Utility is a Measurable Quantity

Dupuit’s contemporaries ignored his analyses. It was Jevons who, in 1879, drew atten­tion to him. The problem was that he stated that Dupuit’s analysis of utility was basi­cally the same as the analysis he himself developed in his Theory of Political Economy.

To illustrate his statement, Dupuit (1849 [2009]: 276) often gave the example of a bridge for pedestrians, the user cost of which is nil. Suppose that an agent h = 1,..., k considers as equivalent the initial situation in which he or she does not cross the bridge and keep an amount of money x^h_m, and a situation in which the bridge is crossed and a diminished amount (x^h_m — ∆x^h_n) of money is left. The absolute utility of the bridge for agent h is ∆x^h_m. For all the agents, it is _χDX_m- This presentation does not entail any interpersonal comparison of utility, nor any hypothesis about the marginal utility of money. Dupuit simply adds up the amounts of money that the agents are ready to pay to cross the bridge (Allais 1981 [1989]: 165).

The relative utility - the benefit that consumer h draws from buying the good - is the difference between the absolute utility of the good and the price p₁ he or she must pay to obtain it - a price for the time being supposed to be uniform: Dx^h_m — p₁.

Dupuit generalizes the result to the case where each agent can consume several units of the good. “Each consumer attaches a different utility to the same object, according to the quantity he can consume. So, a consumer who bought 100 bottles at 10 cents would only buy 50 at 15 cents, and only 30 at 20 cents” (Dupuit 1844 [2009]: 209). He introduces the cost of production (ibid.: 215) and analyses the effects of a technical progress that diminishes the cost: it increases utility because it allows new consumers to buy the good and the former consumers to buy more of the good. He studies the effects of an indirect tax and shows that the measure of the loss incurred is not the sum of the tax collected. What constitutes a loss is the fact that some agents stop consuming the good, which is now too expensive for them.

In the case of a bridge, for example, the problem of the determination of the level of the toll is of the utmost importance to Dupuit. If it is too high, a considerable part of the utility the public investment could have would be lost, and if it is too low it would be impossible to get back the sums invested. To fix it properly, it is necessary to know what it is aimed at. A private company would try to maximize its profit. The state would fix the level of the toll to get back a sum equal to the interest of the investment, the maintenance and the depreciation. The amount fixed by the private company would be larger than that requested by the state. However, for Dupuit, this is not important. The sum that “the company gets in excess over that obtained by the State form a profit for the shareholders who gain exactly what the users lose. Had the tariff no other result, one could say that it has no influence on the public wealth, it just modifies its distribution” (Dupuit 1849 [2009]: 280). The public investment is nevertheless preferable because the higher private tariff would entail a loss for all: those who would stop using the service would be deprived of something, which would have cost nothing, in Dupuit’s example, to the company.

However, if the uniform public toll is preferable, it is not the best one. What would be a rational toll? If it were nil, relative and absolute utilities would be equal, but the financ­ing of the investment should be made through taxes; and if the investment is of no utility to those who pay taxes, this would be unjust. One must in consequence establish a toll and fix it at such a level that those who use it pay a sum proportional to the utility they get (Dupuit 1844 [2009]: 286). It is thus necessary to abandon the idea of a uniform price, and differentiate the toll in order not to prevent its use by anyone who is ready to pay a duty, which exceeds the cost of the service.