# Optimal Referral Reward Considering Customer’s Budget Constraint

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Referral Reward Programs

#### 2.2. Budget Constraints

## 3. The Model and Result

#### 3.1. Friend R’s Problem

#### 3.2. Customer S’s Problem

#### 3.3. The Firm’s Problem

#### 3.4. Analysis and Results

**Lemma 1.**

Candidates | Price | Reward | Expected Profit |
---|---|---|---|

Candidate1 | $L$ | $0$ | $L$ |

Candidate2 | $v$ | $0$ | $qv$ |

Candidate3 | $H$ | $\frac{\mathsf{\beta}}{q}\left(H-v\right)$ | $qH-\mathsf{\beta}\left(H-v\right)$ |

**Proposition 1.**

**Proposition 2.**

Conditions | Optimal Price and Reward | |
---|---|---|

$v<L$ | $q\le \mathsf{\beta}+\left(1-\mathsf{\beta}\right)\frac{L}{H}$ | ${p}^{*}=L$, ${B}^{*}=\mathsf{\beta}\left(L-v\right)$; |

$q>\mathsf{\beta}+\left(1-\mathsf{\beta}\right)\frac{L}{H}$ | ${p}^{*}=H$,${B}^{*}=\frac{\mathsf{\beta}}{q}\left(H-v\right)$; | |

$v\ge H$ | $q\le \frac{L}{H}$ | ${p}^{*}=L$,${B}^{*}=0$; |

$q>\frac{L}{H}$ | ${p}^{*}=H$, ${B}^{*}=0$. |

**Figure 3.**Optimal solutions according to product valuation ($H=0.8$, $L=0.2$, $\mathsf{\beta}=0.5$ and $q=0.55$).

## 4. Firm Financing

**Figure 4.**RRP and firm financing ($H=0.8$, $L=0.2$, $\mathsf{\beta}=0.5$ and $\mathsf{\alpha}=0.8$).

**Proposition 3.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Proof of Lemma 1.**

**Proof of Proposition 1.**

**Proof of Proposition 2.**

**Result of Problem (6).**

Case | Conditions | Optimal Priceand Reward |
---|---|---|

Case 1 | $v<L<H$,$q\le \mathsf{\beta}+\left(1-\mathsf{\beta}\right)\frac{L}{H}$ | ${p}^{*}=L$, ${B}^{*}=\mathsf{\beta}\left(L-v\right)$, ${m}^{*}=0$ |

Case 2 | $v<L<H$,$q\mathsf{\beta}+\left(1-\mathsf{\beta}\right)\frac{L}{H}$ | ${p}^{*}=H$, ${B}^{*}=\frac{\mathsf{\beta}}{q}\left(H-v\right)$, ${m}^{*}=0$ |

Case 3 | $L\le v<H$, $q>\mathsf{\beta}$, $q>\frac{\left(1-\alpha -\mathsf{\beta}\right)v+\mathsf{\alpha}L+\mathsf{\beta}H}{H}$ | ${p}^{*}=H$,${B}^{*}=\frac{\mathsf{\beta}}{q}\left(H-v\right)$,${m}^{*}=0$ |

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**MDPI and ACS Style**

Zhou, D.; Yao, Z.
Optimal Referral Reward Considering Customer’s Budget Constraint. *Future Internet* **2015**, *7*, 516-529.
https://doi.org/10.3390/fi7040516

**AMA Style**

Zhou D, Yao Z.
Optimal Referral Reward Considering Customer’s Budget Constraint. *Future Internet*. 2015; 7(4):516-529.
https://doi.org/10.3390/fi7040516

**Chicago/Turabian Style**

Zhou, Dan, and Zhong Yao.
2015. "Optimal Referral Reward Considering Customer’s Budget Constraint" *Future Internet* 7, no. 4: 516-529.
https://doi.org/10.3390/fi7040516