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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?
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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)
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Chapter II: Labour Market Policy Section 1: Worker Unions
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).
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Chapter II: Labour Market Policy Section 1: Worker Unions
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
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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)
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Chapter II: Labour Market Policy Section 1: Worker Unions
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 .
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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).
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Chapter II: Labour Market Policy Section 1: Worker Unions
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)
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Chapter II: Labour Market Policy Section 1: Worker Unions
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 .
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Chapter II: Labour Market Policy Section 1: Worker Unions
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.
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Chapter II: Labour Market Policy Section 1: Worker Unions
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?
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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!
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Chapter II: Labour Market Policy Section 1: Worker Unions
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!
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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!
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Chapter II: Labour Market Policy Section 1: Worker Unions
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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