Multi-period pricing and order decisions for fresh produce with option contracts

Abstract

With the construction of a multi-period newsvendor model, this study examines the ordering behaviour and pricing decisions of a fresh produce firm. The firm ordered the fresh produce via a wholesale price contract and option contract to determine the optimal preliminary order quantity, option order quantity, and retail pricing when faced with price-dependent random demand. We first determine the optimal decision that could be made through the use of the single-period model. The results revealed that the firm’s retail price, option order quantity and total order quantity all increased in the circulation loss rate. In contrast, the preliminary order quantity was found to decrease in the circulation loss rate. In the multi-period model, the retail price and total order quantity were unaffected by the price difference between the regular and concessionary prices, while an increase in the price difference caused an increase in the option order quantity and decrease in the preliminary order quantity in the previous period. Further, a lower discount factor for the concessionary price than for the regular price did not have as much of an effect on the retail price and total order quantity, while a rise in the discount factor led to a fall in the option order quantity as well as a rise in the preliminary order quantity in the previous period.

Appendix

1.1 Lemma 1 Proof

When $\left(z,{q}_{o}\right)$ is given, it is easy to find $\frac{dE\left[\pi \left(p,z,{q}_{o}\right)\right]}{dp}=a-2bp+brw+\mu -\Theta \left(z\right)$ and $\frac{{d}^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{d{p}^{2}}=-2b<0$. Then, $E\left[\pi \left(p,q,{q}_{o}\right)\right]$ is a concave function on $p$. This means that given $\left(z,{q}_{o}\right)$, let $\frac{dE\left[\pi \left(p,q,{q}_{o}\right)\right]}{dp}=0$ and the optimal pricing ${p}^{*}\equiv p(z)={p}_{d}^{*}-\frac{\Theta \left(z\right)}{2b}$, where ${p}_{d}^{*}=\frac{a+brw+\mu }{2b}$.

1.2 Lemma 2 Proof

When $p$ is given, $\frac{dE\left[\pi \left(p,z,{q}_{o}\right)\right]}{dz}=-\left(1-\beta \right)\left(e+h\right)F\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]-\left(p+g-e\right)\left(1-\beta \right)F\left[z\left(1-\beta \right)\right]+\left(p+g\right)\left(1-\beta \right)-w$, $\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {z}^{2}}=-{\left(1-\beta \right)}^{2}\left(e+h\right)f\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]-{\left(1-\beta \right)}^{2}\left(p+g-e\right)f\left[z\left(1-\beta \right)\right]<0$, $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}}=\left(1-\beta \right)\left(e+h\right)F\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]-\left[o+\left(1-\beta \right)e-w\right]$, $\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}^{2}}=-{\left(1-\beta \right)}^{2}\left(e+h\right)f\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]<0$, $\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial z\partial {q}_{o}}=\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}\partial z}={\left(1-\beta \right)}^{2}\left(e+h\right)f\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]$. Since $\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {z}^{2}}<0$ and $\left|\begin{array}{cc}\frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {z}^{2}}& \frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial z\partial {q}_{o}}\\ \frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}\partial z}& \frac{{\partial }^{2}E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}^{2}}\end{array}\right|={\left(1-\beta \right)}^{4}\left(e+h\right)\left(p+g-e\right)f\left[z\left(1-\beta \right)\right]f\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]>0$, the Hessian matrix of $E\left[\pi \left(p,z,{q}_{o}\right)\right]$ is negative definite. Thus, $E\left[\pi \left(p,z,{q}_{o}\right)\right]$ is concave in $z$ and ${q}_{o}$ simultaneously. The unique ${\mathrm{z}}^{*}$ and unique ${q}_{o}^{*}$ are set by $\frac{dE\left[\pi \left(p,z,{q}_{o}\right)\right]}{dz}=0$ and $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}}=0$, i.e., ${z}^{*}=r{F}^{-1}\left[1-\frac{o}{\left(1-\beta \right)\left(p+g-e\right)}\right]\mathrm{ and }{q}_{o}^{*}={z}^{*}-r{F}^{-1}\left[\frac{o+\left(1-\beta \right)e-w}{\left(1-\beta \right)\left(e+h\right)}\right]$.

1.3 Proposition 1 Proof

We first prove the condition that there exists an unique optimal retail price ${p}^{*}$. Let ${z}^{*}\equiv z(p)$ and ${q}_{o}^{*}\equiv {q}_{o}(p)$. Substituting ${z}^{*}\equiv z(p)$, ${q}_{o}^{*}\equiv {q}_{o}(p)$ into $E\left[\pi \left(p,z,{q}_{o}\right)\right]$, we can obtain $E\left[\pi \left(p\right)\right]=\left(p-rw\right)y\left(p\right)+\left(p+g-e\right){\int }_{A}^{(1-\beta ){z}^{*}}\varepsilon f\left(\varepsilon \right)d\varepsilon -g\mu -\left(e+h\right){\int }_{A}^{\left(1-\beta \right){z}^{*}-\left(1-\beta \right){q}_{o}^{*}}\varepsilon f\left(\varepsilon \right)d\varepsilon +\left[o+\left(1-\beta \right)e-w\right]{rF}^{-1}\left[\frac{o+\left(1-\beta \right)e-w}{\left(1-\beta \right)\left(e+h\right)}\right]$. Since $\frac{dz(p)}{dp}=\frac{d{q}_{o}(p)}{dp}=\frac{o}{{{\left(1-\beta \right)}^{2}\left(p+g-e\right)}^{2}f\left[{z}^{*}\left(1-\beta \right)\right]}$, then we have $\frac{dE\left[\pi \left(p\right)\right]}{dp}=y\left(p\right)-b\left(p-rw\right)+{\int }_{A}^{(1-\beta ){z}^{*}}\varepsilon f\left(\varepsilon \right)d\varepsilon +{z}^{*}\frac{o}{p+g-e}$, $\frac{{d}^{2}E\left[\pi \left(p\right)\right]}{d{p}^{2}}=-2b+\frac{{\left\{1-F\left[{z}^{*}\left(1-\beta \right)\right]\right\}}^{2}}{\left(p+g-e\right)f\left[{z}^{*}\left(1-\beta \right)\right]}$. Consequently, the prerequisite for an unique optimal retail price ${p}^{*}$ is $\frac{{d}^{2}E\left[\pi \left(p\right)\right]}{d{p}^{2}}<0$, i.e.$h\left[{z}^{*}\left(1-\beta \right)\right]>\frac{1-F\left[{z}^{*}\left(1-\beta \right)\right]}{2b\left(p+g-e\right)}$.

We then prove the condition that there exists an unique stocking factor ${z}^{*}$. Substituting ${p}^{*}\equiv p(z)$ into $\left[\pi \left(p,z,{q}_{o}\right)\right]$, we can obtain $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial z}=\left[{p}_{d}^{*}-\frac{\Theta \left(z\right)}{2b}+g-e\right]\left(1-\beta \right)\left\{1-F\left[z\left(1-\beta \right)\right]\right\}-\left(1-\beta \right)\left(e+h\right)F\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]-\left[w-\left(1-\beta \right)e\right]$, $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}}=\left(1-\beta \right)\left(e+h\right)F\left[\left({z-q}_{o}\right)\left(1-\beta \right)\right]-\left[o+\left(1-\beta \right)e-w\right]$. From Lemma 1, let $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial z}=0$, and $\frac{\partial E\left[\pi \left(p,z,{q}_{o}\right)\right]}{\partial {q}_{o}}=0$, respectively. Then, we obtain that the unique ${z}^{*}\left({z}^{*}\in \left[A,B\right]\right)$ satisfies $\left[{p}_{d}^{*}+g-\frac{\Theta \left({z}^{*}\right)}{2b}-e\right]\left\{1-F\left[{z}^{*}\left(1-\beta \right)\right]\right\}-o=0$.

1.4 Corollary 1 Proof

From Lemmas 1 and 2, it can be seen that the handling cost $h$ has no impact on the optimal pricing ${p}^{*}$, the optimal inventory factor ${z}^{*}$ and the optimal total quantity ${q}^{*}$, but only on the optimal preliminary order quantity ${q}_{w}^{*}$ and the optimal option order quantity ${q}_{o}^{*}$, so we can obtain $\frac{{dq}_{w1}^{*}}{dh}=-\frac{o+\left(1-\beta \right)e-w}{{\left[\left(1-\beta \right)\left(e+h\right)\right]}^{2}f\left[\left({{z}^{*}-q}_{o}^{*}\right)\left(1-\beta \right)\right]}<0$, $\frac{{dq}_{o1}^{*}}{dh}=\frac{o+\left(1-\beta \right)e-w}{{\left[\left(1-\beta \right)\left(e+h\right)\right]}^{2}f\left[\left({{z}^{*}-q}_{o}^{*}\right)\left(1-\beta \right)\right]}>0$.

1.5 Proposition 2 Proof

For the period $i(i=\mathrm{1,2}\cdots \left(N{-}1\right))$, when $\left({z}_{i},{q}_{oi}\right)$ is given, $\frac{dE\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}}=a{-}2b{p}_{i}+br{w}_{i}+\mu {-}{\Theta }_{i}\left({z}_{i}\right)$ and $\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}^{2}}={-}2b<0$. Thus, $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is concave in ${p}_{i}$. Let$\frac{dE\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}}=0$, the period $i$ optimal pricing${p}_{i}^{*}\equiv p\left({z}_{i}\right)={p}_{id}^{*}{-}\frac{{\Theta }_{i}\left({z}_{i}\right)}{2b}$, where ${p}_{di}^{*}=\frac{a+br{w}_{i}+\mu }{2b}$. Then, when ${p}_{i}$ is given, $\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}={-}\left(1{-}\beta \right)[{e}_{i}+\tau +{m}_{i+1}{-}r{w}_{i+1}] F[\left({{z}_{i}{-}q}_{oi}\right)(1{-}\beta)]{-}({p}_{i}+g{-}{e}_{i})(1{-}\beta)F[{z}_{i}\left(1{-}\beta \right)]+\left({p}_{i}+g\right)\left(1{-}\beta \right){-}{w}_{i}$, $\frac{{\partial ^{2} E[ {\pi _{1} \left( {p_{1} ,z_{1} ,q_{{o1}} } \right)} ]}}{{\partial z_{1}^{2} }} = - \left( {1 - \beta } \right)^{2} \left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]f\left[ {\left( {z_{i} - q_{{oi}} } \right)\left( {1 - \beta } \right)} \right] - \left( {p_{i} + g - e_{i} } \right)\left( {i - \beta } \right)^{2} f\left[ {z_{i} \left( {i - \beta } \right)} \right] < 0$,$\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=\left(1{-}\beta \right)\left[{e}_{i}+\tau +{m}_{i+1}{-}r{w}_{i+1}\right] F\left[\left({{z}_{i}{-}q}_{oi}\right)\left(1{-}\beta \right)\right]{-}\left[{o}_{i}+\left(1{-}\beta \right){e}_{i}{-}{w}_{i}\right]$,$\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}}^{2} }} = - \left( {1 - \beta } \right)^{2} \left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]f\left[ {\left( {z_{i} - q_{{oi}} } \right)\left( {i - \beta } \right)} \right] < 0$,$\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i} \partial q_{{oi}} }} = \frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}} \partial z_{i} }} = \left( {1 - \beta } \right)^{2} \left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]f\left[ {\left( {z_{i} - q_{{oi}} } \right)\left( {1 - \beta } \right)} \right]$. Since $\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i}^{2} }} < 0$, $\left| {\begin{array}{*{20}c} {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i}^{2} }}} & {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i} \partial q_{{oi}} }}} \\ {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}} \partial z_{i} }}} & {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}}^{2} }}} \\ \end{array} } \right| = \left( {1 - \beta } \right)^{4} \left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]\left( {p_{i} + g - e_{i} } \right)f\left( {z_{i} } \right)f\left[ {\left( {z_{i} - q_{{oi}} } \right)\left( {1 - \beta } \right)} \right] > 0$, the Hessian Matrix of $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is negative definite. Thus, $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is concave in ${z}_{i}$ and ${q}_{oi}$ simultaneously. The unique ${z}_{i}^{*}$ and unique ${q}_{oi}^{*}$ are set by $\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}=0$ and $\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=0$, i.e., ${z}_{i}^{*}=r{F}^{-1}\left[1-\frac{{o}_{i}}{\left(1-\beta \right)\left({p}_{i}+g-{e}_{i}\right)}\right]\mathrm{ and }{q}_{oi}^{*}={z}_{i}^{*}-r{F}^{-1}\left\{\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{\left(1-\beta \right)\left[{e}_{i}+\tau +{m}_{i+1}-r{w}_{i+1}\right]}\right\}$. Substituting ${p}_{i}^{*}\equiv p({z}_{i})$ into$E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$, let $\frac{\partial E\left[{\pi }_{i}\left(p({z}_{i}),{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}=0$, and $\frac{\partial E\left[{\pi }_{i}\left(p({z}_{i}),{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=0$, respectively. Then, we obtain that the unique ${z}_{i}^{*}({z}_{i}^{*}\in \left[A,B\right])$ satisfies $\left[{p}_{di}^{*}+g-\frac{{\Theta }_{i}\left({z}_{i}^{*}\right)}{2b}-{e}_{i}\right]\left\{1-F\left[{z}_{i}\left(1-\beta \right)\right]\right\}-{o}_{i}=0$.

Let ${z}_{i}^{*}\equiv {z}_{i}\left({p}_{i}\right)$ and ${q}_{oi}^{*}\equiv {q}_{oi}({p}_{i})$ Substituting ${z}_{i}^{*}\equiv {z}_{i}\left({p}_{i}\right)$, ${q}_{oi}^{*}\equiv {q}_{oi}({p}_{i})$ into $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$, we can obtain $E\left[ {\pi _{i} \left( {p_{i} } \right)} \right] = \left( {p_{i} - rw_{i} } \right)y_{i} \left( {p_{i} } \right) + \left( {p_{i} + g - e_{i} } \right)\smallint _{A}^{{\left( {1 - \beta } \right)z_{i}^{*} }} \varepsilon _{i} f\left( {\varepsilon _{i} } \right)d\varepsilon _{i} - g\mu - \left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]\smallint _{A}^{{\left( {1 - \beta } \right)z_{i}^{*} - \left( {1 - \beta } \right)q_{{oi}}^{*} }} \varepsilon _{i} f\left( {\varepsilon _{i} } \right)d\varepsilon _{i} + \left[ {o_{i} + \left( {1 - \beta } \right)e_{i} - w_{i} } \right]rF^{{ - i}} \left\{ {\frac{{o_{i} + \left( {1 - \beta } \right)e_{i} - w_{i} }}{{\left( {1 - \beta } \right)\left[ {e_{i} + \tau + m_{{i + 1}} - rw_{{i + 1}} } \right]}}} \right\} + \left[ {rw_{i} - m_{{i + 1}} } \right]\Lambda _{{i - 1}} \left( {z_{{i - 1}}^{*} ,q_{{o\left( {i - 1} \right)}}^{*} } \right)$, $\frac{dE\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}}={y}_{i}\left({p}_{i}\right)-b\left({p}_{i}-{{r}_{i}w}_{i}\right)+{\int }_{A}^{\left(1-\beta \right){z}_{i}^{*}}{\varepsilon }_{i}f\left({\varepsilon }_{i}\right)d{\varepsilon }_{i}+{z}_{i}^{*}\frac{{o}_{i}}{{p}_{i}+g-{e}_{i}}$, $\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}^{2}}=-2b+\frac{1-F\left[{z}_{i}^{*}\left(1-\beta \right)\right]}{\left({p}_{i}^{*}+g-{e}_{i}\right)h\left[{z}_{i}^{*}\left(1-\beta \right)\right]}$. Consequently, the prerequisite for an unique optimal retail price ${p}_{i}^{*}$ is $\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}^{2}}<0$, i.e.$2b>\frac{1-F\left[{z}_{i}^{*}\left(1-\beta \right)\right]}{\left({p}_{i}^{*}+g-{e}_{i}\right)h\left[{z}_{i}^{*}\left(1-\beta \right)\right]}$

For the $N$ sales period, the proof is the same as above except that ${q}_{oN}^{*}={z}_{N}^{*}-r{F}^{-1}\left\{\frac{{o}_{N}+\left(1-\beta \right){e}_{N}-{w}_{N}}{\left(1-\beta \right)\left[{e}_{N}+h\right]}\right\}$.

1.6 Corollary 2 Proof

From Proposition 2, it can be seen that the freshness-keeping cost $\tau$ has no impact on the optimal pricing ${p}_{i}^{*}$, the optimal inventory factor ${z}_{i}^{*}$ and the optimal total quantity ${q}_{i}^{*}$, but only on the optimal preliminary order quantity ${q}_{wi}^{*}$ and the optimal option order quantity ${q}_{oi}^{*}$, so we can obtain $\frac{{dq}_{wi}^{*}}{d\tau }=-\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{{\left(1-\beta \right)}^{2}{\left[{e}_{i}+\tau +{m}_{i+1}-r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}<0$, $\frac{{dq}_{oi}^{*}}{d\tau }=\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{{\left(1-\beta \right)}^{2}{\left[{e}_{i}+\tau +{m}_{i+1}-r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}>0$.

1.7 Corollary 3 Proof

From Proposition 2, it seen by that the price difference ${m}_{i}$ has no impact on the optimal pricing ${p}_{i}^{*}$, the optimal inventory factor ${z}_{i}^{*}$ and the optimal total quantity ${q}_{i}^{*}$, but only on the optimal preliminary order quantity ${q}_{wi}^{*}$ and the optimal option order quantity ${q}_{oi}^{*}$, so we can obtain $\frac{{dq}_{oi}^{*}}{d{m}_{i+1}}=\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{{\left(1-\beta \right)}^{2}{\left[{e}_{i}+\tau +{m}_{i+1}-r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}>0$, $\frac{{dq}_{wi}^{*}}{d{m}_{i+1}}=-\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{{\left(1-\beta \right)}^{2}{\left[{e}_{i}+\tau +{m}_{i+1}-r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}<0$, and $\frac{{dq}_{oi}^{*}}{d{m}_{i+1}}=-\frac{{dq}_{wi}^{*}}{d{m}_{i+1}}$.

1.8 Proposition 3 Proof

For the period$i(i=\mathrm{1,2}\cdots \left(N-1\right))$, when $\left({z}_{i},{q}_{oi}\right)$ is given, $\frac{dE\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}}=a-2b{p}_{i}+br{w}_{i}+\mu -{\Theta }_{i}\left({z}_{i}\right)-\left(1-{\theta }_{i}\right){\Lambda }_{i-1}\left({z}_{i-1},{q}_{o\left(i-1\right)}\right)$ and$\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}^{2}}=-2b<0$. Thus, $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is concave in${p}_{i}$. Let$\frac{dE\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{d{p}_{i}}=0$, the period $i$ optimal pricing${p}_{i}^{*}\equiv p\left({z}_{i},{z}_{i-1},{q}_{o\left(i-1\right)}\right)={p}_{id}^{*}-\frac{{\Theta }_{i}\left({z}_{i}\right)}{2b}-\frac{\left(1-{\theta }_{i}\right){\Lambda }_{i-1}\left({z}_{i-1},{q}_{o\left(i-1\right)}\right)}{2b}$, where${p}_{di}^{*}=\frac{a+br{w}_{i}+\mu }{2b}$. Then, when ${p}_{i}$ is given, $\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}=-\left(1-\beta \right)\left[(1-{\theta }_{i+1}){p}_{i+1}+{e}_{i}+\tau -r{w}_{i+1}\right]F\left[\left({{z}_{i}-q}_{oi}\right)\left(1-\beta \right)\right]-\left({p}_{i}+g-{e}_{i}\right)\left(1-\beta \right)F\left[{z}_{i}\left(1-\beta \right)\right]+\left({p}_{i}+g\right)\left(1-\beta \right)-{w}_{i}$,$\frac{{\partial }^{2}E\left[{\pi }_{1}\left({p}_{1},{z}_{1},{q}_{o1}\right)\right]}{\partial {z}_{1}^{2}}=-{\left(1-\beta \right)}^{2}\left[(1-{\theta }_{i+1}){p}_{i+1}+{e}_{i}+\tau -r{w}_{i+1}\right]f\left[\left({{z}_{i}-q}_{oi}\right)\left(1-\beta \right)\right]-\left({p}_{i}+g-{e}_{i}\right){\left(i-\beta \right)}^{2}f\left[{z}_{i}\left(i-\beta \right)\right]<0$,$\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=\left(1-\beta \right)[(1-{\theta }_{i+1}){p}_{i+1}+{e}_{i}+\tau -r{w}_{i+1}]F[\left({{z}_{i}-q}_{oi}\right)\left(1-\beta \right)]-[{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}]$, $\frac{{\partial ^{2} E[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} ]}}{{\partial q_{{oi}}^{2} }} = - \left( {1 - \beta } \right)^{2} [ {(1 - \theta _{{i + 1}} )p_{{i + 1}} + e_{i} + \tau - rw_{{i + 1}} } ]f[ {\left( {z_{i} - q_{{oi}} } \right)\left( {i - \beta } \right)} ] < 0$, $\frac{{\partial ^{2} E[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} ]}}{{\partial z_{i} \partial q_{{oi}} }} = \frac{{\partial ^{2} E[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} ]}}{{\partial q_{{oi}} \partial z_{i} }} = \left( {1 - \beta } \right)^{2} [ {( {1 - \theta _{{i + 1}} } )p_{{i + 1}} + e_{i} + \tau - rw_{{i + 1}} } ]f[ ( z_{i} - q_{{oi}} )( 1 - \beta ) ]$. Since$\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i}^{2} }} < 0$, $\left| {\begin{array}{*{20}c} {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i}^{2} }}} & {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial z_{i} \partial q_{{oi}} }}} \\ {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}} \partial z_{i} }}} & {\frac{{\partial ^{2} E\left[ {\pi _{i} \left( {p_{i} ,z_{i} ,q_{{oi}} } \right)} \right]}}{{\partial q_{{oi}}^{2} }}} \\ \end{array} } \right| = \left( {1 - \beta } \right)^{4} \left[ {(1 - \theta _{{i + 1}} )p_{{i + 1}} + e_{i} + \tau - rw_{{i + 1}} } \right]\left( {p_{i} + g - e_{i} } \right)f\left( {z_{i} } \right)f\left[ {\left( {z_{i} - q_{{oi}} } \right)\left( {1 - \beta } \right)} \right] > 0$, the Hessian matrix of $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is negative definite. Thus, $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$ is concave in ${z}_{i}$ and ${q}_{oi}$ simultaneously. The unique ${z}_{i}^{*}$ and unique ${q}_{oi}^{*}$ are set by $\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}=0$ and$\frac{\partial E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=0$, i.e.,${z}_{i}^{*}=r{F}^{-1}\left[1-\frac{{o}_{i}}{\left(1-\beta \right)\left({p}_{i}+g-{e}_{i}\right)}\right]\mathrm{ and }{q}_{oi}^{*}={z}_{i}^{*}-r{F}^{-1}\left\{\frac{{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}}{\left(1-\beta \right)\left[(1-{\theta }_{i+1}){p}_{i+1}+{e}_{i}+\tau -r{w}_{i+1}\right]}\right\}$. Substituting${p}_{i}^{*}\equiv p({z}_{i},{z}_{i-1},{q}_{o\left(i-1\right)})$, ${p}_{i+1}^{*}\equiv p({z}_{i+1},{z}_{i},{q}_{oi})$ into$E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$, let $\frac{\partial E\left[{\pi }_{i}\left(p({z}_{i}),{z}_{i},{q}_{oi}\right)\right]}{\partial {z}_{i}}=0$, and$\frac{\partial E\left[{\pi }_{i}\left(p({z}_{i}),{z}_{i},{q}_{oi}\right)\right]}{\partial {q}_{oi}}=0$, respectively. Then, we obtain that the unique ${z}_{i}^{*}({z}_{i}^{*}\in \left[A,B\right])$ satisfies$\left[{p}_{di}^{*}+g-\frac{{\Theta }_{i}\left({z}_{i}^{*}\right)}{2b}-\frac{\left(1-{\theta }_{i}\right){\Lambda }_{i-1}\left({z}_{i-1}^{*},{q}_{o\left(i-1\right)}^{*}\right)}{2b}-{e}_{i}\right]\left\{1-F\left[{z}_{i}\left(1-\beta \right)\right]\right\}-{o}_{i}=0$.

Let ${z}_{i}^{*}\equiv {z}_{i}\left({p}_{i}\right)$, ${q}_{oi}^{*}\equiv {q}_{oi}({p}_{i})$ and ${q}_{o\left(i-1\right)}^{*}\equiv {q}_{o\left(i-1\right)}({p}_{i})$. Substituting ${z}_{i}^{*}\equiv {z}_{i}\left({p}_{i}\right)$, ${q}_{oi}^{*}\equiv {q}_{oi}({p}_{i})$ and ${q}_{o\left(i-1\right)}^{*}\equiv {q}_{o\left(i-1\right)}({p}_{i})$ into $E\left[{\pi }_{i}\left({p}_{i},{z}_{i},{q}_{oi}\right)\right]$, we can obtain $E\left[ {\pi _{i} \left( {p_{i} } \right)} \right] = \left( {p_{i} - rw_{i} } \right)y_{i} \left( {p_{i} } \right) + \left( {p_{i} + g - e_{i} } \right)\smallint _{A}^{{\left( {1 - \beta } \right)z_{i}^{*} }} \varepsilon _{i} f\left( {\varepsilon _{i} } \right)d\varepsilon _{i} - g\mu - \left[ {\left( {1 - \theta _{{i + 1}} } \right)p_{{i + 1}} + e_{i} + \tau - rw_{{i + 1}} } \right]\smallint _{A}^{{\left( {1 - \beta } \right)z_{i}^{*} - \left( {1 - \beta } \right)q_{{oi}}^{*} }} \varepsilon _{i} f\left( {\varepsilon _{i} } \right)d\varepsilon _{i} + [ o_{i} + ( {1 - \beta } )e_{i} - w_{i} ]rF^{{ - i}} \left\{ {\frac{{o_{i} + \left( {1 - \beta } \right)e_{i} - w_{i} }}{{\left( {1 - \beta } \right)\left[ {\left( {1 - \theta _{{i + 1}} } \right)p_{{i + 1}} + e_{i} + \tau - rw_{{i + 1}} } \right]}}} \right\} - [ {( {1 - \theta _{i} } )p_{i} - rw_{i} } ]\Lambda _{{i - 1}} ( {z_{{i - 1}}^{*} ,q_{{o( {i - 1} )}}^{*} } ) + E [\pi _{{i - 1}} \left( {p_{i} } \right) ]$, $\frac{dE\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}}={y}_{i}\left({p}_{i}\right)-b\left({p}_{i}-{{r}_{i}w}_{i}\right)+{\int }_{A}^{\left(1-\beta \right){z}_{i}^{*}}{\varepsilon }_{i}f\left({\varepsilon }_{i}\right)d{\varepsilon }_{i}+{z}_{i}^{*}\frac{{o}_{i}}{{p}_{i}+g-{e}_{i}}-\left(1-{\theta }_{i}\right){\Lambda }_{i-1}\left({z}_{i-1}^{*},{q}_{o\left(i-1\right)}^{*}\right)$, $\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}^{2}}=-2b+\frac{1-F\left[{z}_{i}^{*}\left(1-\beta \right)\right]}{\left({p}_{i}^{*}+g-{e}_{i}\right)h\left[{z}_{i}^{*}\left(1-\beta \right)\right]}+\frac{{\left(1-\theta \right)}^{2}{F}^{2}\left[\left({z}_{i-1}^{*}-{q}_{o\left(i-1\right)}^{*}\right)\left(1-\beta \right)\right]}{\left[{\left(1-{\theta }_{i}\right){p}_{i}^{*}+e}_{i-1}+\tau -r{w}_{i}\right]f\left[\left({z}_{i-1}^{*}-{q}_{o\left(i-1\right)}^{*}\right)\left(1-\beta \right)\right]}$. Consequently, the prerequisite for an unique optimal retail price ${p}_{i}^{*}$ is $\frac{{d}^{2}E\left[{\pi }_{i}\left({p}_{i}\right)\right]}{d{p}_{i}^{2}}<0$, i.e.$2b>\frac{1-F\left[{z}_{i}^{*}\left(1-\beta \right)\right]}{\left({p}_{i}^{*}+g-{e}_{i}\right)h\left[{z}_{i}^{*}\left(1-\beta \right)\right]}+\frac{{\left(1-{\theta }_{i}\right)}^{2}{F}^{2}\left[\left({z}_{i-1}^{*}-{q}_{o\left(i-1\right)}^{*}\right)\left(1-\beta \right)\right]}{\left[{\left(1-{\theta }_{i}\right){p}_{i}^{*}+e}_{i-1}+\tau -r{w}_{i}\right]f\left[\left({z}_{i-1}^{*}-{q}_{o\left(i-1\right)}^{*}\right)\left(1-\beta \right)\right]}$.

For the $N$ sales period, the proof is the same as above except that ${q}_{oN}^{*}={z}_{N}^{*}-r{F}^{-1}\left\{\frac{{o}_{N}+\left(1-\beta \right){e}_{N}-{w}_{N}}{\left(1-\beta \right)\left[{e}_{N}+h\right]}\right\}$.

For the first sales period, it is the same as the single-period model except that ${q}_{o1}^{*}={z}_{i}^{*}-r{F}^{-1}\left\{\frac{{o}_{1}+\left(1-\beta \right){e}_{1}-{w}_{1}}{\left(1-\beta \right)\left[\left(1-{\theta }_{2}\right){p}_{2}^{*}+{e}_{1}+\tau -r{w}_{2}\right]}\right\}$.

1.9 Corollary 5 Proof

From Proposition 3 and Corollary 4, for the $i(i=\mathrm{1,2}\cdots \left(N-1\right))$ sales period, we can obtain $\frac{{dq}_{oi}{\prime}}{d{\theta }_{i+1}}=-\frac{{p}_{i+1}^{*}\left[{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}\right]}{{\left(1-\beta \right)}^{3}{\left[(1-{\theta }_{i+1}){p}_{i}^{*}+{e}_{i}+\tau -r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}<0$, $\frac{{dq}_{wi}{\prime}}{d{\theta }_{i+1}}=\frac{{p}_{i+1}^{*}\left[{o}_{i}+\left(1-\beta \right){e}_{i}-{w}_{i}\right]}{{\left(1-\beta \right)}^{3}{\left[(1-{\theta }_{i+1}){p}_{i}^{*}+{e}_{i}+\tau -r{w}_{i+1}\right]}^{2}f\left[\left({z}_{i}^{*}-{q}_{oi}^{*}\right)\left(1-\beta \right)\right]}>0$, and $\frac{{dq}_{oi}{\prime}}{d{\theta }_{i+1}}=-\frac{{dq}_{wi}{\prime}}{d{\theta }_{i+1}}$.