Consumers’ preferences, number
of firm dynamics and the factor
shares evolution
Alexander Osharin and Valery Verbus
NRU HSE – Nizhny Novgorod
The motivation
To investigate the capabilities of the extended
two-factor ZKT model in explaining some
observations concerning factor shares dynamics,
markup movements and asymmetry of the
business cycle.
Some papers on factor shares
1. Bentolila (2003): Explaining Movements in the Labor
Share.
2. Jalava, Pohjola, Ripatti and Vilmunen (2005): Biased
Technical Change and Capital-Labor Substitution in
Finland, 1902-2003.
3. Матвеенко (2008): Ресурсы, институты, инновации и
экономический рост: двойственный подход.
4. Ripatti , Vilmunen (2010): Declining labor share –
Evidence of a change in the underlying production
technology?
5. Tipper (2011): One for all? The capital-labor substitution
elasticity in New Zealand.
6. Raurich, Sala, Sorolla (2011): Factor shares, the Price
Markup, and the elasticity of Substitution between Capital
and Labor.
Empirical evidence on factor shares
dynamics
Decline of labor share since the mid-1980s in
most of the OECD countries.
Empirical evidence on labor share short
and medium run movements (1)
Empirical evidence on labor share short
and medium run movements (2)
Empirical evidence on labor share short and
medium run movements (3)
Empirical evidence on labor share short and
medium run movements (4)
Empirical evidence on labor share short and
medium run movements (5)
The Labor Share and Real Wages in 12 OECD countries
Labor share
United States
Canada
Japan
Germany
France
Italy
Australia
Netherlands
Belgium
Norway
Sweden
Finland
Mean
Standard deviation
Source:
1970
69.7
66.9
57.5
64.1
67.6
67.1
64.8
68.0
61.6
68.4
69.7
68.6
66.2
3.6
Levels (%)
1980
68.3
62.0
69.1
68.7
71.7
64.0
65.9
69.5
71.6
66.4
73.6
69.6
68.4
3.3
1990
66.5
64.9
68.0
62.1
62.4
62.6
62.9
59.2
64.0
63.9
72.6
72.3
65.1
4.1
Real wage
Changes (%)
1970-1990
-3.3
-2.0
10.5
-2.0
-5.2
-4.5
-1.9
-8.8
2.4
-4.5
2.9
3.7
-1.1
5.2
Changes (%)
1970-1990
0.4
1.3
3.5
2.0
2.2
2.1
1.2
1.8
2.9
2.2
1.6
3.5
2.1
0.9
OECD Economic Outlook Statistics on Microcomputer Diskette.
Raurich, Sala, and Sorolla (2011) findings:
1. The elasticity of substitution between capital and labor is larger
than one in Spain and smaller than one in the U.S.
2. In Spain the labor income share (LIS) has decreased while the ratio
of capital to GDP has increased.
3. In contrast, both the ratio of capital to GDP and the LIS have
decreased in the U.S., which implies an elasticity of substitution lower
than one.
4. Consideration of the price markup drives the value of the elasticity
of substitution away from one and, therefore, provides a further cause
of rejection of the Cobb-Douglas (CD) specification. This result holds
both for Spain and the U.S. but goes in opposite direction: it yields an
upward bias in Spain and a downward bias in the U.S.
5. Price markup accounts for 63% of the LIS evolution in Spain and
57% in the US, whereas the elasticity of substitution explains,
respectively, 27% and 39% of its variation.
6. Price markup time series in both countries is countercyclical.
Mark-ups and return to capital in Spain
Mark-ups and return to capital in the USA
How markups move, in response to
what, and why, is however nearly terra
incognita for macro. . . . We are a long
way from having either a clear picture or
convincing theories, and this is clearly
an area where research is urgently
needed.
Blanchard (2008)
Markups and firm entry and exit
decisions literature
1. Jaimovichz (2003): Firm Dynamics, Markup Variation and the
Business Cycle.
2. Jovanovic (2005): Asymmetric Cycles.
3. Jaimovichz, Floetotto (2008): Firm dynamics, markup variations,
and the business cycle.
4. Floetotto, Jaimovichz, Pruitt (2009): Markup Variation and
Endogenous Fluctuations in the Price of Investment Goods.
5. Li, Mehkari (2009): Expectation Driven Firm Dynamics and
Business Cycles.
6. Nekarda, Ramey (2010): The Cyclical Behavior of the Price-Cost
Markup.
7. Cheremukhin, Tutino (2012): Asymmetric Firm Dynamics under
Rational Inattention.
Empirical evidence on asymmetry of business
cycle, markups and firm entry and exit decisions
1. Business cycle is asymmetric. The economy
tends to alternate between long periods of slow
expansion and short periods of sharp
contraction.
2. Markups lag the business cycle. Lagged
markups are countercyclical.
3. Firm exit is at list 30% more volatile than firm
entry.
4. Firm exit is strongly countercyclical and
asymmetric.
5. Firm entry is procyclical and symmetric.
Empirical evidence on firm entry and exit rates
(Nekarda and Ramey, 2010)
Two-factor model of monopolistic
competition
Preferences of L consumers are additively separable
(as in ZKT) and utility maximization has the form:
N
N
U   u ( xi )di  max , s.t.  pi xi di  e
0
xi
0
where xi is the demand of a consumer, pi is the variety
price vector and e is the individual expenditure, which
is supposed to be constant, N is the mass of varieties.
Goods market
Each firm produces a unique variety and solves the
following profit maximization problem:
 (qi )  pi qi  mqi  f  max
qi
where qi  Lxi is the output of a firm, m and f are the
marginal and constant production cost, which are
identical across firms.
Goods market SR equilibrium
Since all firms are identical, there exist a continuum of
the symmetric short-run equilibriums with
p

m
1
 e 
1  ru  
 Np 
e
x
Np
 (q)  pq  mq  f
where ru () is the relative love for variety, p and q  Lx
are the equilibrium levels of price and output of a firm.
Capital and labor markets
To get an equilibrium on capital and labor markets each
firm solves the following profit maximization problems:
 (ki , li )  pi qi (ki , li )  Rk i  Wli  f  max
ki ,li
where W and R are the nominal wage and interest
rate of capital, k i and li are capital and employment of
a firm.
Capital and labor market SR equilibrium
SR - equilibrium profit as a function of labor and capital
cost:
 (q)  pq  Rk  Wl  f
 p q
  k  R


 p q  W
  l
p
where  
is a markup of a firm, q  q ( k , l ) is a
m
production function of a firm.
Capital and labor shares (1)
For the Cobb-Douglas (CD) production function
 1
q(k , l )  k l
the capital and labor shares equal to
Rk 
Wl 1  
sk 
 , sl 

pq 
pq

1
sk  sl  ,   const

Where
0    1 is a constant.
Capital and labor shares (2)
For the CES production function
q(k , l )  [k   (1   )l  ]1/
where      1 is a parameter, related with capitallabor substitution  kl by  kl  1 /(1   ) , 0   kl  ,
the capital and labor shares equal to
sk 
1
 W / R  /(1 )
  W / R 
 /(1 )
1
,
sl 
1
1
  W / R  /( 1)  1
1/(1 )
where   [ /(1   ]
is a constant.
Substitution between capital and labor
When
and

is negative,
     0 , then  /(1   )  0
0   kl  1
In this case capital and labor are technical compliments
and labor share will increase with increasing W / R
relation.
When

is positive,
0    1 , then  /(1   )  0 and
0   kl  
In this case capital and labor are technical substitutes and
labor share will decrease with increasing W / R relation.
Constant absolute risk aversion
(CARA) utility function
The following constant absolute risk aversion (CARA)
utility function
u ( x)  (1  exp( ax))
has the relative love for variety (RLV)
ru ( x)  ax
with 0  a  1 . The more is
the consumers.
a
the more risk aversive is
SR equilibrium price and mark-up as a
function of the mass of the firms
Price level for CARA utility function and mark-up in the
SR-equilibrium are inversely dependent on the mass of
the firms:
*
ae
p  m
N
p
ae*
   1
m
mN
which corresponds to the pro-competitive behavior of the
equilibrium price.
Mark-ups counter-cyclical behavior (1)
Since real aggregate income equals to Y  Nq and
Npq  E *, where E * is the total nominal expenditure
level (assumed to be constant in the model), we have
E E
Y  Nq 

p m
It means that mark-ups are countercyclical (at list
towards the shocks changing the number of the firms).
The key question for us: whether the real GDP is
increased or decreased in the period when labor share
decreased?
Mark-ups counter-cyclical behavior (2)
Let
ae*
с
ae*
  1
 1 , c 
0
mN
N
m
and N is shocked, and N
ups
is increased, then for mark-
   c 
c

1     2  0
N N  N 
N
while for the total real income
Y
 E
 E
E 1  E 1 c




0
2
2
2
N N p N m
m  N m  N
*
*
*
It means that mark-up is countercyclical.
*
De-trended GDP for the USA (1950-2005)
1500
1000
500
0
1
-500
-1000
-1500
5
9
13
17
21
25
29
33
37
41
45
49
53
57
De-trended GDP for France (1950-2005)
1500
1000
500
0
1
-500
-1000
-1500
5
9
13
17
21
25
29
33
37
41
45
49
53
57
De-trended GDP for Germany (1970-2005)
2000
1500
1000
500
0
1
-500
-1000
-1500
6
11
16
21
26
31
36
De-trended GDP for the United Kingdom
(1950-2005)
2000
1500
1000
500
0
1
-500
-1000
6
11
16
21
26
31
36
41
46
51
56
The key question for us: what is going with
the number of firms ?
If the number of monopolistically competitive
firms on the market increase, then the mark-ups
fall and labor income share increases.
If the number of monopolistically competitive
firms on the market decrease, then the mark-ups
rise and labor income share decreases.
LR equilibrium price and mark-up as
functions of the exogenous parameters
Price level for CARA utility function in the LR
equilibrium
2


af
af


p *  m1 
 1 
1

 2mL

2mL 



Mark-up level for CARA utility function in the LR
equilibrium
2
p
af
af 

*
   1
 1 
 1
m
2mL
 2mL 
Profit as a function of the number of firms
 ( N )  Lpx  Lmx  f

 p1  ru ( x)   m

 ( N )  Lpxru ( x)  f

d ( N )
 Ndru ( x) / dN  ru ( x) 
 Le 
2

dN
N


Profit as a function of the number of firms
d ( N ) Le 
dx ru ( x) 

ru ( x)


dN
N 
dN
N 
dx
Since
 0 (as it is stated in ZKT), the right hand
dN
side sign depends on the sign of the RLV derivative
ru (x)
Profit as a function of the number of firms
In the price-decreasing (pro-competitive) case:
ru ( x)  0

d ( N )
0
dN
In the price-increasing (anti-competitive) case:
ru ( x)  0
d ( N )  0


dN  0
The sign of the initial value of the profit:
Nf
ru ( x) 
  0
Le
Nf
ru ( x) 
  0
Le
Profit as a function of the number of firms
in the pro-competitive case
YR
145000
144500
144000
143500
143000
142500
142000
141500
141000
140500
140000
139500
139000
138500
138000
137500
137000
t(quarters)
0
2
4
6
YR
145000
144500
144000
143500
143000
142500
142000
141500
141000
140500
140000
139500
139000
138500
138000
137500
137000
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
t(quarters)
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Labor share, %
27,6
Mark-up, %
15
27,4
14
27,2
13
27,0
12
26,8
11
26,6
10
26,4
9
26,2
t(quarters)
t(quarters)
26,0
8
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Thank you for rational attention!