D.K. Christopoulos r Energy Economics 22 2000 569]586 571
To incorporate dynamic adjustments, a general dynamic demand framework is constructed using the translog cost-function. Thus, a dynamic structure of the
demand for energy in Greek manufacturing is obtained, which is both disaggre- gated in energy components, consistent with the neo-classical theory of production,
and flexible enough to accommodate a rich pattern of dynamic behaviour.
The paper is organised as follows. In Section 2 the theoretical model is pre- sented. The dynamic formulation of the adopted model is contained and analysed
in Section 3. In Section 4 the statistical estimation is presented and the empirical results are analysed. Also, in Section 4 we examine whether or not the estimated
relationships are structural or spurious. Section 5 concludes the paper.
2. The theoretical model
The specification of the model starts with the assumption that the technology applied in the production process can be described by a twice differentiable
production function which relates the flow of output to various inputs of produc- tion. In algebraic terms it can be expressed as
Ž .
Ž . Y s F X
,T j s
1,2,3, . . . . . . 1
j
where Y is the output, X is the u dimensional vector of inputs j, j s 1,2, . . . n:
j
4 Ž .
N 1, . . . ,u and T is an index of technological progress. It is assumed in Eq. 1 that
Ž .
F X ,T is finite for every X and T and continuous for all non-negative Y and X.
j
Ž .
It is also assumed that monotonicity is valid for F X ,T and that the production
j
Ž .
Ž .
function is strictly convex, see for example, Diewert 1971 and Hall 1973 . Ž .
Next it is assumed that the production function 1 is weakly separable with 4
respect to partition N , . . . . . . N . This means that the marginal rate of substitu-
1 s
Ž .
4 tion MRS between any two inputs i and j from any subset N s 1, . . . . . . m is
s
independent of the quantities of inputs outside N , see Berndt and Christensen
s
Ž .
1973a . In other words,
f
i
Ž . s
for all i , j g N and k f N . 2
s s
ž
X
f
k j
f
Ž .
where f s is the first derivative of F X ,T .
i j
X
j
Ž . 4
The assumption of weak separability 2 with respect to partition N , . . . . . . N
1 s
ensures that the aggregates exist and permits us to write the production function Ž .
1 in the following form:
1
Ž
i
.
2
Ž
j
.
s
Ž
s
.4 Y s F X
X , X
X , . . . . . . X
X i s
1, . . . . . . r , j s
1, . . . . . . l, Ž .
S s 1, . . . . . . m.
3
1
Ž
i
.
2
Ž
j
.
s
Ž
s
. where X
X , X
X , . . . . . . X X
are aggregated functions with elements labelled by indices i, j and S and X
1
, X
2
, X
s
are aggregate inputs.
D.K. Christopoulos r Energy Economics 22 2000 569]586 572
Ž . Furthermore, we assume that aggregated functions in 3 are homothetic in their
components. This assumption provides a necessary and sufficient condition for an Ž
. underlying two stage optimisation procedure see Denny and Fuss, 1977 : optimise
the mix of components within each aggregate and then optimise the level of each aggregate. The existence of homothetic aggregation functions does not imply that
Ž .
the overall production process is homothetic Apostolakis, 1988 . Ž .
Given the production function 1 and the associated assumptions, the cost Ž
. function can be derived. According to duality principles, Samuelson 1947 , Uzawa
Ž .
Ž .
1964 and Shephard 1970 , there is a cost function equivalent to the production function that can represent the technology of production and vice versa.
To start with, it is assumed that the cost function that corresponds to the production function can be written as
Ž .
Ž . C P
,Y ,T s min P X s TC 4
j
where C stands for total cost, P is the vector of input prices, and TC is the total
j
Ž . Ž .
cost. The cost function 2 is considered, similarly to 1 , to be twice differentiable in P and T, finite for every P , Y G 0 and T, continuous in Y and P , linear
j j
j
homogeneous in P and Y G 0, non-decreasing in Y and P and concave in P .
j j
, j
Ž . Also the dual cost function 4 will be weakly separable if the aggregation in
Ž . Ž .
production function 1 exists and the aggregate functions in 3 are homothetic in Ž
. their components see Berndt and Christensen, 1973b .
Ž . Finally it is assumed that the specifiable factors of production are capital K ,
Ž . Ž
. Ž .
Ž . Ž .
labour L , electricity EL , diesel D and crude oil M . So the cost function 4 can be written as follows:
w Ž
. x
Ž . C s g P
P , P
, P , P , P , Y , T
5
E M
EL D
K L
where P is the price of capital, P is the price of labour, P is the price of crude
K L
M
oil, P is the price of electricity, P
is the price of diesel and P is an aggregate
EL D
E
price index of energy, i.e. a function that aggregates the energy prices of three components types. As assumed previously, this aggregator function is homothetic in
the mix of energy types.
1
Ž . Eq. 5 can be estimated in two stages: First a sub energy homothetic model with
constant returns to scale will be considered to construct an instrumental variable for the price of energy. Second an aggregate non-homothetic cost function associ-
ated with Hicks non-neutral technological progress. So, the firms first minimise the energy cost and then the total cost of production.
Ž .
Ž . Under the hypothesis of minimisation of total cost TC the cost function 5 may
Ž take various forms. The translog form is one, it can be expressed as Christensen et
. al., 1973
1
Analogous aggregate functions can be formulated for the other aggregate inputs: capital and labour. However, the lack of disaggregated data for capital and labour for Greek manufacturing make this
procedure inapplicable.
D.K. Christopoulos r Energy Economics 22 2000 569]586 573
2 1
1
Ž .
lnTC s a q a
ln P q y
ln P ln P q a lnY q a lnY
Ý Ý Ý
o i
i i j
i j
Y Y Y
2 2
i i
j
2 1
Ž . Ž .
q a T q a T
q d
lnY ln P q m
ln P T q u lnYT
6
Ý Ý
T T T
Y i i
T i i
T Y 2
i i
where i, j s K,L,E. Under conditions of perfect competition the logarithmic differentiation of Eq.
Ž . 6 with respect to input prices P , P , P
yields expressions for the demanded
K L
E
Ž .
quantity of the corresponding inputs in terms of cost shares Shephard’s Lemma , i.e.
TC
Ž .
Ž . M
s lnTCrln P s P X rTC s a q
y ln P q d lnY q m T
7
Ý
i i
i i
i i j
j i Y
T i
j
where X is the quantity demanded of the production input i and M is the cost
i i
share of the input i demanded. Ž .
TC
For the cost function 6 to satisfy the adding-up criterion M
s 1, i s K,L,E
Ý
i
i
and the properties of the neo-classical production theory, the following linear restrictions must be satisfied,
Ž . a s
1, y s y
, y s
d s
m s u s
8
Ý Ý
Ý Ý
i i j
ji i j
i Y T i
i
i i
i i
for i, j s K,L,E. Ž .
The restrictions 8 are necessary and sufficient conditions ensuring TC is linearly homogeneous in input prices.
Ž .
However, the price of energy P
is considered also to be an aggregate
E 2
w Ž .x
function see function 5 . This aggregate function can be represented by a
homothetic translog cost function with constant returns to scale
1
Ž . ln P s a q
b ln P q
y ln P ln P
9
Ý Ý Ý
E o
i i
i j i
j 2
i i
j
The following energy cost share equations are implied by Shephard’s Lemma:
E
Ž .
M s b q y
ln P 10
Ý
i i
i j j
j
where i, j s EL,D,M. Again the adding-up criterion and the properties of neo-classical production
theory require the parameter restrictions
2
P it is not a consistent aggregate price index if it is a simple weighted average of the P ,
E i
i s EL,D,M unless the energy components are perfectly substitutable or complements in the production
Ž .
process see Berndt and Christensen, 1974 .
D.K. Christopoulos r Energy Economics 22 2000 569]586 574
Ž .
b s 1,
y s y ,
y s 11
Ý Ý
i i j
ji i j
i i
to be satisfied by the aggregate function. Let us now review the steps involved in estimating the proposed model: first the
Ž .
Ž .
sub-energy model 10 is estimated subject to the constraints 11 . The estimated Ž
. Ž .
parameters in 10 are substituted into 9 to obtain an aggregate price index for Ž .
Ž . energy. Next the factor share Eq. 7 is estimated subject to the restrictions 8
using the estimated energy price index as an instrumental variable.
3. Modelling the dynamic structure of production using the translog cost function
