# Sustainable Product Strategy in Apparel Industry with Consumer Behavior Consideration

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Setting

#### 3.1. Firm

#### 3.2. Consumers

## 4. Model Analyses

#### 4.1. Benchmark Case (B)

#### 4.2. Sustainable Strategy with Homogenous Consumers (O)

**Proposition 1.**

- (i)
- If $a\ge bc$, ${\theta}_{O}^{*}=1$, ${p}_{O}^{*}=\frac{1}{2}[1+a+(1+\frac{b}{2})c]\text{\hspace{0.05em}}$ and ${\pi}_{O}^{*}=\frac{1}{4}{[1+a-(1+\frac{b}{2})c]}^{2}\text{\hspace{0.05em}}$
- (ii)
- If $a<bc$, ${\theta}_{O}^{*}=\frac{a}{bc}$, ${p}_{O}^{*}=\frac{1}{2}[\frac{3{a}^{2}}{2bc}+c+1]\text{\hspace{0.05em}}\u200a\text{\hspace{0.05em}}$ and ${\pi}_{O}^{*}=\frac{1}{4}{[\frac{{a}^{2}}{2bc}-c+1]}^{2}$

**Lemma 1.**

**Lemma 2.**

#### 4.3. Market with Two Consumer Segments (T)

**Proposition 2.**

- (i)
- If $ar-(1-r)d\ge bc$, ${\theta}_{T}^{*}=1$, ${p}_{T}^{*}=\frac{1}{2}[r(1+a)+(1+\frac{b}{2})c+(1-r)(1-d)]$ and ${\pi}_{T}^{*}=\frac{1}{4}{[1+ar-(1-r)d-(1+\frac{b}{2})c]}^{2}$;
- (ii)
- If $0<ar-(1-r)d<bc$, ${\theta}_{T}^{*}=\frac{ar-(1-r)d}{bc}$, ${p}_{T}^{*}=\frac{3{[ar-(1-r)d]}^{2}+2bc(1+c)}{4bc}$ and ${\pi}_{T}^{*}=\frac{1}{4}{[\frac{{(ar+dr-d)}^{2}}{2bc}+1-c]}^{2}$;
- (iii)
- If $ar-(1-r)d\le 0$, ${\theta}_{T}^{*}=0$, ${p}_{T}^{*}=\frac{1}{2}(1+c)$ and ${\pi}_{T}^{*}=\frac{1}{4}{(1-c)}^{2}$.

**Corollary 1.**

**Lemma 3.**

**Lemma 4.**

- (i)
- the firm’s optimal sustainable level ${\theta}_{T}^{*}$ is increasing in $r$;
- (ii)
- the firm’s optimal sustainable level ${\theta}_{T}^{*}$ is decreasing in $b$;
- (iii)
- the firm’s optimal sustainable level ${\theta}_{T}^{*}$ is decreasing in $c$.

**Lemma**

**5.**

**Lemma 6.**

- (i)
- when $b>\frac{2}{3}$, $\sqrt{\frac{2b{c}^{2}}{3}}\le D<bc$, the optimal price ${p}_{T}^{*}$ is decreasing in the unit regular cost $c$;
- (ii)
- otherwise, the optimal price ${p}_{T}^{*}$ is increasing in the unit regular cost $c$.

**Lemma 7.**

- (i)
- the optimal price in Case T is less than that in Case O, i.e., ${p}_{T}^{*}<{p}_{O}^{*}$;
- (ii)
- the marginal profit in Case T is less than that in Case O, i.e., ${s}_{T}<{s}_{O}$;
- (iii)
- the product demand in Case T is less than that in Case O, i.e., ${q}_{T}<{q}_{O}$.

#### 4.4. Consumer Segmentation (S)

**Proposition 3.**

**Proposition 4.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of**

**Proposition 1.**

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

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

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

- (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.**

**Proof of**

**Lemma 2.**

**Proof of**

**Proposition 2.**

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

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

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

- (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.**

**Proof of**

**Lemma 3.**

**Proof**

**of Lemma 4.**

**Proof of**

**Lemma 5.**

**Proof of**

**Lemma 6.**

**Proof of**

**Lemma 7.**

**Proof of**

**Proposition 3.**

**Proof of**

**Proposition 4.**

- (i)
- If $M>\frac{{(a\theta +d\theta )}^{2}}{2}$, we have$${r}_{S}=\frac{(a\theta +d\theta )[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2[M-\frac{{(a\theta +d\theta )}^{2}}{2}]}$$When $M>\frac{(a\theta +d\theta )[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2}$, then$${\pi}_{S}^{*}=\frac{M{[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}}{4[2M-{(a\theta +d\theta )}^{2}]},{p}_{S}^{*}=\frac{1}{2}[1-d\theta +(1+\frac{b{\theta}^{2}}{2})c+\frac{{(a\theta +d\theta )}^{2}(1-d\theta -c-\frac{b{\theta}^{2}}{2})c}{2M-{(a\theta +d\theta )}^{2}}]\phantom{\rule{0ex}{0ex}}\mathrm{and}{r}_{S}^{*}=\frac{(a\theta +d\theta )[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2[M-\frac{{(a\theta +d\theta )}^{2}}{2}]}.$$When $M\le \frac{(a\theta +d\theta )[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2}$, then$${\pi}_{S}^{*}=\frac{1}{4}{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}-\frac{M}{2},{p}_{S}^{*}=\frac{1}{2}[1+a\theta +(1+\frac{b{\theta}^{2}}{2})c]\mathrm{and}{r}_{S}^{*}=1.$$
- (ii)
- If $M=\frac{{(a\theta +d\theta )}^{2}}{2}$, firm’s profit function can be shown as$$Max\u200a\text{\hspace{0.05em}}{\pi}_{S}(r)=\frac{1}{4}{[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}+\frac{r(a\theta +d\theta )}{2}[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c].$$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$${\pi}_{S}^{*}=\frac{1}{4}{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}-\frac{M}{2},{p}_{S}^{*}=\frac{1}{2}[1+a\theta +(1+\frac{b{\theta}^{2}}{2})c]\mathrm{and}{r}_{S}^{*}=1.$$
- (iii)
- If $M<\frac{{(a\theta +d\theta )}^{2}}{2}$, we have$$\frac{{d}^{2}{\pi}_{S}(r)}{d{r}^{2}}=\frac{{(a\theta +d\theta )}^{2}}{2}-M>0\mathrm{and}r=\frac{(a\theta +d\theta )[1-d\theta -(1+\frac{b{\theta}^{2}}{2})c]}{2[M-\frac{{(a\theta +d\theta )}^{2}}{2}]}0.$$Firm’s optimal profit, price and segmentation degree are given by$${\pi}_{S}^{*}=\frac{1}{4}{[1+a\theta -(1+\frac{b{\theta}^{2}}{2})c]}^{2}-\frac{M}{2},{p}_{S}^{*}=\frac{1}{2}[1+a\theta +(1+\frac{b{\theta}^{2}}{2})c]\mathrm{and}{r}_{S}^{*}=1.$$

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**MDPI and ACS Style**

Yang, L.; Dong, S.
Sustainable Product Strategy in Apparel Industry with Consumer Behavior Consideration. *Sustainability* **2017**, *9*, 920.
https://doi.org/10.3390/su9060920

**AMA Style**

Yang L, Dong S.
Sustainable Product Strategy in Apparel Industry with Consumer Behavior Consideration. *Sustainability*. 2017; 9(6):920.
https://doi.org/10.3390/su9060920

**Chicago/Turabian Style**

Yang, Liu, and Shaozeng Dong.
2017. "Sustainable Product Strategy in Apparel Industry with Consumer Behavior Consideration" *Sustainability* 9, no. 6: 920.
https://doi.org/10.3390/su9060920