# Oligopoly Pricing: The Role of Firm Size and Number

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Oligopoly Model

**Lemma**

**1.**

**Lemma**

**2.**

#### 2.2. Price Fluctuation Range

**Proposition**

**1.**

**Corollary**

**1.**

## 3. Results

#### 3.1. Merger and Break Up

#### 3.1.1. Edgeworth Zone

#### 3.1.2. Price Fluctuation Range

**Proposition**

**2.**

**Example**

**1.**

#### 3.2. Investment and Divestment

#### 3.2.1. Edgeworth Zone

#### 3.2.2. Price Fluctuation Range

**Proposition**

**3.**

#### 3.3. Entry and Exit

#### 3.3.1. Edgeworth Zone

#### 3.3.2. Price Fluctuation Range

**Proposition**

**4.**

**Proposition**

**5.**

#### 3.4. Recapitulation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1**.

**Proof**

**of**

**Lemma**

**2**.

**Proof**

**of**

**Proposition**

**1**.

**Proof**

**of**

**Proposition**

**2**.

**Proof**

**of**

**Proposition**

**3**.

**Proof**

**of**

**Proposition**

**4**.

**Proof**

**of**

**Proposition**

**5**.

## Notes

1 | |

2 | Price competition under capacity constraints has been studied by Beckmann (1965) [4] Levitan and Shubik (1972) [5], Kreps and Scheinkman (1983) [6], Osborne and Pitchik (1986) [7], Vives (1986) [8], Allen and Hellwig (1986) [9], Maskin and Tirole (1988) [10], Deneckere and Kovenock (1992) [11], Tasnádi (1999) [12], amongst many others. |

3 | Formally, it is $\Omega \left({p}_{i},{p}_{-i}\right)$ and $\Delta \left({p}_{i},{p}_{-i}\right)$, where ${p}_{-i}$ is the vector of prices of all firms other than i. We use this shorthand notation to emphasize on which value of ${p}_{i}$ these sets are based. In the ensuing analysis, the effect of the complete price vector on demand will be clear from the context. |

4 | |

5 | For a general discussion, see, e.g., Tasnádi (2004) [15]. |

6 | There are also many asymmetric equilibria with a subset of sellers pricing above costs. |

7 | A comparable finding in a three-firm Bertrand–Edgeworth model can be found in Chen and Li (2018) [16]. |

8 | It is worth noting that there is a boundary case where $K=D\left(\underline{p}\right)=D\left({p}_{1}^{\ast}\right)$ prior to investment or entry. In this situation, an increase in the leader’s production capacity induces a shift from the monopolistic to the Edgeworth zone, but its residual profit-maximizing price ${p}_{1}^{\ast}$ does not change. Notice, however, that there is still a price decrease in the sense that the price fluctuation range $[{\widehat{p}}_{1},{p}_{1}^{\ast}]$ expands downward (i.e., ${\widehat{p}}_{1}$ decreases when moving into the Edgeworth zone). |

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Small Capacity | ||||
---|---|---|---|---|

Same Zone | Switch Zone | |||

Regular | Leader | Regular | Leader | |

Merger | - | - | ↑ | ↑ |

Break Up | - | - | x | x |

Investment | ↓ | ↓ | ↓ | ↓ |

Divestment | ↑ | ↑ | x | x |

Entry | ↓ | ↓ | ↓ | ↓ |

Exit | ↑ | ↑ | x | x |

Large Capacity | ||||
---|---|---|---|---|

Same Zone | Switch Zone | |||

Regular | Leader | Regular | Leader | |

Merger | - | - | ↑ | ↑ |

Break Up | - | - | x | x |

Investment | - | - | x | x |

Divestment | - | - | ↑ | ↑ |

Entry | - | - | x | x |

Exit | - | - | ↑ | ↑ |

Intermediate Capacity | ||||
---|---|---|---|---|

Same Zone | Switch Zone | |||

Regular | Leader | Regular | Leader | |

Merger | -/↑ | ↑ | - | - |

Break Up | - | ↓ | - | ↓ |

Investment | ↓ | ↓ | ↓ | x |

Divestment | ↑ | ↑ | ↑ | ↑ |

Entry | ↓ | ↓ | ↓ | ↓ |

Exit | ↑ | ↑ | ↑ | ↑ |

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Bos, I.; Marini, M.A.
Oligopoly Pricing: The Role of Firm Size and Number. *Games* **2023**, *14*, 3.
https://doi.org/10.3390/g14010003

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Bos I, Marini MA.
Oligopoly Pricing: The Role of Firm Size and Number. *Games*. 2023; 14(1):3.
https://doi.org/10.3390/g14010003

**Chicago/Turabian Style**

Bos, Iwan, and Marco A. Marini.
2023. "Oligopoly Pricing: The Role of Firm Size and Number" *Games* 14, no. 1: 3.
https://doi.org/10.3390/g14010003