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A Model of Benchmarking Regulation: Revisiting the Efficiency of Environmental Standards

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The conventional economic argument favors the use of market-based instruments over ‘command-and-control’ regulation. This viewpoint, however, is often limited in the description and characteristics of the latter; namely, environmental standards are often portrayed as lacking structured abatement incentives. Yet contemporary forms of command-and-control regulation, such as standards stipulated via benchmarking, have the potential to be efficient. We provide a first formal analysis of environmental standards based on performance benchmarks. We show that under specific conditions, standards can provide efficient incentives to improve environmental performance.

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  1. Despite significant advances in the use (and understanding) of market-based instruments over the last five decades, command-and-control regulation remains dominant. Explanations for instrument choice can be seen in, for example, Buchanan and Tullock (1975), Oates et al. (1989), and Aidt and Dutta (2004).

  2. A notable exception is Montero (2002), who finds that standards may offer greater R&D incentives than permits because R&D under standards only reduces a firm’s own cost, but not that of its rivals.

  3. Of course, the standard could also be determined by the best available technology or some other benchmark. For example, the EU has recently adopted a benchmarking mechanism to determine free allocation of EU emission allowances (EUAs) under the EU ETS where the benchmark is set to reflect the average performance of the 10 % most efficient producers in a given sector (European Commission 2011). While our theoretical model hinges on the benchmark being defined by the industry’s average performance, we are able to show that such instruments can ensure strong incentives to abate pollution.

  4. Related studies on the political economy of environmental regulation have analyzed both how environmental policy is determined via interest groups (e.g., Heyes 1997; Aidt and Dutta 2004; Aidt 2010) as well as the creation of rents from the environmental regulation and the welfare losses incurred from rent-seeking activity (e.g., Dijkstra 1998; Malueg and Yates 2006; MacKenzie and Ohndorf 2012a).

  5. A number of alternative interpretations are possible. For example, \(P\) could reflect a prize fund that is to be allocated to firms for their relative abatement of pollution (see the discussion section of current policies in Sect. 6). Additionally, given appropriate function specifications, our framework could be extended to a case where \(P\) would be endogenously determined by abatement efforts (see, Chung 1996), in which case such a model framework could resemble, for example, a tax-and-refund scheme.

  6. MacKenzie et al. (2008) briefly discuss this approach in their paper on relative performance mechanisms in pollution permit markets. The relative performance approach will motivate the firms to act strategically. For a general analysis of strategic behavior in contests, see Dixit (1987).

  7. While in a concentrated industry incentives for collusion may exist, yardstick competition can be designed to minimize these incentives, e.g., by imposing restrictions on firms admissible choices (Shleifer 1985). Therefore, we only consider non-cooperative behavior.

  8. Note that a tax/subsidy scheme could provide equivalent incentives. Given our focus on command-and-control regulation and the fact that such policy would be significantly more complicated than a conventional emission tax, this is not considered here.

  9. The restriction to the two-firm case is purely for ease of representation. As shown in the Appendix, our results extend to the case of \(n>2\).

  10. When interpreted as a production license (an assumption we relax in Sect. 6), the natural application of the proposed instrument would be the regulation of the production of a homogeneous good. In these circumstances, it seems plausible to assume that producers have similar production costs and, therefore, earn the same mark-up per unit sold. Multiplied with a given demand for the good (or a legally restricted quantity of the output), this mark-up defines the producer surplus, which we hence assume to be constant and exogenous. We relax the first assumption in Sect. 4.

  11. It is easy to verify that the second-order condition is fulfilled and \(\frac{\partial ^2 \pi _i}{\partial c_i^2} < 0\).

  12. It is straightforward to show that any level of abatement is achieved at a lower total cost when the regulatory instrument is a standard established via benchmarking rather than a conventional command-and-control policy.

  13. Note that we do not consider technological spillovers. If they existed in our model, we would expect investment to decrease as spillover effects would reduce the competitive advantage one could gain by investing in a more efficient technology.

  14. Note that the regulator does not prescribe the use of any specified technology, such as in a regime where best available control technologies (BACT) are used.

  15. Alternatively, to yield the first-best level of abatement, the regulator could simply limit the scope of aggregate operating licenses and thus the rent \(P\).

  16. Obviously, if the tax was truly Pigouvian (i.e., first-best efficient), increased abatement would be undesirable and benchmarking could lead to abatement beyond the socially optimal; however, this would be the standard textbook comparison of an optimal instrument with a much more restrictive policy. On the other hand, it can be argued that environmental taxes are typically well below their Pigouvian levels (e.g. Ciocirlan and Yandle (2003)), which motivates our comparison of two suboptimal regulatory policies.

  17. Rent-seeking efforts are typically viewed as being welfare decreasing. Nevertheless, the second choice variable, \(s_i\), can more generally be interpreted as effort, which could have either positive or negative welfare implications. Yet we show that even if the second instrument is welfare decreasing, its use can still improve welfare overall.

  18. One could think of the regulator having the power to choose \(\alpha \), in which case \(\alpha \) would express the regulators valuation of abatement and rent seeking, respectively. Alternatively, one could view \(\alpha \) as being exogenous and simply show that \(\alpha < 1\) can increase the efficiency of the mechanism. We take the latter approach.

  19. Note that (24) requires a given degree of substitutability between the cost of rent seeking and abatement. For a more general formulation, see Gerigk et al. (2013). An alternative specification would allow both choice variables to be multiplicative; for an example, see Arbatskaya and Mialon (2010).

  20. Note that \(\hat{S}\) only accounts for the direct monetary cost of rent seeking. If there were additional welfare costs related to rent seeking, of course, the range for which the latter has the potential to increase welfare would be reduced.

  21. Note that this follows from the second-best argument that two distortionary factors may achieve an outcome closer to the first-best outcome.

  22. Of course, regulation would not be needed, thus the proposed mechanism would not actually be employed.

  23. It is easy to show that the results extend to the case with investment.

  24. If benchmarking regulation is to allocate production licenses, the assumption of homogeneity on the output market is crucial.

  25. Note, however, that market instruments may also incentivize large abatement projects that could cause similar problems if exogenous shocks occurred. Under any instrument there could then be an incentive to renegotiate the regulatory burden. As benchmarking regulation is based on individual contracts between firms and the regulator, it might be more prone to renegotiation than universal policies.

  26. As is standard in the analysis of environmental regulation, \(a_i\) is defined as the difference between emissions from production under a baseline scenario and a scenario in which firm \(i\) takes measures to abate its pollution. While this is typically measured per firm, the interpretation of \(a_i\) as abatement per unit of production is equally possible.

  27. An extensive description of possible measurement methods is given in Commission Regulation (EU) No. 601/2012.

  28. With market instruments, the marginal benefit of over-reporting of abatement is constant and equal to the tax rate or permit price. Under benchmarking regulation, on the other hand, the incentive to over-state abatement is decreasing because the market share is concave in reported emissions. As a result, any positive level of monitoring achieves a reduction in misreporting while with an emission tax, there is no compliance by the firms unless the expected fine, a function of monitoring effort, exceeds the tax rate.


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The authors are grateful to two anonymous reviewers and the Editor Anthony Heyes for helpful comments and suggestions that significantly improved this paper. Furthermore, we would like to thank Alessio D’Amato, Johannes Manser, and Renate Schubert as well as seminar participants in Zürich (ETH), Paris (26th International Climate Policy Workshop), Toulouse (EAERE 2013), and Ascona (SURED 2014) for their valuable comments and suggestions. Financial support by the Swiss Academy of Humanities and Social Sciences (SAGW) is gratefully acknowledged. All errors and shortcomings are our own.

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Correspondence to Ian A. MacKenzie.


Mathematical Appendix

1.1 Proof of Proposition 1


To solve (13) we derive the following set of first-order conditions:

$$\begin{aligned} \frac{\partial \pi _i}{\partial c_i}= \frac{\mu P \left( \frac{3\left( \frac{1}{\mu I_i + \gamma _i}\right) ^{\frac{3}{2}}}{\mu I_{-i}+\gamma _{-i}} + \frac{\sqrt{\frac{1}{\mu I_{-i}+\gamma _{-i}}}}{\left( \mu I_i+\gamma _i\right) ^2}\right) }{4 \left( \sqrt{\frac{1}{\mu I_i+\gamma _i}}+\sqrt{\frac{1}{\mu I_{-i}+\gamma _{-i}}}\right) ^3} - 1 =0 \quad \text {for} \quad i,-i\in \Phi . \end{aligned}$$

Substitution of the first FOC into the second yields after some simplification

$$\begin{aligned} \mu I_X + \gamma _X = \mu I_Y+\gamma _Y. \end{aligned}$$

As \(k_i = \frac{1}{\gamma _i + \mu I_i}\), it follows

$$\begin{aligned} k_X^*=k_Y^*. \end{aligned}$$

Substituting (35) into (34) yields the optimal level of investment:

$$\begin{aligned} I_i^*=\frac{\mu P-8 \gamma _i}{8 \mu }. \end{aligned}$$

As \(\frac{\partial ^2 \pi _i}{\partial I_i^2}<0\), (36) denotes indeed a maximum.

With \(k_i = \frac{1}{\gamma _i + \mu I_i}\), (36) ultimately defines the cost parameter \(k_i^* = \frac{8}{\mu P}\). Finally, substituting (15) into (13) yields the corresponding payoff:

$$\begin{aligned} \pi _i^* = \frac{P}{4}+\frac{\gamma _i}{\mu }. \end{aligned}$$

\(\square \)

1.2 Proof of Proposition 2


The proof requires solving the game represented by Eq. (17) via backward induction. The solution to the second stage is defined by the following first-order condition:

$$\begin{aligned} \frac{\partial \pi _i}{\partial c_i} = \frac{P q c_X^{\frac{q}{2}} c_Y^{\frac{q}{2}-1} k_X^{\frac{q}{2}} k_Y^{\frac{q}{2}}}{2 \left( c_Y^{\frac{q}{2}} k_X^{\frac{q}{2}}+c_X^{\frac{q}{2}} k_Y^{\frac{q}{2}}\right) ^2} - 1 = 0. \end{aligned}$$

Equating both first-order conditions, we find \(c_i = c_{-i}\). Solving the game is then straightforward and yields:

$$\begin{aligned} \tilde{a}_{i,q}&= \sqrt{\frac{P q k_i^{\frac{q}{2}-1} k_{-i}^{\frac{q}{2}}}{2 \left( k_i^{\frac{q}{2}}+k_{-i}^{\frac{q}{2}}\right) ^2}}, \end{aligned}$$
$$\begin{aligned} \tilde{A}_q&= \frac{\sqrt{P q k_i^{\frac{q}{2}} k_{-i}^{\frac{q}{2}}} \left( \sqrt{k_i}+\sqrt{k_{-i}}\right) }{\sqrt{2 k_i k_{-i}} \left( k_i^{\frac{q}{2}}+k_{-i}^{\frac{q}{2}}\right) }, \end{aligned}$$
$$\begin{aligned} \tilde{c}_{i,q}&= \frac{P q k_i^{\frac{q}{2}} k_{-i}^{\frac{q}{2}}}{2 \left( k_{-i}^{\frac{q}{2}}+k_i^{\frac{q}{2}}\right) ^2}, \end{aligned}$$
$$\begin{aligned} \tilde{C}_q&= \frac{P q k_i^{\frac{q}{2}} k_{-i}^{\frac{q}{2}}}{\left( k_{-i}^{\frac{q}{2}}+k_i^{\frac{q}{2}}\right) ^2}, \end{aligned}$$

where \(k_i = k_i(I_i) = \frac{1}{\gamma _i + \mu I_i}\).

With the solution to the second stage defined by (37)–(40), the objective function reduces to (18). Equating the first-order conditions for \(X\) and \(Y\) yields

$$\begin{aligned} \left( \frac{1}{\mu I_Y+\gamma _Y}\right) ^{\frac{q}{2}} \left( \frac{q+2}{\mu I_X+\gamma _X} \!+\! \frac{q-2}{\mu I_Y+\gamma _Y}\right)&= \left( \frac{1}{\mu I_X + \gamma _X}\right) ^{\frac{q}{2}} \left( \frac{q-2}{\mu I_X+\gamma _X}\!+\!\frac{q+2}{\mu I_Y+\gamma _Y}\right) , \nonumber \\ \text{ which }&\text{ implies } \nonumber \\ \frac{1}{\gamma _X + \mu I_X}&= \frac{1}{\gamma _Y + \mu I_Y}. \end{aligned}$$

To see this, note that there exists no \(\delta \ne 1\) with \(\frac{1}{\gamma _X + \mu I_X} = \delta \cdot \frac{1}{\gamma _Y + \mu I_Y}\), for which the first-order condition holds. From (41) it directly follows that \(k_X = k_Y\).

With this finding, the equilibrium level of total abatement reduces to \(\tilde{A}_q = \sqrt{\frac{Pq}{2k_X}}\) and first-best abatement is given by \(A_I^* = \frac{\bar{E}}{b+k_X}\). Equating \(\tilde{A}_q\) and \(A_I^*\), solving for \(q^*\) is straightforward and results in (19).

Note that to ensure consistency of the mechanism’s abatement incentives, \(q\) is constrained such that total abatement under benchmarking regulation, \(\tilde{A}\), increases with higher levels of \(q\); i.e., abatement incentives are consistent iff \(\frac{\partial \tilde{A}_q}{\partial q}>0\):

$$\begin{aligned} \frac{\partial \tilde{A}_q}{\partial q} = -\frac{P (\sqrt{k_X}+\sqrt{k_Y}) \sqrt{k_X^{\frac{q}{2}} k_Y^{\frac{q}{2}}} \left( k_X^{\frac{q}{2}} \left( -2 + q \cdot \ln \left( \frac{k_X}{k_Y} \right) \right) - k_Y^{\frac{q}{2}} \left( 2 + q \cdot \ln \left( \frac{k_X}{k_Y} \right) \right) \right) }{4 \sqrt{2 P q k_X k_Y} \left( k_X^{\frac{q}{2}}+k_Y^{\frac{q}{2}}\right) ^2}. \end{aligned}$$

Hence, for \(k_Y=\sigma k_X, \frac{\partial \tilde{A}_q}{\partial q}{>}0\) iff

$$\begin{aligned} k_X^{\frac{q}{2}} \left( 2 - q \cdot \ln \left( \frac{k_X}{\sigma k_X} \right) \right)&> (\sigma k_X)^{\frac{q}{2}} \left( -2 - q \cdot \ln \left( \frac{k_X}{\sigma k_X} \right) \right) \nonumber \\ \Longleftrightarrow \left( \frac{1}{\sigma }\right) ^{\frac{q}{2}}&> \frac{q \cdot \ln (\sigma )-2}{q \cdot \ln (\sigma )+2}. \end{aligned}$$

A sufficient condition for the mechanism to yield consistent incentives to abate pollution is \(q \cdot \ln (\sigma )-2 <0\) and thus \(q < \frac{2}{\ln (\sigma )}\). \(\square \)

1.3 Proof of Proposition 4


To derive Proposition 4, we proceed with an unconstrained optimization of (28)—where \(k_Y = \sigma k_X\)—and then derive conditions for potential corner solutions. The unconstrained optimization yields

$$\begin{aligned} \alpha ^*=\frac{2 \bar{E}^2}{P \sqrt{\sigma k_X^2} \left( -2 + \frac{b}{\sqrt{\sigma k_X^2}} + \frac{4 \sqrt{\sigma }}{(1+ \sqrt{\sigma })^2}\right) ^2}. \end{aligned}$$

Hence, for \(\bar{E}>0, \alpha ^* > 0\), thus there is no corner solution at the lower bound of the domain of \(\alpha \). It is also straightforward from (43) that a corner solution \(\alpha ^*=1\) exists iff

$$\begin{aligned} \bar{E} \ge \sqrt{P \cdot \Omega }, \end{aligned}$$


$$\begin{aligned} \Omega =\frac{\sqrt{\sigma k_X^2}}{2} \left( -2 + \frac{b}{\sqrt{\sigma k_X^2}} + \frac{4\sqrt{\sigma }}{\left( 1+\sqrt{\sigma }\right) ^2}\right) ^2. \end{aligned}$$

\(\square \)

1.3.1 Proof of Proposition 5


To prove Proposition 5, we first show that for an interior solution of \(\alpha ^*, A(\alpha ^*)>A^*\). For \(\alpha ^* \in (0,1), A(\alpha ^*)< A^*\) iff

$$\begin{aligned} \underbrace{\frac{\bar{E} (1+\sigma )}{b+b\sigma +2\sigma k_X}}_{A^*}&> \underbrace{\frac{\bar{E}}{\sqrt{\sigma } k_X \left( -2+\frac{b}{\sqrt{\sigma k_X^2}} + \frac{4 k_X \sqrt{\sigma }}{\left( \sqrt{\sigma k_X}+\sqrt{k_X}\right) ^2}\right) }}_{A(\alpha ^*)}. \end{aligned}$$

Condition (46) simplifies to \(1<-\frac{(1+\sigma )^2}{\sqrt{\sigma } (1+\sqrt{\sigma })^2}\), which never holds. Hence, \(\forall \alpha ^* \in (0,1), A(\alpha ^*)>A^*\).

Second, it is straightforward from (44) that \(\alpha ^* = 1\) iff \(\Omega \le \frac{\bar{E}^2}{P}\), where \(\Omega \) is defined in Eq. (32). Finally, from (12) we know that \(A(\alpha ^* \mid \alpha ^*=1)<A^*\) iff \(\Psi < \frac{\bar{E}^2}{P}\), where \(\Psi = \frac{\left( b \sigma + b +2\sigma k_X\right) ^2}{2k_X(1+\sigma )^2 \sqrt{\sigma }}\). This completes the proof. \(\square \)

A Generalized Model with \(n\) Agents

1.1 The Maximization Problem of the Agents

For completeness, we show the results of the model with rent seeking in a generalized framework with \(i = 1, \ldots , n\) firms. For \(i \in \Phi \) and \(-i \in \Phi \backslash \{i\}\), the payoff function of the individual is

$$\begin{aligned} \pi _i = \left\{ \begin{array}{c@{\quad }l} \left( \alpha \frac{m_i c_i^{\frac{1}{2}}}{m_i c_i + \sum _{-i} m_{-i} c_{-i}^{\frac{1}{2}}} + (1-\alpha )\frac{s_i}{s_i+\sum _{-i} s_{-i}}\right) P-c_i-s_i &{} \text {if } max\{c_i, c_{-i}, s_i, s_{-i}\}>0, \\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

First- and second-order conditions allow us to solve for the individual and total equilibrium levels of abatement costs, abatement, and rent seeking.

The first derivative of the payoff function (47) with respect to \(c_i\) is given by

$$\begin{aligned} \frac{\partial \pi _i}{\partial c_i} = \frac{\alpha P c_i^{- \frac{1}{2}} m_i \left( \sum _{-i} c_{-i}^{\frac{1}{2}} m_{-i}\right) }{2 \left( c_i^{\frac{1}{2}} m_i + \sum _{-i} c_{-i}^{\frac{1}{2}} m_{-i}\right) ^2} -1 = 0. \end{aligned}$$

As \(c_1 = c_2 = \cdots = c_i = \cdots = c_n\), we can calculate \(c_i\), which is given by

$$\begin{aligned} \hat{c}_{i,n}&= \frac{\alpha P m_i\left( \sum _{-i} m_{-i}\right) }{2 \left( m_i + \sum _{-i} m_{-i}\right) ^2}. \end{aligned}$$

Total abatement costs \(C\) are simply given by \(\hat{C}_n = n \cdot \hat{c}_{i,n} = \sum _{i=1}^n \hat{a}_{i,n}^2 k_i\).

As we have established that \(a_i = \left( \frac{c_i}{k_i}\right) ^{\frac{1}{2}}\), we can easily compute individual and total abatement levels, i.e., \(\hat{a}_{i,n}\) and \(\hat{A}_n = \sum _{i=1}^{n} \hat{a}_{i,n}\).

The equilibrium level of rent seeking \(\hat{s}_{i,n}\) is derived by using the first-order condition of the payoff function with respect to \(s_i\):

$$\begin{aligned} \frac{\partial \pi _i}{\partial s_i}&= (1-\alpha )P \frac{\sum _{-i} s_{-i}}{s_i+\left( \sum _{-i} s_{-i}\right) ^2}-1 = 0. \end{aligned}$$

Given that \(s_1 = s_2 = \cdots = s_i = \cdots = s_n\), we have

$$\begin{aligned} \hat{s}_{i,n}&= \frac{(1-\alpha )P(n-1)}{n^2}. \end{aligned}$$

Aggregate rent seeking is given by the sum of the \(\hat{s}_{i,n}\): \(\hat{S}_n = \sum _{i=1}^n \hat{s}_{i,n} = n \cdot \hat{s}_{i,n}\).

To summarize, if payoffs are defined by (47), the equilibrium is defined by

$$\begin{aligned} \hat{c}_{i,n}&= \frac{\alpha P \left( \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) }{2 \sqrt{k_i} \left( \frac{1}{\sqrt{k_i}}+\sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}, \\ \hat{C}_n&= \sum _{i=1}^n \frac{\alpha P \sum _{-i} \frac{1}{\sqrt{k_{-i}}}}{2 \sqrt{k_i} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}, \\ \hat{a}_{i,n}&= \sqrt{\frac{\alpha P \left( \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) }{2k_i^{\frac{3}{2}} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}}, \\ \hat{A}_n&= \sqrt{\frac{\alpha P}{2}} \cdot \sum _{i=1}^n \sqrt{\frac{\left( \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) }{k_i^{\frac{3}{2}}\left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}}, \\ \hat{s}_{i,n}&= \frac{(1-\alpha )P(n-1)}{n^2}, \\ \hat{S}_n&= \sum _{i=1}^n \hat{s}_{i,n} = n \cdot \hat{s}_{i,n} = \frac{(1-\alpha )P(n-1)}{n}. \end{aligned}$$

1.2 Welfare Optimization

We now determine the optimal \(\alpha _n\) by minimizing the costs to society:

$$\begin{aligned} \min _{\alpha }&\left[ D(\hat{A}_n)+ \hat{C}_n + \hat{S}_n \right] , \\ \text {s.t. } 0&\le \alpha \le 1, \nonumber \end{aligned}$$


$$\begin{aligned} D(\hat{A}_n)&= \frac{\left( \bar{E} - \sqrt{\frac{\alpha P}{2}} \cdot \sum _{i=1}^n \sqrt{\frac{\left( \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) }{k_i^{\frac{3}{2}}\left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}}b\right) ^2}{2b}, \\ \hat{C}_n&= \frac{\alpha P}{2} \sum _{i=1}^n \frac{\sum _{-i} \frac{1}{\sqrt{k_{-i}}}}{\sqrt{k_i} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}, \\ \hat{S}_n&= \frac{(1-\alpha )P(n-1)}{n}. \end{aligned}$$

Rearranging the first-order condition allows one to solve for the interior solution of the optimal \(\alpha \) in the \(n\)-player case:

$$\begin{aligned} \alpha _n^* = \frac{2 \bar{E}^2 \left( \sum _{i=1}^n \sqrt{\frac{\sum _{-i} \frac{1}{\sqrt{k_{-i}}}}{k_i^{\frac{3}{2}} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}}\right) ^2}{P\left( -4 + \frac{4}{n} + b \left( \sum _{i=1}^n \sqrt{\frac{\sum _{-i} \frac{1}{\sqrt{k_{-i}}}}{k_i^{\frac{3}{2}} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2}}\right) ^2 + 2 \sum _{i=1}^n \frac{\sum _{-i} \frac{1}{\sqrt{k_{-i}}}}{\sqrt{k_i} \left( \frac{1}{\sqrt{k_i}} + \sum _{-i} \frac{1}{\sqrt{k_{-i}}}\right) ^2} \right) ^2}. \end{aligned}$$

It is easy to check that this is equivalent to the results of the case with two firms.  Therefore, the underlying incentives remain unchanged.

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Gerigk, J., MacKenzie, I.A. & Ohndorf, M. A Model of Benchmarking Regulation: Revisiting the Efficiency of Environmental Standards. Environ Resource Econ 62, 59–82 (2015).

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