empirical studies of cost function

This PDF is a selection from an out-of-print volume from the NationalBureau of Economic Research

Volume Title: Cost Behavior and Price Policy

Volume Author/Editor: Committee on Price Determination

Volume Publisher: NBER

Volume ISBN: 0-87014-190-2

Volume URL: http://www.nber.org/books/comm43-1

Publication Date: 1943

Chapter Title: Costs And Rate Of Output: Empirical Studies Of Cost Functions

Chapter Author: Committee on Price Determination

Chapter URL: http://www.nber.org/chapters/c2093

Chapter pages in book: (p. 80 - 115)

Chapter V

COSTS AND RATE OF OUTPUT:EMPIRICAL STUDIES OF COST

FUNCTIONS

THE preceding chapter explored the problems of designat-ing, for any single accounting period, those costs arising fromthe use of resources and equipment which contribute tothe output of more than one period. Such a discussion is anecessary introduction to measurement of the cost functionin particular enterprises. The present chapter will examinesome of the other problems which arise in an empiricalstudy of cost functions, survey the investigations that havebeen made thus far, discuss the techniques utilized, and at-tempt to appraise the significance of the results that havebeen obtained.

How costs are influenced by variations in the rate ofutilization of plant has been the subject of two principaltypes of inquiry. The first depicts a relationship existingover preceding accounting periods; the second, whilenecessarily based on the experience of the firm, attemptsto show the probable relationship in the next period. Stud-ies in the first category consist largely of investigationsundertaken by econometricians with the purpose of "ap-proximating" the cost functions of economic theory; theyusually include break-even charts constructed for invest-ment analysis. Those in the second group are developedby accountants, engineers and business managers; these aredesigned to provide constructive information for theformulation of decisions concerning output, costs andprices.

8o

EMPIRICAL STUDIES 8i

i. Problems in the Measurement of Cost Functions

Since controlled experiments relating to costs have notbeen undertaken on any appreciable scale,1 it has beennecessary for investigators to rely almost entirely on theaccounting records of particular firms. These data show inaccounting form not only the components of cost but thetotal cost for successive accounting periods. The variation inthese costs from one period to the next is ordinarily theresult of a large number of influences. Any statistical ap-proximation of the cost function in specific instances must,therefore, involve these two steps: (i) elimination of theeffects of those factors, other than variations in output,which impinge on costs, and (2) measurement of thisresidual relationship between costs and the rate of utiliza-tion.

The task of approximating a cost function might appearsuperficially to be simply a question of choosing the appro-priate statistical technique whereby the influence of onevariable may be isolated from among many, But beforeany particular statistical methods can be used, a large num-ber of questions of definition and interpretation must berecognized and possible sources of confusion clarified.Issues of this sort arise when one attempts to translateaccounting data into the simplified conceptual scheme of acost function and to adapt them to statistical analysis. In thepages immediately following, these difficulties are discussedand an endeavor is made to indicate the limitations of

'A number of experiments in the field of agriculture have measuredthe relations between such input factors as seed, labor, and fertilizeron the one hand and product on the other. See U. S. Department ofAgriculture, Input-Output Relationships in Milk Production, TechnicalBulletin 8zg (May 1942). From input-output functions with given factorprices, total cost functions can be derived. For a summary of some ofthese studies, see E. H. Phelps Brown, "The Marginal Efficacy of aProductive Factor—First Report of the Econometrica Committee onSource Materials for Quantitative Production Studies," Economet'rica,IV (April 1936), pp. 123-37. It must be remembered that studies of input-output relations are not equivalent to inquiries concerning cost func-tions.

8z COST BEHAVIORpossible solutions in order that particular studies may beevaluated critically in the next section.2 At the outset, coststudies are presumed to refer to a single enterprise as theunit of decision formation in order to avoid arbitrary alloca-tions of cost among component departments.

(i) The familiar cost function of economic theoryapplies only to one point in time and under equilibriumconditions is strictly timeless. All statistical approximationsto such relationships must, however, be based upon ob-servations in different periods. Consequently statistical costfunctions, like statistical demand curves,3 must proceed onthe assumption that the theoretical relations being approxi-mated are either unchanged during the period of observa-tion or change in a definable and measurable way. If thetheoretical cost function does vary during a period, butno correction for a particular pattern of change is madein the statistical analysis, the statistical cost function mustbe an average for the period. The longer the period ofobservation (i.e., the greater the number of values), themore confidence can be placed in the derived statisticalrelationship, but the less relevant are the results for apicture of the cost-output relation to be met in a particulardecision of the enterprise. The tendency on the part ofsome interpreters of statistical demand and cost curves toregard the derived results as valid for every instant in aperiod is an unfortunate one. For a statistically derived costfunction must be either an average relationship for a period,a relationship constant through a period, or one thatchanges according to a specified pattern.

2 The issues discussed here have been treated in a study by Joel Dean,"The Relation of Cost to Output for a Leather Belt Shop," TechnicalPaper 2 (National Bureau of Economic Research, 1941). See also, HansStaehle, "Statistical Cost Functions: Appraisal of Recent Contributions,"American Economic Review, XXXII (June '942), pp. 321-33.

Many of the difficulties and limitations encountered in the derivationof statistical cost functions are common to any attempt to approximatestatistically the concepts of comparative statics. Others are peculiar toempirical cost studies. See Henry Schultz, The Theory and Meamre-ment of Demand (University of Chicago Press, 1938), pp. 3-752, for adiscussion of the measurement of demand curves.

EMPIRICALSTUDIES 83

(2) Related to the problem just described is a secondone—the choice of the period of observation—which arisesin the attempt to determine an empirical counterpart to acost function. The period should be long enough to per-mit a number of observations covering a wide variationof output; but it is true also that the longer the period,the greater is the possibility of technical change, affectingboth processes and products, for which existing statisticalmethods do not make adequate allowance. Furthermorethe relationship between costs and the rate of utilization ofplant for an increase in output may differ from that pre-vailing when output is decreased. Such a cost functionwould be said to be irreversible. If the interpretation ofstatistical cost functions is to be clarified, both the lengthof the period of observation and the reversibility of thefunction require careful study. One may then check thereliability of the results by taking different periods ofobservation and comparing the effects of expanding andcontracting output in particular periods.

Certain restrictions on the interpretation of statisti-cally derived cost functions arise from the fact that ob-servations are not continuous. The accounting data applyto discrete intervals, which in some instances may be aslong as a year, in others as short as a fortnight. Values for asingle observation (e.g., a year) which sum the cost andoutput from day to day cannot reveal the relationshipexisting at any time within the year unless the assumptionis made that the cost function remains constant or changesin a definable pattern during the interval embraced by asingle statistical observation. As was analogously arguedabove in (i) for the total period of observation, the practi-cal result of these alternatives is to make the derived re-lationship an average for the discrete interval. The statisti-cal function must ignore the problem of the extent towhich the value for any discrete interval is determined bythe rate of change of output 'within the interval. It begsthe question of the behavior of total costs at differing

84 COST BEHAVIORrates of output within the interval of observation.4 Twoperiods may have exactly the same output but the costsmay differ simply because of the rate of expansion or con-traction of output during the period. The longer this inter-val the more serious the limitation.

A related problem encountered in the transitionfrom the cost function of economic theory to its statisticalapproximation is the difficulty introduced by variation inthe sizes of orders.5 The output of two periods might bethe same, except that in one case it consisted of a largenumber of small "runs" which might increase costs. Oroutput might be increased from less than "capacity" to"capacity" by gradual steps or by a single step; in eithercase the cost-output relations could hardly be expected toremain unchanged. It is possible to recognize the influenceof this factor by making the size of orders or the rate ofchange of output explicit determinants of costs along withthe level of output itself.

A rather simple, but sometimes neglected, limitationto the interpretation of any statistical cost function is thefact that it can apply only to the actual range of output.It is probably even more dangerous to project cost func-tions to volumes of output that have not been achievedthan to project trends into the future. Furthermore, sinceextremes in the volume of output seldom occur as fre-quently as more "normal" values, and hence do not offeras many points of observation, less confidence can beplaced in the shape of the derived cost function at itsextremes. This conclusion is of particular importance, be-cause the costs for extreme values of output attract the

4To a certain extent the effect on costs of the rate of change of out-put—calculated from outputs in successive periods—may be accountedfor and eliminated from the cost function, but the available techniquesare not adequate to deal with all the variations which can and do takeplace within the accounting period.

On some of the effects of change of output within the interval, seeCaleb A. Smith, "Cost-Output Relation for the United States SteelCorporation," Review of Economic Staiistics (November 1942).

5 point is listed separately because output is ordinarily varied inincrements larger than a single "unit" of product.

EMPIRICAL STUDIES 85

most comment from business executives and persons inter-ested in the consequences of business decisions.

(6) It is often a rather difficult task to make certainthat the observed values of output and costs apply to thesame period. A cost function in which the costs of oneperiod were related to the output of another would ordi-narily be meaningless. The longer the time during whichgoods are in process, the greater is the likelihood of thistype of distortion. Expenditures may be recorded mostheavily while goods are being processed, whereas outputmay not be recorded until the goods have undergonefinishing operations on which expenditures are smaller.Thus costs may be accounted in one period and the outputto which they contributed in another. When the volumeof output is being varied rapidly, this lack of synchroniza-tion in the cost and output records may introduce seriousdistortions unless lag corrections are adopted. Again, lackof comparability may arise because of variations in thetime between the incurrence of expenses or completion ofoutput and the recording of these events.0

(') One of the most perplexing aspects of the attemptto approximate cost functions is the question of the unitsin which output is to be measured. If each firm produceda single, homogeneous product, as is so frequently assumedin simplifications of economic theory, the measurement ofoutput would be only a matter of counting these units.But (as was noted in Chapter HI) in reality firms or evenplants produce many different styles, grades, models, andsizes of a "product" as well as many different "products."When output is heterogeneous and composed of itemswhich pass through substantially the same productionprocess, so that their individual costs cannot be isolatedby separate study of various divisions of the plant, a corn-

6The fundamental difficulty of calculating costs (and income) for aperiod short of the life of an enterprise, discussed in derail in the pre-vious chapter, is again relevant.

TThe problems of calculating the costs of specific "products" arediscussed in Chapter VIII.

86 COST BEHAVIORmon denominator must be if the relation betweencost and "output" is to be examined.8

The major alternatives in the measurement of output forcomparison with costs can be briefly surveyed. (a) Theunits of input factors may be an acceptable index of thevolume of output in some instances. In oil refining, forexample, the input of crude oil is fairly homogeneous,while the output is composed of highly heterogeneous jointproducts produced in varying proportions. If the rate ofactivity of the plant is satisfactorily reflected by the rateof input of the major homogeneous raw matenal, and ifthe range of variation in the proportion of different prod-ucts is not wide, the use of input factors may be taken asan index of output. (b) Since it may be impossible toreduce the measurement of output to a single dimension,more than one measure of output may be used as separateindependent variables in multiple correlation analysis. Forexample, in a study of the relationship between cost andrailway traffic volume, it might be necessary to measure"output" in terms of passenger miles, freight ton miles,movement of mail and express, etc. (c) A composite indexof output necessarily involves the determination of therelative importance or weight to be attached to the variousproducts making up the composite unit. The weights sug-gested in particular cost studies include: the relative pricesof the products, the amount of raw materials entering intothe different products, and their variable or total cost.

For a general discussion of production index problems, see WassilyLeontief, "Composite Commodities and the Problem of Index Numbers,"Econometrica, IV (January 1936), pp. Arthur F. Bums, "TheMeasurement of the Physical Volume of Production," Quarterly Journalof Economics, XLIV (February 1930), pp. 242-62; Edwin Frickey,"Some Aspects of the Problem of Measuring Historical Changes in thePhysical Volume of Production," Explorations in Economics, Notes andEssays contributed in honor of F. W. Taussig (McGraw-Hill, 1936),pp. 477-86, and "The Theory of Index-Number Bias," Review of Eco-nomic Statistics, XIX (November '937), pp. 161-73; 0. N. Anderson,"On the Question of the Construction of an Internationally Compara-ble Index of Industrial Production," Publications of the Statistical insti-tute for Economic Research, No. i (State University of Sofia, 1937),pp. 121—31.


In addition to empirical studies of cost functions, outputindexes have been made to measure "productivity" andindustrial activity, and to serve as a basis for some nationalincome estimates. A system of weights adapted to one ofthese types of study may be quite misleading if applied toanother. For instance, output weighted by the relativeprices of constituent items may be useful for national in-come purposes but inadequate for cost purposes. If relativeprices differ because of variation in the character of com-petition, national income as a flow of services which satisfygiven tastes can be regarded as affected by this fact; but itmakes little sense to argue that output is any different forpurposes of calculating a total cost function. The bestweights for individual products, in the construction ofcomposite indexes of output for cost functions, wouldusually appear to be their direct costs. Only if relativeprices were a reflection of difference in relative costs wouldprice weight be theoretically valid for output in the cal-culation of cost functions.

Regardless of the system of weighting used for indexesof output, the assumption must be made that the weightof each individual element in the composite unit remainsunchanged throughout the period of observation. Thelonger the period of observation the less likelihood thereis that any proportion of component products will remainunchanged.

When indexes of output for all the products of a firmare utilized in the calculation of cost functions, the resultsare not directly applicable to individual products. If anattempt is made to analyze the pricing policy for a singlecommodity, it is necessary to investigate the extent towhich the composite output unit is typical of the fluctua-tions in output of that commodity. Since the index of out-put is a weighted average, it is possible that it will not betypical of any particular commodity. This may be a seriouslimitation where the relative proportion of various com-modities in total output fluctuates a great deal.

(8) If the influence upon costs of variations in the rate

88 COST BEHAVIORof plant utilization is to be isolated statistically, other in-fluences must be measured and eliminated. The designationof the other influences that may be expected to affect costssubstantially, and the choice of the methods for eliminat-ing their influence are both vital steps. The former isimportant because the residual relationship is assumed to bethe cost function. If the impact of certain factors is notremoved, the cost-output function is distorted. Strictlyspeaking, it is impossible to be certain that the effects of allother factors have been eliminated. After corrections havebeen made, the residual variations in cost are assumed tobe solely the result of variations in. output. The effect of"technical changes" can be expected to present particulardifficulties; one cannot be certain that plant, equipment,supervision and products have remained exactly the sameeven over short periods,9 for in many industries they aresubject to frequent and constant modification. As for thechoice of methods for measuring and deflating the effectsof factors other than output, it is important because ofthe different shapes of cost functions that may result. If, forinstance, the influence of factor price movements is to beeliminated from costs, the investigator will have to decideon the extent of detail in the classification of factors, andon what constitutes a change in factor prices.

(ç) The reliability of a measurement of a cost functionmust depend also upon the selection of the equation tosummarize the cost-output relationship. Since marginal costis the slope of the total function, the adoption of a linearregression necessarily results in constant marginal costs.Extreme care must be utilized, therefore, in the choice ofthe mathematical function or the freehand curve usedto summarize the observed relation between costs and out-put. While the adoption of higher order equations willimprove the statistical fit, the crucial question is whetherthe greater number of constants results in a "significantly"improved summary relationship. Statistical tests can beused to weigh the relative merits of proposed equations.

°See Chapter X.


Perhaps the simplest of these tests would plot the paireddifferences between successive observations of costs andoutput. Some clue as to the shape of the marginal costfunction is then derived independently of deductionsfrom a mathematically fitted total function. Or the "nohypothesis test" can be applied; this is a device for calculat-ing the probability of obtaining the observed results fromthe several suggested functions. If the observed resultscould have been obtained only with small probabilitiesfrom a fitted function, the particular hypothesis can be dis-carded. Finally, the analysis of variance technique permitsa judgment on the relative "goodness of fit" of severalfunctions.1°

The preceding points indicate most of the importantquestions involved in an empirical approximation to a costfunction. These problems arise in the process of attemptingto approximate the cost function of economic theory. Theyhave been enumerated, not in order to suggest that it isfutile to try to derive exactly the cost function of theeconomist, but rather to the issues on which empiri-cal studies must be appraised. One study may be adjudgedsuperior to another, or an investigation in one industrymore feasible than in another, according to the way in whichthe issues outlined above are treated.

Since the complexities of experience cannot be com-pressed into the simplified models of economic theory,some way must be found to evaluate statistical results. Theacceptable tests of the significance of such results fall intotwo categories. First, there are the statistical tests of theextent to which the derived relation between output .andcosts fits or "explains" the observed costs. In the periodof observation the cost function permits a "prediction" ofcosts from output. Tests of this sort indicate the degree ofreliability that may be credited to such "predictions." Thesecond type of test is much less definite, though equally

10 For further discussion and illustration of this technique, see JoelDean, Statistical Cost Functions of a Hosiery Mill (University ofChicago Press, 1941), pp. 37-5g.

90 COST BEHAVIORimportant; it gauges the significance of empirical resultsin terms of consistency with other types of information—the impressions and opinions of business executives, andlogical deductions from information that can be acceptedwith greater confidence.

2. Empirical Studies of Cost Functions

Two distinct types of investigations have attempted todetermine the cost function of economic theory; they areto be differentiated by the methods employed. One groupof studies has applied to the cost records of precedingperiods statistical techniques of varying refinement in orderto isolate the effects of variations in the rate of output fromother factors presumed to influence costs. Kurt Ehrke11in Germany, Theodore 0. Yntema,12 Joel Dean13 andRoswell H. Whitman14 in the United States have deriveda number of these empirical cost functions from theaccounting records of particular firms. Such cost functionswill be designated as "statistically derived" to distinguishthem from another group of empirical cost-output studies.The second type of inquiry lies between that just indicated

11Die Ubererzeugung in der Zeinentindustrie von 1858—1913 (GustavFischer, Jena, 1933).

12United State: Steel Corporation T.N.E.C. Papers, Comprising thePamphlets and Charts Submitted by United States Steel Corporation tothe Temporary National Economic Committee (United States SteelCorporation, 1940), Vol. 1, pp. 223-30!. See also citation in footnote 19,below.

13Statistical Determination of Costs, with Special Reference to Margi-mu Costs (University of Chicago Press, 1936). Chapter II contains asummary of cost studies concerned with the effects of variations inoutput. See also Statistical Cost Functions of a Hosiery Mill and "TheRelation of Cost to Output for a Leather Belt Shop" and "Department-Store Cost Functions," Studies in Mathematical Economics and Eco-nometrics, edited by Oscar Lange, Francis McIntyre, and Theodore 0.Yntema (University of Chicago Press, 1942), pp. 222-54.

14"Cost Functions in the Department Store" (in manuscript). See also"Round Table on Cost Functions and their Relation to Imperfect Com-petition," American Economic Review, Supplement, XXX (March 1940),

400-02.


and break-even charts, discussed in Section Studies ofthis sort are estimates by engineers and accountants, pre-sented as cost functions and focused on past experience.A number of these inquiries have been made in Germanyby E. Schmalenbach,'5 Reinhard Hildebrandt,16 HerbertPeiser'7 and others.'8 Cost functions derived in this fashionwill be labeled "estimated" in order to distinguish themfrom those "statistically derived"; the terms are in no wiseintended to reflect upon the relative accuracy of theempirical cost functions.

Attention will be directed first to the "statistically de-rived" relationships between cost and output, for whichProfessor Yntema's study of the United States Steel Corpo-ration will be used as an illustration. This choice followsfrom the fact that the Steel Study'9 is the largest and oneof the most complex yet undertaken and has received wideattention through the hearings of the Temporary NationalEconomic Committee. If a problem as perplexing as thecost function for the entire United States Steel Corpora-tion can be solved satisfactorily, the same methods maywell serve in simpler cases. The use of Professor Yntema'sstudy will also permit comparisons with two other inquiriesbased only.upon published quarterly and annual statementsof the United States Steel Corporation: the investigation

Selbstkostenrechnung und Preispolitik (Gloeckner, Leipzig, 1934).'°"Geschäftspolitik auf Mathematischer Grundlage," Technik und

Wirtschaft, XXIV (193').Der Einfluss des Reschäftigungsgrades auf die industrielle Kosten-

entwicklung (Gesellschaft für Betriebsforschung, Berlin, 1929).'8H. Müller-Bernhardt, industrielle Selbstkosten bei schwankendem

BescbIftigungsgrad (Berlin, 1925). A Danish author, Ivar Jantzen, hasmade several studies, among which "Voxende Udbytee i Industrien,"Nationaløkonomisk Tidsskrift, LXII (1924) appears to be the most im-portant. Also see the study by Hans Boiza, "Kostenstudien mit Erfahr-ungs-Zablen aus der Praxis," Nordisk Tidsskrift for Teknisk økonomie(June '937), pp. 97-109.

19 "Steel Study" will refer to the investigation presented by the UnitedStates Steel Corporation to the T.N.E.C. For this reference see, Hearing:Before the Temporary National Economic Committee, 76th Congress,3rd Session, Part 26, iron and Steel industry, January 23, 24, and 25,1940.

92 COST BEHAVIORof Kathryn H. Wylie and Mordecai Ezekiel,2° which"statistically derives" a cost function, and the break-evenchart adopted by Walter Rautenstrauch.2' The present dis-cussion deals with the Steel Study as an example of tech-niques that may be used to meet the difficulties (inherentin any empirical study) mentioned in the preceding section,and does not attempt to be a comprehensive critique ofthis particular investigation.

The procedure followed in the Steel Study can be sum-marized in three major steps:

(i) From the annual profit and loss statements of thecorporation, summarized for the Federal Trade Commis-sion (February i 1939), annual total cost figures for theperiod 1927—38 were obtained by the summing of the fol-lowing items:

Operating Costs, including cost of goods sold, operating ex-penses of transportation, etc., selling and administration,taxes (other than income and profit),22 depreciation anddepletion

Idle Plant ExpenseInterest on Mortgages and Bonds

The total cost figure 23 was corrected for the duplicationinvolved in inter-company transactions between subsidiariesof the United States Steel Corporation. This was a neces-sary adjustment, since "the inter-company sales of anyone company constitute the costs of the other, and sinceinter-company profits are eliminated from inventory valua-

20 "The Cost Curve for Steel Production," Journal of Political Economy,XLVIII (December '940), pp. 777-821.

21 The Economies of Business Enterprise (John Wiley and Sons,1939), pp. 303-09.

22 Income and profit taxes are not considered "costs"; they vary withprofits rather than in any independent way with output. In this sectionthe term "taxes" will always refer to nonincome and profit taxes.

28Discounts from purchases were first subtracted from the abovetotal. Accounting practice charges the corporation or its subsidiaries thelist price of purchased materials and supplies, regarding discounts as adebit item. Costs for the present purpose required the correction, pro-viding a net figure on purchases involving discounts.

EMPIRICAL STUDIES 93tions in making inventory adjustments, both costs andsales and revenue are inflated, from a consolidated view-point, by the amount of the inter-company items . •

•" 24

(2) Since the output of the United States Steel Corpo-ration consists of at least fifty thousand steel "products"and a multitude of nonsteel items such as cement, coal andiron ore, and transportation services, a major step in thestudy was the construction of a measure of output withwhich to compare costs. This was accomplished by cor-recting the actual tons of each product shipped in eachyear by the ratio of its average mill costs to the averagemill costs of all rolled and finished steel products in theperiod 1933—37. Each ton of a product whose 1933—37average mill cost was less than the average of all rolledand finished products was weighted proportionately lessthan a full ton, while products whose average mill cost wasabove average counted proportionately more than a fullton. The number of tons of all products was converted intoequivalent tons of "average cost rolled and finished steelproducts." In similar fashion the output of rolled andfinished products was adjusted to include "the equivalenttons of steel represented by the products other than steelwhich are sold on a tonnage basis by the Corporation'ssubsidiaries."25 One further aspect of the measurement ofoutput must be noted; the data used in constructing thiscomposite index refer to tons shipped (rather than pro-duced) in the year.26 This gap was "bridged" in the SteelStudy by the statement that "the amount of inventoryfluctuation is relatively small . . . and the effect on theultimate cost computation is negligible." 27

Since variations in these total costs were obviouslydue in part to fluctuations in the prices and efficiencies ofinput factors, a series of corrections had to be used to re-

24 United States Steel Corporation, T.N.E.C. Papers, I, pp. 237-38.25 Ibid., p. 239. Cement products are apparently excluded.26That such a probiem would arise was noted in Section the cost

data refer to the goods produced in the year.27 Ibid., p. 241.

94 COST BEHAVIORmove the effects of these influences on total costs. To thisend 1938 was adopted as a base, and costs were recon-structed on the assumption that factor prices and efficien-cies then prevailing had been constant through the wholeperiod. Four corrections were made: (a) Total payrollswere adjusted to the 1938 level of average hourly earn-ings.28 (b) "Other expenses" than payrolls and (nonprofit)taxes were presumed to be largely purchased supplies andmaterials, and one half of this total was corrected by theBureau of Labor Statistics index of wholesale commodityprices other than farm and food products related to a 1938base.29 (c) To correct tax costs for variations in tax ratesduring the period, the 193 2—38 relation between taxes and"output" 30 was projected to the 1927—31 period. Twolinear relationships between taxes and output were foundin the data if the period was split at the year içp. The1932—38 relationship was projected backward to the earlierperiod to eliminate the consequences of tax rate changes.Actual values were used in the 1932—38 period. (d) Sincenumerous changes in the techniques of production tookpiace in the period, it was important to eliminate their in-fluence. Total costs, corrected as just indicated, were plot-ted against "output" and the deviations from a least-squaresstraight line were plotted with time as the independentvariable. A straight-line fit through these deviations waspresumed to measure the influence of changes in technique.Elimination of this effect on cost resulted in the final re-lationship between costs and output. A straight line sum-marized the relation with only small deviations, as can be

28 the discussion wage rates and average hourly earningsare identified. The correction is made for variations in wage rates forwhich an index of average hourly earnings is tacitly presumed to be asuitable measure. For instance, see ibid., p. 246 and Table iz, entided"Adjustment of Payroll to 1938 Wage Rates." The caption in the tableis "average hourly earnings."

29 argument for this procedure was given except, "in order notto overadjust, only one half of the 'other expense' items have been ad-justed." ibid., p. 249.

measure of output was discussed under (2) above.


seen from Chart i, indicating a linear empirical cost func-tion or a horizontal empirical marginal cost function,within the range of observation.8'

Substantially similar results were found by Kathryn H.Wylie and Mordecai Ezekiel. "Even with the more accu-

Chart IRELATIONSHIP BETWEEN TOTAL COSTS OF OPERATION

Millions of dollars

AND VOLUME OF BUSINESS - 1938 CONDITIONS

U. S. Steel Corporation end Subsidiaries

1,200

1,100

1,000

900

800

700

600

500

400

300

200

100

00 1 2 3 4 5 6 7 8 9 10 II 12 13 14 $5 16 17 18

Millions of weighted tons of all tonnage pro4ucts shipped

Note: Total costs adjusted to 1938 interest,tax, pension, and wage rates; to 1938 price level; andto 1938 efficiency.

Reproduced from TN(C Hearingt, Part 26, Iron and Steal Industry (January 23—25. 1940), p. 13773

rate data . . . the results secured by the Corporation[Yntema's study] were very similar to those secured byour study." Perhaps the only significant difference is the

311t is important to note that the effect of all these corrections didnot substantially alter the shape of the cost function calculated fromunadjusted data. In the form y = a + bx, the latter was $120,530,000+ $54.gIx, where x is a "weighted ton" of output. The corrected func-tion was $182,000,000 + $55.73x. The corrected function fits the ob-served values better, but the difference in the fit between the twofunctions is not striking. Compare Chart i (ibid., p. 240) with Chart(p. which is reproduced above.

1929

96 COST BEHAVIORfact that the Wylie-Ezekiel investigation found "there wasevidence of a sharp downward trend in labor requirementsper ton, for given rates of output from 1920 to 1934, withlittle change thereafter." Even this difference must beviewed as tentative since the regressions of costs on wagerates and efliciency in the 'Wylie-Ezekiel study were sub-ject to a wide range of error.82' 33

While a comprehensive critique of the procedure of theSteel Study is beyond the scope of the present survey,34it is important to form some judgment as to the reliabilityof the results. The present concern is with the degree ofconfidence that can be placed in the shape of the statisti-cally derived cost function. There is evidence that thetechniques used in the Steel Study to meet the problemswhich arise in every such inquiry (Section i) have intro-duced some linear bias into the cost function. Hence, themethods of investigation alone may account for a linearcost function in place of the conventionally assumed, in-verted ogive. The possible sources of this sort of bias canbe examined briefly.

82Two of the Wylie-Ezekiel analyses (based on annual data) showslightly declining marginal costs throughout the range of observedoutput; the quarterly data reveal a linear total cost function.

The study of Bernard H. Topkis and H. 0. Rogers, "Man-Hours ofLabor per Unit of Output in Steel Manufacture," Monthly LaborReview (May 1935), p. x i6i, is interesting in this connection. Manhoursplotted against percentage of capacity in the range of zo-6o percentyield a negative average relationship but a rising incremental (marginal)relationship.

United States Coal Commission developed a formula in 1913for the determination of "incremental" costs. Although apparently noelaborate calculations are made, there is reason to believe that variationsin the rate of output in certain ranges would materially affect costsbecause of the importance of pumping charges and ventilation. SeeW. C. Trapnell and Ralph lisley, The Bituminous Coal industry with aSurvey of Competing Fuels, Federal Emergency Relief Administration(multigraphed, December 1935), pp. 84-86.

a detailed discussion of the techniques used in the study, seethe testimony of Melvin G. de Chazeau, Louis H. Bean, MordecaiEzekiel and Martin Taitel before the Temporary National EconomicCommittee, January 23, 1940. See also Caleb A. Smith, bc. cit., wherethe linearity bias is discussed at length.

EMPIRICAL STUDIES 97(a) In order to correct costs for changes in the efficien-

cies of input factors, a straight line was fitted to the devia-tions from a cost-output regression line. Employment ofthis device is equivalent to assuming that changes in "tech-nology" affect costs in a constant fashion through allphases of the business cycle.35 Chapter VII will demon-strate that this is probably not the case; technical changescan be expected to reduce costs by a larger extent at some

- phases of the cycle than at others. If these changes reducecosts most when output is largest, as there seems somereason to believe, then the observed cost values at highlevels of output are "too low" compared to what theywould have been without the technical change. To makelarger corrections for technical change at higher levels ofoutput would introduce an increased slope into the costfunction.

(b) The influence of variations in wage rates on pay-rolls was removed by an index of average hourly earnings.This presumes that both series show equivalent patternswith respect to variations in output. A consistent tendencyfor average hourly earnings to increase more than rateswith increases in output (and vice versa for decreases inoutput) would result in an overcorrection of payrolls andan underestimate of costs at the higher output levels. Thatsuch a relative pattern of earnings and rates is probablefor the United States Steel Corporation follows from thediscussion in Chapter VI. Earnings may be expected to risefaster than rates because of better coordination betweenpiece workers at higher output levels, graduated bonuspayments for increasing output (important in the steelindustry), greater effort of wage earners when large back-logs of orders exist, and technical changes which canoperate directly on earnings under piece rates. Variations inthe composition of the working force between different

35The scatter in Chart 6, ibid., p. 253, where the technical changecorrection is made, fits- the regression line very poorly. The regressionline drawn in the chart is markedly influenced by the single year 2927.

98 COST BEHAVIORpay levels at different rates of output are certain to affectpayrolls. But no simple pattern can be deduced.36

(c) The effects of changes in tax rates were eliminatedby extending the linear relationship for the period 1932—38to the years 1927—31. Such a technique really assumes alinear function for at least this single element of cost andtends to introduce linearity into the total cost function byrestricting the segment of costs in which any nonlinearrelationships can be found. If the same methods had beenused to deflate all "factor" prices, the total function wouldnecessarily have to be linear.37 The aggregate effect of thismethod of eliminating tax rate variations is probably notlarge in the Steel Study.

(d) The acceptance of the Steel Corporation's alloca-tion of overhead over time can also be expected to intro-duce a bias into the cost function, and in general a linearbias. As was noted in Chapter II, any specification of"fixed" costs to accounting periods shorter than the lifeof the enterprise must involve estimates of the future per-formance of the firm, and hence rule-of-thumb methodsin the calculation of costs. The conventional methods of

361t is commonly supposed that as output expands the occupationalcomposition of the working force contains a smaller proportion ofhigher paid positions, an idea probably derived from the notion of askeleton work force that is not discharged at low levels of output. Butthis presumption overlooks the extent to which a large plant can layoff men in each pay level and still maintain skeleton crews withoutchanging the relative proportion of men at each pay interval. Further-more, account must be taken of promotion and "degrading" policies,seniority or division of work provisions of trade union agreements, etc.A comparison of the proportion of wage earners in various occupationalgroups reveals no clear cyclical pattern for the industry as a whole.See Bureau of Labor Statistics, "Wages and Hours of Labor in the Ironand Steel Industry," 1929 and 193!, Bulletin 513 and 567.

Chart E-i7, United States Steel Corporation, op. cit., II, p. 187, showsa slightly negative relation between average hourly earnings and output.The payrolls data, however, include the salaries of management. At lowlevels of output the weight of management salaries is higher than atcapacity output. Average hourly earnings consequently show theslightly negative relation to output.

under the circumstances that any element of cost was zeroor negative before the common lower limit of output.

EMPIRICAL STUDIES 99straight line allocation tend to underestimate total costs inperiods of large output and to overestimate them in timesof low output. This bias may be balanced partially by thefact that on some occasions of large output the corporationhas written off larger amounts as reserves for amortizationof investments. Only half of the usual depreciation wastypically charged to completely idle plant. But suchcorrections are partial and sporadic and make allowancesin one period that should be spread over a longer time. Thecumulative effect of these four factors is to suggest a moreconventional cost function than that derived statisticallyby Professor Yntema. On the other hand it is impossibleto say how much curvature these factors would introduceinto the marginal cost function. In view of the similaritybetween the adjusted and unadjusted functions, the firstthree of the limitations to the Steel Study just noted proba-bly would not greatly change the derived function,although they do detract somewhat from the confidenceto be placed in the empirical results.

Further difficulties of interpretation arise in the SteelStudy from the way in which other problems listed inSection i were handled. Crucial among these are the meas-urement of output, the number and frequency of observa-tions, the selection of the form of the cost function, andthe comparability between periods of cost and output data.The output of all of the diversified products of the SteelCorporation was measured in terms of tons, weighted bythe ratio of average mill costs of any product to the aver-age mill costs of all rolled and finished products. Becauseof the change in the relative importance of such productsas "sheets" through the period 1927—39, this measure ofoutput is not entirely satisfactory. The independent varia-tion in the output of such nonsteel products as coal, trans-portation, and iron ore raise essentially similar difficulties,although the direction of the impact of these ambiguitiesupon "output" is not readily apparent.

The Steel Study does not reveal whether any tests weremade of the reliability of the selection of a linear regression

100 COST BEHAVIORbetween the twelve annual values for output and totalcosts. A cubic or higher order equation might have beenmore appropriate. The fact that a higher order equationmust necessarily fit the observed values of cost and outputwithin narrower limits does nor, however, render the linearregression invalid. The difference in "closeness of fit" ofthe linear and higher order equations must be large enoughto be statistically "significant." Perhaps of even greaterimportance, the difference must be sufficiently large so thatthe higher order equation indicates more accurately theview of cost behavior which figures in decision formation.

On the basis of the foregoing critical appraisal, it wouldappear that while a linear function may be the best singleestimate of the cost function, the degree of confidence thatmay be placed in the results leaves quite a range for possi-ble curvature in the total cost function and, a fortiori, inthe marginal cost function.38 The statistical function maygive a reasonably accurate estimate of the effect on costsof a variation in output from year to year on the average.It should not be presumed, of course, that the derivedfunction is necessarily relevant to decisions regarding out-put and price for particular products at specific times.

Other studies of cost functions have yielded, in manycases, very similar results.3° Although methods analogousto those utilized in the Steel Study have been adopted, thepeculiar features of each firm and industry have substan-tially influenced the character of the available data andthe techniques employed in each case. For instance, thestudy of the cement industry by Kurt Ehrke and E.

had a fairly homogeneous product with which88Richard Ruggles, "Linear Total Cost-Output Regression," American

Economic Review, XXXI (June '941), pp. 332-34.39J. M. Clark, "Basing Point Method of Price Quoting," Canadian

Journal of Economics and Political Science, IV (November 1938), p. 479n., remarks that in the cement industry marginal cost "does not appearto change much with changing rates of utilization."

40Kurt Ehrke, op. cit., pp. 275-3 zo. Dr. Hans Staehle reports thatcalculations based on these cement data reveal rising marginal costsat levels of output above rated capacity.


to deal, whereas Joel Dean's investigations covering firmsin the furniture, leather belting, and hosiery industries anda firm in the department store field4' presented problems inoutput measurement more complex than those in cementbut less involved than in the case of the Steel Corporation.Roswell Whitman's study of a department store is one ofthe few to show other than a completely linear cost func-tion; he found marginal costs rising sharply at the Christ-mas season peak in output.42 Throughout the rest of therange of fluctuation in output there appeared to be a virtu-ally linear cost function.

In addition to "statistically derived" cost-output rela-tions, as was noted at the beginning of this section, a num-ber of studies of "estimated" cost functions have beenmade, particularly by German writers. These inquiriesdiffer from those typified by the Steel Study in that theyutilize cost estimates rather than the direct accountingexperience of a firm. The estimates must rely, of course,on the actual cost experience of the enterprise, but not inany simple and automatic fashion. In many ways "esti-mated" cost functions resemble break-even charts, and forthis reason much of the discussion in the next sectionwill apply to them. Both characteristically measure outputin terms of percent of "capacity," identifying, at least im-plicitly, "economic" and "physical" capacity; both fre-quently require the collaboration of engineers and account-ants. The estimated functions are to be distinguished,however, from break-even charts, since they are cast in theform of a cost function, usually are unaccompanied byestimates of revenue so that they cannot indicate a break-even point, and are strictly historical.

The studies of estimated cost functions invariably arriveat a linear Reinhard Hildebrandt, for example,

cit.42American Economic Review, Supplement, XXX (March r94o),

p. 401.See estimates of the way in which costs for power stations vary

with rate of utilization of plant, in Walter Rautenstrauch, op. cit.,pp. 109-U.

102 COST BEHAVIORregards this relationship as typical: "The repeatedly ob-served phenomenon that, in addition to materials and laborcosts, common indirect costs also show themselves as alinear function of rate of plant utilization, and always atsimilar rates and accompanied by similar values, cannot beregarded as an isolated phenomenon characteristic of onlya few plants."44 The linearity conclusion is somewhat sur-prising in that the estimated functions, like break-evencharts, cover the range of output from zero to full capacity.It is difficult to tell whether this is simply a matter of pro-jecting the relationship expected in the "normal" range offluctuation to upper and lower limits, or a result of carefulcalculation, in the absence of extensive experience withlevels of output at either extreme, the former possibilityappears the more plausible.

3. Relationships Between Sales and Costs

The relationship between sales and costs is frequently pre-sented in break-even charts. They are a relatively recentaddition to the tools of business analysts, engineers andexecutives, and are widely used as an aid to many differ-ent sorts of decision. Such charts are of interest here becausethey contain estimates of the way in which costs vary (orare expected to vary) with sales at constant prices and hencewith output. The X-axis shows output (usually measuredin percentage of capacity); the Y-axis is labeled in dollars.A line indicating total costs and another indicating revenueat various rates of output are plotted according to thissystem of coordinates. This kind of chart takes its namefrom the fact that the lines intersect where costs equalrevenue, that is, the "break-even" point of plant utilization.Total cost is sometimes broken down into its componentelements, at least into fixed and variable costs. The SteelStudy discussed in the preceding section is presented in

"Op. Cit., p. 127.


the form of a break-even chart by the addition of tworevenue lines as is shown in Chart

At least two types of break-even charts are constructed;4°

Chart 2

RELATIONSHIP BETWEEN SALES AND COSTS:EFFECT OF REDUCTION FROM AVERAGE 1938 PRICES

U. S. Steel Corporation and Subsidiaries

Millions of dollars1,200

- Sales and revenues (1938 prices)1.100 —— —— Sales and revenues (1938 prices less 10%)

1,000

900

________

Break—even

800 points

700

600

500Total costs

400

300

200

100 -

0 —4 a . ...

0 I 2 3 4 5 6 7 8 9 10 Il 12 13 14 IS 16 17 18

Millions of weighted tons of all tonnage products shipped

Note: Costs are based on 1927—1938 experience, adjusted to 1938 conditions.Reproduced from Hearings, Part 26, Iron and Steel Industry (January 23-25, 1940), p. 13777

one, the historical, is based on the income and expense ofan enterprise over a period of years as shown in the profit

United States Steel Corporation, op. cit., Vol. II, p.461n addition, some firms draw up detailed budgets showing costs and

realizations for different outputs, and consequently the break-even point,without any chart being drawn. See, for instance, Samuel H. Selman,"Going to Make a Profit?", Factory Management and Maintenance,96 (1938), p. 92. Similarly, the rate of utilization of a particular machineis recognized as relevant to the decision whether or not to buy newequipment. See, for example, E. M. Richards, "To Buy or Not to BuyEquipment," Factory Management and Maintenance, 91 (December1933), p. 499.

104 COST BEHAVIORand loss statement.47 The other, the budgeting break-evenchart, is based on an estimate of the relationship whichmay be expected to prevail for an ensuing period. Thehistorical break-even chart is used by investors in choosingbetween alternative issues, as is evidenced by the fact thatthe term "break-even point" has become a commonplacein investment analysis.48 Adjustments for factor pricechanges and certain problems of measuring output areskirted by the use of dollar sales to measure output. Break-even charts of this type must be employed with extremecaution. In particular, long periods of observation shouldbe avoided, especially when there is a rather steady increasein the volume of output, as the chart may become in effecta curve showing how costs behave under conditions ofexpanding plant rather than of the fluctuations of output ina relatively constant plant. A break-even chart of this sortis most valuable when the enterprise has not introducedany fundamental changes in the scale of operations in aperiod during which there has been sufficient variation inoutput to establish the expense line clearly. The two fol-lowing break-even charts, one for the First National Stores(Chart 3),49 and the second for the Granite City SteelCompany (Chart 4),50 illustrate respectively a case inwhich output was expanded virtually throughout the period,presumably through an expansion in the number of operat-ing units, and a case of apparently unaltered plant untilthe last year covered.

The budgeting type of break-even chart based on esti-47Break-even charts summarizing profit and loss statements do not

make explicit corrections for changes in factor prices. The use of dollarsales as a measure of output serves as a partial corrective. For illustra-tions of historical break-even charts, see Walter Rautenstrauch, op. cit.,Ch. 7.

term is at least four times in the Standard Trade andSecurities Analysis of Steel and Iron Companies, Basic Survey, Part II(March 3', 1939), Vol. 91, No. 26, Sec. 3.

Rautenstrauch, Cit., p. 311.601b1d., p. 308. The most significant change was the installation of a

continuous strip mill in 1936. The point for this year is the farthestfrom the regression line.


mates is used by executives in making decisions. Such achart provides a simple and convenient way in which toestimate the effects on profits of changes in wage rates ormaterial prices; the cost line need only be moved appro-priately. One set of cost and revenue lines must be drawnfor each level of wage rates, material prices, and pricerealizations from goods for sale. It has been suggested thatthese charts are useful also for comparisons of the per-formance of plants of different companies.5' When variousprice structures are being considered, break-even chartsprovide a convenient way of presenting the effects onprofits of different revenue lines (arising from differentprices) Such charts purport to provide direct answersto a question like the following: WTith given factor prices,what must sales return to provide a profit of a specifiedamount? Charts relating output and specific elements incosts may afford important help when budgets are drawnup for a subsequent period, in view of expected levels ofoutput.58

The present. interest in break-even charts is centered onthe shape of the cost lines—the way in which costs varywith different amounts of output. Almost all break-evencharts, whether historical or budgeting, appear with lineartotal cost-output relations. The historical break-even chartsshow straight line relationships partly because of the biasof investigators in looking for a simple relation, such as afixed cost plus a constant variable cost per unit of output,but also because the points plotted on the chart for a period

61 example, a chart comparing Chrysler, General Motors andPackard is presented in Rautenstrauch, op. cit., p. 303.

52 2 (supra) from the Steel Study is drawn to show the effectof a to percent price change. It should be noted that revenue lines areordinarily drawn as if the price of goods sold were independent of theamount sold, in spite of the fact that interest in the break-even pointimplies a realization that the market will not automatically take anyoutput the firm can produce.

Charles Reitell, "Merchandising Operations," in CorporateStatements, Proceedings of the Accounting institute, 1940 (ColumbiaUniversity Press, £940), pp. 3 1-32, illustrates the estimation of deliverycosts from a scatter of costs and volume.

io6 COST BEHAVIORduring which the plant has not been fundamentally changedusually reveal an essentially linear relationship. In the caseof budgeting break-even

Millions of dollars120

hO

l00

90

80

70

60

50

40

30

20

I0

0

Chert 3

THE BREAK-EVEN CHARTOF FIRST NATIONAL STORES, INC.

Moody's financial manuals.

Millions of dollars

solid line represents sales plotted against sales for reference purposes. Th broken linesho*s expenses on the vertical axis and sales on the horiient& axis.

Reprinted by permission from THE ECONOMICS OF BUSINESS ENTERPRISE by W. Rautenstreuch,published by John Wiley I Sons, Inc.

to be determined in large part by the way in which theyare constructed. Total costs (either estimated or historical)are very sharply divided between the fixed and the variable.The former are presumed to be absolutely invariable with

charts this result would appear

Relerence:Note: The

0 tO 20 30 40 50 60 70 80 90 tOO flO 120


Chørt 4THE BREAK-EVEN CHART

OF THE GRANITE CITY STEEL COMPANY

Millions of dollars

this linearity points out, "The conventionalchart is based on the assumption thathas a straight line relationshipto activity, and that all the items classed

The broken in.

16

break-even

as fixed expense

respect to output, whereas the variable costs, in the absenceof more specific and detailed information to the contrary,are held to be directly proportional to output. As a critic of

Mitflons of dollars16

IS

14

13

12

II

I0

9

a

7

6

S

4

3

2

I

00 1 2 3 4 S 6 7 8 9 10 II 12 13 14 IS

Reference: Moody's financial manuals.Note: The solid line represents sales plotted against sales for reference purposes.

shoes expenses on the vertical axis and sales on the horizontal axis.Reprinted by permission from THE ECONOMICS OF BUSINESS ENTERPRISE by W. Rautenstrauth,published by John Wiley & Sons, Inc.

all variable expenseor is in direct proportion

io8 COST BEHAVIORremain as constant fixed expense for the whole range ofplant activity."

The budgeting break-even chart in the hands of a carefulcost executive becomes a delicate tool which must beadjusted frequently for greatest accuracy.

The modified break-even chart constitutes a variable budget.As labor rates change and methods are improved, costschange so that the chart should be revised at least twice ayear at six-month intervals, and more frequently if necessary.During the period of time for which the chart is effective,the relationship between expense, income, and activity shouldbe as shown on the modified break-even chart, provided theaverage of selling prices remains the same. When the aver-age of selling prices is changed, it is a simple matter to super-impose a new income line on the break-even chart.55

It would be very helpful to the study of cost-price rela-tionships if the part played in pricing decisions by thesecharts could be appraised in a number of firms. Unfortu-nately such charts are extremely confidential business docu-ments and are not as readily available as historical break-evencharts.

The fact that linearity is implicit in the customarymethods of making budgeting break-even charts is in itselfsignificant to the purposes of this study. If decisions aremade on the assumption of a linear relationship betweencosts and output, this fact alone is basic to any explanationof such decisions as pricing, quite apart from the "real"

H. R. Mallory, "A Silk Mill," Mechanical Engineering (August1933). According to this executive, variable costs start out at a certainrate and then decrease at a declining rate, approaching a constant perunit of output figure. Rautenstrauch, who uses straight lines, points out(op. cit., pp. 268-69) that the results obtained by this varied variablecost are substantially linear for a range of output from about percentof capacity to capacity. In the chart for the United States Steel Corpora-tion given on page of his study the points for the years 1928—33clearly present a slightly curved line for the cost relationship, withvariable costs decreasing as output increases. Rautenstrauch draws noline for these years alone.

p. 27!.


shape of the cost function. Although the actual cost func-tion of an enterprise at a given time may be of the invertedogive shape, as is customarily assumed by economists, unlessthe curvature is sufficient not only to be recognized in theexperience of the business but also to alter a decision sub-stantially, the linearity assumption on the part of thebusiness executives may be entirely valid. The implicitassumption in most break-even charts that the cost functionis linear may be simply a reflection of the fact that for allpractical purposes of decision formation such an assumptionis adequate, In view of the uncertainties and range of inde-terminacy surrounding decisions, it would require a ratherlarge curvature in the total cost functions to modify anydecision. Except for extreme levels of output, linearity maybe an adequate working conception of the cost functionfor the business executive.

4. Significance of the Empirical Studies

Empirical studies of the relation between variations inoutput and costs, of the types just examined—statisticalcost functions, estimated functions, and cost-revenuecharts—indicate with few exceptions a linear covariation,that is, a linear cost-output relationship. These results areof particular interest, since economic theory has usuallypresumed an inverted ogive for total costs and marginalcosts that are markedly U-shaped. This view has been sup-ported by rather common-sense propositions. As variablefactors are applied first to a fixed factor, the incrementalcost may be expected to decline with improvement in thecombination of factors; after a great many units of onefactor have been applied the combination is bound toworsen and the incremental costs will increase. The beliefthat marginal costs between these extremes of output de-scribe a smooth U-shaped curve reflects a bias in economictheory. If marginal costs approach the horizontal over largeranges of output, small changes in demand make for largerchanges in output. An economic system with linear cost

110 COST BEHAVIORfunctions is more unstable than one with markedlyU-shaped marginal cost functions.56

The marginal cost curve was generally held to beU-shaped throughout the whole range of output on theground that the conditions implicit in discussions of the"law of diminishing returns" were typical of industrial pro-duction. (a) That is, the fixed factor or combination offixed factors were assumed to be entirely indivisible; plantand equipment had to be operated as a whole or not at all.(b) Within the framework of the Marshallian "short run,"as output varied plant and equipment (fixed factors) werethought to be invariable except for adaptations to insurethe optimum combination of all factors. The "form" of thefixed factors was envisaged as changing to permit the tech-nical optimum combination, but the "quantity" of the"fixed" factors was unchanged. With given factor prices,marginal costs would be U-shaped with variations in out-put under these two conditions.57

The exammation of empirical studies in Sections 2 and 3concluded that in the cases explored a linear cost functionwas the most probable relation within the observed rangeof fluctuations of output, although basic difficulties in the

"Economists who make use of the competitive analysis of valuehave a strong unconscious bias in favor of rising and falling supplyprice, simply because, if supply price is constant, their analysis hasnothing interesting to say." Joan Robinson, Economics of ImperfectCompetition (Macmillan, London, p. ii8.

general acceptance of U-shaped Cost curves was supported byempirical studies of production functions in agriculture. E. H. PhelpsBrown, op. cit., Econometric; IV (April 1936), pp. 123-27. Marshall(Principles, Book IV, Ch. Ill) refers to a study of the Arkansas experi.ment station comparing yields per acre with the amount of ploughingand harrowing. In this connection see Bernard F. Haley, "A PreliminaryStudy of the Laws of Varying Costs" (Harvard Ph.D. thesis, November193 i).

It is also probably true that this view of a U-shaped marginal costcurve has been supported by arguments not strictly applicable to thecost function. For instance, some writers have pointed to the importanceof overtime rates, poorer workers, etc., at high levels of output as proofof the U-shape of the curve. These may be induced changes in cost, butthey are not precisely related to the cost function.

EMPIRICAL STUDIES IIImethod and techniques made it impossible to place sufficientconfidence in the solution to preclude marginal cost curveswith considerable curvature. This conclusion certainly doesnot justify the statement that all cost functions are linear,but it does suggest that the conditions underlying discus-sions of "diminishing returns" not only need to be re-examined, but may not be as typical as presumed. Indeed,even so cautious a conclusion as this must be qualified. Theresults of the several types of empirical studies designed tomeasure cost-output relations may be substantially influ-enced by the accounting conventions for allocating costsover different periods (see Chapter IV).

It is quite possible, however, to construct a simplifiedmodel of production that is consistent with linear cost func-tions. In fact, the ordinary discussion of the conditionsaffecting diminishing returns may be so generalized andmodified as to include both linear and U-shaped marginalcost functions as special cases.58 Only empirical research,of course, can indicate the relative representativeness of thevarious possible cases.5° One element in the explanation oflinear cost functions may be the fact that fixed "factors"are more or less divisible. The more divisible the plant andequipment, the more it is likely that the variable costs maybe linear. When a small volume of output is being pro-duced, it is possible in many firms to shut down completelyparts of the plant or groups of machines.60 Within widelimits the fixed factors may be compared to small unitswhose combined costs are simply proportional. But divisi-bility may be achieved in yet another way with fixed

58George Stigler, "Production and Distribution in the Short Run,"Journal of Political Economy, XLVII (June pp. 305-27. Thediscussion uses the terminology suggested in this article. See also JoelDean, Statistical Cost Functions of a Hosiery Mill, Ch. I.

Such reformulation of theoretical tools and designation of the typi-cality of special cases ifiustrate the importance and usefulness of thecoordination of empirical studies and theoretical analysis.

60 political or humanitarian reasons the firm may refrain from acomplete shutdown., particularly when plants are in a one-factory town.

112 COST BEHAVIORplant and equipment that cannot be operated in "units."Where starting or stopping costs are not large it may bepossible to operate a plant more or fewer hours per week,with one or more shifts. As a matter of practice this factoris very important, although it really involves a cost functionwith a meaning different from that implied in the timelessfunctions of comparative statics.

A second element in an explanation of linear cost func-tions has been suggested most clearly by George Stigler,but like the second type of divisibility just discussed thisanalysis involves a concept of a cost function differentfrom that implied in comparative statics. Plant and equip-ment are frequently constructed to produce either a fluc-tuating output or different types of output rather than asingle specified volume or product. "Were it not for theflexibility built into plants, outputs in excess of optimumwould involve prohibitive marginal costs, while those atless-than-optimum outputs would be very unprofitable." 61As output changes, minor variations are made in the plantand its equipment. With the expansion of output in a shoefactory, for instance, it may be possible within wide limitsto add identical machines.62 Or, in a paper mill, insteadof making only one width of paper, a machine may beflexible to the extent of making several widths, thicknessesand types, or may run at three different speeds. Alathe may be used to turn out material of varying lengths.If a single type of product or a specified amount werecertain to be produced, no flexibility would be required. Itis the possibility of other outputs or types of productswhich introduces the need for flexibility. The flexibleequipment could not produce the designated output orproduct as cheaply as equipment entirely specialized, butfor other outputs or products costs would be lower. The

61 Op. Cit., p. 316.02 An electric power generating system may be so built as to combine

hydroelectric power and steam, permitting relatively more constantcosts as output varies. See A. H. Markwart, "Aspects of Steam Powerin Relation to a Hydro Supply," Transactions, American Society ofMechanical Engineers (1926), p. 187.


expectation of variations in output introduces flexible (andless specialized) fixed factors and hence relatively morelinear cost functions than could be anticipated for morespecialized plant and equipment.

Several of the explanations just offered with referenceto possible linearity in the cost function involve a conceptof a cost function different from that presumed by com-parative statics. This traditional function of economictheory shows the way in which costs vary with output,when all other relations are assumed to be unchanged. Onecould formulate still another concept, which would simplybe the way in which business executives at any one timeexpected costs to vary with output, regardless of the sourceof the impact on costs. The concept would include changesin cost due to factor price changes arising from outputvariation, the variation in costs attributable to flexibleplant, minor technical changes, etc. In any system of deter-minants of cost, the concept would include the effects oncosts of cross-derivatives with respect to output. It couldbe argued that such a function would be more relevant toactual decisions. But there can be little doubt that this con-cept would also be difficult to measure statistically, becausethe part of the change in costs which would be correlatedwith output would have to be separated from any totalchange in costs.

For the present it is important to note that the statisticallinearity of cost functions could be explained by the divisi-bility and flexibility of fixed plant and equipment. Thiswould mean, strictly speaking, that the U-shaped marginalcost function of economic theory is not invalid. However,if empirical studies should show that conditions of divisi-bility and flexibility were typical, only a linear functioncould be both pertinent and useful for the interpretation ofeconomic phenomena.63

63The significance of a marginal cost function of this shape forpricing in an individual firm and in various types of market structuresis discussed in Chapter XI.

114 COST BEHAVIOR

5. Research PossibilitiesSince only a limited number of studies have been madeof the effects of variations in the rate of plant utilization oncosts, and because of the importance ascribed in economictheory to this relationship, the field appears to be a veryimportant and rewarding area for research. The survey ofdifliculties inherent in the problem, and the appraisal of theSteel Study of Professor Yntema in Sections and z, sup—port this impression. The sample of studies that have beenmade so far depends largely on the chance of personal con-tacts, the cooperation of particular firms, and the conven-ient availability of data. A large number of useful inquiriescould be made by investigators with some statistical sophis-tication in a wider and more representative group of enter-prises. There is need for additional studies to test thepresent meager sample in firms confronted by differenttypes of circumstances. For instance, the new studies shouldinclude firms with wider fluctuations in output, thoseallowing more frequent observations, those in service andmining industries; and more attention should be paid tosmaller scaled industries and to areas of industry- with lesstechnical change.

As research projects these individual studies of cost func-tions would have the advantage of narrow scope. A numberof such inquiries might profitably be undertaken by univer-sity candidates for degrees where business contacts areavailable, or indeed by business enterprises themselves. Inthe latter case, there would certainly be an occasion formore cooperative research between universities and busi-ness finns.

It is appropriate here to make a suggestion with respectto the method of research which will be applicable to allthe topics of succeeding chapters. Since costs and pricesare both influences on, and results of, business decisions,much more than we now know can be learned about theprocess of reaching these decisions. Explanations could bepitched on various levels of abstraction, from economic


theory to psychoanalysis. But in view of the present stateof knowledge of the motivation and direction of conduct,it appears most advisable that studies of business decisionsproceed along the following lines: (a) Case studies of par-ticular firms should combine qualitative material from in-terviews with statistical techniques. (b) Attempts shouldbe made to ascertain what information is available to thosewho make decisions. (c) In every enterprise certain per-sistent patterns of policy or working rules-of-thumb can beidentified. (d) Individual instances of action should be ex-amined, and not merely issues of broad "policy." Whilesuch studies cannot give definite answers to all questions,or "explain" all actions of the firm, they can provide auseful and almost indispensable counterpart to statisticalinquiries.

In conclusion, we may venture several suggestions whichare even more specific. Since the measurement of outputpresents one of the most difficult tasks in cost functionstudies, those types of industries in which firms producerelatively homogeneous products that are not altered sub-stantially through time offer the simplest cases for investi-gation. Although it is true that a firm seldom, if ever,produces for any considerable period a product completelyhomogeneous with respect to size, model, style or orders,there are enterprises in industries like cement, electricpower, gold mining, grey goods, bread baking, etc., whichpresent much simpler cases than firms producing agricul-tural implements or women's dresses. It would be valuable,too, to extend such studies to nonmanufacturing industriessuch as mining, the service trades, wholesaling and retail-ing. In some cases attempts to measure output would en-counter insuperable obstacles, but in mining or trucking,for instance, the problems ought not to be too difficult.In this field a rather useful study would be the collectionand analysis of a group of engineering estimates of theway in which costs vary with output. The assumptionsunderlying such calculations might prove significant forexplanations of business behavior.

empirical studies of cost function

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