Behavior-based pricing and consumer fairness concerns with green product design

Abstract

Technological innovation enables firms not only to produce green and energy-efficient products but also to more conveniently collect consumer data for price discrimination. However, such price discrimination can easily lead to unfairness when consumers find that they pay a higher price for the same product than others. This paper considers a two-period model with two firms differentiated with respect to greenness and studies the impact of behavior-based pricing (BBP) on the firm when consumers exhibit fairness concerns and the products differ in the improvement in their greenness. We find that when the greening improvement level is exogenous, the existence of consumer fairness concerns and differentiation in the level of greenness make the firm’s practice of BBP profitable. When the greening improvement level is endogenous, fairness concerns can increase the differentiation in the improvement in greenness. In addition, consumers’ fairness concerns do not always lead to BBP practices yielding a higher consumer surplus. The differentiation in the greenness level and fairness concerns mean that the use of BBP reduces social welfare.

Appendix

Definitions

The following aggregate is introduced in Sect. 4.2.2:

$$\begin{aligned} \Delta \subseteq \left\{ \begin{aligned}&0<\delta<\frac{10\lambda ^3-17\lambda ^2+17 \lambda +14}{6\lambda ^2+30\lambda +12},\quad 0\le \lambda \le \lambda _1 \\&\frac{-10\lambda ^3+23\lambda ^2+13\lambda -2}{6\lambda ^2+30 \lambda +12}<\delta \\ {}&\quad<\frac{10\lambda ^3-17\lambda ^2+17 \lambda +14}{6\lambda ^2+30\lambda +12} ,\quad \lambda _1<\lambda \le \frac{1}{16}(\sqrt{105}-3). \\&\frac{-10\lambda ^3+47\lambda ^2+22\lambda -11}{54 \lambda ^2+48\lambda -6}<\delta \\&\quad<\frac{10\lambda ^3+7\lambda ^2+26\lambda +5}{54\lambda ^2+48\lambda -6} , \quad \frac{1}{16}(\sqrt{105}-3)<\lambda \le 1 \end{aligned} \right. \end{aligned}$$

(A1)

where $\lambda _1=\text {Root}[10 \#1^3-23 \#1^2-13 \#1+2 \& ,2,0]$. It can be confirmed that $\delta <\frac{7}{6}.$ $\square$

Proof of $p_{Ho}>p_{Hn}$ and $p_{Lo}>p_{Ln}$

With symmetric second-period pricing policy. Suppose that $p_{Ho}<p_{Hn}$ and $p_{Lo}<p_{Ln}$, the switching consumers face fairness concerns. The indifferent consumer at $\theta _H$ is:

$$\begin{aligned}{} & {} v+\theta _H \alpha _H-p_{Ho}=v+\theta _H \alpha _L-p_{Ln}-\lambda (p_{Ln}-p_{Lo}) \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} \theta _H=\frac{p_{Ho}+\lambda p_{Lo}-(\lambda +1) p_{Ln}}{\alpha _H-\alpha _L} \end{aligned}$$

(A3)

Similarly, the indifferent consumer at $\theta _L$ is:

$$\begin{aligned} \theta _L=\frac{(\lambda +1)p_{Hn}- p_{Lo}-\lambda p_{Ho}}{\alpha _H-\alpha _L} \end{aligned}$$

(A4)

In view of the assumption that $p_{Ho}<p_{Hn}$ and $p_{Lo}<p_{Ln}$, the following equation holds in equilibrium

$$\begin{aligned} \theta _H^*-\theta _L^*=-\frac{(\lambda +1) \left( p_{Hn}^*-p_{Ho}^*+p_{Ln}^*-p_{Lo}^* \right) }{\alpha _H-\alpha _L}<0 \end{aligned}$$

(A5)

However, since both firms have switching customers, $\theta _H^*>\theta _1>\theta _L^*$ is required, which is inconsistent with the above equation $p_{Ho}<p_{Hn}$ and $p_{Lo}<p_{Ln}$.

With asymmetric second period pricing policy, $\frac{1}{3}<\theta _1<\frac{2}{3}$. Suppose that $p_{Ho}<p_{Hn}$ and $p_{Lo}\ge p_{Ln}$, the switching consumers of H firm and old consumers of L firm exhibit fairness concerns. The indifferent customers at $\theta _H$ and $\theta _L$ are:

$$\begin{aligned}{} & {} v+\theta _H \alpha _H-p_{Ho}=v+\theta _H \alpha _L-p_{Ln} \end{aligned}$$

(A6)

$$\begin{aligned}{} & {} \theta _H=\frac{p_{Ho}-p_{Ln}}{\alpha _H-\alpha _L} \end{aligned}$$

(A7)

and

$$\begin{aligned}{} & {} v+\theta _L \alpha _H-p_{Hn}-\lambda (p_{Hn}-p_{Ho})=v+\theta _L \alpha _L-p_{Lo}-\lambda (p_{Lo}-p_{Ln}) \end{aligned}$$

(A8)

$$\begin{aligned}{} & {} \theta _L=\frac{(\lambda +1) (p_{Hn}-p_{Lo})-\lambda (p_{Ho}-p_{Ln})}{\alpha _H-\alpha _L} \end{aligned}$$

(A9)

Focus on the interior solutions. The firms’ second-period profits are $\Pi _{H2}=(1-\theta _H)(p_{Ho}-c_H)+(\theta _1-\theta _L)(p_{Hn}-c_H)$ and $\Pi _{L2}=\theta _L(p_{Lo}-c_L)+(\theta _H-\theta _1)(p_{Ln}-c_L)$.

The first-order conditions w.r.t. $p_{Ho}^*$, $p_{Hn}^*$, $p_{Lo}^*$ and $p_{Ln}^*$ jointly lead to

$$\begin{aligned} p_{Ho}^*(\theta _1)= & {} \dfrac{{k(\lambda +1) ((\lambda (5-2\lambda )+6) \alpha _H^2+(4\lambda +3)\alpha _L^2)}{+ ((\lambda +1)((6-\lambda )\lambda +6)-\theta _1 (-3\lambda ^2+\lambda +3)) (\alpha _H-\alpha _L)}}{(\lambda +1) (\lambda (9-2\lambda )+9)} \end{aligned}$$

(A10)

$$\begin{aligned} p_{Hn}^*(\theta _1)= & {} \frac{k(\lambda +1)((\lambda (7-2\lambda )+6) \alpha _H^2+(2\lambda +3)\alpha _L^2)+(7\theta _1 \lambda +6 \theta _1+\lambda ^2+\lambda )(\alpha _H-\alpha _L)}{(\lambda +1)(\lambda (9-2\lambda )+9)} \end{aligned}$$

(A11)

$$\begin{aligned} p_{Lo}^*(\theta _1)= & {} \frac{k(\lambda +1)((2\lambda +3) \alpha _H^2+(\lambda (7-2\lambda )+6)\alpha _L^2)+(\theta _1(2(1-\lambda )\lambda +3)-\lambda ^2-\lambda )(\alpha _H-\alpha _L)}{(\lambda +1)(\lambda (9-2\lambda )+9)}\nonumber \\ \end{aligned}$$

(A12)

and

$$\begin{aligned} p_{Ln}^*(\theta _1)=\dfrac{{k(\lambda +1)((4\lambda +3) \alpha _H^2+(\lambda (5-2\lambda )+6)\alpha _L^2)}{+((\lambda +1)((3-\lambda )\lambda +3)-\theta _1(\lambda (\lambda +8)+6))(\alpha _H-\alpha _L)}}{(\lambda +1)(\lambda (9-2\lambda )+9)} \end{aligned}$$

(A13)

For $p_{Ho}^*(\theta _1)<p_{Hn}^*(\theta _1)$, we need $\theta _1>\frac{2}{3}+\frac{-\lambda ^3+6\lambda ^2-2k\lambda ^2-2k \lambda +\frac{17\lambda }{3}}{-3\lambda ^2+8\lambda +9}$ which can not hold if $\theta _1<\frac{2}{3}$ because $\frac{2}{3}+\frac{-\lambda ^3+6\lambda ^2-2k\lambda ^2-2k \lambda +\frac{17\lambda }{3}}{-3\lambda ^2+8\lambda +9}\ge \frac{2}{3}$.

By the same logic, suppose that $p_{Ho}\ge p_{Hn}$ and $p_{Lo}< p_{Ln}$. $p_{Lo}^*(\theta _1)<p_{Ln}^*(\theta _1)$ can not hold if $\theta _1>\frac{1}{3}$.

Following a symmetric first-period equilibrium, there exists a pure strategy second-period pricing equilibrium where $p_{Ho}\ge p_{Hn}$ and $p_{Lo}\ge p_{Ln}$. In this case, the indifferent customers at $\theta _H$ and $\theta _L$ are:

$$\begin{aligned}{} & {} v+\theta _H \alpha _H-p_{Ho}-\lambda (p_{Ho}-p_{Hn})=v+\theta _H \alpha _L-p_{Ln} \end{aligned}$$

(A14)

$$\begin{aligned}{} & {} \theta _H=\frac{(1+\lambda )p_{Ho}-p_{Ln}-\lambda p_{Hn}}{\alpha _H-\alpha _L} \end{aligned}$$

(A15)

and

$$\begin{aligned}{} & {} v+\theta _L q_H-p_{Hn}=v+\theta _L \alpha _L-p_{Lo}-\lambda (p_{Lo}-p_{Ln}) \end{aligned}$$

(A16)

$$\begin{aligned}{} & {} \theta _L=\frac{p_{Hn}+\lambda p_{Ln}-(1+\lambda )p_{Lo}}{\alpha _H-\alpha _L} \end{aligned}$$

(A17)

According to the firms’ second-period profits, the first-order conditions w.r.t. $p_{Ho}^*$, $p_{Hn}^*$, $p_{Lo}^*$ and $p_{Ln}^*$ jointly lead to the results in Table 2. The second order conditions of the second-period prices are $\frac{\partial ^2 \Pi _{H2}}{\partial p_{Ho}^2}=\frac{\partial ^2 \Pi _{L2}}{\partial p_{Lo}^2}=-\frac{2 (\lambda +1)}{\alpha _H-\alpha _L}<0$ and $\frac{\partial ^2 \Pi _{H2}}{\partial p_{Hn}^2}=\frac{\partial ^2 \Pi _{L2}}{\partial p_{Ln}^2}=-\frac{2}{\alpha _H-\alpha _L}<0$. For $p_{Ho}^*\ge p_{Hn}^*$, $p_{Lo}^*\ge p_{Ln}^*$ and $0<\theta _L<\theta _1<\theta _H<1$, we require that $\max \{\frac{6d\lambda ^2+6d\lambda -\lambda ^3+\lambda ^2+3\lambda +3}{3\lambda ^2+12\lambda +9},\frac{-3d\lambda -3 d+\lambda ^3-\lambda ^2-3\lambda }{3\lambda ^2-9\lambda -6}\}<\theta _1<\min \{\frac{6 d\lambda ^2+6d\lambda +\lambda ^3-4\lambda ^2+3\lambda +6}{3 \lambda ^2+12\lambda +9},\frac{-3d\lambda -3d-\lambda ^3+4\lambda ^2-3 \lambda -3}{3\lambda ^2-9\lambda -6}\}$. $\square$

Proof of Tables 2 and 3

The above proof details the results of the second period equilibrium given in Table 2. Here, we show the proof of the equilibrium results shown in Table 3.

In period 1, the indifferent consumer at $\theta _1$ is :

$$\begin{aligned} {v+\theta _1 \alpha _H-p_{H1}+(v+\theta _1 \alpha _L-p_{Ln}^e)=v+\theta _1 \alpha _L-p_{L1}+(v+\theta _1 \alpha _H-p_{Hn}^e)} \end{aligned}$$

(A18)

where $p_{Ln}^e$ and $p_{Hn}^e$ are the consumers’ rational expectation of second-period switching prices given the first-period prices. Based on rational expectations conditions, we assume that $p_{Ln}^e=p_{Ln}^*$ and $p_{Hn}^e=p_{Hn}^*$, where $p_{Ln}^*$ and $p_{Hn}^*$ are equilibrium results of firms’ second-period prices for new customers as a function of $\theta _1$. The expression of $\theta _1$ is as follows:

$$\begin{aligned} {\theta _1=\frac{(\alpha _H-\alpha _L) \left( k \left( 2 \lambda ^2+\lambda -1\right) (\alpha _H+\alpha _L)+1\right) +\left( -2 \lambda ^2+3 \lambda +3\right) p_{H1}+\left( 2 \lambda ^2-3 \lambda -3\right) p_{L1}}{4 (\lambda +1) (\alpha _H-\alpha _L)}}\nonumber \\ \end{aligned}$$

(A19)

Then, firms maximize the total profits of the two periods to obtain the first-period prices. Two firms’s profit functions are as follows:

$$\begin{aligned}{} & {} {\pi _{H}=\pi _{H1}+\pi _{H2}^*(\theta _1)=(1-\theta _1) \left( p_{H1}-k\alpha _H^2 \right) +\pi _{H2}^*(\theta _1)} \end{aligned}$$

(A20)

$$\begin{aligned}{} & {} {\pi _{L}=\pi _{L1}+\pi _{L2}^*(\theta _1)=\theta _1 \left( p_{L1}-k\alpha _L^2 \right) +\pi _{L2}^*(\theta _1)} \end{aligned}$$

(A21)

Thus, we can get the optimal results shown in Table 3. The second order condition satisfies the following inequality

$$\begin{aligned} {\frac{\partial ^2 \pi _{H}}{\partial p_{H1}^2}=-\frac{-7 \lambda ^3+\lambda ^2+15 \lambda +7}{8 (\lambda +1)^2 (\alpha _H-\alpha _L)}<0} \end{aligned}$$

(A22)

The Eq. (A22) holds since $0\le \lambda \le 1$. $\square$

Proof of Proposition 1

We first analyze the impact of $\theta _1$ on second-period prices. The equilibrium results are given in Table 2. Taking the partial derivative with respect to $\theta _1$, we have

$$\begin{aligned}{} & {} \frac{\partial p_{Lo}^{*}}{\partial \theta _1}=-\frac{\partial p_{Ho}^{*}}{\partial \theta _1}=\frac{(1-\lambda ) (\alpha _H-\alpha _L)}{3-2 \lambda ^2+3 \lambda }>0 \end{aligned}$$

(A23)

$$\begin{aligned}{} & {} \frac{\partial p_{Ln}^{*}}{\partial \theta _1}=-\frac{\partial p_{Hn}^{*}}{\partial \theta _1}=-\frac{2(1+\lambda ) (\alpha _H-\alpha _L)}{3-2 \lambda ^2+3 \lambda }<0 \end{aligned}$$

(A24)

Therefore, as $\theta _1$ ($1-\theta _1$) increases, L (H) firm’s first-period market share increases, the prices for repeated consumers increase and poaching prices decreases. Then, we focus on the impact of $\lambda$ and $\alpha _i$ on this results.

$$\begin{aligned}{} & {} \frac{\partial p_{Lo}^{*}}{\partial \theta _1 \partial \lambda }=-\frac{\partial p_{Ho}^{*}}{\partial \theta _1\partial \lambda }=-\frac{2 (\lambda ^2-2 \lambda +3)(\alpha _H-\alpha _L)}{(3-2\lambda ^2+3 \lambda )^2}<0 \end{aligned}$$

(A25)

$$\begin{aligned}{} & {} \frac{\partial p_{Ln}^{*}}{\partial \theta _1 \partial \lambda }=-\frac{\partial p_{Hn}^{*}}{\partial \theta _1\partial \lambda }=-\frac{4(\lambda ^2+2 \lambda )(\alpha _H-\alpha _L)}{(3-2\lambda ^2+3 \lambda )^2}<0 \end{aligned}$$

(A26)

So, as $\lambda$ increases, both $\frac{\partial p_{Lo}^{*}}{\partial \theta _1}$ and $\frac{\partial p_{Ln}^{*}}{\partial \theta _1}$ decreases. Similarly, we define the difference of green level $\Delta \alpha$ as $\alpha _H-\alpha _L$. We can obtain that as $\Delta \alpha$ increases, $\frac{\partial p_{Lo}^{*}}{\partial \theta _1}$ (i.e., $\frac{\partial p_{Lo}^{*}}{\partial \theta _1 \partial \Delta \alpha }>0$) increases and $\frac{\partial p_{Ln}^{*}}{\partial \theta _1}$ (i.e., $\frac{\partial p_{Ln}^{*}}{\partial \theta _1 \partial \Delta \alpha }<0$) decreases. $\square$

Proof of Proposition 2

In equilibrium, the total profits $\Pi _{H}^*$ and $\Pi _{L}^*$ as shown in Table 3. Partial derivative with respect to $\lambda$ shows that

$$\begin{aligned} \frac{\partial \pi _{H}^{*}}{\partial \lambda }&= \left( \left( \left( 2000 \lambda ^{12}-8000 \lambda ^{11}-89160 \lambda ^{10}+34540 \lambda ^9+348821 \lambda ^8 +274551 \lambda ^7-488725 \lambda ^6\right. \right. \right. \nonumber \\&\quad -872813 \lambda ^5-64239 \lambda ^4 +682629 \lambda ^3-36 \delta ^2 \left( -2 \lambda ^3+\lambda ^2+6 \lambda +3 \right) ^2 \nonumber \\ {}&\quad \left( 35 \lambda ^4-55 \lambda ^3-381 \lambda ^2-217 \lambda +26 \right) +517329 \lambda ^2 \nonumber \\&\quad +12 \delta (\lambda +1)^2 \left( 5020 \lambda ^8-5520 \lambda ^7-21957 \lambda ^6+19427 \lambda ^5+35484 \lambda ^4-18630 \lambda ^3\right. \nonumber \\ {}&\quad \left. -31385 \lambda ^2-6381 \lambda +1158 \right) \nonumber \\&\quad \left. \left. +140229 \lambda +11286t) (\alpha _H-\alpha _L))/(144 \left( 4-5 \lambda ^2+5 \lambda \right) ^3 \left( -2 \lambda ^3+\lambda ^2+6 \lambda +3 \right) ^2 \right) \right) \end{aligned}$$

(A27)

$$\begin{aligned} \frac{\partial \pi _{L}^{*}}{\partial \lambda }&= \left( \left( -2000 \lambda ^{12}+8000 \lambda ^{11}+33960 \lambda ^{10}-101740 \lambda ^9-89021 \lambda ^8+154689 \lambda ^7\right. \right. \nonumber \\ {}&\quad \left. \left. +263533 \lambda ^6+149765 \lambda ^5-195369 \lambda ^4 \right. \right. \nonumber \\&\quad -468957 \lambda ^3+36 \delta ^2 \left( -2 \lambda ^3+\lambda ^2+6 \lambda +3 \right) ^2 \left( 35 \lambda ^4-55 \lambda ^3-381 \lambda ^2-217 \lambda +26 \right) \nonumber \\&\quad -366825 \lambda ^2+12 \delta (\lambda +1)^2 \left( 4180 \lambda ^8-1680 \lambda ^7-16143 \lambda ^6-7567 \lambda ^5+16428 \lambda ^4\right. \nonumber \\ {}&\quad \left. +23454 \lambda ^3+13093 \lambda ^2+2529 \lambda -246 \right) \nonumber \\&\quad -128061 \lambda -16758) (\alpha _H-\alpha _L))/ \left( 144 \left( 5 \lambda ^2-5 \lambda -4 \right) ^3 \left( -2 \lambda ^3+\lambda ^2+6 \lambda +3 \right) ^2 \right) \end{aligned}$$

(A28)

$\frac{\partial \pi _{H}^{*}}{\partial \lambda }>0$ and $\frac{\partial \pi _{L}^{*}}{\partial \lambda }>0$ hold when $\lambda$ and $\delta$ meets the condition of Eq. (A1). So, a higher $\lambda$ leads to higher profits. Similarly, we focus on the impact of $\delta$ on profits. It is shown that

$\frac{\partial \pi _{H}^{*}}{\partial \delta }=\frac{(100 \lambda ^6-852 \lambda ^5+617 \lambda ^4+1696 \lambda ^3-606 \lambda ^2+6 \delta (14 \lambda ^5-39 \lambda ^4-56 \lambda ^3+90 \lambda ^2+138 \lambda +45)-1468 \lambda -447) (\alpha _H-\alpha _L)}{12 (-5 \lambda ^2+5 \lambda +4)^2 (3+3 \lambda -2 \lambda ^2)}<0$ and

$\frac{\partial \pi _{L}^{*}}{\partial \delta }=\frac{(-100 \lambda ^6+768 \lambda ^5-383 \lambda ^4-1360 \lambda ^3+66 \lambda ^2+6 k (14 \lambda ^5-39 \lambda ^4-56 \lambda ^3+90 \lambda ^2+138 \lambda +45)+640 \lambda +177) (\alpha _H-\alpha _L)}{12 (-5 \lambda ^2+5 \lambda +4)^2(3+3 \lambda -2 \lambda ^2)}>0$ also hold when $\lambda$ and $\delta$ meets the condition of Eq. (A1). A higher $\delta$ leads to a lower profit of H firm and a higher profit of L firm. As $\delta$ increases, H firm becomes less efficient and L more so. Therefore, as the products become more efficient, the firm’s profit increases. $\square$

Proof of Proposition 3

Comparing the profits of the two firms with consumer cognition and without consumer cognition, we have

$$\begin{aligned} \pi _{H}^{*}-\pi _{H}^{N}&= - \left( \left( 200 \lambda ^8-2900 \lambda ^7+13966 \lambda ^6+3595 \lambda ^5-38975 \lambda ^4-16202 \lambda ^3 \right. \right. \nonumber \\&\quad +36 \delta ^2 (\lambda +1)^2 \left( 14 \lambda ^4-53 \lambda ^3-3 \lambda ^2+93 \lambda +45 \right) \nonumber \\&\quad +30788 \lambda ^2+12 \delta \left( 100 \lambda ^7-752 \lambda ^6-235 \lambda ^5+2313 \lambda ^4+1090 \lambda ^3-2074 \lambda ^2-1915 \lambda -447 \right) \nonumber \\&\quad +25875 \lambda +5541)(\alpha _H-\alpha _L))/ \left( 144(\lambda +1) \left( -5 \lambda ^2+5 \lambda +4 \right) ^2 \left( 2 \lambda ^2-3 \lambda -3 \right) \right) \nonumber \\ {}&\quad -\frac{2}{9}(\delta -2)^2 \left( \alpha _H-\alpha _L \right) \end{aligned}$$

(A29)

$$\begin{aligned} \pi _{L}^{*}-\pi _{L}^{N}&= - \left( \left( 200 \lambda ^8-1700 \lambda ^7+5446 \lambda ^6-125 \lambda ^5-14639 \lambda ^4-1898 \lambda ^3 \right. \right. \nonumber \\&\quad +36 \delta ^2 (\lambda +1)^2 \left( 14 \lambda ^4-53 \lambda ^3-3 \lambda ^2+93 \lambda +45 \right) \nonumber \\&\quad +14108 \lambda ^2-12 \delta \left( 100 \lambda ^7-668 \lambda ^6-385 \lambda ^5+1743 \lambda ^4+1294 \lambda ^3-706 \lambda ^2-817 \lambda -177 \right) \nonumber \\&\quad +9483 \lambda +1797)(\alpha _H-\alpha _L))/ \left( 144 (\lambda +1) \left( -5 \lambda ^2+5 \lambda +4 \right) ^2 \left( 2 \lambda ^2-3 \lambda -3 \right) \right) \nonumber \\ {}&\quad -\frac{2}{9}(\delta +1)^2 (\alpha _H-\alpha _L) \end{aligned}$$

(A30)

When both Eqs. (A29) and A30 are positive, we have $\underline{\lambda _1}<\lambda <\underline{\lambda _2}$ and ${\underline{\delta }}<\delta <{\overline{\delta }}$, or $\lambda \ge \underline{\lambda _2}$, where

$\underline{\lambda _1}=Root[2 \#1^4+17 \#1^3+41 \#1^2+15 \#1-3 \& ,2,0]$,

$\underline{\lambda _2}=Root[400 \#1^9-1116 \#1^8+3394 \#1^7+3370 \#1^6+1627 \#1^5+16501 \#1^4+17004 \#1^3+2536 \#1^2-1833 \#1-411 \& ,5,0]$,

${\underline{\delta }}=\frac{1300 \lambda ^6-3044 \lambda ^5-411 \lambda ^4+3712 \lambda ^3+442 \lambda ^2-972 \lambda -195}{2 \left( 400 \lambda ^6-1274 \lambda ^5+9 \lambda ^4+1696 \lambda ^3+466 \lambda ^2-102 \lambda +21\right) }$ $+\sqrt{2} \sqrt{M}$,

${\overline{\delta }}=\frac{-500 \lambda ^6+496 \lambda ^5+429 \lambda ^4-320 \lambda ^3+490 \lambda ^2+768 \lambda +237}{2 \left( 400 \lambda ^6-1274 \lambda ^5+9 \lambda ^4+1696 \lambda ^3+466 \lambda ^2-102 \lambda +21\right) }$ $-\sqrt{2} \sqrt{N}$,

$M=-(10000 \lambda ^{14}-68100 \lambda ^{13}+119200 \lambda ^{12}-317365 \lambda ^{11}+384190 \lambda ^{10}+1254926 \lambda ^9-1192070$ $\lambda ^8-2298964 \lambda ^7+382078 \lambda ^6+1820982 \lambda ^5+822718 \lambda ^4-52071 \lambda ^3-146532 \lambda ^2-49104$ $\lambda -6336)/((\lambda +1) \left( 400 \lambda ^6-1274 \lambda ^5+9 \lambda ^4+1696 \lambda ^3+466 \lambda ^2-102 \lambda +21\right) ^2)$

and

$N=-(10000 \lambda ^{14}-68100 \lambda ^{13}+119200 \lambda ^{12}-317365 \lambda ^{11}+384190 \lambda ^{10}+1254926 \lambda ^9-1192070$ $\lambda ^8-2298964 \lambda ^7+382078 \lambda ^6+1820982 \lambda ^5+822718 \lambda ^4-52071 \lambda ^3-146532 \lambda ^2-49104 \lambda -6336)/((\lambda +1)$ $\left( 400 \lambda ^6-1274 \lambda ^5+9 \lambda ^4+1696 \lambda ^3+466 \lambda ^2-102 \lambda +21\right) ^2)$. $\square$

Proof of Lemma 1

The object functions of H and L firms are shown in Table 3. H firm’s first-order condition leads to

(A31)

For L firm, we have

(A32)

Since $c'({\underline{\alpha }})\ge \frac{1}{12}$, we have $\frac{\partial \pi _{L}^{*}}{\partial \alpha _L}<0$ for $\alpha _L\in [{\underline{\alpha }},+\infty )$ when $\delta \in \Delta$. Therefore, in equilibrium H firm selects $\alpha _H^{*}$ according to Eq. (A31) and L firm chooses ${\underline{\alpha }}$. $\square$

Proof of Proposition 4

Compare the green level with and without consumer recognition. We find $\alpha _L^{*}=\alpha _L^{N}={\underline{\alpha }}$, $\alpha _H^{*}$ is characterized by Eq. (A31) and $\alpha _H^{N}=\frac{k {\underline{\alpha }}+2}{3 k}$ (that is, $c'(\alpha _H^{N})=\frac{2+\delta }{2}$). It can obtain that $c'(\alpha _H^{*})>c'(\alpha _H^{N})$ when $\delta \in \Delta$. Furthermore, $\frac{\partial c'(\alpha _H^{*})}{\partial \lambda }>0$ when $\delta \in \Delta$. Thus, fairness concerns can increase the greening improvement level differentiation. $\square$

Proof of Proposition 5

Let CS and $CS^N$ denote the consumer surplus with and without consumer cognition respectively.

$$\begin{aligned} CS&=\int _{\theta _H}^1 (v+\theta \alpha _H-p_{Ho}-\lambda (p_{Ho}-p_{Hn})+v+\theta \alpha _{H}-p_{H1}) \, d\theta \nonumber \\&\quad + \int _{\theta _L}^{\theta _1}(v+\theta \alpha _H-p_{Hn}+v+\theta \alpha _{L}-p_{L1}) \, d\theta \nonumber \\&\quad +\int _{0}^{\theta _L} (v+\theta \alpha _L-p_{Lo}- \lambda (p_{Lo}-p_{Ln})+v+\theta \alpha _{L}-p_{L1}) \, d\theta \nonumber \\&\quad +\int _{\theta _1}^{\theta _H}(v+\theta \alpha _L-p_{Ln}+v+\theta \alpha _{H}- p_{H1}) \, d\theta \end{aligned}$$

(A33)

$$\begin{aligned} CS^N&=2 \int _{\theta _1^N}^1 (v+\theta \alpha _H-p_H^N) \, d\theta +2 \int _0^{\theta _1^N} (v+\theta \alpha _L-p_L^N) \, d\theta \end{aligned}$$

(A34)

$\square$

We compare CS and $CS^N$, and have the following equation.

(A35)

which is weakly positive when $\lambda <\lambda _{cs}\approx 0.08$ and $\max \{0,\delta _1,\delta _2\}\le \delta <\min \{\frac{10 \lambda ^3-17 \lambda ^2+17 \lambda +14}{6 \lambda ^2+30 \lambda +12},\delta _2\}$, where $\delta _1=\sqrt{\frac{50 \lambda ^9+75 \lambda ^8+20 \lambda ^7+1175 \lambda ^6-2878 \lambda ^5-2623 \lambda ^4+2976 \lambda ^3+3005 \lambda ^2+552 \lambda -48}{182 \lambda ^7-419 \lambda ^6-466 \lambda ^5+857 \lambda ^4+658 \lambda ^3-169 \lambda ^2-54 \lambda +51}}+\frac{1}{2}$ and

$\delta _2=\frac{1}{2}-\sqrt{\frac{50 \lambda ^9+75 \lambda ^8+20 \lambda ^7+1175 \lambda ^6-2878 \lambda ^5-2623 \lambda ^4+2976 \lambda ^3+3005 \lambda ^2+552 \lambda -48}{182 \lambda ^7-419 \lambda ^6-466 \lambda ^5+857 \lambda ^4+658 \lambda ^3-169 \lambda ^2-54 \lambda +51}}$.

Proof of Proposition 6

Let SW and $SW^N$ denote the social welfare with and without consumer cognition respectively. Since $SW=\pi _H+\pi _L+CS$, we then have

(A36)

which is negative when $\delta \in \Delta$. Therefore, SW is lower than $SW^N$. $\square$