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# Economics, cultural transmission, and the dynamics of the sex ratio at birth in China

Edited by Paul R. Ehrlich, Stanford University, Stanford, CA, and approved October 24, 2008 (received for review July 15, 2008)

### This article has a correction. Please see:

## Abstract

In rural China, the ratio of newborn boys to newborn girls [sex ratio at birth (SRB)] has been rising for several decades, to values significantly above its biological norm. This trend has a number of alarming societal consequences, and has attracted the attention of scholars and politicians. The root of the problem lies in a 2,500-year-old culture of son preference. This culture is intricately linked with the economic reality of each couple's life, so that there are financial and psychological repercussions to parents who have no sons. To bring greater clarity and understanding to this issue, we present a quantitative framework that describes the interaction between economics and cultural transmission. We start with an explicit mechanism by which economic incentives can change cultural beliefs of a given individual, and go on to include a mechanism of cultural inheritance from generation to generation. We then show how economic conditions can affect the dynamics of cultural change in an entire society, and may lead to a decrease in the country's sex ratio at birth.

In most human populations, sex ratio at birth (SRB) (the ratio of newborn boys per newborn girls) is close to 1.05. SRB may vary with the number of children per birth, paternal age (1), and even season (2). Today, however, the most significant deviation of SRB from its biological value is due to a combination of socioeconomic factors in a number of countries of South and East Asia, including India and the People's Republic of China (3⇓⇓–6). The chief factor responsible for this deviation is a strong preference for sons (7⇓⇓–10). Although part of this preference may be due to economic necessity, for instance in rural areas where the need for manual labor is great, it is also traceable to deeply rooted practices and cultural beliefs (11). Examples are the continuation of the family lineage, restriction of certain funeral practices to sons of the deceased, and the widespread practice of virilocal marriage (in which a bride goes to live in her husband's household) (12, 13). Due to this preference for sons, many couples in South and East Asia face a difficult choice: either to risk having no sons and lose face in front of their friends and family, or to take deliberate steps to ensure that at least one of their children is male. Many couples make the second choice by means of sex-selective abortions and infanticide, thus contributing to the abnormally high ratio of male to female children in their countries (14⇓–16).

In China, the problem has been exacerbated by two additional factors: availability of ultrasound technology that allows the parents to determine the sex of a fetus (although this technology is illegal for sex determination, its use for this purpose is widespread) (17, 18), and governmental policies that have significantly reduced fertility (10, 19, 20). These factors contribute to an increasing SRB, and make modification of the culture of son preference all of the more important (21⇓⇓–24). China's government has been working to solve this problem in several ways: for instance, by discouraging the illegal use of ultrasound technologies, by providing financial incentives to couples that give birth to girls, or by directly facilitating cultural change away from son preference (12, 22). Here, we aim to model the relationship between cultural transmission and economic incentives to suggest conditions under which each of the above methods might be effective.

In models of transmission of son preference, a certain percentage of women choose to pursue sex selection to ensure that at least one of their children is male (25, 26). In the language of Li and colleagues (26), these women have cultural value π. The remaining women do not carry out sex selection (they have cultural value π_{0}). Thus, π and π_{0} correspond to the behavior of women with respect to sex selection.

Li *et al.* (26) suggest that if the transmission of π_{0} from parents to children or via the media is sufficiently strong, the extent of son preference might eventually decrease. If this occurred, then the strongly male-biased sex ratio in some Asian countries would tend back to normal. However, the model of Li and colleagues (26) does not incorporate the difference in economic value between sons and daughters, and thus does not address how fiscal incentives might affect the dynamics of cultural change.

Explicit economic values assigned to daughters and sons form the basis of the framework developed by Bhattacharjya *et al.* (27). However, this framework does not allow for cultural change that can alter these values. Rather, it describes the state of economic equilibrium to which the system tends under static conditions.

In this article, we propose a quantitative framework for the influence of financial incentives on cultural transmission of son preference. We discuss the application of this model to socioeconomic dynamics of sex ratio in China, and the conditions under which it reduces to a purely economic model (27).

## Results

### Cultural and Economic Value: σ and *m*.

As son preference can be traced to both cultural beliefs and economic necessity, we partition the value of a child into two separate components: economic and cultural. Economic value is that which is readily translatable into money. It includes the value of the child's manual labor over his or her lifetime, the support of parents in their old age, and any governmental subsidies the parents would receive toward raising the child.

Cultural value, in contrast, cannot be immediately converted into money and is dependent on the parents' cultural beliefs. It includes the value of continuing the family's surname in future generations, and the value of knowing that the family's ancestors will be remembered for years to come. We assume that a mother can translate the cultural value of her child into the currency of economic value and that shortly after its birth she accurately assesses the child's total value—cultural and economic combined.

We use σ and *m* to denote cultural and economic value, respectively, and measure the two types of value in the same monetary units. We also assume that cultural value (σ) is entirely marital—in other words, it is realized only if the child finds himself or herself a spouse. Conversely, economic value (*m*) is assumed to be completely non-marital, i.e., not contingent on his or her marital success. So far, all of the examples of economic and cultural value we have given fit these criteria. For a discussion of marital and non-marital value, please see *SI Appendix*, section A.

Let the cultural value of a married son be σ_{s}. Because cultural value is completely marital, for an unmarried son σ_{s} = 0. We assume that the ratio of males to females at birth (*r*) is >1, and equal to the male/female ratio at marriageable age. In other words, we ignore sex-differential mortality before marriage. We also assume that all marriages are monogamous, and that the marriage market is panmictic, so that every woman finds herself a husband, but a number of men do not find themselves wives. In this environment, the probability that a son gets married is 1/*r*. It follows that the expected amount of cultural value a son brings to his household is σ_{s}/*r*. This amount would be different if we had assumed a non-panmictic marriage market, but that would not change any of our qualitative results (*SI Appendix*, section A).

We assume that for mothers who do not follow the sex selection strategy, the cultural value of married sons is equal to that of married daughters: σ_{s} = σ_{d} = σ_{0}. For those who do follow the sex selection strategy, σ_{s} = σ_{1}, where σ_{1} > σ_{d} = σ_{0}. The probability that a daughter gets married in a panmictic marriage market is equal to 1. Consequently, the average amount of cultural value she will bring to her household is equal to σ_{0}.

The expected difference between the cultural values of sons and daughters to their mothers is Δσ, where
Note that σ_{1} and σ_{0} are constants, but the proportion of women with σ_{s} = σ_{1} can vary in time. For women with σ_{s} = σ_{0}, Δσ ≤ 0, and approaches zero as *r* approaches 1. This follows basic intuition: when *r* = 1 (i.e., sons and daughters are equally likely to marry), and a given woman believes that the cultural value of a married son is equal to that of a married daughter, we expect her difference in cultural value between sons and daughters to be zero. For women with σ_{s} = σ_{1}, the expected difference in cultural value can be either positive or negative, depending on the values of σ_{1} and *r*.

The economic value of sons, *m*_{s}, does not depend on cultural beliefs of individuals. Instead, *m*_{s} is determined by society-wide economic factors. As a result, its value is homogeneous in the society at any given time. Thus, we can define Δ*m*—a society-wide difference between the economic value of sons and that of daughters:
where *m*_{d} is the economic value of daughters. Because we assume that economic value is entirely non-marital, neither *m*_{s} nor *m*_{d} need to be discounted by the sex ratio.

Finally, we define total value difference (TVD) as the sum of Δσ and Δ*m*:

### Prejudice Toward Son Preference: ρ.

A mother's behavior with respect to sex selection is correlated with her σ_{s}—the cultural value she places on sons. We now introduce ρ—a separate, inherited marker of prejudice toward son preference. Unlike σ_{s}, ρ does not determine the mother's behavior directly.

After the birth of her children, a mother assesses the economic reality in the surrounding society, combines that with her own cultural beliefs, and evaluates the TVD between a son and a daughter (Eq. **3**). If TVD is greater than some threshold, she forms prejudice toward son preference (ρ = 1). If it is less than that threshold, her prejudice will be against son preference (ρ = 0). We normalize Δ*m* in such a way that this threshold value is zero:
Every woman holds values for σ_{s} and ρ, and every man—a value for ρ. Fig. 1 depicts the full inheritance scheme for σ_{s} and ρ. We can use it to trace one path in the inheritance process through 3 generations, starting with σ_{s} of woman *A*. The σ_{s} of *A* is translated to her daughter's ρ via *TVD* during the early stage of the daughter's life, when much of her learning is intuitive and direct (28). During a later, conceptual learning stage, the σ_{s} that the daughter of *A*learned from her mother is modified according to her own ρ, her husband's ρ, and the societal ρ (i.e., prevalence of ρ = 1 in the entire parental generation). She then transfers this modified σ_{s} to her own daughter–the grandmother of *A*.

### Translation from σ_{s} to ρ.

TVD can take on one of two possible values: one for σ_{s} = σ_{1}, and one for σ_{s} = σ_{0} (see equations 1 and 3). Consequently, there are three possible modes of translation from σ_{s} to ρ via Eqs. **3** and **4**. When *r*, the sex ratio, is close to 1, both possible values of TVD are greater than zero, so that all children receive ρ = 1 (Eqs. **3** and **4**). We call this mode M_{1}. When *r* is large, many men cannot find wives, so that both values of TVD are less than zero, and all children receive ρ = 0. This is mode M_{0}. Finally, in mode M_{σ}, TVD ≥0 for mothers with σ_{s} = σ_{1} and TVD <0 for those with σ_{s} = σ_{0}, so that the cultural belief of a mother is fully correlated with the prejudice value inherited by her child.

To obtain equations for the boundaries between the three modes of translation, we let TVD = 0 with σ_{s} = σ_{0} and σ_{s} = σ_{1}:
and
Values of *r* less than the right hand side of Eq. **5** correspond to mode M_{1}, values of *r* between the lines in equations 5 and 6 correspond to M_{σ}, and those greater than the right hand side of Eq. **6** to M_{0} (Fig. 2).

### Dynamic Variables.

We study the dynamics of change in *r* for a given value of Δ*m*. To do so, we define variables *x*_{10}, *x*_{01}, *x*_{00}, *x*_{11}, *y*_{0}, *y*_{1}, and *z*_{1}. The first 6 variables keep track of the proportions of women and men with different combinations of cultural beliefs (σ_{s} = σ_{0} or σ_{s} = σ_{1}) and prejudice values (ρ = 0 or ρ = 1) (see Table 1). *z*_{1} is the overall proportion of people that have ρ = 1:

### Cultural Conversion.

A mother's σ_{s} can be modified to follow with her own, her husband's, and the societal ρ during transmission to her daughter (Fig. 1). The probability of such conversion is determined by parameters C_{01}, C_{10}, V_{01} and V_{10}. V_{01} and V_{10} come into play in families where the prejudice of both parents is in conflict with the mother's behavior. For instance, in a family where ρ = 1 for both parents, and the mother's σ_{s} = σ_{0}, the σ_{s} she transfers to her daughter will be modified to σ_{1} with probability V_{01}. Conversely, in a family where both parents hold ρ = 0 and the mother holds σ_{s} = σ_{1}, the probability that her daughter's σ_{s} will be σ_{0} is V_{10}. Parameters C_{01} and C_{10} are relevant when prejudice values of the two parents are at odds with each other (i.e., one parent holds ρ = 1 and the other ρ = 0), so that the probability of cultural conversion is determined by the prevailing societal prejudice (*z*_{1}). Specifically, if the mother's σ_{s} is equal to σ_{0}, the probability that it will change to σ_{1} is C_{01} × (1 − *z*_{1}). Conversely, if the mother's σ_{s} is σ_{1}, the probability of change in σ_{s} as it is transmitted from mother to daughter is C_{10} × *z*_{1}. Definitions of C_{01}, C_{10}, V_{01} and V_{10} are summarized in Table 2.

### Recursion Equations in M_{σ}.

Suppose the proportion of women with σ_{s} = σ_{1} and ρ = 1 is *x*_{11}^{t} at time *t*, and the proportion of men with ρ = 1 is *y*_{1}^{t}. Then the proportion of marriages in which the bride has σ_{s} = σ_{1} and ρ = 1, and the groom has ρ = 1, is the product *x*_{11}^{t} × *y*_{1}^{t}. We follow a similar procedure to write down the proportions of the other 7 possible combinations of male and female traits in marriages at time *t* (Table 3, column 4). We then calculate the proportions of male and female progeny with different combinations of σ_{s} and ρ for each marriage type using the definitions of parameters C_{01}, C_{10}, V_{01} and V_{10} (Table 3, columns 5–10). Finally, we obtain *x*_{11} at time *t* + 1 by multiplying the probabilities of the 8 possible marriage types (Table 3, column 4) by the corresponding probabilities that a female child will have ρ = 1 and σ_{s} = σ_{1} (Table 3, column 5), and adding up the resulting products. We obtain *x*_{10}, *x*_{01} and *y*_{1} via a similar process, by cross-multiplying the probabilities of marriage types (Table 3, column 4) with the corresponding probabilities of offspring types (Table 3, columns 6, 7 and 9, respectively):
*z*^{t}_{1} in Eq. **8** can be expressed in terms of other variables (for the derivation, see *SI Appendix*, section B):
Given a number of simplifying assumptions, *r* at time *t* + 1 can be expressed as a function of x_{11} and x_{01} at time *t* (*SI Appendix*, section B):
Together, equations 8 through 10 define the behavior of the system in mode M_{σ}, and can be numerically iterated for different sets of parameters C_{01}, C_{10}, V_{10} and V_{01}. When the parameters are such that the probability of conversion to σ_{0} is relatively high and the probability of conversion away from σ_{0} is relatively low (in other words, C_{10} and V_{10} are large, whereas C_{01} and V_{01} are small (see Fig. S1), *r* tends to increase with time. However, if the former probability is decreased and the latter is increased, *r* can decrease instead (Fig. S2).

Recall that in mode M_{0} (Fig. 2), *r* is so high that everyone has intuitive preference ρ = 0. Thus, we do not expect any conversion to σ_{s} = σ_{1}. In addition, if the rate of conversion to σ_{0} is non-zero, we expect the proportion of women with σ_{s} = σ_{1}, and sex ratio *r*, to decrease. Accordingly, either the society will reenter mode M_{σ}, or *r* will reach 1 (see Fig. S2). Similarly, if the society finds itself in mode M_{1}, it will tend toward increasing values of *r*. In this case, either the society will reach mode M_{σ}, or *r* will reach the maximum value of 5/3. For a quantitative treatment of these processes, see *SI Appendix*, section C.

## Discussion

### Reducing to a Model of Economic Equilibrium.

Bhattacharjya *et al.* (27) constructed a purely economic model that predicts the equilibrium sex selection potency (i.e., the proportion of women who have behavior π) and SRB (i.e., *r*), given a set of values attributed to married and unmarried sons and daughters (27). These authors distinguish between the value *v*_{m} of a married son, the value *v*_{u} of an unmarried son, and the value *v*_{d} of a daughter. In our model, these values are equal to *m*_{s} + σ_{s}, *m*_{s}, and *m*_{d} + σ_{d}, respectively. Bhattacharjya's model (27) does not allow for the possibility of cultural variation and transmission. In terms of our model, this means that σ_{s} = σ_{0} for all women. Thus, we can rewrite *v*_{m}, *v*_{u} and *v*_{d} as *m*_{s} + σ_{0}, *m*_{s} and *m*_{d} + σ_{0}. As a result, cultural transmission mode M_{σ} in Fig. 2 collapses to a single curve, and the society tends to an equilibrium SRB defined by this line (Fig. S3).

In (27), the equilibrium sex selection potency *p** is defined as the equilibrium proportion of women with behavior π, and
where *T* is the total fertility rate and *y* is the probability that a woman is in a sex selection situation. In our model, *T* = 2, and *y* is the probability that a woman's first child is a girl, i.e., 1/2. By substituting 2 for *T*, 1/2 for *y*, *m*_{s} + σ_{0} for *v*_{m}, *m*_{s} for *v*_{u} and *m*_{d} + σ_{0} for *v*_{d} (see above) in Eq. **11**, we obtain
When the width of mode M_{σ} is zero (Fig. S3), the boundary between modes M_{0} and M_{1} is defined by Eq. **6**:
where *r** is the equilibrium SRB to which the system will tend.

Also, from Eq. **10**, we can deduce that
under equilibrium conditions. The inverse of Eq. **14** is
By substituting Eq. **13** into Eq. **15**, we obtain
which is the same as the equilibrium potency derived in (27) (Eqs. **11** and **12**).

### Resemblance to a Model of Cultural Transmission.

When the society is in mode M_{σ}, our model closely resembles a model of cultural transmission described in ref. 26. In both models, the total fertility is equal to 2, and the biological sex ratio at birth is assumed to be 1:1. Unlike Li *et al.* (26), however, we include the possibility of conversion from behavior π_{0} to behavior π, and do not allow cultural transmission due to mass media. Our cultural conversion parameters V_{10} and C_{10} are similar to the parameters that determine the rates of vertical and oblique transmission of Li *et al.* (26).

### Interpreting Societal Observations in Terms of the Model.

We now present a handful of socioeconomic observations related to sex ratio at birth in China, and interpret them in the context of our model.

#### Recent studies on the patterns of marriage and son preference in China have focused on a pair of counties in Shaanxi province: Sanyuan and Lueyang (26, 29, 30).

Sanyuan is a densely populated county where large family clans have a strong influence on village life, and villages maintain a strict patrilineal family system. Lueyang, however, is sparsely populated, and the patrilineal family system is much more relaxed. SRB in Lueyang between 1990 and 1996 was at 105.0, below the national average and close to the biological norm. At the same time, in Sanyuan SRB was at 117.2, much higher than in Lueyang and above the national average.

One possible reason for this discrepancy is that in Lueyang, the perceived cultural value of sons does not vary as much as it does in Sanyuan (i.e., σ_{1} − σ_{0} is smaller, see Fig. S4). As a result, the boundary between M_{0} and M_{σ} may be farther to the right, which may push SRB to lower values. Additionally, SRB in Lueyang may be lower due to a lower value of Δ*m* (Fig. S5).

#### Widely available ultrasound technologies that can determine the sex of a fetus exacerbate China's sex ratio problem (10, 17).

In response, the Chinese government has instituted punishment for the use of ultrasound to exercise sex selection. To explore this phenomenon in terms of our model we can define an arbitrary threshold value difference *C* in Eq. **4**:
*C* is the cost of sex-selection. When ultrasound technologies are made available, *C* is decreased. When the use of these technologies is made punishable by law, *C* is increased, the boundaries of cultural transmission mode M_{σ} shift to the right, and the value of SRB is pushed downwards (Fig. S6).

#### Between the years 2000 and 2003, a number of Chinese and international agencies participated in a program entitled “Chaohu Experimental Zone Improving Girl-Child Survival Environment.”

Its purpose was to improve the chances of survival for girls in Chaohu city of Anhui province via a number of reproductive-health training and social-development activities. As a result of this program, SRB in Chaohu decreased from >125 in 1999 to 114 in 2002 (22). Its greatest effect was likely to raise the awareness of the total cost and benefits of sons, daughters, and of following the strategy of sex selection—thus making the purely economic model (Fig. S3) appropriate for the women who gave birth between 2000 and 2003.

It also could have decreased SRB by reducing Δ*m* (e.g., via the enhancement of the social security system), by shifting the boundaries between M_{0} and M_{1} in the direction of increasing Δ*m* (e.g., via punishing those who commit sex-selective abortions and infanticide (Fig. S6), or via a combination of these factors.

## Conclusion

Our findings help conceptualize the socioeconomic processes that can alter the sex ratio in today's China, and hence have implications on the future governmental policies directed to bring the requisite societal change. We show that policy makers wishing to reduce female selective abortion or infanticide have two levers. One is economic incentives that alter Δm and the other is cultural changes operating on Δσ. It seems reasonable that the framework developed here can be modified to become more appropriate to the situation in India, where the dowry custom is prevalent in many areas and where the sex ratio is also extremely male-biased. Future work will build stronger connections between theoretical models of economics and cultural transmission and real-life observations.

## Materials and Methods

Simulations of this article's dynamic model were performed using Mathematica software (31). The resulting plots were annotated using Graphic Converter X (32). *SI Appendix*, section A contains a brief discussion of the model behavior under a localized marriage market. *SI Appendix*, section B contains a derivation of SRB and the proportion of people with prejudice toward son preference in terms of other dynamic variables. *SI Appendix*, section C contains a derivation of the recursion equations under dynamic modes M_{0} and M_{1}. Figs. S1–S6 provide additional insight into the behavior of the theoretical model.

## Acknowledgments

We thank Shripad Tuljapurkar, Jeremy Van Cleve, Deborah Rogers, Daniel Weissman, Laurent Lehmann, Amanda Casto, and Peter Graham for useful comments on the mathematical model and Zhongshan Yue, Melissa Brown, Arthur Wolf, and Hill Gates for insight into the anthropology of sex ratio in the People's Republic of China. This work was supported by National Institutes of Health Grant GM28016 (to M.W.F.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: lipatov{at}stanford.edu

Author contributions: M.L., S.L., and M.W.F. designed research; M.L. and M.W.F. performed research; and M.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0806747105/DCSupplemental.

- Received July 15, 2008.

- © 2008 by The National Academy of Sciences of the USA

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