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Supply of Child Labour: A Parental OptimizationProblem in the Framework of Household Time AllocationJEL Classification: J20,J22,J24,K31,H53,I20,I28Keywords: Education, Government, Labour Economics,Labour Law, Stigma, Time Allocation.2Abstract:The paper tries to investigate the determinants of supply of child labour through the parent’sproblem of household time allocation for a child in the face of a trichotomy of working in themarket, working at home, and study time. Time spent in home production yields home goodsdeemed as perfect substitute for market goods, produces under conditions of diminishingmarginal productivity. Home production by a child would also depend on some householdattributes. While an increase in the child market wage rate is expected to reduce work athome and increase child labour, an increase in the value of household attribute would reducechild labour. A ceteris paribus increase in parental income, if it delinked with child wage,may not affect child labour under some specific altruistic parental preference pattern. Wefinally investigate the effect of a socio-psychological cost borne by the parent in sending hischild to work and show that optimality may necessitate a reduction of child labour even whenchild’s income contributes to increasing the family consumption. Appropriate role of thegovernment is also discussed in these contexts.3Section-IIntroduction:Even though child labour is looked down upon almost everywhere in the world irrespectiveof the cultural differences, its very own existence is like a slap on the face of a civilizedsociety and a reminder of our hapless failure in tackling this social evil. As per the NationalCensus 2011, there are close to 4.3 million child labourers in India, in the age group of 5 to14 years, and the numbers become more disturbing when we consider the world as a whole.If we accept the simple fact that child labour occurs not because of parents’ greed orselfishness, but because of the parents’ concern for the household’s survival, the popularargument for banning child labour as a solution to this social issue loses much of its force.Child labour being a sensitive issue, with several linkages and externalities, cannot have sucha head on solution. An attempt to abolish child labour would be very unbecoming if it paresdown the income sources of the impoverished mob right away without providing anyalternative option for survival.Thus economic theories have long recognized poverty as a major determinant of child labourand theoretical models have been developed based on this assumption. At macro level,studies have analysed the effect of GDP per capita on child labour, whereas micro levelstudies have focused on low parental wage or household income to determine the supply ofchild labour.In the micro-theoretic framework, a simple ‘Household Decision Making Model’ is based onutility maximization objective of a household with assumptions that the father allocates timebetween market work and leisure; the mother allocates time between market work and childrearing; and children allocate time among market work, education and leisure. Empiricalfindings of this kind of model show that, an increase in parents’ wage will increase thehousehold income and if a child’s education is a normal good, then it will also rise. But ifchild wage rises, it raises the demand for children, which leads to an increase in familyincome, reducing children’s education due to lack of financial resources per child.In general, utility function of the unitary household model often points out towards the trade-off between quantity and quality of children. As number of children, schooling per child,consumption of goods and leisure all enters into the utility function of the household together;any change in the child wages will lead to trade-off between education and labour. The trade-off depends on the household budget constraint, personal ability of the child and on thereturns to home production. Poor parents cannot invest equally to all children because of thelimited household income. There are empirical evidences that suggest that parents investdifferently on the human capital formation of the first and the last born.Another interesting stance is observed in terms of ‘The Human Capital Theory’ whichassumes people to be productive resources and that higher education leads to higher4productivity. Parents make trade-offs between child’s education and labour and their decisionis based on the socio-economic condition of the family. Parents’ decision regarding theinvestment in their child’s human capital depends on return to schooling. If the return fromschooling is high, the number of working children would reduce.Bonnet (1993) argues that failure of the education system is an important explanation for theprevalence of child labour. When parents do not expect children to learn much in school, theydecide to give them informal education in terms of work.Strauss and Thomas (1995) stressing on the parental education argue that parental years ofschooling plays a significant role in reducing the child labour incidence. Brown et.al, arguethat there is an empirical evidence that suggests that a parent’s education affects their futuregeneration. Educated parents value education, and hence invest in the human capitalformation of their children while parents who are illiterate themselves do not value educationfor their children as well. Education affects the future income streams of the household aswell. This also motivates parents to invest in the human capital formation. This implies thathuman capital formation is dependent on the return to schooling which in turn depends on thequality of schooling.Several works in the field of child labour assume altruistic parents, i.e., parents do not wanttheir children to work in the market, trading off study time against work. Humphries (2003)illustrates that even when parents are altruistic, child labour could arise in the presence ofextreme poverty. In economic literature, the role of parents is somewhat mischaracterized. Inthis regard Basu and Van (1998) explain, when children work in a society as a massphenomenon, it is much more likely that this reflects not a difference in the attitude of theparents but the problem of stark poverty. Many researchers have based their empiricalfindings on the theoretical arguments placed by Basu and Van model (1998) which criticizedtrade sanctions. They argue that these trade sanctions will only increase poverty and worsenthe situation of child labour instead of reducing it.In section II, we present a theoretical attempt to provide an explanation for the generation ofchild labour in a time allocation model in which a child’s total time is divided into threecategories, namely study time, work at home and work in the market. Against the backdrop ofparental altruism, this trichotomy of choice marks an extension of the standard householdmodel.Section III contains the comparative static results and geometric representations of the same.Section IV presents a totally independent behavioral stance posited through the mentionedneoclassical framework by considering a socio-psychological cost incurred by parents whenthey send their child to work in the market. This consideration generates interesting policyimplications that are summarised along with other prescriptive conclusions in the section V.5Section-IIA Simple Parental time allocation model involving child labour:We begin with a theoretical description of a time allocation model (following Becker, 1965)for the child. The parent allocates the child’s total time among three broad categories: forstudies (including schooling), for work in the household (including working at her parentalfarm) and for working in the market against remuneration. It is assumed that each householdconsists of one rational utility maximizing parent and one child. The parent is assumed tohave an altruistic attitude towards his child’s study time as it plays an important role in thefuture well-being of the child. This is a digression from the standard household model as weassume a single parent who acts as a benevolent dictator towards the child.If total time is normalized to 1, the time constraint for the child becomes:e: Share of time spent as child labour.s: Share of time spent studying.h: Share of time spent on household work.No explicit mention of the leisure time is made which is also a significant departure from thestandard approach. Aggregating leisure and study time can be justified on the fact that boththe elements react similarly to changes in the socioeconomic environment of poor householdsand studying them separately would not yield a different conclusion. Moreover, as theirrelative price is remaining constant, we can look at them as a composite input, and disregardthe composition for the time being.A parent’s utility function takes the following formwhere C: Household consumption2)Assuming the utility function to be additively separable, we haveAssuming that the parent has no scope to borrow from the capital market nor does he leaveany bequest, the household consumption depends upon parent’s income Y, child’s incomefrom the market (if any) and the production by the child at home captured by a function thatdepends on ‘h’ and household attribute ‘A’.The household production function is:6It is assumed to show diminishing marginal productivity in both the arguments:The value of marginal productivity at home decreases not only because of fatigue or changesin input proportions but also due to a change in the composition of F, as any of the argumentsincrease, a cheaper market substitute of F becomes available. The cross-partial term ispositive as the marginal productivity of the home labourer increases if she has more or betterimplements to work with.The parent inelastically sells his total time endowment (normalized to 1) into the labourmarket to earn wage income. If he has no other source of income, ‘Y’ has to equal adultwage. So the household budget constraint becomes:Incorporating constraints (1) and (5) in the utility function in (3), we obtain the followingunconstrained optimization problem:FOC entails:On solving ,Allowing for the preference to be Quasi-linear and taking suitable forms we haveTherefore, the utility function is now of the form:Here ” signifies the degree of altruism of the parent.So the problem of the household (more specifically of the parent, in absence of any intra-household bargaining power of the child) is to7Incorporating the constraints into the objective function we again end up having anunconstrained maximization problem:The first order conditions are the same as (7). Plugging in the specific values, we get:This is the equilibrium condition which helps us analyse the optimal amount of householdwork is decided upon by the parent for the child.Second order conditions are satisfied as:To investigate the effect of an incremental change of the child wage ‘w’, parent’s income ‘Y’and the household attribute ‘A’ on child labour and labour within household ‘h’, respectively,we turn to comparative static analysis. We also observe how changes in parental incomeaffect the endogenous variables.Section-IIIComparative Static Analysis:Effect of a change in child wage ‘w’:8Here,It implies that an increase in child wage increases the share of time spent by a child as childlabour. The intuitive explanation of this result is quite simple. As child wage increases,parents want to send their children to work in order to gain the extra utility associated withthe extra income from the child’s work as a labourer, and thus the time spent by a child as alabourer increases.It follows from the above result that an increase in child wage reduces the share of time spenton household work by a child. As child wage increases, the altruistic parents make a trade-offbetween child’s time shared by household work and market work and since market work nowgives them extra income, they reduce the child’s time spent in household work to increase theshare of time spent for market work. The opportunity cost of working at home increases.Diagrammatic exposition:9Fig. 1We measure consumption on the Y-axis and time on the X-axis. The household productionfunction is represented by the curve M1.In absence of any market opportunities to sell labour,equilibrium is attained at E 0 where s*=OA and h*=A1. Clearly market labour effort for thechild is zero as no market exists. The level of utility associated with this set up is U 0.Introduction of a child labour market is represented by the line PP whose absolute slope isequal to the child wage rate. The equilibrium is attained at E 1 . Opening up of marketopportunities leads to a reallocation of total time with optimal study time reduced to OC andhousehold work to B1. Child labour supplied to the market is CB (>0).The level of utility atequilibrium associated with the introduction of the child labour market is U 1 (>U 0 ), i.e., thereis an increase in utility derived from the earning potential of the child.Now suppose that the wage rate of the child increases. This change can be captured by asteeper line P’P’. Equilibrium is attained at E 2 . There is a drop of s* to OF and also a drop inhousehold work. Substitution effect dominates and market labour increases to FD with anincrease in child wage.Effect of a change in household attribute ‘A’:We see that,10Due to rise in the household attribute ‘A’, it’s more beneficial for the altruistic parents toallocate a lesser portion of their child’s time for market work in order to increase thehousehold work so that it raises consumption and increases utility. That is why the share oftime spent by a child as child labourer decreases as ‘A’ increases.As the household attribute ‘A’ rises, the productivity from household work increases, and toraise consumption for higher utility, parents decide to increase the share of time spent by achild on household work. The above result is a reflection of the same logic.Diagrammatic exposition:11Fig. 2Here, initial equilibrium is at E 0 . Introduction of a child labour market has the same effect asin the previous case. Now we consider a ceteris paribus change in the household attribute ‘A’.This swings the production function upwards as the productivity of the child’s householdwork increases. Following the change, optimal amount of household labour (h*) increases toF1(;C1). Due to an increase in the productivity, there is an unambiguous fall in marketlabour of the child as gain in consumption due to household work is now higher. The changein study time for the child is ambiguous as there is a fall in market labour but rise in housework. Here the income effect is not zero as there has been a parametric change in the utilityfunction due to the change in ‘A’. It is worthy to point out that the relevant indifference curvein this case does not belong to the indifference map shown by U 0 (?) and U'(?). Henceoptimal study time undergoes change. If the increase in household work outweighs thereduction in market labour, amount of study time decreases. But practically this is highlyunlikely and study time increases from OB to OD as depicted above.Effect of change in parental income ‘Y’:Now,12It follows that an increase in parents’ income decreases the share of time spent by their childas a child labourer. This result is due to the assumption of altruistic parents and that parentssend their children for work only because of their concern for the survival of the family. Asparents’ income increases, the need for the child to work as a child labourer decreases, andthe altruistic parents allocate lesser time for the child to work outside.Household work, in all practical purposes, doesn’t depend upon parents’ income. So if theparent’s income increases, there is no change in the allocation of time for household workand we get the above result. This is in accordance of our supposition of a quasi-linear utilityfunction.Clearly here, we get an unambiguous result whereinThis is in line with our assumption that parents care about the study time of their child. Anincrease in the parental income tends to compel the parent to increase the study time of theirchild, thereby reducing the time spent as a labourer.Diagrammatic exposition:13Fig. 3Child wage has no relation with adult wage, a case considered to be different from that ofBasu and Van in which child labour is a substitute for adult labour. In this case, increase inadult wage, ceteris paribus, has no effect on household work of the child but interestinglychild labour goes down as a different set of indifference map is obtained after householdconsumption undergoes a parametric shift. The unchanged child wage line T’ gets tangent toone such new indifference curve at E 1 showing that study time indeed has increased.Section-IV14Considering the Socio-psychological cost of Parents:Armed with the effects of parametric shifts on the household time allocation problem, wemay proceed to assume that the parent incorporates in his preference certain norms thatessentially imply that child labour is inherently “bad”. A feeling of anxiety or shame may beassociated with violating this norm. If the members of the group to which the person belongsdo not approve the person’s nonconformist behaviour, the person’s welfare is adverselyaffected through a damage done in his sense of belongingness and his identity (Elster, 1989;Akerlof and Kranton, 1998).This is the type of effect that can be modelled as a “stigma.”The imposition of a social stigma on the parent in sending his child to work may be justifiedon two grounds. The first one is based upon a general consensus reflecting a social andparental morality which may be strengthened by governmental efforts in spreading awarenesscampaign on child education and the future adverse effects of child labour. Secondly, adultsmay realize that child labour as a custom would work as a depressant of adult wages in theeconomy and thus would affect the labour market conditions badly. If we indeed assume thatthe child wage is only a fraction of adult wage rate as done in Basu and Van (1998), the latterargument cannot be totally refuted. These two perspectives, the moral principle and thepractical rationale, do not necessarily oppose each other.Following these arguments, we may assume that the parent incurs a “stigma cost” in sendinghis child to the work in the market, as assumed by Fan (2011). The “stigma cost” may dependon the aggregate level of child labour, E, which is taken as a parameter by the households asthe effect of individual’s decision on aggregate variable is negligible. The stigma costfunction is assumed to have the following specifications: M e ;0,M ee ?0,M(0,E) =0, eM e ?0and M eE ;0 which implies that the marginal disutility from child’s effort is decreasing in thetotal amount of child labor in the economy.M(0,E) = 0 implies that stigma cost vanishes when the child is not working. The latterimplies that if e = 0, then M E = 0. The condition implies that if e > 0, . Inother words, an increase in aggregate child labor weakly diminishes the stigma cost, providedthat the child is working in the first place. Note that thsese assumptions imply that M(e,E) > 0whenever e > 0. Therefore, it is being assumed that even if E is very large, as long as onechild works the stigma cost does not vanish.Here the utility function is of the form:15Incorporating the changes, the UMP:FOC entails:Solving,The above condition simply states that the marginal benefit of an extra unit of child laboursupplied in the market, measured in terms of utility from extra consumption, has to equal themarginal cost, as given by the stigma to be borne by the parent, as a function of individualand aggregate child labour supply. From (30), it is possible to obtain the optimal amount ofchild labour hours supplied by the individual household, given by. Hence, theagent considers the wage rates and the expected level of child labour in the economy, E, inorder to optimally choose the number of hours that her child should work.Diagrammatic representation:16Fig. 4E 0 is the point of tangency obtained between the household production function M1 and theindifference curve U* which has no child labour and no stigma cost. As market opportunityopens up, a new tangency point is obtained at E 1 between M1 and a new indifference curve V ‘which in effect signifies a lower level of utility than U*. This is because of a positive amountof child labour and hence stigma cost which drags down the level of utility.Section-VPolicy implications and Conclusion:17In light of our simple theoretical exercise, the government may be expected to play animportant role even in a non-paternalistic way. A lot is discussed in the child labour literatureabout the efficacy of the government’s active intervention in the form of imposing a blanketban on child labour. Again, there are issues like increasing the return from schooling too—anissue we didn’t treat in our analysis while deriving the supply of child labour.In our perspective, the government can always focus upon favourably affecting the householdattribute “A”. Ushering in institutional changes in the economy through measures likeensuring LPG connections to households, enabling them possess zero-balance bank account(as in Jan Dhan Yojana), or opening mobile wallets and making other banking servicesavailable through mobile phones, or even through doling out free bicycles to girl studentsunder schemes like “Kanyashree” practised in West Bengal. The resultant increase of theseon household productivity can hardly be ignored and that in effect may really go a long wayin arresting the growth of incidence of child labour.Further, the government should intensify its continual mission to build up a generalconsensus among the less privileged portion of the population about the future adverse effectsof sending children to work in the market, and ultimately help in building up a social stigmaagainst the same, at least in cases when everything hasn’t boiled down to just surviving dueto pressures of poverty. Literacy campaigns often incorporate in them a conceptual idea of along run trade-off between child labour and child education and the same feature may bevigorously emphasized. This in effect would raise the “stigma cost” borne by the parent. Andhis resultant optimizing exercise would lead to an efficiency point that would contain less ofchild labour.

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