Asymmetric Information and Race in the labour market

7/29/2019 Asymmetric Information and Race in the labour market

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STATISTICAL DISCRIMINATION

IN THE LABOUR MARKET

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ECONOMICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF SOUTHAMPTON UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Luis Pinedo Caro

June 2013


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c Copyright by Luis Pinedo Caro 2013

All Rights Reserved

ii


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I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy.

(John Knowles) Principal Advisor




(...

(Another Department))




(...)

Approved for the University Committee on Graduate Studies

iii


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Preface

This thesis tells you all you need to know about...

iv


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Acknowledgments

I want to thank my supervisor John Knowles for his wisdom and patience.

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Contents

Preface iv

Acknowledgments v

Introduction 1

1 Double screening in the labour market 3

1.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Demographic structure . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Investment technology . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 Production technology . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.5 Detection technology . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.6 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.7 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Firms problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Profit maximization . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Firms constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Workers problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Screening behaviour with n groups 152.1 N groups and statistical discrimination . . . . . . . . . . . . . . . . . . 15

2.2 Changes in the baseline model . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Firms problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Solving for the Competitive Equilibrium. . . . . . . . . . . . . . . . . . 17

2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.5 Dynamics and rection functions . . . . . . . . . . . . . . . . . . . . . . 19

3 Policy 21

Conclussions 22

A Workers utility maximization problem. 23

B Joint CDF and expected value 25

C Model with perfect information -PI- 27

D Moro & Norman with concave utility 29

D.0.1 Moro & Norman wages . . . . . . . . . . . . . . . . . . . . . . . 29

D.0.2 Difference with respect FA model . . . . . . . . . . . . . . . . . 30

E Firms Assignment model -FA- 31

F Self-Selection model -SS- 33

G Welfare 35

H Identification procedure in model with 2 races 37

I Toy model of double screening 38

I.1 Utility maximization problem . . . . . . . . . . . . . . . . . . . . . . . 38I.2 Investment costs and screening variables . . . . . . . . . . . . . . . . . 39

I.3 Firms problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Bibliography 40

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List of Tables

1.1 Detection technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Parameters of the economy: . . . . . . . . . . . . . . . . . . . . . . . . 12

G.1 Different model specification . . . . . . . . . . . . . . . . . . . . . . . . 35

viii


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List of Figures

1.1 Double screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Firms decisions and human capital access . . . . . . . . . . . . . . . . 13

1.3 Assymmetric vs perfect information: Utility distribution . . . . . . . . 14

2.1 Expected profit when group B is thought to be worse. . . . . . . . . . . 18

2.2 Hiring thresholds for blacks and whites . . . . . . . . . . . . . . . . . . 19

2.3 Firms decisions and human capital access . . . . . . . . . . . . . . . . 19

2.4 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Convergence of beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

G.1 Welfare in the labour market: Utility distribution . . . . . . . . . . . . 36

G.2 GDP performance, SS vs FA vs DS . . . . . . . . . . . . . . . . . . . . 36

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2

impossition of such a policy decreases firms output and may lead the firm to change the

labour prices such that a separating equilibrium is reached. In fact, the imposition of

quotas (affirmative action) under pooling is less efficient in terms of output than under

truth-telling (i.e. separating). Therefore, whether the reduction in output is of enough

scope so as to justify a change in the equilibrium is the main focal point of this paper.

Two avenues for further research are being also considered. The first one aims at

generalizing Moro & Norman (2004) general equilibrium model of statistical discrim-

ination with performance pay and explicit firms beliefs. Comparing to the current

paper this generalization would add wages that depend on signals. This allow us to

show a continuum of equilibria where pure pooling (Moro & Normans result) and pure

separation (as explained in this paper) are just two extreme cases. Henceforth firms,

trough the price system, would be indirectly choosing the optimal level of truth-telling.

The second one has to do with an empirical application of statistical discrimination

to explain current wage differentials.

Literature development

Performance

pay and

Signalling

Endogenous

wages

Exogenous

wagesSignalling

Signalswith

Performance

pay

ex-ante

bunching

Adverse selection


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Chapter 1

Double screening in the labour

market

In this chapter we present an overlapping generations model with asymmetric informa-

tion in the labour market. The asymmetry is produced by an employer not knowing

the type of his employees, giving rise to adverse selection. In particular, the principal

cannot distinguish a high skilled worker from an unskilled one. This situation is fol-

lowed by no worker being willing to become high skilled thus leading to an inefficient

outcome. The economy, in terms of modelling, has the spirit of Coate & Loury (1993)

although wages are endogenous following Moro & Normans (2004) general equilibrium

model.

A new key feature of the model is the possibility of double screening by the firm.

On the one hand there exists ex-ante screening via the workers signals, as follows from

Coate & Lourys research, where the rationale behind investing is based on workers

expecting a stronger signal after investing in their human capital. On the other hand

we add the possibility for the firm to observe, ex-post, and with some degree of accuracy,

its workers type. The latter opens the door to provide performance pay in order to

incentivize truth telling.

On top of that the model features a continuum of profit maximizing firms, each of

which has measure 0, living in a competitive setting and a continuum of three-period-lived workers. The description of the model starts with the definition of the enviroment

in which these actors perform their actions and then continues explaining their statregic

interactions. Lastly a definition of equilibrium is provided.

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CHAPTER 1. DOUBLE SCREENING IN THE LABOUR MARKET 4

1.1 Environment

1.1.1 Demographic structure

There is a continuum of workers born every period with mass normalized to 1. All

workers belong to the same group. The workers lifes is splitted in 3 stages; first, when

they are young they decide whether to invest in their human capital or not; then, in

their adulthood they work and in the last period of their lifes they quit from their job

and spend the remaining of their income.

1.1.2 Endowments

Every worker of a newly born generation is endowed with one unit of time that is

inelastically supplied to the labour market.

1.1.3 Investment technology

All workers can invest in their human capital and thus become high skilled if they are

willing to assume a disutility . Disutilities are idiosyncratic investment costs following

a truncated normal distribution T N

, , l, h

, with l > 0. The effect of

the investment is twofold; in terms of productivity, becoming high skilled entails a

shift in complex tasks productivity from Al to Ah (Ah > Al). But it also offers the

investors higher chances of being well regarded by firms. Workers obtain a signal ,

drawn from the distribution Fh for investors and from Fl for non-investors, with Fh

first-order stochastically dominating Fl.

Definition We define the economys investment rate in human capital as the cu-

mulative density up to a given incentive to invest R+. This rate in denoted

= F(, , , l, h).

1.1.4 Production technology

Firms have access to a production technology that uses labour to produce output Y.

The technology uses the efficiency units of workers carrying out complex C and simple

S tasks. The production function is a mapping {Y : 2+ } represented by a CES

function.

Y = A [C + (1 )S]1/ (1.1)


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Where A stands for the total factor productivity and is the income share asso-

ciated to the complex task. In addition determines the degree of substitutability

between efficiency units at each task. The production function also fulfills the Inada

conditions. The efficiency units add up the number of workers times the respective

productivities. Workers in the simple task are assumed to produce the same regardless

of their background. On the contrary high skilled H workers do better the complex

tasks.

C = AhHc + AlL

c (1.2)

S = Hs + Ls (1.3)

1.1.5 Detection technology

After the tasks have been carried out, firms detect whether a worker is high or low

skilled with a certain degree of accuracy. The probability of a high skilled of being

detected as such is normalized to 1. On the contrary, the probability of a low skilled

being mismatched as a high skilled is idiosyncratic to each low skilled worker and is

distributed truncated normal T N(, , l, h).

ObservedH L

R

eal H 1 0

L 1

Table 1.1: Detection technology

The probabilities of fooling the firm are randomly drawn right after a non-investor

is offered a job dealing with complex tasks. The rationale is that a worker only gets to

know his chances of fooling a particular firm once he is inside.

1.1.6 Preferences

Workers have a discounted preference over consumption when they are adults and old.

The utility given by the consumption of goods is represented by an isoelastic utility

function u(c) = c1

1. It is assumed consumption when they are old might be uncertain

and take over two states, high, with probability q and low, with probability (1 q).


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Therefore, the lifetime utility, assumed to be additive separable, becomes:

U

ca, co,h, co,l

=ca

1 1

1 +

q

co,h1

1

1 + (1 q)

co,l1

1

1

I() (1.4)

and the utility function (net of investment costs) evaluated at the optimum is denoted

as

V = U(ca

, co,h

, co,l

) (1.5)

where (0, 1) is the discount factor and is the coefficient of relative risk aversion.

In addition ca and co are, respectively, consumption levels when the worker is an adult

and when he gets old. The indicator function I() takes the value 1 when the worker

has invested in his human capital. Also note that, for simplicity, the cost of investing

in human capital enters linearly the lifetime utility function.

1.1.7 Information

Some information is publicly known. The firms announcement on wages, the hiring

threshold and the workers signals i.

Workers information

Workers privately know the disutility for investing in their human capital, i, and their

chance to fool the employer i. The former is known as soon as they are born, the

latter once they get to know their workplace (after investment has taken place).

Firms information

Firms know the workers cost and fooling probabilities cumulative distribution func-

tions although they cannot observe the type of an individual worker, (i, i). Firms, in

addition, use the signal to obtain the posterior probability of having invested as follows:

p() =

fh()

fh() + (1 )fl() (1.6)

1.2 Firms problem

The productive sector of the economy is made of a continuum of firms, each of measure 0

producing a composite good. Because there exists a large number of firms producing the

same homogeneous good we assume they operate under perfect competition. From now


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on, and without loss of generality since the information and the technology available to

each of these firms is identical, we examine the behaviour of a representative firm.

First of all, the firm offer a menu of contracts for the workers to choose upon. These

contracs take the following form:

Definition A contract {wa

(), wo

(I)} is an agreement between the worker and the firmby which the firm offers two payments, one ex-ante that depends on the signal, and one

ex-post, that depends on the detected1 performance I.Contracts offered to workers doing simple tasks are normalized such that wo = 0

and wa() = ws, as their productivity is known from the beginning. On the contrary,

workers doing complex tasks are offered an unconditional payment wa() = p()wc

based on the posterior plus a perfomance payment conditioned on being declared high

skilled wo = wp|I = 1. The rationale on how to use the performance pay is simple,the firm will set a relatively high performance pay with an also relatively high wage toworkers doing simple tasks so as to disincentivize potential foolers.

In addition to the contracts the firm sets a hiring threshold, (, ), repre-

senting the signal upon which workers are given a chance to perform complex tasks.

The timing of announcement of contracts and thresholds is crucial; before the workers

investment decisions take place the firm will set and announce2 contracts for complex

task workers {p()wc, wp|

I = 1}, simple task workers {ws, 0} and the hiring threshold

.

Finally it is worth explaining how competition affects3 our representative firm

choices.

Definition The outside value of a worker, denoted U, is defined as the utility a worker

could achieve by putting his services back in the market.

It follows from the definition of outside value that workers needs to be offered at least,

as much they could get elsewhere. That value is denoted Us for workers offered to

perform simple tasks and Uc to workers offered to do complex tasks.

1The performance pay does not depend on the signal as no new information arises from the unionof the detected performance and the signal.

2In addition it is in the best interest of the firm to respect the contracts since any deviation wouldyield a worse outcome.

3That is an abstraction exercise since having only firm cannot lead to competition of any kind, butthe logic follows in our original setup on many small firms.


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1.2.1 Profit maximization

The objective of the representative firm is to maximize its profit. The firm, thus,

chooses wages and a hiring threshold to maximize the following function:

Max,wc,wp,ws

P = Y E[p()]wcNc wp (Hc + LcE[|D = 1]) wsNs (1.7)

Where it should be noted performance pay wp is given to all high skilled in the

complex task plus to those who succesfully managed to deceive (D) the firm. The inner

shape of the production function is:

Y = A [C + (1 )S]1/ (1.8)

where C = AhHc + AlL

c and S = Hs + Ls represent the efficiency units at each task.

The heads count of workers doing complex tasks is the proportion of expected investors and non-investors (1) willing to remain doing complex tasks, times the probabilityof being assigned to these tasks. The same logic applies to the number of workers doing

simple tasks.

Nc = Hc + Lc = [1 Fh()] + [1 Fl()](1 )FD (1.9)Ns = Hs + Ls = Fh() + Fl()(1 ) + [1 Fl()](1 )[1 FD] (1.10)

where FD is the proportion of low skilled workers offered a complex task that deceive

the firm and try to fool it.

1.2.2 Firms constraints

The maximization program of the firm is constrained by two Individual Rationality

constraints and two Incentive Compatibility constraints. The two IR constraints reflect

that the firm must offer at least the market utility for a worker to choose that firm as

his workplace, i.e. it is an ex-ante constraint.

UH [1 Fh()]Vh,cE + Fh()V

s IR.H (1.11)

UL [1 Fl()]Vl,cE FD + [1 Fl()]V

s[1 FD] + Fl()Vs IR.L (1.12)


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On the other hand, the two IC constraints state the conditions upon which workers

will not pretend to be a different type inside the firm. These constraints are aimed at

workers who are given the chance to do complex tasks. Workers assigned to do simple

tasks have no choice. These ex-post constraints work in two directions: Firstly the

IC.H makes sure high skilled individuals do not pretend to be low skilled and do simple

tasks. Then, the IC.L states how many low skilled workers are deterred to deceive the

firm. Those with low enough signals and fooling probabilities will reveal their type.

Vh,c Vs IC.H (1.13)

Vs Vl,c(, ), for some (, ) IC.L (1.14)

Note the importance of the IC.L constraint is in the growing cost of separation (i.e.

making all potential foolers to reveal their type would imply Vh,c = Vs and no worker

would invest).

1.3 Workers problem

Workers take two decisions during their life; First they decide whether to become high

skilled or not. Then, after they have entered a firm, low skilled workers given a chance

to do complex tasks decide whether to deceive his employer or not.

A new generation

0

is born

decisionInvestment

Fooling

1

decision

Workersconsume ca

Look-alike

2

investorsreceive wp

Workersconsume co

The generation

3

dies

Value function: To explain these decisions we need to know how they value the

contracts offered by the firm. Given a contract (wa(), wo) and a probability q4 of

obtaining wo, the workers indirect utility is given by

V(wa(), wo, q) = Maxca,co,h,co,l

U. (1.15)

4Note q= 1 for a high skilled worker doing complex tasks.


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where we assume workers face liquidity5 constraints.

Low skilled doing complex tasks. After the realization of their fooling probabilities

i and knowing their signal i, low skilled workers compare the utility they would get

by revealing their type and doing simple tasks (D=0, no deceive) and the expected

conditional utility if they stay doing complex tasks (D=1, deceive).

E[U|D = 1] = Vl,cE (p(i)wc, wp, i) (1.16)

E[U|D = 0] = Vs(ws, 0) (1.17)

the incentive to fool the firm is

= E[U|D = 1] E[U|D = 0] (1.18)

We can define the function that relates signals and fooling probabilities such that a

worker will accept a complex task. This probability is computed as

i =

Vs

Vl,cE (p(i)wc, wp)

(1.19)

therefore workers with combinations of (i, i) such that Vl,cE > Vs will attempt to fool

the firm (D=1) and do complex tasks. The amount of workers deceiving is given by

the joint CDF of the signals for low skilled and the fooling probabilities denoted FD.

FD is calculated as:

FD =

maxmin

maxmin()

fl(, , )f(, l, h)dd (1.20)

In addition we calculate the expected probability of fooling given deceive E[|D = 1]

as:

E[|D = 1] = max

minmaxmin()

fl(, min

, )f(, min

, h

)dd (1.21)

Note in the last equation that the lower truncation of f is moving with min.

5See the appendix for the complete development of the maximization problem.


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Investment decisions. Individuals makes decisions on their human capital as soon

as they know their investment costs. A worker compares the expected utility of invest-

ing and not investing. Given information on wages and the hiring threshold workers

calculate the incentive to invest. This is done by weighting all the potential utilities by

their respective probabilities.

E[U|I = 1] = [1 Fh()]Vh,cE + Fh()V

s (1.22)

E[U|I = 0] = [1 Fl()]Vl,cE FD + [1 Fl()]V

s[1 FD] + Fl()Vs (1.23)

the incentive to invest is defined as

= E[U|I = 1] E[U|I = 0] (1.24)

and workers with i < decide to invest I = 1. The investment rate is thus calculated

by evaluating the incentive to invest in F which gives us the percentage of investors

in the economy = F().


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1.4 Nash equilibrium

Definition Allocations {ca,i, co,i} i (0, 1), workers strategies {Ii, Di} i (0, 1),

firms strategies { }, outside values { UH, UL} and wages {wc, wp, ws} constitute a Nash

equilibrium if they are such that:

i Given market outside values { UH, UL}, firms strategies { } and wages {wc, wp, ws}

solve the firms problem:

ii i (0, 1), given firms strategies { } and wages {wc, wp, ws}, workers strategies

{Ii} solve the workers investment decisions:

iii i (0, 1), given firms strategies { } and wages {wc, wp, ws}, workers strategies

{Di} solve the workers deceiving decisions:

iv All markets clear.

1.5 Simulation

We simulate the economy with the following parameters with respect the preferences

and the available productive technology:

Table 1.2: Parameters of the economy:

Exogenous Value Meaning

parameters

0.6 Discount factor 0.99 Intertemporal substitutionA 10 Aggregate level of the technology

Ah 1 High skilled productivity doing complex tasksAl 0.5 Low skilled productivity doing simple tasks 0 Substitution coefficient between tasks 0.5 Complex tasks share in output

The first figure remarks the trade-off between two competing screening mechanisms,

signals and incentives for truth-telling.

Conclussions:


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Figure 1.1: Double screening

0

1

Fl

Dif in mean between Fh and Fl

Screening usage: Low skilled detection as signals improve

0 2 8 10

0

1

1FD

Caught by the firm

Selfincriminated

Figure 1.2: Firms decisions and human capital access

0 0.2 0.4 0.6 0.8 1 1.2 1.4

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

Salaries depending on investment costs

Mean of investment costs distribution

Salaries

wc

wp

ws

0 0.2 0.4 0.6 0.8 1 1.2 1.40.5

0

0.5

1

1.5Screening usage, thresholds

Mean of investment costs distribution


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Figure 1.3: Assymmetric vs perfect information: Utility distribution

2 1 0 1 2 30

0.5

1

1.5

2

2.5

3

Utility spreaded over investment cost

Investment cost

Utility

Assymmetric information

Perfect information

Remaining low skilled workers should be happier if the investment cost are lower

(easier access to education).

There is a trade-off between the screenings procedures.

3 different screening usage, only signalling, only detection technology, both sig-

nalling and detection technology.

Everyone is happier for having less investment costs.

Low skilled and high skilled workers achieve similar utility in the beginning of

their working lifes, but high skilled are much better later on.

The more useless are low skilled in complex tasks the more the firm uses signalling,

contrary to detection.


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Chapter 2

Screening behaviour with n groups

2.1 N groups and statistical discrimination

In this chapter we generalize the model presented in chapter one by giving space to

the co-existence of several groups. These groups have a characteristic that makes their

identification from each other inmmediate (i.e. skin colour). We add this feature

because under some conditions, the presence of different groups in an economy with

informational assymmetries may lead to statistical discrimination.

Statistical discrimination is, alongside taste for discrimination, one of the prevailing

explanations for treating individuals with similar skills in different ways. As a theory,

statistical discrimination is based on a principal whose beliefs with regards certain

groups do not reflect the reality, meaning a negative bias arises towards the group(s)

that generated less favourable beliefs.

The appearance of statistical discrimination in an economy has 2 consequences; On

the one hand generates allocative inefficiencies in the economy. On the other hand

a negative belief may become a self-fulfilling prophecy converting statistical discrimi-

nation into rational behaviour. It is with regards to the latter that this model, and

contrary to other authors research, might escape from the self-fulfulling prophecy trap

and predicts convergence in workers earnings even though this convergence is not in-

mediate.The addition of different groups in the economy require us to slightly modify the

environment of the economy. That includes modifying the demographic structure and

specifying the beliefs firms hold about each group. Then we state the firms new

maximization problem and the conditions under which statistical discrimation arises.

15


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CHAPTER 2. SCREENING BEHAVIOUR WITH N GROUPS 16

2.2 Changes in the baseline model

Now the economy is populated by n groups, each with size j for j (1, n), j N. The

addition of several groups has an effect on the decision making of the firms.

2.2.1 Information.

On the one hand firms hold a belief for each group in the economy regarding the mean

of the cost distribution, j. These beliefs might differ from each other. The relationshipbetween the different beliefs may cause some groups to face a disadvantage with respect

to other. These sets of beliefs are called discriminatory beliefs.

Definition We define a discriminatory belief as a set of firms beliefs such that j > jfor some j and

k =

k for all k = j.

2.2.2 Firms problem.

On the other hand the firms maximization problem needs to be revisited. The firm

now might choose different hiring thresholds for each group j. Note we assume the

existence of an equal pay act that constrains the firm to pay the same wage to workers

sharing the same signal and doing the same task.

Given {UH,j , UL,j}nj=1, j N, the representative firm chooses hiring thresholds j

and wages to maximize:

Maxj ,wc,wp,ws

P = Ywc

j

EjI[p()]Hc,j +

j

EjN[p()]Lc,j

wp

Hc +

j

Lc,jEj[|D = 1]

(2.1)

where the main difference is in the way we add up the number of workers at each task.

We treat workers labour from different groups as being perfect substitutes.

Nc =n

j=1j

j[1 Fh(

j)] +n

j=1j[1 FjD](1

j)[1 Fl(

j)] (2.2)

Ns =n

j=1

jjFh(j) + nj=1

j(1 j)Fl(j) + nj=1

jFjD(1 j)[1 Fl(

j)] (2.3)

Note different hiring thresholds will now restrict the access to perform complex tasks

for certain groups. Finally, the firms choice of different hiring thresholds for each group


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produces the existence of 2 n Individual Rationality constraints. For all j (1, n) we

have

UH,j [1 Fh(j)]Vh,c + Fh(

j)Vs IR.Hj (2.4)

UL,j [1 Fl(j)]Vl,cE F

jD + [1 Fl(

j)]Vs[1 FjD] + Fl(j)Vs IR.Lj (2.5)

On the other hand, we also have 2 n IC constraints stating the conditions upon

which workers will not pretend to be a different type inside the firm. These constraints

are aimed at workers who are given the chance to do complex tasks. Workers assigned

to do simple tasks have no choice. These ex-post constraints work in two directions:

Firstly the IC.H make sure high skilled individuals do not pretend to be low skilled and

do simple tasks. Then the IC.L states the degree of separation of the economy. Those

with a high enough combination of signals and fooling probabilities will give it a shot.

Vh,c(j) Vs IC.Hj (2.6)

Vs Vl,c(, ), for some (, ) IC.Lj (2.7)

Note the importance of the IC.L constraint is in the growing cost of separation (i.e.

making all potential foolers to reveal their type would imply Vh,c(, ) = Vs and no

worker would invest).

Proposition 2.2.1 When paired, a discriminatory belief and the profit maximizing

behaviour of the firm give rise to statistical discrimination.

2.3 Solving for the Competitive Equilibrium.

We process to find the competitive equilibrium of this economy works as follows:

Find the equilibrium outside utilities for high skilled under a non-discriminatorybeliefB = W.

For any B > W find the expected profit under the old outside values. If the expected profit found is positive shift UH,W up until profit becomes 0. If

the profit is negative shift UH,B up until profit becomes 0.

The new pair {UH,W, UH,B} is the new equilibrium pair.


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2.4 Simulation

We simulate the competitive equilibrium for a 2 group economy j {B, W} for a range

of B group firms beliefs

B (0.8, 1.3) while normalizing the belief for the W group

W = 0.8. That means the firm always believe group B members have, on average, a

higher cost of investing in their human capital.In the first graph we see the expected profit of the firm out of equilibrium. Given a

pair of outside values {Uh,w, Uh,b} obtained in equilibrium under a non-discriminatory

belief we compute again the economy just adding a negative firms belief towards the

B group. We see the expected profit is not always negative. It turns out that for cer-

tain parameterizations the firms ability to select different screening levels for different

groups outweigths the -expected- higher investment costs of certain workers.

Figure 2.1: Expected profit when group B is thought to be worse.

0.8 0.95 1.15 1.3

0.1

0.05

0

0.05

0.1

B

Expected

Profit

Expected profit under raceblind eq outside values

Al=0

Al=0.5

The second graph shows equilibrium behaviour of a representative firm under dif-

ferent beliefs for the B group. The firm is setting a more restrictive hiring threshold

to the W group while it allows for a comparatevly looser threshold for group B when

the belief toward the B group becomes more negative. The firm is giving workers from

an -ex-ante- believed better group a harder time joining the complex task. Why would

the firm do that?

What the firm is actually doing is swaping the screening methods for the whites as

we see in the two graphs below. It holds the screening usage for blacks while relying

on assingment rather than on self-selection for whites. The rationale is as follows, the


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Figure 2.2: Hiring thresholds for blacks and whites

0.8 0.9 1 1.1 1.2 1.3

0

0.4

0.8

1.2

Firms belief towards the B group,

B.

HiringthresholdsW

,

B

Hiring threshold for W

Hiring threshold for B

Figure 2.3: Firms decisions and human capital access

0 0.15 0.35 0.50

0.3

0.7

1

Fl

Dif in mean between FB

and FW

Screening usage: Whites low skilled detection.

0 0.15 0.35 0.50

0.3

0.7

1

1FD

Selfselected

Assigned by the firm

0 0.15 0.35 0.50

0.3

0.7

1

Fl

Dif in mean between FB and FW

Screening usage: Blacks low skilled detection.

0 0.15 0.35 0.50

0.3

0.7

1

1FD

Selfselected

Assigned by the firm

firm is not worried about whites deceiving, in any case most of them are good, but

it is particularly cautious with the blacks as their group is expected to invest less.

Therefore, even though most blacks are offered a position in the complex task few of

them are expected to accept it.

2.5 Dynamics and rection functions


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Figure 2.4: Fixed points

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fixed point and reaction function for FA and DS

Only Signalling

Double Screening45 degree line

Figure 2.5: Convergence of beliefs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W

B

B

B group RF

W group RF

45 degree line

Convergence Zone

Divergence Zone

FA model

DS model


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Chapter 3

Policy

Policies:

No equal wage act. w

c,w

, w

p,w

, w

c,b

, w

p,b

, w

s,w

, w

s,b

.

Quotas w = b and the like.

Subsidy to perfomance pay wages.

21


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Conclussions

...

22


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Appendix A

Workers utility maximization

problem.

Under certain payments

The utility obtained by high skilled workers (w1 = wc, w2 = wp) in the complex task

and any worker in the simple task (w1 = ws, w2 = 0) is known ex-ante, given wages.

To solve for the optimal allocations workers need to maximize the following function:

Max U(ca, co) =ca

1 1

1 +

co1

1

1 (A.1)

subject toca w1 (A.2)

ca + co = w1 + w2 (A.3)

The utility function evaluated at the optimum takes two values:

V =

w11

11

+ w211

1if w1 w

1+w2

1+1/

w1+w2

1+1/1

1

1 + w

1+w2

1+

1/1

1

1 if w1 > w1+w21+1/

(A.4)

Under uncertain payments

If the performance pay is uncertain, as it happen to the low skilled workers when joining

the complex task, some precautionary savings will be taken in order to insure themselves

against the risk of not having anything when they are old. The maximization program

23


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APPENDIX A. WORKERS UTILITY MAXIMIZATION PROBLEM. 24

is:

Max U

ca, co,h, co,l

=ca

1 1

1 +

co,h1

1

1 + (1 )

co,l1

1

1

(A.5)

subject to

co,h = wc + wp ca (A.6)

co,l = wc ca (A.7)

The value function analytical expression is not easy to find so we provide, on the

one hand, the first order condition that gives the optimum:

ca

=

(wc + wp ca) + (1 )(wc ca)

(A.8)

Once ca is known we can pin down future consumption using the constraints of the

problem. The value function is the utility function evaluated at the optimum V =

U(ca

, co,h

, co,l

).


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Appendix B

Joint CDF and expected value

In chapter 1 the model developed needs to calculate the joint CDF of two disjoint

truncated normal random variables and the expected value of one of the variables.

With regards to the former the joint CDF is defined as:

FD =

maxmin

maxmin()

fl(, , )f(, l, h)dd (B.1)

but a word must be said about the integration limits. The upper limits of both integrals

are given by the high end of both truncated normal distributions i.e. max = 1,

max = h. On the contrary the lower bounds calculation is more subtle. On the one

hand the lower bound of the outer integral min might depend on the higher bound of

the inner integral max as follows2:

min = max

Vs

(Vl,c(max))1,

(B.2)

On the other hand the lower bound of the inner integral depends on the outer vari-

able. The idea is that we must ensure the pairs , satisfy the incentive compatibility

constraint for a low skilled to accept a complex task.

min =Vs

(Vl,c())1(B.3)

In addition we calculate the expected probability of fooling given a worker deceives

1fl is not truncated from above2Provided the inverse ofVl,c exists.

25


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APPENDIX B. JOINT CDF AND EXPECTED VALUE 26

E[|D = 1] as:

E[|D = 1] =

maxmin

maxmin()

fl(, min, )f(,

min, h)dd (B.4)

where it has to be noted, in addition to what was explained before, that the lower

truncation of the distributions follows the lower bounds of the respectives integrals.

This is because we must ensure the volume adds up to one. In addition the order of

integration (first , then ) is not trivial; Since the signal occurs first in time, we need

to know what probabilities of fooling are IC and not vice-versa.


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Appendix C

Model with perfect information -PI-

Here we work out the details of the model presented in chapter 1, if the firms were able

to tell whether a worker has invested in his human capital straight away. That clears

out the informational asymmetry from the model and also a source of inneficiency. The

maximization problem that the representative firm faces varies slightly. In particular

note the firm does not choose hiring thresholds, nor it worries about the low skilled

workers potentially doing complex tasks as there are no deceivers. In addition we

simplify the model further by eliminating the performance payment wp, because the

firm knows ex-ante the performance of every worker and there is no reason to withhold

payments.

Maxwc,ws P = Y wcNc wsNs (C.1)

The production function reflects the change in the available information:

Y = A [ (AhNc) + (1 )(Ns)]

1/(C.2)

Where all workers doing complex tasks have productivity Ah (all invested). The heads

count of workers doing complex tasks is just the amount of expected investors while for

simple tasks is the amount of non-investors.

Nc = (C.3)Ns = (1 ) (C.4)

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APPENDIX C. MODEL WITH PERFECT INFORMATION -PI- 28

Constraints There are also changes with regards to the firms constraints. Both

Incentive Compatibility constraint are now missing in action because workers cannot

pretend there are a different type. We are left with the 2 Individual rationality con-

straints:

UH Vh,c IR.H (C.5)

UL Vs IR.L (C.6)


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Appendix D

Moro & Norman with concave

utility

This model replicates a special version of Moro & Normans signalling model. Contrary

to our approach, they assume the base wages wages wc, ws are equal to the marginal

products YC,YS . Since paying the marginal product entails 0 profit the problem sim-

plifies to an output maximization problem where the choice variables are and thehiring threshold .

Max, Y = A [C + (1 )S]1/ (D.1)

where the efficiency units are given by

C = Ah[1 Fh()] + Al(1 )[1 Fl)] (D.2)S = Fh() + (1 )Fl() (D.3)

where the investment rate is given by = F() as before. We need to add an IncentiveCompatibility constraint for high skilled

Vh,c() Vs IC.L (D.4)

D.0.1 Moro & Norman wages

The marginal products for the complex and the simple tasks are:

MPs =A

(C + (1 )S)

1 (1 )S1; (D.5)

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APPENDIX D. MORO & NORMAN WITH CONCAVE UTILITY 30

MPc =A

(C + (1 )S)

1 C1; (D.6)

and the wages actually paid to the workers performing each fo the tasks are {p(, )wc, 0}for the complex task and {ws, 0} for those doing simple tasks.

D.0.2 Difference with respect FA model

We do not have Individual Rationality constraints as we do in the FA model. That is

because zero profit is automatically achieved when assuming the firm is paying out the

marginal products. Still we can compute the outside utilities as follows:

UH = [1 Fh]Vh,cE + FhV

s (D.7)

UL = [1 Fl]Vl,cE + FlV

s (D.8)


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Appendix E

Firms Assignment model -FA-

The idea of this section is to collapse the model of double screening to a particular

version of the signalling model used by Moro & Norman (2003). We delete the detection

tecnology from the environment even though utility is still convex. We maintain the

convex utility function as it is not a crucial feature of Moro & Normans signalling

model and it allows for welfare comparisons among models of the same class.

Firms problem No detection technology means no new information is obtained afer

the tasks have been carried out. If all the relevant information is obtained ex-ante there

is no need for an ex-post payment and so the firm chooses upon a payment for complex

task workers wc, simple task workers ws and a hiring threshold .

Maxwc,ws,

P = Y wc (E[pI()Hc + E[pN()L

c]) wsNs (E.1)

The production function is as defined in the environment of the economy:

Y = A [ (AhHc + AlL

c) + (1 )(Ns)]1/

(E.2)

Where workers doing complex tasks have different producivities depending on their

investment decisions. The heads count of workers doing complex tasks eliminates the

self-selection shown in the main model. Now all workers who are offered to do a complextask will accept.

Nc = [1 Fh] + (1 )[1 Fl] (E.3)Ns = Fh + (1 )Fl (E.4)

31


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APPENDIX E. FIRMS ASSIGNMENT MODEL -FA- 32

Constraints With regards to the Individial Rationality constraints no difference is

shown apart from what arises as a result of the performance pay deletion.

UH [1 Fh]Vh,cE + FhV

s IR.H (E.5)

UL [1 Fl]Vl,cE + FlV

s IR.L (E.6)

But an important change occurs in the Incentive Compatibility constraints. Since

now there is no way to prevent low skilled workers with signals above the threshold

from deceiving the firm IC.L is always broken.

Vh,c Vs IC.H (E.7)

Vs Vl,c IC.L Broken! (E.8)

IC.L is broken for all > as a direct result of IC.H being impossed. The latter implies

a pure pooling equilibrium.


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Appendix F

Self-Selection model -SS-

This appendix describes an economy with adverse selection. For the firm to succesfuly

screen the workers it needs a variable that is correlated with workers productivity. In

this case the firm knows, after output has been realiazed, and with a certain degree of

accuracy, whether the worker invested in his human capital or not.

Contrary to the model of Firms Assingment -FA- the firm does not observe any sig-

nal positively correlated with workers investment decision when they join the company.

Still, the firm might set a hiring threshold if the detection technology is too weak.

Max,wc,wp,ws

P = Y wcNc wp (Hc + E[|D = 1]Lc) wsNs (F.1)

The production function is the standard one:

Y = A [C + (1 )S]1/ (F.2)

Note now (0, 1). The efficiency units are given by:

C = + (1 )[1 F()] (F.3)

S = (1 ) + (1 )()1 + (1 )F() (F.4)Constraints: Finally we need to define 2 individual rationality and 2 incentive com-

patibility constraints. With regards to the Individial Rationality constraints no differ-

ence is shown apart from what arises as a result of the performance pay deletion.

UH Vh,c + (1 )Vs IR.H (F.5)

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APPENDIX F. SELF-SELECTION MODEL -SS- 34

UL Vl,cE [1 F()] + VsF() + (1 )V

s IR.L (F.6)

A key difference arises on the IC.L constraint compared to the FA model. The IC.L

constraint is not broken for a certain subset of the low skilled workers invited to perform

complex tasks and, therefore, some will refuse to carry them out.

Vh,c Vs IC.H (F.7)

Vs = Vl,c() IC.L (F.8)

The IC.L holds for all < meaning there will be a semi-separating equilibrium.


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Appendix G

Welfare

Here we compare a godesque firm that knows the type of each worker with a range of

models specification featuring performance pay and signalling:

1. Firms Assingment -FA-, when the model only use signals.

2. Self Selection -SS-, when performance pay deter foolers.

3. Double screening with ex-ante flat wage -DSflat-, because wa() = wc.

4. Double Screening -DS-, combining signals and performance pay.

The characteristics of each of the specifications is summarized in the table below:

FeaturesSignals wc() Perf. pay

Models

FA SS

DS-flat DS

Table G.1: Different model specification

On the one hand we focus on the welfare distribution sorted by investment cost.

Another measure of welfare is GDP. In the next graph we show GDP in each model

for different signals quality and different detection technologys quality.

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APPENDIX G. WELFARE 36

Figure G.1: Welfare in the labour market: Utility distribution

2 1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

Utility spreaded over investment cost

Investment cost

Utility

Double Screening

Perfect information

Firms Assingment

SelfSelection

Figure G.2: GDP performance, SS vs FA vs DS


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Appendix H

Identification procedure in model

with 2 races

There is an identification problem when solving for competitive equilibrium whenever

there is more than 1 group. Even after we get rid of the IR constraints for low skilled 1

we still have to iterate two numbers UH,w, UH,b until expected profit is 0. How? Who

we are to decide some will recive more than other...!?

Idea for computing competitive equilibrium:

We have, in principle 4 IR constraints: IR.Lw, IR.Hw, IR.Lb and IR.Hb. We know

2 of them do not bind, IRLs. From the IRHs we have:

UH,w

w

Vh,c

+ (1 w

)Vs

IR.Hw

UH,b bVh,c + (1 b)Vs IR.Hb

Identification procedure:

Find the pair {UH,weq , UH,beq } that obtains 0 profit in the model without discrimi-

nation.

Set a new

b >

w .

If the resulting profit is negative make UH,b

smaller until P=0 holding UH,w

=UH,weq .

If the resulting profit is positive make UH,w higher until P=0 holding UH,b = UH,beq .

1Because they dont bind

37


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Appendix I

Toy model of double screening

This appendix offers a toy version of the double screening model developed in chapter

1. The version offered here has the advantage of an analytic solution while maintaining

the main properties of the adult model.

When aiming at an analytic solution we need workable functional forms. We simplify

the production function Y = CS and the utility function U = caco. In addition

we impose uniform distribution functions for the investment costs as well as for the

screening variables: high and low skilled signals and fooling probabilities.

We start working out the value function of a worker receiving a contract wc(), wp(I).I.1 Utility maximization problem

As before agents consume when they are adults ca and old co given a flow of payments

(wa, wo(q)) where q is the probability1 of attaining wo. The maximization problem is:

Max,ca,co,h,co,l

U = ca

qco,h + (1 q)co,l

(I.1)

subject to

co,h = wa + wo ca (I.2)

co,l = wa ca (I.3)

The marshallian demands of this worker are: ca = wa+qwo

2, co,h = w

a+(2q)wo

2and

co,l = waqwo

2if he is not borrowing constrained; if that were the case ca = wa, co,h = wo.

1Note we are generalizing, for some workers q is one (i.e. high skilled doing complex tasks), forothers wo is 0 (i.e. workers doing simple tasks).

38


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APPENDIX I. TOY MODEL OF DOUBLE SCREENING 39

The actual value function of this problem depend, thus, on whether wa is greater or

smaller than qwo.

V(wa, wo, q) =

(wa+qwo)2

4if wa > qwo

waqwo if wa < qwo(I.4)

I.2 Investment costs and screening variables

All random variables in the model are distributed uniformly for simplicity thus we have:

Investment costs U(a, b). Fooling

Fooling probabilities U(a, b)

Signals for investors h U(ah, bh)

Signals for non-investors l U(al, bl)

The cdf of an uniform distribution is F(x) = xaba

and the pdf f(x) = 1/(b a).

I.3 Firms problem


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[3] G. Borjas & M. Goldberg. Biased screening and discrimination in the labour market.

The American Economic Review, vol.68, No 5 (December 1978), pages 918-922.

[4] H. Fang & A. Moro. Theories of Statistical Discrimination and Affirmative Action:

A survey. Handbook of Social Economics, Vol I, North-Holland, (2010)

[5] A. Moro & P. Norman. Affirmative action in a competitive economy. Journal of

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[6] A. Moro & P. Norman. A general equilibrium model of statistical discrimination.

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