Chapter 3: COMPETITIVE EQUILIBRIUM...J. Ignacio Garc´ıa Perez´ Universidad Pablo de Olavide -...

Chapter 3: COMPETITIVEEQUILIBRIUM

J. Ignacio Garcıa Perez

Universidad Pablo de Olavide - Department of Economics

BASIC REFERENCE: Cahuc & Zylberberg (2004), Chapter 5

October 2013 LABOUR ECONOMICS J. Ignacio Garcia-Perez – p. 1/32

INTRODUCTION

In this chapter we will see:

Describe the basic model of the labor market incompetitive equilibrium

See how this model offers insight into the problem offiscal incidence

Understand why, in a situation of perfect competition,the hedonic theory predicts that wage differentialscompensate for the laboriousness or danger of tasks

Use the assortative matching model to explain thesoaring remuneration of superstars and CEOs


INTRODUCTION

Why does John earn a lower wage than Jane? A number of possible reasons come tomind:

Jane stayed in school longer, or obtained a more prestigious diploma.

Jane’s work is more demanding, with heavy responsibilities. Jane is older, or hasbeen with her company longer.

She is more highly motivated and efficient. John works in a region where theaverage wage is lower, or

Jane works in a firm with higher productivity or in a region where the demand forlabor is stronger.

One of the purposes of labour economics is to assess how relevant, and howsignificant, each of these explanatory factors is.

On the theoretical level, we must specify which hypotheses are being used to justifyevery answer.

The answers to this question are not trivial, and without elaborating a simple yet

rigorous conceptual framework, they cannot be given.


INTRODUCTION

The basic frame of reference adopted by economic analysis is the model of PERFECTCOMPETITION.

When applied to labour economics, it explains the formation of wages by assumingthat they match all labour supply with all labour demand;

The basic hypotheses are that agents have no market power because there is freeentry into the market and information is perfect .

This frame of reference leads to positive conclusions about the setting ofcompensation 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 marketin competitive equilibrium.

As we shall see, the interface between supply and demand in a market whereagents are price takers leads to an efficient allocation of resources .

We shall see as well that the model of perfect competition is very useful forevaluating the consequences of taxation.

We shall see how the impact of taxes on employment and wages depends on theinterplay between labour supply and demand.


INTRODUCTION

In section 2, we shall see that the hypothesis of perfect competition yields a very richtheory of wage setting when working conditions are taken into account.

Differences that arise from hard working conditions are explained by the hedonictheory of wages.

This theory proposed by Rosen (1974) accounts for wage heterogeneity arisingfrom compensating differentials .

It shows that the mechanism of perfect competition provides "compensations" forthe workers who hold the hardest jobs.

Section 3 of this chapter describes the competitive functioning of the labour market ina context where agents and jobs are heterogeneous .

The fact is that for certain occupations the heterogeneity of the services traded ispersistent 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 tosteeply unequal compensation packages, which are socially efficientinasmuch as they ensure an optimal allocation of talent.


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 andservices exchanged on that particular market.

All agents are price takers.

Perfect competition with identical workers and jobs of equa l difficulty

Here we will illustrate the functioning of a market on which a perfectly homogeneousservice is traded: every worker offers a service of the same quality , and theworking conditions are the same everywhere .

Let us consider a market in which a representative firm produces a consumption goodwith 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 receivea wage w if they are hired.




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 whenhe is not.

Parameter e measures the effort (or the disagreeability) attached to each of thejobs.

We assume that this disagreeability is identical for all jobs, and without any loss ofgenerality, we shall assume that parameter e is equal to 1 if there is a hire andequal to 0 if not.

The parameter θ ≥ 0 represents the disutility (or the opportunity cost) oflabour for the individual considered. The cumulative distribution function of thisparameter will be denoted by G(.).

In this model, all the jobs thus have the same “intrinsic” difficulty e, but individualsreact differently to the difficulty of the tasks confronting them: Those with a low θ

accept it more easily than those with a high θ.




Finally, in order to simplify, we shall assume that an agent’s utility function takes alinear 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 resultsfrom 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 ifshe is hired, and 0 if she does not work. Consequently, only individuals whoseopportunity 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 toG(w).




The functioning of the labour market is represented in figure ?? , in which the quantityof 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, atwhich 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 equilibriumwage w∗ decide to work.

In the competitive equilibrium model, nobody is unemployed against his will: everyworker 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".




One of the most striking results of microeconomic analysis is that the equilibrium ofperfect competition yields a collective optimum .

at market equilibrium, the allocation of individuals between employment and inactivityis efficient .



The question of tax incidence

The model of perfect competition is grounded in over-simplified hypotheses and is thusan imperfect representation of the functioning of many labour markets.

Still, it is highly useful for analyzing the consequences of shocks such asalterations in the tax regime on wages and employment.

The model of perfect competition allows us to understand such interactions, which arein 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 borneby 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-earnerswho 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 thetax or the end recipient of the subsidy is .




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, itdesignates a subsidy paid to the firm in the form, for example, of a reduction in socialsecurity contributions.

Labour supply remaining equal to G(w), the equilibrium wage on the labour market isalways 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 Et.

We see that the upshot of this payroll tax reduction is a rise in both the w age andthe level of employment .




We see also that the respective amplitudes of these rises depend on the slope sof the curves of labour supply and demand .

This observation can be enhanced by differentiating both sides of relation (3) withrespect to (1 + t) and to w.

After several calculations, we find that the elasticity of the net equilibrium wage withrespect to (1 + t), denoted ηw

t, is given by the formula:

ηwt =ηdw

ηsw − ηdw(4)

where ηsw and ηdw < 0 represent labour supply and labour demand elasticities.

We saw in chapter 1 that under many circumstances labour supply has low elasticity.

Let us take the extreme case of totally inelastic labour supply (ηsw = 0).

In our model, this situation arises when all individuals have the same parameter θrepresenting the opportunity cost of labour.




Put another way, all individuals have the same reservation wage , denoted wA, andthey all offer an indivisible unit of labour for every wage that exceeds the reservationwage.

For w > wA, overall labour supply is then represented by a straight horizontal line, theordinate of which is the size of the active population, denoted N in figure 2.

In this situation, we have ηwt

= −1, which means that any reduction in payroll taxesis fully passed on, in the form of a rise in the equilibrium wag e that leaves thelevel of employment unchanged .




This situation, portrayed in figure 2, is a good illustration of the main point regardingfiscal incidence:

it is not the agent to whom the tax is charged (or the subsidy aw arded) whois the real payer (or beneficiary) .

The equilibrium wage goes from w∗ to w∗∗ but the level of employment remainsthe same.

When labour supply is inelastic, any lowering of payroll taxes meant in principle toaid the firm actually benefits the employee through a wage rise.

In practical terms, then, knowledge of the elasticities of labour supply anddemand proves to be of primary importance , since, as this example has just shownus, a policy of lowering payroll taxes may lead in the end to a wage rise that leaves thelevel of employment where it was.

In more general terms, knowledge of the elasticities of labour supply and demandmakes it possible to calculate the impact of a change in payroll taxes on wages andemployment.



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 theelasticity 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 awage variation of -0.37%.

Employment (or hours of work) therefore shrinks by 0.63 ·0.3=0.19%, since theelasticity of labour demand is equal to −0.3.

The presence of a minimum wage, however, changes these outcomes. To the extentthat labour supply exceeds labour demand due to a minimum wage, the impact ofpayroll taxes on employment is entirely determined by changes in labour demand, forthe same net wage.

Under these conditions, an increase in payroll taxes leading to an ex ante rise of1% in the cost of low-skilled labour entails a fall of 1% in the employment oflow-skilled persons , since the elasticity of labour demand is of the order of −1 for thiscategory of workers.


COMPENSATING WAGE DIFFERENTIALS

We have studied how a labour market would function if labour services are all perfectlyhomogeneous, 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 aremore willing to accept hardship than others.

Perfect competition would ensure that these differences were compensated for bywage differentials .

This is the essence of the hedonic theory of wages.

Equilibrium is still identified as a social optimum, and any measures aimed at reducingthe difficulty of jobs do not ameliorate welfare.

We will study an equilibrium model of the labour market where jobs are arduous tovarying 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 ofresources, with those workers whose tolerance for hardship is greatest holdingthe hardest jobs and receiving higher wages .



Let us introduce heterogeneity among jobs arising from the difficulty of the w orkto be done .

To that end, we tangibly alter the way the production sector is formalized in theprevious model:

we now assume that there exists a continuum of jobs, each requiring one unit oflabour but a different level of effort e > 0.

This effort variable is a synthetic measure of the difficulty of jobs, and so covers anumber of dimensions like accident risk,environment, etc.

Strictly speaking, e should thus be a vector with as many coordinates as there arecharacteristics 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, ory = f(e) with f ′(e) > 0, f ′′(e) < 0 and f(0) = 0.

Productivity y here corresponds to production net of any costs occasioned byemployment, except wages.



For example, if we interpret e as a measure of industrial accident risk, it is possible toreduce these risks by reducing the intensity of work.

In this case, jobs that offer lower risk have less productivity in our mode l.

As previously, we assume that the utility function of an agent takes the linear formu(R, e, θ) = R− eθ, where θ measures aversion to effort, and that effort e is strictlypositive 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 oneunit 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 thatcorrespond to each of these degrees of effort.

In a setting of perfect competition, entrepreneurs keep on entering all markets until, forevery type of work, profits fall to zero.

If w(e) denotes the equilibrium wage that applies to jobs that demand effort e, thenwage 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.



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 thatmaximizes her satisfaction u[f(e), e, θ] = f(e)− eθ.

The first-order condition of this problem gives:

f ′(e) = θ ⇔ e = e(θ)(5)

An additional requirement is to ensure that the participation constraintu(w, e, θ) ≥ u(0, 0, θ) = 0 is met.

This constraint shows that the worker accepts a job if doing so makes her situationpreferable 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 participationconstraint 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.



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 toeffort 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 tow [e(θ)] = f [e(θ)] , the counterpart of tough jobs is a compensating wagedifferential , since wages increase with effort.

This point is illustrated graphically in figure ?? which represents the choices of twotypes 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 increasedsatisfaction.

Hence each worker chooses a level of effort e such that one of her indifferencecurves is tangent to f(e).



In consequence, individuals with a strong aversion to effort choose low-effort jobs withcorrespondingly low wages.

More generally, at equilibrium wages are given as a function of the θ type of eachindividual.

The hd function is called the hedonic wage function: It gives the equilibrium value ofthe wage of a worker in line with that worker’s characteristi cs .



ESTIMATION

The method used to test the predictions of the hedonic theory of wages consists ofestimating the wage w received by an individual as a function of his personalcharacteristics, represented by a vector x, and the non-wage characteristics of the job,represented by a vector e

lnw = xβ + eα+ ε(6)

Vector x of personal characteristics generally includes age, sex, number of years ofstudy or degree obtained, experience, seniority at work, ethnic origin, place ofresidence, family status, and trade-union membership.

Vector e of the non-wage characteristics of jobs incorporates variables like theduration and the flexibility of hours worked, the repetitive aspect of tasks, the risk ofinjury, the level of ambient noise, the physical strength required by the job, the risk ofjob loss, the cost of health insurance, the cost of saving for retirement, etc.

But, we will surely have a problem with unobserved characteristics .



ESTIMATION

Individual efficiency depends on factors such as motivation or talent that as ageneral 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 onremuneration, for the non-wage characteristics of the job, represented by vector e, arecorrelated 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 themost efficient individuals choose the less laborious jobs, which entails a negativerelation between wages and the laboriousness of jobs.

To escape this type of difficulty, it is preferable to make estimates usinglongitudinal data that allow us to follow individuals and thus control for theirobservable and unobservable time invariant personal characteristics.


ASSORTATIVE MATCHING

The models examined so far have assumed the existence of a large potential numberof 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 thesemarkets 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 theassumption that it is possible to transform jobs in order to adapt them to thepreferences of workers.

Such adjustments are pointing to a long-term phenomenon, the potentialtransformation of jobs.

In the shorter term, it is also of interest to gain an understanding of the functioning of amarket where jobs and workers all have different characteristics, and where thedistributions of these characteristics are exogenous functions.

Under these circumstances, we must account not only for how wages are formed, butalso for how workers distribute themselves into the array of jobs they hold.

In other words, we must explain how the characteristics of workers areassociated with the characteristics of jobs .



To analyze this problem, we resort to assortative matching models.

These models are relevant for understanding the functioning of a market in which theheterogeneity of actors is enduring and plays an important r ole .

Such is the case in particular for the markets for "superstars" , whether they besports 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 assortativematching model that associates chief executive officers (CEOs) who have differenttalents with firms of varying size.

This model explains how the remuneration of CEOs is formed, as well as the mannerin which they are allocated among firms.

As we shall see, the model allows us to understand why the remunerations of CEOs ofclosely similar talents may vary steeply, and why their wage can be extremely high andyet be socially efficient.

The reason is that the most "talented" managers are to be found in the largestcompanies, which maximizes the global output of the economy.



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 eachfirm, 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. Thismass 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 inorder to measure his productivity.



Formally we may denote a the rank of a CEO in the distribution of abilities. Bydefinition, the rank falls in the interval [0, 1].

Similarly, we can index each firm by its rank, denoted by s, in the distribution of firmsizes.

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 andthe 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 matchingfunction) α(s) which defines the talent of the CEOs who head firms of size s, and by acompensation 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 compensationfunction w(a), taken as given by each firm and each CEO, and an assignment functionα(s), such that no CEO-firm pair could do better by matching up with each other thanthey are doing with their current partners, and no CEO and no firm prefers to remainsingle.



THE EQUILIBRIUM ASSIGNMENT FUNCTION

The assortative matching model assumes that the mobility of CEOs occurs withoutfriction 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 oftalent 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 pairthat could do better by matching amongst themselves than they are doing with theircurrent 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) withrespect to a, or

Y1(a, s) = w′(a).(8)




At the competitive equilibrium, the assignment function, which describes the relationbetween a and s, must verify (8) for all s. We thus have:

Y1 [α(s), s] = w′ [α(s)] , ∀s(9)

Deriving this equation with respect to s, we have:

α′(s) =Y12 [α(s), s]

w′′ [α(s)]− Y11 [α(s), s], ∀s(10)

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 thecross derivative of the production function.

The latter is said to be supermodular if Y12 ≥ 0 and submodular if Y12 ≤ 0.




In assignment models of CEOs with firms of different sizes, it is assumed that theproduction function is supermodular over the whole of its support.

This amounts to stating that the marginal productivity of talent increases with the sizeof 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 themost talent) is assigned to the largest firm, the one whose talent ranks just below isassigned to the firm whose size ranks just below, and so on down to the least talentedCEO, 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 competitiveequilibrium, 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 CEOcan find another CEO of greater talent willing to change places with him.




The compensation function w(a) defined by equation (9) shows that the wage isincreasing with talent, for Y1 > 0.

Note that this result holds good whatever hypotheses are adopted about the crossderivative Y12.

Thus greater talent is always compensated by more wage, whether the productionfunction is supermodular or submodular.

We may go a bit further by integrating this wage rule. It writes as follows, denoting byσ(·) the reciprocal of function α:

w(a) = w0 +

∫a

0

Y1[x, σ(x)]dx,(11)

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 marginalproductivity, as well as on the marginal productivity of all the CEOs of less talent.


Chapter 3: COMPETITIVE EQUILIBRIUM...J. Ignacio Garc´ıa Perez´ Universidad Pablo de Olavide -...

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