Wolpin s Model of Fertility Responses to Infant/Child Mortality Economics 623 J.R.Walker March 20, 2012 Suppose that births are biological feasible in the first two periods of a family s life cycle, but that the woman is infertile in the third period. Each offspring may die in either of the first two periods of life, as an infant or as a child, with probabilities given by p 1 (infant mortality) and p 2 (child mortality). Within periods, deaths occur subsequent to to decisions 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. Such a death can not be replaced by a birth in the third period, because the woman is infertile. For the same reason, a birth in period 2 is not replaceable even if its death occurs in that period (infant). 1 It is assumed that conditional on surviving the first two periods non adult periods, is unity. The family is assumed to derive utility only from those offspring who survive to adulthood. 2 Let n j = 1 indicate a birth at the beginning of period j = 1, 2 of the family s life cycle and zero otherwise. Likewise, let d k j = 1 indicate a 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 zero (in its first period of life), and a child of age one (in its second period of life). Thus, letting N j be the number of surviving offspring at the beginning of period j = 1, 2, 3 of the family s life cycle. where Mj k j. N 0 =0, N 1 =n 1 (1 d 0 1) = M 0 1, N 2 =M 0 1 (1 d 1 2) + n 2 (1 d 0 2) = M 1 2 + M 0 2, N 3 =M 1 2 + M 0 2 (1 d 1 3) = M 2 3 + M 1 3. = {0, 1} indicates the existence of an offspring of age k = 0, 1, 2 at the end of period Further let c be the fixed exogneous cost of a birth and Y income per period. Finally, utility in period 1 is just period 1 consumption, Y cn 1, utility in period 2 is that period s consumption, Y cn 2, and period 3 utility is consumption in that period plus the utility from the number of surviving children in the period, Y + U(N 3 ). Lifetime utility is not discounted and income is normalized to zero for convenience. 1 Moreover, a birth in the second period must survive child mortality in period 3. 2 One 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 infancy (in the couple s second period) and its childhood (the couple s third period), i.e., d 0 2, d 1 3 = 0, will survive to adulthood while the couple is still alive. 1
Solve by backward recursion. Let Vt n (N t 1 ) be the expected lifetime utility at time t if fertility is n t = 0 or n t = 1 is made at the beginning of period t, given that there are N t 1 surviving offspring at the end of period t 1. Further, define V t (N t 1 ) = max [ Vt 1 (N t 1 ), Vt 0 (N t 1 ) ] to be 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 lifetime expected utility functions at period 2 conditional on the number of surviving children: V 0 2 (0) =U(0) V 1 2 (0) =(1 p 1 )(1 p 2 )U(1) + [(1 p 1 )p 2 + p 1 ]U(1) c V 0 2 (1) =(1 p 2 )U(1) + p 2 U(0) V 1 2 (1) =(1 p 1 )(1 p 2 ) 2 U(2) + (1 p 2 )[2p 2 (1 p 1 ) + p 1 ]U(1) + p 2 [p 2 (1 p 1 ) + p 1 )U(0) c For each state (number of children at the beginning of the period) the couple will decide to birth in period 2 if For s = 0, V 2 (s) equals V 2 (s) = V 1 2 (s) V 0 2 (s) 0, s = 0, 1 (1 p 1 )(1 p 2 )[U(1) U(0)] c 0 c U 1 (1 p 1 )(1 p 2 ) U 1 is the marginal utility of the first child. The first order condition states that the incremental benefit must outweigh the effective cost c/(1 p 1 )(1 p 2 ). The probability that the child will live to adulthood is (1 p 1 )((1 p 2 ). 3 c/(1 p 1 )(1 p 2 ) represents the effective cost of one surviving child. A couple will have a birth in period 2 with a child surviving from the first period (s = 1) if V 2 (s) = V2 1(1) V 2 0 (1) 0 hence (1 p1)(1 p 2 )(1 p 2 )[U(2) U(1)] + p 2 [U(1) U(0)] c 0 The extent to which the gain from a birth in period 2 is increased by the death of an infant born in period 1, equals [ V 1 2 (0) V 0 2 (0) ] [ V 1 2 (1) V 0 2 (1) ] (1 p 2 ) 2 (1 p 1 ) {[U(1) U(0)] [U(2) U(1)]} 3 This may be confusing as a child born in period 2 must survive period 2 (as an infant) and period 3 as a young adult. Utility accrues at the end of period 3. 2
This gain is clearly larger the more rapid is the decline in the marginal utility of surviving offspring and the smaller are the age specific mortality probabilities. Following Wolpin s discussion we can explore the effects of mortality by setting p 2 = 0 and consider only infant mortality. The point is that mortality influences decision making (as it should) even if there is only infant mortality. In this case, the birth decision in period 2 is governed by the sign of (1 p 1 ) U 2 c if there are survivng first period birth and by the sign of (1 p 1 ) U 1 c if there is not. There is no hoarding (p 2 = 0). Yet mortality risk still affects fertility. An increase in p 1 has two effects on fertility. First because an offspring born in period 1 may die during infancy, the family may enter the second period without a surviving offspring (N 1 = 0). In this case, as we have seen 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 children (assuming non satiation). The effect of a (unit) change in the infant mortality probability on the gain from having a second period birth is [U(N 1 + 1) U(N 1 )]. Call this the direct effect of mortality risk. If at an initial level p 1 it were optimal for the household to have a second birth even if the first survived infancy, (1 p 1 )(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 birth died in infancy. Future increases in the infant mortality rate would eventually lead to optimally have zero births (at some level of p 1, (1 p 1 )(U(1) U(1)) c < 0). To illustrate most clearly 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, p 1 = p 2 = 0. Contrast this effect relative to the risk of increasing p 1 from the same state. Further assume that in the zero mortality risk environment it is optimal to have only one surviving child, i.e., [U(2) U(1)] c < 0, [U(1) U(0)] c > 0. Taking derivatives d[v2 1(1) V 2 0(1)] p1 =p dp 2 =0 = [U(2) U(1)] 1 d[v2 1(1) V 2 0(1)] p=1=p2 =0 = [U(2) U(1)] + [(U(1) U(0)) (U(2) U(1))] dp 2 The effect of a change in the infant mortality rate is, as previously derived, the direct effect, which, as noted is negative if there is no satiation at one surviving offspring. The effect of a change in the child mortality rate is the negative direct effect plus an additional non negative term whose mangitude depends on the degress of concavity of the utility function. The positive offset arises because survival of the first offspring to adulthood is now uncertain and the decision about the second birth must be made prior to that realization. This hoarding effect generalizes to any level 3
of mortality risk in the sense that concavity of the utility function is a necessary condition for its existence. Now consider the decision in the first period. We have to consider only s = 0 as it is the only feasible state at the start of the reproductive process. Moreover, remember Wolpin s assumption that utility accrues only in period 3 (after the decision process). Implicit as well is the normalization of income, Y = 0. V1 0 = max [ V2 0 (0), V2 1 (0) ] = V 2 (0). V1 1 =(1 p 1 ) max [ V2 0 (1), V2 1 (1) ] + p 1 max [ V2 0 (0), V2 1 (0) ] c =(1 p 1 )V 2 (1) + p 1 V 2 (0). Hence, the couple will have a birth in period 1 if V 1 1 V 0 1 0 or V 1 1 V 0 1 = [(1 p 1 )V 2 (1) + p 1 V 2 (0) c] V 2 (0) 0 = (1 p 1 )V 2 (1) (1 p 1 )V 2 (0) c = (1 p 1 ) V 2 c = V 2 c 1 p 1 Wolpin states (p.496) in order to characterize the decision rules in period 1, it is necessary to 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. There are three scenarios to consider: 1. it is optimal to have a birth in period 2 regardless of the value of N 1 ; i.e., V2 1(0) > V 2 0(0), V 2 1(1) > V2 0(1). 2. it is optimal to have a birth in period 2 only if N 1 = 0; i.e., there is no surviving offspring (V2 1(0) > V 2 0(0), V 2 1(1) < V 2 0(1) 3. it is not optimal to have a birth in period 2 regardless of the value of N 1, i.e., V2 1(0) < V2 0(0), V 2 1(1) < V 2 0(1). Without providing the details, which are straightforward, the optimal behavior in period 1 is as follows: 1. if it is optimal to have a birth in period 2 when there is a surviving offspring, then it will be optimal to have a birth in period 1, 2. if it is optimal to have a birth in period 2 only if there is no surviving offspring, then it will be optimal to have a birth in period 1; 3. if it is not optimal to have a birth in period 2 regardless of whether there is a surviving offspring, then it will not be optimal to have a birth in period 1. Wolpin reminds us (footnote 10) that these results are special do to the assumption that offspring do not yield contemporaneous utility flows. 4
Together, these results imply that increasing infant mortality risk can not increase fertility. At very low mortality risk, it will be optimal (assuming the birth cost is low enough as well) to have a second birth independently of whether there is surviving first births implying that case (1) holds. As infant mortality risk increases, it will eventually become optimal to have a second birth only if there is not a surviving first birth, implying case (2) holds. Finally, there is some higher level of mortality risk, it will to be optimal to have a seoncd birth regardless of whether there is a surviving first birth, implying case (3) holds. The case of child mortality risk is more complex. Assume that p 1 = 0, so there is only child mortality. Then if p 2 is such that case (2) holds, then increasing child mortality will at some point produce case (3) have two children regardless of the outcome of the mortality experience of the first child; i.e., hoarding behavior. Further increases in child mortality risk will lead eventually to a decline in births back to case (2) and then to case (1). Yet, if true this has implications for my application of trending declines in infant and child mortality. That is, fertility may increase with a decline in infant and child mortality. Thus, may push childbearing earlier in the life cycle and may increase the number of children born. I think. Wolpin s General Model Denote by t = 1 the beginning of the decision horizon, by t = τ +1 the onset of infecundity, and by t = T + 1 the end of life. There are exactly τ fecund periods (the last fecund period begins at t = τ) and T period of life (the last period of life begins at t = T ). For convenience all three dates are taken as deterministic and exogenous. In the fertility contact, the exogeneity assumption for the onset of the decision making (t = 1) is probability the most problematic. If it is taken to be the age of menarche, then it will be necessary in most settings to model the decision to marry explicitly. Assuming the decision process begins at marriage, however, will be correct only if marrage timing does not signify optimizing childbearing behavior. 4. The period length is assumed to be one month and the decision or control variable is whether or not to contracept,. Let κ 0 t = 1 if at t the woman is not pregnant and contraception is not being used and equal to zero otherwise, κ 1 t = 1 if at t the woman is not pregant and the couple chooses to contracept and equal to zero otherwise, and κ 2 t = 1 if the woman is pregnant at t and zero otherwise. Note these three states are mutually exclusive and collectively exhaustive, k κk t = 1. Because the woman becomes infertile at t = τ + 1, κ 0 t = 1, t τ + g where g is the known (non random) gestation period. Define κ t = (κ 0 t, κ 1 t, κ + t = κ t, κ 2 t ), and K t = (κ + t 1,..., κ+ 1 ). Thus, K t is the month by month history of contraceptive use and pregnancies as of t. A birth is assumed to occur at t, n t = 1, if the woman has been pregnant for the pre determined gestation period (g). The history of births is denoted by B t = (n t 1,..., n 1 ). N t is the number of children who are alive 4 Translating Wolpin, the last sentence is supposed to mean if women do not time marriage to begin childbearing. 5
at the end of period t. The evolution of surviving children is given by N 0 =0, N 1 =n 1 (1 d 0 1) = M 0 1, N 2 =M 0 1 (1 d 1 2) + n 2 (1 d 0 2) = M 1 2 + M 0 2, N t =. t k=1 M t k t, where Mt t k is equals to one if a child of age t k is alive at the end of period t and is zero otherwise. The age distribution of children alive at the end of t is given by the vector M t = (Mt 0, Mt 1,..., Mt t 1 ) and the history of the age distribution by M t + = (M t, M t 1,..., M 1 ). Technology consists of mortality and pregnancy risk functions. The probability that a child of age a (a = 0,..., t 1) at time t will die at the beginning of the period of t, p a t, is assumed to depend on the timing and spacing of prior births, B, on a couple specific specific frailty endowment common to all children born to a couple, µ 1, and on a child specific frailty endowment, µ 2. The pregnancy probably at time time (the age of the couple) t, q t, is assumed to depend on the entire history of contraceptive choices, allowing for efficiency of use, and of pregnancies, allowing for the biological dependencies, and on a couple specific fecundity parameter, φ. The frailty parameter of the mortality risk function and the fecundity parameter of the pregnancy risk function are assumed to be drawn prior to t = 1 and are permanent. 5 The couple at each time t is assumed to maximize the remaining discounted lifetime utility with respect to the contraception variables at time t κ t, subject to a budget constraint. Per period utility is state dependent, namely U t = κ 0 t U 0 t (N t, X t, ɛ 0 t ) + κ 1 t U 1 t (N t, X t, ɛ 1 t ) + κ 2 t U 2 t (N t, X t, ɛ 2 t ), (1) where X t is a composite consumption good and ɛ j t is a random timing varying taste parameter for each of the contraception and pregnancy states. The preference parameters may have a permanent component, reflecting population taste variation for children relative to goods. Even ignoring tractability, it is unclear exactly how to model the couples budget constraint. The issue of whether and in what ways couples are liquidity constrained is not resolved in the literature and surely depends on the society under study. To reduce the already cumbersome notation, assume an ad hoc consumption function of the form X t = X t (Y t, κ t, M t, n t ), where Y t is a stochastic exogenous income flow drawn from a known distribution. Notice that contraception is assumed to be costly as are children. To close the model, it is necessary to specify how families forecast future economic and health conditions. For example, what do couples know about future mortality risk, an issue that is particularly relevant if health and medical technologies are evolving? Do couples know their innate frailty and fecundity endowments as of the initial period or do they learn about the as children are born? How do couples forecast future income? Given imperfect foresight, do all couples use the 5 They could also depend on the mother s age at conception for biological reasons. (2) 6
same forecasting rules? Although one may think that these questions are only relevant if structural estimation is pursued such a view would be incorrect. If, for example, couples learn about their infant and child mortality risk in part through experience, then the effect of a death on births will reflect not only replacement behavior but also what they learn and how they adapt to the new information (Mira 1995). The general model represents, in my view, an appropriate direction for future empirical research. What components are tractable to estimate structurally given current technology is unclear. Solution and estimation methods are advancing and limits on computation are receding. 7