Abstract
A Century after Cournot von Stackelberg proposed a modification of the model. He suggested that any duopolist may learn the competitor’s Cournot reaction function, and so search for a better solution as a “leader”. This, of course, works provided the other, the “follower”, actually adheres to the proper reaction function. In this way two feasible leader/follower pairs can be formed. Both can also attempt leadership, in which case there would be economic warfare, from which presumably the firm with better financial backing might come out as winner. It was proved that Stackelberg equilibrium always is more profitable for the leader than Cournot equilibrium. Stackelberg never thought in dynamic terms—he could, for instance, have considered a variable supply policy over time to check if it might render a higher profit than a constant supply ∩ over time, but he did not. If we want to combine the static Stackelberg case with the dynamic Cournot process, things become really interesting. There may even seem to be a contradiction buried here. Stackelberg equilibrium is more profitable, no question about that, but, even the successful Stackelberg leader will ripe a higher temporary profit from jumping to Cournot action. This is so because the Cournot action was defined as the best move in any situation. However, this will not last for long—if the Cournot process is stable, then the orbit will approach Cournot equilibrium—and that is worse than Stackelberg leadership. So the stage is set for designing a rule for switching between Cournot and Stackelberg actions, which seems not to have been attempted before. The natural format is to let the competitors switch to Stackelberg leadership whenever the resulting profit is higher than, not the expected profit from Cournot action, because it never is, but, say, exceeds a fraction, say 50 or 75% of it. Incorporating such a rule of switching results in interesting scenarios, including both duopolists choosing Stackelberg leadership or both using Cournot best reply. The dynamics shows different periodic scenarios, multiplicity of attractors, and bifurcations. Some results are really unexpected, for instance that it may be more profitable than any other action to be a follower if the other becomes a leader, which, of course is not a choice under the follower’s control.
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Notes
- 1.
It has been argued that the realization that there can only be one leader in a stable system was the cause for Stackelberg becoming a “follower” of the “F ührer”, but to the present author this has the flavour of a too simplistic construct. We have no reason to believe that Stackelberg was that naïve—there were more substantial benefits from joining the political leadership.
- 2.
It is tempting to generalize the Stackelberg model from duopoly to oligopoly, for instance triopoly. This brings possibilities to define different levels of leadership, for instance being an intermediate leader, a follower to someone but leader to another. There have been published contributions in this direction, even by the present author. But it seems difficult to achieve anything really interesting. All tends to result in a sterile taxonomy of cases.
- 3.
In one of the stubs we propose a dynamic version set up as a dynamic programming problem. It is immediately apparent that the optimality conditions are not satisfied by the stationary conditions for traditional Stackelberg leadership. How much the stationary profits fall short of the dynamic remains a research agenda.
- 4.
Stackelberg also uses linear demand as an alternative, but linearity has its severe snags and limitations, so the exponential case is potentially more interesting.
- 5.
The assumption of zero production costs is not an oddity only due to Stackelberg—it was used by many other good economists, such as Hotelling, Palander, Wald, and even Cournot himself. Occasionally the production of natural mineral water was proposed—as if the bottling were costless. Not quite convincing!
- 6.
The reaction function of the competitor is not the only thing to learn. It may even seem easier to learn the outcome of the interactive dynamic, particularly if it is a low periodic orbit. This involves the inherent complication that learning periodicity and adapting is always bound to alter this very periodicity, to higher period and even more difficult to learn. This calls in doubt the economists’ favourite idea of intertemporal equilibrium with perfect foresight of the future, now baptized with the absurd term “rational expectations”. It seems that the fixed point is the only orbit about which one can have “rational expectation” and this brings us back to stationary situations and equilibrium theory. The problem is there with all the dynamic models, but we will only deal with it in the present chapter.
- 7.
Note that these expressions may seem to be nonnegative. But if q 2 > 1/c 1 or q 1 > 1/c 2, i.e., the constraints for the first branches of (6.22)–(6.23) are violated, then this is due to the fact that negative costs dominate over negative revenues, which in terms of subject matter is nonsense. Anyhow, we already restricted the map (6.20)–(6.21) to (6.22)–(6.23), so we need not be further concerned.
- 8.
Incidentally, there is no substantial reason to choose the same k = k 1 = k 2 for the two competitors. We just wanted to display bifurcations in the parameter plane in 2D, and, as c 1, c 2 only influence the system through their ratio, there remains just one dimension for k 1, k 2.
- 9.
In terms of what we see in Fig. 6.1 the bounds are the vertical lines left and right in the picture. Note that when \(\frac {c_{2}}{c_{1}}=3+2\sqrt {2}\) then \(\frac {c_{1}}{c_{2}}=3-2\sqrt {2}\). This is so because \(\left ( 3-2\sqrt {2 }\right ) \left ( 3+2\sqrt {2}\right ) =9-8=1\).
- 10.
We also kept the five construction curves for the fixed point regions this time.
- 11.
The reason is that under iso-elastic demand market revenue is a constant. Hence a monopolist (or a pair of collusive duopolists) could retrieve the whole revenue as profit without incurring any production costs if they produce nothing and sell this nothing at an infinite price. Such a solution is purely mathematical and has no meaning in terms of economics; it is just a shortcoming of one assumption.
- 12.
As we know, it is only the ratio \(\frac {c_{2}}{c_{1}}=3\) that counts for the dynamics. The absolute values of unit production costs only have importance for the level of profits.
References
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Puu, T. (2018). Stackelberg. In: Disequilibrium Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-74415-5_6
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