Public economics (1): the measurement of utility and its consequences for economic calculation

Undoubtedly the name of Jules Dupuit belongs to the hall of fame of economic the­ory on account of his contributions to public economics. His attempt to address a practical problem - that of the refinement of utility measurement for public works - led him to develop a new approach which was deeply to influence the subsequent developments of economic analysis.

A path-breaking conception of the measurement of utility

Dupuit presented his conception of utility extensively in two well-known essays (1844, 1849) devoted to the measurement of utility applied to public works. The 1844 article aroused little interest among his fellow engineers, - apart from Louis Bordas (1815-1905),^[74] whose criticism incited Dupuit to improve his reasoning in his 1849 paper. He restated his demonstration, without graphics, in a long paper published in 1853 in the Journal des economistes, intended to reach a wider readership.^[75] Two remarkable contributions appear in these texts: (i) a new way of measuring utility, which results in (ii) a theory-based formulation of the demand curve.

(i) Dupuit’s cutting-edge 1844 article begins with a discussion about the theory of value and an appraisal of Jean-Baptiste Say’s method regarding utility and its measurement.^[76] Like Say, Dupuit considered that the foundation of value is utility, in concordance with many of his contemporaries.^[77] Say focused on the definition of utility and considered that the utility of a commodity could be measured through its market price. To Dupuit this assertion leads to “serious mis­takes” (1844, 206):

If society pays 500 million for the services rendered by roads, it only dem­onstrates one thing: that their utility is of 500 million at least.

measure, and not as a lower boundary of an amount that you do not exactly know, you act as a man eager to measure the height of a wall in the dark, who, seeing that he cannot reach the top by raising his arm, says: this wall is two metres high, because if it was not two metres, my hand would pass over the top. If you assert that this wall is at least two metres high, we agree. But you go further and you say that this is its measure; we do not agree anymore. When daylight comes and when you have a ladder, you will admit that this supposed two-metre wall is actually fifty metres high.

(1844, 207)

In the end, “J.-B. Say’s essential inaccuracy is not to have ignored utility or value in use, but to have excluded it from science, by replacing it with the value in exchange, which he considered as its measure” (1853a, 317). Dupuit preferred to follow the path opened by Pellegrino Rossi (1787-1848) on the theory of value. He introduced a distinction between “utilite absolue” (absolute utility) and “utilite relative” (relative utility): the absolute utility of an object is measured “by the max­imum sacrifice that each consumer would be willing to make to obtain it” (Dupuit 1844, 213), while the relative utility is “the difference between the sacrifice the consumer would be willing to make to get it and the purchasing price he must pay in exchange” (1844, 214).

Dupuit takes an example (which is his favourite method for being understood by the majority) to describe absolute and relative utility, by imagining that a price change occurs as the result of a new tax:

For example, let us assume an object whose market price, roughly equivalent to the production costs, is 20 fr. The utility of this product, depending on the circumstances in which it is consumed, may have the following values:

30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20 fr.

Its utility will therefore be, in the corresponding circumstances,

10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.

If a tax of 5 fr. is imposed, the utility of this product will decrease by 5 fr. under the same circumstances, for all consumers who used to find in it a util­ity of 10, 9, 8, 7, 6, 5 fr. Now they will only find a utility of 5, 4, 3, 2, 1, 0. The loss is equal for all. As for those who found in this consumption a utility of only 4, 3, 2, 1, 0 fr., and who will no longer consume, because of the tax, they will lose precisely the utility they would have found there; their loss will therefore be different for all, and equal to 4, 3, 2, 1, 0 fr.

(Dupuit 1844, 215)

As a consequence, any increase in the price of a commodity ends in a decrease in relative utility. This reasoning allows Dupuit to establish a general formula:

any rise or fall in the price decreases or increases utility by an amount equal to this variation for the consumers who remain in both circumstances; for those who leave or arrive, the utility lost or gained is equal to the old or new relative utility found by them in the product.

(1844, 216)

He also insisted on the variability of utility: the degree of utility of a commodity changes from one person to another and varies according to the quantity already possessed - thus taking up the idea, widespread since Daniel Bernoulli’s 1738 essay, that the same amount of money does not have the same utility for everyone. He particularly insisted on the decrease in marginal utility when the amount of commodity increases, since each incremental quantity satisfies new (secondary) needs.

(ii) Dupuit’s 1844 paper is innovative from another viewpoint: it contains the first explicit formulation of a theory-based demand curve. Obviously, the rela­tion between the quantity demanded and the price had been well-known for many decades, and six years before, Antoine-Augustin Cournot had already expressed his “loi du debit” (1838) from an inductive method based on an interpolation of observed economic data. But the first theory-based demand curve, called “courbe de consommation”, was produced by Dupuit in the Appendix of his 1844 article.

Dupuit’s curve Nnn’P represents a (global) demand curve,^[78] which is a decreasing function of the price P. If the price paid is 0p, then the relative utility is measured by the area of “triangle” npP, which is nothing other than the consumer’s surplus.^[79] Dupuit remarks that “utility decreases as the price of a commodity increases, but more and more slowly, and it increases more and more quickly as the price decreases, since it is expressed by a triangle which expands or contracts” (1844, 238). His formalisation also allows to high­light the utility loss (“utilite perdue”) suffered by the consumer in the event of a price rise (whether resulting from a tax rise or a toll), which enables him to obtain very important results in terms of cost-benefit calculations or tax analysis (see below).

Figure 5.1 Dupuit’s consumption curve (1844)

Dupuit gives his demand function an important property: convexity, explained by the pyramidal structure of society:

the increase [in consumption] due to a lowering of the price is all the greater as the price itself is lower.... This property is due to the structure of society, which, when divided into categories by order of income, and superimposed starting with the poorest, presents the image of one of those pyramids of can­nonballs that one sees in artillery parks, the layers of which are all the more numerous as they are lower. Thus, when the price of an object goes down, its use spreads to more and more consumers, not to mention the fact that existing consumers consume it in greater quantities, as we have explained several times.

(1849, 232-3)

It should be noted that Dupuit’s analysis is a geometrical one, with no transcription in algebraic form. His refinement of the definition of utility led to his being consid­ered the inventor of the notion of surplus, which would be a genuine exemplifica­tion of a “discovery” in economics (Etner 1983).^[80]

Economic calculation a la Dupuit

Dupuit’s new method of measuring utility has an important consequence as regards public economics, since it changed the way in which the cost-benefit ratio of public infrastructure projects is assessed.

Dupuit’s conception of utility led him to change the method of economic cal­culation in the corps of Ponts et Chaussees. During the 1830s, the engineers relied on a rule established by the senior engineer Claude-Louis Navier (1785-1836) to decide on public works issues, especially canals. This rule, published in 1830 in a paper of the Journal du genie civil, stated that the expenditure related to any public infrastructure must not be higher than the transportation cost savings.^[81] This method put into practice Say’s definition of utility measurement: the social benefit of digging a canal was estimated by the difference between the cost of transporting commodities using the new canal and the cost incurred with the previous means of transportation, multiplied by the quantity of commodities transported on the waterway. Dupuit disapproved of Navier’s method, considering it to be subject to the same bias as that of Say, that is to say, it would lead to a patent overestimation of the gains; it would in the end distort the social choice since the infrastructures would appear more profitable than they actually are.

In his 1844 and 1849 papers, Dupuit proposed a revision of Navier’s principle. Any project of public infrastructure (especially in network industries) intended to reduce transportation costs certainly causes an increase in traffic; but the util­ity enjoyed by the last users is inevitably lower than that of the first ones. In both papers, Dupuit establishes a monetary appraisal of utility variations, based on his demand curve presented above: rather than Navier’s inaccurate estimation, the benefit accruing to the users must be evaluated by considering their relative utili­ties, based on the “law of consumption” and their individual willingness to pay. Dupuit criticised the previous method because it leads to “calculation errors” due to “the erroneous principle that the same product, the same service rendered, has the same utility for each consumer and in each circumstance” (1849, 270).

Dupuit proposes a limpid numerical example which is summarised below. Let us consider, he writes, a canal without tolls on which 100 tons of goods circulate; and let us suppose that a toll of one franc per ton is established. If the traffic falls from 100 tons to 70 tons, then “it is clear that for these thirty tons, the utility does not exceed 30 francs” (1849, 264). If the toll is increased to 2 francs per ton and there is a reduction in tonnage from 70 to 50 tons, then it can be said that “for those 20 tonnes that have disappeared from circulation, the utility is 2 francs per ton and 40 francs for the 20” (1849, 264). By replicating this reasoning, Dupuit manages to draw up a table compiling the data in Table 5.1.

Columns (1) and (2) represent the law of consumption, that is, “the relationship between the price of products and the amount consumed, a different law for each product” (1849, 265). Column (3) expresses the decrease in consumption as a result of successive toll increases; the fourth expresses the partial utility^[82] of different

Table 5.1 Utility of a canal at different toll levels

portions of the tonnage, while the last one represents the “possible utility” (1849, 265) of the canal, that is, the utility in the absence of any toll. Then,

to find out the utility of an object, the following method of calculation can be used: suppose that all similar products are subject to a tax which is increased by slight amounts. With each increase in this tax, a certain quan­tity of goods will disappear from consumption. This quantity, multiplied by the rate of the tax, will give its utility evaluated in money. By increasing the tax in this way by slight amounts, until there are no more consumers, and adding all the partial products, we will have the total utility of the object considered.

(1849, 273)

The opposite reasoning is also made by Dupuit, based on a reduction in transport costs as a result of infrastructure improvements or toll reductions. In both cases, Dupuit’s method evaluates the gains and losses of utility as precisely as possible, whereas Navier’s method overestimates them. Dupuit’s reassessment of the meas­urement of public utility,^[83] defined as the utility of public goods, has important consequences, both in terms of economic analysis and for the determination of criteria for public decision-making. In this respect, our engineer is credited with elaborating what would become classical principles in public economics: a cost­benefit analysis based on his way of calculating public utility (Etner 2011, 594).

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