- Email: [email protected]
Branching restrictions and banking costs
Journal
of Banking
and Fmance
BRANCHING
14 (1990)
1151-1162.
RESTRICTIONS
North-Holland
AND BANKING
COSTS
Mark J. BUONO Federal Home Loan Mortgage Corporation, Reston, VA 22090, USA
B. Kelly EAKIN* University of Oregon, Eugene, OR 97403, USA Received
February
1989, final version
received November
1989
Previous research of banking costs has been limited by the choice of the functional form, irregularities in the estimated cost functions, and a failure to rigorously examine the difference between banks in branching states and banks in unit-banking states. This paper addresses these shortcomings. Banks are modelled as three-input-three-output cost minimizing firms. A three equation system is estimated using 1985 Functional Cost Analysis data. The findings indicate that banks in unit-banking states experience diseconomies of scale. Banks in branching states experience diseconomies of scale at the bank level but economies of scale at the branch level. We also find evidence of economies of scope for branch-state banks but not for unit-state banks.
1. Introduction
Increasing deregulation of the financial sector and innovations in the study of multiproduct firms have revived interest in the cost structure of the U.S. banking industry. This paper avoids several problems with existing cost studies of depository institutions and analyzes the effect of state branching restrictions on bank costs. To this end, we model the bank as a costminimizing, multiproduct firm, and estimate a system of total cost and factor share equations for a national sample of banks. Duality of the cost function allows us to infer information about production technology, including economies of scale and economies of scope. Four problems plague existing cost studies of depository institutions. First, these studies either ignore the multiproduct aspect of banks or acknowledge multiple outputs but employ a cost function that is inconsistent with multiproduct technology. Second, only the total cost equation is typically estimated rather than the system of cost and factor shares equations. Neglecting share information leads to inefficient estimates, and our evidence University of Oregon, Eugene OR *Correspondence address: Department of Economics, 97403, USA. Thanks are due to Joe Stone, Stephen Haynes and Chris James for their helpful comments. 0378-4266/90/%03.50
0
1990-Elsevier
Science Publishers
B.V. (North-Holland)
1152
M.J.
Buono and B.K.
Eakin,
Branching
restrictions
and banking
costs
suggests that this omission contributes to theoretical irregularities in the estimated cost function. Third, most studies employ incorrect measures for scope economies and fail to report approximate standard errors for technological measures. Finally, studies that do allow for technological differences between unit-state and branch-state banks do not analyze the cost structure differences beyond simply reporting the different estimates. . We avoid these problems in the following ways. In section 2 we model the bank as a cost-minimizing firm producing three outputs using three inputs. Our choice of a hybrid translog cost function allows us to fully investigate multiproduct technology. Using a varying-parameters structure we have the hypothesis of identical technologies for unit-state and branch-state banks as a nested case. With our specification we are also able to test formally for statistical differences in the relevant technological measures between the unitstate and branch-state banks. In section 3 we describe the data and the construction of the variables, discuss the estimation procedure, and present and interpret the results. In particular, we emphasize the regularity conditions, the relevant technological measures, and an examination of cost structure differences between branchstate and unit-state banks. We summarize our major conclusions in section 4.
2. Cost function and cost concepts Modelling depository institutions as multiproduct firms has been problematic. Clark (1984) and Hunter and Timme (1986) are examples of studies that ignore the multiproduct aspects of banks. They justify a single output approach by showing a high correlation between competing measures of output and demonstrating robustness of results across single output measures. However, their approach is correct only if outputs are produced in fixed proportions. Benston (1965) estimates separate cost functions for different banking services, implicitly assuming separability between bank outputs. His approach also has the problem of the ex ante allocation of costs associated with shared inputs. Benston et al. (1982) approach the multiple output issue by using a Divisia index to construct an aggregate measure of bank output. The effect on cost of altering the mix of outputs can not be investigated in an aggregate output model and separability between outputs is required for such an aggregation to be consistent.’ Benston et al. (1983), Gilligan et al. (1984), Murray and White (1983), Lawrence and Shay (1986), Mester (1987), and Berger et al. (1987) model banks as multiproduct firms, but, with the exception of Berger et al., use a translog cost function which ‘For (1978).
a discussion
of separability
and
its relationship
to aggregation,
see Blackorby
et al.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1153
lacks the flexibility to investigate economies of scope and related measures. Berger et al. estimate an overhead cost equation and a direct cost equation for each output which requires the ex ante allocation of costs associated with shared inputs. With these problems in mind, we posit a twice continuous, dual cost function of the general form
(1)
C=C(I:fqB),
where Y is a vector of M outputs, W is a vector of N input prices, and B is the number of branch offices. For empirical implementation we represent total cost with a hybrid translog function.2 The hybrid translog can be evaluated even if some of the output values are zero, which is a desirable property if the multiple output measures of scope are to be investigated. The hybrid translog cost function and the factor shares are
i=l j=l
I
+/JOB+
k=l
2 6&B+ f
i=l
?,ln w,B++p,B’
(2)
k=l
and M,=fi,+
; n=l
flk,ln
w,+ 5 nikx+rkB,
k=l,...,n,
(3)
i=l
where M, is the kth factor cost share. We are interested in several cost measures that describe technology. The measures of interest are marginal costs (MCiS), output cost elasticities (SCis), and overall economies of scale (SCE). We evaluate these measures at the ‘branch’ and ‘bank’ levels. Cost measures at the branch level are evaluated *Our translog
hybrid translog and the translog developed by Caves et al. (1980).
are special
cases
of the generalized
multiproduct
1154
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
holding the number of branches constant, while, at the bank level, the number of branches is allowed to vary as output changes.3 The measures are:
Ui+ F Uij~+ i
MCi=X/aY;:=C
j=1
rriklnW,+6,B
k=l
,
MC: = MCi+aB/dx,
SC,=LJlnC/aln
(4)
>
(5)
~=[~/C]MC,,
(6)
SC;= [x/C]MC:,
(7)
SCE=
(8)
5 Sci, i=l
SCE*=
;
SC:,
(9)
i=l
where an asterisk denotes the cost concept at the bank level. We are also interested in the measures developed in Baumol et al. (1982) to investigate multiproduct industries. We construct measures of average incremental costs (AZCis), partial economies of scope (SPis), and overall economies of scope (SPE). We evaluate the multiproduct measures at the mean level of branches, because we assume firms producing one or two of the outputs would have the same number of offtces as the consolidated firm. We make this assumption so that the level of service remains constant.4 The multiproduct measures are:
AZCi=[C(Y, w B)-C(YM_i,
w B)]/Y,,
(10)
Sf’i=[C(xyW,B)+C(YM-i,WBB)-C(I:wB)]/C(Y;PVB),
WE=
5 C(x,WB)-C(I:wB) i=l
Ii
(11)
C(y,wB),
(12)
where Y,_.i is the output vector with a zero replacing the quantity of the ith output. The subscript i refers to a single output in eqs. (10) and (12), and to ‘The (aB/aq)‘s are estimated from an auxiliary ‘See Berger et al. (1987, pp. 514-515).
regression
equation
as in Benston
et al. (1983).
M.J. Buono and B.K. Eakin, Branching restwtions
and banking costs
1155
any subset of outputs in eq. (11). We are able to evaluate the concepts given by eqs. (10)-(12) because of our choice of the hybrid translog cost function.5
3. The empirical results 3.1. The data
We use the 1985 Functional Cost Analysis (FCA) survey of 635 member banks of the Federal Reserve System to estimate the cost function. We delete banks with missing or inconsistent data leaving a sample of 613 banks. The data include 387 banks located in branching states and 226 banks located in unit-banking states.6 We model the bank as a three product firm. The outputs are loans (Y,), investments (Y,), and transactions deposits (YJ. We use the ‘intermediation approach’ by measuring each output in dollars.’ Our use of loans, investments, and transactions deposits to measure bank output is based on the classification of Hancock (1985). We construct Divisia index measures of loans and investments by weighting their components by their revenue shares. A Divisia index is not constructed for Y, because the necessary data for the construction are not available. We note here that there is considerable variation in the output ratios indicating that outputs are not produced in fixed proportions. This suggests that the single output approach of Clark (1984) and Hunter and Timme (1986) is inappropriate. Banks are assumed to use capital, labor, and time deposits and purchased funds, to produce the three bank outputs. The treatment of time deposits as an input differs from previous bank studies. * The price of time deposits and purchased funds is measured by a Divisia price index using cost shares to weight the components of time deposits and purchased funds. The price of labor is the average yearly wage rate based on salaries and employee benefits. The price of capital is measured by the sum of occupancy, furniture, and equipment expenses divided by the book value of the building and equipment, less depreciation reserves. The number of branches is the sum of the number of full-service offices sThe multiproduct translog cost function cannot be evaluated when any output is zero. Consequently, studies employing this functional form have not calculated scope economies, or have arbitrarily calculated scope economies by inserting a small value of the relevant output in place of a zero output level. 6A data appendix describing construction of the variables and descriptive statistics is available from the authors. ‘See Humphrey (1985) for a discussion of the difference between the ‘intermediation’ approach and the ‘production’ approach. aSavings and time deposits have been used as an input in cost studies of savings and loans [Mester (1987)] and credit unions [Murray and White (1983)].
1156
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
and the number of limited-service offtces.g Total cost is the sum of the costs associated with the three inputs. The factor cost shares are calculated as the fraction of total cost associated with each of the inputs.
3.2. Estimation We estimate the three equation system given by eqs. (2) and (3). We use a varying parameters structure to determine if restrictions on branching affect the choice of technology. We omit the share equation for capital in the estimation without any loss of economic information because only two of the cost shares are independent. We assume the logarithm of total cost and the input cost shares have additive disturbance terms, which are assumed to come from a joint normal distribution with non-zero correlation for a particular bank, but with zero correlation across different banks. We obtain our estimates of the cost system via the method of iterative seemingly unrelated regressions. Parameter estimates are not presented here but are available upon request. We reject the null hypothesis of identical technologies for branch-state and unit-state banks.” Differences in parameter estimates have also been observed by Benston et al. (1982) and Gilligan et al. (1984). We also reject the hypothesis of no branching effects and the hypothesis of separability among outputs. ‘I The last finding suggests that estimating separate cost functions for different outputs is inappropriate and that the use of an aggregate output measure is inconsistent.
3.3. Regularity conditions A cost function dual to a well-behaved multiproduct production function inherits several properties called regularity conditions. To be regular the cost function must be continuous, twice differentiable, and non-negative. These conditions are guaranteed by our choice of the hybrid translog function. Linear homogeneity in input prices and symmetry are imposed by parameter restrictions. The regularity conditions of monotonicity (positive input demands), concavity, and non-negative marginal costs, however, cannot be ‘Using the natural logarithm of the number of branches instead of the number of branches does not affect the qualitative results that follow but does sightly reduce the fit of the cost system. “‘The likelihood ratio test of identical technologies yields a chi-squared statistic of 327.53 with 28 degrees of freedom. “The likelihood ratio test of no branch effects yields a chi-squared statistic of 74.73 with nine degrees of freedom. The likelihood ratio test of the restrictions implied by separability yields a chi-squared stastistic of 25.26 with six degrees of freedom.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1157
parsimoniously imposed. These conditions must be verified at the means of the data and at each data point. If irregularities are common, or occur at the means of the data, then the estimated cost function cannot be a dual cost function and inferring technological information is not theoretically justified. Despite the fundamental importance of regularity, previous banking studies typically ignore this issue.” Our estimated cost functions satisfy all the regularity conditions at the means of the relevant sample. All regularity conditions are satisfied for 564 of the 613 observations. Estimates of the demand for time deposits and purchased funds are positive for all 613 observations, while the estimated demands for labor and capital are each positive for 610 and 611 observations, respectively. Concavity holds for more than 97% of the observations. Most irregular observations result from a negative estimate for the marginal cost of transactions deposits (26 of the 49 violations). The very high percentage of observations satisfying the regularity conditions gives us reason to conclude that the total cost function is dual to a well-behaved production technology for both unit-state banks and branchstate banks. In contrast to this study, single-equation models of banks have often resulted in irregular cost functions.13 Estimating only the cost equation is inefficient because of unused information, but estimates are unbiased and consistent. Guilkey et al. (1983) do not find much efficiency gain when adding the share equations to the estimation. However, they only introduce moderate levels of multicollinearity into their Monte Carlo experiments. In our data the correlation coefficients for output pairs are 0.68, 0.25, and 0.59. Including the cost shares introduces no new parameters but does give additional degrees of freedom, which are helpful in reducing the ill effects of multicollinearity. We examine the consequences of estimating only the cost equation by estimating eq. (2) without the factor shares. When we estimate only the cost equation, not all of the regularity conditions are satisfied at the means of either the branch-state or unit-state samples. 53% of the unit-state bank sample and only 16% of the branch-state sample satisfy all of the regularity conditions. Several of the cost concepts have significantly different values when calculated from the cost equation estimates that where obtained by the estimation of only the total cost equation than when calculated from the cost system estimates. This finding reduces our confidence in analyses of scale and scope that are based on only estimating the cost function. ‘*Of the cost studies reviewed in this paper only Benston et al. (1982,1983), Murray and White (1983), Clark (1984) and Berger et al. (1987) mention regularity conditions implied by duality. r3Benston et al. (1982,1983), Clark (1984) and Berger et at. (1987) note cost function irregularities. Gilligan et al. (1984) do not discuss regularity conditions, but their parameter estimates strongly suggest that monotonicity and concavity do not hold in their study.
1158
M.J.
Buono and B.K.
Eakin,
Branching
restrictions
and banking
costs
Table 1 Marginal costs.’ Branch-state banks
Unit-state banks
Marginal cost (!I)
Marginal cost (S)
Difference
Branch level
Bank level
Branch level
Bank level
Branch level
Loans
0.040* (0.0050)
0.065* (0.0054)
0.176 (0.0365)
0.175’ (0.0364)
0.136* (0.0368)
0.110, (0.0368)
Investments
0.141* (0.0195)
0.121* (0.0203)
0.169* (0.0377)
0.167’ (0.0378)
0.028 (0.0424)
0.046 (0.0429)
Transactions deposits
0.106’ (0.0150)
0.202; (0.0158)
0.176; (0.0230)
0.188* (0.0237)
0.070* (0.0275)
-0.014 (0.0286)
output
‘An * denotes significance at the 1% level. The approximate parentheses.
Bank level
standard errors are given in
3.4. Production technology 3.4.1.
Marginal costs
We present marginal cost estimates in table 1. The marginal costs for branch-state banks are generally lower than those of unit-state banks. However, only the marginal cost estimate of transactions deposits at the branch level and the marginal cost estimates for loans are significantly lower than their corresponding estimates for unit-state banks. The marginal cost of transactions deposits suggests that branch-state banks should expand deposits at the existing branches rather than expanding deposits via new branches. 3.4.2. Output cost elasticities and overall economies of scale We report our estimates of output cost elasticities and overall economies of scale in table 2. The output cost elasticities sum to a measure of overall economies of scale, which indicates the percentage increase in costs if all outputs were increased by 1%. SCE measures scale economies while holding the number of branches constant and SCE* allows for the number of branches to expand along with output. The SCEs indicate that the typical unit-state bank experiences diseconomies of scale while the typical branchstate bank experiences economies of scale at the branch level and diseconomies of scale at the bank level. These findings are at odds with those of Berger et al. (1987) who find neither economies nor diseconomies of scale for either the branch-state or unit-state bank samples.14 Diseconomies of scale for unit-state banks suggest that branching restrici4Berger et al. (1987) use the 1983 FCA data while we use the 1985 FCA data. There were a number of regulatory changes over this period. Our study also differs from their’s in the choice of bank inputs and outputs.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1159
Table 2 Output cost elasticities and overall economies of scale.’ Branch-state banks
Unit-state banks
Difference
output cost elasticities
Branch level
Bank level
Branch level
Bank level
Branch level
Bank level
Loans
0.221* (0.0262)
0.363* (0.0272)
0.392* (0.0752)
0.390: (0.0752)
0.171** (0.0796)
0.027 (0.0799)
Investments
0.3018 (0.0411)
0.259* (0.0430)
0.335* (0.0719)
0.332* (0.0721)
0.034 (0.0828)
0.073 (0.0839)
Transactions deposits
0.321; (0.0440)
0.610* (0.0429)
0.615* (0.0786)
0.656* (0.0798)
0.294* (0.090)
0.046 (0.0906)
Overall economies of scale
0.843*** (0.0548)
1.231*** (0.0322)
1.342*** (0.0570)
1.378*** (0.0558)
0.499* (0.079)
0.147** (0.0644)
“An * denotes significance at the 1% level, l * denotes significance at the 5% level, and *** denotes significant difference from one at the 1% level. Approximate standard errors are given in parentheses. A scale measure greater (less) than one indicates diseconomies (economies) of scale.
tions result in banks that are inefficiently large for only having one full service branch. The scale measure for unit-state banks at the bank level indicates that expanding the number of limited-service branches does not reduce the scale inefficiency. The existence of such scale inefficiency is circumstantial evidence of effective entry barriers. In unit states, branching restrictions may help enforce an implicit cartel arrangement by greatly increasing the cost of expansion by existing banks. On the other hand, barriers to entry for new banks might exist in branching states, but a more competitive outcome may prevail because of the low cost of expansion by existing banks. The scale measures in table 2 imply that banks in branching states should expand output at existing branches rather than expand output by increasing the number of branches. This finding is consistent with the fact that many large banks in branching states have reduced the number of their offices since 1985. 3.4.3. Average incremental costs Average incremental costs indicate the extra cost per unit of producing a certain amount of one of the outputs versus not producing that output, while holding the level of the other outputs constant. AICs are required for the calculation of economies of scope. Table 3 presents the MC estimates.” We find the AIC estimates are lower for the branch-state banks relative to the unit-state banks. For both the unit-state and branch-state banks the AlCs are somewhat less than the corresponding marginal costs, indicating product 15All of the regularity conditions are satisfied at all the points used to calculate the average incremental costs and overall and partial economies of scope for the branch-state and unit-state banks reported in tables 3 and 4.
1160
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
Table 3 Average incremental costs (AIC). Average incremental cost Loans
Branch-state banks
Unit-state banks
Difference
0.016* (O.ooo2)
0.069* (0.0127)
0.0.53* (0.0128)
,s:zr,
0.059* (0.0070)
0.015* (0.0076)
0.094* (0.0034)
0.137* (0.0039)
0.042* (0.0051)
Investments Trnsactions deposits
‘An * denotes significance at the 1% level. Approximate standard errors are given in parentheses. Table 4 Scope economies.” Product sets CYII, cy21,c~31
CY 11,CYZY31
CY21,CYl, Y33 EY31,EYl, Y21
Branch state banks
Unit-state banks
Difference
Overall economies of scope 0.642* 0.092 (0.0682) (0.063 1)
-0.550* (0.0929)
of scope 0.002 (0.0466) - 0.055 (0.0510) - 0.023 10.0778)
-0.3151 (0.0601) -0.331* (0.0646) -0.317* 10.0896)
Partial economies 0.318* (0.0379) 0.276* (0.0396) 0.294* 10.04451
“An * denotes significance at the 1% level. Approximate standard errors are given in parentheses. diseconomies of scale. We note here that the typical branch-state bank does experience product-set specific diseconomies of scale, yet experiences overall economies of scale at the branch level. This results from scope economies.
specific
3.4.4. Economies of scope The various measures of economies of scope are presented in table 4. The measure of overall economies of scope indicates the percentage cost saving due to the current joint production compared with complete specialization. This is about 9.2% for unit-state banks and about 64% for branch-state banksal Our results differ greatly from those of Berger et al. (1987) who find ‘%ome previous studies, e.g., Murray and White (1983), associate interproduct cost compiementarities with economies of mope. Interproduct cost compiementa~ti~ at all possible data points are sufficient for economies of scope, but the finding of local cost complementarities is neither sufficient nor necessary. We find no evidence of cost complementarities for either sample.
M.J. Buono and B.K. Eakin, Branching
restrictions
and banking
costs
1161
evidence of diseconomies of scope for branch-state banks. Their results are similar to ours with respect to unit-state banks. Our results suggest branching restrictions may prevent banks from fully exploiting scope economies. As shown in Baumol et al. (1982), decreasing AZCs for each product along with economies of scope are sufficient for subadditivity. Subadditivity is the necessary and sufficient condition for a natural monopoly in a multiproduct industry. While we do find evidence of economies of scope, as reported in table 4, the marginal costs in table 1 imply increasing AICs. The existence of increasing AICs suggests that banking is not a natural monopoly.
4. Summary and conclusions
In this paper we addressed several problems in the existing literature on the cost structure of depository institutions. We chose the hybrid translog cost function so we could undertake a more complete analysis of scope economies than was possible in earlier studies which used the less flexible translog function. By estimating a system of cost and share equations, we avoided the theoretical irregularities and insignificance of parameter estimates common to the single-equation studies of banking. We also investigated the impact that branching restrictions have on bank cost structures. We added to the analysis of cost structure differences by calculating the statistical significance of these differences, and also by paying careful attention to regularity conditions, which assured us we were measuring the technological concepts. We found that unit-state and branch-state banks do indeed have different technologies. Comparing these different technologies, we concluded that branch-state banks enjoy economies of scope, while the scope measure for unit-state banks is not statistically significant. Increasing average incremental costs imply that banking is not a natural monopoly. We also found diseconomies of scale at the means of the unit-bank sample and at the bank level for banks in branching states. However, we found that banks in branching states have economies of scale at the branch level. The scale estimates suggest that branch-state banks should either increase output at existing branches or decrease the number of branches.
References Baumol, W., J. Pantar and R. Willig, 1982, Contestable markets and the theory of industry structure (Harcourt, Brace, Jovanovich, New York). Benston, G., 1965, Economies of scale and marginal costs in banking operations, National Bank Review 2, 507-549.
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M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
Benston, G., A. Berger, G. Hanweck and D. Humphrey, 1983, Economies of scale and scope in banking, Research papers in banking and financial economics (Board of Governors of the Federal Reserve System). Benston, G., G. Hanweck and D. Humphrey, 1982, Scale economies in banking: A restructuring and reassessment, Journal of Money, Credit, and Banking 14, 435-456. Berger, A., G. Hanweck and D. Humphrey, 1987, Competitive viability in banking: Scale, scope, and product mix economies, Journal of Monetary Economics 20, 501-520. Blackorby, C., D. Primont and R. Russell, 1978, Duality, separability and functional structure: Theory and economic applications (North-Holland, Amsterdam). Caves, D., L. Christensen and M. Tretheway, 1980, Flexible cost functions for multiproduct firms, Review of Economics and Statistics 62, 477-481. Clark, J., 1984, Estimation of economies of scale in banking using a generalized functional form, Journal of Money, Credit, and Banking 16, 53-68. _ Gilligan, T., M. Smirlock and W. Marshall, 1984, Scale and scope economies in the multiproduct banking firm, Journal of Monetary Economics 13, 393-405. Guilkey, D., C.A.K. Lovell and R. Sickles, 1983, A comparison of the performance of three flexible functional forms. International Economic Review 24, 591-616. Hancock, D., 1985, The financial firm: Production with monetary and nonmonetary goods, Journal of Political Economy 93, 859-880. Humphrey, D., 1985, Costs and scale economies in banking intermediation, in: R. Aspinwall and R. Eisenbeis, eds., Handbook for banking strategy (Wiley, New York). Hunter, W. and S. Timme, 1986, Technical change, organizational form, and the structure of bank oroduction, Journal of Money, Credit, and Banking 18, 152-166. Lawrence; C. and R. Shay, 1986, Technology and linancial intermediation in a multiproduct bankine firm: An econometric studv of U.S. banks. 1979-1982, in: C. Lawrence and R. Shay, eds., T&hnological innovation, regulation and the monetary economy (Ballinger, MA). Mester, L., 1987, A multiproduct cost study of savings and loans, Journal of Finance 42, 423-445. Murray, J. and R. White, 1983, Economies of scale and economies of scope in multiproduct financial institutions: A study of British Columbia credit unions, Journal of Finance 38, 887-902.
of Banking
and Fmance
BRANCHING
14 (1990)
1151-1162.
RESTRICTIONS
North-Holland
AND BANKING
COSTS
Mark J. BUONO Federal Home Loan Mortgage Corporation, Reston, VA 22090, USA
B. Kelly EAKIN* University of Oregon, Eugene, OR 97403, USA Received
February
1989, final version
received November
1989
Previous research of banking costs has been limited by the choice of the functional form, irregularities in the estimated cost functions, and a failure to rigorously examine the difference between banks in branching states and banks in unit-banking states. This paper addresses these shortcomings. Banks are modelled as three-input-three-output cost minimizing firms. A three equation system is estimated using 1985 Functional Cost Analysis data. The findings indicate that banks in unit-banking states experience diseconomies of scale. Banks in branching states experience diseconomies of scale at the bank level but economies of scale at the branch level. We also find evidence of economies of scope for branch-state banks but not for unit-state banks.
1. Introduction
Increasing deregulation of the financial sector and innovations in the study of multiproduct firms have revived interest in the cost structure of the U.S. banking industry. This paper avoids several problems with existing cost studies of depository institutions and analyzes the effect of state branching restrictions on bank costs. To this end, we model the bank as a costminimizing, multiproduct firm, and estimate a system of total cost and factor share equations for a national sample of banks. Duality of the cost function allows us to infer information about production technology, including economies of scale and economies of scope. Four problems plague existing cost studies of depository institutions. First, these studies either ignore the multiproduct aspect of banks or acknowledge multiple outputs but employ a cost function that is inconsistent with multiproduct technology. Second, only the total cost equation is typically estimated rather than the system of cost and factor shares equations. Neglecting share information leads to inefficient estimates, and our evidence University of Oregon, Eugene OR *Correspondence address: Department of Economics, 97403, USA. Thanks are due to Joe Stone, Stephen Haynes and Chris James for their helpful comments. 0378-4266/90/%03.50
0
1990-Elsevier
Science Publishers
B.V. (North-Holland)
1152
M.J.
Buono and B.K.
Eakin,
Branching
restrictions
and banking
costs
suggests that this omission contributes to theoretical irregularities in the estimated cost function. Third, most studies employ incorrect measures for scope economies and fail to report approximate standard errors for technological measures. Finally, studies that do allow for technological differences between unit-state and branch-state banks do not analyze the cost structure differences beyond simply reporting the different estimates. . We avoid these problems in the following ways. In section 2 we model the bank as a cost-minimizing firm producing three outputs using three inputs. Our choice of a hybrid translog cost function allows us to fully investigate multiproduct technology. Using a varying-parameters structure we have the hypothesis of identical technologies for unit-state and branch-state banks as a nested case. With our specification we are also able to test formally for statistical differences in the relevant technological measures between the unitstate and branch-state banks. In section 3 we describe the data and the construction of the variables, discuss the estimation procedure, and present and interpret the results. In particular, we emphasize the regularity conditions, the relevant technological measures, and an examination of cost structure differences between branchstate and unit-state banks. We summarize our major conclusions in section 4.
2. Cost function and cost concepts Modelling depository institutions as multiproduct firms has been problematic. Clark (1984) and Hunter and Timme (1986) are examples of studies that ignore the multiproduct aspects of banks. They justify a single output approach by showing a high correlation between competing measures of output and demonstrating robustness of results across single output measures. However, their approach is correct only if outputs are produced in fixed proportions. Benston (1965) estimates separate cost functions for different banking services, implicitly assuming separability between bank outputs. His approach also has the problem of the ex ante allocation of costs associated with shared inputs. Benston et al. (1982) approach the multiple output issue by using a Divisia index to construct an aggregate measure of bank output. The effect on cost of altering the mix of outputs can not be investigated in an aggregate output model and separability between outputs is required for such an aggregation to be consistent.’ Benston et al. (1983), Gilligan et al. (1984), Murray and White (1983), Lawrence and Shay (1986), Mester (1987), and Berger et al. (1987) model banks as multiproduct firms, but, with the exception of Berger et al., use a translog cost function which ‘For (1978).
a discussion
of separability
and
its relationship
to aggregation,
see Blackorby
et al.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1153
lacks the flexibility to investigate economies of scope and related measures. Berger et al. estimate an overhead cost equation and a direct cost equation for each output which requires the ex ante allocation of costs associated with shared inputs. With these problems in mind, we posit a twice continuous, dual cost function of the general form
(1)
C=C(I:fqB),
where Y is a vector of M outputs, W is a vector of N input prices, and B is the number of branch offices. For empirical implementation we represent total cost with a hybrid translog function.2 The hybrid translog can be evaluated even if some of the output values are zero, which is a desirable property if the multiple output measures of scope are to be investigated. The hybrid translog cost function and the factor shares are
i=l j=l
I
+/JOB+
k=l
2 6&B+ f
i=l
?,ln w,B++p,B’
(2)
k=l
and M,=fi,+
; n=l
flk,ln
w,+ 5 nikx+rkB,
k=l,...,n,
(3)
i=l
where M, is the kth factor cost share. We are interested in several cost measures that describe technology. The measures of interest are marginal costs (MCiS), output cost elasticities (SCis), and overall economies of scale (SCE). We evaluate these measures at the ‘branch’ and ‘bank’ levels. Cost measures at the branch level are evaluated *Our translog
hybrid translog and the translog developed by Caves et al. (1980).
are special
cases
of the generalized
multiproduct
1154
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
holding the number of branches constant, while, at the bank level, the number of branches is allowed to vary as output changes.3 The measures are:
Ui+ F Uij~+ i
MCi=X/aY;:=C
j=1
rriklnW,+6,B
k=l
,
MC: = MCi+aB/dx,
SC,=LJlnC/aln
(4)
>
(5)
~=[~/C]MC,,
(6)
SC;= [x/C]MC:,
(7)
SCE=
(8)
5 Sci, i=l
SCE*=
;
SC:,
(9)
i=l
where an asterisk denotes the cost concept at the bank level. We are also interested in the measures developed in Baumol et al. (1982) to investigate multiproduct industries. We construct measures of average incremental costs (AZCis), partial economies of scope (SPis), and overall economies of scope (SPE). We evaluate the multiproduct measures at the mean level of branches, because we assume firms producing one or two of the outputs would have the same number of offtces as the consolidated firm. We make this assumption so that the level of service remains constant.4 The multiproduct measures are:
AZCi=[C(Y, w B)-C(YM_i,
w B)]/Y,,
(10)
Sf’i=[C(xyW,B)+C(YM-i,WBB)-C(I:wB)]/C(Y;PVB),
WE=
5 C(x,WB)-C(I:wB) i=l
Ii
(11)
C(y,wB),
(12)
where Y,_.i is the output vector with a zero replacing the quantity of the ith output. The subscript i refers to a single output in eqs. (10) and (12), and to ‘The (aB/aq)‘s are estimated from an auxiliary ‘See Berger et al. (1987, pp. 514-515).
regression
equation
as in Benston
et al. (1983).
M.J. Buono and B.K. Eakin, Branching restwtions
and banking costs
1155
any subset of outputs in eq. (11). We are able to evaluate the concepts given by eqs. (10)-(12) because of our choice of the hybrid translog cost function.5
3. The empirical results 3.1. The data
We use the 1985 Functional Cost Analysis (FCA) survey of 635 member banks of the Federal Reserve System to estimate the cost function. We delete banks with missing or inconsistent data leaving a sample of 613 banks. The data include 387 banks located in branching states and 226 banks located in unit-banking states.6 We model the bank as a three product firm. The outputs are loans (Y,), investments (Y,), and transactions deposits (YJ. We use the ‘intermediation approach’ by measuring each output in dollars.’ Our use of loans, investments, and transactions deposits to measure bank output is based on the classification of Hancock (1985). We construct Divisia index measures of loans and investments by weighting their components by their revenue shares. A Divisia index is not constructed for Y, because the necessary data for the construction are not available. We note here that there is considerable variation in the output ratios indicating that outputs are not produced in fixed proportions. This suggests that the single output approach of Clark (1984) and Hunter and Timme (1986) is inappropriate. Banks are assumed to use capital, labor, and time deposits and purchased funds, to produce the three bank outputs. The treatment of time deposits as an input differs from previous bank studies. * The price of time deposits and purchased funds is measured by a Divisia price index using cost shares to weight the components of time deposits and purchased funds. The price of labor is the average yearly wage rate based on salaries and employee benefits. The price of capital is measured by the sum of occupancy, furniture, and equipment expenses divided by the book value of the building and equipment, less depreciation reserves. The number of branches is the sum of the number of full-service offices sThe multiproduct translog cost function cannot be evaluated when any output is zero. Consequently, studies employing this functional form have not calculated scope economies, or have arbitrarily calculated scope economies by inserting a small value of the relevant output in place of a zero output level. 6A data appendix describing construction of the variables and descriptive statistics is available from the authors. ‘See Humphrey (1985) for a discussion of the difference between the ‘intermediation’ approach and the ‘production’ approach. aSavings and time deposits have been used as an input in cost studies of savings and loans [Mester (1987)] and credit unions [Murray and White (1983)].
1156
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
and the number of limited-service offtces.g Total cost is the sum of the costs associated with the three inputs. The factor cost shares are calculated as the fraction of total cost associated with each of the inputs.
3.2. Estimation We estimate the three equation system given by eqs. (2) and (3). We use a varying parameters structure to determine if restrictions on branching affect the choice of technology. We omit the share equation for capital in the estimation without any loss of economic information because only two of the cost shares are independent. We assume the logarithm of total cost and the input cost shares have additive disturbance terms, which are assumed to come from a joint normal distribution with non-zero correlation for a particular bank, but with zero correlation across different banks. We obtain our estimates of the cost system via the method of iterative seemingly unrelated regressions. Parameter estimates are not presented here but are available upon request. We reject the null hypothesis of identical technologies for branch-state and unit-state banks.” Differences in parameter estimates have also been observed by Benston et al. (1982) and Gilligan et al. (1984). We also reject the hypothesis of no branching effects and the hypothesis of separability among outputs. ‘I The last finding suggests that estimating separate cost functions for different outputs is inappropriate and that the use of an aggregate output measure is inconsistent.
3.3. Regularity conditions A cost function dual to a well-behaved multiproduct production function inherits several properties called regularity conditions. To be regular the cost function must be continuous, twice differentiable, and non-negative. These conditions are guaranteed by our choice of the hybrid translog function. Linear homogeneity in input prices and symmetry are imposed by parameter restrictions. The regularity conditions of monotonicity (positive input demands), concavity, and non-negative marginal costs, however, cannot be ‘Using the natural logarithm of the number of branches instead of the number of branches does not affect the qualitative results that follow but does sightly reduce the fit of the cost system. “‘The likelihood ratio test of identical technologies yields a chi-squared statistic of 327.53 with 28 degrees of freedom. “The likelihood ratio test of no branch effects yields a chi-squared statistic of 74.73 with nine degrees of freedom. The likelihood ratio test of the restrictions implied by separability yields a chi-squared stastistic of 25.26 with six degrees of freedom.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1157
parsimoniously imposed. These conditions must be verified at the means of the data and at each data point. If irregularities are common, or occur at the means of the data, then the estimated cost function cannot be a dual cost function and inferring technological information is not theoretically justified. Despite the fundamental importance of regularity, previous banking studies typically ignore this issue.” Our estimated cost functions satisfy all the regularity conditions at the means of the relevant sample. All regularity conditions are satisfied for 564 of the 613 observations. Estimates of the demand for time deposits and purchased funds are positive for all 613 observations, while the estimated demands for labor and capital are each positive for 610 and 611 observations, respectively. Concavity holds for more than 97% of the observations. Most irregular observations result from a negative estimate for the marginal cost of transactions deposits (26 of the 49 violations). The very high percentage of observations satisfying the regularity conditions gives us reason to conclude that the total cost function is dual to a well-behaved production technology for both unit-state banks and branchstate banks. In contrast to this study, single-equation models of banks have often resulted in irregular cost functions.13 Estimating only the cost equation is inefficient because of unused information, but estimates are unbiased and consistent. Guilkey et al. (1983) do not find much efficiency gain when adding the share equations to the estimation. However, they only introduce moderate levels of multicollinearity into their Monte Carlo experiments. In our data the correlation coefficients for output pairs are 0.68, 0.25, and 0.59. Including the cost shares introduces no new parameters but does give additional degrees of freedom, which are helpful in reducing the ill effects of multicollinearity. We examine the consequences of estimating only the cost equation by estimating eq. (2) without the factor shares. When we estimate only the cost equation, not all of the regularity conditions are satisfied at the means of either the branch-state or unit-state samples. 53% of the unit-state bank sample and only 16% of the branch-state sample satisfy all of the regularity conditions. Several of the cost concepts have significantly different values when calculated from the cost equation estimates that where obtained by the estimation of only the total cost equation than when calculated from the cost system estimates. This finding reduces our confidence in analyses of scale and scope that are based on only estimating the cost function. ‘*Of the cost studies reviewed in this paper only Benston et al. (1982,1983), Murray and White (1983), Clark (1984) and Berger et al. (1987) mention regularity conditions implied by duality. r3Benston et al. (1982,1983), Clark (1984) and Berger et at. (1987) note cost function irregularities. Gilligan et al. (1984) do not discuss regularity conditions, but their parameter estimates strongly suggest that monotonicity and concavity do not hold in their study.
1158
M.J.
Buono and B.K.
Eakin,
Branching
restrictions
and banking
costs
Table 1 Marginal costs.’ Branch-state banks
Unit-state banks
Marginal cost (!I)
Marginal cost (S)
Difference
Branch level
Bank level
Branch level
Bank level
Branch level
Loans
0.040* (0.0050)
0.065* (0.0054)
0.176 (0.0365)
0.175’ (0.0364)
0.136* (0.0368)
0.110, (0.0368)
Investments
0.141* (0.0195)
0.121* (0.0203)
0.169* (0.0377)
0.167’ (0.0378)
0.028 (0.0424)
0.046 (0.0429)
Transactions deposits
0.106’ (0.0150)
0.202; (0.0158)
0.176; (0.0230)
0.188* (0.0237)
0.070* (0.0275)
-0.014 (0.0286)
output
‘An * denotes significance at the 1% level. The approximate parentheses.
Bank level
standard errors are given in
3.4. Production technology 3.4.1.
Marginal costs
We present marginal cost estimates in table 1. The marginal costs for branch-state banks are generally lower than those of unit-state banks. However, only the marginal cost estimate of transactions deposits at the branch level and the marginal cost estimates for loans are significantly lower than their corresponding estimates for unit-state banks. The marginal cost of transactions deposits suggests that branch-state banks should expand deposits at the existing branches rather than expanding deposits via new branches. 3.4.2. Output cost elasticities and overall economies of scale We report our estimates of output cost elasticities and overall economies of scale in table 2. The output cost elasticities sum to a measure of overall economies of scale, which indicates the percentage increase in costs if all outputs were increased by 1%. SCE measures scale economies while holding the number of branches constant and SCE* allows for the number of branches to expand along with output. The SCEs indicate that the typical unit-state bank experiences diseconomies of scale while the typical branchstate bank experiences economies of scale at the branch level and diseconomies of scale at the bank level. These findings are at odds with those of Berger et al. (1987) who find neither economies nor diseconomies of scale for either the branch-state or unit-state bank samples.14 Diseconomies of scale for unit-state banks suggest that branching restrici4Berger et al. (1987) use the 1983 FCA data while we use the 1985 FCA data. There were a number of regulatory changes over this period. Our study also differs from their’s in the choice of bank inputs and outputs.
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
1159
Table 2 Output cost elasticities and overall economies of scale.’ Branch-state banks
Unit-state banks
Difference
output cost elasticities
Branch level
Bank level
Branch level
Bank level
Branch level
Bank level
Loans
0.221* (0.0262)
0.363* (0.0272)
0.392* (0.0752)
0.390: (0.0752)
0.171** (0.0796)
0.027 (0.0799)
Investments
0.3018 (0.0411)
0.259* (0.0430)
0.335* (0.0719)
0.332* (0.0721)
0.034 (0.0828)
0.073 (0.0839)
Transactions deposits
0.321; (0.0440)
0.610* (0.0429)
0.615* (0.0786)
0.656* (0.0798)
0.294* (0.090)
0.046 (0.0906)
Overall economies of scale
0.843*** (0.0548)
1.231*** (0.0322)
1.342*** (0.0570)
1.378*** (0.0558)
0.499* (0.079)
0.147** (0.0644)
“An * denotes significance at the 1% level, l * denotes significance at the 5% level, and *** denotes significant difference from one at the 1% level. Approximate standard errors are given in parentheses. A scale measure greater (less) than one indicates diseconomies (economies) of scale.
tions result in banks that are inefficiently large for only having one full service branch. The scale measure for unit-state banks at the bank level indicates that expanding the number of limited-service branches does not reduce the scale inefficiency. The existence of such scale inefficiency is circumstantial evidence of effective entry barriers. In unit states, branching restrictions may help enforce an implicit cartel arrangement by greatly increasing the cost of expansion by existing banks. On the other hand, barriers to entry for new banks might exist in branching states, but a more competitive outcome may prevail because of the low cost of expansion by existing banks. The scale measures in table 2 imply that banks in branching states should expand output at existing branches rather than expand output by increasing the number of branches. This finding is consistent with the fact that many large banks in branching states have reduced the number of their offices since 1985. 3.4.3. Average incremental costs Average incremental costs indicate the extra cost per unit of producing a certain amount of one of the outputs versus not producing that output, while holding the level of the other outputs constant. AICs are required for the calculation of economies of scope. Table 3 presents the MC estimates.” We find the AIC estimates are lower for the branch-state banks relative to the unit-state banks. For both the unit-state and branch-state banks the AlCs are somewhat less than the corresponding marginal costs, indicating product 15All of the regularity conditions are satisfied at all the points used to calculate the average incremental costs and overall and partial economies of scope for the branch-state and unit-state banks reported in tables 3 and 4.
1160
M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
Table 3 Average incremental costs (AIC). Average incremental cost Loans
Branch-state banks
Unit-state banks
Difference
0.016* (O.ooo2)
0.069* (0.0127)
0.0.53* (0.0128)
,s:zr,
0.059* (0.0070)
0.015* (0.0076)
0.094* (0.0034)
0.137* (0.0039)
0.042* (0.0051)
Investments Trnsactions deposits
‘An * denotes significance at the 1% level. Approximate standard errors are given in parentheses. Table 4 Scope economies.” Product sets CYII, cy21,c~31
CY 11,CYZY31
CY21,CYl, Y33 EY31,EYl, Y21
Branch state banks
Unit-state banks
Difference
Overall economies of scope 0.642* 0.092 (0.0682) (0.063 1)
-0.550* (0.0929)
of scope 0.002 (0.0466) - 0.055 (0.0510) - 0.023 10.0778)
-0.3151 (0.0601) -0.331* (0.0646) -0.317* 10.0896)
Partial economies 0.318* (0.0379) 0.276* (0.0396) 0.294* 10.04451
“An * denotes significance at the 1% level. Approximate standard errors are given in parentheses. diseconomies of scale. We note here that the typical branch-state bank does experience product-set specific diseconomies of scale, yet experiences overall economies of scale at the branch level. This results from scope economies.
specific
3.4.4. Economies of scope The various measures of economies of scope are presented in table 4. The measure of overall economies of scope indicates the percentage cost saving due to the current joint production compared with complete specialization. This is about 9.2% for unit-state banks and about 64% for branch-state banksal Our results differ greatly from those of Berger et al. (1987) who find ‘%ome previous studies, e.g., Murray and White (1983), associate interproduct cost compiementarities with economies of mope. Interproduct cost compiementa~ti~ at all possible data points are sufficient for economies of scope, but the finding of local cost complementarities is neither sufficient nor necessary. We find no evidence of cost complementarities for either sample.
M.J. Buono and B.K. Eakin, Branching
restrictions
and banking
costs
1161
evidence of diseconomies of scope for branch-state banks. Their results are similar to ours with respect to unit-state banks. Our results suggest branching restrictions may prevent banks from fully exploiting scope economies. As shown in Baumol et al. (1982), decreasing AZCs for each product along with economies of scope are sufficient for subadditivity. Subadditivity is the necessary and sufficient condition for a natural monopoly in a multiproduct industry. While we do find evidence of economies of scope, as reported in table 4, the marginal costs in table 1 imply increasing AICs. The existence of increasing AICs suggests that banking is not a natural monopoly.
4. Summary and conclusions
In this paper we addressed several problems in the existing literature on the cost structure of depository institutions. We chose the hybrid translog cost function so we could undertake a more complete analysis of scope economies than was possible in earlier studies which used the less flexible translog function. By estimating a system of cost and share equations, we avoided the theoretical irregularities and insignificance of parameter estimates common to the single-equation studies of banking. We also investigated the impact that branching restrictions have on bank cost structures. We added to the analysis of cost structure differences by calculating the statistical significance of these differences, and also by paying careful attention to regularity conditions, which assured us we were measuring the technological concepts. We found that unit-state and branch-state banks do indeed have different technologies. Comparing these different technologies, we concluded that branch-state banks enjoy economies of scope, while the scope measure for unit-state banks is not statistically significant. Increasing average incremental costs imply that banking is not a natural monopoly. We also found diseconomies of scale at the means of the unit-bank sample and at the bank level for banks in branching states. However, we found that banks in branching states have economies of scale at the branch level. The scale estimates suggest that branch-state banks should either increase output at existing branches or decrease the number of branches.
References Baumol, W., J. Pantar and R. Willig, 1982, Contestable markets and the theory of industry structure (Harcourt, Brace, Jovanovich, New York). Benston, G., 1965, Economies of scale and marginal costs in banking operations, National Bank Review 2, 507-549.
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M.J. Buono and B.K. Eakin, Branching restrictions and banking costs
Benston, G., A. Berger, G. Hanweck and D. Humphrey, 1983, Economies of scale and scope in banking, Research papers in banking and financial economics (Board of Governors of the Federal Reserve System). Benston, G., G. Hanweck and D. Humphrey, 1982, Scale economies in banking: A restructuring and reassessment, Journal of Money, Credit, and Banking 14, 435-456. Berger, A., G. Hanweck and D. Humphrey, 1987, Competitive viability in banking: Scale, scope, and product mix economies, Journal of Monetary Economics 20, 501-520. Blackorby, C., D. Primont and R. Russell, 1978, Duality, separability and functional structure: Theory and economic applications (North-Holland, Amsterdam). Caves, D., L. Christensen and M. Tretheway, 1980, Flexible cost functions for multiproduct firms, Review of Economics and Statistics 62, 477-481. Clark, J., 1984, Estimation of economies of scale in banking using a generalized functional form, Journal of Money, Credit, and Banking 16, 53-68. _ Gilligan, T., M. Smirlock and W. Marshall, 1984, Scale and scope economies in the multiproduct banking firm, Journal of Monetary Economics 13, 393-405. Guilkey, D., C.A.K. Lovell and R. Sickles, 1983, A comparison of the performance of three flexible functional forms. International Economic Review 24, 591-616. Hancock, D., 1985, The financial firm: Production with monetary and nonmonetary goods, Journal of Political Economy 93, 859-880. Humphrey, D., 1985, Costs and scale economies in banking intermediation, in: R. Aspinwall and R. Eisenbeis, eds., Handbook for banking strategy (Wiley, New York). Hunter, W. and S. Timme, 1986, Technical change, organizational form, and the structure of bank oroduction, Journal of Money, Credit, and Banking 18, 152-166. Lawrence; C. and R. Shay, 1986, Technology and linancial intermediation in a multiproduct bankine firm: An econometric studv of U.S. banks. 1979-1982, in: C. Lawrence and R. Shay, eds., T&hnological innovation, regulation and the monetary economy (Ballinger, MA). Mester, L., 1987, A multiproduct cost study of savings and loans, Journal of Finance 42, 423-445. Murray, J. and R. White, 1983, Economies of scale and economies of scope in multiproduct financial institutions: A study of British Columbia credit unions, Journal of Finance 38, 887-902.