Environmental subsidy and the choice of green technology in the presence of green consumers

Abstract

In this paper, we present a study on a government using subsidy policy to motivate firms’ adoption of green emissions-reducing technology when consumers are environmentally discerning. We consider two profit-maximizing firms selling two products in a price and pollution sensitive market. The products differ only in their manufacturing costs, selling prices and the amount of pollutant emissions per unit of product. The objective of each firm is to determine the selling prices of the products, taking into account the impact of green technology on costs and customer demands. Two cases are considered: (1) the government has limited budget and can choose only one firm at most to provide subsidy; (2) the government has sufficient budget and can choose both firms to provide subsidy. We discuss which firm should be selected in each case and in which situation the firm has incentive to invest in the green technology. We also show that the green technology level, environmental improvement coefficient and unit cost increase coefficient play important roles in the government subsidy strategy.

Appendices

Appendix 1

Proof of Proposition 5

• (1) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy and it chooses $F_{1}$ to subsidize, the necessary condition of $F_{1 }$ accepting the subsidy policy and investing in technology is $F_{1}$’s profit satisfy: $\pi _{11} >\pi _{10}$ and $\pi _{11} >\pi _{12}$, equivalently,

  $$\begin{aligned} \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {\Delta e-\delta t+\Delta c-\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2} \end{aligned}$$

  and

  $$\begin{aligned} \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {\Delta e-\delta t+\Delta c-\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {\Delta e+\delta t+\Delta c+\gamma t} \right) ^{2}. \end{aligned}$$

  Simplifying these conditions we get

  $$\begin{aligned} s>-\frac{2}{9}\left( {\frac{\delta \left( {\delta ^{2}t^{2}-\left( {\Delta e} \right) ^{2}+t^{2}\delta \gamma \left( {2\delta +\gamma } \right) -2\Delta e\gamma \left( {\Delta e+\Delta c} \right) +\delta \left( {\Delta c} \right) ^{2}} \right) }{\delta ^{2}t^{2}-\left( {\Delta e} \right) ^{2}}} \right) +\xi t. \end{aligned}$$

• (2) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy and it chooses $F_{2}$ to subsidize, the necessary condition of $F_{2 }$ accepting the subsidy policy and investing in technology is $F_{2}$’s profit satisfy: $\pi _{22} >\pi _{20}$ and $\pi _{22} >\pi _{21}$, equivalently,

  $$\begin{aligned} \frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {2\Delta e+2\delta t-\Delta c+\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2} \end{aligned}$$

  and

  $$\begin{aligned} \frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {2\Delta e+2\delta t-\Delta c+\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {2\Delta e-2\delta t-\Delta c+\gamma t} \right) ^{2}. \end{aligned}$$

  Simplifying these conditions we get $s>-\frac{2\delta }{9}\left( {4+\frac{\left( {\gamma t-\Delta c} \right) ^{2}}{\delta ^{2}t^{2}-\left( {\Delta e} \right) ^{2}}} \right) +\xi t$.

• (3) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy and it only chooses $F_{1}$ to subsidize, the necessary condition of $F_{1 }$ accepting the subsidy policy and investing in technology is $F_{1}$’s profit satisfy: $\pi _{11} >\pi _{10}$ and $\pi _{11} >\pi _{12}$, equivalently

  $$\begin{aligned} \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {\Delta e-\delta t+\Delta c-\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2} \end{aligned}$$

  and

  $$\begin{aligned} \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {\Delta e-\delta t+\Delta c-\gamma t} \right) ^{2}-\xi t^{2}+ts>\frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {\Delta e+\delta t+\Delta c+\gamma t} \right) ^{2}. \end{aligned}$$

  Simplifying these conditions we get

  $$\begin{aligned} s>-\frac{2}{9}\left( {\frac{\delta \left( {\delta ^{2}t^{2}-\left( {\Delta e} \right) ^{2}+t^{2}\delta \gamma \left( {2\delta +\gamma } \right) -2\Delta e\gamma \left( {\Delta e+\Delta c} \right) +\delta \left( {\Delta c} \right) ^{2}} \right) }{\delta ^{2}t^{2}-\left( {\Delta e} \right) ^{2}}} \right) +\xi t. \end{aligned}$$

• (4) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy and it chooses both firms to subsidize, the necessary condition of both firms accepting the subsidy policy and investing in technology is the profits of $F_{1}$ and $F_{2}$ satisfy: $\pi _{13} >\max \left\{ {\pi _{10} ,\pi _{12} } \right\}$ and $\pi _{23} >\max \left\{ {\pi _{20} ,\pi _{21} } \right\}$, equivalently

  $$\begin{aligned} \frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2}-\xi t^{2}+ts>\max \left\{ {\frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2}, \frac{1}{9\left( {\Delta e\hbox {+}\delta t} \right) }\left( {\Delta e\hbox {+}\delta t+\Delta c\hbox {+}\gamma t} \right) ^{2}} \right\} \end{aligned}$$

  and $\frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2}-\xi t^{2}+ts>\max \Big \{ \frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2}, \frac{1}{9\left( {\Delta e-\delta t} \right) }( 2\Delta e-\hbox {2}\delta t-\Delta c\hbox {+}\gamma t )^{2} \Big \}$ Simplifying these conditions we get

  $$\begin{aligned} s>\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e\left( {\Delta e+\delta t} \right) }\left( {\left( {\Delta e} \right) ^{2}\delta +2\left( {\Delta e} \right) ^{2}\gamma +\Delta e\delta ^{2}t+2\Delta e\delta t\gamma +2\Delta e\Delta c\gamma +\Delta e\gamma ^{2}t-\left( {\Delta c} \right) ^{2}\delta } \right) +\xi t, \\ \frac{1}{9\Delta e\left( {\Delta e-\delta t} \right) }\left( {-4\left( {\Delta e} \right) ^{2}\delta +4\left( {\Delta e} \right) ^{2}\gamma +4\Delta e\delta ^{2}t-4\Delta e\delta t\gamma -2\Delta e\Delta c\gamma +\Delta e\gamma ^{2}t+\left( {\Delta c} \right) ^{2}\delta } \right) +\xi t \\ \end{array}} \right\} . \end{aligned}$$

Proof of Proposition 6

• (1) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy, $F_{1 }$is chosen if the total change in pollutant emissions satisfy$\Delta E_1 >0$ and $\Delta E_1 >\Delta E_2$, equivalently, $\frac{1}{3}t\left( {\delta +\gamma } \right) >0$and $\frac{1}{3}t\left( {\delta +\gamma } \right) >\frac{1}{3}t\left( {2\delta -\gamma } \right)$. Simplifying these conditions we get $\delta <2\gamma$.

• (2) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy, $F_{2 }$is chosen if the total change in pollutant emissions satisfy $\Delta E_2 >0$ and $\Delta E_2 >\Delta E_1$, equivalently $\frac{1}{3}t\left( {2\delta -\gamma } \right) >0$ and $\frac{1}{3}t\left( {2\delta -\gamma } \right) >\frac{1}{3}t\left( {\delta +\gamma } \right)$. Simplifying these conditions we get $\delta >2\gamma$.

• (3) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy, only $F_{1 }$is chosen if the total change in pollutant emissions satisfy $\Delta E_1 >0$, $\Delta E_1 >\Delta E_2$ and $\Delta E_1 >\Delta E_3$, equivalently $\frac{1}{3}t\left( {\delta +\gamma } \right) >0$, $\frac{1}{3}t\left( {\delta +\gamma } \right) >\frac{1}{3}t\left( {2\delta -\gamma } \right)$, and $\frac{1}{3}t\left( {\delta +\gamma } \right) >\delta t$. Simplifying these conditions we get $\delta <\gamma /2$.

• (4) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy, both firms are chosen if the total change in pollutant emissions satisfy $\Delta E_3 >0$, $\Delta E_3 >\Delta E_1$, and $\Delta E_3 >\Delta E_2$, equivalently $\delta t>0$, $\delta t>\frac{1}{3}t\left( {\delta +\gamma } \right)$ and $\delta t>\frac{1}{3}t\left( {2\delta -\gamma } \right)$. Simplifying these conditions we get $\delta >\gamma /2$.

Appendix 2

Proof of Proposition 10

• (1) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy and it chooses $F_{2}$ to subsidize, the necessary condition of $F_{2 }$ accepting the subsidy policy and investing in technology is $F_{2}$’s profit satisfy: $\pi _{22} >\max \left\{ {\pi _{20} ,\pi _{21} } \right\}$, equivalently,

  $$\begin{aligned} \frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {2\Delta e+2\delta t-\Delta c-\gamma t} \right) ^{2}-\xi t^{2}+ts> \max \left\{ {\begin{array}{l} \frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2}, \\ \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {2\Delta e-2\delta t-\Delta c-\gamma t} \right) ^{2} \\ \end{array}} \right\} . \end{aligned}$$

  Simplifying these conditions we get

  $$\begin{aligned} s>\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e\left( {\Delta e+\delta t} \right) }\left( {\left( {\Delta c} \right) ^{2}\delta -4\left( {\Delta e} \right) ^{2}\delta +4\left( {\Delta e} \right) ^{2}\gamma -4\Delta e\delta ^{2}t+4\Delta e\delta t\gamma -2\Delta e\Delta c\gamma -\Delta e\gamma ^{2}t} \right) +\xi t, \\ \frac{1}{9\left( {\left( {\Delta e} \right) ^{2}-\delta ^{2}t^{2}} \right) }\left( {-4\left( {\Delta e} \right) ^{2}+\left( {\Delta c} \right) ^{2}+\gamma ^{2}t^{2}+2\Delta c\gamma t+4\delta ^{2}t^{2}} \right) +\xi t \\ \end{array}} \right\} . \end{aligned}$$

• (2) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy and it only chooses $F_{2}$ to subsidize, the necessary condition of $F_{2 }$accepting the subsidy policy and investing in technology is $F_{2}$’s profit satisfy: $\pi _{22} >\max \left\{ {\pi _{20} ,\pi _{21} } \right\}$ and we get

  $$\begin{aligned} s>\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e\left( {\Delta e+\delta t} \right) }\left( {\left( {\Delta c} \right) ^{2}\delta -4\left( {\Delta e} \right) ^{2}\delta +4\left( {\Delta e} \right) ^{2}\gamma -4\Delta e\delta ^{2}t+4\Delta e\delta t\gamma -2\Delta e\Delta c\gamma -\Delta e\gamma ^{2}t} \right) +\xi t, \\ \frac{1}{9\left( {\left( {\Delta e} \right) ^{2}-\delta ^{2}t^{2}} \right) }\left( {-4\left( {\Delta e} \right) ^{2}+\left( {\Delta c} \right) ^{2}+\gamma ^{2}t^{2}+2\Delta c\gamma t+4\delta ^{2}t^{2}} \right) +\xi t \\ \end{array}} \right\} \end{aligned}$$

• (3) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy and it chooses both firms to subsidize, the necessary condition of both firms accepting the subsidy policy and investing in technology is the profits of $F_{1}$ and $F_{2}$ satisfy: $\pi _{13} >\max \left\{ {\pi _{10} ,\pi _{12} } \right\}$ and $\pi _{23} >\max \left\{ {\pi _{20} ,\pi _{21} } \right\}$, equivalently,

  $$\begin{aligned} \frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2}-\xi t^{2}+ts_0 >\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e}\left( {\Delta e+\Delta c} \right) ^{2}, \\ \frac{1}{9\left( {\Delta e+\delta t} \right) }\left( {\Delta e+\delta t+\Delta c-\gamma t} \right) ^{2} \\ \end{array}} \right\} \end{aligned}$$

  and

  $$\begin{aligned} \frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2}-\xi t^{2}+ts_0 >\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e}\left( {2\Delta e-\Delta c} \right) ^{2}, \\ \frac{1}{9\left( {\Delta e-\delta t} \right) }\left( {2\Delta e-2\delta t-\Delta c-\gamma t} \right) ^{2} \\ \end{array}} \right\} . \end{aligned}$$

  Simplifying these conditions we get

  $$\begin{aligned}s>\max \left\{ {\begin{array}{l} \frac{1}{9\Delta e\left( {\Delta e+\delta t} \right) }\left( {\left( {\Delta e} \right) ^{2}\delta -\left( {\Delta c} \right) ^{2}\delta -2\left( {\Delta e} \right) ^{2}\gamma +\Delta e\delta ^{2}t-2\Delta e\delta t\gamma -2\Delta e\Delta c\gamma +\Delta e\gamma ^{2}t} \right) +\xi t, \\ \frac{1}{9\Delta e\left( {\Delta e-\delta t} \right) }\left( {-4\left( {\Delta e} \right) ^{2}\delta -4\left( {\Delta e} \right) ^{2}\gamma +\left( {\Delta c} \right) ^{2}\delta +\Delta e\gamma ^{2}t+4\Delta e\delta t\gamma +2\Delta e\Delta c\gamma +4\Delta e\delta ^{2}t} \right) +\xi t, \\ \\ \xi t \\ \end{array}} \right\} \end{aligned}$$

Proof of Proposition 11

• (1) Because $\Delta E_2 >\Delta E_1$, $F_{1}$ will never be chosen when the government has limited budget and can choose at most one firm to provide subsidy

• (2) Under the condition that the government has limited budget and can choose at most one firm to provide subsidy, $F_{2 }$will be chosen because $\Delta E_2 >\Delta E_1$ and $\Delta E_2 >0$.

• (3) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy, only $F_{2 }$will be chosen if the total change in pollutant emissions satisfy $\Delta E_2 >\Delta E_3$, equivalently, $\frac{1}{3}t\left( {2\delta +\gamma } \right) >\delta t$. Simplifying these conditions we get $\delta <\gamma$.

• (4) Under the condition that the government has enough budget and can choose at most both firms to provide subsidy, both firms are chosen if the total change in pollutant emission satisfy $\Delta E_3 >0$ and $\Delta E_3 >\Delta E_2$, equivalently $\delta t>0$ and $\delta t>\frac{1}{3}t\left( {2\delta +\gamma } \right)$. Simplifying these conditions we get $\delta >\gamma$.