2. The model

This section briefly reports the basic ingredients of the PTY model. In a Cournot duopoly, firm i and firm j ($i = \{{1,\,2} \}$, $i \ne j$) produce homogeneous goods, ${q_i}$ and ${q_j}$, respectively, and sell them in a market where the aggregate normalised inverse demand is $p = 1 - Q$, where p denotes the marginal willingness to pay of consumers, and $Q = {q_i} + {q_j}$ is the total supply.

The cost function of firm i is ${C_i}({{q_i},\,{x_i}} )$, where ${x_i}$ represents the ‘green’ R&D effort (abatement) that firm i engages in. Following PTY, we assume:

(1)\begin{equation}{C_i}({{q_i},\; {x_i}} )= c{q_i} + ({\gamma x_i^2/2} ),\quad i,\; j = \{{1,\; 2} \},\; i \ne j,\end{equation}

where $0 \le c < 1$ is the marginal cost of production, and $\gamma > 0$ is a parameter measuring the R&D efficiency. A decrease in $\gamma$ represents a technological development so that investing in green R&D is cheaper (i.e., the efficiency of the green R&D investment increases).

The net emissions following the industrial production of each firm are

(2)\begin{equation}{e_i}({{q_i},\; {x_i}} )= {q_i} - {x_i} \ge 0.\end{equation}

Combining the indirect linear demand with equations (1), (2), the profit function of firm i is:

(3)\begin{equation}{\Pi _i} = ({1 - Q} ){q_i} - c{q_i} - \frac{\gamma }{2}x_i^2 - t({{q_i} - {x_i}} ).\end{equation}

Without loss of generality, we assume that $c = 0$ henceforth by directly following PTY. Total emissions ($E$) are measured by the index $E = \mathop \sum \nolimits^ {e_i}({{q_i},\,{x_i}} )$ and the environmental damage ($ED$) is assumed to be a convex function given by $ED = ({1/2} ){E^2}$.

Owners delegate the choice of output and abatement to managers by offering them a ‘take-it-or-leave-it’ linear retribution scheme (e.g., Fershtman and Judd, Reference Fershtman and Judd1987), whose structure is ${\mathrm{\Omega }_i} = {\beta _i} + {B_i}{\textrm{O}_i}$, with ${\beta _i},{B_i} > 0,$ and with ${O_i}$ the incentive part. As in PTY, we analyse the following alternative contracts:

(4)\begin{align}O_i^{ed} & = {\alpha _i}{\mathrm{\Pi }_i} + (1 - {\alpha _i})t{x_i}\; (\textrm{environmental delegation},\; ed),\end{align}

(5)\begin{align}O_i^{sd} & = {\alpha _i}{\mathrm{\Pi }_i} + (1 - {\alpha _i})p{q_i}\; (\textrm{sales delegation},\; sd),\end{align}

where ${\alpha _i} \in ({0,\,1} )$ is the size of the bonus (chosen by the owner at the bonus stage), $t{x_i}$ is the tax savings with respect to which the delegation is based in scenario (4), and $p{q_i}$ is the revenue with respect to which the delegation is based in scenario (5). Owners fix the compensation scheme such that the manager obtains his/her reservation utility, normalized to zero. Therefore, managers take their decisions by maximizing ${O_i}$, depending on whether the contract includes an environmental component.

The timing of the game resembles PTY.Footnote ² Unlike PTY, however, we add the analysis of the asymmetric sub-games, in which one firm hires an $ed$-oriented manager and the rival an $sd$-oriented one, and the contract decision stage, in which owners endogenously choose the market structure. Therefore, firms engage in a four-stage non-cooperative managerial decision game with homogenous products and complete information. At stage one, each owner chooses to design a contract based on environmental or sales incentives (the contract decision stage). At stage two (the bonus stage), each owner chooses the extent of the bonus that should be set to his/her manager (either in the $ed$ or in the $sd$ case). As in PTY, each contract cannot be re-negotiated and becomes common knowledge. At stage three, the manager hired in each firm decides on the extent of the abatement effort and, simultaneously, the regulator (government) sets the emission tax. Finally, at stage four (the market stage) each manager chooses the output in the product market. The game is solved according to the backward induction logic.

2.1 Symmetric sub-games and exogenous equilibria

The equilibrium outcomes of the symmetric sub-games developed by PTY, in which both owners (i.e., firms) universally design either the $ed$ contract or the $sd$ contract, are summarised in table 1 (the $ed$ contract) and table 2 (the $sd$ contract). The results obtained and discussed in PTY hold here.Footnote ³

Table 1. Equilibrium outcomes in the PTY model when the owners of both firms offer the $ed$ contract

Table 2. Equilibrium outcomes in the PTY model when the owners of both firms offer the $sd$ contract

All the quantities reported in table 1 (including net emissions ${q^{ed}} - {x^{ed}}$) are positive for any $\gamma > 0$, $0 < {\alpha ^{ed}} < 1$ for any $\gamma > 0$, and ${t^{ed}} = 0$ if and only if $\gamma = 0$. Equilibrium social welfare $SW$ is given by the algebraic sum of consumers' surplus ($CS = {Q^2}/2$), producers' surplus ($PS = {\mathrm{\Pi }_i} + {\mathrm{\Pi }_j}$), tax revenues ($TR = t({{e_i} + {e_j}} )$) and environmental damage ($ED$), i.e., $SW = CS + PS + TR - ED$.

Likewise, all the quantities reported in table 2 (including net emissions ${q^{sd}} - {x^{sd}}$) are positive for any $\gamma > 0$, $0 < {\alpha ^{sd}} < 1$ for any $\gamma > 2/3$, and ${t^{sd}} = 0$ if and only if $\gamma = 0$. Therefore, to compare the sub-game $ed$ with the sub-game $sd$, one must assume that $\gamma > 2/3$.

The following lemma summarises the key findings of PTY.

Lemma 1. ${\mathrm{\Pi }^{ed}} > {\mathrm{\Pi }^{sd}}$, ${t^{ed}} < {t^{sd}}$ always hold; ${\alpha ^{ed}} \frac{\gt}{\lt} {\alpha ^{sd}}$, ${x^{ed}} \frac{\gt}{\lt} {x^{sd}}$, ${q^{ed}} \frac{\gt}{\lt} {q^{sd}}$, ${\left(\frac{E}{Q}\right)^{ed}} \frac{\gt}{\lt} {\left(\frac{E}{Q}\right)^{sd}}$ and $S{W^{ed}} \frac{\gt}{\lt} S{W^{sd}}$ if $\gamma \frac{\gt}{\lt} \tilde{\gamma }$, where $\tilde{\gamma } = 2.735$.

3. Asymmetric sub-games and endogenous equilibria

This section studies the asymmetric sub-game in which the owner of one firm, say firm 1, offers his/her manager the $ed$ contract and the owner of the rival, say firm 2, offers the standard $sd$ contract (based on revenues). Thereby, it evaluates the firms' profits emerging in each possible strategic profile, and then determines the SPNE of the game at the contract decision stage, in which the owners strategically choose the contract that should be proposed to their managers (first stage of the game), i.e., the endogenous market structure. The stages of this asymmetric sub-game are identical to those analysed so far for the symmetric ones (see also the online appendix).

3.1 The asymmetric sub-game

First, we note that the formulation of the $sd$ contract adopted by PTY resembles the combination of profits and revenues used by the pioneering works of Fershtman and Judd (Reference Fershtman and Judd1987) and Sklivas (Reference Sklivas1987). Unfortunately, in a delegation model with pollution externalities and environmental tax (set at a timing of the game discussed so far), the revenue formulation of the $sd$ contract does not allow us to solve for ${\alpha _1}$ and ${\alpha _2}$ in closed form at the bonus stage in the asymmetric sub-game. Then, we need to resort to numerical simulations to determine which SPNE endogenously emerges at the contract decision stage in this non-cooperative game. We pinpoint that it is possible to overcome this lacuna and have closed-form expressions allowing analytical characterisations by following the early formulation of the $sd$ contract outlined by Vickers (Reference Vickers1985) and subsequently adopted in several works, for example, van Witteloostuijn et al. (Reference van Witteloostuijn, Jansen and van Lier2007), Jansen et al. (Reference Jansen, van Lier and van Witteloostuijn2009) and Fanti et al. (Reference Fanti, Gori and Sodini2017a, Reference Fanti, Gori and Sodini2017b), in which the performance measure is based on the sales volume instead of revenues. Indeed, as Jansen et al. (Reference Jansen, van Lier and van Witteloostuijn2007) show, the combination of profits and revenues used by Fershtman and Judd (Reference Fershtman and Judd1987) and Sklivas (Reference Sklivas1987) can be rewritten as a combination of profits and sales volume. We carry out this analysis in section 4 to get closed-form expressions as well as to deepen and clarify (analytically and geometrically) the results presented here. By using the $sd$ contract based on the sales volume, section 5 also provides first a comparison of the $ed$ contract and then of the $sd$ contract with profit maximisation (i.e., the $pm$ contract), resembling the case in which the owner does not hire any managers and directly chooses all the relevant variables as a profit maximising agent. This will be useful to get a rationale for the designing of the $ed$ or $sd$ contract and thus for comparison purposes.

We now turn to the study of the stages of the asymmetric sub-game $ed/sd$ and then add the (first) decision stage.

At the fourth stage of the asymmetric sub-game, manager 1 chooses ${q_1}$ by maximising $O_1^{ed/sd}$ as given by the expression in (4). Likewise, manager 2 chooses ${q_2}$ by maximising $O_2^{ed/sd}$ as given by the expression in (5). The optimal value of output just obtained by manager 1 (resp. manager 2) is substituted into the incentive part of his/her managerial contract $O_1^{ed/sd}$ (resp. $O_2^{ed/sd}$) to get an expression that should be maximised by manager 1 (resp. manager 2) at the third stage of the game by choosing the abatement effort ${x_1}$ (resp. ${x_2}$), taking the tax rate t as given. Simultaneously, by replicating the timing schedule used by PTY in the symmetric sub-games, the government sets the emission tax rate by maximising social welfare and by taking the abatement efforts ${x_1}$ and ${x_2}$ as given.

We now move from the narrative to the mathematics of the $ed/sd$ sub-game. The first step is manager 1's utility maximisation $O_1^{ed/sd}$ with respect to ${q_1}$ and manager 2's utility maximisation $O_2^{ed/sd}$ with respect to ${q_2}$. Then, one gets the following downward-sloping output reaction functions of firm 1 and firm 2 in the $({{q_i},\,{q_j}} )$ space as a function of the environmental tax rate t and the incentive parameter of the $sd$ firm (the incentive parameter of the ed firm does not affect the system of best response functions as ${q_1}$ and ${x_1}$ enter additively in $O_1^{ed/sd}$), that is:

\[\frac{{\partial O_1^{ed/sd}}}{{\partial {q_1}}} = 0 \Leftrightarrow {q_1}({{q_2},\; t} )= \frac{{1 - {q_2} - t}}{2},\]

and

\[\frac{{\partial O_2^{ed/sd}}}{{\partial {q_2}}} = 0 \Leftrightarrow {q_2}({{q_1},\; t,\; {\alpha_2}} )= \frac{{1 - {q_1} - {\alpha _2}t}}{2}.\]

These two expressions reveal that the output reaction functions of both firms in the asymmetric sub-game are similar, differing however in one crucial respect: the incentive parameter applied by the owner designing the $sd$ contract mitigates the negative effects of environmental taxation allowing the manager of the $sd$ firm to produce more than the manager of the $ed$ firm. The solution of the system of output reaction functions allows us to get the equilibrium output of firm 1 and firm 2 at the fourth stage of the asymmetric sub-game $ed/sd$ as a function of the environmental tax rate and the incentive parameter ${\alpha _2}$, that is:

\[\bar{q}_1^{ed/sd}({t,\; {\alpha_2}} )= \frac{{1 - t({2 - {\alpha_2}} )}}{3}.\]

and

\[\bar{q}_2^{ed/sd}({t,\; {\alpha_2}} )= \frac{{1 - t({2{\alpha_2} - 1} )}}{3}.\]

A direct comparison of the last two equations reveals that the production of the $ed$ is negatively affected by the taxation, and the production of the $sd$ firm is negatively affected by the taxation if and only if manager 2 is highly rewarded (${\alpha _2} > 1/2$); otherwise, if manager 2 is poorly rewarded (${\alpha _2} < 1/2$), an increase in the tax rate increases the production of the $sd$ firm. Substituting out the intermediate equilibrium values $\bar{q}_1^{ed/sd}$ and $\bar{q}_2^{ed/sd}$ into $O_1^{ed/sd}$, $O_2^{ed/sd}$ and social welfare, one gets the expressions that manager 1, manager 2 and the regulator should maximise. At the third stage of the game, they simultaneously choose the optimal amount of the R&D abatement effort (by taking the tax rate as given) and the optimal tax rate (taking the R&D abatement effort as given), respectively. Then,

\begin{align}\frac{{\partial O_1^{ed/sd}}}{{\partial {x_1}}} & = 0 \Leftrightarrow {x_1}({{\alpha_1},\; t} )= \frac{t}{{\gamma {\alpha _1}}},\nonumber\end{align}

\begin{align}\frac{{\partial \textrm{O}_2^{ed/sd}}}{{\partial {x_2}}} & = 0 \Leftrightarrow {x_2}(t )= \frac{t}{\gamma },\nonumber\end{align}

and

\[\frac{{\partial SW}}{{\partial t}} = 0 \Leftrightarrow t({{x_1},\; {x_2},\; {\alpha_2}} )= \frac{{1 - 3({{x_1} + {x_2}} )}}{{2({1 + {\alpha_2}} )}}.\]

An increase in the incentive parameter ${\alpha _1}$ reduces the amount of pollution abatement chosen by the manager of the $ed$ firm, and the incentive parameter of the $sd$ does not affect the amount of pollution abatement chosen by manager 2, but ${x_1}({{\alpha_1},t} )> {x_2}(t )$. In addition, as expected, an increase in the abatement of both firms reduces the extent of the optimal tax rate. Making use of the last three equations, one obtains the (intermediate) equilibrium values of ${x_1}$, ${x_2}$ and t as a function of ${\alpha _1}$, ${\alpha _2}$ computed at the third stage of the game. That is,

\[\overline {\bar{x}} _1^{ed/sd}({{\alpha_1},\; {\alpha_2}} )= \frac{1}{{3(1 + {\alpha _1}) + 2\gamma {\alpha _1}({1 + {\alpha_2}} )}},\]

\[\overline {\bar{x}} _2^{ed/sd}({{\alpha_1},\; {\alpha_2}} )= \frac{{{\alpha _1}}}{{3(1 + {\alpha _1}) + 2\gamma {\alpha _1}({1 + {\alpha_2}} )}},\]

and

\[{\overline {\bar{t}} ^{ed/sd}}({{\alpha_1},\; {\alpha_2}} )= \frac{{\gamma {\alpha _1}}}{{3(1 + {\alpha _1}) + 2\gamma {\alpha _1}({1 + {\alpha_2}} )}}.\]

Now, substituting out $\overline {\bar{x}} _1^{ed/sd}({{\alpha_1},\,{\alpha_2}} )$, $\overline {\bar{x}} _2^{ed/sd}({{\alpha_1},\,{\alpha_2}} )$, and ${\overline {\bar{t}} ^{ed/sd}}({{\alpha_1},\,{\alpha_2}} )$ into $\bar{q}_1^{ed/sd} ({t,\,{\alpha_2}} )$ and $\bar{q}_2^{ed/sd}({t,\,{\alpha_2}} )$, one gets:

\[\overline {\bar{q}} _1^{ed/sd}({{\alpha_1},\; {\alpha_2}} )= \frac{{1 + {\alpha _1}({1 + \gamma {\alpha_2}} )}}{{3(1 + {\alpha _1}) + 2\gamma {\alpha _1}({1 + {\alpha_2}} )}},\]

and

\[\overline {\bar{q}} _2^{ed/sd}({{\alpha_1},\; {\alpha_2}} )= \frac{{1 + {\alpha _1}({1 + \gamma } )}}{{3(1 + {\alpha _1}) + 2\gamma {\alpha _1}({1 + {\alpha_2}} )}}.\]

The $ed$ delegation contract incentivizes manager 1 to produce less and abate more than manager 2, whose contract is based on the $sd$ delegation (revenue version). Considering the optimal values computed at the third and fourth stages of the game, the owner of firm 1 (resp. firm 2) maximises profits $\mathrm{\Pi }_1^{ed/sd}$ (resp. $\mathrm{\Pi }_2^{ed/sd}$) at the second stage by choosing ${\alpha _1}$ (resp. ${\alpha _2}$). These values do not exist in closed form when the $sd$ contract is based on a combination of profits and revenues (PTY). However, they only depend on $\gamma$, representing the efficiency of the R&D abatement technology.Footnote ⁴

4. A reformulation of the $sd$ contract based on sales volume and (endogenous) Nash equilibria

The main purpose of this section is to make the previous analysis robust, deepening the analytical characterisation of the managerial decision game with pollution externalities and emissions taxes.Footnote ⁶ The lacuna that follows the revenue-based contract can be overcome through an appropriate formulation of the $sd$ contract, as was originally formulated in the influential work of Vickers (Reference Vickers1985) by introducing the sales volume instead of revenues in the incentive part of the contract. Therefore, other things being equal, equation (5) modifies as follows:

(6)\begin{equation}O_i^{sd} = {\alpha _i}{\Pi _i} + (1 - {\alpha _i}){q_i}\; (\textrm{sales delegation},{\kern 1pt} sd).\end{equation}

The stages of the (sub)game are the same as those outlined in the previous sections. Therefore, the equilibrium outcomes of the sub-game in which both firms play $ed$ remain those summarised in table 1. Table 3 details the main equilibrium outcomes of the symmetric sub-game in which both firms offer the $sd$ contract in (6), whereas table 4 presents the corresponding outcomes of the asymmetric sub-game in which the owner of firm 1 designs the $ed$ contract and the owner of firm 2 designs the $sd$ contract. All the quantities reported in table 3 (including net emissions ${q^{sd}} - {x^{sd}}$) are positive for any $\gamma > 0$, $0 < {\alpha ^{sd}} < 1$ for any $\gamma > 0$, and ${t^{sd}} = 0$ if and only if $\gamma = 0$. Likewise, all the quantities reported in table 4 (including net emissions by firm 1, $q_1^{ed/sd} - x_1^{ed/sd}$, and net emission by firm 2, $q_2^{ed/sd} - x_2^{ed/sd}$) are positive, $0 < \alpha _1^{ed/sd} < 1$ for any $\gamma > 0.78077 = ((\sqrt {17} - 1)/4)$, $0.8 < \alpha _2^{ed/sd} < 1$ for any $\gamma > 0$, and ${t^{ed/sd}} = 0$ if and only if $\gamma = 0$. The quantity produced by firm 1 is always positive as $\alpha _2^{ed/sd} > 2/3$ always holds. Definitively, to compare the symmetric and asymmetric sub-game we need to assume that $\gamma > 0.78077$ holds for feasibility.

Table 3. Equilibrium outcomes when the owners of both firms offer the $sd$ contract based on the sales volume instead of revenues

Table 4. Equilibrium outcomes when the owner of firm 1 offers the $ed$ contract and the owner of firm 2 offers the $sd$ contract based on the sales volume instead of revenues

At the first stage of the game, the owners engage in the designing of the contract to offer their managers. The outcome of this choice is summarised in proposition 1.

Proposition 1. [1] If $0.78077 < \gamma < 7.00647$ then ($sd$, $sd$) is the unique Pareto inefficient Nash equilibrium of the game (prisoner's dilemma). [2] If $\gamma > 7.00647$ then ($sd$, $sd$) and ($ed$, $ed$) are two pure-strategy Nash equilibria and the $ed$ payoff dominates $sd$ (coordination game).

Proof. First, we note that $\mathrm{\Delta }{\mathrm{\Pi }_A}(\gamma )< 0$ and $\mathrm{\Delta }{\mathrm{\Pi }_C}(\gamma )< 0$ for any $\gamma > 0.78077$, whereas the sign of $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )$ changes depending on the size of $\gamma$. In particular, $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )> 0$ for any $0.78077 < \gamma < 7.00647$ and $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )< 0$ for any $\gamma > 7.00647$. Then, we have that [1] $\mathrm{\Delta }{\mathrm{\Pi }_A}(\gamma )< 0$, $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )> 0$ and $\mathrm{\Delta }{\mathrm{\Pi }_C}(\gamma )< 0$ for any $0.78077 < \gamma < 7.00647$, and [2] $\mathrm{\Delta }{\mathrm{\Pi }_A}(\gamma )< 0$, $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )< 0$ and $\mathrm{\Delta }{\mathrm{\Pi }_C}(\gamma )< 0$ for any $\gamma > 7.00647$. Q.E.D.

Figure 1 represents the geometrical projection of proposition 1. Figure 2 also shows the equilibrium values of environmental damage and social welfare corresponding to the emergence of Nash equilibria as in proposition 1.

Figure 1. Managerial decision game with pollution externalities and emissions taxes: SPNE when $\gamma$ varies.

Notes: The red region represents the parametric area of unfeasibility. When the abatement technology is relatively efficient ($0.78077 < \gamma < 7.00647$), the game is a prisoner's dilemma. When the abatement technology is relatively inefficient, $\gamma > 7.00647,$ there is indeterminacy, i.e., there exist symmetric multiple SPNE in pure strategies (coordination game) and $ed$ payoff dominates $sd$ (coordination game). The vertical black line at $\gamma = 7.00647$ divides the region in which the game is a prisoner's dilemma (left) from the region in which it is a coordination game (right). The profit differentials of firm i are defined as $\Delta {\Pi _A}(\gamma ): = \Pi _i^{ed/sd} - \Pi _i^{sd}$, $\Delta {\Pi _B}(\gamma ): = \Pi _i^{sd/ed} - \Pi _i^{ed}$ and $\Delta {\Pi _C}(\gamma ): = \Pi _i^{sd} - \Pi _i^{ed}$.

Figure 2. Welfare and environmental damage.

Notes: The white region corresponds to the area of figure 1 in which ($sd$, $sd$) emerges as the unique Pareto inefficient SPNE of the contract decision game (prisoner's dilemma). The sand-coloured region corresponds to the area of figure 1 in which ($sd$, $sd$) and ($ed$, $ed$) emerge as multiple pure strategy SPNE of the contract decision game (coordination game).

Figure 2 clearly shows that the $sd$ contract can never lead to a Pareto efficient outcome for the society, whereas the $ed$ contract can lead to a Pareto efficient outcome for society if firms cooperate to design an environmental managerial contract when the R&D abatement technology is not efficient ($\gamma \uparrow$). Indeed, the $sd$ contract leads to the highest social welfare and the lowest total and marginal environmental damage when the R&D abatement technology is efficient ($\gamma \downarrow$), a scenario that can possibly be observed in the future; however, firms would be better off with the $ed$ contract, which however is not the SPNE of the game.

5. Environmental (and sales) delegation versus profit maximisation

This section goes one step further and considers the comparison of the environmental delegation contract with the profit maximisation ($pm$) contract (i.e., ${\alpha _i} = 1$) to study whether there exists a strategic incentive for the designing of an environmental-oriented contract. The section does also explicitly analyse the case of the $sd$ contract (based on sales) versus the $pm$ contract showing that the endogenous game-theoretic outcome continues to be the Pareto inefficient Nash equilibrium ($sd,\,sd$), as in the literature led by Vickers, Fershtman, Judd and Sklivas without pollution externalities, abatement and environmental taxes. This provides a rationale for the comparison of $ed$ versus $sd$ in a game-theoretic setting.

First, we compare the $ed$ scheme outlined in equation (4) versus the $pm$ scheme, for which the owner (instead of the manager) chooses the quantity at the later stage and the abatement effort at the third stage – simultaneously with the choice of the government about the optimal emission tax rate. The profit function in the symmetric sub-game $pm$ with pollution externalities, abatement and environmental taxes is:

(7)\begin{equation}{\mathrm{\Pi }_i} = ({1 - {q_i} - {q_j}} ){q_i} - \frac{\gamma }{2}x_i^2 - t({{q_i} - {x_i}} ).\end{equation}

By also considering the asymmetric sub-game $ed/pm$, in which only one firm (e.g., firm 1) chooses the $ed$ contract and the rival (e.g., firm 2) the $pm$ contract, the payoff matrix is reported in table 5, in which $\mathrm{\Pi }_i^{ed} = \mathrm{\Pi }_j^{ed} = {\mathrm{\Pi }^{ed}}$ is defined in table 1, $\mathrm{\Pi }_i^{pm} = ({({2{\gamma^2} + 9\gamma + 8} )/8{{({2\gamma + 3} )}^2}} )$, $\mathrm{\Pi }_i^{ed/pm} = ({({2{\gamma^2} + 9\gamma + 6} )/2({16{\gamma^2} + 46\gamma + 27} )} )$ and $\mathrm{\Pi }_j^{ed/pm} = ({16{\gamma^4} + 120{\gamma^3} + 284{\gamma^2} + 260\gamma + 81} )/{({16{\gamma^2} + 46\gamma + 27} )^2}$.

Table 5. Payoff matrix of the managerial decision game $ed$ versus $pm$

Then, the following proposition holds:

Proposition 2. The unique Pareto efficient Nash equilibrium of the game $ed$ versus $pm$ is ($ed$, $ed$) for any $\gamma > 0$ (anti-prisoner's dilemma).

Proof. The sign of the profit differentials is $\mathrm{\Delta }{\mathrm{\Pi }_A}(\gamma )> 0$, $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )< 0$ and $\mathrm{\Delta }{\mathrm{\Pi }_C}(\gamma )< 0$ for any $\gamma > 0$. Q.E.D.

Second, we compare the $sd$ scheme (based on sales) outlined in equation (6) versus the $pm$ scheme summarised in equation (7). By considering the asymmetric sub-game $sd/pm$, in which only one firm (e.g., firm 1) chooses the $sd$ contract based on sales and the rival (e.g., firm 2) the $pm$ contract, the payoff matrix is reported in table 6, in which $\mathrm{\Pi }_i^{sd} = \mathrm{\Pi }_j^{sd} = {\mathrm{\Pi }^{sd}}$ is defined in table 3, $\mathrm{\Pi }_i^{pm} = ({2{\gamma^2} + 9\gamma + 8} )/8{({2\gamma + 3} )^2}$, $\mathrm{\Pi }_i^{sd/pm} = {(2 + \gamma )^2}/8(2{\gamma ^2} + 5\gamma + 4)$ and $\mathrm{\Pi }_j^{sd/pm} = ({2{\gamma^4} + 9{\gamma^3} + 20{\gamma^2} + 20\gamma + 8} )/8{(2{\gamma ^2} + 5\gamma + 4)^2}$.

Table 6. Payoff matrix of the managerial decision game $sd$ versus $pm$

Then, the following proposition holds:

Proposition 3. The unique Pareto inefficient Nash equilibrium of the game $sd$ versus $pm$ is ($sd$, $sd$) for any $\gamma > 0$ (prisoner's dilemma).

Proof. The sign of the profit differentials is $\mathrm{\Delta }{\mathrm{\Pi }_A}(\gamma )> 0$, $\mathrm{\Delta }{\mathrm{\Pi }_B}(\gamma )< 0$ and $\mathrm{\Delta }{\mathrm{\Pi }_C}(\gamma )> 0$ for any $\gamma > 0$. Q.E.D.

The outcomes of proposition 2 and proposition 3 provide the rationale for the designing of the $ed$ contract in a managerial Cournot duopoly, as an alternative to the $sd$ contract. Both managerial schemes emerge as Nash equilibrium outcomes in a game-theoretic setting when each of them is separately compared with profit maximisation. In fact, each profile ($ed$ and $sd$) represents a dominant strategy when contrasted with the $pm$ profile. The only difference is that the Nash equilibrium emerging in the game $ed$ (resp. $sd$) versus $pm$ is Pareto efficient (resp. inefficient). This is because in the former case the increase in production and abatement following the managerial behaviour also favours a tax savings in comparison with the owner's behaviour, whereas in the latter case the same remarks emerging in the literature led by Vickers, Fershtman, Judd and Sklivas hold.

However, this $ed$ contract is ineffective when it is contrasted with the $sd$ contract for the reasons discussed above. Definitively, though there exists an expected incentive for the designing of an environmental-oriented managerial contract as it is strictly preferred to the $pm$ scenario, this is not in the selfish interest of the owners (when the R&D abatement technology is efficient) that unilaterally prefer to set up a sales contract to their manager to avoid being the only ones to make the lowest possible profit following the lowest production.