Abstract
In paper we examine the conditions under which aggregate overall Farrell efficiency decomposes by both observation (i.e., firms) and source (i.e., technical and allocative efficiency) using the same set of aggregation weights. These conditions require firms to produce a single output with a homogenous of degree \(r > 0\) technology. Then, Farrell decomposition at the firm level is also preserved at the industry level.
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Notes
Following Diewert (2005), metrics that are in difference form are called indicators and metrics that take the form of ratios are called indices.
For the relation between the denominator rule and geometric aggregation see Färe and Karagiannis (2020).
The last equality in the first relation in (8) is obtained by applying the denominator rule in \(TE^{k} = \left( {y^{k} /\tilde{y}^{k} } \right)^{1/r}\), which results in \(TE = \mathop \sum \nolimits_{k = 1}^{K} \left[ {{{\left( {\tilde{y}^{k} } \right)^{1/r} } \mathord{\left/ {\vphantom {{\left( {\tilde{y}^{k} } \right)^{1/r} } {\mathop \sum \nolimits_{k = 1}^{K} \left( {\tilde{y}^{k} } \right)^{1/r} }}} \right. \kern-\nulldelimiterspace} {\mathop \sum \nolimits_{k = 1}^{K} \left( {\tilde{y}^{k} } \right)^{1/r} }}} \right)(y^{k} /\tilde{y}^{k} )^{1/r} = {{\left[ {\mathop \sum \nolimits_{k = 1}^{k} \left( {\tilde{y}^{k} } \right)^{1/r} } \right]} \mathord{\left/ {\vphantom {{\left[ {\mathop \sum \nolimits_{k = 1}^{k} \left( {\tilde{y}^{k} } \right)^{1/r} } \right]} {\left[ {\mathop \sum \nolimits_{k = 1}^{K} \left( {y^{k} } \right)^{1/r} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\mathop \sum \nolimits_{k = 1}^{K} \left( {y^{k} } \right)^{1/r} } \right]}}\).
In the case where output-oriented efficiencies are defined to be less than or equal to one, the appropriate aggregation weights are in terms of potential revenue shares.
This is actually how the last equality in the first relation in (12) is obtained: using the denominator rule for \(TE^{k} = y^{k} /\tilde{y}^{k}\) result in \(TE = \mathop \sum \nolimits_{k = 1}^{K} \left( {{{\tilde{y}^{k} } \mathord{\left/ {\vphantom {{\tilde{y}^{k} } {\mathop \sum \nolimits_{k = 1}^{K} \tilde{y}^{k} }}} \right. \kern-\nulldelimiterspace} {\mathop \sum \nolimits_{k = 1}^{K} \tilde{y}^{k} }}} \right)\left( {y^{k} /\tilde{y}^{k} } \right) = {{\left[ {\mathop \sum \nolimits_{k = 1}^{k} \tilde{y}^{k} } \right]} \mathord{\left/ {\vphantom {{\left[ {\mathop \sum \nolimits_{k = 1}^{k} \tilde{y}^{k} } \right]} {\left[ {\mathop \sum \nolimits_{k = 1}^{K} y^{k} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\mathop \sum \nolimits_{k = 1}^{K} y^{k} } \right]}}\).
When applying to output-oriented efficiency measures are less than or equal to one, the Färe and Karagiannis (2014) results imply that they can also be aggregated with input-based weights, namely actual cost shares.
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We would like to thank two anonymous reviewers for useful suggestions in an earlier version of this paper.
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Färe, R., Karagiannis, G. Aggregation and decomposition of Farrell efficiencies. Oper Res Int J 22, 5675–5683 (2022). https://doi.org/10.1007/s12351-022-00721-1
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DOI: https://doi.org/10.1007/s12351-022-00721-1