# Pricing Problems in the Pharmaceutical Supply Chain with Mixed Channel: A Power Perspective

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## Abstract

**:**

## 1. Introduction

- (1)
- What effects do power structures have on pricing, performance and social welfare in the pharmaceutical supply chain where the pharmaceutical manufacturer is regulated?
- (2)
- How does the restricted wholesale price cap affect the financial performance and social welfare in three different power structures?

## 2. Literature

**Price cap regulation in pharmaceutical supply chain**

**Power structure in the supply chain**

**Social welfare in the supply chain**

## 3. Model Descriptions

## 4. Mathematical Models and Equilibrium Analysis

#### 4.1. Base Models

#### 4.1.1. Pharmaceutical Manufacturer Stackelberg Model

#### 4.1.2. Pharmacy Stackelberg Model

#### 4.1.3. Vertical Nash Model

#### 4.2. Price Cap Regulation Models

- (1)
- If the wholesale price cap is higher than the optimal wholesale price in the MS model ($\overline{w}>{w}^{ms}$), the price cap regulation has no effect on the pharmaceutical supply chain under three power structures.
- (2)
- If the wholesale price cap is higher than the optimal wholesale price in the VN model and lower than that in the MS model (${w}^{vn}<\overline{w}<{w}^{ms}$), the price cap regulation only has effects on the pharmaceutical supply chain when the pharmaceutical manufacturer is dominant in the market.
- (3)
- If the wholesale price cap is higher than the optimal wholesale price in the PS model and lower than that in the VN model (${w}^{ps}<\overline{w}<{w}^{vn}$), the price cap regulation affects the pharmaceutical supply chain when the pharmaceutical manufacturer is dominant in the market or the pharmaceutical manufacturer and pharmacy are in a balanced power structure.
- (4)
- If the wholesale price cap is lower than the optimal wholesale price in the PS model ($\overline{w}<{w}^{ps}$), the pharmaceutical supply chain under three different power structures will be affected.

## 5. Effects of the Power Structures

#### 5.1. Effects of Power Structures on the Pricing and Performance

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

#### 5.2. Effects of Power Structure on Social Welfare

**Proposition**

**5.**

## 6. Effects of the Price Cap Regulation

#### 6.1. Effects of Price Cap Regulation on Performance

**Proposition**

**6.**

#### 6.2. Effects of Price Cap Regulation on Social Welfare

**Proposition**

**7.**

## 7. Conclusions and Future Research

- (1)
- When the price cap is in different ranges, the influence of the power structure on the economic and social benefits of the pharmaceutical supply chain is different. When the wholesale price cap is higher than the optimal wholesale price in the MS model, the price cap regulation will not work in the supply chain in three different power structures. The pharmaceutical manufacturer and pharmacy can gain more profits when they dominate the market, whereas the balanced power relationship between supply chain members can improve the total profit and social welfare. When the wholesale price cap is lower than the optimal price cap, the pharmaceutical manufacturer will still gain higher profits when it has more market power over other supply chain members. However, for pharmacies, market power cannot ensure that they can make more profits with the influence of the restricted wholesale price cap. Therefore, leading pharmacies may resist the price cap regulation; as a result, this increases the risk of policy failure. The government may consider subsidizing dominated pharmacies when implementing price cap regulation. When the price cap is very low, an interesting discovery is presented. Price cap regulation might harm financial performance and social welfare when pharmaceutical firms are in a balanced power structure. Thus, when the wholesale price cap is lower than a certain threshold, price cap regulation might be detrimental to creating a fair market environment for enterprises.
- (2)
- We find the wholesale price cap is an important parameter that affects financial performance and social welfare of the pharmaceutical supply chain. Regardless of the power structures, the pharmaceutical manufacturer’s profit is positively correlated with restricted wholesale price. The pharmacy’s profit is negatively correlated with the restricted wholesale price. In addition, social welfare in pharmaceutical manufacturer Stackelberg and pharmacy Stackelberg markets is negatively correlated with the restricted wholesale price, and social welfare in the balanced market is dependent on the restricted wholesale price.
- (3)
- Overall, this paper provides insights into the pricing decisions of pharmaceutical companies under different power structures. In addition, it also provides a decision-making basis for the government to make price limit policies in different power markets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of the Table 1

#### Appendix A.1.1. MS Model

#### Appendix A.1.2. PS Model

#### Appendix A.1.3. VN Model

#### Appendix A.2. Proof of Table 2

#### Appendix A.2.1. MM Model

- (i)
- If $\overline{w}>{w}^{ms}$, the manufacturer price cap regulation has no effect on the medical supply chain, so the optimal prices are ${w}^{mm}={w}^{ms}$, ${p}_{1}^{mm}={p}_{1}^{ms}$ and ${p}_{2}^{mm}={p}_{2}^{ms}$.
- (ii)
- If $\overline{w}<{w}^{ms}$, the price cap regulation has effects on the pharmaceutical manufacturer and pharmacy, so the optimal wholesale price is ${w}^{mm}=\overline{w}$, ${p}_{1}^{mm}=\frac{[\theta \beta +(1-\theta )\gamma ]\alpha}{2({\beta}^{2}-{\gamma}^{2})}+$$\frac{\overline{w}+{c}_{1}}{2}$, ${p}_{2}^{mm}=\frac{\left[\left(1-\theta \right)\beta +\theta \gamma \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{2}}{2}$.

#### Appendix A.2.2. PM Model

- (i)
- If $\overline{w}>{w}^{ps}$, then the manufacturer price cap regulation has no effect on the pharmaceutical manufacturer and pharmacy, so the optimal prices are ${w}^{pm}={w}^{ps}$, ${p}_{1}^{pm}={p}_{1}^{ps}$ and ${p}_{2}^{pm}={p}_{2}^{ps}$.
- (ii)
- If $\overline{w}<{w}^{ps}$, then the manufacturer price cap regulation has effects on the pharmaceutical manufacturer and pharmacy, so the optimal prices are ${w}^{pm}=\overline{w}$, Replace ${w}^{pm}$ in Formula (5), we need solve $\underset{{p}_{1},{p}_{2}}{Max}{\pi}_{r}\left({p}_{1},{p}_{2}\right)=\left({p}_{1}-\overline{w}-c\right)\left(\theta a-\beta {p}_{1}+\gamma {p}_{2}\right)+\left({p}_{2}-\overline{w}-c\right)\left(\theta a-\beta {p}_{2}+\gamma {p}_{1}\right)$. $\frac{\partial {\pi}_{r}\left({p}_{1},{p}_{2}\right)}{\partial {p}_{1}}=\theta \alpha -\beta \left(2{p}_{1}-\overline{w}-{c}_{1}\right)+\gamma \left(2{p}_{2}-\overline{w}-{c}_{2}\right)$, $\frac{\partial {\pi}_{r}\left({p}_{1},{p}_{2}\right)}{\partial {p}_{2}}=\left(1-\theta \right)\alpha -\beta \left(2{p}_{2}-\overline{w}-{c}_{2}\right)+\gamma \left(2{p}_{1}-\overline{w}-{c}_{1}\right)$, $\frac{{\partial}^{2}{\pi}_{r}({p}_{1},{p}_{2})}{\partial {p}_{1}{}^{2}}=$$\frac{{\partial}^{2}{\pi}_{r}\left({p}_{1},{p}_{2}\right)}{\partial {p}_{2}{}^{2}}=-2\beta ,\text{}\frac{{\partial}^{2}{\pi}_{r}\left({p}_{1},{p}_{2}\right)}{{\partial}^{2}{p}_{1}{p}_{2}}=\frac{{\partial}^{2}{\pi}_{r}\left({p}_{1},{p}_{2}\right)}{{\partial}^{2}{p}_{2}{p}_{1}}=2\gamma $.

#### Appendix A.2.3. VM Model

- (i)
- If $\overline{w}>{w}^{vn}$, then the manufacturer price cap regulation has no effect on the pharmaceutical manufacturer and pharmacy, so the optimal prices are ${w}^{vm}={w}^{vn}$, ${p}_{1}^{vm}={p}_{1}^{vn}$ and ${p}_{2}^{vm}={p}_{2}^{vn}$.
- (ii)
- If $\overline{w}<{w}^{vn}$, then the manufacturer price cap regulation has effects on the pharmaceutical manufacturer, so the optimal prices are ${w}^{vm}=\overline{w}$, ${p}_{1}^{vm}={p}_{1}^{vn}$ and ${p}_{2}^{vm}={p}_{2}^{vn}$.

#### Appendix A.3. Proof of Proposition 1

**Proof of Proposition**

**1.**

#### Appendix A.4. Proof of Proposition 2

**Proof of Proposition**

**2.**

#### Appendix A.5. Proof of Proposition 3

**Proof of Proposition**

**3.**

#### Appendix A.6. Proof of Proposition 4

**Proof of Proposition**

**4.**

#### Appendix A.7. Proof of Proposition 5

**Proof of Proposition**

**5.**

#### Appendix A.8. Proof of Proposition 6

**Proof of Proposition**

**6.**

#### Appendix A.9. Proof of Proposition 7

**Proof of Proposition**

**7.**

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MS Model | PS Model | VN Model | |
---|---|---|---|

$w$ | $\frac{\alpha}{4\left(\beta -\gamma \right)}+\frac{2c-{c}_{1}-{c}_{2}}{4}$ | $\frac{\alpha}{8\left(\beta -\gamma \right)}+\frac{6c-{c}_{1}-{c}_{2}}{8}$ | $\frac{\alpha}{6\left(\beta -\gamma \right)}+\frac{4c-{c}_{1}-{c}_{2}}{6}$ |

${p}_{1}$ | $\frac{\left[\left(1+4\theta \right)\beta +\left(5-4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c+3{c}_{1}-{c}_{2}}{8}$ | $\frac{\left[\left(1+4\theta \right)\beta +\left(5-4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c+3{c}_{1}-{c}_{2}}{8}$ | $\frac{\left[\left(1+6\theta \right)\beta +\left(7-6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c+5{c}_{1}-{c}_{2}}{12}$ |

${p}_{2}$ | $\frac{\left[\left(5-4\theta \right)\beta +\left(1+4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c-{c}_{1}+3{c}_{2}}{8}$ | $\frac{\left[\left(5-4\theta \right)\beta +\left(1+4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c-{c}_{1}+3{c}_{2}}{8}$ | $\frac{\left[\left(7-6\theta \right)\beta +\left(1+6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c-{c}_{1}+5{c}_{2}}{12}$ |

${\pi}_{m}\left(w\right)$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{16\left(\beta -\gamma \right)}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{32\left(\beta -\gamma \right)}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{18\left(\beta -\gamma \right)}$ |

${\pi}_{r}\left({p}_{1},{p}_{2}\right)$ | $\frac{4A+5\beta -3\gamma}{32({\beta}^{2}-{\gamma}^{2})}{\alpha}^{2}+\frac{B}{8}$$-\frac{{2\mathrm{c}+\mathrm{c}}_{1}\left(8\theta -3\right)+{c}_{2}\left(5-8\theta \right)}{16}\alpha $$+\frac{\beta \left(5{c}_{1}{}^{2}-6{c}_{1}{c}_{2}+5{c}_{2}{}^{2}\right)}{32}$$+\frac{\gamma \left(3{c}_{1}^{2}-10{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)}{32}$ | $\frac{2A+3\beta -\gamma}{16({\beta}^{2}-{\gamma}^{2})}{\alpha}^{2}+\frac{B}{4}$$-\frac{{2\mathrm{c}+\mathrm{c}}_{1}\left(4\theta -1\right)+{c}_{2}\left(3-4\theta \right)}{8}\alpha $$+\frac{\beta \left(3{c}_{1}{}^{2}-2{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)}{16}$$+\frac{\gamma \left({c}_{1}^{2}-6{c}_{1}{c}_{2}+{c}_{2}{}^{2}\right)}{16}$ | $\frac{9A+13\beta -5\gamma}{72({\beta}^{2}-{\gamma}^{2})}{\alpha}^{2}+\frac{2B}{9}$$-\frac{{8\mathrm{c}+\mathrm{c}}_{1}\left(18\theta -5\right)+{c}_{2}\left(13-18\theta \right)}{36}\alpha $$+\frac{\beta \left(13{c}_{1}{}^{2}-10{c}_{1}{c}_{2}+13{c}_{2}{}^{2}\right)}{72}$$+\frac{\gamma \left(5{c}_{1}^{2}-26{c}_{1}{c}_{2}+5{c}_{2}{}^{2}\right)}{72}$ |

Models | $\mathit{w}$ | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | |
---|---|---|---|---|

MM model | $\overline{w}>{w}^{ms}$ | $\frac{\alpha}{4\left(\beta -\gamma \right)}+\frac{2c-{c}_{1}-{c}_{2}}{4}$ | $\frac{\left[\left(1+4\theta \right)\beta +\left(5-4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c+3{c}_{1}-{c}_{2}}{8}$ | $\frac{\left[\left(5-4\theta \right)\beta +\left(1+4\theta \right)\gamma \right]\alpha}{8\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{2c-{c}_{1}+3{c}_{2}}{8}$ |

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\overline{w}$ | $\frac{\left[\left(1-\theta \right)\gamma +\theta \beta \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{1}}{2}$ | $\frac{\left[\left(1-\theta \right)\beta +\theta \gamma \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{2}}{2}$ | |

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\overline{w}$ | $\frac{\left[\left(1-\theta \right)\gamma +\theta \beta \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{1}}{2}$ | $\frac{\left[\left(1-\theta \right)\beta +\theta \gamma \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{2}}{2}$ | |

$\overline{w}<{w}^{ps}$ | $\overline{w}$ | $\frac{\left[\left(1-\theta \right)\gamma +\theta \beta \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{1}}{2}$ | $\frac{\left[\left(1-\theta \right)\beta +\theta \gamma \right]\alpha}{2\left({\beta}^{2}-{\gamma}^{2}\right)}+\frac{\overline{w}+{c}_{2}}{2}$ | |

PM model | $\overline{w}>{w}^{ms}$ | $\frac{\alpha}{8\left(\beta -\gamma \right)}+\frac{6c-{c}_{1}-{c}_{2}}{8}$ | ||

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\frac{\alpha}{8\left(\beta -\gamma \right)}+\frac{6c-{c}_{1}-{c}_{2}}{8}$ | |||

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\frac{\alpha}{8\left(\beta -\gamma \right)}+\frac{6c-{c}_{1}-{c}_{2}}{8}$ | |||

$\overline{w}<{w}^{ps}$ | $\overline{w}$ | |||

VM model | $\overline{w}>{w}^{ms}$ | $\frac{\alpha}{6\left(\beta -\gamma \right)}+\frac{4c-{c}_{1}-{c}_{2}}{6}$ | $\frac{\left[\left(1+6\theta \right)\beta +\left(7-6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c+5{c}_{1}-{c}_{2}}{12}$ | $\frac{\left[\left(7-6\theta \right)\beta +\left(1+6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c-{c}_{1}+5{c}_{2}}{12}$ |

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\frac{\alpha}{6\left(\beta -\gamma \right)}+\frac{4c-{c}_{1}-{c}_{2}}{6}$ | $\frac{\left[\left(1+6\theta \right)\beta +\left(7-6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c+5{c}_{1}-{c}_{2}}{12}$ | $\frac{\left[\left(7-6\theta \right)\beta +\left(1+6\theta \right)\gamma \right]\alpha}{12\left({\beta}^{2}-{\gamma}^{2}\right)}$$+\frac{4c-{c}_{1}+5{c}_{2}}{12}$ | |

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\overline{w}$ | |||

$\overline{w}<{w}^{ps}$ | $\overline{w}$ |

Models | ${\mathit{\pi}}_{\mathit{m}}\left(\mathit{w}\right)$ | ${\mathit{\pi}}_{\mathit{r}}\left({\mathit{p}}_{1},{\mathit{p}}_{2}\right)$ | |
---|---|---|---|

MM model | $\overline{w}>{w}^{ms}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{16\left(\beta -\gamma \right)}$ | $\frac{16\left(\beta -\gamma \right){\theta}^{2}-16\left(\beta -\gamma \right)\theta +5\beta -3\gamma}{32\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{2c+{c}_{1}\left(8\theta -3\right)+{c}_{2}\left(5-8\theta \right)}{16}\alpha $$+\frac{c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{8}$$+\frac{\beta \left(5{c}_{1}{}^{2}-6{c}_{1}{c}_{2}+5{c}_{2}{}^{2}\right)+\gamma \left(3{c}_{1}{}^{2}-10{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)}{32}$ |

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\frac{\left(\overline{w}-c\right)\left[a-\left(2\overline{w}+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)\right]}{2}$ | $\left[\frac{[\left(\theta -1\right)\theta}{2\left(\beta +\gamma \right)}+\frac{\beta}{4\left({\beta}^{2}-{\gamma}^{2}\right)}\right]{\alpha}^{2}$$+\frac{\beta -\gamma}{2}{\overline{w}}^{2}-\frac{\alpha -\left({c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{2}\overline{w}$$+\frac{{c}_{1}{}^{2}+{c}_{2}{}^{2}}{4}\beta -\frac{\theta \left({c}_{1}-{c}_{2}\right)+{c}_{2}}{2}\alpha $ | |

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\frac{\left(\overline{w}-c\right)\left[a-\left(2\overline{w}+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)\right]}{2}$ | $\left[\frac{[\left(\theta -1\right)\theta}{2\left(\beta +\gamma \right)}+\frac{\beta}{4\left({\beta}^{2}-{\gamma}^{2}\right)}\right]{\alpha}^{2}$$+\frac{\beta -\gamma}{2}{\overline{w}}^{2}-\frac{\alpha -\left({c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{2}\overline{w}$$+\frac{{c}_{1}{}^{2}+{c}_{2}{}^{2}}{4}\beta -\frac{\theta \left({c}_{1}-{c}_{2}\right)+{c}_{2}}{2}\alpha $ | |

$\overline{w}<{w}^{ps}$ | $\frac{\left(\overline{w}-c\right)\left[a-\left(2\overline{w}+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)\right]}{2}$ | $\left[\frac{[\left(\theta -1\right)\theta}{2\left(\beta +\gamma \right)}+\frac{\beta}{4\left({\beta}^{2}-{\gamma}^{2}\right)}\right]{\alpha}^{2}$$+\frac{\beta -\gamma}{2}{\overline{w}}^{2}-\frac{\alpha -\left({c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{2}\overline{w}$$+\frac{{c}_{1}{}^{2}+{c}_{2}{}^{2}}{4}\beta -\frac{\theta \left({c}_{1}-{c}_{2}\right)+{c}_{2}}{2}\alpha $ | |

PM model | $\overline{w}>{w}^{ms}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{32\left(\beta -\gamma \right)}$ | $\frac{8\left(\beta -\gamma \right){\theta}^{2}-8\left(\beta -\gamma \right)\theta +3\beta -\gamma}{16\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{2c+{c}_{1}\left(4\theta -1\right)+{c}_{2}\left(3-4\theta \right)}{8}\alpha $$+\frac{c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{4}$$+\frac{\beta \left(3{c}_{1}{}^{2}-2{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)+\gamma \left({c}_{1}{}^{2}-6{c}_{1}{c}_{2}+{c}_{2}{}^{2}\right)}{16}$ |

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{32\left(\beta -\gamma \right)}$ | $\frac{8\left(\beta -\gamma \right){\theta}^{2}-8\left(\beta -\gamma \right)\theta +3\beta -\gamma}{16\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{2c+{c}_{1}\left(4\theta -1\right)+{c}_{2}\left(3-4\theta \right)}{8}\alpha $$+\frac{c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{4}$$+\frac{\beta \left(3{c}_{1}{}^{2}-2{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)+\gamma \left({c}_{1}{}^{2}-6{c}_{1}{c}_{2}+{c}_{2}{}^{2}\right)}{16}$ | |

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\frac{8\left(\beta -\gamma \right){\theta}^{2}-8\left(\beta -\gamma \right)\theta +3\beta -\gamma}{16\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{2c+{c}_{1}\left(4\theta -1\right)+{c}_{2}\left(3-4\theta \right)}{8}\alpha $$+\frac{c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{4}$$+\frac{\beta \left(3{c}_{1}{}^{2}-2{c}_{1}{c}_{2}+3{c}_{2}{}^{2}\right)+\gamma \left({c}_{1}{}^{2}-6{c}_{1}{c}_{2}+{c}_{2}{}^{2}\right)}{16}$ | ||

$\overline{w}<{w}^{ps}$ | $\left[\frac{[\left(\theta -1\right)\theta}{2\left(\beta +\gamma \right)}+\frac{\beta}{4\left({\beta}^{2}-{\gamma}^{2}\right)}\right]{\alpha}^{2}+\frac{\beta -\gamma}{2}{\overline{w}}^{2}$$-\frac{\alpha -\left({c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{2}\overline{w}+\frac{{c}_{1}{}^{2}+{c}_{2}{}^{2}}{4}\beta -\frac{\theta \left({c}_{1}-{c}_{2}\right)+{c}_{2}}{2}\alpha $ | ||

VM model | $\overline{w}>{w}^{ms}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{18\left(\beta -\gamma \right)}$ | $\frac{36\left(\beta -\gamma \right){\theta}^{2}-36\left(\beta -\gamma \right)\theta +13\beta -5\gamma}{72\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{8c+{c}_{1}\left(18\theta -5\right)+{c}_{2}\left(13-18\theta \right)}{36}\alpha $$+\frac{2c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{9}$$+\frac{\beta \left(13{c}_{1}{}^{2}-10{c}_{1}{c}_{2}+13{c}_{2}{}^{2}\right)+\gamma \left(5{c}_{1}{}^{2}-26{c}_{1}{c}_{2}+5{c}_{2}{}^{2}\right)}{72}$ |

${w}^{vn}<\overline{w}<{w}^{ms}$ | $\frac{{[\alpha -\left(\beta -\gamma \right)\left(2c+{c}_{1}+{c}_{2}\right)]}^{2}}{18\left(\beta -\gamma \right)}$ | $\frac{36\left(\beta -\gamma \right){\theta}^{2}-36\left(\beta -\gamma \right)\theta +13\beta -5\gamma}{72\left({\beta}^{2}-{\gamma}^{2}\right)}{\alpha}^{2}$$-\frac{8c+{c}_{1}\left(18\theta -5\right)+{c}_{2}\left(13-18\theta \right)}{36}\alpha $$+\frac{2c\left(c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)}{9}$$+\frac{\beta \left(13{c}_{1}{}^{2}-10{c}_{1}{c}_{2}+13{c}_{2}{}^{2}\right)+\gamma \left(5{c}_{1}{}^{2}-26{c}_{1}{c}_{2}+5{c}_{2}{}^{2}\right)}{72}$ | |

${w}^{ps}<\overline{w}<{w}^{vn}$ | $\frac{\left(\overline{w}-c\right)\left[\alpha -\left(2c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)\right]}{3}$ | $\left[\frac{\left(7-6\theta \right)\left(5\beta -\gamma \right)+\left(1+6\theta \right)\left(5\gamma -\beta \right)}{144({\beta}^{2}-{\gamma}^{2})}+\frac{\left(2\theta -1\right)\theta}{4\left(\beta +\gamma \right)}\right]{\alpha}^{2}$$+\frac{\left(4c-7{c}_{1}-{c}_{2}\right)\left(6\theta -1\right)+\left(4c-{c}_{1}-7{c}_{2}\right)\left(5-6\theta \right)}{144}\alpha $$-\frac{\left(4c+5{c}_{1}-{c}_{2}\right)\left(1+6\theta \right)+\left(4c-{c}_{1}+5{c}_{2}\right)\left(7-6\theta \right)}{144}\alpha $$-\frac{\left(4c-7{c}_{1}-{c}_{2}\right)\left(4c-{c}_{1}+5{c}_{2}\right)}{144}\gamma -\frac{\left(4c-{c}_{1}-7{c}_{2}\right)\left(4c+5{c}_{1}-{c}_{2}\right)}{144}\gamma $$-\frac{\left(4c-7{c}_{1}-{c}_{2}\right)\left(4c+5{c}_{1}-{c}_{2}\right)}{144}\beta -\frac{\left(4c-{c}_{1}-7{c}_{2}\right)\left(4c-{c}_{1}+5{c}_{2}\right)}{144}\beta $$-\left[\frac{1}{3}\alpha -\frac{2c+{c}_{1}+{c}_{2}}{3}\left(\beta -\gamma \right)\right]\overline{w}$ | |

$\overline{w}<{w}^{ps}$ | $\frac{\left(\overline{w}-c\right)\left[\alpha -\left(2c+{c}_{1}+{c}_{2}\right)\left(\beta -\gamma \right)\right]}{3}$ | $\left[\frac{\left(7-6\theta \right)\left(5\beta -\gamma \right)+\left(1+6\theta \right)\left(5\gamma -\beta \right)}{144({\beta}^{2}-{\gamma}^{2})}+\frac{\left(2\theta -1\right)\theta}{4\left(\beta +\gamma \right)}\right]{\alpha}^{2}$$+\frac{\left(4c-7{c}_{1}-{c}_{2}\right)\left(6\theta -1\right)+\left(4c-{c}_{1}-7{c}_{2}\right)\left(5-6\theta \right)}{144}\alpha $$-\frac{\left(4c+5{c}_{1}-{c}_{2}\right)\left(1+6\theta \right)+\left(4c-{c}_{1}+5{c}_{2}\right)\left(7-6\theta \right)}{144}\alpha $$-\frac{\left(4c-7{c}_{1}-{c}_{2}\right)\left(4c+5{c}_{1}-{c}_{2}\right)}{144}\beta -\frac{\left(4c-{c}_{1}-7{c}_{2}\right)\left(4c-{c}_{1}+5{c}_{2}\right)}{144}\beta $$-\left[\frac{1}{3}\alpha -\frac{2c+{c}_{1}+{c}_{2}}{3}\left(\beta -\gamma \right)\right]\overline{w}$ |

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## Share and Cite

**MDPI and ACS Style**

Yang, X.; Liu, L.; Zheng, Y.; Yang, X.; Sun, S.
Pricing Problems in the Pharmaceutical Supply Chain with Mixed Channel: A Power Perspective. *Sustainability* **2022**, *14*, 7420.
https://doi.org/10.3390/su14127420

**AMA Style**

Yang X, Liu L, Zheng Y, Yang X, Sun S.
Pricing Problems in the Pharmaceutical Supply Chain with Mixed Channel: A Power Perspective. *Sustainability*. 2022; 14(12):7420.
https://doi.org/10.3390/su14127420

**Chicago/Turabian Style**

Yang, Xiaojie, Li Liu, Yi Zheng, Xue Yang, and Shanlin Sun.
2022. "Pricing Problems in the Pharmaceutical Supply Chain with Mixed Channel: A Power Perspective" *Sustainability* 14, no. 12: 7420.
https://doi.org/10.3390/su14127420