Description
Commmercial Bank Play a vital role in the economy for two reasons: they provide a major source of financial intermediation and their checkable deposit liabilities represent the bulk of the nation’s money stock. Evaluating their overall performance and monitoring their financial condition is important to depositors, owners, potential.
31
Piyu Yue
Piyu Yue, a research associate at the IC
2
Institute, University of
Texas at Austin, was a visiting scholar at the Federal Reserve
Bankof St. Louis when this article was written. Lynn Dietrich
providedresearch assistance. The author would like to thank A.
Charnes, Roll Fare and Shawna Grosskopf for theirconstructive
comments and useful suggestions. Their DEA computer code led
to a significant improvement ofthe paper
Data Envelopment Analysis and
Commercial Bank Performance:
A Primer With Applications to
Missouri Banks
OMMERCIAL BANKS PLAY a vital role in the
economy for two reasons: they provide a major
source of financial intermediation and their check-
able deposit liabilities represent the bulk of the
nation’s money stock. Evaluating their overall
performance and monitoring their financial condi-
tion is important to depositors, owners, potential
investors, managers and, of course, regulators.
Currently, financial ratios are often used to
measure the overall financial soundness of a bank
and the quality of its management. Bank regu-
lators, for example, use financial ratios to help
evaluate a bank’s performance as part of the
CAMEL system.
1
Evaluating the economic perfor-
mance of banks, however, is a complicated
process. Often a number of criteria such as
profits, liquidity, asset quality, attitude toward
risk, and management strategies must be consi-
dered. The changing nature of the banking
industry has made such evaluations even more
difficult, increasing the needfor more flexible
alternative forms of financial analysis.
This paper describes a particular methodology
called Data Envelopment Analysis (DEA), that has
been used previously to analyze the relative effi-
ciencies of industrial firms, universities, hospitals,
military operations, baseball players and, more
recently, commercial banks.2 The use of flEA is
demonstrated by evaluating the management of
60 Missouri commercial banks for the period from
1984 to 1990.~
1
For more details, see Booker (1983), Korobow (1983) and
Putnam (1983).
2
The name DEA is attributed to Charnes, Cooper and Rhodes
(1978), for the development of DEA, see Charnes, et al.(1 985)
and Charnes, et at. (1978); for some applications of DEA, see
Banker, et al. (1984), Charnes, et al. (1990) and Sherman and
Gold (1985).
3
Although there is vast literature analyzing competition and
performance in the U.S. banking industry (e.g., Gilbert (1984),
Ehten (1983), Korobow(1983), Putnam (1983), Wall (1983)
and Watro (1989)), actual banking efficiency has received
limited attention. Recently, a few publications have used IDEA
or asimilar approach to study the technical and scale efficien-
cies of commercial banks (e.g., Sherman and Gold (1985),
Charnes etal. (1990), Rangan et al. (1988), Aty et al. (1990),
and Etyasiani and Mehdian (1990)).
a4~sIcs
flEA represents a mathematical programming
methodology that can be applied to assess the effi-
ciency of a varietyof institutions using a variety of
data. This section provides an intuitive explana-
tion of the DEAapproach. A formal mathematical
presentation of flEA is described in appendix A; a
slightly different nonparametric approach is
described in appendix B.
flEA is based on a concept of efficiency that is
widely used in engineering and the natural
sciences- Engineering efficiency is defined as the
ratio of the amount of work performed by a
machine to the amount of energy consumed in the
process. Since machines must be operated
according to the law of conservation of energy,
their efficiency ratios are always less than or equal
to unity.
This concept of engineering efficiency is not
immediately applicable to economic production
because the value of output is expected to exceed
the value of inputs due to the “value added” in
production. Nevertheless, under certain circum-
stances, an economic efficiency standard—similar
to the engineering standard—can be defined and
used to compare the relative efficiencies of
economic entities. For example, a firm can be said
to be efficient relative to another if it produces
either the same level of output with fewer inputs
or more output with the same or fewer inputs. A
single firm is considered “technically efficient” if it
cannot increase any output or reduce any input
without reducing other outputs or increasing
other inputs.
4
Consequently, this concept of tech-
nical efficiencyis similar to the engineering
concept. The somewhat broader concept of
“economic efficiency,”on the other hand, is
achieved when firms find the combination of
inputs that enable them to produce the desired
level of output at minimum cost.
5
The discussion of the flEA approach will be
undertaken in the context of technical efficiency
in the microeconomic theory of production. tn
microeconomics, the production possibility set
consists of the feasible input and output combina-
tions that arise from available production tech-
nology. The production function (or production
transformation as it is called in the case of multiple
outputs) is a mathematical expression for a
process that transforms inputs into output. In so
doing, it defines the frontier of the production
possibility set. For example, consider the well-
known Cobb-Douglas production function:
(1) Y = AK~Ll~a,
where Y is the maximum output for given quanti-
ties of two inputs: capital (K) and labor (Ii. Even if
all firms produce the same good (Y) withthe same
technology defined by equation 1, they may still
use different combinations of labor and capital to
produce different levels of output. Nonetheless, all
firms whose input-output combinations lie on the
surface (frontier) of the production relationship
definedby equation 1 are said to be technologi-
cally efficient. Similarly, firms with input-output
combinations located inside the frontier are tech-
nologically inefficient.
DEAprovides a similar notion of efficiency. The
principal difference is that the flEA production
frontier is not determined by some specific equa-
tion like that shown in equation 1; instead, it is
generated from the actual data for the evaluated
firms (which in flEA terminology are typically
called decision-making units or DMU5).
6
Conse-
quently, the flEA efficiency score for a specific
firm is not defined by an absolute standard like
equation 1. Rather, it is defined relative to the other
firms under consideration. And, similar to engi-
neering efficiency measures, DEA establishes a
“benchmark” efficiency score of unity that no
individual firm’s score can exceed. Consequently,
efficient firms receive efficiency scores of unity,
while inefficient firms receive DEA scores of less
than unity.
4
See Koopmans (1951).
~This is also named “allocative efficiency” because a profit
maximizing firm must allocate its resources such that the
technical rate of substitution is equal to the ratio of the prices
of the resources. Theoretical considerations of atlocative eff i-
ciency can be found in the articles by Banker (1984) and
Banker and Maindiratta (1988).
mated production function represents the average behavior of
firms in the sample. Hence, the estimated production function
depends upon the data for both efficient and inefficient firms.
By imposing suitable constraints, these statistical procedures
can be modified to orient the estimates toward frontiers. In
this manner, the frontier of the production set can be esti-
mated econometrically.
°ltis common to estimate production functions using regres-
sion analysis. When cross-section data are used, the esti-
33
In microeconomic analysis, efficient production
is defined by technological relationships with the
assumption that firms are operated efficiently.
Whether or not firms have access to the same
technology, it is assumed that they operate on the
frontier of their relevant production possibilities
set; hence, they are technically efficient by defini-
tion. As a result, much of microeconomic theory
ignores issues concerning technological ineffi-
ciencies.
flEA assumes that all firms face the same
unspecified technology which defines their
production possibilities set. The objective of flEA
is to determine which firms operate on their effi-
ciency frontier and which firms do not. That is,
flEApartitions the inputs and outputs of all firms
into efficient and inefficient combinations. The
efficient input-output combinations yield an
implicit production frontier against which each
firm’s input and output combination is evaluated.
If the firm’s input-output combination lies on the
DEA frontier, the firm might be considered effi-
cient; if the firm’s input-output combination lies
insidethe DEA frontier, the firm is considered
inefficient.
An advantage of DEA is that it uses actual
sample data to derive the efficiencyfrontier
against which each firm in the sample can be
evaluated.~As a result, no explicit functional form
for the production function has to be specified in
advance. Instead, the production frontier is gener-
ated by a mathematical programming algorithm
which also calculates the optimal DEA efficiency
score for each firm.
To illustrate the relationship between DEA and
economic production in its simplest form,
consider the example shown in figure 1, in which
firms use a single input to produce a single output.
In this example, there are six firms whose inputs
are denoted as x, and whose outputs are denoted
as y~(i = 1,2,..,6); their input-output combinations
are labeled by F
8
(s = 1,2 6). While the produc-
tion frontier is generated by the input-output
combinations for the firms labeled F
1
, F
3
, F
5
and F
6
,
the efficient portion of the production frontier is
shown by the connected hne segments. F
2
and F
4
are clearly flEA inefficient because they lie inside
the frontier; F
6
is flEA inefficient because the
same output can be produced with less input.
hc. ..lmporiance of .rni.’eis in fJE~1
“Facets” are an important concept used to
evaluate a firm’s efficiency in flEA. The efficiency
measure in DEAis concerned withwhether a firm
can increase its output using the same inputs or
produce the same output with fewer inputs.
Consequently, only part of the entire efficiency
frontier is relevant when evaluating the efficiency
of a specific firm. The relevant portion of the effi-
ciency frontier is called a facet. For example, in
figure 1, onlythe facet from F, to F
3
is relevant for
evaluating the efficiency of the firm designated by
F
2
. Similarly, only the facet from F
3
to F
5
is used to
evaluate the firm denoted by F
4
.
8
The use of facets with flEA enables analysts to
identify inefficient firms and, through comparison
with efficient firms on relevant facets, to suggest
ways in which the inefficient firms might improve
their performance. As illustrated in figure 1, F
2
can become efficient by rising to some point on
the F,-F
3
facet. In particular, it could move to A by
simply using less input, to B by producing more
output or to C by both reducing input and
increasing output- Of course, in this example, the
analysis is obvious and the recommendation
trivial. In more complicated, multiple input-
multiple output cases, however, the appropriate
efficiency recommendations would be much more
difficult to discover without the flEA
methodology.~
7
DEA has two theoretical properties that are especially use-
ful for its implementation. One is that the IDEA model is
mathematically related to a multi-objective optimization
problem in which all inputs and outputs are defined as
multiple objectives such that all inputs are minimized and
all outputs are maximized simultaneously under the tech-
nology constraints. Thus, IDEA-efficient DMUs represent
Pareto optimal solutions to the multi-objective optimization
problem, while the Pareto optimal solution does not neces-
sarily imply DEA efficiency.
Another important property is that IDEA efficiency scores
are independent of the units in which inputs and outputs
are measured, as tong as these units are the same tor all
IDMUs. These characteristics make the IDEA methodology
highly flexible. The only constraint set originally in the
CCR model is that the values of inputs and outputs must
be strictly positive.
This constraint, however, has been abandoned in the new
additive IDEA formulation. As a consequence, the additive
IDEA model is used to compute reservation prices for new
and disappearing commodities in the construction of price
indexes by Lovell and Zieschang (1990).
8
1n a multiple dimensional space, the efficiency frontier
forms a polyhedron. In geometry, a portion of the surface
of a polyhedron is called a facet; this is why the same
term is used in IDEA. These facets have important implica-
tions in empirical studies, such as identification of compe-
titors and strategic groups in an industry. See Day, Lewin,
Salazar and Li (1989).
°Foralternative measures of efficiency, see appendix B.
34
Figure 1
Production Frontier and Efficiency Subset
Output Y
K
Scale E/!iciencv
In addition to measuring technological effi-
ciency, flEA also provides information about scale
efficiencies in production. Because the measure of
scale efficiency in DEA analysis varies from model
to model, care must be exercised. The scale effi-
ciency measured for the flEA model used in this
study, however, corresponds fairly closely to the
microeconomic definition of economics of scale in
the classical theory of production.’°
To illustrate, consider the F,-F
3
facet in figure 2.
Firms located on this facet exhibit increasing
returns to scale because a proportionate rise in
their input and output placesthem inside the
production frontier. A proportionate decrease in
their input and output is impossible because it
would move them outside of the frontier. This is
illustrated by a ray fromthe origin that passes
through the F,-F
3
facet at F’
2
.
Firms located on the F
3
-F
3
facet exhibit
decreasing returns to scale because a propor-
tionate decrease in their input and output places
them inside the production frontier. A propor-
tionate increase in their input and output is impos-
sible because it would move them outside of the
frontier.
Constant returns to scale occur if all propor-
tionate increases or decreases ininputs and
outputs move the firm either along or above the
production frontier. In figure 2, for example, F,
exhibits constant returns to scale because propor-
tionate increases or decreases would place it
outside the production frontier.
Since the facets are generated by efficient firms,
the scale efficiency of these firms is determined by
the properties of their particular facet. Scale effi-
ciencies for inefficient firms are determined by
their respective reference facets as well. Thus, F
2
and F
4
in figure 1 exhibit increasing and
decreasing returns to scale, respectively.
.LIEA and Ft’nr,mnk’ I1/
1
/ic1t?nr~t.~
While the discussion of flEA in the context of
technological efficiency of production is useful for
illustrative purposes, it is far too narrow and
limiting. flEA is frequently applied to questions
and data that transcend the narrow focus of tech-
nical efficiency in production. For example, DEA is
frequently applied to financial data when
addressing questions of economic efficiency. In
this regard, its application is somewhat more
problematic. For example, when firms face
different marginal costs of production due to
regional or local wage differentials, one firm may
appear inefficient relative to another. Given the
potential differences in relative costs that a firm
may face, however, it might be equally efficient.
Alternatively, differences that appear to be due to
economic inefficiencies may in fact be due to cost
differences directly attributable to the non-
homogeneity of products. Because of problems
like these, flEA must be applied judiciously.
.na1. g.J/j,,j
0~1
To this point, the discussion of flEA has been
concerned with evaluating the relative efficiency
of different firms at the same time. Those who use
flEA, however, frequently employ a type of sensi-
tivity analysis called“window analysis.” The
performance of one firm or its reference firms
‘°SeeFare, Grosskopf and Lovell (1985). Different DEA
models employ different measures of scale efficiency. See
F,
F, /
F,
F,
0
Input X
appendixes A and B for details,
Figure 2
An Illustration of Scale Efficiencies
OutputY
for a firm over time. Of course, comparisons of
flEA efficiency scores over extended periods may
be misleading (or worse) because of significant
changes in technology and the underlying
economic structure.
0
may be particularly “good” or “bad” at a given time
because of factors that are external to the firm’s
relativeefficiency. In addition, the number of
firms that can be analyzed using the flEA model is
virtually unlimited. Therefore, data on firms in
different periods can be incorporated into the
analysis by simply treating them as if they
represent different firms. In this way, a given firm
at a given time can compare its performance at
different times and with the performance of other
firms at the same and at different times. Through
a sequence of such “windows,” the sensitivity of a
firm’s efficiency score can be derived for a partic-
ular year according to changing conditions and a
changing set of reference firms.” A firm that is
flEA efficient in a given year, regardless of the
window, is likelyto be truly efficient relative to
other firms. Conversely, a firm that is only flEA
efficient in a particular window may be efficient
solely because of extraneous circumstances.
In addition, window analysis provides some
evidence of the short-run evolution of efficiency
APPLYING DEE TO BANKING:
AN EVALU.zVI’ION (IF’ 60 MISSOURI
(X)MJ.~IERCI.ALB
To demonstrate flEA’s use, it is applied to
evaluate relative efficiency in banking. Financial
data for 60 of the largest Missouri commercial
banks for 1984 (determined by their total assets in
1990) are used. Initially, the relative efficiency of
these banks is examined using two alternative
flEA models: the CCII model and the additive flEA
model. A discussion of these alternative DEA
models appears in appendix A. In extending the
discussion and analysis, however, we focus solely
on the CCII model.
Measuring inputs and Outputs
Perhaps the most important step in using flEA to
examine the relative efficiency of any type of firm
is the selection of appropriate inputs and outputs.
This is partially true for banks because there is
considerable disagreement over the appropriate
inputs and outputs for banks. Previous applica-
tions of flEA to banks generally have adopted one
of two approaches to justify their choice of inputs
and outputs.’
2
The first “intermediary approach” views banks
as financial intermediaries whose primary busi-
ness is to borrow funds from depositors and lend
those funds to others for profit. In these studies,
the banks’ outputs are loans (measured in dollars)
and their inputs are the various costs of these
funds (including interest expense, labor, capital
and operating costs).
A second approach views banks as institutions
that use capital and labor to produce loans and
deposit account services. In these studies, the
banks’ outputs are their accounts and transac-
tions, while their inputs are their labor, capital
and operating costs; the banks’ interest expenses
are excluded in these studies.
‘
1
This is called “panel data analysis” in econometrics.
‘
2
Some studies have adopted the simple rule that if it
produces revenue, it is an output; if it requires a net ex-
F, F,
F,
F’,
F,
Input X
penditure, it is an input. For example, see Hancock (1989).
36
Our analysis of 60 Missouri banks uses a variant
of the intermediary approach. The banks’ outputs
are interest income (IC), non-interest income (N1C)
and total loans (1’L). Interest income includes
interest and fee income on loans, income from
lease-financing receivables, interest and dividend
income on securities, and other income. Non-
interest income includes service charges on
deposit accounts, income fromfiduciary activities
and other non-interest income. Total loans consist
of loans and leases net of unearned income. These
outputs represent the banks’ revenues and major
business activities.
‘I’he banks’ inputs are interest expenses (IE),
non-interest expenses (NIE), transaction deposits
(Tfl), and non-transaction deposits (NTD). Interest
expenses include expenses for federal funds and
the purchase and sale of securities, and the inter-
est on demand notes and other borrowed money.
Non-interest expenses include salaries, expenses
associated with premises and fixed assets, taxes
and other expenses. Bank deposits are disaggre-
gated into transaction and non-transaction depos-
its because they have different turnover and cost
structures. These inputs represent measures for
the banks’ labor, capital and operating costs. De-
posits and funds purchased (measured by their
interest expense) are the source of loanable funds
to he invested in assets.”
~ ~( fI••tissn’t.:ri. .tInnk
~/ I 1
The flEA scores and returns to scale measures
resulting fromapplying the CCII and additive flEA
models ar’e presented in table 1.” Althoughthe
overall results are similar across the two models,
there are minor differences in the individual effi-
ciency scores that may provide information about
the relative efficiency of these banks.
The two models differ fundamentally in their
definition of the efficiency frontier. In particular,
the CCII model assumes constant returns to scale,
while the additive model allovvs for the possibility
of constant (C), increasing (I) or decreasing (fi)
returns. Because of this, banks that are efficient in
the CCII model must also he efficient in the addi-
tive model. As table I illustrates for our Missouri
banks, the converse, however, is not true.
The overall efficiency score is composed of
“pure” technical and “scale” efficiencies. In the
CCR model, a firmwhich is technologically effi-
cient also uses the most efficient scale of opera-
tion. In the additive model, however, the score
represents only “pure” technical efficiency. By
comparing the results of the CCII and additive
models, we can see that while five of our Missouri
banks were technologically efficient, they were
not operating at the most efficient scale of opera-
tion. The reader is cautioned, however, that this
analysis excludes a number of factors (such as
demographic characteristics of the markets in
which they operate) that may be important in
determining the most economically efficient scale
of operation.
Since the efficiency scores are defined differ-
ently inthe CCII and the additive flEA models, it is
not possible to generate a measure of scale ineffi-
ciency using the results in table 1. Nevertheless,
the fact that the efficiency scores from the two
models are quite similar suggests that the scale
inefficiency is not a major source of overall ineffi-
ciency for these banks. It appears that the ineffi-
cient banks simply used too many inputs or
produced too fewoutputs rather than chose the
incorrect scale for production.”
~ *‘flw( CI)
‘.1.. ~ I.).’~ ~ .~
An illustration of the use of flEA analysis can he
obtained by considering the data for the bank
with the lowest efficiency score, bank 59. The
results for this hank are summarized in table 2.
The reference banks making up the facet to which
bank 59 is compared and “lambda,” a measure of
the relative importance of each reference bank in
the facet, are given. The table shows that three
reference banks compose the facet for bank 59.
Banks 51 and 39 play the major role and the other
bank is relatively unimportant.
“This is controversial, however. Some researchers specify
deposits as outputs, arguing that treating deposits as inputs
makes banks that depend on purchased money look artifi-
ciallyefficient (see Berg et al., 1990).
“The results from solving the DEA model also include informa-
tion about DEA scale efficiencies, the efficient projection on
the efticiency frontier, slack variables s,’ and s, - and the dual
variables Yr and u,. The “dual” variables represent “shadow
prices” for each input and output. That is, they represent the
marginal effects of the input and output variables on the
bank’s DEA efficiency score. See appendixA for details.
“Similar results of insignificant scale-inefficiency of U.S. banks
have been reported by Aly et al. (1990).
Table 1
Overall Performance of 60 Missouri Commercial Banks
Evaluated by the CCR and Additive DEA Models (1984)
EfficiencyRatio Efficiency Ratio
Bank CCR Additive Type of Bank CCR Additive Type of
no. model -. model scale’ no. model model scale’
8545 .8825 0 31 .8568 9310 D
2 .9228 I 0000 I 32 .9305 9537 D
3 .9033 9129 I 33 .8509 .8642 D
4 .8588 9498 I 34 8392 .9554
5 1.0000 1.0000 C 35 .8596 8986
6 .8766 9042 I 36 1.0000 1 0000 C
7 .8709 9144 I 37 8712 .9813
8 8841 9323 I 38 .8707 .9150 I
9 8735 .9857 I 39 1.0000 1 0000 C
10 8115 9116 I 40 1.0000 1.0000 C
11 9086 9856 I 41 8500 9453
12 7852 8388 I 42 .8867 .9656 I
13 8338 .9927 I 43 .8220 .8965
14 9739 .9024 I 44 .8254 9069
15 .8937 9829 I 45 1 0000 1 0000 C
16 .8292 8492 I 46 .9124 9889
17 8705 8211 I 47 10000 10000 C
18 9684 .9783 I 48 10000 1.0000 C
19 .8439 1.0000 D 49 .9507 9890 I
20 .9527 9930 I 50 1 0000 1 0000 C
21 9746 10000 I 51 1.0000 1.0000 C
22 8681 .8888 I 52 1.0000 1.0000 C
23 .9744 9642 I 53 .8992 9705 I
24 9003 9646 54 9443 1.0000
25 1 0000 1.0000 C 55 .9303 .9931
26 8714 .8406 I 56 8889 1.0000 0
27 1.0000 1 0000 C 57 8434 9338
28 1.0000 1.0000 C 58 10000 10000 C
29 8753 9351 I 59 7600 7824
30 9003 .93I9 D 60 .861~ .9541 I
Scale efficiency is measured by the CCR model
C = corlsla’it returns to scale
= ,ncroasinq returns to scale
D = decreasing returns to scale
Determined by the CCR model.
I he ~aIur niraswr in the lint column in he I able 2 ako prescols LI rnea~ure or h,mk ill
lonri half cit the table ~ius the ~aloe of We denoli’d as he dual -l his measure is impurtani
outputs and the input— Inc bank .19 in 11151. liii- berau~r he ratio ol the duals or- outpuK and
second rultnoo gi~es thr~aloe mr.hurr that hank inputs ~ s the I radent I of increments Or derre—
.111 noold ha~rIn arhre~ein order to hi’ RI. \ dli— merits in inputs and otilputs to Dl. \ rIljcienc~
he tlilierence brlueen Ihese numbers i~ I his H ~ jIb lilt’ as~uniptinn that hid hank is tree to
presented in the third column ‘‘ Rank 513 shnukl ~an oil it its inputs anci outputs. I he lad that the
ocr-ease its total loans In 113 percent arid its non— dual lot \ll. is larue relalke lu the nIliei’,~i’’Psls
mien’,! income In Ii per ol Rank .59 should that the biggest elIicienr~gains hit hank .19 u ill
r edut-e its I our inpots h~2U.ti perrent oF interest comet rum derrea~ing non—inti’resi e\penses.
e\penses and In 2-I perce nl of the other inputs, similar anal\ sk can hr cundot-ted for each joel Ii
‘i-i the case ci oarpus this c,if
4
ere-ice is a measup of
slacK In tie case o’ rpuIs nowev~r.tie sacK variaDle
~, ri~recomplicated
38
cient bank to determine its reference banks and
the way in which it can become DEAefficient.
Table 2
4 (/ I
3 I—,
implies decreasing returns to scale; I A
1
< I
implies increasing returns to scale.
4. An adjustment can be made in order to move
(or project) inefficient DM1.)
0
onto the efficiency
frontier. The projection (1, y) in the CCR model is
formed by the following formulas:
= 0
0
x,
0
— 5; i = 1, .., in
y,.
0
’ = y~
0
+ s r = I, .., S.
The differences (x,
0
x,,,), i = I,.., m, represent
amounts of inputs to be reduced; (y,.
0
— yr,,),
r = 1 ,..,s, represent the amounts of outputs to be
increased in order to move DML’,, onto the effi-
ciency frontier. Hence, these differences can
provide diagnostic information about the ineffi-
ciency of DMU
0
.
‘This also opens the way for many different DEA models
which are refined, more flexible or more convenient for
computations. These DEA models (BCC model, additive
DEA model, cone ratio DEA model, CCW model) and their
mathematical characteristics are beyond this paper.
2
For the c-Method, see Zukhovitskiy et al, (1966), pp. 46-51.
42
5. Problem 1 is defined as the “primal” problem
while problem 2 is the “dual.” The dual variables
have the economic interpretation of “shadow
prices.” The value of i-’, indicates the marginal
effect of input x,,, on the PEA efficiency score. ‘[he
value of Mr indicates the marginal effect of output
Yr on the PEAefficiency score. A comparison of
these dual variables provides information on the
relative importance of inputs and outputs in the
PEA evaluation.
6. In the CCR model, problem I (or problem 2) is
solved for each DMU. ‘rheoretically, ther’e is no
limitation on how many DMUs can enter the PEA
model. Hence, the DEA model can perform an effi-
ciency diagnosis for many DMUs.
Why is this approach referred to as data
envelopment analysis? The two inequalities in
conclusion 1,
y,.° I y,.,A
1
and O,,xr,, ~
forr = I s;i = 1,..., in
are constraints to be satisfied for the optimal solu-
tion. The fit-st inequality implies that the output of
DMU,, should not exceed the linear combination of
all observed output yr
1
thus, the optimal solutions
will create a hyperplane to envelop the output of
DMU,, from above. Similarly, the second constraint
can be interpreted such that the optimal solutions
create another hyperplane which envelops the
input of DMU
0
from below. Since bothoutputs
and inputs of the DMU evaluated are enveloped
from above and below, the name PEAexactly
matches the geometric interpretation of the
procedure.
To see how this works, assume that there is a
group of DMUs that produces the same outputs
using the same inputs, but in varying amounts. In
ranking their efficiencies of DMUs, PEAassigns
weights to the outputs and inputs of each DMU.
These weights are neither predetermined not’
based on prior information or preferences of the
decision makers. Instead, each DMU receives a set
of “optimal” weights that are determined by
solving the above mathematical programming
problem. This procedure generates a PEAeffi-
ciency score for the DMU evaluated based on the
solution value for the input and output weights.
A set of constraints guarantees that no DMU,
including the one evaluated, can obtain an effi-
ciency score that exceeds unity. In this way, PEA
derives a measure of the relative efficiency rating
for each DMU in the cases of multiple input and
output.
~J•:%iJiitii.i~ 4:Ittfl.1f?i
Among PEA models, the additive model has
been important in applications. The additive
model can be formalized as the following two
problems, which are dual to each other.~
Problem 3:
Max I s;/[x
0
[ + I S~/])J~~]
1=3
subject to
x,
0
— I x,
1
A
1
— s; = 0, 1 YriAl_ s; YrO’
I A~ = 1, A~ °, 5; 0, 57 0,
for i = 1,.., m; r = 1,.., s; 5 = 1,.., n.
Problem 4:
Mm r~l M,Yro + I V
1
X
13
+
subject to:
r~i ~ + I v,x,
1
+ u
0
0,
V
1
1/ [x
5
[, Mr 11 ~y,.,,],
fori= l~.,m;r= 1,..,s;j= 1,..,n.
Compared with the CCR model, the additive
model has introduced another constraint
A
1
= I and a new variable u,,. The new
constraint in problem 3 ensures that the efficiency
frontier is constructed by the convex combina-
tions of original data points rather than a convex
cone as in the CCR model. The new variable u
0
in
problem 4 is used to identify returns to scale. The
other variables in the additive model have
interpretations similar to the CCR model.
In addition, there is a difference in the way the
additive model and the CCR ratio model locate the
efficient reference point on the facet. In figur-e
Al, an output isoquant consists of input combina-
tions for five firms (F, F
2
, F
3
, F,and F
3
) in the case
of one-output
and two-input (x, and x
3
). Point F,
represents an inefficient DMLI which uses more of
x, and x,to produce the same amount of output as
3
See Charnes et al. (1985).
4 ‘2
Input x,
to the intersection point B divided by the length
from the origin to F,. In the additive model,
however, the reference efficient point on facet
F,-F,is denoted by A, which is determined by
maximizing the sum of the slacks, s, +5,. Geometri-
cally, the slackvariables are expressed by the
horizontal line starting from F, and the vertical
line extending to the facet F,-F,. Point A is selected
such that the sum of the lengths of the horizontal
and vertical lines are maximized. The PEAeffi-
ciency score in the additive model that we used is
computed by the following formula:
(I, 4 + r=3 ~ x,
0
+ r~1 yr,, + r~ 2s7).
where 4 and y,, are corresponding inputs and
outputs of the efficient reference point, such as
point A.
The PEA scale efficiency in the additive model is
identified by a variable u,, in problem 4 in accor-
dance with the followingcriteria:
If u
0
= 0, DMU,, has constant returns to scale;
otherwise,
its efficient reference DMUs, F, and F
3
. By the CCR
ratio model, the efficiency score is determined via
a value h
0
, which can be interpreted interms of
the ray from the origin to F,. That is, h
0
is ex-
pressed by the length of the ray from the origin
,‘i, n~n
1
erFsdh’v ii~t
4
4
4
4
,rlt,tA4.,1 24.
.,r .,,
u
0
> 0 implies decreasing returns to scale;
u
0
< 0 implies increasing returns to scale.
The value of variable u,, is part of an optimal solu-
tion of the additive model and is produced by the
computer code such that facet rate =
C0( (~?7C7i7’f’~ ~ ~2’= \~t (;;i=;o,fl
In measuring and evaluating technical and scale
efficiencies there are two basic approaches: the
PEA technique developed by Charnes, Cooper and
others in operations research and the approach
developed by Farrell, Fare and Grosskopf, among
others, in economics.’ The latter approach is
based upon a set of axioms on production tech-
nology to define the concept of efficiency. Some
connections of the two approaches have been
investigated by Banker, Charnes and Cooper
(1984) and by Fare and Hunsaker (1986).
Both approaches share the characteristics that
there is no need to specify a production function
or cost function and to estimate the parameters.
Therefore, they are nonparametric, nonstochastic
techniques that canbe used toconstruct a
multiproduct frontier relative to which the effi-
ciency measures of the entities in the sample are
calculated. Because the frontier in these
approaches is generated by data and all observa-
tions are enveloped by the frontier, both
approaches can be viewed as Data Envelopment
Figure A.1
The Difference Between CCR and Additive DEA Models
Input xl,
F,
F,
0
F, F,
1
5ee Fare and Hunsaker (1986); Fare, Grosskopf and
LovelI (1985).
44
Analysis. In this appendix, some of the differences
and similarities among the CCR and the additive
models and the Farrell or Russell models are
discussed.
The choice of efficiency reference on the rele-
vant frontier is a major difference among these
PEA models. In the Farrell or Russell models,
three measures of technical efficiency can be
defined: input, output and graph efficiency
measures.
Using the input efficiency measure, the ob-
served output vector is fixed and the search for
efficient reference is constrained to proportion-
ally reducing inputs until the efficient frontier is
reached. The “ratio of contraction,” as it is called,
is the ratio of the particular input to be efficient to
the current level of inputs (in the Farrell input
model).
Using the output efficiency measure, the ob-
served input vector is fixed and the outputs pro-
portionally expanded until the efficient frontier is
reached. ‘The “stretch ratio” of the output, as it is
called, is the ratio of efficient output to the current
level of output (in the Farrell output model).
For the graph efficiency measure, both input
and output vectors are varied. Inputs are reduced
and outputs are expanded, both proportionally,
with the input ratio reciprocal to the output ratio.
In the case of figure 1 in the text, A is the refer-
ence point for the input efficiency measure, Bis
the reference point for the output efficiency
measure and C might be the reference point for
the graph efficiency measure. These three effi-
ciency measures can be classified as radial
because proportional changes of inputs and/or
outputs are used in defining them.
To illustrate the input efficiency measure, ray
OF
3
in figure 1 of the text is used to represent the
optimal scale that would be generated by long-run
competitive equilibrium. The overall input effi-
ciency measure is defined with respect to the ray
OF:,, while the input pure technical efficiency is
defined with respect to the line segment connect-
ing F,, F, and F,. The measure of input overall
technical efficiency, KP/KF,, can be decomposed
into the measure of pure technical input efficiency
given by the ratio KA/KF, and the measure of input
scale efficiency given by the ratio KD/KA. When
the scale efficiency equals unity, the constant re-
turns to scale occur; otherwise non-increasing or
varying returns to scale hold.
It is clear from these examples that, in general,
these tadial efficiency measures will be different.
Moreover, there is nothing to guarantee that a
firm that is output efficient by this measure is also
input efficient or vice versa. For example, the firm
denoted by F
6
in figure 1 of the text is output effi-
cient by the output efficiency measure, but is not
input efficient (see Fare, Grosskopf and Lovell
(1985)). Howevem-, the Farrell input efficiency
measure is reciprocal to the Farrell output effi-
ciency measure, if and only if, the technology is
homogeneous degree one. Because this condition
is satisfied by constant returns to scale tech-
nology, the Farrell input and output efficiency
measures are “identical” in this case. For models
with other technologies, simple relationships
between input and output efficiency measures do
not hold.
An improvement of the Farrell or Russell models
over the others is the use of non-radial efficiency
measures. The use of proportional changes of
inputs and/or outputs in searching for efficient
reference is abandoned.
Moreover, different piecewise linear technology
can be accommodated in both Farrell and Russell
models to meet the needs of various users. For
example, to measure scale efficiency we can use
constant returns to scale, non-increasing returns
to scale or varying returns to scale technologies.
Thesetechnology constraints canbe easily imposed
by corresponding restrictions on the “intensity
parameters” in the Farrell or Russell models.
In the CCR or additive PEA model discussed in
appendix A, however, only one efficiency measure
is defined: the CCII model uses the radial measure
of efficiency while the additive model uses the
non-radial measure.
Geometrically, the efficiency frontier with cons-
tant returns to scale technology is a convex cone,
but it is a convex hull in cases of both non-increas-
ing and varying returns to scale. In general, these
constraints on technology form a chain such that
one efficiency frontier is enveloped by another.
Consequently, the associated efficiency measures
are compatible and nested.’
2
5ee Grosskopf (1986).
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doc_201288161.pdf
Commmercial Bank Play a vital role in the economy for two reasons: they provide a major source of financial intermediation and their checkable deposit liabilities represent the bulk of the nation’s money stock. Evaluating their overall performance and monitoring their financial condition is important to depositors, owners, potential.
31
Piyu Yue
Piyu Yue, a research associate at the IC
2
Institute, University of
Texas at Austin, was a visiting scholar at the Federal Reserve
Bankof St. Louis when this article was written. Lynn Dietrich
providedresearch assistance. The author would like to thank A.
Charnes, Roll Fare and Shawna Grosskopf for theirconstructive
comments and useful suggestions. Their DEA computer code led
to a significant improvement ofthe paper
Data Envelopment Analysis and
Commercial Bank Performance:
A Primer With Applications to
Missouri Banks
OMMERCIAL BANKS PLAY a vital role in the
economy for two reasons: they provide a major
source of financial intermediation and their check-
able deposit liabilities represent the bulk of the
nation’s money stock. Evaluating their overall
performance and monitoring their financial condi-
tion is important to depositors, owners, potential
investors, managers and, of course, regulators.
Currently, financial ratios are often used to
measure the overall financial soundness of a bank
and the quality of its management. Bank regu-
lators, for example, use financial ratios to help
evaluate a bank’s performance as part of the
CAMEL system.
1
Evaluating the economic perfor-
mance of banks, however, is a complicated
process. Often a number of criteria such as
profits, liquidity, asset quality, attitude toward
risk, and management strategies must be consi-
dered. The changing nature of the banking
industry has made such evaluations even more
difficult, increasing the needfor more flexible
alternative forms of financial analysis.
This paper describes a particular methodology
called Data Envelopment Analysis (DEA), that has
been used previously to analyze the relative effi-
ciencies of industrial firms, universities, hospitals,
military operations, baseball players and, more
recently, commercial banks.2 The use of flEA is
demonstrated by evaluating the management of
60 Missouri commercial banks for the period from
1984 to 1990.~
1
For more details, see Booker (1983), Korobow (1983) and
Putnam (1983).
2
The name DEA is attributed to Charnes, Cooper and Rhodes
(1978), for the development of DEA, see Charnes, et al.(1 985)
and Charnes, et at. (1978); for some applications of DEA, see
Banker, et al. (1984), Charnes, et al. (1990) and Sherman and
Gold (1985).
3
Although there is vast literature analyzing competition and
performance in the U.S. banking industry (e.g., Gilbert (1984),
Ehten (1983), Korobow(1983), Putnam (1983), Wall (1983)
and Watro (1989)), actual banking efficiency has received
limited attention. Recently, a few publications have used IDEA
or asimilar approach to study the technical and scale efficien-
cies of commercial banks (e.g., Sherman and Gold (1985),
Charnes etal. (1990), Rangan et al. (1988), Aty et al. (1990),
and Etyasiani and Mehdian (1990)).
a4~sIcs
flEA represents a mathematical programming
methodology that can be applied to assess the effi-
ciency of a varietyof institutions using a variety of
data. This section provides an intuitive explana-
tion of the DEAapproach. A formal mathematical
presentation of flEA is described in appendix A; a
slightly different nonparametric approach is
described in appendix B.
flEA is based on a concept of efficiency that is
widely used in engineering and the natural
sciences- Engineering efficiency is defined as the
ratio of the amount of work performed by a
machine to the amount of energy consumed in the
process. Since machines must be operated
according to the law of conservation of energy,
their efficiency ratios are always less than or equal
to unity.
This concept of engineering efficiency is not
immediately applicable to economic production
because the value of output is expected to exceed
the value of inputs due to the “value added” in
production. Nevertheless, under certain circum-
stances, an economic efficiency standard—similar
to the engineering standard—can be defined and
used to compare the relative efficiencies of
economic entities. For example, a firm can be said
to be efficient relative to another if it produces
either the same level of output with fewer inputs
or more output with the same or fewer inputs. A
single firm is considered “technically efficient” if it
cannot increase any output or reduce any input
without reducing other outputs or increasing
other inputs.
4
Consequently, this concept of tech-
nical efficiencyis similar to the engineering
concept. The somewhat broader concept of
“economic efficiency,”on the other hand, is
achieved when firms find the combination of
inputs that enable them to produce the desired
level of output at minimum cost.
5
The discussion of the flEA approach will be
undertaken in the context of technical efficiency
in the microeconomic theory of production. tn
microeconomics, the production possibility set
consists of the feasible input and output combina-
tions that arise from available production tech-
nology. The production function (or production
transformation as it is called in the case of multiple
outputs) is a mathematical expression for a
process that transforms inputs into output. In so
doing, it defines the frontier of the production
possibility set. For example, consider the well-
known Cobb-Douglas production function:
(1) Y = AK~Ll~a,
where Y is the maximum output for given quanti-
ties of two inputs: capital (K) and labor (Ii. Even if
all firms produce the same good (Y) withthe same
technology defined by equation 1, they may still
use different combinations of labor and capital to
produce different levels of output. Nonetheless, all
firms whose input-output combinations lie on the
surface (frontier) of the production relationship
definedby equation 1 are said to be technologi-
cally efficient. Similarly, firms with input-output
combinations located inside the frontier are tech-
nologically inefficient.
DEAprovides a similar notion of efficiency. The
principal difference is that the flEA production
frontier is not determined by some specific equa-
tion like that shown in equation 1; instead, it is
generated from the actual data for the evaluated
firms (which in flEA terminology are typically
called decision-making units or DMU5).
6
Conse-
quently, the flEA efficiency score for a specific
firm is not defined by an absolute standard like
equation 1. Rather, it is defined relative to the other
firms under consideration. And, similar to engi-
neering efficiency measures, DEA establishes a
“benchmark” efficiency score of unity that no
individual firm’s score can exceed. Consequently,
efficient firms receive efficiency scores of unity,
while inefficient firms receive DEA scores of less
than unity.
4
See Koopmans (1951).
~This is also named “allocative efficiency” because a profit
maximizing firm must allocate its resources such that the
technical rate of substitution is equal to the ratio of the prices
of the resources. Theoretical considerations of atlocative eff i-
ciency can be found in the articles by Banker (1984) and
Banker and Maindiratta (1988).
mated production function represents the average behavior of
firms in the sample. Hence, the estimated production function
depends upon the data for both efficient and inefficient firms.
By imposing suitable constraints, these statistical procedures
can be modified to orient the estimates toward frontiers. In
this manner, the frontier of the production set can be esti-
mated econometrically.
°ltis common to estimate production functions using regres-
sion analysis. When cross-section data are used, the esti-
33
In microeconomic analysis, efficient production
is defined by technological relationships with the
assumption that firms are operated efficiently.
Whether or not firms have access to the same
technology, it is assumed that they operate on the
frontier of their relevant production possibilities
set; hence, they are technically efficient by defini-
tion. As a result, much of microeconomic theory
ignores issues concerning technological ineffi-
ciencies.
flEA assumes that all firms face the same
unspecified technology which defines their
production possibilities set. The objective of flEA
is to determine which firms operate on their effi-
ciency frontier and which firms do not. That is,
flEApartitions the inputs and outputs of all firms
into efficient and inefficient combinations. The
efficient input-output combinations yield an
implicit production frontier against which each
firm’s input and output combination is evaluated.
If the firm’s input-output combination lies on the
DEA frontier, the firm might be considered effi-
cient; if the firm’s input-output combination lies
insidethe DEA frontier, the firm is considered
inefficient.
An advantage of DEA is that it uses actual
sample data to derive the efficiencyfrontier
against which each firm in the sample can be
evaluated.~As a result, no explicit functional form
for the production function has to be specified in
advance. Instead, the production frontier is gener-
ated by a mathematical programming algorithm
which also calculates the optimal DEA efficiency
score for each firm.
To illustrate the relationship between DEA and
economic production in its simplest form,
consider the example shown in figure 1, in which
firms use a single input to produce a single output.
In this example, there are six firms whose inputs
are denoted as x, and whose outputs are denoted
as y~(i = 1,2,..,6); their input-output combinations
are labeled by F
8
(s = 1,2 6). While the produc-
tion frontier is generated by the input-output
combinations for the firms labeled F
1
, F
3
, F
5
and F
6
,
the efficient portion of the production frontier is
shown by the connected hne segments. F
2
and F
4
are clearly flEA inefficient because they lie inside
the frontier; F
6
is flEA inefficient because the
same output can be produced with less input.
hc. ..lmporiance of .rni.’eis in fJE~1
“Facets” are an important concept used to
evaluate a firm’s efficiency in flEA. The efficiency
measure in DEAis concerned withwhether a firm
can increase its output using the same inputs or
produce the same output with fewer inputs.
Consequently, only part of the entire efficiency
frontier is relevant when evaluating the efficiency
of a specific firm. The relevant portion of the effi-
ciency frontier is called a facet. For example, in
figure 1, onlythe facet from F, to F
3
is relevant for
evaluating the efficiency of the firm designated by
F
2
. Similarly, only the facet from F
3
to F
5
is used to
evaluate the firm denoted by F
4
.
8
The use of facets with flEA enables analysts to
identify inefficient firms and, through comparison
with efficient firms on relevant facets, to suggest
ways in which the inefficient firms might improve
their performance. As illustrated in figure 1, F
2
can become efficient by rising to some point on
the F,-F
3
facet. In particular, it could move to A by
simply using less input, to B by producing more
output or to C by both reducing input and
increasing output- Of course, in this example, the
analysis is obvious and the recommendation
trivial. In more complicated, multiple input-
multiple output cases, however, the appropriate
efficiency recommendations would be much more
difficult to discover without the flEA
methodology.~
7
DEA has two theoretical properties that are especially use-
ful for its implementation. One is that the IDEA model is
mathematically related to a multi-objective optimization
problem in which all inputs and outputs are defined as
multiple objectives such that all inputs are minimized and
all outputs are maximized simultaneously under the tech-
nology constraints. Thus, IDEA-efficient DMUs represent
Pareto optimal solutions to the multi-objective optimization
problem, while the Pareto optimal solution does not neces-
sarily imply DEA efficiency.
Another important property is that IDEA efficiency scores
are independent of the units in which inputs and outputs
are measured, as tong as these units are the same tor all
IDMUs. These characteristics make the IDEA methodology
highly flexible. The only constraint set originally in the
CCR model is that the values of inputs and outputs must
be strictly positive.
This constraint, however, has been abandoned in the new
additive IDEA formulation. As a consequence, the additive
IDEA model is used to compute reservation prices for new
and disappearing commodities in the construction of price
indexes by Lovell and Zieschang (1990).
8
1n a multiple dimensional space, the efficiency frontier
forms a polyhedron. In geometry, a portion of the surface
of a polyhedron is called a facet; this is why the same
term is used in IDEA. These facets have important implica-
tions in empirical studies, such as identification of compe-
titors and strategic groups in an industry. See Day, Lewin,
Salazar and Li (1989).
°Foralternative measures of efficiency, see appendix B.
34
Figure 1
Production Frontier and Efficiency Subset
Output Y
K
Scale E/!iciencv
In addition to measuring technological effi-
ciency, flEA also provides information about scale
efficiencies in production. Because the measure of
scale efficiency in DEA analysis varies from model
to model, care must be exercised. The scale effi-
ciency measured for the flEA model used in this
study, however, corresponds fairly closely to the
microeconomic definition of economics of scale in
the classical theory of production.’°
To illustrate, consider the F,-F
3
facet in figure 2.
Firms located on this facet exhibit increasing
returns to scale because a proportionate rise in
their input and output placesthem inside the
production frontier. A proportionate decrease in
their input and output is impossible because it
would move them outside of the frontier. This is
illustrated by a ray fromthe origin that passes
through the F,-F
3
facet at F’
2
.
Firms located on the F
3
-F
3
facet exhibit
decreasing returns to scale because a propor-
tionate decrease in their input and output places
them inside the production frontier. A propor-
tionate increase in their input and output is impos-
sible because it would move them outside of the
frontier.
Constant returns to scale occur if all propor-
tionate increases or decreases ininputs and
outputs move the firm either along or above the
production frontier. In figure 2, for example, F,
exhibits constant returns to scale because propor-
tionate increases or decreases would place it
outside the production frontier.
Since the facets are generated by efficient firms,
the scale efficiency of these firms is determined by
the properties of their particular facet. Scale effi-
ciencies for inefficient firms are determined by
their respective reference facets as well. Thus, F
2
and F
4
in figure 1 exhibit increasing and
decreasing returns to scale, respectively.
.LIEA and Ft’nr,mnk’ I1/
1
/ic1t?nr~t.~
While the discussion of flEA in the context of
technological efficiency of production is useful for
illustrative purposes, it is far too narrow and
limiting. flEA is frequently applied to questions
and data that transcend the narrow focus of tech-
nical efficiency in production. For example, DEA is
frequently applied to financial data when
addressing questions of economic efficiency. In
this regard, its application is somewhat more
problematic. For example, when firms face
different marginal costs of production due to
regional or local wage differentials, one firm may
appear inefficient relative to another. Given the
potential differences in relative costs that a firm
may face, however, it might be equally efficient.
Alternatively, differences that appear to be due to
economic inefficiencies may in fact be due to cost
differences directly attributable to the non-
homogeneity of products. Because of problems
like these, flEA must be applied judiciously.
.na1. g.J/j,,j
0~1
To this point, the discussion of flEA has been
concerned with evaluating the relative efficiency
of different firms at the same time. Those who use
flEA, however, frequently employ a type of sensi-
tivity analysis called“window analysis.” The
performance of one firm or its reference firms
‘°SeeFare, Grosskopf and Lovell (1985). Different DEA
models employ different measures of scale efficiency. See
F,
F, /
F,
F,
0
Input X
appendixes A and B for details,
Figure 2
An Illustration of Scale Efficiencies
OutputY
for a firm over time. Of course, comparisons of
flEA efficiency scores over extended periods may
be misleading (or worse) because of significant
changes in technology and the underlying
economic structure.
0
may be particularly “good” or “bad” at a given time
because of factors that are external to the firm’s
relativeefficiency. In addition, the number of
firms that can be analyzed using the flEA model is
virtually unlimited. Therefore, data on firms in
different periods can be incorporated into the
analysis by simply treating them as if they
represent different firms. In this way, a given firm
at a given time can compare its performance at
different times and with the performance of other
firms at the same and at different times. Through
a sequence of such “windows,” the sensitivity of a
firm’s efficiency score can be derived for a partic-
ular year according to changing conditions and a
changing set of reference firms.” A firm that is
flEA efficient in a given year, regardless of the
window, is likelyto be truly efficient relative to
other firms. Conversely, a firm that is only flEA
efficient in a particular window may be efficient
solely because of extraneous circumstances.
In addition, window analysis provides some
evidence of the short-run evolution of efficiency
APPLYING DEE TO BANKING:
AN EVALU.zVI’ION (IF’ 60 MISSOURI
(X)MJ.~IERCI.ALB
To demonstrate flEA’s use, it is applied to
evaluate relative efficiency in banking. Financial
data for 60 of the largest Missouri commercial
banks for 1984 (determined by their total assets in
1990) are used. Initially, the relative efficiency of
these banks is examined using two alternative
flEA models: the CCII model and the additive flEA
model. A discussion of these alternative DEA
models appears in appendix A. In extending the
discussion and analysis, however, we focus solely
on the CCII model.
Measuring inputs and Outputs
Perhaps the most important step in using flEA to
examine the relative efficiency of any type of firm
is the selection of appropriate inputs and outputs.
This is partially true for banks because there is
considerable disagreement over the appropriate
inputs and outputs for banks. Previous applica-
tions of flEA to banks generally have adopted one
of two approaches to justify their choice of inputs
and outputs.’
2
The first “intermediary approach” views banks
as financial intermediaries whose primary busi-
ness is to borrow funds from depositors and lend
those funds to others for profit. In these studies,
the banks’ outputs are loans (measured in dollars)
and their inputs are the various costs of these
funds (including interest expense, labor, capital
and operating costs).
A second approach views banks as institutions
that use capital and labor to produce loans and
deposit account services. In these studies, the
banks’ outputs are their accounts and transac-
tions, while their inputs are their labor, capital
and operating costs; the banks’ interest expenses
are excluded in these studies.
‘
1
This is called “panel data analysis” in econometrics.
‘
2
Some studies have adopted the simple rule that if it
produces revenue, it is an output; if it requires a net ex-
F, F,
F,
F’,
F,
Input X
penditure, it is an input. For example, see Hancock (1989).
36
Our analysis of 60 Missouri banks uses a variant
of the intermediary approach. The banks’ outputs
are interest income (IC), non-interest income (N1C)
and total loans (1’L). Interest income includes
interest and fee income on loans, income from
lease-financing receivables, interest and dividend
income on securities, and other income. Non-
interest income includes service charges on
deposit accounts, income fromfiduciary activities
and other non-interest income. Total loans consist
of loans and leases net of unearned income. These
outputs represent the banks’ revenues and major
business activities.
‘I’he banks’ inputs are interest expenses (IE),
non-interest expenses (NIE), transaction deposits
(Tfl), and non-transaction deposits (NTD). Interest
expenses include expenses for federal funds and
the purchase and sale of securities, and the inter-
est on demand notes and other borrowed money.
Non-interest expenses include salaries, expenses
associated with premises and fixed assets, taxes
and other expenses. Bank deposits are disaggre-
gated into transaction and non-transaction depos-
its because they have different turnover and cost
structures. These inputs represent measures for
the banks’ labor, capital and operating costs. De-
posits and funds purchased (measured by their
interest expense) are the source of loanable funds
to he invested in assets.”
~ ~( fI••tissn’t.:ri. .tInnk
~/ I 1
The flEA scores and returns to scale measures
resulting fromapplying the CCII and additive flEA
models ar’e presented in table 1.” Althoughthe
overall results are similar across the two models,
there are minor differences in the individual effi-
ciency scores that may provide information about
the relative efficiency of these banks.
The two models differ fundamentally in their
definition of the efficiency frontier. In particular,
the CCII model assumes constant returns to scale,
while the additive model allovvs for the possibility
of constant (C), increasing (I) or decreasing (fi)
returns. Because of this, banks that are efficient in
the CCII model must also he efficient in the addi-
tive model. As table I illustrates for our Missouri
banks, the converse, however, is not true.
The overall efficiency score is composed of
“pure” technical and “scale” efficiencies. In the
CCR model, a firmwhich is technologically effi-
cient also uses the most efficient scale of opera-
tion. In the additive model, however, the score
represents only “pure” technical efficiency. By
comparing the results of the CCII and additive
models, we can see that while five of our Missouri
banks were technologically efficient, they were
not operating at the most efficient scale of opera-
tion. The reader is cautioned, however, that this
analysis excludes a number of factors (such as
demographic characteristics of the markets in
which they operate) that may be important in
determining the most economically efficient scale
of operation.
Since the efficiency scores are defined differ-
ently inthe CCII and the additive flEA models, it is
not possible to generate a measure of scale ineffi-
ciency using the results in table 1. Nevertheless,
the fact that the efficiency scores from the two
models are quite similar suggests that the scale
inefficiency is not a major source of overall ineffi-
ciency for these banks. It appears that the ineffi-
cient banks simply used too many inputs or
produced too fewoutputs rather than chose the
incorrect scale for production.”
~ *‘flw( CI)
‘.1.. ~ I.).’~ ~ .~
An illustration of the use of flEA analysis can he
obtained by considering the data for the bank
with the lowest efficiency score, bank 59. The
results for this hank are summarized in table 2.
The reference banks making up the facet to which
bank 59 is compared and “lambda,” a measure of
the relative importance of each reference bank in
the facet, are given. The table shows that three
reference banks compose the facet for bank 59.
Banks 51 and 39 play the major role and the other
bank is relatively unimportant.
“This is controversial, however. Some researchers specify
deposits as outputs, arguing that treating deposits as inputs
makes banks that depend on purchased money look artifi-
ciallyefficient (see Berg et al., 1990).
“The results from solving the DEA model also include informa-
tion about DEA scale efficiencies, the efficient projection on
the efticiency frontier, slack variables s,’ and s, - and the dual
variables Yr and u,. The “dual” variables represent “shadow
prices” for each input and output. That is, they represent the
marginal effects of the input and output variables on the
bank’s DEA efficiency score. See appendixA for details.
“Similar results of insignificant scale-inefficiency of U.S. banks
have been reported by Aly et al. (1990).
Table 1
Overall Performance of 60 Missouri Commercial Banks
Evaluated by the CCR and Additive DEA Models (1984)
EfficiencyRatio Efficiency Ratio
Bank CCR Additive Type of Bank CCR Additive Type of
no. model -. model scale’ no. model model scale’
8545 .8825 0 31 .8568 9310 D
2 .9228 I 0000 I 32 .9305 9537 D
3 .9033 9129 I 33 .8509 .8642 D
4 .8588 9498 I 34 8392 .9554
5 1.0000 1.0000 C 35 .8596 8986
6 .8766 9042 I 36 1.0000 1 0000 C
7 .8709 9144 I 37 8712 .9813
8 8841 9323 I 38 .8707 .9150 I
9 8735 .9857 I 39 1.0000 1 0000 C
10 8115 9116 I 40 1.0000 1.0000 C
11 9086 9856 I 41 8500 9453
12 7852 8388 I 42 .8867 .9656 I
13 8338 .9927 I 43 .8220 .8965
14 9739 .9024 I 44 .8254 9069
15 .8937 9829 I 45 1 0000 1 0000 C
16 .8292 8492 I 46 .9124 9889
17 8705 8211 I 47 10000 10000 C
18 9684 .9783 I 48 10000 1.0000 C
19 .8439 1.0000 D 49 .9507 9890 I
20 .9527 9930 I 50 1 0000 1 0000 C
21 9746 10000 I 51 1.0000 1.0000 C
22 8681 .8888 I 52 1.0000 1.0000 C
23 .9744 9642 I 53 .8992 9705 I
24 9003 9646 54 9443 1.0000
25 1 0000 1.0000 C 55 .9303 .9931
26 8714 .8406 I 56 8889 1.0000 0
27 1.0000 1 0000 C 57 8434 9338
28 1.0000 1.0000 C 58 10000 10000 C
29 8753 9351 I 59 7600 7824
30 9003 .93I9 D 60 .861~ .9541 I
Scale efficiency is measured by the CCR model
C = corlsla’it returns to scale
= ,ncroasinq returns to scale
D = decreasing returns to scale
Determined by the CCR model.
I he ~aIur niraswr in the lint column in he I able 2 ako prescols LI rnea~ure or h,mk ill
lonri half cit the table ~ius the ~aloe of We denoli’d as he dual -l his measure is impurtani
outputs and the input— Inc bank .19 in 11151. liii- berau~r he ratio ol the duals or- outpuK and
second rultnoo gi~es thr~aloe mr.hurr that hank inputs ~ s the I radent I of increments Or derre—
.111 noold ha~rIn arhre~ein order to hi’ RI. \ dli— merits in inputs and otilputs to Dl. \ rIljcienc~
he tlilierence brlueen Ihese numbers i~ I his H ~ jIb lilt’ as~uniptinn that hid hank is tree to
presented in the third column ‘‘ Rank 513 shnukl ~an oil it its inputs anci outputs. I he lad that the
ocr-ease its total loans In 113 percent arid its non— dual lot \ll. is larue relalke lu the nIliei’,~i’’Psls
mien’,! income In Ii per ol Rank .59 should that the biggest elIicienr~gains hit hank .19 u ill
r edut-e its I our inpots h~2U.ti perrent oF interest comet rum derrea~ing non—inti’resi e\penses.
e\penses and In 2-I perce nl of the other inputs, similar anal\ sk can hr cundot-ted for each joel Ii
‘i-i the case ci oarpus this c,if
4
ere-ice is a measup of
slacK In tie case o’ rpuIs nowev~r.tie sacK variaDle
~, ri~recomplicated
38
cient bank to determine its reference banks and
the way in which it can become DEAefficient.
Table 2
4 (/ I
3 I—,
implies decreasing returns to scale; I A
1
< I
implies increasing returns to scale.
4. An adjustment can be made in order to move
(or project) inefficient DM1.)
0
onto the efficiency
frontier. The projection (1, y) in the CCR model is
formed by the following formulas:
= 0
0
x,
0
— 5; i = 1, .., in
y,.
0
’ = y~
0
+ s r = I, .., S.
The differences (x,
0
x,,,), i = I,.., m, represent
amounts of inputs to be reduced; (y,.
0
— yr,,),
r = 1 ,..,s, represent the amounts of outputs to be
increased in order to move DML’,, onto the effi-
ciency frontier. Hence, these differences can
provide diagnostic information about the ineffi-
ciency of DMU
0
.
‘This also opens the way for many different DEA models
which are refined, more flexible or more convenient for
computations. These DEA models (BCC model, additive
DEA model, cone ratio DEA model, CCW model) and their
mathematical characteristics are beyond this paper.
2
For the c-Method, see Zukhovitskiy et al, (1966), pp. 46-51.
42
5. Problem 1 is defined as the “primal” problem
while problem 2 is the “dual.” The dual variables
have the economic interpretation of “shadow
prices.” The value of i-’, indicates the marginal
effect of input x,,, on the PEA efficiency score. ‘[he
value of Mr indicates the marginal effect of output
Yr on the PEAefficiency score. A comparison of
these dual variables provides information on the
relative importance of inputs and outputs in the
PEA evaluation.
6. In the CCR model, problem I (or problem 2) is
solved for each DMU. ‘rheoretically, ther’e is no
limitation on how many DMUs can enter the PEA
model. Hence, the DEA model can perform an effi-
ciency diagnosis for many DMUs.
Why is this approach referred to as data
envelopment analysis? The two inequalities in
conclusion 1,
y,.° I y,.,A
1
and O,,xr,, ~
forr = I s;i = 1,..., in
are constraints to be satisfied for the optimal solu-
tion. The fit-st inequality implies that the output of
DMU,, should not exceed the linear combination of
all observed output yr
1
thus, the optimal solutions
will create a hyperplane to envelop the output of
DMU,, from above. Similarly, the second constraint
can be interpreted such that the optimal solutions
create another hyperplane which envelops the
input of DMU
0
from below. Since bothoutputs
and inputs of the DMU evaluated are enveloped
from above and below, the name PEAexactly
matches the geometric interpretation of the
procedure.
To see how this works, assume that there is a
group of DMUs that produces the same outputs
using the same inputs, but in varying amounts. In
ranking their efficiencies of DMUs, PEAassigns
weights to the outputs and inputs of each DMU.
These weights are neither predetermined not’
based on prior information or preferences of the
decision makers. Instead, each DMU receives a set
of “optimal” weights that are determined by
solving the above mathematical programming
problem. This procedure generates a PEAeffi-
ciency score for the DMU evaluated based on the
solution value for the input and output weights.
A set of constraints guarantees that no DMU,
including the one evaluated, can obtain an effi-
ciency score that exceeds unity. In this way, PEA
derives a measure of the relative efficiency rating
for each DMU in the cases of multiple input and
output.
~J•:%iJiitii.i~ 4:Ittfl.1f?i
Among PEA models, the additive model has
been important in applications. The additive
model can be formalized as the following two
problems, which are dual to each other.~
Problem 3:
Max I s;/[x
0
[ + I S~/])J~~]
1=3
subject to
x,
0
— I x,
1
A
1
— s; = 0, 1 YriAl_ s; YrO’
I A~ = 1, A~ °, 5; 0, 57 0,
for i = 1,.., m; r = 1,.., s; 5 = 1,.., n.
Problem 4:
Mm r~l M,Yro + I V
1
X
13
+
subject to:
r~i ~ + I v,x,
1
+ u
0
0,
V
1
1/ [x
5
[, Mr 11 ~y,.,,],
fori= l~.,m;r= 1,..,s;j= 1,..,n.
Compared with the CCR model, the additive
model has introduced another constraint
A
1
= I and a new variable u,,. The new
constraint in problem 3 ensures that the efficiency
frontier is constructed by the convex combina-
tions of original data points rather than a convex
cone as in the CCR model. The new variable u
0
in
problem 4 is used to identify returns to scale. The
other variables in the additive model have
interpretations similar to the CCR model.
In addition, there is a difference in the way the
additive model and the CCR ratio model locate the
efficient reference point on the facet. In figur-e
Al, an output isoquant consists of input combina-
tions for five firms (F, F
2
, F
3
, F,and F
3
) in the case
of one-output
and two-input (x, and x3
). Point F,
represents an inefficient DMLI which uses more of
x, and x,to produce the same amount of output as
3
See Charnes et al. (1985).
4 ‘2
Input x,
to the intersection point B divided by the length
from the origin to F,. In the additive model,
however, the reference efficient point on facet
F,-F,is denoted by A, which is determined by
maximizing the sum of the slacks, s, +5,. Geometri-
cally, the slackvariables are expressed by the
horizontal line starting from F, and the vertical
line extending to the facet F,-F,. Point A is selected
such that the sum of the lengths of the horizontal
and vertical lines are maximized. The PEAeffi-
ciency score in the additive model that we used is
computed by the following formula:
(I, 4 + r=3 ~ x,
0
+ r~1 yr,, + r~ 2s7).
where 4 and y,, are corresponding inputs and
outputs of the efficient reference point, such as
point A.
The PEA scale efficiency in the additive model is
identified by a variable u,, in problem 4 in accor-
dance with the followingcriteria:
If u
0
= 0, DMU,, has constant returns to scale;
otherwise,
its efficient reference DMUs, F, and F
3
. By the CCR
ratio model, the efficiency score is determined via
a value h
0
, which can be interpreted interms of
the ray from the origin to F,. That is, h
0
is ex-
pressed by the length of the ray from the origin
,‘i, n~n
1
erFsdh’v ii~t
4
4
4
4
,rlt,tA4.,1 24.
.,r .,,
u
0
> 0 implies decreasing returns to scale;
u
0
< 0 implies increasing returns to scale.
The value of variable u,, is part of an optimal solu-
tion of the additive model and is produced by the
computer code such that facet rate =
C0( (~?7C7i7’f’~ ~ ~2’= \~t (;;i=;o,fl
In measuring and evaluating technical and scale
efficiencies there are two basic approaches: the
PEA technique developed by Charnes, Cooper and
others in operations research and the approach
developed by Farrell, Fare and Grosskopf, among
others, in economics.’ The latter approach is
based upon a set of axioms on production tech-
nology to define the concept of efficiency. Some
connections of the two approaches have been
investigated by Banker, Charnes and Cooper
(1984) and by Fare and Hunsaker (1986).
Both approaches share the characteristics that
there is no need to specify a production function
or cost function and to estimate the parameters.
Therefore, they are nonparametric, nonstochastic
techniques that canbe used toconstruct a
multiproduct frontier relative to which the effi-
ciency measures of the entities in the sample are
calculated. Because the frontier in these
approaches is generated by data and all observa-
tions are enveloped by the frontier, both
approaches can be viewed as Data Envelopment
Figure A.1
The Difference Between CCR and Additive DEA Models
Input xl,
F,
F,
0
F, F,
1
5ee Fare and Hunsaker (1986); Fare, Grosskopf and
LovelI (1985).
44
Analysis. In this appendix, some of the differences
and similarities among the CCR and the additive
models and the Farrell or Russell models are
discussed.
The choice of efficiency reference on the rele-
vant frontier is a major difference among these
PEA models. In the Farrell or Russell models,
three measures of technical efficiency can be
defined: input, output and graph efficiency
measures.
Using the input efficiency measure, the ob-
served output vector is fixed and the search for
efficient reference is constrained to proportion-
ally reducing inputs until the efficient frontier is
reached. The “ratio of contraction,” as it is called,
is the ratio of the particular input to be efficient to
the current level of inputs (in the Farrell input
model).
Using the output efficiency measure, the ob-
served input vector is fixed and the outputs pro-
portionally expanded until the efficient frontier is
reached. ‘The “stretch ratio” of the output, as it is
called, is the ratio of efficient output to the current
level of output (in the Farrell output model).
For the graph efficiency measure, both input
and output vectors are varied. Inputs are reduced
and outputs are expanded, both proportionally,
with the input ratio reciprocal to the output ratio.
In the case of figure 1 in the text, A is the refer-
ence point for the input efficiency measure, Bis
the reference point for the output efficiency
measure and C might be the reference point for
the graph efficiency measure. These three effi-
ciency measures can be classified as radial
because proportional changes of inputs and/or
outputs are used in defining them.
To illustrate the input efficiency measure, ray
OF
3
in figure 1 of the text is used to represent the
optimal scale that would be generated by long-run
competitive equilibrium. The overall input effi-
ciency measure is defined with respect to the ray
OF:,, while the input pure technical efficiency is
defined with respect to the line segment connect-
ing F,, F, and F,. The measure of input overall
technical efficiency, KP/KF,, can be decomposed
into the measure of pure technical input efficiency
given by the ratio KA/KF, and the measure of input
scale efficiency given by the ratio KD/KA. When
the scale efficiency equals unity, the constant re-
turns to scale occur; otherwise non-increasing or
varying returns to scale hold.
It is clear from these examples that, in general,
these tadial efficiency measures will be different.
Moreover, there is nothing to guarantee that a
firm that is output efficient by this measure is also
input efficient or vice versa. For example, the firm
denoted by F
6
in figure 1 of the text is output effi-
cient by the output efficiency measure, but is not
input efficient (see Fare, Grosskopf and Lovell
(1985)). Howevem-, the Farrell input efficiency
measure is reciprocal to the Farrell output effi-
ciency measure, if and only if, the technology is
homogeneous degree one. Because this condition
is satisfied by constant returns to scale tech-
nology, the Farrell input and output efficiency
measures are “identical” in this case. For models
with other technologies, simple relationships
between input and output efficiency measures do
not hold.
An improvement of the Farrell or Russell models
over the others is the use of non-radial efficiency
measures. The use of proportional changes of
inputs and/or outputs in searching for efficient
reference is abandoned.
Moreover, different piecewise linear technology
can be accommodated in both Farrell and Russell
models to meet the needs of various users. For
example, to measure scale efficiency we can use
constant returns to scale, non-increasing returns
to scale or varying returns to scale technologies.
Thesetechnology constraints canbe easily imposed
by corresponding restrictions on the “intensity
parameters” in the Farrell or Russell models.
In the CCR or additive PEA model discussed in
appendix A, however, only one efficiency measure
is defined: the CCII model uses the radial measure
of efficiency while the additive model uses the
non-radial measure.
Geometrically, the efficiency frontier with cons-
tant returns to scale technology is a convex cone,
but it is a convex hull in cases of both non-increas-
ing and varying returns to scale. In general, these
constraints on technology form a chain such that
one efficiency frontier is enveloped by another.
Consequently, the associated efficiency measures
are compatible and nested.’
2
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doc_201288161.pdf