Scale elasticity versus scale efficiency in banking.

I. Introduction

In the early bank cost literature many of the studies found scale elasticities significantly different from unity. As a result, the authors suggested that changes in industry structure could produce cost savings through increased efficiency. Recent bank cost studies improved upon previous methodologies by utilizing flexible functional forms, accounting for multiproduct production processes, estimating scale measures at both the branch and firm level, distinguishing between branch and unit bank technologies resulting from regulatory restrictions, etc. The typical finding from the recent studies is that relatively minor scale economies exist in banking since the scale elasticity measure differs little from a value of unity. This reported finding is usually followed by a general statement that banks operate relatively efficiently with respect to the scale of production and that the potential cost gains from exploiting scale advantages via merger or growth activities appear to be relatively minor. The implication from the conclusions drawn by the authors of numerous studies is that scale elasticity and scale efficiency are essentially synonymous; the derivation of one automatically provides an accurate or approximate value for the other.

The purpose of this article is to bring attention to a common confusion in the literature between two relatively straightforward concepts: scale elasticity and scale efficiency. The bank production process is one of the most extensively researched aspects of bank behavior. Until recently, however, studies have not typically evaluated scale efficiency.(10) Instead, scale elasticity estimates have been used as a proxy for efficiency, and elasticity measures close to 1.0 are taken to imply that scale inefficiency is trivial. Scale inefficiency is typically assumed to be linearly related to the scale elasticity measure; i.e., equal to one minus the elasticity measure. Empirically, it is also assumed that scale elasticities which are found to be insignificantly different from one in a statistical sense imply scale efficiency. Both statements are incorrect. Yet, failure to distinguish between the two concepts is common in the banking literature. For example:

1. Humphrey [16, 47] states that technical inefficiencies (the inefficient use of inputs) are on the order of . . . "31 to 34 percent. Such a cost reduction would be equivalent to a scale economy value of .69 to .66."

2. Mester [21, 439] finds the estimated scale elasticity for a sample of California S&Ls to be insignificantly different from one indicating that "from the standpoint of costs alone, the typical S&L would not benefit from changing the levels of (output)."(2)

These statements, however, are either incorrect or the basis used to make the statements is insufficient to support them. Scale elasticity and scale efficiency are two distinct concepts. An elasticity measure near one does not necessarily imply small scale inefficiency; nor does a large difference imply substantial scale inefficiency. Below we briefly formalize the scale inefficiency measure and show the relationship between scale elasticity and efficiency. For illustrative purposes we empirically apply the new efficiency measure to a group of large U.S. banks, and also apply it to the results of previous studies to highlight the distinction between the two concepts, The findings reenforce the point that using elasticity alone to determine or approximate scale efficiency is inappropriate and can produce misleading conclusions concerning inefficiency.(3) This is particularly true in an industry, such as banking, in which there is a broad range in firm size.

II. Elasticity and Efficiency Measures

The scale elasticity measure, [Epsilon] = [Delta] ln C/[Delta] ln Q, where C is cost and Q output, is a point elasticity associated with a particular output level and indicates the relative change in cost associated with an incremental change from this output level. Scale inefficiency, I, can be measured as the aggregate cost of F inefficient firms ([Epsilon] [not equal to] 1.0) relative to the cost of a single efficient firm ([Epsilon] = 1.0), where F = the size of the efficient relative to the inefficient one. That is, I = [F [center dot] [C.sub.I]/[C.sub.E]] - 1.0, where [C.sub.I] and [C.sub.E] are the cost of production at the inefficient and efficient firms, respectively.

Intuitively, the two concepts differ because they measure different things: elasticity is related to incremental changes in output, and inefficiency to the change in output required to produce at the minimum efficient scale. The inefficiency measure is typically associated with significantly larger output changes as one measures the difference in total or average cost at distinct output levels. The scale elasticity at the inefficient level of output suggests the initial path to the efficient output level. However, the initial path itself is inadequate to determine the efficient output. In Figure 1, the average cost relationships for three production technologies are shown. Although each produces the same degree of scale inefficiency, the path from the inefficient level of production to the efficient one, and the scale elasticity measure at the inefficient output level, are significantly different. The scale elasticity measure at output [Q.sub.I] gives little information concerning the level of scale inefficiency found in these three technologies. Alternatively, Figure 2 presents average cost relationships for three technologies which have the same point elasticity at output level [Q.sub.I]. The three technologies, however, exhibit significantly different levels of scale inefficiency for production at this output level. The cost savings realized by an incremental increase in output by a scale inefficient firm is irrelevant for measuring inefficiency since this is not the savings realized by producing at the efficient scale. The elasticity measure is important in determining scale inefficiency only to the extent that it can be used to derive the cost differential over a broader range of outputs, i.e., between the output of the scale efficient and inefficient firms. The elasticity value is not even needed to calculate scale inefficiency if direct information is available on average cost at the efficient and inefficient levels of output.

More formally, below we derive a general measure of scale inefficiency employing a standard translog cost function. Let

[Mathematical Expression Omitted],

where P denotes factor prices, Z exogenous variables relevant to the particular industry's production process, and the other variables are as previously defined. For simplification, we rearrange equation (1):

[Mathematical Expression Omitted]

and allow the terms in each set of brackets in equation (2) to be replaced by the coefficients a, b, and c, respectively. Therefore:

ln C = a + b(ln Q) + .5c[(ln Q).sup.2] (3)

represents the cost relationship. For simplicity, we normalize output levels around the level produced by the inefficient firm so that [Q.sub.I] = 1.0 and the output of the scale efficient firm, [Q.sub.E], is a multiple, F, of [Q.sub.I]. For the inefficient firm,(4)

ln [C.sub.I] = a + b(ln [Q.sub.I]) + .5c[(ln [Q.sub.I]).sup.2] = a, (4)

and the scale elasticity,

[[Epsilon].sub.I] = [Delta] ln [C.sub.I]/[Delta] ln [Q.sub.I] = b. (5)

For the scale efficient firm,

ln [C.sub.E] = a + b ln(F [center dot] [Q.sub.I]) + .5c[[ln(F [center dot] [Q.sub.I])].sup.2], (6)

and

[[Epsilon].sub.E] = [Delta] ln [C.sub.E]/[Delta] ln(F [center dot] [Q.sub.I]) = 1.0. (7)

Realizing that [Q.sub.I] = 1.0, by taking the difference between equation (4) and equation (6), and with substitution, it can be shown that

I = [F [center dot] [C.sub.I]/[C.sub.E]] - 1.0 = [F.sup..5(1 - b)] - 1.0. (8)

Since b is the elasticity coefficient resulting from the normalization of output around that of the inefficient firm, the inefficiency measure in equation (8) can be generalized:(5)

I = [F [center dot] [C.sub.I]/[C.sub.E]] - 1.0 = [F.sup..5(1 - [[Epsilon].sub.I])] - 1.0. (9)

The scale inefficiency measure, in general, is obviously not equal to 1 - [[Epsilon].sub.I]. In fact, information about scale elasticity alone is inadequate to derive the inefficiency measure because of the integral role played by the output differential between the efficient and inefficient firms.

Alternatively, since F is determined by the characteristics of the cost function, we can solve for the level of scale inefficiency in terms of the cost parameters only. Solving for F in terms of c (the second derivative of ln C) from equations (6) and (7), and substituting into equation (9) gives

I = [e.sup.(.5/c)[(1 - [[Epsilon].sub.I]).sup.2]] - 1.0. (10)

That is, scale inefficiency is a function of the first and second derivatives of the cost function with respect to output. Setting equation (10) equal to 1 - [[Epsilon].sub.I] we can solve for c to see when the two measures are equal:

[Mathematical Expression Omitted].

For [[Epsilon].sub.I] [not equal to] 1.0, there is only one point on the cost function corresponding to I = (1 - [[Epsilon].sub.I]). For [Mathematical Expression Omitted], larger elasticities produce inefficiencies less than [Mathematical Expression Omitted]; and elasticities less than [Mathematical Expression Omitted] produce inefficiencies greater than [Mathematical Expression Omitted]. Similar conjectures can be made for [Mathematical Expression Omitted]. Figure 3 presents the relationship between scale inefficiency and elasticity for three different values of the second derivative.(6)

Similarly, empirical differences exist in the two cost concepts. An estimated scale elasticity value which is insignificantly different from unity does not imply scale inefficiency is insignificantly different from zero. That is, not only do the measures differ, but the calculated standard errors also differ. For example, from equation (5), at the sample mean the statistical difference of the elasticity measure from a value of unity depends entirely on the standard error of the estimated coefficient b. The standard error for the scale inefficiency measure,

I = [e.sup.(.5/c)[(1 - b).sup.2]] - 1.0, (12)

differs as it depends on the variance and covariance of the estimates for coefficients b and c. That is:

var(I) [approximately equal to] [([Delta]I/[Delta]b).sup.2] [center dot] var(b) + [([Delta]I/[Delta]c).sup.2] [center dot] var(c) + ([Delta]I/[Delta]b) [center dot] ([Delta]I/[Delta]c) [center dot] cov(b, c). (13)

Thus, tests for statistical significance for the two concepts can also differ substantially.

III. Supporting Evidence

Having shown that scale efficiency and elasticity are distinct concepts, we next present evidence of differences in the two measures. A translog cost function, equation (1), with traditional parameter restrictions was estimated using 1987 data for 164 banks which were holding company members and were ranked in the top 500 U.S. banks for the previous twenty years. Exogenous variables include holding company affiliation and the number of branches. The variable definitions, estimation results, and properties of the estimates are presented in Table I.(7) Scale elasticities for output [TABULAR DATA FOR TABLE I OMITTED] quartiles are presented in Table II. The estimated relationship is a well behaved cost function having all the desirable properties, and similar scale estimates to that found in the bank cost literature, suggesting that our findings are not driven by unique cost characteristics.(8) However, more is required to evaluate scale inefficiency. Table III presents point scale elasticities for various observations. Viewing specific data points obviously reveals detail not available in the calculations based on output quartiles. Significant economies of scale are found for the smaller banks, and diseconomies set in at approximately $3.3 billion in output. From equation (10), calculated values for scale inefficiency for these same observations are presented in column 4. As implied by equation (11), scale inefficiency is greater than 1 - [[Epsilon].sub.I] for some observations, and less for others. Obviously, as suggested in Figure 3, the magnitude of the differences in other cost studies will vary with the relevant cost characteristics.

Table II. Scale Elasticity Estimates

Output Quartile            Output (billions)         Mean [Epsilon]

1                          [less than]1.2               .85
2                                 1.2-2.5               .92
3                                 2.5-5.3               .99
4                       [greater than]5.3              1.13
entire sample                                           .98

Notes: [Epsilon] denotes scale elasticity.

As further evidence of the distinction between the two concepts, Table IV presents the findings of a number of bank cost studies, and presents estimates of scale inefficiency based on the assumptions listed in the note to the table. The distinction between the two measures is more pronounced in some of these studies.(9) This occurs because of the sample range and particular [TABULAR DATA FOR TABLE III OMITTED] cost function characteristics. The findings again suggest that inefficiency for specific firms can be substantially greater than that typically referred to in the bank cost literature. Even minor deviations from a value of one for the scale elasticity measure can be associated with significant inefficiency. Again, the elasticity measure is calculated based on incremental changes in output. To generate scale efficiency measures, the output change required to reach the efficient scale of production, as shown in Tables III and IV, may be quite large.

IV. Concluding Comments

The purpose of this note has been to clarify a common confusion in the banking literature. Use of the scale elasticity measure alone to approximate the extent of scale efficiency is inadequate. They are two distinct concepts. We derive the inefficiency measure and provide an empirical illustration to distinguish between the two concepts. A review of earlier studies also shows significantly more scale inefficiency than implied by the elasticity measure.

What do the results imply about the propensity for merger activity in banking?(10) The evidence suggests that, for certain banks, there are significant scale efficiency gains to be achieved by growing via internal means or by merger.(11) Rhoades [25] found that between 1960-83, over [TABULAR DATA FOR TABLE IV OMITTED] 93 percent of acquired banks had assets less than $100 million. Nearly all bank cost studies find scale advantages up to this size. Additionally, equation (9) indicates that efficiency gains increase as the difference between the output levels of the efficient and inefficient firms increases, i.e., F. Viewing bank merger data for the second quarter of 1991, for example, the largest size differential between the acquiring and acquired firm was 651; i.e., the acquirer was 651 times as large as the acquiree. The mean differential was 69; see Matthews [20]. Thus, the size differentials in bank acquisitions appear to be quite large suggesting significant potential gains.(12)

The efficiency gains, however, may not be realized by the larger banks which recently have been so aggressive in pursuing acquisitions. In fact, the larger banks in most cost studies exhibit constant returns to scale or inefficiency resulting from operating under diseconomies of scale. Instead, it is the acquisition of small, inefficient banks, which will improve industry inefficiency.(13) However, as we have shown here, these potential gains cannot be detected by simply evaluating the scale elasticity measure.

1. Recent exceptions include Betger, Hunter, and Timme [6] and some of the accompanying articles, Ferrier and Lovell [13], and Aly, Grabowski and Pasurka [1]. For a review of the banking cost literature see Evanoff and Israilevich [11]. For an early general discussion of multiproduct cost analysis see Cowing and Holtmann [9].

2. Numerous additional examples exist. Comparing scale and non-scale related inefficiencies in banking, Berger and Humphrey [4] evaluate scale efficiency by contrasting the scale elasticity estimate to a value of 1.0. Based on the finding of scale elasticity measures which typically exceed .95 they state that non-scale inefficiencies (technical and allocative inefficiency) of approximately 25% "dominate scale . . . effects, which are measured to be on the order of 5% or less." Mester [23, 558] finds slight scale economies and claims that "the result indicates banks are operating at a scale slightly less than minimum efficient." Analyzing savings and loans, Mester [22, 270] finds a scale elasticity of .95 for the representative firm and asserts that it "is operating within 5% of minimum efficient scale. Thus, only small efficiency gains are possible from increasing the scale of operations . . ." Other examples include Clark [8, 67], Dowling and Philippatos [10, 245], Humphrey [17, 36], Berger, Hanweck, and Humphrey [3, 513], or Benston [2, 541].

3. It should be emphasized that we are evaluating potential efficiency gains from a production technology perspective only. There may indeed be impediments to actually achieving the efficient scale of production, e.g., local market demand may be insufficient to warrant the expansion of output. However, past studies relating scale elasticity and scale efficiency have taken this same perspective.

4. Neither the simplification in equation (2) nor the output normalization will alter the analysis. The chosen functional form, however, is important. The translog function is discussed because it is the form most commonly used in cost studies.

5. As with the scale elasticity measure, the inefficiency measure is functional form specific. For the quadratic form, C = a + b [center dot] Q + .5c [center dot] [Q.sup.2], the inefficiency measure will equal [(b + c [center dot] [Q.sub.1])/(b + c [center dot] [Q.sub.E])] [1/[[Epsilon].sub.1]] and can similarly be derived for alternative forms.

6. The middle relationship corresponds to the empirical results discussed later in the text.

7. The empirical$example should be considered a pedagogical device for illustrative purposes only. Use of an aggregate output measure can be criticized for incorrectly specifying bank output, and can result in biases toward finding greater scale advantages. However, our purpose is to illustrate the difference between the two concepts; not to accurately capture the intricacies of the bank production process. The aggregate output measure simplifies the model and in no way distorts the distinction between scale elasticity and scale inefficiency. However, multiproduct production does make the analysis of scale economies more complex since the product mix, and resulting optimal input mix, can vary with bank size. The ray scale measure, [[Sigma].sub.i] [Delta] ln C/[Delta] ln [Q.sub.i], assumes banks expand along an output ray with product-mix held constant. While this enables us to use the formulas developed here to analyze scale inefficiency by using the ray scale measure and summing across outputs to obtain the second derivative, this is obviously a simplification and brings into question the usefulness of the ray scale measure in evaluating output expansion to levels very far removed from the point at which it is evaluated.

8. All regularity conditions are satisfied: positivity and homogeneity by model construction, monotonicity by having all predicted factor shares positive, and concavity by having factor shares range between four and 96 percent (well within our predicted range). See Evanoff, Israilevich, and Merris [12], particularly footnote 10.

9. The studies considered in Table IV are each relatively current and utilize a flexible functional form. However, the apparent misconception between scale elasticity and efficiency is also present in earlier bank cost studies. Benston [2] states that the elasticities estimated for deposit accounts and loans were close to one implying that "efficiency of operations is not largely a function of bank size." He reports cost elasticities for the sample means in 1960, the middle year of the analysis, for demand deposits, time deposits, mortgage loans, installment loans, and business loans as 0.986 and 0.955, 0.959, 0.881 and 0.978, respectively. However, comparing the per unit cost of the average bank to that of smaller banks reveals scale inefficiency for the services of approximately 28, 10, 10, 30, and 41 percent, respectively. In some instances these figures understate the extent of the inefficiency because data on the smaller banks were not available. Significantly greater inefficiency existed in the sample if comparisons were made relative to banks other than the average bank and for other years. For example, for business loans in 1960 the within sample inefficiency was over 100 percent.

10. Whereas we emphasize the importance of accounting for inefficiency instead of elasticity alone, Caves, Khalilzadeh-Shirazi and Porter [7] took a somewhat similar approach in analyzing entry barriers. They argued that viewing minimum efficient scale as a measure of the barrier was inadequate. One also had to account for the extent of the disadvantage to potential entrants; their cost disadvantage ratio. Simply viewing the output level at which efficient scale is achieved does not indicate the true extent of the barrier. Similarly, simply viewing the elasticity measure to detect the extent of potential gains from achieving scale efficiency is inadequate.

11. The banking literature also suggests that potential gains from the elimination of X-inefficiency exists in most bank mergers, but are frequently not realized [26; 29; 14; 5].

12. It has been argued that the scale elasticity measure may fairly accurately indicate the marginal gains from changes in bank size that actually occur [4]. This may or may not be accurate. However, if the merger data discussed here is representative of the population of mergers, it casts doubt on the contention. If one is interested in evaluating scale efficiency, the direct measure (of which the scale elasticity measure is one component) is a more appropriate measure. It should also be emphasized that the efficiency gains we are discussing are those resulting from altering the size of the inefficient firm to achieve an appropriate scale of production, e.g., by merger. One could also view the gains of the acquiring firm in a merger or the net gains resulting from gains (losses) by both parties.

13. The public policy implications carry a number of caveats. It is assumed, for example, that the cost relationship and outputs are properly modeled in these studies, input prices are held constant, etc. However, these are the same caveats which have existed in the past when cost studies were used to make public policy recommendations based on the belief that no scale inefficiency existed. It may be that analysis of balance sheet information does not allow the researcher to capture the true characteristics of bank production. It is also possible that the gains from scale may be partially offset by other factors. The average cost across various size groups of banks have been shown to be remarkably similar; see Humphrey [16]. In evaluating perspective acquisitions, the parties involved and regulators obviously need to consider more than scale advantages alone.

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