3.1. Fourth stage – output

In this period, firms observe the emission fee $t>0$ that the regulator sets in the third stage, their own abatement efforts in the second stage, $z_{i}$ and $z_{j}$, and the EG's collaboration effort. Every firm $i$ then solves problem (1). The following lemma identifies equilibrium output and profits in this stage.

Intuitively, profits are increasing in the abatement effort that the firm chooses in the third stage, $z_{i}$, given that abatement provides: (i) a tax-saving benefit, since $z_{i}$ reduces emissions for a given emission fee $t$; and (ii) a public image benefit, since a cleaner production attracts more customers. Similarly, an increase in public image, $\lambda$, increases profits, since the firm's demand increases. In contrast, profits decrease in the abatement effort from firm $i$'s rival, $z_{j}$, since a better public image reduces firm $i$'s sales, i.e., a business stealing effect. Finally, note that firm $i$'s output $q_{i}(t)$ is positive as long as its abatement is relatively larger than its rival's.

When analyzing the EG's collaboration effort in the first stage of the game, we show that it chooses a symmetric collaboration profile, $b_{i}=b_{j}$, which entails that both firms invest the same amount in abatement, $z_{i}=z_{j}$. Inserting this property in our results from lemma 1, we can claim that, along the equilibrium path, every firm chooses an output level $q_{i}(t)=({a+\lambda z_{i}-(c+t)}/{3})$ which is positive if and only if $z_{i}>({t-( a-c)}/{\lambda })$.Footnote ²⁰

Firms’ output decisions in the last stage are not directly affected by the presence of the EG in previous periods. However, the EG affects the firm's incentives to invest in green R&D in the second stage, thus impacting its equilibrium profits in the fourth stage.

3.2. Third stage – emission fee

In the third stage, the regulator anticipates the output function $q_{i}(t)$ that firms will choose in the subsequent stage, and solves problem (2). Lemma 2 identifies the equilibrium emission fee.

Lemma 2 In the third stage, the regulator sets an emission fee,

\[ t(Z)=\frac{2(a-c)(4d-1)-Z\left[ \lambda +4d(3-\lambda )\right] }{4(1+2d)}, \]

which is positive if and only if $Z<({(a-c)(4d-1)}/{4d(3-\lambda )+\lambda })\equiv \tilde {Z}$. In addition, $t(Z)$ is unambiguously decreasing in the production cost, $c$, and in aggregate abatement, $Z$, unambiguously increasing in public image, $\lambda$, but increasing in the regulator's weight on environmental damage, $d$, if and only if $Z<({2(a-c)}/{ 2-\lambda })\equiv \overline {Z}$, where cutoff $\overline {Z}$ satisfies $\overline {Z}>\tilde {Z}$ under all parameter conditions.

Therefore, the emission fee $t(Z)$ becomes less stringent in the aggregate investment in abatement, $Z$. Intuitively, the regulator anticipates that a higher abatement reduces net emissions in the subsequent stage, thus requiring a less stringent emission fee.Footnote ²¹ This result gives rise to free-riding incentives in firms’ abatement decisions, as every firm can benefit from the tax-savings effect of other firms’ abatement.

In addition, the emission fee is decreasing when firms become more inefficient (higher $c$), since in that context the regulator expects lower production (and pollution) levels in the subsequent stage, calling for less stringent policies. The opposite argument applies when the regulator assigns a larger weight to environmental damage (higher $d$), inducing her to set a more stringent emission fee.Footnote ²² Finally, $t(Z)$ increases in public image since, for a given aggregate abatement $Z$, a higher value of $\lambda$ expands demand. A stronger demand leads firms to increase output in the last stage of the game, thus increasing net emissions. Anticipating this output expansion due to public image, the regulator sets a more stringent fee in the third stage.

3.3. Second stage – abatement

Every firm $i$ anticipates the equilibrium profits it obtains in the fourth stage, $\pi _{i}(t)=( q_{i}(t)) ^{2}+tz_{i}$, and evaluates them at the equilibrium emission fee that the regulator sets in the third stage, $t(Z)$, to obtain equilibrium profit $\pi _{i}(Z)\equiv \pi _{i}(t(Z))$. We can now insert $\pi _{i}(Z)$ in the firm's problem in this stage, as follows:

(3)\begin{equation} \underset{z_{i}\geq 0}{\max }\,\pi _{i}(Z)-\frac{1}{2}\left( \gamma -\theta b_{i}\right) \left( z_{i}\right) ^{2}\text{,} \end{equation}

where $z_{i}+z_{j}<\overline {Z}$, to guarantee that the emission fee is increasing in $d$. Parameter $\gamma \geq 1$ denotes firm $i$'s initial cost of investing in abatement, while term $\gamma -\theta b_{i}$ represents firm $i$'s final (or net) cost of abatement, after reducing it by the EG's collaboration effort, $b_{i}$. Intuitively, when $\theta =0,$ firms’ abatement costs are unaffected by the EG activity, while when $\theta >0,$ firms’ abatement costs decrease in the EG's collaboration effort $b_{i}$. Therefore, parameter $\theta$ captures how sensitive the firm's abatement costs are to the EG's collaboration effort or, alternatively, how effective collaboration is. Finally, note that abatement costs are increasing and convex in $z_{i}$.

Differentiating with respect to $z_{i}$ in problem (3) yields

\[ \frac{\partial \pi _{i}}{\partial t}\frac{\partial t}{\partial Z}=\left( \gamma -\theta b_{i}\right) z_{i}, \]

since $({\partial Z}/{\partial z_{i}})=1$ given that $Z\equiv z_{i}+z_{j}$. Intuitively, firm $i$ increases its abatement until its marginal profit gain (tax savings) coincides with the marginal cost of investing in abatement.

To investigate how firm $i$'s abatement is affected by its rival's decision, $z_{j}$, we differentiate this expression with respect to $z_{j}$, to obtain $({\partial \pi _{i}}/{\partial t})/({\partial t}/{\partial Z})$, where $({\partial t}/{\partial Z})$ is unambiguously negative (see lemma 2), but $({\partial \pi _{i}}/{\partial t})$ is also negative, as shown in lemma 1, if and only if $z_{j}$ is sufficiently low. Therefore, every firm $i$ increases its abatement when its rival increases its own (implying abatement decisions are strategic complements) when $z_{j}$ is relatively low. In this setting, a more severe fee makes firm $j$ less competitive, inducing firm $i$ to further increase $z_{i}$. In contrast, when $z_{j}$ is relatively high, firm $i$ responds to a marginal increase in $z_{j}$ by decreasing its investment in abatement, entailing that investment decisions are strategic substitutes.

We next identify firm $i$'s best response function $z_{i}(z_{j})$ and examine whether abatement efforts are strategic substitutes or complements.

Lemma 3 Firm $i$'s best response function in the second stage is

\[ z_{i}(z_{j})=\frac{2(a-c)\left[ 4d(2A+\lambda )-2+3\lambda \right] -\left[ 2\lambda +\widehat{A}\left[ \lambda ^{2}+4d(2+\lambda (\lambda -1))\right] \right] z_{j}}{16d\left[ 3+5d+2(1+d)\gamma \right] -32dA\lambda -\widehat{A} ^{2}\lambda ^{2}+4(2\gamma +\lambda )-8A^{2}\theta b_{i}}, \]

where $A\equiv 1+2d$ and $\widehat {A}\equiv 3+4d$. In addition:

1. (1) when $b_{i}=0$ and $\lambda =0$, $z_{i}(z_{j})$ is unambiguously decreasing in $z_{j}$;

2. (2) when $b_{i}=0$ and $\lambda >0$, $z_{i}(z_{j})$ is decreasing in $z_{j}$ if and only if $\gamma > \overline {\gamma }$; and

3. (3) when $b_{i},\lambda >0$, $z_{i}(z_{j})$ is decreasing in $z_{j}$ if and only if $\gamma >\overline {\gamma } +\theta b_{i}$ when $b_{i},\lambda >0$,

where $\overline {\gamma }\equiv ({\lambda (9\lambda -4)+8d[ \lambda (4+3\lambda )-6] -16d^{2}(1-\lambda )(5+\lambda )}/{8(1+2d)^{2}})$. Cutoff $\overline {\gamma }$ decreases in the weight that the regulator assigns to environmental damage, $d$, but increases in public image $\lambda$.

Therefore, when the EG is absent and consumers ignore public image, $b_{i}=0$ and $\lambda =0$, firms’ abatement efforts are strategic substitutes for all parameter values. In this setting, an increase in firm $j$'s abatement, $z_{j}$, only produces a benefit on firm $i$'s profits (tax savings), since an increase in abatement efforts today reduce emission fees tomorrow on both firms. Firm $i$, hence, responds to an increase in $z_{j}$ reducing its own abatement $z_{i}$, indicating that firms free-ride each other's abatement efforts.

When the EG is absent but there are public image effects, $b_{i}=0$ and $\lambda >0$, abatement efforts are strategic substitutes if the initial abatement cost is sufficiently high, $\gamma >\overline { \gamma }$. In this context, an increase in $z_{j}$ produces two opposite effects on firm $i$'s profits: (i) a positive tax-savings effect as discussed above; and (ii) a negative business-stealing effect since $\lambda >0$. Hence, when the positive effect in (i) is larger than the negative effect in (ii), abatement efforts remain strategic substitutes. This occurs, in particular, when firms face a relatively high initial abatement cost, yielding a minor business-stealing effect. Otherwise, firms’ abatement efforts become strategic complements. A similar argument applies when both EG and public image are present, but abatement efforts are now strategic substitutes under more restrictive conditions. Intuitively, the EG's collaboration effort enlarges the business-stealing effect, making it less likely that free-riding incentives dominate.

Finally, abatement efforts become strategic substitutes under larger conditions when $d$ increases. In this context, the regulator sets a more stringent emission fee, enlarging the tax saving benefits, which ultimately makes free riding more pervasive. The opposite argument applies when $\lambda$ increases, as the business stealing effect is more significant, making it more difficult for abatement efforts to be strategic substitutes.

The following proposition identifies the equilibrium abatement effort.

Proposition 1 In the second stage, every firm $i$ selects an equilibrium abatement effort,

\[ z_{i}(b_{i},b_{j})=\frac{(a-c)\left[ 4d(2A+\lambda )+3\lambda -2\right] \left[ 4\gamma +\lambda (1-6\lambda )+B-4A\theta b_{j}\right] }{\left[ 4\gamma +\lambda (1-6\lambda )+B\right] F-2A\theta \left[ \lbrack D-32dA\lambda +C\right] b_{j}+b_{i}[D-32dA\lambda +C-8A^{2}\theta b_{j}]]}, \]

where $A\equiv 1+2d$, $B\equiv 4d[ 3+2\gamma -\lambda (3+2\lambda )]$, $C\equiv 4(2\gamma +\lambda )-(3+4d)^{2}\lambda ^{2}$, $D\equiv 16d(3+5d+2(1+d)\gamma )$ and $F\,{\equiv}\, 4\gamma \,{+}\,8d^{2}(7\,{+}\,2\gamma \,{-}\,5\lambda )+3\lambda (1-\lambda )+ 2d[ 18+ 8\gamma -\lambda (11+2\lambda )]$. In addition, $z_{i}(b_{i},b_{j})+z_{j}(b_{i},b_{j})<\overline {Z}$, which guarantees that the emission fee increases in $d$, holds if $\,b_{i}< \overline {b}_{i}$, where cutoff $\ \overline{b}_{i}$ is provided, for compactness, in the appendix.

When we analyze the EG's collaboration effort, we demonstrate that it selects a symmetric collaboration across firms, $b_{i}=b_{j}=b$ (see section 3.4). Inserting this property into our results from proposition 1, we can state that, along the equilibrium path, abatement effort simplifies to

\[ z_{i}(b,b)=\frac{(a-c)\left[ 2-3\lambda -4d(2+4d+\lambda )\right] }{ (3+4d)\lambda ^{2}+(40d^{2}+22d-3)+4b\theta (1+2d)^{2}-4d\left[ 9+14d+4\gamma (1+d)\right] -4\gamma }. \]

Equilibrium abatement is clearly decreasing in the firm's production cost, $c$. Comparative statics for the EG's collaboration efforts, $b_{i}$ and $b_{j}$, and the public image effect, $\lambda$, are, however, less tractable; so figure 1 evaluates equilibrium abatement $z_{i}(b_{i},b_{j})$ at parameter values $a=\gamma =d=1$, $c=0$, $\theta =0.25$, $b_{j}=1$ and $\lambda =0.1$.Footnote ²³ The positive slope of $z_{i}(b_{i},b_{j})$ in figure 1a indicates that firm $i$ increases its abatement in the EG's collaboration, $b_{i}$. In addition, the upward shift in this figure illustrates that, as public image increases (from $\lambda =0.1$ to $\lambda =0.2$), individual abatement effort also increases. Figure 1b indicates that firm $i$ decreases its abatement effort when its rival is more generously helped by the EG (higher $b_{j}$).

Figure 1. (a) Effect of $\lambda$ on $z_{i}(b_{i},b_{j}).$ (b) Effect of $b_{j}$ on $z_{i}(b_{i},b_{j})$.

In the previous section, we found that the optimal emission fee $t(Z)$ is decreasing in aggregate abatement $Z$. Since $z_{i}(b_{i},b_{j})$ is increasing in the EG's collaboration effort $b_{i}$, emission fee $t(Z)$ is then decreasing in $b_{i}$. In short, a more generous collaboration effort from the EG induces a less stringent emission fee or, in other words, collaboration effort and emission fees are strategic substitutes.

3.4. First stage – collaboration effort

In the first stage, the EG anticipates the equilibrium abatement $z_{i}(b_{i},b_{j})$ from proposition 1, and inserts it into the regulator's emission fee from lemma 2, $t(Z)$, obtaining, $t^{\ast }\equiv t(z_{i}(b_{i},b_{j}),z_{j}(b_{i},b_{j}))$. We can then insert this emission fee $t^{\ast }$ into firm $i$'s output function from lemma 1, yielding $q^{\ast }(b_{i},b_{j})\equiv q_{i}(t^{\ast })$. Firm $i$'s net emissions when EG is present can then be expressed as $e_{i}^{EG}\equiv q^{\ast }(b_{i},b_{j})-z_{i}(b_{i},b_{j})$. We also consider firm $i$'s net emissions when EG is absent, $e_{i}^{NoEG}\equiv q^{\ast }(t^{NoEG})-z_{i}(t^{NoEG})$, where emission fee and abatement effort are evaluated at $b_{i}=b_{j}=0$ and superscript $NoEG$ denotes that the EG is absent. The difference,

\[ \textrm{ER}_{i}\equiv e_{i}^\textrm{{NoEG}}-e_{i}^\textrm{{EG}}, \]

represents the emission reduction in firm $i$'s pollution that can be attributed to the EG's presence. Therefore, the EG chooses a collaboration level $b_{i}$ and $b_{j}$ towards firms $i$ and $j$ that solves

(4)\begin{equation} \underset{\frac{\gamma }{\theta }\geq b_{i},b_{j}\geq 0}{\max }\, \left[ \beta \left( \textrm{ER}_{i}\right) ^{{1}/{2}}-c_\textrm{{EG}}\left( b_{i}\right) ^{2}\right] +\left[ \beta \left( \textrm{ER}_{j}\right) ^{{1}/{2}}-c_{\textrm{EG}}\left( b_{j}\right) ^{2}\right], \end{equation}

where the first term captures the benefit to the EG in the form of emissions reduction from firm $i$, which is increasing and concave in $ER_{i}$, indicating that the emission reduction benefit from a larger collaboration effort is increasing in $b_{i}$, but at a decreasing rate. This benefit is scaled by $\beta >0$, which denotes the weight that the EG assigns to emission reduction. The second term measures the cost of exerting collaboration effort, which is increasing and convex in $b_{i}$, and $c_{EG}>0$ represents the cost of effort. To guarantee weakly positive abatement costs, $\gamma -\theta b_{i}\geq 0$, we set an upper bound on the collaboration effort so that $b_{i}$ cannot exceed $\frac {\gamma }{\theta }$.

Differentiating with respect to $b_{i}$ in problem (4) yields an intractable expression, which does not allow for an explicit solution of $b_{i}^{\ast }$. We, nonetheless, provide an analytical discussion of the first-order conditions. The EG's marginal cost of increasing $b_{i}$ is $MC_{i}=2c_{EG}b_{i}$, which is unambiguously positive and increasing in $b_{i}$. The marginal benefit is

\begin{align*} \textrm{MB}_{i} & \equiv \frac{\partial \left[ \beta \left( \left( \textrm{ER}_{i}\right) ^{ {1}/{2}}+\left( \textrm{ER}_{j}\right) ^{{1}/{2}}\right) \right] }{\partial b_{i}} \\ & =\frac{\beta }{2}\left[ \frac{\partial z_{i}}{\partial b_{i}}-\frac{ \partial q^{{\ast} }}{\partial t}\left( \frac{\partial t}{\partial z_{i}}\frac{ \partial z_{i}}{\partial b_{i}}+\frac{\partial t}{\partial z_{j}}\frac{ \partial z_{j}}{\partial b_{i}}\right) \right] ^{{-}3/2} \\ & \quad +\frac{\beta }{2}\left[ \frac{\partial z_{j}}{\partial b_{i}}-\frac{ \partial q^{{\ast} }}{\partial t}\left( \frac{\partial t}{\partial z_{j}}\frac{ \partial z_{j}}{\partial b_{i}}+\frac{\partial t}{\partial z_{i}}\frac{ \partial z_{i}}{\partial b_{i}}\right) \right] ^{{-}3/2}, \end{align*}

since $e_{i}^{NoEG}=q(t^{NoEG})-z_{i}(t^{NoEG})$ do not depend on $b_{i}$. Because $Z\equiv z_{i}+z_{j}$, we have that $({\partial t}/{\partial z_{i}})=({\partial t}/{\partial z_{j}})$, which helps us simplify the above expression of $\textrm {MB}_{i}$ to

\begin{align*} \textrm{MB}_{i} & =\frac{\beta }{2}\left[ \frac{\partial z_{i}}{\partial b_{i}}-\frac{ \partial q^{{\ast} }}{\partial t}\frac{\partial t}{\partial z_{i}}\left( \frac{ \partial z_{i}}{\partial b_{i}}+\frac{\partial z_{j}}{\partial b_{i}}\right) \right] ^{{-}3/2} \\ & \quad +\frac{\beta }{2}\left[ \frac{\partial z_{j}}{\partial b_{i}}-\frac{ \partial q^{{\ast} }}{\partial t}\frac{\partial t}{\partial z_{j}}\left( \frac{ \partial z_{j}}{\partial b_{i}}+\frac{\partial z_{i}}{\partial b_{i}}\right) \right] ^{{-}3/2}. \end{align*}

In a symmetric equilibrium, $z_{i}=z_{j}$, so $\textrm {MB}_{i}$ simplifies to

\[ \textrm{MB}_{i}=\beta \left[ \frac{\partial z_{i}}{\partial b_{i}}\left( 1-2\frac{ \partial q^{{\ast} }}{\partial t}\frac{\partial t}{\partial z_{i}}\right) \right] ^{{-}3/2}, \]

where $({\partial q^{\ast }}/{\partial t})<0$ from lemma 1 and $({ \partial t}/{\partial z_{i}})<0$ from lemma 2, implying that $1-2({ \partial q^{\ast }}/{\partial t})({\partial t}/{\partial z_{i}})>0$. Therefore, if $z_{i}$ increases in $b_{i}$, as shown in section 3.3, $\textrm {MB}_{i}$ is positive.

Intuitively, the first term captures the direct effect of a marginal increase in $b_{i}$, as $b_{i}$ produces an increase in $z_{i}$. The second term, however, measures the indirect effect of $b_{i}$, since it affects both $z_{i}$ and $z_{j}$, which then change the emission fee $t$, and ultimately firms’ output decisions in the last stage. The indirect effect is positive if the increase in $z_{i}$ is smaller (in absolute value) than the decrease in $z_{j}$, implying that the sum of the first and second terms is positive.

To understand if $\textrm {MB}_{i}$ is decreasing in $b_{i}$, we next differentiate it with respect to $b_{i}$, finding

\[ \frac{\partial \textrm{MB}_{i}}{\partial b_{i}}={-}\frac{3\beta }{4}\left[ \frac{ \partial ^{2}z_{i}}{\partial b_{i}^{2}}\left( 1-2\frac{\partial q^{{\ast} }}{ \partial t}\frac{\partial t}{\partial z_{i}}\right) \right] ^{{-}5/2}. \]

If firm $i$'s abatement, $z_{i}$, is convex in $b_{i}$, $({\partial ^{2}z_{i}}/{\partial b_{i}^{2}})>0$, as suggested in figure 1, the $\textrm {MB}_{i}$ is decreasing in $b_{i}$, suggesting that the EG's help exhibits diminishing returns, which guarantees a unique crossing point with the marginal cost, $\textrm {MC}_{i}$.

Figure 2 considers the same parameter values as in figure 1, and depicts the marginal benefit that the EG obtains from collaborating with firm $i$, $\textrm {MB}_{i}$ (the derivative of the first term in (4) with respect to $b_{i}$), and the marginal cost of its collaboration effort, $\textrm {MC}_{i}$ (the derivative of the second term in (4) with respect to $b_{i}$).Footnote ²⁴

Figure 2. $\textrm {MB}$ and $\textrm {MC}$ of the EG.

Marginal benefit $\textrm {MB}_{i}$ is positive but decreasing in $b_{i}$, but marginal cost $\textrm {MC}_{i}$ is increasing in $b_{i}$, suggesting that additional units of effort become more costly for the EG. In our parametric example, $\textrm {MB}_{i}$ and $\textrm {MC}_{i}$ cross at $b_{i}^{\ast }=0.31$, and similar results emerge for other parameter values as reported in table 1.Footnote ²⁵ Other parameter values yield similar results and can be provided by the authors upon request.

Table 1. Equilibrium collaboration effort

Overall, equilibrium collaboration $b_{i}^{\ast }$ increases in the benefit that the EG obtains from emission reductions, $\beta$, in the strength of demand, $a$, in firm sensitivity to the EG's collaboration, $\theta$, and in the public image, $\lambda$. However, $b_{i}^{\ast }$ decreases in the firm's production cost, $c$, its initial abatement cost, $\gamma$, in the weight the regulator assigns to environmental damage, $d$, and the EG's collaboration cost, $c_{EG}$.

The last row evaluates equilibrium results in the special case where consumers ignore public image, $\lambda =0$, showing that the firm has less incentive to invest in abatement, to which the regulator responds with a more stringent fee. The EG in this case anticipates that the firm benefits from the tax-saving effect, but not from business-stealing effects, which makes the firm less receptive to collaboration.

For completeness, the last columns of table 1 report the equilibrium abatement effort, $z_{i}^{\ast }$, emission fee, $t_{i}^{\ast }$, and output level, $q_{i}^{\ast }$, evaluated at each vector of parameter values. As expected, when the EG's collaboration effort $b_{i}^{\ast }$ increases, the firm responds by increasing the investment in abatement, $z_{i}^{\ast }$, which is subsequently responded to by the regulator setting a less stringent fee $t_{i}^{\ast }$ in the third stage, except when the regulator assigns a larger weight to environmental damages (higher $d$) where fees becomes more stringent.

A natural question is whether collaboration effort and emission fees help reduce net emissions, as evaluated in the last column of table 1. Overall, net emissions increase in the EG's collaboration cost, $c_{EG}$, since the EG reduces $b_{i}^{\ast }$; and in the demand strength, $a$, since production (and pollution) increase. In contrast, net emissions decrease in the EG's weight on emissions, $\beta$, since collaboration effort is more intense; in the firm's sensitivity to the EG's collaboration, $\theta$, as its investment in abatement becomes less costly; and in the regulator's weight on environmental damage, $d$, since the regulator sets more stringent fees leading the firm to invest more in abatement.

3.4.1. Exclusive contracts

We now explore the EG's collaboration effort in ’exclusive contracts,’ namely, industries where the EG collaborates with only one firm, $b_{i}>0$ but $b_{j}=0$. In this setting, the EG solves

(4')\begin{equation} \underset{\frac{\gamma }{\theta }\geq b_{i}\geq 0}{\max }\,\left[ \beta \left( \textrm{ER}_{i}\right) ^{{1}/{2}}-c_{EG}\left( b_{i}\right) ^{2} \right] +\beta \left(\textrm{ER}_{j}\right) ^{{1}/{2}}, \end{equation}

which still includes the emission reductions of both firms $i$ and $j$, but the EG's costs now only originate from collaborating with firm $i$.

As in section 3.4, the EG's marginal cost of increasing $b_{i}$ is $\textrm {MC}_{i}=2c_{\textrm {EG}}b_{i}$, which is unambiguously positive and increasing in $b_{i}$. The marginal benefit in this context is still

\[ \textrm{MB}_{i}\equiv \frac{\partial \left[ \beta \left(\textrm{ER}_{i}\right) ^{{1}/{2} }+\beta \left(\textrm{ER}_{j}\right) ^{{1}/{2}}\right] }{\partial b_{i}}. \]

As shown in section 3.4, this marginal benefit simplifies to

\[ \textrm{MB}_{i}=\beta \left[ \frac{\partial z_{i}}{\partial b_{i}}\left( 1-2\frac{ \partial q^{{\ast} }}{\partial t}\frac{\partial t}{\partial z_{i}}\right) \right] ^{{-}3/2}. \]

To understand this result, note that the EG's costs of collaborating with each firm are additively separable. While the benefits of this collaboration are not, the EG internalizes the effect of increasing $b_{i}$ in firm $j$'s emission reduction, yielding the same marginal benefit of increasing $b_{i}$ as in section 3.4 – which holds both when the EG chooses the collaboration pair $(b_{i},b_{j})$ and when it only selects $b_{i}$. Therefore, the EG chooses the same equilibrium collaboration effort when it helps both firms and when it exclusively helps firm $i$. As a consequence, the welfare effects that we identify in subsequent sections apply to exclusive and non-exclusive contracts.

4.1 Benchmark $A$ – regulator, but no EG

To better understand the effect of the EG in the setting of an emission fee, we now consider a context where the EG is absent, but still allow firms to invest in abatement, $z_{i}\geq 0$. That setting is strategically equivalent to our model, but assuming that the EG's collaboration effort is $b_{i}=0$. Equilibrium results in the fourth stage (output decisions) and in the third stage (emission fees) are unaffected since they were not a function of collaboration efforts $b_{i}$ and $b_{j}$, while the abatement decision in the second stage, $z_{i}(b_{i},b_{j})$, is now evaluated at $b_{i}=b_{j}=0$, yielding

\[ z_{i}^{NoEG}\equiv z_{i}(0,0)=\frac{(a-c)\left[ 4d(2+4d+\lambda )+3\lambda -2 \right] }{4\gamma +8d^{2}(7+2\gamma -5\lambda )+3(1-\lambda )\lambda +2d \left[ 18+8\gamma -\lambda (11+2\lambda )\right] }. \]

Equilibrium abatement when the EG is present, $z_{i}(b_{i},b_{j})$, is increasing in $b_{i}$ but decreasing in $b_{j}$; see section 3.3. As a consequence, we cannot analytically rank abatement levels when the EG is present, $z_{i}(b_{i},b_{j})$, and absent, $z_{i}^{NoEG}$. In the parameter values considered in previous sections, we find $z_{i}^{NoEG}=0.19$ and $z_{i}(b_{i},b_{j})=0.22$, thus indicating that firms invest less in abatement when the EG is absent.

Inserting equilibrium abatement, $z_{i}^{NoEG}$, into the regulator's emission fee from lemma 2, we obtain $t^{NoEG}\equiv t(z_{i}^{NoEG},z_{j}^{NoEG})$. For consistency, figure 3 plots this emission fee considering the same parameter values as in previous figures. For comparison purposes, we also depict the fee when the EG is present, $t^{\ast }$, as found in the previous section. Figure 3 indicates that the presence of the EG induces the regulator to set less stringent emission fees. Intuitively, she free-rides off the EG, as the latter helps curb pollution, making environmental policy less necessary.

Figure 3. Effect of EGs on emission fees.

4.2 Benchmark $B$ – EG, but no regulator

Let us now consider an alternative benchmark, where the regulator is absent but the EG is present. In the fourth stage of the game, we obtain the same results as in section 3.1, but evaluated at an emission fee $t=0$ since the regulator is absent, i.e., output $q_{i}(0)=({[ a+\lambda ( 2z_{i}-z_{j}) ] -c}/{3})$ and profits $\pi _{i}(0)=( q_{i}(0)) ^{2}$. The third stage is inconsequential since the regulator is absent. In the second stage, every firm $i$ chooses its investment in abatement by solving a problem analogous to (3), but without the effect of future taxes, as follows:

(3')\begin{equation} \underset{z_{i}\geq 0}{\max }\,\pi _{i}(0)-\frac{1}{2}\left( \gamma -\theta b_{i}\right) \left( z_{i}\right) ^{2}. \end{equation}

Differentiating with respect to $z_{i}$ we obtain firm $i$'s best response function $z_{i}(z_{j})=({4\lambda (a-c-\lambda z_{j})}/{9(\gamma -\theta b_{i})-8\lambda ^{2}})$, thus indicating that abatement efforts of firm $i$ and $j$ are strategic substitutes if $\gamma >({8\lambda ^{2}}/{9})+\theta b_{i}$, thus exhibiting a similar interpretation as in lemma 2. Firm $j$'s best response function, $z_{j}(z_{i})$, is symmetric. Simultaneously solving for $z_{i}$ and $z_{j}$ in $z_{i}(z_{j})$ and $z_{j}(z_{i})$ yields abatement effort

\[ z_{i}^{NoReg}(b_{i},b_{j})=\frac{4\lambda (a-c)\left[ 3(\gamma -\theta b_{j})-4\lambda ^{2}\right] }{27\gamma ^{2}-48\gamma \lambda ^{2}+16\lambda ^{4}+3\theta (8\lambda ^{2}-9\gamma )b_{j}+3\theta b_{i}\left[ 8\lambda ^{2}-9(\gamma -\theta b_{j})\right] }, \]

which collapses to zero when $\lambda =0$. Intuitively, when the regulator is absent and public image effects are nil, firms do not experience either of the two possible benefits of investing in abatement (tax savings and public image), leading them to abstain from investing.

In the first stage, the EG anticipates $z_{i}^{NoReg}(b_{i},b_{j})$ and solves problem (4) to find the equilibrium collaboration effort with firm $i$, $b_{i}^{\ast }$. As in the model with a regulator, the EG's first-order condition yields non-linear expressions that do not provide an explicit solution for $b_{i}^{\ast }$. It is straightforward to numerically show, however, that collaboration efforts are generally higher in this context than when the regulator is present. Intuitively, the EG increases its collaboration to compensate for the void left by the regulator. We use our numerical results below to evaluate welfare gains from regulation alone, from the EG alone, and from both.

4.3 Benchmark $C$ – no EG and no regulator

Finally, we consider a setting in which both the regulator and the EG are absent. In the fourth stage, firms choose the same output as in Benchmark $B$, that is, $q_{i}(0)=({[ a+\lambda ( 2z_{i}-z_{j})] -c}/{3})$, which yields profits $\pi _{i}(0)=( q_{i}(0)) ^{2}$.

In the second stage, every firm solves problem (3) but evaluated at $b_{i}=b_{j}=0$ since the EG is absent, which produces a best response function $z_{i}(z_{j})=({4\lambda (a-c-\lambda z_{j})}/{9\gamma -8\lambda ^{2}})$. Abatement efforts of firm $i$ and $j$ are then strategic substitutes in this setting if $\gamma >({8\lambda ^{2}}/{9})$. Firm $j$'s best response function, $z_{j}(z_{i})$, is symmetric. Simultaneously solving for $z_{i}$ and $z_{j}$ in $z_{i}(z_{j})$ and $z_{j}(z_{i})$ yields abatement effort $z_{i}^{NoReg,NoEG}=({4\lambda (a-c)}/{9\gamma -4\lambda ^{2}})$, which is increasing in public image, $\lambda$, but decreasing in the initial cost of abatement, $\gamma$.

4.4 Welfare comparison

In this section, we evaluate the welfare that emerges in equilibrium when the EG is present and absent, to measure the welfare gain of environmental regulation in each context. In particular, when the EG is absent, the welfare gain from the introduction of environmental regulation is

\[ \textrm{WGR}_{NoEG}=W_{NoEG,R}-W_{NoEG,NR}, \]

where subscript $NoEG$ denotes that the EG is absent, while $R$ ($NR$) indicates that regulation is present (absent, respectively).Footnote ²⁶ A similar definition applies for the welfare benefit from introducing environmental regulation in a setting where the EG is present, $\textrm {WGR}_{EG}=W_{EG,R}-W_{EG,NR}$.

Alternatively, we can evaluate the welfare gains of introducing an EG in an industry not subject to regulation,

\[ \textrm{WGEG}_{NR}=W_{EG,NR}-W_{NoEG,NR}\text{,} \]

or subject to regulation, $\textrm {WGEG}_{R}=W_{EG,R}-W_{NoEG,R}$. Evaluating these welfare gains at our ongoing parameter values, we obtain table 2, which indicates that the introduction of emission fees produces a welfare benefit when the EG is absent, i.e., $\textrm {WGR}_{NoEG}>0$ in the first column for all values of $d$. However, introducing environmental regulation in a context where an EG is already present produces a smaller welfare gain, as illustrated in the second column. Intuitively, firms now face free-riding incentives that they did not have in the absence of regulation. Specifically, the abatement of firm $i$'s rivals decreases the emission fee that the regulator sets in the third stage of the game (tax-saving effect), inducing every firm to reduce its investment in abatement.

Table 2. Welfare gains from regulation and from EG

The third column indicates that introducing an EG in a setting where the regulator is absent improves social welfare since abatement increases. A similar argument applies to the fourth column which examines the welfare gain of introducing an EG where regulation is already present. In this context, the EG's collaboration ameliorates firms’ free-riding incentives from tax savings, yielding a small welfare gain.

In summary, introducing an EG is welfare improving regardless of whether pollution was being tackled with environmental regulation or not. The welfare gain is, as expected, larger when regulation is absent than when firms were already subject to emission fees; and generally larger than that from introducing environmental regulation. Environmental regulation produces a large welfare gain when no EG was present in the industry, and a small welfare gain when an EG is present, since in this case emission fees introduce new free-riding incentives in abatement that firms did not experience in the absence of regulation. Therefore, while the introduction of emission fees should be promoted in all regulatory settings, the introduction of EGs may only be considered when no environmental regulation exists in that industry or pollution damages are particularly severe.

Comparative statics. Table 3 evaluates our results in table 2 at a higher public image ($\lambda =0.2$ rather than $\lambda =0.1$), showing that the welfare gains of introducing regulation (both when the EG is absent and present) are smaller than in table 2 (first and second columns). Intuitively, public image provides firms with stronger incentives to invest in abatement, even when regulation and EG are absent, reducing the amount of pollution that regulation needs to curb. In contrast, the introduction of an EG yields larger welfare gains than in table 2, both when firms are not subject to regulation (third column) and when they are (fourth column).

Table 3. Welfare gains from regulation and from EG, change in $\lambda$

Table 4 examines how results are affected when market size increases from $a=1$ to $a=1.5$, showing that welfare gains are augmented relative to table 2, but maintain their relative ranking, as well as their comparative statics as $d$ increases.

Table 4. Welfare gains from regulation and from EG, change in $a$

Table 5 considers that marginal production cost $c$ increases, from $c=0$ to $c=0.5$, showing that welfare gains are all smaller relative to table 2. Intuitively, the margin $a-c$ shrinks for a given $a$, leading to lower output levels and pollution in the absence of environmental regulation and EGs, implying that the presence of either agent yields smaller welfare effects.

Table 5. Welfare gains from regulation and from EG, change in $c$

In table 6, we increase parameter $\gamma$ in table 2 from $\gamma =1$ to $\gamma =1.5$, keeping all other parameter values unaffected. Table 6 indicates that the welfare gains from introducing regulation are larger than in table 2 while the welfare gains of the EG are smaller, although all keep their relative ranking unaffected.

Table 6. Welfare gains from regulation and from EG, change in $\gamma$

Finally, table 7 decreases parameter $\theta$ from $\theta =1/4$ to $\theta =1/10$. Relative to table 2, the EG yields lower welfare gains, both when regulation is present and absent. The introduction of regulation produces the same welfare gain when the EG is absent (so $\theta$ is inconsequential for the firm's investment decision) but yields a larger welfare gain when the EG is already present than in table 2. Intuitively, when the effectiveness of the EG's collaboration effort, $\theta$, decreases, the EG collaborates less with the firms, as shown in section 3.4, ultimately making regulation more necessary.

Table 7. Welfare gains from regulation and from EG, change in $\theta$