economies of scale vs returnstoscale

8/7/2019 economies of scale vs returnstoscale

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Economies of, versus Returns to, Scale: A ClarificationAuthor(s): Christopher Ross BellSource: The Journal of Economic Education, Vol. 19, No. 4 (Autumn, 1988), pp. 331-335Published by: Heldref PublicationsStable URL: http://www.jstor.org/stable/1182344

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Economies o f , v e r s u s R e t u r n s t o ,S c a l e : A Clarification

ChristopherRoss Bell

Although the authors of many leading introductory and intermediateeconomics textbooks use them interchangeably, the terms economies ofscale and returns to scale are synonymous only under the very special condi-tion of production-function homotheticity. It is important to rememberthat this is indeed a special condition: in general, we would expect a firm tochoose a different input mix at low levels of output than it would at higherlevels.' By introducing both the concept of returns to scale and the conceptof economies of scale when discussing the shape of the long-run averagecost curve and then implying that these concepts are equivalent, withoutdiscussing the special condition that can make this so, textbook authorscommit at least two cardinal sins: they teach material that must be un-learned later, and they confound their student's natural (and correct) intui-tion regardingthe source of many of the advantages of large-scale produc-tion-the substitution of capital-intensive for labor-intensive productiontechniques. These points will be amplified in the following note.

ECONOMIES OF, VERSUS RETURNS TO, SCALEA production process is said to exhibit economies (constant economies,diseconomies) of scale over a particularrange of output per unit of time ifthe long-run average production cost over this range falls (remains the

same, rises) as output rises.2The term returns to scale refers to the effect onoutput of a proportionate change in the level of use of all inputs. A produc-tion process is said to be characterizedby increasing (constant, decreasing)returns to scale over a particularrange of input use per unit of time if, as theamount of all inputs is increased proportionately, output rises by a greater(equal, lesser) proportion.3Most textbook authors introduce the concept of returns to scale when ex-plaining the shape of the long-run average cost curve. Doing so ismisleading, for it fosters in the student's mind the belief that the shape ofthe long-run average cost curve and the presence or absence of returns toscale are inextricably linked. This is, of course, not so: returns to scale,

ChristopherRoss Bell is an assistant professor of economics at Davidson College. He thanksPaul M. Renner for his contributions to the survey of economics textbooks and OrleyAshenfelter, Claudia Goldin, Edwin Mansfield, his colleagues in the Davidson Collegeeconomics department, and two anonymous refereesfor their comments on an earlierdraft.Fall 1988 331


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although a sufficient condition for economies of scale, are not a necessarycondition. Thus, while a nonhomothetic increasing-returns-to-scaleproduc-tion function implies the presence of economies of scale, it is not implied bythem. Similarly, the presence of diseconomies of scale implies, but is notimplied by, a nonhomothetic decreasing-returns-to-scale production func-tion. These propositions are easy to prove.4

PROPOSITION1. A production function characterized by increasingreturnsto scale implies the presence of economies of scale. This is trueforany production function, homothetic or not.Proof. Obvious.PROPOSITION2. For processes characterized by nonhomothetic produc-tionfunctions, the presence of economies of scale does not imply that thecorrespondingproduction function is characterizedby increasing returnsto scale.Proof. Consider Figure 1. Figure 1 depicts the firm's cost-minimizingchoice of production technique. The least-cost production technique for

producing qo units of output occurs at point a and utilizes the input combi-nation (Lo, Ko). Proportionate changes in this input combination are repre-sented by movement along ray OA. Note that the point Xa (representing aproportionate increase of magnitude Xover the original input combination)lies inside the isoquant Xqo(which represents a proportionate increase ofmagnitudeXover the original output level). Thus the production technologydepicted in this illustration exhibits decreasing returns to scale: q(XLo,XKo)< Xqo(Lo,Ko). Note as well, however, that the least-cost production tech-nique for producing qo units of output occurs at the more capital-intensivepoint b. Point b lies on the isocost line yy. Because isocost line yy lies insideisocost line XxXx(which representsa proportionate increase in total produc-tion cost of magnitude X), production of the good depicted in this illustra-tion exhibits economies of scale: C(Xqo)/Xqo< XC(qo)/Xqo,where C(Xqo)and C(qo)represent the total cost of producing Xqoand q0 units of output,respectively. Q.E.D.

PROPOSITION3. The presence of diseconomies of scale implies that thecorrespondingproduction function is characterizedby decreasing returnsto scale. This is truefor any production function, homothetic or not.Proof. Suppose not. Then both diseconomies of scale and constant (or

even increasing) returns to scale could be found together. The presence ofconstant (or increasing) returns to scale would imply that a proportionateincrease in the use of all inputs would result in (at least) a proportionate in-crease in output. This would result in (at worst) a proportionate increase intotal cost, which implies that averagecost would (at worst) remainconstant.This is, of course, inconsistent with the presence of diseconomies ofscale. Q.E.D.332 JOURNAL OF ECONOMIC EDUCATION


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FIGURE 1The Firm's Choice of Production Technique

Capital

y

x

qoL, Lo x XLo y Xx

Labor

PROPOSITION4. A nonhomothetic production function characterizedbydecreasing returns to scale does not imply the presence of diseconomiesof scale.Proof. Follows directly from the proof of proposition 2.

EXPLAINING THE SHAPE OFTHE LONG-RUN AVERAGE COST CURVEAlthough the interrelationships among the concepts economies of scale,returns to scale, and production-function homotheticity are clearly beyondthe scope of material appropriate for a typical introductory economicscourse, it is not difficult to present the gist of this subject correctly: simplyexplain the shape of the long-run average cost curve using the same exam-ples many textbooks alreadyuse (incorrectly)to illustrate the concepts of in-

Fall 1988 333


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creasing and decreasing returns to scale (e.g., workforce specialization, in-put indivisibilities, and problems of communication and control in large or-ganizations) and omit discussions of returns to scale altogether. Althoughthis method does have the disadvantageof failing to emphasize that produc-tion functions are the primatives underlying cost functions, this point canbe brought out through examples presented in class or on problem sets.The advantages of omitting returns to scale from discussions of the shapeof the long-run averagecost curve far outweigh the disadvantages. Doing soavoids fostering the impression-which must be unlearned later-that theconcepts returns to scale and economies of scale refer to two sides of thesame coin. Perhaps more important, avoiding mention of returns to scaleallows an author to stress instead that unit-cost decreases are often realizedthrough the substitution of capital-intensive for labor-intensive productiontechniques. Doing so reinforces many students' natural (and correct) intui-tion regarding the advantages of mass production, an intuition that canonly be confounded by the common practice of invoking changes in produc-tion technique to explain the origins of returns to scale.Given the advantages of omitting returns to scale from discussions of theshape of the long-run average cost curve, it is difficult to understand theemphasis given to this topic in most textbooks. Aside from a convenientmathematical interpretation (is a production function greater than, lessthan, or first-degree homogeneous?), both casual and formal empiricism5suggest that the concept has very little economic content: it is seldom mean-ingful to ask how output responds to a proportionate change in the use ofall inputs because, in general, a firm producing a good at a low output levelwill choose a different input mix from one producing at a higher level. Fur-ther, advances in econometric techniques no longer requirethat productionfunctions be homothetic in order to estimate cost functions. It would seemthat returnsto scale-at least in introductory textbooks-is a concept whosetime should pass.

Despite the ease with which the subject could be handled, the discussionsof the shape of the long-run averagecost curve found in most textbooks are,at best, muddled. Of thirty-three introductory and intermediate economicstextbooks received by members of the Davidson College economics depart-ment between May 1985 and May 1986, only six contained clear, concise,and technically correct discussions of the shape of the long-run averagecostcurve.6 Pitfalls await students using even the most respected textbooks:Baumol and Blinder use the terms-economies of scale and returnsto scale assynonyms, writing that "[plroduction is said to involve economies of scale,also referred to as increasing returns to scale, if, when all input quantitiesare doubled, the quantity of output is more than doubled (1985, p. 413, em-phasis in original)"; Mansfield asks, "What determines the shape of thelong-run average cost function in a particular industry?" and answers,"[Ilts shape must depend on the characteristics of the productionfunction- specifically, upon whether there are increasing, decreasing, orconstant returns to scale" (1986, p. 525).334 JOURNAL OF ECONOMIC EDUCATION


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Perhaps the greatest source of confusion in the textbooks I examined wasthe use of legitimate illustrations of the sources of economies of scale to ex-plain the origins of returns to scale, a practice adopted by both Samuelsonand Nordhaus (1985, p. 37) and Mansfield (1986, p. 525), among others.Mansfield's presentation is illuminating. After correctly defining returns toscale in terms of the effect on ouput of an increase in "the amount of all in-puts by the same proportion," he goes on to argue that increasing returnsare often realized by using "techniques and inputs that smaller factoriescannot use (1985, p. 525)." The confusion such illustrations create makesmy point nicely: It is precisely because larger factories may choose to usetechniques and inputs different from those used by smaller factories thatreturns to scale should not be used to explain the shape of the long-runaveragecost curve. The sooner our textbooks cease this practice, the soonerour students' muddled understanding of this subject will clear.

NOTES1. A recent article in the Charlotte Observer (Associated Press 1986) provides a nice illustra-tion. The article suggests that the manufacture of the small number of bats used by profes-sional baseball players is relatively labor intensive, stating that they are "produced bycraftsmen" on general purpose lathes "at a rate of one every 20 minutes." The productionof the far-largernumber of bats sold to the general public, on the other hand, is relativelycapital intensive: "The majority of Louisville Sluggers are turned on automatic lathes thatcan make a bat in eight seconds." See also Scherer's (1980, pp. 81-82) discussion of the

manufacture of ball bearings.2. Note that input prices are assumed to remain constant.3. Use of the term returnsto scale in reference to the mathematicalproperty of the productionfunction described above (homogeneity) is virtually universal; agreement on the use of theterm economies of scale is less comprehensive. Note, however, that the usage above is con-sistent with both Scherer's authoritative industrialorganization textbook (1980, pp. 81-118)and Christensen and Green's classic empirical study (1976). Hanoch (1975, p. 492) tracesthe distinction made between these concepts to Edgeworth.4. See Hanoch (1975) for alternative proofs of related propositions.5. See, for example, Christensen and Green (1976), whose estimation of a translog cost func-tion for electric power generation allowed them to test-and reject-production functionhomogeneity.6. It is noteworthy that of these six, only one (Solberg 1982)mentioned returnsto scale. Both asummaryof the shortcomings encountered and an annotated list of the textbooks examinedare available from the author.REFERENCES

Associated Press. 1986. One of baseball's constants: The Louisville Slugger. Carried in theCharlotte Observer, April 13: 8B.Baumol, W. J., and A. S. Blinder. 1985. Economics: Principles and policy. 3d ed. San Diego:Harcourt Brace Jovanovich.Christensen, L. R., and W. H. Greene. 1976. Economies of scale in U.S. power generation.Journal of Political Economy 84 (1, pt. 1): 655-76.Hanoch, G. 1975. The elasticity of scale and the shape of average costs. American EconomicReview 65 (3): 492-97.Mansfield, E. 1986. Economics: Principles, problems, decisions. 5th ed. New York: W.W.Norton.Samuelson, P. A., and W. D. Nordhaus. 1985. Economics. 12th ed. New York: McGraw-Hill.Scherer, F. M. 1980. Industrial market structure and economic performance. 2d ed. Boston:Houghton Mifflin.Solberg, E. J. 1982. Intermediate microeconomics. Plano, Texas: Business Publications, Inc.Fall 1988 335

economies of scale vs returnstoscale

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