Labor Economics
Chapter 3
Competitive Equilibrium and
Compensating Wage Differentials
Pierre Cahuc, Stéphane Carcillo and André Zylberberg
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This chapter will:
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Describe the basic model of the labor market in competitive
equilibrium
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Analyze the interactions between supply and demand given
fiscal incidence issues
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Understand how the hedonic theory (within the perfect
situation) predicts that wage differentials compensate for the
laboriousness or danger of tasks
I
Understand how the assortative matching model shows that
the small difference in capacity leads to huge wage differntials
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Table of contents
The competitive equilibrium
Perfect comptition
Compensating wage differentials and the hedonic theory of wages
A simple model of compensating wage differentials
Estimation
Assortative matching
A competitive equilibrium with assignment
Wage rule and the superstars phenomenon
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Introduction
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Section 1 describes the basic model of the labour market in
competitive equilibrium.
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Section 2 presents how the hypothesis of perfect competition
is related to the theory of wage setting.
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I
I
Through the model of perfect competition, we shall see the
impact of taxes on employment
The hedonic theroy of wages, sketched by Adam Smith and
recently formalized by Rosen, defines differences that arise
from hard working conditions.
We will present how wage earners can choose among different
jobs with different degrees to jobs adapted to the preferences
of workers.
Section 3 describes the competitive functioning of the labor
market in a context where agents and jobs are heterogeneous.
I
In this market, the competitive functioning may lead to steeply
unequal compensation package.
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The competitive equilibrium
Perfect comptition
Compensating wage differentials and the hedonic theory of wages
A simple model of compensating wage differentials
Estimation
Assortative matching
A competitive equilibrium with assignment
Wage rule and the superstars phenomenon
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The competitive equilibrium (1)
Perfect competition with identical workers and jobs of equal
difficulty.
I Supply and demand in a simple model of the labor market:
I Production function: F (L)
I Utility function: u (R, e, θ ) = R − eθ with R income, e
I
I
employment dummy variable and θ the disutility of labor for
the individual
Those with a low θ accept it more easily than those with a
high θ
Firm’s profit: F (L) − wL and its FOC: F 0 (Ld ) = w
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The competitive equilibrium (2)
Equilibrium of perfect competition yields a collective optimum.
The planner’s choice at equilibrium entails:
F 0 [(G (w ∗ )] = w ∗
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In the competitive equilibrium model, there is no involuntary
unemployment
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It can be shown that a perfectly competitive market yields the
same allocation of resources that an omniscient planner would
have chosen
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At the competitive equilibrium, the allocation of individuals
between employment and non-participation is efficient
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Despite simplifying assumptions, perfect competition is a
simple and tractable benchmark to analyze various shocks
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Market equilibrium with perfect competition
Figure: Market equilibrium with perfect competition
I
I
Effects of a reduction of social security contributions, t < 0
Labor demand increases and so at equilibrium both labor and
wages increase
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The question of tax incidence (1)
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The essential point about tax incidence is knowing who the
end payer of the tax or the end recipient of the subsidy is
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The labor demand is F 0 (Ld ) = w (1 − t ), with t the rate of
payroll tax on the net wage w
Labor demand: F 0 (Ld ) = w (1 + t )
I t > 0, t designates a tax paid by the firm
I t < 0, t designates a subsidy paid to the firm
I
LD [w (1 + t )] = LS (w )
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The question of tax incdience (2)
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We also see that the respective amplitudes of these rises
depend on the slopes of the curves of labour supply and
demand
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The elasticity wage-tax is:
ηtw =
I
ηwd
ηws − ηwd
ηwd < 0 represents labor demand elasticity
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Illustration of the tax incidence (1)
Figure: The effects of a reduction in payroll taxes with inelastic labor
supply
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Illustration of the tax incidence (2)
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In this situation, ηtw =-1, which means any reduction in payroll
taxes is fully passed on Consequently, the level of employment
is unchanged
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Fiscal incidence is the situation in which the agent to whom
tax is charged is not the real payer
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Knowledge of the the elasticities of labor 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
Perfect comptition
Compensating wage differentials and the hedonic theory of wages
A simple model of compensating wage differentials
Estimation
Assortative matching
A competitive equilibrium with assignment
Wage rule and the superstars phenomenon
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Compensating wage differentials and the hedonic theory of
wages
In the previous section, the labor market was perfectly
homogeneous.
In reality, there is an extremely wide range of working conditions
across all jobs. Within the hedonic theory of wages, perfect
competition would ensure that such differences were compensated
by wage differentials.
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I
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Workers who expect to be recalled by their previous employers
search substantially less than the average unemployed workers
Across the 50 US states, the time spent looking for a job is
inversely correlated to the level of unemployment benefits,
with an elasticity between -1.6 and -2.2
Job seekers who likely have less access to financial resources
(e.g. because they do not have a working spouse) tend to
respond more to UI benefits than do those with greater
financial wherewithal
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Compensating Wage differentials (1)
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In the previous section, labor services were all perfectly
homogeneous and so did difficulties in a task
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Perfect competition in the labor market ought to lead to a
wage heterogeneity
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The essence of the hedonic theory of wages: job differences
were compensated for by wage differentials
Wages and the difficulty of jobs
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The effort variable e allows to measure the difficulty of jobs
The productivity, which is the production net of any costs
except wages, of every sort of job is an increasing and concave
function of effort
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Compensating Wage differentials (2)
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In this section, let us assume that there is a market for each
of the kinds of jobs that coresponds to each of these degrees
of effort
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If w (e ) denotes the equilibrium wage that applies to jobs that
demand effort e, then we have,
w (e ) = f (e )
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The problem for a worker of type θ consists of selecting a
value of effort that maximizes her satisfaction, f 0 (e ) = θ
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As f 0 (e ) < 0, e 0 (θ ) diminishes with θ. As a consequence, 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
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The hedonic theory of wages
Figure: The hedonic theory of wages
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Estimation (1)
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The main prediction of the hedonic theory of wages is that
wage differentials compensate for the conditions in which a
job is performed
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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 = x β + eα + e
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α and β are vectors of parameters to be estimated and e is a
disturbance term with zero mean (normally distributed).
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But, this model will surely have a problem of unobserved
characteristics.
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Estimation (2)
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All jobs have same productivity if the work performance is
identical
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Hence, considering the efficiency of workers and jobs constant,
wage differences reflect differences in working conditions
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If talent is unobservable and if it influences the choice of
working conditions, the model is biased, for the nonwage
characteristics of the job
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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
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Estimation (3)
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The other dedicated issue is the heterogeneity of individual
preference
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There is not necessarily unanimous agreement that certain
characteristics of jobs are disagreeable
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Without these disagreements, the predictions of the hedonic
theory of wages can only focus on certain elements that are
clearly identifiable as drawbacks or advantages for all worker
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The competitive equilibrium
Perfect comptition
Compensating wage differentials and the hedonic theory of wages
A simple model of compensating wage differentials
Estimation
Assortative matching
A competitive equilibrium with assignment
Wage rule and the superstars phenomenon
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Assortative matching (1)
I
The models examined so far assumed the existence of a large
potential number of suppliers and demanders for every type of
services
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In the hedonic wage model of the previous sectionis, there are
many markets where are a multitude of suppliers and
demanders who are price takers
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We resort to assortative matching models to analyze how the
characteristics of each workers are associated with their jobs
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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
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Assortative matching (2)
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A simple model
I F (·) the CDF of talents and G (·) the CDF of firm’s size
The equilibrium assignment function
I profit π (a, s ) = Y (a, s ) − w (a )
The wage rule and the superstars phenomenon
I The compensation function w (a ) shows that the wage is
I
increasing with talent
The remuneration function of a CEO is given by:
w (a) = w0 +
Z a
0
Y1 [x, σ (x )]dx
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Assortative matching (2) - Equlibrium assignment function
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The assortative matching model assumes that the mobility of
CEOs occurs without friction and without cost, and that
information is perfect for all agents.
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A CEO of talent a obtains a wage w (a) and the firm of size s,
that employs this CEO, obtains π (a, s ) = Y (a, s ) − w (a)
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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.
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FOC: Y1 (a, s ) = w 0 (a)
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Assortative matching (3)
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At the competitive equilibrium, the assignment function is:
Y1 [α(s ), s ] = w 0 [α(s )]
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Given the SOC, we have α0 (s ) S 0 ⇔ Y1 2[α(s ), s ] ≶ 0
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This last inequality links the direction of variation of the
assignment function with the cross derivative of the
production function
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The increasing assignment function means that the most
talented CEO is assigned to the largest firm and so on down
to the least talented CEO
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This kind of allocation is called positive assortative matching
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Assortative matching (4) - Wage rule and superstar
phenomenon
Wage is increasing with talent. Thus, greater talent is always
compensated by more wage.
w (a) = w0 +
Z a
0
Y1 [x, σ (x )] dx
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 least talent.
The equilibrium wage function of the assignment model entails
that some small differences in talent may give rise to large
differences in wage.
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An illustration: The upswing in CEO remuneration (1)
Figure: Median compensation of CEOs and other top officers from 1936
to 2005
The CEO is identified as the president of the company in firms where the CEO title is
not used. ”Other top executives” include any executives among the three highest paid
who are not the CEO. Source: Frydman and Saks (2010)
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An illustration: The upswing in CEO remuneration (1bis)
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The previous figure shows the median level of total
compensation, composed of salary, bonuses, long-term bonus
payment and stock option grants
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Gabaix and Landier (2008) explained this upswing in the
remuneration of CEO utilizing explicit functions for the
distributions of talent and firm size
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The growth of their remunerations comes from the fact that
the elasticity of average CEOs’ compensation to average firm
size is equal to 1
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Indeed, CEO’s productivity increases by 500% and equilibrium
CEO’s pay incresases by 500%
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An illustration: The upswing in CEO remuneration (2)
Figure: Total compensation and the Standard and Poor’s index
Total compensation is composed of salary, bonuses, long-term bonus payments, and
stock option grants. Based on three highest-paid officers in the largest 50 firms in
1940, 1960, and 1990 (a total of 101 firms)
Source: Frydman and Saks (2010)
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An illustration: The upswing in CEO remuneration (2bis)
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In the previous figure, we can clearly see that the 1950s and
1960s were marked by a substantial increase in the size of
firms without a simultaneous increase in CEO remuneration
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Hence, in other complementary or competing theories like
managerial rent extraction, greater power in the managerial
labor market, or increased incentive-based compensation,
partly explains the formation of CEO remuneration, depending
on the epoch in question
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Summary conclusion
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A perfectly competitive equilibrium on the labor market is
characterized by wages that match supply and demand. If all
jobs have equal laboriousness, labor supply is principally
determined by the disutility of work
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The hedonic theory of wages shows that the mechanisms of
perfect competition allow agents to choose different working
conditions, and that wage differentials compensate the
laboriousness or the danger of tasks
I
The assortative matching model explains how far firms and
workers with different characteristics match each other in the
same market. This model shows that the process of matching
may provoke very steep inequalities of remuneration among
workers of closely similar characteristics
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