Price-Based Aggregation
If marginal product is related to its price, energy quality can be measured by using the price of fuels to weight their heat equivalents. The simplest approach defines the weighting factor (l) in Eq. (2) as
p.
lit = p; (4)
p 1t
where Pit is the price per Btu of fuel. In this case, the price of each fuel is measured relative to the price of fuel type 1.
The quality index in Eq. (4) embodies a restrictive assumption—that fuels are perfect substitutes—and the index is sensitive to the choice of numeraire. Because fuels are not perfect substitutes, an increase in the price of one fuel relative to the price of output will not be matched by equal changes in the prices of the other fuels relative to the price of output. For example, the increase in oil prices in 1979-1980 would cause an aggregate energy index that uses oil as the numeraire to decline dramatically. An index that uses coal as the numeraire would show a large decline in 1968-1974, one not indicated by the oil - based index.
To avoid dependence on a numeraire, a discrete approximation to the Divisia index can be used to aggregate energy. The formula for constructing the discrete Divisia index E is
ln E*- ln E*_!
// PitEit Pit-xEit-x
tT 2£ ”=1 PitEit + 2£ ”=1 Pit-Eit-)
x (ln Eit - ln Eit-1)) (5)
where P are the prices of the n fuels, and E are the quantities of Btu for each fuel in final energy use. Note that prices enter the Divisia index via cost or expenditure shares. The Divisia index permits variable substitution among material types without imposing a priori restrictions on the degree of substitution. This index is an exact index number representation of the linear homogeneous translog production function, where fuels are homothetically weakly separable as a group from the other factors of production. With reference to Eq. (3), f() = g() = Dln(), whereas 1it is given by the average cost share over the two periods of the differencing operation.
