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Abstract

We analyze how $N$ firms producing a commodity good with a polluting by-product respond to environmental taxes. These firms vary in operational and environmental efficiency. Under Cournot competition, we examine two demand functions: iso-elastic and linear. We establish the existence and uniqueness of the equilibrium and examine how it changes with tax levels and environmental efficiency. Our findings suggest that taxes may not always benefit firms with green, low-cost technologies, as they erode the advantages of operational efficiency. Additionally, total and per-unit emissions may increase with higher taxes or improved environmental efficiency. Finally, the observed effects can qualitatively differ based on the form of market demand.

Keywords

Environment Pollution Regulation Tax Competition Technology Heterogeneity Market Demand Function

1. Introduction

Environmental taxes refer to charges levied on activities which are considered to be harmful to the environment. As summarized by the European Commission (Remeur, 2020): “The aim of environmental taxation, in principle, is to factor environmental damage, or negative externalities, into prices in order to steer production and consumption choices in a more eco-friendly direction.” A common example is the carbon tax: as of 2023, it has been adopted in 37 countries (Worldbank, 2023); other examples include taxes on industrial waste and water disposal, and charges levied on specific pollutants (Harrington and Morgenstern, 2004).

In this article, we investigate the extent to which the environmental taxation can incentivize the use of cleaner production technologies by heterogenous competing firms. The reason to be skeptical is that several earlier sustainable operations studies demonstrated that environmental taxes can, under some conditions, incentivize dirty technologies. First, in the general monopolistic case, Krass et al. (2013) showed that this could be driven by fixed costs. As the tax increases, the cleaner technologies become more profitable, but the increase in total costs leads to higher equilibrium prices and lower production volumes. Thus, the total additional profit may be insufficient to offset the fixed costs of cleaner technologies, leading the firm to select the dirtier technology instead. Subsequently and in more specific contexts, Sunar and Plambeck (2016) showed that in the context of border adjustment taxes the same effect could be driven by the rules allocating emissions among co-products, and Aflaki and Nettesine (2017) showed that in the context of electricity markets this effect could be driven by the intermittency of renewable energy sources.

This article extends the general-context model of Krass et al. (2013) to oligopolistic competition and shows that while environmental taxes can be an effective mechanism under many conditions, the limitations and potential for undesirable outcomes are even stronger in this context. Moreover, the negative effects are more subtle and arise from the nature of the oligopolistic competition itself; fixed costs play no role in the current article, and hence the previously proposed remedies will not work either.

We consider an industry with $N$ firms producing a commodity-type product in a Cournot setting: firms set production quantities and the market determines the equilibrium price. Producing one unit of product emits a certain amount of pollutant which depends on the production technology used; greener technologies emit less pollutant per unit of production, and dirtier emit more. Firms are heterogeneous in technologies they use and in their base-line costs. This heterogeneity reflects the fact that in many polluting industries (e.g., steel, aluminum, and cement), firms’ locations, energy sources, and other characteristics lead to significant variations in operational and environmental efficiencies, as reflected in the so-called cost curves (ANFRE, 2016; SteelOnTheNet, 2019). Importantly, the operational and environmental parameters are not necessarily coupled: a firm that uses fossil-fuel energy may have both, high emissions, and high fuel costs, while the one that uses solar energy likely has lower emissions and minimal fuel-related costs. We refer to this setting, where no a priori relationship is assumed between environmental and operational parameters, as full heterogeneity.

Firms pay environmental tax per unit of pollutant emitted, with the tax level set exogenously by the regulator. We analyze how changes in the tax rate affect the structure of the market equilibrium (i.e., which firms participate in it), the equilibrium production quantities, market shares, profits of individual firms, as well as industry-wide measures such as the total production quantity, total industry profits and total emissions, as well as per-unit emissions (the emission intensity). We analyze these effects under two different market demand functions: iso-elastic (log-log) and linear, which cover the spectrum of effects one might observe with many other demand functions, see Section 1.2.

1.1. Key Findings and Insights

Our main findings can be summarized as follows:

Finding 1: Structure of the Equilibrium

In Section 3, we show that at any tax level, $t$ , the unique equilibrium exists and corresponds to a partition into active firms with positive production quantities and inactive firms with zero production. Inactive firms are important: active firms strategically soften competition by choosing quantities that prevent inactive firms from entering the equilibrium.

Finding 2: Asymptotic Sensitivity Analysis

In Section 4, we analyze the case of the k-clean equilibrium—a situation where the active production set consists of $k$ firms operating the cleanest technologies. We focus on two related questions: is k-clean equilibrium guaranteed to emerge as $t$ gets large, and whether the regulator can control the value of $k$ (in particular, can $k = 1$ be induced, ensuring that only the cleanest firm is active). We show that the answers dramatically depend on the demand function. In the iso-elastic case, while the answer to the first question is “yes,” the answer to the second question is “no”: above a certain taxation level, $k$ does not depend on $t$ and thus the regulator may never exclude certain dirtier firms from the active production set. The key concepts here are operational efficiency (low production cost) and environmental efficiency (clean technology). The “best” firms (with high operational and environmental efficiency) may see lower profits and market shares as the tax increases. In the linear demand case, the size of the active production set $k$ does shrink with $t$ , and is thus under regulator’s control. However a $k$ -clean equilibrium may not exist: at every tax level, the positive production set may include firms not operating the cleanest technologies; even when, at some sufficiently large $t$ , this set shrinks to just one active firm. The clean and efficient firms generally benefit from higher taxes in the linear case, but their market share may decrease with $t$ .

Finding 3: Local Sensitivity Analysis

In Section 4, we also analyze the impact of “local” changes in $t$ , ensuring that the active set remains invariant. We find that with iso-elastic demand, the best (operationally and environmentally efficient) firms may be hurt by higher tax rates (the profits and market shares may decline with $t$ ), while they are guaranteed to benefit in case of linear demand. Both the industry-wide profits and, paradoxically, even the emission intensity, or per-unit emissions (PUEs), may increase with $t$ . This finding echoes results observed by Gao et al. (2023), though in a substantially different context.

Finding 4: Sensitivity With Respect to Environmental Efficiency

In Section 5, we analyze the impact of varying the environmental efficiency of one of the firms while holding $t$ constant. For the most part the sensitivity effects are similar to those with respect to $t$ , with one notable exception: as the technology of one of the firms becomes cleaner, enabling it to capture higher market share and increase production quantity, not only the PEU, but even the total emissions can increase.

Our work generates four insights, linked to the technical conditions for when the effects arise:

Insight 1: Environmental Taxation Erodes the Benefits of operational Efficiency [Theorems I.4 and L.4, Corollary I.2, and Examples 2 and 4]

Firms that are both operationally and environmentally efficient may see their profits and market shares decreased when environmental taxes are increased. This is because taxes erode benefits of operational efficiency—which is particularly important for the operations scholars to emphasize.

Insight 2: Industry Profits and PUEs May Increase With Tax [Theorems I.6 and L.6 and Examples 3 and 5]

The fact that industry profits may increase confirms prior findings from Anand and Giraud-Carrier (2020) and Fu et al. (2023) and extends them to oligopolistic competition case with full heterogeneity and non-linear demands, where this effect emerges over much wider regions. The potential increase in PUEs is a new insight, which is particularly troubling. By increasing costs, environmental taxes naturally increase prices and thus reduce production. The hope, however, is that due to market share gains of environmentally cleaner firms the resultant emissions decline even more. We show that this is not always the case.

Insight 3: Per Unit and Total Emissions May Increase as Technologies Become Greener [see Section 5]

This new insight is also troubling. The effect stems from the fact that when one firm’s technology becomes greener, its own emissions intensity decreases, but the total production increases, which can offset the emissions reduction at the focal firm.

Insight 4: How the Industry Responds to Tax May Qualitatively Differ Depending on the Demand Function Used [Examples 2 and 4; the ConditionsUnder Which Different Forms of the Demand Function May Arise Are Discussed in the Last Paragraph of Section 1.2]

This is another new and important insight for sustainable operations. Very few papers consider alternative demand models; we show that such analysis is crucial.

1.2. Literature Review

While the literature on various regulatory aspects of sustainable operations, including e-waste regulation, disclosures, eco-labels, and so on, is quite extensive, there are only a few studies that focus on environmental taxation, see recent reviews by Atasu et al. (2020) and Sunar and Swaminathan (2022). Moreover, these studies often make restrictive assumptions, such as that the industry consists of $n$ players with one technology and $m$ players with another (Anand and Giraud-Carrier, 2020; Drake, 2018) or a further simplification to $n = m = 1$ (Fu et al., 2023; Huang et al., 2021; Mohammadi et al., 2024); quality choice and eco-labeling (Murali et al., 2019) could also be viewed as “green technology.” All firms with identical technologies are assumed to incur equal variable costs, and those that have invested in greener technologies (lower PUEs) to have higher fixed and variable costs. All these studies assume linear demand.

Anand and Giraud-Carrier (2020) and Fu et al. (2023) are the two papers that are closest to ours and merit a more detailed discussion. The former considers the impact of cap-and-trade and emissions tax on the competitive industry with $n$ firms with low-slope emission abatement curves and $m$ high-slope firms (their Section 6 mentions heterogeneous costs and emissions, but no details are provided). Linear demand with a cross-elasticity parameter (reflecting the degree of substitutability between products) is assumed; the classical Cournot model emerges when this parameter is set to $1$ . Anand and Giraud-Carrier (2020) established two main results. First, they show that industry profit with some tax $t > 0$ could exceed that with $t = 0$ , echoing our Insight 2. Second, they shows that the cap-and-trade and tax mechanisms are equivalent; it would be interesting to verify whether this equivalence extends to our setting.

Fu et al. (2023) considered the production quantity decisions in response to an environmental tax in an industry with $N = 2$ firms with technologies that have symmetric costs but asymmetric emission intensities. Supplemental Material presents two extensions, and most transparent results are obtained for a special case with one clean firm and $N - 1$ identical dirty firms. A fully heterogeneous case is not considered, and linear demand is assumed throughout. Their main result echoes (Anand and Giraud-Carrier, 2020): the industry profit may increase with tax rate $t$ .

Also relevant to our work are the economics studies on the impact of regulation on green technology adoption. They utilize a different set of assumptions: both the “classical” (Carlsson, 2000; Katsoulacos and Xepapadeas, 1995; Levi and Nault, 2004; Requate, 2006) and the recent papers, for example, (Bian et al., 2020; Buccella et al., 2021), typically assume that the quantities of interest, such as demands, emissions, production, and abatement costs, are represented by continuous twice-differentiable functions that jointly possess a set of desirable properties. This approach is convenient for subsequent derivations, but misses important operational details, such as that in practice managers rarely deal with a continuous “abatement functions”; rather, they are faced with distinct technological alternatives that exhibit full heterogeneity. For this reason, most of the effects listed in Section 1.1 are invisible within the frameworks adopted in this literature.

Lastly, to the best of our knowledge (Cohen et al., 2016) is the only other paper in sustainable operations literature to investigate the impact of the form of market demand function—a key aspect of our paper. Three arguments support our focus on the iso-elastic (log-log) and linear forms. First, both functions are very popular in the management science literature. The linear demand function is ubiquitous in analytical investigations due to its simple algebraic form; the iso-elastic function is more popular in empirical studies, because the demand elasticity can be directly estimated by running a log-log regression. These two functional forms are used in six out of nine studies mentioned by Cohen et al. (2021) and in the meta-analyses of hundreds of studies referenced in Table 6.3 of Bodea and Ferguson (2014). Second, Cohen et al. (2021) showed that for many commonly used demand functions (including quadratic, monomial, and semi-log) optimizing the price under the appropriately selected linear demand incurs a very small profit loss. In contrast, approximating an iso-elastic demand curve with a linear one leads to unbounded profit losses. Therefore, by considering the linear and iso-elastic forms, we effectively cover the spectrum of effects one might observe with many other functions. Third, the linear and iso-elastic demand functions are based on profoundly different assumptions about the underlying consumer preferences: the former corresponds to a uniform distribution of consumers’ willingness to pay (WTP) over some bounded interval of prices $[0, p^{\max}]$ , while the latter corresponds to the power distribution of WTP, with positive demand at any price. Demand elasticity, the rate of reduction in demand as price increases, reflects the degree of substitutability as consumers switch from the focal to alternative goods. With iso-elastic demand, elasticity is constant at any price. With linear demand, the focal good becomes more substitutable as price increases, in effect intensifying competition with alternatives.

We note that in the current article, we treat technology choices as given and study the production decisions of competing firms; a situation where firms simultaneously choose technologies and production quantities presents a rather different set of technical challenges.

2. The Model

We consider $N$ competing firms and a government regulator; unless ambiguous, we use $N$ to represent both, the set of firms and their number. Our model of the firm closely follows that by Krass et al. (2013); we only provide a brief overview here. Firm $j \in N$ is characterized by a variable cost $c_{j}$ per unit (“operational efficiency”), a fixed cost $K_{j}$ , and a PUE cost $t / S_{j}$ , where $t$ is the environmental tax rate (set exogenously by the regulator) and $S_{j} \geq 1$ is the “environmental efficiency”—the quantity of output that generates one unit of emissions. Let $β_{j} (t) = c_{j} + \frac{t}{S_{j}}$ denote the total variable production cost of firm $j$ ; observe that it is a linear function of $t$ .

When firm $j$ produces $x_{j}$ units of the good, it incurs the total cost of $K_{j} + (c_{j} + t / S_{j}) x_{j} = K_{j} + β (t)_{j} x_{j}$ and emits $x_{j} / S_{j}$ units of pollutant. Firms are thus heterogenous along three dimensions: per unit emissions, $1 / S_{j}$ , fixed costs, $K_{j}$ , and variable costs, $c_{j}$ .

We consider Cournot competition with the equilibrium market price $p$ and demand function $D (p)$ satisfying $D (p) = x^{T}$ , where $x^{T} = \sum_{j \in N} x_{j} \equiv x_{j} + x_{- j}$ . The firm’s only decision is the production quantity, $x_{j} \geq 0$ , and using the standard game-theoretic notation, $x_{- j}$ denotes the total production quantity of all firms other than $j$ . Throughout the article, we consider two specific demand functions:

Iso-elastic demand: $D (p) = A p^{- δ}$ , where $δ \geq 1$ is demand elasticity and $A > 0$ is the scaling constant that can be interpreted as the market size. Assuming $x^{T} > 0$ , the equilibrium market price equals:

\begin{aligned} p (x^{T}) = {(\frac{A}{x^{T}})}^{\frac{1}{δ}} . \end{aligned}

(1)

Linear demand: $D (p) = a - b p$ , where $b$ is the demand slope (price sensitivity) and $a$ is the intercept (market size). Assuming $p \in [0, a / b]$ , the equilibrium market price equals:

\begin{aligned} p (x^{T}) = \frac{a - (x^{T})}{b} . \end{aligned}

(2)

The profit of firm

j

producing

x_{j}

units when other firms produce a total of

x_{- j}

units equals:

\begin{aligned} π_{j} (x_{j}, x_{- j}) = x_{j} \times [p (x_{j} + x_{- j}) - β_{j}] - K_{j} . \end{aligned}

(3)

3. Existence, Uniqueness, and Structure of the Equilibrium

For $j \in N$ let $x_{j}^{*} (x_{- j}) = \arg max_{x_{j} \geq 0} π_{j} (x_{j}, x_{- j}) .$ Functions $x_{j}^{*} (\cdot), j \in 1, \dots, N$ represent Cournot best response, that is, the optimal reaction of firm $j$ to a given production vector by other firms $x^{*} = (x_{1}^{*} (\cdot), \dots, x_{N}^{*} (\cdot))$ . Vector $x^{*}$ is a Nash equilibrium iff $π_{j} (x_{j}^{*}, x_{- j}^{*}) \geq π_{j} (x, x_{- j}^{*})$ $\forall x \geq 0, j = 1, \dots, N .$ We refer to this game as $Γ^{I} (t)$ for the case of iso-elastic demand given by (1), and by $Γ^{L} (t)$ for the linear case given by (2). The parameter $t$ emphasizes the dependence on the taxation rate $t$ , but whenever the tax rate $t$ is fixed, we drop it to simplify notation. We designate the iso-elastic demand results with $I$ (e.g., Theorem I.1), and the linear ones with $L$ (e.g., Theorem L.1). All proofs are presented in the E-Companion.

3.1. Equilibrium for Iso-Elastic Demand

Theorem I.1
For a fixed tax level $t > 0$ , let $x = (x_{1}, \dots, x_{n}), x_{i} \geq 0$ be a vector of production quantities. Let $N^{+} = {j \in {1, \dots, N} | x_{j} > 0}$ and $N^{0} = {j \in {1, \dots, N} | x_{j} = 0} = N - N^{+}$ . We will refer to $N^{+}$ as the set of positive production or active firms, and to $N^{0}$ as the zero production or inactive firms. Then: (i)
The vector $x$ is an equilibrium for game $Γ^{I} (t)$ iff it satisfies the following system of inequalities:
$\begin{aligned} δ \sum_{l \in N^{+}} β_{l} + β_{j} - δ | N^{+} | β_{j} {\begin{cases} > 0 & if j \in N^{+}, \\ \leq 0 & if j \in N^{0} . \end{cases} \end{aligned}$
(4)
(ii)
For $j \in N^{+}$ , the equilibrium production quantity, the total equilibrium production, and the equilibrium profit are given by the following equations:
$\begin{aligned} x_{j} & = \frac{A (| N^{+} | - 1 / δ)^{δ}}{(\sum_{l \in N^{+}} β_{l})^{δ + 1}} (δ \sum_{l \in N^{+}} β_{l} + β_{j} - δ | N^{+} | β_{j}), \end{aligned}$
(5)

$\begin{aligned} x^{T} & = \sum_{j \in N^{+}} x_{j} = A {[\frac{| N^{+} | - \frac{1}{δ}}{\sum_{l \in N^{+}} β_{l}}]}^{δ}, \end{aligned}$
(6)

$\begin{aligned} π_{j} & = \frac{A (| N^{+} | - 1 / δ)^{δ - 1}}{δ (\sum_{l \in N^{+}} β_{l})^{δ + 1}} \\ \times {(δ \sum_{l \in N^{+}} β_{l} + β_{j} - δ | N^{+} | β_{j})}^{2} - K_{j} . \end{aligned}$
(7)
(iii)
For $j \in N^{0}, x_{j} = 0, π_{j} = - K_{j}$ .

The first summand in the RHS of (7), which equals $π_{j} + K_{j}$ , represents the “operating profit” and the equilibrium only requires that the operating profit be positive. The total profit in (7), $π_{j}$ , may be negative (e.g., by part (iii) above, $π_{j} < 0$ for any $j \in N^{0}$ with $K_{j} > 0$ , but this may also be the case for some $j \in N^{+}$ ). We implicitly assume that a firm will not cease production if the operating profit does not offset the fixed production costs; we treating the latter as “sunk costs.”

It is intuitive that for a given $t$ the positive production set must consist of firms with the lowest total variable costs. This intuition is confirmed by the following result.
Theorem I.2
For a fixed $t > 0$ , arrange all firms by non-decreasing total variable costs: $β_{1} (t) \leq β_{2} (t) \leq \dots \leq β_{N} (t)$ . Then there exists $1 \leq k (t) \leq N$ such that the unique equilibrium in $Γ^{I} (t)$ is given by the partition $N^{+} (t) = {1, \dots, k (t)}$ and $N^{0} (t) = {k (t) + 1, \dots, N}$ . The corresponding equilibrium production vector $x^{}$ is given by (5).

That $k (t) \geq 1$ for any $t$ implies that even as the environmental tax (and thus the equilibrium price) approaches infinity, at least one firm is able to achieve positive operating profit. In fact, as we will see in Section 4.1, the asymptotic positive set typically consists of several firms.

Importantly, the tax level $t$ affects the structure of the equilibrium in three separate ways: (1) it increases the variable cost* $β_{j} (t)$ for each firm, which directly affects the equilibrium operating profits, total production $x^{T}$ , and the equilibrium price, (2) it affects the ordering of firms by $β_{j} (t)$ , which determines the composition of $N^{+} (t)$ and $N^{0} (t)$ , and (3) it affects the size $k (t)$ of the positive production set $N^{+} (t)$ . The latter two effects are structural. The next result characterizes these, showing that the equilibrium structure can only change at a discrete number of critical taxation levels, which allows for efficient computation of the equilibrium and sensitivity analyses.
Theorem I.3
There exist a discrete set $T^{c}$ of at most $O (N^{3})$ critical taxation levels, such that the ordering of $β_{j} (t)$ and the composition of $N^{+} (t), N^{0} (t)$ stay invariant for $t$ between any consecutive values in $T^{c}$ ; the composition can change only when $t$ crosses a value in $T^{c}$ .

At a first glance, the preceding result may seem obvious as the variable costs $β_{j} (t), j \in N$ are linear in $t$ and thus their ordering can only change when these linear functions cross—clearly a discrete set. However, the size of $N^{+}$ is also a function of $t$ , making the whole function nonlinear and necessitating a more careful analysis. The set $T^{c}$ indeed contains the crossing points of $β_{j} (t)$ , but also additional points where the membership of $N^{+}$ changes, as the next example illustrates.
Example 1
Structure of the Equilibrium in the Iso-Elastic Demand Case. Consider $N = 5$ firms with the following characteristics: $c_{1} = 3, S_{1} = 6$ , $c_{2} = 2.05, S_{2} = 4$ , $c_{3} = 2, S_{3} = 3$ , $c_{4} = 1.5, S_{4} = 2$ , and $c_{5} = 1, S_{5} = 1$ , with $A = 10^{6}$ and $δ = 1.2$ . Observe that firms are ordered by $S_{j}$ , that is, firm $1$ has the most environmentally efficient (“cleanest”) technology and firm $5$ the “dirtiest.” On the other hand, the variable costs $c_{j}$ are ordered in the opposite direction. Thus, firm $5$ has the lowest total variable cost $β_{j} (t)$ when the tax is close to $0$ and the highest $β_{j} (t)$ when the tax level is high. We will use this basic setting, which is illustrated in Figure 1, in examples throughout the article.
Figure 1.
Critical taxation levels and the structure of the equilibrium in the iso-elastic demand case.

Consider some $t$ value in the shaded area of Figure 1, for example, $t = 1.75$ . At this $t$ , the ranking of the firms by total variable cost is $4, 2, 3, 5, 1$ . Applying Theorems I.1 and I.2, we can compute $k (1.75) = 4$ , that is, $N^{+} = {2, 3, 4, 5}$ consists of the four firms with lowest total variable costs, and $N^{0} = {1}$ .

We now illustrate how to compute the set $T^{c}$ from Theorem I.3. As mentioned earlier, $T^{c}$ consists of two kinds of breakpoints. First, are the points where the linear functions $β_{j} (t)$ cross and thus the ranking of the firms changes; we label them as $T^{R} = {0.6, 1.0, 1.4, 1.5, 2.2, 2.4, 3.0}$ and indicate them with dashed vertical lines in Figure 1. The corresponding firm rankings are also given.

To these we must add breakpoints at which $N^{+}$ could change. To illustrate these, consider what happens when tax is varied between two consecutive points in $T^{R}$ , for example, between $t_{0} = 1.5$ and $t_{m} = 2.2$ —the shaded area in Figure 1. For $t \in [1.5, 2.2]$ , the ordering of firms by total variable costs is fixed at $4, 2, 3, 5, 1$ . Since at $t = 1.75$ $N^{+} = {2, 3, 4, 5}$ , by Theorem I.2, there are two possible changes that could happen: firm $5$ could depart $N^{+}$ , or firm $1$ could join it. We compute the respective left-hand sides of (4). For firm 5, it equals $δ \sum_{i \in N^{+}} β_{i} (t) + β_{5} (t) (1 - 4 δ) = 1.2 (\sum_{i \in N^{+}} c_{i} + \sum_{i \in N^{+}} \frac{t}{S_{i}}) - 3.8 (1 + t) = 4.06 - 1.3 t$ which equals zero at $t = 3.123 \notin [1.5, 2.2]$ . Therefore, firm $5$ (and thus also firms $2, 3, 4$ ) will remain in $N^{+}$ for all $t \in [1.5, 2.2]$ . In contrast, for firm 1, we obtain $δ \sum_{i \in N^{+}} β_{i} (t) + β_{1} (t) (1 - 4 δ) = \dots = - 3.54 + 1.867 t$ which equals zero at $t = 1.896 \in [1.5, 2.2]$ . Thus, at $t = 1.896$ (which we label as $t^{+}$ in Figure 1 and add to $T^{c}$ ) firm $1$ enters $N^{+}$ , which then remains invariant for $t \in [1.896, 2.2]$ .

Proceeding iteratively, we find that firm $5$ exits $N^{+}$ at $t = 2.809$ which falls between successive breakpoints $2.4$ and $3$ in $T^{R}$ ; we label it $t^{-}$ and add to $T^{C}$ as illustrated in Figure 1. The remaining breakpoints can be computed similarly.

To finish this section, we explore an important special case of N-positive equilibrium—a situation where all firms are active ( $| N^{+} (t) | = N$ ), which may represent the status quo and thus be politically attractive to the regulator.
Corollary I.1
The N-positive equilibrium $N^{+} (t) = N$ exists for some $t = t^{'}$ iff $\forall j \in N$ , $β_{j} (t^{'}) \in [\bar{β} - ϵ, \bar{β} + ϵ]$ for some $\bar{β} > 0$ and $0 < ϵ \leq \bar{β} / (2 N δ - 1)$ . Moreover, if $N^{+} (t^{'}) = N$ then there exist $0 \leq t_{m i n} \leq t^{'} < t_{m a x}$ such that for any $t \in [t_{m i n}, t_{m a x}), N^{+} (t) = N$ .

This corollary links the existence of the N-positive equilibrium with the total variable costs all falling within a “band” around some $\bar{β}$ , which may be interpreted as the industry-wide “normalized” production cost at some $t$ . Two observations are in order. First, the width of this band may be fairly large. This happens when the $\bar{β}$ is large and the number of firms $N$ is small. Thus, the positive equilibrium may not be an unusual case: to have a non-positive equilibrium, with $N^{+} (t) \subset N$ , requires sufficiently large variation in production and/or environmental efficiencies. Second, if all firms have similar variable production costs $c_{j}$ , but different environmental efficiencies $S_{j}$ , the positive equilibrium condition will hold for lower values of $t$ , but not for higher values. Conversely, if the environmental efficiency of all technologies is similar, then we would see a positive equilibrium for higher values of $t$ , but not necessarily for lower ones. We further explore the impact of $t$ on the equilibrium in Section 4.
3.2. Equilibrium for Linear Demand

In this section, we extend the results of the previous section to the linear demand function (2).

Theorem L.1
With sets $N^{+}, N^{0}$ defined as in Theorem I.1: (i)
The vector $x$ is an equilibrium for game $Γ^{L} (t)$ iff it satisfies the following system of inequalities:
$\begin{aligned} a + b [\sum_{l \in N^{+}} β_{l} - (1 + | N^{+} |) β_{j}] {\begin{cases} > 0 & if j \in N^{+}, \\ \leq 0 & if j \in N^{0} . \end{cases} \end{aligned}$
(8)
(ii)
For $j \in N^{+}$ , the equilibrium production quantity, the total equilibrium production, and the equilibrium profit are given by the following equations:
$\begin{aligned} x_{j} & = \frac{a + b [\sum_{l \in N^{+}} β_{l} - (1 + | N^{+} |) β_{j}]}{1 + | N^{+} |}, \end{aligned}$
(9)

$\begin{aligned} x^{T} & = \sum_{j \in N^{+}} x_{j} = \frac{| N^{+} | a - b \sum_{l \in N^{+}} β_{l}}{1 + | N^{+} |}, \end{aligned}$
(10)

$\begin{aligned} π_{j} & = \frac{{(a + b [\sum_{l \in N^{+}} β_{l} - (1 + | N^{+} |) β_{j}])}^{2}}{b (1 + | N^{+} |)^{2}} - K_{j} . \end{aligned}$
(11)
(iii)
For $j \in N^{0}$ , $x_{j} = 0$ and $π_{j} = - K_{j}$ .

Theorem L.2
For a fixed $t > 0$ , arrange all firms by non-decreasing total variable costs: $β_{1} (t) \leq β_{2} (t) \leq \dots \leq β_{N} (t)$ . Then there exists $k (t) \leq N$ such that the unique equilibrium in $Γ^{L} (t)$ is given by the partition $N^{0} (t) = {k (t) + 1, \dots, N}$ and $N^{+} = N - N^{0}$ . When $N^{+} \neq \emptyset$ (i.e., $k (t) > 0$ ), the corresponding equilibrium production vector $x^{*}$ is given by (9). Moreover, for $j \in N$ , let $t_{j}^{m a x} = (\frac{a}{b} - c_{j}) S_{j}$ and $t^{m a x} = max_{j \in N} t_{j}^{m a x}$ . Then $t \geq t_{j}^{m a x}$ implies that $k (t) < j$ (i.e., ${j, \dots, N} \subset N^{0} (t)$ ), and $k (t) > 0$ iff $t < t^{m a x}$ .

Both previous results are very similar to their iso-elastic counterparts, with the exception of the last clause in Theorem L.2, which indicates that the positive production set is empty for sufficiently large $t$ , specifically, for $t > t^{\max}$ .
Theorem L.3
Theorem I.3 holds for the linear demand: there exists a discrete set $T^{c} \sim O (N^{3})$ s.t. the structure of the equilibrium in $Γ^{L} (t)$ is invariant between any two consecutive values in $T^{c}$ .

As in the iso-elastic case, $T^{C}$ consists of two subsets: the set $T^{R}$ consisting of crossing points of linear functions $β_{j} (t)$ changes, which is the same as in the iso-elastic case, and the set of tax levels where the membership of $N^{+} (t)$ changes, which is determined by the left-hand side of (8) and, therefore, differs from its iso-elastic counterpart. Finally, the positive equilibrium Corollary I.1 has a linear demand equivalent as well:
Corollary L.1
The positive equilibrium $N^{+} (t) = N$ exists for some $t = t^{'}$ iff $\forall j \in N$ , $β_{j} (t^{'}) \in [\bar{β} - ϵ, \bar{β} + ϵ]$ for some $\bar{β} > 0$ and $0 < ϵ \leq a - b \bar{β} / b (1 + 2 N)$ . Moreover, if $N^{+} (t^{'}) = N$ then there exist $0 \leq t_{m i n} \leq t^{'} < t_{m a x}$ such that for any $t \in [t_{m i n}, t_{m a x}), N^{+} (t) = N$ .

The key difference between this result and its iso-elastic counterpart is the expression for the upper bound on $ϵ$ . This upper bound is increasing in $\bar{β}$ for the iso-elastic case, and decreasing for the linear case. Interpreting $\bar{β}$ as the industry-wide normalized cost, we see that a high normalized cost favors the existence of positive equilibrium in the iso-elastic case (widening the band in which all $β_{j} (t)$ must fall), while having the opposite effects in the linear case. Noting that increasing the tax rate $t$ tends to increase $\bar{β}$ , we see that larger taxes may lead to increasing the market participation (i.e., the size of $N^{+} (t)$ ) in the iso-elastic case, but narrowing it in the linear case. This intuition will be made more precise by the results in the following section.
4. Impact of Environmental Taxation on the Equilibrium

In this section, we examine how the composition of the equilibrium, as well as equilibrium market shares, profits, and emissions, change in response to changes in $t$ . We are interested in the extent to which environmental taxation can be used to incentivize firms that adopted environmentally efficient technologies. We start with an asymptotic analysis of the structure of the equilibrium as $t$ gets large and $N^{+} (t)$ changes with $t$ , and then turn our attention to the “local” effects of changes in tax level within a relatively small range of values around a given $t$ where $N^{+} (t)$ is constant. We first consider the iso-elastic demand case, turning to the linear case in Sections 4.3 and 4.4.

4.1. Asymptotic Impact of $t$ in the Iso-Elastic Demand Case

A natural question stemming from Theorems I.2 and I.3 is whether it is possible, by selecting a sufficiently high tax rate, to ensure that $N^{+}$ contains only the firms with “acceptably clean” technologies. To this end, we make the following definition:

Definition 1
For $k \geq 1$ an equilibrium is said to be “ $k$ -clean” if $N^{+} = {1, \dots, k}$ when firms are ordered by non-increasing environmental efficiencies $S_{j}$ , $j \in N$ .
Theorem I.4
Order firms by non-increasing values of $S_{j}, j \in N$ and assume that $t$ is large enough so that $β_{j} (t) \leq β_{j + 1} (t)$ implies $S_{j} \geq S_{j + 1}$ . Define as follows:
$\begin{aligned} {\bar{e}}_{j} = \frac{1}{j} \sum_{l = 1}^{j} \frac{1}{S_{l}} and e_{j} = \frac{1}{S_{j}} [1 - \frac{1}{δ j}] . \end{aligned}$
(12)
Let $t^{c}$ be the largest critical value in set $T^{c}$ defined in Theorem I.3 and let $\tilde{j} = min {j \in N : {\bar{e}}_{j} \leq e_{j}} - 1$ . Then $\tilde{j} > 0$ , ${\bar{e}}_{j} \leq e_{j}$ for all $j > \tilde{j}$ , and ${\bar{e}}_{j} > e_{j}$ for $j \leq \tilde{j}$ . Moreover, $\exists \tilde{t} \leq t^{c}$ s.t. for any $t > \tilde{t}$ , we have ${1, \dots, \tilde{j}} \subseteq N^{+} (t)$ . Furthermore, if ${\bar{e}}_{\tilde{j} + 1} < e_{\tilde{j} + 1}$ then $\forall t > \tilde{t}$ , $N^{+} (t) = {\tilde{N}}^{+} = {1, \dots, \tilde{j}}$ , and $N^{0} (t) = {\tilde{N}}^{0} = {\tilde{j} + 1, \dots, N}$ .

This result implies that a $k$ -clean equilibrium is induced for all large enough $t$ , but the value of $k$ is not under regulator’s control—it is fully determined by the technological efficiencies of the firms. The intuition is as follows. The quantity ${\bar{e}}_{j}$ can be interpreted as the average environmental impact (per unit produced) of the $j$ cleanest firms, and $e_{j}$ is the “adjusted” environmental impact of firm $j$ . Once $t$ is sufficiently large to ensure that variable costs are ordered by environmental efficiencies (i.e., $S_{1} \geq \dots \geq S_{N}$ ), the asymptotic positive production set ${\tilde{N}}^{+}$ contains first $\tilde{j}$ firms for which the adjusted environmental impact $e_{j}$ is lower than the average ${\bar{e}}_{j}$ . We call such firms “relatively efficient.” The ones that are not relatively efficient are typically excluded, with some additional conditions required for firms that are “on the cusp” (i.e., ${\bar{e}}_{j} = e_{j}$ ); these conditions are listed in the more complete statement of Theorem I.4 provided in the e-companion. Note that due to ordering of the firms, the actual (i.e., unadjusted) impact $1 / S_{j}$ of firm $j$ cannot be lower than ${\bar{e}}_{j}$ . However, the adjustment term $(1 - \frac{1}{δ j}) < 1$ , and thus $e_{j} < 1 / S_{j}$ , making ${\bar{e}}_{j} > e_{j}$ possible. The term $(1 - \frac{1}{δ j})$ can thus be interpreted as the relative impact threshold—its value defines the possible gap between the cleanest and the dirtiest firms included in ${\tilde{N}}^{+}$ . Note that this threshold decreases when the demand elasticity $δ$ is high, and/or when there are many firms possessing cleaner technologies, leading to large $\tilde{j}$ . Both of these factors are not influenced by $t$ , and thus the value of $\tilde{j}$ is outside the regulator’s control.

While Theorem I.4 established only the structure of the asymptotic positive production set, it also has implications for other measures of interest, such as market shares, profits, and emissions.
Corollary I.2
Assume firms are ordered by non-increasing $S_{j}, j \in N$ . Let $\tilde{t}, \tilde{j}$ , and ${\tilde{N}}^{+} = {1, \dots, J}$ be as defined in Theorem I.4, ${\tilde{N}}^{+} = N^{+} (t)$ for $t > \tilde{t}$ and $J \geq \tilde{j}$ . For $t \geq 0$ , let $M S_{j} (t) = x_{j} (t) / x^{T} (t)$ be the market share of firm $j$ . Then, for $j \in {\tilde{N}}^{+}$ , as $t \to \infty$ : (i)
The market shares converge to ${\tilde{M S}}_{j} = δ (1 - \frac{e_{j}}{{\bar{e}}_{J}} \frac{S_{J}}{S_{j}}),$ with $e_{j}, {\bar{e}}_{j}$ defined in Theorem I.4. ${\tilde{M S}}_{j}$ s are non-increasing in $j$ and may be larger or smaller than $M S_{j} (t)$ for some $t \leq \tilde{t}$ .
(ii)
The profit $π_{j} (t)$ of firm $j$ converges to ${\tilde{π}}_{j} (t) = \frac{A δ}{t^{δ - 1}} {(\frac{J δ - 1}{δ J {\bar{e}}_{J}})}^{δ - 1} {[1 - \frac{(δ J - 1)}{δ J S_{j} {\bar{e}}_{J}}]}^{2} - K_{j}$ as $t \to \infty$ . These quantities are non-increasing in $j$ and may be larger or smaller than the operating profit $π_{j} (t)$ for some $t \leq \tilde{t}$ .
(iii)
The PUEs converge to ${\tilde{P U E}}_{j} = \sum_{j = 1}^{J} \frac{{\tilde{M S}}_{j}}{S_{j}},$ which may be smaller or larger than the PUEs at $t \leq \tilde{t}$ . However, the total emissions $(T E s)$ are converging to $0$ as $t \to \infty$ .

Thus, similar to the composition of the asymptotic positive production set, the asymptotic market shares and PUEs are fully determined by firms’ environmental technologies, particularly by the relationship between each firm’s adjusted environmental impact $e_{j}$ and the average impact ${\bar{e}}_{J}$ for $j \in {\tilde{N}}^{+}$ and $J = | {\tilde{N}}^{+} |$ . The regulator has no ability to influence these quantities once the tax rate $t$ exceeds $\tilde{t}$ . While the asymptotic profit is decreasing with $t^{δ - 1}$ being in the denominator, the rate of decrease is identical for all firms in ${\tilde{N}}^{+}$ , and for $δ$ close to $1$ is quite slow. The profit impact on an individual firm is solely determined by the ratio of $e_{j}$ and ${\bar{e}}_{j}$ . Moreover, a firm possessing the cleanest technology may have higher market share, higher operating profit and lower PUEs at some smaller (possibly zero) tax level than at a higher one. This effect is due to the difference between operational efficiency $c_{j}$ and environmental efficiency $S_{j}$ . Asymptotic market shares and PUEs are driven only by the latter, while the same quantities at lower tax levels are affected by both. Thus a firm with high operational efficiency (i.e., low $c_{j}$ ) may have a high market share at low tax levels, but at higher tax level the advantage due to operational efficiency is lost, which may cause the market share, as well as profit to decline. For the same reason, the per-unit total emissions may increase when such a firm loses market share to relatively dirtier firms (though total emissions are guaranteed to fall). This effect is illustrated in the following example.
Example 2
Asymptotic Behavior With Iso-Elastic Demand. Continuing with the setting of Example 1, for $t \geq 11.4$ the ranking of firms by non-increasing $β_{j}$ coincides with the ordering of firms by non-increasing $S_{j}$ : $1, 2, 3, 4, 5$ . Applying Theorem I.4, we compare ${\bar{e}}_{j}$ and $e_{j}$ for all $j \in N$ and observe that the first firm for which the condition ${\bar{e}}_{j} > e_{j}$ fails is firm 4 ( ${\bar{e}}_{4} = 0.104, e_{4} = 0.198$ ), and thus, $\tilde{j} = 3$ . Since the inequality ${\bar{e}}_{4} < e_{4}$ is strict, the “moreover” clause of Theorem I.4 applies and for all $t \geq 11.4$ , we have $J = \tilde{j} = 3, {\tilde{N}}^{+} = {1, 2, 3}$ and ${\tilde{N}}^{0} (t) = {4, 5}$ . That is, even though firm 3 is twice as dirty as the cleanest firm 1 (since $S_{3} / S_{1} = .5$ ), it is nevertheless impossible to exclude it from the equilibrium via environmental taxation. This is because the elasticity $δ = 1.2$ is relatively low; it is not hard to verify that firm 3 would be excluded for $δ \geq 1.334$ .

We can also compute the tax threshold $\tilde{t}$ required for the asymptotic results to set in. In this example, $\tilde{t} = 11.4$ coincides with the value of $t$ above which $β_{j}$ are ranked by environmental technology, but in general the asymptotic positive production set may emerge well before the values of $β_{j}$ are ranked that way. For example, changing $S_{1}$ from 6 to 5 leads to the non-increasing ordering of $β$ s for $t \geq 19$ , yet ${\tilde{N}}^{+} = {1, 2, 3}$ emerges at $\tilde{t} = 12.7$ .

We also note that in both cases above, the value of $\tilde{t}$ is much higher than the ones discussed in Example 1. Indeed, the total equilibrium output is greatly reduced for such high tax levels: in our example, the total production $x^{T} (11.4) = \sum_{l \in N^{+} (\tilde{t})} x_{l} (\tilde{t}) = 93, 583$ , while $x^{T} (0) = 418, 038$ , a four-fold reduction in the total market size over the no-tax regime. The equilibrium price increases by almost $3.5$ times from $2.07$ at $t = 0$ to $7.20$ at $t = 11.4$ . That is, compared to $t = 0$ regime, not all of the tax increase is passed on to consumers; the total production is not just reduced but is also re-allocated to firms with cleaner technologies, which are less sensitive to the tax.

The results of Corollary I.2 are illustrated in Figure 2. Panel (a) shows that firm 1, which possesses the cleanest technology, benefits from increasing $t$ at the expense of the less technologically efficient firms 2 and 3. Note that even though the asymptotic ${\tilde{N}}^{+}$ emerges at $t = 11.4$ , the asymptotic market shares are clearly not reached even at $t = 20$ . Panel (b) of Figure 2 shows that the profit of firm 1 also increases with $t$ . Importantly, the total industry profit is seen to be increasing in $t$ over a large interval. This observation echoes the results of Anand and Giraud-Carrier (2020), who also showed the possibility of industry profit increasing with $t$ (though they only consider linear demand; see Section 4.4). Finally, panel (c) shows the total emissions of each firm (left axis) as well as the PUEs (solid blue line, right axis). Total emissions are decreasing with $t$ for all firms; it is interesting to note that the largest emitter is firms 2, even though at higher tax rates it has lower market share and profit than firm 1; in fact the volume of emissions of all three firms in the asymptotic equilibrium are very close for all $t \geq \tilde{t}$ . Similarly, the PUEs are near-constant around 0.22, translating to average $S = 1 / .22 = 4.54$ , which is well below the best available technology’s $S_{1} = 6$ .
Figure 2.
Market shares, profits, and emissions as a function of $t$ for the iso-elastic demand Example 2. Panels (a) to (c): the original setting. Panels (d) to (f): modified setting.

The previous discussion shows that the three-dimensional impact of the tax level (reduction in production quantities, increase in prices, and reallocation of production from dirtier to cleaner firms) can result in a situation where high tax rates are providing benefits to firm(s) operating the cleanest technologies. On the other hand, the limitations of the taxation mechanism are also evident: from the failure to exclude firms with inefficient technology from the positive production set, to PUEs being far above the level of the best available technology.

However, as suggested in Corollary I.2, increasing $t$ could also “punish” the clean(est) firm(s) and “reward” the dirtier one(s). Illustrating this situation requires only a small modification of our setting: changing the variable costs of firm $1$ to $c_{1} = 0.5$ from the original value of $3$ , making it both, the most operationally and environmentally efficient (lowest $c$ and highest $S$ ).

The results for this modified setting, depicted in panels (d) to (f) of Figure 2, are quite dramatic. Panel (d) shows that as $t$ is increased, the market share of firm 1 falls from the maximum value of $\approx 85 %$ reached at a low tax rate $t \approx 1.8$ to the asymptotic market share of $72.4 %$ , while the substantially dirtier firm 2 ( $S_{2} = 4$ vs. $S_{1} = 6$ ) is the clear beneficiary. Similar negative impacts of taxation can be seen in panel (e), where the profits of firm 1, as well as of the industry as a whole, decline with $t$ . Finally, panel (c) shows that PUE increases for $t > 4$ (even though each individual firm’s emissions decrease). As discussed earlier, the reason for these phenomena is that while firm 1 is more environmentally efficient than the other firms, it is even more operationally efficient, and thus its competitive advantage erodes as $t$ is increased.
4.2. Local Impact Around a Given $t$ in the Iso-Elastic Demand Case

From Theorem I.3, we know that for any $t > 0$ there exist values $t_{\min}, t_{\max}$ such that the set $N^{+} (t)$ is invariant for $t \in [t_{\min}, t_{\max}]$ . For the remainder of this section, we limit $t$ to this range (and therefore treat $N^{+}$ as constant), and examine the “local” impact of changes in $t$ on firms $j \in N^{+}$ . We introduce the following four quantities that have intuitive interpretations:

\begin{aligned} \bar{e} & = \frac{1}{| N^{+} |} \sum_{l \in N^{+}} \frac{1}{S_{l}}, Δ_{j}^{e} = \bar{e} - \frac{1}{S_{j}} (1 - \frac{1}{| N^{+} | δ}), \\ \bar{c} & = \frac{1}{| N^{+} |} \sum_{l \in N^{+}} c_{l}, Δ_{j}^{c} = \bar{c} - c_{j} (1 - \frac{1}{| N^{+} | δ}) . \end{aligned}

(13)

Here set $N^{+}$ plays the same role as the asymptotic set ${\tilde{N}}^{+}$ in the previous section. Thus, $\bar{e}$ can be interpreted as the average environmental impact of firms in the positive production set, and $Δ_{j}^{e}$ is the gap between $\bar{e}$ and the adjusted impact of firm $j$ , with the term $1 - \frac{1}{| N^{+} | δ}$ representing the relative impact threshold, as before. We start by examining the impact $t$ on profits.

Theorem I.5

Consider firm $j \in N^{+}$ . (i)

If $Δ_{j}^{e} \leq 0$ , then the profit $π_{j} (t)$ is a non-increasing function of $t$ .

(ii)

If $Δ_{j}^{e} > 0$ , let

\begin{aligned} t_{j}^{c r i t} = \frac{2 Δ_{j}^{e} \bar{c} - Δ_{j}^{c} \bar{e} (δ + 1)}{Δ_{j}^{e} \bar{e} (δ - 1)} . \end{aligned}

(14)

Then

π_{j} (t)

is increasing for

t \in [t_{m i n}, min {t_{j}^{c r i t}, t_{m a x}})

and decreasing for

t \in (max {t_{m i n}, t_{j}^{c r i t}}, t_{m a x}]

The intuition behind case (i) is clear: here firm $j$ is “dirty” relative to the average firm in $N^{+}$ ; increasing $t$ decreases its profit, thus “punishing” the dirty producer. Case (ii) is less clear: here firm $j$ is “clean” relative to the average firm in $N^{+}$ ; its profit benefits from increasing $t$ , but only for $t \leq t_{j}^{c r i t}$ ; beyond that it is “punished” by increasing taxes. The second effect seems counter-intuitive. Examining (14), we see that for $Δ_{j}^{c} > 0$ , $t_{j}^{c r i t}$ is increasing in $Δ_{j}^{e}$ and decreasing in $Δ_{j}^{c}$ . Again, the first part seems quite intuitive: as the environmental efficiency improves relative to $N^{+}$ , the range of tax rates where increasing the tax is beneficial for firm $j$ expands. The second part is more perplexing: why would improvement in relative operational efficiency $Δ_{j}^{c}$ make the range of taxes benefitting the firm narrower? The answer lies in the two different effects of the environmental taxation. The cost advantage effect reduces variable costs of firms with cleaner technologies as $t$ increases. On the other hand, the market shrinkage effect increases market prices as $t$ is increased, thus shrinking the total production, which negatively impacts the profits of all firms. Which of these effects is stronger for a given firm $j$ depends on the relationship between $Δ_{j}^{c}$ and $Δ_{j}^{e}$ .

Next, we analyze the impact of the tax rate on the market shares and emissions of firms in $N^{+}$ .

Theorem I.6

For firm $j \in N^{+}$ , (i)

The market share $M S_{j} (t)$ , $j \in N^{+}$ increases in $t \in [t_{m i n}, t_{m a x}]$ iff $\frac{\bar{c}}{\bar{e}} < \frac{c_{j}}{1 / S_{j}} .$

(ii)

The per-unit emissions $P U E (t)$ increase in $t \in [t_{m i n}, t_{m a x}]$ iff $\frac{\bar{c}}{\bar{e}} < \frac{\sum_{l \in N^{+}} c_{l} / S_{l}}{\sum_{l \in N^{+}} 1 / S_{l}^{2}}$ . The total emissions $T E (t)$ decrease when $P E U (t)$ is decreasing, but may increase or decrease when $P E U (t)$ increases.

Focusing first on part (i), note that in order for a firm to gain market share as a result of an increase in $t$ , its ratio of per-unit operational cost $c_{j}$ over its environmental efficiency $1 / S_{j}$ must be higher than the corresponding ratio for average quantities in the positive production set $N^{+}$ . This implies that firms that certainly lose market share (“losers”) are ones that are relatively operationally efficient ( $c_{j} < \bar{c}$ ) and environmentally inefficient ( $1 / S_{j} > \bar{e}$ ). Similarly, clear “winners” are the relatively operationally inefficient firms that are relatively environmentally efficient (high $c$ and high $S$ ). The impact on market share of the “best firms,” which are both operationally and environmentally efficient (low $c_{j}$ and high $S_{j}$ ) and the “worst” firms (operationally and environmentally inefficient, high $c_{j}$ and low $S_{j}$ ) is less clear—their market share may go up or down. Again, the issue here is the interplay of cost advantage and market shrinkage effects.

Turning our attention to part (ii) of Theorem I.6, we can interpret the numerator on the right-hand side of the inequality as the operational cost weighted by the environmental efficiency: this term is large when larger $c$ values are combined with smaller $S$ values and vice versa, that is, when $N^{+}$ contains firms that are operationally and environmentally inefficient, as well as ones that are operationally and environmentally efficient (the “best” and ”worst” firms above). In this case, the overall PUEs may increase with $t$ . On the other hand, if $N^{+}$ consists mostly of “winners” and “losers” above, the $PEU (t)$ will likely decrease with $t$ . Since the total emissions $TE (t) = x^{T} (t) PEU (t)$ and the total production $x^{T} (t)$ is always decreasing with $t$ , when $PEU (t)$ is decreasing, so will the TEs. The behavior of $TE (t)$ when $PEU (t)$ is increasing is quite complex (see expressions for $\partial TE (t) / \partial t$ in the e-companion) and, under some conditions, $TE (t)$ may increase with $t$ .

Example 3

Local Impact of $t$ on Profits, Market Shares, and Emissions in Iso-Elastic Demand Case. We continue with the modified setting of Example 2, depicted on panels (d) to (f) of Figure 2. The set $N^{+} = {1, 2, 3}$ is invariant with $t$ for $t \geq 3.91$ , which is also the asymptotic positive production set. Recall that in this setting firm $1$ is the “best” ( $c_{1} = 0.5, S_{1} = 6$ ), while firms $2$ and $3$ are mediocre ( $c_{2} = 2.05, c_{3} = 2$ , while $S_{2} = 4, S_{3} = 3$ ).

The profit of each firm as a function of $t$ is illustrated in Figure 2(e). From Theorem I.5, we compute: $t_{j}^{c r i t} = {- 37.4, 54.9, - 25.1}$ for $j = 1, 2, 3$ . Thus profit of firm $2$ increases with $t$ until $t = 54.9$ , and then decreases with $t$ , while the profits of firms 1 and 3 decrease with $t$ for all $t \geq 3.91$ . Therefore, increasing $t$ punishes the best firm 1, as well as firm 3, with the benefits accruing to the mediocre firm 2 (in fact, the best firm 1 suffers a steep decline in profits for $t > 3.91$ ).

Turning our attention to market shares, we compute the ratios in part (i) of Theorem I.6: $\bar{c} / \bar{e} = 6.07, c_{j} / (1 / S_{j}) = {6, 8.2, 3}$ for $j = 1, 2, 3$ . Thus, the market shares of firms $1$ and $3$ decrease with $t$ , while that of firm $2$ increases with $t$ for $t > 3.91$ ; this is confirmed by panel (d) in Figure 2.

Finally, we compute the condition of part (ii) in Theorem I.6: the left-hand side is $\bar{c} / \bar{e} = 6.07$ , while the right-hand side is $6.27$ . Since the inequality holds, we expect $P U E$ to increase with $t$ for $t > 3.91$ , see Figure 2(f). Interestingly, $T E$ is decreasing in this case, in spite of increasing $P U E$ .

4.3. Asymptotic Impact of

t

in the Linear Demand Case

From Theorem L.2, we already know that in the linear demand case the positive production set is empty for $t \geq t^{\max}$ . Thus we focus on changes in the positive production set as $t \to t^{\max}$ , particularly on the extent to which increasing $t$ can “reward” firms with cleaner technology.

As in Definition 1, we order firms by environmental efficiency (non-increasing order for $S_{j}, j \in N$ ; ties are broken by non-decreasing variable costs $c_{j}$ ) and ask whether a $k$ -clean equilibrium can be induced for some $k \leq N$ by setting $t$ sufficiently high, that is, $t_{k} \leq t \leq t^{\max}$ for some $t_{k} < t^{\max}$ ? In particular, is it possible to induce a $1$ -clean equilibrium consisting of the most environmentally efficient firm? As shown in the following result, in general, the answer to both questions is “no”: the firms’ variable costs and environmental efficiencies have to be sufficiently differentiated in order to allow for the tax rate $t$ to separate the cleanest $k$ firms from the rest. Here we see another major difference from the iso-elastic case: whereas in the former only environmental efficiencies $S_{j}$ played a role in the asymptotic regime, in the linear case costs $c_{j}$ also matter.

Before stating the result we define the following quantities that are closely related to their counterparts for the iso-elastic case defined in equation (13). For $k, j \in N$ , let

\begin{aligned} {\bar{e}}_{k} & = \frac{1}{k} \sum_{l = 1}^{k} \frac{1}{S_{l}}, Δ_{j k}^{e} = \frac{k + 1}{k} \frac{1}{S_{j}} - {\bar{e}}_{k}, \\ {\bar{c}}_{k} & = \frac{1}{k} \sum_{l = 1}^{k} c_{l}, Δ_{j k}^{c} = \frac{k + 1}{k} c_{j} - {\bar{c}}_{k} . \end{aligned}

(15)

As before, $\bar{e_{k}} and {\bar{c}}_{k}$ represent the average environmental impacts (respectively, variable costs) for the $k$ cleanest firms, while $Δ_{j k}^{e} and Δ_{j k}^{c}$ are the differences between the adjusted and average impacts (respectively, variable costs) for firm $j \in N$ . We have the following result:

Theorem L.4

Assume firms are ordered by non-increasing order of $S_{j}$ with ties broken by non-decreasing variable costs $c_{j}$ for $j \in N$ . For $k \in N$ tax rate $t \in [0, t^{m a x})$ induces a $k$ -clean equilibrium $N^{+} (t) = {1, \dots, k}$ iff there exists $t$ satisfying the following inequalities:

\begin{aligned} t \geq t_{k}^{s} & = max_{1 \leq j \leq k} max_{k < r \leq N} \frac{c_{r} - c_{j}}{\frac{1}{S_{j}} - \frac{1}{S_{r}}}, t \geq t_{k}^{0} = max_{k < j \leq N} \frac{\frac{a}{b k} - Δ_{j k}^{c}}{Δ_{j k}^{e}}, \end{aligned}

(16)

\begin{aligned} t < t_{k}^{+} & = min {\frac{\frac{a}{b k} - Δ_{j k}^{c}}{Δ_{j k}^{e}} : 1 \leq j \leq k, Δ_{j k}^{e} > 0}, \\ t > t_{k}^{-} & = max {\frac{\frac{a}{b k} - Δ_{j k}^{c}}{Δ_{j k}^{e}} : 1 \leq j \leq k, Δ_{j k}^{e} < 0} . \end{aligned}

(17)

Moreover,

1

-clean equilibrium exists iff there exists

t

satisfying

t_{1}^{m a x} \geq t > max {t_{1}^{s}, t_{1}^{0}}

, where

t_{1}^{m a x} = S_{1} (\frac{a}{b} - c_{1})

t_{1}^{s} = max_{1 < j \leq N} \frac{(c_{1} - c_{j})}{\frac{1}{S_{j}} - \frac{1}{S_{1}}}

, and

t_{1}^{0} = max_{1 < j \leq N} \frac{\frac{a}{b} - 2 c_{j} + c_{1}}{\frac{2}{S_{j}} - \frac{1}{S_{1}}} .

The conditions above come directly from the equilibrium existence conditions of Theorems L.1 and L.2. Indeed, for a $k$ -clean equilibrium to exist, the remaining $N - k$ firms must satisfy conditions (8) for inclusion in $N^{0} (t)$ , resulting in the two inequalities (16). Also, for $j \in {1, \dots, k}$ , conditions (8) for inclusion in $N^{+} (t)$ must be met, leading to (17).

While (16) always imposes lower bounds on $t$ , (17) imposes a mixture of upper and lower bounds, depending on the sign of $Δ_{j k}^{e}$ . From (15), $Δ_{j k}^{e} > 0$ means that firm $j \leq k$ is relatively environmentally inefficient compared to other firms in ${1, \dots, k}$ , thus $t$ must be sufficiently low to allow this firm to remain in $N^{+} (t)$ , leading to an upper bound $t_{k}^{+}$ in (17). Similarly, if $j$ is relatively efficient (i.e., $Δ_{j k}^{e} < 0$ ), then $t$ needs to be sufficiently high to ensure that $j$ is part of the equilibrium, leading to the lower bound $t_{k}^{-}$ in (17). Conditions for the existence of $1$ -clean equilibrium arise directly from applying (16, 17) to the $k = 1$ case.

Theorem L.4 has several important implications. One immediate consequence is that if firms $j$ and $j + 1$ possess the same environmental technology (i.e., $S_{j} = S_{j + 1}$ ), then they cannot be separated by $t$ since the denominator in (16) will be $0$ . More generally, if $S_{j} \approx S_{j + 1}$ then $t_{j}^{s}$ will be very large (possibly higher than $t^{\max}$ ), unless the difference in the variable costs $c_{j + 1} - c_{j}$ is also very large. Thus, in order for firm $j$ to be included in a $k$ -clean equilibrium, while firm $j + 1$ is excluded, there must be a sufficiently large separation between them with respect to environmental efficiencies and/or variable costs. For a specific $k \in N$ the existence $k$ -clean equilibrium is not guaranteed as we illustrate below.

Example 4

Asymptotic Effect of $t$ on the Equilibrium Structure in the Linear Demand Case. We continue with the same parameters as in Examples 1 and 2. To make the two settings comparable, we normalize the linear demand function parameters $a, b$ so that at $t = 2$ the equilibrium is the same as in the iso-elastic case; this leads to $a \approx 506$ , and $b \approx 81$ .

We next compute $t_{k}^{s}, t_{k}^{0}, t_{k}^{+}, t_{k}^{-}$ for $k = 5, \dots, 1$ (ordering firms from the dirtiest to the cleanest), obtaining $t_{k}^{s} = {0, 1, 3, 2.2, 11.4}, t_{k}^{0} = {0, 2.61, 5.83, 9.07, 15.43}$ , $t_{k}^{+} = {2.61, 5.83, 9.07, 15.24, 19.46}$ , and $t_{k}^{-} = {1.77, 0.50, - 15.51, \emptyset, \emptyset}$ , where $\emptyset$ corresponds to cases when $Δ_{j k}^{e} > 0$ for all $1 \leq j \leq k$ . Taking the maximum of $t_{k}^{s}, t_{k}^{0}, t_{k}^{-}$ as the lower bound, and $t_{k}^{+}$ as the upper bound we see that $k$ -clean equilibria exist for all $k = 1, \dots, 5$ , with the corresponding ranges of $t$ given by $[1.77, 2.61), [2.61, 5.83), [5.83, 9.07), [9.07, 15.43), and [15.43, 19.46)$ . Thus, in this example, the regulator can induce any $k$ -clean equilibrium by setting $t$ to the appropriate range—having far more control than in the iso-elastic case in Example 2. It is interesting to note that $k$ -clean equilibrium is not induced for $t \in [0, 1.77)$ , where $N^{+} (t) = {2, 3, 4, 5}$ , which excludes the cleanest firm $1$ .

However, the situation where any $k$ -clean equilibrium can be induced is quite fragile. For example, changing the environmental efficiency of firm 1 from $S_{1} = 6$ to 4.5 (we call this the modified setting for this example) prevents the existence of $1$ -clean equilibrium (here $t_{1}^{s} = 34.2$ , while $t_{1}^{+} = 14.59$ , and thus the lower bound exceeds the upper bound). In this case for $t \in [11.79, 16.77)$ , we have $N^{+} (t) = {2}$ . Intuitively, with this change in $S_{1}$ , firms $1$ and $2$ are no longer sufficiently differentiated with respect to the environmental efficiency to allow for separation by the tax rate. Interestingly, for $k = 2, \dots, 5$ , the $k$ -clean equilibria do exist in this case. Importantly, a similar effect can be also induced by changing $c$ (recall that $c$ did not impact asymptotic behavior in the iso-elastic case): for example, keeping $S_{1} = 6$ but changing $c_{2}$ from $2.05$ to $2.95$ leads to a situation where a $3$ -clean equilibrium exists for $t \in [6.56, 9.57)$ , but for $t \in [9.57, 10.48)$ the equilibrium becomes $N^{+} (t) = {1, 3}$ , that is, a $2$ -clean equilibrium does not exist. The $1$ -clean equilibrium then emerges for $t \in [10.48, 19.46)$ .

To summarize, we obtained several insights about the asymptotic behavior of the equilibrium with linear demand. First, provided the firms are sufficiently well-separated either with respect to the environmental efficiencies or with respect to the variable costs (or both), the regulator has full control over the number of firms, $k$ , in the $k$ -clean equilibrium by setting $t$ to an appropriate range. Second, the existence of a $k$ -clean equilibrium for a given $k$ depends on both, the environmental efficiencies and on operational efficiencies of the firms. When environmental efficiencies of two or more firms are sufficiently close it may not be possible to separate them into positive and zero production sets by changing $t$ (unless close environmental efficiencies are offset by a large gap in operational efficiencies). Note that these insights are quite different to those under the iso-elastic demand, where the $k$ -clean equilibrium is guaranteed to exist for some $k$ , but its composition does not depend on operational efficiencies at all, and the regulator has no control over $k$ .

4.4. Local Impact Around a Given

t

in the Linear Demand Case

In parallel to Section 4.2, we next analyze the local impact of environmental tax in the case with linear demand, that is, where $N^{+} (t)$ is invariant for $t$ in some range $[t_{\min}, t_{\max}]$ . As we will see, while the general structure of the results is similar to the iso-elastic case, there are also clear differences.

Recall that $\bar{e} = \frac{1}{| N^{+} |} \sum_{l \in N^{+}} \frac{1}{S_{l}}, \bar{c} = \frac{1}{| N^{+} |} \sum_{l \in N^{+}} c_{l}$ are the average environmental impact and production cost, respectively. Similarly to (13), we define as follows:

\begin{aligned} Δ_{j}^{e - lin} = \bar{e} - \frac{1}{S_{j}} (1 + \frac{1}{| N^{+} |}), \end{aligned}

(18)

which can be interpreted (identically to the iso-elastic counterpart) as the extent to which firm

j

is “relatively environmentally efficient” versus the average firm in

N^{+}

. Observe that the relative efficiency threshold term depends only on the size of

| N^{+} |

in the linear case.

Theorem L.5

Consider firm $j \in N^{+}$ . The profit $π_{j} (t)$ is non-increasing (non-decreasing) in $t \in [t_{m i n}, t_{m a x}]$ if and only if $Δ_{j}^{e - l i n} \leq 0$ ( $\geq 0$ ).

This result is quite intuitive: firms that are relatively environmentally inefficient, that is, with $Δ_{j}^{e - lin} \leq 0$ , see their profits fall, and vice versa. This is in contrast with Theorem I.5 in the iso-elastic case, where impact of $t$ on profits is more complex, and where the relative operational efficiency, in addition to environmental efficiency, plays a role. Next we present the counterpart to Theorem I.6:

Theorem L.6

For firm $j \in N^{+}$ , (i)

The market share $M S_{j} (t)$ , $j \in N^{+}$ increases in $t \in [t_{m i n}, t_{m a x}]$ iff $\frac{a - b c_{j}}{1 / S_{j}} > \frac{a - b \bar{c}}{\bar{e}}$ .

(ii)

The per-unit emissions $P U E (t)$ increase in $t \in [t_{m i n}, t_{m a x}]$ iff $\frac{\bar{c}}{\bar{e}} < \frac{\sum_{l \in N^{+}} c_{l} / S_{l}}{\sum_{l \in N^{+}} 1 / S_{l}^{2}}$ . However, the total emissions $T E (t)$ are always decreasing with $t$ .

The impact of environmental efficiency on market share (part (i) above) is similar to the iso-elastic case: a cleaner firm (with higher $S_{j}$ ) will see its market share increase with $t$ . However, the impact of operational efficiency is the opposite: an operationally efficient (low $c$ ) firm may lose market share in the iso-elastic case, but not in the linear case since lower $c$ means higher $a - b c_{j}$ . There are also important differences from the iso-elastic case in the impact of $t$ on per-unit and total emissions (part (ii) above). While $PUE (t)$ may increase or decrease with $t$ , the total emissions $TE (t)$ in the linear case are guaranteed to decrease even when per-unit emissions are increasing.

Overall, the impact of local tax increases are much more intuitive in the linear case: the best firms (with low $c_{j}$ and high $S_{j}$ ) will gain market share, while the worst firms (high $c_{j}$ and low $S_{j}$ ) will lose it. The impact on the “mixed” firms (those with high $S_{j}$ and high $c_{j}$ ) is less clear. Recall that the situation was the opposite in the iso-elastic case: the “mixed” firms were the clear winners of tax increases, while the “best” firms could lose market share and profits.

Example 5

Local Effect of $t$ in the Linear Demand Case. We use the original and modified settings in Example 4; as illustrated in Figure 3. Examining the profit plots in Figure 3(b) and (e), we observe that profits of the cleanest firms $1, 2, 3, 4$ increase with $t$ over certain ranges of $t$ . In fact, for $t \in [2.0, 2.65]$ , we have the $N$ -positive equilibrium case, and the profits of firms $1, 2, 3$ increase in $t$ . The total industry profit is generally decreasing in $t$ , but could be locally increasing, for example, approaching $t = 2.61$ when firm $5$ leaves $N^{+}$ . This is consistent with the results by Anand and Giraud-Carrier (2020). Recall that under the iso-elastic demand industry profit could be increasing for a much wider range of tax levels.

Figure 3.

Linear demand Example 5: profits, market shares, and emissions as a function of $t$ ; original setting in panels (a) to (c) and modified setting in panels (d) to (f).

Next, comparing the market shares in panels (a) and (d), we observe that at each $t$ the market shares are increasing for all firms, except for the firm that will be “next to leave” the positive production set. For the original setting depicted in panel (a), where the $k$ -clean equilibria are always present, this firm is always the dirtiest one in the current set $N^{+} (t)$ . However, in the modified setting depicted in panel (d) this is not always the case: for example, the cleanest firm 1 leaves the equilibrium at $t = 11.79$ .

Finally, comparing the emissions plots on panels (c) and (f) we note that while total emissions $T E$ are decreasing, over certain regions of $t$ , emissions of some firms, including the cleanest ones, are increasing—a direct consequence of their increasing market shares. This situation is potentially troubling, as it might damage the public perception of the cleanest firms and disincentivize them from investing in cleaner technologies. Even more provocatively, the PUEs could also be increasing with $t$ . This case is illustrated for $t \in [10.05, 11.79)$ on the magnified fragment of panel (f), where the $N^{+} (t) = {1, 2}$ and the “if” condition in part (ii) of Theorem L.6 holds.

5. Sensitivity to the Environmental Efficiency Parameter

S_{j}

In this section, we study the impact of the technological environmental efficiency parameter, $S_{j}$ , on the structure of the equilibrium, market shares, profits, and emissions. More specifically, we consider a firm $j \notin N^{+}$ with some initial value of the environmental efficiency parameter $S_{j}^{0}$ and analyze the impact of increasing $S_{j}$ , while all other model parameters (including $t$ ) remain constant. The main insights, illustrated in Figure 4, are as follows:

Structure of the equilibrium: Three regions. There exist two breakpoints $S_{j}^{M i n C r i t}$ and $S_{j}^{M a x C r i t} \geq S_{j}^{M i n C r i t} > S_{j}^{0}$ that divide the interval of possible values of $S_{j}$ into up to three regions:

Region (non-competitive) occurs for $S_{j} \in [0, S_{j}^{M i n C r i t})$ ; note $S_{j}^{0}$ is also in NC by definition. Here the environmental efficiency of firm $j$ is too low to enter the positive production set $N^{+}$ . Its market share $M S_{j}$ and profit $π_{j}$ are zero, other firms are unaffected by $S_{j}$ .

Region (competitive) occurs when $S_{j} \in [S_{j}^{M i n C r i t}, S_{j}^{M a x C r i t})$ so that firm- $j$ enters the positive production set, but is not the only firm there. This region is empty if $S_{j}^{m i n C r i t} = \infty$ .

Region (monopolistic), occurs when $S_{j} \geq S_{j}^{M a x C r i t}$ (this region may also be empty when $S_{j}^{M a x C r i t} = \infty$ ). In this region, $j$ is the only producing firm with market share $M S_{j} = 1$ .

Figure 4.

Impact of increase in environmental efficiency $S_{1}$ of firm $1$ on market shares, profits, and emissions in Example 2 for the iso-elastic demand (panels (a) to (c); tax rate $t = 16$ ) and in Example 4 for the linear demand case (panels (d) to (f); $t = 12$ ).

These regions are represented by the three shades of gray in Figure 4. Note, for the iso-elastic case in panels (a) to (c), $S_{j}^{M a x C r i t} = \infty$ and thus region does not exist.

The conditions for regions and to exist are $c_{j} < δ \sum_{i \in N^{+}} β_{i} / (δ | N_{j}^{+} | - 1)$ and $c_{j} < (δ - 1) / δ \times min_{k} β_{k}$ , respectively, for the iso-elastic demand case, and $c_{j} < a + b \sum_{i \in N^{+}} β_{i} / (| N^{+} | + 1)$ and $c_{j} < min {\frac{a}{b}, 2 min_{k} β_{k} - \frac{a}{b}}$ for the linear demand case. Intuitively, these expressions imply that for regions and to exist, the base-line cost of firm $j$ measured by $c_{j}$ has to be sufficiently small relative to the variable costs $β_{k}$ of other firms (note that the latter are increasing with $t$ and thus the conditions above hold for sufficiently large $t$ ). The details of the proof, as well as the closed-form expressions for $S_{j}^{M i n C r i t}, S_{j}^{M a x C r i t}$ are given in Theorem ec.1 in the E-Companion.

Market shares and profits. Theorem ec.2 of the E-Companion shows that in region the profit $π_{j}$ and market share $M S_{j}$ of firm $j$ are increasing with $S_{j}$ (profit $π_{j}$ also increases with $S_{j}$ in region “M”). For another firm $k \in N^{+}$ in region , while the market share is always decreasing with $S_{j}$ , the behavior of the profit depends on the demand function. For the iso-elastic case, profits $π_{k}$ may increase until $S_{j}$ reaches a certain threshold, after which the profit will decline; the increase happens if the variable cost $β_{k}$ is sufficiently low relative to other firms in $N^{+}$ . Thus, with iso-elastic demand, for a certain range of $S_{j}$ , firm $j$ may not be the sole beneficiary of improvement in its technological efficiency. For linear demand, the profit of firm $k \in N^{+} - {j}$ is always decreasing with $S_{j}$ .

Total and PUEs. While both total and PUEs of firm $j$ (and the industry overall) decrease with $S_{j}$ in region , the behavior in region is more complex. Both per-unit and total emissions may initially increase with $S_{j}$ , for example, see panel (c) of Figure 4 for the iso-elastic demand case. Moreover, total emissions may be increasing while the PUEs are decreasing, for example, see panel (f) of Figure 4 for the linear demand case. Intuitively, this occurs when firm $j$ gains the market share very rapidly, leading to the drop in the equilibrium price and the increase in the equilibrium production quantity. This increase may off-set gains in environmental efficiency $S_{j}$ until the latter becomes sufficiently high. See Theorem ec.3 of the E-Companion for details.

6. Directions for Future Research

We study how an industry consisting of fully heterogenous firms responds to environmental taxes under linear and iso-elastic demands. Our results and insights are summarized in Section 1.1. Below we discuss potential extensions and future work.

As noted earlier, we assumed that firms come endowed with technologies and study their “market,” that is, production quantity, response. A situation where firms simultaneously select their technologies and production levels is analyzed in the companion working paper. We believe that both settings are of practical importance: while technology selections do occur, they are rare; much of the response to environmental taxation is likely to be of market type studied here. Both market and technological responses provide incentives to develop greener technologies, for example, see Wang et al. (2020), and it would be of interest to extend our fully heterogeneous competition model to study such incentives. Broadly, welfare implications of both the market-only response considered in this article, and a more general technology and market response to environmental taxation in a fully heterogeneous competitive industry, are interesting directions for future research.

Second, we consider the setting where each firm produces a single product using a single technology. In practice, firms sometimes operate portfolios of technologies (e.g., use different technologies at different production plants) and produce portfolios of products. Kök et al. (2016) considered multiple firms in the context of investing in a portfolio of technologies, and the portfolio approach is also adopted by Drake et al. (2016) and Wang et al. (2013). Considering competition between portfolios would be of interest.

Third, while we considered asymmetries between firms, other asymmetries could be important as well. For example, regulation may be asymmetric; think of domestic and foreign firms as in Drake (2018), Huang et al. (2021), and Sunar and Plambeck (2016). Likewise, industries may be asymmetric; for example, better technologies may exist to decarbonize the production of steel than cement. Such asymmetries could lead to non-trivial substitutions within supply chains (e.g., see Wang et al., 2022), and studying how they would interact with environmental taxation is also an interesting avenue for future research.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478251331345 - Supplemental material for Competitive Industry’s Response to Environmental Taxation

Supplemental material, sj-pdf-1-pao-10.1177_10591478251331345 for Competitive Industry’s Response to Environmental Taxation by Dmitry Krass and Anton Ovchinnikov in Production and Operations Management

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article draws on research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and Sydney Cooper Business and Technology Chair funds held by Dmitry Krass, and by the Social Sciences and Humanities Research Council of Canada (SSHRC), grant # 435-2023-0466, held by Anton Ovchinnikov.

ORCID iDs

Dmitry Krass

Anton Ovchinnikov

Supplemental Material

Supplemental material for this article is available online (doi: ).

How to cite this article

Krass D and Ovchinnikov A (2025) Competitive Industry’s Response to Environmental Taxation. Production and Operations Management 34(10): 3214–3229.

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Competitive Industry’s Response to Environmental Taxation

Abstract

Keywords

1. Introduction

1.1. Key Findings and Insights

Finding 1: Structure of the Equilibrium

Finding 2: Asymptotic Sensitivity Analysis

Finding 3: Local Sensitivity Analysis

Finding 4: Sensitivity With Respect to Environmental Efficiency

Insight 1: Environmental Taxation Erodes the Benefits of operational Efficiency [Theorems I.4 and L.4, Corollary I.2, and Examples 2 and 4]

Insight 2: Industry Profits and PUEs May Increase With Tax [Theorems I.6 and L.6 and Examples 3 and 5]

Insight 3: Per Unit and Total Emissions May Increase as Technologies Become Greener [see Section 5]

Insight 4: How the Industry Responds to Tax May Qualitatively Differ Depending on the Demand Function Used [Examples 2 and 4; the ConditionsUnder Which Different Forms of the Demand Function May Arise Are Discussed in the Last Paragraph of Section 1.2]

1.2. Literature Review

2. The Model

3.1. Equilibrium for Iso-Elastic Demand

4.1. Asymptotic Impact of t in the Iso-Elastic Demand Case

Supplemental Material

sj-pdf-1-pao-10.1177_10591478251331345 - Supplemental material for Competitive Industry’s Response to Environmental Taxation

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iDs

Supplemental Material

How to cite this article

References

Supplementary Material

4.1. Asymptotic Impact of $t$ in the Iso-Elastic Demand Case