The Gender Division of Labor Revisited∗
Silvia Martı́nez Gorricho†
November, 2003.
Abstract
A coordination model characterized by heterogeneity and non-random matching is developed. Heterogeneity among individuals comes solely from their innate aptitude to perform one of the two tasks
available. Potential spouses choose first the level of investment in each of the two types of human capital,
and subsequently, engage in the search for a marriage partner. Premarital investments and aptitudes
determine an individual’s ability in the accomplishment of both tasks and hence, her desirability on
the marriage market. In this environment, and under certain restrictions on the parameters, we show
that gender segregations can be supported as decentralized equilibria in a society which treats the sexes
equally. Furthermore, we claim that the sex ratio together with the extent of the market constitute crucial
factors in the determination of both existence and efficiency of the sexual division of labor. In small or
intermediate marriage markets, characterized by a high risk associated to mismatches in the population,
the sexual division of labor can help coordinate people efficiently in the choice of premarital investments
by relying exclusively on gender considerations. The sexual division of labor in which the gender who is
scarce in the society specializes in the most productive task is efficient only if the sex ratio is sufficiently
high. Otherwise, the alternative sexual division of labor emerges as the efficient rule. When the marriage
market is large, the risk associated to mismatches vanishes and the training-according-to-sex is Pareto
dominated.
Keywords: Sexual Division of Labor; Human Capital; Coordination; Heterogeneity; Nonrandom
Matching; Sex ratio; Extent of the Market
JEL Classification: D13; J16; J24; P41
∗I
am indebted to Aloysius Siow and Ettore Damiano for their guidance through this project. All remaining errors are mine.
of Economics, University of Toronto, 150 St. George St., Toronto, Ontario, M5S 3G7, CANADA.
† Department
Email:smartine@chass.utoronto.ca
1
1
Introduction
The sexual division of labor is a phenomenon found in virtually all human societies. Due to its importance,
economists have attempted to model the causes and implications of gender roles across societies.1
Economic theory suggests that occupational segregation by gender might be due to supply side factors,
demand side factors, or a combination of both. For instance, the major supply-side theory consists on the
human capital explanation which states that since women generally anticipate shorter and less continuous
work lives than men, it will be in their interest to choose occupations which require smaller human capital
investments and have lower wage penalties for time spent out of the labor market (Polachek, 1981). The fact
that women may face barriers to obtaining education and pre-job training in traditionally male fields is called
“societal discrimination” and constitutes an alternative supply effect explanation (Blau, Ferber and Winkler,
1988). On the demand side, the sexual division of labor is supported by discrimination against women within
the paid labor force (eg. Aigner and Cain (1977), Danziger and Katz (1996) and Francois (1998)).
In Becker’s Chapter 2 of his “Treatise on the Family” (1991), the gender division of labor arises due
to gains from specialized investments and intrinsic differences between the sexes. He assumes two types of
human capital: the household and the market sector. Women’s inherent comparative advantage in household
activities and the presence of increasing returns to human capital investment prompt women to specialize in
household work while their spouses specialize in market work. Thus, the model relies on biological differences
across the sexes to support the sexual division of labor.
However, as Hadfield (1999) points out, the sexual division of labor is fairly constant across societies, i.e.
all tasks in a society tend to be gendered (easily identifiable as either women’s work or men’s work) but a
particular division is not (to which sex, a task is specific, varies considerably across cultures). The biological
model does not account for this fact. Furthermore, as time passes, technological developments contribute
to the reduction of the comparative advantage enjoyed by any gender in the performance of a particular
activity. For instance, through the ages, technological changes have lessened the brute physical component of
labor. As a result, the absence of any significant biological advantage also calls into question the continued
applicability of the biological model.
Hadfield proposes a model where an individual’s incentive to coordinate her human capital investments
with those of a future but yet undetermined partner of the opposite sex, leads to a correlation between the
1 For
recent studies about occupational segregation by gender, refer to Blau et all (1998) who analyze the trends in the US
over 1970 and 1980’s while Dolado et all (2002) analyze so in the EU countries vis-à-vis the US.
2
person’s sex and her economic role. In her model, potential spouses choose first the level of investment in
each of the two types of human capital, and subsequently engage in the search for a marriage partner. Until
human capital investments are made, all individuals of a given sex are a priori identical. In her set up,
mates are valued only in terms of the household utility obtained with them. Thus, premarital investments
determine an individual’s desirability on the marriage market.
Engineer and Welling (1999) attempt to introduce heterogeneity in the environment. As in Hadfield’s
paper, they use a nontransferable utility framework without frictions and assume an equal number of males
and females in the population. However, in their model, males and females can either have an aptitude for
market work or household work albeit not for both. In addition, human capital training is modelled as a
discrete binary choice variable since it can be done only in one sphere of work. Finally, the marriage market
is structured as a pure random matching market so that there is no sorting available. These ingredients give
rise to a coordination problem in choosing complementary specializations. Under a large set of circumstances,
the training-according-to-sex equilibria and the training-by-aptitude equilibrium are the only pure strategy
equilibria. The authors rank the equilibria when they coexist and demonstrate that affirmative action policies
can increase average welfare by eliminating some equilibria.
Our paper contributes to the previous literature by modelling human capital investments on both sides
of the market under heterogeneity and nonrandom matching. As it is cited above, Engineer and Welling’s
key assumption consists on families being formed randomly on the basis of “true love”. Nonetheless, when
individuals are heterogeneous, a common observation is that of assortative matching. Hence, in contrast to
the assumptions of Engineer and Welling, this paper provides a framework where matches are done according
to economic criteria and human capital training is no longer a binary choice. In this environment, and
under certain restrictions on the parameters, we show that the gender division of labor can be supported as
equilibria in a society which treats the sexes equally. Furthermore, we claim that the sex ratio together with
the extent of the market constitute crucial factors in the determination of both existence and efficiency of
the sexual division of labor. In the Wealth of Nations, Adam Smith stated that “the extent of the division
of labor is limited by the extent of the market”. Smith’s insight could be applied to this context if the
the sexual division of labor is interpreted as a particular type of labor division. The intuition behind such
statement goes as follows: in small or intermediate marriage markets, characterized by a high risk associated
to mismatches in the population, the sexual division of labor can help to coordinate people efficiently in
the choice of premarital investments by relying exclusively on gender considerations. Nonetheless, when the
marriage market is large, a continuum number of each type of individuals will be expected to exist on both
3
sides of the market so that the risk associated to mismatches vanishes. In such a case, the training-accordingto-sex is Pareto dominated. In a similar fashion, the sex ratio defined as the number of available men to
available women, is expected to alter the allocation of marital output between spouses and thereby, influence
premarital investments. In large marriage markets, when the sex ratio is relatively high, the mix training
according-to-both- aptitude-and-sex emerges as the efficient mechanism but as the sex ratio approaches unity,
it is replaced by the training-according-to-aptitude which satisfies the comparative advantage principle. In
small marriage markets instead, the sexual division of labor in which the gender who is scarce in the society
specializes in the most productive task is optimal only if the sex ratio is sufficiently high. Otherwise, the
alternative sexual division of labor emerges as the efficient coordination mechanism.
The paper is organized as follows; In Section 2, the assumptions and the basic structure of the model
are described. Section 3 characterizes the matching process. Section 4 develops the conditions under which
the sexual divisions of labor are decentralized equilibria in a transferable utility environment. Section 5
investigates whether they constitute ex ante constrained pareto optima. Furthermore, it examines the impact
of the extent of the market and the sex ratio. Section 6 analyzes evolution. Finally, Section 7 concludes by
summarizing the main results and discussing possible extensions.
2
The Static Model
The marriage market has two distinct group of risk-neutral agents, men (M) and women (W). Let N denote
the total number (measure) of available men and F the total number (measure) of available women in the
market. Two spheres of work must be covered. We euphemistically label them task A and task B. Assume
that each individual is characterized by a real number, a ∈ [a, ā] ⊆ [0, 1], which represents the individual’s
unobservable aptitude to perform task A. The closer a is to ā, the more potential the individual has in the
accomplishment of task A. Let G(â) denote the probability that an agent’s aptitude is no greater than â and
R ā
let µ be the expected value of a, i.e. µ ≡ a a dG(a). The distribution G(·) is assumed nondegenerate and
common knowledge to the public. In addition, assume that all agents have the same aptitude to perform
task B which is given by b ∈ (0, 1]. Hence, heterogeneity among individuals comes solely from their innate
aptitude to perform task A. This framework implicity assumes that nature does not discriminate due to the
fact that aptitudes are equally distributed across the sexes.2
2 This
model can be easily extended to a more general framework where Gi (a) constitutes the aptitude probability distribution
for task A and bi the aptitude associated to task B, for gender i = m, w. This modification does not alter the substance of the
4
Each person’s life goes through four stages. In Stage 1, an individual chooses a level of investment in each
of the two types of human capital, ta and tb . For convenience, the units of investment in task A and B are
normalized to 1 (ta + tb = 1). Hence, in this model, there is not an explicit cost of investment in either task
but an implicit opportunity cost: investing more in one task irremediably translates in a lower investment
for the other task. Since the agent is assumed to know her aptitude, she will invest human capital in terms
of it. That is, a acts as an anchor for her human capital investment decision.
In Stage 2, the agent’s ability or skill for both tasks is determined by her human capital investment as
well as her aptitude. Hence, each individual becomes one of four possible types according to the following
process:
Individual Type
Ability in Task A
Ability in Task B
Probability
Type HH
High
High
ata b(1 − ta )
Type HL
High
Low
ata [1 − b(1 − ta )]
Type LH
Low
High
(1 − ata )b(1 − ta )
Type LL
Low
Low
(1 − ata )[1 − b(1 − ta )]
In other words, the probability that an agent becomes high type in task A is given by her aptitude in task
A, a, times her human capital investment in that task, ta . Similarly, the probability that an agent becomes
high type in task B is given by the general aptitude b for task B, times her human capital investment in that
task, tb = 1 − ta . Both probabilities are mutually independent.
In Stage 3, the individual enters the marriage market in search of an optimal mate with whom to set up
a household. The type of an agent is perfectly observed by all the participants. The choice of a partner will
be based on economic considerations.
In Stage 4, households engage in production, deciding about the job assignment for the completion of
both tasks. For expositional ease, we will henceforth assume (as has been done in the literature) that
optimal output is obtained with complete specialization by both members of the household. Specialization
in a particular task comes from the fact that each individual is more productive working predominantly
in a particular sphere than dividing her time between the two spheres. Based on this assumption, when
two individuals with different abilities in one task, match together, specialization leads the individual whose
conclusions that follow.
5
ability is higher to perform that task. Otherwise, job assignment is made according to the training received
in each task. In case that both of them have received the same amount of training, job assignment is random.
Thus, for instance, if a type HL man and a type LL woman marry each other, the type HL man will specialize
in task A while the type LL woman will specialize in task B. If instead, a type HH man gets together with a
type LL woman, this family will assign the partner with the highest human capital in task A to accomplish
that task (in case that both of them received an equal amount of training in A, the assignment would be
random). Hence, specialization follows the basic rule of comparative advantage in ability. Let an individual
be, in this stage, high (low) type if she has high (low) ability in the task she specializes in. In consequence,
and in terms of job assignment, four possible types of marriages can arise in this particular society: M T H if
both tasks are performed by high type individuals; M T HA if task A is performed by a high type individual
whereas task B is performed by a low type individual; M T HB if task A is performed by a low type individual
whereas task B is performed by a high type individual; and M T L if both tasks are performed by low type
individuals. This classification implicity assumes equality of sexes, that is, the type of marriage that will
result when two agents match together depends only on the skill of the individuals for the task in which they
specialize, not on their gender. In addition, both MTHA and MTHB belong to the category of mixed type
of marriages, MTM.
When a couple marry, the married household generates marital output. Let M T be the set of marriage
types and X = {xh , xm , xl } be the set of possible marital outputs. The marriage production function is a
mapping f : M T →X such that:
f (M T L) = xl
f (M T HA) = f (M T HB) ≡ f (M T M ) = xm = xl + c
f (M T H) = xh = xl + d
where xl > 0, c > 0, and d > 2c.
The assumption f (M T HA) = f (M T HB) asserts that task A is as valuable as task B to the family
mutatis mutandis. The assumption d > 2c implies the following: Suppose that there are two individuals
on each side of the market. If their types are such that a MTH and a MTL marriage could be generated,
the total production obtained under this arrangement, exceeds the total production that could be generated
otherwise, i.e. with two MTM type marriages : xh + xl > 2xm . It implicitly leads to positive assortative
6
matching among high and low types individuals. Trade between households is not allowed 3 . We also assume
that individuals who remain single obtain zero utility.
In terms of marital output only, this scenario completely determines the agents’ heterogeneous tastes for
the types of their potential partners when they enter into the marriage market. An agent whose ability is
high in only one of the tasks, will rank agents of the opposite sex according to their ability in the task for
which she has low ability. An agent with the same skill for both tasks, will consider both of the traits instead.
Thus, a type HH woman would be indifferent among matching a type HH, type HL or type LH man, since in
the three cases, the marital output obtained in Stage 4 would be xh . Nonetheless, she would strictly prefer
them to any type LL man due to the fact that xh > xm . Similarly, a type HL woman would be indifferent
among a type HH or a type LH man but she would strictly prefer them to any type HL or type LL man. In
the same fashion, type HH and type HL men would be preferred to type LH and type LL men by any type
LH woman. Finally, a type LL woman would strictly prefer any men whose type differs from hers.
To analyze the matching and investment process, we first start with the equilibrium for the matching
process and then, proceed to the investment stage assuming that the participants in the market foresee the
impact that their investment will have on the outcome of the matching process.
3
The Matching Process
Without loss of generality, assume that N > F so that the total number of marriages in the society will be
equal to F 4 . Assume also that utility is transferable so that marital output is perfectly divisible between
a married couple. For a marriage between type i male and type j female, ∀i, j = HH, HL, LH, LL, the
couple will bargain about the division of the total marital output generated by them, Xij . An individual
may attract a spouse by promising to allocate more of the divisible marital output to that potential spouse.
Let denote τi 5 the share of marital output kept by the male so that the female keeps Xij − τi for herself.
3 The
main focus of this paper consists on examining the isolated effects of the sex ratio and the extend of the market on
the gender division of labor. Albeit we agree that trade between families is clearly important, its consideration goes beyond the
scope of this paper.
4 The case N < F is implicitly analyzed by interchanging the roles of males and females throughout the paper. The case
N = F is approached by taking the limit of the sex ratio to unity.
5 All men belonging to the same type are perfect substitutes among the women. Therefore, if they marry to different type
of women in equilibrium, it must be the case that they must be indifferent among these women, so that the equilibrium shares
satisfy τij = τi ∀j such that men of type i marry type j women.
7
The market clears when given τi for every type of marriage, every individual can find a spouse of his or her
choice if he or she wants to.
Definition 1. The marriage market clears if and only if the shares of marital output kept by the males
are such that, for each profile of types in the second stage, the marriage configuration is stable, i.e. if:
(1) No current married individual wants to leave her marriage in order to be single or to marry any single
individual.
(2) No two individuals from different marriages want to leave their spouses to marry each other.
At this point, let Nlk denote the number of type l agents whose gender is k, ∀l = HH, HL, LH, LL and
∀k = m, w. Since women are scarce in this society, some men will not be married. Type LL men will at
least be some of these single men since they have the lowest possible ability in both tasks which makes them
be the least attractive candidates for women. Therefore, those type LL men who remain single will obtain
zero utility. Since all type LL men are perfect substitutes as spouses, all type LL men will end up receiving
zero payoff and all women married to them will keep for themselves the total marital output generated by
m
the couple. In concreteness, suppose that the profile of types is such that N − F < NLL
(ie. some type LL
w
m
w
m
w
m
. Under these circumstances, a stable
≥ NHL
& NLH
≥ NLH
= 0, NHL
= NHH
men are married), NHH
m
− (N − F ) type LL men to get married
marriage configuration requires N − F type LL men to be single, NLL
only to type LL women, all type HL women to marry only type LH men, all type LH women to get married
solely to type HL men and the rest of the type HL and LH men to marry type LL women. Since some
type LL women marry type LL men, by the argument mentioned above, these women will obtain a share of
marital output equivalent to xl . Nonetheless, some of them will also marry type HL and LH men. For these
women to be indifferent among all type of males, they must keep xl as their share of marital output in all
types of marriages. As a result, type HL and LH men will receive a share of c in case of getting married
to these women. Nonetheless, some of these men will also marry type LH and HL women respectively. In
equilibrium, all men of the same type (since they are identical) should receive the same share of marital
output independently of the type of women they end up marring to. Thus, both type HL and LH men will
keep c for themselves and therefore, the type LH and HL women who get married to them respectively, obtain
a share of marital output equal to xl + (d − c) > xm . These matches satisfy simultaneously conditions (1)
and (2) of Definition 1 and hence, they are in the core.
Assume henceforth that when the equilibrium shares of marital output are not completely determined, the
couples will divide the total marital output according to the Nash-Bargaining solution with equal bargaining
8
power. Thus, for example, in case of a society constituted only by type HL and LL men together with solely
m
type LH women where N − F = NLL
, all type LL men will remain single while all type HL men will marry
all type LH women. The marital output generated by each of these couples is xh . The share of the marital
output kept by the HL males is undetermined, albeit it must be greater or equal to zero (since otherwise,
they would prefer to be single), and it cannot exceed xm (if so, any woman will prefer to get married to any
single type LL men). Using the Nash-Bargaining solution with equal bargaining power, the threat point for
type HL men is zero while for women is xm . Hence, τHL = argmax{τHL (xh − τHL − xm )} or τHL = (d − c)/2.
4
Pre-Marital Investments and Equilibrium
In the first stage, given the strategies of all men and the rest of women, a woman with aptitude aw will
hold beliefs about the expected number of each type of men and women in the market and hence, about
the equilibrium shares that will clear the market. Therefore, she will choose the level of human capital that
maximizes her ex ante payoff given the strategies of the others.
4.1
The Sexual Division of Labor
Gender roles clearly emerge in a society where all members of one gender are trained only for one task while
the opposite gender is trained solely for the complementary task. In such a case, individuals of the same
gender receive training-according-to-sex and given the complementarity in training between the partners, the
sexual divisions of labor (SDL) emerge: all males perform task A while all females perform task B or vice
versa.
Definition 2. There are two possible training-according-to-sex strategies : one in which all males invest
their entire human capital in task A while all females invest exclusively in task B (MAFB), and one in which
all males specialize in task B while all females specialize in task A (MBFA).
Under training-according-to-sex, each individual with aptitude a for task A and whose gender specializes
in task A, becomes a type HL agent with probability a and a type LL agent with probability 1 − a. On the
other hand, those individuals who belong to the opposite sex, and therefore, invest all her human capital in
task B, become type LH agents with probability b and type LL with 1 − b. The aim of this section consists
on finding the conditions for which the training-according-to-sex strategies, and hence, the sexual division of
labor, constitute decentralized equilibria. Two cases are distinguished depending on the extent of the market.
9
First, we study large marriage markets and then, proceed with small/intermediate markets. The problem
is solved by using backward induction. Proposition 1 fully characterizes the equilibrium shares of marital
output that clear any marriage market under these strategies.
Proposition 1 . When women are scarce in the society,
• The equilibrium shares of marital output that clear the market under MAFB are such that:
m
(1) τHL = τLL = 0 if NHL
> F;
m
(2) 0 ≤ τHL ≤ c and τLL = 0 if NHL
= F ( the Nash Bargaining solution is τHL = (1/2)(θ(d − c) +
(1 − θ)c) ≡ τ (θ) where θ is the proportion of type LH women in the female population);
w
m
< NHL
< F;
(3) τHL = c and τLL = 0 if NLH
m
w
< F ( the Nash Bargaining solution is τHL = d/2);
= NHL
(4) c ≤ τHL ≤ d − c and τLL = 0 if NLH
w
m
.
< NLH
(5) τHL = d − c and τLL = 0 if NHL
• The equilibrium shares of marital output that clear the market under MBFA are such that:
m
(1) τLH = τLL = 0 if NLH
> F;
m
= F ( the Nash Bargaining solution is τLH = τ (α) where α is
(2) 0 ≤ τLH ≤ c and τLL = 0 if NLH
the proportion of type HL women in the female population);
w
m
(3) τLH = c and τLL = 0 if NHL
< NLH
< F;
m
w
< F ( the Nash Bargaining solution is τLH = d/2);
= NLH
(4) c ≤ τLH ≤ d − c and τLL = 0 if NHL
w
m
.
< NHL
(5) τLH = d − c and τLL = 0 if NLH
Proof. In the Appendix.
As it has been mentioned in the previous section, the equilibrium shares of marital output for type LL
men are always zero since at least, some of them remain single. On the other hand, the share of marital
output kept by type HL men under MAFB (or LH men under MBFA) changes discretely with the total
number of these men relative to the number of women in the population. Thus, as the number of type HL
(LH) men decreases, the competition among women to attract this type of men intensifies. As a result, the
high demand for these men leads to an increase in the share of marital output captured by them.
10
4.1.1
Large Marriage Markets
This section considers a large marriage market with a continuum of individuals on each side of the market.
By the Law of Large Numbers, the expected number of type HL men and type LH women under MAFB
is N µ and F b respectively. Similarly, the expected number of type LH men and type HL women under
MBFA is N b and F µ respectively. If these values are substituted into Proposition 1, then we obtain that the
market clearing shares change discretely with the sex ratio. In terms of welfare, notice that when the agents’
expected aptitude to perform task A coincides with their aptitude to perform task B, so that on average
they are equally productive in both tasks, both genders are indifferent about which of the sexual divisions
of labor emerges in the society. Nonetheless, the gender segregations are not Pareto rankable otherwise. In
fact, ex ante both genders prefer to specialize in the task for which they are less productive. To see this,
suppose that individuals are born with more aptitude to perform task B than task A on average. If so, ex
ante all women prefer (or at least be indifferent) the sexual division of labor in which they invest all their
human capital in task A, MBFA, because if men specialize in task B, the expected number of high type men
will be greater than in the case they specialize in task A. As a result, competition among women for these
high type men is not so intense so that the share of marital output captured by these men is relatively low
and hence, all women enjoy a higher payoff. Under same circumstances, and by the same argument, ex ante
all males prefer the sexual division of labor in which they specialize in task A.
Proposition 2 characterizes the conditions under which the gender divisions of labor constitute decentralized equilibria. It underlines the trade off faced by any agent when considering the human capital investment
decision in the first stage: acquiring training in a task for which her/his aptitude is low but that is in high
demand or alternatively, acquiring training in a task for which he/she has a higher aptitude but that it is
less demanded and hence, worse remunerated.
Proposition 2. When women are scarce in the society,
• MAFB is a decentralized equilibrium if and only if :
b
1
if N
F < µ − Fµ;
d
b
1
ā ≤ b ≤ a2 d−2c
+ a 2c
if N
2c
F = µ − Fµ;
b ≤ a 1 + a d−2c
& b2 d−2c
+ b ≥ ā if
2c
2c
d
b
b ≤ a & b2 d−2c
+ b 2c
≥ ā if N
2c
F = µ +
c
1
ā d−c
≤ b ≤ a if µb + F1µ < N
F < µ;
(1) ā ≤ b ≤ a
(2)
(3)
(4)
(5)
d−c
c
11
N
F
=
1
Fµ;
b
µ;
(6)
1−ā
2F
d−2c d−c
+ ā
τ (θ)
d−c
≤b≤a
(7) It is always an equilibrium
if
if
N
F
>
N
F
=
1
µ;
1
µ.
• MBFA is a decentralized equilibrium if and only if :
µ
c
(1) ā d−c
≤ b ≤ a if N
F < b;
N
(2) b2 d−2c
+ b ≥ ā & b ≤ a 1 + a d−2c
if F = µb ;
2c
2c
1
if µb < N
(3) ā ≤ b ≤ a d−c
c
F < b;
d−c−( d−2c
2F )
1
F −1
(4) ā ≤ b ≤ a τ (α)−a d−2c
if N
F = b and µ < F ;
( 2F )
d−c−( d−2c
2F )
1
c
if N
(5) ā 2τ (α) ≤ b ≤ a τ (α)−a d−2c
F = b and µ =
( 2F )
(6) It is always an equilibrium if
N
F
F −1
F ;
> 1b .
Proof. In the Appendix.
Under MAFB, men execute task A while women perform task B. If the sex ratio is large enough or
alternatively, for a given value of the sex ratio, if men’s aptitude to perform task A is high on average, so
that the number of type HL men in the population exceeds the total number of females, the sexual division
of labor MAFB is always an equilibrium regardless of the aptitude of the individuals to perform task B. This
is because under such circumstances, competition among men to get a partner leads them to transfer the
entire marital output to their wives so that they obtain a zero payoff independently of their type and hence,
this erases any possible incentives they may have to deviate. On the other hand, given that all women marry
type HL men and keep for themselves the total marital output generated from the marriage, they will obtain
xh in case of becoming type HH or LH and xm otherwise. In consequence, they will invest exclusively in
task B since by doing so, the probability of becoming high type in task B is maximized. Conversely, if the
sex ratio is not large enough or alternatively, if agents’ aptitude to perform task A is not so high on average,
in order for MAFB to be a decentralized equilibrium, the individuals’ aptitude for task B is restricted to lie
in a range of possible values 6 . Otherwise, at least, the lowest fit men and/or the highest fit women for task
6 If
the aptitude to perform task B were to differ among the sexes, so that nature did discriminate, the required conditions
consist on the probability of any man becoming type LH being fairly low whereas the probability of any woman becoming such
type being fairly high.
12
A will find it profitable to invest a positive amount of human capital in the task assigned to the opposite
gender. Notice also that, as the sex ratio increases, the required possible values must be lower 7 .
For the intuition behind these facts, think about any man with the lowest possible aptitude for task A,
a. When the sex ratio is lower or equal to a given threshold (b/µ), the equilibrium share of marital output
obtained by a type LH man is lower than the one received by a type HL man. This is so because in this case,
type HL men are in short number compared to type LH women in the market. Hence, competition among
women to obtain the highest able men leads type LH women to attract these men by offering a higher share
of marital output to them. A type LH man is considered as a perfect substitute to type LL men by type LH
women so that a type LH man will end up being matched with a type LL woman who values him strictly
more than a type LL man. However, the marital output generated by this couple is not as high as the one
generated by a MTH couple. Thereby, the share of marital output offered to a type LH man by a type LL
woman cannot be as high as the share offered to a type HL man by a type LH women. Given this, if the
aptitude to perform task B were equal to the lowest possible aptitude for task A (b = a), the lowest fit man
for task A would face the same probability of becoming high type in either task but since being high type in
task A is better remunerated by the market, he would certainly not deviate and invest all his human capital
in task A. Then, we conclude that the aptitude for task B could indeed exceed the lowest possible value that
the aptitude for task A can take. Nonetheless, this is true up to certain limit, since if the former exceeds
the latter considerably, the higher probability of being high type in task B and obtaining a lower share will
result in a greater expected payoff than one associated to a lower probability of becoming high type in task
A but obtaining a higher share. In contrast, when the sex ratio is greater than the previous threshold (b/µ),
given the strategies of the others, if the lowest fit man for task A deviates by investing all his human capital
in task B and becomes types LH, he will obtain exactly the same payoff as if he were to become type HL by
not deviating. This is because in such a case, the number of type HL men will exceed the number of type
LH women so that some type HL men will end up being married to type LL women. Given that type HL
and LH men are perfect substitutes from a type LL woman’s perspective, both men will receive the same
marital output share. Hence, if the aptitude for task B were higher than his aptitude for task A (i.e. b > a),
his chance of becoming type LH by specializing in B were greater than his chance of becoming type HL by
investing all his human capital in task A. As a result, he would deviate. Thus, we must require all individuals
7 If
nature did discriminate, the conditions placed on males’ aptitude to perform such task would become more restrictive (ie.
we need a lower and lower bm ) while the conditions on females’ aptitude would become less restrictive (ie. the required bw does
not need to be so high).
13
to have a lower aptitude for task B than for task A (i.e. b ≤ a).
A similar logic applies to women but it works in opposite directions. Lets focus our attention on the most
fit woman for task A in the society, characterized by ā. When the sex ratio is lower than the threshold cited
above (b/µ), given the strategies of the others, the share assigned to a woman who deviates and becomes
type HL would be exactly the same one as for a type LH woman, xm . Since the questioned woman is very
capable in task A, she will find it optimal to deviate unless the probability of becoming type LH is greater
than the probability of becoming type HL. In consequence, for the sexual division of labor MAFB to be an
equilibrium, we must require such condition (i.e. b ≥ ā). Recall that as the sex ratio increases, the scarcity
of women makes them more able to extract a higher share from their husbands. Nonetheless, any type HL
woman was already extracting the total marital output from her marriage (due to the indifference condition
for type LL men between being single or married). As a result, now, as the sex ratio increases, it becomes
more attractive to be a type LH woman relative to a type HL one and therefore, the condition placed on
the possible values that the aptitude for task B can take becomes less restrictive, or in other words, the
individuals’ aptitude for task B need not be so high.
Under the sexual division of labor MBFA, men execute task B while women perform task A. If the sex
ratio is large enough or alternatively, if agents’ aptitude to perform task B is high (i.e. b > F/N ), the number
of type LH men in the market will exceed the total number of females, so that this particular sexual division
of labor will always be a decentralized equilibrium. The logic behind it is similar to the one mentioned for the
MAFB case. Instead, if the sex ratio is not so large or alternatively, if agents are not so capable to perform
task B, several conditions must be placed on the agents’ aptitude for task B to support this sexual division
of labor as an equilibrium. Otherwise, at least, the highest fit men and/or the lowest fit women in task A
will find it profitable to deviate. On the other hand, now, as the sex ratio increases, the required possible
values taken by the individuals’ aptitude for task B must be higher 8 . The argument for that statement is
similar to the one provided for MAFB but now we must think in terms of the most productive man and
least productive woman for task A. If the sex ratio is lower or equal to a given threshold (µ/b), being a type
LH man is better remunerated by the market than being type HL man. To make the highest fit man in A
indifferent between investing all his human capital in task A or investing it all in task B, agents’ aptitude
for task B must be lower than the highest possible aptitude value for task A. Nevertheless, as the sex ratio
increases, the previous advantage in terms of the share of marital output kept by any type LH relative to
8 If
nature did discriminate, as the sex ratio increased, the conditions on bw would be relaxed while the ones placed on bm
would become more restrictive (ie. now we need a higher bm and a not so low bw ).
14
type HL diminishes since both remunerations in fact become equal and now, for the most capable man in
task A not to deviate, the agents’ aptitude for task B must be at least as high as his aptitude for task A.
The logic for the least fit woman in task A works similarly but in opposite directions with respect to the sex
ratio. We could summarize all these findings in the following corollary.
Corollary 1. When women are scarce in the society, the gender divisions of labor constitute decentralized
equilibria in large marriage markets, if males are (on average) highly productive in the task they are assigned
for, or alternatively, if the sex ratio is sufficiently high, or finally, if the support of the distribution for the
agents’ aptitude for task A is relatively narrow.
4.1.2
Small Marriage Markets
Suppose now that we have a finite number of males (N) and females (F ) in the market with N > F . Lets
focus first on the MAFB. Consider a man with aptitude am for task A and who is deliberating about the
ˆm
amount of human capital that he should invest in such task. Given the strategies of the others, let NHL
denote the number of type HL men out from the N − 1 pool of men. His share of marital output will be
zero in case of becoming type LL since at least some of these men will be single. If he became a type HH,
HL or LH man, his share would vary depending on the total number of men and women of each type. Since
this man cannot observe the aptitude for task A of each men in the market, he expects the probability that
any men from the pool became type HL male to follow a Bernoulli distribution with parameter µ. Similarly,
the realization of a certain number of type LH females will follow a binomial distribution with parameter b.
Given that, this man will choose the level of human capital which maximizes his expected share of marital
m
m
output. Let wH
(wLH
) denote his expected payoff in case of becoming either type HH or HL (type LH).
Then, his problem consists on:
m
m m
m
m
max am tm
a wH (1 − a ta )b(1 − ta )wLH
0≤tm
a ≤1
The objective function is convex in tm
a so that we have a corner solution. This man will invest exclusively
m
m
m
m
in task A if and only if am wH
≥ bwLH
. Thus, for all men, the required condition is given by, b ≤ a(wH
/wLH
).
As in the large market case, the gender division of labor MAFB imposes an upper bound on the possible
value of individuals’ aptitude for task B.
Consider now a woman with aptitude aw for task A who is deliberating about her human capital investˆw denote the number of type LH women out from
ment decision. Given the strategies of the others, let NLH
15
the F − 1 pool of women. Her share of marital output will depend on the total and relative number of each
type of men and women.
On this basis, and given the pool of N men and F − 1 women, the event of having a certain number of
type HL males and type LH females in the market will follow a binomial distribution. Given the above cases,
w
w
w
if we denote by wHL
, wH
and wLL
her expected share of marital output in case of becoming type HL, type
HH or LH, and type LL respectively, her problem can be formulated as follows:
w
w
w
w
w w
w
w
max aw tw
a [1 − b(1 − ta )]wHL + +b(1 − ta )wH + (1 − a ta )[1 − b(1 − ta )]wLL
0≤tw
a ≤1
Not surprisingly, this function is also convex in tw
a . Hence, for MAFB to be a decentralized equilibrium,
w
w
w
w
we need b ≥ (wHL
− wLL
)/(wH
− wLL
). Thus, as in the large marriage market, this condition imposes a
lower bound on the possible value of individuals’ aptitude for task B.
We can carry out exactly the same analysis for MBFA and as in the large market case, new lower and
upper limits on the aptitude for task B will be required.
4.1.3
Mechanical vs. Nonmechanical task
In the previous sections, we have implicitly considered that task B is a “nonmechanical” task in the sense
that investing the entire amount of human capital in task B does not guarantee to become high type in the
cited task (b < 1).
Definition 3. A task is considered to be mechanical if every agent could perfectly perform it as long as
complete training is received by the agent.
Instead, if task B is considered a “mechanical” task so that b = 1, then under the sexual division of
labor, all the members of the gender who specializes in task B become high type with certainty. As a result,
the competition among these members to marry high type agents of the opposite sex becomes extreme. We
could execute the same analysis as the one developed in the previous section to check for the conditions under
which the sexual division constitute equilibria. This leads us to establish proposition 3.
Proposition 3. If task B is considered a mechanical task, the gender division of labor constitute always
decentralized equilibria regardless of the extent of the market and sex ratio.
Proof. In the Appendix.
16
5
Ex ante Incentive Efficiency
In this model, the seldom nature of a transferable utility framework guarantees sorting efficiency in the second
stage through the shares of marital output that arise in equilibrium. In consequence, our focus in this section
is concerned with the efficiency of premarital investments9 . Ideally, a central authority who knew each agent’s
private information, would choose the amounts of human capital investment which maximize the expected
total marital output obtained in the society according to the joint distribution of types. The authority is not
concerned about the specific shares obtained by each agent. In practice, however, a central authority may
be no more able to observe agents’ private information than are market participants. If this is the case, the
Social Planner will just have access to public information: measure and the aptitude distribution function
associated to each gender. In this environment, we could ask ourselves whether the sexual division of labor
equilibria are ex ante constrained Pareto optima or alternatively, whether it is possible to device a market
intervention that raises aggregate surplus relative to the gender segregation equilibria. Thus, henceforth, we
will restrict attention to a cut-off strategy for each gender, ãi ∀i = m, w, such that if an agent’s aptitude for
task A exceeds this threshold, she will invest all her human capital in that task. Otherwise, she will specialize
in task B. That is,

 0 if a ≤ ãi
tai (a) =
 1 otherwise
∀i = m, w & a ≤ a ≤ ā.
In this case, the sexual division of labor will arise in the society as long as the cutoff assigned to one
gender is a, whereas so is ā to the other. Under these circumstances, capital investment decisions will be
perfectly correlated with gender roles. Instead, the training will be according to aptitude if the thresholds
chosen for both genders are interior and coincide: a < ãm = ãw < ā. In this case, the comparative advantage
allocation is satisfied: the partner with the highest aptitude for task A specializes in such task in all the
marriages. Thus, gender roles are completely absent. Conversely, if the cutoffs for both genders are interiors
9 Peters
and Siow (2002) argue that families make investments in education that are Pareto optimal once marital matching is
endogenized. According to their results, large marriage markets, assortative matching and bilateral efficiency together guarantee
that the equilibrium distribution of premarital investments is efficient.
17
but differ, we will have a mix training strategy in the sense that capital investment decisions are imperfectly
correlated with both gender and aptitude. The comparative advantage is still satisfied within a sex but it is
not longer satisfied across the sexes: a couple consisted of a male trained for task B and a female trained for
task A could arise in such society, in spite of the male having a higher aptitude for task A than the female.
On the other hand, since the authority can identify whether an individual belongs to one sex or the other,
but cannot distinguish directly among different types of males and females, the optimal cut-off rule for each
gender must provide individuals the incentives to truly reveal whether their aptitude for task A is lower or
by contrary, greater than the established threshold for his/her gender (in case it is an interior one).
In this scenario, if the training-according-to-sex are competitive equilibria, the three possible training
strategies equilibria could be ranked according to aggregate welfare. This section also proceeds to examine
how the efficiency property of the gender division of labor is affected by the size of the market and the sex
ratio.
The aggregate welfare in the society depends on the distribution of types HL, LH and LL males and
females in the market. Hence, nine different cases must distinguished by the authority:
m
• Under N − F < NLL
(ie. some type LL men are married),
w
m
w
m
,
≥ NHL
& NLH
≥ NLH
(1) NHL
w
w
+ NHL
Number of MTH marriages: NLH
m
w
m
w
)
Number of MTM marriages: (NHL
− NLH
) + (NLH
− NHL
m
m
Number of MTL marriages: F − NHL
− NLH
m
m
w
w
)c.
+ NLH
)(d − c) + (NHL
+ NHL
Aggregate Welfare: F xl + (NLH
m
w
m
w
w
(2) NHL
< NLH
& NLH
≥ NHL
+ NLL
,
m
w
Number of MTH marriages: NHL
+ NHL
w
m
Number of MTM marriages: F − NHL
− NHL
m
w
Aggregate Welfare: F xm + (NHL
+ NHL
)(d − c).
m
w
m
w
w
(3) NHL
< NLH
& 0 ≤ NLH
− NHL
< NLL
m
w
Number of MTH marriages: NHL
+ NHL
18
m
w
m
w
Number of MTM marriages: NLH
+ NLH
− NHL
− NHL
m
w
Number of MTL marriages: F − NLH
− NLH
.
m
w
m
w
Aggregate Welfare: F xl + (NHL
+ NHL
)(d − c) + (NLH
+ NLH
)c.
m
w
m
w
w
(4) NLH
< NHL
& NHL
≥ NLH
+ NLL
m
w
Number of MTH marriages: NLH
+ NLH
m
w
Number of MTM marriages: F − NLH
− NLH
m
w
Aggregate Welfare: F xm + (NLH
+ NLH
)(d − c).
m
w
m
w
w
(5) NLH
< NHL
& 0 ≤ NHL
− NLH
< NLL
w
m
+ NLH
Number of MTH marriages: NLH
m
w
m
w
Number of MTM marriages: NHL
+ NHL
− NLH
− NLH
w
m
.
− NHL
Number of MTL marriages: F − NHL
w
m
w
m
)c.
+ NHL
)(d − c) + (NHL
+ NLH
Aggregate Welfare: F xl + (NLH
w
m
w
m
,
< NHL
& NLH
< NLH
(6) NHL
m
m
+ NLH
Number of MTH marriages: NHL
w
m
w
m
)
− NLH
) + (NHL
Number of MTM marriages: (NLH
− NHL
w
w
Number of MTL marriages: F − NHL
− NLH
.
w
w
m
m
)c.
+ NLH
)(d − c) + (NHL
+ NLH
Aggregate Welfare: F xl + (NHL
m
(ie. at least all type LL men are single),
• Under N − F ≥ NLL
m
w
m
w
(7) NHL
≥ NLH
& NLH
≥ NHL
,
w
w
Number of MTH marriages: NHL
+ NLH
w
w
Number of MTM marriages: F − NHL
− NLH
w
w
Aggregate Welfare: F xm + (NHL
+ NLH
)(d − c).
19
m
w
m
w
(8) NHL
< NLH
& NLH
≥ NHL
,
m
w
Number of MTH marriages: NHL
+ NHL
m
w
Number of MTM marriages: F − NHL
− NHL
m
w
Aggregate Welfare: F xm + (NHL
+ NHL
)(d − c).
m
w
m
w
(9) NHL
≥ NLH
& NLH
< NHL
m
w
Number of MTH marriages: NLH
+ NLH
m
w
Number of MTM marriages: F-NLH
− NLH
m
w
Aggregate Welfare: F xm + (NLH
+ NLH
)(d − c).
5.1
Large Marriage Markets
Under this human capital investment decision, if there is a continuum of individuals on each side of the
market, the final number of males according to each type, is given by:
m
NHL
m
NHH
=0
Z ā
=N
a dG(a) ≡ N µ̃m
ãm
m
NLH
= N b G(ãm )
m
NLL
= N (1 − µ̃m − b G(ãm ))
Similarly for females.
Hence, the global Social Planner’s problem of choosing the set {a˜i }w
i=m that maximize the objective welfare
function can be divided into nine different subproblems. The Social Planner will solve each subproblem and
finally, choose the cutoff values, among the nine candidates, which provide the highest total welfare.
For instance, the first subproblem can be formulated as:
max F xl + F (µ̃w + b G(ãw ))(d − c) + N (µ̃m + b G(ãm )) c
ãm ,ãw
20
st.
N µ̃m ≥ F b G(ãw )
N b G(ãm ) ≥ F µ̃w
N (µ̃m + b G(ãm )) < F
a ≤ ãm ≤ ā
a ≤ ãw ≤ ā
+ IC constraints10
The tables below illustrates the results under the assumption of uniform aptitude distributions for task
A with support [0.2, 0.8]. Hence, the expected aptitude value for task A is 0.5. All the panels displayed are
obtained by setting xl = 5, c = 2 and d = 6. “Optimal” corresponds to the optimal cutoff that would be
chosen by the central authority whereas “Welfare” represents the aggregate welfare obtained in the society
if such a rule would be imposed. “WMAFB” and “WMBFA” are equivalent to the aggregate welfare under
MAFB or MBFA respectively.
T able I : Large M arriage M arket with b = 0.3
Population
Sex
Optimal
Males
Females
Ratio
Males
Females
150000
50000
3.00
0.53
0.30
125000
50000
2.50
0.59
0.30
100000
50000
2.00
0.53
75000
50000
1.50
66500
50000
62500
Welfare
MAFB
WMAFB
Males
Females
451666.7
0.2
0.8
451666.7
0.2
0.8
0.49
438249.1
0.2
0.57
0.55
408047.3
1.33
0.59
0.58
50000
1.25
0.59
60000
50000
1.20
58500
50000
55000
52500
MBFA
WMBFA
Males
Females
410000
0.8
0.2
440000
410000
0.8
0.2
425000
0.8
410000
0.8
0.2
410000
0.2
0.8
385000
0.8
0.2
390000
397497.8
0.2
0.8
376500
0.8
0.2
379800
0.59
392482.1
0.2
0.8
372500
0.8
0.2
375000
0.69
0.54
388277.0
0.2
0.8
370000
0.8
0.2
372000
1.17
0.68
0.55
386647.8
0.2
0.8
368500
0.8
0.2
370200
50000
1.10
0.60
0.60
383000.0
0.2
0.8
365000
0.8
0.2
366000
50000
1.05
0.60
0.60
379833.3
0.2
0.8
362500
0.8
0.2
363000
10 Deriving
the IC constraints is an easy but cumbersome task due to multiple possible scenarios. They are available by
request.
21
T able II : Large M arriage M arket with b = 0.7
Population
Sex
Optimal
Welfare
Males
Females
Ratio
Males
Females
150000
50000
3.00
0.50
0.70
125000
50000
2.50
0.50
0.70
100000
50000
2.00
0.50
75000
50000
1.50
66500
50000
62500
MAFB
WMAFB
Males
Females
491666.7
0.2
0.8
491666.7
0.2
0.8
0.70
491666.7
0.2
0.50
0.50
485000.0
1.33
0.50
0.50
50000
1.25
0.50
60000
50000
1.20
58500
50000
55000
52500
MBFA
WMBFA
Males
Females
490000
0.8
0.2
450000
490000
0.8
0.2
450000
0.8
490000
0.8
0.2
450000
0.2
0.8
465000
0.8
0.2
450000
485000.0
0.2
0.8
453000
0.8
0.2
443100
0.50
485000.0
0.2
0.8
445000
0.8
0.2
437500
0.50
0.50
485000.0
0.2
0.8
440000
0.8
0.2
434000
1.17
0.50
0.50
485000.0
0.2
0.8
437000
0.8
0.2
431900
50000
1.10
0.50
0.50
485000.0
0.2
0.8
430000
0.8
0.2
427000
50000
1.05
0.44
0.53
455591.9
0.2
0.8
425000
0.8
0.2
423500
T able III : Large M arriage M arket with b = 1
Population
Sex
Optimal
Males
Females
Ratio
Males
Females
150000
50000
3.00
0.20
0.80
125000
50000
2.50
0.20
0.80
100000
50000
2.00
0.20
75000
50000
1.50
66500
50000
62500
Welfare
MAFB
WMAFB
Males
Females
550000.0
0.2
0.8
550000.0
0.2
0.8
0.80
550000.0
0.2
0.34
0.60
529327.5
1.33
0.42
0.51
50000
1.25
0.47
60000
50000
1.20
58500
50000
55000
52500
MBFA
WMBFA
Males
Females
550000
0.8
0.2
450000
550000
0.8
0.2
450000
0.8
550000
0.8
0.2
450000
0.2
0.8
500000
0.8
0.2
450000
516476.7
0.2
0.8
483000
0.8
0.2
450000
0.46
508325.1
0.2
0.8
475000
0.8
0.2
450000
0.48
0.45
502435.2
0.2
0.8
470000
0.8
0.2
450000
1.17
0.47
0.44
498876.2
0.2
0.8
467000
0.8
0.2
450000
50000
1.10
0.49
0.49
491428.4
0.2
0.8
460000
0.8
0.2
450000
50000
1.05
0.49
0.50
485714.3
0.2
0.8
455000
0.8
0.2
450000
When the marriage market is large and task B is considered a “nonmechanical task”, none of the sexual
divisions of labor are efficient: the Social Planner will always choose an interior cutoff for both genders.
In case that the sex ratio is high so that the number of males exceeds fairly the number of females in the
population, the authority will assign most of the women to perform the task for which individuals are more
productive on average. As a result, the thresholds chosen for both genders differ so that the optimal strategy
is the mix training-according-to-sex-and-aptitude. Nonetheless, as the sex ratio decreases and approaches
unity, the training-according-to-aptitude arises as the efficient allocation strategy. Conversely, if task B is
considered a “mechanical task” and sex ratio is high (2 or more men per woman), the Social Planner optimally
chooses the sexual division of labor in which the gender who is scarce in the society specializes in task B
and the opposite gender does so in task A. This is because forcing women, who are in minority, to execute
task B will guarantee a high type individual in at least all marriages and since men are abundant, there are
big chances that a considerable number of them become high type in task A. Nevertheless, as the sex ratio
decreases, the authority will switch to choose an interior cut off for each gender. As the sex ratio approaches
22
unity, both thresholds gets closer and closer and finally, all individuals are trained according-to-aptitude
regardless of their gender. In this case, the sexual division of labor does not achieve the highest aggregate
welfare and a central intervention which imposes the comparative advantage rule will be Pareto improvement.
5.2
Small Marriage Markets
When the innate aptitude of the individuals for task A is not public information, the sexual division of labor
can be viewed as a benchmark which helps the individuals to coordinate each other in the election of human
capital investments. In this case, a training-according-to-aptitude rule may not be as efficient. To see this,
suppose that there are only one man and one woman in the society. In the case that both individuals have
higher or lower aptitude than the assigned thresholds, both of them will end up investing all their human
capital in the same activity. But then, complementarity in tasks, which is high valued by the society, will
not be achieved. Hence, the training-according-to-sex depicts itself as the most promising allocation strategy
which minimizes the risk of mismatches in the society.
As we cited above, the aggregate welfare function depends on the number of each type of males and
females in the population, which leads us to distinguish nine different cases in this respect. Since we have a
finite number of men and women, let denote by h̃m (x, z), the probability that there are x type HL men and
z type LH men in the marriage market. Given the Social Planner’s rule, this probability is given by:
h̃m (x, z) ≡
N
−x
X
N
T
T
z
N −T
x
(bm )z (1 − bm )T −z (p̃m )T (µ̃m )x (1 − µ̃m − p̃m )N −T −x
T =z
where
p̃m ≡ Gm (ãm )
Similarly, h̃w (u, v) will, henceforth, denote the probability that the number of type HL and LH females
in the market is u and v respectively. Thus, when N ≥ F the expected aggregate welfare (W) can be
decomposed in nine terms according to the different cases. For instance, the first one is given by:
W1 ≡
F
−1 F −z−1
x X
z
X
X X
z=0
x=0
q̃ m (x, z)q̃ w (u, v){F xm + (u + v)(d − c) + (x + z)c}
v=0 u=0
As a result, the Planner’s problem can be stated as follows:
23
max
m w
ã ,ã
9
X
Wi
i=1
subject to
a ≤ ãm ≤ ā
a ≤ ãw ≤ ā
+ IC constraints
The results are posted in tables V, VI, VII and VIII. Now, contrary to the large market case, trainingaccording-to-sex constitute efficient coordination mechanisms.
Table V displays the results under b = 0.3. In that case, MBFA is optimal when the sex ratio is high, (ie.
2 men per woman) whereas MAFB is optimal when the sex ratio is lower. Notice that under the assumption
of a uniform distribution of aptitudes on [0.2, 0.8], µ takes a value of 0.5. Hence, on average, the individuals
are expected to be more productive in task A rather than task B. When the sex ratio is very high, assigning
women to the most productive task and men to the least productive task provides a higher welfare than the
alternative gender division of labor. This result is very intuitively: since there are more men than women in
the market, the number of marriages will be equal to F . Assigning men to the most productive task will be
a waste of resources since several men who turn to be high type in task A will probably end up being single.
On the other hand, due to the general low aptitude for task B, most of the women will turn to be low type
in that task. As a result, MAFB does not result very attractive to central the authority. However, if we
force women to do the most productive task (MBFA), the society will be characterized by a greater number
of high type women under this gender assignment than under the previous one. In case that the sex ratio is
high (ie. 2), since there are several men per woman, even if we allocate all men to the least productive task,
at least, some of them will become high type in such task so that some of the best marriages will occur. It
is true that if we compare MBFA to MAFB, under the former, less men will end up being high type than
under the latter. Nonetheless, the Social Planner does care about the number of high type men who get
married and not about the total number of these men in the society due to the fact that single men are
useless or not productive at all. Hence, the difference in the number of married high type men under both
gender divisions of labor may be small so that, given that MBFA results in more married women being high
24
type, such gender segregation becomes definitely more attractive to the central authority. Conversely, when
the sex ratio is not so high (1.5 or lower), assigning men to the most productive task becomes efficient. Since
now there are not so many men per woman, allocating them to task A will not result in a waste of resources
because only low type men will remain single. Hence, this training strategy maximizes the number of married
men who become high type and thus, most of the married couples in the society will contain at least a high
type individual. If alternatively, we were to assign women to task A, on average, only half of them would be
expected to become high type so that there would be less high type couples in the society leading to a lower
aggregate welfare.
Table VI shows the results when b = 0.5. Now, on average, individuals are equally productive in both
tasks, A and B. As a result, both sexual divisions of labor constitute ex ante constrained pareto optima. This
makes the Social Planner indifferent among which one should be implemented.
Table VII displays the results when b = 0.7. Under such a case, individuals are on average more productive
in task B than task A so that by a similar (but reversed) argument to the one provided for table V, women are
assigned to the most productive task so that MAFB is preferred to MBFA when the sex ratio is high (more
than 1.25) while they are assigned to the least productive task otherwise. All these results are compiled in
the next corollary.
Corollary 2. If task B is considered a “nonmechanical task”,
• When the market is large, the sexual division of labor never constitutes a ex ante constrained efficient
coordination mechanism. The mix training strategy is optimal albeit as the sex ratio approaches unity,
the training-according-to-aptitude emerges as the efficient training rule.
• When the market is sufficiently small, the sexual division of labor where females specialize in the most
productive task constitutes ex ante constrained pareto optimal only if the sex ratio is sufficiently high.
Otherwise, the sexual division of labor where females specialize in the least productive task is efficient.
25
T able V
Population
Males
Females
Sex
Ratio
Optimal
Males
: Small M arriage M arket with b = 0.3
Welfare
MAFB
Females
Males
WMAFB
MBFA
Females
Males
WMBFA
Females
2
1
2
0.8
0.2
8.02
0.2
0.8
7.9
0.8
0.2
8.02
12
6
2
0.8
0.2
55.08
0.2
0.8
52.33
0.8
0.2
55.08
3
2
1.5
0.2
0.8
16.54
0.2
0.8
16.54
0.8
0.2
16.51
9
6
1.5
0.2
0.8
54.28
0.2
0.8
54.28
0.8
0.2
51.74
4
3
1.33
0.2
0.8
25.72
0.2
0.8
25.72
0.8
0.2
24.89
12
9
1.33
0.2
0.8
83.53
0.2
0.8
83.53
0.8
0.2
74.98
5
4
1.25
0.2
0.8
35.11
0.2
0.8
35.11
0.8
0.2
32.98
15
12
1.25
0.2
0.8
111.87
0.2
0.8
111.87
0.8
0.2
97.32
6
5
1.2
0.2
0.8
44.53
0.2
0.8
44.53
0.8
0.2
40.84
18
15
1.2
0.2
0.8
139.68
0.2
0.8
139.68
0.8
0.2
119.22
7
6
1.17
0.2
0.8
53.91
0.2
0.8
53.91
0.8
0.2
48.53
8
7
1.14
0.2
0.8
63.22
0.2
0.8
63.22
0.8
0.2
56.11
T able V I : Small M arriage M arket with b = 0.5
Population
Sex
Optimal
Males
Females
Ratio
2
1
2.00
SDL
6
3
2.00
SDL
3
2
1.50
SDL
9
6
1.50
SDL
4
3
1.33
12
9
5
15
6
18
7
14
8
16
Welfare
MAFB
WMAFB
Males
Females
8.50
0.2
0.8
27.36
0.2
0.8
17.56
0.2
0.8
57.46
0.2
0.8
SDL
27.05
0.2
1.33
SDL
87.50
4
1.25
SDL
12
1.25
SDL
5
1.20
SDL
15
1.20
SDL
6
1.17
SDL
12
1.17
SDL
7
1.14
SDL
14
1.14
SDL
MBFA
WMBFA
Males
Females
8.50
0.8
0.2
8.50
27.36
0.8
0.2
27.36
17.56
0.8
0.2
17.56
57.46
0.8
0.2
57.46
0.8
27.05
0.8
0.2
27.05
0.2
0.8
87.50
0.8
0.2
87.50
36.68
0.2
0.8
36.68
0.8
0.2
36.68
116.37
0.2
0.8
116.37
0.8
0.2
116.37
46.28
0.2
0.8
46.28
0.8
0.2
46.28
144.50
0.2
0.8
144.50
0.8
0.2
144.50
55.78
0.2
0.8
55.78
0.8
0.2
55.78
113.99
0.2
0.8
113.99
0.8
0.2
113.99
65.18
0.2
0.8
65.18
0.8
0.2
65.18
132.44
0.2
0.8
132.44
0.8
0.2
132.44
T able V II : Small M arriage M arket with b = 0.7
Population
Sex
Optimal
Males
Females
Ratio
Males
Females
2
1
2.00
0.2
0.8
6
3
2.00
0.2
0.8
3
2
1.50
0.2
0.8
9
6
1.50
0.2
0.8
4
3
1.33
0.2
12
9
1.33
5
4
15
6
Welfare
MAFB
Males
Females
9.10
0.2
0.8
29.07
0.2
0.8
18.44
0.2
0.8
59.28
0.2
0.8
0.8
28.00
0.2
0.2
0.8
88.34
1.25
0.8
0.2
12
1.25
0.8
5
1.20
0.8
18
15
1.20
7
6
8
7
WMAFB
MBFA
WMBFA
Males
Females
9.10
0.8
0.2
8.82
29.07
0.8
0.2
27.15
18.44
0.8
0.2
17.96
59.28
0.8
0.2
56.14
0.8
28.00
0.8
0.2
27.56
0.2
0.8
88.34
0.8
0.2
87.75
37.57
0.2
0.8
37.57
0.8
0.2
37.57
0.2
120.92
0.2
0.8
115.54
0.8
0.2
120.92
0.2
47.91
0.2
0.8
46.99
0.8
0.2
47.91
0.8
0.2
154.39
0.2
0.8
141.54
0.8
0.2
154.39
1.17
0.8
0.2
58.46
0.2
0.8
56.23
0.8
0.2
58.46
1.14
0.8
0.2
69.14
0.2
0.8
65.29
0.8
0.2
69.14
26
Table VIII shows the results under the assumption that task B is a “mechanical task”. In such a case,
when the sex ratio is relatively high, the efficient allocation consists on women specializing in task B and
men in task A. This is because the scarcity of women makes it sociable desirable to allocate them into the
performance of task B for which, they will become high type with certainty. On the other hand, for a given
sex ratio value, the gender division of labor emerges as the appropriate mechanism to coordinate people on
both sides of the market when the latter is small.
Nonetheless, as the marriage market expands, this strategy looses its efficiency property deriving in the
mix training according-to-aptitude-and-sex as the optimal mechanism when the market reaches a certain size.
The extent of the market needed for such switch in strategy is increasing in the sex ratio. In other words,
as the sex ratio decreases and gets closer to one, the optimal solution in the intermediate market converges
to the large market solution at a faster rate. Thus, for example, when the sex ratio is 1.5 and the market
is given by 3 males and 2 females, the central authority is in favor of segregation by gender. To change his
mind, a market with a 5 times larger population than the original one is required. Instead, when the sex
ratio is equal to 1.33 and there are 4 males and 3 females in the market, the Planner will be willing to adopt
the mix training criterium if the population just doubled its size. Furthermore, as in the large market case,
the training according to aptitude depicts itself as the optimal mechanism when the sex ratio approaches
unity. These results are summarized in the next corollary.
Corollary 3. If task B is considered a “mechanical task”,
• When the market is large, the sexual division of labor where females specialize in task B constitutes
a ex ante constrained efficient coordination mechanism only if the sex ratio is high. Otherwise, the
mix training strategy is optimal. As the sex ratio approaches unity, the training-according-to-aptitude
emerges as the efficient rule.
• When the market is sufficiently small, the sexual division of labor where females specialize in task B
constitutes a ex ante constrained pareto optimal strategy but this property vanishes as the extent of the
market increases or as the sex ratio approaches unity.
27
T able V III : Small M arriage M arket with b = 1
Population
Males
6
Females
Sex
Ratio
Optimal
Males
Welfare
Females
MAFB
Males
WMAFB
Females
MBFA
Males
WMBFA
Females
2
1
2.00
0.20
0.80
10.00
0.2
0.8
10.00
0.8
0.2
9.00
14
7
2.00
0.20
0.80
74.07
0.2
0.8
74.07
0.8
0.2
63.00
3
2
1.50
0.20
0.80
19.50
0.2
0.8
19.50
0.8
0.2
18.00
9
6
1.50
0.20
0.80
59.55
0.2
0.8
59.55
0.8
0.2
54.00
12
8
1.50
0.20
0.80
79.62
0.2
0.8
79.62
0.8
0.2
72.00
15
10
1.50
0.34
0.60
101.06
0.2
0.8
99.68
0.8
0.2
90.00
4
3
1.33
0.20
0.80
28.75
0.2
0.8
28.75
0.8
0.2
27.00
8
6
1.33
0.39
0.56
58.45
0.2
0.8
57.84
0.8
0.2
54.00
5
4
1.25
0.20
0.80
37.88
0.2
0.8
37.88
0.8
0.2
36.00
10
8
1.25
0.42
0.50
78.38
0.2
0.8
75.95
0.8
0.2
72.00
6
5
1.20
0.43
0.51
47.62
0.2
0.8
46.94
0.8
0.2
45.00
7
6
1.17
0.45
0.48
57.53
0.2
0.8
55.97
0.8
0.2
54.00
8
7
1.14
0.46
0.45
67.57
0.2
0.8
64.98
0.8
0.2
63.00
Evolution
In this section, the model is used to motivate evolution from well-defined gender roles to training by aptitude,
an emerging feature of modern societies.
A simple explanation for such stylized fact consists on a size-enlargement of the marriage market through
time. In the previous section, we concluded that if the common aptitude task is considered as “nonmechanical”, the training-according-to-sex and hence, the sexual division of labor are optimal allocation mechanisms
in small markets. Nonetheless, through the ages, cities started to develop from originally small towns and
consequently, the local marriage markets began to expand. We showed above that when the dimension of the
market gets large enough, training-according-to-aptitude becomes an optimal strategy as long as the sex ratio
does not differ substantially from unity. Hence, under such circumstances, the observed evolution pattern
arises naturally.
On the other hand, this change in training strategy could also be supported by just an exogenous decrease
(increase) in the sex ratio. Recall that if the common aptitude task is “mechanical”, the sexual division of
labor constitute ex ante constrained efficient competitive equilibria under a relatively high (low) sex ratio
independently of the size of the marriage market. Nonetheless, as the sex ratio falls (rises) and gets relatively
close to unity, the gender division of labor looses its optimality attractiveness in favor of the training according
to aptitude. From there, the evolution pattern follows.
In general both, a fall in the sex ratio and an increase in the size of the marriage market, tend to work in
the same direction, thereby constituting potential reasons for such training evolution.
28
7
Conclusions
The main goal of this paper consists on analyzing whether occupational segregation by gender constitute
decentralized equilibria in an environment where individuals differ in their aptitude to perform one task and
the marriage market is characterized by a non-random matching process. If the task with a common general
aptitude is considered a “mechanical task”, so that it can be executed to perfection as long as complete
training is received by the individual, the sexual division of labor always constitute decentralized equilibria
independently of the extent of the market and the sex ratio. Instead, if we allow certain uncertainty so that
complete training may not be sufficient to become high type in such task, the sexual divisions of labor are
decentralized equilibria in large marriage markets only if the size of male and female population differs so
that the society is characterized by a relatively high or low sex ratio, or alternatively, if the gender, who is
in majority, is quite productive in the performance of the task assigned to it. These conditions underlie the
trade off faced by individuals in the first stage of their life when deciding about the human capital investment
decision to make: investing in the low demanded task for which they are highly productive or in the high
demanded task for which they have a low aptitude.
Furthermore, we examine whether the sexual divisions of labor are ex ante constrained pareto optima. We
conclude that their optimality depends on three important factors: the seldom nature of the homogeneous
task, the sex ratio and the extent of the market. Under the assumption of “mechanical task”, the gender
segregation in which the sex who is in minority specializes in the mechanical task is ex ante constrained
efficient regardless of the extent of the market only when the sex ratio is relatively high. Otherwise, as
the marriage market expands and/or the sex ratio approaches unity, this gender division of labor looses its
optimality power and is progressively replaced by the training-according-to-aptitude strategy. The extent
of the market needed for such change to occur is increasing in the sex ratio. Instead, when the task with
common aptitude is considered a “nonmechanical task”, the gender divisions of labor never constitute ex ante
constrained pareto optimal mechanisms in large marriage markets. Moreover, as the sex ratio approaches
to unity, the training-according-to-aptitude strategy depicts itself as the efficient coordination mechanism in
such markets. The latter satisfies the comparative advantage principle since the partner with the highest
aptitude for the “heterogeneous” task specializes in such task in all the marriages. On the other hand, small
marriage markets support segregations by gender. Thus, the sexual division of labor in which the gender
who is scarce in the society specializes in the most productive task is constrained optimum in small marriage
markets only if the sex ratio is sufficiently high. Otherwise, the alternative sexual division of labor emerges
29
as the efficient rule. This underlies the importance that the gender division of labor possesses in terms of
coordination since it helps to minimize the number of mismatches in the marriage market when there is not
a continuum number of individuals on each side of the market.
There are several directions for future research. For instance, one could consider the case where the
agents not only differ in the aptitude for one of the tasks but for both. Trade between families could also be
incorporated into the model. Finally, in another interesting extension, the marriage market could be modelled
such that individuals do not perfectly observe the type of the other agents but instead, an informative signal
about their productivity is realized. This modification would affect the equilibrium in the matching process
and consequently, in the investment stage.
References
[1] Aigner, D. and Cain G. 1977. “Statistical Theories of Discrimination in Labor Markets.” Industrial
and Labor Relations Review 30:(2), 175-187.
[2] Becker, G. S. 1991. “A Treatise on the Family, Enlarged Edition”. Cambridge, Massachusetts and
London: Harvard University Press.
[3] Blau, F., Simpson, P., Anderson D. 1998. “Continuing Progress? Trends in Occupational Segregation
in the United States over the 1970s and 1980s”. Working Paper 6716. National Bureau of Economic Research.
[4] Danziger, L. and Katz. E. 1996. “A Theory of Sex Discrimination”. The Journal of Economic Behavior
and Organization, 31: 57-66.
[5] Dolado, J.J., Felgueroso, F., Jimeno, J.F. 2002. “Recent Trends in Occupational Segregation by
Gender: A Look Across the Atlantic”. Discussion Paper, No. 524. IZA.
[6] Engineer, M. and Welling, L. 1999. “Human Capital, True Love, and Gender Roles: Is Sex Destiny?”
The Journal of Economic Behavior and Organization, 40: 155-178.
[7] Francois, P. 1998. “A Theory of Gender Discrimination Based on the Household”. Journal of Public
Economics, 68:(1), 1-32.
30
[8] Hadfield, G. 1999. “A Coordination Model of the Sexual Division of Labor”. The Journal of Economic
Behavior and Organization, 40: 125-153.
[9] Peters, M. and Siow A. 2002. “Competing Pre-Marital Investments”. Journal of Political Economy,
110:(3), 592-608.
[10] Polachek, S. 1981. “Occupational Self-Selection: A Human Capital Approach to Sex Differences in
Occupational Structure”. The Review of Economics and Statistics 63:(1), 60-69.
31
Appendix
Proof of Proposition 1: Since N > F , the total number of marriages is F so that some men are single.
Consider first the sexual division of labor of type MAFB. Since type HL men are strictly preferred to type
LL men by both types of women, type LL men are going to be the first ones in being single.
m
(1) If NHL
> F , all type LL men and some type HL men remain unmarried. These men will get zero utility.
Since some type HL men also get married, for any type HL man to be indifferent between being single
or married, they must get zero as the share of marital output when matched to any woman. Thus,
τHL = τLL = 0.
m
(2) If NHL
= F , all type LL men remain single earning zero utility whereas all type HL get married.
The share of marital output received by these men must be at least zero since otherwise, they would
prefer to remain single. On the other hand, it cannot exceed c since otherwise, type LL women would
prefer to propose to type LL men and get a share of marital output equal to xl instead of xm − τHL .
Under Nash Bargaining solution with equal bargaining power, the threat point for type HL men is
zero (i.e. being single) whereas for type LH women and type LL women are xm and xl respectively
(i.e. being married to type LL men). Thus, denoting by θ the proportion of type LH women in the
female population, τHL = argmax(θτHL (xh − τHL − xm ) + (1 − θ)τHL (xm − τHL − xl )) and hence,
τHL = (1/2)(θ(d − c) + (1 − θ)c) ≡ τ (θ).
w
m
(3) If NLH
< NHL
< F , all type HL men will be married: some to type LH women whereas others to
type LL women. Some type LL men will be married as well but only to type LL women. Hence, type
LL women receive the total marital output from these marriages, xl . For a type LL woman to be
indifferent between being married to either a type HL or LL man, she should receive the same utility
independently of the man she marries. Thus, type HL men should get a share of marital output equal
to c and hence, women of type LH will keep xh − c.
w
m
(4) If NLH
= NHL
< F , all type LH women marry all type HL men while all type LL women marry type LL
men. This is a case of perfect assortative matching. Since some type LL men remain single, these men
will get zero utility and women type LL will keep the entire marital output generated by the couple,
xl . As a result, the share of marital output kept by type HL men must be at least c since otherwise,
they will propose to these women. In addition, it must not exceed d − c because if so, type LH women
32
would be better off by proposing to type LL men. Hence, under the Nash Bargaining solution with
equal bargaining power, the threat point for type HL men is c whereas for type LH women is xm . The
solution is τHL = argmax{(τHL − c)(xh − τHL − xm )} or τHL = d/2.
m
w
(5) If NHL
< NLH
, some type LH women are married to type HL men while the rest to type LL men. For
these women to be indifferent among these men, they must receive a share of marital output equal to
xm . Thereby, men of type HL keep xh − xm = d − c for themselves.
m
m
The share of marital transfers that clear the market under MBFA are obtained by replacing NHL
by NLH
w
w
and NLH
by NHL
in the above analysis.
This completes the proof for Proposition 1.
Proof of Proposition 2: Consider first MAFB. By Law of Large Numbers (LLN),
w
m
⇔
< NLH
(1) Assume NHL
N
F
<
b
µ.
By proposition 1, the intra-household allocations are d − c for type HL
men, 0 for type LL men, xm for type LH women and xl for type LL women.
Consider now a man with aptitude am who is reflexing about the optimal premarital investment decision.
Given the strategies of the others, suppose that by investing all his human capital in task A, he becomes
type HL. In that case, if he were to deviate and by doing so became: (a) type HH or HL man, he would
w
>1⇔
obtain d − c; (b) type LH man, thereby marrying type LL women, he would obtain c if NLL
F >
1
1−b
w
= 1); (c) type LL man, he would obtain 0. Instead, if by
(c/2 in the improbable case NLL
investing all his human capital in task A, becomes type LL man, then if he were to deviate and became:
w
m
(a) type HH or HL man, he would obtain d − c if NLH
− NHL
>1⇔
m
w
=1⇔
− NHL
d/2 if NLH
N
F
=
b
µ
−
1
µF
N
F
<
b
µ
−
1
µF
or alternatively,
w
>1⇔F >
; (b) type LH man, he would obtain c if NLL
1
1−b
w
= 1); (c) type LL man, he would obtain 0.
(c/2 if NLL
His expected utility under
N
F
<
b
µ
−
1
µF
&F >
1
1−b ,
is:
EU = am ta (d − c) + (1 − am ta )b(1 − ta )c,
which is convex in ta . For t∗a = 1 to be optimal for all males, we need the following condition am (d−c) ≥
1
. In contrast, if F = 1−b
, by the same argument, we
bc to be satisfied ∀ am , and hence, b ≤ a d−c
c
d−c
need b ≤ 2a c .
Consider now a woman with aptitude aw . Given the strategies of the others, if she deviates from the
gender division and by investing ta > 0, she became either type HH, HL or LH, she would receive a
33
share of marital output equal to xm . Instead, if she became type LL woman, then she would receive
xl . Her expected utility is convex in ta and for t∗a = 0 to be optimal for all females, we need ∀ aw the
following condition to be satisfied xl + bc ≥ xl + aw c, or equivalently, b ≥ ā.
(2) From the analysis in (1), under
N
F
=
b
µ
−
1
µF
& F >
1
1−b ,
a man with aptitude am has an expected
utility of
EU = am ta [am (d − c) + (1 − am )(d/2)] + (1 − am ta )b(1 − ta )c,
which is convex in ta . For t∗a = 1 to be optimal for all males, we need the following condition
d
am [am (d − c) + (1 − am )(d/2)] ≥ bc to be satisfied ∀ am , and hence, b ≤ a2 d−2c
+ a 2c
. In
2c
1
d
2 d−2c
contrast, if F = 1−b , by the same argument, we need b ≤ a
+ a c . The required condition
c
for women not to deviate keeps being b ≥ ā, for the same reasons as in (1).
w
m
⇔
= NLH
(3) Assume NHL
N
F
=
b
µ.
By proposition 1, the intra-household allocations are d/2 for type HL
men, 0 for type LL men, xh − (d/2) for type LH women and xl for type LL women.
Consider now a man with aptitude am . Given the strategies of the others, suppose that by investing
all his human capital in task A, he becomes type HL. In that case, if he were to deviate and by doing
so became: (a) type HH or HL man, he would obtain d/2; (b) type LH man, thereby marrying type
w
LL women, he would obtain c if NLL
>1⇔F >
1
1−b
w
(c/2 in the improbable case NLL
= 1); (c) type
LL man, he would obtain 0. Instead, if by investing all his human capital in task A, becomes type LL
w
>1⇔
man, then if he were to deviate and became: (a) type HH or HL man, he would obtain c if NLL
F >
1
1−b
w
w
>1⇔
= 1); (b) type LH man, he would obtain c if NLL
(τ (θ) in the improbable case NLL
F >
1
1−b
w
(c/2 if NLL
= 1); (c) type LL man, he would obtain 0. His expected utility under
&F >
1
1−b ,
N
F
=
b
µ
1
− µF
is:
EU = am ta [am (d/2) + (1 − am )c] + (1 − am ta )b(1 − ta )c,
which is convex in ta . For t∗a = 1 to be optimal for all males, we need the following condition
am [am (d/2) + (1 − am )c] ≥ bc to be satisfied ∀ am , and hence,b ≤ a 1 + a d−2c
. In contrast, if
2c
τ (θ)
1
2 d
F = 1−b , by the same argument, we need b ≤ 2a 2c − τ (θ) + 2a c .
Consider now a woman with aptitude aw . Given the strategies of the others, suppose that by investing
all her human capital in task B, she becomes type LH. In that case, if she were to deviate and by
deviating and investing ta > 0, she became: (a) type HH or LH, she would receive a share of marital
output equal to xh − (d/2); (b) type HL, she would obtain xm ; (c) type LL woman, she would receive
34
xl . Instead, suppose now that by investing all her human capital in task B, she becomes type LL. In
that case, if by deviating, she became: (a) type HH, HL or LH, she would receive a share of marital
output equal to xm ; (c) type LL woman, she would receive xl . Her expected utility is convex in ta
and for t∗a = 0 to be optimal for all females, we need ∀ aw the following condition to be satisfied
xl + b2 (d/2) + b(1 − b)c ≥ xl + aw c, or equivalently,b2 d−2c
+ b ≥ ā.
2c
m
w
m
(4) Assume NHL
= NLH
+ 1 & N − F < NHL
⇔
b
µ
+
1
µF
=
N
F
<
1
µ.
By proposition 1, the intra-household
allocations are c for type HL men, 0 for type LL men, xl + (d − c) for type LH women and xl for type
LL women.
Consider now a man with aptitude am . Given the strategies of the others, if he deviates from the gender
division and by investing ta < 1, he became either type HH, HL or LH, he would receive a share of
marital output equal to c and zero otherwise. His expected utility is convex in ta and for t∗a = 1 to be
optimal for all males, we need ∀ am the following condition to be satisfied bc ≤ am c , or equivalently,
b ≤ a.
Consider now a woman with aptitude aw . Given the strategies of the others, suppose that by investing
all her human capital in task B, she becomes type LH. In that case, if she were to deviate and by
deviating and investing ta > 0, she became: (a) type HH or LH, she would receive a share of marital
output equal to xl + (d − c); (b) type HL, she would obtain xm ; (c) type LL woman, she would receive
xl . Instead, suppose now that by investing all her human capital in task B, she becomes type LL. In
that case, if by deviating, she became: (a) type HH or LH, she would receive a share of marital output
equal to xh − (d/2); (b) type HL, she would obtain xm ; (c) type LL woman, she would receive xl . Her
expected utility is convex in ta and for t∗a = 0 to be optimal for all females, we need ∀ aw the following
d
condition to be satisfied b2 d−2c
+ b 2c
≥ ā.
2c
m
w
m
(5) Assume NHL
> NLH
+ 1 & N − F < NHL
⇔
b
µ
+
1
µF
<
N
F
<
1
µ.
The intra-household allocations are
still as above: c for type HL men, 0 for type LL men, xl + (d − c) for type LH women and xl for type
LL women.
Regarding a man with aptitude am , the analysis is exactly the same one as in the previous item so that
the required condition is thereby b ≤ a.
Consider now a woman with aptitude aw . Given the strategies of the others, suppose that by investing
all her human capital in task B, she becomes type LH. In that case, if she were to deviate and by
35
deviating and investing ta > 0, she became: (a) type HH or LH, she would receive a share of marital
output equal to xl + (d − c); (b) type HL, she would obtain xm ; (c) type LL woman, she would receive
xl . Instead, suppose now that by investing all her human capital in task B, she becomes type LL. In
that case, if by deviating, she became: (a) type HH or LH, she would receive a share of marital output
equal to xl + (d − c); (b) type HL, she would obtain xm ; (c) type LL woman, she would receive xl . Her
expected utility is convex in ta and for t∗a = 0 to be optimal for all females, we need ∀ aw the following
c
condition to be satisfied xl + b(d − c) ≥ xl + aw c, or equivalently, b ≥ ā d−c
.
m
(6) Assume N − F = NLL
⇔
N
F
=
1
µ.
By proposition 1, the intra-household allocations are τ (θ) for type
HL men, 0 for type LL men, xh − τ (θ) for type LH women and xm − τ (θ) for type LL women.
Consider now a man with aptitude am . Given the strategies of the others, suppose that by investing all
w
his human capital in task A, he becomes type HL. Assume NLL
> 1. In that case, if he were to deviate
and by investing ta < 1, he became either type HH, HL or LH, he would receive a share of marital
output equal to τ (θ) and zero otherwise. Instead, suppose now that by investing all his human capital
in task A, he becomes type LL. In that case, by deviating, he would not be able to change his payoff
and still obtain 0. His expected utility is convex in ta and for t∗a = 1 to be optimal for all males, we
w
need ∀ am the following condition to be satisfied b ≤ am , or equivalently, b ≤ a. In case that NLL
> 1,
the only change in the payoffs consists on obtaining c/2, instead of τ (θ), if he were to become type HL
but by deviating becomes type LH. The required condition is then b ≤ 2τ (θ)a/c.
With respect to a woman with aptitude aw , given the strategies of the others, suppose that by investing
all her human capital in task B, she becomes type LH. In that case, if she were to deviate and by investing
ta > 0, she became: (a) type HH or LH, she would receive a share of marital output equal to xh − τ (θ);
(b) type HL, she would obtain xm ; (c) type LL woman, she would receive xm − τ (θ) + (d − 2c)/2F .
Instead, suppose now that by investing all her human capital in task B, she becomes type LL. In that
case, if by deviating, she became: (a) type HH or LH, she would receive a share of marital output
equal to xh − τ (θ) − (d − 2c)/2F ; (b) type HL, she would obtain xm ; (c) type LL woman, she would
receive xm − τ (θ). Her expected utility is convex in ta and thereby, in order for t∗a = 0 to be optimal
for all females, the following condition b(d − c) ≥ aw τ (θ) + (1 − aw )(d − 2c)/2F ∀aw or alternatively,
d−2c τ (θ)
b ≥ 1−ā
+
ā
is required.
2F
d−c
d−c
m
(7) Assume NHL
> F ⇔
N
F
>
1
µ.
By proposition 1, the intra-household allocations are 0 for type HL
36
and LL men, xh for type LH women and xm for type LL women. Given the strategies of the other
participants, no men will have incentives to deviate since he would obtain a zero payoff independently
of any possible type that he could become. With respect to women, they would obtain a payoff of xh
if by deviating they became class one (i.e. type HH or LH) or xm if they became class two (i.e. type
HL or LL). Therefore, their expected utility, which is given by xm + b(1 − ta )(d − c), is decreasing in
the amount of the investment for task A and as a result, specializing in task B (t∗a = 0) is an optimal
strategy.
The proof for the MBFA gender division of labor can be easily established by following similar steps to
the ones done previously for the MAFB case.
This completes the proof for Proposition 2.
Proof of Proposition 3: We prove this proposition through a series of lemmas.
Lemma 1:
m
NHL
The equilibrium shares that clear the market under MAFB are: (1) τHL = τLL = 0 if
m
m
< F < N;
= F ; (3) τHL = d − c and τLL = 0 if NHL
> F ; (2) 0 ≤ τHL ≤ (d − c) and τLL = 0 if NHL
(4) d − c ≤ τHL ≤ xh and τLL = τHL − (d − c) if N = F ; (5) τHL = xh and τLL = xm if N < F .
Lemma 2: The gender division of labor are always competitive equilibria in large markets regardless of
the sex ratio.
Claim 1: If the sex ratio is such that (N/F ) < (1/µ), the gender division of labor by which, men invest
solely in task A while women do so in task B, MAFB, constitutes a competitive equilibrium whereby women
capture xm .
Proof. In case that µN < F , and given the strategies of all men, the total number of females is expected
to exceed the number of type HL men so that the latter plus some type LL men will get married. For type
LL men to be indifferent between being matched or remaining single, they should obtain zero utility. Hence,
if women invest all their human capital in task B and become type LH , those who marry type LL men will
receive xm as their share of marital output. For all women to be indifferent among men, type HL males
should keep d − c for themselves. Given this, would any man or woman have incentives to deviate?. The
answer is not. Consider a woman with aptitude equal to aw . If by deviating she became either a type HH or
LH woman, the equilibrium shares would remain being the same ones. Instead, if she were to become a type
HL or LL woman, she would then marry a type LL man with certainty and get either xm or xl respectively.
37
Thus, there are not gains in case of deviation but loses associated to the possibility of becoming a type LL
woman. As a result, women will not deviate from the equilibrium strategies.
Given the strategies of all women and rest of men, if a man with aptitude am deviated, and became type
HH or HL, he would get married to a type LH women and thereby, obtain d − c. On the other hand, if as a
victim of the random process, he became a type LH or type LL man, he would be either single or married
but in both cases, he would receive a zero payoff. Hence, the expected utility associated to a human capital
m m
m
investment of tm
a is a ta (d − c) which is increasing in ta and as a result, any man would optimally choose
∗
to invest (tm
a ) = 1 in the first stage, independently of his aptitude . Q.E.D.
Claim 2: If the sex ratio is such that (N/F ) = (1/µ), the gender division of labor by which, men invest
solely in task A while women do so in task B, MAFB, constitutes an equilibrium whereby women capture
xh − τHL with 0 ≤ τHL ≤ (d − c).
Proof. Under the fact that µN = F , and given men’s strategy, any female will have expectations of an
equal number of women and type HL men. Thus, all type HL men will get married to women while all type LL
men will be unmarried. Any woman with aptitude aw does not have incentives to deviate since by investing
exclusively in task B, ensures the maximum possible output generated by a marriage. The equilibrium share
of marital output kept by type HL males must be at least equal to zero since, otherwise, they would prefer
to remain single. On the other hand, it cannot exceed d − c because in that case, women will deviate in the
third stage by proposing to single type LL men, and splitting the surplus xm − τHL between them.
Consider now a man with aptitude am . Given the others strategies, if he were to invest only in task A,
he would become type II man with probability am and type LL man with probability 1 − am . Suppose that
m
this man becomes type HL if he invests tm
a = 1. If now, he deviated by investing ta < 1, he could become
any of the four possible types. If he turned out to be type HH or type HL, the equilibrium shares would not
change so that he would still obtain a payoff of τHL . Instead, if he became type LH or LL, the equilibrium
shares would result in type HL men getting d − c whereas the other men get 0. This is because one of the
females should marry a type LH or LL man. Given that she is indifferent among them, and that all of them
except one will remain single, competition among men will lead her to obtain the total share generated by
the marriage, xm . For women to be indifferent to marry type HL men or these lower class men, all females
should get the same utility so that at the end, type HL men keep d − c for themselves. Suppose now that this
man were to become type LL if he invested tm
a = 1. His utility would remain being zero in case he decided to
deviate. This is because if he became either type HH or type HL, the population of class 1 men will exceed
38
the female population which grants full bargaining power to women and allow them to keep the entire marital
output. If he were to become type LH or LL, he would not get married and consequently, obtain zero payoff.
Hence, by investing tm
a , he would obtain a total expected utility given by :
m
m 2 m
EU = am [am tm
a τ23 ] + (1 − a )0 = (a ) ta τ23
∗
which is increasing in task A investment choice variable so that he optimally chooses (tm
a ) = 1. Q.E.D.
Claim 3: If the sex ratio is such that (N/F ) > (1/µ), the gender division of labor by which, men invest
solely in task A while women do so in task B, constitutes an equilibrium whereby women capture the total
share of marital output.
Proof. In case that µN > F , and given the strategies of all men, women will hold beliefs about the
proportions of type HL and type LL men in the market. In particular, they will expect the number of type
HL men to exceed the total number of females so that all type LL men and some type HL men will remain
single. Hence, for type HL men to be indifferent between being single or married, they must get a zero
∗
payoff so that women receive the total output obtained from marriage. Since by investing (tw
a ) = 1, females
become type LH women in the third stage, they ensure the maximum possible marital output achievable.
Thus, women do not have incentives to deviate. Consider now a man with aptitude a. Given the strategies
of his counterparts, all women will marry some of the existing type HL men in the market so that he will
receive a zero payoff independently of his investment. As a result, neither does he have incentives to deviate
and the gender division of labor constitutes an equilibrium. Q.E.D.
Claim 4: The sexual division of labor where men invest solely in task B and women solely in task A,
MBFA, is always an equilibrium. In that case, the equilibrium shares of marital output that clear the market
are such that type HL women get xh , type LL women xm and all men receive zero utility.
Proof.
Consider a man with aptitude am . If he deviated and invested ta > 0, then this man would
become any of the four possible types in the second stage. Nonetheless, given the strategies of all remaining
men and women, in case that he were to become either type HH or type LH male, he would turn out to
be a perfect substitute among other men from a woman’s perspective. Consequently, and due to the fact
that women are scarce in the society, he would obtain a zero share of marital output. If instead, he were to
become type HL or type LL, women will strictly prefer any other man to this male so that he would certainly
remain single and get a zero payoff. As a result, we can conclude that this man does not have any incentives
39
to deviate. Consider now a woman with aptitude aw . Given the strategies of all men, there will be only
type LH males in the market. Since the number of men exceeds the number of women and all men become
homogenous a posteriori, competition among men will lead women to capture the entire output generated
from marriage. If she deviated by investing ta < 1, and the second stage process transformed this woman
into a type HH or type HL one, she would obtain the maximum attainable share, xh whereas if she became
a type LH or type LL woman, she would obtain only xm . Thus, her expected utility in stage 1 would be an
increasing function of the human capital investment made in task A, xm + aw ta(d − c) and therefore, choosing
not to invest exclusively in task A would not be optimal. Q.E.D.
This completes the proof of Lemma 2.
Lemma 3. In case of an equal population of males and females so that the sex ratio is set to unity, and
under µ < (F − 1)/F , there always exists a sexual division of labor equilibrium where one gender specializes
in task A and get a share of τ if the agents become type HL or τ − (d − c) if they become type LL in the second
stage, and where the other gender specializes in task B, becoming all its members type LH in the second stage
and obtaining xh − τ , where d − c ≤ τ ≤ xh .
Proof. Consider the gender division of labor where all members whose gender is j (for j= males,females)
are fully trained for task A whereas the opposite sex is trained exclusively for task B. If µ < (F − 1)/F , then
the expected number of type LL individuals of gender j exceeds one. As a result, if any gender j agent were
to deviate by not fully specializing in task A and became class 1 (i.e. high type in task A), he/she would
receive a share of marital output equal to τ whereas if conversely, he/she became class 2 (i.e. low type in task
A), he/she would get τ − (d − c). The expected payoff is increasing in the amount of human capital invested
in task A. Thereby, he/she would not deviate. On the other hand, if any member of the opposite sex were
to deviate and became type HH, HL, or LH, given that more than one type LL agents of the opposite sex
are expected to be in the marriage market, she/he would receive a payoff equal to xh − τ . Instead, if she/he
were to become type LL, then she/he would get xh + (d − c) − τ . Therefore, given that xh > xh + (d − c),
investing all human capital in task B provides the highest expected utility and consequently, she/he would
not deviate. Q.E.D.
Lemma 4. The sexual divisions of labor always constitute competitive equilibria in intermediate markets.
40
Proof. In intermediate markets, there is a finite number of males (N ) and a finite number of females (F ).
Under these circumstances, would any man or woman have incentives to deviate?. Consider first MAFB.
• If N < F , given that all women become type LH and that they are in the majority of the population,
men will keep for themselves the total marital output. Thus, a man who becomes type HH or HL will
receive xh whereas a man who becomes type LH or LL, xm . His expected utility is then increasing in
the amount of investment: xm + am tm
a (d − c) so that no men will deviate. Given that the rest of women
invest exclusively in task B, a woman with aw will be indifferent about investing any amount of human
capital in B , since in case of marriage, she will not receive any transfer from her partner.
• If N = F , given the strategies of the others, a man with aptitude am who invest tm
a will obtain a share
m
)=
of marital output equal to τLL (NHL
xm
2
−(
m
NHL
d−c
N )( 2 )
if he becomes class 2 man (ie. type LH or
HL), or alternatively, τLL + (d − c), if he becomes class 1 (ie. type HH or HL). Hence, his expected
utility is given by,
N
−1
X
N −1
x
(µ)x (1 − µ)N −1−x {τLL (x) + am tm
a (d − c + τLL (x + 1) − τLL (x))}
x=0
w
which is increasing in tm
a and hence, this man will have no incentives to deviate. A woman with a ,
will receive xm − τLL if she becomes either type HH, HL or LH and xh − τLL
˜ in case she becomes type
LL where τLL(N
m) =
˜
II
N
X
N
x
xm
2
−(
m
NHL
d−c
N )( 2 )
−
c
2N .
Her expected utility in case she deviates is given by,
w w
(µ)x (1 − µ)N −x {xm − τLL (x) − tw
˜ (x))}
a (1 − a ta )(c − τLL (x) + τLL
x=0
w ∗
which is a convex function in tw
a being optimal to invest exclusively in task B (ie. (ta ) = 0).
• If N > F , now the females constitute the scarce gender. Given the strategies of the others, a man who
invest tm
a in A, will obtain (d-c)/2 if the number of other males who become type HL is equal to F − 1
and he were to become class 1 or alternatively, (d-c) if the total number of other type HL men is less
than F − 1 and he were to become class 1 as well. Thus, his expected utility is:
(F −2
)
X
1
N −1
N −1
am tm
(µ)x (1 − µ)N −1−x +
(µ)F −1 (1 − µ)N −F
a (d − c)
x
2 F −1
x=0
which is increasing in tm
a and therefore, he will not deviate. Women will clearly not deviate. Given
that all men invest ta = 1, there will only be type HL and type LL men in the market. For both
types of men, type HH and type LH women are perfect substitutes since the marital output resulted
41
from those matches would be the same. Nonetheless, investing ta > 0, makes it also possible for her
to become a type HL or type LL woman. If she became type HL woman, her share of marital output
m
would be the same as a type LH woman’s, xm , if NHL
< F − 1. Nonetheless, for the remaining cases
m
(ie. NHL
≥ F − 1) , she will get a strictly lower transfer than the one assigned to a type LH female.
Furthermore, if she became type LL woman, she would always enjoy a lower share of marital output
than a type LH woman does. In consequence, there is no additional gain in becoming a type HH woman
but there is a loss associated to the possibility of becoming a type HL or a type LL woman. Thus, all
women will optimally invest all their human capital in task B, ta∗ = 0, independently of their aptitude
to perform task A.
The proof for MBFA follows similarly by changing above the roles of males and females. Q.E.D.
This completes the proof of Proposition 3.
42