Profits and Productivity

In this study we consider the linkage between productivity change and profit change. We develop an analytical framework in which profit change between one period and the next is decomposed into three sources: (i) a productivity change effect (which includes a technical change effect and an operating efficiency effect), (ii) an activity effect (which includes a product mix effect, a resource mix effect, and a scale effect), and (iii) a price effect. We then show how to quantify the contribution of each effect, using only observed prices and quantities of products and resources in the two periods. We illustrate our analytical decomposition of profit change with an empirical application to Spanish banking during the period 1987-1994.


Background
The objective of this study is to analyze the linkage between business productivity and business profit.The analysis begins at the theoretical level, continues with a discussion of the analytical techniques required to implement the analysis, and concludes with an empirical application.Although the empirical application is to a collection of independent businesses observed over a number of years, the analysis is equally suitable to other applications, such as a collection of regional offices, branches, or profit centers within a particular business, each observed over a sequence of possibly shorter (e.g., quarterly) accounting periods.The techniques can also be applied to a single business, provided only that a sufficient number of accounting periods is available to implement the analysis, or to a sufficiently large collection of businesses in a single accounting period.In each of these applications the analysis can provide a sort of benchmarking technique, in which a business or one of its constituent parts benchmarks against the remaining observations.The great virtue of these techniques, when applied in a benchmarking exercise, is that they utilize determinants of profit change (such as an improvement in operating efficiency), rather than consequences of profit change (such as a change in return on assets).This makes them amenable for use by management in an effort to improve future performance, which would then lead to future profit gains.Finally, although we analyze the linkage between productivity and profit, it is not necessary that profit be the sole objective of the business units being analyzed.The analysis simply provides a linkage between productivity and the bottom line.
It is clear that productivity gains have the potential to contribute to an increase in business profit, but it is equally clear that other factors (e.g., a more favorable price structure) can also contribute to an increase in business profit.It is of interest, therefore, to develop an analytical model of the determinants of business profit change, among them being productivity change.The development and implementation of such an analytical framework is the objective of this study.
The linkage between productivity change and profit change has been explored previously in the business literature (see Genescà and Grifell (1992) for citations).magnitude of efficiency change, and scale economies.They have not, however, sought to relate these components to profit change.What remains is to merge the two branches of the economics literature with the business literature.The result will be a deeper insight into the determinants of business profit change between one period and the next, one based as much on the economic theory of production as on business accounting relationships.
It should be apparent from this brief review that the business and economics literatures, while having different institutional structures, different motivations and different objectives, have been discussing different aspects of the same problem: how business profit change can be allocated to its constituent sources.In this study we continue the tradition of those who have sought to establish a linkage between business profit change and productivity change.We pull together salient contributions from the two literatures, and we extend them to develop a new model of the linkage.Our strategy is to embed a productivity change decomposition similar to that developed in the economics literature within the profit/productivity linkage developed in the recent business literature.The new model highlights the contribution of productivity change and its components, while at the same time not neglecting the contribution of other determinants of profit change.Our analysis sheds new light on four aspects of the linkage.
First, we provide a three-stage decomposition of profit change.In the first stage we decompose profit change into a price effect and a quantity effect.In the second stage we decompose the quantity effect into a productivity effect and an activity effect.The combination of these two decompositions is in the spirit of the relevant business literature developed by Kurosawa (1975) and Eldor and Sudit (1981).In the third stage we extend the previous business literature by exploiting both branches of the economics literature on production and profit.This enables us to decompose the productivity effect into a technical change effect and an operating efficiency effect.Eilon (1985) properly stressed the impact of the efficiency of resource use on business performance, and although the contribution of efficiency has been largely ignored in the subsequent literature, it plays a key role in our decomposition of profit change.We also decompose the activity effect into a product mix effect, a resource mix effect, and a scale effect.Within a very different framework, the impact of cost-effective adjustments to the product mix on profitability has been explored by Thanassoulis (1995) and Soteriou and Zenios (1996).These six components of profit change are mutually exclusive and exhaustive.Second, we achieve this three-stage decomposition without imposing restrictive assumptions on the behavioral objective of the business or on the environment in which it operates.We do not assume profit maximizing behavior, we do allow the business to set some or all of its prices, we do not require that the business operate efficiently, and we do allow the business to produce a variety of products.
Third, we provide computational guidance for implementing the three-stage profit decomposition.The computational technique consists of a sequence of linear programs.
These programs modify and extend a technique known as data envelopment analysis (DEA), which is widely used in the management science and economics literatures to analyze business performance.Fourth and finally, we believe that the ex ante determinants of business profit change we identify are better suited to a useful benchmarking exercise than are the more frequently used ex post financial ratios, primarily because they are forward-looking rather than backward-looking. 1   The study is organized as follows.In Section 2 we develop our analytical framework for the decomposition of business profit change.The decomposition involves unobserved as well as observed quantities of the products a business produces and the resources it consumes in their production, and so in Section 3 we show how to express all unobserved quantities as scalar multiples of observed quantities.We employ distance functions widely used in production economics to provide a theoretical expression for these scalar multiples, and we show how to calculate these distance functions empirically.In Section 4 we illustrate the working of our analytical decomposition with an empirical application to the recent performance of Spanish banking.We measure profit change among a sample of Spanish commercial banks during the period 1987-1994, and we obtain for each bank an empirical decomposition of its measured profit change into six determinants, for each pair of adjacent years and through the entire period.Section 5 contains a summary and our conclusions.

The Production Technology
We consider a business using N resources represented by the nonnegative input quantity vector x = (x 1 ,...,x N ) to produce M products represented by the nonnegative output quantity vector y = (y 1 ,...,y M ).The business pays resource prices represented by the strictly positive input price vector w = (w 1 ,...,w N ), and receives product prices represented by the strictly positive output price vector p = (p 1 ,...,p M ).These prices may be exogenously determined by the forces of market competition, or by a regulatory agency, or they may be endogenously determined by the business itself.Business profit in period t, π t , is defined as the difference between total revenue and total cost, and so π ), which can be positive, zero or negative.We require that the determinants satisfy two criteria: (i) their selection must be grounded in the economic theory of production; and (ii) they must be observable in conventional data sources, such as business financial statements.
We begin with a presentation of some basic concepts from production economics, which provide the necessary grounding for our decomposition.The concepts to be developed in equations ( 1) -( 7) below represent a fairly conventional way of modeling the structure of production technology, and of describing the efficiency of observed resource use and observed output provision in light of the constraints imposed on managerial choice by the structure of production technology. 1.

The Profit Change Decomposition
We are now prepared to decompose business profit change between periods t and t+1.
Our strategy is to proceed in three stages, which are described schematically in Figure 1.In the first stage we decompose the profit change resulting from a movement from ( ) into a pure quantity effect which holds prices constant, and a pure price effect which holds quantities constant.This decomposition appears in Proposition 1, and is similar to decompositions previously obtained by Kurosawa (1975) and illustrated by Eldor and Sudit (1981).In the second stage we decompose the quantity effect into its two basic components: a productivity effect and an activity effect.This decomposition appears in Proposition 2, and is illustrated in Figure 2. It is similar to decompositions previously obtained by Kurosawa (1975) and Miller (1984Miller ( , 1987)).In the third stage we decompose the productivity effect into its two components: a technical change effect and an operating efficiency effect.We also decompose the activity effect into its three components: a product mix effect, a resource mix effect, and a scale effect.We decompose the productivity effect in Proposition 3, and we decompose the activity effect in Proposition 4.
Both third stage decompositions are illustrated in Figures 2 -4.The third stage decompositions appear to have no precedent in the business literature, although they do have precedents in the economics literature.
Proposition 1: The profit change between period t and period t+1 decomposes as The quantity effect shows the impact on profit of an expansion or contraction of the business, holding prices fixed.The quantity effect uses base period prices to weight the quantity changes, and so it can be interpreted as the difference between a Laspeyres type of output quantity index and a Laspeyres type of input quantity index, both being expressed in difference form rather than in the conventional ratio form.The price effect shows the impact on profit of changes in the price structure of the business, holding quantities fixed.
The price effect uses comparison period quantities to weight the price changes, and so it can be interpreted as the difference between a Paasche type of output price index and a Paasche type of input price index, both being expressed in difference form rather than in ratio form. 2 The following three propositions show how the quantity effect decomposes into a total of five determinants of profit change.All five determinants use base period prices to weight quantity changes, and so all five determinants are Laspeyres type quantity indexes, or differences between Laspeyres type quantity indexes.

Proposition 2:
The quantity effect between period t and period t+1 decomposes as ) can be decomposed into three components, each of which exerts an influence on profit change, with magnitude and direction depending on prevailing prices and how they change from the base period to the comparison period. 3 The productivity effect compares the path from y B to y t in period t with the path from y C to y t+1 in period t+1.Part of the productivity effect is measured along the path from y The final component of the activity effect, the resource mix effect, is illustrated in Figure 4, which provides a third partial illustration, partial because the output quantity vector is fixed, either at y t or at y t+1 .In Figure 4, which allows N=2, the input side of the scale effect holds the resource mix fixed along the path from x t to x E , using base period input prices to evaluate the adjustment.The resource mix effect is measured along the path from x E to x t+1 , also using base period input prices to evaluate the adjustment.The sum of the scale effect and the resource mix effect produces a movement from x t to x t+1 , which corresponds to the input side of the activity effect introduced in Proposition 2 and illustrated in Figure 2. 4   The product mix effect and the input mix effect can be due to any number of factors.
Perhaps the most significant factor is the ability of a business to react to product price changes by adjusting its product mix, and to react to resource price changes by adjusting its resource mix.The ability to substitute toward products whose prices are rising and resources whose prices are falling, and away from products whose prices are falling and resources whose prices are rising, contributes to profit gain through the product mix effect and the resource mix effect.Thus the two mix effects capture the substitution possibilities permitted by the structure of production technology, as well as the ability of management to exploit these possibilities.A second factor is a changing regulatory environment which allows business more or less freedom to optimize its product and resource mixes.This factor is particularly significant in the empirical example we use to illustrate our profit ) into several sources.
The first source is an improvement or a deterioration in the price structure of the business, which may have both external and internal causes.A second source is an increase in output not requiring any increase in resource use, due to technical change.A third source is an improvement or a deterioration in operating efficiency.These two sources make up the productivity effect.A fourth source is a change in output that can be proportionately greater than or less than the change in input, due to the presence of economies or diseconomies of scale which characterize the production technology.A fifth source is a change in the product mix, and a sixth is a change in the resource mix.The product mix effect and the resource mix effect encompass a number of phenomena, as we noted above.
The last three sources comprise the activity effect.The six sources are mutually exclusive and exhaustive sources of profit change between period t and period t+1.

Implementing the Profit Decomposition
The price effect in Proposition 1 is expressed in terms of observed base period and ).Our strategy should be apparent from an inspection of Figures 3 and 4, where it is clear that each unobserved quantity vector appears as either a radial expansion or a radial contraction of an observed quantity vector.The distance functions introduced in equations ( 4) and ( 7), being radial distance measures, provide the tools with which to recover the unobserved quantity vectors.).The distance functions employed in Proposition 5 can be calculated empirically using a linear programming technique which provides a modification of data envelopment analysis (DEA).Originally developed by Charnes et al. (1978Charnes et al. ( , 1981) ) and Banker et al. (1984), DEA is a widely used operations research technique for measuring business performance.
Although DEA was originally intended for use in public sector and other not-for-profit environments, it is ideally suited to the problem at hand.Nonetheless, we believe this is the first application of DEA to the problem of decomposing profit change.
In conventional DEA period t technology is constructed from input and output quantity data describing the operations of all producers in period t.In this approach technologies in place in previous periods are "forgotten" in period t, since period t technology is constructed from period t activities only.Our modification of DEA allows period t technology to be constructed from input and output quantity data of all producers in all periods prior to and including period t.In our modified approach, technologies in place in previous periods are "remembered," and remain available for adoption in the current period.This modification influences the way we set up the linear programming problems. 6  We assume that producers use N inputs to produce M outputs.We also assume that there are t time periods, and that in time period s we have I s producers, s = 1,...,t.Let (y ot ,x ot ) be the "contemporaneous" M×1 output quantity vector and the N×1 input quantity vector of producer "o" in period t.Also let (y os ,x os ) be the "sequential" M×s matrix of M outputs produced and N inputs used by producer "o" in each of periods s = 1,...,t.Finally ] be an N×Σ s t =1 I s matrix of N inputs used by all I s producers in each of periods s = 1,...,t.Thus the data matrices Y s and X s are "sequential," since they include output and input quantity data for all producers from the beginning of the sample through period t.Consequently period t technology is determined not just by period t production activities, but by past production activities as well.
Unobserved output quantity vector y oC can be recovered from the solution to the linear programming problem where λ s+1 is a Σ s t =1 I s+1 ×1 activity vector.From Proposition 5 (iii), y oC = θ oC y ot+1 .
Unobserved output quantity vector y oD can be recovered from the solution to the linear programming problem where λ s+1 is a Σ s t =1 I s+1 ×1 activity vector.From Proposition 5 (iv), y oD = θ where λ s+1 is a Σ s t =1 I s+1 ×1 activity vector and y ^ot = y ot / D o t+1 (x ot+1 ,y ot ).From Proposition 5 To summarize, recovery of the five unobserved quantity vectors requires the solution of a series of five linear programming problems for each producer.For any reasonable number of producers and time periods, and for any reasonable number of inputs and outputs, this task can easily be handled on any personal computer.Once the five unobserved quantity vectors have been recovered, empirical implementation of the profit decomposition presented in Propositions 1 -4 is straightforward. 7

An Application to Spanish Banking
In this Section we report results of an empirical investigation into the sources of profit change within the Spanish commercial bank sector during the period 1987 -1994.Annual data for commercial banks are reported in Anuario Estadístico de la Banca Española.The commercial bank sample consists of roughly two-thirds of all commercial banks in existence during the period 1987-1994.However the sample does contain 92% of all commercial bank assets in 1993, so the missing banks are very small.The sample size varies from 59 in 1987 to a high of 61 in 1990 and a low of 56 in 1993.A detailed discussion of the data describing the recent history of Spanish banking is available in Grifell and Torrent (1995). 8 Annual profit consists of operating profit, or profit from intermediation activities, and is defined as gross profit less gains and losses from trading in stocks and public debt instruments, and less extraordinary profit.Extraordinary profit typically comes from sales of fixed assets, but during the 1987-1990 period extraordinary losses also arose from the legally mandated establishment of employee pension plans.On balance, operating profit has accounted for 76% of gross profit among the banks in our sample.
Well over 90% of revenue consists of net loan and investment income, defined as gross loan and investment income less provision for bad debt.We decompose this income into quantity and price components by specifying the quantity component as the average of the beginning-of-period and the end-of-period value of all loans and investments, and by specifying the price component as the ratio of net loan and investment income to the average value of all loans and investments.Thus y 1 t is expressed in pesetas, and p 1 t is expressed as a per cent. 9 The remaining source of revenue consists of net commission income, the difference between commission income generated and commission expenses incurred.There is no natural way of decomposing net commission income into price and quantity components.
However on the assumption that net commission income is a function of the number of deposit accounts, we proxy the quantity component of net commission income by the average of the beginning-of-period and end-of-period number of deposit accounts.The price component is then the ratio of net commission income to the average number of deposit accounts.Thus y 2 t is expressed as a pure number, and p 2 t is expressed in pesetas. 10 Approximately two-thirds of cost is financial expense, consisting of interest paid on deposit accounts and other liabilities.The quantity component of financial expense is defined as the average of the beginning-of-period and end-of-period value of all deposits and other liabilities which generate financial expense.The price component is the ratio of financial expense to the average value of all deposits and other liabilities.Thus x 1 t is expressed in pesetas, and w 1 t is expressed as a per cent.
Labor expense accounts for approximately 20% of cost.The quantity component of labor expense is defined as the average of the beginning-of-period and end-of-period number of employees.The price component is the ratio of labor expense to the average number of employees.Thus x 2 t is expressed as a pure number, and w 2 t is expressed in pesetas.
The remaining source of cost is non-financial, non-labor expense, consisting of non-labor operating expense, direct expenditure on buildings, and amortization expense.
The quantity component of this expense category is proxied by the average of the beginning-of-period and end-of-period value of all fixed assets.The price component is calculated as the ratio of non-financial, non-labor expense to the average value of all fixed assets.Thus x 3 t is expressed in pesetas, and w 3 t as a per cent. 11 Summary statistics for all variables are collected in Table 1.Average operating profit doubled from 1987 to 1990, declined dramatically through 1993, and recovered somewhat in 1994, despite the fact that margins (p 1 -w 1 ) remained above 4% until 1991.
We also include in Table 1 average annual asset values.Bank asset size is defined as the average value of loans and other financial investments (y 1 ) plus the average value of fixed assets (x 3 ).Although asset size is not used in the empirical analysis, it is included in Table 1 to provide an indication of the magnitude of the increase in average bank size during the period.By this measure average bank size nearly doubled during the period, and although a part of this increase is attributable to mergers and acquisitions, the majority of the growth was internally generated.
Results of implementing the profit change decomposition are summarized in Tables 2 -4.Table 2 provides a summary of the initial decomposition of profit change into a productivity effect, an activity effect, and a price effect, averaged over the number of banks indicated in the final column. 12On average, operating profit increased during the first three years of the sample, declined during the next three years, and increased again in 1994.Over the entire period, operating profit increased by an average of 2.4% per year.The productivity effect made a large positive contribution on average, and in six of seven years.
The activity effect made a larger positive contribution on average, and in all seven years.
However these two positive contributions to profit change were nearly offset by the price effect, which was very large and negative, on average and in six of seven years.
The sources of the negative price effect are apparent from Table 1.The average loan rate remained stable through 1993, and while the average commission price more than doubled, its impact was relatively minor because commission revenue was less than 10% of total revenue.Moreover all three input prices increased through 1993, with average deposit rates increasing by 16% and employee compensation increasing by 73%.Gradual deregulation of the Spanish banking system and the consequent increase in competition took its toll on the price structure and the profitability of commercial banks.
We now turn to an analysis of the productivity effect, the decomposition of which appears in Table 3.The technical change component was large and positive on average, and was positive in every year.We attribute a positive technical change component to an improvement in the productivity of the best practice banks.Perhaps because their price structure was deteriorating, best practice banks responded by becoming more efficient, increasing service provision more rapidly than they increased resource usage.The operating efficiency component was very small and negative on average, and was negative in all seven years.We attribute a negative operating efficiency component to a failure of the remaining banks to keep pace with the improved performance of best practice banks.
Thus a second consequence of deregulation and increased competition was an increse in the dispersion of bank performance.
A decomposition of the activity effect appears in Table 4.The large positive activity effect was primarily attributable to a very large product mix effect, which was positive on average, and was positive in every year.The favorable product mix effect is attributable to a doubling of the value of loans and other financial investments, the return on which held steady through 1993, and also to a rise in commission income as reflected in a decline in deposits which was proportionately smaller than the rapid increase in deposit rates.A large positive scale effect also contributed to the positive activity effect, although as we pointed out in note 4 a positive scale effect is not necessarily evidence of increasing returns to scale.Indeed Grifell and Lovell (1997a) have found a wide range of service provision over which Spanish commercial banks experience roughly constant returns to scale.However in the absence of decreasing returns to scale, positive profit is enhanced by expansion, holding prices constant, as the scale effect does.The resource mix effect was large and negative on average, and was negative in all seven years.The source of the large negative resource mix effect is apparent from Table 1; the average value of deposits and other liabilities nearly doubled as their price increased through 1993, and the value of fixed assets more than doubled as their price increased throughout the period.These two effects were only partly offset by modest labor shedding in the face of rising wage rates.
The operating profit change among Spanish commercial banks during the period was thus the consequence of a number of factors, three positive and three negative.The positive contributions came from an improvement in the performance of best practice banks (the technical change effect), a continuing emphasis on loans and other financial investments having high and relatively stable rates of return (the product mix effect), and a general expansion in average bank size, which either exploited scale economies or at least was not offset by diseconomies of scale (the scale effect).However these positive contributions to profit change were nearly offset by the negative contributions of a deterioration in the banks' price structure brought on by deregulation and increased competition (the price effect), a failure of the remaining banks to keep pace with the improving performance of the best practice banks (the operating efficiency effect), and a rapid growth in deposits and other liabilities and in fixed assets when prices of both were increasing (the resource mix effect).
In an effort to examine the sources of profit change more thoroughly, we disaggregated the sample into three categories of bank: the ten banks having the largest profit change in each year, the ten banks having the smallest profit change in each year, and the remaining banks.We found three interesting results.First, the ten most successful banks managed to increase their aggregate profit in every year, while the ten least successful banks experienced an aggregate profit decline in every year.Deregulation and increased competition have created two groups of banks, and they are moving in opposite directions.
Second, the ten most successful banks always experienced a much more favorable (usually positive) aggregate price effect than did the ten least successful banks, who always had a negative aggregate price effect.The difference between the two sets of aggregate price effects accounted for the vast majority of the difference between the two sets of aggregate profit changes.This suggests that one characteristic of the most successful banks is an ability to manage their price structures to a much greater degree than exists at less successful banks.Third, the ten most successful banks had a larger aggregate productivity effect (in every year but 1990-1991) than did the ten least successful banks.Although this difference is small in comparison with the price effect difference, it reaffirms our previous finding that the most successful banks owe a part of their success to efficiency improvements and productivity gains. 13

Summary and Conclusions
Business profit changes from year to year, increasing in some years and declining in others.The business and economics literatures have adopted somewhat different approaches to an analysis of the sources of profit change, although we have found evidence of both considerable overlap and substantial divergence.This has motivated us to develop a three stage decomposition of profit change which draws from, and extends, both literatures.In the initial stage profit change is decomposed into a quantity effect and a price effect, as expressed in Proposition 1.In the second stage the quantity effect is decomposed into a productivity effect and an activity effect, as expressed in Proposition 2. The first two stages are broadly consistent with some decompositions appearing in the business literature.
In the third stage the productivity effect is further decomposed into a technical change effect and an operating efficiency effect, and the activity effect is further decomposed into a product mix effect, a resource mix effect, and a scale effect.The third stage is broadly consistent with the traditional focus in the economics literature on technical change, efficiency change, and the structure of technology as characterized by the nature of scale and scope economies.The third stage decompositions obtained in Propositions 3 and 4 appear to be new, however.
The profit decompositions we have derived are based on observed prices, but they are based on both observed and unobserved quantities, of the resources the business employs and the services it provides.This makes it necessary to obtain expressions for the unobserved quantities in terms of observed quantities.This we achieve in Proposition 5, in which the distance functions introduced in equations ( 4) and ( 7) are used to express unobserved quantities as radial expansions or contractions of observed quantities.Once these relationships have been analytically derived, we show in equations ( 8) -( 12) how to use linear programming techniques to empirically calculate the requisite distance functions, and hence to obtain solutions for the five unobserved quantities.This enables us to obtain the desired profit decomposition.
We have illustrated the profit decompositions using data describing the operations of a sample of Spanish commercial banks.The data cover a difficult period of adjustment to a changing regulatory environment in which competition was increasing.The raw data show a slight improvement in profitability among commercial banks during the sample period, despite two very difficult years in 1992 and 1993.We have attributed the observed profit change to six effects, three positive and three negative.When the various effects are grouped into endogenous and exogenous influences, the picture brightens somewhat.The combined productivity and activity effect, which arguably reflects factors largely under the control of bank management, made a positive contribution to profit change in all seven years.However the price effect, which presumably captures the impact of macroeconomic and other influences beyond the control of bank management, made a negative contribution problem occurs on the input side, where we have information on values of checking and saving deposits, but we do not have information on either prices or quantities of these two types of deposits.
10. Our use of the number of deposit accounts as a proxy for the unobservable quantity component of net commission income is admittedly crude.However since net commission income accounts for only 9.1% of bank revenue during the period, the impact of any approximation error is likely to be small.11.A referee has pointed out that neither our model nor our data explicitly incorporate surrogates for customer service (such as the number of branch offices and the number of ATMs), which contribute to profit.While this is true, it is also true that both the costs and the benefits of these and other customer service surrogates are implicitly included in our data, since they contribute both to operating cost and to revenue.We have not incorporated them explicitly because it is not possible to decompose either their costs or the revenues they generate into separate price and quantity components.The relationship between customer service quality and operating efficiency (but not profitability) at a group of Greek bank branches is explored by Athanassopoulos (1997aAthanassopoulos ( , 1997b)).
12. The number of commercial banks listed in the final column of Table 2 is smaller than the number listed in the final row of Table 1.This is due to three factors.Mergers occuring in period t eliminate two or more banks which existed at the end of period t-1, and create a new bank at the end of period t.For these merging banks none of the linear programs can be solved for both periods, and the profit decomposition cannot be implemented.In addition, banks occasionally appear in only one of a pair of adjacent years, due to data problems.In this case adjacent-period linear programming problems cannot be solved for these banks, and the profit decomposition cannot be implemented for them.Finally, the mixed period linear programs ( 9) and ( 11) are not guaranteed to have solutions for the smallest banks.This is because, for the smallest banks, a radial expansion of y ot may not intersect period t+1 technology.Consequently the profit change indicated in the first column of Table 2 does not correspond to the annual change in average operating profit calculated from the entries in the first row of Table 1.
13. Berger and Humphrey (1997) note that a consensus estimate indicates that, as a result of various operating inefficiencies, US banks realize approximately two-thirds of profit potentially available to them.While our study is an examination of the determinants of Figure 2; changes in scope are not, and are discussed in Proposition 4. With this qualification in mind, the activity effect is measured along the path from y B to y C , and decomposition.A third factor is a consequence of the way we have measured technical change and scale economies.Both the technical change effect and the scale effect involve equiproportionate changes in variables, as Figures 3 and 4 make clear.If technical changeis not neutral with respect to outputs, any unmeasured bias shows up in the product mix effect.If technical change is not neutral with respect to resources (for example, if it is labor-saving), any unmeasured bias shows up in the resource mix effect.Similarly, if efficient expansion or contraction of the business involves nonproportionate expansion or contraction of outputs of inputs, the disproportionate features of scale economies shows up in the product mix effect and the resource mix effect.5The intent of Propositions 1-4 and Figures2-4is to demonstrate that even in the multiple input, multiple output case, it is in principle possible to decompose the profit change resulting from a producer's movement from ( comparison period price vectors, and observed comparison period quantity vectors.The technical change effect in Proposition 3 is expressed in terms of an observed base period output price vector and two unobserved output quantity vectors y Proposition 3 is expressed in terms of an observed base period output price vector, observed base period and comparison output quantity vectors, and two unobserved output quantity vectors y A and y C .The scale effect in Proposition 4 is expressed in terms of observed base period output and input price vectors, the observed base period input quantity vector, and two unobserved output quantity vectors y the resource mix effect in Proposition 4 is expressed in terms of an observed base period input price vector, an observed comparison period input quantity vector, and an unobserved input quantity vector to make the profit change decomposition analysis empirically useful.We now show how to recover each of these unobserved quantity vectors from the observed quantity vectors ( in Proposition 5 into the six components of profit change identified in Propositions 2 -4 enables one to recover each of the unobserved quantity vectors, and thus to conduct an empirical analysis of the sources of profit change from one period to the next.Even though the profit change decomposition involves five unobserved quantity vectors, it can nonetheless be undertaken.The key element in the decomposition is the distance functions.Since these distance functions must be calculated from observed data, we now show how to calculate them.All that is required is input and output quantity data for a sample of producers over a period of time.Price data are not required to obtain the unobserved quantity vectors; price data are used in the price effect, and as weights in each of the five quantity effects.Proposition 5 provides the theoretical foundation required to recover the matrix of M outputs produced by all I s producers in each of periods s = 1,...,t, and let recovered for each producer by solving each of the following linear programming problems I t times, once for each producer in the sample in period t. ×1 activity vector.From Proposition 5 (i), y oA = θ oA y ot .