Plant capacity utilization with piecewise Cobb-Douglas technology: Definition and interpretation

Convex and nonconvex nonparametric technologies have been applied to estimate plant capacity utilization. However, they face challenges in capturing the production characteristic that involves the simultaneous consideration of increasing, constant, and decreasing marginal production rates along the production surfaces, which may be inconsistent with the standard microeconomic production theory. This inconsistency raises concerns regarding the potential bias of plant capacity utilization estimates. Thus, this paper introduces a new plant capacity utilization measure based on the piecewise Cobb–Douglas technology, aligned with the standard microeconomic production theory. This measure is defined using two multiplicative directional distance functions with optimal endogenous directions. It can be interpreted by the Euclidean distance between two points associated with optimal capacities and the Euclidean norm of one optimal endogenous direction vector. The new measure captures potential production slacks to measure plant capacity in the sense of Pareto–Koopmans efficiency. It also suggests a remedy of non-existence of a maximal plant capacity in the Cobb–Douglas function by utilizing the piecewise Cobb–Douglas technology. The proposed plant capacity utilization measure is validated using a secondary dataset of 19 Chilean hydroelectric power plants.