# Random Informative Advertising with Vertically Differentiated Products

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## Abstract

**:**

## 1. Introduction

#### Literature Review

## 2. The Model

## 3. The Equilibrium Outcomes

**Proposition**

**1**

**.**Whenever an equilibrium exists, it is unique. Depending on the position of $(\alpha ,y)$ relative to the zones depicted in Figure 1 and defined analytically in Appendix A, there are four main cases in terms of the existence and type of equilibrium.

- 1
- Zone IE (interior equilibrium): both firms have positive natural markets and charge prices lower than the revenue of the consumers. This zone is divided into three sub-zones, depending on whether or not the firms reach all consumers.
- IE(i)
- Both firms reach all consumers (${\Psi}_{1}^{\ast}={\Psi}_{2}^{\ast}=1$), and relative prices are ${v}_{1}^{\ast}=2/3,$ ${v}_{2}^{\ast}=1/3.$
- IE(ii)
- The high-quality firm reaches all consumers, while the low-quality firm reaches only a fraction of them: ${\Psi}_{1}^{\ast}=1$, ${\Psi}_{2}^{\ast}={\left(\frac{1}{9\alpha}\right)}^{1/3}$; the equilibrium relative prices are: ${v}_{1}^{\ast}=2{\left(\frac{\alpha}{3}\right)}^{1/3},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{v}_{2}^{\ast}={\left(\frac{\alpha}{3}\right)}^{1/3}.$
- IE(iii)
- Both firms reach only a fraction of consumers.9

- 2
- Zone CIE (constrained interior equilibrium): Both firms have positive natural markets, but the high-quality firm charges the revenue of consumers. This zone is also divided into three sub-zones, depending on whether or not firms reach all consumers.
- CIE(i)
- Both firms reach all consumers: ${\Psi}_{1}^{\ast}={\Psi}_{2}^{\ast}=1$, and charge the relative prices ${v}_{1}^{\ast}=y,$ ${v}_{2}^{\ast}=y/2.$
- CIE(ii)
- The high-quality firm reaches all customers (${\Psi}_{1}^{\ast}=1$), the low-quality firm only a fraction ${\Psi}_{2}^{\ast}={y}^{2}/4\alpha $ of them and relative prices are given by: ${v}_{1}^{\ast}=y$, ${v}_{2}^{\ast}=y/2;$
- CIE(iii)
- Both firms reach only a fraction of customers; the high-quality firm charges the relative price ${v}_{1}^{\ast}=y$ and the low-quality firm charges a lower price.10

- 3
- Zone COR (CORner equilibrium): the low-quality firm has a zero natural market, with the following relative prices and advertising intensities:$${v}_{1}^{\ast}={v}_{2}^{\ast}=y;\phantom{\rule{1.em}{0ex}}{\Psi}_{1}^{\ast}=\frac{y}{\alpha},\phantom{\rule{1.em}{0ex}}{\Psi}_{2}^{\ast}=\frac{y}{\alpha}(1-\frac{y}{\alpha}).$$
- 4
- Zone X: There is no equilibrium in pure strategies.

## 4. Comparative Statics

**Corollary**

**1**

**.**At the equilibrium (whenever it exists), the high-quality firm charges a higher price and advertises more (in a broad sense) than the low-quality one.

**Corollary**

**2**

**.**The high-quality firm makes a strictly larger profit than the low-quality one. Both profits converge to zero as α goes to infinity.

**Corollary**

**3**

**.**In all cases of consumers’ income, the profits’ ratio ($\frac{{\pi}_{1}^{\ast}}{{\pi}_{2}^{\ast}}$) is increasing for low enough α and decreasing above some critical level of α, converging to 1, as α goes to infinity.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Definition of Zones

- (i)
- $IE\left(i\right)=\left\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}\alpha \le 1/9\mathrm{and}y\ge (2/3)\right\}.$
- (ii)
- $IE\left(ii\right)=\left\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}\alpha \in (1/9,8/9]\mathrm{and}y\ge 2{(\alpha /3)}^{1/3}\right\}.$
- (iii)
- Let ${\Psi}_{1}^{\ast}\left(\alpha \right)$ be the equilibrium value of ${\Psi}_{1}$ as a function of $\alpha $, and write the equilibrium profit of Firm 2 as $\Delta {\pi}_{2}^{\ast}\left(\alpha \right)$,$$IE\left(iii\right)=\left\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}\alpha 8/9\mathrm{and}y\le \frac{\sqrt{2\alpha {\pi}_{2}^{\ast}\left(\alpha \right)}}{1-{\Psi}_{1}^{\ast}\left(\alpha \right)}\right\}.$$

- (i)
- $CIE\left(i\right)=\left\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}2\sqrt{\alpha}\le y\le 2/3\right\}$.
- (ii)
- $CIE\left(ii\right)=\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}y-{y}^{4}/8\alpha -\alpha \ge 0,y\le 2\sqrt{\alpha}$ and $y\le 2{(\alpha /3)}^{1/3}\}.$
- (iii)
- $CIE\left(iii\right)=\{(\alpha ,y)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+\ast},\mathrm{such}\mathrm{that}y-{y}^{4}/8\alpha -\alpha \le 0,y\ge \frac{1}{2}(-1+\sqrt{1+4\alpha})$ and $y\le {v}_{1}^{IE}\left(\alpha \right)\}$,where${v}_{1}^{IE}\left(\alpha \right)$ corresponds to the price equilibrium of case IE (iii).

**Proposition**

**A1.**

## Appendix B. Proofs

**Proof**

**of**

**Proposition**

**1.**

**Figure A2.**Representation of the expressions of ${\Psi}_{1}(\alpha ,{v}_{2}\left(\alpha \right))$ and ${\Psi}_{2}(\alpha ,{v}_{2}\left(\alpha \right))$ as functions of $\alpha $.

**Figure A3.**Representation of the RHS of Equation (A16).

**Proof of Proposition 1,**

**Case 2 (CIE).**

**Proof of Proposition 1, Case 3**

**(corner equilibrium).**

## Notes

1 | We assume that this information, notably the one on product quality, can be trusted, for instance because false advertising is banned or because quality amounts to some verifiable characteristics. |

2 | While here this limit comes from the fact that prices cannot exceed consumers’ income, it could alternatively be derived from the existence of a maximum utility level from consuming the good. |

3 | In contrast, in horizontal preferences models, consumers have different choices between variants sold at the same price. This is the case in the spatial competition models, such as the well-known Hotelling (1929) [3] model. |

4 | and also in terms of the “perceived consumer effectiveness” (PCE), i.e., their perceived belief in that her/his purchase will prove to have an actual effect (Nurse et al., 2012) [5]. |

5 | the natural market of a firm is composed of consumers who, when informed of both products, will choose the firm’s product. |

6 | We rule out the possibility of using misleading advertisements. We think that firms have all the less interest in deceiving consumers as, in real life, consumers increasingly may seek information on past experiences and have the option of returning products when they are not satisfied. |

7 | The same proviso obviously applies. |

8 | This is not the only possible option. We could have adopted a two-step game, where advertising intensities are chosen prior to prices, or the other way around. We may think that advertising is a decision as flexible as prices, at least in some industries, for which the static game adopted is appropriate. |

9 | Appendix A provides more details on the equilibrium outcome. |

10 | More details may be found in Appendix A. |

11 | The only one identified in the literature. |

12 | In this sense one can say that more differentiated industries advertise more. |

13 | Such that the high-quality firm reaches all customers for values below the critical one |

14 | Implying that $\widehat{\theta}\in \left(0,1\right).$ |

15 | This implies ${p}_{2}^{\ast}\le Y$, because ${p}_{2}^{\ast}<{p}_{1}^{\ast}$ |

16 | This can be observed by using Mathematica’s RegionPlot function. |

17 | It is too long to be reproduced here. |

18 | $g(y,\alpha )$ is too long to be reproduced here. |

19 | Using the RegionPlot function of Mathematica. |

20 | No profitable deviation exists among the prices, ensuring for the firm the whole market as a natural market. |

21 | The Hessian matrix is a definite negative. |

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**Figure 6.**Comparative statics for advertising intensities: the case of intermediate consumers’ income.

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**MDPI and ACS Style**

Lahmandi-Ayed, R.; Laussel, D.
Random Informative Advertising with Vertically Differentiated Products. *Games* **2024**, *15*, 10.
https://doi.org/10.3390/g15020010

**AMA Style**

Lahmandi-Ayed R, Laussel D.
Random Informative Advertising with Vertically Differentiated Products. *Games*. 2024; 15(2):10.
https://doi.org/10.3390/g15020010

**Chicago/Turabian Style**

Lahmandi-Ayed, Rim, and Didier Laussel.
2024. "Random Informative Advertising with Vertically Differentiated Products" *Games* 15, no. 2: 10.
https://doi.org/10.3390/g15020010