Measuring potential gains from reallocation of resources

Abstract

In this work we consider the case when efficient operation of individual economic units does not necessarily imply efficiency for a group of these units. Merging theoretical findings of Li and Ng (Int Adv Econ Res, 1995, 1, 377.) and Färe and Zelenyuk (Eur J Oper Res, 2003, 146, 615), we develop new group-wise efficiency indexes that measure the extent to which the performance of a group of economic units can be enhanced, even if all these units are individually efficient. The existence of such potential improvement is attributed to non-optimal allocation of inputs across the individual economic units from the point of view of a group of these units.

Appendices

Appendix 1

Proof of Lemma 2

We want to prove that ${\overline{P}(x^{1},\ldots,x^{K})\subseteq P^{g}\left( {\sum_{k=1}^K {x^{k}} } \right)}$. Let

$$ Y^{0}\in \overline{P}(x^{0,1},\ldots,x^{0,K}). $$

(*)

Then, ${Y^{0}\in \overline{P}(x^{0,1},\ldots,x^{0,K})\Rightarrow Y^{0}\in \sum_{k=1}^K {P^{k}(x^{0,k})}}$ (by definition (3.1)), and therefore ${\exists y^{0,k}:y^{0,k}\in P^{k}(x^{0,k}), k=1,\ldots,K,\sum_{k=1}^K {y^{0,k}} =Y^{0}}$. Moreover, under standard regularity conditions, P ^k(x ^k) (∀ x ^k∈ℜ ^N₊ ) and T ^k are equivalent characterizations of technology (see Färe and Primont (1995)), i.e., ${y^{0,k}\in P^{k}(x^{0,k})\Leftrightarrow (x^{0,k},y^{0,k})\in T^{k}, \forall k=1,\ldots,K}$. Furthermore, ${\sum_{k=1}^K {(x^{0,k},y^{0,k})} \in \sum_{k=1}^K {T^{k}} \Leftrightarrow \left( {\sum_{k=1}^K {x^{0,k}} ,\sum_{k=1}^K {y^{0,k}} } \right)\in \sum_{k=1}^K {T^{k}} \Leftrightarrow (X^{0},Y^{0})\in T^{g}}$, where ${X^{0}= \sum_{k=1}^K {x^{0,k}} }$. Now, similar to the individual case, under the same regularity conditions, P ^g(X ⁰) (X ⁰∈ℜ ^N₊ ) and T ^g are equivalent characterizations of group potential technology, i.e. ${(\hbox{X}^{0},Y^{0})\in T^{g}\Leftrightarrow Y^{0}\in P^{g}\left( {\sum_{k=1}^K {x^{0,k}} } \right)}$. Combining this with (*) we get the desired result. Q.E.D.

Appendix 2

Proof of Lemma 3

A set S is convex if and only if α₁ S + α₂ S = (α₁ + α₂)S for all α₁ ,α₂ > 0 (Li and Ng (1995)). This implies that for T ^k = T, ∀ k = 1,...,K, ${T^{g}\equiv\sum_{k=1}^K {T^{k}} =KT}$. Using this result, Li and Ng (1995) showed that ${TE^{g}(X,Y)\equiv\mathop {\hbox{max}}\limits_\theta \{\theta : (X,\theta Y)\in T^{g}\}\equiv TE(\tilde {x},\tilde {y})}$, where ${\tilde {x}\equiv K^{-1}\sum_{k=1}^K {x^{k}}}$ and ${\tilde {y}\equiv K^{-1}\sum_{k=1}^K {y^{k}} }$, x ^k∈ℜ ^N₊ , y ^k∈ℜ ^M₊ . Using the same logic, we obtain

$$ \begin{aligned} P^{g}(X)&=\{y: (X,y)\in T^{g}\}=\{y: (X,y)\in KT\}\\ &=\{y:(X,y)/K\in T\} =K\cdot \{y/K: (\tilde {x},y/K)\in T\}=K\cdot P(\tilde {x}). \end{aligned} $$

And, therefore:

$$ \begin{aligned} R^{g}(X,&p)=\mathop {\hbox{max}}\limits_Y \{pY: Y\in P^{g}(X)\}=\mathop{\hbox{max}}\limits_Y \{pY: Y/K\in P(\tilde {x})\}\\ &=K\cdot \mathop {\hbox{max}}\limits_{Y/K} \{pY/K: Y/K\in P(\tilde {x})\}=K\cdot \mathop {\hbox{max}}\limits_{\tilde {y}} \{p\tilde {y}: \tilde {y}\in P(\tilde {x})\}\equiv K\cdot R(\tilde {x},p). \end{aligned} $$

Hence,

$$ \begin{aligned} RE^{g}(X&,Y,p)\equiv R^{g}(X,p)/pY=K\cdot R(\tilde {x},p)/pY=R(\tilde {x},p)/(pY/K)\\ &=R(\tilde {x},p)/p\tilde {y}\equiv RE(\tilde {x},\tilde {y},p), \end{aligned} $$

which proves (5.1). In its turn,

$$ \begin{aligned} AE^{g}(X&,Y,p)\equiv R^{g}(X,p)/(pY\cdot TE^{g})=K\cdot R(\tilde {x},p)/(pK\cdot \tilde {y}\cdot TE)\\ &=R(\tilde {x},p)/(p\tilde {y}\cdot TE)\equiv AE(\tilde {x},\tilde {y},p). \end{aligned} $$

Q.E.D.