Stackelberg Duopoly

In the Cournot model, both firms made their decisions simultaneously and without knowing the other’s decision. In the Stackelberg model, they decide one after the other. We call the one that chooses first, the Leader and the other one the Follower.

• We have two firms.
• They set quantities (and the price is set by the market).
• Leader first decides on her quantity, and then Follower decides on hers.

We will use the same reaction function as in the Cournot model, but the analysis will now be different since they do not choose simultaneously. Leader, who sets her quantity first, has an advantage. She knows that Follower will later set her quantity according to her reaction function. Therefore, Leader sets her quantity to maximize her own profit, given Follower’s optimal response.

One way to illustrate this game is presented in Figure 14.3. We have drawn the reaction functions, r₁ and r₂, but we have also added a few curves indicating Leader's profit, π₁, π₂, and π₃; so-called isoprofit curves. Such curves show different combinations of q₁ and q₂ that give Leader the same profit. For instance, all combinations along π1 give Leader a profit of π₁, etc. Note that Leader's profit increases inwards, the closer to the monopoly quantity (the point where r₁ intersects the X-axis) we get. The profit at π₂ is consequently higher than at π₁, and even higher at π₃. Leader knows that Follower will choose her quantity along the reaction function r₂. Leader therefore finds an isoprofit curve that touches r₂ and that is as close to the monopoly quantity as possible. In the figure, the isoprofit curve π₂ touches r₂ in point A. Leader then chooses the quantity that corresponds to point A, i.e. q₁*. As a response, Follower later chooses the quantity q₂*.
Note that every other choice of quantity for Leader, higher or lower, must result in a lower profit for her. If she, for instance, would choose the quantity q₁' instead, Follower's reaction would be to choose q₂' and Leader's profit would be π₁, which is less than π₂.

Bertrand Duopoly

The Bertrand model assumes that

• We have two firms.
• They set prices (and quantities are set by the market).
• They set prices simultaneously, without knowing which price the other one sets.

The previous models produced results that were very favorable for the firms but less so for the consumers. The Bertrand model, however, puts the two firms in a Prisoner’s Dilemma- type of situation, and forces them to set p = MC, i.e. they set the same price as firms would do in a perfectly competitive market. This is, of course, unfavorable for the firms, but an improvement for consumers and society. To see that the firms will set p = MC, suppose that we know that the other firm has set a high price. Which is then the best price we can set? Remember that we have homogenous (meaning identical) goods, so the consumers will not care from whom they buy it. Furthermore, they have perfect information about all prices. If we choose a price that is just below our competitor’s, all customers will buy from us. This is a good situation for us, but far from optimal for the other firm. If they reason in the same way, they will want to set a price just below ours. Then we would lose all customers… and so forth. No price above MC can consequently be an equilibrium. Regardless of which price the firm has set, the other will always want to undercut it and set a price just below its competitor. The only price that can be an equilibrium is then p = MC. At that price, none of the firms can lower their price since they would then make a loss. None of them would be able to make a profit by increasing the price either, since they would then lose all customers. The surprising result is then that, since p = MC, we get the same outcome as in a perfectly competitive market, even though there are only two firms. If society is able to construct an oligopoly such that it becomes a Bertrand duopoly, there will be no loss of efficiency. One way for the firms in a Bertrand market to increase profits anyway, is to try to differentiate their products. The customers will then not be indifferent between from whom they buy and the firms become two monopolists, however with goods that are very close substitutes.

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