Decision-making of fresh agricultural product supply chain considering the manufacturer’s fairness concerns

Abstract

By considering the characteristics of fresh agricultural products, this article introduces the behavioral tendency of fairness concerns of manufacturer to examine the fresh agricultural product supply chain and the optimal decision-making problem. The effects of the manufacturer’s behavioral tendency of fairness concerns on the fresh agricultural product supply chain is analyzed within the traditional and Nash bargaining frameworks, respectively. Results indicate that the manufacturer’s behavioral tendencies can reduce the freshness-keeping effort, the freshness of the produce, and the market demand of fresh agricultural products. The optimal wholesale price and retail price also increase or decrease to meet the manufacturer’s fairness concerns. These are determined by the price elasticity and the level of fairness concern, along with the fairness concern condition. Whether within the traditional or Nash bargaining fairness concern framework, market demand declines significantly with the increase in the level of fairness concern, thereby leading to the reduction of the utility of retailer in the supply chain. Therefore, the revenue sharing contract is introduced to achieve the Pareto improvement of both the manufacturer and the retailer.

Appendix

Proof of Proposition 1

Proof On the basis of the utility function of the retailer, we obtain the following Hessian matrix of $\cup_{R}^{{}}$:

$$ H_{R} = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \cup_{R}^{{}} }}{{\partial p^{2} }}} & {\frac{{\partial^{2} \cup_{R}^{{}} }}{{\partial p\partial e_{2} }}} \\ {\frac{{\partial^{2} \cup_{R}^{{}} }}{{\partial e_{2} \partial p}}} & {\frac{{\partial^{2} \cup_{R}^{{}} }}{{\partial e_{2}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2b} & {k\gamma_{2} } \\ {k\gamma_{2} } & { - \beta_{2} } \\ \end{array} } \right] $$

From the formula above, we obtain $\left| {H_{R} } \right| = 2b\beta_{2} - k^{2} \gamma_{2}^{2}$. When $2b\beta_{2} - k^{2} \gamma_{2}^{2} > 0$, clearly, we can further conclude that $\left| H \right| > 0$. Furthermore, the first-order sequential principal minor $\left| {H_{R1} } \right| = - 2b < 0$, so that the Hessian matrix is negative definite. That is, $\cup_{R}^{{}}$ is a joint concave function of $p$ and $e_{2}$. Taking the first-order derivative on $\cup_{R}^{{}}$ with respect to $p$ and $e_{2}$, and letting them be zero, that is, ${{\partial \cup_{R}^{{}} } \mathord{\left/ {\vphantom {{\partial \cup_{R}^{{}} } {\partial p}}} \right. \kern-0pt} {\partial p}} = 0$ and ${{\partial \cup_{R}^{{}} } \mathord{\left/ {\vphantom {{\partial \cup_{R}^{{}} } {\partial e}}} \right. \kern-0pt} {\partial e}}_{2} = 0$, we obtain the best response function of the retail price $p$ and the freshness-keeping effort $e_{2}$ to the wholesale price $w$ and the freshness-keeping effort $e_{1}$ as follows:

$$ p^{\prime } = \frac{{\beta_{2} (a + bw + k\theta_{0} + k\gamma_{1} e_{1} ) - k^{ 2} \gamma_{2}^{2} w}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}. $$

$$ e_{2}^{\prime } = \frac{{k\gamma_{2}^{{}} (a - bw + k\theta_{0} + k\gamma_{1} e_{1} )}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}. $$

By combining the equations above, we obtain the Hessian matrix of $\cup_{S}^{{}}$:

$$ H_{S} = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial w^{2} }}} & {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial w\partial e_{1} }}} \\ {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial e_{1} \partial w}}} & {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial e_{1}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \frac{{2b^{2} \beta_{2} }}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} & {\frac{{b\beta_{2} k\gamma_{1}^{{}} }}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} \\ {\frac{{b\beta_{2} k\gamma_{1}^{{}} }}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} & { - \beta_{1} } \\ \end{array} } \right]. $$

By solving the equation above, we conclude that $\left| {H_{S} } \right| = \frac{{b^{2} \beta_{2} (4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )^{2} }}$. When $4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$ clearly, we can obtain $\left| {H_{S} } \right| > 0$. In addition, the first-order sequential principal minor meets the condition that $\left| {H_{S1} } \right| = - 2b^{2} \beta_{2} /(2b\beta_{2} - k^{2} \gamma_{2}^{2} ) < 0$, so that the Hessian matrix above is negative definite. Taking the first-order derivative on $\cup_{S}$ with respect to $e_{1}$ and $w$, and letting them be zero, we obtain the optimal freshness-keeping effort $e_{1}^{ \wedge }$ and the optimal wholesale price $w^{ \wedge }$:

$$ w^{ \wedge } = \frac{{\beta_{1} (2b\beta_{2} - k^{2} \gamma_{2}^{2} )(a + k\theta_{0} ) + bc(2b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{b(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}. $$

$$ e_{1}^{ \wedge } = \frac{{\beta_{2} k\gamma_{1}^{{}} (a - bc + k\theta_{0} )}}{{4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} }}. $$

Then, we can obtain the optimal retail price $p^{ \times }$ as follows:

$$ p^{ \wedge } = \frac{{\beta_{1} (3b\beta_{2} - k^{2} \gamma_{2}^{2} )(a + k\theta_{0} ) + bc(b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{b(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}. $$

$$ e_{2}^{ \wedge } = \frac{{k\beta_{1} \gamma_{2} (a - bc + k\theta_{0} )}}{{4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} }}. $$

When $4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$ and $2b\beta_{2} - k^{2} \gamma_{2}^{2} > 0$, we can get the above optimal solution, which would be greater than zero considering practical significance. Obviously, $A{ = }4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$ and $a - bc + k\theta_{0} > 0$ can be obtained because of $U_{S}^{ \wedge } > 0$ and $D^{ \wedge } > 0$, respectively. If $4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$, clearly, $2b\beta_{2} - k^{2} \gamma_{2}^{2} > \frac{{\beta_{2} k^{2} \gamma_{1}^{2} }}{{2\beta_{1} }} > 0$. If $a - bc + k\theta_{0} > 0$, clearly, $a + k\theta_{0} > bc$, we would obtain $\begin{aligned} p^{ \wedge } = & \frac{{\beta_{1} (3b\beta_{2} - k^{2} \gamma_{2}^{2} )(a + k\theta_{0} ) + bc(b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{b(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}} \\ & > \frac{{\beta_{1} (3b\beta_{2} - k^{2} \gamma_{2}^{2} )bc + bc(b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{b(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}} \\ & = \frac{{bc(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}}{{b(4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} )}} = c > 0 \\ \end{aligned}$. Similarly, $w^{ \wedge } > 0$ can also be proven.

In a word, the unique optimal decision can be obtained and can be practically significant provided that $A= 4b\beta_{1} \beta_{2} - 2\beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$. The proof is complete.□

Proof of Proposition 2

Proof: Similar to Proposition 1, we can obtain the best response function of the retail price $p$ and the freshness-keeping effort $e_{2}$ to the wholesale price $w$ and the freshness-keeping effort $e_{1}$ as follows:

$$ p^{\prime } = \frac{{\beta_{2} (a + bw + k\theta_{0} + k\gamma_{1} e_{1} ) - k^{ 2} \gamma_{2}^{2} w}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}. $$

$$ e_{2}^{\prime } = \frac{{k\gamma_{2}^{{}} (a - bw + k\theta_{0} + k\gamma_{1} e_{1} )}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}. $$

By combining these equations, we obtain the Hessian matrix of $\cup_{S}^{{}}$:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial w^{2} }}} & {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial w\partial e_{1} }}} \\ {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial e_{1} \partial w}}} & {\frac{{\partial^{2} \cup_{S}^{{}} }}{{\partial e_{1}^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \frac{{b^{2} \beta_{2} (3\lambda + 2)}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} & {\frac{{b\beta_{2} k\gamma_{1}^{{}} (2\lambda + 1)}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} \\ {\frac{{b\beta_{2} k\gamma_{1}^{{}} (2\lambda + 1)}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} & { - \frac{{\beta_{1} (2b\beta_{2} - k^{2} \gamma_{2}^{2} )(\lambda + 1) - \beta_{2} k^{2} \gamma_{1}^{2} \lambda }}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}}} \\ \end{array} } \right]. $$

From the formula above, we obtain $\left| H \right| = \frac{{b^{2} \beta_{2} (\lambda + 1)[A + B\lambda ]}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )^{2} }}$. Combining this equation with $b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} - \beta_{2} k^{2} \gamma_{1}^{2} > 0$, we can conclude that $\left| H \right| > 0$. Furthermore, the first-order sequential principal minor $\left| {H_{1} } \right| = - \frac{{b^{2} \beta_{2} (3\lambda + 2)}}{{(2b\beta_{2} - k^{2} \gamma_{2}^{2} )}} < 0$, so that the Hessian matrix is negative. That is, $\cup_{S}^{{}}$ is a joint concave function. Taking the first-order derivative on $\cup_{S}^{{}}$ with respect to $w$ and $e_{1}$, and letting them be zero, we obtain the optimal wholesale price $w^{ \times }$ and the optimal freshness-keeping effort $e^{ \times }$:

$$ w^{ \times } = \frac{{(2b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} )[(a + k\theta_{0} )(2\lambda + 1) + bc(\lambda + 1)] - bc\beta_{2} k^{2} \gamma_{1}^{2} (\lambda + 1)}}{b(A + B\lambda )}. $$

$$ e_{1}^{ \times } = \frac{{k\beta_{2} \gamma_{1} (\lambda + 1)(a - bc + k\theta_{0} )}}{A + B\lambda }. $$

We can further obtain the optimal retail price $p^{ \times }$ as follows:

$$ p^{ \times } = \frac{{(b\beta_{1} \beta_{2} - \beta_{1} k^{2} \gamma_{2}^{2} )[(a + k\theta_{0} )(5\lambda + 3) + bc(\lambda + 1)] - bc\beta_{2} k^{2} \gamma_{1}^{2} (\lambda + 1)}}{b(A + B\lambda )}. $$

$$ e_{2}^{ \times } = \frac{{k\beta_{1} \gamma_{2} (a - bc + k\theta_{0} )}}{A + B\lambda }. $$

The proof is complete.□