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Working Paper Series
Scope Economies: Fixed Costs,
Complementarity, and Functional Form
WP 91-03
Lawrence B. Pulley
College of William and Mary
David B. Humphrey
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 91-3
Scope Economies: Fixed Costs, Cgmplementarity,
and Functional Form
Lawrence B. Pulley**
and
David B. Humphrey***
February, 1991
This is a preprint of an article published in the Journal of Business , July 1993, v. 66, iss.3,
pp. 437-62
*The opinions expressed are those of the authors and do not necessarily reflect
the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
The authors thank John Boschen for his comments on an earlier version of this
paper. Support for Lawrence Pulley was received from the Summer Research Grant
Program at the College of William and Mary.
*School of Business Administration, College of William and Mary,
Williamsburg, Virginia, 23187.
***FederalReserve Bank of Richmond, Richmond, Virginia, 23261.
Abstract. Bank scope economies have been derived from either the standard or
generalized (Box-Cox) multiproduct translog (or other logarithmic) functional
form. Reported results have ranged from strong economies to diseconomies and
are far from conclusive. The problem is functional form. An alternative
composite form is shown to yield stable SCOPE results both at the usual point
of evaluation and for points associated with quasi-specialized production
(QSCOPE). Unstable results are obtained for the other forms. Scope economies
are shown to exist for large U.S. banks in 1988 and to depend on the number of
banking outputs specified. The scope estimates are also separated into their
two sources--fixed-cost and cost-complementarity effects.
Scope Economies: Fixed Costs, Complementarity, and Functional Form
1.
Introduction.
Results from previous studies of economies of scope in banking have been
inconclusive. Some researchers have found large scope economies (Gilligan,
Smirlock, and Marshall (1984), Kim (1986), Buono and Eakin (1990)): others
have found only small scope economies--or even diseconomies (Lawrence and Shay
(1986), Mester (1987b), Berger, Hanweck, and Humphrey (1987)). One reason no
consensus exists on this important issue is the use of the popular
multiproduct translog cost function. Even with outputs generalized through
the use of Box-Cox transformations, the translog is inherently nonrobust when
used to evaluate scope economies.'
There are at least three sources of misinterpretation or error in
existing studies of scope economies in banking. First, a number of
researchers have chosen to measure only cost complementarities and use these
results to infer the existence of scope economies. While cost
complementarities are more easily obtained from the translog than are direct
estimates of economies of scope, the sufficient conditions for scope economies
require that weak complementarities exist for all pairs of outputs and a
all
output levels below the initial point of evaluation (usually the means).
As
pointed out by Mester (1987b), these conditions are rarely tested proper Y
because of the effort involved. Unfortunately, researchers pursuing simpler
alternatives to this strict test have often tested inappropriate hypotheses
such as jointness rather than economies of scope (Gilligan, Smirlock, and
Marshall (1984)).
' In contrast, scale economy estimates using these forms are reasonably
robust. See the surveys of Mester (1987a), Clark (1988), and Humphrey (1990).
-2-
Second, even when scope economies are computed directly, they cannot be
measured using the standard translog specification since the logarithmic
function approaches -a as its argument approaches zero.
This difficulty has
prompted a number of ad hoc modifications. One modification involves
substituting a small positive value for zero output levels both for estimation
and scope economy evaluation. Another is the use of a Box-Cox transformation
on output (and other variables) to derive a generalized translog form which
can admit zero values (Lawrence (1989); Buono and Eakin (1990)). As we will
show below, these modifications do not eliminate the problem of bias in using
the translog form and are unsatisfactory.
Third, measuring scope economies for banks requires evaluating the
estimated cost function for production levels that generally lie well outside
the sample observations. Unlike other industries, few banks produce zero
levels of certain outputs and fewer still produce only one output.
Thus, bank
scope estimates are typically based on extrapolated cost measurements having
little empirical support.
We address these problems in the following way.
While cost
complementarities are an interesting concept, we do not compute them in lieu
of measuring scope economies directly, as is frequently done in the
literature. Rather, the functional form we use allows us not only to compute
scope economies but also to assess the actual contribution of cost
complementarities to the scope value obtained.
Indeed, the two sources of
scope economies--spreading fixed cost over a broader product mix and cost
complementarity--can be formally separated. This has not been done before and
is important in understanding why scope economies vary with the degree of
-3-
output aggregation specified. Functions that are multiplicative (or log
additive), such as the translog, make it impossible to identify and measure
these two effects.
The serious problem of bias from using either the standard or the
generalized (Box-Cox) translog form in obtaining scope economies is overcome
by using an alternative form--a composite function. This form exhibits the
quadratic output structure recommended by Baumol, Panzar, and Willig (1982)
for examining economies of scope and is related to the specification used
recently by Roller (1990) to address problems-with the translog form.
Lastly, although we can not alleviate the need to extrapolate beyond the
confines of our bank data set to derive scope estimates, we can evaluate the
relative stability and sensitivity of scope estimates to this extrapolation
problem. This is done through a suitable modification to the standard formula
for measuring economies of scope that enables us to examine joint cost effects
relative to differing degrees of output specialization. The resulting "quasiscope" values should not vary significantly for small departures from
completely specialized production. Stability over this region is demonstrated
for our composite function and shown not to exist for either the standard or
the generalized translog forms.
Economies of scope, and measures of their two sources--fixed costs and
cost complementarity, are formally defined and derived in the next section.
In Section III,
the problems and biases in using the standard and generalized
translog forms are discussed and our composite model, which overcomes these
difficulties, is introduced. Scope and quasi-scope estimates for large U.S.
banks in 1988 are presented in Section IV. These are obtained from estimated
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composite, translog, and generalized translog cost functions. The stability
of these results is assessed and illustrated for a number of different output
aggregations.
In addition, the contributions of fixed cost and cost
complementarities to scope economies are shown for the composite form.
This
permits us to explain why scope economies vary with the level of output
aggregation. Summary and concluding comments are .contained in Section V.
II. Measurinq Economies of Scone: Fixed Costs and Comolementaritv.
Definition And Measurement. Economies of scope reflect cost savings
resulting from simultaneously producing several outputs in the same firm,
rather than producing each one separately in a specialized firm.
These
savings arise from two sources: reducing excess capacity by producing a
broader product mix, thereby lowering the fixed costs allocated to existing
products, and reducing the costs of joint production through production
complementarities.
In banking, daily deposit accounting for different types of accounts
jointly or the processing of different types of loans jointly often requires
much the same type and level of overhead as each would if produced separately.
Thus, excess computer, branch-network, and loan-production office capacity can
be reduced as more banking products are offered, lowering the cost of each
product compared to a smaller product line.
Cost complementarities arise when
account and credit information developed for deposit products can be used to
reduce the information and monitoring requirements for installment, mortgage,.
and other specialized loan products for the same customer base.
Thus,
expansion of a deposit base, on either the corporate or retail side, can lower
-5-
the cost of providing loan products.
Overall economies of scope are measured as the percentage cost savings
from producing all outputs jointly as opposed to producing each output
separately:
(1)
SCOPE = [C(q,,O,.,O;r)t C(O,q,,O,.,O;r) t...t C(O,.,O,q,;r)
- m,,q,, . ,q,;Jwwl,9q,.
>.4,x)
where C(e) refers to the cost function, qi (i=l,.,m)
is a vector of n input prices.
refers to outputs, and r
Since SCOPE is derived from an estimated
multiproduct cost function, choice of functional form is important. In
addition to being a "proper" cost function (nondecreasing in outputs and
nonegative, nondecreasing, concave, and linearly homogeneous in input prices),
the function must be capable of representing cost relationships adequately
over a region which includes zero levels of production in some outputs.2
While the translog form can be altered to meet this latter requirement, the
fix-ups involved stil‘lyield scope economy estimates that are nonrobust (as is
demonstrated below).
ADDeal Of A Ouadratic SDecification For OutDuts.
It is interesting to
note that when Baumol, Panzar, and Willig (1982) discussed the SCOPE measure,
they recommended estimating it with a cost function having a quadratic
specification for outputs. This is a direct and proper method to account for
zero outputs in estimation and evaluation. In addition, such a form allows us
to identify separately the fixed-cost and complementarity effects contained in
the scope measure.
' For the other outputs, SCOPE estimates are generally evaluated at the
sample means or medians of the produced output and input-price variables.
-6-
The appeal of such a form is evident from the following example of a
two-output cost function that is quadratic in outputs and (for simplicity)
strongly separable in input prices:
ubr)
=
=
bo + qq, + a$42+ &,q2 + q,q,q, + +~22q22l f(r)
l
hWf(r)
l
Although illustrative, this example is precisely the form of the multiproduct
cost function used recently by Roller (1990) to reexamine the multiproduct
structure of the Bell System.
And, it leads to very nearly the same
expression for measuring economies of scope as the composite cost function
introduced in the next section.
The estimate of economies of scope from the cost function above is given
by:
SCOPE = {[(a0 +arq,+4a,,q,‘)-f(r)
1 + 1(~o+a2q2+4a22q22)
*f(r) 1
2+a,2q,q2+%a22922)
-f(r) 1~/C(~I~Y- 1(ao+qv-a2q2+%al,q,
Simplifying,
SCOPE =
(2)
(a, - a,,q,q,)/h(sd.
The calculation generalizes in a straightforward manner to the case of m
outputs:
(3)
SCOPE = [(m-l)a,
- I;i=,Xj,iaijqiqjl/h(q)*
The Seoaration Of Fixed-Cost And Complementaritv Effects. The separate
contributions of the two factors determining economies of scope--fixed costs
and complementarity--are evident in (2) and (3). The first term in the
formulas measures the savings resulting from reduced excess capacity, in which
fixed costs are spread across a broader product line.
In calculating
economies of scope the formula implies that specialized production of each
-7
-
output would require the same level of fixed costs as joint production of all
m outputs.
In comparing joint production with specialized production for an
m-output firm, the level of fixed costs will therefore be "saved" m-l times.
The second term in the SCOPE formulas shows how cost complementarities between
outputs contribute to economies of scope.
If variable inputs are shared by
different product lines, as when deposit and other information serves both
deposit and loan products, the aij coefficients would be expected to be
negative and economies of scope would increase.
As expression (3) makes clear, the contribution to economies of scope
resulting from saved fixed costs will in all likelihood increase with
increases in the number of outputs specified in the cost function. This could
produce differences in measured economies of scope across banking studies that
result solely from differences in output specification or level of
aggregation. Since there remains a debate in the literature regarding the
appropriate number and specification of banking outputs, we derive SCOPE
estimates using three different levels of output aggregation.
The portion of economies of scope attributable to cost complementarities
only is expressed as:
(3')
SCOPE,, = - ~i=1Hj>iaijqiqj/hO
while that attributable to fixed cost is:
(3")
SCOPE,, = SCOPE - SCOPE,, = (m-l)aJh(q).
Multiplicative or log-additive cost functions such as the popular translog
specification do not allow the fixed-cost and complementarity components of
economies of scope to be separated. Consider the translog cost function for
the case of m outputs and n input prices:
-8-
(4)
In C = a0 t ti,lnoi t $XQijlnoilngj t XZ6iklngilnrk
t CB,lnr, t QXB,,lnr,lnr,
where i,j refer to the m outputs and k,l refer to the n input prices.
To
compute economies of scope from (1) using the translog cost function, we must
first exponentiate both sides of the cost function in (4). This yields a cost
specification of the following form:
C = eao.ea’L~‘.ea2L~. . . ..
(5)
The problem of handling zero output values aside, we see that "fixed costs"
(eao) enter as a scaling factor and will therefore cancel out of the SCOPE
formula in (1) so we are unable to isolate the two.3 In our application
below, we provide separate estimates of the fixed-cost and cost complementary
components of economies of scope for large U.S. banks in 1988.
III.
Choice of an Annronriate Cost Function Soecification.
Problems In Using The Translog Form.
It is well known that there are
problems with using the translog cost function to examine scope economies;
nonetheless, the translog remains the most popular model in banking studies.
Three alternatives are available to researchers who prefer the multiproduct
translog model.
The first alternative is to avoid computing scope economies
directly but instead to infer their existence using sufficient conditions
based on cost complementarities. Baumol, Panzar, and Willig (1982) have shown
3 A similar result holds for the Box-Cox variant of the translog form, as
well as for the (logarithmic) Minflex Laurent form recently applied by LeCompte
and Smith (1990) and Hunter, Timme, and Yang (1990). Thus for neither of these
two forms, even when used in a frontier estimation framework (Ferrier and Love11
(1990)), is it possible to separate fixed-cost from complementarity effects.
-9-
that a twice-differentiable cost function will display economies of scope at
g* if it exhibits weak cost complementarities, i.e., if 62C(q',r)/6qiSqj_(0
ifj, for all output pairs at all output vectors g' with Q I 9’ I g* (with the
inequality holding strictly for some subset). As discussed in Mester (1987b),
particular care must be exercised when using this approach. To conduct the
test properly the condition must be examined for a very broad range of output
vectors, g', and this makes testing so cumbersome that it rarely is done
correctly. Furthermore, some authors (Gilligan and Smirlock (1984) and
Gilligan, Smirlock, and Marshall (1984)) have tested the estimated translog
cost function for jointness and mistakenly interpreted their results as
implying economies of scope.b
However, if we view the translog cost function as a log-quadratic
approximation to the true unknown cost function, it is possible to develop a
test of the complementarity condition that is based on cost function
coefficients only and does not involve variable values (Denny and Pinto
(1978)).
In this case, the condition 62C(q',r)/6qi6qj5 0 reduces to the
requirement that output coefficients satisfy aiaj<aij for all i and j, j#i.5
4 By restricting the cost function parameters, they impose the condition
s2c(s9E)/sqisqe = 0, i#j (which for the translog specification is given by
s21nC(q,r)/Glnh-61nq.t (&lnC(q,r)/Slnq,)(6lnC(g,~)/6lnq-)= 0 i#j). Likelihoodratio tests shoi that the restrictions are rejected by thk data and this is taken
by the authors as evidence of economies of scope. However, the test performed
is a test of jointness only: rejection of the condition is consistent with either
62c(a,r)/6qi6qj< 0 (implying economies Of scope) x 62C(q,r)/6qi6qj> 0 (impTying
increased costs for joint production). Without further testing it is impossible
to determine which alternative has empirical support.
5 Lawrence (1989) employs a variant of this procedure to test for economies
of scope in banking. It is not clear from his discussion whether he conducted
his likelihood ratio tests against the null hypothesis that there are no scope
economies (aiaj2aij)or the null that there are no economies x diseconomies
- 10 -
The Denny and Pinto condition (aiaj<aij)
is based on the relationship between
the parameters of the true cost function and those of the log-quadratic
(translog) approximation at the point of expansion. However, White (1980) has
shown that the translog coefficients estimated using OLS need not coincide
with the parameters of the approximation, even at the point of expansion.
From a practical standpoint, therefore, this test is not very useful.
The second alternative is to estimate the translog cost function and
substitute some small positive value in place of the zero outputs necessary to
evaluate the SCOPE measure (1).
This practice should not continue.
Since the
logarithmic function becomes arbitrarily steep and its value approaches -a as
its argument approaches zero, the translog cost function is badly behaved not
only precisely at zero, but in a region around zero.
Furthermore, the size of
this region and the extent of the problem depend on parameter estimates and
thus it is impossible to determine ex ante at which point cost behavior
improves. The procedure of substituting small positive values for zero in the
translog is apt to produce large estimates of scope economies (if most of the
a ii coefficients on squared outputs are positive) or large estimates of scope
diseconomies (if most of the aii coefficients are negative).
This
characteristic of the translog is referred to as the "flip-flop" property by
Riiller(1991) and has been evident in a number of banking studies (e.g.,
Berger, Hanweck, and Humphrey (1987)).
The third alternative for measuring economies of scope using the
At any rate, it would have been preferable if he had conducted a
joint test over all output pairs rather than test individual output pairs
separately.
(aiQj’Q.i
j)
.
- 11 -
translog methodology is to estimate a generalized version of the translog
model.
Caves, Christensen, and Tretheway (1980) obtained a generalized
translog cost function by applying a Box-Cox transformation to the output
variables in the translog model.
(4)
q
The Box-Cox transformation is defined as:
=
(49-w~
if 4#0
=
lnq
if #=O.
The generalized multiproduct translog cost function of Caves, Christensen, and
Tretheway is therefore given by:
In C = a, + D!iqi"' + ~~ijqi'~'qj'~'+ XX6ikqi'"'lnrk
(6)
t XBklnr, t 4XZB,,lnr,lnr,
The standard translog cost function results if 4 = 0.
The generalized
translog specification in (6) does admit zero output values (except in the
case of 4 = 0) and can be examined for economies of scope using the SCOPE
formula (l).6
From an estimation standpoint, the generalized translog often turns out
to be less of an improvement than anticipated. In their study of the
structure of the Bell System, Evans and Heckman (1984) estimate a generalized
translog cost function and find the optimal value of 4 to be fairly close to
zero, the value corresponding to the translog. Similarly, when Lawrence
(1989) estimates 4 using banking data, it is found to be close to zero.
Upon further reflection, this result is not unexpected. The translog
6 Some studies, rather than estimating the Box-Cox transformation parameter
on output, have instead assumed that4 = 1 (Berger, Hanweck, and Humphrey (1987);
Buono and Eakin (1990)).
While this alleviates the problem of zero output for
scope estimation, the reported results suggested scopediseconomies and economies
respectively.
- 12 -
cost function is a second-order Taylor expansion'of the logarithm of costs
(1nC) in terms of logarithms of outputs and input prices (lnq and lnr).
We
should therefore expect that the "optimal" Box-Cox transformation of q would
turn out to be very close to lnq, i.e. a # very close to zero, since the log
transformation is still applied to the cost and input-price variables.
If
this is the case, then the estimated generalized translog model will be very
close to the estimated translog model.
Although we may then compute cost
estimates for zero outputs (as long as the optimal 6 is not exactly zero), the
difficulties with the translog cost behavior in the neighborhood of zero will
remain.
The behavior of scope economies for the generalized translog at
output points at or close to zero, illustrated below, confirms this
observation. Thus, there appear to be no straightforward procedures for
obtaining reliable estimates of economies of scope from the translog model.7
A More Arxwowiate
Specification:
The Comoosite
Form.
A more satisfying
way to address the problems associated with scope measurement is to develop a
multiproduct cost function by adding a suitable input price structure to the
quadratic output structure recommended for examining multiproduct technologies
by Baumol, Panzar, and Willig (1982) [hereafter, BPW].
One such family of
7 Yet another possibility is to generalize completely the log-quadratic
(translog) cost specification by applying Box-Cox transformations not only to
outputs, but also to input prices and the dependent variable, costs, as well.
Such a cost function specification was examined in Lawrence (1989). However,
since the transformation parameter estimates obtained by Lawrence are not
statistically different from those of the standard translog form, little
advantage is gained in using this approach to estimate economies of scope. In
general, the procedure can produce quite complicated expressions for the SCOPE
formula, particularly because of the transformation applied to the dependent
variable. Another significant drawback is that it is difficult to restrict the
cost function to be linearly homogeneous in input prices, as required by theory.
- 13 -
cost functions discussed by BPW (Chapter 15) is that suggested by Pulley and
Braunstein (1982, 1984):
wbr,)
(7)
= (a0 + %iqi +
fBijqiqj)
' f CL)
where f(e) must be easily restricted to be linearly homogeneous in r: Pulley
and Braunstein use a simple Cobb-Douglas form for f(e).
A member of the same
class of cost functions has been used recently by Roller (1990) who shows that
such cost functions satisfy the condition of "output flexibility".' The
problem with cost functions of the form of expression (7), however, is that
they are strongly separable in outputs and input prices. As BPW point out,
separable cost functions are not well suited to model the effects of input
prices on industry structure because of the restrictions imposed on
potentially important input price-output quantity interactions. At a minimum,
separability should be framed as a testable hypothesis rather than imposed 2
priori.
The cost function model we use in this study is a variant of the
composite cost function described in Pulley and Braunstein (1990).
It retains
the output flexibility of the specification in expression (7) but adds output
and input-price interaction terms so that separability is no longer imposed.
Specifically, the composite cost function combines the quadratic output
specification recommended for examining multiproduct technologies by BPW with
the log-quadratic input price structure exhibited by the translog cost
function--an input price structure that is easily restricted to be linearly
* Roller estimates a CES-quadratic form where f(c) is a constantelasticity-of-substitution (CES) specification while outputs follow a quadratic
specification.
- 14 -
homogeneous. The composite cost function will be estimated in logarithmic
form:9
(8)
In C = ln(a, t ~iqi
t P~ijqiqj
t XZSi,Cjilnr,)
t XB,lnr, t QXB,,lnr,lnr,
= ln[h(q;lnr)] t f(m).
The specification (8) cannot be obtained through generalizations of the sort
examined by Lawrence (1989)."
In what follows, we demonstrate that the
composite form (8) is robust to points of evaluation at or close to zero while
the standard or generalized translog forms are not.
IV. Estimates of Scone and Ouasi-Scone for Large U.S. Banks.
Outnuts, InDuts, And Estimation. Banks produce a variety of payment,
safekeeping, intermediation, and accounting services for deposit and loan
customers (Benston and Smith (1976); Mamalakis (1987)). However, some have
argued that banks primarily produce loans.
With this (asset) approach, the
production of deposit services is viewed as merely payment in kind for the use
of funds from which to make loans (Sealey and Lindley (1977)).
In effect,
this is a reduced-form model of the banking firm: the production of deposit
services is treated as an intermediate output to depositors who provide
loanable funds inputs for loans, so deposit services are netted out.
9 See Pulley and Braunstein (1990) for a more general specification of the
composite model.
lo The formula for economies of scope, derived by substituting the composite
cost function into the SCOPE formula in (l), simplifies to yield the formulas in
(2) for two outputs and (3), (3'), and (3") for m outputs, the only change being
h(q;m)
replaces h(g) in the denominator.
- 15 -
But there is no need to focus on only a single type of banking output
like loans, especially since the production of deposit services accounts for
half of all physical capital and labor input expenditures. Because deposit
services are such a large component of bank value added, explicit modeling of
their productive structure, along with that for loans, will yield a more
accurate structural description of the bank as a whole.
This objective can be
achieved using a structural model of a multiproduct banking firm.
In such a
model, the production of deposit services is included among the set of bank
outputs.
For our purposes, banks are considered to produce payment and
safekeeping outputs (associated with the value of demand deposits (DO) and
savings and small denomination time deposits (TS)) as well as intermediation
and loan outputs (associated with the value of real estate (RE), installment
and credit card (IN), and commercial and industrial loans (CI)). Over the
last decade, these 5 output categories accounted for 75 to 80 percent of value
added in banking (Berger and Humphrey (1991), Table 1).
Such a categorization
of bank output, with one exception, is consistent with that identified in the
user cost approach to determine bank inputs from outputs (Hancock (1986);
Fixler and Zieschang (1991)).
Three inputs are specified: physical capital,
labor, and funds (composed of core deposits plus purchased funds) and their
prices enter the model directly."
" There is no inconsistency in specifying the value of core deposits a.s an
output category and, at the same time, including interest expenses on these funds
as contributing to total cost. The value of the stock of these deposits is taken
as an indicator of the underlying payment and deposit and withdrawal transactions
that comprise the actual flow of bank demand deposit and time and savings output,
which is not available for large banks. And, using the same logic, the value of
- 16 -
Bank Call ReDort data on 205 large U.S. banks, all with assets greater
than $1 billion in 1988, are used to illustrate the effect of employing
different functional forms to obtain scope estimates. These banks account for
almost 45 percent of all bank assets.
Our preferred specification
incorporates the 5-output, 3-input grouping noted above.12 To illustrate the
effect of aggregation on these results, scope estimates are also presented
where first deposits and then deposits and loans are separately summed so that
the 5 outputs become 4 and then 2, respectively.'3
Standard procedures were used to estimate the composite cost function
(8), the standard translog cost function (4), and the generalized translog
(6). Sheppard's Lemma was used to develop the three input cost shares, and
the cost functions were estimated jointly with the labor and interest cost
share equations. The form of the share equation for input k from the
the stock of loans is used as an indicator of the flow of new loan outputs.
Since interest expenses, including those of purchased funds, comprise some 70%
of bank total costs, it is clear that the interest cost of loanable funds is an
important component of total cost and needs to be included. The controversy in
the literature over whether deposits are an input or an output is not an issue
here as both aspects are incorporated in our model. In any event, recent studies
treating deposits first as an input then as an output have found that scale
economy estimates and subadditivity are little affected either way (Hunter,
Timme, and Yang (1990)).
l2 The input prices were: (1) new-contract replacement cost per square foot
of bank and office building space for 9 U.S. regions; (2) expenditures on labor
divided by the number of full-time-equivalent workers; and (3) the average
nominal interest rate paid on core deposits.
l3 Specifically, the 4-output model is (DDtTS),RE,IN,CI while the e-output
(DDtTS),(REtINtCI). A model containing only bank assets was specified,
but we experienced convergence problems and dropped it as an illustration. The
problems with convergence are not too surprising since loan assets directly
account for less than half of all bank costs (so a reduced-form specification is
more difficult to identify locally than a structural model).
case is
- 17 -
composite model is:
Sk = [a0 t ~iqi t ~~ijqiqj t Z%i~q'lnr,]-' L'6ikqi
t B, t XB,,lnr,.
l
The form of the share equation for the translog model is:14
‘k
= 86iklnqi t Bk t 28,,lnr,.
All models were estimated using the nonlinear-least-squares procedure LSQ of
the Time Series Processor (TSP) package.'5 If the errors are normally
distributed, the estimates will be maximum likelihood estimates.
Ouasi-Scone: A Measure Of ExtraDolation Sensitivitv. Although the
composite model--unlike the translog--imposes no arbitrary restrictions on
estimated cost behavior for zero outputs, the points examined in computing
economies of scope for banks almost always lie outside the sample. We can
have only limited confidence in estimated costs using these points, no matter
what model is used for the cost function.
More confidence could be obtained if the SCOPE formula in (1) were
generalized to examine a broader range of output values and shown to be stable
at many points of evaluation, including values circumscribed by the sample
observations. Therefore, instead of defining the costs of specialized
production as the costs associated with producing a given amount of one output
and none of the others, i.e. C(O,.,qi,.,O;r), the effects of quasi-specialized
production are considered in which small amounts of all the other outputs are
l4 The share equation for the generalized translog is identical except the
Box-Cox transformation qiu) replaces lnq,.
I5 Parameter restrictions for symmetry and linearly homogeneity in input
prices were imposed. The restrictions are identical for the composite, translog,
and generalized translog cost functions and are given by: Symmetry: "ij = o-j;
0 , for all i; xB,=I; X[Bk[=Q, for all 'k.
8kl = B,,; and Linear Homogeneity: xk6'k=
- 18 -
produced as we11.16 Production of the specialized output is adjusted so that
the total amount produced via quasi-specialized production is equal to the
quantity produced in joint production. Defining the parameter c to be the
proportion of the other outputs produced, the SCOPE formula in (1) may be
modified to measure what can be referred to as "quasi" economies of scope:17
(9)
QSCOPE
= [C({1-(m-l)~}s,,Eq2,.
,q,,;r) +
+ ... + wq
C(~q,,(l-(m-l)E}q,,~q~,
.,q,,x)
l,.,rq,_,,(l-(m-l)~}q,;r)
- C(q,,q,,.,q,;~)]/C(q,,q,,.,q,;r).
As E increases, the production points examined become less and less extreme
relative to the sample observations. Using QSCOPE we can demonstrate the
relative stability and sensitivity of scope estimates from the composite,
standard translog, and generalized translog forms to the common problem of
extrapolation beyond the confines of the banking data set.
When c = 0, QSCOPE becomes the traditional SCOPE measure in (l),
capturing the fixed-cost savings from single-firm, as opposed to m-firm,
production and the cost-complementarity effects from joint as opposed to
specialized production. Strictly speaking, for e > 0 QSCOPE is an empirical
subadditivity measure examining both scope and scale effects for a particular
m-firm division of total output. The maximum value for E:in the QSCOPE
calculations is l/m.
When 6 = l/m, the quantity (l-(m-l)c}
equals l/m and the
'6 If positive amounts of only some of the other outputs are produced, our
concept is similar to the diversification measure developed by Grosskopf, Hayes,
and Yaisawarng (1990), an approach discovered subsequent to our analysis here.
I7 This concept is closely related to expansion path subadditivity (Berger,
Hanweck, and Humphrey (1987)), where there are only two firms producing quasispecialized production. One of them produces the output level and mix of the
average firm observed at a point on the industry expansion path while the other
(synthetic) firm produces output equal to the difference between this firm's
production and that of an average firm further out on the expansion path.
- 19 -
costs of quasi-specialized production, C(cq,,.,{l-(m-l)r}qi,.,cqm;E),equal
the costs of producing the proportion l/m of all outputs,
C(q,/m,. ,qi/m,. 4&w).
At this point the distinction of specialized
production is lost and QSCOPE is a measure of the fixed costs effects just
described and scale economies. Since banks have been shown to exhibit
approximately constant returns to scale,'* QSCOPE should reflect only fixedcost savings for 6 approaching l/m.
SCOPE And QSCOPE Estimates. Table 1 contains the results of evaluating
QSCOPE for the 5-output specifications of the composite, translog, and
generalized translog cost models.
The evaluation takes place at the sample
medians.19 QSCOPE values are obtained for eight values of E (O-corresponding to SCOPE, .OOOl, -001, .Ol,
.05, .lO, .15, and .20).*'
Asymptotic standard errors are computed for each of the QSCOPE estimates using
the procedure described in the Appendix.
In addition to the QSCOPE estimates
and standard errors, Table 1 also contains summary statistics from the
estimation of the cost models.
The optimal value of the Box-Cox
transformation parameter, 6, is also reported for the generalized translog
model.
Tables 2 and 3 contain the results from the 4-output (DDtTS,RE,CI,IN)
and 2-output (DDtTS,REtCItIN) models.
The results for the composite model in Tables 1, 2, and 3 show
l8 See the surveys referenced in footnote 1.
l9 The sample medians of the five outputs (DD,TS,RE,CI,IN) are 681170,
1588700, 810860, 833870, and 444460, respectively--measured in $000'~.
'O Since 6 = 0 cannot be used with the translog cost function, we use c =
.OOOOOlin place of E = 0.0 in this case only.
- 20 -
significant economies of scope for all output specifications. The estimates
of traditional economies of scope (corresponding to c = 0.0) indicate that the
costs of specialized production of the five (four, two) banking outputs would
be 50 (36, 30) percent greater than the costs of joint production.
Importantly, for small E equal to .OOOl, .OOl, or .Ol the QSCOPE estimates are
virtually identical to the traditional SCOPE estimate.
This consistency of
the estimated values for such modifications in the evaluation procedure
mitigates our concern over the extrapolation problem discussed above.
As e
increases, the composite function QSCOPE estimates--though still significantly
different from zero--decline as specialized production becomes less and less
concentrated in the given output; therefore, the advantages to specialization
diminish.
For example, when 6 = .20, the QSCOPE values are roughly half as
large as the values when c = 0."
As expected, the QSCOPE estimates for the translog models for f values
close to zero cannot be taken seriously. Large estimates of economies of
scope are obtained because the coefficients on squared outputs (the
positive in sign.
a,j’S)
are
The procedure of computing economies of scope by inserting
small positive values in place of zero in the estimated translog function is
unsatisfactory. Although the estimated asymptotic standard errors reflect the
imprecision of the procedure, it does not seem prudent to attempt to estimate
scope economies using a methodology that is expected a oriori to produce
meaningless estimates.
Furthermore, the magnitudes of the estimates for the
2' When 6 = .20, the QSCOPE estimate in the 5-output model measures the cost
savings from producing the median values of the five outputs jointly in one firm
as opposed to producing one-fifth of the medians of each of the five outputs in
five identical firms.
- 21 -
translog models do not result from taking the logarithm of arbitrarily small
numbers. Even for small 6 values, the arguments that actually enter the
logarithmic functions are not trivial.22
The optimal value of the Box-Cox parameter in the generalized translog
model is estimated to be .20 in the 5-output and 4-output models, and .lO in
the E-output model.
Our expectation that the optimal value would be fairly
close to zero--the value that results in the log-quadratic translog
specification--is confirmed. Similarly, the QSCOPE estimates derived from the
generalized translog cost function for small 6 values exhibit the same
unstable patterns as those derived from the translog.
For larger 6 values the translog and generalized translog specifications
yield QSCOPE estimates that are both significantly different from zero and
similar to the estimates obtained from the composite model.
Therefore, for E
2 .lO the QSCOPE estimates obtained from the composite cost function are
robust across the three types of cost function models examined here.
This
finding buttresses the arguments made earlier regarding the appeal of the
composite model in examining economies of scope. Furthermore, the differences
between the estimates from the composite and the two translog models and the
dramatic fluctuations in scope estimates for the latter models with small
changes in 6 (when E is itself small) buttress earlier arguments against the
use of the translog and generalized translog models in such examinations.
While the composite form is clearly the most reliable and stable one for
investigating scope economies, all three forms yield very similar estimates of
22 For example, given the medians of the banking outputs reported above, the
value of ~qi ranges from 44 to 158 when 6 = .OOOl.
- 22 -
banking scale economies. This agreement is anticipated since scale estimates
are based on curvature conditions at points were the sample information is
complete: nonetheless, it is important since greater accuracy in identifying
scope economies should not be offset by inaccuracies or biases in measuring
scale economies. All three forms gave point estimates that indicate slight
scale economies for the set of large U.S. banks examined here, when evaluated
at the medians of their 5 outputs in 1988."
Other studies of large banks,
both cross-section (Noulas, Ray, and Miller (1990)) and over time (Hunter,
Timme, and Yang (1990); Humphrey (1991)),
find average costs to have a slight
U-shape, which is consistent with weak economies or diseconomies of scale,
depending on the exact point of evaluation.
Marqinal Costs And Statistical Fit.
Well-behaved cost functions exhibit
nonnegative marginal costs at all output vectors. As Berger, Hanweck, and
Humphrey (1987) point out, a number of previous studies of scope economies
obtained negative marginal costs for some banking outputs using the translog
cost function.
Since nonnegative marginal costs are part of the maintained
hypothesis for using the estimated cost function to measure economies of
scope, the validity of findings from such studies can be questioned.
For all
output specifications of the composite, translog, and generalized translog
models, we evaluated the marginal costs of each output for each of the 205
banks in our sample. The marginal costs for the first output (demand
deposits--DD) in the 5-output models were negative for the majority of the
23 The scale economy values for the composite, standard translog, and
generalized translog forms are, respectively, .941, .952, and .938 (indicating
slight economies).
- 23 -
banks far all three cost function models.
Except in a handful of cases,
however, the marginal costs for all outputs under all three cost function
specifications are positive in the 4-output and 2-output models.
Thus, while
the 5-output model is our preferred one, the theoretical conditions are more
fully met in the more aggregative models.
The summary statistics for the models in Tables 1, 2, and 3 indicate
that all three cost functions and associated share-equations provide fairly
comparable descriptions of the banking data.
Since the translog is a
constrained version of the generalized translog, log-likelihood ratios can be
used to test whether it provides a statistically indistinguishable fit.
The
translog model is rejected in favor of the generalized translog model for the
5-output and 4-output specifications, while the two models are not
statistically different in the 2-output case. Judged on the basis of sum-ofsquared-error and adjusted-R2 criteria, the composite specification provides a
slightly better overall fit for the cost function and share equations for the
5-output and 4-output models in Tables 1 and 2, while the translog and
generalized translog provide slightly better descriptions when only two
outputs are specified. The generalized translog cost function provides the
better fit for the cost function considered separately, although the
differences in adjusted R2 values are trivial.24
Fixed-Cost And Comolementaritv Effects. Estimates of economies of scope
from the composite model increase in magnitude from 30 to 50 percent as the
number of outputs increases. As discussed in Section II, this is to be
24 To examine this issue correctly, single-equation estimation must be
performed.
- 24 -
expected given the treatment of fixed costs in the SCOPE and QSCOPE formulas.
Table 4 presents estimates of the fixed-cost (SCOPE,,)and costcomplementarity (QSCOPE,,)components of the scope and quasi-scope economies
estimated for the composite function.25
The fixed-cost component increases from 6 to 15 to 26 percent as the
number of outputs rises from 2 to 4 to 5.
Thus the proportion of scope
economies accounted for by fixed costs rises from 20 to 44 to 52 percent,
respectively, illustrating that scope economies rise with reductions in the
level of output aggregation. Since the fixed cost component of economies of
scope is independent of the level of output, SCOPE,, is not a function of E.
The cost complementarity component of scope economies behaves
differently with respect to the level of output aggregation. Unlike the
effect of spreading fixed costs across a broader product mix, the cost
complementarity effects should be relatively stable regardless of the number
of outputs specified. When E = 0, cost complementarities are between 20 to 24
percent for all output specifications,26indicating that complementarities
are responsible for making the costs of joint production some 20 to 24 percent
lower than the costs of specialized production. For the 5-output case, this
is around the same percentage cost reduction that is attributable to reducing
fixed costs.
However, as the point of evaluation moves away from the zero
25 It is not possible to separate fixed-cost from complementarity effects
in either the standard or the generalized translog forms.
26 This consistency is all the more impressive when it is recognized that
the number of output interaction terms (the (lij,where j>i) on which the
calculation is based ranges from only one in the P-output model to ten in the 5output model.
- 25 -
output levels associated with scope economies, the cost-complementarity effect
becomes smaller and smaller. When E = .20, the cost-complementarity effect on
QSCOPE is zero or close to it, and the only remaining effect is from spreading
fixed costs (i.e. QSCOPE = SCOPE,, at large e values). Recall from the
earlier discussion of the QSCOPE measure that as E approaches l/m, the costcomplementarity effect converges to a pure scale effect. Since the estimated
cost function shows almost no scale economies, the only influence left to
affect QSCOPE derives from fixed cost.
To sum up, increases in SCOPE
associated with reductions in the level of output aggregation are due to the
effects of spreading fixed costs over a larger product mix while reductions in
QSCOPE associated with moving the point of evaluation away from zero outputs
are due to the diminishing effects of cost complementarity and the absence of
scale economies.27
V. Summarv and Conclusions.
To date, scope economies in banking have been derived from logarithmic
functional forms.
This includes the standard multiproduct translog form (with
its ad hoc fix-ups to accommodate zero outputs), the generalized translog
(with a Box-Cox transformation on outputs), and the (logarithmic) Minflex
Laurent form.
Banking studies have reported scope results that have ranged
from strong economies to diseconomies and are far from conclusive. The
27 This decomposition can be generalized to the expansion path subadditivity
measure (EPSUB) of Berger, Hanweck, and Humphrey (1987) and the diversification
measure (DIVERS) of Grosskopf, Hayes, and Yaisawarng (1990). Like QSCOPE, the
levels of EPSUB and DIVERS are determined from the net effect of both expanding
the number of outputs produced at all specializing firms and by the level of
quasi-specialized production.
-
26 -
problem is largely attributable to the choice of functional form, as noted in
another context by Roller (1990). The instability of the logarithmic
functional form is demonstrated using data on 205 large banks for 1988,
accounting from almost 45 percent of total U.S. banking assets. An
alternative functional form--the composite form--is introduced, which
addresses these problems and is shown to yield stable scope results.
This
holds both at the usual point of evaluation (where non-specialized outputs are
zero--SCOPE) and for points associated with quasi-specialized production
(QSCOPE). When measurement is restricted to output levels much greater than
zero, all the functional forms estimated here yield similar estimates of
quasi-scope. This result, along with our other findings, suggests that the
translog form (and its variants) should not be used to determine scope
economies in banking, although they can be used to determine other efficiency
measures such as QSCOPE and scale economies.
Using our composite functional form, significant scope economies on the
order of 30 to 50 percent are shown to exist for large U.S. banks.
The size
of these economies depends on the number of banking outputs specified,
becoming larger with increased disaggregation. Because the composite form
permits us to decompose scope economies into their two sources--spreading
fixed cost over a broader product mix and cost complementarity, this result
can be investigated further. The component of economies of scope attributable
to spreading fixed costs across product lines is a function of the number of
banking outputs, and this is the reason why larger scope economies are
measured when more, less-aggregated outputs are included in the cost function.
In contrast, the component of scope economies attributable to cost
- 27 -
complementarities is relatively stable with respect to the number of banking
outputs specified. This effect accounts for between 20 and 24 percentage
points of the 30 to 50 percent values obtained for overall scope economies.
However, as the point of evaluation moves away from specialized production
(the point of evaluation for SCOPE) and toward proportional production, the
cost complementarity effect becomes smaller and approaches a measure of scale
economies. Since banking scale economies are very slight, the net effect is
to reduce our quasi-scope (QSCOPE) measure down to the point where it is
almost solely comprised of cost reductions associated with spreading fixed
costs over a larger product mix.
Such insights into the reasons why the
efficiency of joint production varies positively with output disaggregation
but negatively with less-specialized production are not possible using the
functional forms applied in current banking studies.
APPENDIX
Procedure for Computing Asymptotic Standard Errors for Estimates
of Economies of Scope Derived from Multiproduct Cost Functions
Asymptotic variances for a (twice-differentiable) nonlinear function
f(b,,...,b,) of n random coefficients may be approximated by Var(f(b,,...,b,))
= (Sf/bb)'q(Sf/6b), where &f/&b is the n-dimensional column vector
representing the gradient of f(e) with respect to the random coefficients and
Z$,is the variance-covariance matrix of the coefficients.
If the random
coefficients are themselves maximum likelihood estimates, f(e) will have a
limiting normal distribution. Here f(e) is the SCOPE or QSCOPE formula and
the random coefficients are the estimated parameters of the cost function. The
procedure is described in Thiel (1971) Chapter 8, especially problems 3.1-3.3,
3.6, and 3.7.
Note that the medians of the output and price data in the SCOPE
and QSCOPE formulas are treated as constants. Although this practice is
standard and conforms to the procedure used in Mester (1987b),
the sample variability associated with computing medians.
it does ignore
Table 1
QSCOPE Estimates Based on Composite, Translog,
And Generalized Translog Cost Functions
(5-Output Specification: DD,TS,RE,CI,IN)
QSCOPE VALUES
(Asymptotic Standard Errors in Parentheses)
Value of c
Comnosite Model
O.Oa (=SCOPE)
.OOOl
Transloq Model
6.41Et09
3239.5
33.6
1.64
.OOl
.Ol
.05
.lO
.15
.20
(1.32Etll)
(27960.)
(144.9)
(3.44)
Generalized Transloq
(Optimal 6: .20)
4.14 (4.67)
1.46 (1.44)
.85 ( .80)
.43 ( .47)
.27 ( .04)
.14 ( .06)
MODEL SUMMARY STATISTICS
Cost Function:
SSEb
Adj. R2
Labor Share Equation:
SSEb
Adj. R2
4.78
.975
4.71
5.00
.975
.974
.341
.295
.416
.325
.323
.360
Share Equation:
SSEb
.722
.531
Adj. R2
.872
.434
.817
.470
Interest
Log Likelihood
792.68
764.72
777.07
a ~=.000001 was used in place of c=O.O in the translog model.
b SSE refers to sum of squared errors.
Table 2
QSCOPE Estimates Based on Composite, Translog,
And Generalized Translog Cost Functions
(4-Output Specification: DDtTS,RE,CI,IN)
QSCOPE VALUES
(Asymptotic Standard Errors in Parentheses)
Value of E
Comoosite Model
O1;;o\=SCOPE)
Transloq Model
Generalized Transloq
(Optimal 4: .20)
4.51Et09 (3.90EtlO)
3460.5 (13557.)
38.9 (93.0)
.OOl
.Ol
.05
.lO
.15
.20
754. (1867)
75.0 (130.)
24.2 (33.0)
MODEL SUMMARY STATISTICS
Cost Function:
SSEb
Adj. R2
5.60
.971
4.84
,975
.363
.359
.285
.347
.310
Share Equation:
SSEb
1.03
Adj. R2
.330
1.09
.293
1.08
.300
Labor Share Equation:
SSEb
Adj. R2
5.42
.972
.322
Interest
Log Likelihood
748.54
724.86
743.51
a c=.OOOOOlwas used in place of E=O.O in the translog model.
b SSE refers to sum of squared errors.
Table 3
QSCOPE Estimates Based on Composite, Translog,
And Generalized Translog Cost Functions
(2-Output Specification: DDtTS,REtCItIN)
QSCOPE VALUES
(Asymptotic Standard Errors in Parentheses)
Value of E
Composite Model
O.Oa (=SCOPE)
.OOOl
.OOl
.Ol
.05
.lO
.15
.20
.30 (.lO)
.30 (.lO)
Transloa Model
Generalized Transloq
(Optimal QI: .lO)
8.34Et14 (3.88Et15)
3.03Et06 (6.17Et06)
3626 (4109)
4.86E+09 (1.93EtlO)
1327 (1682)
111 (94.2)
MODEL SUMMARY STATISTICS
Cost Function:
SSEb
Adj. R2
Labor Share Equation:
SSEb
Adj. R2
Interest
.969
5.61
,971
5.59
.971
.327
.352
,313
.380
.311
,384
1.01
,346
1.01
.346
6.02
Share Equation:
1.11
SSEb
Adj. R2
Log Likelihood
.281
726.49
742.54
743.83
a c=.OOOOOlwas used in place of c=O.O in the translog model.
b SSE refers to sum of squared errors.
Table 4
Estimates of Fixed Costs and Cost Complementarity Effects
SCOPE,,: Economies of Scope Resulting from Fixed Cost Only
5Outout Model
4-Outout Model
2-Outout Model
.26 (.08)
.15 (.04)
.06 (.Ol)
QSCOPE,,: Economies of Scope Resulting from Complementarities Only
(Asymptotic Standard Errors in Parentheses)
Value of e
0 :O&SCOPE,,)
.OOl
.Ol
.05
.lO
.15
l20
5-Outnut Model
4-Output Model
2-Output Model
.20 (.08)
.24 (.lO)
.12 (.05)
.09 (.04)
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@FedFRASER