Lecture 5: Labour Economics and Wage-Setting Theory

Spring 2013

Lars Calmfors

Literature: Chapter 7 Cahuc-Zylberberg (pp 393-403) OECD Employment Outook (2006)

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Topics

• Collective bargaining, continued • General equilibrium • Weakly efficient bargaining

• Strongly efficient bargaining

• Wage dispersion

• Insiders and outsiders

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General equilibrium model

(1 )i

w wγ α γ

α

+ −=

• Assume mobility in the labour market. An unemployed in a given

firm (labour pool) can either find a job in another firm (labour pool) or become unemployed.

• Symmetric economy with a large number of firms.

• Look at wage-setting in firm i.

• Probability of getting a job in another firm = l = the economy-wide employment rate = employment/labour force.

• Probability of not finding a job elsewhere = 1-l.

• A worker who finds a job elsewhere receives the wage w.

• If unemployed, the worker receives the unemployment benefit b. w = the expected income if not employed in firm i = alternative income

(1 )w w b= + −

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[ ]

[ ]

Hence:

(1 ) (1 )

In a symmetric equlibrium

(1 )Denote the mark-up factor

Then:

(1 )

(1 ) (B)

1

i

i

w w b

w w

m

w m w b

mw b

m

γ α γ

α

γ α γ

α

+ −= + −

=

+ −=

= + −

−=

−

• The wage is still a mark-up over the unemployment benefit as

(1 ) 1 > 1

m m m− > − ⇔

• The overall wage in the economy, w, is positively related to employment as:

2

( 1) 0

(1 )

w m m

m

∂ −= >

∂ −

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w = f(l) is called a wage-setting schedule

It shifts upwards if:

(1) γ↑ (2) b↑ • Equilibrium employment is given by intersection between the

wage-setting schedule and the labour-demand schedule.

• Shift of labour-demand schedule affects the equilibrium employment rate.

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Key question: How is the unemployment benefit determined?

1. Constant in real terms

2. Constant replacement rate r, so that b = rw Constant replacement rate:

2

(1 )

1

(1 )

1

(1 )1

1

1

(1 )

(1 ) 0

( )

mw b

m

mw rw

m

mr

m

rm

m r

m m

r m r

−=

−

−=

−

−=

−

−=

−

∂ −= <

∂ −

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• Vertical wage-setting schedule determined by labour-market institutions only (here r and γ)

• An increase in the replacement rate reduces the employment rate

• Shifts in labour demand have no effect on the equilibrium employment rate.

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Efficient contracts

• Bargaining over the wage only and letting employers determine employment (right to manage) is not efficient.

• An efficient solution can be found by bargaining over both the wage and employment.

[ ] [ ]1

,

( ) ( ) ( )

s.t. 0 and

Maxw L

R L wL w w L

L N w w

γ γ γν ν−

− −

≤ ≤ ≥

Interior solution

'( )(1 ) 0 (I)

( )

'( )(1 ) 0 (II)

( ) ( ) ( )

R L w

R L wL L

L w

R L wL w w

γγ

γνγ

ν ν

−− + =

−

− − + =− −

Eliminate γ between the two equations to get

(III)( ) ( )

'( ) '( )

w ww R L

w

ν ν

ν

−− =

8

This is the equation of a contract curve (Pareto-efficient combinations of w, L) connecting tangency points of indifference and isoprofit curves.

The same equation would be obtained by maximising [ ]( ) ( ) s.t. L w wν ν π π− =

Differentiation of the contract curve equation gives:

[ ]"( )

"( ) '( )

dw R L

dL w w R Lν=

−

0 '( ) according to (I)R L wγ = ⇒ =

'( ) ( ) ( ) and according to (III)R L w w w w wν ν= ⇒ = = Hence the contract curve starts on the labour demand schedule at w w=

If '( )w R L> and workers are risk averse, i.e.

" 0, then / 0 for '( ).dw dL w R Lν < > >

0γ = gives the competitive level of employment ( )L L w=

With 0γ > , the union uses its bargaining power to raise both the wage and employment over the competitive levels.

If workers are risk-neutral, then " 0 and .dw

dLν = → ∞ Hence the contract

curve is vertical. Employment is at the competitive level.

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Overemployment if workers are risk-averse – “weak efficiency” as

'( ) due to employment being higher than defined by '( )c c

R L w L R L w< =

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Strongly efficient contracts

• Efficiency gain for union if utility of employed and unemployed are equated

• Incentive to bargain with firm over unemployment benefit paid by the firm

Union objective

( ) ( ) ( )L w N L b wν ν+ − +

Firm profit

[ ]0

0

,

,

( ) ( )

Max ( ) ( ) ( )

s.t.

Max ( ) ( ) ( ) ( ) ( )

w b

w b

R L wL N L b

L w N L b w

L w N L b w R L wL N L b

π

ν ν

π π

ν ν λ π

= − − −

+ − +

=

+ − + + − − − −

FOC

'( ) 0

( ) '( ) ( ) 0

'( )

'( )

L w L

N L b w N L

w

b w

ν λ

ν λ

ν λ

ν λ

− =

− + − − =

=

+ =

11

Hence:

'( ) '( )

w b w

w b w

ν ν= +

= +

• Pareto efficiency requires a wage for the employed that is equal to the income as unemployed.

• The firm pays a benefit b to all unemployed.

• It pays a wage w b+ to the employed.

• Employment does not matter to the union, since members are insured against unemployment.

The bargaining problem

[ ] [ ]

[ ]

1 ( *) * ( ) ( )

FOC:

( ) ( ) ( *) *

'( ) 1

with

'( *)

Max b

R L wL bN w b w

w b w R L wL bN

w b N

w w b

R L w

γ γν ν

ν ν γ

ν γ

−− − + −

+ − − −=

+ −

= +

=

• Employment equals the competitive level

• Union members appropriate a share of the firm’s profit without this having negative effects on employment

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Diagrammatical illustration Indifference curves:

1

1

( )

0

0

0

sw

dw

dw

dLdw

dL

ν

ν

ν

υ =

=

=

=

The indifference curves are horizontal lines. Isoprofit curve

( ) ( ) ( )

0 '( )

'( )

R L wL bN R L wL N w w

d R L dL wdL Ndw

dw R L w

dL N

π

π

= − − = − − −

= = − −

−=

• Tangency points between isoprofit curves and indifference curves

give a vertical contract curve (at the competitive level of employment)

• Bargaining over wages, employment and unemployment benefits from firms is strongly efficient.

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Collective bargaining and wage dispersion

• Heterogeneous workers

• Collective bargaining reduces wage dispersion

• Two types of workers, indexed by i = 1, 2

• Revenue of the firm = R(L1, L2)

• Type -1 workers are more productive with a higher reservation wage

1 2w w>

• Ni workers of type i in the firm’s labour pool

• The union utility function

2

i=1

( ) ( ) ( ) s i i i i i i i i

L w N L w b L Nυ υ υ= ∑ + − + ≤

• Strongly efficient bargaining over employment, wages and

unemployment benefits

• Optimal contract implies i i i

w w b= +

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Bargaining problem

{ }1 2 1 2

12

1 2

1

2

1, , ,

( , ) ( )

s.t. 0 = 1, 2

( ) ( )Max i i i i

i

i i

i i i i

ib b L L

R L L w L b N

L N i

N w b wγ γ

ν ν−

= =

− +

≤ ≤

+ −⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑

FOCs

[ ]

1 2

1

2

i=1

i 2

1 2

1

(11)

(12)

( , )

( ) ) ( )1

'(w ) ( , ) ( )

i

i i i i

i

i i i i

i

R L Lw

L

N w b wb

R L L w L b N

ν νγ

νγ

γ=

∂=

∂

+ −−

+ =

− +

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

∑

∑

• Equation (11): Productive efficiency, i.e. the marginal productivity

of each type of worker equals the reservation wage. This implies the competitive level of employment.

• Equation (12): RHS is independent of i. Hence the same wage for the two types of labour.

• Wage equality follows from the assumption of a utilitarian union and identical preferences.

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1 2 1 2

1 2 1 2

1 2 1 2 1 2 1 2

( ) ( ) N N N N

w w w wN N N N N N N N

ν ν ν+ ≤ ++ + + +

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

Because of concavity the union is better off with a wage

1 2

1 2

1 2 1 2

N N

w wN N N N

++ +

for everyone than with separate wages w1 and w2.

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Two-stage bargaining over employment (Manning 1987)

Stage 1: Bargaining over the wage

Stage 2: Bargaining over employment

Different bargaining strengths in the two negotiations Bargaining over employment (given the wage)

[ ] [ ]1 ( ) ( ) ( ) s.t. 0Max L L L

LR L wL w w L L N

γ γ γν ν−

− − ≤ ≤

The solution gives ( , , )L

L L w wγ=

Bargaining over the wage (takes the outcome of second-stage bargaining over employment into account)

[ ] [ ]1 ( ) ( ) ( )

= ( , , ) and

Max

s.t. L

wR L wL w w L

L L w w w w

γ γ γν ν

γ

−− −

≥

Different cases

• and 0 0L

γ γ= > gives the right-to-manage model

• L

γ γ= gives (weakly) efficient bargain model

• Otherwise solution on neither labour-demand schedule nor contract curve

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Motivations

• Efficient bargaining is complex

• Wage bargaining precedes employment bargaining

• Wage bargaining is often at more centralised level

• Strongly efficient bargaining is improbable because of moral hazard problems: unemployed being fully insured will not search effectively for jobs

- argument for partial insurance - individual firm (sector) offering full insurance would be

swamped by labour inflow

• One does not find many examples of contracts with unemployment benefits paid by firms

• Unclear empirical results on right-to-manage model and (weakly efficient) bargaining

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Insiders and outsiders

• Unions negotiate on behalf of insiders (the already employed those with a strong affiliation to the labour market)

• Unions do not negotiate on behalf of outsiders (the unemployed in general or the long-term unemployed)

An insider-outsider model

• LO insiders

• The firm decides on how many insiders LI ≤ LO it wants to retain.

• It also decides on how many outsiders LE it wants to hire.

• Revenue function R(LI + LE)

• The firm’s profit: π = R(LI + LE) - w(LI + LE)

• Employment of insiders, LI, and of outsiders, LE, is found by maximising profits s. t. LI ≤ LO and LE > 0.

• Define wO by R′(LO) = wO.

• Define L as the employment level such that R′(L ) = w, where w is the current wage.

Labour demand

and = 0 if

and if

If we have , so some insiders are fired.

I E O

I O E O O

O I O

L L L w w

L L L L L w w

w w L L L

= ≥

= = − ≤

≥ = <

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Wage bargaining

IV = expected utility of an insider

O ( ) (1 ) ( ) = Min (1, / )I w w L LV ν ν= + −

w = the reservation wage

[ ] [ ]{ }1Max ( ) ( ) ( )

with ( ) ( )w

w w w

w R L wL

γ γπ ν ν

π

−−

= −

• Let w1 be the solution when /

OL L= (interior solution

with some unemployed insiders).

• The solution is the same as in the standard right-to-manage model but with LO = N.

1

1

(10)( ) ( )

'( ) (1 )L

w w

w w

w w π

ν ν γ

ν γη γ η

−=

+ −

Solution with = 1

• Set 0L

wη = in (10); employment of insiders cannot increase

2

2 2

( ) ( )

'( ) (1 )w

w w

w w π

ν ν γ

ν γ η

−=

−

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Different solutions

1 0

2 0

1 2

Nash bargaining product when , i.e. some employed outsiders

Nash bargaining product when , i.e. some unemployed insiders

We have:

B L

B L L

B Bw w

L= >

= <

∂ ∂>∂ ∂

Larger gain from wage increase if only outsiders lose their jobs than if also insiders do. Second-order conditions for a maximum

2

1 12

22 22

( / ) 0

( / ) 0

B B www

B B www

∂ ∂ ∂ ∂= <∂∂

∂ ∂ ∂ ∂= <∂∂

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(1) Interior solution with 0 0 and w w L L> ≤

1 2 > 0 > 0B Bw w

∂ ∂∂ ∂

(2) Corner solution with 0 0 and w w L L= =

1 2 > 0 < 0B Bw w

∂ ∂∂ ∂

(3) Interior solution with 0 w w<

1 2> 0 < 0B Bw w

∂ ∂∂ ∂

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Conclusion

• A fall in the number of insiders results in an unchanged wage or in an increase in the wage

• Explanation of the persistence of unemployment

• No incentive to reduce the wage as the union does not care about the unemployed

• Empirical research has had problems finding that a reduction in lagged employment has a positive effect on the wage.

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