Jules Dupuit and the Early Theory of Marginal Cost Pricing Author(S): R
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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 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]. The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Political Economy. http://www.jstor.org 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 contribu- tionto 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 con- sumers."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 P 4 P 3 p2 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