Journal of Bioeconomics (2006) 8:253–268 DOI 10.1007/s10818-006-9006-x

© Springer 2006

Gender Imbalance: The Male/Female Sex Ratio Determination YONG J. YOON Department of Economics and Center for Study of Public Choice, George Mason University, Fairfax, VA 22030, USA Department of Economics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061, USA ([email protected])

Synopsis: The aim of this paper is to develop a rational choice model for male/female sex ratio determination. The equilibrium sex ratio at birth is determined by the preference for sons and birth rates. The conventional theory predicts that low birth rate will increase the disparity between male and female births. However, our model predicts that low birth rate will decrease the sex ratio at birth. The argument has relevance both for positive and normative aspects of demographic composition, as well as other applications. Key words:

sex ratio at birth, fertility, rational choice, tragedy of the commons, optimal sex ratio

JEL classifications: D10, D71, J10, J13, J18

1.

The male/female sex ratio problem

Other than in arid academic discussions, the issue of the male/female sex ratio at birth did not attract much attention until the 1970s when new fertility medicine made gender selection possible. The aim in this paper is to develop a rational choice model for sex ratio determination under these circumstances. The sex ratio of boys to girls at birth (B/G ratio, henceforth) strikes a biologically natural balance with consistency in all human populations: about 103 to 106 boys are born for every 100 girls.1 But increasingly, there are exceptions since 1980s. The B/G ratios in some East Asian countries—China, India, South Korea and Taiwan—have been unusually high.2 Most authors agree that the skewed ratios can be accounted for by a strong preference for sons and the low cost of medical technology (e.g., sonogram) that is used to enable sex-selective abortions (Park & Cho 1995, Fukuyama 2002). We recognize that the problem of gender imbalance is an example of the tragedy of the commons problem. The argument has relevance both for positive and normative aspects of demographic composition. The natural B/G ratio generates a stable equilibrium for matching in marriage. But the persistence of the high B/G ratio for two decades causes concern for the unmarried males. The imbalance in the marriage market could be worsened

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by the decreasing birth rates in the region, so the argument goes. Some authors (Hudson & den Boer 2002, Wilson 2002) predict social and even international disturbance of peace as a consequence of a high B/G ratio (in China) and the existence of many unmarried men. This paper offers no opinion about such concerns. Instead, we note that the misery of unattached males, men who cannot find wives though they want to marry, is an obvious result of this gender imbalance. We also note that there is no unanimous agreement among authors on whether it is bad to rely on gender selection technologies (Singer & Wells 1985). In the following sections, I show that, aside from ethical considerations, the B/G ratio determination is more complex than simple extrapolation techniques would predict. There are two opposing predictions about the persistence of a high B/G ratio. Some economists and demographers accept the argument that the B/G ratio is self-correcting when the marriage market works effectively (Simpson 1979, Singer & Wells 1985, Becker 1991, Park & Cho 1995). According to this theory, if the B/G ratio at birth becomes unbalanced, the under-represented sex will be in relatively greater demand as marriage partners (Angrist 2002). The parents will then respond to this demand by producing more offspring of the under-represented sex. If the B/G ratio is self correcting in this manner, there is no need for concern. However, others (Chu 2001, Arnold et al. 2002) concern empirical questions whereas Edlund (1999) describes a model with some parallel to that presented below in this paper.3 Demographers also claim that, when males marry younger females, a low birth rate increases the imbalance of B/G ratio (Park & Cho 1995, Eberstadt 2004). In contrast, my model predicts that the sex ratio imbalance will be lower, rather than higher, when birth rate is lower. Parents hope their sons will marry and know that too many male births will limit this chance. My model incorporates parents’ preference over the marital status of children while conventional theory considers preferences for sons per se. I develop, in the following section, a rational choice model for determination of the B/G ratio and show that, when parents have a preference for sons and gender selection technologies are available, the ratio is not self correcting. This is true even when parents have a preference for married children over unmarried children. The equilibrium sex ratio is supported by a stable Nash equilibrium in which each couple uses a mixed strategy. Based on the model, I show that the Pareto superior B/G ratio is the natural sex ratio at birth that is balanced. The high B/G ratio is a consequence of individuals acting independently analogous to the tragedy of the commons (Hardin 1968, Ostrom 1990, Buchanan & Yoon 2000). If couples in the polity were to choose a rule, they would agree unanimously to maintain the natural sex ratio at birth. The externality argument points to Hardin’s (1968) alternative solution to the problem of the tragedy of the commons: government regulation outlawing gender selection technology.

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2.

255

Model for sex ratio determination

There is some evidence that son preference, partly biology-based, is a phenomenon observed across time and cultures (Winston 1932, Singer & Wells 1985). However, even if parents can control gender of offspring, Simpson (1979) and others argue that the sex ratio imbalance will be temporary. If parents expect that the underrepresented sex will become more in demand as marriage partners, the balance will be restored rather quickly. If indeed the anomaly of a high B/G ratio were of a temporary nature, it would be best to leave the issue to the parents’ discretion. The mechanism supporting Simpson’s self-correction hypothesis is the altruism of the parents who care about the marital status of their children. From the sociobiological perspective, parents may want to continue their genes, either through a son or a daughter. Thus, in an environment of a high B/G ratio, some parents would prefer to give birth to girls who are more likely to marry. In this sense, ‘altruism’ is equivalent to maximizing reproductive success. Furthermore, the selfcorrection adjustment will not take long if parents’ expectations are rational and forwardlooking. The self-correction hypothesis, however, is not consistent with our observations in East Asia where high B/G ratios have continued over twenty years. Conventional theory interprets son preference as preferences for male children per se, but I claim that parents prefer married sons. That is, if parents knew that the child would never marry and would not carry their genes into the future, preferences would be similar to those observed in child adoption, and, according to personal communication with the staff at Holt International, adoptive parents tend to prefer girls over boys.4 Thus, I formulate a rational choice model for son preference which is influenced both by cultural and biological factors. Parents have a utility function u over gender and marital status of the children. If parents prefer a married son (m∗ ) to a married daughter (f∗ ), and a married daughter (f∗ ) to an unmarried son (m) or daughter (f), then the preference ordering is: u(m∗ ) > u(f∗ ) > u(m) ? u(f)

(1)

The parental preferences for unmarried children, m or f, are left undetermined. For the sake of argument, I assume that every man and woman wants to marry, and they marry by random mating. For simple exposition, I also assume that the natural B/G ratio is 100:100; equal number of boys and girls are born in each period. I first calculate the equilibrium B/G ratio for the case of replacement birth rate that maintains a constant population. To understand the effect of birth rate on the B/G ratio, I then analyze birth rates other than the replacement level.

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Replacement level birth rate

Consider a population consisting of many identical couples who produce at replacement rate. Given the population B/G ratio, we can calculate the probability that a son will marry. We let s denote the fraction of boys from children at birth and use it as a measure of the B/G ratio: the ratio s of males to (1 − s) females. Then, if there are more boys than girls, the probability p that a boy will marry would be equal to the ratio of girls to boys; if there are more girls than boys, boys will marry with certainty: If s ≤ ½, then p(s) = 1; and If s > ½, then p(s) = (1 − s)/s.

(2)

The corresponding probability q(s) that a girl will marry is If s ≤ ½, then q(s) = s/(1 − s); and If s > ½, then q(s) = 1.

(2.a)

Figure 1 illustrates probability functions p(s) and q(s) over the interval [0, 1]. The parents’ expected utility from a male child is Eu(m) = p(s)u(m*) + (1 − p(s))u(m)

(3)

where the italic m denotes a random variable that assumes either a married son (m∗ ) or an unmarried son (m). The expected utility Eu(f ) from a female child is similarly defined: Eu(f ) = q(s)u(f*) + (1 − q(s))u(f).

(3.a)

If parents let nature take its course, the probability of producing a son is one half which is the natural B/G ratio. The expected utility in this case is Eu(m)/2 + Eu(f )/2.

(4)

If the expected utility from a male child is greater than the expected utility in (4), parents will use gender selection technologies to secure a son: Eu(m) > Eu(m)/2 + Eu(f )/2, or equivalently

(5)

Eu(m) > Eu(f ). Note that condition (5) is equivalent to the convenient form Eu(m) > Eu(f ) : parents will try to secure a boy when the expected utility from a male child is greater than that from a female child. Parents’ decision may depend on the psychological barrier of aborting fetus as well as the costs of using reproductive technology. It may also depend on the legal penalty if such practice is banned by law. Thus, the inequality in (5) is a necessary, rather than a sufficient, condition for securing a son. For a simple exposition, I ignore the cost aspects of gender selection.

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p(s)

1

1/2

1

s

q(s)

1

1/2

1

s

Figure 1. Probability of marrying: p (sons) and q (daughters).

An equilibrium in this model can be characterized by a population B/G ratio s∗ at which the couples are ambivalent about using gender selection methods; or, equivalently, parents become indifferent between a male child and a female child, Eu(m) = Eu(f ).

(6)

We may treat Eu(m), the left side of (6), as a function of s the B/G ratio, and denote it by M(s). Then, from (2) and (3), for s ≤ 0.5, M(s) = u(m*) is a constant; and for s > 0.5, M(s) = p(s)u(m*) + (1 − p(s))u(m) is a decreasing function of s. Likewise, we consider Eu(f ) as a function of s and denote it by F(s). Then, for s ≤ 0.5, F(s) = q(s)u(f*) + (1 − q(s))u(f) is increasing in s; and for s > 0.5, F(s) = u(f*) is a constant. The equilibrium B/G ratio s∗ occurs when the two curves M(s) and F(s) cross. As is illustrated in Figure 2, F(s) crosses M(s) from below at a B/G ratio higher than 0.5. To calculate the population B/G ratio s∗ at equilibrium, I first calculate p∗ , the probability of sons marrying when condition (6) is satisfied or parents become indifferent between producing a male child and a female child. Since the B/G ratio

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utils

U(m*)

M(s)

F(s)

U(f*)

U(f)

U(m) 1/2 s*

Sex ratio 1

Figure 2. M(s) and F(s) cross at s∗ .

s∗ is greater than one half, more boys than girls will be born and girls will be sure to marry. The expected utility from a girl child is Eu(f ) = u(f*), and p∗ can be obtained from (3) and (6), p*u(m*) + (1 − p*)u(m) = u(f*), or p* = [u(f*) − u(m)]/[u(m*) − u(m)].

(7)

Note that the term [u(m*) − u(m)]/[u(f*) − u(m)] is a measure of son preference, which we denote by a parameter π : π = [u(m*) − u(m)]/[u(f*) − u(m)].

(8)

Parameter π increases in u(m*) − u(m) and decreases in u(f*) − u(m), indicating that the opportunity cost of producing a boy is the forgone opportunity of producing a girl that marries. This suggests that parameter π incorporates the preference for married children over unmarried children, and not a preference for boys per se. We also note that the son preference π and the results we obtain do not depend on parents’ preferences over unmarried children; whether u(m) is greater or lesser than u(f) is irrelevant. By combining (2) and (8), we can obtain the equilibrium ratio s∗ corresponding to p∗ . Since p∗ = 1/π < 1, the first part of inequality in (2) applies and we obtain5 s* = 1/(1 + p*) = π/(1 + π ).

(9)

The ratio s∗ in (9) is greater than 0.5 (π being greater than 1), as we already predicted from the discussion of Figure 2. To justify that s∗ is an equilibrium B/G ratio, it suffices to show two things. First, once the B/G ratio reaches s∗ , the utility maximizing behavior of the couples will be such that the ratio will remain at s∗ . Second, if the ratio is other than s∗ it will converge to s∗ .

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Suppose the current B/G ratio is s∗ . Then, by condition (6), the couples (parents) become indifferent between a male and a female child, and do not have the incentive to control gender of offspring. Consequently, the B/G ratio will start falling toward the natural ratio of 0.5. While the B/G ratio falls, the probability p(s) of sons marrying increases, and the utility maximizing parents will start to produce more sons than girls, etc. This possibility asks for an equilibrium concept in which individual couples use a mixed strategy. With probability r, 0 < r < 1, each couple uses gender selection technologies to secure a male child. With probability 1 − r they let nature take its course. Since the natural B/G ratio is balanced at birth, each couple will end up producing a male child with probability r + (1 − r)/2. Thus, to be consistent with the population B/G ratio at equilibrium, the mixed strategy r satisfies, r + (1 − r)/2 = s* or r = 2s* − 1.

(10)

Thus the equilibrium characterized by the mixed strategy, r = 2s* − 1, is a sustainable equilibrium. The dynamics of the B/G ratio can be analyzed by using (5), the condition for controlling gender of offspring. If the current ratio is below 0.5, boys will marry for certain and parents will be willing to produce boys. The B/G ratio rises. At a ratio between 0.5 and s∗ , females are under-represented, yet parents might produce boys by using gender selection methods. The B/G ratio is bound to rise toward s∗ . At a ratio above s∗ , the utility maximizing behavior is either to produce girls or to follow nature’s course, until the B/G ratio drops to s∗ . We summarize these results as propositions. Proposition 1. Suppose parents have a preference for sons as represented by (1) and gender selection technologies are available. Then the B/G ratio will converge to the equilibrium ratio s ∗ = π/(1 + π), where π is the son preference parameter in (9). More boys than girls will be born. The equilibrium ratio s∗ is sustained by couples using a mixed strategy in (10). We note that the mixed strategy in Proposition 1 allows an alternative interpretation. A population may be polymorphous, consisting of different types that use pure strategies, i.e. some family types have boys and others have girls.

2.2.

Low birth rate

Most demographers (Park & Cho 1995, Eberstadt 2004) argue that lower fertility will worsen the sex ratio imbalance. Males usually marry younger females and thus, when birth rate remains below replacement level, fewer females are available for males to marry. However, my model predicts that lower fertility will be

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associated with a decreased equilibrium B/G ratio. The different result obtains because parents in my model recognize that too many male births will reduce the chance that their sons will marry. In my model, the B/G ratio is determined as a result of choices made by parents who recognize that males generally marry females two to five years younger. Suppose, at each period t, Nt children are born and fewer children are born next period, Nt+1 = kNt (k < 1), such that the birth rate is below replacement level. Let Mt and Ft denote the number of male and female children, respectively, born at period t: Nt = Mt + Ft . We assume that males marry females one period younger. Then the probability pt that a boy will marry is equal to the number of girls Ft+1 born next period divided by the number of boys Mt born this period. In particular, if Mt ≤ Ft+1 , then boys are sure to marry, pt = 1, while Mt > Ft+1 implies pt < 1. In expressing the probability pt as a function of the B/G ratio, we note that males born this period is Mt = st Nt , where st is the fraction of boys born in period t, and females born one period later is Ft+1 = (1 − st+1 )Nt+1 = (1 − st+1 )kNt . Thus pt is a function of st and st+1 . However, for a steady state analysis, we consider stationary B/G ratio s = st = st+1 . Then, the condition Mt ≤ Ft+1 implies sNt ≤ (1 − s)kNt or s < k/(1 + k), and the relationship between the stationary s and the probability for sons marrying p(s) becomes If s < k/(1 + k), then p(s) = 1; and If s > k/(1 + k), then p(s) = k(1 − s)/s.

(11)

At equilibrium parents become indifferent between producing a male and a female child, and the probability p∗ of sons marrying is again determined by (7). Let s∗ (k) denote the B/G ratio at equilibrium when the birth rate parameter is k. Then, since the probability p∗ = 1/π is less than 1, equilibrium values of s∗ (k) and p∗ satisfy the second part of (11): p* = k[1 − s*(k)]/s*(k) or s*(k) = k/(k + p*) = kπ/(kπ + 1).

(12)

If k = 1 in (12), we obtain the ratio s* in (9) for the replacement case. If the birth rate is as low as k = 1/π , the equilibrium ratio becomes the natural sex ratio at birth. In this case parents will have no incentive to control the gender of offspring. Regardless of the fertility rate, parents behave as if they try to maintain a constant level of probability for sons marrying (p∗ = 1/π), so that the male/female sex ratio at marriage will remain constant. It is no more difficult for boys to marry when birth rate is lower or higher than the replacement level. Also note that the equilibrium B/G ratio s∗ (k) in (12) decreases as birth rate (k) falls. This result is against the conventional wisdom in demography. For a comparison with my model, Appendix A provides a simple model for conventional theory, which illustrates that the B/G ratio rises as birth rate falls. Subsection 2.3. provides a numerical example that contrasts the two models.

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Table 1. Fertility and the sex ratio at birth

Year

Sex ratio1 at birth below age 5

Population2

Births3

1981 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

107.2 109.4 111.7 108.8 113.3 111.8 116.5 112.4 113.6 115.3 115.3 113.2 111.7 108.4 110.1 105.6 106.2 109.0 —

— 3,845 — — — — — — — — — 3,545 — 3,524 — 3,349 3,259 3,161 3,048

865 663 — — — — 659 — — — — 727 — — — — 605 569 495

Note: Unit for 1: boys per hundred girls; Unit for 2 and 3: thousand persons. Source: National Statistics Office, Republic of Korea.

The current birth statistics in South Korea is worth mentioning. As shown in Table 1, the birth rate in South Korea has been falling last twenty years. We compare the birth statistics for the two periods: 1985 through 1994 and 1995 through 2002. During the first ten years, 1985 through1995, the birth rate was falling at one percent per year or about 30 000 newborns per year. The ten year average B/G ratio is 112.8. Between 1995 and 2002, the birth rate has been falling at two percent per year or 71 000 newborns per year. The average B/G ratio in this period is 109. The B/G ratio decreases as the birth rate falls! The numbers, however, could be accidental and it is premature to make any conclusion, thus I do not necessarily claim that the Korean data support the model’s prediction. The result in this subsection is stated as a proposition Proposition 2. Suppose parents have a preference for sons as represented in (1) and gender selection technologies are available. The B/G ratio converges to the equilibrium ratio s ∗ (k) = kπ/(kπ + 1); and B/G ratio s∗ (k) decreases as the birth rate (k) falls. Proof. Proposition 1 outlined basic argument for convergence for the replacement case, k = 1. Thus, for a detailed proof, I restrict consideration to the B/G ratio

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st, st+1

g(s)

45˚ s

s*

sex ratio

Figure 3. The sex ratio dynamics. Note: g(s) is the sex ratio change as a result of the effort to produce a male child.

s over the interval [0.5, s∗ (k)] and birth rate parameter k > 1/π. In this interval, though females are under-represented and are sure to marry, the expected utility from a female child will be less than that from a male child: u(f*) < Eu(m) = p(s)u(m*) + (1 − p(s))u(m). The effort to produce a son will depend on the utility gap Eu(m) − u(f*), and the resulting increase in the B/G ratio g(s) will depend on this effort. (More effort can be interpreted as involving a mixed strategy that uses gender selection methods with a higher probability.) For current ratio st , the next period ratio is st+1 = st + g(st ), and st+2 = st+1 + g(st+1 ), etc. As is illustrated in Figure 3, we have an increasing sequence st < st+1 < st+2 < . . . Since Eu(m) is decreasing in s, the utility gap Eu(m) − u(f*) and ratio increase g(s) decrease in s over the interval of interest and become zero when s = s*(k). The difference in the B/G ratio between two adjacent periods, st+1 − st = g(st ), converges to zero in the limit when s = s*(k). Thus the ratio converges to s∗ (k). Q.E.D.

2.3.

Numerical example

Conventional theory claims that, when parents have a preference for sons, declining birth rate will increase the sex ratio at birth. Appendix A provides a simple model that supports such a possibility. The behavioral assumption is that parents try to secure at least one son. The B/G ratio is 0.54 for the high birth rate case when each couple produces three children; and the ratio is 0.625 for the low birth rate case when each couple produces two children. Indeed, the B/G ratio increases as the birth rate falls.

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To illustrate my model’s predictions for the same population, we need reasonable numbers for numerical manipulation. For this purpose, I assume that the population consists of two groups: half of the couples in the population produce two children per couple and the other half produce three children per couple. Then, the birth rate parameter is k = 1.25, and the population sex ratio at birth is s = 0.5 × 0.54 + 0.5 × 0.625 = 0.575. Based on this data, the corresponding son preference parameter π can be obtained by substituting k = 1.25 and s = 0.575 into (12): s π= = 1.08. k(1 − s) For a population consisting of couples with son preference π = 1.08, I calculate equilibrium B/G ratios for different birth rates. If each couple in the population produces three children, the high birth rate case, the birth rate parameter is k = 1.5 and the B/G ratio is from (12): s*(1.5) = (1.5 × 1.08)/(1 + 1.5 × 1.08) = 0.61. For the low birth rate case in which each couple produces two children, the B/G ratio becomes: s* = π/(1 + π) = 0.519.

3.

Optimal sex ratio

There is no point in imagining contracts between parents and unborn infants regarding gender selection. However, the model developed in this paper suggests that the currently producing couples have an incentive to agree to a rule that requires a lower B/G ratio than the Nash equilibrium ratio s∗ . We may imagine the population as a common marriage pool. If each couple (parents) decides independently, the outcome is a high B/G ratio of s∗ at which parents’ valuation of the marriage pool is lower than that at the balanced B/G ratio of 0.5. This situation is analogous to ‘the tragedy of the commons’ that has been analyzed by Gordon (1954), Hardin (1968), Ostrom (1990), and Buchanan & Yoon (2000, 2001) among others.6 To be more formal, suppose that the B/G ratio is set at s by a binding rule. The value of the marriage pool to an individual couple is the expected utility from using a mixed strategy that assigns the couple a probability of s to having a son and a probability (1 − s) to having a daughter. Then the expected utility v(s) is v(s) = sEu(m) + (1 − s)Eu(f ).

(13)

The function v(s) is increasing over the interval [0, 0.5] and decreasing over interval [0.5, 1]; thus v(s) obtains a maximum at s = 0.5. See Figure 4. Appendix B provides an algebraic derivation of the result. For the parents, as well as offspring, the optimal B/G ratio is the natural and balanced ratio 0.5. Thus unless ‘demography is destiny’ as Auguste Comte claimed, parents would

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v(s)

1/2

s* v(s) = sEu(m) + (1-s)Eu(f)

Sex ratio

Figure 4. Optimal sex ratio and value of the common marriage pool.

unanimously agree that the B/G ratio be regulated to equal the balanced natural ratio. The Pareto superior move is possible if all agree on outlawing gender selection technology.

4.

Son preference and legislation

It is well known that any culture with an agricultural tradition has the tendency to prefer male children to help the parents.7 But in industrialized societies, like South Korea or Taiwan, economic condition do not seem to explain son preference. A son is no more valuable than a daughter to the parents. As an alternative explanation, cultural settings should be considered. In East Asian countries like South Korea and Taiwan, we may argue that son preference is based on the Confucian tradition of family lineage; getting a male child to carry on the name. In Confucian societies, as Mote (1989, p. 22) notes, ‘even in the eyes of the state, filial responsibility had priority over loyalty to rulers and state.’ A similar cultural condition at one time existed in the West as we see from Hume’s (1758) discussion of primogeniture as an institution for resolving rent-seeking within the family.8 But there is a twist to this argument. Japan shares the same Confucian heritage, yet is free from gender imbalance. We note that in the Japanese society, through adult adoption, the family line can be continued by married females, as well as by males (Posner 1992). Furthermore, current Japanese family law allows children to assume either the mother’s or father’s surname. The Korean family law specifies that children’s surname be that of the father. Social change and people’s behavior is difficult to predict, but Japan’s experience provides a lesson. Legislation that allows flexible use of family names may

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influence the son preference parameter in (8) and reduce the B/G ratio. If parents understand that the family line can continue through daughters as well as sons, there will be less reason for preferring a male child.

5.

Conclusion

Though biologists and social scientists recognize son preference phenomenon, the B/G ratios differ across countries and cultures.9 For instance, in America while ‘many couples express the desire to select the sex of their offspring, very few would abort a healthy fetus of the wrong sex’ (Singer & Wells 1985). In this respect, son preference in East Asian countries can be interpreted as a socially evolved preference. I develop a rational choice model for such preferences and attempt to understand the basics of demographic destiny for countries with strong son preference. I show that the B/G ratio at equilibrium is endogenously determined by preference for married sons and birth rate, and offer implications for changes in the B/G ratio in an environment of falling fertility. A conventional wisdom in demography is that low fertility will increase the B/G ratio. I offer a different prediction that the B/G ratio will adjust downward to the falling birth rate. This result follows, in a crucial way, from the assumption of inequalities in equation (1); parents prefer married daughter to an unmarried son. The assumption is almost necessary for parents’ decision problem, though I am not aware of any empirical work on this issue. My model has the merit of allowing discussion of an optimal B/G ratio. I prove that the natural balanced ratio is optimal, not only for the offspring, but for the parents who prefer married sons. Pareto superior move is possible if all are willing to sacrifice liberty of choice to produce a child.

Acknowledgements I benefited from discussions with Jim Buchanan, Deby Cassill, Sandy Pearce, and Djavad Saleihi-Isfhani among others. I remain thankful to the participants in the Public Choice Society meetings in San Diego, in Tokyo, and the participants in the seminar at the Korean Ministry of Gender Equality, Seoul, Korea. Special thanks go to the staff of Holt International who provided statistics on gender preference in child adoption. The research was financially supported by the Earhart Foundation. I acknowledge helpful refereeing comments by Michael Ghiselin, Janet Landa, and an anonymous reviewer.

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Appendix A: Fertility and the B/G ratio: a conventional model By a simple model I summarize the conventional argument that, when parents have a preference for sons, low fertility will worsen the sex ratio imbalance. The behavioral assumption is that parents want to have at least one son, and I consider two different fertility scenarios. The first is the high fertility case in which each couple produces three children. In the second and low fertility case, each couple produces two children. There are N couples in both cases. In the high fertility case, for the first and second children parents let nature take its course. On average, the number of boys born as a first child is N/2 and the same number of girls are born as a first child. And N/2 boys and the same number of girls are born as a second child. Couples who produced girls as the first and the second child will use sex selective methods to secure a boy for the third child. There is N/4 such couples, who produce N/4 boys as the third child. Parents who have at least one boy either as a first or second child will let nature take its course for the third child. There are (3/4)N such couples and they produce (3/4)×(N/2) number of boys as the third child. Thus the number of boys as a third child is N/4 + 3N/8 = 5N/8. The total number of boys is N/2 (1st child) + N/2 (2nd child) + 5N/8 (3rd child) = (13/8)N out of 3N children. Thus, the fraction of boys is 13/24 = 0.54 or the B/G ratio is 118 boys for 100 girls. In the low fertility case, N/2 boys and the same number of girls are born as a first child. Couples who had a girl as the first child will secure a boy as the second child. There is N/2 such couples and they produce N/2 boys as second children. Couples who had a boy as the first child will let nature take its course for the second child. There is N/2 such couples and they produce N/4 boys on average as the second child. Thus, there are N/2(1st child) + N/2(2nd child) + N/4(2nd child) = 5N/4 boys and the total number of children born is 2N. The fraction of boys is 5/8 = 0.625, or the B/G ratio is 167 boys to 100 girls. The example demonstrates that the B/G ratio is higher for the lower fertility case. B: Valuation v(s) of the marriage pool The expected utility of valuation function v(s) depends on the probability p of sons marrying. When the B/G ratio s is below one half, 0 < s < 0.5, sons are sure to marry, p = 1; and the probability q of daughters marrying is less than 1, q = s/(1 − s). Then, from (13) v(s) = sEu(m) + (1 − s)Eu(f ) = su(m*) + (1 − s)[qu(f*) + (1 − q)u(f)] = su(m*) + (1 − s)u(f) + (1 − s)q[u(f*) − u(f)] = u(f) + s[u(m*) − u(f) + u(f*) − u(f)]. Since u(m∗ ) and u(f∗ ) are greater than u(f), the sign of the term inside the bracket is positive, and v(s) is increasing over the interval [0, 0.5].

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For 0.5 < s < 1, the probability p for sons marrying is less than one: p(s) = (1 − s)/s, and daughters are sure to marry, q = 1. Thus, v(s) = s[pu(m*) + (1 − p)u(m)] + (1 − s)u(f*) = (1 − s)u(m*) + su(m) − (1 − s)u(m) + (1 − s)u(f*) = [u(m*) + u(f*) − u(m)] − s[u(m*) + u(f*) − 2u(m)] Since u(m∗ ) and u(f∗ ) are greater than u(m), the sign of the terms inside the brackets is positive, and v(s) is decreasing over the interval [0.5, 1]. Thus, v(s) obtains the maximum at s = 0.5. Notes 1. See Fisher (1930) and Maynard Smith (1982) for an evolutionary game theoretic explanation of the balanced natural sex ratio. Fisher argued that, if parents could control the sex ratio, the balanced sex ratio will be the stable state. Fisher’s theory of sex ratio depends on the fact that each sex must supply half the ancestry of all future generations. Parental investment in the scarcer sex results in a higher return on investment. If males are in excess of females, then females are the scarcer sex and have a higher reproductive value. 2. See Sen (1990), Park & Cho (1995), Tuljapurka et al. (1995), Goodkind (1996), and Eberstadt (1998, 2004); also see the coverage on this issue in The New York Times (June 21, 2002) and Newsweek (January 26, 2004). 3. Edlund (1999) discusses steady state behavior for a population whose size is constant over time, while I consider population growth at a positive or negative rate. 4. According to the statistics provided by Holt International in year 2002 through 2004, of the families who applied to the Korean program, 58% of the families requested a child of either gender, 35% of the families requested a female child, and 7% of the families requested a male child. Holt International has a policy on choice of gender. A childless family or a family which already has a girl are required that they are open to adopting a child of either gender. 5. My approach in obtaining Proposition 1 is analogous to the method of evolutionary stable strategy used by Maynard Smith (1982, p. 23–25). An individual couple is ‘playing the field’ against the population as a whole. We may let W(1, s∗ ) denote the payoff to a couple by producing a male child and W(0, s*) denote the payoff by producing a female child. Maynard Smith requires the condition W(1, s*) = W(0, s*) at equilibrium. This condition is equivalent to (6) in my model when we note that W(1, s*) = Eu(m) and w(0, s*) = Eu(f). 6. The familiar tragedy of the commons involves the over-utilization of a resource as multiple users are allowed open access. 7. Becker (1991) and others argue that sex preference reflects the economic conditions. 8. Also see Buchanan (1983) for rent seeking behavior in noncompensated transfers. 9. Norberg (2004) attempts to explain the recent decline in the sex ratio at birth in some developed countries including the U.S.

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