The theory of prices: “concurrence indefinie” and cooperation
From duopoly to “unlimited” competition
The effects of competition reached their limit when any firm’s production could stop without creating a significant change in price. Let’s consider the firm i.
Its profit function is:
The manager sets the production level Di that maximises profit: 
If, as Cournot (1838, 69) argues, “each of the partial productions Di is unresponsive, not only to the total production D = f (p), but also to the derivative f' (p)”, then the equilibrium condition is that the price is equal to the marginal cost of each firm.
The price of the good and the production of each firm are the solutions of the following system:
System (6) enables to determine the production of each firm as a function of the price. Total production, here called aggregate supply, appears to be a function Ω of the price determined by the equality between supply and demand:
Cournot mentioned that “it is... clear that, under the hypothesis of unlimited competition, and where at the same time the function φi' should be a decreasing one, nothing would limit the production of the good” (1838, 70). Simply put, unlimited competition is incompatible with increasing returns.
Cournot drew upon this analysis to study the effects of taxation and measure the deadweight loss caused by a tax. In Figure 4.2, the equilibrium is initially determined by the intersection between the demand curve D and the supply curve Ω0.
The price is p0. Suppose the production costs of the firms rise by the same quantity as it would be the case when an indirect tax is created. The increase in cost is represented by the shift of the supply curve from Ω0 to Ω1 with NE1 = t. The new equilibrium price exceeds the initial price, but the price increase is less than the amount of the tax. Let ∆p denote the increase in price, if t is small, one can write:
Ω'(p) is positive and f'(p0) is negative; therefore, ∆p and ∆p-t have opposite signs, while ∆p and t have the same sign. One can conclude that a tax hike raises prices, but the rise in price is less than the tax hike.
To analyse the effects of taxation on the interests of producers and consumers, Cournot hypothesised that a tax leading to a rise in price which is less than the tax increase would necessarily lead to a lower output. After the tax is applied, producer i loses the following amounts: 
Figure 4.2 The effects of taxation
Source: (Cournot 1838, 70)
The total loss borne by producers is 
Of the mutual relations ofproducers
Though producers compete against each other while selling the same good, they can cooperate to supply inputs that are required to produce another good. For
instance, copper is alloyed with zinc to make brass. In other words, the firm producing copper cooperates with the one producing zinc in order to produce brass. Cournot (1838, 80) shows that, in that case, “the association of monopolists, working for their own interest... will also work for the interest of consumers, which is exactly the opposite of what happens with competing producers”.
His demonstration, discussed at length by Edgeworth (1897), brings to light several important analytical problems because it does not necessarily lead to an equilibrium.Cournot’s starting point is a scenario where copper and zinc have no use other than as alloying agents to produce brass. Production costs are neglected, and it is assumed that two monopolists handle the production of copper and zinc separately. Let pl be the price of brass, pc that of copper, and pz that of zinc. The coefficients of fabrication are fixed: a quantity ac of copper and a quantity az of zinc are combined to produce one unit of brass. The price of a unit of brass is given by the following equation:
The demand for brass D is a function of its price:
The demands for copper and zinc are respectively:
Each producer determines its profit-maximising price while regarding the price of the other monopolist as given. The conditions are respectively:
Copper and zinc prices are determined by these two equations.
Cournot represented the solution of the system graphically. The first equation of the system (7) is represented by m1n1, which shows the price of copper set by its producer for a given price of zinc. Similarly, m2n2 represents the price of zinc as a function of the price of copper. We observe that when the price of one commodity is equal to zero, the optimal price of the other has a finite value. When the price of one commodity goes up, the optimal price of the other can rise or fall. These results are shown in Figure 4.3.
One can conclude that the equilibrium is stable.Cournot indicated that the model could have no solutions or lead to economically impractical solutions. One can imagine that the demand function for brass is in such a way that the curves m1n1 and m2n2 would not intersect. That is the case, for
Figure 4.3 The mutual relations of producers
Source: (Cournot 1838, 79)
instance,
. We can also imagine that the roots of system (7) render
values implying that producers provide quantities of copper and zinc which exceed their production capacities. Cournot qualified his remarks, however. He stated that this singular result stems from an abstract hypothesis of the nature of those that we can discuss in this essay: it is very clear that, in the order of actual facts, and where all the conditions of an economic system are taken into account, there is no item whose price is not completely determined.
(1838, 82)
The main difference between this model and the model dealing with the competition of producers is that
the composite commodity will always be made more expensive, by reason of separation [of monopolies] than by reason of the fusion of monopolies. The association of monopolists, working for their own interest, in this instance will also work for the interest of consumers, which is exactly the opposite of what happens with competing producers.
(1838, 80)
4.