Question

Collusion
Consider two firms producing homogenous goods and choosing prices in each period for an
infinite number of periods. Each of the two firms owns a share α of its rival. This share is small enough for each firm to keep full control of its own activities and decisions: the rival
is a minority shareholder, who is not represented in the board and just receives a share α of
the firm’s profits. Write the no deviation condition for collusion under trigger strategies and
discuss how the likelihood of collusion is affected by this cross-ownership.

Answer 1

Answer 2

In the context of oligopolistic competition, collusion refers to a situation where two or more firms coordinate their pricing and output decisions to maximize joint profits instead of competing with each other. Trigger strategies are a common way to sustain collusion over multiple periods, where firms continue to cooperate as long as the other firm also cooperates and "trigger" a punishment if the other firm deviates from the collusive agreement.

In this scenario, we have two firms with cross-ownership, each owning a share α of the other firm. Let's consider Firm A and Firm B, where α represents the ownership share of each firm in the rival company. The profits of Firm A and Firm B in a given period can be represented as ${�}_{�}$ and ${�}_{�}$, respectively.

The no deviation condition for collusion under trigger strategies states that neither firm has an incentive to deviate from the collusive agreement as long as they are receiving non-negative profits from cooperation. Mathematically, the condition can be expressed as follows:

${�}_{�}\ge {�}_{�}^{\ast }$${�}_{�}\ge {�}_{�}^{\ast }$

where ${�}_{�}^{\ast }$ and ${�}_{�}^{\ast }$ are the profits that Firm A and Firm B would earn if they were to deviate from the collusive agreement and revert to a competitive behavior.

The trigger strategies work based on the threat of punishment in case of deviation. If either Firm A or Firm B decides to undercut the agreed-upon price, the other firm will respond by also deviating from the collusive agreement and reducing its price to punish the deviating firm. This punishment can lead to lower profits for both firms compared to the collusive profits, acting as a deterrent to deviation.

The likelihood of collusion is affected by the cross-ownership between the firms. When firms have cross-ownership, they are more likely to consider the long-term implications of their actions on their own profits as well as their share of profits in the rival firm. This can strengthen the commitment to the collusive agreement as the firms have a vested interest in the profitability of the other firm.

However, cross-ownership can also lead to potential conflicts of interest. The minority shareholders may prioritize their own short-term profits over the long-term sustainability of the collusive agreement. If the shareholders in either firm push for higher individual profits, it may lead to a breakdown in collusion and deviation from the agreed-upon prices.

In conclusion, the no deviation condition for collusion under trigger strategies ensures that firms have an incentive to stick to the collusive agreement as long as their profits remain non-negative compared to the profits they would earn from deviation. Cross-ownership can enhance the likelihood of collusion as firms have a mutual interest in each other's profitability, but it can also introduce potential conflicts of interest that may impact the stability of collusion over time.

Answer 3

In the context of trigger strategies and collusion in a duopoly, the no deviation condition is a crucial concept to ensure that firms have no incentive to cheat or deviate from their collusive agreements. Trigger strategies are a way to sustain collusion by imposing severe punishments on firms that break the agreement and undercut prices.

The no deviation condition states that a firm should not find it profitable to unilaterally deviate from the collusive agreement and lower its price to gain a larger market share. In other words, if both firms adhere to the collusive price and production levels, they can collectively earn higher profits compared to a scenario where one firm deviates and takes advantage of the situation.

Now, let's consider how cross-ownership affects the likelihood of collusion. In the given scenario, each firm owns a share α of its rival. This cross-ownership can have both positive and negative effects on collusion.

Positive Effect: Cross-ownership can enhance trust between the two firms. Since each firm is a minority shareholder in the other, they have a mutual interest in maintaining high profits for both entities. This mutual interest can strengthen the commitment to the collusive agreement and make it more likely for the firms to stick to the agreed-upon prices and output levels. The fear of triggering a retaliation from the other firm through lower prices (and lower profits for both) can act as a strong deterrent against deviation.

Negative Effect: On the other hand, cross-ownership may also lead to a weaker incentive for retaliation. If a firm deviates from the collusive agreement and lowers its price, the cross-owned rival might be less inclined to retaliate with a price cut, as it would harm its own profits as well. This weaker threat of retaliation can reduce the effectiveness of trigger strategies in deterring deviations.

Overall, the effect of cross-ownership on collusion depends on the specific circumstances and the extent of cross-ownership (i.e., the value of α). If the cross-ownership is significant, it may weaken the trigger strategies' ability to deter deviations and make collusion less likely. Conversely, if cross-ownership is relatively small, it may enhance trust and commitment, increasing the likelihood of sustaining collusion.

In conclusion, the no deviation condition and trigger strategies are essential in maintaining collusion in a duopoly. Cross-ownership between the firms can influence collusion by affecting trust, commitment, and the effectiveness of retaliation threats. The impact of cross-ownership on collusion is complex and needs to be analyzed in each specific case to understand its implications fully.

Answer 4

In a scenario of collusion between two firms producing homogenous goods, trigger strategies are commonly used to sustain the collusive behavior over an infinite number of periods. Trigger strategies involve the threat of punishment if one of the firms deviates from the collusive agreement. The "no deviation condition" refers to the situation where, under trigger strategies, it is not profitable for any firm to deviate from the collusive agreement.

To better understand the no deviation condition, let's define the parameters:

• π represents the profits of each firm when they both adhere to the collusive agreement.

• π_c represents the profits of each firm when they both deviate from the collusive agreement.

• α is the share that each firm owns of its rival.

Under the assumption of trigger strategies, the no deviation condition can be expressed as follows:

π ≥ π_c

In other words, collusion is sustained if the firms find it more profitable to maintain the collusive agreement (π) rather than deviating from it (π_c). This condition acts as a deterrent against defection from the collusive behavior because any attempt to deviate from the agreement would result in reduced profits for both firms.

Now, let's discuss how the likelihood of collusion is affected by cross-ownership, where each firm owns a share (α) of its rival.

Cross-ownership can have both positive and negative effects on the likelihood of collusion:

Positive Effects:

1. Mutual Interest: When each firm owns a share of its rival, they have a mutual interest in each other's success. This shared ownership can foster a sense of cooperation and reduce the temptation to deviate from the collusive agreement.

2. Alignment of Goals: Cross-ownership may align the interests of the firms more closely. This alignment can strengthen the commitment to maintain the collusive behavior, as defection would harm the profits of both firms.

Negative Effects:

1. Reduced Incentive for Punishment: Cross-ownership might reduce the incentive for punishment in the trigger strategy. If one firm deviates, the other firm's profits will be affected due to the cross-ownership, which may lead to less willingness to punish the defector.

2. Coordination Difficulties: Cross-ownership could complicate coordination between the firms. Since each firm has a minority share in the other, it may be challenging to agree on strategies and pricing decisions.

Overall, the impact of cross-ownership on collusion is complex and depends on various factors such as the level of α, the firms' strategic behavior, and the structure of the market. In some cases, cross-ownership may enhance collusion by fostering cooperation, while in other cases, it may create challenges and reduce the effectiveness of trigger strategies.