- Email: [email protected]
On pollution permits and abatement
Economics Letters 119 (2013) 302–305
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
On pollution permits and abatement Hamid Beladi a,∗ , Lu Liu b , Reza Oladi c a
Department of Economics, University of Texas at San Antonio, USA
b
School of Economics, Southwestern University of Finance and Economics, China
c
Department of Applied Economics, Utah State University, USA
highlights • We construct a holistic model of pollution, permits, and abatement. • We characterize a steady-state equilibrium. • We show emission and pollution stock are increasing in natural absorption rate.
article
info
Article history: Received 19 January 2013 Received in revised form 25 February 2013 Accepted 8 March 2013 Available online 15 March 2013
abstract We construct a dynamic general equilibrium model of pollution and study in a holistic way the environmental policies, whereby government sets up emission caps and sells emission permits at a competitive price which can be viewed as an emission tax. Then, it uses the collected tax revenues to finance public abatement activities. We characterize a steady-state equilibrium and show how it changes with the size of the economy and with a change in natural pollution absorption rate. © 2013 Elsevier B.V. All rights reserved.
JEL classification: F1 Q5 Keywords: Pollution permits Abatement International trade
1. Introduction Environmental degradation and its relationship with economic activities have been a subject of intense debates in the academic community and among policy makers. While some aspects of these debates such as the impacts of economic activities on global warming remains unsettled, there are little disagreements as to whether some economic activities (e.g., some industrial production processes that emit industrial waste) lead to negative impacts on the environment. From the policy perspective, various solutions have been proposed from ‘‘do nothing’’ to different forms of interventions. A branch of literature considers Pigovian tax (for examples, see Baumol, 1971, among others), while other studies suggest the use
∗ Correspondence to: Department of Economics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249-0633, USA. Tel.: +1 210 458 7038; fax: +1 210 458 7040. E-mail address: [email protected] (H. Beladi). 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.03.016
of pollution permits (see Crocker, 1966; Copeland and Taylor, 1995). Yet others propose abatement in the construct of environmental policies (see Chao and Yu, 2000, among others). Notwithstanding the immense contributions in these studies, a holistic study of these policy tools in a dynamic context is missing in the literature.1 The purpose of this paper is take a holistic view of environmental policies where government chooses an optimal emission cap, issues permits and takes pollution abatement activity to clean up the environment. As the government sells emission permits, in a sense it can be viewed as imposing taxes on emission. So our view takes all three major facets of environmental policies into consideration together. To do so, we construct a dynamic general equilibrium model of pollution permits and abatement for an economy. We assume that our economy produces and consumes two commodities.
1 See also Batabyal (1998, 2012), Chao et al. (2012), Choi (in press), and Marjit et al. (2013) for other related studies.
H. Beladi et al. / Economics Letters 119 (2013) 302–305
Both production sectors use labor and pollution (as a by-product). We view emission as a flow variable that adds to the stock of pollution. The government interferes by using three instruments. On the one hand, it imposes a cap on emission and sells permits to the production sectors (i.e., taxes the production sectors). On the other hand, it reduces the stock of pollution through abatement activities financed by the collected taxes from the production sectors. To our knowledge, no attempt has been made in the literature to investigate optimal pollution permits and abatement in a holistic dynamic general equilibrium setup. We study the conditions under which a steady-state equilibrium exists at which abatement level, emission flow, and pollution stock is positive. We then characterize the steady-state equilibrium by showing that the abatement activities, as expressed in terms of employment in the abatement sector, is proportional to the labor force. We also indicate that larger economies emit more pollutants and have a higher steady-state level of pollution stocks. Finally, we derive a relationship between steady-state pollution stock, as well as the emission level, with the natural rate of pollution absorption. The organization of the rest of the paper is as follows. We present our dynamic general equilibrium model in Section 2. Section 3 derives our steady-state equilibrium properties and draws results with regard to optimum levels of permits and pollution treatment. We conclude the paper in Section 4. 2. The model Assume an economy that produces two goods, 1 and 2, using labor and pollution with Cobb–Douglas production technology given by: α β
Yi = Ei i Li i ,
i = 1, 2
(1)
where Yi , Ei , and Li are the production level, emission, and labor usage of sector i. We assume that both sectors exhibit constant returns to scale, i.e., αi + βi = 1, and that α1 < α2 and β1 > β2 implying that sector 1 is labor intensive and sector 2 is emission intensive. Assume that the household preference is represented by the utility function: θ1 θ2
U = C1 C2 X
−γ
(2)
where, Ci , i = 1, 2, is the consumption of good i and X is the accumulated stock of pollution, γ ∈ (0, 1), θi ∈ (0, 1), i = 1, 2, and θ1 + θ2 = 1. The last assumption guarantees the concavity of our indirect utility function that we will derive shortly. We further assume that a benevolent government that maximizes the household’s welfare by issuing pollution permits and cleaning up the environment. The abatement technology is represented by: A = LνA X u
(3)
where LA is the labor employment by the government in its abatement facilities. We assume that ν ∈ (0, 1) and u ∈ (0, 1), implying that marginal productivity of labor in these facilities is positive but decreasing and that such labor marginal productivity is increasing in pollution stock but a decreasing rate.2 We maintain that the government runs a balanced budget. That is, it chooses
2 It is noteworthy to state that both production sectors of the economy also respond to factor market (labor and permit market) conditions, as it will be clear shortly, and adjust their emission due to imperfect substitutability of emission and labor in production processes. Thus, producers also engage in pollution abatement by possibly choosing cleaner technologies. However, throughout the paper we refer to reduction of pollution stock by the government as pollution abatement.
303
a level of abatement activity that leads to zero profit.3 Thus, the government faces the following restriction.
w LA = τ E
(4)
where w and τ are the economy wide wage rate and the pollution permit price. We assume that pollution stock accumulates over time and obeys the following law of motion. X˙ = E − ηX − A
(5)
where X˙ is the accumulation rate of pollution and η is the natural rate of pollutant decay or absorption. Finally, assume that both good markets are competitive. Then, the competitive market assumption implies that: Ei =
Pi αi Yi
τ Pi βi Yi Li = w
i = 1, 2
(6)
i = 1, 2
(7)
where Pi is the price of good i. We normalize the initial good prices to unity throughout the paper. The labor and emission permit market clearing conditions are: L1 + L2 + LA = L
(8)
E1 + E2 = E
(9)
where L is the constant stock of labor and E is the total amount of pollution permits to be determined by the government. Turning now to the household and producers’ problems. The household face the budget constraint P1 C1 + P2 C2 = w L. The household maximization problem leads to the demand functions Ci = θi w L/P1 , i = 1, 2. Similarly, the producers’ demand for labor and permits can be derived as Ei = αi Pi Yi /τ and Li = βi Pi Yi /w, i = 1, 2. Using market clearing conditions Ci = Yi , i = 1, 2, the demand functions and input demand functions, we obtain that E1 /E2 = θ1 α1 /θ2 α2 and L1 /L2 = θ1 β1 /θ2 β2 . Next use these relationships between equilibrium input usage as well as Eqs. (8) and (9) to obtain:
θi αi E θ1 α1 + θ2 α2 θi βi Li = (L − LA ). θ1 β1 + θ2 β2
Ei =
(10) (11)
By substituting Eqs. (10) and (11) in Eq. (1), using the market clearing conditions once again, and then substituting the resulting equilibrium consumption levels in Eq. (2), we obtain the indirect utility function: V = Ω (L, LA )E σ X −γ
(12)
where σ = θ1 α1 + θ2 α2 and Ω (L, LA ) = Λ(L − LA )θ1 β1 +θ2 β2 ,
Λ = ([θ1 α1 /(θ1 α1 +θ2 α2 )]α1 [θ1 β1 /(θ1 β1 +θ2 β2 )]β1 )θ1 [θ2 α2 /(θ1 α1 + θ2 α2 )]α2 [θ2 α2 /(θ1 α1 + θ2 α2 )]α2 .
Before we conclude this section, it may be of interest to derive a relationship between equilibrium production/consumption combination and emission level. Again by substituting Eqs. (10) and (11) in Eq. (1) and after some simplification, we obtain: Y1 Y2
= Φ E α1 −α2 (L − LA )β1 −β2
(13)
where Φ = [θ1 α1 /(θ1 α1 + θ2 α2 )]α1 [θ1 β1 /(θ1 β1 + θ2 β2 )]β1 [θ2 α2 / (θ1 α1 + θ2 α2 )]−α2 [θ2 β2 /(θ1 β1 + θ2 β2 )]−β2 > 0. Since α2 > α1
3 One can view and rationalize this assumption slightly differently. While the government is the sole agent that cleans up the environment, it acts as if the market for environmental cleanup were competitive, leading to zero-profit level of abatement.
304
H. Beladi et al. / Economics Letters 119 (2013) 302–305
and β1 > β2 (i.e., good 1 is labor intensive and good 2 is pollution intensive), an increase in pollution permits issued by the government decreases Y1 /Y2 while this output ratio is increasing in (L − LA ), ceteris paribus. Similarly, by using demand functions we derived earlier as well as market clearing conditions for good 1 and 2 along with Eq. (13), we obtain:
θ2 (14) Φ E α1 −α2 (L − LA )β1 −β2 θ1 where p ≡ P2 /P1 is the relative price of good 2. Thus, the relative price of good 2 is decreasing in E and increasing in (L − LA ). p=
Next, use the demand function and the good market clearing condition for good 1, Eq. (11), as well as the pollution permit demand for sector 1, to obtain τ = (α1 θ1 + α2 θ2 )w L/E. Similarly, by using the demand function and the good market clearing condition for good 1, Eq. (12), as well as the labor demand for sector 1, we get w = (β1 θ1 + β2 θ2 )w L/(L − LA ). By using these factor price relationships, we obtain:
ΨE (15) ω= L − LA where ω ≡ w/τ is the relative wage rate and Ψ = (β1 θ1 + β2 θ2 )/(α1 θ1 + α 2θ2 ) > 0. We therefore conclude that the relative wage rate is increasing in pollution permits issued by the government and decreasing in (L − LA ). 3. Optimal abatement and permit We assume that the government chooses levels of abatement and pollution permit to maximize welfare. That is, the government faces the following optimization problem. ∞
max E ,LA
Ω (L, LA )E σ X −γ exp(−ρ t )dt
0
(16)
s.t. X˙ = E − ηX − A X (0) = X0
where ρ is the discount rate and X0 is the initial pollution stock. Before we proceed with the above optimal control problem, we will take a closer look at the optimal abatement activities. Since we assumed by Eq. (4) that the government maintains a balanced budget, the choice of LA is somewhat tied to the choice of pollution permits. In fact, as the following proposition states, the choice of optimal labor employment in the abatement sector is intimately tied with the labor endowment.
where Λ is defined as in the preceding section. Therefore, the government’s problem reduces to:
E
1 1+Ψ
L
(17)
where Ψ is defined as in Eq. (15).
It follows also from Proof of Proposition 1 that the level of equilibrium abatement for a given stock of pollution is A = (L/(1 + Ψ ))v X u . That is, given the fixed endowment of labor, the equilibrium level of abatement is a function of pollution stock only. It also follows from the above proposition that the amount of labor left for the production sectors 1 and 2 will also be proportional to labor endowment. Particularly, we use Eq. (17) to obtain L − LA = Ψ L/(1 + Ψ ). Therefore, function Ω in the indirect utility function, given an equilibrium in the abatement sector, will be reduced to:
Υ (L) = Ω (L, LA ) = Λ
Ψ 1+Ψ
θ1 β1 +θ2 β2
Lθ1 β1 +θ2 β2
(18)
(19)
X (0) = X0 . The present value Hamiltonian for the above control problem is: H = Υ (L)E σ X −γ exp(−ρ t ) + ζ (E − ηX − LνA X u )
(20)
where ζ is the costate variable. The corresponding first order conditions are given by:
σ Υ (L)E σ −1 X −γ exp(−ρ t ) + ζ = 0 ζ˙ = γ Υ (L)E σ X −γ −1 exp(−ρ t ) + ζ (η + uLνA X u−1 ).
(21) (22)
As it is easier to work with current value first order conditions, we rewrite Eqs. (21) and (22), respectively, as:
λ = −σ Υ (L)E σ −1 X −γ λ˙ = (ρ + η + uLνA X u−1 )λ + γ Υ (L)E σ X −γ −1
(23) (24)
where λ ≡ ζ exp(ρ t ) is the corresponding current value costate ˙ = 0, and simplify variable. Use Eqs. (5), (17), (23) and (24), X˙ = λ to obtain:
σX ρ + η + u E − ηX −
ν
L 1+Ψ
ν
L 1+Ψ
X u −1
− γE = 0
(25)
Xu = 0
(26)
Eqs. (25) and (26) fully characterize the steady-state equilibrium of our dynamical system. Next, in the result that follows, we address the conditions under which a meaningful steady state equilibrium exists. Proposition 2. There exists a steady-state equilibrium at which 0 < X < ∞ and 0 < E < ∞ if u < γ /σ < 1 + ρ/η. Moreover, the steady state equilibrium X and E is increasing in labor endowment L. Proof. Solve Eqs. (25) and (26) simultaneously for X and E to obtain:
s
Proposition 1. The level of labor employment in abatement sector is proportional to the labor endowment in the economy.
LA =
Υ (L)E σ X −γ exp(−ρ t )dt
0
s.t. X˙ = E − ηX − A
X =
Proof. Rewrite Eq. (4) as LA = E /ω. Next, use this relationship and Eq. (15) to obtain:
∞
max
(γ − σ u)
1 1+Ψ
σ (ρ + η) − γ η
σ η(1 − u) + σ ρ E = (γ − σ u) s
ν 1−1 u
ν
L 1−u
(γ − σ u)
(27)
1 1+Ψ
ν 1−1 u
σ (ρ + η) − γ η
ν
L 1−u .
(28)
On the one hand, if u < γ /σ then clearly γ − σ u > 0. On the other hand, γ /σ < 1 + ρ/η implies that σ (ρ + η) − γ η > 0. Therefore, condition u < γ /σ < 1 + ρ/η guarantees a steady state equilibrium of the type stated in this proposition. It also follows from Eqs. (27) and (28) that dX s /dL > 0 and dE S /dL > 0 since u < 1. As we are addressing a general theory of production, pollution, abatement efforts, and pollution cap and permits, it is crucial to address the possible differences that arise from differing degree of natural absorption rates for various pollutants. Nuclear wastes are clearly different from carbon emission as it could take decades for nuclear waste to be absorbed by nature. Our framework allows for such a difference in pollutants. Therefore, our results may be different when we apply it to various type of pollutants with different natural rate of absorption. The following proposition formally addresses this issue.
H. Beladi et al. / Economics Letters 119 (2013) 302–305
Proposition 3. The steady state level of emission permit and pollution stock are increasing in η if σ > γ . Proof. Differentiate Eqs. (27) and (28) with respect to η and simplify the result to obtain:
σ −γ (1 − u)[σ (ρ + η) − γ η] ν 1−1 u (γ − σ u) 1+1Ψ ν × L 1−u σ (ρ + η) − γ η dE s σ −γ σ ( 1 − u) = + dη (1 − u)[σ (ρ + η) − γ η] γ − σu 1 ν 1−1 u (γ − σ u) 1+Ψ ν L 1−u . × σ (ρ + η) − γ η dX s dη
=
(29)
(30)
Given our earlier assumptions on the existence of steady state equilibrium, γ − σ u > 0 and γ /σ < 1 + ρ/η. Then, since 0 < u < 1, it follows from Eqs. (29) and (30) that both X s and E s are increasing in η if σ > γ . This result seems interesting. An increase in absorption rate can lead to an increase in the steady state pollution stock. To make sense of economic intuition of this result, recall that σ ≡ θ1 α1 + α2 θ2 . This can be interpreted as some sort of emission intensity measure for the overall economy, which is the weighted average of pollution shares of outputs where consumption (or production) shares have been used as weights.4 It can be used as a measure of extent of how intensely an economy is emitting pollutants. In other words, it is a measure of how dirty an economy is. The proposition states that if such a measure of aggregate pollution intensity exceeds a threshold, then an increase in absorption rate will increase the stock of pollution as well as emission. Interestingly, this threshold turns out to be γ , the parameter of the damage accrued by the consumers due to pollution. Another way to view this is that the pollution stock is increasing in absorption rate if the extent of the pollution damage γ is less than a threshold, where the threshold is the overall measure of the dirtiness of the economy, i.e., σ . 4. Conclusion We construct a dynamic Ricardian general equilibrium model of production and pollution to study a holistic view of environmental policies. The government sets an optimal emission level for the
4 Clearly this is a departure from the notion of factor intensity which in essence is an inter-sectoral notion rather than an aggregate notion.
305
economy and sells emission permits at a competitive price, thus in a sense it taxes emission. The collected taxes (i.e., pollution permit revenues) are used to abate pollution stock. We show that the equilibrium level of abatement activities is proportional to the labor force. We also derive the equilibrium permit price and the steady-state levels of emission and pollution stock. The conditions under which a larger economy will have greater steady-state levels of emission and pollution stock were also studies. In addition, we explain how natural pollution absorption rate affects our steadystate equilibrium pollution stock and emission flow. Our model has various features that can be extended and used in future research. One can use our framework to study issues such as global pollution and international environmental agreements. Yet another interesting extension is to look at the dynamics of comparative advantage using our setup. In particular, one can address questions such as whether the dynamics of pollution permits and abatement would alter a country’s comparative advantage and pattern of trade. Acknowledgments We thank an anonymous referee for insightful comments. The usual caveats apply. Hamid Beladi acknowledges supports from IBC Bank and Reza Oladi is thankful for financial support from Utah Agricultural Experiment Station. References Baumol, W.J., 1971. Environmental Protection, International Spillovers and Trade. Almqvist and Wiksell, Stockholm. Batabyal, A.A., 1998. Developing countries and international environmental agreements: the case of perfect correlation. International Review of Economics and Finance 7, 85–102. Batabyal, A.A., 2012. Project financing, entrepreneurial activity, and investment in the presence of asymmetric information. North American Journal of Economics and Finance 23, 115–122. Chao, C-C., Laffargu, J-P., Sgro, P.M., 2012. Environmental control, wage inequality and national welfare in a tourism economy. International Review of Economics and Finance 22, 201–207. Chao, C-C., Yu, E.S.H., 2000. TRIMs, environmental taxes, and foreign investment. Canadian Journal of Economics 33, 799–817. Choi, K., 2012. Genetic contamination of traditional products. International Review of Economics and Finance (in press). Copeland, B.R., Taylor, M.S., 1995. Trade and transboundary pollution. American Economic Review 85, 716–737. Crocker, T.D., 1966. The structuring of atmospheric pollution control system. In: Wolozin, H. (Ed.), The Economics of Air Pollution. W.W. Norton, New York, pp. 61–86. Marjit, S., Kar, S., Hazari, B.R., 2013. Emigration, unemployment and welfare The role of non-traded sector. North American Journal of Economics and Finance 24, 298–305.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
On pollution permits and abatement Hamid Beladi a,∗ , Lu Liu b , Reza Oladi c a
Department of Economics, University of Texas at San Antonio, USA
b
School of Economics, Southwestern University of Finance and Economics, China
c
Department of Applied Economics, Utah State University, USA
highlights • We construct a holistic model of pollution, permits, and abatement. • We characterize a steady-state equilibrium. • We show emission and pollution stock are increasing in natural absorption rate.
article
info
Article history: Received 19 January 2013 Received in revised form 25 February 2013 Accepted 8 March 2013 Available online 15 March 2013
abstract We construct a dynamic general equilibrium model of pollution and study in a holistic way the environmental policies, whereby government sets up emission caps and sells emission permits at a competitive price which can be viewed as an emission tax. Then, it uses the collected tax revenues to finance public abatement activities. We characterize a steady-state equilibrium and show how it changes with the size of the economy and with a change in natural pollution absorption rate. © 2013 Elsevier B.V. All rights reserved.
JEL classification: F1 Q5 Keywords: Pollution permits Abatement International trade
1. Introduction Environmental degradation and its relationship with economic activities have been a subject of intense debates in the academic community and among policy makers. While some aspects of these debates such as the impacts of economic activities on global warming remains unsettled, there are little disagreements as to whether some economic activities (e.g., some industrial production processes that emit industrial waste) lead to negative impacts on the environment. From the policy perspective, various solutions have been proposed from ‘‘do nothing’’ to different forms of interventions. A branch of literature considers Pigovian tax (for examples, see Baumol, 1971, among others), while other studies suggest the use
∗ Correspondence to: Department of Economics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249-0633, USA. Tel.: +1 210 458 7038; fax: +1 210 458 7040. E-mail address: [email protected] (H. Beladi). 0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.03.016
of pollution permits (see Crocker, 1966; Copeland and Taylor, 1995). Yet others propose abatement in the construct of environmental policies (see Chao and Yu, 2000, among others). Notwithstanding the immense contributions in these studies, a holistic study of these policy tools in a dynamic context is missing in the literature.1 The purpose of this paper is take a holistic view of environmental policies where government chooses an optimal emission cap, issues permits and takes pollution abatement activity to clean up the environment. As the government sells emission permits, in a sense it can be viewed as imposing taxes on emission. So our view takes all three major facets of environmental policies into consideration together. To do so, we construct a dynamic general equilibrium model of pollution permits and abatement for an economy. We assume that our economy produces and consumes two commodities.
1 See also Batabyal (1998, 2012), Chao et al. (2012), Choi (in press), and Marjit et al. (2013) for other related studies.
H. Beladi et al. / Economics Letters 119 (2013) 302–305
Both production sectors use labor and pollution (as a by-product). We view emission as a flow variable that adds to the stock of pollution. The government interferes by using three instruments. On the one hand, it imposes a cap on emission and sells permits to the production sectors (i.e., taxes the production sectors). On the other hand, it reduces the stock of pollution through abatement activities financed by the collected taxes from the production sectors. To our knowledge, no attempt has been made in the literature to investigate optimal pollution permits and abatement in a holistic dynamic general equilibrium setup. We study the conditions under which a steady-state equilibrium exists at which abatement level, emission flow, and pollution stock is positive. We then characterize the steady-state equilibrium by showing that the abatement activities, as expressed in terms of employment in the abatement sector, is proportional to the labor force. We also indicate that larger economies emit more pollutants and have a higher steady-state level of pollution stocks. Finally, we derive a relationship between steady-state pollution stock, as well as the emission level, with the natural rate of pollution absorption. The organization of the rest of the paper is as follows. We present our dynamic general equilibrium model in Section 2. Section 3 derives our steady-state equilibrium properties and draws results with regard to optimum levels of permits and pollution treatment. We conclude the paper in Section 4. 2. The model Assume an economy that produces two goods, 1 and 2, using labor and pollution with Cobb–Douglas production technology given by: α β
Yi = Ei i Li i ,
i = 1, 2
(1)
where Yi , Ei , and Li are the production level, emission, and labor usage of sector i. We assume that both sectors exhibit constant returns to scale, i.e., αi + βi = 1, and that α1 < α2 and β1 > β2 implying that sector 1 is labor intensive and sector 2 is emission intensive. Assume that the household preference is represented by the utility function: θ1 θ2
U = C1 C2 X
−γ
(2)
where, Ci , i = 1, 2, is the consumption of good i and X is the accumulated stock of pollution, γ ∈ (0, 1), θi ∈ (0, 1), i = 1, 2, and θ1 + θ2 = 1. The last assumption guarantees the concavity of our indirect utility function that we will derive shortly. We further assume that a benevolent government that maximizes the household’s welfare by issuing pollution permits and cleaning up the environment. The abatement technology is represented by: A = LνA X u
(3)
where LA is the labor employment by the government in its abatement facilities. We assume that ν ∈ (0, 1) and u ∈ (0, 1), implying that marginal productivity of labor in these facilities is positive but decreasing and that such labor marginal productivity is increasing in pollution stock but a decreasing rate.2 We maintain that the government runs a balanced budget. That is, it chooses
2 It is noteworthy to state that both production sectors of the economy also respond to factor market (labor and permit market) conditions, as it will be clear shortly, and adjust their emission due to imperfect substitutability of emission and labor in production processes. Thus, producers also engage in pollution abatement by possibly choosing cleaner technologies. However, throughout the paper we refer to reduction of pollution stock by the government as pollution abatement.
303
a level of abatement activity that leads to zero profit.3 Thus, the government faces the following restriction.
w LA = τ E
(4)
where w and τ are the economy wide wage rate and the pollution permit price. We assume that pollution stock accumulates over time and obeys the following law of motion. X˙ = E − ηX − A
(5)
where X˙ is the accumulation rate of pollution and η is the natural rate of pollutant decay or absorption. Finally, assume that both good markets are competitive. Then, the competitive market assumption implies that: Ei =
Pi αi Yi
τ Pi βi Yi Li = w
i = 1, 2
(6)
i = 1, 2
(7)
where Pi is the price of good i. We normalize the initial good prices to unity throughout the paper. The labor and emission permit market clearing conditions are: L1 + L2 + LA = L
(8)
E1 + E2 = E
(9)
where L is the constant stock of labor and E is the total amount of pollution permits to be determined by the government. Turning now to the household and producers’ problems. The household face the budget constraint P1 C1 + P2 C2 = w L. The household maximization problem leads to the demand functions Ci = θi w L/P1 , i = 1, 2. Similarly, the producers’ demand for labor and permits can be derived as Ei = αi Pi Yi /τ and Li = βi Pi Yi /w, i = 1, 2. Using market clearing conditions Ci = Yi , i = 1, 2, the demand functions and input demand functions, we obtain that E1 /E2 = θ1 α1 /θ2 α2 and L1 /L2 = θ1 β1 /θ2 β2 . Next use these relationships between equilibrium input usage as well as Eqs. (8) and (9) to obtain:
θi αi E θ1 α1 + θ2 α2 θi βi Li = (L − LA ). θ1 β1 + θ2 β2
Ei =
(10) (11)
By substituting Eqs. (10) and (11) in Eq. (1), using the market clearing conditions once again, and then substituting the resulting equilibrium consumption levels in Eq. (2), we obtain the indirect utility function: V = Ω (L, LA )E σ X −γ
(12)
where σ = θ1 α1 + θ2 α2 and Ω (L, LA ) = Λ(L − LA )θ1 β1 +θ2 β2 ,
Λ = ([θ1 α1 /(θ1 α1 +θ2 α2 )]α1 [θ1 β1 /(θ1 β1 +θ2 β2 )]β1 )θ1 [θ2 α2 /(θ1 α1 + θ2 α2 )]α2 [θ2 α2 /(θ1 α1 + θ2 α2 )]α2 .
Before we conclude this section, it may be of interest to derive a relationship between equilibrium production/consumption combination and emission level. Again by substituting Eqs. (10) and (11) in Eq. (1) and after some simplification, we obtain: Y1 Y2
= Φ E α1 −α2 (L − LA )β1 −β2
(13)
where Φ = [θ1 α1 /(θ1 α1 + θ2 α2 )]α1 [θ1 β1 /(θ1 β1 + θ2 β2 )]β1 [θ2 α2 / (θ1 α1 + θ2 α2 )]−α2 [θ2 β2 /(θ1 β1 + θ2 β2 )]−β2 > 0. Since α2 > α1
3 One can view and rationalize this assumption slightly differently. While the government is the sole agent that cleans up the environment, it acts as if the market for environmental cleanup were competitive, leading to zero-profit level of abatement.
304
H. Beladi et al. / Economics Letters 119 (2013) 302–305
and β1 > β2 (i.e., good 1 is labor intensive and good 2 is pollution intensive), an increase in pollution permits issued by the government decreases Y1 /Y2 while this output ratio is increasing in (L − LA ), ceteris paribus. Similarly, by using demand functions we derived earlier as well as market clearing conditions for good 1 and 2 along with Eq. (13), we obtain:
θ2 (14) Φ E α1 −α2 (L − LA )β1 −β2 θ1 where p ≡ P2 /P1 is the relative price of good 2. Thus, the relative price of good 2 is decreasing in E and increasing in (L − LA ). p=
Next, use the demand function and the good market clearing condition for good 1, Eq. (11), as well as the pollution permit demand for sector 1, to obtain τ = (α1 θ1 + α2 θ2 )w L/E. Similarly, by using the demand function and the good market clearing condition for good 1, Eq. (12), as well as the labor demand for sector 1, we get w = (β1 θ1 + β2 θ2 )w L/(L − LA ). By using these factor price relationships, we obtain:
ΨE (15) ω= L − LA where ω ≡ w/τ is the relative wage rate and Ψ = (β1 θ1 + β2 θ2 )/(α1 θ1 + α 2θ2 ) > 0. We therefore conclude that the relative wage rate is increasing in pollution permits issued by the government and decreasing in (L − LA ). 3. Optimal abatement and permit We assume that the government chooses levels of abatement and pollution permit to maximize welfare. That is, the government faces the following optimization problem. ∞
max E ,LA
Ω (L, LA )E σ X −γ exp(−ρ t )dt
0
(16)
s.t. X˙ = E − ηX − A X (0) = X0
where ρ is the discount rate and X0 is the initial pollution stock. Before we proceed with the above optimal control problem, we will take a closer look at the optimal abatement activities. Since we assumed by Eq. (4) that the government maintains a balanced budget, the choice of LA is somewhat tied to the choice of pollution permits. In fact, as the following proposition states, the choice of optimal labor employment in the abatement sector is intimately tied with the labor endowment.
where Λ is defined as in the preceding section. Therefore, the government’s problem reduces to:
E
1 1+Ψ
L
(17)
where Ψ is defined as in Eq. (15).
It follows also from Proof of Proposition 1 that the level of equilibrium abatement for a given stock of pollution is A = (L/(1 + Ψ ))v X u . That is, given the fixed endowment of labor, the equilibrium level of abatement is a function of pollution stock only. It also follows from the above proposition that the amount of labor left for the production sectors 1 and 2 will also be proportional to labor endowment. Particularly, we use Eq. (17) to obtain L − LA = Ψ L/(1 + Ψ ). Therefore, function Ω in the indirect utility function, given an equilibrium in the abatement sector, will be reduced to:
Υ (L) = Ω (L, LA ) = Λ
Ψ 1+Ψ
θ1 β1 +θ2 β2
Lθ1 β1 +θ2 β2
(18)
(19)
X (0) = X0 . The present value Hamiltonian for the above control problem is: H = Υ (L)E σ X −γ exp(−ρ t ) + ζ (E − ηX − LνA X u )
(20)
where ζ is the costate variable. The corresponding first order conditions are given by:
σ Υ (L)E σ −1 X −γ exp(−ρ t ) + ζ = 0 ζ˙ = γ Υ (L)E σ X −γ −1 exp(−ρ t ) + ζ (η + uLνA X u−1 ).
(21) (22)
As it is easier to work with current value first order conditions, we rewrite Eqs. (21) and (22), respectively, as:
λ = −σ Υ (L)E σ −1 X −γ λ˙ = (ρ + η + uLνA X u−1 )λ + γ Υ (L)E σ X −γ −1
(23) (24)
where λ ≡ ζ exp(ρ t ) is the corresponding current value costate ˙ = 0, and simplify variable. Use Eqs. (5), (17), (23) and (24), X˙ = λ to obtain:
σX ρ + η + u E − ηX −
ν
L 1+Ψ
ν
L 1+Ψ
X u −1
− γE = 0
(25)
Xu = 0
(26)
Eqs. (25) and (26) fully characterize the steady-state equilibrium of our dynamical system. Next, in the result that follows, we address the conditions under which a meaningful steady state equilibrium exists. Proposition 2. There exists a steady-state equilibrium at which 0 < X < ∞ and 0 < E < ∞ if u < γ /σ < 1 + ρ/η. Moreover, the steady state equilibrium X and E is increasing in labor endowment L. Proof. Solve Eqs. (25) and (26) simultaneously for X and E to obtain:
s
Proposition 1. The level of labor employment in abatement sector is proportional to the labor endowment in the economy.
LA =
Υ (L)E σ X −γ exp(−ρ t )dt
0
s.t. X˙ = E − ηX − A
X =
Proof. Rewrite Eq. (4) as LA = E /ω. Next, use this relationship and Eq. (15) to obtain:
∞
max
(γ − σ u)
1 1+Ψ
σ (ρ + η) − γ η
σ η(1 − u) + σ ρ E = (γ − σ u) s
ν 1−1 u
ν
L 1−u
(γ − σ u)
(27)
1 1+Ψ
ν 1−1 u
σ (ρ + η) − γ η
ν
L 1−u .
(28)
On the one hand, if u < γ /σ then clearly γ − σ u > 0. On the other hand, γ /σ < 1 + ρ/η implies that σ (ρ + η) − γ η > 0. Therefore, condition u < γ /σ < 1 + ρ/η guarantees a steady state equilibrium of the type stated in this proposition. It also follows from Eqs. (27) and (28) that dX s /dL > 0 and dE S /dL > 0 since u < 1. As we are addressing a general theory of production, pollution, abatement efforts, and pollution cap and permits, it is crucial to address the possible differences that arise from differing degree of natural absorption rates for various pollutants. Nuclear wastes are clearly different from carbon emission as it could take decades for nuclear waste to be absorbed by nature. Our framework allows for such a difference in pollutants. Therefore, our results may be different when we apply it to various type of pollutants with different natural rate of absorption. The following proposition formally addresses this issue.
H. Beladi et al. / Economics Letters 119 (2013) 302–305
Proposition 3. The steady state level of emission permit and pollution stock are increasing in η if σ > γ . Proof. Differentiate Eqs. (27) and (28) with respect to η and simplify the result to obtain:
σ −γ (1 − u)[σ (ρ + η) − γ η] ν 1−1 u (γ − σ u) 1+1Ψ ν × L 1−u σ (ρ + η) − γ η dE s σ −γ σ ( 1 − u) = + dη (1 − u)[σ (ρ + η) − γ η] γ − σu 1 ν 1−1 u (γ − σ u) 1+Ψ ν L 1−u . × σ (ρ + η) − γ η dX s dη
=
(29)
(30)
Given our earlier assumptions on the existence of steady state equilibrium, γ − σ u > 0 and γ /σ < 1 + ρ/η. Then, since 0 < u < 1, it follows from Eqs. (29) and (30) that both X s and E s are increasing in η if σ > γ . This result seems interesting. An increase in absorption rate can lead to an increase in the steady state pollution stock. To make sense of economic intuition of this result, recall that σ ≡ θ1 α1 + α2 θ2 . This can be interpreted as some sort of emission intensity measure for the overall economy, which is the weighted average of pollution shares of outputs where consumption (or production) shares have been used as weights.4 It can be used as a measure of extent of how intensely an economy is emitting pollutants. In other words, it is a measure of how dirty an economy is. The proposition states that if such a measure of aggregate pollution intensity exceeds a threshold, then an increase in absorption rate will increase the stock of pollution as well as emission. Interestingly, this threshold turns out to be γ , the parameter of the damage accrued by the consumers due to pollution. Another way to view this is that the pollution stock is increasing in absorption rate if the extent of the pollution damage γ is less than a threshold, where the threshold is the overall measure of the dirtiness of the economy, i.e., σ . 4. Conclusion We construct a dynamic Ricardian general equilibrium model of production and pollution to study a holistic view of environmental policies. The government sets an optimal emission level for the
4 Clearly this is a departure from the notion of factor intensity which in essence is an inter-sectoral notion rather than an aggregate notion.
305
economy and sells emission permits at a competitive price, thus in a sense it taxes emission. The collected taxes (i.e., pollution permit revenues) are used to abate pollution stock. We show that the equilibrium level of abatement activities is proportional to the labor force. We also derive the equilibrium permit price and the steady-state levels of emission and pollution stock. The conditions under which a larger economy will have greater steady-state levels of emission and pollution stock were also studies. In addition, we explain how natural pollution absorption rate affects our steadystate equilibrium pollution stock and emission flow. Our model has various features that can be extended and used in future research. One can use our framework to study issues such as global pollution and international environmental agreements. Yet another interesting extension is to look at the dynamics of comparative advantage using our setup. In particular, one can address questions such as whether the dynamics of pollution permits and abatement would alter a country’s comparative advantage and pattern of trade. Acknowledgments We thank an anonymous referee for insightful comments. The usual caveats apply. Hamid Beladi acknowledges supports from IBC Bank and Reza Oladi is thankful for financial support from Utah Agricultural Experiment Station. References Baumol, W.J., 1971. Environmental Protection, International Spillovers and Trade. Almqvist and Wiksell, Stockholm. Batabyal, A.A., 1998. Developing countries and international environmental agreements: the case of perfect correlation. International Review of Economics and Finance 7, 85–102. Batabyal, A.A., 2012. Project financing, entrepreneurial activity, and investment in the presence of asymmetric information. North American Journal of Economics and Finance 23, 115–122. Chao, C-C., Laffargu, J-P., Sgro, P.M., 2012. Environmental control, wage inequality and national welfare in a tourism economy. International Review of Economics and Finance 22, 201–207. Chao, C-C., Yu, E.S.H., 2000. TRIMs, environmental taxes, and foreign investment. Canadian Journal of Economics 33, 799–817. Choi, K., 2012. Genetic contamination of traditional products. International Review of Economics and Finance (in press). Copeland, B.R., Taylor, M.S., 1995. Trade and transboundary pollution. American Economic Review 85, 716–737. Crocker, T.D., 1966. The structuring of atmospheric pollution control system. In: Wolozin, H. (Ed.), The Economics of Air Pollution. W.W. Norton, New York, pp. 61–86. Marjit, S., Kar, S., Hazari, B.R., 2013. Emigration, unemployment and welfare The role of non-traded sector. North American Journal of Economics and Finance 24, 298–305.