The demand is then equal to

$$d_M^m = \frac{\alpha(-ce_M + \bar{\theta}e_M + \rho q_B)}{2\bar{\theta}e_M} \qquad (D2)$$

Firm M's profit is therefore defined as $\pi_M^m = \bar{\theta}e_M d_M^m/\alpha$, with $d_M^m$ and $\pi_M^m$ positive if $ce_M \le \bar{\theta}e_M + \rho q_B$.

The conditions for the existence of a fragmented market are as follows:

* $\alpha \ge \alpha_0 \equiv -\bar{\theta}e_B/(\rho q_B - \bar{\theta}e_B)$ ensures that $p_B^{BMu} < \rho q_B$.

* $\alpha_0 < \alpha \le \alpha_3 \equiv \frac{-\bar{\theta}e_M e_B}{(ce_M e_B + \rho q_B(e_M - e_B))}$ ensures that the market is fragmented (i.e., $\tilde{\theta}_B < \tilde{\theta}_M$).

* $c > (\rho q_B - \bar{\theta}e_M)/e_M$ ensures that $p_M^m > \rho q_B$.

If $c \le (\rho q_B - \bar{\theta}e_M)/e_M$, the market is segmented with a corner solution because $p_M^m \le \rho q_B$: Firms M and B set the limit price $\rho q_B$. We then assume that all uninformed consumers consume Product B while all informed consumers consume Product M.

If $\alpha < \alpha_0$, the market is segmented. Firm B sets a price $p_B^m$ and earns a profit $\pi_B^m$. Firm M still supplies Product M at price $p_M^m$ and earns profit $\pi_M^m$.

## D.2 | Complete coverage

### D.2.1 | Covered market with an interior solution

The market is covered with an interior solution if $\tilde{\theta}_M < \tilde{\theta}_{BM} < \tilde{\theta}_B$. In this case, maximization of profits $\pi_B = p_B((1 - \alpha) + \alpha\tilde{\theta}_{BM}/\bar{\theta})$ and $\pi_M = (p_M - ce_M)(\alpha(\bar{\theta} - \tilde{\theta}_{BM})/\bar{\theta})$ leads to the equilibrium prices $p_B^{BMc}$ and $p_M^{BMc}$ defined in Equations (19) and (20). Using these prices, the conditions $\tilde{\theta}_M < \tilde{\theta}_{BM} < \tilde{\theta}_B$ boil down to

$$\bar{\theta}(e_M - 2e_B) \le 3\rho q_B \text{ that requires that } \bar{\theta} < -\frac{3}{2} \frac{\rho q_B}{e_B},$$

$$c < \frac{(e_M - e_B)(-\bar{\theta}(e_M - 2e_B) + 3\rho q_B)}{(e_M - 2e_B)e_M},$$

$$\alpha_4 \equiv \frac{\bar{\theta}(2e_M - e_B)(e_M - e_B)}{-ce_M(e_M - 2e_B) + (e_M - e_B)(\bar{\theta}(e_M + e_B) + 3\rho q_B)} < \alpha \le 1.$$

The demand for Product M, $d_M^{BMc}$ defined in Equation (21), is positive if $ce_M \le 2\bar{\theta}(e_M - e_B)$ or, otherwise, if $ce_M > 2\bar{\theta}(e_M - e_B)$ and $\alpha < \frac{\bar{\theta}(e_M - e_B)}{ce_M - 2\bar{\theta}(e_M - e_B)}$, while the demand for Product B is always positive. Under these conditions, firms earn positive profits: $\pi_B^{BMc} = -\bar{\theta}(e_B - e_M)d_B^{BMc2}/\alpha$ and $\pi_M^{BMc} = \bar{\theta}(e_B - e_M)d_M^{BMc2}/\alpha$.

### D.2.2 | Covered market with a corner solution

We have to distinguish two cases of covered market with corner solution, depending on the values of c and $\alpha$. In both cases, the firms set a limit price because neither the conditions for a fragmented market nor the conditions for a covered market are satisfied.

If $c \le (\rho q_B - \bar{\theta}e_M)/e_M$ and $\alpha_0 \le \alpha \le \alpha_4$, Firms M and B set the limit price $\rho q_B$ that allows them to capture the demand of all consumers in their market segments (as explained above).

If $c > (\rho q_B - \bar{\theta}e_M)/e_M$ and $\alpha_3 \le \alpha \le \alpha_4$, there is a corner solution where the prices are such that $\tilde{\theta}_M = \tilde{\theta}_{BM} = \tilde{\theta}_B$ for all $\alpha \in [\alpha_3, \alpha_4]$. All consumers with a lower marginal WTP than $\tilde{\theta}_i$ buy Product B while all the other informed consumers buy Product M. To find the equilibrium, we calculate $p_B^f$ for $\alpha = \alpha_3$, and we define the limit price of Product B as $p_B^{BMcc}$ characterized in Equation (18). The price of Product M remains the monopoly price $p_M^m$.
The demand is then equal to

$$d_M^m = \frac{\alpha(-ce_M + \bar{\theta}e_M + \rho q_B)}{2\bar{\theta}e_M} \qquad (D2)$$

Firm M's profit is therefore defined as $\pi_M^m = \bar{\theta}e_M d_M^m/\alpha$, with $d_M^m$ and $\pi_M^m$ positive if $ce_M \le \bar{\theta}e_M + \rho q_B$.

The conditions for the existence of a fragmented market are as follows:

* $\alpha \ge \alpha_0 \equiv -\bar{\theta}e_B/(\rho q_B - \bar{\theta}e_B)$ ensures that $p_B^{BMu} < \rho q_B$.

* $\alpha_0 < \alpha \le \alpha_3 \equiv \frac{-\bar{\theta}e_M e_B}{(ce_M e_B + \rho q_B(e_M - e_B))}$ ensures that the market is fragmented (i.e., $\tilde{\theta}_B < \tilde{\theta}_M$).

* $c > (\rho q_B - \bar{\theta}e_M)/e_M$ ensures that $p_M^m > \rho q_B$.

If $c \le (\rho q_B - \bar{\theta}e_M)/e_M$, the market is segmented with a corner solution because $p_M^m \le \rho q_B$: Firms M and B set the limit price $\rho q_B$. We then assume that all uninformed consumers consume Product B while all informed consumers consume Product M.

If $\alpha < \alpha_0$, the market is segmented. Firm B sets a price $p_B^m$ and earns a profit $\pi_B^m$. Firm M still supplies Product M at price $p_M^m$ and earns profit $\pi_M^m$.

## D.2 | Complete coverage

### D.2.1 | Covered market with an interior solution

The market is covered with an interior solution if $\tilde{\theta}_M < \tilde{\theta}_{BM} < \tilde{\theta}_B$. In this case, maximization of profits $\pi_B = p_B((1 - \alpha) + \alpha\tilde{\theta}_{BM}/\bar{\theta})$ and $\pi_M = (p_M - ce_M)(\alpha(\bar{\theta} - \tilde{\theta}_{BM})/\bar{\theta})$ leads to the equilibrium prices $p_B^{BMc}$ and $p_M^{BMc}$ defined in Equations (19) and (20). Using these prices, the conditions $\tilde{\theta}_M < \tilde{\theta}_{BM} < \tilde{\theta}_B$ boil down to

$$\bar{\theta}(e_M - 2e_B) \le 3\rho q_B \text{ that requires that } \bar{\theta} < -\frac{3}{2} \frac{\rho q_B}{e_B},$$

$$c < \frac{(e_M - e_B)(-\bar{\theta}(e_M - 2e_B) + 3\rho q_B)}{(e_M - 2e_B)e_M},$$

$$\alpha_4 \equiv \frac{\bar{\theta}(2e_M - e_B)(e_M - e_B)}{-ce_M(e_M - 2e_B) + (e_M - e_B)(\bar{\theta}(e_M + e_B) + 3\rho q_B)} < \alpha \le 1.$$

The demand for Product M, $d_M^{BMc}$ defined in Equation (21), is positive if $ce_M \le 2\bar{\theta}(e_M - e_B)$ or, otherwise, if $ce_M > 2\bar{\theta}(e_M - e_B)$ and $\alpha < \frac{\bar{\theta}(e_M - e_B)}{ce_M - 2\bar{\theta}(e_M - e_B)}$, while the demand for Product B is always positive. Under these conditions, firms earn positive profits: $\pi_B^{BMc} = -\bar{\theta}(e_B - e_M)d_B^{BMc2}/\alpha$ and $\pi_M^{BMc} = \bar{\theta}(e_B - e_M)d_M^{BMc2}/\alpha$.

### D.2.2 | Covered market with a corner solution

We have to distinguish two cases of covered market with corner solution, depending on the values of c and $\alpha$. In both cases, the firms set a limit price because neither the conditions for a fragmented market nor the conditions for a covered market are satisfied.

If $c \le (\rho q_B - \bar{\theta}e_M)/e_M$ and $\alpha_0 \le \alpha \le \alpha_4$, Firms M and B set the limit price $\rho q_B$ that allows them to capture the demand of all consumers in their market segments (as explained above).

If $c > (\rho q_B - \bar{\theta}e_M)/e_M$ and $\alpha_3 \le \alpha \le \alpha_4$, there is a corner solution where the prices are such that $\tilde{\theta}_M = \tilde{\theta}_{BM} = \tilde{\theta}_B$ for all $\alpha \in [\alpha_3, \alpha_4]$. All consumers with a lower marginal WTP than $\tilde{\theta}_i$ buy Product B while all the other informed consumers buy Product M. To find the equilibrium, we calculate $p_B^f$ for $\alpha = \alpha_3$, and we define the limit price of Product B as $p_B^{BMcc}$ characterized in Equation (18). The price of Product M remains the monopoly price $p_M^m$.
# APPENDIX E: PROOFS OF NASH EQUILIBRIA WITH PRODUCTS M AND G

We define the informed consumer of type $\tilde{\Theta}_{MG}$ as the consumer indifferent between Products M and G:

$$ \tilde{\Theta}_{MG} = \frac{p_G - p_M + \rho(q_B - q_G)}{e_G - e_M}. $$

Informed consumers with $\theta \ge \tilde{\Theta}_{MG}$ prefer Product G to Product M, while consumers with $\theta < \tilde{\Theta}_{MG}$ prefer Product M to Product G.

## E.1 | Fragmentation

The market with only Products M and G cannot be fragmented. To prove this, we note that informed consumers with $\theta \le \tilde{\Theta}_M$ do not consume, those with $\theta \ge \tilde{\Theta}_M$ accept to buy Product M, and those with $\theta \ge \tilde{\Theta}_G$ accept to buy Product G. The market is fragmented if $\tilde{\Theta}_M > 0$ and $\tilde{\Theta}_G > 0$, that requires $p_M > \rho q_B$ and $p_G > \rho q_G$. In this case, the uninformed consumers do not accept to buy M or G, but they are still willing to buy Product B at a price lower than or equal to $p_B^m = \rho q_B$. Therefore, the market structure can be a triopoly supplying the three Products B, M, and G if $\tilde{\Theta}_B < 0 < \tilde{\Theta}_M < \tilde{\Theta}_G < \tilde{\Theta}_{MG} < \bar{\Theta}$, or a duopoly supplying Products B and M if $\tilde{\Theta}_B < 0 < \tilde{\Theta}_M < \tilde{\Theta}_G < \bar{\Theta} < \tilde{\Theta}_{MG}$), or a duopoly supplying Products B and G if $\tilde{\Theta}_{MG} < 0 < \tilde{\Theta}_G < \tilde{\Theta}_M < \bar{\Theta}$.

## E.2 | Complete coverage

The market is covered if $\tilde{\Theta}_M \le 0 \le \tilde{\Theta}_G \le \tilde{\Theta}_{MG} < \bar{\Theta}$. In this case, the demands are then defined as follows: $d_G^{MGc} = \alpha(\bar{\Theta} - \tilde{\Theta}_{BG})/\bar{\Theta}$ and $d_M^{MGc} = 1 - \alpha + \alpha\tilde{\Theta}_{BG}/\bar{\Theta}$. Maximization of profits $\pi_M = (p_M - ce_M)d_M^{MGc}$ and $\pi_G = p_G d_G^{MGc} - F_G$ leads to the equilibrium prices $p_M^{MGc}$ and $p_G^{MGc}$ defined in Equations (23) and (24) and the demand for Product M, $d_M^{MGc}$ defined in Equation (25). The conditions for the positivity of prices and demands and for the uninformed consumers to consume Product M ($p_M^{MGc} \le \rho q_B$ and $\rho q_B - p_M^{MGc} > \rho q_G - p_G^{MGc}$) are expressed as follows:

$$ \begin{aligned} \alpha > \alpha_5 &\equiv \frac{\bar{\Theta}(e_G - e_M)}{-2ce_M + 2\bar{\Theta}(e_G - e_M) + \rho(q_B - q_G)}, \\ \alpha > \alpha_6 &\equiv \frac{2\bar{\Theta}(e_G - e_M)}{-2ce_M + \bar{\Theta}(e_G - e_M) + \rho(2q_B + q_G)}, \end{aligned} $$

where the lower bounds can be negative or positive, depending on the level of the unit production cost of Product M, c, compared with the WTP for environmental and intrinsic qualities.

The effects of a higher fraction of informed consumers on prices and demands are given by the following derivatives:

$$ \begin{aligned} \frac{\partial p_M^{MGc}}{\partial \alpha} &= \frac{-2\bar{\Theta}(e_G - e_M)}{(3\alpha^2)} < 0, \\ \frac{\partial p_G^{MGc}}{\partial \alpha} &= \frac{-\bar{\Theta}(e_G - e_M)}{(3\alpha^2)} < 0, \\ \frac{\partial d_M^{MGc}}{\partial \alpha} &= \frac{-ce_M - \bar{\Theta}(e_G - e_M) + \rho(q_B - q_G)}{3\bar{\Theta}(e_G - e_M)}, \\ \frac{\partial d_G^{MGc}}{\partial \alpha} &= \frac{ce_M + \bar{\Theta}(e_G - e_M) - \rho(q_B - q_G)}{3\bar{\Theta}(e_G - e_M)}. \end{aligned} $$

# APPENDIX F: PROOFS OF SUBGAME PERFECT NASH EQUILIBRIA

In stage 3, one of the firms decides to replace its product with Product M if it will increase its profits or to continue offering the same product otherwise. The resulting SPNE depends on the market structure (segmented, fragmented, or covered) determined by $\alpha$ and $e_M$, and satisfies the following nondeviation conditions:

* The SNPE with Products B and G requires $\pi_G^{BG} > \pi_M^{BM}$ and $\pi_B^{BG} > \pi_M^{MG}$.
## APPENDIX E: PROOFS OF NASH EQUILIBRIA WITH PRODUCTS M AND G

We define the informed consumer of type $\tilde{\Theta}_{MG}$ as the consumer indifferent between Products M and G:

$$ \tilde{\Theta}_{MG} = \frac{p_G - p_M + \rho(q_B - q_G)}{e_G - e_M}. $$

Informed consumers with $\theta \ge \tilde{\Theta}_{MG}$ prefer Product G to Product M, while consumers with $\theta < \tilde{\Theta}_{MG}$ prefer Product M to Product G.

### E.1 | Fragmentation

The market with only Products M and G cannot be fragmented. To prove this, we note that informed consumers with $\theta \le \tilde{\Theta}_M$ do not consume, those with $\theta \ge \tilde{\Theta}_M$ accept to buy Product M, and those with $\theta \ge \tilde{\Theta}_G$ accept to buy Product G. The market is fragmented if $\tilde{\Theta}_M > 0$ and $\tilde{\Theta}_G > 0$, that requires $p_M > \rho q_B$ and $p_G > \rho q_G$. In this case, the uninformed consumers do not accept to buy M or G, but they are still willing to buy Product B at a price lower than or equal to $p_B^m = \rho q_B$. Therefore, the market structure can be a triopoly supplying the three Products B, M, and G if $\tilde{\Theta}_B < 0 < \tilde{\Theta}_M < \tilde{\Theta}_G < \tilde{\Theta}_{MG} < \bar{\Theta}$, or a duopoly supplying Products B and M if $\tilde{\Theta}_B < 0 < \tilde{\Theta}_M < \tilde{\Theta}_G < \bar{\Theta} < \tilde{\Theta}_{MG}$), or a duopoly supplying Products B and G if $\tilde{\Theta}_{MG} < 0 < \tilde{\Theta}_G < \tilde{\Theta}_M < \bar{\Theta}$.

### E.2 | Complete coverage

The market is covered if $\tilde{\Theta}_M \le 0 \le \tilde{\Theta}_G \le \tilde{\Theta}_{MG} < \bar{\Theta}$. In this case, the demands are then defined as follows: $d_G^{MGc} = \alpha(\bar{\Theta} - \tilde{\Theta}_{BG})/\bar{\Theta}$ and $d_M^{MGc} = 1 - \alpha + \alpha\tilde{\Theta}_{BG}/\bar{\Theta}$. Maximization of profits $\pi_M = (p_M - ce_M)d_M^{MGc}$ and $\pi_G = p_G d_G^{MGc} - F_G$ leads to the equilibrium prices $p_M^{MGc}$ and $p_G^{MGc}$ defined in Equations (23) and (24) and the demand for Product M, $d_M^{MGc}$ defined in Equation (25). The conditions for the positivity of prices and demands and for the uninformed consumers to consume Product M ($p_M^{MGc} \le \rho q_B$ and $\rho q_B - p_M^{MGc} > \rho q_G - p_G^{MGc}$) are expressed as follows:

$$ \begin{aligned} \alpha > \alpha_5 &\equiv \frac{\bar{\Theta}(e_G - e_M)}{-2ce_M + 2\bar{\Theta}(e_G - e_M) + \rho(q_B - q_G)}, \\ \alpha > \alpha_6 &\equiv \frac{2\bar{\Theta}(e_G - e_M)}{-2ce_M + \bar{\Theta}(e_G - e_M) + \rho(2q_B + q_G)}, \end{aligned} $$

where the lower bounds can be negative or positive, depending on the level of the unit production cost of Product M, c, compared with the WTP for environmental and intrinsic qualities.

The effects of a higher fraction of informed consumers on prices and demands are given by the following derivatives:

$$ \begin{aligned} \frac{\partial p_M^{MGc}}{\partial \alpha} &= \frac{-2\bar{\Theta}(e_G - e_M)}{(3\alpha^2)} < 0, \\ \frac{\partial p_G^{MGc}}{\partial \alpha} &= \frac{-\bar{\Theta}(e_G - e_M)}{(3\alpha^2)} < 0, \\ \frac{\partial d_M^{MGc}}{\partial \alpha} &= \frac{-ce_M - \bar{\Theta}(e_G - e_M) + \rho(q_B - q_G)}{3\bar{\Theta}(e_G - e_M)}, \\ \frac{\partial d_G^{MGc}}{\partial \alpha} &= \frac{ce_M + \bar{\Theta}(e_G - e_M) - \rho(q_B - q_G)}{3\bar{\Theta}(e_G - e_M)}. \end{aligned} $$

## APPENDIX F: PROOFS OF SUBGAME PERFECT NASH EQUILIBRIA

In stage 3, one of the firms decides to replace its product with Product M if it will increase its profits or to continue offering the same product otherwise. The resulting SPNE depends on the market structure (segmented, fragmented, or covered) determined by $\alpha$ and $e_M$, and satisfies the following nondeviation conditions:

* The SNPE with Products B and G requires $\pi_G^{BG} > \pi_M^{BM}$ and $\pi_B^{BG} > \pi_M^{MG}$.
<figure><img src="image_33.png" /></figure>

* The SNPE with Products B and M requires $\pi_M^{BM} > \pi_G^{BG}$ and $\pi_M^{BM} > \pi_G^{MG}$.

* The SNPE with Products M and G requires $\pi_M^{MG} > \pi_B^{BG}$ and $\pi_G^{MG} > \pi_B^{BM}$.

## F.1 | Subgame perfect Nash equilibria with Products B and M

According to the strategies $\alpha$ and $e_M$ of the NGO, Firm G decides to replace Product G with Product M when its competitor still produces Product B if $\pi_M^{BM} > \pi_G^{BG}$. To detail all the conditions according to $e_M$ and $\alpha$ would be laborious. For example, the conditions obtained for $0 < \alpha \le \alpha_0$ are

* If $e_M < \rho q_B / (\bar{\theta} + c)$: $\alpha < \frac{4\bar{\theta} e_G F_G}{(\bar{\theta}e_G + \rho q_G)^2 - 4\bar{\theta} e_G (\rho q_B - c e_M)}$.

* If $e_M > \rho q_B / (\bar{\theta} + c)$:

$$ \alpha < \frac{4\bar{\theta} e_G e_M F_G}{(e_M e_G (-c^2 e_M + \bar{\theta}^2 (e_G - e_M) - 2\bar{\theta} \rho (q_B - q_G) + 2c(\bar{\theta} e_M + \rho q_B)) + \rho^2 (e_M q_G^2 - e_G q_B^2)} $$

The boundary such that $\pi_M^{BM} = \pi_G^{BG}$ (illustrated in Figure 4) is a decreasing and convex function because, in all configurations of the market, $\pi_M^{BM}$ is a decreasing and convex function of $e_M$ under the following conditions:

$$ \frac{\partial \pi_M^{mc}}{\partial e_M} = -\alpha c e_M < 0, $$

$$ \frac{\partial \pi_M^m}{\partial e_M} = \frac{-c e_M + \bar{\theta} e_M - \rho q_B}{2 e_M} d_M^m < 0 \quad \text{if} \quad -\bar{\theta} e_M + \rho q_B < c e_M < \bar{\theta} e_M - \rho q_B, $$

$$ \frac{\partial^2 \pi_M^m}{\partial e_M^2} = \frac{\alpha \rho^2 q_B^2}{2 \bar{\theta} e_M^3} > 0, $$

$$ \frac{\partial \pi_M^{BMc}}{\partial e_M} = \frac{-\alpha c (e_M - 2e_B) + (1 + \alpha) \bar{\theta} (e_M - e_B)}{3\alpha (e_M - e_B)} d_M^{BMc} < 0 \quad \text{if } \alpha > \frac{\bar{\theta} (e_M - e_B)}{c (e_M - 2e_B) - \bar{\theta} (e_M - e_B)}, $$

$$ \frac{\partial^2 \pi_M^{BMc}}{\partial e_M^2} = \frac{2\alpha c^2 e_B^2}{9\bar{\theta}(e_M - e_B)^3} > 0. $$

Moreover, $\pi_M^{BM}$ and $\pi_G^{BG}$ are both increasing functions of $\alpha$ when the market is fragmented or covered with a corner solution and decreasing functions when the market is covered with an interior solution:

$$ \frac{\partial \pi_M^{mc}}{\partial \alpha} = \rho q_B - c e_M > 0, $$

$$ \frac{\partial \pi_M^m}{\partial \alpha} = \frac{(-c e_M + \bar{\theta} e_M + \rho q_B)^2}{4 \bar{\theta} e_M} > 0, $$

$$ \frac{\partial \pi_M^{BMc}}{\partial \alpha} = \frac{-\alpha c e_M - (1 - \alpha) \bar{\theta} (e_M - e_B)}{3\alpha^2} d_M^{BMc} < 0, $$

$$ \frac{\partial \pi_G^m}{\partial \alpha} = \frac{(\bar{\theta} e_G + \rho q_G)^2}{4 \bar{\theta} e_G} > 0, $$

$$ \frac{\partial \pi_G^{BGc}}{\partial \alpha} = \frac{-(1 - \alpha) \bar{\theta} (e_G - e_B) - \alpha \rho (q_B - q_G)}{3\alpha^2} d_G^{BGc} < 0. $$

As a result, the boundary $\alpha$ such that $\pi_M^{BM} = \pi_G^{BG}$ is a decreasing and convex function of $e_M$ as soon as the conditions ensuring that $\pi_M^{BM}$ is a decreasing function of $e_M$ are satisfied. The condition for the nondeviation of Firm B, such that $\pi_B^{BM} > \pi_G^{MG}$, does not affect the shape of the boundary.
<figure><img src="image_33.png" /></figure>

* The SNPE with Products B and M requires $\pi_M^{BM} > \pi_G^{BG}$ and $\pi_M^{BM} > \pi_G^{MG}$.

* The SNPE with Products M and G requires $\pi_M^{MG} > \pi_B^{BG}$ and $\pi_G^{MG} > \pi_B^{BM}$.

## F.1 | Subgame perfect Nash equilibria with Products B and M

According to the strategies $\alpha$ and $e_M$ of the NGO, Firm G decides to replace Product G with Product M when its competitor still produces Product B if $\pi_M^{BM} > \pi_G^{BG}$. To detail all the conditions according to $e_M$ and $\alpha$ would be laborious. For example, the conditions obtained for $0 < \alpha \le \alpha_0$ are

* If $e_M < \rho q_B / (\bar{\theta} + c)$: $\alpha < \frac{4\bar{\theta} e_G F_G}{(\bar{\theta}e_G + \rho q_G)^2 - 4\bar{\theta} e_G (\rho q_B - c e_M)}$.

* If $e_M > \rho q_B / (\bar{\theta} + c)$:

$$ \alpha < \frac{4\bar{\theta} e_G e_M F_G}{(e_M e_G (-c^2 e_M + \bar{\theta}^2 (e_G - e_M) - 2\bar{\theta} \rho (q_B - q_G) + 2c(\bar{\theta} e_M + \rho q_B)) + \rho^2 (e_M q_G^2 - e_G q_B^2)} $$

The boundary such that $\pi_M^{BM} = \pi_G^{BG}$ (illustrated in Figure 4) is a decreasing and convex function because, in all configurations of the market, $\pi_M^{BM}$ is a decreasing and convex function of $e_M$ under the following conditions:

$$ \frac{\partial \pi_M^{mc}}{\partial e_M} = -\alpha c e_M < 0, $$

$$ \frac{\partial \pi_M^m}{\partial e_M} = \frac{-c e_M + \bar{\theta} e_M - \rho q_B}{2 e_M} d_M^m < 0 \quad \text{if} \quad -\bar{\theta} e_M + \rho q_B < c e_M < \bar{\theta} e_M - \rho q_B, $$

$$ \frac{\partial^2 \pi_M^m}{\partial e_M^2} = \frac{\alpha \rho^2 q_B^2}{2 \bar{\theta} e_M^3} > 0, $$

$$ \frac{\partial \pi_M^{BMc}}{\partial e_M} = \frac{-\alpha c (e_M - 2e_B) + (1 + \alpha) \bar{\theta} (e_M - e_B)}{3\alpha (e_M - e_B)} d_M^{BMc} < 0 \quad \text{if } \alpha > \frac{\bar{\theta} (e_M - e_B)}{c (e_M - 2e_B) - \bar{\theta} (e_M - e_B)}, $$

$$ \frac{\partial^2 \pi_M^{BMc}}{\partial e_M^2} = \frac{2\alpha c^2 e_B^2}{9\bar{\theta}(e_M - e_B)^3} > 0. $$

Moreover, $\pi_M^{BM}$ and $\pi_G^{BG}$ are both increasing functions of $\alpha$ when the market is fragmented or covered with a corner solution and decreasing functions when the market is covered with an interior solution:

$$ \frac{\partial \pi_M^{mc}}{\partial \alpha} = \rho q_B - c e_M > 0, $$

$$ \frac{\partial \pi_M^m}{\partial \alpha} = \frac{(-c e_M + \bar{\theta} e_M + \rho q_B)^2}{4 \bar{\theta} e_M} > 0, $$

$$ \frac{\partial \pi_M^{BMc}}{\partial \alpha} = \frac{-\alpha c e_M - (1 - \alpha) \bar{\theta} (e_M - e_B)}{3\alpha^2} d_M^{BMc} < 0, $$

$$ \frac{\partial \pi_G^m}{\partial \alpha} = \frac{(\bar{\theta} e_G + \rho q_G)^2}{4 \bar{\theta} e_G} > 0, $$

$$ \frac{\partial \pi_G^{BGc}}{\partial \alpha} = \frac{-(1 - \alpha) \bar{\theta} (e_G - e_B) - \alpha \rho (q_B - q_G)}{3\alpha^2} d_G^{BGc} < 0. $$

As a result, the boundary $\alpha$ such that $\pi_M^{BM} = \pi_G^{BG}$ is a decreasing and convex function of $e_M$ as soon as the conditions ensuring that $\pi_M^{BM}$ is a decreasing function of $e_M$ are satisfied. The condition for the nondeviation of Firm B, such that $\pi_B^{BM} > \pi_G^{MG}$, does not affect the shape of the boundary.
## F.2 | Subgame perfect Nash equilibria with Products M and G

According to the strategies $\alpha$ and $e_M$ of the NGO, Firm B decides to replace Product B with Product M when its competitor still produces Product G if $\pi_M^{MG} > \pi_B^{BG}$. First of all, note that $\pi_M^{MGc} < \pi_B^{BGc}$ as soon as $\rho(q_B - q_G)(e_M - e_B) < ce_M(e_G - e_B)$. Moreover, from the definitions of profits, we can see that the conditions for $\pi_M^{MG} > \pi_B^{BG}$ can all be expressed as conditions for quadratic polynomials in $\alpha$ to be positive. The conditions on $\alpha$ and $e_M$ for polynomials to be positive can be deduced from the analysis of the sign of the discriminant and the sign of the coefficient of $\alpha^2$ of each polynomial function (which depends on $e_M$). Details of all conditions are not of interest to the analysis. Illustrations are given in Figure 4.

## APPENDIX G: PROOF OF PROPOSITION 3

When $\alpha < \alpha_0$, the market is segmented (denoted with superscript $m$) and also covered if the production cost of Product M is low (denoted with superscript $mc$). The associated levels of environmental quality are

$$ 
\begin{aligned}
E^{BMmc} &= (1 - \alpha)e_B + \alpha e_M, \\
E^{BMm} &= (1 - \alpha)e_B + \alpha \frac{ce_M(e_G - e_M) + \bar{\theta} e_M(e_G + e_M) - \rho q_B(e_G - e_M)}{2\bar{\theta} e_M}.
\end{aligned}
$$

Because $E^{BGm} = (1 - \alpha)e_B + \alpha e_G$, it is clear that $E^{BGm} > E^{BMmc}$. Moreover, $E^{BGm} > E^{BMm}$ because $\alpha e_G$ is higher than the second term of $E^{BMm}$: the difference between both terms is characterized by $\alpha(e_G - e_M)(-ce_M + \bar{\theta} e_M + \rho q_B))/(2\bar{\theta} e_M) > 0$.

## APPENDIX H: PROOFS OF NGO'S STRATEGIES IN FRAGMENTED AND COVERED MARKETS

### H.1 | The environmental quality with Products B and M

For $\alpha \ge \alpha_0$, the levels of environmental quality associated with a fragmented market ($f$), a covered market with a corner solution ($cc$) and a covered market with an interior solution ($c$) are

$$ E^{BMf} = \frac{e_G + e_B}{2} + \alpha \frac{(e_M - e_B)(\bar{\theta} e_M e_B + \rho q_B e_G) + ce_M e_B (e_G - e_M)}{2\bar{\theta} e_B e_M}, \quad (H1) $$

$$ E^{BMcc} = e_B + \alpha \frac{(e_M - e_B)(\bar{\theta} e_M + \rho q_B - ce_M)}{2\bar{\theta} e_M}, \quad (H2) $$

$$ E^{BMc} = \frac{e_M + 2e_B}{3} + \alpha \frac{\bar{\theta}(e_M - e_B) - ce_M}{3\bar{\theta}}, \quad (H3) $$

We compare the environmental quality $E^{BMk}$ with $E^{BGk}$ ($k = m, mc, f, c, cc$) in the only cases where there is an SPNE with Products B and M. The analysis is therefore limited to the cases where $e_M < \frac{\rho q_B}{c + \bar{\theta}}$.

$$ (1) \text{ The cases with } \alpha_0 \le \alpha < \alpha_1 $$

* For $\alpha_0 < \alpha < \min\{\alpha_4, \alpha_1\}$, the market with B and M is segmented with a corner solution ($mc$), while the market with B and G is fragmented ($f$). A necessary condition for

$$ E^{BMmc} = e_G + (e_B - e_G)d_B^{mc} + (e_M - e_G)d_M^{mc} > E^{BGf} = e_G + (e_B - e_G)d_B^f $$

is $d_B^{mc} = 1 - \alpha < d_B^f$, that is satisfied when $\bar{\theta} e_G > \rho q_G$ and $\alpha > \alpha_0$. However, we have
## F.2 | Subgame perfect Nash equilibria with Products M and G

According to the strategies $\alpha$ and $e_M$ of the NGO, Firm B decides to replace Product B with Product M when its competitor still produces Product G if $\pi_M^{MG} > \pi_B^{BG}$. First of all, note that $\pi_M^{MGc} < \pi_B^{BGc}$ as soon as $\rho(q_B - q_G)(e_M - e_B) < ce_M(e_G - e_B)$. Moreover, from the definitions of profits, we can see that the conditions for $\pi_M^{MG} > \pi_B^{BG}$ can all be expressed as conditions for quadratic polynomials in $\alpha$ to be positive. The conditions on $\alpha$ and $e_M$ for polynomials to be positive can be deduced from the analysis of the sign of the discriminant and the sign of the coefficient of $\alpha^2$ of each polynomial function (which depends on $e_M$). Details of all conditions are not of interest to the analysis. Illustrations are given in Figure 4.

# APPENDIX G: PROOF OF PROPOSITION 3

When $\alpha < \alpha_0$, the market is segmented (denoted with superscript $m$) and also covered if the production cost of Product M is low (denoted with superscript $mc$). The associated levels of environmental quality are

$$ 
\begin{aligned}
E^{BMmc} &= (1 - \alpha)e_B + \alpha e_M, \\
E^{BMm} &= (1 - \alpha)e_B + \alpha \frac{ce_M(e_G - e_M) + \bar{\theta} e_M(e_G + e_M) - \rho q_B(e_G - e_M)}{2\bar{\theta} e_M}.
\end{aligned}
$$

Because $E^{BGm} = (1 - \alpha)e_B + \alpha e_G$, it is clear that $E^{BGm} > E^{BMmc}$. Moreover, $E^{BGm} > E^{BMm}$ because $\alpha e_G$ is higher than the second term of $E^{BMm}$: the difference between both terms is characterized by $\alpha(e_G - e_M)(-ce_M + \bar{\theta} e_M + \rho q_B))/(2\bar{\theta} e_M) > 0$.

# APPENDIX H: PROOFS OF NGO'S STRATEGIES IN FRAGMENTED AND COVERED MARKETS

## H.1 | The environmental quality with Products B and M

For $\alpha \ge \alpha_0$, the levels of environmental quality associated with a fragmented market ($f$), a covered market with a corner solution ($cc$) and a covered market with an interior solution ($c$) are

$$ E^{BMf} = \frac{e_G + e_B}{2} + \alpha \frac{(e_M - e_B)(\bar{\theta} e_M e_B + \rho q_B e_G) + ce_M e_B (e_G - e_M)}{2\bar{\theta} e_B e_M}, \quad (H1) $$

$$ E^{BMcc} = e_B + \alpha \frac{(e_M - e_B)(\bar{\theta} e_M + \rho q_B - ce_M)}{2\bar{\theta} e_M}, \quad (H2) $$

$$ E^{BMc} = \frac{e_M + 2e_B}{3} + \alpha \frac{\bar{\theta}(e_M - e_B) - ce_M}{3\bar{\theta}}, \quad (H3) $$

We compare the environmental quality $E^{BMk}$ with $E^{BGk}$ ($k = m, mc, f, c, cc$) in the only cases where there is an SPNE with Products B and M. The analysis is therefore limited to the cases where $e_M < \frac{\rho q_B}{c + \bar{\theta}}$.

$$ (1) \text{ The cases with } \alpha_0 \le \alpha < \alpha_1 $$

* For $\alpha_0 < \alpha < \min\{\alpha_4, \alpha_1\}$, the market with B and M is segmented with a corner solution (mc), while the market with B and G is fragmented (f). A necessary condition for

$$ E^{BMmc} = e_G + (e_B - e_G)d_B^{mc} + (e_M - e_G)d_M^{mc} > E^{BGf} = e_G + (e_B - e_G)d_B^f $$

is $d_B^{mc} = 1 - \alpha < d_B^f$, that is satisfied when $\bar{\theta} e_G > \rho q_G$ and $\alpha > \alpha_0$. However, we have
<figure><img src="image_34.png" /></figure>

$$E^{BMmc} - E^{BGf} = \frac{\bar{\theta} e_B (e_G - e_B) + \alpha [(\bar{\theta} e_B + \rho q_B)e_G - (2\bar{\theta} e_M + \rho q_B - \bar{\theta} e_B)e_B]}{-2\bar{\theta} e_B},$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_0} = \frac{\bar{\theta} e_B (e_G - e_B)}{\rho q_B - \bar{\theta} e_B} < 0,$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_1} = \frac{e_B(\bar{\theta} e_G (e_G - 2e_M + e_B) + \rho q_G (e_G - e_B))}{2\rho(q_B e_G - q_G e_B)},$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_1} < 0 \quad \text{if } e_M < \frac{e_G + e_B}{2} + \frac{\rho q_G (e_G - e_B)}{2\bar{\theta} e_G},$$

$$\text{with } \frac{e_G + e_B}{2} + \frac{\rho q_G (e_G - e_B)}{2\bar{\theta} e_G} < \frac{\rho q_B}{c + \bar{\theta}} \quad \text{if } c > -\bar{\theta} + \frac{2\rho q_B \bar{\theta} e_G}{\bar{\theta} e_G (e_G + e_B) + \rho q_G (e_G - e_B)}$$

and $E^{BMmc} - E^{BGf}$ is a linear function of $\alpha$. Therefore, $E^{BMmc} < E^{BGf}$ as soon as $e_M$ is lower than the above thresholds and $c$ sufficiently high. However, an increase in $e_M$ improves $E^{BMmc}$ (since $\partial E^{BMmc}/\partial e_M = \alpha$) and can then lead to $E^{BMmc} > E^{BGf}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).

* For $\max\{\alpha_0, \alpha_4\} < \alpha < \alpha_1$, the market with B and M is covered (c) while the market with B and G is fragmented (f). A necessary condition for $E^{BMc} > E^{BGf}$ is $d_B^{BMc} < d_B^f$, that is satisfied if $\alpha > \hat{\alpha} \equiv \frac{-\bar{\theta} e_B (e_M - e_B)}{(\bar{\theta} e_B + 3\rho q_B)(e_M - e_B) + 2ce_B e_M} > 0$, with

$$\begin{aligned} \hat{\alpha}|_{e_M=0} &= \alpha_4|_{e_M=0} = \frac{-\bar{\theta} e_B}{\bar{\theta} e_B + 3\rho q_B}, \\ \left. \frac{\partial \hat{\alpha}}{\partial e_M} \right|_{e_M=0} &= \frac{\bar{\theta}(-2ce_B + 3\bar{\theta} e_B + 6\rho q_B)}{(\bar{\theta} e_B + 3\rho q_B)^2}, \\ \left. \frac{\partial \alpha_4}{\partial e_M} \right|_{e_M=0} &= \frac{-2c\bar{\theta} e_B}{(\bar{\theta} e_B + 3\rho q_B)^2} > \left. \frac{\hat{\alpha}}{\partial e_M} \right|_{e_M=0} \quad \text{if } \bar{\theta} e_B + 2\rho q_B > 0. \end{aligned}$$

Therefore, $\alpha > \alpha_4 > \hat{\alpha}$ and $d_B^{BMc} < d_B^f$ if $\bar{\theta} e_B + 2\rho q_B > 0$. However, we have

$$\begin{aligned} E^{BMc} - E^{BGf} &= \frac{-\bar{\theta} e_B (e_B + 2e_M - 3e_G) + \alpha [-2e_B (\bar{\theta} e_M + \rho q_B - ce_M) + (\bar{\theta} e_B + \rho q_B)(3e_G - e_B)]}{-6\bar{\theta} e_B}, \\ E^{BMc} - E^{BGf}|_{\alpha=\alpha_0} &= \frac{\bar{\theta} e_B (3e_G - 2e_M - e_B) + \rho q_B (e_M - e_B) + ce_B e_M}{-3\bar{\theta} e_B + 3\rho q_B}, \end{aligned}$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_0} < 0 \quad \text{if } e_M < \frac{-e_B(3\bar{\theta} e_G - \bar{\theta} e_B - \rho q_B)}{-2\bar{\theta} e_B + \rho q_B + ce_B},$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_1} = \frac{(\bar{\theta} e_G + \rho q_G)(3e_G - 2e_M - e_B)e_B + 2\rho q_B(e_M - e_B)e_G + 2ce_B e_M e_G}{6(\rho q_B e_G - \rho q_G e_B)},$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_1} < 0 \quad \text{if } e_M < \frac{-(\bar{\theta} e_G + \rho q_G)(3e_G - e_B)e_B + 2\rho q_B e_B e_G}{-2e_B(\bar{\theta} e_G + \rho q_G) + 2\rho q_B e_G + 2ce_B e_G}$$

and $E^{BMc} - E^{BGf}$ is a linear function of $\alpha$. Therefore, $E^{BMc} < E^{BGf}$ as soon as $e_M$ is lower than both of the above thresholds, which increase with $c$. However, an increase in $e_M$ can improve $E^{BMc}$:

$$\frac{\partial E^{BMc}}{\partial e_M} = \frac{\alpha(\bar{\theta} - c) + \bar{\theta}}{3\bar{\theta}} > 0$$

if $\alpha(\bar{\theta} - c) + \bar{\theta} > 0$. This can then lead to $E^{BMc} > E^{BGf}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).
<figure><img src="image_34.png" /></figure>

$$E^{BMmc} - E^{BGf} = \frac{\bar{\theta} e_B (e_G - e_B) + \alpha [(\bar{\theta} e_B + \rho q_B)e_G - (2\bar{\theta} e_M + \rho q_B - \bar{\theta} e_B)e_B]}{-2\bar{\theta} e_B},$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_0} = \frac{\bar{\theta} e_B (e_G - e_B)}{\rho q_B - \bar{\theta} e_B} < 0,$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_1} = \frac{e_B (\bar{\theta} e_G (e_G - 2e_M + e_B) + \rho q_G (e_G - e_B))}{2\rho (q_B e_G - q_G e_B)},$$

$$E^{BMmc} - E^{BGf}|_{\alpha=\alpha_1} < 0 \quad \text{if } e_M < \frac{e_G + e_B}{2} + \frac{\rho q_G (e_G - e_B)}{2\bar{\theta} e_G},$$

$$\text{with } \frac{e_G + e_B}{2} + \frac{\rho q_G (e_G - e_B)}{2\bar{\theta} e_G} < \frac{\rho q_B}{c + \bar{\theta}} \quad \text{if } c > -\bar{\theta} + \frac{2\rho q_B \bar{\theta} e_G}{\bar{\theta} e_G (e_G + e_B) + \rho q_G (e_G - e_B)}$$

and $E^{BMmc} - E^{BGf}$ is a linear function of $\alpha$. Therefore, $E^{BMmc} < E^{BGf}$ as soon as $e_M$ is lower than the above thresholds and $c$ sufficiently high. However, an increase in $e_M$ improves $E^{BMmc}$ (since $\partial E^{BMmc}/\partial e_M = \alpha$) and can then lead to $E^{BMmc} > E^{BGf}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).

* For $\max\{\alpha_0, \alpha_4\} < \alpha < \alpha_1$, the market with B and M is covered (c) while the market with B and G is fragmented (f). A necessary condition for $E^{BMc} > E^{BGf}$ is $d_B^{BMc} < d_B^f$, that is satisfied if $\alpha > \hat{\alpha} \equiv \frac{-\bar{\theta} e_B (e_M - e_B)}{(\bar{\theta} e_B + 3\rho q_B)(e_M - e_B) + 2ce_B e_M} > 0$, with

$$\begin{aligned} \hat{\alpha}|_{e_M=0} &= \alpha_4|_{e_M=0} = \frac{-\bar{\theta} e_B}{\bar{\theta} e_B + 3\rho q_B}, \\ \left. \frac{\partial \hat{\alpha}}{\partial e_M} \right|_{e_M=0} &= \frac{\bar{\theta}(-2ce_B + 3\bar{\theta} e_B + 6\rho q_B)}{(\bar{\theta}e_B + 3\rho q_B)^2}, \\ \left. \frac{\partial \alpha_4}{\partial e_M} \right|_{e_M=0} &= \frac{-2c\bar{\theta} e_B}{(\bar{\theta}e_B + 3\rho q_B)^2} > \left. \frac{\hat{\alpha}}{\partial e_M} \right|_{e_M=0} \quad \text{if } \bar{\theta} e_B + 2\rho q_B > 0. \end{aligned}$$

Therefore, $\alpha > \alpha_4 > \hat{\alpha}$ and $d_B^{BMc} < d_B^f$ if $\bar{\theta} e_B + 2\rho q_B > 0$. However, we have

$$\begin{aligned} E^{BMc} - E^{BGf} &= \frac{-\bar{\theta} e_B (e_B + 2e_M - 3e_G) + \alpha [-2e_B (\bar{\theta} e_M + \rho q_B - ce_M) + (\bar{\theta} e_B + \rho q_B)(3e_G - e_B)]}{-6\bar{\theta} e_B}, \\ E^{BMc} - E^{BGf}|_{\alpha=\alpha_0} &= \frac{\bar{\theta} e_B (3e_G - 2e_M - e_B) + \rho q_B (e_M - e_B) + ce_B e_M}{-3\bar{\theta} e_B + 3\rho q_B}, \end{aligned}$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_0} < 0 \quad \text{if } e_M < \frac{-e_B(3\bar{\theta} e_G - \bar{\theta} e_B - \rho q_B)}{-2\bar{\theta} e_B + \rho q_B + ce_B},$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_1} = \frac{(\bar{\theta} e_G + \rho q_G)(3e_G - 2e_M - e_B)e_B + 2\rho q_B (e_M - e_B)e_G + 2ce_B e_M e_G}{6(\rho q_B e_G - \rho q_G e_B)},$$

$$E^{BMc} - E^{BGf}|_{\alpha=\alpha_1} < 0 \quad \text{if } e_M < \frac{-(\bar{\theta} e_G + \rho q_G)(3e_G - e_B)e_B + 2\rho q_B e_B e_G}{-2e_B(\bar{\theta} e_G + \rho q_G) + 2\rho q_B e_G + 2ce_B e_G}$$

and $E^{BMc} - E^{BGf}$ is a linear function of $\alpha$. Therefore, $E^{BMc} < E^{BGf}$ as soon as $e_M$ is lower than both of the above thresholds, which increase with $c$. However, an increase in $e_M$ can improve $E^{BMc}$:

$$\frac{\partial E^{BMc}}{\partial e_M} = \frac{\alpha (\bar{\theta} - c) + \bar{\theta}}{3\bar{\theta}} > 0$$

if $\alpha(\bar{\theta} - c) + \bar{\theta} > 0$. This can then lead to $E^{BMc} > E^{BGf}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).
(2) The case with $\max\{\alpha_1, \alpha_4\} \le \alpha < \alpha_2$

The market with B and M is covered (c), while the market with B and G is covered with a corner solution (cc). We have

$$ 
\begin{aligned}
E^{BMc} - E^{BGcc} &= \frac{2\bar{\theta} e_G(e_M - e_B) - \alpha(\bar{\theta} e_G(3e_G - 2e_M - e_B) + 3\rho q_G(e_G - e_B) + 2ce_M e_G)}{6\bar{\theta} e_G}, \\
E^{BMc} - E^{BGcc}|_{\alpha=\alpha_1} &= \frac{(\bar{\theta} e_G + \rho q_G)(3e_G - 2e_M - e_B)e_B + 2\rho q_B(e_M - e_B)e_G + 2ce_B e_M e_G}{6(\rho q_B e_G - \rho q_G e_B)} < 0, \\
\text{if } e_M < \frac{-(\bar{\theta} e_G + \rho q_G)(3e_G - e_B)e_B + 2\rho q_B e_B e_G}{-2e_B(\bar{\theta} e_G + \rho q_G) + 2\rho q_B e_G + 2ce_B e_G}.
\end{aligned}
$$

$E^{BMc} - E^{BGcc}$ is a decreasing function of $\alpha$, negative for $\alpha = \alpha_1$ as soon as $e_M$ is lower than the above threshold, which increases with c. In this case, $E^{BMc} < E^{BGcc}$. However, an increase in $e_M$ improves $E^{BMc}$ if $\alpha(\bar{\theta} - c) + \bar{\theta} > 0$ and can then lead to $E^{BMc} > E^{BGcc}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).

(3) The case with $\max\{\alpha_2, \alpha_4\} \le \alpha \le 1$

The markets with B and M and with B and G are covered (c). We have

$$ E^{BMc} - E^{BGc} = \frac{-\bar{\theta}(e_G - e_M) - \alpha(\bar{\theta} e_G + \rho q_G - (\bar{\theta} e_M + \rho q_B - ce_M))}{3\bar{\theta}} < 0 \quad \text{if } \bar{\theta} e_G + \rho q_G > \bar{\theta} e_M + \rho q_B. $$

## H.2 | The environmental quality with Products M and G

The environmental quality in the covered market with Products M and G is

$$ E^{MGc} = \frac{e_G + 2e_M}{3} + \alpha \frac{\bar{\theta} e_G + \rho q_G - \bar{\theta} e_M - \rho q_B + ce_M}{3\bar{\theta}}. $$

(1) The case with $\alpha_0 < \max\{\alpha_5, \alpha_6\} < \alpha < \alpha_1$

The market with Products B and G is initially fragmented. We have

$$ 
\begin{aligned}
E^{MGc} - E^{BGf} &= \frac{-e_G + 4e_M - 3e_B}{6} + \alpha \frac{-\bar{\theta} e_B(e_G + 2e_M - 3e_B) - \rho q_B(3e_G - e_B) + 2\rho q_G e_B + 2ce_M e_B}{6\bar{\theta} e_B}, \\
E^{MGc} - E^{BGf}|_{\alpha=\alpha_0} &= \frac{\bar{\theta} e_B e_G + \rho q_B e_G - \rho q_G e_B - (c + \bar{\theta})e_B e_M + 2\rho q_B(e_M - e_B)}{3(\rho q_B - \bar{\theta} e_B)}, \\
E^{MGc} - E^{BGf}|_{\alpha=\alpha_1} &= \frac{\bar{\theta} e_G e_B(e_G + 2e_M - 3e_B) - \rho q_G e_B(e_G + 4e_M - 3e_B) + 2\rho q_B e_G(e_G + 2e_M - 2e_B) - 2ce_G e_M e_B}{6(\rho q_B e_G - \rho q_G e_B)}.
\end{aligned}
$$

As $E^{MGc} - E^{BGf}$ is monotonous and positive when $\alpha = \alpha_0$ and $\alpha_1$ as soon as $\bar{\theta} < \bar{\theta}_1$, it is positive for any $\alpha \in [\alpha_0, \alpha_1]$ and any $e_M \ge 0$.

(2) The case with $\max\{\alpha_1, \alpha_5, \alpha_6\} < \alpha < \alpha_2$

The market with Products B and G is initially covered with a corner solution. We have
(2) The case with $\max\{\alpha_1, \alpha_4\} \le \alpha < \alpha_2$

The market with B and M is covered (c), while the market with B and G is covered with a corner solution (cc). We have

$$ 
\begin{aligned}
E^{BMc} - E^{BGcc} &= \frac{2\bar{\theta} e_G(e_M - e_B) - \alpha(\bar{\theta} e_G(3e_G - 2e_M - e_B) + 3\rho q_G(e_G - e_B) + 2ce_M e_G)}{6\bar{\theta} e_G}, \\
E^{BMc} - E^{BGcc}|_{\alpha=\alpha_1} &= \frac{(\bar{\theta} e_G + \rho q_G)(3e_G - 2e_M - e_B)e_B + 2\rho q_B(e_M - e_B)e_G + 2ce_B e_M e_G}{6(\rho q_B e_G - \rho q_G e_B)} < 0, \\
\text{if } e_M < \frac{-(\bar{\theta} e_G + \rho q_G)(3e_G - e_B)e_B + 2\rho q_B e_B e_G}{-2e_B(\bar{\theta} e_G + \rho q_G) + 2\rho q_B e_G + 2ce_B e_G}.
\end{aligned}
$$

$E^{BMc} - E^{BGcc}$ is a decreasing function of $\alpha$, negative for $\alpha = \alpha_1$ as soon as $e_M$ is lower than the above threshold, which increases with c. In this case, $E^{BMc} < E^{BGcc}$. However, an increase in $e_M$ improves $E^{BMc}$ if $\alpha(\bar{\theta} - c) + \bar{\theta} > 0$ and can then lead to $E^{BMc} > E^{BGcc}$ for a high $e_M$ (in the limit of the region $(e_M, \alpha)$ compatible with an SNPE with B and M).

(3) The case with $\max\{\alpha_2, \alpha_4\} \le \alpha \le 1$

The markets with B and M and with B and G are covered (c). We have

$$ E^{BMc} - E^{BGc} = \frac{-\bar{\theta}(e_G - e_M) - \alpha(\bar{\theta} e_G + \rho q_G - (\bar{\theta} e_M + \rho q_B - ce_M))}{3\bar{\theta}} < 0 \quad \text{if } \bar{\theta} e_G + \rho q_G > \bar{\theta} e_M + \rho q_B. $$

## H.2 | The environmental quality with Products M and G

The environmental quality in the covered market with Products M and G is

$$ E^{MGc} = \frac{e_G + 2e_M}{3} + \alpha \frac{\bar{\theta} e_G + \rho q_G - \bar{\theta} e_M - \rho q_B + ce_M}{3\bar{\theta}}. $$

(1) The case with $\alpha_0 < \max\{\alpha_5, \alpha_6\} < \alpha < \alpha_1$

The market with Products B and G is initially fragmented. We have

$$ 
\begin{aligned}
E^{MGc} - E^{BGf} &= \frac{-e_G + 4e_M - 3e_B}{6} + \alpha \frac{-\bar{\theta} e_B(e_G + 2e_M - 3e_B) - \rho q_B(3e_G - e_B) + 2\rho q_G e_B + 2ce_M e_B}{6\bar{\theta} e_B}, \\
E^{MGc} - E^{BGf}|_{\alpha=\alpha_0} &= \frac{\bar{\theta} e_B e_G + \rho q_B e_G - \rho q_G e_B - (c + \bar{\theta})e_B e_M + 2\rho q_B(e_M - e_B)}{3(\rho q_B - \bar{\theta} e_B)}, \\
E^{MGc} - E^{BGf}|_{\alpha=\alpha_1} &= \frac{\bar{\theta} e_G e_B(e_G + 2e_M - 3e_B) - \rho q_G e_B(e_G + 4e_M - 3e_B) + 2\rho q_B e_G(e_G + 2e_M - 2e_B) - 2ce_G e_M e_B}{6(\rho q_B e_G - \rho q_G e_B)}.
\end{aligned}
$$

As $E^{MGc} - E^{BGf}$ is monotonous and positive when $\alpha = \alpha_0$ and $\alpha_1$ as soon as $\bar{\theta} < \bar{\theta}_1$, it is positive for any $\alpha \in [\alpha_0, \alpha_1]$ and any $e_M \ge 0$.

(2) The case with $\max\{\alpha_1, \alpha_5, \alpha_6\} < \alpha < \alpha_2$

The market with Products B and G is initially covered with a corner solution. We have
<figure><img src="image_35.png" /></figure>

$$E^{MGc} - E^{BGcc} = \frac{e_G + 2e_M - 3e_B}{3} - \alpha \frac{2e_G(\bar{\Theta} e_M + \rho q_B - ce_M) + (e_G - 3e_B)(\bar{\Theta} e_G + \rho q_G)}{6\bar{\Theta} e_G},$$

$$E^{MGc} - E^{BGcc} > 0 \quad \text{if } \alpha < \tilde{\alpha} \equiv \frac{2\bar{\Theta} e_G (e_G + 2e_M - 3e_B)}{2e_G (\bar{\Theta} e_M + \rho q_B - ce_M) + (e_G - 3e_B)(\bar{\Theta} e_G + \rho q_G)},$$

$$\text{with } \frac{\partial \tilde{\alpha}}{\partial e_M} > 0 \quad \text{and} \quad \tilde{\alpha} < 1 \quad \text{if } e_M < \frac{\rho q_B}{\bar{\Theta} + c} - \frac{(\bar{\Theta} e_G - \rho q_G)(e_G - 3e_B)}{2(\bar{\Theta} + c)e_G} < \frac{\rho q_B}{\bar{\Theta} + c} \quad \text{if } \bar{\Theta} e_G > \rho q_G,$$

$$E^{MGc} - E^{BGcc}|_{e_M=0} > 0 \quad \text{if } \alpha < \tilde{\alpha}|_{e_M=0} \quad \text{when } \bar{\Theta} < \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B} \quad \text{or if } \bar{\Theta} \ge \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B},$$

$$\frac{\partial(E^{MGc} - E^{BGcc})}{\partial e_M} = \frac{(2 - \alpha)\bar{\Theta} + \alpha c}{3} > 0.$$

Therefore $E^{MGc} - E^{BGcc}$ is positive when $\max\{\alpha_1, \alpha_5, \alpha_6\} < \alpha < \min\{\alpha_2, \tilde{\alpha}\}$. It can be negative only in the specific case where $\tilde{\alpha} < \alpha < \alpha_2$, which requires that $e_M$ is close to zero and $\bar{\Theta} < \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B} < \bar{\Theta}_1$.

(3) The case with $\max\{\alpha_2, \alpha_5, \alpha_6\} < \alpha$

The market with Products B and G is initially covered with an interior solution. We have

$$E^{MGc} - E^{BGcc} = \frac{(2 - \alpha)\bar{\Theta}(e_M - e_B) + \alpha c e_M}{3\bar{\Theta}} > 0 \quad \forall \alpha, \forall e_M.$$

The resulting covered market with Products M and G is unambiguously of better environmental quality, except in the specific case of low $\bar{\Theta}$ and $e_M$ close to 0, and $\tilde{\alpha} < \alpha < \alpha_2$. In addition, an increase in $\alpha$ and $e_M$ improves environmental quality $E^{MGc}$ if

$$\frac{\partial E^{MGc}}{\partial \alpha} = \frac{\bar{\Theta} e_G + \rho q_G - \bar{\Theta} e_M - \rho q_B + c e_M}{3\bar{\Theta}} > 0 \quad \text{if } \bar{\Theta} e_G + \rho q_G > \bar{\Theta} e_M + \rho q_B,$$

$$\frac{\partial E^{MGc}}{\partial e_M} = \frac{1}{3} \left( 2 + \alpha \left( \frac{\bar{\Theta} - c}{\bar{\Theta}} \right) \right) > 0 \quad \text{if } \alpha > 0 \quad \text{when } c < \bar{\Theta},$$

$$\text{or } \alpha < \alpha_7 \equiv 2\bar{\Theta} / (c - \bar{\Theta}) \quad \text{when } c > \bar{\Theta} \text{ with } \alpha_7 \le 1 \quad \text{if } c > 3\bar{\Theta}.$$
<figure><img src="image_35.png" /></figure>

$$E^{MGc} - E^{BGcc} = \frac{e_G + 2e_M - 3e_B}{3} - \alpha \frac{2e_G(\bar{\theta} e_M + \rho q_B - ce_M) + (e_G - 3e_B)(\bar{\theta} e_G + \rho q_G)}{6\bar{\theta} e_G},$$

$$E^{MGc} - E^{BGcc} > 0 \quad \text{if } \alpha < \tilde{\alpha} \equiv \frac{2\bar{\theta} e_G (e_G + 2e_M - 3e_B)}{2e_G (\bar{\theta} e_M + \rho q_B - ce_M) + (e_G - 3e_B)(\bar{\theta} e_G + \rho q_G)},$$

$$\text{with } \frac{\partial \tilde{\alpha}}{\partial e_M} > 0 \quad \text{and} \quad \tilde{\alpha} < 1 \quad \text{if } e_M < \frac{\rho q_B}{\bar{\theta} + c} - \frac{(\bar{\theta} e_G - \rho q_G)(e_G - 3e_B)}{2(\bar{\theta} + c)e_G} < \frac{\rho q_B}{\bar{\theta} + c} \quad \text{if } \bar{\theta} e_G > \rho q_G,$$

$$E^{MGc} - E^{BGcc}|_{e_M=0} > 0 \quad \text{if } \alpha < \tilde{\alpha}|_{e_M=0} \quad \text{when } \bar{\theta} < \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B} \quad \text{or if } \bar{\theta} \ge \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B},$$

$$\frac{\partial(E^{MGc} - E^{BGcc})}{\partial e_M} = \frac{(2 - \alpha)\bar{\theta} + \alpha c}{3} > 0.$$

Therefore $E^{MGc} - E^{BGcc}$ is positive when $\max\{\alpha_1, \alpha_5, \alpha_6\} < \alpha < \min\{\alpha_2, \tilde{\alpha}\}$. It can be negative only in the specific case where $\tilde{\alpha} < \alpha < \alpha_2$, which requires that $e_M$ is close to zero and $\bar{\theta} < \frac{\rho q_G}{e_G} + \frac{2\rho q_B}{e_G - 3e_B} < \bar{\theta}_1$.

(3) The case with $\max\{\alpha_2, \alpha_5, \alpha_6\} < \alpha$

The market with Products B and G is initially covered with an interior solution. We have

$$E^{MGc} - E^{BGcc} = \frac{(2 - \alpha)\bar{\theta}(e_M - e_B) + \alpha c e_M}{3\bar{\theta}} > 0 \quad \forall \alpha, \forall e_M.$$

The resulting covered market with Products M and G is unambiguously of better environmental quality, except in the specific case of low $\bar{\theta}$ and $e_M$ close to 0, and $\tilde{\alpha} < \alpha < \alpha_2$. In addition, an increase in $\alpha$ and $e_M$ improves environmental quality $E^{MGc}$ if

$$\frac{\partial E^{MGc}}{\partial \alpha} = \frac{\bar{\theta} e_G + \rho q_G - \bar{\theta} e_M - \rho q_B + c e_M}{3\bar{\theta}} > 0 \quad \text{if } \bar{\theta} e_G + \rho q_G > \bar{\theta} e_M + \rho q_B,$$

$$\frac{\partial E^{MGc}}{\partial e_M} = \frac{1}{3} \left( 2 + \alpha \left( \frac{\bar{\theta} - c}{\bar{\theta}} \right) \right) > 0 \quad \text{if } \alpha > 0 \quad \text{when } c < \bar{\theta},$$

$$\text{or } \alpha < \alpha_7 \equiv 2\bar{\theta} / (c - \bar{\theta}) \quad \text{when } c > \bar{\theta} \text{ with } \alpha_7 \le 1 \quad \text{if } c > 3\bar{\theta}.$$