Proceedings of Applied International Business Conference 2008
MODELING DISTORTION EFFECT IN SCALE ECONOMIES
Sohail Zafar ψ
International Islamic University, Islamabad, Pakistan
Abstract
Pakistan’s banking industry has gone through remarkable transformation on account of privatization and
restricting of state owned banks foreign and domestic private banks since 1991 it matches that of many
other developing countries. The financial reforms in Pakistan envoy the structural change, it turns out that
other developing countries may be compelled to carry out in order to liberalize their banking sector to cope
with the challenges of globalization. Factor markets often produce scale inefficiencies in the banking of
developing countries like Pakistan such inefficiencies are common due to distortion in factor markets
leading to the use of inappropriate factor proportions.
Keywords: Scale economies; Shadow cost; Actual cost; Distortion effect.
JEL Classification Codes:G21; D57; D61.
1. Introduction
Lack of effective competition in factor markets often produces scale inefficiencies in banking sector of the
developing countries like Pakistan. Such inefficiencies are common due to distortion in factor markets
leading to the use of inappropriate factor proportions Nisar A (1999) Burki et al (1997) Burki and
Mehmood (2004) Burki and Shaabir (2007) Isik and Hassan (2002) Kwan (2006) Patti and Hardy (2005)
Ataullah et al (2004) Havrylchyk (2006). Pakistan is also one of the countries where labor is abundant but
capital and raw material are scarce. Distortions in the factor markets in the banking sector of Pakistan
produce inefficiencies leading to biased economies.
In view of this fundamental restructuring of the banking sector in terms of organization and production a
central question is the degree to which these banks enjoy economies of scale. The presence of scale
economies mean large banks have a cost advantage over small ones, while evidence of economies of scope
implies that multi product banks are more efficient than financial boutiques, eos in banking sector has been
studied extensively in the US and Canada and other developed countries but relatively few papers have
focused on banks in developing countries.
As indicated by Boumol et al (1982) there are two types of production economies that may be achieved by
any firm in any industry eos and scope, eos are associated with firm size and exist if average production
cost decline as output rises. Conversely a firm exhibits diseconomies of scale if average production cost
increases with output.
Two issues arise in assessment of whether a bank enjoys eos. The first relates to measurement the second is
concerned with estimation. There are two types of measure standard and expansion path. The assumptions
associated with the standard measures are very restrictive. Banks must have identical product mixes and
specialize completely and increase one output while holding the other constant. However banks rarely if
ever satisfy criteria. Berger et al (1987) addressed these problems by using expansion path measures which
capture the impact of changing scale and product mix simultaneously.
Return to scale is a property explains a proportionate change in inputs and the resulting change in output
Allen and Liu (2005) found increasing return to scale in models of large Canadian banks. Typically results
ψ
Corresponding author. Sohail Zafar. Department of Business Administration, Faculty of Management
Sciences, International Islamic University, Sector H-10, Islamabad, Pakistan. Email: [email protected]
Proceedings of Applied International Business Conference 2008
show eos for small banks and economies of scope for all banks Mertens and Urga (2001) Allen and Rai
(1996) Mester (1996) found small banks have significant economies.
Recently more flexible functional forms have been developed and used with in parametric and
nonparametric framework see figure 1. One of the most common formulations is the translog form, which
is quadratic in nature second order tailor series approximated. It has linear and nonlinear output terms, like
Cobb Douglas, but an addition of squared terms of output. Berndt and Christensen (1973) develops an
empirical framework for using the translog function to investigate the permissibility of aggregating diverse
among the input factors in U.S banking.
Figure1: Functional forms in measuring EOS
Measuring EOS
Frontier
Approach
Parametric
Estimation
Stochastic
Frontier
Non Frontier
Approach
Non Parametric
Estimation
DEA
Parametric
Estimation
Non
Parametric
Estimation
Average
Response
Function Time
Invariant
Translog
Divisa
Index
Bayesian
Approach
It turns out translog form can estimate a U-shaped cost curve reflected by data. If a U-shaped cost curve
will estimate, eos at smaller firms and diseconomies at larger ones Lawrence and Sahy (1986) or Benston et
al (1982) Humphery (1990) found bank cost curves are weakly U-shaped. Translog cost function is derived
from second order tailor series approximation translog form estimates well known U-shaped cost curve
reflected by data. U-shaped cost curve will estimate, scale efficiencies at smaller firms and diseconomies at
larger ones.
Distortion component in cost function capture variation of scale magnitude across different size of firms
studies such as Lawrence and Shay (1986) and Benston et al (1982) Humphery (1990) found bank cost
curves are weakly U-shaped, eos existed in banking seem to be limited relatively in comparison to smaller
banks. eos appear as an average cost curve indicating low costs varying with output.
2. The model
We use cost function, which is more suitable than production function due to several reasons. First the cost
function expresses minimal total cost in terms of factor prices and the level of output while the production
function expresses output in terms of factor inputs. Factor price is treated as exogenous in both the cost and
production function. The second reason for using cost function is that it yields direct estimates of AllenUzawa elasticities of substitution. These parameters are the key to describe the pattern and degree of
substitutability and complementarity among the factors of production. The third advantage of the cost
function approach is that duality theory can be exploited without imposing any restriction on the returns to
scale in the underlying technology for example by Shepherd’s lemma. We start with simple translog cost
function assuming that the classical assumption of perfect competition in input market holds that is there is
no allocative inefficiencies. The translog cost function is obtained as the second order taylor series
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Proceedings of Applied International Business Conference 2008
approximation. The function is highly flexible it allows the scale economies and input proportions to vary
with the level of output. For the translog cost function we denote factor prices by p and their respective
quantities by x and the level of output by y while total cost is denoted by c given by:
c=
∑p x
(1)
i i
Symbolically the translog cost function can be written as follows:
ln c = α o +
n
∑α ln p
i
i =1
n
i
+ 1/ 2
n
∑∑ β
ij
ln pi ln p j + α y ln y + 1 / 2β y (ln y ) 2 +
i =1 j =1
n
∑β
iy
ln pi ln y
(2)
i =1
As a bank may not be able to have the optimal combination of inputs due to imperfections in decision
making and imposition of distortionary government regulations. The role of these potential sources of
misspecifications behavior can be incorporated by introducing the concept of shadow prices instead of
actual prices, using either production or cost function approach assuming imperfection results in Nisar A
(1999) Burki et al (1997) an opportunity cost of the input that differs from its market price i.e. assuming
distortion function Ri(xi,wi)=ciwixi.
The opportunity cost of the input that differs from its market price i.e. assuming distortion function as nonzero rows pi , pi −1 , pi −2 ,.... p1 of a system are linearly dependent it means one of the rows say p j is linear
combination of the preceding ones i.e.
p j = τ j +1 p j +1 + τ j + 2 p j + 2 + ....... + τ i pi
i≠ j
(3)
where τ i is the distortion in factor prices the kth component of p j is its first nonzero element hence kth
components of the system p j +1 , p j + 2 ,....., pi are all zero kth component of p j is given by:
τ j +1 ( 0) + τ j + 2 ( 0) + .......τ i ( 0) = 0
(4)
contrary to the assumption that kth component of p j is nonzero. The concept of shadow prices here after
represented by the following equation pi* = f (τ i pi ) The firms dual total shadow cost function is defined as
c s = c s (τp, y ) . Where τp is the vector of input specific shadow price. We can derive actual input demand
function with the help of shadow cost function by applying Shepherd’s lemma.
∂c s
= xi
∂τ i pi
(5)
total actual cot is given by:
ca =
n
∑p x
(6)
i i
i =1
To incorporate the shadow prices in the function we substitute the values of input demand obtained by
shadow cost function given by:
ca =
n
∑
i =1
pi
∂c s
∂τ i pi
(7)
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Proceedings of Applied International Business Conference 2008
The shadow translog cost function can be written as:
ln c s = α 0 +
n
∑
α i ln(τ i pi ) + 1 / 2
i =1
n
+
∑β
ij
n
n
∑∑ β
ij
ln(τ i p i ) ln(τ j p j )
i =1 j =1
(8)
ln(τ i y i ) ln y + α y ln y + 1 / 2 β yy (ln y ) 2
i =1
Symmetry condition require that β ij = β ji . Monotonicity of the shadow cost function entail that when all
shadow prices increase shadow cost also increase. Homogeneity of the shadow cost function requires the
following restriction:
n
∑
n
αi = 1 ,
∑
β ij =
i =1
i =1
n
∑
β ji = 0 ,
i =1
n
∑β
iy
=0
(9)
i =1
Differentiation of equation yields parametric expression for the shadow cost shares i.e.
τ p ∂c s
τ px
∂ ln c s
= i si
= i si i = ς is
∂ ln(τ i pi )
c ∂τ i pi
c
ς is = α i +
n
∑β
ij
(10)
ln(τ j p j ) + β iy ln y
(11)
i =1
The curvature condition for the cost function requires Hessian matrix H, of the second order partial
derivates with respect to factor prices, should be negative semi definite. The share equation of the shadow
cost function can be written as:
ς is =
τ i pi x i
(12)
cs
Solving above for x i we have:
xi = ς is c s (τ i p i ) −1
c a = cs
n
∑τ
(13)
−1 s
i ςi
(14)
i =1
Actual cost function is given by:
ln c a = ln c s + ln
n
∑ (τ
−1 s
i ςi )
(15)
i =1
substitution yield the actual cost function:
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Proceedings of Applied International Business Conference 2008
ln c a = α 0 +
n
∑
α i ln(τ i pi ) + 1 / 2
i =1
n
n
∑∑
β ij ln(τ i pi ) ln(τ j p j ) +
i =1 j =1
2
+ 1 / 2 β yy (ln y ) + ln(
n
n
∑β
iy
ln(τ i pi ) ln y + +α y ln y
i =1
∑
τ i−1 (α i
n
+
i =1
∑β
ij
(16)
ln(τ i pi ) + β iy ln y ))
j =1
In above expression if τ i = τ j firm’s total actual cost function reduces to total shadow cast function share
equations of the shadow cost function are defined as:
(α i +
ς ia =
n
∑β
ij
ln(τ i pi ) + β iy ln y )τ i−1
j =1
n
n
∑ (α + ∑ β
i
i =1
(17)
ij
ln(τ i pi ) + β iy ln
y )τ i−1
j =1
Elasticity of cost with respect to proportional change infer scale inefficiencies of outputs holding the input
prices unchanged, differentiating the translog cost function with respect to output we get eos.
n
a
= β y + β yy ln y +
η cy
n
∑β
i =1
iyτ i
ln pi +
∑β τ
i =1
n
∑
−1
ij i
(18)
ς isτ i−1
i =1
a −1
ξ a = 1 − (ηcy
)
(19)
3. Conclusion
The initial specification of conditions included, perfectly functioning markets. This assumption is useful,
simply because it allows us to concentrate on distribution questions given otherwise efficient production. It
turns out if this condition is relaxed two questions must be considered. First can a government role be
defined with respect to the treatment of these imperfections? Second does the existence of these
imperfections change. The fourier functional form, seem very much worth further investigation, the issue of
whether the modeling of distortion would exploit strength or share the weakness.
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