Anticipated environmental policy and transitional dynamics in an endogenous growth model

Environmental and Resource Economics 25: 233–254, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Anticipated Environmental Policy and Transitional Dynamics in an Endogenous Growth Model JHY-HWA CHEN1, CHING-CHONG LAI2,∗ and JHY-YUAN SHIEH3

1 Department of Economics, Tamkang University, Taiwan; 2 Sun Yat-Sen Institute for Social Sciences and Philosophy, Academia Sinica, Taiwan; Institute of Economics, Academia Sinica, Taiwan; Department of Economics, National Taiwan University, Taiwan; 3 Department of Economics, Chinese Culture University, Taiwan (∗ Author for correspondence: E-mail: [email protected])

Accepted 4 November 2002 Abstract. This paper makes a new attempt to investigate how an anticipated environmental policy governs the transitional dynamics of an economy when pollution externality is taken into account. The modeling strategy we use is an AK technology endogenous growth framework with an endogenous leisure-labor choice. It is found that, unlike inelastic labor supply framework, a rise in public abatement expenditure will stimulate the balanced economic growth rate. It is also found that public abatement technology plays an important role in determining the transitional adjustment of the economic growth rate in response to a pre-announced environmental policy. Key words: endogenous growth, elastic labor supply, pollution, public abatement JEL classifications: O40, Q20

1. Introduction Pollution is an important source for generating negative externalities on the private sector. Government interventions, such as Pigouvian taxes, subsidies, quantity regulations on the environment, and public abatement policies, are proposed to remedy these negative externalities. However, conventional wisdom argues that environmental abatement expenditure has an adverse effect on economic growth since it crowds out private expenditure. Many studies embodying endogenous growth theories with a pollution externality have recently consented to this assertion. For example, given that the government collects income tax revenue to finance public abatement expenditure, Huang and Cai (1994) and Ligthart and van der Ploeg (1994) develop an AK type endogenous growth model with pollution externality. They show that a rise in the income tax rate will deter the steady-state growth rate.

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Running in contrast to the aforesaid viewpoints, some studies propose that an environmental policy may stimulate economic growth via production and health channels. For example, using the Lucas (1988) type growth model, Gradus and Smulders (1993) show that the endogenous growth rate is stimulated by increased environmental care if the pollution damage affects the marginal return to education. Bovenberg and Smulders (1995), Smulders and Gradus (1996), and Bovenberg and de Mooij (1997) argue that environmental quality not only governs consumer’s preference, but also has a productive role. They show that an ambitious environmental policy may stimulate economic growth by means of an effective improvement on the quality of the environment. This paper uses an endogenous growth model with an endogenous labor-leisure choice and stresses that public abatement expenditures will boost the balanced growth rate even without a direct positive environmental externality in production. Most, if not all, existing literature on environmental endogenous growth theories are concerned with the effects of permanent changes in environmental policy on the steady-state growth rate, but ignore the transitional dynamics.1 Given the fact that environmental authorities usually undertake policies with a pre-announcement, it seems that the analysis dealing with anticipated policies may be more realistic in the real world. Based on such an observation, this paper develops an endogenous growth model embodying the endogenous labor-leisure choice and pollution externality, and uses it to examine the announcement effect of the environmental policy on the balanced growth rate and the transitional responses of economic growth. Within this framework, we show that an ambitious environmental policy will stimulate the steady-state growth rate if labor-leisure is chosen endogenously. Furthermore, we also find that the important factor determining the transitional effect of an anticipated public abatement policy on the consumption-capital ratio, the pollution-capital ratio, and the economic growth rate is the public abatement technology. The organization of the paper is as follows. The analytical framework is outlined in section 2. Section 3 discusses the transitional property of the dynamic system and examines the steady-state effect of an expansionary abatement policy. Section 4 discusses the transitional effect of an anticipated public abatement policy. Finally, some concluding remarks are presented in section 5.

2. The Model Consider an economy consisting of a representative household and a government.2 The household produces a single composite commodity, which can be consumed, accumulated as capital, and paid for as a lump-sum tax. The government collects its tax revenue and provides public abatement to lessen pollution damage. The representative household derives positive utility from both consumption C and leisure l, and derives negative utility from the stock of pollution, P . The

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objective of the representative household is to maximize the discounted sum of future instantaneous utilities:
∞ U (C, P , l)e−ρt dt, (1) 0

where U is the utility function and ρ is the subjective time preference rate. In line with Gradus and Smulders (1993), Huang and Cai (1994), and Smulders and Gradus (1996), the instantaneous utility function is specified as follows: U (C, P , l) =

(CP −θ l η )1−σ − 1 , 1−σ

(2)

where the parameters η and θ measure the impact of leisure and pollution on the welfare of the household, respectively. Following Keeler et al. (1972) and Tahvonen and Kuuluvainen (1991), to satisfy the requirement that private consumption and leisure yield a positive but diminishing marginal utility and that the pollution flow yields a negative and diminishing marginal utility, we impose the restrictions η, θ > 0 and σ > (1 + θ)/θ > 1. Moreover, σ + (θ − η)(1 − σ ) < 0 is imposed to ensure that the utility function is concave in the quantities, C, P , and l.3 In common with existing literature, pollution flow is specified to be positively related to private capital K and negatively related to public abatement M. For ease of presentation, we assume the stock pollution is not affected by the environment’s self-cleaning capacity.4 Hence, the accumulation of pollution stock is given by:5 P˙ = M −α K β ,

(3)

where an overdot denotes the rate of change with respect to time, and α and β are the pollution production elasticity with respect to public abatement and private capital, respectively. At each instant of time, the representative household is bound by a flow constraint linking capital accumulation to any difference between its disposable income and expenditure. We assume that the household is endowed with a unit of time, such that 1 − l is the working time (hence, 0 < l < 1). As a consequence, the household budget constraint can be described as: K˙ = AK(1 − l)ε − C − T ,

(4)

where T is a lump-sum tax, Q = AK(1 − l)ε is the production function, and A is the technology parameter. To ensure positive but diminishing marginal productivity of labor, the restriction 0 < ε < 1 is imposed. Following Ligthart and van der Ploeg (1994), Michel and Rotillon (1995), Elbasha and Roe (1996), and Bovenberg and de Mooij (1997), we assume that the household treats the stock of environmental pollution as given since the household

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feels that its activities are too insignificant to affect the pollution stock. With this understanding, the representative household chooses consumption and leisure so as to maximize the discounted sum of utility defined in equation (1), subject to equations (2) and (4), and given the initial capital. The optimal conditions necessary for this optimization problem are as follows: C −σ P −θ(1−σ ) l η(1−σ ) = λ,

(5a)

ηC 1−σ P −θ(1−σ ) l η(1−σ )−1 = λεAK(1 − l)ε−1 ,

(5b)

λ˙ /λ = ρ − A(1 − l)ε ,

(5c)

together with equations (3) and (4) and the transversality condition lim λKe−ρt = t →∞ 0. Term λ is the co-state variable, which can be interpreted as the shadow value of private capital stock, measured in utility terms. Equation (5a) defines that the co-state variable λ is equal to the marginal utility of consumption. Equation (5b) states that the marginal utility of leisure is equal to the marginal cost of leisure. The differential equation (5c) shows that the change in the shadow value of capital depends upon the difference between the rate of time preference and the marginal product of capital. The government is assumed to collect the lump-sum tax revenue to finance its public abatement expenditure. In order to sustain an equilibrium with balanced growth, following Devereux and Love (1995), Turnovsky (1995) and Bruce and Turnovsky (1999), we assume that the government sets its public abatement expenditure as a fixed fraction of output, that is: M = φAK(1 − l)ε ,

(6)

where 0 < φ < 1. The parameter φ is the public abatement expenditure share. Assuming that the government balances its budget at any instant time, the government’s budget constraint thus can be expressed as: T = M = φAK(1 − l)ε .

(7)

Plugging equation (7) into (4), the resource constraint for the whole economy is given by: K˙ = (1 − φ)AK(1 − l)ε − C.

(8)

3. Transitional Dynamics The evolution of the system can proceed as follows. First of all, following Futagami et al. (1993), Barro and Sala-i-Martin (1995), and Faig (1995), we

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define the following two transformed variables: x = P 1/(β−α)/K and y = C/K. From equations (5a) and (5b) we can then derive the following instantaneous relationship: l = l(y, A, η),

(9)

where ly = η/ > 0, lη = y/ > 0, lA = −[ε(1 − l)ε−1 l]/ < 0, and

= εA(1 − l)ε−2 (1 − εl). Using equations (3), (5a)–(5c), (6), and (8), the optimal change in consumption is given by:6 1 C˙ = {(1 − εl)[A(1 − l)ε − ρ − θ(1 − σ )(φA)−α (1 − l)−αε x α−β ] C  (10) −η(1 − σ )(1 − l)[A(1 − φ)(1 − l)ε − y]}, where  = σ (1 − εl) − η(1 − σ )(1 − l) > 0. We then can use equations (3), (6), (8), and (10) to derive a dynamic system in terms of the transformed variables, x and y, as follows: 1 x˙ = x α−β (φA)−α (1 − l)−αε − A(1 − φ)(1 − l)ε + y, x (β − α) (1 − εl) y˙ = {A(1 − l)ε [1 + σ (1 − φ)] − y  ρ − θ(1 − σ )(φA)−α (1 − l)−αε x α−β + σy}.

(11)

(12)

At steady-growth equilibrium, the economy is characterized by x˙ = y˙ = 0, and x and y are at their stationary levels, namely x ∗ and y ∗ . Moreover, this paper only deals with the shock of public abatement expenditures (i.e., a change in φ). Hence, substituting equation (9) into (11) and (12), and then linearizing the resulting equations around the steady-state equilibrium, we have:7        x˙ a11 a12 x − x∗ a13 = + dφ, (13) a21 a22 y − y∗ a23 y˙ where a11 = −(φA)−α (1 − l)−αε (x ∗ )α−β , a12 =

a13 =

x∗ {β − α + εA(1 − l)ε−1 ly [(β − α)(1 − φ) + β −α αφ −α A−(1+α) (1 − l)−ε(1+α)(x ∗ )α−β ]}, A(1 − l)ε x ∗ [(β − α) − α(x ∗ )α−β (φA)−(1+α) (1 − l)−ε(1+α)], β−α

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a21 = a22 =

a23 =

θy ∗ (x ∗ )α−β−1 (1 − σ )(β − α)(φA)−α (1 − εl)(1 − l)−αε ,  (1 − εl)y ∗ {σ − εA(1 − l)ε−1 ly [1 − σ (1 − φ) +  αθφ −α A−(1+α) (1 − σ )(1 − l)−ε(1+α)(x ∗ )α−β ]}, A(1 − l)ε (1 − εl)y ∗ [σ + αθ(1 − σ )(φA)−(1+α)(1 − l)−ε(1+α)(x ∗ )α−β ]. 

Let s1 and s2 be the two characteristic roots of the dynamic system. From equation (13), we then have: ≥ s1 + s2 = a11 + a22 < 0, s1 s2 = a11 a22 − a12 a21 =

(14a) −(x ∗ )α−β y ∗ (φA)−α (1 − εl)(1 − l)−αε , 

(14b)

where = + εA(1 − l)ε−1ly [(1 − φ) − 1] and = σ + θ(β − α)(1 − σ ) > 0.8 As addressed in the literature of dynamic rational expectation models, including Burmeister (1980), Buiter (1984), and Turnovsky (1995), the dynamic system has a unique perfect-foresight equilibrium if the number of unstable roots equals the number of jump variables.9 Since the dynamic system reported in equation (13) has one jump variable, y, the restriction > 0 should be imposed to ensure s1 s2 < 0, and hence the dynamic system is assured to display a unique perfect-foresight equilibrium. In addition, we assume φ < ( − 1)/ < 1 to assure the condition > 0.10 We now consider the steady-state effect of a rise in public abatement expenditure on the long-run growth rate. First, it follows from equation (13) with x˙ = y˙ = 0 that the following steady-state relationship is derived: ∂x ∗ −φ α A1+α x ∗1−α+β (1 − l)ε(1+α) = {α(φA)−(1+α)(1 − l)−ε(1+α)(x ∗ )α−β ∂φ (β − α) +εA(1 − l)ε−1 ly [β − α + αφ −α A−(1+α) (1 − l)−ε(1+α)(x ∗ )α−β ]} < 0, (15a) ∂y ∗ −A(1 − l)ε = < 0. ∂φ

(15b)

Along a balanced growth path, the fraction of time devoted to leisure is constant, i.e., l˙ = 0. Let the superscript “*” denote the steady-state value. Moreover, at steady∗ ∗ ˙ ˙ = (K/K) and (P˙ /P )∗ = (β − growth equilibrium, x˙ = y˙ = 0 implies (C/C) ∗ ˙ . Given the production technology Q = AK(1 − l)ε and equation (6), α)(K/K) ∗ ∗ ∗ ∗ ˙ ˙ ˙ ˙ = (K/K) and (M/M) = (K/K) should hold in the it is clear that (Q/Q) ∗ steady-state equilibrium. Accordingly, letting γ denote the steady-state growth

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∗ ∗ ∗ ∗ ˙ ˙ ˙ ˙ = (C/C) = (M/M) = (K/K) = [1/(β − α)](P˙ /P )∗ rate, γ ∗ = (Q/Q) holds in the steady-growth equilibrium.11,12 With the feature of stationary growth equilibrium, from equations (8), (9), and (15b) we can easily obtain:

∂y ∗ ∂l ∗ = ly < 0, ∂φ ∂φ

(16a)

∂γ ∗ εA2 (1 − l)2ε−1 ly = > 0. ∂φ

(16b)

It is clear from equations (16a) and (16b) that a permanent rise in the share of public abatement expenditures will increase both labor supply and the steady-state growth rate. Intuitively speaking, public abatement expenditure’s impact on the growth rate reflects the joint consequence operating through both the resources withdrawal effect and the intertemporal substitution effect. The resources withdrawal effect indicates that a rise in public abatement expenditure will reduce the amount of resources available to the private sector through taxes. This will induce the household to lower private consumption and raise labor supply, and in turn encourages capital accumulation since an increase in the marginal product of private capital is associated with a rise in labor supply. The intertemporal substitution effect expresses that a higher value of public abatement expenditure will reduce pollution damage, leading to a decrease in the marginal utility of future consumption C −σ P −θ(1−σ ) l η(1−σ ) and future leisure ηC 1−σ P −θ(1−σ ) l η(1−σ )−1,13 given that the substitution between intertemporal consumption is inelastic (σ > 1). With this adjustment, the representative household tends to raise current consumption and current leisure, thereby discouraging current capital accumulation since the marginal product of private capital decreases. In steady-state equilibrium, as reported in equation (16b), the resources withdrawal effect outweighs the intertemporal substitution effect, and thus a rise in the share of public abatement expenditure will stimulate the steady-state growth rate. It should be noted that the efficacy of the public abatement expenditure on the balanced growth rate is crucially related to whether the decision of labor supply is endogenous or not. If the labor supply is exogenous (i.e., ε = η = 0), as specified in Lighart and van der Ploeg (1994), then equation (16b) indicates that the public abatement expenditure does not affect the sustained growth rate. Equation (14b) indicates that s1 s2 < 0. For expository convenience, in what follows let s1 be the negative root and s2 be the positive root (i.e., s1 < 0 < s2 ). From equation (13), the general solution for x and y can thus be expressed as: x = x ∗ + A1 es1 t + A2 es2 t , y = y∗ +

s1 − a11 s2 − a11 A1 es1 t + A2 es2 t , a12 a12

(17a) (17b)

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Figure 1. Phase diagram.

where A1 and A2 are as yet undetermined coefficients. A graphical solution of the system is provided in Figure 1. From equations (11) and (12), both the x˙ = 0 locus and y˙ = 0 locus are upward sloping,14 and the x˙ = 0 locus is steeper than the y˙ = 0 locus due to s1 s2 < 0. Furthermore, the SS curve and U U curve represent the stable and unstable branches, respectively. As indicated by the direction of arrows, both the SS curve and the U U curve are upward sloping, and the U U locus is steeper than the x˙ = 0 locus while the SS locus is flatter than the y˙ = 0 locus.15

4. Dynamics of a Shock in Public Abatement Expenditure By using a graphical apparatus like Figure 1, this section proceeds to trace the possible adjustment patterns of the consumption-capital ratio and pollution-capital ratio in response to an anticipated shock in public abatement expenditure. The experiment we conduct is that, at time t = 0, the authority announces that the public abatement expenditure share will permanently rise from φ0 to φ1 at t = T in the future. From equation (13) we have:  a13 ≥ ∂x  ≥ =− < 0; if a13 < 0,  ∂φ x=0 a 11 ˙

(18a)

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Figure 2. Transitional dynamics: α is relatively small.

 a23 ≥ ∂x  ≥ =− 0; if a23 < 0. <  ∂φ y=0 a21 ˙

(18b)

In response to a rise in φ, both x˙ = 0(φ0 ) and y˙ = 0(φ0 ) may shift either rightward or leftward depending upon the sign of a13 and 23 . To trace the component of a13 and a23 , we find that both a13 and 23 are positive (negative) if the elasticity of pollution production with respect to public abatement, α, is sufficiently small (large).16 Thus, in what follows two cases will be considered: (1) α is relatively small (a13 > 0 and a23 > 0); (2) α is relatively large (a13 < 0 and a23 < 0).17 (1) α is relatively small (a13 > 0 and a23 > 0) In Figure 2 the initial equilibrium where x˙ = 0(φ0 ) intersects y˙ = 0(φ0 ) is established at E0 ; the initial pollution-capital ratio and consumption-capital ratio are x0 and y0 , respectively. In response to an anticipated permanent rise in φ, both x˙ = 0(φ0 ) and y˙ = 0(φ0 ) shift rightward to x˙ = 0(φ1 ) and y˙ = 0(φ1 ), respectively. The new steady-state equilibrium is at point E∗ , with x and y being x ∗ and y ∗ , respectively. Before proceeding to study the economy’s dynamic adjustment, three points should be addressed. First, for expository convenience, in what follows 0− and 0+ denote the instant before and after the policy announcement, respectively, while T − and T + denote the instant before and after the policy implementation, respectively.

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Second, during the dates between 0+ and T − , the public abatement expenditure remains at its initial level φ0 , and point E0 should be treated as the reference point that governs the dynamic adjustment of x and y. Third, since the public knows that the public abatement share will increase from φ0 to φ1 at the moment of T + , the transversality condition requires that the economy moves to a point on the convergent stable branch associated with φ1 , SS(φ1 ), at that instant of time.18 Based on these considerations, as depicted in Figure 2, two adjustment patterns are possibly present depending upon the length of lead-time between policy announcement and implementation T . If T is relatively short, then at the instant 0+ , y will immediately fall from y0 to y0+ , while x is fixed at x0 since it is predetermined. As a consequence, the economy will instantaneously jump from point E0 to a point like E1 on impact. From 0+ to T − , as the arrows indicate, both x and y continue to decrease, and the economy moves from E1 to ET . At time T + , when the public abatement expenditure is enacted, the economy exactly reaches point ET on the convergent stable path SS(φ1 ). Thereafter, from T + onwards, both x and y continue to decrease as the economy moves along the SS(φ1 ) curve towards its stationary equilibrium E∗ . On the other hand, if T is relatively large, then at the instant 0+ , y will 1 , while x is fixed at x0 since it is predetermined. immediately fall from y0 to y0+ Consequently, the economy will jump discretely from point E0 to a point like E11 on impact. From time 0+ to T − , as the arrows indicate, the system will move from E11 to ET1 in which both x and y keep falling. At the moment of policy implementation T + , the economy exactly reaches point ET1 on the convergent stable path SS(φ1 ). Thereafter, from T + onwards, both x and y keep rising as the economy moves along the SS(φ1 ) curve towards its stationary equilibrium E∗ . (2) α is relatively large (a13 < 0 and a23 < 0) Figure 3 depicts the situation where α is relatively large. In response to an increase in φ, both x˙ = 0(φ0 ) and y˙ = 0(φ0 ) shift leftward to x˙ = 0(φ1 ) and y˙ = 0(φ0 ), respectively. The new steady-state equilibrium is at point E∗ , with x and y being x ∗ and y ∗ , respectively. We can draw a line connecting the initial steady state E0 and new steady state E∗ . This line is named the LL locus. As is evident in Figure 3, the relative steepness between the LL schedule and the convergent branch SS(φ1 ) is ambiguous. If the SS(φ1 ) locus is flatter than the LL line, namely SS1 (φ1 ), then at the instant of policy announcement, y will immediately fall from y0 to y0+ , while x is fixed at x0 since it is predetermined. Accordingly, the economy will instantaneously jump from point E0 to a point like E1 on impact. From time 0+ to T − , as the arrows indicate, both x and y continue to decrease. At time T + , as the public abatement expenditure increases, the economy exactly reaches point ET on the convergent stable path SS1 (φ1 ). Subsequently, from T + onwards, both x and y continue to decline as the economy moves along the SS1 (φ1 ) curve towards its stationary equilibrium E∗ .

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Figure 3. Transitional dynamics: α is relatively large.

On the other hand, if the SS(φ1 ) locus is steeper than the LL line, namely 1 and the SS2 (φ1 ), then at the instant 0+ , y will discontinuously rise from y0 to y0+ 1 economy will immediately jump from point E0 to a point like E1 on impact. From 0+ to T − , as the arrows indicate, both x and y continue to increase. When the public abatement expenditure actually increases at time T + , the economy exactly reaches the point ET1 on the convergent stable path SS2 (φ1 ). Thereafter, from T + onwards, both x and y will continue to fall as the economy moves along the SS2 (φ1 ) curve towards its stationary equilibrium E∗ . It is now important for us to understand what the transitional behavior of the capital growth rate is in response to an anticipated expansion in public abatement expenditure. Let γK be the growth rate of capital stock. From equations (8) and (9) we have: ˙ l˙ = ly y,

(19)

˙ γ˙K = −[1 + ε(1 − φ)A(1 − l)ε−1 ly ]y.

(20)

It is clear to infer from equations (19) and (20) that the changes in the capital growth rate are negatively related to the changes in the fraction of time devoted to leisure. This result is quite intuitive. Given the production function AK(1 − l)ε , a rise in the fraction of time devoted to leisure implies a fall in the marginal product of private capital. The representative agent thus tends to decrease investment in response.

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Figure 4. Time paths of γK : α is relatively small.

Based on the dynamic paths of (x, y) exhibited in Figure 2, Figure 4 depicts how the growth rate of capital described in equation (20) will adjust over time following an anticipated expansion in public abatement expenditure. Intuitively, transitional dynamic adjustment paths of the growth rate in response to a pre-announced environmental policy also clearly reflect the joint outcome operating through the resources withdrawal effect and the intertemporal substitution effect. The degree of public abatement technology plays no role in the resources withdrawal effect, but has a crucial role in the intertemporal substitution effect. Specifically, the lower (higher) the public abatement technology is, the less (more) the intertemporal substitution effect will be. Based on this information, if the public abatement technology is relatively low, then the resources withdrawal effect will outweigh the intertemporal substitution effect at the instant of policy announcement and during the period following the anticipation, but prior to the implementation of the public abatement expenditure. Under such a situation, the representative household tends to boost investment instantly; the growth rate hence discretely rises at the instant of policy announcement. In Figure 2 it is found that y immediately falls and undershoots its long-run steady-state value y ∗ at time 0+ , implying that γK discretely rises and undershoots its steady-state value, γK∗ , on impact. Subsequently, the resources withdrawal effect continuously outweighs the intertemporal substitution effect, and the representative

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household continues to increase investment, such that a continuous rise in the economic growth rate will prevail during the period following the anticipation, but prior to policy implementation. The dynamic adjustment pattern of y and γK reveals that in this period y continues to decrease and γK continues to increase. At the instant of policy implementation, the household’s investment decreases since the amount of resources available to the public actually turns down. The growth rate hence immediately falls.19 After policy realization, the economy moves along the convergence curve towards its stationary equilibrium. It should be stressed that, as exhibited in Figure 2, if the time frame is long enough (T = T2 ), then y overshoots its steady-state value and hence the growth rate overshoots its steady-state value. Under such a situation, the intertemporal substitution effect surmounts the resource withdrawal effect after policy implementation. Hence, as exhibited in Figure 2, y continues to increase as the economy moves along the convergence curve towards its stationary equilibrium. Corresponding to the evolutional adjustment of y, in Figure 4 the growth rate continues to decrease toward its new stationary level (γK∗ ) after public abatement is implemented. On the other hand, if the time frame is relatively short (T = T1 ), then the resources withdrawal effect outweighs the intertemporal substitution effect after policy implementation. Hence, y continues to decrease as the economy moves along the convergence curve towards its stationary equilibrium, and thereby the growth rate continues to increase toward its new stationary level. We can utilize a similar procedure to infer the transitional behavior of γK associated with the situation where α is relatively large. To save space, we do not repeat the inference here and only present the time path of γK in Figure 5a and Figure 5b. An interesting result is worth mentioning from the graphical illustration. If the public abatement technology is relatively high, then the intertemporal substitution effect may exceed the resources withdrawal effect at the instant of policy announcement and during the period following the anticipation, but prior to policy implementation. As a consequence, the representative household tends to lessen its investment and hence a fall in the economic growth rate will prevail before the policy implementation. This result confirms Bovenberg and Smulders’ (1996, p. 863) finding: “the transition reveals that short-run impacts may contrast quite sharply with long-run effects. For example, environmental policy may hurt short-run growth, but enhance growth in the long run.” Before ending our discussion, it should be noted that the framework we have adopted in this paper is well suited to discussing the case of an inelastic labor supply. From equation (16b) with ε = η = 0, it is clear that an ambitious environmental policy has no effect on the steady-state growth rate. However, a pre-announced environmental policy also has significant effects on the adjustment patterns of economic growth.20

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Figure 5a. Time paths of γK : α is relatively large and the SS locus is steeper than the LL locus.

5. Concluding Remarks It goes without saying that labor is also an important input in the modern capitalistic economy. However, as claimed in Eriksson (1996, p. 533), “the choice between work and leisure has been remarkably neglected in the theory of economic growth.” The recent development of real business cycle theories emphasizes not only the choice between leisure and work, but also the intertemporal substitution of work. Turnovsky (2000, p. 186) recognizes that “recent endogenous growth models have stressed the role of fiscal policy as a key determinant of long-run growth. One limitation of these new models is that with few exceptions, they treat labor supply as inelastic, thereby abstracting from the decision to allocate time between work and leisure. This treatment severely limits certain aspects of fiscal policy, implying for example, that both a consumption tax and a tax on labor income operate as nondistortionary lump sum taxes.” Based on the fact that the existing contribution on endogenous growth with an environment externality treats labor supply to be fixed, this paper considers a more general model to include the endogenous labor-leisure choice behavior. We hence set up an endogenous growth model with an endogenous laborleisure choice to examine how a pre-announced environmental policy governs the transitional dynamics of the economy when pollution externalities are taken into

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Figure 5b. Time paths of γK : α is relatively large and the SS locus is flatter than the LL locus.

account. Within such a framework, it is found that a government environmental policy favors the economic growth rate when the endogenous labor-leisure choice is allowed. It is also found that the important factor determining the transitional effect of an anticipated public abatement policy on the consumption-capital ratio, the pollution-capital ratio, and the economic growth rate is the public abatement technology. Acknowledgements The authors are deeply indebted to two anonymous referees for their helpful suggestions and insightful comments on an earlier version of this paper. We alone are responsible for all the views and any remaining shortcomings herein. Appendix A The purpose of this Appendix is to show the optimal change in consumption described in equation (10). Given the fact that the representative household is endowed with a unit of time to allocate to either leisure or work, we have the following relationship: l l˙ 1 − l˙ =− . 1−l 1−l l

(A1)

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Substituting equation (5a) into (5b) and then taking logarithms and differentiating the resulting equation with respect to time, we have: K˙ l˙ 1 − l˙ C˙ − = − (1 − ε) . C K l 1−l

(A2)

Plugging equation (6) into (3) yields: P˙ = (φA)−α (1 − l)−αε x α−β . P

(A3)

Taking logarithms of equation (5a) and differentiating the resulting equation with respect to time, we have:   −1 λ˙ P˙ l˙ C˙ = + θ (1 − σ ) − η(1 − σ ) . C σ λ P l

(A4)

Plugging equations (A1)–(A3) and (5c) into (A4), it is easy to obtain the optimal change in consumption described by equation (10).

Appendix B In this Appendix we intend to examine whether there exists a unique steady state for positive values of x and y. At the steady-growth equilibrium, the economy is characterized by x˙ = y˙ = 0, and x and y are at their stationary levels, namely x ∗ and y ∗ , respectively. From equations (11) and (12) we then have: 1 (x ∗ )α−β (φA)−α (1 − l)−αε = A(1 − φ)(1 − l)ε − y ∗ , (β − α) A(1 − l)ε − θ (1 − σ )(φA)−α (1 − l)−αε (x ∗ )α−β − σ A(1 − φ)(1 − l)ε + σy ∗ = ρ.

(B1)

(B2)

Equation (B1) can be alternatively written as: x ∗ = {(β − α)[A(1 − φ)(1 − l)ε − y ∗ ](φA)α (1 − l)αε }1/(α−β).

(B3)

Substituting equation (B3) into (B2), the resulting equation (B2) can be alternatively expressed as: (y ∗ ) = ρ,

(B4)

where (y ∗ ) = [1 − (1 − φ) ]A(1 − l)ε + y ∗ . We can first solve y ∗ from equation (B4), and then substitute the solution of y ∗ into equation (B3) to obtain x ∗ . A graphical illustration is helpful for us to understand how x ∗ and y ∗ can be uniquely determined. In Figure B1 a horizontal line which has a positive

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Figure B1. Determination of steady-state values of x ∗ and y ∗ .

value ρ is depicted. Furthermore, the function (y ∗ ) has the following properties: (i) (y ∗ ) is zero when y ∗ is equal to −[1 − (1 − φ) ]A(1 − l)ε / (> 0); (ii) (y ∗ ) is equal to [1 − (1 − φ) ]A (< 0) as y ∗ → 0; (iii) (y ∗ ) → ∞ as y ∗ → ∞; and (iv) (y ∗ ) is positively related to y ∗ with an increasing rate.21,22 Therefore, we can draw the locus (y ∗ ). It is quite clear from equation (B4) that the intersection of both the (y ∗ ) locus and the horizontal line determines y ∗ . Next, let the XX locus depict the loci of x ∗ and y ∗ which satisfy equation (B3). It clear from equation (B3) we can infer that XX locus has the following properties: (i) x ∗ → ∞ as y ∗ → A(1 − φ)(1 − l)ε ; (ii) x ∗ = {(β − α)A(1 − φ)(φA)α }1/(α−β) > 0 as y ∗ → 0; and (iii) x ∗ is positively related to y ∗ .23 From the XX locus we can then find a positive value x ∗ associated with y ∗ , and the combination of x ∗ and y ∗ is the solution that satisfies equations (B3) and (B4).

Appendix C We intend to examine the growth effect of environmental policies under the situation β = α in this Appendix. Substituting equation (6) into (3) in the text, the accumulation of pollution stock is given by: P˙ = [φA(1 − l)ε ]−α K β−α .

(C1)

On the other hand, using equations (8) and (10) in the text yields: K˙ C = (1 − φ)A(1 − l)ε − , K K   1 P˙ λ˙ C˙ l˙ = η(1 − σ ) − θ (1 − σ ) − . C σ l P λ

(C2)

(C3)

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Along the asymptotical growth path, the fraction of time devoted to leisure is constant, i.e., l˙ = 0. Let l¯ denote the steady-state value of l. Given β = α, it is quite easy to infer that the stock of pollution at time t is equal to P0 + [φA(1 − l)ε ]−α t. Accordingly, the asymptotically-balanced growth rate of P (i.e., lim (P˙ /P )∗ is zero. Moreover, the t →∞

asymptotical growth rates of output, consumption, public abatement expenditures, and the capital stock are constant along the asymptotical growth path. Let γ¯ be the asymptotically∗ ˙ ˙ ˙ balanced growth rate, and hence γ¯ = lim (Q/Q) = lim (C/C) = lim (M/M) = t →∞ t →∞ t →∞ ˙ lim (K/K). Substituting equation (5a) and (5b) in the text into (C2) and equation (5c)

t →∞

in the text into (C3), we obtain: ¯ε− γ¯ = (1 − φ)A(1 − l) γ¯ =

¯ − l) ¯ ε−1 εAl(1 , η

¯ ε −ρ A(1 − l) . σ

(C4)

(C5)

¯ The effect of changes in the public Equations (C4) and (C5) jointly determine γ¯ and l. abatement expenditure share on the economic growth rate is: ∂ γ¯ ηA(1 − εl)(1 − l)1+ε = , ¯ ∂φ

(C6)

¯ = σ + η(1 − l)[σ (1 − φ) − 1]/(1 − εl). Given the situation β = α, the stability where ¯ > 0. Consequently, we can infer that a condition in the main text > 0 degenerates to permanent increase in the public abatement expenditure share will stimulate the economic growth rate. It is quite easy to learn that the economy also exhibits an asymptotical growth path when β < α. Under such a situation, the asymptotically-balanced growth rate of P is zero. As a consequence, similar to the situation β = α, equations (C4) and (C5) jointly determine the asymptotically-balanced growth rate and the steady-state leisure. Therefore, a higher public abatement expenditure share is associated with a higher economic growth rate.

Notes 1. To our knowledge, Bovenberg and Smulders (1996) fully work out the dynamic effects of environmental policies, but only deal with an unanticipated action. 2. We assume that the economy consists of constant identical individuals and the population remains fixed over time; then all the aggregate quantities are equal to the individual quantities multiplied by the population size. Given that the purpose of this paper is to highlight the role of endogenous leisure-labor choice, we hence normalize the population size to one for ease of presentation. In fact, the assumption of exogenous population change as made in most growth studies does not change our qualitative results. 3. It is noted that the value of utility is negative if (CP −θ l η )1−σ > 1. However, this CRRA utility function is an ordinal utility function in nature. As is evident, we can monotonically transform equation (2) to a utility function to assure that positive utility represents the same preferences as the original utility function.

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4. It can be easily shown that our results are valid even when the environment performs a selfcleaning capacity. 5. Some studies make an alternative assumption. They argue that many kinds of pollution such as noise, smoke, and air pollution are flow variables rather than stock variables in nature, e.g., Ligthart and van der Ploeg (1994), Gradus and Smulders (1993), and Smulders and Gradus (1996). However, even with such a specification our conclusions concerning the steady-state equilibrium do not change. The only difference between the two specifications is the transitional behavior. 6. Appendix A provides a detailed derivation. 7. It should be noted that our analysis is appropriate for a shock of sufficiently small magnitude around the equilibrium. We are grateful to an anonymous referee for bringing this point to our attention. 8. Supposing ≤ 0 is true, we then have [1 + (1 − φ)εA(1 − l)ε−1 ly ] ≤ εA(1 − l)ε−1 ly since εA(1 − l)ε−1 ly > 0 and 1 + (1 − φ)εA(1 − l)ε−1 ly > 0. Using such a relationship, we know ≤ 0, but it contradicts the restriction > 0. Thus, > 0 must hold to guarantee > 0. 9. Burmeister (1980, pp. 802–803) provides three rationalizations for examining the regular saddlepoint equilibrium. “First, the problem may be viewed as one of intertemporal planning in which the maximization of some social welfare function necessitates convergence via the transversality condition. Second, under some conditions all nonconvergent paths eventually may be inconsistent with competition (and hence are dynamic inefficient in the Samuelson sense . . .). Third, we may interpret the model as one in which agents maximize their intertemporal utility functions, and under some conditions, often restrictive, this maximization behavior may exclude nonconvergent paths.” Based on Burmeister’s (1980) argument, we only discuss the unique stable equilibrium. Given that K and P are state variables and that C is a control variable, the transformed variable x is hence a predetermined variable and y is a jump variable. If we assume < 0 instead of assuming > 0, then the system may be characterized by either two negative roots or two positive roots. The former situation indicates that the number of positive roots is less than the number of jump variables, and the analysis will involve the problem of nonuniqueness. The latter situation implies that the number of positive roots is greater than the number of non-predetermined variables, and the perfect foresight equilibrium does not exist. Consequently, s1 s2 < 0 thus is required to ensure a unique perfect-foresight equilibrium. 10. Huang and Cai (1994, p. 399) indicate that the values of government abatement expenditure (φ) “in most OECD member countries (e.g., U.S.A., Japan, Netherlands, France, Germany, etc.) were no more than 10% in 70s and 80s. Now the values in these countries are even smaller (generally below 5%).” Given that the empirical results reveal that the value of φ is significantly small, therefore the assumption φ < ( − 1)/ seems reasonable. We are grateful to an anonymous referee for suggesting this point. 11. In this paper we restrict our analysis to the situation where the economy exhibits a positive sustained growth rate. Hence, β > α is imposed to ensure that pollution also exhibits a positive sustained growth rate. Huang and Cai (1994), Ligthart and van der Ploeg (1994), Michel and Rotillon (1995), and Mohtadi (1996) make a similar specification. Some studies, for example, Bovenberg and Smulders (1995), Elbasha and Roe (1996), Smulders and Gradus (1996), Bovenberg and de Mooij (1997), and Grimaud (1999), make an alternative assumption. They argue that the pollution damage must exhibit a zero (or negative) growth rate. In such a scenario, the economy exhibits an asymptotically-balanced growth path. In fact, our main steady-state results remain true if pollution exhibits a zero or negative growth rate. Please see Appendix C for a detailed derivation. 12. Following Barro and Sala-i-Martin (1995, p. 142), we restrict (1 − σ )[1 − θ(β − α)]γ ∗ < ρ to assure the agent’s discounted sum of bounded utility.

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13. It follows from equation (2) with σ > 1 that: UCP = −θ(1 − σ )C −σ P −θ(1−σ )−1 l η(1−σ ) > 0,

UlP = −ηθ(1 − σ )C 1−σ P −θ(1−σ )−1 l η(1−σ )−1 > 0. 14. Substituting the definition of into the aforesaid relationship, we have 1 − (1 − φ)σ < θ(1 − σ )(1 − φ)(β − α) < 0. Thus, a22 > 0 is derived. Accordingly, we have (∂y/∂x)|x˙ =0 = = −a21 /a22 > 0. −a11 /a12 > 0 and (∂y/∂x)|y=0 ˙ 15. It is clear from equations (17a) and (17b) that (∂y/∂x)|SS = (s1 −a11 )/a12 > 0, (∂y/∂x)|U U = (s2 −a11 )/a12 > 0, (∂y/∂x)|SS −(∂y/∂x)|y˙ =0 = a11 s1 /a22 (s1 −a22 ) < 0, and (∂y/∂x)|U U − = s2 /a12 > 0. Moreover, all other unstable trajectories in the figure correspond (∂y/∂x)|x=0 ˙ to the values with A1 = 0 and A2 = 0 in equations (17a) and (17b). From equations (17a) and (17b), we have: lim x = ±∞; if A1 ≷ 0; lim y = ±∞; if A1 ≷ 0, t →−∞

t →−∞

lim x = ±∞; if A2 ≷ 0; lim y = ±∞; if A2 ≷ 0.

t →∞

t →∞

In addition, we have: ˙ x) ˙ = (s2 − a11 )/a12 ; lim (y/ ˙ x) ˙ = (s1 − a11 )/a12 . lim (y/

t →∞

t →−∞

As a consequence, the common feature of these divergent paths is that they start from ∞ (−∞) with the slope of SS, and asymptotically approaches to −∞ (∞) with the slope of U U as time goes by. The dynamic system thus is represented by the phase diagram in Figure 1 when the parameters satisfy the condition that guarantees the unique perfect-foresight equilibrium. 16. It is clear from the definition of a13 and a23 that: β ≥ ≥ , a13 < 0 if α < −(1+α) 1 + (φA) (1 − l)−ε(1+α)(x ∗ )α−β −σ ≥ ≥ . a23 < 0 if α < −(1+α) θ(1 − σ )(φA) (1 − l)−ε(1+α)(x ∗ )α−β 17. Using equations (18a) and (18b), we have the following relationship:   ∂x  ∂x  a13 a23 φ α A1+α (1 − l)ε(1+α)(x ∗ )1−α+β − = − + = < 0. ∂φ  ∂φ  a a σ (β − α)(1 − σ ) x=0 ˙

18.

19.

20.

21.

y=0 ˙

11

21

Equipped with this information, equations (18a) and (18b), and footnote 8, three cases should be considered: (i) when α < β/[1 + (φA)−(1+α) (1 − l)−ε(1+α) (x ∗ )α−β ], a13 > 0 and a23 > 0 are valid; (ii) when β/[1 + (φA)−(1+α) (1 − l)−ε(1+α)(x ∗ )α−β ] < α < −σ/θ(1 − σ )(φA)−(1+α)(1 − l)−ε(1+α) (x ∗ )α−β , a13 < 0 and a23 > 0 are valid; and (iii) when α > −σ/θ(1 − σ )(φA)−(1+α)(1 − l)−ε(1+α) (x ∗ )α−β , a13 < 0 and a23 < 0 are valid. Since the dynamics of the case a13 < 0 and a23 > 0 are quite similar to that of the case a13 > 0 and a23 > 0, we do not discuss it in order to save space. Applying Burmeister’s (1980) viewpoints stated in footnote 9, in order to avoid the economy from being nonconvergent (i.e., x and y does not diverge as t → ∞), the representative household will choose an appropriate composition of consumption and leisure so as to maximize their intertemporal utility subject to their budget constraint. With such a rationalization, the economy should exactly move to the saddle path at time T to prevent an exploding solution. ˙ = (1 − φ)A(1 − l)ε − C/K in equation (8), and With the definition y = C/K, γK = K/K l = l(y, A, η) in equation (9), we can infer that the jump of γK at the moment of increasing φ is ∂γK /∂φ = −A(1 − l)ε < 0. It is of interest to examine the relationship between social welfare and public abatement expenditures. However, a space constraint does not allow us to address here. This matter is a subject for future research. From equation (5a) and (5b) in the text, we obtain ηy = εAl(1 − l)ε−1 . Given ε, A, η > 0 and 0 < l < 1, we can infer that l → 0 as y ∗ → 0 and l → 1 as y → ∞ both hold.

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22. Differentiating (y ∗ ) defined in equation (B4) with respect to y ∗ , we have: ∂(y ∗ ) = > 0. ∂y ∗ [1 − (1 − φ) ]ε(ε − 1)A(1 − l)ε−2 (ly )2 ∂ 2 (y ∗ ) = > 0. 1 − εl) ∂y ∗2 23. Using equations (B3), the slope of locus XX is given by:  1 + ε(1 − l)ε−l ly [A(1 − φ) + α(x ∗ )α−β (φA)−α (1 − l)−ε(1+α)/(β − α)] ∂x ∗  = > 0.  ∗ ∂y (φA)−α (x ∗ )α−β−1 (1 − l)−αε XX

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