American Economic Association

A Test for Subadditivity of the Cost Function with an Application to the Bell System Author(s): David S. Evans and James J. Heckman Reviewed work(s): Source: The American Economic Review, Vol. 74, No. 4 (Sep., 1984), pp. 615-623 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/1805127 . Accessed: 14/05/2012 18:04 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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A Test for Subadditivity of the Cost Function with an Application to the Bell System By DAVID S.

EVANS AND JAMES J. HECKMAN*

The recentliteratureon multiproductfirms demonstratesthat when all firms in an industry (actual or potential)have access to a common technology,propertiesof the firm cost functions reveal the most efficient industry structure(see, for example,Elizabeth Bailey and Ann Friedlaender,1982). An important issue addressedin this literatureis the derivationof conditionswhich guarantee that an industryis a naturalmonopoly. Despite the relevanceof this issue to discussions concerningthe desirabilityof competitionin regulated industries, few empirical studies offer reliableevidenceon this question.The reason for this is that the theory of multiproductindustries(see, for example,William Baumol, John Panzar, and Robert Willig, 1982) suggeststhat global informationabout cost functions is requiredto determinethe presence of natural monopoly. Such informationis seldomavailable. This articleproposes a new test of necessary conditions for natural monopoly that does not requireglobal informationon firm cost functions.Our test does not requirethe extrapolationof estimatedcost functionswell outside the range of the availabledata that are requiredin tests currentlyproposed in the literature. To see why global informationis required to test for necessaryand sufficientconditions for naturalmonopoly,considera firmwhich produces an output vector q = (ql,..., qn)

with cost function C(q). A firm is a natural monopolyif and only if C(q) is subadditive * Departmentsof Economics,Fordham University, Bronx,NY 10458 and Universityof Chicago,Chicago, IL 60637,respectively.We areindebtedto GeorgeYates for programmingthe subadditivitytest, ThomasColeman andJohnBenderfor researchassistance,and Kevin M. Murphy,RodneySmith,and LesterTelserfor helpful comments.The paper has benefitedfrom seminar presentationsat the AppliedPriceTheoryWorkshopat the Universityof Chicagoand the GraduateSchool of Business,New York University.

615

over the relevant range of output levels.2 The function C(q) is subadditiveat the output level c if and only if n

(1)

C(q) < E c(qi, i=1

where for nonnegativecr n

(2) i =1

with at least two vectors i nonzero, for all ci satisfying(2). Thus an industryis a natural monopolyif a single firmcan produceall relevant output vectors more cheaply than two or more firms. To test this condition requiresknowing the cost of all output vectors qi smaller than c and thus requires global informationabout the cost function. Such knowledgeabout C(q) is requiredfor all possible industryequilibriumoutput values. Less empiricallydemandingnecessaryand sufficientconditionsfor subadditivityremain to be developed. Baumol et al. derive separate necessary and sufficient conditions which requireless informationthan the joint necessary and sufficientconditions for subadditivity.3For example, the presence of economies of scope is a necessarycondition 'Subadditivityof the firmcost functionis a necessary and sufficientconditionfor naturalmonopolyonly when all firmshave access to the same technologyand when marketcoordinationbetweenseparatefirmsis unableto achievethe sameeconomies(say by networkingor pooling arrangements)as internal coordinationwithin a single firm. These assumptionsmay not characterize many real-worldindustries.See our (1983a) articlefor furtherdiscussion. 2The relevantrange of outputs dependson the demand and cost conditionsprevailingover the periodof interestto the analystor policymaker. 3See Baumol et al. or Bailey and Friedlaenderfor formalstatementsof these conditions.

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THE AMERICAN ECONOMIC REVIEW

for subadditivity.Economiesof scope exist if the costs of producing each product separately exceed the costs of producing all products jointly. Economies of scope and declining averageincrementalcost for each productare sufficientconditionsfor subadditivity. The averageincrementalcost of output i, when the total output vector is q, is the cost of producing output vector 4 less the cost of producing all of c~ excluding product i dividedby the output of i, qi. The average incrementalcost for product i declines if the averageincrementalcost of producing i decreaseswith increasesin the level

of ji.4 Baumol et al. recommendtesting the necessary and sufficientconditionsfor subadditivity separately.If the necessaryconditionis rejected,subadditivityis rejected.If the sufficient conditions are accepted, subadditivity is accepted.Unfortunately,even this restrictive procedure for testing subadditivityrequires more information than is usually available for most industries.Their test requires observationson the stand-alonecosts of productionfor each productin the output vector and observationson the costs of production for all output vectors containing positive quantities of n -1 outputs and a zero quantityof the other output. To illustrate the problems that arise in implementingtheir test, consider testing for subadditivityin the postal industry.This industryprovidesseveraltypes of mail service, some of whichhave been opened to competition in recent years. It would be interesting to determine whether a single firm could provide all postal servicemore cheaply than could two or more firms. But data on the stand-alonecosts of providingfirst-classmail service are not available. Consequently, calculating economies of scope between first-classand other postal services requires extrapolationof estimatedcost functionsfar outside the rangeof observationsover which they are estimated.5Similarproblemsbeset 4Economiesof scope and averageincrementalcosts can also be defined for subsets of the output vector containingmultipleproducts.See Baumolet al. 5MelvynFuss and LeonardWaverman(1981) have used this test to determinewhetherBell Canadahas a naturalmonopolyover local, toll, and privateline tele-

SEPTEMBER 1984

the calculationof averageincrementalcosts. Moreover,the Baumol et al. test may prove inconclusiveeven when data are availablefor calculatingreliableestimatesof the relevant quantities. Acceptance of their necessary condition but rejection of their sufficient conditionsmay occur. This article proposes a new test for subadditivity within a region that avoids the need to extrapolateoutside the range of the available data. The test is local and not global. It is based on the idea that if subadditivity is rejected in one region, global subadditivitymust be rejected.Section I describes our test for a two-productindustry. (Generalization to n-product industries is straightforward.)Section II applies the test to data from the U.S. Bell System. I. Subadditivity Test We begin by refiningthe terminologyused in the introduction.Considera two-product industryin which all firmshave access to the same technology.The cost functionC(q1,q2) is subadditiveat q = (c1, q2) if and only if for nonnegative 41, q2

(3)

I:iC(ai4l, bi42) > C014, 2), i=1,...,n,

(4) Yai = ,

Ebi = ,

ai 20,

bj20,

for at least two ai or bi not equal to zero. It

is superadditive at q if and only if " >" is

replacedwith " <" in (3). It is additiveat q if and only if " >" is replaced with " =" in

(3). A firm with superadditivecosts could save money by breaking itself into two or more divisions.Unless there are firm-specific fixed factorswhich precludesuch decentralization, we would not expect to observe profit-maximizingfirms operating with superadditivecosts. A firm with additivecosts phone services.They rejectthe hypothesisthat thereare economies of scope between private line service and local and toll service.But, as they readilyadmit, their test is unreliablesince they have to extrapolatethe cost function far outside their sample in order to calculate stand-alonecosts for local, toll, and privateline services.

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EVANS AND HECKMAN: SUBADDITIVITY OF THE COST FUNCTION

may have decentralizeditself into the optimal configuration.Consequently,in many situations the interestingstatisticalquestion concerns whether the cost function is additive or subadditiveat observedoutput levels. The cost function is quasi-globallysubadditive, superadditive,or additiveif and only if the cost function is subadditive,superadditive, or additive,respectively,at all relevant outputvectors4. The relevantoutputvectors are those which are consistentwith industry equilibriumgiven demand and cost conditions for alternative possible organization patternsof the industry(for example,multifirmvs. single firm). Our test computes(3) for an "admissible range"of outputs.For simplicity,we restrict our evaluation of (3) to n = 2 so that we compare two-firm configurationswith the monopolyconfiguration.We also confineour attention to the case where the analyst has time-seriesdata on a single firm assumedto be a monopoly, although the test can be readilyappliedto situationswherethe analyst has cross section or panel data on a sample of firms.Denote each time-seriesobservation by an output qt, t =1,..., T. Denote the first hypothetical firm by A and the second hypotheticalfirmby B. Consider output level 4 and let 4A + 4B = 4. Figure 1 illustratesour test. We define an "6admissible region"representedby an area on the floor of the diagram. The cost of producing qA iS CA, the cost of producing iS GB, and the cost of producing 4 4B is C. If CA + CB > C, then the cost function is subadditiveat 4 with respectto the particular two-firm industry configuration described by (4A,

4B), a situation which holds

true in Figure 1. We consider all two-firm configurations (4A, 4B) in the admissible region which sum to 4. If CA + CB> C for all

such configurations,then the cost functionis subadditiveat 4 over the admissibleregion, the situationdepictedin Figure1. Our choice of an admissible region for each output 4 is dictated by our desire to

avoid "excessive"extrapolationoutside the data. It is difficultto make this notion precise withoutknowingin advanceexactlywhat it is we seek to estimate. Approximation theory provides bounds for particular ap-

617 C~~~~~~~~~A+CBl

C

-A

q2

FIGURE1

TESTFORSUBADDITIVITY

proximations to known functions. But these

bounds are of little use in applied work since, in practice, the true cost function is unknown.6In this paper we define the admissibleregionto keep hypotheticalindustry output configurationswithin the range of output configurationsactually observed in the data. Specifically,we define the admissible region so that it satisfies two constraints.The first is that no hypotheticalfirm in the twofirm industryis permittedto produceless of either output than the firm for which we have data.Define qM as the vectorof minimal sizes of the output of the firmspermittedby this criterion: qM

(qlM, q2M) = (minqlt,

minq2t).

6See, for example,Philip Davis (1975) for a discussion of approximationtheory and bounds for various functions.

SEPTEMBER 1984

THE AMERICAN ECONOMIC REVIEW

618

q

Firm A produces + qlM, wq* +

qt = (pq;t

(5)

q2M).

Ru

Firm B produces q = ((1

(6)

ADMISSIBLE

0)q*t+

RL

REGION

qjm,

(1- w)q* +

q2M). 2q9M

The parameters(p, w) satisfy 0<

0 < .<1,

(7)

<1. OBSERVEDOUTPUTLEVELS

lM

AggregatingacrossfirmsA and B, we obtain (8)

41,

q* + 2qm=

q2*+ 2q2M=2

so that FIGURE 2.

(9) for

q 2M

q2t

ql*=qt2qlM, 4ij > 2qiM.

We restrict the test to values

RL < wq2

RL
+ q2M

< RU

-(p-)q*t

+ qlm

q2

+ q2M

(I1-')

< RU

where (11)

RL=

minqlt/q2t,

q2

DETERMINATIONOF ADMISSIBLEREGION

q2t2q2M,

of ct which satisfythis inequality.We test in years for which the output of both goods is at least twice the lowest output level in the sample. The second constraintis that both firm A and B produce q1 and q2 in a ratio within the range of ratios observed in the data. Thus we require (10)

2q 2M

RU = maxq1t/q2 t

This constraint precludes our hypothetical firms from specializingin either output to a greaterextent than has the firmfor whichwe have data.7 71t is possible to define a broaderadmissibleregion by also includingoutputconfigurationsthat are not too For example, far removedfromobservedconfigurations. we could modify the first constraintso that neither

Figure 2 illustrateshow these constraints restrict the admissible sample region. The smallesthypotheticalfirmsize we consideris qM. The hypothetical firm must also have output ratios between the vectors RL and RU, where RL is the lowest ratio of output 1 to output 2 in the sample and RU is the highest ratio of output 1 to output 2 in the sample. In addition, the test only applies to Q, which satisfy (9) and thereforelie in the northeastquadrantof the box drawnin Figure 2.

hypotheticalfirmis allowedto produceless than 1/a of the minimumamountof outputobservedin the sample or more thana of the amountof outputobservedin the sample,where a ?1. We could modify the second constraintso that the rangeof specializationfor hypothetical firmlies between RL/f and PRu whereA > 1. The admissibleregiondescribedin the text is for the special case where a = / =1. We have chosen a conservative admissibleregionfor two reasons.First,since this study reportsthe firstapplicationof our test for subadditivity, we believe it is best to be conservativeand restrictthe test to observed output combinations. Second, the estimatesreportedbelow deteriorateas we get further from the sample mean. Expandingthe admissibleregion, at least for our example,does not providemuch additional information.We would like to thank the refereefor the suggestionfor expandingthe admissible region.

VOL. 74 NO. 4

EVANS AND HECKMAN: SUBADDITIVITY OF THE COST FUNCTION

619

We measurethe degreeof subadditivityby Sub, (p, o)

q~~~~~~~~~~~~ [t

qA

q2 FIGuRE 3.

OF SUBADDITIVITYTEST IMPLEMENTATION

Figure 3 illustratesan implementationof the test for output level c, in the relevant output

set. Let qt

=

(oq*, wq2*t). Firm A

producesqM plus an incrementqt and firm B producesqM plus an incrementqtB where q, and qt' are chosen so that qt + qt + 2qm sum to the outputlevel 7tat whichthe test is being performed.All possiblevectorsqt and qt/ within the admissible region, and thus all possible two-firm output configurations within the admissibleregion, are evaluated. In practice,the q't and qtB are constructed by varying 4 and w over a grid given by p=i/g, w= k/g, i, k = O,..., g where g is the grid size. We now formally describe our test for subadditivity.Let (

Ct &

O

=) Ceq(qm+ qt)

f e(.

B)

- B

Ct

(

)

t

(+

/t-

If Sub,(p, w) is less than zero, the cost function is subadditivewith respect to the industry configuration given by (4, w). If Sub,(O,w) is equal to zero, the cost function is additivewith respect to the industryconfigurationgiven by (4, w). If Subt(@, A) is greater than zero, the cost function is superadditiveover the industry configuration given by (p, o). We calculate Max(, (,)Subt(O,o). If this quantity is negative and statisticallysignificantly differentfrom zero, we reject the hypothesis that the cost function is additiveat q, over the admissibleregion. Values of the test statistic that are negative and statistically significantlydifferentfrom zero do not contradictthe hypothesisthat the cost function is subadditiveat c, relative to the admissibleregion. If we do not rejectthe composite hypothesis that the cost function is subadditiveat all feasibleoutput levels, then we do not rejectthe hypothesisthat the cost function is quasi-globallysubadditive relative to these output levels. A test of the hypothesis of subadditivityover the whole sample is based on Max(,, , T*)Subt(@,Xo),

where T * is the set of sample years for which feasible partitions exist.8 Note that our criteriontests only necessaryconditions for subadditivityas long as the admissible region is a proper subset of the possible 8Note that becausethe maximizationis with respect to a continuousset of exogenousexplanatoryvariables and becausein generalthereis only one stationarypoint for the maximum problem, our proceduredoes not create an order-statisticsproblem,and so we can use conventionaltest statistics.We makeno claimaboutthe optimality of our test. A better test, that grew out of discussions with Kevin M. Murphy, computes Subj(c, w) for all pointsin the admissibleregion.Thisis an infinitedimensionalstatisticbut its distributioncan be derived.A one-sidedtest of the hypothesisthat this functionis everywherenegativecan be constructedthat utilizesmoreinformationthan the test proposedin this paper.Whilethis test is no worsethanthe one proposed in the text, it is also more complicatedto compute,and is not developedhere.

620

SEPTEMBER 1984

THE AMERICAN ECONOMIC REVIEW

region of output configurations.Failure to find subadditivitywithin the admissibleregion is informativein rejectingthe hypothesis of subadditivity; evidence supporting subadditivity within the admissible region obviouslydoes not indicate supportfor that hypothesisoutsidethe admissibleregion. 11. Subadditivity Tests for the U.S. Bell System

We use data on the Bell System from 1947-77 to test whetherthe Bell Systemhad a subadditive cost function at the output levels producedduring those years. We assume the Bell Systemproduceslocal and toll telephone services with capital, labor, and materials.We estimateseveralalternativecost function specificationsunder alternativeassumptions concerning the structureof the disturbances.Using likelihood ratio statistics, we find that the preferredspecification is a generaltranslogcost function with firstorder autoregressivedisturbancesin the cost and factor share equations. The estimated cost functionis monotonicand concavewith respect to input prices in all years. The estimated own-price elasticities of demand for capital,labor, and materialsare negative in all years. Thereforeour estimates satisfy the conditions requiredof an economically valid cost function.9The Appendix reports the estimatedcost function. Between 1947 and 1977, cost increased more than fourteenfold,toll serviceincreased almost fourteenfold, and local service increased fivefold. The smallest quantities of local and toll servicewere both producedin

9Like other economistswho estimate translogcost functions, we require our cost function to be linear homogeneousin input prices and to have a symmetric Hessian matrix with respect to input prices. See, for example, Laurits Christensen and William Greene (1976); Fuss and Waverman;Friedlaender,Clifford Winston,and KungWang(1983).Unlike theseanalysts, we report tests of these restrictions.We resoundingly rejectthem.Thisphenomenonoccursin all otherspecifications of the model that we estimate.We conjecture that these restrictionswould be rejectedin other translog cost functionstudies.In orderto makeour estimates consistentwith the translogestimatesreportedby other economists,we impose homogeneityand symmetryrestrictionson the cost function.

PERCENT GAIN FROM MULTIFIRM TABLE 1 -MAXIMUM PRODUCTION VS. SINGLE-FIRM PRODUCTIONa Percent Year

1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 'Entnes

Gain

13 20 25 25 33 40 44 48 53 58 51 50 39 36 39 41 42 45 59 51

Standard Error

X

15 14 14 14 14 15 15 16 23 23 26 30 22 21 21 20 21 20 20 19

equal Max SubX100.

production that multifirm indicates than single firm production.

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.6 0.5 0.3 0.5 0.4 0.4 0.4 0.4 0.4 0.5 0.5 A positive is

more

0.0 0.2 0.4 0.6 0.5 0.5 0.6 0.6 0.9 0.7 0.8 0.9 0.7 0.6 0.6 0.6 0.6 0.6 0.5 0.5 number efficient

1947. Output doubled by 1958 making this year the first feasible one for our test. Over the sample period, the ratio of local to toll varies between0.5 and 1.3. For each year t = 1958,...,1977, we calculate Sub,(4, w) over a grid with a g = 10 for (4, w) lying in the admissibleregion.Table 1 reportsthe estimatesof MaxSubt(, o). Table 2 reports the estimate of Subt(:, c) for t = 1961, a year near the center of our data. We find that Max Sub, is alwaysgreaterthan and often statisticallysignificantlydifferent from zero for output configurationsproduced between 1958 and 1977. Thereforewe reject the hypothesis that the Bell System's cost function is subadditiveat any of these output levels. We also reject the hypothesis that the Bell System'scost functionis quasiglobally subadditiveover these output levels. The fact that Max Sub,(4, o) is neversignificantly less than zero and the fact that Sub, is frequentlypositive suggest that our finding that the Bell System cost function is not quasi-globallysubadditiveis robust. The frequentpositive and sometimesstatistically significantpoint estimates of Subt

VOL. 74 NO. 4

EVANS AND HECKMAN: SUBADDITIVITY OF THE COST FUNCTION

621

TABLE 2-PERCENT GAIN OR LOSS FROMMULTIFIRMVS. SINGLE-FIRMPRODUCTION ALTERNATIVEINDUSTRYCONFIGURATIONS,1961 DATAa

S=

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.1

0.2

8 (21) 8 (19) 9 (18) 12

8 (20) 8 (18) 10

8 (19) 9

9

(16)

(17)

13 (13) 16

(18)

10 (17) 13

(19)

9 (18) 11

9 (18) 9

9

(14)

(15)

(16)

(17)

(18)

(18)

(15)

(16)

(17)

(17)

23 (16)

18 (16)

15 (17) 20 (17)

15 (15) 20 25 (14)

21

17

0.8

0.3

14

0.4

11

0.9

0.5

10 (18) 12 (18) 16 (18)

0.6

0.7

0.8

9 (18) 10 (18) 12 (18) 17 (19)

9 (18) 10 (19) 13 (19)

8 (19) 10 (20)

1.0

0.9

1.0

8 (20) 10 (21)

8 (21)

aEntriesequal Sub1961X1OO. A positive number indicates that multifirmproductionis more efficient than single-firmproduction. Standarderrorsare reportedin parentheses. The numbersreportedhere are based on an autoregressivetranslogcost functionreportedin our (1983b) article.

suggest that the Bell System was not optimally decentralizedduring our sample period. This finding is consistent with anecdotal evidence on the Bell System. Fortune Magazine reported that William L. Weiss, head of the new operatingcompany for the Midwest region, "thinks AT&T'smarketing was too centralized with the result that managers became 'less creative and more dependent on the signal caller"' (June 27, 1983, p. 64). The Midwest operating company will reportedlyoperateas a decentralized confederationof the fivecompaniesfrom which it is to be constituted. III. Conclusions

This articlepresentsa new test for subadditivityof the cost functionthat requiresless informationon the cost function than does the test proposed by Baumol, Panzar, and Willig.The test providesan easily computed rejectioncriterionfor the hypothesisof sub-

additivity.Applying this test to the Bell System using 1947-77 time-seriesdata, we reject the hypothesis that the Bell System's cost function is subadditiveat the output levels producedbetween 1958-77. We find limited evidence that the Bell System did not optimally decentralizeitself during these years and was thereforeoperatinginefficiently. APPENDIX

We estimatethe Bell Systemcost function using aggregatetime-seriesdata on the Bell System for 1947-77. We estimate a cost function rather than a production function in order to make our approach consistent with previousstudiesof the productioncharacteristics of the telecommunicationsindustry, and because the theoreticalliterature on naturalmonopolyrelies on the cost function ratherthan the productionfunction.We disaggregateoutputs into local and long distance servicesand inputs into capital, labor,

622

THE A MERICAN ECONOMIC RE VIE W

SEPTEMBER 1984

TABLE 3-ESTIMATED COST FUNCTION and materials.As a proxy for technological change, we follow previous studies of the Coefficient Standard productioncharacteristicsof the Bell System Parameter Estimate Error (see, for example,LauritsChristensen,Diane Constant 9.054 (.005) Cummings and Philip Schoech, 1981; M. Capital .535 (.008) Ishaq Nadiri and Mark Shankerman,1981; Labor .355 (.007) and HrishikeshVinod, 1976), by using an Toll .260 (.309) index of lagged research and development Local .462 (.226) Technology -.193 (.086) expendituresby Bell Laboratories. Capital2 .219 (.024) Following conventionalpractice (see, for Labor2 .174 (.027) example, Christensenand William Greene, Capital-Labor -.180 (.019) 1976), we use the translogapproximationto Toll2 - 8.018 (2.170) - 4.241 Local2 (1.314) the cost functionand estimatea systemcon11.663 (3.144) sisting of the cost equation,the capital share Local-Toll Technology2 -.176 (1.033) equation,and the labor shareequationusing .337 Capital-Toll (.138) maximumlikelihoodtechniques.We find that - .359 Capital-Local (.122) a specificationwith first-orderautoregressive Labor-Toll - .179 (.083) Labor-Local .164 (.071) disturbancesmaximizesthe likelihood func.083 Capital-Technology (.053) tion and purges the residualsof serial corLabor-Technology -.057 (.047) " relation.'0 The estimated cost function is - 1.404 Toll-Technology (1.497) reported in Table 3. This cost function is Local-Technology 1.207 (1.431) Autocorrelation monotonicallyincreasingand concave with Parameter for: respect to all input prices in all years. The .187 Equation (.105) own-priceelasticitiesof demand for capital, Cost .712 Share Equations (.094) labor and materialsare negativein all years. The estimatesthus satisfy the sufficientconNotes: Degrees of ditions for an economicallyvalid cost funcR2 Freedom Durbin-h Summary Statistics tion." We rejectthe hypothesisthat the cost .9999 Cost Function 15 .65 function is separablein local and long dis27 1.50 tance output. Thus it is not valid to aggre- Capital Share Equation .9756 Share Equation .9835 27 1.37 gate local and long distancetelephoneservice Labor Generalized Variance for System = 10.568 into a single measure of telephone service and estimate a single-productcost function as is done in the Christensenet al., NadiriREFERENCES Shankerman,and Vinod studies.Our(1983b) articleprovidesfurtherdetailson our estima- Bailey, Elizabeth E. and Friedlaender,Ann E., tion proceduresand resultsand on our data "Market Structure and Multiproduct Insources. dustries," Journal of Economic Literature,

l?We also estimate two alternativesto the translog cost function.The first alternativeapplies a Box-Cox transformationto the output variables.The second alternativeappliesa Box-Coxtransformationto all righthand side variables.We are unable to reject the hypothesis that the correct specificationis translog.See our (1983b)article. "We assumethat outputlevels and factorpricesare exogenous.Althoughboth assumptionsare questionable on a priorigroundsfor the telephoneindustry,using a Durbin(1954)-Wu(1973) test, we rejectthe hypothesis that outputlevels and input pricesare endogenous. 12But see fn. 9.

September 1982, 20, 1024-41. Baumol, William, Panzar, John C. and Willig, Robert D., Contestable Markets and the Theory of Industry Structure, San Diego: Harcourt, Brace, Jovanovich, 1982. Christensen, Laurits, Cummings, Diane and Schoech, Philip, "Econometric Estimation of Scale Economies in Telecommunications," Working Paper No. 8124, SSRI,

University of Wisconsin-Madison,September1981. and Greene, William, "Economies of Scalein U.S. ElectricPowerDistribution," Journal of Political Economy, October 1976,

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EVANSAND HECKMAN: SUBADDITIVITYOF THE COST FUNCTION

84, 655-76. Davis, PhilipJ., Interpolation and Approximation, New York: Dover, 1975. Durbin,J., "Errors in Variables," Review of the International Statistical Institute, 1954, 20, 22-32. Evans,DavidS. and Heckman,James J., (1983a) "Natural Monopoly," in D. S. Evans, ed., Breaking Up Bell: Essays on Industrial Organization and Regulation, New York: North-Holland, 1983, 127-56. , (1983b) "Multiproduct and Cost Function Estimates and Natural Monopoly Tests for the Bell System," in D. S. Evans, ed., Breaking Up Bell, New York: North-Holland, 1983, 253-82. Friedlaender,Ann, Winston, Cliffordand Wang, Kung,"Costs, Technology, and Productivity in the U.S. Automobile Industry," Bell Journal of Economics, Spring 1983, 14, 1-20. Fuss, Melvynand Waverman,Leonard,The Reg-

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ulation of Telecommunications in Canada, Ontario: Economic Council of Canada, 1981. Nadiri, M. Ishaq and Shankernan,Mark, "The Structure of Production, Technological Change and the Rate of Growth of Total Factor Productivity in the U.S. Bell System," in T. Cowing and R. Stevenson, eds., Productivity Measurement in Regulated Industries, New York: Academic Press, 1981. Vinod,Hrishikesh,"Application of New Ridge Regression Methods to a Study of Bell System Scale Economies," Journal of the American Statistical Association, December 1976, 71, 835-41. Wu, De Min., "Alternative Tests For Independence Between Stochastic Regressors and Disturbances," Econometrica, December 1973, 41, 733-50. FortuneMagazine, "Ma Bell's Kids Fight for Position," June 27, 1983, 107, 62-69.