness of
future investment to current investment
It
follows
marginal
ht+\.
from equation 8 that as long as
utility,
any increase
in family
XJ
M
and
Us
are subject to diminishing
income (say ht goes up) necessarily must increase
Moreover, parental preferences, given by the nature of the
clearly play a role.
Moreover,
it
nature and form of altruism that
is
evident that the result here
we have assumed,
17
UM
is
and
Us
functions,
uninfluenced by the
since these terms
do not enter the
above condition. The key condition
reduces to equation
we
4, i.e.,
is
that U^
^
When U^ =
0:
benchmark
collapse back to the
0,
the above condition
case.
Once
again, this
remains true whether we have perfect or imperfect altruism and whether we have no
=
contracting (A
Moreover since symbolic consumption
0) or bi-generational contracting.
of other people's material
consumption
is
effectively just
a form of altruism, adding this
type of symbolic consumption does not alter the result. Finally, while not shown here,
it is
easy to show that the same result (that we get same results as the benchmark case
unless there
symbolic consumption of investment
is
in
human
capital) holds in the case
of multigenerational contracting including the case of complete contracts.
This leads us to the following.
Observation
Given perfect credit markets, there can be income
4:
parental preference effects on investment in
bolic
It
consumption of investment
in
follows from this observation
human
human
capital
if
and only
if
effects
there
is
and
sym-
capital.
and the previous one that the evidence of income
effects
on educational investment (see Jacoby, 1994; Glewwe and Jacoby, 2000; and Carvalho,
2000) can only be interpreted as evidence for credit constraints
assume that there
is
if
we
are prepared to
no symbolic consumption of educational investment
in itself.
Beyond the Beckerian Model: Steady State Prop-
5
erties
Credit Constraints in the Beckerian Model
5.1
5.1.1
We
Case
1:
Altruism Without Contracting
are interested in the long run properties of the Beckerian
ending are ruled out.
We
start with the case
contracting with generation
that there
is
t
where altruism
Model
is
having control over c ,p +i and ht+ i.
t
no symbolic consumption.
18
t
if
borrowing and
perfect but there
is
no
We continue to assume
Using the budget constraint we can rewrite
Ti)w" -(yw^1 + (l-"f)wf )s(ht+i,h )(l-e
J
t
{h
to the sequence
t
(7U& +
}
1 )
Yl'tZo
+e —T
^u
t
{
c t,Pt+\) as YluLo 6 u(wf'
+h
t
(l-
,p +i). Maximizing this with respect
t
yields the first order conditions:
(1
- 7 )^L_
1
)s 1
(/i t
,
^_i)(l
-
ej)
dU(ct-i,Pt)/act-i
=
Using the fact that in the steady state q_i
ct
,
Pt
—
1
Pt+iiwf
— wH
= wL
and w\
\/t,
this reduces to the condition
Sl (h, h)
which
+
be recognized as Equation
will
-
6s 2 (h, h)
6,
~! l)/{ } ~ 6l =
7+ (1 -~t)g(h,a)\
{1
6
0,
the steady state condition under altruism and
perfect credit markets. It follows that under the assumption that S\2
equalization in the long-run, even though the rich invest
capital of their children. Moreover, the steady state
and consumption
The
what they would be
0,
there
more than the poor
unique and the
in a first-best world.
case^
—lump-sum taxes and subsidies have no
and subsidies do have an
effect,
is
complete
human
investment
18
same
as in the
and while proportional
no taxes and no subsidies remain optimal. The
general policy bias that comes from this model
matters
effect,
is
in the
level of
long-run effects of policy in this model, not surprisingly, are the
benchmark
taxes
in the limit are
is
=
is
that, at least in the long run,
what
removing barriers that prevent the returns to human capital from being as
high as possible. 19
Case
5.1.2
Complete Contracts with
2:
In this subsection
we consider
This
is
Altruism
the exact opposite model
pletely selfish in the sense that parents
18
No
do not care at
essentially a deterministic version of the result in
— a model where people are comall
about what happens to their
Loury (1981) showing that
in the long
run
dynasties have the same distribution of consumption even in the presence of credit constraints.
all
1
lar,
The
short-run effects of policy are. however, potentially different from the first-best case. In particu-
because
human
capital
is
not instantly equalized, a subsidy financed by a tax on
human
capital does
redistribute and the resulting increase in the income of the poor will typically increase their investment
in
human
capital
and thus hasten convergence.
19
where there
children, but
out
is
a complete contract covering
all
forms of symbolic consumption. In this world, parents
all
rule
stil
education
will invest in the
an imperfect substitute
of their children, because the family here acts as
We
generations.
for the missing
credit market. In other words, parents invest in the education of their children in return
There are several plausible reasons
age care.
for old
be possible
typically
in
lot
more about
selection; second, there
their parents, but
who do
ties
may be
their children,
social
which may
limit the
may
a transaction
an economy where there are no other credit transactions:
know a
be emotional
why such
for
First, parents
amount
of adverse
norms that punish children who do not take care
do not apply to those who
fail
to honor their debts; finally, there
of
may
between parents and children which enable parents to "punish" children
not take care of them.
Given these assumptions, we can now write down the exact maximization problem
facing a particular dynasty at time 0, given that
and an obligation
capital
T,fl Q X
t
U(c
w\ +
To
see
t
- T X )wf - (7wf +
what happens
(1
A[(l
human
maximize
+e -
ei)
r
- p = c u Vt.
t
,p t+1 )/dc
= dU(c _ u p
t
t
+ {\- 1 )wL) s ,(h t+l ,h ){\-e
t
- ri)wf+1 - (7^+1 +
(1
5! (h,
h)
(9)
t
above
interior solution exists) are:
l
t
)/dp t
)dU{c u p t+l )/dc
t
- 7M+i)s 2 (^+2,/it+i)(l -
In a steady state the second equation reduces to the
The
of
in the long-run, observe that the first order conditions for the
XdU{c
=
amount
and po and the budget constraint
/i
- 7 X)s(/i (+1 A)(l -
maximization problem (assuming an
{~iw?
started with ho
of po towards its parent. It has to choose {/it,Pi}t=i° t°
,pt+i) given the initial values
ht{\
it
+ Xs 2 (h, h) = X
(1
expression defining the steady state level of
much more compact
~ Tl)/( 1
,
human
condition:
\
capital
8.
.
Bl
7,
1
dition as in the case of true altruism, with A replacing
20
e 1 )}dU(ct+ i,Pt+2)/dct
A
is
fall
exactly the
in A,
much
same
like
con-
a
fall
in 8, discourages
investment in
generation leads to
less
capital
—greater power
in the
hands of the older
education.
other properties of the steady state are relatively unsurprising: Under the main-
The
tained assumption that
is
human
su =
0,
there
is
no inequality
in the long-run,
and the steady state
unique. Moreover, the changes in policy parameters have effects in the
same
direction
as they would have in the case of true altruism.
However, a number of new policy issues
arise: First, if
as-you-go social security system, investment in education
will
no longer
There
is
is likely
to collapse since parents
the need for transfers from their children. Second,
feel
of whether the
the government initiates a pay-
government should try to
no a priori reason why
A,
alter the
it
raises the issue
balance of power within the family:
which measures the bargaining power of the young,
should also be the weight children get in the social welfare function.
For example, the
government may want to maximize steady state welfare rather than the maximand that
actually gets maximized. In this case, unlike in the case of true altruism, a proportional
subsidy to education or a mandatory schooling law
the government
may push
may be
optimal.
In other words,
people to invest more in education in order to safeguard the
interests of the young.
Observation
5:
The introduction
of credit constraints alone do not alter the
properties of the steady state as long as there
we have
either perfect altruism
is
no symbolic consumption, and
and no contracting or complete contracting
and no altruism.
One might wonder why
the results described here do not extend to the case where
we have both complete contracting and
clear that there will
be a steady state
perfect altruism.
in this case,
complete contracting the weight on generation
on generation
t
+
1.
21
t is
The problem
is
that
it is
not
because once we combine altruism and
no longer a constant multiple of weight
Beyond the Beckerian Model: Symbolic Consumption of
5.2
Human
An
5.2.1
To
see
Capital
example
what changes with symbolic consumption of human
capital, let us start with
a
case where the analytics are particularly straightforward: Consider a model where parents
symbolically consume the
issue of
amount spent on the
human
altruism and no contracting. Finally,
and that the cost
of education
from the s(h t+ i,h
utility function of
t
)
constraint (Equation 7). It
we assume
we assume no
that there
is
With
is
avoid the
by assuming
credit markets,
i.e.,
that
no
age
in old
we can drop
these assumption, each generation maximizes a
U M (c + U s ((-fwf + (1 - j)w^)s(h +i))
t
start
no consumption
unaffected by parental education,
is
function.
the form
In addition,
To
education.
whether we should consider pre-tax expenditure or post-tax, we
that there are no taxes and subsidies.
ht
child's
)
t
easy to see the
subject to the budget
order conditions for this problem
first
is
u,s {{lwf + (i-tKX^+i))
U>M( W L
+ hw H _
(
F
fl
+
(1
1,
- j)w L )s{ht+1 ))
which yields the steady state condition
s
H
U' ((jw + (1 U' M (w L + hw H - (jw H
Parental preference
effect: It
is
j)w L )s{h))
+
(1
- j)w L )s(h))
1.
(10)
evident from the form of equation 10 that parental
preferences, represented in the intensity of desire for symbolic
material consumption will affect the accumulation of
human
consumption
relative to
capital even in the long run.
Inequality: Both the numerator and the denominator of the expression on the
of equation 10 are typically going to be decreasing as a function of h. w
left
Whether the
curve given by this expression slopes up or down depends on which of the two decreases
In particular,
faster.
This
is
it is
not hard to find cases where
not always true because
for
high enough values of h,
increases with h. This requires, however, that the family
words,
it
it
must be the case that an increase
in
is
goes up and down, generating
it
is
conceivable that the numerator
heavily over-investing in education. In other
h actually reduces output net of educational expenditure.
22
—
the possibility of several solutions, which imply that there can be inequality in the steady
state.
To
more
see
precisely
what
constant relative risk aversion,
going on here, consider
is
u(c)
i.e.,
= j^,
the case where there
first
and moreover
v(-)
=
is
6u(-). In this case,
condition 10 can be rewritten in the form:
w L + hw H
,l.i
= ! + (->.
L
11
S
{-yw
+ (1 ~i)w )S{h)
using the fact that the function S(h) =
_
,
,
checked,
It is easily
convex, and the fact that S(0)
h,
=
0,
and therefore the value of h that
that the left-hand side
solves this equation
is
ni
.
(11)
y
s(h, h)
is
increasing and
decreasing as a function of
is
unique.
Consider next the other standard formulation for preferences, namely the constant
absolute risk aversion family:
i.e.,
u(c)
—
1
— e~ ac and
v(-)
=
6u(-). In this case,
Equation
10 can be written in the form:
[w
The
'
+ hw H + 2{{ 1W H +
)
- 7 )w L )S(/i)) = a
left-hand side of this equation defines a convex function of h.
starts
than
Assuming
h).
and h
=
marginal
^ log e 8 and S(h)/h
-
bequests
one at h 2
In both the
effect of
>
oo as h
Equation 10 does not actually hold at h
utility of
states, the
—
—
*
oo,
There are three potential stationary values of h
1.
h2
-w L >
strictly less
is
since bequests cannot be negative, h
If
8.
S'(0)
(12)
=
0, it
always
is
CARA
stable
and
—
=
0:
than the marginal
is
we have the
in this case^
case that
=
h
What happens
utility of
0,
is
h
=
is
h\
that the
consumption, but
the optimal choice. Of the other two steady
and therefore perhaps more worthy of
CRRA cases the basic forces
are the
interest.
—there
same
is
an income
being more educated, which makes educated parents more willing to invest. This,
however,
make
log e
by being downward-sloping but eventually slopes up (because S(h) grows faster
in Figure
is
counteracted by the fact that an educated parent has to invest more just to
sure that his child
of investment
is rising.
one or more stationary
21
1
(1
is
not
Which
less
educated than he
is,
and, moreover, the marginal cost
of these effects dominates
level of h,
This source of long run inequality
and therefore whether there
depends on the exact curvature of the
is
closely related to that in Galor
23
utility functions.
and Moav (1999), though
is
21
their
Uniqueness: The argument
inequality in the long run for
capital will also tend to
it is
high
likely that
human
There
preferences. Dynasties that start with
will also fail:
Given that there
capital.
An economy
we choose
CRRA
Moreover assume (purely
for
why
=
inequality,
is
number
human
uniqueness might
preferences so that in the long run there
convenience) that 7
more human
that starts with a large
capital dynasties will tend to have a high long-run level of
actually a second, independent reason
is
this reason
CARA
end up with more human
uniqueness
paragraphs shows that there can be
in the previous
is
capital.
To
fail:
of
highlight
perfect equality.
Under these conditions we can
1.
rewrite Equation 11 in the form:
=1 +
S(h)
(
Both the denominator and the numerator of the
know
5
K
ratio
S
that the ratio h/S(h) declines as h goes up since
possible that
9
£ 'y increases in
h at
least over a
(13)
range
a
g^
+
A
are increasing in h.
convex. However,
is
—
',
g
this will
degree of substitutability between skilled and unskilled labor
is
in Figure 2.
There are
relatively small.
+
intuition:
Teachers are cheap in an economy where there
therefore the
same
high level of
human
capital
Taxes and Subsidies:
dies enter the preferences.
e i)
bequests buys more
level of
— e o)- This assumption
is
g'fA
clearly three steady states in this case, of
extreme ones are stable. The multiplicity of steady states
human
is
entirely
be the case when the
therefore quite straightforward to construct examples where the function
form given
it is
capital,
lot of
It is
has the
which the two
reflects the following
a
We
human
simple
capital
and
and as a
result the initial
how
taxes and subsi-
reproduced. 22
We now need
One
possibility
to address the question of
is
that v
= v^w" + (1 — 7)iUjL )s(/i t+ i, h
says that people count as their bequest the entire
t
)(l
—
amount they
have spent on education, but not the taxes they have paid which are then spent on subsidizing education.
model
22
is
An
alternative possibility
based on Stone-Geary type preferences
for
would be that people count the taxes they
bequest and
is
This type of multiplicity of steady states, sustained by price
Banerjee and
Newman
(1993).
24
therefore analytically
effects, is similar
more demanding.
to the multiplicity in
this case v
have paid as part of their bequest. In
e i)
~
the
first
^o
+
t
+
1
Tihtwf
While the second assumption
).
on the grounds that
of these assumptions
we
Under the
+
from
effects in this
h{\
-
)w H -
Tl
)
is
)(l -
prefer
imagine that peo-
difficult to
going to pay for the educational
hw H + (1
model:
An
- j)w L )S(h)(l - ~/)w L )S{h){l - ej) - e
(1
this expression that
Moreover, an increase in
At
very
more conventional, we
t
assumption, the steady state condition turns out to be:
v'{(jw H +
It is clear
is
- j)w^)s(h t+ i,h
(1
allow for both.
first
u'{w L
it is
payments that
ple can identify the part of their tax
subsidies, but
= v((jwf +
r
)
)
and a cut
in To
promote investment
and r that leaves the government
human
increases investment in
-
e
changes in lump-sum taxes and subsidies have long-run
increase in eo
e
+
ei)
capital. Indeed,
deficit
any increase
in education.
=
unchanged (Ae
in subsidies
and taxes
that keep everyone's net income unchanged (at the original steady state value of h) will
increase investment.
This
between the increases
because people
is
in subsidy
and the increase
Under the second assumption, the steady
u'{w L
+ h{\ v'((-yw H
)w" -
Tl
+
(1
-
(~fw
H +
(1
-y)w L )S(h)(l
model do not make the connection
in this
in taxes.
state condition
is
going to be:
- j)w L )S{h){\ - ei + e - e - e + r + t^w?)
)
x
r
)
_
)
In this case, changes in taxes and subsidies that do not change the public deficit
have no
effect
on investment as long as there
the other hand, an increase
increase investment in
If,
however, there
is
in eo
human
1,
if
where we have
no inequality
in the steady state.
capital (at the expense of increasing the public deficit).
inequality in the steady state, changes in proportional taxes
there
is
CARA
—and as a
no change
result there
the public
in
deficit.
may be
is
preferences and multiple steady states.
a balanced budget,
i.e.,
e
=
25
Tihw H where h
,
an
effect
is
To
i.e., e\
the
and
on average
Consider the case
assume that subsidies are lump-sum but taxes are proportional,
and that there
On
without any other changes in taxes or subsidies does
subsidies have distributional effects
investment even
is
in
Figure
simplify matters,
=
mean
and tq
level of
=
0,
human
With
capital.
these assumptions
we can
rewrite Equation 12 as
-w L - hw H + 2(>yw H + (1 - >y)w L )S(h) + 2(h First consider
Tx
is
down
7"i
shown
left.
in Figure
1.
An
for fixed values of
we
high values of
for
the stable steady state at h
If
wL
and the
=
However,
for larger increases in t\, the
entire population
by making the
fixed.
effect of
8.
an increase
ends up at
This
hi-
reduce the
elasticity of substitution in
in
human
and
h\ vanish,
us that higher taxes can bring about
tells
—so
far
we have kept the
in
w L and w H
capital holds even
when
wL
and
and
wH
capital,
are endogenous.
this leads to a large increase
capital.
fact that the steady state
is
not unique even when there
steady state suggests a different possibility for a virtuous cycle.
"bad" steady state, the government borrows
subsidy to
values
production high enough. Therefore, the same
Essentially, redistribution here helps start a virtuous cycle
The
human
level of
=
steady states at h
incomplete
is
mean
namely that a lump-sum subsidy financed by a proportional tax on human
human
in
For small changes in T\ starting at
h.
However, note that we can minimize the changes
can increase investment
in
log e
think of the population as being initially distributed across these two steady
w L and w H
result,
.
G
remains and the stable "good" state as hi moves
more investment. However, the argument
of
w H The
and
states, it is clear that the effect of increasing T] is to
capital.
H = -
increase in T\ shifts the left-hand side of the above equation
low values of h and up
for
= 0,
what happens
h)T lW
human
where there
is
capital. If the subsidy
more and more human
money abroad and
large enough, this
is
capital
no inequality
richer,
in the
Suppose starting
invests
can set
it
at a
in a proportional
off
and therefore investment
becomes cheaper and cheaper. As the economy gets
and repay the
is
a virtuous cycle
in
human
capital
the government can raise taxes
initial debt.
Reusing returns to education: The
effect of
changes in returns to education are
CRRA
best seen by looking at the steady state condition with
g(h,a)
+
h
=
S(h)
26
.
1
,,1^1
+
(
5
K
preferences (with
7
=
1):
It is
clear that a shift in
a
that shifts g(h, a)
down
(a skill-biased change), has to reduce
investment in education, since at any stable steady state the function
9
'
a
/
a declining function of
investment in
education
is
human
h.
An
human
increase in the rewards for
capital in this case, because
it
this
from the
rewards from education
extreme assumption,
clearly an
is
capital always reduces
this result derives
fact that given the specification of preferences, the size of the
While
has to be
makes teachers more expensive and
produced using teachers. Clearly the strength of
do not matter to the parents.
+
|,
it
not
is
implausible that parents will care less about the returns to education than about the
costs, given that
they pay the costs and their children get the returns.
Symbolic Consumption of
5.2.2
The problem with
this
example
is
no credit markets, no altruism)
now
Let us
— 7)w L
(
.
(
s
)s(/i( +s+1 ))]
it
The General Case
Capital:
builds in a
in addition to the
consider the case where
kets into this scenario.
(1
that
Human
we introduce
number
of assumptions (in particular,
assumption of symbolic consumption.
and perfect credit mar-
perfect altruism
Each generation now maximizes Yl s =o P s [U M ( c t+s)
subject to the budget constraint
As already noted, one
2.
of the first-order conditions for this problem
which, in steady state reduces
It is
+ (1 - ~f)w L )s(h))
+ hw H - (yw H + (1 - -y)w L )s{h))
implies that in a steady state r
=
ii
is
(jw H
+
(1
that U' A1 (c
t
)
- y)w L )s'(h)
=
r t (3U'
M (c
t
'
+i),
^
which
1/6.
Note that the left-hand side of equation 14
is
exactly what
we had
same reasons may be non-monotonic. The expression on the
however clearly increasing as a function of h (because s"
likely
Equation 8
ii-
r
easy to show that another first-order condition
for the
is
to:
s
H
U' {(jw
U' M {xv L
+ U s (('yiu^+S +
>
0).
in
equation 10 and
right of equation 14
This probably makes
is
it less
that the curves representing the two sides will intersect more than once (compared
to equation 10,
where the right-hand side
In particular, since
we
is
a constant), but
is
by no means impossible:
are essentially free to choose the shape of the U' S /U' M function
we
can always generate cases where there are multiple steady sates and therefore inequality
27
in the steady state.
In terms of the question of the uniqueness of the average level of
fact that
w L /w H
is
human
capital, the
an increasing function of h does make the right-hand side of Equation
14 steeper, which probably goes against multiplicity; but on the other hand, at least in
CRRA case,
the
it
also
makes the left-hand
up more
side go
which tends to favor
steeply,
multiplicity.
The
income
fact that there are these multiplicities is clearly related to the presence of strong
effects,
even in the steady state, and given that these income effects are impor-
tant, the distribution of
investment.
income
will
matter for the steady state
level of
human
capital
Hence, policies that redistribute can improve the efficiency of investment,
exactly as in the example above.
Observation
6:
In the presence of symbolic consumption of
steady state
may
run
investment in
levels of
not be unique.
The
human
history of each dynasty
capital as will their preferences.
subsidies will affect long run investment in
subsidies
human
may
human
capital,
increase the efficiency of investment in
capital, the
matter for their long-
Lump sum
taxes
and
and proportional taxes and
human
capital.
Higher returns to
capital are not necessarily associated with higher investment.
consistent with there being perfect credit markets
5.3
may
human
These
results are
and perfect altruism.
Beyond the Beckerian Model: Imperfect Altruism, Incomplete Contracts
The
steady state analysis has so far avoided the problems that arise from the fact that
the current generation's preference over future outcomes
preferences of the generations
who have
possible.
The problem obviously
Finally, in the
not be the same as the
control over those outcomes.
of perfect altruism avoids this problem because
preferences.
may
all
The assumption
generations, in effect, have the
also does not arise
when a complete
example of symbolic consumption
same
contract
in the previous subsection,
parents only care about the amount spent on the education of their children, which
28
is
is
—
entirely
under their control.
However, this issue becomes unavoidable once we depart from the assumptions of
perfect altruism
and complete contracting and allow each generation to have preferences
over outcomes controlled by future generations.
To
the different generations.
what
see
up a game between
In effect, this sets
might imply, consider a particularly simple
this
example which combines symbolic consumption and imperfect altruism:
generation
reduces to the case where there
generation
contrast
tion
t
t
+
cares only about U([ic t )
t
6U(1
—
In the case where
/x)c t+ i).
\i
=
1/2, this
imperfect altruism but no symbolic consumption, since
is
if \x
^
1/2 there
is
By
necessarily an element of symbolic consumption since genera-
cares about the next generation's consumption but differently from
how
generation
own consumption. 23 Note, however, that the symbolic consumption
here involves the material consumption of others, rather than
how much human
they have, as in the previous example. As we saw above, this distinction
understanding short run behavior
interested in whether this
is still
in a
human
without symbolic consumption of
It is
assume that
values the next generation's consumption exactly as they themselves do.
t
cares about their
in
We
true
is
very important
model without credit constraints
capital
when
capital
was much better behaved.
—the model
We
are
now
there are credit constraints.
evident that the preferences assumed here build in a conflict of interest between
the generations
—the current generation cares only about the next, while the next gener-
ation cares about the one after.
the generations
dollar that
invest).
it
Ex
This raises the issue of strategic interactions between
ex ante each generation would like to promise that
gets from
post,
24
it
will
its
it
will
consume every
parent's investment (this maximizes the parent's willingness to
want to share some of that money with
their
own
children. Since
contracts are ruled out, each generation's maximization problem should take account of
23
A
potential interpretation of these preferences
on the next generation
anything to
in old
24
age
its
is
for old
that each generation
age support. The next generation
parent, but consumption
is
is
is
also selfish
purely selfish but depends
and does not want to give
a family public good and therefore the parent's consumption
proportional to his son's consumption in middle age.
See Bernheim and Ray (1987)
difficulties of
is
for
a different model of this class, which, once again illustrates the
working with such models.
29
the next generation's equilibrium decision rule for allocating their earnings between con-
sumption and human capital investment. Unfortunately,
this gives rise to the possibility
To
of multiple equilibria, which differ in the decision rule adopted by each generation.
limit the
number
we
of equilibria,
confine ourselves to looking only at the set of Markov-
ian equilibria. These are equilibria where each generation uses a decision rule of the form
h +i(h
t
t
).
In other words, the
human
capital investment by each generation depends only
on the amount of human capital that
an equilibrium
easily checked that such
However,
economy
this
by
itself
since everything
does not
and
A
5.3.1
parent and the index of time. 25
its
We
tell
much about
us very
t
show that
for
),
the steady states of this
which
more
u(n{w^
+
+6u((l -
=
x — ^4>x 2
ht(l
-
ri)tuf
m)K+i +
.
Assume
in
we need
to
detail.
also that s(h t +i,h t )
=
(1
- j)wf)s(ht+1 ,h )(l t
- T1M+1 - (7^+i +
(1
"
(1
-
«W((1 - ^)c t+1 )(l -
ei )
7KL+i)
rx ) -
human
of time
is
capital, there
is
+e -
r
1.
))
s (^+2, fcn-i)(l
/i t+1
7KL M1 - ei)
/xJK^l -
=
as:
-
eij
+e -
turns out to be:
(15)
s{
1W ?+x +
(1
- 7 )<i)(l " ei)§^].
claim, that given our assumptions about preferences
technology for
The index
h t+i and that 7
s
order condition for utility maximization with respect to
We now
25
- (Twf +
fh+iO-
u'(^t)M7< +
=
determined
Linear-quadratic Example
Using the budget constraint, we rewrite the family's maximand
first
itself is
a specific choice of parameters, h t+ i(ht)
this allows us to characterize the solution in
Assume that u(x)
The
It is
exists.
depends on the function h t+ i(h
look at a parametric example:
linear
got from
In order to understand the possibilities that arise in this case,
equilibrium.
is
it
and the production
a linear solution to this maximization problem. In
there to allow for the possibility that
path.
30
wH
and
wL
change along the equilibrium
r
)).
+
»
other words, there
that
=
^^
rj
and
t+1
fact that u"(c t )
=
=
-
(1
differentiate
=
u"(c t+ \)
ftwf^w? +
(1
—^
This equation defines
property that
will also
^^ =
r/
t
l-ei
this,
assume
t
-
Tl )
ei)(l
"
ri)
-
ei
)(7< +
5(1
- 7
(
<+
-
(1
(1
-
7^(1 - ^^g
tKK+i]
terms of a set of constants and
in
i]
t+1
be a constant, which confirms that there
is
1
]
2
(16)
-
As long
.
as
rj
t+1 is
a solution with the
Vt.
t,
Using the steady state conditions (including the fact that
reduces
To show
,Vt.
r)
Equation 15 with respect to h to get (using also the
7K Ml -
-
=
</>):
^^[^(1 -
a constant, ^j^1
-^
a solution that takes the form
is
=
rj
t
=
7] t+1
77),
Equation 16
to:
*rr^i (7
(i
-
l-/z
^r^'") *"<—
1-ei
1
f
7)9W
)
=
l
1
2
i
(7
+
- ,),(*»
-
r
~
ei
\
12
"r^r'"'
or
S
where h
is
the average
The two
8r)
_
2
"(l^) +
amount
of
1
-
Ti
^l-e
human
1
l
(7
1
+ (l-7)#))
economy
capital in the
is
close to zero
and increasing
zero for high values of
for
77
Therefore,
7/.
in the
equation there must be a second. In Figure
77
and
77.
Note that both
more education
The
77
will
77
and
77
for
The right-hand
any
side,
fixed value of h. It
neighborhood of zero but decreases towards
there
if
tj
steady state.
in the
sides of this equation are represented in Figure 3.
represented as H(rj), defines a non-monotonic function of
(17)
1
is
3,
are positive:
at least
one value of
77
that solves this
these two steady states are identified as
As
in the previous models, parents
with
have children with more education.
multiplicity of possible values for
means that each generation responds
77
to
is
not particularly surprising
an increase
stantially increasing investment in its children's
31
in its
human
— a high value of
own human
capital.
capital by sub-
This amounts to saying
that the current generation of the middle-aged get a low rate of return on their investment
in children
which
may
responsive to their
We do not
lead
them
own human
may
to invest less, but
capital.
take a stand on which of
This
is
and
rj
r\
what
is
also
make
their investment
gives us the multiplicity.
The
the right solution.
using the stability properties to eliminate solutions
is
more
usual
method
of
problematic here given the complex,
forward-looking dynamics associated with Equation 16.
However we can say something more about the properties of
rj.
Rewrite steady state
condition 15 in the form
=
S
and combine
it
8
1
e£M
I
with equation 17 to
Sn
1
— T\
~ ei
1
(7
«V)
77
>
1 if
_jl,
A
an even larger increase
if ji
'
>
1/2.
positive shock to
h t+ i, and so
in
which corresponds to the case, not implausible
where the balance favors the
(19)
y
l-fi
and only
dynastic trajectories will tend to diverge:
state, will generate
(1
get:
V((l-/i)c)
This immediately implies that
+
~ 7)gW)
elderly, is better
on.
In the case where
h
t
,
will focus
where
/j
=
on the case where
1/2,
which
is
fj,
<
1/2 and
the case where there
consumption. In this particular case
it is
is
1
The
case where
/i
<
1/2
in the context of developing countries
behaved since
77
>
starting at the steady
it
implies
77
case, dynasties will return to their steady state values following a shock.
we
77
<
1.
Note that
<
1.
In this
Henceforth,
this rules out the case
only imperfect altruism and no symbolic
evident that
7/
=
1
and there
is
no symbolic
consumption.
Finally, observe that the steady state condition 18
is
not unlike the corresponding
condition for the first-best case, the one difference being that 8 has been replaced by
naLLLnzl
It is
not clear
how
these numbers compare since there
is
no reason why
:+S V
(1-M)u'«l-M)c)
the 8's should be the same, but
from 1/2,
it
can be shown that
if
they were the same, then, at least for
6
u ,. (
>
,
8.
\i
not too different
In other words, a result of the strategic
:+Sr)
(1-m)u'((1-m)c)
interaction between the generations
is
that there
32
is
underinvestment in education.
Parental Preference Effect: Parental preferences
mination of
human
,"' (mc
through the term
T)
\
,
clearly play a role in the deter-
and therefore have an
,
on investment
effect
in
capital even in steady state.
Inequality: These different values of
will typically lead to
\i
<
They
long-run inequality in the sense that those dynasties will have
ferent steady state levels of
as long as
represent different dynastic strategies.
77
consumption and human
=
1/2, u u^°\ c)
^^Zfz^
19 that in this case, high values of
77
\
is
c
capital.
increasing in
c.
To
It
dif-
see this, observe that
follows from Equation
be associated with low values of steady state
will
consumption. 26
Uniqueness: As suggested
steady state, the presumption
in the previous sub-section,
is
77
is
inequality in the
that there will also be multiple steady states: Intuitively,
an economy which starts with only high
which starts with only low
once there
dynasties will be different from an economy
77
dynasties, even in the long-run. However, as before,
it is
also
worth checking whether there can be multiple steady states without any inequality. In
equation 18, there
is
yet another possible source of multiplicity.
more involved discussion than we
strating multiplicity will require a
here.
However actually demonfeel is
appropriate
We therefore confine ourselves to explaining the logic of the construction:
Suppose h
were to go down for some reason. This pushes the curve up in Figure 3 because g(h) goes
down. As we know, the
all
dynasties started with
has to go up (assuming
tells
\i
77
<
=
77,
77
77
depends on the
initial
initially
1/2), contradicting our premise that
everyone was at
pushing h down. There
The
is
77.
The
reasoning
"positive feed-back"
basic intuition for the multiplicity
there are two equilibrium values of
Since c
is
value of
Assuming
77.
goes down, which means that in the steady state, h
us that there cannot be multiple steady states in this case.
where
2e
has on
effect this
77.
An
is
,
is
exactly the
h had gone up. This
Now
consider the case
same but now
goes up,
77
creating the possibility of multiplicity.
closely related to the explanation for
increase in h lowers the rate of return to
why
human
typically increasing as a function of h, this implies that high values of the marginal
propensity to invest in education will go with
this implies that high values of
77
will
less overall
investment. Moreover, since h t+ i(h t )
go with low values of h
:va
.
=
ho+rjht,
capital,
when
which has both an income
and a substitution
effect
the income effect dominates.
human
Taxes, subsidies and returns to
capital:
do not enter Equations 17 and 19 and therefore have no
values, of
and
77
L
c.
However
+ e — To,
7)tu )(l -ei)
lead to a
fall in
be to increase
be a
An
18,
fall
c
in
77
since in steady state c
h
for
any fixed
and an increase
in
c
and
which
c,
\i
who
started at
those
who
started at
<
know whether
this
Figure
in
7:
human
+
(1
—
None
This will
77,
the effect will
initially at
there
77,
fall in
raises
and reduces
<
77.
For
1/2).
For
h though we do not yet
77
the effect
=
is
an
always positive, but
77.
of the long run properties of the Beckerian
we introduce imperfect
\jl
effect.
logic also applies to the case of
77,
when
77
h (assuming
initial fall in
The same
negative
the standard disincentive
which
capital. Starting at
it is
is
uniqueness, the effects of taxes and changes in the return on
hold once
sh{^/w H
which can be seen from equation
/7,
3,
a further
would counteract the
rule the possibility that
Observation
initially at
who were
1/2). For those
the net effect can be positive.
increase in the return to
we cannot
on the steady state
direct effect
who were
are fixed. This
this will lead to
77,
77,
77
down
also pushes the curve
those
subsidies
h.
under the assumption that c and
it
and
raises the curve in Figure 3.
increase in T\ has a direct negative effect on
However
taxes
= w L + hw H (l —T\) —
For those
77.
and h (again assuming
both
Lump-sum
an increase in the net (lump-sum) subsidy to the household sector
will lead to a decrease in
will
Investment goes up
effect.
human
altruism, incomplete contracts
model
(equality,
capital) necessarily
and symbolic consump-
tion of other people's consumption.
This result
namics of
unstable.
this
We
is,
however, relatively tentative, since
model
—the steady
we have
yet to understand the dy-
state with the perverse properties
also looked at a very special
may be
example and we do not know how
results generalize.
34
globally
far these
Conclusion
6
The main point
of this paper was to underscore the important differences between the
implications of alternative ways of modeling the family in a world where educational
investment
is
largely financed by the family.
In part, this suggests that
we need more
theoretical research
aimed toward putting
some structure on the nature of family decision-making. Rangel (1999)
is
one example
of this type of work, arguing that the requirement that the implicit contract within the
family be enforceable imposes important restrictions on the family's choices. In part,
it
suggests an empirical agenda aimed toward identifying and testing the peculiar implications of these
and other models of the
family.
As pointed out
already, because
many
of
these models display income effects, their short run implications tend to be quite similar.
However, there
may be
other implications which are quite different, say with regard to
the reaction to an increase in the child's income (see Ermisch, 1996, for example). 2
this
to
'
Both
kind of empirical work and related experimental work have an enormous contribution
make
to the
way we think about education
policy.
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+ 2{h-h)r w H
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