The Model

The model used in this paper is based on that of Boschini and Sjögren (2007), and for the purpose of simplicity and comparison, the notations too are similar. Suppose a chairperson has to choose and name one of many aspirants as the CEO of the company.

In the pool of potential CEO candidates,

φ

^C is the fraction of females, and 1−φ^C is the fraction of males. In addition,

φ

^P represents the fraction of female chairpersons in all companies, and 1−

φ

^P is the fraction of male chairpersons. During the matching process, there are two groups of agents in this model (i.e. the group of potential CEOs, and the group of all chairpersons). Boschini and Sjögren (2007) considered only one group of agents (i.e. all authors) in their random matching model. Thus, the following equations will have slightly different expressions.

Every chairperson decides whether to hire a CEO or not.

U is used to denote the

_ia utility of the chairperson cooperating with different team types, i indicates the gender of

Figure: 1 Classification of observed groups

the chairperson, i.e. i∈

{

Male, Female

}

, while different team types are shown by a, and

{

S M C

}

a∈ , , . The details can be written as:

S: One person team. The chairperson him/herself is also CEO.

M: Mixed team. The chairperson hires an opposite sex CEO.

C: Same sex team. The chairperson hires a same sex CEO.

There are different perceptions of outcome of teamwork. We assume that some chairpersons always rank teamwork higher than working alone. This fraction is

μ

_i, and

} {

Male Female

i∈ , . On the other hand,

σ

_i means the proportion of gender i that Male

(1－φ)

Female

ψ

μ i

--gender i, i∈{Male, Female}, who prefers team-work.

σ

i-- gender i, i∈{Male, Female}, who prefers to work alone.

k i

-- gender i, i∈{Male, Female}, who chooses team types depends on the performance.

v i

-- gender i, i∈{Male, Female}, who is gender neutral.

v

m

k m

σ f

k f

μ m

μ f

σ m

v

f

1－

v

m ^1－

v

_f

always rank single-working higher than teamwork. Consequently,

k

_i = 1−

μ

_i −

σ

_i is the number of chairpersons who view the outcome as a more important consideration than the team type. In terms of gender preference,

v denotes the fraction of gender i

_i who are gender neutral, and (1−

v

_i) are those who have gender preferences. Further,

v

_i is assumed independent of

μ

_i and

σ

_i. Figure 1 summarizes the above notations, and depicts the classification of the observed groups.

Based on the model structure, we can compute several probabilities for different team types under random matching assumptions. First, if both the chairperson and the CEO of the company are female, the probability is:

^P

( )

P C⁽¹ f^)2.

Cf =φ φ −σ (1) Equation (1) shows the probability of a female chairperson cooperating with a female

CEO. On the right side of the equation, we use

φ

^Pand

φ

^Cto denote the proportion of female chairpersons and CEOs in each group, respectively. The term

(

¹−

σ

f

)

² means that none of them prefers working alone.

The same idea can be applied to the probability of a male chairperson cooperating with a male CEO, which is:

^P

( ) (

¹ P

)(

¹ C

)

⁽¹ m^)2.

C

m = −

φ

−

φ

−

σ

(2)

(

¹⁻

^φ

^P

)

denotes the fraction of male chairpersons and

(

¹⁻

^φ

^C

)

denotes the fraction of

male CEOs. The last term,

(

^1−σ_m

)

², denotes both the male chairperson and the male CEO willing to work with others.

Equation (3) describes the probability of a mixed team, which means the chairperson cooperates with an opposite sex CEO, i.e. a male chairperson teams with a female CEO or a female chairperson works with a male CEO.

( )

M =

[ (

1−φ^P

)

φ^C +φ^P

(

1−φ^C

) ] (

μ_f +k_fv_f

) (

μ_m +k_mv_m

)

P (3)

The first two terms in (3) are the probabilities of a firm having a male chairperson and a

female CEO

[ (

¹⁻

^φ

^P

) ^φ

^C

]

or a firm having a female chairperson and a male CEO

( )

[

¹⁻

^φ

^C

^φ

^P

]

. The last term,

( μ

_f +

k

_f

v

_f

) ( μ

_m+

k

_m

v

_m

)

, represents the probability of both teaming up with others, or both not having any particular team preference and being gender neutral at the same time.

Another possibility is that the chairperson is also named the CEO of the company, which may imply that he/she does not want to have close cooperation with another person. Or, at least one of the two (chairperson and the CEO) has a gender preference, i.e. they don’t like to team with an opposite sex colleague. The probability of the same person being the chairperson and the CEO is (Equation 4):

( ) [ ( ) ]

P

(

C

) [ (

f f f

) (

m m m

) ]

f C

P

f =

φ φ

1− 1−

σ

+

φ

1−

φ

1−

μ

+

k v μ

+

k v

S

P ² (4)

In Equation (4), the first part shows that both the chairperson and the CEO are female (i.e.

φ

^P

φ

^C ), and at least one of them does not like to work with others (i.e.

( )

[

¹−¹−

σ

f ²

]

). The second part depicts that either the chairperson or the CEO likes to work alone, or has gender preference (i.e.

( ) ( [

f f f

) (

m m m

) ]

C

P −φ − μ +k v μ +k v

φ 1 1 ). A

similar case in a situation when both the chairperson and the CEO are male, is described in Equation (5).

( ) ( )( ) [ ( ) ] (

P

)

C

[ (

m m m

) (

f f f

) ]

m C

P

m = 1−

φ

1−

φ

1− 1−

σ

+ 1−

φ φ

1−

μ

+

k v μ

+

k v

S

P ² (5)

Next, several conditional probabilities are computed by using equations (1)-(5).

First, the conditional probability of a female chairperson to name a female CEO is given in Equation (6).

( ) ( ) ( )

P f ^C

Cf

f P FC

P σ φ

φ

1− 2

=

= (6) The term FC (female CEO) is used to describe a team of a female chairperson and a female CEO. From the definition of conditional probability, the numerator means that a female chairperson teams with a female CEO (i.e. P(Cf), see Equation (1) for details).

Therefore, with the proportion of female chairpersons as the denominator (i.e.

φ

^P), the conditional probability of a female chairperson cooperating with a female CEO can be calculated as in (6).

The probability of a male chairperson cooperating with a female CEO is similar to the above case, described in Equation (7):

In addition, the probability of the same person being the chairperson and the CEO of a company is considered as a single team (S). The probability that a male chairperson himself functions as the CEO is as follows (Equation 8):

( ) ( ^{( )}

P

)

m

⁽

m

^{) (} [

m

⁾ (

f f f

) (

m m m

) ]

^C

In Equation (8), the numerator is the probability of a male chairperson working alone (i.e. P(Sm) (see Equation (5) for details). The fraction of male chairpersons is the denominator. The case for females is as in Equation (9), and it works the same way as Equation (8).

We found that the probability of a chairperson co-working with a female CEO increases as the proportion of female CEOs

φ

^C increases, as shown in equations (6) and (7). In

conjunction with equations (6) to (9) introduced above, two hypotheses are formulated.

Proposition 1: Gender Irrelevance

.

The proof can be found in Appendix 1. Since team preferences (σ and μ) of both sexes are the same and gender preferences (v) are also the same, gender would not be considered as an important element here, which means gender is irrelevant in this proposition.

Proposition 2: Gender Neutral

.

f and v v then implies that and

If σ σ σ σ β β α α β β

different preferences (σ_f ≠σ_m ) of team formation of the two sexes are assumed in Proposition 2. It allows gender neutrality to be sustained even when team preferences of the two genders are different. For example, we might observe that female chairpersons have a higher propensity to cooperate with female CEOs than males (i.e. β_m^FC <β_f^FC), and the gender neutrality hypothesis (v_f =v_m =1) can still hold if men are more likely

to work alone than women (i.e.

σ

_m >

σ

_f ).

Based on these two propositions, we will first test whether there is difference of the partnership between chairman and chairwoman. And, if there is a difference, the single team type can then be tested in order to find support for the gender neutral hypothesis.

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