PRICE-BASED VS. QUANTITY-BASED POLICIES

AND ECONOMIC GROWTH:

An application to pollution taxes vs. pollution targets

by∗

George Economides and Apostolis Philippopoulos

Athens University of Economics and Business

September 21, 2000

Abstract: This paper compares optimally chosen pollution taxes with optimally chosenpollution targets. Specifically, we study the joint dynamics of economic growth, naturalresources and second-best environmental policy, when the latter is conducted eithervia pollution taxes or via pollution targets. We focus on the different implications ofthese two types of environmental policy for long-run behavior and dynamic stability.The setup is a general equilibrium growth model, in which pollution is a by-product ofoutput and environmental quality is a public good. We show that there are advantagesand disadvantages from each type of policy. Thus, in a dynamic setup, the choicebetween price-based and quantity-based policies is nontrivial even under certainty.

Keywords: Prices vs. limits. Optimal policy. Growth. Dynamics. Environment.JEL classification: H2, C61, O13.

Correspondence to: A. Philippopoulos, Athens University of Economics andBusiness, 76 Patission street, Athens 10434, Greece. Tel. +301-8203413, Fax: +301-8214122. Email: aphil@aueb.gr

∗ We thank P. Hatzipanayotou, T. Kollintzas, S. Lahiri, T. Moutos, H. Park, M. Petrakis and T.Xepapadeas for comments and discussions. All errors are ours.

1

I. INTRODUCTION

It is known that in the presence of externalities and public goods, there is a

role for economic policy. The problem is to select the appropriate policy instrument.

An important question (see Weitzman [1974]) is: Is it better to use price-based

policies (e.g. taxes and tariffs) or quantity-based policies (e.g. quotas and other

quantitative targets)? In the area of environmental economics, the relevant question

is:1 Is it better to charge pollution taxes or to set pollution standards? For instance, the

Kyoto Protocol, negotiated in December 1997, emphasized the need for ceilings on

emissions (known as pollution targets) which is a quantity-based policy. 2

What is the relevant literature? Weitzman’s [1974] seminal paper shows that the

kind of uncertainty is important for the choice of policy instrument (taxes vs. limits).

Hoel [1998] studies the pros and cons of pollution taxes when the focus is on

information and unemployment. Schmutzler and Goulder [1997] compare pollution

taxes and output taxes under imperfect monitoring. Finkelshtain and Kislev [1997] focus

on the political aspects (rent-seeking activities between interest groups) of taxes vs.

limits. Helm [1998] reviews the role of information and political constraints when he

discusses pollution taxes and permits. Barrett [1998] evaluates the recent international

agreements. Requate [1998] investigates the incentive to innovate under emission taxes

and tradable permits. Ulph [1997] compares pollution taxes to pollution limits when

national governments set minimum standards under asymmetric information. However,

to our knowledge, there is no literature on the comparison between price-based policies

and quantity-based policies on the grounds of dynamics.3

This paper compares optimally chosen pollution taxes with optimally chosen

pollution targets. Specifically, we study the joint dynamics of economic growth, natural

resources and second-best environmental policy, when the latter is conducted either

1 In the area of environmental economics, price-based policy instruments include pollution taxes,output taxes, input taxes, etc. Quantity-based policy instruments include economy-wide pollutiontargets, regulation of an input, allowing firms to pollute freely up to a limit, etc. Other regulationsinvolve the choice of production process, forced shutdown of plants, etc. On the spending side, there ispollution abatement (i.e. cleanup activity). For a survey, see e.g. Bohm and Russell [1985].2 The Kyoto Protocol specifies maximum national emission levels relative to 1990 emission levels. Italso specifies dates by which these goals must be met. See Barrett [1998]. In general, the currentapproach to environmental policy emphasizes the need for quantity-based policies. See Helm [1998].3 The literature on growth and the environment is very rich. For growth and the optimal use of naturalresources, see e.g. Kolstad and Krautkraemer [1993], Smulders [1995] and Aghion and Howitt [1998,chapter 5]. For environmental tax policy and growth, see e.g. Bovenberg and de Mooij [1994],Bovenberg and Smulders [1995], Lopez [1994], Ligthart and van der Ploeg [1994], Bovenberg and vander Ploeg [1994] and Tahvonen and Kuuluvainen [1993].

2

via pollution taxes or via pollution targets.4 The focus is on the different implications of

these two types of second-best environmental policy for long-run behavior and dynamic

stability. Although the paper is motivated by environmental issues, we believe that our

modeling approach can be used by a large class of dynamic optimization models with

price-based and quantity-based policies.

The model is as follows. Households purchase goods, work and save in the

form of capital. They get direct utility from consumption and environmental quality.

Firms produce goods by using a linear, AK technology. In doing so, they pollute the

environment. Households and firms take environmental quality as a public good. This

setup justifies government intervention. We study two types of environmental policy.

According to the first type, the government imposes taxes on polluting firms, and then

uses the collected tax revenues to finance its clean-up policy. According to the second

type, the government sets economy-wide pollution targets. Specifically, current pollution

is a fraction of pollution in the previous time-period. This is consistent with international

practice (see e.g. Barrett [1998]) where the goal is to stabilize national emissions at a

previous level, or to reduce national emissions by a certain percentage from a

previous level. Policy instruments (taxes and targets) are chosen optimally by a

benevolent government that plays Stackelberg vis-à-vis private agents. We establish

conditions for the existence and uniqueness of a long-run equilibrium, in which the

economy can grow without damaging the environment. Such long-run equilibria are

known as Sustainable Balanced Growth Paths (SBGPs).5 We focus on SBGPs in which

natural resources grow at the same constant positive rate with capital and consumption.

Concerning long-run properties, we show that the SBGP is independent of

environmental policy when the latter is conducted via pollution targets. Also, pollution

targets lead to a higher SBGP than pollution taxes. Therefore, pollution targets are better

for both long-run growth and the environment than pollution taxes. Concerning

transitional dynamics, we show that when policy is conducted via pollution taxes, the

SBGP is locally determinate. That is, there is a unique path to the steady state. In

contrast, when policy is conducted via pollution targets, the SBGP can be locally

indeterminate. In other words, there can be an infinite number of self-fulfilling time-

paths, each of which is consistent with a given initial condition and with convergence to

4 In practice, both taxes and limits are used to protect the environment (see Smith [1995], Helm [1998]and Barrett [1998]). Here, we model them separately for expositional reasons. Also, we assume thattargets are directly chosen. In practice, they can be met indirectly.

3

a unique steady state. Therefore, when policy is conducted via pollution targets,

identically endowed countries with similar initial conditions can grow at completely

different rates and enjoy completely different environmental quality over time, even if

they eventually converge to the same growth rate.6

But what causes this difference in transitional dynamics among the two regimes?

Obviously, by restricting current pollution to depend directly upon last period pollution

in the second regime, it is the same as restricting government policies to depend upon

last period policies. This is equivalent to solving for Markov policies and hence

transforming the structure of the model to a Markov one. This makes the dynamics of

the model in the second regime different from those in the first regime where such

dependence is not imposed and where we solve a standard second-best dynamic policy

problem.

Therefore, there are advantages and disadvantages. Pollution taxes lead to a

lower long-run growth rate of consumption, capital and natural resources than pollution

targets. On the other hand, quantity-based policies in the form of targets can lead to

many different transition paths to the same long-run equilibrium; each of these paths can

have different welfare implications. Also, conditions for the existence and uniqueness of

a long-run equilibrium differ depending on whether we use taxes or targets.

Consequently, when we add dynamics, the choice between price-based and quantity-

based policies is nontrivial even under certainty. This generalizes Weitzman [1974].

The rest of the paper is as follows. Section II sets up the model. Section III solves

for optimal policy conducted via pollution taxes. Section IV solves for optimal policy

conducted via pollution targets. Section V closes the paper. Proofs are in an Appendix.

II. COMPETITIVE EQUILIBRIUM GIVEN ENVIRONMENTAL POLICY

Consider a closed economy populated by private agents (a representative

household and a representative firm) and a government. Households purchase goods,

work and save in the form of capital. They get direct utility from consumption and

environmental quality. Firms produce goods by using a linear, AK technology. In

5 For SBGPs, see e.g. Smulders [1995] and Aghion and Howitt [1998, chapter 5].6 See e.g. Benhabib and Farmer [1994] and Benhabib and Perli [1994].

4

doing so, they pollute the environment.7 Households and firms take environmental

quality as a public good. This setup justifies environmental policy. We first model the

case in which policy takes the form of pollution taxes. We then model the case in which

policy takes the form of economy-wide pollution targets.8 This section solves for a

competitive equilibrium given environmental policy. We assume continuous time,

infinite horizons and perfect foresight.

Households

The representative infinite-lived household maximizes intertemporal utility:

dteNcu t∫∞

−

0

)],([ ρ (1)

where c is private consumption, N is the stock of natural resources and the

parameter ρ> 0 is the rate of time preference. The instantaneous utility function (.)u

is increasing and concave in its two arguments, and also satisfies the Inada conditions.

For algebraic simplicity, we assume that (.)u is additively separable and logarithmic:

NcNcu loglog),( ν+= (2)

where the parameter 0>ν is the weight given to environmental quality relative to

private consumption.

Households save in the form of capital. When they rent out k to firms, they

receive a rate of return, r . They also supply inelastically one unit of labor services

7 Thus, pollution is a by-product of production. Our results do not change if pollution is also a by-product of consumption. On the other hand, modeling pollution as a by-product of economic activity(production and consumption) differs from the case in which natural resources are extracted frompreserved natural environments to be used as inputs in production. In that case, “pollution” is modeledas the flow of recently extracted natural resources which enter the private firms’ production function asa positive externality. In a more general model, all of them (i.e. resource extraction, production andconsumption activities) affect the environment (see Kolstad and Krautkraemer [1993]).8 We do not model pollution targets as (binding) constraints on private agents. For instance, we assumethat private firms can pollute freely. Instead, we study economy-wide optimal pollutiontargets/standards when private agents (firms and households) treat the environment as a public good.As we argue in the paper, this is not inconsistent with recent international practice. Also, as long aspollution is modeled as a public good (as we do here), allowing firms to pollute freely only up to alimit, does not change any of our results. Of course, the algebra would change if pollution were achoice variable from the firms’ point of view.

5

and get a labor income w . Further, they receive profits π . Thus, the budget constraint

of the representative household is:

π++=+•

wrkck (3)

where a dot over a variable denotes time derivative. The initial stock, 0k , is given.

The household acts competitively by taking prices and evironmental quality,

N , as given. The latter is justified by the open-access and public-good features of the

environment. The control variables are c and k , so that the first-order conditions for

a maximum are equation (3) as well as the familiar Euler condition:

crc )( ρ−=•

(4)

Firms

Output is linear in capital, as in the well-known AK model of growth. Thus,

firms have a linear production function of the form:

Aky = (5)

where 0>A is a parameter. As is known, this model generates endogenous growth.

Let 10 <≤θ be the tax rate on polluting firms’ output.9 Then, net profits of

the representative firm are:

wrky −−−= )1( θπ (6)

The firm acts competitively by taking prices and tax policy as given. This is a

static problem. The control variable is k , so that the first-order condition for a

maximum is simply:

Ar )1( θ−= (7)

9 Our results do not change if taxes are imposed on households.

6

which equates the rate of return to the after-tax marginal product of capital. Note that

(5), (6) and (7) imply 0=−=+ rkywπ . This is a well-known result in the AK

model. Namely, all realized income goes to capital.10

Competitive Equilibrium with Pollution Taxes

This subsection characterizes a Competitive Equilibrium (CE) for any feasible

pollution tax rates, θ .

We start by modeling pollution, p . We assume that p is a by-product of

output produced, y .11 Specifically, for notational simplicity, we assume a one-to-one

link between p and y . Then, by using (5), we have in a CE:

Akyp == (8)

that is, one unit of output generates one unit of pollution. This also implies that, in our

model, pollution and output taxes coincide.

Using 0=+πw and (7), equations (4) and (3) give respectively:

cAc ])1([ ρθ −−=•

(9a)

ckAk −−=•

)1( θ (9b)

which are the private agents’ optimal rules for consumption and savings in a CE.

It remains to specify natural resources. The stock of natural resources, N ,

evolves over time according to:

ypNN θδ +−=•

(10a)

10 Thus, if capital is paid its realized marginal product, there is nothing left for labor or profits. AsBarro and Sala-i-Martin [1995, pp. 141-2] point out, this is because in the AK model, “capitalencompasses human capital, knowledge and public infrastructure”, while “the zero wage rate can bethought as applying to raw labor which has not been augmented by human capital”.11 This assumption is consistent with evidence that the world’s largest emitters are the rich countries,the US, Japan and Germany. A clear example is 2CO emissions (see e.g. Barrett [1998]). This is why

reduction in emissions is expected to result in serious harm to the industrial countries, especially theUS (see e.g. Barrett [1998] for the debate on the Kyoto Protocol negotiated in December 1997). Notethat this does not include e.g. land-use and forestry, which have to do with resource extraction. In thecase of resource extraction, most of the environmental damage takes place in the developing countries.

7

so that natural resources regenerate themselves by a rate 0≥δ , decrease with

pollution emission, p , and increase with clean-up policy being financed by taxes on

polluting firms’ output, yθ . The initial stock, 0N , is given.

Using (8) into (10a), the motion of natural resources in a CE is:12

AkNN )1( θδ −−=•

(10b)

We summarize. Equations (9a), (9b) and (10b) summarize a Competitive

Equilibrium (CE), when environmental policy is conducted via pollution tax rates, θ .

In this equilibrium: (i) households maximize utility and firms maximize profits; (ii) all

constraints are satisfied and all markets clear. This CE holds for given initial

conditions and any feasible θ . Section III below will endogenize the choice of θ .

Competitive Equilibrium with Pollution Targets

This subsection characterizes a Competitive Equilibrium (CE) for any feasible

pollution targets, τ . The government sets the current flow of pollution, p , to be a

fraction, 0>τ , of the flow of pollution in the previous time-period, p′ (where a

prime denotes previous-period values). That is, pp ′= τ .13

Equations (1)-(9) still hold but now 0=θ . That is, the private agents’ optimal

decision rules in a CE in (9a)-(9b) above are reduced to:

cAc )( ρ−=•

(11a)

cAkk −=•

(11b)

while the motion of natural resources changes from (10a) to

pNN ′−=•

τδ (11c)

12 For clarification, we also present the discrete-time analogue of (10b). (10a) is

tttttt ypNNN θδ +−=−+1 . Then, using (8), we get ttttt kANNN )1(1 θδ −−=−+ , where tk denotes thepredetermined capital stock at time t . This is equation (10b).

8

By using (8), equations (11b) and (11c) can be written respectively as:14

kc

A′

−−+=•

τττ 1 (12a)

kANN ′−=•

τδ (12b)

where k ′ denotes the previous-period capital stock.

We summarize. Equations (11a), (12b) and (12c) summarize a Competitive

Equilibrium (CE), when environmental policy is conducted via pollution targets, τ .

This is for given initial conditions and any feasible τ . Section IV below will

endogenize the choice of τ .

III. OPTIMAL POLLUTION TAXES AND ECONOMIC GROWTH

This section studies optimal (second-best) environmental policy conducted via

pollution taxes, θ . We endogenize θ by assuming that the government is benevolent

and plays Stackelberg vis-a-vis private agents. Thus, the government chooses the

paths of Nkc ,,,θ to maximize (1)-(2) subject to the Competitive Equilibrium,

summarized by (9a), (9b) and (10b). The current-value Hamiltonian, H , is:

[ ]ρθλν −−++≡ )1(loglog AcNcH [ ]ckA −−+ )1( θγ [ ]AkN )1( θδµ −−+ (13)

where λ , γ and µ are the multipliers associated with (9a), (9b) and (10b)

respectively. That is, λ is the social value of private marginal utility of assets, γ is

the social value of physical capital and µ is the social value of natural resources.15

13 The discrete-time analogue is 1−= ttt pp τ .14 To understand (12a), we also present its discrete-time analogue. Since at any time t , 1−= ttt pp τ and

tt pAk = , we have ttt kk 11 ++ =τ . Then, tttt cAkkk −=−+1 in (11b) is written as ttt ckA −=−−+ )1( 1τ or

11 )1(

−+ −=−−

tt

tt k

cA

ττ . Subtracting tτ from both sides, we have:

11 1

−+ −−+=−

tt

tttt k

cA

ττττ , or in continuous

time kc

A′

−−+=•

τττ 1 . This is (12a). See below in Section IV for choice variables under this policy

regime.15 Throughout, we assume commitment technologies on behalf of the government so that policies arechosen once-and-for-all. Thus, we do not study time-consistency issues.

9

The first-order conditions with respect to µγλθ , , , , , , Nkc are respectively:

( ) kkc µγλ =+ (14a)

( )[ ] γρθλρλλ +−−−−=•

11

Ac

(14b)

( )[ ]ρθ −−=•

1Acc (14c)

( ) )1(1 θµθγργγ −+−−=•

AA (14d)

( ) ckAk −−=•

θ1 (14e)

δµν

ρµµ −−=•

N (14f)

( )AkNN θδ −−=•

1 (14g)

These necessary conditions are completed with the addition of a transversality

condition that guarantees utility is bounded. This happens when:

( )[ ] ρδρθ <+−−1A (14h)

so that the growth rate of consumption, ( ) ]1[ ρθ −−A , plus the rate of regeneration of

natural resources, δ , is less than the rate of time preference, ρ . Note that if

0 ,0 ,0 ≥≥≥ µγλ , and since the objective function and the constraints in (13) are

concave in Nkc , , ,θ , the necessary conditions are also sufficient for optimality (see

Kamien and Schwartz [1991, pp. 132-5]).

Following usual practice, we transform the variables so as to facilitate

analytical tractability. Let define zck

≡ , ψ µ≡ k and φ µ≡ N . Then, Appendix A

shows that the dynamics of (14a)-(14g) are equivalent to the dynamics of (15a)-(15d):

zzz )( ρ−=•

(15a)

( ) ψδφν

ρθψ

−−+−−=

•

zA 1 (15b)

10

( )φ

φψθ

φν

ρφ

−−−=

• A1 (15c)

1=

++ ψδ

φν

z (15d)

where (15a)-(15d) constitute a system in θφψ , , ,z . Note that (15d) is static. That is,

the dynamics of θ follow from the dynamics of φψ, ,z (see also below).

Definition of Long-run Equilibrium

We now investigate the long-run properties of (15a)-(15d). We will study

Sustainable Balanced Growth Paths (SBGPs). The steady state is characterized by

0===•••

φψz .16 This implies that consumption, capital and natural resources can

grow at the same constant positive rate.17 Hence, this is a properly defined SBGP. Let

us denote the resulting steady state values of ( )θφψ , , ,z by ( )θφψ ~ ,~ ,~ ,~z . In what

follows, we solve for ( )θφψ ~ ,~ ,~ ,~z .

Solution of Long-run Equilibrium

Setting (15a) equal to zero, the consumption-to-capital ratio, ~z , is:

~z = ρ (16a)

Setting (15b) equal to zero, we get for the social value of natural resources, ~φ :

δθν

φ−−

=)

~1(

~

A (16b)

16 When 0=•z in (15a), c and k grow at the same rate by following (14c) and (14e) respectively.

When 0=•ψ in (15b), the relative social value of the stock of capital, ψ , does not change. When

0=•φ in (15c), the social value of the stock of natural resources, φ , does not change either. Note that

0===•••φψz , together with (15d), give also a solution for the long-run tax rate, θ

~.

11

Setting (15c) equal to zero, and using (16b), we get for the relative social

value of capital, ~ψ :

])~

1([)~

1(

)]~

1([~δθθ

θδρνψ

−−−

−−+=

AA

A (16c)

In turn, using (16a), (16b) and (16c) into (15d), we get:

])~1()[~1()]~1()][~1([ δθθθδρθρν −−−=−−+−+ AAAA (16d)

which is a quadratic equation in θ~

only. Once we solve (16d) for θ~

, (16b) and (16c)

respectively can give the values of ~φ and ~ψ . So the main task is to solve (16d) for

1~

0 <≤θ and then check whether the resulting steady state solution is well-defined.

A well-defined solution requires: (i) the economy to grow without damaging the

environment; (ii) natural resources to be valued positively, i.e. 0>µ ; (iii) the

transversality condition (14h) to hold.

Then, Appendix B proves the following Proposition:

PROPOSITION 1: If the parameter values satisfy the following restrictions:

δρ+>A (17a)

δρ 2> (17b)

δρνδ −>2 (17c)

there exists a unique well-defined long-run pollution tax rate, ~θ , which is a solution

to (16d) and lies in the region:

11~

10 <−<<+

−<AAρ

θδρ

(18)

In turn, this pollution tax rate supports a unique well-defined steady state in which

consumption, capital and natural resources grow at the same constant positive rate.

Hence, the steady state is a Sustainable Balanced Growth Path (SBGP).

17 When 0==••φψ , and since kµψ ≡ and Nµφ ≡ , we have

NN

kk

••

= . Also, 0=•z implies

kk

cc

••

= .

Therefore, consumption, capital and natural resources grow at the same rate.

12

Conditions (17a)-(17c) are jointly sufficient for a well-defined and unique

long-run equilibrium to exist. The algebra is in Appendix B. Conditions (17a) requires

the productivity of private capital, A , to be higher than the rate of time preference,

ρ , plus the regeneration rate of natural resources, δ . This is a condition for long-

term growth. Notice that we require a stronger condition than usually (see e.g. Barro

and Sala-i-Martin [1995, p. 142]) because here the economy must also devote

resources to environmental quality. Condition (17b) guarantees that (14h) holds and

so the attainable utility is bounded (see e.g. Barro and Sala-i-Martin [1995, p. 142] for

similar conditions). Finally, (17c) is a condition for existence. It implies that existence

obtains more easily when the rate of regeneration of natural resources is high (i.e. a high

δ helps existence), we care about the environment/public good (i.e. a high ν helps

existence) and we care about the future (i.e. a low ρ helps existence).

Properties of Long-run Equilibrium

Total differentiation of (16d) implies:18

=

+−−−

A, , ,~ρνδθθ (19)

Therefore, we have the following comparative static results: (i) When natural

resources regenerate themselves (i.e. δ is high), the need for pollution taxes is

smaller. (ii) When private agents themselves value environmental standards (i.e. ν is

high), the need for environmental policy is less acute. (iii) The more we care about the

future (i.e. the lower is ρ), the higher the chosen pollution tax rate. (iv) When private

capital is productive (i.e. A is high), we can afford higher pollution taxes.

These are intuitive results for the long-run pollution tax rate, θ~

. In turn, the

properties of the long-run growth rate, SBGP, can easily follow. In particular, the

properties of the SBGP are symmetrically opposite to those of θ~

(in equation (14c),

the SBGP is decreasing in θ~

). Finally, note that all comparative static properties are

in logical accordance with the results for existence in the previous subsection.

18 Signs above parameters give equilibrium properties.

13

Transitional Dynamics

We finally study the transitional dynamics of (15a)-(15c). We study stability

properties around steady state. Linearizing (15a)-(15c) around (16a)-(16d) implies

that the local dynamics are approximated by the linear system:

−−

−=

•

•

•

φψ

ρθφψν

ψ

ρ

φ

ψz

A

z

)~

1(0

~~

0~00

2 (20)

where the elements of the Jacobian matrix have been evaluated at the steady state.

The determinant of the Jacobian matrix in (20) is ( )2~~

)~

1(detφψν

θρ −= AJ . This

is positive. Hence, there are two possibilities: Either there are three positive roots, or

one positive and two negative roots. Since all three variables ),,( φψz are jump

variables, the former possibility (i.e. three positive roots) implies local determinacy,

while the latter possibility (i.e. one positive and two negative roots) implies local

indeterminacy. As we show in Appendix C by applying Descartes’ Theorem, the

characteristic equation of the Jacobian matrix excludes the latter possibility. Hence,

there is local determinacy.

What does it mean? Without predetermined variables, determinacy means that

the jump variables jump immediately, and in a unique way, to take their long-run

values and stay there (until the system is disturbed in some way). There are no

transitional dynamics and the saddle-path solution is equivalent to the steady state.

This is as in the basic AK model (see e.g. Barro and Sala-i-Martin [1995]).

Therefore, we have:

PROPOSITION 2: Under the conditions in Proposition 1, the unique long-run

pollution tax rate and the associated SBGP are locally determinate. This means that,

when environmental policy is conducted via pollution taxes, the economy jumps

immediately and in a unique way to its steady state, as in the basic AK growth

model.

14

IV. OPTIMAL POLLUTION TARGETS AND ECONOMIC GROWTH

Now environmental policy is conducted via pollution targets, τ . As before,

we endogenize τ by assuming that the government is benevolent and plays

Stackelberg vis-a-vis private agents. Thus, the government chooses the paths of

Nc,,τ to maximize (1)-(2) subject to the Competitive Equilibrium, summarized by

(11a), (12a) and (12b). The current-value Hamiltonian, H , is:

)(loglog ρλν −++≡ AcNcH

′−−++

kc

Aτ

τε 1 )( kAN ′−+ τδµ (21)

where λ , ε and µ are the new multipliers associated with (11a), (12a) and (12b)

respectively. That is, λ is the social value of private marginal utility of assets, ε is

the marginal efficiency cost of pollution limits and µ is the social value of natural

resources.19

The first-order conditions with respect to µλετ , , , , , Nc are respectively:20

′−+′+=

•

kc

kA2

1τ

εµρεε (22a)

kc

A′

−−+=•

τττ 1 (22b)

kA

c ′+−−−=

•

τε

ρλρλλ )(1

(22c)

)( ρ−=•

Acc (22d)

δµν

ρµµ −−=•

N (22e)

kANN ′−=•

τδ (22f)

19 Obviously, λ and µ in (21) can differ from λ and µ in (13). However, we use the same notationfor expositional convenience so as to make the results under targets comparable to the results undertaxes.20 The predetermined capital stock, k′ , is not a choice variable. Then, given k′ , the choice of thepollution target, τ , fully determines the capital stock k , where kk ′= τ . See the substitutions infootnote 14 above.

15

These necessary conditions are completed with the addition of the

transversality condition that guarantees utility is bounded. Thus, as in (14h) above,

ρδρ <+− )(A (22g)

Note that if 0 ,0 ,0 ≥≥≥ εµλ , and since the objective function and the constraints in

(21) are concave in τ , c , N , the necessary conditions are also sufficient for

optimality.

As before, to facilitate analytical tractability, we transform the variables. Let

define kc

z′

≡ , k ′≡ µψ , φ µ≡ N and cλχ ≡ . Appendix D shows that the dynamics

of (22a)-(22f) are equivalent to the dynamics of (23a)-(23f) below:

zzz )( ρ−=•

(23a)

τττ

zA −−+=

•

1 (23b)

ψδφν

ρψ

−−+−=

•

zA (23c)

φφψτ

φν

ρφ

−−=

• A (23d)

−++=

•

21

τεψρεε

zA (23e)

χτχε

χρχ

+−=

• z1 (23f)

where (23a)-(23f) constitute a dynamic system in χεφψτ , , , , ,z .

16

Definition of Long-run Equilibrium

We now investigate the long-run properties of (23a)-(23f). The steady state is

characterized by 0======••••••χεφψτz .21 Again, this implies that consumption,

capital and natural resources grow at the same constant positive rate. Hence, this is a

properly defined SBGP. Let denote the resulting steady state values of

( )χεφψτ , , , , ,z by ( )χεφψτ ~ ,~ ,~ , ,~ ,~z . In what follows, we solve for ( )χεφψτ ~ ,~ ,~ , ,~ ,~z .

Solution of Long-run Equilibrium

Setting (23a) equal to zero, the solution for z~ is again:

~z = ρ (24a)

Setting (23b) equal to zero, and using (24a), we get:

+=+

2~1~1τρ

τA (24b)

which is an equation in τ~ only. Note that (24b) implies 0~ >τ .

Setting (23c) equal to zero, and using (24a), the solution for φ~ is:

δν

φ−

=A

~ (24c)

Setting (23d) equal to zero, and using (24c), we have for ψ~ :

21 When 0=•z in (23a), consumption and capital grow at the same rate. When 0=

•τ in (23b), the

pollution target does not change. When 0=•ψ in (23c), the social value of capital, ψ , does not change.

When 0=•φ in (23d), the social value of natural resources, φ , does not change. When 0=

•ε in (23e), the

social cost of pollution limits, ε , does not change. When 0=•χ in (23f), the social value of private

consumption, χ , does not change.

17

)(~~

δτδρν

ψ−

)−+(=

AAA

(24d)

Setting (23e) equal to zero and using (24a), we get for ε~ :

−+

−=

2~1

~~

τ

ρρ

ψε (24e)

Finally, setting (23f) equal to zero and using (24a), we have for χ~ :

τε

ρχ ~

~1~ −= (24f)

Therefore, once we solve (24b) for τ~ , (24d) can give ~ψ . In turn, (24e) can

give ε~ , and finally (24f) gives χ~ . So the main task is to solve (24b) for 0~ >τ , and

check whether the resulting steady state solution is well-defined. Recall that a well-

defined steady state solution requires: (i) the economy to grow without damaging the

environment; (ii) natural resources to be valued positively, i.e. 0>µ ; (iii) the

transversality condition (22g) to hold.

Then, Appendix E proves the following Proposition:

PROPOSITION 3: If the parameter values satisfy the following restrictions:

ρ>A (25a)

δρ 2> (25b)

δρ+<A (25c)

there exists a unique well-defined long-run pollution target, τ~ , which is a solution to

(24b) and lies in the region:

ρρ

τ+

<<1

~0 (26)

In turn, this pollution target supports a unique well-defined steady state in which

consumption, capital and natural resources grow at the same constant positive rate.

Hence, the steady state is a Sustainable Balanced Growth Path (SBGP).

18

(25a)-(25c) are jointly sufficient conditions for a well-defined and unique

long-run equilibrium to exist. Specifically, condition (25a) requires the productivity of

private capital, A , to be higher than the rate of time preference, ρ . Without clean-up

policy, this is the usual condition for long-term growth in AK models (see e.g. Barro

and Sala-i-Martin [1995, p. 142]). Condition (25b) ensures that utility is bounded (see

also (17b) above). Finally, condition (25c) acts as an upper barrier on the productivity

of private capital, A . This condition was not present when policy was conducted by

pollution taxes. By contrast, when policy is conducted by pollution targets, and since

pollution is a by-product of output produced, we have to restrain output and pollution,

and this can be ensured by not allowing private capital to be too productive and hence

too polluting. When policy is conducted by pollution taxes, such a condition is not

required because taxes can prevent output and pollution from becoming too high.

Properties of Long-run Equilibrium

Total differentiation of (24b) implies:

),(~ −+

= Aρττ (27)

That is, the more we care about the future (i.e. the lower is ρ), the lower the

optimal pollution limit, i.e. the tougher is environmental policy. Also, the higher is A ,

the lower the optimal pollution limit. Thus, we can afford a tougher environmental

policy when private capital is productive. Therefore, the properties of the optimal

pollution target are qualitatively similar to those of the optimal pollution tax rate.

Also, observe from (22d) that the SBGP is equal to )( ρ−A , which is

independent of τ~ . In other words, when policy is conducted via pollution targets, the

long-term growth rate of capital, consumption and natural resources is independent of

environmental policy. Put it differently, targets do not distort long- term growth. Also,

)( ρ−A , which is the SBGP under pollution targets, is always higher than

])1([ ρθ −−A , which is the SBGP under pollution taxes (this is because θ is between

zero and one). In other words, pollution targets lead to a higher long- term growth than

pollution taxes.

19

Transitional Dynamics

We finally study the transitional dynamics of (23a)-(23f). We study stability

properties around steady state. Linearizing (23a)-(23f) around (24a)-(24f) implies that

the local dynamics are approximated by the linear system:

−−

−

−

−

−−−

=

•

•

•

•

•

•

εψ

τερ

τε

ρτψτρ

ρτερ

τε

φψν

ψ

τρ

τ

ρ

ε

φ

χ

ψτ

~~

001~~2

~~

00~~0~00~

~~~

0~~

000~

0000~1~1

00000

32

2

2

2

AA

z

εφχψτz

(28)

where the elements of the Jacobian have been evaluated at the steady state.

There are six variables ) , , , , ,( εψχφτz , which include five jump variables

) , , , ,( ψχφτz and one predetermined )(ε .22 Hence, for determinacy, we need five

positive (i.e. unstable) roots and one negative (i.e. stable) root. It easily follows that

the determinant of the Jacobian matrix in (29) is ( )2

2

22

~~~~

~1detφε

τψντρ

ρA

J

−= . This

is negative because 0~ >ε and 0~12

<

−

τρ

(see Appendix E). Hence, with six roots,

there is an odd number (one, three or five) of negative roots. This automatically

excludes the possibility of instability (i.e. it is not possible to have six positive roots).

Then, there are two possibilities. (i) There is one negative and five positive roots. In

this case, the model will exhibit local determinacy, i.e. saddle-path stability. (ii) There

are three or five negative roots, and the rest are positive. In this case, the model will

exhibit local indeterminacy. To proceed further, we have to check the characteristic

equation in (28). Then, as we argue in Appendix F by applying Descartes’ Theorem,

22 ε is predetermined because it is the co-state variable of a jump variable, τ . See also Ploeg [1987].

20

both cases are theoretically possible. However, as Appendix F shows, indeterminacy

is the case most likely to occur for empirically plausible parameter values.23

The above results can be summarized by:

PROPOSITION 4: Under the conditions in Proposition 3, the unique long-run

pollution target and the associated SBGP can be locally indeterminate. That is, when

optimal environmental policy is conducted via pollution targets, there exists an

infinite number of equilibrium trajectories all of which are consistent with a given

initial condition and a unique steady state.

By restricting current pollution to depend directly upon last period pollution, it is

the same as restricting government policies to depend upon last period policies. This is

equivalent to solving for Markov policies and hence transforming the structure of the

model to a Markov one. This makes the dynamics of the model in this case different

from those in the case presented in section III. Recall that in section III, where such

dependence is not imposed, we solve a standard second-best dynamic policy problem in

which the SBGP is locally determinate.

What does indeterminacy mean? It means there are many possible paths of

consumption, output and natural resources, each of which is consistent with a given

initial condition and with convergence to a unique steady state. All these paths are

possible to occur.24 That is, identically endowed economies with the same initial

conditions may enjoy completely different rates of growth and environmental

standards over time. Only in the long run, they will converge to the same growth rate

of consumption, output and natural resources (although not to the same level).

Inspection of (22a)-(22f) and (23a)-(23f) reveals the above results. Although

the values of the consumption-to-capital ratio, kc

z′

≡ , and the growth rate of

23 See Appendix F for details. Here we do want to pursue any further the issue of how empiricallyplausible indeterminacy is. We just want to stress that, under pollution targets, indeterminacy ispossible to arise. By contrast, under pollution taxes, there is always determinacy.24 Therefore, the indeterminacy result can provide a general explanation of why some countries, whichinitially were at similar levels of wealth, they develop differently over time. See Benhabib and Perli[1994].

21

consumption, cc•

, will be unchanged along the optimal path, 25 the levels of

consumption, output and pollution need not to be the same (this can happen if

different economies choose different levels of initial consumption which is a jump

variable). These different levels can have very different welfare implications.

V. CONCLUSIONS AND EXTENSIONS

This paper has compared optimally chosen pollution taxes with optimally

chosen pollution targets. We studied the different implications of these two types of

environmental policy for long-run properties and dynamic stability. We showed that

there are advantages and disadvantages from each policy, so that the choice between

price-based and quantity-based policies in nontrivial even under full certainty. Thus,

our dynamic analysis generalizes the result of Weitzman [1974].

We feel that our modeling approach can be used by a large class of growth

models, in which second-best policy takes the form of price-based or quantity-based

policy instruments. For instance, we can apply the same methodology to a public finance

model, in which stabilization of public debt can take place via tax increases and/or

quantitative rules of the Maastricht Treaty type.

We close with two extensions in the context of environmental economics.

First, we can extend the model to a multi-country world economy to study the

dilemma of taxes versus limits when there are cross-country spillover effects. This

would enable us to investigate the rationality of the recent international agreements

for national pollution targets (see e.g. Barrett [1998] for the Kyoto Protocol). It would

also help us to study issues like free-riding, especially if pollution targets are

collective and aggregate (see e.g. Barrett [1998]). Second, here we have assumed that

targets are directly chosen. However, in practice, chosen targets can be met only

indirectly (for instance, by means of fiscal measures, pollution taxes or conservation

programs). We leave these extensions for future work.

25 z does not change over time because equation (23a) is an unstable differential equation. Hence, zjumps to its steady state value and equals this value all the time. Note that the same holds for equation(23b). That is, this is also an unstable equation, and so τ equals its steady state value all the time.

22

APPENDICES

APPENDIX A: From equations (14a)-(14g) to equations (15a)-(15d).

Taking logs on both sides of (14a) and differentiating with respect to time, we

get:

+=

++++ ••

••••

kkkkc

kkccµµ

µγλγγλλ 1

(A.1)

Substituting (14b), (14c), (14d), (14e) and (14f) for the rates of growth of

µγλ ,, , , kc respectively into (A.1), we obtain after some manipulation:

Nk

kcν

δµµ ++=1 (A.2)

If zck

≡ , (14c) and (14e) give (15a) in the text. Also, if ψ µ≡ k and φ µ≡ N ,

(14c), (14f) and (14g) give (15b) and (15c) in the text. Finally, (A.2) gives (15d) in

the text.

APPENDIX B: Proof of Proposition 1.

We work in steps. In the first step, we specify the region in which a well-

defined solution (if any) for θ~

should lie. A well-defined solution requires: (i)

0)~1( >−− ρθ A , i.e. Aρ

θ −< 1~

. This is required for the economy to grow in the

long-run. (ii) ρδθ 2)~1( <+− A , i.e. θδρ ~2

1 <−

−A

. This is required for the

transversality condition (14h) to hold. (iii) 0)~1( >−−+ Aθδρ , i.e. θδρ ~

1 <+

−A

.

This follows from inspection of (16b)-(16d) in the text. (iv) 0)~1( >−− δθ A , i.e.

Aδ

θ −< 1~

. Again this follows from inspection of (16b)-(16d) in the text. (iv)

0)~1(2 >−− δθ A , i.e. A2

1~ δθ −< . This is required for the left-hand side of (16d) to

be monotonically increasing in θ~

(see below why we need this). Now, if we combine

(i)-(iv), and given the parameter restrictions in (17a) and (17b) in Proposition 1, it

23

follows that the “binding” lower boundary for θ~

is A

δρ+−< 10 ,26 while the

“binding” upper boundary for θ~

is 11 <−Aρ

.27 Thus,

11~

10 <−<<+

−<AAρ

θδρ

(B.1)

which gives the region in which a well-defined solution (if any) for θ~

should lie.

(B.1) is equation (18) in the main text.

Consider now the second step. We study whether such a solution for θ~

actually exists and it is unique. Recall that θ~

solves (16d). Define the left-hand side

of (16d) by )~(θL and the right-hand side by )~(θR . Then, 0)~

( >θθL (see condition

(iv) above) and 0)~

( <θθR . Concerning the lower boundary, i.e. A

δρ+−1 , we have

01 =

+

−A

Lδρ

which is always smaller than 01 >

+

−A

Rδρ

. Concerning the

upper boundary, i.e. Aρ

−1 , we have >

−

AL

ρ1 01 >

−

AR

ρ, if the parameter

values satisfy condition (17c) in the text. Since 0)~

( >θθL and 0)~

( <θθR

monotonically, these values of )~(θL and )~(θR at the lower and upper boundaries

mean that )~

(θL and )~

(θR intersect once, as it is shown in Figure 1 below.

Figure 1 here

Therefore, a unique, well-defined solution for θ~

exists. This in turn supports -

via (16b) and (16c) - a unique well-defined solution for φ~ and ψ~ .

26 In particular, if we assume δρ 2> [which is condition (17b)], it follows θ

δρδρ ~1

21 <

+−<

−−

AA.

That is, when θδρ ~

1 <+

−A

, it also always holds θδρ ~2

1 <−

−A

, so that the transversality condition is

always satisfied. In turn, we assume δρ +>A [which is condition (17a)] so that 01 >+

−A

δρ.

Therefore, the binding lower boundary for θ~

is A

δρ+−1 which is positive.

24

APPENDIX C: Transitional Dynamics of (20).

The characteristic equation of the Jacobian evaluated at the steady state is:

0~~)

~1(

~~)

~1(

222

223 =−

−

−++−

φψνθρ

εφ

ψνθρρεε

AA (C.1)

where ε is an eigenvalue. The coefficient on ε2 is negative, the coefficient on ε is

positive, and the constant term is negative (i.e. 0~~)

~1(

2<

−−

φψνθρA

). That is, there are

three sign alterations in (C.1). Now, we use Descartes’ Theorem (see e.g. Azariadis

[1993]) which states that the number of positive roots cannot be higher than the

number of sign alterations. Hence, we cannot have more than three positive roots.

Next define ′ ≡ −ε ε . In this case, there are no sign alterations in (C.1). Hence, we

cannot have a negative root. Therefore, there are three positive roots and local

determinacy.

APPENDIX D: From equations (22a)-(22f) to equations (23a)-(23f).

If kc

z′

≡ , (22d) and zAkk

−=′′

•

give (23a). If k ′≡ µψ , (22e) and zAkk

−=′′

•

give (23c). If Nµφ ≡ , (22e) and (22f) give (23d). Also using these definitions, (23b)

follows directly from (22b), while (23e) follows directly from (22a). Finally, if

cλχ ≡ , (23f) follows from the combination of (22c) and (22d).

APPENDIX E: Proof of Proposition 3.

We work as in Appendix B. A well-defined steady state requires: (i)

0>− ρA . This is required for the economy to grow. (ii) δρ −< 2A . This is

required for the transversality condition (22g) to hold. (iii) δ>A . This follows from

inspection of (24c)-(24d) in the text. (iv) δρ+<A . Again, this follows from

inspection of (24c)-(24d) in the text. (v) 0~12

<

−+

τ

ρρ . This is required for ε~ to

be positive and so the second-order conditions to be satisfied (this is because the

27 In particular, δρ 2> implies AA 2

11δρ

−<− and also AAδρ

−<− 11 . Therefore, the binding upper

boundary for θ~

is Aρ

−1 .

25

associated constraint, (12a), is concave in τ ). Now, it can be easily seen that if

conditions (25a)-(25c) in Proposition 3 hold, then conditions (i)-(iv) are satisfied.

Also, condition (v) is satisfied for 1~0 <+1

<<ρ

ρτ .

Next, consider (24b), which is an equation in τ~ only. If 0~12

<

−τρ

, i.e. if

ρτ << ~0 , the right-hand side of (24b), defined as )~(τT , is monotonically

decreasing in τ~ . That is, there is a unique τ~ as shown in Figure 2 below.

Figure 2 here

But if ρ

ρτ

+1<~ , then it must also be ρτ <~ . Therefore, the solution of

(24b) should lie in the region 11

~0 <+

<<ρ

ρτ .

APPENDIX F: Transitional Dynamics of (28).

The characteristic equation of the Jacobian evaluated at the steady state is

written as:

0012

23

34

45

56 =++++++ HHHHHH ηηηηηη (F.1)

where η is an eigenvalue and 012345 ,,,,, HHHHHH are coefficients. Since these

coefficients are too complicated, here we will just sketch the solution to save on space

(all details are available upon request). It is convenient to work with 5H . If 05 <H ,

we cannot draw any conclusions, so everything is possible to happen. On the other

hand, if 05 >H , then 0,0,0,0 0123 <><> HHHH . If, in addition, 04 <H , there

are five sign alterations in (F.1) and by using Descartes’ Theorem, it follows that there

are five positive and one negative root. That is, there is determinacy. Note that

05 >H and 04 <H , when 31

>ρ and

−+

−+

−> τε

δρτρ

ρ

ρτεν

φ ~~1~3

,13

~~max

~ 2

AA

A.

On the other hand, if 05 >H and 04 >H , there are three sign alterations in

(F.1) and by using Descartes’ Theorem, it follows that there are three positive and

three negative roots. That is, now there is indeterminacy.

26

Note that 31

>ρ , which is required for determinacy, implies a value of the

discount factor, ρ , that is too high. Hence, indeterminacy seems to be the likely case.

27

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29

FIGURE 1

)(θR )(θL

)(θL

)(θR

0 α

δρ+−1 θ

~

αρ

−1 1

30

FIGURE 2

)~(τT

α+1

0 τ~ τ