DOCUMENT DE TRAVAIL WORKING PAPER N°08-19.RS RESEARCH SERIES

LABOR MARKET DISCRIMINATION AS AN AGENCY COST Pierre-Guillaume Méon Ariane Szafarz

Avenue F.D. Roosevelt, 50 - CP-140 l B-1050 Brussels l Belgium

DULBEA l’Université Libre de Bruxelles

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Labor market discrimination as an agency cost1

October 2008

Pierre-Guillaume Méon Université Libre de Bruxelles (ULB)

DULBEA CP 140 avenue F.D. Roosevelt, 50 1050 Bruxelles, Belgium

Ariane Szafarz Université Libre de Bruxelles (ULB)

Centre Emile Bernheim CP 145/1 avenue F.D. Roosevelt, 50 1050 Bruxelles, Belgium

Abstract: This paper studies labor market discriminations as an agency problem. It sets up a principal-agent model of a firm, where the manager is a taste discriminator and has to make unobservable hiring decisions that determine the shareholder’s profits because workers differ in skills. The paper shows that performance-based contracts may moderate the manager’s propensity to discriminate, but that they are unlikely to fully eliminate discrimination. Moreover, the model predicts that sectors with high skill leverages discriminate less. Finally, the impacts of unknown taste for discrimination, of a wage gap between groups, and of a diversity premium are investigated. Keywords: discrimination, agency theory, hiring. JEL codes: J71, D21, M12, M51

1 We thank Beatriz Armendariz, Patricia Garcia-Prieto, Danièle Meulders, Marc Labie, Kim Oosterlinck, Khalid Sekkat, seminar participants at CERMI, Université Libre de Bruxelles, and Large, Université Robert Schuman and Université Louis Pasteur, Strasbourg, for useful comments. Without twisting the truth no one but the two authors can claim the responsibility for all remaining errors or approximations.

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1. Introduction 

Differences on the labor market are despairingly ubiquitous and persistent. By and

large, women have been, and still are, discriminated against men, and ethnic minorities

against ethnic majorities virtually everywhere, as surveys by Altonji and Blank (1999) or Blau

and Kahn (2000) emphasize.

When it comes to explaining this stylized fact, economic theory usually rejects the

idea that discrimination finds its roots in employers’ prejudices. The influential argument, put

forward by Becker (1957), is that in a competitive environment, firms that discriminate would

get smaller profits, and eventually be driven out of the market. This argument does not rule

out prejudices, but stresses that hiring workers in a prejudiced manner is costly. Employers

should therefore face a strong incentive not to discriminate. This argument is particularly

appealing for large firms, where shareholders never even meet workers and should therefore

not seek anything but profit maximization.

The persistence of discrimination has subsequently been blamed on statistical

discrimination. According to Phelps (1971), employers hold biased beliefs on the skills of

discriminated workers, who in turn rationally validate those beliefs by not acquiring these

skills, because their return would be too low on a discriminatory labor market. The only

prejudice that may affect firms’ behavior is therefore those of consumers, as documented by

Holzer and Ihlanfeld (1998), and those of employees, as in Bertrand and Hallock (2001). That

view of discrimination does nevertheless not apply to large modern corporations where

shareholders maximize profit, and should therefore ban any form of costly discrimination.

However, in this industrial context, the firm’s owners do not hire workers.

In contrast, managers do work with employees, and hire them. If they are biased

against a given minority, they may consequently be tempted to behave in a discriminatory

way. Their temptation is moreover not mitigated by lower profits, since managers, unlike

shareholders, earn wages, not dividends. As shareholders can furthermore not precisely

monitor hiring processes, hiring decisions therefore have all the features that result in a

classical principal-agent problem that stems from the separation of ownership and control

emphasized in the classic works of Berle and Means (1932) or Jensen and Meckling (1976).

Surprisingly, while the corporate governance literature has extensively stressed that

managers’ incentives interfere with profit maximization, and found evidence of such a

behavior, as Shleifer and Vishny (1997) emphasize, it has not extended to discrimination.

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Conversely, the literature on discrimination has not taken managerial incentives into account.

Namely, all theories that, to our knowledge, analyze discrimination assume that decision

makers either maximize profits or weigh profit maximization against the psychological

benefits of discrimination. In other words, those theories implicitly consider that ownership

and control are not separated.

Yet, it is important to take true incentives into account when designing public policies

to curb discrimination. The discrimination literature has already underlined that the incentives

of discriminated groups matter, as in Phelps (1971) or more recently Coate and Loury (1993a,

b). Nevertheless, the literature has so far overlooked the incentives faced by managers and

shareholders or implicitly assumed an ownership structure that is at odds with the

organization of most firms. Managers’ specific incentives should also be taken into account,

because they control the hiring process. The present paper therefore focuses on the incentives

of managers when they hire workers for a firm that they do not own. It thus offers a model

acknowledging the agency problem pointed out by Arrow (1998, p. 95): “It is hardly in the

stockholders' interests to discriminate under the postulated condition, and competition in the

capital market should be effective in eliminating discrimination. (…) An alternative

hypothesis is that labor market discrimination is due to discriminatory tastes of other

employees. In the case of large corporations, for example, it would be those of the executives

(…)”.

With that end in view, this paper sets up a principal-agent model of a firm, where the

manager is a taste discriminator, and has to make unobservable hiring decisions that

determine the shareholder’s profits. It is to our knowledge the first theoretical attempt to set

up a model of discrimination that takes into account the principal-agent relationship at work

in hiring decisions. Solving the model shows that if performance-based contracts may

mitigate the manager’s propensity to discriminate, they may not always be able to fully

eliminate costly discrimination, and are innocuous to discrimination that does not affect the

firm’s performance. Moreover, the equilibrium level of discrimination depends on the

composition, in terms of skill and groups of workers, of the pool of workers where the firm

recruits, and on the impact of skills on profits.

To reach those conclusions, the paper is organized as follows. Section 2 presents the

baseline model, where a manager, chooses workers according to two unrelated characteristics,

skills and group membership, while only the former matters to the firm’s shareholder. Section

3 investigates the model’s equilibrium outcome and the impact of the model parameters on

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this outcome. Section 4 offers a series of realistic extensions, that all leave the model’s

qualitative conclusions unaltered. Section 5 concludes.

2. The model 

The model considers a firm facing a hiring decision. Applicants for the job can be

either skilled (κ = S) or unskilled (κ = U), and belong to an identifiable group i (i = M, W).

Following Coate and Loury (1993), we assume that both skills and group membership are

observable, and that the worker’s pay-off is F, irrespective of his/her characteristics.2 A

rejected applicant’s pay-off is zero.

Productivity R is related to skills, but not to group membership. We assume that a

skilled employee yields to the firm revenue 0sR > , while an unskilled employee’s yield is

normalized such that Ru = 0.

The hiring process is the following. Two candidates drawn randomly from a large pool

of workers apply for the open position. Within the pool of potential workers, applicants can

be either skilled or unskilled, and belong to group M or W. The full population of the pool of

workers features the following proportions of each of the four categories: MSγ , MUγ , WSγ , WUγ ,

with ( )0 , ; ,i i M W S Uκγ κ> = = and , ,

1ii M F S U

κκ

γ= =

=∑ ∑ . The outcome of the hiring process is

described by the worker’s vector of characteristics (i, κ) chosen by the manager from the two

applicants, on the basis of their characteristics.

The manager is a taste discriminator. Namely, he/she is biased against the applicants

from the W group, and is therefore reluctant to work with members of that group. That

behavior may be due to the manager’s affinity with his/her own group. Several recent pieces

of empirical evidence support the affinity theory. For instance, experiments from Garcia and

Ibarra (2007) show that, all other things being equal, Californians tend to hire Californian

candidates. Bertrand and Mullainathan (2005) uncovered racial discrimination on the US job

market by sending fictitious résumés to US firms, and observing that white-sounding names

received 50% more call-backs than African American ones.

However, the manager may even discriminate against his/her own group, when that

group is discriminated. For instance, Steinpreis et al. (1999) report experimental evidence that

2 The assumption that workers get the same wage regardless of group membership is at odds with reported wage gaps between non-discriminated and discriminated groups. One way to motivate it is to say that the model is concerned with hiring a worker for a particular job, and not with the equilibrium wage on the labour market. In any case, we will drop that assumption in section 4.

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not only male but also female judges prefer male over female applicants when their records

were identical. More generally, the observed propensity of individually successful women to

be reluctant to support other women in male-dominated environments has attracted attention

under the name “queen bee syndrome” since Staines et al. (1974) coined the expression.

The manager’s group membership notwithstanding, what matters here is that he/she

would rather not hire a member of the W group. As a result, whenever the skills of the two

applicants are identical, a biased manager will never choose a worker of the F group. Thus, in

this setting “costless” discrimination is always at work.3

The manager’s decision may however become ambiguous if the more skilled applicant

belongs to the discriminated group. The manager’s distaste for that group is assumed large

enough for him/her to rather hire an unskilled member of the other group than a competent

member of the discriminated group. In other words, the manager’s prejudice is detrimental to

the firm’s efficiency. We therefore explicitly model discrimination to be costly to

shareholders.

For reasons that will appear below, the manager’s decision is modeled as follows.

When one of the two applicants belongs to the discriminated group is the more skilled, and

the other applicant belongs to the non discriminated group, the manager hires the latter with

probability (1 – λ), λ ∈ [0, 1]. Under the same circumstances, his/her decision is therefore

only based on skills, regardless of group membership, with probability λ. This variable may

be interpreted as a measure of the manager’s propensity not to let his/her prejudices interfere

with the hiring decision. Alternatively, if one considers that the manager has a large number

of hiring decisions to make, then λ measures the proportion of instances where the manager

chooses a W skilled worker when the other applicant is an unskilled M worker.

A key feature of the model is that the manager’s prejudice needs not systematically be

relevant in the hiring decision. Namely, that prejudice does not affect that decision when the

two candidates belong to the same group, discriminated or not. In that case, which occurs with

a positive probability, the manager can only base the hiring decision on skills. The result is

that an outsider who would not observe the skills of applicants would be unable to determine

whether the outcome of the hiring process is due to discrimination, or simply luck. This is

precisely the situation of shareholders.

3 Costless here is understood to mean that discrimination comes at no cost for the shareholder.

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Table 1: outcomes of the hiring process (i, κ)

Applicant 1

Applicant 2 (W, S) (W, U) (M, S) (M, U)

(W, S) (W, S) (W, S) (M, S) (W, S) with prob. λ (M, U) with prob. (1−λ)

(W, U) (W, S) (W, U) (M, S) (M, U)

(M, S) (M, S) (M, S) (M, S) (M, S)

(M, U) (W, S) with prob. λ (M, U) with prob. (1−λ) (M, U) (M, S) (M, U)

The distribution of outcomes of the hiring decision is summarized in table 1, given

that the discriminated population is group W. The characteristics of one applicant are

displayed in the first column of table 1, while the other applicant’s characteristics are

displayed in the table’s first row. Each cell of that table displays the characteristics of the

hired worker, and, whenever relevant, their probabilities.

Depending on the hiring decisions, the firm’s revenue will differ. Table 2 displays the

revenue for the firm in each possible configuration.

Table 2: Revenues associated to the outcomes of the hiring Applicant 1

Applicant 2 (W, S) (W, U) (M, S) (M, U)

(W, S) Rs Rs Rs Rs with prob. λ 0 with prob. (1−λ)

(W, U) Rs 0 Rs 0

(M, S) Rs Rs Rs Rs

(M, U) Rs with prob. λ 0 with prob. (1−λ) 0 Rs 0

The revenue of the firm is thus the random variable defined by:

( ) ( )( )

2 20 with probability : 2 2 1

with probability :1MU WU MU WSWU MU

s

RR

λ γ γ γ γ λ γ γ

λ

⎧ Ω = + + + −⎪= ⎨− Ω⎪⎩

(1)

Note that ( )λΩ is a linear function of λ:

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( ) ( )2 2 2 2 2WU MU MU WU MU WS MU WSa b

λ γ γ γ γ γ γ λγ γ

λ

Ω = + + + −

= − (2)

where: 2 2 2 2

2WU MU MU WU MU WS

MU WS

ab

γ γ γ γ γ γγ γ

⎧ = + + +⎪⎨

=⎪⎩

Parameter a is the probability that a fully discriminating manager hires an unskilled

worker. This occurs when both applicants belong to the same group (M or W) and are

unskilled, or when an unskilled M applicant is opposed to a skilled W group member.

Parameter b is the probability of having to choose from a pair of applicants consisting of an

unskilled M worker and a skilled W worker. It is useful to remember that b is the probability

of the only situation where discrimination between the two applicants affects the firm’s

revenue. This situation occurs each time the skills of the two applicants differ, and the more

skilled worker belongs to the discriminated group. As both a and b are probabilities, they are

positive and smaller than one. Moreover, the definition of b makes it clear that b ≤ a.

Given his/her preferences, the key decision that the manager has to make is the value

of probability λ, namely the probability that he/she will not act according to his/her prejudices

when the better candidate belongs to the discriminated group. We must therefore specify

his/her objective and constraint. The manager gets consumption utility from his/her wage ω.

At the same time, increasing λ entails a psychological cost due to his/her bias against the

discriminated group. The manager’s expected utility therefore decreases with λ. We assume

the following risk-neutral expected utility function:

[ ] [ ] ( )21 02

E U E d dω λ= − ≥ (3)

As d increases, the manager’s expected disutility of hiring a skilled W worker in lieu

of an unskilled M worker increases. Parameter d gauges the aversion for the discriminated

group relative to the utility of consumption. It thus measures the intensity of the manager’s

gender or ethnic bias. An unbiased manager is characterized by d = 0, but there is no upper

limit on that parameter.

Finally, the manager has an outside option that will provide him/her utility U0,

normalized here to zero.

The wage ω that is paid to the manager is determined by the shareholder. The latter is

assumed risk-neutral, and seeks to maximize expected profits. The key characteristic of the

shareholder’s position is that he/she owns the firm but does not run it. Consequently, he/she

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can only observe the firm’s performance, and not the decisions that have been made by the

manager. The shareholder does in particular not observe the hiring process, and cannot

therefore pinpoint discriminatory behaviors. In particular, the shareholder may observe the

characteristics of the hired workers, but not those of the other applicants. The best he/she can

do is therefore to pay a wage that is contingent on the firm’s performance R. We assume a

standard linear contract of the following form:

[ ]. , 0,1C s R sω = + ∈ , C > 0 (4)

where C and s are parameters that are determined before the hiring decision. They are

therefore chosen by the shareholder who is assumed to maximize expected profits, E[π]. The

value of C is chosen to ensure fulfilling of the manager’s participation constraint.4

Accordingly, the shareholder’s expected utility reads:

[ ] [ ] ( )E V E E R Fπ ω= = − − (5)

To close the model, we specify the timing of the game. The shareholder first chooses

the parameters of the wage schedule, under the participation constraint, which states that the

manager’s expected utility must exceed that provided by his/her outside option. The manager

then determines the value of λ. The hiring process subsequently takes places. Once workers

have been hired, the firm’s performance is observed and the manager’s wage paid. Finally,

profits are distributed. This timing is summarized by the timeline in figure 1.

Figure 1: the timing of the game

The shareholder designs the wage contract

s, C

The managers sets his/her propensity to

discriminate λ

Hiring takes place κ, i

Performance is realized R, ω, π

This timing allows describing one of the key features of hiring decisions by managers

who do not own the firm. Namely, their principals do not participate in hiring decisions,

which is surrounded by much uncertainty, pertaining in particular to the skills of applicants.

Managers can therefore always deny discriminating by arguing that decisions were made only

on the basis of skills, but that the quality of applicants was such that it resulted in hiring fewer

members of the W group. We now investigate the equilibrium of the game.

4 Note that when s = 1 and C = 0, the manager’s incentives mimic those of a biased shareholder. This special case of the model can thus be used to describe the behaviour of a firm run by its owner.

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3. Equilibrium discrimination 

We solve the model by backward induction. As the manager is the last player to make

a decision, we therefore start by describing his/her behavior. We then describe the manager’s

reaction, which determines the game’s outcome. A graphical interpretation is developed in the

last sub-section.

3.1. The manager’s reaction function The manager chooses the value of λ that maximizes his/her utility:

[ ] [ ] ( )21 02

Max E U E d dω λ= − ≥ (6)

Replacing [ ]E ω , R, and ( )λΩ by their expressions derived from (4), (1), and (2), the

manager’s maximization problem becomes:

( ) 21. 12s

Max C s R a b dλ

λ λ⎧ ⎫+ − + −⎨ ⎬⎩ ⎭

(7)

The first order condition accordingly reads:

. . 0ss R b d λ− = (8)

which yields:

. .ss R bd

λ = . (9)

As it is a product of non-negative terms, the above expression is non-negative and

increasing in s, the shareholder’s incentive instrument. However, as λ is a probability, one

must also make sure that it is smaller than or equal to one, which eq. (9) does not guarantee.

Depending on the values of the parameters, the manager’s optimum may therefore lead to

corner solutions.

Two configurations may arise. The first arises when λ does not exceed one, i.e. when

.sd R b> that is when the manager’s taste for discrimination is large. Then eq. (9) provides

the interior solution of the manager’s problem. In that case, the manager’s optimal reaction,

( )* sλ , is positive and strictly smaller than one. It is moreover a linear and increasing

function of s, the parameter that determines the performance-dependent part of the wage

schedule. The second configuration arises when λ in eq. (9) can exceed one, i.e. whenever

.sd R b≤ . In that case, the manager’s bias is small enough to lead to a corner solution for

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some values of s. The manager’s optimal λ will then be given by (9) for small values of s

(.s

dsR b

≤ ), and be equal to one for larger values of s.

The manager’s reaction function is accordingly a function of s from [0, 1] to [0, 1] that

reads:

( )

. . if .

*1 if

.

s

s

s

s R b dsd R b

sds

R b

λ

⎧ 0, if that ratio is small enough, that is if the premium is small enough or the

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taste for discrimination large enough, then the manager chooses a value of λ that is smaller

than one. In that case, increasing the monetary benefit of hiring discriminated workers gives

the manager an incentive to discriminate less.

On the other hand, once the incentive to hire skilled workers from the discriminated

group becomes large enough for the manager to always base his/her decision on skills only,

that is once λ = 1, he/she cannot react to further increases of the premium. The manager’s

behavior then becomes independent from s.

3.2. The shareholder’s optimal contract Being the Stackelberg leader, the shareholder designs the performance contract by

anticipating its effects on the manager’s behavior. He/she therefore maximizes expected profit

taking the manager’s reaction as a constraint. Namely:

[ ] ( ) [ ]1s s

Max E R F Max s E R C Fω ⎡ ⎤− − = − − −⎣ ⎦ (11)

Replacing E[R] by its value, [ ] ( ) ( )1 1s sE R R R a bλ λ= − Ω = − +⎡ ⎤⎣ ⎦ , and taking the

value of λ that is optimal for the manager as given by expression (10), one gets the

shareholder’s program.

Depending on the parameters values, there can be an interior as well as a corner

solution to the manager’s program. Accordingly, the two cases are treated separately.

In the first case, i.e. when .sd R b≥ , the manager’s program leads to interior solutions.

In the second one, i.e. whenever .sd R b< , the manager’s bias is small enough for corner

solutions to exist. For simplicity sake, we start by investigating the second case.

When the manager’s bias is large enough, i.e. whenever .sd R b≥ , the shareholder’s

program reads:

( )2. .1 1 sss

b s RMax s R a C Fd

⎡ ⎤⎛ ⎞− − + − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦ (12)

The first order condition therefore is: 2 2. 2 . .1 0s sb R b s Rad d

− + − =

It leads to the following solution:

01 11 .2 . s

a dsb b R

⎛ ⎞−= −⎜ ⎟

⎝ ⎠ (13)

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For this solution to be realistic, the performance premium, s, must be non-negative.

We therefore impose s ≥ 0. Two sub-cases must then be considered depending on whether this

additional constraint bites or not.

The constraint that s ≥ 0 does not bite whenever1 . 1. s

a db b R−

< . This condition is

fulfilled if the manager’s taste for discrimination is not too large, namely if .1 s

bd b Ra

<−

.

In that case, s0 ∈ [0, 1], and the optimal value of s is given by (13):

01 1* 12 . s

a ds sb b R

⎛ ⎞−= = −⎜ ⎟

⎝ ⎠ (14a)

This optimal performance premium is decreasing in the manager’s bias and increasing

in the productivity of skilled workers. The manager’s optimal reaction is then given by the

first line of eq. (10). By plugging the value of s* in that expression, one gets:

( )0

1.1*2

s aR bd b

λ λ−⎛ ⎞

= = −⎜ ⎟⎝ ⎠

(14b)

It follows that the equilibrium probability λ* of hiring a skilled W worker competing

against an unskilled M worker is increasing in skilled workers’ output, Rs, and decreasing in

the manager’s bias, d. Solution (14a, b) however only applies when the manager’s bias is not

too large.

If the manager’s taste for discrimination is very large, that is .1 s

bd b Ra

≥−

, then the

value of s0 in eq. (13) is negative, and the shareholder’s program leads to the corner solution:

s* = 0. The manager’s optimal response is again found by replacing s* by its value in the first

line of eq. (10). The outcome of the game accordingly reads:

s* = 0 (15a)

λ* = 0 (15b)

In that configuration, the risk of the manager having the possibility to make a decision

according to his bias is so small that the shareholder is better off not paying a performance

premium, which would be wasted most of the time.

Let us now turn to the case where the manager’s taste for discrimination is small

enough for the reaction function to be kinked (d < Rs b). The resulting shareholder’s optimal

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contract can still be either an interior point or a corner solution, depending on the position of

s0 defined by eq. (14a) with respect to . s

db R

.

Consequently, the game results in an interior solution, if s0 ≤ . s

db R

. To check if it the

case, we replace s as defined by (13) by its value in the manager’s reaction function (10). This

gives:

01 11 .2 . .s s

a d dsb b R b R

⎛ ⎞−= − ≤⎜ ⎟

⎝ ⎠

Solving the above inequality reveals that the game leads to interior solutions when the

manager’s taste for discrimination remains larger than a given threshold:

.1 2 s

bd b Ra b

≥− +

In that case, the equilibrium values of λ and s are respectively given by expressions

(14a) and (14b).

The last sub-case arises when the manager’s taste for discrimination becomes very

small, namely, whenever:

.1 2 s

bd b Ra b

<− +

In this specific case, the manager’s program leads to a corner solution. His/her

behavior is accordingly described by the second part of expression (10). Namely, he/she does

not discriminate skilled W workers (λ* = 1). The shareholder’s best response is therefore to

pay the manager the smallest premium that will give the latter the incentive not to

discriminate. The equilibrium of the game therefore reads:

*. s

dsb R

= (16a)

* 1λ = (16b)

The outcomes of the game may be ranked according to the intensity of the manager’s

taste for discrimination, which takes us to our first three propositions:

Proposition 1: If the manager’s taste for discrimination is small ( .1 2 s

bd b Ra b

<− +

), then:

i. there is full discrimination of unskilled workers;

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ii. the probability of choosing a skilled worker from the discriminated group against

an unskilled worker from the other group is equal to one;

iii. the manager’s wage is an increasing function of the firm’s profits, an increasing

function of his/her taste for discrimination, and a decreasing function of the

probability that discrimination between the two applicants affects the firm’s

revenue.

Proof: (i) results from the fact that an unskilled W applicant is rejected when facing any other

applicant. We have moreover shown that whenever .1 2 s

bd b Ra b

<− +

, the outcome of

the game is described by expressions (16a) and (16b). Therefore, λ is equal to one,

which proves (ii). As s is positive, the manager’s wage is an increasing function of the

firm’s profits. Equation (16a) shows that s is increasing in d, and decreasing in b and

Rs, which proves (iii).

As the firm’s profits are unaffected by the rejection of unskilled W applicants, the

shareholder has no incentive to take it into account when designing the wage contract, which

explains the first part of the proposition.

The rationale of the rest of the proposition is the following. As long as the manager is

biased against the W group, an incentive is needed to reduce discrimination, and the larger

his/her bias, the larger the incentive must be. However, if the manager’s bias is sufficiently

small, the shareholder can find it optimal to eliminate its effects through wage incentives. The

latter therefore pays the manager just the premium necessary to get rid of costly

discrimination.5

The manager’s taste for discrimination is however not always sufficiently low for the

previous configuration to arise, which takes us to the next proposition.

5 The performance contract only eliminates the discrimination of skilled W workers competing against unskilled M workers. Costless discrimination remains. Regardless of skills no W applicant will ever by chosen over a skilled M one. To make this point clear, it suffices to compare the proportions of unskilled W workers in the firm

and in the original pool: 2

1WU WUWU

γ γγ

= < . Moreover, when there is no costly discrimination, the proportion of

skilled W workers in the firm over the proportion in the pool is given by: 2 2 2 2 2WS MU WS WU WS WS MU WU

WS

γ γ γ γ γ γ γ γγ

+ += + + .

This ratio is smaller than one when the pool does not include too large a proportion of unskilled workers, i.e., when: MU WU MSγ γ γ+ < .

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Proposition 2: If the manager’s taste for discrimination is intermediate

( . .1 2 s s

b b R d b Ra b

≤ <− +

), then:

i. there is full discrimination of unskilled workers;

ii. the probability of choosing a skilled worker from the discriminated group

against an unskilled worker from the other group is smaller than one. It

decreases with the manager’s taste for discrimination, and increases with the

skilled workers’ impact on profits;

iii. the manager’s wage is an increasing function of the firm’s profits, a decreasing

function of his/her taste for discrimination, and an increasing function of the

probability that discrimination between the two applicants affects the firm’s

revenue.

Proof: This configuration occurs in two cases: 1) the manager’s taste for discrimination is

small enough ( .sd R b< ) to make him/her choose a value of * 1λ < , but still large

enough ( .1 2 s

bd b Ra b

≥− +

) to result in * 0λ > , and 2) the manager’s reaction function

is kinked ( .sd R b≥ ), but the shareholder’s optimum lies on the increasing part of that

curve, which occurs when the manager’s taste for discrimination is not too large

( .1 s

bd b Ra

<−

). In both instances, the outcome of the interaction between the manager

and the shareholder is described by expressions (14a)-(14b).

According to (14b), *λ is strictly smaller than one, increases with Rs and b,

and decreases with d, which proves (ii). Expression (14a) shows that s is strictly

positive, increasing in b and Rs, and decreasing in d which proves (iii).

The intuition of that result is the following. When the manager’s taste for

discrimination is neither too small nor too large, the outcomes of both the manager’s and the

shareholder’s programs lead to interior solutions. The manager weighs the psychological

benefits of discriminating W workers against increasing his/her expected wage. The trade-off

depends on the values taken by the structural parameters: the manager’s taste for

discrimination (d), the firm’s additional profit brought by a skilled worker ( sR ), and the

probability of facing an unskilled M applicant and a skilled W applicant (b), i.e., the only case

of potential costly discrimination.

16

The manager discriminates more when 1) his/her taste for discrimination is larger, 2)

the impact of discrimination on the firm’s profits is smaller, and 3) the occurrence of a pair of

applicants (MU, WS) has a lower probability. The shareholder increases the performance-

related part of the manager’s wage when 1) the impact of discrimination on the firm’s profits

is larger, and 2) the occurrence of a pair of applicants (MU, WS) has a higher probability,

knowing that the manager’s decision has a larger impact on the firm’s profits. However, when

the manager’s bias, d, increases, the marginal impact of the performance premium on the

firm’s profits becomes smaller. The shareholder therefore reduces that premium.

Unfortunately, the manager’s taste for discrimination may not always be small enough

for the performance contract to dissuade him/her from fully discriminating skilled W workers.

In that case, the outcome of the game is described by the third proposition.

Proposition 3: If the manager’s taste for discrimination is large ( . , .1s s

bd Max b R b Ra

⎛ ⎞≥ ⎜ ⎟−⎝ ⎠),

then: i. there is full discrimination against all W workers;

ii. the manager’s wage is independent from the firm’s profits.

Proof: If . , .1s s

bd Max b R b Ra

⎛ ⎞≥ ⎜ ⎟−⎝ ⎠, then the outcome of the game is described by expressions

(15a)-(15b). According to (15a), s* = 0. The manager’s wage is constant, which proves

(i).

Moreover, λ* = 0 according to (15b). The probability to hire a skilled W worker facing

an unskilled M worker is zero. Accordingly, there is full discrimination in equilibrium.

The rationale for this result is that when the manager’s bias is too large, then the

marginal expected benefit of increasing the probability of hiring skilled W workers against

unskilled M workers is smaller than its marginal expected cost. In that case, the shareholder

prefers to pay the manager a fixed wage, even though the former knows that the latter will

systematically waste the skills of W applicants.

To put the last touches on the picture, we investigate the sensitivity of the equilibrium

to the composition of the pool of workers, and more precisely to the proportion of skilled

workers in the discriminated group. To do so, we compute partial derivatives of the optimal

premium s* and discriminating factor λ* with respect to probability WSγ , keeping constant the

17

total number of W members of the pool. We thus impose WS WU Wγ γ γ+ = , with γW fixed. The

fourth proposition relates discrimination of skilled workers to the composition of the W group.

Proposition 4a: The probability of choosing a skilled W worker against an unskilled M

worker:

i. is independent from the share of skilled workers in the W group if the

manager’s taste for discrimination is either small ( .1 2 s

bd b Ra b

<− +

) or large

( . , .1s s

bd Max b R b Ra

⎛ ⎞≥ ⎜ ⎟−⎝ ⎠);

ii. is an increasing function of the share of skilled workers in the W group if the

manager’s taste for discrimination is intermediate ( . .1 2 s s

b b R d b Ra b

≤ <− +

).

Proof: When the manager’s taste for discrimination is small, the equilibrium value of λ, as

described by (16a) is equal to one. When that taste is large, the equilibrium value of λ

is equal to one, as (15b) shows.

When the manager’s taste for discrimination is intermediate, the equilibrium value of

λ is given by expression (14b). The effect of raising the number of skilled W workers

is obtained by first replacing WUγ by W WSγ γ− in (14b), then deriving the result with

respect to Wγ :

( )2

1 2* 04

WU WS

WU WSsMU

WS MU WSC

aRd

γ γ

γ γλ γγ γ γ

+ =

− +∂= + >

∂,

which proves (ii).

To complement proposition 4a, proposition 4b describes how the manager’s contract

changes with the composition of the W group.

Proposition 4b: the manager’s performance premium:

i. is independent from the share of skilled W workers if the manager’s taste for

discrimination is large ( . , .1s s

bd Max b R b Ra

⎛ ⎞≥ ⎜ ⎟−⎝ ⎠);

18

ii. is an increasing function of the share of skilled W workers if the manager’s

taste for discrimination is either small ( .1 2 s

bd b Ra b

<− +

) or intermediate

( . .1 2 s s

b b R d b Ra b

≤ <− +

).

Proof: If the manager’s taste for discrimination is large then his/her contract entails no

performance premium, as (15a) shows. If the manager’s taste for discrimination is

moderate or small, then the equilibrium wage contract is described by expression

(14a). Replacing WUγ by W WSγ γ− in that expression, and then deriving it with respect

to Wγ gives:

2* 0

WU WS

MUWS sC

s db Rγ γ

γγ

+ =

∂= >

∂,

which proves (ii).

Intuitively, when the share of skilled workers is higher in the W group of the

population, costly discrimination becomes more probable leading the shareholder to enhance

the performance remuneration scheme, which in turn pushes the manager towards less

discriminative attitudes.

3.3. A graphic interpretation The three configurations of the model may be clarified thanks to a graphic

representation. To do so, we complement figures 2a and 2b, which graph the manager’s

reaction function, by the shareholder’s objective. The shareholder maximizes expected profits,

which depend positively on λ and negatively on s, being at their maximum when λ = 1 and

s = 0. The shareholder’s objective can thus be described by a series of increasing and convex

iso-expected-profit curves centered on bliss point (0, 1). The shareholder thus chooses the

point on the manager’s reaction function that lies on the rightmost iso- expected-profit curve.

The smaller the manager’s taste for discrimination, d, the steeper his/her reaction

function. When the manager’s bias exceeds the threshold . sb R , the kink in the reaction

function disappears. Figure 3 depicts the four possible cases presented by order of increasing

d.

19

Figure 3: Configurations of the model

Fig. 3a: .1 2 s

bd b Ra b

<− +

Fig. 3b: . .1 2 s s

b b R d b Ra b

≤ <− +

λ*(s) λ*(s)

.sR bd

1 1

.s

dR b

1 s .s

dR b

1 s

* 1λ = ( )1.1*2

s aR bd b

λ−⎛ ⎞

= −⎜ ⎟⎝ ⎠

*. s

dsb R

= 1 1* 12 . s

a dsb b R

⎛ ⎞−= −⎜ ⎟

⎝ ⎠

Fig. 3c: . .1s s

bb R d b Ra

≤ <−

Fig. 3d: .1 s

bd b Ra

≥−

λ*(s) λ*(s)

1 1

.sR bd

1 s 1 s ( )1.1*

2s aR bd b

λ−⎛ ⎞

= −⎜ ⎟⎝ ⎠

* 0s =

1 1* 12 . s

a dsb b R

⎛ ⎞−= −⎜ ⎟

⎝ ⎠ * 0λ =

The manager’s equilibrium propensity to discriminate, λ*, increases with d. When d is

very small (fig. 3a), the manager’s reaction function is very steep and kinked. In that case, the

shareholder’s optimum therefore lies exactly on the kink of the manager’s reaction function.

The latter therefore pays the former the smallest premium to eliminate discrimination. The

20

equilibrium value λ* is then equal to one, and s* is given by the kink’s abscissa, .s

dR b

, which

increases with the manager’s bias (see proposition 1).

When d takes intermediate values (fig. 3b and 3c), the manager’s reaction function

becomes steeper, and the shareholder’s optimum is to be found on the increasing part of the

graph. Probability λ* is therefore still a decreasing function of d, but the premium decreases

with it (see proposition 2).

Finally, when d is very large (fig. 3d), the manager’s reaction function becomes very

flat, and the shareholder’s optimum lies on the origin. Both s* and λ* are then equal to zero

(see proposition 3).

Figure 4: Equilibrium probability of hiring skilled W workers, λ*, and wage premium, s*.

λ*

1

d s*

ba

b21 +−

⎟⎠⎞

⎜⎝⎛ −+

21

21min ,

bba

.1 2b

R bsa b− + max . , .1bR b R bs sa

⎛ ⎞⎜ ⎟⎝ ⎠−

d

No discrimination

of skilled W workers

Partial

discrimination

of skilled

W workers

Full discrimination

of all W workers

Figure 4 summarizes the manager’s equilibrium behavior by describing the evolution

of λ as a function of d. For low values of the manager’s bias, there is no discrimination of

skilled workers in equilibrium. For larger values, discrimination starts increasing. For even

larger values, there is full discrimination of W workers. Figure 4 shows that the larger the

cost, sR , of hiring an unskilled worker instead of a skilled one, the larger the range of values

21

of the manager’s bias for which the performance-based contract is able to eliminate

discrimination of skilled W workers.

Discrimination of skilled workers is less likely where profits are very dependent on

skills. This occurs when the productivity gap between skilled and unskilled workers is large

and the probability b that discrimination between applicants affects the firm’s revenue is

large, as proposition 2 underlines. As a result we should expect to see less discrimination in

sectors and occupations where skills matter more. This is consistent with Meulders et

al.’s (2003) finding that, although women are still underrepresented among industrial

researchers, there is a trend for firms to hire more highly qualified young women. One can

indeed hardly think of a sector where skills matter more than research. It is therefore a sector

where discrimination of skilled workers is bound to be costly.

Second, as, according to proposition 4a, b is increasing in the proportion of skilled

discriminated workers in the population, the model predicts less discrimination in societies

where discriminated groups are more skilled.

It should be stressed that, while discrimination of skilled workers may depend on the

firm’s characteristics and on the pool of workers to which it has access, there is always full

costless discrimination, namely discrimination of unskilled workers and discrimination of

equally skilled workers. That private incentives affect discrimination differently for different

skills is an important contribution of the model.

4. Extensions 

This section checks the robustness of our main results along three directions. The first

sub-section investigates the consequences of the shareholder ignoring the magnitude of the

manager’s bias, while the second one introduces a wage gap between the two groups of

workers. The third sub-section modifies the incentive scheme offered to the manager to

directly reward hiring workers from the discriminated group.

4.1. Unknown taste for discrimination Up to now, the model rested on the assumption that the manager’s bias was known to

the shareholder. However, people’s prejudices differ, and some managers’ bias may be larger

than others. As the bias is unobservable, the shareholder must design the performance

contract without knowing the actual value of that parameter. This is bound to affect the

resulting contract, but also the behavior of managers.

22

To address this issue, we now modify the model to allow managers to be of two types:

either very biased ( . 0sd d R b= ≥ > ) with probability ψ, or completely unbiased ( 0d d= = )

with probability (1 – ψ). This implies that the manager’s reaction function is:

( )( )

. . with probability *

1 with probability 1

ss R bs d

ψλ

ψ

⎧⎪= ⎨⎪ −⎩

The firm’s expected revenue is then the sum of expected profits with a biased and an

unbiased manager weighted by their probabilities:

[ ] ( )1. .1 1s s

E R E R d d E R d d

s R ba b Rd

ψ ψ

ψ ψ

⎡ ⎤= = + − ⎡ = ⎤⎣ ⎦⎣ ⎦⎡ ⎤⎛ ⎞= − + + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(17)

The shareholder therefore maximizes expected profits that read:

( ) . .1 1 1ssss R bMax s R a b C F

dψ ψ

⎧ ⎫⎡ ⎤⎛ ⎞− − + + − − −⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭ (18)

The first order condition then gives: 2 2. 2 . . .1 0s sb R b s Ra b bd d

ψ ψψ− + − + − =

As a result, we can express the equilibrium wage contract and the manager’s

probability of hiring a skilled W worker competing against an unskilled M worker:

( )0 21 1* 1 .2 s

a b ds sRb

ψψ

⎛ ⎞− += = −⎜ ⎟

⎝ ⎠ (19a)

( )( )

21 1 . . with probability 2*

1 with probability 1

ss

a bd R bs b R

ψλ ψ

ψ

⎧ ⎛ ⎞− +−⎪ ⎜ ⎟

= ⎨ ⎝ ⎠⎪ −⎩

(19b)

We thus reach our fifth proposition, which complements the previous ones:

Proposition 5: When the manager may be either biased ( . 0sd d R b= ≥ > ) or unbiased

( 0d d= = ), then:

i. the probability that a biased manager chooses a skilled worker from the

discriminated group against an unskilled worker from the other group is

decreasing in the probability that the manager be unbiased;

23

ii. the manager’s wage is an increasing function of the firm’s profits, and an

increasing function of the probability that the manager be unbiased.

Proof: Expression (19b) shows that λ* is a decreasing function of ψ, which proves (i).

Expression (19a) shows that s* is an increasing function of ψ, which proves (ii).

Intuitively, if most managers are unbiased then there is no incentive to relate their

wage to performance. Indeed, unbiased managers spontaneously maximize the firm’s profits.

However, if the probability of the manager being biased is large, then the shareholder has a

strong incentive to relate the manager’s wage to performance, so as to limit the impact of the

latter’s bias. As a result, the equilibrium performance premium increases in the probability

that the manager is biased.

A corollary is that unbiased managers benefit from the existence of biased managers.

They get a free lunch in the guise of a performance-related wage, without having to change

their behavior to maximize their wage. The wage schedule is more stimulating the more

common biased managers are. This increases unbiased managers’ free lunch.

On the flip side, the wage contract is more stimulating when the probability of having

a biased manager increases. Consequently, biased managers are more induced to refrain from

discriminating in environments where biased managers are frequent. Ironically,

discrimination may thus be less prevalent on markets where biased managers are more

numerous.

4.2. A wage gap Until now, we have assumed for simplicity that all workers got the same wage,

regardless of group membership. This assumption is however at odds with empirical

evidence, as Altonji and Blank (1999) or Blau and Kahn (2000) point out. One may therefore

question the validity of the model’s outcome. This section explicitly addresses this caveat and

assumes a wage gap δ between workers of the two observable groups. In other words, W

workers earn δ less than M workers.6 To avoid a trivial equilibrium where the gap would be

so large that the manager would only hire W workers, we assume that: 0 sRδ< < . A skilled

worker is thus always more profitable to the firm than an unskilled one. A biased manager

will consequently always prefer a skilled M worker to an unskilled W worker.

6 In our model, this is equivalent to a subsidy offered to the firm for every hired W worker. Moreover, this configuration of the model nests the previous one for δ = 0.

24

The difference between the present configuration and the previous one is that even the

manager now faces a trade-off even when two equally skilled workers belonging to different

groups apply for the offered position. By hiring the W worker, the manager affects positively

the firm’s profits, hence his/her own performance premium.

Table 3: outcomes of the hiring process Applicant 1

Applicant 2 (W, S) (W, U) (M, S) (M, U)

(W, S) (W, S) (W, S) (W, S) with Prob. μ

(M, S) with Prob. (1−μ)

(W, S) with Prob. λ

(M, U) with Prob. (1−λ)

(W, U) (W, S) (W, U) (M, S) (W, S) with Prob. μ

(M, U) with Prob. (1−μ)

(M, S) (W, S) with Prob. μ

(M, S) with Prob. (1−μ) (M, S) (M, S) (M, S)

(M, U) (W, S) with Prob. λ

(M, U) with Prob. (1−λ)

(W, U) with Prob. μ

(M, U) with Prob. (1−μ) (M, S) (M, U)

Table 4: Revenues associated to the outcomes of the hiring process Applicant 1

Applicant 2 (W, S) (W, U) (M, S) (M, U)

(W, S) sR δ+ sR δ+ sR δ+ with Prob. μ

sR with Prob. (1−μ) sR δ+ with Prob. λ

0 with Prob. (1−λ)

(W, U) sR δ+ δ sR δ with Prob. μ

0 with Prob. (1−μ)

(M, S) sR δ+ with Prob. μ

sR with Prob. (1−μ) sR sR sR

(M, U) sR δ+ with Prob. λ 0 with Prob. (1−λ)

δ with Prob. μ 0 with Prob. (1−μ) sR 0

We define μ, the probability of hiring a W worker competing against an M worker of

equal skills. That probability is the same regardless of the applicants’ skills, because the

decision is the same from the manager’s point of view, since the performance premium is

independent from the applicants’ qualification. Table 3 summarizes the outcome of the hiring

25

process. It is equivalent to table 1 if μ is set to zero, which happens whenever there is no wage

gap.

As before, the firm’s profits are determined by the skills of the hired workers. This is

reflected in table 4. Again, that table boils down to table 2 whenever there is no wage gap

between the two groups of workers. In this case, the probability for a W applicant to be

selected when facing an M applicant of equal skills is zero.

We must define the manager’s and the shareholder’s programs according to their new

constraints. The first step is to determine the probability distribution of the firm’s revenue,

which can now take four different values. Namely, the firm’s revenue is still zero when an

unskilled M worker is hired. When an unskilled W worker is hired, the wage gap is equivalent

to a revenue of equal size. By the same token, the firm’s revenue is Rs when a skilled M

worker is hired but Rs + δ when a skilled W worker gets the position. The following

expression associates the relevant probability to those profits:

( ) ( ) ( )( )( ) ( )

( )

2

2

2

2

0 with proba : 2 1 2 1

with proba : 2

with proba : 2 2 2 1

with proba : 2 2 2

MU MU WU MU WS

WU MU WU

s MS MS WU MU MS MS WS

s WS WS WU MS WS MU WS

RR

R

γ μ γ γ λ γ γ

δ γ μγ γ

γ γ γ γ γ μ γ γ

δ γ γ γ μγ γ λγ γ

⎧ + − + −⎪⎪ +⎪= ⎨

+ + + −⎪⎪

+ + + +⎪⎩

(20)

As the manager dislikes hiring W workers regardless of their skills, but faces different

incentives to do so, we rewrite his/her objective. Namely, the manager chooses two distinct

probabilities of hiring a W worker: λ when a skilled W worker is facing an unskilled M

worker, μ when any W worker is facing an M worker of equal skills. The agent’s objective

thus reads:

( ) [ ] ( )2 2,

1, .2

Max f C s E R dλ μ

λ μ λ μ⎧ ⎫= + − +⎨ ⎬⎩ ⎭

where expected profits are:

[ ] ( ) ( ) ( )

( ) ( )

2 2

2

2 2 2 2 1

2 2 2

WU MU WU s MS MS WU MU MS MS WS

s WS WS WU MS WS MU WS

E R R

R

δ γ μγ γ γ γ γ γ γ μ γ γ

δ γ γ γ μγ γ λγ γ

⎡ ⎤ ⎡ ⎤= + + + + + −⎣ ⎦ ⎣ ⎦⎡ ⎤+ + + + +⎣ ⎦

[ ] ( ) ( ) ( )( ) ( )( ) [ ]

2 2 22 2 2

2 2WU s MS s MS WU s WS s WS WU s MS WS

s MU WS MU WU MS WS

E R R R R R R

R

δ γ γ γ γ δ γ δ γ γ γ γ

λ δ γ γ μδ γ γ γ γ

= + + + + + + +

+ + + +

The manager’s first order condition thus reads:

26

( ) ( )

( ) ( )

,2 0

,2 0

s MU WS

MU WU MS WS

fs R d

fs d

λ μδ γ γ λ

λλ μ

δ γ γ γ γ μμ

∂⎧= + − =⎪ ∂⎪

⎨∂⎪ = + − =⎪ ∂⎩

As neither λ nor μ can exceed one, the first order condition leads to the following

reaction function of the manager:

( ) ( )

( )

* min 2 ,1

* min 2 ,1

s MU WS

MU WU MS WS

Rs s

d

s sd

δ γ γλ

γ γ γ γμ δ

⎧ +⎧ ⎫=⎪ ⎨ ⎬

⎪ ⎩ ⎭⎨+⎧ ⎫⎪ = ⎨ ⎬⎪ ⎩ ⎭⎩

(21)

As before, the shareholder maximizes expected profits taking into account the

manager’s reaction. His/her program is accordingly given by:

( ) ( ) ( )( ) ( ) ( ) ( )

2 2 22 2 21

* 2 2 *WU s MS s MS WU s WS s WS WU s MS WS

ss MU WS MU WU MS WS

R R R R RMax s

s R s

δγ γ γ γ δ γ δ γ γ γ γ

λ δ γ γ μ δ γ γ γ γ

⎡ ⎤+ + + + + + +− ⎢ ⎥

+ + + +⎢ ⎥⎣ ⎦

To save on space, we only consider the interior solutions of the manager’s program.

The shareholder’s program then becomes:

( )( ) ( )

( ) ( )

2 2 2

2 22 2 2

2 2 21 4

WU s MS s MS WU s WS s WS WU s MS WS

sMU WS s MU WU MS WS

R R R R RMax s s R

d

δγ γ γ γ δ γ δ γ γ γ γ

γ γ δ δ γ γ γ γ

⎧ ⎫+ + + + + + +⎪ ⎪− ⎨ ⎬⎡ ⎤+ + + +⎪ ⎪⎣ ⎦⎩ ⎭

The shareholder’s first order solution reads:

( ) ( ) ( ) ( )

( ) ( )

2 2 2 22 2 2 2 2 2

2 2 2

4 8

2 2 2 0

MU WS s MU WU MS WS MU WS s MU WU MS WS

WU s MS s MS WU s WS s WS WU s MS WS

sR Rd d

R R R R R

γ γ δ δ γ γ γ γ γ γ δ δ γ γ γ γ

δγ γ γ γ δ γ δ γ γ γ γ

⎡ ⎤ ⎡ ⎤+ + + − + + +⎣ ⎦ ⎣ ⎦

− − − − + − + − =

Solving this equation for s yields the equilibrium value of the performance premium:

( )( ) ( )2 22 2

. 11* 12 4

s

MU MU MU MU s

d R as

R bδ γ γ γ γ δ

⎡ ⎤−= −⎢ ⎥

+ + +⎢ ⎥⎣ ⎦ (22a)

By replacing s* by its value in the reaction functions of the manager, i.e. equation (21),

one gets the equilibrium values of λ and μ:

( ) ( ) ( )[ ]( ) ( )

( ) ( ) ( )[ ]( ) ( )2222

2222

2222

441

421

δγγγγδγγγγδδγγδ

δγγγγδγγδγγγδλ

+++++++

+

+++++−+

−=

sWSMSWUMU

WSMSWUMUsWSMUs

sWSMSWUMU

WUWSWSsWSMUs

RbRbR

d

RbaRR*

(22b)

27

( ) ( ) ( )[ ]( ) ( )

( ) ( ) ( )[ ]( ) ( )2222

2222

2222

2

441

41

δγγγγδγγγγδδγγγγδ

δγγγγδγγδγγγγδμ

+++++++

+

+++++−+

−=

sWSMSWUMU

WSMSWUMUsWSMSWUMU

sWSMSWUMU

WUWSsWSMSWUMU

RbRb

d

RbaR*

(22c)

Those expressions boil down to those of the first section if δ is set to zero. More to the

point, they confirm previous results, as the following proposition underlines.

Proposition 6: If the wages of the W group are smaller than those of the M group and the

manager’s bias is sufficiently large, then:

i. the probability of choosing a W worker against an M worker of equal skills can

take any value between zero and one, and is increasing in the wage gap;

ii. the probability of choosing a skilled W worker against an unskilled M worker

can take any value between zero and one, and is increasing in the wage gap;

iii. the manager’s wage is an increasing function of the firm’s profits, an

increasing function of his/her taste for discrimination, and a decreasing

function of the wage gap.

Proof: Equation (22a) shows that *s is unambiguously increasing in d and sR and decreasing

in δ, which proves (iii).

The probability of choosing a W worker against an M worker of equal skills is given

by equation (22c). Its first term is unambiguously negative, while its second term is

positive and proportional to the inverse of d. As a result, μ* decreases from one to zero

as d increases from zero to infinity. Moreover, the derivative of that expression with

respect to δ, which is given in the appendix is unambiguously positive, which proves

(i).

By the same token, the probability *λ , given by (22b), of choosing a skilled W worker

against an unskilled M worker is decreasing in d and can take any value between zero

and one.

From (21), µ* is function of both δ and s, while s is a function of δ according to (22a).

The total effect of δ on µ* is therefore the sum of a direct effect measured by 0>∂

∂δ

μ * ,

28

and an indirect effect measured by 0>∂∂

∂∂

δμ *s

*s* , and is accordingly positive, which

proves (i).

By the same token, from (21), *λ is function of both δ and s, while s is a function of

δ according to (22a). The total effect of δ on λ* is therefore the sum of a direct effect

measured by 0>∂

∂δ

λ * , and an indirect effect measured by 0>∂∂

∂∂

δλ *s

*s* , and is

accordingly positive, which proves (ii).

Proposition 6 provides two new insights. The first is that unskilled W workers are not

fully discriminated anymore. Since hiring an unskilled W worker is cheaper than hiring an

unskilled M worker, the firm’s profits will be larger each time the manager will choose an

unskilled W worker instead of an unskilled M worker. This gives the manager an incentive to

hire some unskilled W workers even when the other candidate is of equal skills. The

probability of hiring an unskilled W worker may then be greater than zero.

Second, the equilibrium values of s, λ, and μ are all increasing in the wage gap.

Indeed, the greater the wage gap, the costlier discrimination, the greater is the shareholder’s

incentive to increase the performance premium to limit discrimination. Accordingly, s is an

increasing function of the wage gap. This gives the manager two incentives to cut down

discrimination. On the one hand, the share of profits that accrues to him/her is larger when the

output gap is larger. On the other hand, the impact on profits of reducing discrimination is

also larger. Both concur in giving the manager an incentive to mitigate further his/her taste for

discrimination. Both λ and μ are therefore increasing in the wage gap.

However, the wage gap may not be sufficient to completely compensate the manager’s

taste for discrimination. If the latter is sufficiently large, discrimination will not disappear

completely.

4.3. A diversity premium In the previous sections, the manager was indirectly induced to hire W workers

through a performance contract. The shareholder could do nothing better to maximize profits

than to find the optimal share of profits to pay to the manager. However, if shareholders

identify discrimination as an agency issue in their firm, they may also tackle it directly.

Instead of rewarding performance, they may thus reward diversity by offering to the manager

29

a premium directly related to his/her probability of hiring W workers. In this section, we

replace the performance contract by such an antidiscrimination contract.7

Analytically, the new contract is reminiscent of the previous sub-section’s wage gap. It

includes a bonus B that is equal to τ each time a discriminated worker is hired, and zero

otherwise. The difference here is that τ is paid and chosen by the shareholder, whereas δ was

exogenously determined in previous section. This results in the following revenue:

( ) ( )

( )

2

2

2

2

0 with proba : 2 1 2 1

with proba : 2

with proba : 2 2 2 1

with proba : 2 2 2

MU WU MU WSMU

MU WUWU

s MS WU MU MS MS WSMS

s WS WU MS WS MU WSWS

-τR

R

R - τ

γ μ γ γ λ γ γ

γ μγ γ

γ γ γ γ γ μ γ γ

γ γ γ μγ γ λγ γ

⎧ + − + −⎪⎪ +⎪= ⎨

+ + + −⎪⎪

+ + +⎪⎩

(23)

As before, the manager chooses probabilities λ and μ so as to maximize expected

utility, which can be written as:

( ) [ ] ( )2 2,

1,2

Max f C E B dλ μ

λ μ λ μ= + − +

The expected bonus is:

[ ] ( )2 22 2 2 2MU WU WS WU MS WS MU WSWU WSE B τ γ μγ γ γ γ γ μγ γ λγ γ= + + + + + The manager’s first order condition reads:

( )

( ) ( )

,2 0

,2 0

MU WS

MU WU MS WS

fd

fd

λ μλ τ γ γ

λλ μ

μ τ γ γ γ γμ

∂⎧= − + =⎪⎪ ∂⎨∂

⎪ = − + + =∂⎪⎩

This leads to the following optimal responses:

( ) 2* MU WSdτλ τ γ γ= (24a)

( ) ( )2* MU WU MS WSdτμ τ γ γ γ γ= + (24b)

Taking the manager’s reaction functions as constraints, the shareholder determines the

value of τ that maximizes expected profits. To avoid being overly taxonomic, we only

consider interior solutions of the manager’s problem. Accordingly, the shareholder’s program

is obtained by replacing the optimal values of λ and μ in expected profits, and maximizing for

τ. Expected profits thus read: 7 This contract can be coupled with a performance contract. We distinguish them in this section for clarity sake, but the main results would remain unchanged if the manager’s contract included both a diversity and a performance premium.

30

[ ] ( ) ( )( )( )

2 2

2

2 2 2 2 1

2 2 2MU WU s MS WU MU MS MS WSWU MS

s WS WU MS WS MU WSWS

E R R

R

τ γ μγ γ γ γ γ γ γ μ γ γ

τ γ γ γ μγ γ λγ γ

⎡ ⎤= − + + + + + −⎣ ⎦+ − + + +

Taking the first order condition and solving for τ, gives the equilibrium value of the

diversity premium:

( ) ( )

2 2 2 2*

2 22 2 2 2

22 8

MU WS WS WU WS WU

MU WU MS WS MU WS MU WU MS WS MU WS

R dγ γ γ γ γ γτγ γ γ γ γ γ γ γ γ γ γ γ

+ += −

⎡ ⎤ ⎡ ⎤+ + + +⎣ ⎦ ⎣ ⎦

(25a)

One finally obtains the equilibrium values of λ and μ by replacing τ* by its value in

the manager’s reaction functions:

( )( )

( )

23 3*

2 22 2 2 24WS WU MU WSMU WS

MU WU MS WS MU WS MU WU MS WS MU WS

Rd

γ γ γ γγ γλγ γ γ γ γ γ γ γ γ γ γ γ

+= −

⎡ ⎤ ⎡ ⎤+ + + +⎣ ⎦ ⎣ ⎦

(25b)

( )( )

( )( )( )

22 2*

2 22 2 2 24MU WS MU WU MS WS MU WU MS WS WU WS

MU WU MS WS MU WS MU WU MS WS MU WS

Rd

γ γ γ γ γ γ γ γ γ γ γ γμ

γ γ γ γ γ γ γ γ γ γ γ γ

+ + += −

⎡ ⎤ ⎡ ⎤+ + + +⎣ ⎦ ⎣ ⎦

(25c)

Proposition 7: When the shareholder pays a diversity premium:

i. the probability of choosing a worker from the discriminated group against a

worker of equal skills from the other group can be larger than zero and

smaller than one;

ii. the probability of choosing a skilled worker from the discriminated group

against an unskilled worker from the other group can be larger than zero and

smaller than one;

iii. the manager’s wage is an increasing function of the firm’s profits, an

increasing function of his/her taste for discrimination.

Proof: The probability of choosing a worker from the discriminated group against a worker of

equal skills is described in expression (25c). Its first term is unambiguously negative,

while its second term is positive and proportional to the inverse of d. As a result, μ*

decreases from one to zero as d increases from zero to infinity, which proves (i).

By the same token, one sees that the probability of choosing a skilled worker from the

discriminated group against an unskilled worker from the other group, which is given

by (25b) is decreasing in d, and can take any value between zero and one, which

proves (ii).

31

Finally, s is described by (25a), which shows that it is unambiguously increasing in

both d and Rs, which proves (iii).

The rationale of those results is the same as those of previous section. The manager

trades-off a bonus against the psychological cost of hiring discriminated workers. If his/her

taste for discrimination is large enough, it will not pay-off to shareholders to offer a premium

that eliminates discrimination. The innovation here is that the premium does not depend on

profits, but directly on discrimination. Those results show that even a specific diversity

premium may not be able to get rid of discrimination, when the manager’s taste for

discrimination is large enough.

A key difference between a diversity premium and a standard performance premium is

that the latter has no effect on discrimination of unskilled workers. Since hiring an unskilled

worker of the discriminated group never increases the firm’s profit, the manager has no

incentive to do so when his/her wage depends on profits. In contrast, when it depends on

diversity per se, it pays off for the manager to hire workers from the discriminated group,

even when this does not affect profits.

5. Conclusion 

In this paper, we set up a model where labor discrimination can be costly, but where

its cost is not borne by the agent who discriminates. Hiring decisions thus result in a

principal-agent problem between shareholders and managers. In that model, performance-

based contracts may fail to avoid costly discrimination, namely discrimination where

unskilled workers are hired instead of skilled workers of the discriminated group. Moreover,

such contracts have no impact on the kind of discrimination that is costless to shareholders,

namely discrimination of equally skilled workers. Profit maximization is therefore not

sufficient to preclude labor discrimination.

The model suggests avenues for empirical investigation. First, it predicts that

discrimination is lighter in owner-managed firms than in those where shareholders delegate

their decision power. Second, it implies that discrimination should be less prevalent in

industries where skills matter more, like the high-tech sector, and where skilled workers are

harder to find in the privileged group. Both implications are testable, and could be taken to the

data.

32

Moreover, since the model suggests that private incentives do not prevent all

discriminative practices, public anti-discrimination policies may be required. Our model

allows investigating how those policies may perform when shareholders’ and managers’ are

distinct individuals. This has, to our knowledge, been overlooked in the literature on

discrimination and affirmative action, which has so far stuck to the assumption that firms are

profit, or utility maximizers. Our model may be used to compare the merits of quotas and

other antidiscrimination measures in a more realistic framework.

Other extensions may also reveal fruitful. For instance, the number of applicants could

be taken as random, and more sophisticated hiring processes could be investigated, for

instance by including a human resources department and/or head hunters. The model could

also shed light on the impact of testing on discrimination that Autor and Scarborough (2008)

have observed. The labor market on which the firm operates could be modeled explicitly, so

as to endogenize the wage gap. The model could also be adapted to fit non-profit

organizations which may also be affected by principal-agent issues. This could then help

tackling the persistent gender bias observed in academia, notably among economists (see,

e.g., Ginther and Kahn, 2004).

Regardless of the extension considered, using a more realistic description of firm

behavior and organization is likely to provide new insights in the consequences of anti-

discrimination policies. Our model provides a framework to investigate those consequences.

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