In this article we show equivalence between the input oriented and output oriented scale elasticity measures for multi-output, multi-input technologies. We show the necessary and sufficient condition for this equivalence. We also provide a Lagrange multiplier (or shadow price) interpretation of the scale elasticity measure.

Valentin Zelenyuk

English

Research Papers in Economics

 
 
 
 
 
 
Centre for Efficiency and Productivity Analysis 
 
 
 
 
 
Working Paper Series 
No. WP05/2011 
 
 
 
 
 
 
A Note on Equivalences in Measuring Returns to Scale in Multi-output-multi-input 
Technologies 
 
Valentin Zelenyuk 
 
 
 
Date: May 2011 
 
 
 
School of Economics 
University of Queensland 
St. Lucia, Qld. 4072 
Australia 
 
 
ISSN No. 1932 - 4398  
A Note on  
Equivalences in Measuring Returns to Scale in  
Multi-output-multi-input Technologies 
 
 
Valentin Zelenyuk
* 
 
Abstract 
 
In  this  article  we  show  equivalence  between  the  input  oriented  and  output  oriented  scale  elasticity 
measures for multi-output, multi-input technologies.  We show the necessary and sufficient condition 
for this equivalence.  We also provide a Lagrange multiplier (or shadow price) interpretation of the scale 
elasticity measure. 
 
Key words:  Scale elasticity, production theory, distance functions. 
 
JEL Code: D20, D24. 
 
 
May 26, 2011 
 
 
 
 
*  School  of  Economics  and  Centre  for  Efficiency  and  Productivity  Analysis,  The  University  of 
Queensland,  Australia,  530,  Colin  Clark  Building  (39),  St  Lucia,  Brisbane  Qld  4072,  AUSTRALIA; 
v.zelenyuk@uq.edu.au; tel: + 61 7 3346 7054;  Definitions 
Let 
N
N x x x )' ,..., ( 1  be a vector of inputs and 
M
M y y y )' ,..., ( 1  be a vector of outputs, while 
the production technology be characterized by the technology set T , 
 
  } : ) , {(
N M M N x from producible is y y x T .        (1) 
 
For a single output case, one can use the production function 
1 :
N f  defined as 
 
} ) , ( : max{ ) ( T y x y x f ,                (2) 
 
to completely characterize the technology set T under fairly mild regularity conditions on T.  In this case, 
a commonly used measure of scale elasticity used in economics is given by  
 
) ( / ) (
) (
) (
ln
) ( ln
) (
'
1 1
x f x x f
x f
x
x
x f x f
x e x
i
N
i i
i .        (3) 
 
where  ) (x f  is the production function defined in (2) that is assumed to be differentiable (at least once) 
and where  ) / ) ( ,..., / ) ( ( ) ( 1
'
N x x x f x x f x f ) its gradient.   
  The definition of scale measure in (3) is designed for a single output case only.  For the multiple 
output case, Färe and Primont (1995), modifying the ideas of Hanoch (1975), Panzar and Willig (1977) 
suggested deriving the scale elasticity measures from the Shephard‟s distance functions. Following them, 
recall that the Shephard‟s (1970) output distance function  } { :
1 M N
o D  is defined as  
 
    } ) / , ( : 0 inf{ ) , ( T y x y x Do ,            (4)  
while the Shephard‟s (1953) input distance function  } { :
1 N M
i D  is defined as 
  
    } ) , / ( : 0 sup{ ) , ( T y x x y Di .            (5) 
 
It  is  well  known  that  under  fairly  mild  conditions  on  technology,  both  functions  possess  useful 
properties,
1 in particular, they both completely characterize technology set T in the sense that 
 
T y x x y D y x D i o ) , ( 1 ) , ( 1 ) , (           (6) 
 
as well as are homogeneous of degree 1 in the second argument, i.e., 
 
) , ( ) , ( y x D t y t x D o o ,   
M N y x t ) , ( ), , 0 ( ,        (7) 
and 
) , ( ) , ( x y D t x t y D i i ,   
M N y x t ) , ( ), , 0 ( .      (8) 
 
Also note that under the constant returns to scale (CRS) technology, the two functions are reciprocals to 
each other, i.e., we have 
  
M N
i o y x x y D y x D ) , ( ), , ( / 1 ) , ( .            (9) 
 
In general, however, if technology is not of CRS type, equality (9) does not hold and, in general, there is 
no explicit equality relationship between the two.   
                                                            
1 In particular, we assume standard regularity axioms in production theory, such as “no free lunch”, “doing nothing is 
possible”, T is closed and the output sets } ) , ( : { : ) ( T y x y x P  , 
N x  are closed, as well as weak disposability 
for all inputs and outputs (see Färe and Primont (1995) for details of these axioms and resulting properties). Following Färe and Primont (1995), one can use a measure of local returns to scale based on one 
of these distance functions.  In particular, the output oriented measure of scale elasticity can be defined by 
 
1 , 1 ln
ln
) , ( y x eo ,  such that     1 ) , ( y x Do .        (10) 
 
Or, alternatively, one could use the input oriented measure of scale elasticity, defined as 
 
1 , 1 ln
ln
) , ( x y ei ,    such that     1 ) , ( x y Di .        (11) 
 
Intuitively, both formulas are trying to measure  scale elasticity by looking at the re lationship between 
equi-proportional  changes  in  all  inputs  with  equi-proportional  changes  in  all  outputs,  but  do  it  by  using 
different characterizations of technology. Specifically, measure (10) is based on the function that focuses 
on the output spaces of T at given input levels, while measure in (11) focuses on the input spaces of T at 
given output levels.  Both functions completely characterize technology, in the sense of (6), but do it in 
different and, in some sense, „orthogonal‟ ways.   
 
Several natural questions arise: „What is a relationship between these two alternative measures of scale 
elasticity?” Are (10) and (11) equivalent? Always? Under what conditions? Indeed, in the light of the fact 
that, in general, the input distance function and output distance function do not have explicit one-to-one 
equality relationship with each other (except for peculiar cases such as CRS) do they still give the same 
information about the scale properties of a given technology, at least under some conditions? If so, what 
are these conditions?  Intuitively, familiar with the subject reader should expect a positive result to these 
questions, yet a proof in the multi-input-multi-output context is not trivial.  It is also not immediately clear,  before  the  proof  is  seen,  what  the  necessary  and  sufficient  conditions  are,  if  any,  for  such 
relationship. Answering these questions is the goal of the next section.  
 
Results 
 
In the theorem below we provide the main result of this paper, showing exact relationship between the 
input  oriented  and  output  oriented  scale  elasticity  measures  for  the  multi-output-multi-input 
technologies, stating required (necessary and sufficient) conditions and then we give an example that 
illustrates a case when the theorem does not hold. 
 
Theorem. Given definitions (1), (4), (5), (10) and (11), standard regularity conditions of production 
theory and assuming that  ) , (
' x y Di y ,  ) , (
' x y Di x  and  ) , (
' y x Do x ,  ) , (
' y x Do y  exist, we have 
 
) , ( / 1 ) , ( x y e y x e i o ,              (12) 
if and only if  
0 ) , (
' y x y Di y    and   0 ) , (
' x y x Do x  .          (13) 
 
Proof.   First of all, note immediately that by using the implicit function theorem, we can rewrite 
definitions in (10) and (11) in a more compact form 
 
x y x D y x e o x o ) , ( ) , (
' ,              (14) 
and    
y x y D x y e i y i ) , ( ) , (
' .              (15)
2 
                                                            
2 To be precise, note that  0 ) , ( M o x D 0  and  N M i x x D 0 0 , ) , ( , as well as  ) , ( y D N o 0  and  
M N i y y D 0 0 , 0 ) , ( , but these peculiar cases are ruled out from our consideration because the definition 
of the output and input scale elasticity measures require that  1 ) , ( y x Do  and  1 ) , ( x y Di , respectively.  
Now, to prove the necessity part, assume that  ) , ( / 1 ) , ( x y e y x e i o , then (14) and (15) imply that 
 
x y x Do x ) , (
' = y x y Di y ) , ( / 1
'  
 
which clearly can hold only if (13) is true. 
To prove the sufficiency part, assume (13) is true. Further, note that, due to (6), we can rewrite the 
output distance function as following:  
 
} 1 ) , / ( : 0 inf{ ) , ( x y D y x D i o .             (16)
   
The resulting Lagrangian function for this optimization problem can be written as  
 
) 1 ) , / ( ( ) , | , ( x y D y x L i ,              (17) 
 
and so the system of equations defined by the first order condition for (17) is given by  
 
  0 ) ) /( 1 ( ) , / ( 1
2 * * '
/
*
*
* y x y D L i y ,          (18) 
and       
0 1 ) , / (
*
*
* x y D L i ,              (19)
     
where  ) , ( ), , (
* * y x y x  are the optimal solutions to (17).  Now, note immediately that from 
(18) it follows that   
] ) /( ) , / ( /[ 1
2 * * '
/
* y x y Di y .               (20) 
 
Also note that since 
* is a solution to optimization problem (17), its value must be equal to unity, and 
therefore (20) reduces to 
 
y x y Di y ) , ( / 1
' * .                 (21) 
 
Furthermore, note that the r.h.s. in (21) is non-zero due to our assumption (13), implying that  0
* , 
i.e., the constraint in (16) is binding at the optimal values, i.e.,  1 ) , ( x y Di . Therefore, according to (15), 
the r.h.s. is just the reciprocal of  ) , ( x y ei , i.e., (21) reduces to 
 
* ) , ( / 1 x y ei .                  (22) 
 
On the other hand, note that the envelope theorem applied to (17), tells us that 
 
) , / ( ) , | , ( ) , (
* ' * * * ' ' x y D y x L y x D i x x o x .         (23) 
 
Now, post-multiplying both sides of (23) by the vector of inputs and by ( -1), and using  again our 
knowledge that  1
* ,  we get 
 
x x y D x y x D i x o x ) , ( ) , (
' * '               (24) 
 Note  that the  l.h.s. in  (24)  is  assumed  to be non-zero according to  (13), and is equal  to  ) , ( y x eo , 
according to (14), since  1 ) , ( y x Do  at the optimum.  Also recall that the input distance function is 
homogeneous of degree one in inputs , as was  stated  in (8),  and  so  the  Euler’s  rule  tells  us  that 
x x y Di x ) , (
' ) , ( x y Di , which as was concluded above, is equal to unity.  Thus, (24) becomes 
 
) , ( y x eo  
*.               (25) 
 
Combining this last result in (24) with (22), we get the desired expression, i.e., 
 
) , ( / 1 ) , ( x y e y x e i o .              (26) 
    Q.E.D.
3 
 
Discussion 
 
Intuitively, as one might expect, the theorem tells us that the two scale elasticity formulas measure the 
same  property  of  technology  equivalently  and  we  can  get  one  from  the  other  by  just  taking  the 
reciprocal, provided that certain condition is satisfied.  Importantly, this condition is the necessary and 
sufficient condition for this equivalence result to hold and it states that, at the points where elasticity is 
measured, the gradients of the input and output distance functions shall not be orthogonal to the output 
and input vectors, respectively.  In turn, this technical requirement ensures not running into situation of 
division by zero (as can be seen among the steps in proof of the theorem). By this condition, we ensure 
that at a given point of measurement, none of the two measures of scale elasticity explodes and does not 
degenerate  to  zero,  and  then  (and  only  then)  they  give  equivalent  information  about  the  scale  of 
technology  at that point.  On the  other hand,  the  necessary and sufficient condition  (13) also have 
                                                            
3 Note that similar proof can be established by starting from the definition of  ) , ( x y Di in (16). economic meaning: it requires that an increase in all inputs by the same proportion shall cause some 
proportional, non-zero change in all outputs and, naturally, it should not be an infinite increase.  In turn, 
this condition also implies that, for the equivalence of the two measures, the technology must be such 
that  1 ) , ( 1 ) , ( x y D y x D i o , at a point where the scale elasticity is measured, as can also be seen 
from the steps of the proof of the theorem.  
 
An example where the theorem above does not hold at some points is i llustrated in Figure 1. From this 
figure, one can see that the main condition of the theorem is violated for some points, e.g., the point 
) , (
o o y x  located on the portion of the frontier orthogonal to the output space. Indeed, for this point we 
have  1 ) , (
o o
o y x D  but  1 ) , (
o o
i x y D  and so  0
* , thus  0 ) , (
' x y x Do x , while  ) , ( x y ei  would be 
undefined. The theorem, however, will still hold at other points   for this technology (or its convex 
analogue) where condition (13) is satisfied. In other words, condition (13) is not a global restriction on 
technology for the theorem to hold but is a local requirement at which elasticity is to be measured and 
this carries important empirical implication, broadening the class of technologies for consideration by an 
applied researcher. 
 
 
                y                         
                     T   
                         
            (x
o,y
o) 
 
 
                           x 
 
Figure 1. Technology where condition (13) is not always true. 
 One of the values of this theorem is the empirical one.  Indeed, to model multi-output-multi-input 
technologies, it became very popular to estimate the Shephard‟s distance functions.
  4 Sometimes the 
output-oriented  Shephard‟s  distance  function  is  used  and  sometimes  the  input-oriented  Shephard‟s 
distance function is used and it is not always clear whether and which results based on these alternative 
characterizations of technology would or should be the same or similar, at least qualitatively.
5  The main 
message  of this paper  outlines  a  more-or-less clear  theoretical  answer  on when such alternative 
approaches to modelling the production process should yield equivalent  (or similar,  due  to some 
statistical noise) results, for the particular case of measuring sc ale elasticity.  In some sense, it is also 
desirable that this answer turns out to be  in terms of necessary and sufficient condition for technology 
(at the point of measurement). 
 
It might be also worth noting that  a side fruit of the proof  of the theorem above is that it provides an 
interesting economic or shadow-price intuition of the scale elasticity measures. Specifically, equation (25) 
tells us that the output oriented scale elasticity is in fact the Lagrange multiplier, that is, it is the shadow price 
of relaxing the constraint, in the optimization problem (16).  Similar argument can be made about the 
input oriented scale elasticity measure being the Lagrange multiplier or the shadow price of relaxing the 
constraint in optimization of the input distance function defined in terms of the constraint on the output 
distance  function,  formulated  analogous  to  (16).  Such  Lagrange  multiplier  interpretation  of  scale 
elasticity measures also gives a simple way of  obtaining and analyzing a measure of scale elasticity, 
without taking derivatives as was required in (10), (11), (14) and (15). 
 
Finally, it should also be noted that similar relationship can be derived for the dual framework—between 
the cost-based and revenue-based scale elasticity measures, given additional assumptions (convexity of 
                                                            
4 E.g., see the fundamental works on estimation of distance functions by Atkinson and Primont (2002), Atkinson, 
Cornwell and Honerkamp (2003), O’Donnell and Coelli (2005), Färe, Grosskopf, Hayes and Margaritis (2008) and 
references cited there in, to mention just a few here. 
5 In fact, for some measurements it is known that results would not be equivalent in general: e.g., efficiency measurement 
would give equivalent results only under the case of a constant return to scale technology—a trivial case for our context. input  and  output  sets,  cost  and  revenue  optimization  and  strictly  positive  input  and  output  prices, 
respectively).  This relationship can be done directly, as we performed above, or indirectly, by using the 
duality equivalence results for the output oriented scale elasticity with the revenue-based scale elasticity 
on one hand and for the input oriented scale elasticity with cost-based scale elasticity measure on the 
other hand, as was derived in Färe, Grosskopf and Lovell (1986) and reinstated in Färe and Primont 
(1995). 
 
 
Acknowledgement:  I would like to thank Rolf Färe, Knox Lovell, Antonio Peyrache, Chris O‟Donnell 
and Prasada Rao for fruitful discussions and valuable feedback to this paper. I remain to be solely 
responsible for the views expressed in this article. 
 
References 
Atkinson,  S.W., Cornwell,  C. and O.  Honerkamp (2003) "Measuring and  decomposing productivity 
change: stochastic distance function estimation versus data envelopment analysis" Journal of Business 
and Economic Statistics 84, pp.284-294. 
 
Atkinson,  S.,  and  D.  Primont  (2002)  “Stochastic  estimation  of  firm  technology,  inefficiency,  and 
productivity growth using shadow cost and distance functions,” Journal of Econometrics 108:2, pp. 203-
225 
 
Färe, R. S. Grosskopf, K. Hayes, D. Margaritis (2008) “Estimating demand with distance functions: 
Parameterization in the primal and dual,” Journal of Econometrics, 147:2, pp. 266-274. 
 
Färe,  R.,  S.  Grosskopf  and  C.A.K.  Lovell  (1986),  “Scale  Economies  and  Duality”  Zeitschrift  für 
Nationalökonomie 46:2, pp. 175-182.  
 
Färe,  R.,  and  D.  Primont  (1995)  Multi-Output  Production  and  Duality:  Theory  and  Applications.  Kluwer 
Academic Publishers, Boston. 
 
Shephard, R.W. (1953) Cost and production functions. Princeton, NJ: Princeton University Press. 
 
Shephard, R.W. (1970) Theory of cost and production functions. Princeton, NJ: Princeton University Press.  
Hanoch, G. (1975) “The Elasticity of Scale and the Shape of Average Costs,” American Economic Review 
65, pp. 492-497. 
 
O'Donnell  J.  and  T.  Coelli  (2005)  “A  Bayesian  Approach  to  Imposing  Curvature  on  Distance 
Functions,” Journal of Econometrics, 126, pp. 493-523. 
 
Panzar, J.C. and R.D. Willig (1977) “Economies of Scale in Multi-Output Production, Quarterly Journal of 
Economics 91, pp. 481-493. 

A Bayesian Approach to Imposing Curvature on Distance Functions,”

Cost and production functions.

Economies of Scale in Multi-Output Production,

Measuring and decomposing productivity change: stochastic distance function estimation versus data envelopment analysis&quot;

Multi-Output Production and Duality: Theory and Applications.

Stochastic estimation of firm technology, inefficiency, and productivity growth using shadow cost and distance functions,”

The Elasticity of Scale and the

Theory of cost and production functions.

A Note on Equivalences in Measuring Returns to Scale in Multi-output-multi-input Technologies

http://www.uq.edu.au/economics/cepa/docs/WP/WP052011.pdf

A Note on Equivalences in Measuring Returns to Scale in Multi-output-multi-input Technologies

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics