The Sexual Division of Labor Revisited

Transcription

1 The Sexual Dvson of Labor Revsted Slva Martínez Gorrcho Aprl, Abstract The am of ths paper s to analyze whether tranng-accordng-to-sex can be supported n equlbrum n a transferable utlty envronment n whch ndvduals are heterogeneous n terms of ther apttudes to perform one task (I call t task A) and n whch matchng s done accordng to economc crtera. If a task can be executed perfectly by an ndvdual who receves complete tranng n that task, then ths task s mechancal. I show that f the task at whch ndvduals have a homogenous apttude (I call t task B) s mechancal, then tranng-accordng-to-sex s a Nash equlbrum ndependently of the sze of the market and the sex rato. Instead, f task B s nonmechancal, then the exstence of tranngaccordng-to-sex Nash equlbra n large marrage markets requres ether a suffcently hgh sex rato or that the average ndvdual has a hgh apttude for some task. Fnally, the tranng-accordng-to-sex can help coordnate people effcently n the choce of premartal nvestments n small markets, whch are characterzed by a hgh rsk assocated to msmatches n the populaton. In general, as the marrage market expands and/or the sex rato approaches unty, tranng-accordng-to-apttude eventually replaces tranng-accordng-to-sex as the optmal strategy. Keywords: Sexual Dvson of Labor; Human Captal; Coordnaton; Heterogenety; Nonrandom Matchng; Sex rato; Extent of the Market JEL Classfcaton: D13; J16; J24; P41 I am ndebted to Ettore Damano, Martn J. Osborne and Aloysus Sow for ther gudance through ths project. All remanng errors are mne. Department of Economcs, Unversty of Toronto, 150 St. George St., Toronto, Ontaro, M5S 3G7, CANADA. Emal:smartne@chass.utoronto.ca 1

2 1 Introducton The sexual dvson of labor s a phenomenon found n vrtually all human socetes. Due to ts mportance, economsts have attempted to model the causes and mplcatons of sex roles across socetes. 1 Economc theory suggests that occupatonal segregaton by sex mght be due to supply sde factors, demand sde factors, or a combnaton of both. For nstance, the most promnent supply sde theory based on human captal accumulaton explans that, gven women generally antcpate shorter and less contnuous work lves than men, they wll prefer occupatons requrng smaller human captal nvestments because these jobs have lower wage penaltes for tme spent out of the labor market (Polachek, 1981). The fact that women may face barrers to obtanng educaton and pre-job tranng n tradtonally male felds, somethng referred to as socetal dscrmnaton, consttutes an alternate supply sde explanaton (Blau, Ferber and Wnkler, 1988). On the demand sde, the sexual dvson of labor s supported by dscrmnaton aganst women wthn the pad labor force (eg. Agner and Can (1977), Danzger and Katz (1996) and Francos (1998)). In Chapter 2 of Becker s Treatse on the Famly (1991), the sexual dvson of labor arses due to gans from specalzed nvestments and ntrnsc dfferences between the sexes. He assumes two types of human captal, the household and the market human captal. Women s nherent comparatve advantage n household actvtes and the presence of ncreasng returns to human captal nvestment prompt women to specalze n household work whle ther spouses specalze n market work. Thus, the model reles on bologcal dfferences across the sexes to support the sexual dvson of labor. Despte ntutve appeal, Becker s explanaton s not perfect. Though tasks n all socetes tend to be easly dentfable as ether women s work or men s work, the dvson vares consderably across cultures (Murdock and Provost (1973)). Becker s bologcal model does not account for ths fact. Furthermore, as tme passes, technologcal developments contrbute to the reducton of the comparatve advantage enjoyed by any sex n the performance of a partcular actvty. For nstance, technologcal change has lessened the brute physcal component of labor. As a result of the absence of any sgnfcant bologcal advantage, the contnued applcablty of the bologcal model s questonable. In lght of the weaknesses of the bologcal model, Hadfeld (1999) proposes a theory n whch the sexual dvson of labor can be derved solely from ncentves to coordnate human captal nvestments wth those of future, but yet undetermned, partners of the opposte sex. In her model, potental spouses choose frst the level of nvestment n each of the two types of human captal, and subsequently engage n the search for a 1 For recent studes about occupatonal segregaton by sex, refer to Blau et all (1998) who analyze the trends n the US over the 1970 s and 1980 s whle Dolado et all (2002) analyze the trends n EU countres vs-à-vs the US. 2

3 marrage partner. Untl human captal nvestments are made, all ndvduals are homogeneous and there s an equal number of males and females n the populaton. In her set up, mates are valued only n terms of the household utlty obtaned wth them. The utlty s nontransferable. The marrage market s ncomplete, that s, t s composed of a randomly drawn subset of the full populaton and matches are made accordng to the procedure proposed by Gale and Shapley (1962). Thus, premartal nvestments determne an ndvdual s desrablty and her success on the marrage market s determned by her relatve desrablty n the pool of market canddates. For a suffcently large marrage market, there are two possble equlbrum outcomes: ether labor s dvded along sex lnes or the dvson of labor s gender-free. Engneer and Wellng (1999) ncorporate heterogenety n a smlar framework. In ther model, males and females can ether have an apttude for market or household work albet not for both. In addton, human captal tranng s modeled as a dscrete bnary choce varable snce t can be done only n one sphere of work. Fnally, the marrage market s structured as a pure random matchng market so that there s no sortng. These ngredents gve rse to a coordnaton problem n choosng complementary specalzatons. Under a large set of crcumstances, the tranng-accordng-to-sex equlbra and the tranng-by-apttude equlbra are the only pure strategy equlbra. The authors rank the equlbra when they coexst and demonstrate that affrmatve acton polces can ncrease average welfare by elmnatng some equlbra. Ths paper contrbutes to the prevous lterature by modelng human captal nvestments on both sdes of the market n a transferable utlty envronment under heterogenety and non-random matchng. As cted above, Engneer and Wellng s key assumpton s that famles are formed randomly on the bass of true love. However, when ndvduals are heterogeneous, a common observaton s that of assortatve matchng. Hence, n contrast to the assumptons of Engneer and Wellng, ths paper provdes a framework n whch matches are made accordng to economc crtera and human captal tranng s no longer a bnary choce. Furthermore, we clam that the sex rato together wth the sze of the market consttute crucal factors n the determnaton of both the exstence and the effcency of tranng-accordng-to-sex equlbra. The sex rato, defned as the number of avalable men to avalable women, s expected to alter the allocaton of martal output between spouses and thereby nfluence premartal nvestments. In addton, tranng-accordng-tosex could help people coordnate effcently n the choce of premartal nvestments n small or ntermedate marrage markets characterzed by a hgh rsk assocated to msmatches n the populaton. When the marrage market s large, a contnuum number of each type of ndvduals wll be expected to exst on both sdes of the market so that the rsk assocated to msmatches vanshes. We show that tranng-accordng-to-sex can be supported as equlbra n a socety whch treats the sexes 3

4 equally under certan condtons. If a task can be executed perfectly by an ndvdual who receves specalzed tranng n that task, then ths task s mechancal. If the task at whch ndvduals have a homogenous apttude (we call t task B) s mechancal, tranng-accordng-to-sex s a Nash equlbrum ndependently of the sze of the market and the sex rato. Instead, f task B s nonmechancal, then the exstence of tranng-accordng-tosex Nash equlbra n large marrage markets requres ether a suffcently hgh sex rato or that the average ndvdual has a hgh apttude for some task. Furthermore, we examne numercally the ex ante optmalty of tranng-accordng-to sex. We conclude that ts optmalty depends on three mportant factors: the nature of task B, the value of the sex rato and the sze of the market. If task B s mechancal, the tranng-accordng-to-sex strategy n whch the mnorty sex specalzes n task B s effcent for all market szes only f the sex rato s relatvely hgh. The alternatve tranng-accordng-to-sex strategy s not an effcent mechansm for any market sze and sex rato. If task B s nonmechancal, tranng-accordng-to-sex s an effcent allocaton mechansm n small marrage markets. In general, as the marrage market expands and/or the sex rato approaches unty, tranng-accordng-toapttude eventually replaces tranng-accordng-to-sex as the optmal strategy. The market sze requred for ths to occur s ncreasng n the sex rato. The paper s organzed as follows: In Secton 1, the assumptons and the basc structure of the model are descrbed; Secton 2 characterzes the stable matches; Secton 3 characterzes the premartal nvestment game and t defnes the equlbrum concept n use; Secton 4 develops the condtons under whch the tranngaccordng-to-sex strateges are Nash equlbra; Secton 5 examnes the mpact of the extent of the market and the sex rato on the effcency of the tranng-accordng-to-sex strateges; Secton 6 analyzes evoluton; Secton 7 concludes by summarzng the man results and dscussng possble extensons. All the proofs are contaned n Secton 8, the appendx. 2 The Statc Model The marrage market has two dstnct groups of rsk-neutral agents, men and women. Let M denote the total number of avalable men and W the total number of avalable women n the market. Assume that M > W so that the total number of marrages n the socety wll be equal to W. 2 Two spheres of work must be undertaken. We euphemstcally label them task A and task B. Assume 2 The case M < W s mplctly analyzed by nterchangng the roles of males and females throughout the paper. The case M = W s approached by takng the lmt of the sex rato to unty. 4

5 that each ndvdual s characterzed by a real number, a [a, ā] (0, 1), whch represents the ndvdual s apttude to perform task A. The closer a s to ā, the more potental the ndvdual has n accomplshng task A. Assume that an ndvdual s apttude s only known by the ndvdual; that s, t s prvate nformaton. Let G(â) denote the probablty that an agent s apttude s no greater than â. The dstrbuton G( ) s assumed to be non-degenerate, dentcal and ndependent across ndvduals and common knowledge to the publc. In addton, assume that all agents have an dentcal apttude to perform task B whch s gven by b (0, 1). Hence, heterogenety among ndvduals comes solely from ther nnate apttude to perform task A. Ths framework mplctly assumes that nature does not dscrmnate due to the fact that apttudes are equally dstrbuted across the sexes. 3 Each person s lfe goes through three stages: (1) In Stage 1, an ndvdual chooses a level of nvestment n each of two types of human captal. For convenence, the total nvestment n the two types of human captal s normalzed to 1. We denote the human captal nvestment made n task A by y. In ths model, there s no explct cost of nvestment n ether task. Instead, an mplct opportunty cost exsts; nvestng more n one task translates nto lower nvestment for the other task. An agent s nvestment n each type of human captal mght depend on her apttude. (2) In Stage 2, the agent s ablty or skll for both tasks s determned by her human captal nvestment as well as her apttude. For each task, there are two levels of ablty: hgh ablty (H) and low ablty (L < H). Let t J denote the ablty of an ndvdual n task J, where J = A,B and t J T J = {H,L}. The probablty that an agent develops a hgh ablty n a gven task s determned by her apttude and her human captal nvestment n the performance for a partcular task. These probabltes are mutually ndependent across tasks. The set of types s denoted by T = {HH,HL,LH,LL} and t s an element of T: t T. The acquston of ablty s modeled accordng to the followng producton functon: 3 Ths model can be easly extended to a more general framework where G (a) consttutes the apttude probablty dstrbuton for task A and b the apttude assocated to task B, for gender = m, w. Ths modfcaton does not alter the substance of the conclusons that follow. 5

6 Indvdual Type (t) Ablty n Task A (t A ) Ablty n Task B (t B ) Probablty Type HH Hgh Hgh ayb(1 y) Type HL Hgh Low ay[1 b(1 y)] Type LH Low Hgh (1 ay)b(1 y) Type LL Low Low (1 ay)[1 b(1 y)] (3) In Stage 3, the ndvdual enters the marrage market n search of a mate wth whom to set up a household. The type of an agent s perfectly observed by all the partcpants. When a couple marres, the marred household generates martal output. The outcome of the marrage market (.e. who marres whom and how much output each agent gets) depends on the soluton concept. We consder a frctonless marrage market n whch the matchng assgnment s stable gven the dstrbuton of abltes n the populaton together wth a specfcaton of the household producton functon. We further assume a transferable utlty between the spouses to characterze the stable assgnment. In our envronment, both tasks, A and B, must be completed and trade between households s not allowed. 4 Thus, ths model s best able to account for the sexual dvson of labor n economes wth lttle technology and market trade for whch coordnaton between husband and wfe s paramount. Many actvtes can be categorzed as such. Growng wheat, grdng t and bakng bread s an example of such actvtes. An solated household that s composed of ndvduals who only know how to grow wheat may starve from hunger before the sklls of grndng wheat and bakng bread can be learned. The marrage producton functon of a type t man and a type t woman s gven by: X(t,t ) = x l x h x m f t = t = LL f max{t A, t A} = max{t B, t B} = max{t A, t B} = max{t A, t B} = H otherwse for all t T, t T and t J T J wth J = A,B, 0 < x l < x m < x h and max{h,l} = H. 4 The man focus of ths paper s to examne the solated effects of the sex rato and the sze of the market on the gender dvson of labor. Albet we agree that trade between famles s clearly mportant, ts consderaton goes beyond the scope of ths paper. 6

7 One embedded assumpton n the producton functon s that complete specalzaton by both members of the household s optmal. Optmalty n specalzaton n a partcular task comes from the fact that each ndvdual s more productve workng predomnantly n a partcular sphere than dvdng her tme between the two spheres. Based on ths assumpton, when two ndvduals wth dfferent abltes for one task match together, specalzaton leads the ndvdual whose ablty s hgher to perform that task. The model makes no assumptons about the job assgnment between partners wth the same abltes for each task or between a hghly able partner and a lowly able partner for both tasks. The producton functon assumes equalty of sexes, that s, the type of martal output produced by a couple depends only on the ablty of the ndvduals at the task they specalze n, and not on ther sex. It also assumes that task A s as valuable as task B to the household. Gven ths, those couples that are composed of ndvduals who have a hgh ablty for the task they specalze n, are able to produce a hgh martal output (x h ). Those couples that are composed of only one ndvdual wth a hgh ablty for the task she specalzes n, able to produce a medum martal output (x m ). Fnally, those couples composed of ndvduals wth low abltes for both tasks, able to produce a low martal output (x l ), where 0 < x l < x m < x h. We further assume that the ndvduals types complement each other so that x h + x l > 2x m. Ths nequalty mplctly leads to postve assortatve matchng among hgh and low type ndvduals. Indvduals who reman sngle obtan zero utlty ndependently of ther type. Marred partners dvde ther martal output and ther utltes are gven by ther share of the output. An ndvdual may attract a spouse by promsng to allocate more of the dvsble martal output to the potental spouse. To analyze the matchng and nvestment process, we frst examne the equlbrum for the matchng process. We then proceed to the nvestment stage and assume that the market partcpants foresee the mpact that ther nvestment wll have on the outcome of the matchng process. 3 Stable Matches An assgnment of the marrage market s a set of marrages. Each man may be matched wth at most one woman and each woman may be matched wth at most one man. Stayng sngle s an opton for each ndvdual. Let M = {m 1,m 2,...m M } be the set of men and W = {w 1,w 2,...w W } the set of women n the marrage market. A matchng µ of any set of ndvduals splts the set nto pars, each consstng of a man and a woman, and sngle ndvduals. We refer to µ(z) as the mate of z and we denote a person who remans sngle by µ(z) =. 7

8 Defnton 1. A matchng µ s a one-to-one correspondence from the set M W nto M W { } such that () µ(m j ) W { } for all m j M () µ(w k ) M { } for all w k W () µ(m j ) = w k f and only f µ(w k ) = m j for all m j M and w k W. Notce that each of these ndvduals n the market s characterzed by a partcular type of realzaton. Ths mples that the matchng µ also defnes a mappng from the set of realzed men s types n the market (denoted by T m ) nto the set of realzed women s types n the market (denoted by T w ). We refer to µ m (t) as the set of types of women who are the partners of type t men under the matchng µ and µ w (t ) as the set of types of men who are marred to type t women under the matchng µ. In addton, µ (t) ncludes f a person of type t and gender remans sngle. Thus, µ (t) { } T j for all = m,w and j = w,m. A sharng rule specfes the shares of the husband and wfe n every marrage. Let X m (t,t ) denote the martal share for a husband whose type s t and who s marred to a type t woman and X w (t,t ) the martal share for a wfe whose type s t and who s marred to a type t man. We call ((X m,x w );µ) an outcome of the marrage market. Defnton 2. An outcome of the marrage market ((X m,x w );µ) s feasble f X m (t,t ) + X w (t,t ) X(t,t ) for all t T m and t µ m (t) Defnton 3. An outcome of the marrage market ((X m,x w );µ) s stable f () X m (t,t ) 0, X w (t,t ) 0 for all t T m and t µ m (t) () X m (t,t ) + X w (t,t ) X(t,t ) for all t T m, t µ m (t), t T w and t µ w (t ). Condton () (ndvdual ratonalty) reflects that an ndvdual always has the opton of remanng sngle and hence, no current marred ndvdual should want to leave her marrage n order to be sngle. Condton () requres that the outcome s not blocked by any par of agents. That s, no current marred ndvdual wants to leave her marrage n order to marry any sngle ndvdual and no two ndvduals from dfferent marrages want to leave ther spouses to marry each other. Lemma 1 follows from the defnton of feasblty and stablty. 8

9 Lemma 1. Let ((X m,x w );µ) be a feasble and stable outcome of the marrage market. Then () X m (t,t ) + X w (t,t ) = X(t,t ) for all pars (t,t ) such that t µ m (t) and t T m () X m (t,t ) = X m (t,t ) X m (t), X w (t,t ) = X w (t,t ) X w (t ) for all t T m, t,t µ m (t) and t µ w (t ). () X j (HH) = max{x j (HL),X j (LH)} for all j = m,w. Condton () asserts that all agents of a gven type must receve the same share of martal output no matter whom they marry because they are consdered perfect substtutes by the ndvduals on the other sde of the market. Lkewse, condton () states that any type HH ndvdual must obtan a share of martal output equal to the share of martal output receved ether by an ndvdual of type HL of the same sex or by a type LH ndvdual of the same sex or by both. Ths s because type HH agents are perfect substtutes for type HL ndvduals regardng ther ablty to perform task A and are also perfect substtutes for type LH ndvduals regardng ther ablty to perform task B. Notce that some men wll not be marred n ths market snce women are scarce. In a stable assgnment, t must be the case that type LL men are some, f not all, of these sngle men. Otherwse, there would be unmarred men wth hgh ablty n at least one task who could bd away a marred type LL man s spouse. Ths mples that none of the type LL men can get any surplus from marrage and hence, all women marred to them must keep for themselves the total martal output generated by the couple: X m (LL) = 0. In general, the stable outcome wll depend on the number of dfferent types of agents. Let M t denote the number of type t men and W t the number of type t women n the market. For nstance, suppose that the profle of types s such that M W < M LL (.e. some type LL men are marred), M HH = W HH = 0, M HL W LH and M LH W HL. Under these crcumstances, a stable outcome requres M W type LL men to be sngle, M LL (M W) type LL men to get marred only to type LL women, all type HL women to marry only type LH men, all type LH women to get marred solely to type HL men and the rest of the type HL and LH men to marry type LL women. Snce some type LL women marry type LL men, by the argument mentoned above, these women wll obtan a share of martal output equvalent to x l. Nonetheless, some of them wll also marry type HL and LH men. As a result, type HL and LH men wll receve a share of x m x l and therefore type LH and HL women obtan a share of martal output equal to x h + x l x m > x m. There are crcumstances n whch the agents equlbrum shares wll be undetermned by stable matchng. For example, consder a market populated only by type HL and LL men and type LH women such that 9

10 M W = M LL. A stable marrage confguraton requres all type LL men to be sngle whle all type HL men marry all type LH women. The share of martal output kept by the type LH females s undetermned, albet t must be at least equal to x m (otherwse, any type LH woman wll prefer to get marred to any sngle type LL men) and t cannot exceed x h (otherwse, any type HL man wll prefer to be sngle): x m X w (LH) x h. 4 Pre-Martal Investments and Equlbrum As mentoned prevously, competton over mates determnes who marres whom and the shares of the martal output of each type of man and woman dependng on the confguraton of realzed types. In turn, these shares gude the ndvduals decsons to nvest and marry. Human captal nvestments are chosen to maxmze each ndvdual s ex ante expected payoff. Thus, ths game s strategc n the frst stage. A strategy for a player of gender j = m,w s a functon that maps apttude types nto human captal nvestments: y j ( ) : [a,ā] [0,1]. We can formulate ths stuaton as a Bayesan game as follows. Players The set of all men and all women n the marrage market. States The set of all profles (a m 1,...,a m M,aw 1,...,a w W ) (am,a w ) of apttudes at task A of men and women, where a a j ā for all and for all j = m,w. Actons Each player s set of actons s the set of hs/her possble human captal nvestments n task A (nonnegatve numbers that are at most 1). Sgnals The set of sgnals that each player may observe s the set of possble apttudes. The sgnal functon τ j of each player of gender j s gven by τ j (am,a w ) = a j (each player knows hs/her own apttude). Belefs Every type of man assgns probablty G(a m 1 )G(a m 2 )...G(a m 1 ) G(am +1 )...G(am M ) G(aw 1 )...G(a w W ) Ĝ m (am,aw ) to the event that the apttude of every other player k of gender j s at most a j k. Every type of woman assgns probablty G(a m 1 )...G(a m M )G(aw 1 )G(a w 2 )...G(a w 1 ) G(aw +1 )...G(aw W ) Ĝw (am,a w ) to the event that the apttude of every other player k of gender j s at most a j k. Payoff functons Gven a strategy profle (y m ( ),y w ( )), let (y m (a m ),y w (a w )) denote the value of ths profle at (a m,a w ). Gven the strategy profle (y m ( ),y w ( )), the Bernoull payoff to a man n state (a m,a w ) s an average of four expected payoffs, each assocated wth one of the four possble abltes of ths man. In turn, each of these expected payoffs depends on the realzaton of abltes of the other market 10

11 partcpants. Gven the strategy profle (y m ( ),y w ( )), the Bernoull payoff to a man n state (a m,a w ) s gven by: u m [y m (a m ),y w (a w ),(a m,a w )] = = a m y m (a m )b[1 y m (a m )]EX m [HH y (a m m ),y w (a w ),(a m,a w )]+ + a m y m (a m )[1 b(1 y m )(a m )]EX m [HL y (a m m ),y w (a w ),(a m,a w )]+ + [1 a m y m (a m )]b[1 y m (a m )]EX m [LH y (a m m ),y w (a w ),(a m,a w )]+ + [1 a m y m (a m )][1 b(1 y m (a m ))]EX m [LL y (a m m ),y w (a w ),(a m,a w )] where EX m[t (ym (am ),yw (a w ),(a m,a w ))] denotes the expected martal share that a man of type t would obtan by stable matchng n stage 3 condtonal on the profle of apttudes (a m,aw ) n task A and the profle (y m (am ),yw (a w )) of human captal nvestments n task A. Gven the strategy profle (y m ( ),y w ( )), the Bernoull payoff to a woman n state (a m,a w ) can be obtaned smlarly. Defnton 4. A profle (y m ( ),y w ( )) s a pure-strategy Bayesan Nash equlbrum f for each man, y m (a m ) arg max u m y [y m m,y (a m m ),y w (a w ),(a m,a w )]dĝm (a m,a w ) a m a w and f for each woman, and y w (a w ) arg max y w a m a m a w Let s defne EX m [t y ( ),y m w ( )] EX w [t y m ( ),y ( )] w u w [y m (a m ),y w,y w (a w ),(a m,a w )]dĝw (a m,a w ) a w. a m a m a w EX m [t y m (a m ),y w (a w ),(a m,a w )]dĝ(am,a w ) a w EX w [t y m (a m ),y w (a w ),(a m,a w )]dĝ(am,a w ) That s, EX m [t ym ( ),yw ( )] denotes the ex ante expected martal share that a man of type t would obtan by stable matchng n stage 3 condtonal on the strategy profle (y m ( ),yw ( )) of human captal nvestments n task A. The same holds for women. 11

12 Proposton 1. For every player of gender j n any Bayesan Nash equlbrum there exsts ã j such that the Bayesan Nash equlbrum nvestment functon has ths form: y j ( ) = 1 f a j > ãj {0,1} f a j = ãj 0 f a j < ãj for all a j [a,ā] and j = m,w. Every equlbrum profle satsfes that each player s strategy s a weakly ncreasng and possbly dscontnuous functon of her apttude. We confne attenton to Sex-Symmetrc Bayesan Nash Equlbra (SSBNE). In a sex-symmetrc equlbrum, all the agents of the same sex play the same strategy, that s, y j ( ) = yj ( ) for every player and j = m,w. Thus, the equlbrum strateges for each sex are such that f an agent s apttude at task A exceeds a certan threshold, ã j, she wll nvest all her human captal n task A. Otherwse, she wll be traned n task B. The tranng strateges wll be accordng-to-sex when the threshold assocated wth one sex s a and the threshold assocated wth the other sex s ā. Under these crcumstances, captal nvestment decsons wll be perfectly correlated wth sex. If the thresholds assocated wth both sexes are nteror and concde, then tranng wll be accordng-to-apttude; that s, a < ã m = ã w < ā so that sex roles are completely absent. Conversely, f the cutoffs are nteror but dffer, we have a mx-tranng-strategy n the sense that captal nvestment decsons are mperfectly correlated wth both sex and apttude. Characterzng all equlbra s not the scope of ths chapter. Instead, we are nterested n answerng the followng questons: Can tranng-accordng-to-sex be supported as equlbra n ths envronment? If so, are they effcent? 5 Tranng accordng to sex Gender roles could emerge n a socety n whch all members of one sex are traned exclusvely n one task whle the opposte sex s traned n the complementary task. Gven the producton functon, a sexual specalzaton n tranng could lead to a sexual specalzaton n tasks. The am of ths secton s to fnd the condtons under whch tranng-accordng-to-sex strateges consttute Bayesan Nash equlbra and the condtons under whch the sexual dvsons of labor emerge n the socety. 12

13 There are two possble tranng-accordng-to-sex strateges: one n whch all males nvest ther entre human captal n task A whle all females nvest exclusvely n task B ( AB ) and, the other, n whch all males specalze n task B whle all females specalze n task A ( BA ). In tranng-accordng-to-sex equlbra, each ndvdual wth apttude a for task A and whose sex receves tranng n task A, becomes a type HL agent wth probablty a and type LL agent wth probablty 1 a. On the other hand, those ndvduals who belong to the opposte sex, and therefore, nvest all ther human captal n task B, become type LH agents wth probablty b and type LL wth 1 b. Thus, specalzaton n tranng by sex leads to only two types of agents for each gender. The problem s solved by usng backward nducton. 5.1 The Thrd Stage Proposton 2 characterzes the stable matchng n stage 3 gven the sexual specalzaton n tranng n stage 1. Ths characterzaton depends on the confguraton of realzed types n stage 2. Proposton 2. Gven AB specalzaton n tranng n stage 1, any stable outcome satsfes the followng condtons: X m (HL) = X 0 f M HL > W x m x l f W LH < M HL < W m (LL) = 0 x h x m f M HL < W LH X w (LH) = X x h f M HL > W x h + x l x m f W LH < M HL < W w (LL) = x m f M HL < W LH x m x l f M HL > W f M HL < W µ m (HL) = {LH, LL, } {LH, LL} {LH} f M HL > W f W LH < M HL < W f M HL < W LH 13

14 M W M W M W HL LL LH LL HL LL LH LL HL LL LH LL M HL > W W LH < M HL < W M HL < W LH Fgure 1: The stable matchng n stage 3 gven the AB tranng-accordng-to-sex strateges n stage 1. µ m (LL) = { } {LL, } {LH, LL, } f M HL > W f W LH < M HL < W f M HL < W LH Gven BA specalzaton n tranng n stage 1, any stable outcome s analogous to the one above: the roles of type HL males are replaced by those of type LH males whle the roles of type LH females are replaced by those of type HL females. Fgure 1 shows the stable matchng assgnment n stage 3 gven the AB tranng-accordng-to-sex strateges n stage 1. If the profle of types s such that the number of type HL men exceeds the number of women n the market, then at least some of these men wll reman sngle. The abundance of men wth a hgh ablty n task A make men unable to capture any of the martal output produced by the couple and transfer t entrely to ther wves. Suppose that the profle of types s such that type HL men are relatvely abundant n the market n the sense that they exceed n number type LH women but not the total number women n 14

15 the market. Under these crcumstances, a stable outcome requres all type LH women to get marred only to type HL men. The remanng type HL men and some of type LL men wll get marred to all type LL women. Gven that some type LL men reman sngle, these men cannot get any surplus from marrage and type LL women keep for themselves the total martal output generated by the couple, x l. As a result, type HL men obtan a share of martal output equal to x m x l. Consequently, type LH women capture the remanng of the hgh martal output, x h (x m x l ). Fnally, suppose a profle of types such that type HL men are relatvely scarce n the marrage market. That s, the number of type HL men fall short of the number of type LH women n the market. Under these crcumstances, a stable outcome requres all type HL men to get marred only to type LH women. The remanng type LH women and all type LL women wll get marred to type LL men. Gven that some type LL men reman sngle, these men cannot get any surplus from marrage and all women keep for themselves the total martal output generated by the couple. Consequently, type HL men capture the remanng of the hgh martal output, x h x m. Gven AB specalzaton n tranng, the stable matchng for the non-generc case M HL = W requres all men of type HL to marry all women n the marrage market. The agents equlbrum shares are undetermned by the stable outcome. However, stablty mposes an upper and lower bound on the shares of martal output kept by the type HL males. These shares must not be lower than zero snce otherwse, they would prefer to be sngle and they cannot be greater than x m x l, snce otherwse the type LL women would prefer to leave ther spouses and marry type LL men. Smlarly, the stable matchng for the non-generc case M HL = W LH requres all men of type HL to marry all women of type LH n the market. The agents equlbrum shares are also undetermned by the stable outcome. Stablty mposes that these shares must not be lower than x m x l snce otherwse, type HL men would prefer to leave ther spouses and marry type LL women, and they cannot be greater than x h x m, snce otherwse the type LH women would prefer to marry type LL men. Any dvson of the martal output that makes the outcome stable for the non-generc cases nconsequental for the rest of the analyss. As a result, Proposton 2 does not consder these generc cases. Defne the market sex rato as the number of avalable men to women n the populaton, M W. Notce that for a gven sex rato, as the number of type HL (LH) men decreases n the populaton, the competton among women to attract ths type of men ntensfes. As a result, the hgher demand for these men leads to an ncrease n the share of martal output captured by them. 15

16 5.2 The Frst Stage Large Marrage Markets Ths secton consders a large marrage market whch s defned as a market populated by a contnuum of ndvduals on each sde of the market. Here M denotes the measure of all men n the populaton and W the measure of all women. Let α be the expected value of a, (.e. α ā a dg(a)). By the Law of Large Numbers a (LLN), almost surely the mass of type HL men and type LH women are known and under AB, they are αm and bw respectvely. Smlarly, the mass of type LH men and type HL women under BA are almost surely bm and αw respectvely. If we substtuted these values nto Proposton 2, then we would obtan the stable marrage confguraton and equlbrum shares of martal output. Notce that prevously, the equlbrum market shares were determned by the relatve number of type HL or LH men n the populaton. Now, the equlbrum payoffs are fully determned by the value of the market sex rato. Proposton 3 characterzes the condtons under whch the tranng-accordng-to-sex strateges consttute equlbra n a large marrage market. We focus our attenton only on generc values of the sex rato. Recall that the par (a, b) represents the ndvdual s unobservable apttudes to perform task A and task B. The equlbrum n tranng s determned by both the value of the sex rato and the ncentves to devate by the ndvduals wth the hghest and lowest apttude to perform task A. Ths proposton hghlghts the possble trade off faced by any agent when consderng the human captal nvestment decson n the frst stage. Here the agent trades off acqurng tranng n a task at whch she has a lower apttude for but that s hghly sought and hence, endogenously well remunerated, aganst acqurng tranng n a task for whch she has a hgher apttude for but that s less n demand and hence, endogenously poorly remunerated. Proposton 3. In large marrage markets, (1) AB s a Nash equlbrum n tranng f and only f: M W > 1 α ( ) or f b α < M W < 1 α and ā xm x l x h x m b a ( ) or f M W < b α and ā b a xh x m x m x l (2) BA s a Nash equlbrum n tranng f and only f : M W > 1 b. 16

17 ( ) or f α b < M W < 1 b and ā b a xh x m x m x l or f M W < α b and ā ( xm x l x h x m ) b a Under AB, men are traned n task A whle women are traned n task B. Gven the sex rato s large enough or alternatvely, for a gven value of the sex rato, f the apttude to perform task A s on average hgh enough, the AB tranng-accordng-to-sex consttutes an equlbrum ndependently of the ndvduals apttude to perform task B. Ths s because under such crcumstances, men who develop a hgh ablty at performng task A become abundant, exceedng the number of women n the market. Competton among these men to get a partner leads them to transfer the entre martal output to ther wves. Gven that all men obtan a zero payoff ndependently of ther ex ante type and of whether they get marred or reman sngle, no man has an ncentve to devate and nvest n task B. On the other sde of the market, the severe scarcty of women enables them to capture the entre martal output and obtan x h f they develop a hgh ablty at performng task B and x m otherwse. The probablty of developng a hgh ablty n task B s maxmzed by nvestng all the human captal avalable n that task. As a result, no woman has an ncentve to devate and nvest n task A. Ths specalzaton n tranng leads ultmately to a sexual dvson of labor n whch all men perform task A and all women perform task B. Gven the sex rato s not large enough or, alternatvely, f the agents apttude to perform task A s not suffcently hgh on average, then AB can be supported n equlbrum f and only f the ndvduals apttude at task B s restrcted to le wthn a range of values. Otherwse, at least, the lowest ft men and/or the hghest ft women for task A wll fnd t proftable to nvest a postve amount of human captal n the task n whch the opposte sex s beng traned. Notce also that, as the sex rato decreases, the requred possble values for b must be larger. 5 Ths s because as the sex rato decreases, the payoff to a man who has a hgh ablty at performng at least one task ncreases. In partcular, the payoff assocated wth havng a hgh ablty at performng task A gets relatvely larger than the payoff assocated wth havng a hgh ablty at performng task B snce the former type of men become relatvely scarce and n hgh demand. Thus, the lowest ft man for task A faces now lower ncentves to devate and the requred upper bound value of b becomes less restrctve. On the other hand, as the sex rato decreases, the payoff to a woman who has a hgh 5 If nature dd dscrmnate, the condtons placed on males apttude to perform such task would become less restrctve (.e. the requred b m does not need to be so hgh) whle the condtons on females apttude would become more restrctve (.e. we need a lower and lower b w ). 17

18 ablty at performng task B decreases and gets relatvely closer to the payoff assocated wth havng a hgh ablty at performng task A. Thus, the hghest ft woman for task A faces now hgher ncentves to devate and the requred lower bound value of b becomes more restrctve. Ths specalzaton n tranng strategy leads to a sexual dvson of labor among all the couples except for the (LL, LL) matches for whch the model has no predctons n terms of job assgnments. If we assumed that the job assgnment n these marrages s made accordng to the tranng that each ndvdual receved n each task, then the prevous sexual dvson of labor would emerge n socety. Otherwse, f the measure of (LL, LL) matches whch s gven by W αm, s relatvely small, then task A would tend to be dentfable as men s work. Under BA, men are traned n task B whle women are traned n task A. If the sex rato s large enough or alternatvely, for a gven value of the sex rato, f the apttude to perform task B s hgh enough, the BA tranng-accordng-to-sex consttutes an equlbrum. The ntuton behnd ths result s smlar to the one provded for the AB case. Ths specalzaton n tranng leads ultmately to a sexual dvson of labor n whch all men perform task B and all women perform task A. Conversely, f the sex rato s not large enough or, alternatvely, f the agents apttude to perform task B s not suffcently hgh, then BA can be supported n equlbrum f and only f the ndvduals apttude at task B s restrcted to le wthn a range of values. Otherwse, at the least, the hghest ft men and/or the lowest ft women for task A wll fnd t proftable to nvest a postve amount of human captal n the task assgned to the opposte sex. Notce also that, as the sex rato decreases, the requred possble values for b must be lower. 6 Ths s because as the sex rato decreases, the payoff to a man who has a hgh ablty at performng task B gets relatvely larger due to the scarcty of ths type of men. Thus, the hghest ft man for task A faces now lower ncentves to devate and the requred lower bound value of b becomes less restrctve. On the other hand, as the sex rato decreases, the payoff to a woman who has a hgh ablty at performng task A decreases and gets relatve closer to the payoff assocated wth havng a hgh ablty at performng task B. Thus, the lowest ft woman for task A faces now hgher ncentves to devate and the requred upper bound value of b becomes more restrctve. As prevously, f we assumed that the job assgnment among couples s made accordng to the tranng receved n each task, then the sexual dvson of labor n whch men perform task B and women task A would emerge n ths socety. Otherwse, task B would tend to be dentfable as men s work as long as the measure of (LL, LL) matches, whch s gven by W bm, s relatvely small. 6 If nature dd dscrmnate, as the sex rato decreased, the condtons on b m would be relaxed whle the ones placed on b w would become more restrctve. 18

19 Corollary 1. The sexual dvson of labor s an equlbrum outcome n large marrage markets f the malefemale sex rato s suffcently hgh or, alternatvely, f men have on average a hgh apttude for the task for whch they are traned n equlbrum. Proposton 3 makes t clear that the tranng-accordng-to-sex equlbra do not exst over the entre parameter space. Fgure 2 llustrates that these equlbra coexst for some range of parameter values, that nether equlbrum exsts over part of the parameter space and that there s a unque tranng-accordng-to-sex equlbrum for the remanng range of parameter values. Remark 1. When both tranng-accordng-to-sex equlbra coexst, the average ndvdual prefers all agents of hs/her sex to be traned n the task at whch he/she has a lower apttude f and only f the fracton of males s suffcently close to the fracton of females. Otherwse, the average man s ndfferent about both equlbrum outcomes and the average woman prefers the equlbrum outcome n whch all women are traned n the task at whch she has a hgher apttude. The ntuton behnd ths remark s straghtforward. Suppose frst that on average ndvduals are born wth more apttude to perform task B than task A and consder the AB and BA equlbrum outcomes. If women specalze n gettng traned n task B, the probablty that women develop a hgh ablty at performng task B n stage 2 wll on average be hgher than the probablty of developng a hgh ablty at performng task A had they specalzed n gettng traned n task A. Lkewse, f men specalze n gettng traned n task A, the probablty that men develop a hgh ablty at performng task A n stage 2 wll on average be lower than the probablty of developng a hgh ablty at performng task B f they specalzed n gettng traned n task B. If the sex rato s suffcently low, M W < 1 α, the relatve abundance of the ex post hgh type women and the relatve scarcty of the ex post hgh type men leads to hgh competton among women to get marred to men wth hgh ablty n task A. Ths hgher competton for men among women results n a lower transfer of martal output from husbands to wves. Overall, ths competton effect domnates and despte the hgher probablty of becomng a hgh type woman ex post, the average woman s expected payoff s lower under the AB equlbrum outcome than under the BA equlbrum outcome. Under the same crcumstances, and by the same argument, the average man (ex ante all men) prefers the equlbrum outcome n whch all men are traned n task A. If the sex rato s suffcently hgh, M W > 1 α, the number of men who develop a hgh ablty at performng the task they are traned n, exceeds the number of women n the populaton under both AB and BA. These 19

20 M W ( ) xm xl ā x < a < α 2 < ( ) xh xm α < ā < a h xm xm x l α b AB Both Equlbra 1 α Both Eq. No Equlbrum BA Both Eq. b α 1 0 ( ) xm xl ā x h xm a α 2 α α ā ( ) xh xm a xm x l 1 b 1 b M W ( ) xm xl ā x < α 2 < a < α < ā < ( ) xh xm α < a h xm xm x l α b AB Both Equlbra 1 α Both Equlbra No Equlbrum BA b α 1 0 ( ) xm xl ā x h xm α 2 a α ā Both Equlbra α ( ) xh xm a xm x l 1 b 1 b Fgure 2: Tranng-accordng-to-sex equlbra. 20

21 men are unable to capture any postve share of the martal output due to the hgh competton for women that exsts among them. Snce women capture the entre martal output n both equlbrum outcomes and the probablty of becomng a hgh type woman ex post s on average greater under AB tranng strategy than under BA tranng strategy, the average woman prefers the AB equlbrum outcome. Fnally, when the agents expected apttude to perform task A concdes wth ther apttude to perform task B, both sexes are ndfferent about the two tranng-accordng-to-sex equlbrum outcomes Small Marrage Markets Ths secton analyzes a small marrage market wth a fnte number of men and women. Lets focus frst on the tranng-accordng-to-sex AB. Consder a man wth apttude a m for task A who s delberatng about the amount of human captal that he should nvest n task A. Gven the strateges of the others, hs share of martal output wll be zero n case he becomes type LL. Ths s because at least some of these men wll be sngle. If he became type HH, HL or LH, hs share would vary dependng on the realzed number of each type of man and woman n the market. The realzed number of type HL males wll follow a bnomal dstrbuton wth parameter α. Smlarly, the realzed number of type LH females wll follow a bnomal dstrbuton wth parameter b. We abuse notaton by lettng EX m [t 1,0] denote the expected share of martal output obtaned by hm n case he becomes type t gven the strateges of the other partcpants. Ths expected share would be the same f he became type HH or type HL. Ths man chooses the level of human captal whch maxmzes hs ex ante expected share of martal output. max 0 y m 1 a m y m The objectve functon s convex n y m EX m [HL 1,0] + (1 a m y m )b(1 y m )EX m [LH 1,0] to ensure a corner soluton. Ths man wll nvest exclusvely n task A f and only f a m EXm [HL 1,0] bex m [LH 1,0]. Consequently, all men wll nvest exclusvely n task A f and only f b a(ex m [HL 1,0]/EX m [LH 1,0]) Ths condton wll not be bndng f the rght hand sde of the nequalty s greater than one. Notce that the rato EX m [HL 1,0]/EX m [LH 1,0] has a value of at least one. If the realzed number of type LL women were postve and none of these women got marred to type LL men, then a type LH man would capture 21

22 the same share of martal output captured by a type HL man. Instead, f some of these women got marred to type LL men, then the share of martal output captured by a LH man would be lower than the share of martal output captured by a type HL man. In the lmt, as b 0, almost all women become type LL n stage 2 and the value of ths rato approaches one. As the common apttude at task B ncreases, the relatve number of type LH women n the populaton rses, and as the result, the rato EX m [HL 1,0]/EX m [LH 1,0] also ncreases. In the lmt, as b 1, almost all women become type LH n stage 2 so that any type LH man s unable to capture any postve share of martal output. Consequently, the rato approaches nfnty and the above condton s satsfed. Consder now a woman wth apttude a w for task A who s delberatng about her human captal nvestment decson. Her share of martal output wll also depend on the total and relatve number of each type of men and women n the market. We abuse notaton and denote by EX w [HL 1,0], EX w [LH 1,0] and EX w [LL 1,0] her expected share of martal output n case she becomes type HL, type HH or LH, and type LL respectvely gven the strateges of the other partcpants. Her problem can be formulated as follows: max 0 y w 1 a w y w [1 b(1 y w )]EX w [HL 1,0]+ +b(1 y w )EX w [LH 1,0] + (1 a w y w )[1 b(1 y w )]EX w [LH 1,0] Ths functon s also convex n y w. Ths woman wll nvest exclusvely n task B f and only f a w (EXw [HL 1,0] EX w [LL 1,0]) b(ex w [LH 1,0] EX w [LL 1,0]). All women wll nvest n task B f and only f: b ā(ex w [HL 1,0] EX w [LL 1,0])/(EX w [LH 1,0] EX w [LL 1,0]) Notce that the rato EXw [HL 1,0] EX w [LL 1,0] EX w [LH 1,0] EX w [LL 1,0] has a nonnegatve value of at most one and therefore, ths condton wll be bndng. If we carry out exactly the same analyss for the BA tranng-accordng-to-sex strateges, potental lower and upper lmts on the apttude at task B are obtaned as necessary and suffcent condtons for the BA tranng-accordng-to-sex to be a Nash equlbrum Mechancal vs. Nonmechancal task In the prevous sectons, we mplctly assumed that task B was a nonmechancal task. That s, an ndvdual was not guaranteed ex post to have a hgh ablty at performng task B f she nvested her entre amount of human captal n the cted task. A task s mechancal f every agent s guaranteed to have a hgh ablty at 22

23 performng that task as long as complete tranng were to be receved by the agent. Task B s mechancal f and only f b = 1. If ths s the case and under the tranng-accordng-to-sex strateges, any sex who specalzes n acqurng tranng n task B wll develop a hgh ablty at performng task B. As a result, the competton among these members to marry agents of the opposte sex wth hgh ablty at performng task A becomes extreme. The same analyss as the one developed n the prevous secton can be performed. The results are stated n proposton 4. Proposton 4. If b = 1, both tranng-accordng-to-sex strateges are Nash equlbra for all generc values of the sex rato and for all market szes, leadng to the emergence of the sexual dvsons of labor n the socety. Conversely, f every ndvdual were guaranteed ex post to develop a low ablty at performng task B even f she nvested her entre amount of human captal n the cted task, then only the AB tranng-accordng-tosex could be supported n equlbrum n large marrage markets. A necessary and suffcent condton s a hgh enough sex rato ( M W > 1 α). 7 Under ths condton, type HL men are hghly abundant n the populaton. Competton among them leads them to obtan a zero payoff. No man faces ncentves to devate snce he would be unable to capture any of the martal surplus by devatng and nvestng a postve amount of human captal n task B. Wth certanty, women become type LL by nvestng exclusvely n task B, an acton that leads them to capture a share of martal output equal to x m. By nvestng a postve amount of human captal n task A, they could become ether type HL or type LL women. Both types of women are perfect substtutes from a type HL man s perspectve so that ther share of martal output would stll be x m ndependently of the type realzaton. Thus, no woman faces any ncentves to devate. Women face ncentves to devate f the sex rato s not hgh enough or f the marrage market s small. In these cases, competton among type LL women to attract type HL men leads them to transfer a share of martal output equal to x m x l. Some type LL men now get marred. However, type HL or type LL women are no longer perfect substtutes from these men s perspectve. Any of these women would keep for herself the total martal output produced by the couple f they marred a type LL man. Thus, any woman would obtan a hgher expected martal output f she nvested a postve amount of human captal n task A and therefore, she would certanly devate. In a smlar ven, men face ncentves to devate under the BA tranng-accordng-to-sex strategy. 7 If b = 0 and G(a) were a degenerate dstrbuton ether at a = 1 or at a = 0, then both tranng-accordng-to-sex would always be Nash equlbra. 23