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Vertical aspects of male–female disparity in the labor market
Economics Letters 107 (2010) 136–141
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Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Vertical aspects of male–female disparity in the labor market Olivier Baguelin ⁎ Centre d'études de l'emploi, “Le Descartes I”, 29 promenade Michel Simon, 93166 Noisy-le-Grand Cedex, France
a r t i c l e
i n f o
Article history: Received 28 October 2008 Received in revised form 8 December 2009 Accepted 7 January 2010 Available online 13 January 2010
a b s t r a c t The organization based economy model presented by Garicano and Rossi-Hansberg (G&RH, 2004) is used to study the vertical dimensions of male–female disparity in the labor market. This sheds a new light on the major aspects of the phenomenon. © 2010 Elsevier B.V. All rights reserved.
JEL classification: J21 J23 J31 J41 J7 Keywords: Sex discrimination Gender-segregation Occupational choice
1. Introduction There are many different aspects to male–female disparity in the labor market — differences in workforce participation, in unemployment and part time work, occupational segregation, differences in wages. In all of these dimensions, women appear to be at a disadvantage compared to men (OECD, 2002). A structuring dimension is vertical occupational segregation by gender i.e. the fact that women are underrepresented in higher-paid jobs, a particular expression of which is the comparative rarity of women in the upper ranks of management (often referred to as the “glass ceiling” phenomenon). Possible explanation of these disparities can be arranged along two main axes — human capital and discrimination. Lazear and Rosen (L&R, 1990) address vertical occupational segregation according to gender from a human capital perspective. They argue that women have a lower promotion probability due to a higher propensity to quit their job for children rearing and not to discrimination. Although this line of argument has proved valuable for some aspects of male–female disparity, the test of the L&R's model using Austrian data led to a negative conclusion: “neither the risk of childbearing nor different productive characteristics can explain the crowding of females in lower hierarchical positions” (Winter-Ebmer and Zweimüller, 1997). This paper aims to provide a discrimination-based analysis of male– female disparity in the labor market to complement analyses based on
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the human capital argument. Three aspects are studied together: low workforce participation among low-skilled women (compared to men with equivalent skills), occupational segmentation according to gender and the male–female earnings gap. To do this, we have built upon the organizations based economy model introduced by Garicano and RossiHansberg (2004) which enables us to capture the hierarchical or, more generally, vertical dimensions of male–female disparity. 2. The model 2.1. The economy — G&RH (2004) Production requires operational time and the ability to solve problems arising during operations. Workers are endowed with an ability z which they use to deal with problems; they draw a particular problem (one per period) and, possibly, solve it, in which case their operational time is useful. We consider a population described by a given distribution of ability levels, Γ(.), with density function γ(.) over [0, 1]. Problems are distributed over the same segment, problem z occurring with probability ϕ(z); Φ(.) denotes the associated CDF. An agent of ability z can solve all problems over [0, z]. Agents can form teams so that whenever they fail to solve a problem on their own they can submit it to a more able agent. This allows the latter to specialize on handling some problems and not others. It is assumed that higher ability agents spend a fraction h b 1 of their time communicating their knowledge about each problem submitted to them, irrespective of whether they can actually solve it
O. Baguelin / Economics Letters 107 (2010) 136–141
or not. An agent asking for help does not know who holds the solution: he first tries to solve the problem himself and, upon failing, submits it to a more able agent in his team. An employee of ability z, therefore submits a problem with probability 1−Φ(z). The focus is on the case where organizations only have one or two layers, and agents can choose to work on their own (to be a selfemployed, the organization has one layer) or work in a team (an organization of two layers). Teams are formed by managers, whose time is entirely spent problem-solving, and employees, who use all their time in production.1 Hence, a team of n employees and 1 manager has n units of production time available and 1 unit of knowledge communication time. The production of such a team is then simply Φ(z)n, where z is the ability of its manager, and is subject to this manager's time constraint. The number of employees matched to some given manager is limited by their ability. Since each employee of ability z• fails to solve his problem with probability 1−Φ(z•), the managerial time constraint is given by: (1−Φ(z•))nh= 1. Denoting w(z•) the wage of an ability z• employee, the rent of a manager of ability z is given by r =Φ(z)n −w(z•)n. 2.2. Male, female, and communication There are two types θ of agents, females (F) and males (M), assumed to be identically distributed over the ability segment [0, 1]. Yet, differences exist. A first difference comes from the cost of communicating in a team: if hθθ ̂ denotes the fraction of time necessary for a manager of gender θ̂ to communicate his/her knowledge to an M F 2 employee of gender θ, it is assumed that: hFF = hM M = hF = h ≤ hM. A second difference is that the least able females may fail to participate in the labor force whereas males always do. 3 Females of ability below a given exogenous parameter z̲F ≥ 0 are assumed not to participate in the labor force. In addition to its recipient ability, the wage function may therefore depends on gender: wθ(.), θ 2 {F, M}. 2.3. Characterization and properties of the equilibrium An equilibrium is characterized by: an allocation of agents of each gender to occupational positions (employees, managers, or selfemployed); the ability compositions of teams (i.e. the matching between employees and managers); the gender composition of teams (i.e. the matching between employees' and manager's gender and the representation of employees of each gender within teams); two earnings functions, one per gender, such that agents do not want to switch either teams or occupational positions. Proposition 1. Garicano and Rossi-Hansberg, 2004 Such an equilibrium exists, is unique, and exhibits positive sorting: higher ability employees are matched to higher ability managers.
137
male employees of ability zM, managers therefore devote a fraction tθ̂ of their time to the former, and 1 − tθ̂ to the latter. For this to be an equilibrium, the assignment must be such that managers could not be better off: matching with either lower or higher ability employees; changing the gender composition of their work team (and thus the allocation of their time). This requires that the assignment solves: max
ˆ
ðzF ;nF Þ;ðzM ;nM Þ;t θ
ðΦðzÞ−wF ðzF ÞÞnF + ðΦðzÞ−wM ðzM ÞÞnM ;
subject to ˆ
ˆ
ð1−ΦðzF ÞÞhFθ nF = t θ ; θˆ ð1−ΦðzM ÞÞhM nM
θˆ
= 1−t ;
ˆθ
t ∈½0; 1: We can rewrite this as: max
ΦðzÞ−wM ðzM Þ ˆ
zF ;zM ;t θ
ˆ
θ ð1−ΦðzM ÞÞhM
+
ΦðzÞ−wF ðzF Þ ˆ
ð1−ΦðzF ÞÞhFθ
−
ΦðzÞ−wM ðzM Þ
!
ˆ
θ ð1−ΦðzM ÞÞhM
ˆθ
ˆθ
t ; t ∈½0; 1: ð1Þ
The lemma below states that if two managers with equal ability but of different gender were to be matched to employees of a given gender, these employees would be of identical ability. Lemma 2. Ability assignment functions do not depend on managers' gender. Proof. From Eq. (1), it appears that communication costs act as scale factors: the technical aspects of the assignment (complementarity between manager's and employees' abilities) are independent from cost minimization. Ability assignment functions, thus, are only indexed by employees' gender. □ The previous lemma makes the writing of the problem much more simple. Let mθ(.) be defined, for any employee's ability z, from the first order optimality condition, that is: ′
wθ ðzÞ =
Φðmθ ðzÞÞ−wθ ðzÞ ϕðzÞ: 1−ΦðzÞ
In order for labor markets4 to clear, it must be the case that the supply of employees for any measurable set of abilities be equal to the demand managers have for these employees. This labor markets equilibrium condition splits according to gender as follows. For all z a [z̲F, z̃F], z
―F
F
M
t ð xÞ t ð xÞ m ðzÞ γðxÞdx + ∫mF z γðxÞdx; F ð― F Þ h h
ð2Þ
M 1−t F ðxÞ m ðzÞ 1−t ðxÞ γðxÞdx: γðxÞdx + ∫mMM ð0Þ h hFM
ð3Þ
m ð zÞ z FÞ F ð―
∫z ð1−ΦðxÞÞγðxÞdx = ∫mF
while, for all z ≤ z̃M, The equilibrium can therefore be characterized by two pairs (z̃θ, z̃θ)θ2{F, M} such that z̃θ ≤ z̃θ and, for θ 2 {F, M}: gender θ agents of ability z ≤ z̃θ are employees, gender θ agents of ability z ≥ z̃θ̃̃ are managers — those in between are self-employed.
z
m ðzÞ
∫0 ð1−ΦðxÞÞγðxÞdx = ∫mMM ð0Þ
3. A horizontally segmenting equilibrium
The only differences between conditions (2) and (3) are a tightened female labor force zF N 0 and a higher communication cost in female manager/male employees teams hFM N h.
Let us consider the optimal composition of (two-layers) organizations. Suppose that a mass 1 of θ̂ managers of ability z are matched with a mass nF of female employees of ability zF, and a mass nM of
Proposition 3. The equilibrium is gender-segmenting: male employees are matched to male managers and female employees to female managers.
1
Garicano (2000) has proved that such a specialization pattern is optimal. Experiments in social psychology reveal that, in group discussion, men talk more than women, more often assume leadership position, receive more positive statements and fewer negative statements. Smith-Lovin and Brody (1989) found that, in their interruptions, men discriminate by sex whereas women interrupt and yield the floor to males and females equally. As regards access to leadership, Lucas (2003) finds that male leaders attain higher influence than female leaders. 3 For some empirical supports to this assumption, see OECD (2002, table D, p.318). 2
Proof. The result derives from wage adjustment. If a male manager had an interest in matching a female employee this would also be the case for a female manager, and ability-matching male employees would remain unemployed. In labor market equilibrium there are no unemployment but also no “mixed-matches”. □ 4
There is one for each pair (gender, ability level).
138
O. Baguelin / Economics Letters 107 (2010) 136–141
From previous proposition, it follows that for all z a [z̃F, 1], tF(z) = 1 whereas, for all z a [z̃M, 1], tM(z) = 0. Therefore, labor markets equilibrium conditions simply state: for all z a [z̲F, z̃F], z
∫z ð1−ΦðxÞÞγðxÞdx = ―F
1 mF ðzÞ ∫ γðxÞdx; h mF ð―z F Þ
4.2. Male–female earnings difference The tightening of the female workforce (as compared to that of males) entails a wage adjustment. Indeed, the combination of horizontal gender-segmentation on the one hand, and low workforce participation of the least able female employees, on the other hand, affects the overall distribution of female earnings.
while, for all z ≤ z̃M, z ∫0 ð1−ΦðxÞÞγðxÞdx
Proposition 6. Let us assume that both abilities and problems are uniformly distributed over [0, 1]. Then, z̲F N 0 entails: (a) for all z a [z̲F, z̃F [, wF (z) N max{wM(z), Φ(z)}; (b) for all z a ]z̃ M, 1], r M(z) N max{r F(z), Φ(z)}.
1 m ð zÞ = ∫mMM ð0Þ γðxÞdx: h
In the remaining section, we focus on the special case where both problems and abilities are uniformly distributed.
With a given ability, female employees earn more than male employees whereas male managers earn more than female ones.
4. The properties of the segmented labor market 5. Intuition behind the results The maximum number of layers is set to 2 which, in the case where abilities and problems are uniformly distributed requires h ≥ 0.75. The results below require building the equilibrium which we do in Appendix A. Proofs are provided in Appendix A. 4.1. Male–female differences in occupational position The combination of horizontal gender-segmentation and low workforce participation of the least able female employees affects the occupational positions of all women in the labor market when compared to that of men. Proposition 4. Let us assume that both abilities and problems are uniformly distributed over [0, 1]. Then, z̲F N 0 entails: z̃F N z̃M, and z̃ F N z̃M. The latter proposition yields two results. First, that women remain employees while possessing higher ability levels than men (or, likewise, that women become self-employed from a higher ability standard than men). Second, that women become managers from a higher ability standard than men. Proposition 5. Let us assume that both abilities and problems are F 1− z˜ M b1− z˜ , uniformly distributed over [0, 1]. Then, zF N 0 entails: (a) 1−z ―F
the fraction of managers is higher among men than among women; F z˜ − z˜F M N z˜ − z˜ M , the fraction of self-employed is higher among (b) 1−z ―F
women than among men; and (c)
z˜F −z ―F b z˜ M , 1−z ―F
the fraction of employees is
higher among men than among women.
Note that result (5.a) replicates the glass ceiling concern of the relative absence of women from management ranks.
Let us comment on the equilibrium features of the female labor market taking the male labor market as a reference. When removing the least able women from the labor market, we do two different things: we reduce the mass of available female employees (a quantitative impact); we tighten the range of available female abilities (from the bottom of the ability scale, a qualitative impact). These changes induce two adjustment processes: one is a wage adjustment, the second is an ability matching adjustment. Wages and ability matching adjust so as to guarantee labor market equilibrium. We can view this ability matching adjustment as playing a particular role, namely, guaranteeing the “positive sorting” result (see proposition 1). Quantitatively, the reduced mass of available female employees requires positive wage variations to encourage the switching of some female potential self-employed into employees; it also entails a decrease in the rents of female managers, some of whom (the least able) switch to self-employed. Qualitatively, wage positive adjustments insert more able agents in the mass of female employees. These agents are absorbed, through the ability matching adjustment, by the best female managers (“positive sorting”). This adjustment spreads down the (managers') ability scale so that, at a given ability, female managers are matched to better employees than male managers. Furthermore, it makes it possible for female managers to form upsized production teams. Finally, since female managers match upsized employee teams, covering a given mass of employees requires a smaller mass of female managers. The mass of female potential self-employed who become employees is lower than that of potential managers who become selfemployed.
Appendix A A.1. Characterizing equilibrium Note first that, since labor markets equilibrium conditions hold for a continuum of values, we can derive them with respect to z. For uniform distributions over [0, 1], this leads to the following differential equations: h i ′ z ; z˜F ; mF ðzÞ = ð1−zÞh for all z∈ ―F ′ mM ðzÞ = ð1−zÞh for all z∈½0; z˜ M ;
and therefore: 2 h i 1 2 F h + z˜ for all z∈ ―F − ð 1−z Þ z ; z˜F ; 1−z ―F 2 1 2 M mM ðzÞ = 1−ð1−zÞ h + z˜ for all z∈½0; z˜ M : 2
mF ðzÞ =
O. Baguelin / Economics Letters 107 (2010) 136–141
139
We now turn to wage functions. For all z a [z̲F, z̃F], ′
wF ðzÞ =
ΦðmF ðzÞÞ−wF ðzÞ ϕðzÞ; 1−ΦðzÞ
that is, for uniform distributions, ′
wF ðzÞ =
mF ðzÞ−wF ðzÞ : 1−z
With the expression of mF(.) obtained below, we need to solve the following differential equation:
′
wF ðzÞ +
2 2 F 1−z −ð1−zÞ h + z˜ ―F
1 2
wF ðzÞ = 1−z
1−z
:
The result is, for all z a [z̲F, z̃F]: F 2 ˜F z wF ―F z −z z˜ −wF ―F 1 ―F z wF ðzÞ = h+ z+ : z−z ―F 1−z 1−z 2 ―F ―F The determining of wM(.) goes through the same stages, leading to: wM ðzÞ =
1 2 M z h− wM ð0Þ− z˜ z + wM ð0Þ: 2
To fully characterize equilibrium, it just remains to find (z̃F, z̃F), (z̃M, z̃M), wM(0) and wF(zF) such that: first, mF(z̃F) = 1, wF (z̃F) = Φ(z̃F), and rF(z̃F) = Φ(z̃F); second, mM(z̃M) = 1, wM(z̃M) = Φ(z̃M), and rM(z̃M) = Φ(z̃M). This leads to: 2 1 1− + ρ h;―F ; z z ; −1 + ―F z −ρ h;―F h h 2 wF ―F ; z z −1 + ―F z −ρ h;―F = 1− 1−z ―F h h 1 2 M z˜ M ; z˜ = 1− + ρðh; 0Þ; −1−ρðh; 0Þ ; h h 2 wM ð0Þ = ð1−hÞ −1−ρðh; 0Þ : h
z˜F ; z˜F
=
where ρ h;―F z =
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 1−h 3 4 − +1+2 z +― z 2F ⇒ρðh; 0Þ = − +1 . ―F 2 2 h
h
h
h
h
A.2. Occupational positions
Proof 4. For z̲F N 0, ρ(h, z̲F) N ρ(h, 0) which directly induces z̃F N z̃M. Furthermore, given z̃ F–z̃ M = z̲F –(ρ(h, z̲F)–ρ(h, 0)), it is easy to verify that □ arg maxh2[0,75;1]ρ(h, z̲F) − ρ(h, 0) = 1 so that, for all h 2 [0, 75; 1[, ρ(h, z̲F) − ρ(h, 0) b ρ(1, z̲F) − ρ(1, 0) = z̲F which ensures that z̃ F N z̃ M. F
Proof 5. (a)
1− z˜ 1−z ―F
b 1− z˜
M
if and only if
1 ρ h;―F z −z ―F + 2 1− h
1−z ―F
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1−h 2 b ρðh; 0Þ + 2 1− . With ρðh;―F z + z2F , one easily obtains z Þ = ρðh; 0Þ + 2 ―F h
h
that, for h 2 [0.75, 1[, the previous condition holds if and only if z b2
―F
2 1−h −1−ρðh; 0Þ ρðh; 0Þ− h h : 2 2 1− −1−ρðh; 0Þ h
For h 2 [0.75, 1[, the right side term is higher than 1 and thus higher than z̲F. F
(b) With
M 1− z˜ b1− z˜ , 1−z ―F
proving that the fraction of employees is higher among men than among women is sufficient to showthatthe fraction ˜ 1−z ―F
z −z 1 of self-employed is higher among women than among men - and easier too. z˜M N F ―F if and only if 1− + ρðh; 0Þ b
1 z : ρ h;―F z bρðh; 0Þ− ρðh; 0Þ− ―F h
h
1 h
1− + ρ h;―F z −z ―F 1−z ―F
or
140
With ρ h;―F z = z ―F
b
1−h 2 h
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−h z h ―F
2
ρðh; 0Þ + 2
O. Baguelin / Economics Letters 107 (2010) 136–141
+ z 2 , for h 2 [0.75, 1[, the previous condition holds if and only if F
+ ð1−ρðh; 0ÞÞρðh; 0Þ : 1 2 1− ρðh; 0Þ− h
For h 2 [0.75, 1[, the right side term of this inequality is higher than 1 and thus higher than z̲F. (c) is a direct consequence of the latter two results.
□
A.3. Earnings
Proof 6. (a) For all z 2 [z̲F, z̃M], earnings are given by 2 1 F F h + z˜ h z + ð1−hÞ z˜ ; and z−z ―F 2 1 2 M M wM ðzÞ = z h + z˜ h z + ð1−hÞ z˜ : 2 wF ðzÞ =
With z̃F–z̃M = z̲F –(ρ(h, z̲F)–ρ(h, 0)), wF(z) − wM(z) N 0 if and only if:
1 N −1 + z: h −ρðh; 0Þ
1 −1 ―F z h
ρ h;―F z
1 2
2
+ ― zF
We have
1 1 2 h i −1 ―F z + ― z 1 1 h 2 F z ; z˜ M ; N ð −1Þ + z˜ M ⇒ N −1 + z; for all z∈ ―F h h ρ h;―F z −ρðh; 0Þ −ρðh; 0Þ
1 −1 ―F z h
ρ h;―F z
1 2
2
+ ― zF
and the premise holds. For all z 2 [z̃M, z̃F[, female earnings remains wage wF(z) whereas male are self-employed and earn Φ(z). Yet, by construction, wF(z) N Φ(z) so that female employees earn more than male self-employed.(b) For all z 2 [z̃F, 1], earnings are given by F
r ðzÞ =
z−wF ðzF ðzÞÞ z−wM ðzM ðzÞÞ M and r ðzÞ = ; ð1−zF ðzÞÞh ð1−zM ðzÞÞh
where zθ(.) is defined, for all z 2 [z̃θ, 1], by mθ(zθ(z)) = z. That is:5,6 h i 2 2 1 F 2 ˜ zF ðzÞ = 1− 1−z − for all z∈ z˜ F ; 1 ; z− z ―F h h i 1 2 M 2 M zM ðzÞ = 1− 1− for all z∈ z˜ ; 1 : z− z˜ h Using the expression of z̃F and z̃M we can check that 1 h i 2 1 2 2 F + ð1−zÞ for all z∈ z˜ ; 1 ; −ρ h;―F z zF ðzÞ = 1− h h 2 1 h i 1 2 2 zM ðzÞ = 1− for all z∈ z˜ M ; 1 : −ρðh; 0Þ + ð1−zÞ h h As a consequence 0 12 1 2 2 h 3 2 2 B 1 C + ð1−zÞ + ρ h;―F z − A −ρ h;―F z + ρ h;―F z − + 1; wF ðzF ðzÞÞ = @ h h h 2 2h 0
12 2 1 2 2C h 3 2 B 1 wM ðzM ðzÞÞ = @ + ρðh; 0Þ− A −ρðh; 0Þ + ð1−zÞ + ρðh; 0Þ− + 1: h h h 2 2h
2 F h zF(.) is defined for z such that zb z˜ + 1−z . This condition is always verified for z ≤ 1. Indeed, given ρ h;― zF = ―F 2 2 1 2 F h ≥1⇔ ρ h;― zF − ≥0 which, obviously is true. z˜ + 1−z ―F 2 h 6 To check that it is the case, read the previous footnote substituting 0 to z̲F. 5
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 h2
−
4 h
+1+2
1−h z h ―F
+― z 2F , note that
O. Baguelin / Economics Letters 107 (2010) 136–141
141
Earnings functions are expressed as: 0 12 1 2 1 2 2A h 2 @ z−1− −ρ h;―F z + ð1−zÞ + ρ h;―F z − −ρ h;―F z + h
h
h
2
F
r ð zÞ =
;
1 2 1 2 2 −ρ h;―F z + ð1−zÞ h h
h
0 12 1 2 1 2 2 h 2 z−1−@ −ρðh; 0Þ + ð1−zÞ + ρðh; 0Þ− A −ρðh; 0Þ + h
h
h
2
M
r ð zÞ =
3 2h
2 1 −ρðh; 0Þ h
+
1 2 2 ð1−zÞ h h
3 2h
;
which simplifies into:
r F ð zÞ =
2 − h
1 2 1 2 2 + ð1−zÞ −ρ h;―F z ; −ρ h;―F z h h
2 r ð zÞ = − h M
2 1 1 2 2 −ρðh; 0Þ + ð1−zÞ −ρðh; 0Þ: h h
Let us consider the function α(.) defined by
αðρÞ =
2 1 2 1 2 2 − −ρ + ð1−zÞ −ρ: h h h
Deriving α(.) with respect to ρ leads to ′
α ðρÞ =
1 −ρ h
2 1 −ρ h
+
1 2 2 ð1−zÞ h
−1;
which is strictly negative. It follows that for all z 2 [z̃F, 1] and z̲F N 0, ρ(h, 0) b ρ(h, zF) ⇒ rM(z) N rF(z). As for z 2 ]z̃M, z̃F], the fact that men have chosen □ to produce as managers rather than self-employed entails rM(z) N Φ(z).
References Garicano, L., 2000. Hierarchies and the organization of knowledge in production. Journal of Political Economy 108 (5), 874–904. Garicano, L., Rossi-Hansberg, E., 2004. Inequality and the organization of knowledge. Papers and Proceedings: American Economic Review, vol. 94, No. 2, pp. 197–202. Lazear, E.P., Rosen, S., 1990. Male–female wage differentials in job ladders. Journal of Labor Economics 8 (1), S106–S123.
Lucas, J.W., 2003. Status processes and the institutionalization of women as leaders. American Sociological Review 68, 464–480. OECD, 2002. Women at work: who are they and how are they faring? OECD Employment Outlook, pp. 61–123. Smith-Lovin, L., Brody, C., 1989. Interruptions in group discussions: the effect of gender and group composition. American Sociological Review 54, 424–435. Winter-Ebmer, R., Zweimüller, J., 1997. Unequal assignment and unequal promotion in job ladders. Journal of Labor Economics 15 (No 1, Part 1), 43–71.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Vertical aspects of male–female disparity in the labor market Olivier Baguelin ⁎ Centre d'études de l'emploi, “Le Descartes I”, 29 promenade Michel Simon, 93166 Noisy-le-Grand Cedex, France
a r t i c l e
i n f o
Article history: Received 28 October 2008 Received in revised form 8 December 2009 Accepted 7 January 2010 Available online 13 January 2010
a b s t r a c t The organization based economy model presented by Garicano and Rossi-Hansberg (G&RH, 2004) is used to study the vertical dimensions of male–female disparity in the labor market. This sheds a new light on the major aspects of the phenomenon. © 2010 Elsevier B.V. All rights reserved.
JEL classification: J21 J23 J31 J41 J7 Keywords: Sex discrimination Gender-segregation Occupational choice
1. Introduction There are many different aspects to male–female disparity in the labor market — differences in workforce participation, in unemployment and part time work, occupational segregation, differences in wages. In all of these dimensions, women appear to be at a disadvantage compared to men (OECD, 2002). A structuring dimension is vertical occupational segregation by gender i.e. the fact that women are underrepresented in higher-paid jobs, a particular expression of which is the comparative rarity of women in the upper ranks of management (often referred to as the “glass ceiling” phenomenon). Possible explanation of these disparities can be arranged along two main axes — human capital and discrimination. Lazear and Rosen (L&R, 1990) address vertical occupational segregation according to gender from a human capital perspective. They argue that women have a lower promotion probability due to a higher propensity to quit their job for children rearing and not to discrimination. Although this line of argument has proved valuable for some aspects of male–female disparity, the test of the L&R's model using Austrian data led to a negative conclusion: “neither the risk of childbearing nor different productive characteristics can explain the crowding of females in lower hierarchical positions” (Winter-Ebmer and Zweimüller, 1997). This paper aims to provide a discrimination-based analysis of male– female disparity in the labor market to complement analyses based on
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the human capital argument. Three aspects are studied together: low workforce participation among low-skilled women (compared to men with equivalent skills), occupational segmentation according to gender and the male–female earnings gap. To do this, we have built upon the organizations based economy model introduced by Garicano and RossiHansberg (2004) which enables us to capture the hierarchical or, more generally, vertical dimensions of male–female disparity. 2. The model 2.1. The economy — G&RH (2004) Production requires operational time and the ability to solve problems arising during operations. Workers are endowed with an ability z which they use to deal with problems; they draw a particular problem (one per period) and, possibly, solve it, in which case their operational time is useful. We consider a population described by a given distribution of ability levels, Γ(.), with density function γ(.) over [0, 1]. Problems are distributed over the same segment, problem z occurring with probability ϕ(z); Φ(.) denotes the associated CDF. An agent of ability z can solve all problems over [0, z]. Agents can form teams so that whenever they fail to solve a problem on their own they can submit it to a more able agent. This allows the latter to specialize on handling some problems and not others. It is assumed that higher ability agents spend a fraction h b 1 of their time communicating their knowledge about each problem submitted to them, irrespective of whether they can actually solve it
O. Baguelin / Economics Letters 107 (2010) 136–141
or not. An agent asking for help does not know who holds the solution: he first tries to solve the problem himself and, upon failing, submits it to a more able agent in his team. An employee of ability z, therefore submits a problem with probability 1−Φ(z). The focus is on the case where organizations only have one or two layers, and agents can choose to work on their own (to be a selfemployed, the organization has one layer) or work in a team (an organization of two layers). Teams are formed by managers, whose time is entirely spent problem-solving, and employees, who use all their time in production.1 Hence, a team of n employees and 1 manager has n units of production time available and 1 unit of knowledge communication time. The production of such a team is then simply Φ(z)n, where z is the ability of its manager, and is subject to this manager's time constraint. The number of employees matched to some given manager is limited by their ability. Since each employee of ability z• fails to solve his problem with probability 1−Φ(z•), the managerial time constraint is given by: (1−Φ(z•))nh= 1. Denoting w(z•) the wage of an ability z• employee, the rent of a manager of ability z is given by r =Φ(z)n −w(z•)n. 2.2. Male, female, and communication There are two types θ of agents, females (F) and males (M), assumed to be identically distributed over the ability segment [0, 1]. Yet, differences exist. A first difference comes from the cost of communicating in a team: if hθθ ̂ denotes the fraction of time necessary for a manager of gender θ̂ to communicate his/her knowledge to an M F 2 employee of gender θ, it is assumed that: hFF = hM M = hF = h ≤ hM. A second difference is that the least able females may fail to participate in the labor force whereas males always do. 3 Females of ability below a given exogenous parameter z̲F ≥ 0 are assumed not to participate in the labor force. In addition to its recipient ability, the wage function may therefore depends on gender: wθ(.), θ 2 {F, M}. 2.3. Characterization and properties of the equilibrium An equilibrium is characterized by: an allocation of agents of each gender to occupational positions (employees, managers, or selfemployed); the ability compositions of teams (i.e. the matching between employees and managers); the gender composition of teams (i.e. the matching between employees' and manager's gender and the representation of employees of each gender within teams); two earnings functions, one per gender, such that agents do not want to switch either teams or occupational positions. Proposition 1. Garicano and Rossi-Hansberg, 2004 Such an equilibrium exists, is unique, and exhibits positive sorting: higher ability employees are matched to higher ability managers.
137
male employees of ability zM, managers therefore devote a fraction tθ̂ of their time to the former, and 1 − tθ̂ to the latter. For this to be an equilibrium, the assignment must be such that managers could not be better off: matching with either lower or higher ability employees; changing the gender composition of their work team (and thus the allocation of their time). This requires that the assignment solves: max
ˆ
ðzF ;nF Þ;ðzM ;nM Þ;t θ
ðΦðzÞ−wF ðzF ÞÞnF + ðΦðzÞ−wM ðzM ÞÞnM ;
subject to ˆ
ˆ
ð1−ΦðzF ÞÞhFθ nF = t θ ; θˆ ð1−ΦðzM ÞÞhM nM
θˆ
= 1−t ;
ˆθ
t ∈½0; 1: We can rewrite this as: max
ΦðzÞ−wM ðzM Þ ˆ
zF ;zM ;t θ
ˆ
θ ð1−ΦðzM ÞÞhM
+
ΦðzÞ−wF ðzF Þ ˆ
ð1−ΦðzF ÞÞhFθ
−
ΦðzÞ−wM ðzM Þ
!
ˆ
θ ð1−ΦðzM ÞÞhM
ˆθ
ˆθ
t ; t ∈½0; 1: ð1Þ
The lemma below states that if two managers with equal ability but of different gender were to be matched to employees of a given gender, these employees would be of identical ability. Lemma 2. Ability assignment functions do not depend on managers' gender. Proof. From Eq. (1), it appears that communication costs act as scale factors: the technical aspects of the assignment (complementarity between manager's and employees' abilities) are independent from cost minimization. Ability assignment functions, thus, are only indexed by employees' gender. □ The previous lemma makes the writing of the problem much more simple. Let mθ(.) be defined, for any employee's ability z, from the first order optimality condition, that is: ′
wθ ðzÞ =
Φðmθ ðzÞÞ−wθ ðzÞ ϕðzÞ: 1−ΦðzÞ
In order for labor markets4 to clear, it must be the case that the supply of employees for any measurable set of abilities be equal to the demand managers have for these employees. This labor markets equilibrium condition splits according to gender as follows. For all z a [z̲F, z̃F], z
―F
F
M
t ð xÞ t ð xÞ m ðzÞ γðxÞdx + ∫mF z γðxÞdx; F ð― F Þ h h
ð2Þ
M 1−t F ðxÞ m ðzÞ 1−t ðxÞ γðxÞdx: γðxÞdx + ∫mMM ð0Þ h hFM
ð3Þ
m ð zÞ z FÞ F ð―
∫z ð1−ΦðxÞÞγðxÞdx = ∫mF
while, for all z ≤ z̃M, The equilibrium can therefore be characterized by two pairs (z̃θ, z̃θ)θ2{F, M} such that z̃θ ≤ z̃θ and, for θ 2 {F, M}: gender θ agents of ability z ≤ z̃θ are employees, gender θ agents of ability z ≥ z̃θ̃̃ are managers — those in between are self-employed.
z
m ðzÞ
∫0 ð1−ΦðxÞÞγðxÞdx = ∫mMM ð0Þ
3. A horizontally segmenting equilibrium
The only differences between conditions (2) and (3) are a tightened female labor force zF N 0 and a higher communication cost in female manager/male employees teams hFM N h.
Let us consider the optimal composition of (two-layers) organizations. Suppose that a mass 1 of θ̂ managers of ability z are matched with a mass nF of female employees of ability zF, and a mass nM of
Proposition 3. The equilibrium is gender-segmenting: male employees are matched to male managers and female employees to female managers.
1
Garicano (2000) has proved that such a specialization pattern is optimal. Experiments in social psychology reveal that, in group discussion, men talk more than women, more often assume leadership position, receive more positive statements and fewer negative statements. Smith-Lovin and Brody (1989) found that, in their interruptions, men discriminate by sex whereas women interrupt and yield the floor to males and females equally. As regards access to leadership, Lucas (2003) finds that male leaders attain higher influence than female leaders. 3 For some empirical supports to this assumption, see OECD (2002, table D, p.318). 2
Proof. The result derives from wage adjustment. If a male manager had an interest in matching a female employee this would also be the case for a female manager, and ability-matching male employees would remain unemployed. In labor market equilibrium there are no unemployment but also no “mixed-matches”. □ 4
There is one for each pair (gender, ability level).
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O. Baguelin / Economics Letters 107 (2010) 136–141
From previous proposition, it follows that for all z a [z̃F, 1], tF(z) = 1 whereas, for all z a [z̃M, 1], tM(z) = 0. Therefore, labor markets equilibrium conditions simply state: for all z a [z̲F, z̃F], z
∫z ð1−ΦðxÞÞγðxÞdx = ―F
1 mF ðzÞ ∫ γðxÞdx; h mF ð―z F Þ
4.2. Male–female earnings difference The tightening of the female workforce (as compared to that of males) entails a wage adjustment. Indeed, the combination of horizontal gender-segmentation on the one hand, and low workforce participation of the least able female employees, on the other hand, affects the overall distribution of female earnings.
while, for all z ≤ z̃M, z ∫0 ð1−ΦðxÞÞγðxÞdx
Proposition 6. Let us assume that both abilities and problems are uniformly distributed over [0, 1]. Then, z̲F N 0 entails: (a) for all z a [z̲F, z̃F [, wF (z) N max{wM(z), Φ(z)}; (b) for all z a ]z̃ M, 1], r M(z) N max{r F(z), Φ(z)}.
1 m ð zÞ = ∫mMM ð0Þ γðxÞdx: h
In the remaining section, we focus on the special case where both problems and abilities are uniformly distributed.
With a given ability, female employees earn more than male employees whereas male managers earn more than female ones.
4. The properties of the segmented labor market 5. Intuition behind the results The maximum number of layers is set to 2 which, in the case where abilities and problems are uniformly distributed requires h ≥ 0.75. The results below require building the equilibrium which we do in Appendix A. Proofs are provided in Appendix A. 4.1. Male–female differences in occupational position The combination of horizontal gender-segmentation and low workforce participation of the least able female employees affects the occupational positions of all women in the labor market when compared to that of men. Proposition 4. Let us assume that both abilities and problems are uniformly distributed over [0, 1]. Then, z̲F N 0 entails: z̃F N z̃M, and z̃ F N z̃M. The latter proposition yields two results. First, that women remain employees while possessing higher ability levels than men (or, likewise, that women become self-employed from a higher ability standard than men). Second, that women become managers from a higher ability standard than men. Proposition 5. Let us assume that both abilities and problems are F 1− z˜ M b1− z˜ , uniformly distributed over [0, 1]. Then, zF N 0 entails: (a) 1−z ―F
the fraction of managers is higher among men than among women; F z˜ − z˜F M N z˜ − z˜ M , the fraction of self-employed is higher among (b) 1−z ―F
women than among men; and (c)
z˜F −z ―F b z˜ M , 1−z ―F
the fraction of employees is
higher among men than among women.
Note that result (5.a) replicates the glass ceiling concern of the relative absence of women from management ranks.
Let us comment on the equilibrium features of the female labor market taking the male labor market as a reference. When removing the least able women from the labor market, we do two different things: we reduce the mass of available female employees (a quantitative impact); we tighten the range of available female abilities (from the bottom of the ability scale, a qualitative impact). These changes induce two adjustment processes: one is a wage adjustment, the second is an ability matching adjustment. Wages and ability matching adjust so as to guarantee labor market equilibrium. We can view this ability matching adjustment as playing a particular role, namely, guaranteeing the “positive sorting” result (see proposition 1). Quantitatively, the reduced mass of available female employees requires positive wage variations to encourage the switching of some female potential self-employed into employees; it also entails a decrease in the rents of female managers, some of whom (the least able) switch to self-employed. Qualitatively, wage positive adjustments insert more able agents in the mass of female employees. These agents are absorbed, through the ability matching adjustment, by the best female managers (“positive sorting”). This adjustment spreads down the (managers') ability scale so that, at a given ability, female managers are matched to better employees than male managers. Furthermore, it makes it possible for female managers to form upsized production teams. Finally, since female managers match upsized employee teams, covering a given mass of employees requires a smaller mass of female managers. The mass of female potential self-employed who become employees is lower than that of potential managers who become selfemployed.
Appendix A A.1. Characterizing equilibrium Note first that, since labor markets equilibrium conditions hold for a continuum of values, we can derive them with respect to z. For uniform distributions over [0, 1], this leads to the following differential equations: h i ′ z ; z˜F ; mF ðzÞ = ð1−zÞh for all z∈ ―F ′ mM ðzÞ = ð1−zÞh for all z∈½0; z˜ M ;
and therefore: 2 h i 1 2 F h + z˜ for all z∈ ―F − ð 1−z Þ z ; z˜F ; 1−z ―F 2 1 2 M mM ðzÞ = 1−ð1−zÞ h + z˜ for all z∈½0; z˜ M : 2
mF ðzÞ =
O. Baguelin / Economics Letters 107 (2010) 136–141
139
We now turn to wage functions. For all z a [z̲F, z̃F], ′
wF ðzÞ =
ΦðmF ðzÞÞ−wF ðzÞ ϕðzÞ; 1−ΦðzÞ
that is, for uniform distributions, ′
wF ðzÞ =
mF ðzÞ−wF ðzÞ : 1−z
With the expression of mF(.) obtained below, we need to solve the following differential equation:
′
wF ðzÞ +
2 2 F 1−z −ð1−zÞ h + z˜ ―F
1 2
wF ðzÞ = 1−z
1−z
:
The result is, for all z a [z̲F, z̃F]: F 2 ˜F z wF ―F z −z z˜ −wF ―F 1 ―F z wF ðzÞ = h+ z+ : z−z ―F 1−z 1−z 2 ―F ―F The determining of wM(.) goes through the same stages, leading to: wM ðzÞ =
1 2 M z h− wM ð0Þ− z˜ z + wM ð0Þ: 2
To fully characterize equilibrium, it just remains to find (z̃F, z̃F), (z̃M, z̃M), wM(0) and wF(zF) such that: first, mF(z̃F) = 1, wF (z̃F) = Φ(z̃F), and rF(z̃F) = Φ(z̃F); second, mM(z̃M) = 1, wM(z̃M) = Φ(z̃M), and rM(z̃M) = Φ(z̃M). This leads to: 2 1 1− + ρ h;―F ; z z ; −1 + ―F z −ρ h;―F h h 2 wF ―F ; z z −1 + ―F z −ρ h;―F = 1− 1−z ―F h h 1 2 M z˜ M ; z˜ = 1− + ρðh; 0Þ; −1−ρðh; 0Þ ; h h 2 wM ð0Þ = ð1−hÞ −1−ρðh; 0Þ : h
z˜F ; z˜F
=
where ρ h;―F z =
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 1−h 3 4 − +1+2 z +― z 2F ⇒ρðh; 0Þ = − +1 . ―F 2 2 h
h
h
h
h
A.2. Occupational positions
Proof 4. For z̲F N 0, ρ(h, z̲F) N ρ(h, 0) which directly induces z̃F N z̃M. Furthermore, given z̃ F–z̃ M = z̲F –(ρ(h, z̲F)–ρ(h, 0)), it is easy to verify that □ arg maxh2[0,75;1]ρ(h, z̲F) − ρ(h, 0) = 1 so that, for all h 2 [0, 75; 1[, ρ(h, z̲F) − ρ(h, 0) b ρ(1, z̲F) − ρ(1, 0) = z̲F which ensures that z̃ F N z̃ M. F
Proof 5. (a)
1− z˜ 1−z ―F
b 1− z˜
M
if and only if
1 ρ h;―F z −z ―F + 2 1− h
1−z ―F
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1−h 2 b ρðh; 0Þ + 2 1− . With ρðh;―F z + z2F , one easily obtains z Þ = ρðh; 0Þ + 2 ―F h
h
that, for h 2 [0.75, 1[, the previous condition holds if and only if z b2
―F
2 1−h −1−ρðh; 0Þ ρðh; 0Þ− h h : 2 2 1− −1−ρðh; 0Þ h
For h 2 [0.75, 1[, the right side term is higher than 1 and thus higher than z̲F. F
(b) With
M 1− z˜ b1− z˜ , 1−z ―F
proving that the fraction of employees is higher among men than among women is sufficient to showthatthe fraction ˜ 1−z ―F
z −z 1 of self-employed is higher among women than among men - and easier too. z˜M N F ―F if and only if 1− + ρðh; 0Þ b
1 z : ρ h;―F z bρðh; 0Þ− ρðh; 0Þ− ―F h
h
1 h
1− + ρ h;―F z −z ―F 1−z ―F
or
140
With ρ h;―F z = z ―F
b
1−h 2 h
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−h z h ―F
2
ρðh; 0Þ + 2
O. Baguelin / Economics Letters 107 (2010) 136–141
+ z 2 , for h 2 [0.75, 1[, the previous condition holds if and only if F
+ ð1−ρðh; 0ÞÞρðh; 0Þ : 1 2 1− ρðh; 0Þ− h
For h 2 [0.75, 1[, the right side term of this inequality is higher than 1 and thus higher than z̲F. (c) is a direct consequence of the latter two results.
□
A.3. Earnings
Proof 6. (a) For all z 2 [z̲F, z̃M], earnings are given by 2 1 F F h + z˜ h z + ð1−hÞ z˜ ; and z−z ―F 2 1 2 M M wM ðzÞ = z h + z˜ h z + ð1−hÞ z˜ : 2 wF ðzÞ =
With z̃F–z̃M = z̲F –(ρ(h, z̲F)–ρ(h, 0)), wF(z) − wM(z) N 0 if and only if:
1 N −1 + z: h −ρðh; 0Þ
1 −1 ―F z h
ρ h;―F z
1 2
2
+ ― zF
We have
1 1 2 h i −1 ―F z + ― z 1 1 h 2 F z ; z˜ M ; N ð −1Þ + z˜ M ⇒ N −1 + z; for all z∈ ―F h h ρ h;―F z −ρðh; 0Þ −ρðh; 0Þ
1 −1 ―F z h
ρ h;―F z
1 2
2
+ ― zF
and the premise holds. For all z 2 [z̃M, z̃F[, female earnings remains wage wF(z) whereas male are self-employed and earn Φ(z). Yet, by construction, wF(z) N Φ(z) so that female employees earn more than male self-employed.(b) For all z 2 [z̃F, 1], earnings are given by F
r ðzÞ =
z−wF ðzF ðzÞÞ z−wM ðzM ðzÞÞ M and r ðzÞ = ; ð1−zF ðzÞÞh ð1−zM ðzÞÞh
where zθ(.) is defined, for all z 2 [z̃θ, 1], by mθ(zθ(z)) = z. That is:5,6 h i 2 2 1 F 2 ˜ zF ðzÞ = 1− 1−z − for all z∈ z˜ F ; 1 ; z− z ―F h h i 1 2 M 2 M zM ðzÞ = 1− 1− for all z∈ z˜ ; 1 : z− z˜ h Using the expression of z̃F and z̃M we can check that 1 h i 2 1 2 2 F + ð1−zÞ for all z∈ z˜ ; 1 ; −ρ h;―F z zF ðzÞ = 1− h h 2 1 h i 1 2 2 zM ðzÞ = 1− for all z∈ z˜ M ; 1 : −ρðh; 0Þ + ð1−zÞ h h As a consequence 0 12 1 2 2 h 3 2 2 B 1 C + ð1−zÞ + ρ h;―F z − A −ρ h;―F z + ρ h;―F z − + 1; wF ðzF ðzÞÞ = @ h h h 2 2h 0
12 2 1 2 2C h 3 2 B 1 wM ðzM ðzÞÞ = @ + ρðh; 0Þ− A −ρðh; 0Þ + ð1−zÞ + ρðh; 0Þ− + 1: h h h 2 2h
2 F h zF(.) is defined for z such that zb z˜ + 1−z . This condition is always verified for z ≤ 1. Indeed, given ρ h;― zF = ―F 2 2 1 2 F h ≥1⇔ ρ h;― zF − ≥0 which, obviously is true. z˜ + 1−z ―F 2 h 6 To check that it is the case, read the previous footnote substituting 0 to z̲F. 5
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 h2
−
4 h
+1+2
1−h z h ―F
+― z 2F , note that
O. Baguelin / Economics Letters 107 (2010) 136–141
141
Earnings functions are expressed as: 0 12 1 2 1 2 2A h 2 @ z−1− −ρ h;―F z + ð1−zÞ + ρ h;―F z − −ρ h;―F z + h
h
h
2
F
r ð zÞ =
;
1 2 1 2 2 −ρ h;―F z + ð1−zÞ h h
h
0 12 1 2 1 2 2 h 2 z−1−@ −ρðh; 0Þ + ð1−zÞ + ρðh; 0Þ− A −ρðh; 0Þ + h
h
h
2
M
r ð zÞ =
3 2h
2 1 −ρðh; 0Þ h
+
1 2 2 ð1−zÞ h h
3 2h
;
which simplifies into:
r F ð zÞ =
2 − h
1 2 1 2 2 + ð1−zÞ −ρ h;―F z ; −ρ h;―F z h h
2 r ð zÞ = − h M
2 1 1 2 2 −ρðh; 0Þ + ð1−zÞ −ρðh; 0Þ: h h
Let us consider the function α(.) defined by
αðρÞ =
2 1 2 1 2 2 − −ρ + ð1−zÞ −ρ: h h h
Deriving α(.) with respect to ρ leads to ′
α ðρÞ =
1 −ρ h
2 1 −ρ h
+
1 2 2 ð1−zÞ h
−1;
which is strictly negative. It follows that for all z 2 [z̃F, 1] and z̲F N 0, ρ(h, 0) b ρ(h, zF) ⇒ rM(z) N rF(z). As for z 2 ]z̃M, z̃F], the fact that men have chosen □ to produce as managers rather than self-employed entails rM(z) N Φ(z).
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