Humph ey, Da id B.; Vale, Ben
Wo king Pape
Scale Economies, Bank Me ge s, and Elec onic Paymen s:
A Spline Func ion App oach
Wo king Pape , No. 2003/5
P o ided in Coope a ion wi h:
No ges Bank, Oslo
Sugges ed Ci a ion: Humph ey, Da id B.; Vale, Ben (2003) : Scale Economies, Bank Me ge s, and
Elec onic Paymen s: A Spline Func ion App oach, Wo king Pape , No. 2003/5, ISBN 82-7553-213-2,
No ges Bank, Oslo,
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Wo king Pape
Resea ch Depa men
Scale economies, bank me ge s, and elec onic paymen s:
A spline unc ion app oach
by
Da id B. Humph ey and Ben Vale
ISSN 0801-2504
ISBN 82-7553-213-2
Wo king pape s om No ges Bank can be o de ed by e-mail:
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SCALE ECONOMIES, BANK MERGERS, AND
ELECTRONIC PAYMENTS: A SPLINE
FUNCTION APPROACH∗
Da id B. Humph ey
Flo ida S a e Uni e si y
Ben Vale
No ges Bank (The cen al bank o No way)†
May 20, 2003
Abs ac
This pape demons a es he impo ance o using a lexible cos unc ion
speci ica ion when analyzing economies o scale and es ima ing he cos effec
o banking me ge s. The in lexibili y o he anslog cos unc ion is illus a ed
and esul s a e compa ed o mo e lexible spline and Fou ie cos unc ions.
Using hese diffe en app oaches we p edic he ex an e effec on a e age
cos om me ge s o e 1987-1998 using a balanced panel o 130 No wegian
banks. On a e age me ge s a e p edic ed o lowe cos s. P edic ions using
he Fou ie o spline app oach a e in o e all ag eemen wi h compu ed ac ual
a e age me ge -cos changes ex pos . Cos effec s o elec onic paymen s a e
also es ima ed and exceed cos educ ions associa ed wi h me ge s.
JEL code: G21, L11
Keywo ds: Economies o scale, unc ional o m, me ge s.
∗Thanks o Øy ind Ei heim, Claudia Gi a done, Øy ind Ho eid, Kje s i-G o Lindquis , Jon
Williams, an anonymous e e ee, semina pa icipan s a No ges Bank, Bank o Finland, Uni e si y
o Gö ebo g, pa icipan s a he 39 h Bank S uc u e Con e ence a he Fede al Rese e Bank o
Chicago, and a he EARIE 2001 con e ence in Dublin o help ul commen s o p e ious e sions
o he pape . Views and conclusions exp essed in his pape a e hose o he au ho s and canno
be a ibu ed o any o he pe sons o ins i u ions men ioned abo e.
†Co esponding au ho : Ben Vale, No ges Bank, C51, Box 1179, Sen um, N-0107 Oslo, No -
way. Fax: +47 22 42 40 62. E-mail: ben . ale@no ges-bank.no
1. In oduc ion.
The as majo i y o he pas g ow h o la ge banking ins i u ions in he U.S. and
Eu ope has been achie ed h ough me ge s and acquisi ions (c. ., Rhoades (1985),
o he U.S.). This me hod o expansion is iewed as being cheape han de no o
en y and quicke han elying on demand-de e mined g ow h. The ecen emo al
o es ic ions on na ionwide b anching in he U.S., he opening up o domes ic
and c oss-bo de banking and inancial ma ke s in Eu ope, and he oppo uni y o
mo ing o mo e cos -efficien elec onic se ice deli e y me hods — wi h inhe en ly
mo e scale economies1— ha e accele a ed his his o ical end o me ge -dependen
g ow h.
Public s a emen s by me ging banks in a iably ocus on hei expec ed cos sa -
ings, planned expansions in hei se ice mix, and he need o ha e a sufficien ly
la ge p esence in ce ain ma ke s o compe e effec i ely wi h o he domes ic o in-
e na ional playe s. These posi i e ex an e s a emen s con as sha ply wi h ex pos
analyses ha : (a) compa e p e- and pos -me ge cos s o me ging banks (Rhoades
(1993)); (b) assess me ge - ela ed changes in on ie cos efficiency le els o ank-
ings (Be ge (1998); Be ge and Humph ey (1992)); o (c) summa ize case s udies
o indi idual me ging banks (Rhoades (1998)). Ex pos s udies o banking me ge s
consis en ly ind ha uni cos s o efficiency le els a e essen ially unchanged o
he a e age banking me ge almos ega dless o b anch ma ke o e lap (whe e he
expec a ion o cos sa ings is highes ). While some banks expe ience educ ions in
uni cos , o he s expe ience inc eases. Fo he small subse o me ge s ha u n
ou o be ”success ul”, howe e , uni cos s may all by up o 5%.2
Equi y ma ke e en s udies o Eu ope (Cybo-O one and Mu gia (2000)), bu
no he U.S. (Hawawini and Swa y (1990); Hannan and Wolken (1989)), ind sig-
ni ican ly posi i e abno mal equi y e u ns associa ed wi h me ge announcemen s.
Equi y e en s udies should e lec he po en ial o signi ican cos educ ions as
well as possible inc eases in ma ke powe and p o i s. Pe haps due o diffe en ma -
1See o ins ance Lindquis (2002), who documen s inc eased scale economies in banking ol-
lowing he adop ion o elec onic paymen s echnology.
2Reduc ions in o al expenses, common in me ge announcemen s, need o be adjus ed o
associa ed educ ions in asse s. This is why uni cos is a mo e use ul me ic.
2

ke s uc u es and an i us en o cemen , he expec ed me ge bene i s ha e been
posi i e o Eu ope. So a , when examined a e he ac , his expec a ion has no
been ealized in he ew ex pos Eu opean s udies ha ha e been done (e.g., Ca bo,
Humph ey, and Fe nandez (2003)).
Appea ances o he con a y, a 5% educ ion in uni cos om a success ul me ge
is consis en wi h s anda d p ojec ions by banking consul an s o o e all cos sa ings
in a me ge co esponding o 30% o 40% o an acqui ed bank’s ope a ing cos . Since
bank ope a ing expenses (depending on he in e es a e cycle) a e a ound o y
pe cen o o al cos s and he a e age acqui ed bank comp ises a ound one- hi d
o he consolida ed o ganiza ion, a 30% o 40% educ ion in ope a ing cos o an
acqui ed ins i u ion ansla es in o a 4% o 5% educ ion in uni cos (e.g., 40% ×
.40 ×.33 = 5.3%) o heme ged i m.O e hal o hecos educ ionachie edina
success ul me ge is due o educing employmen (Rhoades (1998)) while addi ional
sa ings come om closing edundan b anch offices. In some coun ies, howe e ,
public opposi ion o bank me ge s has led managemen o p omise ha labo o ce
educ ions and b anch closings will be limi ed, which educes p e iously announced
expec ed gains om a me ge . Al hough a 5% educ ion in uni cos (which only
applies o success ul me ge s) may sound small, i can ansla e in o a 23% o 35%
imp o emen in e u n on asse s.3
In his pape we i s show ha a mo e de ailed analysis o scale effec s in
banking can lead o mo e accu a e p edic ions ega ding he po en ial cos effec s
o me ge s. This is achie ed by compa ing he es ima ed scale economies om a
es ic i e speci ica ion like he anslog o he mo e lexible speci ica ions o a spline
o a Fou ie unc ion. The anslog speci ica ion will in mos cases p edic a e age
cos as a s anda d U-shaped quad a ic cu e whe e scale economies a e exhaus ed
o all bu ela i ely small o medium-sized banks. By cons uc ion a quad a ic
unc ion can only ha e one u ning poin , hence he ypical U-shape o a anslog
cos unc ion can in many cases be conside ed an a i ac . Ne e heless, his has
been he s anda d unc ional o m used in mos bank cos s udies. In con as
3A 5% educ ion in uni cos implies ha he a io o o al cos o o al asse s (which a e ages
.07) will all by .0035. Since he e u n on asse s o en anges be ween a ound .01 o .015, a cos
educ ion o .0035 (i en i ely e lec edinap o i inc ease) sugges s ha e u n on asse s (ROA)
could ise by 35% o 23%, espec i ely.
3
o he anslog, bo h he spline and he Fou ie speci ica ions—whichbo hallow
mo e han one u ning poin — indica e ha p edic ed a e age cos is no quad a ic.
Indeed, ins ead o a ypical U-shaped a e age cos cu e, he spline and Fou ie
o ms bo h show some hing simila o an M-shaped cu e so ha economies and
diseconomies exis o bo h smalle and la ge banks.4
In o de o p edic he cos effec o me ge s, we ma ch me ging ins i u ions wi h
hei p e- and pos -me ge bank-speci ic scale measu es a he ime hey me ged o
de e mine he likely effec on cos o each me ge indi idually and as a g oup.
These esul s a e epo ed using he anslog, spline, and Fou ie cos unc ions.
On a e age, he anslog p edic s ha a e age cos will all by less han a i h o
one pe cen while he Fou ie and spline unc ions p edic a wo o h ee pe cen
dec ease. F equency dis ibu ions o hese se s o p edic ions show ha all h ee
unc ions p edic bo h cos inc eases and dec eases o indi idual me ge s, wi h
mo e dec eases o he spline and Fou ie o ms. These me ge cos p edic ions
a e compa ed o how a e age cos has ac ually changed p e- and pos -me ge a
acqui ing banks ela i e o he es o he indus y. This cos change ela ed o
me ge s is hen con as ed wi h cos sa ings esul ing om ad ances in paymen
echnology which educe back office ope a ing cos s and se ice deli e y expenses.
The accu acy o ou analysis in p edic ing he di ec ion o change in a e age cos
is on pa wi h ha o me ge pa icipan s who in a iably p edic cos educ ions bu ,
due o un o eseen e en s, ealize his goal only abou hal he ime. The alue o ou
app oach lies in p o iding a mo e accu a e es ima e o he a e age effec o me ge s
on cos . This gi es an expec ed alue which compe i ion au ho i ies may use o
assess he likelihood ha me ge cos sa ings announced by pa icipan s may in ac
be ealized. Fo his ask, he spline and Fou ie unc ions gi e mo e eliable esul s
han he anslog.
In wha ollows, sec ion 2 ou lines ou use o spline and Fou ie unc ions o
mo e accu a ely de e mine a ia ions in scale economies ac oss all banks. This
4This poin was demons a ed on a da a se o U.S. banks by McAllis e and McManus (1993).
Also, in a special case, Shaffe (1998) shows ha es ima ing a anslog cos unc ion can esul in
a spu iously U-shaped a e age cos cu e when da a a e gene a ed om a p ocess wi h declining
a e age cos s. The cos unc ion sugges ed as a eplacemen , howe e , has lexibili y p oblems o
i s own.
4
analysis is illus a ed by i s applying a single ou pu model o a balanced panel o
131 No wegian banks o e 1987-1998 (12 yea s), gi ing 1,572 obse a ions. These
esul s a e con as ed wi h hose ob ained using a anslog cos unc ion, p o iding
a benchma k o assessing he imp o emen in iden i ying scale economy and me ge
cos effec s. Then ou single ou pu model is expanded in o a wo-ou pu model
whe e we addi ionally con ol o he ansi ion om pape based o elec onic pay-
men s. In sec ion 3 his wo-ou pu model is applied o p edic cos effec s om
bank me ge s. These p edic ions a e con as ed in sec ion 4 wi h es ima es o he
ac ual me ge - ela ed cos change. The pu pose is o de e mine how accu a e ou
ex an e me ge cos p edic ions a e ela i e o me ge analyses which ely on ex
pos da a. In sec ion 5, he effec on bank expenses om he shi o lowe cos
elec onic paymen s is es ima ed and compa ed o he me ge cos effec . Sec ion
6 con ains a summa y and conclusions.
2. Spline and Fou ie Func ion Measu es o Scale Economies.
In his sec ion we i s discuss how o bes a ange he da a in o de o cap u e
effec s om scale economies. Then we demons a e he impo ance o he choice o
unc ional o m when scale economies a e o be e alua ed using an econome ically
es ima ed cos unc ion. To acili a e a g aphic illus a ion o his issue we use
single ou pu models. Nex , howe e , we expand ou models o con ain wo ou pu s
— business loans and consume loans — and con ol o he e olu ion in paymen
echnology ha ook place du ing ou sample pe iod. This wo-ou pu model is he
model applied in sec ion 3 when we p edic me ge cos effec s ex an e.
2.1. A anging he Da a o Bes Cap u e Scale Effec s.
Scale effec s ha e been es ima ed using c oss-sec ion and panel da a. Wi h panel
da a, i can make a diffe ence how he da a a e a anged. The a e age cos igu e
below illus a es wo ways o doing his when bank a( he la ge o he wo banks)
acqui es bank b, o ming he new bank a+b.
The usual way is o c ea e a balanced panel by backwa d agg ega ing all p e-
me ge da a on banks aand bp io o hei me ge , gene a ing poin 3 in he Figu e
2.1. Once he me ge occu s, yielding poin 4, he ou pu le el will no change by
5
1234567891011
6
6.5
7
7.5
8
8.5
9
9.5
10
Bank b (acqui ed bank)
Bank a (acqui ing bank)
Me ged bank a+b (pos -me ge )
Me ged bank a+b (p e-me ge )
o
AVERAGE COST
OUTPUT
1
2
3
4
Figu e 2.1:
much (since he banks ha e al eady been a i icially me ged) bu cos s may change i
he e a e scale economies. This is a common app oach and we ha e done i ou sel es.
Upon e hinking he issue, we eel a mo e de ensible app oach (and he one adop ed
below) would be o ins ead use he obse ed p e-me ge da a poin 2 along wi h
poin 4. Wi h hisda aa angemen ,when2iscompa ed o4inapanel eg ession
bo h cos and ou pu a e allowed o change in esponse o he me ge . The me ge
p ocess is hus ep esen ed as a se ies o ou pu and cos ”jumps” which is how i is
also iewed by he me ge pa icipan s.
2.2. App oxima ing he Cu a u e o A e age Cos .
A spline unc ion is a eg ession-based me hod de eloped o i a polynomial equa-
ion o a gi en o de o a sca e plo o da a poin s. A one ex eme, he polynomial
i ed can be a s aigh line equal o wha is ob ained wi h a simple linea eg es-
sion. A he o he ex eme, a se ies o line segmen s would connec each poin in
a sca e plo . The bene i o a spline is ha i can ep esen nume ous cu a u es
in e media e o hese wo ex emes a he disc e ion o he esea che , which is how
i is applied he e.
Take hecaseo as anda da e agecos (AC)cu e. I ACislinea ,loglinea ,
6
o m. As elec onic bill paymen s and pay oll disbu semen s a e au oma ed and
debi ca ds subs i u e o cash a he poin o sale (Snellman, Vesala, and Humph ey
(2000)), he e is less need o di ec bank- o-cus ome in e ac ion, allowing banks
o educe he numbe and size o b anch offices and hus lowe se ice deli e y
expenses. This apid change o echnology ook place wi hin he pe iod co e ed by
ou sample, and con olling o i in es ima ing a cos unc ion is impo an gi en
he cos sa ings i has implied.
No e ha he a iable ELE only a ies ac oss ime no ac oss banks. As all
banks in No way a e connec ed o he same clea ing sys em o e ail paymen s, ELE
ep esen s a common change in echnology affec ing all banks. Econome ically his
implies ha o he indi idual bank his a iable can be conside ed as exogenous.
In o de o assess he use ulness o ou spline unc ion model, he scale economy,
paymen echnology, and me ge cos effec s om he spline unc ion a e compa ed
wi h simila esul s om anslog and Fou ie speci ica ions es ima ed join ly wi h
he cos sha es:
T anslog Cos Func ion:
lnTC =α0+
2
X
i=1
αilnQi+1/2
2
X
i=1
αi,i(lnQi)2+A0+B+γ1lnELE +γ21/2(lnELE)2
Sk=βk+
4
X
m=1
βk,mlnPk+
2
X
i=1
δi,klnQi(2.2)
Fou ie Cos Func ion:14
lnTC =T anslog Cos Func ion
+
2
X
i=1
(τ1isin(lnQ∗
i)+τ2isin(2lnQ∗
i)+τ3isin(3lnQ∗
i))
+
2
X
i=1
(τ4icos(lnQ∗
i)+τ5icos(2lnQ∗
i)+τ6icos(3lnQ∗
i))
+τ7sin(lnQ∗
i+lnQ∗
j)+τ8cos(lnQ∗
i+lnQ∗
j)
+τ9sin(2lnQ∗
i+lnQ∗
j)+τ10 cos(2lnQ∗
i+lnQ∗
j)
+τ11 sin(lnQ∗
i+2lnQ∗
j)+τ12 cos(lnQ∗
i+2lnQ∗
j)
Sk=βk+
4
X
m=1
βk,mlnPk+
2
X
i=1
δi,klnQi(2.3)
14The e ms lnQ∗
iand lnQ∗
ja e compu ed as de ined ea lie in sec ion 2.2
13

F om he es ima ion o he h ee cos unc ion speci ica ions (2.1) o (2.3) he
scale elas ici ies, o he elas ici ies o o al cos w. . . o al loans (consume loans
plus business loans), can be calcula ed as:
ETC =
2
X
i=1
Ei,T C ,
whe e Ei,T C is he elas ici y o o al cos w. . . ou pu i. The scale elas ici ies ETC
o se en diffe en sub-samples acco ding o bank size15 a e epo ed in Table 2.1 as
sample means.16
Table 2.1: Elas ici ies o o al cos s o o al loans
To al asse s,
Size class NOK billions T anslog Fou ie Spline N
1 0—0.25 0.92 0.93 0.93 255
20.25—0.5
0.94 0.92 0.91 458
3 0.5 — 1.0 0.95 0.96 0.96 403
4 1.0 — 5.0 0.96 0.97 0.95 265
55.0—10.0
1.00 1.04 1.10 55
6 10.0 — 25.0 1.01 1.05 0.97 62
725.0— 1.04 0.94 0.92 50
To al sample 0.95 0.95 0.94 1548
In es ima ing he scale elas ici ies in Table 2.1, we ha e con olled o (a) cos
changes associa ed wi h he inc eased use o elec onic paymen s (by speci ying he
a io o elec onic o o al non-cash ansac ions) and (b) he e ogenei y be ween
business and consume loans (by speci ying wo sepa a e ou pu s). These scale
elas ici ies show a simila pa e n as ha ep esen ed by he cu es in Figu e 2.2
using only one ou pu ( o al asse s) and no sepa a ely con olling o changes in
paymen echnology. Wi h he wo-ou pu (business and consume loans) anslog
speci ica ion and con olling o changes in paymen s echnology, scale economies a e
seen o be ully ealized o almos all banks wi h an asse size o only NOK 10 (1.3
Eu os) billion o la ge . In sec ion 2.2 whe e we used a single ou pu ( o al asse s —
15These se en size classes a e based on he alue o o al asse s and co epond o he se en
segmen s used in es ima ing he spline unc ion illus a de in Figu e 2.2.
16Two banks om ou o iginal sample o 131 banks did no epo loans by bo owe ca ego ies.
Hence when es ima ing models wi h wo ou pu s we a e le wi h 129 banks and a o al o 1548
obse a ions.
14
a he han wo loan ca ego ies), scale economies we e ully ealized o banks wi h
an asse size i e imes la ge (NOK 50 billion) in he anslog speci ica ion. This
diffe ence e lec s he ac ha among banks wi h o al asse s la ge han NOK 10
billion, loans — a labo in ensi e asse — cons i u e a smalle ac ion o a bank’s
asse s he la ge is he bank.
As in sec ion 2.2 a log likelihood a io es o he Fou ie speci ica ion e sus
he mo e es ic ed anslog speci ica ion indica ed ha he added igonome ic
e ms in he Fou ie a e join ly signi ican .17 Thus, econome ically he Fou ie
speci ica ion is clea ly p e e able o e he anslog.18 Bo h he Fou ie and he spline
speci ica ions show economies o scale o many smalle banks, some diseconomies o
medium-sized ins i u ions, and (qui e diffe en ly om he anslog) scale economies
o la ge banks.
Ou pu speci ic cos elas ici ies as well as ma ginal cos s o business and con-
sume loans a e shown in Table 2.2. On a e age o e he whole sample, expanding
business loans by 10% gene a es a 2.3% ise in associa ed o al cos s while a 10%
expansion in consume loans gene a es a 7.1% ise ( i s wo columns in Table 2.2).
This diffe en ial cos effec e e s only o he diffe en slopes o he implied business
and consume loan a e age cos cu es — no o hei le el. O e he whole sample,
o each ex a NOK 100 o business loans ex ended NOK 16 goes o co e he ma -
ginal in e es , capi al, and ope a ing expenses while a simila ex ension o consume
loans gene a es NOK 12 o ex a expenses (las wo columns in Table 2.2). Business
loans usually in ol e mo e sc eening and moni o ing han do consume loans, and
hence a e mo e cos ly on he ma gin.
The app oxima e M-shaped a e age cos cu e seen o he Fou ie and spline
unc ions in Figu e 2.2 is also seen when he spline unc ion is es ima ed wi h wo
loan ou pu s in Figu e 2.3.19 The igu e plo s he diffe en p edic ed a e age cos s
17The log likelihood a io s a is ics is 56.122 wi h 18 deg ees o eedom, gi ing a p- alue less
han 0.001.
18Since he spline and he anslog a e no nes ed, we canno es he spline e sus he anslog
using a log-likelihood a io es . Ins ead we ha e ouseanon-nes ed es p ocedu elike heJ- es .
Howe e ,as equen lycanhappenwi h heJ- es , i was inconclusi e. I was e en inconclusi e
be ween he Fou ie and he anslog, indica ing he es has low powe . In addi ion we calcula ed
he Akaike in o ma ion c i e ion, which (al hough i has no i m basis in s a is ical heo y) indica ed
ha he spline is mo e in o ma i e han he anslog.
19The i ed Fou ie o m wi h wo loan ou pu s is simila o he spline and is hus no shown.
15
10 11 12 13 14 15 16 17 18 19
0.09
0.10
0.11
0.12
0.13
0.14
0.15
P edic ed a e age cos
Fi ed cubic spline
(solid line)
ln(business loans + consume loans)
Figu e 2.3: Poin s : P edic ed a e age cos s o he wo-ou pu spline cos unc ion,
using sample mean alues o inpu p ices and ELE, plo ed agains log o loans.
Solid line: a i ed cubic spline o he poin s.
Table 2.2: Ou pu -Speci ic Cos Elas ici ies and Ma ginal cos s, Business and Con-
sume Loans
To al asse s, Cos elas ici ies Ma ginal cos s
NOK billions Business Consume Business Consume
0 — 0.25 0.16 0.77 0.188 0.140
0.25 — 0.5 0.21 0.70 0.157 0.114
0.5 — 1.0 0.22 0.75 0.158 0.111
1.0 — 5.0 0.25 0.70 0.151 0.105
5.0 — 10.0 0.38 0.72 0.143 0.157
10.0 — 25.0 0.29 0.67 0.103 0.114
25.0 — 0.55 0.37 0.146 0.102
To al sample 0.23 0.71 0.158 0.117
Ma ginal cos s a e millions o NOK in inc eased o al cos s pe millions o NOK in
inc eased lending. In calcula ing he elas ici ies and ma ginal cos s all ou inpu
p ices and he a io o elec onic paymen s a e se a hei ac ual alues. Only
esul s om he spline speci ica ion a e shown. The Fou ie speci ica ion gi es
simila esul s.
16
agains he log o business plus consume loan alues (on he X-axis).20 The solid
line shown in he igu e is a cubic spline i ed o hese p edic ed alues. As seen,
he esul is simila o a M-shaped a e age cos cu e. Fo he small banks wi h
loans be ween NOK 40 millions and NOK 100 millions ( hose wi h he log o loans
be ween 10.6 and 11.25), he solid line ises showing ini ial diseconomies o scale.
The eason o hese ini ial diseconomies is wo- old. Fi s , as no ed abo e, business
loans cos mo e o p oduce han he same alue o consume loans. Second, as hese
small banks become la ge , hey s a o issue mo e business loans. The shi owa d
p oducing mo e o he mo e expensi e business loans gene a es he ini ial scale
diseconomies seen in he igu e. Bu as hese banks become la ge s ill, g ea e
scale economies o business loans s a o domina e. The la e ”humps” in a e age
cos shown in he igu e a e associa ed wi h high ma ginal cos s o consume loans
o he medium sized banks, banks wi h loans la ge han NOK 5 billions. Finally,
he educ ion in a e age cos seen o he e y la ges banks is he di ec esul o
signi ican ly smalle ma ginal cos s o consume loans o hese la ge ins i u ions,
as shown in Table 2.2. This indica es ha he smalle banks and in pa icula he
la ge banks ha e been able o ex end consume loans a low cos s.21
No ice also ha he ma ginal cos o business lending is highe o he la ges
banks, almos as la ge as i is o small banks. The highe cos s associa ed wi h la ge
banks can be due o mo e complex moni o ing and sc eening o he la ge i ms ha
ypically bo ow a la ge banks. Fu he mo e, he la ge co po a e cus ome s o
la ge banks usually pu chase ela ed inancial se ices such as o eign exchange,
sho - e m in es men s, and inancial de i a i es om hei banks. These se ices
equi e physical inpu s ha a e epo ed as highe o al cos s in ou model, bu we
can no speci y hese addi ional se ices as ou pu s due o a lack o da a.
20All inpu p ices and he sha e o elec onic paymen s a e held cons an a hei mean alues.
21One explana ion o his pa e n in consume lending is ha small and ai ly local banks ha e
good in o ma ion abou hei consume bo owe s and need no spend a lo o esou ces on sc een-
ing and moni o ing hese loans. Once banks each a medium size and encompass egions a he
han communi ies, hese local in o ma ion ad an ages a e educed as loan equi emen s become
mo e s anda dized and less pe sonal. Hence hey ha e o spend mo e esou ces on sc eening and
moni o ing he consume loans hey make. Al hough e y la ge na ionwide banks also ha e he
same in o ma ion disad an age, hey a e geog aphically much mo e di e si ied and can o some
deg ee subs i u e di e si ica ion o in ensi e moni o ing. Fu he mo e, mos o he moni o ing
and sc eening o consume loans can be done h ough hese banks’ IT-sys ems, sys ems associa ed
wi h la ge ‘ ixed’ cos s making hem mo e a ac i e o la ge banks han o smalle banks.
17
While scale economies exis , scope economies do no . Indeed, sepa a e p oduc-
ion o consume and business loans appea s o ha e ma ginally lowe cos s han
does join p oduc ion. The a e age scope economy alue wi h he Fou ie o m was
-0.01 while i was -0.12 wi h he anslog.22 Thus he join p oduc ion we obse e
in p ac ice is mo e likely due o e enue bene i s han cos educ ions. In any case,
ew me ging banks ac ually changed hei composi ion o business and consume
loans p e- o pos -me ge and so e en i scope economies we e posi i e (indica ing
lowe cos s) he effec would ha e been small.23
In he nex sec ion pa ame e es ima es om he h ee cos unc ion speci ica-
ions (2.1) o (2.3) a e used o gi e ex an e p edic ions o he cos sa ings om
26 bank me ge s ha ac ually occu ed in ou da a se be ween 1988 and 1997.24
As ou scale es ima es a e independen o back office cos educ ions associa ed
wi h changes in paymen echnology — by he e m con olling o his echnological
change — so (app op ia ely) a e ou p edic ed cos effec s om me ge s.
3. P edic ing Me ge Cos Effec s Ex An e.
When banks aand bme ge in o bank a+bin yea , by bank aacqui ing bank b
in ha yea , he da a we use o yea will con ain he balance shee i ems, cos ,
and income s a emen s o he new la ge bank, bank a+b.Fo yea −1and
all p e ious yea s he epo ed da a con ains sepa a e in o ma ion o bank aand
bank bas hey we e p io o hei me ge . In o de o p edic ex an e he cos
effec s o his me ge we a i icially me ge banks aand bin only yea −1in a
22The o m o he spline unc ion makes i e y difficul o de i e a scope measu e and one is
no p esen ed. In unc ional no a ion, ou scope calcula ion was: SCOPE =[C(0.9∗Q1,0.1∗Q2)
+C(0.1∗Q1,0.9∗Q2)−C(Q1,Q
2)] /C(Q1,Q
2)whe e Q1(Q2)=business (consume ) loans
and C(·)is he es ima ed cos unc ion e alua ed wi h ou pu le els as shown bu wi h all inpu
p ices and he sha e o elec onic paymen s held cons an a hei mean alues. As is now known,
de e mining scope economies om a loga i hmic cos unc ion should no be done wi h comple e
o almos comple e specializa ion in p oduc ion. Hence ou use o 0.9and 0.1poin s o e alua ion
a he 1.0and 0.0o 0.99 and 0.01). O he mo e sub le biases also exis , c. . Pulley and Humph ey
(1993).
23Fo he 26 me ging banks, he a e age a io o business o o al loans was .36 and he a e age
pe cen age poin change in his a io p e- o pos -me ge was .011 (wi h a s anda d de ia ion o
.036).
24Ou da a co e s 1987-1998 bu we need a leas one obse a ion bo h one yea p io o a
me ge and one yea a e a me ge o calcula e and con as ou p edic ed and ac ual change in
me ge - ela ed a e age cos . This elimina es some me ge s ha occu ed in 1987.
18

sepa a e da a se . We hen compa e he p edic ed a e age cos om each o he
h ee cos unc ions o a bank wi h ou pu le els he size o bank ain yea −1
o he p edic ed a e age cos o bank abu using he sum o ou pu o bank aand
bank bin yea −1.
Mo e o mally, his cos compa ison can be s a ed as ollows: Le (·)be he
symbol o an es ima ed mapping o inpu p ices, ou pu s, and he paymen echnol-
ogy con ol a iable ELE o he o al cos o a bank he size o bank aalone in yea
−1and hen o hesizeo bankaand bank bas i hey had me ged in o bank
a+bin yea −1. Using his es ima ed mapping he p edic ed a e age cos s a e:
ACa, −1= (Q1,a,, −1,Q
2,a, −1,Pa, −1,ELE
−1)
Q1,a,, −1+Q2,a, −1
ACa+b, −1= (Q1,a,, −1+Q1,b,, −1,Q
2,a, −1+Q2,b,, −1,Pa, −1,ELE
−1)
(Q1,a,, −1+Q1,b,, −1)+(Q2,a, −1+Q2,b,, −1).
The pe cen change in a e age cos p edic ed o be associa ed wi h he scale effec
o a me ge is:
PCAC =(ACa+b, −1−ACa, −1)/ACa, −1·100 (3.1)
whe e Qi,h, −1,i=1,2,h=a, b, is he he quan i y o ou pu io bank hin yea −1,
and Pa, −1is he ec o o inpu p ices o bank ain yea −1. Hence ACa, −1is he
p edic ed a e age cos o bank abe o e he me ge , and ACa+b, −1is he p edic ed
a e age cos o bank aassuming i —one yea ahead o he ac ual me ge —had eached
he join size o he wo me ging banks. In calcula ing bo h ACa, −1and ACa+b, −1
we use he inpu p ice ec o Pa, −1and so dis ega d any effec on inpu p ices om
he me ge due o inc eased ma ke powe o change in unding composi ion. Thus
wha we conside he e is a pu e economies o scale effec . PCAC is calcula ed using
each o he h ee cos unc ion speci ica ions, anslog, Fou ie , and spline. Nega i e
alues p edic a educ ion in a e age cos while posi i e alues p edic a ise.
Ou da a se con ains usable da a on 26 (ou o 33) bank me ge s ha occu ed
o e 1987-1998.25 Fo hese 26 me ge s we calcula e he p edic ed pe cen change
in a e age cos o all me ging banks as a g oup due o scale effec s (PCAC)as
desc ibed in equa ion (3.1). Ou a e age esul s a e epo ed in Table 3.1.26
25Asse enme ge soccu edin he i s yea o he da a se (1987), we a e unable o p edic
19
Table 3.1: P edic ed cos effec om me ge s (PCAC)
T anslog Fou ie Spline
Pc . change AC (PCAC)
(ACa+b−ACa)/ACa−0.16% −2.06% −3.09%
No. AC inc ease (+) 16 11 7
No. AC dec ease (-) 10 15 19
Pc . change o al cos 28.8% 26.3% 25.0%
Pc . change o al loans 29.0% 29.0% 29.0%
Ma ginal scale economy .99 .91 .86
The p edic ed pe cen change in weigh ed a e age cos (AC) due o
scale effec s om all 26 me ge s oge he is shown in he i s ow
(calcula ed om (ACa+b, −1−ACa, −1)/ACa, −1.Thenumbe o
nega i e (posi i e) changes indica e he numbe o bank me ge s ha
a e p edic ed o educe (inc ease) a e age cos . Pe cen changes in
o al cos and o al business and consume loans e e o he sum o
all 26 me ge s. Di iding he o me by he la e gi es a ma ginal
scale economy alue ha e e s o he p edic ed o e all me ge effec .
As may be expec ed om he ea lie almos L-shaped a e age cos cu e in
Figu e 2.2 and scale economy alues epo ed in Table 2.1 o he anslog speci-
ica ion, only a sligh educ ion in a e age cos (-.16%) is p edic ed as a weigh ed
a e age o all me ge s in Table 3.1.27 Ou o 26 me ge s, 10 a e p edic ed o lowe
a e age cos . As he p edic ed change in o al cos (28.8%) is sligh ly lowe han
he change in o al business and consume loans (29.0%), he scale economy effec
o hese me ge s as a g oup is .99. On balance, use o he anslog o m sugges s
ha me ge s in No way would yield some hing close o cons an a e age cos wi h
li le bene i o cos o use s o banking se ices.
In con as , bo h he Fou ie and spline unc ions sugges ha me ge s would
gene a e la ge educ ions in a e age cos o all me ge s oge he . The p edic ed
educ ions ange om -2.06% (Fou ie ) o -3.09% (spline) wi h co esponding scale
hei cos effec one yea p io o hei me ge .
26To e lec he diffe en p edic ed o al cos (TC) and ou pu le els (Q) among me g-
ing banks, he weigh ed a e age cos in Table 3.1 is compu ed om a a io o a e ages:
(P26
=1 TC
a+b, −1, /P26
=1 Qa+b, −1, )−(P26
=1 TCa, −1, /P26
=1 Qa, −1, )
(P26
=1 TC
a, −1, /P26
=1 Qa, −1, )whe e e e s o he me ge num-
be . A simple a e age o 26 a e age cos alues (an a e age o a ios as in P26
=1[(ACa+b, −1, −
ACa, −1, )/ACa, −1, ]/26 ) would weigh all me ging banks equally e en hough hei impac on
indus y cos is diffe en .
27A simple a e age o he p edic ed pe cen changes in a e age cos o he 26 me ge s sugges s
ha hese cos s would ise (no all) by 0.27% using he anslog o m.
20
Table 3.2: P edic ed cos effec o 26 indi idual me ge s
Pc . change 0-10% 10-20% 20-50% 50-100% >100%
o al loans 10 8 4 3 1
F equency:
Pe cen T anslog Fou ie Spline
6 o11 1
2 o6 22 2
1 o2 23 1
0 o1 12 5 4
0 o-1 911 9
-1 o -2 12 5
-2 o -6 3
-6 o -11 22
economies o .91 and .86, espec i ely.28 He e om 15 o 19 me ge s a e expec ed
o educe cos compa ed o only 10 wi h he anslog. On balance, use o ei he he
Fou ie o he spline unc ion p edic s ha me ge s in No way could bene i use s
o banking se ices.29
In la ge coun ies (such as he U.S.) he a e age banking me ge inc eased he
size o he acqui ing bank by 50% o mo e (Be ge and Humph ey (1992)). Thus
he acqui ed bank accoun s o one- hi d o mo e o he consolida ed ins i u ion.
Table 3.2 shows he size dis ibu ion o acqui ed banks in No way in e ms o hei
p e-me ge expec ed change in loan ou pu .30 Fully 18 o he 26 me ge s we e
compa a i ely small and we e no expec ed o inc ease he size o he acqui ing
bank by mo e han 20%. Only 4 me ge s inc eased he size o he acqui ing bank
by 50% o mo e. Wi h so many ela i ely small me ge s we would also expec
ha he p edic ed impac on indi idual banks’ a e age cos would also be ela i ely
small. The equency dis ibu ion o he p edic ed pe cen change in a e age cos
28Using simple a e ages sugges s ha a e age cos would ise by a miniscule 0.02% wi h he
Fou ie o m bu all by 1.07% wi h he spline.
29O cou se, lowe a e age cos s would be passed on o businesses and consume s only i he
ma ke is compe i i e. I so, use s o banking se ices could expec o see ma ginally lowe p ices
a e a lag o o expe ience a smalle ise o e ime han would o he wise occu .
30This is om [(Q1,a +Q1,b +Q2,a +Q2,b)/(Q1,a +Q2,a)]−1and ep esen s he pe cen change in
loan ou pu expec ed om he me ge i no hing else changes. The usual si ua ion wi h a me ge ,
howe e , is ha a ew bo owe s swi ch banks while o he s may no ha e hei loans enewed so
he simple sum in he nume a o he e may o e s a e somewha he ac ual change in loan ou pu .
21
shown in Table 3.2 con i ms his esul . Indeed, ou o 26 me ge s, 21 ( anslog), 16
(Fou ie ), and 13 (spline) a e p edic ed o change a e age cos —ei he up o down—by
less han 1%. As small me ge s will likely ha e ela i ely small effec s on banking
cos s, ou ocus will be on he ew bank me ge s ha a e la ge o his is whe e he
g ea es impac on he banking indus y and po en ially on hei cus ome s will be.
4. Es ima ing Me ge Cos Effec s Ex Pos .
One difficul y in es ima ing he ac ual cos effec o me ge s lies in sepa a ing he
effec o he me ge om possible concu en cos changes affec ing he banking
indus y as a whole. Such concu en changes include (cos push) effec s om
changes in in e es a es and wages, effec s om he shi o elec onic paymen s,
and (demand pull) effec s due o changes in he business cycle ha may affec
p oduc i i y and al e cos s pe uni o ou pu o e longe pe iods o ime. We
app oach his h ee ways, al hough each me hod has i s own p oblems.
4.1. Compa ing P e- and Pos -Me ge Cos Effec s o Indi idual Me g-
e s.
We i s looka howa e agecos a ame gingbankhaschangedoneyea p io o
and one yea a e hei me ge , excluding he me ge yea i sel . Use o such a sho
pe iod should minimize cos changes un ela ed o me ge s. The pe cen change in
a e agecos o heme gingbankis hencompa ed o he pe cen change in a e age
cos o he banking indus y as a whole o e he same h ee yea s. I a e age cos
a a me ging bank alls by 5% (say du ing a pe iod when in e es a es a e alling)
while a e age cos o he indus y as a whole alls by 3% hen i is p esumed ha
ac ual a e age cos a he me ging bank expe ienced a ne educ ion o 2% (ac ually,
wo pe cen age poin s). The assump ion is ha changes in cos s expe ienced by
he indus y will also be expe ienced o an equal deg ee by he me ging bank.
This p ocedu e has been applied o each me ge o e he 8 diffe en yea s ha
hese me ge s occu ed. The equency dis ibu ion o he es ima ed ac ual change
in a e age cos is shown in Table 4.1, along wi h he equency dis ibu ions o
he p edic ed changes om he anslog, Fou ie and spline unc ions.31 While he
31As he ange o he ac ual changes is la ge han he ange o he p edic ed changes, he
22
— due o poo planning, poo execu ion, managemen con lic , o o he p oblems —
do no expe ience signi ican cos educ ions. Academics poin o (selec ed) s udies
o bank scale economies which show ha scale economies exis and sugges ha
expec ed me ge cos sa ings may be simila o he a e age scale effec es ima ed.
Ou goal is o demons a e an imp o ed way o es ima e bank scale economies and
o de elop mo e accu a e p edic ions o expec ed me ge cos effec s.
Following he wo k o McAllis e and McManus (1993), we demons a e he im-
po ance o using a cos unc ion speci ica ion ha is mo e lexible han he anslog
o m when es ima ing scale economies. Bo h a linea spline and a Fou ie cos unc-
ionp o ideamo eaccu a e ep esen a iono howa e agecos a iesbysizeo
bank han does he anslog o m and hus imp o es he accu acy o p edic ions o
hecos effec o me ge s.44 Using a balanced panel o 131 No wegian banks o e
12 yea s (1987-1998), he anslog o m effec i ely yields an L-shaped a e age cos
cu e while he linea spline and Fou ie o ms gi e an M-shaped cu e. Ins ead
o anslog-de e mined scale economies o only smalle banks and cons an a e age
cos s o only la ge ins i u ions, he spline and Fou ie models show economies and
diseconomies o bo h smalle and la ge ins i u ions. This occu s when o al asse s
a e used as a single measu e o banking ou pu as well as a wo-ou pu case using
business and consume loans. Al hough business loans ha e a highe ma ginal cos
han consume loans, hey also ha e g ea e scale economies and hus a g ea e
po en ial o uni cos educ ions om a me ge .
Using h ee cos unc ions, we compu ed he p edic ed change in a e age cos
o each o 26 me ge s. When bank a( he la ge bank) acqui es bank b o o m
he me ged bank a+b, he p edic ed change in a e age cos equals he p edic ed
a e age cos o a bank he size o bank aone yea p io o i s me ge minus he
p edic ed a e age cos o a bank he size o bank a+balso one yea p io o he
me ge . Fo all me ge s oge he , he anslog p edic ion is o a e age cos o
all sligh ly by -0.16% while la ge cos educ ions a e p edic ed wi h he Fou ie
(-2.06%) and spline (-3.09%) models. The o e all change in ac ual a e age cos o
all yea s p e-me ge o all yea s pos -me ge o ou 26 me ge e en s was -2.81%.
44Howe e , i one is only in e es ed in he mean scale effec and no conce ned wi h how cos s
a y wi h size, hen any one o hese h ee o ms may be used since hey all gi e almos iden ical
mean scale economy alues.
29

In con as wi h o he s udies, bank me ge s in No way ha e — on a e age — esul ed
in lowe cos s.
In he U.S., he a e age acqui ed bank is hal he size o he acqui ing ins i u ion.
In No way, he a e age me ge is smalle : in 18 o 26 me ge s he acqui ed bank
is less han one- i h he size o he acqui ing ins i u ion. Indeed, he dis ibu ion
is qui e skewed. One me ge accoun s o hal o he o al change in he alue
o loan ou pu om me ge s while he six la ges me ge s accoun o 87% o his
change. Fo hese six me ge s, he accu acy o he di ec ion o he ou p edic ed
change in a e age cos is less han one-hal wi h he Fou ie o spline unc ions. This
sounds like a poo showing un il i is compa ed o he accu acy o me ge pa icipan s
hemsel es. Me ge pa icipan s almos always p edic ha hei me ge will educe
cos bu ex pos analyses ypically ind ha his occu s in hal o less han hal o
he cases. O e all, ou ex an e cos p edic ions o indi idual me ge s a e abou
as good (o poo ) as hose o me ge pa icipan s hemsel es. The e is g ea e
ag eemen when he a e age o p edic ed ex an e cos effec s (-3.09% o he spline)
a e compa ed wi h he a e age o ex pos changes (-2.81%). Thus he es ima ed
expec ed alue o he cos effec om a me ge can play a use ul ole in assessing
he likelihood ha cos sa ing es ima es by me ge pa icipan s may be ealized in a
p obabilis ic sense. In his a ea, he spline and Fou ie unc ions gi e mo e eliable
esul s han he anslog.
We also es ima ed how bank a e age cos has been affec ed by he on-going shi
om expensi e pape -based paymen ins umen s (checks, pape gi o ansac ions)
o lowe cos elec onic paymen subs i u es (debi ca ds, elec onic gi o paymen s).
This swi ch om pape o elec onic paymen s has been subs an ial. The sha e o
elec onic paymen s in all non-cash ansac ions ose om 16% in 1987 o 74% in
1998. This shi is associa ed wi h a mean yea ly educ ion in all bank a e age
cos o 1.3%, o -13% o he indus y o e ou 12 yea pe iod. This cos educ ion
exceeds ha o all 26 me ge s and, in e ms o consume and bank bene i s, policies
ha encou age elec onic paymen s a e p e e ed o e hose associa ed wi h me ge s.
30
Re e ences
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p elimina y look a he 1990s da a,” in Bank Me ge s and Acquisi ions,ed.by
Y. Amihud, and G. Mille . Kluwe Academic, Bos on, MA.
Be ge , A. N., and D. B. Humph ey (1992): “Megame ge s in banking and he use
o cos efficiency as an an i us de ense,” An i us Bulle in, 37, 541—600.
Be ge , A. N., and L. J. Mes e (1997): “Inside he black box: Wha explains
diffe ences in he efficiencies o inancial ins i u ions,” Jou nal o Banking and
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Ca bo, S., D. Humph ey, and F. Fe nandez (2003): “Bank de egula ion is be e
han me ge s,” Jou nal o In e na ional Financial Ma ke s, Ins i u ions & Money,
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Da id B. Humph ey and Ben Vale: Scale economies, bank me ge s, and elec onic paymen s: A spline unc ion app oach Wo king Pape 2003/5
KEYWORDS:
Economies o scale
Func ional o m
Me ge s
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