The theory of price: “loi du debit”, monopoly and duopoly

Cournot (1877, 91) considered his theory of prices his main contribution to eco­nomic theory. He broke not only with his predecessors because he used mathemat­ics and proposed new tools, but also because his whole approach was different.

On the theory of value, Cournot referred to “only one axiom... that every­body tries to get the greatest possible value from his thing or labour” (1838, 35). He applied this axiom to explain the behaviour of the producers who wanted to maximise their profits, but he refused to use it with regard to the demand for com­modities or to estimate the benefit an individual received from exchange. “There is nothing in common between the feeling of pleasure or pain and the mathematical notion of quantity in mathematics” (1851, 233).

But the axiom that anybody tried to get the maximum income from his or her resources was not Cournot’s sole hypothesis. In several instances, he affirmed the principle of the uniformity of a price, which originated in the very notion of market understood as “the whole territory, the parts of which are united through free trade, so that prices are easily and quickly levelled” (1838, 40).

Cournot’s framework was partial equilibrium. While it is true that “the eco­nomic system is an entity, the parts of which are linked together and act on each other” (1838, 99), yet it is possible, as a first approximation, to abstract from such effects: the change in price of a good causes producers’ income to change and thus make up for the change in consumers’ spending, so that “the same [amount of] funds remains available for the demand of all other merchan­dise” (l838, 101).

The “loi du debit”

Cournot posited that the relationship between price and quantity produced - the “debit” - cannot be established theoretically.

Given that the function f (p) is continuous, the function pf (p) which expresses the total value of demand must also be continuous. Since the consump­tion of any good remains finite even on the hypothesis that it is absolutely free, the total value of demand is zero when p = 0. A value can always be assigned to p so great that the demand for the article would cease. “Consequently, since the function pf (p) at first increases, and then decreases as p increases, there is a value ofp that makes this function a maximum, which is given by the [following] equation” (1838, 40):

The root of equation (1) corresponds to a maximum under the following condition:

If f'(p) is negative, whenever f"(p) is negative, it is impossible that there should be a minimum, nor more than a maximum. In the contrary case, Cournot admitted that the existence of several maxima or minima was not proved to be impossible. He would later concede that, with regard to taxation, this scenario could make sense:

a tax cut, even modest, causes tax revenue to decline, which indicates that a maximum revenue can be reached by raising the tax; on the other hand, with a significant tax cut, say by half or by a quarter, it is possible that tax revenue rises a great deal.

(1877, 95)

Capitalising on his “loi de debit”, Cournot profoundly renewed price theory. While classical economists were almost exclusively interested in competition, the starting point of Cournot’s analysis was monopoly, which he viewed as the simplest way to analyse the formation of prices. By increasing the number of competitors, he then studied duopoly, oligopoly, and finally concluded his analysis with the study of unlimited competition, described as a situation where the change in output of one firm does not affect the market price.

Monopoly

Cournot wrote:

suppose that a man finds himself owner of a mineral spring which has just been found to possess salutary properties possessed by no one else.... After various trials, he will end up adopting the value ofp which renders the prod­uct pf (p) maximum.

(1838, 43)

represents the revenue of the owner of the source, and this revenue only depends on the type of demand function.

To make equation (1) applicable, it must be supposed that for the value of p obtained from it, there is a corresponding value of demand D that the owner of the spring can deliver; otherwise, the owner will derive the price of mineral water p from the maximum quantity, say q_l,_la,., that the spring can produce: p = f (q_mx ).

Let us now assume that an individual possesses the secret of an artificial min­eral water, for which the materials and labour must be paid for. If the total cost is a function φ(D) of the quantity produced, the producer maximises the net product (or net revenue) pf (p ) - j ( D). Since D is a function of p, the net product can be regarded as “depending implicitly on the single variable p, although generally the cost of production is an explicit function, not of the price of the good produced, but of the quantity produced” (Cournot 1838, 44). Consequently, the pricep* to which the producer should sell his good is the root of the equation:

Though Cournot did not use the term “marginal cost”, he nonetheless intro­duced this concept in his analysis by noting that the form of function φ' (D) “exerts the greatest influence on how to solve the principal problems of economic sci­ence” (1838, 45).

The marginal cost is capable of increasing, decreasing or remaining stable, as production grows. For manufactured goods, it is generally the case that the cost becomes proportionally less as production increases, due to a fall in general expense.

As a result, powerful capitalists or large firms are able to... kill competi­tion and artificially create a true monopoly which allows them to earn an above-normal rate of profit, that is, a rent... in some industries that would not allow it under normal circumstances. This type of monopolies usu­

ally entails a fall in prices favorable to consumers; though the influence of the monopoly always persists in that the price does not fall as much as in the competitive case.

(Cournot 1863, 79-80)

It may happen, however, that when production is carried beyond certain limits, it induces higher prices for raw materials and labour to the point where φ' (D) again begins to increase with D. Thus, the idea that the marginal cost could successively rise and fall is clearly discussed in Cournot’s writings.

In the case of productions using agricultural lands, mines or quarries, the func­tion φ'(D) increases with D. In discussing Ricardo’s theory of differential rent in Principes de la theorie des richesses (1863, 76) and in Revue sommaire des

Cournot and the emergence of mathematical economics 85 doctrines economiques, Cournot admitted that land displayed various degrees of fertility.

This fact is indisputable, but it is not the reason that explains the principle and the existence of rent. All plots of land of equal fertility would eventu­ally produce the same level of rent Total rent would be shared between

landowners in proportion to the amount of land they possess, instead of being unequally distributed based on the degree of fertility.............................................................

reason [that explains the formation] of rent always consists in the fact that, according to the law of demand, the value of the product exceeds the cost of production.

(Cournot 1877, 99-100)

Rent can appear because plots of land of different quality are put into cultivation to meet demand; but rent can also appear on homogenous land due to increasing marginal cost, because only a limited quantity of produce can be farmed or because the same plot of land can satisfy multiple uses.

There are two particular cases: the marginal cost could remain constant or could be zero. In the latter case, the price does not depend on cost. Such a scenario is more frequent than usually thought at first sight. By way of examples, Cournot mentioned a theatre play or the tariff of a bridge. In such cases, the price is set as if production costs were nonexistent.

It seems natural to admit that, when production costs climb, the price set by the monopolist grows by the same amount. Cournot sought to demonstrate that this conjecture is incorrect: depending on the form of the demand function, an increase in costs could lead to a price increase of a smaller or larger proportion. In particular, this reasoning applies to taxation, which can be of two types: direct or indirect. If the monopoly is subject to a fixed tax or a tax proportional to its income, this tax will affect neither the price of the commodity nor the quantity produced. The burden of this direct tax falls entirely on the monopolist. Though the burden of this direct tax does not fall on consumers, it could nevertheless be detrimental to the general interest since the State may use the tax revenue less well than the monopolist would have done. Furthermore, such a tax can be an obstacle to capital accumulation.

No one will use his capital in new investments, nor in the improvement of existing investments, if he can no longer obtain the ordinary interest brought in by capital in businesses of the same kind, on account of the tax imposed on the net income from his investment.

(Cournot 1838, 52)

Indirect taxation is another alternative.

evaluate the loss borne by each type of agents once such a tax is established. The pecuniary loss borne by consumers who continue to buy the commodity in spite of the increased price is

The revenue of the tax perceived by the State is

So if the price increase exceeds the tax, p₁ - p₀ > t, the loss of consumers alone will be higher than the amount of tax revenue perceived.

Cournot noted that consumers who stop purchasing a good upon which a tax has been imposed and transfer their spending to other goods experience a loss. How­ever, Cournot contended, “this kind of loss cannot be estimated numerically.... It is in one of those relations of order and not of size, which numbers can indicate indeed, but cannot measure” (1838, 103). Cournot deliberately closed a door that Dupuit would later open to measure the consumer surplus.

The loss borne by the monopolist is

Since p₀ is the price which maximises the profit of the monopolist, the expression is necessarily positive. Thus, the loss to the monopolist alone exceeds the amount of tax revenue perceived by the State. Cournot concluded that the doctrine of the physiocrats was perfectly applicable to the monopoly regime. It is better to levy a direct tax on the net income of the monopolist than to lay a specific tax on the commodity.

If, instead of levying a tax, the State subsidises the monopolist, the gain for the monopolist would be less than the consumers’ burden.

Duopoly

Departing from monopoly, the second step of Cournot’s price theory consists of introducing a second producer. Two firms producing the same good could have an interest in agreeing with each other to set a monopoly price. But each of them can increase its profit through an increase in its production. The cooperative equi­librium, Cournot stressed, is unstable: “it could not persist unless a formal link is established; because one cannot... suppose... men making no errors or not being careless” (1838, 62).

Rather than cooperating - a case examined below - each producer tries to increase his revenue and must take into account the behaviour of his competitor. He supposes that his competitor’s output is given: this is called Cournot’s conjec­ture. Instead of using the demand function, Cournot used the reciprocal function which determines the price in function of the quantity of water demanded. Let D₁ denote the flow of spring 1 and D₂ the flow of spring 2, the price is a function of the total flow:

Each firm seeks to maximise its profit independently (that is, with no coopera­tion). If the cost of production is zero, the profits of the firms are respectively expressed by:

To determine the quantities firms ought to produce, additional assumptions are required. Cournot admitted that the owner of spring 1 has no direct influence on the determination of the quantity produced by its rival. “All he can do, when D₂ has been determined by owner (2), is to choose for D₁ the value which is best for him, which he can accomplish by properly adjusting his price” (1838, 60). The quantity D₁ is that which maximises profit of firm 1 for a given level of output D₂ generated by its rival:

Symmetrically, the quantity D₂ is that which maximises profit of firm 2 for a given level of output D₁ generated by its rival:

In Figure 4.1, the m_in_i curve represents the reaction function of firm i (Equa­tions (2) and (3) above), i = 1, 2. An equilibrium is reached at point E where each firm sets its output at the level that its competitor took into account by setting its own output.

Cournot could have supposed that the decision variable is the price instead of the quantity. Each firm would thus have fixed the price in such a way that is maximal, D_j being the production level of its competitor. The result would have been the same. In fact, each firm behaves as a monopolist, the demand for its product being the residual demand.

To study stability, Cournot described a dynamic process where each firm sets its production in turn, supposing that its rival will maintain its output at the level of the previous period (Figure 4.1). If output of firm 1 is ox_l, firm 2 will produce

Figure 4.1 Cournot’s duopoly

Source: (Cournot 1838, 60)

oy₁, that is, a quantity such as to maximise its profits for a given ox₁. But for the same reason, firm 1 will produce ox₁₁. Cournot maintained that this process leads firms 1 and 2 to produce in the end, respectively, ox and oy. Equilibrium is stable. “If either of the two producers... momentarily departs [from equilibrium], he will be brought back to it by a series of reactions, the magnitude of which will be constantly diminishing” (Cournot 1838, 61).

The two firms could have colluded, instead of competing against each other. Cournot showed that if that was the case, producers’ revenue would have been higher. If the firms compete against each other, by summing up equations (2) and (3) based on the total flow D, we obtain:

If the producers had colluded, the price would have been yielded by the following equation:

To compare the equilibrium price in both cases, Cournot transformed equation (4) as follows:

and equation (5) as follows:

Given that, ifis a decreasing function ofp, the price is higher

under collusion.

Cournot’s duopoly theory and stability analysis were criticised almost fifty years after the publication of Recherchesd Joseph Bertrand (1883, 503) maintained that the situation described by Cournot is not an equilibrium. “Whatever could be the common price adopted, if one competitor diminishes its price, he would attract... the whole sales and double his revenues if his competitor does not react”. He dis­carded the assumption of a uniform price and replaced Cournot’s conjecture with the assumption that each firm took the price ofits rival as given. Edgeworth (1897, 117-8) picked up the idea and introduced a simple form of decreasing returns in the model: the production capacity of each firm is limited. He concluded that the outcome was indeterminate: the price will continuously fluctuate between a minimum required by the firm’s capacity, and a maximum, that is, the monopoly price. Edgeworth claimed having showed that Cournot’s conclusion was erroneous; but nothing could be further from the truth, however. What Edgeworth showed is that, under differ­ent assumptions, not used by Cournot, one could reach different conclusions. For his part, Irving Fisher criticised all his predecessors and stressed that, under certain conditions, Cournot’s conclusions are correct. Nonetheless, Fisher explained that these conditions are not applicable to competition between two producers.

A more natural hypothesis, and one often tacitly adopted, is that each assumes his rival’s price will remain fixed, while his own price is adjusted. Under this hypothesis each would undersell the other as long as any profit remained, so that the final result would be identical with the result of unlimited competition.

(Fisher 1898, 126)

William Fellner added to the list of analysts who claimed that Cournot’s reasoning is mistaken. His argument is different from Bertrand and Edgeworth’s, however.

7 On this topic, see Magnan de Bornier (1992, 2000). Fellner advanced that Cournot’s analysis of duopoly was unacceptable because the assumption upon which his model was designed - each firm assumes that the output of its rival is given - is not compatible with Cournot’s description of the process leading to the equilibrium, a process during which each firm changes its position based on the decisions of its competitor. The intersection of the reaction curves constitutes an equilibrium only if the firms react as these curves indicate, and it is unreasonable to conclude that they behave in such a way.

To be sure, that firms should assume of one another that the other follows a policy of fixed output is conceivable, but on the way to the Cournot solution they would necessarily realise that their assumptions were incorrect and they would change their assumptions. This would, of course, destroy the validity of the Cournot reaction functions and of any analysis based on them.

(Fellner 1949, 65)

These are the leading interpretations of Cournot’s duopoly model before the emer­gence of game theory. Most authors admitted that the analysis takes place in real time, where both firms react simultaneously until a state of equilibrium is achieved. No one suggested that producers make their decisions sequentially, with one firm setting its output at t time while the other firm reacts in the subsequent period.

Later, the work of John Nash led to a better understanding of Cournot, which discards dynamics: Cournot’s solution is interpreted as an application of the the­ory of non-cooperative equilibria to the analysis of duopoly. The profit of a firm depends on the quantity produced by its rival. To make a decision, the manager must consider and anticipate the behaviour of its competitor. For this reason, the model can be represented as a one-shot game. An equilibrium is a pair of quantities (D_i, D₂) so that each firm maximises its profit based on the expectations of what its rival will do, and assuming that the expectations about the behaviour of its rival are correct. This set-up may look like the problem that Cournot aimed to solve, but it is different, however.

3.