In this blog post I will explain the Cournot oligopoly model. If you have followed a basic micro economics course you are most likely already familiar with it. The Cournot model is part of a range of oligopoly models that try to capture the strategic interaction between firms that arises if there is only a limited amount of competition (in the sense that there are only a small number of sellers of a product).
I have divided this blog post up in to three parts. Part 1 will setup the model, Part 2 will solve the model, Part 3 will generalize the model and compare the outcomes to a monopoly. At the end you should have a good understand of the Cournot model and its implications.
Warning: long post will follow!
Cournot’s model of oligopoly, named after the French economist Auguste Cournot, originally described the behaviour of two firms that sold bottled water from mineral springs. He did this in his book Researches on the Mathematical Principles of the Theory of Wealth (1838). The model therefore is also known as the Cournot duopoly model.
The model uses game theory to model the firms behaviour. Game theory is used to model situations where people strategically interact with each other. We solve the model by finding the Nash equilibrium. Do not panic if you do not know game theory or Nash equilibrium, solving the model only requires basic algebra.
The central assumption of the model is that companies compete on quantities. The companies choose their output simultaneous and treats the others output as a fixed number. Thus the companies do not react on each other’s production decisions, in the sense of first and second mover interactions, for such a situation see Stackelbergs model. The companies also do not compete on price, see Bertrands model for this.
Setup of the model: assumptions
In game theory Cournot models falls under the section of static games of complete and perfect information. In this case this means that:
- The companies choose their action (the amount of products to produce) simultaneous (static game)
- All the information is known to both companies. So they know each other’s cost function, the amount of companies active in the market, each others preferences etc. (complete and perfect information).
- Companies are rational, in the sense that they will try to maximise their profits.
Other assumptions of the model (that follow more or less from the oligopoly market) are:
- The amount of companies active in the market is fixed. In reality there are often barriers of entry in an oligopoly market.
- Companies have market power, indeed there are only few suppliers. The quantities they produce have an effect on the market price. If this would not be the case we would not have any strategic interaction, as the output of one company does not has any effect anymore on the profit of the other company.
- It also assumed that companies do not collude. If this would be the case we would basically be in a monopoly market.
- Lastly, companies do not compete trough product differentiation. In other words you might say that the companies produces homogenous goods.
Before going on to solving the model we need to actually setup the model. In game theory this means that you describe three things: the players, the actions for each player, and their preferences over the different kind of actions. For the Cournot model this means:
- Players: the firms ( for now we assume the original case of two firms)
- Actions: the amount of output to produce
- Preferences: the amount of profit, hence the company wants to choose the amount of output that maximises profits
Setup of the model: functions
First we need to describe the demand function, that will give us the market price. We also need the cost function of the companies.
Suppose that the cost function is given by:
\[\ C_i(q_i) = c_iq_i \mbox { with }\ i \{1,2\} \]
Where $c_i$ are the constant ‘unit cost’ and $q_i$ is the output of the given firm. Furthermore, $i \{1,2\}$ simply denotes that there are two companies, company 1 and company 2.
Note: for simplicity it is often assumed that $c_i$ is zero, or that it is identical for each firm. I will use a more general form were $c_i$ might be different between firms.
Next we will need the demand function, in our example this is given by:
\[ P(Q) \left\{\begin{array}{11} \alpha - Q \qquad \mbox {if $Q \le \alpha$}\\ 0 \qquad \qquad \quad \mbox {if $Q > \alpha$} \end{array} \right. \]with $Q = q_1 + q_2$
First of do not get confused about the array form presentation. The first line gives a 'normal' demand function. However to be complete we state in the second line that the demand function cannot go negative. Hence their would be no companies if they would have to sell for negative price.
$P(Q)$ states the market price, for a given Q
$Q$ is the total production of the market
$\alpha$ denoted the y intercept, in this case this may be interpreted as the maximum amount consumers are willing to pay for the product. As there is no demand for the product if the price is above $\alpha$.
Lastly, we have to assume that $c<\alpha$. It states that there is some quantatity where the unit cost $c$ is lower than the maximum amount consumers are willing to pay. If this would not be the case companies could never make a profit as the cost of making the product always would be greater than the maximum price they could sell it for.
Now that everything is setup we will start solving the model in part 2.
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