Doshisha University Center for the Study of the Creative Economy
Discussion Paper Series No. 2015-03

Inequality and conditionality in cash transfer:
Demographic transition and economic development

Koji Kitaura (Faculty of Social Sciences, Hosei University)
Kazutoshi Miyazawa (Faculty of Economics, Doshisha University)

Discussion Paper SeriesInequality and conditionality in cash transfer:
Demographic transition and economic development
Koji Kitauraa,*a Faculty of Social Sciences, Hosei University,

4342 Aihara, Machida, Tokyo 194-0298, Japan

Kazutoshi Miyazawabb Faculty of Economics, Doshisha University,

Kamigyo, Kyoto 602-8580, Japan

Abstract
This paper examines the effects of conditionality in cash transfer on growth andinequality.

We consider an overlapping generations model where the poor household

faces a trade-off between schooling and child labor.

We show that the growth rate inattaching conditions to cash transfer is greater than that in the case of no condition
because the cash transfer policy stimulates education.

However, adding conditionality

may be a source of income inequality between different income groups due to thefertility differential.

Keywords: Conditional Cash Transfer; Child Labor; Differential Fertility; Inequality
JEL Classification: D91; I28; J13; O11* Corresponding author.
E-mail addresses: kitaura@hosei.ac.jp (K. Kitaura).
11. Introduction
The government of poor countries started to adopt cash transfer (CT) programs
focusing on poverty alleviation and inequality over the last several decades.

It iswell-known that CT programs are conditional or unconditional; conditional cashtransfers (CCTs) transfer cash to poor households to invest in their children’s humancapital, while unconditional cash transfers (UCTs) provide benefits to all eligible
beneficiaries.

Both of them are at anti-poverty programs, but CCTs typically requires

school enrolment and regular attendance1.

As Bourguignon et al (2003) was suggested,

this condition plays important roles in encouraging the human capital of the children
due to the change in their time-allocation decisions.

The problem of whether

conditionality should be attached or not has been discussed as one of the mostimportant issues in developing economy (see, for example, Fiszbein et al. 2009; Adato
and Hoddinott, 2010; Arnold et al. 2011 for an excellent survey).

Behrman andSkoufias (2006) pointed out that CCTs may contribute to policy objectives of reducing
inequality, but they are not necessarily superior to UCTs.

This issue still seems

empirically controversial (Skoufias and Di Maro, 2008; Fiszbein et al., 2009; Samson,
2009; Baird et al, 2011; UNESCO, 2015, p90)2.

The purpose of this paper is to examine

the effects of conditionality in cash transfer on economic growth and inequality.
CCTs are one of the most popular programs to focus on the long-term human capital
accumulation to break the inter-generational transmission of poverty (Hall, 2006)3.
Since the pioneering Mexico’s PROGRESA (renamed Oportunidades) was launched in
1997, many researchers have evaluated the impact of CCTs on educational attainment.
Using the data from the PROGRESA randomized experiment, Schultz (2004), Behrman
et al (2005), Todd and Wolpin (2006), De Janvry and Sadoulet (2006) and Attanasio et al(2012) demonstrated that these programs have a positive impact on education outcomes.
Randomized experiments in Latin America consistently found that poor children
eligible for CCTs are more likely to enroll in school over short periods.

Recently,

Behrman et al. (2009, 2011) empirically showed that CCTs have both medium- andIn terms of education conditions, almost all CCTs require enrollment and attendanceon 80 or 85 percent of school days (see, for example, Ayala Consulting, 2003).
2 Skoufias and Di Maro (2008) found that CCTs had poverty reduction effects whichwere stronger on the poverty gap and severity of poverty measures. Fiszbein et al.
(2009) also suggested that CCTs generally helped reduce national poverty of Mexico.
In contrast, Samson (2009) pointed out that UCTs also significantly reduce inequality inSouth Africa.
3 For example, the goals of Bolsa Escola (renamed Bolsa Familia) in Brazil are toincrease education attainment, reduce both short-term and long-term poverty, reducechild labor and provide a social safety net for times of economic crisis (World Bank,
2001).
12

long-term impacts in increasing schooling enrollment and decreasing child labor.
These evidences suggest that the impact of CCTs on poverty is robust with time4.
Theoretical analysis showed the relative merits of CCTs to UCTs in terms of welfare(Del Rey and Estevan, 2013).

In a political economy, Estevan (2013) examined theimpact of CCTs compared with UCTs on the level of public education.

However, they

did not consider the effects of conditionality in cash transfer on evolution of growth andinequality.

Thus, we compare the policy implications of CCTs and UCTs programs for

economic growth and inequality.
For our purpose, we use the overlapping generations model in which poor parents
allocate their children’s time between schooling and child labor.

The empirical study ofthe linkage between CT programs and child labor has been developed by many authors.
Skoufias et al (2001), and Edmonds and Schady (2012) showed that PROGRESA had aclear negative impact on children's work. Using the data of Bolsa Escola, Bourguignon
et al (2003) and Cardoso and Souza (2003) showed that CCTs were critical andsuccessful in increasing school participation and UCTs would have no impact on school
enrollment rates and child labor.
Another important assumption in our model is that there are heterogeneous
individuals with endogenous and differential fertility.

De la Croix and Doepke (2003,

2004) among others examined the effect of fertility differential between the rich and thepoor on economic growth and income inequality in analyzing education policy5.

These

effects lead to the different time allocations between child education and working
accompanied with a quality and quantity trade-off in the decision on children, and thus
the evolution of inequality.

Recently, Simone and Fioroni (2013) extended thisframework by introducing the role of child labor.

They demonstrated the emergence ofa vicious cycle between child labor and inequality.
This paper is also related to the literature on the effect of policy option on inequality.
Many theoretical studies have attempted to explain the relationship between child labor
regulations (CLRs) and inequality6.

Emerson and Knabb (2006) showed that childlabor ban will not reduce poverty or income inequality in the future if the government
did not provide the appropriate education resources for children and opportunities in
Reimers et al (2006) pointed that CCTs are effective instruments to alleviate poverty
in the long term, and that they induce families to support the education of their children
in ways that will make them less likely to be poor in the future.
5 See, for example, Lam (1986), Dahan and Tsiddon (1998), Morand (1999), Kremer andChen (2002), Moav (2005), Sarkar (2008).
6 Dessy and Knowls (2008) take compulsory education and child labor regulations(CLRs) to be equivalent. See, for example, Krueger and Donohue (2003) and Strulik(2004).
43

the labor market.

Bell and Gersbach (2009) demonstrated that, whereas

introduction of compulsory education made temporary inequality unavoidable, long-run
inequality was avoidable if school attendance is unenforceable.

Recently, Simone andFioroni (2013) demonstrated that child labor regulations (CLRs) policy lowers the levelof inequality in the long run if enforced. In this paper, we present an alternativeeducation policy; CT program such as CCTs and UCTs to reduce both short and long run
poverty.

Baird et al (2014) found that both CCTs and UCTs improve schooling

outcomes compared to no cash transfer program, using data from 75 reports that cover
35 different studies.

More recently, both CCTs and UCTs programs have been

introduced by several developing countries 7 .

For example, in the Burkina Faso

experiment, Akresh et al (2013) found that CCTs are more effective than UCTs in
improving the attendance of the children who are initially not enrolled in school or areless likely to go to school.

They evaluated the relative effectiveness of the following

four cash transfer schemes; CCTs given to fathers, CCTs given to mothers, UCTs given
to fathers, and UCTs given to mothers. To focus on the difference between the CCTsand UCTs, we do not consider the heterogeneity within the couples.
The results of this study are as follows.

Comparing the CCTs schemes with UCTs

schemes, it is shown that the growth rate under the CCTs scheme is greater than that
under the UCTs scheme because the cash transfer policy stimulates education.
increases not only the steady state income but also the speed of convergence.

It

However,adding conditionality may be a source of income inequality between different income
groups because a higher rate of growth favors a higher income group.

Under the CCTs

schemes, education transfer induces the sharp fertility differential between the groups,
which is accompanied with a quality and quantity trade-off of children, and thus theincome inequality may be widen.

However, the inequality improves at a relatively

high speed, and the income difference becomes smaller than the initial difference.

Onthe other hand, under the UCTs schemes, the inequality continues to worsen for a longtime.
The remainder of this paper is organized as follows. In Section 2 a basic model ispresented and the growth rate is derived.
characterized.

In Section 3 the properties of inequality are

A numerical example is offered in Section 4.

Section 5 offers someconclusions.

For example, in Sub-Sahara Africa, nine countries implement both CCTs and UCTs
programs in 2010 (see, for example, Garcia et al, 2012).
7

4

2. The ModelWe

consider

a

small

open

overlapping

generations

two-period-lived individuals (childhood and parenthood).
idifferent in their initial human capital h0 .

model

populatedbyIndividuals of type i are

They go to school and work in theirchildhood, and work and rear children in their parenthood.

2.1 Individuals
Consider a child born at t  1 (called generation t ), with human capital inheritedby his parents. The human capital of the children, hti1 , depends on his/her schoolingtime, eti .

hti1   ( e ti )  ,(1)

where   0 , 0    1 .
The utility function is assumed to be quasi-linear utility function of the form8,

U ti  cti   ln(nti hti1 ) ,

(2)

i

i

i

where ct is the consumption in the parenthood; nt is the number of children; ht 1 ishuman capital of children; 0    1 is the preference parameter attached to altruism.
i

Parent allocates the time endowment of children between schooling, et , and working,1  eti . Let   (0, hti ) be the wage rate of child labor and hti is his/her own humancapital.

They supply (1  et )ht units of efficient labor as child labor in childhood.
iiThey devote nt units of time to rearing n t children and the remaining 1  nt unitsiiiof time to working in parenthood. Thus, their inter-temporal budget equation can be
written as

cti  (1  nti )hti  nti (1  eti )  CTt i ,(3)This setting means that there are no income effects on the consumption.
income effects are discussed after the main analysis.

85Introducing

iwhere CTt is the cash transfer.

2.2 Cash transfer schemes
The government is assumed to adopt the following cash transfer schemes ( , Tt ) ,i

CTt i  Tt  nti eti ,(4)where Tt

is transfer which is not dependent on type i ;   [0,1) is a rate of education

subsidy.

When   0 , the CT program is called "unconditional cash transfers" (UCTs).

When   0 , that is called "conditional cash transfers" (CCTs).

This specification oftransfer schemes is consistent with findings by Baired et al (2011) and Akresh et al(2013).

Skoufias (2005) mentioned that the design feature of the PROGRESA programis that the level of transfer was set with the aim of compensating for the opportunity
cost of children’s school attendance.

In this paper, it also followed by Adato andHoddinott (2010) who pointed out that one of the characteristics of CCTs is to be madeas a lump-sum or determined based on the number of children.

2.3 Utility maximization problem
Substituting equations (1), (3), and (4) into equation (2), the utility maximization
problem can be rewritten as

max
U ti  (1  nti )hti  nti (1  eti )  Tt  nti eti   ln nti   ln eti .
i i
et , nt

The first-order conditions require that

U ti
n

i
t

U ti
e

i
t






n

i
t


e

i
t

 hti   (1  eti )  eti  0 ,(5)

 nti  nti  0 ,(6)

where equation (6) holds with inequality when et  1 .
i

The optimal schooling time is:6  (hti   )e   (1   )(1   )

1
i
t


 hti  h( )hti  h( )

if

(7)

where the threshold human capital level is given by :h( ) (1     )



.

(8)

The optimal number of children is :  (1   ) h i  
int   t iht  
 hti  h( )hti  h( )

if

(9)

Substituting equation (7) into equation (1), the accumulation of human capital is given
by





(hti   ) if
hti1  H (hti ,  )    (1   )(1   )  hti  h( )hti  h( )(10)Given an initial human capital, h0i , equation (10) determines the path of human capitalh , and equation (9) determines the path of fertility rate n . In the following, we
i
t

i
t

assume that

 (1   )  (1   )1 2  2

   h( ) .

(11)

With this assumption, we can show that the curve hti1  H ( hti ,  ) intersects with 45
degree line twice in an interval hti  (  , h ( )) (See below).

Denoting two steadystate values by h  ( ) and h( ) ( h  ( )  h ( ) ), this implies lim h  h ( ) , given
i



t 

that h0i ( )  h( ) .
A main focus of this paper is the time path of the growth rate of human capital
because a high growth rate could worsen income inequality in transition.
analyzed by checking whether hti1 hti increases or not.
7This can be

The following proposition

summarizes the result.

Proposition 1 (Growth Rate).

Assume that equation (11) is satisfied. Then, hti1 hti

increases when hti  h( ),  (1   )  , and decreases when hti   (1   ) , h  ( ) .
If the initial condition satisfies h0i   (1   )  , then the growth rate of humancapital increases in the first several period, and then decreases toward one.

If (1   )   h0i  h  ( ) , then the growth rate of human capital decreases
monotonically toward one.
Proof.

From equation (10), we obtain




hti1

 
(hti   ) (hti ) 1 ,iht
 (1   )(1   ) 
if    hti  h ( ) .

(12)

Let us define a function f ( h )  (h   )  h 1 , h    .

function has a unique maximum at h   (1   )  .

This

Therefore, the right-hand side ofequation (12) has a maximum, 2 (1   )1 2
 (1   ) ,which is greater than one from equation (11). In this case, we have two steady state,h  ( )  h( ) , and given that h0i  ( h ( ), h  ( )) , human capital monotonicallyincreases and converges to h  ( ) .

The growth rate of human capital increases when,hti   (1   )  and decreases when hti   (1   )  .

[Figure 1 is here]
Figure 1 illustrates the evolution of human capital in equation (12).

8

A solid curve

in the figure is a case of   0.3 , and a dashed curve is a case of   0 9.
The growth rate under the CCTs schemes is greater than the UCTs schemes because
the subsidy policy stimulates education.
but also the speed of convergence.

It increases not only the steady state income

However, adding conditionality may be a source ofincome inequality between different income groups because a higher rate of growthfavors a higher income group.

We analyze this possibility numerically in the next

section.
To close the model, we introduce the government budget constraint.

We assumethat they are supported by the development banks and other international development
agencies10.

Then the government budget constraint is given byNt

N t g t  N t Tt    nti eti ,

(13)

i 1

where g t  0 stands for per capita grant aid.

From this, the lump-sum transfer canbe written as

Tt  g t  E[ nti eti ] ,(14)

where E[  ] stands for the average schooling time.
From equation (11), hti  h ( ) for all t  1 and all i 11.

nti eti Then, we obtain,(1   )

for all t  1 and all i from equation (7) and (9).

Substituting this into equation (14),the lump-sum transfer becomes9

The other parameters are   0.5 ,   0.25 ,   0.2 , and   0.1 .

So far the World Bank and the Inter-American Development Bank (IDB) haveencouraged their adoption in many low and middle income countries. As Handa andDavis (2006) and Reimers et al (2006) were pointed out, many CCTs program have been
implemented through World Bank and IDB loans. For example, Colombia’s program isfinanced through IDB and World Bank loans and in Honduras, CCTs will probably
continue to be supported through soft loans from the IDB. Although Progresa andBolsa Escola were initially designed and financed without the help of the developmentbanks. However, in both cases subsequent expansion was financed through loans(Handa and Davis, 2006). In fact, the Mexican government was supported theimplementation of Oportunidades until 2008.

10

11

If h0i  h( ) , then h1i    h( ) .
9

Tt  g t .
(1   )

Finally, the consumption level is given bycti  hti  Tt   hti  g t .
(1   )

(15)

3. Inequality
In this section, we examine the evolution of inequality under both CCTs and UCTs
schemes qualitatively.

We assume that there is a two-class economy, i  H , L where

H is the group with more human capital: htL  htH . The initial human of each group,h0H and h0L are different, and the population size of each group, N tH and N tL are also
different because the fertility rates are different.
To understand the evolution of inequality, we first analyze a between-groupinequality in sub-section 3.1.

This inequality measure may not be sufficient because

the population size itself changes over time.

Then we investigate the Gini index as an

economy-wide inequality measure in sub-section 3.2.

3.1. Between-group inequality
We first define a between-group inequality by

t 

htL
.
htH

(16)

Using equation (10), this inequality index evolves according to  h    , t 1   t t
 ht   (17)where we have used htH  ht for notational simplicity.

[Figure 2 is here]

10Figure 2 illustrates a phase diagram of ( ht ,  t ) (The derivation is put aside toStarting from an initial state ( h0 ,  0 ) , the inequality increases at first,

Appendix A).

and then decreases over time.

Adding conditionality plays an important role in thetime path of income inequality in the sense that a higher  widens the region ofd t dht  0 for  t  1 . Then we have the following proposition.

Proposition 2. (Inequality)
An increase in the amount of the cash transfer attached to education condition leads tothe initial widening income gap between the groups with low and high human capital.
The intuition of this proposition is as follows. From (10), an increase in the amount of
the cash transfer attached to the education condition induces the human capital levelsof the poor and the rich to diverge. As can be seen from (9) and (10), it leads to the
sharp fertility differential, which is accompanied with a trade-off between quality andquantity of children.

Thus, between-group inequality is widened.

Together with Proposition 1 and Proposition 2, we can also see the speed ofconvergence under both CCTs and UTCs schemes.
the higher the growth rate.

From Proposition 1, the higher  ,Thus, the speed of convergence under the CCTs schemes isfaster than that under the UCTs schemes.
This inequality measure is not considered the population size which changes overtime.

In the next subsection, we take into account the Gini index as an economy-wide

inequality measure.

3.2. Gini index
Next, let us define the population differential between the two groups by

st 

N tL
.
N tH

Taking s0  N 0L N 0H as given, we get

11

N 0L i 0 ni
t 1

st 

N

H
0



 n

i 1 is0 n0.
nt

Substituting equation (9) into this, the population differential is given by h    .
st  s0  t
 h0   (18)Appendix shows that the Gini index is given bygt 

st (1   t ).
(1  st )(1  st  t )(19)

We can trace the time path of Gini index in equation (19) by combining the time pathof  t in equation (17) and the time path of st in equation (18).
A general characteristic of the Gini index in the two-class economy is summarized in
the following lemma.
Lemma 2.

(Simone and Fioroni, 2013)(i) The Gini index decreases if the between-group inequality decreases: g t  t  0 .
(ii) The Gini index increases with the population differential when st   t0.5 anddecreases when st   t0.5 .

g tmax 

1  t1  t

The maximum is given by

.

(20)

Proof
By totally differentiating equation (19), we obtain(1   t )(1  st2 t )st
dg t  d t 
dst .
(1  st  t ) 2(1  st ) 2 (1  st  t ) 2
Obviously g t  t  0 .

Also, we know

12

(21)

g t 0  st  t0.5 .
st Substituting st   t0.5 into equation (19), we have equation (20).

The first term on the RHS of (21) is the between-group effect, which is negative. The
second term is the effects produced through the population differential between the twogroups, which are not determined as positive or negative.

Thus, the relationshipbetween the population differential and the Gini index is inverted-U shaped due tofertility differential.

This can be interpreted intuitively as follows.

CT policy

stimulating education leads to a greater fertility differential, and the between-income
gap increases.

Once inequality reaches a peak, and then begins to improve because

both of the groups decline the fertility rate sharply, and thus the Gini index reduces.
It should be noted that this peak under the CCTs schemes is earlier than one under
the UCTs schemes.

In the presence of the fertility differential, education transfer, such

as CCTs, affects the population difference.

In contrast, it is smaller than that under

the CCTs schemes.

4. Numerical analysis
So far we focused on the evolution of inequality as a consequences of CT described in
the previous section.

In this section, we intend to present some numerical examples toillustrate our analytical results under the two different schemes.

Suppose a situationin which the policy provides with cash transfer to the two groups subsequently. First,
the group named H receives the education subsidy in one period.

In the next period,

the other group named L does.
In the numerical analysis, we set parameter values as follows:   0.5 ,   0.25 ,  0.2 ,   0.1 , and s0  0.1 . From equation (11), we need   0.5 . In thefollowing, we analyze two cases,   0 and   0.3 .

The former represents a UCTs

schemes, and the latter a CCTs schemes.
Equation (10) gives two steady state values ( h , hˆ )  (3.5841,2.2430 ) when   0 ,

and(h  , hˆ)  (4.3023,2.413) when   0.3 .

13

To compare the two schemes, weassume the initial human capital is h0  2.2440 , and that the per capita grant aid is
10 per cent of income ( g t  0.2244 ) .

[Table 1 and Figure 3-7 are here]
The first and second column in Table 1 stand for the time path of human capital.
These are indicated graphically in Figure 3. The time path of the higher income groupis given by shifting the time path of the lower income group leftward ( htH

 ht 1 ) .

Figure 3 shows that human capital under the CCTs schemes increases faster than the
UCTs schemes.
The third and fourth column stand for the time path of consumption. Figure 4
shows that the figure looks like Figure 3. One exception is that consumption under the
CCTs schemes is smaller than the UCTs schemes in the first period.

This is attributed

to the fact that the lump-sum transfer under the CCTs schemes is smaller than the
UCTs schemes (See the third term in equation (15)).
The fifth and sixth column stand for the time path of the fertility rate.

Figure 5shows that the fertility rate under the CCTs schemes decreases sharply in a few periods,
while the fertility decline is moderate under the UCTs schemes.
The seventh and eighth stand for the time path of the population differentialst  N tL N tH . Figure 6 shows that, under the CCTs schemes, the ratio of the lower
income group increases sharply because the fertility difference is fairly large in the first
several generations.

On the other hand, under the UCTs schemes, the populationdifference becomes large after the fifth generations under the UCTs schemes.
The ninth and tenth column stand for the time path of the between-group inequality

 t  htL htH . Figure 7 shows that the between-group inequality under the CCTs
schemes worsens in the first two generations according to increases in the growth rateof human capital.

After that, the inequality improves at a relatively high speed, andthe income difference in the fourth generation becomes smaller than the initialdifference. Under the UCTs schemes, however, the inequality continues to worsen fora long time.
Finally, the eleventh and twelfth column stand for the time path of the Gini index.
14

Figure 8 shows that, under the CCTs schemes, the Gini index has a peak at the first
generation, which is different from the between-group index.
population differential matters in the Gini index.

This is because theUnder the UCTs schemes, themovement of the Gini index is similar to the between-group inequality because thepopulation difference is fairly small.

5. Discussions

5.1. Income effect
In this section we discuss a possible extension.
function in the basic model.

We used a quasi-linear utility

This implies that the income effect of transfer policy isneglected. In this subsection, we show that this assumption is not essential.
Let us assume that the utility function is given by

U ti  (1   ) ln cti   ln(nti hti1 ) .
Individuals maximize this subject to equations (1), (3), and (4).

Assuming interior

solutions, the first-order conditions require that1  it ,
i
ct


i
t

n


i
t

e





 it hti  (1  eti )  eti , it (nti  nti ) ,where it is a multiplier attached to equation (3).

Solving them, we get

cti  (1   )( hti  Tt ) ,eti  (hti   ),(1   )(1   )nti 

 (1   )(hti  Tt ).
hti  

The optimal schooling time is the same as the basic model, which implies the processof human capital accumulation is also the same.
15

A main difference is an income effectof Tt on the fertility rate. Under the UCTs schemes, the grant aid increases fertility
because children are normal goods.

Under the CCTs schemes, this effect would besmall because the increased share of education subsidy makes the lump-sum transfer
small by the budget constraint.
ii

To show this formally, substituting nt and et into equation (13), we obtainTt (1   ) g t  E[hti ].
1  (1   )

(21)

From equation (21), we know Tt  g t when   0 and that Tt is decreasing in  , as
the basic model. In addition to the basic model, the subsidy rate affects Tt by way ofhuman capital accumulation.
which decreases

For a large  , human capital grows at a fast rate,

Tt because the expenditure of education subsidy increases.

Therefore, the income effect on fertility under the CCTs schemes becomes smaller overtime.

5. Concluding remarks
Most of the literature on CT programs such as CCTs and UCTs has concentrated on theeffectiveness of both programs in improving education outcomes.

There is muchdebate about whether transfers should be made conditional on enrolment or attendance.
In this paper we explore the dynamic evolution of human capital, fertility and childlabor when attaching conditions to cash transfers.
We analytically demonstrate that the growth rate under the CCTs schemes is greaterthan that under the UCTs schemes.
also the speed of convergence.

It increases not only the steady state income but

However, adding conditionality may be a source ofincome inequality between different income groups because a higher rate of growthfavors a higher income group.

Under the CCTs schemes, although the income

inequality may be widened, the inequality improves at a relatively high speed, and theincome difference becomes smaller than the initial difference.

On the other hand,under the UCTs schemes, the inequality continues to worsen for a long time.
In this paper, we assume that the government is financed by external support when16the cash transfer programs are implemented.

It would be important to investigate

whether debt-financed policy leads to a higher or lower growth rate in comparison to an
external aid-financed one.

This is indeed an interesting problem which goes beyond

the scope of our analysis and must be left to future research.

Acknowledgements
The authors thank seminar participants at the Economic Theory and Policy Workshop
for their useful comments.

17Appendix A. Phase diagram
Suppose a two-class economy where human capital of type i  H , L evolves accordingtohti1  A( )(hti   )  ,(A1)where A( ) is given byA( )    .
 (1   )(1   ) 
Obviously, A( ) is increasing in  .

t 

Let us define the between group inequality by

htL
.
htH

(A2)

H

To simplify notations, we use ht instead of ht from here on.

From equations (A1) and(A2), human capital of group H and the between-group inequality evolves according to
the following two equations:ht 1  A( )(ht   )  ,

(A3)



  h    .
 t 1   t t
 ht   (A4)Human capital of group L is given by  t ht .
First, define the increment in human capital between period t and t  1 by

ht  ht 1  ht  A( )(ht   )  ht .
Then, Figure 1 shows ht  0 if h( )  ht  h  ( ),ht  h( ), h  ( )  ht ,ht  0 if
where h ( ) and h ( ) are the solutions of ht  0 .


Second, define the increment in between-group inequality by

18

(A5)



  h      t .
 t   t 1   t   t t
 ht   
Obviously,  t  1 is a solution of  t  0 .

Assume that  t  1 . Then,1

 t  0

ht 

 1t H ( t ) .
1
 t   t
(A6)

The function H ( t ) in equation (A6) has the following characteristics: H ( t )  0
and lim 1 H ( t )   (1   )  .
tFinally, differentiating  t with respect to ht , we get  t  0  t  0

ifif

ht  H ( t )ht  H ( t )

and
and

0   t  1,

 t  1.

Combining equations (A5) and (A7), we get the phase diagram in Figure 2.

19

(A7)

Appendix B. Proof of equation (19)
By definition, the Gini index is given by[h i ]g

2 E[ h i ],where E[ht ] and [ht ] stand for the mean of human capital and the mean difference,
respectively.

We omit the time script for simplicity.

In the two-class economy of the main body, we haveE[ h i ] [ h ] iwhere s  N

L

N tH h H  N tL h L
N tH  N tLh H (1  s )2 N tH N L (h H  h L )( N tH  N tL ) 2

1 s



,

2 sh H (1   )(1  s ) 2,

N H and   h L h H .

Substituting them into the above equation, we obtaings(1   ).
(1  s)(1  s )20

References
Adato, M., Hoddinott, J., 2010. Conditional cash transfers in Latin America. Intl Food
Policy Res Inst.
Akresh, R., De Walque, D., Kazianga, H., 2013. Cash transfers and child schooling:
Evidence from a randomized evaluation of the role of conditionality. World Bank
Policy Research Working Paper, 6340.
Arnold, C., Conway, T., Greenslade, M., 2011. Cash transfers literature review. London:
Department for International Development.
Attanasio, O. P., Meghir, C., Santiago, A., 2012. Education choices in Mexico: using a
structsural model and a randomized experiment to evaluate Progresa. The Review of
Economic Studies, 79 (1), 37-66.
Ayala Consulting, 2003. Workshop on conditional cash transfer programmes (CCTs):
operational experiences, report. Quito : Ayala Consulting.
Baird, S., McIntosh, C., Özler, B., 2011. Cash or condition? evidence from a cashtransfer experiment. The Quarterly Journal of Economics, 126, 1709-1753.
Baird, S., Ferreira, F. H., Özler, B., Woolcock, M., 2014. Conditional, unconditional andeverything in between: a systematic review of the effects of cash transfer
programmes on schooling outcomes. Journal of Development Effectiveness, 6 (1),1-43.
Behrman, J. R., Sengupta, P., Todd, P., 2005. Progressing through PROGRESA: an
impact assessment of a school subsidy experiment in rural Mexico. Economic
Development and Cultural Change, 54 (1), 237-275.
Behrman, J. R., Skoufias, E., 2006. Mitigating myths about policy effectiveness:evaluation of Mexico’s antipoverty and human resource investment program. The
Annals of the American Academy of Political and Social Science, 606 (1), 244-275.
Behrman, J. R., Parker, S. W., Todd, P. E., 2009. Schooling impacts of conditional cashtransfers on young children: Evidence from Mexico. Economic Development andCultural Change, 57 (3), 439.
Behrman, J. R., Parker, S. W., Todd, P. E., 2011. Do conditional cash transfers forschooling

generate

lastingbenefits?

A

five-year

followupof

PROGRESA/Oportunidades. Journal of Human Resources, 46 (1), 93-122.
Bell, C., Gersbach, H., 2009. Child labor and the education of a society. Macroeconomic
Dynamics, 13 (02), 220-249.
Bourguignon, F., Ferreira, F. H., Leite, P. G., 2003. Conditional cash transfers,
schooling, and child labor: Micro-simulating Brazil's Bolsa Escola program. The
World Bank Economic Review, 17(2), 229-254.
21

Cardoso, E., Souza, A. P., 2004. The impact of cash transfers on child labor and school
attendance in Brazil.
Dahan, M., Tsiddon, D., 1998. Demographic transition, income distribution, andeconomic growth. Journal of Economic Growth, 3 (1), 29-52.
De La Croix, D., Doepke, M., 2003. Inequality and growth: why differential fertility
matters. The American Economic Review, 93 (4), 1091-1113.
De la Croix, D., Doepke, M., 2004. Public versus private education when differential
fertility matters. Journal of Development Economics, 73 (2), 607-629.
De Janvry, A., Sadoulet, E., 2006. Making conditional cash transfer programs more
efficient: designing for maximum effect of the conditionality. The World Bank
Economic Review, 20 (1), 1-29.
Del Rey, E., Estevan, F., 2013. Conditional cash transfers and education quality in thepresence of credit constraints. Economics of Education Review, 34, 76-84.
Dessy, S., Knowles, J., 2008. Why is child labor illegal?. European Economic Review, 52(7), 1275-1311.
Edmonds, E., Schady, N., 2012. Poverty alleviation and child labor. American Economic
Journal: Economic Policy 2012, 4 (4), 100-124.
Emerson, P. M., Knabb, S. D., 2006. Opportunity, inequality and the intergenerationaltransmission of child labour. Economica, 73 (291), 413-434.
Estevan, F., 2013. The impact of conditional cash transfers on public education
expenditures: A political economy approach. European Journal of Political Economy,
32, 268-284.
Fiszbein, A., Schady, N. R., Ferreira, F. H., 2009. Conditional cash transfers: reducing
present and future poverty. World Bank Publications.
Garcia, M., Moore, C. G., Moore, C. M., 2012. The cash dividend: the rise of cashtransfer programs in sub-Saharan Africa. World Bank Publications.
Hall, A., 2006. From Fome Zero to Bolsa Família: social policies and poverty alleviation
under Lula. Journal of Latin American Studies, 38 (04), 689-709.
Handa, S., Davis, B., 2006. The experience of conditional cash transfers in Latin
America and the Caribbean. Development policy review, 24(5), 513-536.
Kremer, M., Chen, D., 1999. Income-distribution dynamics with endogenous fertility.
American Economic Review, 155-160.
Krueger, D., & Donohue, J. T. (2005). On the distributional consequences of child labor
legislation. International Economic Review, 46 (3), 785-815.
Lam, D. 1986. The dynamics of population growth, differential fertility, and inequality.
The American Economic Review, 1103-1116.
22

Moav, O., 2005. Cheap children and the persistence of poverty. The Economic Journal,
115 (500), 88-110.
Morand, O. F., 1999. Endogenous fertility, income distribution, and growth. Journal of
Economic Growth, 4 (3), 331-349.
Reimers, F., Da Silva, C. D., Trevino, E., 2006. Where is the" education" in conditional
cash transfers in education? . Montreal: UNESCO Institute for Statistics.
Samson, M., 2009. The impact of social transfers on growth, development, poverty andinequality in developing countries. Building decent societies: Rethinking the role ofsocial security in development.
Sarkar, J., 2008. Mortality, fertility, and persistent income inequality. Southern
Economic Journal, 332-350.
Schultz, T. P., 2004. School subsidies for the poor: evaluating the Mexican Progresa
poverty program. Journal of Development Economics, 74 (1), 199-250.
Simone, D., Fioroni, T., 2013. Child Labour and Inequality (No. 17/2013).
Skoufias, E., Parker, S. W., Behrman, J. R., Pessino, C., 2001. Conditional cashtransfers and their impact on child work and schooling: Evidence from the progresa
program in Mexico [with comments]. Economia, 45-96.
Skoufias, E., Di Maro, V., 2008. Conditional cash transfers, adult work incentives, andpoverty. The Journal of Development Studies, 44 (7), 935-960.
Strulik, H., 2004. Child mortality, child labour and economic development. The
Economic Journal, 114 (497), 547-568.
Todd, P. E., Wolpin, K. I., 2006. Assessing the impact of a school subsidy program in
Mexico: Using a social experiment to validate a dynamic behavioral model of childschooling and fertility. The American Economic Review, 1384-1417.
UNESCO, 2015. EFA Global Monitoring Report 2010-15: achievements and challenges.
Paris, UNESCO.
World Bank, 2001. Brazil: An Assessment of the Bolsa Escola Programs, World Bank
Report No. 20208-BR.

23

Figure 1. Human capital accumulation

45 degree line

1

μ
,

1

O

α

∗

24

α

α

t

Figure 2. Phase diagram

1

O

α

∗
1

25

α

t

Table1. Parameter value and Results

Human capital

Consumption

Fertility

Population differential Human capital inequality

Gini index

t

α=0.3

α=0

α=0.3

α=0

α=0.3

α=0

α=0.3

α=0

α=0.3

α=0

α=0.3

α=0

0

2.24402.24402.15842.18848.60668.60661.00001.00000.91420.99940.02240.000112.45472.24532.36912.18974.61848.56091.86351.00530.85590.9987

0.03610.00032

2.86812.24832.78252.19272.41918.45753.55781.01760.8507

0.99690.02890.000833.37132.25523.28572.1996

1.53148.22885.6201

1.04590.89200.99320.01530.00174

3.77962.27073.69402.21511.18007.75777.2934

1.10940.93690.98540.00710.003754.03402.3044

3.94842.24881.0324

6.89888.3361

1.24750.9671

0.97110.00320.00736

4.17112.37304.08552.31740.96735.63008.89801.52870.98380.95050.00150.0122

7

4.23962.49674.15402.44110.93774.22799.17872.0357

0.99230.93090.00070.016084.27272.68204.18712.62640.92403.07929.31432.7951

0.99630.92380.00030.015794.28842.90324.20282.84760.91772.32519.37873.7016

0.99830.93220.00020.01209.40994.56720.9992

0.94880.00010.00799.43576.49221.00001.00000.00000.000010

4.29583.11444.2102

3.05880.91471.884411

4.29923.28244.21363.22680.9134

1.6376∞

4.30233.5841

4.21673.52850.91211.325726

Figure 2. Human capital
4.5

4.0

Low income, CCT

3.5

High income, CCT
Low income, UCT
3.0High income, UCT

2.5

2.0
0

1

2

3

4

5

6

7

8

9

10

Period8910PeriodFigure 3. Consumption
4.5

4.0

Low income, CCT

3.5

High income, CCT
Low income, UCT
3.0High income, UCT

2.5

2.0
0

1

2

3

4

5

27

6

7

Figure 4. Fertility
10

9

Low income, CCT

8High income, CCT

7

Low income, UCT
6High income, UCT

5

4

3

2

1

0
0

1

2

3

4

5

6

7

8

9

10

Period910PeriodFigure 5. Population differential
10

9

8

7

CCT

6

UCT
5

4

3

2

1

0
0

1

2

3

4

5

28

6

7

8

Figure 6. Between-group inequality
1.05

1.00

0.95

0.90

0.85

CCT

0.80

UCT

0.75
0

1

2

3

4

5

6

7

8

9

10

Period910PeriodFigure 7. Gini index
0.04

0.04

CCT

0.03

UCT

0.03

0.02

0.02

0.01

0.01

0.00
0

1

2

3

4

5

29

6

7

8