Scale elasticity versus scale efficiency in banking.

• Author: Evanoff, Douglas D.

1. 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.

2. 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...