Stackelberg duopoly

The Stackelberg model is a strategic game in economics , which is characterized by the fact that the market-leading company draws first and then the following companies decide. If there are only two companies, one speaks of a Stackelberg duopoly .

It is named after the German economist Heinrich Freiherr von Stackelberg , who published his work Market Form and Equilibrium in 1934, in which the model was described, and represents a further development of Cournot's duopoly model . The two players are referred to as Stackelbergführer and Stackelbergfolder , and they compete in units of measure. The Stackelbergführer is sometimes referred to as the market leader.

Explanation

For the existence of an equilibrium in the Stackelberg duopoly there are some further conditions:

The Stackelbergführer must know that the Stackelberg follower is watching his action. The Stackelberg follower must not have the opportunity to commit to a future action in front of the Stackelbergführer (i.e. also not on an action out of balance in the Stackelberg model), and the Stackelbergführer must be aware of this. If this were possible, the Stackelberg follower would commit himself to the set that the Stackelberg guide chooses in the Stackelberg model, and the Stackelberg guide's best answer to this would be to choose the set that the Stackelberg follower chooses in the Stackelberg model (so the whole thing would turn around!).

Companies can find themselves in the Stackelberg competition if one of them has some kind of advantage that enables them to decide first. Usually the Stackelbergführer should be able to make a decision. Openly choosing your action first is the most obvious form of doing this; As soon as the Stackelbergführer has chosen his action, he cannot undo it, he is bound to it. The possibility of the first move can be given, for example, in a situation in which the Stackelbergführer has a monopoly and the Stackelberg follower is new on the market.

Nash equilibrium

The Stackelberg model can be solved to find one (or more) Nash equilibria (s), i.e. the strategy configuration (s) in which each of the players has chosen the optimal set given the choice of the other players' sets.

In general, the inverse demand function for the market in a duopoly is given by , where the demand amount of the Stackelberg leader and the demand amount of the Stackelberg follower denotes. Furthermore applies . The price is thus a function of the total output. The company i have the cost function . The model is released by backward induction. Company 1 determines the best response from Company 2; H. how this will react as it watches the choice of crowd . Company 1 (the Stackelberg guide) then chooses a set in such a way that, anticipating the response from Company 2 (the Stackelberg follower), it will maximize its payout. Company 2 observes this and in equilibrium actually chooses the expected amount as the answer. ${\displaystyle P(Q)}$ ${\displaystyle q_{1}}$ ${\displaystyle q_{2}}$ ${\displaystyle Q=q_{1}+q_{2}}$ ${\displaystyle C_{i}(q_{i})}$ ${\displaystyle q_{1}}$

In order to calculate the Nash equilibrium, the best-answer function of the Stackelberg follower must first be calculated (→ backward induction ).

The profit of company 2 (Stackelberg Successor) is its revenue minus its costs; the revenue is the product of price and quantity of companies 2 produced and the cost is determined by the cost structure of the company, the profit is: . The best answer is the value of the given maximized , the output of the Stackelbergführers (Company 1). This value indicates the output that maximizes Company 2's profit. So the maximum of below has to be found. First derive from: ${\displaystyle \Pi _{2}=P(q_{1}+q_{2})\cdot q_{2}-C_{2}(q_{2})}$ ${\displaystyle q_{2}}$ ${\displaystyle \Pi _{2}}$ ${\displaystyle q_{1}}$ ${\displaystyle \Pi _{2}}$ ${\displaystyle q_{2}}$ ${\displaystyle \Pi _{2}}$ ${\displaystyle q_{2}}$

${\displaystyle {\frac {\partial \Pi _{2}}{\partial q_{2}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot q_{2}+P(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}\cdot }$

According to the sufficient condition for an extremum, this must result in 0 (then it must be checked whether the second derivative is negative or there is a change in sign from + to -):

${\displaystyle {\frac {\partial \Pi _{2}}{\partial q_{2}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot q_{2}+P(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0\cdot }$

The values ​​of that satisfy this equation are in the set of the best answers. Now consider the profit function of company 1. It is calculated by using Company 2's best answer function when calculating the price. ${\displaystyle q_{2}}$

The profit of company 1 (the Stackelbergführer) results in , with the output of company 2 as a function (namely the best answer function from above) of company 1. The value of sought that maximizes is now given . This means that given the reaction function of the stacker successor (company 2), the output must be found that maximizes the profit of company 1. So the maximum of below has to be found. First derive from: ${\displaystyle \Pi _{1}=P(q_{1}+q_{2}(q_{1}))\cdot q_{1}-C_{1}(q_{1})}$ ${\displaystyle q_{2}(q_{1})}$ ${\displaystyle q_{1}}$ ${\displaystyle \Pi _{1}}$ ${\displaystyle q_{2}(q_{1})}$ ${\displaystyle \Pi _{1}}$ ${\displaystyle q_{1}}$ ${\displaystyle \Pi _{1}}$ ${\displaystyle q_{1}}$

${\displaystyle {\frac {\partial \Pi _{1}}{\partial q_{1}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot {\frac {\partial q_{2}(q_{1})}{\partial q_{1}}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}.}$

According to the sufficient condition for an extreme point, this must result in 0 (see above):

${\displaystyle {\frac {\partial \Pi _{1}}{\partial q_{1}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{2}}}\cdot {\frac {\partial q_{2}(q_{1})}{\partial q_{1}}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0.}$

example

The following example is characteristic. It assumes a linear demand curve and places some conditions on the cost structures for the sake of simplicity so that the problem can be solved.

${\displaystyle {\frac {\partial ^{2}C_{i}(q_{i})}{\partial q_{i}\partial q_{j}}}=0,\forall j}$ other ${\displaystyle {\frac {\partial C_{i}(q_{i})}{\partial q_{j}}}=0,j\neq \ i}$

to simplify the calculation. The cost structure of one company is therefore independent of the output of the other company.

Company 2 (Stackelberg Successor) profit is:

${\displaystyle \Pi _{2}={\bigg (}a-b(q_{1}+q_{2}){\bigg )}\cdot q_{2}-C_{2}(q_{2}).}$

The maximization problem is generally solved as follows ( necessary condition ):

${\displaystyle {\frac {\partial {\bigg (}a-b(q_{1}+q_{2}){\bigg )}}{\partial q_{2}}}\cdot q_{2}+a-b(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0,}$

${\displaystyle \Rightarrow \ -bq_{2}+a-b(q_{1}+q_{2})-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}=0,}$

${\displaystyle \Rightarrow \ q_{2}={\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}}.}$

Let us consider the problem of company 1 (Stackelbergführer):

${\displaystyle \Pi _{1}={\bigg (}a-b(q_{1}+q_{2}(q_{1})){\bigg )}\cdot q_{1}-C_{1}(q_{1}).}$

Inserting the response function that we got from the maximization problem of company 2: ${\displaystyle q_{2}(q_{1})}$

${\displaystyle \Pi _{1}={\bigg (}a-b{\bigg (}q_{1}+{\frac {a-bq_{1}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}}{\bigg )}{\bigg )}\cdot q_{1}-C_{1}(q_{1}),}$

${\displaystyle \Rightarrow \Pi _{1}={\bigg (}{\frac {a-b\cdot q_{1}+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}){\bigg )}\cdot q_{1}-C_{1}(q_{1}).}$

The maximization problem is generally solved as follows (necessary condition):

${\displaystyle {\frac {\partial \Pi _{1}}{\partial q_{1}}}={\bigg (}{\frac {a-2bq_{1}+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2}}{\bigg )}-{\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}=0.}$

By dissolving according to results , the optimal choice of the Stackelberg guide: ${\displaystyle q_{1}}$ ${\displaystyle q_{1}^{*}}$

${\displaystyle q_{1}^{*}={\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}.}$

This is the best choice of the Stackelberg guide anticipating the answer of the Stackelberg follower in balance. The action of the Stackelberg successor can now be found by inserting the output of company 1 into the reaction function obtained above:

${\displaystyle q_{2}^{*}={\frac {a-b\cdot {\frac {a+{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}-2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{2b}}-{\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}}{2b}},}$

${\displaystyle \Rightarrow q_{2}^{*}={\frac {a-3\cdot {\frac {\partial C_{2}(q_{2})}{\partial q_{2}}}+2\cdot {\frac {\partial C_{1}(q_{1})}{\partial q_{1}}}}{4b}}.}$

The Nash equilibria are all . Obviously (if you leave out the costs) the Stackelbergführer has a big advantage. If it weren't for that, he could just as easily choose the amount from the Cournot equilibrium. Since he does not do this, although he has the opportunity, he gains an advantage from his position as a market leader. ${\displaystyle (q_{1}^{*},q_{2}^{*})}$

Economic analysis

A representation in extensive form is often used to analyze the Stackelberg duopoly. The model, also known as the decision tree, shows the output combinations and payouts of both companies in the Stackelberg game.

The example is pretty simple. The cost structure only includes marginal costs (there are no fixed costs ). The demand function is linear and the amount of its price elasticity is 1. Nevertheless, it shows the advantage of the Stackelbergführer.

The Stackelberg follower chooses an amount that will maximize his payout . By deriving this and setting it to zero (to determine the maximum), we get the value of that does exactly that. ${\displaystyle q_{2}}$ ${\displaystyle q_{2}(5000-q_{1}-q_{2}-c_{2})}$ ${\displaystyle q_{2}={\frac {5000-q_{1}-c_{2}}{2}}}$ ${\displaystyle q_{2}}$

The Stackelbergführer wants to choose an amount that will maximize his payout . He knows that the Stackelberg follower will choose the one from above in balance . So the Stackelbergführer will actually maximize his payout ( as a reaction function of the Stackelberg follower was used). Deriving this shows that the maximum payout occurs for . Insertion into the reaction function of the Stackelberg follower results . Assuming the marginal costs of the two curves are identical (so that the Stackelbergführer has no other advantage besides being first on the train) and in particular . The Stackelbergführer would produce 2000 units and the Stackelbergfollower 1000. This would generate the Stackelbergführer a profit of 2 million and the Stackelbergfollower a profit of one million. Only by being first on the move has the Stackelbergführer achieved twice as much profit as the Stackelbergfollower. In the Cournot competition, the winnings would be around 1.78 million each, which means in comparison the Stackelberg leader has gained relatively little, while the Stackelberg follower has lost a lot. However, this is not generally the case. There can also be cases in which the Stackelbergführer gains a comparatively large amount compared to the Cournot competition that comes close to monopoly profits (for example, if the Stackelbergführer also has a great advantage in the cost structure, for example through a better production function ) . It can also be the case that the Stackelbergfollower even makes a higher profit than the Stackelbergführer, but only if he has much lower costs. ${\displaystyle q_{1}}$ ${\displaystyle q_{1}(5000-q_{1}-q_{2}-c_{1})}$ ${\displaystyle q_{2}}$ ${\displaystyle q_{1}(5000-q_{1}-{\frac {5000-q_{1}-c_{2}}{2}}-c_{1})}$ ${\displaystyle q_{2}}$ ${\displaystyle q_{1}={\frac {5000-2c_{1}+c_{2}}{2}}}$ ${\displaystyle q_{2}={\frac {5000+2c_{1}-3c_{2}}{4}}}$ ${\displaystyle c_{1}=c_{2}=1000}$

Incredible threats from the Stackelberg follower

If after the choice of the equilibrium amount by the Stackelbergführer the Stackelberg follower deviated from the equilibrium and chose a non-optimal amount, it would not only harm himself but also the Stackelbergführer. If the Stackelberg follower chose a far larger amount than his best answer, the market price would go down and the Stackelbergführer's profit would go down considerably, possibly below the profit that would be achievable in the Cournot competition. In this case, the Stackelberg follower could announce to the Stackelbergführer before the start of the game that if the Stackelbergführer does not choose the Cournot set, it will deviate from the equilibrium, so that the Stackelbergführer's profit suffers considerable losses. The reason for the consideration is the fact that the set chosen by the Stackelberg leader in equilibrium is only optimal if the Stackelberg follower also chooses the equilibrium set. However, the Stackelbergführer is in no danger. Once he has chosen his equilibrium set, it is irrational for the Stackelberg follower to deviate; because a deviation would reduce his payout, which he is currently trying to maximize. As soon as the Stackelbergführer has chosen, the Stackelbergfollower is well advised to choose the equilibrium path. Therefore, such a threat, as expressed by the Stackelberg follower above, would be implausible (see also sub-game perfect balance ).

In a (infinitely) repeated Stackelberg game, however, the Stackelberg follower would possibly play a punishment strategy ( tit for tat ), which punishes the Stackelberg leader in the respective period for playing the Stackelberg equilibrium. This threat is credible since it is rational for the market follower not to leave his threat empty in order to get the Stackelbergführer to play the crowd out of Cournot equilibrium in the coming periods.

Stackelberg compared with Cournot

The Stackelberg model and the Cournot model are similar to one another, since in both cases there is competition in units of measure. However, the first move gives the Stackelbergführer a decisive advantage. The assumption of the existence of perfect information in the Stackelberg duopoly is essential: the Stackelberg follower must observe the set chosen by the Stackelberg guide, otherwise the Cournot model is to be applied. If the information is imperfect, the threats described above can be credible. If the Stackelberg follower cannot observe the choice of the Stackelbergführer, it is no longer irrational for him to choose, for example, the set that he would play in the Cournot model (which is actually a balance here). However, imperfect information must exist to the effect that the Stackelberg follower is not able to follow the action of the Stackelbergführer, because it would be irrational for the Stackelberg follower not to do this if he were able to: To make an optimal decision he will watch the Stackelbergführer. Any threat by the successor of the Stackelberg not to observe the action of the Stackelbergführer, although he would be able to do so, is just as implausible as the others described so far. This is an example of how the presence of information can harm a player. In the Cournot competition, it is the simultaneity of the game that means that neither player is at a disadvantage cp.

Game theory considerations

See also : game theory

As mentioned earlier, imperfect information leads to Cournot competition. In the Stackelberg duopoly, however, some Cournot equilibria have been preserved as Nash equilibria, which, however, can be identified as implausible threats (as described above) by applying the solution concept of partial game perfection. It turns out that the very reason that the Cournot equilibrium is a Nash equilibrium in the Stackelberg game is responsible for the fact that it is not subgame perfect.

Let us consider a Stackelberg game (i.e. one that fulfills the conditions for the existence of a Stackelberg equilibrium described above) in which for some reason the Stackelberg guide believes that the Stackelberg follower will choose the Cournot set, no matter which action he chooses. (Perhaps the Stackelberg leader thinks the Stackelberg follower is irrational.) If the Stackelbergführer plays the crowd out of Stackelberg equilibrium, he believes that the Stackelberg follower would react with the crowd out of Cournot equilibrium. Therefore it is not ideal for the Stackelbergführer to play the Stackelberg crowd. In fact, its best answer (according to the Cournot equilibrium definition) is to choose the Cournot set. Once he does that, the Stackelberg successor's best answer is to play the Cournot crowd as well.

So let's consider the following combination of strategies:

The Stackelbergführer plays the Cournot crowd. The Stackelberg follower plays the Cournot crowd, no matter what the Stackelbergführer plays.

This strategy configuration is a Nash equilibrium, since each of the players, given the other player's strategy, reacts optimally. Choosing the Cournot set, however, would not be optimal for the Stackelberg leader if the Stackelberg follower would also react to the Stackelberg set with the Stackelberg set. In this case, the Stackelberg guide's best answer would be to choose the Stackelberg set. So what makes this combination of strategies a Nash equilibrium is the fact that the Stackelberg follower does not choose the Stackelberg set if the Stackelberg leader does so.

Precisely this fact means, however, that this strategy combination is not a Nash equilibrium of the sub-game that starts at the point where the Stackelberg guide has already chosen the Stackelberg set. (This subgame is outside the equilibrium path.) As soon as the Stackelberg leader has chosen the Stackelberg set, it is the best answer of the Stackelberg follower to also choose the Stackelberg set (and thus it is the only action that a Nash equilibrium in this Subgame generated). This strategy combination, which results in the Cournot equilibrium, is therefore not subgame-perfect.

Comparison with other oligopoly models

In comparison with other oligopoly models, the following applies in equilibrium:

• The total output in the Stackelberg duopoly is greater than in the Cournot duopoly, but less than in the Bertrand competition
• The price in the Stackelberg duopoly is lower than in the Cournot duopoly, but higher than in the Bertrand competition
• The consumer surplus in the Stackelberg duopoly is greater than in the Cournot duopoly, but less than in the Bertrand competition
• The total output in the Stackelberg duopoly is greater than in a monopoly or cartel, but less than in perfect competition
• The price in the Stackelberg duopoly is lower than in a monopoly or cartel, but higher than in perfect competition.

These results are to be seen as model predictions, the occurrence of which depends on the type of cost functions and the size of the market in individual cases (see e.g. Steckelbach (2002)).

See also

• Cournot oligopoly
• Bertrand competition
• Extensive form of a game
• Duopoly

literature

• D. Fudenberg and J. Tirole : Game Theory. MIT Press, 1993 (especially Chapter 3, Section 1)
• R. Gibbons: A primer in game theory. Harvester-Wheatsheaf, 1992 (especially Chapter 2, section 1B)
• MJ Osborne and A. Rubenstein : A Course in Game Theory. MIT Press, 1994 (especially pp. 97-98)
• L. Steckelbach: Effects of competition policy regulations on oligopolistic markets. Hamburg 2002
• J. Tirole: The Theory of Industrial Organization. Cambridge, 1988