- Email: [email protected]
Sustainable growth with irreversible stock effects of renewable resources
Accepted Manuscript Sustainable growth with irreversible stock effects of renewable resources Ram´on E. L´opez, Sang W. Yoon PII: DOI: Reference:
S0165-1765(16)30348-2 http://dx.doi.org/10.1016/j.econlet.2016.08.044 ECOLET 7320
To appear in:
Economics Letters
Received date: 31 May 2016 Revised date: 29 August 2016 Accepted date: 31 August 2016 Please cite this article as: L´opez, R.E., Yoon, S.W., Sustainable growth with irreversible stock effects of renewable resources. Economics Letters (2016), http://dx.doi.org/10.1016/j.econlet.2016.08.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights.
Stock effects of pollution is examined in a model of sustainable growth. Irreversible threshold of renewable resource stock is considered. Maximum critical level of initial emissions for sustainability is derived. Sustainability is feasible via pollution tax depending on the initial emissions.
*Title Page
Sustainable Growth with Irreversible Stock Effects of Renewable Resources
Ramón E. Lópeza,*, Sang W. Yoonb,** a
b
Department of Economics, University of Chile Department of Economics and Finance, Southern Connecticut State University
Abstract This paper examines sustainability conditions when the stock effects of renewable resources prevail, and characterizes the maximal critical level of initial pollution emissions such that Pigovian tax alone ensures sustainable growth.
JEL classification number: O1, Q2 Keywords: sustainable growth, stock effects, optimal pollution tax, renewable resources
*
Corresponding author. Universidad de Chile, Diagonal Paraguay 257, Santiago, Chile. Tel: +1301-405-1281. Email: [email protected]. ** 501 Crescent Street, New Haven, CT 06515, United States.
*Manuscript Click here to view linked References
1. Introduction López and Yoon (2014) shows that continuous economic growth with eventually decreasing pollution emissions is achievable with Pigouvian tax as long as either the consumer or producer’s elasticity of substitution is sufficiently large or if the consumer elasticity of marginal utility of income is greater than one in an economy with two final goods and inputs.1 However, they do not consider stock accumulation effects of pollution and ignore the possibility of irreversible threshold occurring as pollution accumulates in the atmosphere. We develop here a two-sector endogenous growth model explicitly accounting for the existence of irreversible thresholds affecting the stock of renewable natural resources such as the stock of clean air in the atmosphere. We show that for each level of clean air stock, there exists a corresponding critical level of initial pollution emissions such that Pigovian tax alone can still ensure environmentally sustainable economic growth if the initial pollution level lies below the critical level. 2. Framework of the analysis We consider an economy producing a clean and a dirty good. The dirty good production emits pollution, x , while production of the clean good emits no pollution. The dirty output yd is modeled by a constant elasticity of substitution:
yd F (kd , x) kd
1
(1 )( x)
1
1
,
(1)
where k d is the amount of capital employed; is the elasticity of substitution between capital and pollution and is a fixed distribution coefficient. The output of the clean good, which is
1
For an earlier pioneering research, see Figueroa and Pasten (2013).
1
assumed to be used as a final good as well as new capital, is modeled by the linear production technology:
yc A(k kd ). where A is the return to capital in the clean sector. Let p pd / pc denote the relative price of dirty good. Then the gross capital accumulation, which is equal to net savings (income less consumption), can be expressed in units of the clean good,
k Ac (k kd ) pAd F (kd , x) c k ,
(2)
where k dk / dt ; c is the total consumption expenditure (e.g., c cc pcd ); denote the net capital accumulation and depreciation rate. Let’s denote the stock of renewable resource (e.g., clean air) in the upper atmosphere as E , the threshold of minimal stock of clean air below which an environmental catastrophe occurs as E , the pristine stock level by E . Following Acemoglu, et al. (2012), we assume that before an
environmental catastrophe occurs, the rate of natural atmospheric purification is given as constant, 0 1 . Thus,
E E x for E E E .
x
(3)
for E E .
The welfare of the representative consumer is comprised of two parts, a utility derived from the consumption of goods and the disutility generated by pollution. When E E , the economy
2
is in catastrophic calamity, and the consumption falls to zero (e.g., Weitzman, 2009). We specify the consumer’s total welfare function as: 2 1 a
1 c U (c, x; E ) 1 a e(1, p)
x1 when E E 1 when E E ,
where a 0 is a parameter equal to elasticity of marginal utility ( EMU ) of consumption, and 1
e(1, p ) is the unit (dual) expenditure function, given as e(1, p) c d p1 1 , where is the
consumption elasticity of substitution between the dirty and clean goods, and c 0 and d 0 are fixed parameter. Assuming a fixed pure time discount rate ( ) and socially optimal intervention, the competitive economy is modeled “as if” it maximizes the present discounted value of the utility function:
U (c, x; E ) exp( t )dt, 0
subject to the budget constraint (e.g., Equation (2)), clean air stock level constraint E E (Equation (3)) and the initial conditions k k0 and E E0 . Assuming that both goods are always produced (e.g., kd (t ) k (t ) ), the current value Hamiltonian function is: 2
As in Weitzman (2009), the consumption utility becomes for
c 0 when a 1 . Since it is not possible to
describe individual optimization when the economy falls toward the catastrophic calamity, we assume that the environmental damage itself is greater than any finite conceivable magnitude (and hence minus infinity) even when
a 1 .
3
H E U (c, x, E) A(k kd ) pF (kd , x) c k E x E E
(4)
where and denote co-state variables each representing the shadow price of man-made capital and natural capital, respectively while 0 is a time-varying Lagrange multiplier associated with the stock constraint. When the stock constraint is not binding, the first-order necessary conditions for maximization of (4) and instantaneous market clearing conditions implies the following dynamical systems for ^
k pˆ , d and xˆ :3 x
pˆ M 1 Sk / a 1 / W 0 , ^
kd / x M / a 1 / W
(5)
0,
(6)
xˆ MV (t ) / W ,
(7)
where V (t ) 1 s(t ) s(t )Sk (t ) (1/ a) ( )Sk (t ) and 1 W (1 Sk (t ))(1 z(t ) ) Sk (t ) Sk (t ) 0 where s( p) d / ( c p d ) is the 1
1 consumer’s budget share of the dirty final good and Sk (1 ) kd / x is the cost
share of capital in production. López and Yoon (2014) has shown that the growth rate of real consumption expenditure, which ^
c 1 is given as M s( p) pˆ , remains positive throughout the equilibrium dynamic path for e a any positive 3
and
.
Then if there exists a finite time, T 0 , such that xˆ 0 , for any t T ,
See López and Yoon (2014) for derivations.
4
and that E (t ) E for all t , the economy becomes environmentally sustainable in a growing economy over time.4 In an environmentally sustainable economy, we have that lim xˆ 0 , t
3. Stock Effects Since Equation (7) gives the (optimal) rate of change of x that depends on the relative magnitude of a, , , the full path of x is entirely determined by the initial level of pollution emissions, x0 . The question is whether along this path the stock of clean air ever reaches the catastrophic level. In order to identify such a critical level of initial emissions, we first note that for any given initial level of man-made capital, a unique optimal growth path for p , kd / x and
x is
derived.5 In particular, we can define the pollution emissions at a point in time as: t
x(t ) x0 exp( g ( ) ) d , 0
where g ( ) is the rate of change of pollution at time
, which is a function of all parameters and
the predetermined variable, k0 . As we show below, the effect of the initial clean air stock on
x(t ) occurs entirely through its effect on x0 . From Equation (4)
t
E (t ) exp( t ) E0 x( ) exp( )d 0
(4’)
4
See Barbier & Markandya (1990) for similar notion for sustainable economic growth.
5
Since the system of Equations (5), (6) and (7) is a system of continuous autonomous differential equations there
exists a unique solution for each set of initial values. The solution for emissions,
x,
including initial level of
emissions constitutes an optimal control for dynamic optimization in the absence of stock constraints.
5
for E (t ) E ; E0 is the initial, predetermined level of the stock of clean air. We can then define the unique path of pollution emission flows and stock of clean air as conditional functions of the (endogenous) initial values of pollution emissions as well as of the (predetermined) initial stock of clean air as follows:
x(t ) G(t , x0 ; k0 , ) and E (t ) J (t , x0 ; k0 , E0 , ) , where the function J (t , x0 ; k0 , E0 , ) is defined in (4’) and (a, , ) denotes a vector of structural parameters with x(0) G(0, x0 ; k0 , ) x0 and E (0) J (0, x0 ; k0 , E0 , ) E0 . From Equation (4’) it is clear that unless the pollution emissions x(t ) eventually starts falling over time the stock constraint, E (t ) E for all t 0 , cannot be satisfied. López and Yoon (2014) characterizes the set of (a, , ) which guarantees eventual decline of pollution emissions, so that lim xˆ 0 . Then for any in , and man-made stock t
of capital, we can define the admissible set, D( , k0 ) of initial values of clean air stock and flow level of pollution which assures sustainable growth. Thus,
D( , k0 ) E0 , x0 J (t , x0 ; k0 , E0 , ) E , for all t 0 . Thanks to continuity of J (t , x0 ; k0 , E0 , ) , it is bounded above and closed. Thus, for given E0 , c there exists the maximal element of pollution level, x0 ( E0 ) above which an environmental
c disaster occurs. We can then define the envelope set C ( , k0 ) E0 , x0 ( E0 ) E E0 .
Additionally, we note that for any eventually declining pollution path, there exists a time
T 0 after which pollution decreases in a monotonic way. It follows that there exists a critical turnaround time t* T such that
6
x(t*) G (t*, x0c ; k0 , ) E ,
(8)
E (t*) J (t*, x0c ; k0 , E0 , ) E ,
(9)
where x0c is the maximum initial level of pollution emissions that corresponds to any given
E0 E
consistent with avoiding environmental disaster and t * is the critical turnaround time
at which the stock of clean air reaches the minimum level necessary to avoid a catastrophe. The c c two Equations (8) and (9) solve for the two endogenous variable, x0 x0 ( E0 ; k0 , E , , ) and
t* N ( E0 ; k0 , E, , ) . For illustration purposes, let us assume that pollution emissions follow an inverted U-shaped pattern as economy grows where the envelope C reaches E at the turnaround time t * .6 The thick curve, denoted as C in Figure 1, is the envelope of set D as defined above. It provides an envelope for all trajectories of ( E , x ) that start from ( E0 , x0 ) with x0 x0c and satisfy the constraint E (t ) E at all times, which is called set D in Figure 1. The uniqueness property of the adjustment paths guarantees that any two different trajectories starting from different initial positions move in parallel and never cross each other. Hence, any trajectory starting below
x0c ( E0 ) never reaches the catastrophic stock level, while any trajectory starting above C is bound to eventually violate the stock constraint. That is, any trajectory lying outside (above) the envelope C , denoted as a complement of set D (set D c ) in Figure 1 (which is shaded), leads to an environmental catastrophe.
6
For the necessary and sufficient conditions for emergence of U-shaped pollution curve from equations (5), (6) and
(7), see López and Yoon (2014).
7
Fig. 1 The admissible set D and the Envelop C in E-x space In Figure 1 the curve labeled OO represents the optimal trajectory while the curve SS shows an arbitrary suboptimal but sustainable trajectory associated with a suboptimal tax. The suboptimal tax satisfies two conditions: first, it is sufficiently high to permit the initial pollution level to be c
below the critical level ( x0 ) as defined earlier and second, it adjusts over time to allow for an c optimal rate of change of pollution according to Equation (7). In general, finding x0 is easier and
demands much less information than determining the optimal initial pollution level. It must be noted that pollution levels within the trajectory OO are lower than those within trajectory SS at each point in time. Thus, the previous analysis implies the following proposition.
Proposition 1. For any given level of initial stock of clean air, there exists a critical level of pollution emissions such that Pigovian tax alone can ensure environmental sustainability of dynamic competitive equilibrium unless the initial level exceeds the critical level.
8
Example: Cobb-Douglas Economy Given a Cobb-Douglas specification of technology and preferences, S k , and s are both constant. We then obtain from Equation (7) that x(t ) x0 exp(t ) , where 0 . Let x0c be the maximum initial level of pollution emissions that corresponds to any given E0 E consistent with avoiding environmental disaster and t * , the critical turnaround time at which the stock of clean air reaches the minimum level necessary to avoid a catastrophe. Let us normalize the stock of c clean air so that E 1 . To solve for the corresponding critical level of emission, x0 , we first
note that
x0c exp( t*) E
(10)
Also, from Equation (4’), we have, t*
E (t*) exp( t*)( E0 x0 exp( t )dt ) E 1 0
.
It follows that
x0c exp( t*) E0 (exp(( )t*) 1) 1
(11)
Equations (10) and (11) describe an envelope C of the Cobb-Douglas economy.
5. Concluding Comments An imperfect government may not be able to determine the optimum initial pollution emissions that leads to an optimal sustainable pollution path. However, as long as the government allows the initial emissions below the maximal critical level it can still induce environmentally sustainable growth indefinitely as long as it follows the myopic growth rule dictated by the
9
necessary conditions for dynamic optimization that would arise in the absence of stock constraints. The result would be a suboptimal rule, implying higher level of emissions than the optimum at all points in time, but one that assures sustainable and positive economic growth thus preventing environmental disaster.
10
Reference Acemoglu, D., Aghion, P., Bursztyn, L., Hemous, D., 2012. The environment and directed technical change. Am. Econ. Rev. 102(1), 131-166. Barbier, E. B., Markandya, A., 1990. The conditions for achieving environmentally sustainable development. Eur. Econ. Rev. 34(2), 659-669. Figueroa, E., Pasten, R., 2013. A tale of two elasticities: A general theoretical framework for the environmental Kuznets curve analysis. Econ. Lett. 119(1), 85-88. López, R. E., Yoon, S. W., 2014. Pollution–income dynamics. Econ. Lett. 124(3), 504-507. Weitzman, M. L., 2009. On modeling and interpreting the economics of catastrophic climate change. Rev. Econ. Stat. 91(1), 1-19.
11
S0165-1765(16)30348-2 http://dx.doi.org/10.1016/j.econlet.2016.08.044 ECOLET 7320
To appear in:
Economics Letters
Received date: 31 May 2016 Revised date: 29 August 2016 Accepted date: 31 August 2016 Please cite this article as: L´opez, R.E., Yoon, S.W., Sustainable growth with irreversible stock effects of renewable resources. Economics Letters (2016), http://dx.doi.org/10.1016/j.econlet.2016.08.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights.
Stock effects of pollution is examined in a model of sustainable growth. Irreversible threshold of renewable resource stock is considered. Maximum critical level of initial emissions for sustainability is derived. Sustainability is feasible via pollution tax depending on the initial emissions.
*Title Page
Sustainable Growth with Irreversible Stock Effects of Renewable Resources
Ramón E. Lópeza,*, Sang W. Yoonb,** a
b
Department of Economics, University of Chile Department of Economics and Finance, Southern Connecticut State University
Abstract This paper examines sustainability conditions when the stock effects of renewable resources prevail, and characterizes the maximal critical level of initial pollution emissions such that Pigovian tax alone ensures sustainable growth.
JEL classification number: O1, Q2 Keywords: sustainable growth, stock effects, optimal pollution tax, renewable resources
*
Corresponding author. Universidad de Chile, Diagonal Paraguay 257, Santiago, Chile. Tel: +1301-405-1281. Email: [email protected]. ** 501 Crescent Street, New Haven, CT 06515, United States.
*Manuscript Click here to view linked References
1. Introduction López and Yoon (2014) shows that continuous economic growth with eventually decreasing pollution emissions is achievable with Pigouvian tax as long as either the consumer or producer’s elasticity of substitution is sufficiently large or if the consumer elasticity of marginal utility of income is greater than one in an economy with two final goods and inputs.1 However, they do not consider stock accumulation effects of pollution and ignore the possibility of irreversible threshold occurring as pollution accumulates in the atmosphere. We develop here a two-sector endogenous growth model explicitly accounting for the existence of irreversible thresholds affecting the stock of renewable natural resources such as the stock of clean air in the atmosphere. We show that for each level of clean air stock, there exists a corresponding critical level of initial pollution emissions such that Pigovian tax alone can still ensure environmentally sustainable economic growth if the initial pollution level lies below the critical level. 2. Framework of the analysis We consider an economy producing a clean and a dirty good. The dirty good production emits pollution, x , while production of the clean good emits no pollution. The dirty output yd is modeled by a constant elasticity of substitution:
yd F (kd , x) kd
1
(1 )( x)
1
1
,
(1)
where k d is the amount of capital employed; is the elasticity of substitution between capital and pollution and is a fixed distribution coefficient. The output of the clean good, which is
1
For an earlier pioneering research, see Figueroa and Pasten (2013).
1
assumed to be used as a final good as well as new capital, is modeled by the linear production technology:
yc A(k kd ). where A is the return to capital in the clean sector. Let p pd / pc denote the relative price of dirty good. Then the gross capital accumulation, which is equal to net savings (income less consumption), can be expressed in units of the clean good,
k Ac (k kd ) pAd F (kd , x) c k ,
(2)
where k dk / dt ; c is the total consumption expenditure (e.g., c cc pcd ); denote the net capital accumulation and depreciation rate. Let’s denote the stock of renewable resource (e.g., clean air) in the upper atmosphere as E , the threshold of minimal stock of clean air below which an environmental catastrophe occurs as E , the pristine stock level by E . Following Acemoglu, et al. (2012), we assume that before an
environmental catastrophe occurs, the rate of natural atmospheric purification is given as constant, 0 1 . Thus,
E E x for E E E .
x
(3)
for E E .
The welfare of the representative consumer is comprised of two parts, a utility derived from the consumption of goods and the disutility generated by pollution. When E E , the economy
2
is in catastrophic calamity, and the consumption falls to zero (e.g., Weitzman, 2009). We specify the consumer’s total welfare function as: 2 1 a
1 c U (c, x; E ) 1 a e(1, p)
x1 when E E 1 when E E ,
where a 0 is a parameter equal to elasticity of marginal utility ( EMU ) of consumption, and 1
e(1, p ) is the unit (dual) expenditure function, given as e(1, p) c d p1 1 , where is the
consumption elasticity of substitution between the dirty and clean goods, and c 0 and d 0 are fixed parameter. Assuming a fixed pure time discount rate ( ) and socially optimal intervention, the competitive economy is modeled “as if” it maximizes the present discounted value of the utility function:
U (c, x; E ) exp( t )dt, 0
subject to the budget constraint (e.g., Equation (2)), clean air stock level constraint E E (Equation (3)) and the initial conditions k k0 and E E0 . Assuming that both goods are always produced (e.g., kd (t ) k (t ) ), the current value Hamiltonian function is: 2
As in Weitzman (2009), the consumption utility becomes for
c 0 when a 1 . Since it is not possible to
describe individual optimization when the economy falls toward the catastrophic calamity, we assume that the environmental damage itself is greater than any finite conceivable magnitude (and hence minus infinity) even when
a 1 .
3
H E U (c, x, E) A(k kd ) pF (kd , x) c k E x E E
(4)
where and denote co-state variables each representing the shadow price of man-made capital and natural capital, respectively while 0 is a time-varying Lagrange multiplier associated with the stock constraint. When the stock constraint is not binding, the first-order necessary conditions for maximization of (4) and instantaneous market clearing conditions implies the following dynamical systems for ^
k pˆ , d and xˆ :3 x
pˆ M 1 Sk / a 1 / W 0 , ^
kd / x M / a 1 / W
(5)
0,
(6)
xˆ MV (t ) / W ,
(7)
where V (t ) 1 s(t ) s(t )Sk (t ) (1/ a) ( )Sk (t ) and 1 W (1 Sk (t ))(1 z(t ) ) Sk (t ) Sk (t ) 0 where s( p) d / ( c p d ) is the 1
1 consumer’s budget share of the dirty final good and Sk (1 ) kd / x is the cost
share of capital in production. López and Yoon (2014) has shown that the growth rate of real consumption expenditure, which ^
c 1 is given as M s( p) pˆ , remains positive throughout the equilibrium dynamic path for e a any positive 3
and
.
Then if there exists a finite time, T 0 , such that xˆ 0 , for any t T ,
See López and Yoon (2014) for derivations.
4
and that E (t ) E for all t , the economy becomes environmentally sustainable in a growing economy over time.4 In an environmentally sustainable economy, we have that lim xˆ 0 , t
3. Stock Effects Since Equation (7) gives the (optimal) rate of change of x that depends on the relative magnitude of a, , , the full path of x is entirely determined by the initial level of pollution emissions, x0 . The question is whether along this path the stock of clean air ever reaches the catastrophic level. In order to identify such a critical level of initial emissions, we first note that for any given initial level of man-made capital, a unique optimal growth path for p , kd / x and
x is
derived.5 In particular, we can define the pollution emissions at a point in time as: t
x(t ) x0 exp( g ( ) ) d , 0
where g ( ) is the rate of change of pollution at time
, which is a function of all parameters and
the predetermined variable, k0 . As we show below, the effect of the initial clean air stock on
x(t ) occurs entirely through its effect on x0 . From Equation (4)
t
E (t ) exp( t ) E0 x( ) exp( )d 0
(4’)
4
See Barbier & Markandya (1990) for similar notion for sustainable economic growth.
5
Since the system of Equations (5), (6) and (7) is a system of continuous autonomous differential equations there
exists a unique solution for each set of initial values. The solution for emissions,
x,
including initial level of
emissions constitutes an optimal control for dynamic optimization in the absence of stock constraints.
5
for E (t ) E ; E0 is the initial, predetermined level of the stock of clean air. We can then define the unique path of pollution emission flows and stock of clean air as conditional functions of the (endogenous) initial values of pollution emissions as well as of the (predetermined) initial stock of clean air as follows:
x(t ) G(t , x0 ; k0 , ) and E (t ) J (t , x0 ; k0 , E0 , ) , where the function J (t , x0 ; k0 , E0 , ) is defined in (4’) and (a, , ) denotes a vector of structural parameters with x(0) G(0, x0 ; k0 , ) x0 and E (0) J (0, x0 ; k0 , E0 , ) E0 . From Equation (4’) it is clear that unless the pollution emissions x(t ) eventually starts falling over time the stock constraint, E (t ) E for all t 0 , cannot be satisfied. López and Yoon (2014) characterizes the set of (a, , ) which guarantees eventual decline of pollution emissions, so that lim xˆ 0 . Then for any in , and man-made stock t
of capital, we can define the admissible set, D( , k0 ) of initial values of clean air stock and flow level of pollution which assures sustainable growth. Thus,
D( , k0 ) E0 , x0 J (t , x0 ; k0 , E0 , ) E , for all t 0 . Thanks to continuity of J (t , x0 ; k0 , E0 , ) , it is bounded above and closed. Thus, for given E0 , c there exists the maximal element of pollution level, x0 ( E0 ) above which an environmental
c disaster occurs. We can then define the envelope set C ( , k0 ) E0 , x0 ( E0 ) E E0 .
Additionally, we note that for any eventually declining pollution path, there exists a time
T 0 after which pollution decreases in a monotonic way. It follows that there exists a critical turnaround time t* T such that
6
x(t*) G (t*, x0c ; k0 , ) E ,
(8)
E (t*) J (t*, x0c ; k0 , E0 , ) E ,
(9)
where x0c is the maximum initial level of pollution emissions that corresponds to any given
E0 E
consistent with avoiding environmental disaster and t * is the critical turnaround time
at which the stock of clean air reaches the minimum level necessary to avoid a catastrophe. The c c two Equations (8) and (9) solve for the two endogenous variable, x0 x0 ( E0 ; k0 , E , , ) and
t* N ( E0 ; k0 , E, , ) . For illustration purposes, let us assume that pollution emissions follow an inverted U-shaped pattern as economy grows where the envelope C reaches E at the turnaround time t * .6 The thick curve, denoted as C in Figure 1, is the envelope of set D as defined above. It provides an envelope for all trajectories of ( E , x ) that start from ( E0 , x0 ) with x0 x0c and satisfy the constraint E (t ) E at all times, which is called set D in Figure 1. The uniqueness property of the adjustment paths guarantees that any two different trajectories starting from different initial positions move in parallel and never cross each other. Hence, any trajectory starting below
x0c ( E0 ) never reaches the catastrophic stock level, while any trajectory starting above C is bound to eventually violate the stock constraint. That is, any trajectory lying outside (above) the envelope C , denoted as a complement of set D (set D c ) in Figure 1 (which is shaded), leads to an environmental catastrophe.
6
For the necessary and sufficient conditions for emergence of U-shaped pollution curve from equations (5), (6) and
(7), see López and Yoon (2014).
7
Fig. 1 The admissible set D and the Envelop C in E-x space In Figure 1 the curve labeled OO represents the optimal trajectory while the curve SS shows an arbitrary suboptimal but sustainable trajectory associated with a suboptimal tax. The suboptimal tax satisfies two conditions: first, it is sufficiently high to permit the initial pollution level to be c
below the critical level ( x0 ) as defined earlier and second, it adjusts over time to allow for an c optimal rate of change of pollution according to Equation (7). In general, finding x0 is easier and
demands much less information than determining the optimal initial pollution level. It must be noted that pollution levels within the trajectory OO are lower than those within trajectory SS at each point in time. Thus, the previous analysis implies the following proposition.
Proposition 1. For any given level of initial stock of clean air, there exists a critical level of pollution emissions such that Pigovian tax alone can ensure environmental sustainability of dynamic competitive equilibrium unless the initial level exceeds the critical level.
8
Example: Cobb-Douglas Economy Given a Cobb-Douglas specification of technology and preferences, S k , and s are both constant. We then obtain from Equation (7) that x(t ) x0 exp(t ) , where 0 . Let x0c be the maximum initial level of pollution emissions that corresponds to any given E0 E consistent with avoiding environmental disaster and t * , the critical turnaround time at which the stock of clean air reaches the minimum level necessary to avoid a catastrophe. Let us normalize the stock of c clean air so that E 1 . To solve for the corresponding critical level of emission, x0 , we first
note that
x0c exp( t*) E
(10)
Also, from Equation (4’), we have, t*
E (t*) exp( t*)( E0 x0 exp( t )dt ) E 1 0
.
It follows that
x0c exp( t*) E0 (exp(( )t*) 1) 1
(11)
Equations (10) and (11) describe an envelope C of the Cobb-Douglas economy.
5. Concluding Comments An imperfect government may not be able to determine the optimum initial pollution emissions that leads to an optimal sustainable pollution path. However, as long as the government allows the initial emissions below the maximal critical level it can still induce environmentally sustainable growth indefinitely as long as it follows the myopic growth rule dictated by the
9
necessary conditions for dynamic optimization that would arise in the absence of stock constraints. The result would be a suboptimal rule, implying higher level of emissions than the optimum at all points in time, but one that assures sustainable and positive economic growth thus preventing environmental disaster.
10
Reference Acemoglu, D., Aghion, P., Bursztyn, L., Hemous, D., 2012. The environment and directed technical change. Am. Econ. Rev. 102(1), 131-166. Barbier, E. B., Markandya, A., 1990. The conditions for achieving environmentally sustainable development. Eur. Econ. Rev. 34(2), 659-669. Figueroa, E., Pasten, R., 2013. A tale of two elasticities: A general theoretical framework for the environmental Kuznets curve analysis. Econ. Lett. 119(1), 85-88. López, R. E., Yoon, S. W., 2014. Pollution–income dynamics. Econ. Lett. 124(3), 504-507. Weitzman, M. L., 2009. On modeling and interpreting the economics of catastrophic climate change. Rev. Econ. Stat. 91(1), 1-19.
11