Price and cold-chain service decisions versus integration in a fresh agri-product supply chain with competing retailers

Abstract

This paper studies the price and cold-chain service decisions of a fresh agri-product supply chain with competing retailers and explores the impacts of (horizontal/vertical) integration on the decisions and profits. When the cold-chain service price is exogenous, we find that vertical integration is more effective to decrease quantity/quality loss of agri-products; the third-party logistics provider is better off with vertical integration while horizontal integration hurts both the third-party logistics provider and the supplier. We identify the conditions under which a retailer benefits from vertical or horizontal integration. Whether horizontal integration can improve the retailer’s profit depends on product substitutability. The profit of a retailer can be enhanced by vertical integration between the other retailer and the supplier if product substitutability is low. The endogenization of the cold-chain service price does not change the main results qualitatively.

Appendix

Proof of Table 3

Under NI, from (3), we obtain $\partial^{2} \pi_{ri} /\partial p_{i}^{2} = - 2 < 0$; that is, $\pi_{ri}$ is concave in $p_{i}$. Solving the first-order conditions $\partial \pi_{r1} /\partial p_{1} = 0$ and $\partial \pi_{r2} /\partial p_{2} = 0$ for $(p_{1} ,p_{2} )$, we derive

$$ p_{1} (w_{1} ,w_{2} ,e) = (2a_{1} + \gamma a_{2} + 2c_{1} + \gamma c_{2} + 2w_{1} + \gamma w_{2} )/(4 - \gamma^{2} ) $$

(4)

$$ p_{2} (w_{1} ,w_{2} ,e) = (2a_{2} + \gamma a_{1} + 2c_{2} + \gamma c_{1} + 2w_{2} + \gamma w_{1} )/(4 - \gamma^{2} ) $$

(5)

where $a_{1} = \rho (a + \beta e)$, $a_{2} = (1 - \rho )(a + \beta e)$, $\rho_{1} = \rho$, $\rho_{2} = 1 - \rho$.

Substituting (4) and (5) into (2), we obtain $\pi_{s} (w_{1} ,w_{2} ,e)$. Further, the Hessian matrix of $\pi_{s} (w_{1} ,w_{2} ,e)$ over $(w_{1} ,w_{2} )$ is as follows:

$$ \begin{aligned} {\mathbf{H}} = & \left[ {\begin{array}{*{20}l} { - 2(2 - \gamma^{2} )/(4 - \gamma^{2} )} \hfill & {2\gamma /(4 - \gamma^{2} )} \hfill \\ {2\gamma /(4 - \gamma^{2} )} \hfill & { - 2(2 - \gamma^{2} )/(4 - \gamma^{2} )} \hfill \\ \end{array} } \right] \\ \left| {{\mathbf{H}}_{1} } \right| = & - 2(2 - \gamma^{2} )/(4 - \gamma^{2} ) < 0, \\ \left| {{\mathbf{H}}_{2} } \right| = & 4(2 - \gamma^{2} - \gamma )(2 - \gamma^{2} + \gamma )/(4 - \gamma^{2} )^{2} > 0. \\ \end{aligned} $$

Thus, H is negatively definite. Solving the first-order conditions $\partial \pi_{s} (w_{1} ,w_{2} ,e)/\partial w_{1} = 0$ and $\partial \pi_{s} (w_{1} ,w_{2} ,e)/\partial w_{2} = 0$ for $(w_{1} ,w_{2} )$, we derive

$$ w_{1} (e) = (a_{1} + \gamma a_{2} )/[2(1 - \gamma^{2} )] + (t + sr - c_{1} - sre)/2 $$

(6)

$$ w_{2} (e) = (a_{2} + \gamma a_{1} )/[2(1 - \gamma^{2} )] + (t + sr - c_{2} - sre)/2 $$

(7)

Substituting (6) and (7) into (1), we obtain $\pi_{l} (e)$.

In addition, we can derive $\partial^{2} \pi_{l} (e)/\partial e^{2} = - \lambda < 0$; that is, $\pi_{l} (e)$ is concave in e. Solving the first-order condition $\partial \pi_{l} (e)/\partial e = 0$ for e, we have

$$ e^{NI * } = t[\beta + 2(1 - \gamma )sr]/[2\lambda (2 - \gamma )] $$

(8)

$e^{NI * } \le 1$ is equivalent to $\lambda \ge \hat{\lambda }^{NI} = t[\beta + 2(1 - \gamma )sr]/[2(2 - \gamma )]$. Substituting (8) into (4)–(7), we derive $(w_{1}^{NI * } ,w_{2}^{NI * } ,p_{1}^{NI * } ,p_{2}^{NI * } )$. Similarly, we can derive equilibrium outcomes under VI1, VI2 and HI, as shown in Table 3. □

Proof of Proposition 2 Part (1)

Based on $e^{NI * }$, we can derive $\partial e^{NI * } /\partial \gamma = t(\beta - 2sr)/[2\lambda (2 - \gamma )^{2} ]$. Obviously, if $\beta > 2sr$, we obtain $\partial e^{NI * } /\partial \gamma > 0$; otherwise, $\partial e^{NI * } /\partial \gamma < 0$. Likewise, we can derive other results in Proposition 2. □

Proof of Proposition 3

From $e^{NI * }$ and $e^{HI * }$, we obviously derive $e^{HI * } < e^{NI * }$. From $e^{NI * }$ and $e^{VI2 * }$, we obtain

$$ \begin{aligned} e^{VI2 * } - e^{NI * } = & t[(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]/(4\lambda ) - t(\beta + 2(1 - \gamma )sr)/(2\lambda (2 - \gamma )) \\ = & t\{ \beta [(2 - \rho + \gamma \rho )(2 - \gamma ) - 2] + sr[(2 - \gamma )(3 - 2\gamma - \gamma^{2} ) - 4(1 - \gamma )]\} /[4\lambda (2 - \gamma )] \\ = & t\{ (1 - \gamma )(2 - 2\rho + \gamma \rho )\beta + (1 - \gamma )^{2} (2 + \gamma )sr\} /[4\lambda (2 - \gamma )] > 0. \\ \end{aligned} $$

Then, we have $e^{NI * } < e^{VI2 * }$. Likewise, we can show $e^{NI * } < e^{VI1 * }$.

In addition, under VI2 and VI1, it is easy to obtain $e^{VI1 * } \le e^{VI2 * }$ for $\rho \in (0,0.5]$, and $e^{VI2 * } < e^{VI1 * }$ for $\rho \in (0.5,1)$. □

Proof of Proposition 4 Part (1)

From $(w_{1}^{NI * } ,w_{1}^{VI2 * } ,w_{1}^{HI * } )$ we have

$$ w_{1}^{NI * } - w_{1}^{VI2 * } = [\beta (\rho + \gamma - \gamma \rho ) - (1 - \gamma^{2} )sr](e^{NI * } - e^{VI2 * } )/[2(1 - \gamma^{2} )] $$

According to Proposition 3, we have $e^{NI * } - e^{VI2 * } < 0$. Hence, we obtain $w_{1}^{VI2 * } \le w_{1}^{NI * }$ for $\beta \le sr(1 - \gamma^{2} )/(\rho + \gamma - \gamma \rho )$. Likewise, we can derive $w_{1}^{NI * } \le w_{1}^{HI * }$ for $\beta \le sr(1 - \gamma^{2} )/(\rho + \gamma - \gamma \rho )$. In addition, we can obtain $w_{1}^{HI * } < w_{1}^{NI * } < w_{1}^{VI2 * }$ for $\beta > sr(1 - \gamma^{2} )/(\rho + \gamma - \gamma \rho )$.

Similarly, we can demonstrate that $w_{2}^{VI1 * } \le w_{2}^{NI * } \le w_{2}^{HI * }$ for $\beta \le sr(1 - \gamma^{2} )/(1 - \rho + \gamma \rho )$, and $w_{2}^{HI * } < w_{2}^{NI * } < w_{2}^{VI1 * }$ for $\beta > sr(1 - \gamma^{2} )/(1 - \rho + \gamma \rho )$.

Part (2)

Based on Table 3, we obtain

$$ \begin{aligned} D^{NI * } - D^{HI * } & = \{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr \\ & \quad + \beta ]e^{NI * } \} /[2(2 - \gamma )] - \{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{HI * } \} /4 \\ & \quad > [a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{NI * } ]/4 \\ & \quad - [a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{HI * } ]/4 \\ & = [2(1 - \gamma )sr + \beta ](e^{NI * } - e^{HI * } )/4 \\ \end{aligned} $$

As $e^{NI * } > e^{HI * }$ in Proposition 3, we can show $D^{NI * } > D^{HI * }$.

Likewise, we obtain

$$ \begin{aligned} D^{VI2 * } - D^{NI * } & = \{ (2 - \rho + \gamma \rho )a - (1 - \gamma )c_{1} - (2 - \gamma - \gamma^{2} )c_{2} \\ & \quad - (3 - 2\gamma - \gamma^{2} )(t + sr) + [(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]e^{VI2 * } \} /4 \\ & \quad - \{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{NI * } \} /[2(2 - \gamma )] \\ & \quad > \{ (2 - \rho + \gamma \rho )a - (1 - \gamma )c_{1} - (2 - \gamma - \gamma^{2} )c_{2} - (3 - 2\gamma - \gamma^{2} )(t + sr) \\ & \quad + [(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]e^{VI2 * } \} /4 \\ & \quad - \{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{VI2 * } \} /[2(2 - \gamma )] \\ \end{aligned} $$

for $e^{VI2 * } > e^{NI * }$ according to Proposition 3.

Thus,

$$ \begin{aligned} D^{VI2 * } - D^{NI * } > & (1 - \gamma )[(2 - 2\rho + \rho \gamma )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} - (2 - \gamma - \gamma^{2} )(t + sr) \\ & + [(2 - 2\rho + \rho \gamma )\beta + (2 - \gamma - \gamma^{2} )sr]e^{VI2 * } ]/[4(2 - \gamma )] = (1 - \gamma )D_{2}^{VI2 * } /(2 - \gamma ) > 0, \\ \end{aligned} $$

i.e., $D^{VI2 * } > D^{NI * } .$ Overall, we obtain $D^{HI * } < D^{NI * } < D^{VI2 * }$.□

Proof of Proposition 5 Part (1)

From Table 3, we obtain

$$ \begin{aligned} \pi_{l}^{NI * } - \pi_{l}^{HI * } & = t\{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{NI * } \} /[2(2 - \gamma )] \\ & \quad - t\{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{HI * } \} /4 + \lambda [(e^{HI * } )^{2} - (e^{NI * } )^{2} ]/2 \\ & \quad > t\{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{NI * } \} /[2(2 - \gamma )] \\ & \quad - t\{ a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) + [2(1 - \gamma )sr + \beta ]e^{HI * } \} /[2(2 - \gamma )] + \lambda [(e^{HI * } )^{2} - (e^{NI * } )^{2} ]/2 \\ & = t[2(1 - \gamma )sr + \beta ](e^{NI * } - e^{HI * } )/[2(2 - \gamma )] + \lambda [e^{HI * } - e^{NI * } ][e^{HI * } + e^{NI * } ]/2 \\ & = (e^{NI * } - e^{HI * } )\{ t[2(1 - \gamma )sr + \beta ]/[2(2 - \gamma )] - \lambda (e^{HI * } + e^{NI * } )/2\} \\ & \quad > (e^{NI * } - e^{HI * } )\{ t[2(1 - \gamma )sr + \beta ]/[2(2 - \gamma )] - \lambda e^{NI * } \} \\ & \quad > (e^{NI * } - e^{HI * } )\{ t(2(1 - \gamma )sr + \beta ) - t(\beta + 2(1 - \gamma )sr)\} /[2(2 - \gamma )] = 0 \\ \end{aligned} $$

Hence, we derive $\pi_{l}^{HI * } < \pi_{l}^{NI * }$.

$$ \begin{aligned} \pi_{l}^{VI2 * } - \pi_{l}^{NI * } & = t \cdot \{ [(2 - \rho + \gamma \rho )a - (1 - \gamma )c_{1} - (2 - \gamma - \gamma^{2} )c_{2} - (3 - 2\gamma - \gamma^{2} )(t + sr) \\ & \quad + [(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]e^{VI2 * } ]/4 \\ & \quad - [a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) \\ & \quad + [2(1 - \gamma )sr + \beta ]e^{NI * } ]/[2(2 - \gamma )]\} + \lambda [(e^{NI * } )^{2} - (e^{VI2 * } )^{2} ]/2 \\ & \quad > t\{ [(2 - \rho + \gamma \rho )a - (1 - \gamma )c_{1} - (2 - \gamma - \gamma^{2} )c_{2} - (3 - 2\gamma - \gamma^{2} )(t + sr) \\ & \quad + [(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]e^{VI2 * } ]/4 \\ & \quad - [a - (1 - \gamma )(c_{1} + c_{2} ) - 2(1 - \gamma )(t + sr) \\ & \quad + [2(1 - \gamma )sr + \beta ]e^{VI2 * } ]/[2(2 - \gamma )]\} + \lambda [(e^{NI * } )^{2} - (e^{VI2 * } )^{2} ]/2 \\ = & t(1 - \gamma )[(2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} \\ & \quad - (2 - \gamma^{2} - \gamma )(t + sr) + [(2 - 2\rho + \gamma \rho )\beta \\ & \quad + (2 - \gamma^{2} - \gamma )sr]e^{VI2 * } ]/[4(2 - \gamma )] + \lambda (e^{NI * } + e^{VI2 * } )(e^{NI * } - e^{VI2 * } )/2 \\ & \quad > t(1 - \gamma )[(2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} \\ & \quad - (2 - \gamma^{2} - \gamma )(t + sr) + [(2 - 2\rho + \gamma \rho )\beta \\ & + (2 - \gamma^{2} - \gamma )sr]e^{VI2 * } ]/[4(2 - \gamma )] + \lambda e^{VI2 * } (e^{NI * } - e^{VI2 * } ) \\ & = t(1 - \gamma )[(2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} \\ & \quad - (2 - \gamma^{2} - \gamma )(t + sr) + [(2 - 2\rho + \gamma \rho )\beta \\ & \quad + (2 - \gamma^{2} - \gamma )sr]e^{VI2 * } ]/[4(2 - \gamma )] + t[(2 - \rho + \gamma \rho )\beta \\ & \quad + (3 - 2\gamma - \gamma^{2} )sr](e^{NI * } - e^{VI2 * } )/4 \\ & = t(1 - \gamma )[(2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} \\ & \quad - (2 - \gamma^{2} - \gamma )(t + sr)]/[4(2 - \gamma )] \\ & \quad + t\{ [(2 - \rho + \gamma \rho )\beta + (3 - 2\gamma - \gamma^{2} )sr]e^{NI * } /4 - [\beta + 2(1 - \gamma )sr]e^{VI2 * } /[2(2 - \gamma )]\} \\ & = t(1 - \gamma )[(2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} - (2 - \gamma^{2} - \gamma )(t + sr)]/[4(2 - \gamma )] \\ \end{aligned} $$

In this paper, we assume that the market demand of retailer 2 is positive under VI2 without the cold-chain service level, i.e.,

$$ (2 - 2\rho + \gamma \rho )a + \gamma c_{1} - (2 - \gamma^{2} )c_{2} - (2 - \gamma^{2} - \gamma )(t + sr) > 0. $$

Hence, we obtain $\pi_{l}^{NI * } < \pi_{l}^{VI2 * }$. Overall, we obtain $\pi_{l}^{HI * } < \pi_{l}^{NI * } < \pi_{l}^{VI2 * }$.

Part (2)

$$ \begin{aligned} \pi_{s}^{NI * } - \pi_{s}^{HI * } = & (w_{1}^{NI * } - t - sr + sre^{NI * } )D_{1}^{NI * } + (w_{2}^{NI * } - t - sr + sre^{NI * } )D_{2}^{NI * } \\ & - (w_{1}^{HI * } - t - sr + sre^{HI * } )D_{1}^{HI * } - (w_{2}^{HI * } - t - sr + sre^{HI * } )D_{2}^{HI * } \\ \end{aligned} $$

First, we obtain

$$ \begin{aligned} (w_{1}^{NI * } - t - sr + sre^{NI * } ) - (w_{1}^{HI * } - t - sr + sre^{HI * } ) = & (\rho + \gamma - \gamma \rho )(a + \beta e^{NI * } )/[2(1 - \gamma^{2} )] + (sre^{NI * } - t - sr - c_{1} )/2 \\ & - (\rho + \gamma - \gamma \rho )(a + \beta e^{HI * } )/[2(1 - \gamma^{2} )] - (sre^{HI * } - t - sr - c_{1} )/2 \\ = & (\rho + \gamma - \gamma \rho )(\beta e^{NI * } - \beta e^{HI * } )/[2(1 - \gamma^{2} )] + (sre^{NI * } - sre^{HI * } )/2 > 0. \\ \end{aligned} $$

Like the proof of Proposition 4(2), we can derive $D_{1}^{NI * } > D_{1}^{HI * }$. Thus, we have

$$ (w_{1}^{NI * } - t - sr + sre^{NI * } )D_{1}^{NI * } - (w_{1}^{HI * } - t - sr + sre^{HI * } )D_{1}^{HI * } > 0. $$

Likewise, we can obtain $(w_{2}^{NI * } - t - sr + sre^{NI * } )D_{2}^{NI * } - (w_{2}^{HI * } - t - sr + sre^{HI * } )D_{2}^{HI * } > 0$.

Consequently, we have $\pi_{s}^{HI * } < \pi_{s}^{NI * }$.□

Proofs of Propositions 6 and 7

The proofs of Propositions 6 and 7 are similar to those of Propositions 4 and 3, respectively. Here, we omit them. □

Proof of Table 4

Under NI, the Hessian matrix of $\pi_{l} (t,e)$ over $(t,e)$ is as follows:

$$ {\mathbf{\rm H}} = \left[ {\begin{array}{*{20}l} { - 2(1 - \gamma )/(2 - \gamma )} \hfill & {[\beta + 2(1 - \gamma )sr]/[2(2 - \gamma )]} \hfill \\ {[\beta + 2(1 - \gamma )sr]/[2(2 - \gamma )]} \hfill & { - \lambda } \hfill \\ \end{array} } \right]. $$

If $\lambda > \hat{\lambda}_{E1}^{NI} = [\beta + 2(1 - \gamma )sr]^{2} /[8(1 - \gamma )(2 - \gamma )]$, then the Hessian matrix of $\pi_{l} (t,e)$ is negatively definite. Solving the first-order conditions $\partial \pi_{l} (t,e)/\partial t = 0$ and $\partial \pi_{l} (t,e)/\partial e = 0$ for $(t,e)$, we obtain $e_{E}^{NI * }$ and $t_{E}^{NI * } .$

$e_{E}^{NI * } \le 1$ is equivalent to $\lambda > \hbox{max} \{ \hat{\lambda}_{E2}^{NI} ,\hat{\lambda}_{E3}^{NI} \}$, and $t_{E}^{NI * } < s$ is equivalent to $\hat{a}_{E1}^{NI} < a < \hat{a}_{E2}^{NI}$. Further, similar to Table 3, we can complete the proof of Table 4. □