# Pharmaceutical Supply Chain in China: Pricing and Production Decisions with Price-Sensitive and Uncertain Demand

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivations

- How can a mathematical model be developed to depict pricing and production decisions in China’s pharmaceutical supply chain with price-sensitive and uncertain demand?
- How do factors such as price sensitivity, ex-factory cost or governmental discounts influence the optimal decisions of drug retailers?
- Which factors have contributed to the drug shortage problem?
- Why do drugstores sometimes charge much more for a drug than hospitals do?

- Describing the drug pricing competition between drugstores and hospitals in China’s pharmaceutical supply chain.
- Obtaining the optimal strategies of participants in pharmaceutical supply chain and analyzing the influential factors of optimal prices, order quantities, profits and satisfaction rates.
- Identifying the main reasons for drug shortage and price disparity problems in China and providing suggestions for solving the problems.

#### 1.2. Contributions

- Building a pricing model in China’s pharmaceutical supply chain with price-sensitive and uncertain demand considering governmental discount.
- Proving the existence and uniqueness of a pure-strategy Nash equilibrium in the game and deriving the closed form of two sellers’ optimal prices under the assumption of linear uniformly distributed demand.
- Analyzing the impacts of ex-factory price and government discounts on optimal prices and satisfaction rates to obtain insights on the drug shortage problem.
- Analyzing the impacts of price sensitivity on optimal prices, order quantities, expected profits, and satisfaction rates in two special cases of linear demand to provide insights into the price disparity problem.
- Providing suggestions for how the government can act to avoid drug shortage and price disparity problems.

## 2. Literature Review

#### 2.1. Long-Term Decision Problems on Pharmaceutical Supply Chain

#### 2.2. Short-Term Decision Problems on Pharmaceutical Supply Chain

#### 2.3. Mid-Term Decision Problems on Pharmaceutical Supply Chain

#### 2.4. Newvendor Problem with Different Approaches

#### 2.5. Research Gap

## 3. Research Methods

#### 3.1. Problem Description

#### 3.2. Assumptions

**Assumption**

**1.**

- ${\xi}_{d}$ and ${\xi}_{h}$ are independent of ${p}_{d}$ and ${p}_{h}$, furthermore, $E\left[{\xi}_{d}\right]=E\left[{\xi}_{h}\right]=1$. So $E\left[R{D}_{d}\right]={D}_{d}({p}_{d},{p}_{h},{A}_{d})$, $E\left[R{D}_{h}\right]={D}_{h}({p}_{d},{p}_{h},{A}_{h})$.
- ${\xi}_{d}$ and ${\xi}_{h}$ are uniformly distributed on $[1-{\sigma}_{d},1+{\sigma}_{d}]$ and $[1-{\sigma}_{h},1+{\sigma}_{h}]$ respectively, where ${\sigma}_{d}\in [0,1]$ and ${\sigma}_{h}\in [0,1]$.

**Assumption**

**2.**

**Assumption**

**3.**

- ${D}_{d}$ is twice continuous differentiable in ${p}_{d}$ on $[c,{\overline{p}}_{d}]$, so is ${D}_{h}$ in ${p}_{h}$ on $[\varphi c,{\overline{p}}_{h}]$.
- $\frac{\partial {D}_{d}({p}_{d},{p}_{h},{A}_{d})}{\partial {p}_{d}}<0$, $\frac{\partial {D}_{h}({p}_{d},{p}_{h},{A}_{h})}{\partial {p}_{h}}<0$.
- $\frac{\partial {D}_{d}({p}_{d},{p}_{h},{A}_{d})}{\partial {p}_{h}}>0$, $\frac{\partial {D}_{h}({p}_{d},{p}_{h},{A}_{h})}{\partial {p}_{d}}>0$.
- $\frac{\partial {D}_{d}({p}_{d},{p}_{h},{A}_{d})}{\partial {A}_{d}}>0$, $\frac{\partial {D}_{h}({p}_{d},{p}_{h},{A}_{h})}{\partial {A}_{h}}>0$ and ${A}_{h}\u2a7e{A}_{d}$.

**Assumption**

**4.**

- ${e}_{d}=-\frac{\partial {D}_{d}/\partial {p}_{d}}{{D}_{d}/{p}_{d}}$ and ${e}_{h}=-\frac{\partial {D}_{h}/\partial {p}_{h}}{{D}_{h}/{p}_{h}}$ are increasing with ${p}_{d}$ and ${p}_{h}$, respectively. i.e., $\partial {e}_{d}/\partial {p}_{d}>0$ and $\partial {e}_{h}/\partial {p}_{h}>0$.
- ${e}_{d}$ and ${e}_{h}$ are non-increasing with ${p}_{h}$ and ${p}_{d}$ respectively, i.e., $\partial {e}_{d}/\partial {p}_{h}\le 0$ and $\partial {e}_{h}/\partial {p}_{d}\le 0$.

**Assumption**

**5.**

- The linear form:$${D}_{d}={A}_{d}-{a}_{d}{p}_{d}+{b}_{d}{p}_{h},\phantom{\rule{3.33333pt}{0ex}}{D}_{h}={A}_{h}-{a}_{h}{p}_{h}+{b}_{h}{p}_{d}.$$
- The logarithmic form:$${D}_{d}=\frac{{A}_{d}{e}^{-{a}_{d}{p}_{d}}}{{A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}}},\phantom{\rule{3.33333pt}{0ex}}{D}_{h}=\frac{{A}_{h}{e}^{-{a}_{h}{p}_{h}}}{{A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}}}.$$

## 4. Model Formulation and Analysis

#### 4.1. The Optimal Order Quantities of Hospital and Drugstore

**Lemma**

**2.**

#### 4.2. Equilibrium Analysis

#### 4.2.1. Existence of Nash Equilibrium

**Lemma**

**3.**

**Theorem**

**1.**

#### 4.2.2. Uniqueness of Nash Equilibrium

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Theorem**

**2.**

#### 4.3. Pricing Analysis with Linear Demand Functions

**Theorem**

**3.**

- ${s}_{d}^{*}$ is monotonically decreasing with c and increasing with ϕ.
- ${s}_{h}^{*}$ is monotonically decreasing with c and ϕ.

#### 4.3.1. Symmetric Linear Demand

**Theorem**

**4.**

#### 4.3.2. Seller-Reliant Linear Demand

**Theorem**

**5.**

- ${p}_{d}^{*}$ is monotonically decreasing with d and increasing with h, ${p}_{h}^{*}$ is monotonically decreasing with h and increasing with d.
- ${s}_{d}^{*}$ is monotonically decreasing with d and increasing with h, ${s}_{h}^{*}$ is monotonically decreasing with h and increasing with d.

- If $d\u2a7e2h$, that is, in the market, the drugstore has an influence at least twice that of the hospital. We obtain that $(d-2h)(dc+{A}_{d})$ is non-negative and $(2d-h)(h\varphi c+{A}_{h})$ is positive. Thus in this situation, the game will end up with ${p}_{h}^{*}>{p}_{d}^{*}$.
- If $d\u2a7d\frac{h}{2}$, the drugstore has an influence of half or less than half that of the hospital. Under this circumstance. We obtain that $(d-2h)(dc+{A}_{d})$ is non-positive and $(2d-h)(h\varphi c+{A}_{h})$ is negative. Thus, at the equilibrium point, the hospital will have a lower selling price than the drugstore.
- If $\frac{h}{2}<d<2h$, we rewrite Equation (13) as$${p}_{h}^{*}-{p}_{d}^{*}=\frac{c{d}^{2}+({A}_{d}+2{A}_{h}+2h\varphi c-2hc)d-{h}^{2}\varphi c-h{A}_{h}-2h{A}_{d}}{3dh}.$$Define $u\left(d\right)\triangleq c{d}^{2}+({A}_{d}+2{A}_{h}+2h\varphi c-2hc)d-{h}^{2}\varphi c-h{A}_{h}-2h{A}_{d}$. Investigate the sign of Equation (13) is then equivalent to discuss the sign of $u\left(d\right)$ on $(\frac{h}{2},2h)$.Taking the derivative of $u\left(d\right)$ on d, we have ${u}^{\prime}\left(d\right)=2cd+{A}_{d}+2{A}_{h}+2h\varphi c-2hc$.So the minimum point of $h\left(a\right)$ is ${d}^{*}=h-\frac{{A}_{d}+2{A}_{h}+2h\varphi c}{2c}$.As ${d}^{*}<h$, we know by the symmetry of $h\left(d\right)$ that $u\left({d}^{*}\right)\u2a7du\left(\frac{h}{2}\right)<u\left(2h\right)$. In case 1 and 2, we have already shown that $u\left(\frac{h}{2}\right)<0$ and $u\left(2h\right)>0$, there must exist a unique ${d}_{0}$ such that $u\left({d}_{0}\right)=0$. We could obtain ${d}_{0}=\frac{\sqrt{{K}^{2}+4cL}-K}{2c}$ by solving the equation $u\left(d\right)=0$, where $K={A}_{d}+2{A}_{h}+2h\varphi c-2hc$ and $L={h}^{2}\varphi c+h{A}_{h}+2h{A}_{d}$. The expression of ${d}_{0}$ is not important, what we are concerned about is that when $\frac{h}{2}<d<{d}_{0}$, the game will have ${p}_{d}^{*}>{p}_{h}^{*}$ at the equilibrium point, and when ${d}_{0}\u2a7dd<2h$, it will end up with ${p}_{d}^{*}\u2a7d{p}_{h}^{*}$.

## 5. Numerical Analysis

#### 5.1. Symmetric Linear Demand

#### 5.2. Seller-Reliant Linear Demand

## 6. Conclusions

#### 6.1. Implications

#### 6.2. Suggestions for Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Lemmas and Theorems

#### Appendix A.1. Proof of Lemma 1

**Proof**

**of**

**Lemma**

**1.**

_{d}s satisfy Assumptions 3 and 4.

- Linear Form:For Assumption 3, $\frac{\partial {D}_{d}}{\partial {p}_{d}}=-{a}_{d}<0,\phantom{\rule{3.33333pt}{0ex}}\frac{\partial {D}_{d}}{\partial {p}_{h}}={b}_{d}>0,\phantom{\rule{3.33333pt}{0ex}}\frac{\partial {D}_{d}}{\partial {A}_{d}}=1>0$.For Assumption 4, ${e}_{d}=\frac{{a}_{d}{p}_{d}}{{A}_{d}-{a}_{d}{p}_{d}+{b}_{d}{p}_{h}}$, thus $\frac{\partial {e}_{d}}{\partial {p}_{d}}=\frac{{a}_{d}({b}_{d}{p}_{h}+{A}_{d})}{{({A}_{d}-{a}_{d}{p}_{d}+{b}_{d}{p}_{h})}^{2}}>0$ and $\frac{\partial {e}_{d}}{\partial {p}_{h}}=\frac{-{a}_{d}{b}_{h}{p}_{d}}{{({A}_{d}-{a}_{d}{p}_{d}+{b}_{d}{p}_{h})}^{2}}\u2a7d0$.
- Logarithmic Form:For Assumption 3, we have $\frac{\partial {D}_{d}}{\partial {p}_{d}}=-\frac{{a}_{d}{A}_{d}{A}_{h}{e}^{-{a}_{d}{p}_{d}-{a}_{h}{p}_{h}}}{{({A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}^{2}}<0$,$\frac{\partial {D}_{d}}{\partial {p}_{h}}=\frac{{a}_{h}{A}_{d}{A}_{h}{e}^{-{a}_{d}{p}_{d}-{a}_{h}{p}_{h}}}{{({A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}^{2}}>0$ and $\frac{\partial {D}_{d}}{\partial {A}_{d}}=\frac{{A}_{h}{e}^{-{a}_{d}{p}_{d}-{a}_{h}{p}_{h}}}{{({A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}^{2}}>0$.For Assumption 4, we have ${e}_{d}=\frac{{a}_{d}{A}_{h}{p}_{d}{e}^{-{a}_{h}{p}_{h}}}{{A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}}}$, then $\frac{\partial {e}_{d}}{\partial {p}_{d}}=\frac{{a}_{d}{A}_{h}{e}^{-{a}_{h}{p}_{h}}({a}_{d}{A}_{d}{p}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}{{({A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}^{2}}>0$,$\frac{\partial {e}_{d}}{\partial {p}_{h}}=-\frac{{a}_{d}{a}_{h}{A}_{d}{A}_{h}{p}_{d}{e}^{-{a}_{d}{p}_{d}-{a}_{h}{p}_{h}}}{{({A}_{d}{e}^{-{a}_{d}{p}_{d}}+{A}_{h}{e}^{-{a}_{h}{p}_{h}})}^{2}}\u2a7d0$

#### Appendix A.2. Proof of Lemma 2

**Proof**

**of**

**Lemma**

**2.**

#### Appendix A.3. Proof of Lemma 3

**Proof**

**of**

**Lemma**

**3.**

#### Appendix A.4. Proof of Theorem 1

**Proof**

**of**

**Theorem**

**1.**

#### Appendix A.5. Proof of Lemma 4

**Proof**

**of**

**Lemma**

**4.**

#### Appendix A.6. Proof of Lemma 5

**Proof**

**of**

**Lemma**

**5.**

- NecessityAccording to definition, a function $f\left(x\right)$ is called quasi-concave if $f(t{x}_{1}+(1-t){x}_{2})\u2a7e\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$ holds for all $t\in (0,1)$.Assume f is neither monotonic nor first non-decreasing and then non-increasing. It means that $f\left(x\right)$ is either first decreasing and increasing or first non-decreasing and then non-increasing but at last decreasing and increasing again. As $f\left(x\right)$ is twice continuously differentiable, it is equivalent to say ${f}^{\prime}\left(x\right)$ will cross 0 more than once.Let ${a}_{1}$, ${a}_{2}$ be the two adjacent crossing points that satisfy ${f}^{\prime}\left({a}_{i}\right)=0$. There exist a $\delta >0$, for any $0<\epsilon <\delta $, we have ${f}^{\prime}({a}_{i}+\epsilon )\xb7{f}^{\prime}({a}_{i}-\epsilon )<0$, where $i=1,2$. Without loss of generality, we assume ${a}_{1}<{a}_{2}$.
- If $f\left({a}_{1}\right)<f\left({a}_{2}\right)$Because $f\left({a}_{1}\right)=f\left({a}_{2}\right)=0$, we know that there exist a ${\delta}_{1}>0$, such that for all $x\in [{a}_{1}-{\delta}_{1},{a}_{1}]$, $f\left(x\right)$ is strictly monotonic decreasing and all $x\in [{a}_{1},{a}_{1}+{\delta}_{1}]$, $f\left(x\right)$ is strictly monotonic increasing. This means that $f\left(x\right)$ is convex on $[{a}_{1}-{\delta}_{1},{a}_{1}+{\delta}_{1}]$, which contracts the quasi-concavity of $f\left(x\right)$.
- If $f\left({a}_{1}\right)>f\left({a}_{2}\right)$Similarly, we could say that there exist a ${\delta}_{2}>0$, such that $f\left(x\right)$ is convex on $[{a}_{2}-{\delta}_{2},{a}_{2}+{\delta}_{2}]$, which also contracts the quasi-concavity of $f\left(x\right)$.

Thus our assumption is wrong, which means $f\left(x\right)$ is either monotonic or first non-decreasing and then non-increasing. - SufficiencyTake any ${x}_{1}$, ${x}_{2}$, without loss of generality, we assume ${x}_{1}<{x}_{2}$.
- If $f\left(x\right)$ is monotonic increasing, we have $f\left({x}_{1}\right)\u2a7df\left({x}_{2}\right)$, then $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{1}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$, thus $f\left(x\right)$ is quasi-concave.
- If $f\left(x\right)$ is monotonic decreasing, we have $f\left({x}_{1}\right)\u2a7ef\left({x}_{2}\right)$, then $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{2}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$, thus $f\left(x\right)$ is quasi-concave.
- If $f\left(x\right)$ is first non-decreasing and then non-increasing. Let ${x}_{0}$ denote the turning point of $f\left(x\right)$, that is, for $x<{x}_{0}$, $f\left(x\right)$ is non-decreasing and $x>{x}_{0}$, $f\left(x\right)$ is non-increasing.
- (a)
- If ${x}_{1}<{x}_{2}\u2a7d{x}_{0}$, we have $f\left({x}_{1}\right)\u2a7df\left({x}_{2}\right)$, then $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{1}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$.
- (b)
- If ${x}_{0}<{x}_{1}<{x}_{2}$, we have $f\left({x}_{1}\right)\u2a7ef\left({x}_{2}\right)$, then $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{2}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$.
- (c)
- If ${x}_{1}<{x}_{0}<{x}_{2}$, we will discuss the position of ${x}_{3}=t{x}_{1}+(1-t){x}_{2}$.
- If ${x}_{1}<{x}_{3}\u2a7d{x}_{0}<{x}_{2}$, we have $f\left({x}_{3}\right)\u2a7ef\left({x}_{1}\right)$, which means $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{1}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$.
- If ${x}_{1}<{x}_{0}<{x}_{3}<{x}_{2}$, we have $f\left({x}_{3}\right)\u2a7ef\left({x}_{2}\right)$, which means $f(t{x}_{1}+(1-t){x}_{2})\u2a7ef\left({x}_{2}\right)=\mathrm{min}(f\left({x}_{1}\right),f\left({x}_{2}\right))$.

As discussed above, the sufficiency is proved.

#### Appendix A.7. Proof of Lemma 6

**Proof**

**of**

**Lemma**

**6.**

#### Appendix A.8. Proof of Theorem 2

**Proof**

**of**

**Theorem**

**2.**

#### Appendix A.9. Proof of Theorem 3

**Proof**

**of**

**Theorem**

**3.**

#### Appendix A.10. Proof of Theorem 4

**Proof**

**of**

**Theorem**

**4.**

#### Appendix A.11. Proof of Theorem 5

**Proof**

**of**

**Theorem**

**5.**

## Appendix B. Equation Derivations

#### Appendix B.1. The Derivation of Equations (7) and (8)

#### Appendix B.2. The Derivation of Equation (A4)

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Symbol | Description |
---|---|

c | Ex-factory price for the supplier |

$\varphi $ | Discount factor for the hospital, where the hospital gets a marginal cost of $\varphi c$, $\varphi \in (0,1]$ |

${p}_{d}$ | Selling price for the drugstore |

${p}_{h}$ | Selling price for the hospital |

${Q}_{d}$ | Order quantity for the drugstore |

${Q}_{h}$ | Order quantity for the hospital |

${p}_{d}^{*}$ | Optimal selling price for the drugstore |

${p}_{h}^{*}$ | Optimal selling price for the hospital |

${Q}_{d}^{*}$ | Optimal order quantity for the drugstore |

${Q}_{h}^{*}$ | Optimal order quantity for the hospital |

${A}_{d}$ | Reliability factor of the drugstore |

${A}_{h}$ | Reliability factor of the hospital |

${D}_{d}$ | Deterministic part of the drugstore’s demand, also represented as ${D}_{d}({p}_{d},{p}_{h},{A}_{d})$ |

${D}_{h}$ | Deterministic part of the hospital’s demand, also represented as ${D}_{h}({p}_{d},{p}_{h},{A}_{h})$ |

${e}_{d}$ | Price elasticity of the drugstore’s demand |

${e}_{h}$ | Price elasticity of the hospital’s demand |

${\xi}_{d}$ | Random part of the drugstore’s demand |

${f}_{{\xi}_{d}}$ | Density function of ${\xi}_{d}$ |

${F}_{{\xi}_{d}}$ | Cumulative distribution function of ${\xi}_{d}$ |

${\xi}_{h}$ | Random part of the hospital’s demand |

${f}_{{\xi}_{h}}$ | Density function of ${\xi}_{h}$ |

${F}_{{\xi}_{h}}$ | Cumulative distribution function of ${\xi}_{h}$ |

${r}_{d}^{*}\left({p}_{h}\right)$ | Best response function of the drugstore |

${r}_{h}^{*}\left({p}_{d}\right)$ | Best response function of the hospital |

$R{D}_{d}$ | Drugstore’s demand, where $R{D}_{d}={D}_{d}\xb7{\xi}_{d}$ |

$R{D}_{h}$ | Hospital’s demand, where $R{D}_{h}={D}_{h}\xb7{\xi}_{h}$ |

${\prod}_{d}$ | Expected payoff of the drugstore, also represented as ${\prod}_{d}({p}_{d},{p}_{h},{Q}_{d})$ |

${\prod}_{h}$ | Expected payoff of the hospital, also represented as ${\prod}_{h}({p}_{d},{p}_{h},{Q}_{h})$ |

${\prod}_{s}$ | Expected payoff of the supplier, also represented as ${\prod}_{s}\left(c\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Wu, S.; Luo, M.; Zhang, J.; Zhang, D.; Zhang, L.
Pharmaceutical Supply Chain in China: Pricing and Production Decisions with Price-Sensitive and Uncertain Demand. *Sustainability* **2022**, *14*, 7551.
https://doi.org/10.3390/su14137551

**AMA Style**

Wu S, Luo M, Zhang J, Zhang D, Zhang L.
Pharmaceutical Supply Chain in China: Pricing and Production Decisions with Price-Sensitive and Uncertain Demand. *Sustainability*. 2022; 14(13):7551.
https://doi.org/10.3390/su14137551

**Chicago/Turabian Style**

Wu, Suhan, Min Luo, Jingxia Zhang, Daoheng Zhang, and Lianmin Zhang.
2022. "Pharmaceutical Supply Chain in China: Pricing and Production Decisions with Price-Sensitive and Uncertain Demand" *Sustainability* 14, no. 13: 7551.
https://doi.org/10.3390/su14137551