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Jules Dupuit and the Early Theory of Marginal Cost Pricing Author(s): R. B. Ekelund, Jr. Source: Journal of Political Economy, Vol. 76, No. 3 (May - Jun., 1968), pp. 462-471 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1829307 . Accessed: 05/05/2014 19:39

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This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions Jules Dupuit and the Early Theory of Marginal Cost Pricing

R. B. Ekelund,Jr.* Texas A and M University

I. Introduction The name of JulesDupuit, the nineteenth-centuryFrench engineer, has been frequentlyinvoked in contemporaryeconomic literature concerned with marginalcost pricing(Hotelling, 1938, pp. 242-44; Nelson, 1964, pp. vii-viii) and cost-benefitanalysis (Prest and Turvey,1965, p. 683). Althoughhis contributionsin the area of utilitytheory (Stigler, 1950), consumers'surplus (Houghton, 1958), and price discrimination(Edge- worth,1912) were,by any standard,remarkable for the time,his role as proclaimedmentor of the moderntheory of marginalcost pricingand, more generally,of cost-benefittheory has been largelyunexplored and often misunderstood.The resulthas been a general confusionamong moderntheorists concerning his achievementin this area.1 Most writers have not botheredto investigateDupuit's originalworks and, following Hotelling'soriginal attribution, have simplyaccepted Dupuit as the first marginalcost theorist.Ragnar Frisch, Hotelling's firstcritic, may be placed in thiscamp (Frisch,1939, p. 145). Such neglecthas probablybeen nurturedby the relativeobscurity of his writingsand by the fact that, untilrecently, only two of his economicarticles have been translatedinto English(Dupuit, 1844, 1849b). The purposeof thisarticle is to assess the natureof Dupuit's contributionto thewelfare theory of marginalcost pricing.It willbe concludedthat although Dupuit has rightfulclaims as the firstcost-benefit economist, * This paper has grownout of a largerstudy of the economictheory of Dupuit. I would like to acknowledgemy appreciation for a generousgrant from the L.S.U. Foundationwhich made thetranslation of Dupuit's workspossible. I would like to thankProfessor J. P. Payne,Jr., R. F. Hebert,and L. H. Falk, as wellas theeditors of thisJournal, for helpful comments on earlierdrafts of thispaper. Finally, a very specialthanks to ProfessorW. J.Stober, not only for useful suggestions, but for having furnishedthe proofof Dupuit's theorem.I am solelyresponsible for final content, however. I Hotelling,for example, who originally(1938, p. 242) ascribedthe originsof the argumentto Dupuit,later modified his position.

462

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions JULES DUPUIT AND MARGINAL COST PRICING 463 he was not a progenitorof theprinciple. Discovery of a short-runmarginal cost principlein Dupuit's writings,in brief,would requirea contrivedand incorrectinterpretation of his remarkson costs and on the efficacyof subsidies.The issues are especiallytimely in view of the renascenceof interestin both the theoryand applicationof cost-benefitanalysis.

II. The Principleof UtilitePerdue Dupuit was thefirst economist explicitly to statethe principleof marginal utility2and to associate the area underthe demand curvewith a welfare measureutilize absolue. Dupuit, as did Cournotearlier, believed quantity demandedto be a decreasingfunction of price and, as earlyas 1844, he developedthe conceptwhich Marshall later called "consumers'surplus." Dupuit called thissurplus "relative utility" or "utilityremaining to consumers."The demand curvewas used by Dupuit as a utilitymeasure to analyze the welfareeffects of tolls, tariffs,costs, or prices; and it is here that the welfareeconomics of Marshall findsits origins.3 Dupuit set out to assess the effectsof taxes and tolls,though not speci- ficallyrelating them to costs,on whathe called utiliteperdue, which is the differencebetween utilityproduced (producers' costs and consumers' surplus) at any quantityand the total area under the demand curve. Increasesin prices,taxes, and tolls would reduce outputand the "utility available to society" (consumers'and producers'surplus), but Dupuit was even moreprecise. He pointedout that"where a tax is smallrelative to thecost of manufacture. .. it is legitimateto suppose a uniformrate of decrease [in quantityconsumed]," and, further,that "it may thus be said that the loss of utilityis proportionalto the square of the tax" (Dupuit, 1844,p. 104).5(See Fig. 1).

2 W. F. Lloyd (1833) discoveredby E. R. A. Seligman(1903, pp. 356-63)-is oftenattributed with the earliest exposition of the theoryof marginalutility, but no lessan authoritythan Alfred Marshall credited Dupuit with first "formally describing ... small incrementsof price as measuringcorresponding small incrementsof pleasure,"relegating to Lloyd the role of having "anticipated" utilityanalysis (Marshall,1920, p. 101). Lloyd's statement,according to Stigler,was adventitious (1950,pp. 312-13). 3 Marshall'smeasures, as contrastedto those of Dupuit, were protectedon all sides. Marshallassumed constancy of the marginalutility of money,to the con- sternationof contemporary theorists, so thatthe area underthe demand curve would representan unambiguouswelfare measure. 4 This utiliteperdue later became associated with reductionsin "net benefit," whichwas thesum of producers'and consumers'surplus. 5 Assumingthe marginalutility of moneyconstant, the area underthe demand curvein Figure I representsa moneymeasure of utility.Dupuit's theoremstates thatthe loss of utility,AUm, is proportionalto thesquare of thetax or price,Pm. In termsof Figure1, utiliteperdue may be written: AUm = ;AQmPm. (1) Now, by construction, Pm = mP1 and A Qm= mAQ, for a negatively sloped

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions 464 JOURNAL OF POLITICAL ECONOMY

The rationalefor marginalcost pricingas a welfaretool clearlyfinds its roots in "Dupuit's theorem."Prices above marginalcosts resultin utiliteperdue, and, as Hotelling(1938, p. 245) was later to point out, per-unitor excise taxes, by raisingthe marginalcost curve,have similar effectson "net benefit."Dupuit himselfnoted the desirability of spreading taxes over largenumbers of commodities,but he did not linktolls, taxes, or priceswith marginal costs or with increasesin marginalcosts in the utiliteperdue argument. Here we simplyfind the generalproposition that tolls,taxes, and so forth,effected changes in welfare.

P 4

P 3

0 Q Q Q Q A 4 3 2 1

FIG. 1

III. The Case of Bridges It is oftenthought that in his theoreticalcost-benefit studies of bridges Dupuit invoked the marginalcost dictum as a governmentalpricing guideline.Such conclusionsare not warranted,however, when one con- sidersDupuit's writings.In thisconnection, it is also necessaryto notethe ambiguityinvolved in referringto Dupuit's "bridge," since thereare no less than six bridge examples in his writings,some of them not even remotelysuggestive of a marginalcost argument(Dupuit, 1849b,p. 15). Several of the theoreticalbridges, however, do brushthe argument.An adaptationof a representativebridge (bridge "C") fromDupuit's article "On Tolls and TransportCharges" is presentedas Table 1 (1849b,p. 9).

lineardemand function. Making use of theserelations, and multiplyingnumerator and denominator of (1) by P1, the result becomes AUUm= aP', where a = AQ1/2P1 is the constantfactor of proportionality.This resultholds for any lineardemand functionand approximatesthe loss of utilityfor small incrementsin price for a non-lineardemand function.

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions JULES DUPUIT AND MARGINAL COST PRICING 465

The demand curve(or "curve of consumption")for bridgepassage is givenin columns(1) and (2). Column (3) showsthe marginalreduction in bridgecrossings due to unittoll rateincreases. The totalutility lost at any toll rate (column4) is calculatedfor any giventoll as the sum of utility lost at thatrate [(1) x (3)] and the total utilitylost at the previousrate. Dupuit termedcolumn (5) "the yieldof the toll," and it simplyrepresents total revenueor receipts.Column (6), representingconsumers' surplus, was not includedby Dupuit, but it is calculated here for convenience.

TABLE 1 A THEORETICAL TOLL BRIDGE

Reduction of Utility Crossings Corre- Number Due Utility Con- spending Toll of to Rate Lost Yield sumers' to Toll Rate Crossings Increase by Toll of Toll Surplus (7) (1) (2) (3) (4) (5) (6) [(5) + (6)]

0 . . . 100 0 0 0 445 445 1 80 20 20 80 345 425 2 63 17 54 126 265 391 3 . . . 50 13 93 150 202 352 4 . . . 41 9 129 164 152 316 5 . . . 33 8 169 165* 111 276 6 . . . 26 7 211 156 78 234 7 . . . 20 6 253 140 52 192 8 . . 14 6 301 112 32 144 9 . 9 5 346 81 18 99 10 . 6 3 376 60 9 69 11 . . 3 3 409 33 3 36 12 . 0 3 445 0 0 0

* Maximumyield.

Column (7) [(5) + (6)] is net benefit(actually gross benefitwithout costs of production),that is, the sum of producers'and consumers'surplus. It should be observedthat Dupuit did not,at the outset,include a provi- sionfor costs. The informationcontained in Table 1 is solelyin the province of demand. The utilityproduced, evidently,is divided between the monopolist and his consumers.In order to determineconsumers' surplus produced at any rate, total receiptsmust be subtractedfrom the "total utility produced" correspondingto thatrate, and it is apparentthat consumers' surplus will vary inverselywith the toll. Producers' surplus or total receipts (in the absence of costs) increase up to rate 5 and diminish thereafter.Dupuit, referringto this example, noted that "the distribution of utilityis very different"with differentrates (Dupuit, 1849b, p. 10); yet

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions 466 JOURNAL OF POLITICAL ECONOMY it was theeffect of rateson thetotal utilityproduced (" utilitycorrespond- ing to the toll") whichwas his consistentmaximand. Dupuit made it clear that the total utilityof the bridgewould depend on whereownership resided. "If the road or bridgeor canal is private property,"he pointedout that "the ownercompany has only one aim, and that is to get the largestpossible income fromthe toll." The profit maximizingrate would be 5 in Table 1, producinga total utilityof 276 and a utiliteperdue of 169. The total utilityproduced of 276 would be partitionedinto 111 of consumers'surplus and 165 of producers'surplus. Departingfrom principles of profitmaximization Dupuit analyzed a pricingpolicy under social ownershipof the bridge: If ... the bridgeis public property,the governmentwill want to recoverfrom the toll merelya fixedsum representinginterest on the capital spent for construction,maintenance cost and perhapsamortization. Suppose, forexample, that bridge C cost 150,000francs to build and thatthe relative figures shown in the table forcrossings are one-hundredthof the real trafficfigures; the governmentwill rest contentwith toll rate 1, because the proceedsof 8000 are enoughto cover interestat 4 per cent and leave over 2000 francsfor upkeep and amortization.The com- pany would charge 5, the governmentonly 1.... Surely,the extra 8500 francsto be paid by the consumers[under private monopoly]are reason enough to declare for public operation, yetthis is a secondaryconsideration in thelight of a comparison of the utilityof the bridge in the two cases [Dupuit, 1849b, p. 11]. Dupuit's opinion on distributionis crystallizedin his last statement. He would not be so opposed to tariffsif theyhad no othereffect than to change the distributionof utility.But tariffsdid positiveharm if they diminishedthe total utilitywhich commodities were capable of producing above costs. This point has been jaded all too oftenby a generalpre- occupation with Dupuit's statementof consumers'surplus. Producers' surplus in the nature of profitscould also be consideredpart of the maximandwere it not for the fact that profitmaximization reduced the total utilityafforded (the sum of producers'and consumers'surplus) by the bridge.But, apart fromthis issue, it is not at all clear wheremarginal cost pricingemerges from the statement. A clear analysisof costs was one of Dupuit's most seriousweaknesses, as Stiglerhas pointedout (1950, p. 314). Interestand amortization,as in the above case, cannot be easily construedas marginalwith respectto the numberof travelerswho cross the bridge.Maintenance costs have a betterclaim, and in an example of the Parisian Pont des Arts bridge, whichfollowed closely on the heelsof thetheoretical bridges, he indicated

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions JULES DUPUIT AND MARGINAL COST PRICING 467 that,when a privatecompany could double the numberof crossingsby cuttingits rate in half,and at the same time "still earn enough extrato coverthe slightincrease in maintenanceexpenses and the costs of collec- tion" (Dupuit, 1849, p. 12), it should do so. This would suggestthat maintenanceexpense is indeed marginalwith respectto quantity,but the statement,taken by itself, is notconvincing, particularly in view of the prior insistencethat the governmentrecover a "fixed sum" fromthe bridgeusers. Dupuit seemsto have intendedthis fixed sum to be inclusive of fixedcosts and to be independentof quantity,a situationnot particularly evocativeof an incrementalcost, either constant or changingwith quantity. Otherreferences in his writingsshed some lighton thisimportant issue. Dupuit made severalinteresting statements concerning costs of production in the 1844 article. Juxtaposingthe utilityargument over the pricing principlesof yet anotherbridge, he concludedthat, with a high enough tariff,it was possibleto renderthe bridge useless. He thenqueried whether this means "that thereshould only be verylow tolls or even that there should be none at all?" (Dupuit, 1844,p. 97). The rhetoricalquestion is answered in the negative,and the reader is directed to study tariffs " accordingto rationalprinciples, in orderto producethe greatest possible utilityand at thesame timea revenuesufficient to coverthe cost of upkeep and intereston capital" (Dupuit, 1844, p. 98). If Dupuit were using a short-runmarginal cost argument,he would have proposed a negligible toll or none at all. At such points,Dupuit appearsto have been intuitively gropingtoward a long-periodconcept of marginalcosts (implicitly assum- ingconstant returns to scale), but thecloudy notion of thenature of costs, eithershort or long run, casts strongdoubts on this possibility.The interestcharge would be marginalif the constructionof the bridgewere the issue, since thereare obviouslyno fixedcosts at such a time. Un- fortunately,Dupuit's studiesare couched in termsof alternativesbetween governmentaland privateoperation after a facilityhas been builtand is in operation.

IV. The Issue of Subsidies The key to understandingDupuit's concreterecommendation that fixed costs be recouped under governmentoperation lies in his bias against subsidies paid out of taxation. In an earlier discussion, Dupuit made referenceto a statementmade by Navier, an engineer,in an article appearingin the Annalesdes Pontset Chausseesof 1832. Navier was also concernedwith measuring the "utility"of publicworks, but he was under the spell of Say's dictum that costs of productionequaled price and thatthis was the measureof utility.This point was, of course,criticized by Dupuit, but Navier had also indicatedthat in order for government "operation not to be a burden on the taxpayer,the annual economy

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions 468 JOURNAL OF POLITICAL ECONOMY effectedby the transportmust be at least equal to the intereston the capital expendedtogether with the costs of maintenance"(Dupuit, 1844, p. 92). Dupuit consistentlytook Navier's proposal as his own, although one of his statementsspuriously implies the advocacy of subsidy.Con- sideringthe consequencesof a tariffreduction for the state,he pointed out that the moneywhich escapes the fiscstays in the pocketsof the old users,together with all the profitthey have made throughthe rate reduction,and new users have profitedin theirturn; the governmentcan thereforerecover its loss by levyingin other formsthe money it lost by loweringthe toll [Dupuit, 1849b, p. 12]. Here Dupuit was notsuggesting that the government must recover revenue foroperational losses in public projects,that is, due to pricingbelow full costs; rather,it should recoverin otherforms the monopolyrevenue it would lose by "lowering the toll." It is the prcfit-maximizingrevenue whichhas "escaped the fisc."Dupuit, then,could not have been thinking in termsof marginalcost pricingunder conditionsof decreasingcost, and it is thisapplication which has been the stockand trade of modern theoristsin the area. He could have advocated marginalcost pricing only in constantand increasingcost industries,for, in the latter,full costs are always covered. There is no clear indicationof the natureof productioncosts in Dupuit's writings,however, and the restrictionis a limitationwhich would not ordinarilybe placed on the tool. The analyticaldetails contained in Dupuit's developmentof several otherpricing tools do not lend supportto a marginalcost argumentin the case of the bridge.In one of the most distinctivecontributions to the early theory and application of price discrimination(Dupuit, 1865), Dupuit advocated just that amount of discriminationon the part of municipalgovernments which would be necessaryto coverthe full costs of waterdistribution. Although Dupuit notedthe high fixed costs of supplying waterto communities,he insistedthat water should not be made freebut thatdifferential subscription should be so designedas to cover fullcosts exactly(Dupuit, 1865,pp. 13-14). The goal in thiscase was to make use of all the wateravailable whileseeking to indemnifythe exploiter'sinitial and recurringexpenses. Monopoly in transportationwas also one of Dupuit's major concerns, buthere, as in the case of waterdistribution, we finda markeddistaste for subsidiesfrom tax revenue.Although governmental operation would be desirableif full costs could be covered and public utilityenhanced, a subsidywould be proofpositive that the enterprisewas ill-founded.In a contributionto the Dictionnairede l'TconomiePolitique, he points out that "the subsidy which it [the State] gives correspondsalways to an

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions JULES DUPUIT AND MARGINAL COST PRICING 469 equivalenttax whichit collects.Not onlyis thereno wealthproduced, but thereis wealthlost by virtueof the subsidy" (Dupuit, 1853,pp. 15-16). Dupuit saw a diminutionof wealthas stemmingfrom the real cost of the transfer.Capital investmentwill be "directednaturally to the mostlucra- tiveopportunities of the moment," that is, toward those most in demandby the public. "The subsidy device," continued Dupuit, "detours them [capitals]from these practicalusages and relatesthem to otherswhich are farless so" (Dupuit, 1853,p. 16). Thus, in the cases of waterdistri- butionand transportmonopoly, Dupuit was emphaticallyopposed to the subsidization of publicly owned or operated enterprises.Although Dupuit there entertaineda discriminatorypricing policy, there is no evidenceto indicatethat he was of a differentmind in the matterof toll bridges. Objections of a non-economiccharacter also account for Dupuit's aversionto subsidization.Dupuit, for example,assailed the grantingof privilegeby the state in severalcontributions to theJournal des tconomistes. Postmasters,because of theirestablishment, demanded an "indemnity" of 25 centimeson railroad travel where railroad routes were parallel. Dupuit foundthat the practiceled to misdirectionand to poor establish- mentof railroadtraffic, adding that the state should "let those die who cannotlive withtheir receipts," (Dupuit, 1851,p. 151). Here and in other statements(Dupuit, 1849a, p. 219), he was emphaticallyopposed to the capriceof politicalinfluence in the grantingof subsidies.His opposition was based on the beliefthat the politicalselection of projectsto be subsi- dized would not be groundedin economiccriteria. It is noteworthythat Harold Hotelling,among those who haveattributed the marginalcost principleto Dupuit, later qualifiedhis position in a 1949 letterto BurnhamP. Beckwith.Hotelling pointed out that"Dupuit mentions... theidea of a zero toll forwhich I arguedin my 1938 paper" but that "he fails to endorse it explicitly"(Beckwith, 1955, p. 83).6 Dupuit's views on subsidies,quite apart fromthe fecklesstreatment of costs in his writings,lends supportto the qualification.

V. Conclusions This paper has shown that a short-runmarginal cost theorycannot be attributedto Dupuit,at least as thatargument is mostcommonly applied. Had Dupuit been equipped with Marshallianperiod analysisor had he specifiedor indicatedthat his bridgestudies involved a long-runmarket period, the fixed and maintenanceexpenses would have indeed been marginal. A long-runapproach, moreover,would have possibly been more in keepingwith contemporary practice in cost-benefitstudies. But

6 Houghton(1958, p. 50, n. 5) questionsHotelling's earlier attribution, unaware of thequalification.

This content downloaded from 128.97.27.21 on Mon, 5 May 2014 19:39:54 PM All use subject to JSTOR Terms and Conditions 470 JOURNAL OF POLITICAL ECONOMY the practitionermay yet hold the achievementof Dupuit in highregard. His early and originalinsights into welfaretheory have provided the necessarybackdrop against which an importantand fruitfularea of moderneconomics is beingenacted. The clearenunciation and application of the utilityprinciple and the demonstrationthat society's welfare could be improvedby public action in a privateeconomy when conditionsof competitionare not ubiquitouslyeffective leave Dupuit unchallengedas the mostimportant early precursor of moderndoctrine and practicein the area.

References Beckwith, Burnham P. Marginal-Cost Price-Output Control. New York: Columbia Univ. Press, 1955. Dupuit, Jules. "On the Measurementof the Utilityof Public Works." Trans- lated by R. H. Barback from the Annales des Ponts et Chaussees, 2d ser., Vol. VII (1844) in the InternationalEconomic Papers, No. 2. London: Macmillan Co., 1952. Page referencesare to the translation. . "On Present Transport Legislation," J. Economists, 1st ser., Vol. XXIII (1849). (a) - "On Tolls and TransportCharges." Translated by Elizabeth Hender- son from the Annales des Ponts et Chaussees, 2d ser., Vol. XVII (1849) in theInternational Economic Papers, No. 11. London: Macmillan Co., 1962. Page referencesare to the translation.(b) . "About the Tax Paid to Postmastersby Owners of Public Vehicles," J. Economists, 1st ser., Vol. XXVII (1851). . "Ways of Communication." Translated by Candace Uter and edited by R. B. Ekelund, Jr.,from C. Coquelin and C. Guillaumin (eds.). Diction- naire de l'Economie Politique. Vol. II. Paris: Guillaumin & Co., 1853. Page referencesare to the translation available only in the library of Louisiana State University,Baton Rouge. - -. "Disadvantages of the Method of Intermittent Distribution." Trans- lated by Candace Uter and edited by R. B. Ekelund, Jr.,from Mario de Bernardi (ed.). De I' Utilite et sa Mesure. Torino: La Riforma Sociale, 1934. The selection is the fourth chapter of Dupuit's Theoretical and Practical Treatise on the Conductionand Distributionof Waters. 2d ed. Paris, 1865. Page referencesare to the translation available only in the libraryof Louisiana State University,Baton Rouge. Edgeworth,F. Y. "A Contributionto the Theory of Railway Rates," Econ. J., Vol. XXII (1912). Frisch, Ragnar. "The Dupuit Taxation Theorem," Econometrica,Vol. VII (1939). Hotelling,Harold. "The General Welfarein Relation to Problemsof Taxation and Railway and UtilityRates," Econometrica,Vol. VI (1938). Houghton, R. W. "A Note on the Early History of Consumer's Surplus," Economica, Vol. XXV (February, 1958). Lloyd, William F. "A Lecture on the Notion of Value as DistinguishableNot Only from Utility,But Also from Value in Exchange." Delivered before the Univ. of Oxford in Michaelmas Term, 1833. Marshall, Alfred.Principles of Economics. 8th ed. London: Macmillan Co., 1920.

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Nelson, James R. (ed.). Marginal Cost Pricingin Practice. Englewood Cliffs, N.J.: Prentice-Hall,Inc., 1964. Prest, A R., and Turvey, R. "Cost-Benefit Analysis: A Survey," Econ. J., Vol. LXXV (December, 1965). Seligman, E. R. A. "On Some Neglected British Economists," Econ. J., Vol. XIII (1903). Stigler,George J. "The Development of UtilityTheory," J.P.E., Vol. LVIII (August, 1950).

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