Complementarity-Substitution Relationships in the Demand for Time-Differentiated Inputs under Time-of-Use Pricing *

Abstract

In this paper we incorporate the non-synchronic responses of different inputs to changes in relative factor prices and develop sufficient conditions under which time-differentiated (over the day) electricity inputs are complements or substitutes. Similar sufficient conditions are developed for time-differentiated labour inputs. We also examine the strong and sometimes one-directional, relationships between the distributions over the day of the demands for labour and electricity. These relationships depend, among other factors, on the objective function of the firm (profit maximization, cost minimization) and on the specific time-of-use (TOU) schedules (of labour, electricity, etc.). Our results are also dependent on the assumption that firms can adjust inputs to changes in input prices on an hourly basis; more specifically, the underlying technology is assumed to be given by an hourly production function. Two issues are emphasized in the analysis. First, we show that short-run cost minimization may be an inappropriate procedure for cost-benefit analysis. Second, under the model developed in this paper, the commonly used weak separability assumption (between electricity and other inputs) implies radically different relationships among the time-differentiated inputs under profit maximization and cost minimization.