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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

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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).

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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).

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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

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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."

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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.
=

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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

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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

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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

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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)

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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~.

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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

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.

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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.

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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.

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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).

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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.

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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.

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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

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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.
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