Scope of Economic Incentives and Abatement Technologies to Regulate a Natural System’s Resilience in a General Equilibrium Model

David Tobón Orozco♣, Carlos Molina Guerra,♦ and Carlos Vasco Correa♠

Abstract:

This paper discusses a general equilibrium model consisting of a productive sector generating

externalities on another sector having clean production and consumers, and affecting the property of

resilience of a natural system that feeds economic system. The scope of efficiency of economic

incentives is analyzed simultaneously with production activities in the polluting sector in the cases

of: market equilibrium in which nature is a private good, market efficiency amendment through the

use of Pigouvian taxes and the simultaneous use of a pollution abatement technology. Our model

predicts that: the polluting sector could find itself in a worse situation in the equilibrium with

externalities (boomerang effect); one could imagine that inside it, some firms will benefit while

others will be markedly disadvantaged regarding indirect effects of pollution. Additionally,

incentives to acquire abatement technologies are not homogeneous. In any case, the use of

economic incentives helps keep pollution levels steady so as not to affect the resilience property.

Keywords: Natural ecosystem, resilience, externalities, Pigouvian taxes, Abatement technologies,

applied general equilibrium models.

Introduction The instruments that regulate externalities may be grouped into two categories: command and

control (CAC), when referring to emission caps —performance standards— and technological

constraints —design standards—, or incentives, such as taxes, subsidies, tradable permits and

deposits. Economic incentives are preferable, but their effectiveness depends mainly on the ability

to measure all indirect marginal damages, low market transaction costs, the possibility of defining

and protecting property rights, their ease of implementation, regulation and surveillance, the state’s

fiscal appetite, technical possibilities and the costs of using abatement technologies. Also, there

may be classification criteria, such as equity and ethical dilemmas (Eskeland, 1999; Fullerton and

♣ Professor of Economics and Director Applied Microeconomics Group, Department of Economics, University of Antioquia, email: davidtobon@gmail.com, david.tobon@udea.edu.co, phone: +57(4)2195837. ♦ Research assistant, Applied Microeconomics Group, Department of Economics, University of Antioquia. ♠ Research assistant, Applied Microeconomics Group, Department of Economics, University of Antioquia.

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Heutel, 2007). Under conditions of uncertainty concerning the costs of pollution or the benefits of

its reduction, and considering monitoring and enforcement costs and the regulator’s commitment,

the use of hybrid instruments has been suggested: for example those combining instruments of

command and control, which help to reduce the risk of high emissions but entail high costs for the

emitters, with the opportunity to pay a tax or purchase a permit on their excesses over the standards,

thus reducing costs (Montero, 2011). These additional emissions or the quantities of permits have a

cap, but they may reduce incentives for innovation that would motivate the non-payment of taxes or

permits thanks to the performance standards (Perino, 2008).

This paper presents a theoretical general equilibrium model with externalities based on Laffont

(1988), which allows us to analyze the scopes of efficiency of economic incentives (taxes) whose

application must be studied in parallel with the activities of production and investment in abatement

technologies. Our model also incorporates a natural system constraint (nature).

The proposed model was constructed taking the economy of a representative consumer, two

productive sectors and an environmental service derived from nature as a basis in order to analyze

the following cases: market equilibrium with nature as a private good, the differences between the

assignment of efficiency and the market’s general equilibrium, the payment of Pigouvian taxes1, the

possibility of using technologies that allow for pollution abating or reducing tax payment.

Moreover, nature is subject to losing its property of resilience, that is, the persistence and ability to

absorb changes and shocks, and to allow for the supply of environmental services fundamental to

economic functioning (Holling, 1973; Doak, 1998; Groffman et al., 2006; Brand, 2009).

Efficiency implications of using economic incentives by agents or regulating authorities are

examined simultaneously with the possibility of investing in abatement technologies. We assume

that there are no transaction costs, information is perfect, the world is static and technologies are

already given, both for production and pollution mitigation.

The main result of inefficient equilibrium with externalities is that the situation is worsened for the

polluting sector. Within this sector there will always be some who benefit and some who are

harmed in greater proportion, since they will be more affected by the drop in both relative demand

1 Pollution permit transactions are not analyzed here, since given the characteristics of the model (a representative consumer who owns all the firms), there are no additional income effects and the results of efficiency and welfare are equivalent to the use of taxes.

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for their products and/or their relative prices. Therefore, incentives for investment in abatement

technologies would not be homogeneous. This finding is supported by the widely accepted

affirmation that research and development activities adopt the figure of a public good, and that their

results are probabilistic, thus reducing incentives to invest in them. Nonetheless, incentives to adopt

abatement technologies are higher when taxes are paid, without the need of granting subsidies.

The remainder of this article is organized as follows: section 1 presents the theoretical framework.

Section 2 develops the general equilibrium model proposed. In section 3, the simulation of each

situation modeled is given, and their main results are presented. The conclusion explains the paper’s

main findings.

1. Theoretical Framework

General equilibrium models focus on analyzing the difference between efficiency allocation and

market equilibrium in the presence of environmental externalities that affect different sectors, and

on the effectiveness of policy instruments. This approach agree on economic incentives, such as

Pigouvian taxes or permit sales, and on the fact that these incentives are preferable to CAC

regulations. When the option of pollution abatement technologies is included, there is an abuse of

assumptions which leads to improper conclusions. This could include, for example, assuming fixed

proportions on abatement technologies, or supposing the decontamination activity is subsequent to

the production activity, rather than occurring simultaneously. This is an incorrect assumption, since

the producer’s decisions regarding the good’s final price must include both production and

decontamination costs (Pindyck & Rubinfeld, 2013). In addition, the incidence of economic

incentives has been studied, assuming that the research and development effort is made by the

polluting company itself (Popp et al., 2009).

In practice, instrument effectiveness depends on a series of factors including the party responsible

for innovation development, whether polluting companies are heterogeneous or whether they have

market power; whether technology is adapted or there are direct investments on research and

development, with underlying disincentives due to the probabilistic character of innovation and its

becoming a public good once it is produced; whether the abatement technology completely reduces

pollution —which is pure fiction— or slightly mitigates it; whether technology helps reduce

pollution per product, per input, saves power consumption per unit produced or whether

replacement for a cleaner energy source is approved (Löschel, 2002). Lastly, some regulator

behaviors (such as commitment or short-sightedness) have been researched once the abatement 3

technology is already available or under development (depending on the phase and success of

innovation) in order to foster its diffusion (Goulder and Parry, 2009; Montero, 2011; Requate,

2004).

In addition, Magat (1979) and Popp et al. (2010) have stated that when abatement technologies are

available market based regulations encourage behavior through market signals rather than through

explicit directives regarding CAC, thus allowing firms some flexibility to choose or identify the

lowest-cost solutions to meet the policy goal. However, this suggestion is obviously not repeatable

in a second-best world. In particular, the ranking of policy instruments also depends on the number

and the type of interactions, and on the innovator’s ability to appropriate spillover benefits. An

initial motivation to include subsidies is that market-based policies also reduce production, and a

mix of tax pollution and subsidies may be better suited to overcoming the joint market failure: a

negative externality from pollution and a positive externality or spillover effect of R&D, given the

public nature of innovation (Popp et al., 2010). However, the presence of a successful innovator

could motivate the necessity of formulating discriminatory regulations since even this innovator

would prefer CAC regulations so as to raise costs for competitors.

A combination of certain features of both price-based (market signals) and quantity-based (CAC)

regulations in their pure form, usually named hybrids, is also recommended in second-best

situations. These include asymmetric information and principal-agent problems between regulators

and polluting firms, insufficiency of market signals and R&D policies for promoting investment in

high potential technologies, behavioral factors, uncertainty, capital constraints and risk aversion,

imperfect property rights, externalities with other market failures and distributional aspects.

Administrative costs, cost impacts from fiscal interactions (offsetting reductions in other fiscal

taxes) and political feasibility have also been included. These restrictions may provide further

justification for setting performance standards or mandating a particular suite of technologies on

certain sector failures (Goulder & Parry, 2009; Benner & Stavins, 2007; Goulder, 2008).

Labandeira and Linares (2010), for example, emphasize the necessity of mixed market-based

regulations with subsidies or standards to account for uncertainty, bounded rationality and social

acceptability: “Carbon taxes may be more attractive theoretically. However, auctioned cap-and-

trade systems, while retaining the rent-capturing feature of taxes, also allow for redistributing more

explicitly and more easily than taxes a part of the cost, and may therefore be more politically

acceptable. Their acceptability would even be higher if they are combined as hybrid instruments,

such as safety valves, to hedge against unexpected high costs. These more efficient instruments

should probably be coupled in some sectors – those closer to the final customer – with technology 4

standards to account for bounded rationality and also to improve acceptability; with technology

policies (both market-pull and market-push, depending on their situation in the learning curve) to

counteract knowledge spillovers; with education and training policies to reduce bounded rationality

and to decrease perceived costs, and with voluntary approaches when performance is not easily

observable”. Goulder and Parry (2009) also argue: “Abatement costs uncertainties would be

accompanied of provisions such as banking and borrowing or subsidies, however the instruments

and the level of support are less clear, and lesser when technologies are available”.

However, the whole is not necessarily the sum of its parts; the use of multiple instruments is not the

same as the use of a hybrid instrument. As the number of policy instruments grow, so does the

interaction, being either detrimental or beneficial. Fankhauser et al. (2010) show that in order to

achieve an adequate price on carbon and limits on its fluctuations to combat climate change, some

European policy makers are combining Cap and Trade with carbon taxes or with Feed in Tariff —

renewable energy obligations—. Adding a carbon tax or feed-in tariff to the existing carbon system

(EU ETS) also reduces the carbon price to such an extent that the overall price signal would remain

unaffected. It aims to shift the burden of payment and depress the carbon price, rather than to

achieve any additional emission reductions, unless the tax is so high that it replaces and thereby

intensifies the price signal from the trading scheme. In any case, it must be made clear that the best

policy would be to auction original permits (caps) to pollute rather than to grant them free. For

example, they could be assigned in an options market since the spot market may lower the prices of

a license for using such technology and, consequently, may reduce incentives for innovation

(Laffont-Tirole, 1996; Fairley, 2009).

Lastly, the type of innovation must be distinguished. If it reduces marginal pollution, it is called an

end-of-pipe solution —such as installing a water treatment plant adjacent to the production plant.

On the other hand, it might involve changing the production process, thereby making the marginal

abatement cost steeper. Note that in this last case, CAC policies would be more efficient than

market-based policies for promoting innovation, since they also imply diminishing production.

Nonetheless, if high levels of abatement are required, marginal abatement costs will rise acutely, so

it is better to invest in new production processes (Bauman et al., 2008; Popp et al., 2010). In this

sense, moderate efficiency gains in conventional technologies will have a great impact on the

economy, since these technologies are widely used.

5

Resilience of a Natural System

A natural system has two fundamental properties: Stability, or its ability to return to equilibrium

after a disturbance, and Resilience, or its persistence and its capacity to absorb changes and shocks

while maintaining the same relation between its populations and functions, such that it allows for

the provision of environmental goods and services (Holling, 1973; Doak, 1998). A natural system

can eventually return to a state similar to the one that existed prior to the disturbance, and

depending on the distance to its ecological threshold, the response will be smooth or abrupt

(Groffman et al.., 2006). Therefore, if the disturbances are of sufficient magnitude or duration, they

can deeply affect reach and force a threshold beyond the original steady state obtaining another

regime with different processes and structures,; less desirable and degraded conditions are obtained

and may take up to critical points where recovery is not feasible. As the system undergoes these

shocks2, its stable states become increasingly transient. It becomes increasingly difficult for the

system to return to its initial state, reaching critical points more easily and changing from one state

to another3.

Resilience is inversely proportional to the degree of threat, i.e., the frequency of endogenous and

exogenous effects that attack the ecosystem (Brand, 2009). It is also related to the vulnerability, i.e.,

susceptibility to the biophysical effects and its natural ability to deal with them; susceptibility

reflects the potential of the system to be affected, while the resilience and resistance are determinant

to the stability of the system against disturbance (Klein, 1998; De Groot, 2003). The farther is a

system of its ecological threshold, the less threatened it will be (Briguglio et al., 2008)4.

Consideration of the mechanics of a natural system should be incorporated into general equilibrium

models (GEM) that incorporate natural capital, given the increasing environmental problems

associated to pollution, resource extraction and removal of functional groups (Simal & Ortega,

2011; Schouten, et al., 2009; Rose et al., 2005; Rose, 2009; Elbourne et al., 2008). Particularly, in

Computable General Equilibrium (CGE) models, a strategy for modeling resilience has recently

2 Those shocks can occur in two ways: i) not very intensive but constant, which will gradual bring the system to the critical point and finally will change its steady state; and ii) punctual, one disturbance of sufficient strength will bring the system beyond the critical point arriving at a less desirable steady state. 3 Resilience depends on the functional diversity of species (competitors and substitutes) that support critical processes of structuring a natural system. For example, specialization gains simplifying farming systems involve a reduction of resilience. Reducing the costs of resilience include: herbicides, pesticides, fertilizers, irrigation and other inputs needed to keep production on a simplified system, the costs of relief when output falls, relocation when soil and water resources have been irreversibly affected, rehabilitation where damage is reversible and insurance against crop damage by pests or diseases. 4 When speaking of resilience, spatial interactions are also important; they may change as a result of changes in areas surrounding the ecosystem and it is longed that it can be maintained at some level by human actions (Groffman et al., 2006). Other aspects related to the ability of adaptation, renewal, regeneration, transformation, reorganization and development in an ecosystem are of great importance for sustainability such that components, relationships, innovation and continuity will be maintained (Cumming, 2005; Folke, 2006; Huitric et al., 2009).

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been found. According to Rose and Liao (2005), this is a promising approach for analyzing the

impact of disasters on resilience, as it is possible to model the behavioral response to the shortage of

inputs and changing market conditions.

Rose and Liao (2002) and Chang et al. (2000) highlight the importance of avoiding overlooking

costs and natural hazards that can be disrupted by terrorist attacks; for example, the provision of

public services, because such interruptions trigger a reaction or propagation in the whole economic

system. Rose and Liao (2005) analyze natural disasters in regional economies, conceiving resilience

as an inherent property of the system that enables it to return to its previous situation within a

reduced time horizon after a crash caused by an unforeseen event (an earthquake, a terrorist attack,

a natural disaster). For this preparation, they considered essential investing in actions to mitigate the

effects and maintain or increase this capacity for resilience. Making an analysis of Portland, the

conclusion is that the substitution of inputs that maintain a stable state of nature is expensive, and it

is preferable to apply preventive measures to reduce losses when disasters like earthquakes occur,

that affects the provision of public services (water supply network). Meanwhile, Fadali et al. (2012)

found that the value of water in the U.S. is in constant motion and that this is because resilience is

affected by changes in water supply, demand, changes in prices of inputs and factors, prices of

production, income, government policies and institutions.5

2. The Model We propose formalizing a general equilibrium model splitting the world into a polluting industrial

sector X, a clean industrial sector Y, a representative consumer not generating externalities and

nature as a supplier of a fundamental environmental service to all agents (Christopherson et al.,

2010; Wing, 2011)6. This model admits a simultaneous selection between production activities,

pollution, the use of economic incentives (Pigouvian taxes) and the option of adapting abatement

technologies. We include an end-of-pipe solution for a given production process, available in the

market, in a first best world. This model is static and there are no transaction costs or the figure of a

regulator explicitly coordinating agents.

5 See Schouten et al. (2009) for a discussion of the issues brought about by resilience in rural areas. On a macro level, resilience can be linked to financial institutions and norms, and to the role of scarce resources (Jeroen et al., 2010). 6 For the sake of clarity, we can imagine an economy grouped in either polluting or non-polluting sectors and heavily relying on a natural system. See Wing (2011) for a proposal of a general equilibrium model incorporating varied interactions of nature and the remaining sectors, even though nature appears as receiving negative externalities and as a provider of common use public goods (positive externalities). This author also analyses the incidence of taxes, CAC measures and the option of acquiring abatement technologies, but without establishing any prioritization among them.

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Nature is considered to be a private good, since property rights are defined. This economy might

allocate some resources to N, i.e. to increase its production or improve quality. Yet, it is also

affected by X negative externalities, disturbing particularly its resilience, as well as its production

capacity from a productive steady state up to other less productive steady state. In such an

environment, our goal is both to analyze Pigouvian taxes properties and to see the marginal effect

of this technology for reducing the possibility of affecting nature’s property of resilience.

Preferences and technologies are neoclassical and are given by the following equation system:

𝑈 = 𝑋𝐶𝛼𝑌𝐶𝛽𝑁𝐶

1−𝛼−𝛽𝑋−𝛾 , 𝑋 = 𝐻.𝑁𝑋𝛿𝑌𝑋𝜀 , 𝑌 = 𝑀.𝑁𝑌𝜋𝑋𝑌𝜃𝑋−𝜌 (1)

𝐽𝑖 suggests the firm produces good i using input j. In its turn, 𝑖 represents the production of good i.

H and M are the technology of sectors X and Y.

In order to model N, a specific functional form is proposed considering: 1) an initial natural capital

stock A; 2) a marginal contribution of human resources intended to try to maintain, increase or shift

such a stock (𝑌𝑁); and 3) a component accounting for the resilience capacity, the shift of steady

state from productive to less productive due to contamination by X affecting supply by N.

Be

𝑁 = 𝐴𝑌𝑁𝜎𝑓(𝑋) (2)

Where 𝒀𝑵 has a positive marginal productivity, small and decreasing to reflect the fact that it is

possible to reproduce a certain N’s level or quality, but in a constantly decreasing proportion

(Brand, 2009).

In terms of environment, it is clear that a natural system will never be the same as a system rebuilt

by humankind. We are assuming that N should be transformed to produce a profit-generating

environmental service, but with costly reproduction, which might become impossible depending on

the level of X.

Regarding externalities, the first derivative should be negative, meaning that increases in the level

of affectation have a diminishing effect on N. However, the marginal level of the externality has

some properties, particularly considering that all of natural systems do not exhibit the same 8

response when faced with a level of affectation. At this point, resilience becomes a determining

factor to try to explain this relationship in depth.

Graph 1: Affectation on resilience caused by an economic externality (pollution)

Given the characteristics of resilience, N reacts to negative externalities striving to keep balance.

The dotted line represents the involving function and associates the way N would respond to levels

of externality X, where every oscillation in the function represents less valuable steady state.

The envelope function characterizes the nature´s long run trajectory, and from it the analysis of the

critical point is made. Finally, it is a function that is initially convex as in graph (it maintains its

characteristics). Besides having this property, this function also changes concavity (it turns from

convex to concave), reflecting that N has a level of externality from which it becomes untenable.

This level is precisely the (cp) location, where the change in concavity appears. The characteristics

for the function are:

𝑑𝑁𝑑𝑋

< 0 𝑎𝑛𝑑 𝑑2𝑁𝑑𝑋2

> 0 𝑖𝑓 𝑥 < 𝑐𝑝, 𝑑2𝑁𝑑𝑋2

< 0 𝑖𝑓 𝑥 > 𝑐𝑝,𝑑2𝑁𝑑𝑋2

= 0 𝑖𝑓 𝑥 = 𝑐𝑝

The cubic function is able to present such properties. Therefore, we may represent N as:

𝑁 = 𝐴𝑌𝑁𝜎(𝑎𝑋3 + 𝑏𝑋2 + 𝑐𝑋 + 𝑑 + 𝜆(𝑥)) (3)

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So that: 𝜎 → 0

This function exhibits concavity changes and, under certain restraints on parameters, it allows us to

model the relevant characteristics of resilience.7 Finally, 𝜆(𝑥) = 𝑠 ∗ sin (𝑤 ∗ 𝑋) denotes the

associated function to the long run trajectory deviations or to different steady states8.

If a natural system is sufficiently disturbed by an environmental externality, it may be necessary to

loosen the convexity assumption of the production sets. This means the second welfare theorem

cannot be applied here, which would in turn suggest the impossibility for the market to replicate

some efficient, socially desirable allocations; that is, it is not possible to decentralize decisions

through a price mechanism (Brock, Maler & Perrings, 2001; Murty, 2006). This suggests that,

ceteris paribus, the relationship between the input and output is not affected when the externality

level rises. That is, when the externality level changes, N changes, but it is considered that the

relationship between this level and the input continues to be positive but decreasing (Laffont, 1996).

The mechanics call for building several general equilibrium models to analyze the following cases:

differences between efficiency allocation and market general equilibrium; payment of taxes9, and

the option of acquiring pollution abatement technologies; and making a hybrid with them both. As

property rights on N are supposed to be allocated, this system allows for three markets and three

price levels: 𝑃𝑋 ,𝑃𝑌,𝑃𝑁.

Market equilibrium:

Each agent solves its particular problem:

Max 𝐵𝑋 = 𝑃𝑋𝑋 − 𝑃𝑁𝑁𝑥 − 𝑃𝑌𝑌𝑥 s.t. 𝑋 = 𝐻𝑁𝑥𝛿𝑌𝑥𝜀 (4)

Therefore, 𝑁𝑥𝛿𝑋

= 𝑃𝑋𝑃𝑁

and 𝑌𝑥𝜀𝑋

= 𝑃𝑋𝑃𝑌

(5)

Where the derived demands and production of X are: 𝑁𝑋∗,𝑌𝑋∗,𝑋∗

7 Generally, under the same nomenclature, the point of inflexion is called cp, the characteristics requested from the function and imposing constraints on the parameters are: 1) that the point of inflexion is in x=cp; 2) the function is always decreasing; 3) when x=0, the function is positive, and 4) when x=cp, that the function is positive. In the same order, the previous items call for:

3b ap= − ; 20 y 3a ap c< > ; 0d > (a redundant condition if p>0) y

2(2 )d p ap c> − . 8 This continuous function allows to model required oscillations, parameters s and w are associated with the amplitude and period of the function. 9 The non-existence of transaction costs and the fact that the firms are owned by the representative consumer cause the allocation of contamination permit payments to be equivalent to taxes, in terms of efficiency and welfare.

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Y solves:

Max 𝐵𝑌 = 𝑃𝑌𝑌 − 𝑃𝑁𝑁𝑦 − 𝑃𝑋𝑋𝑦 s.t. 𝑌 = 𝑀𝑁𝑦𝜋𝑋𝑦𝜃𝑋−𝜌 (6)

Then, 𝜋𝑃𝑌𝑌𝑁𝑦

= 𝑃𝑁𝑃𝑌

and 𝜃𝑃𝑌𝑌𝑋𝑦

= 𝑃𝑋𝑃𝑌

Where the derived demands and production of Y are: 𝑁𝑌∗,𝑋𝑌∗ ,𝑌∗

Max 𝐵𝑁 = 𝑃𝑁𝑁 − 𝑃𝑌𝑌𝑁 s.t. 𝑁 = 𝐴𝑌𝑁𝜎𝑓(𝑋) (7)

Then, 𝜎𝑁𝑌𝑁

= 𝑃𝑌𝑃𝑁

Determining: 𝑌𝑁∗,𝑁∗

And finally the representative customer solves:

Max 𝑈 = 𝑋𝑐𝛼𝑌𝑐𝛽𝑁𝑐

1−𝛼−𝛽𝑋−𝛾 s.t. 𝑅 = 𝐵𝑋 + 𝐵𝑌 + 𝐵𝑁 = 𝑃𝑋𝑋𝑐 + 𝑃𝑁𝑁𝑐 + 𝑃𝑌𝑌𝑐 (8)

Such that: 𝛼𝑁𝑐

(1−𝛼−𝛽)𝑋𝑐= 𝑃𝑋

𝑃𝑁 and

𝛼𝑌𝑐𝛽𝑋𝑐

= 𝑃𝑋𝑃𝑌

Determining: 𝑋𝑐∗,𝑌𝑐∗ ,𝑁𝑐∗

Equilibrium prices 𝑃𝑋∗,𝑃𝑁∗ are found when supply is equal to demand in each market, being 𝑃𝑌∗ = 1

by Walras Law.

𝑋∗ = 𝑋𝑐∗ + 𝑋𝑦∗

𝑁∗ = 𝑁𝑐∗ + 𝑁𝑥∗ + 𝑁𝑦∗

The allocation of efficiency is obtained by solving:

Max 𝑈 = 𝑋𝑐𝛼𝑌𝑐𝛽𝑁𝑐

1−𝛼−𝛽𝑋−𝛾

s.t. (9)

𝑋 = 𝐻𝑁𝑥𝛿𝑌𝑥𝜀 → 𝜆𝑥

𝑌 = 𝑀𝑁𝑦𝜋𝑋𝑦𝜃𝑋−𝜌 → 𝜆𝑦

𝑁 = 𝐴𝑌𝑁𝜎𝑓(𝑋) → 𝜆𝑁

𝑓(𝑋) = 𝑎𝑋3 + 𝑏𝑋2 + 𝑐𝑋 + 𝑑

𝑋 = 𝑋𝑐 + 𝑋𝑦 → 𝜇𝑥

𝑌 = 𝑌𝑐 + 𝑌𝑥 + 𝑌𝑁 → 𝜇𝑦

𝑁 = 𝑁𝑐 + 𝑁𝑥 + 𝑁𝑦 → 𝜇𝑁

With 𝜆𝑖 and 𝜇𝑖 being the shadow prices.

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The optimum marginal rates of substitutions of consumer, firms Y, N and X are respectively given

by: 𝛼𝑌𝑐𝛽𝑋𝑐

= 𝜇𝑋𝜇𝑌

, 𝛼𝑁𝑐(1−𝛼−𝛽)𝑋𝑐

= 𝜇𝑋𝜇𝑁

, 𝛽𝑁𝑐(1−𝛼−𝛽)𝑌𝑐

= 𝜇𝑦𝜇𝑁

(10)

𝜋𝑌𝑁𝑦

= 𝜇𝑁𝜇𝑌

and 𝜃𝑌𝑋𝑦

= 𝜇𝑋𝜇𝑌

(11)

𝜎𝑁𝑌𝑁

= 𝜇𝑌𝜇𝑁

(12)

And,

𝑌𝑥𝜀𝑋

+ �𝛾𝑋

1𝛽𝑌𝑐

+ 𝜌 𝑌𝑋− 𝐴𝑌𝑁𝜎𝑓𝑥′(X) 𝑌𝑁

𝜎𝑁� , 𝑓′(X) < 0 (13)

Observe that the level of activity in the sector X is regulated here by ensuring that the marginal

social damage is measured and equal to the sum of individual marginal damages (13) opposite to

the inefficient market equilibrium (5), i.e., We must add the sum of the marginal damages provoked

to the consumer, weighted by the marginal benefit derived from consumption of good Y, to sector

Y and to N, also weighted by the inverse of the marginal productivity of YN input.

Sector X has the choice to abate pollution by acquiring technology 𝐿 = h(𝑁𝑥𝑙), so that pollution

will be 𝑍 = 𝑋 − 𝐿. We assume that, in order to make this plant work (end-of-pipe solution) there is

a need to acquire input Nxl, nature itself, at a market price Pn (Laffont, 1988; Popp et al., 2010). The

new problem of assignment of efficiency in this economy will be:

Max 𝑈 = 𝑋𝑐𝛼𝑌𝑐𝛽𝑁𝑐

1−𝛼−𝛽𝑍−𝛾

s.t. (9)

𝑍 = 𝑋 − 𝐿

𝑋 = 𝐻𝑁𝑥𝛿𝑌𝑥𝜀 → 𝜆𝑥

𝑌 = 𝑀𝑁𝑦𝜋𝑋𝑦𝜃𝑍−𝜌 → 𝜆𝑦

𝑁 = 𝐴𝑌𝑁𝜎𝑓(𝑍) → 𝜆𝑁

𝐿 = 𝑁𝑥𝑙𝜓 → 𝜆𝑙

𝑓(𝑋) = (𝑎𝑋3 + 𝑏𝑋2 + 𝑐𝑋 + 𝑑 + 𝜆(𝑥))

𝑋 = 𝑋𝑐 + 𝑋𝑦 → 𝜇𝑥

𝑌 = 𝑌𝑐 + 𝑌𝑥 + 𝑌𝑁 → 𝜇𝑦 12

𝑁 = 𝑁𝑐 + 𝑁𝑥 + 𝑁𝑦 + 𝑁𝑥𝑙 → 𝜇𝑁

Which, in turn, gives rise to a new term in the assignment of efficiency in addition to (13)

concerning the use of 𝑁𝑥𝑙.

�𝛾𝑍1𝛽𝑌𝑐

+ 𝜌 𝑌𝑍− 𝐴𝑌𝑁𝜎𝑓𝑙′(Z) 𝑌𝑁

𝜎𝑁�𝜓𝑁𝑥𝑙

𝜓−1 =𝜇𝑁𝜇𝑦

(14)

Equation (14) shows how the economy assesses a new activity consisting of mitigating the impact

of pollution (𝑍) on affected agents. This will depend on the marginal productivity of the resource

employed in the abating activity.

Now, market inefficient assignment is corrected with Pigouvian taxes and the use of abatement

technologies (with distinct levels of productivity), so that the market becomes efficient.

a. When X sector pays taxes:

Max 𝐵𝑋 = (𝑃𝑋 − 𝑡∗)𝑋 − 𝑃𝑁𝑁𝑥 − 𝑃𝑌𝑌𝑥 s.t. 𝑋 = 𝐻𝑁𝑥𝛿𝑌𝑥𝜀 (6)

𝑡∗ = �𝛾𝑋

1𝛽𝑌𝑐

+ 𝜌𝑌𝑋− 𝐴𝑌𝑁𝜎𝑓′(X)

𝑌𝑁𝜎𝑁

�𝑃𝑌

With t∗ being the sum of marginal damages on consumer, Y and N, valued at PY, such that the

allocation of efficiency is replicated. The amount 𝑡∗𝑍 will be added as a monetary transfer to the

consumer´s income.

b. When X simultaneously pays taxes and invests in available pollution abatement

technologies:

We can define: 𝑧 = 𝑋 − 𝐿, as net pollution and L as the quantity of X that can be cleaned, such

that: 𝐿 = 𝑁𝑥𝑙𝜓, i.e., it is a technology using N to reduce pollution per product unit.

Max 𝐵𝑋 = 𝑃𝑋𝑋 − 𝑃𝑁𝑁𝑥 − 𝑃𝑌𝑌𝑥 − 𝑃𝑁𝑁𝑥𝑙 − 𝑡∗𝑧

(𝑃𝑋 − 𝑡∗)𝑋 − 𝑃𝑁𝑁𝑥 − 𝑃𝑌𝑌𝑥 − 𝑃𝑁𝑁𝑥𝑙 + 𝑡∗𝐿

s.t. (7)

𝑋 = 𝐻𝑁𝑥𝛿𝑌𝑥𝜀

13

𝑧 = 𝑋 − 𝐿

𝐿 = 𝑁𝑥𝑙𝜓

What is new here is the fact that, given the choice of decontaminating, abatement technologies will

be used insomuch as their marginal productivity is equal to the cost of taxes t*. Again, tax

collection is transferred to the consumer, and the pollution that affects Y and N and consumer

utility is reduced from X to Z. We then expect that social welfare will increase.

The virtue of an economic model is that it is characterized by an emphasis in the different trade off

arising when the sector causing negative externalities is also important in terms of added value

generation and its contribution to material welfare. The economic instruments to correct

externalities may have different effects in terms of distribution and social welfare; and as abatement

technologies arise, incentives to invest in them are given, and as the productivity of this technology

increases all the agents improve their welfare. However, this is a model of representative agents or

sectors in which the interactions and distributive effects that might arise inside it are not detailed

neither the scarce incentives to invest in abatement technologies when they adopt the figure of a

public good and their results are probabilistic.

3. APPLICATION AND RESULTS

The systems of equations resulting from the various processes of optimization were solved using

“Wolfram Mathematica” software. The command used, “Findroot,” is interpreted by Mathematica

as an instruction to simultaneously employ the Newton-Raphson method, the secant and Brent’s

method in order to find the roots of the non-linear equation system, that is, the prices of the

equilibrium. Prior to executing this command, initial values were established for the various

parameters, such as productivity, preferences, affectation degrees of externalities on consumers,

nature and the less contaminating sector, the stock of nature and productivity of pollution abatement

technology (Nettleton, 2010).

Parameter Description Restraint Initial

value

α,β Consumer’s preferences

towards X and Y α,β > 0 0.33

ε Marginal productivity of Y

input to produce X ε+δ<1 0.1

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δ Marginal productivity of

input N to produce X ε+δ<1 0.7

θ Marginal productivity of

input X to produce Y θ+π<1 0.1

π Marginal productivity of

input N to produce Y θ+π<1 0.7

σ Marginal productivity of

input Y to produce N σ<<1 0.001

ψ Marginal productivity of

input N to clean X ψ<<1 0.001

ρ Externality measure over Y

by X sector ρ>0 0.1

γ Externality measure over

the consumer by X sector γ>0 0.33

f(x):

aX3+bX2+cX+d

Externality measure over

nature by X sector a=-0.0003, b=0.006, c=-0.1, d=1

A Natural capital stock 100

(Huge) or

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M,H Level of technology of

each sector (X and Y) M,H >1 2,3

Below are the results obtained by all four models presented regarding variations in production

levels, prices, benefits and welfare for various parameter values.

Returning to equation 3, critical point X=6,7 (-b/3a) is considered to be the one in which function N

turns from a productive steady state to another less productive.

Aiming to explain findings from simulating the model, the following comparisons were made. No

numeric analyses were presented since the extent of variations more closely follows a parameter

magnitude issue; parameters are not deriving from data resulting from information on a particular

region or case, but obeying to magnitudes representing a typical economy from a purely theoretical

point of view.

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First, a comparison between two models is presented: a base scenario in which sector X does not

issue externalities on the others, compared to a situation where X affects the others. Secondly, a

third model is analyzed, where a pigouvian taxe on sector X is introduced, modifying X’s

production choice, and the impact of this choice on the other agents. Thirdly, we consider the

possibility that X acquires a pollution abatement technology, thus reducing its tax payments and

being able to increase its production. Subsequently, trends related to changes in supply by three

sectors, X, Y and N, benefits to each sector, and changes on welfare are examined, measured both

in the consumer’s indirect utility changes and firms’ benefits10.

Model 1: without externalities & Model 2: with externalities

In an economy, going from a scenario where sector X does not impose externalities on the other

sectors to a scenario where X affects all other sectors is a comparative statics exercise intending to

analyze the move from a completely ideal world where there are no affectations or externalities to a

world where they do exist.

The model shows that in such a transition, the welfare of all agents in the economy drops as a

consequence of externalities, which may be reflected in the demand shrinking in all productive

inputs. Furthermore, price behavior reflects that N increases cost in relation to the other goods,

suggesting that, in the presence of externalities, spoilage of natural resources may be quite

dangerous in that externality levels may be reached, where nature loses its resilience at increasingly

higher rates. This is reflected in such an overproduction that the critical point threshold is exceeded.

Secondly, when considering the externality in the production of X on the other sectors, a reduction

of relative prices PX/PY was found in the simulation, which indicates that even though sector X

affects the other sectors, its benefits are reduced because of the fall in the demand of good X and

the drop in price. Thus, X causes harm to the other sectors, but this affectation is reverted in a fall in

its own benefits, like a “boomerang effect.”

Model 3: Tax payment

10 In this context, “social welfare” refers to the aggregation of industry benefits and the consumer’s indirect utility. It should be noted here that the consumer’s income, through which an optimal basket of consumer goods is chosen, is equal to the sum of firms’ benefits, which are supposed to be owned by the consumer, and therefore that consumer is directly receiving their benefits. Accordingly, “social welfare” may be considered a measurement of social general welfare, as long as all economic agents included in the model (both production and consumer sectors) are added.

16

In model 3, Pigouvian taxes applied to sector X may have any allocation, but we assume they are

transferred in their entirety to the consumer. In this case, following the application of a tax on

contaminating sector X, its benefits are greatly reduced and non-contaminating agents’ benefits

(such as N, Y and consumers) are compensated. Consumers are reimbursed to a great extent given

that they are benefitted by taxes. It is important to note that this model (when compared transition

from model 2 to 3) predicts that the economy will undergo a structural change in the way it utilizes

its production and consumption resources after the externality is corrected, precisely because the tax

greatly increases X’s price (in relation to Pn and Py), leading productive sectors and consumers to

make a marginal replacement for the remaining goods, that is, Y and N. Social welfare results point

out that if it is not possible to cross out externalities, the market by itself will not function

efficiently, and that correcting this through Pigouvian taxes may be an excellent solution. This is

reflected in the fact that in the equilibrium, nature keeps its resilience properties because affectation

by X has not yet reached the threshold or critical point. Thus, we may state that the economy, when

the externality is corrected through tax levy, enters a “safe” or more resilient “zone.”

A social idea derived a priori from this analysis is that, in practice, it is possible that sector X is

highly labor-intensive, so that imposing Pigouvian taxes would result in unemployment, poverty

and other undesirable consequences, which should be weighted when correcting the externality.

It is important to note that marginal damage affecting N explains almost everything all the tax. That

is, the high increase resulting from taxes levied in the price paid by a good X is explained mainly by

marginal affectation inflicted on nature.

Models 4: Introduction of a pollution abatement technology in sector X

The existence of an abatement technology allows access to the effects of a hybrid environmental

policy, a mix of tax payment and decontamination. In this model, the abatement technology is

supposedly not advanced enough to clean everything, for if this were the case, aside from being an

unreal assumption, such a reading would certainly be unimportant. Therefore, in sector X, a portion

of the production could be decontaminated, while a tax is levied to another portion. It is obviously

that X’s benefits will be a monotonously increasing function to productivity of abatement

technology, since an increase in this factor would help X decontaminate in a larger proportion, and

consequently pay less taxes. This is evidenced, in this case, in an increase in X profits in relation to

the previous model.

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Additionally, given the interconnectedness between the level of activity in different sectors and the

“reduction of the externality,” augmenting productivity in sector X via decontaminating

technologies translates into an increase in the other agents’ profits. Thus, this model predicts that

any abatement technology will also generate a higher Pareto equilibrium for any social agent and

thereby for society as a whole.

Concerning this new equilibrium, nature becomes much more resilient, which is reflected in the fact

that, with abatement technologies, the economy moves increasingly farther from the critical point

(graph 2).

Graph 2: Regulating non-resilience caused by an economic externality (pollution)

Stressing the model:

By varying the relative importance which nature has in consumer preferences and the initial natural

capital stock, for all possible values of parameters, we can dimension the impact that the generation

of externalities has on the economy (graph 3): If A and consumer preferences are smaller, not only

is there a reduction in the overall well-being, but nature loses its resilience property, and this also

greatly affects the same generator sector of externalities (what we have named a “Boomerang

effect”). The correction of these damages through the Pigouvian tax surpasses the nature to a

resilient area and increases social welfare, but remember that X falls dramatically, and increasing

Px is an exaggeration due mainly to the included tax. This last phenomenon can be understood as

the insurance that society undertakes to maintain the property of resilience of N, across Px.

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Graph 3: Changing nature’s relative importance

CONCLUSIONS

We have represented an economy with a sector that, while of importance in terms of value creation

as an input and consumer good, negatively affects the entire economy: the sector of

environmentally friendly goods, nature and representative consumer. The remaining sectors also

use nature to use environmental services, but they are regulated since property rights have been

defined. The introduction of economic incentives to correct externalities has differentiated effects in

terms of the benefits for companies, the consumer’s indirect utility and social welfare.

Even though the model does not inquire how the economy comes to obtain abatement technologies,

or about the formation of production technologies or consumer preferences, this model shows how

incentives to come to the path of adopting abatement technologies are generated. These incentives

even help to increase benefits to polluters, and as productivity increases, all the agents within an

economy enhance their own welfare.

It is obviously a model of representative agents or sectors whose inner interactions are not detailed.

For instance, the main result of the inefficiency of market equilibrium with externalities is that the

sector generating externalities as a whole is in a worse situation, but we may believe that inside the

model, if it would be detailed, there will always be those who benefit and those who are harmed in

a higher proportion, which will depend on the relative demand for that sector’s products by other

sectors of the economy. Therefore, incentives to invest in abatement technologies would not be

homogeneous. This finding may be complemented by the common knowledge that research and

development activities adopt the form of a public good and their result is probabilistic, which

reduces the incentives to invest in them.

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The equation representing environmental service aims to incorporate the complex definition of

resilience as a version of workings within a natural ecosystem and shows its interrelation with the

economic system. Thus, knowledge about this production function in a feature summarizing an

ecosystem’s properties and its affectations is a result of resource extraction, pollution or random

endogenous and exogenous events. While our model states that the use of taxes allows for

maintaining the polluting sector’s level of production so that resilience N is not affected, it is also

true that the tax is quite high in relative terms (compared to the prices of economy) and consumers

take on much of it because of price increases, which could also be interpreted a kind of insurance

for resilience.

Finally, it would be interesting to make some additional considerations within the bounds of this

paper. First, to model the inclusion of uncertainty in resilience, reflecting the fact that it is virtually

impossible to know to which extent nature would be able to endure affectation made on itself.

Second, to include other social aspects, such as unemployment or poverty, when making the

decision on how to correct the externality by introducing work input and labor market. Third, to

analyze incentives to innovate in abatement technologies within the bounds of a model of

interaction between several polluting agents instead of supposing available technologies for a

representative sector. Fourth, to study the differences when pollution abatement technologies are

not end-of-pipe solutions, but rather modify the production process of the polluting good. Last, to

expand the model to a dynamic character so that, for example, accumulative or de-accumulative

functions, representing best nature inner functioning and how accumulating pollution disturbs it.

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