Chapter 3: COMPETITIVE
EQUILIBRIUM
J. Ignacio Garcı́a Pérez
Universidad Pablo de Olavide - Department of Economics
BASIC REFERENCE: Cahuc & Zylberberg (2004), Chapter 5
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INTRODUCTION
In this chapter we will see:
Describe the basic model of the labor market in
competitive equilibrium
See how this model offers insight into the problem of
fiscal incidence
Understand why, in a situation of perfect competition,
the hedonic theory predicts that wage differentials
compensate for the laboriousness or danger of tasks
Use the assortative matching model to explain the
soaring remuneration of superstars and CEOs
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INTRODUCTION
Why does John earn a lower wage than Jane? A number of possible reasons come to
mind:
Jane stayed in school longer, or obtained a more prestigious diploma.
Jane’s work is more demanding, with heavy responsibilities. Jane is older, or has
been with her company longer.
She is more highly motivated and efficient. John works in a region where the
average wage is lower, or
Jane works in a firm with higher productivity or in a region where the demand for
labor is stronger.
One of the purposes of labour economics is to assess how relevant, and how
significant, each of these explanatory factors is.
On the theoretical level, we must specify which hypotheses are being used to justify
every answer.
The answers to this question are not trivial, and without elaborating a simple yet
rigorous conceptual framework, they cannot be given.
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INTRODUCTION
The basic frame of reference adopted by economic analysis is the model of PERFECT
COMPETITION.
When applied to labour economics, it explains the formation of wages by assuming
that they match all labour supply with all labour demand;
The basic hypotheses are that agents have no market power because there is free
entry into the market and information is perfect.
This frame of reference leads to positive conclusions about the setting of
compensation for labour, which empirical studies allow us to confirm or reject.
In the first section of this chapter, we will describe the basic model of the labour market
in competitive equilibrium.
As we shall see, the interface between supply and demand in a market where
agents are price takers leads to an efficient allocation of resources.
We shall see as well that the model of perfect competition is very useful for
evaluating the consequences of taxation.
We shall see how the impact of taxes on employment and wages depends on the
interplay between labour supply and demand.
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INTRODUCTION
In section 2, we shall see that the hypothesis of perfect competition yields a very rich
theory of wage setting when working conditions are taken into account.
Differences that arise from hard working conditions are explained by the hedonic
theory of wages.
This theory proposed by Rosen (1974) accounts for wage heterogeneity arising
from compensating differentials.
It shows that the mechanism of perfect competition provides "compensations" for
the workers who hold the hardest jobs.
Section 3 of this chapter describes the competitive functioning of the labour market in
a context where agents and jobs are heterogeneous.
The fact is that for certain occupations the heterogeneity of the services traded is
persistent and plays an important role.
This holds particularly true of the markets for "superstars".
We shall see that the competitive functioning of this type of market may lead to
steeply unequal compensation packages, which are socially efficient
inasmuch as they ensure an optimal allocation of talent.
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THE COMPETITIVE EQUILIBRIUM
A market works according to the principles of perfect competition if:
Agents are perfectly informed about the quality and the price of all the goods and
services exchanged on that particular market.
All agents are price takers.
Perfect competition with identical workers and jobs of equal difficulty
Here we will illustrate the functioning of a market on which a perfectly homogeneous
service is traded: every worker offers a service of the same quality, and the
working conditions are the same everywhere.
Let us consider a market in which a representative firm produces a consumption good
with a production function F (L) where labour, denoted by L, is the sole input.
There is a large number of workers, all of whom supply one unit of labour and receive
a wage w if they are hired.
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THE COMPETITIVE EQUILIBRIUM
Perfect competition with identical workers and jobs of equal difficulty
The welfare of a worker is evaluated using a utility function u(R, e, θ).
Income R is equal to wage w when the worker is employed, and equal to 0 when
he is not.
Parameter e measures the effort (or the disagreeability) attached to each of the
jobs.
We assume that this disagreeability is identical for all jobs, and without any loss of
generality, we shall assume that parameter e is equal to 1 if there is a hire and
equal to 0 if not.
The parameter θ ≥ 0 represents the disutility (or the opportunity cost) of
labour for the individual considered. The cumulative distribution function of this
parameter will be denoted by G(.).
In this model, all the jobs thus have the same “intrinsic” difficulty e, but individuals
react differently to the difficulty of the tasks confronting them: Those with a low θ
accept it more easily than those with a high θ.
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THE COMPETITIVE EQUILIBRIUM
Perfect competition with identical workers and jobs of equal difficulty
Finally, in order to simplify, we shall assume that an agent’s utility function takes a
linear form equal to the difference between income and the opportunity cost of labour,
or u(R, e, θ) = R − eθ.
In a competitive market, firms regard the wage as a given, and labour demand results
from the maximization of profit F (L) − wL. It is thus defined by:
F ′ (Ld ) = w
(1)
On the assumption that the marginal productivity of labour is decreasing (F ” < 0),
labour demand is a decreasing function of the wage.
In addition, a worker with an opportunity cost θ attains a level of utility equal to w − θ if
she is hired, and 0 if she does not work. Consequently, only individuals whose
opportunity cost θ is less than the wage decide to work.
If we normalize the measure of the labour force to one, then labour supply is equal to
G(w).
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THE COMPETITIVE EQUILIBRIUM
Perfect competition with identical workers and jobs of equal difficulty
The functioning of the labour market is represented in figure ??, in which the quantity
of labour is shown on the vertical axis and the wage on the horizontal axis.
Labour demand is represented by the decreasing curve Ld (w) and labour supply,
equal to G(w), is represented by an increasing curve passing through the origin.
At labour market equilibrium, supply is equal to demand. The equilibrium wage, at
which labour demand and labour supply meet, is thus defined by the relation:
F ′ [G(w∗ )] = w∗
(2)
and the equilibrium level of employment is equal to L∗ = Ld (w∗ ) = G(w∗ ).
Note that only individuals for whom the disutility of work θ is less than the equilibrium
wage w∗ decide to work.
In the competitive equilibrium model, nobody is unemployed against his will: every
worker who wishes to hold a job at the equilibrium wage w∗ can do so.
Those who choose not to work should be classified as "inactive".
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THE COMPETITIVE EQUILIBRIUM
Perfect competition with identical workers and jobs of equal difficulty
One of the most striking results of microeconomic analysis is that the equilibrium of
perfect competition yields a collective optimum.
at market equilibrium, the allocation of individuals between employment and inactivity
is efficient.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence
The model of perfect competition is grounded in over-simplified hypotheses and is thus
an imperfect representation of the functioning of many labour markets.
Still, it is highly useful for analyzing the consequences of shocks such as
alterations in the tax regime on wages and employment.
The model of perfect competition allows us to understand such interactions, which are
in fact similar in models of imperfect competition.
The fact that a tax is a charge upon the revenue of an agent (the payroll
(Social Security) taxes paid by firms, for example) does not entail that the cost is borne
by that agent.
A firm might offset a rise in payroll taxes by lowering wages.
In that case, the cost of labour to the firm remains the same, and it is the wage-earners
who finance the larger social security contributions by taking home smaller paychecks.
The essential point about tax incidence is this: knowing who the end payer of the
tax or the end recipient of the subsidy is.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence
Let us consider a firm subject to a rate t of payroll tax on the net wage w.
Its labour demand is defined by the equality F ′ (Ld ) = w(1 + t).
When t is positive, it designates a tax paid by the firm; when t is negative, it
designates a subsidy paid to the firm in the form, for example, of a reduction in social
security contributions.
Labour supply remaining equal to G(w), the equilibrium wage on the labour market is
always characterized by the equality of supply and demand which now writes:
Ld [w(1 + t)] = Ls (w).
(3)
Figure 1 illustrates the effect of a reduction in social security contributions (t < 0).
Such a reduction corresponds to an upward shift in labour demand.
labour market equilibrium then goes from E ∗ to point E t .
We see that the upshot of this payroll tax reduction is a rise in both the wage and
the level of employment.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence
We see also that the respective amplitudes of these rises depend on the slopes
of the curves of labour supply and demand.
This observation can be enhanced by differentiating both sides of relation (3) with
respect to (1 + t) and to w.
After several calculations, we find that the elasticity of the net equilibrium wage with
respect to (1 + t), denoted ηtw , is given by the formula:
ηtw
(4)
d
ηw
= s
d
ηw − ηw
s and η d < 0 represent labour supply and labour demand elasticities.
where ηw
w
We saw in chapter 1 that under many circumstances labour supply has low elasticity.
s = 0).
Let us take the extreme case of totally inelastic labour supply (ηw
In our model, this situation arises when all individuals have the same parameter θ
representing the opportunity cost of labour.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence
Put another way, all individuals have the same reservation wage, denoted wA , and
they all offer an indivisible unit of labour for every wage that exceeds the reservation
wage.
For w > wA , overall labour supply is then represented by a straight horizontal line, the
ordinate of which is the size of the active population, denoted N in figure 2.
In this situation, we have ηtw = −1, which means that any reduction in payroll taxes
is fully passed on, in the form of a rise in the equilibrium wage that leaves the
level of employment unchanged.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence
This situation, portrayed in figure 2, is a good illustration of the main point regarding
fiscal incidence:
it is not the agent to whom the tax is charged (or the subsidy awarded) who
is the real payer (or beneficiary).
The equilibrium wage goes from w∗ to w∗∗ but the level of employment remains
the same.
When labour supply is inelastic, any lowering of payroll taxes meant in principle to
aid the firm actually benefits the employee through a wage rise.
In practical terms, then, knowledge of the elasticities of labour supply and
demand proves to be of primary importance, since, as this example has just shown
us, a policy of lowering payroll taxes may lead in the end to a wage rise that leaves the
level of employment where it was.
In more general terms, knowledge of the elasticities of labour supply and demand
makes it possible to calculate the impact of a change in payroll taxes on wages and
employment.
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THE COMPETITIVE EQUILIBRIUM
The question of tax incidence: SOME NUMBERS
We know that on average the elasticity of labour supply is of the order of 0.5, while the
elasticity of labour demand is of the order of −0.3.
This means that an increase in social security contributions that ex ante augments
( i.e. at given net wage w) the cost of labour (equal to w(1 + t)) by 1% leads to a
wage variation of -0.37%.
Employment (or hours of work) therefore shrinks by 0.63·0.3=0.19%, since the
elasticity of labour demand is equal to −0.3.
The presence of a minimum wage, however, changes these outcomes. To the extent
that labour supply exceeds labour demand due to a minimum wage, the impact of
payroll taxes on employment is entirely determined by changes in labour demand, for
the same net wage.
Under these conditions, an increase in payroll taxes leading to an ex ante rise of
1% in the cost of low-skilled labour entails a fall of 1% in the employment of
low-skilled persons, since the elasticity of labour demand is of the order of −1 for this
category of workers.
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COMPENSATING WAGE DIFFERENTIALS
We have studied how a labour market would function if labour services are all perfectly
homogeneous, and the work are equally arduous no matter what job one held.
In reality, there is an extremely wide range of working conditions across all jobs.
Perfect competition in the labour markets ought to lead to wage heterogeneity,
inasmuch as some jobs are harder to do than others and some suppliers of labour are
more willing to accept hardship than others.
Perfect competition would ensure that these differences were compensated for by
wage differentials.
This is the essence of the hedonic theory of wages.
Equilibrium is still identified as a social optimum, and any measures aimed at reducing
the difficulty of jobs do not ameliorate welfare.
We will study an equilibrium model of the labour market where jobs are arduous to
varying degrees, and workers also vary in their willingness to tolerate hard labour.
In this setting, the equilibrium of perfect competition leads to an optimal allocation of
resources, with those workers whose tolerance for hardship is greatest holding
the hardest jobs and receiving higher wages.
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COMPENSATING WAGE DIFFERENTIALS
Let us introduce heterogeneity among jobs arising from the difficulty of the work
to be done.
To that end, we tangibly alter the way the production sector is formalized in the
previous model:
we now assume that there exists a continuum of jobs, each requiring one unit of
labour but a different level of effort e > 0.
This effort variable is a synthetic measure of the difficulty of jobs, and so covers a
number of dimensions like accident risk,environment, etc.
Strictly speaking, e should thus be a vector with as many coordinates as there are
characteristics to any job, but we will reduce heterogeneity to a single dimension.
The productivity of every sort of job is an increasing and concave function of effort, or
y = f (e) with f ′ (e) > 0, f ′′ (e) < 0 and f (0) = 0.
Productivity y here corresponds to production net of any costs occasioned by
employment, except wages.
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COMPENSATING WAGE DIFFERENTIALS
For example, if we interpret e as a measure of industrial accident risk, it is possible to
reduce these risks by reducing the intensity of work.
In this case, jobs that offer lower risk have less productivity in our model.
As previously, we assume that the utility function of an agent takes the linear form
u(R, e, θ) = R − eθ, where θ measures aversion to effort, and that effort e is strictly
positive when the worker is employed, and amounts to 0 when he is not participating.
Let us assume that every firm may be thought of as an occupational slot requiring one
unit of labour with its own particular degree of effort.
Let us assume further that there is a market for each of the kinds of job that
correspond to each of these degrees of effort.
In a setting of perfect competition, entrepreneurs keep on entering all markets until, for
every type of work, profits fall to zero.
If w(e) denotes the equilibrium wage that applies to jobs that demand effort e, then
wage equals productivity and we have w(e) = f (e).
A worker with information about all available jobs, and with perfect mobility, is able to
“visit” different markets and choose the job that gives her the greatest satisfaction.
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COMPENSATING WAGE DIFFERENTIALS
If she chooses a job in which effort equals e, she will receive wage f (e).
Hence the problem for a worker of type θ consists of selecting a value of effort that
maximizes her satisfaction u[f (e), e, θ] = f (e) − eθ.
The first-order condition of this problem gives:
(5)
f ′ (e) = θ ⇔ e = e(θ)
An additional requirement is to ensure that the participation constraint
u(w, e, θ) ≥ u(0, 0, θ) = 0 is met.
This constraint shows that the worker accepts a job if doing so makes her situation
preferable to non-participation (where R = e = 0 ).
When the effort function verifies relation 5, we have u(w, e, θ) = f (e) − ef ′ (e).
The latter quantity is positive, since function f is concave and thus the participation
constraint is met.
Consequently individuals with "weak" aversion to effort, i.e. those for whom θ < f ′ (0),
do participate in the labor market while the rest stay home.
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COMPENSATING WAGE DIFFERENTIALS
The size of the active population is thus equal to G [f ′ (0)] .
Equation (5) indicates that an agent chooses the job in which the marginal return to
effort f ′ (e) is equal to the disutility θ that it gives rise to.
As f ′ (e) is decreasing with e, optimal effort e(θ) diminishes with parameter θ.
Given that the equilibrium wage received by a worker of type θ amounts to
w [e(θ)] = f [e(θ)] , the counterpart of tough jobs is a compensating wage
differential, since wages increase with effort.
This point is illustrated graphically in figure ?? which represents the choices of two
types of worker.
Type θ+ is characterized by a stronger aversion for effort than type θ− < θ+ .
The effort is on the horizontal axis and the wage on the vertical axis.
The indifference curves are straight lines with slope θ.
For given θ, an upward shift of the indifference curve corresponds to increased
satisfaction.
Hence each worker chooses a level of effort e such that one of her indifference
curves is tangent to f (e).
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COMPENSATING WAGE DIFFERENTIALS
In consequence, individuals with a strong aversion to effort choose low-effort jobs with
correspondingly low wages.
More generally, at equilibrium wages are given as a function of the θ type of each
individual.
The hd function is called the hedonic wage function: It gives the equilibrium value of
the wage of a worker in line with that worker’s characteristics.
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COMPENSATING WAGE DIFFERENTIALS
ESTIMATION
The method used to test the predictions of the hedonic theory of wages consists of
estimating the wage w received by an individual as a function of his personal
characteristics, represented by a vector x, and the non-wage characteristics of the job,
represented by a vector e
ln w = xβ + eα + ε
(6)
Vector x of personal characteristics generally includes age, sex, number of years of
study or degree obtained, experience, seniority at work, ethnic origin, place of
residence, family status, and trade-union membership.
Vector e of the non-wage characteristics of jobs incorporates variables like the
duration and the flexibility of hours worked, the repetitive aspect of tasks, the risk of
injury, the level of ambient noise, the physical strength required by the job, the risk of
job loss, the cost of health insurance, the cost of saving for retirement, etc.
But, we will surely have a problem with unobserved characteristics.
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COMPENSATING WAGE DIFFERENTIALS
ESTIMATION
Individual efficiency depends on factors such as motivation or talent that as a
general rule are not observed by the econometrician.
If talent is unobservable, and if it influences the choice of working conditions, equation
(6) does not permit us to estimate correctly the impact of working conditions on
remuneration, for the non-wage characteristics of the job, represented by vector e, are
correlated with the error term ε.
For instance, good working conditions are likely to be normal goods, the “consumption”
of which increases as income rises. If the income effect is sufficiently strong, then the
most efficient individuals choose the less laborious jobs, which entails a negative
relation between wages and the laboriousness of jobs.
To escape this type of difficulty, it is preferable to make estimates using
longitudinal data that allow us to follow individuals and thus control for their
observable and unobservable time invariant personal characteristics.
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ASSORTATIVE MATCHING
The models examined so far have assumed the existence of a large potential number
of suppliers and demanders for every type of service traded.
So, there are as many markets as there are degrees of hardship, and on each of these
markets there are implicitly a multitude of agents who are price takers.
In addition, in this model the hypothesis of free entry into each market amounts to the
assumption that it is possible to transform jobs in order to adapt them to the
preferences of workers.
Such adjustments are pointing to a long-term phenomenon, the potential
transformation of jobs.
In the shorter term, it is also of interest to gain an understanding of the functioning of a
market where jobs and workers all have different characteristics, and where the
distributions of these characteristics are exogenous functions.
Under these circumstances, we must account not only for how wages are formed, but
also for how workers distribute themselves into the array of jobs they hold.
In other words, we must explain how the characteristics of workers are
associated with the characteristics of jobs.
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ASSORTATIVE MATCHING
To analyze this problem, we resort to assortative matching models.
These models are relevant for understanding the functioning of a market in which the
heterogeneity of actors is enduring and plays an important role.
Such is the case in particular for the markets for "superstars", whether they be
sports figures, artists, etc, people who possess specific talents hard to replicate.
We will study the functioning of a market of this type on the basis of an assortative
matching model that associates chief executive officers (CEOs) who have different
talents with firms of varying size.
This model explains how the remuneration of CEOs is formed, as well as the manner
in which they are allocated among firms.
As we shall see, the model allows us to understand why the remunerations of CEOs of
closely similar talents may vary steeply, and why their wage can be extremely high and
yet be socially efficient.
The reason is that the most "talented" managers are to be found in the largest
companies, which maximizes the global output of the economy.
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ASSORTATIVE MATCHING
Take the case of a continuum of workers (CEOs for present purposes) who differ in
"talent" and productivity (ability), denoted p ≥ 0.
The distribution of talents is characterized by a cumulative distribution function (CDF)
F (.).
Take as well a continuum of firms with varying capacities to produce wealth.
We may assume that this capacity is represented by the stock market value of each
firm, which we shall call its "size," denoted γ > 0, in order to simplify the vocabulary.
Their size distribution is characterized by a CDF G(.).
There is the same number, or more exactly the same mass, of workers and firms. This
mass is normalized to 1.
Most of the time the talent of a CEO is not objectively measurable.
In practice, it is convenient to use the CEO’s position in a rank rather than his talent in
order to measure his productivity.
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ASSORTATIVE MATCHING
Formally we may denote a the rank of a CEO in the distribution of abilities. By
definition, the rank falls in the interval [0, 1].
Similarly, we can index each firm by its rank, denoted by s, in the distribution of firm
sizes.
A firm of size s matched to a CEO of talent a produces an output Y (a, s) ≥ 0
It is assumed that production function Y (a, s) is increasing with the size of the firm and
the talent of the CEO.
We also assume that CEOs who do not get matched obtain a payoff of zero.
The equilibrium of this model is described by an assignment function (or matching
function) α(s) which defines the talent of the CEOs who head firms of size s, and by a
compensation function w(a) which defines the remuneration of a CEO of talent a.
More precisely, in this model a competitive equilibrium is made up of a compensation
function w(a), taken as given by each firm and each CEO, and an assignment function
α(s), such that no CEO-firm pair could do better by matching up with each other than
they are doing with their current partners, and no CEO and no firm prefers to remain
single.
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ASSORTATIVE MATCHING
THE EQUILIBRIUM ASSIGNMENT FUNCTION
The assortative matching model assumes that the mobility of CEOs occurs without
friction and without cost, and that information is perfect for all agents.
The talent of CEOs and the size of firms in particular are perfectly observable.
A CEO of talent a gets a wage w(a) and the firm of size s which employs a CEO of
talent a obtains a profit
π(a, s) = Y (a, s) − w(a)
(7)
The composite of functions {w(a), α(s)} is an equilibrium if there is no CEO-firm pair
that could do better by matching amongst themselves than they are doing with their
current partners.
The assignment function is obtained by maximizing profit (7) with respect to a.
The first order condition is then obtained by canceling the derivative of π(a, s) with
respect to a, or
Y1 (a, s) = w′ (a).
(8)
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ASSORTATIVE MATCHING
THE EQUILIBRIUM ASSIGNMENT FUNCTION
At the competitive equilibrium, the assignment function, which describes the relation
between a and s, must verify (8) for all s. We thus have:
(9)
Y1 [α(s), s] = w′ [α(s)] , ∀s
Deriving this equation with respect to s, we have:
(10)
α′ (s) =
Y12 [α(s), s]
, ∀s
w′′ [α(s)] − Y11 [α(s), s]
Given the second order condition, we have that α′ (s) ≶ 0 ⇔ Y12 [α(s), s] ≶ 0, ∀s.
This last inequality links the direction of variation of the assignment function with the
cross derivative of the production function.
The latter is said to be supermodular if Y12 ≥ 0 and submodular if Y12 ≤ 0.
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ASSORTATIVE MATCHING
THE EQUILIBRIUM ASSIGNMENT FUNCTION
In assignment models of CEOs with firms of different sizes, it is assumed that the
production function is supermodular over the whole of its support.
This amounts to stating that the marginal productivity of talent increases with the size
of the firm, that is, that talent and firm size are complementary factors of production.
That is, the assignment function is increasing: the "best" CEO (the one with the
most talent) is assigned to the largest firm, the one whose talent ranks just below is
assigned to the firm whose size ranks just below, and so on down to the least talented
CEO, who is assigned to the firm of smallest size.
Allocation of this kind is called positive assortative matching.
In this context, the assignment function and the wage function define a competitive
equilibrium, since each firm possesses a CEO whose talent maximizes its profit.
No firm then has an interest in separating from the CEO it has. Reciprocally, no CEO
can find another CEO of greater talent willing to change places with him.
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ASSORTATIVE MATCHING
THE EQUILIBRIUM ASSIGNMENT FUNCTION
The compensation function w(a) defined by equation (9) shows that the wage is
increasing with talent, for Y1 > 0.
Note that this result holds good whatever hypotheses are adopted about the cross
derivative Y12 .
Thus greater talent is always compensated by more wage, whether the production
function is supermodular or submodular.
We may go a bit further by integrating this wage rule. It writes as follows, denoting by
σ(·) the reciprocal of function α:
(11)
w(a) = w0 +
Z
a
Y1 [x, σ(x)]dx,
0
where w0 is a constant representing the remuneration of the CEO of least talent.
This equation shows that the remuneration of each CEO depends on his own marginal
productivity, as well as on the marginal productivity of all the CEOs of less talent.
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