Efficiency and Productivity Measurement:
Multi-output Distance and Cost functions
D.S. Prasada Rao
School of Economics
The University of Queensland Australia
1
Duality and Cost Functions
• So far we have been working with the production
technology and production function – this is known as the
primal approach.
• Instead we could recover all the information on the
production function by observing the cost, revenue or
profit maximising behaviour of the firms.
• This is known as the dual approach to the study of
production technology.
• In our lectures, we have looked at the possibility of
describing the technology using input and output distance
functions.
• Here we will focus on cost functions – see the textbook for
revenue and profit functions (Chapter 2).
2
Duality and Cost Functions
• The cost function is defined as:
c ( w , q )  m in
x
 w x
such that ( q , x )  S 
C.1
Nonnegativity:
Costs can never be negative.
C.2
Nondecreasing in w:
An increase in input prices will not decrease costs.
More formally, if w 0  w 1 then c ( w 0 , q )  c ( w 1 , q ).
C.3
Nondecreasing in q:
It costs more to produce more output. That is, if
0
1
0
1
q  q then c ( w , q )  c ( w , q ).
C.4
Homogeneity:
Multiplying all input prices by an amount k> 0 will
cause a kfold increase in costs (eg., doubling all
input prices will double cost). Mathematically,
c(kw, q) = kc(w, q) for k > 0.
C.5
Concave in w:
c ( w  (1   ) w , q )   c ( w , q )  (1   ) c ( w , q )
for all 0    1.
This statement is not very
0
1
0
1
intuitive. However, an important implication of the
property is that input demand functions cannot slope
upwards.
3
Duality and Cost Functions
• When dealing with multi-output and mlti-input
technologies, we can use the cost-function to derive the
input demand functions. From Shepherd’s Lemma, we
have:
xn ( w , q ) 
c (w , q )
wn
.
• The input demand function satisfies all the expected
properties: (i) non-negativity; (ii) non-increasing in w;
(iii)non-decreasing in q; (iv) homogeneous of degree zero
in input prices; and (v) Symmetry
• Estimation of cost function – either as a single equation or
a system of equations including input-share equations.
4
Cost frontiers
• Advantages:
– captures allocative efficiency
– can accommodate multiple outputs
– suits case where input prices exogenous and
input quantities endogenous
– suits case where input quantity data
unavailable
• Disadvantages:
– requires sample input price data that varies
– biased if frontier firms are not cost minimisers
5
Estimation of cost frontier
• Before we examine the estimation of multi-output and
multi-input distance function, we briefly consider the
estimation of cost-frontier.
ci ≥ c(w1i, w2i, …, wNi, q1i, q2i, …, qMi)
• If we assume that the cost function is modelled using
Cobb-Douglas functional form (we use this since it is
possible to decompose cost efficiency into technical and
allocative efficiency. We use
N
ln c i   0 

n 1
M
n
ln w ni 

m
ln q m i  v i  u i
m 1
where ui is a non-negative random variable representing
inefficiency.
6
Estimation of cost frontier
N
• Imposing linear homogeneity in input prices:   1. and
re-writing the model after imposing this constraint we
have:
n
n 1
ln  c i w N i    0 
N 1

n 1
n
ln  w ni w N i  
M

m
ln q m i  v i  u i .
m 1
which can be written in a standard frontier model as:
ln  c i w N i   x i β  v i  u i
 ln  c i w N i    x i β  v i  u i .
• This model can be estimated using the standard frontier
methods and the Frontier program.
7
Cost efficiency and decomposition
• Cost efficiency is measured in a way similar to what we
did for technical efficiency using
C E i  exp(  u i ).
and other formulae to find firm-specific efficiencies.
• Decomposition of Cost Efficiency:
– If we have input quantities or cost shares, cost efficiency can be
decomposed into technical and allocative efficiency components.
– In this case it is possible to model a system of cost-share equations
for different inputs
– The cost frontier has both allocative and technical efficiency
combined and share equations have information on allocative
efficiency but the relationships between these is quite complex
– A simpler approach is possible in the Cobb-Douglas since in this
case both the production function and cost function have the same
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functional form.
Cost efficiency (CE) decomposition
• Translog is difficult - because the function is not
self-dual. In this case the options are:
– solve a non-linear optimisation problem for
each observation to decompose CE
– estimate a system of equations
• Important references:
– Kumbhakar (1997)
– Kumbhakar and Lovell (2000)
9
SFA model as a basis for estimating
distance functions
Suppose we want to estimate the input distance function di(xi,
qi) and suppose we assume that the distance function is of loglinear form, then
N
ln d
I
i
 0 

M
n
ln x ni 
n 1

m
ln q m i  v i
m 1
The main problem in estimating this model is that the distance
function is unobserved. But we know that:
• the distance function is non-decreasing, linearly homogeneous and
concave in inputs; and
• non-increasing and quasi-concave in outputs.
• Linear homogeneity in inputs gives us the condition
N

n
 1.
n 1
• Then the model can be rewritten as
N 1
ln x N i   0 

n 1
 n ln  x ni x N i  
M
I
  m ln q m i  v i  u i u i  ln d i
m 1
TE i 
1
I
di
 exp(  u i ).
10
SFA model as a basis for estimating
distance functions
There are several issues that need further consideration and
resolution:
• It is possible that the explanatory variables may be correlated
with the composite error term – this can lead to inconsistent
estimators.
• It may be necessary to use an instrumental variable
framework.
•Coelli (2000) argues that in the case of Cobb-Douglas and
translog specifications, this is not an issue provided revenue
maximisation or cost minimisation behaviour is assumed.
• The distance functions need to satisfy the concavity or quasiconcavity properties implied by economic theory. Otherwise, it
may lead to strange results.
• This requires a Bayesian approach – see O’Donnell and Coelli
(2005).
•Coelli and Perleman (1999) makes a comparison of parametric
and non-parametric approaches to distance functions.
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