ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A PARTIAL EQUILIBRIUM MODEL OF DERIVED
DEMAND FOR PRODUCTION FACTOR INPUTS
Nalin Kulatilaka
WP 1176-80
October 1980
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A PARTIAL EQUILIBRIUM MODEL OF DERIVED
DEMAND FOR PRODUCTION FACTOR INPUTS
Nalin Kulatilaka
October 1980
WP 1176-80
This paper is part of a larger study attempting to reconcile
disparate results in the
KLEI-I
literature, headed by David 0.
Wood at the Energy Laboratory at MIT.
I
thank Professors
Robert Pindyck, Tom Stoker, Ernie Berndt, and David Wood for
comments on an earlier draft, and also Professor Joanne Yates
for editorial assistance.
Abstract
This paper presents
a
model of aggregate production in the conjugate
The model provides a technique to infer
dual, cost function framework.
full equilibrium properties of
equilibrium data.
In
a
technology from the observed partial
particular, we use a three-factor (KLE) model,
assuming capital to be quasi-fixed in the short run, and therefore in
disequilibrium, while assuming that labor and energy adjust rapidly to
equi librium values.
The partial equilibrium model
is estimated
using two sets of data:
time series data on U.S. manufacturing (Berndt-Wood) and pooled data on
OECD countries (Pindyck).
properties.
The results are used to infer full equilibrium
These equilibrium levels of capital are compared with the
observed disequilibrium data.
actual
Results indicate divergences between
and full equilibrium levels of capital.
This casts serious doubt
on the validity of the equilibrium specification used by previous
authors.
Furthermore, substitution elasticities between the factors were
computed for the partial-equilibrium (short-run) and full-equilibrium
(long-run) situations.
These values are then compared and contrasted
with the original work of Pindyck, Berndt-Wood, and others which assumes
all
factors to be in equilibrium.
We conclude that under an equilibrium
framework, time series data give elasticity estimates that are close to
short-run values obtained via the disequilibrium model, while pooled data
give values close to long-run values.
This analysis does not resolve the E-K substitutability/
However, we are able to eliminate the
complementarity controversy.
short-run versus long-run nature of the results as a cause of the
controversy.
An extension of this analysis,
where the data construction
and specification differences are taken into account,
method to resolve this issue.
r^L4-\ M-H-M-
is suggested as a
1.
Introduction
In
recent years economists have shown
a
renewed interest in the
structure of industrial production, and in particular the substitution
possibilities between input factor prices.
The sudden price increases of
input factors, specifically those of energy, were largely instrumental
motivating these studies.
But the major analytical
tool
in
that made these
studies possible was the introduction of the flexible functional forms by
The ability
Diewert (1971) and Christenson, Jorgenson, and Lau (1971).
of these functional forms to specify and estimate the technology without
placing
a
priori restrictions on substitution possibilities made them
attractive for impirical work.
The numerous empirical
studies that
information regarding the relationship
followed, while providing valuable
among input factors, have also created conflicting evidence.
From
a
policy point of view the most disturbing difference relates to the degree
of substitutability between capital
and energy.
A reconciliation of these estimates requires
a
careful
procedures and the data used by the different studies.
look at the
Most of these
estimates were based on the assumption that observed factor demands were
in
equilibrium.
In this study we
use
a
partial equilibrium factor demand
model of the technology and thereby test the validity of the equilibrium
assumption.
The results are also used to reconcile the question of
whether earlier results pertain to the short run or the long run.
A full equilibrium static model
to their
in the
assumes that firms adjust all inputs
long-run cost-minimizing value within one time period.
real world there
is
However,
little reason to believe that factors adjust
that rapidly in response to exogenous changes in price or output.
fact, for sticky and irreversible factors such as capital,
In
the observed
4
utilization level will most certainly depart from the equilibrium levels,
especially during periods of rapid factor price changes.
One way to
remedy this is to use dynamic models where the factor adjustment
processes are made explicit.
These models, however, suffer from the ad
For example, Berndt,
hoc nature of the hypothesized adjustment process.
Fuss and Waverman (1977,
1979) model the intertemporal
to minimize the cost of adjustment.
disequilibrium
is
In
adjustment so as
their framework the only cause of
assumed to be the costs of adjustment.
They neglect
many other sources that could create stickiness in factor adjustments,
and therefore such models could lead to incorrect results.
The model
used in this paper does not explicitly specify the adjustment process.
Instead, we arrive at the familiar dichotomy between the short run (where
long run (where all
some of the factors are in disequilibrium) and the
factors are in full equilibrium).
A partial equilibrium model shall
therefore provide an accurate and convenient way to model the production
process.
We use this model to perform empirical tests on two different data
sets:
time series data on U.S. manufacturing from 1947 to 1971 and a set
of pooled time series-cross section data for OECD countries.
technology is modeled via
energy (E)
— translog
quasi-fixed.
a
three-factor
— capital
cost function where capital
(K),
is
The
labor (L), and
assumed to be
The results indicate very small departures between observed
and equilibrium levels of capital for the time series data, but larger
differences in the pooled data sample.
support of the conventional
claiiTi
The results also provide some
that time series data reflect short-run
behavior while cross-section data reflect long-run behavior.^
5
The paper proceeds as follows.
the model
In
Section
2,
a
verbal description of
and its uses is presented as motivation for the more formal
development that follows later on in the section.
Section 3 presents an
outline of a framework for using the partial equilibrium analysis towards
the reconciliation of disparate results in the KLEM literature.
In
Section 4 we present and interpret results from applications to
Berndt-Wood time series data and the Pindyck pooled data.
Section
2.
Finally,
in
we make some concluding remarks.
5,
The Model
and Its Uses
Our study differs from previous industrial factor demand studies in
its partial
equilibrium specification of the technology via a restricted
cost function.
the model
and
In
its
Section 2.1, we describe the assumptions underlying
mechanics to prepare the reader for the more formal
development of the model
in
consider
a
a
special case,
Section 2.2.
in
Section 2.3, we
KLE translog technology where capital
assumed to be quasi-fixed, which will
2.1
Finally,
is
later be used in empirical work.
Motivation and Description of the Model
Traditionally, economists have modeled the production technology of
firms in
a
static full-equilibrium framework.
In
the event of changes in
factor prices or the level of output, these models assume instantaneous
adjustment of all input factors such that total costs are minimized.
However,
is very
in
the real world where factor adjustment
unlikely that such
technology.
a
is
slow and costly,
model would be an adequate description of
it
6
To remedy this, more recently,
some researchers have used dynamic
models, where the factor adjustment processes are explicitly specified.
adjustment mechanism, of course,
The exact nature of the intertemporal
depends on the condition that
is
assumed to cause disequilibrium.
such models can be found in Berndt, Fuss and Waverman (BFW)
1979).
Two
(1977,
The first rrethod assumes lags in adjustment to be the cause of
disequilibrium and therefore the process is modeled by
distribution.
In the more
a
Koyck lag
sophisticated second method, they postulate
adjustment costs as the cause of slow adjustment.
Then the adjustment
takes place along a path that minimizes the costs of adjustment.
Although conceptually appealing, both versions suffer from the ad hoc
choices of the lag structure and adjustment costs.
Even more
importantly, other considerations stemming from government regulation,
the differential treatment of investment and costing decisions within
organizations, and the physical inability to change inputs (due to
irreversibility or lags) are neglected
in
such dynamic models by not providing
complete treatment of the causes
a
their analysis.
Therefore
of disequilibrium can lead to misleading results.
An alternative approach is to consider a dichotomous situation
between the firm's technology in the short run (the immediate time
period) and the long run (some unspecified time after the immediate time
period).
This partial equilibrium approach is more general than the
extremely restrictive full equilibrium approach, but does not provide
a
detailed description of the intertemporal behavior as given by the
dynamic models.
However, unlike che dynamic .models described earlier,
the adjustment process in partial equilibrium models can accommodate any
or all of the causes generating. the short-run disequilibrium.
7
Therefore, we feel such
a
model would provide a simple but realistic
framework for aggregate production analysis.
The theory used in these partial equilibrium models is well
in the traditional
developed
microeconomic literature where costs are broken down
into fixed and variable parts.
(1980) use such a model
In
a
recent study Brown and Christenson
to study the substitution
factor inputs in U.S. agriculture.
possibilities among
Their model of the farm sector
assumes self-employed labor and farm land to be quasi-fixed.
The results
indicate that the observed levels of these quasi-fixed factors differ
from the full equilibrium values predicted by the model.
They also
obtain short-run elasticities that are significantly different from the
long-run values, thus indicating slow adjustment of these factors.
These
results, by exhibiting the power of the partial equilibrium technique,
provided the initial motivation for this paper.
In
this paper, we will be more precise about the mechanics bringing
about the partial equilibrium model.
consists of two stages.
In
The firm's input choice decision
the first stage, firms adjust the variable
factors to minimize variable costs.
This minimization takes place within
the time period in which changes in factor prices or output levels
occur.
The other factors need not be rigidly fixed but they do not
adjust to the long-run cost-minimizing equilibrium levels.
them quasi-fixed factors.
Hence we call
The resulting technology is in a temporary
(short-run) equilibrium, which is described by the restricted (short-run)
cost function.
In the
second stage, firms aojust quasi-fixed factors until the total
cost is minimized.
initial
time period.
This adjustment takes place after the end of the
The resulting cost function characterizes the
8
Hence, we arrive at the
long-run (full-equilibrium) technology.
dichotomy between the short run and the long run.
This two-stage optimization model
features.
If the observed
interesting
leads to several
levels of the quasi-fixed factors are equal to
the constructed full equilibrium levels, then a full equilibrium model
would adequately describe the technology.
However,
other
in all
situations, the full equilibrium specification will result in an
inaccurate description.
Hence, by using
a
partial equilibrium
specification, we will do at least as well as the full equilibrium
specification.
Even though we can differentiate between short-run and long-run
properties, this model is essentially in
a
static framework.
The
second-stage optimization does not feed into later results.
cost function is only
a
The long-run
theoretical construct used to describe the
situation where all factors are utilized in the most efficient manner.
2.2
A Formal Development of the Model
The theoretical
basis of restricted cost- and profit functions were
developed by McFadden (1978) and Lau (1976) in their seminal work on the
uses of duality theory in production economics.
the relevant results that are to be used in
Consider
a
Here
I
only summarize
later applications.
n+m factor production technology where in the short run
only n factors are variable.
The other m factors are quasi-fixed in the
short run and these will be called the quasi-fixed inputs.
observed technology will be in partial equilibrium.
Then the
This short-run
technology can be characterized by solving the following cost
minimization problem:
CR = Min.
P
.
X
s.t.
Y = F(X,
Z)
where
X and P
9
levels and prices of the variable factors, Z is a m
are vectors of the
vector of quasi-fixed factor levels,
the production function.
is
function can be written as:
However,
in
long run,
the
Y
is the
level
of output,
and F(.)
When minimized over X, the restricted cost
CR = CR(P,
Z,
Y).
the firms can adjust the quasi-fixed
factors and thereby reach the full equilibrium which is characterized by
the following total cost minimization problem;
Min CT(P, Z, Y) + Q
Z,
.
C = Min CT(P, Q,
Z, Y) =
where Q is the vector of the quasi-fixed factor
prices.
The first-order condition of this optimization problem is
ICR
(1)
Eq.
(4)
+ Q =
can be solved to obtain the full equilibrium values of the
quasi-fixed factors, Z*.
in terms of the
Then the
long-run cost function can be written
restricted cost function and the cost of the quasi-fixed
factors as:
C
(2)
= CR(P,
where Z* will be
a
Z*, Y) + Q
.
Z*
function of P,
Q,
and Y.
We now have complete characterizations of the technology in both the
short run and the
long run.
These cost functions can be used to evaluate
expressions for the substitution elasticities between the input factors.
The short-run Allen elasticity of substitution between factors
i
and
j
is
given by:
CR CR
^
'
.
.
°ij " CR. CRj
where the subscripts of the cost function denote its partial derivatives
with respect
to.
the factor prices.
Similarly, the long-run Allen
10
elasticity of substitution between factors
C
,
and
i
can be written as:
j
C.
The most interesting feature of this model
is
its ability to
represent long-run properties via the short-run (restricted) cost
function which can be estimated using observable data.
C,
C^.
and C
.
can be written in terms of CR,
In other words,
its derivatives,
and Z*.
These results are given in Brown and Christenson (1980) and we
summarize
them below.
S=TPT-^
'='
where
b^.
=
if
i
e
V
=1
if
i
c
F
and we define the set F such that
set V such that
(6)
i
e
-^
V if X^
=
C.
3P.3Pj IJ
.
^-S
*i^jr^
is a
L?CR_ ^
3P.3Pj.
F
c
i
p,^,
if X.
.b..z.
1
is a fixed
1
factor and the
variable factor.
V
^
'^k
3P.3Z*^
^ W
^y
'^k
3^CR ^
^iZ^^p,
3^CR
aZ*
3Z^*3Z* iPT^
+ b
At the full equilibrium the terms in the curly brackets
in equation
(6) will
go to zero since
equation (1)) for all
k
e
|§¥
+ P^ =
(from the first-order condition in
F.
To complete the computation of a
method of evaluating these values
^'s
L
,
•
vie
.
,
.
need to evaluate
aZ*
^,
A general
given in Brown and Christenson, but
for the special case considered in this study we were
able to derive a
simpler method.
The re?t of
'
,s
paper will focus on this special form
of the problem which still capt-res the gist of ^he
disequilibrium
11
concepts and also provides
set of interesting results when applied to
a
particular data sets.
A Special Case
2.3
To proceed any further with this analysis it is necessary to
characterize the restricted cost function by
form.
(L)
In this study
deal with a three-factor technology where
we
and short-run variables whicle capital
and energy (E)
be quasi-fixed.
particular functional
a
labor
is assumed to
By modeling the industrial production technology by only
these three factors we are making an implicit assumption about the
separability of K, L, and
E
from non-energy materials and other inputs.
As a suitable functional form for the restricted cost function,
we choose
the translog form (see equation (7)) due to its ability to make a good
second-order approximation while maintaining flexibility of
substitutability between factors.
(7)
In CR = a
+ oy
In Y +
Y
+
iZ Z
^
+
i=L,E j=L,E
£
i=L,E
S
a.
In P.
1
1
Y.. In p.
In p.
1
J
.^^^^
iJ
p^.
In Y In P.
^'
'
+
S
i=L,E
+
In K +
B.
K
- Yw^n
^
YY
Y)^
'TVi^nnK)2
KK
2
p_
In P.
'^
'
In K +
t:
In Y In K
For purposes of this study, we introduce a number of constraints on the
cost function.
In
addition to the usual neoclassical properties of
homogeneity of degree one in factor prices and output, symmetry of the
second order cross terms, we also impose the further simplifying
assumption of long-run constant returns to scale.
Under these
assumptions the cost furction can be written as (see Appendix
derivation)
1
for
12
In CR = a^ + ay ln(Y/K) + a^
(8)
+
^
ln(PL/P^) -
y^\_{'^n{PJP^))^ + Pyl InCP^/P^)
j n(ln(Y/K))^
ln(Y/K) + In P^ + In K
An application of the Shephard's Lemma yields the variable factor share
equations:
(9)
\
(10)
Sg =
=
- Yll T^(Pl/Pe) " f^YL
\
-
1
-
aj^
^^^"^'^^
Yll ln(PL/''E^ " ^YL
P.X.
"^^^^
^-
=
P.X.
=
P.X.
Z
ieV
^
"•"^Y^'^)
"CR-
^
The full equilibrium cost function for this special case simplifies
to:
(11)
C
= CR(Pl, P^, Y, K*) + P^ K*
and the first order condition of cost minimizing becomes
where K* is the equilibrium level of the capital input.
aCR
Evaluating the partial derivative, -^[^ for the translog function in
equation
(7)
and substitution in equation (12) gives:
(13)
CR(P, K*, Y).
where
Aj =
1
Equation (13),
- oy +
a
(Aj -
tt
ir
In Y -
In K*)
Py,^
+ P^ K* =
ln(P|_/P^).
non-linear equation in K* can be solved to obtain the
equilibrium values K*.
For this particular functional form (or for many
other commonly used cost functions)
form solution to (13).
used in solving for K*.
it
is not
possible to obtain a closed
Therefore, numerical simulation techniques are
13
As
seen in Section 2.3,
the partial
problem
is
the next step in computing a--
derivatives of
^p-
.
is to
obtain
For the single fixed factor case the
greatly simplified from the general case mentioned earlier.
By implicitly partial differentiating equation (13) with respect to the
factor prices and solving for
aK*
N .
(S.
_^,
aK*
—r-
we
get:
V.P^^)
for
(14)
-
N'^
where
N = A,
and
V.
-
ir
In
TT
i
= L, E
- N
K*
if
i
= L
if
i
= E
=
1
+1
and^
(15)
aK* _ JO^
^''k
^K
N
(N^ -
it
- N)
Once the equilibrium capital
Remaining terms
in
level
is known
the expression of c
th-;se
,
values can be evaluated,
are summarized below:
14
=0
if
i
^
Observe now the following two properties (partially differentiating (14)
with respect to K* and P.):
2
(20)
-^
/9i\
^ a^CR
3K aP.
^,,2
=
1
3^CR
^
^
|^+
Substituting these
r
fPoN
^^^^
Sj
=
3K
_
- n
3P.
equation (6) yields:
in
9^CR
aP.aP.
^
a^CR
aP.aK*
iKl
aP.
^_Ar_^
if
n
i
i,
if
i
i
c
j
c
V
v
V
j
e
F
aP.aK* aP^
if i, j
= K*
e
F
Similarly from (5)
C.
(23)
=p^S.
if
i
e
V
= K*
if
i
e
F
We can summarize the computations as follows:
obtain K*.
Then compute
^
First, solve (13) to
and the partial derivatives of CR from
i
e quations (14) - (19).
Finally,
3.
Compute C
•
,
C-j using equations (22) and (23).
substitute these in (3) and [7) to get the elaticities.
Framework to Reconcile Disparate KLE Results
approach
One of the most interesting uses of this partial equilibrium
stems from its ability to distinguish between the short-run and the
long-run characteristics of this technology.
This provides us with a
15
framework to address some of the issues which are thought to generate
disparities in aggregate factor demand studies.
to review the
Here, we do not attempt
literature on industrial factor demand studies.
Instead in this section we develop a framework by focusing on two aspects
which are likely to contribute towards the disparities.
The first
concerns specification of an equilibrium cost function, common to all
studies generating the initial E-K substititability/complementarity
debate.
The second focuses on the short-run versus
long-run nature of
the results obtained by previous researchers.
Specification Test
3.1. A
The validity of the equilibrium specification can be tested by
comparing the actual
level of capital, K, with the full equilibrium
level, K*, obtained by solving eq.
13.
If departures between the two
paths occur, then that would invalidate an equilibrium specification,
thus casting doubt on the results stemming from such a model.
A formal
test of the hypothesis K = K* requires information on the distribution of
K*.
But since eq.
13 does not have a closed-form solution for K*,
and
can only be solved via numerical methods, we are unable to compute
o
standard errors of the estimates for K*.
The results of thi
specification test are, however,
still
conditional on the assumption that some of the factors reach long-run
equilibrium instantaneously.
For example,
in the
KLE framework, if we
allow only capital to be quasi-fixed while labor and energy are at their
full equilibrium levels, then the results are conditional on this
assumption.
capital
is
Therefore, finding that the full-equilibrium level of
not significant
V,
different from the actual observed level may
16
validity of the full-equilibrium
not be sufficient to conclude on the
specification.
the actual
However,
if we
do find significant disparities between
conditional on the
and equilibrium values under this model,
levels, violation of
assumption of observed equilibrium labor and energy
conclusion.
this assumption will only enhance the
In
other words, we can
but not to accept it.
use this test to reject the hypothesis
3.2
9
Short-Run/Lonq-Run Controversy
econometricians has been to
The conventional wisdom among applied
cross-section data as capturing
interpret elasticity estimates based on
time series data as
long-run effects, while those estimates based on
capturing short-run effects.
Pindyck (1977) and Griffin and Gregory
paper by Houthakker (1965).
(1975) trace the origins of this claim to a
Stapleton (1980) makes references to
a
paper by Kuh (1959) where he
equilibrium is slow, estimates
points out that, if adjustment to long-run
estimates are likely to reflect
based on cross-section demand-function
reflect only partial
long-run elasticities while time series estimates
adjustments.
series data is used
Evidence from pooled cross section-time
differences in
by both Kuh and Houthakker in showing the considerable
cross section as compared with
coefficient values obtained from a single
those based on time averages.
Pindyck (1977), using pooled cross
tests this hypothesis
section-time series data from ten OECD countries,
informally, and finds the results to be inconclusive.
earlier addresses this issue
The adjustment cost literature mentioned
by explicitly modeling the adjustment path.
But, as pointed out earlier,
conditions about the causes and
these studies assume rather restrictive
nature of the disequilibrium.
17
this paper, we only consider the polar cases:
In
is
within one time interval, and
a
time after the end of the period.
a short run which
long run which is at
some undefined
To address the reconciliation issue
via these estimates requires careful
consideration of the data and the
validity of the original estimates which assumed all factors to be in
equilibrium.
If the previous analysis does not
indicate any significant
differences between the actual and equilibrium levels of the short-run
fixed factors, thereby justifying the specifications used by previous
studies, then we can make direct comparison between the corresponding
elasticity estimates obtained via these two specifications.
For example,
we should check the following relationships:
TS
S
L
^ij " °ij < °ij
xs
a
.
.
1J
= a
s
.
.
1J
> a
.
.
IJ
where a..'s are the elasticities of substitution between factors
j.
The superscripts TS, XS,
S,
and
L
i
and
denote times series,
cross-sectional, short-run (restricted), and long-run (full-equilibrium)
cases, respectively.
Since these estimates are random variables, the
above relationships should be checked via significance tests.
as mentioned earlier,
the procedure used
in
However,
this paper does not provide a
facility to readily compute standard errors of the elasticities.
However, if the discrepancies from the equilibrium are very large and
significant, then the elasticity estimates of previous models (hereafter
I
will
refer to them as "equilibrium" models or "original" models)
difficult to interpret.
In
such a case,
it
is
necessary to evaluate the
elasticities on different data sets (time series and pooled data) on the
18
same or at least similar technologies,
if one
is to to
resolve the
long-run versus short-run debate.
4.
Empirical Application
conveniently lends itself to
The methodology developed thus far
established data on aggregate
empirical applications using previously
industrial production.
In this section we
use the restricted cost
technology for two data sets, one time
function to estimate the short-run
section-time series, to estimate.
series and the other pooled cross
The
then used to compute the full
procedure described in Section 2.3 is
elasticities of
equilibrium levels of the quasi-fixed factors and
long run.
substitution in both the short run and the
4.1
Data
Because this paper is part of
a
larger reconciliation study, the most
that created the initial
logical data to be used here are those
controversy.
features
Convenience as well as some characterizing
(B-W)
(explained later) led to the choice of Bernt-Wood
(1975)
time
to 1971 and Pindyck (1977) pooled
series data on U.S. industries for 1947
for 1963-1973.
cross section-time series data on 10 OECD countries
detail
sets of data are explained in considerable
original papers.
Carson (1978).
Both
in their respective
found in
Further details on the Pindyck data can be
several other data
A critical analysis of these two and
data construction methods, can
sets, paying much needed attention to the
by Hirsch (1980).
be found in an unpublished memo
19
Several caveats with special relevance to this particular technique
need mentioning.
The first concerns the data on
level of output.
In the
orginal B-W study the cost function specification did not include cross
terms in factor prices and output.
Therefore, the output term did not
enter into the estimation of the factor share equations.
Pindyck incuded
such cross terms in his cost function and used the "net value of output"
for
Y
in the
share equations.
This amounts to using real
values as instrumental variables for the level of output.
(deflated)
In this study
we estimate the entire short-run cost function and thus need a measure of
the level of output.
We obtain this by constructing output price as a
divisia index using arithmetic value weights and price of the input
factors.
Diewert (1977) has shown that the divisia index forms and exact
index in the case of a translog technology.
Therefore we have an exact
index of the
input prices and the divisia
'output'
The
of the KLE input.
index for B-W data are plotted in Figures
1
The
and 2 respectively.
divisia indexes for each of the 10 countries in the Pindyck data are
reported in Table
1.
The
level of output is obtained by dividing the
value of output by this divisia index.
Second,
aggregation.
it
should be kept in mind that the data 1s at a high level of
The conditions for the existence of aggregate production
functions are extremely stringent and
it
is very
conditions would be met by the data used here.-^^
unlikely that these
Although most
studies have this problem, researchers find that the 'reasonable'
empirical results obtained by such aggregate analysis provide some
justification for using such data.
7 Thirdly, the price of capital
short run and in the long run.
that this price is
a
rental
P.
is assumed to be the same
In this
in the
analysis it is crucial to note
price of capital
and not an asset price.
20
Figure
'i)d
e.-40
1:
B-W Prices for K, L and
E.
21
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22
4.2
Estimation of the restricted cost function
Due to the full equiibrium nature of the specification used in the
original papers by B-W and Pindyck,
it
was sufficient to estimate the
factor share equations to obtain the relevant substituion properties.
However, as seen in Section 2.3,
in
this partial equilibrium set-up we
need to estimate the entire restricted cost function to obtain the
Although there does not
long-run values of the quasi fixed factors.
exist simulataneity between the cost function and the share equations,
the error terms of these equations are correlated.
To account for the
contemporaneous correlation of the errors, we use Zellner's seemingly
unrelated regression (SURR) to obtain estimates of the short-run
parameters.
Details of the error structure and the estimation technique
are given in Appendix 2.
are reported
in Table
In the case
The estimated coefficients for both data sets
2.
the translog form described
of B-W data,
16 provide a good fit of the short-run technology.
in eqs.
15 and
However, for the
Pindyck data, the fit was not that good (see column 2, Table 2)
coud be attributed to a number of reasons.
This
First, as Pindyck notes in
his paper, the assumption of a homogenous production function across the
10
countries in the sample, is not justified.
To remedy this,
he
allows
the intercept terms in the share equations to vary across countries.
Fuerthermore, he finds that
a
better specification is obtained if U.S.
and Canada are pooled separately from Japan and Europe.
allowing such
difficulties.
a
On this study
specification would create significant computational
A more sohisticated computational procedure that is
currently being developed, would enable us to carry out more realistic
specifications in the next stage of the study.
results
\/ery
soon.)
(I
hope to have these
23
Table 2
SURR Estimation Results:
(1) Restricted cost function:
(2)
Labor share equation:
eq.
eq.
(10)
(9)
24
Second, even within the specification used here, we notice that the
coefficient
ir,
has a low t-statistic,
indicating its insignificant
contribution towards explaining the dependent variable (see column
Table 2).
2,
An alternative specification, where the cross terms between K
and Y are omitted (i.e. with
column 3, Table
2.
tt
These results are in
= 0), was estimated.
To test the null
likelihood ratio test as follows;
hypothesis of
it
= 0, we
construct a
The likelihood ratio test statistic,
can be written as:
-2 In
where
= N In
^)
(
are the determinants of the estimated error
and
covariance matrices for the restricted and unrestricted models
respectively,
observations.
is the
likelihood ratio and
N
the number of
Under the null hypothesis, this statistic will be
distributed as chi-squared with degrees of freedom equal to the number
of restrictions.
2
X.
The computed test statistic of 0.0936 is
value at 75 percent of 0.1015.
rejected.
In what follows we use
lower than the
Thus the null hypothesis cannot be
this constrained specification with all
10 countries pooled together.
4.3
Equilibrium levels of capital, K*
Once the parameters of the short-run cost function is estimated, we
use eq.
13 to solve for K*,
fixed capital.
the full equilibrium values of the quasi
Values for K and K* are compared in Figure 3 (for B-W
data) and Figures 4a-4j (for Pindyck data).
Before commenting on these results, we define "over capitalization"
25
R5
o
I
CO
ta
Q.
<j
«»-
o
v>
>
E
=3
•I—
i-
JQ
3
C7
=3
4->
<U
CO
<u
J-
a
26
Figure 4.
Actual and Equilibrium levels of Capital: P ndyck data
4a.
CANADA
155.
135.
115
1968
1963
4 b.
1973
FRANCE
190.
i7e.
ISO.
•
130.
lie.
1963
1968
1973
27
Figure
4.
cont.
Actual and Equilibrium levels of capital: Pindyck data.
4 c.
25©.
ITALY
28
Figure^, cont.
Actual and equilibrium levels of capital: Pi'ndyck data.
4e.
NETHERLANDS.
sa.
65.
59<
3S.
1963
1968
4f.
1973
NORWAY
25.0
ee.e
15.
le.e
1963
1908
1973
29
Figure 4. cont.
Actual and Equilibrium levels of capital: Pindyck data.
4g.
SWEDEN
S9,
82.
1963
1968
4h.
2S9.
-
ise.
lee.
-
U.K.
1973
30
Figure 4. cont. Actual and equilibrium levels of capital: Pindyck data.
4i.
1969.
176e.
1560.
1360.
iiee.
-
U.S.A.
31
and "under capitalization" as situations where K > K* and K < K*
If K = K* then the observed
respectively.
technology will be in full
equilibrium and the short-run and long-run properties will be identical.
In
K
the case of B-W data, over most of the range, differences between
and K* (see Figure 3)
are small.
During 1958-1961 we observe
over-capitalization, while in 1953-1958 and in 1968 we observe
under-capitalization.
the range,
an
Upon casual observation we feel that, over most of
equilibrium specification would provide an adequate
representation.
To formally test the hypothesis of K = K*
it
is
necessary to obtain the standard errors of these estimated levels of K*.
Unfortunately, since closed form solutions do not exist for eq. 13, we
are unable to compute the standard errors of K* without resorting to non
linear numerical techniques such as monte-carlo simulations.
In
the cae of Pindyck data, the differences between K and K* are
considerably larger than in B-W case, for most countries in the sample.
We observe over-cpaitalization for France (Figure 4b), Japan (fig 4d) and
W.
Germany Figure 4j); for Sweden (Figure 4g, U.K.
U.S.A.
(Figure 4h) and the
(Figure 4i) the situation is reversed, to give
under-capitalization; Canada (Figure 4a) and Italy (Figure 4c) exhibit
over-capitalization
in the
first half of the sample period and
under-capitaliztaon
in the
second half; the two variables are closest
together and cross over several times
The Netherlands (Figure 4e).
in
cases of Norway (Figure 4f) and
These results indicate significant
departures between full equilibrium and the observed data.
Thus, an
equilibrium specification, as the one used in the original study may lead
to incorrect results.
Cut before concluding on this issue,
necessary to keep in mind
tli-
it
is
inadequacy of the partial equilibrium
32
specification which assumed
countries,
as
a
homogenous technology across all 10
a
model of the true underlying technology.
The errors thus
introduced could have caused the observed discrepencies between K and
K*.
However, since the partial equilibrium model
possibility of full equilibrium
as
a
includes the
special case, this method will
always be superior to the full equilibrium specification.
4.4
Substitution elasticities
Short run and the long run
:
The elasticities of substitution computed via the algorithm in
Section 2.3 are given in Tables 3 and 4 f(or the short run and the long
run respectively) for B-W data.
A summary, giving the mean over the
sample period for each of the 10 countries in the Pindyck data are
reported in table
in Appendix 3.
6.
The inter temporal details f the Pindyck case are
As with the K* values earlier, here too we are unable to
compute the standard errors of the elasticity estimates.
For the B-W data, the elasticity estimates of L, E own- and
cross-effects show very little difference between the short run and the
long run (see Tables 3 and 4).
This result is to be expected due to the
closeness of K and K* values discussed
short run LE show substitutability.
the previous section.
in
In the
substitutable with both enregy and capital.
In the
long run labor is found to be
But,
as with the original
B-W study, we still find EK to be complements.
The long-run elasticities,
than the short-run values,
o,
s
o.
.
o.
p
and orp respectively.
,
is large only in the case of LL.
a,
^
s
s
.
and Orr are consistently greater
,
This difference
Since the elasticities are dependent
not only on the parameters of the static cost function but also on the
data
it
is
possible to computp estimates for each time period (see
33
Tables
3
For
There are some noticeable trends in these results.
and 4).
example, all of the short-run elasticities have
decreasing trend over
long-run values in table 4, show a downward
The
time (see Table 3).
a
trend for LL, LE and EE, but have an upward trend for LK, EK and KK.
To test the hypothesis that "time series data captures short-run
effects," we consider the following relationship described in Section 3.
s
a
•
— a
•
TS
•
•
L
<
a
IJ
IJ
•
IJ
where a-- refers to the elasticities resutling from an equilibrium
specification of
a
LKE model using the B-W time series data.
results together with
Tables
3
a
summary of the short-run and long-run values from
are given in Table 5.
and 4,
standard errors in columns
variation over time.
These
2
that the
It should be noted
and 3, Table
5,
only take into account the
The estimation error component of the variance in
the elasticities are not computed due to the problems mentioned earlier.
Therefore, we are unable to perform any formal tests on the above
hypothesis.
Hence, we form a heuristic test by considering the interval
within two or three standard deviations around each variable.
left hand side of the above relation, we compare columns
To test the
1
and 2 of Table 5.
overlap within
2
The hypothesis can be accepted (i.e.
standard errors) only in the case of
the
ar-c-
intervals
^or
o.
.
and
o.r the short-run values are much smaller than the time series estimates
in column 1.
However, the right hand inequality (compare cols.
1
and 3,
Table 5) is found to be satisfied in all except a^^ (where the time series
values are larger than the long-run values).
In the
case of
0^.
,
the
inequality will be true only if we form intervals using 3 standard
errors.
But here again,
if estimation error
in the partial
equilibrium
34
Table 3:
Short-run Allen Elasticities:
SZLL
1947
—J sua
-0. 071907
--.0-085 28 1_
-
1949
-0.03558
j.s.5a
-^tO-077.7J_._
1951
1952
1953
-U.
-0.072615
1954 ___
1955
1956 .„.
_-
0.417835
—0.-4530.23
0. 453 74
—0.. 4 33093.
0.419868
_0^ 4.23 3.77._
-0. 071191
.-0. 07 03 53,.
.^.0.413 309._
-0.070215
0.412899
_-C..071795
^0.4175.13
3. 407 83
-0.066519
0,0 6,6.3 63_
-0. 60018
.-0.06 024 4
..-
1959
1S6
1961
-0.058737
—1962
•ZLE
.C.A7335.6,.
1957
.._.1.S5 8
B-W data.
_-.0.056.823.„|
0.415763
__0.._4
0A22.1_
0.380411
—0.331 137_
0.375965
_0.. 35 9.153_
1963
-0. 052741
0.353912
„_.1.96.4
--C. 052965
1965
_.0..354.77J_
-0. 050243
= C. 048324
-0. 047155
...J.966
__
.
1967
__1S6a
1969
-0, 044525
-0. 44 162
_ ...IS 7.0
.r0.042S83
197
-0.051165
1
0.344073
_ 0.336 2 35
0.331339
.0.319 953
0.
310345
_0...:O30 32..
C.
347757
SZEE
J-
-2.42796
-^2.-4
653
-2.40571
-•^.-42269.
-2.42772
^2.JJ2J
-2.42809
-.-2.
.4
2 80 9
-2.42807
._-2. 42799
-2.42744
I'
_ji2.it255J. _1
-2.41117
..-2.4 1193
-2.40648
--2.39822
-2.3749
^.2*JJ.64J__
-2.35635
33958
-2.32821
_.r2.
^.-2.29917
-2.29477
I
i
J
^j:.2,.2737J__j
-2.36361
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36
Table 5:
Summary of Substituion Elasticities:
B-W data
Partial Equi librium Model
2
37
model
taken into account, the results would be even stronger.
is
In
summary, we conclude that an equilibrium time series model captures some
but not all of the long-run substitution effects.
For the Pindyck data, the long-run own elasticities are consistently
larger than the short-run values (compare row
row 9, Table 6).
1
with row 7 and row 3 with
But the cross elasticity between L and E is indicating
larger values for the short-run than in the long-run.
between
a.
and a.r are still small.
^
The differences
As with the B-W case, LE and LK
show substitutability, but unlike the B-W data, we find EK also to be
substitutes in the long run.
Thus, the EK substitutability/
complementarity controversy remains unresolved.
Nevertheless, we have
found an interesting result by being able to conclude that the causality
of the controversy does not stem from the time series and the
cross-section nature of the data.
We now go on to use these results to test the following hypothesis;
elasticity estimates derived from an equilibrium model using pooled
cross-section - time series data, captures the long-run effects in the
As described
underlying technology.
in Section 3, this test can be
conducted by checking the following relationship:
cs
s
CT
.
.
IJ
<
a
L
- a
1J
.
•
1J
CS
where a., is obtained by estimating an equilibrium model on the Pindyck
cross section - time series data.
cs
Results for a., obtained by Pindyck
*
are given in Table 7.
u
As with the B-W case, the test will
be conducted
in a heuristic manner.
In the short-run owfi E and L
elasticities are much lower than those
obtained via the cross-section specification used by Pindyck.
Thus,
38
39
o
OO
u->
CO
i-l
oo
O
oo
tM
CO
On
>-i
r^
i-t
CO
O
o
t-t
I
<
to
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o
oo
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OO
CO
1-1
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od
o
oo
rH CO
CM <r«
to
oo
-3-
tM
CM x-*
t
CO
CM
vo 00
*0 iH
r».
O
oo
oo
to lO
CO
o
oo
r^ CO
o
co
tr>
oo
oo
_J
CO
CO rH
<7\
CM
-<f
'-A
O
I
^-'
O
OO
r-i
CO rH
O
OO
CM
CO
O^
oo
\o
CO CM
vo CO
00
<-(
o
o o-
o
oo
—
\0 CO
o
oo
cv
OO
u->
CO
CO
o
1-^
r-i
oo
oo
I
I
-a
o
o
o
u-l
r-t
•H
O
oo
m CO
O
oo
VC
CO -»
«
o
c
D
O
O
c
u
1^ CM
CO
p.
oo
O
oo
0^
V»
CM
I
7^
H
.J
<:
<
.-I
CM \0
O
40
we have evidence supporting the
relationship.
inequality in the above
The LE values for most countries lie within 2 standard
cs
deviations of a.r,
these results.
left hand
but since
In the
L
s
o. r
a.
it
r,
is
difficult to interpret
long run, own elasticities are much closer to the
cross-section results than with the short-run cases.
countries, fall within
2
or
3
Values for most
standard deviations of each other.
long-run cross elasticities are even better agreement.
one standard deviation for many of the countries.
The
They fall within
Therefore, on this
heuristic level, the evidence indicates convincing support for accepting
the hypothesis.
5.
Concluding Remarks
In this
paper we modeled the industrial production process in
a
partial equilibrium framework where some of the factor inputs are assumed
to be quasi-fixed
to vary in the
in the short-run.
By allowing the quasi fixed factors
long-run, we can make inferecnes about the full
equilibrium (long-run) behavior.
In particular, we
KLE production function where capital
is
consider
a 3-f actor
assumed to be quasi-fixed in the
observed data.
A restricted cost function
is used
to estimate the
technology, in two empirical applications.
short-run
One of the most interesting
results stem from the differences between actual and equili^brium levels
of capital.
be small.
For the B-W time series data, these differences are found to
But for the pooled Pindyck data, we notice large departures
between actual and equilibrium capital
levels.
Although the tests used
here are not formal, we employ heuristic methods that indicate these
differences to be significant enough to invalidate the equilibrium
41
Furthermore, since the
specification used by previous researchers.
partial equilibrium model contains the full equilibrium model as a
special case,
it
is
always safe to use such a partial equilibrium
specification in modeling a technology.
The fact that actual capital
departed significantly from equiibrium
levels also has interesting policy implications.
For example, the "under
capitalization" observed in certain countries could be remedied by
policies which would increase incentives for higher capital
investment.
Increasing investment tax credits or permitting firms to depreciate
capital costs more rapidly (e.g., to expense all capital costs with one
year lag, as suggested in the U.K.) could provide such incentives.
On
the other hand, when "over capitalization" is present the policies should
aim at achieving the opposite effect.
However, specific policy
recommendations require the use of more sophisticated models.
These
models show disaggregate to identify particular sectors where the effects
are strongest.
Substitution elasticities between the input factors provided another
host of intersting results.
First we observe that the long-run
elasticities are almost always larger than the short-run values (i.e., no
over-shooting).
This can be explained by a slow adjustment process.
Second, the time series models such as B-W, seem to underestimate the
long-run elasticities but overestimate slightly the short-run values.
The pooled data gave full equilibrium elasticities that were close to the
corresponding values obtained by Pindyck.
results in
a
It
is
hard to interpret these
straightforward way, because of the large departures from
the equilibrium.
However, we can say this:
the long-run estimates
obtained via ful equilibrium models, are closer to short-run values when
42
using time series data but give estimates closer to long-run values when
using pooled data.
This can be thought of as the appropriate
modification to the claim made by Pindyck (1977) and Griffin-Gregory
(1976).
This analysis was not able to resolve the E-K complementarity/
substitutabi lity debate.
E
Even under the partial equilibrium assumptions,
and K were found to be substitutes when using Pindyck data but
complements when using B-W data.
However, we can now eliminate the
short-run/long-run nature of the result as the cause of the disparity.
Thus, data construction and coverage issues may be at the heart of the
controversy.
An extension of this study where the data would be adjusted
to acccommodate accounting and other differences would
convenient vehicle to address the above issue.
serve as a
43
Footnotes-
Isimilar disparities were observed in the literature on
Berndt (1976), in an attempt
Capital-Labor production function studies.
alternative
estimates
to reconcile between
of elasticity of substitution,
found that the results were very sensitive to data construction
techniques.
^Griffin-Gregory (1976) and Pindyck (1977) make this claim.
•^See Berndt, Morrison and Watkins (1980) for a comparative
assessment of dynamic energy models.
^Although we refer to it as a disequilibrium, this can also be
This is made clear in the
thought of as a temporary equilibrium.
description of the model which follows.
^Here we do not attempt to review the literature on industrial
factor demand studies.
^Computing such values necessitates the use of Monte Carlo
However, we
simulations, which were too costly to be carried out here.
can make subjective statements by comparing the time paths of K and K*.
^Although I do not have a rigorous proof, the following heuristic
Suppose, contrary
reasoning might be helpful in providing some insight.
Then,
to assumption, observed labor is not at the equilibrium level.
solving the long-run total cost minimization problem we would have extra
In other
degrees of freedom over which the optimization can be done.
in
labor levels
changes
for
adjust
words, capital quantities will have to
as well as their own changes.
o
Note that here we assume separabiity of KLE from M and therefore
the value of output will refer to the sum of the KLE inputs in this value
added framework.
^Hirsch (1980) has taken a closer look at the validity of some of
the assumptions behind the existence of aggregate production functions In
the context of the KLEM literature.
He finds the aggregation problems to
be extremely serious but reasonable empirical results are considered as
providing some justification.
•^^It is straightforward to compute the usual own and cross price
elasticities of demand from the Allen elasticities:
Ti
.
.
11
= o
.
.
11
S
.
1
and
n
.
.
ij
= a
.
.
1J
i
•
J
this paper we only deal with the Allen elasticities which should be
interpreted as indicating substitutabi lity if positive and
complementarity if negative.
In
44
References
Berndt, E.R. (1976), "Reconciling Alternative Estimates of the Elasticity
of Substitution," Review of Economics and Statistics ( REStat ) , Feb. 1976,
58, pp. 59-68.
Fuss, M.A., and Waverman, L. (1977), "Dynamic Models of the
_,
Industrial Demand for Energy," Electric Power Research Institute Report
EA-500, Palo Alto, Nov. 1977.
(1979), "Dynamic Model of Costs of Adjustment and
Interrelated Factor Demands, with an Empirical Application to Energy
Demand in U.S. Manufacturing," Discussion paper No. 79-30, University of
British Columbia.
Catherine Morrison and G.C. Watkins, (1980), "Dynamic Models
,
of Energy Demand: An Assessment and Comparison," University of British
Columbia, Resources Paper No, 49.
and Wood, D.O.
Demand for Energy," REStat
,
(1975), "Technology, Prices,
Aug. 1975, 56, pp. 259-268.
and the Derived
(1979), "Engineering and Economic Interpretations of
,
Energy-Capital Complementarity," AER vol. 69, no. 3, June 1979, pp.
342-354.
.
Brown, R.S. and Christensen, L.R. (1980), "Estimating Elasticities of
Substitution in a Model of Partial Static Equilibrium: An Application ot
U.S. Agriculture, 1947-1974," forthcoming in Berndt and Field, eds..
Measuring and Modeling Natural Resources Substitution , MIT Press (1981).
Carson, J. (1978), "A User's Guide to the World Oil Project Demand Data
Base," MIT Energy Labortory Working Paper MIT-EL 78-016WP.
Christensen, L.R., Jorgenson, D.W., and Lau, L.J. (1973), "Conjugate
Duality and the Transcendental Logarithmic Production Function,"
Econometrica , July 1971, pp. 255-256.
Diewert, W.E. (1974), "Applications of Duality Theory," in M.D.
Intrilligator and D.A. Kendrick (eds.), Frontiers of Quantitative
Economics , v. 2, Amsterdam:
North-Holland, 1974.
Econometrics ,
(1976), "Exact and Superlative Index Numbers," Journal of
4, 1976, pp. 115-145.
Field, B.C. and Grebenstein, C. (1980), "Capital-Energy Substitution in
U.S. Manufacturing," forthcoming as note in REStat
.
Griffin, J.M. (1979), "The Energy-Capital Complementarity Controversy:
Progress Report on Reconciliation Attempts," an unpublished memo.
A
45
Griffin, J.M. and Gregory, P.R. (1976), "An Intercountry Translog Model
of Energy Substitution Process," American Economic Review, vol. 66, Dec.
1976, pp. 847-857.
Hirsch, R.B. (1980), "The Capital-Energy Substitution Controversy:
Critical Commens on Translog Aggregate Factor Demand Studies,"
unpublished memo, MIT Energy Laboratory.
Houthakker. H.S. (1965), "New Evidence on Demand Elaticities,"
Econometrica , April 1965, vol. 33, pp. 277-288.
Johansen, L. (1972), Prod'jction Functions:
An Integration of Micro and
Macro, Short-Run and Long-Run Aspects , North-Holland, 1972.
Kuh, E., "The Validity of Cross-Sectionally Estimated Behavior Equations
in Time Series Applications," Econometrica , 27, 1959, pp. 197-214.
(1976), "A Characterization of the Normalized Restricted Profit
Function," Journal of Economic Theory , 12^, 1976, pp. 131-163.
Lau, L.J.
McFadden, D. (1978), "Cost, Revenue and Profit Functions" in Fuss, M. and
McFadden, D. (eds.). Production Economics: A Dual Approach to Theory and
Applications vol. 1, North-Holland, 19/8.
,
Nadiri, M.I. and Rosen S., A Disequilibrium Model of Demand for Factors
of Production , Columbia University Press, 1973.
Pindyck, R.S. (1977), "Interfuel Substitution and the Industrial Demand
for Energy:
An International Comparison," MIT Energy Labortory Working
Paper, MIT-EL 77-026WP, and REStat, vol. 61, no. 2, May 1979, pp. 169-179.
Stapleton, David
(1980), "Inferring Long term substitution possibilities
from cross-section and time-series data", unpublished memeo. University
of British Columbia.
46
APPENDIX
I
Imposing restrictions on KLE translog function with one fixed input (k)
:
Consider the following translog restricted cost-function.
(A.l)
In CR =
+
u
^^
+
Y
E Z
Z)
D^.
equation
is
a,
In P,
J
J
In P.
y
^J
i=L,E j=L,E
J
L
+
i=L,E
^
+ B^ In K + |
K
2
In P. +
^
In Y In P
^^
i=L,E
If this
E
.^^^^
+ a. In Y +
Of.
I
^
6(ln
KK
In P.
p
"^
^^(1^
2
Y)
YY
K)^
In K +
TT
In Y In K
^
directly estimated, there are 16 free parameters that
have to be estimated.
However,
homogeneity of degree one
in
imposing theoretical restrictions of
input prices and the symmetry conditions
gives the following linear restrictions:
(A. 2)
symmetry
(A. 3)
homogeneity of degree one,
y-^.
j:Y.j=i:Yij
= Yj^
"V"
=
i,
i,
J
j =
L,E
c
and
y]a. =
^7"
1
1
1
For the cost function in equation (1), these constraints can be explicitly
written as:
(A.4)
Yle = YEL
\e
^
^EE " ^EL
"^
^LL
Thus in compact notation
^LL = ^EE = "^EL = "^LE
(A. 5)
Pyl^
= -Pyp
(A. 6)
p^,^
= -p^^
47
(A. 7)
\
"
°E
=
^
Imposing restrictions (A. 4) - (A. 8) in the cost function we getl
(A. 8)
In CR = ttQ +
+
a^ In
Y
+
In P^ +
y^^_{'ir^{P^/P^)f +
I
+
In K
P|^|^
I
a,_
ln(P|_/P^)
+
b^ In K + ^
^^^
y,^k^^"
6^^(ln K)^ + py^ In Y InCPL/P^)
ln(P|_/P^) +
TT
In
Y In K.
Now the number of free parameters is 10.
As a further simplification, we can
impose constant returns to scale on
the technology by introducing the following linear restrictions:
+
(A. 9)
=
B,^
^Y
1
PyL " ^LK =
.
°
^EK =
=
"*"
YyV
""
''YE
If
- 6^^ =
Rewriting equation (A, 9) we get
(A. 10)
B,^
=
1
- Oy
^LK = -^YL
^YY "
" ""
'^KK
The resulting cost function (now with only 6 free parameters) can be
written as
(A. 11)
(In CR - In P^ - In K) = a^ + ay ln(Y/K) + a^ ^n{P^IP^)
-
^
n
(ln(Y/K))^ +
i y^_^{^MPJP^)f
Note the following property, (ln(^))^ = (In x)
+ PyL
^n{PJ^
- 2 In x
)
In (Y/K)
In y + (ln(y))'
48
Appendix
II
Estimating the Restricted Cost Function
The estimation method chosen here involves simultaneous estimation of
and one of the two share equations.
the restricted cost function, CR,
Due to the adding-up of property of the share equations,
one of the share
equations can be left out while still extracting all the information.
In
general, the system of equations to be estimated can be written as
follows,
(B.l)
In CR = Oq + ay In Y +
E
B-
Z.
In
ieV
*
IZE^ij
i
+
2
I
1"
Pi'" "i*
Yyy (In Y)^
+^
o^.
In p^
ieV
^I>ij
i
J
a^
+X)
'"
'"
"i
*
iSBij!"
i
J
In Y
^i
^^^.
In P.
z,
1" z.
J
In Y In Z- +
1
(B.2)
S.
=a. ^PYi
1"Y^ LrijlnPj^Z^-jl^Pj^^i
j
j
for all except one ieV.
The estimation method should be one that is capable of incorporating
both the cross equation parameter restrictions and the specific
structural relationships of the error terms.
further,
I
Before pursuing these any
will present the simple single fixed factor KLE model
considered in this study, thereby, the analysis and description will be
much simplified while maintaining the concepts.
.
2
{B.3)
(In CR - In K - In P^) = oq + ay ln(Y/K) + a^ InCP^/P^) -| it(ln Y/K)
49
Since the share equation is derived by applying Shepard's Lemma to
the cost function, the ei^ror terms of the two equations will be related
as follows:
— —n— from
=
S,
T
L
3
In
Shepard's Lemma
P,
By integrating equation (B.4) with respect to In P._ ^g „q^
(8.5)
In CR =
A^
d(ln P^) =
J
{a^ +
=
aj_
In
P^_
+
+
G.
In
P^^
+ function (P^,
I
^
^" ^^ ^^^
y^L ^"(Pl/Pe^
^YL
y^^ In (Pl/P^) In Pl
Y,
"•
"^
^^^^^'^
PyL ^" ^"^'^^ ^" ^L
K)
Similarly, by integrating S^ with respect to
In
P^, we get
'^E
~ ^YL
^IL
(B.6)
In
CR =
-
-^
(1
- a^)
In Pf -
e
In
+ function
P
1"
(P|_,
^^L^^E^
Y,
^"
"""
^'^^^^
""^
K)
For consistency of equations (B.5), (B.6) with (B.3),
(B.7)
£(>
where plim
= e^ ^"
(g.
.
*"
('\/''e
°
^C =
and plim np =
n-) =
The most efficient estimator of the system of equations (B.3) and
(B.4) should take into account this explicit relationship between the
error terms as well as the cross equation restrictions.
Furthermore,
if the
error terms are assumed to be serially
uncorrelated, then by using iterative Zellner estimation, we take into
account the cross equation restrictions and the contemporaenous
correlation between the error terms.
The only loss of efficiency of
using Zellner as opposed to FIML is its inability to incorporate the
specific error structure.
''e
^l
50
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1986
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ACME
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NOV
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CO., INC.
1983
100 CAMSRIDGE STREET
CHARLESTOWN, MASS.
BASEMENT
HD28.M414 no.l175- 80
Medoff, James /The role
seniority
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TOAD DDl TTb
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HD28.M414 no.l176- 80
Kulatilaka. Na/A partial eauilibrmm
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741444
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H028.IV1414 no.ll77- 80
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HD28.IVI414 no.ll81- 81
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