Halving the rate of productivity growth for making e-capital causes a failure of the second WACM test, because e-capital is then not cheap enough in 1998 to comport with the large amount of e-capital in use.
The only restriction that production theory imposes on the movements of factors and their prices is the WACM, tested in the previous section.
In the first part of this investigation, Hall chooses technological parameters for which it is logically possible that e-capital does not violate the weak axiom of cost minimization (WACM).
If there is a violation of the WACM, then C(|W.sup.i~,|X.sup.i~) |is greater than~ C(|W.sup.i~,|X.sup.j~) for all |Y.sup.j~ |is greater than or equal to~ |Y.sup.i~.
The factor demands that are consistent with the conditions of the WACM in equation (1) and the Cobb-Douglas production function in equation (3) are:
Equation (6) is the explicit functional form of the cost function consistent with the necessary and sufficient conditions of cost-minimizing behavior implied by the WACM. It is also noted that the natural logarithm of the actual expenditures by a water utility on labor, energy, materials, and capital, lnC(|W.sup.i~,|X.sup.i~), is equal to the natural logarithm of the expenditure minimizing amount, lnC(|W.sup.i~,|X.sup.j~), plus an error term, u, representing the optimization error.
Once these estimates are obtained, one can determine the cost of the production process implied by the WACM, C(|W.sup.i~,|X.sup.j~), in equation (6) and then calculate the efficiency index, |e.sup.ij~, in equation (2) to examine the cost behavior of private and public water utilities.
Based on the critical value of the chi-square statistic with 1 degree of freedom at the 5 percent level of significance of 3.841, the maintained hypothesis of constant returns to scale in deriving equation (8) under the WACM is accepted.
The last step is to empirically examine the fundamental question of whether the cost behavior of individual water utilities in each ownership category is consistent with the WACM explained in Section II.
It should be emphasized that the calculated values of the efficiency index are positive for each individual water utility because the actual cost of production by the firm, C(|W.sup.i~,|X.sup.i~), to produce a given level of output must always be greater than the minimum cost of production, C(|W.sup.i~,|X.sup.j~), for producing the same level of output if there is a violation of the WACM.
The values of the efficiency index in excess of 5 percent for each individual water utility are also examined to determine the possible causes of these departures from the cost-minimizing behavior consistent with the WACM. The important observation is that for each individual water utility with an efficiency index in excess of 5 percent, it has a high portion of its observed cost in terms of the expenditures on one of the four factor demands: labor, energy, materials, and capital.