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*Correspondence address: School of Accounting, Banking and Economics, University of Wales,

Bangor Gwynedd, Bangor, LL57 2DG, U.K. Tel.: (01248) 382170; fax: (01248) 364760.E-mail address: [email protected] (P. Molyneux).

European Economic Review 45 (2001) 1931}1955

E$ciency in European banking

Y. Altunbas7, E.P.M. Gardener, P. Molyneux*, B. Moore

The Business School, South Bank University, London, UK

University of Wales, Bangor, UK

Erasmus University, Rotterdam, UK

Downing College, University of Cambridge, Cambridge, UK

Received 1 September 1997; accepted 1 April 2000

Abstract

This paper extends the established literature on modelling the cost characteristics of

banking markets by applying the #exible Fourier functional form and stochastic cost

frontier methodologies to estimate scale economies, X-ine$ciencies and technical change

for a large sample of European banks between 1989 and 1997. The results reveal that

scale economies are widespread for smallest banks and those in the ECU 1 billion to

ECU 5 billion assets size range. Typically, scale economies are found to range between

5% and 7%, while X-ine$ciency measures appear to be much larger, between 20% and

25%. X-ine$ciencies also appear to vary to a greater extent across di!erent markets,

bank sizes and over time. This suggests that banks of all sizes can obtain greater cost

savings through reducing managerial and other ine$ciencies. This paper also shows that

technical progress has had a similar in#uence across European banking markets between

1989 and 1997, reducing total costs by around 3% per annum. The impact of technical

progress in reducing bank costs is also shown to systematically increase with bank size.

Overall, these results indicate that Europe's largest banks bene"t most from technicalprogress although they do not appear to have scale economy advantages over their

smaller counterparts. 2001 Elsevier Science B.V. All rights reserved.

JEL classixcation: G21; D21; G23

Keywords: Banking; E$ciency; Frontiers

0014-2921/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 0 1 4 - 2 9 2 1 ( 0 0 ) 0 0 0 9 1 - X


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1. Introduction

The Commission of the European Communities (1988) has stressed in its 1992single market programme that substantial bene"ts would accrue to those sectors

that can bene"t from positive supply-side e!ects. In particular, &price reductions

occasioned by competitive pressures will force "rms to look actively for reduc-

tion in costs through the elimination of areas of low productivity or by a greater

exploitation of scale economies' (European Economy, 1988, p. 162). Despite the

importance of cost e$ciencies, however, only a few studies have investigated

cost characteristics in European banking and no studies, as far as we are aware,

have provided comparable cross-country comparisons of scale and X-e$cien-

cies. This paper aims to redress this imbalance by using the #exible Fourier

functional form and stochastic cost frontier approach to evaluate evidence ofscale and X-ine$ciencies, as well as technical change, across 15 European

banking markets between 1989 and 1997.

This paper extends the established literature on modelling the cost character-

istics of banking markets by applying the #exible Fourier functional form and

stochastic cost frontier methodologies to estimate scale economies, X-ine$cien-

cies and technical change for a large sample of European banks between 1989

and 1997. The results reveal that scale economies are widespread for smallest

banks and those in the ECU 1 billion to ECU 5 billion assets size range.

Typically, scale economies are found to range between 5% and 7%, while

X-ine$ciency measures appear to be much larger, between 20% and 25%.

X-ine$ciencies also appear to vary to a greater extent across di!erent markets,

bank sizes and over time. This suggests that banks of all sizes can obtain greater

cost savings through reducing managerial and other ine$ciencies. This paper

also shows that technical progress has had a similar in#uence across European

banking markets between 1989 and 1997, reducing total costs by around 3% per

annum. The impact of technical progress in reducing bank costs is also shown to

systematically increase with bank size. Overall, these results indicate that

Europe's largest banks bene"t most from technical progress although they do

not appear to have scale economy advantages over their smaller counterparts.

2. E7ciency in banking markets } A brief literature review

Over recent years the structure of European banking has been changing

rapidly and a main motivation has been the drive for greater e$ciency. A

substantial US literature has emerged (for example, see Berger et al., 1993;

Kaparakis et al., 1994; Mester, 1996; Mitchell and Onvural, 1996) which "nds

that X-e$ciencies, brought about by superior management, technologies and

other factors, exceed those e$ciencies resulting from scale and scope economies.In their review of the US literature, Berger et al. (1993) "nd that X-ine$ciencies

1932 Y. Altunbas7 et al. / European Economic Review 45 (2001) 1931}1955


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account for around 20% or more of costs in banking, while scale and product-

mix ine$ciencies, &when accurately estimated', are usually found to account for

less than 5% of costs.Although European research on cost e$ciency has not matched the volume of

US studies this has begun to change in recent years. The majority of European

studies have focused on the issue of scale and scope economies in individual

countries and for particular types of banks. The earliest researchers used

Cobb}Douglas and CES cost function methodologies to model underlying cost

functions, whereas from the mid-1980s onwards, most studies have used the

translog functional form to estimate the cost characteristics of the banking

industry. Levy-Garboua and Renard (1977), Dietsch (1988, 1993) and Martin

and Sassenou (1992) have examined cost economies in French banking and

found mixed results, although the studies suggest stronger evidence of scalee$ciencies, especially for the smallest banks. Studies of the Italian market

undertaken by Cossutta et al. (1988), Baldini and Landi (1990) and Congliani

et al. (1991) also "nd strong evidence of scale e$ciencies. Lang and Welzel (1996)

also used the standard translog cost function methodology to estimate cost

economies for German cooperative banks and they "nd evidence of scope

economies for the largest banks. Cost studies in the UK have focused on the

building society sector mainly because of the limited number of domestic

commercial banks with similar business pro"les. These include studies by

Gough (1979), Cooper (1980), Barnes and Dodds (1983), Hardwick (1989, 1990),

Drake (1992, 1995) and McKillop and Glass (1994). The UK studies use a range

of competing methodologies and report con#icting results. Evidence of scale

economies has also been found in Finland for the cooperative and savings bank

sector (Kolari and Zardkoohi, 1990), Ireland (Glass and McKillop, 1992), Spain

(Fanjul and Maravall, 1985; Rodriguez et al., 1993) and Turkey (Fields et al.,

1993). Finally, Vennet (1993) uses the translog approach to investigate a sample

of 2600 credit institutions operating in the EU for the year 1991. He "nds that

optimal scale is situated in the $3}10 billion asset range and there also appears

to be scope economies for the largest banks. (See Molyneux et al. (1996, Chapter

6) for a comprehensive review of this literature.)The above literature focuses on scale and scope economies, whereas more

recent literature has attempted to evaluate X-ine$ciencies and technical change

in various European banking markets. Berg et al. (1991, 1992), Berg et al. (1993)

and Berg et al. (1995) have made an important contribution to the literature in

their studies of Scandinavian banking markets. The earlier studies focus on the

Norwegian market and the later work analyses e$ciency di!erences across

Scandinavian banking markets. For example, Berg et al. (1993) uses Data

Envelopment Analysis (DEA) to measure the X-ine$ciency of banks in three

Nordic countries (Finland, Norway and Sweden). An innovative technique

(a form of Malmquist productivity index) was used to model the bankingfrontier technologies. Overall they found that Swedish banks tended to be more

Y. Altunbas7 et al. / European Economic Review 45 (2001) 1931}1955 1933


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e$cient than their Nordic counterparts. The most recent study, by Berg et al.

(1995) uses DEA to investigates ine$ciencies in the banking industries of

Denmark, Finland, Norway and Sweden. The study "nds that the largestDanish and Swedish banks were among the most e$cient units in their pooled

sample and only one large Finnish bank and one large Norwegian bank were

more than 90% e$cient. They concluded that the Danish and Swedish banks

were in the best position to expand in a common Nordic banking market.

Various studies of the Spanish banking market have used DEA techniques to

investigate productivity, evaluating improvement in cost e$ciency by measur-

ing total factor productivity and technical change. Perez and Quesada (1994),

for example, provide estimates of changes in productivity for the main savings

and commercial banks between 1986 and 1992 and show that the productivity

gains of the largest banks have been substantial. They also show that a group ofcommercial banks representing up to 40% of the sector operate with e$ciencies

20% or lower than the most e$cient banks (see also Pastor et al., 1994). More

recent studies undertaken by Gri!ell-Tatje and Lovell (1995a,b, 1996) use sim-

ilar linear programming techniques and their own &generalised Malmquist

productivity index' to investigate productive e$ciency and total factor produc-

tivity in Spanish savings banks between 1986 and 1991. They conclude that

neither branching nor mergers provide an adequate explanation for the nature

of the productivity decline over the period studied. Other studies using DEA

techniques to model bank productivity and e$ciencies include Drake and

Howcroft (1994) for UK building societies and Gobbi (1995) on Italian banks.

While the aforementioned European studies use non-parametric techniques,

such as DEA, to estimate e$ciencies in banking markets, there appears to be

limited evidence of the use of stochastic cost frontier techniques to model these

relationships. This is surprising given that much of the recent US literature (for

example, see Berger et al., 1993; Kaparakis et al., 1994; Mester, 1996) use

parametric techniques and Resti (1997), in his study of the Italian banking

market, shows that both linear programming and stochastic cost frontier ap-

proaches tend to provide similar cost e$ciency results. (A "nding also con"rmed

by Drake and Weyman-Jones (1992)). This paper aims to add to the establishedliterature by modelling e$ciencies using the stochastic cost frontier and #exible

Fourier (FF) functional form to approximate the underlying cost characteristics

of the EU banking industry.

3. Methodology

While there continues to be debate about the de"nition of outputs used in cost

e$ciency studies we follow along the lines of the traditional intermediation

approach as suggested by Sealey and Lindley (1977), where the inputs, labour,physical capital and deposits are used to produce earning assets. Two of our



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Thanks to a referee for pointing out the implications of incorporating risk variables in the costfunction speci"cation.

outputs, total loans and total securities are earning assets and we also include

total o!-balance sheet items (measured in nominal terms) as a third output.

Although the latter are technically not earning assets, this type of businessconstitutes an increasing source of income for banks and therefore should be

included when modeling banks' cost characteristics, otherwise, total output

would tend to be understated (Jagtiani and Khanthavit, 1996). Various recent

studies (such as those by Hughes and Mester (1993), Hughes et al. (1995), Mester

(1996) and Clark (1996)) have drawn attention to the fact that bank e$ciency

studies typically ignore the impact of risks on banks' costs, and they suggest that

risk characteristics need to be incorporated in the underlying industry cost

function because, &unless quality and risk are controlled for, one might easily

miscalculate a bank's level of ine$ciency' (Mester, 1996, p. 1026). As suggested

in Hughes and Mester (1993) and Mester (1996) we include the level of equitycapital in our cost frontier to control for di!erences in banks risk preferences.

Ine$ciency measures are estimated using the stochastic cost frontier ap-

proach. This approach labels a bank as ine$cient if its costs are higher than

those predicted for an e$cient bank producing the same input/output combina-

tion and the di!erence cannot be explained by statistical noise. The cost frontier

is obtained by estimating a cost function with a composite error term, the sum of

a two-sided error representing random #uctuations in cost and a one-sided

positive error term representing ine$ciency.

Frerier and Lowell (1990) have shown that ine$ciency measures for

individual "rms can be estimated using the stochastic frontier approach as

introduced by Aigner et al. (1977), Meeusen and van den Broeck (1977). The

single-equation stochastic cost function model can be given as

C"C(QG,P

G)#

G(1)

where TCis observed total cost,QG

is a vector of outputs, and PG

is an input-price

vector. Following Aigner et al. (1977), we assume that the error of the cost

function is

"u#v (2)

where u and v are independently distributed; u is assumed to be distributed as

half-normal, u&N(0,S

), i.e., a positive disturbance capturing the e!ects of

ine$ciency, and v is assumed to be distributed as two-sided normal with zero

mean and variance, T

, capturing the e!ects of the statistical noise.

Observation-speci"c estimates of the ine$ciencies, u, can be estimated by

using the conditional mean of the ine$ciency term, given the composed error

term, as proposed by Jondrow et al. (1982). The mean of this conditional



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See Bauer (1990) for an excellent review of the frontier literature and how di!erent stochastic

assumptions can be made. Cebenoyan et al. (1993), for example, uses the truncated normal model.

Mester (1993) in common with many studies uses the half-normal distribution. Stevenson (1980) and

Greene (1990) have used the normal and gamma model, respectively. Altunbas7 and Molyneux (1994)

"nd that e$ciency estimates are relatively insensitive to di!erent distributional assumptions when

testing the half normal, truncated normal, exponential and gamma e$ciency distributions, as alldistributions yield similar ine$ciency levels for the German banking market.

distribution for the half-normal model is shown as

E(uG"

G)"

1# f(G/)

1!F(G/)#

G

(3)

where "S

/T

and total variance, "S#

T; F( . ) and f( . ) are the stan-

dard normal distribution and density functions, respectively. E(u " ) is an unbias-

ed but inconsistent estimator of uG, since regardless of N, the variance of the

estimator remains non-zero (see Greene, 1993, pp. 80}82). Jondrow et al. (1982)

have shown that the ratio of the variability for u and v can be used to measure

a banks' relative ine$ciency, where "S

/T

, is a measure of the amount of

variation stemming from ine$ciency relative to noise for the sample. The

X-ine$ciency term, u, is assumed to remain constant over time for each bank.Estimates of this model can be computed by maximising the likelihood function

directly (Olson et al., 1980). Previous studies modelling international bank

ine$ciencies such as, Allen and Rai (1996) and those which examine US banks,

such as Kaparakis et al. (1994) and Mester (1996), all use the half-normal

speci"cation to test for ine$ciency di!erences between banking institutions.

The next step, given the choice of the half-normal ine$ciency stochastic

frontier approach, relates to choosing the underlying cost function speci"cation.

In this paper, we use the #exible Fourier (FF) form to examine the speci"cation

which best "ts the underlying cost structure of EU banking systems. Gallant

(1981, 1982), Berger et al. (1994) and Mitchell and Onvural (1996) have stated

that the FF is the global approximation which can be shown to dominate the

conventional translog form. It has been widely accepted that the global property

is important in banking where scale, product mix and other ine$ciencies are

often heterogeneous. Therefore, local approximations (such as those generated

by the translog function) may be relatively poor approximation to the underly-

ing true cost function.

The FF is a semi-nonparametric approach used to tackle the problem arising

when the true functional form of the relationships are unknown. This methodo-

logy was "rst proposed by Gallant (1981, 1982), was discussed by Elbadawi et al.(1983), Chalfant and Gallant (1985), Eastwood and Gallant (1991), Gallant and

Souza (1991) and was applied to the analysis of bank cost e$ciency by Berger



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Note that the "nancial capital variable (E) is fully interactive with the output (Q) and input pricevariables.

et al. (1994), Spong et al. (1995) and more recently by Mitchell and Onvural

(1996) used this methodology. It has been shown (Tolstov, 1962), that a linear

combination of the sine and cosine function, namely the Fourier series, can "texactly any well behaved multivariate function. This is due to the mathematical

behaviour of the sine and cosine functions which are mutually orthogonal over

the [0, 2] interval and function space-spanning. The FF form, therefore, pro-

vides a better approximation of the true form of the unknown cost function

without misspeci"cation.

To calculate the ine$ciency measures, the FF form, including a standard

translog and all "rst-, second- and third-order trigonometric terms, as well as

a two-component error structure is estimated using a maximum likelihood

procedure. Note also that a variable for "nancial capital is included to control

for risk. This is shown as

lnC"#

G

G

lnQG#

J

J

lnPJ#

#

lnE

#1

2G

H

GH

lnQG

lnQH#

J

K

JK

lnPJ

lnPK

#

lnE lnE##

G

K

GK

lnQG

lnPK

#G

G

lnPG

lnE#G

G

lnQH

lnE#G

G lnQ

G

#J

J lnP

J#

G

[aG

cos(zG)#b

Gsin(z

G)]

#G

H

[aGH

cos(zG#z

H)#b

GHsin(z

G#z

H)]# (4)

where

ln

C"

natural logarithm of total costs (operating and "nancial cost),lnQ

G"natural logarithm of bank outputs,

lnPJ"natural logarithm of ith input prices (i.e. wage rate, interest rate

and physical capital price),

lnE"the natural logarithm of equity capital,

"time trend,



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Mitchell and Onvural (1996, p. 181) did not impose restrictions on the trigonometric input price

coe$cients for computational reasons. Gallant (1982), however, has shown that this should not

prevent an estimated FF cost equation from closely approximating the true cost function.

Berger et al. (1997) restricted zG

to span [0.1:2, 0.9; 2], however, the use of this interval

provided inconsistent results in the present study. Following Mitchell and Onvural (1996),

we adopted a second trigonometric order in our study. According to Gallant (1982), increasing the

number of trigonometric orders, relative to sample size, reduces approximation errors. Eastwood

and Gallant (1991) show that the FF cost function produces consistent and asymptotically

normal parameter estimates when the number of parameters estimated is set to the number of

e!ective observations raised to the two thirds power. However, Gallant (1981) advocates that even

a limited number of trigonometric orders is su$cient to obtain global approximations. The choice of

the range used by di!erent researchers is, however, subjective and relative to the size of data setanalysed.

ZG"the adjusted values of the log output ln Q

Gand lnE such that

they span the interval [0, 2],

,, , , , ,,,, ,, , a and b are coe$cients to be estimated.

Following Berger et al. (1994), the study applies Fourier terms only for the

outputs, leaving the input price e!ects to be de"ned entirely by the translog

terms. The primary aim is to maintain the limited number of Fourier terms for

describing the scale and ine$ciency measures associated with di!erences in

bank size. Moreover, the usual input price homogeneity restrictions can be

imposed on logarithmic price terms, whereas they cannot be easily imposed on

the trigonometric terms.

In addition, the scaled log-output quantities, zG , are calculated aszG"

G(lnQ

G#w

G), lnQ

Gare unscaled log-output quantities;

Gand w

Gare scaled

factors, writing the periodic sine and cosine trigonometric functions within one

period of length 2 before applying the FF methodology (see Gallant, 1981). The

G's are chosen to make the largest observations for each scaled log-output

variable close to 2; wG's are restricted to assume the smallest values close to

zero. In this study, we restricted the zG

to span between 0.001 and 6 to reduce

approximation problems near the endpoints as discussed by Gallant (1981) and

applied by Mitchell and Onvural (1996).

Since the duality theorem requires that the cost function is linearly homo-

geneous in input prices and second-order parameters are symmetric, the follow-ing restrictions apply to the parameters of the cost function in Eq. (4):

J

J"1;

J

JK"0;

J

J"0;

K

GK"0,

GH"

HGand

JK"

KJ. (5)



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The cost frontiers are estimated using the random e!ects panel data approach

(as in Lang and Welzel, 1996). We use the panel data approach because technical

e$ciency is better studied and modelled with panels (Baltagi and Gri$n, 1988;Cornwell et al., 1990; Kumbhakar, 1993). The random e!ects model is preferred

over the "xed e!ects model because the latter is considered to be the more

appropriate speci"cation if we are focusing on a speci"c set ofN "rms. More-

over, and ifN is large, a "xed e!ects model would also lead to a substantial loss

of degrees of freedom (Baltagi, 1995).

Within sample scale economies are calculated as in Mester (1996) and are

evaluated at the mean output, input price and "nancial capital levels for the

respective size quartiles. A measure of economies of scale (SE) is given by the

following cost elasticity by di!erentiating the cost function in Eq. (4) with

respect to output. This gives us

SE"G

* ln

* ln

C

QG

"G

G#

G

H

GH

lnQH#

G

K

GK

lnPK

#G

G#

G

G

[!aG

sin(ZG)#b

Gcos(Z

G)]

#2G

G

H

[!aGH

sin(ZG#Z

H)#b

GHcos(Z

G#Z

H)]. (6)

IfSE(1, then we have increasing returns to scale (implying economies of scale);

if SE"1 then we have constant returns to scale; if SE'1 then we have

decreasing returns to scale (implying diseconomies of scale). Following McKil-

lop et al. (1996) and Lang and Welzel (1996) the rate of technical progress may

be inferred from changes in a "rm's cost function over time. A time trend

variable, , serves as a proxy for disembodied technical change. The time-trend

is a &catch-all' variable that captures the e!ects of technological factors: i.e.

learning by doing and organisational changes allowing for the more e$cient use

of existing inputs, together with the e!ects of other factors, such as changing

environmental regulations (Baltagi and Gri$n, 1988; Nelson, 1984). Technical

progress allows the "rm to produce a given output, Q, at lower levels of total

cost over time, holding input prices and regulatory e!ects constant. In order to

estimate the impact of technical change we calculate the variation in the average

cost due to a given change in technology. This can be measured by the partial

derivative of the estimated cost function with respect to the time trend () and

can be shown as follows:

* lnC

*"##

J

J lnPJ#G

G lnQG . (7)



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Thanks to one referee for pointing out the problems associated with using nominal data in

estimating technical change.

Various structural tests were undertaken to test for data poolability and heteroscedasticity and

these con"rmed the applicability of the panel data approach. These results are available upon

request from the authors.

These results di!er from those obtained by estimating the same cost function without controlling

for risk (i.e. excluding the equity capital variable). While the yearly banking system estimates present

a similar picture of widespread scale economies (ranging between 5% and 10%), the results for

di!erent size categories of banks was almost the complete opposite. In all countries, apart from in

Belgium, Greece, the Netherlands, Portugal and Spain, the smallest banks exhibited constant

returns. Scale economies typically become larger with size and optimal bank size was inexhausted.These results are available from the authors on request.

4. Data and results

This study uses banks' balance sheet and income statement data for asample of European banks between 1989 and 1997, obtained from the London-

based International Bank Credit Analysis Ltd.'s &Bankscope' database. Table

1 reports the de"nition, mean and standard deviation of the input and output

variables in real terms used in the cost frontier estimations (all data are in real

1990 terms and they have been converted using individual country GDP

de#ators). The descriptive statistics along with parameter estimates are shown

in the appendix.

Scale economies are estimated for all the banks and the mean of the overall

economies of scale are reported for each country and for di!erent sizes of banks

over the years 1989 to 1997 are shown in Table 2.Table 2 presents a mixed picture. The top of the table suggests that scale

economies are prevalent across EU banking systems, with the notable exception

of the Finnish market. Typically scale economies range between 5% and 7%.

Thus a 100% increase in the level of all outputs, on average, would lead to about

a 93% to 95% increase in total costs, respectively. However, the lower part of

Table 2 reveals that the aforementioned "ndings are mainly a result of wide-

spread scale economies found for the smallest banks (banks with assets size

between ECU 1 to ECU 200 million) and those in the ECU 1 to ECU 5 billion

assets size range. The largest banks, with the exception of those in Denmark,

Germany, Netherlands and the UK, exhibit constant or diseconomies of scale

(as do the majority of European banks in the ECU 200 million } 1 billion asset

size category). The magnitude of the scale economy estimates for the overall

banking system are in accordance with previous studies of the US banking

system, as is evidence on widespread economies for the smallest banks (see

Berger et al., 1993). The lack of evidence of scale economies for the largest banks

is to a certain extent corroborated by the "ndings of Vennet's (1993) study on

EU banking.



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Table

1

Descriptivestatisticsoftheoutputsand

inputvariablesusedinthemodel,1997

Variables

Description

Mea

n

Median

StDev

Min

Max

C

Totalcost(operatingand

"nancialcost)(ECUmil)

331.1

31.3

1506.0

0.6

25804.6

P

Priceoflabour(ECUmil)(totalpersonnelexpenses/

totalasset)

0.0148

0.0141

0.0086

0.0006

0.1340

P

Priceoffunds(%)(totalinterestexpenses/totalfunds

(demand,saving,timeinte

rbankdeposits,long-term

debt,subordinateddebtandother))

0.0499

0.0423

0.0311

0.0079

0.4133

P

Priceofphysicalcapital(%

)(totaldepreciationand

othercapitalexpenses/total"xedassets)

0.5083

0.4741

0.2330

0.0697

2.4000

Q

Thevalueoftotalaggrega

teloans(alltypesofloans)

(ECUmil)

2665

.0

253.0

12971.0

2.0

254630.0

Q

Thevalueoftotalaggrega

tesecurities(short-term

investment,equityandoth

erinvestmentsandpublic

sectorsecurities)(ECUmil)

2431

.0

187.0

11972.0

1.0

243006.0

Q

Thevalueoftheo!-balancesheetactivities

(nominalvalues)(ECUmil)

1632

.8

45.2

12143.3

0.5

337864.4

E

Thevalueofthetotalaggregateequities(ECUmil)

237.2

28.8

1009.9

1.4

19447.5

Numb

erofobservedbanks:4104.

The

"gureshavebeende#atedusingc

ountryspeci"cGDPde#atorswith1990asabaseyear.

We

de"nethepriceoflabourastotal

personnelexpensesdividedbytotalassetsbecausetheIBCAbank

databasedoesnotincludecompre

hensive

inform

ationonbanksta!

numbers.Asonerefereepointedoutgiventhat

theratiooftotalassetstonumberofemployeesforeachbankingsystemis

unlike

lytobeconstantovertheyearsun

derstudy.Apriceoflabourmeasu

reusingsta!

expensestonumbero

femployeesmayyielddi!erentparameter

estima

tesfromthosereportedinthisstu

dy.



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Table 2

Scale economies for European banks 1989}1996

1989 1990 1991 1992 1993 1994 1995 1996 1997

Austria 0.981 0.973 0.960 0.948 0.957 0.965 0.954 0.957 0.954

Belgium 0.947 0.948 0.950 0.966 0.950 0.958 0.958 0.965 0.984

Denmark 0.936 0.934 0.941 0.892 0.890 0.891 0.892 0.898 0.918

Finland 1.010 1.018 1.024 1.029 1.007 1.008 1.014 1.011 1.018

France 0.974 0.975 0.981 0.974 0.971 0.967 0.969 0.968 0.972

Germany 0.966 0.969 0.963 0.930 0.922 0.921 0.921 0.941 0.940

Greece 0.952 0.949 0.904 0.919 0.913 0.935 0.941 0.947 0.964

Ireland 0.955 0.967 0.960 0.970 0.974 0.957 0.955 0.961 0.954

Italy 0.962 0.968 0.970 0.968 0.930 0.924 0.924 0.932 0.946

Luxembourg 0.964 0.930 0.934 0.932 0.919 0.919 0.921 0.928 0.963Netherlands 0.947 0.925 0.943 0.935 0.934 0.927 0.923 0.929 0.943

Portugal 0.987 0.954 0.961 0.980 0.994 0.989 0.999 1.021 1.032

Spain 0.951 0.940 0.932 0.925 0.927 0.931 0.936 0.940 0.945

Sweden 1.011 0.959 0.960 0.959 0.955 0.956 0.960 0.960 0.986

UK 0.906 0.924 0.920 0.943 0.952 0.956 0.958 0.954 0.948

All 0.962 0.958 0.957 0.946 0.934 0.931 0.932 0.945 0.949

Asset sizes (ECU Mil)

1}99.9 100}

199.9

200}

299.9

300}

499.9

500}

999.9

1000}

2499.9

2500}

4999.9

5000#

Austria 0.873 0.940 1.009 1.013 1.005 0.987 0.968 1.059

Belgium 0.879 0.974 1.019 1.025 1.003 1.004 1.019 1.031

Denmark 0.832 0.983 1.011 1.026 0.998 0.916 0.912 0.967

Finland 0.857 0.942 } 1.029 0.974 0.951 1.012 1.081

France 0.900 0.978 1.007 1.020 1.005 0.975 0.980 0.994

Germany 0.832 0.915 0.985 1.016 1.008 0.958 0.930 0.964

Greece 0.857 0.978 1.029 1.024 0.984 0.897 0.949 1.081

Ireland 0.940 0.944 0.989 0.995 0.975 0.931 0.882 1.016

Italy 0.805 0.981 1.018 1.013 0.974 0.929 0.957 1.014

Luxembourg 0.885 0.986 0.997 1.004 1.003 0.994 1.041 1.045

Netherlands 0.853 0.936 0.982 0.970 0.979 0.958 0.923 0.940

Portugal 0.911 1.014 1.038 1.036 0.983 0.964 1.015 1.109

Spain 0.848 0.961 0.993 0.995 0.961 0.925 0.930 0.996

Sweden 0.879 0.934 0.965 0.976 0.979 0.951 0.934 0.990

UK 0.864 1.003 0.986 0.992 0.978 0.980 0.984 0.951

All 0.849 0.936 0.995 1.013 0.997 0.960 0.961 0.992

Note: Bold typeface for values indicates signi"cantly di!erent from one at the 5% level.



13/25

Ine$ciency results derived from the cost frontier speci"cation excluding the risk variable tended

to yield similar results. For instance, Austria, Denmark, Germany and Italy were also found to be

the most e$cient banking sectors, although e$ciency levels were found to be slightly lower. In

estimates derived from the standard cost frontier we also found more systems exhibiting increasingine$ciency over time. These results are also available from the authors on request.

Ine$ciency measures are given in Table 3 and they show greater variation

across time, countries and bank sizes than the scale economy estimates. The

country estimates reveal that the relative ine$ciency of various banking markets(Finland, Luxembourg, Netherlands, and Sweden up to 1996) have increased

over time. The results also show that the banks in Sweden and the UK have

been on average, relatively ine$cient compared with other European banks.

The most e$cient banking sectors are those of Austria, Denmark, Germany and

Italy.

Table 3 also shows ine$ciency measures for di!erent sizes of banks. Apart

from in Austria, there appears to be no strong evidence that the largest banks

are systematically more e$cient than smaller banks. While there are observable

di!erences between size categories no trend is apparent. On average, X-ine$c-

iencies appear to range between 20% and 25% across di!erent size classes, andthis suggests that the same level of output could be produced with 75}80% of

current inputs if banks were operating on the e$cient frontier. This is in the

same range as those found in Resti (1997) and Gri!ell-Tatje and Lovell (1996).

Overall the above "ndings indicate that X-ine$ciencies are more important

than scale economies across European banking markets. The policy implication

therefore is that greater cost savings are to be obtained if banks focus their

attentions on reducing managerial, technological and other ine$ciencies, com-

pared with increasing size. Nevertheless, there are still cost savings of between

5% and 7% that can be realised for small and some medium sized banks

through increasing output size.Estimates of technical change are shown in Table 4. This shows that technical

change, has made a positive contribution across all banking markets, reducing

the real annual cost of production by about 3%. The impact of technical change

on reducing costs is shown to systematically increase with bank size. These

estimates, however, should be treated with caution given the problems asso-

ciated with using a time trend to measure technical change (Hunter and Timme,

1991).

5. Conclusion

This paper extends the established literature on modelling the cost character-

istics of banking markets by applying the #exible Fourier functional form and



14/25

Table 3

Average X-ine$ciency levels of banks in the EU 1989}1997

1989 1990 1991 1992 1993 1994 1995 1996 1997

Austria 0.209 0.212 0.189 0.202 0.201 0.186 0.200 0.205 0.181

Belgium 0.369 0.355 0.356 0.320 0.239 0.228 0.234 0.236 0.322

Denmark 0.222 0.215 0.227 0.202 0.196 0.202 0.195 0.194 0.191

Finland 0.193 0.192 0.208 0.239 0.294 0.259 0.294 0.292 0.296

France 0.288 0.264 0.262 0.265 0.270 0.269 0.266 0.275 0.244

Germany 0.218 0.209 0.202 0.186 0.170 0.162 0.161 0.158 0.135

Greece 0.280 0.256 0.227 0.249 0.247 0.242 0.235 0.236 0.238

Ireland 0.166 0.266 0.269 0.252 0.278 0.289 0.289 0.303 0.323

Italy 0.217 0.218 0.231 0.219 0.205 0.211 0.217 0.192 0.126

Luxembourg 0.234 0.246 0.246 0.251 0.245 0.229 0.254 0.243 0.330Netherlands 0.213 0.204 0.206 0.222 0.227 0.248 0.257 0.252 0.238

Portugal 0.335 0.314 0.298 0.308 0.319 0.265 0.289 0.280 0.289

Spain 0.234 0.234 0.238 0.219 0.220 0.227 0.246 0.237 0.237

Sweden 0.194 0.232 0.288 0.313 0.328 0.431 0.410 0.439 0.165

UK 0.298 0.324 0.312 0.333 0.314 0.302 0.303 0.289 0.298

All 0.245 0.241 0.241 0.233 0.209 0.200 0.202 0.202 0.179

Asset sizes (ECU Mil)

1}99.9 100}

199.9

200}

299.9

300}

499.9

500}

999.9

1000}

2499.9

2500}

4999.9

5000#

Austria 0.313 0.207 0.159 0.171 0.148 0.112 0.098 0.160

Belgium 0.291 0.264 0.239 0.213 0.257 0.283 0.262 0.254

Denmark 0.182 0.191 0.184 0.211 0.217 0.202 0.251 0.287

Finland 0.379 0.190 } 0.179 0.348 0.373 0.212 0.257

France 0.288 0.269 0.286 0.260 0.267 0.236 0.264 0.249

Germany 0.164 0.152 0.159 0.165 0.167 0.169 0.163 0.167

Greece 0.234 0.297 0.256 0.209 0.192 0.253 0.233 0.252

Ireland 0.432 0.238 0.206 0.457 0.255 0.250 0.302 0.310

Italy 0.163 0.181 0.208 0.205 0.188 0.282 0.254 0.210

Luxembourg 0.244 0.254 0.243 0.263 0.290 0.305 0.289 0.231

Netherlands 0.255 0.211 0.195 0.173 0.242 0.195 0.255 0.294

Portugal 0.239 0.306 0.314 0.270 0.344 0.289 0.313 0.309

Spain 0.272 0.229 0.208 0.212 0.207 0.228 0.206 0.266

Sweden 0.383 0.346 0.272 0.303 0.326 0.371 0.300 0.345

UK 0.337 0.301 0.280 0.343 0.310 0.229 0.264 0.333

All 0.212 0.182 0.192 0.197 0.206 0.215 0.232 0.249



15/25

Table4

OveralltechnicalprogressforEuropeanbanks1

989}1997

1989

1990

1991

1992

1993

1994

1995

1996

1997

Austria

!

0.0

39

!

0.0

38

!

0.0

40

!

0.0

38

!

0.0

36

!

0.0

37

!

0.0

3

8

!

0.0

36

!

0.0

38

Belgium

!

0.0

40

!

0.0

43

!

0.0

44

!

0.0

42

!

0.0

41

!

0.0

40

!

0.0

4

1

!

0.0

40

!

0.0

40

Denmark

!

0.0

29

!

0.0

32

!

0.0

33

!

0.0

33

!

0.0

32

!

0.0

27

!

0.0

2

8

!

0.0

26

!

0.0

23

Finland

!

0.0

41

!

0.0

45

!

0.0

48

!

0.0

50

!

0.0

45

!

0.0

46

!

0.0

4

8

!

0.0

46

!

0.0

42

France

!

0.0

32

!

0.0

36

!

0.0

38

!

0.0

39

!

0.0

39

!

0.0

37

!

0.0

3

9

!

0.0

39

!

0.0

36

Germany

!

0.0

27

!

0.0

31

!

0.0

33

!

0.0

33

!

0.0

32

!

0.0

32

!

0.0

3

3

!

0.0

34

!

0.0

36

Greece

!

0.0

39

!

0.0

42

!

0.0

36

!

0.0

39

!

0.0

39

!

0.0

43

!

0.0

4

1

!

0.0

41

!

0.0

40

Ireland

!

0.0

37

!

0.0

35

!

0.0

36

!

0.0

35

!

0.0

38

!

0.0

38

!

0.0

4

3

!

0.0

44

!

0.0

44

Italy

!

0.0

26

!

0.0

28

!

0.0

30

!

0.0

33

!

0.0

33

!

0.0

31

!

0.0

3

4

!

0.0

36

!

0.0

37

Luxembourg

!

0.0

48

!

0.0

49

!

0.0

49

!

0.0

48

!

0.0

44

!

0.0

43

!

0.0

4

5

!

0.0

44

!

0.0

45

Netherlands

!

0.0

39

!

0.0

41

!

0.0

41

!

0.0

43

!

0.0

42

!

0.0

43

!

0.0

4

4

!

0.0

43

!

0.0

48

Portugal

!

0.0

38

!

0.0

44

!

0.0

43

!

0.0

44

!

0.0

43

!

0.0

43

!

0.0

4

4

!

0.0

43

!

0.0

42

Spain

!

0.0

26

!

0.0

30

!

0.0

32

!

0.0

33

!

0.0

34

!

0.0

33

!

0.0

3

6

!

0.0

37

!

0.0

35

Sweden

!

0.0

49

!

0.0

57

!

0.0

61

!

0.0

60

!

0.0

60

!

0.0

56

!

0.0

5

7

!

0.0

61

!

0.0

61

UK

!

0.0

33

!

0.0

37

!

0.0

35

!

0.0

35

!

0.0

33

!

0.0

33

!

0.0

3

7

!

0.0

38

!

0.0

42

All

!

0.0

31

!

0.0

35

!

0.0

37

!

0.0

37

!

0.0

35

!

0.0

34

!

0.0

3

5

!

0.0

36

!

0.0

37

Assetsizes(ECUMil)

1}99.9

100}

199.9

200}

299.9

300}

499.9

500}

999.9

1000}

2499.9

2500}

4999.9

5000#

Austria

!

0.0

26

!

0.0

34

!

0.0

38

!

0.0

40

!

0.0

36

!

0.0

5

1

!

0.0

50

!

0.0

45

Belgium

!

0.0

37

!

0.0

39

!

0.0

40

!

0.0

43

!

0.0

45

!

0.0

4

6

!

0.0

49

!

0.0

43

Denmark

!

0.0

23

!

0.0

26

!

0.0

27

!

0.0

28

!

0.0

31

!

0.0

3

5

!

0.0

48

!

0.0

47

Finland

!

0.0

38

!

0.0

38

}

!

0.0

43

!

0.0

53

!

0.0

4

6

!

0.0

52

!

0.0

43



16/25

Table

4(Continued).

1989

1990

1991

1992

1993

1994

1995

1996

1997

France

!0

.032

!

0.0

38

!

0.0

41

!

0.0

38

!

0.0

37

!

0.0

36

!

0.0

42

!

0.0

47

Germany

!0

.030

!

0.0

33

!

0.0

33

!

0.0

35

!

0.0

35

!

0.0

36

!

0.0

36

!

0.0

46

Greece

!0

.037

!

0.0

40

!

0.0

38

!

0.0

38

!

0.0

42

!

0.0

53

!

0.0

42

!

0.0

43

Ireland

!0

.039

!

0.0

29

!

0.0

38

!

0.0

42

!

0.0

40

!

0.0

42

!

0.0

47

!

0.0

37

Italy

!0

.029

!

0.0

29

!

0.0

29

!

0.0

29

!

0.0

36

!

0.0

40

!

0.0

42

!

0.0

42

Luxem

bourg

!0

.042

!

0.0

49

!

0.0

53

!

0.0

49

!

0.0

53

!

0.0

53

!

0.0

51

!

0.0

47

Netherlands

!0

.034

!

0.0

42

!

0.0

40

!

0.0

41

!

0.0

49

!

0.0

45

!

0.0

56

!

0.0

47

Portugal

!0

.041

!

0.0

49

!

0.0

48

!

0.0

48

!

0.0

39

!

0.0

39

!

0.0

40

!

0.0

41

Spain

!0

.031

!

0.0

34

!

0.0

31

!

0.0

33

!

0.0

31

!

0.0

33

!

0.0

35

!

0.0

42

Swede

n

!0

.036

!

0.0

38

!

0.0

43

!

0.0

53

!

0.0

45

!

0.0

61

!

0.0

73

!

0.0

67

UK

!0

.030

!

0.0

36

!

0.0

37

!

0.0

41

!

0.0

39

!

0.0

37

!

0.0

36

!

0.0

42

All

!0

.031

!

0.0

34

!

0.0

35

!

0.0

36

!

0.0

37

!

0.0

38

!

0.0

42

!

0.0

46

Note:Boldtypefaceforvaluesindicatessigni"cantlydi!erentfromzeroat

the5%

level.



17/25

Table

5

Numb

erofbanksandaverageassetsize

saccordingtoyears

EUco

untries

1989

1990

1991

1992

1993

1994

1995

1996

1997

Numbe

rofbanks

Austria

17

19

21

32

39

57

93

84

81

Belgiu

m

21

21

25

43

75

86

95

91

67

Denm

ark

24

25

29

54

77

92

107

103

88

Finlan

d

7

7

9

9

11

12

11

11

12

France

145

157

171

354

414

426

423

394

328

Germany

137

153

199

55

4

1436

1859

1853

1543

1588

Greece

7

8

10

14

17

21

21

20

24

Ireland

3

4

5

8

12

15

14

13

26

Italy

121

130

135

158

266

286

329

304

313

Luxem

bourg

37

68

77

94

129

141

138

130

119

Netherlands

10

11

12

43

50

58

62

55

48

Portugal

8

15

18

36

37

38

44

43

40

Spain

93

105

117

124

135

139

139

179

146

Swede

n

7

12

17

21

21

26

28

25

17

UK

23

29

37

105

132

138

137

131

120

All

660

764

882

1649

2851

3394

3494

3126

3017

Assets

ize

Austria

1693

1507

1391

1072

2008

2939

3054

3459

4304

Belgiu

m

10923

11312

11126

979

6

6172

5753

7421

8406

14040

Denm

ark

6540

6199

6447

3188

4011

3526

3212

3563

2440

Finlan

d

10264

10182

11356

10960

9114

7774

11405

11196

19870

France

13669

13913

13888

7770

7420

7182

7817

8912

10848

Germany

10435

10573

8847

383

5

1947

1706

1857

2451

2942



18/25

Table

5(Continued).

EUco

untries

1989

1990

1991

1992

1993

1994

1995

1996

1997

Greece

1155

931

752

1238

2191

2267

2400

2520

2708

Ireland

5926

5320

4399

6144

4837

4518

6005

7435

5397

Italy

5585

6021

6121

5786

4624

4564

4040

3963

5046

Luxem

bourg

3410

2018

1770

1551

1204

1170

1241

1404

3226

Netherlands

10800

14495

14866

1251

9

14437

12825

13249

17007

23641

Portugal

1534

917

1646

276

7

3641

3387

3795

4468

5362

Spain

4314

4708

4541

4525

4751

4778

4887

4008

5290

Swede

n

18931

19476

17525

14182

14070

11411

11138

13117

22270

UK

30702

26449

22820

11796

10725

10808

11958

13350

15059

All

9222

9046

8623

5739

4036

3601

3852

4573

5452



19/25

Table 6

Number of banks according to asset sizes (ECU Mil)

EU countries 1}99.9 100}

199.9

200}

299.9

300}

499.9

500}

999.9

1000}

2499.9

2500}

4997.9

5000# All

Austria 31 68 54 71 62 77 37 43 443

Belgium 79 69 40 48 46 96 35 111 524

Denmark 202 96 74 72 29 24 38 64 599

Finland 0 2 1 5 17 12 4 48 89

France 245 266 207 241 394 558 423 478 2812

Germany 1000 1869 1300 1399 1697 1224 398 435 9322

Greece 12 30 14 12 24 15 19 16 142

Ireland 0 1 4 2 11 46 13 23 100

Italy 244 204 151 239 391 351 144 318 2042Luxembourg 270 96 74 124 114 112 54 89 933

Netherlands 16 35 26 40 35 66 45 86 349

Portugal 19 18 16 31 47 46 46 56 279

Spain 118 81 89 78 191 272 126 222 1177

Sweden 3 8 9 8 13 30 17 86 174

UK 74 71 54 68 83 127 131 244 852

All 2313 2914 2113 2438 3154 3056 1530 2319 19 837

Note that in estimates derived from the standard cost frontier speci"cation that excludes the

equity capital variable, we "nd evidence of large scale economies for large banks. These results,

available from the authors, suggest that controlling for risk in the cost estimation can havea substantial impact on scale economy estimates for di!erent size banks.

stochastic cost frontier methodologies to estimate scale economies, X-ine$cien-

cies and technical change for a large sample of European banks between

1989 and 1997. The results reveal that scale economies are widespread

for smallest banks and those in the ECU 1 billion to ECU 5 billion assets size

range. Typically, scale economies are found to range between 5% and 7%,

while X-ine$ciency measures appear to be much larger, between 20% and

25%. X-ine$ciencies also appear to vary to a greater extent across di!erent

markets, bank sizes and over time. This suggests that banks of all sizes can

obtain greater cost savings through reducing managerial and other ine$cien-

cies. This paper also shows that technical progress has had a similar in#uence

across European banking markets between 1989 and 1997, reducing total costsby around 3% per annum. The impact of technical progress in reducing bank

costs is also shown to systematically increase with bank size. Overall, these

results indicate that Europe's largest banks bene"t most from technical progress

although they do not appear to have scale economy advantages over their

smaller counterparts.



20/25

Table 7

Descriptive statistics of total assets 1989}1997

EU countries N Mean Median StDev Min Max

Austria 443 2912 494 8237 9 96 452

Belgium 524 8654 713 22 512 6 160 461

Denmark 599 3722 201 10 945 8 65 950

Finland 89 11 512 6946 14 675 105 88 505

France 2812 9175 1092 33 914 4 341 303

Germany 9322 2150 344 14 886 9 456 292

Greece 142 2050 540 3858 28 22 821

Ireland 100 5 571 1668 8692 198 40 747

Italy 2042 4823 724 14 028 10 127 957

Luxembourg 933 1 719 342 3848 2 23 768

Netherlands 349 15 305 1551 44 143 19 338 751

Portugal 279 3557 1141 5560 34 31 232

Spain 1177 4641 1124 11 445 5 117 330

Sweden 174 14 785 4203 18 672 84 82 802

UK 852 13 682 1744 34 899 18 235 175

Table 8Maximum likelihood parameter estimation of the cost frontier

Variables Parameters Coe$cients Standard error t-Ratio

Constant !0.2113 0.01004 !21.042lnQ

0.4939 0.00368 134.056lnQ

0.4474 0.00424 105.407lnQ

0.0044 0.00358 1.229ln E

0.0109 0.00527 2.070

lnP

0.2393 0.00524 45.648lnP

0.7283 0.00601 121.172lnQ

lnQ

/2

0.0297 0.00038 78.127

lnQ

lnQ

!0.0609 0.00072 !84.944lnQ

lnQ

0.0135 0.00050 26.834lnQ

lnE

0.0341 0.00107 31.915

lnQ

lnQ

/2

0.0454 0.00052 87.521lnQ

lnQ

0.0117 0.00043 27.155

lnQlnE 0.0120 0.00113 10.665lnQ

lnQ

/2

0.0034 0.00039 8.627

lnQ

lnE !0.0008 0.00070 !1.147

LnE lnE/2 !0.0138 0.00111 !12.403lnP

lnP

/2

0.1138 0.00201 56.634

lnP

lnP

!0.2190 0.00328 !66.699lnP

lnP

/2

0.1841 0.00320 57.445

lnP

lnQ

!0.0199 0.00093 !21.394lnP

lnQ

!0.0045 0.00099 !4.563lnP

lnQ

!0.0012 0.00091 !1.314lnP

lnE

0.0255 0.00148 17.249

lnP

lnQ

0.0321 0.00094 34.296lnP

lnQ

0.0242 0.00084 28.977

lnPlnQ !0.0098 0.00085 !11.542lnP

lnE

!0.0353 0.00166 !21.292



21/25

Table 8 (Continued).

Variables Parameters Coe$cients Standard error t-Ratio

!0.0142 0.00279 !5.098

H/2 !0.0016 0.00022 !7.146

lnQ

!0.0028 0.00040 !6.959

lnQ

0.0020 0.00038 5.290

lnQ

!0.0010 0.00038 !2.651

LnE 0.0016 0.00057 2.824

lnP

0.0110 0.00062 17.826

lnP

!0.0149 0.00073 !20.301

cos(z

) a

!0.0618 0.00364 !16.968

sin(z

) b

!0.0170 0.00254 !6.695

cos(z) a!

0.0096 0.00414!

2.318sin(z

) b

!0.0264 0.00300 !8.805

cos(z

) a

!0.0024 0.00310 !0.775

sin(z

) b

!0.0023 0.00409 !0.562

cos(z

) a

0.0163 0.00275 5.918

sin(z

) b

!0.0280 0.00346 !8.083

cos(z#z

) a

0.0015 0.00241 0.623

sin(z#z

) b

0.0247 0.00244 10.139

cos(z#z

) a

!0.0043 0.00272 !1.582

sin(z#z

) b

0.0070 0.00253 2.769

cos(z#z

) a

!0.0226 0.00254 !8.907

sin(z#z

) b

!0.0011 0.00281 !0.391

cos(z#z) a !0.0024 0.00296 !0.811sin(z

#z

) b

!0.0183 0.00260 !7.046

cos(z#z

) a

!0.0258 0.00228 !11.308

sin(z#z

) b

0.0081 0.00195 4.145

cos(z#z

) a

0.0035 0.00238 1.468

sin(z#z

) b

!0.0040 0.00251 !1.596

cos(z#z

) a

!0.0099 0.00254 !3.898

sin(z#z

) b

!0.0102 0.00265 !3.850

cos(z#z

) a

!0.0004 0.00245 !0.163

sin(z#z

) b

!0.0153 0.00228 !6.712

cos(z#z

) a

0.0146 0.00258 5.656

sin(z#z

) b

0.0104 0.00281 3.699

cos(z#z) a 0.0143 0.00216 6.616sin(z

#z

) b

!0.0060 0.00257 !2.336

u/v 3.1954 0.02138 149.488

v 0.3006 0.00112 267.262

lnP

0.0324

lnP

lnP

0.1052

lnP

lnP

0.0349

lnP

lnP

/2

!0.1401

lnP

lnQ

!0.0122

lnP

lnQ

!0.0197

lnP

lnQ

/2

0.0110

lnP

lnE 0.0099

lnP

0.0039



22/25

Given that only a limited number of studies have investigated X-ine$ciencies

in European banks we suggest that a possible area for future research could be

to investigate whether similar relationships hold for banks which have di!erentownership characteristics, such as mutual and public banks. It may also be

interesting to evaluate the impact of alternative risk and output quality factors

as well as macro-economic cycles on the production characteristics of European

banks.

Appendix A

This appendix provides details on asset sizes and cost frontier (Tables 5}8).

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