using a Divisia index of input where s
i
is replaced with the relevant revenue share. This calculation
makes the neoclassical assumption of competitive profit — maximizing price-taking behavior where
in equilibrium the marginal products of inputs are equated to their prices. Substituting Eq. 2 into
3 it is found that generalized unit cost is also given by:
U :
G
= − s
i
A :
i
− s
R
A :
R
− s
R
R : − s
j
N :
j
4 Thus moves in the generalized unit cost indicator
are the sum of the four terms in Eq. 4, respectively:
1. Technical change: s
i
A :
i
. 2. Resource depletion or augmentation: s
R
A :
R
. 3. Change in uncontrolled natural resource in-
puts such as rainfall and temperature in agri- culture:
s
j
N :
j
. 4. Change in the dimension of the resource stock
e.g. area farmed: s
R
R : .
The traditional calculation of unit cost as per- formed by Barnett and Morse 1963 has been
criticized by energy analysts for not accounting for the historical substitution of energy for other
inputs in many extractive industries Hall et al., 1986; Cleveland, 1991. This shortcoming can be
demonstrated using Eq. 4. The simple labor only unit cost for a constant returns to scale
production function for gross output can be shown to be:
U :
B
= − s
i
A :
i
− s
R
A :
R
− s
j
N :
j
− s
R
R : −s
E
E :
− s
M
M : −s
K
K : +1−s
L
L :
5 where K is capital, E is energy, and M is materials
assuming a KLEM type specification of the pro- duction function such as in Berndt and Wood
1975. As the coefficient of labor is greater than those of the other inputs — it is positive and the
others are negative — the equation demonstrates that substitution of energy or capital or materi-
als for labor reduces unit cost, though nothing has happened to the terms representing the qual-
ity or availability of resources or the efficiency with which they can be extracted and processed.
Results for a labor and capital only unit cost are very similar. Generalized unit cost, Eq. 4 does
not suffer from this problem.
4. Implications for energy cost
Energy cost suffers from a similar shortcoming to unit cost in that it can be distorted by changes
in factor ratios. Energy cost is defined by: U
E
= I + E
Q 6
where I is indirect energy and E is the direct energy used. Indirect energy is given by:
I =
n − 1 i = 1
y
i
X
i
+ y
E
E 7
where the y
i
are the energy intensities i.e. energy used in the production of a unit of the commod-
ity of each of the inputs X
i
and y
E
is the direct and indirect energy used to extract a unit of
energy — the nth input. Resource inputs are the n + 1th and greater inputs. The inputs X
i
include all inputs other than direct energy, resource
stocks, and uncontrolled inputs such as rainfall. From Eqs 2 and 6 the change in the logarithm
of energy cost is given by:
U :
E
= −
n − 1 i = 1
s
i
A :
i
− s
E
A :
E
− s
R
A :
R
−
n − 1 i = 1
s
i
X :
i
− s
E
E : −s
j
N :
j
− s
R
R : +T:
8 where T
: is the change in the logarithm of total energy use I + E. It can be seen immediately
that the condition for U :
E
to be equal to the change in the logarithm of generalized unit cost
U
G
, Eq. 4 is: T
: =
n − 1 i = 1
s
i
X :
i
+ s
E
E :
9 Without this restriction, Eq. 8 depends on
changes in the scale of production, and changes in relative factor ratios as well as changes in resource
quality and availability. So, for example, if the price of energy rises relative to the prices of other
inputs and as a result producers use less direct energy and more non-energy inputs, U
E
can change.
The restriction, Eq. 9, will be met if the output elasticities of the inputs including energy
are proportional to the inputs’ shares of total embodied and direct energy:
s
i
= RTSy
i
X
i
I + E Ö
i = 1, . . . , n − 1 10
and: s
E
= RTS1 + y
E
EI + E 11
where RTS is the returns to scale when all inputs including energy but excluding the resource stock
and uncontrolled inputs are increased or reduced by a uniform percentage. In Eqs. 10 and 11
relative marginal products are proportional to relative embodied energy.
However, this result is derived on the assumption that there is a smoothly differentiable production
function with positive marginal products. But, as was argued in Section 2, the energy analysis ap-
proach implies a Leontief input – output model which consists of production functions that are not
smooth and where marginal products are all zero for increases in inputs from a profit-maximizing
equilibrium. Therefore, in contrast to the neoclas- sical
model neither
marginal products
nor prices need to be proportional to embodied energies
for energy cost to be an accurate resource quality indicator under the Leontief model. In the
following, the above result is developed for a model of the latter type. The production function is given
by:
Q = mina
1
B
1
X
1
,…, a
n
B
n
X
n
12 where the B
i
are augmentation factors associated with the respective factors of production, the a
i
are fixed coefficients, and the other symbols are as
defined above. In the resource extracting sector the augmentation factors reflect changes in the quality
and size of the resource stock as well as changes in the quality of the inputs, pure changes in technol-
ogy, and changes in the uncontrolled environmen- tal inputs. These augmentation factors are not
necessarily equal to the neoclassical augmentation factors A
i
. At a profit-maximizing optimum: X
i
= Q
a
i
B
i
Ö i = 1, . . . , n
13 From Eqs. 6, 7, and 13 energy cost is given by:
U
E
=
n − 1 i = 1
y
i
a
i
B
i
+ 1 + y
E
a
E
B
E
14 which does not involve the quantities of any inputs
or the scale of production. U
E
is purely a function of the state of technology and hence Eq. 14 is
equivalent to Eq. 4 under the non-substitutability assumption.
What are the consequences of changes in factor ratios likely to be on energy cost under the neoclas-
sical model? This can be seen by taking the deriva- tive of the RHS of Eq. 6 with respect to direct
energy use and dividing the numerator and denom- inator by Q:
U
E
E =
1 + y
E
− QEU
E
Q 15
Then: U
E
E \
if E1 + y
E
I + E \
ln Q ln E
16 That is, direct energy use has a positive effect on
energy cost, ceteris paribus, if the share of direct energy and energy used to extract that energy in
total energy used is greater than the output elastic- ity of energy and vice versa. Differentiating Eq. 6
for the other inputs X
i
: U
E
X
i
= y
i
− QX
i
U
E
Q 17
Then: U
E
X
i
\ if
y
i
X
i
I + E \
ln Q ln X
i
18 Inputs which are relatively energy intensive com-
pared to their marginal productivity in producing Q will have a positive effect on energy cost and vice
versa. From the neoclassical ecological economics viewpoint these are likely to be inputs with rela-
tively low embodied information and knowledge. Condition 16 would be expected to be true as
stated as not only might y
E
be relatively high — if energy extraction is relatively energy intensive —
but also the actual energy content of E is taken into account. Table 1 shows the energy intensities of the
inputs used in the empirical example in the next section. Energy, fertilizer, and pesticides all have
very high energy intensities, while labor and the other inputs which include agricultural services
have very low energy intensities. Capital occupies an intermediate rank.
5. Empirical investigation
