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CHAPTERS
ESTIMATES OF PRODUCTION FUNCTION AND RETURNS TO SCALE
Chapter 5
ChapterS
ESTIMATES OF PRODUCTION FUNCTION AND RETURNS TO SCALE
5.1 Introduction
The objective of this chapter is to estimate alternative specifications of production
functions in the banking industty in India over the period 1985 to 1995-96. Production function
can be estimated by imposing the restriction of Constant Returns to Scale (CRS). If the
production is characterised by non-constant returns to scale - increasing returns to scale
(decreasing returns to scale) then larger banks will appear more (less) efficient. This problem
can be overcome by estimating a production fi.mction, which allows for increasing returns to
scale (decreasing returns to scale).
The plan of this chapter is as follows. The next section discusses the choice of
functional form in the production analysis. Section 5.3 presents the concepts of returns to scale.
The fourth section specifies and estimates the production function Section 5.5 presents and
analysed the empirical results both for the Translog production function and Cobb-Douglas
production function. The last section summarises the findings.
5.2 Choice of Functional Form in the Production Analysis
There are at least two approaches by which we can find the productivity, efficiency,
economies of scale, etc. of an establishment or industty. They are - i) econometric approach and
ii) mathematical prograrruning analysis. Each one has certain advantages and limitations. In
mathematical prograrruning approach, though no explicit functional forms need to be imposed,
the calculations may be warped if the data are contaminated by statistical noise (Bauer, 1990).
Again, no test can be made of how well the production function fits the data because the
estimates resulting from the mathematical programming techniques do not have known
statistical properties. On the other hand, econometric approach removes the above problems
and therefore, we proceed with the econometric approach for our study.
Though, from duality theoty, cost function and production function represent the same
technology, the choice between them is a matter of analytical objective and statistical
convenience. The main contention is whether the level of output should be considered
endogenous or exogenous. Our decision to estimate production function rather than cost
function is based on the following considerations: i) Production maximisation appears to be a
reasonable assumption in case of Indian banking industty which is a service industty. In India it
74
Chapter5
provides social service, the aim being service delivery and universality of service. These services
are required to be done by the public sector banks as well as private banks including the foreign
banks. As far as deposits and advances, that is, businesses are concerned, the Indian banking
industry is dominated by the public sector banks. In the process of providing services any deficit
incurred by them (public sector banks) would be absorbed by the government, 1•2 ii) Banking is
a regulated service industry where defining the variables (output, inputs, etc.) and getting the
data of those variables are problematic. Estimation of cost fimction is more demanding in terms
of data requirement as compared to the estimation of production fimction This is because, for
the estimation of cost fimction we require to obtain the data (construct the variable) of the
prices of factor inputs besides the prices of output, iii) It may be argued that defining output is
controversial in a banking industry. However, there has been a long debated problem regarding
the definition and measurement of cost of a bank. A controversy arises about the treatment of
interest expense. Under the 'production approach' for the measurement of output and cost,
interest expenses are excluded from the cost whereas under the 'intermediation approach' cost
is defined to include both interest and other operating expense3.
Choice of Functional Form
A production fimction represents the maximum output that can be produced from a
given set of inputs and production technology. In general, we suppose that output y of a
production process depends on the input x, i.e., y = f(x), where x is a vector of input. For
empirical analysis it is necessary to specifY the fimctional form of the production process which
meets the economically reasonable restrictions4. In the literature on production analysis, one
can find the different forms of production fimctions. Among the various forms available, three
forms are widely used for the empirical estimation. They are i) Cobb-Douglas (CD) fimction, ii)
Constant Elasticity of Substitution (CES) fimction, and iii) Transcendental Logarithmic
1 It may, however, be argued that this is not a reasonable asswnption for banking industry in India because it may focus on the cost (deficit) minimisation by asswning that input prices are exogenous.
2 In his analysis of non-structural test of competition in Indian banking, Subramaniyam (1995, p.31 ), said that the period 1985-87 and 1990-91 led to a market environment of maximising (business maximising) behaviour on the part of the bank business.
3 It may be argued that definition and measurement of banking output are also controversial. But by using the production function approach we avoid another controversy.
4 For a discussion on the general principles to be satisfied by a production function, see Fuss, McFadden and Mundlak (1978).
75
Chapter 5
(Translog) function. Lau (1986) has proposed that any functional form needs to be evaluated
on the basis of the criteria of theoretical consistency, domain of applicability, flexibility, factual
conformity and computational facility. It has been shown that since all the above conditions are
not met by any ftmctional form, trade-o:ffs have to be made.
i) CD function: It has the following form
Y = Akalf:l
A> 0, 0 <a< 1, 0 < f3 < 1
where y is output, k is capital input, 1 is labour input, A is efficiency parameter, a and f3 are the
parameters representing partial elasticities of capital and labour respectively and (a+ f3) implies
the elasticity of scale. For example, if a+ f3 = 1, there is constant returns to scale. This fi.mction
is homogenous of degree (a+ f3). The elasticity of substitution parameter cr is constant and is
restricted to be unity for this fi.mction Its isoquants are negatively sloped throughout and
strictly convex for positive values of k and I and it is strictly quasi-concave for positive k and I
(Chiang, 1984).
Though this ftmction is convenient for estimation since it is linear in the parameter A, a
and f3, however, it imposes serious and unrealistic restrictions on the production process (e.g.,
elasticity of substitution between each pair of factors must be identically one) because it is
desirable to have a functional form for a production fi.mction that places fewer restrictions on
the nature of the technology.
ii) The CES fi.mction relaxes the assumption of unitary elasticity of substitution and
restricts pairwise elasticities to be constant and equal for all input levels. The form of CES
function is as follows:
y =A [8 k"P + o -8) 1-pr/p
A> 0, 0 < 8 < 1, p ~ -1, v > 0
where A is efficiency parameter, 8 is the labour intensity (distribution) parameter, p is
substitution parameter, v is the degree of homogeneity (returns to scale) parameter. Here,
elasticity of substitution, cr = 1/ (1 + p ), which is constant but not necessarily equal to one. As p
varies from -1 to infinity, cr should lie between zero and infinity.
The CES function is more general than CD fi.mction in the sense that the latter is a
special case of the former with the elasticity of substitution equal to one (p = 0 or cr = 1 ). If cr
equals to infinity then it reduces to linear production fi.mction and when cr equals to zero then it
corresponds to Leontief production fi.mction.
76
Chapter 5
The problem with the CES function is that its estimation is difficult unlike the CD
function because it is intrinsically non-linear. Thus to estimate the function directly a non-linear
estimation procedure is required. One way out suggested by Kmenta (1967) is the Taylor series
logarithmic approximation around p = 0. His approximation to the CES function can be written
as:
Logy := ao + a1 log I + a2log k + a3 log (Vk)2
where ao =log A, a1 = v.8, a2 = v (1 - 8), aa = -112 p.v.8 (1 - 8)
The term aa log (llki accounts for non-unitary elasticity of substitution The closer the
elasticity of substitution to unity, the better the approximation. It can be checked that when in a
special case, p = 0, it reduces to CD fimction
Another approach, which is not direct, to the estimation of CES function, is through
SMAC relation. It assumes perfect competition in both product and factor market, and constant
returns to scale.
Both CD and CES functions are restrictive functional forms as they restrict returns to
scale and elasticity of substitution co-efficient to remain invariant over input points.
iii) Translog function5: There are many flexible functional forms like Generalised
Leontief, Translog, Quadratic, Generalised Box-Cox, Generalised CD, Generalised Square
Root Quadratic, etc. Translog function has larger domain of theoretical consistencl. However,
Translog function is not well behaved globally, except under stringent conditions. This function
is flexible in the sense it imposes a few a priori restrictions on the scale and substitution
properties of the underlying technology. The elasticity of scale parameter and the Allen Partial
Elasticities of Substitution (APES) could be estimated at each data point (Allen, 1938). It
represents a production structure by functions that are quadratic in log of input quantities. It is a
flexible form of production function in the sense that many of the technical characteristics,
which it summarises, are themselves functions of the levels of inputs and thus vary from
observation to observation Constant returns to scale and separability can be imposed by
testable restrictions on the parameters. It is theoretically consistent and computationally
tractable. This function is more general and it enables us to test for CD form and Kmenta's
approximation to CES form (Berndt and Christensen, 1973). Thus the Translog function is
theoretically consistent, flexible and parsimonious in parameters. It does not make a priori
5 Translog production function is developed by Christensen, Jorgenson and Lau (1971, 1973).
6 See Lau (1986).
77
Chapter 5
assumptions regarding separability, substitution and transformation, returns to scale,
homogeneity, homotheticity of input structure, neutrality of technical change. However, it may
not be well behaved because monotonocity and concavity (isoquants are convex) may not
always be satisfied. The general form ofTranslog function withy as output and Xi as n inputs is
as follows:
Log y = cx0 + L <Xi log Xi + Y2 L L f3ij log xi log Xj j
f3ij = f3ji, the symmetric condition; ~ j = 1, 2, ... , n
If f3ij = 0, then it is multi-input CD function and thus CD fimction is a special case of the
Translog production function. It can be shown that this function allows for variable scale
elasticity and elasticity of substitution If L f3ij = 0, for j = 1, 2, ... , n, then the production
function is homothetic. It is linear homogeneous if L <Xi= 1 and L, f3ij = 0, for j = 1, 2, ... , n. I I
A production function is considered to be well-behaved if it has a positive marginal
product for each input [monotonicity (8y/8Xi > 0)], i.e., if output increases monotonically with
all inputs and if its isoquants are convex (Berndt and Christensen, 1973), i.e., if production
function is quasi-concave which requires that bordered Hessian matrix of the function to be
negative semi-definite. The CD and CES functions are globally well behaved when the
parameters satisfY the usual restrictions. However, the Translog production function is not
globally well behaved as it does not satisfY the above two conditions at all points in the input
space. If we can find sufficient number of points in the input space, where the restrictions are
satisfied then the Translog production function is considered as well behaved. Hence,
monotonocity and quasi-concavity have to be verified at each data point before estimating the
technical co-efficient.
So after discussing the different advantages and limitations of CD, CES and Translog
functional forms, our choice is in favour of Translog function. However, there is not much
difference between the CD and CES production functions. First, we will try to estimate the
Translog production function for the banking industry in India for the period 1985 to 1995-96
and check whether the function satisfies the requirements of monotonicity and quasi-concavity.
If it does not fulfil the required conditions, CD production function can then be estimated for
the banking industry in India using panel data.
78
Chapter 5
5.3 Concepts ofRetums to Scale
Returns to scale is a micro concept applicable at plant/firm level. Nevertheless, the
concept is generally estimated in the empirical studies at a highly aggregative leveL that is, at
industry level across firms.
The concept of returns to scale refers to the variation in the quantity of output resulting
from an equiproportional variation of all inputs. It is measured by the coefficient of returns to
scale which is defined as the elasticity of output with respect to any of the inputs when all inputs
change in the same proportion from a given input point. Mathematically, we can write it as
e = L a In y ...... ~. (i) ; 8lnX;
where, e = returns to scale coefficient,
Y = output quantity,
xi = quantity of i-th input
The proportional variation in all inputs leads to a change in the scale of production
leaving the relative input combinations unchanged and can be written as
X~ = X~ = ____ = X~ . . . . . . . . . . . (ii) XI x2 xn h X 0 X 0 X 0 -· .• W ere I , 2 ,. . • • . n - IDpUt quantities,
XI' X 2 , .•. X n =Increments in input quantities.
The above equation gives a straight line through the origin in n dimensional input space
and this line has been called as input ray or 'factor beam' (Frish, 1965, p.69). The returns to
scale coefficient can be measured as the sum of partial elasticities of output with respect to
inputs. Standard production functions like CD and CES implicitly assume that e is invariant to
changes in scale and hence, they are homogenous production functions.
If we define average productivity of an input as the output quantity calculated per unit
of a given input, the elasticity of average productivity of any input with respect to scale can be
derived as (e-1) (Frish, 1965, p.71). Accordingly, average productivities of all factors increase,
remain constant or decrease depending on whether e >,or=, or< 1. At any given input point, if
e > 1, it means that output quantity changes proportionately more rapidly than changes in input
quantities (or scale factor) and average productivity of all factors increase. This condition is
termed as increasing returns to scale. If e = 1, the proportional increment in output equals the
proportional increment in input quantities and average productivities of all inputs reach their
79
Chapter 5
maximum value and remain constant. This condition is referred to as constant returns to scale.
The output quantity increases less than proportionately to input quantities when e <1 and
average productivities of all inputs decrease at the given input point. This situation is called
decreasing returns to scale.
The estimation of returns to scale coefficient for an industry with its constituent firms as
observations involves firms of different sizes and input combinations. Then the estimated
coefficient may have to be taken with reference to the average size and input combination of the
industry. Thus, if returns to scale coefficient is greater (less) than one or equal to one, we infer
that the specific industry is operating, at its average size and input combination, with increasing
(decreasing) returns to scale or constant returns to scale.
5.4 Specification and Estimation
The concept of returns to scale can be given an unambiguous meaning only at the micro
level. The estimates of the production function based on the banking industry (aggregate) level
data may not correctly show the extent of economies of scale or diseconomies of scale
associated with branch size. For a proper measurement of economies of scale, bank branch
level data should be used for the estimation of production function, because aggregate data
tend to combine economies of branch size with the economies of the size of the market 7.
However, the branch level data for the Indian banking industry are not available. Therefore, the
figures on output and inputs have been divided by the number of standard bank branches to
obtain the average values ofthe variables per standard bank branch, and production functions
are estimated from such data For getting an estimate of returns to scale parameter this appears
to be a better procedure than using aggregate data
We have discussed in chapter two that the proportion of rural and semi-urban branches
to total branches is higher for public sector banks. And even within the public sector banks this
proportion is not the same across the public sector banks. The volumes of business in terms of
deposits and advances are lower in case of rural and semi-urban branches compared to the
urban and metropolitan branches. For instance, one bank with the same number ofbranches but
with a higher proportion of metropolitan branches tends to generate more business and income
than other. Following Subramaniam and Swami (1994) we convert all the bank offices into
7 Goldar (1997) explains \\by plant level data should be used for the estimation of production ftmction in case of manufacturing sector.
80
Chapter 5
urban office equivalence based on their respective business estimates for public, domestic
private and foreign banks. Therefore, with a view to minimise the extent of bias in the
regression estimates, we normalise all the variables in our models after dividing by the
respective standard bank branches8.
Translof!
Now we present the model of the banking industry for the estimation of returns to
scale. Estimates are attempted to derive first from the Translog production :fimction. The model
for the four inputs Translog production ftmction for the banking industry is of the following
form:
In Yit = ao + aK In ~~ + <XF In Fit + <XL In Lit + aE In Eit + PKK (In ~ti + PFF (In Fiti + PLL (In
~+~~~+~~~~~+~~~~~+~~~~~+~~~
(In Lit) + PFE (In FiJ (In EiJ + PLE (In LiJ (In EiJ + Vit
i stands for banks and t for time period 1985 to 1995-96, where a 0 = log A, Y is the output.
The factors of production considered are capital (K), loanable ftmd (F), officer (L) and 'other
employees,' that is clerks and sub-staff (E). The variables are either in nu~t~L or. ~orrected for ---price changes over time. The conditions for constant returns to scale are aK + aF + aL + aE = 1;
~+~+~+~=~~+~+~+~=~~+~+~+~=~~+~+~
+ PLE = 0. Using the estimates of parameters these restrictions can be tested.
We have three different Translog :fimctions depending on the version of the output and
accordingly, the loanable ftmd, we are using. The three different·versions of dependent variable,
i.e., outputs are:
i) Y 1 = gross income per standard branch
ii) Y 2 = earning assets per standard branch, and
iii) Y 3 = earning assets plus deposits per standard branch.
8 The conversion of all bank offices into urban office equivalence are made on the basis of average business (deposits plus advances) estimates of each population group category of branches, i.e., rural, semi-urban, urban and metropolitan branches for public, domestic private and foreign bank branches. In terms of average business for foreign banks one metropolitan branch is equivalent to 5.75 to 6.25 urban branches and one urban branch is equivalent to, on average, 1.75 to 2.25 semi-urban branches. It is in case of public sector banks and domestic private banks, on average, one metropolitan branch is equivalent to 2 to 2.5 urban branches and one urban branch is equivalent to 1.25 to 1.75 semi-urban branches and 4 to 4.5 rural branches respectively. This equivalence is used for arriving at an urban equivalence of all branches for foreign banks and other banks.
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Chapter5
The inputs are capital (K), officer (L), 'other employees' (E), apart from two different types of
loanable fund, per standard branch. The loanable fund, borrowing (F2) corresponds to the
output Y 3 and it is deposit plus borrowing (F 1), otherwise, i.e., when output is either Y 1 or Y l. Method of Estimation
There are two ways by which we can estimate the Translog production function. First,
the direct estimation of it by the OLS method and second, the joint estimation of the Translog
production function along with the factor share equations using some multi-variate estimation
technique. It may be argued that the second method is better than the first since it provides
more efficient estimates of the parameters. However, the derivation of the income share
equation involves the assumption of competitive equilibrium, which implies that factors are paid
according to their marginal products. But these assumptions are not reasonable for the banking
industry in India because of the market imperfections. Therefore, estimation of the production
function jointly with income share equations may yield biased estimate of the parameters.
Hence, OLS have directly been applied for the estimation of Translog production function
instead of joint estimation of production function and factor share equations.
However, it may be noted that the estimation of the production function by the OLS
may yield biased and inconsistent estimates because the disturbance term in the production
function may not be independent ofthe choice of inputs. However, Zellener et al. (1966) have
portrayed that under reasonable assumptions about the disturbance term and the behavioural
relations, the inputs can be shown to be independent of the disturbance term of the production
function. Therefore, direct application ofOLS method gives consistent and unbiased estimates.
5.5 Empirical Results
Because of the panel nature of our data, we have sufficient number of observations. In
this chapter all through the presentation of the estimated results, the t-statistic for each co
efficient, when presented, is in the next columns of the co-efficient. The term "statistical
significance" is used to indicate that the co-efficient is significantly different from zero. In cases
where Durbin-Watson test rejects the hypothesis of zero correlation, the equation is re
estimated correcting for the first order auto-correlation using the Cochrane-Orcutt procedure
for correction.
9 For details of the definitions of variables see Chapter Three.
82
ChapterS
The next subsection deals with the empirical results obtained from the Translog
production function. The results from the Cobb-Douglas production function are presented in
the subsection 5.5.2.
5. 5.1 Translog Production Function
The estimates of the Translog production function for three alternative measures of
output for the banking industry in India for the period 1985 to 1995-96 are presented here in
Tables 5.1-A, -B and -C for the first (Y1), second (Y2) and third (Y3) measures of output
respectively.
Before explaining the above results we first test the regularity conditions which a
Translog function needs to satisfY. Monotonicity requires that OY/oXi > 0, where Y is output
and X is input. Since inputs and output levels are always positive, this equivalent requires that
the logarithmic marginal products, i.e., olnY/olnXi be positive at each data point. Concavity of
the production function requires that the matrices of the second order partial derivatives be
negative semi-definite. Necessary and sufficient conditions for the Hessian matrix to be negative
semi-definite are that the matrices of share elasticities are themselves negative semi-definite.
Since the partial derivatives of the production function depend on the values of inputs and
output as well as on the co-efficients of the estimated production function, both monotonicity
and quasi-concavity may be verified at each data point.
Table5.1-A
OLS Estimate ofthe Translog Production Function for the Banking Industry: 1985 to 1995-96
Dependent Variable: In Y1
Variable Co-efficient 't'- statistic Intercept -2.3004 ** -6.5383 InK -0.0458 -0.3490 In F1 0.6602 ** 3.8333 In L 0.8024 ** 4.9585 In E 0.2844 ** 3.4291 (In K)2 -0.0416 -1.3545 (In F1)" 0.1385 ** 2.6383 (In Lt -0.0166 -0.1728 (In E)2 -0.0267 -1.1775 (In K) (In F1) -0.0330 -0.8653 (InK) (In L) 0.0911 ** 2.6527 (InK) (In E) 0.0191 1.2172 (In F1) (In L) -0.0952 * -1.9970 (In F1) (In E) -0.0122 -0.5268 (In L) (In E) -0.1202 * -2.4682 Adjusted R" = 0.9788 Durbin-Watson Statistic= 2.0494 F-statistic = 2191.319 . .
Note:** and* denote stat1st1cally s1gn1ficant at 1 %and 5% level respectively .
83
Table5.1-B
OLS Estimate of the Translog Production Function for the Banking Industry: 1985 to 1995-96
D d tV . bl I Y epen en ana e: n 2 Variable Co-efficient 't'- statistic Intercept -0.7600- -3.1547 InK 0.0143 0.1634 In F1 1.0914- 9.3872 In L 0.5724- 4.7133 In E 0.1525 * 2.5147 (In K)2 -0.0120 -0.5738 (In FS' 0.0003 0.0091 (In L)2 -0.3118- -4.6825 (In E)" -0.0516- -3.0977 (In K) (In F1) -0.0048 -0.1900 (InK) (In L) 0.0024 0.1049 (InK) (In E) -0.0011 -0.1036 (In F1) (In L) -0.0178 -0.5277 (In F1) (In E) -0.0564- -3.3580 (In L) (In E) 0.0254 0.7347 Adjusted R" = 0.9896 Durbin-Watson Statistic= 2.2554 F-statistic = 4507.58
Note: Same as Table 5.1-A
Table5.1-C
OLS Estimate of the Trans log Production Function for the Banking Industry: 1985 to 1995-1996
Dependent Variable: In Y 3
Variable Co-efficient 't'- statistic Intercept 1.5871- 4.4172 InK 0.3216 ** 3.1338 In F2 -0.0251 -0.5141 In L 0.9156- 3.8967 In E 0.3881- 2.8635 (In K)2 -0.0203 -0.8534 (In F2)" 0.0204- 3.0618 (In L)2 -0.1989 -1.6715 (In E)" -0.1383- 3.7683 (In K) (In F2) -0.0219 * -2.1219 (InK) (In L) -0.0594 -1.6116 (InK) (In E) 0.0103 0.4667 (In F2) (In L) 0.0334 1.6322 (In F2) (In E) -0.0052 -0.4140 (In L) (In E) -0.0251 -0.3974 Adjusted R" = 0.9431 Durbin-Watson Statistic= 2.1874 F-statistic = 788.14
Note: Same as Table 5.1-A
Chapter 5
84
Chapter 5
The Translog production ftmction for three alternative measures of output are evaluated
for monotonicity and the results of the evaluation are reported in Table 5.2.
Table5.2
Tests of Monotonicity for the Four-factor Translog Production Function for Three Alternative Definitions of Output (percentage)
Output measure Observations failing test of monotonicity First measure 38.46 Second measure 48.04 Third measure 30.23
Though the adjusted R2 s are quite high for all the three measures of output, they fail
the check for monotonicity in the neighbourhood represented by the sample. More than 30 per
cent of the data points do not satisfy the monotonicity condition for all the three alternative
outputs and thus the Translog form is not well behaved in the region represented by the sample
of all the three measures of output. As this production ftmction fails to satisfy the monotonicity
condition, we do not go for testing quasi-concavity requirements.
Therefore, we can say the estimated Translog ftmctional form is not sufficiently well
behaved to describe a production ftmction for the banking industry in India during the period
1985 to 1995-96. Any estimates of the elasticity, homotheticity and a number of other
characteristics of production technologies based on the parameters of the Translog production
ftmction would be wholly unreliable. Hence, alternatively we take the Cobb-Douglas form as
the preferred form of the production ftmction for the banking industry.
5.5.2 Cobb-Douglas Production Function
The model for the four inputs CD production ftmction for the banking industry is of the
following kind:
where a 0 = ln A, i stands for banks 1, 2, ... , 65, and t stands for time periods 1985, 1986, ... ,
1995-96. Y is the output and the inputs are capital (K), loanable ftmd (F), officer (L) and 'other
employees' (E). The returns to scale is measured by aK + aF + aL + a£.
We have three different CD production fimctions corresponding to the three alternative
definitions of the output (Y ~, Y 2 and Y 3). We have two versions of loanable fimds - i) F 1 =
deposit plus borrowing, and ii) F 2 = borrowing. When we take Y 3 as output, the borrowing (F2)
is the loanable fimd input, otherwise deposits plus borrowing, i.e., F 1 is the loanable fimd input.
85
Chapter 5
There are two ways by which we can estimate the parameters of the above production
function. Firstly, we can estimate it by fitting directly and secondly, by the marginal productivity
conditions. Due to the regulated nature of the banking industry in India, marginal productivity
conditions do not hold and hence, we estimate it by applying OLS directly.
The estimates of the CD production function for the banking industry in India for the
period 1985 to 1995-96 are presented here in Tables 5.3-A, -B and -C for the first (Y 1), second
(Y 2) and third (Y 3) measures of output respectively.
In all the three sets of results, the regression procedure performs reasonably well. The
adjusted R2 s are uniformly quite high and provide good fit to the data The co-efficients ofthe
input variables are all of expected signs, i.e., positive except for the coefficient of the 'other
employees,' i.e., clerks and sub-staff for the first version of output (gross income), and less than
one.
Table5.3-A
OLS Estimates of the Cobb-Douglas Production Function for the Banking lndustry:1985 to 1995-96
Dependent Variable: In Y1
Variable Co-efficient 't' -statistic Intercept -1.2858 ** -17.5166 InK 0.1773 ** 12.4159 In F1 0.8790** 43.1395 In L. 0.0228 0.9077 In E -0.0376 * -2.4795 Adjusted RL = 0.9770 Durbin-Watson statistic= 2.0460 F-statistic = 6054.34
Note: Same as Table 5.1-A
The co-efficients of officer, and employee other than officer are quite low for the first
two versions of the output. The negative sign ofthe co-efficient ofthe other employees, when
output is gross income, might indicate the fact that the banking industry employs more
employees other than officers (clerk and sub-staft) than they actually need and support the claim
that the banking industry is over-staffed and employees inefficient. Further, there is the
possibility of measurement error, as our measure of "other employees" does not distinguish
between clerk and sub-staff. The deposits plus borrowings inputs have the highest estimated co
efficients (0.88 for the first case and 0.97 for the second case). Moreover, capital and loanable
funds are the only explanatory variables, which are significant at one per cent level of
significance for all the above three cases. When deposits are included as a measure of output
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Chapter 5
(third case), the co-efficients of officers are the highest (0.62) among all the explanatory
variables and significant.
Table5.3-B
OLS Estimates of the Cobb-Douglas Production Function for the Banking Industry: 1985 to 1995-96
Dependent Variable: In Y 2
Variable Co-efficient 't' -statistic Intercept -0.2291** -3.7443 InK 0.0298- 2.7711 In F1 0.9735 ** 60.0578 In L 0.0780 ** 3.6483 In E 0.0209 1.5945 Adjusted R" = 0.9873 Durbin-Watson statistic= 2.2313 F-statistic = 11045.55
Note: Same as Table 5.1-A
Table 5.3-C
OLS Estimates of the Cobb-Douglas Production Function forthe Banking Industry: 1985 to 1995-96
Dependent Variable: In Y3
Variable Co-efficient 't' -statistic Intercept 2.0810- 13.9962 InK 0.3593** 20.1268 In F2 0.0494- 5.8816 In L 0.6150 ** 13.3181 In E 0.1170 ** 3.5380 Adjusted RL = 0.9396 Durbin-Watson statistic = 2.1139 F-statistic = 2218.505
Note: Same as Table 5.1-A
The OLS estimation of the CD production fimction for three alternative de:fini1
output give us the estimates of <XK, <Xp, <XL and <XE. The sum of the estimates of <Xk, <Xp,
<XE is a measure of the degree ofhomogeneity of the production fimction. Thus, returns
are increasing, constant, or decreasing, depending on whether the degree of homoge
greater than one, equal to one, or less than one. It is essential that one should consider
testing of a hypothesis of returns to scale and not merely rely on the estimated co-efficie
test whether the degree of homogeneity is significantly different from unity or that the re1
scale are significantly different from being constant for all the three versions of the outp1
the three cases, observed values, seen in the regression, are statistically different from 1
the null hypotheses (<XK + <Xp +<XL+ <XE = 1) of constant returns to scale are rejected
per cent level of significance. They are found to be significantly above one for all thi
Chapter 5
indicating thereby the presence of economies of scale. The magnitude of economies of scale
coefficient is estimated to range from 0.04 to 0.15 for three alternative variants of output.
Therefore, we can say that the banking industry in India is characterised by increasing returns to
scale for the period 1985 to 1995-96. Our finding of increasing returns to scale in the banking
industry is consistent with the results obtained by Noulas and Ketkar (1996) who reports
majority of public sector banks operate under increasing returns to scale in his study of Indian
public sector banks.
5.6 Summary and Conclusions
In this chapter, we have discussed the choice of functional form in the production
analysis. Here, the properties of the three most commonly used and well known production
functions are stated. Then we have discussed the concepts of returns to scale -- increasing
returns to scale, constant returns to scale and decreasing returns to scale. After that, we have
specified and estimated the four factors - Translog and Cobb-Douglas production function
using three alternative variants of outputs - gross income (Y 1), total earning assets (Y 2) and
total deposits plus earning. assets (Y3) per standard branch at constant price.
Our findings of this chapter may be summarised as follows. The estimated Translog
production function has not satisfied the regularity conditions at most of the data pdint,
therefore, CD production function has been used for estimation. The hypothesis of constant
returns to scale has been rejected for the banking industry in India in all the three variants of
outputs. The magnitude of economies of scale coefficient is estimated to range from 0.04 to
0.15 for three types of output. This indicates the existence of increasing returns to scale in all
the three variants of outputs, implying that the banking industry in India is characterised by
increasing returns to scale.
It is not appropriate to impose constant returns to scale, as our findings are supporting
increasing returns to scale production function. Therefore, to estimate productivity and relative
technical efficiency, we have chosen to impose the unrestricted CD production function on the
underlying production structure of the banking industry in India
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