How Corporate–NGO Partnerships Affect Eco-Label Adoption and Diffusion

Abstract

In a proposed multistage game, a firm and an environmental nongovernmental organization (NGO) initially partner to develop an eco-label, to attain both the environmental benefits and profits. The partnership enables a compromise, and the partner firm, as well as other competitors in the market, subsequently decide whether to adopt the eco-label. Finally, the firms compete on price or quantity in a homogeneous or vertically differentiated market. By analyzing firms’ incentives to adopt the eco-label, this study predicts outcomes related to the eco-labeling strategies of the NGO and the corporate partner, the diffusion of the eco-label, environmental quality, and welfare, depending on both the nature of the competition (Bertrand or Cournot) and the degree of consumer heterogeneity. The results of the game show that it is always in the interest of the environmental NGO and the firm to form a partnership. The corporate partner’s incentive to adopt the eco-label is stronger in Bertrand than in Cournot competition, but diffusion occurs only with Cournot competition. Paradoxically, the NGO may prefer a less stringent eco-label, to promote its diffusion and the consumption of eco-labeled products. Whereas in Cournot competition, eco-labeling improves all components of welfare, in Bertrand competition, it degrades consumer surplus, even if it improves overall profits and environmental quality.

Appendices

Appendices

All the proofs were performed using Mathematica software. The Mathemica file is available on request from the author.

1.1 Appendix 1: Proof of Proposition 1

To demonstrate Proposition 1, we analyze the profits $\pi _L^{B1}(q_g)$ and $\pi _F^{B1}(q_g)$.

1.1.1 The leader’s profit

The derivative of $\pi _L^{B1}(q_g)$ is defined as follows:

$$\begin{aligned} \dfrac{\partial \pi _L^{B1}(q_g)}{\partial q_g}=\dfrac{-24q_g^3+2(8{\overline{\theta }}+11q_b)q_g^2-(12{\overline{\theta }}+5q_b)q_bq_g+8{\overline{\theta }}q_b^2-2q_b^3}{2(4q_g-q_b)^2}x_g^{B1} \end{aligned}$$

(14)

The sign of the derivative is equal to the sign of the polynomial of degree 3, denoted $P_L^B(q_g)$, in the numerator of Eq. 14. To study the sign of $P_L^B(q_g)$, we first investigate whether the polynomial has one or three real roots by studying the sign of its discriminant $\Delta _L^B$, characterized by:

$$\begin{aligned} \Delta _L^B=4(8{\overline{\theta }}-3q_b)q_b^2(5997 q_b^3 - 18592 {\overline{\theta }} q_b^2+ 2704 {\overline{\theta }}^2 qb - 2944 {\overline{\theta }}^3). \end{aligned}$$

Because $\Delta _L^B<0$,²⁹$P_L^B(q_g)$ has only one real root.

We then investigate whether the real root is in the interval of admissible values for $q_g$, that is $]q_b,2{\overline{\theta }}]$, by studying the function $P_L^B(q_g)$. The derivative of $P_L^B(q_g)$ is defined as follows:

$$\begin{aligned} {P_L^B}'(q_g)= -72 q_g^2 + (44 q_b + 32 {\overline{\theta }})q_g- 12 {\overline{\theta }} q_b-5 qb^2 \end{aligned}$$

The discriminant of ${P_L^B}'(q_g)$ is ${\Delta _L^B}'=16(64 {\overline{\theta }}^2-40{\overline{\theta }}q_b+31q_b^2)>0$. ${P_L^B}'(q_g)$ has therefore two real roots, denoted $r_1$ and $r_2$ (with $r_1<r_2$) defined by $(8{\overline{\theta }}+11q_b\pm \sqrt{\Delta })/36$, such that $r_1<q_b$ and $r_2<2{\overline{\theta }}$. $r_2 \ge q_b$ only if $q_b \le 20{\overline{\theta }}/33$. So $P_L^B(q_g)$ is a decreasing function for $q_g>r_2$. Moreover, $P_L^B(q_b)=3(4{\overline{\theta }}-3q_b)q_b^2>0$ and $P_L^B(2{\overline{\theta }})=-2(64{\overline{\theta }}^3-32{\overline{\theta }}q_b+{\overline{\theta }}q_b^2+q_b^3)<0$. Therefore, the real root of $P_L^B(q_g)$ is in the interval $[r_2,2{\overline{\theta }}]$. We deduce that $\pi _L^{B1}(q_g)$ is a bell-shaped function of $q_g$ that reaches its maximum at $q_g^{BL^*} \in ]q_b,2{\overline{\theta }}[$, such that $\partial \pi _L^{B1}/\partial q_g=0$.

1.1.2 The follower’s profit

The derivative of $\pi _F^{B1}(q_g)$ is defined as follows:

$$\begin{aligned} \dfrac{\partial \pi _F^{B1}(q_g)}{\partial q_g}=\dfrac{8q_g^3-8q_bq_g^2+(4{\overline{\theta }}+q_b)q_bq_g+2{\overline{\theta }}q_b^2-q_b^3}{2(4q_g-q_b)^2q_g}q_b x_b^{B1} \end{aligned}$$

(15)

The sign of the derivative is equal to the sign of the polynomial of degree 3, denoted $P_F^B(q_g)$, in the numerator of Eq. 15. We have:

$$\begin{aligned} {P_F^B}'(q_g)=24q_g^2-16q_bq_g+4{\overline{\theta }}q_b+q_b^2 \end{aligned}$$

of which the discriminant is ${\Delta _F^B}'=-32(12{\overline{\theta }}q_b-5q_b^2)<0$. Therefore, ${P_F^B}'(q_g)>0$ and $P_F^B(q_g)$ is an increasing function of $q_g$. Moreover, $P_F^B(q_b)=6{\overline{\theta }}q_b^2>0$ and $P_F^B(2{\overline{\theta }})=64{\overline{\theta }}^3-24{\overline{\theta }}^2q_b+4{\overline{\theta }}q_b^2-q_b^3>0$. As a result, $P_F^B(q_g)>0$ and $\pi _F^{B1}(q_g)$ is an increasing function of $q_g$.

1.2 Appendix 2: Proof of Lemma 1

The derivative of $\pi _F^{C1}$ is defined as follows:

$$\begin{aligned} \frac{\partial \pi _F^{C1}}{\partial q_g}=\dfrac{2(2q_g^2-q_bq_g-{\overline{\theta }}q_b+q_b^2)q_b}{(4q_g-q_b)^2}x_b^{C1} \end{aligned}$$

(16)

The quadratic function, denoted $P_F^C(q_g)$, in the numerator of Eq. 16 has a discriminant $\Delta _F^C = (8{\overline{\theta }}-7 q_b)qb$. With $q_b<{\overline{\theta }}$, $\Delta _F^C>0$ and $P_F^C(q_g)$ has two real positive roots of which only one, equal to $(q_b+\sqrt{8{\overline{\theta }}-7q_b})/4$, is higher than $q_b$. $P_F^C(q_g)$ is a decreasing function for $q_g \le (q_b+\sqrt{8{\overline{\theta }}-7q_b})/4$ and increasing otherwise. Therefore, $\pi _F^{C1}$ is a U-shaped function of $q_g$ in the interval $[q_b,2{\overline{\theta }}]$.

The derivative of $\pi _F^{C2}$ is equal to $(2{\overline{\theta }}-3q_g)/6$. $\pi _F^{C2}$ is then a bell-shaped function of $q_g$ which reaches its maximum at $q_g^{C2^*} \equiv 2{\overline{\theta }}/3$.

Knowing that the leader adopts the eco-label, the follower has an incentive to adopt it too if $I_F^C(q_g)=\pi _F^{C2}(q_g)-\pi _F^{C1}(q_g) > 0$. $I_F^C(q_g)$ is defined as follows:

$$\begin{aligned} I_F^{C}(q_g)=\dfrac{(q_g-q_b)q_g(16 q_g^3-(64 {\overline{\theta }}+q_b)q_g^2+4(16{\overline{\theta }}^2-17{\overline{\theta }}q_b+9q_b^2) q_g-4{\overline{\theta }}^2q_b)}{36{\overline{\theta }}(4q_g-q_b)^2} \end{aligned}$$

(17)

$I_F^C(q_g)$ is a bell-shaped function of $q_g$ when $q_b<10{\overline{\theta }}/17$, because $\pi _F^{C1}(q_g)$ becomes a U-shaped function of $q_g$, $\pi _F^{C2}(q_g)$ is a bell-shaped function of $q_g$, $I^C_F(q_b)=0$ and

$$\begin{aligned} \left. \dfrac{\partial I_F^C}{\partial q_g}\right| _{q_g=q_b}=\dfrac{(2{\overline{\theta }}-q_b)(10{\overline{\theta }}-17q_b)}{108{\overline{\theta }}}>0 \text { if } q_b<10{\overline{\theta }}/17 \end{aligned}$$

Moreover, $I_F^C(q_g)$ is positive for $q_g \in ]q_b,{\overline{q}}_g^{CF}]$, where ${\overline{q}}_g^{CF}$ is such that $I_F^C({\overline{q}}_g^{CF})=0$. Threshold ${\overline{q}}_g^{CF}$ is the only real root (in the real numbers and greater than $q_b$) of the following polynomial of degree 3:

$$\begin{aligned} 16 q_g^3-(64 {\overline{\theta }}+q_b)q_g^2+4(16{\overline{\theta }}^2-17{\overline{\theta }}q_b+9q_b^2) q_g-4{\overline{\theta }}^2q_b=0 \end{aligned}$$

(18)

The discriminant of the polynomial (18), denoted $P^{CF}(q_g)$, is defined by:

$$\begin{aligned} \Delta ^{CF}=-144q_b(8{\overline{\theta }}-7q_b)^2(2{\overline{\theta }}-q_b)(423 q_b^2 - 608 {\overline{\theta }} q_b+ 1024 {\overline{\theta }}^2). \end{aligned}$$

with $\Delta ^{CF}>0$. The polynomial has three real roots. Its derivative is:

$$\begin{aligned} P^{CF'}(q_g)=48 q_g^2 - 2(64 {\overline{\theta }} +q_b)q_g+4(9q_b^2-17q_b{\overline{\theta }}+16{\overline{\theta }}^2) \end{aligned}$$

The discriminant of $P^{CF'}(q_g)$ is:

$$\begin{aligned} \Delta ^{CF'}=-4 (1727 q_b^2 - 3392 q_b {\overline{\theta }} - 1024 {\overline{\theta }}^2)>0. \end{aligned}$$

$P^{CF'}(q_g)$ has therefore two real roots, $r_1<r_2$, such that $r_i=(64{\overline{\theta }}+q_b\pm \sqrt{\Delta ^{CF}}$)/48, $r_1<{\overline{\theta }}$, $r_2>2{\overline{\theta }}$, and $r_1>q_b$ if $q_b<16{\overline{\theta }}/41$. Moreover, $P^{CF}(q_b)=3(2{\overline{\theta }}-q_b)(10{\overline{\theta }}-17q_b)q_b>0$ for $q_b<10{\overline{\theta }}/17$ (required for $I_F^C(q_g)>0$). Therefore, $P^{CF}(q_g)$ is positive and increases for $q_g \in [q_b,r_1]$. It decreases for $q_g \in [r_1,2{\overline{\theta }}]$. Moreover, $P^{CF}({\overline{\theta }})=(16{\overline{\theta }}-9q_b)({\overline{\theta }}-4q_b){\overline{\theta }}>0$ if $q_b<{\overline{\theta }}/4$ and $P^{CF}(2{\overline{\theta }})=-75(2{\overline{\theta }}-q_b){\overline{\theta }}<0$. Therefore, $P^{CF}(q_g)$ has a unique real root: ${\overline{q}}_g^{CF} \in [{\overline{\theta }},2{\overline{\theta }}]$ if $q_b<{\overline{\theta }}/4$ or ${\overline{q}}_g^{CF} \in [\max \{r_1,qb\},{\overline{\theta }}]$ if $q_b \in [{\overline{\theta }}/4,10{\overline{\theta }}/17]$.

Therefore, the follower’s best response to adoption by the leader is to adopt only if $q_g<{\overline{q}}_g^{CF}$ and $q_b<10{\overline{\theta }}/17$.

1.3 Appendix 3: Proof of Proposition 2

The derivative of $\pi _L^{C1}$ is defined as follows:

$$\begin{aligned} \frac{\partial \pi _L^{C1}}{\partial q_g}=\dfrac{ -24q_g^3+2(8{\overline{\theta }}+5q_b)q_g^2-4({\overline{\theta }}+q_b)q_b q_g+2{\overline{\theta }}q_b^2-q_b^3}{2(4q_g-q_b)^2}x_g^{C1} \end{aligned}$$

(19)

The sign of the derivative is equal to the sign of the polynomial of degree 3, denoted $P_L^C(q_g)$, in the numerator of Eq. 19. The discriminant of $P(q_g)$ is characterized by:

$$\begin{aligned} \Delta _L^C=32(8{\overline{\theta }}-7q_b)q_b^2(149 q_b^3 - 66 {\overline{\theta }} q_b^2 -30 {\overline{\theta }}^2 qb - 112 {\overline{\theta }}^3). \end{aligned}$$

$\Delta _L^C<0$ when $q_b<{\overline{\theta }}$ (by assumption). So $P_L^C(q_g)$ has only one real root.

The derivative of $P_L^C(q_g)$ is defined as follows:

$$\begin{aligned} {P_L^C}'(q_g)=-72q_g^2+4(8{\overline{\theta }}+5q_b)q_g-4{\overline{\theta }}q_b-4q_b^2 \end{aligned}$$

The discriminant of ${P_L^C}'(q_g)$ is:

$$\begin{aligned} \Delta =16(64{\overline{\theta }}^2+8{\overline{\theta }}q_b-47q_b^2)>0. \end{aligned}$$

${P_L^C}'(q_g)$ has two real positive roots (denoted $r_1$ and $r_2$), defined by $(8{\overline{\theta }}+5q_b \pm \sqrt{64{\overline{\theta }}^2+8{\overline{\theta }}q_b-47q_b^2})/36$, such that $r_1<q_b$ and $r_1<r_2<2{\overline{\theta }}$. $r_2>q_b$ only if $q_b<{\overline{\theta }}/2$. Therefore ${P_L^C}'(q_g)>0$ if $q_b<q_g<r_2$ and ${P_L^C}'(q_g) \le 0$ otherwise. Moreover, $P_L^C(q_b)=(14{\overline{\theta }}-19q_b)q_b^2>0$ if $q_b<14{\overline{\theta }}/19$ and $P_L^C(2{\overline{\theta }})=-128{\overline{\theta }}^3+32{\overline{\theta }}^2q_b-6{\overline{\theta }}q_b^2-q_b^3<0$. Therefore, the real root of $P(q_g)$ is in the interval $[r_2,2{\overline{\theta }}]$ if $q_b\le {\overline{\theta }}/2$ and in $[q_b,2{\overline{\theta }}]$ if ${\overline{\theta }}/2<q_b<14{\overline{\theta }}/19$. We deduce that $\pi _L^{C1}$ is a bell-shaped function of $q_g$ that reaches its maximum at $q_g^{CL^*} \in ]q_b,2{\overline{\theta }}[$.

If $q_b \ge 14{\overline{\theta }}/19$, the real root of $P_L^C(q_g)$ is lower than $q_b$, $\pi _L^{C1}$ is a decreasing function of $q_g$ in the interval $[q_b,2{\overline{\theta }}]$ and $P_L^C(q_b)\le 0$. Therefore, the leader has no interest in adopting the eco-label; it still produces the brown product of quality $q_b$.

The necessary condition for the leader to adopt the eco-label is therefore $q_b<14{\overline{\theta }}/19$.

The leader has an incentive to adopt the eco-label if $I_L^C(q_g)=\pi _L^{C1}(q_g)-\pi _L^{C0}(q_g) > 0$. $I_L^C(q_g)$ is defined as follows:

$$\begin{aligned} I_L^{C}(q_g)=\dfrac{(q_g-q_b)(36 q_g^4-36(4 {\overline{\theta }}- q_b)q_g^3 + 72(2 {\overline{\theta }}- q_b){\overline{\theta }} q_g^2-16(2{\overline{\theta }}-q_b)^2q_b q_g+(2{\overline{\theta }}-q_b)^2q_b^2)}{36{\overline{\theta }}(4q_g-q_b)^2} \end{aligned}$$

(20)

$I_L^C$ is a bell-shaped function of $q_g$ when $q_b<14{\overline{\theta }}/19$, because $\pi _L^{C1}(q_g)$ is then a bell-shaped function of $q_g$, $\pi _L^{C0}$ is independent of $q_g$, $I^C_L(q_b)=0$ and

$$\begin{aligned} \left. \dfrac{\partial I_L^C}{\partial q_g}\right| _{q_g=q_b}=\dfrac{(2{\overline{\theta }}-q_b)(14{\overline{\theta }}-19q_b)}{108{\overline{\theta }}}>0 \text { if } q_b<14{\overline{\theta }}/19 \end{aligned}$$

Furthermore, $I_L^C(q_g)$ is positive for $q_g \in ]q_b,{\overline{q}}_g^{CL}]$, where ${\overline{q}}_g^{CL}$ is such that $I_L^C({\overline{q}}_g^{CL})=0$. Threshold ${\overline{q}}_g^{CL}$ is the only real root (in the real numbers and greater than $q_b$) of the following polynomial of degree 4:

$$\begin{aligned} 36 q_g^4-36(4 {\overline{\theta }}- q_b)q_g^3 + 72(2 {\overline{\theta }}- q_b){\overline{\theta }} q_g^2-16(2{\overline{\theta }}-q_b)^2q_b q_g+(2{\overline{\theta }}-q_b)^2q_b^2=0 \end{aligned}$$

(21)

The discriminant of this polynomial is positive; the polynomial has four distinct roots which are either all real or form two pairs of conjugate complex numbers. The analytical proof of the uniqueness of ${\overline{q}}_g^{CL}$ in $[q_b,2{\overline{\theta }}]$ is therefore tedious. Using a numerical demonstration, with $q_b$ from 0 to ${\overline{\theta }}$, we can show that there is always a unique real root, ${\overline{q}}_g^{CL} \in [q_b,2{\overline{\theta }}]$.

Moreover, $I_L^C(q_g)>I_F^C(q_g)$ for all $q_g \in [q_b,2{\overline{\theta }}]$, because $I^C_F(q_b)=0$, $I^C_L(q_b)=0$ and $I^C_L(q_g)$ grows faster than $I_F^C(q_g)$ when $q_g$ is close to $q_b$ if $q_b<10{\overline{\theta }}/17<14{\overline{\theta }}/19$. In addition, $I_L^C(q_g)-I_F^C(q_g)=(q_g-q_b)P^{I}(q_g)/(36{\overline{\theta }}(4q_g-q_b)^2)$ has the same sign as the polynomial function $P^I(q_g)$, defined as follows:

$$\begin{aligned} & P^I(q_g)=20 q_g^4-(80 {\overline{\theta }}- 37q_b)q_g^3 + 4(20 {\overline{\theta }}^2-{\overline{\theta }}q_b- 9q_b^2) q_g^2\\ & -4(5{\overline{\theta }}-2q_b)(3{\overline{\theta }}-2q_b)q_b q_g+(2{\overline{\theta }}-q_b)^2q_b^2 \end{aligned}$$

To show that $P^I(q_g)>0$, we analyze the discriminant of the second derivative of ${P^I}(q_g)$, defined by ${\Delta ^I}^{''}=12(6400{\overline{\theta }}^2-17120{\overline{\theta }}q_b+9867q_b^2)$:

• If $q_b>0.545{\overline{\theta }}$, ${\Delta ^I}^{''}<0$ and ${P^I}^{''}(q_g)>0$ because the coefficient of $q_g^2$ is equal to $20\times 4 \times 3=240>0$. Therefore, ${P^I}^{'}(q_g)$ and ${P^I}(q_b)$ are positive (since ${P^I}^{'}(q_b)>0$ and ${P^I}(q_b)>0$) and increasing in the interval $[q_b,2{\overline{\theta }}]$.

• If $q_b<0.545{\overline{\theta }}$, ${\Delta ^I}^{''}>0$ and ${P^I}^{''}(q_g)$ has two real roots defined by $r_i\equiv {\overline{\theta }}-\frac{37q_b}{80} \pm \frac{\sqrt{{\Delta ^I}^{''}}}{80\sqrt{3}}$, with $q_b<r_1<r_2<2{\overline{\theta }}$. ${P^I}^{''}(q_g)$ is positive outside the roots and negative between the roots. In addition, ${P^I}^{'}(q_b)>0$ for all $q_b$ and ${P^I}^{'}(r_2)>0$ for $q_b \in [0.198{\overline{\theta }},0.545{\overline{\theta }}]$. In this case, ${P^I}(q_g)$ is increasing and positive on the interval $[q_b,2{\overline{\theta }}]$.

• If $q_b<0.198{\overline{\theta }}$, we can show that the discriminant of ${P^I}^{'}(q_g)$ is positive. ${P^I}^{'}(q_g)$ has then three real roots, including two roots, denoted $r_3\in ]r_1,r_2]$ and $r_4 \in ]r_2,2{\overline{\theta }}]$. ${P^I}^{'}(r_2)<0$ is the minimum of ${P^I}^{'}(q_g)$. We can show that $\partial r_2/\partial q_b>0$ and $\partial {P^I}^{'}(r_2)/\partial q_b>0$: A higher $q_b$ reduces the slope of ${P^I}(q_g)$ in its decreasing part, and thus increases ${P^I}(q_g)$ for $q_g \ge r_2$. In addition $P^{I}(q_g)=20(2{\overline{\theta }}-q_g)^2q_g^2>0$ for $q_b=0$ and ${P^I}(r_2)>0$ $\forall q_b, q_g$. Therefore ${P^I}(q_g)>0$ for $q_g \in [r_2,2{\overline{\theta }}]$ and thus for all $q_g$.

As a result, $I_L^C(q_g)>I_F^C(q_g)$ for all $q_g \in [q_b,2{\overline{\theta }}]$. Moreover, denoting ${\overline{q}}_g^{CL}$ and ${\overline{q}}_g^{CF}$ the environmental qualities such that, $I_L^C({\overline{q}}_g^{CL})=0$ and $I_F^C({\overline{q}}_g^{CF})=0$, we have ${\overline{q}}_g^{CF}<{\overline{q}}_g^{CL}$.

1.4 Appendix 4: Proof of Proposition 3

To prove that $I_L^B(q_g)>I_L^C(q_g)$, we note that $I_L^B(q_b)=0$ and $I_L ^C(q_b)=0$. Moreover, we can express the incentive difference as follows:

$$\begin{aligned} I_L^B(q_g)-I_L^C(q_g)=\dfrac{q_b(q_g-q_b) \Big [4(16q_g-q_b){\overline{\theta }}^2-4(16q_g-q_b)q_b{\overline{\theta }}+(9q_g^2+16q_bq_g-q_b^2)q_b\Big ]}{36{\overline{\theta }}(4q_g-q_b)^2} \end{aligned}$$

The discriminant of the polynomial into brackets, equal to $\Delta =-144(16q_g-q_b)q_bq_g^2$, is negative, so the polynomial takes the sign of $4(16q_g-q_b)$, which is positive. Therefore, $I_L^B(q_g)(q_g)>I_L^C(q_g)$.

1.5 Appendix 5: The Nash bargaining model and the leader’s take-it-or-leave-it eco-label

1.5.1 The Nash bargaining model

Negociations between the NGO and the firm can be described as a Nash bargaining problem. The bargaining over the eco-label stringency consists of the following maximization program:

$$\begin{aligned} \max \limits _{q_g}\left\{ (E^{st}-E^{s0})^\alpha (\pi _L^{st}-\pi _L^{s0}) ^{(1-\alpha )} \text { st } E^{st}>E^{s0} \ \text {and} \ \pi _L^{st}>\pi _L^{s0}\right\} \end{aligned}$$

where $E^{s0}$ and $\pi _L^{s0}$ are the status quo or disagreement outcomes for the NGO and the leader respectively. $E^{st}$ represents environmental quality in Stages t ($t=0,1,2$) when firms use strategies s ($s=C,B$). $\alpha$ measures the NGO’s bargaining power and $1-\alpha$ measures the firm’s bargaining power, such that $\alpha \in [0,1]$. In the case of Bertrand competition, the negotiations between the partners have a short-term horizon ($t=1$) because the follower never adopts the eco-label; in the case of Cournot competition, which allows the diffusion of the eco-label, the horizon can be $t=2$ if the leader does not impose an exclusivity clause on the eco-label in the negotiations, or $t=1$ otherwise.

The first-order condition for the maximization program can be expressed as follows:

$$\begin{aligned} \dfrac{\alpha }{E^{st}-E^{s0}}\dfrac{\partial E^{st}}{\partial q_g}+\dfrac{1-\alpha }{\pi ^{st}-\pi ^{s0}}\dfrac{\partial \pi ^{st}}{\partial q_g}=0 \end{aligned}$$

Therefore, the environmental quality resulting from eco-label bargaining, denoted $q_g^{s^*}$, is intermediate between green quality $q_g^{sL^*}$ that maximizes the profit of the leader (when $\alpha =0$) and green quality $q_g^{sN^*}$ that maximizes the quality of the environment (when $\alpha =1$).

1.5.2 The leader’s take-it-or-leave-it eco-label

When the leader offers a take-it-or-leave-it eco-label, its objective is to maximize its profit $\pi _L^{s1}(q_g)$, while ensuring that environmental quality improves. Its program (where $s=B,C$) therefore is:

$$\begin{aligned} \max \limits _{q_g} \pi _L^{s1}(q_g) \text { st } E^{s1}(q_g) \ge E^{s0} \text { and } \pi _L^{s1}(q_g) \ge \pi _L^{s0} \end{aligned}$$

In the case of Bertrand competition, there exists a unique eco-label $q_g^{BL^*}\in [q_b,2{\overline{\theta }}]$ that maximizes the leader’s profit. The NGO’s participation constraint, $q_g^{BL^*} \le {\overline{q}}_g^{BN}$, is fulfilled if $q_b \le 2{\overline{\theta }}/3$. When $q_b$ is higher, the quality of the environment cannot be improved (compared with the initial situation), so no negotiation takes place between the firm and the NGO, and no eco-label emerges. The leader’s eco-label is less stringent than NGO’s eco-label if the market is large ($q_b \le 0.494 {\overline{\theta }}$) and more stringent otherwise. In a large market the leader prefers a less demanding eco-label, which leads to a lower price and greater consumption of the green product and, through these effects, to higher profits, without compromising environmental quality. The closer $q_b$ is to ${\overline{\theta }}$, the more demanding the leader’s eco-label. Thus, the leader tries to keep product differentiation substantial enough to mitigate price competition and ensure its profits are as high as possible.

In the case of Cournot competition, the leader earns a higher profit if the follower chooses not to adopt the eco-label, so the leader implements either a deterrence strategy, by setting $q_g$ just above ${\overline{q}}_g^{CF}$ when the market is sufficiently large ($q_b \le 0.444{\overline{\theta }}$), or a standard profit maximization strategy, by setting $q_g$ to $q_g^{CL^*}>{\overline{q}}_g^{CF}$ in the case of a narrower market ($q_b\in [0.444{\overline{\theta }},{\overline{\theta }}]$). As a result, the leader’s eco-label is more stringent with Cournot than with Bertrand competition when the market is large ($q_b \le 0.378{\overline{\theta }}$) and less stringent otherwise. Moreover, the leader’s eco-label is more stringent than NGO’s eco-label if the market is large ($q_b \le {\overline{\theta }}/4$) and less stringent otherwise ($0.419{\overline{\theta }},{\overline{\theta }} \big ]$). In a medium market, where $q_b \in \big [ {\overline{\theta }}/4,0.419{\overline{\theta }} \big ]$, the NGO and the leader set environmental qualities very close to ${\overline{q}}_g^{CF}$, such that $q_g^{CN^*}<{\overline{q}}_g^{CF}< q_g^{CL^*}$. The interests of the NGO and the leader are opposed, as long as the market is not so narrow as to preclude any diffusion of the eco-label. Whereas the NGO wants to promote the diffusion of the eco-label, the leader prefers to prevent it. Therefore, in a large market, the leader sets a very demanding eco-label, which discourages adoption by the follower, while keeping its own profit higher than it would be in the case of diffusion. Conversely, the NGO sets a less demanding eco-label that leads to the highest possible quality of the environment through the diffusion of the eco-label. In a medium-sized market, paradoxically, the NGO and leader propose similar eco-labels, yet their conflicting interests in terms of diffusion prevent them from agreeing on a single eco-label ${\overline{q}}_g^{CF}$. Therefore, if Cournot competition exists, unless the market is narrow, negotiations between the partners for eco-labeling are particularly challenging.

1.6 Appendix 6: Proof of Proposition 4

1.6.1 Bertrand Competition

The first-order condition (FOC) for environmental quality maximization is characterized as follows:

$$\begin{aligned} \frac{\partial E^{B1}}{\partial q_g}=- \frac{ 16q_g^3-2(8{\overline{\theta }}+3q_b)q_g^2+8{\overline{\theta }}q_bq_g+2{\overline{\theta }}q_b^2-q_b^3}{2{\overline{\theta }}(4q_g-q_b)^2}=0 \end{aligned}$$

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The numerator of the FOC implicitly defines the preferred eco-label of the NGO, $q_g^{BN^*}$, as the real solution of the polynomial of degree 3 in $q_g$.

The discriminant of the polynomial, denoted $P_N^B(q_g)$, is defined by:

$$\begin{aligned} \Delta _N^B=96q_b^2(8{\overline{\theta }}-3q_b)^2(2{\overline{\theta }}-3q_b)(4{\overline{\theta }}+3q_b) \end{aligned}$$

Under the condition that $q_b<2{\overline{\theta }}/3$ (required for participation of both partners), $\Delta _N^B>0$ and the polynomial has three distinct real roots.

The derivative of $P_N^B(q_g)$ is:

$$\begin{aligned} P{_N^B}'(q_g)=4(4q_g-qb)(2{\overline{\theta }}-3q_g) \end{aligned}$$

such that $P{_N^B}'(q_g)>0$ for $q_b<q_g<2{\overline{\theta }}/3$ and $P{_N^B}'(q_g)\le 0$ for $2{\overline{\theta }}/3<q_g<2{\overline{\theta }}$. Because $P_N^B(q_b)=3q_b^2(2{\overline{\theta }}-3q_b)>0$ and $P_N^B(2{\overline{\theta }})=-(4{\overline{\theta }}-q_b)(16{\overline{\theta }}^2+2q_b{\overline{\theta }}=q_b^2)<0$, there is only one real root in the interval $[2{\overline{\theta }}/3,2{\overline{\theta }}]$.

As a result, $q_g^{BN^*}$ is in the interval $[\frac{2{\overline{\theta }}}{3},2{\overline{\theta }}]$. We cannot give an analytical definition of $q_g^{BN^*}$, but we have proved the existence of $q_g^{BN^*}$ when $q_b<2{\overline{\theta }}/3$.

1.6.2 Cournot Competition

Assuming $q_b<{\overline{\theta }}$, we have:

$$\begin{aligned} E^{C1}-E^{C0}=\dfrac{q_g-q_b}{{\overline{\theta }}(4q_g-q_b)} \Big [4{\overline{\theta }}q_g(3q_g-q_b)-(6q_g^2+3q_bq_g-2q_b^2)\Big ] \end{aligned}$$

The term in brackets is positive for $q_g \in \big [q_b,\dfrac{12{\overline{\theta }}-3q_b+\sqrt{3(48{\overline{\theta }}^2-56{\overline{\theta }}cq_b+19{\overline{\theta }}^2}}{12} \Big ]$. We also have:

$$\begin{aligned}&E^{C2}-E^{C1}=\dfrac{q_g-q_b}{6{\overline{\theta }}(4q_g-q_b)} \Big [4{\overline{\theta }}-2q_g-3q_b)\Big ] \end{aligned}$$

The term in brackets is positive for $q_b<4{\overline{\theta }}/5$ and $q_g \in \big [q_b,(4{\overline{\theta }}-3q_b])2 \big ]$, with:

$$\begin{aligned} \dfrac{4{\overline{\theta }}-3q_b}{2}<\dfrac{12{\overline{\theta }}-3q_b+\sqrt{3(48{\overline{\theta }}^2-56{\overline{\theta }}cq_b+19{\overline{\theta }}^2}}{12}. \end{aligned}$$

Therefore, if $q_b$ and $q_g$ fulfill the conditions for $E^{C2}>E^{C1}$, then $E^{C1}>E^{C0}$.

The maximum of $E^{C2}$ is reached at $q_g={\overline{\theta }}$. According to "Appendix 2", ${\overline{q}}_g^{CF} > {\overline{\theta }}$ if $q_b<{\overline{\theta }}/4$ and ${\overline{q}}_g^{CF} \le {\overline{\theta }}$ when ${\overline{\theta }}/4 \le q_b < 10{\overline{\theta }}/17$.

The FOC for maximizing $E^{C1}$ is as follows:

$$\begin{aligned} \frac{\partial E^{C1}}{\partial q_g}=-\frac{16q_g^3-2(8{\overline{\theta }}+5q_b)q_g^2+2(4{\overline{\theta }}+q_b)q_bq_g-q_b^3=0}{2{\overline{\theta }}(4q_g-q_b)^2}=0 \end{aligned}$$

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The numerator of the FOC implicitly defines the eco-label, $q_g^{CN^*}$, as the real solution of the polynomial, denoted $P_N^C(q_g)$. The discriminant of $P_N^C(q_g)$ is defined by:

$$\begin{aligned} \Delta _N^C=16q_b^2(8{\overline{\theta }}-7q_b)(47q_b^3-48q_b^2{\overline{\theta }}-48q_b{\overline{\theta }}^2+128{\overline{\theta }}^3) \end{aligned}$$

Because $\Delta _N^C>0$, $P_N^C(q_g)$ has three distinct real roots.

The derivative of $P_N^C(q_g)$ is:

$$\begin{aligned} {P_N^C}'(q_g)=-4(4q_g-qb)(6q_g-q_b-4{\overline{\theta }}) \end{aligned}$$

such that: ${P_N^C}'(q_g)>0$ for $q_b<4{\overline{\theta }}/5$ and $q_g<(q_b+4{\overline{\theta }})/6$; ${P_N^C}'(q_g) \le 0$ otherwise. Because $P_N^C({\overline{\theta }})=(q_b^2-2q_b{\overline{\theta }}+2{\overline{\theta }}^2)>0$ and $P_N^C(2{\overline{\theta }})=q_b^3-4q_b^2{\overline{\theta }}+24q_b{\overline{\theta }}^2-64{\overline{\theta }}^3<0$, there is only one real root in the interval $[{\overline{\theta }},2{\overline{\theta }}]$.

As a result, $q_g^{CN^*}$ is in the interval $[{\overline{\theta }},2{\overline{\theta }}]$. We cannot give an analytical definition of $q_g^{CN^*}$, but we have proved the existence of $q_g^{CN^*}$.

1.6.3 Methodology for numerical simulations

To find the threshold of $q_b$ above which $E^{C1}(q_g^{CN^*})\ge E^{C2}({\overline{q}}_g^{CF})$, even though $q_g^{CN^*}$ and ${\overline{q}}_g^{CF}$ have no analytical expressions, we turn to numerical simulations:

• we set ${\overline{\theta }} \ge 1$;

• we compute $q_g^{CN^*}$ and ${\overline{q}}_g^{CF}$ for all admissible values of $q_b=\gamma {\overline{\theta }}$, with $\gamma \in [0,1]$, by using the FindRoot function of the Mathematica software, which allows us to numerically find local roots of the conditions $\partial E^{C1}/\partial q_g=0$ and $\pi _F^{C2}-\pi _F^{C1}=0$, starting from a value $q_g={\overline{\theta }}$;

• we compute the corresponding values for $E^{C1}(q_g^{CN^*})$ and $E^{C2}({\overline{q}}_g^{CF})$;

• we deduce that, regardless the value set for ${\overline{\theta }}$, we have $E^{C1}(q_g^{CN^*}) \ge E^{C2}({\overline{q}}_g^{CF})$ for $\gamma \ge 0.41899$.

We thus prove that $E^{C1}(q_g^{CN^*})\ge E^{C2}({\overline{q}}_g^{CF})$ for $q_b \ge 0.41899{\overline{\theta }}$ for all ${\overline{\theta }} \ge 1$.

We follow the same methodology to compare $q_g^{CN^*}$ and $q_g^{BN^*}$ (Proposition 4 and Fig. 2). Figure 2 plots $q_g^{BN^*}$, $q_g^{CN^*}$ and ${\overline{q}}_g^{CF}$ as a function of $q_b$. We have checked that these results are robust regardless of the value of ${\overline{\theta }}$; in Fig. 2, only the scales of $q_g^{BN^*}$ and $q_g^{CN^*}$ change with ${\overline{\theta }}$. We also follow the same methodology to compare $E^{Ct}$ and $E^{Bt}$ (Proposition 5 and Fig. 3), $CS^{Ct}$ and $CS^{Bt}$, and $\Pi ^{Ct}$ and $\Pi ^{Bt}$ (Proposition 6 and Fig. 4).

1.7 Appendix 7: Consumer Surplus

In the case of Bertrand competition, consumer surplus in Periods 0 and 1 are defined by:

$$\begin{aligned} CS_b^{B0}=&\int _{\tilde{\theta }_b}^{{\overline{\theta }}} \dfrac{u_b(\theta )}{{\overline{\theta }}} \,d\theta \ =2{\overline{\theta }}q_b {x_b^{B0}}^2;\\ CS_b^{B1}=&\int _{0}^{{\hat{\theta }}} \dfrac{u_b(\theta )}{{\overline{\theta }}} \,d\theta \ =\dfrac{{\overline{\theta }}q_b}{2} {x_b^{B1}}^2;\\ CS_g^{B1}=&\int _{{\hat{\theta }}}^{{\overline{\theta }}} \dfrac{u_g(\theta )}{{\overline{\theta }}} \,d\theta \ =\dfrac{-2q_g^2+(4{\overline{\theta }}+q_b)q_g+4{\overline{\theta }}q_b-2q_b^2}{4(4q_g-q_b)}q_g x_g^{B1}. \end{aligned}$$

We have:

$$\begin{aligned} \dfrac{\partial CS_b^{B1}}{\partial q_g}=&\dfrac{(4q_g^2-2q_bq_g-2{\overline{\theta }}q_b+q_b^2)q_b}{2(4q_g-q_b)^2};\\ \dfrac{\partial CS_g^{B1}}{\partial q_g}=&\dfrac{q_g}{8{\overline{\theta }}(4q_g-q_b)^3} \Big ( 48q_g^4-4(32{\overline{\theta }}+5q_b)q_g^3+4(16{\overline{\theta }}^2+8{\overline{\theta }}q_b+3q_b^3)q_g^2\\&-3(16{\overline{\theta }}^2-8{\overline{\theta }}q_b+3q_b^2)q_b q_g-4(4{\overline{\theta }}-q_b)(2{\overline{\theta }}-q_b)q_b^2 \Big ). \end{aligned}$$

We deduce that $\partial CS_b^{B1} / \partial q_g > 0$ if $q_g>\Big ( q_b+\sqrt{(8{\overline{\theta }}-3q_b)q_b} \Big )/4$ and $q_b<2{\overline{\theta }}/3$ (such that $x_b^{B1}$ increases) and negative otherwise. Because the polynomial of degree 4 into bracket in the expression of $\partial CS_g^{B1} / \partial q_g$ cannot be explicitly solved, we turn to numerical simulations. Simulations show that $\partial CS_g^{B1} / \partial q_g < 0$ and that $CS_g^{B1}+CS_b^{B1}$ is a decreasing function of $q_g$ for all for all admissible values of $q_b$ and ${\overline{\theta }}$.

In the case of Cournot competition, consumer surplus in Periods 0 and 1 are defined by:

$$\begin{aligned} CS_b^{C0}=&\int _{\tilde{\theta }_b}^{{\overline{\theta }}} \dfrac{u_b(\theta )}{{\overline{\theta }}} \,d\theta \ =2{\overline{\theta }}q_b {x_b^{C0}}^2;\\ CS_b^{C1}=&\int _{0}^{{\hat{\theta }}} \dfrac{u_b(\theta )}{{\overline{\theta }}} \,d\theta \ =\dfrac{{\overline{\theta }}q_b}{2} {x_b^{C1}}^2;\\ CS_g^{C1}=&\int _{{\hat{\theta }}}^{{\overline{\theta }}} \dfrac{u_g(\theta )}{{\overline{\theta }}} \,d\theta \ =\dfrac{-2q_g^2+2(2{\overline{\theta }}+q_b)q_g+2{\overline{\theta }}q_b-3q_b^2}{4(4q_g-q_b)}q_g x_g^{C1}. \end{aligned}$$

We have:

$$\begin{aligned} \dfrac{\partial CS_b^{C1}}{\partial q_g}=&\dfrac{(2q_g^2-q_bq_g-{\overline{\theta }}q_b+q_b^2)q_b}{(4q_g-q_b)^2};\\ \dfrac{\partial CS_g^{C1}}{\partial q_g}=&\dfrac{1}{8{\overline{\theta }}(4q_g-q_b)^3} \Big ( 48q_g^5-4(32{\overline{\theta }}+13q_b)q_g^4 +32({\overline{\theta }}+q_b)(2{\overline{\theta }}+q_b)q_g^3\\&-12(4{\overline{\theta }}^2+2{\overline{\theta }}q_b+q_b^2)q_b q_g^2+8(2{\overline{\theta }}^2-{\overline{\theta }}q_b+q_b^2)q_b^2 q_g +(2{\overline{\theta }}-q_b)(2{\overline{\theta }}-3q_b)q_b^3 \Big ) \end{aligned}$$

We deduce that $\partial CS_b^{C1} / \partial q_g > 0$ if $q_g>\Big ( q_b+\sqrt{(8{\overline{\theta }}-7q_b)q_b} \Big )/4$ and $q_b<2{\overline{\theta }}/3$ (such that $x_b^{C1}$ increases) and negative otherwise. Because the polynomial of degree 5 into bracket in the expression of $\partial CS_g^{C1} / \partial q_g$ cannot be explicitly solved, we turn to numerical simulations. Simulations show that $\partial CS_g^{C1} / \partial q_g$ is U-shaped function of $q_g$ and that $CS_g^{C1}+CS_b^{C1}$ is a bell-shaped function of $q_g$ for all for all admissible values of $q_b$ and ${\overline{\theta }}$.