chapter ii: labour market policy...chapter ii: labour market policy section 1: worker unions the...

Chapter II: Labour Market Policy Section 1: Worker Unions

Chapter II: Labour Market Policy

Section 1: Worker unions

Literature:

Pierre Cahuc and André Zylberberg: Labour Economics

Chapter 7: Section 3

Christopher Pissarides: Equilibrium Unemployment

Chapter 3: Section 1

Christian Bauer and Jörg Lingens: Does Collective Wage Bargaining

Restore Effciency in a Search Model with Large Firms?

Prof. Dr. Christian Holzner Page 279


The role of unions:

Unions attract only workers, if collective bargaining implies a higher utility for itsmembers than individual bargaining.

Unions maximize the utility of their members, e.g. all workers or only employedworkers.

To investigate the impact of collective bargaining, we have to analyse large firms (notsingle vacancy firms).

Research question:

What are the implications for wages and employment, if unions bargain instead ofindividual workers?

What are the efficiency properties of collective versus individual bargaining.



Framework:

- Like in the Mortensen-Pissarides framework (Ch. I, Sec. 2).

- The output for a firm depends on the number of workers employed, i.e., y = f (l)with f ′ (l) > 0 and f ′′ (l) < 0.

- For simplicity we will assume a Cobb-Douglas production function, i.e., y = plα.

- Individual bargaining is indexed by a superscript I.

- Union bargaining is indexed by a superscript C for collective.



5.1 Social optimum in a large firm model

The social optimum solves as a benchmarkt in order to evaluate the impact of unions(collective bargaining) compared to individual bargaining.



Hiring of workers:

A firm that has v unfilled vacancies hires workers at the rate m (θ) v.

Given that workers exist employment at an exogenous rate q, the labor input of afirm evolves according to

dl

dt= l̇ = m (θ) v − ql.

We normalize the number of firms to unity, so that l = 1 − u.



The social planner’s maximization problem:

The social planner maximizes aggregate social welfare, i.e.,

maxv

∫

∞

0

[f (l) + [1 − l] z − hv] e−rtdt,

subject to the constraint implied by matching frictions, i.e.,

l̇ = m (θ) v − ql.



Hamiltonian:

H = [f (l) + [1 − l] z − hv] e−rt + µ [m (θ) v − ql]

FOC:

∂H

∂v= 0 ⇐⇒ he−rt = µm (θ)

[

1 +θm′ (θ)

m (θ)

]

(1)

∂H

∂l= µ̇ ⇐⇒ [f ′ (l) − z] e−rt + µ

[

m′ (θ) θ2 − q]

= µ̇ (2)

Transversality condition:

limt→∞

µ l = 0.



Differentiating equation (1) with respect to time t implies: µ̇ = rµ

Substituting µ using equations (1) and (2) implies the following condition for theoptimal labour market tightness θ, i.e.

h

m (θ)=

[f ′ (l) − z][

r + q − m′ (θ) θ2]

[

1 +θm′ (θ)

m (θ)

]

orh

m (θ)=

[1 − η (θ)] [f ′ (l) − z]

[r + q + η (θ) θm (θ)](3)

where η (θ) equals the elasticity of the matching function with respect to the unem-ployment rate u, i.e.

η (θ) = −θm′ (θ)

m (θ)



5.2. Individual bargaining in a large firm model

Idea:

Wages are bargaining each time a worker enters or leaves a firm.

When deciding on the number of vacancies, firms take into account that the numberof workers employed will have an influence on the marginal product of a worker.

=⇒ Firms increase employment beyond the social optimal level to reducewage costs (by reducing the marginal product).



The firm’s maximization problem:

The firm maximizes profits given the wage function w (l) that comes out of thebargaining game, i.e.,

Π = maxv

∫

∞

0

[f (l) − w (l) l − hv] e−rtdt

and subject to the constraint implied by matching frictions, i.e.

l̇ = m (θ) v − ql



Hamiltonian:

H = [f (l) − w (l) l − hv] e−rt + µ [m (θ) v − ql]

FOC:

∂H

∂v= 0 ⇐⇒ he−rt = µm (θ) (4)

∂H

∂l= µ̇ ⇐⇒ [f ′ (l) − w (l) − w′ (l) l] e−rt − µq = µ̇ (5)

Transversality condition:

limt→∞

µ l = 0.



Vacancy creation condition:

Differentiating equation (4) with respect to time t implies: µ̇ = rµ

Substituting µ using equations (4) and (5) implies the following vacancy creationcondition, i.e.,

h

m (θ)=

f ′ (l) − w (l) − w′ (l) l

r + q(6)

The number of vacancies that a firm creats

• decreases with the wage level w (l),

• and increases, if additional labor input decreases the wage, i.e., if w′ (l) < 0.



Worker’s and firm’s match surplus:

Match suplus of a worker (see Ch.1, Sec. 2, p. 38),

Ve (w) − Vu =w (l) − rVu

r + q.

equals the discounted difference between the wage and the flow value of being un-employed.

Firm’s suplus of employing one additional worker (see equation (5) of thefirm’s optimization problem),

∂Π

∂l=

f ′ (l) − w (l) − w′ (l) l

r + q.

equals the discounted difference between the wage and the flow value of being un-employed.



Individual wage bargaining:

The relative bargaining power for workers is given by γ and for firms by (1 − γ).

The bargaining wage maximizes the Nash-Product, i.e.,

w = arg max [Ve (w) − Vu]γ

[

∂Π

∂l

](1−γ)

FOC:

0 = γV ′e (w)

Ve (w) − Vu+ (1 − γ)

∂2Π/∂l∂w

∂Π/∂l

where

V ′e (w) =1

r + qand

∂2Π

∂l∂w=

−1

r + q



Wage curve:

Substituting and rearranging implies

w (l) = (1 − γ) rVu + γ [f′ (l) − w′ (l) l]

The solution to this first order differential wage equation is given by

w (l) = (1 − γ) rVu + l−

1γ

∫ l

0

x1−γγ f ′ (x) dx,

= (1 − γ) rVu + γα

1 − γ (1 − α)plα−1,

where the last equality follows for a Cobb-Douglas production function.

• The wage curve is identical to the MP-framework with one vacancy, if the firmhas constant returns to scale, i.e., α = 1.

• With a concave production function, i.e., α < 1, firms have an incentive toincrease the number of open vacancies in order to reduce the marginal productof labor, i.e., w′ (l) < 0, reduces the wage sum.



Equilibrium market tightness:

The wage equation implies

w (l) + w′ (l) l = (1 − γ) rVu + γα

1 − γ (1 − α)αplα−1

Substituting into the vacancy creation condition (6) implies

h

m (θ)=

αplα−1 − (1 − γ) rVu − γα

1−γ(1−α)αplα−1

r + q

Substituting

rVu =(r + q) z + θm (θ)

[

(1 − γ) rVu + γαplα−1

1−γ(1−α)

]

r + q + θm (θ)

=(r + q) z + γθm (θ) αpl

α−1

1−γ(1−α)

r + q + γθm (θ)



implies the following vacancy creation condition:

h

m (θ)=

(1 − γ)[

11−γ(1−α)αpl

α−1− z

]

r + q + γθm (θ)(7)

Comparing with the vacancy creation condition of the social planner (3), i.e.,

h

m (θ)=

[1 − η (θ)][

αplα−1 − z]

[r + q + η (θ) θm (θ)]

and enforcing the Hosios condition, i.e., γ = η (θ), implies that firms create morevacancies under individual bargaining in large firms than socially optimal.

The higher market tightness leads to a higher matching rate for unemployedworkers, i.e., θIm

(

θI)

> θSPm(

θSP)

, a lower unemployment rate and a higheremployment rate, i.e.,

uI < uSP and lI > lSP .



5.3. Collective bargaining with firm level unions

Idea:

Unions maximize the value of being employed for all employed workers, i.e.,l [Ve (w (l)) − Vu].

The union and the firm bargain over the wage and the firm determines thenumber of employed workers, i.e., the number of vacancies created. This is the”right to manage” approach.

The vacancy creation condition is the same as under individual bargaining, i.e.,equal to equation (6), with the exeption of different wages, i.e., w (l) will be different.



Worker’s and firm’s match surplus:

Firm level unions maximize the gain from employment for their members,i.e., employed workers,

l [Ve (w) − Vu] = lw (l) − rVu

r + q.

Firm level unions take the value of being unemployed rVu as given, i.e., they do nottake into account that the bargaining wage will influence the labor markettightness.

Firm’s suplus of having no strike,

SΠ = f (l) − w (l) l.

equals output minus wage costs. This is identical to assuming that firms continueto recruit workers even if there is a strike, i.e., they still pay the cost for theiropen vacancies.



Collective wage bargaining with firm level unions:

The bargaining wage maximizes the Nash-Product, i.e.,

w = arg max [l (Ve (w) − Vu)]γ [f (l) − w (l) l](1−γ)

FOC:

0 = γ1

w (l) − rVu+ (1 − γ)

−l

f (l) − w (l) l

Wage curve:

w (l) = (1 − γ) rVu + γf (l)

l

The collective bargaining wage depends on the average product not the marginalproduct, since the threat of the union is to strike and stop the whole production(not like under individual bargaining to produce with one worker less).



Equilibrium market tightness:

The wage equation implies

w (l) + w′ (l) l = (1 − γ) rVu + γf′ (l)

Substituting into the vacancy creation condition (6) implies

h

m (θ)=

f ′ (l) − (1 − γ) rVu − γf′ (l)

r + q

Substituting

rVu =(r + q) z + θm (θ)

[

(1 − γ) rVu + γf(l)

l

]

r + q + θm (θ)

=(r + q) z + γθm (θ) f(l)l

r + q + γθm (θ)



implies the following vacancy creation condition:

h

m (θ)=

(1 − γ) [f ′ (l) − z]

r + q + γθm (θ)−

(1 − γ)

r + q

γθm (θ)[

f(l)l − f

′ (l)]

r + q + γθm (θ)(8)

Comparing with the vacancy creation condition of the social planner (3), i.e.,

h

m (θ)=

[1 − η (θ)][

αplα−1 − z]

[r + q + η (θ) θm (θ)]

and enforcing the Hosios condition, i.e., γ = η (θ), implies that firms create lessvacancies under collective bargaining in large firms than socially optimal.

The lower market tightness leads to a lower matching rate for unemployed wor-kers, i.e., θCm

(

θC)

< θSPm(

θSP)

, a higher unemployment rate and a loweremployment rate, i.e.,

uC > uSP and lC < lSP .



Do employed workers gain from joining a union?

Comparing the wages between individual and collective bargaining at a fixedvalue of being unemployed (individual workers do not take into account theequilibrium effects of their choice), i.e.,

wI = (1 − γ) rVu + γα

1 − γ (1 − α)p(

lI)α−1

,

wC = (1 − γ) rVu + γp(

lC)α−1

shows that at a given labor input, i.e., lI = lC, workers will always join a union,since

α

1 − γ (1 − α)< 1, (9)

which is always satisfied, since γ < 1. If workers take the labor input choice of thefirm into accout, i.e., lI > lC, collective bargaining always leads to a higher marginal

product, i.e., p(

lC)α−1

> p(

lI)α−1

, and higher wages.



5.4. Can unions improve efficiency?

Idea:

What happens, if unions and firms bargain over wages and employment (or vacan-cies)?

What happens, if national unions take the effect on the market tightness into accountthen bargaining with employers’ association?

What are the consequences, if agreements between national unions and employers’association are not binding?

What are the implications of union bargaining on general or firm-specific training?



Bargaining over wages and employment:

McDonald and Solow (1981) show in a competitive market that efficiency canbe restored, if unions and firms bargain over wages and employment.

Intuition:

Bargaining over wages and employment interalizes the effect that wages have onemployment, i.e., unions and firms take into account that higher wages will lead tounemployment.

Search framework:

Collective bargaining with firm level unions is expected to reduce the negative effecton employment. It is likely that this does not restore efficiency in a search framework,since unions and firms do not take the equilibrium effect on the market tightness intoaccount. But, this is still an open research equation!



What happens, if the national union and the employers’ associations take

the equilibrium effect into account?

• This is likely to restore efficiency, if they bargain over wages and employment,since Nash-Bargaining always maximize the joint surplus.

• This is still an open research equation!



What are the consequences, if agreements between national unions and

employers’ association are not binding?

• Some firms will exit the employers’ association and some workers will exit theunion.

• The threat to exit the employer’s association or exit the union will discipline thebargaining parties, even if they just bargain over wages.

• Dobbelaere and Luttens (2011) show in a competitive enviroment, that the threatof some agents to exit can restore efficiency, even if unions and firms only bargainover wages.

• The outcome within a search framework has not yet been studied!



What are the implications of union bargaining on general or firm-specific

training?

• Unions also try to reduce the wage dispersion across skill groups within a firm.

• This increases a firm’s value of employing a skilled worker relative to an unskilledworker.

• This additional rent provides an incentive for firms to invest in the human capitalof their workers.

• This policy of worker unions contributes to a more educated workforce.


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