A PARAMETRIC ESTIMATION OF BANK EFFICIENCIES USING A
FLEXIBLE PROFIT FUNCTION WITH PANEL DATA
TAI-HSIN HUANG
Tamkang UniversityThis paper generalises Kumbhakar's (1996a) single product shadow pro®t function to a multiproduct one, which in contrast to Berger, Hancock, and Humphrey (1993) is consistent with a ®rm's pro®t maximising behaviour. By estimating a parametric translog pro®t function, which does not require special assumptions about the error distribution, and using panel data from Taiwan's banking industry, the following conclusions can be drawn: (i) Parameter estimates from the translog functional form are more robust than those from the Fuss (non-logarithmic) form. (ii) More than half of all potential variable pro®ts are found to be lost due to inef®ciencies. (iii) Greater reduction in pro®t results from de®cient output revenues than from a suboptimal input mix. (iv) The model ®nds technical progress during the sample period. (v) A type of `weakly' optimal scope economies is detected, which suggests that the joint production of the two products can increase pro®ts for some banks while not hurting the pro®ts of others.
I. In t ro duc t i o n
Studies on bank ef®ciencies generally include cost ef®ciencies, as well as economic ef®ciencies. The former mainly examine scale and scope economies, and other related issues pertaining to banks. Benston (1965, 1972) and Bell and Murphy (1968) consider a bank as a collection of separate and independent Cobb-Douglas production functions. This approach suffers from eliminating important information about the effects of the separate outputs on total cost and the potential existence of product mix economies. It was not until the 1980s that a ¯exible multiproduct translog functional form became widely used to investigate cost ef®ciencies in banking, as exempli®ed by Murray and White (1983), Gilligan et al. (1984), Kim (1986), Mester (1987), Berger et al. (1987), Hunter et al. (1990), Humphrey (1993), Dietsch (1993), Mulder and Sassenou (1993), Lang and Welzel (1996) and Jagtiani and Khanthavit (1996), among others. However, these studies presume that ®rms are always on their ef®cient cost frontier and therefore ignore possible economic inef®ciencies.1
# Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden MA 02148, USA and the University of Adelaide and Flinders University of South Australia 1999. The author is indebted to Prof. C.A.K. Lovell for insightful comments on earlier drafts, during and after the Taipei International Conference on Ef®ciency and Productivity Growth, 1997.
1 Berger and Humphrey (1991), Bauer et al. (1993), McAllister and McManus (1993), Mester (1993)
and BHH have pointed out that the assumption ofwhether or not a ®rm is on its ef®cient frontier does not affect much the quality of the estimation of scale economies. However, evaluating data without being on the cost frontier would confound substantially scope economies with economic ef®ciencies.
Economic ef®ciencies are often referred to as x-ef®ciencies as distinguished from cost ef®ciencies, and relate to both the technical and allocative ef®ciencies of individual ®rms. FaÈre and Lovell (1978) show that there are two distinct measures of technical ine f®ciency, namely, output technical inef®ciency and input technical inef®ciency. An output technically inef®cient ®rm is characterised by its failure to produce maximal output given a set of inputs. An input technically inef®cient ®rm is described as one with either under- or over-utilisation of inputs given output and the input mix. The two measures are equivalent only if the technology exhibits constant returns to scale. Allocative inef®ciency occurs if a ®rm fails to equate the marginal rate of technical substitution between any two of its inputs to the ratio of corresponding input prices. Such inef®ciency may be due to a ®rm's regulatory environ-ment, or its slow adjustment to price changes.
Berger, Hunter and Timme (1993) classify four different approaches that have been utilised in evaluating ®rms' x-ef®ciencies.2 However, they overlook an important approach;
that is, the parametric approach, ®rst proposed by Lau and Yotopoulos (1971) and later generalised by Toda (1976), Atkinson and Halvorsen (1980, 1984), Lovell and Sickles (1983), Atkinson and Cornwell (1993, 1994a, 1994b), and Kumbhakar (1996a, 1996b, 1997). In the parametric approach, Atkinson and Cornwell (1993) refer to output technical inef®ciency as a factor which scales output down from the production frontier and input technical inef®ciency as a factor that scales down input usage. Atkinson and Cornwell (1994a) use shadow prices to represent allocative inef®ciency, and assume that the ®rm minimises shadow cost, rather than actual cost, since the ®rm may face additional constraints imposed by its regulatory environment.
Berger, Hancock, and Humphrey [henceforth BHH] (1993) ®rst derive x-ef®ciency estimates by utilising a distribution-free approach to the pro®t function. Although the pro®t function has several advantages over the cost function in estimating ef®ciency, it is not commonly utilised in the ef®ciency literature, with the exception of Ali and Flinn (1989), Yotopolous and Lau (1973), Kumbhakar (1987, 1996a) and Kumbhakar and Bhattacharya (1992).3 BHH devise new measures of allocative and technical inef®ciencies which differ
substantially from the standard de®nitions of Farrell (1957) and nest the standard measures as special cases.4 Despite the fact that their new measures of allocative and technical
inef®ciencies are, as they claim, potentially easier to compute than the standard measures, and are more useful because the new decomposition focuses on the source of inef®ciency, nevertheless their derived pro®t function and the corresponding netput equations seem to be inconsistent with the standard behavioural assumption of pro®t maximisation.
The above inconsistency results from the fact that they introduce allocative and technical inef®ciencies into the actual pro®t function in two steps, forcing the allocative and technical inef®ciencies to be independent of each other. In the ®rst step, a shadow pro®t function is used to incorporate allocative inef®ciency. Next, an intercept representing the difference between actual and desired netputs, obtained by differentiating the shadow pro®t function
2 The four approaches are (i) the econometric frontier approach, see Ferrier and Lovell (1990),
Kumbhakar (1987, 1991), and Timme and Yang (1991); (ii) the thick frontier approach, see Berger and Humphrey (1991, 1992a); (iii) data envelopment analysis, see English et al. (1993), Rangan et al. (1988), Aly et al., Elyasiani and Mehdian (1990), Fixler and Zieschang (1991), and Drake and Weyman-Jones (1996); (iv) the distrubution-free approach, see Berger (1993), and Berger and Humphrey (1992b).
3 See Berger, Hunter and Timme (1993), pp.317±318.
4 Farrel (1957) associates the proportionate (radial) overuse of all inputs with technical inef®ciency
with respect to shadow prices, is added to each netput equation. A more theoretically appealing approach would have both allocative and technical inef®ciencies simultaneously entering the pro®t function, thus preventing technical inef®ciency from being separated completely from allocative inef®ciency.
This paper extends the Kumbhakar's (1996a) single product pro®t function, which includes allocative and technical inef®ciencies, to the multiproduct case in the context of a panel data model. Our derived pro®t function is similar to Kumbhakar (1996b). Using a parametric translog pro®t function, we show that the model developed by BHH may yield inconsistent parameter estimates and that models excluding either, or, both technical, or allocative inef®ciencies will lead to biased and inconsistent estimates, with the measure of technical change also being biased. As an application, we collect panel data from 22 of Taiwan's domestic banks and implement an empirical study of their ef®ciency. As Berger, Hunter and Timme (1993) have mentioned, except Berg et al. (1993), there are few studies on the x-ef®ciency of banks and other ®nancial institutions outside of the U.S. Hence, more research on the measurement of the x-ef®ciency for such ®nancial intermediaries across international borders is needed.
During the forty years prior to 1991, ®nancial institutions in Taiwan were tightly regulated by the government. No new domestic banks were allowed to enter during this period. Moreover, the then existing 22 banks were not free to expand their operations through the island by increased branching. Since 1991 the government has taken steps to liberalise the banking industry. There are now 18 new private banks competing with the original 22 banks. Moreover, 11 of these 22 banks are substantial public enterprises when compared in terms of total assets with the remaining 11 private banks. The entry of new private banks has intensi®ed the degree of competition in the banking industry. A natural question arises as to the effect these developments have had on the ef®ciency of the original 22 banks.
The remainder of the paper is organised as follows. In Section II, we derive our pro®t function and corresponding pro®t share equations and various inef®ciency measures. Section III exposits the estimation methodology, and Section IV brie¯y discusses the 'optimal scope economies' concept proposed by BHH. In Section V, we describe our data set and report sample statistics. Section VI analyses our empirical results and compares them with previous ®ndings, while Section VII concludes the paper.
II. Th e P ro f i t Fu nc t i o n a n d M e a s u r e s o f X-In e f f ic i e nc i e s We assume that the ®rm employs m variable inputs,5 denoted by ~X (X
1, . . . , Xm)9, to
produce n outputs, denoted by Y (Y1, . . . , Yn)9. The corresponding shadow price vectors
are W (W1, . . . Wm) and P (P1, . . . , Pn), respectively. All ®rms' decisions are assumed to be based on the shadow prices, which are different from the actual (realized) prices due to regulation or sluggish adjustment to past changes in input and output prices. A ®rm is said to be allocatively inef®cient when the marginal rate of technical substitution between any two of its inputs or the marginal rate of transformation between any two of its outputs is not equal to the ratio of corresponding input or output prices. The former may be referred to as input allocative inef®ciency, while the latter output allocative inef®ciency.
5 For simplicity, we drop ®xed inputs out of the model for the time being. They will be considered in
Following Jorgenson and Lau (1974a, 1974b), we solve for a variable input as the left-hand-side variable of the production function. For a speci®c production plan, each element in the input vector is non-positive and each element in the output vector is non-negative, by convention. The production function is given by
L ÿXm F(Y9, X9)6
X (X1, . . . , Xmÿ1)9: (1)
A feasible production plan (Y9, X9, ÿL) can be achieved by using the minimum value of L for given values of Y and X. The short-run normalised shadow pro®t function can be expressed as
~
Ð(~P, ~W) sup
Y,Xf~PY ~WX ÿ F(Y9, X9)jY, X 2 R n
Rÿmÿ1g,7 (2)
where ~Ð, ~P, and ~W are normalised by the shadow price of Xm, Wm. Using a Legendre
transformation,8the following dual relationships may be deduced
@ ~Ð
@ ~Pi Yi(^P, ~W), i 1, . . . , n, (3) @ ~Ð
@ ~Wj Xj(^P, ~W), j 1, . . . , m ÿ 1: (4) The above model implicitly assumes that ®rms are technically ef®cient. We now incorporate technical inef®ciency in the normalised shadow pro®t function to show the consequences of ignoring such inef®ciency. The production function of an input technically inef®cient ®rm can be written as
~L ÿbXm F(Y9, bX9),9 (5)
where b, 0 , b < 1, is a parameter scaling input usage and representing the extent to which actual and optimal input mix differ. The term ÿbXjrepresents the ef®ciency units of
input j, j 1, . . . , m ÿ 1. The closer the value of b is to unity, the more technically ef®cient is the ®rm. When b 1, the ®rm is said to be fully technically ef®cient. Conversely, the smaller the parameter b, the less technically ef®cient it will be. The corresponding ef®ciency-adjusted normalised short-run pro®t function becomes
^
Ð(^P, ~W) sup
Y,bXf^PY ~WbX ÿ F(Y9, bX9)jY, bX 2 R n
Rÿmÿ1g (6)
where
6 F(Y, X) is assumed to possess 7 properties. They are (i) domain, (ii) continuity, smoothness, (iv)
monotonicity, (v) convexity, (vi) twice differentiability, and (vii) boundedness. See Lau (1978).
7 Ð(~P, ~~ W) has the corresponding properties relative to F(Y, X) and they are implied by each
other. Among them, two of the properties will be tested once the parameters of the normalised pro®t function have been estimated, viz. monotonicity and convexity.
8 See Lau (1978) and Rockafeller (1970).
9 The production function of an output technically inef®cient ®rm is expressed as
Yn B F Y9 B, X9 ,
where Y (Y1, . . . , Ynÿ1)9, X (X1, . . . , Xm)9, and B is a technical inef®ciency measure which
^
Ð Ð
Wm=b b ~Ð, and ^P P
Wm=b b~P:
Equation (6) differs from equation (49) of Kumbhakar (1996b), in that an output is de®ned as the numeraire instead of an input. Analogous dual relationships are seen to hold
@ ^Ð @ ^Pi
Yi(^P, ~W), i 1, . . . , n, (7)
@ ^Ð
@ ~Wj bXj(^P, ~W), j 1, . . . , m ÿ 1, (8) The output supply functions in equation (7) show that the presence of technical inef®ciency (b , 1) reduces normalised shadow output prices by the factor b. Such reductions in relative prices caused by the existence of technical inef®ciency result in decreasing supply of all outputs as a consequence. However, the effects of technical inef®ciency on the input demand functions in equation (8) are indeterminate, a priori. It can only be argued that an input technically inef®cient ®rm cannot increase all its input usages to maximise its ef®ciency adjusted normalised shadow pro®t.
Since shadow pro®t is unobservable and, hence, not estimable, we operationally relate it to actual pro®t ~Ðaas ~ ÐaÐa Wm ~PY(^P, ~W) ~W X(^P, ~W) ÿ 1 bF(Y(^P, ~W), bX(^P, ~W)), (9)
where ~P (P1=Wm, . . . , Pn=Wm) and ~W (W1=Wm, . . . , Wmÿ1=Wm) are actual
normal-ised output and input price vectors, respectively. Multiplying b on both sides of (9), we obtain ~ Ða b ~Ða Ða Wm=b ^PY(^P, ~W) ~W bX(^P, ~W) ÿ F(Y9(^P, ~W), bX9(^P, ~W)) ^Ð(^P, ~W) Xn i1 (^Piÿ ^Pi)Yi Xmÿ1 j1 ( ~Wjÿ ~Wj)bXj ^Ð Xn i1 (^Piÿ ^Pi)^Si ^ Ð ^Pi Xmÿ1 j1 ( ~Wjÿ ~Wj) ^SjÐ^ ~ Wj ^Ð 1 Xn i1 (ëÿ1 i ÿ 1)^Si Xmÿ1 j1 (èÿ1 j ÿ 1)^Sj 2 4 3 5, (10) where ~Pi ëi~Pi, i 1, . . . , n, (11)
~ Wj èjW~j, j 1, . . . , m ÿ 1, (12) ^Si(^P, ~W) @ ln ^@ ln ^PÐ i Yi(^P, ~W)^ ^Pi Ð(^P, ~W), i 1, . . . , n, (13) ^Sj(^P, ~W) @ ln ^@ ln ^ÐW j bXj(^P, ~W) ~ Wj ^ Ð(^P, ~W), j 1, . . . , m ÿ 1, (14) Equations (11) and (12) follow Atkinson and Halvorsen (1980) and Atkinson and Cornwell (1994a), in which ëi(i 1, . . . , n) and èj( j 1, . . . , m ÿ 1) are positive but unknown
parameters. If ëi èj 1, for all i and j, there will be no allocative inef®ciency. ^Si and ^Sj
are shadow pro®t shares of the ith output and jth input, respectively. Taking logarithms of both sides of (10) and rearranging terms, we obtain
ln ^Ða ln ^Ð(^P, ~W) ln G(^P, ~W) ÿ ln b, (15) where G(^P, ~W) 1 Xn i1 (ëÿ1 i ÿ 1)^Si Xmÿ1 j1 (èÿ1 j ÿ 1)^Sj, (16)
The actual and the shadow pro®t shares are related by ^Sa i ^PÐ^iYai^PÐ^iYi ^ Ð ^ Ða ^Pi ^Pi ^Si(ëiG)ÿ1, i 1, . . . , n, (17) ^Sa j ~ WjbXj ^ Ða ~ WjbXj ^ Ð ^ Ð ^ Ða ~ Wj ~ Wj ^Sj(èjG)ÿ1, j 1, . . . , m ÿ 1: (18) Equation (15) differs from Kumbhakar's (1996a) equation (16) in two respects. First, the technical inef®ciency term lnb is subtracted from the right hand side of (15), rather than added to (15). Second, (15) incorporates the output shadow prices, whereas Kumbhakar (1996a) assumes single output ®rms and uses output as the numeraire. Except the term lnb, the technical inef®cient parameter b enters (15) in a more complicated form (to be shown shortly) than BHH.
When a functional form of ln ~Ð(^P, ~W) is speci®ed, all terms appearing in equations (15), (17), and (18) are either data or unknown parameters. If random disturbance terms, with zero means and constant variances, are appended to these equations, the unknown parameters can be estimated.
We next employ a ¯exible translog function for ln ^Ð to derive the corresponding actual pro®t function and pro®t share equations for empirical study later, i.e.
ln ^Ð a0 Xn i1 ailn ^Pi 12 Xn i1 Xn k1 aikln ^Pi ln ^Pj Xmÿ1 j1 bjln ~Wj 12 Xmÿ1 j1 X mÿ1 r1 bjrln ~Wj ln ~Wr Xn i1 Xmÿ1 j1 áijln ^Pi ln ~Wj, (19) which gives
^Si @ ln ^@ ln ^PÐ i ai Xn k1 aikln ^Pk Xmÿ1 j1 áijln ~Wj, i 1, . . . , n, (20) ^Sj @ ln ~@ ln ^ÐW j bj X mÿ1 r1 bjrln ~Wr Xn i1 áijln ^Pi, j 1, . . . , m ÿ 1, (21) and G(^P, ~W) 1 Xn i1 (ëÿ1 i ÿ 1) ai Xn k1 aikln ^Pk X mÿ1 j1 áijln ~Wj 0 @ 1 A Xmÿ1 j1 (èÿ1 j ÿ 1) bj Xmÿ1 r1 bjrln ~Wr Xn i1 áijln ^Pi ! : (22)
Substitution of equations (19) through (22) into equations (15) through (18) yields ln ~Ða ln ^Ð(^P, ~W) ln G(^P, ~W) ÿ ln b ln ~Ð(~P, ~W) ln[G(~P, ~W) H(b, ô)] ÿ ln b 1 ÿXn i1 aiÿ Xn i1 Xn k1 aikln ~Pi ÿ12ln b Xn i1 Xn k1 aikÿ Xn i1 Xmÿ1 j1 áijln ~Wj 2 4 3 5,(23) H(b, ô) ln b Xn i1 (ëÿ1 i ÿ 1) Xn k1 aik X mÿ1 j1 (èÿ1 j ÿ 1) Xn i1 áij 2 4 3 5, ô (ë1, . . . , ën, è1, . . . , èmÿ1), ^Sa i ~Si Xn k1 aikln b ëi G(~P, ~W) Xn i1 (ëÿ1 i ÿ 1) Xn k1 aik Xmÿ1 j1 (èÿ1 j ÿ 1) Xn i1 áij 2 4 3 5ln b 8 < : 9 = ; i 1, . . . , n, (24) ^Sa j ~Sj Xn i1 áijln b èj G(~P, ~W) Xn i1 (ëÿ1i ÿ 1)Xn k1 aik Xmÿ1 j1 (èÿ1 j ÿ 1) Xn i1 áij 2 4 3 5ln b 8 < : 9 = ; , j 1, . . . , m ÿ 1: (25)
It is worth noting that the ®rst two terms (excluding H(:)) of the (log) realized pro®t function (23) include only the parameters of allocative inef®ciency, while the last term (and H (.)) captures the effects of both technical and allocative inef®ciencies on normalised pro®t. Moreover, it is unlikely that technical inef®ciency b and allocative inef®ciency ô can be
separated from each other. This result differs from BHH, in which they deduced a normal-ised pro®t function with separable technical and allocative inef®ciencies. It is clear that allocative and technical inef®ciencies are independent only if Pn
k1aik Pni1áij 0,
indicating that the pro®t function (19) is homogenous in output prices.10 Therefore, if the
term lnb is simply appended to the pro®t function (23), as BHH, Ali and Flinn (1989), and Kumbhakar and Bhattacharyya (1992), the parameter estimates may be inconsistent due to speci®cation error.
Fixed inputs can easily be considered by adding those terms of Xq k1 ckln Zk12 Xq k1 Xq j1 ckjln Zkln Zj Xn i1 Xq k1 âikln ^Pi ln Zk X mÿ1 j1 Xq k1 öjkln ~Wjln Zk
to the right hand side of (2-19). Furthermore, if technical change is to be examined, the term dtt 12dttt2Pni1äitln ^Pi t
Pmÿ1
j1ìjtln ~Wjt must be introduced into (19). Certainly,
equations (20) through (25) have to be modi®ed in a straightforward way.
III. E st i m at i o n St r at e gy
Equations (23) through (25) form a system of simultaneous equations comprising a pro®t function and pro®t shares and can be estimated after random disturbance terms are added to each of those equations.11To make the model manageable and estimable, some additional
structure needs to be imposed. Technical inef®ciency b is assumed to vary across ®rms, but not over time. In practice, it is possible that no ®rm is fully technically ef®cient. Thus, for the purpose of estimation, we elect to normalise b to unity for the most ef®cient ®rm (say ®rm e) in the sample. It follows that the relative input technical ef®ciency of ®rm k(k 6 e) is de®ned as bk=be bk, which can be estimated for the rest of the ®rms in the sample.
Allocative inef®ciency terms ëiand èj, by contrast, are restricted to be constant over time
and across ®rms. They capture persistent or average allocative inef®ciency in the use of netput i and j. It is important to note that this assumption imposes only that the ratios of shadow prices to the corresponding actual prices be constant across ®rms and over time, rather than requiring that the levels of the shadow prices are invariant. Similar to the estimation of b, we measure n m ÿ 1 relative allocative ef®ciencies, as only relative prices matter in our pro®t function. Since the mth input is the chosen numeraire, the corresponding èmis normalised to unity. ëi, 1(ëi. 1) implies under (over-) production of output i relative
to the mth input. Furthermore, èj, 1(èi. 1) indicates that the ®rm mistakenly desires to
employ more (less) of input j relative to the numeraire.
A nonlinear iterative seemingly unrelated regression technique is utilised to estimate equations (23) through (25), simultaneously, using panel data on Taiwan's banking industry as described in Section V. This technique is equivalent to maximum likelihood when convergence is achieved.
10 By referring to Fuss and McFfadden (1978) on page 173, vol. I, under certain assumptions, a
normalised pro®t function is homogeneous of degree 1=(1 ÿ k), k , 1, in the normalised output prices, if and only if the production function is almost homogeneous of degree 1 and k.
11 There are m n equations to be estimated. One of the pro®t share equations has been automatically
excluded by normalisation. Therefore, the problem of singularity in the variance-covariance matrix of the random disturbances is avoided. Note that which input (output) is chosen as the numeraire and dropped from the system will not in¯uence the results.
To show that the parameter estimates from our model are comparatively more robust than those from BHH, we shall estimate three different models. Model I contains a ®xed input, whereas Model II includes two ®xed inputs. Model III contains no ®xed inputs, but rather incorporates a time trend to investigate technical change.
Once the parameters of the pro®t function are obtained, they can be used to determine several inef®ciency measures. The loss of pro®t due to technical inef®ciency alone can be calculated as the difference between the ®tted values of ln ~Ða with and without the
assumption b 1. Unfortunately, such pro®t reductions depend on allocative inef®ciency terms ëi and èjand, therefore, vary with allocative inef®ciency. Similarly, the loss of pro®t
caused by allocative inef®ciency alone will not be independent of technical inef®ciency either. This is why we shall compute two sets of pro®t reductions due to either technical or allocative inef®ciency, viz.,
Ð1 TI(ô) ln ~ða(^P, ~W)jb1ÿ ln ~ða(^P, ~W) ln ~ða(~P, ~W) ÿ ln ~ða(^P, ~W) (26) Ð1 AI(b) ln ~ða(^P, ~W)jô1ÿ ln ~ða(^P, ~W) ln ~ða(^P, ~W) ÿ ln ~ða(^P, ~W) (27) Ð2 TI ln ~ða(^P, ~W)jb1ÿ ln ~ða(^P, ~W) ln ~ða(~P, ~W) ÿ ln ~ða(^P, ~W) (28) Ð2 AI ln ~ða(~P, ~W)jô1ÿ ln ~ða(~P, ~W) ln ~ða(~P, ~W) ÿ ln ~ða(~P, ~W) (29) where ô (ë1, . . . , ën, è1, . . . , èmÿ1)9. Ð1
TI(ô)(Ð2TI) represents the pro®t reductions due to technical inef®ciency alone without
(with) the assumption that allocative ef®ciency has been achieved. Ð1
AI(b)(Ð2AI) is the pro®t
reductions caused by allocative inef®ciency alone without (with) the assumption that technical ef®ciency has been achieved. It is clear that Ð2
TI(Ð2AI) is independent of the
allocative (technical) inef®ciency terms, but Ð1
TI(ô)(Ð1AI(b)) is not. Therefore,
Ð1
TI(ô)(Ð1AI(b)) may not necessarily equal Ð2TI(Ð2AI).
Most previous studies on technical progress in banking are within the context of a general translog cost function. These studies evaluate technical change on the ef®cient cost frontier, that is, they implicitly assume that ®rms can always achieve x-ef®ciency.12 However, since
empirical evidence suggests that evaluating data without being on the cost frontier can seriously confound scope economies with x-ef®ciencies, as has been found by Berger and Humphrey (1991), Bauer et al. (1993), and Mester (1993), failure to consider x-inef®ciencies may bias the measure of technical change as well.
Rewriting (23) with a time trend, we obtain
ln ~Ða(^P, ~W, t) ln ~Ð(~P, ~W, t) ln[G(~P, ~W, t) H(b, ô)] I(b, ~P, ~Wt),
(30) where
G(~P, ~W, t) 1 Xn i1 (ëÿ1i ÿ 1) ai Xn k1 aikln ~Pk Xmÿ1 j1 áijln ~Wj äitt 0 @ 1 A Xmÿ1 j1 (èÿ1 j ÿ 1) bj Xmÿ1 r bjrln ~Wr Xn i1 áijln ~Pi ìjtt ! , and I(b, ~P, ~W, t) ÿln b 1 ÿXn i1 aiÿ Xn i1 Xn k1 aikln ~Pi ÿ12ln b Xn i1 Xn k1 aik " ÿXn i1 X mÿ1 j1 áijln ~Wj ÿ t Xn i1 äit
Technical change ( _Ðt) is typically de®ned as the rate of change in pro®t (cost) over time,
other things being equal. More speci®cally, under the assumption that b is time-invariant and ®rm-speci®c, _Ðt@ ln ~Ð a @t @ ln ~Ð(~P, ~W, t) @t @ ln[G(~P, ~W, t) H(b, ô)] @t @I(b, ~P, ~W, t) @t dt dtt Xn i1 äitln ~Pi Xmÿ1 j1 ìjtln ~Wj 0 @ 1 A ln bXn i1 äit Xn i1 (ëÿ1 i ÿ 1)äit Xmÿ1 j1 (èÿ1 j ÿ 1)ìjt 0 @ 1 A G(~P, ~W, t) H(b, ô) 8 > > > < > > > : 9 > > > = > > > ; Xn i1 äitln ëi Xmÿ1 j1 ìjtln èj, (31)
which shows that _Ðt can be decomposed into three parts. The terms in parentheses is
nothing but the traditional measure of technical change; the contribution of technical inef®ciency alone to technical change is contained in the second term, while the last term in braces represents the effects of both technical and allocative inef®ciencies on _Ðt.
Since technical inef®ciency b and allocative inef®ciency are both present in (31), it is clear that ignoring any one of them or both would cause a biased measure of technical change, except when b 1, and/or ô 1. Such a bias is attributed to the use of a wrong formula and/or the use of biased parameter estimates, as has been pointed out by Kumbhakar (1996a). There is one more possibility that technical change will not depend on the presence of x-inef®ciencies, that is, if technical change is neutral (i.e., äit ìjt 0, 8i, j). Equation
(31) then reduces to
_Ðt dt dttt: (32)
However, even if this is the case, _Ðtof (32) may still be poorly measured if parameters dt
and dtt, obtained from estimating a model precluding x-inef®ciencies, are biased and
IV. S c op e E c o n o m i e s (SC)
When a bundle of outputs could be more cheaply produced by a single enterprise than by independent specialised ®rms, economies of scope exist. Almost all earlier studies on SC are based upon cost function with various modi®cations.13 Recently, BHH proposed a novel
concept of SC called `optimal scope economies' which is de®ned within a pro®t function framework. Their optimal SC concept takes into account both the revenue effects of output decisions and the cost effects of input decisions, and therefore seems more general than the conventional SC concept de®ned from the cost function alone. By assuming ®rms to be fully x-ef®cient, the new concept can be tested empirically by evaluating whether the optimal level of every output is greater than zero at all observed prices and ®xed netputs. If this were the case, we may conclude that it is pro®table for ®rms to produce all outputs whenever prices and ®xed netputs do not exceed their observed ranges. By contrast, if the opposite condition holds, we can conclude that it is pro®table for some ®rms to specialise. This approach makes some recognised problems in the estimation of SC more tractable.14
We perform the test using the optimal output share equations ~Si ai Xn K1 aikln ~PK Xmÿ1 j1 áijln ~Wj Xq K1 âikln Zk, i 1, . . . , n, (33)
which is equivalent to the employment of output functions as utilised by BHH. We substitute the ®rst quartile of each prices and ®xed netputs into equation (33) to see if each ~Si is
signi®cantly different from zero.15It follows that if each calculated optimal output share is
signi®cantly positive, then optimal SC exist in the data. An alternative result would imply that some ®rms should specialise in some of the products to increase their pro®ts.
V. Data D e s c r i p t i o n
Based on the ®nancial intermediation approach, we identify two variable outputs, two variable inputs, and two ®xed inputs. The output categories include investments (Y1), which
consists of government and corporate securities, and loans (Y2), which are composed of short
and long term loans. All deposits and borrowed money (X1) and labour (X2) are regarded as
inputs. We arbitrarily choose labour as our numeraire and, therefore, all variable pro®t and prices are normalised by the wage. In addition, net ®xed assets (Z1) and number of branches
(Z2) are viewed as ®xed inputs.16In this study, we use panel data on 22 of Taiwan's domestic
banks, of which 11 are public banks, over the period 1981 to 1995. There are two main sources in collecting the data set. Before 1991, the number of workers and labour costs of each bank are obtained from a survey conducted in 1992, while all others are taken from publications of the Central Bank as well as the Ministry of Finance.
13 When a translog cost function is used to examine SC, since log of zero is unde®ned, Gilligan et al.
(1984), Kim (1986), and Mester (1993,1996), for example, substitute either a very small number (but different from zero) or the sample minimum value for zero.
14 See Berger, Hunter and Timme (1993).
15 The reason of using the ®rst quartile values of prices and ®xed netputs is to avoid the effect of
extreme values on ~Si, i 1, . . . , n.
16 Keeping the number of branches in the objective function means that we are investigating at the
In the sample period, two banks have negative pro®ts for two and four years, respectively, and another two banks have missing values in 1981. Hence, the actual number of observa-tions being studied is 322. Sample statistics for each variable are summarised in Table I.
VI. E st i m at i o n a n d R e s u lt s Parameter estimates
Parameter estimates of model I to III are reported in Table II. The estimated ®rm-speci®c technical inef®ciency measures of b are reported in Appendix A,17 where only the
parameters from Model I are presented since all estimates from Model II and III are small and insigni®cant, though all have their expected positive signs. It is most likely that the translog pro®t functions contain too many parameters to be estimated (at least 39 para-meters) and the number of periods for each bank are at most 15 years. Admittedly, this is a potential weakness inherent in the parametric approach. Moreover, we estimate the three similar models using the Fuss pro®t function, as used by BHH, and the results are listed in Appendix B.
In Table II, although the corresponding parameter estimates from the three models are different, they are nevertheless quite close and moreover have the same signs. By contrast, the same estimates derived from the Fuss pro®t function are not only quite different from one another, but also have different signs. In addition, some of the three allocative inef®ciency parameters are even negative,18and one of them is decisively signi®cant. These
results seem to reveal that the translog functional form is more robust against different model speci®cations and is more appropriate in describing the data than the Fuss functional form.
Substituting each observation into the translog pro®t function, both monotonicity and Table I Sample statistics
Variable name Mean Standard deviation
Variable pro®t 3066.137 3944.417
Net real assets 4152.830 5193.492
Wages 0.569 0.275
log-normalized variable pro®t 7.772 1.339
Normalized price of Y1 0.172 0.472 Normalized price of Y2 0.241 0.186 Normalized price of X1 0.145 0.109 Pro®t share of Y1 0.948 2.856 Pro®t share of Y2 6.5562 11.102 Pro®t share of X1 5.313 10.027 Pro®t share of X2 1.191 3.933
Notes: :Millions of new Taiwans dollars Base Year: 1991
Number of observations: 322
17 Since all those technically inef®cient parameters are small relative to the ef®cient ®rm, we would
expect that technical inef®ciency may be serious. We shall return to this point in the next subsection.
18 BHH obtains two negative parameter estimates (though not signi®cant) for the unit banking sample.
convexity conditions are satis®ed by most observations.19This indicates that these estimates
can properly re¯ect our representative bank's technology. Because the standard errors adjusted for heteroskedasticity do not change considerably, we merely present the unadjusted standard errors in Table II and implicitly assume that there is no conditional heteroskedasti-city.
As expected, estimates of ëiand èjare all signi®cantly positive. It is noteworthy that ^ë2is
Table II Parameter estimates
Variable Name Model Model Model
Constant 9.8480a 8.9216a (0.081) 11.4752a (0.456) (0.335) ln P1 0.0976c 0.0802c (0.046) 0.2800a (0.054) (0.075) ln P2 2.9187a 2.2974a (0.254) 0.9319a (0.307) (0.153) ln P1lnP1 0.0247b 0.0088 (0.005) 0.0620a (0.011) (0.017) ln P2lnP2 ÿ0.0027 0.1423b (0.057) ÿ0.0566 (0.033) (0.047) ln P1lnP2 ÿ0.0166 0.0068 (0.004) ÿ0.0617a (0.011) (0.018) ln w1 ÿ1.4580a ÿ0.7591a (0.187) ÿ0.4154a (0.266) (0.156) ln1ln w1 ÿ0.118a ÿ0.0780a (0.030) ÿ0.0437b (0.042) (0.018) ln P1lnw1 ÿ0.0421a ÿ0.0141c (0.0088) ÿ0.0057 (0.012) (0.005) ln P2lnw1 ÿ0.0039 ÿ0.0869b (0.040) 0.0071 (0.017) (0.007) . 0.2933 (0.310) ln z2 0.1734 ÿ0.4735 (0.231) (0.427) ln z1ln z1 ÿ0.0774 (0.074) ln z2ln z1 ÿ0.0774 (0.074) ln z2ln z2 ÿ0.1067 ÿ0.5223a (0.098) (0.190) ln z1ln z2 0.2539b (0.109) ln P1lnz1 0.0027 (0.006) ln P1lnz2 ÿ0.0234 ÿ0.0105 (0.017) (0.008) ln P1lnz1 0.0513 (0.034) ln P2lnz2 ÿ0.4136a ÿ0.2443a (0.088) (0.075) ln w1ln z1 ÿ0.0917a (0.033) ln w1ln z2 0.0238 0.0897c
19 It is a common dif®culty encountered by most empirical studies, for example, Gropper (1995), Glass
et al. (1995), and Rller (1990), that the regularity conditions can only be passed by most, but not all, of the sample. This is possibly due to large adverse random disturbances.
very close to unity in Model I and II. Therefore, the null hypothesis of ë2 1 cannot be
rejected, implying that the loan product is allocatively ef®cient relative to labour. That ^è is consistently smaller than unity in all three models indicates that borrowed money is over utilised relative to labour. This ®nding is supported by referring to Table I, where the mean pro®t share of borrowed money exceeds the average pro®t share of labour to a great extent. Due to the fact that ^ë1=^ë2 is less than unity, we could conclude that the product of
investments is under produced, as opposed to the product of loans. This can also be noted by checking Table I in which the mean pro®t share of loans is a multiple of the investment share.
In summary, to improve allocative ef®ciency, banks should increase the production of investments, keep the level of loans unchanged, hire more labour, and reduce the amount of borrowed money. The reduction in borrowed money implies that banks should increase capital and/or decrease loans, which in turn would increase the capital adequacy ratio (CAR) and lessen the chance of bank insolvency. In 1989, the Banking Law of Taiwan was modi®ed to require banks to maintain a CAR of at least eight per cent, with those banks failing to do so denied permission to open new branches.
Effects of X-inef®ciencies on pro®t
The parameter estimates from Model I can now be applied to determine the inef®ciency measures using equations (26) through (29). If a bank is technically inef®cient, its pro®t must be lower than an ef®cient bank facing the same input and output prices. Equation (26) calculate such pro®t losses due to technical inef®ciency alone by subtracting the predicted value of ln ~Ða with technical inef®ciency from the predicted value of ln ~Ða without
technical inef®ciency. Equation (28) does the same job, but further assumes that banks are Table II (cont.)
Variable Name Model Model Model
(0.056) (0.048) ë1 0.2444a 0.1257b (0.051) 0.4531a (0.068) (0.139) ë2 0.8017a 0.1257b (0.051) 0.4531a (0.183) (0.093) è 0.4770a 0.4191a (0.115) 0.1315b (0.128) (0.053) t 0.0460 (0.037) t2 0.0002 (0.003) ln P1t ÿ0.0104a (0.003) ln P2t ÿ0.0187b (0.008) ln w1t ÿ0.0029c (0.002) log-likelihood ÿ2560.81 ÿ2514.96 ÿ2513.90
Notes: a: signi®cant at one per cent level. b: signi®cant at ®ve per cent level. c: signi®cant at ten per cent level. Numbers in parentheses are standard errors
allocatively ef®cient. Equations (27) and (29) compute pro®t losses caused by allocative inef®ciency alone with and without technical inef®ciency.
The results are shown in Appendix C. It is obvious that the average losses of pro®t due to technical inef®ciency is much higher than the average pro®t reductions caused by allocative inef®ciency, consistent with our previous expectation. This can be seen by comparing either Ð1
AI(b) with Ð1TI(ô), or by comparing Ð2AI with Ð2TI. If the entire sample is divided into
public and private banks, the same conclusion holds. This implies that de®cient output revenues cause a larger pro®t reductions than excessive input costs. The main source of x-inef®ciencies comes from technical side, rather than input misallocation. This ®nding is consistent with, for example, Berger and Humphrey (1991), BHH, Kumbhakar (1991), and English et al. (1993), among others.
The geometric mean actual normalised pro®t of private banks is about 19.9 per cent of that of public banks. The ratio of the mean ®tted pro®t of private relative to public banks lowers slightly to 17.5 per cent provided that the both types of banks are allocatively ef®cient without being technically ef®cient. However, if the reverse is true, i.e., if these banks were technically ef®cient but allocativelly inef®cient, the average level of pro®t for private banks would be even slightly higher than that of public banks. The optimal ®tted pro®ts for public and private banks are also very close to each other. We conclude that private banks seem to have larger room to improve their economic inef®ciencies especially technical inef®ciency. In other words, private banks are farther below their pro®t frontiers than public banks due to the lower ef®ciency units of inputs. As public banks are larger in terms of total assets than private banks, Appendix C suggests that large banks are substantially more ef®cient from the technical perspective, but only slightly less ef®cient from the allocative perspective, than are small banks in Taiwan.
The same inef®ciency measures can be expressed in terms of the percentage loss of pro®t and are reported in Table III. The table shows that inef®ciencies in banking are possibly quite large. The overall estimated mean inef®ciencies consume 67.3 per cent, 93.4 per cent, and 85.2 per cent of the potential variable pro®ts of public, private, and all banks, respectively, which are close to the estimates of DeYoung and Nolle (1996), Akhaven et al. (1997a, 1997b), Hughes and Moon (1995), and Hughes et al. (1996a, 1996b) ranging from 24 per cent to some 85 per cent of the optimal pro®ts for US banks. These numbers con®rm the results shown in Appendix C. The relatively small role of allocative inef®ciency is not surprising in the light of parameter estimates, as well as results, shown in Appendix C.
As stated in Section I, our sample banks are subject to rigorous regulation for the past few decades. Without competition from new entrants, we would expect that the sample banks are most likely to be lacking of ef®ciencies. Therefore, our ®ndings appear to be acceptable. To improve ef®ciencies, banks should direct their efforts to raising the productivity of all their inputs to reach production frontiers. Financial deregulation is suggested to enhance competi-tion and lessen allocative inef®ciency.
Table III Pro®t losses (per cent) due to x-inef®ciencies
Type of Bank Ð1
AI(b) Ð2AI Ð1TI(ô) Ð2TI Total
Public banks 15.73 24.61 56.58 61.16 67.27
Private banks 10.44 14.96 92.18 92.58 93.35
Measures of technical change
Estimates of technical change utilising (31) are summarised in column 1 of Table IV (labelled actual). These values are computed using the parameter estimates from Model III and sample mean values of the independent variables. Moreover, we calculate technical change under the assumption that banks are either allocatively ef®cient alone, labeled as intermediate in Table IV, or x-ef®cient, labeled as optimal in the same table. All the three models exhibit strong technical progress, but differ in their magnitudes. The intermediate model, which omits the contribution of technical inef®ciency, has the largest values. Despite the fact that the actual and optimal models have similar estimates of technical change for all banks, there are quite different measures of technical progress for public and private banks. Private banks are seen to be faster in improving their technology than public banks, according to the actual model.
Similar estimates of technical change over time are reported in Appendix D, in which the estimates are sample mean values rather than being evaluated at average values of the explanatory variables. Both the intermediate and optimal models show signi®cant technical progress and capture an increasing trend. Although the actual model is not so accurately estimated, compared with the corresponding standard deviations, as to the other two models, it does appear to reveal an increasing trend. In addition, estimates from the actual model for each year are less than the intermediate model, but greater than the optimal model. We should note that the conclusions drawn from the intermediate and optimal models are in accord with Hunter and Timme (1986), Humphrey (1993) and Lang and Welzel (1996), which examined technical change by estimating a translog cost function without considering x-inef®ciencies and which found moderate technical progress.
The foregoing seems to suggest that neglecting x-inef®ciencies in the formulation of the regression equations may introduce considerable bias into the estimates of technical change. Estimation of `Optimal Scope Economies'
The two output share equations of (33) are evaluated by applying the parameter estimates from Model I and II, letting b : 1 for each observation. If the calculated shares are all positive, then the data exhibit optimal scope economies for the sample banks. Otherwise, the data reveal that it may be pro®table for some ®rms to specialise. There are 47 and 297 observations having positive ®tted ~S1 (investments) for models I and II, respectively, while
the ®tted ~S2(loans) are all positive for the two models. This indicates that the two models
appear to reach different conclusions on scope economies.
We further conduct the same analysis in a slightly different way. The ®rst quartiles of all the exogenous variables are substituted into the two optimal output share equations. The Table IV Measures of technical change
Type of banks Actual (b, ô) Intermediate (b, ô 1) Optimal (b 1, ô 1)
Public banks 0.0736a (0.029) 0.1552 a (0.023) 0.1061 a (0.025) Private banks 0.1946a (0.036) 0.2494 a (0.039) 0.0923 a (0.027) All banks 0.1005a (0.026) 0.1668 a (0.023) 0.0985 a (0.026) Notes: a: signi®cant at one per cent level
reason for utilising the ®rst quartile values in place of the exogenous variables is to avoid the effect of extreme values on the ®tted output shares.The results are shown in Table V.
Model I cannot reject the null hypothesis that ~S1equals zero, while Model II does reject
this hypothesis. The null of ~S2 0 is statistically rejected for both models. Results from
Model I suggest that the so-called `asymmetric jointness', which is referred to by BHH in describing the situation that although joint production is preferable for most banks, it may be optimal for other banks to specialise in production of loans and to neglect production of investments entirely, be detected from the sample. However, results from Model II suggest there are optimal scope economies throughout the observed data. Based on Table V, we may conservatively conclude that `weakly' optimal scope economies appear to be found. The `weakly' optimal scope economies suggest that the investments always be produced with loans. By so doing, most banks will be better off, while other banks will at least not be worse off, given the observed prices and ®xed netputs.
VII. C o nc lu di n g R e m a r k s
This paper develops a variable pro®t function, which accommodates both technical and allocative and technical inef®ciencies, along with the corresponding pro®t shares derived from the assumptions that multi-product ®rms attempt to maximise their pro®ts subject to several endogenous and exogenous constraints and that ®rms are input technically inef®cient. Assuming that the pro®t function can be speci®ed as having a translog form, we show that some popular methods used to analyse x-inef®ciencies are inappropriate, such as Atkinson and Halvorsen (1980) and Atkinson and Cornwell (1994b) who completely ignored technical inef®ciency, and Ali and Flinn (1989), Kumbhakar and Bhattacharyya (1992), and BHH who simply added a negative ®rm-speci®c term to the pro®t function, which is inconsistent with pro®t maximising behaviour. Failure to model x-inef®ciencies correctly can lead to biased parameter estimates, and furthermore, all inferences based on these biased parameter estimates, for example, the estimation of scale and scope economies, measures of pro®t losses (or cost increases) caused by inef®ciencies and technical change, will be incorrect as well.
Based on the empirical results, the following conclusions are reached. First, the parameter estimates of the translog pro®t functions seem to be more robust than those of the Fuss pro®t functions employed by BHH. Second, the economic inef®ciencies in Taiwan's banking sector appear to be serious since more than half of the industry's potential variable pro®ts are Table V Tests of `optimal scope economies'
Model Model
Type of bank Optimal share of Y1 Optimal share of Y2 Optimal share of Y1 Optimal share of Y2 Public banks ÿ0.0733 (0.150) 1.6725 a (0.235) 0.2537 a (0.051) 1.7563 a (0.173) Private banks ÿ0.0160 (0.133) 1.7275 a (0.247) 0.2551 a (0.047) 1.9272 a (0.186) All banks ÿ0.0227 (0.137) 1.6875 a (0.239) 0.2761 a (0.048) 1.7859 a (0.177) Notes: Numbers in parentheses are standard errors. a. signi®cant at one per cent level
estimated to be lost due to such inef®ciencies. Third, technical inef®ciency dominates allocative inef®ciency, indicating that banks tend to overuse their inputs given outputs. Fourth, public banks are found to be more (technically) ef®cient than private banks. Fifth, technical progress is evident. Lastly, a type of `weakly' optimal scope economies is detected.
A pp e n di x A
Parameter estimates of the ®rm-speci®c technical inef®ciency
Bank Number Parameter Estimates
1. 1 2. 0.342a (0.066) 3. 0.291c (0.176) 4. 0.285c (0.151) 5. 0.229a (0.051) 6. 0.169a (0.050) 7. 0.165a (0.047) 8. 0.160a (0.045) 9. 0.105a (0.036) 10. 0.096a (0.032) 11. 0.039 (0.028) 12. 0.033b (0.015) 13. 0.029b (0.013) 14. 0.022b (0.011) 15. 0.022b (0.011) 16. 0.021b (0.011) 17. 0.019c (0.010) 18. 0.019c (0.010) 19. 0.006 (0.006) 20. 0.002 (0.002) 21. 0.120E-3 (0.230E-3) 22 0.621E-4 (0.144E-3)
Notes: a: signi®cant at one per cent level b: signi®cant at ®ve per cent level c: signi®cant at ten per cent level
A pp e n di x B Parameter estimates from the fuss pro®t functions
Variable Model Model Model
constant 600.71 (720.7) 1333.8 b (627.6) 18095.3a (1906.6) ln P1 33409.3a (3058.1) 2716.4 b (1293.5) 34619.0a (5691.3) ln P2 122285a (10286.8) 3224.9 (3034.9) 89099.2 a (11844.6) ln P1ln P1 3935.1 (2590.6) 1912.2 (1643.3) (2801.7)1788.0 ln P2lnP2 ÿ11775.3 (8523.7) ÿ35181.6 b (17553.2) ÿ5130.6 (9222.1) ln P1lnP2 ÿ35660.9 (25104.6) ÿ3076.4 (5004.9) (24181.7)25964.4 ln w1 ÿ150554a (12470.5) 593.29 (4433.2) ÿ136631 a (15784.8) ln w1ln w1 552699a (106790) ÿ177.08 (2636.7) 341421 a (75135.9) ln P1lnw1 5327.8 (33152.7) ÿ636.44 (4528.7) ÿ34285.1(25364.1) ln P2lnw1 ÿ108692 (72004) ÿ2239.8 (16049.3) 264369 a (94584.6) ln z1 ÿ0.7233a (0.116) ln z2 89.348a (19.73) (19.64)20.856 ln z1ln z1 0.36E-4 (0.60E-5) ln z2ln z2 ÿ1.2340a (0.290) (0.369)0.0977 ln z1ln z2 ÿ0.0010 (0.001) ln P1lnz1 6.5355a (0.395) ln P1lnz2 ÿ17.652 (33.04) ÿ196.06 a (37.05) ln P2lnz1 28.380a (0.920) ln P2lnz2 671.19a (68.39) ÿ8.8067(71.94) ln w1ln z1 ÿ33.675a (1.192) ln w1ln z2 ÿ917.19a (86.82) ÿ142.69(89.99) 1 0.2593a (0.063) 1.4057 a (0.501) 0.1923b (0.080) 2 0.1397a (0.044) 1.3552 a (0.343) ÿ0.2658a (0.069) è 0.1729a (0.053) ÿ37.014 (283.8) 0.2828a (0.058) t 0.0016 ÿ1959.0a (305.86) t2 137.54a (28.88) ln P1t ÿ9.7902 (418.8) ln P2t 5761.9a (734.9) ln w1t ÿ5008.3a (1070.0) log-likelihood ÿ14891.2 ÿ14658.2 ÿ14934.2
Ap pe n d ix C
Allocative and technical inef®ciencies by type of banks Type of Bank Sample Size Actual Pro®t
(1) Fitted Pro®t (b,.)(2) (b, : 1) (3)Fitted Pro®t (b 1, :) (4)Fitted Pro®t Optimal ®tted Pro®t(b : 1) (5) Loss of Pro®t due tox-ineff. (5)±(2) Public Banks 162 8.5753 8.7423 8.9135 9.5776 9.8591 1.1168 Private Banks 160 6.9584 7.0594 7.1697 9.6083 9.7704 2.7110 All Banks 322 7.7719 7.9061 8.0470 9.5923 9.8150 1.9089 Type of Banks Ð1 AI (3)±(2) Ð2AI (5)±(4) Ð1TI(c) (4)±(2) Ð2TI (5)-(3) Public Banks 0.1712 0.2825 0.8343 0.9456 Private Banks 0.1103 0.1621 2.5489 2.6007 All Banks 0.1409 0.2227 1.6862 1.7680 A PARAMETRIC ESTIMA TION OF BANK EFFICIENCI ES 441 Blackw ell Publishers Ltd/Uni versity of Adelaide and Flinders Uni versity of South Australia 1999.
A pp e n di x D
R e f e r e nc e s
Akhavein, J.D., Berger, A.N., and Humphrey, D.B. (1997a), `The Effects of Megamergers on Ef®ciency and Prices: Evidence from a Bank Pro®t Function', Review of Industrial Organiza-tion, vol. 12, pp. 95±139.
Akhavein, J.D., Swamy, P.V.A.D., Taubman, S.B., and Singamsetti, R.N. (1997b), `A General Method of Deriving the Ef®ciencies of Banks from a Pro®t Function', Journal of Productivity Analysis, vol. 8, pp. 71±93.
Ali, M. and Flinn, J. C. (1989), `Pro®t Ef®ciency Among Basmat Rice Producers in Pakistan Punjab', American Journal of Agricultural Economics, vol. 71, pp. 303±10.
Aly, H.Y., Grabowski, R., Pasurka, C., and Rangan, N. (1990), `Technical, Scale and Allocative Ef®ciencies in U. S. Banking: An Empirical Investigation', Review of Economics and Statistics, vol. 72, pp. 211±218.
Atkinson, S.E. and Cornwell, C. (1993), `Measuring Technical Ef®ciency with Panel Data: A Dual Approach', Journal of Econometrics, vol. 59, pp. 257±261.
Atkinson, S.E. and Cornwell, C. (1994a), `Parametric Estimation of Technical and Allocative Inef®ciency with Panel Data', International Economic Review, vol. 35, pp. 231±243.
Atkinson, S.E. and Cornwell, C. (1994b), `Estimation of Output and Input Technical Ef®ciency Using a Flexible Functional Form and Panel Data', International Economic Review, vol. 35, pp. 245±56.
Atkinson, S.E. and Halvorson, R. (1980), `A Test of Relative and Absolute Price Ef®ciency in Regulated Utilities', Review of Economics and Statistics, vol. 62, pp. 185±196.
Atkinson, S.E. and Halvorsen, R. (1984), `Parametric Ef®ciency Tests, Economies of Scale, and Input Demand in U.S. Electric Power Generation', International Economic Review, vol. 25, pp. 647±662.
Bauer, P.W., Berger, A.N. and Humphrey, D. B. (1993), `Ef®ciency and Productivity Growth in U.S. Banking, in H.O. Fried, C.A.K. Lovell and S.S. Schmidt, (eds.), The Measurement of Productive Ef®ciency: Techniques and Applications (Oxford: Oxford University Press), pp. 386±413.
Bell, F. and Murphy, N. (1968), `Cost in Commercial Banking: A Quantitative Analysis of Bank Behavior and its Relation to Bank Regulations', Research report no. 41, Federal Reserve Bank of Boston, M.A.
Measures of technical change over time
Year Actual (b,.) Intermediate (b, : 1) Optimal (b 1, : 1)
1981 0.098 (0.100) 0.195 (0.065) 0.080 (0.023) 1982 0.106 (0.099) 0.201 (0.059) 0.077 (0.017) 1983 0.132 (0.097) 0.214 (0.063) 0.090 (0.011) 1984 0.108 (0.164) 0.216 (0.062) 0.092 (0.014) 1985 0.139 (0.100) 0.223 (0.060) 0.096 (0.014) 1986 0.170 (0.087) 0.235 (0.062) 0.107 (0.012) 1987 0.168 (0.134) 0.239 (0.061) 0.115 (0.015) 1988 0.194 (0.065) 0.240 (0.057) 0.116 (0.017) 1989 0.177 (0.072) 0.238 (0.058) 0.114 (0.016) 1990 0.150 (0.084) 0.232 (0.060) 0.108 (0.009) 1991 0.162 (0.087) 0.241 (0.062) 0.117 (0.009) 1992 0.174 (0.087) 0.247 (0.063) 0.123 (0.007) 1993 0.176 (0.084) 0.251 (0.061) 0.127 (0.008) 1994 0.177 (0.085) 0.254 (0.062) 0.129 (0.009) 1995 0.165 (0.101) 0.256 (0.065) 0.132 (0.008)
Benston, G.J. (1965), `Economies of Scale and Marginal Costs in Banking Operations', National Banking Review, vol. 2, pp. 507±549.
Benston, G.J. (1972), `Economies of Scale in Financial Institutions', Journal of Money, Credit and Banking, vol. 4, pp. 319±354.
Berg, S.A., Forsund, F., Hjalmarsson, L., and Suominen, M. (1993), `Banking Ef®ciency in the Nordic Countries', Journal of Banking and Finance, vol. 17, pp. 371±388.
Berger, A.N. (1993), `"Distribution-free" Estimates of Ef®ciency in the U.S. Banking Industry and Tests of the Standard Distributional Assumptions', Journal of Productivity Analysis, vol. 4, pp. 261±292.
Berger, A.N., Hanweck, G.A., and Humphrey, D.B. (1987), `Competitive Viability in Banking: Scale, Scope and Product Mix Economies', Journal of Monetary Economics, vol. 20, pp. 501± 520.
Berger, A.N., Hancock, G.A., and Humphrey, D.B. (1993), `Bank Ef®ciency Derived from the Pro®t Function', Journal of Banking and Finance, vol. 17, pp. 317±347.
Berger, A.N. and Humphrey, D.B. (1991), `The Dominance of Inef®ciencies Over Scale and Product Mix Economies in Banking', Journal of Monetary Economics, vol. 28, pp. 117±148. Berger, A.N. and Humphrey, D.B. (1992a), `Measurement and Ef®ciency Issues in Commercial
Banking', in Z. Griliches, (ed.), Output Measurement in the Service Sectors, National Bureau of Economic Research (Chicago, IL: University of Chicago Press), pp. 245±279.
Berger, A.N. and Humphrey, D.B. (1992b), `Megamergers in Banking and the Use of Cost Ef®ciency as an Antitrust Defense', Antitrust Bulletin, vol. 37, pp. 541±600.
Berger, A.N., Hunter, W.C. and Timme, S.G. (1993), `The Ef®ciency of Financial Institutions: A Review and Preview of Research Past, Present, and Future', Journal of Banking and Finance, vol. 17, pp. 221±249.
DeYoung, R. and Nolle, D. (1996), `Foreign-owned Banks in the US: Earning Market Share or Buying it?', Journal of Money, Credit, and Banking, vol. 28, pp. 622±636.
Dietsch, M. (1993), `Economies of Scale and Scope in French Commercial Banking Industry', The Journal of Productivity Analysis, vol. 4, pp. 35±50.
Drake, L. and Weyman-Jones, T.G. (1996), `Productive and Allocative Inef®ciencies in U.K. Building Societies: A Comparison of Non-parametric and Stochastic Frontier Techniques', The Manchester School, pp. 22±37.
Elyasian, E. and Mehdian, S.M. (1990), `A Nonparametric Approach to Measurement of Ef®ciency and Technological Change: The case of Large U.S. Commercial Banks', Journal of Financial Services Research, vol. 4, pp. 157±168.
English, M., Grosskopf, S., Hayes, K., and Yaisawarng, S. (1993), `Output Allocative and Technical Ef®ciency of the Financial Services Sector', Journal of Banking and Finance, vol. 17, pp. 349±366.
FaÈre, R. and Lovell, C.A.K. (1978), `Measuring the Technical Ef®ciency of Production', Journal of Economic Theory, vol. 19, pp. 150±162.
Farrel, H.J. (1957), `The Measurement of Productive Ef®ciency', Journal of the Royal Statistical Society A, vol. 120, pp. 253±281.
Ferrier, G.D. and Lovell, C.A.K. (1990), `Measuring Cost Ef®ciency in Banking: Econometric and Linear Programming Evidence', Journal of Econometrics, vol. 46, pp. 229±45.
Fixler, D.J. and Zieschang, K.D. (1991), `Pricing Imputation Problems in Commercial Banking: A Production Function Approach', Working paper (Bureau of Labor Statistics).
Fuss, M. and McFfadden, D. (1978), Production Economics: A Dual Approach to Theory and Applications, (eds.), (Amsterdam: North-Holland Publishing Company).
Gilligan, T., Smirlock, M., and Marshall, W. (1984), `Scale and Scope Economies in the Multiproduct Banking Firm', Journal of Monetary Economics, vol. 13, pp. 393±405.
Glass, J.D., McKillop, D.G. and Hyndman, N. (1995), `Ef®ciency in the Provision of University Teaching and Research: An Empirical Analysis of UK Universities', Journal of Applied Econometrics, vol. 10, pp. 61±72.
Gropper, D.M. (1995), `Product-line Deregulation and the Cost Structure of U. S. Savings and Loan Associations', Applied Economics, vol. 27, pp. 183±191.
Humphrey, D.B. (1993), `Cost and Technical Change: Effects from Bank Deregulation', The Journal of Productivity Analysis, vol. 4, pp. 9±34.
Hughes, J. and Moon, C.G. (1995), `Measuring Bank Ef®ciency when Managers Trade Return for Reduced Risk', Working paper, Department of Economics, Rutgers University.
Hughes, J., Lang, W., Mester, L.J., and Moon, C.G. (1996a), `Ef®cient Banking under Interstate Branching', Journal of Money, Credit, and Banking, vol. 28, pp. 1045±1071.
Hughes, J., Lang, W., Mester, L.J., and Moon, C.G. (1996b), `Safety in Numbers? Geographic Diversi®cation and Bank Insolvency Risk', Working paper 96±14, Federal Reserve Bank of Philadelphia.
Hunter, W.C. and Timme, S.G. (1986), `Technical Change, Organizational Form and the Structure of Bank Production', Journal of Money, Credit, and Banking, vol. 18, pp. 152±166.
Hunter, W.C., Timme, S.G., and Yang, W.K. (1990), `An Examination of Cost Subadditivity and Multiproduct Production in Large U.S. Banks', Journal of Money, Credit and Banking, vol. 22, pp. 504±525.
Jagtiani, J. and Khathavit, A. (1996), `Scale and Scope Economies at Large Banks: Including Off-balance Sheet Products and Regulatory Effects (1984±1991)', Journal of Banking and Finance, vol. 20, pp. 1271±87.
Jorgenson, D.W. and Lau, L.J. (1974a), `The Duality of Technology and Economic Behavior', Review of Economic Studies, vol. 41, pp. 181±200.
Jorgenson, D.W. and Lau, L.J. (1974b), `Duality and Differentiability in Production', Journal of Economic Theory, vol. 9, pp. 23±42.
Kim, H.Y. (1986), `Economies of Scale and Economies of Scope in Multiproduct Financial Institutions: Further Evidence from Credit Unions', Journal of Money, Credit and Banking, vol. 18, pp. 220±226.
Kumbhakar, C.S. (1987), `The Speci®cation of Technical and Allocative Inef®ciency in Stochastic Production and Pro®t Frontier', Journal of Econometrics, vol. 34, pp. 335±48.
Kumbhakar, S.C. (1991), `The Measurement and Decomposition of Cost Inef®ciency: The Translog Cost System', Oxford Economic Papers, vol. 43, pp. 667±683.
Kumbhakar, S.C. (1996a), `A Parametric Approach to Ef®ciency Measurement Using a Flexible Pro®t Function', Southern Economic Journal, pp. 473±487.
Kumbhakar, S.C. (1996b), `Ef®ciency Measurement with Multiple Outputs and Multiple Inputs', Journal of Productivity Analysis, vol. 7, pp. 225±255.
Kumbhakar, S.C. (1997), `Modeling Allocative Inef®ciency in a Translog Cost Function and Cost Share Equations: An Exact Relationship', Journal of Econometrics, vol. 76, pp. 351±356. Kumbhakar, S.C. and Bhattacharyya, A. (1992), `Price Distortions and Resource-use Ef®ciency
in Indian Agriculture: A Restricted Pro®t Function Approach', Review of Economics and Statistics, pp. 231±239.
Lang, G. and Welzel, P. (1996), `Ef®ciency and Technical Progress in Banking Empirical Results for a Panel of German Cooperative Banks', Journal of Banking and Finance, vol. 20, pp. 1003±1023.
Lau, L.J. (1978), `Applications of Pro®t Functions, in: Production Economics: A Dual Approach to Theory and Applications, M. Fuss and D. McFfadden, (eds), (Amsterdam: North-Holland Publishing Company), pp. 113±216.
Lau, L.J. and Yotopoulos, P.A. (1971), `A Test for Relative Ef®ciency and an Application to Indian Agriculture', American Economic Review, vol. 61, pp. 94±109.
Lovell, C.A.K. and Sickles, R.C. (1983), `Testing Ef®ciency Hypothesis in Joint Production: A Parametric Approach', Review of Economics and Statistics, vol. 65, pp. 51±58.
McAllister, P.H. and McManus, D.A. (1993), `Resolving the Scale Ef®ciency Puzzle in Banking', Journal of Banking and Finance, vol. 17, pp. 389±405.
Mester, L.J. (1987), `A Multiproduct Cost Study of Savings and Loans', Journal of Finance, vol. 42, pp. 423±45.
Mester, L.J. (1993), `Ef®ciency in the Savings and Loan Industry', Journal of Banking and Finance, vol. 17, pp. 267±286.
Mester, L.J. (1996), `A Study of Bank Ef®ciency Taking into Account Risk-preferences', Journal of Banking and Finance, vol. 20, pp. 1025±1045.
Mulder, U. and Sassenou, M. (1993), `Economies of Scale and Scope in French Banking and Savings Institutions', The Journal of Productivity Analysis, vol. 4, pp. 51±72.
Murray, J.D. and White, R.W. (1983), `Economies of Scale and Economies of Scope in Multi-product Financial Institutions: A Study of British Columbia Credit Unions', Journal of Finance, vol. 38, pp. 887±902.
Rangon, N., Grabowski, R., Aly, H., and Pasurka, C. (1988), `The Technical Ef®ciency of U.S. Banks', Economics Letters, vol. 28, pp. 169±75.
Rockafellar, R.T. (1970), Convex Analysis, (Princeton, N.J.: Princeton University Press).
Rller, L. (1990), `Modeling Cost Structure: The Bell System Revisited', Applied Economics, vol. 22, pp. 1601±1674.
Timme, S.G. and Yang, W.K. (1991), `On the Use of a Direct Measure of Ef®ciency in Testing Structure-performance Relationships', Working paper (Georgia State University).
Toda, Y. (1976), `Estimation of a Cost Function when Cost is not a Minimum: The Case of Soviet Manufacturing Industries, 1958±71', Review of Economics and Statistics, vol. 58, pp. 259±68. Yotopolous, P.A. and Lau, L.J. (1973), `A Test for Relative Economic Ef®ciency: Some Further
