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.

81

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

86

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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