Stackelberg


 * Stackelberg Model**

The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially.

This econimic theory was founded by the German economist Heinrich Freherr von Stackelberg. Born in Moscow in 1905, Stackelberg studied mathematics and economics at the University of Cologne. He then became a professor of economics at the University of Bonn. Stackleberg then moved on to Spain where he also taught. His life was cut short at the age of forty-one by lymphoma in 1946. After his death, he became recognized for his work on 'duopoly models.'

In game theory terminaology, the players of this game are a leader and a follower and they compete on quantity. The Stackelberg leader is sometimes referred to as the Market Leader.

In simple terms, the Stackelberg model involves two players, 'Company A' and 'Company B'. Both companies produce a product that is completely undifferentiated with know demand. The two companies compete on output quantity, Qa and Qb. In this model, one company must make the first move. We will say that Company A will make the first move in the market. Company A chooses an output quantity, Qa, and Company B observes this. Company B will then base it's output quantity, Qb, on the information obtained from observing Company A. It will produce an amount that will optimize profitability. Company A is completely aware that Company B will act according to it's initial move into the marketplace. As a result, Company A can react and increase production to capture remaining market demand. This puts Company A into the position of 'Market Leader'. Acting first gives them a distinct advantage.

An extensive-form representation is often used to analyze the Stackelberg leader-follower model. This model shows the combination of outputs and payoffs both firms have in the Stackelberg game The image on the left depicts in extensive form a Stackelberg game. The payoffs are shown on the right. This example is fairly simple. The basic cost structure involves only marginal cost(there is no fixed cost for simplification purposes). The demand function is linear and price elasticity of demand is 1. However, it illustrates the leader's advantage. The follower wants to choose //q//2 to maximise its payoff 5000 − //q//1 − //q//2 − //c//2. Taking the first order derivative and equating it to zero (for maximisation) yields q²=5000-q¹-c² / 2 as the maximum value of //q//2. The leader wants to choose //q//1 to maximise its payoff 5000 − //q//1 − //q//2 − //c//1. However, in equilibrium, it knows the follower will choose //q//2 as above. So in fact the leader wants to maximise its payoff 5000-q¹-((5000-q¹-c²) / 2))-c¹ (by substituting //q//2 for the follower's best response function). By differentiation, the maximum payof is given by q1=(5000-2c¹-c²) / 2 . Feeding this into the follower's best response function yields q2=(5000+2c¹-3c²)/4 . Suppose marginal costs were equal for the firms (so the leader has no market advantage other than first move) and in particular //c//1=c2=1000. The leader would produce 2000 and the follower would produce 1000. This would give the leader a profit of two million and the follower a profit of one million. Simply by moving first, the leader has accrued twice the profit of the follower. There may be cases where a Stackelberg leader has huge gains that approach monopoly profits (for example, if the leader also had a large cost structure advantage, perhaps due to a better production function). There may also be cases where the follower actually enjoys higher profits than the leader, but only because it, say, has much lower costs.


 * What happens if the follower deviates?**

If, after the leader had selected its equilibrium quantity, the follower deviates and chooses some non-optimal quantity, it would hurt both the leader and itself. If the follower choses an excess quantity, the market price would lower and the leader's profits would be less. The follower could announce to the leader before the game starts that the leader must choose a Cournot equilibrium quantity or the follower will choose a deviant quantity that will hit the leader's profits. After all, the quantity chosen by the leader in equilibrium is only optimal if the follower also plays in equilibrium. The leader should not be in danger. Once the leader has chosen its quantity, it would be foolish for the follower to deviate because it too would be hurt. Once the leader has chosen, the follower is better off by playing in equilibrium. Hence, a threat by the follower is not credible.

However, if a follower were in a better financial position than the leader, it is possible that they may threaten to deviate at later time if the leader does not act in a way that the follower seems appropriate.

a. Company A - Acting first b. Company B - Acting as follower c. Neither
 * Q1: In general which company has an advantage in a Stackelberg model?**

Answer: a. Typically the company acting first has an advantage over the follower.

a. The company acting first has higher fixed costs b. The company acting as follower has higher fixed costs c. The company acting first has lower market share.
 * Q2: What items could lead the company acting first in a Stackelberg model to realizing enormous gains over the company acting as follower when marginal costs are the same?**

Answer: b. One reason that the company acting first may see enormous gains is that they may have lower fixed costs.

Answer: False, It would hurt the leaders by choosing such a strategy, however, it would also hurt the follower.
 * Q3: True or False: It be advisable for a follower to choose an irrational quantity outside of equilibrium to hurt the leader.**

a. produce excess quanitity in the current segment of the game b. produce a smaller quantitiy in the current segment of the game c. threaten to produce an excess quantity in later segment of the game. d. threaten to produce an excess quantity in the current segment of the game.
 * Q4: How is it feasible for a follower to influence the decision of the leader?**

Answer c. A follower could threaten a later move that would hurt the leader.

a. sequentially b. simultaneousley c. randomly
 * Q5: In a Stackelberg game the leader and follower move ___________________.**

Answer a. The leader and follower must move sequentially in a Stackelberg game.

Stackelberg Competition. Accessed Oct. 21, 2007 http://www.answers.com/topic/stackelberg-competition?cat=technology,

'The Basics of Game Theory'. Accessed Oct. 22, 2007 http://ocw.mit.edu/NR/rdonlyres/Sloan-School-of-Management/15-010Fall-2004/8DAD5D4B-4BBF-43D7-BD41-413ECE07DB19/0/the_bsc_game_thy.pdf

Stackelberg Competition. Accessed Oct. 24, 2007 http://en.wikipedia.org/wiki/Stackelberg_competition,

Stackelberg Competition. Accessed Oct. 28, 2007 http://www.painreliefchat.com/arthritis-pain-relief/Stackelberg_competition