Ž .Economic Modelling 19 2002 487�507

Environmental policy in an endogenousgrowth model with human capital and

endogenous labor supply

Walid OueslatiTHEMA, Uni�ersite Paris X Nanterre, 200 a�. de la Republique, 92001 Nanterre cedex, France´ ´

Accepted 20 April 2001

Abstract

This paper analyses environmental fiscal policy within a two-sector endogenous growthmodel with elastic labor supply. Pollution is modeled as a side product of production. Theframework allows us to analyze the consequences of an environmental tax on the economicdynamics. Both transitional dynamics and balanced growth path are computed and theresponse to an environmental tax change is explored. Short- and long-run welfare costs arealso computed. We show that an environmental tax change induces a sharp contrast betweenshort- and long-run effects. The magnitude of this contrast depends on the agents’ aptitudeto substitute studying time for leisure. � 2002 Elsevier Science B.V. All rights reserved.

JEL classifications: E62; I21; H22; Q28; O41; D62

Keywords: Endogenous growth; Human capital; Environmental externality; Environmental tax; Transi-tional dynamics; Welfare cost

1. Introduction

Recent growth theory has made progress in analyzing the dynamic effects oftaxes. Until recently, economic models that could offer insight into this questionwere lacking. The bulk of the growth literature focused on steady states withconstant per capita output, whilst those that did consider sustained growth focusedon exogenous trends. By definition, such taxation cannot impact on this long-run

Ž .E-mail address: walid.oueslai@u-paris10.fr W. Oueslati .

0264-9993�02�$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 2 6 4 - 9 9 9 3 0 1 0 0 0 7 4 - 8

( )W. Oueslati � Economic Modelling 19 2002 487�507488

exogenous growth path. It is only since the development of endogenous growththeory that a tool has existed for investigating how taxation affects growth. Thesenew models explicitly model the processes through which growth is generated and,by doing so, can trace out the effects of taxation upon the underlying individualdecisions. Thus, taxation incidences on growth can be rigorously understood andpredicted. This is also true for efficient instruments, such as a Pigouvian tax, thatinternalize environmental externalities.

How environmental tax affects economic growth is an ambiguous issue. In thesimplest endogenous growth model, the AK model, the growth effect of environ-

Ž .mental policy is negative. This is shown by both Gradus and Smulders 1993 for acentrally planned economy with varying pollution weight in the utility function and

Ž .Ligthart and van der Ploeg 1994 for a decentralized economy. In the literature onendogenous growth with human capital, it is shown that a tighter environmentalpolicy might have a stimulating growth effect. In a Uzawa-Lucas set-up augmented

Ž .with an explicit treatment of the environment, Gradus and Smulders 1993 findthat the optimal growth rate is independent of environmental care. Only byassuming that pollution also negatively affects the efficiency in the human-capitalsector did they detect positive growth effects.

Ž .Bovenberg and Smulders 1995 consider a two-sector model consisting of aconsumption�capital good and a research and development sector generatingknowledge about pollution-augmenting techniques. Since better environmentalquality improves factor productivity in the consumption�good sector, positivegrowth effects of a tighter environmental policy are possible. In a pure humancapital variant of the two-sector Lucas model, van Ewijk and van WijnbergenŽ .1994 also find positive growth effects of a tighter environmental policy byassuming that pollution negatively affects the production process.

Hence, the existing literature can only explain positive growth effects of tighterenvironmental policy by assuming direct positive productivity effects � positiveenvironmental externalities in production � either in the education or in theconsumption�good sector.

In contrast with this conclusion, we show in this paper that, in a two-sectorendogenous growth model with leisure, a higher environmental tax might affect thelong-run growth rate. The reason for this is as follows. Due to an increasedenvironmental tax, firms increase their abatement activities, which reduces finaloutput net of abatement at the expense of households’ consumption. Householdssubstitute education time for leisure time so as to counteract reduced consump-tion, and this finally boosts growth.1

Whereas most endogenous growth models dealing with environmental concernsrestrict the analysis to the steady state, little has been said so far on the short-runeffects of taxation. There are a few exceptions in the literature. Ligthart and van

Ž .der Ploeg 1994 derive the transitional dynamics of linear growth model aug-mented with a renewable environmental resource. By increasing the disutility

1 Ž .In a similar model, Hettich 1998 shows analytically that changes in leisure may create a linkbetween growth and environment, but he does not study the short-run effects in this structure.

( )W. Oueslati � Economic Modelling 19 2002 487�507 489

parameter of pollution of the representative agent, they find that the fall in theshort-run growth rate in the centrally planned economy is bigger than the long-rungrowth rate. Note that, in this linear framework, the evolution of the environmen-tal stock is responsible for and solely determines the transitional dynamics of the

Ž .economy. Bovenberg and Smulders 1996 analytically compute the transitionaldynamics of a two-sector model consisting of a consumption�capital goods sectorand a research and development sector that generates knowledge about pollution-augmenting techniques. The renewable environmental resource acts both as publicconsumption and as a public input into production, where the latter is identical toa productive environmental spillover. They find that if the environment acts mainlyas a consumption good, then a tighter environmental policy reduces growth in boththe long- and short-run. But if the environment acts mainly as a public investmentgood, then long-run growth rises, while short-run growth declines.2

In this paper, we compute the entire dynamic adjustment path towards abalanced growth path. Our analysis of the transition reveals the short-run impactof environmental tax. Furthermore, this analysis of the dynamic adjustment pathenables us to perform welfare calculations. In particular, we make explicit thetrade-off between the short- and long-run costs of environmental policy.

The remainder of the paper is organized as follows. In Section 2 the generalmodel is laid out and both market and central-planner solutions are derived. Theoptimal environmental tax rate in a first-best setting is analyzed. Section 3proposes a numerical exercise: we calibrate the model at the steady state, computethe transitional dynamics and comment the short-run dynamics. Section 4 com-putes short- and long-run welfare costs of taxation. Section 5 summarizes the mainfindings.

2. The model

We consider an economy populated with an infinitely-lived, representativehousehold. The household owns the stock of physical capital in the economy, K ,t

Ž .and is endowed with a normalized unit time. The time endowment can beŽ .allocated between work remunerated at the current competitive wage rate ,

leisure and schooling. The capital stock used in production causes a negativeenvironmental externality as a side product. Pollution is assumed to affect individu-als’ utility. We introduce government in minimal fashion; its task consists solely ofcorrecting the market failure caused by the environmental externality.

2.1. Preferences, technology and pollution

The behavior of the rational household is guided by the maximization of the

2 Ž .Without taking the environment into account, Mulligan and Sala-i-Martin 1993 , Devereux andŽ . Ž .Love 1994 and Ladron-de-Guevara et al. 1997 investigate the transitional dynamics within similar

models.

( )W. Oueslati � Economic Modelling 19 2002 487�507490

discounted lifetime utility:

�t Ž . Ž .W � � u C ,l ,P 1Ý0 t t t

t�0

where

Ž . Ž .u C ,l ,P � logC � � log l � � log P 2t t t t l t P t

and C is consumption, l represents hours spent away from leisure, 0 � � � 1 ist tthe discount factor and P is the net pollution flow. The parameters � and �t l Prepresent the weights of leisure and pollution in utility. The consumer budgetconstraint can be written as follows:

Ž . Ž .K � 1 � r � � K � w u H � C � T 3t t K t�1 t t t�1 t t

where r is the return to physical capital and w is the gross wage rate per effectivet tunit of human capital u H . T represents transfers from the public sector, and ut t�l t tis the supply of working time. � denotes the rate of depreciation for physicalKcapital.

The representative agent can increase his human capital stock, H , by devotingttime to schooling. We assume that this activity takes place outside the market, andnew human capital can only be obtained by spending time. Thus, the law of motionfor human capital is given by the constraint:

Ž . Ž .H � H � B� � � H 4t t�1 t H t�1

where B is the marginal productivity of schooling time � and � denotes the ratet Hof human capital depreciation.

Ž .The household is endowed with a normalized time unit, which can be allocatedto work, leisure or schooling:

Ž .1 � u � � � l 5t t t

The physical capital used in production is the source of the pollution flow P. Thisflow can be reduced by means of private abatement activities D, which in turnconsume a part of output, in line with the flow resource constraint. The netpollution function has the form:3

�Kt�1 Ž .P � 6t ž /Dt

where � � 0 is the exogenous elasticity of P with respect to K�D.

3 Ž .The same specification is used by Gradus and Smulders 1993 .

( )W. Oueslati � Economic Modelling 19 2002 487�507 491

2.2. Firms

The economy consists of a large number of identical and competitive firms. Theyrent capital and hire effective labor from the households at the interest rate r andthe wage rate w, respectively. They use the following constant-returnsCobb�Douglas technology:

1��Ž . Ž .�Y � AK u H 7t t�1 t t�1

where A � 0 and 0 � � � 1.Firms must pay a pollution tax P according to their net pollution P . Abate-t t

ment is assumed to be a privative good, which enables firms to increase outputwithout causing more pollution. Firms are assumed to maximize their market value,which is equal to the appropriately discounted sum of profit flows, with the lattergiven by:

� Y � r K � w u H � D � PPt t t t�1 t t t�1 t t t

Profits maximization implies that, in equilibrium, firms pay each production factorat its marginal productivity.

Y Pt tP Ž .r � � � � 8t tK Kt�1 t�1

YtŽ . Ž .w � 1 � � 9t u Ht t�1

P Ž . � P � D 10t t t

Without a pollution tax, firms would neglect the negative side product of physicalcapital in the production process and abatement activities would be zero. Themarket clearing condition for the goods market is:

Ž . Ž .Y � C � D � K � 1 � � K 11t t t t K t�1

The government budget constraint implies that all revenue is lump-sum transferredŽ P .back to households in every period P � T .t t t

2.3. The market solution

Definition 1. A competitive equilibrium for this economy is a set of allocations� 4 � 4 PC , D , u , l , K , H , P , T , a price system r , w and an environmental tax ,t t t t t t t t t t

� 4such that, taking the price system and fiscal policy as given, C , D , u , lt t t tŽ . Ž . Ž . � 4maximizes Eq. 1 , subject to Eqs. 3 � 5 , and the path u , K , H , r , w satisfiest t t t tŽ . Ž .equations Eqs. 8 � 11 .

( )W. Oueslati � Economic Modelling 19 2002 487�507492

So as to characterize the competitive equilibrium, let us focus on the differenttrade-offs faced by the household. After eliminating the shadow prices for physicaland human capital, the first-order conditions for the household problem are givenby:

� Cl t Ž .l � 12t w Ht t�1

Ct�1 � � Ž .� � 1 � r � � 13t�1 KCt

C wt�1 t�1 � Ž . � Ž .� � 1 � B 1 � l � � 14t�1 HC wt t

Ž .Eq. 12 equates the marginal rate of substitution between consumption and leisureŽ . Ž .to the real wage. Eqs. 13 and 14 are the Euler conditions determining the

optimal accumulation of physical and human capital. It is obvious that environmen-tal tax affects only the intertemporal incentive to invest in physical capital, as

Ž .described by Eq. 13 .Ž . Ž . Ž . Ž .These conditions, along with Eqs. 3 � 5 , 8 � 11 constitute a dynamical system

in C, D, u, �, l, K and H which, together with the transversality conditions4 andŽ . Ž .initial K 0 and H 0 , fully describe the dynamic behavior of the economy along an

interior equilibrium.

2.4. The central-planner solution

In contrast to a market solution, the central planner maximizes the utility of therepresentative economic agent and takes pollution into account. The centralplanner maximizes lifetime utility by choosing time paths for C, D, K, H, u and l,

Ž .subject to the flow-resource constraint Eq. 11 and human capital accumulationŽ .constraint Eq. 4 . After eliminating the shadow prices, the first-order conditions of

the central-planner solution are given by:

Ž .D � � �C 15t P t

Ž .C 1 � � lt t Ž .� 16Y � ut l t

Ž . Ž .K � Y � 1 � � K � D � C 17t t K t�1 t t

4 These conditions are standard and impose that the present discounted value of both capital stockstends to zero at the infinity.

( )W. Oueslati � Economic Modelling 19 2002 487�507 493

� Ž . � Ž .H � 1 � B 1 � l � u � � H 18t t t H t�1

C Y Dt�1 t�1 t�1 Ž .� � 1 � � � � � 19KC K Kt t t

Ž .C Y � u Ht�1 t�1 t�1 t � Ž . � Ž .� � 1 � B 1 � l � � 20t�1 HŽ .C Y � u Ht t t t�1

The central-planner solution differs only from the market solution through Eq.Ž .15 . It shows that, for a social optimum, the marginal utility of consumption and

Ž .abatement must be equalized. Eq. 19 is also known as the Keynes�Ramsey rule,describing the optimal consumption path over time. The right-hand side consists ofthe private marginal product of physical, corrected by the term D�K, the deprecia-tion rate of the physical capital stock � , and the rate of time preference �. ThereKis a wedge between private and social return to physical capital. The term D�Kcan be seen as the marginal damage of physical capital. Consumption grows,remains constant, or declines if the social return to physical capital is larger than,equal to, or smaller than, respectively, the sum of the rate of depreciation and therate of time preference.

2.5. Optimal en�ironmental tax rate

To determine the first-best environmental tax, we compare the first-orderconditions of the market solution and their central-planner counterparts. Particu-

Ž . Ž .larly, by comparing Eqs. 10 and 15 , we can compute the optimal pollution-taxrule:

1��1�� 1��Ž .1 D � � Ct P top Ž . � K � K 21t t�1 t�1ž / ž /� K � Kt�1 t�1

The optimal environmental tax op must be equal to the product of the currenttŽ .physical capital stock with the optimal consumption�capital ratio C�K of the

central-planner solution. The ratio C�K is constant along a balanced growth path,but K increases over time.

For that reason, the Pigouvian tax rate must increase over time with the growthrate of the economy. This result becomes intuitive by remembering that P must beconstant along a balanced growth path. To keep the level of pollution constant, op

must rise over time, because the physical capital stock, which is responsible for thepollution, accumulates over time. Firms only increase abatement activities overtime if they have an incentive to do so via an increasing pollution tax. Therefore,we can separate trend and level of the pollution tax rate. To do so, we normalize it

op opby the physical capital stock and define � �K , which is constant along at t�1Ž .balanced growth path. Eq. 21 will be used later to calculate � .P

( )W. Oueslati � Economic Modelling 19 2002 487�507494

2.6. The balanced growth path

In this section we will focus on the dynamic properties of the balanced growthpath.

Ž . �Definition 2. A balanced growth path or steady state is an allocation C , D , u ,t t t4 � 4 P� , l , K , H , P , T , a price system r , w and an environmental tax satisfyingt t t t t t t t

Ž . Ž .Definition 1, and such that, for some initial conditions K 0 � K and H 0 � H ,0 0� 4the paths C , D , K , H , T grow at a constant rate g, and u , � , l and P remaint t t t t t t t t

constant.

For analytical convenience we use the following transformed variables: h �tP PH �K , � �K , c � C �K , y � Y �K , d � D �K and g �t t t t�1 t t t�1 t t t�1 t t t�1 t

K �K .t t�1Using this change of variables, we obtain the following dynamic system:

l � ct l t Ž .� 22Ž .u 1 � � yt t

Ž .r � � y � d 23t t

ytŽ . Ž .w � 1 � � 24t u ht t�1

1�1��PŽ . Ž .d � � 25

Ž .g � 1 � y � d � c � � 26t t t K

ht Ž . Ž .g � 1 � B 1 � u � l � � 27t t t Hht�1

ct�1 � � Ž .g � � 1 � r � � 28t t�1 Kct

c wt�1 t�1 � Ž . � Ž .g � � 1 � B 1 � l � � 29t t�1 Hc wt t

Steady-state values c, d, l, u, P and g are obtained by eliminating the index t.5

5 PThe environmental tax rate is fixed by the government. It has no a transitional dynamic.

( )W. Oueslati � Economic Modelling 19 2002 487�507 495

From the linearization of the above system, it can be shown that, independently ofthe size of tax rate, the model displays a saddle-path dynamic structure. Thus,

Žunlike other models presented in the literature Benhabib and Perli, 1994; Xie,.1994; Bond et al., 1996 , our model is unable to generate the indeterminacy

phenomenon typical of distorted economies.6

3. Numerical results

3.1. Calibration

In this section, we derive a full numerical solution for the model. For thiscalibration exercise, we cannot really hope to be as precise as those who employthe same model without environmental externality, since we lack strong empiricalevidence concerning the nature of the environmental preferences and pollutionfunction. Nevertheless, to the greatest possible extent, we follow the recentliterature.

Ž . Ž .The parameter values needed are: i preferences parameters, � , � , and � ; iil PPŽ .technology parameters � , A, B, � , � and � ; and iii environmental tax rate .K H

We proceed by choosing parameters according to the arguments below to pin downa benchmark economy.

We calibrate the model in two different steps. First, we consider that theeconomy is initially on the equilibrium growth path where polluted emissions aretaxed at a lower rate.7 To compute the steady-state variable values, we resort tocommon parameter values already used in two-sector endogenous growth models.Additionally, the calibration is carried out so as to capture a pollution abatementas a percentage of GDP of 1.8%, which corresponds to the average of environmen-tal protection expenses in OCED countries. Second, we compute the final steady-state values corresponding to the optimal rate of environmental tax. We supposethat this optimal rate is twice as high as that initially used. This optimal rate is then

� Ž .� 8used to determine the environmental preferences’ weight see Eq. 21 .

3.1.1. Parameter choiceThe calibration is carried out in order to capture a quarterly equilibrium growth

rate of 0.3%, which corresponds to 1.2% per annum. This rate is plausible for mostdeveloped countries. Our choice of parameters closely follows the literature on

6 Ž .In Bond et al. 1996 , indeterminacy emerges from the presence of taxes in a model with physicalcapital as an input in the educational sector. As we assume that physical capital is only productive in theoutput sector, the condition for general instability or indeterminacy is never satisfied. In Benhabib and

Ž . Ž .Perli 1994 and Xie 1994 , indeterminacy arises from knowledge spillovers.7 The environmental tax rate is exogenously fixed by the government. Both abatement expenses and

pollution are induced by this tax rate.8 Obviously, this is an arbitrary choice, which means that current abatement efforts do not corre-

spond to the environmental agents’ preferences.

( )W. Oueslati � Economic Modelling 19 2002 487�507496

Table 1Baseline parameter values

Parameter Value Description

� 0.99 Discount factorg 1.003 Quarterly growth rate� � � 0.0125 Depreciation rateK H� 0.25 Physical capital share in productionD�Y 0.018 Share of pollution abatement in production� 1.24 Weight of leisure in utilityl� 0.1 Elasticity of pollution with respect to the ratio K�D

simulated two-sector endogenous growth models that are similar to ours. Accord-� Ž .�ing to Barro and Sala-i-Martin 1995, p. 37 , the measured depreciation rate for

the overall stock of structure and equipment is approximately 5% per annum,which corresponds to 1.25% per quarter. Taking this as a proxy for the industrial-ized economies, the rate of depreciation of the physical and human capital stockare assumed to be � � � � 0.0125. The share of physical capital in final goodK H

Ž .production � � 0.25 is taken from Lucas 1988 . In addition, we set the discountfactor to 0.99, so that the quarterly equilibrium interest rate is equal to 1%.

Ž .Following Devereux and Love 1994 , we set the weight of leisure in utility � tolŽ .1.24. Finally, we consider a share of pollution abatement in production D�Y of 1.

8% and, in an ad hoc manner, we set the elasticity of pollution with respect to theŽ . Ž .ratio K�D , � to 0.1 Table 1 .

We suppose that, in period t � 0, the government decides to double theenvironmental tax rate in order to reach an optimal level. Thus, the share of

Ž .pollution abatement expenses D�Y � 1.8% does not match the agent’s prefer-Pences. We suppose that doubling allows reaching an optimal share D�Y.

3.1.2. ResultsWe have chosen the following variables and parameters values: � , g, � , � , � ,K H

� , and d�y. Values of the remaining parameters and variables are solutions to thelŽ . Ž . Ž .system Eqs. 22 � 29 . Initial steady-state SS1 values are summarized in Table 2.

We suppose that in period t � 0 the government doubles the environmental taxŽ .rate. This environmental policy change affects the steady state SS1 and initiates

transitional dynamics to a new steady state. During the transitional dynamics,variables grow differently, which reflects the responses of agents to the environ-

Ž .mental policy shock. The new steady state SS2 is directly deduced from theŽ . Ž . Ž .equilibrium system Eqs. 22 � 29 Table 2 .

We get a slight growth rate increase, and both consumption and leisure decrease.Both the share of pollution abatement in GDP and the ratio y � Y�K increase.Thus, environmental tax increase stimulates long-run growth rate, since laborsupply is endogenous. Therefore, an abatement increase negatively affects bothconsumption and investment. In order to compensate the decrease in the share ofconsumption in GDP, agents reduce their leisure and devote more time to

( )W. Oueslati � Economic Modelling 19 2002 487�507 497

Table 2Calibration results at SS1

Parameter Value Description

y 0.1105 Final output per unit of physical capital stockP 0.0107 Environmental tax rate per unit of physical capital stock

c 0.0930 Consumption per unit of physical capital stockB 0.0397 Human capital productivityl 0.3549 Leisureu 0.2550 Working hoursh 15.3346 H�K ratio

g y c�y d�y u lSteady-state changeSS1 1.0030 0.1105 0.8417 0.0180 0.2550 �0.3549SS2 1.0031 0.1178 0.8360 0.0317 0.2550 0.3525

Ž .variation % 0.0111 0.7430 �0.5720 1.3690 0.0024 �0.2379

schooling, which finally improves human capital accumulation and boosts thelong-run growth rate.

Additionally, the environmental tax change induces a factorial substitutionŽ .process, whereby production becomes more human capital intensive clean factor .

Notice that the working time u is almost not sensitive to the environmental taxvariation.

3.2. Transitional dynamics

To compute the transitional dynamics, we log-linearize the dynamic system Eqs.Ž . Ž .22 � 29 to make the equations approximately linear in the log-deviations from

Ž .the steady state see Appendix A . After doing this, we solve the recursiveequilibrium law of motion via the methods of undetermined coefficients.

The recursive equilibrium law of motion can be defined by two relations:

� On the one hand, a relation between state variables in t and their values inŽ .t � 1 state dynamics .

� On the other hand, a relation between jump variables in t and state variables inŽ .t � 1 jump-state dynamics .

Ž .Let us collect the state variables in the vector X , i.e. X � h , g , and the jumpt t t tŽ .variables in the vector Y , i.e. Y � y ,c ,u ,l ,r ,w ,P . The recursive equilibriumt t t t t t t t t

law of motion then becomes:

X � PPXt t�1 Ž .30Y � QQXt t�1

where PP and QQ are matrices of partial elasticities.

( )W. Oueslati � Economic Modelling 19 2002 487�507498

The linear dynamic system can be represented in the following matrix formula:

0 � AAX � BBX � CC Yt t�1 t Ž .310 � FF X � GGX � HHX � JJ Y � KK Yt�1 t t�1 t�1 t

Using the method of undetermined coefficients we can compute the PP and QQ

matrices.9

The simulation of the transitional dynamics starts in period 0, where thegovernment suddenly doubles the environmental tax rate. This environmentalpolicy shock induces an instantaneous reaction of all economic variables. We then

Ž .observe different impacts on the variables, which leave their initial level at SS1Ž .and reach their new level at SS2 at different rates. Table 2 summarizes the

changes in values.In fact, the environmental tax change induces three effects:

� A crowding out effect caused by the increase in abatement expenses, whichnegatively affects consumption and investment.

� A factorial reallocation effect, which reduces the intensity of physical capital inproduction.

� A reallocation of available time, whereby schooling time increases.

Ž .The pace at which the economy reaches the new steady state SS2 is the resultof the interaction between these three effects. In the short run, the stock ofphysical capital decreases, but inherits an increased trend after a while, and finallyits growth rate reaches a new level on SS2, which is slightly higher than its initialSS1 level.

The short-run behavior of the economy is described by the growth-rate transition-Ž .al dynamics Figs. 1 and 2 .

A higher environmental tax reduces the physical capital�human capital ratioŽ .k�h , because the clean input factor H is substituted for the dirty input factor K.

� Ž .The productivity of physical capital increases and boosts growth see Eqs. 8 andŽ .�28 . However, there is also an effect working in the opposite direction: a higherenvironmental tax reduces the productivity of physical capital and lowers growth.Due to the increased environmental tax, firms increase their abatement activities,which reduces final output net of abatement at the expense of households’consumption. Households increase their marginal utility of leisure by substituting

Ž .schooling time for leisure to counteract reduced consumption. From Eq. 27 , itcan be seen that a higher studying time enhances growth. A higher environmental

Ž .tax increases the ratio d�y, and hence lowers pollution see Fig. 1d and Fig. 2f .In the beginning of the transitional dynamics, the crowding out effect of

Ž .abatement reduces both the growth rate Fig. 1a and the ratio of physicalŽ .capital�production Fig. 1b . As mentioned before, increased environmental tax

9 We use the Matlab routines to compute PP and QQ. Starting from this solution, the time series caneasily be reconstituted from the SS1 variable values.

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Fig. 1. Transitional dynamics 1.

leads to a more human capital intensive final output and to a higher studying time.The immediate response to the environmental tax increase is a sectorial realloca-tion of resources, which reduces the physical capital�human capital ratio. House-

Ž .holds reduce their leisure time Fig. 1f and increase their studying time. The timespent at work decreases in the beginning of the transitional dynamics, but increases

Fig. 2. Transitional dynamics 2.

( )W. Oueslati � Economic Modelling 19 2002 487�507500

Fig. 3. Welfare decomposition.

after some periods to reach a slightly higher level than before. As soon as thesectorial reallocation becomes important, the crowding out effect is reduced. As afinal result, the higher environmental tax boosts long-run growth.

The main conclusion to be drawn is that an unanticipated increase in theenvironmental tax leads to a slight increase in the long-run growth rate, but alsoleads to a negative physical capital growth rate in the short run. The aptitude ofagents to substitute their schooling time for leisure determines the amplitude ofthe short- and long-run effects.

4. Welfare costs

The results presented in the previous section demonstrate the role played by anenvironmental tax change in the dynamic behavior of the economy. In this section,we compute the welfare cost associated with this environmental tax change. Wesuppose that at t � 0, the economy transits from an initial optimal steady state

P op 10Ž . Ž . Ž . Ž .SSop with � to a non-optimal situation SSnop Fig. 3 . Evidently, thisenvironmental policy shock changes the steady state, initiates transitional dynamicsand generates a welfare cost. Two types of welfare variations can be distinguished:a first variation is associated with the short-run dynamics, and a second is relatedto the change of steady state.

4.1. Welfare decomposition

ŽWe decompose welfare into transitional welfare also referred to as the short-run. Ž . Ž .welfare W corresponding to the economy’s transition from SSop to SSnop ,1� 2

and welfare related to the new steady state W . To obtain a numerical result, we2

10 Ž .We supposed earlier that, by doubling the environmental tax, an optimal ratio D�Y could bereached. This assumption allows us to calculate � .P

( )W. Oueslati � Economic Modelling 19 2002 487�507 501

suppose that the transition from one steady state to another is achieved in a finitenumber of periods, and we simply denote T as the date at which we consider thatthe economy has numerically reached its new rest point. The total welfare associ-ated with the environmental policy change W Tot is equal to the sum of utility flowsfrom t � 0 to �, which can be written as the sum of W and W :1� 2 2

T ot Ž .W � W � W 321� 2 2

Note that the economy converges only asymptotically to the steady state, and wetherefore truncate the transitional dynamics in the effective computation at thehorizon T. This horizon is chosen so that, for all t � T , the difference between the

Ž . Ž .value of physical capital stock at T k and its value at SSnop k is numericallyT 2very small.11

Formally, the transitional welfare can be written as:12

T t�1t Ž . Ž . Ž .W � � log c � log g � � log l � � log P 33Ý Ý1� 2 t i l t P t

t�0 i�0

Ž .the welfare related to the new steady state SSnop is given by:

T�1 T�P P PŽ . Ž . Ž .W � logc � log g � � log l � � log P Ý2 i l P1 � � i�0

PŽ .�log g Ž .� 34

1 � �

Ž . 13and the welfare related to the initial steady state SSop is given by:

op op op opŽ . Ž . Ž . Ž .logc � � log l � � log P �log g l P Ž .W � � 351 21 � � Ž .1 � �

Fixed at its optimal level, the environmental tax has zero welfare cost. Conversely,Ž P .the transition from a non-optimal tax rate to an optimal tax rate generates a

welfare benefit.To obtain a meaningful evaluation of the welfare cost associated to our policy

change, we express all welfare measures as a percentage point of the permanentconsumption that generates an equivalent welfare in the benchmark case. Thus,our welfare cost measures the compensation in consumption terms that leaves the

11 Ž P . �1 0We tolerate a difference between k and k smaller than 10 .T12 The formal computation of welfare decomposition is available on request.13 We assume that K � 1.�1

( )W. Oueslati � Economic Modelling 19 2002 487�507502

consumer indifferent between the non-optimal taxed stationary consumption pathand the consumption path corresponding to the optimal tax rate.14 We propose adissociation of welfare cost related to transitional dynamics from the total welfarecost.

4.1.1. Total welfare costPŽ .Let us define c as the constant flow of consumption that gives a welfare˜

Tot PŽ . Ž .W when agents work, as in the initial steady state SSop , pollution disutilityand growth rate are constant.

�P Tot P op opŽ . Ž . Ž . Ž . Ž .c � exp 1 � � W � log g � � log l ˜ l1 � �

opŽ . Ž .�� log P 36P

The total welfare cost is given by:

opŽ .c ˜Ž .� � � 1 37PŽ .c ˜

where

�op Tot op op opŽ . Ž . Ž . Ž . Ž .c � exp 1 � � W � log g � � log l ˜ l1 � �

opŽ . Ž .�� log P 38P

4.1.2. Transitional welfare costWe suppose that the economy can instantaneously jump on the new steady state

rp PŽ . Ž .without transition . Let us define c as the constant flow of consumption that˜gives the same welfare in this hypothetical scenario. We obtain:

�rp P op op opŽ . Ž . Ž . Ž . Ž .c � exp 1 � � W � log g � � log l � � log P ˜ 2 l P½ 51 � �

The welfare cost of transition expressed in consumption terms is then:

rp P PŽ . Ž .c � c ˜ ˜dyn Ž .� � 39opŽ .c ˜

14 Ž .See Hairault et al. 2001 .

( )W. Oueslati � Economic Modelling 19 2002 487�507 503

Fig. 4. Environmental tax welfare costs.

4.2. Welfare costs simulation

We propose to compute these two measures of the welfare cost for manyop op� �environmental tax ratio belonging to the interval 0.5 � , 1.5 � . When

environmental tax is equal to its optimal level, the total welfare cost is zero.Ž .Therefore, with environmental tax lower higher than the optimal rate, welfare

Ž .cost decreases increases . A ‘U-shaped’ curve can be obtained, which relates theŽ .total welfare cost and different tax rate levels near the optimal rate Fig. 4a . In the

same manner, we compute the transitional welfare cost for different environmentaltax rates near the optimal one. We obtain an increasing relation: the higher theenvironmental tax, the higher is the transitional welfare cost.

Now we propose to compute total and transitional welfare costs for differentŽ .values of the weight of leisure in households’ preferences � . Thus, for a share ofl

PŽ .abatement expenses in production of 1.8%, e.g. � 0.0107 , we obtain thewelfare costs values summarized in Table 3.

Ž .From this sensitivity analysis we note that total welfare cost � is positive and

Table 3Welfare costs sensitive to leisure preferences weight

� � 0 � � 1.24 � � 2.48 � � 3.24 � � 4.56l l l l l

Ž .� % 0.6627 0.6559 0.6487 0.6442 0.6360dyn Ž .� % �6.9142 �7.2640 �7.6361 �7.8760 �8.3157

( )W. Oueslati � Economic Modelling 19 2002 487�507504

very low. When the government applies an optimal environmental policy, theeconomy can realize a slight welfare gain.15 In the short-run, the transitionalwelfare cost is negative and relatively important. Thus, reaching the optimalenvironmental policy induces an important welfare cost in the short term.16

We note also that � has a lower sensitivity with respect to leisure preferenceŽ . dynweight � and that � has a higher sensitivity to � . When agents’ preferencesl l

for leisure are important, an increased environmental tax will induce sizableshort-run movements, which induce an important transitional welfare cost.

When government increases the environmental tax rate to reach an optimal one,the economy experiences a slight welfare gain in the long run and a relativelyimportant welfare cost in the short run. The magnitude of the welfare cost isrelated to the marginal utility of leisure. In an economy where agents give a higherweight to leisure, the short-run welfare cost will be very important.

5. Conclusion

We have studied in this paper the short- and long-run behavior of an economyresponding to environmental fiscal policy effects. The model we used is a version ofa two-sector endogenous growth model. This model allows us to consider theenvironmental tax effect on the growth rate according to the preferences’ weightfor leisure.

Our ambition was to bring further development to results in the literatureconcerning the transitional effects of an environmental tax. This literature empha-sized the effect of an environmental tax on the equilibrium growth rate by showingthat pollution affects factors productivity or human capital accumulation. Byintroducing leisure into the utility function, we have established a link betweenenvironmental tax and long-run growth rate without making the above-mentionedassumptions. Furthermore, we have contributed to the short-run dynamic study ofsuch a tax reform.

The existing calibration exercises concerning two-sector endogenous growthmodels inspired us to develop our numerical study. Two steps were necessary tocalibrate our model: first, we calculated the environmental tax rate in order to be

Ž .sure that the pollution abatement in the product D�Y is equal to 1.8%. Then, wesupposed that, to obtain an optimal level of D�Y, we should double the environ-mental tax rate.

The next step of our work was to simulate and comment on the transitionaldynamics associated with an environmental policy change. Along the transitional

15 Note that our computational method concerns a transition from an optimal to a non-optimalsituation, which induces a welfare cost. Conversely, it will induce a welfare gain when we start from thedistorted economy and reach the optimal steady state.

16 Since the transitional welfare cost is negative, it is synonymous with a welfare gain.

( )W. Oueslati � Economic Modelling 19 2002 487�507 505

dynamics, agents face different trade-offs. On the one hand, firms initiate afactorial substitution process that progressively neutralizes the crowding-out effectcaused by pollution abatement. On the other hand, households substitute educa-tion time for leisure. These effects finally combine into a higher long-run growthrate.

Our third and final step was an assessment of the welfare cost induced byvariations in the environmental tax rate. We have made a clear difference betweenthe total welfare cost and that associated with the transitional dynamics. Measuringthese different welfare costs caused by the environmental tax impact on growthconfirms the results found elsewhere in the literature, and emphasizes both thesizable welfare cost in the short-run and the overall welfare benefit in the long-run.The magnitude of the short-term cost is an increasing function of the weights ofleisure in utility. If the agents’ labor supply is elastic, an environmental tax increasecan stimulate growth, improve welfare in the long run, but causes an importantwelfare decline in the short run.

Acknowledgements

I am grateful to an anonymous referee. I wish to thank Gilles Rotillon, SjakSmulders, Alain Ayong Le Kama, Julien Matheron and Tristan-Pierre Maury forhelpful and stimulating discussions. The paper has benefited greatly from thecomments of Zeina Kouatli. All errors are my own.

Appendix A. The log-linear system

Once the system has been stationarized, we can linearize it in the neighborhoodof its rest point. Let any lower-case letter without time subscript denote the steadystate value of the associated stationary variable, and in the same manner, let alower-case letter with a hat denote the log-deviation of the associated stationary

Pvariable. We surmise that is a parameter fixed by the government. We resort toŽ .the method proposed by Uhlig 1995 , and the approximate log-linear system is

given by:

ˆ Ž .l � u � c � y � 0 E1ˆ ˆ ˆt t t t

Ž .rr � � yy � 0 E2ˆ ˆt t

ˆ Ž .w � y � u � h � 0 E3ˆ ˆ ˆt t t t�1

( )W. Oueslati � Economic Modelling 19 2002 487�507506

ˆŽ . Ž . Ž .y � 1 � � u � 1 � � h � 0 E4ˆ ˆt t t�1

Ž .gg � yy � cc � 0 E5ˆ ˆ ˆt t t

Bu Blˆ ˆ ˆ Ž .g � h � h � u � l � 0 E6ˆ ˆt t t�1 t tg g

�rŽ .g � c � c � r � 0 E7ˆ ˆ ˆ ˆt t�1 t t�1g

�Bl ˆ Ž .g � c � c � w � w � l � 0 E8ˆ ˆ ˆ ˆ ˆt t�1 t t�1 t t�1g

This log-linearized system can be written in the following matrix form:

P P PŽ . Ž . Ž .0 � AA X � BB X � CC Yt t�1 t

P P P P PŽ . Ž . Ž . Ž . Ž .0 � FF X � GG X � HH X � JJ Y � KK Yt�1 t t�1 t�1 t

Then we use Matlab routines to compute the matrices PP and QQ from the recursiveequilibrium law of motion.

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