# Oligopoly Pricing: The Role of Firm Size and Number

^{1}

^{2}

^{*}

## 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). |

## References

- Edgeworth, F.Y. The Pure Theory of Monopoly. Pap. Relat. Political Econ.
**1925**, 1, 111–142. [Google Scholar] - Tirole, J. The Theory of Industrial Organization; MIT Press: Cambridge, MA, USA, 1988. [Google Scholar]
- Vives, X. Oligopoly Pricing: Old Ideas and New Tools; The MIT Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Beckmann, M. Edgeworth-Bertrand Duopoly Revisited. In Operations Research-Verfahren; Rudolf, H., Ed.; Sonderdruck, Verlag Anton Hain: Meisenheim, Germany, 1965; Volume III, pp. 55–68. [Google Scholar]
- Levitan, R.; Shubik, M. Price Duopoly and Capacity Constraints. Int. Econ.
**1972**, 13, 111–122. [Google Scholar] [CrossRef] - Kreps, D.; Scheinkman, J. Quantity Precommitment and Bertrand competition yields Cournot outcomes. Bell J. Econ.
**1983**, 14, 326–337. [Google Scholar] [CrossRef] - Osborne, M.J.; Pitchik, C. Price Competition in a Capacity-Constrained Duopoly. J. Econ. Theory
**1986**, 38, 238–260. [Google Scholar] [CrossRef] - Vives, X. Rationing rules and Bertrand–Edgeworth Equilibria in Large Markets. Econ. Lett.
**1986**, 21, 113–116. [Google Scholar] [CrossRef] - Allen, B.; Hellwig, M. Bertrand–Edgeworth Oligopoly in Large Markets. Rev. Econ. Stud.
**1986**, 53, 175–204. [Google Scholar] [CrossRef] - Maskin, E.; Tirole, J. A Theory of Dynamic Oligopoly, II: Price Competition, Kinked Demand Curves and Edgeworth Cycles. Econometrica
**1988**, 56, 571599. [Google Scholar] [CrossRef][Green Version] - Deneckere, R.J.; Kovenock, D. Price Leadership. Rev. Econ. Stud.
**1992**, 59, 143–162. [Google Scholar] [CrossRef] - Tasnádi, A. Existence of pure strategy Nash equilibrium in Bertrand–Edgeworth oligopolies. Econ. Lett.
**1999**, 63, 201–206. [Google Scholar] [CrossRef][Green Version] - Bos, I.; Marini, M.A.; Saulle, R.D. Myopic Oligopoly Pricing; Working Paper; Nota di Lavoro 32.2021; Fondazione Eni Enrico Mattei: Milano, Italy, 2021. [Google Scholar]
- Edwards, R.A.; Routledge, R.R. Information, Bertrand–Edgeworth Competition and the Law of One Price. J. Math. Econ.
**2022**, 101, 1–7. [Google Scholar] [CrossRef] - Tasnádi, A. Production in Advance versus Production to Order. J. Econ. Organ.
**2004**, 54, 191–204. [Google Scholar] [CrossRef] - Chen, Z.; Li, G. Horizontal Mergers in the Presence of Capacity Constraints. Econ. Inq.
**2018**, 56, 1346–1356. [Google Scholar] [CrossRef][Green Version] - Bos, I.; Vermeulen, D. On Pure-Strategy Nash Equilibria in Price-Quantity Games. J. Math. Econ.
**2021**, 96, 1–13. [Google Scholar] [CrossRef] - Montez, J.; Schutz, N. All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices. Rev. Econ. Stud.
**2021**, 88, 2407–2438. [Google Scholar] [CrossRef] - Myatt, D.P.; Ronayne, D. A Theory of Stable Price Dispersion; Working Paper, Economics Series Working Papers 873; University of Oxford, Department of Economics: Oxford, UK, 2019. [Google Scholar]
- Gabszewicz, J.; Marini, M.A.; Zanaj, S. Random Encounters and Information Diffusion about Product Quality. J. Econ. Manag.
**2022**, forthcoming. [Google Scholar] [CrossRef] - Aas, Ø.N.; Wulfsberg, F.; Moen, E.R. Price Dispersion and the Role of Stores; Working Paper; BI Norwegian Business School, Oslo Business School and Université Libre de Bruxelles: Bruxelles, Belgium, 2018. [Google Scholar]
- Gorodnichenko, Y.; Sheremirov, V.; Talavera, O. Price Setting in Online Markets: Does IT click? J. Eur. Econ.
**2018**, 16, 1764–1811. [Google Scholar] [CrossRef][Green Version] - Eckert, A. Retail Price Cycles and the Presence of Small Firms. Int. J. Industrial Organ.
**2003**, 21, 151–170. [Google Scholar] [CrossRef] - Noel, M.D. Edgeworth Price Cycles: Evidence from the Toronto Retail Gasoline Market. J. Ind. Econ.
**2007**, 55, 69–92. [Google Scholar] [CrossRef] - Noel, M.D. Edgeworth Price Cycles, Cost-based Pricing and Sticky Pricing in Retail Gasoline Markets. Rev. Econ. Stat.
**2007**, 89, 324–334. [Google Scholar] [CrossRef][Green Version] - Wang, Z. Collusive Communication and Pricing Coordination in a Retail Gasoline Market. Rev. Ind. Organ.
**2008**, 32, 35–52. [Google Scholar] [CrossRef][Green Version] - Foros, Ø.; Steen, F. Vertical Control and Price Cycles in Gasoline Retailing. Scand. J. Econ.
**2013**, 115, 640–661. [Google Scholar] [CrossRef] - Linder, M. Price Cycles in the German Retail Gasoline Market Competition or Collusion? Econ. Bull.
**2018**, 38, 593–602. [Google Scholar] - Hauschultz, F.P.; Munk-Nielsen, A. The Role of Demand in Price Cycles: Evidence from Danish Pharmaceutical Markets; Working Paper; University of Copenhagen: Copenhagen, Denmark, 2020. [Google Scholar]
- Zhang, X.; Feng, J. Cyclical Bid Adjustments in Search-Engine Advertising. Manag. Sci.
**2011**, 57, 1703–1719. [Google Scholar] [CrossRef][Green Version] - Kruse, J.B.; Rassenti, S.; Reynolds, S.S.; Smith, V.L. Bertrand–Edgeworth Competition in Experimental Markets. Econometrica
**1994**, 62, 343–372. [Google Scholar] [CrossRef] - Fonseca, M.A.; Normann, H.-T. Excess Capacity and Pricing in Bertrand–Edgeworth Markets: Experimental Evidence. J. Inst. Theor. Econ.
**2013**, 169, 199–228. [Google Scholar] [CrossRef]

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 | ↑ | ↑ | ↑ | ↑ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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