Estimating global frontier shifts and global Malmquist indices

Abstract

The Malmquist index is a measure of productivity changes, of which an important component is the frontier shift or technological change. Often technological change can be viewed as a global phenomenon, and therefore individual or local measures of technological changes are aggregated into an overall measure, traditionally using geometric means. In this paper we propose a way of calculating global Malmquist indices and global frontier shift indices which provides a better estimation of the true frontier shift and furthermore is easy to calculate. Using simulation studies we show how this method outperforms the traditional aggregation approach, especially for sparsely populated production possibility sets and for frontiers that also change shape over time. Furthermore, our global indices can be used for unbalanced panels without disregarding any information. Finally, we show how the global indices are meaningful for calculating differences between frontiers from different groups rather than different time periods as illustrated in a small case study of bank branches in different countries.

Appendix

1.1 T1. Strong identity test

$$M^{\rm G}(t, t+i;(\overline X ,\overline Y)) =1\quad\hbox{if}\;X^{t}=X^{t+i}\;\hbox{and}\;Y^{t}= Y^{t+i}$$

Proposition 1

The global adjacent productivity change index satisfies the strong identity test.

Proof 1

$$ \hbox{Since}\;X^{t}= X^{t+i}\;\hbox{and}\; Y^{t}= Y^{t+i}\;\hbox{then}\;D^{t}(.,.)=D^{t+i}(.,.) $$

$$ M^{\rm G}(t,t+i;(\overline X,\overline Y))=\left( {\frac{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^t(x_k^\tau,y_k^\tau)} }{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^t(x_k^\tau,y_k^\tau)} }} \right)^{1/(K\times T)} \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^t(x_k^t ,y_k^t)}{D^t(x_k^t,y_k^t)}} \right]} ^{1/K}=1 $$

□

Proposition 2

The global base period productivity change index satisfies the strong identity test.

Proof 2

$$ \tilde {M}^{\rm G}(t,t+i;(\overline X,\overline Y))=\left( {\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^t,y_k^t)}} } \right)^{1/(K\times T)}=1 $$

□

1.2 T2. Time reversal test

$$ M^{\rm G}(t, t+i;(\overline X,\overline Y)) = [M^{\rm G}(t+i, t;(\overline X ,\overline Y))]^{-1}. $$

Proposition 3

The global adjacent productivity change index satisfies the time reversal test.

Proof 3

$$ \begin{array}{l} M^{\rm G}(t,t+i;(\overline X,\overline Y))=\hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y )){}\times \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})) \\ =[\hbox{TC}^{\rm G}(t+i,t;(\overline X,\overline Y))]^{-1}{}\times [\hbox{EC}^{\rm G}(t+i,t;(X^{t+i},Y^{t+i}),(X^t,Y^t))]^{-1} \\ =[M^{\rm G}(t+i,t;(\overline X,\overline Y))]^{-1} \\ \end{array} $$

□

Proposition 4

The global base period productivity change index satisfies the time reversal test.

Proof 4

$$ \begin{array}{l} \tilde {M}^{\rm G}(t,t+i;(\overline X,\overline Y))=\tilde {\rm T}{\rm C}^{\rm G}(t,t+i;(\overline X,\overline Y))\times \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})) \\ =[\tilde {\rm T}{\rm C}^{\rm G}(t+i,t;(\overline X,\overline Y))]^{-1}{}\times [\hbox{EC}^{\rm G}(t+i,t;(X^{t+i},Y^{t+i}),(X^t,Y^t))]^{-1} \\ =[\tilde {M}^{\rm G}(t+i,t;(\overline X,\overline Y))]^{-1} \\ \end{array} $$

□

1.3 T3. Circular test

$$ M^{\rm G}(t, t+h;(\overline X,\overline Y)) M^{\rm G}(t+h, t+i;(\overline X,\overline Y)) = M^{\rm G}(t, t+i;(\overline X,\overline Y )). $$

Proposition 5

The global adjacent productivity change index satisfies the circular test.

Proof 5

$$ \begin{array}{l} M^{\rm G}(t,t+h;(\overline X,\overline Y))M^{\rm G}(t+h,t+i; (\overline X,\overline Y)) \\ =\left({\frac{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^{t+h}(x_k^\tau,y_k^\tau)} }{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}}{D^t(x_k^\tau,y_k^\tau)} }} \right)^{1/(K\times T)} \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^t(x_k^t ,y_k^t)}{D^{t+h}(x_k^{t+h},y_k^{t+h})}} \right]} ^{1/K} \left({\frac{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^{t+i}(x_k^\tau,y_k^\tau)} }{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^{t+h}(x_k^\tau,y_k^\tau)} }} \right)^{1/(K\times T)} \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^{t+h}(x_k^{t+h},y_k^{t+h} )}{D^{t+i}(x_k^{t+i},y_k^{t+i})}} \right]} ^{1/K} \\ =\left({\frac{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^{t+i}(x_k^\tau,y_k^\tau)} }{\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^t(x_k^\tau,y_k^\tau)} }} \right)^{1/(K\times T)} \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^t(x_k^t ,y_k^t)}{D^{t+i}(x_k^{t+i},y_k^{t+i})}} \right]} ^{1/K}=M^{\rm G}(t,t+i;(\overline X,\overline Y)) \\ \end{array} $$

□

Proposition 6

The global base period productivity change index satisfies the circular test.

Proof 6

$$ \begin{array}{l} \tilde {M}^{\rm G}(t,t+h;(\overline X,\overline Y))\tilde {M}^{\rm G}(t+h,t+i;(\overline X,\overline Y))= \\ =\left({\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+h},y_k^{t+h} )}}}\right)^{1/(K\times T)} \left({\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^{t+h},y_k^{t+h})}{D^\tau (x_k^{t+i} ,y_k^{t+i})}} } \right)^{1/(K\times T)} \\ =\left({\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},y_k^{t+i} )}} } \right)^{1/(K\times T)}=\tilde {M}^{\rm G}(t,t+i;(\overline X ,\overline Y)) \\ \end{array} $$

□

1.4 T4. Commensurability test

$$ \begin{array}{l} M^{\rm G}(t, t+i;((X^{1},X^{2}, \ldots, X^{t-1},(\alpha ^{1}, \ldots,\alpha^{N})X^{t}, X^{t+1},\ldots,X^{T}),(Y^{1},Y^{2}, \ldots,Y^{t-1}, (\beta ^{1}, \ldots,\beta^{S})Y^{t}, Y^{{t}+1},\\ \ldots,Y^{T}))= M^{\rm G}(t, t+i;(\overline X,\overline Y))\quad \hbox{for}\,\alpha^{1}, \ldots,\alpha^{N} > 0\,\hbox{and}\,\beta^{1}\ldots,\beta^{S} > 0.\\ \end{array} $$

Proposition 7

The global adjacent productivity change index satisfies the commensurability test.

Proposition 8

The global base period productivity change index satisfies the commensurability test.

Proofs 7 and 8

D ^t(x ^t, y ^t) = D ^t((α ¹, ... ,α ^N)x ^t, (β ¹, ... ,β ^S)y ^t) for α ¹, ... ,α ^N > 0 and β ¹, ... ,β ^S > 0, cf. e.g. Althin (2001). Therefore

$$ \begin{array}{l} M^{\rm G}(t, t+i;((X^{1},X^{2}, \ldots, X^{t-1},(\alpha ^{1}, \ldots,\alpha^{ N})X^{t}, X^{t+1}, \ldots,X^{T}),(Y^{1},Y^{2},\ldots,Y^{t-1}, (\beta ^{1},\ldots,\beta^{S})Y^{t},Y^{t+1},\\ \ldots,Y^{T}))= M^{\rm G}(t, t+i;(\overline X,\overline Y))\quad \hbox{for}\,\alpha^{1}, \ldots,\alpha^{N} > 0\,\hbox{and}\,\beta^{1},\ldots,\beta^{S} > 0\\ \end{array} $$

and

$$ \begin{array}{l} \tilde {M}^{\rm G} (t, t+i;((X^{1},X^{2}, \ldots, X^{t-1},(\alpha^{1},\ldots,\alpha^{ N})X^{t}, X^{t+1}, \ldots,X^{T}),(Y^{1},Y^{2}, \ldots,Y^{t-1}, (\beta^{1}, \ldots,\beta^{S})Y^{t},\\ Y^{t+1}, \ldots,Y^{T}))= M^{G}(t, t+i;(\overline X,\overline Y))\quad \hbox{for} \,\alpha ^{1}, \ldots,\alpha^{N} > 0\,\hbox{and}\,\beta^{1},\ldots,\beta^{S} > 0.\\ \end{array} $$

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1.5 T5. Determinateness test

$$ M^{\rm G}(t, t+i;(\overline X,\overline Y )) \in \Re _{++}. $$

Proposition 9

The global adjacent productivity change index does not fulfill the determinateness test.

Proposition 10

The global base period productivity change index does not fulfill the determinateness test.

Proofs 9 and 10

As shown in Althin (2001) D ^t(x ^t + 1,y ^t + 1) = 0 in certain cases, whereby both the global adjacent productivity change index and the global base period productivity change index become indeterminate.□

1.6 T6. Proportionality test of inputs

$$ M^{\rm G}(t, t+i;((X^{1}, \ldots ,X^{t},\ldots, X^{{t+i}-1},\alpha X^{t+i},X^{{t+i}+1}, \ldots, X^{T}),\overline Y)=\alpha M^{\rm G}(t, t+i;(\overline X,\overline Y))\quad \hbox{for}\,\alpha > 0 $$

.

Proposition 11

The global adjacent productivity change index does not fulfill the proportionality test of inputs.

Proof 11

$$ \begin{array}{l} M^{\rm G}(t,t+i;(X^1,\ldots,X^t,\ldots,X^{t+i-1},\alpha X^{t+i},X^{t+i+1},\ldots,X^T),\overline Y)) \\ =\hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \left( {\frac{\prod\limits_{k=1,\ldots,K} {D^{t+i}(\alpha x_k^{t+i},y_k^{t+i} )}}{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^t(\alpha x_k^{t+i},y_k^{t+i})} }} \right)^{1/(K\times T)}{}\times \\ \; \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})){}\times \prod\limits_{k=1,\ldots,K}{\left[ {\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i})}{D^{t+i}(\alpha x_k^{t+i} ,y_k^{t+i})}} \right]} ^{1/K} \\ =\hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \left({\frac{\alpha ^K\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})} }{\alpha ^K\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})}}} \right)^{1/(K\times T)}{}\times \\ \; \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})){}\times \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i})}{\alpha D^{t+i}(x_k^{t+i} ,y_k^{t+i})}} \right]} ^{1/K} \\ =M^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \frac{1}{\alpha } \\ \end{array} $$

□

Proposition 12

The global base period productivity change index does not fulfill the proportionality test of inputs.

Proof 12

$$ \begin{array}{l} \tilde {M}^{\rm G}(t,t+i;(X^1,\ldots,X^t,\ldots,X^{t+i-1},\alpha X^{t+i},X^{t+i+1},\ldots,X^T),\overline Y))=\left( {\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (\alpha x_k^{t+i} ,y_k^{t+i})}}}\right)^{1/(K\times T)}\\ =\left({\frac{1}{\alpha^{K\times T}}\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}}{\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},y_k^{t+i} )}}}\right)^{1/(K\times T)}=\frac{1}{\alpha }\tilde {M}^{\rm G}(t,t+i;(\overline X,\overline Y)) \\ \end{array} $$

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1.7 T6’. Proportionality test of outputs (given constant returns to scale)

$$ M^{\rm G}(t, t+i; (X^{t},Y^{t}), (X^{t+i},\beta Y^{t+i})) =\beta [M^{\rm G}(t, t+i; (X^{t},Y^{t}),(X^{t+i},Y^{t+i}))], \beta > 0.$$

Proposition 13

The global adjacent productivity change index satisfies the proportionality test of outputs if and only if the technology exhibits constant returns to scale.

Proof 13

The input distance function is homogenous of degree −1 in outputs if and only if the technology exhibits constant returns to scale (Färe 1988, Althin 2001).

First assume that the technology exhibits constant returns to scale:

$$ \begin{array}{l} M^{\rm G}(t,t+i;(\overline X,(Y^1,\ldots,Y^t,\ldots,Y^{t+i-1},\beta Y^{t+i},Y^{t+i+1},\ldots,Y^T))) \\ =\hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \left( {\frac{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},\beta y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i} ,y_k^{t+i})} }\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})} }} \right)^{1/(K\times T)}{}\times \\ \quad \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})){}\times \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i})}{D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}} \right]} ^{1/K} \\ =\hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \left({\frac{\left[ {\frac{1}{\beta }} \right]^K\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})} }{\left[ {\frac{1}{\beta }} \right]^K\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})} }} \right)^{1/(K\times T)}{}\times \\ \quad \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})){}\times \prod\limits_{k=1,\ldots,K} {\left[{\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i} )}{\frac{1}\beta D^{t+i}(x_k^{t+i},y_k^{t+i})}} \right]} ^{1/K} \\ =M^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \beta \\ \end{array} $$

Therefore, in the case of a constant returns to scale technology, the global adjacent productivity change index satisfies the proportionality test. To prove that the index only satisfies the proportionality test if the technology is constant returns to scale:

$$ \begin{array}{l} M^{\rm G}(t,t+i;(\overline X,(Y^1,\ldots,Y^t,\ldots,Y^{t+i-1},\beta Y^{t+i},Y^{t+i+1},\ldots,Y^T)))=\beta M^{\rm G}(t,t+i;(\overline X,\overline Y)) \\ \Leftrightarrow \hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \left({\frac{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}}{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i} ,y_k^{t+i})}}\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i})}}{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})}}} \right)^{1/(K\times T)}{}\times \\ \quad \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})){}\times \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i})}{D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}} \right]} ^{1/K} \\ \quad =\beta {}\times \hbox{TC}^{\rm G}(t,t+i;(\overline X,\overline Y)){}\times \hbox{EC}^{\rm G}(t,t+i;(X^t,Y^t),(X^{t+i},Y^{t+i})) \\ \Leftrightarrow \left({\frac{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},\beta y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} }\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})} }} \right)^{1/(K\times T)}{}\times \prod\limits_{k=1,\ldots,K} {\left[ {\frac{D^{t+i}(x_k^{t+i} ,y_k^{t+i})}{D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}} \right]} ^{1/K}=\beta \\ \Leftrightarrow \frac{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i} ,\beta y_k^{t+i})} }{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i} ,y_k^{t+i})}}\frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i})}}{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})}}\frac{\left[{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})}}\right]^T}{\left[ {\prod\limits_{k=1,\ldots,K}{D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}}\right]^T}=\beta ^{KT} \\ \Leftrightarrow \frac{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i})}}{\prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})} }\frac{\left[ {\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} } \right]^{T-1}}{\left[ {\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},\beta y_k^{t+i})}}\right]^{T-1}}=\beta ^{KT} \\ \Leftrightarrow \prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i}){}\times}\left[{\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} } \right]^{T-1}=\beta ^K{}\times \prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i},\beta y_k^{t+i})}{}^ \ast \beta ^{K{}\times (T-1)}\left[ {\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},\beta y_k^{t+i})} } \right]^{T-1} \\ \Leftrightarrow \prod\limits_{k=1,\ldots,K} {D^t(x_k^{t+i} ,y_k^{t+i}){}\times } \left[ {\prod\limits_{k=1,\ldots,K} {D^{t+i}(x_k^{t+i},y_k^{t+i})} } \right]^{T-1}=\prod\limits_{k=1,\ldots,K} {D^t(\beta x_k^{t+i} ,\beta y_k^{t+i})}{}\times \left[ {\prod\limits_{k=1,\ldots,K} {D^{t+i}(\beta x_k^{t+i},\beta y_k^{t+i})}}\right]^{T-1} \\ \end{array} $$

(16)

Assume that the technology L ^t+i does not exhibit constant returns to scale (CRS) everywhere (i.e. is variable returns to scale). Choose x ^t+i, y ^t+i such that D ^t+i(x ^t+i, y ^t+i) =  1, i.e. (x ^t+i, y ^t+i) is on the frontier of L ^t+i, and L ^t+i does not exhibit CRS locally at (x ^t+i, y ^t+i). The proof will be shown for the case where (x ^t+i, y ^t+i) exhibits increasing returns to scale (IRS). The proof for decreasing returns to scale (DRS) can be derived analogously.

Define γ _xy =  1/D ^t+i−1(x ^t+i,y ^t+i). Thus D ^t+i−1(γ _xy x ^t+i,y ^t+i) = 1, or equivalently, (γ _xy x ^t+i,y ^t+i) is on the frontier for production technology L ^t+i−1. Let L ^t+i−1_γ denote the production technology for time t+i-1 where all inputs are multiplied by γ _xy , i.e. (x,y) ∈L ^t+i−1_γ corresponds to (γ _xy x,y) ∈L ^t+i−1.

Based on the definition of returns to scale, the following relationships are true for (x,y) on the frontier of L ^τ:

$$ \mathop {\lim }\limits_{\alpha \to 1} D^\tau (\alpha x,\alpha y) > 1\quad \hbox{for}\,\hbox{IRS}, $$

and

$$ \mathop {\lim }\limits_{\alpha \to 1} D^\tau (\alpha x,\alpha y) < 1\quad \hbox{for}\,\hbox{DRS}. $$

Returning to Eq. (16):

$$ \begin{array}{l} D^{t+i}(\beta x^{t+i},\beta y^{t+i})D^{t+i-1}(\beta x^{t+i},\beta y^{t+i})=D^{t+i}(x^{t+i},y^{t+i})D^{t+i-1}(x^{t+i},y^{t+i}) \\ \Leftrightarrow D^{t+i}(\beta x^{t+i},\beta y^{t+i})D^{t+i-1}(\beta \gamma _{xy} x^{t+i},\beta y^{t+i})=D^{t+i}(x^{t+i},y^{t+i})D^{t+i-1}(\gamma _{xy} x^{t+i},y^{t+i}) \\ \Leftrightarrow D^{t+i}(\beta x^{t+i},\beta y^{t+i})D^{t+i-1}(\beta \gamma _{xy} x^{t+i},\beta y^{t+i})=1\times 1=1 \\ \end{array} $$

(17)

Since (x ^t+i,y ^t+i)L ^t+i exhibits IRS in

$$ \mathop {\lim }\limits_{\beta \to 1} D^{t+i}(\beta x^{t+i},\beta y^{t+i}) > 1\\ $$

The production frontiers for L ^t+i and L ^t+i−1_γ coincide at (x ^t+i,y ^t+i). This point of intersection between the frontiers can either represent a crossover point (i.e. the frontier for L ^t+i is above/below the frontier for L ^t+i−1_γ to the left and below/above to the right), or a non-crossover point.

Case (i) crossover

If the crossover for the frontier of L ^t+i is from below to above, choose β = 1 − h, where h is a very small positive number. At slightly lower output levels than y ^t+i, the frontier for L ^t+i−1_γ dominates that of L ^t+i, i.e. requires fewer inputs. Hence:

$$ D^{t+i-1}(\beta \gamma _{xy} x^{t+i},\beta y^{t+i})\ge D^{t+i}(\beta x^{t+i},\beta y^{t+i}) > 1 $$

which leads to a contradiction in Eq. (17). If the crossover is from above to below, choose β = 1 +  h.

Case (ii) non-crossover

If the frontiers do not crossover, then the elasticities (slopes) at (x ^t+i,y ^t+i), for fixed input and output mixes, are the same, i.e.

$$ \mathop {\lim }\limits_{\beta \to 1} D^{t+i-1}(\beta \gamma _{xy} x^{t+i},\beta y^{t+i})=\mathop {\lim }\limits_{\beta \to 1} D^{t+i}(\beta x^{t+i},\beta y^{t+i}) > 1 $$

This again leads to a contradiction in Eq. (17). Thus for the global adjacent productivity change index to satisfy the proportionality test of outputs in the general case, the production technology must exhibit constant returns to scale.□

Proposition 14

The global base period productivity change index satisfies the proportionality test of outputs if and only if the technology exhibits constant returns to scale.

Proof 14

Assuming first that the technology satisfies constant returns to scale:

$$ \begin{array}{l} \tilde {M}^{\rm G}(t,t+i;(\overline X,(Y^1,\ldots,Y^t,\ldots,Y^{t+i-1},\beta Y^{t+i},Y^{t+i+1},\ldots,Y^T)))= \\ =\left({\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},\beta y_k^{t+i})}} } \right)^{1/(K\times T)} =\left( {\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{\frac{1}{\beta }D^\tau (x_k^{t+i},y_k^{t+i})}} } \right)^{1/(K\times T)} \\ =\left({\beta ^{KT}\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},y_k^{t+i} )}} } \right)^{1/(K\times T)}=\beta {}\times \tilde {M}^{\rm G}(t,t+i;(\overline X,\overline Y)) \\ \end{array} $$

To prove that the global base period productivity change index only satisfies the proportionality test if the technology exhibits constant returns to scale:

$$ \begin{array}{l} \tilde {M}^{\rm G}(t,t+i;(\overline X,(Y^1,\ldots,Y^t,\ldots,Y^{t+i-1},\beta Y^{t+i},Y^{t+i+1},\ldots,Y^T)))=\beta \tilde {M}^{\rm G}(t,t+i;(\overline X ,\overline Y)) \\ \Leftrightarrow \left({\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},\beta y_k^{t+i})}} } \right)^{1/(K\times T)}=\beta {}\times \left( {\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {\frac{D^\tau (x_k^t,y_k^t)}{D^\tau (x_k^{t+i},y_k^{t+i} )}} } \right)^{1 /(K\times T)} \\ \Leftrightarrow \prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^\tau (x_k^{t+i},y_k^{t+i})} =\beta^{{K\times} T}{}\times \prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^\tau (x_k^{t+i},\beta y_k^{t+i})} \\ \Leftrightarrow \prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^\tau (x_k^{t+i},y_k^{t+i})} =\prod\limits_{\begin{array}{l} k=1,\ldots,K \\ \tau =1,\ldots,T \\ \end{array}} {D^\tau (\beta x_k^{t+i},\beta y_k^{t+i})} \\ \end{array} $$

Letting this be Eq. (16) the rest of the proof is now similar to Proof 13.□