Diversity in the Workplace

Felix J Vardy; John Morgan

doi:10.5089/9781451864977.001.A001

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Diversity in the Workplace

Author: Mr. Felix J Vardy and Mr. John Morgan

Contributor Notes

Authors’ E-Mail Addresses: morgan@haas.berkeley.edu and fvardy@imf.org

Publication Date:: 01 Oct 2006

eISBN:: 9781451864977

ISBN:: 9781451864977

Language:: English

Keywords:: WP; lower bound; cost of capital; unselective employer; optimal strategy; employer's belief; minority status; candidate population

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We study a model where an employer, trying to fill a vacancy, engages in optimal sequential search by drawing from two subpopulations of candidates who differ in their "discourse systems": during an interview, a minority candidate with a discourse system not shared with the employer conveys a noisier unbiased signal of ability than does a majority candidate. We show that, when the employer is "selective," minority candidates are underrepresented in the permanent workforce, fired at greater rates, and underrepresented among initial hires, even though the employer has no taste for discrimination and the populations are alike in their average ability. Furthermore, workplace diversity is increased if: (1) the cost of firing is reduced, (2) the cost of interviewing is increased, (3) the opportunity cost of leaving the position unfilled is increased, or (4) the prior probability that a candidate can perform the job is increased. Indeed, if the prior probability is sufficiently high, or the cost of firing sufficiently low, then minority candidates may be overrepresented in the permanent workforce.

Abstract

I. Introduction

A central social, political, and economic challenge confronting the European Union today arises from the tensions created by the growth of immigrant populations, particularly those from predominantly Muslim countries. These tensions have manifested themselves in sometimes dramatic fashion—the murder of Theo van Gogh and its aftermath in the Netherlands, the widespread unrest over Danish anti-Muslim cartoons, and the weeks-long violence and rioting in the outskirts of Paris in the Fall of 2005. Many have argued that these events are mere symptoms of a broad underlying discontent caused, in large part, by a lack of economic opportunities. Indeed, generally high unemployment in the EU, often attributed to labor market rigidities, affects immigrant populations particularly severely: unemployment rates for minorities remain stubbornly higher than for the majority, and grow especially severe during economic downturns.

What accounts for the disparity in the employment experiences of Europe’s majority populations versus its minority populations? Skill and age differences are surely part of the explanation. Minority populations are, on average, less educated and younger than the majority, and unemployment rates tend to be higher among the low-skilled and the young. Nevertheless, while the employment disadvantage of minorities is reduced once differences in educational attainment and age are taken into account, it does not disappear. (See, e.g., Tesser, Merens, and Van Praag, 1999; and Dagevos, 2006). Moreover, the disadvantage does not disappear over time either: in the Netherlands, even second-generation Muslims display considerably higher unemployment rates than their majority counterparts. In fact, controlling for education and age, the employment disadvantage of second-generation Muslims is even greater than that of the first generation (Dagevos, 2006).

Of course, it may be that employers simply have a taste for discrimination and that the underrepresentation of Muslims in the EU workforce reflects the strength of these tastes. While it is hard to rule out this explanation, one would expect that the cost of indulging in a taste for discrimination has been raised with the increased globalization of the EU economy. Thus, one would expect to see the unemployment gap between Muslims and the rest of the population shrink through competitive pressures, when, in fact, the opposite has occurred in the Netherlands over the past couple of years (Dagevos, 2006).

An alternative explanation for higher minority unemployment may be gleaned from the intercultural communication and sociolinguistics literatures. (See, for instance, Scollon and Scollon, 2001.) According to this hypothesis, minority job candidates struggle to make themselves understood due to differences in “discourse systems.” For example, a candidate’s behavior during a job interview may be quite revealing to an employer if they share the same social or cultural background. But if they do not, it can be much harder for the employer to form an accurate opinion about the applicant. In other words, the signals conveyed by minorities during interviews may be so garbled that they fail to convince (majority) employers of their qualities, even when they are perfectly competent and employers have no taste for discrimination. In contrast, by virtue of sharing the same discourse system as employers, majority job candidates do not face this problem. Thus, for them it tends to be easier to convey an accurate impression of their quality. As a consequence, minority populations find greater difficulty in securing employment than majority populations.²

This hypothesis raises several questions. Can differences in discourse systems alone explain differences in unemployment rates between majority and minority populations, absent any differences in underlying ability of the two populations? If so, what policy prescriptions could remedy this? Should employment protection be increased or decreased? What about other rigidities—are these helpful or harmful to workplace diversity? What about macro implications—can the EU simply grow itself out of the problem?

To examine these questions, we study a model in which an employer tries to fill a vacancy by sequentially interviewing job candidates from a pool of potential employees. The pool consists of two subpopulations. One subpopulation may be thought of as the majority population, the other as the minority population. The employer has no inherent taste for discrimination, and the only thing he cares about is whether a candidate can do the job. On average, candidates from both subpopulations are equally likely to be able to do the job. This means that there is no role for the standard type of statistical discrimination in our model. Candidates do, however, differ, in their discourse systems. The majority population has the same discourse system as the employer, while the minority has a different discourse system. To capture this difference, we suppose that when the employer interviews a minority candidate he receives a noisier signal of that candidate’s true ability than when he interviews a majority candidate.

Our main result shows that, when an employer is “selective,” equilibrium always entails underrepresentation of the minority population in the permanent workforce. Here, “selective” means that candidates are hired only when the post-interview probability that they can do the job exceeds the prior probability. More surprisingly, when an employer is sufficiently “unselective,” equilibrium entails overrepresentation of the minority population. Sufficiently “unselective” means that a candidate is hired provided he does not disappoint too much during the interview. Finally, regardless of the selectivity of the employer, the firing rate of minority candidates always exceeds that of majority candidates.

The intuition for the main result may be seen in the following example. Suppose that the prior probability that a random candidate can do the job is 50% and assume that the employer is very selective, such that only candidates about whom the employer is at least 95% certain after the interview that they can do the job are hired. Such a high threshold is optimal when firing costs are very high. In that case, the relative uninformativeness of a minority candidate’s signal about his qualifications makes it extremely hard to change the employer’s 50% prior belief of “success” to a posterior belief of at least 95%. Therefore, it is very unlikely that a minority candidate is going to fill the position. As a result, selective hiring practices lead to severe underrepresentation of minorities, even though minorities are as competent as the majority and employers are not prejudiced against them. On the other hand, if the employer is not selective at all, such that any candidate is hired provided that the posterior probability that he can do the job is no less than 5%, then the relative uninformativeness of a minority candidate’s signal about his qualifications is an advantage. It makes it virtually impossible for the employer’s 50% prior belief of success to be downgraded to less than 5%. Under these circumstances, virtually all minority candidates are given a chance and remain in the job if they turn out to be good. At the same time, in relative terms, many majority candidates are turned away at the gate, because the informativeness of their signals does make significant belief revisions possible. As we show, this leads to “reverse discrimination:” minorities will be overrepresented in the workforces of unselective employers. For similar reasons, the model also predicts that the degree of underrepresentation of minorities depends on the prior probability that random candidates can do the job. Specifically, minorities will be most severely underrepresented in positions that demand rare skills, such that the employer’s priors are very pessimistic. In contrast, minorities will be overrepresented in positions that nearly anyone can do.

Next, the model predicts that the relative representation of minorities in the workplace varies over the business cycle. Specifically, if employers are at all selective, diversity is predicted to be procyclical, increasing during economic upturns and decreasing during downturns. Intuitively, when the economy is booming, recruiting job candidates is more costly. At the same time, the opportunity cost of leaving the position unfilled is higher. Both effects make the employer less picky, encouraging employers to “take a chance” on job candidates whose quality is uncertain. This reduces the underrepresentation of minorities.

This prediction is roughly consistent with the Dutch experience over the past decade. During the second half of the 1990s, a period of rapid economic expansion, unemployment among Muslim minorities in the Netherlands fell quite spectacularly, from over 30% in 1995 to around 9% in 2001. During the same period, the unemployment rate among the nonimmigrant Dutch fell from around 6.5% to 3%. Since then, the trend has largely reversed. By 2005, unemployment among Muslims was again as high as 24%, while unemployment among the nonimmigrant Dutch had only risen to 5%. (Dagevos, 2006.)

Finally, we turn to policy solutions to the “diversity problem.” Our main finding in this regard is that high firing costs harm diversity. Intuitively, protections that raise the cost of firing lead the employer to guard more vigilantly against Type II errors (hiring of incompetent candidates). The employer achieves this by becoming more selective, which exacerbates the underrepresentation of minorities. This suggests that EU labor market rigidities such as high costs of firing contribute to the economic and social exclusion of Muslim minorities in Europe.

To conclude, the model implies that differences in discourse systems can indeed generate differences in unemployment across otherwise homogeneous populations. Going beyond the model, it suggests a feedback system between cultural and economic barriers to integration: the lack of a shared discourse system leads to few opportunities for minorities to land demanding jobs with selective employers. Instead, minorities are more likely to be unemployed, or stuck at the lower end of the labor market. This, in turn, implies that they are less likely to be in close contact with the dominant discourse system and, therefore, the cultural segregation across populations is self-reinforcing and may, in fact, harden over time, perhaps explaining the experience of second-generation Muslims in the Netherlands.

While the model presented in this paper is motivated by the plight of immigrant populations in the EU, it does seem to have wider applicability. For instance, in a U.S. context, the model may cast some light on the heated discussion about the lack of diversity among Supreme Court law clerks. When members of Congress asked why the justices did not cast their nets more widely to find more minority candidates, they responded that they could ill afford to take a chance that even one of their clerks might not be a top performer. (Peppers, 2006). This line of reasoning fits well with our model, in which high costs of making a mistake lead to severe underrepresentation of minorities.

From here, the paper proceeds as follows. We begin with an overview of the related literature in Section II. In Section III we develop the model. In Section IV the employer’s payoff-maximizing hiring strategy is derived. In Section V we study the consequences of optimal hiring for minority representation. Section VI discusses potential policy responses and Section VII concludes. We have relegated most proofs to the Appendix.

II. Related Literature

The nearest antecedent to the current paper is Cornell and Welch (1996). Cornell and Welch look at the probability that a minority candidate is hired when an employer chooses the best prospect from a fixed number of candidates. As in our paper, Cornell and Welch assume that the minority population is equally skilled as the majority population but that (majority) employers are better at assessing the quality of majority candidates than that of minority candidates. When the number of candidates is large, they show that the employer is overwhelmingly more likely to hire a majority candidate than a minority candidate. The intuition relies on an order-statistic argument: the higher accuracy of majority candidate evaluations makes the variance of their inferred qualities higher than the variance of the inferred quality of minority candidates. This makes it much more likely that outliers—in particular, the first-order statistic—come from a majority candidate than from a minority candidate. And it is the outlier who gets hired. In contrast, our results do not rely on order-statistic effects.

Our paper differs from Cornell and Welch in a number of ways. Most important, we employ a sequential search approach in the spirit of McCall (1970), as compared with Cornell and Welch’s fixed-sample-size-approach along the lines of Stigler (1961). Indeed, the difference in the two models is analogous to fixed sample versus sequential search in the pricing literature. For a summary of the differences in the predictions and optimality between the two see, e.g., Baye, Morgan, and Scholten (forthcoming).

Our optimal sequential search approach allows us to explicitly model and analyze the effects of what Cornell and Welch call “ex ante screening” versus “on-the-job performance measurement.” Also, we differentiate between skill levels and show that this distinction matters in an important way: while discrimination of minorities tends to be strong for jobs that require rare skills, it is much less so for common skill jobs. In fact, if firing costs are low, corresponding to cheap “on-the-job performance measurement,” minorities will be overrepresented in common skill positions.

In a broader context, the first to analyze discrimination from an economic perspective was Becker (1957). He studied the economic consequences of people’s intrinsic dislike of (interacting with) other races. See Arrow (1998) for a survey. In the literature, this kind of discrimination is known as taste- or preference-based discrimination. Somewhat related to our work are Black (1995), who examines this motive in a search-theoretic setting, and Rosen (1997), who combines search with a match-specific payoff.

Closer to our work is the statistical discrimination literature beginning with the seminal paper of Phelps (1972). In this literature, discrimination is information based. Majority and minority populations are assumed to differ statistically with respect to some relevant characteristic, such as average labor productivity. Because interviews and tests can only imperfectly predict the labor productivity of a particular job candidate, belonging to one subpopulation or another is statistically significant and taken into account by a potential employer. It is used as an imperfect proxy influencing the employer’s belief about a candidate’s expected ability, in addition to the information gathered through interviews and test scores. More recently, by endogenizing human capital acquisition, Coate and Loury (1993) as well as Lundberg and Startz (1998) have shown how statistical discrimination can arise even with ex ante homogeneous populations.

Aigner and Cain (1977) extended Phelp’s analysis by showing that lower wages can result not only from lower expected productivity, but also from higher variance in inferred productivity. Less accurate testing, or higher intrinsic quality variation, depresses wages for high-scoring minorities and boosts wages for low-scoring minorities. The intuition behind their result is similar to the intuition underlying ours: beliefs about the quality of minority candidates are less sensitive than those about majority candidates.

Finally, our work is also somewhat related to other “language” theories of discrimination, such as Lang (1986) or Athey, Avery, and Zemsky (2000). Tension in these models stems from interactions between workers, where workers who speak the same language are more productive. In our model, worker interactions do not play a role. Rather, we focus on problems of communication between an employer and potential job candidates.

III. Model

We study a labor market search problem in which the employer does the searching. In order to fill a vacancy, an employer takes random draws at a cost k > 0 per draw from a population of job candidates with the power of the continuum. Each draw can be thought of as the employer conducting a job interview with a candidate. Each candidate has two characteristics: what subpopulation he belongs to, which is observable to the employer at the time of the interview; and whether he can do the job, which only becomes observable if the candidate is actually hired. We shall refer to the former characteristic as a candidate’s kind and to the latter as a candidate’s type.

A candidate’s kind is denoted by κ ∈ {A, B}. A fraction m_A of the candidates is from subpopulation A; which consists of members of the “dominant” culture—i.e., candidates with the same discourse system as the employer/evaluator. The remaining fraction m_B = 1 − m_A of the candidates is from subpopulation B; which consists of members not belonging to the dominant culture. As shorthand for differences between the dominant and nondominant cultures, we shall sometimes refer to candidates of kind A as “majority” candidates and candidates of kind B as “minority” candidates—although, as the description above makes clear, majority candidates do not necessarily have to be more numerous than minority candidates.

A candidate’s type, denoted by π, equals 1 if he can do the job and equals zero if he cannot. Let p_κ denote the probability that a randomly drawn candidate of kind κ can do the job; that is, p_κ ≡ Pr (∏ = 1| κ). We assume that the two subpopulations are equally qualified to do the job; that is,

p_{A} = p_{B} = p .

Hence, none of the results in the paper are driven by differences between the type distributions in the subpopulations.

In advance of the interview, the employer does not know, or does not act upon, a candidate’s minority status.³ However, at the interview stage, a candidate’s kind —A or B— is perfectly revealed to the employer through some easily observable characteristic such as dialect or skin color. In addition, the interview also reveals to the employer a signal S_κ as to the competence of the candidate, where

S_{κ} = π + ε_{κ} .

That is, the signal is equal to the candidate’s type π plus an error term ε_κ, which is assumed to be Normally distributed with zero mean and variance $σ_{κ}^{2}$ .

The key difference between candidates of different kinds is that the employer finds it easier to assess the competence of candidates from the same culture compared with those from a different culture. To model this difference, we assume that σ_B > σ_A. That is, from the perspective of the employer, there is more noise in the signal of a minority candidate than in the signal of a majority candidate.

The timing of the employer’s decision problem is as follows. In period 1, the employer draws a random candidate and conducts an interview at a total cost k. On the basis of the candidate’s interview signal s, and taking into account his kind κ, the employer calculates the candidate’s “success probability” q. That is, q is the employer’s posterior belief about the probability that the candidate can do the job. Given q, the employer then decides whether to hire the candidate, and period 1 ends.

In period 2 and all subsequent periods, if the employer did not hire in the previous period, he interviews a new candidate and the game proceeds as before. If, however, the employer did hire in the previous period, the employee’s type π is perfectly revealed to the employer. If the employee can do the job—i.e., π = 1—he is retained forever, and all search ceases. In that case, the employer enjoys a payoff with a net present value of υ > 0. If, however, the employee cannot do the job—i.e., π = 0—then by retaining the employee the employer earns a payoff with a net present value of −w < 0. Alternatively, the employer can fire the employee in period 2 and incur a cost of c > 0: Throughout, we assume that c < w; hence it is optimal to fire incompetent employees. Finally, we assume that the employer has a discount factor δ ∈ (0, 1) between periods.

Posterior Beliefs

As we shall see, the employer’s optimal strategy is to impose a success probability threshold, ${\underline{q}}^{*}$ , when deciding whether to hire a candidate. That is, a candidate is hired if and only if the probability that he can do the job is at least ${\underline{q}}^{*}$ . The optimal threshold turns out to be the same for both kinds of candidates. It depends on the posterior distribution of the employer’s beliefs as to the competence of a candidate. Thus, it is useful to summarize key features of this posterior distribution.

Define q_κ (s) to be the employer’s posterior belief that a candidate of kind κ with signal s can do the job. Formally,

q_{κ} (s) \equiv P (\prod = 1 | S_{κ} = s) .

By Bayes’ rule, we can rewrite this expression as

q_{κ} (s) = \frac{ϕ (\frac{s - 1}{σ_{κ}}) p}{ϕ (\frac{s - 1}{σ_{κ}}) p + ϕ (\frac{s}{σ_{κ}}) (1 - p)},

where ϕ (·) denotes the density of a standard Normal random variable.

It will sometimes be useful to determine the signal realization s corresponding to a given success probability q, which we shall denote by s_κ (q). Since q_κ (s) is a monotone function, it is invertible in the extended reals and s_κ (q) is well-defined. Using that $ϕ (t) \equiv \frac{1}{\sqrt{2 π}} \exp [- \frac{1}{2} t^{2}]$ , it may be readily shown that

s_{κ} (q) = \frac{1}{2} - σ_{κ}^{2} \ln (\frac{1 - q}{q} \frac{p}{1 - p}) .

Prior to the realization of the signal but after having observed a candidate’s kind, the success probability Q_κ = q_κ (S_κ) is a random variable. Now, let G_κ (·) denote the cumulative distribution function (cdf) of Q_κ. Formally,

\begin{array}{l} G_{κ} (q) = & Pr (Q_{κ} \leq q) \\ = & p Φ (\frac{s_{κ} (q) - 1}{σ_{κ}}) + (1 - p) Φ (\frac{s_{κ} (q)}{σ_{κ}}), \end{array}

where Φ(·) denotes the cdf of a standard Normal distribution. The associated density of G_κ (q) is

g_{κ} (q) = (p ϕ (\frac{s_{κ} (q) - 1}{σ_{κ}}) + (1 - p) ϕ (\frac{s_{κ} (q)}{σ_{κ}})) \frac{σ_{κ}}{q (1 - q)} .

Similarly, let G (·) denote the cdf of success probability Q prior to observing the candidate’s kind or signal, and g (·) denote the associated density. Formally,

G (q) = (1 - m_{B}) G_{A} (q) + m_{B} G_{B} (q) .

Finally, it is useful to establish the following stochastic dominance relations for G (·) and G_κ (·).

Lemma 1

For all p > p’, G (·; p) first-order stochastically dominates G (·; p’). That is,

\frac{d}{dp} G (q) < 0, for all q \in (0, 1) .

Lemma 2

G_A (·) is a mean-preserving spread of G_B (·). And, for all m_B < m‘_B, G (·; m’_B) is a mean-preserving spread of G (·; m‘_B).

IV. Optimal Search and Hiring

In this section, we show that there exists a unique solution to the employer’s optimization problem. The optimal hiring strategy is to set an identical success probability threshold, ${\underline{q}}^{*}$ , for all candidates, irrespective of their kind. That is, after observing a signal s from a candidate of kind κ, the candidate is hired if and only if the posterior probability that he can do the job, q_κ (s), is at least ${\underline{q}}^{*}$ .

To see this, let V* denote the employer’s expected payoff if he follows an optimal search and hiring strategy. In any optimal strategy, the employer hires a candidate if and only if his belief q that the candidate can do the job is such that the payoff from hiring, which we denote by H (q, V*), exceeds the payoff from not hiring and moving to the next period.

Hence, we may write the value function as

\begin{matrix} V^{*} = δ \int_{0}^{1} \max [H (q, V^{*}), V^{*}] dG (q) - k, & (1) \end{matrix}

where

H (q, V^{*}) = q υ + (1 - q) (- c + V^{*}) .

Note that, according to our timing convention, cost k is incurred immediately, while the payoff from hiring, H (q, V*), is received in the next period.

The following observation is crucial.

Lemma 3

Under the standard Euclidean metric, equation (1) specifies a contraction mapping T: ℝ → ℝ in V*.

Lemma 3, together with the well-known Contraction Mapping Theorem (see, for example, Stokey, Lucas, and Prescott, 1989), implies that there exists a unique, optimal value V* for the employer’s problem.

Since the employer’s problem is stationary, any strategy attaining V* must be a threshold strategy (see, for example, McCall, 1970). Moreover, the threshold must be the same for both kinds of candidates. The reason is that, conditional on q, a candidate’s kind κ is completely irrelevant: the only thing that matters is the probability of success itself, and not whether the candidate is mainstream or minority.

Finally, it remains to show that the threshold strategy attaining V* is unique. Under a generic threshold strategy, which we denote by $\underline{q}$ , the value function given in equation (1) reduces to

V (\underline{q}) = δ [G (\underline{q}) V (\underline{q}) + \int_{\underline{q}}^{1} H (q, V (\underline{q})) dG (q)] - k .

Substituting for H and solving for $V (\underline{q})$ , we obtain

V (\underline{q}) = \frac{δ \int_{\underline{q}}^{1} (q υ + (1 - q) (- c)) dG (q) - k}{1 - δ (1 - \int_{\underline{q}}^{1} qdG (q))} .

Thus, the employer’s problem reduces to choosing $\underline{q}$ to maximize $V (\underline{q})$ . Proposition 1 characterizes the unique optimum.

Proposition 1

The optimal threshold strategy, ${\underline{q}}^{*}$ , is the unique interior solution to

\begin{matrix} {\underline{q}}^{*} = \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ + k} . & (2) \end{matrix}

The next proposition shows that every possible threshold success probability can be an optimum.

Proposition 2

For all q ∈ [0, 1), there exist parameter values such that ${\underline{q}}^{*} = q$ .

V. Performance Metrics

Recall that the optimal hiring strategy established in Proposition 1 is “color-blind” in the sense that the employer sets the same threshold success probability for both kinds of candidates, and that the optimal hiring threshold can be at any level. In this section, we study the implications of a uniform hiring threshold for observable performance metrics of diversity.

Permanent Workforce Composition

Perhaps the most important performance metric of diversity is the fraction of minorities in the permanent workforce of an organization, relative to their share in the underlying population. In terms of our model, this corresponds to the probability that a permanently hired candidate is a minority.

Formally, let r_κ denote the probability that the vacancy is permanently filled by a candidate of kind κ, when the employer uses the, not necessarily optimal, threshold strategy $\underline{q}$ . Then, r_κ can be expressed recursively as follows.

\begin{matrix} r_{κ} = m_{κ} (p ((1 - G_{κ} (\underline{q} | \prod = 1)) + G_{κ} (\underline{q} | \prod = 1) r_{κ}) + (1 - p) r_{κ}) \\ + (1 - m_{κ}) ((1 - p (1 - G_{- κ} (\underline{q} | \prod_{- κ} = 1))) r_{κ}) . \end{matrix}

We can write this expression much more compactly if we define G_κπ to be the probability that a candidate of kind κ and type π induces a posterior success probability less than or equal to $\underline{q}$ . Formally,

G_{κ π} \equiv G_{κ} (\underline{q} | \prod = π) .

Solving for r_κ, we obtain, in our more economical notation,

r_{κ} = \frac{m_{κ} (1 - G_{κ 1})}{1 - m_{κ} G_{κ 1} - (1 - m_{κ}) G_{- κ 1}} .

We want to compare minority representation in the workplace, r_B, with the minority share of the underlying population, m_B. Minorities are proportionally represented in the workplace when $\frac{r_{κ}}{m_{κ}} = 1$ . It is easily verified that this is equivalent to the condition that G_A1 = G_B1. In other words, minorities are proportionally represented if and only if the probability of Type I error (rejection of competent candidates) is the same for both kinds of candidates.

When does equality of Type I error hold?

Lemma 4

There exists a unique threshold, ${\underline{q}}^{1} \equiv \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{{2 σ}_{A} σ_{B}}}} < p$ , where the probability of Type I errors is the same for both kinds of candidates.

Unsurprisingly, the optimal threshold ${\underline{q}}^{*}$ given in Proposition 1 is generically not equal to ${\underline{q}}^{1}$ . The next proposition shows that, depending on the relationship between ${\underline{q}}^{*}$ and ${\underline{q}}^{1}$ , minorities may be under or overrepresented in the workplace.

Proposition 3

Minorities are overrepresented in the workplace (i.e., $\frac{r_{B}}{m_{B}} > 1$ ) if and only if $0 < {\underline{q}}^{*} < {\underline{q}}^{1}$ .
Minorities are underrepresented (i.e., $\frac{r_{B}}{m_{B}} < 1$ ) if and only if ${\underline{q}}^{1} < {\underline{q}}^{*} < 1$ .
Minorities are proportionately represented (i.e., $\frac{r_{B}}{m_{B}} = 1$ ) if and only if ${\underline{q}}^{*} \in {0, {\underline{q}}^{1}}$ .

If ${\underline{q}}^{1} < {\underline{q}}^{*}$ , then minority candidates are more subject to Type I error than majority candidates. That is, competent minority candidates are rejected at a higher rate than competent majority candidates. This results in underrepresentation of minorities in the workplace relative to their share in the underlying population. On the other hand, if ${\underline{q}}^{1} > {\underline{q}}^{*}$ , then it is the majority candidates who are more subject to Type I error. This results in minority candidates being overrepresented in the workplace. Hence, the outcome depends on how “choosy” the employer is.

The following figure illustrates how the difference in Type I errors for majority and minority candidates varies with the threshold strategy of the employer. It displays the ratio $\frac{1 - G_{A 1}}{1 - G_{B 1}}$ of hiring probabilities for competent majority versus competent minority candidates as a function of the employer’s threshold strategy $\underline{q}$ . The parameter values used to draw the figure are: p = .3, σ_A = 1, $σ_{B} = \sqrt{2}$ . Notice that at low thresholds $(\underline{q} < {\underline{q}}^{1} < p)$ minority candidates are overrepresented in the workforce, and this disparity grows as the threshold increases from $\underline{q} = 0$ . Since the workforce proportions exactly reflect those of the candidate population at $\underline{q} = {\underline{q}}^{1}$ , minority overrepresentation must reverse itself for a sufficiently choosy employer. In the figure, the degree of minority overrepresentation is greatest at $\underline{q} = 0.18$ and declines thereafter. For thresholds $\underline{q} > {\underline{q}}^{1}$ , the effect of the difference in Type I errors can be quite severe for competent minority candidates. By the time the threshold reaches 0.7, a competent majority candidate stands an almost 140 times better chance of being hired than a competent minority candidate. Indeed, as the figure shows, the ratio of hiring probabilities increases without bound as the threshold approaches 1.

The figure illustrates that it becomes exceedingly unlikely that a minority candidate will fill the position as the threshold increases. Put differently, the workplace composition becomes increasingly homogeneous. As we show in the next proposition, the positive relationship between the choosiness of an employer and the homogeneity of the workplace is a general property of the model.

Proposition 4

Suppose that the employer is “selective” in its hiring policy, i.e., $\underline{q} > p$ , then:

As the employer becomes more selective, minority representation in the workplace decreases. Formally, r_B is decreasing in $\underline{q}$ .
As the employer becomes arbitrarily selective, minorities vanish from the workplace. Formally, $\lim_{\underline{q} \to 1} r_{B} = 0$ .

One may wonder what conditions on primitives guarantee that an employer will indeed be selective in the sense described in Proposition 4. A useful lower bound on the optimal threshold may be derived from the case of a “myopic” employer who only derives benefit one period into the future. Such an employer would choose a “break-even” threshold where υq −(1 − c) q = 0 or, equivalently, $q = \frac{c}{c + υ}$ . Employers who value payoffs in periods beyond the next will optimally raise the threshold above the break-even level to capture some of the option value of waiting. Hence, ${\underline{q}}^{*} > \frac{c}{c + υ}$ . As a result, a sufficient condition for an employer to be selective is that $p < \frac{c}{c + υ}$ .

Initial Hiring Rates

We have shown that differences in Type I errors can lead to gross differences between the share of minorities in the permanent workforce compared with their share of the candidate population. Given the “color blind” threshold strategy of the employer, one might speculate that the fraction of minorities among initial hires would reflect the underlying population. As we shall see, this is not typically the case. Define the fraction of initial hires who are of kind κ as

h_{κ} = \frac{m_{κ} (1 - G_{κ})}{m_{κ} (1 - G_{κ}) + m_{- κ} (1 - G_{- κ})} .

Notice that the probability that a candidate of kind κ will be hired, 1 − G_κ, consists of the probability of two separate events: (i) the joint event that the candidate is competent and passes the interview; and (ii) the joint event that the candidate is incompetent and passes the interview. Event ii is equivalent to the probability of Type II error.

Having previously established a threshold, ${\underline{q}}^{1}$ , where Type I error is equalized across the two kinds of candidates, it is useful to determine the analogous threshold where Type II error is equalized. That is, define ${\underline{q}}^{0}$ to be the threshold such that

G_{A 0} = G_{B 0},

which has as its solution

{\underline{q}}^{0} = \frac{1}{1 + \frac{1 - p}{p} e^{- \frac{1}{{2 σ}_{A} σ_{B}}}} > p .

When $\underline{q} < {\underline{q}}^{0}$ , notice that incompetent minority candidates have a greater chance of being hired than incompetent majority candidates, while for $\underline{q} > {\underline{q}}^{0}$ the opposite holds. Furthermore, notice that the threshold at which Type II error is equalized always lies above that where Type I error is equalized. That is, ${\underline{q}}^{1} < {\underline{q}}^{0}$ .

Finally, we turn our attention to the threshold, ${\underline{q}}^{θ}$ , where the initial hiring proportions are equal to the underlying population proportions. That is, ${\underline{q}}^{θ}$ solves

G_{A} = G_{B} .

Unlike for the thresholds for equal Type I and Type II errors, there exists no closed-form solution for ${\underline{q}}^{θ}$ . However, from the fact that G_A is a mean-preserving spread of G_B (Lemma 2), it follows that ${\underline{q}}^{θ}$ exists and is unique. Moreover, since ${\underline{q}}^{θ}$ represents a trade-off between Type I and Type II errors, ${\underline{q}}^{1} < {\underline{q}}^{θ} < {\underline{q}}^{0}$ .

As was the case for the composition of the permanent workforce, depending on the optimal threshold ${\underline{q}}^{*}$ , minorities may be under- or overrepresented among initial hires. Using arguments identical to those in Proposition 3, it may be readily shown that

Proposition 5

Minorities are overrepresented among initial hires (i.e., $\frac{h_{B}}{m_{B}} > 1$ ) if and only if $0 < {\underline{q}}^{*} < {\underline{q}}^{θ}$ .
Minorities are underrepresented (i.e., $\frac{h_{B}}{m_{B}} < 1$ ) if and only if ${\underline{q}}^{θ} < {\underline{q}}^{*} < 1$ .
Minorities are proportionately represented (i.e., $\frac{h_{B}}{m_{B}} = 1$ ) if and only if ${\underline{q}}^{*} \in {0, {\underline{q}}^{θ}}$ .

It is interesting to note that, since ${\underline{q}}^{1} < {\underline{q}}^{θ}$ , it may well be that an employer’s optimal policy leads to favorable initial hiring rates for minorities, while their greater firing rates lead to underrepresentation in the permanent workforce. We now turn to formally analyzing firing rates.

Firing Rates

We saw that minority over- or underrepresentation among initial hires and in the permanent workforce depends on the threshold strategy of the employer. In the case of firing rates, by contrast, the model delivers unambiguous predictions. The main result of this section is that minority hires are fired at higher rates than majority hires for all (interior) threshold strategies $\underline{q} \in (0, 1)$ .

The firing rate for hires of kind is equal to the probability that a candidate of kind κ is incompetent conditional on his being hired in the first place. Formally, define the firing rate as

\begin{matrix} f_{κ} & = Pr (\prod = 0 | Q_{κ} \geq \underline{q}) \\ = \frac{(1 - G_{κ 0}) (1 - p)}{1 - G_{κ}} . \end{matrix}

To see how firing rates reflect the trade-off between Type I and Type II errors, it is helpful to write f_κ as follows

f_{κ} = \frac{(1 - p) Pr (Type II)}{(1 - p) Pr (Type II) + p (1 - Pr (Type I))} .

When ${\underline{q}}^{1} \leq \underline{q} \leq {\underline{q}}^{0}$ , minorities suffer greater Type I and Type II errors than do majorities. As a consequence, the firing rate of minorities is higher than for majorities.

When $\underline{q} < {\underline{q}}^{1}$ , minorities continue to experience greater Type II error; however, Type I error is now higher for majorities than for minorities. As a consequence, the ordering of majority and minority firing rates becomes ambiguous and depends on the relative magnitude of the two types of errors. Similarly, when $\underline{q} > {\underline{q}}^{0}$ , Type II error is smaller for minorities than for majorities but Type I error is greater. Hence, also in this case, the ordering could go either way. As the next proposition shows, however, the trade-off between Type I and Type II errors is always resolved in the direction of higher firing rates for minorities.⁴

Proposition 6

For all $\underline{q} \in (0, 1)$ , minority hires are fired at a higher rate than majority hires.

Summary

The following figure summarizes the various performance metrics of diversity as a function of the success probability threshold $\underline{q}$ .

VI. Policy Implications

In this section, we examine how the optimal threshold—and, by implication, the diversity metrics described above—varies with changes in the parameters of the model. Some of these parameters are likely to be under policy control; hence, there is the possibility of influencing workplace diversity. Throughout this section, we shall use the term “workplace diversity” as being synonymous with the minority representation ratio $\frac{r_{B}}{m_{B}}$ . The closer this ratio is to unity, the more diverse is the workplace.

Diversity and Worker Protections

There has been considerably debate, especially in Europe, over the appropriate level of worker protections against summary dismissal. The mass street protests in France during the Spring of 2006 against the contrat première embauche are a salient example. This new law would have allowed for summary dismissal of employees below the age of 26 during the first two years of their contract. By reducing the risk of hiring, it was hoped that the contrat première embauche would lead to a reduction in the very high youth unemployment. Whether it would have achieved its goal shall remain unknown, since the law was retracted in response to the protests.

In many U.S. organizations, there are various restrictions in interviewing practices to ensure a “level playing field” between majority and minority candidates. For example, for NFL head coaching vacancies, the league rule is that a minimum number of minority candidates must be interviewed before filling the position (NFL, 2003). Similarly, the University of California has many rules and restrictions governing interviewing practices to ensure fairness. For example, in filling a position at the University of California, the interviewer is obligated to fill out forms for each of the interviewed individuals stating the precise reasons that they were not selected for the position. (See, for example, Search Activity Statement UCI-AP-80, available at http://www.ap.uci.edu/Forms/APforms/UCI-AP-80.pdf.)

In terms of our model, EU worker protection policies may be thought of as increasing the cost of firing, c, while the University of California interviewing practices may be thought of as increasing the cost per interview, k. Obviously, increases in both c and k raise the “frictions” associated with the hiring process, yet, as we shall see, they have opposite implications for diversity.

Implication 1

Suppose that the employer is “selective” in its hiring policy—i.e., ${\underline{q}}^{*} > p$ —then:

An increase in the cost of firing, c, reduces workplace diversity.
An increase in the cost of interviewing, k, increases workplace diversity.

Intuitively, raising the cost of firing increases the cost of Type II errors for the employer. As a result, he becomes more reluctant to take a chance on whether a candidate can do the job and, consequently, raises the threshold for hiring. As we have shown in the previous section, when the employer is at all selective, increased hiring thresholds have the effect of differentially raising Type I errors to the disadvantage of minorities. As a result, workplace diversity decreases.

In contrast, raising the cost of interviewing makes it more expensive for the employer to be choosy. As a result, the employer lowers his threshold for hiring and this, in turn, reduces the difference in Type I errors between minorities and majorities. As a result, workplace diversity increases.

Diversity over the Business Cycle

Next, we consider how the employer’s optimal threshold varies with the business cycle. At a peak in the business cycle, job candidates become more scarce and, hence, the cost of recruiting increases. As we have shown above, this has the effect of raising workplace diversity. In addition, the value-added of a competent employee is also likely to be higher at the peak of the business cycle than during a recession. In terms of our model, this corresponds to an increase in υ.

Implication 2

Suppose that the employer is “selective” in its hiring policy— i.e., ${\underline{q}}^{*} > p$ —then diversity is procyclical. Formally, ${\underline{q}}^{*}$ is decreasing in υ (and k).

Intuitively, as a competent employee’s value-added increases, it becomes more costly to leave the position unfilled. As a consequence, the employer is more willing to take a chance by hiring possibly incompetent employees and, hence, the optimal threshold falls. A lower threshold reduces the difference in Type I errors between minorities and majorities. Consequently, workplace diversity increases. As mentioned in the introduction, the procyclicality of diversity is indeed consistent with the Dutch experience over the last decade.

Diversity and the Cost of Capitals

Another testable implication of the model is that variation in the riskiness of firms leads to differences in workplace diversity. If we interpret the discount parameter δ as representing an employer’s cost of capital, which presumably varies with the riskiness of his business, then we have the following implication:

Implication 3

Suppose that the employer is “selective” in its hiring policy, i.e., ${\underline{q}}^{*} > p$ , then riskier firms are more diverse. Formally, ${\underline{q}}^{*}$ is increasing in δ.

Intuitively, the option value of waiting is worth less for risky firms than for safe firms. Since the degree to which the optimal threshold lies above the break-even threshold positively depends on this option value, the optimal threshold for a riskier firm is lower than that for a less risky firm. In turn, this lower threshold reduces the difference in Type I errors between minorities and majorities, and, consequently, workplace diversity increases. As mentioned in the Introduction, the procyclicality of diversity is indeed consistent with the Dutch experience over the past decade.

Diversity and the Scarcity of Competence

As we highlighted above, the key determinant of minority over- or underrepresentation is the relationship between the optimal threshold ${\underline{q}}^{*}$ and the thresholds for equating Type I and Type II errors across the two populations— ${\underline{q}}^{1}$ and ${\underline{q}}^{0}$ , respectively. These two thresholds bracket the prior probability that a candidate is competent; that is, ${\underline{q}}^{1} < p < {\underline{q}}^{0}$ ; moreover, the thresholds depend on p. Thus, a question that naturally arises is how under- or overrepresentation varies with the underlying probability that a candidate can do the job.

When few candidates can do the job—i.e., when p is low—the results of the interview must be sufficiently convincing to induce the employer to take a chance on the candidate given the costs of firing. A candidate with a very noisy signal is going to have a difficult time in making this case. In the limit, imagine a situation where B candidates have arbitrarily noisy signals and where the employer is selective. Clearly, there is virtually no possibility of overcoming the employer’s prior belief about the low likelihood that the candidate is qualified. In contrast, a candidate with a very precise signal faces no such handicap. In this extreme case, one would expect (and the model predicts) severe underrepresentation of minority candidates both at the hiring stage and in the permanent workforce.

By contrast, when most candidates can do the job, i.e., when p is high, an imprecise signal in the interview stage can be an advantage for a candidate. Suppose that p is sufficiently high such that the employer is predisposed to give most candidates a chance to prove themselves on the job. In that case, having an arbitrarily noisy signal virtually guarantees that the candidate will not greatly disappoint in the interview and, hence, will be offered the position. In contrast, a more precise signal exposes the candidate to a greater possibility of making a bad impression in the interview and hence being rejected for the job—even in the case where the candidate is in fact competent. In this situation, overrepresentation of minority candidates, both in hiring and in the permanent workforce, is the more likely outcome. The next implication formalizes this intuition.

Implication 4

In jobs that are sufficiently selective, minorities will be underrepresented. In jobs that are sufficiently nonselective, minorities will be overrepresented. Formally, there exists 0 < p₀ < p₁ < 1 such that, for all p ∈ (0, p₀), $\frac{r_{B}}{m_{B}} < 1$ while for all p ∈ (p₁, 1), $\frac{r_{B}}{m_{B}} > 1$ .

VII. Conclusions

In this paper we have investigated the implications of assuming that employers finds it easier to evaluate majority job candidates, with whom they tend to share a cultural and social background, than minority job candidates, whose background is quite different from the employers’. In terms of sociolinguistics, employers and minority job candidates fail to share a discourse system enabling clear communication.

We have shown that this basic premise implies that there exists a tension between job security, scarcity of skills, and workplace diversity. When job security is high—i.e., firing nonperforming staff is expensive—minorities are likely to be severely underrepresented in selective positions. At the other extreme the converse holds. When job security is low, minorities are overrepresented in nonselective positions. These distortions occur even though majority and minority populations have identical skill levels.

On a fundamental level, our results are driven by Bayes’ law, which implies that employers’ posterior beliefs about majority candidates respond more strongly to new information than their beliefs about minority candidates. When the information received is better than expected, this high belief-sensitivity works to the advantage of majority candidates. On the other hand, when the information is worse than expected, high belief-sensitivity works to the disadvantage of majority candidates.

While the occurrence of “reverse discrimination” may be interesting from a theoretical perspective, from a policy perspective, the underrepresentation of minorities in selective positions seems the more relevant model prediction. Given that minorities are indeed grossly underrepresented in many organizations, what can be done about it?

In our model, the lack of workplace diversity arises because of a postulated information or communication mismatch between the majority employer/interviewer and minority job candidates. Obviously, matching the background of the interviewer with the background of the candidate would solve this problem. However, more often than not, this may be quite difficult to implement. First, in organizations lacking diversity, minorities are scarce to begin with. Second, it should not be forgotten that the various minorities are culturally highly diverse, thus requiring a very careful matching between the evaluator and the evaluee. For instance, while a French-speaking West African and an African-American are both people of color, it seems quite clear that they do not share the same discourse system.

A second, and probably more realistic, option to increase workplace diversity is to lower firing costs. We have shown that high costs of firing induce employers to impose extreme threshold success probabilities. The relatively low informativeness of minority candidates’ signals makes it virtually impossible to pass such high thresholds, irrespective of their skills. This is especially true for very selective jobs, where employers’ prior beliefs that a random candidate can do the job are very low. When firing costs are reduced, threshold success probabilities come down to more realistic levels. This lessens the disadvantage of minority candidates, levels the playing field, and leads to a more diverse workplace. Finally, policies that increase the cost of interviewing (and hence optimally reduce selectivity by employers) are also diversity enhancing.⁵

This discussion would not be complete without pointing out the limitations of the model. From a technical standpoint, one limitation is the one-sided search, or partial equilibrium nature of the analysis. It might be worthwhile extending the model to a general equilibrium framework in which candidates choose what kind of positions they apply to. Also, the binary nature of competence in our model—candidates either can do the job or they cannot—is clearly restrictive. Other limitations are of a less technical nature, such as the assumptions of equal average skill levels, identical firing costs across subpopulations, and no “naked” racism and no “directed search” on the part of the employers. Also, we have assumed that employers only care about technical competence, and not about how a candidate “fits” into the dominant culture of the organization. Some or even all of these assumptions do not hold in practice; however, most realistic deviations point in the same direction: towards more rather than less discrimination than predicted by the model. As such, the model puts a lower bound on the problem and shows that, even under the best of circumstances, competent minority candidates are likely to have a much harder time securing a coveted job than equally competent majority candidates, in particular when job security is high.

APPENDIX

Appendix: Proofs of Lemmas, Propositions and Implications

Proofs of Lemmas

Lemma 1

For all p > p’, G (·; p) first-order stochastically dominates G (·; p’). That is,

\frac{d}{dp} G (q) < 0, for all q \in (0, 1) .

Proof. Recall that

G (q) = (1 - m_{B}) G_{A} (q) + m_{B} G_{B} (q),

where

G_{κ} (q) = p Φ (\frac{s_{κ} (q) - 1}{σ_{κ}}) + (1 - p) Φ (\frac{s_{κ} (q)}{σ_{κ}}),

κ = A, B.

\begin{array}{l} Now, \frac{d}{dp} G_{κ} (q) = \\ = Φ (\frac{s_{κ} (q) - 1}{σ_{κ}}) - Φ (\frac{s_{κ} (q)}{σ_{κ}}) + (p ϕ (\frac{s_{κ} (q) - 1}{σ_{κ}}) + (1 - p) ϕ (\frac{s_{κ} (q)}{σ_{κ}})) \frac{\partial s_{κ} (q)}{\partial p} < 0, \\ because \frac{\partial s_{κ} (q_{κ})}{\partial p} = - \frac{σ_{κ}^{2}}{p (1 - p)} < 0 and Φ (\frac{s_{κ} (q) - 1}{σ_{κ}}) < Φ (\frac{s_{κ} (q)}{σ_{κ}}) . \end{array}

Since G (q) is a convex combination of G_A (q) and G_B (q), it follows that $\frac{d}{dp} G (q) < 0$ for all q ∈ (0, 1). This proves the lemma. ■

Lemma 2

G_A (·) is a mean-preserving spread of G_B (·). And, for all m_B < m‘_B, G (·; m_B) is a mean-preserving spread of G (·; m‘_B).

Proof. First, we verify that $E_{G_{A}} [Q_{A}] = E_{G_{B}} [Q_{B}] = p$ .

By definition,

E_{G_{κ}} [Q_{κ}] = \int_{0}^{1} {qg}_{κ} (q) dq,

where κ ∈ {A, B}. Changing the integration variable from probability q to signal s, we get

E_{G_{κ}} [Q_{κ}] = \int_{- \infty}^{\infty} q_{κ} (s) g_{κ} (s) \frac{{dq}_{κ} (s)}{ds} ds,

where $q_{κ} (s) = \frac{p ϕ (\frac{s - 1}{σ_{κ}})}{p ϕ (\frac{s - 1}{σ_{κ}}) + (1 - p) ϕ (\frac{s}{σ_{κ}})}$ , $\frac{\partial q_{κ} (s)}{\partial s} = \frac{p (1 - p)}{σ_{κ}} \frac{ϕ (\frac{s}{σ_{κ}}) ϕ (\frac{s - 1}{σ_{κ}})}{{(p ϕ (\frac{s - 1}{σ_{κ}}) + (1 - p) ϕ (\frac{s}{σ_{κ}}))}^{2}}$ and $g_{κ} (s) = (p ϕ (\frac{s - 1}{σ_{κ}}) + (1 - p) ϕ (\frac{s}{σ_{κ}})) \frac{σ_{κ}}{q_{κ} (s) (1 - q_{κ} (s))}$ . Hence,

\begin{matrix} E_{G_{κ}} [Q_{κ}] & = \int_{- \infty}^{\infty} q_{κ} (s) g_{κ} (s) \frac{{dq}_{κ} (s)}{ds} ds \\ = p \int_{- \infty}^{\infty} ϕ (\frac{s - 1}{σ_{κ}}) ds \\ = p . \end{matrix}

This proves that $E_{G_{A}} [Q_{A}] = E_{G_{B}} [Q_{B}] = p$ . For later use, note that $E_{G (\cdot; m_{B})} [Q] = E_{G (\cdot; m'_{B})} [Q] = p$ .

To prove that G_A (·) is a mean-preserving spread of G_B (·) it now suffices to show that, on the interval (0, 1), G_B (·) crosses G_A (·) only once and from below. We do this by establishing that the difference D (q) ≡ G_A (q) − G_B (q) has two extrema: starting from zero at q = 0, D (q) first reaching a maximum—at which D (q) is strictly positive—and then a minimum—at which D (q) is strictly negative.

Let

ζ = \ln (\frac{1 - q}{q} \frac{p}{1 - p})

such that

\begin{array}{l} D = G_{A} (q) - G_{B} (q) \\ = p Φ (\frac{- \frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}}) + (1 - p) Φ (\frac{\frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}}) \\ - p Φ (\frac{- \frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}}) - (1 - p) Φ (\frac{\frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}}) . \end{array}

Relying on the fact that ζ is a monotone function of q, we now ask when $\frac{dD}{d ζ} = 0$ :

\begin{array}{l} \frac{dD}{d ζ} = - σ_{A} p ϕ (\frac{- \frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}}) - σ_{A} (1 - p) ϕ (\frac{\frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}}) \\ + σ_{B} p ϕ (\frac{- \frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}}) + σ_{B} (1 - p) ϕ (\frac{\frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}}) \\ = 0 \end{array}

⇔

\begin{array}{l} \frac{σ_{A}}{σ_{B}} = \frac{ϕ (\frac{- \frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}}) + \frac{1 - p}{p} ϕ (\frac{\frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}})}{ϕ (\frac{- \frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}}) + \frac{1 - p}{p} ϕ (\frac{\frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}})} \\ = \frac{e^{- \frac{1}{2} {(\frac{- \frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}})}^{2}} + \frac{1 - p}{p} e^{- \frac{1}{2} {(\frac{\frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}})}^{2}}}{e^{- \frac{1}{2} {(\frac{- \frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}})}^{2}} + \frac{1 - p}{p} e^{- \frac{1}{2} {(\frac{\frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}})}^{2}}} . \end{array}

Now consider the right-hand side, which we denote by Ψ, as a function of ζ.

\begin{array}{l} Ψ \equiv \frac{e^{- \frac{1}{2} {(\frac{- \frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}})}^{2}} + \frac{1 - p}{p} e^{- \frac{1}{2} {(\frac{\frac{1}{2} - σ_{B}^{2} ζ}{σ_{B}})}^{2}}}{e^{- \frac{1}{2} {(\frac{- \frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}})}^{2}} + \frac{1 - p}{p} e^{- \frac{1}{2} {(\frac{\frac{1}{2} - σ_{A}^{2} ζ}{σ_{A}})}^{2}}} \\ = e^{- \frac{1}{2} (\frac{1}{4 σ_{B}^{2}} + σ_{B}^{2} ζ^{2}) + \frac{1}{2} (\frac{1}{4 σ_{A}^{2}} + σ_{A}^{2} ζ^{2})} \\ = e^{(\frac{1}{8} (\frac{1}{σ_{A}^{2}} - \frac{1}{σ_{B}^{2}}) + \frac{1}{2} (σ_{A}^{2} - σ_{B}^{2}) ζ^{2})} \end{array}

Thus, D takes on extrema at values of ζ that solve

\frac{σ_{A}}{σ_{B}} = e^{(\frac{1}{8} (\frac{1}{σ_{A}^{2}} - \frac{1}{σ_{B}^{2}}) + \frac{1}{2} (σ_{A}^{2} - σ_{B}^{2}) ζ^{2})} .

Taking logs,

\ln \frac{σ_{A}}{σ_{B}} = \frac{1}{8} (\frac{1}{σ_{A}^{2}} - \frac{1}{σ_{B}^{2}}) + \frac{1}{2} (σ_{A}^{2} - σ_{B}^{2}) ζ^{2} .

Therefore, the solutions to ζ are roots of the function

\frac{1}{8} (\frac{1}{σ_{A}^{2}} - \frac{1}{σ_{B}^{2}}) + \frac{1}{2} (σ_{A}^{2} - σ_{B}^{2}) ζ^{2} - \ln \frac{σ_{A}}{σ_{B}} .

These roots are

ζ = \frac{- 1}{2 σ_{A} σ_{B} \ln \frac{σ_{B}}{σ_{A}}}, ζ = \frac{1}{2 σ_{A} σ_{B} \ln \frac{σ_{B}}{σ_{A}}} .

The existence of exactly two distinct roots for ζ (and hence for q) implies that G_A and G_B cross each other exactly once. It remains to verify that G_B crosses G_A from below and not from above. Now,

\begin{array}{l} D = G_{A} (q) - G_{B} (q) \\ = p (G_{A 1} - G_{B 1}) + (1 - p) (G_{A 0} - G_{B 0}) . \end{array}

At $q = {\underline{q}}^{1} = \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{2 σ_{A} σ_{B}}}}$ , G_A1 − G_B1 = 0 while G_A0 − G_B0 > 0. Hence, $D ({\underline{q}}^{1}) > 0$ .

At $q = {\underline{q}}^{0} = \frac{1}{1 + \frac{1 - p}{p} e^{- \frac{1}{2 σ_{A} σ_{B}}}}$ , G_A0 − G_B0 = 0 while G_A1 − G_B1 < 0. Hence, $D ({\underline{q}}^{0}) < 0$ .

Now, because ${\underline{q}}^{1} < {\underline{q}}^{0}$ , this implies that G_B crosses G_A from below.

This completes the proof that G_A (·) is a mean-preserving spread of G_B (·).

Finally, to prove that G (·; m_B) is a mean-preserving spread of G (·; m’_B) for all m_B < m’_B, it remains to show that G (·; m_B) second-order stochastically dominates G (·; m‘_B). Or,

\int_{0}^{\overset{⌃}{q}} G (q, m_{B}) dq - \int_{0}^{\overset{⌃}{q}} G (q, m'_{B}) dq \leq 0

for all $\overset{⌃}{q} \in (0, 1)$ , with strict inequality for some $\overset{⌃}{q}$ . Now,

\begin{array}{l} \int_{0}^{\overset{⌃}{q}} G (q, m_{B}) dq - \int_{0}^{\overset{⌃}{q}} G (q, m'_{B}) dq \\ = (m'_{B} - m_{B}) \int_{0}^{\overset{⌃}{q}} (G_{A} (q) - G_{B} (q)) dq \leq 0, \end{array}

where the weak inequality for all $\overset{⌃}{q}$ , and the strict inequality for some $\overset{⌃}{q}$ , follow from the fact that G_B (·) second-order stochastically dominates G_A (·).

This completes the proof. ■

Lemma 3

Under the standard Euclidean metric, ^{equation (1)} specifies a contraction mapping T: ℝ → ℝ in V*.

Proof. Let T: ℝ → ℝ be given by

T (υ) = δ \int_{0}^{1} \max [H (q, υ), υ] dG (q) - k .

Then, we have to show that for all υ, w ∈ ℝ and for some 0 ≤ β < 1,

| | T (υ) - T (w) | | \leq β | | υ - w | | .

\begin{array}{l} Now, | | T (υ) - T (w) | | = \\ = | | δ \int_{0}^{1} \max [H (q, υ), υ] dG (q) - k - δ \int_{0}^{1} \max [H (q, w), w] dG (q) + k | | \\ = δ | | \int_{0}^{1} (\max [H (q, υ), υ] - \max [H (q, w), w]) dG (q) | | \\ = δ | | \int_{0}^{1} (\max [q υ + (1 - q) (- c + υ), υ] - \max [q υ + (1 - q) (- c + w), w]) dG (q) | | \\ \leq δ | | \int_{0}^{1} (υ - w) dG (q) | | = δ | | υ - w | | . \end{array}

To see that the last inequality holds, assume, without loss of generality, that υ > w.

Now, if υ < H (q, υ) = qυ + (1 − q) (−c + v), then qυ − c (1 − q) > qυ > qw. Hence, w < qυ + (1 − q) (−c + w) = H (q, w). Therefore,

\begin{array}{l} \max [H (q, υ), υ] - \max [H (q, w), w] \\ = H (q, υ) - H (q, w) \\ = (1 - q) (υ - w) < υ - w . \end{array}

If υ H (q, υ), then υ > H (q, w). Therefore,

\begin{array}{l} \max [H (q, υ), υ] - \max [H (q, w), w] \\ = υ - \max [H (q, w), w] \\ < υ - w . \end{array}

This completes the proof. ■

Lemma 4

There exists a unique threshold, ${\underline{q}}^{1} \equiv \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{2 σ_{A} σ_{B}}}} < p$ , where the probability of type I errors is the same for both kinds of candidates.

Proof.

\begin{matrix} G_{A 1} (\underline{q}) = G_{B 1} (\underline{q}) \\ \Leftrightarrow & Ф (\frac{s_{A} (\underline{q}) - 1}{σ_{A}}) = Φ (\frac{s_{B} (\underline{q}) - 1}{σ_{B}}) \\ \Leftrightarrow & \frac{s_{A} (\underline{q}) - 1}{σ_{A}} = \frac{s_{B} (\underline{q}) - 1}{σ_{B}} \\ \Leftrightarrow & \frac{\frac{1}{2} - σ_{A}^{2} \ln ((\frac{1}{\underline{q}} - 1) \frac{p}{1 - p}) - 1}{σ_{A}} = \frac{\frac{1}{2} - σ_{B}^{2} \ln ((\frac{1}{\underline{q}} - 1) \frac{p}{1 - p}) - 1}{σ_{B}} \\ \Leftrightarrow & \underline{q} = \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{{2 σ}_{A} σ_{B}}}} . \end{matrix}

■

Lemma 5

Suppose $\underline{q} > p$ . Then:

The distribution G_A1 dominates G_B1 in terms of the likelihood ratio.
The distribution G_A0 dominates G_B0 in terms of the likelihood ratio.

Proof. To establish this, it is sufficient to show that $\frac{\partial^{2} \ln g_{A 1}}{\partial σ \partial q} > 0$ .

\begin{array}{l} \frac{\partial^{2} \ln g_{A 1}}{\partial σ \partial q} = \frac{\partial^{2} \ln ϕ (\frac{s (q) - 1}{σ}) \frac{σ}{q (1 - q)}}{\partial σ \partial q} \\ = \frac{\partial^{2} \ln (\frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} {(\frac{s (q) - 1}{σ})}^{2}} \frac{σ}{q (1 - q)})}{\partial σ \partial q} \\ = \frac{2}{q} σ \frac{\ln (\frac{1 - q}{q} \frac{p}{1 - p})}{1 - q} > 0, \end{array}

where the inequality holds since q > p: The proof of part 2 of the Lemma is virtually identical. ■

Lemma 6

Suppose $\underline{q} > p$ . Then:

The distribution G_A1 dominates G_B1 in terms of the hazard rate.
The distribution G_A0 dominates G_B0 in terms of the hazard rate.

Proof. Lemma 5 implies that

\frac{g_{B 1} (q')}{g_{B 1} (q)} < \frac{g_{A 1} (q')}{g_{A 1} (q)}

for all p < q < q′.

Hence,

\begin{matrix} \int_{q}^{1} \frac{g_{A 1} (t)}{g_{A 1} (q)} dt > \int_{q}^{1} \frac{g_{B 1} (t)}{g_{B 1} (q)} \\ \frac{1 - G_{A 1} (q)}{g_{A 1} (q)} > \frac{1 - G_{B 1} (q)}{g_{B 1} (q)}, \end{matrix}

or, equivalently,

\frac{g_{A 1} (q)}{1 - G_{A 1} (q)} < \frac{g_{B 1} (q)}{1 - G_{B 1} (q)} .

The proof for part 2 of the lemma is virtually identical. ■

Proofs of Propositions

Proposition 1

The optimal threshold, ${\underline{q}}^{*}$ , is the unique interior solution to

{\underline{q}}^{*} = \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ + k} .

Proof. Recall that

\begin{array}{l} V (\underline{q}) & = \frac{δ \int_{\underline{q}}^{1} (q υ + (1 - q) (- c)) dG (q) - k}{1 - δ (1 - \int_{\underline{q}}^{1} qdG (q))} \\ = \frac{δ υ \int_{\underline{q}}^{1} qdG (q) - δ c (1 - G (\underline{q})) + δ c \int_{\underline{q}}^{1} qdG (q) - k}{1 - δ (1 - \int_{\underline{q}}^{1} qdG (q))} . \end{array}

It is useful to represent this as numerator and denominator components for purposes of differentiation. Hence, define

N \equiv δ \int_{\underline{q}}^{1} (q υ + (1 - q) (- c)) dG (q) - k,

and

D \equiv 1 - δ (1 - \int_{\underline{q}}^{1} qdG (q)) .

Thus, the first-order necessary condition for optimality, $\frac{\partial V (\underline{q})}{\partial \underline{q}} = 0$ , may be expressed as

\frac{DN' - ND'}{D^{2}} = 0 .

Therefore,

\begin{array}{l} \frac{\partial V (\underline{q})}{\partial \underline{q}} = \frac{D (- δ g (\underline{q}) ((υ + c) \underline{q} - c)) - N (- δ \underline{q} g (\underline{q}))}{D^{2}} \\ = δ g (\underline{q}) \frac{- D (υ + c) \underline{q} + Dc + N \underline{q}}{D^{2}} . \end{array}

Hence,

- D (υ + c) \underline{q} + Dc + N \underline{q} = 0,

and this implies that

{\underline{q}}^{*} = \frac{Dc}{D (υ + c) - N} .

Substituting for D and N, and simplifying, we get the following implicit characterization of ${\underline{q}}^{*}$ :

\begin{array}{l} {\underline{q}}^{*} = \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) (υ + c) - δ \int_{{\underline{q}}^{*}}^{1} (q υ + (1 - q) (- c)) dG (q) + k} \\ = \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ + k}, \end{array}

and this yields the expression in Lemma 1.

Having derived the necessary first-order condition for an interior solution ${\underline{q}}^{*} \in (0, 1)$ , we now prove its actual existence.

At ${\underline{q}}^{*} = 0$ , LHS < RHS. At ${\underline{q}}^{*} = 1$ , LHS > RHS. Hence, by continuity and the intermediate value theorem, there must be a ${\underline{q}}^{*} \in (0, 1)$ such that LHS = RHS.

Next, we prove uniqueness by showing that there is at most one ${\underline{q}}^{*} \in (0, 1)$ that satisfies the necessary first-order condition.

To see this, first notice that we may rewrite the first-order condition as follows:

{\underline{q}}^{*} (c + (1 - δ) υ + k) = c - c δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q)) + δ G ({\underline{q}}^{*}) c {\underline{q}}^{*} .

Integrating by parts, we obtain

{\underline{q}}^{*} (c + (1 - δ) υ + k) = c - c δ \int_{{\underline{q}}^{*}}^{1} G (q) dq .

Adding and subtracting $c δ \int_{0}^{{\underline{q}}^{*}} G (q) dq$ to the right-hand side yields

{\underline{q}}^{*} (c + (1 - δ) υ + k) = c - c δ \int_{0}^{1} G (q) dq + c δ \int_{0}^{{\underline{q}}^{*}} G (q) dq .

Finally, noting that $\int_{0}^{1} G (q) dq = 1 - p$ and substituting, we obtain

{\underline{q}}^{*} (c + (1 - δ) υ + k) = c (1 - δ) + c δ (p + \int_{0}^{{\underline{q}}^{*}} G (q) dq) .

Hence,

{\underline{q}}^{*} = \frac{(1 - δ) c + c δ p}{(c + (1 - δ) υ + k)} + \frac{c δ}{(c + (1 - δ) υ + k)} \int_{0}^{{\underline{q}}^{*}} G (q) dq .

Note that the right-hand side is monotonically increasing in ${\underline{q}}^{*}$ at a speed < 1, for all ${\underline{q}}^{*} \in (0, 1)$ . This implies, however, that the right-hand side can cross the 45-degree line, which corresponds to the left-hand side, at most once. Hence, there is at most one ${\underline{q}}^{*} \in (0, 1)$ that satisfies the necessary first-order condition.

Finally, we show that at the unique interior ${\underline{q}}^{*}$ , the value function reaches a global maximum. This follows from the observation that $\lim_{\underline{q} \to 1} V (\underline{q}) \to - \infty$ , and that there exists an ε > 0 such that for all $0 < \underline{q} < ε$ , $\frac{\partial V (\underline{q})}{\partial \underline{q}} > 0$ . To see that the latter assertion is indeed true, recall that

V (\underline{q}) = \frac{δ \int_{\underline{q}}^{1} (q υ + (1 - q) (- c)) dG (q) - k}{1 - δ (1 - \int_{\underline{q}}^{1} qdG (q))}

and that

\frac{\partial V (\underline{q})}{\partial \underline{q}} = δ g (\underline{q}) \frac{- D (υ + c) \underline{q} + Dc + N \underline{q}}{D^{2}},

where N and D denote the numerator and the denominator of $V (\underline{q})$ , respectively.

Now we rewrite $\frac{\partial V (\underline{q})}{\partial \underline{q}}$ to get

\frac{\partial V (\underline{q})}{\partial \underline{q}} = δ g (\underline{q}) (\frac{c}{D} + \frac{V (\underline{q}) - (υ + c)}{D} \underline{q}) .

Written in this form, it is obvious that, for sufficiently small $\underline{q} > 0$ , both factors in the last expression are strictly positive. This proves the proposition. ■

Proposition 2

For all q ∈ [0; 1); there exist parameter values such that ${\underline{q}}^{*} = q$ .

Proof. Fix k = 0. In that case, the employer will always wish to participate by interviewing candidates rather than eschewing the employment market. When c = 0; the right-hand side of ^{equation (2)} equals zero; hence, ${\underline{q}}^{*} = 0$ . When c → ∞, the right-hand side of ^{equation (2)} goes to 1 as the following argument shows:

\begin{array}{l} \lim_{c \to \infty} \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ} \\ \geq \lim_{c \to \infty} \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} dG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ} \\ = \lim_{c \to \infty} \frac{(1 - δ G ({\underline{q}}^{*})) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ} = 1 . \end{array}

Hence, $\lim_{c \to \infty} {\underline{q}}^{*} = 1$ . Finally, since the right-hand side of ^{equation (2)} is continuous in c; it follows that there exist parameter values such that ${\underline{q}}^{*} = q$ for all q ∈ [0, 1). ■

Proposition 3

Minorities are overrepresented in the workplace (i.e., $\frac{r_{B}}{m_{B}} > 1$ ) if and only if $0 < {\underline{q}}^{*} < {\underline{q}}^{1}$ .
Minorities are underrepresented (i.e., $\frac{r_{B}}{m_{B}} < 1$ ) if and only if ${\underline{q}}^{1} < {\underline{q}}^{*} < 1$ .
Minorities are proportionately represented (i.e., $\frac{r_{B}}{m_{B}} = 1$ ) if and only if ${\underline{q}}^{*} \in {0, {\underline{q}}^{1}}$ .

Proof. Under a uniform threshold success probability $\underline{q}$ , $\frac{r_{B}}{m_{B}} = 1$ if and only if $G_{A 1} (\underline{q}) = G_{B 1} (\underline{q})$ . As we saw in Lemma 4, this corresponds to $\underline{q} = {\underline{q}}^{1} = \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{{2 σ}_{A} σ_{B}}}}$ . To prove the proposition, we show that at the critical point ${\underline{q}}^{1}$ , raising $\underline{q}$ leads to strict underrepresentation of minorities. That is, we calculate the derivative of

G_{A 1} (q) - G_{B 1} (q) = Φ (\frac{s_{A} (q) - 1}{σ_{A}}) - Φ (\frac{s_{B} (q) - 1}{σ_{B}})

with respect to q, evaluate it at ${\underline{q}}^{1} = \frac{1}{1 + \frac{1 - p}{p} e^{\frac{1}{{2 σ}_{A} σ_{B}}}}$ and show that it is strictly negative.

The derivative is equal to

g_{A 1} (q) - g_{B 1} (q) = ϕ (\frac{s_{A} (q) - 1}{σ_{A}}) \frac{σ_{A}}{q (1 - q)} - ϕ (\frac{s_{B} (q) - 1}{σ_{B}}) \frac{σ_{B}}{q (1 - q)} .

Multiplying by q (1 − q) and evaluating at ${\underline{q}}^{1}$ , we get

\begin{array}{l} = ϕ (\frac{\frac{1}{2} \frac{σ_{B} - σ_{A}}{σ_{B}} - 1}{σ_{A}}) σ_{A} - ϕ (\frac{- \frac{1}{2} \frac{σ_{B} - σ_{A}}{σ_{A}} - 1}{σ_{B}}) σ_{B} \\ = ϕ (- \frac{1}{2} \frac{σ_{B} + σ_{A}}{σ_{A} σ_{B}}) σ_{A} - ϕ (- \frac{1}{2} \frac{σ_{B} + σ_{A}}{σ_{A} σ_{B}}) σ_{B} \\ = (σ_{A} - σ_{B}) ϕ (\frac{1}{2} \frac{σ_{B} + σ_{A}}{σ_{A} σ_{B}}) < 0 . \end{array}

This proves the proposition. ■

Proposition 4

Suppose that the employer is “selective” in its hiring policy, i.e., $\underline{q} > p$ , then:

As the employer becomes more selective, minority representation in the workplace decreases. Formally, r_B is decreasing in $\underline{q}$ .
As the employer becomes arbitrarily selective, minorities vanish from the workplace. Formally, $\lim_{\underline{q} \to 1} r_{B} = 0$ .

Proof. To prove part 1, differentiate r_B with respect to $\underline{q}$ :

\begin{array}{l} \frac{\partial r_{B}}{\partial \underline{q}} = \frac{- m_{B} g_{B 1} (1 - m_{B} G_{B 1} - m_{A} G_{A 1}) - (- m_{B} g_{B 1} - m_{A} g_{A 1}) m_{B} (1 - G_{B 1})}{{(1 - m_{B} G_{B 1} - m_{A} G_{A 1})}^{2}} \\ = \frac{m_{B} m_{A} (g_{A 1} (1 - G_{B 1}) - g_{B 1} (1 - G_{A 1}))}{{(1 - m_{B} G_{B 1} - m_{A} G_{A 1})}^{2}} . \end{array}

Notice that the sign of $\frac{\partial r_{B}}{\partial \underline{q}}$ depends only on the hazard rates of G_A1 and G_B1. And by Lemma 6 it then follows that $\frac{\partial r_{B}}{\partial \underline{q}} < 0$ .

To prove part 2 of the proposition, notice that (via L’Hôpital’s rule)

\lim_{\underline{q} \to 1} r_{B} = \lim_{\underline{q} \to 1} \frac{m_{B}}{m_{B} + m_{A} \frac{g_{A 1}}{g_{B 1}}},

and this limit depends solely on the limit of the likelihood ratio, $\frac{g_{A 1}}{g_{B 1}}$ . Finally, it may be readily shown that:

\begin{array}{l} \lim_{\underline{q} \to 1} \frac{g_{A 1}}{g_{B 1}} = \lim_{\underline{q} \to 1} \frac{ϕ (\frac{s_{A} (q) - 1}{σ_{A}}) σ_{A}}{ϕ (\frac{s_{B} (q) - 1}{σ_{B}}) σ_{B}} \\ = \lim_{\underline{q} \to 1} e^{\frac{1}{8 σ_{A}^{2} σ_{B}^{2}} (4 σ_{A}^{2} σ_{B}^{2} \ln^{2} (\frac{q}{1 - q}) - 1) (σ_{B}^{2} - σ_{A}^{2})} \frac{σ_{A}}{σ_{B}} \to \infty . \end{array}

Hence,

\lim_{\underline{q} \to 1} r_{B} = 0 .

■

Proposition 5

For all $\underline{q} \in (0, 1)$ , minority hires are fired at a higher rate than majority hires.

Proof. Because hires are fired if and only if they turn out to be incompetent, we have to prove that

Pr (π_{A} = 0 | q_{A} \geq \underline{q}) = \frac{(1 - G_{A 0}) (1 - p)}{1 - G_{A}} < \frac{(1 - G_{B 0}) (1 - p)}{1 - G_{B}} = Pr (π_{B} = 0 | q_{B} \geq \underline{q})

for all $\underline{q} \in (0, 1)$ .

This is equivalent to showing that

\begin{matrix} \frac{1 - G_{A 0}}{1 - G_{A}} < \frac{1 - G_{B 0}}{1 - G_{B}}, \end{matrix}

\begin{matrix} \frac{1 - G_{B}}{1 - G_{B 0}} < \frac{1 - G_{A}}{1 - G_{A 0}} . \end{matrix}

Now,

\begin{matrix} \frac{1 - G_{B}}{1 - G_{B 0}} < \frac{1 - G_{A}}{1 - G_{A 0}} \Leftrightarrow \\ \frac{1 - p G_{B 1} - (1 - p) G_{B 0}}{1 - G_{B 0}} < \frac{(1 - p G_{A 1} - (1 - p) G_{A 0})}{1 - G_{A 0}} \Leftrightarrow \\ \frac{1 - G_{B 1}}{1 - G_{B 0}} < \frac{1 - G_{A 1}}{1 - G_{A 0}} . \end{matrix}

Hence, showing that $Pr (π_{A} = 0 | q_{A} \geq \underline{q}) < Pr (π_{B} = 0 | q_{B} \geq \underline{q})$ is equivalent to showing that the ratio of good hiring decisions over bad hiring decisions, $\frac{1 - G_{κ 1}}{1 - G_{κ 0}}$ , is greater for kind A hires than for kind B hires. To prove the latter, we show that

\frac{d}{d σ_{κ}} [\frac{1 - G_{κ 1} (\underline{q})}{1 - G_{κ 0} (\underline{q})}] < 0 .

Now, $\frac{d}{d σ_{κ}} [\frac{1 - G_{κ 1} (\underline{q})}{1 - G_{κ 0} (\underline{q})}]$

\begin{array}{l} = \frac{d}{d σ_{κ}} [\frac{\int_{\underline{q}}^{1} g_{κ 1} (q) dq}{\int_{\underline{q}}^{1} g_{κ 0} (q) dq}] \\ = \frac{d}{d σ_{κ}} [\frac{\int_{\underline{q}}^{1} ϕ (\frac{s_{κ} (q) - 1}{σ_{κ}}) \frac{σ_{κ}}{q (1 - q)} dq}{\int_{\underline{q}}^{1} ϕ (\frac{s_{κ} (q)}{σ_{κ}}) \frac{σ_{κ}}{q (1 - q)} dq}] . \end{array}

Using that $\frac{{ds}_{κ} (q)}{d σ_{κ}} = \frac{2 (s_{κ} (q) - \frac{1}{2})}{σ_{κ}}$ , straightforward algebra leads to the conclusion that the sign of $\frac{d}{d σ_{κ}} [\frac{1 - G_{κ 1} (\underline{q})}{1 - G_{κ 0} (\underline{q})}]$ is equal to the sign of

\int_{\underline{q}}^{1} g_{κ 1} (q) dq \int_{\underline{q}}^{1} s_{κ} (q) (s_{κ} (q) - 1) g_{κ 0} (q) dq - \int_{\underline{q}}^{1} g_{κ 0} (q) dq \int_{\underline{q}}^{1} s_{κ} (q) (s_{κ} (q) - 1) g_{κ 1} (q) dq .

Changing variables of integration from q to s, we get

\begin{array}{l} \int_{s_{κ} (\underline{q})}^{1} g_{κ 1} (s) \frac{\partial q_{κ} (s)}{\partial s} ds \int_{s_{κ} (\underline{q})}^{1} s (s - 1) g_{κ 0} (s) \frac{\partial q_{κ} (s)}{\partial s} ds \\ - \int_{s_{κ} (\underline{q})}^{1} g_{κ 0} (s) \frac{\partial q_{κ} (s)}{\partial s} ds \int_{s_{κ} (\underline{q})}^{1} s (s - 1) g_{κ 1} (s) \frac{\partial q_{κ} (s)}{\partial s} ds . \end{array}

Substituting for g_κ0, g_κ1, and $\frac{\partial q_{κ} (s)}{\partial s}$ ,

\int_{s_{κ} (\underline{q})}^{1} ϕ (\frac{s - 1}{σ_{κ}}) ds \int_{s_{κ} (\underline{q})}^{1} s (s - 1) ϕ (\frac{s}{σ_{κ}}) ds - \int_{s_{κ} (\underline{q})}^{1} ϕ (\frac{s}{σ_{κ}}) ds \int_{s_{κ} (\underline{q})}^{1} s (s - 1) ϕ (\frac{s - 1}{σ_{κ}}) ds .

Expanding s (s − 1),

\begin{array}{l} \int_{s_{κ} (\underline{q})}^{1} ϕ (\frac{s - 1}{σ_{κ}}) ds (\int_{s_{κ} (\underline{q})}^{1} s^{2} ϕ (\frac{s}{σ_{κ}}) ds - \int_{s_{κ} (\underline{q})}^{1} s ϕ (\frac{s}{σ_{κ}}) ds) \\ - \int_{s_{κ} (\underline{q})}^{1} ϕ (\frac{s}{σ_{κ}}) ds (\int_{s_{κ} (\underline{q})}^{1} s^{2} ϕ (\frac{s - 1}{σ_{κ}}) ds - \int_{s_{κ} (\underline{q})}^{1} s ϕ (\frac{s - 1}{σ_{κ}}) ds) . \end{array}

Writing in terms of conditional expectations,

\begin{array}{l} (1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})) (1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}})) (E [S_{κ 0}^{2} | S_{κ 0} \geq s_{κ} (\underline{q})] - E [S_{κ 0} | S_{κ 0} \geq s_{κ} (\underline{q})]) \\ - (1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}})) (1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})) (E [S_{κ 1}^{2} | S_{κ 1} \geq s_{κ} (\underline{q})] - E [S_{κ 1} | S_{κ 1} \geq s_{κ} (\underline{q})]) . \end{array}

Dividing by the common positive factor $(1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})) (1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}}))$ :

E [S_{κ 0}^{2} | S_{κ 0} \geq s_{κ} (\underline{q})] - E [S_{κ 0} | S_{κ 0} \geq s_{κ} (\underline{q})] - E [S_{κ 1}^{2} | S_{κ 1} \geq s_{κ} (\underline{q})] - E [S_{κ 1} | S_{κ 1} \geq s_{κ} (\underline{q})] .

Now, the moment generating function, mgf, of a left-truncated standard normal random variable U with truncation point d is (see, for example, Heckman and Honoré, 1990):

mgf (β) = e^{\frac{1}{2} β^{2}} \frac{\int_{d - β}^{\infty} \frac{1}{\sqrt{2 π}} \exp (- \frac{1}{2} u^{2}) du}{\int_{d}^{\infty} \frac{1}{\sqrt{2 π}} \exp (- \frac{1}{2} u^{2}) du} .

Hence,

\begin{array}{l} E [U | U \geq d] & = \frac{\partial mgf}{\partial β} |_{β = 0} \\ = \frac{ϕ (d)}{1 - Φ (d)}, \end{array}

while

\begin{array}{l} E [U^{2} | U \geq d] & = \frac{\partial^{2} mgf}{\partial β^{2}} |_{β = 0} \\ = 1 + d \frac{\partial mgf}{\partial β} |_{β = 0} \\ = 1 + \frac{d ϕ (d)}{1 - Φ (d)} . \end{array}

For X ~ N (μ, σ²), this implies

\begin{array}{l} E [X | X \geq d'] = μ + \frac{σ ϕ (\frac{d' - μ}{σ})}{1 - Φ (\frac{d' - μ}{σ})} \\ E [X^{2} | X \geq d'] = σ^{2} + (μ + d') \frac{σ ϕ (\frac{d' - μ}{σ})}{1 - Φ (\frac{d' - μ}{σ})} + μ^{2} . \end{array}

Now, recall that S_κ0 ~N (0, σ_κ) and S_κ1 ~N (1, σ_κ). Hence,

E [S_{κ 0}^{2} | S_{κ 0} \geq s_{κ} (\underline{q})] - E [S_{κ 0} | S_{κ 0} \geq s_{κ} (\underline{q})] - E [S_{κ 1}^{2} | S_{κ 1} \geq s_{κ} (\underline{q})] - E [S_{κ 1} | S_{κ 1} \geq s_{κ} (\underline{q})]

\begin{array}{l} = σ_{κ}^{2} + s_{κ} (\underline{q}) \frac{σ_{κ} ϕ (\frac{s_{κ} (\underline{q})}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}})} - \frac{σ_{κ} ϕ (\frac{s_{κ} (\underline{q})}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}})} \\ - (σ_{κ}^{2} + (1 + s_{κ} (\underline{q})) \frac{σ_{κ} ϕ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})} + 1 - 1 - \frac{σ_{κ} ϕ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})}) . \end{array}

Dividing by σ_κ and collecting terms, we get

(s_{κ} (\underline{q}) - 1) \frac{ϕ (\frac{s_{κ} (\underline{q})}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q})}{σ_{κ}})} - s_{κ} (\underline{q}) \frac{ϕ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})}{1 - Φ (\frac{s_{κ} (\underline{q}) - 1}{σ_{κ}})} .

Hence, the question is whether

(s - 1) \frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})} - s \frac{ϕ (\frac{s - 1}{σ})}{1 - Φ (\frac{s - 1}{σ})} < 0,

\frac{s - 1}{σ} \frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})} - \frac{s}{σ} \frac{ϕ (\frac{s - 1}{σ})}{1 - Φ (\frac{s - 1}{σ})} < 0,

for all s ∈ ℝ and σ > 0.

Denote hazard rate $\frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})}$ by $λ (\frac{s}{σ})$ . The expression then becomes

(s - 1) λ (\frac{s}{σ}) - s λ (\frac{s - 1}{σ}) .

Graphically, when s − 1 < 0,

Hence, for all s − 1 < 0, it is obvious that

(s - 1) λ (\frac{s}{σ}) - s λ (\frac{s - 1}{σ}) < 0 .

When s − 1 > 0, graphically,

Here, in principle, it could go either way.

Now, for s − 1 > 0,

\begin{array}{l} (s - 1) λ (\frac{s}{σ}) - s λ (\frac{s - 1}{σ}) \\ = (s - 1) (λ (\frac{s}{σ}) - λ (\frac{s - 1}{σ})) - (s - (s - 1)) λ (\frac{s - 1}{σ}) \\ \leq \int_{λ (\frac{s - 1}{σ})}^{λ (\frac{s}{σ})} λ^{- 1} (l) dl - \int_{s - 1}^{s} λ (\frac{x}{σ}) dx, \end{array}

where the inequality follows from the convexity of $λ (\frac{s}{σ})$ .

Changing the variable of integration in the first term from hazard rate l to signal x, the last expression becomes

\begin{array}{l} = \int_{s - 1}^{s} x \frac{\partial l}{\partial x} dx - \int_{s - 1}^{s} λ (\frac{x}{σ}) dx \\ = \int_{s - 1}^{s} \frac{x}{σ} λ' (\frac{x}{σ}) dx - \int_{s - 1}^{s} λ (\frac{x}{σ}) dx \\ = \int_{s - 1}^{s} (\frac{x}{σ} λ' (\frac{x}{σ}) - λ (\frac{x}{σ})) dx . \end{array}

Finally, we show that the integrand, which we write as

z λ' (z) - λ (z),

is negative for all z ≥ 0.

First, note that

\begin{array}{l} λ' (\frac{s}{σ}) = \frac{d}{d \frac{s}{σ}} λ (\frac{s}{σ}) = \frac{d}{d \frac{s}{σ}} [\frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})}] \\ = \frac{- \frac{s}{σ} ϕ (\frac{s}{σ}) (1 - Φ (\frac{s}{σ})) + ϕ^{2} (\frac{s}{σ})}{{(1 - Φ (\frac{s}{σ}))}^{2}} \\ = \frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})} (\frac{ϕ (\frac{s}{σ}) - \frac{s}{σ} (1 - Φ (\frac{s}{σ}))}{(1 - Φ (\frac{s}{σ}))}) \\ = \frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})} (\frac{ϕ (\frac{s}{σ})}{1 - Φ (\frac{s}{σ})} - \frac{s}{σ}) \\ = λ (\frac{s}{σ}) (λ (\frac{s}{σ}) - \frac{s}{σ}) . \end{array}

Hence, the integrand can be written as

\begin{array}{l} z λ' (z) - λ (z) \\ = z λ (z) (λ (z) - z) - λ (z) \\ = λ (z) (z (λ (z) - z) - 1) . \end{array}

Dividing by λ (z), The question becomes whether

z (λ (z) - z) < 1

for z ≥ 0.

Now, note that λ’ (z) < 1 for all z, as the derivative of the hazard rate of the standard Normal distribution converges to 1 from below when z → ∞. Hence, it suffices to show that

z (λ (z) - z) \leq λ (z) (λ (z) - z) = λ' (z) .

Now,

z (λ (z) - z) \leq λ (z) (λ (z) - z)

is equivalent to

0 \leq {(λ (x) - x)}^{2},

where the last inequality is obviously true. ■

Proofs of Implications

Implication 1

Suppose that the employer is “selective” in its hiring policy, i.e., $\underline{q} > p$ , then:

An increase in the cost of firing, c, reduces workplace diversity.
An increase in the cost of interviewing, k, increases workplace diversity.

Proof. To establish part 1 of the implication, we show that ${\underline{q}}^{*}$ is increasing in c: Recall that optimality of the threshold strategy implies that

\begin{matrix} (V ({\underline{q}}^{*}) - υ) {\underline{q}}^{*} + (1 - {\underline{q}}^{*}) c = 0 . & (3) \end{matrix}

Implicitly differentiating with respect to c while noting that $\frac{\partial V ({\underline{q}}^{*})}{\partial {\underline{q}}^{*}} = 0$ gives

(V ({\underline{q}}^{*}) - υ) \frac{d {\underline{q}}^{*}}{dc} + \frac{\partial V ({\underline{q}}^{*})}{\partial c} {\underline{q}}^{*} + (1 - {\underline{q}}^{*}) - c \frac{d {\underline{q}}^{*}}{dc} = 0 .

Solving for $\frac{d {\underline{q}}^{*}}{dc}$ ,

\frac{d {\underline{q}}^{*}}{dc} = \frac{(\frac{\partial V ({\underline{q}}^{*})}{\partial c} - 1) {\underline{q}}^{*} + 1}{υ + c - V ({\underline{q}}^{*})} .

It is easily checked that

\frac{\partial V ({\underline{q}}^{*})}{\partial c} = \frac{- δ \int_{{\underline{q}}^{*}}^{1} (1 - q) dG (q)}{1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))} .

Substituting into the expression for $\frac{d {\underline{q}}^{*}}{dc}$ and simplifying, one obtains

\frac{d {\underline{q}}^{*}}{dc} = (\frac{δ G ({\underline{q}}^{*}) - 1}{1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))}) {\underline{q}}^{*} + 1 .

To establish that the right-hand side of this expression is positive requires that we show that

(1 - δ G ({\underline{q}}^{*})) {\underline{q}}^{*} - (1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) < 0 .

To see this, notice that

\begin{array}{l} (1 - δ G ({\underline{q}}^{*})) {\underline{q}}^{*} - (1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) \\ < (1 - δ G ({\underline{q}}^{*})) {\underline{q}}^{*} - (1 - δ (1 - {\underline{q}}^{*} (1 - G ({\underline{q}}^{*})))) \\ = - (1 - δ) (1 - {\underline{q}}^{*}) \\ < 0 . \end{array}

To establish part 2 of the implication, we show that ${\underline{q}}^{*}$ is decreasing in k. Implicitly differentiating equation (3) with respect to k while noting that $\frac{\partial V ({\underline{q}}^{*})}{\partial {\underline{q}}^{*}} = 0$ , we obtain

(V ({\underline{q}}^{*}) - υ) \frac{d {\underline{q}}^{*}}{dk} + \frac{\partial V ({\underline{q}}^{*})}{\partial k} {\underline{q}}^{*} - c \frac{d {\underline{q}}^{*}}{dk} = 0 .

Solving for $\frac{d {\underline{q}}^{*}}{dk}$ ,

\frac{d {\underline{q}}^{*}}{dk} = \frac{\frac{\partial V ({\underline{q}}^{*})}{\partial k} {\underline{q}}^{*}}{υ + c - V ({\underline{q}}^{*})} .

Hence, $\frac{d {\underline{q}}^{*}}{dk}$ and $\frac{\partial V ({\underline{q}}^{*})}{\partial k}$ have the same sign, while it is easily checked that $\frac{\partial V ({\underline{q}}^{*})}{\partial k} < 0$ . ■

Implication 2

Diversity is procyclical. Formally, q* is decreasing in υ (and k).

Proof. From Implication 1, we already know that q* is increasing in k.

Implicitly differentiating equation (3) with respect to υ while noting that $\frac{\partial V ({\underline{q}}^{*})}{\partial {\underline{q}}^{*}} = 0$ , we obtain

(V ({\underline{q}}^{*}) - υ) \frac{d {\underline{q}}^{*}}{d υ} + (\frac{\partial V ({\underline{q}}^{*})}{\partial υ} - 1) {\underline{q}}^{*} - c \frac{d {\underline{q}}^{*}}{d υ} = 0 .

Solving for $\frac{d {\underline{q}}^{*}}{d υ}$ :

\frac{d {\underline{q}}^{*}}{d υ} = \frac{(\frac{\partial V ({\underline{q}}^{*})}{\partial υ} - 1) {\underline{q}}^{*}}{υ + c + V ({\underline{q}}^{*})} .

It is easily checked that

\frac{dV ({\underline{q}}^{*})}{d υ} = \frac{δ \int_{{\underline{q}}^{*}}^{1} (q) dG (q)}{1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))} .

Substituting this back into $\frac{d {\underline{q}}^{*}}{d υ}$ and simplifying, one obtains

\begin{array}{l} \frac{d {\underline{q}}^{*}}{d υ} = - \frac{\frac{1 - δ}{1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))} {\underline{q}}^{*}}{υ + c - V ({\underline{q}}^{*})} \\ < 0 . \end{array}

■

Implication 3

Suppose that the employer is “selective” in its hiring policy, i.e., ${\underline{q}}^{*} > p$ , then riskier firms are more diverse. Formally, ${\underline{q}}^{*}$ is increasing in δ.

Proof. Implicitly differentiating equation (3) with respect to δ while noting that $\frac{\partial V ({\underline{q}}^{*})}{\partial {\underline{q}}^{*}} = 0$ , we obtain

(V ({\underline{q}}^{*}) - υ) \frac{d {\underline{q}}^{*}}{d δ} + (\frac{dV ({\underline{q}}^{*})}{d δ} - 1) {\underline{q}}^{*} - c \frac{d {\underline{q}}^{*}}{d δ} = 0 .

Solving for $\frac{d {\underline{q}}^{*}}{d δ}$ ,

\frac{d {\underline{q}}^{*}}{d δ} = \frac{(\frac{dV ({\underline{q}}^{*}) {\underline{q}}^{*}}{d δ} - 1) {\underline{q}}^{*}}{υ + c - V ({\underline{q}}^{*})} .

It is easily checked that:

\frac{dV ({\underline{q}}^{*})}{d δ} = \frac{Z (1 - δ X) + X (δ Z - k)}{{(1 - δ X)}^{2}},

where

\begin{array}{l} Z \equiv \int_{\underline{q}}^{1} (q υ + (1 - q) (- c)) dG (q) \\ X \equiv (1 - \int_{\underline{q}}^{1} qdG (q)) . \end{array}

To show that $\frac{d {\underline{q}}^{*}}{d δ} > 0$ , it is sufficient to show that $\frac{dV ({\underline{q}}^{*})}{d δ} - 1 > 0$ , or, equivalently,

Z (1 - δ X) + X (δ Z - k) - {(1 - δ X)}^{2} > 0 .

To see this, simplify the left-hand side of the above expression and recall that, since the employer finds it optimal to search in the first place, δZ − k ≥ 0. This yields

\begin{array}{l} Z - Xk + {(1 - X δ)}^{2} \\ \geq Z - X δ Z + {(1 - X δ)}^{2} \\ = (1 - X δ) (Z + 1 - X δ) \\ > 0, \end{array}

where the last inequality follows from the fact that Z > 0 and X, δ∈ (0, 1). ■

Implication 4

In jobs that are sufficiently selective, minorities will be underrepresented. In jobs that are sufficiently non-selective, minorities will be overrepresented. Formally, there exists 0 < p₀ < p₁ < 1 such that, for all p ∈ (0; p₀), $\frac{r_{B}}{m_{B}} < 1$ while for all p ∈ (p₁, 1), $\frac{r_{B}}{m_{B}} > 1$ .

Proof. First, we establish that $\lim_{p ↑ 1} {\underline{q}}^{*} < 1$ and $\lim_{p ↓ 0} {\underline{q}}^{*} > 0$ . To see this, note that ${\underline{q}}^{*}$ is monotone in p since, by implicitly differentiating equation (3),

\frac{d {\underline{q}}^{*}}{dp} = \frac{\frac{\partial V ({\underline{q}}^{*})}{\partial p} {\underline{q}}^{*}}{υ - V ({\underline{q}}^{*}) + c} > 0,

where the inequality follows from the fact that $υ > V ({\underline{q}}^{*})$ and, by Lemma 1, $\frac{\partial V ({\underline{q}}^{*})}{\partial p} > 0$ .

Since ${\underline{q}}^{*}$ is bounded and monotone function of p we know that both limits must exist.

To establish that $\lim_{p ↑ 1} {\underline{q}}^{*} < 1$ , suppose, to the contrary, that $\lim_{p ↑ 1} {\underline{q}}^{*} = 1$ . Then the right-hand side of ^{equation (2)} becomes

\begin{array}{l} \lim_{p ↑ 1} \frac{(1 - δ (1 - \int_{1}^{1} qdG (q))) c}{(1 - δ G (1)) c + (1 - δ) υ + k} \\ = \frac{(1 - δ) c}{(1 - δ) c + (1 - δ) υ + k} \neq 1, \end{array}

which is a contradiction.

To establish that $\lim_{p ↓ 0} {\underline{q}}^{*} > 0$ , recall that ${\underline{q}}^{*}$ is implicitly defined by ^{equation (2)}. Taking limits,

\begin{array}{l} \lim_{p ↓ 0} {\underline{q}}^{*} = \lim_{p ↓ 0} \frac{(1 - δ (1 - \int_{{\underline{q}}^{*}}^{1} qdG (q))) c}{(1 - δ G ({\underline{q}}^{*})) c + (1 - δ) υ + k} \\ > \lim_{p ↓ 0} \frac{(1 - δ) c}{c + (1 - δ) υ + k} > 0 . \end{array}

To complete the proof, it remains to show that ${\underline{q}}^{0}$ and ${\underline{q}}^{1}$ are monotone in p with limits $\lim_{p ↓ 0} {\underline{q}}^{0} = 0$ and $\lim_{p ↓ 1} {\underline{q}}^{1} = 1$ . Monotonicity may be readily verified by differentiating the expressions for ${\underline{q}}^{0}$ and ${\underline{q}}^{1}$ . Likewise, the limit results are trivial to obtain. ■

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John Morgan is at U.C. Berkeley. The authors would like to thank Mary Amity, Burkhard Drees, Bob Feldman, Andrew Feltenstein, Harold Houba, Keith Takeda and, especially, Johan Walden for extremely useful comments. Morgan gratefully acknowledges the financial support of the National Science Foundation.

Of course, matching the background of the interviewer with the background of the candidate would solve this problem. However, more often than not, this may be quite difficult to implement. First, in organizations lacking diversity, minorities are scarce to begin with. Second, it should not be forgotten that the various minorities are culturally highly diverse, thus requiring a very careful matching between the evaluator and the evaluee. For instance, while a French-speaking West African and an African-American are both people of color, it seems quite clear that they do not share the same discourse system.

In reality, an employer may be able to guess a potential candidate’s minority status from his name or address. On the basis of that information, the employer might decide not to invite him for an interview. Even though in most countries this is clearly against the law, there is evidence that it does happen. See, for example, Bertrand and Mullainathan (2004). The assumption in our model is that employers do abide by the law and, therefore, do not discriminate in this way. Technically speaking, our model is one of undirected search.

^{Proposition 6} ignores the cases where $\underline{q} \in {0, 1}$ since, for these degenerate cases, either everyone is hired or no one is hired, and the firing rate problem is trivial.

Outside of policies affecting the ecomomic incentives of the employer, policies that reduce of eliminate the difference in signal precision between the minority and non-minority candidates are diversity enhancing as well.

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Diversity in the Workplace

Author: Mr. Felix J Vardy and Mr. John Morgan

Volume/Issue:: Volume 2006: Issue 237

Publisher:: International Monetary Fund

ISBN:: 9781451864977

ISSN:: 1018-5941

Pages:: 45

DOI:: https://doi.org/10.5089/9781451864977.001

Keywords
I. Introduction
II. Related Literature
III. Model
IV. Optimal Search and Hiring
V. Performance Metrics
VI. Policy Implications
VII. Conclusions
- APPENDIX
  - Appendix: Proofs of Lemmas, Propositions and Implications
References

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Hiring Probability Ratios of Competent Candidates of Kind A versus $B : \frac{1 - G_{A 1} (q)}{1 - G_{B 1} (q)}$ .

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Over- and Underrepresentation of Minorities.

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Abstract

I. Introduction

II. Related Literature

III. Model

Posterior Beliefs

IV. Optimal Search and Hiring

V. Performance Metrics

Permanent Workforce Composition

Initial Hiring Rates

Firing Rates

Summary

VI. Policy Implications

Diversity and Worker Protections

Diversity over the Business Cycle

Diversity and the Cost of Capitals

Diversity and the Scarcity of Competence

VII. Conclusions

APPENDIX

Appendix: Proofs of Lemmas, Propositions and Implications

Proofs of Lemmas

Proofs of Propositions

Proofs of Implications

References

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