Cournot oligopoly

Cournot-Nash equilibrium, which results from the intersection of two reaction functions

In economics , the Cournot oligopoly is a model market situation that was first described and analyzed by Antoine-Augustin Cournot . It appears in the literature under the names Cournot- Dyopol and Nash- Cournot equilibrium.

The Cournot oligopoly describes the behavior of two or more competitors in an imperfect market in which the supply volume is the “strategic variable” (see Bertrand competition , in which the price is the “strategic variable”). Cournot's oligopoly model represents a simple and fundamental model in market and price theory, which is characterized by the fact that it includes the market situations of monopoly and polypol as borderline cases.

Well-known variants of the model depend on the number of providers: two providers form the Cournot duopoly , if there is only one provider, one speaks of the Cournot monopoly .

Basic model: Cournot duopoly

initial situation

Consider a market with the following characteristics:

• Duopoly : There are only two providers on the market, I and J.
• Homogeneous goods : The products offered by these two suppliers are of identical nature and quality.
• Complete information : The customers are informed about the offer prices of the providers at all times and try to buy from the cheapest provider.
• Infinitely fast response times: Each of the two providers has knowledge of its own offer price and that of its competitor at all times and can react infinitely quickly to price changes. Furthermore, every provider knows the initial situation in full at the beginning of the market, especially the marginal costs of his competitor.
• Maximization calculation : Both providers want to maximize their profit (= (sales price - marginal costs) offer quantity) and know that their counterpart also wants this. ${\displaystyle \cdot }$
• Simultaneous decision : Both providers decide simultaneously on their supply quantities without being aware of the competitor's supply quantity. You do this using best-answer functions (also called reaction functions).
• Demand follows a well-known linear price-sales function (price = K - supply quantity).
• Both providers have identical and constant marginal costs (= provision costs per unit of the product); there are no fixed costs .
• The supply quantities are set by the two providers before the start of the market and cannot be changed during the period under consideration. Unsold goods are worthless after the market ends. Changes to the offer price are possible at any time. The providers will vary their prices until the demand exactly matches the supply; H. after all, they choose exactly the price at which they get everything sold. Thus, the supply volume is the “strategic variable”, the choice of which determines the market result (and thus also the market price).

Mathematical derivation of the market result

• Let the demand function with p = market price, = supply quantities of providers I and J, a and b be constants> 0. ${\displaystyle p=a-b(x_{i}+x_{j})}$ ${\displaystyle x_{i},x_{j}}$
• The profit functions of the providers are , with = profit of the provider (I or J); let be the offer quantity of the supplier and let be the variable costs per unit of measure ( marginal costs ). ${\displaystyle G_{n}=px_{n}-cx_{n}}$ ${\displaystyle G_{n}}$ ${\displaystyle n}$ ${\displaystyle x_{n}}$ ${\displaystyle n}$ ${\displaystyle c}$

The demand function can be inserted into the profit functions of the providers by replacing: ${\displaystyle p}$

${\displaystyle G_{i}=(a-bx_{i}-bx_{j})x_{i}-cx_{i}}$and . ${\displaystyle G_{j}=(a-bx_{i}-bx_{j})x_{j}-cx_{j}}$

Now the profit functions are derived, by and by , and equal to 0 to determine the maximum of profit functions: ${\displaystyle G_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle G_{j}}$ ${\displaystyle x_{j}}$

${\displaystyle {\frac {\partial G_{i}}{\partial x_{i}}}=a-2bx_{i}-bx_{j}-c=0\iff x_{i}={\frac {a-bx_{j}-c}{2b}}}$, one comes up analogously . ${\displaystyle x_{j}={\frac {a-bx_{i}-c}{2b}}}$

An equilibrium is reached when both equations are fulfilled, so one can insert:

${\displaystyle x_{i}={\frac {a-b{\frac {a-bx_{i}-c}{2b}}-c}{2b}}\iff x_{i}^{*}={\frac {a-c}{3b}}}$, analogously one obtains . ${\displaystyle x_{j}^{*}={\frac {a-c}{3b}}}$

This only applies if a> c, since otherwise the unit costs are higher than the highest achievable sales price (prohibitive price), nothing is of course offered. In balance, both providers offer the same amount. This set is called the Cournot set when comparing different models for oligopolistic market situations .

Inserted into the price-sales function results in the market price . ${\displaystyle p=a-b(x_{i}+x_{j})={\frac {a+2c}{3}}}$

Note: The mathematical model can be set up with both a simpler and a more general starting situation (by normalizing existing or adding additional variables). There are many other representations in textbooks that i. A. Generalizations or special cases of the formulation mentioned here are.

Verbal explanation of the result

The two competitors know of each other that they include each other's options for action in their calculations. Therefore, they try to choose the supply volume that is the "best answer" to the competitor's anticipated supply volume. The only combination of supply quantities, both of which are the " best answer " to the opponent's supply quantity, are the quantities described above. If one of the providers chooses the Cournot set, it is optimal for the opponent to also choose the Cournot set, even if he knows his opponent's decision on the amount. If he chose more than the Cournot set, he would lose more from the falling market price than he would gain from increasing sales. If he chose less, he would lose more from the lower sales volume than he would gain from the then higher market price. It is therefore a stable equilibrium that represents a Nash equilibrium , i.e. it meets JF Nash's requirements for a stable equilibrium in strategic decision-making situations.

Generalization to the oligopoly

We now consider a market with suppliers and set b = 1 for simplification. ${\displaystyle n\in \mathbb {N} }$

The price-sales function is now:

${\displaystyle p=a-\sum _{i=1}^{n}x_{i}}$ ,

the profit function of a provider i is:

${\displaystyle G_{i}=(a-(x_{i}+\sum _{j=1;j\neq i}^{n}x_{j}))x_{i}-cx_{i}}$ ,

whose derivation is: ${\displaystyle x_{i}}$

${\displaystyle {\frac {\partial G_{i}}{\partial x_{i}}}=a-2x_{i}-\sum _{j=1;j\neq i}^{n}x_{j}-c=0\iff x_{i}={\frac {a-\sum _{j=1;j\neq i}^{n}x_{j}-c}{2}}}$.

From this and from the consideration that in an equilibrium of equal players the sets of players must be equal, one can derive the equilibrium supply sets:

${\displaystyle x_{i}={\frac {a-(n-1)x_{i}-c}{2}}\iff x_{i}={\frac {a-c}{n+1}}}$ .

If the number of suppliers n = 2 is inserted into this equation, the result is the Cournot duopoly set calculated above; with a number of suppliers of n = 1, i.e. a monopoly, one obtains the Cournot monopoly set, which Cournot described in his price theory of monopoly ( Cournot point ). For an almost infinite number of providers, the model converges against the market result of the polypole predicted by the theory .

The following table provides an overview of the market results for different numbers of providers (with a = 1 and b = 1):

Number of providers and market results

Number of providers	1	2	3	...	infinite
Amount offered per provider	${\displaystyle {\frac {1-c}{2}}}$	${\displaystyle {\frac {1-c}{3}}}$	${\displaystyle {\frac {1-c}{4}}}$	...	near 0
Total supply quantity	${\displaystyle (1-c)\cdot {\frac {1}{2}}}$	${\displaystyle (1-c)\cdot {\frac {2}{3}}}$	${\displaystyle (1-c)\cdot {\frac {3}{4}}}$	...	${\displaystyle (1-c)\cdot 1}$
Market price	${\displaystyle c+{\frac {1-c}{2}}}$	${\displaystyle c+{\frac {1-c}{3}}}$	${\displaystyle c+{\frac {1-c}{4}}}$	...	c

One recognises,

• that the sales of the individual provider dwindle as the number of providers increases.
• that the market supply (total supply volume) improves with an increasing number of providers.
• that the market price decreases with increasing number of providers in the direction of marginal costs.

Interpretation of the oligopoly model

Some further results of the model (or of extensions of the model):

• The total amount of corporate profits is a maximum of one provider ( monopoly ). Even with two providers, the sum of corporate profits drops considerably, and this is also divided between two providers. With 4–5 providers, the profit per company drops to a small fraction of the monopoly profit. With an infinite number of providers, corporate profits drop to zero.
• The consumer surplus (measure of the prosperity gains of the consumer through the market) corresponds in the monopoly case to the business profit of the monopoly. As the number of providers increases, it increases, and more so than corporate profits decrease. The economically most favorable case (sum of corporate profits and consumer surplus is maximum) is the polypol (infinite number of providers).
• If the companies 'marginal costs are different, the companies' profits will be different. It can happen that the market price (especially with a high number of providers) falls below the marginal costs of individual companies. They stop selling and leave the market. It can therefore be beneficial for an individual company to lower its marginal costs in two ways: First, the spread between market price and costs increases due to the lower costs themselves, and thus profits. Second, if the profit margin increases, it is optimal for this company to slightly increase its supply in the market. This reduces the market price, which is why u. Competing companies could possibly reduce their supply volume or withdraw from the market entirely. If, as a result, there are fewer providers on the market, the profit is distributed among fewer companies.
• If the companies have to bear fixed costs for participating in the market , it may be that the company profits are insufficient to cover the fixed costs. If the companies did not already incur the fixed costs ( sunk costs ), only a limited number of companies will enter the market. In this case, it is also economically optimal that there are not an infinite number of companies active in the market, since from a certain number of providers the increase in consumer surplus is no longer sufficient with an increase in the number of providers, the falling corporate profits on the market and the fixed costs of the additional company cover up. Extreme cases are conceivable (and also realistic) in which, for reasons of very high fixed costs, it is economically optimal if only one company serves the entire market (i.e. is a monopoly).

Further oligopoly models

There are very well-known basic models for some other market situations:

• the Stackelberg competition (after Heinrich von Stackelberg ), in which the market participants do not make their decisions on quantities simultaneously, but one after the other,
• the Bertrand competition (after Joseph Bertrand ), in which price, not quantity, is the strategic variable.
• the Kreps-Scheinkman model (after David M. Kreps and J. Scheinkman ), in which price and quantity competition are combined through decisions about the development of production capacities.
• the Hotelling location model (after Harold Hotelling ), which deals with price differences due to spatial distances or product differentiation .

literature

• Augustin A. Cournot: Recherches sur les principes mathématiques de la théorie des richesses. L. Hachette, Paris 1836 ( gallica.bnf.fr ).
• Wilhelm Pfähler, Harald Wiese: Corporate strategies in competition - A game theory analysis. Springer-Verlag, Heidelberg, second edition 2006, ISBN 3-540-28000-6 .