# Duopoly and Endogenous Single Product Quality Strategies

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Model

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Assumption**

**6.**

**Assumption**

**7.**

**Proposition**

**1.**

- I.
- In equilibrium, the market is fully served. That is, ṯ = a.
- II.
- In equilibrium, the consumer surplus of consumer a from the choice of pairs $({Q}_{1},{p}_{1})\in {s}_{1}^{*}$ offered by firm 1 is positive. In particular, $CS\left(a\right)>0$8.

- Note 1

**Proposition**

**2.**

- Note 2

**Proposition**

**3.**

- Note 3

- The main result—part I

- Note 4

- i.
- Economically, firm 1 aims to distinguish itself from firm 2 by reducing its quality in order to relax competition.
- ii.
- The intuition for the latter equation is that to maximize profit per unit, the slope of the indifferent consumer (a) and the slope of the cost function (${C}_{1}^{{}^{\prime}}\left({Q}_{1}\right)$) must be equal in an interior solution9.

#### Market Competition in Qualities and Prices

- The main result—part II

- i.
- When $b\ge $${C}_{2}^{{}^{\prime}}\left(\overline{Q}\right)$, then ${\overline{\mathrm{q}}}_{2}=\overline{Q}$; in other words, firm 2 will offer top of the line quality10.
- ii.
- When $b<$${C}_{2}^{{}^{\prime}}\left(\overline{Q}\right)$, then firm 2 will offer ${\overline{\mathrm{q}}}_{2}<\overline{Q}$, where $b=$${C}_{2}^{{}^{\prime}}({\overline{\mathrm{q}}}_{2})$.

- Note 5

**Proposition**

**4.**

- i.
- For both firms, there is a constant relationship between profit per unit and market share. That is, $\Delta Q=\frac{{p}_{1}-{C}_{1}\left({Q}_{1}\right)}{\Delta \mathrm{\u1e6f}}=\frac{{p}_{2}-{C}_{2}\left({\overline{Q}}_{2}\right)}{\Delta \overline{\mathrm{t}}}$.
- ii.
- The profit per unit, market share and total profit of firm 1 are larger than firm 211.
- iii.
- The following condition must be satisfied: $b\le 2a-\frac{3{C}_{2}\left({Q}_{1}\right)-2{C}_{1}\left({Q}_{1}\right)-{C}_{2}\left({\overline{Q}}_{2}\right)}{{\overline{Q}}_{2}-{Q}_{1}}$, where ${\overline{Q}}_{2}=Min\{{\overline{q}}_{2},\overline{Q}\}$12.

## 4. A Numerical Example

**Example**

**1.**

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of Proposition 1 part I.**

**Proof**

**of Proposition 1 part II.**

**Lemma**

**A1.**

**Proof**

**of Lemma A1.**

**Proof**

**of Proposition 2.**

**Proof**

**of Proposition 3.**

**Proof**

**of the main result—part I.**

**Proof**

**of the main result—part II.**

**Proof**

**of Proposition 4.**

#### An Explanation for Example 1

## Notes

1 | In the monopoly case, we will use the results of Mussa and Rosen (1978) [3] in order to compare them to our results, under the duopolistic case. |

2 | Assumption 1 states that net income is non-negative for all the consumers, since for the case where $p>t\xb7Q$, there is no consumer t who would choose the pair $(Q,p)$ for every $t\in T$. Therefore, ${e}_{t}\ge t\xb7\overline{Q}\ge t\xb7Q\ge p$ implies that ${y}_{t}={e}_{t}-p\ge 0$. |

3 | The intuition is that the marginal utility of consumer a, who can buy ${\underset{\xaf}{\mathrm{q}}}_{1}$, cannot be less than the corresponding slope of cost function—${C}_{1}^{{}^{\prime}}({\underset{\xaf}{\mathrm{q}}}_{1})$. Similarly, the marginal utility of consumer b who buys ${\overline{\mathrm{q}}}_{2}$ cannot be below the corresponding slope of the cost function—${C}_{2}^{{}^{\prime}}({\overline{\mathrm{q}}}_{2}).$ |

4 | Assumptions 4 and 5 indicate that there will be an intersection of production costs. Intuitively speaking, this intersection leads to a set of qualities where each firm maintains a superior technology over the other. Otherwise, there will be no equilibrium with more than one active firm. |

5 | If $p>b\overline{Q}$, the consumer surplus of b and all other potential consumers of this quality price pair will be negative, which means that none of the consumers will choose this pair. |

6 | By contrast, if we have two quality price pairs $(\tilde{Q},\tilde{p})$ and $(\hat{Q},\hat{p})$, where $\tilde{Q}\ge $$\hat{Q}$ and $\tilde{p}<\hat{p}$, none of the consumers will choose the pair $(\hat{Q},\hat{p})$. |

7 | This includes the possibility of not buying at all. |

8 | This result is contrasts with an equilibrium with a monopoly, in which $CS\left(a\right)=0$. The intuition is that in the duopoly case, it is not beneficial for firm 1 to increase its price until $CS\left(a\right)=0$ due to the “burden” of competition because of the existence of firm 2. See also in the following Example 1. |

9 | This equation also holds in the case of a monopoly when the market is fully served. |

10 | See also Example 1. |

11 | This result contrasts with the monopoly case, where the profit per unit increases with the quality index. |

12 | See also Shaked and Sutton (1982) [4], who assumed that there were no production costs, and accordingly had $b\le 2a$. |

13 | The rationale why the value of the quality offered by firm 1 is 1.12 is proven in the last subsection of Appendix A. Note that in this example, there are numerical rounding offs. |

14 | See also work by Shitovitz et al. (1989) [24], who derived similar results in the case of a monopoly with a finite number of consumers. |

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Gayer, A.
Duopoly and Endogenous Single Product Quality Strategies. *Games* **2023**, *14*, 56.
https://doi.org/10.3390/g14040056

**AMA Style**

Gayer A.
Duopoly and Endogenous Single Product Quality Strategies. *Games*. 2023; 14(4):56.
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**Chicago/Turabian Style**

Gayer, Amit.
2023. "Duopoly and Endogenous Single Product Quality Strategies" *Games* 14, no. 4: 56.
https://doi.org/10.3390/g14040056