F. Yi r Energy Economics 22 2000 285]297 287
Ž .
Electricity is measured in kilowatt hours kwh ; its price is calculated as the purchase cost divided by the quantity. Fuel is measured in kwh-equivalents, and its
price is calculated similarly. Labor is measured in hours, and its price is calculated as the total annual compensation to labor divided by the total working hours in the
year. The quantity of capital stock is measured as the net capital stock of buildings, construction and machinery, and its price is the user’s cost. All the prices are
adjusted to 1980 levels.
On average, paper, non-metallic minerals, and base metals each consumed about twice the energy per unit of output as the chemical industry did, the latter using
about twice the energy per unit of output as the other industries. Of all the industries, printing used the least energy. Two-thirds of energy input are fuels in
non-metallic minerals but one-third in paper, printing, and chemicals. Inputs of fuels and labor per unit of output were generally decreasing in the period, whereas
the inputs of electricity and capital were generally increasing; exceptions were the electricityroutput ratios in chemicals and base metals. The data suggest a substitu-
tion of electricity and capital for fuels and labor; the substantial increases in capital in all the industries, even when output decreased sharply or consistently, also
suggest a general substitution of capital for other factors.
In the next section, we formulate the energy-demand models.
3. Models
This section formulates TL and GL models. For each model, we have a number of specifications to test.
3.1. TL model TL cost function is often used for empirical analyses; a few examples are
Ž .
Ž .
Ž .
Moroney and Trapani 1981 , Gollop and Roberts 1983 , Holly and Smith 1986 , Ž
. Ž
. Vlachou and Samouilidis 1986 , and Tsai and Norsworthy 1991 among others.
We first assume that the production is represented by a TL cost function lnC s a q Ý a ln p q a ln y q a T q a ln D
t j
j t j
y t
T d
t
Ž .
q Ý Ý b ln p ln p q b ln y ln y q b
TT q b ln D ln D r2
i j
i j t i
t j y y
t t
T T d d
t t
q Ý g ln y ln p q Ý g T ln p q Ý g ln D ln p q m T ln y
j y j
t t j
j T j
t j j
d j t
t j T y
t
Ž . q m
ln y ln D q m T ln D 1
y d t
t T d
t
in which, C , p , y, T, and D are total costs, exogenous factor price, output,
t t i
t
time-index to catch technical change, and degree days, respectively, while a , a ,
j
a , a , a , b , b , b , b
, g , g , g , m , m , and m
are parameters in
y T
d i j
y y T T
d d y j
T j d j
T y y d
T d
which subscript t stands for time, while i s 1,2,3,4 and j s 1,2,3,4 stand for inputs: 1 s electricity, 2 s fuels, 3 s labor, and 4 s capital. Considering symmetry and
F. Yi r Energy Economics 22 2000 285]297 288
linear homogeneity of cost function in factor prices, we constrain the parameters so that b s b , S a s 1, S b s S b s 0, S g s 0, S g s 0, and S g s 0.
i j ji
j j
j i j
i i j
j y j
j t j
j d j
According to Shephard’s lemma, the ith cost share is S
U
s lnC rln p s a q Ý b ln p q g ln y q g T q g ln D
i t t
i t i
j i j
jt y i
t T i
d i t
in which, the asterisk marks long-run. Here the adding-up condition Ý S
U
s 1
i i t
holds. The long-run factor demand is x
U
s S
U
C rp and the short-run factor
i t i t
t i t
demand is adjusted from the long-run’s
U
Ž .
Ž .
Ž . x s Bx q I y B x
; t 2
t t
ty 1
in which, t denotes time and I is an identity matrix, while B is an adjustment matrix for factor vector x with 0 F B F 1. For vintage or putty]clay production
t
Ž .
Ž . Ž
. technology see Førsund and Hjalmarsson, 1987 , B s 1 y 1 y d r 1 q g , in
i
which d is the depreciation rate and g the capacity growth. The null value of B implies no adjustments, whereas an identity adjustment matrix corresponds to
instantaneous adjustments. The long-run inputroutput ratio is obtained as a
U
s x
U
r y
and the short-run
i t i t
t
adjustment is
U
Ž .
Ž . a s b a q
1 y B a 3
t t
ty 1
Ž . Ž .
in which a is a vector of inputroutput ratios. Eq. 2 or Eq. 3 together with Eq.
t
Ž . 1 constitutes the two variations of basic TL dynamic model with either factor]in-
put adjustments or inputroutput ratio adjustments. We follow ad-hoc adjustments for simplicity and assign B
s 0. The long-run
i j
elasticity of factor i with respect to price j is «
U
s ln x
U
r ln p s S
U
q b r S
U
i jt i t
jt jt
i j i t
and with respect to its own price is «
U
s ln x
U
r ln p s S
U
q b r S
U
y 1 s yÝ
«
U
i i t i t
i t i t
i i i t
j i i jt
Ž Ž
. Ž
. Ž
. see Berndt and Wood 1975 , Berndt et al. 1977 , Fuss 1977 , Siddayao et al.
Ž .
Ž ..
1987 , and Hogan 1989 . The short-run price elasticity is Ž
U
.
U
« s
ln x rln p s x rx B «
i jt i t
i t i t
i t i
i jt
in which B s B , and Allen partial-elasticity of substitution between inputs i and
i i i
U U
Ž .
j is calculated as s s « rS
see Chamberg, 1988 .
i jt i jt
jt
3.2. GL model Ž
. Following Walfridson 1992 , we now assume that the short-run costs are jointly
Ž . determined both by capacity and actual output. With the defined output
y ,
t
F. Yi r Energy Economics 22 2000 285]297 289
Ž .
Ž .
inputs x and prices p
above, we now formulate a GL model beginning with a
i t i t
short-run GL cost function
1r2 v
Ž 1yv .
Ž .
Ž .
Ž . C s y Q
Ý Ý b p p
exp dT 4
t t
t i
j i j
i t jt
in which, Q is a proxy for long-run output and is here defined as capacity; v is
t
Ž .
cost flexibility; b b s b
is a parameter; d is an indicator of disembodied
i j i j
ji
technical change and T is a time index. In the long-run, y s Q , the production is
t t
constant returns to scale. In the formulation, y
v
Q
Ž 1yv .
may be replaced by
t t
y U
Žvy 1.
, in which, U is capacity utilization. The inclusion of capacity utilization is
t t
t
meaningful since it affects the costs through factor hoarding and returns to scale. As capacity data are not directly available, we expect it to be given by the capital
stock of the previous period divided by the short-run optimal capital input coeffi- Ž
. cient of the same period, as Walfridson 1992 did. Using Shephard’s lemma,
short-run optimal factor demand is obtained as x s C r p s y
v
Q
Ž 1yv .
a
U
i t t
i t t
t i t
Ž .
U
in which x is jointly determined both by y and Q
alternatively by U a s
i t t
t t
i t
Ž .
1r2
Ž .
Ý b p rp
exp dT is long-run input coefficient of factor i. We assume
j i j
jt i t
different elasticities v of factor i with respect to output:
i v
i
Ž 1yv
i
.
U
Ž . x s y Q
a .
5
i t t
t i t
Ž . Together with Eq. 4 , it constitutes the basic GL model. Moreover, we can allow
for non-neutral technical change by replacing d with d :
i 1r2
U
Ž .
Ž .
a s Ý b p rp
exp d T .
i t j
i j jt
i t i
The specification of neutral technical change will be tested against non-neutral in Section 4. Since long-run output equals capacity, the long-run factor demand is
obtained as
x
U
s Q a
U
.
i t t
i t
The adjustment in inputroutput ratio is
U
Ž .
a s B a q 1 y B a
.
i t i
i t i
i ty 1
The interpretation of B is the same as that for the TL model. The long-run
i
elasticity of factor i with respect to price j is
1r2
U U
U
Ž .
Ž .
« s
ln x rln p s b p rp
r 2 a
i jt i t
jt i j
jt i t
i t
and with respect to its own price is «
U
s yÝ «
U
.
i i t k i
k i t
F. Yi r Energy Economics 22 2000 285]297 290
The short-run cross-price elasticity of factor i with respect to price j is Ž
U
.
U
« s
a ra B «
i jt i t
i t i
i jt
and the Allen partial elasticity of substitution between inputs i and j is s s «
U
r S
U
.
i jt i jt
jt
The next section tests some specifications for the models.
4. Hypotheses and tests
