2.1 Within-cohort fertility differentials

The TFR in the USA was decreasing at the beginning of the 19th century (Figure 1). At the beginning of 1800s, the TFR was approximately seven children (for white women—the rate was higher for black women). The TFR was approximately two children at the beginning of the second quarter of the 20th century (Figure 2). After a brief period of increase (the “Baby Boom”), the TFR returned to roughly two children (the “Baby Bust”), and remained relatively constant for the last 30 years of the 20th century. The periods of interest for this study are the 1930s and the 1970s, during which time the TFR was approximately two children. For these periods, consider another indicator of fertility behavior: the distribution of women by number of children (completed fertilities). Figure 4 depicts the distribution of women by number of children ever born for US women aged 45–49, born in 1912–1916 and 1946–1950, obtained from the National Vital Statistics System (NVSS). Despite the similarity of the average fertility (proxied by TFR) for these cohorts, their fertility behavior (fertility stopping rules) differed significantly. In the earlier cohort, some 60% of women had 1–3 children, less than a quarter of women had exactly two children, and roughly 20% of women had more than four children, making the average fertility a weak predictor of the completed fertility for any given mother. Women of the later cohort were more homogeneous in their fertility behavior. Almost 70% of women had 1–3 children, more than 35% had exactly two children, and large families became much more rare. However, the proportion of childless women, as well as women who had just one child in their lifetime, also decreased. Overall, the standard deviation of completed fertility decreased from 1.9 to 1.4, resulting in a distribution more concentrated around two children. A more complete picture of this concentration around two children (with multiple cohorts of women born between 1912 and 1961) can be observed in Figure 7. Note that in Figure 4 (as well as Figure 7), the distribution is presented for all women, while the analytical derivations (section 4) and numerical exercise (section 5) are performed for married women only. This raises concerns that some features observed in Figure 4 may be related to changes in celibacy rates, and that data on married women would be more appropriate for the purpose of this study. These data are available and are presented in studies such as Jones and Tertilt (Reference Jones and Tertilt2008). This study is interested in marital fertility and presents the distribution of married women (aged 40–45) by parity, obtained from the US Census data. The cohorts from 1908 and 1948 in Jones and Tertilt (Reference Jones and Tertilt2008), seen in their Figure A3, correspond most closely to those from NVSS that I use in my study. Changes in the distribution of fertility depicted in their Figure A3 are similar to the changes depicted in Figure 4 of my study; namely, we both observe a reduction in variability, as well as a reduction in the proportions of women with extreme parities (no children, one child, and three or more children) for the later cohort. Despite the availability of data on marital fertility, I decided to use NVSS data due to imperfections observed in the pre-1950 US Census data on childlessness. Namely, Baudin et al. (Reference Baudin, de la Croix and Gobbi2015) mention issues (relating to the classification of having no children) with the pre-1950 US Census data, which can potentially cause overestimation of childlessness rates. These issues are not present in NVSS, and the methodology of calculating the proportion of childless women is differentFootnote ¹. Concerning changes in celibacy rates, Figure 7 in Fitch and Ruggles (Reference Fitch and Ruggles2000), using Current Population Survey (CPS), presents data on never married women. They show that between 1960 and 1995, among women aged 45–54, there was approximately a 1 percentage point decrease in the proportion of unmarried white women, although black women saw a 6 percentage point increase. Given that the black population was roughly 12% of the total population, the weighted effect implies that, for the total female population, the proportion of unmarried women was lower in 1995 compared to 1960. Due to this, I think that the change in marriage pattern cannot comprehensively explain the observed dynamics of childlessness and single-child women (although it undeniably had some effect). Fortunately, the data coming from NVSS has distribution by race, so I was able to obtain the likes of Figure 4 for each race separately (Figure 3) and it can be seen that the dynamics observed in Figure 4 are still present. In fact, despite the fact that there was an increase in the proportion of unmarried black women between 1960 and 1995, they experience a much more drastic decrease in childlessness than white women.

Figure 1.

TFR, USA from 1800 to 2000 for White and Black women.

Source: M.R. Haines and J.D. Hacker.

Figure 2.

Average fertility of women aged 50, US birth cohorts from 1911 to 1955, all races.

Source: Cumulative birth rates, by live-birth order, exact age, and race of women in each cohort from 1911 through 1991: United States, 1961–2006 (CDC/NCHS, National Vital Statistics System).

Figure 3.

Distribution of women by number of children ever born, US women aged 45–49 by races (White and Black), born in 1912–1916 and 1946–1950.

Source: CDC/NCHS, National Vital Statistics System.

Figure 4.

Distribution of women by number of children ever born, US women aged 45–49 of all races born in 1912–1916 and 1946–1950.

Source: CDC/NCHS, National Vital Statistics System.

Another potential explanation for this childlessness may be the Great Depression. However, Sobotka et al. (Reference Sobotka, Skirbekk and Philipov2011) found that, during the Great Depression, the secular downward trend in fertility observed for the previous 100 years did not show significant signs of acceleration. This implies that the Great Depression did not necessarily have a major effect on completed fertility (as opposed to period fertility, which may change due to postponement). In fact, both of my cohorts of interest, those having children in the 1930s and the 1970s, were having children during a depression/recession (the mid-1970s saw a significant recession in the USA too, although not as harsh as the Great Depression). I would also suggest that one be careful to distinguish between the postponement of childbearing due to unfavorable economic conditions (although some may argue that a drop in wages could imply a reduced opportunity cost of having children) compared to the decision to have fewer children (completed fertility). The postponement of childbearing may have an effect on completed fertility if women tend to have children until they are no longer able to do so (natural limit). Because of this, it is obvious that postponement leaves fewer years until menopause, so fewer children can be born as a result. However, even for cohorts born in the 1850s, as evident from Jones and Tertilt (Reference Jones and Tertilt2008), such high fertility was very rare. Thus, if the households wanted to have six to eight children (which is a TFR observed in the early 1800s), postponing childbearing by several years should not significantly affect completed fertility.

Concerning the decrease in the proportion of women with many children, it is possible that some part of the decrease in the right tail may result from more efficient fertility control methods. However, note that even households which stopped at seven to eight children, let alone those which stopped at three to four, were, by no means, were close to the natural limit of fertilityFootnote ². So, I assume that even households with highest parities households were able, somehow, to control fertility. Bailey and Hershbein (Reference Bailey and Hershbein2018) nicely summarize the argument on the control of fertility, noting that the drop in fertility started with women born in the 1850s and continued until the cohort in the 1910s. Moreover, that decline is explained, not by the change in the age of first marriage, but rather by the delaying and spacing of births, which implies significant control over fertility. Overall fertility now is not much lower than it was for the 1910s cohort, during which time “the pill” obviously did not exist. Thus, it is not obvious that earlier cohorts were unable to control fertility, or that fertility control was not very widespread. Note also that the proportion of childless and single-child households, as shown in Figure 4, was higher for earlier cohorts, implying that a significant part of the population was able to limit its fertility.

2.2 Empirical evidence of son-preferring differential stopping behavior

Childbearing, by nature, is a consecutive decision-making process. There is some evidence that the decision to have another child depends on the gender of the existing children in the household. In particular, the US households which had boy(s) at a lower parity (first, second, third child) were much less likely to have another child. For example, McDougall et al. (Reference McDougall, DeWit and Ebanks1999) cites a number of 1970s US studies that show that boys were preferred as a first or only child, and having more boys in the family was also preferred.

Dahl and Moretti (Reference Dahl and Moretti2008) conducted a more thorough empirical study of the probability of progression to another child, conditional on having girls vs. having boys. They used 1940–2000 US Censuses and found that households whose first two, three, or four children are girls have a higher probability of having another child compared to those with two, three, or four boys (the gender effect). The gender effect was stronger for women aged 20–30 born in the 1950s than for those born in the 1970s. However, women born earlier than the 1940s seemed to have a weaker gender effect. Note that they only report the effect in the case of two girls vs. two boys. A potential explanation for the weaker gender effect in the earlier cohorts may lie in the fact that earlier cohorts tended to have more children on average. No matter what, if a household wants to have three children, on average, one should expect a weak gender effect for the first two children. This idea is supported in Dahl and Moretti (Reference Dahl and Moretti2008) by the fact that, in developing countries, where average fertility is much higher than in the USA, the gender effect gets stronger at higher parities. To get a better picture over time, ideally one may need to have longer time-span data to identify women born in the early 20th century. However, the only data available for the USA that goes back in time to that extent is the Census and it unfortunately only reports children residing with their families. If one chooses women over the age of 40 with a hope of including only those who are close to finishing their fertility cycle, there is a risk that some children left the household.

The CPS fertility supplement can be used to cope with this problem. Supplements do not have a complete fertility record, as they report gender of only the first four children and the last child. However, as shown in Rosenblum (Reference Rosenblum2013) the gender of the first child (in the absence of gender-selective abortions practiced at the first birth) can be used to test for the presence of the SP-DSB). I use CPS fertility supplements for 1990 and 1995 obtained from IPUMS CPS, King et al. (Reference King, Ruggles, Alexander, Flood, Genadek, Schroeder, Trampe and Vick2010). In these supplements, fertility information is recorded only for those who were below the age of 66. Thus, from 1990, I select women who are married, aged 60–65, who had obviously finished childbearing, and have at least one child. From the 1995 sample, I choose married woman who are between 45 and 50 years old. Additionally, to ensure that they have finished childbearing, I choose those who stated that they did not intend to have additional children. Thus, the first cohort of women was having children in the mid-1940s and the mid-1950s, while the second cohort was having children in the 1970s. Having married women is important as the dissolution of marriage or loss of a spouse can potentially alter childbearing plans of a woman for a variety of reasons (like the cultural stigma of having children out of wedlock, or the deterioration of their financial situation due to staying single, etc.).

The following equation tests the presence of the son-preferring fertility stopping rule:

(1)$$y_i = \alpha + \gamma X_i + \theta t + \tau Z_i + \varrho Z_i\cdot t_i + u_i,$$

using the pooled data of CPS 1990 and 1995, where y _i is the number of live births, X _i is a vector of variables, which includes race, age at first birth, labor market status of a women, female labor force participation rate in the state of residence, and her education levelFootnote ³; t _i is an indicator (dummy) variable for women of the younger cohort (women from the CPS 1995 sample aged 45–50), Z _i is an indicator variable of a first-born male child and Z _i · t _i is an interaction term. Results of the regression (1) are shown in Table 1. It is clear that women of the earlier cohort had fewer children if the first child was a boy. The Ordinary Least Squares (OLS) estimates (first column) indicate that women with a male first-born had approximately 0.2 fewer children. For comparison, Rosenblum (Reference Rosenblum2013) obtains 0.35 fewer children for India where son-preference is well-documented. Given that I count data where a count of 0 is impossible (I included women who had at least one child), I also estimate equation (1) using zero-truncated Poisson regression, which implies (second column) approximately 5 percentage points fewer children for those with first-born sons. This effect is statistically significant. Moreover, the effect of the first-born child's gender on the later cohort is negative too, but has a smaller magnitude. Other variables in X _i have expected signs: higher age at first birth, being white, being in the labor force, and being more educated tend to reduce completed fertility. Interestingly, the education variables are highly non-significant, which might be explained by the fact that I control for age at first birth, which is likely connected to the level of education (the more a woman studies, the more likely she will have a child later in life). Indeed, if one omits the woman's age at the first birth, the education variables become significant. Labor force participation at the state level reduces the effect of the first-born's gender, but only marginally, and itself is highly non-significant, a result probably stemming from the issues discussed above. Being limited to only 1990 and 1995, the data allow me to go back no further than the 1925–1930 cohort, which is some 15–20 years later than the earlier cohort in Figure 4. Even this is enough to see that the smaller gender effect for the pre-1940s cohorts in Dahl and Moretti (Reference Dahl and Moretti2008) can be due to the problems with the data discussed above. In summary, there is empirical evidence that early cohorts of US households exhibited son-preferring fertility stopping behavior. For the later cohorts, there is weak, if any, evidence for such fertility behavior.

Table 1.

Estimation of gender stipulated fertility stopping behavior

Notes: The sample consist of married women who had at least one live birth and at the moment of the survey has completed fertility (testified by being in the age of menopause as well as directly stating absence of intention to have more children). Independent variable is the number of live births a women ever had. Data come from CPS Fertility Supplements 1990 and 1995 (obtained from IPUMS-CPS). OLS uses final basic weights. For zero-truncated Poisson e ^τ is the average ratio of number of children of women with male first born to the number of children of a women with female first born. ***significant at less than 1%; **significant at 5%; *significant at 10%.

2.3 Evolution of the labor force participation of the married women over time

The 20th century also saw significant changes in the composition of the workforce in the USA (Figure 5) and other developed nations. An important phenomenon was the emergence of married working women. Women were employed in the formal labor market throughout 19th century in quite large numbers, but these were mostly unmarried women. After marriage, most of these employed women tended to leave labor market. However, from the early 20th century onward, that trend started to change. As shown in Figure 5, the proportion of married women in the labor force (FLFPR) constantly rose during all of the 20th century. If less than 5% of married women in the 1900s were in the labor force, by the 1990s, this number rose to approximately 70%.

Figure 5.

Percentage of married women in labor force.

Source: US Census.

The choice of the FLFPR as a factor affecting parental choice is not random. Mills and Begall (Reference Mills and Begall2010), in their study of gender preference in Europe, come to the conclusion that, in places where there is a lower gender equity, clear preference for sons exists. I think it is not unreasonable to use the FLFPR as a metric of gender equity, ultimately affecting child gender preferences. After all, the presence of women in leading positions in business, politics, academia, sports, military, etc. is perceived as a sign of gender equality, making women socially and economically indistinguishable from men. Another often-cited measure of equality between sexes is the gender gap in wages. However, the precondition for the existence of a gender gap is the existence of employed women. The FLFPR of married women was steadily increasing and I believe this captures the changes in gender roles and movement toward gender equality that occurred over the past century. In this context, I see the FLFPR as a reasonable proxy for the advances in the status of women in society, which should positively affect parental satisfaction from children, whether we view parents as altruistic or egoistic. From an altruistic point of view, being in the labor market and having one's own income source may mean more freedom in choosing a partner (the choice is not limited to those who can “provide”); higher bargaining power within a family (a financially independent female's value of outside (to marriage) option is higher in the case of divorce); potential well-being of grandchildren (e.g., working daughters would not appear in sharp poverty in case of the death of the husband), and, of course, the pride of the daughters’ self-realization in the professional domain. For egoistic parents, working daughters may mean a higher chance of receiving old-age support, either because daughter's family will have more income to share, or due to the above-mentioned stronger bargaining position to insist on supporting her parents. Note that many of the factors favoring daughters in the labor force are linked to a notion of girls becoming good mothers (ability to avoid poverty in the very common case of early death of a husband, education to help their own children with schooling, etc.). So, even in a society where the middle class may maintain “patriarchal views” on the role of women, working should not be viewed as incompatible with that role. After all, even in rural communities, which usually tend to be more patriarchal, women from early ages are expected to work both at home and on the farm.

3.1 Modeling approach

Given that women's status in society is proxied by the likelihood of being in the labor force as a married woman, it seems natural to model parental preferences such that they derive utility from the income of their children. In such an approach, the parental utility from children would be dependent on the expected labor market status of their children. Fortunately, modeling parental preferences, such that they derive utility from the income of children, is quite common in the literature, e.g., Galor and Weill (Reference Galor and Weill2000), de la Croix and Doepke (Reference de la Croix and Doepke2003), Kalemli-Ozcan (Reference Kalemli-Ozcan2003), and Hazan and Zoabi (Reference Hazan and Zoabi2015). Note, however, that by total income of children, I do not mean a discounted lifetime income earned by children (too complex to model, as discussed below), but rather the sum of wages of all children earned per unit of time, where these wages depend on the amount of human capital. In this sense, these wages, which can be earned per unit of time due to the presence of a child in the labor market, demonstrate the child's “earning potential.” One may also say that a child's income (per unit of time) is a measure of indirect utility, as the child's “earning potential” captures the tightness of the budget constraint to be faced to the future.

3.2 Model setup

The society consists of a single cohort of an arbitrary large number of households, which are ex-ante identical (the same age, income “earning potential”, preferences, budget constraints, etc.)Footnote ⁴. A household consists of a male and a female who jointly behave as one economic agent, and make decisions on having children and investment in the human capital formation of children. Each household has four uncertainties affecting its decision: (1) uncertainty over the labor force status of its female: each household has the equal probability α (that is the current FLFPR) of the female of the household participating in the labor market; (2) uncertainty over its level/valuation of “outside” option to childbearing ξ _j; (3) the gender of each consecutive child (once they start childbearing); and (4) the labor force status of their daughters (if any): probability of being employed is proxied by the current FLFPR (α). The first two uncertainties are realized at the onset of the childbearing decision, i.e., the household learns the labor force status of its female and its level of “outside” option and only then starts to plan childbearing. The third uncertainty is resolved at childbirth, and the forth uncertainty is resolved only when the children grow up, that is when all household decisions concerning human capital investment and fertility progression have been made.

I study the effects of the increased FLFPRFootnote ⁵ on the example of a fertility model whose variation can be found in Galor and Weill (Reference Galor and Weill2000), de la Croix and Doepke (Reference de la Croix and Doepke2003), Kalemli-Ozcan (Reference Kalemli-Ozcan2003), Hazan and Zoabi (Reference Hazan and Zoabi2015), etc. It is a one-time model where households, observing the exogenously given FLFPR, determine whether or not to have children, and if they decide to have children, when to stop having them. A household derives utility from their own consumption and from the expected income of its children if it has child(ren), or from the level of “outside” options available to it, if it chooses not to have any child. This means that parents, who have children, derive utility only from children who will be employed. This statement requires clarification. Altruistic parents enjoy children even if they are not employed. Thus, in the initial version of the study, parents derived utility from the total income of children who were expected to work, and children not in the labor market contributed some constant χ to the parental utility. That χ was to stand for the “pure joy” or “intrinsic happiness” of having a child regardless of its labor market status. However, this addition made analytical derivation difficult. Assuming that there is no reason why this “pure joy” of having offspring should be different between sexes, I “normalized” χ to zero. This simplification comes with the additional benefit of making the FLFPR similar to child survivability, thus allowing the use of theoretical results obtained in studies such as Sah (Reference Sah1991) and Kalmli-Ozcan (2005). For further discussion on the effect of the choice of the utility function on the main results of this study, please refer to Appendixes A-10 and A-11, where numerical estimation using several more general utility function specifications yield similar fertility stopping rules.

In the simplified model, all boys are assumed to become employed adultsFootnote ⁶. Each of the girl's probability of being employed is given by the current FLFPR (in this model, it is the best estimate of that probability in the future). As all girls have the same probability of being an employed woman, the expected number of employed girls is characterized by a binomial distribution. As a result, the household's expected utility functionFootnote ⁷ is

(2)$$\left\{ {\matrix{ {U_j\,(c,e_b,e_g,b,g) = u\,(c) + \psi \mathop \sum \limits_{i = 0}^g \alpha^i\left( {\matrix{ g \cr i \cr}} \right)\,{(1-\alpha )}^{g-1}v\,(bh_bw + ih_gw),\,{\rm if}\,b \gt 0\,{\rm and/or}\,g \gt 0} \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,U_j(c,e_b,e_g,b,g) = u\,(c) + \psi v\,(\xi_j),\,{\rm if}\,b = 0\,\,{\rm and}\,\,g = 0,} \cr}} \right.$$

where u (.) and v (.) are two differentiable concave functions. The c is the lifetime consumption, b is the number of boys in the household, each of whom gets e _b of household resources invested in human capital formation, g is number of girls in the household, each of whom gets e _g of household resources invested in human capital formation. I make a simplification that all households are fertile. In this case, a household stays (voluntarily) childless if the “outside” option to being parentsFootnote ⁸, which I denote by ξ _j, delivers more utility to the household j than having a child (described in detail in the next subsection). If households decide to have a child, then parents are obviously “inside” parenting and do not derive utility from an “outside” option ξ _j. The ψ is the level of parental altruism. A child's human capital is denoted by h _k,  k = {b,  g} and the human capital formation function is $h_k = e_k^\gamma $, where k = {b ,  g}, 0 < γ < 1 and the (per unit of time) wage per unit of human capital is w. The FLFPR is the probability of a girl becoming an employed women and is denoted by α. Note that, when households start developing their contingency plan (fertility stopping rules), they face uncertainties (3) and (4). This last source of uncertainty is incorporated in the expected utility function similar to the incorporation of child survivability in studies like Sah (Reference Sah1991), Portner (Reference Portner2001), and Kalemli-Ozcan (Reference Kalemli-Ozcan2003). Thus, the utility function used in this study is a combination of a utility function that has uncertainty over child gender and a utility function that has uncertainty over the “survivability” of the children. The budget constraint is

(3)$$\matrix{ {c + be_b + ge_g = y_m-p(b + g) + y_f\,(1-z\,(b + g)),\,y_f = \lsqb {0,{\overline y}_f} \rsqb } \cr}. $$

The male and female of the household are each endowed with a unit of time, which can earn income y _m for the male and income $y_f = \overline y _f$ for the female, if she is in the labor market and y _f = 0 if she is not part of the labor forceFootnote ⁹. If the female of the household is part of the labor force, then the household is considered to be of type I, otherwise it is considered to be of type II. I assume that females specialize in raising children, so the time opportunity cost z is incurred only by them. The p ∈ (0,  1) is the fixed cost of having a child in terms of goods. The presence of p (and z) is crucial, otherwise households can continue having children until they get the required number of boys, and a household with, for example, three boys will be equivalent to a household with three boys and any number of girls. I assume that, like the cost of having a child (p), the cost of education is also in terms of consumption goods. However, the nature of costs of child-bearing and education does not play any significant role in this model as α does not enter u(.), so the results hold if any of these costs, or both, are in terms of parental time or consumption goodsFootnote ¹⁰.

3.3 Description of the household's problem

After uncertainties (1) and (2) are resolved, households proceed to plan for childbearing, which is done following a two-stage maximization procedure.

In the first stage, for any number of boys and girls (b,  g), a household determines how much investment will be made into the human capital of each boy and each girl (determination of e _b and e _g). This is done by maximizing utility (2) subject to constraint (3) for each possible pair of (b,  g). In the second stage, given the human capital investment plan at every possible gender outcome of each consecutive pregnancy, the household develops a fertility stopping rule. At any possible pair (b,  g)Footnote ¹¹, if the household decides to have a child, with equal probability, it will have (b + 1,  g) or (b,  g + 1) children. Let V (b, g) stand for maximized utility for a given (b,  g), while V (b + 1, g) and V(b, g + 1) be maximized utilities for (b + 1,  g) or (b,  g + 1) children. The household will find it optimal to have another child only if the expected utility from having another child is greater than staying with current number of children (b,  g); that is MU(b, g)—marginal utility from having another child (4) at parity (b,  g) must be positive. Thus, the household can determine whether it will continue childbearing at every physically conceivable parity (b,  g). So, at the moment the household starts having children, it already has the complete plan of action regarding educational and fertility choices—the fertility stopping rule:

(4)$$MU(b,g) = 0.5V(b + 1,g) + 0.5V(b,g + 1)-V(b,g).$$

In essence, households develop a contingency plan, before starting childbearing, but after the resolution of uncertainties (1) and (2). Think of that contingency plan as an action tree. At the beginning, a household with no children knows that the first child will be a boy or a girl with equal probability. It can determine the optimal investment in human capital in case the child is a boy, and in case it is a girl. If the expected utility from having a child outweighs the utility of staying childless (“outside” option) the household will have the first child. Note that, for a given level of the FLFPR (α) and the labor status of the female in the household (type I or type II), there are two threshold levels $\tilde{\xi} _I^\alpha $ and $\tilde{\xi} _{II}^\alpha $ of constant ξ _j, above which households do not have children. The assumption that ξ _j is irrelevant for those who have already started childbearing avoids increased complexity due to a new dimension of heterogeneity. So all type-I households, who started childbearing, are identical (thus, they have the same contingency tree), and the only thing distinguishing them is the fertility history that they will have. The same is true for type-II households. This simple assumption allows the model to generate childlessness and avoid any external factor (except the FLFPR), changing the contingency tree. After having the first child, the household continues as described above: for a given node (b,  g) on the contingency tree, it determines the optimal education investment for the (b + 1,  g) and (b,  g + 1) nodes and determines whether to proceed with an additional child based on the sign of the marginal utility (4). The node at which the household stops childbearing will totally depend on its fertility history, i.e., the series of random resolutions of gender uncertainty (3). Thus, the complete contingency plan for type-I and type-II households is a tree with branches of various lengths. Given the contingency trees, it is clear that, as a result of childbearing, the initially identical mass of type-I and type-II households will be distributed over the nodes of trees generating fertility variation. The main task of this study is to see how trees for type-I and type-II households change when the households appear in environments with a different FLFPR.

3.4 Son-preferring differential stopping behavior (SP-DSB)

The fact that, at a FLFPR<1, not all girls may end up being employed women implies that households, whose first-born children are girls, will tend to have another child in the hopes that it will be a boy. To see this in a simple setup, assume an exogenous human capital case where each child is born with a unit of human capital earning w = 1. Assume also a household A, which has (b + 1,  g) children, and a household B, which has (b,  g + 1) children. For the simplicity of this exposition, I assume that both households are of type II (actually, as long as households A and B are of the same type, the exact type does not matter). The marginal utilities from having another child for these households (household type is denoted by subscript) at an extremely low FLFPR (I consider the case of α = 0) areFootnote ¹²

$$\eqalign{MU_A\,(b + 1,g) &= u\,(y_m-(b + g + 2)p)-u\,(y_m-(b + g + 1)p) + 0.5\psi \,(v(b + 2) \cr &\quad -v(b + 1)),}$$

$$\eqalign{MU_B\,(b,g + 1) &= u\,(y_m-(b + g + 2)p)-u\,(y_m-(b + g + 1)p) + 0.5\psi (v(b + 1) \cr &-v(b)).}$$

As MU _A (b + 1, g) <MU _B (b, g + 1)Footnote ¹³, even in the case of MU _A (b + 1, g) = 0 (utility maximizing fertility achieved), household B will have another child as MU _B(b, g + 1) > 0, so household B will exhibit SP-DSB. Thus, even if we have ex-ante identical households, at low FLFPR levels, households A and B will end up with different levels of completed fertility simply because, at a low FLFPR, they will exhibit SP-DSB and child gender determination is a stochastic process.

4.1 Marginal households

Before studying the effect of changes in the FLFPR, I need to define the Marginal Households. I will be studying the effect of changes in the FLFPR on the behavior of these households. The Marginal Household MH (b,  g,  α ₀) is a household, which at a given number of children (b,  g)Footnote ¹⁴ and FLFPR (α ₀), is indifferent to having another child or abstaining from having one that is, the marginal utility of having another child (5) is zero.

4.2 Childless households

I start the study of household behavior with households which, at lower FLFPR, decide to stay childless. Unlike the general case of a household which has b > 0 boys and g > 0 girls, in this simple case, the effect of the FLFPR on education and the fertility decision is mathematically tractableFootnote ¹⁵.

A household chooses to have the first child if having that child delivers more utility than the “outside” option of choosing to stay childless, ξ _j (MU (5) is positive).

(5)$$\eqalign{MU &= 0.5\,(u(y_m-p-e_b-y_f\,(1-z)) + v\,(e_b^\gamma ) \cr & \quad + 0.5\,(u(y_m-p-e_g-y_f)\,(1-z)) + \alpha v(e_g^\gamma )) \cr & \quad -u(y_m + y_f)-v\,(\xi _j).} $$

Proposition 1. In an environment with α ₁ such that α ₁ > α ₀ the MH (0,  0,  α ₀) will definitely have another child as MU (5), for a given ξ _j, will be positive.

Proof: See Appendix A-1.

This result is similar to the “essential complementarity” discussed in Aaronson, Lange, and Mazumder (Reference Aaronson, Lange and Mazumder2014). To gain from an increased FLFPR, parents must have at least one child. Note that Proposition 1 implies that the threshold for the level of “outside” options at which households prefer to stay childless should be higher at α ₁, i.e., $\tilde{\xi} _j^{\alpha _1} \gt \tilde{\xi} _j^{\alpha _0} $. Thus, a higher FLFPR implies a smaller proportion of households staying childless. This result is consistent with the situation observed in Figure 4, where the later cohort contains 5 percentage points fewer childless women than the earlier cohort.

4.3 Change in the education decision for a household with an arbitrary number of boys and girls

Starting from this subsection, to facilitate analytical derivations, all results are obtained assuming logarithmic u (.) and v (.) functions.

Let us take a household with an arbitrary number of boys and girls (b,  g)Footnote ¹⁶.

Substituting

$$c = y_m-(b + g)\,p-be_b-ge_g-y_f\,(1-z\,(b + g))$$

and

$$E\lsqb {v\,(e_b,e_g,b,g)} \rsqb = gi = 0\mathop \sum \nolimits \alpha ^i\left( {\matrix{ g \cr i \cr}} \right)\,(1-\alpha )^{g-i}v\,(be_b^\gamma w + ie_g^\gamma w),$$

into utility function (2) the First Order Conditions (FOCs) for investments in human capital are:

(6)$$e_g{\kern 1pt} :\,\,u_{e_g}\,(e_b,e_g,b,g) + E\lsqb {v_{e_g}\,(e_b,e_g,b,g)} \rsqb = 0$$

(7)$$e_b{\kern 1pt} :\,\,u_{e_b}\,(e_b,e_g,b,g) + E\lsqb {v_{e_b}(e_b,e_gb,g)} \rsqb = 0.$$

To know how the investment in human capital changes when the FLFPR changes, I take the full derivative of FOCs (6) and (7) with respect to α which allows the establishment of the following proposition:

Proposition 2. For a given number of boys and girls (b,  g), ∂e _b/∂α < 0 and ∂e _g/∂α > 0.

Proof: See Appendix A-2.

Within a household, an increase in the FLFPR makes parents relocate some of their resources from boys to girls. The intuition is simple: the human capital function is concave. At a higher FLFPR, an investment in girls becomes less risky, the expected marginal return from education increases, so an investment in girls becomes more attractive. Obviously, when the FLFPR is 1, boys and girls are identical “assets” and all children, regardless of gender, will receive the same education. The same result is obtained numerically for relatively concave utility functions.

4.4 Change in the fertility decision: the “stock” and “flow” effects

The effect of changing the FLFPR on the investment in human capital of boys and girls is not hard to predict. In the extreme case of the FLFPR being 0, it is clear that no investment is made into the human capital of girls, while when the FLFPR is 1, boys and girls, being identical, are treated equally. The previous section helped determine the trajectories at which investments in human capital for boys and girls will converge when the FLFPR increases.

To study the effect of an increased FLFPR on the decision to have another child, assume that at an arbitrary α = α ₀ there is a MH (b,  g,  α ₀). My goal is to see whether this marginal household (which is indifferent to having another child, or not, at the current α ₀) definitely will or will not have another child in case it appears in an environment with α = α ₁, such that α ₁ > α ₀. To do this, I totally differentiate MU (4) with respect to α. Obviously, if the derivative of the marginal utility with respect to α (MU _α) is positive, the household will have an additional child, as the marginal utility from having another child becomes a positive number. If it is negative, the household will abstain from having another child, as the marginal utility from having another child becomes a negative number. As V(b + 1,  g),  V(b,  g + 1), and V(b, g) are all utilities maximized by the optimal level of investment in human capital of boys and girls, the Envelope Theorem states that the change in marginal utility is equal to the direct effect of the FLFPR on the marginal utility function, which is the following expression (8):

(8)$$\eqalign{MU_\alpha &= 0.5\,(\partial E\,[v\,(e_b(b + 1,\,g,\,\alpha _0),\,e_g(b + 1,\,g,\,\alpha _0),\,b + 1,\,g)]/\partial \alpha ) \cr & \quad -0.5\,(\partial E\,[v\,(e_b(b,\,g,\,\alpha _0),\,e_g(b,\,g,\,\alpha _0),\,b,\,g)]/\partial \alpha ) \cr & \quad + 0.5(\partial E\,[v\,(e\,(b,\,g + 1,\,\alpha _0),\,e_g(b,\,g + 1,\,\alpha _0),\,b,\,g + 1)]/\partial \alpha ) \cr & \quad -0.5(\partial E\,[v\,(e(b,\,g,\,\alpha _0),\,e_g(b,\,g,\,\alpha _0),\,b,\,g)]/\partial \alpha ) \lessgtr 0,} $$

where e _b(b + 1, g, α ₀),  e _g (b + 1, g, α ₀),  e _b(b, g + 1, α ₀),  e _g(b, g + 1, α ₀),  e _b (b, g, α ₀),  and e _g(b, g, α ₀) are optimal levels of investment in human capital for a household with (b + 1,  g),  (b,  g + 1),  and (b,  g) children at α = α ₀ (see Appendix A-3 for details of expression (8)). Unfortunately the sign of expression (8) is hard to identify. One may think that different levels of education are behind the ambiguity of expression (8). However, even in case of exogenous human capital (see Appendix A-4), the sign of expression (8) remains ambiguous.

I identify two effects which can intuitively explain why we have ambiguous results. These effects are themselves a complex interaction of certain phenomena, and I can separate these effects based on the different directions in which they move the MU _α. Intuitively, I call them the “flow” and “stock” effects. Increasing the FLFPR makes the expected “flow” of utility from an additional child bigger, so I call this the “flow effect.” On the other hand, if the household already has girls, a change in the FLFPR will affect each girl, increasing the expected “stock” of utility from the existing “portfolio” of children, so I call this the “stock effect.” Note that the “stock effect” is bigger when a household has more girls. Both the “flow” and the “stock” effects, generated by increases in the FLFPR, tend to increase utility, but their interplay, which determines the effect on marginal utility, is more complex. This is why I propose to consider two polar cases in which only one of the effects will be present to see in which direction they move the MU. Below, I present the case of a household which has only boys, and the case of a household which has only girls.

4.5 Case 1: A household with only boys

To start, assume there is a household that has b boys, that is considering having another child. In case that they know the child is a boy, the marginal utility from that child does not depend on the FLFPR. Thus, changes in the FLFPR have no effect on the decision of whether or not to have another child. The “flow” effect is illustrated when this household knows that the next child will be a girl, as changes in the FLFPR affect only that girl. In that case, the marginal utility from that additional girl is expression (9):

(9)$$MU = V(b,1)-V(b,0).$$

Intermediate result 1. In an environment with a higher FLFPR, a household with b boys has greater incentive to have another child.

Proof: Appendix A-5.

This is due to MU _α>0. As the existing “stock” of children is not affected, only the expected gain from an additional child increases, while the marginal cost of having that child stays the same. This is what we have referred to as the “flow effect.” The same result holds true for the case of exogenous human capital of children (Appendix A-6), indicating that education is not the main driving force behind this result.

What if the gender of the next child is unknown? As described in subsection 4.4, the marginal utility from another child whose gender is unknown is just an average of marginal utilities from having a boy and a girl (10):

(10)$$0.5\,(V(b + 1,0)-V(b,0)) + 0.5\,(V(b,1)-V(b,0)).$$

Proposition 3. In an environment with α ₁ such that α ₁ > α ₀ the MH (b,  0,  α ₀) will definitely have another child as the MU (10) will be positive.

Proof. The additive nature of the MU (10) allows us to look at the MU _α as a sum of derivatives of the marginal utility from Intermediate result 1 and MU of a household whose next child is surely a boy. In an environment with a higher FLFPR, the MU (10) of the MH (b,  0,  α ₀) will be a positive number at a higher FLFPR (α ₁). This is due to the fact that the derivative of marginal utility from having a boy is zero and, from Intermediate result 1, the derivative of the marginal utility from having a girl is positive.

4.6 Case 2: A household with only girls

Another special case is when a household has g girls and considers having another child. Assume that it knows that the next child will be a boy. Here, only the “stock” effect is in play as changes in the FLFPR do not affect additional utility generated by this boy. The marginal utility from having a boy for this household is

(11)$$MU = V(1,g)-V(0,g).$$

Intermediate result 2. In an environment with a higher FLFPR, a household with g girls has less incentive to have the expected boy.

Proof: see Appendix A-7.

This is because the MU (11) is a decreasing function of the FLFPR (MU _α < 0). It is tempting to think that once the FLFPR increases, the investment in human capital of girls becomes more valuable, so the household decides to abstain from having another child in order to invest more in each girl. However, this is not the case. Even in the case of exogenous human capital of children, this result holds, and parents abstain from having another child (see Appendix A-9). The observed phenomenon reflects the “prudence” of the household [see Leland (Reference Leland1968), for instance]. Increases in the FLFPR mean that both the expected return (from children) and the variance of that return change. This is due to the fact that, in the case of a binomial distribution of the return on assets (children), the FLFPR affects both the mean and the variance of that distribution. Due to the logarithmic utility function, income and substitution effects caused by an increased return cancel each other out, leaving only the effect of the changed variance, that is changed uncertainty. If the FLFPR is low, the household will engage in precautionary saving, i.e., it will have many children. As the uncertainty over the number of employed girls decreases with a higher FLFPR, the household reduces its precautionary demand for children.

The same “prudence” is at work in another case, when a household with g girls considers having another child and it is known to be a girl.

Intermediate result 3. In an environment with a higher FLFPR, a household with g girls has less incentive to have the expected girl.

Proof. Appendix A-8.

This is because the MU (11) is a decreasing function of the FLFPR (MU _α < 0). Note that the model with exogenous human capital of children delivers the same resultFootnote ¹⁷. As previously mentioned, the FLFPR is employed in my model as child survivability/mortality is employed in many fertility models, which makes results comparable across models. More specifically, when an all-girl household knows that it will have a girl, my model is similar to the model developed by Kalemli-Ozcan (Reference Kalemli-Ozcan2003). Kalemli-Ozcan shows that the increase in the child survivability rate reduces the marginal utility from another child, implying a reduction in fertility, which is caused by a reduction in the precautionary demand for children. Similar also is a study by Portner (Reference Portner2001) where marginal utility (with no restriction on the form of the utility function) from another child is more likely to be decreasing in child survivability the more parents are risk averse and the more positive the third derivative of the utility function is (prudence). Additionally, Rosati (Reference Rosati1996) states that an increase in variance of the infant mortality rate increases the demand for children.

Now assume there is a household with g girls, which considers having another child, whose gender is ex-ante unknown. Given the stochastic nature of child gender determination, two states of the world are possible. In one, the next child is a boy and in another, it is a girl. Marginal utility from having that child is the expected utility from having a child (equal-weighted sum of the utilities in two states of the world) minus the current utility (12):

(12)$$MU = 0.5\,(V\,(1,g)-V\,(0,g)) + 0.5\,(V(0,g + 1)-V(0,g)).$$

Proposition 4. In an environment with α ₁ such that α ₁ > α ₀ the MH (0,  g,  α ₀) will definitely abstain from having another child as MU (12) will be negative.

Proof. The additive nature of the MU (12) allows us to look at the MU _α as the sum of the derivatives of the marginal utility from Intermediate results 2 and 3. Intermediate result 2 states that, for those who have only girls and the next child is a boy, an increase in the FLFPR has a negative effect on the marginal utility. Intermediate result 3 states that if the next child is a girl, the change in marginal utility is also negative, so that the MU (12) of the MH (0,  g,  α ₀) is negative for a higher FLFPR (α ₁).

4.7 Changes in the distribution of women by number of children

The study of the all-boy and all-girl household polar cases in subsections 4.4 and 4.5 is important for us to understand the concentration of the distribution of women by number of children, as observed in Figure 4. As previously stated, low fertility households tended to be all-boy households, or ones where the first-born children were boys, while households with larger families tended to be all-girl households, or ones where girls were the first-born children. Thus, from Case 1 and Case 2, we can conclude that as the FLFPR increases, households on one extreme (few children) increase their fertility, while households on the other extreme (many children) decrease their fertility, so that the distribution becomes more concentrated.

It is reasonable to conjecture that households with children of both genders will resemble the behavior of households in Case 1 and Case 2 depending on the relative number of boys and girls in the household. This is an important feature of this study. The fertility stopping rules depend not only on the number of children, but also on the fertility history (gender mix of children) in the household. This is why MU _α is ambiguous in subsection 4.3 for a general case of a household with b boys and g girls.

To verify the conjecture on the behavior of household with an arbitrary number of boys and girls, as well as to illustrate the evolution of the fertility stopping rules as the FLFPR increases, I conduct a numerical exercise. Note that the numerical exercise is not a calibration/simulation exercise (although parameter values are assumed so that resulting TFR resembles those observed in Figure 4), but rather a way to visualize fertility stopping rules and their evolution when the FLFPR increases. Thus, the parameters are chosen to capture the major features of Figure 4, rather than precisely replicating it (e.g., the model generates a higher proportion of single child households than observed in Figure 4).

5.1 Parameters

As described in section 3, before childbearing, households learn their type. In type-I households, the female is in the labor force, while in type-II households, the female is not in the labor force. The current FLFPR is the proportion of type-I households in society. Knowing their type, households observe the level of available “outside” options to being parents. The expected utility functions for households which decide to have children, that is whose levels of “outside” options to being parents are below threshold values of $\tilde{\xi} _I^\alpha $ and $\tilde{\xi} _{II}^\alpha $, are (13) for type-I and (14) for type-II households:

(13)$$\eqalign{U_I &= u(y_m-(b + g)p-be_b-ge_g + y_f\,(1-z(b + g))) \cr & \quad + \psi \sum\limits_{i = 0}^g {\alpha ^i} \left( {\matrix{ g \cr i \cr}} \right)(1-\alpha )^{g-i}v\,(b(e_b)^\gamma w + i(e_g)^\gamma w)} $$

(14)$$\eqalign{U_{II} &= u((y-(b + g)p-be_b-ge_g)) \cr & \quad + \psi \sum\limits_{i = 0}^g {\alpha ^i} \left( {\matrix{ g \cr i \cr}} \right)\,(1-\alpha )^{g-i}\,v(b(e_b)^\gamma \,w + i + (e_g)^\gamma w).} $$

The u(.) and v(.) functions are of the Constant Relative Risk Aversion (CRRA) type with a coefficient of relative risk aversion equal to 0.5. The y _m and y _f are male and female lifetime earnings. The cost of child p, investments in human capital e _b and e _g are all in terms of goods, which are paid from the total income of the household. In addition to the fixed cost of p, for type-I households, each child comes with an additional cost of z, which is a fraction of the female's lifetime income forgone due to having a child. Echevarria and Merlo (Reference Echevarria and Merlo1999) estimate that, in Canada, raising a child costs women 5% of their lifetime income, while in de la Croix and Doepke (Reference de la Croix and Doepke2003), the cost of a child is estimated to be 7.5% of their lifetime income. I stay close to these numbers and assume p = 0.05 (5%) and the rest is expenses on education. The opportunity cost of having a child is assumed to be z = 0.05 (5%), if we assume that a women works for 40 years, and with every child, she loses approximately 2 years of earnings. Lifetime earnings of a male are normalized to 1 and given that there was always a wage gender gap, female lifetime income is assumed to be 70% (y _f) of male income. The γ of the human capital accumulation function is assumed to be 0.5 as in Hazan and Zoabi (Reference Hazan and Zoabi2015). The altruism parameter ψ is set equal to 0.25 to obtain an average fertility rate of approximately 2, which was the TFR observed in the 1930s when the FLFPR was approximately 0.2. The distribution of “outside” options ξ _j is assumed to be log-normal with μ = −4.567 and σ = 2.643. These parameters are chosen such that the proportion of childless women at a FLFPR of 0.2 $(\tilde{\xi} _I^{0.2} = 0.112,\,\tilde{\xi} _{II}^{0.2} = 0.106)$ and 0.5 $(\tilde{\xi} _I^{0.5} = 0.206,\,\tilde{\xi} _{II}^{0.5} = 0.188)$, which correspond to the values of the FLFPR observed during the reproductive lifetime of the cohorts from 1912–16 to 1946–50, are comparable with the childlessness rates observed in Figure 4.

5.2 Fertility stopping rules and the resulting distribution of women by number of children

Tables 2 and 3 depict the fertility stopping rules for type-I [(13)] and type-II [(14)] households. Note that these are households whose realization of ξ _j in the environment of the FLFPR α is below $\tilde{\xi} _j^\alpha $ (changes in the proportion of childless households are discussed below in the description of Figure 6). At a very low FLFPR, the fertility stopping rule of both types is the extreme case of the SP-DSB and is described quite simply as: “have children until the first boy”. The desire to have a boy is expected as, at a low FLFPR, the return from having a girl is minimal, thus parents with only girls desperately want to have a boy. What is less expected is that households stop having children as soon as they have a boy. However, in this model, this behavior is reasonable, because if a household has a boy, it is “lucky” and does not want to risk having another child, which may turn out to be a girl. The existence of this phenomenon is mentioned in McClelland (Reference McClelland1979), who states that the fear of having another child of an undesired gender may force women to abstain from having that child even if the current gender composition of children is not desirable. This is because such households have an equal chance of getting a desired gender composition of children or further worsening it by having a child of an undesired gender.

Figure 6.

Distribution of women by number of children, model results.

Table 2.

Fertility stopping rules of type-I households

Notes: The “x” indicates that household with (b, g) children will have another child.

Table 3.

Fertility stopping rules of type-II households

Notes: The “x” indicates that household with (b, g) children will have another child.

As the FLFPR increases slightly, households with a large number of children start to disappear. At a higher FLFPR, girls become more desirable, or rather more “acceptable.” However, this is still not so “acceptable” as to make a parent with a boy risk having another child, but “acceptable” enough for a household with 7 or 8 or 9, etc. girls to stop attempting to have a boy. With a FLFPR equal to 0.6, a “lucky” type-I household, whose first child is a boy, will risk having another child, while a “lucky” type-II household will risk having another child at α = 0.4. On the other hand, those with girls, who previously desperately wanted to have at least one boy, will get more satisfaction from having girls and will not engage in a “child gamble,” potentially accumulating many girls. This is consistent with the conjecture of subsection 4.6 that the marginal benefit from an additional child increases if the household has few girls and decreases if the opposite is true, which is stipulated by the interaction of the “flow” and the “stock” effects. Note that, at α = 1, all households have the same number of children, this is because at α = 1 boys and girls become completely identical so that the fertility stopping rules cease to be fertility history-dependent.

The fertility stopping rules for type-I and type-II households result in the economy-wide distribution of women by number of children depicted in Figure 6. Although each generation of households in the model are ex-ante identical, the model generates heterogeneous childbearing histories, resulting in fertility differentials. First, note that due to an increase in $\tilde{\xi} _j^\alpha $ resulting from increases in α, the proportion of childless women decreases for both types of households. Second, as in its empirical counterpart, the early cohorts depicted in Figure 6 have larger fertility differentials. For the later cohort, differentials are smaller, the distribution loses mass on the left and right tails, and becomes more concentrated around two children, which is also observed in Figure 4. The model transition from one distribution (due to increases in the FLFPR) to another is depicted in Figure 7 and follows the general pattern observed empirically in Figure 8 for women born between 1912 and 1961Footnote ¹⁸.

Figure 7.

Distribution of women by number of children ever born, US women aged 45–49 of all races born from 1912 to 1961.

Source: CDC/NCHS, National Vital Statistics System.

Figure 8.

Distribution of women by number of children ever born, model results for Female Labor Force Participation rates ranging from 0.2 to 0.7.

5.3 Robustness of the results

For a curvature parameter (coefficient of relative risk aversion) close to 1, that is closer to the logarithmic utility function, the household behavior is consistent with the baseline results. However, they are sharper. At a low FLFPR, the SP-DSB is observed as in the baseline case, generating a dispersed distribution. At a higher FLFPR, the SP-DSB virtually disappears: due to the “flow” effect, all-boy households increase their completed fertility, and all-girl households reduce it due to the “stock” effect. These results are well predicted by the model, where most of the results are derived for the logarithmic case. However, for curvature parameters close to 0, the behavior changes. At a lower FLFPR, the SP-DSB is still observed as households are “prudent” and, given the high levels of uncertainty, are ready to “accumulate” girls. Unlike more concave cases, at a higher FLFPR, the “stock” effect does not necessarily dominate the “flow” effect, so for some values of parameters, an all-girl household may decrease then increase, increase, or maintain unchanged fertility. However, as expected, all-boy households increase fertility as they are affected only by the “flow” effect. The variation of the effect of an increased FLFPR on all-girl households is due to the fact that income and substitution effects coming from an increased FLFPR do not cancel each other out as in the logarithmic case and there is a complex interaction between them. This interaction is further complicated by the presence of the opportunity to invest in the human capital of children. This opportunity certainly plays a role as, for example, in the exogenous human capital case the “flow” effect dominates the “stock” effect at any level of the FLFPR. Note that, in the case of an almost flat utility function, the distribution of women by number of children, instead of becoming more concentrated, moves toward the right extreme and the average fertility increases.

5.4 Results on education

Although it is shown that the presence of endogenous human capital does not change the main results, it is still interesting to note what happens to the investment into human capital of children in the model. As expected at an extremely low FLFPR, the average investment in a girl's human capital is minimal, while boys get much more (almost 33 times more investment). This is true for both type-I and type-II households. The majority of girls are born with many siblings, thus in addition to small investment due to a low expected return from a girl's human capital, households with girls spread available resources over more children. As the FLFPR increases, the average investment in a boy's human capital decreases while investment in a girl's human capital increases. This continues until α = 1, at which point the average investment in human capital is identical for both genders. Figure 9 depicts the evolution of children's human capital investment in society consisting of type-I and type-II households.

Figure 9.

Investment in human capital of children, model results.