## Cournot Duopoly

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# Cournot Duopoly

The Cournot model is a model of duopolies and is developed in line with the game theoretical approach we presented in last chapter. The Cournot model assumes that:

- We have two firms.
- They set quantities (and the price is then set by the market, given the quantity).
- They choose simultaneously, without knowing which quantity the other chooses.

How would these two firms reason? Both of them want to maximize their own profit. However, each firm’s profit partly depends on the quantity set by the other firm, as total quantity determines the market price. If a firm knows the quantity the other firm has chosen, then it is able to decide exactly which quantity that would maximize their own profit. There is an ptimal

response to each choice of the other firm. Let us use that observation, and determine that best response for each choice of quantity the other firm can possibly make. If we do that, we get a so-called reaction function. In Figure 14.2, r_{1} is firm 1's reaction function and r_{2} is firm 2's.

To give an example of how to interpret the reaction function, suppose that firm 2 chooses to produce the quantity q_{2,1}. Which is firm 1’s optimal response? Indicate q_{2,1} on the Y-axis, go to line r_{1} (point A) and read off the corresponding

value on the X-axis: q_{1,1} is firm 1's optimal response. Note however that if firm 1 chooses the quantity q_{1,1}, then the quantity q_{2,1} is not optimal for firm 2. Instead, the quantity q_{2,2}, at point B, is optimal for firm 2. However, then q_{1,1} is not optimal for firm 1… and so on. It is possible to show that the only point where both firms simultaneously respond optimally to the other’s choice is point C, where the two reaction curves intersect each other. As no agent can achieve a better outcome by unilaterally changing her strategy, we have a Nash equilibrium (see Section 13.3). The conclusion of the Cournot model is then that, both firms will choose the Nash equilibrium quantities, q_{1}* and q_{2}*.

Note that, if you continue to use the method of finding successive optimal responses as we did above, you will tend to get closer and closer to the Nash equilibrium in each round. One should also note another thing in Figure 14.2. If firm 2 would produce nothing at all, firm 1 would be a monopolist in the market. The optimal quantity would then be the monopoly quantity. Similarly for firm 2. The reaction function of each firm must consequently hit the firm’s own axis at the monopoly quantity. In the figure, these points are labeled q_{1M} and q_{2M}.

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