Introducing Uncertainty into Baland and Robinson’s
Model of Child Labour
William Pouliot1
Carleton University
Jean-Marie Baland and James A. Robinson (2000) develop a model of child labour in which child
labour can be inefficiently high when parents face credit constraints or bequests are at a corner.
We show here that, when period-2 parental income is uncertain, Propositions 1, 2 and 3 of Baland
and Robinson remain valid. However, when returns to human capital are uncertain and insurance
markets do not exist, we show that the level of child labour can be inefficiently high even when
parents are not credit constrained and bequests are interior. (JEL D61, D80, D91, J13, J24)
I. Introduction
Jean-Marie Baland and James A. Robinson (2000) construct a model of child labour with the
purpose of studying how household decisions affect child labour, the child’s ability to acquire
human capital, and the welfare implications of this. In essence, it is a two period model
without uncertainty that relies, though not exclusively, on market imperfections in the form
of credit constraints to generate conditions under which child labour can be inefficiently
high. This theme of market imperfections is continued here by first allowing uncertainty in
period-2 parental income and then follows with the introduction of uncertainty in returns
to human capital along with insurance markets, both complete and incomplete.
In our first consideration, we show that the introduction of uncertainty in period-2
parental income does not alter Propositions 1, 2 and 3 of Baland and Robinson. The fact
that Proposition 1 continues to hold is unexpected, as one would expect parents to increase
period-1 levels of child labour beyond what is efficient in order to offset uncertainty in period2 income. This, as will be shown via Corollary 1, does not occur as parents can compensate
for uncertianty in period-2 income through transfers between periods that may occur by
increasing savings or alternatively by reducing bequests. For our next consideration, we
permit uncertainty in returns to human capital, where we establish that, when insurance
markets are complete, Propositions 1, 2 and 3 of Baland and Robinson (2000) remain valid.
However, under the scenario of uncertainty in returns to human capital and the absence
of insurance markets, we find, unexpectedly, that these two assumptions create conditions
under which child labour can be inefficiently high. More specifically, we find Proposition 1
of Baland and Robinson (2000) no longer holds, i.e. child labour is inefficiently high even
when both savings and bequests are interior. This result is unexpected as the nature of
this uncertainty can no longer be offset through reallocation of savings across periods or
1
I would like to thank Chris Worswick and Jeff Bernstein for providing many valuable comments.
1
altering the amount of bequests, as was the case for uncertainty in period-2 income.
Some might consider our assumption regarding returns to human capital to be questionable. More specifically, why should returns to human capital be uncertain? If one were
to interpret human capital as education, and accumulation of human capital as higher education, it then seems rather obvious; returns to education will depend on labour market
conditions, the overall state of the economy, and on advances in technology. All of these
factors affect labour market conditions, its productivity, and thereby, affect future returns
to education. Interpreting accumulation of human capital along these lines provides some
motivation for incorporating uncertainty into Baland and Robinson’s model of child labour,
and in so doing, we see how uncertainty in returns to human capital affects household
choices. In particular, we shall see that, via Proposition 10, households substitute out of
the risky asset, human capital, which causes child labour to become inefficiently high. These
results indicate that it is not uncertainty in itself that is important for households’ decisions
but rather the way in which it affects different variables involved in these decisions.
II. Baland and Robinson’s Model
We limit description of Baland and Robinson’s (2000) model to that of the household’s
maximization problem. Their model is comprised of two periods, referred to as 1 and 2.
At the beginning of period 1, parents decide how to allocate their children’s unit time
endowment between child labour lc and human capital accumulation, where lc represents
the fraction of a child’s unit labour endowment allocated to work. For simplicity, we assume
that parents earn income, A, in periods 1 and 2.
In period 2, each child supplies h(1 − lc ) units of labour, where h(·) represents human
0
00
capital, and h (·) > 0 and h (·) < 0. We normalize the return to human capital to
1. The parent’s utility function is separable in c1p and c2p , consumption in periods 1 and 2
respectively, and is denoted by
Wp (c1p , c2p , Wc (cc )) = U (c1p ) + U (c2p ) + Wc (cc ),
(1)
where U (·) and Wc (·) are both twice continously differentiable and strictly concave,
and where Wc (·) denotes the child’s utility function. Parents can also choose to provide
transfers, denoted b and referred to as bequests, to their children in period 2 and can transfer
income between periods by saving, denoted s. Capital markets are imperfect which prevents
parents from borrowing. They can, nevertheless, save.
The household maximization problem can be represented as,
max U (c1p ) + U (c2p ) + Wc (cc )
s,b,lc
(2)
subject to
c1p = A + lc − s,
c2p = A − b + s,
(3)
cc = h(1 − lc ) + b.
After substituting the constraints for each variable in the maximization problem, we arrive
at the following first-order conditions,
2
0
0
U (A − b + s) = Wc (h(1 − lc ) + b) b > 0,
0
0
U (A + s) > Wc (h(1 − lc ))
0
0
(4)
b = 0,
(5)
0
U (A + lc − s) = Wc (h(1 − lc ) + b)h (1 − lc ),
0
0
U (A + lc − s) = U (A − b + s),
0
0
U (A + lc ) > U (A − b),
(6)
s > 0,
(7)
s = 0.
(8)
Remark 1 We shall define optimal values of lc , s and b as determined by equations
(4) - (8) to be lcc , sc and bc respectively, where the superscript c refers to certainty.
Using equations (4) through (8), Baland and Robinson (2000) establish the following
propositions.
Proposition 1 If bequests and savings are interior, then the laissez-faire level of child
labour is efficient, i.e. h(1 − lc ) = 1.
Proposition 2 If bequests are at a corner, then the laissez-faire level of child labour, lc , is
inefficiently high, i.e. h(1 − lc ) > 1.
Proposition 3 If savings are at a corner, then the laissez-faire level of child labour, lc , is
inefficiently high, i.e. h(1 − lc ) > 1.
III. Uncertainty in Parental Period-2 Income
For our first consideration, we introduce uncertainty by allowing parental period-2 income,
Θ, to be an absolutely continuous random variable, and note here that all corollaries and
propositions obtained herein apply when parental income is in the form of a discrete random
variable. Under this specification, we introduce the expectation operator E[ · ]. Accordingly,
we set E[Θ] = A. Households’ now face the following intertemporal expected utility
maximizing problem,
n
o
max E U (A + lc − s) + U (Θ − b + s) + Wc (h(1 − lc ) + b)
s,b,lc
(9)
subject to
c1p = A + lc − s,
c2p = Θ − b + s,
cc = h(1 − lc ) + b.
(10)
Again, Propositions 1 through 3 hold for the case where Θ is an absolutely continuous
random variable. They are restated below as Propositions 4 - 6.
3
We have the following first order conditions,
0
0
E[U (Θ − b + s)] = E[Wc (h(1 − lc ) + b)]
0
0
E[U (Θ + s)] > Wc (h(1 − lc ))
0
0
b > 0,
(11)
b = 0,
(12)
0
U (A + lc − s) = E[Wc (h(1 − lc ) + b)h (1 − lc )],
0
0
U (A + lc − s) = E[U (Θ − b + s)]
0
0
U (A + lc ) > E[U (Θ − b)]
(13)
s > 0,
(14)
s = 0.
(15)
Proposition 4 When parental period-2 income, Θ, is an absolutely continuous random
variable with savings and bequests interior, i.e. s > 0 and b > 0, then the laissez-faire
level of child labour is efficient.
Proof.
Using equations (11) and (14), we arrive at
0
0
U (A + lc − s) = E[Wc (h(1 − lc ) + b)].
(16)
Substituting equation (16) into equation (13), we have
0
0
0
E[Wc (h(1 − lc ) + b)] = E[Wc (h(1 − lc ) + b)]h (1 − lc )
(17)
which implies
0
h (1 − lc ) = 1,
(18)
from which we can conclude that child labour is efficiently allocated.
Proposition 5 When parental period-2 income, Θ, is an absolutely continuous random
variable, and bequests are at a corner, i.e. b = 0, then the laissez-faire level of child
labour is inefficiently high.
Proof.
Using equations (12) and (14), we arrive at
0
U (A + lc − s) > Wc (h(1 − lc )).
(19)
Substituting equation (19) into equation (13), we arrive at
0
0
0
Wc (h(1 − lc )) < Wc (h(1 − lc ))h (1 − lc ).
(20)
which implies
0
h (1 − lc ) > 1,
from which we can conclude that child labour is inefficiently high.
4
(21)
Proposition 6 When parental period-2 income, Θ, is an absolutely continuous random
variable and savings are at a corner, then the laissez-faire level of child labour is inefficiently
high.
Proof.
Using equations (11) and (15), we arrive at the following relation
0
0
U (A + lc ) > E[Wc (h(1 − lc ) + b)],
(22)
this along with equation (13) implies
0
0
0
E[Wc (h(1 − lc ) + b)]h (1 − lc ) > E[Wc (h(1 − lc ) + b)],
(23)
and hence
0
h (1 − lc ) > 1,
(24)
from which we conclude that child labour is inefficiently high.
Remark 2 We shall define optimal values of lc , s and b as determined by equations (11)
- (15) to be lcuc , suc and buc respectively, where the superscript uc refers to uncertainty.
We make the following observation regarding the two values of child labour, lcuc and
0
0
and note this follows from the assumption that h (·) > 0 and that h (·) = 1 at the
optimum.
lcc ,
Remark 3 lcuc = lcc .
We now compare the optimal values, lcc , sc and bc found under certainty with the
optimal values, lcuc , suc and buc found under uncertainty. In order to do so, we make use
of Jensen’s inequality.
Jensen’s Inequality: Let X be a random variable and f (·) a strictly convex function of this
random variable. Then E[f (X)] > f (E[X]).
0
Consider equation (11), using Jensen’s inequality and strict convexity of U (·), we can
establish the following inequality,
0
0
E[U (Θ − bc + sc )] > U (E[Θ] − bc + sc )
0
=
U (A − bc + sc )
=
Wc (h(1 − lcc ) + bc )
=
E[Wc (h(1 − lcc ) + bc )]. (25)
0
0
Since equality does not hold in (25), which is the optimal condition given in equation (11),
this implies that lcc , sc , and bc are not optimal values under uncertainty. In order for
equality to hold, we must have, then, that either the difference −bc + sc rises or that lcc rises.
Since either one or both of these can happen, it is no longer clear what the relationship will
be between the optimal values, lcuc , suc and buc , and lcc , sc and bc . One might expect the
5
optimal values, that are determined under uncertainty, of savings and child labour to rise
and bequests to fall relative to the corresponding optimal values determined under certainty.
This, however, cannot be supported as bequests could rise, with savings and child labour
rising more than enough to offset this rise in bequests.
Even though the relationship between corresponding members of the two sets of optimal
values is not as clear as intuition suggests, we can, nevertheless, make one statement in this
regard. The details of this statement are provided in the following corollary.
Corollary 1 If parental period-2 income, Θ, is an absolutely continuous random variable,
0
with the values of sc , bc ,
suc and buc all interior, and 2 U (·) is a strictly convex
function, then at least one of the following must hold:
bc > buc or suc > sc .
Proof.
Assume not, then
bc ≤ buc and sc ≥ suc .
(26)
sc − bc ≥ suc − buc .
(27)
Equation (26) implies
Then
0
0
U (Θ − bc + sc ) ≤ U (Θ − buc + suc )
(28)
00
which follows by (27) and U (·) < 0. This implies
0
0
E[U (Θ − buc + suc )] ≥ E[U (Θ − bc + sc )]
(29)
which uses a property of the Riemann-Stieltjes integral. Using Jensen’s inequality along
0
with the strict convexity of U (·), we arrive at the following inequality,
0
0
0
0
E[U (Θ − bc + sc )] > U (E[Θ] − bc + sc ) = U (A − bc + sc ) = Wc (h(1 − lcc ) + bc ), (30)
where the last equality follows from the first-order condition given in equation (4). Using
00
buc ≥ bc , Wc (·) < 0 and Remark 3, we establish that
0
0
Wc (h(1 − lcc ) + bc ) ≥ Wc (h(1 − lcuc ) + buc ).
(31)
But this implies
0
0
E[Wc (h(1 − lcc ) + bc )] ≥ E[Wc (h(1 − lcuc ) + buc ))].
Combining equations (29), (30) and (32), we can conclude that
0
0
E[U (Θ − buc + suc )] > E[Wc (h(1 − lcuc ) + buc )].
2
This holds for the CARA class of utility functions.
6
(32)
But this is a contradiction, as lcuc , suc and buc are optimal.
Corollary 1 provides the necessary explanation as to why child labour remains efficiently
allocated when parental period-2 income is uncertain. Specifically, we see that uncertainty
in parental period-2 income causes at least one of savings to rise or bequests to fall relative
to their corresponding values found under certainty. Through these adjustments, parents
are able to compensate for uncertainty in period-2 income and can smooth consumption.
IV. Uncertainty in Returns to Human Capital and Parental Period-2 Income
For our second consideration along these lines, we now permit the return to human capital to
be an absolutely continuous random variable and insurance markets to exist. With regards
to the latter, we incorporate uncertainty through the following parametrization. Let
cc = ∆h(1 − lc ) + b,
(33)
where ∆ is a random variable with positive support, E[∆] = 1. We note here that cc
increases with ∆, a result that will be used in the proof of Proposition 10.
The generic household’s maximization problem now becomes
n
o
max E U (c1p ) + U (c2p ) + Wc (cc ) ,
(34)
s,b,lc
subject to
c1p = A + lc − s,
c2p = Θ − b + s,
(35)
cc = ∆h(1 − lc ) + b.
Replacing c1p , c2p and cc in the objective function given by (34) with the corresponding
values that are detailed in the budget constraint found in (35), and then differentiating the
objective function in (34) with respects to lc , b and s results in the following first order
conditions,
0
0
E[U (Θ − b + s)] = E[Wc (∆h(1 − lc ) + b)]
0
0
E[U (Θ + s)] > E[Wc (∆h(1 − lc ))]
0
0
b > 0,
(36)
b = 0,
(37)
0
U (A + lc − s) = E[Wc (∆h(1 − lc ) + b)∆h (1 − lc )],
0
0
U (A + lc − s) = E[U (Θ − b + s)]
0
0
U (A + lc ) > E[U (Θ − b)]
(38)
s > 0,
(39)
s = 0.
(40)
Our next three propositions, Propositions 7, 8 and 9, establish equivalent versions of
Propositions 1, 2 and 3 but now for the case when returns to human capital are random
and insurance markets are complete. In order to establish the propositions that follow, we
assume that the two parties enter into a binding insurance contract, where ∆ and h(1 − lc )
are common knowledge. Moreover, we assume that the child purchases insurance in period
2, sometime before the resolution of uncertainty, i.e. before ∆ is revealed, and that parents
are aware in period 1 of the child’s future purchase.
7
Proposition 7 If children have access to competitive insurance markets where they can
contract at rate
π = ∆ − 1,
per unit of human capital, returns to human capital are an absolutely continuous random
variable, and both savings and bequest are interior, i.e. s > 0 and b > 0, then child labour
will be efficiently allocated.
Proof.
To establish this, we show first that expected profit of the insurance company is zero. The
profit Π of the insurance company is defined as
Π = π h(1 − lc ).
(41)
Taking expectations in (41), we conclude that
E[Π] = E[π] h(1 − lc ) = 0,
(42)
as E[π] = 1 − E[∆] = 1 − 1 = 0, and h(1 − lc ) is predetermined.
We now show that the household’s budget constraint with insurance reduces to that
given in (3). Without insurance, the child’s consumption is given by
cc = ∆h(1 − lc ) + b.
(43)
With insurance, the child’s consumption becomes
cc = ∆h(1 − lc ) + b − π h(1 − lc )
= ∆h(1 − lc ) + b + (1 − ∆)h(1 − lc )
= h(1 − lc ) + b,
which is the consumption given in equation (10), and the maximization problem reduces
to that given in (9) and (10). According to Proposition 4, we can conclude that Proposition
7 holds.
Proposition 8 If children have access to competitive insurance markets where they can
contract at rate
π = ∆ − 1,
per unit of human capital, returns to human capital are an absolutely continuous random
variable, bequests are at a corner and savings are interior, i.e. b = 0 and s > 0, then
child labour will be inefficiently high.
Proof.
We saw in the proof of Proposition 7 that the budget constraint of the children given in
(35) reduces to the budget constraint in (10). As a direct consequence of Proposition 5, we
can conclude that Proposition 8 holds.
8
Proposition 9 If children have access to competitive insurance markets where they can
contract at rate
π = ∆ − 1,
per unit of human capital, returns to human capital are an absolutely continuous random
variable, savings are at a corner and bequests are interior, i.e. s = 0 and b > 0, then
child labour will be inefficiently high.
Proof.
We saw in the proof of Proposition 7 that the budget constraint of the children reduces to
the budget constraint in (10). As a direct consequence of Proposition 6, we can conclude
that Proposition 9 holds.
For our second consideration along these lines, we use the above first-order conditions
detailed in equations (36) - (40) and equation (33) to establish Proposition 10, which is
detailed below. It states that when the returns to human capital are random and insurance
markets are incomplete, then even when both bequests and saving are interior, child labour
can be inefficiently high. We see that such an outcome did not occur under certainty; as
long as both bequests and savings are interior, child labour is efficiently allocated. This
proposition emphasizes the importance of complete insurance markets in the sense that, in
their absence, parents compensate for future income loss by allocating out of the risky asset,
human capital, and increase the level of child labour. This, as expected, leds to inefficiently
high levels of child labour. With this introduction, we now formalize this in the following
proposition.
Proposition 10 When returns to human capital, ∆, are an absolutely continuous random
variable, insurance markets do not exist and parents to do not use bequests as a form of
insurance, and bequests and savings are interior, i.e. s > 0 and b > 0, then child labour
is inefficiently high.
Proof.
Upon substitution of equation (36) into (39), we conclude that
0
0
E[U (Θ + lc − s)] = E[Wc (∆h(1 − lc ) + b)].
(44)
Substituting equation (44) into (38), we have
0
0
0
E[Wc (∆h(1 − lc ) + b)∆h (1 − lc )] = E[Wc (∆h(1 − lc ) + b)].
0
0
(45)
Subtracting E[Wc (∆h(1 − lc ) + b)]h (1 − lc ) from the left-hand side of equation (45), we
obtain,
0
0
0
0
E[Wc (∆h(1 − lc ) + b)∆h (1 − lc )] − E[Wc (∆h(1 − lc ) + b)]h (1 − lc )
0
0
0
0
= E[Wc (∆h(1 − lc ) + b)∆h (1 − lc ) − Wc (∆h(1 − lc ) + b)h (1 − lc )]
9
0
0
0
= E[Wc (∆h(1 − lc ) + b)(∆h (1 − lc ) − h (1 − lc ))]
0
0
= E[Wc (∆h(1 − lc ) + b)(∆ − 1)]h (1 − lc ).
0
(46)
0
Now, subtracting E[Wc (∆h(1 − lc ) + b)]h (1 − lc ) from the right-hand side of equation (45),
we arrive at
0
0
0
E[Wc (∆h(1 − lc ) + b)] − E[Wc (∆h(1 − lc ) + b(B))]h (1 − lc )
0
0
0
= E[Wc (∆h(1 − lc ) + b) − Wc (∆h(1 − lc ) + b)h (1 − lc )]
0
0
= E[Wc (∆h(1 − lc ) + b)(1 − h (1 − lc ))]
0
0
= E[Wc (∆h(1 − lc ) + b)](1 − h (1 − lc )).
(47)
Replacing the left-hand side of equation (45) with equation (46), and the right-hand side
of equation (45) with (47), results in the following equality,
0
0
0
0
h (1 − lc )E[Wc (∆h(1 − lc ) + b)(∆ − 1)] = (1 − h (1 − lc ))E[Wc (∆h(1 − lc ) + b)].
(48)
Subtracting
0
h
0
i
h (1 − lc )E E[Wc (∆h(1 − lc ) + b)](∆ − 1)
= 0
h
i
from the left-hand side of (48), where we used the result, E E[X](∆ − 1)
1] = 0, we conclude with the following representation of (48),
0
0
0
0
h (1 − lc )Cov(Wc , ∆) = E[Wc (∆h(1 − lc ) + b)](1 − h (1 − lc )).
0
= E[X]E[∆ −
(49)
0
Since cc increases with ∆, we have that Cov(Wc , ∆) < 0, and since h (1 − lc ) > 0, we
have that the left-hand side of (49) is negative. For equality to hold in equation (49), the
0
0
right-hand side must also be negative. We see that Wc (cc ) > 0 implies E[Wc (cc )] > 0.
0
But for the left hand side to be negative, it must be that 1 − h (1 − lc ) < 0, or
0
h (1 − lc ) > 1.
As a result of these considerations, we now conclude that child labour is inefficiently high.
We see from Proposition 10 that, when returns to human capital are uncertain and
insurance markets are incomplete, then child labour will be inefficiently high even when
both bequests and savings are interior. In this case, parents, aware that returns to human
capital are risky, now allocate more of their child’s time to earning certain period-1 income,
rather than permit the child to accumulate more of the risky asset human capital.
One can remove the assumption in Proposition 10 that parental period-2 income is
uncertain and still conclude that child labour will be inefficiently high. Corollary 2, found
below, details this result.
Corollary 2 When return to human capital, ∆, is an absolutely continuous random variable, insurance markets do not exist, parents to do not use bequests as a form of insurance,
and bequests and savings are interior, i.e. s > 0 and b > 0, then child labour is inefficiently high.
10
Proof. This follows as a special case of Proposition 10.
V. Conclusion
Our first consideration has shown that, when parental period-2 income is uncertain, Propositions 1, 2 and 3 of Baland and Robinson (2000) remain valid. Moreover, our second
consideration in this regard has also confirmed, for the case when returns to human capital
are uncertain and insurance markets are complete, Propositions 1, 2 and 3 of Baland and
Robinson. However, when returns to human capital are uncertain and insurance markets
do not exist, Proposition 1 no longer holds; even when savings and bequests are interior,
child labour is inefficiently high. Moreover, we see from our analysis that uncertainty in
period-2 income does not affect the optimal level of child labour but uncertainty in returns
to human capital leads to inefficient high levels of child labour. This suggests that households are affected not by uncertainty alone but also by the good involved, in our case human
capital.
References
Baland, Jean-Marie and Robinson, James A. ‘Is Child Labour Inefficient?’ J.P.E.
108, no. 4 (2000): 663-681.
Varian, Hal. Microeconomic Analysis. New York: W. W. Norton & Company, 1992.
11
vol.