An Option Value Explanation for the

Existence and Persistence of Discrimination

Claudia A. González Mart́ınez∗

Department of Economics

University of Wisconsin–Madison

cgonzale@ssc.wisc.edu

Premilinary and Incomplete

December 8, 2001

Abstract

I introduce a dynamic statistical discrimination model where no investment in human capital

is considered and inference regarding expected productivity is based on a noisy ability signal.

The accuracy of the signal is endogenously determined but taken as given by each generation

of individuals. Discrimination, understood as different average wages for two groups of workers

that have the same initial distribution of ability, arises as an optimal response to the existence

of an outside option available to dropouts from the labor market. The outcome indicates that,

as long as the outside option is the same for both types, a stationary equilibrium where both

groups are equally treated does not exist; moreover, if the majority type is in the advantaged

position for at least one period, the type with the smaller presence in the Economy will always

be in disadvantage.

1 Introduction

Working requires effort; but, when too much effort is needed or the wage is too low in comparison

to the effort required and there is an outside option available, people may, optimally, decide to drop∗Usual comments and disclaimers here.

1

out from the labor market and take the alternative option.1 Obviously, the attractiveness of the

alternative option depends on the value of the expected benefits of staying in the labor market. If

these benefits differ depending on an identifying external characteristic (e.g. race, gender, religion,

etc.), then the optimal decision of whether or not to work is characterized differently depending

upon the group each individual belongs to.

What could explain different expected benefits of staying in the labor market to different demo-

graphic groups with the same distribution of ability? In a world where productivity is not directly

observable and wages depend on the expected productivity of each worker, high ability individuals

may want to differentiate themselves in order to increase their expected payoffs by signaling their

condition (see Spence [18], Stiglitz [19]). This process requires that employers are able to interpret

correctly the ability signals emitted by workers. This communication process is more likely to be

successful when employers and workers share a similar sociocultural background (see Lang [10]).

Considering that the world is heterogenous; and, for historic reasons, the possibility of the majority

of employers belonging to a single group (say, white males), in equilibrium there will be different

wage schedules for people with the same ability indicators but coming from different gender and/or

demographic groups. In this case, dropping out from the labor market is not equally characterized

for each group in the economy. In equilibrium, this leads to a situation where groups with the same

initial distribution of productivity behave differently and receive different average wages.

Surprisingly, I have not seen this outside option argument exploited in the theoretical literature on

discrimination, and this is the gap I plan to, at least in part, fill in with this paper.

In the context of this paper, an outcome will be considered discriminatory if members of two groups

with the same innate ability distributions are, on average, paid different wages.

Formally, I present a simple dynamic model with no investment in human capital or, equivalently,

where investment in education is costless.2 In this economy, labor is the sole productive factor

and the total surplus produced only depends on the ability of the worker hired. It is assumed

that there are two ex-ante identically productive groups that differ in an external, non-productive

characteristic (say, race or gender). Right before entering the labor market, applicants provide

an imperfect, costless and publicly-observed ability signal, which determines their future wage.

Successful interpretation of the signal is the result of a learning process, thus I endogenize the

accuracy of the signal by assuming individuals who live for two periods: they may work during the1Actually, this has been a big share of the criticism associated to the incentives structure provided by unem-

ployment insurance and anti-poverty programs. On this line, some scholars (Brown [6], Smith and Welch [17]) have

empirically associated the existence of these programs to the persistence of discrimination.2This assumption translates into pre-hiring identical distributions of ability in the labor market.

2

first one, in which case they learn how to communicate to other members of the economy. During

the second stage, they become employers (firms) and apply the knowledge from the previous period

in order to interpret the signals provided by the new generation. Firms compete for the worker in

a Bertrand fashion. As assumed by classical models of statistical discrimination (see Aigner and

Cain [1]), a noisier signal for the minority group members is obtained. Hence, compared to the

favored group situation, firms rely less on the signal and more in the prior information they have

as talent indicators for the minority group members. This situation affects the desirability of the

outside option driving the discriminatory outcome.

The main result indicates that, when the overall participation of each type of individual in the

economy is not equal but the value of the outside option is, a stationary equilibrium where both

types are equally treated does not exist; moreover, if the majority group is in the advantageous

position for at least one period, the type with the smaller presence in the Economy will always be

in disadvantage.

The paper is organized as follows. The second section presents some background on the previous

literature on statistical discrimination and discuss what is added by this paper. The third section

presents the model and the fourth develops its equilibrium. This section is split into four parts.

Subsection 4.1 explores the equilibrium of the within generations game for a generic group, and

subsection 4.2 analyzes how this equilibrium is affected when the signal becomes noisier. Parts 4.3

and 4.4 consider the intergenerational equilibrium endogenizing the value of the accuracy of the

signal and study the implications about discrimination of the results obtained. The next section

endogenizes the value of the outside option, which will depend on the proportion of members from

each group who remained unemployed in the previous generation. The final section of the paper

concludes and discuss how this model may be extended and its drawbacks.

2 Related Previous Research

Statistical discrimination was introduced by Arrow [3] and Phelps [15]. These models postulate

discrimination as a rational response to non-observability of productive characteristics. Thus, em-

ployers use race, gender, etc., as proxies to assess the value of these characteristics and construct

productivity expectations by using the score of some imperfect screening device (the signal) and

prior beliefs.

There are two major branches of literature regarding statistical discrimination. The first one as-

sumes exogenous differences in the quality and unbiasedness of the information regarding productive

3

characteristics, which is similar to the line I follow in this paper.3,4 The second line of research pos-

tulate that, “when workers must decide whether or not to make productivity enhancing investments,

discrimination can also arise in the presence of multiple, self-confirming equilibria”.5 In these mod-

els, discrimination is the result of a coordination failure and not an outcome of exogenous group

differences.

Regarding the first of the two lines of research, the basic model was proposed by Phelps [15]

and Aigner and Cain [1] who assume two groups with the same average productivity, but an

ability signal which is more accurate for the favored group. The result is a steeper wage schedule

for the non-discriminated type. However, this outcome can not be considered discriminatory as

types with the same average productivity are (1) paid the same average wage, and (2) the amount

that above-average-productivity minority individuals are underpaid (in relation to their majority

counterparts) is the same amount below-average-productivity workers are overpaid. Aigner and

Cain add employers’ risk aversion to generate different mean wages for both groups, which is not a

particularly appealing assumption.

From this point of departure, other two main strands of models have developed. These two lines

are differentiated by whether or not investment in human capital is considered as a force driving

the discriminatory outcome.

Among the models that do not consider investment in human capital and are able to obtain different

average wages paid to groups with the same average productivity, are Lang [10] and Cornell and

Welch [8]. Lang develops a model based on different speech communities with different verbal and

non-verbal languages. This model recognizes that communication is more costly between dissimilar

groups. In equilibrium, this cost will be minimized by segregation, but still borne by minorities.

Thus, discrimination persists even though no human capital investment is considered. Cornell

and Welch [8] suggest a tournament-style model where the distribution of ability is initially the

same for both types of worker and employers hire the highest expected productivity from a pool

of applicants. If the signal is noisier for one group, the most likely result is to pick an individual

from the other type since the variance of inferred abilities increases as the variance of the signal

decreases. The results from this model are questionable because identical ability distributions in3See, for example, Phelps [15], Borjas and Goldberg [5], Aigner and Cain [1], Lundberg and Startz [12], Cornell

and Welch [8], Lang [10] and [11], and Oettinger [14], who also introduces dynamics to the model and empirically

tests its implications.4The difference between this line of research and what I propose consists of endogenous generation of the differences

in the quality and unbiasedness of the signal which is not found in previous research.5E.g., Akerlof [2], Coate and Loury [7], Antonovics [4].

4

the pool of applicants are sustainable only if individuals live for one period or search for a new job

every period of their lives: clearly, if only applicants from one group are hired, then the members

from the other group must be of a higher expected ability on equilibrium.

Another model where education is not considered is introduced and empirically tested by Oettinger

[14]. He expands the model of Aigner and Cain [1] to include dynamics assuming that uncertainty

about the worker’s ability is resolved as on-the-job performance is observed. This model assumes

that productivity is match specific; thus, uncertainty affects both the worker and the firm equally.

The theoretical results indicate that at labor force entry there should be no black-white wage

gap, but at higher levels of experience a wage gap should emerge. I have one main objection to

this model: by supposing that uncertainty for all workers is resolved after one period, Oettinger

implicitly assumes that the learning rate about the uncertain productivity is faster for a member

from the discriminated group than for a favored type worker. This is not credible and various of

his theoretical conclusions and empirical implications depend strongly on this assumption.

None of the previous models consider that individuals may have outside alternatives to the job

market and that the existence of those affect their reservation wages. As result of these alternatives,

workers may decide to dropout from the labor market affecting the mean ability and wage of those

who remained working. This issue is addressed in this paper by assuming the existence of an outside

option available to workers. This option value is binding for low earners leading them to dropout

from the labor market. Thus, if this option value differs for different groups, or, if different types of

workers face wage schedules with different slopes; then this outside alternative influences individuals

decisions of staying or leaving the labor market differently. The result is a lower average wage for

the group with the less accurate signal; thus, a statistically discriminatory outcome is obtained even

though no investment on human capital is considered.

Another issue that is has not been clearly addressed previously is the fact that if the problem is

exogenous differences in the quality of communication, then minority employers able to successfully

communicate with minority workers would have arbitrage opportunities that majority type employ-

ers would not. In equilibrium, competition among the former should lead to segregation and wage

parity between both groups. I address this issue by recognizing that successful communication is

the result of a learning process, thus I endogenize the accuracy of the signal and introduce dynamics

to the classical model of statistical discrimination.

5

3 The Model.

I consider an infinite horizon, dynamic economy with one productive sector. There is a continuum

of individuals who live for only two periods: during the first one they may work and, if this is the

case, they become employers (the “firms”) during the second stage of their lives. In this case, they

compete for newly born workers. If they did not work during the first stage, then they have an

outside option available that gives them a present value of k ∈ < over the two periods of their lives.6

I assume the population to be conformed by two perfectly identifiable groups: A and B at propor-

tions ρ and (1− ρ) respectively. The identifying characteristic separating A and B is not correlatedwith ability (e.g. gender or skin color). Productivity only depends on innate ability α ∈

with during this period. Individuals apply this knowledge when they become employers and are

required to make inferences about the unobserved productivity of possible new workers.

The signal θ is therefore assumed to have the following form:

θi = α + ²i

with i ∈ {A,B}, where ²i is a random noise variable uncorrelated with the ability level α; ²i isnormally distributed with mean equal to zero and variance σ2i . These assumptions imply that the

distribution of the signal θ conditional on the value of ability level, is normal with mean α and

variance σ2i , this is, θ|α,σ2i à N(α, σ2i

)with i ∈ {A,B}.

It is assumed that the variance of the signal is a decreasing function of the proportion of individuals

from each type present in the labor market at the moment employers were workers. This is, denoting

the noise of the signal for type i at time t as σ2it and the proportion of type i workers as γit, then:

σ2it+1 = f (γit)

dσ2it+1dγit

< 0 and∂2σ2it+1

∂γ2it> 0 with lim

γit→0σ2it+1 = ∞

with i ∈ {A, B} .

4 The Equilibrium

This is an overlapping generations game. Each generation participates in two stage games: the

first one as workers and in the second one as employers. This section describes two types of equi-

librium: the Within Generations Equilibrium and the Intergenerational (Stationary) Equilibrium.

The within generations equilibrium takes the variance of the signal for each type of worker in the

economy as given. Nevertheless, the values of these variances change over generations as the out-

come of the within generation equilibrium affects the proportions of workers from each group in

the next generation. The stationary equilibrium corresponds to the one in which the proportions

of workers from each type and, thus, the variances of the signal, are constant over time.

4.1 Within-Generations Equilibrium

Acknowledging the inexistence of externalities between individuals that belong to the same gener-

ation but to different groups, I consider the partial equilibrium for one generic type and analyze

7

afterwards how this equilibrium is affected when the variance of the signal θ increases.9

The equilibrium is found using backwards induction and the concept of Perfect Bayesian Equilibrium

(PBE).

Definition 1 (An Individual’s Strategy) An individual’s strategy is a combination of two de-

cision rules. The first one applies to the first period of his/her life, and the second to the second

stage:

1. Which wage offer to accept, if any. If the individual does not accept any offer, he/she takes

the outside option.

2. If the individual worked during the first period of his/her life then he/she becomes an employer

in the second period. If this is the case, given the employer’s prior beliefs about the expected

ability level of job-applicants and given the strategies followed by the other firms in the market,

this is a decision rule that indicates for every possible realization of the signal θ ∈ < the valueof the wage offer to be made to each newly born individual applying for the job: w : θ ∈< × [0, 1] 7−→

the conditional expectation function of ability coincides with its best linear predictor:

E(α|θ; σ2) = σ

2

σ2α + σ2m +

σ2ασ2α + σ2

θ (2)

The conditional expectation function of ability, E(α|θ; σ2), is continuous and monotonic on thesignal θ. Intuitively, this is because the signal is an imperfect index of ability, and it is easier for an

individual with a high ability level to provide a high value signal.

During the first period of their lives, individuals face wage offers after providing the ability signal

θ. If offers are higher than the present value of the outside option k, they take the highest offer

that is being presented to them and choose randomly in case there is more than one. Since the

expected benefits of the second period of their lives are zero and whatever offer they select is not

going to affect their payoffs during that second stage, they choose independently of their future

possible actions. If the highest wage offer is less than the present value of the alternative option,

k, they dropout from the labor market and take it. Therefore, the optimal course of action for an

individual of ability α at this point is:

s = max {w ∈ max {W} , k}

The monotonicity of the conditional expectation function and the possibility of the expected total

surplus being lower than the alternative option value k lead to postulate the existence of a minimum

threshold signal θ∗ such that, in equilibrium, workers stay in the labor market only when a signal

equal or larger than this value is provided:

Proposition 1 (Minimum Threshold Signal) There exists a unique threshold signal θ∗(k; σ2

),

such that individuals are willing to take a job only if a signal higher than or equal to this value is

provided. Given the distributional assumptions of the model, the threshold signal value θ∗(k; σ2

)is

defined by:

θ∗(k;σ2

)= k + (k −m) σ

2

σ2α(3)

Proof. Unless otherwise indicated, this and all upcoming proofs can be found in the Appendix.

As expected, the standard θ∗(k; σ2

)is monotonically increasing in the value of k. I define now the

random variable x = θ−m√σ2α+σ

2, then x à N (0, 1). Hence, the signal threshold θ∗

(k, σ2

)is redefined

in terms of this new variable as:

x∗(k, σ2

) ≡√

σ2α + σ2

σ2α(k −m) (4)

9

Thus, denoting the unconditional standard normal pdf as n (·) and its cdf as N (·), and recallingthat E (x|x ≥ x∗) = n(x∗)1−N(x∗) ; then the expected ability of those individuals who start employmentis:10

E(α|θ ≥ θ∗; σ2) = m + σ

2α√

σ2α + σ2E

(x|x ≥ x∗ (k, σ2)) (5)

As expected, this expression indicates that as the attractiveness of the outside value increases (k

increases), those individuals who start employment are expected to be of a higher ability level since

the ones who decide to drop out from the labor market are in expectation located at the bottom of

the ability distribution.

The average wage of the entire population of each group is equal to:

E(w (θ) ; σ2

)= kN

(x∗

(k, σ2

))+

(m +

σ2α√σ2α + σ2

E(x|x ≥ x∗ (k, σ2))

)(1−N (x∗ (k, σ2))) (6)

This measure is increasing and convex in the value of the alternative option:

∂E(w (θ) ; σ2

)

∂k= N

(x∗

(k, σ2

))> 0

∂2E(w (θ) ; σ2

)

∂k2=

√σ2α + σ2

σ2αn

(x∗

(k, σ2

))> 0

The unconditional probability that an individual stays out of the labor market, which coincides

with the unemployment rate, is given by:

λx∗(k,σ2t ) ≡ λ(k, σ2t

)= N

(x∗

(k, σ2t

))(7)

4.2 Discrimination in the Within-Generations Equilibrium: When the

Signal gets Noisier

This paper hypothesizes that communication failure among people from different backgrounds af-

fects the desirability of available outside options, which in turn results into different average wages

for different types of workers. Discrimination in the within generations equilibrium is then ad-

dressed assuming that the distribution of the signal for the minority group, conditional on ability

level α, is a Mean-Preserving-Spread of the distribution of the signal for the favored group (see

Rothschild and Stiglitz [16]). Therefore, my aim in this part of the paper is to look at how the

static equilibrium is affected by a change in the noise parameter σ2.10See Appendix.

10

4.2.1 The Conditional Expected Ability

When the signal loses accuracy, firms will give more weight to the information they have prior to

observe the signal and less to the specific information providing by the ability indicator. Mathe-

matically, in the conditional expected ability function, this is translated into the slope on the signal

θ being decreasing in the variance σ2 and the weight given to the unconditional expected ability m

being increasing in the same parameter. Thus, as the noise of the signal increases, the conditional

expected ability reverses towards the mean:

∂E(α|θ;σ2)

∂σ2= − σ

2α

(σ2α + σ2)2 (θ −m) :

> 0 ∀θ < m= 0 θ = m

< 0 ∀θ > m

This mean reversion process also implies that the variance of the inferred values of ability is de-

creasing in the noise of the signal:

V(E

(α|θ; σ2)) = σ

4α

σ2α + σ2

Reciprocally, the variance of the ability values that may have produced the observed value of the

signal is given by V(α|θ;σ2) = σ2σ2ασ2α+σ2 : as σ

2 increases, the range of possible values that may have

generated the observed value widens and, thus, the signal is less informative.

4.2.2 The Threshold θ∗(k, σ2

)

Proposition 2 The standard θ∗(k, σ2

)will be increasing in the noise parameter σ2 only if k > m

and decreasing otherwise.

When the value of the outside option is high (k > m), the fact that the ability signal is more

accurate for one group than for the other leads to a higher signal standard being imposed on the

group for which the signal is less informative. Nevertheless, since when the signal gets noisier high

ability individuals are comparatively more likely to provide low signals, when the value of k is small

the opposite is true. In the latter case, the gains in terms of expected ability compensates for the

larger variance of the signal, and bad signal providers are given “the benefit of the doubt”.

4.2.3 The Unemployment Rate λ(k, σ2

)

Proposition 3 The unemployment rate is decreasing in the noise of the signal whenever k < m,

and increasing otherwise.

11

The value of the unemployment rate is mainly determined by those for whom the outside option is

relevant at the moment of deciding whether or not to participate in the labor market, i.e., by those

more likely to provide signal values close to the signal threshold. If the opportunity cost is small

(k < m), the signal threshold decreases when the accuracy of the signal decreases since the expected

ability of low signal providers increases; and thus, the expected wage of these individuals increases

(see section 4.2.5). Therefore, those who, in expectation, were indifferent between whether or not

to take a job, now prefer to start employment. As result, the unemployment rate decreases.

On the other side, when k > m, the signal threshold increases as the signal lose accuracy, and

the expected benefits of those who start employment decrease for levels of the signal provided.

Therefore, the unemployment rate increases.

4.2.4 The Wage Distribution

Proposition 4 In the presence of an outside option k, identical for both types of workers, the

expected ability inside the labor market, E(α|θ ≥ θ∗ (k; σ2) ; k, σ2) is strictly decreasing in the noise

of the signal σ2.

Proposition 5 In the presence of an outside option k, identical for both types of workers, the

average wage of the entire population, E(w (θ) ; k, σ2

), is strictly decreasing in the noise of the

signal σ2.

Is this a (statistically) discriminatory outcome? Considering that the distribution of ability is the

same for both types of individual, the answer is “yes” since both groups are paid a different average

wage. Thus,

Corollary 1 ∀ σ2A < σ2B and ∀ k > −∞, a discriminatory outcome favoring group A membersholds in this economy. Formally,

If σ2A < σ2B =⇒ E

(w (θ) ; k, σ2A

)> E

(w (θ) ; k, σ2B

)

In summary, the average wage of those who are hired in the labor market and the entire population

always decreases when the noise of the ability signal increases. This result holds despite the fact

that both groups of workers have exactly the same distribution of ability and the same opportunity

cost (k). It is also perfectly possible that members from the favored group face a higher signal

threshold than that faced by minority workers, and this will happen whenever the outside option

12

is lower than the mean ability level of the population. In this case, the unemployment rate will be

higher for the favored group and lower for the discriminated type.

What is the intuition behind this result? the existence of an outside option imposes a lower bound

in the inferred ability values which is identical for both types of individuals. Nevertheless, the

conditional expected ability function of minority workers lies below the favored type’s equivalent

function for all values of the signal higher than m driving the discriminatory outcome. In absence

of this outside option, when the value of the ability indicator is low the conditional expected ability

is larger for the group with a noisier signal and this compensates for the lower values of inferred

ability in the upper tail of the signal distribution; this is why in such case both groups receive the

same average compensation.

4.2.5 Who benefits and who is harmed by an increase in the noise of the signal?

In this model, not all workers are worse-off in presence of discrimination. In fact, in expectation

high ability individuals are harmed by noisier signals and low ability workers are better-off when

communication worsens. Since, as the noise of the signal increases, high ability workers are more

likely to provide bad signals, bad signal providers are given “the benefit of the doubt” and their

actual expected ability increases when the variance of the signal increases. Similarly, low ability

individuals face now a higher probability of providing good signals, which is reflected in a decreased

expected ability for high signal providers; thus, these individuals are given the “punishment of the

doubt”.

I define now the variable x(α; σ2

)= θ−α√

σ2, then x

(α;σ2

)Ã N (0, 1) . Therefore,

x∗(α; k, σ2

) ≡ k − α√σ2

+k −m

σ2α

√σ2 ≡ x∗α (8)

will be addressed as the Normalized Individual Signal Threshold of a Worker of Ability Level α.

Then, the expected wage of an individual of ability level α is given by:

E(w

(θ|α; k, σ2)) = kN (x∗α)+

(σ2

σ2α + σ2m +

σ2ασ2α + σ2

(α +

√σ2E (x|x ≥ x∗α)

))(1−N (x∗α)) (9)

which is increasing in α and strictly larger than k for all levels of ability.

By defining the variable x(α, σ2), I normalize the distribution of the signal as a standard normal

for all ability levels. As a consequence, the normalized individual threshold x∗(α, σ2) takes a

different value for each α, moreover, it decreases as ability raises its level reflecting the fact that the

probability of fulfilling this standard is higher for high values of α than for low values of it. Thus,

13

∀α, σ2 :∂x∗

(α; σ2

)

∂α= − 1√

σ2< 0

Regarding the effect of an increase in the noise of the signal over the individual threshold, and

noticing that the probability of fulfilling the signal standard for an individual of ability α is given

by N (x∗α) :

Proposition 6 When the noise of the signal increases, given the values of k and m, fulfilling the

signal standard becomes easier for low ability levels (α < k− σ2σ2α (k −m)). This situation is reversedwhen higher ability levels are considered (α > k − σ2σ2α (k −m)).

Proof.∂x∗

(α;σ2

)

∂σ2=

12

(k −mσ2α√

σ2− k − α

σ3

):

≤ 0 ∀α ≤ k − σ2σ2α (k −m)> 0 ∀α > k − σ2σ2α (k −m)

There are two terms in the R.H.S. of the expression in this proof. The first one affects all individuals

equally and, as seen in the previous section, it induces an increase in the signal threshold only if

k > m and a decrease otherwise. The second term depends on each individual’s level of ability and,

as expected, it could lead to an increase in the individual’s threshold for high ability individuals.

This result implies that when α ≤ k − σ2σ2α (k −m), an increase in the noise of the signal actuallyfacilitates meeting the threshold for that individual, and makes it harder if α > k − σ2σ2α (k −m) .Therefore, in expectation, when the communication process worsens a higher proportion of low

ability workers is hired, and the amount of high ability individuals that start employment decreases.

Finally, considering an individual of ability level α, the effect over his/her expected wage of an

increase in the noise of the signal is given by:

∂E(w

(θ|α; k, σ2))

∂σ2= − σ

2α

(σ2α + σ2)2

((α−m) (1−N (x∗α)) +

σ2 − σ2α2√

σ2n (x∗α)

)

which leads to,

Proposition 7 ∀σ2 ∈ 0 ∀α < α∗ (k, σ2)

< 0 ∀α > α∗ (k, σ2)

Where the threshold α∗(k, σ2

)is defined by the solution to the following expression:

α∗ ≡ m− σ2 − σ2α2√

σ2E (x|x ≥ x∗α∗)

14

Proposition 8 The value of the threshold α∗(k, σ2

)is decreasing in the noise of the signal σ2.

The first of these two propositions indicates that, in expectation, individuals of ability lower than

α∗(k, σ2

)are actually better-off when the quality of communication worsens; however, this situation

reverses for high ability levels. In summary, this also implies that the situation for the whole group

of individuals worsens in expectation since high ability individuals are more likely to be in the labor

market than low ability workers. The second proposition implies that as the noise of the signal

increases, the proportion of individuals that benefits from that increase decreases.

4.3 Intergenerational (Stationary) Equilibrium

Following the previous results, the intertemporal transition in this model is given by:

λAt ≡ N(√

σ2α + σ2At (λAt−1, λBt−1)σ2α

(k −m))

∀t (10)

λBt ≡ N(√

σ2α + σ2Bt (λAt−1, λBt−1)σ2α

(k −m))

∀t (11)

There exist intertemporal negative externalities between the two groups and intertemporal positive

externalities between workers inside of each group. To see this, recall that λit represents the

proportion of type i individuals that stays out of the labor market at time t, then:

γAt (λAt, λBt) =ρ (1− λAt)

ρ (1− λAt) + (1− ρ) (1− λBt) (12)γBt (λAt, λBt) = 1− γAt (λAt, λBt) (13)

Therefore, for i, j ∈ {A,B} such that j 6= i:∂γit (λAt, λBt)

∂λit< 0

∂γit (λAt, λBt)∂λjt

> 0

Notice then that, for i, j ∈ {A,B}

σ2it+1 ≡ σ2it+1 (γit (λAt, λBt))

with ∂σ2it+1(γit(λit,λjt))

∂λit> 0 and ∂σ

2it+1(γit(λit,λjt))

∂λjt< 0.

This implies that, since the labor market participation of both groups is affected when an individual

decides whether or not to take a job, the variance of the signal of each of the groups is affected when

an individual decides to start employment: he/she will make the signal more accurate for his/her

own group during the next period, and comparatively noisier for the other type of individuals.

15

In this context, a Stationary Equilibrium is defined as the equilibrium where the proportion of

workers from each group in the labor market is constant over time and the proportion inside each

group that takes a job remains constant in time as well. This is represented by:

σ2it = σ2it−1 = σ

2i i ∈ {A,B} , ∀t (14)

Then, if condition (14) holds:

λit = λit−1 = λi i ∈ {A,B} , ∀tγit = γit−1 = γi i ∈ {A,B} , ∀t

Definition 2 Given the values of σ2α, k, m and δ, the intergenerational (stationary) equilibrium is

given by the values λA ∈ [0, 1] and λB ∈ [0, 1] such that these values solve the following system oftwo equations and two unknowns:

λA ≡ N(√

σ2α + σ2A (λA, λB)σ2α

(k −m))

∀t, λB (15)

λB ≡ N(√

σ2α + σ2B (λA, λB)σ2α

(k −m))

∀t, λA (16)

Does equilibrium exists? Given the continuity of the standard normal cdf, a fixed point argument

ensures that, for each given value of λi, a solution for λj always exists with i, j ∈ {A,B}, inparticular for the equilibrium values these two variables may take.

However, the existence of equilibrium does not ensure its stability or eventual convergence over

time, in order to have this the following condition is imposed:

Condition 1 (Stability of the Stationary Equilibrium) a stationary equilibrium λ∗ ≡ {λA, λB}is locally stable or recurrent if and only if there exists ε ≡ {εA, εB} > {0, 0} such that ∀ {λAt−1, λBt−1} ∈[λA − εA, λA + εA]× [λB − εB , λB + εB ] , the following condition holds:

∂λit∂λit−1

=n (x) (k −m)

2σ2α√

σ2α + σ2it (λAt−1, λBt−1)∂σ2it (γit−1 (λit−1, λjt−1))

∂λit−1∈ (−1, 1) , i, j ∈ {A,B}

This condition implies that the distance between λit and λit−1 for i ∈ {A, B} shrinks as t increasesonce the values of the unemployment rates are close to their stationary values. Notice also that the

value of ∂λit∂λit−1 is negative when k < m, this implies that in such a case the stationary equilibrium

is unique (the intersection of the R.H.S. of expressions (15) and (16) with the 45o line is unique).

This is not necessarily true when k > m.

16

4.4 Discrimination in the Intergenerational Equilibrium

Given that the option value is the same for both types of individual and the average wage is strictly

decreasing in the noise of the signal, then equality is reached when the variance of the signal is

the same and both groups face equal wage schedules this is when σ2At = σ2Bt, which is equivalent

to γAt−1 = γBt−1. Therefore, using expression (12), the required condition such that the signal is

equally noisier for both types of individual and, thus, both types are equally treated in this economy

at time t is given by:ρ

1− ρ =1− λBt−11− λAt−1 (17)

For each generation, the favored group will be that with a higher participation in the labor market

during the previous period since a higher participation is translated into a more accurate signal.

This in turn leads to a higher expected ability in the labor market and overall higher wages for the

entire population of that group. Thus,

Condition 2 Type A individuals will be favored over type B individuals at time t if and only if:

ρ

1− ρ >1− λBt−11− λAt−1

since in such case γAt−1 > γBt−1

Proposition 9 If ρ 6= 12 and the outside option k takes the same value for both types of individual,then no stationary equilibrium where both groups are equally treated exists.

Intuitively, the assumption of an equilibrium where equality between the two groups exists implies

that, if ρ > 12 , then the participation of the minority group (B) has to beρ

1−ρ > 1 times larger than

that of the majority group in the labor market. If that is the case, then the noise of the signal is

the same for both groups and both face the same wage schedule; thus, if the outside option is also

the same, the proportion of workers inside of each group has to be the same for both types as well.

However, this implies then that γAt > γBt, therefore σ2B > σ2A during the following period and this

is a contradiction with the assumption that an equilibrium where both groups are equally treated

exists.

Therefore, a stationary equilibrium where both groups face the same wage schedules is possible only

when the outside option value is lower for type B individuals than for members of group A, but

in such case the average wage for the entire population of type B individuals is lower than that of

17

group A.11

The next proposition follows from this result:

Proposition 10 If ρ > 12 , then:

1. If k < m, type A individuals are favored over type B individuals at any stationary equilibrium.

2. If k > m and type A is the favored group at any period t, then they will be in advantage at all

periods thereafter.

This proposition indicates that, as long as one group is in majority in the economy, the other type

of individuals will almost always be in disadvantage.

5 Endogenizing the Value of k: Effect over the Within-

Generations Equilibrium.

The constant k represents the opportunity cost for workers to stay in the labor market. Is it possible

that the value of this outside option is, at least at some levels, increasing on the noise of the signal?

If this is the case, it may explain the higher rate of black dropouts from the labor market. The

argument would be as follows: product of discrimination and lower expected wages, minorities

may need to build a parallel, informal economy and help-networks that may allow them to survive

(e.g. crime).12 It is possible that, as the number of members from an specific group engaged in

activities in this informal economy increases, the social acceptance of it inside the neighborhood

will also increase. There may also exist complementarities, such that the returns of being involved

in the informal economy activities may increase (at a decreasing rate and up to certain scale) as

more individuals participate. This informal economy should be stronger for those who have been

historically discriminated, as the network and organization involved are also stronger, therefore, it

is possible that the outside option value (k) is higher for minorities than for the favored group,

which in turn would lead to a further increase in the unemployment rate of the discriminated type11Recall from the static model results that an increase in the noise of the signal always induces a lower expected

ability inside of the labor market and lower wages for the overall group population. Therefore, on average, the type

with the noisier signal is always worse-off than the other group.12Actually and not surprisingly, the empirical literature indicates that higher rates of unemployment and nonpar-

ticipation in the labor market is correlated to higher crime rates.

18

and, as a consequence, a reinforcement of the informal economy leading to a self-perpetuation of

discrimination. A formal analysis of this argument is presented in what remains of this section.

I assume now a dynamic economy where the noise of the signal is exogenous and larger for group

B. Each generation of workers takes the value of k as given. However, the value of k depends

upon the proportion of workers that stays outside of the labor market in the previous generation.

This assumption implies that the within-generation equilibrium does not change with respect to

what is presented in the previous section of this paper; nevertheless, the value of the outside option

may change from generation to generation as the proportion of workers who remains unemployed

changes.

Therefore, denoting the proportion of individuals from a certain group that drops out from the

labor market in period t as λt,

k ≡ k (λt−1) (18)

Where the behavior of this function is defined as k : [0, 1] → Θ ⊂ < such that

k(0) > −∞ k(1) < ∞

and,

∃λ ∈ (0, 1] such that k′ (λt−1) :≥ 0 ∀λt−1 ≤ λ< 0 ∀λt−1 > λ

k′′ (λt−1) < 0 ∀λt−1 ∈ [0, 1]

The intertemporal transition of the proportion of individuals that remained unemployed is derived

from the results of section (4) and given by:

λt ≡ N(√

σ2α + σ2

σ2α(k (λt−1)−m)

)(19)

Clearly, the R.H.S. of this expression is increasing in the value of k, which intuitively means that,

as the outside option becomes more attractive, the proportion of individuals that stays out of the

labor market also increases.

In this model, the Intergenerational (Stationary) Equilibrium is defined as the situation where

the value of the outside option k is constant across generations. This is, the Within-Generations

Equilibrium is the same for every generation of individuals. It is characterized by:

λt = λt+1 = λ∗ ∀t

Formally,

19

Definition 3 Given the values of σ2α, σ2 and m, a Stationary Equilibrium will be given by the value

λ∗ ∈ (0, 1) , such that this value solves the following identity:

λ∗ = N

(√σ2α + σ2

σ2α(k (λ∗)−m)

)

Does this equilibrium exists? Given continuity of the normal cdf, a fixed point argument ensures

the existence of at least one equilibrium. Also, the border conditions (k(0) > −∞ and k(1) < ∞)imposed on the function k (λ) ensure the interiority of any equilibrium in this model (λ∗ ∈ (0, 1)).

However, the existence of equilibrium does not ensure stability or eventual convergence over time.

In order to have this, the following condition is required:

Condition 3 (Stability of the Stationary Equilibrium) a stationary equilibrium λ∗ is locally

stable or recurrent if and only if there exists ε > 0 such that ∀λt−1 ∈ [λ∗ − ε, λ∗ + ε] , the followingcondition holds:

dλtdλt−1

=

√σ2α + σ2

σ2αn (x∗ (λt−1)) k′ (λt−1) ∈ (−1, 1)

This convergence condition implies that, in a neighborhood of the equilibrium λ∗, the distance

between λt−1 and λt shrinks as t increases.

Is it possible the existence of multiple equilibria? Looking at the behavior of the function λt (λt−1)

and recalling that x∗ (λ) =√

σ2α+σ2

σ2α(k (λ)−m) ∂λt∂λt−1 =

√σ2α+σ

2

σ2αn (x∗ (λt−1)) k′ (λt−1) > 0 ⇔

k′ (λt−1) > 0Considering that the function k (λ) is increasing for small levels of λ and may eventually

become decreasing, the function λt (λt−1) is increasing in its argument as long as k′ (λt−1) > 0,

which is true for at least small values of the unemployment rate, and decreasing otherwise. This

allows for the possibility of multiple equilibria.

Looking at the concavity/convexity of λt (λt−1) ,

∂2λt∂λ2t−1

=

√σ2α + σ2

σ2αn (x∗ (λt−1))

(k′′ (λt−1)−

√σ2α + σ2

σ2αx∗ (λt−1) k′ (λt−1)

2

)

therefore, ∂2λt

∂λ2t−1≷ 0 ⇔

√σ2α+σ

2

σ2αx∗ (λt−1) ≶ k

′′(λ)k′(λ)2 . As a result of the concavity of k (λ) ,

k′′(λ)k′(λ)2 <

0. Thus, λt (λt−1) is concave for at least all values of k (λ) above m. If k (λ) has an interior

maximum, then λt (λt−1) is also concave in the proximity of the values for the unemployment rate

that maximizes the outside option.

Given the border conditions for k (λ) , if λt (λt−1) is either convex or concave in all of its range,

then the equilibrium is unique. If λt (λt−1) alternates between concavity and convexity, then it is

possible to have more than one stable equilibrium.

20

The possibility of multiple equilibria suggests another possible explanation to the persistence of

discrimination: coordination failure. If there are two stable equilibria consistent with the accuracy

of the signal, this is, two stationary and stable equilibrium values for the outside option k, then it

is possible that discrimination is sustainable even if the signal is equally accurate for both groups.

This result implies that, one of the two stable values for the alternative option behaves as a poverty

trap. A coordination failure equilibrium is characterized by the group with the higher k having

a higher expected ability in the labor market and a higher overall wage than the group with the

smaller opportunity cost. This first group will also present a higher unemployment rate than that

with a smaller alternative value:

dE(α|θ ≥ θ∗; k∗, σ2)

dk∗=

√σ2α + σ2√

σ2E (x|x ≥ x∗) (E (x|x ≥ x∗)− x∗) > 0

and,dE

(w (θ) ; k∗, σ2

)

dk∗= λ∗ > 0

This alternative explanation to the persistence of discrimination will not be pursued further in this

paper.

5.1 Discrimination: how the equilibrium is affected when the signal be-

comes noisier

Proposition 11 The stationary unemployment rate (λ∗) is affected as follows when the noise of

the signal increases.

dλ∗

dσ2:

< 0 if k (λ∗) < m

> 0 if k (λ∗) > m

Proof.dλ∗

dσ2=

12

n(x∗)x∗

σ2α+σ2

1− n (x∗)√

σ2α+σ2

σ2αk′ (λ∗)

The stability condition around the stationary equilibrium implies that the denominator of this expres-

sion is always positive, therefore the sign of this expression depends solely on the sign of x∗ (λ∗) ≡√σ2α+σ

2

σ2α(k (λ∗)−m) .

This proposition indicates that when the opportunity cost is small (x∗ < 0), the proportion of

workers that stays out of the labor market decreases when the signal loses accuracy regardless of

what happens to the equilibrium value of the outside option. This is explained because, in this

case, the gain in expected wage for low signal providers more than compensates those individuals

21

who previously were indifferent with respect to whether or not to take a job, even if the value of the

outside option may also increase. On the other side, the workers damaged by this increased variance

do not alter their previous decision since they correspond to those to whom the outside option is

almost irrelevant at the point of deciding whether or not to participate in the labor market.

On the other hand, when the opportunity cost is high (x∗ > 0), an increase in the variance of the

signal damages all individuals who were willing to work previously, so an increase in the unemploy-

ment rate follows unambiguously.

This result indicates that, in a stable equilibrium, a change in the opportunity cost as a response to a

change in the stationary unemployment rate (k′ (λ∗)) does not affect the direction of the movement

of the stationary unemployment rate, only its magnitude. Recall that:

∂λ(k, σ2

)

∂σ2=

12

n (x∗)x∗

σ2α + σ2

Hence, the magnitude effect over the unemployment rate of the change in the stationary outside

option value is:

dλ∗

dσ2=

∂λ(k,σ2)∂σ2

1− n (x∗)√

σ2α+σ2

σ2αk′ (λ∗)

:

>∂λ(k,σ2)

∂σ2 if k′ (λ∗) > 0

<∂λ(k,σ2)

∂σ2 if k′ (λ∗) < 0

Now, regarding the value of the stationary opportunity cost,

Proposition 12 When the signal becomes noisier, the effect over the stationary opportunity cost,

k∗, is:

• If k′ (λ∗) > 0, thendk∗

dσ2:

< 0 if k (λ∗) < m

> 0 if k (λ∗) > m

• If k′ (λ∗) < 0, thendk∗

dσ2:

> 0 if k (λ∗) < m

< 0 if k (λ∗) > m

The value of the stationary outside option will move in the same direction as the unemployment

rate only if the former is increasing in the value of such rate (k′ (λ∗) > 0); i.e., only if there are

complementarities in the informal sector such that the stationary value of staying unemployed is

increasing in the proportion of individuals who decide not to take a job.

These results affect the expected ability inside the labor market as follows:

22

dE(α|θ ≥ θ∗; k∗, σ2)

dσ2=

σ2αE (x|x ≥ x∗)(E (x|x ≥ x∗) x∗ − 1− x∗2)

(σ2α + σ2)32

+

√σ2α + σ2E (x|x ≥ x∗) (E (x|x ≥ x∗)− x∗)√

σ2dk∗

dσ2

With respect to how the average wage of the entire population is affected when the variance of

signal increases:

dE(w (θ) ; k∗, σ2

)

dσ2= − σ

2α

(σ2α + σ2)32n (x∗)

(1 + 2x∗2

)+ λ∗

dk∗

dσ2

In these two expressions, the first term is negative, indicating the decrease in expected ability and

average wage for the case when the value of k is exogenous. The second term of both expressions

determines the effect added when the outside option is regarded as endogenous.

Overall, the average wage of the entire population will be decreasing in the noise of the signal only

when:

dE(w (θ) ; k∗, σ2

)

dσ2< 0 ⇐⇒ dk

∗

dσ2<

σ2α

(σ2α + σ2)32E (x|x ≥ x∗) (1 + 2x∗2)

which is equivalent to:

k′ (λ∗) <2σ2α√

σ2α + σ2(λ∗ x∗1+2x∗2 + 2n (x

∗))

The R.H.S. of this expression is always weakly positive (recall that λ∗ = N (x∗)), therefore the case

where the average wage is decreasing in the noise of the signal, even though the value of the outside

option is increasing in the unemployment rate (k′ (λ∗)) and the opportunity cost is yet larger than

m (x∗ > 0) is possible. In this situation, the unemployment rate and the outside option in the

stationary equilibrium will be larger for the discriminated group but the average wage of the entire

group population will not.

6 Conclusions

In the presence of an outside option, the average ability level of those who are hired in the labor

market and the average wage of the entire population are decreasing in the variance of the ability

signal. Contrary to previous models of statistical discrimination, it is possible that the signal

23

threshold is higher for the favored group members. This result is driven just by the existence of a

reservation price, regardless of its value and the fact that the distribution of ability in both groups

is identical. When the opportunity cost k is small, the proportion of those who stay in the labor

market is increasing in the noise of the signal as a response to a higher expected benefit for low

ability individuals. This increase in expected benefits results from two effects:

1. The “benefit of the doubt”. When the accuracy of the signal decreases, high ability workers

are more likely to provide bad signals, thus the expected productivity for signal values lower

than the unconditional expected ability (m) increases.

2. The “Insurance” effect. All workers face an increased likelihood of providing extreme value

signals. However, the expected payoff function of low ability individuals is positively affected.

This is explained since the increased probability of providing signals less than their actual

ability level does not affect them as at those levels they preferred to take the outside option

anyway; and, they have the positive effect added by the increased likelihood of providing good

signals.

These same two effects affect negatively the expected benefits of the high ability workers, since

for these individuals the outside option is not binding. However and regardless of the decreased

expected benefits, these workers are more likely to stay in the labor market because expected wages

are always increasing in ability level. As a consequence, there exists a critical ability level such that

the expected wage of those whose ability level is lower than this critical value are actually better-off

when the signal becomes noisier. The opposite is true when high ability individuals are considered.

When the noise of the signal is endogenized, the result indicates that if the participation of each

type of individual in the economy are not equal but the value of the outside option is, a stationary

equilibrium where both types are equally treated does not exist, moreover, the minority group will

almost always be in a disadvantageous position (the contrary is possible only when k > m and only

if the majority group has never been in the favored position).

When the value of the outside option is regarded as endogenous, the result of the model indicates

that a stationary equilibrium where the opportunity cost and the unemployment rate are higher for

minorities, and this group receives an overall lower average than the majority type is sustainable

as well.

This models is too simple and it lacks from many features necessary to make it more realistic, in

particular, it is necessary to introduce a costly skill investment decision. This will be done in a next

paper.

24

References

[1] Aigner, Dennis and Glen Cain, “Statistical Theories of Discrimination in Labor Markets”.

Industrial and Labor Relations Review, January 1977, 30(1), 175-187.

[2] Akerlof, George, “The Economics of Caste and of the Rat Race and Other Woeful Tales”. The

Quarterly Journal of Economics, November 1976. 90(4), 599-617.

[3] Arrow, Kenneth, “Models of Job Discrimination” in A.H. Pascal, ed., Racial Discrimination

in Economic Life (Lexington, MA: D.C. Heath, 1972)

[4] Antonovics, Kate, “Persistent Racial Discrimination” University of Wisconsin-Madison, work-

ing paper.

[5] Borjas, George and Matthew Goldberg, “Biased Screening and Discrimination in the Labor

Market”. The American Economic Review, December 1978, 68(5), 918-922.

[6] Brown, “Black-White Earnings Ratios Since the Civil Rights Act of 1964: The Importance of

Labor Market Dropouts”. The Quarterly Journal of Economics, February 1984. 99(1), 31-44.

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The American Economic Review, December 1992, 83, 1220-1240.

[8] Cornell, Bradford and Ivo Welch, “Culture, Information, and Screening Discrimination”. Jour-

nal of Political Economy, 1996, 104(3), 542-571.

[9] Hogg, Robert and Allen Craig. “Introduction to Mathematical Statistics”. Fifth Edition, 1995.

Prentice Hall, Inc. New Jersey, USA.

[10] Lang, Kevin, “A Language Theory of Discrimination”. The Quarterly Journal of Economics,

May 1986, 101(2), 363-382.

[11] Lang, Kevin, “A Sorting Model of Statistical Discrimination”. Boston University, 1990.

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[12] Lundberg, Shelly and Richard Startz, “Private Discrimination and Social Intervention in Com-

petitive Labor Markets”. The American Economic Review, June 1983, 73(3), 340-347.

[13] Mailath G., Samuelson L., Shaked A., “Endogenous Inequality in Integrated Labor Markets

with Two-Sided Search”. Journal of Economic Literature, Forthcoming (1999).

25

[14] Oettinger, Gerald, “Statistical Discrimination and the Early Career Evolution of the Black-

White Wage Gap”, Journal of Labor Economics, January 1996, 14(1), 52-78.

[15] Phelps, Edmund, “The Statistical Theory of Racism and Sexism”. The American Economic

Review, September 1972, 62(4), 659-661.

[16] Rothschild, Michael and Joseph Stiglitz, “Increasing Risk: I. A Definition”. Journal of Eco-

nomic Theory, 1970, 2, 225-243.

[17] Smith, James and Finis Welch, “Race and Poverty: A Forty Year Record”. The American

Economic Review, May 1987, 77(2), 152-158.

[18] Spence, Michael, “Job Market Signaling”. Quarterly Journal of Economics, August, 1973,

87(3), 355-374.

[19] Stiglitz, Joseph, “The theory of “Screening”, Education and the Distribution of Income.” The

American Economic Review, June 1975, 65(3), 283-300.

Appendix

A Proofs

The truncated standard normal distribution expected value comes from:

E (x|x ≥ x∗) =∫∞

x∗ x∗n (x∗)

1−N (x∗) = −∫∞

x∗dn(x∗)

dx∗ dx∗

1−N (x∗) =n (x∗)

1−N (x∗)

Proof Proposition 1. The indifference condition that defines the threshold is given by:

E(α|θ∗; σ2) = σ

2

σ2α + σ2m +

σ2ασ2α + σ2

θ∗ ≡ k (20)

Therefore, isolating the value of θ∗, I obtain θ∗(k; σ2

)= k + (k −m) σ2σ2α

Proof Proposition 2.

dθ∗(k; σ2

)

dσ2=

k −mσ2α

:

< 0 if k < m

> 0 if k > m

Proof Proposition 3.

dλ(k, σ2

)

dσ2=

dN(x∗

(k, σ2

))

dx∗dx∗

(k, σ2

)

dσ2= n (x∗)

k −m2σ2α

√σ2α + σ2

:

< 0 if k < m

> 0 if k > m

26

Proof Proposition 4. Recalling that x∗ =√

σ2α+σ2

σ2α(k −m):

dE(α|θ ≥ θ∗; k, σ2)

dσ2=

σ2α

(σ2α + σ2)32E (x|x ≥ x∗) (E (x|x ≥ x∗)x∗ − 1− x∗2) < 0 (21)

This is clearly negative for all x∗ < 0 (.i.e., ∀θ∗ (k; σ2) < m), since E (x|x ≥ x∗) > 0 for all possiblevalues of x∗.

For the case x∗ > 0, numeric simulation shows that the term(E (x|x ≥ x∗) x∗ − 1− x∗2) is increasing

in x∗ until it reaches its maximum level (−0.10000567, when x∗ = 4.4623) and then decreases, reachingthe level of −1 when x∗ →∞. Therefore, this expression is always negative.

Proof Proposition 5. Recalling that x∗ =√

σ2α+σ2

σ2α(k −m):

dE(w (θ) ; k, σ2

)

dσ2= − σ

2α

(σ2α + σ2)32n (x∗)

(1 + 2x∗2

)< 0 (22)

Clearly, this expression is always negative.

Proof Proposition 7. Clearly,∂E(w(θ|α;k,σ2))

∂σ2 < 0 if (α−m) (1−N (x∗α)) +σ2−σ2α2√

σ2n (x∗α) > 0.

This last requirement is equivalent to∂E(w(θ|α;k,σ2))

∂σ2 < 0 ⇔ α > m −σ2−σ2α2√

σ2E (x|x ≥ x∗α) . I need to

prove now that once the requirement is met for a certain level of α (from now on referred to as α∗), then it

will be met for all ability levels superior to this value. So, let’s define the value of α∗ as α ∈ < such that:

α∗ ≡ m− σ2 − σ2α2√

σ2E (x|x ≥ x∗α∗)

There two possible situations when this is true:

1. m− σ2−σ2α2√

σ2E (x|x ≥ x∗α) is decreasing in α.

2. m− σ2−σ2α2√

σ2E (x|x ≥ x∗α) is increasing in α but at a rate less than 1.

The first of these two situations applies when σ2 < σ2α, and the second one applies otherwise.

1. σ2 < σ2α.

∂

∂α

(m− σ

2 − σ2α2√

σ2E (x|x ≥ x∗α)

)=

σ2 − σ2α2σ2

(∂E (x|x ≥ x∗α)

∂x∗α

)< 0 since σ2 < σ2α

This result implies also that the intersection between the curve f (α) = α and the curve h (α) =

m− σ2−σ2α2√

σ2E (x|x ≥ x∗α) is unique. Thus the value α∗ ≡ m− σ

2−σ2α2√

σ2E (x|x ≥ x∗α∗) also is.

27

2. σ2 > σ2α.

∂

∂α

(m− σ

2 − σ2α2√

σ2E (x|x ≥ x∗α)

)=

σ2 − σ2α2σ2

(∂E (x|x ≥ x∗α)

∂x∗α

)> 0

Thus, I need to prove that

12

(1− σ

2α

σ2

)

︸ ︷︷ ︸∈(0,1)

(E (x|x ≥ x∗α) (E (x|x ≥ x∗α)− x∗α)) < 1

Recall now that E (x|x ≥ x∗α) = n(x∗α)

1−N(x∗α) =1√2π

e−12 x∗2α

1−N(x∗α) . A numeric simulation of the term

(E (x|x ≥ x∗α) (E (x|x ≥ x∗α)− x∗α))

shows that it never reaches the value of 1. It is increasing and reaches a maximum when x∗α = 4.0164.

The maximum takes the value of 0.944555 and then decreases and converges to zero in the limit when

x∗α →∞. Therefore, (E (x|x ≥ x∗α) (E (x|x ≥ x∗α)− x∗α)) < 1 and the result holds.

Proof Proposition 8. I need to prove that dα∗

dσ2 < 0. Recall that α∗ ≡ m − σ2−σ2α

2√

σ2E (x|x ≥ x∗α∗) ,

then

dα∗

dσ2≡ −

14

(1√σ2

+ σ2α

σ3

)E (x|x ≥ x∗α∗)

1− 12(1− σ2ασ2

)(E (x|x ≥ x∗α∗) (E (x|x ≥ x∗α∗)− x∗α∗))

The numerator of this expression is clearly positive. The denominator is also positive for all σ2 < σ2α.

When σ2 > σ2α, the previous proof shows that12

(1− σ2ασ2

)(E (x|x ≥ x∗α) (E (x|x ≥ x∗α)− x∗α)) < 1 for

all values of α; hence, in this case the denominator is also always positive. Consequently,

dα∗

dσ2< 0 ∀σ2 > 0

Proof Proposition 9. Suppose this is not true, and assume that σ2At = σ2Bt is a stationary equilibrium.

The within generation equilibrium result implies that both groups face the same wage schedules and,

since both face the same opportunity cost k, then λAt = λBt = λt. From here, γAt = ρ (1− λt) andγBt = (1− ρ) (1− λt) . Now, given that ρ 6= 12 , then γAt 6= γBt and therefore σ2At+1 6= σ2Bt+1, which isa contradiction with the assumption that σ2A = σ

2B is a stationary equilibrium.

Proof Proposition 10. There are two cases to consider:

1. If k < m. Suppose this is not true and assume that type B is the favored group at time t, thenρ

1−ρ >1−λBt−11−λAt−1 which implies that the average wage of type B is larger than the equivalent for type

28

A during period t. Now, given that the value of the outside option is the same for both groups and

the average wage is strictly decreasing on the noise of the signal, then σ2Bt < σ2At. Recall that, in

this case, k < m implies that λBt > λAt and as result,1−λBt1−λAt < 1 <

ρ1−ρ . Therefore, group A will

also be the favored type at time t + 1.

2. If k > m. If type A individuals are the favored group at any time t, then ρ1−ρ >1−λBt−11−λAt−1 . As

consequence, σ2Bt > σ2At and type A population has a higher average wage than group B individuals

in this period. Since k > m, then λBt > λAt and as result,1−λBt1−λAt < 1 <

ρ1−ρ . Therefore, group A

will also be the favored type at time t + 1.

Proof Proposition 12.

dk∗

dσ2= k′ (λ∗)

12

n(x∗)x∗

σ2α+σ2

1− n (x∗)√

σ2α+σ2

σ2αk′ (λ∗)

As before, the stability condition around the equilibrium implies that the denominator of this expres-

sion is always positive, therefore the sign of this expression will depend solely on the sign of x∗ ≡√σ2α+σ

2

σ2α(k (λ∗)−m) and the sign of k′ (λ∗)

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