1 J Popul Econ 2001) 14: 119± Cildren as insurance Claus Cr. PoÈrtner University of Copenagen, Institute of Economics, Studiestraede 6, 1455 Copenagen K, Denmark Fax: , Received: 8 September 1998/Accepted: 9 June 1999 Abstract. Tis paper presents a dynamic model of fertility decisions in wic cildren serve as an incomplete insurance good. Te model incorporates uncertainty about future income and te survival of cildren as well as a discrete representation of te number of cildren. It contributes to te understanding of te negative relation between fertility and education, sows wy parents may demand cildren even if te return is negative, and explains wy fertility migt rise wit increasing income wen income is low and decrease wen income is ig. Furtermore, te model can account for te decline in fertility wen te risk of infant and cild mortality decreases. Finally, te implications for empirical tests of te demand for cildren are also examined. JEL classi cation: J13, O12, D11 Key words: Fertility, mortality, insurance, uncertainty 1. Introduction [A]n important question is weter aving many cildren and/or a large extended ouseold is an optimizing strategy allowing ouseolds to derive bene ts oterwise lost due to poorly functioning markets... Birdsall 1988, p 502) In less developed countries insurance and credit markets are examples of poorly functioning or absent markets tat may a ect te demand for cildren. I would like to tank Alessandro Cigno, Birgit Grodal, Rasmus Heltberg, Jens Kovsted, James McIntos, U e Nielsen, Stepen O'Connel, Oded Stark, and Finn Tarp for elpful comments. Responsible editor: Jon F. Ermisc.
2 120 C. Cr. PoÈrtner Wit little possibility for risk diversi cation, te need for insurance as to be satis ed by oter means. Te ypotesis of tis paper is tat cildren, under tese circumstances, can serve as an incomplete insurance good. Most teoretical and empirical analyses of cildren as security assets ave focused on te old-age security aspect of cildren.1 Cain 1981, 1982, 1983) as, owever, empasised tat cildren can also provide insurance against sortfalls in income under oter circumstances. Similarly, Jones 1987, capt. 1) argues tat te near universal marriage pattern and iger fertility found in Asia compared wit Europe in pre-modern time was not due to old-age security considerations. Instead it arose from te need to ``... command as muc labour as possible to elp recover from te e ects of recurrent disaster'' Jones 1987, p 17). Yet, formal modelling of decisions wen cildren serve as a general substitute for insurance is lacking. Appelbaum and Katz 1991) analysed fertility decisions wen te income of individual ouseold members including cildren) is uncertain. Yet, teir model is static, and tey do not explicitly examine te causes of uncertainty in cildren's income. In tis paper, I analyse fertility decisions of a ouseold wen it faces uncertainty wit respect to future income and cild survival. Cildren are costly in te rst period of teir life and provide a positive net income in subsequent periods. Te total net income from a cild over te parents' lives is assumed to be at most zero and is uncertain because of mortality. Hence, parents use cildren as a means to sift income from a period wit certain income to future periods wit uncertain income, tereby insuring temselves against te possibility of low income. Assuming realistically tat te number of cildren can only take integer values a binomial distribution is used to model mortality. Tis was rst suggested by Sa 1991). He used te approac to analyse te e ects of mortality canges on fertility and parents' utility in a model were parents derive direct utility from te number of surviving cildren. Te major advantages of using a binomial distribution are tractability and te realism of a discrete number of cildren. Te model as two novel aspects: Cildren are modelled as a general insurance and saving asset, and te dynamic aspects of income and fertility are explicitly examined. Tis leads to alternative explanations of a number of stylised facts about fertility. Tey include te strong negative correlation between moters' education and fertility, te positive relation between income and fertility at low levels of development, and te negative relation at iger levels of development. Te model also demonstrates wy parents may demand cildren even if te monetary return is negative and te fact tat fertility is likely to fall as infant and cild mortality decrease. Section 2 discusses cildren as insurance and alternative strategies. Te two-period model wit uncertain income in one period is presented and analysed in Sect. 3. Possible extension to amulti-period model is also discussed. Sect. 4 concludes and sums up. 2. Insurance, alternative strategies and cildren Disruption of a ouseold's income stream may result from disability or deat of a person, wo provides a signi cant labour input, as well as from adverse weater conditions, suc as ooding or drougt. Oter causal factors are te risk of depredation and patriarcal risks,2 wic are primarily ± but not
3 Cildren as insurance 121 exclusively ± faced by te rural population in developing countries. Altoug ouseolds in developed countries and in parts of te urban areas in developing countries ave ready access to insurance, eiter from private companies or troug state-funded initiatives, poor ouseolds in te urban areas and most of te people living in te rural areas of developing countries do not.3 Wit absent or incomplete insurance markets, ouseolds need to rely on alternative income and consumption smooting strategies. Wile it is unlikely tat a ouseold will rely solely on one strategy tey are presented separately ere. Five possibilities are considered: Saving, borrowing, public sector support, ``traditional'' systems of support and cildren.4 For saving or borrowing to be a viable means of insurance, a ouseold needs a surplus in te oter periods. If tere is a surplus, accumulation can take place in cas, commodities, livestock or land. Te rst tree are subject to depreciation of value, teft and costly storage, and since te duration of te adverse condition is normally unknown tere is a risk of using up savings and borrowings before conditions improve. Hence, borrowing and saving can in many cases only provide relatively sort-term relief. Land can generate income, but yields vary wit te weater. Furtermore, markets for land tend to be tin or nonexistent and sale of land leads to lower future income. Finally, land must be closely managed and acquiring large amounts of land means tat te ouseold must eiter be large or ire outside labour.5 If a ouseold relies on borrowing and uses land as collateral it faces extra ardsip if it defaults on te loan since its earnings capacity will diminis. Te ouseold can also rely on te public sector, but public support varies from setting to setting and may be very unreliable or directly absent Cain 1981). Te fourt strategy is to use te ``traditional'' systems of support. Tese include te village, te commons, and te extended family. Townsend 1994) examines weter te village as an institution can insure its inabitants against bad weater conditions or oter adverse conditions, but fails to nd any strong support for te ypotesis. For bot te village and te commons a ig degree of co-variation in risks is likely, making it di½cult to provide support wen it is most needed. Wit respect to te extended family, Cain 1981) nds tat a large part of te distress sale of land is to closely related kin, suc as a broter. Since e could buy te land, it is also likely tat e would ave te money to elp te relative in need, but decided not to. Rosenzweig and Stark 1989) nd tat daugters migrate to oter villages to marry to mitigate teir families' income risk. Te nal possibility considered ere is to use cildren as a substitute for insurance. Cildren can elp eiter by working at ome or as wage labour,6 and older cildren wo eiter ave teir own ouseold or ave migrated can make transfers to teir parents.7 It is important to note tat even if wages are depressed, aouseold still gains from alarge number of working cildren, provided tat income covers costs. A cild's consumption can also be reduced in case of adverse conditions, implying tat te net return need not decrease muc even wit lower wages.8 Te most obvious reason wy cildren wit teir own ouseold or migrated family members want to remit money is family ties, also refered to as altruism or wat Nugent 1985) calls loyalty. It follows tat cildren are likely to be more reliable as a means for insurance tan more distant family.9 Finally, if te ouseold is in dire straits te parents may actually ``sell'' teir cildren as bounded labour. Tere is usually
4 122 C. Cr. PoÈrtner an underlying presumption tat cildren sould be of a certain age and in some societies of aspeci c sex to serve as asubstitute for insurance. Neverteless, te argument tat only boys can act as insurance carries less weigt if one accepts te point of Rosenzweig and Stark 1989) refered to above, because a larger number of girls leads to more connections wit oter ouseolds. Cildren, wen seen as a substitute for insurance, ave tree special properties. First, te expected net return of an additional cild need not be positive for risk averse parents to ave anoter cild, since by de nition tey are willing to give up some of teir income in order to reduce te risk. Hence, te insurance argument can contribute to understanding wy studies, suc as Cain 1982) and Lindert 1983), of te net return to cildren ave failed to nd any large positive return to cildren. Secondly, cildren are a very general means of risk diversi cation and are not ``used up'' to te same extent as savings or borrowings. Tis means tat cildren are in some aspects more like an annuity tan an insurance policy. Bot consumption and work e ort of te cildren can cange, owever, making tem closer to standard insurance. If parents derive utility from teir cildren's consumption and education it is likely tat if te family is well o te cildren will work less, consume more and possibly go to scool. Te parents can ten increase te workload of te cildren and decrease te consumption as discussed above if needed. Tirdly, cildren are only an incomplete substitute for insurance. Tey ave a long maturing time, during wic tey are potentially very expensive, tey may die before being able to provide any return to teir parents, and tere is no way a priori of knowing te sex or ability of te cild. Furtermore, te number of cildren can take only discrete values. Hence, cildren are a crude substitute for insurance, but possibly better tan te alternatives. Tree studies provide empirical support for te ypotesis tat cildren provide general insurance against various risks. De Vany and Sancez 1977) analyse te e ect of land reform in rural Mexico and nd tat uncertain land tenure rigts associated wit te ejido system, in wic land is granted to individual families on a usufruct basis and were land cannot be bougt, sold, leased or mortgaged, leads to ig fertility. Tey conclude tat: ``Cildren function as surrogate capital instruments, or securities, wic permit parents to partially bridge te incompleteness of markets in claims to uncertain, future states'' De Vany and Sancez 1977, p 761). Cain 1990) analyses te relation between risk and fertility in two villages in Nortern India. It is sown tat altoug weater induced risk is relatively small and common property resources are available tere are considerable ``predatory'' political depredation) and patriarcal risks. Tis combined wit semi-feudal social relations, wic mean poor access to credit and little e ect of state interventions, lead to a iger demand for sons compared wit te villages in Soutern India, studied in Cain 1981), were te risk environment is more benign and access to insurance substitutes easier. Finally, Das Gupta 1995) examines fertility decline in te Ludiana District, Punjab. Total fertility began to decline around 1940; well before te onset of family planning programmes and te initiation of te Green Revolution in According to Das Gupta tis decline in fertility came about as a result of increased security against mortality peaks and food sortages. Te improvement is partially due to te expansion of irrigation,
5 Cildren as insurance 123 wic meant tat ``... bot te level and te variance of yields were improved'' Das Gupta 1995, p 484). 3. Te model Consider atwo-period decision problem for aouseold tat faces acertain income in te rst period and uncertainty about income and cild survival in te second period. Te ouseold decides on te number of birts in period one. In te second period te ouseold's income is revealed togeter wit te number of surviving cildren. Te number of birts and te number of surviving cildren are assumed to be discrete variables. Let N A f0; 1; 2;...g denote te number of birts and n A f0; 1;...; Ng te number of surviving cildren in period two. It is assumed tat te survival probability of eac cild is s A 0; 1Š, wit s independent of te number of cildren and taken as given by te ouseold. Hence, te probability tat n cildren out of N birts survive follows abinomial distribution wit te density function b n; N; s 1 N n s n 1 s N n : 1 First period ouseold income is given by y 1. In te second period tere are two states of te world x A f1; 2g, and ouseold income is y 2 x ˆ y if x ˆ 1 y if x ˆ 2 Te probability of state 1 is p 1 and te probability of state 2 is p 2 ˆ1 p 1. Eac birt carries wit it a constant cost, so tat te total cost of N birts in te rst period is N. In te second period income minus expenditures for eac surviving cild is. Hence, total income from n surviving cildren is n. Since te cost and income factors are assumed to be equal, tere can never be a pecuniary gain from aving cildren even if tey all survive. Tis corresponds to a stocastic rate of interest tat is eiter zero or negative. If te second period income is known tis implies tat te ouseold demands cildren only if te second period income is su½ciently lower tan te rst period income, assuming tat te two period utility functions are identical.10 It follows tat if te expected second period income is equal to te rst period income ten any demand for cildren is due to te uncertainty of future income, again assuming tat te two period utility functions are identical. It is in tis sense tat cildren serve as insurance.11 Te coice of N determines consumption in period one as c 1 ˆ y 1 N: 2 Te maximum number of birts te ouseold can ave in te rst period is y i 1 or te biological maximum, wic for simplicity is assumed to be iger
6 124 C. Cr. PoÈrtner tan te budget constrained maximum. Consumption in te second period is te stocastic variable c 2 x; n ˆy 2 x n: 3 Te ouseold is assumed to ave a von Neumann-Morgenstern expected utility function U c 1 ; c 2 x; n ˆ X u c 1 ; c 2 x; n p x b n; N; s : 4 x; n Assuming additive separability in consumption in te two periods, expected utility is ~U c 1 ; c 2 x; n ˆ u 1 c 1 X x p x X n b n; N; s u 2 c 2 x; n : 5 Furtermore, bot te rst u 1 c 1 and second period utility function u 2 c 2 x; n, de ned on sure amounts of consumption in eac period, are assumed to be strictly increasing and concave in consumption. Te ouseold decides on te number of birts rater tan directly on consumption. Terefore, te expected utility of N birts, for given s and p, is U N; s; p ˆu 1 y 1 N X x p x X n b n; N; s u 2 y 2 x n : 6 Te ouseold maximises 6) subject to te rst period budget constraint 2) Deciding on te optimal number of birts To analyse te optimal coice of te ouseold one needs te discrete equivalent of te rst and second order derivatives. Te ``marginal rst period utility'' of an additional birt is denoted by u 1 N y 1 N 1 u 1 y 1 N 1 u 1 y 1 N : 7 Since u 1 c 1 is strictly increasing in c 1 and taking account of 2) tis is negative. Te cange in te marginal rst period utility due to an extra birt is given by u 1 NN y 1 N 1 u 1 N y 1 N 1 u 1 N y 1 N : 8 Tis is also negative due to te concavity assumption of te rst period utility wit respect to consumption. Since te rst period utility is de ned over te continuous variable c 1, y1 N 1 y 1 N 2 d dm u1 m dm ˆ u 1 y 1 N 1 u 1 y 1 N 2 ˆ u 1 N y 1 N 1
7 Cildren as insurance 125 and y1 N y 1 N 1 d dm u1 m dm ˆ u 1 y 1 N u 1 y 1 N 1 Togeter wit 8) tis leads to ˆ u 1 N y 1 N : y1 unn 1 N 1 y d 1 N ˆ dm u1 m dm y 1 N 2 y1 N y 1 N 1 By assumption d dc 1 u 1 c 1 is non-increasing in c 1. Hence u 1 NN y 1 N U 0: d dm u1 m dm: Tat is, rst period utility is decreasing and concave in te number of birts for given rst period income. Note tat un 1 and u1 NN are de ned only for N U y i 1. Te marginal second period utility of an extra surviving cild, given by u 2 n y 2 x n 1 u 2 y 2 x n 1 u 2 y 2 x n ; 9 is always positive. Tis follows from te assumption tat second period utility is strictly increasing in consumption. Te cange in marginal second period utility due to an extra surviving cild, u 2 nn y 2 x n 1 u 2 n y 2 x n 1 u 2 n y 2 x n ; 10 is negative or equal to zero following te same arguments as for te cange in marginal rst period utility. Wit respect to te expected utility function de ne te marginal cange in expected utility from an additional birt as U N N; s; p 1 U N 1; s; p U N; s; p : 11 Let te cange in tis marginal expected utility due to one more birt be denoted as U NN N; s; p 1 U N N 1; s; p U N N; s; p : 12 Tese two expressions are de ned for N V 0andN U y i 1. Te following two relationsips emerge12
8 126 C. Cr. PoÈrtner U N N; s; p ˆu 1 N y 1 N s X x p x XN nˆ0 b n; N; s u 2 n y 2 x n 13 U NN N; s; p ˆu 1 NN y 1 N s 2 X x p x XN nˆ0 b n; N; s u 2 nn y 2 x n U 0: 14 Te interpretation of 13) is close to te standard rst order derivative in a maximisation problem. An additional birt leads to a cost in foregone rst period utility, i.e. is te rst part of 13). If tis additional cild does not survive to te second period tere is no second period utility gain. If te cild survives te ouseold as one extra cild in eac of te possible outcomes of cild survival and income states, captured by P x p x P N nˆ0 b n; N; s u2 n. Te assumption tat te cild as a survival probability of s leads to 13). Equation 14) sows tat te expected utility function is concave. Proposition 1. Tere exists a solution to te ouseold's maximisation problem and te optimal number of birts is eiter a unique number or tere are two neigbouring numbers tat are bot optimal. Proof. Since te number of birts N is bounded from below by zero and from above by y i 1 and can only take integer values tere must be a nite number of possible coices. Hence asolution must exist. Let N s; p denote te largest optimal number of birts given s and p. Owing to te concavity of U, it must satisfy te following conditions U N N s; p ; s; p < 0; and 15a U N N s; p 1; s; p V 0: 15b If U N N s; p ; s; p < 0; and U N N s; p 1; s; p > 0; it follows tat N s; p is te only optimal number of birts given tat U NN < 0. If U N N s; p ; s; p < 0; and U N N s; p 1; s; p ˆ0 bot N s; p and N s; p 1 are optimal. Finally, if
9 Cildren as insurance 127 U N 0; s; p < 0 te optimal solution is N s; p ˆ0. 9 Having two optimal numbers of birts migt be seen as problematic. It is unlikely, owever, tat tis situation arises given tat a very small cange in te parameter values would instead lead to a single optimal number of birts. For brevity, it is assumed in te following proofs tat te optimal solution is an interior one Risk aversion and canges in income Te optimal number of birts depends on, among oter variables, te ouseold's present and future income and its degree of risk aversion. Wit respect to future income two e ects are of interest ere: Te e ect of a cange in te level of income and te e ect of a cange in te dispersion of income. Proposition 2. Te optimal number of birts is non-increasing for increasing probability of iger second period income. Proof. Since p 1 and p 2 must sum to one, te marginal utility from an additional birt is U N N; s; p ˆu 1 N y 1 N s p XN nˆ0 1 p XN b n; N; s un 2 y n nˆ0 b n; N; s u 2 n y n : 16 Di erentiating tis expression wit respect to p, wic indicates te probability of ig second period income, leads to q qp U N N; s; p ˆs XN nˆ0 b n; N; s un 2 y n u2 n y n Š: 17 Since te second period utility is increasing and concave in consumption and y is larger tan y, te derivative must be negative for all N. Tat is, q qp U N N; s; p < 0: 18 Evaluating tis at N ˆ N s; p leads to U N N s; p ; s; p 0 < U N N s; p ; s; p for p 0 > p: 19 Using te optimality condition 15a) yields
10 128 C. Cr. PoÈrtner U N N s; p ; s; p 0 < 0: 20 Because of te concavity of U U N N; s; p 0 < 0 for N V N s; p : Furtermore, U N s; p ; s; p 0 > U N s; p 1; s; p 0 > Hence, N > N s; p cannot be optimal at p 0 and N s; p must terefore be locally non-increasing in p. Te global result is13 N s; p 0 U N s; p for p 0 > p: 9 Ruling out te case were canges in te level of expected income as no e ect on te number of birts te interpretation of Proposition 2 is tat an increased probability of ig future income leads to less demand for insurance and terefore fewer birts. A similar e ect can be sown to arise if te probability distribution remains te same, but eiter te income in te low income state, te income in te ig income state or bot are increased. Te iger te expected future income, relative to te present income, te more willing te ouseold is to take te risk of a low future income. Hence, tere is less need for insurance. Wile an increase in te probability of ig income or an increase in eiter low or ig income may increase or decrease te variance of income, tis e ect is always dominated by te level e ect, at least as long as te lower income is not decreased. Neverteless, as indicated by te following proposition te dispersion of future income can also a ect te demand for cildren. Proposition 3. A mean-preserving spread of future income cannot lead to a lower optimal number of birts. Proof. By assumption te expected future income is E y 2 Šˆpy 1 p y: Let d be an arbitrary constant and let p A 0; 1Š be te probability tat d p is d added to second period income and 1 p te probability tat 1 p is substracted from second period income in eac state, ten a mean-preserving spread of future income is E y 2 Šˆp p y d 1 p y d p 1 p 1 p p y d 1 p y d : p 1 p
11 Cildren as insurance 129 Let ~U N N; s; p denote te marginal cange in expected utility from an additional birt wen te distribution of expected income is te mean-preserving spread. Ten ~U N N; s; p U N N; s; p ˆ s p X b n; N; s pun 2 y d p n 1 p un 2 y d 1 p n u 2n y n 1 p X b n; N; s pu 2 n y d p n 1 p un 2 y d 1 p n u 2n y n V 0: Te inequality follows from te assumption tat u 2 is increasing and concave. Since te marginal expected utility from an additional birt is iger for te mean-preserving spread distribution tan for te original distribution it follows from Proposition 1 tat it cannot be optimal to ave fewer birts. 9 Proposition 3 demonstrates tat a mean-preserving spread in future income cannot lead to a lower demand for cildren. Clearly, te result would be te same if te ig income is increased and te low income is decreased, keeping te mean constant. Te proof also indicates tat te more risk averse te ouseold is i.e. te more concave te second period utility function is), te iger is te likeliood tat te optimal number of birts will increase. It is likely tat adverse conditions in developing countries can lead to a future income so low tat it treatens te very survival of te ouseold. Te e ect of tis possibility on te demand for cildren depends on te caracteristics of te utility function as consumption approaces zero. Assuming tat te marginal utility goes to in nity as future consumption goes to zero it would appear tat te ouseold would demand an in nite number of cildren or in te real world ave as many cildren as biologically possible. Te maximum number of birts is, owever, also constrained by te rst period budget constraint, so te marginal utility of consumption in te rst period y i would also increase substantially as N approaces 1. Proposition 2 still olds but it is less likely tat an increase in te ig income would generate any observable e ect on te observed number of birts. Te implication is tat even families wo are relatively ricer in te sense tat teir ig second period income is iger tan oters would tend to ave a large number of cildren if tey faced a risk of zero or very low income in some periods. Tis would seem to support te conclusion by Cain 1986) tat in rural Banglades, were te important sources of risk are endemic, ``... one sould not expect fertility to vary systematically across region or economic status''. If everybody experiences aig risk of avery low income no matter teir status tere would not be muc di erence between fertility levels due to security considerations.
12 130 C. Cr. PoÈrtner Families di er not only wit respect to teir expected income but also wit respect to teir present income. Te model can also be used to analyse te impact of present income on fertility. Tis is done in Propostion 4. Proposition 4. For a given expectation of second period income te optimal number of birts cannot be iger for a lower rst period income tan for a iger rst period income. Proof. Use 13) to nd U N N; y 1 U N N; y 1 ˆ u 1 N y 1 N u 1 N y 1 N : Tis is positive because of te concavity assumption and terefore U N N; y 1 > U N N; y 1 : 21 From te optimality condition U N N y 1 ; y 1 < 0: Using tis and 21) yields U N N y 1 ; y 1 < U N N y 1 ; y 1 : Tis leads, togeter wit te concavity assumption, to U N N; y 1 < 0 for N V N y 1 : Tus, it can never be optimal to ave more cildren wen te present income is lower tan te number one would ave if income were iger. 9 Ruling out te uninteresting case were rst period income as no e ect on N, te optimal number of birts is lower if te present income is lower. Mostly, in empirical analyses of te demand for cildren, only present income or some proxy for income is observed togeter wit te number of cildren. According to Proposition 4 tere sould be apositive relation between income and fertility in a given period, but Proposition 2 predicts a negative relation between future expected income and te number of birts. Hence, to determine te demand for cildren it is not su½cient to observe present income, one also needs people's own assesment of future expected income and its variability. In poverty striken environments wit little possibility for investment in cildren and assuming tat te variation in expected future income across ouseolds is not too big tere will be apositive relationsip between present income and fertility according to bot te teory presented ere and te standard beckerian model. In te insurance model te reason for tis is tat since everybody as approximately te same distribution of expected income only proposition 4 will be in e ect and any di erence in fertility must ten be a result of di erences in present or past) income. For te beckerian model
13 Cildren as insurance 131 fertility will be positively correlated wit income if te quality of te cildren is assumed to be basically constant. Hence, simply regressing fertility against present or some measure of past) income may lead to accepting a ypotesis of cildren as a standard consumer good. Tus, one concludes tat better-o people will tend to ave more cildren. Yet, te true underlying ypotesis may be tat cildren serve as a substitute for insurance. Te result is tat as people's expected future income increase teir fertility does in fact decline. Te above argument assumes as stated tat te variation in expected income across te ouseolds in te sample is not too large. Te oter extreme is tat tere is little variation in present income but variation in expected income. In tis case a family wit a ig expected future income will ave fewer cildren tan a family wit low expected future income, even if teir present income is te same. Terefore, wen analysing te demand for cildren it is sensible to control for expected future income. Tis variable is di½cult to observe. Tus, a proxy, suc as education, needs to be found. Education is a possible proxy for tree reasons. First, it is typically correct tat te more educated a person is, te iger te expected future income.14 Secondly, more education is likely to lead to less variation in expected future income, even if it does not substantially increase expected income. Finally, since education provides people wit te ability to collect and process information, tey are in a better position to asses teir future income and to take steps to prevent a very low income state occuring. If education is a good proxy for future income and its variation tis model could elp explain part of te inverse relationsip between fertility and education. Empirical studies of te relation between education and fertility ave found tat te moter's education as te strongest impact on ouseold fertility Scultz 1997). Tis corresponds wit te prediction of tis model if cildren are primarily demanded as insurance by women as argued by Cain 1982) and Nugent 1985). Furtermore, various studies ave sown a negative relation between infant and cild mortality and te education of moters.15 Hence, education as two e ects tat bot tend to lower te number of birts. First, iger education means less need for insurance because of iger expected income and lower variation in income. Secondly, te ouseold needs fewer birt since cild mortality decreases wit education. Te opposite e ects arising from an increase in present income and an increase in future income are especially important if te ouseold does not fully realise tese canges wen tey take place. Consider an exogenous increase in te probability of ig income in future periods tat is not fully realised by te ouseold wen it occurs. Te result is a iger expected) fertility, since te ouseold is now likely to ave more periods wit ig income, wile tey act as if tey ave a lower expected income during te updating of teir beliefs. Taking account of tis learning e ect may explain part of te experienced increase in fertility in te beginning of a country's development. Clearly te iger fertility is sub-optimal for te ouseold. Tere is terefore a potential welfare gain if te ouseold knew te true distribution E ects of canges in survival probability Beside te level and variance of income discussed above, te survival probability of te cildren is also important in determining te optimal number of
14 132 C. Cr. PoÈrtner birts, since it in uences te return on cildren. Te following proposition examines te e ect on fertility. Proposition 5. For a su½ciently risk averse ouseold, te optimal number of birts is rst non-decreasing and ten non-increasing in te survival probability. Proof. From te proof of Proposition 2, asu½cient condition for te optimal number of birts to be non-increasing in te survival probability is tat te marginal utility of an additional birt is decreasing in te survival probability, wen evaluated at te optimal number of birts. Di erentiating U N wit respect to s leads to16 q qs U N N; s; p ˆX x X N 1 p x nˆ0 fsnb n; N 1; s B n; N; s g unn 2 y 2 x n un 2 y 2 x N : 22 Te derivative of te marginal utility of an additional birt wit respect to te survival probability depends on te sign of te expression inside te curled brackets and te curvature of te second period utility function. Tere are tree possible cases. First, for low s, a nd N not too large, te expression inside te brackets is negative for all n. Since te second period utility is concave by assumption, an increase in te survival probability will unambigously lead to a positive value of te derivative. Hence, te optimal number of cildren must be non-decreasing in te survival probability for low values of te survival probability. Secondly, for medium range values of s, te expression can be bot positive and negative. For low values of n te expression will be positive and for iger values it will be negative. Terefore, te derivative can be bot positive and negative, depending on te concavity of te second period utility function. A negative value is more likely te more risk averse te ouseold is and te more positive te tird order derivative is. Finally, for iger values of s, te expression is positive for all n.17 Again, te more risk averse te ouseold is te more likely it is tat te derivative is negative and terefore tat te optimal number of birts is non-increasing in te survival probability.18 9 An increase in te survival probability of cildren as two e ects on te demand for cildren as insurance. First, increased survival probability is equivalent to a iger return to birts less wasted resources). Secondly, te iger expected number of survivers leads to a iger expected consumption in te future. Wile te substitution e ect tends to raise te optimal number of birts te income e ect as te opposite e ect. If te income e ect dominates te optimal number of birts will decrease. Proposition 5 states tat te more risk averse a ouseold is te more likely it is tat te income e ect will at one point dominate te substitution e ect wen te survival probability is increased. Hence, te model is able to illustrate te observed fall in fertility following a decline in infant and cild mortality, provided ouseolds are indeed risk averse.
15 Cildren as insurance Extension to a multi-period model Te ouseold may ave some overall idea about te number of surviving cildren it wants, but te decisions on timing and number of birts are in uenced by present income and te number of surviving cildren. It follows tat te ouseold's fertility coice is potentially better described as a stocastic dynamic programme. As sown in PoÈrtner 1998) te results for te twoperiod model do carry over into te tree-period model, were te parents can ave cildren in te rst two periods and were income is uncertain for te last two periods.19 Besides te propositions equivalent to te ones in te two-period model, one can sow tat te optimal number of second period birts is nonincreasing in te number of surviving cildren from te rst period. Tere are two e ects from an extra surviving cild in te second period. First, te cild will increase te expected income in bot te low and te ig income states. Tis would tend to reduce te demand for cildren in te second period. Secondly, wit te additional cild te ouseold as a iger present income, wic implies aiger demand for cildren. Wen te utility functions for period two and tree are identical te rst e ects dominates te second. Finally, it is possible to sow tat te optimal number of birts in te rst period is non-increasing in bot te survival rate of te second period birts and in te probability of a ig tird period income. Tis assumes tat a unit decrease in te number of surviving cildren from te rst period does not lead to more tan a unit increase in te optimal number of birts in te second period. Tese results are te natural extensions of te propositions dealing wit te e ects of canges in expected income and survival in te next period. Tey indicate tat a cange in expectations will ave te same qualitative e ect weter te cange concerns te next period or one of te following periods. 4. Conclusion Te main ypotesis of tis paper is tat cildren can act as a security asset wen insurance and credit markets are eiter absent or poorly functioning. Incorporating tis into a model wit uncertain future income and cild survival and a discrete representation of te number of cildren tree main results emerge. First, for risk-averse ouseolds, te number of birts is decreasing in te survival probability for realistic values of te survival probability. Secondly, a iger expected future income leads to a lower number of birts. Tirdly, for agiven expectation of future income te number of birts in a period is increasing in income. Togeter, te second and tird result imply tat standard empirical analysis of te demand for cildren were te number of cildren is regressed against income will not be able to capture te entire e ect. Two ouseolds wit similar present income may ave di erent expected income and ence di erent number of cildren. Te simple model put forward ere provides a di erent perspective on a variety of stylised facts wit respect to fertility in developing countries. First, te model indicates wy a negative return to cildren does not necessarily imply tat parents do not demand cildren. Secondly, te observed positive relation between income and fertility at low income levels, and te negative
16 134 C. Cr. PoÈrtner relation between fertility and income at iger income levels can bot be accomodated witin te model. Tirdly, parallel wit te results on income, te model provides a possible reason for te negative relation between fertility and education, witout resorting to te assumption tat cildren are necessarily very time intensive for te moter, someting wic may not be true in many developing countries.20 Finally, te model can also elp explain te decline in fertility wen infant and cild mortality decrease. Wit respect to te policy implications of te above results, any policy tat could reduce te variability of income or raise te level of income or bot would be bene cial to te ouseolds and reduce fertility. One way of acieving tis could be irrigation as discussed in Das Gupta 1995). Better functioning markets, bot for credit, insurance and goods would also raise te utility of te ouseold and reduce fertility. Te same would be te case wit reliable state intervention in times of crisis, more secure property rigts and access to an impartial and well-functioning judicial system. Better ealt care will also improve te situation of te ouseolds by reducing te risk of incapacitating illnesses and lowering infant and cild mortality. Te reduced risk of illness would reduce te need for insurance and ence cildren, and at te same time improve te quality of life for te parents. Policies tat create more opportunities for women, especially by raising teir earnings capacity, would also lead to lower fertility and improved welfare. One suc policy tat is likely to be e ective is providing education to women. Future work on tis topic could progress along bot teoretical and empirical lines. On te teoretical side an extension of te model to include endogenous mortality risk or oter forms of investments in cild ``quality'' could be of interest. On te empirical tere is so far very little direct evidence on weter cildren serve as a substitute for insurance. Tis igligts te need for formulating furter testable results and collecting te needed data. At te moment tere is at least one result tat is testable wit data. An implication of Proposition 3 is tat ouseold wit less expected variation in teir income would ave lower fertility for a given income. Hence, one direct test of te model presented ere is to analyse te e ects of a mean-preserving spread of future income on fertility. Tis can be done using, for example, a combination of ouseold and agricultural data. Endnotes 1 See Nugent 1985), Nugent and Anker 1990), Erlic and Lui 1991, 1997, 1998), Scultz 1997) and references erein. 2 Patriarcal risk is te special risk faced by women and includes risk of widowood, divorce or abandonment. Te risk of depredation is te risk of a direct treat to one's property, eiter troug fraud or expropriation. Cain 1981) consideres te di erent types of risks in more detail. 3 Nugent 1985) discusses wy insurance companies nd it unpro table to operate in tese areas. 4 Cain 1981) and Nugent 1985) examine tese strategies and teir e ectiveness in more detail. 5 Cain 1983, 1985) examines te problems associated wit investing in land for security purposes. 6 In many less developed countries cildren from poor families begin to work a substantial number of ours per day from age 5±6 Cain 1977, 1982; Dasgupta 1993, p 359).
17 Cildren as insurance Alternatively, a parent can migrate wile te cildren stay beind. Wit respect to migration te ypotesis of tis paper can be seen as an extension of Stark's 1991, c. 4 and 5) suggestion tat migration is in part a response to risk aversion. 8 Empirical evidence on te e ects of adverse conditions on te consumption of cildren is not conclusive. Te intra-ouseold distribution of food varies between di erent locations as well as wit te severity of te situation Rosenzweig and Scultz 1982, Harriss 1990, Dasgupta 1993). 9 For furter discussion of wy migrants remit, suc as altruism and self-enforcing contracts, see Lucas and Stark 1985), Stark 1991, c. 15), Cox and Stark 1996) and Lillard and Willis 1997). 10 Tere would also be a demand for cildren if te income factor was su½ciently iger tan te cost factor or if te utility function allowed for direct utility of cildren. 11 Tis is also known as precautionary saving, wic is de ned ``... as te extra saving caused by future income being random...'' Leland 1968, p 465). See also Kimball 1990). 12 Te details of te derivations are available from te autor on request. 13 Te details of tis proof are available from te autor. 14 Foster and Rosenzweig 1996) analyse te return to scooling during te green revolution in India. See also te discussion of te e ects of scooling in Rosenzweig 1995). 15 Examples are Buiya and Streat eld 1991) and Sandiford, Cassel, Montenegro, and Sancez 1995). 16 Te derivation is available from te autor on request. 17 Sa 1989, p 23) nds tat te expression is positive if s V 0:81 and N U Using simulation it can be sown tat for y 1 ˆ 500; y ˆ 900; y ˆ 100, ˆ 25; p ˆ 0:5, and utility functions u i ˆ c1 g i 1 g ; i ˆ 1; 2; q qs U N N; s; p < 0 for g ˆ 1:67 and s close to one. Furtermore, te optimal number of birts is decreasing in s for g ˆ 2: Tis also assumes tat tere cannot be a pecuniary return to aving cildren. Furtermore, it is assumed tat cildren are only at risk in teir rst period of life. 20 Reasons for tis include economies of scale in cild-rearing wit older siblings taking care of younger ones; te fact tat many women work at ome or at te family farm, wic means tat cild-rearing and work is not exclusive activities; and tat older family members often live wit teir cildren and ence can participate in te cild-rearing if te moter works outside te ome. References Appelbaum E, Katz E 1991) Te Demand for Cildren in te Absence of Capital and Risk Markets: A Portfolio Approac. Oxford Economic Papers 43 2):292±304 Buiya A, Streat eld K 1991) Moters' Education and Survival of Female Cildren in a Rural Area of Banglades. Population Studies 45 2):253±264 Birdsall N 1988) Economic Approaces to Population Growt. In: Cenery H, Srinivasan T ed) Handbook of Development Economics, vol. I, Elsevier Science Publisers BV, Amsterdam, 477±542 Cain MT 1977) Te Economic Activities of Cildren in a Village in Banglades. Population and Development Review 3 3):201±27 Cain MT 1981) Risk and Insurance: Perspectives on Fertility and Agrarian Cange in India and Banglades. Population and Development Review 7 3):435±474 Cain MT 1982) Perspectives on Family and Fertility in Developing Countries. Population Studies 36 2):159±175 Cain MT 1983) Fertility as an Adjustment to Risk. Population and Development Review 9 4):688±702 Cain MT 1985) On te Relationsip between Landolding and Fertility. Population Studies 39 1):5±15 Cain MT 1986) Risk and Fertility: A Reply to Robinson. Population Studies 40 2):299±304 Cain MT 1990) Risk and Fertility in a Semi-Feudal Context: Te Case of Rural Madya Prades. Researc Division Working Paper 19, Te Population Council
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