Public economics (2): monopoly, price differentiation and taxation

The exposition of the demand curve by Dupuit and the identification of a surplus enjoyed by the consumer has numerous consequences for his economic analysis. We focus on two issues: price discrimination in a monopoly situation and taxa­tion principles.

Natural monopoly and differentiated pricing

In his study on tolls (1849) and later in the entry “Peage” of the Dictionnaire de l^,economie politique (Dupuit 1853b), Dupuit wondered what optimal pric­ing should be applied to communication routes. To study the effects of tolls, he used numerical examples from which he drew general principles. The calcula­tion of utility is based on the “law of consumption” presented above, which is established on (observed or hypothetical) data about the use of infrastructures. According to him,

ignorance of the law of consumption is not an obstacle to rational speculation on tolls; probability, by substituting itself for certainty, makes the problem more difficult without doubt; but it adds a new attraction to the solution.... [With] the uncertainty in which the producer finds himself, the solution depends both on the sagacity with which he guesses the needs of the consum­ers, and on his imagination, which suggests to him the means of obtaining from them the greatest possible sacrifice.

(1849, 281)

For the measurement of the overall effects of a toll, Dupuit draws up the same kind of table as above using a different example (1849, 276, example A) - that of a bridge, but this time including the gains and losses for the producer (either the toll concessionary or the State).

Table 5.2 gives important indications for measuring the effect of every toll level on the utility of the bridge.

At this stage, the situation is different, he says, depending on whether the toll is implemented by the State or by a private company. He continues: “if the road is privately owned, it is clear that the owning company will have no other aim

Table 5.2 Gains and losses associated with a bridge toll

Notes: ⁽*) Maximum of the toll

than to make as much money as possible from the tolls” (1849, 279), which in the above example would result in the choice of a toll of 3 fr. per passage, which is a monopoly price. But “if the State, on the other hand, is the owner of the bridge, it will only want to get a fixed sum from the toll, representing the interest on the construction costs, the maintenance costs and perhaps a depreciation” (280); in the previous example, this could be achieved by opting for a one-franc toll, which would provide the State with annual resources of 55,000 francs, enabling it to cover all or part of its costs.

There are clearly many advantages to be gained from public management of the toll: the toll will be lower, since there are no shareholders to pay - part of the money collected by the companies “is a profit for the shareholders who earn pre­cisely what the users lose” (1849, 280); and consequently, the total utility will be higher due to wider use of the infrastructure. Dupuit touches on the issue of natural monopoly: where a private company (free to set its own tariffs) would favour quan­tity rationing and a higher passage price, the State could lower the toll in order to reduce utility losses:

There will only be a fiscal loss; the money it [the State] has lost in this way has remained in the pockets of the existing users with all the profit they have made through the reduction of the toll; moreover, the lowering of the fare has given other profits to new users; the State is therefore in a position to make good its loss, by demanding in another form the sum which it has lost by the lowering of the fare.

(1849, 281)

But Dupuit’s reasoning on the pricing issue goes further: rather than think­ing in terms of a unique toll price, thought should be given to the possibility of “tatonnements” (1849, 283) to define categories of users.

The bridge owner could insert a clause in his tariff as follows: For the user in cap, smock or jacket, the toll is reduced to 0.01 fr. If he has thus succeeded in defining in a sufficient manner the workers he wants to enjoy the reduc­tion, he will necessarily get the 3,000 francs revenue that the new crossings must yield.

(1849, 285)

This is where the notion of price differentiation comes in, as it was called later by Arthur Cecil Pigou and Joan Robinson. As a consequence, the price paid by the user must be disconnected from the production cost, contrary to the majority view at the Ponts et Chaussees,^[84] but should mirror the willingness to pay of each user: “[t]he best of all pricing systems would be the one which charges the user of any trans­portation route a toll proportionate to the utility enjoyed by using it” (1849, 286).

The practical way to enforce differential pricing consists in offering a variety of services or qualities to the users through some kind of classification. This classifi­cation can be done externally, for example, by defining categories of goods trans­ported by rail or by ship: an ad valorem toll could be charged, which would affect the goods more or less according to their price (thus depending on whether the transport concerns necessities or luxury goods). Classification can also be done by the users themselves, for example, by offering them several classes (for instance, several classes for train rides or theatre tickets^[85]), so that each user, by choosing one or another level of service, expresses his ability to pay, which reflects his per­sonal utility gain. Dupuit took up his reasoning again in later writings, applying it to other network industries; in a sense, his method of tariff determination “recalls what today is known as Ramsey’s pricing theorem” (Mosca 1998, 262), which states that a monopoly achieves second-best when it charges a tariff according to the elasticity of demand for its services.

Of course, Dupuit was aware of the practical and moral difficulties of such a system:

it is clear that, by multiplying the classes indefinitely, one could make travel­lers pay for all the utility they derive from the road. but for this to happen, it is necessary to be able to distinguish between the classes that attach a different utility to their transport and to oblige them to voluntarily classify themselves

in one or another category of the tariff. Now, this is a great difficulty, which gives rise to the establishment of a host of measures that are generally poorly understood by the public. It would cost so little, it is said, to put a few metres of leather here and a few kilograms of horsehair there, that there is nothing but avarice in refusing them. Certainly, no one believes less than I do in the philanthropy of the companies We strike at the poor, not because we

want to make them suffer personally, but to frighten the rich.... [It] is to keep in the first [class] those who would be willing to make this expenditure, if the second did not exist.

(1849, 296-7)

The practical relevance of price discrimination lies in the ability for each seller to determine which proportion of his customer base is willing to purchase at any price, as if submarkets could be identified and separated according to the willing­ness to pay of consumers. The engineer also recognised the risks associated with the “universal unscrupulousness of users” (1849, 286). This is why in some cases, Dupuit suggested the possibility of a unique toll, mainly for reasons of simplifica­tion, in spite of the inevitable loss of utility (Etner 1983, Mosca 1998). Marginal cost pricing would certainly generate losses, but Dupuit imagined that the State could easily compensate for them: “the State is in a position to make good its loss, by requesting in another form the sum it lost through the lowering of the tariff” (1849, 281). For this reason, Hotelling credited him with the development of mar­ginal cost pricing for natural monopolies, probably going a little beyond Dupuit’s own intentions.²⁸

Dupuit’s pricing analysis did not arouse the interest of his fellow engineer col­leagues until the 1880s, particularly with regard to railroads (Vatin 2003).

Dupuit’s views on taxation

Dupuit’s developments on utility and demand also go in another direction: the anal­ysis of the effects of taxation, with two main points: the analysis of the effect of a tax increase, and the criterion for an optimal tax system.

Dupuit’s analysis, based on his decreasing and convex demand function, gave him the opportunity to develop a singular but luminous analysis of tax changes (Figure 5.2). He considers a commodity whose initial pricep is subject to several successive tax increases.

28 Hotelling introduces his article as follows:

In this paper we shall bring down to date the revised form of an argument due essentially to the engineer Jules Dupuit, to the effect that the optimum of the general welfare corresponds to the sale of everything at marginal cost.

(1938, 242)

Figure 5.2 Tax and loss of utility

Source: This figure is a simplified reproduction of Dupuit’s third graph in his 1844 Appendix (1844, 239)

Going fromp top’, the deadweight utility loss is measured by the triangle area nn’q. Not only do those who continue to consume the good despite the increase in its price suffer a loss - the same as that highlighted by Cournot - but some of those who previously consumed the good have to give it up, since the price they would have to pay is higher than the additional utility they would get from it: “the tax acts not only on those who pay it, but also on all those who would have consumed with­out the tax” (Dupuit 1844, 215). The introduction of the tax results in a net loss for consumers. In his 1838 Recherches sur les principes mathematiques de la theorie des richesses, Cournot came close to this outcome: he perceived the double effect of such a tax which weighs upon consumers at the same time as it excludes others; he, however, did not attempt to estimate the latter effect.^[87] Dupuit succeeded in giving this loss a monetary measure.

Dupuit’s argument goes further. Doubling the tax, bringing the new price to p ’’, causes a multiplication by four of the utility loss (nn ''q'), while tripling it would multiply the loss by nine (nn ’’’q''), with this conclusion: “the higher the tax, the lower the revenue it yields relatively. The lost utility increases as the square of tax” (1844, 239). Thus, the amount of the loss to consumers depends on the slope of the demand curve; and doubling the tax would only lead to a proportionally smaller increase in State resources. In the end, Dupuit’s analysis seems from this point of view to argue for the broadest possible tax base and the lowest possible tax rate.

Whereas the French liberals showed a clear preference for direct taxes,^[88] Dupuit considered that there is an “analogy between capital tax, income tax and consump­tion tax” (1865a, 275), which always end up hitting personal income. Dupuit held a singular position among his colleagues of the Societe d’economie politique who strongly rejected consumption tax, on the grounds that it would be inequitable. To Dupuit, direct taxes also have major shortcomings:

Seeking justice in the tax base is a mistake; justice is nothing but a mirage which vanishes when approached; what is to be found in the tax base as in wealth repartition is public utility. The best tax is the one that is least damag­ing to society’s wealth.

(1865a, 277)

As a consequence, the best tax system is the one with the least negative impact on public utility: “each type of tax must be settled so that it causes the smallest dam­age to society. Here is the rule that must be followed by legislators” (1859a, 580), without worrying about its consequences on wealth distribution. Tobacco taxes fit this criterion perfectly, and Dupuit considers them as the most desirable ones. To him, admittedly “tobacco tax is... supremely inequitable” (1859a, 578) since it affects the poorest while the rich are spared. However, he is “neither for its sup­pression nor for its decrease. Because it is outstandingly utile” (1859a, 578) since it encourages a reduction in the use of a harmful good satisfying a dispensable and damageable need.^[89] Finally, tobacco tax must not be suppressed or diminished because “it is easy to levy it; taxpayers pay it without trouble; it does not harm the consumption of anything that is useful for health” (1859a, 579). Unsurprisingly, he also approved of differential tax rates, especially on tobacco,^[90] since the objective is to “extract from the tobacco tax all the revenue it can yield” (1859a, 582).

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