A Sexually Unbalanced Model of Current Account Imbalances

Transcription

1 University of Zurich Department of Economics Center for Institutions, Policy an Culture in the Development Process Working Paper Series Working Paper No. 105 A Sexually Unbalance Moel of Current Account Imbalances Qingyuan Du an Shang-Jin Wei March 010

2 A Sexually Unbalance Moel of Current Account Imbalances Qingyuan Du (Columbia University) an Shang-Jin Wei (Columbia University an NBER) y This version: November 3, 010 Abstract Large savings an current account surpluses by China an other countries are sai to be a contributor to the global current account imbalances. This paper proposes a theory of excess savings base on a major transformation in many of these societies, albeit one that is selomly emphasize by macroeconomists, namely, a steay increase in the surplus of men relative to women. We construct an OLG moel with two sexes an a esire to marry. We show conitions uner which an intensi e competition in the marriage market can inuce men to raise their savings rate, an prouce a rise in the aggregate savings an current account surplus. This e ect is economically signi cant if the biological esire to have a partner of the opposite sex is strong. A calibration of the moel suggests that this factor coul generate economically signi cant current account responses, or more than 1/ of the actual current account imbalances observe in the ata. Key wors: surplus men, trae surplus, global imbalances JEL coe: F3, F4, J1 an J7 Corresponing author: Shang-Jin Wei, Columbia University, Grauate School of Business, 619 Uris Hall, 30 Broaway, New York, NY shangjin.wei@columbia.eu. y Acknowlegement: We thank Stefania Albanesi, V. Bhaskar, Pierre-Anre Chiappori, E Hopkins, Aloysius Siow, an especially Xiaobo Zhang, an seminar participants at the IMF, Worl Bank, University of California-Berkeley, Columbia University, University of Washington, Hong Kong University an Chinese University of Hong Kong for very helpful iscussions, an Joy Glazener for eitorial assistance. All errors are our responsibilities.

3 1 Introuction High savings rates in excess of omestic investment rates in many countries in East an Southeast Asia have prouce a massive current account surplus as a share of GDP, an are sai to be a major contributor to the global current account imbalances, to the unusually low long-term interest rates, an possibly to the onset of the global nancial crisis. As to theories of savings behavior, the existing literature has highlighte the roles of life-cycle consierations (Moigliani, 1970), precautionary savings (Kimball, 1990), habit formation (Carroll, Overlan, an Weil, 008), culture (Belton an Uwaifo Oyelere, 008), an nancial uner-evelopment (Caballero, Farhi, an Gourinchas, 008; Ju an Wei, 006, 008 an forthcoming; Menoza, Quarini an Rios-Rull 007). The aim of the current paper is to propose an alternative theory that gives prominence to a major, albeit insu ciently recognize by macroeconomists, social transformation in many economies, namely an increasing gap in the numbers of men an women in the marriage market. The basic thesis is that as competition intensi es in the marriage market, men or parents with sons raise their savings rates with the hope of improving their relative staning in the marriage market. Because the biological esire to have a partner of the opposite sex is strong, this e ect is quantitatively important enough to reveal itself in the aggregate savings rate an the current account balance. The irect inspiration for the theory is an empirical paper by Wei an Zhang (009), which stuies househol savings behavior in China. They provie both cross-regional an cross-househol evience that is consistent with the notion that a worsening prospect for men in the marriage market has motivate them an their parents to raise their savings rates substantially. They call this the "competitive saving motive." Chinese househol savings as a share of isposable income rose from 16% in 1990 to 30% in 007. Wei an Zhang suggest that the rise in the sex ratio imbalance coul account for half the total increase in the savings rate. Because their paper oes not have a formal theory, there is a nee to construct a moel to see if the hypothesis can work in a general equilibrium, an whether a calibration of the moel can prouce an e ect whose magnitue is economically signi cant. In this paper, we aim to ll these important vois. The core part of the paper is to analyze theoretically whether an how a sex ratio imbalance will in uence the economy-wie savings rate an the current account. We construct a simple overlapping generations (OLG) moel with two sexes an a esire to marry. To focus on the macroeconomic implications of sex ratio imbalances, we intentionally shut own channels such as the usual precautionary savings motive, habit formation, culture, an nancial evelopment. Because it is an OLG moel, there are still life-cycle consierations, which, however, o not lea to current account imbalances on their own. Uner reasonable conitions, we show that men respon to a rise in the sex ratio by raising their savings rates. Moreover, the increment in their savings is always enough to o set any ecrease in women s savings when it happens. As a result, the aggregate savings rises with the sex ratio. To check if the moel can eliver an e ect that is economically signi cant, we go to quantitative calibrations. In the benchmark case, for a small open economy, as sex ratio rises from 1 to 1.15, the economy-wie 1

4 savings rate an the current account will rise by more than 10%. We also consier the case of two large economies, whose relative sizes an income levels are calibrate to mimic China an the Unite States. The synthetic Unite States is assume to always have a balance sex ratio, while the synthetic China experiences a rise in the sex ratio from 1 (balance) to 1.5 (very unbalance). The rise in China s sex ratio prouces a rise in its current account surplus, an a corresponing rise in the current account e cit for the Unite States. The magnitues of the current account imbalances in the simulations (about 7.7% of GDP for China an -.6% of GDP for the Unite States) are such that they are more than one-half of the actual current account imbalances observe in the ata. While the sex ratio imbalance is not the sole reason for the global current account imbalances in recent years, it coul be one of the signi cant, an yet thus far unrecognize, factors. A esire to enhance one s prospect in the marriage market through a higher level of wealth coul be a motive for savings even in countries with a balance sex ratio. But such a motive is not as easy to etect when the competition is moest. When the sex ratio gets out of balance, obtaining a marriage partner becomes much less assure. A host of behaviors that are motivate by a esire to succee in the marriage market may become magni e. But sex ratio imbalances so far have not been investigate by macroeconomists. This may be a serious omission. A sex ratio imbalance at birth an in the marriage age cohort is a common emographic feature in many economies, especially in East, South, an Southeast Asia, such as Korea, Inia, Vietnam, Singapore, Taiwan an Hong Kong, in aition to China. In many economies, parents have a preference for a son over a aughter. This use to lea to large families, not necessarily an unbalance sex ratio. However, in the last three ecaes, as the technology to etect the gener of a fetus (Ultrasoun B) has become less expensive an more wiely available, many more parents engage in selective abortions in favor of a son, resulting in an increasing relative surplus of men. The strict family planning policy in China, introuce in the early 1980s, has inuce Chinese parents to engage in sex-selective abortions more aggressively than their counterparts in other countries. The sex ratio at birth in China rose from 106 boys per hunre girls in 1980 to 1 boys per hunre girls in 1997 (see Wei an Zhang, 009, for more etail). It may not be a coincience that the Chinese current account surplus starte to garner international attention aroun 003 just when the rst cohort born after the implementation of the strict family planning policy were entering the marriage market. Throughout the moel, we assume an exogenous sex ratio. While the sex ratio is enogenous in the long-run as parental preference shoul evolve, the assumption of an exogeous sex ratio can be efene on two grouns. First, the technology that enables the rapi rise in the sex ratio has only become inexpensive an wiely accessible in eveloping countries within the last 5 years or so. As a result, it is reasonable to think that the rising sex ratio a ects only the relatively young cohorts savings ecisions, but not those who have past half of their working careers. Secon, in terms of cross country experience, Korea is the only major economies whose sex ratio appeare to have starte to revert back from a very skewe level. Most other countries still exhibit a rising sex ratio. This suggests that the mean reversion process of the sex ratio is a very slow one. A systematic reversal of the sex ratio is unlikely to happen in most economies in the short run.

5 To see if the theoretical preicion has any support in the ata, we check if a country s private sector savings rate (e ne as current account minus government savings, ivie by GDP) is systematically linke to its sex ratio. After controlling for the e ects on the savings rate from income, the share of working age people in the population (i.e., a proxy for the life cycle theory), the ratio of private bank creit to GDP (a proxy for nancial evelopment), an social security expeniture as a share of GDP (a proxy for the precautionary savings motive), we n that a rise in the sex ratio from a balance level to 1.15 (the sex ratio in China) is associate with a higher current account (excluing government savings) by over 10% of GDP. There are three boies of work that are relate to the current paper. First, the literature on status goos, positional goos, an social norms (e.g., Cole, Mailath an Postlewaite, 199, Corneo an Jeanne, 1999, Hopkins an Kornienko, 004 an 009) has o ere many useful insights. One key point is that when wealth can improve one s social status (incluing improving one s staning in the marriage market), in aition to a oring a greater amount of consumption goos, there is an extra incentive to save. This element is in our moel as well. However, all existing theories on status goos feature a balance sex ratio. Yet, an unbalance sex ratio presents some non-trivial challenges. In particular, while a rise in the sex ratio is an unfavorable shock to men (or parents with sons), it is a favorable shock to women (or parents with aughters). Coul the latter group strategically reuce their savings so as to completely o set whatever increments in savings men or parents with sons may have? In other wors, the impact on aggregate savings from a rise in the sex ratio appears ambiguous. Our moel will aress this question. In any case, the literature on status goos has no iscernible impact in policy circles. For example, while there are voluminous ocuments prouce by the International Monetary Fun or speeches by U.S. o cials on China s high savings rate an large current account surplus, no single paper or speech thus far has pointe to a possible connection with its high sex ratio imbalance. A secon relate literature is the economics of family, which is too vast to be summarize here comprehensively. One interesting insight of this literature is that a marrie couple s consumption has a partial public goos feature (Browning, Bourguignon an Chiappori, 1994; Donni, 006). We make use of this feature in our moel as well. None of the papers in this literature explores the general equilibrium implications for aggregate savings from a change in the sex ratio. The thir literature examines empirically the causes of a rise in the sex ratio. The key insight is that the proximate cause responsible for a majority of the recent rise in the sex ratio imbalance is sex-selective abortions, which have been mae increasingly possible by the sprea of Ultrasoun B machines. There are two eeper causes for the parental willingness to isproportionately abort female fetuses. The rst is the parental preference for sons, which in part has to o with the relatively inferior economic status of women. When the economic status of women improves, sex-selective abortions appear to ecline (Qian, 008). The secon is either something that leas parents to voluntarily choose to have fewer chilren than the earlier generations, or a government policy that limits the number of chilren a couple can have. In regions of China where the family planning policy is less 3

6 strictly enforce, there is also less sex ratio imbalance (Wei an Zhang, 009). Bhaskar (009) examines parental sex selections an their welfare consequences. The rest of the paper is organize as follows: in Section, we present a benchmark moel with no intra-househol bargaining. In Section 3, we consier a number of extensions. One such extension allows for intra-househol bargaining, an show that the key propositions still hol. In Section 4, we calibrate the moel to see if the sex ratio imbalance can prouce changes in the aggregate savings rate an current account whose magnitues are economically signi cant. In Section 5, we provie some international evience that sex ratio may have signi cant impact on a country s current account. Finally, in section 6, we o er concluing remarks an iscusses possible future research. The Benchmark Moel We construct an overlapping generations moel with two sexes. Both men an women live two perios: young an ol. An iniviual (of either sex) receives an exogenous enowment in the rst perio an nothing in the secon perio. She or he consumes a part of the enowment in the rst perio an saves the rest for the secon perio. A marriage can only take place between a man an a women in the same generation an at the beginning of their secon perio. Once marrie, the husban an the wife pool their rst-perio savings together an consume an ientical amount in the secon perio. The secon perio consumption within a marriage has a partial public goo feature. In other wors, the husban an the wife can each consume more than half of their combine secon perio income - the exact proportion is an exogenous parameter to be explaine below. Everyone is enowe with an ability to give his/her spouse some emotional utility (or "love" or "happiness"). This emotional utility is a ranom variable in the rst perio with a common an known istribution across all members of the same sex, an its value is realize an becomes public information when the iniviual enters the marriage market. Each generation is characterize by an exogenous ratio of men to women (>= 1). All men are ientical ex ante, an all women are ientical ex ante. Men an women are symmetric in all aspects except that the sex ratio may be unbalance. We escribe the equilibrium in this economy in six steps. First, we start with a representative woman s optimization, followe by a representative man s optimization problem. Secon, we escribe how the marriage market works. Thir, we perform comparative statics, in particular, on how the savings rates change in response to a rise in the sex ratio. Fourth, we consier a small open economy with prouction an iscuss the current account response to a change in the sex ratio. Fifth, we solve for a two-country moel in which the global interest rate is enogenous. Sixth, we use numerical calibrations to see if the moel can eliver current account responses that are economically signi cant. 4

7 .1 A Representative Woman s Optimization Problem A representative woman makes her consumption/saving ecisions in her rst perio, taking as given the choices by men an all other women. If she is not marrie, her secon-perio consumption is c w;n = Rs w y w where R, y w an s w are the gross interest rate, her rst perio enowment, an savings rate, respectively. If she is marrie (at the beginning of the secon perio), her secon-perio consumption is c w = (Rs w y w + Rs m y m ) where y m an s m are her husban s rst perio enowment an savings rate, respectively. ( 1 =< =< 1) represents the notion that consumption within a marriage is a public goo with congestion. As an example, if a couple buys one car, both spouses can use it. When = 1 ; the husban an the wife only consume private goos. In contrast, when = 1, all the consumption is a public goo with no congestion 1. The representative woman chooses her savings rate to maximize the following objective function: subject to the buget constraints that V w = max s w u(c 1w) + E [u(c w ) + m ] c 1w = (1 ( s w )y w (.1) c w = (Rs w y w + Rs m y m ) if marrie Rs w y w otherwise (.) where V w is her value function, an E is the expectation operator. m is the emotional utility (or "love") she obtains from her husban, which is a ranom variable with a istribution function F m. The exact value of emotional utility is reveale at the beginning of the secon perio an becomes a common knowlege at that time. Bhaskar (009) also introuces a similar "love" variable.. A Representative Man s Optimization Problem 1 By assuming the same for the wife an the husban, we abstract from a iscussion of bargaining within a househol. In an extension later in the paper, we allow to be gener specifc, an be a function of the sex ratio an the relative wealth levels of the two spouses, along the line in Chiappori (1988 an 199) an Browning an Chiappori (1998). This tens to make the response of the aggregate savings stronger to a given rise in the sex ratio. 5

8 A representative man has a similar optimization problem as the representative woman. In particular, if he is not marrie, his secon-perio consumption is c m;n = Rs m y m If he is marrie, his secon-perio consumption is c m = (Rs w y w + Rs m y m ) He chooses his savings rate to maximize the following value function: subject to the buget constraints that V m = max s m u(c 1m) + E [u(c m ) + w ] c 1m = (1 ( s m )y m (.3) c m = (Rs w y w + Rs m y m ) if marrie Rs m y m otherwise (.4) where V m is his value function. w is the emotional utility he obtains from his wife, which is rawn from a istribution function F w. We assume w an m are inepenent..3 The Marriage Market In the marriage market, every woman (or man) ranks all members of the opposite sex by a combination of two criteria: (1) the level of wealth (which is etermine solely by the rst-perio savings), an () the size of "love" he/she can obtain from his/her spouse. The weights on the two criteria are implie by the utility functions speci e earlier. More precisely, woman i prefers a higher ranke man to a lower ranke one, where the rank on man j is given by u(c w;i;j )+ m j. Symmetrically, man j assigns a rank to woman i base on the utility he can obtain from her u(c m;j;i )+ w i. (To ensure that the preference is strict for men an women, when there is a tie in terms of the above criteria, we break the tie by assuming that a woman prefers j if j < j 0 an a man oes the same.) Note that "love" is not in the eyes of beholer in the sense that every woman (man) has the same ranking over men (women). The marriage market is assume to follow the Gale-Shapley algorithm, which prouces a unique an stable equilibrium of matching (Gale an Shapley, 196; an Roth an Sotomayor, 1990). The algorithm speci es the following: (1) Each man proposes in the rst roun to his most preferre choice We use the wor" market" informally here. The pairing of husbans an wives in this moel is in fact not one through prices. 6

9 of woman. Each woman hols the proposal from her most preferre suitor an rejects the rest. () Any man who is rejecte in roun k-1 makes a new proposal in roun k to his most preferre woman among those who have not have rejecte him. Each available women in roun k "hols" the proposal from her most preferre man an rejects the rest. (3) The proceure repeats itself until no further proposals are mae 3. With many women an men in the marriage market, all women (an all men) approximately form a continuum an each iniviual has a measure close to zero. Let I w an I m enote the continuum forme by women an men, respectively. We normalize I w an let I w = (0; 1). Since the sex ratio is, the set of men I m = (0; ). Men an women are orere in such a way that a higher value means a higher ranking by members of the opposite sex. In equilibrium, there exists a unique mapping ( w ) for women in the marriage market. w : I w! I m That is, woman i (i I w ) is mappe to man j (j I m ), given all the initial wealth an emotional utility raws. This implies a mapping from a combination (s w i, w i ) to another combination (sm j, m j ). In other wors, for woman i, given all her rivals (s w i, w i ) an all men s (sm, m ), the type of husban j she can marry epens on her (s w i, w i ). Before she enters the marriage market, she knows only the istribution of her own type but not the exact value. As a result, the type of her future husban (s m j, m j ) is also a ranom variable. Woman i s secon perio expecte utility is where w i = Z Z w i h max u (cw;i;j ) + m w ( ijs w i ;w i ;sw i ;w i ;sm ; m ) ; u (Rsw i yw i ) i F w ( w i ) h i u (c w;i;j ) + m w ( ijs w i ;w i ;sw i ;w i ;sm ; m ) F w ( w i ) + Z w i u (Rs w i y w i ) F w ( w i ) is her threshol ranking on men such that she is ini erent between marriage or not. Any lower-ranke man, or any man with w i < w i, won t be chosen by her. Since we assume there are (weakly) fewer women than men, we expan the set I w to I ew so that ei w = (0; ). In the expane set, women in the marriage market start from value 1 to. The measure for women in the marriage market remains to be one. In equilibrium, there exists an unique mapping for men in the marriage market: m : I m! e I w where m maps man j (j I m ) to woman i (i I w ). Those men who are matche with a low value i < 1 in set e I w will not be marrie. In that case, w m (j) = 0 an c m;j;i = Rs m j ym j. In general, 3 If only women can propose an men respon with efere acceptance, the same matching outcomes will emerge. What we have to rule out is that both men an women can propose, in which case, one cannot prove that the matching is unique. 7

10 man j s secon perio expecte utility is = Z Z h max u (cm;j;i ) + w m ( jjs m j ;m j ;sm j ;m j ;sw ; w ) ; h i u (c m;j;i ) + w m ( jjs m j ;m j ;sm j ;m j ;sw ; w ) F m u Rsm j ym j m j Z m j + i F m m j u Rs m j y m j F m m j m j where m j is his threshol ranking on all women. Any woman with a poorer rank, m j < m j be chosen by him., will not We assume that the ensity functions of m an w are continuously i erentiable. Since all men (women) in the marriage market have ientical problems, they make the same savings ecisions. In equilibrium, a positive assortative matching emerges for those men an women who are matche. In other wors, there exists a mapping M from w to m such that 1 F w ( w ) = (1 F m (M( w ))), M( w ) = (F m ) 1 F w ( w ) + 1 For simplicity, we assume that w an m are rawn from the same istribution, F w = F m = F. Furthermore, the lowest possible value of the emotional utility min is su ciently large that everyone esires to be marrie. We also assume that there exists a small an exogenous possibility p that a woman may not n a marriage partner ue to "natural" frictions in the marriage market that are not relate to the sex ratio. The last assumption plays no role in the analytical part of the moel but will simplify the quantitative calibrations later. In equilibrium, given all her rivals saving ecisions an w, woman i s secon perio utility is Z (1 p) u( (Rs w i y w + Rs m y m )) + M(~ w i ) F ~ (~ w i ) + pu(rs w y w ) min where ~ w i = u( (Rs w i yw + Rs m y m )) u( (Rs w y w + Rs m y m )) + w. Due to the symmetry, we rop the sub-inex i for women. For a woman who chooses to enter the marriage market, given men s savings ecisions, the rst orer conition to her optimization problem is u 0 1wy w + (1 p) " u 0 w R M (~ w ) F ~ # w (~ w w + pu 0 w;ny w = 0 (.5) R M (~ w ) ~ F (~ w w Z = u 0 wry w M 0 ( w )F ( w ) + M( min )f( min ) Similarly, for a man who ecies to enter the marriage market, given women s savings ecisions, 8

11 the rst orer conition to his optimization problem is (1 p) " ~ m j u Rs w y w + Rs m j y m Z + M 1 M( min ) ~ m j # h i F ~ (~ m j ) + (1 p)(1 ~ m j ) + p u(rs m j y m ) where ~ m j = u Rs w y w + Rs m j ym u( (Rs w y w + Rs m y m )) + m j an ~ m j is the probability he gets marrie ~ m j = Pr u(c w (j)) u(c w ) + m j M( min ) Rs w y w ; Rs m y m = 1 F M( min ) u(c w (j)) + u(c w ) (.6) If a man chooses to enter the marriage market, the rst orer conition for his optimization problem is u 0 1my m +(1 p) where Z " M( min ) m u + 1 (~ m ) m M( min F ~ m (~ m ) m +f M( min ) u u m m u m;n + 1 (~ m Z m F ~ m (~ m ) = u 0 wry m M( min ) # +[(1 m ) (1 p) + p] u 0 m;ny m = 0 M 1 0 ( m ) F ( m ) (.7).4 Equilibrium Savings Rates An equilibrium is e ne as a collecton of savings rates by men an women that solve their respective optimalization problems, taking all other men an women s ecisions as given. De nition 1 Equilibrium fs w ; s m j y w ; y m ; F w ; F m g satis es the following conitions: s w i = arg max Vi w j s w i; s m ; y w ; y m ; F w ; F m an s m j = arg max V m s w ; s m j; y w ; y m ; F w ; F m j where i an j stan for a representative woman an man, respectively, an i an j represent all women other than i an all men other than j, respectively. s w = s w i ; sw i an s m = s m j ; sm j are the sets of women s an men s savings rates respectively. We assume that population growth rate is zero, an women an men receive the same rst perio income (y w = y m = y). Before perio t, the economy has a balance sex ratio. In this case, s w = s m = s, an s can be obtaine from solving the set of rst orer conitions (.5) or (.7): u 0 1y + (1 p) [] + + pyu 0 9

12 where we use the fact that at = 1, M( min ) = min. We rewrite the rst orer conition as u (1 p)u 0 + M( min )f( min ) + pu 0 n = 0 (.8) an can solve for the optimal savings rates. The rst key proposition is about the e ect of a rise in the sex ratio on the aggregate savings rate. The thought experiment assumes that people in the ol cohort have mae their savings ecision when the sex ratio is balance. When the sex ratio rises, any change in the aggregate savings is riven by a change in the savings by the young cohort. This simplifying assumption is motivate by the reality: A rise in the sex ratio in almost all economies is a recent phenomenon, since large-scale sex-selective abortions are a recent phenomenon. More precisely, while the iagnostic sonography use for prenatal checkups was available in the 1960s, the proceure became graually more a orable to people in countries that currently have a high sex ratio only since the 1980s. (The strict version of the Chinese family planning policy, another contributor to the sprea of sex-selective abortions, was also put in place in the early 1980s.) For this reason, the savings pattern for the currently ol was largely ecie when there was no severe sex ratio imbalance. In what follows, whenever we say a man (or a woman), we mean a young man, or a young woman, unless otherwise speci e. We rst state the propostion formally, an then explain the intuition behin the key parts of the proposition. A etaile formal proof is provie in Appenix A. Proposition 1 As the sex ratio rises, a representative man weakly increases his savings rate while a representative woman weakly reuces her savings rate. However, the economy-wie savings rate increases unambiguously. Proof. See Appenix A. Three remarks are in orer. First, it is perhaps not surprising that the representative man raises his savings rate in response to a rise in the sex ratio because the nee to compete in the marriage market becomes greater. Why oes the representative woman reuce her savings rate? Because she anticipates that a higher sex ratio woul inuce a higher savings rate from her future husban. As a result, she oes not nee to sacri ce her rst-perio consumption as much as she otherwise woul have to. Secon, why oes the aggregate savings rate rise in response to a rise in the sex ratio? In other wors, why is the increment in men s savings greater than the ecline in women s savings? Intuitively, a representative man raises his savings rate for two reasons: in aition to improving his relative staning in the marriage market, he raises his savings rate to make up for the lower savings rate by his future wife. The more he anticipates his future wife to cut own her savings, the more he woul have to raise his own savings rate to compensate. This ensures that his incremental savings is more 10

13 than enough to o set any reuction in her future wife s savings. In aition, a rise in the sex ratio implies a change in the composition of the population - relatively more men with a higher savings rate than women with a lower savings rate, which woul also raise the aggregate savings rate. While both channels contribute to a rise in the aggregate savings rate, it is easy to verify that the rst channel (the incremental competitive savings by any given man) is more important than the secon e ect (a change in the composition of the population with i erent saving propensities). Thir, as the sex ratio rises, women experience a welfare gain while men have to enure a welfare loss. Women gain both because they can free rie on their future husbans higher savings rates, an because they now face an improve probability of marrying a man with a higher level of emotional utility. In comparision, men lose both because they face a lower probability of marriage, an because the reuctions in their rst-perio consumption o not in the en alter their probability of marriage in the aggregate. (The aggregate number of unmarrie men is inepenent of the savings ecision.) We will present an extension in which the ecision to enter or exit the marriage market is enogenous. We will also iscuss another extension in which women may not reuce their savings rate in response to a rise in the sex ratio, when their bargaining power within a marriage epens in part on their savings rate. In both cases, the aggregate savings rate rises in response to a rise in the sex ratio (uner some conitions)..5 A Prouction Economy To analyze how the sex ratio imbalance a ects a country s current account imbalance, we nee to compare economy-wie savings with investments. In this subsection, we introuce a prouction sector. We assume that both the nal goo market an the factor markets are perfectly competitive. The prouction function is Cobb-Douglas: Q t = Kt L 1 t (.9) where K t is the capital stock an L t is the labor input. is the share of capital input to total output an is the total factor prouctivity (TFP). Everyone in the economy inelastically supplies one unit of labor an earns the same income 4. A representative rm maximizes the pro t max Q t R t K t W t L t K t;l t 4 Allowing men an women to earn i erent wages (with a xe proportional gap) woul not change our results. 11

14 The capital return an the wage rate are etermine by R t 1 t 1 = t K t W t t = (1 ) K t (.11) where we normalize the aggregate labor supply in the economy to be 1, i.e., L t = 1. For simplicity, we assume no tax or government expeniture; then y t = W t where y t is the corresponing rst perio isposable income in the enownment economy. We also assume complete epreciation in each perio. The aggregate capital supply in perio t + 1 is preetermine by the aggregate savings in perio t K s t+1 = 1 + sm t W t sw t W t (.1).6 Current Account in a Small Open Economy In a small open economy, we assume that capital can ow freely among countries an the gross interest rate R is exogenously etermine by the rest of the worl. By (.10) an (.11), the wage rate is also a constant, an the aggregate investment in the economy is K t = Substituting (.10) an (.13) into the prouction function, we have W t (1 )R t (.13) Q t = W t 1 The current account in perio t equals the increase in net foreign assets, NF A t = Q t + (R 1) NF A t 1 C 1t C t K t+1 where (R 1) NF A t 1 is the factor income from abroa. C 1t an C t represent the aggregate consumptions by young an ol people respectively. Then NF A t = 1 + sm t W t sw t W t NF A t 1 K t+1 We e ne the economy-wie savings rate as the aggregate private savings to GDP ratio; then s P t = Q t + (R 1) NF A t 1 C 1t C t Q t (.14) 1

15 We assume that the country has a balance sex ratio in perio t 1, an the sex ratio in the young cohort in perio t, rises from one to (> 1). Then the ratio of the current account to GDP is where the secon equality hols because 5 ca t = Q t + (R 1) NF A t 1 C 1t C t Kt+1 (.15) Q t = (1 ) 1 + sm t sw t s t 1 NF A t 1 = s t 1 W t 1 K t where s t 1 is the savings rate by the cohort born in perio t 1: Since the sex ratio is balance at that time, both the women an the men will have the same savings rate. Since the wage rate is constant in the small open economy, we can show that a country s current account rises as its sex ratio rises (up to a point). Proposition The economy-wie savings rate an the current account in perio t are increasing in. Proof. See Appenix B. The assumption of an exogenous interest rate hols only for a small open economy. But some of the countries that motivate this stuy are large. An increase in the savings rate in such economies coul lower the worl interest rate, which coul alter investment an savings ecisions in all countries. We examine the large country case in the next subsection..7 Two Large Countries Consier a worl consisting of only two countries. The two countries are ientical in every respect except for their sex ratios in perio t (they both have balance sex ratios in perio t 1). Country 1 s sex ratio 1 is smaller than Country s sex ratio. There are no barriers to either goos trae or capital ows (although labor is not mobile internationally). We can show the following result: Proposition 3 Country 1 (with a more balance sex ratio) runs a current account e cit while country runs a current account surplus. Proof. See Appenix C. 5 In overlapping generations moels, net foreign asset is equal to the i erence between the savings by the young cohort an the omestic investment eman. 13

16 To see the intuition, let us x 1 = 1 (i.e., Country 1 has a balance sex ratio). If Country were to have a balance sex ratio, the current account must be zero for both countries since they are ientical in every respect. In other wors, within each country, the investment must be equal to the aggregate savings. However, the sex ratio imbalance in Country causes it to have a higher aggregate savings for a given worl interest rate. This epresses the worl interest rate. The lower interest rate raises the investment level in both economies (an reuces the savings rate a little bit). This must imply that the esire investment level in Country 1 is now greater than its esire savings rate. As a result, capital ows from Country to Country 1. That is, Country 1 runs a current account e cit, an Country a surplus..8 Welfare There are two sources of market failure in the moel economy. On one han, a part of the savings in the competitive equilibrium is motivate by a esire to out-save one s competitors in the marriage market. The increment in the savings, while iniviually rational, is not useful in the aggregate, since when everyone raises the savings rate by the same amount, the ultimate marriage market outcome is not a ecte by the increase in the savings. In this sense, the competitive equilibrium prouces too much savings. On the other han, because the savings contribute to a pubic goo in a marriage (an iniviual s savings raises the utility of his/her partner), but an iniviual in the rst perio oes not take this into account, he/she may uner-save relative to the social optimum. Note that these sources of market failure exist even with a balance sex ratio. They also have opposite e ects on the aggregate savings rate. (In the calibrations that will be reporte later, these two e ects cancel each other out when the sex ratio is balance.) As the sex ratio rises, the importance of the over-saving e ect also increases, which gives rise to our rst proposition. We now consier what a hypothetical welfare-maximizing central planner woul o. The central planner gives equal weight to each man an women. He assigns the marriage market matching outcome an optimally chooses women s an men s savings rates to maximize the following social welfare function, The rst orer conitions are u 0 1m + max U = U w U m u 0 1w + (1 p)u 0 w + pu 0 w;n = 0 (.16) (1 p) u 0 1 w + 1 (1 p) u 0 w;n = 0 (.17) Comparing (.16), (.17) to (A.1) an (A.), in general, it is not obvious whether women or men will save at a higher rate in a ecentralize equilibrium than that uner central planning ue to the two opposing sources of market failure. However, when = 1, since women an men have the same 14

17 optimization problem, by (.8), if f( min )M( min ) > 0, then women an men will save more in the competitive equilibrium. If f( min )M( min ) is su ciently small, the competitive equilibrium is very close to the central planner s economy. In calibrations with a log utility function, we show that men s welfare uner a ecentralize equilibrium relative to the central planner s economy eclines as the sex ratio increases. In comparision, women s relative welfare increases as the sex ratio goes up. The social welfare (a weighte average of men s an women s welfare) goes own as the sex ratio rises. 6 As a thought experiment, one may also consier what the central planner woul o if she can choose the sex ratio (in aition to the savings rates) to maximize the social welfare, the new rst orer conition with respect to is U m U w = 0 (.18) The only sex ratio that satis es (.18) is = 1. In other wors, the central planner woul have chosen a balance sex ratio. Deviations from a balance sex ratio represent welfare losses. 3 Extensions While the benchmark moel is simple with clear preictions, we now consier a series of extensions that woul a some more realisms to the moel. First, We consier enogenous ecisions to enter an exit the marriage market. This allows agents to choose to quit (or not enter) the marriage market. Secon, we iscuss potential intra-family bargaining between husban an wife, with their relative bargaining power epening in part on their relative savings rate. Thir, we incorporate parental savings in aition to agents own savings. An nally, we consier a setting in which each agent has 50 perios in his/her ault life, an has to make savings ecisions in multiple perios. 3.1 Marriage market with enogenous entries an exits In the rst extension, we allow both men an women to choose to enter or exit the marriage market together with the savings rates. For a representative woman, if she chooses to be single throughout her life, the optimization problem is V w n = maxu(c 1w;n ) + u(c w;n ) s w n where Vn w is the value function of a life-time spinster. We still use V w to enote her value function if she chooses to enter the marriage market. Agents can play a mixe strategy in terms of whether or not to enter the marriage market. More precisely, the representative woman chooses the probability of entering the marriage market w, a 6 The results are similar if we change the utility function to a CRRA form. 15

18 savings rate if she ecies to enter, an a (potentially) separate savings rate if she ecies to abstain from marriage. The optimization problem is max w ;s w ;s w n w V w + (1 w )V w n Obviously, she will choose w = 1 if an only if V w > V w n. For a representative man, similarly, if he chooses to be a life-time bachelor, his optimization problem is V m n = maxu(c 1m;n ) + u(c m;n ) s m n where Vn w is the corresponing value function. The representative man also chooses the probability of entering the marriage market m as well as two potentially separate savings rates. The optimization problem is max m ;s m ;s m n m V m + (1 m )V m n He ecies to enter the marriage market with probability one if an only if the expecte utility of oing so is greater than otherwise, or V m > V m n : We now e ne an equilibrium in this case. De nition 1 An equilibrium is a set of savings rates an probabilities of entering the marriage market, fs w ; w ; s m ; m j y w ; y m ; F w ; F m g that satis es the following conitions: (s w i ; w i ) = arg max Vi w j s w i; w i; s m ; m ; y w ; y m ; F w ; F m an s m j ; m j = arg max V m j s w ; w ; s m j; m j; y w ; y m ; F w ; F m where i an j stan for a representative woman an man, respectively, an i an j represent all women other than i an all men other than j, respectively. (s w ; w ) = s w i ; sw i ; w i ; w i an (s m ; m ) = s m j ; sm j ; m j ; m j are the sets of women s an men s choices, respectively. At = 1, i.e., when the sex ratio is balance, uner the assumption that the mean value of emotional utility is large enough, it is easy to verify that the ominant strategy for both women an men is to enter the marriage market. Moreoever, we can show the following proposition. Proposition 4 There exist two threshol values, 1 an, where 1 <, that satisfy V m = Vn m an U w = 0 respectively. For <, both women an men choose to enter the marriage market with probability one. In aition, (i) for any < 1, as the sex ratio rises, a representative man weakly increases his savings rate while a representative woman weakly reuces her savings rate. However, the economy-wie savings rate increases unambiguously; (ii) For 1 <, as the sex ratio rises, a 16

19 representative man keeps his (high) savings rate constant but a representative woman raises her savings rate. The aggregate savings rate increases. Proof. See Appenix D. A few remarks are in orer. First, the ominant strategy for both men an women is to enter the marriage market unless the sex ratio is extremely high. To see the intuition, we note rst that when the sex ratio is balance, both men an women strictly prefer to be marrie than to be single. As both the utility function an the ensity function of are assume to be continuously i erentiable, in the neighbourhoo of = 1, entering the marriage market must also be a ominant strategy for men an women. Secon, 1 is the threshol level of sex ratio at which the representative man is i erent between entering the marriage market or not, V m = V m n. The value of the threshol can be very high if a man s biological esire to be matche with a female is strong enough. In the calibrations that we will report later, the threshol is always higher than the highest sex ratio one observes in the ata. In other wors, the sex ratios in the real worl appear to be always below this threshol. Finally, beyon the rst threshol, 1 ; if the sex ratio continues to rise, up to another threshol ( ), men woul switch to a mixe strategy (that assigns some probability of not entering the marriage market) if women o nothing. Due to the bene ts associate with marriage, women woul not n it optimal to o nothing an let some goo men to skip the marriage market. By raising their own savings rate, women can succee in enticing men to come back to the marriage market. However, beyon the secon threshol, if the sex ratio continues to rise, the require sacri ce for women in terms of reuce rst-perio consumption becomes too large for them to continue to raise their savings rate. At that point, women woul keep their (high) savings rate constant, an let men play their mixe strategy. More formally, is the level of sex ratio at which U w = 0. In other wors, can be solve by equation y u 0 1w + (1 p)u 0 w + pu 0 s w w;n + E [] = 0 Since the sex ratios in the real worl are unlikely to reach the rst threshol 1, it is even less likely to reach the secon threshol : To summarize, as the sex ratio rises, as long as <, the aggregate savings rate rises unambiguously. Base on the same reasoning, we can show that all results in Propositions an 3 (uner the conition that < ) also hol in this extension. 3. Intra-househol Bargaining Chiappori (1988 an 199) an Browning an Chiappori (1998), among others in the literature on the economics of family, have consiere intra-househol bargainings, an emphasize that the relative 17

20 income between the husban an the wife matters for their respective bargaining power. In this section, we exten the benchmark moel by allowing the relative bargaining power to be a function of both the sex ratio an the relative wealth. We assume that everyone consumes two goos in the secon perio, a public goo (e.g., a house) an a private goo. The aggregate secon perio consumption inex is z i h1 c i = (1 ) 1 i = w; m where z w an z m are private goos consumption by women an men respectively. h is the public goo consumption. A representative househol maximizes the weighte sum of the utilities of the husban an the wife. Let enote the weight on the wife s utility, which represents her bargaining power in the family. Then the househol s optimization problem is max u(c w ) + (1 )u(c m ) h;z w;z m with the resource constraint z w + z m + h = Rs w y w + Rs m y m (3.1) If we assume u(c) = c1 1 1 ( 1), solving the househol s maximization problem, we have c w = c m = Rs w y w + Rs m y m Rs w y w + Rs m y m If = 1, this is the case in our benchmark moel an 1 < = < 1. More generally, similar to Browning et al (1994), we assume to be an increasing function in the sex ratio, the relative wealth, an w, a ecreasing function in m, then c w = w (Rs w y w + Rs m y m ) c m = m (Rs w y w + Rs m y m ) where w = m = " " # 1 + (3.) 1 # 1 + (3.3) 18

21 We enote the erivatives we will use as following: w 1 0, m 1 w w 0, Rsw y w = Rs m m w w 0, m 3 w m 0, m m w m Rsm y m Rs w y w 0 For simplicity, we assume that w = m, if = 1, Rs w y w = Rs m y m an w = m. We assume y w = y m = y. To simplify the iscussions, we make several aitional assumptions. Assumption j j i j u 0 i c i This means that the elasticity of the bargaining power for househol member i is small enough given the relative wealth between women an men. This assumption ensures that u( i (Rs w y w + Rs m y m )) + j is increasing in j, i.e., everyboy wants her/his spouse to have a higher. Therefore, in equilibrium, the mapping that results in an assortative matching M still hols. i Rs i y i Rs j y j Rs i y i i Rs j y j Rs j y j Rs i y i + Rs j y j This ensures that each person prefers a wealthy spouse even though this may reuce his/her bargaining power within the househol. Assumption E3: M( min ), u(c m ) + min = u(c m;n ). In the secon perio, for the man who enters the marriage market with type This implies that the marginal marrie man is ini erent between getting marrie an staying single. This assumption is reasonable because ex ante men o not know their type an make the same saving ecision. In the secon perio, the woman matche to the man of type M( min ) makes an intra-househol allocation o er that woul make the man slightly happier than being single. If he oes not accept the o er, the next ranke man may accept the o er, which woul have resulte in a welfare loss to the man of type M( min ). Hence, he accepts the o er. Assumption E4: i 1 is either positive or small in its absolute value, i = w; m. 19

22 Without this assumption, as a man s incentive to raise savings rate is so strong, the contribution of a woman s own savings to her bargaining power within the family woul be small. This woul not be very interesting as the result woul be similar to the benchmark moel. Proposition 5 Uner assumptions E1-E4, as the sex ratio rises, a representative man s savings rate rises, but the change in a woman s savings rate is ambiguous. However, the aggregate savings rate rises unambiguously. Proof. (See Appenix E). The ambiguity in a woman s response to a rise in the sex ratio comes from two opposing forces. On one han, she wishes to free rie on her future husban s higher savings rate by reucing her savings rate. On the other han, she oes not wish to see an erosion in her intra-househol bargaining power, an one way to protect her bargaining power is to raise her own savings rate. This ambiguity on the part of the women s savings oes not a ect the comparative statics on the aggregate savings. Even if the women o not raise their savings rate to preserve intra-househol bargaining power, we alreay know from Proposition 1 that in the aggregate, the increment in men s savings is always more than enough to o set any ecline in women s savings. Any increase in women s savings out of concern for intra-househol bargaining power can only further increase the response of the aggregate savings to a rise in the sex ratio. As for the aggregate savings rate, the qualitative results are similar to Proposition 1. As a result, it is easy to verify that all open-economy conclusions still hol with this extension. 3.3 Parental savings As ocumente in Wei an Zhang (009), parental savings for chilren are likely to be an important part of househol savings. To incorporate this feature, we consier an OLG moel in which every cohort lives three perio (young, mile-age, an ol). Everyone works an earns labor income in the rst two perios. If one gets marrie, the marriage takes place at the beginning of the secon perio, an the couple prouces a single chil right away. The chil is a boy with a probability of 1+, an a girl with a probability of If the chil ever gets marrie, the marriage takes place at the beginning of the chil s secon perio, which is also the beginning of the parents thir perio. They erive irect emotional utility from having a chil, although the value of this emotional utility coul epen on the gener of the chil. Parents are also altrustic towar their chil an can save for their chilren (in aition to for themselves). For simplicity, we assume that chilren o not make a reverse nancial transfer to their parents. With this setup, the optimization problem for a representative woman who enters the marriage 0