Cournot competition

Cournot's economic theory was little noticed until Léon Walras credited him as a forerunner. This led to an unsympathetic review of Cournot's book by Joseph Bertrand which in turn received heavy criticism. Irving Fisher found Cournot's treatment of oligopoly "brilliant and suggestive, but not free from serious objections".[4] He arranged for a translation to be made by Nathaniel Bacon in 1897.[5]

Reactions to this aspect of Cournot's theory have ranged from searing condemnation to half-hearted endorsement. It has received sympathy in recent years as a contribution to game theory rather than economics. James Friedman explains:

In current language and interpretation, Cournot postulated a particular game to represent an oligopolistic market...[5]

The maths in Cournot's book is elementary and the presentation not difficult to follow. The account below follows Cournot's words and diagrams closely. (The diagrams were presumably included as an oversized plate in the original edition, and are missing from some modern reprints.)

Cournot's curve of 'demand or sales'

Cournot was wary of psychological notions of demand, defining it simply as the amount sold of a particular good (helped along by the fact that the French word débit meaning 'sales quantity' has the same initial letter as demande [6]). He formalised it mathematically as follows:

We will regard the sales quantity or annual demand D, for any commodity, to be a function F (p) of its price.[7]

It follows that his demand curves do some of the work of modern supply curves, since producers who are able to limit the amount sold have an influence on Cournot's demand curve.

Cournot remarks that the demand curve will usually be a decreasing function of price, and that the total value of the good sold is p F (p) which will generally increase to a maximum and then decline towards 0. The condition for a maximum is that the derivative of p F (p), i.e. F (p) + p F '(p), should be 0 (where F '(p) is the derivative of F (p)).

Cournot insists that each duopolist seeks independently to maximise profits, and this restriction is essential, since Cournot tells us that if they came to an understanding between each other so as each to obtain the maximum possible revenue, then completely different results would be obtained, indistinguishable from the consumer's point of view from those entailed by monopoly.

Cournot presents a mathematically correct analysis of the equilibrium condition corresponding to a certain logically consistent model of duopolist behaviour. However his model is not stated and is not particularly natural (Shapiro remarked that observed practice constituted a "natural objection to the Cournot quantity model"[8]), and "his words and the mathematics do not quite match".[9]

His model can be grasped more easily if we slightly embellish it. Suppose that there are two owners of mineral water springs, each able to produce unlimited quantities at zero price. Suppose that instead of selling water to the public they offer it to a middle-man. Each proprietor notifies the middle-man of the quantity he or she intends to produce. The middle-man finds the market clearing price, which is determined by the demand function F and the aggregate supply. He or she sells the water at this price, passing the proceeds back to the proprietors.

The consumer demand D for mineral water at price p is denoted by F (p); the inverse of F  is written f ; and the market-clearing price is given by p = f (D) where D = D₁ + D₂ and D_i is the amount supplied by proprietor i.

Each proprietor is assumed to know the amount being supplied by his or her rival, and to adjust his or her own supply in the light of it to maximise his or her profits. The position of equilibrium is one in which neither proprietor is inclined to adjust the quantity supplied.

It needs mental contortions to imagine the same market behaviour arising without a middle-man.

A feature of Cournot's model is that a single price applies to both proprietors. He justified this assumption saying that "... dès lors le prix est nécessairement le même pour l'un et l'autre propriétaire".[10] Magnan de Bornier expands on this saying that "the obvious conclusion that only a single price can exist at a given moment" follows from "an essential assumption concerning his model, [namely] product homogeneity".[11]

Later on Cournot writes that a proprietor can adjust his supply "en modifiant correctement le prix".[12] Again this is nonsense: it is impossible for a single price to be simultaneously under the control of two suppliers. If there is a single price, then it must be determined by the market as a consequence of the proprietors' decisions on matters under their individual control.

Cournot's account threw his English translator (Nathanial Bacon) so completely off balance that his words were corrected to "properly adjusting his price".[13] Edgeworth regarded equality of price in Cournot as "a particular condition, not... abstractly necessary in cases of imperfect competition".[14] Magnan de Bornier says that in Cournot's theory "each owner will use price as a variable to control quantity" without saying how one price can govern two quantities. A. J. Nichol claimed that Cournot's theory makes no sense unless "prices are directly determined by buyers".[15] Shapiro, perhaps in despair, remarked that "the actual process of price formation in Cournot's theory is somewhat mysterious".[8]

Cournot's duopolists are not true profit maximisers. Either supplier could increase his or her profits by cutting out the middle-man and cornering the market by marginally undercutting his or her rival; thus the middle-man can be seen as a mechanism for restricting competition.

Cournot's model of competition is typically presented introduced for the case of a duopoly market structure, the following example provides a straightforward analysis of the Cournot model for the case of Duopoly. Therefore, suppose we have a market consisting of only two firms which we will call firm 1 and firm 2. For simplicity, we assume each firm faces the same marginal cost, that is for a given firm ${\displaystyle i}$'s output quantity, denoted ${\displaystyle q_{i}}$ where ${\displaystyle i\in \{1,2\}}$, firm ${\displaystyle i}$'s cost of producing ${\displaystyle q_{i}}$ units of output is given by,

$${\displaystyle C(q_{i})=\chi q_{i}}$$

.

This assumption tells us that both firms face the same cost-per-unit produced. Therefore, as each firm's profit is equal to its revenues minus costs, where revenue equals the number of units produced multiplied by the market price, we can denote the profit functions for firm 1 and firm 2 as follows.

$${\displaystyle \Pi _{1}(Q)=p(Q)q_{1}-\chi q_{1}}$$

$${\displaystyle \Pi _{2}(Q)=p(Q)q_{2}-\chi q_{2}}$$

Note that in the above profit functions we have price as a function of total output which we denote as ${\displaystyle Q}$ and for two firms we must have ${\displaystyle Q=q_{1}+q_{2}}$. For examples sake, let us assume that price (inverse demand function) is linear and of the form ${\displaystyle p=a-bQ}$. So the inverse demand function can then be rewritten as ${\displaystyle p=a-bq_{1}-bq_{2}}$.

Now substituting our equation for price in place of ${\displaystyle p(Q)}$ we can write each firm's profit function as

$${\displaystyle \Pi _{1}(q_{1},q_{2})=(a-bq_{1}-bq_{2}-\chi )q_{1}}$$

$${\displaystyle \Pi _{2}(q_{1},q_{2})=(a-bq_{1}-bq_{2}-\chi )q_{2}}$$

As firms are assumed to be profit maximisers, the first order condition's (hereafter F.O.C.) for each firm are

$${\displaystyle {\dfrac {\partial \Pi _{1}(q_{1},q_{2})}{\partial q_{1}}}=a-2bq_{1}-bq_{2}-\chi =0}$$

$${\displaystyle {\dfrac {\partial \Pi _{2}(q_{1},q_{2})}{\partial q_{2}}}=a-bq_{1}-2bq_{2}-\chi =0}$$

The F.O.C's state that firm ${\displaystyle i}$ is producing at the profit maximising level of output when the marginal cost (MC) is equal to the marginal revenue (MR). Intuitively, this suggests that firms will produce up to the point where it remains profitable to do so, as any further production past this point will mean that ${\displaystyle MC>MR}$, and therefore production beyond this point results in the firm losing money for each additional unit produced. Notice that at the profit maximising quantity where ${\displaystyle MC=MR}$, we must have ${\displaystyle MC-MR=0}$ which is why we set the above equations equal to zero.

Now that we have two equations describing the states at which each firm is producing at the profit maximising quantity, we can simply solve this system of equations to obtain each firm's optimal level of output, ${\displaystyle q_{1},q_{2}}$ for firms 1 and 2 respectively. So we obtain

$${\displaystyle q_{1}={\dfrac {a-\chi }{2b}}-{\dfrac {q_{2}}{2}}}$$

$${\displaystyle q_{2}={\dfrac {a-\chi }{2b}}-{\dfrac {q_{1}}{2}}}$$

These functions describe each firm's optimal (profit maximising) quantity of output given the price firms face in the market, ${\displaystyle p}$, the marginal cost, ${\displaystyle \chi }$, and output quantity of rival firms. The functions can be thought of as describing a firm's "Best Response" to other firms level of output.

We can now find a Cournot-Nash Equilibrium using our "Best Response" functions above for the output quantity of firms 1 and 2. Recall that both firms face the same cost-per-unit (${\displaystyle \chi }$) and price (${\displaystyle p}$). Therefore, using this symmetrical relationship between firms we find the equilibrium quantity by fixing ${\displaystyle q_{1}=q_{2}=q^{*}}$. We can be sure this setup gives us the equilibrium levels as neither firm has an incentive to change their level of output as doing so will harm the firm at the benefit of their rival. Now substituting in ${\displaystyle q^{*}}$ for ${\displaystyle q_{1},q_{2}}$ and solving we obtain the symmetric (same for each firm) output quantity in Equilibrium as

$${\displaystyle q^{*}={\dfrac {a-\chi }{3b}}}$$

This equilibrium value describes the optimal level of output for firms 1 and 2, where each firm is producing an output quantity of ${\displaystyle q^{*}}$. So at equilibrium, the total market output ${\displaystyle Q}$ will be,

$${\displaystyle Q=q_{1}^{*}+q_{2}^{*}={\dfrac {2(a-\chi )}{3b}}}$$

.

The revenues accruing to the two proprietors are p D₁ and p D₂, i.e. f (D₁ + D₂) D₁ and f (D₁ + D₂) D₂. The first proprietor maximises profit by optimising over the parameter D₁ under his control, giving the condition that the partial derivative of his profit wrt. D₁ should be 0; and the mirror-image reasoning applies to his or her rival. We thus get the equations

f (D₁ + D₂) + D₁ f '(D₁ + D₂) = 0   and   f (D₁ + D₂) + D₂ f '(D₁ + D₂) = 0.

The equlibirum position is found by solving these two equations simultaneously. This is most easily done by adding and subtracting them, turning them into

D₁ = D₂   and   2f (D) + Df '(D) = 0

where D = D₁ + D₂. Thus we see that the two proprietors supply equal quantities, and that the total quantity sold is the root of a single non-linear equation in D.

Cournot goes further than this simple solution, investigating the stability of the equilibrium. Each of his original equations defines a relation between D₁ and D₂ which may be drawn on a graph. If the first proprietor was providing quantity x_l then the second proprietor would adopt quantity y_l from the red curve to maximise his or her revenue. But then, by similar reasoning, the first proprietor will adjust his supply to x_ll to give him or her the maximum return as shown by the blue curve when D₂ is equal to y_l . This will lead to the second proprietor adapting to the supply value y_ll and so forth, until equilibrium is reached at the point of intersection i whose coordinates are (x,y).

Since proprietors move towards the equilibrium position it follows that the equilibrium is stable, but Cournot remarks that if the red and blue curves were interchanged then this would cease to be true. He adds that it is easy to see that the corresponding diagram would be inadmissible since, for instance, it is necessarily the case that m₁ > m ₂. To verify this, notice that when D₁ is 0 the two equations reduce to

f (D₂) = 0   and   f (D₂) + D₂ f '(D₂) = 0.

The first of these corresponds to the quantity D₂ sold when the price is zero (which is the maximum quantity the public is willing to consume), while the second states that the derivative of D₂ f (D₂) wrt. D₂ is 0; but D₂ f (D₂) is the monetary value of an aggregate sales quantity D₂, and the turning point of this value is a maximum. Evidently the sales quantity which maximises monetary value is reached before the maximum possible sales quantity (which corresponds to a value of 0), so the root m₁ of the first equation is necessarily greater than the root m ₂ of the second.

We have seen that Cournot's system reduces to the equation 2f (D) + Df '(D) = 0. D is functionally related to p  via f  in one direction and F  in the other. If we re-express this equation in terms of p, it tells us that F (p) + 2p F '(p) = 0, which can be compared with the equation F (p) + p F '(p) = 0 obtained earlier for monopoly.

If we plot another variable u against p, then we may draw a curve of the function u = –F (p) / F '(p). The monopoly price is the p for which this curve intersects the line u = p , while the duopoly price is given by the intersection of the curve with the steeper line u = 2p. Regardless of the shape of the curve, the intersection with u = 2p occurs to the left – i.e. at a lower price – than the intersection with u = p. Hence prices are lower under duopoly than under monopoly, and quantities sold accordingly higher.

When there are n proprietors the price equation becomes F (p) + n p F '(p) = 0; the price can be read from the diagram from the intersection of u = n p with the curve. Hence the price diminishes indefinitely as the number of proprietors increases. With an infinite number of proprietors the price becomes zero; or more generally, if we allow for costs of production, the price becomes the marginal cost.