taxes or permits, to prevent an infinite increase of pollution in a growing economy. ... consumption (C) and abatement (D), an intermediate good (x) used to pro-.
Environmental Policies, Pollution and Growth in a Model with Vertical Innovations Andr´e Grimaud1 Mars 2000
1 GREMAQ,
IDEI and LEERNA Universit des Sciences Sociales de Toulouse I. I would like to thank P. de Donder for his comments and suggestions.
Abstract
The paper introduces environmental considerations and public policies in a Schumpeterian model with vertical innovations and ”creative destruction”. It is shown that environmental policies affect growth through several partial and general equilibrium mechanisms. In particular, they affect profits in the intermediate good sector. Thus through this channel they influence the value of firms, and finally research and growth. These mechanisms lead to a trade-off between environment and growth.
1
1
Introduction
Economists are convinced that it is necessary to use specific tools, such as taxes or permits, to prevent an infinite increase of pollution in a growing economy. But they are also generally convinced that these tools have effects not only on pollution but also on many other macroeconomic variables, in particular aggregate output. The main purpose of this paper is precisely to study these types of effects, and in particular to identify the channels through which public policies can modify the output growth rate. It is obviously necessary to use an endogenous growth model to study this type of question. In this paper, we have chosen to use a schumpeterian model in which there is a “creative destruction” mechanism : vertical innovations arrive randomly and new inventions make old technologies obsolete.1 More precisely, we start from the elementary schumpeterian model presented in chapter 2 of Aghion-Howitt’s book (1998), in which we introduce environmental considerations. In fact, our model is close to the Aghion-Howitt’s chapter 5. But the two analysis are different on several points. First, Aghion and Howitt make only a welfare analysis when the main point of this paper is to study the equilibria associated to different pairs of public tools: a pollution tax and a subsidy to research. Second, we wish to present a technically simple model. For that, contrary to Aghion-Howitt, we assume that there is no accumulation of capital and that there is only one intermediate good used to produce the final output.2 The relative simplicity of the basic model allows to identify the main mechanisms by which public tools modify the steady state variables. The model is presented in section 2. As we said above, it is an extension of the elementary model of Aghion and Howitt, in which we introduce environmental considerations, as for instance Stokey (1998) and Aghion-Howitt (chapter 5). Equilibrium is analyzed in section 3. Once it is defined (see proposition 1), it is possible to study the effects of public policies, in particular on environment and growth. Some of these effects are direct, as in a partial equilibrium analysis. For instance, increasing the pollution tax induces firms to choose (more quickly) cleaner technologies, which in turn 1
This type of question has been already studied in models with horizontal innovations. See for instance Verdier (1993), Hung, Chang and Blackburn (1993), Elbasha and Roe (1996), and Grimaud (1999). 2 The model studied by P. Aghion and P. Howitt is an aggregated model which can be disaggregated by introducing vertical innovations or horizontal innovations (as in Romer (1990)). Our choice of a simple model allows to give proofs which are not too complicated: see proposition 1 and proposition 2 below. A consequence of this choice is that there are discontinuities in the path of some variables: see 3.4, remark 1.
2
decreases pollution (at a higher pace). On the other hand, these choices are costly and output growth is lower. Other effects are more complex because they are linked to the general equilibrium structure of the model. Fundamentally, these effects concern the profitability of the intermediate good sector and thus the financing of research. More precisely, they concern the quantity of labor which is used in research, and finally the innovations rate.3 We then show that there is a tradeoff between environment and growth at equilibrium. Finally, in section 4, we make a welfare analysis of the model. We obtain here results which are very similar to those obtained by Aghion and Howitt.
2
The Model
There are five goods in the economy : An homogenous good (Y ) used for consumption (C) and abatement (D), an intermediate good (x) used to produce Y , labor (L) used to invent and to produce (x), pollution (P ), and environment (E). • At each date t, the final output is produced by a competitive sector according to Yt = At xαt
0 0. Each innovation τ replaces the old one τ − 1 (it is the creative destruction process) and is such that Aτ = θAτ −1 ,
θ > 1,
for all τ
(8)
(Note that τ is a discrete index for innovations, when t is a continuous variable for time). Remark : consider the elementary interval of time (t, t+∆t), and let be nt the total quantity of labor devoted to research on this interval. At t, the level of technology is At . At t + ∆t, this level can take the value θAt with probability λnt ∆t (one innovation occurs on (t, t + ∆t)), or the value At with probability 1 − λnt ∆t (no innovation). The average value of A at t + ∆t is At+∆t = λnt ∆t θAt + (1 − λnt ∆t)At = At + λnt ∆tAt (θ − 1). Then, (At+∆t − At )/∆t = (θ − 1)λnt At . If ∆t leads to zero, the left side is the derivative of At with respect to t. Thus, in average, the law of motion of At is A˙ t = (θ − 1)λnt At ,
for all t
(9)
As P. Aghion and P. Howitt, we assume that the amount of labor devoted to research is determined by an arbitrage condition saying that the cost of one unit of labor (i.e., the wage) is equal to the expected value of this hour in research (see (18) later). Once a new good has been invented, the firm that has succeeded in innovating monopolizes the intermediate sector until replaced by the next innovator. We assume that the production of one unit of x requires one unit of labor. Thus we have L = xt + nt ,
for all t
(10)
where xt is the amount of labor used in manufacturing and nt the amount of labor used in research. • Finally, the utility function of the infinitely lived representative agent R is 0∞ u(c, E)e−ρt dt, where ρ is a positive rate of time preference and u(c, E) has an additive isoelastic form, such that ∂u(c, E) = c− ∂c
and
∂U (c, E) = (−E)ω , > 0, ω > 0 (11) ∂E 5
3
Equilibrium
If our objective were to implement the optimal path, it would be necessary to use several tools in order to correct the distorsions which prevent the equilibrium path to be optimal. First, pollution is a negative externality. Second, there is a positive externality in research coming from the intertemporal spill over since the knowledge embedded in each innovation is used by all future researchers. Third, there is a negative externality on research coming from the business-stealing effect by which the private research firm does not internalize the loss to the previous monopolist caused by an innovation. Finally, there is an appropriability effect that reflects the private monopolist’s inability to appropriate the whole output flow. In fact, we are not so ambitious and we introduce only two public tools in the model. First, we assume that the government uses a pollution tax.4 Second, we assume that, as for instance in Barro-Sala-i-Martin (1995),it uses a subsidy to research.
3.1
Final good (Y )
At each time t, the profit in the final sector is πtY = Yt − Dt − pt xt − νt Pt where pt is the price of the intermediate good and νt is the unit tax on pollution. Using (1), (3) and (5), we have πtY = At xαt zt (1 − νt ztγ ) − pt xt Differentiating πtY with respect to zt and equating to zero gives zt =
1 νt (γ + 1)
!1/γ
, for all t
(12)
Differentiating πtY with respect to xt and equating to zero gives
xt =
αAt zt (1 − pt
4
! νztγ ) 1
1 −α
Alternatively, it would be possible to introduce pollution permits. See for instance Stokey (1998) and Grimaud (1999) on this point.
6
From (12) we have νt ztγ = 1/(γ + 1), and thus zt (1 −
νt ztγ )
"
1 = νt (γ + 1)
#1/γ
γ γ+1
Then we obtain the demand of intermediate good from the final sector :
xt =
"
αγAt 1/γ
#
1 1−α
pt νt (γ + 1)(γ+1)/γ
(13)
From (12), we see that an increase of the tax on pollution νt leads to a decrease in zt . Then, for a given level of production Yt , it leads to an increase of abatement Dt = Yt (1−zt ) (see (3)) and a decrease in consumption Ct = Yt zt (see (2)). This is the first direct channel by which environmental policies affect growth (see 4.1 later). Moreover, (13) shows that if the tax νt increases, the demand of intermediate good xt decreases. As we see later, this leads to a decrease of the profitability in the intermediate good sector, an thus to more complex general equilibrium effects of environmental policies on growth that we examine in section 4.
3.2
Intermediate good (x)
At each time t, the τ th incumbent innovator chooses the price pτ t (or equivalently the quantity xτ t ) which maximizes the profit πτ t = pτ t xτ t − wt xτ t , where the demand from the final sector xτ t is given by (13) and wt is the wage. The first order condition of the above maximization program yields immediately 2
xτ t =
α γAτ 1/γ νt (γ
+ 1)(γ+1)/γ wt
!
1 1−α
(14)
or, equivalently pτ t =
wt α
(15)
Observe that in (14) the level of technology Atτ = Aτ is constant for this incumbent, until it disappears. We show later that, at the steady state, x is constant ; then for a given Aτ , wt progressively decreases if the tax νt increases. When there is a new innovation, Aτ jumps to Aτ +1 = θAτ , and wt jumps also. Afterwards, wt again progressively decreases until a new innovation occurs. 7
Using (15), we have πτ t = wt xτ t (
1−α ), and thus α 2
πτ t =
1−α α γAt wt 1/γ α νt (γ + 1)(γ+1)/γ wt
!
1 1−α
(16)
Let be Vτ =
Z
0
∞
πτ s e−δns e−rs ds
(17)
the discounted expected payoff to the τ th innovator, where πτ t is given by (16) and where r is the interest rate (r is constant at the steady state). The arbitrage conditions is (see Aghion-Howitt (1998), chapter 2) : wτ (1 − σ) = λVτ +1
(18)
where σ is the rate of subsidy to research, and where wτ is the level of wage when the level of technology is Aτ .
3.3
Government and representative household
The budget constraint of the government is Z
0
∞
(σwt nt − νt Pt − Tt )e−rt dt = 0
(19)
where Tt is a lump sum tax (or a subsidy) used to finance the difference between the subsidy on research (σwt nt ) and the tax on pollution (νt Pt ). Finally the maximization of the intertemporal utility (11) leads to the standard condition : c˙t rt − ρ = ct
3.4
(20)
Balanced equilibrium growth path
Assume that the rate of growth gν of the tax on pollution is constant, and that the subsidy σ does not depend on t. Then, with each doublet (gν , σ) we can associate a particular balanced growth path equilibrium (gy is the rate of growth of any variable y). Proposition 1 A balanced growth path equilibrium is a set of quantities, prices, and rates of growth that take the following values : 8
Quantities :
n =
x = zt = At Yt Ct Pt
= = = =
θλL(1 − α) gν ( − 1) + −ρ α(1 − σ) γ ! θ(1 − α) λ + (θ − 1) + 1 α(1 − σ) L−n 1 ! 1 γ νt (γ + 1) A0 egA t (A0 : initial value) At x α At xα zt Yt zt1+γ
(21)
(22) (23) (24) (25) (26) (27)
Prices : r = gc + ρ
(28) 2
wt = pt =
α γAt 1/γ νt (γ wt
+ 1)(γ+1)/γ x(1−α)
(29) (30)
α
Rates of growth : gn = gx = 0 gν gz = − γ gA = gY = (θ − 1)λn gν gc = gA − γ γ+1 gP = gE = gA − gν γ gr = 0 gν gw = gp = gc = gA − γ 9
(31) (32) (33) (34) (35) (36) (37)
Remark 1 : we know that between two innovations (for instance τ and τ + 1), A is constant. If we assume that the tax on pollution νt increases continuously at rate gν , we can distinguish the behavior of two types of variables. Some of them (n, x, Y, and r) remain constant. The others vary continuously : zt (gz < 0); wt , pt and Ct (gw = gp = gc = −gν /γ < 0) ; Pt (gP = −(γ + 1)gν /γ). When A jumps (for instance, from Aτ to Aτ +1 = θAτ ), some variables remain constant : n, x and r. One varies continuously : z. All the others jump : Y, C, P, w, p. This explains why the rates of growth of A, Y, C, P, w and p in proposition 1 are in fact average rates of growth. These discontinuities in trajectories come from the fact that we assume that there is only one good (and thus, one firm) in the intermediate good sector. If we would use a more general Schumpeterian model with a continuum of intermediate goods, these discontinuities would disappear, but the analysis would be more complex that is contrary to our objective here. Remark 2 : a first glance to proposition 1 shows that public tools have strong effects on equilibrium. The subsidy to research σ has a direct effect on n (labor used in research: see (21)), and thus an effect on growth of consumption (see (33) and (34)) and pollution (see (35)). The rate of growth gν of the tax on pollution affects not only n but also gz (the speed of technological modifications for the ”intensity of pollution”: see (32)), and thus it affects also growth (see (34)). All these points will be investigated in section 3. Remark 3 : the rate of growth of the total tax on pollution νt Pt is equal to γ+1 gν = gA −gν /γ = gc . the one of consumption Ct , since gν +gP = gν +gA − γ Proof of proposition 1 : Assume n (and thus x = L − n) constant. First, from (1), (2), (5), (6), (9), (12), and (13), we obtain a balanced growth path where gY = gA , gc = gA + gz , gP = gY + (γ + 1)gz , gE = gP , gA = (θ − 1)λn, gz = −gν /γ, 0 = gx = gA /(1 − α) − gp /(1 − α) − gν /γ(1 − α). Now, the main difficulty is to calculate n, the labor used in research. This can be done by using the arbitrage condition (18). This condition can be written wt (1 − σ) = λVt = λ
Z
∞
t
where, using (16), we have 10
πs e−(λn+r)(s−t) ds
(38)
2
πs =
α γAs 1−α 1−α ws xs = ws 1/γ α α νs (γ + 1)(γ+1)/γ ws
!
1 1−α
(39)
Assume that an innovation occurs at t. Then, immediately, A jumps from At to θAt : therefore, we have As = θAt for all s > t. Since x is constant, we see from (14) that at the same time the wage w jumps from wt to θwt . Afterwards, for s > t, we obtain ws = θwt egw s = θwt e−gν s/γ (see again (14)). Thus, Vt can be written !
γAs
Vt = (1 − α)α(1+α)/(1−α)
1/γ νs (γ
Vt = (1 − α)α(1+α)/(1−α) θwt
+
1)(γ+1)/γ ws
1 Z 1−α
t
!
γAs 1/γ νs (γ
+
∞
1 1−α
1)(γ+1)/γ ws
θwt e−gν s/γ e−(λn+r)(s−t) ds
1 λn + r + gν /γ
(40)
1 where the term (·) 1 − α is constant because x is constant (see (14)). The arbitrage condition (38) becomes (1 − σ) = λ(1 − α)α(1+α)/(1−α) θ
·
1 1−α
1 λn + r + gν /γ
and thus
!
γAs 1/γ νs (γ
+
1 1−α
1)(γ+1)/γ ws
=
(1 − σ)(λn + r + gν /γ) , θλ(1 − α)α(1+α)/(1−α)
for all s
(41)
From (10) and (14), we have 2
L=n+
α γAt 1/γ νt (γ
!
1 1−α
+ 1)(γ+1)/γ wt
Using (41), this equality becomes
L=n+
α(1 − σ)(λn + r + gν /γ) θλ(1 − α) 11
(42)
From (2), (9) and (12) we have gc = gA + gz = (θ − 1)λn − gν /γ. From (20), we have also gc = (r − ρ)/, and thus, r = gc + ρ. Eliminating gc , we obtain r = (θ − 1)λn − gν /γ + ρ. Replacing r by this expression gives finally θλL(1 − α) gν ( − 1) + −ρ α(1 − σ) γ ! n= θ(1 − α) λ + (θ − 1) + 1 α(1 − σ) This is exactly the equilibrium value of n given by (21). All the others variables can be immediately obtained. Remark : we know that we must have 0 ≤ n ≤ L. Using (21), this condition leads to θλL(1 − α) gν ( − 1) + −ρ α(1 − σ) γ
≥0
(43)
and gν ( − 1) ≤ λL((θ − 1) + 1) + ρ γ
4
(44)
Public policies, environment and growth
Our objective now is to study the effects of ν (tax on pollution) and σ (subsidy to research) on the steady state variables.
4.1
Tax on pollution on trade-off environment-growth
A) Formal analyses Assume that gν , the rate of growth of the pollution tax, increases. From proposition 1, we see that this increase has two directs effects. The first one concerns the choice of the ”intensity of pollution” z (or, equivalently, the choice of abatement D) by the final sector. From (32), gz = −gν /γ, we see that gz decreases. The second one concerns the quantity of labor n used in research (see (21)). If < 1 ( is the elasticity of marginal utility), a higher gν leads 12
to a lower n, and then to a lower gA = gY = (θ − 1)λn (see (33)). If > 1, the reverse is true : is this case, we obtain a paradoxal result according to which a higher gν leads to more research, and thus to more growth before abatement.5 Examine now the effects of gν on gc (rate of growth of consumption) and gP (rate of growth of pollution). From (33) and (34), we have gc = gA − gν /γ = (θ − 1)λn − gν /γ. Using (21), we obtain finally gc =
H − Bgν D
(45)
where !
θλL(1 − α) H = (θ − 1) −ρ α(1 − σ) ! θ(1 − α) B = + θ /γ α(1 − σ) θ(1 − α) D = + (θ − 1) + 1 α(1 − σ)
Note that if we assume σ = 0 and gν = 0, the condition n ≥ 0 implies H ≥ 0 (see (43)). From (34) and (35), we have gP = gc − gν =
H B − gν 1 + D D
(46)
Finally, (45) and (46) show that a higher gν leads to a lower gc (even in the case > 1, where gA increases, the negative effect on growth prevails) and to a lower gP . Now we give a more detailed economic interpretation of these results. B) Economic interpretation Consider again an increase of gν . This increase has two types of effects : directs effects on the final sector behavior, and indirect effects (or ”general equilibrium effects”) on the profitability of the intermediate good sector. 5
The same result has been obtained by Grimaud (1999) in a model ” la Romer” with horizontal innovations. It can be put closer the arguments of M.E Porter and C. Van der Linden (1995) which claim that ”regulation might act as a spur to innovation”.
13
The direct effects on the final sector can be easily located. The first one has been already pointed out. It concerns the choice of z (see (12), and thus (32)) : an increase of gν leads to choose less polluting technologies (or, equivalently, to more abatement), that is more costly and then unfavorable for growth. The second one concerns the demand of intermediate good given by (13) : the tax on pollution weakens this demand and this is the origin of indirect effects on the intermediate good sector. The indirect effects concern precisely the profitability of this sector, and thus the labor used in research. To examine this point, we study now the impact of gν on the discounted expected payoff V of innovators. The tax on pollution has two opposite effects on V . The first effect, which is negative, is the following. Consider the situation of an innovator which produces the intermediate good. Its profits are given by (39). When the tax νs increases, x remains constant whereas the wage ws and the price ps decrease. Thus the profit πs decreases also at the rate gπ = gw = gp = −gν /γ. If gν increases, then gπ decreases and V decreases : formally, this effects corresponds to the term gν /γ at the denominator of (40). Finally, the decrease of V implies a decrease of n, and thus a decrease of gA = (θ − 1)λn (see (33)) : this effect is clearly unfavorable to growth. The second effect, which is positive, corresponds to a negative effect on r (the interest rate) in the term 1/(λn + r + gν /γ), inside (40). We know that an increase of gν leads to a decrease of gc (see (45)). Thus, it leads to a decrease of r = gc + ρ (see (28)), and finally an increase of V : this effect is favorable to growth. Observe that more is high, more this effect is important. In particular we have observed that, if > 1, an increase of gν leads to an increase of n : from (21), we have ∂n/∂gν > 0. We have now an explanation of this apparently paradoxal result. C) The trade-off environment/growth In the previous paragraph, we have studied the channels by which the tax on pollution affects the growth rates of pollution and consumption. We can now go further. We have seen that an increase of gν leads to a simultaneous decrease of gP and gc . But gP decreases more quickly than gc : this is why there is a trade-off between environment and growth. This result follows directly from (34) and (35) which give gP = gc − gν , and thus ∂gP /∂gν = ∂gc /∂gν − 1. Therefore, we have ∂gP /∂gν < ∂gc /∂gν . 14
The trade-off can appear more clearly if we express directly gP as a function of gc . From (45), we have gν = (H − Dgc )/B. Using (46), we obtain gP =
−H D + gc 1 + B B
(47)
In figure 2, point E corresponds to the case where gν = 0, and thus gP = gc : in this case, the pollution tax has a level effect but no growth effect. When gν increases, the equilibrium moves to the south-west along the line : gP decreases but gc decreases also. The channels by which the increase of gν works on gP and gc have been identified in B) above. gP 6
sE % %
H/D % %
- gc
%
0
% %
H/D
%
−H/B %
%
% % %
F Figure 2 : the trade-off environment/growth
4.2 Subsidy to research and trade-off environment/ growth From proposition 1, it is easy to derive the effects of an increase of σ, the subsidy to research. First, differentiating n with respect to σ in (21), and using (44), it can be shown that n is an increasing function of σ. Thus a higher σ leads to higher gA = gY = (θ − 1)λn (see (33)), higher gc = gA − gν /γ (see (34)), and higher gP = gA − ((γ + 1)/γ)gν (see (35)). The economic intuition is clear : when the labor cost in research decreases, there is more research, and then more growth and 15
more pollution. Note that, contrarily to the tax on pollution ν, σ has no direct effect on the technological choice z. The links between an increase of σ and the trade-off environment/growth can be seen in Figure 3, which comes back to Figure 2. If σ increases, the straight line (47) moves to the right. If ν is constant (i.e., gν = 0) the common rate of growth of gc and gP increases : in Figure 3, E moves north-east along the 45◦ line. More generally, for any given positive gν , we have seen that gc and gP increase. gP 6
E2 s
E1 s
P q P q P
- gc
Figure 3 : increase in subsidy to research and trade-off environment/growth
5
Welfare
Our main objective was to study the effects, at equilibrium, of public policies on growth and pollution. Moreover,it is possible to characterize the optimal path. That is the purpose of this section. 16
Proposition 2 A balanced optimal growth path is a set of quantities and rates of growth that take the following values :
n = x = gY = gz = gc = gP =
L/α − ρ/(θ − 1)λ ω + + γ(ω + 1) 1 − α + ω + + γ(ω + 1) α L−n gA = (θ − 1)λn > 0 −(ω + ) gA < 0 ω + + γ(ω + 1) γ(ω + 1) >0 gA + gz = gA (ω + ) + γ(ω + 1) γ(1 − ) gE = gA + (γ + 1)gz = gA 1. This condition was already found by Aghion and Howitt (1998) in a more complex model with capital accumulation. It can be noted that, along this optimal path, consumption increases and pollution decreases. This is feasible because the intensity of pollution z decreases quickly enough (or, equivalently, abatement increases quickly enough). Note that, if we would implement this path in a decentralized economy, it would be necessary that the pollution tax νt increase quickly enough : see for instance (32). We can also note that this optimal path corresponds to a particular point in Figure 2. If we assume that the social planner chooses optimal tools, the optimum path is a point of the segment EF where we have simultaneously gc > 0 and gP < 0. Proof of proposition 2 : The program of the social planner is to maximize the utility Z
∞
0
−ρt
U (ct , Et )e
dt =
Z
0
∞
U (At (L − nt )α zt , Et )e−ρt dt
subject to A˙ t = (θ − 1)λnt At E˙ t = −At (L − nt )α ztγ+1 − ηEt and the threshold condition (7): E min ≥ Et ≥ 0.
17
(µt ) (χt )
The Hamiltonian in current value is H = U (At (L − nt )α zt , Et ) + µt (θ − 1)λnt At − χt (At (L − nt )α ztγ+1 + ηEt ) First order conditions : ∂H ∂U = − αAt (L − nt )α−1 zt + µt (θ − 1)λAt + χt αAt (L − nt )α−1 ztγ+1 ∂nt ∂c =0 (54) ∂H ∂U = At (L − nt )α − χt (γ + 1)At (L − nt )α ztγ =0 (55) ∂zt ∂c ∂U ∂H = (L − nt )α zt + µt (θ − 1)λnt − χt (L − nt )α ztγ+1 = ρµt − µ˙ t ∂At ∂c (56) ∂H ∂U = − χt η = ρχt − χ˙ t ∂Et ∂E
(57)
(55) can be written ∂U − χt (γ + 1)ztγ = 0, ∂c
where
∂U = c− ∂c
(see (11))
(58)
From (54), we have α−1
µt (θ − 1)λ = α(L − nt )
zt
∂U − χt ztγ ∂c
!
that, using (58), becomes µt (θ − 1)λ = αγ(L − nt )α−1 χt ztγ+1 From (56), we have gµ =
µ˙ t (L − nt )α zt (∂U/∂c − χt ztγ ) =ρ− − (θ − 1)λnt µt µt
Using (54), this condition becomes gµ = ρ −
(θ − 1)λ (L − nt ) − (θ − 1)λnt α 18
(59)
and thus gµ = ρ + (θ − 1)λnt
!
1 − α) (θ − 1)λL − α α
(60)
∂U = (−E)ω ∂E
(61)
Finally, (57) becomes gχ = ρ + η −
∂U/∂E , χt
where
(see (11))
To obtain a balanced growth path it suffices to start from equations (1), (2), (5), (6), (9), (10), (11), (58), (59), (60), (61). They lead immediately to the following system concerning the rates of growth : gY gc gA gE −gc gµ gµ gχ
= = = = = =
gA gA + gz (θ − 1)λn gP = gA + (γ + 1)gz gχ + γgz gχ + (γ + 1)gz 1−α (θ − 1)λL = ρ + (θ − 1)λn − α α = ωgE
From this system, one can calculate the rates of growth gY , gA , gc , gz , gE = gP , gµ , gχ , and the quantity of labor n. This quantity and the rates of growth of Y, A, c, z, E and P are those of proposition 2. The first transversality condition is lim e−ρt µt At = 0,
t→∞
1−α (θ − 1)λL that gives here ρ + (θ − 1)λn − + (θ − 1)λn − ρ < 0. α α This gives (θ − 1)λ(n − L)/α < 0, and thus n < L, that imposes a condition on the parameters of the model (see (48)).
The second one is lim e−ρt χt Et = 0,
t→∞
Since gχ = ωgE , we have gχ + gE = (ω + 1)gE , that is negative since gE < 0. Then the second transversality condition holds. 19
6
Conclusion
The main purpose of the paper was to study the effects of public policies on growth and pollution, and to identify the channels through which these policies operate (at equilibrium). To do this, we have introduced environmental considerations in a Schumpeterian model with vertical innovations and ”creative destruction”. We have shown that environmental policies have direct partial equilibrium effects, but also more complex indirect effects through general equilibrium mechanisms. In fact, these policies affect the demand of intermediate good by the final sector, and thus the profitability of the intermediate good sector. That is the main reason why they influence research, and thus growth. However, we have shown that this ”profitability effect” can be decomposed in several sub-effects. One of them may appear paradoxal, enouncing that an increase in the rate of growth of pollution tax leads to more research, that is favorable to growth. However, in spite of this particular effect, there is a trade-off between environment and growth at equilibrium in the whole model.
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