Environmental Injustice: The Effects of Environmental Taxes on Income Distribution in an Oligopolistic General Equilibrium Model

Abstract

We apply a static oligopolistic general equilibrium model to investigate the effects of an environmental tax on labor incomes, capital incomes, profits, and the distribution of income. The study is motivated by the fact that environmental taxation is one main political tool to realize environmental sustainability and support sustainable development. However, to ensure social and economic sustainability, the taxes applied must be perceived as fair by the majority of the civil society. Moreover, efforts to determine a fair taxation policy would ensure, inter alia, responsible consumption and production, and lower inequality in the economy, which are one of the two priorities of the United Nations Sustainable Development Goals (SDG 10 and 12). Therefore, it is necessary to determine the tax incidence to inform policymakers regarding the distribution of the tax burden. To examine environmental policy, we assume the government applies a policy objective to realize strong environmental sustainability, as proposed by the Dutch economist Rofie Hueting. The main result is that oligopolistic firms can shift the whole tax burden resulting from environmental taxes to workers and capital owners. Consequently, we show that environmental taxes can lead to more income inequality, and the more concentrated the markets, the bigger the social and economic inequality. Noting that addressing environmental problems is a priority of the UN SDGs, our analysis shows that approaching the issue using just environmental tax propositions is not advisable. These results of the analysis also provide a justification of why many members of the society tend to oppose environmental taxes.

1. Introduction

It is an acknowledged fact that all countries still need to make extreme efforts to achieve the goals of the Paris Climate Agreement. Significant financial resources are necessary to realize these efforts, and it is well known that an environmental tax, for example, on energy consumption has a regressive distribution effect as a consequence. In particular, the Yellow Vest movement in France in 2018 showed that this can lead to significant resistance from those affected. In the field of economics, climate change and the natural environment primarily play a role in connection with negative externalities, so the focus on internalizing externalities has been and is directed toward allocative efficiency, but the resulting distributional consequences play no or only a minor role. As important as allocative efficiency is, one cannot ignore the regressive impact of environmental taxes, particularly on energy consumption. The regressivity of taxes on energy arises from the fact that poorer individuals must spend a bigger part of their income on energy than richer individuals. Kenner [1] highlighted a significant disparity in greenhouse gas emissions (GHGs) between the super-rich and the average person within countries. Recent data correlating emissions with income and wealth distribution speak volumes. For example, Khalfan et al. [2] estimated that the per capita emissions of CO₂ of 20 billionaires were more than 8000 tons of CO₂ per year in 2019. This high level of CO₂ emissions resulted from motor yachts and private jets. Barros and Wilk [3] obtained similar results for the year 2018. Collins et al. [4] stated that one out six flights are private jet flights, which are associated with 10 times more CO₂ emissions per passenger than the usual passenger flights. Khalfan et al. [2] estimated that the wealthiest 10% of individuals on the planet are responsible for 50% of all CO₂ emissions, while the poorest 50% of the world’s population contributes only 8% to total CO₂ emissions. It is worth noting that Khalfan et al. [2] also considered investment-related emissions, arguing that the wealthiest individuals are the ones who determine whether investments are made in clean or dirty industries. This unequal distribution is not only observed at a global scale but also at the national level, as indicated in

Table 1. CO₂ emissions (World Inequality data and own calculations).

Country	Average CO2e Emissions Tons per Capita	Share Top 10%	Share Top 1%	Share Bottom 10%	Ratio Top 10% to Bottom 10%	Ratio Top 1% to Bottom 10%	Ratio Top 10% to Average	Top 1% to the Average
USA	21.1	61	197.7	7.3	8.4	27.1	2.9	9.4
France	9	21.7	65.1	3.2	6.8	20.3	2.4	7.2
Germany	12.4	31.1	108.7	3.9	8.0	27.9	2.5	8.8
China	7.7	33	133.7	2.1	15.7	63.7	4.3	17.4
Korea	15	48.2	168.8	4.4	11.0	38.4	3.2	11.3
Russia	12.7	41.1	173.2	4.4	9.3	39.4	3.2	13.6
UK	10.3	24.7	70.3	3.4	7.3	20.7	2.4	6.8

. The last column highlights that the top one percent of a society emits between 7 and 17 times more GHGs than the average citizen, and a staggering 20 to 60 times more than the poorest 10% of citizens. Chancel [5] estimated that 64% of global carbon inequality stems from within-country disparities, whereas this share stood at only 38% in 1990. This dramatic change can be directly related to the increasing inequality of income distribution, which has been observed since the 1980s (Autor et al. [6,7]; Elsby et al. [8], Karabarbounis and Neiman [9,10]). The increasing inequality of incomes can be mostly attributed to increasing market concentration in developed countries (Akcigit et al. [11], De Loecker et al. [12], Philippon [13], Barkai [14], Barkai and Benzell [15], Affeldt et al. [16], Velasquez [17]).

The numbers in Table 1 indicate the per capita GHGs of selected countries based on production emissions. The second column displays emissions, followed by columns representing the share of emissions among top 10%, top 1%, and bottom 10% income earners. The last four columns illustrate selected ratios of income groups. For instance, in the USA, the top 10% income earners emit 8.4 times more GHGs than the bottom 10% earners, while the top 1% emit 9.4 times more GHGs than the average US citizen does. This implies a very uneven distribution of environmental benefits and damage (Hsiang et al. [18]).

Another problem in this context is that environmental taxes have a regressive impact on income distribution. This results from the fact that the share of income poor people spent for fossil consumption (heating, transport, electricity) exceeds the income share rich people spent for fossil consumption. For example, Johne et al. [19] demonstrate through their analysis of the distributional effects of a potential nitrogen tax in Germany that this environmental tax has a regressive impact on income distribution. Thus, Starr et al. [20] and Chancel et al. [21] propose not only taxing environmental externalities but also implementing a progressive tax on income and investments, considering the carbon intensities related to the latter. This approach aims to address the unequal distribution of emissions among income groups.

Therefore, in this study, we complement these explanations by introducing another reason why conventional environmental (or Pigouvian) taxes fail to resolve environmental issues equitably. We will theoretically show that in an economy which is governed by oligopolistic markets, firm owners are able to shift the environmental tax burden fully to workers and capital owners. To validate this statement, we propose considering a general equilibrium model with imperfect competition to analyze the distributional effects of environmental taxation. A fundamental issue in neoclassical general equilibrium models is the assumption of perfectly competitive markets, neglecting corporate profits (c.f. Aubert and Chiroleu-Assouline [22], Chiroleu-Aussouline and Fodha [23]). This assumption, contrary to reality, fails to account for the observed income and wealth inequality, which results primarily from imperfect competition and associated profit income, rather than an inhomogeneous workforce as previously suggested in neoclassical theory.

Thus far, the literature on the distributional effects of environmental taxation under imperfect competition are scant. It is our aim to fill part of this gap by considering an economy with imperfect competition. The challenge in analyzing imperfect competition in equilibrium models was the technical complexity, making it difficult, if not impossible, for a long time to develop such models. Even existing oligopolistic general equilibrium models lack universality compared to perfect competition models. However, we will attempt to investigate the impacts of environmental taxes on distribution of income using a simple general oligopolistic equilibrium model introduced by Stauvermann and Kumar [24]. As we will show by using this approach, oligopolistic companies manage to put the entire tax burden resulting from environmental taxes onto workers and capital providers.

We structure the rest of the paper in the following way. We present in Section 2 the literature related to the model and the background of the sustainable national income. In Section 3, we present the model and analyze the consequences of environmental taxes viz. income distribution. Finally, in Section 4, we compare the optimal environmental tax rates of an oligopoly with the optimal environmental tax rates in a perfectly competitive market. In Section 5, we conclude with appropriate policy recommendations.

2. Relevant Literature and Sustainable National Income

In Section 2.1, we give an overview of the relevant literature, and explain why it is so sparse, and in Section 2.2, will explain the idea of the sustainable national income (SNI).

2.1. Oligopolistic General Equilibrium Models

Stauvermann and Kumar’s [24] model is inspired by Laitner [25], although Laitner’s model consists an oligopolistic sector and a perfectly competitive sector. In an earlier study, Stauvermann and Kumar [26] have simplified the approach of Laitner and slightly changed the dynamic structure of the model. Nevertheless, their model also includes an oligopolistic sector alongside a perfectly competitive sector within the economy.

In this study, we utilize a static version derived from Stauvermann and Kumar [24]. An advantage of using this general oligopolistic equilibrium model is that, assuming a sufficiently large number of oligopolistic firms, it corresponds to a model with perfect competition. Thus, this approach offers the benefit of facilitating a direct comparison between economies characterized by different market structures.

Although there is a huge amount of research on monopolistic competition, the literature delving into oligopolies within a general equilibrium framework is scarce (e.g., Hart [27,28], Neary [29,30,31,32], Neary and Tharakan [33], Leahy and Neary [34], Kreickemeier and Meland [35], Richter [36], Colaciccio [37,38]). This scant literature is primarily due to technical challenges (Hart [27,28]; Neary [29,32]) in terms of addressing how to prevent that oligopolies, which influence prices in goods markets, exercise also market power in factor markets. Neary [29] provides an excellent overview on the problem and the history of unsatisfactory proposals for solving the problem. Colaciccio [37] presents a very good and comprehensive survey on Neary’s General Oligopolistic Equilibrium (GOLE) model.

One significant distinction between our model and Neary’s GOLE model [29,31,32] is that we consider two input factors—capital and labor—while the GOLE model exclusively focuses on labor as the input factor. Moreover, the model we employ can be smoothly transformed into one of perfect competition by assuming an infinite number of firms within the oligopolistic sectors. Another difference is that the GOLE model assumes an additively separable utility function over a continuum of consumption goods of unit mass, where the consumption goods are produced in oligopolistic markets. Furthermore, the GOLE model makes use of a linear production function, while we use a neoclassical production function with diminishing returns of production factors.

Undoubtedly, the model presented by Stauvermann and Kumar [24] shares many equilibrium characteristics with Neary’s (GOLE) model, especially if all firms in the GOLE model utilize identical technologies, although the assumptions of both approaches are quite different. However, there are just a few studies which focus on environment in a GOLE model (Colacicco [38]; Richter [36]), which primarily investigate the effect of strategic environmental policy and international trade relations.

Further, there are an uncountable number of partial equilibrium models, which accounts for the effects of environmental taxes in imperfect markets. Because it is far beyond this paper to survey the excessive literature on partial equilibrium models analyzing the relationship between oligopolies and the environment, we refer to the surveys provided by Requate [39], Lambertini [40], Sturm [41], and Ulph [42,43]. The key result of these studies is that the optimal or welfare maximizing environmental tax rate under imperfect competition is lower than under perfect competition, and in the extreme, the optimal tax rate becomes negative under imperfect competition.

2.2. Roefie Hueting’s Concept of Sustainable National Income

For our analysis, a crucial decision revolves around defining the environmental objectives of society. In this regard, we have opted for an approach rooted in the work of the Dutch economist Rofie Hueting, an early pioneer who developed a well-founded theory on environmental sustainability. Regrettably, Hueting’s approach did not receive the recognition it deserves, potentially due to his radical concept of strong sustainability and his focus on national accounting, which has waned in popularity within economic education. Despite this, we find Hueting’s fundamental ideas compelling, particularly in advocating for the application of the concept of strong sustainability. Recognizing the lack of widespread acknowledgment of his work, we believe it is useful to revisit his core ideas.

We present a short review on sustainable national income based on Hueting and de Boer [44], Hueting and Reijnders [45,46], van Dieren [47], Hueting [48,49,50,51,52,53,54], Hueting et al. [55,56], Colignatus [57], and Stauvermann [58]. The Dutch economist Roefie Hueting began exploring the intersection of economics and the environment as early as 1967, paralleling with studies on standard national income studies. His main work, “New Scarcity and Economic Growth”, published in 1974, advocated for extending the System of National Accounts (SNA) to include environmental losses. Hueting strongly advocated for the incorporation of a pragmatic sustainability concept within the national accounting system.

Hueting proposes measurable factors that impact welfare, particularly the (net) income per capita, deflated to real values using observed market prices. The statistical challenge does not merely lie in income measurement but rather in examining income development concerning the available quantity of environmental goods. These goods, which comprise our physical environment, form the basis for production and the livelihood of citizens.

His interpretation of sustainability goes back to John Stuart Mill’s [59] statements on the “steady state” and “stationary state”. This framework allows the assumption that the preservation of the natural environment becomes a responsibility of the present generation, reflecting the principle of presumed preferences for intergenerational equity.

To operationalize Hueting’s concept it required the calculation of the expenses related to conserving the nature and deducting the conservation costs from the Net National Income (NNI). In his view, establishing an optimal maximum environmental burden to align with these preferences for strong sustainability becomes a task assigned to natural scientists and not to policymakers. Based on Hueting’s presumptions regarding sustainability preferences, the value of environmental degradation equates to the conservation costs. With these costs quantified, it becomes feasible to calculate the so-called Sustainable National Income (SNI). This value is derived by subtracting the aggregated costs necessary for preserving the natural environment from the Net National Income (NNI). Or in the words of Hueting and de Boer [44], p. 19):

“The SNI according to Hueting is the maximum net income which can be sustained on a geological time scale, with future technology progress assumed only in the development of substitutes for non-renewable resources, where such substitution is indispensable for sustaining environmental functions, in turn essential for sustaining income.”

The difference between these incomes serves as an indicator, often used to measure the distance between the current state of the economy and an ideal state, which is the sustainable economy. The gap between the NNI and SNI delineates the unsustainable part of economic activities. Obviously, a widening gap indicates an increasing level of economic growth that is unsustainable, whereas a narrowing gap indicates progress towards more sustainable growth.

One main advantage of Hueting’s approach is that the calculation method of the SNI circumvents the necessity of foreseeing the future, which otherwise poses insurmountable challenges. Nevertheless, several issues need resolution before computing the Sustainable National Income (SNI). Firstly, accounting for the environment presents a significant obstacle when it serves various functions for the economy. For instance, Hueting [50] highlights water, which embodies multiple roles—such as drinking, cooling, transport, agriculture, recreation, and waste disposal. These distinct functions of environmental goods compete with each other, matching with the definition of economic scarcity and being interpreted as economic goods.

As noted by Hueting and Reijnders [45], environmental functions comprise the potential uses of humanity’s biophysical environment (e.g., water, air, soil, natural resources, plants, animals and so on). Moreover, sustainability, within Hueting’s framework, entails utilizing these environmental functions in a manner that ensures their perpetual availability. Accordingly, the supply curve for that environmental function is depicted by the function describing the cost of conserving the environment. Equipped with this knowledge, Figure 1 explains Hueting’s ideas about the demand and supply side for an environmental function.

In Figure 1, B denotes the current availability of environmental functions, while D is the minimum of environmental functions to prevent nature from degradation. The dashed line depicting an incomplete demand curve is derived from expenditures compensating for the loss of environment and financial damages. The vertical line shows the (perfectly inelastic) demand curve based on the preferences for strong sustainability. Meanwhile, the elimination costs curve serves as an interpretation of the supply curve for environmental functions.

Achieving sustainability requires society’s refraining from consuming BD physical units of environmental functions. In monetary terms, this equates to society foregoing AC units of money. With the knowledge of elimination cost curves for all environmental goods and demand curves for strong sustainability, determining the environmental burden becomes straightforward.

Tinbergen and Hueting [60] applied this concept and estimated that around 50% of the world GDP was unsustainable at that time. Verbruggen et al. [61] arrived at similar results for the Netherlands, and a series of estimations were conducted by Hofkes et al. [62] for the period 1990–2000, during which they observed an increase in SNI.

To summarize Hueting’s idea regarding the development of environmental functions in an equation, we can write it as:

$$E_{t+1}=E_t\left(1+\delta \right)-D\left(P\left(Y_t\right)\right)$$

(1)

where $E_{t+1}$ and $E_t$ represent the states of environmental functions in t + 1 and t, respectively; and $P\left(Y_t\right)$ represent the emissions generated by economic activities. The parameter $\delta >0$ represents the ability of environmental functions to absorb environmental damages and to recover to some extent. According to (1), environmental functions are in a steady state or sustainable state, only if:

$$E_t\delta -D\left(P_t\right)\ge 0,$$

(2)

For our purposes, we assume without the loss of generality that an abatement technology exists so that:

$$D=D(P,T_r), \mathrm{with} D_P>0 \mathrm{and} D_{T_r}<0.,$$

(3)

where $T_r$ is the real tax revenue used to reduce the impact of environmental damages. Examples of such abatement technologies include sewage treatment plants or forest reforestation. Regarding climate change, we can consider Climeworks’ “Orca” project in Iceland, capable of capturing 4000 tons of CO₂ from the air annually and storing it underground by injecting it into basalt rock formations. The CO₂ reacts with the basalt, undergoing mineralization, effectively locking the carbon dioxide away and preventing its release into the atmosphere. According to Interclimate Network [63], the current market price for direct air capture of one ton of CO₂ by Orca is USD 1200, while the costs to do so range between USD 600 and USD 800. Just for the sake of argument, if only this Orca technology existed, it would follow from Hueting’s SNI idea that USD 1200 per ton of CO₂ would have to be deducted from GDP. In other words, the current CO₂ emissions, which amount to around 38 billion tons of CO₂ (2022), would have to be multiplied by USD 1200, and then this cost would have to be deducted from global GDP. This results in an environmental damage of USD 45,600 billion, which corresponds to around 45% of the global gross national product.

3. The Model

As noted above, the main difficulty with setting up a general oligopolistic equilibrium model is that each oligopolist has market power in the goods markets but is a price taker in the factor market. To address this problem, we construct a relatively complex model with different production stages and make a few simplifying assumptions to keep the model tractable. Therefore, we explain at first the structure of the model.

The model assumes a perfectly competitive market for final goods. This assumption allows us to omit a utility function in this static approach. In other words, the quantity of final goods received by individuals are interpreted as an index for the level of well-being.

To capture the essence of oligopolistic interactions, we initially establish a complex model with three production stages (Figure 2). This complexity reflects the need for a relatively large number of oligopolistic goods markets, ensuring each oligopolist has a small share of the overall economy. We then simplify the model to two production stages for analytical tractability.

In the first stage, we have the final goods ($Q$) sector, which utilizes two intermediate goods ($Q_1$ and $Q_2$) in the production of final goods. The intermediate goods markets are assumed to be perfectly competitive. The firms at this stage use $m$ preliminary goods ($x_{1,j }$ and $x_{2,j }, \mathrm{r}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y})$ to produce the two intermediate goods. The firms in the intermediate good sectors have access to a symmetric Cobb–Douglas function using $m$ preliminary goods. At the production stage of the preliminary goods, we have $m$ markets, and all these markets are oligopolies with either $n_1$ firms or $n_2$ firms, depending on if they produce for the intermediate goods market 1 or 2. All oligopolistic firms have access to the same Cobb–Douglas function, which requires labor and capital as factor inputs. This means that we have in total $m\left(n_1+n_2\right)$ identical oligopolistic firms. If $m$ is sufficiently large, we can have a duopoly in all preliminary goods markets. The duopolistic firms’ individual demands for capital and labor remain small in relation to the aggregate labor and capital supply. Labor and capital is assumed to be fixed in supply. Therefore, we can assume perfect competition in the input factor markets, where the supply of both labor and capital is price inelastic.

In the Appendix A, we show that we can reduce the number of production stages without changing the outcomes of the complex model. The reduction is possible because of the symmetry assumptions. The reduction induces two main changes regarding the intermediate goods sector., Firstly, it appears that the intermediate good sectors are oligopolies with $n_1$ firms and $n_2$ firms, respectively, which demands labor and capital as input factors. Secondly, we can assume without loss of generality that the firms behave like price takers in the factor markets. In other words, each oligopolistic firm in one of the intermediate markets of the reduced model represents $m$ small oligopolistic firms of the complex model, and it behaves like a small oligopolistic firm. Figure 3 illustrates the structure of the reduced version of the model.

Additionally, we assume that producing intermediate goods creates emissions that harm the environment. In the complex model (Figure 2), these emissions come from firms that produce preliminary goods. Since the results obtained from the reduced model are identical to those that can be derived from the complex model, we will focus on the simpler version.

3.1. The Final Good Sector

The quantity of final goods $Q$ is produced in a perfectly competitive sector by using a Cobb–Douglas production function. The intermediate goods 1 and 2 of quantities $Q_i$ ($i=\mathrm{1,2})$ are used as input factors:

$$Q=B{\left(Q_1\right)}^{\beta }{\left(Q_2\right)}^{1-\beta },$$

(4)

where $B>0$ and $0<\beta <1$. A representative firm, producing the final goods, maximizes its profit as:

$$\underset{Q_1,Q_2}{\mathit{max}}{p}_{Q}Q-p_1{Q}_1-p_2{Q}_2,$$

(5)

where $p_Q$ is the nominal price of the final good, and $p_1$ and $p_2$ represent the nominal prices of the intermediate goods, respectively. Combining the necessary conditions for a profit maximum and the zero-profit condition results in the inverted demand functions for both intermediate goods as follows:

$$p_1\left(Q_1\right)=p_Q\beta B{\left(\frac{Q_2}{Q_1}\right)}^{1-\beta },$$

(6)

$$p_2\left(Q_2\right)=p_Q\left(1-\beta \right)B{\left(\frac{Q_1}{Q_2}\right)}^{\beta },$$

(7)

The respective price elasticities are $\frac{1}{1-\beta }>1$ and $\frac{1}{\beta }>1$, which means that both demand functions are price elastic. From Equations (6) and (7), we obtain the price ratio of the intermediate goods as follows:

$$\frac{p_1}{p_2}=\frac{\beta }{\left(1-\beta \right)}\frac{Q_2}{Q_1}.$$

(8)

3.2. The Intermediate Good Sectors

Starting with intermediate goods sector 1, we consider a symmetric oligopoly consisting of $n_1$ companies. To justify this oligopolistic market structure, we presume that legal barriers to entry hinder potential competitors from entering the market. Each company i in this sector employs capital $K_{1,i}$, where the depreciation rate is 100% per period, and labor $L_{1,i}$. All companies use an identical a Cobb–Douglas production function. Hence,

$$Q_{1,i}=A{\left(K_{1,i}\right)}^{\alpha }{\left(L_{1,i}\right)}^{1-\alpha },\forall i=1,..,n_1,$$

(9)

where $A>0$ is a constant technology parameter, $0<\alpha <1$ is the capital share. The numbers of firms in sector is $n_1$. We define the aggregate of capital used in sector 1 as the sum of the capital of all $n_1$ firms: $K_1=\sum _{i=1}^{n_1}{K}_{1,i}$. Accordingly, we aggregate labor force and aggregate output as $L_1=\sum _{i=1}^{n_1}{L}_{1,i}$ and $Q_1=\sum _{i=1}^{n_1}{Q}_{1,i}$. We assume that the $n_1$ companies compete according to Nash–Cournot behavior and that companies are price takers in the factor markets. Further, we assume that the production is associated with environmentally harming emissions $P_{1,i}=Q_{1,i}$, and these emissions are taxed with a tax rate of ${\tau }_1$. Accordingly, company i of sector 1 maximizes its profit as follows:

$$\underset{L_{1,i},K_{1,i}}{\max }{p}_1\left(Q_{1,i}, Q_{1,-i}\right)Q_{1,i}-w{L}_{1,i}-R{K}_{1,i}-{\tau }_1{Q}_{1,i}$$

(10)

Following usual calculations to determine the Nash–Cournot equilibrium and aggregation (see Appendix A), we obtain the following necessary conditions with respect to the wage rate $w$ and interest factor $R$:

$$\left(p_1\left(\frac{n_1-1}{n_1}\right)-{\tau }_1\right)\left(1-\alpha \right)A{\left(K_1\right)}^{\alpha }{\left(L_1\right)}^{-\alpha }=w,$$

(11)

$$\left(p_1\left(\frac{n_1-1}{n_1}\right)-{\tau }_1\right)\alpha A{\left(K_1\right)}^{\alpha -1}{\left(L_1\right)}^{1-\alpha }=R.$$

(12)

The aggregate profits result from the difference between sector 1’s revenue, represented by $p_1\left(Q_1\right)Q_1$, and the total production costs, including tax payments:

$${\mathsf{\Pi }}_1=\left(1-\frac{n_1-1}{n_1}\right)p_{1}A{\left(K_1\right)}^{\alpha }{\left(L_1\right)}^{1-\alpha }=\frac{p_1{Q}_1}{n_1}.$$

(13)

Having in mind that the marginal costs of a linear-homogenous production function are equal to one in terms of output, the nominal price of an intermediate good 1 is:

$$p_1^{*}=\left(1+{\tau }_1\right)\frac{n_1}{n_1-1}.$$

(14)

The production function used in sector 2 is identical to the one used in sector 1. Applying the same procedures as for sector 1, the first-order conditions for a profit maximum become:

$$\left(p_2\left(\frac{n_2-1}{n_2}\right)-{\tau }_2\right)\left(1-\alpha \right)A{\left(K_2\right)}^{\alpha }{\left(L_2\right)}^{-\alpha }=w.$$

(15)

$$\left(p_2\left(\frac{n_2-1}{n_2}\right)-{\tau }_2\right)\alpha A{\left(K_2\right)}^{\alpha -1}{\left(L_2\right)}^{1-\alpha }=R.$$

(16)

Therefore, the aggregate profits in sector 2 are given by:

$${\mathsf{\Pi }}_2=\left(1-\frac{n_2-1}{n_2}\right)p_{2}A{\left(K_2\right)}^{\alpha }{\left(L_2\right)}^{1-\alpha }=\frac{p_2{Q}_2}{n_2}.$$

(17)

and the nominal price of good 2 is:

$$p_2^{*}=\left(1+{\tau }_2\right)\frac{n_2}{n_2-1}.$$

(18)

3.3. Market Equilibrium

With the symmetry assumptions and Equations (11), (12), (15), and (16), we derive that the capital intensities are identical for of all intermediate good firms; i.e.,

$$\frac{K_1}{L_1}=\frac{K_{1,i}}{L_{1,i}}=\frac{K_2}{L_2}=\frac{K_{2,j}}{L_{2,j}}, \forall i=1,..,n_1 \mathrm{and} \forall j=1,..,n_2.$$

(19)

We define the input factor ratio of sector 2 and sector 1 as $\theta$:

$$\theta =\frac{K_2}{K_1}=\frac{L_2}{L_1}.$$

(20)

We aggregate the total capital stock $K=\sum _{i=1}^2{K}_i$ and total labor $L=\sum _{i=1}^2{L}_i$. Then, we can rewrite the factor inputs in the two sectors as:

$$K_1=\frac{K}{1+\theta }, L_1=\frac{L}{1+\theta }, K_2=\frac{\theta K}{1+\theta }, L_2=\frac{\theta L}{1+\theta }.$$

(21)

Taking the price ratio from Equation (8), the nominal prices from Equations (10) and (13), and the input factor ratio $\theta$, we obtain the ratio of input factors:

$${\theta }^{*}\left(n_1,n_2\right)=\frac{p_1^{*}}{p_2^{*}}\frac{\left(1-\beta \right)}{\beta }=\left(\frac{1+{\tau }_1}{1+{\tau }_2}\right)\frac{n_1}{\left(n_1-1\right)}\frac{\left(n_2-1\right)}{n_2}\frac{\left(1-\beta \right)}{\beta }.$$

(22)

Differentiating the input factor ratio ${\theta }^{*}$ with respect to ${\tau }_1 \mathrm{a}\mathrm{n}\mathrm{d} {\tau }_2$, respectively, leads to:

$$\frac{\partial {\theta }^{*}}{\partial {\tau }_1}=\left(\frac{1}{1+{\tau }_2}\right)\frac{n_1}{\left(n_1-1\right)}\frac{\left(n_2-1\right)}{n_2}\frac{\left(1-\beta \right)}{\beta }>0 \mathrm{and} \frac{\partial {\theta }^{*}}{\partial {\tau }_2}=-\left(\frac{1+{\tau }_1}{{\left(1+{\tau }_2\right)}^2}\right)\frac{n_1}{\left(n_1-1\right)}\frac{\left(n_2-1\right)}{n_2}\frac{\left(1-\beta \right)}{\beta }<0.$$

(23)

From the derivatives (23), we derive Lemma 1:

Lemma 1.

Raising the environmental tax rate in one oligopolistic sector will lead to a reduction in capital and labor employed in that sector, while the capital and labor used in the other oligopolistic sector will increase.

Accordingly, the increase in the tax rate in one intermediate goods sector results in an increase in capital and labor in the other sector. The reason is that the higher tax rate in a sector leads accordingly to a higher product price and results in a decline in the demand in that sector. Obviously, a tax rate increase influences the output ratio of the two sectors. For an examination of this, we define a pseudo production function Z = $A{\left(K\right)}^{\alpha }{\left(L\right)}^{1-\alpha }$, and accordingly, the equilibrium quantities of the two sectors can be expressed in terms of the pseudo production function Z:

$$Q_1^{*}=A{\left(\frac{K}{1+{ \theta }^{*}}\right)}^{\alpha }{\left(\frac{L}{1+{ \theta }^{*}}\right)}^{1-\alpha }=\frac{Z}{1+{\theta }^{*}},$$

(24)

$$Q_2^{*}=A{\left(\frac{{\theta }^{*}K}{1+\theta }\right)}^{\alpha }{\left(\frac{{\theta }^{*}L}{1+{\theta }^{*}}\right)}^{1-\alpha }=\frac{{\theta }^{*}Z}{1+{ \theta }^{*}}.$$

(25)

Not surprisingly, the output ratio between sector 2 and sector 1 equals the input factor ratio $\frac{Q_2^{*}}{Q_1^{*}}={\theta }^{*}$. Therefore, we can state the following:

Proposition 1.

An increase in the tax rate in one sector, leads to decrease in this sector’s output and to an increase in the other sector’s output.

If the tax rate to be paid in sector 1 increases, the price of the intermediate good produced in sector 1 will increase. The higher price results in a substitution of intermediate goods produced in sector 1 with those produced in sector 2. Consequently, the demand for intermediate good 2 will increase, while the demand for intermediate good 1 will decrease.

The output produced in the final goods sector $Q^{*}$ is calculated by plugging Equations (24) and (25) into production function (4):

$$Q^{*}=B{\left(Q_1^{*}\right)}^{1-\beta }{\left(Q_2^{*}\right)}^{\beta }=\frac{B{\left({\theta }^{*}\right)}^{1-\beta }}{1+{\theta }^{*}}Z=\frac{{B\left({\theta }^{*}\right)}^{1-\beta }}{1+{\theta }^{*}}A{K}^{\alpha }{L}^{1-\alpha }.$$

(26)

3.4. The Natural Environment

From Equation (26), it becomes evident that capital and labor is always fully employed, regardless of the size of the tax rate. The reason for this outcome is the assumption of perfect factor price inelasticity in the supply of labor and capital. This implies that both factors can be employed at any positive market price. Additionally, the production of both intermediate goods is similarly environmentally harmful, so the reduction in the quantity of one intermediate good and the increase in production of the other intermediate good do not influence the total amount of pollution. As noted above, we assume for the sake of simplicity a linear relationship between production and emissions:

$$P^{*}=P_1^{*}+P_2^{*}=Q_1^{*}+Q_2^{*}=Z=A{K}^{\alpha }{L}^{1-\alpha }.$$

(27)

This means that the typical result found in partial equilibrium models, where a pure environmental tax reduces emissions, does not hold in this straightforward general equilibrium model. If the tax rate increases in one sector in a partial equilibrium model, the quantity produced in that sector decreases, consequently reducing its emissions. Conversely, in a general equilibrium model, the price of goods in that sector rises, lowering the relative price of goods produced in the other sector, leading in general to an increase in demand and quantity of production in the latter. However, the latter effect is omitted in a partial equilibrium model. Due to the symmetry assumptions, the emission reduction in one sector caused by an environmental tax is exactly offset by an increase in emissions in the other sector. There are two reasons for this outcome. Firstly, there are no environmentally friendly substitutes for existing goods, and secondly, the input factors are supplied factor price inelastic. Consequently a reduction in emissions can only be realized by investments in abatement technologies. Further, we assume that the environmental damage caused by the quantity of equilibrium emissions exceeds the absorption capacity of the environment $D\left(E^{*},0\right)>\delta E_t$, because otherwise, the economy faces no issue with the natural environment and emissions. Additionally, we assume the existence of tax revenue measured in real terms to be $T_r^{*}=\frac{T^{*}}{p_Q^{*}}$ to fulfill the environmental sustainability condition (3). The variable $T^{*}={\tau }_1{Q}_1^{*}+{\tau }_2{Q}_2^{*}$ is the nominal tax revenue. Sustainability requires that the resulting emissions do not exceed the absorption capacity of the natural environment:

$$D\left(P^{*},T_r^{*}\right)\le \delta E_t$$

(28)

From the assumptions above, it follows that in the absence of abatement technologies, reducing emissions can only be achieved by decreasing the aggregate output. This involves implementing control and command policies that constrain the use of input factors. Such a drastic policy measure becomes necessary if $T_r^{*}$ cannot be generated through economic activities. For the rest of the paper, we assume that it is feasible to generate $T_r^{*}$ and this leads to the restriction that the tax revenue has always to be equal to $T_r^{*}$. Given this restriction is fulfilled, the environment is in a sustainable state.

3.5. The Factor Incomes

For the determination of real incomes, we calculate the price of the final good using the zero-profit condition of the final goods sector. This means the revenue of the final goods sector, $p_Q{Q}^{*}$, is equal to expenditures on intermediate goods, $p_1^{*}{Q}_1^{*}+p_2^{*}{Q}_2^{*}$. Hence, taking $Q^{*}$, which represents the real national income, the final good’s price is calculated by using Equations (14), (18), (22), (24), and (25):

$$p_Q^{*}=\frac{p_1^{*}+{\theta }^{*}{p}_2^{*}}{B{\left({\theta }^{*}\right)}^{1-\beta }}=\frac{{\left(\left(1+{\tau }_1\right)\frac{n_1}{n_1-1}\right)}^{\beta }{\left(\left(1+{\tau }_2\right)\frac{n_2}{n_2-1}\right)}^{1-\beta }}{B{\beta }^{\beta }{\left(1-\beta \right)}^{1-\beta }}=\sigma {\left(\left(1+{\tau }_1\right)\frac{n_1}{n_1-1}\right)}^{\beta }{\left(\left(1+{\tau }_2\right)\frac{n_2}{n_2-1}\right)}^{1-\beta },$$

(29)

where we define $\sigma ={\left(B{\beta }^{\beta }{\left(1-\beta \right)}^{1-\beta }\right)}^{-1}.$ Differentiating the price level $p_Q^{*}$ with respect to tax rates gives us:

$$\frac{\partial p_Q^{*}}{\partial {\tau }_1}=\frac{\beta }{1+{\tau }_1}{p}_Q^{*}>0,\frac{\partial p_Q^{*}}{\partial {\tau }_2}=\frac{\beta }{1+{\tau }_2}{p}_Q^{*}>0 \mathrm{and} {\left.\frac{\partial p_Q^{*}}{\partial \tau }\right|}_{{{\tau }_1=\tau }_2=\tau }=\frac{p_Q^{*}}{1+\tau }>0.$$

(30)

From results (30), we derive Lemma 2.

Lemma 2.

If one or both tax rates are increased, the price of the final output will rise.

Utilizing Equations (11)–(17), we derive the nominal incomes of workers, capital owners, and entrepreneurs:

$$wL=\left(1-\alpha \right)Z,$$

(31)

$$RK=\alpha Z$$

(32)

$$\mathsf{\Pi }={\mathsf{\Pi }}_1+{\mathsf{\Pi }}_2=\left[\left(1+{\tau }_1\right)\left(\frac{1}{n_1-1}\right)\frac{1}{1+{\theta }^{*}}+\left(1+{\tau }_2\right)\left(\frac{1}{n_2-1}\right)\frac{{\theta }^{*}}{1+{\theta }^{*}}\right]Z.$$

(33)

Equations (31) and (32) show the presence of environmental taxes do not impact the nominal labor and capital incomes. Conversely, the aggregate nominal profits of the firm owners increase with an increasing tax rate, regardless of which tax rate is increased.

$$\frac{\partial {\Pi }_1}{\partial {\tau }_1}=\frac{{\beta }^2{n_2}^2{\left(1+{\tau }_2\right)}^2{\left(n_1-1\right)}^2}{{\left(n_1\left(\beta \left(n_2\left({\tau }_1-{\tau }_2\right)-{\tau }_1-1\right)-\left(n_2-1\right)\left(1+{\tau }_1\right)\right)+n_2\beta \left(1+{\tau }_2\right)\right)}^2}>0,$$

(34)

$$\frac{\partial {\Pi }_2}{\partial {\tau }_1}=\frac{{\left(1-\beta \right)}^2{n_1}^2{\left(1+{\tau }_1\right)}^2{\left(n_2-1\right)}^2}{{\left(n_1\left(\beta \left(n_2\left({\tau }_1-{\tau }_2\right)-{\tau }_1-1\right)-\left(n_2-1\right)\left(1+{\tau }_1\right)\right)+n_2\beta \left(1+{\tau }_2\right)\right)}^2}>0,$$

(35)

$$\frac{\partial {\Pi }_1}{\partial {\tau }_2}=\frac{\beta \left(1-\beta \right){\left(1+{\tau }_1\right)}^2\left(n_1-1\right)\left(n_2-1\right)n_1{n}_2}{{\left(n_1\left(\beta \left(n_2\left({\tau }_1-{\tau }_2\right)-{\tau }_1-1\right)-\left(n_2-1\right)\left(1+{\tau }_1\right)\right)+n_2\beta \left(1+{\tau }_2\right)\right)}^2}>0,$$

(36)

$$\frac{\partial {\Pi }_2}{\partial {\tau }_2}=\frac{\beta \left(1-\beta \right){\left(1+{\tau }_2\right)}^2\left(n_1-1\right)\left(n_2-1\right)n_1{n}_2}{{\left(n_1\left(\beta \left(n_2\left({\tau }_1-{\tau }_2\right)-{\tau }_1-1\right)-\left(n_2-1\right)\left(1+{\tau }_1\right)\right)+n_2\beta \left(1+{\tau }_2\right)\right)}^2}>0.$$

(37)

From the four derivatives above, we can immediately deduce that the nominal aggregate profits increase with an increase in one of the tax rates.

To obtain the real incomes, we divide the nominal incomes by the price level $p_Q^{*}$:

$$w^{r}L=\left(1-\alpha \right)\frac{Z}{p_Q^{*}},$$

(38)

$$R^{r}K=\alpha \frac{Z}{p_Q^{*}},$$

(39)

$${\mathsf{\Pi }}_1^r=\left(1+{\tau }_1\right)\left(\frac{1}{n_1-1}\right)\frac{1}{1+{\theta }^{*}}\frac{Z}{p_Q^{*}},$$

(40)

$${\mathsf{\Pi }}_2^r=\left(1+{\tau }_2\right)\left(\frac{1}{n_2-1}\right)\frac{{\theta }^{*}}{1+{\theta }^{*}}\frac{Z}{p_Q^{*}},$$

(41)

$${\mathsf{\Pi }}^r={\mathsf{\Pi }}_1^r+{\Pi }_1^r=\left[\left(\frac{1}{n_1-1}\right)\frac{1}{1+{\theta }^{*}}+\left(\frac{1}{n_2-1}\right)\frac{{\theta }^{*}}{1+{\theta }^{*}}\right]\frac{Z}{p_Q^{*}}$$

(42)

Proposition 2.

An increase in one or both environmental tax rates will lead to a decrease in real labor and real capital incomes.

Proof of Proposition 2.

Considering Equations (39) and (40), the real labor and real capital incomes will decline if one or both tax rates are increased due to an increasing price level (see Lemma 1). □

Later, we will investigate the effect of environmental taxes on the profit incomes. The rationale behind this proposition is simple: environmental taxes raise the prices of intermediate goods. Consequently, the overall price level increases while nominal incomes remain constant. As a result, if the tax rates increase, the respective incomes of the workers and capital owners will decline.

In examining the income distribution, we focus on income shares and tax revenue share. To derive the income shares of labor (LS), capital (CS), the profit shares of sector 1 (${PS}_1$) and sector 2 (${PS}_1$), the aggregate profit share (PS), and the tax share (TS), we calculate these by dividing the nominal incomes of workers, capital owners, aggregate profits of sectors 1 and 2, and the nominal tax revenue by the nominal national product.

$$LS=\left(1-\alpha \right)\frac{\beta \left(1+{\theta }^{*}\right)}{p_1^{*}},$$

(43)

$$CS=\alpha \frac{\beta \left(1+{\theta }^{*}\right)}{p_1^{*}},$$

(44)

$${PS}_1=\frac{\beta }{n_1},$$

(45)

$${PS}_2=\frac{1-\beta }{n_2},$$

(46)

$$PS={PS}_1+{PS}_2=\frac{\beta n_2+\left(1-\beta \right)n_1}{n_1{n}_2},$$

(47)

$$TS=\frac{{\tau }_1+{\theta }^{*}{\tau }_2}{p_1^{*}+{\theta }^{*}{p}_2^{*}}.$$

(48)

Considering Equations (45)–(47), it becomes evident that profit shares are independent of tax rates. Regarding the influence of environmental taxes on the income shares, we state the following:

Proposition 3.

If one or both tax environmental tax rates increase, the capital and labor income share will decline, while the profit shares remain constant.

Proof of Proposition 3.

We need to demonstrate that the derivatives of labor’s income share and capital’s income share with respect to the two tax rates are negative: $\frac{\partial LS}{\partial {\tau }_1}=-\frac{\left(1-\alpha \right)\beta }{\left(1+{\tau }_1\right)p_1^{*}}<0; \frac{\partial LS}{\partial {\tau }_2}=-\frac{\left(1-\alpha \right)\left(1-\beta \right)}{\left(1+{\tau }_2\right)p_2^{*}}<0;\frac{\partial CS}{\partial {\tau }_1}=-\frac{\alpha \beta }{\left(1+{\tau }_1\right)p_1^{*}}<0; \frac{\partial CS}{\partial {\tau }_2}=-\frac{\alpha \left(1-\beta \right)}{\left(1+{\tau }_2\right)p_2^{*}}<0$. The fact that the profit income shares are not affected follows directly from Equations (45)–(47). □

The impact of increasing tax rates on the tax revenue is obviously positive. Having in mind that the multiplication of the income shares with the real income $Q^{*}$ delivers the respective real income, it is obvious that the influence of tax rates on profit incomes only depends on the effect of tax rates on real income $Q^{*}$.

3.6. Optimal Tax Structure and Tax Rates

To determine the optimal tax structure, we maximize the real income with respect to the tax rates, and we obtain the following necessary conditions:

$$\frac{\partial Q^{*}}{\partial {\tau }_1}=\frac{\partial Q^{*}}{\partial {\theta }^{*}}\frac{\partial {\theta }^{*}}{\partial {\tau }_1}=0,$$

(49)

$$\frac{\partial Q^{*}}{\partial {\tau }_2}=\frac{\partial Q^{*}}{\partial {\theta }^{*}}\frac{\partial {\theta }^{*}}{\partial {\tau }_2}=0.$$

(50)

From (49) and (50), we obtain the optimality condition:

$$\frac{1+{\tau }_1}{1+{\tau }_2}=\left(\frac{n_1-1}{n_1}\right)\left(\frac{n_2}{n_2-1}\right).$$

(51)

Solving Equation (51) for the tax rate 2 delivers the following optimal tax structure:

$${\tau }_2=\left(\frac{n_1}{n_1-1}\right)\left(\frac{n_2-1}{n_2}\right){\tau }_1+\frac{n_2-n_1}{n_2\left(n_1-1\right)}.$$

(52)

Taking tax rate 2 from (52) and substituting in the optimal input ratio delivers:

$${\theta }^{opt}=\frac{\left(1-\beta \right)}{\beta }.$$

(53)

The input factor allocation ${\theta }^{opt}$ equals the input factor allocation realized in this model with perfectly competitive intermediate markets and without taxes, which can be written as ${\left.{\theta }^{*}\left(\infty ,\infty \right)\right|}_{{\tau }_2={\tau }_2=0}=\frac{\left(1-\beta \right)}{\beta }$.

The maximum national income and the optimal price level are then given by:

$$Q^{opt}=\frac{Z}{\sigma },$$

(54)

$$p_Q^{opt}=\sigma \left(1+{\tau }_1\right)\left(\frac{n_1}{n_1-1}\right)=\sigma \left(1+{\tau }_2\right)\left(\frac{n_2}{n_2-1}\right).,$$

(55)

Therefore, selecting the optimal tax structure described by Equation (52) ensures allocative efficiency.

To derive the optimal size of the tax rates, we use the optimal tax structure together with the requirement that $T_r^{*}={\tau }_1{Q}_1^{*}+{\tau }_2{Q}_2^{*}$, where the latter can be written as:

$${\tau }_1{Q}_1^{*}+{\tau }_2{Q}_2^{*}=\frac{\left(n_1\left({\tau }_1\left(n_2-1+\beta \right)+\beta -1\right)-n_2\left(\beta \left(1+{\tau }_1\right)-1\right)\right)\sigma Z}{n_1{n}_2\left(1+{\tau }_1\right)}.$$

(56)

Now, solving Equation (56) for ${\tau }_1$ delivers:

$${\tau }_1^{opt}=\frac{\left(n_1-n_2\right)\left(1-\beta \right)Z+n_1{n}_2\sigma T_r^{*}}{Z\left(n_1\left(n_2-1+\beta \right)-n_2\beta \right)-n_1{n}_2\sigma T_r^{*}}.$$

(57)

Substituting this result in Equation (53) delivers the optimal rate of ${\tau }_2^{*}$:

$${\tau }_2^{opt}=\frac{n_1{n}_2{\sigma T}_r^{*}-\left(n_1-n_2\right)\beta Z}{Z\left(n_1\left(n_2-1+\beta \right)-n_2\beta \right)-n_1{n}_2{\sigma T}_r^{*}}.$$

(58)

Proposition 4.

If the market for intermediate goods 1 is more competitive than the market for intermediate goods 2 ($n_1>n_2$ ), then the optimal environmental tax rate on intermediate good 1 is higher than the tax rate on good 2.

Proof of Proposition 4.

Let us assume $n_2<n_1$, then ${\tau }_1^{opt}-{\tau }_2^{opt}=\frac{\left(n_1-n_2\right)Z}{Z\left(n_1\left(n_2-1+\beta \right)-n_2\beta \right)-n_1{n}_2{\sigma T}_r^{*}}>0$. □

The latter proposition aligns with the results obtained from partial equilibrium models. According to these models, the optimal environmental tax rate in an imperfectly competitive market is lower than that in a perfectly competitive market. To clarify this, let us assume that sector 1 is perfectly competitive ($n_1=\infty$), and sector 2 is an oligopoly, then it follows from proposition 4 that the optimal tax rate in sector 1 exceeds the optimal tax rate in sector 2.

Proposition 5.

The difference between the optimal tax rates declines with a decreasing difference in the number of firms in the two intermediate goods sectors.

Proof of Proposition 5.

Let us assume $n_2<n_1$, then $\frac{\partial \left({\tau }_1^{opt}-{\tau }_2^{opt}\right)}{\partial n_1}=\frac{n_{2}Z\left(\left(n_2-1\right)Z-{n_2\sigma T}_r^{*}\right)}{{\left(\left(n_2\left(n_1-\beta \right)-\left(1-\beta \right)n_1\right)Z-{n_2{n}_1\sigma T}_r^{*}\right)}^2}>$0. □

It is important to acknowledge that under specific economic circumstances, the optimal tax rate of the more concentrated sector could be negative. In such scenarios, firms within this sector would receive government subsidies rather than paying taxes. This result coincides fully with the result for imperfect markets derived in a partial equilibrium model regarding an optimal environmental tax (see, e.g., Requate [39]).

Taking the optimal tax rates (57) and (58), and substituting them into Equations (40) and (41), leads to the corresponding profit incomes:

$${\mathsf{\Pi }}_1^{r,opt}=\frac{\beta \sigma Z}{n_1},$$

(59)

$${\mathsf{\Pi }}_2^{r,opt}=\frac{\left(1-\beta \right)\sigma Z}{n_2},$$

(60)

Obviously, the real profits are independent of the tax rates. Hence, we can state:

Proposition 6.

The profit incomes are not affected by environmental taxation or subsidies.

These results show that firm owners’ profit incomes will not be affected by environmental taxation, if the optimal tax structure is applied. Consequently, the entire tax burden falls on workers and capital owners, as firm owners contribute nothing to the environmental tax revenue itself. This outcome is indeed surprising. Intuitively, this seems counterintuitive, and can be explained by the market power of firms, further reinforced by the result within the model that optimal environmental taxation does not influence the level of production. This outcome breaks down if the government deviates from the optimal tax structure, but such a measure will result in an inefficient factor allocation.

4. Comparison of Tax Rates in a Perfectly Competitive Economy and Oligopolistic Economy

To compare outcomes in an oligopolistic economy with the outcomes in a perfectly competitive economy, we consider the special case of two symmetric sectors in the oligopolistic economy, where $n_1=n_2=n$. Then, the optimal environmental tax rate is:

$${\tau }_1^{opt}={\tau }_2^{opt}={\tau }_n^{opt}=\frac{n{\sigma T}_r^{*}}{\left(n-1\right)Z-{n\sigma T}_r^{*}}.$$

(61)

The optimal environmental tax in a perfectly competitive economy ${\tau }_{pc}^{opt}$ can be derived by calculating the limit value of the tax rate as the number of firms approaches infinity:

$${\tau }_{pc}^{opt}=\underset{n\to \infty }{\lim }{\tau }_n^{opt}=\frac{{\sigma T}_r^{*}}{Z-{\sigma T}_r^{*}}.$$

(62)

Proposition 7.

All other things being equal, the optimal environmental tax in an oligopolistic economy exceeds the optimal environmental tax in a perfectly competitive economy.

Proof of Proposition 7.

We have to show that ${\tau }_n^{opt}$ is a function continuously decreasing in the number of firms. Differentiating (61) regarding the numbers of firms delivers the following: $\frac{\partial {\tau }_n^{opt}}{\partial n}=-\frac{{\sigma T}_r^{*}Z}{{\left(\left(n-1\right)Z-{n\sigma T}_r^{*}\right)}^2}<0$. Therefore, the optimal environmental tax will decline if the markets become more competitive. Thus, the lowest optimal environmental tax will be realized if the markets are perfectly competitive. □

The intuitive explanation for why the optimal tax rate in an oligopolistic economy exceeds that in a perfectly competitive market is related to the outcome that the effective tax base—the real labor incomes and real capital incomes—in an oligopolistic market is lower than in a perfectly competitive market. Or in other words, the effective tax base is maximal in a perfectly competitive economy, because then the profit incomes are zero. Conversely, the tax base is minimized in the case of a duopoly.

The implications of environmental taxes on the distribution of income are now clear. Environmental taxes do not affect the incomes of firm owners, but they reduce the real incomes of capital owners and workers. The more concentrated the intermediate goods markets, the higher the environmental tax rates will be, and accordingly, the lower the real incomes of capital owners and workers, and the lower their income share. The latter also means a more unequal distribution of income between firm owners on the one hand and workers and capital owners on the other hand.

Additionally, the income share of firm owners increases with an increasing market concentration. To show the latter, we differentiate the profit share from Equation (47) with respect to the number of firms:

$$\frac{\partial PS}{\partial n_1}=-\frac{\beta }{n_1^2}<0 \mathrm{and} \frac{\partial PS}{\partial n_2}=-\frac{1-\beta }{n_2^2}<0$$

(63)

Both derivatives are negative, implying that an increasing market concentration will increase the profit income share and will lead to a decline in the labor and capital income share. Putting these results together, a higher market concentration induces a more unequal distribution of income. Additionally, the introduction of environmental taxes, as proposed above, will further exacerbate income inequality, with the degree of exacerbation increasing as market concentration rises. From what is stated above, we can directly derive proposition 8.

Proposition 8.

Perfect competition in all markets minimizes the aggregate tax burden and maximizes the real labor and capital incomes in the presence of optimal environmental taxes.

5. Conclusions

Recent research has shown that the richest members of society are responsible for a disproportionately large share of GHGs. There are concerns that environmental taxes may have a regressive effect on income distribution, meaning that poorer citizens would pay a higher proportion of their income to prevent a significant increase in average temperature.

On the other hand, it is clear that the incomes and wealth of the top 10% have grown much faster in the last 50 years compared to the rest of society. This disparity in income growth can be partly explained by the increase in market power observed in most developed countries. This disparity in the developments of incomes and wealth can be an important reason for a significant share within the remaining 90% poorer members of the society to resist environmental taxes, because they perceive the recent income developments and environmental taxes, which amplify the income inequality, as unfair.

In this paper, we have theoretically demonstrated that the situation is even worse than previously described. Using a simple oligopolistic general equilibrium model, we have shown that environmental taxes will not affect the real incomes of firm owners. Consequently, the entire tax burden falls on capital owners and workers. Further, the more concentrated the markets are, the greater the tax burden borne by poorer members of society, leading to a more unequal income distribution.

These findings have significant political implications. The higher the necessary environmental tax rates, the less likely their introduction becomes in a democratic society, leading to stronger political opposition. What is the solution? Two potential solutions exist. First, a strong antitrust policy is recommended to restore competition and reduce market concentration. Second, wealth and corporate taxes targeting the top earners, as proposed by Chancel et al. [21], could be implemented. Otherwise, taking the necessary measures to fight climate change may become politically impossible. A significant challenge in this context is the scarcity of firm-level data on greenhouse gas emissions (GHGs) and other polluting activities. This data gap hinders the development of well-informed proposals for a more appropriate tax system. Regarding firm-specific GHG disclosure, Karim et al. [64] propose a novel approach that merits exploration.

It is important to note that our strong results stem from specific assumptions. We assumed no environmentally friendly alternatives to existing goods and completely inelastic input factor supplies. While relaxing these assumptions might alter the specifics, we are confident that the core finding will hold; i.e., workers and capital owners have to bear the majority of the environmental tax burden, and as a consequence, the gap between the rich and the poor will widen. Or in other words, a more equal distribution of income and wealth make it cheaper from the view of the disadvantaged and poor to realize ecological sustainability. In practice, this means that policymakers should consider reducing as much as possible unnecessary barriers to market entry that hinder competition, and if market concentration is unavoidable because of technological reasons (e.g., network externalities), it is advised to progressively tax the profits.