Impact of demand forecast information sharing on the decision of a green supply chain with government subsidy

Abstract

This paper investigates a green supply chain (GSC) consisting of one manufacturer and one retailer who possesses private demand forecast information. To promote green consumption, the government may provide subsidies to consumers. Within a dynamic game where the manufacturer serves as the leader and the retailer acts as the follower, three cases are examined: centralized decision, decentralized decision with and without demand forecast information sharing between the retailer and the manufacturer. We mainly examine the value of information sharing on the decisions of a GSC in the context of government subsidies for consumers. We find that: (i) demand forecast information sharing benefits the manufacturer but damages the retailer; (ii) if the predicted value is higher than the determinate part of the demand, the manufacturer is willing to choose a higher green degree of products in the case with information sharing compared with that without information sharing; otherwise, the manufacturer is willing to choose a lower green degree of products; (iii) a two-part tariff contract is appropriate to coordinate the GSC and it is effective in increasing the green degree of products; (iv) information sharing benefits the GSC if the green production efficiency is high enough; (v) the ex-ante social welfare always increases with information accuracy. Finally, numerical analyses are conducted to verify the above findings.

Appendices

Appendix A1. Centralized decision (Case 1)

1.1 Proof of Proposition 1

The expected profit of the GSC is expressed as follows.

$$ \pi_{t}^{c} = E\left[ {\left( {\left( {p - c} \right)\left( {a - z + bg} \right) - \frac{1}{2}kg^{2} } \right)\left| \gamma \right.} \right] $$

(A1-1)

The Hessian matrix of Eq. (A1-1) related to $p$ and $g$ is $H = \left[ {\begin{array}{*{20}c} { - 2} & b \\ b & { - k} \\ \end{array} } \right]$. Given $k > b^{2}$, the Hessian matrix is negative, and Eq. (A1-1) is a joint concave function of the retail price and green degree of products. Then, we obtain

$$ p = \frac{{E\left( {a\left| \gamma \right.} \right) + bg + s + c}}{2},g = \frac{{b\left( {p - c} \right)}}{k} $$

(A1-2)

Hence, the equilibrium $p$ and $g$ can be obtained as follows.

$$ p^{c*} = \frac{{k\left[ {\left( {1 - t} \right)a_{0} + t\gamma + s + c} \right] - b^{2} c}}{{2k - b^{2} }},g^{c*} = \frac{{b\left[ {\left( {1 - t} \right)a_{0} + t\gamma + s - c} \right]}}{{2k - b^{2} }} $$

(A1-3)

Then, we can obtain the ex-ante profit of the GSC in the following.

$$ E\left( {\pi_{t}^{c*} } \right) = \frac{{k \left[( {a_{0} + s - c)^{2} + tv} \right]}}{{2\left( {2k - b^{2} } \right)}} $$

(A1-4)

1.2 Proof of Corollary 1

According to (A1-3), the first order derivative of $p^{c*}$ and $g^{c*}$ with respect to $t$ is

$$ \frac{{\partial p^{c*} }}{\partial t} = \frac{{k\left( {\gamma - a_{0} } \right)}}{{2k - b^{2} }} > 0 $$

(A1-5)

$$ \frac{{\partial g^{c*} }}{\partial t} = \frac{{b\left( {\gamma - a_{0} } \right)}}{{2k - b^{2} }} > 0 $$

(A1-6)

If $\gamma \ge a_{0}$, then $\frac{{\partial p^{c*} }}{\partial t} \ge 0$ and $\frac{{\partial g^{c*} }}{\partial t} \ge 0$; if $\gamma < a_{0}$, then $\frac{{\partial p^{c*} }}{\partial t} < 0$ and $\frac{{\partial g^{c*} }}{\partial t} < 0$.

The first order derivative of $E\left( {\pi_{t}^{c*} } \right)$ with respect to $t$ is

$$ \frac{{\partial E\left( {\pi_{t}^{c*} } \right)}}{\partial t} = \frac{k}{{2\left( {2k - b^{2} } \right)}}\left( {v + t\frac{\partial v}{{\partial t}}} \right) = \frac{{k\left( {v + \mu t} \right)}}{{2\left( {2k - b^{2} } \right)\left( {1 - t} \right)}} $$

(A1-7)

The first order derivative of $E\left( {WF^{c*} } \right)$ with respect to $t$ is

$$ \frac{{\partial E\left( {WF^{c*} } \right)}}{\partial t} = \frac{{\left( {3k^{2} - kb^{2} } \right)}}{{2(2k - b^{2} )^{2} }}\left( {v + t\frac{\partial v}{{\partial t}}} \right) = \frac{{\left( {3k^{2} - kb^{2} } \right)\left( {v + \mu t} \right)}}{{2(2k - b^{2} )^{2} \left( {1 - t} \right)}} $$

(A1-8)

Because $t = \frac{v}{v + \mu }$, and $0 < t < 1$, it is easy to obtain $\frac{{\partial E\left( {\pi_{t}^{c*} } \right)}}{\partial t} > 0$ and $\frac{{\partial E\left( {WF^{c*} } \right)}}{\partial t} > 0$.

Appendix A2. The retailer does not share demand forecast information (Case 2)

2.1 Proof of Proposition 2

The expected profits of the manufacturer, the retailer and the GSC are written as

$$ \pi_{m}^{dni} = E\left[ {\left( {w - c} \right)\left( {a - z + bg} \right) - \frac{1}{2}kg^{2} } \right] $$

(A2-1)

$$ \pi_{r}^{dni} = E\left[ {\left( {p - w} \right)\left( {a - z + bg} \right)\left| \gamma \right.} \right] $$

(A2-2)

$$ \pi_{t}^{dni} = E\left[ {\left( {\left( {p - c} \right)\left( {a - z + bg} \right) - \frac{1}{2}kg^{2} } \right)\left| \gamma \right.} \right] $$

(A2-3)

According to backward induction, the retailer first decides the retail price $p$ as follows.

$$ p = \frac{{E\left( {a\left| \gamma \right.} \right) + s + bg + w}}{2} $$

(A2-4)

Because the retailer does not share forecast information, the manufacturer’s expected retail price is expressed as

$$ E\left( p \right) = \frac{{a_{0} + s + bg + w}}{2} $$

(A2-5)

By substituting Eq. (A2-5) into Eq. (A2-1), the expected profit of the manufacturer is obtained.

$$ \pi_{m}^{dni} = \frac{1}{2}\left( {w - c} \right)\left( {a_{0} + s - w + bg} \right) - \frac{1}{2}kg^{2} $$

(A2-6)

We can obtain the wholesale price and green degree of products in Eq. (A2-6)

$$ w = \frac{{a_{0} + bg + c + s}}{2} $$

(A2-7)

$$ g = \frac{{b\left( {w - c} \right)}}{2k} $$

(A2-8)

Then, the equilibrium wholesale price and green degree of products are as follows.

$$ w^{ni*} = \frac{{2k\left( {a_{0} + s} \right) + \left( {2k - b^{2} } \right)c}}{{4k - b^{2} }} $$

(A2-9)

$$ g^{ni*} = \frac{{b\left( {a_{0} - c + s} \right)}}{{4k - b^{2} }} $$

(A2-10)

By substituting Eq. (A2-9) and (A2-10) into Eq. (A2-4), the equilibrium retail price is obtained as follows.

$$ p^{ni*} = \frac{{3k\left( {a_{0} + s} \right) + \left( {k - b^{2} } \right)c}}{{4k - b^{2} }} + \frac{{t\left( {\gamma - a_{0} } \right)}}{2} $$

(A2-11)

Based on the above equilibrium solutions, the ex-ante profits of the manufacturer, the retailer and the GSC can be obtained as follows.

$$ E\left( {\pi_{m}^{dni*} } \right) = \frac{{k(a_{0} - c + s)^{2} }}{{2\left( {4k - b^{2} } \right)}} $$

(A2-12)

$$ E\left( {\pi_{r}^{dni*} } \right) = \frac{{k^{2} (a_{0} - c + s)^{2} }}{{(4k - b^{2} )^{2} }} + \frac{tv}{4} $$

(A2-13)

$$ E\left( {\pi_{t}^{dni*} } \right) = \frac{{k(a_{0} - c + s)^{2} }}{{2\left( {4k - b^{2} } \right)}} + \frac{{k^{2} (a_{0} - c + s)^{2} }}{{(4k - b^{2} )^{2} }} + \frac{tv}{4} $$

(A2-14)

Then, the ex-ante consumer surplus is

$$ E\left( {CS^{ni*} } \right) = \frac{{[k\left( {a_{0} - c + s} \right)]^{2} }}{{2(4k - b^{2} )^{2} }} + \frac{tv}{8} $$

(A2-15)

Hence, the ex-ante social welfare can be expressed as

$$ E\left( {WF^{dni*} } \right) = \frac{{\left( {7k^{2} - kb^{2} } \right)(a_{0} - c + s)^{2} }}{{2(4k - b^{2} )^{2} }} + \frac{3tv}{8} - \frac{{ks\left( {a_{0} - c + s} \right)}}{{4k - b^{2} }} $$

(A2-16)

Appendix A3. The retailer shares demand forecast information (Case 3)

3.1 Proof of Proposition 3

The proof of proposition 3 is similar to that of proposition 1, thus we omit the details here.

Appendix A4. Comparative analysis

4.1 Proof of Proposition 4

We get the following equilibrium solutions.

$w^{i*} = \frac{{2k\left( {E\left( {a\left| \gamma \right.} \right) + s} \right) + \left( {2k - b^{2} } \right)c}}{{4k - b^{2} }}$, $w^{ni*} = \frac{{2k\left( {a_{0} + s} \right) + \left( {2k - b^{2} } \right)c}}{{4k - b^{2} }}$, $g^{i*} = \frac{{b\left( {E\left( {a\left| \gamma \right.} \right) - c + s} \right)}}{{4k - b^{2} }}$, $g^{ni*} = \frac{{b\left( {a_{0} - c + s} \right)}}{{4k - b^{2} }}$, $p^{i*} = \frac{{3k\left( {E\left( {a\left| \gamma \right.} \right) + s} \right) + \left( {k - b^{2} } \right)c}}{{4k - b^{2} }}$, and $p^{ni*} = \frac{{3k\left( {a_{0} + s} \right) + \left( {k - b^{2} } \right)c}}{{4k - b^{2} }} + \frac{{t\left( {\gamma - a_{0} } \right)}}{2}$.

Comparing the solutions in different cases, we get.

$w^{i*} - w^{ni*} = \frac{{2kt\left( {\gamma - a_{0} } \right)}}{{4k - b^{2} }}$, $g^{i*} - g^{ni*} = \frac{{bt\left( {\gamma - a_{0} } \right)}}{{4k - b^{2} }}$, and $p^{i*} - p^{ni*} = \frac{{t\left( {\gamma - a_{0} } \right)\left( {2k + b^{2} } \right)}}{{2\left( {4k - b^{2} } \right)}}$.

Thus, if $\gamma \ge a_{0}$, then $w^{i*} \ge w^{ni*}$, $g^{i*} \ge g^{ni*}$, and $p^{i*} \ge p^{ni*}$; otherwise, we have $w^{i*} \textless w^{ni*}$, $g^{i*} \textless g^{ni*}$, and $p^{i*} \textless p^{ni*}$.

4.2 Proof of Proposition 5

(1) When information is shared, the ex-ante profits of the manufacturer and the retailer are

$$ E\left( {\pi_{m}^{di*} } \right) = \frac{{k\left[( {a_{0} - c + s)^{2} + tv} \right]}}{{2\left( {4k - b^{2} } \right)}},{\text{and}}\quad E\left( {\pi_{r}^{di*} } \right) = \frac{{k^{2} \left[( {a_{0} - c + s)^{2} + tv} \right]}}{{(4k - b^{2} )^{2} }} $$

(A4-1)

When no information is shared, the ex-ante profits of the manufacturer and the retailer are

$$ E\left( {\pi_{m}^{dni*} } \right) = \frac{{k(a_{0} - c + s)^{2} }}{{2\left( {4k - b^{2} } \right)}},{\text{and}}\quad E\left( {\pi_{r}^{dni*} } \right) = \frac{{k^{2} (a_{0} - c + s)^{2} }}{{(4k - b^{2} )^{2} }} + \frac{tv}{4} $$

(A4-2)

Comparing the ex-ante profits in different cases, we can get

$$ E\left( {\pi_{m}^{di*} } \right) - E\left( {\pi_{m}^{dni*} } \right) = \frac{ktv}{{2\left( {4k - b^{2} } \right)}} > 0,{\text{and}}\quad E\left( {\pi_{r}^{di*} } \right) - E\left( {\pi_{r}^{dni*} } \right) = \frac{{\left( {6k - b^{2} } \right)\left( {b^{2} - 2k} \right)tv}}{{4(4k - b^{2} )^{2} }} < 0 $$

(A4-3)

(2) When no information is shared, the ex-ante profit of the GSC is

$$ E\left( {\pi_{t}^{dni*} } \right) = \frac{{k(a_{0} - c + s)^{2} }}{{2\left( {4k - b^{2} } \right)}} + \frac{{k^{2} (a_{0} - c + s)^{2} }}{{(4k - b^{2} )^{2} }} + \frac{tv}{4} $$

(A4-4)

When information is shared, the ex-ante profit of the GSC is

$$ E\left( {\pi_{t}^{di*} } \right) = \frac{{k \left[( {a_{0} - c + s)^{2} + tv} \right]}}{{2\left( {4k - b^{2} } \right)}} + \frac{{k^{2} \left[( {a_{0} - c + s)^{2} + tv} \right]}}{{(4k - b^{2} )^{2} }} $$

(A4-5)

Comparing the ex-ante profits in different cases, we can get

$$ E\left( {\pi_{t}^{di*} } \right) - E\left( {\pi_{t}^{dni*} } \right) = \frac{{\left( { - 4k^{2} + 6kb^{2} - b^{4} } \right)tv}}{{4(4k - b^{2} )^{2} }} $$

(A4-6)

From Eq. (A4-6), we find that if $b^{2} < k < \frac{{\left( {3 + \sqrt 5 } \right)b^{2} }}{4}$, then $E\left( {\pi_{t}^{di*} } \right) > E\left( {\pi_{t}^{dni*} } \right)$; if $k \ge \frac{{\left( {3 + \sqrt 5 } \right)b^{2} }}{4}$, then $E\left( {\pi_{t}^{di*} } \right) \le E\left( {\pi_{t}^{dni*} } \right)$.

(3) When no information is shared, the ex-ante social welfare is

$$ E\left( {WF^{dni*} } \right) = \frac{{\left( {7k^{2} - kb^{2} } \right)(a_{0} - c + s)^{2} }}{{2(4k - b^{2} )^{2} }} + \frac{3tv}{8} - \frac{{ks\left( {a_{0} - c + s} \right)}}{{4k - b^{2} }} $$

(A4-7)

When information is shared, the ex-ante social welfare is

$$ E\left( {WF^{di*} } \right) = \frac{{\left( {7k^{2} - kb^{2} } \right) \left[( {a_{0} - c + s)^{2} + tv} \right]}}{{2(4k - b^{2} )^{2} }} - \frac{{ks\left( {a_{0} + s - c} \right)}}{{4k - b^{2} }} $$

(A4-8)

Comparing the ex-ante social welfares in different cases, we find that if $k \le \frac{{\left( {5 + \sqrt {10} } \right)b^{2} }}{10}$, then $E\left( {WF^{{di{*}}} } \right) \ge E\left( {WF^{{dni{*}}} } \right)$; if $k > \frac{{\left( {5 + \sqrt {10} } \right)b^{2} }}{10}$, then $E\left( {WF^{{di{*}}} } \right) < E\left( {WF^{{dni{*}}} } \right)$.

4.3 Proof of Proposition 6

In the case with the contract, the expected profits of the manufacturer and the retailer are expressed as follows.

$$ \pi_{m}^{tp} = E\left[ {\left( {\left( {w^{tp} - c} \right)\left( {a - z + bg} \right) - \frac{1}{2}kg^{2} } \right)\left| \gamma \right.} \right] + F $$

(A4-9)

$$ \pi_{r}^{tp} = E\left[ {\left( {p - w^{tp} } \right)\left( {a - z + bg} \right)\left| \gamma \right.} \right] - F $$

(A4-10)

According to the coordination conditions, we substitute $g^{tp*} = g^{c*}$ and $p^{tp*} = p^{c*}$ into Eqs. (A4-9) and (A4-10), the equilibrium wholesale price and green degree under the two-part tariff contract are

$$ p^{tp*} = \frac{{k\left[ {\left( {1 - t} \right)a_{0} + t\gamma + s} \right] + \left( {k - b^{2} } \right)c}}{{2k - b^{2} }} $$

(A4-11)

$$ g^{tp*} = \frac{{b\left[ {\left( {1 - t} \right)a_{0} + t\gamma + s - c} \right]}}{{2k - b^{2} }} $$

(A4-12)

Since the ex-ante profits of the manufacturer and the retailer are subject to $E\left( {\pi_{m}^{tp*} } \right) \ge E\left( {\pi_{m}^{dni*} } \right)$, $E\left( {\pi_{r}^{tp*} } \right) \ge E\left( {\pi_{r}^{dni*} } \right)$, then we obtain $F \in \left[ {E\left( {\pi_{m}^{dni*} + \frac{1}{2}k(g^{c*} )^{2} + \left( {w^{tp} - c} \right)q^{c*} } \right),E\left( {\left( {p - w^{tp} } \right)q^{c*} - \pi_{r}^{dni*} } \right)} \right]$.