Polluting Non-Renewable Resources, Tradeable
Permits and Endogenous Growth
André Grimaud1 and Luc Rougé2
December 2003
1 Université
de Toulouse 1 (GREMAQ, IDEI and LEERNA), 21 allée de Brienne, 31000
Toulouse, France, and Groupe ESC Toulouse. E-mail: [email protected]
2 Groupe de Finance, Groupe ESC Toulouse. E-mail: [email protected]
Abstract
We set up an endogenous growth model with vertical innovations in which the use of
a non-renewable resource within the production process generates a flow of pollution.
This flow affects negatively the dynamics of the stock of environment, which is an
argument of the non-separable utility function. We study the general equilibrium
effects of an environmental policy consisting in emissions of tradeable permits. In
particular, we show that a more stringent policy can lead, by the channel of the
permits price dynamics, to more R&D; in all cases it promotes growth.
Keywords: endogenous growth, environmental policy, non-renewable resources,
non-separable utility function, pollution, tradeable permits, vertical innovation.
JEL classification: O32, O41, Q20, Q32
1
Introduction
The question concerning the possibility -as well as the optimality- of long-term
growth in the presence of a finite stock of resources has been widely studied in
the literature. The two main types of models that have been used to assess this
problem, namely Ramsey-type growth models (where we can quote Stiglitz (1974),
Garg and Sweeney (1978) or Dasgupta and Heal (1979)) and the endogenous growth
models (Schou (1996), Aghion and Howitt (1998), Scholz and Ziemes (1999), Barbier
(1999) or Grimaud and Rouge (2003)), have generally shown that long term growth is
feasible, desirable and that it is attainable at equilibrium under certain technological
conditions (and given that, within the second type of models, incentive policies for
research are implemented).
Indeed, the non-renewable character of some of the resource stocks that are
consumed within an economy is no longer considered to be a major problem, and
the Club of Rome’s worries seem to be, at least partly, unjustified. Instead, environmental quality and its deterioration by the different types of pollution issued
from human activity have slowly become the major concern both for public opinion
and within the scientific community. Hence, numerous authors have studied the
compatibility between long-term economic development and the protection of the
environment. Articles that survey the literature on this subject include van der
Ploeg and Withagen (1991) who study the issue using Ramsey-type models, and
Smulders (1995 and 1999) as well as Beltratti (1996) who do the same within an
endogenous growth framework.
Simultaneously, certain countries became aware of the potential dangers arising
from the increase of polluting emissions at a global scale. World summits began to
take place, where nations attempted to reach some degree of co-ordination about
abatement policies for these global emissions. A UN framework convention concerning the reduction of atmospheric CO2 emissions (the main greenhouse gas) as
well as of other greenhouse gases was adopted by 160 countries at the Earth Summit in Rio (in May 1992). The Kyoto Conference (December 1997) managed to
adopt a protocol1 for the reduction of greenhouse gas emissions spanning several
1
This protocol was largely influenced by the IPCC’s conclusions (International Panel of Climate
Change) (1995) which forecasted increases of up to three degrees in the average global surface
temperature according to different scenarios of CO2 concentration in the atmosphere. Since their
publication, these forecatst have been largely debated. Certain authors criticised the IPCC for
having neglected the eventual adoption of cleaner technologies that could considerably reduce CO2
1
decades, and the Johannesburg Conference (August 2002) on sustainable development underlined the imperative character of such measures. Indeed, whereas 96% of
atmospheric CO2 is of ”natural” origin and only 4% originates from human activity,
the latter proportion has not ceased to increase. Carbon dioxide emissions have
been multiplied by a factor of 17 between 1890 and 1990. Yet, despite that 70 to
75% of these emissions are caused by the combustion of fossil fuels (coal and hydrocarbons), most of the models that study the ”growth-environment” problem, and
specially those that focus on greenhouse gas polluting emissions, neglect the fact
that non-renewable resources are the source of that kind of pollution. At the same
time, the above-cited literature on growth in the presence of non-renewable resources
fails to incorporate the effects of the pollution that is generated through their use.
In view of the facts that we have briefly recalled, it appears that a synthesis of the
two approaches (growth-environment and growth-non renewable resources) is called
for.
Some authors have already pointed out the fact that part of the observed polluting emissions proceed from the consumption of non-renewable natural resources.
Kolstad and Krautkraemer (1993) survey that literature and underline the (mostly
technical) complexity of the problem posed. More recently, Hoel and Kverndokk
(1996), and then Tahvonen (1997) studied the optimal paths arising from the use of
a polluting resource when an alternative non-polluting technology is available -the
main differences lie within the cost structure. Schou (2000 and 2002) studies Lucas
and Romer-type growth models when the use of a non-renewable resource creates
a negative externality that affects either technology (erosion, rust, workers’ health)
or households’ welfare. In both cases, he concludes that, at equilibrium, a specific
environmental policy is not necessary in order to implement the optimum. The
general problem’s key issue is that using the resource (hence extracting it) entails
pollution. The resource is thus a dirty input, as opposed to work or knowledge.
Therefore, the questions are not only ”at what rate should the finite resource stock
be extracted?” or ”how to share a finite stock of resource among an infinite number
of generations?” but we must now add the following ones: ”at what rate should a
finite stock of pollution be emitted?” or even ”how should a finite stock of pollution
be shared among an infinite number of generations?” By taking into account the
characteristic of the resource to be the pollutant, we introduce a new dimension to
concentration (see Chakravorty, Roumasset and Tse (1997)).
2
the problem: the choice of a resource extraction path now implies the choice of a
pollution path.
In the present article, we consider that the use of a non-renewable natural resource in the production process generates a flow of pollution. The environmental
quality is a stock whose evolution through time depends negatively upon this flow
of pollution and positively upon its own auto-regenerative capacity. Moreover, in
addition to instantaneous consumption, this flow is an argument of the utility function. In fact, we employ a non-separable utility function (like, e.g. Gradus et
Smulders (1993)) and we will see that this choice conveys important modifications
to the Ramsey-Keynes condition. Indeed, the consumption growth rate depends on
the interest rate, but also on the environmental quality’s growth rate (or equivalently, on the resource’s extraction rate). We set up an endogenous growth model
within which the source of growth is vertical innovation (that is, an enhancement in
knowledge) over a continuum of sectors. Contrarily to that done by Grossman and
Helpman (1991) or Aghion and Howitt (1992), knowledge is not incorporated into
intermediate goods. This is why we suppose that it is directly paid for by those who
use it.
In order to correct the distortion introduced by the pollution that is generated
from the use of the resource, we implement an environmental policy that consists
in an emission of pollution rights. The latter allows us to remain consistent with
respect to the above mentioned observations (the Kyoto protocol). An economic
policy is therefore the announcement to the market of a profile of permits issued
by the government; this automatically determines the pollution profile and thus
the extraction of the resource. Note right away that, due to the modification of
the above cited Ramsey-Keynes condition, the policy will have, outside of general
equilibrium, direct effects on households’ saving behaviour and, therefore, upon
on the consumption growth rate. Additionally, it will have a profound and complex
impact on the economy at general equilibrium, since it generates a profile of prices of
permits that affects equilibrium variables (R&D effort, economic growth...) precisely
via the price channel. In particular, we show that the policy instrument is not the
price of the permits themselves (this only allows a reallocation of income among
sectors) but rather its dynamics. Furthermore, we prove that, within our model, a
more stringent environmental policy entails an increased R&D effort, if preference for
environmental quality is high enough across society; moreover, instead of slowing
down the economy, it favours growth, something which can be interpreted as a
3
possible justification for Porter and Van der Linde’s (1995) hypothesis.
In the second section of this article, we introduce the model and we completely
characterise the equilibrium in section 3. In the fourth section we then study the
effects of the environmental policy by exhaustively presenting all the subsequent
modifications to equilibrium arising from the implementation of that policy. The
final section draws conclusions.
2
The model
At each time t, a quantity Yt of consumption good is produced according to the
following technology:
Yt = Btν LαY t Rt1−α , with 0 < α < 1 and ν > 0,
(1)
where LY t is the amount of labour devoted to production, Bt is the total knowledge
and Rt is the flow of non-renewable resource used within the production process.
In this model, there are no intermediate goods as in more standard endogenous
growth models2 . Innovations are produced by a continuum of sectors, each sector
being denoted by i, i ∈ [0; 1]. At each time t, any sector i is characterized by a
level of knowledge Bit , and it has its own research activity. In order to produce
innovations -that are successive increase in knowledge, the sectors draw on the same
R1
pool of shared knowledge, given by Bt = 0 Bit di.
Let us now make the two following assumptions. First, the Poisson arrival rate
of vertical innovations in any sector i is
ψit = λLRDit,
where λ is a parameter indicating the productivity of research, and LRDit is the
amount of labor devoted to R&D in sector i. Second, denoting by τ (i) the number
of innovations in sector i (τ (i) = 1, 2, ...), we assume that when an innovation occurs
in this sector, the increase in knowledge is given by
Bτ +1(i) = Bτ (i) + σBt(τ (i)) , with σ > 0,
2
See for instance Jones (2003) who uses several models of this type in his survey. A more
complete presentation of the methodology used here is given in Grimaud (2002).
4
where Bt(τ (i)) is the total level of knowledge when the level of knowledge is Bτ (i)
in sector i. As suggested by Aghion and Howitt (1998, ch 3) for instance, this
assumption is a formalization of the intersectoral spillovers, that is to say the fact
that new technologies ”diffuse gradually, through a process in which one sector
gets ideas from the research and experience of others” (p 85). It simply says that
the increase in knowledge in sector i, Bτ +1(i) − Bτ (i) , is a linear function of total
knowledge Bt(τ (i)) . It is easy to verify that it leads to a very simple law of motion of
the average knowledge in sector i: E(Ḃit ) = λσLRDit Bt (see Appendix A1). Since
R1
Bt = 0 Bit di, one gets finally
Ḃt = λσLRDt Bt ,
where LRDt =
R1
0
(2)
LRDit di is the total amount of labor devoted to research.
Note that this is only a technical relation ; we do not need LRDi = LRDj for all
i, j ∈ [0; 1], as for instance in Aghion-Howitt model.
As we mentioned it before, labor is used by the production sector (LY t ) and by
the research sector (LRDt ). We assume that the total flow of labor supply is fixed,
and we normalize it to one. Thus, at each time t,we have
LY t + LRDt = 1.
(3)
The resource is extracted from an initial finite stock S0 , and we have the standard
resource stock law of motion:
Ṡ = −Rt .
(4)
Last, the utility function of the infinitely lived representative agent is
u(Ct , Et ) =
Z
0
+∞
[Ct (−Et )−ω ]1−ε −ρt
e dt, with ε > 0 and ω > 0.
1−ε
(5)
where ρ is a positive rate of time preference. Moreover
Ct = Yt ,
(6)
that is, the whole production flow is consumed by the household; and Et is, as in
Aghion-Howitt (1998 (p 157)), a negative variable measuring the difference between
the actual quality of the environment and its upper limit (reached if all pollution
5
were to cease indefinitely). We assume that the law of motion of Et is3
Ėt = −Pt − φEt , with φ > 0
(7)
Pt = γRt , with γ > 0,
(8)
where
Pt being the flow of pollution generated by the natural resource use within the
production process. Thus E is depleted over time by pollution but it has its own
regeneration capacities.
Note that, for all C > 0 and E < 0, we have (we denote by Ux the derivative
of the utility function relative to any variable x): UC > 0, UCC < 0 and UE > 0.
For UEE to be negative (as for the standard utility function used by Aghion-Howitt
(1998) for example) we need
ε>
ω+1
,
ω
which implies ε > 1. Moreover, we also have UCE = ω(1 − ε)Ct−ε (−Et )−ω(1−ε)−1
which is negative (since 1 − ε is negative).
3
Equilibrium
The price of good Y is normalized to one, and wt , pRt and rt are, respectively, the
wage, the resource price, and the interest rate on a perfect financial market. On
this market, R&D firms sell bonds to the households. We denote by Dt the stock
of bonds at time t.
In order to eliminate the market failure arising from the fact that firms do not
take into account the negative externality due to the use of the non-renewable resource within the production process , i.e., pollution, we use a policy tool: pollution
permits. At each time t, authorities allocate a quantity Qt of tradeable permits to
the firms producing the consumption good which trade them on a perfectly competitive market at price qt .
We could choose a more general regeneration function, for instance Ėt = −Ptγ − φEt . This
would not change the results of the model. We thank one referee for this remark.
3
6
3.1
Behaviour of agents
Consumption good sector:
At each time t, the profit of the representative firm4 is
π Yt = Btν LαY t Rt1−α − wt LY t − pRt Rt − pRt qt Pt + pRt qt Qt
where pRt qt Pt is the value of permits bought by the firm, and pRt qt Qt corresponds
to its initial endowments. This production function can be rewritten as follows
πYt = Btν LαY t Rt1−α − wt LY t − pRt θt Rt + pRt qt Qt
(9)
with, using (8), θt = γqt + 1. We can interpret pRt θt as the global cost of one unit of
resource: pRt is paid to the resource owner and pRt γqt is paid to the permits owners.
Differenciating π Yt with respect to LY t and Rt , and equating to zero, gives the
two following first order conditions:
1−α
wt = αBtν Lα−1
Y t Rt
(10)
pRt θt = (1 − α)Btν LαY t Rt−α .
(11)
R&D sector:
Knowledge is used by two sectors: the consumption good sector and the R&D
sector. As we mentioned above, knowledge is not embodied inside intermediate
goods, thus it cannot be financed by the profits made on the sale of these goods.
We suppose here that it is directly financed. Therefore we formalize ideas already
expressed by authors like Arrow (1962), Scotchmer (1991) or Dasgupta et al. (1996).
For instance, Dasgupta et al. write ”a possible scheme is for society to grant intellectual property rights to private producers for their discoveries, and permit them
to charge (possibly differential) fees for their use by others. This creates private
markets for knowledge. Patents and copyrights protections are means of enforcing
intellectual property rights. It is as well to note that, in this scheme the producer
(or owner) of a piece of information should ideally set different prices for differ4
For computational convenience, we consider one unique firm. We could also consider a de1−α
sagregated sector with n firms whose production function would be Yjt = Btν Lα
Y jt Rjt , j = 1, ..., n.
In this case, we would denote by Qjt the quantity of permits issued to firm j, with Σnj=1 Qjt = Qt
for all t. The results would not be modified.
7
ent buyers, because different buyers typically value the information differently. In
economics, these variegated prices are called Lindahl prices”. Nonetheless, as the
sectors using knowledge exhibit constant returns to scale in the private goods, their
profits are nil. Thus we assume that the government subsidies the consumption
good sector and the R&D sector to buy knowledge.
Remark: This type of decentralization defines an equilibrium that is a benchmark. The main interest of designing such an equilibrium is that it allows us to
focus on one single distorsion: the one caused by pollution. Note that, as seen
above, this leads to assume that the governement finances partially R&D. However,
it is possible, assuming that the two sectors using innovations (final sector and R&D
sector) are imperfectly competitive, to construct an equilibrium in which R&D is
entirely and privately financed by these sectors (we find a similar equilibrium path
in this case).
At each time t, the value of one unit of knowledge is:
Vt =
Z
+∞
vs e−
t
Rs
t
ru du
ds,
(12)
where vt is the sum of the willingnesses to pay for one unit of knowledge of the
homogeneous good sector (vtY ) and the R&D sector (vtRD ): vt = vtY + vtRD .
The expected profit on knowledge at t is5 :
E(π RD
t ) = E(Ḃt )Vt − wt LRDt = λσLRDt Bt Vt − wt LRDt .
(13)
This leads to the following free entry condition:
wt = λσBt Vt
(14)
Remark: Here also we could desagregate this sector into a large number of firms,
but the results would not be changed.
From (9), (13), (14) and (10) we have:
vtY =
∂π Yt
= νBtν−1 LαY t Rt1−α
∂Bt
5
(15)
The way we present the calculations is a shortness; a more complete presentation is given in
appendix A2. The results are the same.
8
and vtRD =
∂E(π RD
t )
1−α
= λσLRDt Vt = αLRDt Btν−1 Lα−1
.
Y t Rt
∂Bt
(16)
Resource sector:
On the competitive natural resource market, the maximization of the profit funcRs
R +∞
tion t pRs Rs e− t ru du ds, subject to Ṡs = −Rs , Ss ≥ 0, Rs ≥ 0, s ≥ t, yields the
standard equilibrium ”Hotelling rule”:
ṗRt
= rt , for all t.
pRt
(17)
Representative household:
At each time t, the representative household maximizes the utility function
R +∞
−ω ]1−ε
t)
e−ρt dt subject to Ḋt = wt + rt Dt + pRt Rt − Tt − Ct ,
u(Ct , Pt ) = 0 [Ct (−E1−ε
where Ḋt is lendings at t, and Tt is a lump-sum tax levied by the government to
finance research. This maximization leads to the following condition:
Ċ
r − ρ − ω(1 − ε)Ė/E
=
, at each time t.
C
ε
(18)
Note that, for a given r, a decrease in Ė/E causes a decrease in Ċ/C since
1 − ε < 0 (this comes from UEE < 0 (see section 2)). At steady-state, where all
growth rates are constant, Ė/E = Ṙ/R = Ṗ /P < 0 (dividing each hand side of
equation (7) by Et yields this result6 ). Therefore, a decrease in gE (we denote by gx
the growth rate of any variable x) means more pollution, i.e. a worse environment,
today (and thus less pollution tomorrow, as the pollution stock is finite). In order to
compensate this disutility, the household consumes more today (and less tomorrow);
that is why the growth rate of consumption decreases (see Figure 1). Moreover, the
higher ε, the higher is the decrease in gC due to a given decrease in gE . This comes
from the fact that a high ε means that the household does not like variations on
his utility path (i.e. the higher ε, the more the household is keen on uniform utility
path). Thus the compensation in terms of present consumption for a initial disutility
caused by a worse environment today is maximum.
Observe that this effect of gE on gC is a direct consequence of the non-separability
of the utility function (see (5)). We see later that, at general equilibrium, it is
6
Since the resource stock is finite, at steady-state the flow of extraction decreases at a constant
rate. Using (8), we see that pollution decreases at the same rate. Finally, equation (7) shows that
the quality of the environment asymptotically tends to its upper limit.
9
Rt (= Pt / γ )
Ct = Yt
t
t
Figure 1: Effect of a change in gE = gR on gC , for a given r
widened by a decrease in r.
Government:
The government’s budget constraint is7
Tt = sYt + sRD
t , for all t
(19)
= vtRD Bt are research subsidies.
where sYt = vtY Bt − pRt qt Qt and sRD
t
Note that part of the financing of the knowledge used by the consumption good
sector comes from the initial permits endowments distributed by the government.
Remark: If the environmental policy tool was a tax on resource purchases these
initial endowments would not appear, and we would have sYt = vtY Bt . The model we
use illustrates one side of the international debate on tradeable permits: indeed, it
appears clearly that the initial allocation of permits determines the ex-post profits.
3.2
Characterization of the equilibrium
Now we give a complete description of the general equilibrium of this economy.
Moreover we focus on the study of steady-state paths, i.e., on paths along which the
growth rate of any variable is constant, as mentioned above.
7
We assume here that the government possesses all the information and thus can perfectly
determine v Y and v RD . Of course this is just a benchmark and informational asymmetries could
be introduced.
10
Proposition 1 A steady-state equilibrium is a set of quantities and growth rates
that take the following values (we drop time subscripts for notational convenience):
LRD =
−αρ + λσν((α + ω)(1 − ε) + ε) + gθ α(1 − ε)(α + ω − 1)
,
λσν(ε(1 − ω) + ω)
LY = 1 − LRD ,
(21)
gB = λσLRD ,
(22)
gR = gQ = gP = gE = gS =
gC = gY =
−ρ + λσν(1 − ε) + gθ (1 − α + εα)
,
ε(1 − ω) + ω
−ρ + λσν(1 + ω − εω) − gθ (1 − α − αω(1 − ε))
,
ε(1 − ω) + ω
gθ =
(20)
−ρ + λσν(1 − ε) − gQ (ε(1 − ω) + ω)
.
(1 − α + εα)
(23)
(24)
(25)
The prices are given in Appendix B.
Remark: At the steady-state equilibrium, given S0 and B0 , the initial quantities
are Y0 = C0 = (LY )α (−S0 gR )1−α (B0 )ν , R0 = P0 /γ = −S0 gR , E0 = −γS0 gR /(−gR −
φ) (dividing each side of (7) yields this result) and at each time t we have xt = x0 egx t
for any variable x.
Basically, equation (25) shows how the intertemporal path of permits issued by
the government (gQ ), determines the growth rate of the permits price: we analyze
this relation later in the analysis (see section 4.2). Once gQ , and thus gθ , are set,
equations (20) and (21) explain how labour is allocated between research and production. Then, (22) gives the growth of knowledge, and (23) at the same time gives
the extraction path of the resource and the evolution of environmental quality. Finally, equation (24) determines the growth of production and thus the growth of
consumption.
Proof. See Appendix B.
3.3
Existence of interior equilibrium
Now we study the equilibrium path we just described, in the absence of any environmental policy (i.e. qt = 0).
We need 0 < LRD < 1, and gR < 0 to have a solution. Proposition 1 shows that
there exist parameter values for which these conditions are not fulfilled. We are thus
11
looking for the set of parameter values in which these conditions are together fulfilled,
in particular in order to avoid corner solutions (with, for example, no research at
equilibrium). To do so, we use equations (20) and (23). The set is described in the
following proposition.
Proposition 2 An interior equilibrium exists if and only if
ρ < (λσν/α)((α + ω)(1 − ε) + ε).
In the case where ω > 1 − α, the condition ε < (α + ω)/(α + ω − 1) needs also to be
fulfilled.
4
Environmental policy
4.1
An exhaustive description of the decentralized equilibrium with tradeable permits
Now we give a complete description of the general equilibrium at steady-state; in
particular, we show how equilibrium flows are modified by the environmental policy.
Remember that, at each time t, the government issues a quantity Qt of permits,
and thus sets the flow Pt (= Qt ) of pollution, and the resource extraction Rt
(= Qt /γ). As we showed in Proposition 1, this environmental policy simultaneously
has an impact on the set of variables of the economy. In order to have an exhaustive
description of these effects, let us come back to the behaviour of the agents and
gather the results in a table depicting the general equilibrium of the economy. First,
let us describe the operations of each agent at each time t (each flow is divided by
Yt ).
• Consumption good sector
Expenditures: wages (wLY /Y = α, see (10)); knowledge (vY B/Y = ν, see
(15)); resource (total payment: pR θR/Y = 1 − α, see (11)). Note that, as
mentioned above in the comments on (9), the payment for the resource can be
decomposed in two parts:
- payment to the resource sector: pR R/Y = (1 − α)/θ,
- payments to the representative firm itself (value of permits): pR qQ/Y =
(1 − α)(θ − 1)/θ.
12
An essential feature of the equilibrium is that a modification in the level of
the prices of the permits θ has no effect on the total payment for the resource,
pR θR/Y = 1 − α. Such a modification only changes the decomposition of
this total payment: an increase in θ leads to a decrease in the payment to the
resource sector (pR R/Y = (1 − α)/θ), and an increase in the payment to the
permits owners (pR qQ/Y = (1 − α)(θ − 1)/θ).
Revenues: consumption good (C/Y = 1, see (6)); permits endowments (pR qQ/Y =
(1 − α)(θ − 1)/θ, see above); subsidy (sY /Y = ν − (1 − α)(θ − 1)/θ). This
subsidy corresponds to the difference between the payment for knowledge (ν)
and the initial permits endowments ((1 − α)(θ − 1)/θ).
• R&D sector
Expenditures: wages (wLRD /Y = αLRD /LY (see (10)); knowledge (vRD B/Y =
αLRD /LY , see (16)); interests (rD/Y = αgY /λσLY + ν (see Appendix C)).
Revenues: knowledge (vB = (vY + vRD )B/Y = αLRD /LY + ν (see above));
subsidy (sRD /Y = vRD B/Y = αLRD /LY (see Appendix C)); borrowings
(Ḋ/Y = αgY /λσLY (see Appendix C)).
• Resource sector
Since there are no extraction costs, the resource sector profit is given by the
payment coming from the consumption good sector. We assume that this
profit is distributed to the households.
Expenditures: profit (pR R/Y = (1 − α)/θ, since there are no extraction costs).
Revenues: resource (pR R/Y = (1 − α)/θ, see equation (11)).
• Households
Expenditures: consumption (C/Y = 1, see (6)); lendings (Ḋ/Y = αgY /λσLY ,
see above); taxes (T /Y = (sY + sRD )/Y = ν + αLRD /LY − (1 − α)(θ − 1)/θ,
see equation (19) and above).
Revenues: wages (w/Y = α/LY , see (10)); profits issued from the resource
sector (pR R/Y = (1 − α)/θ, see equation (11) and above (resource sector));
interests (rD/Y = αgY /λσLY + ν, see above (R&D sector)).
• Government
13
Expenditures: subsidies ((sY + sRD )/Y = ν + αLRD /LY − (1 − α)(θ − 1)/θ,
see above).
Revenues: taxes (T /Y = ν + αLRD /LY − (1 − α)(θ − 1)/θ, see equation (19)).
Table 1 depicts the flows per unit of output. We give expenditures and revenues
for each agent: H is for household, CG for consumption good sector, RD for R&D
sector, R for resource sector and G for government.
EXPENDITURES
H
Good Y
CG
RD
H
CG
RD
1−α
θ
ν
Permits
a
Wages
R
G
1−α
θ
b
ν +b
a
α
α
LY
1−α
θ
b
1−α
θ
Profits
Interests
Subsidies
G
1
Knowledge
Taxes
R
1
Resource
New Bonds
REVENUES
d+ν
d+ν
d
d
ν+b
ν+b
−a
−a
ν−a
ν+b
−a
b
Table 1: General equilibrium
We denote
4.2
(1−α)(θ−1)
θ
by a,
αLRD
LY
by b and
αgY
λσLY
by d.
Effects of the environmental policy
First of all, observe that no environmental policy, that is, qt = 0, and thus θt = 1,
corresponds to to the case where a = 0 in Table 1. Then the permits row disappears,
and the government subsidizes the whole payment for knowledge of consumption
good sector.
In fact, an environmental policy corresponds here to a path {Qt }+∞
t=0 , that is, at
steady-state, to a couple {Q0 , gQ } . This policy then generates a particular path of
14
permit prices, that is, a profile of θt -remember that pRt θt can be seen as the generalized cost of one unit of resource, or equivalently, the unit price which embodies
the value of permits. Equations (25) and (23) shows that gθ is negative if and only
if gQ is higher than gR , i.e. than the equilibrium value of the extraction growth rate
in the absence of any environmental policy (see Figure 2). More generally, we have
∂gθ /∂gQ < 0 (see (25)).
θt
Rt
Case 1:
Qt
t
t
θt
Qt
Case 2:
Rt
t
t
Figure 2: Environmental policy
We will focus now on the case of an initial decrease in pollution emissions, imposed by the government. This corresponds to the first case in Figure 2. As we
said, at steady-state, this means choosing gQ > gR , i.e. a slower extraction of the
resource8 . Since the basic transmission channel of the environmental policy is the
8
In the present model, this implies more pollution for future generations (relative to the initial
situation). However, if the resource was not necessary, and if there were substitutes in the long
15
change in the path of permits prices, we now study the effects of changes in gθ .
Let us consider an increase in gQ (when an environmental policy is implemented,
or or when policy measures become more stringent). As we showed it, this implies
a decrease in gθ . We get the two following propositions.
Proposition 3 An increase in gQ , that is, a decrease in gθ , leads to a decrease in
LRD , and thus in gB , if and only if ω < 1 − α.
Proof. Differentiating (20) with respect to gθ yields the result (remember that
ε > 1).
Let us consider the limit case where ω = 0. In this case, households are indifferent
to their environment, thus they react as in the non-polluting resource standard case
(see Stiglitz (1974), or Schou (1996)). An increase in gQ , implies an initial decrease
in the extraction flows. Thus, if nothing else is changed, the production flows for
the first generations will get lower. The only way to avoid such a decrease in current
consumptions, and thus to avoid this virtual disutility (for a given ρ), is to increase
the amount of another input: labor devoted to production. For this reason, labor
devoted to research decreases, and we get a lower growth rate of knowledge.
If ω is high, households give a strong value to their environment. Then, the
initial decrease in consumptions (due to the decrease in production caused by a less
intense resource use) leads to a virtual disutility that can be totally offset by the
virtual utility gain due to a cleaner environment. We can also consider the case
where LY can even diminish, and thus LRD increase, because households are very
glad to see their environment improved.
Thus, a more stringent environmental policy can lead to higher efforts in research
if households value environment a lot.
Proposition 4 An increase in gQ , that is, a decrease in gθ , entails an increase in
gY : a more stringent environmental policy favours growth.
Proof. Differentiating (24) with respect to gθ yields the result.
Yet we showed in section 3.1 the partial equilibrium positive effect of gE (= gQ )
on gC (= gY ). Now, we have a general equilibrium effect that strengthens this
run, then the resource stock may not be entirely exhausted in the long run. Thus pollution would
not necessarily increase for future generations. This point goes beyond the scope of our analysis.
16
one. We can easily see that gpR gets higher values as gθ is getting lower. Moreover,
we know, by assumption, that gR is increasing, i.e., that initially less ressource is
extracted, and more in the far future. Thus the profile of instantaneous profits for
the resource sector, {pRt Rt }+∞
t=0 , is modified. In fact, this sector will make lower
profits today, and higher profits tomorrow. This means lower revenues today for the
households, and higher revenues in the future. Thus, ceteris paribus, the household
needs to save less, and that explains why r increases. This higher interest rate
leads to more growth. Thus, as we say it in the proposition, in the present model,
the environmental policy promotes growth. This can also be interpreted in light of
arguments of Porter and Van der Linde (1995).
5
Conclusion
The main objective of this paper was to study the effects of an environmental policy
in a general equilibrium framework with the following main characteristics:
- the use of a non-renewable resource within the production process generates a
flow of pollution that affects negatively the evolution of the environmental stock,
- households are characterized by a non-separable utility function whose arguments are consumption and the stock of environment,
- authorities use tradeable pollution permits in order to control the emissions,
- we use an endogenous growth framework in which innovations are vertical, and
knowledge is not embodied inside intermediate goods.
We describe exhaustively the general equilibrium and we show how each flow
is affected by the environmental policy. Moreover, we prove that the price level of
permits does not matter; it is the growth rate of this price that is the key channel
of transmission of this policy. Finally, we show that implementing an environmental
policy (or making it more stringent) leads to more (respectively less) efforts in
research when households value their environment a lot (resp. a little). In all cases,
this favours growth.
17
Appendix
A. Research sector
A1. Consider an time interval (t, t+∆t). If knowledge in sector i is Bit at time t,
its value at t + ∆t is a random variable Bit+∆t which can take two values: Bit+∆t =
Bit + σBt with probability λLRDit ∆t (one innovation during the time interval),
and Bit with probability 1 − λLRDit ∆t (no innovation). One gets E(Bit+∆t ) =
(Bit + σBt )λLRDit ∆t + Bit (1 − λLRDit ∆t) = Bit + λσLRDit Bt ∆t. If ∆t tends to zero,
we have
E(Ḃit ) = λσLRDit Bt .
Recall that, since Bt =
R1
0
Bit di, we have on average
Ḃt = λσLRDt Bt
(see equation (2) in the main text).
A2. The R&D firm in sector i maximizes the sum of present values of current
Rt
R∞
profits 0 (vt Bit − wt LRDit )e− 0 ru du dt, subject to the constraint Ḃit = λσLRDit Bt .
The Hamiltonian of this program is H = (vt Bit − wt Lit )e−
Rt
0
ru du
+ χt λσLRDit Bt ,
where χt is the costate variable. The condition ∂H/∂LRDit = 0 leads to
wt e−
Rt
0
ru du
(26)
= χt λσBt .
The condition ∂H/∂Bit = −χ̇t leads to
vt e−
Rt
0
ru du
= −χ̇t .
Remark 1: Different formulation of (26)
Integrating (27) between t and +∞ gives
the transversality condition
R∞
t
vs e−
(27)
Rs
0
ru du
ds = −χ+∞ + χt . From
lim χt Bit = 0,
t→+∞
we have
lim χt = 0
t→+∞
because Bit is bounded below by Bi0 > 0 and Bit is an increasing function of t. Thus
Rs
R∞
we have χt = t vs e− 0 ru du ds.
18
Let us Vt =
R∞
t
vs e−
Rs
t
ru du
ds the value at t of one unit of knowledge (see (12) in
the main text). Then we have χt = Vt e−
Rt
0
ru du
: it is the present value of Vt at 0.
Finally, (26) can be rewritten wt = Vt λσBt , that is, exactly condition (14).
Remark 2: Willingness to pay (w.p.) for one unit of knowledge
Differentiating H with respect to Bit , one gets the w.p. of i at time 0: ∂H/∂Bit =
χt λσLRDit = λσLRDit Vt e−
Rt
ru du
. Thus the w.p. at time t is vit = λσLRDit Vt .
R1
Finally, the w.p. of the R&D sector is vRDt = 0 vit di = λσLRDt Vt , that is, equation
0
(16).
B. Computation of the equilibrium
• At steady-state, differentiating (1) with respect to time gives
gY = (1 − α)gR + νgB .
(28)
Using (6), (2), (18) and the fact that gE = gR , as we show it in section 3.1, we
get
(ε(1 − α − ω) + ω)gR = r − ρ − ενλσLRD ,
(29)
which is a first relation between gR , r and LRD (or, equivalently gθ , r and
LRD ).
• Differentiating (11) with respect to time yields
gpR = −αgR + νλσLRD − gθ .
This relation, together with (17), gives
r = −αgR + νλσLRD − gθ ,
(30)
which is a second relation between gR , r and LRD (or, equivalently gθ , r and
LRD ).
• Differentiating (14) with respect to time, we get
V̇ /V = gw − gB .
19
(31)
From (10), we know that gw = gY , thus (31) and (28) give
V̇ /V = (1 − α)gR + (ν − 1)gB .
(32)
Moreover, equation (12), (15) and (16) together yield
V̇ /V = (−λσν/α)(1 − LRD ) − λσLRD + r.
(33)
Equations (32), (2) and (33) provide us a third relation between gR , r and
LRD (or, equivalently gθ , r and LRD ):
(1 − α)gR = (λσν/α)(1 − α)LRD − (λσν/α) + r.
(34)
(29), (30) and (34) give the values of gR , r and LRD at the steady-state equilibrium. Moreover, using (2) and (3), we get gB and LY .
The prices are:
r = λσν − gθ (1 − α),
(35)
1−α
wt = αBtν Lα−1
,
Y t Rt
(36)
pRt = (1 − α)Btν LαY t Rt−α (θ0 egθ t )−1 ,
(37)
vtY = νBtν−1 LαY t Rt1−α ,
(38)
1−α
vtRD = αLRDt Btν−1 Lα−1
,
Y t Rt
(39)
1−α
Vt = (α/λσ)Btν−1 Lα−1
,
Y t Rt
(40)
where Rt = −S0 gR egR t and Bt = B0 egB t .
C. Revenues and expenditures in the research sector
First, let us show that vRD B = sRD (subsidy).
At each time t, the debt is D = BV (the unique asset), that gives Ḋ = ḂV +B V̇
and rD = rBV = (V̇ /V + v/V )BV = V̇ B + vB. The total (knowledge sold) +
(borrowings) - (wages) - (interests on the debt), is given by vB + ḂV +B V̇ −wLRD −
V̇ B − vB = ḂV − wLRD , which is nil, since profits are nil. Thus we have the result:
20
v RD B = sRD , which gives
sRD /Y = wLRD /LY .
We can now compute the debt Ḋ/Y and the interests rD/Y .
Debt: From (14), we have D = BV = w/λσ, and thus Ḋ = ẇ/λσ. Using
equation (10), we get Ḋ = αgY /λσLY .
Interests: Above we saw that rD = V̇ B + vB.
We know that vB/Y =
v Y B/Y + v RD B/Y = ν + αLRD /LY . Moreover, we have B V̇ /Y = Ḋ/Y − ḂV /Y =
αgY /λσLY − αLRD /LY . Finally we get rD/Y = αgY /λσLY − αLRD /LY + ν +
αLRD /LY = αgY /λσLY + ν.
21
References
[1] P. Aghion and P. Howitt, AModel of Growth Through Creative Destruction,
Econometrica 60, 323—351 (1992).
[1] P. Aghion and P. Howitt, Endogenous Growth Theory, The MIT Press (1998).
[2] K.J. Arrow, The Economic Implications of Learning-by-Doing, Review of Economic Studies, 29(1), 155-173 (1962).
[3] E.B. Barbier, Endogenous Growth and Natural Resource Scarcity, Environmental and Resource Economics, 14, 51—74 (1999).
[4] R.J. Barro and X. Sala-i-Martin, Economic Growth, McGraw-Hill, Inc (1995).
[5] A. Beltratti, Models of Economic Growth with Environmental Assets, Kluwer
Academic Publishers, Dordrecht (1996).
[6] U. Chakravorty, J. Roumasset et K. Tse, Endogenous Substitution among Energy Resources and Global Warming, Journal of Political Economy, 6, 1201-1234
(1997).
[7] P.S. Dasgupta and G.M. Heal, The Optimal Depletion of Exhaustible Resources,
Review of Economic Studies, 41, 3—28 (1974).
[8] P.S. Dasgupta and G.M. Heal, Economic Theory and Exhaustible Resources,
UK, Oxford University Press (1979).
[9] P.S. Dasgupta, K.G. Mäler, G.B. Navaretti and D. Siniscalo, On Institutions
that Produce and Dissipate Knowledge, Mimeo (1996).
[10] P.C. Garg and J.L. Sweeney, Optimal Growth with Depletable Resources, Resources and Energy 1, 43—56 (1978).
[11] R. Gradus and S. Smulders, The Trade-off between Environmental Care and
Long-term Growth; pollution in three proto-type growth models, Journal of
Economics, 58 , No. 1, 25-51 (1993).
[12] A. Grimaud, Pollution Permits and Sustainable Growth in a Schumpeterian
Model, Journal of Environmental Economics and Management, 38, 249-266
(1999).
22
[13] A. Grimaud, Schumpeterian Growth with Complete Markets, Mimeo IDEI
(2002).
[14] A. Grimaud and L. Rouge, Non-Renewable Resources and Growth with Vertical
Innovations: Optimum, Equilibrium and Economic Policies, Journal of Environmental Economics and Management, 45, 433-453 (2003).
[15] G. Grossman et E. Helpman, Innovation and Growth in the Global Economy,
Cambridge MA, MIT Press (1991).
[16] M. Hoel et S. Kverndokk, Depletion of Fossil Fuels and the Impacts of Global
Warming, Resource and Energy Economics, 18, 115-136 (1996).
[17] International Panel of Climate Change (IPCC), Climate Change, 3 Vols. Cambridge University Press, Cambridge, UK (1995).
[18] C.I. Jones, Population and Ideas: a Theory of Endogenous Growth, Mimeo
(2001).
[19] C.I. Jones, Growth and Ideas, Working Paper, Department of Economics, U.C.
Berkeley and NBER (2003).
[20] C.D. Kolstad and J.A. Krautkraemer, Natural Resource Use and the Environment, in: Handbook of Natural Resources and Energy Economics, vol. III, A.V.
Kneese and J.L. Sweeney (eds.), Elsevier Science Publishers, 1219-1265 (1993).
[21] M.E. Porter and C. van der Linde, Toward a New Conception of the
Environment-Competitiveness Relationship, Journal of Economic Perspectives,
9(4), 97-118 (1995).
[22] C.M. Scholz and G. Ziemes, Exhaustible Resources, Monopolistic Competition
and Endogeneous Growth, Environmental and Resource Economics, 13, 169—185
(1999).
[23] P. Schou, A Growth Model with Technological Progress and Non-renewable
Resources, Mimeo, University of Copenhagen (1996).
[24] P. Schou, Polluting Non-Renewable Resources and Growth, Environmental and
Resource Economics, 16, 211-227 (2000).
23
[25] P. Schou, When Environmental Policy is Superfluous: Growth and Polluting
Resources, Scandinavian Journal of Economics, 104, 605-620 (2002).
[26] S. Scotchmer, Standing on the Shoulders of Giants: Cumulative Research and
the Patent Law, Journal of Economic Perspective, Symposium on Intellectual
Property Rights (1991).
[27] S. Smulders, Entropy, Environment, and Endogenous Economic Growth, International Tax and Public Finance, 2, 319—340 (1995).
[28] S. Smulders, Endogenous Growth Theory and the Environment, dans J.C.J.M.
van den Bergh (ed), Handbook of Environmental and Resource Economics, Edward Elgar, 610-621 (1999).
[29] J. Stiglitz, Growth with Exhaustible Natural Resources : I) Efficient and Optimal Growth, II) The Competitive Economy, Review of Economic Studies Symposium, 41, 123—152 (1974).
[30] O. Tahvonen, Fossil Fuels, Stock Externalities, and Backstop Technology, Canadian Journal of Economics, XXX, 855-874 (1997).
[31] F. van der Ploeg et C. Withagen, Pollution Control and the Ramsey Problem,
Environmental and Resource Economics, 1, 215-236 (1991).
24