Disaggregation of the cost Malmquist productivity index with joint and output-specific inputs

In multi-output settings, different types of inputs are simultaneously used in the production process. On the one hand, some inputs are jointly used to produce all (or a subset of) the outputs. These inputs give rise to economies of scope, which constitutes a prime economic motivation to produce multiple outputs. On the other hand, some inputs are allocated to specific output productions. Using nonparametric output-specific modeling of the production process, we propose a new productivity index for cost minimizing producers in these multi-output settings. The new index takes the form of a cost Malmquist productivity index. The output-specific modeling of the production process naturally allows us to define output-specific cost Malmquist productivity indexes and to disaggregate the cost Malmquist productivity index in terms of output-specific cost efficiency measurements. We also tackle the issue of input price availability and explain how to extend the cost Malmquist productivity index with partial input price information or without assuming observation of the input prices. In the latter case, we establish a duality with a technical productivity index that takes the form of a Malmquist productivity index. The new indexes can be used to evaluate cost-productivity and productivity changes or can be fairly easily combined with existing extensions. We propose an application to the electricity plants.