2. The model

Consider a two-country model where there is one firm operating in each country, and each firm generates pollution which creates external costs to society. Firms compete in an oligopolistic industry. The firm operating in the home (foreign) country produces quantity ${q^{h}}$ (${q^{\,f}}$), and faces linear demand ${p^{k}}=\alpha -\beta ({q^{h}}+{q^{\,f}})$, for country $k=h,f$. Costs are given by a function $C^{k}({q^{k}},{e^{k}},e^{-k})$, where ${e^{k}}$ denotes net emissions (i.e., emissions after abatement) coming from the firm operating in country $k$ and $e^{-k}$ net emissions from the other firm. The properties of the cost function are given below. Each firm pays emission taxes ${e^{k}}{t^{k}}$ where ${t^{k}}$ is a per-unit emission tax. Therefore, the profit function for each firm in country $k=h,f$ is given by

(1)\begin{equation} \pi ^{k}={p^{k}}{q^{k}}-C^{k}({q^{k}},{e^{k}},e^{{-}k})-{e^{k}}{t^{k}}. \end{equation}

To capture aspects of conjectural variation in the output market, we follow Dixit (Reference Dixit1986) where the partial derivative of $\pi ^{h}$ with respect to ${q^{h}}$ incorporates $d{q^{\,f}}/d{q^{h}}$. That is, the firm in the home country is assumed to believe that its own output, ${q^{h}}$, is tied to the foreign firm's output ${q^{\,f}}$ via the conjectural variation relation. In other words, the home firm conjectures about the foreign firm's reaction to a change in the output of the home firm (Figuières et al., 2004). We assume that conjectural variations in output are constant and identical across firms, $d {q^{\,f}}/d{q^{h}}=d{q^{h}}/d{q^{\,f}}={v_{q}}$, where ${v_{q}}$ is a constant between $-1$ and $1$. The case of collusion is captured by ${v_{q}}\simeq 1$, the case of perfect competition by ${v_{q}}\simeq -1$, and in the case of Cournot, ${v_{q}}=0,$ since firms take each other's output as given (Ebert, Reference Ebert1992). The reason for assuming ${v_{q}}$ constant is to facilitate the analysis of the individual (separate) role of conjectural variations in abatement and output.

In addition to conjectural variation in output, we incorporate conjectural variation in abatement. Analogous to the case of the output market, the idea here is that net emissions by, say, the home firm, ${e^{h}}$, are tied to net emissions coming from the foreign firm, ${e^{\,f}}$. For example, the home firm may conjecture that the foreign firm pollutes more resulting from more pollution by the home firm. Alternatively, the home firm may conjecture that the foreign firm pollutes less if the home firm pollutes more. This set-up allows the analysis of zero vis-à-vis nonzero conjectural variations in abatement and output, and the study of the implications for policy design in a unified framework of analysis.

With these points in mind, we consider the following cost function for the home firm (an analogous cost function applies to the foreign firm):

(2)\begin{equation} C^{h}({q^{h}},{e^{h}},{e^{\,f}})={c^{h}}{q^{h}}+(\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}})^{2}/2, \end{equation}

where (subscripts denote partial derivatives) $C_{{q^{h}}}^{h}>0$, $C_{{q^{h}}{q^{h}} }^{h}>0$, $C_{{e^{h}}}^{h}<0$, $C_{{e^{h}}{e^{h}}}^{h}>0$, $C_{{q^{h}}{e^{h}}}^{h}<0$, and $\delta^{h} {q^{h}}$ denotes gross pollution where $\delta^{h}$ is the pollution intensity of the home firm which is assumed constant. In (2) the first term represents production costs with constant marginal cost, ${c^{h}}$, and the second term abatement costs. The third term within the abatement cost component captures the extent to which foreign net emissions affect the home firm's abatement costs and net emissions. Examples include the impact on tourism and fishing industry in the Salton Sea area (Hulbert, Reference Hulbert2008), and the impact of acid rain on local and regional agricultural yields and quality, and surface water (e.g., Menz and Seip, Reference Menz and Seip2004; Fang et al., Reference Fang, Wang, Wang, Zhang, Qu and Wang2013; Zhang et al., Reference Zhang, Cheng, Sun, Wang, Zhou and Huang2013). The idea is that an increase (decrease) in net emissions by the foreign firm lowers (raises) abatement costs of the home firm. This is because home takes into account foreign net emissions in its cost structure. For instance, in the case of water pollution it may be relatively less costly to remove solid waste, but additional investment may be needed to remove contaminants coming from pharmaceuticals (International Joint Commission Newsletter, 2013).

The last term in (2) incorporates $d{e^{\,f}}/d{e^{h}}=d{e^{h}}/d{e^{\,f}}={v_{e}}$, where ${v_{e}}$ is a constant between $-1$ and $1$. If the home firm forms a conjecture that the foreign firm will increase (decrease) its own net emissions as a result of the home firm raising net emissions, then firms collude (do not collude) i.e., $d{e^{\,f}}/d{e^{h}}={v_{e}}>(<)0$. In the zero-conjectural-variation case, ${v_{e}}=0$. A positive (negative) ${v_{e}}$ implies that the home firm's marginal abatement costs increase (decrease), meaning that the extent of a change in its total abatement costs, given a change in home's emissions, is relatively larger (smaller). This is because the home firm's total abatement cost function incorporates the increased (decreased) level of emissions coming from the foreign firm; i.e., $MAC^{h}=-C_{{e^{h}}}^{h}=(1+{v_{e}})(\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}})$.Footnote ²

The order of events is as follows. The government in each country sets the emission tax in a Cournot-Nash fashion, meaning governments simultaneously choose their own tax taking each other's tax as given so as to maximize welfare. Then, firms take policy as given and choose the level of emissions and output simultaneously. The model is solved via backward induction. We assume interior solutions throughout in order to analyze the role of varying conjectural variations on output and abatement. The assumption of Cournot-Nash in the choice of taxes by governments is made to focus the analysis on the role of conjectural variations at the firm level, with implications, in turn, on the characterization of policy. Furthermore, the assumption about the simultaneous choice of emissions and output by firms is to put aside issues of strategic choice of abatement and output (e.g., Carlsson, Reference Carlsson2000; Montero, Reference Montero2002) and to focus the analysis on the role of conjectural variations.

3. The equilibrium

For the home firm, maximization of (1) with respect to ${q^{h}}$ and ${e^{h}}$ yields, respectively, the following first-order conditions:

(3)\begin{align} {p^{h}}-\beta {q^{h}}(1+{v_{q}})-{c^{h}}-\delta^{h} (\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}}) & =0 \end{align}

(4)\begin{align} - {t^{h}} -(\delta^{h}{q^{h}}-{e^{h}}-{e^{\,f}})({-}1-{v_{e}}) & =0, \end{align}

where ${p^{h}}=\alpha -\beta ({q^{h}}+{q^{\,f}})$, ${v_{q}}$ in (3) captures the degree of market power associated with conjectural variation in output, and ${v_{e}}$ in (4) the component of the marginal abatement cost function associated with conjectural variation. Equations (3) and (4), along with an analogous set of equations for the foreign firm, implicitly determine the equilibrium vector $q^{h\ast }$, $q^{f\ast }$, $e^{h\ast }$ and $e^{f\ast }$ as functions of the emission tax ${t^{h}}$, ${t^{\,f}}$ and conjectural variations parameters ${v_{q}}$, ${v_{e}}$.

Substitution of (4) into (3), undertaking a similar substitution using the first-order conditions of the foreign firm, and then solving for the equilibrium level of output for the home firm, gives

(5)\begin{equation} q^{h}=\frac{( \alpha -{c^{h}}-({\delta^{h} {t^{h}} }/{(1+{v_{e}}))}) (2+{v_{q}} )-( \alpha -{c^{\,f}}-({\delta^{\,f}{t^{\,f}}}/{(1+{v_{e}}))}) }{\beta \left( (2+{v_{q}} )^{2}-1\right) }, \end{equation}

where an analogous expression applies to the output level of the foreign firm, ${q^{\,f}}$:

(6)\begin{equation} q^{\,f}=\frac{( \alpha -{c^{\,f}}-({\delta^{\,f}{t^{\,f}}}/{(1+{v_{e}}))}) (2+{v_{q}})-( \alpha -{c^{h}}-({\delta^{h} {t^{h}} }/{(1+{v_{e}}))})}{\beta ( (2+{v_{q}} )^{2}-1) }. \end{equation}

Equations (5) and (6) point to the role of conjectural variations in output and abatement in a unified framework. For example, under zero conjecture in abatement (${v_{e}}=0$), the expression for output still accounts for the role of conjectural variations in output (${v_{q}}\not =0$). And with zero conjectural variaton in output and abatement (i.e., ${v_{q}}=0$, ${v_{e}}=0$), the Cournot closed-form solution is obtained.

Assumption 3.1 (i), (ii) and (iii) ensure, respectively, positive home output, ${q^{h}}$, foreign output, ${q^{\,f}}$, and aggregate output, $Q={q^{h}}+{q^{\,f}}$.

4. Output and conjectural variations

Using definition 2.2 and equation (5), an increase (decrease) in ${v_{q}}$ results in lower (higher) home output due to the relatively higher (lower) degree of collusion in output; that is, $\partial {q^{h}}/\partial {v_{q}}<0$. Analogous results apply to ${q^{\,f}}$. The reason higher ${v_{q}}$ implies lower output is captured by the term $(1+{v_{q}})\beta {q^{h}}$ in (3), which indicates the role which the degree of conjectural variation plays in the market power of the home firm.

The analysis with respect to ${v_{e}}$ indicates that home output may fall or rise; that is, the sign of $\partial {q^{h}}/\partial {v_{e}}$ is ambiguous. This is because of two effects. On the one hand, as the conjectural variation ${v_{e}}$ rises (i.e., starting at any ${v_{e}}$, a higher degree of collusion in abatement), home's marginal abatement costs increase and, as a result, home raises its own output to equate marginal abatement costs to the existing tax, ${t^{h}}$. The key element here is the presence of the emission tax which leads to an increase in output. On the other, as the foreign firm also experiences an increase in ${v_{e}}$ (i.e., by assumption conjectural variation in abatement is identical for the home and foreign firm), foreign also raises its own output to equate its marginal abatement costs to the tax, ${t^{\,f}}$. As a result, home reacts strategically by lowering output because of the oligopolistic interaction. If this latter effect is relatively small, then home output rises with a small increase in ${v_{e}}$. In particular, using (5) gives

(7)\begin{align} \frac{\partial {q^{h}}}{\partial {v_{e}}}& =\frac{1}{(1+{v_{e}})^{2}((2+{v_{q}})^{2}-1)\beta }\left[ \delta^{h} {t^{h}} (2+{v_{q}})-\delta^{\,f}{t^{\,f}}\right] >0\nonumber\\ & \Leftrightarrow\delta^{\,f}{t^{\,f}}<\delta^{h} {t^{h}} (2+{v_{q}}), \end{align}

where $\delta^{h} {t^{h}} (2+{v_{q}})$ denotes the home output-increasing effect, and $-\delta ^{\,f} {t^{\,f}}$ the home output-reducing effect as a result of the oligopolistic interaction. Figure 1 illustrates these results, where the right-hand-side and left-hand-side planes illustrate, respectively, that the home output-increasing and output-decreasing effect dominates. An analogous result applies to the foreign country, where using (6) gives: $\partial {q^{\,f}}/\partial {v_{e}}=(\delta ^{\,f}{t^{\,f}}(2+{v_{q}})-\delta^{h} {t^{h}} )/((1+{v_{e}})^{2}((2+{v_{q}} )^{2}-1)\beta )$. The key mechanism in explaining the sign of (7) is the presence of an emission tax, where an increase in marginal abatement costs (due to an increase in ${v_{e}}$) results in higher emissions and output as the home firm ensures that the existing tax is equal to marginal abatement costs. In section 5 we delve into the implications on abatement and aggregate emissions. Next, we consider aggregate output, $Q={q^{h}}+{q^{\,f}}=((\alpha -{c^{h}}-\delta^{h} {t^{h}} /(1+{v_{e}} ))+(\alpha -{c^{\,f}}-\delta ^{\,f}{t^{\,f}}/(1+{v_{e}})))/\beta (3+{v_{q}})$, where it can be shown that output: (i) falls with an increase in ${v_{q}}$ (i.e., $\partial Q/\partial {v_{q}} <0$); and (ii) rises with an increase in ${v_{e}}$ (i.e., $\partial Q/\partial {v_{e}}>0)$. The intuition for (i) is due to the higher degree of collusion. And the intuition for (ii) is that a higher degree of collusion in abatement means that the output-increasing effect is always large enough. The implication here is that in the presence of nonzero conjectural variation in abatement, equilibrium aggregate output could potentially be larger even in the presence of collusion in the output market.

Figure 1. Home output, $q^{h}$, and conjectural variation in abatement, $v_e$.

4.1. Output, taxation and conjectural variations

Consider the effect of the tax on home output; in particular, differentiation of (5) yields

(8)\begin{equation} \frac{\partial {q^{h}}}{\partial {t^{h}} }=\frac{(-\delta^{h} /(1+{v_{e}}))(2+{v_{q}})}{\beta \left( (2+{v_{q}})^{2}-1\right) }<0\quad \frac{\partial {q^{h}}}{\partial {t^{\,f}}}=\frac{\delta^{\,f} /(1+{v_{e}})}{\beta \left( (2+{v_{q}})^{2}-1\right) }>0. \end{equation}

Results are consistent with the literature (e.g., Requate, Reference Requate, Tietenberg and Folmer2006). First, an increase in the tax in the home (foreign) country renders the home firm relatively less (more) cost competitive and, as a result, home output falls (rises). Second, aggregate output falls with an increase in either the home or foreign tax.

To see the role of ${v_{q}}$, we note that the expression $\partial {q^{h}}/\partial {t^{h}}$ suggests that the effectiveness of an increase in the home tax in reducing home output is lower as the conjectural variation for output, ${v_{q}}$, increases. This is because, with a higher degree of collusion in output, firms enjoy more market power and thus face a demand which is less price sensitive (see appendix B). As a result, it takes a larger tax increase to induce a change in home output. A similar rationale applies to the effect of foreign taxation using the expression $\partial {q^{h}}/\partial {t^{\,f}}$: a higher degree of collusion in output renders the foreign tax less effective in altering home output. As a result, the effects on home output via the oligopolistic interdependence are smaller, too. These results are illustrated in figure 2, where the expressions in (8) are plotted as a function of ${v_{q}}$. For instance, the bottom curve in figure 2 illustrates how a higher degree of collusion is associated with a reduction in the extent to which an increase in the home tax decreases home output. These results will be used in subsequent sections to analyze the relationship between emissions and conjectural variations, and characterize optimal policy. Furthermore, from (8) it can be shown that the effectiveness of a change in the tax in altering output is lower with higher ${v_{e}}$. The reason is that a higher ${v_{e}}$ results in higher marginal abatement costs. As a result, it takes a larger tax increase to induce additional abatement, which reflects on subsequent adjustments in output.Footnote ³ The analysis thus implies that an increase in ${v_{e}}$ reduces the ability of the tax to reduce aggregate output.

Proposition 4.1 For any pollution intensities $\delta^{h}$, $\delta ^{\,f},$ the effectiveness of an increase in the home tax in: (i) decreasing (increasing) per-firm home (foreign) output; and (ii) reducing aggregate output is lower with either an increase in the conjectural variation in output, ${v_{q}}$, or abatement, ${v_{e}}$.

Figure 2. Taxation, home output, $q^{h}$, and conjectural variation, $v_q$.

Proof. See appendix C.

5. Emissions and conjectural variations

Following the analysis in the previous section, changes in aggregate emissions arising from ${v_{q}}$ work through changes in output, which implies that aggregate emissions fall with an increase in ${v_{q}}$ (i.e., $\partial E/\partial {v_{q}}<0$, see appendix D) regardless of any difference in pollution intensities. We expand on the analysis in section 4 and argue that an increase in ${v_{e}}$ lowers per-firm abatement and raises per-firm emissions in the presence of an emission tax, meaning that aggregate emissions rise as ${v_{e}}$ increases (i.e., $\partial E/\partial {v_{e}}>0$, see appendix D).

To see these results, consider marginal abatement costs for the home firm, $MAC^{h}=-C_{{e^{h}}}^{h}=(1+{v_{e}})(\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}})$, where differentiation gives

(9)\begin{equation} dMAC^{h}=(\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}})d{v_{e}}+(1+{v_{e}})( \delta^{h} d{q^{h}}-d{e^{h}}-d{e^{\,f}}) , \end{equation}

where $\delta^{h} {q^{h}}-{e^{h}}-{e^{\,f}}>0$ denotes home-firm abatement. Since the tax remains fixed and the home firm equates marginal abatement costs to the tax, we know that $dMAC^{h}=0$, which implies from (9) that abatement for the home firm falls as ${v_{e}}$ increases; that is, $( \delta^{h} d{q^{h}}/d{v_{e}}-d{e^{h}} /d{v_{e}}-d{e^{\,f}}/d{v_{e}}) <0$. The reduction in home abatement and increase in home emissions arises from the fact that an increase in ${v_{e}}$ raises marginal abatement costs of the home firm, which in the presence of the tax induces the firm to equate marginal abatement costs to the tax. Since an analogous result applies to the foreign firm, the implication here is that aggregate emissions rise given an increase in ${v_{e}},$ i.e., $E={e^{h}}+{e^{\,f}} \Rightarrow \partial E/\partial {v_{e}}>0$. This result holds regardless of any asymmetries in pollution intensities as home and foreign emissions rise.

5.1. Emissions, taxation and conjectural variations

We analyze the role of conjectural variation in output on the extent to which the tax is effective in lowering aggregate emissions by considering the change in total emissions with respect to the tax as a function of ${v_{q}}$. Consider the change in aggregate emissions, $E={e^{h}}+{e^{\,f}}$, with respect to the tax in the home country (i.e., $\partial E/\partial {t^{h}} =\partial {e^{h}} /\partial {t^{h}} +\partial {e^{\,f}}/\partial {t^{h}} )$, where ${t^{h}} /(1+{v_{e}})=\delta^{h} {q^{h}}-{e^{h}} -{e^{\,f}}$, ${t^{\,f}}/(1+{v_{e}})=\delta ^{\,f}{q^{\,f}}-{e^{\,f}}-{e^{h}}$ from firms’ profit-maximization conditions (see appendix E for a detailed derivation):

(10)\begin{equation} \frac{\partial E}{\partial {t^{h}} }=\frac{1}{1+{v_{e}}}\left[ \delta^{h} \frac{\partial {q^{h}}}{\partial {t^{h}} }+\delta^{\,f}\frac{\partial {q^{\,f}}}{\partial {t^{h}} }-\frac{1}{1+{v_{e}}} \right] , \end{equation}

where the first, second and third terms denote, respectively, changes in aggregate emissions via changes in home output, foreign output and abatement induced by the home tax. Next, substitution of $\partial {q^{h}}/\partial {t^{h}}$ and $\partial {q^{\,f}}/\partial {t^{h}}$ into (10) yields $\partial E/\partial {t^{h}}$ as a function of ${v_{q}}$; that is, $f({v_{q}})\equiv \partial E/\partial {t^{h}}$:

(11)\begin{equation} f({v_{q}})=\frac{1}{\beta (1+{v_{e}})^{2}( (2+{v_{q}})^{2}-1) }[ \delta^{h}( -\delta^{h} (2+{v_{q}})+\delta^{\,f}) -\beta ( (2+{v_{q}})^{2}-1)] , \end{equation}

where for any ${v_{q}}$, ${v_{e}}$, equation (11) is negative (positive) if $\delta^{h} \geq \delta ^{\,f}$ ($\delta ^{\,f}>\delta^{h}(2+{v_{q}})+(\beta (2+{v_{q}})^{2}-1)/\delta^{h}$). That is, aggregate emissions fall (rise) with an increase in the home tax if the home (foreign) pollution intensity exceeds foreign's (is sufficiently large). This result arises from (5) and the role of asymmetries in pollution intensities. The reason is that the direction (though not the magnitude) in which foreign and home output change with respect to a tax increase is the same regardless of conjectural variations, thereby pointing to the role of asymmetries in pollution intensities.

Definition 5.2 The foreign firm's pollution intensity coefficient, $\delta ^{\,f}$, is referred to as sufficiently large whenever $\delta ^{\,f}>\bar {\delta }^{\,f}\equiv \delta^{h} (2+{v_{q}}) + \beta ( (2+{v_{q}})^{2} -1)/\delta^{h}$. The home firm's pollution intensity exceeds foreign's whenever $\delta^{h} >\delta ^{\,f}$.

Inspection of (11) gives the following proposition.

Proposition 5.3 For any ${v_{q}}$, ${v_{e}}$, aggregate emissions fall (rise) with an increase in the home tax if the home (foreign) pollution intensity exceeds foreign's (is sufficiently large); i.e., $\delta^{h} >\delta ^{\,f}$ ($\delta ^{\,f}>\bar {\delta }^{\,f}$).

Next, the role of ${v_{q}}$ is examined using (11) through two cases. First, if $\delta^{h} \geq \delta ^{\,f}$, then $f<0$, $f^{\prime }>0$, $f^{\prime \prime }<0$.Footnote ⁴ This means that aggregate emissions fall with taxation at home, but the effectiveness of an increase in the tax in reducing aggregate emissions is lower where there is a higher degree of collusion in output. This is because: (i) home's pollution intensity exceeds foreign's so it offsets any increase in emissions coming from foreign output, which takes place via the increase in ${t^{h}}$; and (ii) the effectiveness of an increase in the tax in reducing emissions is lower due to larger conjectural variations in output (proposition 4.1).

Second, from equation (11) aggregate emissions rise with an increase in the home tax if $\delta ^{\,f}>\bar {\delta }^{\,f}$ regardless of whether the output market is relatively close to the perfectly competitive or collusion case. But aggregate emissions increase by more given an increase in the home tax, if $\delta ^{\,f}>\bar {\delta }^{\,f}$ and the output market is relatively close to the perfectly competitive case, since in this case ${v_{q}}$ is close to $-1$, meaning that output is more sensitive to taxation. This result is illustrated in the top curve of figure 3, where for sufficiently large $\delta ^{\,f}$, lower values of ${v_{q}}$ are associated with higher levels of $f( {v_{q}})$. However, if $\delta ^{\,f}$ is not sufficiently large, then aggregate emissions fall with an increase in the home tax for a range of ${v_{q}}$ relatively close to collusion. In this case foreign's pollution intensity is not sufficiently large and taxation on foreign output is relatively less effective (see appendix F for specific range of $\delta ^{\,f}$). The bottom curve in figure 3 illustrates this result. Importantly, results point to the role of pollution intensities coupled with conjectural variation in output, and suggest the need to look for additional measures to tackle emissions, particularly in cases where there is sufficient asymmetry in pollution intensities.

Proposition 5.4 If conjectural variation in output is small and foreign exhibits a sufficiently large pollution intensity, $\delta ^{\,f}>\bar {\delta }^{\,f}$, then the effectiveness of an increase in the home tax in reducing aggregate emissions is lower.

Figure 3. Aggregate emissions, $E$, taxation and conjectural variation, ${v_{q}}$.

Proof. See appendix F.

6. Welfare

This section looks at the role of conjectural variations in the characterization of the optimal emission tax. The strategy is to first characterize the non-cooperative optimal tax (section 6.1), and then analyze changes in optimal taxation arising from exogenous changes in conjectural variations (sections 6.2 and 6.3). In section 6.2 we show that the non-cooperative tax increases even as output falls with a higher degree of collusion in output (proposition 6.2). The driver is the presence of large asymmetries in pollution intensities, where there are negligible reductions in damages from home emissions (and thus little need to lower the tax), but large enough gains from reduced transboundary pollution. A similar analysis is applied to conjectural variation in abatement in section 6.3, where the non-cooperative tax decreases (proposition 6.3) even as emissions rise in markets with a higher degree of collusion in abatement. The reason is that large asymmetries in pollution intensities allow for control of emissions via lower taxation.

6.1. Non-cooperative tax

The government in the home and foreign countries set taxes simultaneously via welfare maximization, taking each other's taxes as given.

The home country maximizes the following welfare function:

(12)\begin{equation} {W^{h}}={CS^{h}}({q^{h}},{q^{\,f}})+\pi^{h}+{e^{h}}{t^{h}} -\varphi ^{h}({e^{h}}+{e^{\,f}}), \end{equation}

where ${CS^{h}}$ denotes consumer surplus, $\pi ^{h}$ home profits, ${e^{h}}{t^{h}}$ revenue from taxation, and the function $\varphi ^{h}(\cdot)$ damages from foreign (transboundary) and home emissions, where $\varphi ^{h^{\prime }}>0$, $\varphi ^{h^{\prime \prime }}>0$. It is noteworthy that marginal damages are independent from where the emissions occur. An analogous welfare function applies to the foreign country, ${W^{\,f}}$. Differentiation of (12) with respect to ${t^{h}}$ and, analogously, differentiation of ${W^{\,f}}$ with respect to ${t^{\,f}}$ yields two first-order conditions (subscripts denote partial derivatives) $W^{h}_{{t^{h}} }=0$ and $W^{\,f}_{{t^{\,f}}}=0$, and hence a policy vector $t^{h\ast }$, $t^{f\ast }$ as functions of conjectural variations ${v_{q}}$, ${v_{e}}$.

In particular, $W^{h}_{{t^{h}} }=0$ gives

(13)\begin{equation} t^{h\ast }=\varphi ^{h^{\prime }}+\varphi ^{h^{\prime }}e^{\,f}_{{t^{h}} }/e^{h}_{{t^{h}} }+{q^{\,f}}p^{\,f}_{{t^{h}} }/e^{h}_{{t^{h}} }-\beta (1+{v_{q}}){q^{h}}q^{h}_{{t^{h}} }/e^{h}_{t^{h}}, \end{equation}

where consistent with the literature (e.g., Lahiri and Symeonidis, Reference Lahiri and Symeonidis2007; Gautier, Reference Gautier2013) $e^{\,f}_{{t^{h}} }>0$, $e^{h}_{{t^{h}} }<0$, $p^{\,f}_{{t^{h}} }>0$, $q^{h}_{{t^{h}} }<0$ (see derivation of (13) and selected expressions in appendices G and J, respectively). The first and second terms denote, respectively, damages from home emissions which put an upward pressure on the home tax, and transboundary pollution which puts a downward pressure because a lower tax at home lowers emissions coming from the foreign firm. The third and fourth terms denote, respectively, profit-shifting, and output distortion effects, each of which puts a downward pressure on the tax. From (13) the home tax is clearly below marginal damages. We assume positive optimal taxation in each country throughout.

6.2. The non-cooperative tax and conjectural variation, ${v_{q}}$

To explore the role of ${v_{q}}$ on taxation we differentiate the first-order conditions, $W^{h}_{{t^{h}} }({t^{h}} ({v_{q}}),{t^{\,f}}({v_{q}}),{v_{q}})=0$ and $W^{\,f}_{{t^{\,f}}}({t^{h}} ({v_{q}}), {t^{\,f}}({v_{q}}),{v_{q}})=0$, with respect to ${v_{q}}$, which yields the change in the home tax given a small change in conjectural variation in output, $t^{h}_{{v_{q}}}$:

(14)\begin{equation} \eta \hspace{2pt}t^{h}_{{v_{q}}}={-}W^{\,f}_{{t^{\,f}}{t^{\,f}}}W^{h}_{{t^{h}} {v_{q}}}+W^{h}_{{t^{h}} {t^{\,f}}}W^{\,f}_{{t^{\,f}} {v_{q}}}, \end{equation}

where $\eta =W^{\,f}_{{t^{\,f}}{t^{\,f}}}W^{h}_{{t^{h}} {t^{h}} }-W^{\,f}_{{t^{\,f}}{t^{h}} }W^{h}_{{t^{h}} {t^{\,f}}}>0$, $W^{\,f}_{ {t^{\,f}}{t^{\,f}}}<0$, $W^{h}_{{t^{h}} {t^{h}} }<0$ by the concavity of the welfare function and stability of the system. The first term in (14) denotes the adjustment in the home tax resulting from welfare changes in that country. And the second term captures the adjustment in the home tax as a result of adjustments in the foreign country's tax arising from changes in ${v_{q}}$. We assume this latter effect is negligible to focus the analysis on firm interdependence.Footnote ⁵

Assumption 6.1 Cross-country adjustments in the tax are negligible, i.e., $W^{h}_{ {t^{h}} {t^{\,f}}}\simeq 0$.

Under assumption 6.1 the expression in equation (14) reduces to

(15)\begin{equation} -W^{h}_{{t^{h}} {t^{h}} }t^{h}_{{v_{q}}}=\Omega +({t^{h}} -\varphi ^{h^{\prime }})e^{h}_{{t^{h}} {v_{q}} }-\varphi ^{h^{\prime }}e^{\,f}_{{t^{h}} {v_{q}}}-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{q}}}, \end{equation}

where $e^{h}_{{t^{h}} {v_{q}}}>0$, $e^{\,f}_{{t^{h}} {v_{q}}}<0$, $({t^{h}} -\varphi ^{h^{\prime }})<0$, $E_{{v_{q}}}<0$. The sign of $E_{{t^{h}} }$ (i.e., the change in aggregate emissions from a change in the home tax) is ambiguous as previously discussed and depends on relative pollution intensities. $\Omega$ in (15) denotes consumer surplus, profit and output distortion effects. To focus the analysis on the role of asymmetries in pollution intensities and damages from emissions, we assume these effects to be small. The second term denotes the decrease in the tax since damages from home emissions, ${e^{h}}$, fall given an increase in ${v_{q}}$. The third term is positive and captures damages from transboundary pollution. Foreign emissions fall as ${v_{q}}$ increases and so damages from transboundary pollution decrease, thereby reducing the need for lower taxation. If damages from transboundary pollution had risen, then the home tax would have decreased because a lower tax at home lowers foreign emissions. The sign of the fourth term is ambiguous since it depends on the sign of $E_{{t^{h}} }$.

To analyze the aforementioned effects and the sign of $t^{h}_{{v_{q}}}$ in (15), we consider two cases. First, we assume the home firm pollution intensity exceeds foreign's; and second, we assume the foreign firm is relatively pollution intensive.

6.2.1. Case 1: home exceeds foreign's pollution intensity

The home firm's pollution intensity exceeds foreign's, $\delta^{h} >\delta ^{\,f}$, and therefore, from section 5, $E_{{t^{h}} }<0$, $E_{{t^{h}} {v_{q}}}=f^{\prime }>0$. These signs were examined using (11), where $E_{{t^{h}} {v_{q}}}$ denotes the extent to which conjectural variation ${v_{q}}$ affects the effectiveness of an increase in the home tax in reducing aggregate emissions. The focus is on the role of asymmetries in pollution intensities and damages. From (15) the second and third terms are negative and positive, respectively, and on balance negative; i.e., $({t^{h}} -\varphi ^{h^{\prime }})e^{h}_{{t^{h}} {v_{q}}}-\varphi ^{h^{\prime }}e^{\,f}_{{t^{h}} {v_{q}}}<0$. This is because the effects via home's pollution intensity, and thus damages, are large enough since the home country is relatively more pollution intensive. There are two opposing effects. On the one hand, because damages from home emissions are falling, the home tax falls (i.e., $({t^{h}} -\varphi ^{h^{\prime }})e^{h}_{{t^{h}} {v_{q}}}<0$). On the other, as damages from transboundary pollution fall, the need to lower the tax lessens, $-\varphi ^{h^{\prime }}e^{\,f}_{{t^{h}} {v_{q}}}>0$. The former effect completely offsets the latter. As for the fourth term, it is negative since aggregate emissions fall with an increase in ${v_{q}}$ (i.e., $-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{q}}}<0$), thereby putting a further downward pressure on the tax. Overall, the home tax decreases, viz., $t^{h}_{ {v_{q}}}<0$, because of the reduction in damages from emissions. This result is in line with the literature, where lower taxation addresses the output distortion (e.g., Requate, Reference Requate, Tietenberg and Folmer2006). However, here the decrease in the tax arises from considerations of damages from emissions.

6.2.2. Case 2: foreign is relatively pollution intensive

Consider a relatively pollution-intensive foreign firm $\delta ^{\,f}>\hat {\delta } ^{\,f}\equiv 3\delta^{h} +8\beta /\delta^{h}$, whence $E_{{t^{h}} {v_{q}}}=f^{\prime }<0$, $E_{ {t^{h}} }>0$.Footnote ⁶ From (15), because foreign is relatively pollution intensive, damages from transboundary pollution dictate the increase in the tax. Specifically, a higher degree of collusion results in lower foreign output, emissions and thus damages from transboundary polllution. This reduces the need to lower the home tax. The fourth term is positive, also reflecting the smaller need to lower the tax. Overall, the home tax increases (i.e., $t^{h}_{{v_{q}}}>0$) because of the larger role reduced damages from transboundary pollution play; i.e., $-\varphi ^{h^{\prime }}e^{\,f}_{{t^{h}} {v_{q}}}-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{q}}}>-({t^{h}} -\varphi ^{h^{\prime }})e^{h}_{{t^{h}} {v_{q}}}$. Equivalently, the reduction in damages from home emissions, and thus the need to decrease taxation, is negligible. This result is important because: (i) it contrasts with the literature (e.g., Ebert, Reference Ebert1992: 158) where taxation falls with larger ${v_{q}}$ to address the output distortion arising from a higher degree of collusion, and (ii) it underscores the role of asymmetries in pollution intensities in this adjustment. The analysis points to the room governments may have to set taxation in the presence of a relatively high degree of collusion and potentially less competition, but also carbon leakage.

Proposition 6.2 Let the foreign firm be relatively pollution intensive ($\delta ^{\,f}> \hat {\delta }^{\,f}$), and assume small consumer surplus, output distortion and profits effects. Consider an increase in ${v_{q}}$. The non-cooperative tax in the home country increases if reductions in damages from transboundary pollution are sufficiently large.

Proof. See appendix H.

6.3. Optimal tax and conjectural variation in abatement, ${v_{e}}$

The analysis in this section follows the methodology applied in section 6.2. Results indicate that a higher degree of collusion in abatement results in an increase in the non-cooperative home tax if damages from emissions are large enough. This is because, with an increase in ${v_{e}},$ aggregate emissions rise (i.e., $E_{{v_{e}}}>0$) and so higher taxation is needed. However, if foreign exhibits a sufficiently large pollution intensity, then the home non-cooperative tax decreases since a sufficiently large foreign pollution intensity implies that aggregate emissions are controlled with lower taxation (i.e., $E_{{t^{h}} }>0$). This result illustrates the possibility of lower taxation even when there is a higher degree of collusion in abatement and potentially higher emissions. The role of asymmetries in pollution intensities is crucial for this result since it dictates whether a higher/lower tax controls aggregate emissions. But now there is a second effect, absent in section 6.2, which arises from an increase in ${v_{e}}$. In section 4, we argue that an increase in ${v_{e}}$ increases marginal abatement costs, which consequently lowers abatement and the effectiveness of a tax increase in reducing aggregate emissions. Analogous to section 6.2 we consider case 1 (case 2) where home (foreign) pollution intensity exceeds foreign's (is sufficiently large).

6.3.1. Case 1: home exceeds foreign's pollution intensity

Consider a pollution-intensive home firm $\delta^{h} >\delta ^{\,f}$ where $E_{{t^{h}} }<0$. Recall $E_{{v_{e}}}>0$ for any pollution intensity. With these points in mind and using an analogous expression to (14) under assumption 6.1 yields

(16)\begin{equation} {-}W^{h}_{{t^{h}} {t^{h}} }t^{h}_{{v_{e}}}=\hat{\Omega}+{t^{h}} e^{h}_{{t^{h}} {v_{e}}}-\varphi ^{h^{\prime }}E_{{t^{h}} {v_{e}}}-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{e}}}. \end{equation}

$\hat {\Omega }$ in (16) denotes consumer surplus, output distortion and profit effects. As before the focus of the analysis is on the role of pollution intensities and damages, so $\hat {\Omega }$ is assumed small. The term $-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{e}}}>0$ captures the upward pressure on the tax to address rising aggregate emissions. The term ${t^{h}} e^{h}_{{t^{h}} {v_{e}}}$ is also positive. Recall that an increase in ${v_{e}}$ implies higher abatement costs and thus lower abatement. This consequently puts an upward pressure on the tax to address higher emissions. The term $-\varphi ^{h^{\prime }}E_{{t^{h}} {v_{e}}}<0$ captures the decrease in the effectiveness of an increase in the home tax to tackle rising aggregate emissions via higher taxation. Overall, (16) suggests that the non-cooperative home tax increases ($t^{h}_{{v_{e}}}>0$), if damages from aggregate emissions are large enough, i.e., $-\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{{v_{e}}}+{t^{h}} e^{h}_{{t^{h}} {v_{e}} }>\varphi ^{h^{\prime }}E_{{t^{h}} {v_{e}}}$. The reason is that higher taxation addresses higher emissions even in the presence of a less effective tax.

6.3.2. Case 2: foreign is sufficiently pollution intensive

Consider a sufficiently large foreign pollution intensity $\delta ^{\,f}>\bar {\delta } ^{\,f}$, where $E_{{t^{h}} }>0$. Using (16), the non-cooperative home tax decreases ($t^{h}_{{v_{e}}}<0$) if damages from aggregate emissions are large enough. The intuition is that lower home taxation controls rising aggregate emissions (since $E_{{t^{h}} }>0$). The term $-\varphi ^{h^{\prime \prime }}E_{ {t^{h}} }E_{{v_{e}}}<0$ in (16) captures the downward pressure on the tax to address rising aggregate emissions. The term ${t^{h}} e^{h}_{{t^{h}} {v_{e}}}>0$ is positive as explained in section 6.3.1. The term $-\varphi ^{h^{\prime }}E_{{t^{h}} {v_{e}}}>0$ captures the decrease in the effectiveness in the home tax to tackle rising aggrgeate emissions via lower taxation. Overall, (16) points to a decrease in the non-cooperative home tax ($t^{h}_{{v_{e}}}<0$), if damages from aggregate emissions are large enough, i.e., ${t^{h}} e^{h}_{{t^{h}} {v_{e}} }-\varphi ^{h^{\prime }}E_{{t^{h}} {v_{e}}}<\varphi ^{h^{\prime \prime }}E_{{t^{h}} }E_{ {v_{e}}}$. This is because lower taxation controls higher emissions even in the presence of a less effective tax.

Proposition 6.3 Let the foreign firm be sufficiently pollution intensive, and assume small consumer surplus, output distortion and profits effects. Consider an increase in ${v_{e}}$. The non-cooperative tax in the home country decreases if damages from aggregate emissions are sufficiently large.

Proof. See appendix I.

7. Policy discussion and concluding remarks

This paper presents a model which incorporates nonzero conjectural variations in abatement and output in a unified framework of analysis. The inclusion of these two elements is relevant to the debate regarding policy design of environmental policy in an international context because they capture potential responses of firms in one country as a result of decisions made by its competitors on emissions and production elsewhere. We contribute to the environmental policy literature under oligopoly by deriving cases where optimal taxation may be larger (e.g., relative to the zero-conjectural-variation, Cournot case) due to the presence of nonzero conjectural variations and large asymmetries in pollution intensities. For instance, even in a market with a relatively high degree of collusion, where a lower tax may be prescribed, we show that taxation may be higher in the presence of large asymmetries in pollution intensities. Additionally, for any degree of conjectural variation in output, taxation may be lower even if there is a relatively higher degree of collusion in abatement associated with higher pollution because of large asymmetries in pollution intensities. These results point to the role of asymmetries in pollution intensities across countries and nonzero conjectural variation in the characterization of policy.

In the context of international environmental agreements, negotiating countries may have concerns about issues of carbon leakage and their ability to stay competitive. The analysis suggests that in the presence of leakage, a case can be made for lower taxation in the country where firms are sufficiently less pollution intensive. This is a direct application of proposition 6.3: the presence of large asymmetries in pollution intensities implies that lower taxation by the low-polluting country controls rising aggregate (global) emissions and, at the same time, helps the low-polluting country be more cost competitive. Hoel (Reference Hoel1996) also argues that with greater emissions leakage taxes are lower, not because aggregate emissions are controlled via lower taxation, but rather because damages from aggregate emissions are not addressed since taxation simply shifts pollution elsewhere. In our model, in contrast, lower taxation actually helps reduce aggregate emissions via the oligopolistic interdependence of firms. Furthermore, proposition 6.2 suggests that for sufficiently low pollution-intensive countries experiencing increased foreign competition (i.e., a reduction in ${v_{q}}$), a smaller emission tax may be justified in the presence of large damages from transboundary pollution. This reduction in taxation, in the presence of large asymmetries in pollution intensities, addresses concerns about competitiveness and at the same time rising aggregate (global) emissions, particularly damages from transboundary pollution. The key factor behind these results is the presence of large asymmetries in pollution intensities and its implications on emissions via the oligopolistic interdependence.

Specific applications of the results can be illustrated by linking cases (e.g., more/less pollution intensive, more/less foreign competition) to real-life situations and hypothetical scenarios. For instance, a decrease in ${v_{q}}$ could be interpreted as a newly-signed trade treaty (i.e., output would rise as a result of the treaty) between a high- and low-pollution-intensive country (e.g., Norway and China). In this case the model suggests the possibility of lower taxation by the low-polluting country (in this case, Norway) as long as transboundary pollution damages are large; however, absent large asymmetries in pollution intensities, higher taxation could result. Analogously, if an environmental agreement between a high- and low-pollution-intensive country is broken (increase in ${v_{e}}$), then results suggest that lower taxation by the low-polluting country takes place to control emissions, but also in this case lower taxation addresses concerns about cost competitiveness by the low-polluting country. The exogenous change in ${v_{q}}$ and ${v_{e}}$ may help capture effects similar to those from environmental and trade treaties. However, the specific mechanisms causing these changes are not clear in the present model set-up, and so a specific modelling strategy would be required.

Inevitably, the analysis relies on a set of assumptions which curtails the generality of results, but on the other hand it allows for the derivation of closed-form solutions which in turn facilitate tracing of the equilibrium for a myriad of cases. Additionally, results are derived assuming away the response of the policy adjustment in one country as a result of policy behavior in the other country. Further, the analysis is presented using a two-country model which abstracts from issues of coalitions and the fact that environmental agreements normally entail several countries and not just two. This is a particularly important aspect since one of the key factors driving the results is the presence of large asymmetries in pollution intensities. Similarly, exploring the role of multiple firms within each country could yield interesting results, particularly in the presence of free-entry. We conjecture that the case of fixed number of firms may not yield significantly different results as long as pollution intensities within countries are not heterogenous and the difference in the number of firms across countries is not too large. This is because key in the present analysis is the role of asymmetry of pollution intensities across countries. Another extension is to explore the nature of the marginal abatement cost function so that cases such as substantial learning spillovers (solar panels, electric vehicles) or cross-border invasion (invasive species, diseases) are also captured.