## Appendix A

**Proof of** **Proposition 1.**

Firm’s profit in Case O is given by

where

$0\le \theta \le 1$ and

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{O}<1+a\theta $,

${\pi}_{O}({p}_{O},\theta )$ is a function of

${p}_{O}$ and

$\theta $. Firstly, we solved the first-order conditions of

${\pi}_{O}({p}_{O},\theta )$ on

${p}_{O}$, and reached the result

${p}_{O}(\theta )=\frac{1}{2}[a\theta +(1+\frac{b{\theta}^{2}}{2})c+1]$. Since

$\frac{{\partial}^{2}{\pi}_{O}({p}_{O},\theta )}{\partial {p}_{O}^{2}}=-2<0$, we only need to show that

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{O}(\theta )<1+a\theta $. Because

$1+a\theta >(1+\frac{b{\theta}^{2}}{2})c$, we can prove that

${p}_{O}(\theta )$ abides by this condition. When substituting

${p}_{O}(\theta )$ into the firm’s profit function, we have

${\pi}_{O}(\theta )=\frac{1}{4}{[a\theta -(1+\frac{b{\theta}^{2}}{2})c+1]}^{2}$. We proceed by solving the first-order conditions of

${\pi}_{O}(\theta )$ on the

$$, and have

**Table A1.**
Monotonicity judging table of ${\pi}_{O}(\theta ).$

**Table A1.**
Monotonicity judging table of ${\pi}_{O}(\theta ).$

$\mathit{\theta}$ | (−$\mathit{\infty}$, ${\mathit{\theta}}_{\mathit{O}1}$) | ${\mathit{\theta}}_{\mathit{O}1}$ | (${\mathit{\theta}}_{\mathit{O}1}$, ${\mathit{\theta}}_{\mathit{O}2}$) | ${\mathit{\theta}}_{\mathit{O}2}$ | (${\mathit{\theta}}_{\mathit{O}2}$, ${\mathit{\theta}}_{\mathit{O}3}$) | ${\mathit{\theta}}_{\mathit{O}3}$ | (${\mathit{\theta}}_{\mathit{O}3}$, +$\mathit{\infty}$) |
---|

${\pi}_{O}^{(1)}(\theta )$ | - | 0 | + | 0 | - | 0 | + |

${\pi}_{O}(\theta )$ | ↘ | Min | ↗ | Max | ↘ | Min | ↗ |

We notice that ${\theta}_{O1}=\frac{a-\sqrt{{a}^{2}-2b{c}^{2}+2bc}}{bc}=\frac{a-\sqrt{{a}^{2}+2bc(1-c)}}{bc}<\frac{a-\sqrt{{a}^{2}}}{bc}=0.$

In order to get the optimal solutions, we have considered this question from two perspectives.

- (i)
$a\ge bc$. In this case, ${\pi}_{O}(\theta )$ is monotone increasing in $\theta $($0\le \theta \le 1$). Thus, when ${\theta}_{O}^{*}=1$, then ${\pi}_{O}^{*}=\frac{1}{4}{[a-(1+\frac{b}{2})c+1]}^{2}\u200a\text{\hspace{0.05em}}$, and ${p}_{O}^{*}=\u200a\text{\hspace{0.05em}\hspace{0.05em}}\u200a\frac{1}{2}[a+(1+\frac{b}{2})c+1]\text{\hspace{0.05em}}$.

- (ii)
$a<bc$. In this case, we notice that

${\theta}_{O2}=\frac{a}{bc}<1$, and we have

${\theta}_{O3}-1=\frac{a+\sqrt{{a}^{2}-2b{c}^{2}+2bc}}{bc}-1>0$. By combining the cases of

${\pi}_{O}(\theta )$ in

Table A1, when

${\theta}_{O}^{*}=\frac{a}{bc}$, the firm will get optimal profit

${\pi}_{O}^{*}=\frac{1}{4}{[\frac{{a}^{2}}{2bc}-c+1]}^{2}$ and will set the optimal price as

${p}_{O}^{*}=\u200a\text{\hspace{0.05em}\hspace{0.05em}}\u200a\frac{1}{2}[\frac{3{a}^{2}}{2bc}+c+1]\text{\hspace{0.05em}}$.

**Proof of** **Lemma 1.**

If

$a\ge bc$, the price difference between Case B and Case O is presented as

If

$a<bc$, the price difference between Case B and Case O is presented as

Therefore, it is evident that ${p}_{O}^{*}>{p}_{B}^{*}$ for the two above cases.

**Proof of** **Lemma 2.**

If

$a\ge bc$, the profit difference between Case B and Case O is presented as

If

$a<bc$, the difference profit between Case B and Case O is presented as

Thus, $\Delta {\pi}_{OB}^{*}={\pi}_{O}^{*}-{\pi}_{B}^{*}>0$, i.e., ${\pi}_{O}^{*}>{\pi}_{B}^{*}$.

**Proof of** **Proposition 2.**

In case T, firm’s profit function is given by

where

$0\le \theta \le 1$ and

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{T}<1+a\theta $. In this case, we have considered that these two consumer segments are in the market, so

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{T}<1-\mathrm{d}\theta $. We have solved the first-order conditions of

${\pi}_{T}({p}_{T},\theta )$ on

${p}_{T}$, and got the price

Consequently, we can prove that

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{T}(\theta )<1-\mathrm{d}\theta $, and substitute

${p}_{T}(\theta )$ into the Equation (3), resulting in

We solve the first-order conditions of

${\pi}_{T}^{*}(\theta )$ on

$\theta $, and we have

Having in view the conditions in this case, we have

${\theta}_{T1}<{\theta}_{T2}<{\theta}_{T3}$;

**Table A2.**
Monotonicity judging table of ${\pi}_{T}^{*}(\theta ).$

**Table A2.**
Monotonicity judging table of ${\pi}_{T}^{*}(\theta ).$

$\mathit{\theta}$ | (−$\mathit{\infty}$, ${\mathit{\theta}}_{\mathit{T}1}$) | ${\mathit{\theta}}_{\mathit{T}1}$ | (${\mathit{\theta}}_{\mathit{T}1}$, ${\mathit{\theta}}_{\mathit{T}2}$) | ${\mathit{\theta}}_{\mathit{T}2}$ | (${\mathit{\theta}}_{\mathit{T}2}$, ${\mathit{\theta}}_{\mathit{T}3}$) | ${\mathit{\theta}}_{\mathit{T}3}$ | (${\mathit{\theta}}_{\mathit{T}3}$, +$\mathit{\infty}$) |
---|

${\pi}_{T}^{(1)}(\theta )$ | - | 0 | + | 0 | - | 0 | + |

${\pi}_{T}(\theta )$ | ↘ | Min | ↗ | Max | ↘ | Min | ↗ |

Additionally, we will discuss the optimal solution from three perspectives:

- (i)
$ar-(1-r)d\ge bc$. In this case ${\pi}_{T}^{*}(\theta )$ is monotone increasing in $\theta $ ($0\le \theta \le 1$). As a result, when ${\theta}_{T}^{*}=1$, then ${\pi}_{T}^{*}=\frac{1}{4}{[1+ar-(1-r)d-(1+\frac{b}{2})c]}^{2}$, and ${p}_{T}^{*}=\frac{1}{2}[r(1+a)+(1+\frac{b}{2})c+(1-r)(1-d)]$.

- (ii)
$0<ar-(1-r)d<bc$. In this case, we have

$0<{\theta}_{T2}=\frac{ar-(1-r)d}{bc}<1$ and

${\theta}_{T3}-1=\frac{ar-(1-r)d+\sqrt{{[ar-(1-r)d]}^{2}+2\mathrm{bc}(1-\mathrm{c})}}{bc}-1>0$. Combining the cases in

Table A2, we can learn that when

${\theta}_{T}^{*}=\frac{ar-(1-r)d}{bc}$, then

${\pi}_{T}^{*}=\frac{1}{4}{[\frac{{(ar+dr-d)}^{2}}{2bc}+1-c]}^{2}$,

${p}_{T}^{*}=\frac{3{[ar-(1-r)d]}^{2}+2bc(1+c)}{4bc}$.

- (iii)
$ar-(1-r)d\le 0$. From the conditions of this case, we have ${\theta}_{T2}=\frac{ar-(1-r)d}{bc}\le 0$ and ${\theta}_{T3}-1=\frac{ar-(1-r)d+\sqrt{{[ar-(1-r)d]}^{2}+2\mathrm{bc}(1-\mathrm{c})}}{bc}-1>0$. In this case, when ${\theta}_{T}^{*}=0$, then ${\pi}_{T}^{*}=\frac{1}{4}{(1-c)}^{2}$,${p}_{T}^{*}=\frac{1}{2}(1+c)$.

**Proof of** **Corollary 1.**

If

$ar-(1-r)d\ge bc$, the profit difference between Case T and Case B is presented as

If

$0<ar-(1-r)d<bc$, the profit difference between Case T and Case B is presented as

Thus, $\Delta {\pi}_{TB}={\pi}_{T}^{*}-{\pi}_{B}^{*}>0$, i.e., ${\pi}_{T}^{*}>{\pi}_{B}^{*}$.

**Proof of** **Lemma 3.**

If

$ar-(1-r)d\ge bc$, the product demand and firm’s optimal profit are given by

If

$0<ar-(1-r)d<bc$, the product demand and firm’s optimal profit are given by

Therefore, when $ar-(1-r)d>0$, the product demand is increasing in $r$, and the firm’s profit is also increasing in $r$.

**Proof** **of Lemma 4.**

When

$0<ar-(1-r)d<bc$, the firm’s optimal sustainable level is given by

We calculate

${\theta}_{T}^{*}$ first derivative of

$r,b,c$ respectively, and we have

**Proof of** **Lemma 5.**

If

$ar-(1-r)d\ge bc$, the firm’s optimal price, product demand, marginal profit and firm’s profit are given by

Since

$D=ar-(1-r)d$, these functions can be written as

Similarly, if

$0<ar-(1-r)d<bc$, then we can write the firm’s optimal price, product demand, marginal profit and firm’s profit as

and we have

In conclusion, when $ar-(1-r)d>0$, the firm’s optimal price is increasing in $D$; the product demand is increasing in $D$; the marginal profit is increasing in $D$; and the firm’s profit is increasing in $D$.

**Proof of** **Lemma 6.**

When

$ar-(1-r)d\ge bc$, the optimal price is given by

Consequently, we have $\frac{d{p}_{T}^{*}}{dc}=1+\frac{b}{2}>0$

When

$0<ar-(1-r)d<bc$, the optimal price is given by

We can also write the equation as:${p}_{T}^{*}$ as ${p}_{T}^{*}=\frac{3{D}^{2}}{4bc}+\frac{1+c}{2}$, and we have $\frac{d{p}_{T}^{*}}{dc}=\frac{1}{2}-\frac{3{D}^{2}}{4b{c}^{2}}$.

If $b\le \frac{2}{3}$, $\frac{d{p}_{T}^{*}}{dc}=\frac{1}{2}-\frac{3{D}^{2}}{4b{c}^{2}}>0$

Therefore, when $ar-(1-r)d>0$, if $b>\frac{2}{3}$, $\sqrt{\frac{2b{c}^{2}}{3}}\le D<bc$, the optimal price is decreasing in $c$; otherwise, the optimal price is increasing in $c$.

**Proof of** **Lemma 7.**

When

$ar-(1-r)d\ge bc$, the optimal price, marginal profit and product demand are given by

in Case T, and

in Case O. Then, we have

When

$0<ar-(1-r)d<bc\le a$, the optimal price, marginal profit and product demand are given by

in Case T, and

in Case O. As a result, we have

When

$a<bc$, the optimal price, marginal profit and product demand are given by

in Case T, and

in Case O. Then, we have

Eventually we know that the optimal price, marginal profit and product demand in Case T are less than that in Case O.

**Proof of** **Proposition 3.**

In Case S, firm’s profit function is given as

We consider that

${p}_{S}\ge 1-d\theta $. Because regular consumers will not buy sustainable products, there are only environmentally conscious consumers in the market and

$r>0$. Firm’s profit function can be shown as

When solving the first-order conditions of

${\pi}_{S}({p}_{S},r)$ on

${p}_{S}$, we have

From the model setting, we can prove that $(1+\frac{b{\theta}^{2}}{2})c<{p}_{S}<1+a\theta $. Since $\frac{{d}^{2}{\pi}_{S}({p}_{S},r)}{d{p}_{S}^{2}}=-2r<0$, when $\frac{1}{2}[1+a\theta +(1+\frac{b{\theta}^{2}}{2})c]\ge 1-d\theta $, and when substituting ${p}_{S}$ into firm’s profit function, we have ${\pi}_{S}(r)=\frac{r}{4}{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}-\frac{M{r}^{2}}{2}$. By solving the first-order conditions of ${\pi}_{S}(r)$ on $r$, we have ${r}_{S}=\frac{{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}}{4M}$. Because $0<r\le 1$, we need to compare ${r}_{S}$ and 1. Since $\frac{{d}^{2}{\pi}_{S}(r)}{d{r}^{2}}=-M<0$,

If

$M>\frac{{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}}{4}$, then

If

$M\le \frac{{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}}{4}$, then

**Proof of** **Proposition 4.**

Similar to the proof of Proposition 3, when

${p}_{S}<1-d\theta $, the firm’s profit function in Case S can be shown as

Solve the first-order conditions of

${\pi}_{S}({p}_{S},r)$ on

${p}_{S}$, and we have

From pre-conditions, we can prove

$(1+\frac{b{\theta}^{2}}{2})c<{p}_{S}(r)<1+a\theta $. Since

$\frac{{d}^{2}{\pi}_{S}({p}_{S},r)}{d{p}_{S}^{2}}=-2$, when

${p}_{S}(r)<1-d\theta $, substitute

${p}_{S}(r)$ into firm’s profit function, and we have

From this equation, we can calculate $\frac{{d}^{2}{\pi}_{S}(r)}{d{r}^{2}}=\frac{{(a\theta +d\theta )}^{2}}{2}-M$

- (i)
If

$M>\frac{{(a\theta +d\theta )}^{2}}{2}$, we have

from the first-order condition of

${\pi}_{S}(r)$ on

$r$. Because

$0<r\le 1$, we need to compare

${r}_{S}$ and 1.

When

$M>\frac{(a\theta +d\theta )[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2}$, then

When

$M\le \frac{(a\theta +d\theta )[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2}$, then

- (ii)
If

$M=\frac{{(a\theta +d\theta )}^{2}}{2}$, firm’s profit function can be shown as

From this equation, we know that

${\pi}_{S}(r)$ is increasing in

$r$. Thus, in this situation, firm’s optimal profit, price and segmentation degree are given by

- (iii)
If

$M<\frac{{(a\theta +d\theta )}^{2}}{2}$, we have

Firm’s optimal profit, price and segmentation degree are given by