Adjustment costs cause movements of the labour share if the economy experiences demand or wage shocks. With linear adjustment costs and Cobb-Douglas technology, these movements are independent of the size of these shocks and depend only on the size of the adjustment costs. -- Anpassungskosten auf dem Arbeitsmarkt verursachen Schwankungen der Lohnquote, wenn eine Volkswirtschaft Lohn- oder Nachfrageschocks ausgesetzt ist. Für den Fall einer Cobb-Douglas Produktionstechnologie wird gezeigt, dass die induzierten Schwankungen nicht von der Größe der Lohn- oder Nachfrageschocks abhängen, sondern nur von der Höhe der Anpassungskosten.Labour share,adjustment costs,labour demand

Kessing, Sebastian

English

Research Papers in Economics

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Kessing, Sebastian
Working Paper
A note on the determinants of
labour share movementsA note on
the determinants of labour share
movements
Discussion papers // WZB, Wissenschaftszentrum Berlin für Sozialforschung,
Forschungsschwerpunkt Markt und Politische Ökonomie, No. FS IV 02-30
Provided in cooperation with:
Wissenschaftszentrum Berlin für Sozialforschung (WZB)
Suggested citation: Kessing, Sebastian (2002) : A note on the determinants of labour share
movements, Discussion papers // WZB, Wissenschaftszentrum Berlin für Sozialforschung,
Forschungsschwerpunkt Markt und Politische Ökonomie, No. FS IV 02-30, http://
hdl.handle.net/10419/51149 
 
   
WISSENSCHAFTSZENTRUM BERLIN 
FÜR SOZIALFORSCHUNG   
discussion papers 
SOCIAL SCIENCE RESEARCH 
CENTER BERLIN 
  
 
    
 
 
 
 
FS IV 02 – 30 
 
 
A Note on the Determinants of  Labour Share Movements 
 
 
Sebastian Kessing 
 
 
Freie Universität Berlin 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
December 2002 
 
 
 
 
 
ISSN Nr. 0722 - 6748 
 
 
   
 
 Forschungsschwerpunkt 
  Markt und politische Ökonomie   
 
 Research  Area 
  Markets and Political Economy   
 
 
 
 
Zitierweise/Citation: 
 
Sebastian Kessing, A Note on the Determinants of Labour 
Share Movements, Discussion Paper FS IV 02-30, 
Wissenschaftszentrum Berlin, 2002. 
 
Wissenschaftszentrum Berlin für Sozialforschung gGmbH, 
Reichpietschufer 50, 10785 Berlin, Tel. (030) 2 54 91 – 0 
Internet: www.wz-berlin.deABSTRACT 
A Note on the Determinants of Labour Share Movements   
by Sebastian Kessing
* 
Adjustment costs cause movements of the labour share if the economy experiences 
demand or wage shocks. With linear adjustment costs and Cobb-Douglas technology, 
these movements are independent of the size of these shocks and depend only on the 
size of the adjustment costs. 
 
Keywords:  Labour share, adjustment costs, labour demand 
JEL classification numbers: D33, J63  
ZUSAMMENFASSUNG 
Determinanten von Lohnquotenschwankungen  
Anpassungskosten auf dem Arbeitsmarkt verursachen Schwankungen der Lohnquote, 
wenn eine Volkswirtschaft Lohn- oder Nachfrageschocks ausgesetzt ist. Für den Fall 
einer Cobb-Douglas Produktionstechnologie wird gezeigt, dass die induzierten Schwan-
kungen nicht von der Größe der Lohn- oder Nachfrageschocks abhängen, sondern nur 
von der Höhe der Anpassungskosten. 
                                                 
*   I would like to thank Giuseppe Bertola, Dan Hamermesh, Winfried Koeniger and Jarkko Turunen for 
comments. Financial support from DAAD is gratefully acknowledged. 1 Introduction
Time series properties of, as well as cross-country diﬀerences in, labour share
movements have recently been recongnized as a key magnitude for under-
standing the interaction between macroeconomic shocks, institutions and
unemployment, see Blanchard (1997, 1998) and Caballero and Hammour
(1998). Bentolila and St. Paul (1999) have shown empirically, that ad-
justment costs are the single most important factor aﬀecting labour share
movements. This note considers how shocks, either to general business con-
ditions (a demand shock, for example) or to wages, translate into labour
share movements in the case of linear adjustment costs. Two neutrality re-
sults with respect to the size of the shocks aﬀecting business conditions or
wages are derived. If technology is Cobb-Douglas, the size of labour share
ﬂuctuations may only depend on the size of adjustment costs.
2 Factor shares in a Markov chain model
Consider the simple stochastic labour demand model of Bertola (1990). A
representative risk-neutral ﬁrm’s dynamic labour demand problem is given
1by
Max
{Li}
Et
 
∞  
k=0
 
1
1+r
 k
[R(Zt+k,L t+k) − wt+kLt+k − C(Lt+k − Lt+k−1]
 
.
(1)
Ri(Zi,L i) denotes the ﬁrm’s one period revenue, as a function of the amount
of labour employed Li and the prevailing business conditions Zi. It is assumed
either that business conditions follow a two state (good, i = g, and bad, i = b)
Markov chain (in which case the wage is assumed to be constant) or that the
wage rate wi follows a two state (high, i = h,a n dl o w ,i = l)M a r k o vc h a i n ,
in which case Zi is constant and set to unity. Letting H and F represent
the given costs per hired and ﬁred worker respectively, the ﬁrm’s asymmetric
linear costs of adjusting its labour force are
C(Li − Li−1)=

  
  
H(Li − Li−1) if Li − Li−1 > 0
−F(Li − Li−1) if Li − Li−1 < 0.
(2)
Deﬁne Vt ≡ Et
 
∞  
k=0
 
1
1+r
 k Mt+k(Zt+k,L t+k) − wt+k
 
as the shadow product
of labor where Mi(Zi,L i) ≡
∂Ri(Zi,Li)
∂Li is the marginal revenue product of
labor (MRPL). The ﬁrst order conditions of the ﬁrm’s problem are given as
−F ≤ Vt ≤ H always, (3)
Vt = H, if Lt − Lt−1 > 0a n dVt = −F, if Lt − Lt−1 < 0. (4)
2The ﬁrm’s optimal policy is to hire either when business conditions improve
or when wages go down and to ﬁre either when times turn bad or when wages
go up. When conditions stay the same, inaction is optimal, since there are
no voluntary quits. Hence, if adjustment costs are not prohibitively high,
employment itself follows a Markov chain. From (4) it must be that
Vt = M(Zg,L g) − wg +
1
1+r
E [Vt+1]=H (5)
when times turn good. Substituting E [Vt+1]=pgH +( 1− pg)(−F), where
pg is the probability of good times remaining good next period, gives
wg = M(Zg,L g) −
1
1+r
(1 − pg)(H + F) −
r
1+r
H. (6)
Analogous equations can be derived for for bad times and for the case in
which the wage is ﬂuctuating. These equations show the wedges that are
driven between wages and the MRPL by the presence of adjustment costs
and which cause the labour share to vary. They implicitly deﬁne optimum
labour demand as Li = Li(Zi,w i,H,F,p i,r). Comparative statics reveal
that everything that increases the wedges Qj ≡| Mj − wj|, j = g,l, reduces
labour demand and everything that increases the wedges, Qk ≡| Mk − wk|,
k = b,h, increases labour demand in these states. The labour share in state
3i, Si, is given by
Si =
wiLi(Zi,w i,H(wi),F(wi),p i,r)
Ri(Zi,L i(Zi,w i,H(wi),F(wi),p i,r))
,i = g,b,h,l. (7)
The eﬀect of adjustment costs on the labour share is given in
Proposition 1 The labour share is unambigously increased in bad times and
in times of high wages, and reduced in good times and in times with low wages,
by the presence of adjustment costs.
Proof: The eﬀect of an increase in employment for ﬁxed Z and w equals
∂Si
∂Li
=
w(Ri −
∂Ri
∂LiLi)
R2
i
> 0. (8)
Since adjustment costs increase labour demand in bad times and with high
wages, but reduce it in good times or with low wages, the labour share will
be raised in bad times, or with low wages, and reduced in good times, or
with high wages.
In the case of Cobb-Douglas technology with multiplicative shocks rev-
enue equals R(Zi,L i)= 1
1−βZiL
1−β
i and the following neutrality result holds:
Proposition 2 With Cobb-Douglas revenue and multiplicative shocks, the
size of labour share ﬂuctuations is invariant with respect to the size of these
shocks, as long as adjustment costs are not prohibitively high.
4Proof: From (7) it follows that in general
∂Si
∂Zi
=
wLi
RiZi
 
∂Li
∂Zi
Zi
Li
−
∂Ri
∂Zi
Zi
Ri
 
,i = g,b. (9)
Optimal labour demands with adjustment costs equal Lg =[ w + Qg]
− 1
β Z
1
β
g
and Lb =[ w − Qb]
− 1
β Z
1
β
b . Then revenues are Rb = 1
1−βZb
 
[w − Qb]
− 1
β Z
1
β
b
 1−β
and Rg = 1
1−βZg
 
[w + Qg]
− 1
β Z
1
β
g
 1−β
. Taking logs and diﬀerentiating yields
∂Li
∂Zi
Zi
Li = 1
β,i= g,b and
∂Ri
∂Zi
Zi
Ri = 1
β,i= g,b. Thus,
∂Si
∂Zi =0 ,i= g,b, due to
(9).
As is evident from (9), the eﬀect on the size of labor share movements
depends on the relative importance of labour demand and revenue elasticities.
In the case of Cobb-Douglas these exactly balance.
When labour share movements are due to ﬂuctuations in wages an anal-
ogous result can be derived if adjustment costs are proportional to wages.
This is not implausible for severence payments and red tape costs, which are
typically very labour intensive:
Proposition 3 If hiring and ﬁring costs are proportional to wages and tech-
nology is Cobb-Douglas, the size of labour share ﬂuctuations caused by wage
ﬂuctuations is invariant to the size of these ﬂuctuations, as long as adjust-
ment costs are not prohibitively high.
5Proof: With proportional adjustment costs H = cw and F = bw the
wedges become Ql =
w(1−pl)(c+b)
1+r + wrc
1+r and Qh =
w(1−ph)(b+c)
1+r + wrb
1+r.T h e
labour shares equal
Sl =
wl [wl + Ql]
− 1
β
1
1−β [wl + Ql]
1−β =
(1 − β)
(1 +
(1−pl)(c+b)
1+r + rc
1+r)
, (10)
Sh =
wh [wh − Qh]
− 1
β
1
1−β [wh − Qh]
1−β =
(1 − β)
(1 −
(1−ph)(c+b)
1+r − rb
1+r)
.
Obviously, the labour share does not depend on wages in both states. Thus,
∂Si
∂wi =0 ,i= h,l.
If adjustment costs are proportional their relative importance remains
constant, which translates into a constant size of labour share movements.
3 Conclusion
If production is Cobb-Douglas and adjustment costs are linear, factor share
movements do not depend on the size of the demand shocks hitting the
economy, but only the size of the adjustment costs. Similarly, if production
is Cobb-Douglas and adjustment costs are linear and proportional to wages,
factor share ﬂuctuations do not depend on the size of wage shocks, but only
on the size of adjustment costs. Consequently, the size of adjustment costs,
and the labour market institutions that determine them, are more important
6for labour share movements than the size of, and diﬀerences in, demand and
wage shocks.
References
[1] Bentolila, Samuel and Gilles Saint-Paul (1999), ”Explaining Movements
in the Labour Share”, CEMFI Working Paper 9905.
[2] Bertola, Giuseppe (1990), ”Job Security, Employment and Wages”, Eu-
ropean Economic Review, 34, 851-886.
[3] Blanchard, Olivier (1998), ”Revisiting European Unemployment: Unem-
ployment, Capital Accumulation and Factor Prices”, NBER WP 6566.
[4] Blanchard, Olivier (1997), ”The Medium Run”, Brookings Papers on
Economic Activity, Fall, 89-141.
[5] Caballero, Ricardo J. and Mohamad L. Hammour (1998), ”Jobless
Growth: Appropriability, Factor Substitution, and Unemployment”,
Carnegie Rochester Conference on Public Policy, 48, 51-94.
7

Job Security, Employment and Wages”,

Jobless Growth: Appropriability, Factor Substitution, and Unemployment”,

Revisiting European Unemployment: Unemployment, Capital Accumulation and Factor Prices”,

The Medium Run”,

A note on the determinants of labour share movements

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