Generalized unit cost indicators

3. Generalized unit cost indicators

Energy cost belongs to a class of indicators which also includes Barnett and Morse’s 1963 unit cost. Unit cost is measured as the quantity of labor or of a laborcapital aggregate required to extract and process a unit of resource commodity. The most general indicator in this class — generalized unit cost GUC — is defined as the reciprocal of an index of multi-factor productivity. The change in multi-factor productivity is defined as the change in output of the resource commodity holding all controllable inputs, not including the resource stock itself, constant. The more conventional total factor productivity index also holds the quantity of the resource stock constant. 5 These indicators mea- sure resource quality and availability. According to Cleveland and Stern 1999, generalized unit cost, Barnett and Morse’s unit cost, and energy cost belong to the class of use scarcity indicators as opposed to the more familiar indicators of ex- change scarcity such as prices and rents. The higher the quality of resources and the greater their availability, the more use value they are capable of generating for a given commitment of controlled factors of production and vice versa. The restric- tions imposed by the special cases such as energy cost and unit cost can be examined by decomposing GUC. I make the standard neoclassical assumption that there is smooth substitutability between different inputs so that technology can be represented by a differentiable production function for the gross output of a resource commodity Q. The energy analysis approach implies restrictions on this gen- eral model. The production function can be repre- sented by: Q = fA 1 X 1 ,..., A n X n , A R R, N 1 where R is the resource stock for example, the area of agricultural land from which the resource is extracted, and N is a vector of additional uncon- trolled natural resource inputs such as rainfall and temperature. The X i are other factors of production controlled by the extractor such as capital, labor, energy, and materials, and the A i are augmentation factors associated with the respective factors of production. A R is the augmentation or depletion index of the resource base. 6 In theory, the effective units per crude unit of N could also be allowed to vary, though in most applications it will be assumed that their augmentation indices are constant. Equa- tion 1 can obviously be generalized to multiple outputs and multiple resource inputs. A useful simplifying assumption is that the production func- tion exhibits constant returns to scale in all inputs including the resource inputs. Again, generaliza- tions can be made. If N is measured in terms of rainfall, temperature, etc., rather than water, heat, etc., the relevant constant returns relate to the expansion of X and R but not N. There are decreasing returns when more inputs X are applied to a constant amount of the resource stock R. Taking the time derivative of ln Q yields: Q : = s i A : i + s R A : R + s i X : i + s R R : + s j N : j 2 where the s i are the output elasticities of the various inputs. A dot on a variable indicates the derivative of the logarithm of the variable with respect to time. The change in the logarithm of a generalized unit cost indicator U G = XQ is given by: U : G = s i X : i − Q : 3 Typically, the change in ln U G will be calculated 5 This would be a purer indicator of resource quality as the change in resource availability would not appear in 4. But it would a poorer indicator of use value scarcity — more land, mineral deposits, etc. means less scarcity. 6 Factor augmentation is a restriction on the nature of technological change. It specifies that technical change in- creases or decreases the effective quantity of each factor of production available per crude unit of the input used. The rates of change in the augmentation indices for different inputs can vary. Augmentation indices can be interpreted in terms of both qualitative changes in the inputs themselves and disem- bodied changes in the effectiveness of factor inputs. An exam- ple of a positive change of the former kind would be increases in the skills of workers, and an example of a negative change would be land degradation. These changes could also be represented by changes in human capital or natural capital, respectively, if such direct data were available. Treating them as changes in technology is a neoclassical version of the way they would be treated in input – output analysis as changes in input – output coefficients. 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