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American English
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\begin{document}

\title{Discrimination and job-uncertainty}
\author{P. Frijters \and \and Tinbergen Institute Amsterdam}
\maketitle

\begin{abstract}
In this paper I look at the possibility of encorporating group behaviour
into a model of the labour market by showing that discrimination can be the
result of competition between coalitions of workers and bosses for a scarce
amount of jobs. Coalitions can form either on the basis of the productivity
of the individual members or on the basis of a recognisable characteristic.
If the probability of correctly assessing the productivity of individual
workers decreases, coalition-formation on the basis of recognisable
characteristics becomes relatively more rewarding than coalition forming on
the basis of productivity. I thus identify the conditions under which each
individual in the endogeneously defined group actively discriminates persons
with different recognisable characteristics, independent of productivity.
\end{abstract}

\section{Introduction}

Within the economic literature on discrimination there has been a broad
acceptance of the idea that real discrimination is not the result of
competitive forces. The argument is that any employer considering any other
trait than the productivity of a worker will be not be profit-maximizing and
thus will not remain under perfect competition. In this paper I will try to
argue that discrimination can occur between completely identical groups if
we allow for uncertainty for both employers and employees as to whether they
will retain their current jobs in future periods. This uncertainty reflects
an assumed perceived scarcity in the amount of jobs or rents.

Although the idea that discrimination is the result of the struggle over
scarce resources in the presence of uncertainty comes from sociology and
anthropology, this paper tries to add insight into the dynamics of
discrimination by constructing a simple infinite-period game whose defining
feature is that it allows for insecurity over the division of jobs. This is
done by distinguishing the total population between bosses, workers and
unemployed. Each period, the bosses are selected from the entire population
according to a probability-distribution which itself is dependent on the
outcome in the previous period. Each boss must then select one worker from
the rest of the population, whereafter boss and worker share the value of
their relationship, which is a function of their individual productivities.
Because selecting a worker increases the chance of that individual to be a
boss next period, this simple mechanism encourages coalition forming: if a
group of individuals only selects workers from its own group but allows its
members to be workers for bosses from other groups, this group increases the
number of jobs it will probably possess next period. As long as the cost of
selecting workers on the basis of group identity as opposed to
worker-productivity is not too big, a discriminating group will thus prosper
at the expense of others. Each potential group has to solve the problem of
how to recognize its members. Because of this identification problem, the
amount of information that individuals have about each other determines the
eventual outcome of the game. The possibility of mistaking low-productivity
workers for high-productivity workers is what allows coalitions to form on
the basis of recognizable characteristics in stead of on reasons of
productivity: coalitions who form on the basis of productivity are not able
to obtain all the jobs in a market because they will mistake some
low-productivity workers for high-productivity workers, thereby allowing
some of the jobs to flow to others. Coalitions on the basis of recognizable
characteristics will however obtain and keep all the scarce positions and
thus discrimination between groups of workers can occur on no other basis
than on recognizable characteristics. The possibility that any recognizable
trait can be the basis of a group seems to confirm well to reality, where
racial and ethnic identity are often found not to be rooted in a common
genetic or historical basis (Lewontin et al. (1984), Harris (1993)) but are
still used as a focal point for current group identity and group behaviour.

The organisation of the paper is as follows. Section 2 critiques some of the
theoretical models that are currently used to explain observed differences
in wages and job-opportunities for different recognizable groups. Section 3
presents the model and highlights some special cases. Section 4 concludes.\\

\section{Theories of discrimination}

The definition of discrimination I will use in the paper mirrors the
literature: real (job)-discrimination occurs whenever a job-applicant is not
hired for a position because of any characteristic not related to that
applicant's productivity in the current position or any other that may
follow as a result of hiring the worker\footnote{%
This definition does not necessarily imply that employers have a ''taste for
discrimination'' (Becker (1991)), as it does not rule out profit-maximizing
behaviour.}. As standard competitive theory predicts that productivity
related variables are the only relevant variables a profit maximizing
decision maker should consider, the question arises what the reason could be
for discrimination if it is nevertheless observed within competitive markets%
\footnote{%
Although I will make no attempt at rigourously defending my assumption that
real discrimination indeed takes place in competitive markets, references to
the recent empirical literature on this topic can be found in Botwininck
(1993) who argues that no neo-classical theory as yet has explained observed
wage-differentials between different groups of workers.}. How can one
observe discrimination when there should be none? In order to explain the
''anomaly'' of discrimination within competitive markets, a number of
explanations have been suggested. The dominant theories of the moment
explain observed wage-differentials as being the outcome of differences in
expected productivity. One type of these ''statistical discrimination''
models assumes there is something wrong with the discriminated group, be it
a higher probability of women for leaving the labour market (Polachek
(1995)), be it a greater difficulty of observing the quality of the workers,
or be it a comparative advantage in a different field of activity (Renes and
Ridder (1995), Lazear and Rosen (1990) and Becker (1991)). A second type of
these statistical discrimination models explains discrimination as a
self-fulfilling prophesy (Arrow (1973), Lang (1986), Coate and Loury (1993),
Kremer (1993)), whereby low expectations of the average productivity of a
group lead individuals in that group to undertake actions which make the
expectation come true, such as making lower investments in human capital
(Kremer (1993)).

Apart from these general explanations of discrimination, there are many
theories which can explain discrimination as the result of severe market
failures, such as monopsonists taking advantage of different labour-supply
elasticities of different groups by offering one group a lower wage rate
than the other group, higher transaction costs occurring for one group, or
the existence of segmented labour markets. However, even in these models,
the common opinion prevails that discrimination is the optimal market
response to differences in group-characteristics related to productivity,
tastes, labour supply and expectations which are formed outside the labour
market.

There are two problems with this common opinion I want to highlight.
Firstly, current theories of discrimination insufficiently incorporate
group- behaviour: the group which is not ''discriminated'' is not expected
to benefit from the lower wages of the group which is discriminated. This
seems to me to contradict reality, where groups of individuals fight bitter
wars over scarce resources, such as land, rents and high-paying jobs. The
second problem concerns the irrelevance of labour market uncertainty in
current models of discrimination. I argue instead that insecurity on the
part of workers and employers as to whether they will obtain a part of the
scarce resources encourages groups of individuals to form coalitions against
other groups of individuals to ensure future labour market success.

A small step in the direction of incorporating group behaviour and
uncertainty, whilst assuming no differences at all between different groups,
into a theory of discrimination is then the aim of this paper.

\section{A model of discrimination}

\subsection{A simple model}

In this section I will construct a model of a competitive market with
job-insecurity. The way that individuals act within that market given their
own productivity will be determined by the information available to them. To
get the flavor of the general model, I start with an exceedingly unrealistic
small model, which incorporates the notion of group competition over scarce
resources and insecurity over the division of jobs.

Take a population of four individuals, 2 of type A and 2 of type B. These
types denote some recognizable characteristic, whereas individuals are
otherwise completely anonymous for each other: in each period, player one
faces two indistinguishable individuals of type B and one individual of type
A, whom he will recognize merely by virtue of being 'the other one of type
A''. The four players have a fixed identical productivity, which is
described by a one-dimensional quality variable and observable to all
players, termed $q_i$:\\
\[
q_1=q_2=q_3=q_4=1 
\]

The four individuals play an infinitely-repeated game whereby there are only
two jobs available each period; in the first period one of the four workers
is randomly selected to be the boss for the first period. In every
subsequent period, the boss is selected randomly from the worker and the
boss in the previous period. Being a boss or a worker in one period thus
gives one a 50\% chance of becoming boss in the next period. This boss then
has to hire one of the remaining three individuals as his worker. This makes
the probability of being a boss in period t: 
\[
P_{i,t}=\frac{wage_{i,t-1}}2 
\]

The pay-offs in each period for the boss, the worker and the unemployed, are
as follows: 
\begin{eqnarray*}
Wage_{unemployed} &=&0 \\
Wage_{boss} &=&Wage_{worker}=(q_{boss}+q_{worker})/2
\end{eqnarray*}

No time-preference is assumed in this model and throughout. Persons are thus
interested in the undiscounted total flow of pay-offs, which can be
interpreted as an assumption of constant real discounted wages. I now define
a group of N individuals to be a coalition of size N if each individual in
that group will only try to hire individuals from within the group should
they become a boss. The model above gives only one perfect coalition proof
Nash-equilibrium: the two players of type A will form a coalition, as will
the two players of type B. This can easily be seen if we reflect that should
one of the individuals in these two coalitions of two be appointed boss in
period t, that this coalition will keep the two jobs available for all
subsequent periods, as the two other individuals will never get a chance to
become boss. Thus the other two workers will have a long-run expected wage
of zero unless they keep the two jobs between themselves should they become
boss, thus forming a coalition themselves. The ability of keeping a set of
jobs within a coalition is what creates the incentive for individuals to
form coalitions, whereas the impossibility of recognizing individuals makes
it impossible for a stable coalition to form on any other basis than type in
this model.\footnote{%
The superiority of forming a coalition on the basis of type as opposed to
picking a worker at random or of the opposite type, holds even when the
probabilty of being a boss next period is partially random and partially
related to previous wages: as long as $0\leq \varepsilon <1$ , $P_{it}=\frac
\varepsilon 4+\frac{(1-\varepsilon )Wage_{i,t-1}}2$, produces the same
coalitions.}

The general model will try to generalize the circumstances under which this
coalition formation on the basis of type will happen, but will be driven by
the same principle.\\

\subsection{The general model}

Consider a labour market with 2N persons, N of type A and N of type B. Apart
from their type, individuals have a productivity known only to themselves.
The distribution of productivity over the two types is identical, whereby
half of each group has a low-productivity (=1) and the other half a high
productivity (=$\delta >1$):\\

$q_1=q_2=..=q_{N/2}=1$

$q_{N/2+1}=q_{N/2+2}=..=q_N=\delta $

$q_{N+1}=q_{N+2}=..=q_{3N/2}=1$

$q_{3N/2+1}=q_{3N/2+2}=..=q_{2N}$=$\delta $\\

In stead of assuming that productivities are perfectly observed, I assume
that the actual productivity of an individual is correctly observed with a
probability $\gamma ,$ which lies between 0.5 and 1. Thus a high
productivity worker has a probability $\gamma $ of being assessed as a high
productivity worker in one period, and a probability ($1-\delta $) of being
assessed as a low-productivity worker in that period\footnote{%
it doesn't make a difference for the results if we interpret this to mean
that one worker is assessed the same by everybody else in a period, or
whether each individual is assessed differently by different persons. The
first option is however how we interpret this in the text.}. Otherwise, each
player is anonymous.

As before, we obtain a situation of scarcity by restricting the number of
bosses each period to $\frac N2$ which implies that N persons will have a
job each period and N persons will be unemployed. The pay-offs are still:\\

$Wage_{unemployed}=0$

$Wage_{boss}=Wage_{worker}=(q_{boss}+q_{worker})/$2\\

In the first period, every worker has a 25\% chance of becoming a boss, but
in each subsequent period, the chances of becoming a boss for individual i
are completely dependent on his wage relative to the average wage:\\

$P_{i=boss,t}=\frac{Wage_{i,t-1}}{4WAGE_{t-1}}$\ ,

whereby\ $WAGE_{t-1}=\frac 1{2N}\sum_{i=1}^{2N}Wage_{i,t-1}$\\

Given these probabilities and pay-offs, we can represent the model as an
infinite repetition of the following period game:\\\\{\bf Stage 1}. $\frac 2N
$ jobs are devided over the whole population according to the probability
schedule described above.\\{\bf Stage 2}. each person is assessed to either
high-productivity or low-productivity, based on their actual productivity
and $\gamma $\\{\bf Stage 3}. each boss offers a job to one person. The
person accepts or not.\\{\bf Stage 4}. Stage 3 is repeated until all bosses
have found a worker who accepts a job-offer\\{\bf Stage 5}. worker and boss
receive pay-off and the partnership is dissolved\\

Coalitions between workers can now form on the basis of both type and
quality, as quality can be (imperfectly) observed and type is perfectly
observed. The following theorem identifies the nature of these coalitions
and the circumstances under which coalitions on the basis of type will occur.

\begin{theorem}
In the game described above, there are at least two Nash-equilibria for the
set $\delta \in \left\{ 1.05,1.1,..,99.95\right\} \times \gamma \in \left\{
0.51,0.52,..,0.99\right\} $. The first Nash-equilibrium, the quality
equilibrium, is where each of the 2N players, should they become boss in any
period, will try to hire a worker of high-productivity, independent of his
type. The second Nash-equilibrium, the type-coalition, which is a
Nash-equilibrium under any value of $\gamma $ and $\delta $, occurs when
each player of type A (B) will only hire type A (B) workers and will try to
hire high-productivity type A (B) workers. The expected pay-off to the
high-productivity workers in a quality equilibrium equals X$_1(\gamma
,\delta )$, whereas the expected pay-off to high-productivity workers of
type A (B), should all bosses be of type A (B), equals Y$_1(\gamma ,\delta )$%
. If Y$_1$\TEXTsymbol{>}X$_1$ then a type-coalition will not be
coalition-proof. If X$_1>$Y$_1$ then a quality-coalition will not be
coalition proof.
\end{theorem}

The proof is long and is given in the appendix.

The intuition behind the proof is this: simply due to the law of large
numbers, we get that if all individuals of one type form a coalition and the
other group not, the coalition on the basis of type will secure all the jobs
available. For a coalition of quality to emerge, it must thus hold that for
some of the persons in the type-coalition, a quality coalition must give
them a higher expected pay-off than a succesful type-coalition would. The
workers who will most benefit from a quality coalition are going to be the
high-productivity individuals who in a type-coalition would often have to
work together with a low-quality worker. In a quality-coalition however, it
will be impossible for the high-quality workers to secure all jobs due to
the impossibility of perfectly recognizing the productivity of its members.
In a quality-coalition-equilibrium, there will thus be a number of jobs
(depending on $\delta $ and $\gamma $) which will be held by
low-productivity workers whilst some high-productivity workers will be
unemployed. Thus for every $\gamma $, $\delta $ has to be higher than some
number for quality-coalitions to give the higher-productivity workers higher
expected pay-offs then a type coalition would if the individuals of the
other type do not form a coalition on the basis of type. The reason why we
do not have to concern ourselves with the low-quality workers is because
they will, under any coalition, try to hire high-productivity workers should
they become boss. As the lower-productivity workers have a chance of one of
getting a job in a type-coalition, where they will also have higher chances
of working with a high-productivity person, the lower-productivity persons
will prefer the type-coalition if the higher-productivity persons prefer it.
Due to the fact that everyone in a type-coalition will have expected wages
of zero if only one person in that type deviates, the resulting equilibrium
is also optimal for each individual to adhere to.

By calculating the expected pay-offs for the high and low productivity
workers of either a type-coalition or a quality-coalitions for many
combinations of $\delta $ and $\gamma $, we can construct the following
graph, whereby a quality coalition will occur in the area above the line and
a type coalition will occur below the line.

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We may note two things: these simulations show that if $\gamma <0.8$, a
quality coalition cannot be a coalition proof Nash-equilibrium, independent
of $\delta .$ Secondly, the necessary quality-differential between
low-productivity workers and high productivity workers (=$\delta $), rises
quite sharply if the quality coalition is to be preferred above the
type-coalition: if $\gamma =0.99$ , $\delta $ already has to be 1.345 for a
quality coalition to occur. With $\gamma =0.95$ , $\delta $ has to be 2.303,
and with $\gamma =0.90$ , $\delta $ already has to surpass 3.502. Thus even
small probabilities of mistaking high-productivity workers for
low-productivity workers and vice versa, lead to coalition forming on the
basis of type, resulting in overall losses to the economy and discrimination.

There are several ways one can interpret the model. First of all one can
label the model as an extreme case of job-insecurity in the labour market
during the life-time of an individual, as is done in this paper. Secondly
one can view the given model as a representation of the labour dynamics
within an organisation, in which individuals compete for the best jobs
within that organisation and where individuals can help each other to obtain
the best jobs within the organisation. Unemployed in this case would then
mean having a low-paying job within the organisation. Thirdly one could
interpret the different persons not as individuals but as units (e.g.
families). This would describe the case where the progeny of a boss will not
automatically become a boss themselves but may be helped by the actions of
its altruistic parent in a previous period. Lastly, one could see the model
as being no more than an unrealistic game in which one gets discrimination
on the basis of recognizable characteristics.

The analysis thus covers more relationships than strictly that of group
dynamics in an extremely uncertain labour market but also some aspects of
internal labour market dynamics and intergenerational family decision making.

\section{Conclusions}

In this paper I have argued that discrimination can be the result of the
struggle of groups to obtain a scarce amount of resources, called jobs or
rents. In the model of this paper, the defining feature is that each period
there are a group of persons (called bosses) who are in a position to do
others favours by hiring them as workers, thereby making it more likely for
these persons to be bosses next period. This leads to a situation whereby it
is optimal for a group to internalize this favour, i.e., only to give
favours to persons within one's coalition and thereby to capture the entire
market for jobs if other groups do not do the same. As this behaviour is
dependent upon the ability to recognize persons from one's own group,
recognizable traits unrelated to productivity can become important. In the
model, a decrease in the ability to correctly identify someone's
productivity reduces the relative pay-off of forming a coalition on the
basis of productivity. Even very small probabilities (5\%) of mistaking a
low-productivity worker for a high productivity worker, mean that the
productivity differential between high-productivity workers and
low-productivity workers has to be 130\% if coalitions are to be based on
productivity instead of a recognizable characteristic.

In the light of the necessity of coalition members to recognize each other,
we could interpret networking, old-boys-clubs, free-masons, rotary clubs and
other distinction-creating groups as an attempt by individuals to provide
such a recognition for its members, whereby the individuals in such groups
increase their expected pay-offs.

Off course the idea that discrimination is caused by a struggle over scarce
resources in the presence of uncertainty is not new and has been put forward
in other fields of science before and can command some empirical support
(see e.g. Harris (1993), Cohen (1975), North and Thomas (1975)). This paper
highlights the importance of uncertainty for both the employers (called
''bosses'') and employees as to whether they will have a job next period.
This is somewhat in line with well-studied historical examples where
uncertainty in the labour markets has been connected to discrimination and
racism, such as the 1930's in Germany and South Africa (see Lipton (1989)
and Sparks (1990)). \\\\\newpage\ 

{\bf Literature }\\

\begin{enumerate}
\item  Arrow, K.J. (1973), ''The theory of discrimination'', in Ashenfelter,
O. Rees, A. (eds.), Discrimination in Labor Markets: Priceton University
Press.

\item  Becker, G. (1991), {\it A treatise on the family}. Cambridge, Mass,:
Harvard University Press.

\item  Bernheim, B.D., Peleg, B., Whinston, M.D. (1987), ''Coalition-proof
Nash equilibria'', {\it Journal of Economic Theory,} 42, pp. 1-29.

\item  Botwinick, H. (1993), ''Persistent inequalities: wage disparity under
capitalist competition'', Princeton: Princeton University Press.

\item  Coate, S., Loury, G.C. (1993), ''Will affirmative-action policies
eliminate negative stereotypes?'' American Economic Review, 83, pp.
1220-1240.

\item  Cohen, M.N. (1975), ''Population and the origins of Agriculture'' in
Steven Polgar, ed., ''Population, Ecology and social evolution'', pg.79-121.
The Hague: Mouton.

\item  Gottfries, N., McCormick, B. (1995), ''Discrimination and open
unemployment in a segmented labour market'', European Economic Review, vol.
39, pp. 1-15

\item  Harris, M., (1988), ''Culture, people, nature'', 5th edition,
HarperCollinsPublishers Inc.

\item  Kremer, M. (1993), ''The O-ring theory of economic development'',
QJE, pp. 551-575

\item  Lang, K. (1986), ''A language theory of discrimination'', Quarterly
Journal of Economics, 101, 363-382.

\item  Lazear, E., Rosen, S. (1990), ''Male-female wage differentials in job
ladders'', Journal of Labour Economics, 8, pp. S106-S123.

\item  Lewontin, R.S., Rose, S., Kamin, L (1984), {\it Not in our genes:
biology, ideology and human nature}, New York: Pantheon.

\item  Lipton, M. (1984), {\it Capitalism and Apartheid}, Aldershot: Gower.

\item  North, D.C. \& Thomas, R.P. (1973), {\it The rise of the western
world: a new economic history}, Cambridge Univ. Press.

\item  Phelps, E.S. (1973), ''The statistical theory of racism and sexism''.
American Economic Review, 62, pp. 659-661.

\item  Polachek, S. (1995), ''Human capital and the gender earnings gap'',
in Kuiper, E., Sap, J. (eds.), Out of the margin. Feminist perspectives on
economics, London: Routledge, pp. 61- 79.

\item  Renes, G., Ridder, G. (1995), ''Are women overqualified?'' Labour
Economics, 2, pp. 3-18.

\item  Sparks, A. (1990), {\it The mind of South Africa}, New York: Knopf.\\
\end{enumerate}

{\bf Appendix 1}: theorem one.\\

The proof proceeds in four steps. Firstly, I show that a situation where all
the persons of type A form a coalition and all the persons of type B form a
coalition is a Nash-equilibrium. I compare this with the second
Nash-equilibrium, which is the situation when everyone who becomes a boss
tries to hire a high-productivity worker, independent of type. I then show
that the expected pay-offs of following a quality-strategy or a
type-strategy for the four groups (high-productivity type A, low
productivity type A, high-productivity type B, low-productivity type B) have
the following form:

$
\begin{tabular}{c|c|c}
Table 1 & Type B, type-strat. & Type B, qual. strategy \\ \hline
Type A, type-strategy & $\left( 
\begin{array}{cc}
Y_1/2, & Y_2/2 \\ 
Y_1/2, & Y_2/2
\end{array}
\right) $ & $\left( 
\begin{array}{cc}
Y_1, & Y_2 \\ 
0 & 0
\end{array}
\right) $ \\ \hline
Type A, qual. strategy & $\left( 
\begin{array}{cc}
0 & 0 \\ 
Y_1 & Y_2
\end{array}
\right) $ & $\left( 
\begin{array}{cc}
X_1, & X_2 \\ 
X_1, & X_2
\end{array}
\right) $%
\end{tabular}
$

I then identify the conditions under which a type-coalition is not a
coalition-proof Nash-equilibrium (arising out of a prisoner's dilemma
between groups) and when the quality-coalition is not a coalition-proof
Nash-equilibrium (the quality coalition is not coalition proof if $Y_1>X_1$
and $Y_2>X_2)$.

If all type A workers form a coalition and all type B workers form a
coalition, then simply due to the law of large numbers, it will occur once
that all the bosses will be either all of type A or all of type B,
whereafter the type that obtained all the bosses will keep all the jobs till
eternity. If all players are thus only hiring workers from within their own
type should they become boss, each individual i of type X will thus have to
maximize the following:

\[
\lim_{T\rightarrow \infty }\left\{ E[\frac 1T\sum_{t=1}^Twage_i]\right\}
=P(X=winning)E[wage_i|X=winning] 
\]
Which means that each individual will first maximize the probability that
his type will obtain all the jobs and then maximize his wages if his type
has secured all the jobs. Consider now what happens with the probability of
obtaining all the jobs if one individual of type A deviates by not hiring a
worker of his own type. We can immediately reason from this equation that
only when it is in the interest of the type B coalition, i.e. if it
increases the chance that type B will obtain all the jobs, will someone from
type B accept a job from a type A boss. As soon as it is in the interest of
type B however to have some of its individuals working for type A, it must
be in the interest of all type A individuals not to do so. Thus we may say
that deviating behaviour can never occur if one person wants to deviate.
Thus the situation where all bosses hire workers from their own type is a
Nash-equilibrium. As to the division of jobs within a coalition, something
curious will happen in the time when both types still have jobs. As it is in
each person's interest to give the type to which he belongs as big a total
of wages as possible (which increases the probability of obtaining all
jobs), a low-productivity worker will reveal to each boss his true
productivity so that a boss will always find a high-productivity worker if
they are not already employed. Thus, until one type ''wins'',
low-productivity workers will voluntarily be unemployed.

To examine the expected pay-offs of an individual if his type wins, we now
have to look at what the strategies of individuals will be after his type
has won all the jobs. Once a type obtains all the jobs, then it becomes in
each persons interest to hire a high-productivity worker and to be hired by
a high-productivity worker (because the chance of a job is then 1, deviating
behaviour cannot change the chance of a job next period and will only have
an effect on one's own wages this period). As higher productivity persons
receive higher average wages than lower productivity persons, the number of
high-productivity bosses will be higher than low-productivity bosses. From
the remaining persons, those looking for a job, there will be a lower number
of individuals assessed as high-productivity than there are actually bosses
who are taken to be high-quality. Therefore the individuals who are taken to
be high-quality in a period will only accept job-offers from bosses who are
assessed as high quality bosses. Although the chances of being a boss in one
period or of managing to hire a high-productivity worker are dependent on
the labour market outcome last period, we can calculate the long-term
expected pay-off by assuming that expected pay-offs remain constant (we will
cheque whether this model does yield a stable outcome later) . We therefore
term D the fraction of bosses who are assessed as high-quality and who
manage to hire a worker who is assessed to be high-quality.

We name the chance of a high-quality boss to obtain a high-quality worker P.
The chance of a high-quality boss to obtain a low-quality worker is 1-P. The
chance of a high-quality individual to be hired by a high-quality boss is
termed K whereas the chance of a high-quality worker to be hired by a
low-quality boss is 1-K. Similarly, these probabilities for a low quality
boss\TEXTsymbol{\backslash}worker are termed F and G. The probability of
being a boss when one is of high-quality equals $\frac{Y_1}{Y_1+Y_2}$. In
the steady state, this set of probabilities gives expected pay-offs to
high-quality workers of Y$_1$ and to low-productivity workers a pay-off of Y$%
_2$, which are the values arising from$:$

\begin{eqnarray*}
Y_1 &=&\frac{Y_1}{Y_2+Y_1}(P\delta +(1-P)(1+\delta )/2)+\frac{Y_2}{Y_2+Y_1}%
(K\delta +(1-K)(1+\delta )/2) \\
Y_2 &=&\frac{Y_2}{Y_2+Y_1}(F(\delta +1)/2+(1-F))+\frac{Y_1}{Y_2+Y_1}%
(G(\delta +1)/2+(1-G)) \\
D &=&\frac{Y_2\gamma +Y_1(1-\gamma )}{Y_1\gamma +Y_2(1-\gamma )} \\
P &=&\gamma \frac{DY_2\gamma }{Y_2\gamma +Y_1(1-\gamma )}+((1-\gamma
)+\gamma (1-D))\frac{Y_2(1-\gamma )}{Y_2(1-\gamma )+Y_1\gamma } \\
K &=&\gamma \frac{Y_1\gamma }{Y_1\gamma +Y_2(1-\gamma )}+(1-\gamma )(\frac{%
Y_1(1-\gamma )+Y_1\gamma (1-D)}{Y_2(1-\gamma )+Y_1\gamma }) \\
F &=&(1-\gamma )\frac{DY_2\gamma }{Y_2\gamma +Y_1(1-\gamma )}+(\gamma
+(1-\gamma )(1-D))\frac{Y_2(1-\gamma )}{Y_2(1-\gamma )+Y_1\gamma } \\
G &=&(1-\gamma )\frac{Y_1\gamma }{Y_1\gamma +Y_2(1-\gamma )}+\gamma (\frac{%
Y_1(1-\gamma )+Y_1\gamma (1-D)}{Y_2(1-\gamma )+Y_1\gamma })
\end{eqnarray*}

For the moment we will assume this system, which is actually iterative, to
yield stable expected pay-offs.

In the second Nash-equilibrium each boss will try to hire a
high-productivity individual of any type. First we calculate the steady
state outcome if the strategy of each boss is to try to hire a
high-productivity individual and then show that it actually is a
Nash-equilibrium.

Because more individuals who are assessed as being high-productivity can be
hired in a period then there are bosses, one only has a chance to become
employed if one is either a boss or assessed as high-productivity in a
period. To calculate X$_1$,X$_2$, we have to calculate the steady state of
the pay-offs should there be a quality-equilibrium, when each boss will try
to hire a high-productivity worker. This leads to the following set of
equations:

\begin{eqnarray*}
X_1 &=&\alpha (\delta +U)/2+(1-\alpha )\gamma j(\delta +E)/2 \\
X_2 &=&\beta (1+U)/2+(1-\beta )(1-\gamma )j(1+E)/2 \\
\beta &=&\frac{X_2}{2X_2+2X_1} \\
\alpha &=&\frac{X_1}{2X_2+2X_1}=\frac 12-\beta \\
U &=&\frac{\gamma (1-\alpha )}{\gamma (1-\alpha )+(1-\gamma )(1-\beta )}%
\delta +\frac{(1-\gamma )(1-\beta )}{\gamma (1-\alpha )+(1-\gamma )(1-\beta )%
} \\
j &=&2(\gamma (1-\alpha )+(1-\gamma )(1-\beta )) \\
E &=&\frac \alpha {\alpha +\beta }\delta +\frac \beta {\alpha +\beta
}=2\alpha \delta +2\beta
\end{eqnarray*}

whereby U equals the expected quality of the worker when one is a boss, E
the quality of the boss of a high-quality individual and j the chance of
being hired when one is taken to be a high quality worker.

This is a Nash-equilibrium because no single worker can affect the
steady-state if the steady state is stable, and thus each worker will try to
maximize their expected pay-offs given the steady state, which automatically
means they will try to hire high-quality individuals should they become
boss, as that gives them the highest possible pay-off this period and the
highest chance of working next period. The stability of the steady state can
be shown by looking at what a deviation from the steady state does. To see
what happens, consider what happens if the steady-state $\beta $ is
decreased by $\varepsilon ,$ and the steady-state $\alpha $ is thus
increased by $\varepsilon .$ We want to show that if we calculate the
changes for one period, that $\alpha _t$ and $\beta _t$ in the next period,
say $\alpha ^{*}$ and $\beta ^{*}$ deviate less than $\varepsilon ,$ thereby
reducing the distance to the steady-state and thus returning to it. Then
there must hold: 
\begin{equation}
\alpha ^{*}=\frac{X_1-\varepsilon \frac{dX_1}{d\beta }}{2(X_2+X_1)-2%
\varepsilon (\frac{dX_1}{d\beta }+\frac{dX_1}{d\beta })}<\alpha +\varepsilon
=\frac{X_1+2\varepsilon (X_2+X_1)}{2(X_2+X_1)}  \label{a2}
\end{equation}

Which is the case if:

\begin{equation}
0<-(\frac{dX_2}{d\beta })X_1+(X_2+X_1)2(X_2+X_1)+o(\varepsilon )  \label{a3}
\end{equation}

which reduces to $\frac{dX_2}{d\beta }<2(X_1+2X_2+\frac{X_2^2}{X_1})$

Now, if we explicitly calculate this condition, we get a six-degree
polynomial of $\beta $ for which we can only cheque numerically if it indeed
satisfies this condition of stability for particular values of $\delta $ and 
$\gamma $ . Doing this for the set $\delta \in \left\{
1.05,1.1,..,99.95\right\} \times \gamma \in \left\{
0.51,0.52,..,0.99\right\} $, we can indeed confirm that for these values, (%
\ref{a3}) is satisfied. Indeed, a similar excercise for the type-coalition
confirms that that also has stable expected pay-offs. $\alpha $ is thus a
stable parameter under small deviations. Now, as $\alpha =\frac 12-\beta ,$
stability will then also hold for $\beta $. As all other parameters only
vary because of variations in $\alpha ,$ and $\beta $, the pay-offs will
also be stable. Thus, as small errors 'dampen out', the whole
quality-equilibrium is stable and individuals will thus take the
steady-state as given as their individual actions will not change the
steady-state. Thus individuals will maximize their expected wages, which
will mean trying to hire a high-productivity worker should they become boss
and accepting all job-offers. As this is essentially a flow-model with two
distinct groups (high-productivity and low-productivity), who can either
become boss or not, it does not seem likely that this does not hold for all
the intermediate values of $\delta $ and $\gamma $.

We may note in passing that the reason why low-productivity workers will not
form a sub-coalition by trying to hire only low-productivity workers is
because they face a free-riders problem in that each individual's
probability of obtaining jobs in future is not affected by his hiring
decision now, making it optimal to deviate from what otherwise might be
beneficial for the low-productivity individuals as a whole: because workers
of the same productivity cannot perfectly recognize each other, it is not
possible for either the high-productivity workers or the low-productivity
workers to obtain all jobs. A deviation from low-productivity workers from
the quality coalition might thus be an improvement for their pay-offs but is
not self-enforcing.

However, these two models define Y$_1$,Y$_2$,X$_1$, and X$_2$, as
high-degree polynomials of $\gamma $ and $\delta $ which are not easily
solvable or manipulable. Given the structure of this game, we can see that
the quality-coalition cannot be a perfect coalition proof Nash-equilibrium
if Y$_1>$X$_1$, and Y$_2>$X$_2$, as that would mean that a type-coalition
gives higher expected pay-offs to all the individuals of one type should the
workers from the other type not form a type-coalition. It is always the case
that Y$_2>$X$_2$ because in a type-coalition the lower-productivity workers
both have a higher probability of a job and will have a higher probability
of working with a high-productivity individual. If Y$_1>$X$_1$, then the
high-quality individuals can increase their expected pay-offs by forming a
high-quality coalition, leaving the low-quality bosses no better alternative
but also to try to hire high-productivity workers of any type. By
calculating the pay-offs for many combinations, we can however get an idea
of when the conditions for one of the two Nash-equilibria not to be perfect
coalition-proof, prevail, which is done in figure 1.

\end{document}
