Joint optimization of pricing and inventory control for dual-channel problem under stochastic demand

Abstract

We consider an optimization problem for a dual-channel supply chain with one manufacturer and one retailer under stochastic demand. The manufacturer and retailer play a Stackelberg game. The manufacturer, acting as the leader, decides the total inventory capacity, wholesale price and selling price in the direct channel. The retailer, acting as the follower, decides the order quantity and retail price in the traditional channel. By solving the optimization problem, we find that given the selling price in the direct channel and wholesale price, the retailer’s decisions on order quantity is similar to that for the newsvendor problem. Meanwhile, the manufacturer’s decision on the inventory allocated to the direct channel also has the similar structure. Besides, we show that in our analysis framework, the retail price in the traditional channel, wholesale price and total inventory capacity can be presented in closed-forms. Finally, we numerically study the effects of the demand uncertainties and price sensitivities on the optimal solutions and performance. It is interesting to show that the retailer’s expected profit may be unexpectedly increased when the demand variability in the traditional channel increases, due to the competition between two channels.

Appendix

1.1 Proof of Lemma 2

Proof

From Lemma 1, we have

$$\begin{aligned} {{q}_{1}}\left( {{z}_{1}} \right) ={{z}_{1}}+{{y}_{1}}\left( {{p}_{1}}\left( {{z}_{1}} \right) ,{{p}_{2}} \right) ={{z}_{1}}+\frac{{{{\varTheta }}_{1}}\left( {{z}_{1}} \right) }{2}+{{a}_{1}}-{{b}_{11}}p_{1}^{0}+{{b}_{12}}{{p}_{2}}, \end{aligned}$$

then

$$\begin{aligned} \frac{\partial {{q}_{1}}\left( {{z}_{1}} \right) }{\partial {{z}_{1}}}=\frac{1+{{F}_{1}}\left( {{z}_{1}} \right) }{2}>0, \end{aligned}$$

indicating that ${{q}_{1}}\left( {{z}_{1}} \right)$ is increasing in ${{z}_{1}}$. $\square$

1.2 Proof of Proposition 1

Proof

From the chain rule and Lemma 1, we have:

$$\begin{aligned} \frac{d{\mathbb {E}}[{\varPi }_r(z_1,p_1\left( z_1\right) )]}{dz_1}= & {} -w+p_1[1-F_1(z_1)]. \end{aligned}$$

To obtain the values of $z_1$ that satisfy this first-order optimality condition, let $R_1(z_1)\equiv d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]/dz_1$ and consider that

$$\begin{aligned} \frac{dR_1(z_1)}{dz_1}= & {} \frac{d}{dz_1}\left[ \frac{d{\mathbb {E}}[{\varPi }_r(z_1,p_1(z_1))]}{dz_1}\right] \nonumber \\= & {} -\frac{f_1(z_1)}{2b_{11}}\left\{ 2b_{11}p_1^0-{\varTheta }_1(z_1)-\frac{1-F_1(z_1)}{r_1(z_1)}\right\} \end{aligned}$$

where $r_1(\cdot )\equiv f_1(\cdot )/[1-F_1(\cdot )]$ denotes the hazard rate. Also,

$$\begin{aligned} \frac{d^2R_1(z_1)}{dz_1^2}= & {} \left[ \frac{dR_1(z_1)/dz_1}{f(z_1)}\right] \frac{df_1(z_1)}{dz_1}-\frac{f_1(z_1)}{2b_{11}} \nonumber \\&\cdot \left\{ [1-F_1(z_1)]+\frac{f_1(z_1)}{r_1(z_1)}+\frac{[1-F_1(z_1)[dr_1(z_1)/dz_1]}{r_1(z_1)^2}\right\} \end{aligned}$$

Thus, we can obtain,

$$\begin{aligned} \left. {\frac{d^2R_1(z_1)}{dz_1^2}}\right| _{dR_1(z_1)/dz_1=0}= & {} -\frac{f_1(z_1)[1-F_1(z_1)]}{2b_{11}r_1(z_1)^2}\left\{ 2r_1(z_1)^2+\frac{dr_1(z_1)}{dz_1}\right\} \end{aligned}$$

Since we have $2r_1(z_1)^2+{dr_1(z_1)}/{dz_1}>0$, then it follows that $R_1(z_1)$ is monotone or unimodal, implying that $d{\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]/dz_1=0$ has at most two roots. Further, ${{R}_{1}}\left( {{B}_{1}} \right) =-w<0$. Therefore, if ${{R}_{1}}\left( {{z}_{1}} \right)$ has only one root, it indicates a change of sign for ${{R}_{1}}\left( {{z}_{1}} \right)$ from positive to negative, and thus it corresponds to a local maximum of ${\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right]$ ; if it has two roots, the larger of the two corresponds to a local maximum and the smaller of the two corresponds to a local minimum of ${\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right]$. In either case, ${\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right]$ has only one local maximum, identified either as the unique value of ${{z}_{1}}$ that satisfies ${{R}_{1}}\left( {{z}_{1}} \right) =d{\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right] /d{{z}_{1}}\ =0$ or as the larger of two values of ${{z}_{1}}$ that satisfies ${{R}_{1}}\left( {{z}_{1}} \right) =0$. And since ${\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right]$ is unimodal if ${{R}_{1}}\left( {{z}_{1}} \right)$ has only one root, a sufficient condition for unimodality of ${\mathbb {E}}\left[ {{{\varPi }}_{r}}\left( {{z}_{1}},{{p}_{1}}\left( {{z}_{1}} \right) \right) \right]$ is ${{R}_{1}}\left( {{A}_{1}} \right) >0$ or, equivalently, $2{{b}_{11}}{{R}_{1}}\left( {{A}_{1}} \right) >0$, where

$$\begin{aligned} 2{{b}_{11}}{{R}_{1}}\left( {{A}_{1}} \right)&=-2{{b}_{11}}w+\left[ 2{{b}_{11}}{{p}^{0}}-{{{\varTheta }}_{1}}\left( {{A}_{1}} \right) \right] \cdot \left[ 1-{{F}_{1}}\left( {{A}_{1}} \right) \right] \\&=-2{{b}_{11}}w+\left[ {{a}_{1}}+{{b}_{11}}w+{{\mu }_{1}}+{{b}_{12}}{{p}_{2}}-\left( {{\mu }_{1}}-{{A}_{1}} \right) \right] \\&={{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}. \end{aligned}$$

$\square$

1.3 Proof of Corollary 1

Proof

Note from Lemma 2 that $q_1(z_1)$ increases in $z_1$. So the boundary of $q_1$ could be mapped to $z_1$, such that $N=y_1(p_1(z_1), p_2)+z_1$ is satisfied. As shown in Proposition 1, if ${{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}>0$, then ${\mathbb {E}}[{\varPi }_r(z_1, p_1(z_1))]$ is unimodal in $z_1$. So given $q_1(z_1)\ge N$, we have $z_1^*=\bar{z_1}$. If ${{a}_{1}}-\left( {{b}_{11}}w-{{b}_{12}}{{p}_{2}} \right) +{{A}_{1}}\le 0$, there are two $z_1$ satisfy the first-order condition. So we need to compare the values of ${\mathbb {E}}[{\varPi }_r(A_1, p_1(A_1))]$ and ${\mathbb {E}}[{\varPi }_r({\bar{z}}_1, p_1({\bar{z}}_1))]$. If ${\mathbb {E}}[{\varPi }_r(A_1, p_1(A_1))]<{\mathbb {E}}[{\varPi }_r({\bar{z}}_1, p_1({\bar{z}}_1))]$, then $z_1^*={\bar{z}}_1$, otherwise, $z_1^*=A_1$. $\square$

1.4 Proof of Lemma 3

Proof

We first define retailer’s global optimal solutions as $z_1^*\left( N,{{p}_{2}},w \right)$, $q_1^*\left( N,{{p}_{2}},w \right)$ and $p_1^*\left( N,{{p}_{2}},w \right)$, for given $\left( N,{{p}_{2}},w \right)$. Besides, similar to the definition of ${{\hat{z}}}_1$, we let ${{\hat{p}}}_1=p_1(p_2,w)$ and ${{\hat{q}}}_1=q_1(p_2,w)$.

There may exist two cases when the manufacturer sets the optimal N. The first case is $N\ge q_1(p_2,w)$, where the optimal responses of retailer’s decisions are characterized in Proposition 1. The corresponding values are ${\hat{z}}_1$, ${\hat{q}}_1$ and ${\hat{p}}_1$. The second case is $N<q_1(p_2,w)$, where the optimal responses are given in Corollary 1. The corresponding values are $z_1^*\left( N,{{p}_{2}},w \right)$, $q_1^*\left( N,{{p}_{2}},w \right)$ and $p_1^*\left( N,{{p}_{2}},w \right)$. Next we show that it would not be the optimal solution such that $N<q_1(p_2,w)$, when the manufacturer sets N.

We assume that there exists a total inventory capacity $N=n$, i.e., $0\le n<{\hat{q}}_1$. From Corollary 1, we know that the corresponding optimal responses are $z_1^*\left( n,{{p}_{2}},w \right)$, $q_1^*\left( n,{{p}_{2}},w \right)$ and $p_1^*\left( n,{{p}_{2}},w \right)$. It is easy to check that $z_1^*(n,p_2,w)<{\hat{z}}_1$. Besides, from Lemmas 1 and 2, we know that both ${{p}_{1}}\left( {{z}_{1}} \right)$ and ${{q}_{1}}\left( {{z}_{1}} \right)$ are increasing in ${{z}_{1}}$. Then we have $p_1^*(n,p_2,w)<{\hat{p}}_1$ and $q_1^*(n,p_2,w)\le n<{\hat{q}}_1$. The manufacturer’s expected profit is

$$\begin{aligned} {\mathbb {E}}\left[ {{{{\varPi }}'}_{m}}\right] ={\mathbb {E}}\left[ {{p}_{2}}\min \left\{ {{D}_{2}}\left( p_1^*(n,p_2,w),{{p}_{2}},\epsilon _2 \right) ,n-q_1^*(n,p_2,w) \right\} -cn+w{q_1^*(n,p_2,w)}\right] . \end{aligned}$$

Now we consider another feasible solution $N=n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}$. It is easy to verify that $n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}\ge n+ {\hat{q}}_1-q_1^*(n,p_2,w)\ge {\hat{q}}_1> n$. Here, the first inequality follows from $w\ge c$, the second inequality follows from $q_1^*(n,p_2,w)\le n$, and the last inequality follows from $n<{\hat{q}}_1$. Given the optimal responses ${\hat{q}}_1$ and ${\hat{p}}_1$, the manufacturer’s expected profit function is

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}\right]= & {} {\mathbb {E}}\bigg [{{p}_{2}}\min \left\{ {{D}_{2}}\left( {\hat{p}}_1,{{p}_{2}},\epsilon _2 \right) ,n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}-{\hat{q}}_1 \right\} \\&\quad -\,c\left\{ n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}\right\} +w{{\hat{q}}_1}\bigg ]. \end{aligned}$$

Then we have

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}\right] -{\mathbb {E}}\left[ {{{{\varPi }}'}_{m}}\right]= & {} {\mathbb {E}}\bigg [{{p}_{2}}\min \left\{ {{D}_{2}}\left( {\hat{p}}_1,{{p}_{2}},\epsilon _2 \right) ,n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}-{\hat{q}}_1 \right\} \\&\quad -\,{{p}_{2}}\min \left\{ {{D}_{2}}\left( p_1^*(n,p_2,w),{{p}_{2}},\epsilon _2 \right) ,n-q_1^*(n,p_2,w) \right\} \bigg ]\\\ge & {} 0. \end{aligned}$$

The inequality follows from ${\mathbb {E}}[D_2\left( {\hat{p}}_1,{{p}_{2}},\epsilon _2 \right) ]\ge {\mathbb {E}}[{{D}_{2}}\left( p_1^*(n,p_2,w),{{p}_{2}},\epsilon _2 \right) ]$ because ${\hat{p}}_1> p_1^*(n,p_2,w)$, and $n+\frac{w\left[ {\hat{q}}_1-q_1^*(n,p_2,w) \right] }{c}-{\hat{q}}_1\ge n-q_1^*(n,p_2,w)$. It implies that the case $N=n<{\hat{q}}_1$ would not be optimal.

Thus, we can conclude that given the price in the direct channel $p_2$ and wholesale price w, the manufacturer will only set the total inventory capacity N in the region $R_N=\{N: N\ge q_1(p_2,w)\}$. $\square$

1.5 Proof of Corollary 2

Proof

From Proposition 1, we know that the retailer’s profit function satisfies the first-order condition, i.e. $\frac{\partial {\mathbb {E}}[{\varPi }_r(z_1,p_1)]}{\partial z_1}=0$. Thus we can obtain:

$$\begin{aligned} -\frac{1+F_1(z_1)}{2}w+[b_{12}p_2+a_1+\mu _1-{\varTheta }_1(z_1)]\frac{[1-F_1(z_1)]}{2b_{11}}= & {} 0. \end{aligned}$$

By rearranging the above equation, we can obtain

$$\begin{aligned} w=\frac{[a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)][1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]}. \end{aligned}$$

Without loss of generality, we assume that $\mu _1=0$. According to (5), we can obtain that $w=[1-F_1(z_1)]p_1$. Plugging it back to Lemma 1, we can obtain

$$\begin{aligned} p_1= \frac{a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)}{b_{11}[1+F_1(z_1)]}. \end{aligned}$$

$\square$

1.6 Proof of Proposition 3

Proof

To show that the optimization problems ${{{\text {P}}}_{m}}$ and ${{{\text {P}}}_{new}}$ are equivalent, we first show the corresponding regions of their feasible solutions are equivalent, and then we show that the optimal solutions for problem ${{{\text {P}}}_{new}}$ are also optimal for problem ${{{\text {P}}}_{m}}$.

Let ${{{\varOmega }}_{{\text {m}}}}$ be feasible solution region of problem ${{{\text {P}}}_{m}}$, then we have ${{{\varOmega }}_{{\text {m}}}}=\left\{ \left( {{p}_{2}},w \right) :{{p}_{2}}\ge c,w\ge c \right\}$. Let ${{\varOmega }}_{{\text {new}}}$ define the feasible solution region of problem ${{{\text {P}}}_{new}}$, then we have ${{{\varOmega }}_{{\text {new}}}}=\big \{ \left( {{p}_{2}},{{z}_{1}} \right) :{{p}_{2}}\ge c,{{A}_{1}}\le {{z}_{1}}\le {{B}_{1}},p_2\ge g_1(z_1),p_2\ge g_2(z_1)\big \}$, where $g_1(z_1)=\frac{cb_{11}[1+F_1(z_1)]-[a_1+\mu _1-{\varTheta }_1(z_1)][1-F_1(z_1)]}{b_{12}[1-F_1(z_1)]}$ and $g_2(z_1)=\frac{[1-F_1(z_1)][1+F_1(z_1)]-2[a_1+\mu _1-{\varTheta }_1(z_1)]r_1(z_1)}{2b_{12}r_1(z_1)}$, and their corresponding constraints directly follow from (17) and (18), respectively. Define ${\varOmega }=\Big \{\left( {{p}_{2}},w\right) :w = \frac{[a_1+\mu _1+b_{12}p_2-{\varTheta }_1(z_1)][1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]},\left( {{p}_{2}},{{z}_{1}}\right) \in {{{\varOmega }}_{{\text {new}}}}\Big \}$. Then, we have ${{{\varOmega }}_{{\text {m}}}}={\varOmega }$, implying that the corresponding regions of the feasible solutions for the two optimization problems are equivalent,

Next we prove the results that the optimal solutions for problem ${{{\text {P}}}_{new}}$ are also optimal for problem ${{{\text {P}}}_{m}}$. Let $\left( p_{2}^{*},{{w}^{*}} \right)$ be an optimal policy of problem ${{\text {P}}_{m}}$, and $\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right)$ be an optimal policy of problem ${{\text {P}}_{new}}$. Because $\left( p_{2}^{*},{{z}_{1}}\left( p_{2}^{*},{{w}^{*}} \right) \right)$ is a feasible policy for ${{\text {P}}_{new}}$, where $z_1\left( p_2^*,w^*\right)$ can be obtained from (12) and (13), then we have ${\mathbb {E}}\left[ {{{\varPi }}_{new}}\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right] \ge {\mathbb {E}}\left[ {{{\varPi }}_{new}}\left( p_{2}^{*},{{z}_{1}}\left( p_{2}^{*},{{w}^{*}} \right) \right) \right]$.

Besides, by definition we have ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]={\mathbb {E}}[{\varPi }_m(z_2(p_2,w(p_2,z_1)),p_2,w(p_2,z_1))]$ and $w(p_2,z_1(p_2,w))=w$, then we have

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( {{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) ,{{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) \right] \ge {\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( p_{2}^{*},{{w}^{*}} \right) ,p_{2}^{*},{{w}^{*}} \right) \right] . \end{aligned}$$

(19)

On the other hand, $\left( {{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right)$ is a feasible policy for ${{\text {P}}_{m}}$, where $w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right)$ can be obtained from (14), then we have

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( p_{2}^{*},{{w}^{*}} \right) ,p_{2}^{*},{{w}^{*}} \right) \right] \ge {\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( {{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) ,{{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) \right] . \end{aligned}$$

(20)

Combining inequalities (19) and (20), we obtain that

$$\begin{aligned} {\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( {{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) ,{{{\hat{p}}}_{2}},w\left( {{{\hat{p}}}_{2}},{{{\hat{z}}}_{1}} \right) \right) \right] ={\mathbb {E}}\left[ {{{\varPi }}_{m}}\left( {{z}_{2}}\left( p_{2}^{*},{{w}^{*}} \right) ,p_{2}^{*},{{w}^{*}} \right) \right] . \end{aligned}$$

Thus, the optimal solutions for problem ${{{\text {P}}}_{new}}$ are also optimal for the problem ${{{\text {P}}}_{m}}$. $\square$

1.7 Proof of Lemma 4

Proof

Taking the first derivative of the expected profit with respect to $p_2$ , we have,

$$\begin{aligned} \frac{\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]}{\partial p_2}= & {} y_2+(p_2-c)\left\{ -b_{22}+\frac{b_{12}b_{21}}{b_{11}[1+F_1(z_1)]}\right\} -\int ^{B_2}_{z_2}(u-z_2)f_2(u)du \nonumber \\&\quad +\,(y_1+z_1)\cdot \frac{b_{12}[1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]}+(w-c)\left[ b_{12}-\frac{b_{12}}{1+F_1(z_1)}\right] \nonumber \\= & {} \frac{-{\mathbb {M}}p_2+{\mathbb {N}}-{b_{11}[1+F_1(z_1)]}{\varTheta }_2\left( z_2\right) }{b_{11}[1+F_1(z_1)]}, \end{aligned}$$

where $y_2=a_2-b_{22}p_2+b_{21}p_1$, $y_1=a_1-b_{11}p_1+b_{12}p_2$, and $z_2 = F_2^{-1}\left( \frac{p_2-c}{p_2}\right)$.

Further,

$$\begin{aligned} {\mathbb {M}}=&-2\left\{ {{b}_{12}}{{b}_{21}}-{{b}_{11}}{{b}_{22}}[1+{{F}_{1}}({{z}_{1}})]\text {+}\frac{b_{12}^{2}{{F}_{1}}({{z}_{1}})[1-{{F}_{1}}({{z}_{1}})]}{1+{{F}_{1}}({{z}_{1}})} \right\} >0,\\ {\mathbb {N}}=\,&{{a}_{2}}{{b}_{11}}\left[ 1+{{F}_{1}}({{z}_{1}}) \right] +{{b}_{21}}\left[ {{a}_{1}}-{{{\varTheta }}_{1}}\left( {{z}_{1}} \right) \right] -c\left\{ {{b}_{12}}{{b}_{21}}-{{b}_{11}}{{b}_{22} \big [1 +F_1\left( z_1\right) \big ] +b_{11}b_{22}F_1\left( z_1\right) } \right\} \\&+\,{{b}_{12}}\left[ 1-{{F}_{1}}({{z}_{1}}) \right] \left( {{a}_{1}}+{{z}_{1}} \right) -\frac{{{b}_{12}}{{\left[ 1-{{F}_{1}}({{z}_{1}}) \right] }^{2}}\left[ {{a}_{1}}-{{{\varTheta }}_{1}}\left( {{z}_{1}} \right) \right] }{1+{{F}_{1}}({{z}_{1}})}. \end{aligned}$$

Here, we define $p_2^0={{\mathbb {N}}}/{\mathbb {M}}$ as the optimal risk-less price in the direct channel, it is easy to check that when $p_2>p_2^0$, $\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]/\partial p_2<0$, then the optimal price in the direct channel $p_2^*$ is always less than or equal to $p_2^0$. $\square$

1.8 Proof of Proposition 4

Proof

Define $R_2(p_2)=\frac{\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]}{\partial p_2}$, we can obtain,

$$\begin{aligned} \frac{dR_2(p_2)}{dp_2}= & {} \left\{ -b_{22}+\frac{b_{12}b_{21}}{b_{11}[1+F_1(z_1)]}\right\} +\frac{b_{12}b_{21}-b_{11}b_{22}[1+F_1(z_1)]}{b_{11}[1+F_1(z_1)]} \\&\quad +\,\frac{b_{12}[1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]}\left[ b_{12}-\frac{b_{12}}{1+F_1(z_1)}\right] +\frac{b_{12}^2F_1(z_1)[1+F_1(z_1)]}{b_{11}[1+F_1(z_1)]^2} \\&\quad +\,\left( 1-\frac{p_2-c}{p_2}\right) \left[ F_2^{-1}\left( \frac{p_2-c}{p_2}\right) \right] ' \\= & {} \frac{2b_{12}b_{21}-2b_{11}b_{22}[1+F_1(z_1)]}{b_{11}[1+F_1(z_1)]} +\frac{2b_{12}^2F_1(z_1)[1-F_1(z_1)]}{b_{11}[1+F_1(z_1)]^2}+\frac{c^2}{p_2^3f_2(z_2)}. \end{aligned}$$

Define $r_2(\cdot )=\frac{f_2(\cdot )}{1-F_2(\cdot )}$, we can obtain,

$$\begin{aligned} \frac{{{d}^{2}}{{R}_{2}}\left( {{p}_{2}} \right) }{dp_{2}^{2}}= & {} {{\left[ \frac{{{c}^{2}}}{p_{2}^{3}{{f}_{2}}\left( {{z}_{2}} \right) } \right] }^{\prime }} \\= & {} {{\left\{ \frac{{{\left[ 1-{{F}_{2}}\left( {{z}_{2}} \right) \right] }^{2}}}{c}\cdot \frac{1}{{{r}_{2}}\left( {{z}_{2}} \right) } \right\} }^{\prime }} \\= & {} -\left\{ \frac{1-{{F}_{2}}\left( {{z}_{2}} \right) }{c}\cdot \frac{2{{f}_{2}}\left( {{z}_{2}} \right) }{{{r}_{2}}\left( {{z}_{2}} \right) }+\frac{{{\left[ 1-{{F}_{2}}\left( {{z}_{2}} \right) \right] }^{2}}}{c}\cdot \frac{{d{{r}_{2}}\left( {{z}_{2}} \right) }/{d{{z}_{2}}}\;}{{{r}_{2}}{{\left( {{z}_{2}} \right) }^{2}}} \right\} \cdot \frac{d{{z}_{2}}}{d{{p}_{2}}} \\= & {} -\frac{{{\left[ 1-{{F}_{2}}\left( {{z}_{2}} \right) \right] }^{2}}}{c}\cdot \frac{1}{{{r}_{2}}{{\left( {{z}_{2}} \right) }^{2}}}\cdot \frac{d{{z}_{2}}}{d{{p}_{2}}}\cdot \left\{ 2{{r}_{2}}{{\left( {{z}_{2}} \right) }^{2}}+\frac{d{{r}_{2}}\left( {{z}_{2}} \right) }{d{{z}_{2}}} \right\} . \end{aligned}$$

Since we have $2{{r}_{2}}{{({{z}_{2}})}^{2}}+d{{r}_{2}}({{z}_{2}})/d{{z}_{2}}>0$, then it follows that ${{R}_{2}}({{p}_{2}})$ is monotone or unimodal, implying that ${\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]}/{\partial {{p}_{2}}=0}\;$ has at most two roots. Further, ${{R}_{2}}\left( p_{2}^{0} \right) <0$. Therefore, if ${{R}_{2}}({{p}_{2}})$ has only one root, it indicates a change of sign for ${{R}_{2}}({{p}_{2}})$ from positive to negative, and thus it corresponds to a local maximum of ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]$; if it has two roots, the larger of the two corresponds to a local maximum and the smaller of the two corresponds to a local minimum of ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]$. In either case, ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]$ has only one local maximum, identified either as the unique value of ${{p}_{2}}$ that satisfies ${{R}_{2}}({{p}_{2}})={\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]}/{\partial {{p}_{2}}}\;=0$ or as the larger of two values of ${{p}_{2}}$ that satisfies ${{R}_{2}}({{p}_{2}})=0$. And since ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]$ is unimodal if ${{R}_{2}}({{p}_{2}})$ has only one root, a sufficient condition for unimodality of ${\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]$ is ${{R}_{2}}(c)={\partial {\mathbb {E}}[{\varPi }_{new}(p_2,z_1)]}/{\partial {{p}_{2}}}|_{p_2=c}>0$. $\square$