A One-Stage Model of Abatement Innovation in Cournot Duopoly: Emissions vs Performance Standards

Abstract

This paper evaluates the comparative performance of emission and performance standards in a one-stage game of abatement R&D and Cournot duopoly, in terms of R&D propensity, output and social welfare. For each standard, firms simultaneously select R&D and output levels, given the standard’s exogenous constraint. A performance standard generates higher R&D investments and output, but lower profit, than the pollution-equivalent emissions standard. The same conclusion extends to social welfare only under high demand. We also conduct a similar comparison for each of the two instruments across the one-stage and the two-stage models. The two-stage model leads to higher levels of R&D and industry output for both standards. The same conclusion applies to the social welfare comparison for the emissions standard. However, for the performance standard, the same conclusion requires a damage parameter below a given threshhold. When the standards are chosen to maximize welfare, the performance comparison becomes highly parameter-dependent, except that social welfare is higher for the performance standard. Some policy implications are discussed.

Appendices

Appendix A. Proofs

This section provides the proofs for all the results of the paper. Since the proofs are based on simple but sometimes involved computations, only the main lines are presented to allow the interested reader to follow the steps; details are thus mostly left out.

Proof of Proposition 1

• (i) Provided that $e=q_{h}h=\frac{\gamma h[a-c(1-h)]}{3b\gamma -(1-h)^{2}}$, we have

  $$\begin{aligned} x_{e}=\frac{(a-c)}{3b\gamma -1}-\frac{3b}{3b\gamma -1}\frac{\gamma h[a-c(1-h)]}{3b\gamma -(1-h)^{2}} \end{aligned}$$

  and

  $$\begin{aligned} x_{h}-x_{e}=\frac{h[3bc\gamma -a(1-h)]}{[3b\gamma -1][3b\gamma -(1-h)^{2}]}, \end{aligned}$$

  which is clearly (strictly) positive by (A1) and the fact that $0<h<1$.

• (ii) Similarly,

  $$\begin{aligned} q_{h}-q_{e}=\frac{\gamma h[3bc\gamma -a(1-h)]}{[3b\gamma -1][3b\gamma -(1-h)^{2}]}>0. \end{aligned}$$

• (iii) The profit result follows from the following inequalities:

  $$\begin{aligned} \pi _{e} &= {} q_{e}(a-2bq_{e})-(c-x_{e})(q_{e}-e)-\gamma x_{e}^{2}/2 \\ &\ge {} q_{h}(a-b(q_{e}+q_{h}))-(c-x_{h})(q_{h}-e)-\gamma x_{h}^{2}/2 \\ &> {} q_{h}(a-2bq_{h})-(c-x_{h})q_{h}(1-h)-\gamma x_{h}^{2}/2=\pi _{h}. \end{aligned}$$

  The first inequality follows by the Nash property, the second one, by $q_{e}<q_{h}$ (part ii) and $e=q_{h}h$. The equalities are given by definition of the two equilibrium profit levels.

This completes the proof of Proposition 1. $\square$

Proof of Proposition 2

Since we fixed $e=q_{h}h=\frac{\gamma h[a-c(1-h)]}{3b\gamma -(1-h)^{2}}$, we can write $W_{e}$ in terms of h, and then compute the difference

$$\begin{aligned} W_{h}-W_{e}=-C(bc\gamma A-aB)/D, \end{aligned}$$

where

$A=h(6b\gamma -1)+(6b\gamma -2)>0,$ $B=(4b\gamma -1)h(1-h)+2b\gamma (3b\gamma -1)>0,$

$C=\gamma (3bc\gamma -a(1-h))h>0,$ and $D=[3b\gamma -1]^{2}[3b\gamma -(1-h)^{2}]^{2}>0$.

Here the first three inequalities follow by (A1) and the last one is obvious. Then, $W_{h}>W_{e}$ if and only if $\frac{a}{c}>b\gamma \frac{A}{B}$, which leads to the desired result. $\square$

Proof of Proposition 3

Assumption (A2) is required for a well-defined, interior and symmetric subgame-perfect equilibrium in the two-stage games (see details in [2]). To prove the four desired inequalities here, we proceed by direct calculations. As the steps are very simple, we leave the details to the reader.$\square$

Proof of Proposition 4

First, observe that Assumption (A2) is imposed here for the same reasons as in the proof of Proposition 3.

• (i) Using the expressions for $W_{e}$ and $\tilde{W}_{e}$ given in the text, we have after simplification that $\tilde{W}_{e}-W_{e}=\frac{b(e-(a-c)\gamma )^{2}}{(3b\gamma -1)^{2}(9b\gamma -4)}$, which is strictly positive since $9b\gamma >8$.

• (ii) Using the expressions for $W_{h}$ and $\tilde{W}_{h}$ given in the text, we have after simplification

  $$\begin{aligned} \tilde{W}_{h}-W_{h}=\frac{\gamma ^{2}(1-h)^{2}[a-c(1-h)]^{2}[b(9b\gamma -4(1-h)^{2})-h^{2}s(36b\gamma -14(1-h)^{2})]}{[9b\gamma -4(1-h)^{2}]^{2}[3b\gamma -(1-h)^{2}]^{2}}>(<)0 \end{aligned}$$

  if $\frac{b}{h^{2}s}>(<)\frac{36b\gamma -14(1-h)^{2}}{9b\gamma -4(1-h)^{2}}$

Proof of Proposition 5

The reader can easily verify that the FOC for the social welfare maximizing problem (14) is:

$$\begin{aligned} \frac{-2 a ( 4 b \gamma -1) + 2b ( 9 b \gamma -2)(e + c \gamma ) - 4 e (3 b \gamma -1)^2\,s}{(3 b \gamma -1)^2}=0, \end{aligned}$$

which leads to $e^*=\frac{cb\gamma (9b\gamma -2)-a(4b\gamma -1)}{2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)}$. Substituting this result into equations (2) leads to $x^{*}_e=\frac{2(a-c)(3b\gamma -1)s-b(c(9b\gamma -2)-a)}{2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)}$ and $q^{*}_e=\frac{(3b\gamma -1)(2(a-c)\gamma s -a)}{2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)}$.

The SOC of this problem is $\frac{2b(9b\gamma -2)-4\,s(3b\gamma -1)^{2}}{(3b\gamma -1)^{2}}<0,$ equivalent to $2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)>0$ or $s>\frac{b(9b\gamma -2)}{2(3b\gamma -1)^2},$ that is, the denominator of $e^*$, $x^*_e$ and $q^*_e$ must be strictly positive. (A3)(i) and (A3)(ii) imply the last inequality, since $\frac{b(c(9b\gamma -2)-a)}{2(a-c)(3b\gamma -1)}>\frac{b(9b\gamma -2)}{2(3b\gamma -1)^2}$. Notice that (A1) implies that $a-c>0$, $3b\gamma -1>0$ and $9b\gamma -2>0$. If (A3)(ii) also holds, we have that $c(9b\gamma -2)-a>0$, otherwise, it must be that $3b\gamma <1,$ which contradicts (A1).

Clearly, (A3)(i) and (A3)(ii) imply that $e^*>0$; these two assumptions also imply that $x^*_e<c$. To see this, observe that $c-x^*_e=\frac{-a b + 2 ( 3 b \gamma -1) ( 3 b c \gamma -a) s}{ 2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)}>0$ iff $s>\frac{ab}{2(3b\gamma -1)(3bc\gamma -a)}$; by (A1), $3b\gamma -1>0$ and $3bc\gamma -a>0$. (A3)-(ii) implies that $\frac{b(c(9b\gamma -2)-a)}{2(a-c)(3b\gamma -1)}>\frac{ab}{2(3b\gamma -1)(3bc\gamma -a)}$, and thus, (A3)(i) and (A3)(ii) guarantee that $x^*_e<c$.

Finally, observe that (A3)(i) implies that both the numerator and the denominator of $x^*_e$ are strictly positive, hence, $x^*_e>0$. In addition, $q^*_e-e^*=\frac{\gamma [2 (a - c) ( 3 b \gamma -1) s-b(c ( 9 b \gamma -2)-a) ]}{ 2\,s(3b\gamma -1)^{2}-b(9b\gamma -2)}>0$ iff $s>\frac{b(c(9b\gamma -2)-a)}{2(a-c)(3b\gamma -1)}$, corresponding to (A3)(i). $\square$

Proof of Proposition 6

The comparative statics results for $e^{*}$ are straightforward from the (closed-form) expression for $e^{*}$. Since a closed-form expression for $h^{*}$ is not tractable, we shall use [13] theorem for monotone comparative statics [13] or Vives, [14].

(i) The idea is to take $\ln W_{h}(h)$ to turn it into additively separable parts, and then take the cross-partial w.r.t. h and the relevant exogenous variable. In the case of a, we get $\frac{\partial ^{2}\ln W_{h}(h)}{\partial h\partial a}=\frac{-2c}{[a-c(1-h)]^{2}}<0$. Hence, by Topkis’s theorem, $h^{*}$ is decreasing in a.

(ii)-(iii) Similarly, it is easy to check that $\frac{\partial ^{2}\ln W_{h}(h)}{\partial h\partial \gamma }>0$ and $\frac{\partial ^{2}\ln W_{h}(h)}{\partial h\partial s}<0$, which imply that $h^{*}$ is increasing in $\gamma$ and decreasing in s. $\square$

Appendix B

This Appendix provides a comparison of the equilibrium variables of interest in the game with endogenous standards, for a given set of parameters. To distinguish the second-best solution from that of the one-stage game, we will add the superscript $^{*}$ to the variables. Thus, $e^{*}$($h^{*}$) denotes the equilibrium level of emission (performance) standard, while $W_{e}^{*}$ ($W_{h}^{*}$) stands for the corresponding welfare. Throughout this section we assume $b=c=1$, meaning that a accounts exactly for market size, the demand slope is 1, and the initial unit cost of abatement is one.

1.1 Varying Parameter a

We first consider the performance of both policy instruments assuming a variation in the size of the market a. Table 3 provides this comparison as the market size a varies.

Table 3 Second-best solution for $b=c=1$, $s=\gamma =2.5$ and chosen values of a

As seen in Table 3, as higher output leads to more abatement, firms have a higher incentive to invest in R&D (the second row in Table 3). Since higher output raises the pollution level, the regulator tightens the standards (decrease in the level of standard in the first row in Table 3) in order to reduce the damage to the environment. Overall, social welfare increases.

Further details concerning the impact that the change in parameter a has on the comparison between the equilibrium variables under the emission and performance standards are given next. Here and in the next two subsections, the given parameter values are chosen as representative and in respect of Assumption (A3).

Remark 2

Let $b=c=1$, $s=\gamma =2.5$ and $1.59\le a\le 5.69$.¹⁹ Then

• (i) for $1.59\le a<1.79,$ $x_{e}^{*}<x_{h}^{*}$, $q_e^*<q_h^*$, $e^{*}>q_{h}^{*}h^{*}$, and $q_e^*-e^*<q_h^*(1-h^*)$;

• (ii) for $1.79< a<3.09,$ $x_{e}^{*}<x_{h}^{*}$, $q_e^*<q_h^*$, $e^{*}<q_{h}^{*}h^{*}$, and $q_e^*-e^*<q_h^*(1-h^*)$;

• (iii) for $3.09< a<5.19,$ $x_{e}^{*}>x_{h}^{*}$, $q_e^*<q_h^*$, $e^{*}<q_{h}^{*}h^{*}$, and $q_e^*-e^*>q_h^*(1-h^*)$;

• (iv) for $5.19< a \le 5.69,$ $x_{e}^{*}>x_{h}^{*}$, $q_e^*>q_h^*$, $e^{*}<q_{h}^{*}h^{*}$, and $q_e^*-e^*>q_h^*(1-h^*)$;

• (v) $W_{h}^{*}>W_{e}^{*}$.

Remark 2 shows how R&D investment, output, emissions, abatement and welfare change with market size under the two policy regimes. For small market sizes (parts i-ii), the performance standard generates higher R&D, industry output and abatement, and lower pollution levels. If $1.79<a<3.09$ (part ii), all the variables are higher under the performance standard. In part (iii), the performance standard leads in terms of output, but lags in R&D and in abatement level. For a large market size (part iv), the emission standard dominates in terms of R&D, output and abatement. One consistent outcome is that the comparison of R&D investment mirrors that of the abatement level.

Overall, considering all relevant values of a, the comparison of the equilibrium levels of R&D and output under the two policy instruments is highly parameter-dependent, but the performance standard is uniformly superior for social welfare (part v).

1.2 Varying Parameter $\gamma$

We turn to the impacts that an increase in the cost of R&D, $\gamma$, has under both policy instruments. Table 4 demonstrates the results of this comparison for different values of $\gamma$.

Table 4 Second-best solution for $b=c=1$, $s=2.5$, $a=3$ and selected values of $\gamma$

As R&D becomes more costly, firms reduce their output (the third row in Table 4) and their R&D efforts (the second row in Table ), in order to lower abatement cost. In an attempt to limit the resulting consumer surplus loss, the regulator weakens the stringency of the pollution constraint under each policy instrument (the amount of pollution allowed increases as shown in the first row in Table 4). It is intuitive that welfare goes down.

Remark 3 summarizes these findings and gives details concerning the comparison between the equilibrium variables under both policy instruments; $s=2.5$ and $a=3$ were chosen for illustrative purposes.

Remark 3

Let $a=3$, $b=c=1$, $s=2.5$ and $\gamma \ge 1.30$.²⁰ Then (i) for $1.3 \le \gamma < 1.39,$ $x_{e}^{*}>x_{h}^{*},$ $q_{e}^{*}>q_{h}^{*}$ and $q^*_e-e^*>q_{h}^{*}(1-h^{*});$ (ii) for $1.39< \gamma < 2.33,$ $x_{e}^{*}>x_{h}^{*},$ $q_{e}^{*}<q_{h}^{*}$ and $q^*_e-e^*>q_{h}^{*}(1-h^{*});$ (iii) for $2.33 < \gamma ,$ $x_{e}^{*}<x_{h}^{*},$ $q_{e}^{*}<q_{h}^{*}$ and $q^*_e-e^*<q_{h}^{*}(1-h^{*});$ (iv) $e^*<q_{h}^{*}h^{*}$; (v) $W_{h}^{*}>W_{e}^{*}$.

Here, the emission standard delivers more R&D and output at low R&D cost (part i), but less of both at high R&D cost (part iii). This occurs because under the performance standard the fall in output not only has the desired effect of a reduction in abatement cost, but it also leads to a reduction in environmental damage (the term $2s(hq_{h})^{2}$ in Eq. (15)). This effect is absent under the fixed emission standard, where the reduction in output does not directly affect the level of environmental damage (i.e., the term $2se^{2}$ in Eq. (13) does not contain $q_{e}$). Then, smaller reductions in output are required under the performance standard to mitigate the increased R&D cost. Hence, for a high level of R&D cost, output is higher for the performance standard.

Once more, the performance standard dominates in terms of welfare for all our parameter values, but pollution is lower under the emission standard.

1.3 Varying Parameter s

This part evaluates the comparative performance of the two standards as the damage parameter s varies. The comparison is presented in Table 5, in which, in addition to varying values of parameter s, we also assume different values of $\gamma$. In this way, we are able to capture the range of possible results (qualitatively).

Table 5 Second-best solution for $a=3$, $b=c=1$ and A: $\gamma =2.5$, $s=10$; B: $\gamma =s=1.3$

Segments A and B of Table 5 compare the equilibrium values of R&D, output, emissions, abatement and welfare under both policy instruments for different values of parameters. In Segment A, $\gamma =2.5$ and $s=10$ have been chosen for illustrative purposes. Table 5 shows that for these parameter values the performance standard outperforms the emission standard in terms of the incentives to generate higher output and welfare, but not R&D. Remark 4 complements these results by showing how the relationship between R&D, output, emissions, abatement and welfare under the two policy instruments changes with different values of s, while $\gamma$ is kept constant.

Remark 4

Let $a=3,$ $b=c=1$, $\gamma =2.5$ and $s\ge 0.68$.²¹ Then

• (i) for $s<(>)5.86$, $x_{h}^{*}>(<)x_{e}^{*}$ and $q_{h}^{*}(1-h^{*})>(<)q_{e}^{*}-e^{*}$;

• (ii) $q_{h}^{*}>q_{e}^{*}$ and $q_{h}^{*}h^{*}>e^{*}$;

• (iii) $W_{h}^{*}>W_{e}^{*}.$

Remark 4 confirms that the comparison of R&D levels under the two policy instruments depends on the value of s. When environmental damage is not significant (part i), the performance standard generates more R&D incentives; however, high values of s lead to a reversion. This is not the case for output and welfare, which are larger under the performance standard under the present set of parameters.²²

To see that the performance standard does not generally provide more incentives to produce output, we now consider the effects of varying s under a different value of $\gamma$. An example is illustrated in part B of Table 5.

Remark 5

Let $a=3$, $b=c=1$, $\gamma =1.3$ and $s\ge 0.84$.²³ Then

• (i) $x_{h}^{*}<x_{e}^{*}$ and $q_{h}^{*}(1-h^{*})<q_{e}^{*}-e^{*}$;

• (ii) $q_{h}^{*}<q_{e}^{*}$ and $q_{h}^{*}h^{*}>e^{*}$;

• (iii) $W_{h}^{*}>W_{e}^{*}$.

Here, output under the emission standard is larger than under the performance standard, and so is environmental R&D, with the latter reversing the conclusion of Proposition 1. Hence, once more, this confirms that the results of the comparison of R&D and output incentives depends in key ways on whether the two regulatory regimes are exogenously given (on the basis of equal emissions) or endogenous (i.e., set to maximize welfare).