Grimaud, André

English

Research Papers in Economics

Note on the Schou’s Paper : “When Environmental
Policy is Superﬂuous : Growth and Polluting
Resources”
Andr´ e Grimaud1
January 2004
1GREMAQ, IDEI and LEERNA, Universit´ e de Toulouse I, 21 all´ ee de Brienne,
31000 Toulouse, France, and ESCT.Abstract
In a growth model with polluting resources, Schou shows that the pol-
lution externality does not distort the decisions of the market economy, so
that a speciﬁc environmental policy is superﬂuous. We show that this result
is not generally true, and that it stems from the particular speciﬁcations
chosen by Schou.
11 Introduction
In a recent paper, Schou (2002) studies an endogenous growth model in
which a non renewable resource gives rise to pollution. He obtains a very
stimulating result along with the pollution externality does not distort the
decisions of the market economy, so that a speciﬁc environmental policy is
superﬂuous.
The purpose of the present paper is to show that this result is not gene-
rally true, and that is stems from the particular speciﬁcations of the model
chosen by Schou.
To obtain more general results, we study a non speciﬁed version of the
Schou’s model in which we characterize the optimal paths and the equi-
librium ones. Comparing the two shows that an environmental policy is
generally necessary to implement the optimum.
The non speciﬁed model is studied in section 2. In section 3, we consider
two speciﬁed models : the model of Schou and a model with a separable
utility function.
2 The non speciﬁed model
The homogenous good is produced according to the technology (for
convenience, the time index is suppressed)
Y = F(
 A
0
f(xi)di,LY ,R). (1)
LY is the amount of labour, and R is the ﬂow of non-renewable resource; xi
is the amount of intermediate good i, A is the number of existing types of
intermediate goods (the stock of knowledge) and X =
 A
0 f(xi)di is an index
in which one assumes f  > 0a n df

< 0. The partial derivatives FX,F L and
FR are positive. In the R&D sector, knowledge is produced along with
˙ A = q(LA,A),q L > 0,q A > 0, (2)
and LA + LY = 1 (normalized labor force).
The natural resource is extracted without cost from an initial stock So,
and we have the standard resource law of motion :
˙ S = −R. (3)
Pollution is generated by the use of the natural resource within the
production process
P = h(R) ,h   > 0. (4)
2The homogenous good is used both for consumtion, C, and investment
in intermediate goods; one unit of homogenous good can be transformed
into one unit of intermediate good, that gives
Y = C +
 A
0
xidi. (5)
Finally, the utility of the representative household is
 ∞
0
U(C,P)e−ρtdt, UC > 0,U P < 0,ρ>0. (6)
2.1 Welfare
The social planner maximizes
 ∞
0 U(C,P)e−ρt, subject to (1), (2), (3),
(4) and (5). After elimination of the costate variables, the ﬁrst-order condi-
tions reduce to the three following characteristic conditions (see Appen-
dix A) :
Fxi = FXf (xi)=1 , for all i, (7)
that implies xi = x ∀ i, and that allows to note Fx = Fxi, ∀ i.
ρ −
˙ UC
UC
=
˙ FL
FL
−
˙ qL
qL
+
qL
FL
(FXf(x) − x)+qA. (8)
˙ FR
FR
−
UPh 
UCFR
(ρ −
•
UPh 
UPh )=
˙ FL
FL
−
˙ qL
qL
+
qL
FL
(FXf(x) − x)+qA. (9)
Condition (7) (which is the “static eﬃciency” condition (3) of Schou)
says that the marginal productivities of intermediate goods are equal to their
marginal cost, that is to say to one. Equations (8) and (9) are respectively
the conditions of Ramsey-Keynes and of Hotelling. In these conditions, the
RHS (let us denote it H) replaces the marginal productivity of physical
capital (“FK”) of the standard neo-classical model. In order to interpret
(9), we multiply the two sides by FR and we consider an interval of time
(t,t +∆ t).
The term ˙ FR(t)∆t is the increase in production (Y )o n( t,t+∆t)i fo n e
unit of resource is kept in the ground at t and if it is extracted at t +∆ t.
The term FR(t)H(t)∆t is the increase in Y if one unit of resource is
extracted at t and if it is (indirectly) embodied in the R&D sector to produce
more Y.We use the expression “indirectly” because extracting more resource
allows to liberate labor from the ﬁnal good sector (LY decreases), then to
increase labor in R&D (LA increases), and ﬁnally to increase A and Y.
3If there is no pollution, the standard Hotelling condition is ˙ FR = FRH.
In the present model, extracting means also polluting, and extracting at t
or at t +∆ t means also polluting at t or at t +∆ t.
The term [UPh (ρ −
•
UPh /UPh )/UC]∆t is the quantity of consumption
good that has to be given to the household to compensate him for the
transfer of pollution of one unit of resource from t to t +∆ t. Thus the
diﬀerence ˙ FR∆t−[UPh (ρ−
•
UPh /UPh )/UC]∆t is the net increase of good
(after compensation of households) resulting from the transfer of one unit of
extraction from t to t+∆t. At optimum, it is equal to FRH∆t, that explains
the Hotelling condition (9).
2.2 Equilibrium
At equilibrium, there are three market failures (see Schou, p.613). Thus
we use three tools : a subsidy to research (σ), a demand subsidy for the
intermediate goods (s), and a tax on the resource (θ)( s i n c eP and R are
linked by a functional relation, it is equivalent to tax P or R). The price of
good Y is normalized to one, and p = pi ∀ i,w,pR and r are, respectively, the
price of intermediate goods, the wage, the resource price, and the interest
rate.
In the homogenous good sector, the proﬁt is
ΠY = F(
 A
0
f(xi)di,LY ,R) −
 A
0
p(1 − s)xidi − wLY − pR(1 + θ)R.
The ﬁrst order conditions of the maximization are
Fxi − p(1 − s)=0 , where Fxi = FXf (xi), for all i (10)
FR − τpR =0 , where τ =1+θ, (11)
FL − w =0. (12)
From (10), one gets xi = x, for all i. Moreover, diﬀerentiating (10) with
respect to xi gives the slope of the demand of any intermediate good i :
∂xi
∂pi
=
1 − s
Fxx
, where Fxx =
∂2F
∂x2
i
, ∀ i. (13)
The proﬁt of the monopolist which produces xi = x, ∀ i, is
Πm =( p − 1)x =(
Fx
1 − s
− 1)x.
Using (13), the maximization of Πm leads to
Fxx
1 − s
x +
Fx
1 − s
− 1=0 , (14)
4that gives the proﬁt at its maximum level,
Πm =
−x2Fxx
1 − s
. (15)
The value of the ﬁrm at t is Vt =
 ∞
t Πm
s e−
 s
t rududs. Diﬀerentiating with
respect to t and rearranging gives rt = ˙ Vt/Vt +Π m
t /Vt.
In the research sector, the proﬁt is Vq(LA,A)−w(1−σ)LA. The maxi-
mization with respect to LA gives
V =
1 − σ
qL
w. (16)
In the resource sector, we have the standard equilibrium “Hotelling ru-
le” :
˙ pR
pR
= r. (17)
Finally, the maximization of the intertemporal utility gives
ρ −
˙ UC
UC
= r. (18)
Now we eliminate prices in order to obtain conditions which can be
compared to (7)-(8)-(9).
From (14), one gets
Fx =( 1− s) − xFxx (19)
which can be compared to (7).
From (12) and (16), one gets V =( 1− σ)FL/qL, that gives ˙ V/ V =
˙ FL/FL− ˙ qL/qL. Using (15) and (16), we have r = ˙ V/ V+Πm/V = ˙ FL/FL−
˙ qL/qL − x2FxxqL/(1 − s)(1 − σ)FL.
Finally, using (18), one gets
ρ −
˙ UC
UC
=
˙ FL
FL
−
˙ qL
qL
−
x2FxxqL
(1 − s)(1 − σ)FL
, (20)
which can be compared to (8).
From (11) and (17), we have ˙ FR/FR =˙ τ/τ + r. Then, using (18) and
(20), one gets
˙ FR
FR
−
˙ τ
τ
=
˙ FL
FL
−
˙ qL
qL
−
x2FxxqL
(1 − s)(1 − σ)FL
, (21)
which can be compared to (9).
52.3 Optimal policies
Putting together (7)-(8)-(9) and (19)-(20)-(21) allows to compute the
optimal tools.
From (7) and (19), one gets the demand subsidy for intermediate goods :
s = −xFxx. (22)
From (8) and (20), one gets
qL
FL
(FXf(x) − x) − qA =
x2FxxqL
(1 + xFxx)(1 − σ)FL
, (23)
which allows to obtain σ, the subsidy to research.
Finally, from (9) and (21), one gets the rate of growth of the tax τ :
˙ τ
τ
=
UPh 
UCFR
(ρ −
•
UPh 
UPh ). (24)
There is no reason to think that the expression between brackets in (24)
is nil. This implies that it is generally necessary to conduct an environmental
policy to implement the optimal path. Let us note that this policy concerns
only the rate of growth of the tax, and not its level. We come back on this
point below.
3 Speciﬁed models
Here we consider two diﬀerent speciﬁed models : ﬁrst, the model of
Schou; second, a model with an additive utility function.
3.1 The Schou’s model
We consider the following speciﬁcations :
Y = B(
 A
0
xα
i di)L
β
Y Rγ (B>0,α>0,β>0,γ>0,α+ β + γ =1 ) ,
•
A = DLAA (D>0,A(0) = A0 > 0),P = ζRδ (δ>0,ζ>0)
and U(C,P)=[ ( CP−λ)1−ε − 1]/(1 − ε)( ε>0,ε =0 ,ρ>0,λ>0).
Using the three conditions (7)-(8)-(9) which characterize any optimal
path, it is easy to ﬁnd again the solution obtained by Schou (see Appen-
dix B) :
gR =
D(1 − ε) − ρ
λδ(1 − ε)+ε
(25)
6which corresponds to (13) in Schou,
gY =
D[δλ(1 − ε)+1 ]− ρ
λδ(1 − ε)+ε
(26)
which corresponds to (14) in Schou,
and gA =
D[(β + γ)δγ(1 − ε)+β + γε] − βρ
(γ + β)[δλ(1 − ε)+ε]
, (27)
which corresponds to (12) in Schou.
Then, from (22)-(23)-(24), one gets the optimal tools :
s =1− α (28)
σ =
(β + γ)[δλ(1 − ε)+ε] − β[ρ/D − (1 − ε)]
(β + γ[δλ(1 − ε)+ε]+γ[ρ/D − (1 − ε)]
(29)
and
˙ τ
τ
=0 . (30)
(30) conﬁrms the result of Schou. In this particular case, the term bet-
ween brackets of (24), ρ−(
•
UPh /UPh ), is nil, that explains why it is unne-
cessary to conduct environmental policy.
3.2 A model with a separable utility
Now we consider an example that it is exactly the same than Schou,
except for the utility function which is separable. We retain the speciﬁcation
of Aghion and Howitt (1998) : U(C,P)=C1−ε/(1−ε)−P1+ω/(1+ω),ε>
0,ω>0 (see Appendix C).
In this case, the term between brackets in (24) is
ρ −
•
UPh 
UPh  = ρ −
(δ(ω +1 )− 1)(D(1 − α) − βρ)(1 − ε)
δ(ω +1 ) ( β(ε − 1) + 1 − α) − γ(1 − ε)
,
which generally is not nil. This implies that the rate of growth of the optimal
tax is diﬀerent of zero.
We show in Appendix C that the rate of growth of the optimal tax, ˙ τ/τ,
can be positive or negative. Note that in the particular case δ =1 , that
is to say if pollution is a linear function of the ﬂow of extracted resource
(P = ζR), we have ˙ τ/τ <0.
A second basic result is that the tax level has no eﬀect on the equili-
brium, except that it leads to a rent transfer from the resource sector to
the government. To prove this result, let us consider a steady state where
the tax is τt = τ0egτt : τ0 is the level of the tax and gτ its rate of growth.
From (11), one gets τtpRt = FR = γBAtxα
t (LY )βR
γ−1
t . Thus an increase in
τ0 leads to a decrease in pRt, such that the product τtpRt is unchanged.
74 Conclusion
In a model which is exactly the model of Schou, but without particu-
lar speciﬁcations, we have characterized the optimal and the equilibrium
paths. Then we have given the general expression of the rate of growth of
the optimal tax on the natural resource. We have veriﬁed that, with the spe-
ciﬁcations of Schou, this rate of growth is nil. Finally, in the case of a non
separable utility, we have shown that the level of the tax does not matter
and that its rate of growth can be positive or negative.
8Appendix A : Welfare
The hamiltonian of the social planner’s program is :
H = U{F[
 A
0
f(xi)di,LY R] −
 A
0
xidi,h(R)} + µq(A,L − LY ) − νR.
The ﬁrst-order conditions ∂H/∂xi =0 ,∂H/∂L Y =0 , and ∂H/∂R =0
yield
UC(FXf (xi) − 1) = 0, ∀ i (A.1)
UCFL − µqL +0 ( A . 2 )
UCFR + UPh (R) − ν =0 . (A.3)
Moreover, ∂H/∂A = ρµ − ˙ µ and ∂H/∂S = ρν − ˙ ν yield
UC[FXf(x) − x]+µqA = ρµ − ˙ µ (A.4)
ρν − ˙ ν =0 . (A.5)
From (A.1), one gets xi = x ∀i and FXf (x)=1 , that is the static
eﬃciency condition (3) obtained by Schou. Log-diﬀerentiating (A.2) and
using (A.4) gives the Keynes-Ramsey condition :
ρ −
˙ UC
UC
=
˙ FL
FL
−
˙ qL
qL
+
qL
qL
(FXf(x) − x)+qA.
Log-diﬀerentiating (A.3) and using (A.5) gives after some calculations
the Hotelling condition :
ρ −
˙ UC
UC
=
˙ FR
FR
−
UPh 
UCFR
(ρ −
•
UPh 
UPh ).
9Appendix B : The Schou’s model
Equations (1), (2), and (4) are now Y = B(
 A
0 xα
i di)L
β
Y Rγ, ˙ A = DLAA
and P = ζRδ, that gives at steady state : gY = gA +αgx +γgR,g A = DLA,
and gP = δgR.
Moreover, from the utility function U(C,P)=[ ( CP−λ)1−ε − 1]/(1 − ε),
one gets ˙ UC/UC = −εgc − λ(1 − ε)gP.
Then, by eliminating gx, the eﬃciency condition (7) becomes
gY = gA + γgR/(1 − α).
The Keynes-Ramsey condition (8) is
ρ + εgY + λδ(1 − ε)gR = gY + D(1 − α)gA/β.
Finally, the Hotelling condition (9) is
ρ + εgY + λδ(1 − ε)gR = gY − gR +
λδ(1 − α)
γ
[ρ − (1 − ε)gY +( δλ(1 − ε)+1 ) gR].
These three equations allow to compute the three rates of growth, gY ,g A
and gR, given by (25)-(26)-(27), which are the ones obtained by Schou.
Then, from (22)-(23)-(24), one gets the optimal policies (28)-(29)-(30).
10Appendix C : Separable utility function
Now the utility function is U(C,P)=C1−ε/(1−ε)−P1+ω/(1+ω), that
gives ˙ UC/UC = −εgc.
As previously, the eﬃciency condition is gY = gA + γgR/(1 − α).
The Keynes-Ramsey condition is
ρ + εgY = gY + D(1 − α)/β +( 1− α)gA/β.
Finally, the Hotelling condition is
ρ + εgY = gY − gR +
δ(1 − α)(ζRδ)ω+1
γC1−ε [ρ − (δ(ω +1 )− 1)gR],
that implies at steady state δ(ω +1 ) gR =( 1− ε)gY .
As in the previous case, one obtains the three rates of growth gY ,g R and
gA :
gY =
δ(ω +1 ) [ D(1 − α) − βρ]
δ(ω +1 ) ( β(ε − 1) + (1 − α)) − γ(1 − ε)
,
gR =
(1 − ε)gY
δ(ω +1 )
, and gA =
(1 − α)δ(ω +1 )− γ(1 − ε)
δ(1 − α)(ω +1 )
gY .
Since UPh  = −δζω+1Rδ(ω+1)−1, we have
•
UPh /UPh  =[ δ(ω +1)−1]gR.
After substitution, one gets
•
UPh 
UPh  =
(δ(ω +1 )− 1)(1 − ε)(D(1 − α) − βρ)
δ(ω +1 ) ( β(ε − 1) + 1 − α) − γ(1 − ε)
,
which is diﬀerent of ρ.
If D(1 − α) − βρ > 0a n dε>1, one gets gY > 0a n dgR < 0. Then the
sign of
•
UPh /UPh  is given by the sign of 1 − δ(ω +1 ) .
If δ>1/(ω +1 ) , we have
•
UPh /UPh  < 0. In this case, the diﬀerence
ρ −
•
UPh /UPh  is positive, that implies ˙ τ/τ <0.
If δ<1/(ω +1 ) ,
•
UPh /UPh  is positive and two cases can occur. If
ρ−(δ(ω+1)−1)gR is positive, one gets ˙ τ/τ <0. In the other case, we have
˙ τ/τ >0.
Finally, we can observe that if δ =1 , t h a ti st os a yi fP = ζR (pollution
is a linear function of the ﬂow of extraction), the optimal tax decreases along
time : ˙ τ/τ <0.
11References
Aghion, P. and P. Howitt (1998), “Endogenous Growth Theory”, The MIT
Press (1998).
Schou, P. (2000), “Polluting Non-renewable Resources and Growth ”, Envi-
ronmental and Resource Economics, 16, 211–227.
Schou, P. (2002), “When Environmental Policy is Superﬂuous : Growth and
Polluting Resources ”, Scandinavian Journal of Economics, 104(4), 605–
620.
12

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