i> A Λ

b) If the hospitals in the region haven’t achieved the power of self-pricing, then the price of ${H}_{A}$ should refer to the government limit and take the government limit P as the optimal price, that is, ${p}_{A}=P$.

2) When the service capacity of the community hospital is large enough ( ${\mu }_{C}\ge \frac{32}{15}\text{Λ}$ ), the optimal pricing strategy of ${H}_{A}$ should be discussed according to the following scenarios:

a) If the hospital can make self-pricing, and when the ${p}_{A}^{\ast }$ could ensure that the two hospitals are neither in loss nor losing their patients, when

$\frac{{c}_{1}{\mu }_{A}+T}{\theta \alpha \text{Λ}}\le {p}_{A}^{\ast }\le {q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$, the price of ${H}_{A}$ should be as follows:

${p}_{A}={p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\text{Λ}}}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)$

b) If the hospital can make self-pricing, and ${p}_{A}^{\ast }$ has gone beyond ${p}_{A}$, namely, ${p}_{A}^{\ast }>{q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$, then the optimal price of ${H}_{A}$ should be the upper limit of the price range, that is,

${p}_{A}={q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$

c) If the hospital cannot make self-pricing, and P goes beyond the upper limit of ${p}_{A}$ ( $P\ge {p}_{A}$ ), then the optimal price of ${H}_{A}$ should be the upper limit of the price range, that is,

${p}_{A}={q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$

d) If the hospital cannot make self-pricing, and P is between ${p}_{A}^{\ast }$ and the upper limit ( ${p}_{A}^{\ast }\le P<{p}_{A}$ ), then the optimal price of ${H}_{A}$ should be the optimal solution of the price, then,

${p}_{A}={p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\text{Λ}}}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)$

e) If the hospital cannot make self-pricing, and P is smaller than ${p}_{A}^{\ast }$ ( $P<{p}_{A}^{\ast }$ ), then the optimal price of ${H}_{A}$ should be the upper limit of P, namely, ${p}_{A}=P$.

We can see from Proposition 2 that it is only when the government subsidy B reaches a certain value ( $B\ge \left({c}_{2}-{p}_{C}\right)\frac{{\mu }_{A}-\text{Λ}}{1-\alpha }$ ), there will be patients coming to ${H}_{C}$. And also in the scenario of a given ${p}_{C}$ of ${H}_{C}$, if $B\ge \left({c}_{2}-{p}_{C}\right)\frac{{\mu }_{A}-\text{Λ}}{1-\alpha }$, then the service capacity of ${H}_{C}$ will be ${\mu }_{C}^{\ast }=\frac{B}{{c}_{2}-{p}_{c}}$. And at this time, ${\lambda }_{C}^{\ast }>0$. Otherwise the community hospital will be out of the market due no patients, ${\lambda }_{C}^{\ast }=0$. Therefore, when $\frac{{\mu }_{A}-\text{Λ}}{1-\alpha }\le {\mu }_{C}<\frac{32}{15}\text{Λ}$, B should satisfies the condition of:

$\left({c}_{2}-{p}_{c}\right)\cdot \frac{{\mu }_{A}-\text{Λ}}{1-\alpha }\le B<\left({c}_{2}-{p}_{c}\right)\cdot \frac{32}{15}\text{Λ}$

When ${\mu }_{C}\ge \frac{32}{15}\text{Λ}$, B should satisfies the condition of

$B\ge \left({c}_{2}-{p}_{c}\right)\cdot \frac{32}{15}\text{Λ}$

Therefore, in the Stackelberg game model in this study, we take ${H}_{A}$ as the leader and ${H}_{C}$ as the follower. When $\text{Λ}<{\mu }_{A}<\left(\frac{47}{15}-\frac{32}{15}\alpha \right)\cdot \text{Λ}$ and $B\ge \left({c}_{2}-{p}_{c}\right)\cdot \frac{{\mu }_{A}-\text{Λ}}{1-\alpha }$, ${H}_{A}$ get the maximum profit by designing an optimal pricing strategy, that is, the best strategy.

Based on the above conclusions, we can also draw the following Lemmas:

Lemma 2. ${\mu }_{C}^{*}$ of ${H}_{C}$ is related to B and ${c}_{2}$, and it increases with the increase of B and decreases with the increase of ${c}_{2}$.

Lemma 2 shows that, for community hospital, when ${c}_{2}$ is constant, the bigger B is, the higher the service capacity. However, when B is constant, the increasing service cost per unit time will reduce its service capacity. Therefore, in order to avoid the decline in service capacity of the community hospital due to the increase in service costs, community hospital should improve service efficiency and other means to control service costs, so as to ensure its service capacity so that there will be enough patients to come for treatment.

Lemma 3. When $\left({c}_{2}-{p}_{c}\right)\cdot \frac{{\mu }_{A}-\text{Λ}}{1-\alpha }\le B<\left({c}_{2}-{p}_{c}\right)\cdot \frac{32}{15}\text{Λ}$, the price of the

tertiary hospital ${p}_{A}$ will not be affected by it. In contrast, it increases with the increase of ${q}_{A}$ and the increase of ${\mu }_{A}$ of the tertiary hospital, and it decreases with the increase of $\Lambda$ in the system.

Lemma 3 shows that when B is comparatively less, the community hospitals are difficult to contend and compete with the tertiary hospital. At this time, the tertiary hospitals have sufficient autonomy to make self-pricing to achieve its profit maximization. Meanwhile, to improve the medical quality and service capacity of the tertiary hospital will increase its pricing level, thereby increasing its profit values. But with the increase in the total number of patients in the system, patients will have greater autonomy, which will also force the tertiary hospital to reduce their optimal pricing, resulting in lower profits.

Lemma 4. When B is comparatively higher and satisfies $B\ge \left({c}_{2}-{p}_{c}\right)\cdot \frac{32}{15}\text{Λ}$,

if the optimal price of the tertiary hospital is ${p}_{A}={p}_{A}^{\ast }$, the optimal price of the medical expenses in the tertiary hospital will decrease with the increase of B.

Lemma 4 shows that when the subsidy B is higher, the service capacity of the community hospital can be improved. In such a case, the community hospital will have the ability to compete with the tertiary hospital. Therefore, at this time, the tertiary hospital’ blind increase in medical expenses will lead to the loss of patients. So they have to lower ${p}_{A}$ to attract more patients.

Lemma 5. ${\lambda }_{C}^{\ast }$ of the community hospital increases with the increase of ${p}_{A}^{\ast }$ of the tertiary hospital.

Lemma 5 shows that when the medical expenses of the tertiary hospital continue to increase, the patients will pay a too high price, which will lead to a lower utility value. And at this time, they will consider visiting the community hospital for the first diagnosis, increasing to the first diagnosis rate of them.

Lemma 6. ${\lambda }_{C}^{\ast }$ of the community hospital increases with the increase of ${q}_{C}$.

Lemma 6 shows that with the improvement of the medical quality of the community hospital, the non-cure rate due to the too low medical quality will decline and more patients begin to favor the convenient community hospital with relatively low medical expenses. Patients who go to community hospital for first diagnosis will receive a higher utility value relative to the tertiary hospital. Meanwhile, it will also force the tertiary hospital to adjust their pricing strategies to better participate in the market competition.

Lemma 7. When patients go to the community hospital or the tertiary hospital for first diagnosis, their waiting time and waiting costs will increase with the reduction of ${q}_{C}$ of the community hospital.

Lemma 7 shows that with the improvement of the medical quality of the community hospital, the non-cure rate will decline. For the patients who go to the community hospital for first diagnosis, the probability of requiring referral will be reduced and the waiting time and waiting costs caused by referral will decrease. Similarly, for patients who go to the tertiary hospital for first diagnosis, their waiting time will be reduced due to the reduction in the number of referral patients so that the waiting costs will also be reduced.

It is not difficult to find that the profit value of the tertiary hospital does not simply increase with the increase of medical treatment fees, but increases first and then decreases. This also reflects that, in actual situations, reasonable adjustments must be made according to market competition, so as to choose the optimal pricing to achieve the maximum benefits. However, according to the state’s relevant regulations on medical pricing, not all hospitals have the power to make independent pricing, and the tertiary hospital should not only consider their own interests, but also weigh the relationship with community hospital. Therefore, its pricing cannot be too low, so that patients in community hospital are lost to the tertiary hospital, which puts greater pressure on the development of medical work in the tertiary hospital

5. Numerical Analysis

5.1. The Influence of the Non-Cure Rate $\alpha$ of the Community Hospital on the Arrival Rate of Patients ${\lambda }_{C}$ of Them

Assume that the model parameters are:

${q}_{a}=180,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{C}=90,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{C}=90,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Lambda =1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}h=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0.5$

We draw the relationship between ${\lambda }_{C}$ and $\alpha$ when ${\mu }_{C}=0.5,0.8,1$, as shown in Figure 2.

We can see from Figure 2 that with the increase of $\alpha$, ${\lambda }_{C}$ is decreasing. In other words, when the medical quality of the community hospital decreases (non-cure rate increases), the patient will choose not to go there for the first diagnosis due to the too low net utility value, which is consistent with routine thinking patterns and judging criteria of the majority of patients.

In addition, when the non-cure rate does not change, the arrival rate of patients will also increase with the improvement of ${\mu }_{C}$, which proves the conclusion of Lemma 1, the ${\lambda }_{C}^{\ast }$ of ${H}_{C}$ will increase with the increase of ${\mu }_{C}$.

5.2. The Impact of Government Subsidy B on Community Hospitals’ Service Ability of ${\mu }_{C}$

Assume that the model parameters are

$\alpha =0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mu }_{A}=1.6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{a}=180,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{C}=80,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{2}=90,\text{\hspace{0.17em}}\text{\hspace{0.17em}}h=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0.5$

Figure 2. The influence of $\alpha$ on ${\lambda }_{C}$.

We draw the relationship between ${\mu }_{C}$ and B when ${p}_{C}=60,70,100$ respectively, as shown in Figure 3.

We can see from Figure 3, when the medical expenses of the community hospital are fixed, the service capacity changes with the change of government subsidies. From the foregoing assumptions, we can see that community hospital as an important part of primary medical institutions, the medical expenses are comparatively lower. They often need government subsidies to ensure that they will not be out of the competitive market because of losses. Therefore, in most cases, ${p}_{C}$ is lower than ${c}_{2}$. We can easily find through numerical analysis that when ${p}_{C}$ is lower than ${c}_{2}$, the service capacity of community hospital increases with the increase of government subsidy. When ${p}_{C}$ is higher than ${c}_{2}$, the service capacity of community hospital decreases with the increase of government subsidy. And through Figure 1 and the analysis, we can also see that the service capacity of community hospital will affect the first diagnosis rate. Therefore, we can easily infer that the government subsidy to community hospital not only will affect their service capacity, but also their first diagnosis rate. In other words, it will affect the patients’ treatment options, thus affecting the implementation of hierarchical diagnosis and treatment.

5.3. The Effect of the Total Amount of Patients $\Lambda$ on System Performance When Government Subsidy B Is Comparatively Less

Assume that the model parameters are

$\alpha =0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mu }_{A}=1.6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{a}=180,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{C}=80,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{C}=90$

$h=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{1}=1.2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{2}=1.1$

We draw respectively the relationship between $\Lambda$ and ${p}^{*}$ as well as that between $\Lambda$ and $\pi$, when B is comparatively less. As shown in Figure 4(a) and Figure 4(b).

Figure 3. The impact of B on community hospital’ service ability of ${\mu }_{C}$.

(a) (b)

Figure 4. (a) The effect of $\Lambda$ on ${p}^{\ast }$ when the government subsidy is comparatively less; (b) The effect of $\Lambda$ on $\pi$ when the government subsidy is comparatively less.

When B is comparatively less, the service capacity of community hospital will be comparatively smaller and the patient arrival rate will be comparatively lower. At this time, most patients will choose to go to tertiary hospital for first diagnosis. We can see from Figure 4(a) that when B is comparatively less, the total arrival quantity of patients will increase and the optimal price of the tertiary hospital shows a decreasing trend. It decreases gradually in $\left(0,{\Lambda }_{0}\right]$ and rapidly in $\left[{\Lambda }_{0},\Lambda \right)$. That is to say, when the total patient volume in the system gradually increases to ${\text{Λ}}_{0}$, the optimal price of the tertiary hospital decreases gradually. But because the number of patients at this time is showing a growth trend and the price decline is slow, so in general, the profit of the tertiary hospital still shows a state of growth. However, when the total amount of patients is more than ${\text{Λ}}_{0}$, the patients will choose to go directly to the community hospital for they will wait too long time if going to the tertiary hospital for first diagnosis. At this time, due to the sudden leaving of the patients, the price of the tertiary hospital will decline greatly. Then the first diagnosis rate and the price decrease at the same time, the profit of the tertiary hospital will also have a “diving” decrease.

5.4. The Effect of the Total Amount of Patients $\Lambda$ on System Performance When Government Subsidy B Is Comparatively More

Assume that the model parameters are

$\alpha =0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mu }_{A}=1.6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{a}=180,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{C}=80,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{C}=90$

$h=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{1}=1.2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{2}=1.1$

We draw respectively the relationship between $\Lambda$ and ${p}^{*}$ as well as that between $\Lambda$ and $\pi$, when B is comparatively more. As shown in Figure 5(a) and Figure 5(b).

When the government subsidy B is comparatively more, the service capacity of community hospital will be comparatively larger and the patient arrival rate will be comparatively higher. At this time, most patients will choose to go to community hospital for first diagnosis.

We can see from Figure 5(a) that when B is comparatively more, although the optimal price of the tertiary hospital is higher, the profit value will also be at a low level as fewer patients come for the first diagnosis. With the increase in the overall arrival rate of patients, the optimal price will show a slight decline in the trend, and at this time, with the increase in the number of patients, the profit value will be improved. However, when the total amount of patients reaches ${\Lambda }_{0}$, due to service capacity constraints of community hospital, the patients have to wait for too long and the waiting cost increases, the patient begin to consider going to the tertiary hospital for the first diagnosis. At this time, to attract the patients waiting in community hospital, the tertiary hospital must lower the price. So the optimal price of the tertiary hospital at this time will suffer a substantial “diving” decrease and the total profit will reduce significantly due to the price reduction. When the total number of patients is getting bigger and bigger, the arrival rate of patients in the tertiary hospital is also growing rapidly. At this time, in order to coordinate the community hospital and the tertiary hospital on the arrival rate of

(a) (b)

Figure 5. (a) The effect of $\Lambda$ on ${p}^{\ast }$ when the government subsidy is comparatively more; (b) The effect of $\Lambda$ on $\pi$ when the government subsidy is comparatively more.

patients, the tertiary hospital must raise prices to force the patients to choose community hospital for first diagnosis. But because of the impact of the total amount of patients, the profit will still show a dramatic increase.

6. Conclusions

In this paper, the two influencing factors of medical quality and waiting cost that have been focused more are selected as the research object. We combine with the existing related research of the domestic and foreign scholars and build a two-way referral system taking a tertiary hospital and a community hospital as the main body. We study the pricing of the tertiary hospital and the service capacity design issues of the community hospital through theoretical research and numerical simulation. The main work and conclusions are as follows.

1) We construct a utility function model of the patients’ first diagnosis to the community hospital and the tertiary hospital. We compare the utility values of the two and determine what the first choices of the patients are to provide reference for community hospital and tertiary hospital on attracting patients better and improving the medical and health level comprehensively.

2) We analyze how community hospital can set their own service capabilities to accommodate and serve more patients and improving the patients’ utility values.

3) We construct the profit function model of the tertiary hospital, analyze the nature of the function and obtain the maximum profit in what kind of pricing scenario.

4) We take into account the effect of government subsidies on the system performance and analyze the operation conditions of the system under different subsidies

However, in this study, we conduct the research based on assumptions in many places. However, in reality, the situations are often a lot more complicated. Therefore, there are some shortcomings in this paper needed to be discussed and studied in the future research. Some patients feel that medical quality is more important than waiting cost, and vice versa. Therefore, in the future study, we can divide the two with a certain balance. In this paper, we only consider the referral to the upper hospitals, not to the lower ones. We only consider the two-level system, not the three-level or multi-level systems. We can explore the more complex patterns in future research.

Acknowledgements

This research was supported by the National Natural Science Foundation of China under Grants No. 71572154, the Fundamental Research Funds for the Central Universities under Grants No. 26816WTD25, and the Service Science and Innovation Key Laboratory of Sichuan Province under Grants No. KL1705.

Proof

Proposition 1.

The balanced arrival rate of patients in the tertiary hospital ${H}_{A}$ and community hospital ${H}_{C}$ ( ${\lambda }_{A}^{\ast }$, ${\lambda }_{C}^{\ast }$ ) satisfies the following formula:

$\begin{array}{l}{q}_{A}-{p}_{A}-\frac{h}{{\mu }_{A}-\alpha {\lambda }_{C}-{\lambda }_{A}}-\frac{\alpha }{1-\alpha }{p}_{A}\left(1-\theta \right)\\ ={q}_{C}-{p}_{C}-\frac{h}{{\mu }_{C}-{\lambda }_{C}}-\frac{\alpha }{1-\alpha }\cdot \frac{h}{{\mu }_{C}-{\lambda }_{C}}\end{array}$ (1-1)

To further simplify the formula, assuming that:

${A}_{0}=\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)+\left(1-\alpha \theta \right){p}_{A}$

${A}_{1}=\left(1-\alpha \right)h$

${A}_{2}={\mu }_{A}-\text{Λ}$

Then:

${A}_{0}+\frac{{A}_{1}}{{A}_{2}+\left(1-\alpha \right){\lambda }_{C}^{\ast }}-\frac{{A}_{1}}{\left(1-\alpha \right)\left({\mu }_{C}-{\lambda }_{C}^{\ast }\right)}=0$ (1-2)

Then:

$\begin{array}{l}{A}_{0}\left(1-\alpha \right){\lambda }_{C}^{\ast 2}+\left[{A}_{0}{A}_{2}+2{A}_{1}-{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]{\lambda }_{C}^{\ast }\\ \text{ }+\left(\frac{{A}_{1}{A}_{2}}{1-\alpha }-{A}_{0}{A}_{2}{\mu }_{c}-{A}_{1}{\mu }_{C}\right)=0\end{array}$ (1-3)

${\lambda }_{C}^{\ast }=\frac{{A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2{A}_{0}\left(1-\alpha \right)}$

In order to make ${\lambda }_{C}^{\ast }\ge 0$, then:

${A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}\ge 0$ (1-4)

$\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}\ge 2{A}_{1}$

Therefore, it only needs to satisfy ${A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}\ge 0$

That is: when ${\mu }_{C}\ge \frac{{\mu }_{A}-\text{Λ}}{1-\alpha }$, the Formula (1-4) holds, otherwise ${\lambda }_{C}^{\ast }=0$, at this time:

${\lambda }_{A}^{\ast }=\text{Λ}-\frac{{A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2{A}_{0}\left(1-\alpha \right)}$

${\lambda }_{C}^{\ast }=\frac{{A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2{A}_{0}\left(1-\alpha \right)}$

Proposition 2.

$\pi \left({p}_{A}\right)={p}_{A}{\lambda }_{A}+\theta {p}_{A}\alpha {\lambda }_{C}-{c}_{1}{\mu }_{A}-T$ (2-1)

${U}_{A}={q}_{A}-{p}_{A}-\frac{h}{{\mu }_{A}-\alpha {\lambda }_{C}-{\lambda }_{A}}$

When ${U}_{A}\ge 0$, the patient will choose the first visit to a tertiary hospital, then:

${p}_{A}\le {q}_{A}-\frac{h}{{\mu }_{A}-\alpha {\lambda }_{C}-{\lambda }_{A}}$ (2-2)

When the first consultation of all patients in the system chooses to go directly to the t tertiary hospital, the revenue of the tertiary hospital will reach the maximum value. At this time, the pricing of the tertiary hospital can also be intentionally lowered because there is no need to worry about no one coming. In other words, ${p}_{A}$ can take the maximum value. Due to ${\lambda }_{C}=0$, ${\lambda }_{A}=\text{Λ}$, then, (2-2) becomes:

${p}_{A}\le {q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$ (2-3)

If you want the hospital to maintain normal operation without loss, the minimum income must be greater than or equal to the total cost of the hospital, that is, the sum of service costs and taxes, can be expressed as:

$\theta {p}_{A}\alpha \text{Λ}\ge {c}_{1}{\mu }_{A}+T$ (2-4)

Then:

${p}_{A}\ge \frac{{c}_{1}{\mu }_{A}+T}{\theta \alpha \text{Λ}}$ (2-5)

Therefore, by combining (2-3) and (2-5), the value range of ${p}_{A}$ can be obtained:

$\frac{{c}_{1}{\mu }_{A}+T}{\theta \alpha \text{Λ}}\le {p}_{A}\le {q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$ (2-6)

Therefore, the decision model of the tertiary hospital can be expressed as:

${\mathrm{max}}_{{p}_{A}\ge 0}\pi \left({p}_{A}\right)={p}_{A}{\lambda }_{A}+\theta {p}_{A}\alpha {\lambda }_{C}-{c}_{1}{\mu }_{A}-T$ (2-7)

$s.t.\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{c}_{1}{\mu }_{A}+T}{\theta \alpha \text{Λ}}\le {p}_{A}\le {q}_{A}-\frac{h}{{\mu }_{A}-\text{Λ}}$ (2-8)

Formula (2-7) can be simplified as:

$\pi \left({p}_{A}\right)=\text{Λ}{p}_{A}-\left(1-\alpha \theta \right){\lambda }_{C}{p}_{A}-{c}_{1}{\mu }_{A}-T$ (2-9)

${\lambda }_{C}^{\ast }=\frac{{A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2{A}_{0}\left(1-\alpha \right)}$

${A}_{0}=\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)+\left(1-\alpha \theta \right){p}_{A}$

Let:

$k=\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)$

(2-9) becomes:

$\begin{array}{c}\pi \left({p}_{A}\right)=\Lambda {p}_{A}-\frac{1-\alpha \theta }{2}{p}_{A}{\mu }_{C}+\frac{1-\alpha \theta }{2}\cdot \frac{{A}_{2}{p}_{A}}{1-\alpha }+\frac{h\left(1-\alpha \theta \right)}{k+\left(1-\alpha \theta \right){p}_{A}}\cdot {p}_{A}-{c}_{1}{\mu }_{A}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-T-\frac{\left(1-\alpha \theta \right){p}_{A}\sqrt{{\left[k+\left(1-\alpha \theta \right){p}_{A}\right]}^{2}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2\left(1-\alpha \right)\left[k+\left(1-\alpha \theta \right){p}_{A}\right]}\end{array}$ (2-10)

Let:

${A}_{3}=\frac{1-\alpha \theta }{2}$

Then:

$\begin{array}{c}\pi \left({p}_{A}\right)=\text{Λ}{p}_{A}-{A}_{3}{\mu }_{C}{p}_{A}+\frac{{A}_{2}{A}_{3}}{1-\alpha }{p}_{A}+\frac{2{A}_{3}h}{k+2{A}_{3}{p}_{A}}\cdot {p}_{A}-{c}_{1}{\mu }_{A}-T\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{A}_{3}{p}_{A}\sqrt{\left({k}^{2}+4{A}_{3}^{2}{p}_{A}^{2}+4{A}_{3}k{p}_{A}\right){\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{\left(1-\alpha \right)\left(k+2{A}_{3}{p}_{A}\right)}\end{array}$ (2-11)

$\begin{array}{c}\frac{\partial \pi }{\partial {p}_{A}}=\text{Λ}-{A}_{3}{\mu }_{C}+\frac{{A}_{2}{A}_{3}}{1-\alpha }+{\left(\frac{2{A}_{3}h}{k+2{A}_{3}{p}_{A}}\cdot {p}_{A}\right)}^{\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left\{\frac{{A}_{3}{p}_{A}\sqrt{\left({k}^{2}+4{A}_{3}^{2}{p}_{A}^{2}+4{A}_{3}k{p}_{A}\right){\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{3}}}{\left(1-\alpha \right)\left(k+2{A}_{3}{p}_{A}\right)}\right\}\end{array}$

Let:

$m=\frac{2{A}_{3}h}{k+2{A}_{3}{p}_{A}}\cdot {p}_{A}$

$n=\frac{{A}_{3}{p}_{A}\sqrt{\left({k}^{2}+4{A}_{3}^{2}{p}_{A}^{2}+4{A}_{3}k{p}_{A}\right){\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{\left(1-\alpha \right)\left(k+2{A}_{3}{p}_{A}\right)}$

Then:

$\frac{\partial m}{\partial {p}_{A}}={\left(2{A}_{3}h\cdot \frac{{p}_{A}}{k+2{A}_{3}{p}_{A}}\right)}^{\prime }=2{A}_{3}h\cdot \frac{k+2{A}_{3}{p}_{A}-{p}_{A}2{A}_{3}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}=\frac{2{A}_{3}hk}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}$

Let:

$\sqrt{G}=\sqrt{\left({k}^{2}+4{A}_{3}^{2}{p}_{A}^{2}+4{A}_{3}k{p}_{A}\right){\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}$

$\begin{array}{l}\frac{\partial n}{\partial {p}_{A}}=\frac{{A}_{3}}{1-\alpha }\cdot \frac{{\left({p}_{A}\sqrt{G}\right)}^{\prime }\left(k+2{A}_{3}{p}_{A}\right)-\left({p}_{A}\sqrt{G}\right)\cdot 2{A}_{3}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ =\frac{{A}_{3}}{1-\alpha }\cdot \frac{\left[\sqrt{G}+{p}_{A}{\left(\sqrt{G}\right)}^{\prime }\right]\cdot \left(k+2{A}_{3}{p}_{A}\right)-2{A}_{3}{p}_{A}\sqrt{G}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ =\frac{{A}_{3}}{1-\alpha }\cdot \frac{k\sqrt{G}+{p}_{A}{\left(\sqrt{G}\right)}^{\prime }\cdot \left(k+2{A}_{3}{p}_{A}\right)}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ =\frac{{A}_{3}}{1-\alpha }\cdot \frac{k\sqrt{G}+\left(k+2{A}_{3}{p}_{A}\right)\cdot {p}_{A}\cdot \frac{1}{2\sqrt{G}}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}\cdot \left(8{A}_{3}^{2}{p}_{A}+4{A}_{3}k\right)}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ =\frac{{A}_{3}}{1-\alpha }\cdot \left\{\frac{k\sqrt{G}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}+\frac{{p}_{A}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}\cdot \left(8{A}_{3}^{2}{p}_{A}+4{A}_{3}k\right)}{2\left(k+2{A}_{3}{p}_{A}\right)\sqrt{G}}\right\}\end{array}$

Then:

$\begin{array}{c}\frac{\partial \pi }{\partial {p}_{A}}=\Lambda -{A}_{3}{\mu }_{C}+\frac{{A}_{2}{A}_{3}}{1-\alpha }+\frac{2{A}_{3}hk}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{ }-\frac{{A}_{3}}{1-\alpha }\cdot \left\{\frac{k\sqrt{G}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}+\frac{{p}_{A}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}\cdot \left(8{A}_{3}^{2}{p}_{A}+4{A}_{3}k\right)}{2\left(k+2{A}_{3}{p}_{A}\right)\sqrt{G}}\right\}\end{array}$ (2-12)

Further derivate the Formula (2-12).

Let:

$M=\frac{\partial m}{\partial {p}_{A}}=\frac{2{A}_{3}hk}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}$

$N=\frac{\partial n}{\partial {p}_{A}}=\frac{{A}_{3}}{1-\alpha }\cdot \left\{\frac{k\sqrt{G}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}+\frac{{p}_{A}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}\cdot \left(8{A}_{3}^{2}{p}_{A}+4{A}_{3}k\right)}{2\left(k+2{A}_{3}{p}_{A}\right)\sqrt{G}}\right\}$

Then:

$\frac{{\partial }^{2}\pi }{{\partial }^{2}{p}_{A}}={M}^{\prime }-{N}^{\prime }$

And:

${M}^{\prime }=\frac{\partial M}{\partial {p}_{A}}=\frac{-2{A}_{3}hk\left(8{A}_{3}k+8{A}_{3}^{2}\right)}{{\left(k+2{A}_{3}{p}_{A}\right)}^{4}}<0$

Then:

$\frac{{\partial }^{2}\pi }{{\partial }^{2}{p}_{A}}={M}^{\prime }-{N}^{\prime }<0$

Therefore, according to the nature of the function, $\pi \left({p}_{A}\right)$ is a concave function about ${p}_{A}$.

Proposition 3 & Proposition 4

$\begin{array}{c}\frac{\partial \pi }{\partial {p}_{A}}=\Lambda -{A}_{3}{\mu }_{C}+\frac{{A}_{2}{A}_{3}}{1-\alpha }+\frac{2{A}_{3}hk}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{A}_{3}}{1-\alpha }\cdot \left\{\frac{k\sqrt{G}}{{\left(k+2{A}_{3}{p}_{A}\right)}^{2}}+\frac{{p}_{A}\cdot {\left[{A}_{2}+{\mu }_{C}\left(1-\alpha \right)\right]}^{2}\cdot \left(8{A}_{3}^{2}{p}_{A}+4{A}_{3}k\right)}{2\left(k+2{A}_{3}{p}_{A}\right)\sqrt{G}}\right\}\\ =0\end{array}$

We can find a zero boundary point of the function.

Because there are too many variables and parameters involved in this formula, we will optimize the solution during the calculation, otherwise it will affect the final profit function because the formula is too complicated.

Let:

$\alpha =0.5$

$\theta =0.5$

$h=1$

Then:

${A}_{0}=\frac{1}{2}\left({q}_{C}-{q}_{A}-{p}_{C}\right)+\frac{3}{4}{p}_{A}$

${A}_{1}=\frac{1}{2}$

${A}_{2}={\mu }_{A}-\text{Λ}$

${A}_{3}=\frac{3}{8}$

$k={k}_{1}=\frac{1}{2}\left({q}_{C}-{q}_{A}-{p}_{C}\right)$

Then:

$\begin{array}{c}\pi \left({p}_{A}\right)=\Lambda {p}_{A}-\frac{3}{8}{\mu }_{C}{p}_{A}+\frac{3}{16}\left({\mu }_{A}-\Lambda \right){p}_{A}+\frac{3}{4}\cdot \frac{{p}_{A}}{{k}_{1}+\frac{3}{4}{p}_{A}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{3}{16}\cdot {p}_{A}\cdot \frac{\sqrt{{\left({k}_{1}+\frac{3}{4}{p}_{A}\right)}^{2}{\left[{\mu }_{A}-\Lambda +\frac{1}{2}{\mu }_{C}\right]}^{2}+1}}{{k}_{1}+\frac{3}{4}{p}_{A}}-{c}_{1}{\mu }_{A}-T\end{array}$

Let:

$\sqrt{{G}_{1}}=\sqrt{{\left({k}_{1}+\frac{3}{4}{p}_{A}\right)}^{2}{\left[{\mu }_{A}-\Lambda +\frac{1}{2}{\mu }_{C}\right]}^{2}+1}\approx \left({k}_{1}+\frac{3}{4}{p}_{A}\right)\cdot \left({\mu }_{A}-\Lambda +\frac{1}{2}{\mu }_{C}\right)$

Then:

$\begin{array}{c}\frac{\partial \pi }{\partial {p}_{A}}=\Lambda -\frac{3}{8}{\mu }_{C}+\frac{3}{16}\left({\mu }_{A}-\Lambda \right)+\frac{\frac{3}{4}k}{{\left({k}_{1}+\frac{3}{4}{p}_{A}\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{3}{16}\left[\frac{{k}_{1}\sqrt{{G}_{1}}}{{\left({k}_{1}+\frac{3}{4}{p}_{A}\right)}^{2}}+\frac{{p}_{A}{\left({\mu }_{A}-\Lambda +\frac{1}{2}{\mu }_{C}\right)}^{2}\cdot \frac{3}{2}\left({k}_{1}+\frac{3}{4}{p}_{A}\right)}{2\left({k}_{1}+\frac{3}{4}{p}_{A}\right)\sqrt{{G}_{1}}}\right]\end{array}$

When:

${k}_{2}={\mu }_{A}-\Lambda +\frac{1}{2}{\mu }_{C}$

$\beta ={k}_{1}+\frac{3}{4}{p}_{A}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({p}_{A}=\frac{4}{3}\left(\beta -{k}_{1}\right)\right)$

Then:

$\frac{\partial \pi }{\partial {p}_{A}}=\Lambda -\frac{3}{8}{\mu }_{C}+\frac{3}{16}\left({\mu }_{A}-\Lambda \right)+\frac{3}{4}\frac{{k}_{1}}{{\beta }^{2}}-\frac{3}{16}\cdot \frac{{k}_{1}{k}_{2}}{\beta }-\frac{3}{16}\cdot \left(\beta -{k}_{1}\right)\cdot \frac{{k}_{2}}{\beta }$

If $\frac{\partial \pi }{\partial {p}_{A}}=0$, then:

$\Lambda -\frac{3}{8}{\mu }_{C}+\frac{3}{16}\left({\mu }_{A}-\Lambda \right)+\frac{3}{4}\frac{{k}_{1}}{{\beta }^{2}}-\frac{3}{16}\cdot \frac{{k}_{1}{k}_{2}}{\beta }-\frac{3}{16}\cdot \left(\beta -{k}_{1}\right)\cdot \frac{{k}_{2}}{\beta }=0$

And:

${\beta }^{*}=\sqrt{\frac{12{k}_{1}}{6{\mu }_{C}-13\text{Λ}-3{\mu }_{A}+3{k}_{2}}}=\sqrt{\frac{12{k}_{1}}{\frac{15}{2}{\mu }_{C}-16\text{Λ}}}=\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\text{Λ}}}$

Because of ${p}_{A}=\frac{4}{3}\left(\beta -{k}_{1}\right)$

${p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\Lambda }}-{k}_{1}\right)$

Then:

$\begin{array}{c}{\lambda }_{C}^{\ast }=\frac{{A}_{0}{\mu }_{C}\left(1-\alpha \right)-{A}_{0}{A}_{2}-2{A}_{1}+\sqrt{{\left[{A}_{0}{A}_{2}+{A}_{0}{\mu }_{C}\left(1-\alpha \right)\right]}^{2}+4{A}_{1}^{2}}}{2{A}_{0}\left(1-\alpha \right)}\\ \approx {\mu }_{C}-\frac{1}{{A}_{0}}={\mu }_{C}-\frac{1}{{k}_{1}+\frac{3}{4}{p}_{A}^{\ast }}\end{array}$

If ${p}_{A}=\frac{4}{3}\cdot \left(\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\text{Λ}}}-{k}_{1}\right)$

$\begin{array}{c}\pi \left({p}_{A}^{\ast }\right)={p}_{A}^{\ast }{\lambda }_{A}+\theta {p}_{A}^{\ast }\alpha {\lambda }_{C}^{\ast }-{c}_{1}{\mu }_{A}-T\\ ={p}_{A}^{\ast }\left(\Lambda -\frac{3}{4}{\lambda }_{C}^{\ast }\right)-{c}_{1}{\mu }_{A}-T\\ ={p}_{A}^{\ast }\left(\Lambda -\frac{3}{4}{\mu }_{C}+\frac{3}{4{k}_{1}+3{p}_{A}^{\ast }}\right)-{c}_{1}{\mu }_{A}-T\\ =\left(\Lambda -\frac{3}{4}{\mu }_{C}\right)\left(\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\Lambda }}-{k}_{1}\right)+\frac{3\left(\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\Lambda }}-{k}_{1}\right)}{{k}_{1}+3\sqrt{\frac{24{k}_{1}}{15{\mu }_{C}-32\Lambda }}}-{c}_{1}{\mu }_{A}-T\end{array}$

It can be known from the above solution process that the value of ${k}_{1}$ changes with the change of $\alpha$. Therefore, when ${k}_{1}$ is reduced to k and when $\frac{\partial \pi }{\partial {p}_{A}}=0$, ${p}_{A}^{\ast }$ is:

${p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\text{Λ}}}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)$

Then:

$\begin{array}{c}\pi \left({p}_{A}^{\ast }\right)=\left(\Lambda -\frac{3}{4}{\mu }_{C}\right)\left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{3\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}-3\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)+3\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}}-{c}_{1}{\mu }_{A}-T\end{array}$

To ensure that ${p}_{A}^{\ast }$ exists, it must satisfy:

$\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }\ge 0$

The previous article has been obtained, $0\le \alpha \le 1$, ${q}_{C}\le {q}_{A}$, ${q}_{C}-{q}_{A}-{p}_{C}<0$, therefore, to satisfy (5-18), then $15{\mu }_{C}-32\Lambda <0$, that is:

${\mu }_{C}<\frac{32}{15}\Lambda$

If and only if ${\mu }_{C}\ge \frac{{\mu }_{A}-\Lambda }{1-\alpha }$, ${H}_{C}$ will have patients come to the clinic:

$\frac{{\mu }_{A}-\Lambda }{1-\alpha }<\frac{32}{15}\Lambda$

$\Lambda <{\mu }_{A}<\left(\frac{47}{15}-\frac{32}{15}\alpha \right)\cdot \Lambda ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\mu }_{A}-\Lambda }{1-\alpha }\le {\mu }_{C}<\frac{32}{15}\Lambda$

$\frac{{c}_{1}{\mu }_{A}+T}{\theta \alpha \Lambda }\le {p}_{A}\le {q}_{A}-\frac{h}{{\mu }_{A}-\Lambda }$

Lemma 2.

When $B\ge \left({c}_{2}-{p}_{C}\right)\frac{{\mu }_{A}-\text{Λ}}{1-\alpha }$, the optimal service capability of ${H}_{C}$ is:

${\mu }_{C}^{\ast }=\frac{B}{{c}_{2}-{p}_{c}}$

Then:

$\frac{\partial {\mu }_{C}^{\ast }}{\partial B}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {\mu }_{C}^{\ast }}{\partial {c}_{2}}<0$

Lemma 3.

When

$\frac{{\mu }_{A}-\Lambda }{1-\alpha }\le {\mu }_{C}<\frac{32}{15}\Lambda$,

$\left({c}_{2}-{p}_{c}\right)\cdot \frac{{\mu }_{A}-\Lambda }{1-\alpha }\le B<\left({c}_{2}-{p}_{c}\right)\cdot \frac{32}{15}\Lambda$

${p}_{A}={q}_{A}-\frac{h}{{\mu }_{A}-\Lambda }$

Then:

$\frac{\partial {p}_{A}}{\partial {q}_{A}}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {p}_{A}}{\partial {\mu }_{A}}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {p}_{A}}{\partial \Lambda }<0$

Lemma 4.

When ${p}_{A}={p}_{A}^{\ast }$

${p}_{A}={p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}^{\ast }-32\text{Λ}}}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)$

$\frac{\partial {p}_{A}^{\ast }}{\partial {\mu }_{C}^{\ast }}=-10{\mu }_{C}^{-\frac{3}{2}}<0$

$\frac{\partial {\mu }_{C}^{\ast }}{\partial B}>0$

Then:

$\frac{\partial {p}_{A}^{\ast }}{\partial B}=\frac{\partial {p}_{A}^{\ast }}{\partial {\mu }_{C}^{\ast }}\cdot \frac{\partial {\mu }_{C}^{\ast }}{\partial B}<0$

Lemma 5.

Because

${\lambda }_{C}^{\ast }={\mu }_{C}-\frac{1}{k+\frac{3}{4}{p}_{A}^{\ast }}$

We can find that:

$\frac{\partial {\lambda }_{C}^{\ast }}{\partial {p}_{A}^{\ast }}>0$

Lemma 6.

${\lambda }_{C}^{\ast }={\mu }_{C}-\frac{1}{k+\frac{3}{4}{p}_{A}^{\ast }}$

Then

${p}_{A}^{\ast }=\frac{4}{3}\cdot \left(\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}-\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)\right)$

${\lambda }_{C}^{\ast }={\mu }_{C}-\frac{1}{\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}}$

$\frac{\partial {\lambda }_{C}^{\ast }}{\partial \alpha }<0$

Therefore, ${\lambda }_{C}^{\ast }$ decreases as $\alpha$ increases.

Due to

$\alpha =1-\frac{{q}_{C}}{{q}_{A}}$,

$\frac{\partial \alpha }{\partial {q}_{C}}<0$

Then:

$\frac{\partial {\lambda }_{C}^{\ast }}{\partial {q}_{C}}=\frac{\partial {\lambda }_{C}^{\ast }}{\partial \alpha }\cdot \frac{\partial \alpha }{\partial {q}_{C}}>0$

Lemma 7.

The waiting costs for patients are:

${h}_{A}=\frac{h}{{\mu }_{A}-\left(1-\frac{{q}_{C}}{{q}_{A}}\right){\lambda }_{C}-{\lambda }_{A}}$

${h}_{C}=\frac{h}{{\mu }_{C}-{\lambda }_{C}}+\left(1-\frac{{q}_{C}}{{q}_{A}}\right)\cdot \frac{h}{{\mu }_{A}-\left(1-\frac{{q}_{C}}{{q}_{A}}\right){\lambda }_{C}-{\lambda }_{A}}$

${\lambda }_{C}+{\lambda }_{A}=1$

Then:

${h}_{A}=\frac{h}{{\mu }_{A}-1+\left(1-\alpha \right)\cdot \left[{\mu }_{C}-\frac{1}{\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}}\right]}$

$\begin{array}{c}{h}_{C}=h\cdot \sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\alpha \cdot \frac{h}{{\mu }_{A}-1+\left(1-\alpha \right)\cdot \left[{\mu }_{C}-\frac{1}{\sqrt{\frac{24\left(1-\alpha \right)\left({q}_{C}-{q}_{A}-{p}_{C}\right)}{15{\mu }_{C}-32\Lambda }}}\right]}\end{array}$

Can be calculated by:

$\frac{\partial {h}_{A}}{\partial \alpha }>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {h}_{C}}{\partial \alpha }>0$

Due to $\frac{\partial \alpha }{\partial {q}_{C}}<0$, then:

$\frac{\partial {h}_{A}}{\partial {q}_{C}}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {h}_{C}}{\partial {q}_{C}}<0$

Cite this paper
Guan, Z. , Chen, H. , Zhao, N. and Zhang, A. (2020) Pricing and Capability Planning of the Referral System Considering Medical Quality and Delay-Sensitive Patients—Based on the Chinese Medical System. Journal of Mathematical Finance, 10, 96-131. doi: 10.4236/jmf.2020.101008.
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