Revenue sharing contracts of green fresh product considering freshness kee** effort

Abstract

We investigate a two echelon green fresh product supply chain consisting of an upstream farmer who provides green fresh product and a downstream retailer who buys green fresh product and transports, storages, sells it to the consumer and needs to pay freshness kee** effort (FKE). This paper adopts the Stackelberg model under three different scenarios to investigate the decision-making problems for decentralized supply chain. Our results demonstrate that: First, the equilibrium decisions, comprised of wholesale price, greenness improvement level (GIL) and freshness kee** effort, are the largest when under the Nash bargaining revenue sharing scenario (BS), but the least when under the retailer-led no revenue sharing case (NS). Second, the retailer earns more profit when under Nash bargaining revenue sharing scenario than the farmer-led revenue sharing model (RS), and the NS scenario is the least. Whereas, the farmer's profit under the three scenarios depends on the specific parameter values. Third, the whole channel profit is the largest when under BS case, the RS case comes next and the NS case is the least. The findings provide compelling insights into how farmer and retailer can better manage their supply chain strategies when freshness kee** is concerned.

Appendices

Appendix A. The decentralized channel case

For the decentralized channel case, the retailer's and the farmer's optimization problems can be formulated as

$$\underset{m,\eta }{\mathrm{max}}{\pi }_{R}^{\text{NS}}=m\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}L{\eta }^{2}$$

and

$$\underset{w,\theta }{\mathrm{max}}{\pi }_{F}^{\text{NS}}=\left(w-c\right)\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}K{\theta }^{2}$$

respectively.

According to the backward induction, we first solve the farmer's optimization problem. The Hessian Matrix is

$$\left[\begin{array}{cc}-2b& \alpha \\ \alpha & -K\end{array}\right]$$

By algebra, the Hessian Matrix is negative definite when $2bK-{\alpha }^{2}>0$, thus, the farmer's objective function is joint concave with respect to the decision variables. The first order derivative is

$$\frac{\partial {\pi }_{F}^{\text{NS}}}{\partial w}=a-b\left(w+m\right)+\alpha \hspace{0.17em}\theta +\beta \eta -\left(w-c\right)b$$

and

$$\frac{\partial {\pi }_{F}^{\text{NS}}}{\partial \theta }=\left(w-c\right)\alpha -K$$

Equating the first order derivative to 0 and solving the equations, we get

$$\theta \left(m,\eta \right)=\frac{\alpha \hspace{0.17em}\left(\beta \eta -bc-bm+a\right)}{2\hspace{0.17em}Kb-{\alpha }^{2}}$$

(A1)

and

$$w\left(m,\eta \right)=\frac{K\eta \beta +Kbc-Kbm-{\alpha }^{2}c+Ka}{2\hspace{0.17em}Kb-{\alpha }^{2}}$$

(A2)

We substitute the Equations (A1)-(A2) into the retailer's objective function and derive

$$\underset{m,\eta }{\mathrm{max}}{\pi }_{R}^{\text{NS}}=\frac{L{\eta }^{2}{\alpha }^{2}-2bk\left[mb\left(c+m\right)-m\left(\beta \eta +a\right)+L{\eta }^{2}\right]}{4Kb-2{\alpha }^{2}}$$

By computing the first order derivative

$$\begin{array}{c}\frac{\partial {\pi }_{R}^{\text{NS}}}{\partial m}=\frac{bK\left[a+\beta \eta -bc-2mb\right]}{2Kb-{\alpha }^{2}}\\ \frac{\partial {\pi }_{R}^{\text{NS}}}{\partial \eta }=\frac{L\eta {\alpha }^{2}+bk\left(\beta m-2L\eta \right)}{2Kb-{\alpha }^{2}}\end{array}$$

and the second order derivative

$$\begin{array}{c}\frac{{\partial }^{2}{\pi }_{R}^{\text{NS}}}{\partial {m}^{2}}=-L\\ \frac{{\partial }^{2}{\pi }_{R}^{\text{NS}}}{\partial {\eta }^{2}}=\frac{L{\alpha }^{2}-2LbK}{2Kb-{\alpha }^{2}}\\ \frac{{\partial }^{2}{\pi }_{R}^{\text{NS}}}{\partial m\partial \eta }=\frac{\beta bK}{2Kb-{\alpha }^{2}}\end{array}$$

We can write the Hessian Matrix of ${\pi }_{R}$ as

$$\left[\begin{array}{cc}-L& \frac{\beta bK}{2Kb-{\alpha }^{2}}\\ \frac{\beta bK}{2Kb-{\alpha }^{2}}& \frac{L{\alpha }^{2}-2LbK}{2Kb-{\alpha }^{2}}\end{array}\right]$$

Through computing the Hessian Matrix, the retailer's profit function is joint concave with respect to the decision variables when $4LbK-{\beta }^{2}K-2L{\alpha }^{2}>0$. Let the first order derivative be 0. We get

$${\eta }^{{\text{N}}{\text{S}}^{*}}=\frac{\beta K\left(a-bc\right)}{4LbK-{\beta }^{2}K-2L{\alpha }^{2}}$$

and

$${m}^{{\text{N}}{\text{S}}^{*}}=\frac{L\left({\alpha }^{2}bc+2Kab-a{\alpha }^{2}-2K{b}^{2}c\right)}{b\left(4LbK-{\beta }^{2}K-2L{\alpha }^{2}\right)}$$

Then, we have

$$\begin{array}{c}{\theta }^{{\text{N}}{\text{S}}^{*}}=\frac{L\alpha \left(a-bc\right)}{4LbK-{\beta }^{2}K-2L{\alpha }^{2}}\\ {w}^{{\text{N}}{\text{S}}^{*}}=\frac{\left[L\left(3bc+a\right)-c{\beta }^{2}\right]K-2Lc{\alpha }^{2}}{4LbK-{\beta }^{2}K-2L{\alpha }^{2}}\\ {p}^{{\text{N}}{\text{S}}^{*}}={w}^{{\text{N}}{\text{S}}^{*}}+{m}^{{\text{N}}{\text{S}}^{*}}=\frac{KL{b}^{2}c-Kb{\beta }^{2}c-L{\alpha }^{2}bc+3KLab-La{\alpha }^{2}}{b\left(4LbK-{\beta }^{2}K-2L{\alpha }^{2}\right)}\\ {\pi }_{R}^{{\text{N}}{\text{S}}^{*}}=\frac{KL{\left(a-bc\right)}^{2}}{2\left(4LbK-{\beta }^{2}K-2L{\alpha }^{2}\right)}\\ {\pi }_{F}^{{\text{N}}{\text{S}}^{*}}=\frac{K{L}^{2}{\left(a-bc\right)}^{2}\left(2Kb-{\alpha }^{2}\right)}{2{\left(4LbK-{\beta }^{2}K-2L{\alpha }^{2}\right)}^{2}}\\ {\pi }_{SC}^{{\text{N}}{\text{S}}^{*}}=\frac{KL{\left(-bc+a\right)}^{2}\left(6KLb-K{\beta }^{2}-3L{\alpha }^{2}\right)}{2{\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}^{2}}\end{array}$$

1.1 Proof of Proposition 3.1

Proof. We can easily get the results from the computational process.

1.2 Proof of Proposition 3.2

Proof. The above relationships can be proved easily through first order derivative.

Appendix B. Farmer-led revenue sharing model

For the farmer-led revenue sharing case, the farmer's and the retailer's optimization problems are formulated as

$$\underset{w,\theta }{\mathrm{max}}{\pi }_{F}^{\text{RS}}=\left[\left(w-c\right)\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}K{\theta }^{2}\right]\psi$$

and

$$\underset{m,\eta }{\mathrm{max}}{\pi }_{R}^{\text{RS}}=m\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}L{\eta }^{2}+\left[\left(w-c\right)\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}K{\theta }^{2}\right]\left(1-\psi \right)$$

respectively.

By using the backward induction approach, we solve for the farmer's objective function first.

The first order derivative

$$\begin{array}{c}\frac{\partial {\pi }_{F}^{\text{RS}}}{\partial w}=\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta -\left(w-c\right)b\right]\psi\\ \frac{\partial {\pi }_{F}^{\text{RS}}}{\partial \theta }=\left[\left(w-c\right)\alpha -K\theta \right]\psi\end{array}$$

and the second order derivative

$$\begin{array}{c}\frac{{\partial }^{2}{\pi }_{F}^{\text{RS}}}{\partial {w}^{2}}=-2b\psi\\ \frac{{\partial }^{2}{\pi }_{F}^{\text{RS}}}{\partial {\theta }^{2}}=-K\psi\\ \frac{{\partial }^{2}{\pi }_{F}^{\text{RS}}}{\partial w\partial \theta }=\alpha\psi \end{array}$$

The Hessian Matrix can be written as

$$\left[\begin{array}{cc}-2b\psi & \alpha \psi \\ \alpha \psi & -K\psi \end{array}\right]$$

We can ascertain the Hessian Matrix is negative definite if $2Kb{\psi }^{2}-{\alpha }^{2}{\psi }^{2}>0$.

Thus, the farmer's profit function is jointly concave in w and $\theta$. Let the first order derivative be zero, we obtain

$$\theta \left(\eta ,m\right)=\frac{\alpha \left(-bc-bm+\beta \eta +a\right)}{2Kb-{\alpha }^{2}}$$

(B1)

$$w\left(\eta ,m\right)=\frac{Kbc-Kbm+K\beta \eta -{\alpha }^{2}c+ka}{2Kb-{\alpha }^{2}}$$

(B2)

Substituting the Equations (B1-B2) into the retailer's objective function and exerting derivatives.

According to the first order derivative

$$\begin{array}{c}\frac{\partial {\pi }_{R}^{\text{RS}}}{\partial m}=\frac{Kb\left(-bc\psi -bm\psi +\beta \eta \psi +a\psi -bm\right)}{2Kb-{\alpha }^{2}}\\ \frac{\partial {\pi }_{R}^{\text{RS}}}{\partial \eta }=\frac{-Kb\beta c\psi -Kb\beta m\psi +K{\beta }^{2}\eta \psi +2KLb\eta +Ka\beta \psi +Kb\beta c-K{\beta }^{2}\eta -L{\alpha }^{2}\eta -Ka\beta }{{\alpha }^{2}-2Kb}\end{array}$$

and the second order derivative

$$\begin{array}{c}\frac{{\partial }^{2}{\pi }_{R}^{\text{RS}}}{\partial {m}^{2}}=-\frac{{b}^{2}K\left(1+\psi \right)}{2Kb-{\alpha }^{2}}\\ \frac{{\partial }^{2}{\pi }_{R}^{\text{RS}}}{\partial {\eta }^{2}}=-\frac{K{\beta }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}}{2Kb-{\alpha }^{2}}\\ \frac{{\partial }^{2}{\pi }_{R}^{\text{RS}}}{\partial m\partial \eta }=\frac{bK\beta \psi }{2Kb-{\alpha }^{2}}\end{array}$$

Then the Hessian Matrix is negative definite if

$$2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}>0$$

By the first order derivative, we have

$$\begin{array}{c}\eta \left(\psi \right)=\frac{K\beta \left(-bc+a\right)}{2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}}\\ m\left(\psi \right)=\frac{L\psi \left(-2K{b}^{2}c+{\alpha }^{2}bc+2Kab-a{\alpha }^{2}\right)}{b\left(2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\end{array}$$

Substituting m and $\eta$ into w, $\theta$ and the farmer's objective function, we can obtain the farmer's profit function with respect to $\psi$

$${\pi }_{F}^{\text{RS}}\left(\psi \right)=\frac{1}{2}\frac{\psi \left(2Kb-{\alpha }^{2}\right){\left(-bc+a\right)}^{2}{L}^{2}K}{{\left(2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}^{2}}$$

By computing the first order derivative and the second order derivative, we can ascertain the second order derivative is negative if 2KLbψ – Lα²ψ – 4KLb + 2Kβ² + 2Lα² < 0. Thus, π_F^RS(ψ) is concave in $\psi$ for the above condition. Solving the first order condition for $\psi$ gives $\psi =\frac{2KLb-K{\beta }^{2}-L{\alpha }^{2}}{L\left(2Kb-{\alpha }^{2}\right)}$.

Then we obtain

$$\begin{array}{c}{m}^{{\text{R}}{\text{S}}^{*}}=\frac{-bc+a}{2b}\\ {p}^{{\text{R}}{\text{S}}^{*}}={m}^{{\text{R}}{\text{S}}^{*}}+{w}^{{\text{R}}{\text{S}}^{*}}\\ {\eta }^{{\text{R}}{\text{S}}^{*}}=\frac{\left(-bc+a\right)\beta K}{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {w}^{{\text{R}}{\text{S}}^{*}}=\frac{3KLbc-2K{\beta }^{2}c-2L{\alpha }^{2}c+KLa}{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\theta }^{{\text{R}}{\text{S}}^{*}}=\frac{\left(-bc+a\right)L\alpha }{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{R}^{{\text{R}}{\text{S}}^{*}}=\frac{{\left(-bc+a\right)}^{2}LK}{4\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{F}^{{\text{R}}{\text{S}}^{*}}=\frac{{\left(-bc+a\right)}^{2}LK}{8\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{SC}^{{\text{R}}{\text{S}}^{*}}=\frac{3{\left(-bc+a\right)}^{2}LK}{8\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\end{array}$$

2.1 Proof of Proposition 3.3

Proof. We can easily get the results from the computational process.

Appendix C. Bargaining revenue sharing contract model

Due to the farmer and the retailer decide the revenue sharing proportion $\psi$ through bargaining, we formulate the objective function as

$$\underset{\psi }{\mathrm{max }}{\pi }_{F}^{\text{BS}}{\pi }_{R}^{\text{BS}}$$

(C1)

in which

$$\underset{w,\theta }{\mathrm{max}}{\pi }_{F}^{\text{BS}}=\left[\left(w-c\right)\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}K{\theta }^{2}\right]\psi$$

and

$$\underset{m,\eta }{\mathrm{max}}{\pi }_{R}^{\text{BS}}=m\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}L{\eta }^{2}+\left[\left(w-c\right)\left[a-b\left(m+w\right)+\alpha \theta +\beta \eta \right]-\frac{1}{2}K{\theta }^{2}\right]\left(1-\psi \right)$$

According to the game structure, we solve the farmer's optimization problem first and many steps are similarly with the farmer-led revenue sharing scenario. The only difference between BS and RS is that the solving process of $\psi$, i.e., the $\psi$ is determined by solving Equation (C1) while it's obtained via optimizing the farmer's optimization problem under the RS scenario. For the sake of brevity, we omitted the computational process before the solving of sharing proportion.

After get the expressions of $m\left(\psi \right)$, $\eta \left(\psi \right)$, $\theta \left(\psi \right)$ and $w\left(\psi \right)$, we substitute them into the Nash bargaining function (Equation C1). The Equation (C1) can be rewrite as

$$\underset{\psi }{\mathrm{max}} {\pi}^{\text{B}}\left(\psi \right)=\frac{{\left(-bc+a\right)}^{4}{K}^{2}{L}^{3}\psi \left(2Kb-{\alpha }^{2}\right)}{4{\left(2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}^{3}}$$

By computing the first order derivative and the second order derivative of the bargaining model objective function. The first order derivative

$$\frac{{\text{d}}{\pi}^{\text{B}}}{{\text{d}}\psi }=-\frac{{\left(-cb+a\right)}^{4}{K}^{2}{L}^{3}\left(2Kb-{\alpha }^{2}\right)\left(4KLb\psi -2L{\alpha }^{2}\psi -2KLb+K{\beta }^{2}+L{\alpha }^{2}\right)}{4{\left(2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}^{4}}$$

The second order derivative

$$\frac{{\text{d}}^{2}{\pi}^{\text{B}}}{{\text{d}}{\psi }^{2}}=\frac{3{\left(-cb+a\right)}^{4}{K}^{2}{L}^{4}{\left(2Kb-{\alpha }^{2}\right)}^{2}\left(2KLb\psi -L{\alpha }^{2}\psi -2KLb+K{\beta }^{2}+L{\alpha }^{2}\right)}{2{\left(2KLb\psi -L{\alpha }^{2}\psi +2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}^{5}}$$

The second order derivative is negative if

$$2KLb\psi -L{\alpha }^{2}\psi -2KLb+K{\beta }^{2}+L{\alpha }^{2}<0$$

Thus, the objective function is concave in $\psi$ for the above condition. Let the first order derivative be 0. We obtain

$${\psi }^{{\text{B}}{\text{S}}^{*}}=\frac{2KLb-K{\beta }^{2}-L{\alpha }^{2}}{2L\left(2Kb-{\alpha }^{2}\right)}$$

Thus, the equilibrium decisions and profit functions are

$$\begin{array}{c}{\eta }^{{\text{B}}{\text{S}}^{*}}=\frac{2\beta K\left(-bc+a\right)}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {m}^{{\text{B}}{\text{S}}^{*}}=\frac{-bc+a}{3b}\\ {\theta }^{{\text{B}}{\text{S}}^{*}}=\frac{2L\alpha \left(-bc+a\right)}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {w}^{{\text{B}}{\text{S}}^{*}}=\frac{4KLbc-3K{\beta }^{2}c-3L{\alpha }^{2}c+2KLa}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{R}^{{\text{B}}{\text{S}}^{*}}=\frac{LK{\left(-bc+a\right)}^{2}}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{F}^{{\text{B}}{\text{S}}^{*}}=\frac{LK{\left(-bc+a\right)}^{2}}{9\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\\ {\pi }_{SC}^{{\text{B}}{\text{S}}^{*}}=\frac{4LK{\left(-bc+a\right)}^{2}}{9\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\end{array}$$

3.1 Proof of Proposition 3.4

Proof. We can get the results from the computational process.

Appendix D. Analysis regarding the equilibrium outcomes

4.1 Proof of Proposition4.1

Proof. It is can be easily proved through basic algebraic operation.

4.2 Proof of Proposition 4.2

Proof. We prove ${\eta }^{{\text{N}}{\text{S}}^{*}}<{\eta }^{{\text{R}}{\text{S}}^{*}}<{\eta }^{{\text{B}}{\text{S}}^{*}}$ for example. The proofs of other results can be done analogously.

$$\begin{array}{c}{\eta }^{{\text{R}}{\text{S}}^{*}}-{\eta }^{{\text{N}}{\text{S}}^{*}}=\frac{K\beta \left(-bc+a\right)}{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{K\beta \left(-bc+a\right)}{4KLb-K{\beta }^{2}-2L{\alpha }^{2}}=\frac{{K}^{2}{\beta }^{3}\left(-bc+a\right)}{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}\\ {\eta }^{{\text{B}}{\text{S}}^{*}}-{\eta }^{{\text{R}}{\text{S}}^{*}}=\frac{2K\beta \left(-bc+a\right)}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{K\beta \left(-bc+a\right)}{2\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}=\frac{K\beta \left(-bc+a\right)}{6\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\end{array}$$

Since $-bc+a>0$ and $2KLb-K{\beta }^{2}-L{\alpha }^{2}>0$, then ${\eta }^{{\text{N}}{\text{S}}^{*}}<{\eta }^{{\text{R}}{\text{S}}^{*}}<{\eta }^{{\text{B}}{\text{S}}^{*}}$ holds true.

4.3 Proof of Proposition 4.3

Proof.

$$\begin{array}{c}{\pi }_{R}^{{\text{R}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{N}}{\text{S}}^{*}}=\frac{KL{\left(-bc+a\right)}^{2}}{4\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{KL{\left(-bc+a\right)}^{2}}{2\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}=\frac{{K}^{2}L{\beta }^{2}{\left(-bc+a\right)}^{2}}{4\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}\\ {\pi }_{R}^{{\text{B}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{R}}{\text{S}}^{*}}=\frac{KL{\left(-bc+a\right)}^{2}}{3\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{KL{\left(-bc+a\right)}^{2}}{4\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}=\frac{KL{\left(-bc+a\right)}^{2}}{12\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}\end{array}$$

It's easily proved that ${\pi }_{R}^{{\text{B}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{R}}{\text{S}}^{*}}>0$ and ${\pi }_{R}^{{\text{R}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{N}}{\text{S}}^{*}}>0$. Then, ${\pi }_{R}^{{\text{N}}{\text{S}}^{*}}<{\pi }_{R}^{{\text{R}}{\text{S}}^{*}}<{\pi }_{R}^{{\text{B}}{\text{S}}^{*}}$ holds true.

5.1 Proof of Proposition 4.4

Proof. By algebra, we have

$${\pi }_{SC}^{{\text{B}}{\text{S}}^{*}}-{\pi }_{SC}^{{\text{R}}{\text{S}}^{*}}=\frac{4KL{\left(-bc+a\right)}^{2}}{9\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{3KL{\left(-bc+a\right)}^{2}}{8\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}=\frac{5KL{\left(-bc+a\right)}^{2}}{72\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}$$

and

$${\pi }_{SC}^{{\text{R}}{\text{S}}^{*}}-{\pi }_{SC}^{{\text{N}}{\text{S}}^{*}}=\frac{3KL{\left(-bc+a\right)}^{2}}{8\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right)}-\frac{KL{\left(-bc+a\right)}^{2}\left(6KLb-K{\beta }^{2}-3L{\alpha }^{2}\right)}{2{\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}^{2}}=\frac{{K}^{2}L{\beta }^{2}{\left(-bc+a\right)}^{2}\left(8KLb-K{\beta }^{2}-4L{\alpha }^{2}\right)}{8\left(2KLb-K{\beta }^{2}-L{\alpha }^{2}\right){\left(4KLb-K{\beta }^{2}-2L{\alpha }^{2}\right)}^{2}}$$

Owing to $2KLb-K{\beta }^{2}-L{\alpha }^{2}>0$, we have $8KLb-K{\beta }^{2}-4L{\alpha }^{2}>0$ and then ${\pi }_{R}^{{\text{B}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{R}}{\text{S}}^{*}}>0$ and ${\pi }_{R}^{{\text{R}}{\text{S}}^{*}}-{\pi }_{R}^{{\text{N}}{\text{S}}^{*}}>0$. Thus, ${\pi }_{SC}^{{\text{N}}{\text{S}}^{*}}<{\pi }_{SC}^{{\text{R}}{\text{S}}^{*}}<{\pi }_{SC}^{{\text{B}}{\text{S}}^{*}}$ holds true.

5.2 Proof of Proposition 4.5

Proof. The results can be proved easily by using first order derivative.