Sequentiality versus Simultaneity:
Interrelated Factor Demand
Magne Krogstad Asphjell*
Wilko Letterie**
Øivind A. Nilsen***
Gerard A. Pfann****
May 2010
JEL Codes: D92, E22, E24, J23, L60
Key Words: Factor Demand, Interrelation, Nonconvex Adjustment Costs
* Norwegian School of Economics and Business Administration, Department of Economics,
Hellevn. 30, NO-5045 Bergen, Norway; Email: magne.asphjell@nhh.no.
** Maastricht University, Faculty of Economics and Business Administration, Department of
Organization and Strategy, P.O. Box 616, 6200 MD Maastricht, The Netherlands, Email:
w.letterie@maastrichtuniversity.nl; Tel: +31 43 3883645; Fax: +31 43 3884893.
*** Norwegian School of Economics and Business Administration, Department of
Economics, Hellevn. 30, NO-5045 Bergen, Norway; Email: oivind.nilsen@nhh.no.
**** Maastricht University, Faculty of Economics and Business Administration, Department
of Econometrics, P.O. Box 616, 6200 MD Maastricht, The Netherlands, Email:
g.pfann@maastrichtuniversity.nl.
Abstract:
In this paper we investigate a firm that has to decide about the timing of factor demand
adjustments. A structural model is developed to describe a firm’s demand for two production
factors which is subject to the presence of nonconvex adjustment costs. In this model the two
factor demand decisions are interrelated. That means that simultaneous adjustment of these
two production factors may either increase or decrease the total costs incurred by the firm. In
case simultaneous adjustment lowers the total costs, firms are more likely to conduct their
input factor changes simultaneously. On the other hand if simultaneous adjustment increases
the total costs incurred, firms are inclined to change the two input factors sequentially. One
reason for the first case, i.e. synchronisation of the two types of adjustments, is when
simultaneous adjustment reduces the time of disruption to the production process. On the
other hand, one could also think of a case where it would be efficient to implement input
changes subsequently. For instance, when introducing a new technology, it might be
economically reasonable to hire and train new workers prior to investing, such that the new
technology becomes productive as soon as possible after installation. Our model also shows
that flexible production factors, not being subject to fixed costs, may still exhibit intermittent
adjustment lumps, due to the cost of interrelation. Moreover, the importance of interrelation
on the decision thresholds increases when fixed costs are smaller. This implies that when
thresholds are being estimated for a single factor demand model alone, the model’s structural
parameters will be biased, especially if the factor is rather flexible. We apply the model to
investigate empirically the optimal timing of labour and capital demand changes. We develop
a maximum likelihood routine to estimate the structural parameters of the model. Plant level
data from Statistics Norway are used to estimate the cost or benefits of simultaneous
adjustment. The data concern plants operating in manufacturing industries and cover the
period 1993-2005. The estimates indicate that it is advantageous to adjust the stock of labour
and capital simultaneously. The cost advantage of changing labour and capital simultaneously
is small compared to the fixed adjustment cost of capital but is large compared to the fixed
cost of adjusting labour. Moreover, fixed costs of changing labour are much smaller than the
fixed costs of conducting investment. Our empirical results suggest that when estimating
single factor demand models the bias of parameter estimates is likely to be most severe in
case of labour demand.
2
1. Introduction
Firms have been observed to adjust the stock of their most productive factors, such as capital
and labour, in a lumpy fashion. Thus, they tend to concentrate big changes into short periods
while inaction dominates between these spikes. Such a pattern suggests that the smooth
adjustment of the important input factors is precluded by nonconvex, for instance linear or
fixed, costs leading to partial irreversibility of factor input decisions. With a few exceptions,
the existing literature on the irreversibility of production factors, has only considered separate
adjustment of one quasi-fixed production factor alone.1 Abel and Eberly (1998) note that the
observed lumpy employment pattern may not solely be due to a fixed cost component of
labour adjustment. They show that lumpy investment behaviour causes simultaneous large
employment adjustments in a model where labour demand is a fully flexible production
factor. Bloom (2009) finds that ignoring labour adjustment costs, as is typical in the
investment literature, is a reasonable approximation when modelling investment, while a
model with labour adjustment costs only, as is typical in the dynamic labour demand
literature, is problematic in the sense that the estimated parameters are far away from the true
ones found in a model that included both investment and labour adjustment costs. These
results indicate that controlling for investment dynamics is important when analysing the
more flexible labour input decisions.
Earlier research on multivariate factor input decisions suggests that the decisions
about changing several input factors are mutually dependent. Interrelation was initially
addressed using sector-level data in a linear setting by Nadiri and Rosen (1969). This study
was not based on a structural model with adjustment costs but it inspired others to investigate
the issue of interrelated factor demand decisions more deeply. Shapiro (1986) expands upon
1
For capital adjustment, see recent studies by Abel and Eberly (2002), Cooper and
Haltiwanger (2006), Letterie and Pfann (2007) and Nilsen and Schiantarelli (2003). For labour
adjustment, see the seminal contribution by Hamermesh (1989), and the more recent ones of Abowd
and Kramarz (2003), Rota (2004), and Nilsen et al. (2007).
3
Nadiri and Rosen (1969) and estimates a structural dynamic model of factor demand with
interrelation derived from the Euler equations. Galeotti and Schiantarelli (1991), and, more
recently, Merz and Yashiv (2007), have studied the topic of interrelation in a framework
without nonconvex costs of adjustment. Thus, from these findings it is hard to learn much
about the source of the lumpiness often seen in micro data. There is a substantial amount of
inaction observations for both labour and capital adjustments. This lumpiness may reflect the
existence of nonconvexities in the adjustment costs of the input factors. Contreras (2006)
investigates interrelation for a glass mould firm using simulated moments and calibration. He
finds that it is more costly for the firm to adjust capital and employment at the same time
rather than sequentially. Descriptive statistics from simultaneous analyses indicate positive
correlation between hiring and investment decisions. Recent empirical studies based on micro
data by Sakellaris (2004), Letterie et al. (2004), and Nilsen et al. (2009) have indeed revealed
that in the context of lumpy adjustment the dynamics of labour and capital demand are
interrelated. In particular, these papers have shown that at the micro level investment and
labour spikes are synchronized to a large extent. This may indicate complementarities in the
production process. Additionally, it may indicate that firms prefer synchronization, which
may stem from reduced adjustment costs when adjusting input factors simultaneously. Of
course, the described pattern may also reflect the nature of shocks to the shadow values of the
input factors. Note however, the studies by Sakellaris, Letterie et al., and Nilsen et al. op. cit
are all using non-structural and explorative approaches to analyse interrelateness. It is
therefore hard to identify whether the synchronisation is due to the nature of the changes in
the shadow values of the factors inputs, or whether it is caused by interrelated adjustment
costs. One of the goals of this paper is to build a structural model that incorporates
nonconvex adjustment costs together with interrelated adjustment costs that could either be
negative or positive (i.e. reduced costs due to synchronisation) or positive (making
4
simultaneous adjustments more costly). The advantage of a more structural model is that one
could potentially disentangle these two effects. It would also make it possible to identify
whether lumpy investment behaviour for one input is the effect of fixed costs for this factor,
or caused by interrelations in the adjustment cost function.
In this paper we investigate the consequences of interrelation where adjustments of
quasi-fixed input factors involve nonconvex costs. We refer to these factors as capital and
labour but it can also describe the behaviour of a firm deciding about any two different input
types (for instance R&D and natural resources). Our model deviates from work by Eberly and
Van Mieghem (1997), Dixit (1997), Abel and Eberly (1998), and Bloom (2009) in the sense
that we allow for the possibility that adjustment costs decrease or increase when the firm
decides to adjust two factors simultaneously. The occurrence of simultaneous adjustment
(synchronization) depends on the interrelation and especially on the question whether or not
interrelation adds to the costs of changing inputs or lowers those costs. One reason for the
latter case, i.e. synchronisation of the two types of adjustments, is when simultaneous
adjustment reduces the time of disruption to the production process. On the other hand, one
could also think of a case where it would be efficient to implement input changes
subsequently. For instance, when introducing a new technology, it might be economically
reasonable to hire and train new workers prior to investing, such that the new technology
becomes productive as soon as possible after installation.
We apply the model to investigate empirically the dynamics of joint labour and capital
demand decisions. Using Norwegian plant level data covering the manufacturing sector from
1993-2005 we obtain estimates of the adjustment costs parameters of the model by employing
a maximum likelihood routine and assess whether simultaneous adjustment of labour and
capital is beneficial or not. The paper proceeds as follows. In section 2 we develop the
theoretical model. In section 3 we discuss the role of fixed costs in relation with the cost of
5
interrelationship. Next, in section 4 we develop the econometric model. The data are
described in section 5. The empirical results are presented in section 6. Finally, section 7
concludes.
2. The model
Consider a firm that employs two production factors (capital Kt and labour Lt in year t) to
produce a non-storable output. The firm’s management maximizes V(.) denoting the present
discounted value of the cash flow resulting from its business operations. The objective
function is given by the value of the firm represented by


s






V

E
F
A
,
K
,
L

w
L

C
I
,
K
,
H
,
L

t
t
t

s
t

s
t

s
t

s
t

s
t

s
t

s
t

s
t

s
s

0
(1)
The term Et indicates that expectations are taken with respect to information available at time
t. The discount rate is given by  with 0    1. The variable wt denotes the wage paid by
the firm to a full time worker. Investment and hiring (or firing) are denoted by It and Ht
respectively. Sales are given by the expression FAt ,Kt ,Lt  where the term At represents a
variable capturing randomness in technology or stochastic behaviour of the demand
conditions the firm is facing.
When adjusting the stock of capital or the number of workers the firm incurs
adjustment costs defined as:
2
k


I
b
I

I

K
t



It,K







C
,
H
,
L

p
I


I

0

p
I


I

0




I

0


K

t
t t
t t
t
t t
t
t

 t
2
K


t



2
L

 (2)

H
b
H

H

L
t










p
H


H

0

p
H


H

0




H

0


L

t
t
t
t
t
t
t
L
 t
2


t



KL
It 

H




0
0
t
6
It,Kt,H
In the adjustment cost function C
t,L
t  the indicator function  . assumes the value
1 if the condition in brackets is satisfied and equals zero otherwise. The purchase price of
I
capital is given by pt . In case the firm sells capital we assume that the price received for one
I
I
I
unit of capital equals pt . Due to irreversibility of investment decisions pt  pt . This
costly reversibility of factors arises when a firm faces firing costs or when there is lemon’s
market for used capital such that the firm cannot sell its capital for the same price as when
bought. Note that the actual investment costs are incorporated in the adjustment costs
function in its linear part. Linear adjustment costs with respect to hiring and firing are
H
H
denoted by pt when H t  0 and pt when H t  0 . The convex costs are represented by
conventional quadratic functions in
It
Ht 2
and
. The fixed costs associated with adjusting
Kt
Lt
capital and labour are represented by  K and  L respectively and we assume that both
parameters are nonnegative and do not depend on whether I t and H t are positive or negative.
Fixed costs can be a result of the firm’s production processes being disrupted when
adjustment of capital or labour takes place.3 Alternatively, the firm’s management may have
to spend costly time guiding these organizational changes. We also assume that if the firm
adjusts labour or capital, then the adjustment costs of the other factor demand component are
affected. The term  KL is positive if simultaneous adjustment of capital and labour forces the
management to spend additional time and effort on the joint adjustment, or that the
adjustment causes a more severe production interruption compared to when capital or labour
2
It is straightforward to extend this adjustment cost function with a term capturing convex
costs of interrelation like
IH

 



K

Lwhere
0



1
.
 
We abstract from this possibility
K
L


1

to facilitate notation and hence enhance readability of the paper, but the extension is available upon
request. We note that in the figures presented later in this paper the area where simultaneous
adjustment (I≠0 and H≠0) takes place becomes defined by curved boundaries rather than straight lines
if   0.
3
Fixed costs are used as a simple way of introducing non-convexities.
7
adjustments are undertaken at separate points in time. On the other hand, the
expression  KL is negative if the firm’s managers save time or if the firm can save on costs of
disruption when adjusting the factors jointly.4
The firm decides upon the optimal size of the capital stock, K t , by setting
K
investment I t at the appropriate level. Since capital depreciates at rate  , the stock of capital
evolves according to the law of motion
K
K

I
1
K
i,t
1
i
,t
i
,t
(3)
Simultaneously, the firm determines the optimal value for the number of workers Lt by
choosing the desired and hence optimal level of hiring or firing denoted by H t . The amount of
labour evolves according to
L

H
1LL
i,t
1
i,t
i,t
(4)
L
where  measures the autonomous quit rate of workers.5 To obtain the optimal values for
I t and H t equation (1) can be optimized with respect to these decision variables subject to the
laws of motion governing the dynamics of capital and labour as given by equations (3) and
K
L
(4). Before proceeding we note that the variables t and t are the conventional shadow
values of an additional unit of capital and labour, respectively. We provide a derivation of the
shadow values in the appendix. It is easy to show that this is the expected present discounted
value of the marginal product of capital or labour plus the reduced adjustment costs in future
4
Note that we will assume that
K
L
K L



m
in
(

, ). The economic meaning of such an
assumption is that the cost of interrelation  KL do not completely offset the fixed cost of adjustment
 K or  L . If the assumption does not hold, for instance if KLL 0, ( KLK 0), this
implies that the firm will always adjust labour (capital) when it invests (hires or fires). This regularity
is unlikely to hold in reality.
5
We assume without any loss of generality that changes in capital and labour materialize with
a lag.
8
periods. Using the Lagrangean provided in the appendix the first order conditions for
investment and labour adjustment are


I








p


I

0

p


I

0

b

0
and


K


KI

t
t
t
I

t
t
K
t
(5)
t


H








p


H

0

p


H

0

b

0


L


LH

t
t
t
H

t
t
Lt
t
respectively. Similarly to Abel and Eberly (1994, 2002) the optimal amount of investment and
hiring or firing are
I t  tK  ptI

Kt  b K




Ht  tL  ptH

Lt  b L




(6)
I
I

I





p

I
0

p

I
0
In this latter set of equations we use the notation that p
and
t
t
t
t
t
HH

H





p
p

H
0

p

H
0
.6 Due to the presence of fixed costs of adjustment the
t
t 
t
t 
t
firm will not always follow the decision rules presented in equation (6). Sometimes it may be
optimal to abstain from adjusting capital and or adjusting labour. The threshold values for the
K
L
shadow values t and t can be derived by finding the value for which a change in I t and or
H t generates non-negative profits. The equation determining whether to change the stock of
capital and or to adjust labour is



I

H

C
I
,
K
,
H
,
L
K
t t
L
t t
t
t
t
(7)
t
The left hand side of (7) measures the expected benefits of changing capital and or labour,
whereas the right hand side denotes the cost associated with the firm’s decisions. Note that
tK and tL are the Lagrange multipliers (i.e. shadow values of capital and labour) employed in
6
We have made this as a notational simplification. It does not affect the qualitative findings in
this paper.
9
our optimization problem subject to constraints given by equations (3) and (4) depicted in the
appendix. Hence, they measure how the value of the firm changes if the constraints (3) and
(4) are relaxed or, equivalently, if capital and labour are increased by one unit. Therefore, the
K
L
expression t It t Ht represents an appropriate estimate of the benefits of input
adjustments. In a continuous time framework with one production factor a similar expression
holds exactly.
Using equation (6) it can be shown that equation (7) holds if

 

 
2
1K I2
1L H





p

I
0


p

H
0

t K
t
t
t L
t
t
Kt
Lt
2
b
2
b











I

0




H

0




I

0


H

0
K
L
t
KL
t
t
(8)
t
To solve the optimization problem of the firm we derive the conditions necessary for various
adjustment decisions.
3. Factor input decisions
The
firm

regards
adjusting
the
stock
of
capital
goods
to
be
desirable
if

1 K I2
K

p

. Hence, a necessary condition for changing the amount of capital is
t K
t
K t
2
b
2
b


p
A
KK
K
t
I
t
K
(9)
K
t
A similar necessary condition for hiring or firing workers is


1 L H2
L

p
L

implying
t
t
L t
2
b
2
b


p
A
LL
L
t
H
t
L
(10)
L
t
Equations (9) and (10) show that if the net benefits of adjusting capital and labour do
not exceed a certain minimum threshold, the firm decides to abstain adjusting. These two
K
L
thresholds are caused by the existence of the fixed adjustment costs  and  .
10
Consider now the case where both necessary conditions to adjust capital and labour
are satisfied as given in equations (9) and (10). Hence, the firm has an incentive to adjust at
least one factor of production. However, due to the cost of interrelation the firm may need to
select adjusting only one factor to maximise its objective function. It is optimal to adjust the
number
of
workers
rather
than

  

the
stock
of
capital
if

2
2
1
LH
L1
KI
K

p


p

. This holds when
t
t L
t
t
t K
t
L
K
2
b
2
b
K
b
2
b







p



p




L
L
L
L
2
2
LH
LK
t KI
t
t
t
K t
t
t
b
(11)
We can also rewrite this condition as:

K
2
b
b
2
b





p





p


 

L
L
L
2
LH
t
t
L
L
L
2
L
I
K
t K
t
K t
t
t
t
b


(11’)
The first term on the RHS of this expression is positive given our assumptions about the
adjustment costs parameters and the content in the squared bracket is also positive according
to eq. (9). Thus, the sum of the two terms is positive. Hence it is optimal to adjust labour
rather than capital if



 


L
L
K
2
2
b
L Hb
LK
t KI
|t

p
|


p


t
t
K t
L
L
b
t
t
(12)
Under certain conditions it is optimal to adjust only one input factor, because of the
cost of interrelation. It is optimal to adjust an additional factor of production if the net benefits
K
L
associated with that adjustment exceed the fixed costs of that second input (  or  ) plus
KL
the cost of interrelation   0 . Hence, it is worth also adjusting the stock of capital (given
that adjusting labour yields a higher value of the firm if only one input needs to be selected)
as soon as

  
1 K I2
K KL

p
α
t K
t 
Kt
2
b
(13a)
11
Similarly, labour will also be adjusted (given that changing capital yields a higher firm value
if only one input is selected) as soon as

  
2
1L H
L KL

p
L
α
t
t 
L t
2
b
(13b)
Hence, the boundaries determining when the firm will adjust both factors of production are


tK ptI 
2bK K KL
BK
Kt
tL ptH 
2b  
Lt

L
L
3a. Adjustment decisions when 
(14)
B
KL
KL
L
0
The analysis of firm level capital and labour demand decisions is summarized in figure 1.7
Starting with the inaction area, we see that this area is bounded by AK and AL and -AK and -AL.
As already noted, this inaction is caused by the presence of the fixed adjustment costs,  K
and  L , meaning that the shadow value of a new unit of capital (labour) has to move beyond
the thresholds defining area I. The firm only adjusts the two factors of production
simultaneously in the area indicated by III. These areas move further away from the origin if
 KL increases decreasing the area where simultaneous adjustment occurs. In fact, higher
KL
interrelated adjustment costs,  , increase the distance between AK and BK, and between AL
and BL. This means that the net benefits of changes need to be significant before a firm
chooses to change both input factors simultaneously. Demand for both factors is non-zero if
K
I
L
H
the benefits of change (i.e. | t  pt | and | t  pt | ) are high. For instance, a high positive
K
L
demand shock may increase the shadow values t and t simultaneously and hence provide
7
Our figures deviate from the ones by Dixit (1997), Eberly and van Mieghem (1997), and
Bloom (2009) in the sense that we have the marginal value of an additional unit on the axes, while the
authors op. cit. have productivity A normalized with K and L , respectively.
12
the firm an incentive to expand the scale of the firm by increasing both factors of production.
On the other hand, a firm may be increasing one input and decreasing the other input at the
same time if shadow values move in opposite directions. Such a situation may arise due to a
policy change affecting the relative price of the two factors of production or due to a
technology shock changing the optimal share of the inputs to produce a certain level of
output. But whether the adjustments of the input factors are made simultaneously or
sequentially depend on the size of the interrelated adjustment costs.
[Insert Figure 1 about here]
To see the importance of the interrelated costs, it might be helpful to see what
happens when the interrelated costs would be nonexistent, i.e. where  KL  0 .8 That means
that AK  BK , and AL  BL . In this case the areas where only one type of change would take
place, area II, would become much smaller. The do-both-changes area III would at the same
time be larger. Thus, the presence of positive interrelated costs would increase the area where
a firm would only involve in one type of investment activity at the time.
The curved boundary in the upper right corner in figure 1 crosses the rectangular areas
at AL  AK and BL  BK . This curve, corresponding to the right hand side of equation (12),
K
I
L
H
is concave if  K   L in the area where t pt 0 and t pt 0. Hence in figure 1 we
depict the case  K   L where the curved boundary crosses the horizontal axis (i.e.
K K L
2
b
(

)

where  p 0) at x
. If  K   L the right hand side of equation
K
K
t
L
t
H
t
K
I
(12) is convex and the curved boundary crosses the vertical axis (i.e. where t pt 0) at a
8
This is the case analysed by Dixit (1997).
13
L L
K
2
b
(



)
point defined as y
. Note that if  K   L , the boundary determining
L
L
t
whether to invest or adjust labour becomes a straight line. The three other curves in the figure
are analogous to the one just discussed.
Here we illustrate the importance of αKL assuming that  K   L . It is straightforward
i
to show that the distance between the thresholds decreases as  increases:


 



i
i
1
1


B

A
i
KL
i
2
2




0
i



(15)

i
i
lim
B
A
0for i {K, L}. This means that in figure 1 the area where the firm
and that 
i


completely abstains adjusting, area I, and the area where both factors are adjusted
simultaneously, area III, tend to move closer to each other as the fixed costs become larger
relative to the interrelated cost, αKL. Hence, large fixed costs will suppress the importance of
interrelation.9
Omitting the interrelated costs when estimating a single factor demand model with
adjustment costs may introduce a bias in the estimates of the fixed costs. Given a proxy for
the shadow value of capital, when estimating a single factor q model, the threshold will be
located between A K and B K for investment, since it is the presence of zeroes that identifies
the threshold. For labour demand the threshold will lie between A L and B L in a model
considering labour only. This indicates that when interrelation is important then a single
factor model is likely to produce biased and imprecise estimates for the adjustment costs in
i
particular for a flexible production factor with a low adjustment cost  .
Our analysis also shows that a lumpy adjustment pattern may be caused by the
existence of interrelated adjustment costs, and not by fixed adjustment costs for the factor
KL
 i is small with large
In addition, the relative importance of interrelation measured by 
fixed costs. In such a circumstance, interrelation will not play a major role in the dynamics of factor
demand.
9
14
L
K
KL
itself. Suppose for instance that   0 and that   0 and   0 . Note that
L
L
given   0 , A  0 here. Though in this case labour is fully flexible, the firm will not
K
I
K
L
H
L
K
I
always adjust labour when it invests (if |
t p
t |A ; |
t p
t |B and both | t  pt |
L
H
and | t  pt | are located at point (i) and (ii) in figure 1 for instance). Hence, labour
adjustment may appear intermittent with a large number of zero hiring / firing observations
even if it does not involve the firm incurring fixed costs for labour itself. Abel and Eberly
(1998) also argue that a flexible factor can be subject to lumpy dynamics due to large
adjustments of a less flexible factor. Our model reveals that the cost of interrelation is an
additional reason why flexible factors may exhibit intermittent patterns.
3b. Adjustment decisions when  KL  0 (and still  K   L ).
If  KL  0 firms actually benefit from adjusting both input factors simultaneously. The above
analysis can be applied to a large extent here as well. The main difference is that the choice
between investment or labour adjustment as presented below equation (12) has become
irrelevant in this case. This is due to the fact that the thresholds BL and BK are smaller than AL
and AK respectively if  KL  0 (also see equations 9, 10 and 14).
[Insert Figure 2 about here]
Figure 2 indicates that if the firm incurs lower adjustment costs because of
simultaneous adjustment area III where this event occurs becomes larger. If  KL decreases
then area III representing the situation that the firm changes both labour and capital move in
the direction of the origin of the figure. If LKL 0, the horizontal threshold at BL will lie
at the horizontal axis of figure 2. This means if the firm invests it will also change its labour
15
force, i.e. the area I 0,H0disappears. If KKL 0 then the firm will always invest
as soon as it alters its number of workers because the vertical threshold at BK will hit the
vertical axis of the figure and the area I 0,H0 vanishes. If both conditions KKL 0
and LKL 0 hold then the firm will always change the two factors of production at the
same time.
Like in section 3a we find that the distance between the thresholds decreases for
larger fixed costs:


 




i
i
1
1


A

B
i
i
KL
2
2




0
i



(16)

i
i
lim
A
B
0for i {K, L}. Again this implies that interrelation is less likely to
and that 
i


be a main determinant for factors that involve large fixed adjustment costs. But single factor
demand models for flexible inputs may be misspecified when interrelation is important.
4. The econometric specification
The investment decisions are dependent on the shadow values of investment and labour.
These represent the present discounted value of the marginal product of capital plus the
adjustment cost savings in future periods due to the higher level of capital, and
correspondingly for labour, the present discounted value of the marginal product of labour
plus the adjustment cost savings due to an increase in the stock of workers. See the appendix
for a derivation of the shadow values. The econometrician will be able to observe firms’
L
K
investment decisions. However, marginal values it and it are not observable. Thus, in
order to make the model estimable, we need to approximate these shadow values. For
simplicity we assume these to be linear functions of some observables, ZK and ZL.10 In
L
K
addition we add stochastic error terms it and it to the definition of the shadow values to
10
In the appendix a detailed discussion is provided on the variables included in ZK and ZL.
16
capture idiosyncratic factors not observable. It should be noted that this will also introduce
error terms in the two demand equations, equations (6).

p


Z

(17)

p

Z

(18)
K
it
L
it
I
it
H
it
K
0
L
0
KK
1 it
LL
1 it
K
it
L
it
K
L
We will assume that  it and  it are bivariate normally distributed with zero means, variance
 K and  L , and correlation coefficient  :11


  
 






K2

0








,




~ N
,|
Z
,K
,
Z
,L




0



 
KL
it it
K
it it
L
it it
L2
(19)
Thus, with the approximation described in equations (17) and (18), and the thresholds
equations (9), (10) and (14), we have all the necessary information to build up a bivariate
ordered probit model. That is, we simplify the problem by saying that every firm decides
between three options per input factor in period t; no change, decrease, or increase.
To illustrate the derivation of the likelihood function we can consider Figure 1 which
KL
assumes a positive interrelation cost,   0 . The firm has a set of nine possible choices
concerning adjustment of capital and labour. In the area denoted I the firm will abstain from
adjustment of any factor, in the four areas denoted II the firm will adjust one of the two
factors separately, while in the four areas denoted III, the optimal decision is to adjust the
levels of capital and labour simultaneously. The horizontal- and vertical-axis measure the
marginal benefit of capital and labour respectively. The optimal investment decisions for
K
I
L
H
each factor change as it  pit and it  pit move across their respective thresholds.
We note that the value of the correlation coefficient  has a clear economic interpretation. If
this parameter is positive, shocks to marginal values of capital and labour are positively correlated as
we would expect for demand shocks. If  is negative, it would indicate a negative correlation in
shocks to marginal values as we would expect to see for some types of technological shocks.
11
17
The fact that each observation is assigned to one of the regimes, based on the values
of I/K and H/L makes identification of our parameters of interest possible. Specifically, we
are able to estimate two different sets of threshold levels for separate and joint adjustments
because we allow for different thresholds to apply conditional on the adjustments being made
sequentially or simultaneously. Graphically, this means that we need observations such as
that represented by point (i) in Figure 1.
If we, as in figure 1, assume interrelation costs to be positive, we will in some cases
see marginal values for both factors exceed the thresholds for sequential adjustment, but not
for joint adjustment. That is, cases where marginal values exceed threshold level AK and AL,
but not BK and BL. This situation can be represented by the point denoted (ii) in figure 1, and
there will exist four different areas where similar situations can arise. According to equation
(12) the firm will invest in labour rather than in capital if



 


L
L
K
2
2
b
L Hb
KI
LK
it
|it

p
|


p


it
K it it
L
L
b
it
it
The values that give equality in the above expression are represented by curved lines in the
K
I
L
H
figure. Substituting with approximations as given by (17) and (18) for it  pit and it  pit ,
we can rewrite the condition as follows
 





 
K
b
2
b


(

Z
)




(

Z
)


b
L
L
2
LL
K
K
K
L
KL
L
L
i
t K
L
i
t
01
i
t i
t
0
1
i
t
K
i
t
i
t
(20)
L
K
Because the error terms  it and  it both enter in this inequality, the decision rule to be
implemented in a likelihood function would introduce significant complexity. We therefore
18
simplify by assuming that the four curved lines in figure 1 can be approximated by straight
lines.12
When the interrelated adjustment costs turn out to make synchronisation costKL
efficient, i.e.   0 , the limits of the investment regimes become somewhat different. This
is illustrated in Figure 2. Under the conditions in figure 2, we have no areas where the firm
must decide between investments in two adjustments that are separately profitable but not
jointly, and no approximation like the one discussed above needs to be made. Because of the
difference between cases with positive and negative interrelation costs represented by Figure
1 and 2, it is not possible to apply the same likelihood function in both cases. This can be
seen as the thresholds that limit the nine areas of the figure are not the same. This makes it
KL
necessary to specify slightly different likelihood functions conditional on the sign of  .
The main structure of the function though, is identical.
The likelihood function to be used in the ML estimation follows a standard setup for
discrete variables.13 It can be written as a sum of 9 different probability expressions for the
nine different outcomes summarized over periods t and firms i. The probabilities are given by
the standard cumulative bivariate normal distribution function. The expression below is an
abbreviated version of the full log likelihood.
12
When we assign probabilities to observations, we calculate the integrals of the four areas
where the decision rule should be applied, and due to the approximation by a straight line we divide
the integrals by two. Each part is then assigned to the appropriate action space. Thus, it is possible to
write the likelihood using only values of the univariate and bivariate cumulative normal distribution
functions.
13
Admittedly, we could have developed a model to use more information from the data, as in a
Tobit-type model. Our choice is based on reasons of computational simplicity. Regardless, our chosen
model identifies our parameters of interest, so that there would only be efficiency gains from the
alternative approach.
19







 ˆ

ˆ
KK
K
L
L
Lˆ
K
L
ˆ
ˆ
ˆ
b
2
b

2
ˆ
ˆ
Kˆ
K
K
Lˆ
L
L 
ˆ
L
o
g
L

l
o
g




Z



Z
,


2
0
1
i
t,
0
1
i
t




,

K
L
t

1
i


i
t
i
,
t
t




T
T
T
T

l
o
g
.

l
o
g
.

.
.
.

l
o
g
.













,
0
t

1
i


t

,

t

1
i


t

,

t

1
i


t
(21)
 2 denotes the standard cumulative bivariate normal distribution function, where the

, 
,0 
,

,
...,
subscript 2 indicates a bivariate function. 
denote the sets of firms that
t , t , t ,
t
in period t are allocated in the different investment regimes.14 A tilde above the parameters
indicates that as in standard models of discrete outcomes, we estimate the ratio between the
K
L
original parameter and standard deviations   and   . We will refer to these ratios as
normalized parameters.
K
K
L
L
The model allows us to recover normalized estimates of 0 ,
1 ,
0,
1 , pseudo-


 

K
K
L
L
K
L
LL K
L
,
b
,
b

a
n
d
b
KK


in addition to an estimated
thresholds b
correlation coefficient,  .15
We are, in other words, not able to identify convex and
nonconvex cost parameters directly. However, estimates of the normalized pseudo-thresholds
enable us to identify ratios of fixed cost parameters. Say that we have recovered estimates of
the parameters16
14
Superscripts -, 0 or + denote negative, zero and positive adjustments of capital and labor,
,
respectively. I.e. t is the set of observations with negative adjustment of both factors in period t,
t ,0 is the set of observations with negative adjustment of capital and zero adjustment of labour, etc.
15
We call them pseudo thresholds, since the estimates do not include the terms
2
16
K
K
L
L
Lit , while the thresholds A , B , A , and B do.
Asterisks in the superscripts denote that these values are pseudo thresholds.
20
2
K it and
K
K
K
L
L
L

KL
bKK ˆ*L bLL ˆ*K b  
 B
ˆ
ˆ*L b
A*K 
A

B


2
2
and
K 2,
L 2,
 
 
 
K
 
L
(22)
Because of the differences between the thresholds for separate and joint adjustments, we can
find the ratios between fixed cost parameters that are not normalized.


 
 


*
K*
K
K
K
L K
L
b


b


B

A
b

 K
 2


K
K
K K
*
K
K b
b
A

2
K
KK
L K
KK


 
 


*
L*
Lb
L
K
L K
L

b


B

A
b

 L
 2


L
L
K L
*
L
L b
b
A

(23)
2
LL K
L L
L L


*
K *
K *
L
L K
L
L
BA

A



.
L
K
*
L *
L *
K K K
B

A
A
(24)
(25)
Since it is possible to derive these ratios after estimation of pseudo-thresholds, it is
also possible to estimate cost parameters directly as parameters in the likelihood function by
changing the parameterization. We chose to follow a step-wise procedure when carrying out
estimations on simulated data. In the first step, we apply to different likelihood functions, and
obtain two different sets of estimates. One conditional on a positive interrelation cost
KL
KL
parameter  , and one assuming that   0 . By reviewing the estimates of pseudo
thresholds and comparing Aˆ * K with Bˆ * K and Aˆ * L with Bˆ * L , we get an indication of the sign
KL
of  , and thus which is the appropriate likelihood function to apply. In the second step,
after changing the parameterization, we use the applicable likelihood function to obtain direct
estimates of parameter ratios in equations (23) – (25).
5. Data
21
5a. Sample construction
The empirical evidence in this work is based on plant level information from Norway for
plants in the manufacturing industry, covering the period 1993-2005. The data are collected
by Statistics Norway. Focusing on manufacturing gives access to detailed information about
production and production costs, together with detailed information about investment and
employment. Investment is defined as purchases minus sales of fixed capital. Expenditures
related to repairs of existing capital goods are excluded from the definition of investment. We
focus on equipment which includes machinery, office furniture, fittings and fixtures, and
other transport equipment, excluding cars and trucks. We have restricted our attention to
plants with 10 or more employees. Some data might be available for also smaller plants. Note
however, that these observations may be associated with measurement errors since some of
the information from these types of plants often are imputed by Statistics Norway. For our
purpose it should not be critical, rather on the contrary. For these smaller plants it would be
very hard to disentangle the effects of indivisibility from the effects caused by nonconvex
adjustment costs. We have excluded all auxiliary units which do not take part directly in
production, such as separate storage and office units. Plants in which the central or local
governments own more than 50% of the equity have also been excluded from the sample, as
well as observations that are reported as “copied from previous year”. This actually means
that a data entry is missing. The remaining data were trimmed to remove outliers.17
Capital stock was built up using the perpetual inventory formula, using information
about initial capital stock and net investments (see the Data Appendix for details about the
initial capital stock). Finally, we include only plants for which we have 6 or more consecutive
observations. For the analysis we only use the fifth or higher usable observations for each
plant, reserving the first periods for each plant for the initial stock of capital. Our final sample
We excluded observations where the investment rate, I/K, the net workplace changes (L/L),
and wage per employee, w, were outside “reasonable” limits.
17
22
used in the maximum likelihood estimations include 2460 plants with a total of 14759
observations from the period 1997-2005.18
5b. Descriptives
Table 1 presents a number of summary statistics. The average firm in the sample employs
approximately 75 individuals, L. In 1996 prices the average stock of capital, K, is about 60
million NOK or €7.5 million. This reflects that in manufacturing industries firms are
relatively capital intensive. We observe that investment and hiring rates, I/K and ΔL/L, are
significantly positively correlated to the capital and labour productivity respectively. Also the
hiring rate is positively correlated to the wage rate, w, and this correlation is significantly
different from zero. This finding is not so obvious at first sight. However, it may be the case
that firms start hiring workers when the economy is booming and that hence higher wages
reflect that expansion efforts are worthwhile. Wage rates are also positively related to labour
productivity, Y/L which is what one would expect to happen if labour markets operate well.
[Insert Table 1 about here]
In table 2 we report the frequency distribution of the hiring and firing rates. We
observe that hiring is zero in 11.2 percent of the cases, supporting the notion that non convex
costs play a role in determining labour adjustments. A similar argument can be put forward
for capital adjustment for which we report 15.1 percent zeroes. At least one adjustment is
being made very often as both hiring and investment are zero at the same time in only 2.2
percent of the observations. We find that just one adjustment is made (hiring or investment is
18
The criterion that these plants are only included if they have at least six or more useable
observations, might introduce some selection issues. Probably larger and more successful plants are
more likely to be present in our data.
23
non zero) in 21.9 percent of the cases. For sequential adjustment a necessary condition is that
in a period only one adjustment must take place. Hence, this finding is suggestive of the
advantage of simultaneous adjustment. In fact in our data we find that 75.9 percent of the
observations are cases where firms adjust labour and capital simultaneously.
[Insert Table 2 about here]
Table 3 reports the correlations between contemporaneous, lagged and forwarded investment
and hiring rates. It appears that contemporaneous hiring rates and their lagged and forwarded
counterparts are negatively correlated at -0.08 (-0.083 and -0.078 in the table). This indicates
that labour adjustments in two consecutive years are less likely. For investment we find that
contemporaneous investment is positively correlated with lagged and forwarded investment
rates with a coefficient of 0.148 and 0.131 respectively. This means that large investments are
spread over at least two consecutive years. We observe that the correlation coefficient
between contemporaneous investment and hiring rates is about 0.116. In line with cost
advantages of simultaneous adjustment of labour and capital the table indicates that the
correlation of contemporaneous hiring (investment) with lagged or forwarded investment
(hiring) is lower at 0.08 (ranging between 0.070 and 0.090).
[Insert Table 3 about here]
6. Results
The descriptive statistics so far indicate that firms in our sample tend to implement labour
and capital adjustments simultaneously. To get a better understanding of what drives the
synchronization of labour demand and investment the structural model is estimated to
24
identify whether interrelations in the adjustment cost function are causing this phenomenon.
Table 4 presents the estimation results. It shows that the thresholds satisfy: AK  BK and
AL  BL . This result is consistent with the notion that simultaneous adjustment yields cost
KL
advantages:   0 and that interrelation is far more important for labour than for capital.
KL
 KL


0
.
27
In fact the results imply that
and that K  0 . The fixed adjustment costs of
L

capital exceed those for labour a lot. The estimates indicate that
L
 0 . In section 3 we
K
observed that the cost of interrelation matters most when an input factor is flexible or, to put
it differently, is associated with relatively small fixed adjustment costs. The estimates
indicate that primarily the dynamics of the flexible input (i.e. labour demand) is affected by
the cost of interrelation, whereas capital demand remains rather insensitive to it. This is in
line with Bloom (2009) who states that ignoring labour demand when investigating
investment is not very harmful. However, when estimating a demand model for labour,
ignoring capital adjustment yields biased estimates.
[Insert Table 4 about here]
L
K
The shocks to the marginal value of labour and capital  it and  it have a positive correlation
coefficient (ρε=0.17). This means that shocks to the fundamental variables driving labour and
capital demand tend to move in the same direction. This may be due to demand shocks that
induce a firm to adjust labour and capital in the same direction. A precise interpretation of the
shocks is cumbersome as they are picking up misspecification of the model, technology
shocks, or other shocks unaccounted for by the model.
25
We have also estimated the model using the maximum likelihood routine using the
KL
assumption that simultaneous adjustment is costly:   0 . The estimates of this analysis
KL
indicate that the assumption   0 does not hold. The results show that the estimated
thresholds are consistent with cost advantages of simultaneous adjustment of labour and
capital. We do not report the results to save some space.19
7. Conclusion
If it is costly to adjust two factors of production at the same time a firm has an incentive to
conduct a one at a time policy: adjust only one factor. The firm is inclined to adjust
simultaneously if there are cost advantages of doing so. Recent empirical studies indicate that
large adjustments tend to be synchronized in some data sets, but in others they are not. In this
paper we show that this feature of the data may be consistent with the hypothesis that joint
adjustment increases or reduces the overall costs of adjustment. We also find that when fixed
costs of adjustment are large then the role played by the cost of interrelation may be minor.
Hence, synchronization may occur because large fixed costs dampen the relative importance
of interrelation. One prediction from our theoretical model is that when one factor is more
flexible than the other, in a model neglecting interrelated costs serious omission bias will
occur in the estimated adjustment costs for the most flexible production factor. We also show
that a production factor may be subject to intermittent dynamics even though adjusting this
factor does not raise a fixed cost. Lumpiness may just occur due to the presence of cost of
interrelation. If costs of interrelation are high, at times the firm changes the least flexible
production factor, it will leave the flexible factor unchanged.
19
Asphjell et al. (2010) investigate whether the maximum likelihood routine employed
in this paper is capable of retrieving the true parameters from a simulated dataset. All
simulations indicate the routine has satisfactory properties when the data set has the size of
the data used for the estimations in this paper.
26
Using Norwegian plant level data concerning the manufacturing industry in the period
1993-2005 we obtain insight into the deep structural parameters of a labour and capital
demand model. The maximum likelihood routine we employ reveals that simultaneous
adjustment of the two production factors yields cost advantages. The estimates indicate that
the fixed cost of capital is large compared to both the fixed cost of adjusting labour and the
cost advantage of simultaneous adjustment. However, the benefits of synchronizing labour
and capital demand are large relative to the fixed adjustment cost of labour. These results
imply that when estimating single factor demand models the parameters are likely to be
biased most severely in case of labour demand.
27
References
Abel, A.B., Eberly, J.C., 1994, “A unified model of investment under uncertainty”, American
Economic Review, 84, 1369-1384.
Abel, A.B., and Eberly, J.C., 1998, “The Mix and Scale of Factors with Irreversibility and
Fixed Costs of Investment”. Carnegie-Rochester Conference Series on Public Policy,
48, 101-135.
Abel, A.B., and Eberly, J.C., 2002, Investment and Q with fixed costs: an empirical analysis.”
Mimeo, The Wharton School, University of Pennsylvania.
Asphjell, M.K., Ø.A. Nilsen, W. Letterie, “Estimating Interrelated Nonconvex Adjustment
Costs by Maximum Likelihood.” Mimeo
Bloom, N., 2009, “The impact of uncertainty shocks”, Econometrica, 77, 623-685
Contreras, J.M., 2006, “Essays on factor adjustment dynamics.” PhD Thesis, University of
Maryland, Department of Economics.
Dixit, A., 1997, “Investment and employment dynamics in the short and the long run”, Oxford
Economic Papers, 49, 1-20.
Dixit, A., and R. Pindyck, 1994, Investment under Uncertainty, Princeton University Press,
Princeton, New Jersey.
Eberly, J.C., van Mieghem, J.A., 1997, “Multi-factor dynamic investment under uncertainty”,
Journal of Economic Theory, 75, 345-387.
Galeotti, M. and F. Schiantarelli, 1991, “Generalized Q Models for Investment”, Review of
Economics and Statistics, 73, 383-392.
Letterie, W.A., Pfann, G.A., Polder, J.M., 2004, “Factor adjustment spikes and interrelation:
an empirical investigation”, Economics Letters, 85, 145-150.
Merz, M., Yashiv, E., 2007, “Labor and the value of the firm”, American Economic Review,
97, 1419-1431.
28
Nadiri, M.I., Rosen, S., 1969, “Interrelated factor demand functions”, American Economic
Review, 59, 457-471.
Nilsen, Ø.A, A. Raknerud, M. Rybalka and T. Skjerpen, 2009, “Lumpy Investments, Factor
Adjustments and Labour Productivety”, Oxford Economic Papers, 61, 104-127.
Sakellaris, P., 2004, “Patterns of plant adjustment”, Journal of Monetary Economics, 51, 425450.
Shapiro, M., 1986, “The dynamics of demand for capital and labor", Quarterly Journal of
Economics, 51, 513-542.
29
K
L
Appendix: Derivation shadow values t and t :
K
L
The terms t and t represent the Lagrange multipliers for capital and labour respectively,
associated with constraints given by equation (3) and (4). Using equation (1), The
Lagrangean for this optimization problem is:
  


K
L






L

V

E
I

1

K

K

I

1

L

L

t
t
tt
t
K
t
t

1
t
t
L
t
t

1
s

0


To obtain the Lagrange multiplier for capital we solve











L

E
F
K
,
L
,

w
L

C
I
,
K
,
H
,
L
K
K
t
t
t

1
t

1
t

1
t

1
t

1
t

1
t

1
t

1
t

1






E
1


0
t
t
K
t

1

K

K
t

1
t

1
 
In the next, P refers to a probability of an event occurring. For this probability, the
superscript K refers to the event of only capital adjustment, L refers to only labour
adjustment, B refers to adjustment of both labour and capital and N indicates the event of no
adjustment at all.
Lt
tK Et 1KtK1 
Kt1


2


K
 

 It1 
b
K
I
K







P
p
I



K
t1
t1 t1
 t1
FK ,L ,   

2
K
 t1 


t1 t1 t1


 

Kt1
Kt1






2


 

bL Ht1 
L H
L






P
p H   
Lt1


 t1 t1 t1




2  Lt1 





Et




Kt1






2
2
K
L
 





I
H
b
b

 B pI I K   t1  K pH H L   t1  L KL
t1
t1 t1
 P
 t1
 t1
 t1 t1

2
2
Kt1 
 Lt1 




 

Kt1








30


2


K




b  Its1 
K  I
K







P
I
  
ts
1 p
ts
1 
K
 ts1 ts1

2 K
FK ,L ,  

ts
1


ts
1 ts
1 ts
1




K
K


ts
1
ts
1




2





bL H
L  H
L


ts
1





P
p H   
ts
1 
L
 ts1 ts1 ts1



2
L

 ts1 



K
K s s
1

t Et 
1 





K
s0
ts
1






2
K





I
b

I
K


ts
1
p I

  
ts
1

K 
K

 ts1 ts1

2
 ts1
B



P
ts
1
2

L

 H

b H
L
KL 
ts
1




 
 

ts
1
ts
1
L
pts1H



2
ts
1
L






K


ts
1




Similarly we can show

2

K



 Its1 
b
K  I
K





P
p
I



K
ts
1
ts
1 ts
1
ts
1 





2 K
FK ,L , 
ts
1



ts
1 ts
1 ts
1

w

ts
1
L
L

ts
1
ts
1


2




bL H
L  H
L

ts
1




P
p
H



L
ts
1
ts
1 ts
1
ts
1 






2L

ts
1



L
L s s
1
t Et 
1 



L
s0
ts
1



2



bK  Its1 

K

pI I



  
K
ts
1



 ts1 ts1

2 K
ts
1
B



P
ts
1
2

L

 H



b H
L
KL
ts
1






p
H



L




t

s

1
t

s

1
t

s

1
L 




2
 ts1 




L

ts
1



31
As shown in the above equations the shadow values of capital and labour are a function of the
expected value of the future marginal productivity of capital and labour and the changes in
the expected value of future adjustment costs. These features reflect that current decisions
affect future properties of the firm’s optimization problem. Current factor demand affects the
production scale and hence productivity of next periods and it determines the size of the
adjustment costs and the probability of adjustment. Hence, our derivations indicate that to
proxy the shadow values of labour and capital one needs to capture information on how
current decisions affect the future marginal productivity of factor inputs, the future
probability of adjustment and the change in the size of adjustment costs. The future marginal
productivity of input, the probability of future adjustment and the change in the size of future
adjustment costs in
next periods t+s are a function of the state variables

I
H
w
,
p
,
p
,t
,
K
and
L
and future decisions concerning investment and hiring (It+s
t

s
t

s
t

s
s
t

s
t

s
and Ht+s). At the beginning of a period t when the investment and hiring decisions are set,

K
L
information is available on w
t,p
t,p
t,
t,
tand
t. There are no other variables in the
I
H
model that are relevant to the optimization problem of the firm. Of course future realizations
of these quantities are relevant to the firm’s optimization problem, but these realizations are
governed by either a stochastic process (i.e. w
t
s,p
t
s,p
t
s,
t
s) or the optimal decision of the
I
H
Lts ). Hence, the firm derives expectations on these future
firm in the future (i.e. Kts and
realizations using information it has on the current realizations. The history of these variables
is irrelevant as we assume that all past information on them is contained in their current

K
L
realizations. Using the information set available to the firm (i.e. w
t,p
t,p
t,
t,
tand
t) its
I
H
management will determine both the cost of the factor demand decisions and the proceeds
K
L
K
L
given by t and t . Hence, we conjecture that t andt are a function of the state
32

K
L
variables w
t,p
t,p
t,
t,
tand
t. Hence, to obtain a proxy for the two shadow values
I
H
tK andtL we need variables that capture information on these state variables.
Suppose first that the revenue function is Cobb Douglas with constant returns to scale
 1
YvK
L
then

1

YK
YL


v




1



Therefore, we suggest that the shadow values are proxied by


Y
Y



H
time
dummies,
sector
dummies
,
w
, ,


K
L


KK
t
t t
t
t t


Y
Y



H
time
dummies
,
sector
dummies,
w
, ,


K
L


L L
t
t t
t
t t
We use time dummies to capture the dynamics of investment and hiring prices. In
principle we have information on investment prices, but lack a proper variable to measure
hiring prices. To capture both we employ time dummies. Note that due to multicollinearity
we cannot use both the investment price and the time dummies. Together the latter two
variables in these functions contain information on ε, K and L. The variables used to proxy
the shadow values concern beginning of the period information. In the estimations we use
lagged values of these variables. We assume that it is easier for the firm’s management to
observe lagged rather than contemporaneous information.
K
Note that the shadow value of capital t is also a function of the marginal
productivity of labour (Y/L) and the wage rate (w). In single factor models it is usually
K
assumed that this is not the case. However, from the expression above concerning t it can
K
be seen that t is affected by expectations on future realizations of the hiring rate. Hence,
tK is a function of expectations on future outcomes of tL . As a consequence, also from this
33
K
perspective it makes sense to include Y/L and w in t . A similar argument holds for why Y/K
L
is included in t .
Real Replacement Value of Capital Stock at the end of period t (Kt): The replacement value is
computed using the perpetual inventory method. This method requires a starting value for the
capital stock. Our data do not provide information on insurance or book value of the capital
stock. Therefore we estimate the starting value as follows. The real stock of capital at the end
of year m=t0+n, where n is an integer larger than or equal to 1, is given by
Km = Im+(1-)Im-1+...+(1-)nKt0
where  is the rate of depreciation set equal to 0.06. If we assume that in year t the firm
grows at a rate equal to gt then investment in the n periods should be approximately sufficient
to ensure that
Km = (1+gt0)...(1+gm)Kt0.
Given the values gt it is possible to solve for the value of Kt0. We approximate gt by
calculating the firm’s real production growth from time t-1 to t. The value of n is set equal to
5. The reason is that we need a sufficiently long period to estimate the starting value of
capital, because firms tend to concentrate investment in a relatively short period of time. If n
is very small the probability of underestimating the starting value of the capital would be
considerable. This procedure requires an uninterrupted sequence of data on production and
investment during at least five years. As our data start in 1993 and run until 2005, the starting
year for the data needed to construct the capital stock for a plant can vary between 1993 and
2000. We drop observations for which we find a negative stock of capital as these are poorly
estimated. Given the initial value the replacement value of the capital stock at the beginning
of year t is calculated using the perpetual inventory formula Kt= It-1+(1-)Kt-1.
34
KL
Figure 1: Investment Regimes,   0
35
KL
Figure 2: Investment Regimes,   0
36
Table 1: Descriptive statistics
Mean
St.dev
Correlations
I/K
L/L
(Y/K)-1
(Y/L)-1
I/K
L/L
(Y/K)-1
0,134
0,004
7,389
0,251
0,175
8,646
1,000
0,116
0,263
1,000
0,061
1,000
(Y/L)-1
1362,2
1453,3
0,002
0,067
0,179
1,000
294,6
82,6
59127,5 436941,0
75,5
130,0
-0,028
-0,039
0,006
0,106
-0,011
0,030
0,031
-0,079
-0,042
0,341
0,107
0,129
w-1
K
L
Notes: Wages, w , are defined as total wage expenditure per employee, including pay-roll taxes
Capital, K , is given in 1000 NOK (≈ € 125) in 1996 prices
w-1
K
L
1,000
0,075
0,159
1,000
0,216
1,000
Table 2: Distribution of I/K and L/L
I/K
<-0.667,-0.167]
<-0.167,0.000>
=0
<0.000,0.167]
<0.167, +>
L/L
<-0.667,-0.167] <-0.167,0.000>
=0
<0.000,0.167] <0.167, +>
0,2
0,8
1,6
5,0
1,2
0,2
1,5
3,9
23,3
6,6
0,1
0,5
2,2
9,0
3,3
0,1
0,8
2,4
19,6
8,4
0,1
0,2
1,0
4,6
3,3
0,7
3,8
11,2
61,5
22,8
8,8
35,5
15,1
31,3
9,2
100,0
38
Table 3: Correlation in timing
(I/K)+1
(I/K)-1
I/K
(I/K)+1
I/K
(I/K)-1
1,000
0,131
0,071
1,000
0,148
1,000
L/L)+1
L/L
L/L)-1
0,108
0,080
0,043
0,070
0,116
0,090
0,036
0,083
0,117
L/L)+1
1,000
-0,083
0,004
L/L
L/L)-1
1,000
-0,078
1,000
39
Table 4: Estimation Results
Investment
Hiring
0.00757
0.00644
(0.00190)
(0.00131)
(Y/L)-1
0.00000530
-0.0000107
(0.0000123)
(0.00000786)
w-1
0.000540
0.00193
(0.000185)
(0.000143)
constant
1.457
-0.381
(0.0639)
(0.0493)
Threshold Estimates
327.2
AK
(16.72)
(Y/K)-1
AL
0.918
(0.0914)
BK
327.2
(16.72)
BL
0.675
(0.0286)
Parameter Ratios
L
K
 KL
K
Other Parameter
0.000179
(0.0000200)
-0.0000475
(0.0000188)
0.171
(0.0144)
N
14759
Note: Standard errors are in parentheses. Year and sector
ρε
dummies are included in the estimation but not reported here
to conserve space.
40