A dynamic general equilibrium model of pollution abatement under learning by doing

Economics Letters 122 (2014) 285–288

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

A dynamic general equilibrium model of pollution abatement under learning by doing Shoude Li, Xiaojun Pan ∗ Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200052, China

highlights • • • •

Extend the model of Beladi et al. (2013) with learning by doing. Investigate the dynamic general equilibrium model of pollution abatement. Analyze the steady-state equilibrium properties of emission permits and pollution treatment. Derive the steady-state optimal levels of emission permits and pollution treatment.

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Article history: Received 17 October 2013 Received in revised form 28 November 2013 Accepted 1 December 2013 Available online 11 December 2013 JEL classification: D21 L51 Q52

abstract In 2013, Beladi et al. constructed a dynamic general equilibrium model of pollution, and characterized a steady-state equilibrium. In this paper, we extend Beladi et al.’s model to an even more general model in which the pollution abatement costs under learning by doing are taken into account. In our model, the instantaneous abatement costs depend on both the rate of abatement and the experience of using a technology. Our objective is to apply optimal control theory to investigate the dynamic general equilibrium model of pollution abatement, and derive the steady-state equilibrium properties and optimal levels of emission permits and pollution treatment. © 2013 Elsevier B.V. All rights reserved.

Keywords: General equilibrium Pollution abatement Learning by doing

1. Introduction In a recent work, Beladi et al. (2013) constructed a dynamic general equilibrium model of pollution and characterized a steady state equilibrium. A noted feature of the authors’ paper is taking a holistic view of environmental policies where the 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 the authors’ view takes all three major facets of environmental policies into consideration together. In this paper, following the analytical framework of Beladi et al. (2013), we present a dynamic general equilibrium model of pollution abatement under learning by doing. Furthermore, our analysis is in the spirit of the study by Bramoullé and Olson (2005)—the

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Corresponding author. Tel.: +86 21 52302551; fax: +86 21 62932982. E-mail address: [email protected] (X. Pan).

0165-1765/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.12.002

first to obtain the analytical expression for the experience using pollution abatement technology—as well as more recent works by Argotte and Epple (1990) and Greaker and Rosendahl (2008). Another related paper is by Goulder and Mathai (2000), in which the authors investigate the problem of abatement-cost function. Goulder et al.’s abatement-cost function differs from ours, however, in that it focuses on learning that improves abatement technologies and thus reduces abatement costs. In our model, the experience using pollution abatement technology is measured by the cumulative abatement from time 0 to t. This assumption is consistent with empirical studies of learning by doing, in which experience is generally measured as cumulative production. The instantaneous abatement costs depend on both the rate of abatement and the experience of using a technology. This paper is organized as follows. In the next section we present our dynamic general equilibrium model of pollution abatement. Section 3 derives our steady-state equilibrium properties and optimal levels of emission permits and pollution treatment. We summarize the results in Section 4.

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S. Li, X. Pan / Economics Letters 122 (2014) 285–288

2. The basic model We start out with the simple but widely used model in which an economy produces two goods, 1 and 2. Let Yi , Ei and Li be the production level, emission level and labor usage of sector i, respectively, i = 1, 2. According to Beladi et al. (2013), the production level is given by the following Cobb–Douglas production function: α β

Yi = Ei i Li i

(1)

where αi and βi are positive constants. Here 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. Following Beladi et al. (2013), the consumers’ preference or utility function can be represented as θ

θ

U = C1 1 C2 2 X −γ

(2)

where Ci is the consumption of goods i, i = 1, 2; X is the accumulated stock of pollution; γ ∈ (0, 1), θi ∈ (0, 1), i = 1, 2. In order to ensure the concavity of the utility function, we assume that θ1 + θ2 = 1. Let A(t ) be the pollution abatement level at time t. According to Bramoullé and Olson (2005), experience of using pollution abatement technology is measured by the cumulative abatement from time 0 to t and is given by the following form: Z (t ) = Z0 + µ lim

t →∞

A(s)ds

(3)

0

where Z0 denotes the initial level of experience using pollution abatement technology, µ is the positive constant. This assumption is consistent with empirical studies of learning by doing, in which experience is generally measured as cumulative production (Argotte and Epple, 1990). Following Bramoullé and Olson (2005), the instantaneous abatement cost C (A(t ), Z (t )) depends on both the rate of abatement and the experience of using a pollution abatement technology, and the instantaneous abatement cost C (A(t ), Z (t )) has the following properties: (i) C (A(t ), Z (t )) is twice differentiable; (ii) C (A(t ), Z (t )) is increasing in A and decreasing in Z ; (iii) C (A(t ), Z (t )) is convex in (A(t ), Z (t )); (iv) C (0, Z (t )) = 0. The abatement cost is increasing and convex in abatement and decreasing and convex in experience. Learning reduces the abatement cost at a decreasing rate and the gains from experience are higher when experience is low. Following Bramoullé and Olson (2005) and Goulder and Mathai (2000), the instantaneous pollution abatement cost C (A(t ), Z (t )) function is assumed to take the form: C (A(t ), Z (t )) = b1 A(t )(σ −γ ) − b2 (Z (t ) − Z0 )

(4)

where b1 and b2 are constants; σ = θ1 α1 + θ2 α2 , and we assume that γ ≤ σ . The dynamics of pollution stock X˙ (t ) is prescribed by the ordinary differential equation X˙ (t ) = E (t ) − A(t ) − ηX (t )

(5)

where η > 0 is a constant decay rate of pollution. Let ω and τ be the economy wide wage rate and the pollution permit price, Pi be the price of goods i, i = 1, 2, and we assume that both goods markets are competitive. Then, we can get the following expressions: Ei =

Pi αi Yi

,

τ Pi βi Yi Li = , ω

E1 + E2 = E

(8)

L1 + L2 = L

(9)

where L is the constant stock of labor, and E is the total amount of pollution permits to be determined by the government. Now, we consider the consumers and firms’ problems. The consumers face the following budget constraint P1 C1 + P2 C2 = ωL.

(10)

The consumers’ maximization problem leads to the demand functions Ci =

θi ωL Pi

,

i = 1, 2.

(11)

Similarly, the firms’ demand for labor and permits can be PαY PβY derived as Ei = i τi i , Li = i ωi i , i = 1, 2. Using market clearing conditions Ci = Yi , i = 1, 2, we obtain that E1 /E2 = θ1 α1 /θ2 α2 , and L1 /L2 = θ1 β1 /θ2 β2 . Using these relationships between equilibrium input usage as well as Eqs. (10) and (11), one obtains:

θi αi E θ1 α1 + θ2 α2 βi θi L. Li = β1 θ1 + β2 θ2

Ei =

t



We normalize the initial goods prices to unity throughout the paper. The labor and emission permits market clearing conditions are:

i = 1, 2

(6)

i = 1, 2.

(7)

(12) (13)

Substituting Eqs. (12) and (13) into (1), using the market clearing conditions once again, and substituting the resulting equilibrium consumption levels in Eq. (2), we obtain the following money-metric indirect utility function: V = Υ (L)E (t )σ X (t )−γ where Υ (L) = Λ

Λ=



ψ 1+ψ

(14)

β1 θ1 +β2 θ2

β θ +β θ

Lβ1 θ1 +β2 θ2 , ψ = θ 1α1 +θ 2α2 , 1 1 2 2

α1 θ1  β1 θ1 θ1 α1 β1 θ1 θ1 α1 + θ2 α2 β1 θ1 + β2 θ2  α2 θ2  β2 θ2 θ2 α2 β2 θ2 × . θ1 α1 + θ2 α2 β1 θ1 + β2 θ2



According to Jehle and Reny (2011), a consumer’s indirect utility function gives the consumer’s maximal utility when faced with a price level and an amount of income. It represents the consumer’s preferences over market conditions. The indirect utility function for the consumer is analogous to the profit function for the firm. So we can measure the indirect utility function by using moneymetric. Eq. (14) is a money-metric indirect utility function which gives the consumer’s maximal utility when faced with price level Pi of the consumption goods Ci and an amount of income ωL, i = 1, 2. Further, according to Rubio and Casino (2002), the problem for the government consists in maximizing with regard to controls E and A the expected value of the following functional: ∞

 max E ,A

e−rt [Υ (L)E (t )σ X (t )−γ − (b1 A(t )(σ −γ )

0

− b2 (Z (t ) − Z0 ))]dt .

(15)

In the next section, we apply optimal control theory to find the optimal levels of abatement and pollution permits such that the discounted stream of welfare is maximized.

S. Li, X. Pan / Economics Letters 122 (2014) 285–288

3. Optimal solution of the model

Substituting Eq. (23) into (19) gives

Let us assume that the government chooses levels of pollution abatement and pollution permits to maximize welfare. Then the problem for the government is to set E and A so as to maximize following optimization problem over an infinite time horizon. ∞

 max E ,A

0

(16)

H = Υ (L)E (t )σ X (t )−γ − b1 A(t )−γ +σ

+ b2 µ lim

t →∞

(30)

µb2 ((r +η)σ −γ η)(σ −1−γ ) , the initial condition A b1 (σ −γ )−Υ (L)σ σ γ −γ (r +η)(σ −1)

1

1

As (t ) = (1 − γ + σ ) γ −σ −1 (−kt ) γ −σ −1 .

(31)

Substituting the expression (31) into (27) and (28) gives 1

γ (1 − γ + σ ) γ −σ −1 (−kt ) γ −σ −1 r σ + ησ − γ η

(32)

(19) (20)

1

(r + η)σ (1 − γ + σ ) γ −σ −1 (−kt ) γ −σ −1 E (t ) = . r σ + ησ − γ η s

(33)

From expression (31), we have the following proposition. Proposition 1. If σ ≥

(18)

(0) =

0. Furthermore, because γ , σ ∈ (0, 1), so we have γ − σ ̸= −1, namely 2 + γ − σ ̸= 1. Differential equation (30) is a kinematic equation of pollution ˙ t ) = 0. abatement level under steady state conditions X˙ (t ) = λ( Solving the differential equation (30), and identifying the optimal level of pollution abatement under steady state by the superscript ‘‘s’’, then we have

1

(r +η)γ +r , 2r +η

then the steady state equilibrium

A (t ) is increasing in η; and, if σ ≤ equilibrium As (t ) is decreasing in η. s

(r +η)γ +r , 2r +η

then the steady state

Proof. From expression (31), we get 1 1 ∂ As ∂k −1 = [−k(1 − γ + σ )] γ −σ −1 t γ −σ −1 . ∂η ∂η

Because (21)

∂k ∂η

b2γ Υ (L)µ[(r +η)σ −γ η−r (1+γ −σ )]γ γ σ σ (r +η)σ [(r +η)σ −γ η](γ +σ ) , [b1(r +η)(γ −σ )γ γ ((r +η)σ −γ η)σ +Υ (L)((r +η)σ −γ η)γ +1 ((r +η)σ )σ ]2 (r +η)γ +r ∂k ∂k if σ ≥ 2r +η , we have ∂η ≥ 0, otherwise, we have ∂η

=

(22)

then ≤ 0, Furthermore, because (r + η)σ > γ η, k ≤ 0 and γ − σ − 1 < 0. s (r +η)γ +r As Then if σ ≥ , we get ∂∂η ≥ 0, if not, we have ∂∂ηA ≤ 0. 2r +η Finishing the proof.

(23)

Further, from expressions (31)–(33), we have the following Propositions 2 and 3.

From Eq. (18), we get

λ(t ) = −σ Υ (L)E (t )σ −1 X (t )−γ .

where k =

(17)

0

∂H X˙ (t ) = = E (t ) − ηX (t ) − A(t ). ∂λ(t )

(29)

1

where λ(t ) is the dynamic adjoint variable associated with its respective state equation X˙ (t ). The following are the necessary conditions, where the variable subscripts denote partial derivatives.

∂H = σ Υ (L)E (t )σ −1 X (t )−γ + λ(t ) = 0 ∂ E (t ) ∂H µb2 A(t ) = −b1 (−γ + σ )A(t )−1−γ +σ − λ(t ) + =0 ∂ A(t ) A˙ (t ) ∂H = −γ Υ (L)E (t )σ X (t )−γ −1 − ηλ(t ) ∂ X (t ) ∂H ˙ t ) = r λ(t ) − λ( ∂ X (t ) = (r + η)λ(t ) + γ Υ (L)E (t )σ X (t )−γ −1

= σ Υ (L)E (t )σ −1 X (t )−γ .

X s (t ) =

t

A(s)ds + λ(t )[E (t ) − ηX (t ) − A(t )]

µb2 A(t ) A˙ (t )

A˙ (t ) = kA(t )(2+γ −σ )

where r is the social discount rate. The decision problem of the government is to determine optimal levels of abatement and pollution permits over an infinite planning period, such that the objective functional in model (16) is maximal. In order to obtain the optimality conditions for the optimal control problem, we use Pontryagin’s maximum principle. The current value Hamiltonian for this optimal control problem is:



b1 (−γ + σ )A(t )−1−γ +σ −

Substituting the expressions (27) and (28) into (29), and rearranging to get:

e−rt [Υ (L)E (t )σ X (t )−γ − (b1 A(t )(σ −γ )

− b2 (Z (t ) − Z0 ))]dt  X˙ (t ) = E (t ) − ηX (t ) − A(t )    t  A(s)ds s.t. Z (t ) = Z0 + µ lim t →∞ 0    X (0) = X0 , A(0) = 0, Z (0) = Z0

287

Substituting expression (23) into (21), we get Proposition 2. The steady state equilibrium As (t ), X s (t ) and E s (t ) is increasing in labor endowment L.

˙ t ) = −(r + η)σ Υ (L)E (t )σ −1 X (t )−γ λ( + γ Υ (L)E (t )σ X (t )−γ −1 .

(24)

Now, we investigate a system of two variables described by the differential equations (22) and (24). By definition, the steady state ˙ t ) = 0. Then from Eqs. (22) and (24), we have conditions X˙ (t ) = λ( E (t ) − ηX (t ) − A(t ) = 0

(25)

−(r + η)σ Υ (L)E (t )σ −1 X (t )−γ + γ Υ (L)E (t )σ X (t )−γ −1 = 0. (26)

Proof. From expressions (31)–(33), we have 1 1 ∂ As ∂k −1 = [−k(1 − γ + σ )] γ −σ −1 t γ −σ −1 ∂L ∂L 1

γ [−k(1 − γ + σ )] γ −σ −1 ∂Xs = ∂L r σ + ησ − γ η

−1 γ −σ1 −1 t

1

Solving the system of equations (25) and (26), one obtains X (t ) =

γ A(t ) (r + η)σ − γ η

(27)

E (t ) =

(r + η)σ A(t ) . (r + η)σ − γ η

(28)

−1

∂k ∂L 1

∂ Es (r + η)σ [−k(1 − γ + σ )] γ −σ −1 t γ −σ −1 ∂ k = . ∂L r σ + ησ − γ η ∂L  β1 θ1 +β2 θ2 ψ Lβ1 θ1 +β2 θ2 is increasing in Because Υ (L) = Λ 1+ψ labor endowment L, and k =

µb2 ((r +η)σ −γ η)(σ −1−γ ) , b1 (σ −γ )−Υ (L)σ σ γ −γ (r +η)(σ −1)

we have

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S. Li, X. Pan / Economics Letters 122 (2014) 285–288

∂k ∂L

≥ 0. Furthermore, because (r + η)σ > γ η, k ≤ 0 and s s s γ − σ − 1 < 0, we conclude that ∂∂AL ≥ 0, ∂∂XL ≥ 0, ∂∂EL ≥ 0. Finishing the proof.

Proposition 3. The steady state equilibrium As (t ), X s (t ) and E s (t ) is decreasing in µ. Proof. From expressions (31)–(33), we have 1 1 ∂ As ∂k −1 = [−k(1 − γ + σ )] γ −σ −1 t γ −σ −1 ∂µ ∂µ 1

γ [−k(1 − γ + σ )] γ −σ −1 ∂Xs = ∂µ r σ + ησ − γ η

−1 γ −σ1 −1 t

1

∂ Es (r + η)σ [−k(1 − γ + σ )] γ −σ −1 = ∂µ r σ + ησ − γ η Because k = have

∂k ∂µ

∂k ∂µ

−1 γ −σ1 −1 t

µb2 ((r +η)σ −γ η)(σ −1−γ ) b1 (σ −γ )−Υ (L)σ σ γ −γ (r +η)(σ −1)

Acknowledgments

∂k . ∂µ

and k ≤ 0, then we

≤ 0. Furthermore, because (r +η)σ > γ η, γ −σ − 1 < 0,

As we conclude that ∂∂µ proof.

X E ≤ 0, ∂∂µ ≤ 0, ∂∂µ ≤ 0. Finishing the s

derive the optimal levels of emission permits and pollution treatment under steady-state. The present paper discusses the dynamic general equilibrium problems of pollution abatement under emission permits trading. However, emission permits banking and pollution damage are not taken into account. Emission permits banking and borrowing provide the firm with intertemporal flexibility in meeting their abatement responsibilities. A further research direction would be needed to examine the situations under learning by doing, when the firms introduce emission permits banking and pollution abatement investment.

s

4. Conclusions In this paper, following the analytical framework of Beladi et al. (2013), we present a dynamic general equilibrium model of pollution abatement under learning by doing. In our model, the instantaneous abatement costs depend on both the rate of abatement and the experience of using a technology. Applying optimal control theory, we investigate the steady-state equilibrium properties, and

The authors thank the anonymous referees and the editor for their careful reading and their comments on this paper. This research was supported by the National Natural Science Foundation of China (Project No. 71333010). References Argotte, L., Epple, D., 1990. Learning curves in manufacturing. Science 247, 920–924. Beladi, H., Liu, L., Reza Oladi, R., 2013. On pollution permits and abatement. Econom. Lett. 119, 302–305. Bramoullé, Y., Olson, L.J., 2005. Allocation of pollution abatement under learning by doing. J. Public Econ. 89, 1935–1960. Goulder, H., Mathai, K., 2000. Optimal CO2 abatement in the presence of induced technological change. J. Environ. Econ. Manag. 39, 1–38. Greaker, M., Rosendahl, K.E., 2008. Environmental policy with upstream pollution abatement technology firms. J. Environ. Econ. Manag. 56, 246–259. Jehle, G.A., Reny, P.J., 2011. Advanced Microeconomic Theory, third ed. Prentice Hall, pp. 28–33. Rubio, S.J., Casino, B., 2002. A note on cooperative versus non-cooperative strategies in international pollution control. Resour. Energy Econ. 24, 251–261.