Stackelberg competition

The Stackelbergmodell is a strategic game in economics, which is characterized in that the market leader moves first and then the market following companies choose. If it is only two companies is called a Stackelberg duopoly. It is named after the German economist Heinrich Freiherr von Stackelberg who published his work market form and balance in 1934, in which the model has been described, and represents a further development of Cournot duopoly model

We denote the two players as Stackelbergführer or leader and Stackelbergfolger or followers, and they compete in quantity units. The Stackelbergführer is thereby also sometimes referred to as a leader.

For the existence of an equilibrium in the Stackelberg duopoly there are some other conditions: The leader needs to know that the follower observes his action. The follower must have no possibility to commit to a future action against the leader ( ie not on an action outside of equilibrium in the Stackelberg model) this is known, and the leader must. If this were possible, the followers would commit to the amount chosen by the leader in the Stackelberg model, and the best answer to that would be the leader to select the quantity chosen by the follower in the Stackelberg model. ( The whole thing would then turn around! )

Companies can be found in the Stackelberg competition if one of them has an advantage of some kind, which puts it in a position to decide first. Usually, the leader in should be able to define themselves. His action open to choose first, the most obvious form is; as soon as the leader has chosen his action, he can not be undone, it is bound to it. The possibility of the first train may be given, for example, in a situation in which the leader has a monopoly and the follower is new to the market.

• 4.1 Game Theoretical Considerations

Nash equilibrium

The Stackelberg model can be solved to find a (or several) Nash equilibrium ( e ), ie the strategy configuration (s), wherein each of the player has selected the optimum amount for a given choice of the amounts of the other players.

In general, the inverse demand function is for the market in the duopoly is given by where the subscript 1 the leader and the index 2 denotes the followers. The price thus obtained as a function of total output. The company i have the cost function. The model is achieved by reverse induction. Companies 1 to determine the best response of Firm 2, ie as this will react when it observes the choice of the amount. Company 1 ( the leader ) then selects a lot so that it ( the follower) maximizes his payoff under anticipating the response of Firm 2. Firm 2 observes this and elect in equilibrium actually the expected amount as the answer.

To calculate the Nash equilibrium, the best response function of the follower must be calculated first ( → backward induction ).

The profit of Firm 2 ( follower) is its revenue minus its cost; the revenue is the product of price and quantity produced by Firm 2 and the cost is given by the cost structure of the company, the profit is:. The best answer is the value of that maximizes, given the output of the leader (firm 1). This value specifies the output that maximizes the profit of Firm 2. So the maximum of below must be found. Leite to first to rate:

After the sufficient condition for an extremum must have the 0 result ( then still needs to examine whether the second derivative is negative or there is a sign change from to - are ):

The values ​​of that satisfy this equation lie in the set of best responses. Now, the profit function of firms 1 is considered. It is calculated in the in the calculation of the price of the best - response function of Firm 2 is employed.

The profit Firm 1 ( the leader ) is given by, where the output of Firm 2 as a function (namely, the best response function from above) indicating Firm 1. The value of will be sought, the maximized given. That is where the reaction function of the follower (firm 2), the output has to be found which maximizes the profit of companies 1. So the maximum of below must be found. Leite to first to rate:

After the sufficient condition for an extreme point this must be 0 result (see above):

Example

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

In order to simplify the calculation. The cost structure of a company is independent from the output of other companies.

The profit of Firm 2 ( follower) is:

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

Consider the problem of company 1 (Leader):

Onset of the reaction function that we have obtained from the maximization problem of Company 2:

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

Solving for yields, the optimal choice of the leader:

This is the best choice of leader under anticipating the response of the followers in balance. The action of the followers can now be found by substituting the output of Firm 1 in the above-obtained reaction function:

The Nash equilibria are all. Obviously ( if you let the costs of foreign ago) has a great advantage of the leader. If that were not the case, it could also simply select the quantity from the Cournot equilibrium. Since he does not, even though he has the chance, he will receive an advantage by its market leader position.

Economic Analysis

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

The image on the left shows a Stackelberg game in extensive form. The payouts are doing right. The example is quite simple. The cost structure includes only marginal costs (there are no fixed costs ). The demand function is linear and the amount of their price elasticity is 1 Nevertheless, it shows the advantage of the leader.

The follower chooses a quantity that maximizes his payoff. By deriving this and sets zero ( for the determination of the maximum), is obtained as the value of which satisfies precisely this.

The Leader would like to choose a set that maximizes his payoff. He knows this is that the follower in equilibrium, the select from above. So the leader is actually maximize his payoff ( in response function of the Followers was used ). By deriving a result, the maximum payout is set for. Insertion into the reaction function of the follower results. Suppose the marginal cost of the two curves are identical ( so that the leader has no other advantage than the first on the train to be ), and in particular. The leader would produce 2000 units and the followers 1000th This would generate the Leader a profit of 2 million and a profit of one million followers. Only in this way, the first to be on the train, the leader has reached twice as much profit as the follower. In Cournot competition, the gains at about 1.78 million each would be, that is, in comparison, the Leader relatively little is gained, the follower lost for quite a lot. However, this is not generally the case. There may also be cases in which the leader of a relatively large amount compared to the Cournot competition to win that come close to monopoly profits ( for example, if the leader still has in addition a great advantage in the cost structure, such as a better production function ). It may also be the case that the follower even make a higher profit than the leader, but only if he has a much lower cost.

Implausible threats of Followers

If after the election of the equilibrium amount would vary by the Leader of the Followers of balance and would choose a non- optimal amount, it would not only hurt himself, but also the leader. Would the followers choose a much larger amount than its best response, the market price would drop and the profit of the leader would drop significantly, possibly under the profit that would be achieved in Cournot competition. In this case, the follower could announce the leader before the start of the game, that it. , In case that the leader does not choose the Cournot quantity will deviate from equilibrium, so that the profit of the leader suffers considerable losses Reason for the consideration is the fact that the amount is selected from the leader in the balance is only optimal when the follower also selects the equilibrium amount. However, the leader is in no danger. Once he has chosen his equilibrium quantity, it is irrational to depart for the followers; because any deviation would reduce its payout that yes tries to maximize precisely this. Once the leader has chosen the followers would be well advised to choose the equilibrium path. Why would such a threat, pronounced as above from followers, implausible (see also subgame perfect equilibrium).

In an ( infinitely ) repeated Stackelberg game, however, the followers would possibly a punishment strategy ( Tit for Tat ) play that punishes the leader in the respective period for playing the Stackelberg equilibrium. This threat is credible because it is rational for the market follower, not to let his threat look empty to bring the leader to the amount to play in the coming periods in the Cournot equilibrium.

Stackelberg compared with Cournot

The Stackelberg model and the Cournot model are similar to each other, as will compete in both cases in terms of volume. However, the first train gives the leader a decisive advantage. The assumption of the existence of perfect information in the Stackelberg duopoly is essential: the follower must observe the leader chosen by lot, otherwise the Cournot model is to be applied. With imperfect information, the threats described above can be credible. If the Stackelberg follower can not observe the choice of the leader, it is no longer, for example, to select more irrational for him the amount which he would play in the Cournot model (which here actually represents a balance ). However imperfect information must exist to the effect that the follower is not able to follow the action of the leader, because it would be irrational for the follower not to do this if he were in a position: In order to make an optimal decision he is watching the leader. Any threat of followers as to not to observe the action of the leader, even though he would be able, therefore, is as untrustworthy as the other described so far. This is an example that the presence information can be harmful to a player. In Cournot competition, it is the simultaneity of the game, which results in the fact that no player cp is at a disadvantage.

Game Theoretical Considerations

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

Consider a Stackelberg game (ie one that meets the conditions described above for the existence of a Stackelberg equilibrium) in which believes for some reason the leader that the follower will choose the Cournot quantity, no matter what action he chooses. (Maybe believes the leader, the follower is irrational. ) If the leader plays the amount of the Stackelberg equilibrium, he believes, would the followers react to the amount of the Cournot equilibrium. Therefore, it is not optimal for the leader to play the Stackelberg quantity. In fact, his best response ( as defined by the Cournot equilibrium) is to choose the Cournot quantity. Once he has done that, it is the best response of the followers, also playing the Cournot quantity.

Let us consider the following strategy combination:

The leader plays the Cournot quantity. The follower plays the Cournot quantity, no matter what plays the leader.

This strategy configuration is a Nash equilibrium, since each of the player, given the other player's strategy is optimal responding. Choosing the Cournot quantity, but would not be optimal for the leader when the follower would also respond to the Stackelberg quantity with the Stackelberg quantity. In that case, would be the best answer of the leader is to choose the Stackelberg quantity. What makes this strategy combination to a Nash equilibrium, so the fact that the follower does not choose the Stackelberg quantity if the leader does.

, It is this fact means, however, that this strategy combination is not a Nash equilibrium of the sub- game, which starts at the point at which the leader has already chosen the Stackelberg quantity. ( This part of the game is out of the equilibrium path. ) As soon as the leader chosen the Stackelberg quantity, it is the best response of the followers, also to choose the Stackelberg quantity (and thus it is the only action that this is a Nash equilibrium in subgame generated ). This is the strategy combination that results in Cournot equilibrium, not subgame perfect.

Comparison with other oligopoly models

Compared with other oligopoly models is considered in equilibrium:

• The total output is in the Stackelberg duopoly Cournot duopoly is greater than in, but lower than in Bertrand Competition

• The price is in the Stackelberg duopoly Cournot duopoly in less than but greater than in Bertrand Competition

• Consumer surplus is the Stackelberg duopoly Cournot duopoly is greater than in, but lower than in Bertrand Competition

• The total output is in the Stackelberg duopoly is larger than in the monopoly or cartel, but lower than in perfect competition
• The price is in the Stackelberg duopoly lower than in monopoly or cartel, but larger than in perfect competition.

These results can be seen as a pattern predictions whose arrival in the individual case on the nature of the cost functions and the market size dependent (see eg Steckelbach (2002) ).