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p a p e r
Efficient Investment In Children
by S. Rao Aiyagari, Jeremy Greenwood,
Ananth Seshadri
FEDERAL
RESERVE
BANK
OF
CLEVELAND
Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated
to stimulate discussion and critical comment on research in progress. They may not have been subject to
the formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views
stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of
Cleveland or of the Board of Governors of the Federal Reserve System.
Working papers are now available electronically through the Cleveland Fed’s site on the World Wide
Web: www.clev.frb.org.
Working Paper 01-05
May 2001
Efficient Investment in Children
by S. Rao Aiyagari, Jeremy Greenwood, Ananth Seshadri
Many would say that children are society’s most precious resource. So, how should we invest in
them? To gain insight into this question, a dynamic general equilibrium model is developed
where children differ by ability. Parents invest time and money in their offspring, depending on
their altruism. This allows their children to grow up as more productive adults. First, the efficient
allocation is characterized. Next, this is compared with the outcome that arises when financial
markets are incomplete. The situation where childcare markets are also lacking is then examined.
Additionally, the consequences of impure altruism are analyzed.
JEL codes: D1, D31, D58, I2
Keywords: investment in children, lack of childcare markets
Jeremy Greenwood is at the University of Rochester. S. Rao Aiyagari (deceased) was with the
University of Rochester. Ananth Seshadri is at the University of Wisconsin. The authors thank
Edward C. Prescott and an associate editor for helpful comments. This research is based on the
last notes of S. Rao Aiyagari (available on request). His co-authors miss him. All computer
programs are available from Ananth Seshadri.
Jeremy Greenwood may be contacted at gree@troi.cc.rochester.edu
Ananth Seshadri may be contacted at aseshadr@ssc.wisc.edu
Efficient Investment in Children
1
1
Introduction
In the U.S. economy a male in the top 5th percentile earns about 8.6 times the
labor income of one in the bottom 5th. The correlation between a father’s and
a son’s earnings is high, too, somewhere between 0.40 and 0.65. Many take
this as prima facie evidence that markets fail. They believe that differences
in ability cannot be so great as to explain such great differences in income.
They also feel that the transmission of genetic factors across generations
cannot be so high as to explain this low degree of intergenerational mobility.
This may be true.
The problem for the economist is that ability is not well observed. Further, psychometric testing provides ordinal, and not cardinal, measures of
ability. Consequently, they say little about the dispersion in ability.1 Also,
there is evidence suggesting that testing is influenced by family background,
factors such as whether one’s parents went through a divorce early in life.2
Family background in turn is related to family income. Furthermore, even if
a true measure of ability could be found you would need to know how ability
translates into earnings, ceteris paribus. This translation will depend on the
economy’s production technologies, at a minimum. Should a person with
twice the ability of another earn four times as much or one half as much?
Who knows?
An obvious factor influencing earnings may be investments by parents in
the human capital of their children. Poor parents have less wherewithal to
invest in their children than do rich ones. They also can’t borrow against
their offspring’s income in order to finance their kid’s human capital formation. This will lead to parental background being an important determinant
Efficient Investment in Children
2
in income, besides ability.3 It will lead to persistence in income across the
generations of a dynasty. There may be public programs that can alleviate
such imperfections. It is important to invest in children, however, while they
are still young. To quote Heckman [7, p. 96]:
“The reason is this: Cognitive ability is formed relatively early in
life and becomes less malleable as children age. By age 14, basic
cognitive abilities seem to be fairly well set. Since ability promotes academic progress, successful interventions early in the life
cycle of learning lead to higher overall achievement. By the time
individuals finish high school, scholastic ability is determined, and
tuition policy will have little effect on college attendance.”
And, Currie and Thomas [5] find that school test scores at the age of 7 are
significant determinants of future labor market outcomes. Therefore, the
focus of the current analysis is on children where such market imperfections
are likely to weigh the heaviest, as opposed to young adults.
So, what determines parental investment in children? The answer to this
question will depend upon how the world is viewed. To gain some insight into
this issue, an overlapping generations model is constructed where children
differ by ability. Ability has a random component to it. In line with the
classic papers by Becker and Tomes [3] and Loury [14], the productivity
of an adult is determined by his ability and the amount of human capital
investment that his parents undertook when he was a child. The amount
that a parent invests in a child depends on how altruistic parents are toward
children, as well as upon the assumed structure of markets.
Efficient Investment in Children
3
Several different market structures are analyzed. To begin with, the efficient equilibrium is modelled. Then, a world with incomplete financial
markets is entertained. There are two sources of incompleteness. First,
parents are unable to purchase insurance on the ability of their grandchildren. Second, they face borrowing constraints when educating their children.
Specifically, a parent cannot pass on any debts to his offspring. The analysis
here has the flavor of Aiyagari [1] and Laitner [12], who analyze the behavior
of savings in an economy with incomplete financial markets and idiosyncratic
risk. While the focus is different, the work here is also related to Knowles’s
[11] study of the implications of the Becker and Barro [2] fertility model,
where parents decide upon both the quality and quantity of children, for
modeling the distribution of income. Next, the lack of child-care markets is
introduced into the environment with incomplete financial markets. In this
situation a parent must use his own time to improve the human capital of
his child.
The implications of these varying structures on efficiency, output, and the
distribution of income are catalogued. In the general equilibrium model developed here, the absence of insurance markets and the presence of borrowing
constraints does not necessarily lead to underinvestment in children. It can
lead to overinvestment. The investment is inefficient, however, in the sense
that it is not directed toward the children who warrant it the most. Impure
altruism towards children has a big impact on investment in children. This
may be troubling for economists. The fact that tastes are interdependent,
in the sense that a child’s welfare enters a parent’s utility function, does not
imply that an equilibrium lacks Pareto optimality.4 How a parent should love
Efficient Investment in Children
4
his offspring takes one outside of the realm of economics. Tastes may evolve
over time, though, since a little over one hundred years ago children had a
capital asset aspect associated with them; they were expected to work when
young and to provide old-age support to their parents when grown up.
2
Environment
Generational Structure: The environment is a discrete-time infinite-horizon
economy with periods denoted by t ∈ {0, 1, 2, 3, ...}. In each period there is
a continuum of children. Adults live for two periods. In the first period of
life they are young, while in the second period they are old. At the end of
each period t the old adults die. They are replaced by a new generation of
children spawned by the period-t generation of young adults. These children
will become young adults in period t+1 who will then have their own children.
Life goes on in the future in similar fashion.
Ability and Productivity: Children are distributed according to innate
ability, a. A child’s ability may be a function of his parent’s ability, a−1 , in
line with the cumulative distribution function, A(a|a−1 ). The distribution
function A is taken as a primitive. The ability of a child is perfectly known
in period t. The initial distribution for abilities will be given by A0 = A
where A(x) =
R
A(x|a)dA(a). That is, initial abilities are drawn from the
stationary distribution associated with A with the implication that the crosssectional distribution of ability will be given by At = A for all t.
Adults differ according to their productivities, π. Parents can influence
the productivity of their offspring by investing time and money in them.
There is a fixed cost φ associated with educating a child. Now, consider a
Efficient Investment in Children
5
young parent who invests m units of resources (in addition to the fixed cost
φ) and n units of child-care time in his child. The child will grow up next
period with productivity, π 0 , as described by
π 0 = H(a, m, n),
(1)
where H(a, m, n) = a if either n = 0 or m = 0. The function H is taken as
a primitive. Assume that H is strictly increasing in all its arguments and
that H12 , H13 , and H23 > 0. Furthermore, suppose that H is strictly concave
in m and n, both jointly and separately. When resources are invested in a
child he will be labeled as skilled. Otherwise, he will be called unskilled. The
above assumptions guarantee that an efficient allocation will dictate that the
amount of money and time invested in a child will increase with ability.
Goods Production: Each young adult has one unit of time. He can spend
his time either manufacturing goods or supplying services on a child-care
market. If a young adult spends one unit of time making goods then he
can supply π efficiency units of labor in production. Suppose that this adult
had drawn the ability level a−1 last period as a child. A unit of time in
child-care then generates a−1 efficiency units of labor in this activity. Skilled
agents have a comparative advantage in manufacturing goods since for them
π > a−1 while for unskilled agents π = a−1 . Old adults can’t work. Output,
o, is produced according to the constant-returns-to-scale production function
o = O(k, l),
where k and l are the aggregate quantities of capital and labor used in production. Aggregate labor is the sum over the efficiency units of effort supplied
by individuals to manufacturing.
Efficient Investment in Children
6
Output can be used for consumption, c, investment in capital goods, i,
and investment in children, m. In other words
c + i + m = o.
Capital goods accumulate according to the law of motion
k0 = (1 − δ)k + i.
3
(2)
Efficiency
What will an efficient markets equilibrium look like? To answer this question some notation will be introduced. Let Πt (π) denote the distribution of
adults according to productivity in period t. The initial distribution Π0 is
predetermined, while future Π’s will be determined endogenously in a manner discussed below. The amount of time that a young adult of productivity
π spends in production in period t will be represented by Lt (π). In similar
fashion, Mt (a) will specify the amount of goods invested (excluding the fixed
cost φ) in period t on a child of ability a. Likewise Nt (a) will denote the
quantity of young adult time spent in period t on a type-a child. Note that
in an efficient markets equilibrium investment in a child will depend solely on
the child’s ability and nothing else, such as the ability of his or her parents.
Given this notation, the productivity distributions evolve as follows:
Πt+1 (π) = m {a : H (a, Mt (a) , Nt (a)) ≤ π} ,
(3)
where m is the measure on the set of abilities corresponding to the stationary
distribution, A.5 Let
St = {a : H (a, Mt (a) , Nt (a)) > a},
Efficient Investment in Children
7
so that St represents the set of children in period t that become skilled in
t + 1. The set of unskilled children, Ut , will be given by the complement of
this set so that Ut = Stc .
The amount of goods invested in children is given by m =
φ]dA (a), so the resource constraint for this economy reads
c+i+
Z
S
[M (a) + φ]dA (a) ≤ O(k, l).
R
S [M
(a) +
(4)
Finally, the amount of labor that is used in production (measured in efficiency
units) must be less than the total supply of it minus the amount that is used
in child care so that
Z
l=
πdΠ (π) −
Z
N (a)dA (a) .
S
Assuming that only unskilled agents work in the child-care sector (discussed
in the next section), the amount demanded for child care in any period t
must be less than the supply of unskilled agents so that
Z
St
3.1
Nt (a)dA (a) ≤
Z
adA (a) .
(5)
Ut−1
Characterizing Efficient Allocations
Efficiency means that it is not possible to have more consumption at some
date without having less consumption at some other, assuming that leisure
is not valued. The problem of efficient investment in children is to determine
the schedules Lt (π), Mt (a) and Nt (a) in each period given this efficiency
criterion.
Characterizing the schedule Lt is straightforward. Assume that there
are a sufficient number of unskilled agents to meet the economy’s child-care
Efficient Investment in Children
8
requirements. It’s obvious that there should be some π ∗t such that
Lt (π t ) = 1,
if π t ≥ π ∗t ,
Lt (π t ) ≤ 1,
if π t ≤ π ∗t .
(6)
This follows because skilled agents have a comparative advantage in goods
production; that is, their productivity in goods production, π t , exceeds their
productivity in child care, at−1 . Now, since π t+1 = H(at , Mt (at ), Nt (at )), it
transpires that any cutoff rule for π t+1 , or π ∗t+1 will amount to a cutoff rule
for at , or a∗t .
In light of the above, rewrite (4) as
ct +it +
Z
[Mt (a)+φ]dA (a) ≤ O(kt ,
∗
at
Z
πdΠt (π)−
Z
a∗t
Nt (a) dA (a)).(7)
Observe that output next period, ot+1 , can be rewritten to obtain
ot+1 = O(kt+1 ,
Z
H(a, Mt (a), Nt (a))dA(a) +
∗
at
Z
a∗t
adA(a) −
Z
a∗t+1
Nt+1 (a) dA (a)).
The Planning Problem: Let {pt }t≥0 be a sequence of ‘efficiency prices’
with pt > 0 for all t. Then any allocation which maximizes
P
t≥0
pt ct is
efficient. Interpret pt /pt+1 = rt as the gross interest rate from t to t + 1. The
primary interest here is in steady states. Without loss of generality, look
at the problem of maximizing (pt ct + pt+1 ct+1 ) with respect to a∗t+1 , Mt (a),
Nt (a) and kt+1 and then look at the steady-state versions of the first-order
necessary conditions characterizing the solution.
Therefore, the problem of efficient investment in children is to maximize
pt {O(kt ,
Z
πdΠt (π) −
+pt+1 {O(kt+1 ,
−
Z
a∗t+1
Z
a∗t
Z
a∗t
Nt (a) dA (a)) − it −
H(a, Nt (a), Mt (a))dA(a) +
Nt+1 (a) dA (a)) − it+1 −
Z
a∗t+1
Z
Z
a∗t
a∗t
[Mt (a) + φ]dA (a)}
adA(a)
[Mt+1 (a) + φ]dA (a)}.
Efficient Investment in Children
9
subject to (2). The first-order necessary conditions associated with the above
problem are:
a∗t : pt [O2 (·t)Nt (a∗t ) + Mt (a∗t ) + φ] − pt+1 O2 (·t + 1)[H(·t + 1) − a∗t ] = 0, (8)
Mt (a) : −pt + pt+1 O2 (·t + 1)H2 (·t + 1) = 0 (for Mt (a) > 0),
(9)
Nt (a) : −pt O2 (·t) + pt+1 O2 (·t + 1)H3 (·t + 1) = 0 (for Nt (a) > 0), (10)
kt+1: pt = pt+1 [O1 (·t + 1) + (1 − δ)].
(11)
The notation X(·t) signifies that the function X is being evaluated at its
date-t arguments.
The steady state is characterized by the following equations:
M(a∗ ) + φ + N (a∗ )w =
wH(a∗ , M (a∗ ), N(a∗ )) wa∗
−
,
r
r
(12)
wH2 (a, M (a), N(a)) = r (for M (a) > 0),
(13)
H3 (a, M (a), N (a)) = r (for N (a) > 0),
(14)
r = O1 (·) + (1 − δ).
(15)
w = O2 (·)
(16)
Here
represents the wage rate for an efficiency unit of labor. Equation (12) states
that the cost of becoming skilled, M (a∗ ) + φ + N(a∗ )w, should equal the
benefit or the discounted skill premium, wH(a∗ , M (a∗ ), N (a∗ ))/r − wa∗ /r, at
the cutoff level of ability, a∗ . Again note that equation (12) can equivalently
Efficient Investment in Children
10
be thought of as defining a cutoff rule for productivity, π ∗ , which is defined
by π ∗ = H(a∗ , M (a∗ ), N (a∗ )). Next, society should invest time in a child up
until the point where the discounted marginal return wH3 (a, M (a), N(a))/r
equals the cost of the extra child care, w. This is what (14) states. Condition
(13) states a similar condition for resources. Last, (15) is a standard condition
equating the marginal product of capital to the interest rate.
The solution has the following feature. As noted, skilled adults spend
all their time producing. Some unskilled adults will devote their time to
producing, while others will spend it taking care of children. Some children
will have positive amounts of adult time and goods invested in them and
will (when they become young adults) work full time in production as skilled
agents. The rest of the children will have zero adult time and goods invested
in them and will (when they become young adults) work as unskilled agents
either in production or taking care of the next generation of children. Basically, the above conclusion is a result of the assumption that skilled agents
have a comparative advantage in producing manufacturing goods.
4
Market Arrangements
The Setting: Can the efficient allocation be supported as a competitive equilibrium? To answer this question, something has to be said about preferences.
Assume that adults are matched one-to-one with children and that each adult
cares about his child altruistically. Each young parent has preferences of the
form
U(cy ) + βE[U (co0 ) + θV 0 ], 0 < β < 1, 0 < θ ≤ 1,
(17)
Efficient Investment in Children
11
where cy and co0 are his consumptions when young and old. Here V 0 denotes
the expected lifetime utility that his child will realize upon growing up. The
young adult attaches the weight θ to his offspring’s expected lifetime utility
and he discounts the future at rate β.6 The analysis presumes that children
cannot transact for themselves. Hence, there would be no investment in a
child if it was not for his parent’s altruism (i.e., if it wasn’t for the fact that
θ > 0). The case where θ < 1 will be labeled “impure” altruism.7
There are one-period ahead complete insurance markets so that an adult
can insure against the ability level of his grandchild next period. Since the
focus of the analysis is on steady states, all prices will be assumed to be
constant over time. Let q (a0 |a) denote the price of a claim which delivers
one unit of consumption next period if the grandchild’s ability level is a0 and
nothing otherwise, conditional on the young adult having a child of ability a.
The quantity of such claims that the young adult purchases is s(a0 |a). Last,
the young adult can leave, when old, a bequest to his offspring, if he desires.
In particular, if he wants his offspring to receive b0 units of consumption in a
bequest then he will have to put aside b0 /r units of consumption when old,
where r is the market rate of interest on a one-period bond. This bequest
can be negative.
Choice Problems: The dynamic-programming problem facing a young
parent can now be written as
V (π, a, b) = max
0
s(a |a),m,n
½
y
U (c ) + β
Z
0
0
0
0
0
J (π , a , s (a |a) + b) A1 (a |a) da
¾
,(18)
subject to (1) and
cy + m + φI(π 0 , a) + wn +
Z
q (a0 |a) s (a0 |a) da0 = wπ,
(19)
Efficient Investment in Children
12
where the indicator function I is defined so that
0
I(π , a) =
Here
1,
0,
π 0 > a,
otherwise.
J(π 0 , a0 , s(a0 |a) + b) =max
{U (co0 ) + θV (π 0 , a0 , b0 )},
0
b
(20)
subject to
co0 + b0 /r = s(a0 |a) + b.
(21)
When the agent is old he will have a wealth level of s(a0 |a)+b and a grandchild
of ability a0 . At this time the agent will have to decide how much to leave to
his adult child in bequests or b0 . Problem (20) describes the decision making
at this stage of life. Therefore, J(·) is the indirect utility function for an old
adult.8,9
The first-order necessary conditions associated with this problem are:
s(a0 |a) : U1 (cy )q (a0 |a) = βJ3 (π 0 , a0 , s (a0 |a) + b)A1 (a0 |a),
R
m : U1 (cy ) = βH2 (a, m, n) J1 (π 0 , a0 , s (a0 |a) + b) A1 (a0 |a) da0
(when m > 0),
(22)
(23)
R
(24)
b0 : U1 (co0 )/r = θV3 (π 0 , a0 , b0 ).
(25)
n : U1 (cy )w = βH3 (a, m, n) J1 (π 0 , a0 , s (a0 |a) + b) A1 (a0 |a) da0
(when n > 0),
and
Efficient Investment in Children
13
Last, an application of the Benveniste and Scheinkman and envelope theorems to (18) and (20) yields
V1 (π, a, b) = U1 (cy )w,
V3 (π, a, b) = β
Z
J3 (π 0 , a0 , s (a0 |a) + b) A1 (a0 |a) da0 ,
(26)
(27)
J1 (π0 , a0 , s (a0 |a) + b) = θV1 (π 0 , a0 , b0 ),
(28)
J3 (π0 , a0 , s (a0 |a) + b) = U1 (co0 ).
(29)
and
The Perfectly-Pooled Steady State: Now, in a perfectly-pooled steady
state all young agents will consume the same amount, cy .10 Likewise, all old
agents will have the identical level of consumption, co . From (25), (27), and
(29) it then transpires that
r = 1/(βθ).
(30)
If θ = 1 then r = 1/β, the standard result for the neoclassical growth model.
Alternatively, when the parent cares more about his own utility than his offsprings, or when θ < 1, it happens that r > 1/β. Here parents place a higher
weight on present consumption relative to the dynasty’s future consumption.
This dissuades savings and drives up the interest rate. In a perfectly-pooled
equilibrium insurance will sell at its actuarially fair price
q(a0 |a) = A1 (a0 |a)/r.
From (22) this will imply that
U1 (cy ) = βrU1 (co ).
(31)
Efficient Investment in Children
14
Therefore, cy < co when θ < 1.
By using (26), (28), and (30) in (23), and (24), it can be deduced that11
m, n
> 0,
= 0,
if wn + m + φ < w[H(a, m, n) − a]/r,
if wn + m + φ ≥ w[H(a, m, n) − a]/r,
1 = H2 (a, m, n)w/r (when m > 0),
and
1 = H3 (a, m, n)/r (when n > 0).
These are the same conditions as (12), (13), and (14). Therefore, the efficient
allocation can be supported by a competitive equilibrium with complete insurance markets. Markets are still efficient even when parents do not care
about their offsprings as much as themselves.
4.1
Numerical Example One
An example of the efficient markets equilibrium will now be provided. Certain
aspects of this example will be maintained in the subsequent two examples.12
Additionally, some parameter values are chosen so that certain features of
Example Two, which analyzes the economy with incomplete markets, are in
accord with the U.S. data. Take the unit of time for a period to be 20 years.
Tastes: Suppose that parents care about their children as much as they
care about themselves; i.e., let θ = 1. The discount factor is set so that
β = 0.9120 = 0.15. From (30) this implies that in the efficient markets case,
the (annualized) interest rate will be 9.9 percent. This value for the discount
Efficient Investment in Children
15
factor is selected so that the incomplete markets example can replicate the
interest rate and investment-to-GDP ratio observed in the U.S. economy.
Production: Let production be given by a Cobb-Douglas production function so that
o = O(k, l) = zkα l1−α .
In the U.S. economy labor’s share of income is about 64 percent. So, set
α = 0.36. In the U.S. capital depreciates about 10 percent a year implying
that δ = 1 − (1 − 0.10)20 = 0.88.
Ability and Productivity: Assume abilities lie in the discrete set A =
{a1 , a2 , ..., a15 } and evolve in line according with a m-state Markov chain. In
particular, suppose that
Aij = Pr[a0 = aj |a = ai ].
The Markov chain for ability is tuned, following the procedure of Tauchen
√
[23], to match the stochastic process ln a0 = ι(1 − ω) + ω ln a + σ 1 − ω 2 ζ,
where ι = 1.0/(1 − 0.35), ω = 0.35, σ = 0.45 and ζ ∼ N(0, 1). Next, little
is known about the production function for human capital accumulation.
Suppose that
H(a, m, n) = aχ [τ nε + (1 − τ )mε ]ρ/ε + a, ε ≤ 1.
(32)
For now simply assume that χ = 1.55, τ = 0.65, ε = 0.32, and ρ = 0.16. The
fixed cost of becoming skilled is set so that φ = 0.13.
At this stage simply take the choice of parameters values as given for
the stochastic process governing ability and the human capital production
function. They have been picked so that distribution of income arising in
Efficient Investment in Children
16
the incomplete markets model is in congruence with U.S. observation. This
choice of parameter values is discussed in further detail in Example Two.
Algorithm: The equilibrium is computed as follows: To begin with note
from (30) that, given a value for 1/(βθ), the interest rate is known. Since the
production function exhibits constant returns to scale this implies from (15)
that k/l is known too, since O1 (·) is homogeneous of degree zero. Consequently, the equilibrium wage rate w = O2 (·) is also known, since O2 (·) also
depends solely on the k/l ratio. Given w, equations (13) and (14) can then
be used to compute M(a) and N(a) for each value of a. The solutions for w,
M (a) and N(a) are then used to calculate the threshold level of ability, a∗ ,
using (12). Last, for the equilibrium to be meaningful, the child-care market
clearing condition (5) must hold.
4.1.1
Results
The upshot of the example is shown in Figure 1, which plots the ability and
productivity distributions for the population. These distributions are represented by step functions portraying the relevant histogram. The threshold
level of ability lies at about the 6th decile; i.e., only the top 40 percent become
skilled. There is a jump in the productivity distribution at this point. Also,
observe that the productivity distribution is more skewed than the ability
one. For future reference, let W denote the set of productivities that obtains
in the efficient markets equilibrium. The fact that high-ability individuals
have more time and resources invested in them amplifies wage inequality.
This isn’t an issue in an efficient markets equilibrium, since all actors enjoy
the same level of consumption.
Efficient Investment in Children
17
Impure Altruism: Now consider the case where altruism is impure. Specifically, let θ = 0.5. When parents care less about their children they leave
less in bequests. Hence, aggregate savings will be less and the steady-state
interest rate higher. This fact can be seen immediately from (30). The (annualized) interest rate rises from 9.9 to 13.8 percent. The capital-labor ratio
drops by about 110 percent. Additionally, one would expect that parents
will now invest less in their children too. They do. The aggregate amounts
of money and time invested in children fall by 278 percent and 251 percent,
respectively. As a result, output drops by 106 percent. This translates into
a decrease in consumption. When altruism is pure individuals consume an
equal amount in each period, since the interest rate is equal to the rate of
time preference. When altruism is impure their consumption profile slopes
up over time, since the interest rate is higher than their discount factor.
Consumption when young falls by a 120 percent, while consumption when
old drops by 86 percent. While this equilibrium may seem horrifying relative
to the previous one, remember that it is still efficient. Last, observe from
Figure 1 that inequality is reduced.
The standard overlapping generations model: Consider the case where
θ = 0. Now, as θ → 0 equation (30) implies that r → ∞. This isn’t the
standard overlapping generations model, however, as might appear at first
glance. As the old care less about their offspring they borrow more against
their children’s income. This drives up the interest rate. In the standard
overlapping generations model the old can’t borrow against their offspring’s
income; that is, θ = 0 and b0 ≥ 0. In this setting no parent would invest
in his child. Hence, the steady-state supply of labor will be l = E[a]. Next,
Efficient Investment in Children
18
each adult will save according to maxs {U(wa − s/r) + βU(s)}. This yields
the standard efficiency condition U1 (wa − s/r) = βrU1 (s). Now, suppose
that U (c) = [c1−µ − 1]/(1 − µ). Then the solution for savings will be given
by s = S(a; w, r) = wa/[(βr)−1/µ + r−1 ]. The steady-state stock of capital
is given by k = E[S(a; w, r)/r]. This allows the wage and interest rates
to be expressed as w = O2 (E(s)/[rE(a)]) = (1 − α)[E(s)/(rE(a))]α and
r = O1 (E(s)/[rE(a)]) + (1 − δ) = α[rE(a)/E(s)]1−α + (1 − δ). Finally, it is
easy to deduce that E[a]/E[s] = [(βr)−1/µ + r−1 ]/w = {[(βr)−1/µ + r−1 ]/[(1 −
α)r−α ]}1/(1−α) . Therefore, r = r[α/(1 − α)][(βr)−1/µ + r−1 ] + (1 − δ).
In the standard overlapping generations model the interest rate is 6.2
percent (when µ = 2.0), below the 9.9 percent for the efficient markets equilibrium. The capital/labor ratio is higher by 110 percent. The capital stock
is only slightly higher, though, about 23 percent. The reason is that the
aggregate stock of labor is much smaller (87 percent or so), since there is no
investment in children. This translates into aggregate consumption being 50
percent lower. The coefficient of variation in labor income is the same as the
coefficient of variation in ability, or 0.45. Therefore, wage inequality is much
lower in the standard overlapping generations model.
With the efficient markets equilibrium in mind, it is now possible to
discuss various sources of inefficiencies in a decentralized system.
5
Lack of Insurance and Loan Markets
The Setting: The idealized world modeled above assumes that each parent
can buy insurance on the ability of his grandchild. Those parents who draw a
low-ability child are compensated with a cash payment financed by premiums
Efficient Investment in Children
19
paid by parents with a high-ability kid. Further, it also assumes that each
parent can pass on a debt to his child. It’s time to come down from this
rarefied peak.
Suppose that parents can no longer buy or sell insurance. Instead they
are free to trade one-period bonds subject to the proviso that they cannot
pass on any debts to their offspring. Hence, they can self insure against
the ability of their descendents by accumulating a stockpile of assets. Let b
denote the (nonnegative) bequest a young adult inherits upon his parent’s
death and b0 represent the amount that he will leave his child. The amount
of savings that a young adult carries over for his old age will be given by s.
The non-negativity of bequests rules out a credit market. Adults with low
productivity and high-ability children are unable to borrow in order to undertake the efficient amount of investment in their children. Public education
might mitigate this inefficiency somewhat. For instance, if a child’s ability is
currently not known and is independently distributed across generations, or
if the productivity of investment is independent of the child’s ability level,
then efficiency dictates a uniform level of investment in all children regardless
of ability. Borrowing constrained adults may undertake lower investments.13
Choice Problems: After the birth of his child, a young adult’s state of
the world will be given by his productivity, π, the ability of his offspring, a,
and the bequest he will receive from his parent, b. At this stage, the only
randomness in his life will be the ability level of his grandchild, a0 . The
dynamic programming problem facing a young parent is
V (π, a, b) = max {U (cy ) + β
m,n,s≥−b
Z
J(π 0 , a0 , b + s)A1 (a0 |a)da0 },
(33)
Efficient Investment in Children
20
subject to (1) and
cy + m + φI(π 0 , a) + wn + s/r = wπ.
Here
J(π 0 , a0 , s + b) =max
{U (co0 ) + V (π 0 , a0 , b0 )},
0
b ≥0
(34)
subject to
co0 + b0 /r = s + b.
When the individual is old he will have a wealth level of s + b, a grown
child with productivity π 0 , and a grandchild of ability a0 . At this time the
agent will have to decide how much to leave to his adult child in bequests
or b0 . Problem (34) describes the decision making at this time. Therefore,
J(·0 ) is the indirect utility function for the old adult. Denote the decision
rules for s, m, n, and b0 that arise out of these problems by s = S(π, a, b),
m = M (π, a, b), n = N (π, a, b), and b0 = B(π 0 , a0 , s + b).
If the young parent chooses not to educate his offspring then π 0 = a and
m = n = 0. If the individual chooses to educate his offspring then π0 > a
and m, n > 0. The first-order necessary conditions for the young adult are14
R
s : U1 (cy ) = rβ J3 (π 0 , a0 , b0 )A1 (a0 |a)da0 ,
R
m : U1 (cy ) = βH2 (a, m, n) J1 (π 0 , a0 , b0 )A1 (a0 |a)da0 (when m > 0),
and
R
n : U1 (cy )w = βH3 (a, m, n) J1 (π0 , a0 , b0 )A1 (a0 |a)da0 (when n > 0).
Efficient Investment in Children
21
The last two equations imply that
wH2 (a, m, n) = H3 (a, m, n).
(35)
Equation (35) is also implied by (9) and (10). Therefore, while the lack of
insurance might influence the level of investment in a child as measured by
the attained level of productivity, π 0 , it does not distort the decision about
whether to invest cash, m, or time, n.
The Steady State: Again focus on a stationary equilibrium for the economy. In a competitive equilibrium the interest and wage rates will once again
be given by (15) and (16). In a stationary equilibrium the time-series mean of
some variable for the agent will also equal the cross-sectional average across
agents at any point in time. The aggregate supplies of capital and labor will
be given by15
l = E[π] − E[n],
k = E[s/r + b/r].
5.1
Numerical Example Two
Setup: An example of the incomplete markets equilibrium will now be computed. At this point the momentary utility function needs to be parameterized, so let
U(c) =
c1−µ − 1
.
1−µ
Let the coefficient of relative risk aversion assume a standard value of 2
so that µ = 2. Retain the specification of tastes, technology, ability and
Efficient Investment in Children
22
productivity from the previous example. Hence, α = 0.36, β = 0.15, θ = 1.0,
δ = 0.88, ι = 1.0/(1−0.35), ω = 0.35, σ = 0.45, χ = 1.55, τ = 0.65, ε = 0.32,
ρ = 0.16, and φ = 0.13.
Algorithm: Problems (33) and (34) are computed on a discrete space.
Specifically, assume that π ∈ P ≡ {π 1 , ..., π 100 } ⊃ A ∪ W, s + b ∈ S ≡
{υ 1 , ..., υ 125 }, and b ∈ B ≡ {b1 , ..., b125 }.16 Problem (33) can be rewritten as
V (π i , aj , bk ) =
max {U (wπ + bk /r − C(aj , π 0 ; w) − υ/r)
υ∈S,π 0 ∈P
+β
15
X
l=1
(36)
J(π 0 , al , v)Ajl },
where
C(a, π 0 ; w) =
min {m + φ + wn : π 0 = H(a, m, n)}, if π 0 > a,
m,n
0,
(37)
if π 0 = a .
Observe that equations (1) and (35) solve (37). Also, note that it is easy to
recover the solution for s from the above problem since s = υ − bk . Likewise,
problem (34) reads
J(π i , aj , ν k ) =max
{U(υ k − b0 /r) + V (πi , aj , b0 )}.
0
b ∈B
(38)
Now, to compute the solution for J one needs to know the solution for V
and vice versa. This is a fixed-point problem. This problem is solved using
the following iterative scheme. Suppose that one enters some iteration j with
a guess for V , denoted by V j . Given the interest rate, r, and the guess, V j ,
one can then solve (38) to obtain a guess for J, represented by J j . Then a
revised guess for V , or V j+1 , can be obtained by computing the solution to
(36), given J j , r and w. And so the algorithm goes on until V j+1 → V j and
J j+1 → J j . Of course one needs to compute the solutions for the equilibrium
Efficient Investment in Children
23
interest and wage rates, r and w. The details of the algorithm are in the
appendix.
5.1.1
Results
Precautionary Savings: To begin with, the (annualized) interest rate in the
incomplete markets economy is 5.0 percent. This is somewhat shy of the
6.9 percent return on capital reported by Cooley and Prescott [4]. It is also
less than the (annualized percentage) rate of time preference of (1 − β 1/20 ) ×
100% = 9.0 percent. The investment-to-GDP ratio is 0.13, close to the 0.11
observed in the postwar U.S. As has been noted by Aiyagari [1] and Laitner
[12], in economies with uninsured idiosyncratic risk individuals will tend
to engage in precautionary saving. That is, they build up buffer stocks of
financial assets to self insure against a run of bad luck. These precautionary
savings drive down the interest rate. As a result of this precautionary savings,
the capital stock in the incomplete markets economy is 147 percent higher
than in the efficient markets case. In the model, b/(b + s) = 1/3; that is, one
third of total wealth is made up by intergenerational transfers. Modigliani
[16] reports that estimates of this number for the U.S. range from 1/5 to 4/5.
Additionally, individuals invest 60 percent more money in children, but
about the same amount of time as before. Now, 75 percent of children become
skilled as opposed to 41 percent previously. Therefore, borrowing constraints
(in the presence of idiosyncratic risk) do not necessarily lead to underinvestment in children, as is typically presumed.17 It does lead to misinvestment,
however. The total supply of labor in market production is now 1.0 percent lower. This transpires because human capital investment is not directed
Efficient Investment in Children
24
toward the most able individuals.
To see the effect that idiosyncratic risk has on precautionary savings, cut
the standard deviation of the ability shock by half so that σ = 0.22. The
mean level of ability remains unchanged. The interest rate rises from 5.0 to
6.2 percent, while the capital stock drops by 61 percent. Both the money
and time invested in children falls (7.5 percent and 39.8). The number of
children who become skilled also decreases by 3.5 percentage points.
Inequality: Figure 2 plots the distributions of ability and productivity.
The ability distribution is portrayed by a step function while the productivity distribution is illustrated by a discrete density function. The distribution
of productivities is approximately lognormal and resembles the U.S. earnings
distribution – as documented by Knowles [11]. The coefficient of variation
in productivity is about 0.78, close to the 0.77 observed in the data. Likewise,
the Gini coefficient for the distribution of income in the model is 0.39 versus
0.35 in data. Solon [20] reports that for the U.S. the correlation of earnings
across generations is about 0.52; in the model it is 0.64.18 The distribution of
productivities does not arise in a straightforward manner from the distribution of abilities. The distribution of productivities is more skewed than the
distribution of abilities, as can be seen from Figure 2. The match between
the model and the U.S. data is obtained by picking the parameters governing the ability distribution in conjunction with the parameters governing the
production of human capital.
Does the presence of incomplete insurance increase income inequality?
The answer is no. There is less inequality in productivity across individuals
in the incomplete markets world relative to the efficient one. This is readily
Efficient Investment in Children
25
seen by comparing Figures 1 and 2. The Gini coefficient in the efficient markets case is 0.51, as opposed to 0.39 here. The ratio of productivities earned
by the top 5 percent relative to the bottom 5 percent is 20.57, compared with
27.15 for the efficient markets world. In the efficient markets world inequality
isn’t a problem; however, since everybody enjoys the same consumption due
to perfect risk sharing. There may be reasons why inequality may be less
in the incomplete markets world. First, borrowing constraints may reduce
the ability of parents to invest in highly talented children, arguing for lower
dispersion. Second, given the lack of insurance markets, parents may want
to invest more in their children’s human capital (irregardless of ability) to
insure against idiosyncratic risk – recall that the interest rate is lower in
this world.
Welfare Gain from Completing Markets: So, what is the welfare loss that
arises from the uninsured idiosyncratic risk? Some care must be exercised
when assessing this. Steady-state output is 52 percent higher in the incomplete markets economy, as compared with the efficient one. Average consumption is 43 percent higher too. Utility is higher as a consequence. Surely,
the average agent can’t be better off in the incomplete markets economy as
opposed to the efficient one. The answer to this apparent contradiction lies
in the comparison of steady states. Recall that in the incomplete markets
economy there is overaccumulation due to precautionary savings. This leads
to high levels of output, average consumption, and utility.
Now imagine starting the efficient markets economy from the steady-state
capital stock and productivity distribution that obtain in the incomplete
markets economy.19 Over time this economy will converge to the efficient
Efficient Investment in Children
26
markets steady state. Would a young agent prefer the utility realized in this
economy or the average level of expected utility level that obtains in the
incomplete markets economy? Let {cyt , cot }∞
t=0 be the path of consumptions
that will arise in the efficient markets economy and E[V ] denote the average
level of expected utility in the incomplete markets economy. The agent would
be willing to increase his consumption in each period by λ × 100% and still
be happy to live in the efficient markets economy, where
E[V ] + [1 + β]/[(1 − µ)(1 − βθ)]
}1/(1−µ) − 1.
λ = { P∞
y 1−µ
o
t
1−µ
+ β(ct+1 ) ]/(1 − µ)
t=0 (βθ) [(ct )
Observe that as the level of expected utility in the incomplete markets economy, E[V ], increases the fraction of efficient markets consumption that the
agent would be willing to give up, or λ, falls. Clearly solving for λ requires
computing the transitional dynamics for the efficient markets economy. The
algorithm used to do this is detailed in the appendix.
It turns out that λ = −0.63, so that an individual would prefer to live in
the efficient markets economy. Along the transition path from the incomplete
to complete markets economy the individual temporarily increases his consumption as the economy runs down its stocks of physical and human capital.
The time path for aggregate consumption is shown in Figure 3, which also
plots the evolution of the economy’s productivity distribution.20 The rapid
convergence to the efficient-markets steady state should be expected given
that a period is 20 years.
In fact more can be said than this. It is possible to compute the compensating variation for a person starting off from any initial condition, or
(π, a, b)-combination, in the incomplete markets economy. Intuitively, one
would expect that an agent with high values for π, a, and b would gain less
Efficient Investment in Children
27
from such a move than an individual with low values for these variables — recall that in the efficient markets economy all people within a given generation
enjoy the same level of consumption. The distribution of these compensating
variations is plotted in Figure 4. Note that everybody is made better off from
the regime switch, although the person with lowest expected utility in the
incomplete markets economy gains about 3 times as much as the person with
the highest utility.
Impure altruism, again: Once again set θ = 0.5, implying that parents
care less about their children than themselves. How does the new equilibrium
compare with the incomplete markets economy with pure altruism? The
amount of time that parents invest in their childrens’ human capital falls
by 230 percent, while the amount of goods falls by 183 percent. They also
leave 222 percent less in bequests. The fact that parents are investing less
in the future leads to a rise in the equilibrium interest rate from 5.0 to
6.0 percent as the aggregate capital stock drops by 97 percent. The cut in
human capital investment leads to 66 percent less efficiency units of labor
being used in production. The net result of all of this is that output declines
by 77 percent. As θ → 0 the model converges to the standard overlapping
generations structure discussed in the previous section.
6
Lack of Child-Care Facilities
The Setting: The efficient equilibrium presumes that an efficient child-care
market exists. Suppose not. Then, each parent must invest his own time
in his child. Consider a parent of productivity π with bequest b who has a
child of ability a. Assume that a parent of productivity π has a productivity
Efficient Investment in Children
28
P (π) in nurturing his own child. For instance, on the one hand, π = P (π)
represents a “quality-time” world where a parent’s productivity in child care
is the same as in the market. On the other hand,
P (π) =
π,
p∗ ,
π ≤ p∗ ,
(39)
otherwise,
could be thought of as a world where child care is a (relatively) low-productivity
occupation that high-productivity agents have no real advantage at. Now,
for each young parent it must transpire that
n/P (π) + l = 1.
In other words, for a parent of productivity π it costs n/P (π) units of time to
provide n efficiency units of child care. A non-existent (or badly functioning)
labor market in child care will force highly productive adults to devote time
to child care instead of production.
Choice Problems: The dynamic programming problem facing a young
parent is
y
V (π, a, b) = max {U (c ) + β
s≥−b,m,n
Z
J(π 0 , a0 , b + s)A1 (a0 |a)da0 },
subject to (1) and
cy + m + φI(π 0 , a) + s/r = wπ(1 − n/P (π)).
Once again J(·0 ) is defined by (34).
The first-order necessary conditions for the young adult are
R
s : U1 (cy ) = rβ J3 (π 0 , a0 , b0 )dA1 (a0 |a),
Efficient Investment in Children
29
and
R
m : U1 (cy ) = βH2 (a, m, n) J1 (π 0 , a0 , b0 )dA1 (a0 |a) (when m > 0),
R
n : U1 (cy )wπ/P (π) = βH3 (a, m, n) J1 (π 0 , a0 , b0 )dA1 (a0 |a) (when n > 0).
The last two equations imply that
[wπ/P (π)]H2 (a, m, n) = H2 (a, m, n).
(40)
Equation (40) is similar to (35), with one exception. Now, the parent’s
relative productivity level in nurturing, π/P (π), affects the decision about
how much time to invest in child care. The more productive the young parent
is in the market vis à vis at home, the more he will favor investing money as
opposed to time in his child, other things equal.
Before proceeding, note that the quality-time case is just simply uninteresting. If an individual is equally productive in child care as market work
then he would be indifferent between using his own time in child care or using
it at work. Consider a person of productivity π. To buy π units of quality
time in child care on the market (if it was available) would cost w units of
consumption. The agent could supply the same amount of quality time himself and lose w in wage income. Hence, the lack of a child-care market would
be inconsequential. Each parent could easily raise his own child and would
be no cost advantage in letting someone else do it. In the quality-time world
the absence of a child-care market will not matter.
6.1
Numerical Example Three
Setup: The case where child care is a (relatively) low-productivity occupation
is now considered. The parameterization from the incomplete markets case
Efficient Investment in Children
30
(with pure altruism) will be retained. The same numerical algorithm used
to solve the incomplete markets case is employed here. All that remains to
be specified is the threshold level of productivity, p∗ , in (39). It is assumed
that this threshold lies at about the 50th percentile in productivity, implying
that p∗ = 8.5.
6.1.1
Results
Consumption and output both fall by about 6 percent, relative to the incomplete markets case with child care. This is caused by a 78 percent drop
in child-care time. The amount of goods invested in children only decreases
by 7 percent, though. The fraction of children receiving no investment rises
slightly from 25 to 28 percent. Now, the drop in consumption and output
may seem small. This transpires for three reasons. First, the human capital production function (32) is very concave. Second, note that the welfare
loss from an inefficient child-care market arises because high-productivity
individuals must spend their time inefficiently at home raising their kids as
opposed to working. There will be no loss for those agents with π ≤ p∗ . For
an individual with productivity π > p∗ the loss will be w(π/p∗ − 1) per unit
of child-care time. So, a large drop in consumption and output will require
that π − p∗ is large and positive for a significant fraction of the population.
This seems unlikely given the shape of the income distribution and the average earnings of child-care specialists – Figure 5 portrays the situation using
data generated from the model.21 Here the jagged solid line shows the distortion, (π/p∗ − 1), weighted by the number of affected agents. Third, not much
parental time is involved in the human capital development of children.22
Efficient Investment in Children
7
31
Conclusions
When discussing the impact of imperfect financial markets, Arthur Okun [18,
pp. 80-81] once said that “the most important consequence is the inadequate
development of the human resources of the children of poor families – which,
I would judge, is one of the most serious inefficiencies of the American economy today.” A general equilibrium model was developed here where children
differ by ability. Parents could invest time and goods in the development of
their children’s human capital. The model can be used to examine this type
of claim. In a world with perfect financial markets parental investment in a
child would be a function solely of the kid’s ability. Financial markets aren’t
complete, however, in the real world. First, ideally an individual would like
to insure against his grandchild’s ability, as long as there is some randomness
in it. Second, a parent cannot borrow against his child’s future income in
order to educate him today. Given this, the analysis is not as straightforward as Okun [18] and others presume. In fact in the numerical example
presented, the absence of insurance markets and the presence of borrowing
constraints did not lead to underinvestment in children — more money was
invested in kids. The investment was inefficient, however, in that it was not
directed toward the children with the highest ability.
Another market failure may be the lack of child-care markets. This too
is more problematic than is typically believed. For this market failure to be
severe, the returns in terms of a child’s productivity to an extra unit of investment in time cannot fall off too dramatically with the level of investment.
Additionally, there must be a significant number of individuals whose productivity at work is greater than the productivity of the child-care specialist
Efficient Investment in Children
32
who will look after their child. This seems unlikely to be case. As such, it
is likely to be rich people (doctors, lawyers, etc.) and not poor ones (janitors, restaurant waitresses, etc.) that will benefit the most from completing
child-care markets.
Perhaps the problem of underinvestment in children is that altruism is
impure: that is, parents do not care about their children as much as they care
about themselves. Parents invest much less in their children when altruism is
impure. Impure altruism, however, can’t be labelled a market failure in the
traditional sense. The equilibrium may still be Pareto optimal. Over time
the lot of children in society has improved; they no longer work and they
go to school. When analyzing this process, economists often tend to take
agents’ preferences as constant and model it as the outcome of technological
progress. Historians and sociologists often view this process as arising from
shifts in societal attitudes toward children, or changes in preferences. They
arrive at this conclusion by analyzing changes in attitudes towards children
and shifts in childrearing practices, etc. — see Stone [22, chp. 9]. Undoubtedly
both technological and cultural forces are at play in determining the wellbeing of children. There is little an economist can say about how goods (here
children) should (as opposed to do) factor into a person’s tastes. This is a
moral question that society may have to take a stand on.
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Efficient Investment in Children
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Theory of Fertility, Quarterly Journal of Economics, CIII (1988), 125.
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Efficient Investment in Children
A
35
Appendix: Algorithms
Incomplete Markets Steady State: The algorithm used to compute the solution for the incomplete markets case will now be described. The other cases
are computed in a similar manner.
Computing the competitive equilibrium for the incomplete market economy involves the following steps. To begin with, draw a random time series
of T observations for a using the distribution function A. Call this sample
path {at }Tt=0 .
1. Enter iteration j with a guess for the interest and wage rates, r and w,
denoted by rj and wj .
2. Given this guess, solve the choice problems (33) and (34).
3. Simulate the decision rules for (33) and (34) T times using the randomly
generated sample for the a’s. To do this, start at the point (π 0 , a0 , b0 ).
Use the decision rules from problem (33) to get s0 , π1 . Next, use the
decision rule from problem (34) at the point (π 1 , a1 , s0 + b0 ) to obtain
b1 . The decision rules for (33) can now be evaluated at the point
(π 1 , a1 , b1 ) to get s1 , π 2 . Proceed down the rest of the sample path in
similar manner. Collect data on s, b0 , n, and π; that is the sequences
{st }Tt=1 , {bt+1 }Tt=0 , {nt }Tt=1 , and {π t }Tt=1 . Calculate E[s + b0 ], E[π], and
E[n], or the sample means for s + b0 , π, and n.
4. Compute a revised guess for the interest and wage rates, rj+1 and w j+1 .
Since the focus is on a stationary competitive equilibrium, a natural
Efficient Investment in Children
36
way to do this would be to set
rj+1 = O1 (kj , lj ) − δ,
and
wj+1 = O2 (kj , lj ).
Now, in equilibrium aggregate savings will be given by kj = E[s+b0 ]/rj
and lj = E[π] − E[n].
5. Check if metric(rj+1 , rj ) and metric(wj+1 , wj ) fall below some specified
tolerance. If so, stop. If not, go back to step 1.
The child-care market must clear for an equilibrium to prevail. This necessiates checking that the following condition holds:
PT
t=1
nt ≤
PT
t=1
at−1 [1−
I(πt , at−1 )], where again I(π t , at−1 ) = 1 if πt > at−1 and I(π t , at−1 ) = 0 if
π t = at−1 .
Complete Markets Transitional Dynamics: Let the initial aggregate stock
of capital be represented by k0 and the initial distribution of productivities
be denoted by Π0 . Recall that these state variables arise from the incompletemarkets-economy steady state. The goal is to compute the economy’s transition path to the efficient markets steady state. Pick a T , suitably large
enough, so that convergence to the new steady state takes place within T + 1
periods. Therefore, let kt , lt , mt assume their steady-state values for all
t ≥ T + 1. The algorithm works as follows:
1. Enter iteration j with a guess for the time paths {kt }Tt=1 , {lt }Tt=1 , and
{mt }Tt=1 denoted by {kjt }Tt=1 , {ljt }Tt=1 , and {mjt }Tt=1 . This implies a guess
Efficient Investment in Children
37
for {wt }Tt=1 , denoted by {wtj }Tt=1 . Note that k0 and E0 [π] are tied down
by the initial condition.
2. Start off at period 0. Now, given w1j , mj1 , and kj2 solve for a∗0 , M0 (a),
N0 (a), l0 , and k1 using
M0 (a∗0 )
+φ+
N0 (a∗0 )
z
w0
}|
{
(41)
O2 (k0 , l0 )
= w1j [H(a∗0 , M0 (a∗0 ), N0 (a∗0 )) − a∗0 ]/ [O1 (k1 , lj1 ) + (1 − δ)],
|
r0
H2 (a∗0 , M0 (a∗0 ), N0 (a∗0 )) =[O1 (k1 , lj1 ) + (1 − δ)],
|
{z
{z
}
}
r0
w1j H3 (a∗0 , M0 (a∗0 ), N0 (a∗0 )) =[O1 (k1 , lj1 ) + (1 − δ)] O2 (k0 , l0 ), (42)
|
and
{z
r0
[O(k0 , l0 )+(1 − δ)k0 − m0 − k1 ]−µ
}
(43)
= βθ[O1 (k1 , lj1 ) + (1 − δ)][O(k1 , lj1 )+(1 − δ)k1 − mj1 − kj2 ]−µ .
Equations (41) to (42) derive from (8) to (9). Equation (43) is the Euler equation governing capital accumulation and is a rewritten version
of (11). In any perfectly-pooled equilibrium, a young parent’s Euler
equation implies that cyt = (βr)−1/µ cot+1 . Additionally, it can be shown
that for each dynasty cot = θ−1/µ cyt . Aggregating over agents, while
using these two facts, gives ct = (βθr)−1/µ ct+1 . This forms the basis
for (43).
Efficient Investment in Children
38
(a) Solving the above system of equations requires an inner loop. That
is, given a guess for l0 and k1 , first solve for a∗0 , M0 (a), N0 (a) using
the first three equations. Then, revise the guess for l0 and k1 using
the l0 = E0 [π]−
R
a∗0
N0 (a)dA(a) and (43). Iterate until convergence
in the answers for a∗0 , M0 (a), N0 (a), l0 , and k1 is achieved. Exit
the inner loop.
3. Given this solution enter period 1 with the initial condition k1 and Π1 .
Given w2j , mj2 , and kj3 solve for a∗1 , M1 (a), N1 (a), l1 , and k2 using the
updated version of (41) to (43). Travel down the path in this fashion
to get {kt }Tt=1 , {lt }Tt=1 , and {mt }Tt=1 . Use this solution for the revised
}Tt=1 , {lj+1
}Tt=1 , and {mj+1
}Tt=1 .
guess {kj+1
t
t
t
4. Repeat until convergence in {kjt }Tt=1 , {ljt }Tt=1 , and {mjt }Tt=1 is obtained.
Exit the algorithm. Additionally, for the solution to be meaningful, it
must also be checked that the child-care market-clearing condition (5)
always holds along the equilibrium path.
Efficient Investment in Children
39
FOOTNOTES
1. For example, IQ test scores are normalized to have a mean of 100
and a standard deviation of 15. Hence, dispersion in IQ cannot be used to
measure dispersion in ability.
2. See Heckman, Hsse and Rubinstein [8]. Also, Neal and Johnson [17]
find that AFQT scores are influenced by family background and school environments.
3. “(T)he disadvantages young black workers now face in the labor market
arise mostly from the obstacles they faced as children in acquiring productive
human capital”, say Neal and Johnson [17, p. 871].
4. So long as the child’s welfare does not enter into someone else’s utility
there will be no incentive for one benefactor to free ride off another.
5. Let π 0 = H(a, N(a), M (a)) ≡ G(a). Now, suppose that G has a
continuously differentiable inverse and that A has a continuous density, A1 .
0
Then, Π1 (π 0 ) = A1 (G−1 (π 0 ))|G−1
1 (π )|.
6. These are similar to the preferences considered in a classic paper by
Phelps and Pollak [19]. Each generation assigns a more primal role to its
own utility vis à vis its offspring’s. These preferences are non-stationary,
however, since the next generation will assign a primal role to its own utility.
As Phelps and Pollak [19] note, Frank P. Ramsey termed the practice of
discounting the next generation’s utility “ethically indefensible.”
7. The word impure arises from Edgeworth [6, p. 16] who said “(f)or
between the two extremes of Pure Egoistic and Pure Universalistic there
may be an indefinite number of impure methods; wherein the happiness of
others as compared by the agent (in a calm moment) with his own, neither
Efficient Investment in Children
40
counts for nothing, not (sic) yet ‘counts for one,’ but counts for a fraction.”
8. Observe that each parent assumes that his offspring will do what is
in the descendent’s best interest. That is, while the parent doesn’t assign a
primal role to the offspring’s utility he correctly assumes that his offspring
will. The resulting equilibrium is time consistent.
9. The forms of problems (18) and (20) would become more complicated
if children overlapped more periods with their parents, and/or if children
also cared about their parents. Strategic considerations between parents and
children would then emerge. See Laitner [13] for an excellent review of this
literature.
10. If the economy starts out in a perfectly-pooled equilibrium, then it will
remain there forever. The question about how a perfectly-pooled equilibrium
arose to begin with is ignored. A classic application of the perfect-pooling
concept is Lucas’s [15] study on international asset pricing — see his analysis
for more detail on this notion.
11. Deriving the threshold condition is a little less straightforward. Substituting equation (21) into (19) gives a young parent’s lifetime budget constraint.
cy + m + φI(π 0 , a) + wn +
Z
q(a0 |a)[co0 + b0 /r]da0 = wπ +
Z
q(a0 |a)bda0 .
So, all a young agent cares about is the present-value of his income, wπ +
R
q(a0 |a)bda0 , not how it is split up between wages and bequests. Hence,
R
q(a0 |a)bda0 , a). In a perfectly-pooled steady state this further simplifies
the young agent’s value function can be rewritten as V (π, a, b) = W (wπ +
to V (π, a, b) = W (w(π − a) + wa + b/r, a), since q(a0 |a) = A1 (a0 |a)/r. Now
imagine solving problem (18) subject to the additional constraint that m, n >
Efficient Investment in Children
41
0; i.e., that the agent’s child becomes skilled. Let m and n denote the
optimal solutions for money and time. Hence π 0 = H(a, m, n) > a. It costs
wn + m+ φ in terms of current resources to provide an individual’s child
with an extra w[H(a, m, n) − a] units of labor income. Now, given the form
of the value function, w[H(a, m, n) − a] in labor income is worth the same
to the child as rw[H(a, m, n) − a] in bequests. But, as is evident from the
lifetime budget constraint, leaving b0 = rw[H(a, m, n) − a] in bequests costs
only
R
q(a0 |a)w[H(a, m, n) − a]da0 = w[H(a, m, n) − a]/r in terms of current
resources. Therefore, in order to skill the child it must transpire that
wn + m + φ < w[H(a, m, n) − a]/r.
12. The examples presented are intended merely to illustrate the theory.
13. Borrowing constraints may be a factor in limiting college attendance,
too. The situation here is different for two reasons: first, a young adult is
presumably now deciding about his own educational inputs and, second, is
borrowing against his own future income. That is, the young adult is issuing
a claim against his own income and not against his descendents’s incomes.
14. Assume that U1 (0) = ∞ so that the individual will avoid hitting the
borrowing constraint at all cost.
15. Let Dy (π, a, b) represent the stationary distribution across young
agents.
Now, the distribution A(a0 |a) and the decision-rules M (π, a, b),
N(π, a, b), S(π, a, b), and B(π0 , a0 , s+b) define a transition operator T y (π 0 , a0 , b0 |π, a, b).
R
The stationary distribution Dy must solve Dy (π 0 , a0 , b0 ) = T y (π 0 , a0 , b0 |π, a, b)dDy (π, a, b).
Hence, l =
R
[π − N(π, a, b)]dDy (π, a, b). Last, the distribution over old
R
agents, Do (π, a, s−1 +b−1 ), will be defined by Do (π 0 , a0 , s+b) = T o (π 0 , a0 , s+
b|π, a, b)dDy (π, a, b), where the form of transition operator, T o , will depend
Efficient Investment in Children
R
on A, M , N , and S. Therefore, k = [ S(π, a, b)dDy (π, a, b) +
b)dDo (π 0 , a0 , s + b)]/r.
42
R
B(π 0 , a0 , s +
16. Recall that W is the set of productivities that emerges in the efficient
markets equilibrium.
17. This seems to derive from the higher level of physical wealth in
economy. Hence, parents can invest more cash in their kids. Additionally,
as the interest rate falls parents substitute out of physical capital and into
human capital.
18. For a review of this literature, see Stokey [21]. The assumed degree
of persistence in ability (ω = 0.35) is not high. According to Hernnstein and
Murray [9] the intergenerational correlation in AFQT scores lies somewhere
between 0.4 and 0.8. Hence, in the model, about one half of the persistence
in income comes from market structure.
19. As before, assume a perfectly-pooled equilibrium. Hence, within each
generation all actors are equally well off.
20. The left panel shows how the productivity distribution evolves over
time. The initial distribution is portrayed by the discrete density function
shown by the − − lines. The − · · · − line shows, in step-function form, the
productivity distribution that obtains after one period. The solid line gives
the final productivity, again in step-function form.
21. One could argue that the market sector is more efficient at providing
child care than the home sector, say due to economies of scale or specialization. Suppose that the market sector is twice as efficient at looking after
children relative to the home sector. To capture this, let P (π) = π/2, for
π ≤ p∗ , and P (π) = p∗ /2, otherwise. Now, there is a 164 percent drop in
Efficient Investment in Children
43
child-care time, while the amount of goods invested falls by 15 percent, relative to the incomplete markets case with child care. Consumption and output
are both reduced by 13 percent. Of course, one could just as easily argue
that the market sector is less efficient at providing child care than the home
sector, due to incentive and other problems. For instance, a daycare provider
may not care about your children as much as you do. A recent study financed
by the National Institute on Child Health and Human Development found
(as reported by the The New York Times, April 19th 2001) that children
raised in daycare are three times as likely to experience behavioral problems
as those raised primarily by their mothers. The study followed 1,100 children
in 10 cities from a variety of child-care settings.
22. In the U.S., an average mother spends about 3.0 hours a week per
child on direct child care, according to Hill and Stafford [10, Table 17.6]. She
had about 2.5 kids (in the postwar period). Direct child care is defined to be
activities such as “helping/teaching, reading/talking (including ‘yelling at’),
indoor playing, outdoor playing, medical care and other regular child care
such as feeding, dressing, supervising, and other direct interaction” (p. 427).
The time spent drops off dramatically as a child ages. For instance, a highschool educated mother spends about 6.0 hours a week on these activities for
a preschooler, but only about 1.7 hours a week for a child in school. Now, a
parent only raises a child for about 18 of the 40 or so years that he works.
And, the average household puts in about 54 hours of market per week. Thus,
about [2.5 × (18/40) × 3.0]/(2.5 × (18/40) × 3.0 + 54) × 100% = 5.9 percent of
a parent’s time is spent on child care, or about 2.4 percent per child. These
numbers seem low. The estimates exclude any purchased time on direct
Efficient Investment in Children
44
child care. In the incomplete markets model with child-care markets about
E[n]/E[a] × 100% = 2.7 percent of the total feasible time available for child
care is used. In the framework without child-care markets only about 0.9
percent of available parental time is used. Any serious quantitative analysis
would have to obtain accurate statistics on the amount of privately controlled
inputs going into a kid’s human capital production.
Efficient Investment in Children
Pure altruism: θ = 1
Impure Altruism: θ = 0.5
Threshold
Threshold
0.20
0.20
0.15
0.15
Mass
Mass
45
0.10
0.05
0.10
0.05
Productivity
Ability
Ability
Productivity
0.00
0.00
20
50
80
Index -- Ability and Productivity
20
50
80
Index -- Ability and Productivity
Figure 1: Ability and Productivity Distributions — Efficient Markets Case.
Efficient Investment in Children
46
Ability
Productivity
0.10
0.20
0.08
Mass
Mass
0.15
0.10
0.06
LogN(2.25; 0.0.54)
Discrete Density
0.04
0.05
0.02
0.00
0.00
0
20
40
Index -- Ability
60
0
20
40
60
Index -- Productivity
Figure 2: Ability and Productivity Distributions — Incomplete Markets Case.
Efficient Investment in Children
Productivity Distribution
47
Capital and Consumption
4
First
Final
Initial
Mass
0.15
0.10
Capital and Consumption
0.20
3
Consumption
2
1
Incomplete Markets Steady State
0.05
Capital
0
0.00
20
50
80
Index -- Productivity
1
12
Periods
Figure 3: Transitional Dynamics — From Incomplete to Complete.
23
Efficient Investment in Children
48
0.08
Frequency
0.06
0.04
0.02
0.00
-0.920
-0.842
-0.764
-0.686
-0.608
-0.530
-0.452
Compensating Variation
Figure 4: Distribution of Compensating Variations.
-0.374
-0.296
Efficient Investment in Children
49
p*
8
0.06
Mass
Mass
6
0.04
4
2
0.02
[F /p*) - 1]×Mass
0
0.00
10
30
50
70
Index -- Productivity
Figure 5: Lack of Child-Care Market — Distortion.
90
Weighted and Unweighted Wedge
10
F /p*) - 1
0.08
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