Comparison of subsidy strategies on the green supply chain under a behaviour-based pricing model

Abstract

To explore the optimal government subsidy strategy for a green supply chain (GSC) under behaviour-based pricing (BBP), three types of GSC subsidy models under BBP are explored by using game theory, and the influence of different subsidy ratios and subsidy coefficients on pricing and greenness under BBP was analysed for the first time. Additionally, the effects of different strategies are compared. Our results reveal the following: First, when the subsidy is based on output or green degree, the proportion of green product retailers receiving subsidies has only a positive impact on the wholesale price of green products. Second, when the product R&D input cost is subsidized, the proportion of green product retailers receiving subsidies is negatively correlated with greenness, wholesale price, loyalty price, poaching price, market share of green products and GSC profit. Third, when the product R&D input cost is subsidized, the loyalty and poaching prices of green products always increase with an increase in the subsidy coefficient. However, when the subsidy is based on output or green degree, the loyalty and poaching prices of green products increase or decrease with the increased subsidy coefficient. A numerical case example shows that the optimal subsidy strategy of a GSC under a BBP is different from those of previous studies. Subsidies based on green degrees are the optimal strategy for green product retailers, green consumers and governments.

Appendix

Proof of Lemma 1.

Under subsidy strategy 1, the reverse solution method is adopted according to the decision order. First, the profit maximization of nongreen product manufacturers is considered. Let $\left\{\begin{array}{c}\frac{d{\pi }_{Mb}^{{s}_{1}}}{d{p}_{b}^{{s}_{1}}}=0\\ \frac{d{\pi }_{Mb}^{{s}_{1}}}{d{q}_{b}^{{s}_{1}}}=0\end{array}\right.$, and we obtain $\left\{\begin{array}{c}{q}_{b}^{{s}_{1}}=\frac{{p}_{g}^{{s}_{1}}}{2}-\frac{\tau }{2}-\frac{\gamma {\theta }^{{s}_{1}}}{2}+\tau {x}_{0}\\ {p}_{b}^{{s}_{1}}=\frac{{q}_{g}^{{s}_{1}}}{2}+\frac{\tau }{2}-\frac{\gamma {\theta }^{{s}_{1}}}{2}\end{array}\right.$. Substitute the above formula into Formulas (1) and (2) to obtain ${x}_{1}^{g}$ and ${x}_{1}^{b}$ represented by ${p}_{g}^{{s}_{1}}$ and ${q}_{g}^{{s}_{1}}$, substitute them into Formula (5), and then, consider the maximum profit of green product retailers under subsidy strategy 1. Let $\left\{\begin{array}{c}\frac{d{\pi }_{Rg}^{{s}_{1}}}{d{p}_{g}^{{s}_{1}}}=0\\ \frac{d{\pi }_{Rg}^{{s}_{1}}}{d{q}_{g}^{{s}_{1}}}=0\end{array}\right.$, and we obtain $\left\{\begin{array}{c}{p}_{g}^{{s}_{1}}=\frac{\tau }{2}+\frac{{w}_{g}^{{s}_{1}}}{2}+\frac{\gamma {\theta }^{{s}_{1}}}{2}+\tau {x}_{0}-\frac{\alpha {s}_{1}}{2}\\ {q}_{g}^{{s}_{1}}=\frac{3\tau }{2}+\frac{{w}_{g}^{{s}_{1}}}{2}+\frac{\gamma {\theta }^{{s}_{1}}}{2}-2\tau {x}_{0}-\frac{\alpha {s}_{1}}{2}\end{array}\right.$. Substitute the solved ${p}_{g}^{{s}_{1}}$ and ${q}_{g}^{{s}_{1}}$ expressions into ${q}_{b}^{{s}_{1}}$, ${p}_{b}^{{s}_{1}}$ and Eqs. (1) and (2). We obtain ${x}_{1}^{g}$ and ${x}_{1}^{b}$ in terms of ${w}_{g}^{{s}_{1}}$ and ${\theta }^{{s}_{1}}$, substitute them into Eq. (4), and finally, consider the case where the green product manufacturer has the maximum profit. Let $\left\{\begin{array}{c}\frac{d{\pi }_{Mg}^{{s}_{1}}}{d{w}_{g}^{{s}_{1}}}=0 \\ \frac{d{\pi }_{Mg}^{{s}_{1}}}{d{\theta }^{{s}_{1}}}=0\end{array}\right.$; the values of ${w}_{g}^{{s}_{1}}$ and ${\theta }^{{s}_{1}}$ can be obtained by solving the equations, and the other equilibrium results in Table 2 can then be obtained.

The calculation method of the equilibrium results under subsidy strategies 2 and 3 (Tables 3 and 4) is the same as above and will not be described below.

Proof of Lemma 2

According to Assumption 4, to ensure the validity of the solution, i.e. ${p}_{g}^{{s}_{1}*}>{w}_{g}^{{s}_{1}*}>0;{q}_{g}^{{s}_{1}*}>{w}_{g}^{{s}_{1}*}>0,{p}_{b}^{{s}_{1}*}>0;{q}_{b}^{{s}_{1}*}>0,{w}_{g}^{{s}_{1}*}{M}_{g}^{{s}_{1}*}>{C}_{g}^{{s}_{1}*}$,

$$ \frac{{16\mu \tau^{2} + 2\gamma^{2} s_{1} + \gamma^{2} \tau + 12\mu \tau^{2} x_{0} - 3\gamma^{2} \tau x_{0} - 4\mu s_{1} \tau }}{{2\left( { - \gamma^{2} + 8\mu \tau } \right)}} > \frac{{8\mu \tau^{2} + \gamma^{2} s_{1} \left( {1 - \alpha } \right) - 4\mu \tau^{2} x_{0} - 4\mu \tau s_{1} + 8\mu s_{1} \tau \alpha }}{{ - \gamma^{2} + 8\mu \tau }} > 0 $$

(22)

$$ \frac{{32\mu \tau^{2} + 2\gamma^{2} s_{1} - \gamma^{2} \tau - 36\mu \tau^{2} x_{0} + 3\gamma^{2} \tau x_{0} - 4\mu s_{1} \tau }}{{2\left( { - \gamma^{2} + 8\mu \tau } \right)}} > \frac{{8\mu \tau^{2} + \gamma^{2} s_{1} \left( {1 - \alpha } \right) - 4\mu \tau^{2} x_{0} - 4\mu \tau s_{1} + 8\mu s_{1} \tau \alpha }}{{ - \gamma^{2} + 8\mu \tau }} > 0 $$

(23)

$$ \frac{{ - \left( {\tau \left( {4\mu s_{1} - 48\mu \tau - 5\gamma^{2} x_{0} + 7\gamma^{2} + 36\mu \tau x_{0} } \right)} \right)}}{{4\left( { - \gamma^{2} + 8\mu \tau } \right)}} $$

(24)

$$ \frac{{ - \left( {\tau \left( {4\mu s_{1} + 5\gamma^{2} x_{0} + \gamma^{2} - 44\mu \tau x_{0} } \right)} \right)}}{{4\left( { - \gamma^{2} + 8\mu \tau } \right)}} $$

(25)

$$ \left( {\frac{{8\mu \tau^{2} + \gamma^{2} s_{1} \left( {1 - \alpha } \right) - 4\mu \tau^{2} x_{0} - 4\mu \tau s_{1} + 8\mu s_{1} \tau \alpha }}{{ - \gamma^{2} + 8\mu \tau }}} \right)\left( {\frac{{\mu \left( {s_{1} + 2\tau - \tau x_{0} } \right)}}{{ - \gamma^{2} + 8\mu \tau }}} \right) > \frac{\mu }{2}\left( {\frac{{\gamma s_{1} + 2\gamma \tau - \gamma \tau x_{0} }}{{ - \gamma^{2} + 8\mu \tau }}} \right)^{2} $$

(26)

Because ${\pi }_{Mg}^{{s}_{1}*}>0$, we know that $-{\gamma }^{2}+8\mu \tau >0$. If Eq. (22) is true, then ${\gamma }^{2}\tau +20\mu {\tau }^{2}{x}_{0}-3{\gamma }^{2}\tau {x}_{0}+ 2{\gamma }^{2}{s}_{1}\alpha +4\mu {s}_{1}\tau -16\mu {s}_{1}\tau \alpha >0$, so $0<{s}_{1}<\frac{{\gamma }^{2}\tau +20\mu {\tau }^{2}{x}_{0}-3{\gamma }^{2}\tau {x}_{0}}{16\mu \tau \alpha -2{\gamma }^{2}\alpha -4\mu \tau }$.

If Eq. (23) is true, then $16\mu {\tau }^{2}-{\gamma }^{2}\tau -28\mu {\tau }^{2}{x}_{0}+3{\gamma }^{2}\tau {x}_{0}+ 2{\gamma }^{2}{s}_{1}\alpha +4\mu {s}_{1}\tau -16\mu {s}_{1}\tau \alpha >0$, so $0<{s}_{1}<\frac{16\mu {\tau }^{2}-{\gamma }^{2}\tau -28\mu {\tau }^{2}{x}_{0}+3{\gamma }^{2}\tau {x}_{0}}{16\mu \tau \alpha -2{\gamma }^{2}\alpha -4\mu \tau }$.

If Eq. (24) is true, then $4\mu {s}_{1}-48\mu \tau -5{\gamma }^{2}{x}_{0}+7{\gamma }^{2}+36\mu \tau {x}_{0}<0$, so $0<{s}_{1}<\frac{48\mu \tau +5{\gamma }^{2}{x}_{0}-7{\gamma }^{2}-36\mu \tau {x}_{0}}{4\mu }$.

If Eq. (25) is true, then $4\mu {s}_{1}+5{\gamma }^{2}{x}_{0}+{\gamma }^{2}-44\mu \tau {x}_{0}<0$, so $0<{s}_{1}<\frac{-5{\gamma }^{2}{x}_{0}-{\gamma }^{2}+44\mu \tau {x}_{0}}{4\mu }$.

If Eq. (26) is true, Eq. (26) can be simplified to $\frac{-\left(-{\gamma }^{2}+8\mu \tau \right)\left({s}_{1}-2\tau +\tau {x}_{0}-2{s}_{1}\alpha \right)}{2}>0$, so $0<{s}_{1}<\frac{2\tau -\tau {x}_{0}}{1-2\alpha }$. Thus, Lemma 2 is proved.

Proof of Proposition 1

The equilibrium data in Tables 2 and 3 indicate that when the government subsidizes green products by output or green degree, only the wholesale price of green products is related to parameter $\mathrm{\alpha }$. The first partial derivative of the wholesale price of green products with respect to α under the two subsidy modes is obtained $\frac{d{w}_{g}^{{s}_{1}*}}{d\mathrm{\alpha }}={s}_{1}>0$, $\frac{d{w}_{g}^{{s}_{2}*}}{d\mathrm{\alpha }}=\frac{{s}_{2}\tau \left({s}_{2}+\gamma \right)\left({x}_{0}-2\right)}{{{s}_{2}}^{2}+2{s}_{2}\gamma +{\gamma }^{2}-8\mu \tau }>0$

Therefore, Proposition 1① is proved.

Under the model according to the product R&D input cost, the first derivative of the green degree of green products and the wholesale, loyalty and encroachment prices of two types of products with respect to α are derived.

$$ \frac{{d\theta^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{8\mu s_{3} \gamma \tau^{2} \left( {x_{0} - 2} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0,\frac{{dw_{g}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{4\mu s_{3} \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0, $$

$$ \frac{{dp_{g}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{dq_{g}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{6\mu s_{3} \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0,\frac{{dp_{b}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{dq_{b}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{\mu s_{3} \gamma^{2} \tau^{2} \left( {2 - x_{0} } \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} > 0 $$

Therefore, Proposition 1② is proved.

Proof of Proposition 2.

Under subsidy strategy 3, the first partial derivative of the market share of green products, the profit of green product manufacturers and the profit of green product retailers with respect to α are obtained.

$$ \frac{{dM_{g}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{\mu s_{3} \gamma^{2} \tau \left( {x_{0} - 2} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0;\frac{{d\pi_{Mg}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{ - \mu s_{3} \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)^{2} }}{{2\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0; $$

$$ \frac{{d\pi_{Rg}^{{s_{3} *}} }}{{d{\upalpha }}} = \frac{{ - \mu s_{3} \gamma^{2} \tau^{2} \left( {\gamma^{2} + 16\mu \tau s_{3} \alpha } \right)\left( {x_{0} - 2} \right)^{2} }}{{2\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{3} }} < 0 $$

Thus, Proposition 2 is proved.

Proof of Proposition 3.

The first partial derivative of subsidy coefficient ${s}_{1}$ is obtained for the wholesale, loyalty and poaching prices of the two types of products under subsidy strategy 1:

$$ \frac{{dw_{g}^{{s_{1} *}} }}{{d{\text{s}}_{1} }} = \alpha - \frac{{ - \gamma^{2} + 4\mu \tau }}{{ - \gamma^{2} + 8\mu \tau }};\frac{{dp_{g}^{{s_{1} *}} }}{{d{\text{s}}_{1} }} = \frac{{dq_{g}^{{s_{1} *}} }}{{d{\text{s}}_{1} }} = \frac{{\gamma^{2} - 2\mu \tau }}{{ - \gamma^{2} + 8\mu \tau }};\frac{{dp_{b}^{{s_{1} *}} }}{{d{\text{s}}_{1} }} = \frac{{dq_{b}^{{s_{1} *}} }}{{d{\text{s}}_{1} }} = \frac{ - \mu \tau }{{ - \gamma^{2} + 8\mu \tau }} < 0 $$

Thus, when $\gamma >\sqrt{\frac{4-8\alpha }{1-\alpha }\mu \tau }$,$\frac{d{w}_{g}^{{s}_{1}*}}{d{\mathrm{s}}_{1}}>0$, proposition 3① is proved.

When $\gamma >\sqrt{2 \mu \tau }$,$\frac{d{p}_{g}^{{s}_{1}*}}{d{\mathrm{s}}_{1}}=\frac{d{q}_{g}^{{s}_{1}*}}{d{\mathrm{s}}_{1}}>0$, proposition 3② is proved.

Proof of Proposition 4.

When green products are subsidized according to the green degree, the first partial derivative of subsidy coefficient ${s}_{2}$ is obtained for the wholesale, loyalty and poaching prices of the two types of products.

$$ \frac{{dw_{g}^{{s_{2} *}} }}{{d{\text{s}}_{2} }} = \frac{{\tau \left( {2 - x_{0} } \right)\left( {\gamma \left( {1 - \alpha } \right)\left( {\gamma + s_{2} } \right)^{2} + 8s_{2} \mu \tau \left( {2\alpha - 1} \right) + 8\mu \tau \gamma \alpha } \right)}}{{\left( {\left( {\gamma + s_{2} } \right)^{2} - 8\mu \tau } \right)^{2} }};\frac{{dp_{g}^{{s_{2} *}} }}{{d{\text{s}}_{2} }} = \frac{{dq_{g}^{{s_{2} *}} }}{{d{\text{s}}_{2} }} = \frac{{\tau \left( {2 - x_{0} } \right)\left( {4\mu \tau \left( {\gamma - s_{2} } \right) + \gamma \left( {\gamma + s_{2} } \right)^{2} } \right)}}{{\left( {s_{2}^{2} + 2s_{2} \gamma + \gamma^{2} - 8\mu \tau } \right)^{2} }} $$

$$ \frac{{dp_{b}^{{s_{2} *}} }}{{d{\text{s}}_{2} }} = \frac{{dq_{b}^{{s_{2} *}} }}{{d{\text{s}}_{2} }} = \frac{{2\mu \tau^{2} \left( {s_{2} + \gamma } \right)\left( {x_{0} - 2} \right)}}{{\left( {s_{2}^{2} + 2s_{2} \gamma + \gamma^{2} - 8\mu \tau } \right)^{2} }} < 0 $$

Thus, when $\gamma \left(1-\alpha \right){\left(\gamma +{s}_{2}\right)}^{2}+8{s}_{2}\mu \tau \left(2\alpha -1\right)+8\mu \tau \gamma \alpha >0$, $\frac{d{w}_{g}^{{s}_{2}*}}{d{\mathrm{s}}_{2}}>0$ and Proposition 4① is proved. When $4\mu \tau \left(\gamma -{s}_{2}\right)+\gamma {\left(\gamma +{s}_{2}\right)}^{2}>0$,$\frac{d{p}_{g}^{{s}_{2}*}}{d{\mathrm{s}}_{2}}=\frac{d{q}_{g}^{{s}_{2}*}}{d{\mathrm{s}}_{2}}>0$ and Proposition 4② is proved.

Proof of Proposition 5.

When the subsidy is provided according to the R&D input cost of the product, the first partial derivative of the subsidy coefficient ${s}_{3}$ is obtained for the wholesale, loyalty and poaching prices of the two types of products.

$$ \frac{{dw_{g}^{{s_{3} *}} }}{{d{\text{s}}_{3} }} = \frac{{4\mu \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)\left( {\alpha - 1} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} > 0;\frac{{dp_{g}^{{s_{3} *}} }}{{ds_{3} }} = \frac{{dq_{g}^{{s_{3} *}} }}{{ds_{3} }} = \frac{{6\mu \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)\left( {\alpha - 1} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} $$

$$ \frac{{dp_{b}^{{s_{3} *}} }}{{d{\text{s}}_{3} }} = \frac{{dq_{b}^{{s_{3} *}} }}{{d{\text{s}}_{3} }} = \frac{{ - \mu \gamma^{2} \tau^{2} \left( {x_{0} - 2} \right)\left( {\alpha - 1} \right)}}{{\left( { - \gamma^{2} + 8\mu \tau - 8\mu \tau s_{3} \left( {1 - \alpha } \right)} \right)^{2} }} < 0 $$

Therefore, Proposition 5 is proved.