Economic inefficiency measurement of input spending when decision-making units face different input prices

In a recent paper, Kaoru Tone (J Opl Res Soc (2002) 2: 429–444) showed that when the Farrell measure of cost efficiency is estimated for two firms that have different input prices, a firm with higher costs can be deemed more efficient than a firm with lower costs. As an alternative approach, Tone proposed a radial cost efficiency measure that is estimated using levels of spending on each input, rather than input quantities. Thus, firms with higher costs are less efficient than firms with lower costs. In this paper, we extend Tone's approach by allowing for non-radial changes in spending. Our approach builds on earlier work by Luenberger (J Math Econ (1992) 21: 461–481) and Chambers et al (J Econ Theo (1996) 70: 407–419) who use directional distance functions to measure inefficiency. We provide an example and illustration of our approach using Japanese bank data.