Click for next page ( 75


The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 74
The Impact of Infant and Child Mortality Risk on Fertility Kenneth . Wolpin INTRODUCTION The relationship between the infant and child mortality environment and human fertility has been of considerable interest to social scientists primarily for two reasons: (1) The fertility and mortality processes are the driving forces governing population change, so an understanding of the way they are linked is crucial for the design of policies that attempt to influence the course of popula- tion change. (2) The "demographic transition," the change from a high fertility- high infant and child mortality environment to a low fertility-low mortality envi- ronment, which has occurred in all developed countries, has been conjectured to result from the fertility response to the improved survival chances of offspring. Fundamental to either of these motivations is an understanding of the micro foundations of fertility behavior in environments where there is significant infant and child mortality risk. My purpose in this chapter is to clarify and summarize the current state of knowledge. To that end, I survey and critically assess three decades of research that has sought to understand and quantify the impact of infant and child mortality risk on childbearing behavior. To do so requires the explication of theory, estimation methodology, and empirical findings. I begin by posing the basic empirical (and policy relevant) question: "What would happen to a woman's fertility (children born and their timing and spacing) if there was a once-and-for-all change in infant or child mortality risk?" Alterna- tive behavioral formulations, encompassing static and dynamic decision-theo- retic models found in the literature, answer that question and are reviewed. An illustrative three-period decision model, in which actual infant and child deaths 74

OCR for page 74
KENNETH I. WOLPIN 75 are revealed sequentially and behavior is both anticipatory and adaptive, is devel- oped in some detail, and the empirical counterparts for theoretical constructs derived from that model are developed and related to those found in the literature. Specifically, I demonstrate how replacement and hoarding "strategies," which are prominent hypotheses about reproductive behavior in this setting, fit explic- itly into the dynamic model and how these concepts are related to the question posed above. I review a number of empirical methods for estimating the quantitative effect of infant and child mortality risk on fertility, connecting them explicitly to the theoretical framework. I pay particular attention to the relationship between what researchers have estimated and the basic behavioral question. Finally, I present and discuss an overview of empirical results. THEORY Static Lifetime Formulations The earliest formal theoretical models were static and lifetime (i.e., the fam- ily attempts to satisfy some lifetime fertility goal decided at the start of its "life". The "target fertility" model is the simplest variety of such models.1 Suppose a couple desired to have three surviving children. If the mortality risk were zero, they could accomplish their goal by having three births; if instead they knew that one of every two children would die, then they would need six births. Thus, it appeared straightforward that fertility would be an increasing function of mortal- ity risk. Also, quite obviously, the number of surviving children would be invari- ant to the fraction of children who survive because the number of births is exactly compensatory.2 The target fertility model provides the intuitive basis for the concept of "hoarding," that is, of having more births than otherwise would be optimal if mortality risk were zero. The target fertility model ignores the fact that children are economic goods, that is, that they are costly. A number of authors have introduced a budget constraint into the optimization problem (O'Hare, 1975; Ben-Porath, 1976~. A1- though most formulations included additional decision variables, usually follow- ing the quality-quantity trade-off literature (Becker and Lewis, 1973; Rosenzweig and Wolpin, 1980), the essential features of the mortality-fertility link can be 1 The theoretical notion of a target level of fertility, leading to a positive association between fertility and infant and child mortality, was conceptualized at least as early as 1861 (see the quotation from J.E. Wappaus in Knodel, 1978). 2More formally, if the utility function is U(N) and achieves a maximum at N= N (the "target" fertility level) where N. the number of surviving children, is equal to sB, the (actual) survival rate (s) times the number of births (B), then by maximizing utility and performing the comparative statics, it can be shown that dB/ds = -Bls < 0 (sB is constant).

OCR for page 74
76 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL~Y demonstrated in a simpler framework in which the family maximizes a lifetime utility function only over the number of surviving children and a composite consumption good, subject to a lifetime budget constraint.3 In this model, as in the target model, there is no uncertainty; parents know exactly how many chil- dren will survive for any number of births. In this model it is easily shown that an increase in the survival rate will reduce the number of births, as in the target model, only if fertility has an inelastic demand with respect to its price (i.e. to the cost of bearing and rearing a child).4 Thus, it could be optimal to have fewer births at a positive mortality rate than at a zero mortality rate (the opposite of hoarding).5 This result can arise because births per se are costly. At the higher mortality rate, although the number of surviving children is lower for the same number of births, increasing the number of surviving children by having additional births is costly. Depending on the properties of the utility function, the optimal response may be to reduce births. Therefore, the hoarding-type implication of the target fertility model is not robust to the addition of a resource cost to bearing a child. These models have several shortcomings. First, although the family might be assumed to know the survival risk their children face, they cannot know with certainty the survival fraction (realized survival rate) (i.e., exactly how many children will die for any given number of births). Furthermore, if the number of surviving children is a random variable, these formulations are inconsistent with expected utility analysis unless utility is linear. Second, fertility is clearly dis- crete. The number of children can take only integer values. Third, fertility decision making would seem a priori to be best described as a sequential optimi- zation problem in which one child is born at a time and in which there is, there- fore, time to respond to realized deaths (Ben-Porath, 1976; Knodel, 1978; O'Hara, 1975; Williams, 1977~. Sah (1991) considered the case of an expected utility maximizing family choosing the number of discrete births to have. He showed that if there is no ex ante birth cost (a cost that is incurred regardless of whether or not the child 3If x is a composite consumption good, Y is wealth, and c the fixed cost of bearing a child, the problem is to choose the number of births B that will maximize U(sB, X) subject to Y= cB + X. 4The optimal number of births is found by setting the marginal rate of substitution between the consumption good and the number of surviving births, ullu2, equal to the ``real,, price of a surviving birth, cls. An increase in the survival rate reduces this price. The elasticity of fertility with respect to the survival rate, d in B/d in s, is equal to negative one plus the elasticity of fertility with respect to its cost, -(1 + d in B/d in c). 5However, although the number of births may rise or fall as the survival fraction increases, the number of surviving children must increase. The elasticity of the number of surviving children (sB) with respect to s equals minus the elasticity of births with respect to c. Strictly speaking, the result follows if children are not Giffen goods. The target fertility result will arise only if the elasticity of fertility with respect to c is zero (fertility is perfectly inelastic with respect to c).

OCR for page 74
KENNETH I. WOLPIN 77 survives), then the number of births must be a nonincreasing function of the survival risk (as is true of the previous model). Consider the case in which the choice is between having two, one, or no children. In that case, the difference in expected utilities associated with having one versus no child is the survival risk s times the difference in utilities (i.e., sLU(l) - U(O)~. Similarly, the difference in expected utilities between having two children versus having one child is s2{ [U(2) - U(1~] - [U(1) - U(O)] ~ + sLU(l) - U(O)~. Now suppose that for a given s, it is optimal to have one child but not two, a result that requires satiation at one surviving offspring [U(2) - U(1) < 03. Clearly, at a higher s, it will be optimal to have at least one child. However, at the higher value of s it will still not be optimal to have a second child, and indeed the difference in expected utilities between having two and having one cannot increase. As Sah demonstrates, the argument generalizes beyond a feasible set of two children to any discrete num- ber of children. This result, that increasing the mortality risk of children cannot reduce fertil- ity (except in the neighborhood of certain mortality, s = 0), is the obvious analog to the target fertility result. However, unlike the target fertility model, it does not imply that the number of surviving children will be invariant to the survival rate. The reason is due to the discreteness (and the uncertainty). An example may be helpful. Suppose that U(1) - U(O) = 2 and U(2) - U(1) = -1. Now, assuming s is nonzero, it will always be optimal to have at least one child, sLU(l) - U(O)] = s > 0. However, in this example, for any survival rate less than two-thirds, it will be optimal to have two children. At a survival rate just below two-thirds, the expected number of surviving children is close to 1.33, whereas at a survival rate just above two-thirds, the expected number of surviving children is close to 0.67. There is, thus, a decline in the expected number of surviving children as the survival rate increases in the neighborhood of two-thirds. However, the relation- ship is not monotonic; the higher the survival rate within the zero to two-thirds range, and again within the two-thirds to unity range, the more surviving children there will be on average because the number of births is constant within each range. The example also illustrates hoarding behavior. Because utility is actually lower when there are two surviving children as opposed to one, if the survival rate were unity (zero mortality risk) only one child would be optimal. However, when survival rates are low enough, below two-thirds in the example, the couple will bear two children because there is a significant chance that they will wind up with none who survive to adulthood. Indeed, at survival rates above one-half (but below two-thirds), on average the couple will have more than one surviving birth, exceeding the optimal number of births with certain survival. The key to this result, as will be apparent in the dynamic framework considered below, is that the family's fertility cannot react to actual infant and child deaths. Sah (1991) demonstrates, however, that adding a cost of childbearing, as before, leads to ambiguity in the effect of the survival rate on fertility. He

OCR for page 74
78 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY develops two sets of sufficient conditions for fertility (in the general case of any finite number of children) to decline with the survival rate (for hoarding to be optimal) that depend on properties of the utility function: that the utility function is sufficiently concave (in discrete numbers of children), or that for any degree of concavity the marginal utility of the last optimally chosen birth be nonpositive, that is, that the marginal utility of the last child be nonpositive if all of the optimally chosen children were to survive. Obviously, this second condition will fail to hold if there is no target fertility level, that is, if children always have positive marginal utility. Sah shows that these conditions are weaker than those that would be required if fertility were treated as a continuous choice within the same expected utility framework, and it is in that sense that discreteness reduces ambiguity. Sequential Decision Making In formulating the theoretical linkages between infant and child mortality and fertility, the early contributors to this area of research clearly had in mind sequential decision-making models under uncertainty. No biological or eco- nom~c constraints would force couples to commit to a particular level of fertility that is invariant to actual mortality experience. However, as in other areas of economics, the formalization of such dynamic models of behavior, particularly in the context of estimation, awaited further development.6 To illustrate the infor- mal argumentation of that time, consider the following discussion by Ben-Porath (1976:S164~: Let us distinguish between two types of reaction to child mortality: "hoard- ing" and "replacement." Hoarding would be the response of fertility to expect- ed mortality of offspring; replacement would be the response to experienced mortality.... If children die very young and the mother can have another child, the same life cycle can be approximated by replacement. Where the age profile of deaths is such that replacement can reconstitute the family life cycle, replace- ment is superior to hoarding as a reaction, since the latter involves deviations from what would be the optimum family life cycle in the absence of mortality. If preferences are such that people have a rigid target of a minimum number of survivors at a given phase in the life cycle, hoarding involves a large number of births and the existence of more children than necessary who have to be sup- ported in other phases of the life cycle. . . The superiority of replacement is clear, but of course it is not always possi- ble. The risks of mortality are often quite significant beyond infancy. Parents 6Not all researchers believed that it was necessary to specify the optimization problem formally, however. In describing fertility strategy, Preston (1978:10) states, "These are obviously simplifica- tions of what could be exceedingly complex 'inventory control' problems. But it is probably reason- able to apply no more sophisticated reasoning to the problem than parents themselves would."

OCR for page 74
KENNETH I. WOLPIN may be afraid of a possible loss of fecundity or some health hazard that will make late replacement impossible or undesirable. The reaction to mortality which is expected to come at a late phase of either the children's or the parents' life cycle may be partly in the form of hoarding. 79 It is obvious from Ben-Porath's remarks that he viewed the replacement decision as a sequential process made in an environment of uncertain mortality and that the hoarding of births is a form of insurance that depends on forward- looking behavior. Furthermore, Ben-Porath postulated that the essential features of the environment that lead to hoarding behavior as an optimal response are those that make replacement impossible, namely that children may die beyond the period of infancy and that the fertile period is finite (and possibly uncertain). Although several sequential decision-making models of fertility are discussed in the literature, which include nonnegligible infant mortality risk (Wolpin, 1984; Sah, 1991; Mira, 1995), none explicitly model sequential fertility behavior when mortality past infancy is significant (probably because of its intractability in a many-period setting). However the essential behavioral implications of sequen- tial decision making and the intuition for them can be demonstrated in a sequen- tial decision-making model with only three periods. Moreover, a three-period formulation is sufficient to illustrate and operationalize replacement and hoard- ing concepts. Suppose that births are biologically feasible in the first two periods of a family's life cycle, but that the woman is infertile in the third (Figures 3-1 and 3- 2 provide a graphical representation of the structure of the model). Each off- spring may die in either of the first two periods of life, as an infant or as a child, with probabilities given hype (the infant mortality rate) andp2 (the child mortal- ity rate, conditional on first-period survival). Within periods, deaths occur subse- quent to the decision about births. Thus, an offspring born in the first period of the family's life cycle may die in its infancy (its first period of life) before the second-period fertility decision is made. However, that same offspring, having survived infancy, may instead die in its childhood (its second period of life) after the second-period fertility decision is made. Such a death cannot be replaced by a birth in the third period because the woman will be infertile. For the same reason, a birth in the second period is not replaceable even if its death occurs in that period (as an infant). It is assumed for simplicity that the survival probability to the adult period of life, conditional on surviving the first two nonadult periods, is unity. The family is assumed, for ease of exposition, to derive utility from only those offspring who survive to adulthood.7 This corresponds to the notion that children are investment goods as in the old age security hypothesis (e.g., Willis, 1980) that offspring provide benefits only as adults. lone can think of the third period of the couple's life as longer than a single period of life so that a birth in the second period that survives its infancy (in the couple's second period) and its childhood (the couplets third period) (i.e., d2O = d3 = 0) will survive to adulthood while the couple is still alive. =

OCR for page 74
80 PEFI10D 1 _~ FIGURE 3-1 Decision tree: Period 1. THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY No (surYiYir~g) children One sunriv~ng young cold Formally, let nj = 1 indicate a birth at the beginning of period j = 1, 2 of the family's life cycle and zero otherwise. Likewise, let djk= 1 indicate the death of an offspring of age k, k = 0, 1 at the beginning of period j, zero otherwise, given that a birth occurred at the beginning of period j - k. By convention, an infant is age O (in its first period of life) and a child is age 1 (in its second period of life). Thus, letting Nj_~ be the number of surviving offspring at the beginning of period i = 1, 2, 3, 4 of the family's life cycle, No = 0, N~=n~(1-d~)=M~, N2 = MY (1 - do) + no (1 - d O2 ) = MY + MOO, = MY + M3. No = MY +MO2(1-d3) J ~ (1) where Mjk = {0,1 } indicates the existence of an offspring of age k = 0, 1, 2 at the end of periodj (and beginning of j + 1~. Further, let c be the fixed exogenous cost of a birth and Y the income per period. Finally, utility in period 1 is just period 1

OCR for page 74
KENNETH I. WOLPIN 8 PERIOD 2 PERISH:) 3TEERMINAL UTILITY surviving_ ~ U(O) ~ children No (surviving) _~ Q /' children ~OneU(O) 7 ~SU~I~n9 ~ r young child ~ ~U(1) One ,' survhring ,/ child U(1) ~ No / surviving - - -A U(O) / children ~One surviving /`-~)i~ older and oneUps) One sing ~Ail/ surviving ~4 V young child OF < ~younger child ~< to U(2) surviving \ A, older child \ Surviving ~ U(O) \ younger child .,c; \ -it Urn) No saving children ~ U(03 FIGURE 3-2 Decision tree: Periods 2 and 3. consumption, Y- con, utility in period 2 is that period's consumption, Y- cn2, and period 3 utility is consumption in that period plus the utility from the number of surviving children in that period, Y + U(N3).8 Lifetime utility is not discounted and income is normalized to zero for convenienced tion. 8Note that the couple does not care about the age distribution of children. none can normalize income at zero without loss of generality because utility is linear in consump

OCR for page 74
82 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY Because the decision horizon is finite, the problem of optimally choosing a sequence of births so as to maximize lifetime utility is most easily solved back- wards. Define Vie (Nt_~) to be the expected lifetime utility at time t if fertility decision no = 1 or O is made at the beginning of period t, given that there are Nt_i surviving offspring at the end of period t - 1. Furthermore, define Vt(Nt_~) = maxi Vat (Nt_i, V (Nt_~] to be the maximal expected lifetime utility at period t for given surviving offspring at the end of period t - 1. Because no decision is made at the beginning of period 3, consider the lifetime expected utility functions at period 2 conditional on the number of surviving children, namely V2O (0) = = V2O (1) = V21 (1) = U(O) ~ (l-pl) (l-P2)U(l)+[(l-pl)p2 +pl]U(0)-c, (1 - P2)U(1) + P2U() ~ (1-P2)2(1- pl)u(2)+(1-P2) [2P2(1-P1) +pl]U(l) +P2[P2(l-Pl) +pl]U(O)- c. (2) For either of the two states, Ni = 0 or 1, the decision of whether or not to have a birth is based on a comparison of the expected lifetime utilities of the two alternatives. If the family has no surviving offspring at the beginning of period 2, either because there was no child born in period 1 or because the infant did not survive to period 2, then from equation (2), the family will choose to have a birth in period 2 if and only if Vim) > Vale), or (1 -pi -p2~LU(l) - U(O)] - c > 0. If there is a surviving offspring at the beginning of period 2, then the condition for choosing to have a birth is that V2 (1) > V2 (1) or (1 - pi - p2) ~ (1 - p2~LU(2) - U(1~] + p2LU(l) - U(O)] ~ - c > 0. It is easily seen from these expressions that as long as the utility function exhibits diminishing marginal utility in the discrete stock of surviving offspring, that is, U(2) - U(1) < U(1) - U(O), then for all values of pi and P2, the difference between expected lifetime utilities associated with having and not having a birth in period 2 is greater when there is no surviving offspring at the beginning of period 2 (Ni = 0) than when there is a surviving offspring (Ni = 1) (i.e., the gain to have a birth in period 2 is larger if an offspring born in period 1 dies as an infant than if it survives to period 2~. The extent to which the gain from a birth in period 2 is increased by the death of an infant born inperiodl [V2~0~- V2~0~-EV2~1~- V2~1~]isequalto(1-p2~2~1-p~LU(l) - U(O)] - [U(2) - U(1~] ~ . This gain is clearly larger the more rapid the decline in the marginal utility of surviving offspring and the smaller the age-specific mortality probabilities. It is this gain that represents the motivation for "replace- ment" behavior. To isolate the effect of the infant mortality risk on second-period fertility, suppose that the child mortality probability P2 is zero. In this case, the birth

OCR for page 74
KENNETH I. WOLPIN 83 decision in period 2 is governed by the sign of (1 -pl)[U(2] - U(1)] - c if there is a surviving first-period birth and by the sign of (1 -pl)[U(l ) - U(O)] - c if there is not. Clearly, the family would not have a second birth as insurance against the child death of the firstborn (i.e., there would be no hoarding because such a death, given the survival of infancy, is impossible by assumption). However, the ab- sence of such a hoarding motive does not imply that there is no effect of mortality risk on fertility. An increase in infant mortality risk P1 has two effects on fertility. First, because an offspring born in period 1 is more likely to die during infancy, the family is more likely to enter the second period without a surviving offspring (N1 = 0). In this case, according to the previous analysis, the gain from a birth in period 2 would be larger. Second, the value of having a second-period birth is lower in the new mortality environment regardless of the existing stock of chil- dren (assuming nonsatiation). The effect of a (unit) change in the infant mortality probability on the gain from having a second-period birth is - [U(N1 + 1) - U(N1)]. For expositional purposes, call this the "direct" effect of mortality risk. If at an initial level of P1 it were optimal for the household to have a second birth even if the first survived infancy, (1 -pl)[U(2) - U(1)] - c > 0, then increasing infant mortality risk sufficiently would make it optimal to have a second birth only if the first died in infancy. Further increases in the infant mortality rate would eventually lead to optimally having zero births (at some level of P1 (1 - pl)[U(l) - U(O)] - c < 0). If having a second surviving child reduces utility (satiation), then a second birth would only be optimal if the first died during Infancy. To illustrate the effect on second-period fertility of increasing the probability of death in the second period of life, assume that the increase occurs from an initial state in which there is no mortality risk in either period of life, P1 = P2 = 0 It is useful to contrast that effect relative to the effect of increasing the first-period mortality risk from the same state. Furthermore, assume that in the zero mortality environment it is optimal to have only one surviving child (i.e., [U(2) - U(1)] - c 0). Then, taking derivatives of the relevant expres- sions in equation (2) evaluated at zero mortality risk yields dl V2 (1) - V2O (1) 1 1 dp1 ~ PI =P2 = 0 dl V2 (1) - V2O (1) 1 1 = -tU(2) - U(l~l, dp2 IPI=P2=0 = -1 U(2 ) - U(1 ) 1 + ~ 1 U(1 ) - U(O ) -tU(2)-U(l)l). (3)

OCR for page 74
84 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY The effect of a change in the "infant" mortality rate is, as previously derived, the direct effect, which is negative if there is no satiation at one surviving offspring. The effect of a change in child mortality risk is the negative direct effect plus an additional non-negative term whose magnitude depends on the degree of concav- ity of the utility function. As with Sah's result, this positive offset arises because survival of the first offspring to adulthood is now uncertain and the decision about the second birth must be made before that realization. The hoarding effect generalizes to any levels of mortality risk in the sense that concavity is a neces- sary condition for its existence. Both replacement and hoarding behavior depend on the curvature of the utility function. The analysis of the second-period decision, taking the first-period birth deci- sion as given, does not provide a complete picture of the effect of infant and child mortality risk on the family's fertility profile. To see how the decision to have a birth in the first period varies with mortality risk, it is necessary to consider the relevant expected lifetime utilities in period 1, namely, V = max ~V2O (N~ = 0), V2 (N~ = 0) ~ = V2 (0), Vet = (~1 - pi ~ max ~V2O (N~ = 1), V2 (N~ = 1) ~ + pi max ~V2O (N~ = 0), V2 (N~ = 0) ~ - c, = (1 - p~)V2~1) + p~V2~0) - c. (4) The value (expected lifetime utility) of forgoing a first-period birth is simply the maximum of the values attached to entering period 2 without a surviving off- spring. The value attached to having a first-period birth depends on the probabil- ity that the infant will survive. If the offspring survives infancy, the family receives the maximum of the values associated with entering the second period with an offspring and choosing either to have or not to have a birth in that period [see equation (2~. If the offspring does not survive, the family receives the maximum of the values associated with entering the second period without a surviving offspring [see equation (2~. The couple has a birth in period 1 if V, > V. To characterize the decision rules in period 1, consider the types of behavior that would be optimal in period 2 under each of the two regimes, having or not having a surviving offspring at the beginning of period 2. narios to consider: There are three sce (1) It is optimal to have a birth in period 2 regardless of the value of Nil V2 (0) > Veto), Vow) > V2~1~. (2) It is optimal to have a birth in period 2 only if Nit = 0 (i.e., if there are no surviving offspring ~ V2 (0) > V2 (0), V2 (1) < V2 (1~.

OCR for page 74
KENNETH I. WOLPIN = DMCI = E(L I n1 = 1, d1 = id, I tg(tlul=l,dl=l) t=1 LG(L*Inl=l,dl=l) 101 1,L OCR for page 74
102 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY TABLE 3-2 Parity Progression Ratios and Mean Closed Intervals Based on Nineteenth Century Bavarian Village Data Mean Closed Intervals (months) Outcome Mommlingen Schonberg Anhausen All birth intervals No infant deaths 30.0 22.0 19.9 One or more infant deaths 19.4 20.0 19.2 Second to third child First child survives 29.0 23.9 23.4 First child dies 25.4 21.1 17.6 Parity Progression Ratios (percent) Mommlingen Schonberg Anhausen Second to third child No infant deaths 96.3 97.4 84.0 One or more infant deaths 100.0 97.6 90.5 Third to fourth child No infant deaths 93.0 93.1 81.3 One infant death 94.5 87.9 92.9 Two or more infant deaths 85.7 76.5 90.0 SOURCE: Knodel (1978). Table 3-3, for all levels of completed family size and regardless of the birth order of the infant death, retrospectively obtained mean closed intervals are about one year less when an infant death is experienced. PPRs differ by about 16 percent- age points for the movement between first and second births when the firstborn did or did not die, by approximately the same amount for the movement between second and third births given that the secondborn did or did not die, and by about 10 percentage points for higher parities. As was the case for the German histori- cal data, the later French data reveal similarly higher fertility subsequent to an infant death. Numerous other studies report mean closed intervals and PPRs by mortality experience. Most use cross-sectional data where birth and death infor- mation is collected retrospectively. Some report estimates based on regressions that hold individual characteristics constant (e.g., Ben-Porath) and in that sense are not completely nonparametric. The general findings in the literature are qualitatively the same as for the two papers discussed above, namely that the evidence is consistent with the existence of replacement behavior.

OCR for page 74
KENNETH I. WOLPIN Parity TABLE 3-3 Parity Progression Ratios and Mean Closed Intervals Based on 1962 French Survey of Family Structure Mean Closed Intervals No Infant Death Infant Death Total fertility Two Three 4.14 3.17 First birth 3.35 2.43 Second birth 4.19 3.39 Four First birth 2.77 2.16 Second birth 3.43 2.48 Third birth 4.06 3.26 Parity Progression Ratios (n to n + 1) No Infant Death Infant Death (birth n) (birth n) First to second child 68.5 84.7 Second to third child 57.6 72.8 Third to fourth child 56.4 67.4 Fourth to fifth child 57.3 67.9 Fifth to sixth child 59.2 69.2 SOURCE: Vallin and Lery (1978). Estimates Based on Total Births and Deaths 103 Tables 3-4 and 3-5 report estimates of replacement effects based on the use of total births and total deaths. Table 3-4 shows replacement effects obtained by Olsen and Table 3-5 those by Mauskopf and Wallace. Olsen uses data from the 1973 Columbia Census Public Use Sample and reports his results for different age and residential location groups. Only the oldest age group, women who were age 45-49 in 1973, are shown. The uncorrected estimates, that is, the regression coefficient on total deaths, imply a replacement rate of over one for both urban and rural women, regardless of whether controls are added. The corrected esti- mate that assumes a homogeneous mortality rate in the population is negative, implying that there are actually fewer births when there is an infant or child death. This result is consistent with the negative "direct" effect of higher infant mortality. The replacement effect obtained under the assumption that the mortal- ity rate varies in the population (independently from births) yields point estimates of around 0.2.25 Olsen also estimates a replacement effect when the mortality 25The independence assumption is inconsistent with optimizing behavior.

OCR for page 74
104 o a' Cam o . . Cut s~ o Cq Cq a' bC a' Cq o E~ a' a' a' C) a' 1 E~ s~ C) s~ o V C) s~ V ;^ C 4= o Cd s~ o ;^ 4= ~ s~ ~ X o Cd ;^ o Cd o ;^ ~ X o ~ C) s~ s~ o C) ca o ca ca o oo .. OlOl 4= C) s~ C) ~ oo ~o .. oo oo 11 .. 1 o ca .0 C) ~ ~ ~ Ct U:, ~ ;2 ~ 4= ~ o oo ca ~ .g ~ ~ o o s~ . . v o v,

OCR for page 74
KENNETH I. WOLPIN TABLE 3-5 Replacement Effects from Total Births Regressors: Mauss- hopf and Wallace Method Years of Education Mortality Rate All Women None 1-4 5+ Fixed 0.601 0.348 0.592 0.964 (0.03)a (0.04) (0.06) (0.13) Random 0.593 0.437 0.613 0.978 (0.04) (0.04) (0.05) (0.08) aStandard errors in parentheses. SOURCE: Mausshopf and Wallace (1984). 105 rate is correlated with births. Those estimates vary between 0.13 and 0.22 de- pending on the joint distributional assumption for the mortality rate and total fertility.26 The estimates based on the method developed by Mauskopf and Wallace are presented in Table 3-5. Mauskopf and Wallace use data from the 1970 Brazilian census, restricting attention to women who were between 40 and 50 years old at the time of the survey. The replacement rate, assuming the mortality risk to be fixed in the population, was estimated to be 0.6 for the total sample. It was 0.35 for those with zero schooling, 0.6 for women with 1-4 years of schooling, and almost unity for women with 5 or more years of schooling. Allowing the mortal- ity rate to differ in the population, using the method described above, only changed the estimate significantly for the lowest education group. Approximate Decision Rules Mroz and Weir (1989) developed a discrete-time statistical representation of the timing of births that can be viewed as an approximation to the decision rules that arise from a dynamic sequential utility maximizing model. Three stochastic processes are specified as (1) the process generating the probability of resuming ovulation after a birth, (2) the process generating the probability of conception, 26Olsen (1983) adds an estimate of innate mortality risk to the regression of total births and total deaths in combination with his correction method as an attempt to separate replacement and hoarding behavior. However, the effect of early age mortality risk on fertility cannot be called a hoarding response, as hoarding would not exist in an environment without significant mortality risk among older children. Controlling for innate infant frailty, however, would provide an estimate of the replacement rate that is uncontaminated by unobserved mortality risk. Olsen estimates a replace- ment rate of 0.17 using this method.

OCR for page 74
106 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY and (3) the process generating the onset of secondary sterility. The waiting time to a birth is the convolution of the waiting time to the resumption of ovulation and the waiting time to a conception, conditional on the resumption of ovulation and conditional on not becoming infecund. The probability of observing a woman with a particular sequence of births up to any given age is specified in terms of these three stochastic processes. Mroz and Weir allow for unobserved heteroge- neity in each of the three waiting times; women may differ biologically in the postanovulatory and fecund processes, and they may differ biologically and be- haviorally in the conception process. However, there is neither observed nor unobserved heterogeneity in mortality risk (cross sectionally or temporally). Monthly probabilities are modeled as logistic functions. The fecund hazard at any month depends on duration since the start of the interval, age, age at marriage, parity attained by that month (dummy variables for each attained par- ity), dummy variables for whether the particular month is the first month of risk of conception in the interval, a dummy for the first month of marriage, and the number of surviving children during that month. Heterogeneity shifts the monthly probability proportionately and is assumed to take on a small number of discrete values (Heckman and Singer, 1984~. Identification in this model is achieved by a combination of functional form assumptions, assumptions about biological processes (for example, exactly 9 months gestation) and a clever use of data (using the timing of an infant death to tie down the beginning of the fecund period given the cessation of breastfeeding). The reader is referred to their paper for the exact details. The model is estimated using reconstituted data between 1740 and 1819 from 39 French villages based on birth and death histories for women who were married at age 20-24. The results provide evidence on the importance of unob- served heterogeneity (in the fertility process) in the estimation of replacement effects. Mroz and Weir report that simulations conducted prior to estimation, omitting controls for unobserved heterogeneity in the fecund hazard rate and recognizing that they accounted for the cessation of lactation due to an infant death, resulted in the probability of a birth increasing in the number of surviving children (conditional on parity, age, duration, and age at marriage). Controlling for heterogeneity in estimation, however, resulted in a negative effect as is con- sistent with a behavioral replacement effect. Quantitatively, Mroz and Weir found that births increase by 13 percent due to the cessation of lactation alone following an infant' s death and by 17 percent overall. Given an average of about seven births, the absolute behavioral replacement effect is 0.28. Mroz and Weir essentially assume that mortality risk does not vary in the population (given covariates).

OCR for page 74
KENNETH I. WOLPIN The Impact of Infant and Child Mortality Risk on Fertility Structural Estimation 107 Wolpin (1984) illustrates structural estimation. The model has the following characteristics: (1) Per-period utility is quadratic in the number of surviving children in that period and in a composite consumption good, (2) fertility control is costless and perfect, (3) there is a fixed cost of bearing a child and a cost of maintaining a child in its first period of life (if it survives infancy), (4) children can die in only their first period of life subject to an exogenous time-varying (and perfectly forecasted) infant mortality rate, (5) the household has stochastic in- come and consumption net of the cost of children that is equal to income in each period, and (6) the household's marginal utility of surviving children varies sto- chastically over time according to a known (to the household) probability distri- bution. Given this framework, the household chooses in each period whether or not to have a child.27 For the purpose of estimation, Wolpin assumes that the time-varying prefer- ence parameter is drawn independently over both time and across households from a normal distribution. The mortality rate faced by the household is assumed known to the researcher, measured by the state-level mortality rate in each pe- riod, and the researcher is assumed to forecast future mortality rates exactly as the household is assumed to do, namely based on the extrapolated trend in the mor- tality rate at the state level. Future income is forecasted from the time series of observed household income, again under the assumption that the household uses the same forecasting method. The data are drawn from the 1976 Malaysian Family Life Survey that con- tains a retrospective life history on marriages, births, child deaths, household income, etc., of each woman in the sample. Wolpin used a subsample of 188 Malay women who were over age 30 in 1976, currently married, and married only once. The period length was chosen to be 18 months, the initial period was set at age 15 (or age of marriage if it occurred first), and the final decision period was assumed to terminate at age 45. Thus, there were 20 decision periods. In the implementation, the cost of a birth is allowed to be age varying as a way of capturing age variation in fecundity and in marriage rates. In addition, the woman's schooling is allowed to affect the marginal utility of surviving children. Parameter estimates are obtained by maximum likelihood. As already al- luded to, the procedure involves solving the dynamic programming problem for each household (given their income and mortality risk profiles) and calculating 27Mira (1995) recently extended that model to the case in which families learn about the innate mortality risk they face through their realized mortality experience.

OCR for page 74
108 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTIL[lY the probability of the observed birth sequence. Because the woman's fertility is observed from what is assumed to be an exogenous initial decision period (either age 15 or age at marriage, whichever occurs first), the likelihood function is conditioned on the initial zero stock of children, which is the same for all women. The birth probability sequences that form the likelihood function can be written as products of single-period birth probabilities conditional on that period's stock of surviving children, the output of the dynamic programming solution. Given the parameter estimates, the replacement effect is calculated in each period and for each number of surviving children for a representative couple. The replacement effect is estimated to be small, ranging between 0.01 and 0.015 additional children ever born per additional infant death. The reason for the negligible replacement effect is that the actual fertility behavior is best fit in the context of this optimization model with utility parameters that imply essentially a constant marginal utility of children. Wolpin also calculated that an increase in the infant mortality risk by 0.05 would lead to a reduction in the number of births by about 25 percent. (Note that this effect includes the potential replacement of the increased number of infant deaths, which is in this case negligible given the very small estimated replacement effects.) Thus, the impact of a policy that altered infant mortality risk would depend quite heavily on how quickly that policy change was perceived to have been effective.28 Nonstructural Estimation A number of studies have attempted to estimate the effect of mortality risk on fertility using nonstructural estimation methods. As already discussed, ob- taining correct estimates is particularly challenging if there is unobserved mortal- ity risk variation, and more so when mortality risk is endogenous, as when fertil- ity spacing affects mortality risk as discussed above. Mortality risk can also be endogenous if it is affected by behaviors that are subject to choice and if, in addition, there is population heterogeneity in preferences. Several studies, hav- ing recognized this problem, have attempted to estimate the effect of innate family-specific mortality risk on fertility. To do so requires that one estimate the production function for child survival, accounting for all behavioral and biologi- cal determinants.29 Although the credibility of the estimates of the fertility- frailty relationship depends in part on the way frailty estimates are obtained, let 28Interestingly, direct evidence about hoarding comes from my 1984 study, although ~ failed to recognize it at the time. Given the finding there that the marginal utility of surviving children is essentially constant, which led to the negligible estimated replacement rates, the potential hoarding response, if child mortality were significant in that environment, would also be negligible since hoarding also depends on concavity as shown in equation (3). 29See Wolpin (in press) for a discussion of the methods used to estimate the survival technology as well as empirical findings.

OCR for page 74
KENNETH I. WOLPIN 109 us consider the findings of studies that estimate its effect on fertility behavior assuming the frailty estimates to be credible. Rosenzweig and Schultz (1983), using data from the 1967, 1968, and 1969 National Natality Followback Surveys (U.S. Department of Health, Education and Welfare), find that the expected number of children ever born per woman would be 0.17 greater for an infant mortality risk of 0.1 as opposed to zero. Given that in their sample the infant mortality rate is less than 3 percent, this experiment may be within sample variation. At the sample average of 2.5 births per woman, an additional 0.25 deaths per woman leads to 0.17 more births and therefore to 0.08 fewer surviving children. Such a finding, it should be noted, must arise from replacement behavior to be consistent with the dynamic model presented above; in that model an increase in infant mortality risk cannot increase births for the same number of infant deaths. Olsen and Wolpin (1983), also using the 1976 Malaysian Family Life Sur- vey, estimate that a couple faced with a 1 percent higher monthly probability of death within the first 24 months of life will have their first birth approximately 2 weeks earlier. This effect is rather small given that the average interval between births is 30 months. Although seemingly inconsistent with the simple three- period model, that model is not rich enough to capture more complicated behav- iors that might explain this result. For example, it is possible that greater mortal- ity risk induces an earlier first birth so as to increase the time over which to respond to actual mortality. CONCLUSIONS The original question posed in the introduction to this chapter was intended to focus attention on the micro foundations of fertility behavior as a necessary prerequisite to informed population policies. It is fair to ask whether after several decades of empirical research we can confidently report to policy makers the quantitative estimates of the effects of changing infant and child mortality risk on fertility at the individual level. The answer, in my view, is unfortunately no. That assessment does not rest simply on the fact that estimates vary widely or that the empirical approaches are methodologically flawed. Rather it rests more fundamentally on the fact that we do not have a deep enough understanding of behavior to know how to generalize our results beyond the setting within which we obtain estimates. To ultimately accomplish that goal requires that we estab- lish tighter links between theory (behavioral decision rules) and empirical meth- ods (what is estimated). ACKNOWLEDGMENTS Support from National Science Foundation grant SES-9109607 is gratefully acknowledged. This chapter is in part a summarization and condensation of the

OCR for page 74
110 THE IMPACT OF INFANT AND CHILD MORTALITY RISK ONFERTILIIY paper "Determinants and Consequences of the Mortality and Health of Infants and Children," which will appear in the Handbook of Population and Family Economics, M. Rosenzweig and 0. Stark, eds. I have received useful comments from Barney Cohen, Mark Montgomery, and several anonymous reviewers. REFERENCES Becker, G.G., and H.G. Lewis 1973 On the interaction between the quantity and quality of children. Journal of Political Economy 81:S279-S288. Ben-Porath, Y. 1976 Fertility response to child mortality: Micro data from Israel. Journal of Political Economy 84(2):S163-S178. Heckman, J.J. 1982 Statistical models for discrete panel data. Pp 114-178 in C. Manski and D. McFadden, eds., Structural Analysis of Discrete Data with Econometric Applications. Cambridge, Mass.: MIT Press. Heckman, J.J., and B. Singer 1984 A method for minimizing the distributional assumptions in econometric models of dura- tion data. Econometrica 52(2):271-320. Knodel, J. 1978 European populations in the past: Family-level relations. Pp. 21-45 in S.H. Preston, ea., The Effects of Infant and Child Mortality on Fertility. New York: Academic Press. Mauskopf, J., and T.D. Wallace 1984 Fertility and replacement: Some alternative stochastic models and results for Brazil. Demography 21(4):519-536. Mira, P. 1995 Uncertain Child Mortality, Learning, and Life Cycle Fertility. Unpublished Ph.D. disser- tation, University of Minnesota, Minneapolis. Mroz, T.A., and D. Weir 1989 Structural change in life cycle fertility during the fertility transition: France before and after the revolution of 1789. Population Studies 44:61-87. O'Hara, D.J. 1975 Microeconomic aspects of the demographic transition. Journal of Political Economy 83: 1203-1216. Olsen, R.J. 1980 Estimating the effect of child mortality on the number of births. Demography 17(4):429- 443. 1983 Mortality rates, mortality events, and the number of births. American Economic Review 73:29-32. Olsen, R.J., and K.I. Wolpin 1983 The impact of exogenous child mortality on fertility: A waiting time regression with exogenous regressors. Econometrica 51(3):731-749. Preston, S.H. 1978 Introduction. Pp 1-18 in S.H. Preston, ea., The Effects of Infant and Child Mortality on Fertility. New York: Academic Press. Rosenzweig, M.R., and T.P. Schultz 1983 Consumer demand and household production: The relationship between fertility and child mortality. American Economic Review 73:38-42.

OCR for page 74
KENNETH I. WOLPIN 111 Rosenzweig, M.R., and K.I. Wolpin 1980 Testing the quantity-quality model of fertility: Results from a natural experiment using twins. Econometrica 48:227-240. Sah, R.K. 1991 The effects of child mortality changes on fertility choice and parental welfare. Journal of Political Economy 99(3):582-606. Vallin, J., and A. Lery 1978 Estimating the increase in fertility consecutive to the death of a young child. Pp 69-90 in S.H. Preston, ea., The Effect of Infant and Child Mortality on Fertility. New York: Academic Press. Williams, A.D. 1977 Measuring the impact of child mortality on fertility: A methodological note. Demogra- phy 14(4):581-590. In Willis, R.J. 1980 The old age security hypothesis and population growth. Pp. 43-68 in T. Burch, ea., Demographic Behavior: Interdisciplinary Perspectives on Decision Making. Boulder, Colo.: Westview Press. Wolpin, K.I. 1984 An estimable dynamic stochastic model of fertility and child mortality. Journal of Politi- cal Economy 92(5):852-874. Determinants and consequences of the mortality and health of infants and children. In M. press Rosenzweig and O. Stark, eds., Handbook of Population and Family Economics. Amsterdam, Netherlands: North-Holland.