Disaggregation for efficiency analysis

Abstract

Efficiency analysis is a popular technique used to compare the performances of Decision Making Units (DMUs). Procedures to aggregate DMU-level efficiency and related efficiency concepts to obtain group-specific counterparts have been proposed recently. In this paper, we suggest procedures for the opposite direction: disaggregate DMU-level efficiency and related efficiency concepts into output-specific counterparts. For being largely applicable, based on economic optimization behaviour, easy to use and to interpret, these procedures establish a novel toolkit for efficiency analysis practitioners.

Appendix

The proof is inspired by the proof for the aggregation case provided by Färe et al. (2004), which is itself a particular version of the proof of the aggregation of profit function of Mas-Colell et al. (1995).

A first step is to define the technology at the DMU-level. We impose very weak conditions for that technology set. The only requirement is that the input-output combinations are feasible for all output-specific technologies. Formally, we obtain:

$$T = \left\{ {({\mathbf{x}},{\mathbf{y}}){\mathrm{|}}\forall q \in \{ 1,\, \ldots ,\,Q\} :({\mathbf{x}}^q,\,{\mathbf{y}}^q) \in T^q} \right\}.$$

(72)

In words, x can produce y if, and only if, every output quantity y^q can be produced by x^q. In a similar vein, we define the DMU-level input requirement set as follows:

$$I({\mathbf{y}}) = \left\{ {{\mathbf{x}}{\mathrm{|}}\forall q \in \left\{ {1,\, \ldots ,\,Q} \right\}:{\mathbf{x}}^q \in I^q\left( {{\mathbf{y}}^q} \right)} \right\}.$$

(73)

Next, we proof that the minimal cost at the DMU-level, denoted by C(y, w), is obtained as the sum of the output-specific minimal costs: $C({\mathbf{y}},{\mathbf{w}}) = \mathop {\sum}\nolimits_{q = 1}^Q {\kern 1pt} C^q({\mathbf{y}}^q,{\mathbf{w}}^q)$. Note that we assume that the DMU-level actual cost is defined by the sum of the output-specific actual costs, i.e., $\mathop {\sum}\nolimits_{q = 1}^Q {\kern 1pt} {\mathbf{w}}^{q{\prime}}{\mathbf{x}}^q$. This holds true for several input types (see Remark 1). The proof contains two steps:

• 1. For q ∈ {1, …, Q}, let x^q ∈ I^q(y^q) be arbitrary, then x ∈ I(y) by Eq. (73). By construction, minimal cost at the DMU-level cannot exceed actual cost: $C({\mathbf{y}},{\mathbf{w}}) \le \mathop {\sum}\nolimits_{q = 1}^Q {\kern 1pt} {\mathbf{w}}^{q{\prime}}{\mathbf{x}}^q$. As x^q for q ∈ {1, …, Q} are arbitrary, we can choose the cost minimizer levels, which gives:

  $$C({\mathbf{y}},{\mathbf{w}}) \le \mathop {\sum}\limits_{q = 1}^Q {\kern 1pt} C^q({\mathbf{y}}^q,{\mathbf{w}}^q).$$

  (74)

• 2. Let x ∈ I(y) be arbitrary, then there exists, for q ∈ {1, …, Q}, x^q ∈ I^q(y^q). It follows that: $\mathop {\sum}\nolimits_{q = 1}^Q {\kern 1pt} {\mathbf{w}}^{q{\prime}}{\mathbf{x}}^q \ge \mathop {\sum}\nolimits_{q = 1}^Q {\kern 1pt} C^q({\mathbf{y}}^q,{\mathbf{w}}^q)$. As x (and thus x^q, $\forall q$) is arbitrary, we can choose the cost minimizer levels, which gives:

$$C({\mathbf{y}},{\mathbf{w}}) \ge \mathop {\sum}\limits_{q = 1}^Q {\kern 1pt} C^q({\mathbf{y}}^q,{\mathbf{w}}^q).$$

(75)

Combining the two previous equations end the proof.