Documents

de travail

« Fixed cost, variable cost, markups

and returns to scale »

Auteurs

Xi Chen, Bertrand M. Koebel

Document de Travail n° 2013 – 13

Septembre 2013

Faculté des sciences

économiques et de
gestion
Pôle européen de gestion et
d'économie (PEGE)
61 avenue de la Forêt Noire
F-67085 Strasbourg Cedex

Secétariat du BETA
Géraldine Manderscheidt
Tél. : (33) 03 68 85 20 69
Fax : (33) 03 68 85 20 70
g.manderscheidt@unistra.fr
www.beta-umr7522.fr
 Fixed cost, variable cost, markups
and returns to scale
Xi Chen
Bertrand M. Koebel

September 2013

Abstract. This paper derives the structure of a production func-

tion which is necessary and su¢ cient for generating a …xed cost.
We extend the classical production function in order to allow each
input to have a …xed and a variable part. We characterize and es-
timates both …xed and variable components of the cost function
and studies how …xed and variable costs interact and a¤ect …rms’
behavior in terms of price setting and returns to scale.

Keywords: identi…cation, imperfect competition, returns to scale,

unobserved heterogeneity.

JEL Classi…cation: C33, D24, E10, J23, L60.

BETA, CNRS, Université de Strasbourg. Email: xchen@unistra.fr.

Corresponding author: Bureau d’Economie Théorique et Appliquée (BETA), CNRS, Université de Strasbourg,
61 avenue de la Forêt Noire, 67085 Strasbourg Cedex (France). Tel (+33) 368 852 190. Fax (+33) 368 852 071.
Email: koebel@unistra.fr.

We would like to thank Pierre Dehez, Andreas Irmen, François Laisney, Clemens Puppe and the participants
of seminars in Aix-en-Provence, Cergy, Luxembourg, Maastricht, Nancy, Trier and Strasbourg for their helpful
comments.
1. Introduction
A long tradition going back to Viner (1931) considers that …xed costs correspond to the
cost of …xed inputs.1 However, splitting the whole set of inputs into two disjoint sets
(with either …xed or variable inputs) does not provide a faithful description of many
economically interesting technologies. If some variable inputs are substitutable to …xed
inputs, then this sharp distinction vanishes. This paper extends the microeconomic
foundations of production analysis by allowing each input to have a …xed and a variable
part.
Empirical speci…cations of production and cost functions are also shaped by this
dichotomy between …xed and variable inputs. Some speci…cations consider …xed costs to
be the cost of the …xed inputs. Others, like the Cobb-Douglas, the CES, and even ‡exible
functional forms like the Translog, assume that …xed costs are nonexistent. We propose
a generalization of the Translog functional form which is compatible with inputs having
both a …xed and a variable part. Our empirical results support the extended Translog
speci…cation and show that the …xed cost is signi…cant and neglecting it yield estimation
biases, especially on the markup and the rate of returns to scale. Fixed costs, although
not functionally dependent on the output level, are correlated with output, and should
be explicitly considered to avoid these estimation biases. Our …ndings are compatible
with the predictions of models with heterogenous technologies (see e.g. Acemoglu and
Shimer (2000) and Cabral (2012)), in which there is a trade-o¤ between production
functions having a large …xed cost and low variable cost and those with the converse
con…guration.
Despite the challenging result of Baumol and Willig (1981, p.405) according to which
…xed costs “do not have the welfare consequences normally attributed to barriers to
entry”, there is a quite large literature on …xed inputs. Fixed costs are useful for
explaining coordination failure (Murphy et al., 1989) and international trade (Krugman,
1979, Melitz, 2003). Blackorby and Schworm (1984, 1988) and Gorman (1995) have
shown that …xed inputs hamper the aggregation of production (and cost) functions,
whereas a …xed cost does not represent an aggregation problem. Fixed costs are also
1
In the words of Viner (1931, p.26): “It will be arbitrarily assumed that all of the factors can for the short-run be
sharply classi…ed into two groups, those which are necessarily …xed in amount, and those which are freely variable. [...]
The costs associated with the …xed factors will be referred to as the "…xed costs".”

1
considered in general equilibrium theory with imperfect competition, see for instance
Dehez et al. (2003). Contributions in the …eld of industrial organisation on the reasons
and consequences of …xed (and sunk) cost, are so numerous that we cannot survey
them here. Berry and Reiss (2007) discuss some important issues on identi…cation and
heterogeneity of …xed costs. Di¤erences between …xed and sunk cost are commented by
Wang and Yang (2004) and Sutton (2007).
We mainly contribute to the literature in production analysis. One objective is to
characterize and estimate both …xed and variable components of the cost function, to
investigate their heterogeneity over …rms and study how …xed costs a¤ect their behavior
in terms of price setting and returns to scale. Microeconomic textbooks present alter-
native characterizations of …xed costs. We follow Baumol and Willig (1981, p.406) and
consider the long run …xed cost as the magnitude of the total long run cost function
when the production level tends to zero. This paper derives the production technology
which generates the …xed cost, an issue which is usually neglected when dealing with
…xed cost. It is well known (see Mas Colell et al., 1995, p.135) that …ctitious inputs can
be used for imposing constant returns to scale on arbitrary technologies. This paper
shows that the …xed cost of production can be represented as the cost of …ctitious (un-
observed) inputs. We …rst characterize the production technology which generates the
traditional …xed cost and show that it is quite restrictive and given by y = F xv + xf
where xv denotes the vector of variable inputs and xf the …xed inputs. As total input x
can always be additively split into two categories, the structure F may be considered as
perfectly general. However, two physically similar inputs may be technologically di¤er-
ent and we propose to extend the production function to y = G xv ; xf . This extended
production technology generates a …xed cost which is not equal to the cost of inputs xf ;
and identi…cation of …xed inputs is no longer possible. However, the amount of inputs
which allows to initiate production is well identi…ed.
Our theoretical contribution also requires extending the econometric toolbox for esti-
mating cost functions. First, usual cost function speci…cations are not compatible with
a ‡exible speci…cation of the …xed cost. For approximating a cost function with a …xed
cost component, we have to go beyond (locally) ‡exible cost functions, and develop a
cost speci…cation which is a valid approximation at two points: around the actual point

2
of production and around the breakup point which allows a …rm to start production.
Second, as the inputs xv and xf cannot be observed, we have to amend the traditional
estimation method by introducing unobserved and correlated heterogeneity in the …xed
and variable cost speci…cation. We extend Swamy’s (1970) random coe¢ cient estimator
to our nonlinear setup. The empirical part of this paper uses panel data for US man-
ufacturing sectors in order to estimate the height and the type of …xed cost as well as
their implications in terms of markup pricing, returns to scale and technical change.
In Sections 2 and 3 we explore two de…nitions of …xed costs and their microeconomic
foundations. Sections 4, 5, and 6 discuss econometric issues related to …xed costs: biases
when they are neglected, speci…cation issues, and unobserved heterogeneity. Section 7
reports the empirical results, obtained for 462 US manufacturing industries observed
over the years 1958 to 2005.

2. De…ning …xed costs and …xed inputs

The de…nition of …xed costs is central in economics and is brie‡y discussed in most
introductory microeconomic textbooks.2 One di¢ culty with most de…nitions is that
they do not highlight the relationship between the …xed cost and the …xed inputs. Are
…xed inputs physically …xed? Do …xed inputs correspond to nonoptimal choices? This
section shows that it is not necessarily the case: a …xed cost can arise in a context where
all inputs are optimally adjusted.
Most economists agree that the …xed cost u corresponds to the part of the cost which
does not vary with the level of production:

c (w; y) = u (w) + v (w; y) ; (1)

where w denotes the input prices and y the output level. Function v corresponds to the
variable cost of production and satis…es v (w; 0) = 0: Any cost function can uniquely be
written in this way by de…ning

u (w) c (w; 0) (2)

v (w; y) c (w; y) c (w; 0) :

2
It seems somewhat surprising, however, that the New Palgrave dictionary of economics has no entry for the term
"…xed cost". The term is also not commented in Diewert’s (2008) contribution on cost functions.

3
We will comment the following alternative de…nitions for the …xed cost and …xed inputs.

De…nitions 1. For an active …rm, the …xed cost is

a) the accounting cost of the inputs which are physically …xed.
b) the cost of the inputs required for producing an arbitrarily small amount of output.

De…nition 1 does not require (at this stage) that the level of the …xed cost is optimal
(so it does not necessarily correspond to the minimal value of the accounting cost).
In D1b the inputs required for initiating production could be physically …xed but it is
not necessary the case. Since the cost function is related to input demands x by the
accounting relationship c (w; y) = w> x (w; y) for any y 0; we obtain the level of …xed
cost compatible with D1b as

u (w) = lim+ c (w; y) = lim+ w> x (w; y) :

y!0 y!0

This shows that D1b implies that the …xed cost does not change with the production
level, but can change with w. Whereas it is straightforward to de…ne variable inputs as
inputs whose level can be adjusted to minimize their accounting cost and can possibly
be set to zero, the de…nition of …xed inputs is more involved, as they are not necessarily
optimal, nor can they necessarily be set to zero.
We show that the …xed cost u (w) does not necessarily correspond to the cost of the
…xed inputs, but that it also includes a part of the cost of variable inputs when they
are su¢ ciently complementary to the …xed inputs. For instance, if capital is physically
…xed and energy is fully variable, but capital cannot be run without say 1000 KWh of
energy, then the part of the energy input which is necessary to run the …xed capital
input becomes …xed. It is the production technology which determines whether inputs
are variable or …xed and which part of each input is …xed or variable. This remark
has important implications for the speci…cation of …xed and variable cost functions and
these have been largely ignored in the literature.

3. A microeconomic framework for …xed costs

The main result of this section characterizes an extended production function able to
describe …xed inputs in a more general way than the existing literature. A shortcoming

4
of the traditional restricted cost function (see Subsection 3.1), is that it relies on a
partition of all inputs into two disjoint categories: variable and …xed inputs. Actually,
similar inputs can be used for di¤erent types of production activities. Engineers, for
instance, can be allocated to production or to research and development activities. While
engineers’ production increases the current output level, it is not the case when they
are allocated to research and development, which withdraws them from production (like
in Aghion and Howitt, 1992, for instance). Similarly, computers can be used either for
logistics, production management or accounting, activities which do not have the same
impact in terms of production and cost. Before presenting the extended production and
cost function, we shortly overview traditional production analysis.

3.1 On the limitations of traditional production analysis

For modelling …xed inputs, production analysis relies on a partition of the input vector
e)
x into two disjoint categories: those which can be adjusted (variable inputs, denoted x
and those which are …xed or quasi-…xed (x):3
e
x
x= 0: (3)
x
>
The corresponding input prices are denoted by we> ; w> . The output level is given by
y = F (x) where F : RJ ! R denotes the production function which is increasing in x.
The restricted variable cost function is de…ned as:
n o
e x; y) = min w
Vr (w; e> x
e : F (e
x; x) y :
e 0
x

The properties of the restricted cost functions have been investigated by Lau (1976) and
Browning (1983). For empirical implementations see e.g. Caves et al. (1981), Pindyck
and Rotemberg (1983) and Morrison (1988). The total restricted cost function is given
by
e x; y) + w> x;
Vr (w; (4)
where the last term denotes the …xed cost. In the long-run, all the …xed inputs can
be adjusted at their optimal level and this de…nes the long-run or unrestricted cost
function:
n o
>
e x; y) + w x = e
c (w; y) = min Vr (w; c (w; y) + c (w; y) ; (5)
x 0

3
Here, the notation x > 0 means that all J components xj > 0: In contrast x 0 means that xj 0 for all j:

5
where e e x (w; y) ; y) represents the long run variable cost, and c (w; y) =
c (w; y) = Vr (w;
w> x (w; y) is the long run …xed cost. Function x denotes the optimal level of …xed
inputs, which, without further restrictions on Vr ; depends on the production level. As a
consequence, this approach violates (in the long run) both de…nitions given in D1. More
than that, in the long run it is not possible to identify e
c separately from c, unless we
make strong a priori assumptions on which inputs are …xed in the short run. A further
drawback of technology F appears when we impose that Vr be a variable cost function,
e x; 0) = 0: This restriction implies that there are no …xed cost in the long
namely Vr (w;
run: x (w; 0) = 0 (unless we impose a positive lower bound to x).
So, according to traditional production analysis, the only justi…cation for …xed cost
is that physically …xed inputs cannot be optimally adjusted (either for technical reasons
or for lack of rationality). This view excludes a variety of interesting situations in which
…xed and variable inputs are imperfect substitutes and play di¤erent roles in production.

3.2 Another view of the traditional production function

Instead of partitioning x into two disjoint types of inputs, let us assume that each input
comprises a part which can be adjusted and a part which is …xed (in a sense that is
clari…ed in De…nition 2 below):
x = xv + xf ; (6)
with x; xv ; xf 2 RJ+ : This generalizes (3) which is obtained as a special case when xv =
> >
e> ; 0>
x and xf = 0> ; x> . This subsection shows that the variable and …xed cost
functions used in production analysis is generated from an additive production function

y = F xv + xf ; (7)

which requires perfect substitutability between xv and xf :

As our purpose is to describe the production possibilities for a production level close
to zero (in order to be consistent with D1b), we de…ne the input requirement set as
follow.

De…nition 2. In terms of the traditional production function, the …xed cost is the
cost associated to inputs belonging to the input requirement set XF de…ned as

XF lim fz 0 : F (z) = "g :

"!0+

6
 De…nition 2 requires that the limiting isoquant XF exists. De…nition 2 is useful to
characterize the …xed cost in terms of the production function F : it is easy to show
that a …xed cost occurs if the set XF does not include the point x = 0:4 In order to be
compatible with De…nition 1b, we consider in De…nition 2 the isoquant corresponding
to the production level " > 0; instead of " = 0; because with most production functions
compatible with a …xed cost, the condition F (x) = 0 characterizes a thick isoquant, in
the sense that, if it is possible to produce nothing with something (9x > 0 : F (x) = 0),
then it is also possible to produce nothing with even less (there exist x0 < x such that
F (x0 ) = 0). So, only the upper frontier of the set fz 0 : F (z) = 0g is interesting for
identifying a …xed cost. Let us investigate the implications of this additive structure in
terms of the restricted variable and total cost functions:
n o
vr w; xf ; y = min w> xv : F xv + xf y
xv 0
cr w; xf ; y = vr w; xf ; y + w> xf :

The restriction xv 0 is important here, because it can be optimal to use no variable

inputs at all for some levels of xf .

Proposition 1. Let xf 2 XF 6= ? and xv 0: Then cr w; xf ; 0 = w> xf 0;

vr w; xf ; 0 = 0: The restricted cost function cr and the cost minimizing variable inputs
xv satisfy either
(i) xv > 0 and for y > 0,
cr w; xf ; y = C (w; y) > w> xf (8)
and xv w; xf ; y = Xv (w; y) > 0 or
(ii) xv;j = 0 for some j; and
e x; y) + w> x;
cr w; xf ; y = Vr (w; (9)
>
where x is a subvector of xf and we corresponds to the price subvector of w = we> ; w>
corresponding to xv;j > 0.

The proof of this result is given in the Appendix. Proposition 1 states that the
variable cost is zero when production vanishes. This result is driven by the additive
structure of F which ensures that if there exists a point x such that F (x) = 0; then xv
4
For the purpose of exposition we assume that the minimum value of y included in the range of F is zero, but we
could easily generalize to any other value.

7
can be set to zero in the additive decomposition x = xv + xf : In the case of Proposition
1(i), the production function F yields a cost function which is independent of the level
of …xed input, and which is compatible with both De…nitions 1a and 1b. A frustrating
consequence of Proposition 1(i) is that …xed inputs can be seen as if they were set at
their optimal level, as:
@cr @vr
w; xf ; y = 0 , w; xf ; y = w:
@xf @xf
It is the perfect substitutability between the variable and the …xed inputs which is
driving this result. Any mistake in adjusting xf can be perfectly compensated by setting
xv optimally. In summary, technology F is not really suitable for modelling …xed inputs,
as it lacks generality. Proposition 1(ii) gives the general formulation of the cost function
corresponding to F when corner solutions for the variable inputs are allowed. The
structure of the cost function (9) is the same as in (4) and is common in traditional
production analysis (see Chambers, 1988, for instance). So we conclude this section
by noting that production function F with an additive structure between xv and xf
is behind the traditional theory of …xed and variable costs. This additive structure is
restrictive and hides important features of production theory.5

x2
F(xv + x0f) = F(xv + x1f) = y1

x 0f B

x1f
F(0+ x0f) = F(0 + x1f) = 0
0 x1

Figure 1: Isoquants for F xv + xf

Figure 1 illustrates Proposition 1. Endowed with a …xed input vector x0f ; the variable
inputs available to the …rm and satisfying xv 0 are located in the north-east quad of
5
One restriction is that
@ 2 cr
(w; xf ; y) = 0:
ej @wk
@w
For given xf ; y; there is no substitutability between inputs j and k. This is too restrictive because, even for given xf ;
inputs j and k can be substituted for each other because they have a …xed and a variable component.

8
x0f : At given input prices w; the …rm minimizes its variable cost of producing y 1 at the
interior point A. At this point, according to Proposition 1i, the cost function is given by
C (w; y) : With another level of …xed inputs, however, the available set of variable inputs
will be di¤erent. With x1f ; the minimal variable cost for producing y 1 is achieved at B;
n o
on the boundary of the set xv 0 : F xv + x1f y 1 : At point B we have xv1 = 0 and
the optimal level of x2v is restricted by the level of x1f :

3.3 An extended production function

Whereas from the accounting viewpoint both types of inputs xv and xf are similar (the
cost of a unit of the j th …xed and ‡exible input is wj ), technologically they should not
be restricted to play similar roles as it is the case with F xv + xf . We now de…ne
an extended production function G as y = G xv ; xf where G : RJ+ RJ+ ! R+ . For
simplicity, we assume that G is single valued, continuously di¤erentiable, increasing in
its arguments and that G 0; xf = 0. In this context, the restricted variable cost function
now becomes:
n o
vr w; xf ; y = min w> xv : G xv ; xf y : (10)
xv
Now, a given input, say capital, can appears twice in (10): once in vector xv and once in
xf ; their marginal productivities can be di¤erent. This overlapping structure is similar
to the one considered by Blundell and Robin (2000) in consumer analysis. In contrast
to their approach, we do not impose that xv is separable from xf (a structure which
they call latent separability).
Leontief’s (1947) aggregation theorem highlights the restrictions which are implicit
in production function F . The number 2J of inputs xv and xf which appear in G can
be reduced to the J aggregate inputs xv + xf i¤ we have
@G @G
= ; 8i = 1; : : : ; J:
@xvi @xfi
We do not assume in the sequel that these restrictions necessarily apply to G:
One di¢ culty with (10), is that if vr is de…ned for any arbitrary levels of xf ; we can
switch the notation from xf to xv and rewrite vr (w; xv ; y) : So, in order to be able to
identify xf as the …xed inputs, we need to put more structure on vr , and we do this by
introducing restrictions derived from the de…nition of the …xed cost and inputs.

De…nition 3. In terms of the extended production function G, the …xed cost is the

9
cost associated to inputs belonging to the …xed input requirement set XG de…ned as

XG lim fz 0 : G (0; z) = "g : (11)

"!0+

D3 de…nes the set of all …xed input combinations required for starting production.
This de…nition is more general than D2, because it does not assume that …xed and
variable inputs are perfectly substitutable. As for XF ; we impose that xv = 0 belongs
to the …xed input requirement set XG , but get rid of additivity. The next result is a
straightforward extension to technology G of those available for technology F:6

Proposition 2. If xf 2 XG then,
(i) vr w; xf ; 0 = 0; vr w; xf ; y > 0 for any y > 0
(ii) vr is increasing in y
(iii) vr is decreasing in xf .

Proposition 2 means that the restricted variable cost function vr satis…es the proper-
ties of a variable cost function: it vanishes for arbitrarily small production levels. As a
consequence, the restricted …xed cost is given by

ur w; xf lim cr w; xf ; y = w> xf ;
y!0+

and total restricted cost satis…es

cr w; xf ; y = ur w; xf + vr w; xf ; y : (12)

Both production technologies F and G are represented on Figure 2 in the case where
a single input is decomposed into a …xed and a variable component. Figure 2a illustrates
how the introduction of a …xed input xf satisfying F x0f = 0 and the reparameterization
x xv + xf yield the technology F xv + x0f . On Figure 2b, the isoquant corresponding
to the startup production level G xv ; xf = " is not a straight line, which opens the
possibility to choose a …xed input di¤erent from x0f as an admissible value for starting
production. Input x1f for instance, allows to start production with production function
G xv ; x1f 6= G xv ; x0f ; provided that xv is su¢ ciently high for compensating the decline
from x0f to x1f .

6
We only give the properties which are the more interesting for our purpose, see Lau (1976) and Browning (1983) for
other properties.

10
 y y
G(xv ,xf0)
F(xv+xf0)
xf
xf G(xv ,xf1)
xf0
xf0 F(x)
xf1
G(xv ,xf) = 0

0 0
xf0 xv, x xv

(a) Technology F (b) Technology G

Figure 2: Fixed and variable inputs and production possibilities

We illustrate the usefulness of technology G with an example which also illustrates
the claims of Proposition 2.

Example 1. The technology G : R2+ ! R+ is given by

y = G xv ; xf = xv + xf xf

for y 0: Here xf 2 XG , xf = ( = )1=( +1)

: This yields the restricted variable cost
function
y
vr w; xf ; y = wxv w; xf ; y = w (y + ) xf w xf = w ;
xf
which satis…es vr w; xf ; 0 = 0 for xf = ( = )1=( +1)
: The restricted …xed cost function
is ur w; xf = wxf = w ( = )1=( +1)
: For = 0 and = 1 we obtain the traditional
production function as a special case. Example 1 also illustrates that in both cases of
exogenous (physically …xed) and endogenous input xf ; there is no con‡ict between D1a
and D1b.

The structure of the isoquants of F and G is represented in Figure 3 for J = 2,

in the (x1 ; x2 )-plane (with x1 = xv1 + xf 1 ). In Figure 3a the slopes of the isoquants
corresponding to F only depend upon total input use x = xv + xf and not upon the
share of the …xed inputs xf in the composite input x: At point A for instance, it is
possible to produce y 0 using …xed input x0f or x1f : Only the total input quantity matters
and since x0f + x0v = x1f + x1v at point A; the choice of the …xed input is irrelevant. Note
that, contrary to Fig. 2, the isoquants do not necessarily cross the axes on Fig. 3,
because axes now report total input levels for two di¤erent inputs, and not just how a

11
given input is split into variable and …xed amounts.

x2 x2
F(xv + x0f) = y0 G(xv , x0f) = y0
C
A A

x 0f x 0f G(xv , x1f) = y0
F(xv + x1f) = y1 B

x 1f F(0 + xf) = 0 x 1f
G(0 , xf) = 0
0 x1 0 x1

(a) Isoquants for F (b) Isoquants for G

Figure 3: Fixed and variable inputs and substitution possibilities

Figure 3b represents in the (x1 ; x2 )-plane the isoquants for technology G and two dif-
ferent …xed input vectors x0f and x1f : With technology G; the choice of the level of …xed
inputs determines the substitution possibilities between the variable inputs. Although
we have not introduced any distinction between ex-ante and ex-post technologies in our
model, Figure 3 resembles those typically obtained with putty-putty (or putty-clay or
clay-clay) technologies (see e.g. Fuss, 1977). The similarity is due to the fact that we
split x into two (…xed and variable) non-additive components. With technology G the
choice of a particular …xed input level xf coincides with a choice of a particular produc-
tion technology and a speci…c substitution pattern between variable inputs. On Figure
3b, the isoquant corresponding to x0f characterizes inputs which can easily be substi-
tuted the one for the other, whereas for x1f substitution becomes more di¢ cult. Note
that for a given output level, the isoquants for G corresponding to the …xed input level
x0f can cross those obtained for x1f : For instance, at point A the production level y 0 can
be produced using two types of technologies, each one exhibiting a speci…c substitution
pattern.
Figure 3b also illustrates that if …xed inputs are neglected, production function G
is not necessarily quasi-concave in x (at point A). Moreover, optimal choices for input
bundles can be located in the zone violating quasi-concavity in x and so the cost function

12
will not necessarily be concave in w: In the context of …xed cost, imposing simultaneously
concavity in w and xf = 0 on the cost function may end up with worse estimates than
extending the cost function to be compatible with the occurrence of …xed cost (see Lau,
1978, and Diewert and Wales, 1987, for seminal contributions on concavity enforcement).
The next di¢ culty we have to deal with is related to the fact that the level of …xed
inputs can be either exogenous or endogenous. Figure 3b depicts at point C a situation at
which the variable inputs are optimal given the levels of …xed inputs x0f and production
level y 0 ; however, if xf could be chosen, the …rm would set them to x1f and produce y 0
at point B: It is important to note that isoquant and isocost line are not necessarily
tangent at the optimum level x1f for xv = 0:
Whereas variable inputs can by de…nition be adjusted for minimizing costs, the …xed
inputs are not necessarily set at their optimal level. We say that a …xed input xf j is
exogenous when its actual level is not optimal in the sense that the equality between its
shadow value and market price is violated:
@vr
w; xf ; y 6= wj ; (13)
@xf j
for the observed values of w; xf ; y and xf j > 0: The extended framework based on
G xv ; xf is useful as it allows to split the input x into a part xv that is e¢ ciently
allocated, and a part xf which is not necessarily so.7
In the long run, …xed inputs can be determined endogenously by the …rm, and they
may in some case be set to zero. Such a corner solution occurs at xf = 0 if 0 2 XG and:
n o
0 vr (w; 0; y) min w> xv : y G (xv ; 0) < vr w; xf ; y + w> xf ;
xv 0

for any xf > 0: Equivalently, the choice xf = 0 is (locally) optimal if at point (w; 0; y)
the increase in …xed cost is not compensated by a greater reduction of the variable cost:
@vr
wj + (w; 0; y) > 0:
@xf j
Then it is optimal to adopt a production structure without any …xed input. In many
cases however, an inner solution for xf exists. It is characterized by the equality between
the shadow value of the …xed input and its market price:
@vr
w; xf ; y = wj : (14)
@xf j
7
Common explanations for why the level of the …xed inputs is not optimal are related to (i) technological constraints,
(ii) indivisibilities of the …xed inputs, (iii) allocative ine¢ ciencies and (iv) intertemporal dependences.

13
 Example 1 (continuation). For vr w; xf ; y = w (y + ) xf w xf ; we …nd that
(assuming > 0 and < 1)
1
1+
xf (w; y) = (y + )
1
which varies with the level of output. In the traditional case: = 0 and = 1; the re-
stricted variable cost function becomes vr w; xf ; y = wy and we obtain a corner solution
xf = 0; conformably to Section 3.1. The long-run variable cost function becomes:
1
1+ 1+
vr w; xf ; y = w (y + ) (y + ) w (y + )
1 1
and this does not necessarily vanish anymore for a production level going to zero:
1
1+ 1+
vr w; xf (w; 0) ; 0 = w w :
1 1

Example 1 illustrates the fundamental identi…cation problem occurring when inputs

are optimally adjusted: the …xed cost generally di¤ers from the cost of the …xed inputs.
Indeed, after normalizing the variable and …xed cost function according to (2), we obtain
the …xed cost
u (w) = w> xv w; xf (w; 0) ; 0 + w> xf (w; 0) :
When …xed and variable inputs can be imperfectly substituted for each other, the op-
timal amount of …xed input depends upon w and xf (w; 0) is not necessarily included
in the input requirement set XG : This means that the level of …xed input cannot be
determined ex-ante using only the de…nition of XG . When xf can be adjusted, it is no
longer possible to separately identify xf and xv . Fortunately, de…nition D1b of the …xed
cost is fully compatible with this situation, but D1a is violated: u (w) 6= w> xf (w; 0).
Brie‡y, an input cannot be said to be …xed or variable prima facie, using only physical
properties of the inputs. It is the technology which in last instance determines whether
a given input is …xed or variable. This explains why D1b which relies on the technology
provide the more general de…nition of the …xed cost. Few technologies allow to obtain
an optimal level of xf independent of y . We characterize them below.

Proposition 3. Assume that the technology G is increasing and quasi-concave in xv ;

and that xv > 0 at the optimum. Let K : RJ+ ! RJ+ and F : RJ+ ! R+ both be increasing

14
functions.
(i) The restricted cost function is given by

cr w; xf ; y = ur w; xf + v (w; y) ; (15)

with v (w; 0) = 0 if and only if the production function is given by

G xv ; xf = F xv + K xf : (16)

(ii) The optimal level of xf is independent of y if and only if the restricted cost
function is (15) or the production function is (16).

Proposition 3 characterizes the cost and production functions which generate a …xed
cost. Requirement (16) is less stringent than separability of G in xf because it does not
impose that K xf be a unique aggregate …xed input. Here, the vector valued function
K comprises J aggregates for the …xed inputs. Proposition 3 also aggregates additively
some …xed and variable inputs together since F depends upon xv +K xf : As can be seen
by comparing (16) and F xv + xf , the former is also more general than the traditional
production function F for which …xed and variable inputs are perfect substitutes. Figure
4 provides an illustration in the two inputs case (J = 1). It shows that xf does not vary
with y; contrary to xv .8 Figure 4 gives the decomposition of variable input xv into a
fully variable component xv (w; y) xv (w; 0) which can be set to zero when there is no
production, and xv (w; 0) which has to be used for starting production. It also shows
how technology F xv + K xf di¤ers from F xv + xf : With F xv + K xf there is
perfect substitutability between the components of xv and K xf , but not between xv
and xf : For a given input xi ; the slope (@F=@xvi ) = @F=@xf i of the isoquant (Figure
4) is not restricted to be equal to 1 out of the optimum. Moreover, for two di¤erent
inputs, xh and xi ; the slope @F=@xf h = @F=@xf i of the isoquant is not restricted to
be equal to (@F=@xvh ) = (@F=@xvi ) out of the optimum. Fixed inputs can be substituted
according to a di¤erent pattern than variable inputs.

8
We also see why the corner solution xv = 0 has to be excluded, because at this point the level of xf can vary with y.

15
 xf
F(xv +K(xf)) = 0

F(xv +K(xf)) = y

0 xv

Figure 4: The …xed and variable inputs

decomposition

We conclude this section by emphasizing that, even though a separate identi…cation

of xf and xv is not possible without additional restrictions, it is possible to identify
uniquely xf (w; 0) + xv (w; 0) as well as the level of the …xed cost. If we assume that
decomposition (1) is not unique, then there exist ue 6= u and ve 6= v such that:

e (w) + ve (w; y) ;
c (w; y) = u

with ve (w; 0) = v (w; 0) = 0: However, the equality

e (w) + ve (w; y)
u (w) + v (w; y) = u

is satis…ed for any (w; y) i¤ u (w) = ue (w) (obtained for y = 0) and v (w; y) = ve (w; y), and
this proves unicity. It is also interesting to note, that although …xed cost cannot be
observed, because the situation in which …rms produce an output level close to zero is
hypothetical, the level of …xed cost is well identi…ed empirically and can be estimated.

4. Some consequences of neglecting …xed costs

This section discusses three drawbacks arising when …xed inputs are neglected. A …rst
problem of disregarding xf is the oversimpli…cation of various economic relationships,
in particular the relationship between …xed inputs and pricing behavior. Let p = P (y; z)
denote the inverse output demand which depends on exogenous macroeconomic parame-
ters z and the …rm’s own production level. With market power, the …rms’the optimum

16
is characterized by:
@vr @P y
w; xf ; y = p 1 + : (17)
@y @y P
This equation and the discussion above shows that a …xed input xf has an impact on the
marginal cost function unless cr has the speci…c structure given in (15). It also implies
that there is a relationship between the …xed input and the markup @ ln P=@ ln y; via
the marginal cost.
Neglecting the …xed cost is a source of bias. By Shephard’s lemma, we have
@u @v
x (w; y) = (w) + (w; y) :
@w @w
If the …xed cost is neglected, then it enters the residual term which will be correlated
with w; which may bias the estimates.
From a theoretical viewpoint, neglecting the …xed cost by setting u (or ur ) equal
to zero may lead to underestimation of returns to scale. In order to show this point,
we consider the long-run case and assume that the cost function is convex in y . By
convexity we have,
@c
c (w; 0) c (w; y) + (w; y) (0 y)
@y
@c y c (w; 0)
) (w; y) 1 :
@y c (w; y) c (w; y)
As the return to scale is the inverse of the cost elasticity with respect to the output,
imposing zero …xed cost implies imposing decreasing returns to scale. The equation
above also shows that for given level of costs and outputs, neglecting the …xed cost
leads to an overestimation of the marginal cost, which will also cause an underestima-
tion of the markup. As @c=@y (w; y) = w> @x =@y (w; y) ; overestimating the marginal cost
often coincides with the overestimation of the input demand sensitivity to output vari-
ations. In addition, from an empirical viewpoint, setting the …xed cost equal to zero
introduces an omitted-variable bias in the estimation of technology parameters. In the
following sections, we discuss the empirical issues raised by the estimation of the …xed
cost, including suitable functional forms for cost functions, and the treatment of cost
heterogeneity with unobserved levels of xf .

17
5. On ‡exible functional forms
In the 1970’s and 1980’s, several researchers proposed new parametric speci…cations for
the production technology, and introduced so-called ‡exible functional forms, which are
able to approximate locally an arbitrary cost function. These functional forms, still
widely used in production analysis, are not adequate for modelling …xed costs: either
they completely exclude …xed costs, or specify them in an in‡exible way. The variable
t is now introduced for denoting technical change.
In their seminal paper, Diewert and Wales (1987) have introduced several cost func-
tions, many of which can be written as

C DW (w; y; t) = a>
ww +
>
ww at t + V DW (w; y; t) ; (18)

with V DW (w; 0; t) = 0: This identi…es the …xed cost as U DW (w; t) = a>

ww +
>w
w at t;
where aw ; w ; at denote technological parameters. So, the …xed cost function is linear
in w and t and is not a ‡exible speci…cation (in the sense of Diewert and Wales, 1987).
The same can be shown for the variable cost speci…cation V DW .
Let us now consider the Translog functional form (Christensen et al., 1971) with
technology parameters given by :

C T L (w; y; t) = exp( 0 + >w ln w + y ln y + t t (19)

1
+ ln w> Bww ln w + ln w> Bwy ln y + ln w> Bwt t
2
1 1
+ yy (ln y)2 + yt t ln y + t2 );
2 2 tt
where the notation is as in Koebel et al. (2003). One of the main drawbacks of the
Translog functional form is that it is not suitable for modelling …xed cost.

Proposition 4. The Translog functional form implies a …xed cost that is either zero
or in…nite (in which case C T L is decreasing in y for some values of y ).

This result shows that the Translog cost function is badly behaved in some regions,
and especially when production is close to zero, which de…nes the …xed cost of produc-
tion. This proposition illustrates that the Translog is only able to approximate locally
an unknown cost function, but not globally, and justi…es the speci…cation of alternative
functional forms for the purpose of estimating a …xed cost. Proposition 4 points out
a paradox: although the Translog speci…cation is ‡exible (Diewert and Wales, 1987,

18
Theorem 1), it excludes …xed costs. The reason for this apparent contradiction is to
be found in the limitations of the ‡exibility requirement, which just requires that the
cost function be a local approximation, in some neighborhood of y , but not necessarily
at the neighborhood of y = 0 which de…nes the …xed cost. In the sequel we rely on a
functional form which is ‡exible at two points.

De…nition 4. A two-points Flexible Functional Form (2FFF) for a cost function

provides a second order approximation to an arbitrary twice continuously di¤erentiable
cost function C at point where y > 0 and at y = 0+ :

We have seen that a production technology with …xed cost, can be represented by two
di¤erent production technologies: one for initiating production H xf G 0; xf (using
only …xed inputs), and one for reaching the output level y; and given by G xv ; xf : So
it becomes quite natural to specify both technologies in a ‡exible way. Similarly, the
cost function is additively separable in two parts: one part u corresponding to the cost
at zero output level and one part, v; re‡ecting the production cost of the output. So if
our objective is to provide an approximation of the production technology, both parts
should be treated with equal importance, and we suggest here to use a ‡exible functional
form for both the …xed and variable cost functions. De…nition 4 implies that a 2FFF
cost function is the sum of two 1FFF …xed and variable cost functions U and V:
Diewert and Wales (1987, p.45-46) de…ne a one point (1FFF) ‡exible cost function
at the point w0 ; y 0 ; t0 as one being able to approximate an arbitrary cost function C 0
locally, where C 0 is continuous and homogeneous of degree one in w: This de…nition is
satis…ed if and only if C has “enough free parameters so that the following 1 + (J + 2) +
(J + 2)2 equations can be satis…ed”:

C w0 ; y 0 ; t0 = C 0 w0 ; y 0 ; t0 (20)
rC w0 ; y 0 ; t0 = rC 0 w0 ; y 0 ; t0

r2 C w0 ; y 0 ; t0 = r2 C 0 w0 ; y 0 ; t0 ;

where the rC (respectively r2 C ) denotes the …rst (second) order partial derivatives
with respect to all arguments of C . Since the Hessian is symmetric and C is linearly
homogeneous in w; this system includes only J (J + 1) =2 + 2J + 3 free equations. The
requirements (20) have to be ful…lled at a single point y 0 which can be chosen to be

19
positive, so the 1FFF de…nition is compatible with the absence of …xed cost. This
explains why the Translog is ‡exible although U 0: This drawback of 1FFF explains
why we consider 2FFF.
A 2FFF for a cost function has enough free parameters for satisfying the following
1 + (J + 1) + (J + 1)2 + 1 + (J + 2) + (J + 2)2 equations:

U w0 ; t0 = U 0 w0 ; t0 ; (21)
rU w0 ; t0 = rU 0 w0 ; t0 ;

r2 U w0 ; t0 = r2 U 0 w0 ; t0 ;

and for y 0 > 0;

V w0 ; y 0 ; t0 = V 0 w0 ; y 0 ; t0 ; (22)
rV w0 ; y 0 ; t0 = rV 0 w0 ; y 0 ; t0 ;

r2 V w0 ; y 0 ; t0 = r2 V 0 w0 ; y 0 ; t0 :

Since U is linearly homogeneous in w; and its Hessian is symmetric, this imposes the
following additional restrictions 2 + J + (J + 1) J=2 on U :
@U @ 2U @U
w> (w; t) = U (w; t) ; w> (w; t) = (w; t) ;
@w @w@t @t
@ 2U
w> >
(w; t) = 0; r2 U (w; t) = r2 U (w; t)>
@w@w
It turns out the …xed cost function U has at least (J + 1) + J (J + 1) =2 free parameters in
order to be ‡exible. Similarly, the variable cost function V must have at least (J + 2) +
(J + 1) (J + 2) =2 free parameters. In total, a 2FFF cost function must have at least
1 + 3 (J + 1) + J (J + 1) free parameters. Moreover, in order to identify V as a variable
cost function, we impose
V w0 ; 0; t0 = 0:
Note that (21) and (22) imply (20), but not conversely.

6. Econometric treatment of cost heterogeneity

In our most general model, the level of …xed input is not necessarily optimal and has
an impact on both the …xed and variable cost:

cr w; xf ; y; t = ur w; xf ; t + vr w; xf ; y; t ;

20
 which is somewhat embarrassing as we do not observe the level of xf ; but only total
input quantity x: However, our objective is not to estimate …rm speci…c functions vr
and ur but rather their conditional mean given the value of the observed explanatory
variables w; y and t; so we consider:

V (w; y; t) E vr w; xf ; y; t jw; y; t ;

U (w; t) E ur w; xf ; t jw; t :

Here integration is over unobserved heterogeneity with respect to the joint distribution
of xf and the individual cost functions vr and ur : Using these de…nitions, we rewrite the
model as follow:
U V
cr w; xf ; y; t = w; xf ; t U (w; t) + w; xf ; y; t V (w; y; t) ; (23)

where the functions U and V are de…ned by:

U ur w; xf ; t V vr w; xf ; y; t
w; xf ; t ; w; xf ; y; t ;
U (w; t) V (w; y; t)
and satisfy E U jw; t =E V jw; y; t = 1: Note that the covariance between U and V

can a priori take any value. However, we derive an important statistical relationship
between the …xed and variable cost functions UU and V V:

Proposition 5. Under the assumptions that, (a) individual heterogeneity in the …xed
and variable cost functions is independent of xf ; (b) the …xed inputs xf are positive and
are optimally allocated; then:
(i) the conditional covariance cov U ; V jw; y; t is nonpositive;
(ii) the conditional variance matrix V [ jw; y; t] is singular.

When the …xed inputs are unobserved we will not be able to estimate functions ur and
vr ; and we cannot test whether @cr =@xf = 0 is satis…ed or not. However, we will be able
to estimate V [ jw; y; t] and cov U ; V jw; y; t : If the statistical test leads to rejection of
the singularity of V [ jw; y; t] or cov U ; V jw; y; t 0; then we can deduce that either
the …xed inputs are not optimally allocated (Proposition 5), or that the production
technology has the speci…c structure given in (16). The level of the …xed cost UU and
the level of the variable cost VV are certainly positively correlated with any dataset:
both the …xed and the variable cost increase over time, and …rms with a high …xed cost
certainly produce more than smaller …rms and also have a higher variable cost. Hence

21
the positive correlation between UU and V V: Proposition 5, however, states that there
is a tradeo¤ –a negative correlation –between the …xed and the variable cost for given
values of the explanatory variables (w; y; t) : Such a tradeo¤ cannot be directly observed
in a dataset, because it pertains to unobserved heterogeneity.With panel data, the issue
of interrelated heterogeneity is often discarded, one exception is Gladden and Taber
(2009) who considered it in estimating linear wage equations. In contrast to Gladden
and Taber (2009), we derive the sign of the covariance from a structural nonlinear model.
Let us now explain our strategy for estimating this covariance along with other sta-
tistics of interest. We have to explicitly introduce the parameters in the notations of
the cost function and rewrite the observed cost level cnt as follow:
U V
cnt = nt U (wnt ; t; ) + nt V (wnt ; ynt ; t; ) + ent ; (24)

where n = 1; : : : ; N denotes the sector, t = 1; : : : ; T represents time. The random term

ent is iid, satis…es E [ent jwnt ; ynt ; t] = 0 and has constant variance 2. It is also assumed
c

that ent is uncorrelated with U; V > and any right hand side regressors. Equiv-
nt nt nt

alently, we can write our empirical model as:

cnt = U (wnt ; t; ) + V (wnt ; ynt ; t; ) + "cnt : (25)

with the composite error term:

"cnt U
nt 1 U (wnt ; t; ) + V
nt 1 V (wnt ; ynt ; t; ) + ent : (26)

Note that E ["cnt jwnt ; ynt ; t] = 0: We also assume that

2
U UV
V[ nt jw; y; t] = 2 ; (27)
UV V
and V > = 0; for any n 6= m and t 6= s. This model is an extension of
nt ms jw; y; t

Swamy’s (1970) random coe¢ cient model to our nonlinear setup with individual and
time varying random coe¢ cients. The values of nt can be considered as incidental
parameters, because they are not fundamentally interesting (and cannot be identi…ed).
Their distribution however is informative. The joint distribution of nt re‡ects the way
the variable and …xed cost vary together. The covariance between U and V allows to
nt nt

discriminate between the case of optimally and nonoptimally allocated …xed input and
whether …xed cost has an impact on the marginal cost of production and the markup
>
>; >
via (17). The parameters of interest are the technology parameters and

22
the variance matrix :
In principle, all estimates of the technology parameters and the covariance matrix
can be obtained simultaneously by solving (numerically) the likelihood maximization
or the nonlinear least squares problem.9 However, these objective functions are highly
nonlinear in , and it turns out that nonlinear numerical algorithms often do not converge
to a solution. We avoid these numerical problems, and use a two-stage estimation
procedure. First, the technological parameters are consistently estimated (without
identi…cation of and 2) by minimizing the sum of squared residuals:
c
X
b = arg min [cnt U (wnt ; t; ) V (wnt ; ynt ; t; )]2 :
;
n;t
As the random term "cnt exhibits heteroscedasticity and serial correlation, we rely on the
h i
Newey-West (1987) estimator for estimating the variance matrix V b .
In the second-stage, two equivalent estimation methods are again available: Maxi-
mum Likelihood (ML) and Least Squares (LS). The conditional variance of ^"cnt can be
expressed as (using (26)):
h i
E (^"cnt )2 jwnt ; ynt ; t 4nt ( 2
c; ; b) (28)
= 2
c + 2 2
U U (wnt ; t; b ) +
2 2
VV wnt ; ynt ; t; b + 2 UV U (wnt ; t; b ) V wnt ; ynt ; t; b :

It turns out that the parameters 2; 2; 2 and of (28) can be estimated by an

c U V UV

OLS regression of the squared NLS residuals

h i2
2
"cnt ) = cnt
(b U (wnt ; t; b ) V wnt ; ynt ; t; b (29)

on a constant, Ub 2 , Vb 2 and Ub Vb . If we assume that the heterogeneity vector nt and the

error term ent follow some parametric distribution, then the estimated covariance matrix
can be obtained by maximizing the likelihood function. Both second-stage estimation
methods are asymptotically equivalent, but their estimation outcomes may di¤er: …rst,
because the ML is more e¢ cient than OLS if the distribution of the random terms is

9 2
The NLS estimator of ; c; could be obtained (in one step) by minimizing the following sum of squared residuals:
X c2 2 2 2 2 2
"nt ( ) UU (wnt ; t; ) V V 2 (wnt ; ynt ; t; ) 2 UV U (wnt ; t; ) V (wnt ; ynt ; t; ) :
n;t

An alternative estimator of parameters , 2c and is the maximum likelihood estimator. Under the normality assumption
of the random term "cnt N 0; 4nt ; 2c ; ; we can write
1 X n 1
o
logL( ; 2c ; ) = log (2 ) + log 4nt ; 2c ; + 4nt ; 2c ; ("cnt ( ))2 :
2 n;t

23
well speci…ed; second, because the covariance matrix is not restricted to be positive-
de…nite in the OLS regression, but this restriction is imposed in most ML estimation
algorithms.10 As this matrix may well be singular (Proposition 5), we prefer the OLS
approach.
Our estimation approach can be viewed as a sequential two-stage M-estimation, where
in the …rst-stage b is obtained by solving a NLS problem and then, given b, the estimates
b2 ; b are obtained by OLS. This second stage estimator is simple and consistent if
the …rst-stage estimator is consistent for , see Cameron and Trivedi (2005, Section
6.6). However, the asymptotic distribution of ^ given the estimation of b is di¢ cult to
establish. Hence we use the panel bootstrap for deriving the standard deviations of the
second-stage estimator.11

7. The empirical investigation

In this section, we …rst summarize the empirical models and strategies, we then present
brie‡y the data set and discuss the estimation results.

7.1 The empirical models and estimation strategies

For the empirical …xed and variable cost functions U and V; we assume Translog func-
tional forms denoted by U T L and V T L . As seen in Proposition 4, the traditional Translog
cost function C T L satis…es C T L (w; 0; t) = 0 and is not compatible with the occurrence of
a …xed cost (in the best case where yy 0). It is, however, quite simple to generalize
the Translog speci…cation by adding a …xed cost function to the variable Translog cost
function (the two-points ‡exible form):

C T L (w; y; t; ; ) = U T L (w; t; ) + V T L (w; y; t; );

where
1 1
U T L (w; t; ) = expf 0 + >
w ln w + tt + ln w> Aww ln w + ln w> Awt t + tt t
2
g; (30)
2 2

10
It can be shown that the …rst order conditions of ML are identical to the moment conditions of OLS.
11
We assume that the errors are i.i.d. over individuals (but not over time). The panel bootstrap performs a classical
paired bootstrap that resamples only over n and not over t.

24
and
> 1
V T L (w; y; t; ) = expf 0 + w ln w + y ln y + tt + ln w> Bww ln w
2
1 1
+ ln w> Bwy ln y + ln w> Bwt t + yy (ln y)2 + yt t ln y + 2
tt t g:
2 2
We impose linear homogeneity and symmetry in w using the following 2 + J + (J + 1) J=2
parametric restrictions on U T L :
> > >
w = 1; Awt = 0; Aww = 0; Aww = A>
ww : (31)

There are 1 + J + (J + 1) J=2 free parameters left in U T L . Similarly, the variable cost
function V T L has 3 + 2J + (J + 1) J=2 free parameters which satisfy
> > > > >
w = 1; Bwt = Bwy = 0; Bww = 0; Bww = Bww : (32)

Note that the logarithmic transformation of the total cost function is not useful anymore
for linearizing the nonlinear Translog speci…cation (unless U T L 0). For J = 4; the …xed
cost function has 15 free parameters to which are added the 21 free parameters of the
variable cost function.
Given the two-points ‡exible speci…cation, we estimate the parameters and by
using NLS based on (25) in the …rst-stage. The second-stage consists in the estimation
of the variance matrix and 2 by using OLS based on (28) and (29). The clas-
c

sical Translog cost function which includes only the variable cost function V T L (and
assumes that U T L 0) is also considered for comparison. We consider further empiri-
cal models that include the system estimation by adding the input demand equations
(obtained by applying Shephard’s lemma to C T L ), as well as the model estimated in
…rst-di¤erences. Substantial gains in e¢ ciency can be realized by system estimation,
because more observations are available. The …rst-di¤erence estimation model is more
robust against non-stationarity of the series and unobserved individual …xed e¤ects.
Henceforth, Model I denotes the single equation model without any …xed cost. Model II
is the baseline model where the cost function includes both a …xed and a variable part
(the two-points ‡exible form). More e¢ cient frameworks are Model III (in level) and
Model IV (in di¤erence), which include the cost and the input demand functions. We
note that the choice of starting values is crucial for reaching the optimum in the case of

25
system NLS.12

7.2 The data and empirical results

We use the NBER-CES manufacturing industry database for our empirical study.13
This database records annual information on output ynt , output price pnt ; and the input
levels xnt ; together with input prices indices wnt , for 462 U.S. manufacturing industries
(at the six-digit NAICS aggregation level) and covers the period 1958 to 2005. See Chen
(2012) for descriptive statistics and details on the computations made for generating the
depreciation rate, interest rate, and the user cost of capital. Information is available for
four inputs: capital, labor, energy and intermediate materials.
We begin by commenting the …rst-stage estimation results for models I to IV. Instead
of reporting estimates for all Translog parameters, we only select some informative
estimated coe¢ cients and statistics. An important coe¢ cient is the parameter yy ;

which is crucial for Proposition 5. Given the estimated Translog coe¢ cients, we compute
statistics such as the share of the …xed cost in the total cost U=C , the ratio of the
output price to the predicted marginal cost of production p= (@C=@y) which measures
the markup, and the rate of returns to scale 1=" (C; y), where " (C; y) @ ln C=@ ln y
denotes the elasticity of costs with respect to output.
As mentioned in Section 5, neglecting the …xed cost is a source of bias. By comparing
the estimation outcomes of Model I and Model II, we note that the results of the two
models di¤er with respect to several key points. First, the parameters of the …xed
cost function ( ) in Model II are signi…cantly di¤erent from zero, which indicates the
existence of …xed costs in the production process. Second, the model without a …xed
cost (Model I) suggests that the industries exhibit decreasing returns to scale, but the
model with a …xed cost (Model II) suggests increasing returns to scale. The bias on
the degree of returns to scale is due to the overestimation of the elasticity of cost and
neglect of the …xed cost (see Section 4). Finally, the overestimation of marginal costs
by Model I leads to underestimation of the markup: the median of p= (@C=@y) in Model

12
For the single equation estimation (Model I and II), the starting values are set arbitrarily to zero. For the system
estimation in levels (Model III), the starting values are the estimates obtained from Model II. For the system estimation
in …rst di¤erences (Model IV), the starting values are obtained from the estimation of the cost function in …rst-di¤erences.
13
The dataset can be downloaded at: http://www.nber.org/data/nbprod2005.html

26
 Table 1. Summary of estimation results
Model I II III IV
1st q - 0.19 0.27 0.48
U=C median - 0.52 0.51 0.76
3rd q - 0.91 0.77 0.94
1st q 1.19 1.32 1.37 1.58
p= (@C=@y) median 1.36 1.87 1.78 2.63
3rd q 2.47 6.60 2.74 5.73
1st q 0.69 1.01 1.01 1.10
1=" (C; y) median 0.89 1.40 1.37 2.07
3rd q 0.98 6.04 2.71 7.07
yy coe¤ -0.05 -0.14 -0.15 -0.32
t-value -2.05 -4.21 -3.11 -3.85
2 coe¤ - 0.54 30.83 1.09
U
t-value - 1.67 1.29 0.15
2 coe¤ - 0.02 0.27 0.12
V
t-value - 1.95 1.96 1.27
UV coe¤ - -0.32 -13.04 -1.19
t-value - -0.91 -1.43 -0.11
2 coe¤ 3.7e+6 1.9e+6 1.3e+6 2.1e+6
c
t-value - 2.09 0.24 0.66
Notes: Rows 2 to 11 report the estimated parameter values and the
corresponding t-statistic for the hypothesis that the parameter is equal
to zero. Rows 12 to 20 report the median value of the corresponding
statistic over all observations as well as the 1st and 3rd quartiles.

I is about 36% lower than the one predicted by Model II.14

Table 1 also shows that empirical results obtained from models II to IV exhibit some
regularities. First, the estimated coe¢ cient of yy is signi…cantly negative in all cases,
which implies that the limit of the classical Translog variable cost function is zero as y
approaches 0: Second, all models predict that the …xed cost represents a considerable
share of total cost. The median of estimated shares U=C varies in the range between 51%
and 76%. Third, the estimation results also suggest that the industries exhibit increasing
1
returns to scale. The median of the rate of returns to scale, " (C; y) ranges between
1:4 and 2:1. Fourth, there is a signi…cant di¤erence between the selling price and the
predicted marginal cost of production, the median of estimated markup varies from 1:8
to 2:6. However, we note that the results of Model IV (with data in …rst-di¤erences)
di¤er quantitatively from those of Model II and Model III (with data expressed in levels).
14
We also reestimate Model I after appending a linear …xed cost term w> e in the speci…cation. The corresponding
empirical results are not reported, but lie in between those obtained for Model I and II.

27
 Now, we focus on the …xed cost share (U=C ), in particular on its evolution over time.
These series (averaged over all industries) are depicted on Figure 5. We note that for all
the empirical models, the …xed cost shares are decreasing over time. This may re‡ect
…rms’e¤orts to increase production ‡exibility. The series generated by model II (where
the input demands system is not included in the estimation), exhibit a structural break
around 1980. For other models, the decline of …xed cost shares over time is smoother.
However, the decrease is less signi…cant in the …rst-di¤erenced model (Model IV).
When it comes to the second-stage estimation, the estimates of 2; 2; and 2,
U V UV c

are somewhat more divergent across the models. However, we see that the variance of
the …xed cost heterogeneity U is always larger than the variance of the variable cost
heterogeneity V: The covariance between heterogeneities is found to be negative and
the covariance matrix is close to singular for all models, which is conform to what
we expect from Proposition 6. The second-stage estimation results, however, are not
precisely estimated and are not statistically signi…cant. This result may be due to our
overly restrictive assumption of random heterogeneity in the …xed and variable cost
function speci…cation (23). Economically, this heterogeneity may well be correlated
with further explanatory variables which are individual speci…c (like for instance the
level of production, the type of industry, etc.). So we conduct further analysis in the
next subsection.

28
 Figure 5. Fixed costs shares over time

7.3 Estimation with industry speci…c dummies

Although Models II to IV with random heterogeneity yield some interesting results on

the scope of …xed cost and returns to scale, the interaction between …xed and variable
costs was not precisely estimated. This may due to the fact that heterogeneity is not
purely random but correlated with sectorial characteristics as the level of production
or the level of …xed and variable cost. We pursue the investigation a step further and
introduce individual-speci…c dummies into Model IV. The most ‡exible speci…cation
replaces U and V in regression (24) by 2N individual-speci…c parameters. In order
nt nt

to limit overparameterization, we introduce instead dummies for more broadly de…ned

groups of industries. There are di¤erent ways to de…ne these groups, for instance, in
the spirit of Mundlak’s (1978) correlated random coe¢ cient model, individuals can be
grouped w.r.t. the average value of their covariates. For the industry database, however,
a more natural clustering criterion, is to group the 462 manufacturing sectors available
at the six-digit NAICS level into 20 three-digit NAICS sectors. See Table 2 for a list of

29
the 3-digit industries.15
Formally, we introduce the multiplicative dummy variables U and V for j = 1; :::; 20
j j

in place of the random parameters of (24) which becomes:

U TL V TL
cnt = j U (wnt ; t; )+ j V (wnt ; ynt t; ) + ent : (33)

Since the Translog cost function also includes the terms 0 and 0, all the parameters
cannot be identi…ed separately, unless we consider two additional restrictions. Since
by construction, we have E U jw; t = E V jw; y; t = 1, it is natural to impose the
normalization conditions:
20 20
1 X U 1 X V
j = j = 1;
20 20
j=1 j=1
which allow to identify all parameters. In this case, the estimated parameters U and V
j j

represent the industry-speci…c deviation in percentage from the average. For instance,
if the estimated value of U is signi…cantly above one and the estimated value of V is
j j

signi…cantly below one, this indicates that the industry group j incurs more …xed and
less variable costs than average. In this framework, the interaction between the …xed
and variable components of the cost function is characterized by the variation of U
j

and V over industry groups. We examine the empirical correlation between U and V
j j j

along with group-speci…c shares of …xed cost, degree of returns to scale, markups and
rate of technical change.
We estimate the parameters of the extended Model IV and report the estimation
results in Table 2. Column 3 and 4 of Table 2 report the estimated coe¢ cients of U
j

and V. Our estimation results indicate, for instance, that compared to the average, the
j

industry group NAICS 311 (food) operates with 24% less …xed cost and 4% less variable
cost than the average. We also note that industries with lower than average …xed cost
generally have higher than average variable cost and conversely. Contrary to the above
random e¤ect models, the parameters re‡ecting cost heterogeneity are now statistically
signi…cant.
Columns 5 to 10 report the median (for each group) of the …xed cost share U=C ,
the markup p= (@C=@y), returns to scale 1=" (C; y), and technical change measured as

15
At the three-digit NAICS level, there are actually 21 manufacturing industry groups. We merge the smallest (in terms
of the number of subsectors) NAICS 324 industry group (petroleum and coal products manufacturing) with NAICS 325
industry group (chemical manufacturing).

30
@ ln C=@t; @ ln U=@t and @ ln V =@t. For the NAICS 311 industry group, the estimates
indicate that the …xed cost represents 25% of the total production cost, with almost
constant returns to scale and a markup of 68%. In average over all industries, the
results con…rm former …ndings with strong evidence for …xed cost, increasing returns to
scale, and markup pricing. We also …nd evidence for the conjecture brought forward in
Section 4: industries with higher …xed cost also exhibit higher markups and returns to
scale.
Regarding the rate of technical change, our results on @ ln V =@t show that the variable
cost is on average decreasing by 0.9% over time with little variance over industries. In
contrast, the …xed cost increases with time i.e. @ ln U=@t = 0:04: Altogether, our results
are in line with those obtained by Diewert and Fox (2008) who found modest empirical
evidence for technical change in US manufacturing. Our interpretation is that the
deterministic trend only partially captures technical progress, and that one important
part of technical change is stochastic and embodied in the unobserved …xed inputs (the
xf ). These …xed inputs contribute to increase the …xed cost and decrease the variable
cost and, as a consequence of our approach, this random component of technical change
is captured by the negative correlation between U and V.
j j

31
 Table 2. Summary of estimation results with industry dummies
NAICS industry groups U V U U=C p= (@C=@y) 1="(C; y) @ ln C=@t @ ln U=@t @ ln V =@t
j j j
311 Food 0.76 (7.36) 0.96 (26.40) 0.25 1.68 1.05 0.004 0.039 -0.009
312 Beve.&Toba. 1.22 (6.03) 0.37 (6.22) 0.62 4.42 2.07 0.020 0.037 -0.009
313 Textile 0.58 (5.02) 1.35 (18.15) 0.34 1.57 1.11 0.008 0.041 -0.009
314 Textile Prod. 0.51 (7.17) 1.10 (17.52) 0.41 2.14 1.19 0.015 0.042 -0.009
315 Apparel 0.50 (6.55) 1.51 (18.64) 0.28 1.57 1.01 0.008 0.044 -0.009
316 Leather 0.16 (3.76) 1.39 (11.83) 0.27 2.18 0.97 0.005 0.042 -0.010
321 Wood 1.00 (10.22) 1.01 (24.23) 0.41 1.74 1.29 0.009 0.038 -0.009
322 Paper 1.56 (10.03) 0.81 (16.49) 0.61 2.15 1.93 0.019 0.036 -0.010
323 Printing 0.70 (3.16) 1.15 (8.52) 0.31 1.50 1.13 0.005 0.038 -0.009
324-5 Petr.&Chem. 0.46 (2.91) 1.12 (24.15) 0.17 1.35 0.96 -0.001 0.035 -0.009

32
326 Plastic 1.44 (8.86) 1.03 (17.43) 0.52 1.74 1.62 0.012 0.037 -0.009
327 Mineral Prod. 0.64 (9.94) 1.07 (26.75) 0.43 1.83 1.31 0.011 0.036 -0.010
331 Primary Metal 1.08 (6.91) 0.87 (11.44) 0.47 1.85 1.46 0.011 0.036 -0.010
332 Fabricated Metal 0.72 (8.73) 1.13 (28.03) 0.32 1.47 1.13 0.006 0.037 -0.009
333 Machinery 0.81 (14.34) 0.85 (26.75) 0.42 1.88 1.33 0.010 0.036 -0.010
334 Computer 3.96 (9.62) 0.20 (4.10) 0.94 8.81 13.47 0.037 0.042 -0.009
335 Elec.Equipement 0.58 (7.36) 1.05 (32.20) 0.30 1.70 1.10 0.004 0.038 -0.009
336 Transportation 2.37 (10.41) 0.77 (26.46) 0.49 1.84 1.61 0.011 0.035 -0.009
337 Furniture 0.44 (5.06) 1.29 (22.58) 0.28 1.50 1.04 0.004 0.040 -0.009
338 Miscellaneous 0.50 (7.29) 0.97 (15.30) 0.45 2.16 1.32 0.014 0.040 -0.009
Average 1.00 1.00 0.42 2.25 1.91 0.011 0.039 -0.009
U V
Note: t-values are reported in parenthesis for H0 : j = 0 and for H0 : j = 0.
 Figure 6. Scatterplot of bUj and bVj :

Table 3 reports the empirical correlations between di¤erent estimated statistics. The
main result is that the correlation between bUj and bVj is negative, and quite strong
( 0:79). The scatterplot of bUj and bVj is depicted on Figure 6. These results are in
line with Proposition 5. The extension of Model IV to include industry-speci…c …xed
and variable cost heterogeneity now allows us to …nd more precise empirical results
than those obtained with random heterogeneity. The separable structure of Proposition
3, (15), which implies no interaction between …xed and variable cost, is statistically
rejected: technology G xv ; xf …ts the data better than F xv + K xf for any function
K:
We also …nd that the …xed-cost heterogeneity is positively correlated with most of
the statistics especially with the markup and the rate of returns. This coincides with
our discussion of Section 4 on the dangers of neglecting …xed cost. Not surprisingly,
the correlations involving V have the opposite sign to those involving U: The strong
j j

positive correlation between U U=C and p= (@C=@y) seems to be contrary to the pre-
j

diction made by the theory of contestable markets (Baumol, Panzar and Willig, 1982).

33
 Table 3. Correlation matrix
U V UU p 1 @ ln C @ ln U @ ln V
yj crj
j j j
C @C=@y " (C; y) @t @t @t

V -0.79
j

UU 0.86 -0.84
j
C
p
0.80 -0.77 0.83
@C=@y
1
0.86 -0.67 0.79 0.95
" (C; y)
@ ln C
0.81 -0.78 0.97 0.88 0.83
@t
@ ln U
-0.07 0.31 0.01 0.28 0.26 0.20
@t
@ ln V
0.22 0.02 -0.03 0.26 0.29 0.09 0.56
@t
yj 0.58 -0.58 0.29 0.27 0.28 0.18 -0.55 0.22

crj 0.23 -0.39 0.20 0.23 0.07 0.20 -0.16 0.14 0.58

Hj 0.24 -0.45 0.25 0.21 0.01 0.24 -0.20 0.17 0.61 0.92

34
However, it can be explained in the light of our framework: a higher …xed cost reduces
the variable cost (at given level of production), a relationship which is re‡ected by the
negative correlation between U and V: This negative correlation is in turn inherited
j j

by U U=C and @C=@y:

j

These results help to understand why speci…cations neglecting the …xed cost (or
including an in‡exible parameterization of the …xed cost) are likely to overestimate the
marginal cost of production and underestimate the markup and the rate of returns to
scale. The omission of the …xed cost leads to attribute neglected variations in …xed
costs (which according to Table 3 are positively correlated with output) to the variable
cost function which is increasing in y . Like in the case of an omitted variable bias, the
variable cost function (and especially its partial derivative w.r.t. y ) will catch up the
part of the …xed cost function which is correlated with production and so, it will be
biased upwards. The positive correlation corr U; y = 0:58 explains the gap between
j j

the results obtained with the standard and extended Translog speci…cations (see Table
1). In Model I, the neglected …xed cost is directly responsible for the low rate of returns
to scale and moderate markups obtained with this speci…cation.
Regarding technical change, we …nd that @ ln C=@t is positive and highly correlated
with U; V; U U=C and p= (@C=@y) ; which means that …xed cost and market power
j j j

preclude productivity growth (as in Arrow, 1962). Surprisingly, neither @ ln U=@t nor
@ ln V =@t are strongly correlated with market power. This paradox is solved if we go
back to the de…nition of technical change, in which the share of …xed cost plays an
important role: !
U UU
@ ln Cj @ ln U j U @ ln V j
= + 1 ;
@t @t Cj @t C
and introduces correlation between @ ln Cj =@t and U and U U=C . We also investigate
j j j

the link between the …xed cost, the size and the concentration of industries. Table 3
also reports correlations between the …xed cost and the average output level (over time
and subsectors within industry j ), the concentration ratio for the 20 largest …rms crj ;
and the Hirschman-Her…ndahl index Hj :16 We …nd a positive correlation between the
…xed cost share and the industrial concentration. These results suggest that industries
with a higher …xed cost and a lower variable cost, produce more in average, and are
16
The concentration data for 2002 are obtained from the U.S. Census Bureau.

35
more concentrated.

8. Conclusion
This paper investigates technologies in which …xed inputs can be imperfectly substituted
to variable inputs, and we propose extended production and cost functions compatible
with the occurrence of a …xed cost. Many available ‡exible speci…cations, like the
Translog cost function, restrict the …xed cost to be equal to zero. Our extended speci-
…cation of the Translog is compatible with arbitrary levels of …xed cost, and allows for
interactions between the …xed and the variable cost. Our empirical …ndings highlight
the importance of …xed cost which represent about 20% to 60% of total cost in the
manufacturing industries and tend to decline to decline over time. Our estimates also
supports our extended framework which explains why industries with higher …xed cost,
in average have lower variable cost, higher returns to scale and markups. Conformably
to our theoretical prediction, we also …nd that the classical Translog cost function un-
derestimates the rate of returns to scale and the markup.
A natural extension of our framework would be to examine explicitly strategic inter-
actions between …rms in their joint decision on product price and production capacity
(…xed cost). This would potentially allow to revisit the link between …xed cost and
barriers to entry.

9. Appendix: proof of the results

Proof of Proposition 1. From the de…nition of XF and XF 6= ? it directly follows
that
n o
cr w; xf ; 0 = min w> xv + w> xf : F xv + xf 0 = w > xf 0;
xv 0
and so vr w; xf ; 0 = cr w; xf ; 0 w> xf = 0:
(i) The variable inputs must satisfy the nonnegativity constraints xv 0. If these
constraints are not binding at the optimum, we can write
n o
cr w; xf ; y = min w> xv + w> xf : F xv + xf y = vr w; xf ; y + w> xf ;
xv >0

where vr w; xf ; y minx>xf w> x : F (x) y w> xf > 0: Then cr w; xf ; y = C (w; y)

and by Shephard’s lemma xv w; xf ; y = Xv (w; y) :

36
 (ii) If some constraints xv;j 0 are binding at the optimum, the total input x can be
rewritten as
e
x
x = xv + xf = ;
x
with xei = xv;i + xf;i for xv;i > 0 and xj = xf;j for xv;j = 0: Vector w is partitioned
>
accordingly as w = we> ; w> : Then
n o
> >
cr w; xf ; y = min w xv + w xf : F xv + xf y
xv 0
n o
= min w e> x
e + w> x : F (e
x; x) y
e>0
x
n o
>
= min w e x x; x) y + w> x = Vr (w;
e : F (e e x; y) + w> x:
e>0
x

Proof of Proposition 2.
(i) If xf 2 XG then xv = 0 is admissible and so
n o
vr w; xf ; 0 = min w> xv : G xv ; xf 0 = 0:
xv 0
The assumption that G is single valued and increasing implies that G xv ; xf > 0 for
and xv > 0 and xf 2 XG : Then vr w; xf ; y = w> xv w; xf ; y > 0 for y > 0 because
w > 0 at least one element of xv w; xf ; y is strictly positive.
(ii) For y 0 > y; and G increasing in xf ; it implies that xv : G xv ; xf y0 xv : G xv ; xf y
and as a consequence
n o
vr w; xf ; y 0 = min w> xv : G xv ; xf y 0 > vr w; xf ; y :
xv 0

(iii) Similarly, x0f > xf and G increasing in xv ; xf ; implies that xv : G xv ; xf y

n o
xv : G xv ; x0f y and as a consequence
n o
vr w; x0f ; y >
= min w xv : G xv ; x0f y < vr w; xf ; y :
xv 0

Proof of Proposition 3.
Part (i), Necessity. For an exogenous level of xf 2 XG ; we have
n o
vr w; xf ; y = min w> xv : y = F xv + K xf
xv 0
n o
= min w> xv + w> K xf : y = F xv + K xf w > K xf
xv 0
n o
= min w> X : y = F (X) w > K xf
X K(xf )

= vy (w; y) w > K xf :

The last line follows from our assumption that xv (w; y) > 0 at the optimum. De…ning

37
v (w; y) vy (w; y) vy w; 0+ ensures that v w; 0+ = 0: De…ning ur w; xf vy w; 0+
w> K xf + w> xf ensures that cr w; xf ; y = ur w; xf + v (w; y) :
Conversely, we can recover the convex hull of all inputs producing y , for a given level
of xf ; by solving
n o
min w> xv vy (w; y) + w> K xf :
w
The corresponding J …rst order conditions for an inner solution are given by
@vy
xv + K xf (w; y) = 0;
@w
which can be solved with respect to w=wJ and y to obtain

y = F xv + K xf :

If G is quasi-concave in xv ; this convex hull corresponds to the isoquants of G:

Part (ii). Necessity. With (15), the …rst order conditions for an inner solution in xf
to the cost minimization problem are given by
@ur
w; xf = w;
@xf
and do not depend on y and so the solutions xf (w). With (16), the …rst order conditions
for an inner solution in xv are
@F
w = xv + K xf
@xv
y = F x v + K xf ;

where denotes the Lagrange multiplier. The solution in xv to this system takes the
form xv w; xf ; y = X (w; y) K xf and so the restricted cost function (15), with
vy (w; y) w> X (w; y) and ur w; xf = w> xf w> K xf : Then xf is independent of y:
Su¢ ciency. If xf depends only upon w; then the …rst order conditions for an inner
solution, given by
@ur @vr
w; xf + w; xf ; y = 0
@xf @xf
imply that
@ 2 vr
w; xf ; y = 0
@xf @y
and so cr w; xf ; y = ur w; xf + v (w; y) :

Proof of Proposition 4. We rewrite C T L as

>
Bwy + 21
C T L (w; y; t) = b (w; t) y y +ln w yy ln y+ yt t ;

38
with
> 1 1
b (w; t) exp 0 + w ln w + tt + ln w> Bww ln w + ln w> Bwt t + tt t
2
> 0:
2 2
If yy 0; then
lim+ C T L (w; y; t) = 0; (34)
y!0
whereas if yy > 0;
lim+ C T L (w; y; t) = +1:
y!0
The cost function is nondecreasing in y > 0 i¤
@C T L > C T L (w; y; t)
(w; y; t) = y + ln w Bwy + yy ln y + yt t 0:
@y y
If yy > 0; then
@C T L
lim+
(w; y; t) < 0;
y!0 @y
and @C T L =@y becomes positive only for y su¢ ciently large.

Proof of Proposition 5. There are two types of unobserved heterogeneities here:

one due to unobserved xf and one due to heterogenous functional forms for ur and vr
over individuals. For simplicity we use the subscript r for denoting this heterogeneity.
Let fujx denote the conditional density function of ur w; xf ; t jxf : Under Assumption
(a) we can write fujx = fu where fu denotes the marginal density of ur : Let us de…ne the
average …xed and variable cost functions (over all …rms in our sample) as
Z
u w; xf ; t ur w; xf ; t fu (r) dr
Z
v w; xf ; y; t vr w; xf ; y; t fv (r) dr:

These functions still depend on the unobserved heterogeneity in xf ; but individual het-
erogeneity in the cost functions ur and vr has been integrated out. Let us also consider
U u w; xf ; t V v w; xf ; y; t
w; xf ; t ; w; xf ; y; t ;
U (w; t) V (w; y; t)
and (we skip the arguments for simplicity)
U V
c= U+ V:

Using the optimality condition @cr =@xf = 0; and Assumption (a), it follows that @c=@xf =
0: So, conditionnaly on observations (w; y; t) ; we write
h i h i h i V h i
U V V V V V V
cov ; = cov c V =U; = cov V =U; = V 0
U
h i h i V2 h i
V U = V c V
V =U = 2 V V :
U

39
 (i) Under Assumption (a) we can write
Z
U V U V
cov ; = 1 1 fx dxf
Z Z Z
U V
= fuv (r) dr 1 fuv (r) dr 1 fx dxf
R R
Z Z
U V
= 1 1 fuv (r) fx xf drdxf
R
Z Z
U V
= 1 1 fuvjx rjxf fx xf drdxf
R
U V
= cov ; ;

where the fourth equality follows from the fact that under Assumption (a) we have
the independence of individual heterogeneity with respect to the level of …xed inputs:
fuvjx rjxf = fuv (r) : Putting things together, we have cov( U; V ) = cov U; V 0:
(ii) Similarly, the variance matrices satisfy V [ ] = V [ ] and so
" #
V2 V V V
V[ ] = U2 V UV ;
V V V
UV V
whose determinant is zero.

10. References
Acemoglu D. and R. Shimer, 2000, “Wage and Technology Dispersion,” The Review
of Economic Studies, 67, 585-607.
Aghion, P., and P. Howitt, 1992, “A model of growth through creative destruction,”Econo-
metrica, 60, 323-351.
Arrow, K., 1962, Economic welfare and the allocation of resources for invention, in: Nelson,
R. (Ed.), The Rate and Direction of Inventive Activity, Princeton University Press, 609-
625.
Baumol, W. J. and R. D. Willig, 1981, “Fixed costs, sunk costs, entry barriers, and
sustainability of monopoly,”The Quarterly Journal of Economics, 96, 405-431.
Baumol W. J., J. C. Panzar and R. D. Willig, 1982, Contestable Markets and the Theory
of Industry Structure, Harcourt Brace Jovanovich Inc.
Berry, S. and P. Reiss, 2007, “Empirical Models of Entry and Market Structure,” in M.
Armstrong and R. Porter (ed.), Handbook of Industrial Organization, vol. 3, Elsevier.
Blackorby C. and W. Schworm, 1984, “The structure of economies with aggregate measures

40
 of capital: a complete characterization,”Review of Economic Studies, 51, 633-650.
Blackorby, C., and Schworm, 1988, “The Existence of Input and Output Aggregates in
Aggregate Production Functions,”Econometrica, 56, pp.613-643.
Blundell R. and J.-M. Robin, 2000, “Latent Separability: Grouping Goods without Weak
Separability,”Econometrica, 68, 53-84.
Browning, M. J., 1983, “Necessary and su¢ cient conditions for conditional cost functions,”
Econometrica, 51, 851-857.
Cabral, L., 2012, “Technology uncertainty, sunk costs, and industry shakeout,”Industrial
and Corporate Change, 21, 539-552.
Cameron, A. C., P. K. Trivedi., 2005, Microeconometrics: methods and applications, Cam-
bridge University Press.
Caves D. W., L. R. Christensen and J. A. Swanson, 1981, “Productivity growth, scale
economies, and capacity utilization in U.S. railroads, 1955–1974,” American Economic
Review, 71, 994-1002.
Chambers, R. G., 1988, Applied production analysis: the dual approach, Cambridge Uni-
versity Press.
Chen, X., 2012, “Estimation of the CES Production Function with Biased Technical
Change: A Control Function Approach,” BETA Working paper No 2012-20, Depart-
ment of Economics, University of Strasbourg.
Christensen, L.R., D.W. Jorgenson, and L.J. Lau, 1971, “Conjugate duality and the tran-
scendental logarithmic production function,”Econometrica 39, 255-256.
Colander, D., 2004, “On the treatment of …xed and sunk costs in the principles textbooks,”
The Journal of Economic Education, 35, 360-364.
Dehez, P., J. H. Drèze, T. Suzuki, 2003 “Imperfect competition à la Negishi, also with
…xed costs,”Journal of Mathematical Economics, 39, 219-237.
Diewert, W. E., 2008, “Cost functions,”The New Palgrave Dictionary of Economics, sec-
ond edition, S. N. Durlauf and L. E. Blume (eds.), Palgrave Macmillan.
Diewert, W. E. and K. J. Fox, 2008, “On the estimation of returns to scale, technical
progress and monopolistic markups,”Journal of Applied Econometrics 145, 174–193.
Diewert, W. E., and T. J. Wales, 1987, “Flexible functional forms and global curvature
conditions,”Econometrica, 55, 43-68.

41
Fuss, M. A., 1977, “The structure of technology over time: a model for testing the “putty-
clay”hypothesis,”Econometrica, 45, pp.1797-1821.
Gladden, T, C. Taber, 2009, “The relationship between wage growth and wage levels,”
Journal of Applied Econometrics, 24, 914-932.
Gorman, W. M., 1995, Separability and Aggregation, Collected Works of W. M. Gorman,
Volume I, edited by C. Blackorby and A. F. Shorrocks, Clarendon Press: Oxford, 1995.
Koebel, B., M., Falk and F. Laisney, 2003, “Imposing and testing curvature conditions on
a Box-Cox cost function,”Journal of Business and Economic Statistics 21, 319-335.
Krugman, P., 1979, “Increasing returns, monopolistic competition, and international trade,”
Journal of International Economics, 9, 469-479.
Lau, L. J., 1976, “A Characterization of the normalized restricted pro…t function,”Journal
of Economic Theory, 12, 131-163.
Lau, L. J., 1978, “Testing and Imposing Monoticity, Convexity and Quasi-Convexity Con-
straints,”in M. Fuss and D. McFadden (eds.) Production Economics: A dual Approach
to Theory and Applications, Volume 1, North Holland, 409-453.
Leontief, W., 1947, “Introduction to a theory of the tnternal structure of functional rela-
tionships,”Econometrica, 15, 361–373.
Mas-Colell, A., M. D. Whinston, J. R. Green, 1995, Microeconomic theory, Oxford Uni-
versity Press.
Melitz, M., 2003, “The impact of trade on intra-industry reallocations and aggregate in-
dustry productivity,”Econometrica, 71, 1695-1725.
Morrison, C. J., 1988, “Quasi-…xed inputs in U.S. and Japanese manufacturing: a gener-
alized Leontief restricted cost function approach,”Review of Economics and Statistics,
70, 275-287.
Mundlak, Y., 1978, “On the Pooling of Time Series and Cross Section Data,”Econometrica,
46, 69-85.
Murphy, K., A., Shleifer, and R. Vishny, 1989, “Industrialization and the Big Push,”
Journal of Political Economy, 97, 1003-26.
Newey, W. K., K. D. West, 1987, "A simple, positive semi-de…nite, heteroskedasticity and
autocorrelation consistent covariance matrix," Econometrica, 55, 703-708.
Pindyck, R. S., and Rotemberg, J. J., 1983, “Dynamic Factor Demand and the E¤ects of

42
 Energy Price Shocks,”American Economic Review, 73, 1066-1079.
Sutton, J., 2007, “Market Structure: Theory and Evidence,” in M. Armstrong and R.
Porter (ed.), Handbook of Industrial Organization, vol. 3, Elsevier.
Swamy, P. A. V. B.,1970, "E¢ cient inference in a random coe¢ cient regression model,"
Econometrica, 38, 311-323.
Viner, J., 1931, "Costs Curves and Supply Curves," Zeitschrift für Nationalölkonomie, 3,
23-46.
Wang, X. H. and B. Z. Yang, 2001, “Fixed and Sunk Costs Revisited,” The Journal of
Economic Education, 32, 178-185.
Wang, X. H. and B. Z. Yang, 2004, “On the treatment of …xed and sunk costs in principles
textbooks: a comment and a reply,”The Journal of Economic Education, 35, 365-369.

43