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Total Factor Productivity Growth and Its Decomposition

This article considers overall total factor productivity improvement achieved by 68 Indian commercial banks from 1998-99 to 2006-07, and breaks it into its components - technical change, technical efficiency change and scale (efficiency) change factor - using the Data Envelopment Analysis methodology. The results suggest that public sector banks, on an average, adjusted to the changing environment better and improved their performance relative to their counterparts under private and foreign ownership. This finding has important policy implications in that the government should be more cautious in liberalising the Indian banking sector and not blindly invite more foreign players to it.


Total Factor Productivity Growth and Its Decomposition

The Indian Banking Sector during Liberalisation

Anup Kumar Bhandari

This article considers overall total factor productivity improvement achieved by 68 Indian commercial banks from 1998-99 to 2006-07, and breaks it into its components – technical change, technical efficiency change and scale (efficiency) change factor – using the Data Envelopment Analysis methodology. The results suggest that public sector banks, on an average, adjusted to the changing environment better and improved their performance relative to their counterparts under private and foreign ownership. This finding has important policy implications in that the government should be more cautious in liberalising the Indian banking sector and not blindly invite more foreign players to it.

The author would like to thank an anonymous referee for some helpful comments and suggestions on an earlier draft of the paper. The usual disclaimer applies.

Anup Kumar Bhandari ( is with the Centre for Development Studies, Ulloor, Thiruvananthapuram.

1 Introduction

recognition of the need of an effi cient fi nancial sector to promote overall economic development can be traced all the way back to the early 20th century when Joseph Schumpeter (1911) argued in his Theory of Economic Development that scarcity of finance is a serious obstacle to development. Cross-country experience also suggests that the existence of a healthy, efficient and competitive fi nancial sector, which Joseph Stiglitz (1998) termed the “brain” of the economy, is a necessary precondition for rapid economic development. This necessity is more pronounced in the case of backward or so-called developing economies because the opportunity cost of capital is more in them, coupled with underdeveloped financial markets (Smith 1998). Further, inefficiency in financial intermediation carries with it the possibility of misallocation of funds, which could result in more non-performing assets (Barman 2007).

Financial intermediaries such as banks are major players in any financial market, and their overall performance is therefore an important determinant of the performance of the fi nancial sector concerned, in particular, and that of the overall economy, in general. Over time, the banking systems in many developing economies performed poorly, and researchers diagnosed it as a direct consequence of the excessive regulations that were in place. However, the experience with deregulation in the banking sector has been mixed in nature. Empirical studies in the US show that measured cost productivity actually decreased following deregulation (Bauer, Berger and Humphrey 1993; Humphrey and Pulley 1997; Berger and Mester 2001). On the other hand, a study by Chaffai (1997) analysed the deregulation experience in Tunisia and found that total factor productivity (TFP) of banks increased following a liberalisation programme initiated in 1986. However, the rate of technical progress was higher than the rate of productivity growth, implying that the banks, on an average, b ecame less efficient after liberalisation.1 Thus the issue of whether financial deregulation actually helps overall development or sometimes can be so counterproductive as to hinder the process of development may be an interesting subject of debate. The issue becomes more relevant in view of the c ontinuing global financial crisis, which originated in the US mortgage lending market and soon spread to others. As noted by analysts, uncontrolled financial innovations introduced by investment agencies and other banks, as well as by some other

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financial institutions, were one of the m ajor causes of the enhance the quality of credit decisions and facilitate faster
crisis. The objective of the present paper is to study the overall credit delivery.
performance of major Indian commercial banks in the post- However, as pointed out by Barman (2007), two distinct
financial deregulation period through a thorough analysis of phases are discernible in the reform of the Indian banking sys
their TFP growth and its major components. tem. The first phase, 1992-98, can be thought of as a period of
It is useful to briefly recall here the nature of the Indian bank transition from a regulated regime to one in which there was a
ing system at the time financial sector reforms were initiated in gradual adaptation of international standards. The second
the early 1990s. This would facilitate a greater clarity of the ra phase, the post-1998 period, can be considered the “true” post
tionale and basis of reforms. The Indian financial system in the liberalisation period. In this regime, banks were able to enjoy
pre-reform period essentially catered to the needs of planned almost full freedom in pricing their products. In sharp contrast
development in a mixed economy where the government sector to the earlier phase, this regime was perceived as more accom
played a dominant role in economic activity. The strategy of modative towards competition. Further, the entry of new pri
planned economic development required huge expenditure, vate banks and some foreign banks made a signifi cant change
which was met through the government ownership of major in the structure of the Indian banking sector. For one, there has
banks, an automatic monetisation of the fi scal deficit and by been increasing competition among banks (as reflected in their
subjecting the banking sector to large pre-emptions – both in share of expenditure on advertising and publicity as a propor
terms of the statutory holding of government securities (statu tion of total operating cost), and the share of publicly owned
tory liquidity ratio, or SLR) and the administrative direction of banks, though still the largest among the major bank groups,
credit to preferred sectors. Further, a complex structure of ad has been gradually diminishing over time (Table 1). These
ministered interest rates prevailed, guided more by social priori changes necessarily make the individual players more market
ties, necessitating cross-subsidisation to sustain the commercial oriented and call for them to improve their performance. Our
viability of institutions. These not only distorted the interest rate concern in this paper is whether such anticipation holds good
mechanism but also adversely affected development of the for the Indian banking industry in the “true” post-liberalisation
financial market (Rangarajan 2007). period. For that, we have examined TFP changes that have
Contrary to this, financial reforms in India created an ena taken place in the last year we have considered, 2006-07 over
bling environment for banks to overcome external constraints the year 1998-99. We also decompose such TFP changes into its
and operate with greater flexibility. Such measures related to major components such as technical change, change in techni
dismantling the administered structure of interest rates, and cal efficiency of banks and so on to identify the principal driv
the removal of several pre-emptions to do with reserve re ing force(s) of TFP changes in Indian banking over this period.
quirements and credit allocation to certain sectors. Interest Table 1: Some Important Indicators of the Major Indian Commercial
rate deregulation was carried out in stages, allowing suffi - Bank Groups
cient resilience to build up in the system. This was an impor- Important Indicators Year SBI and Its Other Domestic Foreign Associates Nationalised Private Banks
tant component of the reform process, which has made re- Banks Banks
source allocation more efficient. A parallel strengthening of prudential regulation, improved market behaviour, gradual Share in Total Deposits 1999 30.3 58.0 7.2 4.6 2000 30.1 57.0 8.6 4.3 2001 31.2 55.0 9.4 4.5
financial opening and, above all, underlying improvements in 2002 30.5 54.1 11.1 4.4
macro economic management helped the liberalisation proc 2003 29.8 52.9 12.4 4.9
ess run smoothly. Interest rates have now been deregulated. 2004 28.4 52.8 13.9 4.9
Other major objectives of banking sector reforms were enhancing efficiency and productivity through increased com 2005 28.3 52.1 15.1 4.6 2006 25.8 51.3 17.7 5.1 2007 24.1 51.8 18.6 5.5
petition and, for that, modifying the overall legal environ- Share in Total Assets 1999 32.1 54.8 7.0 6.0
ment for conducting banking business in India. Establishment 2000 32.2 53.5 8.4 5.9
of new banks was allowed in the private sector and foreign 2001 32.9 51.6 9.1 6.4
banks were also permitted more liberal entry. Yet another step towards enhancing competition was allowing foreign di 2002 30.6 48.5 14.8 6.1 2003 30.1 48.7 14.8 6.5 2004 28.8 49.0 15.6 6.6
rect investment in private sector banks up to 74% from all 2005 27.4 50.1 16.2 6.3
sources. As for the modification of the legal environment, the 2006 25.6 48.9 18.6 6.9
Securitisation Act was enacted in 2002 to enhance protection 2007 24.0 48.6 19.7 7.7
of creditor rights. To combat the abuse of the fi nancial system for crime-related activities, the Prevention of Money Laundering Act was also enacted in 2002 to provide the enabling Expenditure on Advertisement/ 1999 0.3 0.4 1.5 6.0 Publicity as Percentage of 2000 0.3 0.4 1.6 5.8 Operating Expenditure 2001 0.3 0.3 2.9 7.0 2002 0.3 0.4 1.9 5.1
legal framework. The Negotiable Instruments (Amendments 2003 0.4 0.5 2.4 4.5
and Miscellaneous Provisions) Act 2002 expanded the erst 2004 0.7 0.8 2.6 5.6
while definition of a “cheque” by introducing the concept of “electronic money” and “cheque truncation”. The Credit 2005 0.6 0.9 3.1 6.6 2006 0.8 1.0 3.1 10.9 2007 0.7 1.2 2.8 9.5
I nformation Companies (Regulation) Act 2005 is expected to Source: Reserve Bank of India.
Economic & Political Weekly March 24, 2012 vol xlviI no 12 69

In this connection, we briefly review some of the important recent work on the performance of the Indian banking sector. Using data envelopment analysis (DEA) to analyse data on 70 Indian commercial banks from 1986 to 1991, Bhattacharyya et al (1997) found that publicly-owned Indian banks are the most efficient among all ownership categories considered in the study, followed by foreign-owned banks and Indian private banks, respectively. However, they also found something odd (and almost diametrically opposite) when the inter-temporal behaviour of such performance was considered. Evidence of temporal improvement was seen in the performance of foreignowned banks, virtually no such trend in that of Indian private banks and a temporal decline in that of the publicly-owned banks. They explained these patterns in terms of the government’s evolving regulatory policies. A study by Sarkar et al (1998) – with the motive of evaluating enterprise performance under different ownership patterns – confirmed that in the absence of a well-functioning capital market, there might not be any significant difference in the performance of public and private sector banks. Their analysis highlighted the importance of creating an appropriate institutional background before pushing privatisation in developing economies.

Kumbhakar and Sarkar (2003) analysed the relationship between deregulation and TFP growth in the Indian banking industry using a generalised shadow cost function approach. Analysing disaggregated panel data on a population of public and private banks from 1985 to 1996, they found evidence in favour of a significant decline in regulatory distortions and also non-materialisation of anticipated TFP growth until 1996. Using DEA, Sathye (2003) measured the productive effi ciency of banks in India for 1997-98. The efficiency scores, for three groups of banks – p ublicly-owned, privately-owned and foreign – were measured. The study showed that the mean efficiency score of Indian banks compared well with the world mean efficiency score and the efficiency of private sector commercial banks as a group was paradoxically lower than that of public sector banks and foreign banks in India. The study also recommended that the existing policy of reducing non-performing assets and rationalisation of staff and branches might be continued to obtain effi ciency gains and make Indian banks internationally more competitive. Chakrabarti and Chawla (2005) used DEA to evaluate the relative efficiency of Indian banks during 1990-2002 and observed that on a “value” basis, foreign banks as a group had been considerably more efficient than all other bank groups, followed by Indian private banks. However, from a “quantity” perspective, the Indian private banks seemed to be doing very well while the foreign banks were the worst off. This, as it can be easily understood, might be a reflection of the general policy of foreign banks to “cherry-pick” more profitable businesses, ignoring the social obligation of offering banking services to a wider section of society. Further, public sector banks were seen to be lagging behind their private counterparts in performance.

Das and Ghosh (2006) investigated the performance of the Indian commercial banking sector during the post-reform period 1992-2002. Using DEA, they applied all the three different a pproaches – intermediation, value added and operating – to d ifferentiate how efficiency scores varied with changes in inputs and outputs. The analysis also linked the variation in calculated efficiencies to a set of variables such as bank size, ownership, capital adequacy ratio, non-performing loans, management quality, and so on. Their findings suggested that medium-sized public sector banks performed reasonably well and were more likely to operate at higher levels of technical efficiency. A close relationship was observed between efficiency and soundness as determined by a bank’s capital adequacy ratio. Their empirical results also showed some evidence in favour of the expected r elationship that technically more efficient banks were those that had, on an average, less non-performing loans.

To evaluate the impact of computerisation2 on the productivity and profi tability of Indian banks, Mittal and Dhingra (2007) applied DEA methodology to the Centre for Monitoring Indian Economy (CMIE) data on 27 selected Indian commercial banks in 2003-04 and 2004-05. They observed that private sector banks, which took more information technology (IT) initiatives, were more efficient in terms of the productivity and profi tability p arameters than their counterparts under public ownership. Das, Ray and Nag (2009)3 used DEA to measure the labour-use efficiency of individual branches of a public sector bank with a large network of branches across India. They found considerable variation in the average levels of efficiency of bank branches across the four metropolitan regions considered in the study. They also introduced the concept of area or “spatial effi ciency” for each region relative to the nation as a whole. The results suggested that the policies, procedures, and incentives handed down from the corporate level could not fully neutralise the d etrimental influence of local work culture across different regions. Most of the potential reduction in labour cost appeared to be coming from possible downsizing of the clerical and subordinate staff.

We thus see that the issues raised earlier are yet to be explored to a great extent and that is precisely the objective of this paper. The paper is organised as follows. Section 2 briefl y states the analytical methodology we consider here. Section 3 describes the data set we have used and our major fi ndings from analysing it and Section 4 concludes. Appendix (pp 75-76) shows some further discussions.

2 Analytical Methodology

The productivity of a firm is measured by the quantity of output produced by it per unit of input. In the simplest single-input single-output case, it is merely the ratio of the quantity of the firm’s output to its input. But in a more general case where a number of inputs are used to produce a number of outputs, outputs (in the numerator) as also inputs (in the denominator) are to be meaningfully aggregated so that productivity still r emains the ratio of two scalar values. The productivity index of a firm for a current period relative to a base period measures the relative change in its productivity in the latter period relative to the earlier. Such a productivity index may be of two types – positive and normative. Positive measures are those measurements where one need not know the production technology. The Fisher productivity index and the Törnqvist p roductivity index are two such popular positive measures

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d iscussed in the literature. On the other hand, measurement of the Malmquist productivity index, a normative measure, requires knowledge about the benchmark production technology. Since our objective in this paper is to measure the productivity change of Indian commercial banks over the last eight years and decompose such change into economically meaningful components such as technical change, technical effi ciency change and the scale (efficiency) change factor to get the relative importance of these factors causing changes in TFP, we consider the Malmquist productivity index here.4

3 Data Used and Empirical Findings

A major problem one has to face in empirical banking research is defining the “inputs” and “outputs” of banks. Due to its ambiguous nature of use, an asset/liability may either be considered as an output of a bank or as its input used to produce some other output. For instance, if we view banks as service providers to their customers, as the production approach does,5 deposits of banks should be taken as an output. On the other hand, it should be included in the set of inputs if we consider a bank to be an intermediating entity between savers and investors whose goal is to earn profi t through lending and investing resources collected from customers in the form of d eposits. In view of such complexity, four approaches have come to dominate the literature on banking output – the production approach, the intermediation approach, the operating (income-based) approach and, more recently, the modern approach.6 We use a variant of the intermediation approach (subject to our data availability constraint) where deposits and borrowings and other liabilities, together with real resources such as labour, are defined as inputs whereas the output set includes earning assets such as loans and investments (Model I,

Table 2: TFP Change and Its Components of Indian Banks between 1999 and 2007
Name of Bank Model I Model II Name of Bank Model I Model II
State Bank of India 1.18 1.00 0.89 1.05 1.34 1.00 0.81 1.08 IndusInd Bank* 0.65 1.00 0.83 0.54 0.91 1.00 0.63 0.57
State Bank of Bikaner & Jaipur 0.90 1.07 1.08 1.04 1.06 1.21 0.80 1.02 Jammu and Kashmir Bank 0.98 1.07 1.34 1.40 1.07 1.00 0.82 0.87
State Bank of Hyderabad 0.97 0.95 1.04 0.96 1.17 0.95 0.74 0.82 Karur Vysya Bank 0.98 1.07 1.21 1.27 1.21 0.90 0.84 0.92
State Bank of Indore 0.96 1.03 1.09 1.07 1.11 1.34 0.75 1.12 Lakshmi Vilas Bank 0.98 0.98 1.17 1.12 1.20 1.25 0.89 1.35
State Bank of Mysore 0.94 1.07 1.03 1.04 1.14 0.76 0.77 0.67 Lord Krishna Bank 0.84 1.03 0.95 0.82 1.16 0.67 0.87 0.68
State Bank of Patiala 1.10 1.03 0.97 1.10 1.04 1.00 0.78 0.82 Nainital Bank 0.72 0.74 1.36 0.72 0.81 1.00 1.17 0.95
State Bank of Saurashtra 0.92 1.04 1.11 1.06 0.93 0.97 0.91 0.83 Ratnakar Bank 0.85 0.97 1.07 0.88 0.80 1.08 1.15 1.00
State Bank of Travancore 0.92 1.08 1.07 1.07 1.11 0.88 0.81 0.79 Sangli Bank 0.55 0.96 1.19 0.62 0.82 0.42 1.01 0.35
Average of State Bank Group 0.98 1.03 1.03 1.05 1.11 1.00 0.79 0.88 SBI Comm and Intern Bank 0.71 0.91 1.03 0.67 0.79 1.16 0.86 0.79
Allahabad Bank 0.97 1.04 1.19 1.19 1.24 1.14 0.79 1.12 South Indian Bank 1.01 0.98 1.27 1.26 1.12 0.97 0.87 0.94
Andhra Bank 0.86 1.03 1.16 1.03 1.15 0.96 0.73 0.81 Tamil Nadu Mercantile Bank 0.92 1.09 1.33 1.34 1.06 0.84 0.81 0.72
Bank of Baroda 1.24 1.00 1.05 1.30 1.35 1.00 0.70 0.95 Average of New
Bank of India 1.29 0.98 0.88 1.11 1.23 1.00 0.68 0.83 Private Sector Banks 0.97 0.99 0.66 0.63 1.38 0.93 0.48 0.61
Bank of Maharashtra 0.80 1.00 1.12 0.90 1.08 0.84 0.76 0.70 Average of
Canara Bank 1.13 1.00 1.04 1.16 1.29 1.00 0.77 1.00 Private Sector Banks 0.89 0.97 1.01 0.88 1.10 0.90 0.77 0.76
Central Bank of India 0.79 1.00 1.08 0.85 1.05 1.16 0.93 1.13 ABN Amro Bank** 1.13 0.91 0.60 0.62 1.44 1.43 0.54 1.12
Corporation Bank 1.06 0.98 0.98 1.02 1.37 1.00 0.62 0.84 Abu Dhabi Commercial Bank 0.70 0.75 0.89 0.46 0.84 0.81 0.92 0.62
Dena Bank 0.95 0.98 1.14 1.07 1.17 0.91 0.83 0.88 American Express Bank** 0.82 1.04 1.00 0.85 1.53 0.54 0.77 0.64
IDBI Bank 1.25 1.00 0.47 0.59 1.25 1.00 0.58 0.73 Arab Bangladesh Bank 1.00 1.00
Indian Bank 0.82 1.07 1.00 0.87 1.13 0.92 0.81 0.84 Bank International Indonesia 0.64 1.00 0.89 0.58 2.05 2.07 0.37 1.55
Indian Overseas Bank 1.09 1.02 0.95 1.05 1.18 1.01 0.79 0.94 Bank of America** 0.79 1.00 1.10 0.87 2.27 1.00 1.20 2.72
Oriental Bank of Commerce 0.89 1.00 1.25 1.12 1.47 1.00 0.66 0.97 Bank of Bahrain & Kuwait 0.96 1.10 0.98 1.04 0.76 1.20 0.98 0.90
Punjab & Sind Bank 0.77 1.04 1.17 0.94 1.05 0.84 0.94 0.83 Bank of Ceylon 0.69 0.54 0.98 0.36 1.00
Punjab National Bank 0.99 1.00 1.05 1.04 1.17 1.00 0.79 0.93 Bank of Nova Scotia** 0.82 1.00 0.98 0.80 1.46 1.09 0.86 1.37
Syndicate Bank 1.04 0.97 1.07 1.08 1.28 1.03 0.79 1.05 Barclays Bank** 0.81 1.00 0.87 0.70 1.22 0.53 1.04 0.67
UCO Bank 0.95 1.02 1.01 0.99 1.07 1.07 0.88 1.01 Citibank** 1.21 0.97 0.56 0.66 1.46 1.00 0.64 0.94
Union Bank of India 0.91 1.00 1.06 0.96 1.33 1.05 0.78 1.09 DBS Bank** 0.63 0.84 1.01 0.54 2.06 1.27 1.15 3.02
United Bank of India 0.59 1.00 1.13 0.67 0.86 0.97 0.90 0.75 Deutsche Bank** 0.91 1.00 0.81 0.74 2.00 1.00 0.84 1.67
Vijaya Bank 0.94 1.05 1.15 1.13 1.15 1.17 0.78 1.05 HSBC** 1.16 0.83 0.75 0.72 1.82 1.03 0.65 1.22
Average of Other Krung Thai Bank 1.00 1.00
Nationalised Banks 0.95 1.01 1.03 0.99 1.19 1.00 0.77 0.91 Mashreq Bank 1.12 1.02 0.90 1.02 1.94 0.58 0.59 0.67
UTI/Axis Bank* 1.09 1.00 0.61 0.66 1.48 1.00 0.56 0.83 Oman International Bank 0.63 1.17 1.00 0.74 1.05 0.85 0.71 0.63
Bank of Rajasthan 0.88 1.04 1.10 1.00 1.00 1.17 0.88 1.03 Societe Generale** 0.97 1.08 0.92 0.97 1.29 0.91 1.01 1.18
Catholic Syrian Bank 0.86 1.01 1.12 0.97 0.99 0.90 0.96 0.85 Sonali Bank 0.84 0.85 0.98 0.70 1.00
City Union Bank 0.96 0.98 1.19 1.12 1.10 0.92 0.87 0.88 Standard Chartered Bank** 1.27 0.94 0.64 0.76 1.70 1.29 0.70 1.52
Development Credit Bank* 0.82 0.93 1.04 0.80 1.26 0.71 0.68 0.61 Average of
Dhanalakshmi Bank 0.96 0.84 1.18 0.95 1.07 0.54 0.86 0.50 Big Foreign Banks 0.94 0.96 0.82 0.74 1.63 0.97 0.83 1.30
Federal Bank 1.03 0.97 1.07 1.07 1.17 1.07 0.86 1.07 Average of Foreign Banks 0.87 0.94 0.87 0.71 1.49 0.98 0.78 1.13
HDFC Bank* 1.10 1.00 0.62 0.68 1.77 1.00 0.37 0.66 Bank with an asterisk (*) and two asterisks (**) is a New Private Sector Bank and Big Foreign
ICICI Bank* 1.35 1.00 0.37 0.50 1.67 1.00 0.28 0.46 Bank respectively. By ‘big’ we are referring to a foreign bank having 100 or even more number of employees in 2007.
Economic & Political Weekly March 24, 2012 vol xlviI no 12 71

Figure 2: Scatter Plot of TFP Change and Its Components through One Model Relative to the Other a verage value of the SBI lending Technical Changes Comparison Technical Efficiency Changes Comparison rate; deposits are discounted by


the average value of the deposit rate; and borrowing and other

2.0 1.5 1.0 0.5

0.50 0.70 0.90 1.10 1.30 1.50

Technical change – Model II

Technical change – Model II


liabilities are discounted by the

average value of the bank rate

over the eight-year period

(2000-07). Similarly, variables

used in Model II are also ad




justed by the average values of

0.50 0.70 0.90 1.10 1.30

Technical change – Model I TE change – Model I the proper variables over this Scale Efficiency Changes Comparison TFP Changes Comparison period. For instance, demand

deposits and ST deposits are dis-counted by the short-term (one to three years) and long-term (more than three years) deposit rates respectively, fi xed assets are adjusted by the wholesale price index (WPI) of the ma-SFF – Model II 1.4 1.2 1.0 0.8 0.6 0.4 0.2 TFP Changes – Model II 3.3 2.8 2.3 1.8 1.3 0.8 0.3

0.35 0.85 1.35 1.85 0.40 0.80 1.20 1.60

chinery and equipment group

SCF – Model I TFP Changes – Model I

and the proxy variable for

hereafter).7 We also use the production approach (Model II, hereafter) to see whether or not the basic results of the performance-related issues considered in the present study change drastically due to changes in the approach of defi ning the inputs and outputs of banks.

We use individual bank-level (yearly) data for 68 major I ndian commercial banks for 19998 and 2007. The data is taken from the Reserve Bank of India (RBI) website. We have data on eight State Bank of India (SBI) and its associates and 20 banks each from the other publicly-owned, privately-owned and foreign-owned categories. The input and output variables we have used in our analysis are discussed below.

Model I

As we have already mentioned, the number of employees, total deposits and borrowing and other liabilities are considered as three inputs whereas investments and advances are considered as two outputs.

Model II

As per the production approach, the total number of deposits created by a bank are considered its output. Since we have no information about these numbers for all the three types of deposits a bank creates (viz, demand deposits, saving deposits and term deposits), we have taken their values and consider two different outputs – demand deposits and ST deposits (which is the sum of savings deposits and term deposits). Here we have considered the total number of employees, amount of fixed assets and operating expenses less payments to and provision for employees (as a proxy of materials used by the bank) as three inputs.

We have adjusted the nominal figures of the variables mentioned above by discounting/adjusting them using suitable interest rate/price indicators. For instance, 2007 values of the variables investments and advances are discounted by the material used is adjusted by the WPI of manufactured products.

Empirical Findings

We have used the econometric/statistical package SHAZAM to solve the various DEA LP problems to determine the individual bank-wise scores on TFP change and its components. The results we have obtained are given in Table 2 (p 71). As discussed earlier, one of the objectives of using two alternative models in the present study is to see whether the basic results regarding the performance-related issues of Indian commercial banks changes by simply changing the sets of their inputs and outputs. Our answer to this question is clearly “no”, at least for the sets of inputs and outputs we have considered. For instance, we present scatter plots of TFP changes as well as of three of its components, showing the correlation between these scores experienced by the individual banks through one model relative to the other in Figure 2. In two – those of technical change and scale (effi ciency) change factor – of the total four cases, there is clearly a positive correlation between the two sets of scores. Although there is no such evidence of any positive correlation in the remaining two cases, there is undoubtedly no negative correlation between the two sets of scores. The positive correlation for both technical change and scale (effi ciency) change factor is also confirmed by the Spearman’s rank correlation coefficient between the rankings of the banks on the basis of the two sets of scores obtained through Model I and Model II. We observed that this correlation coefficient is statistically significantly different from zero9 for these two components of TFP change experienced by the 68 Indian commercial banks we have considered.

Now we turn to the overall changes in the performance of the Indian banks over the period 1999 to 2007. As can be easily understood from our methodological discussion, technical change is a measure of the extent of shift of the concerned frontier production function, it is, therefore, collectively determined by all

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the firms and it is unusual that a firm in itself d etermining the (possible upward, if any) shift of the entire frontier and eventually the index of its (and others’) technical change. Rather, firms as a whole play a major role in determining it. Thus, although technical change of a firm is an important component of its TFP change, the firm itself has generally little contribution in determining it. Rather, two other components, namely technical efficiency change and scale (efficiency) change factor, of a fi rm are much more important determinants (in improving its overall performance) and they are influenced by its own activity.

So, in judging the improvement of overall performance of a bank in our study we pay more attention to the two latter components of its TFP change and relatively less to the earlier. Table 2 shows that, on an average, the group comprising the SBI and its associates improved its performance best in the light of overall TFP change or any of its three components under Model I. Other nationalised banks, private-sector banks and foreign banks follow one after another in the same order. Contrary to this, the order becomes foreign banks, other nationalised banks, the SBI group and private-sector banks in the light of overall TFP change under Model II. However, this improvement of the foreign bank group is mainly driven by technical changes among its various members. If we consider only technical effi ciency improvement and improvement in scale (effi ciency) change, the two indicators that are mostly d etermined by the activity of a bank itself, the story becomes almost the same as that which we have observed under Model I. Here again, the SBI group comes first, followed by other nationalised banks, foreign banks and private-sector banks, one after another. Therefore, even in a truly changed liberal economic environment, the nationalised banks have adjusted and i mproved themselves better compared to their counterparts under private or foreign ownership.

We now turn to individual bank-wise performance by conducting some fractile group analysis. To be specific, we order the individual banks according to the change in their overall performance and three of its components and consider only the top 17 banks (that is, 25% or more) and see what their distribution is among the four bank groups. This distribution, given in Tables 3a and 3b (showing a similar distribution as that of the top 34 banks (that is, 50% or more)). These two tables show almost an identical distribution, which demonstrates that public-sector banks have adjusted well to the changed scenario and improved their performance better than their private as well as foreign counterparts under Model I. On the other hand, under Model II, foreign banks followed by private sector banks were doing better relative to their nationalised counterparts. Now the immediate question is why then is the overall performance of foreign banks low as per the latter two indicators of TFP change even under Model II? The obvious answer is that there are a few foreign banks such as Abu Dhabi Commercial Bank, American Express Bank, Mashreq Bank, and Oman International Bank within the foreign group and those like Development Credit Bank, Dhanalakshmi Bank, and Lord Krishna Bank within the private group which pull down the respective group averages for technical effi ciency change and scale (efficiency) change factor to excessively low levels.

Economic & Political Weekly

March 24, 2012 vol xlviI no 12

By simply providing a close look at the output vectors we have considered under the two models, it can be easily understood why nationalised banks are shown to be lagging behind

Table 3a: Distribution of the Top 17 Banks according to the Change in Different Performance Indicators

Bank Group No of Banks Out of the Top 17
Model I Model II
State Bank of India and its associates 2 5 0 3 0 2 1 2
Other nationalised banks 7 4 8 7 3 4 4 4
Private sector banks 3 4 9 7 3 5 6 2
Foreign banks 5 4 0 0 11 6 6 9

Table 3b: Distribution of the Top 34 Banks according to the Change in Different Performance Indicators

Bank Group No of Banks Out of the Top 34
Model I Model II
State Bank of India and its associates 5 7 5 8 2 2 4 3
Other nationalised banks 12 11 14 14 12 10 7 12
Private sector banks 10 9 14 9 7 8 15 8
Foreign banks 7 7 1 3 13 14 8 11

their counterparts under foreign and private ownership under Model II while the scenario is the opposite under Model I. We have already argued that the nationalised banks have more developmental as well as social obligations than the other two groups of banks and distribute their services among more and more economically backward regions, in general, and rural a reas, in particular. Thus, one of their declared objectives, as a representative of the government, is to bring as many people as possible into the formal financial system and relieve them from the credit-cobweb of informal moneylenders. In doing so, they have a large number of small customers but the total deposits collected from them are also relatively (or to be specifi c, proportionately) small. On the other hand, private sector and foreign banks mainly target a fewer number of creditworthy customers and the total deposits collected from them are relatively large. Since we have used the total value of deposits created by a bank instead of the number of deposits created by it, as proposed by the production approach, foreign and private sector banks seem to be better improving themselves when compared to their nationalised counterparts. The picture may show the opposite even under Model II if we were able to use the total number of deposits created by a bank as its output. Similarly, since we have considered investments and advances as the two outputs produced by a bank under Model I, both of which, in general, and the latter, in particular, is a combination of many of the socially desired factors, intuitively it is not very difficult to understand why the public sector banks are seen to be performing relatively better under this model.

Again, as shown in Table 2, nationalised banks show improvement in both technical efficiency and scale effi ciency during this time under Model I while under Model II there is no change in the former and the change is, in fact, in the negative direction for the latter. On a totality, despite an improvement in technical change under Model II overall TFP growth indicator becomes smaller here than that under Model I. On the contrary, all the three components of change in TFP worsen under Model I for the foreign banks, while although the two efficiency components change negatively under Model II technical improvement is so sharp here that it pulls the TFP change index above unity. In fact, although all of the four bank groups experience technical improvement under Model II, the intensity of such change is very high in case of foreign banks relative to their domestic counterparts, which results in highest TFP change index for the foreign banks under this model. Observe that, the economic intuition discussed above considering the specifi c output vectors in question may also be well applicable here in explaining such differences in overall TFP change indices across the bank groups under the two models.

However, one may opine that it is not desirable to consider all the domestic private sector banks in a single group and similarly for the foreign banks as well, in view of their widely different characteristics. For that, we have distinguished fi ve new private banks from the old private banks and 11 relatively big foreign banks having 100 or more employees in 2007, from the others. We have reported the overall performance indicators for these two newly defined bank groups in Table 2. Interestingly, the results we have already discussed remain exactly the same even when we compare the performance of new private banks and relatively bigger foreign banks with the nationalised banks. Therefore, our results seem to be robust.

We want to examine one more feature, i e, whether and to what extent the TFP performance is correlated with fi nancial performance of the banks. In doing so, we have calculated the correlation coeffi cient of TFP indices with fi nancial performance indicators like the average profit per employee (PPE), average business10 per employee (BPE), average profit per unit volume of business (PPB), growth of PPE, BPE and PPB during this period, etc. We find such correlation to be signifi cant11 for the cases of average PPE and TFP under Model II, average BPE and TFP under Model II and growth of BPE and TFP under Model I and these values are 0.33, 0.37 and 0.64, respectively. So, there is some degree of positive correlation between fi nancial performance and TFP performance of the banks.

4 Concluding Remarks

Assessments of the performance of Indian commercial banks are not new in the literature. We have already discussed a few of them earlier in this paper. As evident from our discussion, some earlier studies have observed that nationalised banks perform relatively better than their counterparts under private and foreign ownership, whereas others show an opposite kind of experience. However, most of the earlier studies considered relatively partial measures such as the technical e fficiency of the banks. We have considered overall TFP improvement achieved by the individual banks and decomposed it into the three of its economically meaningful components. Furthermore, we have considered in some sense the true liberalised era of the Indian banking sector as our study period and assessed the extent to which individual banks have adjusted themselves to the new regime and improved in this period. Our results suggest that public sector banks are, on an average, adjusting themselves to the changing environment better and improving their performance relative to their counterparts under private and foreign ownership. The latter were e xpected to do better under the new regime, given their relatively more flexible operating systems as well as their better market orientation.

In an earlier study, Bhaumik and Dimova (2004) show that although the domestic private and foreign banks were better performing, and, hence, more efficient than public sector banks during the initial years of post-financial deregulation in India, competition forced public sector banks to eliminate this performance gap by the financial year 1998-99. Sensarma (2006) also shows that although Indian banks, in totality, have improved their performance during the period 1986 to 2000 in terms of both efficiency and productivity, foreign banks were the worst performers throughout the period as compared with public and domestic private banks. Since we have considered the immediate later period to that considered in the above two studies, our findings, coupled with those of the two mentioned here, have important policy implications for the government’s attitude towards overall market orientation of the Indian banking sector. To be specific, the government should have a more cautious approach liberalising its banking sector and not blindly invite more foreign players to it in view of the fact that the banks under (domestic) private and foreign ownerships may not be necessarily better performers. Of course, their presence may be of immense help to make the overall Indian banking business more and more competitive, which obviously have a positive bearing on the Indian overall fi nancial system to be more effi cient.

However, we have used DEA methodology, which is based on mathematical programming techniques, without considering the possible error structures that may affect the analysis. Since any methodology has its relative advantages as well as disadvantages over its possible alternatives, our analysis is not free from its respective limitations.

Notes time cost associated with each transaction, to 6 Interested readers may look up Mohan (2005), 1 See Casu and Molyneux (2003) for an exten

a huge extent. Berger and Humphrey (1992), Frexias and Rochet sive survey of the relevant literature on per-3 However, this study is to some extent different (1997) for detailed discussions on these approaches. formance of banks. from the others mentioned above in the sense 7 This is also known as the “asset approach”.

2 Indian banks are now investing heavily in that the others deal with different Indian com-8 The year 1999 refers to the financial year becomputerized technologies such as telebank-mercial banks while this one deals with differ-ginning in April 1998 and ending in March

ing, mobile banking, net banking, automated ent branches of a single public-sector bank. 1999. Similarly, the year 2007 refers to the teller machines (ATMs), credit cards, debit 4 Detailed exposition of Malmquist productivity financial year April 2006 – March 2007. We cards, smart cards, call centres, customer index and its decomposition is shown in the adopt this convention throughout the paper. relationship management (CRM), data ware-Appendix. Interested readers may also look up 9 We know that for sample size (n) more than or

housing and the like. All these facilities, which Ray (2004, Chapter 11) for a detailed discus-equal to 40, rs — n – 1 is approximately normally are new innovations in banking technologies, sion on popular productivity indices of both distributed with mean zero and variance unity, help the Indian banking system improve its the positive and normative kind. n

where r = 1 – (6 Ȉdi2/n(n2–1)), is the Spearman's


service quality, particularly by lowering the 5 Which we shall discuss later in details. i=1


man’s rank correlation coefficient between the two sets of ranks of the observations and di is the difference between these two sets of ranks for the ith observation. In our sample, values of this statistic are 3.39 and 3.70 for technical change and scale (efficiency) change factor respectively, which clearly exceed the concerned tabulated value even at 1% level of signifi cance.

10 Sum of the values of total deposits and advances of a bank is defined to be the total volume of business of it.

11 We know that for sample size (n) more than 6,

r — n – 2 have an approximately t-distribution

— 1 – r2

with n – 2 degrees of freedom where r is the sample correlation coefficient between the two variables.

12 Clearly f (•) and R (•) are the north-western boundary of T and TC respectively.

13 Note that scale efficiency does not state anything about the actual scale of production relative to the MPSS, in the sense that one cannot say whether the firm is actually practising more or less than the MPSS by simply observing its scale effi ciency score.

14 V for VRS and C for CRS.

15 Simar and Wilson (1998) decomposed the Malmquist TFP index further and provide more economically meaningful interpretation of both of the technical change and the scale change factor of the Färe et al (1994) and Ray-

Desli (1997) measures. Interested readers may look up the paper for this decomposition. However, we do not consider their decomposition in the present study.

16 Interested readers may look up Ray (2004, Chapters 2, 3) for an explicit discussion on the formation of the respective production possibility set for alternative technological specifi cations and how the associated LP problems are structured from that.


Banker, R D (1984): “Estimating the Most Productive Scale Size Using Data E nvelopment Analysis”, European Journal of Operational Research, 17 (1), pp 35-44.

Barman, R B (2007): “Determinants of Profi tability of Banks in India”, presidential address delivered at the 43rd Annual Conference of the I ndian Econometric Society, Indian Institute of Technology, Bombay (5 January).

Bauer, P W, A N Berger and D B Humphrey (1993): “Efficiency and Productivity Growth in US Banking” in H O Fried, C A K Lovell and S S Schmidt (ed.), The Measurement of Productive Efficiency: Techniques and Applications ( Oxford: Oxford University Press), pp 386-413.

Berger, A N and D B Humphrey (1992): “Measurement and Efficiency Issues in Commercial Banking” in Z Griliches (ed.), Output Measurement in the Service Sector (Chicago: Chicago University Press), pp 245-79.

Berger, A N and L J Mester (2001): “Explaining the Dramatic Changes in Performance of US Banks: Technological Change, Deregulation, and Dynamic Changes in Competition”, Working Paper, University of Pennsylvania.

Bhattacharyya, A, C A K Lovell and P Sahay (1997): “The Impact of Liberalisation on the Productive Efficiency of Indian Commercial Banks”, European Journal of Operational Research, 98 (2), pp 332-45.

Bhaumik, S K and R Dimova (2004): “How Important Is Ownership in a Market with Level Playing Field? The Indian Banking Sector Revisited”, Journal of Comparative Economics, 32 (1), pp 165-80.

Chaffai, M E (1997): “Productivity and Effi ciency Performances of the Tunisian Banking Industry before and after the Economic Liberalisation Program: An Econometric Study using Panel Data” in R Dahel and I Sirageldin (ed.),

Models for Economic Policy Evaluation Theory and Practice: An International Experience. Research in Human Capital and Development

(Greenwich, CT: JAI Press), Vol II, Part B, pp 335-50.

Chakrabarti, R and G Chawla (2005): “Bank Effi ciency in India since the Reforms: An Assessment”, Money Finance, 2 (22-23), pp 31-48.

Das, A and S Ghosh (2006): “Financial Deregulation and Efficiency: An Empirical Analysis of Indian Banks during the Post-Reform P eriod”, Review of Financial Economics, 15 (3), pp 193-221.

Das, A, S C Ray and A Nag (2009): “Labour-Use E fficiency in Indian Banking: A Branch-Level Analysis”, Omega, 37 (2), pp 411-25.

Färe, R, S Grosskopf, B Lindgren and P Roos (1992): “Productivity Changes in Swedish Pharmacies 1980-1989: A Non-parametric Malmquist Approach”, Journal of Productivity Analysis, 3 (1/2), pp 85-101.

Färe, R, S Grosskopf, M Norris and Z Zhang (1994): “Productivity Growth, Technical Progress, and Efficiency Change in Industrialised Countries”, American Economic Review, 84 (1), pp 66-83.

Farrell, M J (1957): “The Measurement of Productive Effi ciency”, Journal of the Royal Statistical Society, Series A, General, 120 (3), pp 253-81.

Frexias, X and J C Rochet (1997): Microeconomics of Banking (Cambridge, Massachusetts: MIT Press).

Frisch, R (1965): Theory of Production (Chicago: Rand McNally).

Humphrey, D B and L B Pulley (1997): “Banks’ Responses to Deregulation: Profi ts, Technology, and Effi ciency”, Journal of Money, Credit and Banking, 29 (1), pp 73-93.

Kumbhakar, S C and S Sarkar (2003): “Deregulation, Ownership, and Productivity Growth in the Banking Industry: Evidence from India”, Journal of Money, Credit and Banking, 35 (3), pp 403-24.

Mittal, R K and S Dhingra (2007): “Assessing the Impact of Computerisation on Productivity and Profitability of Indian Banks: An Application of Data Envelopment Analysis”, Delhi Business Review, 8 (1), pp 63-73.

Mohan, R (2005): “Reforms, Productivity and Effi ciency in Banking: The Indian Experience”, Address Delivered at the 21st Annual General Meeting and Conference of the Pakistan Society of Development Economists, Islamabad.

Rangarajan, C (2007): “The Indian Banking System – Challenges Ahead”, First R K Talwar Memorial Lecture, Indian Institute of Banking and Finance.

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Ray, S C and E Desli (1997): “Productivity Growth, Technical Progress, and Efficiency Change in Industrialised Countries: Comment”, American Economic Review, 87 (5), pp 1033-39.

Sarkar, J, S Sarkar and S K Bhaumik (1998): “Does Ownership Always Matter? Evidence from the Indian Banking Industry”, Journal of Comparative Economics, 26 (2), pp 262-81.

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Malmquist Productivity Index and Its Decomposition

As we have already mentioned, the Malmquist productivity index is a normative measure; an associated benchmark technology has to be taken into account to measure it. Since production technology itself may change over time, either of the technology of the base period and the current period may be used as the benchmark. To be specific, let us assume that (x0, y0) and (x1, y1) are the input-output combinations of a firm in the periods 0 and 1 respectively. Then change in the Malmquist TFP index from period 0 to period 1 can be written as

y y fi(x ) Ri(x )

11 11


x fi(x ) Ri(x ) x

1 111

3 , i = 0, 1

i ii 0 00

y0 y f(x)R(x)


x0 fi(x)Ri(x) x

0 00

Where fi (•) and Ri (•) are the production frontiers of the ith period, assuming that the production technology exhibits variable returns to scale (VRS) and constant returns to scale (CRS) respectively, and the two concerned production possibility set be denoted by T and TC respectively.12 Therefore, Ȇ0 and Ȇ1 may be different if the production technology itself changes from period 0 to period 1. To get rid of such complexity, the conventional way is to measure the index once considering the base period technology as the benchmark and once again considering the technology of the current period, and then take the geometric average of these two measures to obtain the overall

Figure 1






Y1 D1 F

B1 X O AX* X1

March 24, 2012 vol xlviI no 12

change in the Malmquist TFP index. Thus the overall measure of changes of the Malmquist TPF index can be written as follows:

Ȇ= [Ȇ × Ȇ1]½


Let us discuss first the concepts of technical effi ciency (TE) and scale efficiency of a production unit with the help of the diagram below. Let ATBC (in Figure 1, p 75) be the production frontier (exhibiting VRS technology with other usual desirable properties). An (output-oriented) measure of TE of fi rm F, as defined to be the ratio of actually produced amount of output to the frontier level of output for the given level of input used by this

FX1 FX1/OX1firm, is given by = which is equal to the ratio of productivity,


as defined to be the amount of output per unit of input used, at the point F to that at the point B. Note that TE is identical (and equal to unity) at all points on the frontier, but productivity is not. It is easy to see that productivity is the highest at the point T among all feasible points (that is, those that lie within the production possibility set). Hence, OX* is the size relating to the concept of technically optimal production scale (TOPS) (a la F risch 1965), and the widely known, most productive scale size (MPSS) (a la Banker 1984) in the diagram. Output-oriented scale effi ciency of a firm is defined to be the ratio of productivity at its (output-oriented) projection on to the frontier to that at the MPSS. Similarly, input-oriented measure of scale efficiency of a firm is the ratio of productivity at its (input-oriented) projection on to the frontier to that at the MPSS. In other words, scale efficiency is a measure of the relative productivity of a fi rm with respect to productivity at the MPSS, if the firm becomes able to eliminate its technical inefficiency in production and, therefore, naturally it lies between 0 and 1.13 So, scale efficiency of any firm lies on the vertical

BX1/OX1line BX1 is , which is the ratio of productivity at point B to that at


point T, and (the input-oriented) scale effi ciency of any firm that lies on the horizontal line B1F is the ratio of productivity at the point B1 to that at point T. But, productivity at point T is equivalent to that of the hypothetical firms at points D and D1. Although, these points are not feasible under the VRS technology, they are on the graph of the CRS technology. Thus,


= = = and similarly we can show that TX*/OX* DX1/OX1 DX1 FX1/BX1

Y1D1/Y1F the ratio of productivity at B1 to that at D1 is equal to the ratio .


So, scale efficiency of a firm is the ratio of its TE under the CRS technology to that under the VRS technology, irrespective of the orientation of the measurement of technical effi ciency.

Let us now define the concept of output-oriented distance function (a la Shephard 1970) here. The (output-oriented) distance function evaluated y

at any input-output pair (x, y) is given by DV(orC) (x, y) = min į: (x, )T (or TC)

į if production technology is assumed to exhibit VRS (or CRS). So, it can be easily understood that the (output-oriented) TE and the (output-oriented) distance function are the same. However, using distance functions, Ȇcan be shown, a la Ray and Desli (1997), to be the product of three economically meaningful components: technical change (TC), technical efficiency change (TEC) and scale (efficiency) change factor (SCF) and these components can be shown as follows:

f1(x0) f1(x1) ½ DV0(x0,y0) DV0(x1,y1) ½ TC =× = × ,

[][ ]

f0(x0) f0(x1) DV1(x0,y0) DV1(x1,y1)

y1/f1(x1) DV1(x1,y1) TEC == and

[][ ]

y0/f0(x0) DV0(x0,y0) DC1(x1,y1) DC0(x1,y1) ½


f1(x1)/R1(x1) f0(x1)/R0(x1) ½ DV ) DV0(x1,y1) SCF =× =


[ ][]

f1(x0)/R1(x0) f0(x0)/R0(x0) DC1(x0,y0) DC0(x0,y0)

DV1(x0,y0) DV0(x0,y0) where superscript and subscript of D are used to indicate, respectively, the period of technology considered as the benchmark and assumed returns to scale specification for the technology respectively.14 Before Ray-Desli, Färe et al (1992) introduced a decomposition of the Malmquist TFP index assuming that the true production technology exhibits CRS. According to their decomposition, Ȇ can be shown to be the product of two different

R1(x0) R1(x1) ½ components: a measure of technical change × , which is


R0(x0) R0(x1)

the (un-weighted) geometric mean of the shift in the true (CRS) production function at input levels x0 and x1, and technical effi ciency change

y1/R1(x1) , – again using the true (CRS) production function as the



benchmark. Note that, if the production technology truly exhibits CRS, the last component, that is, SCF of Ray-Desli decomposition disappears whereas the other two components exactly match these two components of Färe et al (1992). Since globally CRS is a restrictive assumption about the underlying technology, when CRS does not hold everywhere, Färe et al (1992) decomposition is not particularly meaningful. In an effort to accommodate VRS, Färe et al (1994) proposed the extended decomposition according to which the Malmquist TFP index can be written as a product of three different components: a measure of technical change,

R1(x0) R1(x1) y1/f1(x1) × ; a measure of technical effi ciency change, ;

[ ] ½ [ ]

R0(x0) R0(x1) y0/f0(x0)

f1(x0)/R1(x1)and a measure of scale effi ciency change, . But Ray and



Desli (1997) rightly argued that the first component of Färe et al (1994) is not an appropriate measure of technical change when production technology does not follow CRS globally.

However, one particular disadvantage of the Ray-Desli decomposition is that at most two (namely, the first and the third ones) of their three decomposed components may not be obtained for some observations if the quantity of any individual input of an observation in the base (current) period is smaller than the smallest quantity of the corresponding input across all firms in the current (base) period. However, we follow only the Ray-Desli measure15 in our study.

From the description of the distance function provided earlier, it is easy to see that the Shephard distance function is identical to Farrell’s (1957) measure of (output-oriented) technical efficiency and can, therefore, be obtained straightway by solving the various DEA linear programming (LP) problems for alternative technological specifications. For instance, the “same-period” VRS distance function for the kth production unit can be shown to


be D t (x,y) = 1/ijk where ijk = max ij such that the four constraints:

v tt N N N

t Ȝi t t Ȝt i ”

(i) Ȉ yi t •ijyk, (ii) Ȉxi xt k, (iii) ȈȜit= 1 and (iv) Ȝit • 0 for all i, are

i=1 i=1 i=1

satisfied. Similarly, the “cross-period” VRS distance function for


the Kth production unit can be shown to be Ds (x,y) = 1/ijk where

v ttN

** s Ȝi t

= max ij where such that the four constrains: (i) Ȉ yi s •ijyk ,

ijk i=1


s Ȝis ”

(ii) Ȉxi xt k, (iii) ȈȜis= 1 and (iv) Ȝis • 0, for all i, are satisfi ed. We have

i=1 i=1

to solve these two LP problems without the constraint (iii) to get the CRS distance functions for same period and cross period respectively.16 In the above LP problems, any one of s and t can be used as an indicator of the base period and the other as an indicator of the current period.

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