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Growth, Innovation and Environmental Policy:
Clean vs. Dirty Technical Change
by
Ana Balcão Reisa , Maria A. Cunha-e-Sáa and Alexandra Leitãob,∗
a
Universidade Nova de Lisboa, Faculdade de Economia,
Campus de Campolide, PT-1099-032, Portugal
b
Universidade Católica Portuguesa, Centro Regional do Porto,
Faculdade de Economia e Gestão,
Rua Diogo Botelho, 1327, PT-4169-005, Portugal
February 19, 2008
Abstract
This paper focuses on a two sector endogenous growth model with
environmental quality, with two goods and factors, one clean and
one dirty. Endogenous technological change creates either clean or
dirty innovations, depending on relative profitability. The reduction
of emissions intensity of aggregate output is achieved by changing the
dirty-bias of technology in the economy. The decentralized equilibrium
growth rate can be either below or above the optimal one. Distortions
caused by the monopoly power and the R&D activity tend to decrease
the decentralized equilibrium growth rate relative to the optimal one,
while the environmental externality works in the opposite direction.
We study both first and second-best policies.
Keywords: pollution; endogenous growth; innovation; environmental policy; decentralized equilibrium; optimal equilibrium;
∗
Email: [email protected]
1
1
Introduction
There is substantial agreement upon that climate is changing due to human activities. The last reports of the Intergovernmental Panel on Climate
Change (IPCC) make clear the urgency of the combat to climate change.
The solution for this serious problem implies a significant reduction of the
current level of greenhouse gas emissions (GHG). According to IPCC it is
still possible achieve the strategic objective of limiting the global average
temperature increase to not more than 2o C above pre-industrial levels, a
rise which scientists say represents the limit of safety, beyond which catastrophic changes will occur. To meet this level, the last report of the IPCC
indicates that the peak of global emissions will have to occur at about 2015,
after which global emissions must go down substantially, between 50 and
85% in 2050 compared to 1990 (Fourth Assessment Report of the IPCC
[12]).
International community is answering to this challenge through international agreements, such as the Kyoto Protocol, under which the industrialized countries have compromised to reduce its emissions. However, it is
urgent to speed up the reduction of the emissions and to reach more ambitious goals. The Spring European Council, in March 2007, endorsed an EU
objective of a 30% reduction in GHG emissions by 2020 compared to 1990
(Presidency Conclusions of the Brussels European Council [17]). In June, in
the annual G8 summit, the leading industrialized nations agreed on the need
of joint efforts on climate protection and to establish common goals for the
reduction of GHG. These countries agreed with the importance of diminishing by half GHG emissions up to 2050 and reaffirmed the importance of
United Nations for the negotiation of a new international agreement for the
post-2012 period, which should build upon and broaden the Kyoto Protocol
with the widest possible participation (Chair’s Summary of the G8 Summit
[8]).
The reduction and stabilization of the GHG emissions demand a joint
effort of governments, firms and individuals. There is need for deep changes
in the way we produce and consume, towards a more rational use of energy.
New technology is critical on the response to climate change. It is important
in determining the future GHG emission levels as well as feasibility and costs
2
of emission reduction efforts. With the current technology and equipment, a
reduction of emissions entails a fall in energy inputs, which is likely to lower
output below its potential level. Thus, there is a trade-off between economic
growth and environmental quality. In endogenous growth models, in order
to avoid stagnation along the optimal path, the substitution of clean inputs
for dirty ones either through increased abatement or technological progress
is required, as shown in Bovenberg and Smulders [4], Stokey [21], Elbasha
and Roe [7], and Reis [18], among others. Therefore, technological progress
and the development of environmental friendly technologies will protect the
environment and, at the same time, drive economic growth and increase the
countries’ competitiveness, lowering the conflict between economic growth
and its negative environmental impacts.
In such a context, the adoption of a technology policy that stimulates
R&D is very important. Moreover, technological change is responsive to environmental policy instruments, such as emission constraints, emission trading systems or taxes on emissions, which will induce the development and
adoption on a large scale of new low-emission technologies and the increase
of energy efficiency.1 Studies that compare alternative policy instruments
find that when comparing a policy with only R&D subsidies to an emission
tax, the emissions-based policies performed substantially better (IPCC [12]).
This paper focuses on a two sector endogenous growth model with clean
and dirty innovations, in an economy that cares about the environment. On
the one hand, there is evidence that in a market economy research tends to
be under-provided relative to the social optimum.2 On the other hand, the
extent to which research is environmental oriented and its policy implications
are a matter of discussion.
The aim of the paper is to enhance the importance of technological
change for lowering the environmental consequences of economic growth,
such as the emissions of GHG. Technical change is crucial for economic
growth, however to reconcile growth and environmental concerns, appropriate policies are required to stimulate adequate technological advance. In
1
Carraro and Siniscalco [5] present studies showing that large corporations typically
adjust to environmental policy measures through innovation, rather than by switching
inputs or reducing output. Moreover, they also mention the fact that without innovation
very high taxes are required to curb down CO2 emissions.
2
Jones and Williams [14], [15].
3
particular, we examine how environmental policy affects both the rate and
the type of technological change. The question is not only “Are there limits
to growth?” as in Stokey [21], but also in what kind of innovations are we
interested in and what incentives should policy give.
Other papers have focused on the importance of endogenous research in
market economies with environmental damages, like Elbasha and Roe [7]
and Grimaud [9]. Our paper differs from theirs since we consider both clean
and dirty innovations. More recently, Ricci [19] and Hart [10], [11], analyze,
in a vintage model, the effects of environmental regulation on the quantity
of research effort (how much research is performed), and on its quality (in an
environmental dimension). We follow a different framework which focuses
primarily on the relative productivity of dirty and clean technologies.
The structure of the modelled economy follows the work of Acemoglu [1]
and Acemoglu and Zilibotti [2] where innovations are carried out by profit
maximizing firms. We assume two goods and two factors of production, one
clean and one dirty. The polluting good is produced from the dirty factor
and a range of dirty complementary intermediates. Pollution is a by-product
of the dirty intensive good production. Endogenous technological change
extends the range of intermediate inputs available in the market that can be
either new clean and new dirty complementary intermediates. R&D firms
develop new clean or new dirty complementary intermediates depending on
relative profitability. In particular, we focus our attention on the relative
productivity of dirty and clean technologies, which will be the measure of
the dirty-bias in the economy, as in Acemoglu and Zilibotti [2]. This feature
of the model enables the analysis of the equilibrium versus the efficient
dirty-bias in the economy in the presence of environmental externalities and
the corresponding policy implications. Moreover, since energy efficiency
increase is crucial on the response to environmental problems, we focus on
the reduction of emissions per unit of output and not in absolute levels.3
3
This approach will be followed, for example, by Canada. Government intends to
implement a domestic trading system in which emission reduction credits can be traded
to limit carbon emissions from much of the economy’s energy and industrial sectors. The
emission targets will be expressed in terms of a reduction in emission intensity targets
rather than in absolute levels of emissions reductions. This approach has raised questions
about the international tradability of Canadian emissions credits because they are different
from other credits, such as European credits, which represent an absolute amount of
emissions reduction (Amano and Sedjo [3]).
4
The solutions for the laissez-faire economy, the social planner’s problem
and the regulated economy are considered, assuming three imperfections:
monopoly power in the market for intermediates, surplus appropriability
problem and pollution. We compare the decentralized equilibrium solution
to the efficient one. The corresponding growth rates are, in general, different.
Unlike Acemoglu [1], where the decentralized equilibrium growth rate is
below the optimal one, in this case we may obtain that the equilibrium
growth rate is too high or too low. Distortions caused by the monopoly
power and the R&D activity tend to decrease the decentralized equilibrium
growth rate relative to the optimal one, while the environmental externality
works in the opposite way. This last effect is larger the more consumers value
the environment. As well, emissions intensity of aggregate output is larger
in the decentralized economy than in the optimal solution, as expected.
We identify the first-best regulatory instruments and show how relevant
environmental concerns are to growth and to clean technical change, and,
thus, how environmental policy influences the nature of technical change.
We show that the optimal policy encourages a change in the quality of
research, in favor of the clean sector of the economy. According to our
specification, it is possible to have the decentralized growth rate below the
optimal rate, if the concern with the environmental externality is not too
large. In this case, the optimal solution is characterized by boosting growth
and decreasing the dirty-bias in the economy and, therefore, decreasing the
emissions intensity of aggregate output. We also study a second best context,
when there are multiple imperfections but insufficient policy instruments
available to correct them.
The paper is organized as follows. The model is presented in Section
2. The decentralized equilibrium is derived in Section 3. In Section 4, the
solution to the social planner’s problem is discussed and compared to the
decentralized solution. Section 5 focuses on policy implications. Section 6
summarizes the main conclusions. Technical details are presented in the
appendices.
5
2
The model
The economy is closed and produces an aggregate output from two commodities, which use primary factors of production and a set of differentiated
intermediate inputs. The model builds in Acemoglu [1] and Acemoglu and
Zilibotti [2], introducing preferences for environmental quality.
Consumers There are many identical infinitely lived consumers who get
utility from consumption of aggregate output and environmental quality.
Utility of a representative consumer is given by
Z ∞
U=
(log C + µ log Q) e−ρt dt
(1)
0
where C is consumption of the aggregate output, Q measures the quality of
the environment, µ reflects environmental preferences and ρ is the rate of
time preference. The utility function is increasing and strictly concave in C
and Q as long as µ > 0. Also, we assume the representative consumer values
the consumption more than the environmental quality, such that µ < 1.
Consumers are endowed with two primary factors of production: a clean
(L) and a dirty (Z) factor of production.4
Final output sector The aggregate output Y is produced from two goods,
YL and YZ , through a Cobb-Douglas production function, as in Acemoglu
and Zilibotti [2]5
Y = YLγ YZ1−γ
(2)
where γ ∈ (0, 1) is the elasticity of aggregate output with respect to YL .
YL is intensive in the clean factor, L, while YZ uses the dirty factor, Z,
intensively. YL and YZ production functions are given by
µZ NL
µZ NZ
¶
¶
1
1
1−β
1−β
β
xL (j)
dj L
and YZ =
xZ (j)
dj Z β
YL =
1−β
1
−
β
0
0
(3)
4
We do not allow factor accumulation in order to simplify the analysis.
Our approach deviates from Acemoglu’s approach [1]. We assume elasticity of substitution between the two factors of production (and between the two goods) equal to 1 and
only focus on the relative physical productivity of dirty and clean factors and not on the
corresponding relative factor rewards. Consequently, we focus on factor-augmenting technical change and not in factor-biased technical change. Moreover, we assume a constant
relative supply of primary factors of production, in contrast to Acemoglu [1]. A possible
extension of our work is to assume a general elasticity of substitution between the two
factors. In such case, we can follow the work of Acemoglu more closely.
5
6
where β ∈ (0, 1), and L and Z are assumed to be supplied inelastically.
The L-intensive good is produced from the clean factor and a range of Lcomplementary (L-augmenting) intermediates (xL ). The range of intermediate inputs that can be used with factor L, and available at a given time
period, is denoted by NL . The amount of input j used is denoted by xL (j).
The production function for the other good, YZ , uses the dirty factor intensively and Z-complementary intermediates (xZ ). NZ and xZ (j) are similar.6
The range of intermediates used in the production of the two goods, NL
and NZ , may be different. A greater NL enables the production of a greater
level of YL for a given quantity of L, that is, it improves the productivity of
the clean factor, while an increase in NZ improves the productivity of the
dirty factor. The ratio (NZ /NL ) determines the relative productivity of dirty
and clean technologies, which will be the measure of the dirty-bias in the
economy, as in Acemoglu and Zilibotti [2]. For a given state of technology,
that is, given NL and NZ , the production functions exhibit constant returns
to scale. There will be aggregate increasing returns, when NL and NZ are
endogenously determined.
The markets for YL and YZ are perfectly competitive. The set of differentiated inputs are bought from the intermediate inputs sectors.
Intermediate inputs sectors Each input j ∈ [0, NL ] and j ∈ [0, NZ ] is
supplied by a technology monopolist. Each monopolist faces a marginal
cost, ψ, in terms of the final good, of producing intermediate inputs, which
is the same for all intermediate inputs. The monopolist sets the price for
the intermediate input that supplies in the market, χL (j) or χZ (j) .
R&D sector Research and development are undertaken in this sector. It
is assumed that only the final good is used in generating new innovations (lab
equipment specification), which are added to the set of available intermediate
inputs. Technical change is modeled as the invention of dirty and clean
augmenting intermediates. The production functions for the innovations
are
ṄL = η L RL
and
ṄZ = η Z RZ
(4)
where RL and RZ are spending on R&D for the clean and dirty factor6
For example, we can think of Y as the production of energy, with YL as CO2 free
energy and YZ as fossil fuel energy.
7
intensive good (in terms of final good), respectively. According to this
production function, one unit of final good spent on R&D directed at Lcomplementary intermediates will generate η L new varieties of clean factorcomplementary intermediates. ṄZ is explained similarly. The parameters
η L and η Z allow the costs of the two types of innovations to differ.
Research is motivated by the future benefits which follow from the discovery of a new variety. A firm that discovers a new variety receives a patent
on this intermediate input and becomes the technology monopolist. More
innovations will be generated in response to greater profits and R&D firms
develop new clean or new dirty complementary intermediates depending on
relative profitability.
Budget constraint The budget constraint of the economy is
C + I + R = Y, that is
∙ Z
C+ ψ
NL
xL (j) dj + ψ
0
Z
¸
NZ
xZ (j) dj + (RL + RZ ) = Y
0
(5)
where I denotes investment, and R is total R&D expenditure. Consumption,
investment and R&D expenditure are all the possible competing uses of the
final good.
Environmental quality We model the quality of the environment as a
flow variable. Environmental damages are a by-product of the dirty sector.
Environmental quality is measured by the inverse of emissions7
Q=
1
1
=
,
EZ
δYZ
0< δ<1
(6)
where emissions, EZ , are proportional to the production of YZ . δ is a technology parameter that quantifies the detrimental effect of YZ on the environment. Since energy efficiency increase is crucial on the response to environmental problems, we focus our attention on the reduction of emissions
intensity,
3
EZ
Y .
Decentralized equilibrium
In this section, we determine the decentralized equilibrium without government intervention. Agents take the environmental quality as given.
7
See, for example, Elbasha and Roe [7].
8
Given NL and NZ , an equilibrium is a path for prices and a path for
quantities such that (i) consumers decide on consumption and asset accumulation taking prices, the interest rate and the environmental quality as
exogenous functions of time; (ii) producers of YL and YZ choose quantities of
the clean and dirty factors and corresponding complementary intermediates,
xL (j), j ∈ [0, NL ], and xZ (j), j ∈ [0, NZ ], taking the prices of those goods,
the prices of primary and intermediate factors and the range of intermediate inputs as given; (iii) intermediate inputs producers set prices, χL (j) or
χZ (j), that maximize profits, taking as given the downward-sloping demand
for their products; (iv) all markets clear.
We assume that all agents have perfect foresight, so that at each moment
of time the entire path of prices is known.
The markets for YL and YZ are competitive, so market clearing implies
that their relative price p is given by
pZ
p=
=
pL
∂Y
∂YZ
∂Y
∂YL
∂Y
∂YZ
∂Y
∂YL
where
1−γ
=
γ
µ
YZ
YL
¶−1
(7)
The greater the supply of YZ relative to YL , the lower is its relative price p.
Let the price of the final good be the numéraire, the following relationship
between the prices of the two goods holds
pγL p1−γ
Z
γ γ (1 − γ)1−γ
=1
(8)
Firms in the clean sector solve the following maximization problem
Z NL
M axL,{xL (j)} pL YL − ωL L −
χL (j) xL (j) dj
(9)
0
taking the price of their product, pL , the price of the primary factor, ω L ,
the prices of the intermediate inputs, χL (j), and the range of intermediate
inputs, NL , as given. Firms in the dirty sector face a similar problem. They
maximize profits, treating environmental damages as exogenous.
From the first-order conditions of these problems, the rewards for the
primary factors of production, ω L and ωZ , are obtained, as follows
ωL = β
pL YL
L
and
9
ωZ = β
pZ YZ
,
Z
(10)
and the demand for each intermediate input is given by
xL (j) =
µ
pL
χL (j)
¶1
β
L
and
xZ (j) =
µ
pZ
χZ (j)
¶1
β
Z
(11)
The demand for each intermediate input is the same for all j ∈ [0, NL ] and
j ∈ [0, NZ ] .
Each firm in the intermediate goods sectors, behaving as a monopolist,
chooses prices, χL (j) and χZ (j), that maximize profits, taking into account
the demand functions in (11). The monopolist supplying L-complementary
intermediate j solves the following maximization problem
M axχL (j) π L (j) = (χL (j) − ψ) xL (j)
(12)
Since the demand curve for the intermediate input (11) is iso-elastic, profits
are maximized by following the constant mark-up over marginal cost pricing
rule χL (j) =
ψ
1−β ,
j ∈ [0, NL ]. To simplify the algebra, we normalize the
marginal cost to ψ = 1 − β. In equilibrium, prices are χL (j) = χZ (j) = 1.
Therefore, price, quantity and, hence, the level of profits are the same
for all firms operating in each intermediate inputs sector. Using the prices
and demands above, the profits of technology monopolists are
1
π L = βpLβ L
1
and
π Z = βpZβ Z
(13)
Profit maximizing firms will produce more innovations in response to
greater profits. R&D firms develop new clean or new dirty complementary
intermediates depending on relative profitability. What is relevant for the
monopolists are not the instantaneous profits, but the net present discounted
value of profits.
To maintain asset market equilibrium, the rate of return from holding
equities (dividends plus changes in the value of the firms) must be equal to
the rate of return on a one period loan, r, that is
πL
V̇L
+
=r
VL
VL
and
πZ
V̇Z
+
=r
VZ
VZ
(14)
where r is the interest rate, potentially time varying. These equations relate
the present discounted value of future profits, V (the value of a firm in
the intermediate inputs sector), to the flow of profits, π. The V̇ term takes
10
care of the fact that future profits may be different from current profits, for
example, because prices are changing.
We will mainly focus on the long-run balanced growth path of this economy, where prices and the interest rate are constant. Thus, the V̇ terms are
0. Then, combining (13) and (14), yields
1
1
βp β L
VL = L
r
and
βp β Z
VZ = Z
r
(15)
The greater VZ is relative to VL , the greater are the incentives to develop
dirty factor complementary intermediates, NZ , rather than clean factor complementary intermediates, NL . Since VZ and VL are increasing in prices,
there are greater incentives to invent technologies producing more expensive goods.8
Firms in the R&D sector are allowed to freely enter into or exit from the
sector. Thus, in equilibrium, the value of a firm in the intermediate inputs
sector (Vi , i = L, Z) is equivalent to the cost of a new variety ( η1 , i = L, Z).
i
We assume a balanced growth path (BGP), or steady-state equilibrium,
where prices pL and pZ are constant, and NL and NZ , grow at the same and
constant rate. In equilibrium, along a BGP, all variables grow at the same
rate: Ẏ /Y = ṄL /NL = ṄZ /NZ = Ċ/C = ẎZ /YZ = ẎL /YL = g. In order to
generate growth, the number of new designs must be expanding over time.
This occurs if the spending on R&D is increasing. More spending means
more ideas, sustaining growth in the model. In this case, the growth in ideas
is clearly related to the final output growth (ṘL /RL = ṘZ /RZ = g).
Technology monopolists are willing to innovate in both sectors if the
following condition holds
η L VL = ηZ VZ
or
ηLπL = ηZ πZ
(16)
According to this condition, it is equally profitable to invest money to invent
L and Z-complementary intermediate inputs, so that along the BGP, NL
and NZ can both grow. After substitution of (15), it implies
ηZ
ηL
µ
pZ
pL
¶1
β
8
Z
=1
L
(17)
Another determinant of innovations is the cost of different innovations, as we will see
later.
11
This condition can be solved for the dirty-bias in the decentralized economy,
(NZ /NL ).
Substituting (11) into the production functions (3) we obtain the market
productions of YL and YZ , given NL and NZ . Substituting these into (7) and
solving for p = pZ /pL we get the relative price of the two goods as a function
of the relative productivity of dirty and clean technologies (NZ /NL ), and of
the relative factor supply, Z/L:
p=
µ
1−γ
γ
¶β µ
NZ Z
NL L
¶−β
(18)
From (11) and (18), the relative demand of intermediate inputs as a
function of (NZ /NL ) is
µ
xZ
xL
¶DE
=
µ
1−γ
γ
¶µ
NZ
NL
¶−1
(19)
Substituting the relative price p, (18), in (17), and solving for (NZ /NL )
we obtain
µ
NZ
NL
¶DE
=
µ
ηZ
ηL
¶µ
1−γ
γ
¶
(20)
This equation defines the relative physical productivity of dirty and clean
factors along the BGP, which is positively affected by the relative productivity of the R&D labs and the relative weight in the production of Y.
Substituting (20) in (19), we obtain the relative demand of intermediate
inputs, with endogenous technology
µ ¶DE µ ¶−1
ηZ
xZ
=
xL
ηL
(21)
The solution to the consumer’s problem is obtained by solving the following intertemporal maximization problem
Z ∞
M ax{C} U =
(log C + µ log Q) e−ρt dt
0
s.t. C + ȧ = ra + ω L L + ω Z Z
where a denotes assets, and r is the interest rate. From the optimality
conditions, we obtain
9
Ċ
C
= r − ρ.9
Moreover, the transversality condition must hold: limt−→∞ λae−ρt = 0.
12
Along a BGP, all variables grow at the same rate, g = r − ρ. The
free-entry condition for the technology monopolists working to invent Lcomplementary intermediate inputs, VL =
1
β
1
ηL ,
together with (15), imply
η L βpL L = r, in the steady-state. Using this condition together with (8),
(18), to substitute for pL , we obtain the growth rate of the economy for a
given (NZ /NL ),
³
´ 1 µ γ ¶1−γ µ N Z ¶1−γ
Z
1−γ β
γ
L−ρ
g = η L β γ (1 − γ)
1−γ
NL L
(22)
The long-run growth rate of the economy, in terms of the parameters of the
model and the endowments, is summarized in the following proposition.10
Proposition 1 The long-run growth rate of the decentralized economy is
given by
´1
³
β
(η Z Z)1−γ (η L L)γ − ρ
g DE = β γ γ (1 − γ)1−γ
(23)
Proof. Combining (20) with (22), the result follows.
Consumer’s utility depends on environmental quality, which is not taken
into account by the producer of the polluting good. Thus, the correct price
for the polluting good will have to be determined by a regulator that maximizes consumers’ welfare by internalizing the pollution externality. Besides
the pollution externality there are other distortions that come from the R&D
activity.
Production of intermediates is lower than the efficient level due to the
market power of the technology monopolists. The mark-up of price over
marginal cost (monopoly distortion) reduces sales of intermediate inputs
when compared to the optimum level.
Finally, there is another distortion in the model that causes R&D to differ
from its optimal level (from now on, it will be denoted by R&D distortion):
the surplus appropriability problem or “consumer” surplus effect.11 The
10
We now briefly analyze the stability properties of the model outside the BGP. The
equilibrium condition for the relative profitability of innovations along the BGP is given
by ηηZ VVZ
= 1 (see (16)). If ηηZ VVZ
> 1 (< 1) it will only be undertaken R&D in dirty
L L
L L
complementary (clean complementary) intermediates. Since the relative profitability of
Z
is decreasing in N
creating dirty complementary intermediates VVZ
NL (combine (15) and
L
(18)), the system is stable.
11
See Jones [13], Jones and Williams [15], Romer [20].
13
market values research according to the stream of profits that are earned
from the new design. The inventor of the new design captures this monopolist profit. However, the potential gain to society from inventing the good is
the whole “consumer” surplus, under the demand curve (11), evaluated at
the optimal allocation. Since the inventor of a new design captures only part
of this surplus, there is underinvestment relative to what is socially optimal.
The profits that the market offers to potential innovators are substantial,
however they are still far short of the total gain to society from innovations.
This gap between social and private returns suggests that large gains are
still available from encouraging research.12
We address these issues in the next section, by solving for the social
planner’s problem.
4
The social planner’s problem
The social planner’s problem for this economy can be stated as
Z ∞
M ax
(log C + µ log Q) e−ρt dt
(24)
0
s.t. (2), (3), (4), (5), (6)
L = L̄; Z = Z̄
and NL (0) , NZ (0) > 0 given
12
Empirically, there is evidence that the market tends to provide too little research.
Jones and Williams [14] present empirical evidence, using econometric estimates of returns
to R&D, that socially optimal R&D investment is at least four times greater than actual
spending (accounting for the current patent system and subsidies to research). This
clearly suggests the existence of underinvestment in R&D. However, their approach does
not identify which factors determine that result. Knowing them may have important
policy implications. Jones and Williams [15], using a calibrated model, and in the absence
of taxes and subsidies, find that the decentralized economy typically underinvests in R&D
relative to what is socially optimal and that the main force promoting underinvestment is
the surplus appropriability problem.
14
Thus, the current value Hamiltonian for the social planner’s problem is
h
i
H = [log C + µ log Q] + pC YLγ YZ1−γ − I − RL − RZ − C
(25)
µZ NL
∙
¸
¶
1
+rL
xL (j)1−β dj Lβ − YL
1−β
0
µZ NZ
∙
¸
¶
1
1−β
β
xZ (j)
dj Z − YZ
+rZ
1−β
0
∙ Z NL
¸
Z NZ
xL (j) dj + ψ
xZ (j) dj − I
+pI ψ
0
0
h
i
+λL η L RL + λZ η Z RZ + pQ (δYZ )−1 − Q
where pC , rL , rZ , pI , λL , λZ , pQ denote the shadow prices corresponding to
the relevant constraints.
The first-order conditions for a maximum are given by
∂H
∂YL
∂H
∂YZ
∂H
∂C
∂H
∂Q
∂H
∂xL (j)
∂H
∂xZ (j)
∂H
∂I
∂H
∂RL
∂H
∂RZ
∂Y
− rL = 0
∂YL
∂Y
Q
= pC
− pQ
− rZ = 0
∂YZ
YZ
= pC
(26)
(27)
= C −1 − pC = 0
(28)
= µQ−1 − pQ = 0
(29)
= rL xL (j)−β Lβ + pI ψ = 0
(30)
= rZ xZ (j)−β Z β + pI ψ = 0
(31)
= −pC − pI = 0
(32)
= −pC + λL η L = 0
(33)
= −pC + λZ η Z = 0
(34)
1
x1−β Lβ − pI ψxL
1−β L
1
x1−β Z β − pI ψxZ
= ρλZ − rZ
1−β Z
λ̇L = ρλL − rL
(35)
λ̇Z
(36)
Also, the following transversality conditions must hold:
lim λL NL e−ρt = 0;
lim λZ NZ e−ρt = 0
t−→∞
t−→∞
15
(37)
If we solve (30) and (31) for the demands of the intermediate inputs, and
combine them with (32), (26), (27), (28) and (29), we obtain
x∗L (j)
µ
=
µ
x∗Z (j) =
rL
pC ψ
rZ
pC ψ
¶1
β
¶1
L=
β
Z=
Ã
Ã
∂Y
∂YL
ψ
∂Y
∂YZ
!1
β
L
− µ YCZ
ψ
(38)
!1
β
Z
(39)
The optimal demand for each input is the same for all j ∈ [0, NL ] and
j ∈ [0, NZ ], and it is constant along the BGP.13 ψ, the marginal cost of
producing intermediate inputs, corrects for the monopoly distortion. The
mark-up over marginal cost in the decentralized solution determines a lower
demand than optimal for the intermediate inputs. On the other hand, from
(27), it is clear that the shadow price of the Z-intensive good is now corrected
for the pollution externality. In the decentralized economy, where firms do
not account for the externality, the polluting good is overvalued by the
market.
From conditions (33) and (34),
λ̇L
λL
=
λ̇Z
λZ
follows. Thus, equating (35)
and (36), combined with (30), (31), (33), (34), (38), and (39) implies that
ηZ
ηL
µ
rZ
rL
¶1
β
Z
=1
L
(40)
Equation (40) can be solved for the the optimal dirty-bias, (NZ /NL )∗ . Also,
it shows how this ratio is influenced by the cost of innovations and by the
shadow price ot the two goods. The price of YZ in the decentralized economy,
pZ , is higher than its shadow price, rZ , which internalizes the environmental
externality. Thus, in the decentralized equilibrium, innovation directed at
Z-complementary intermediates is overvalued.
∂Y
From (26), rL = pC ∂Y
. Rearranging (27), and using the optimality
L
conditions (28) and (29), the shadow price for YZ can be written as rZ =
∂Y
κpC ∂Y
, where κ, which captures the pollution externality and is a constant
Z
along the BGP, is given by
Ã
κ=
13
µC
1 − ∂Y
∂YZ YZ
!
,
0<κ61
In the first-order conditions (35) and (36) this is already assumed.
16
(41)
Notice that the lower is κ, the larger the environmental externality. When
there is no externality, as in Acemoglu [1], κ is equal to 1.14
Substituting (38) and (39) into the production functions (3), we obtain
the optimal YL and YZ , given NL and NZ . Substituting these into the relative marginal product of YZ , given by (7), and solving, we get, implicitly, the
efficient marginal rate of transformation between the two goods, as a function of both relative productivity of dirty and clean technologies, (NZ /NL ),
and the relative factor supply, (Z/L), as follows
∂Y
∂YZ
∂Y
∂YL
−(1−β)
=κ
µ
1−γ
γ
¶β µ
NZ Z
NL L
¶−β
(42)
From (38), (39) and (42), it follows that the relative demand of intermediate inputs as a function of (NZ /NL ) is
µ
xZ
xL
¶∗
µ
1−γ
=κ
γ
¶µ
NZ
NL
¶−1
(43)
For a given state of technology, (NZ /NL ), the market economy relative demand of dirty complementary intermediates is higher than optimal (recall
equation (19)).
The efficient dirty-bias is sumarized in the next proposition.
Proposition 2 The efficient dirty-bias in the economy is
µ
NZ
NL
¶∗
=κ
µ
ηZ
ηL
¶µ
1−γ
γ
¶
=κ
µ
NZ
NL
¶DE
(44)
Since κ < 1, the efficient dirty-bias is smaller than the equilibrium one of
the decentralized economy, given by (20).
Proof. Substituting (42) and the above expressions for rL and rZ in
(40), and solving for (NZ /NL ), we get equation (44).
This equation shows that the technology is highly dirty-biased (high
NZ /NL ) in the market economy. The research effort is directed at new dirty
complementary intermediates more than what is efficient, as the benefits of
lower environmental damages are external. In the optimum, there is a higher
relative physical productivity of the clean factor of production. Moreover,
14
κ is studied in detail below.
17
the higher the environmental concerns (the smaller κ), the lower the efficient
dirty-bias. The policy implication of this result is to decrease the degree of
dirty-bias in the economy, promoting the innovation of new clean factor
complementary intermediates.15
With endogenous technology, the relative demand of intermediate inputs
is the same as in the decentralized economy (see (21)). From (43) and (44)
it follows that, in the market economy, the relative production of YZ is
larger than the efficient one, both for a given state of technology and with
endogenous technology. Consequently, the emissions intensity of aggregate
output is higher than optimal. The laissez-faire pollution is higher than
optimal, as expected.
From conditions (26), (28), (30), (33), (35), (38), together with (42) and
the production function, (2), and the relative marginal product of YZ , given
by (7), we obtain, after some algebra, the growth rate of the economy, for a
given (NZ /NL ),
µ ¶1 ³
´ 1 (1−γ)(1−β) µ γ ¶1−γ µ N Z ¶1−γ
1 β
Z
1−γ β
γ
β
β γ (1 − γ)
κ
L−ρ
g = ηL
ψ
1−γ
NL L
(45)
The long-run growth rate (and also the optimal growth rate of innovation)
of this economy is summarized in the following proposition.
Proposition 3 The optimal long-run growth rate of the economy is given
by
µ ¶1 ³
´ 1 1−γ
1 β
1−γ β
γ
g =
β γ (1 − γ)
κ β (ηZ Z)1−γ (η L L)γ − ρ
ψ
∗
(46)
Proof. Combining (44) with (45), the result follows.
The environmental externality influences the growth rate through (i) the
adjustment on the intermediate inputs, and (ii) the incentives to innovation.
The growth rate of the economy, given by (45), before incorporating the
efficient dirty-bias, captures the externality through the adjustment on the
intermediate inputs, for a given (NZ /NL ). However, the externality is only
15
However, after the initial level effect, the relative productivities of the clean and dirty
factors are constant since NZ and NL grow at a common rate g.
18
fully accounted for after incorporating the optimal dirty-bias of technology,
as shown in (46).
The optimal growth rate, (46), corrects for both the environmental exter³ ´1
β
nality (through κ) and the monopoly and R&D distortions (through ψ1 ).
While the environmental externality has a negative impact on the optimal
growth rate (κ < 1), the monopoly and R&D distortions have a positive
³ ´1
β
impact ( ψ1
> 1). The environmental externality contributes to reduce
g ∗ relative to g DE , while the monopoly and R&D distortions work in the
opposite way.
If the correction for the monopoly and R&D distortions prevails, then
g∗
> g DE . However, the optimal growth rate is lower than it would be
without environmental externalities. In such case, when the environmental
externality is not too large (large κ), the optimal solution is characterized
by boosting growth and decreasing the dirty-bias in the economy. If the
correction for the externality prevails, then g ∗ < g DE . The decentralized
growth rate will be larger than the optimal rate, the stronger the effects
of the environmental externality on welfare (the smaller κ). Proposition 4
summarizes the result.
Proposition 4 The decentralized equilibrium growth rate, g DE , can be smaller
or higher than the optimal growth rate, g ∗ .
So far, we have solved the model without taking into account that κ is
endogenously determined. In order to obtain κ, we must derive the conC
Y , at the optimal solution. Notice that (41) can
µ C 16
Substituting the consumption to output
(1−γ) Y .
sumption to output ratio,
be rewritten as κ = 1 −
ratio in κ, we obtain
κ = 1 − ϕ(κ), where
ϕ(κ) =
(47)
µρ
´ 1 1−γ
³ ´1 ³
β
β
γ γ (1 − γ)1−γ
(1 − γ) (1 − µ) ψ1
κ β (ηZ Z)1−γ (η L L)γ + (1 − γ) µρ
Let κ = {κs } represent the set of solutions for this equation.
16
See Appendix A for details. We follow Di Maria and Smulders [6], in a different
context. Appendix A also shows that YI , YR and YC ∈ (0, 1), as expected.
19
In the model the effects of the environmental externality are captured by
µ. When consumers do not value environmental quality (µ = 0), from (47),
it follows that κ is equal to 1. When environmental quality is as important
to consumers’ welfare as consumption (µ = 1), then κ is negative. The more
consumers value the environment (higher µ), and, therefore, the larger the
environmental externality, the lower κ, and the lower the optimal growth
rate of the economy.
Lemma 1 At the equilibrium value of κs , we have
∂κs
∂µ
< 0.
Proof. By analogy with stability analysis, we can show that
1 iff
∂κ(κs )
∂µ
∂(1−ϕ(κs ))
∂κ
<
< 0. In other words, at the equilibrium value of κs , as (1 − ϕ(κ))
is positively sloped, it has to cut the 45-degree line from above. This result
is shown in Appendix B.
The solution can be shown graphically. When 1 − γ 6 β (that is, when
the share of YZ on Y is smaller or equal than the share of Z on YZ ), 1 −ϕ(κ)
is concave. Figure 1 illustrates the left hand side and the right hand side of
equation (47), for a given µ, when 1 − γ 6 β.
When 1 − γ > β, 1 − ϕ(κ) is convex for κ < κ̂, and concave for κ > κ̂.17
Figure 2 illustrates the left and right hand sides of equation (47), for a given
µ, when 1 − γ > β.
In both cases, as µ increases, 1 − ϕ(κ) crosses the horizontal axis at a
higher κ.18 Also, when 1 − γ > β, the inflection point, κ̂, is increasing in µ.
Thus, the right hand side of equation (47) moves to the right.
There is a solution for equation (47) as long as ∃ κ : 1−ϕ(κ) > κ, that is,
³ ´1 ³
´1
1−γ
1−γ β
1 β
γ
(1 − γ) (1 − µ) ψ
γ (1 − γ)
(η Z Z)1−γ (η L L)γ κ β (1 − κ)−(1 − γ) µρκ−
γµρ > 0.
We assume a positive optimal growth rate of the economy, g ∗ > 0, that
³ ´1 ³
´1
h i β
β
β
ρ 1−γ
, where A = ψ1
(η Z Z)1−γ (η L L)γ .
γ γ (1 − γ)1−γ
is, κ > βA
"
17
The inflection point is κ̂ =
18
1 − ϕ (κ) crosses the horizontal axis at
"
κ̄ =
lows.
µρ(1−γ−β)
³ ´1
1
1 β γ γ (1−γ)1−γ β (η Z)1−γ (η L)γ
(1−γ+β)(1−µ) ψ
(
)
Z
L
µργ
³ ´1
1
1 β γ γ (1−γ)1−γ β (η Z)1−γ (η L)γ
(1−γ)(1−µ) ψ
(
)
Z
L
20
#
#
β
1−γ
.
β
1−γ
. Since
∂ κ̄
∂µ
> 0, the result fol-
κ
1
1 − ϕ (κ )
−
κ
κsE
κ
γ
1− γ
Figure 1: Equation (47), when 1 − γ 6 β
κ
1
1 − ϕ (κ )
κsE
κ̂
−
γ
1− γ
Figure 2: Equation (47), when 1 − γ > β
21
κ
This means that µ <
#
³
´ β
1−γ
ρ
(1−γ)A 1− βA
"
#
³
´
³
´ β
´ β
³
1−γ
1−γ
ρ
ρ
ρ
(1−γ)A
1−
(1−γ)ρ+γρ
+ βA
βA
βA
³
ρ
βA
´
"
< 1. In
other words, the solution we are considering is only possible
#
" for environmental preferences described by 0 < µ <
³
³
´ β
1−γ
ρ
(1−γ)A 1− βA
#
"
.
³
³
´
³
´ β
´ β
1−γ
1−γ
ρ
ρ
ρ
+
(1−γ)A
1−
(1−γ)ρ+γρ
βA
βA
βA
ρ
βA
´
A very high environmental concern by consumers would mean either zero or
negative growth.19
As well, from the maximization of the utility function, it has to be the
case that g ∗ <
ρ
1−µ .
If this condition holds for µ = 0, it also holds for
positive values of µ.
The comparative static results of the optimal growth rate, (46), are
summarized in the following proposition.20
Proposition 5 The optimal growth rate is higher (i) the larger the country’s
endowment of L and Z, (ii) the smaller the cost of new intermediate inputs
(larger η i , i = L, Z), (iii) the smaller the rate of time preference (smaller
ρ), and, (iv) the less consumers value the environment (lower µ).
Proof. Differentiating the optimal growth rate (46) with respect to the
relevant parameter, the results follow. The same results apply to the decentralized growth rate.
The comparison of the share of the clean and dirty goods in the aggregate
output in the decentralized equilibrium and in the optimum is summarized
in the following proposition.
Proposition 6 In the long-run, the share of the clean good in the aggregate
output in the decentralized equilibrium is lower than the optimal one, and the
opposite occurs with the share of the dirty good in the aggregate output, that
is, it is higher than optimal in the decentralized economy. Also, the more
consumers value the environment (the higher µ), the higher the difference
between the decentralized and the optimal shares just mentioned.
Proof. See Appendix E.
19
See Appendix C.
Also, we can show that the higher µ, the higher the consumption to output ratio,
C/Y, and the lower the share of final good devoted to investment, I/Y, and to R&D
expenditures, R/Y (see Appendix D).
20
22
Consequently, emissions per unit of output are higher in the laissez-faire
economy than in the optimal solution. Moreover, the more consumers value
the environment (higher µ), the lower the optimal emissions intensity of
³ ´∗
³
´γ ³
´ γ
γ
ηZ Z β
aggregate output, which is given by EYZ = δ YYZ = δκs 1−γ
=
γ
ηL L
³
´
DE
γ
and is constant along the BGP.
κs EYZ
Summarizing, for a given (NZ /NL ), the efficient relative production of
the polluting good, and, thus, the efficient level of emissions intensity, is
attained by adjusting the production of intermediate inputs. The decentralized relative demand for dirty complementary intermediates is higher than
the efficient one, as shown in (43). With endogenous technology, the optimal
dirty-bias, (44), allows for the optimal share of emissions in the aggregate
output. The internalization of the pollution externality and the reduction
of emissions per unit of output is achieved by moving R&D towards new
clean complementary intermediates and, therefore, increasing the relative
physical productivity of the clean factor along the BGP.
The long-run growth rate of the economy can be manipulated by policy
makers using conventional policies. The next section focuses on the policy
implications determined by the fact that the decentralized growth path is
different from the optimal one.
5
Policy implications
Since the decentralized equilibrium growth path is different from the optimal
one, and the the dirty-bias in the decentralized economy is different from
the efficient one, there is scope for government intervention. In this section
we look for policy instruments that align both growth paths and allow for
the efficient dirty-bias, thus, internalizing the environmental externality.
We begin by looking at first-best policies. There are three distortions in
the decentralized economy: (i) the surplus appropriability problem tends to
generate too little research, (ii) intermediate goods are produced by technology monopolists, producing less than optimal, and, finally, (iii) because
of the environmental externality, production of YZ is larger than optimal.
Three market failures require for three policy instruments: a technology
policy instrument that takes the form of a subsidy to the R&D sector to encourage research, a subsidy to the producers of intermediate inputs to equate
23
marginal costs to prices and neutralize the effect of monopoly pricing, and an
environmental policy instrument on producers of YZ to internalize pollution
externalities. In this last case, we assume an emission tax on the polluting
good, which operates by closing the gap between private and social costs of
the polluting activity. It makes the polluting activity less attractive, providing incentives to reduce it. When optimally choosing the level of the policy
instruments, the optimal allocation can be attained through a decentralized
competitive market driven equilibrium.
We first assume that all these instruments are available to policy makers.
However, because there might be insufficient policy instruments available to
correct the multiple imperfections, we also look at second-best solutions. If
growth is the most important target, we only need one policy instrument
to align the two growth paths. However, one single instrument may not be
enough to attain the optimal share of emissions in the aggregate output.
5.1
First-best policies
In order to determine the optimal policies, we introduce the policy changes
and compute the policy-ridden solution. The optimal level of the policy
instruments is obtained by comparison with the social planner’s solution.
Let us consider that the set of first-best instruments available are the
following: (i) the government pays a proportional subsidy, φ∗ > 1, to the
R&D sector for each new design, which is assumed to be the same for new
L and Z-complementary intermediates. The subsidy to the R&D sector
generates incentives to research, since it decreases the costs of a new variety
(given by 1/ (φ∗ ηi ) , i = L, Z). This has impact on growth by increasing
Ṅi , i = L, Z; (ii) the government pays a subsidy, s∗ , to each producer
of intermediate inputs, which is assumed to be the same for technology
monopolists producing xL and xZ , and, finally, since the purpose of the
environmental policy is to discourage production of YZ , (iii) a unit emission
tax, τ ∗Z , is levied on the polluting sector.
The optimal policy is summarized in the following proposition.
Proposition 7 The social optimum can be implemented through (i) a subsidy to the R&D sector equal to φ∗ = 1/ψ, (ii) a subsidy to the producers of
24
intermediate inputs equal to s∗ = β, and, (iii) a tax on emissions equal to
τ ∗Z =
=
C
µ
δ (1 − γ) Y
µρ
∙
¸
´ 1 1−γ
³ ´1 ³
1−γ β
1−γ
γ
β
1 β
γ
γ (1 − γ)
δ (1 − γ) (1 − µ) ψ
κs (η Z Z)
(η L L) + µρ
Proof. Comparing the policy-ridden solution given by
ψ (1 − s)
g=φ
1−β
µ
1−β
ψ(1 − s)
¶1
β
³
´1
1−γ
β
β γ γ (1 − γ)1−γ
(1 − τ δ) β (η Z Z)1−γ (ηL L)γ − ρ
with the social planner’s solution (46), the results follow. κs is the solution
for equation (47) and
∂κs
∂µ
< 0 holds.
The optimal tax increases with µ and decreases with the elasticity of Y
with respect to the polluting good.
The first-best policy, summarized in the previous proposition, implements not only the optimal growth rate of the economy, but also the optimal emissions intensity of aggregate output. In fact, τ ∗Z induces the efficient
demand for intermediates (together with the subsidy paid to the technology monopolists), and it implements the efficient dirty-bias in the economy.
The tax decreases the demand of the dirty intermediates and, therefore,
depresses the value of patents of new dirty complementary intermediates.
This will encourage research towards new clean complementary intermediates, therefore, increasing the clean-bias in the economy. We obtain the
optimal production of YZ , when compared to the production of YL , both for
a given state of technology, and with endogenous technology. Therefore, the
optimal level of emissions per unit of output is achieved.
5.2
Second-best policies
Because policy makers may not have all the instruments available simultaneously, we also specify a second best optimum, where the social planner has
available only one policy instrument. We assume that the primary target of
the policy-maker is to align the market and the optimal growth paths, as
long as the emissions intensity of aggregate output decreases, or, it is at the
most as in the decentralized economy.
25
In this case, we consider a tax on the polluting good and two different
kinds of subsidies to R&D. We compute the policy-ridden growth rate, then
equate it to the optimal growth rate and solve for the optimal level of the
policy variable.
(1) Tax on the polluting good If the pollution externality prevails over
the remaining distortions in the model (that is, κ is small) and the regulator
wants to decrease growth (g DE > g ∗ ), a per unit emission tax, τ Z , on
the polluting sector can be considered. The growth rate of the regulated
economy equates to the optimal one when
"
#
µ ¶ 1
1 1 1 1−γ
−
τZ =
κs > 0
δ δ ψ
(48)
Although the growth rate for the economy is optimal, the emissions
intensity of aggregate output is not. The tax moves the economy towards
the optimum, by adjusting (i) the relative demand of intermediates, for
a given state of technology, (ii) the dirty-bias (NZ /NL ). However, since
this second-best tax is smaller than the first-best one, it achieves neither
the efficient demand of intermediates nor the efficient dirty-bias. There is
a decrease on the relative production of YZ , and, thus, a decrease on the
emissions per unit of output when compared to the laissez-faire economy.
However, it remains above the efficient level.
(2) Subsidy to R&D (i) If we assume a technology policy that takes the
form of a common subsidy to both R&D sectors, that is, the same subsidy
for new L and Z-complementary intermediates (φ = φZ = φL ), the optimal
growth rate of the economy can be implemented through
"µ ¶ 1
#
1 β 1−γ
β
κs
φ=
>1
ψ
(49)
This policy instrument can be used when the monopoly and R&D distortions prevail over the environmental externality (that is, κ is large) and
the regulator wants to boost growth (g DE < g ∗ ). However, since the introduction of φ does not require any adjustment on the demand of intermediate
inputs, for a given (NZ /NL ), nor on the efficient dirty-bias, the relative production of YZ is the same as in the laissez-faire economy. Thus, emissions
per unit of output are the same as in the laissez-faire economy, above the
efficient level.
26
(ii) In contrast to the previous case, the government’s technology policy
may distinguish between the new designs by paying a different subsidy to
each one, that is, φZ 6= φL . In order to achieve g ∗ , by computing the growth
rate of the regulated economy, the following relationship between φZ and
φL must hold:
1
"µ ¶ 1
# 1−γ
1 β 1−γ
φZ =
κs β φ−γ
L
ψ
(50)
φZ = κs φL
(51)
Also, if
holds, the efficient dirty-bias is implemented. Solving this system of equa³ ´ 1 1−γ +γ
³ ´ 1 (1−γ)(1−β)
β
β
κs β
and φL = ψ1
κs β
,
tions ((50) and (51)), we obtain φZ = ψ1
for which both the optimal growth rate of the economy and the efficient
dirty-bias in the economy are obtained. Since κs < 1, φZ < φL , implying
that the new Z-complementary intermediates become relatively more costly
than the new L-complementary intermediates. This will encourage research
towards new clean complementary intermediates, therefore, increasing the
clean-bias in the economy.
However, with this instrument, there is no adjustment on the demand
of intermediate inputs, for a given (NZ /NL ). Moreover, after endogeneizing
for technical change, the relative demand of dirty complementary intermediates is higher than the optimal one. Therefore, although this instrument
implements the efficient dirty-bias, the production of YZ is higher than the
efficient one, and, consequently, the emissions intensity also is. However, the
economy moves towards the optimum and attains a lower share of emissions
in the aggregate output than in the decentralized equilibrium.
Differentiating R&D subsidies towards the clean sector corrects for the
technology externalities and at the same time reduces emissions. However,
using this instrument to achieve the emission reduction entails a welfare
loss as R&D subsidies are a first-best instrument to internalize technology
externalities but a second-best instrument to reduce emissions.
In the absence of this policy instrument both growth and environment
are under-provided in the market economy. It leads to both an increase
in the growth rate of production and to an improvement in environmental
27
quality. This result can be interpreted in the line of Porter’s arguments for
growth enhancing effects of environmental regulation. Hence, in this case, a
kind of Porter-like result emerges (Porter and Van der Linde [16]). não sei
se devo manter a frase pois isto é technology policy e não environmental policy???
In summary, despite that the optimal growth rate can be achieved using
all these second-best instruments, only the tax on emissions, τ Z > 0, or the
subsidies to research, (φZ )s < (φL )s , will decrease the emissions per unit
of output when compared to the decentralized economy, in the long-run. A
common subsidy to both types of research keeps the same emissions intensity
as in the laissez-faire economy.
6
Conclusions
In this paper we develop an endogenous growth model for a closed economy,
that incorporates the welfare effects of environmental quality and innovation. The aggregate output is produced from two goods, and two factors of
production, one clean and one dirty. In this economy, technological change
extends the range of two types of intermediate inputs: new clean and new
dirty complementary intermediates.
We study the trade-off between environmental quality and growth. As
expected, long-run growth decreases with the environmental externality.
The more consumers value the environment, the lower the optimal growth
rate. This suggests that different concerns about the environment can help
explain the differences in environmental policies, and, thus, in growth rates
across countries.
Because there is a negative externality from pollution and distortions
caused by the monopoly power and the R&D activity, the decentralized
growth rate can be higher or smaller than the optimal rate. Eventually, the
two growth rates are equal, if the opposing forces offset each other. Since it
is also possible to have the decentralized growth rate below the optimal one
when the environmental externality is not too large, the optimal solution
implies boosting growth and decreasing the dirty-bias in the economy.
We study the problem of implementing the optimal path chosen by the
social planner. There are three distortions in the decentralized economy: the
surplus appropriability problem, the monopoly power by intermediate inputs
28
producers and pollution. Since our goal is to implement the optimal path,
three market failures require for three policy instruments. The first-best
policy is implemented through a subsidy to the R&D sector to encourage
research, a subsidy to the producers of intermediate inputs to neutralize the
effect of monopoly pricing, and an emissions tax on the polluting good, to
eliminate the distortion due to pollution. We compute the levels of the three
policy instruments which allow implementing the optimal solution.
The regulator wishes to decrease the relative production of dirty intermediates and encourage environmentally oriented research. In the market
economy, research effort directed at new dirty intermediates relative to new
clean intermediates is larger than the efficient one. The environmental policy instrument achieves both. The tax affects both the balance of production between clean and dirty intermediates and the efficient dirty-bias in
the economy. It has a distortionary impact on the two sectors of intermediates. It decreases the demand of the dirty intermediates and, and, therefore,
depresses the value of patents of new dirty complementary intermediates.
Since R&D firms develop new clean or new dirty complementary intermediates depending on relative profitability, this encourages research towards
new clean complementary intermediates, and, therefore, implements the efficient dirty-bias in the economy, internalizing the pollution externality.
The reduction in the emissions intensity of aggregate output is possible by reducing the use of the dirty complementary intermediates, in the
short-run, and by decreasing the dirty-bias in the economy, in the long-run.
The higher the environmental concern in the utility function, the more the
planner makes use of clean complementary intermediates and favours the
innovation of new clean complementary intermediates.
In summary, it is very important the adoption of a technology policy
that stimulates R&D to address environmental problems. However, it must
be complemented with environmental policy to stimulate adequate technological change. The environmental policy instruments encourage research
of new clean factor-complementary intermediates and allow the regulated
economy to choose the efficient dirty-bias. Pigouvian taxes change research
incentives by increasing its environmentally friendliness, which is socially
desirable. Thus, the optimal policy encourages a change in the quality of
research, in favor of the clean sector of the economy. With first-best policies,
29
the optimal level of emissions per unit of output is implemented.
We also discuss second-best policies, assuming that the primary target of
the policy makers is to align the market and the optimal growth paths. However, only a tax on emissions, or a differentiated subsidy to R&D, according
to the type of research, decrease the share of emissions in the aggregate
output, when compared to the decentralized economy. However, although
the economy moves towards the optimum, the optimal emissions intensity
of aggregate output is not achieved.
30
Appendix A - Share of output devoted to C, I
and R
In order to obtain
C
Y,
we need to make use of the final good resource
constraint (5). From this condition,
C
Y
= 1 − YI − YR . Thus, we need to deter-
mine how much of the final good is devoted to investment in intermediate
inputs and to R&D expenditures.
The total cost of producing L and Z-complementary intermediates is
I = ψx∗L (j) NL + ψx∗Z (j) NZ , where ψ is the unitary cost of production
in terms of the final good, and x∗i (j) Ni is the total amount of intermediates purchased (i = L, Z). Substituting (38) and (39) into the production functions (3), and solving for the total
of intermediates pur´
³ amount
chased (x∗i (j) Ni ), we obtain x∗i (j) Ni = pCriψ Yi (1 − β) . Together with
³
´
∂Y
C
∂Y
+
(1
−
β)
Y
−
µ
(26) and (27), we obtain I = (1 − β) YL ∂Y
Z
∂YZ
YZ . As
L
∂Y
∂Y
Y = YL ∂Y
+ YZ ∂Y
, the share of final good invested in intermediate inputs
L
Z
¡
¢
I
is given by Y = (1 − β) 1 − µ YC .
From (4), R&D expenditures in each sector can be written as Ri =
g ∗ Ni /η i .
From (28), (30), (33), (35), (38) and the production
³
´ function for
YL β(1−β)
NL
rL
NZ
∗
YL after substituting for xL , we find that η = g∗ +ρ
pC ψ . η can be
L
Z
obtained similarly. Summing up over both sectors,
³ the total´expenditure in
YL βg ∗ ∂Y
YZ βg ∗
∂Y
R&D can be written as R = g∗ +ρ ∂YL + g∗ +ρ ∂Y
− µ YCZ . The share of
Z
¡
¢
∗
C
.
final good devoted to R&D expenditures is YR = gβg
∗ +ρ 1 − µ Y
Finally, from (5), we obtain the consumption to output ratio as
C
Y
=
=
β−
βg ∗
g ∗ +ρ
h
1 − µ (1 − β) +
βg ∗
g ∗ +ρ
i
(52)
ρ
´ 1 1−γ
³ ´1 ³
β
β
γ γ (1 − γ)1−γ
(1 − µ) ψ1
κ β (η Z Z)1−γ (η L L)γ + µρ
after substituting for g ∗ , given by (46).
31
The optimal share of output devoted to consumption, investment and
R&D ∈ (0, 1), as expected.
∗
β − gβg
∗ +ρ
C
h
=
Y
1 − µ (1 − β) +
βg ∗
g ∗ +ρ
∗
i
βρ
C
(i) The numerator is positive: β − gβg
∗ +ρ = g ∗ +ρ > 0. Thus, Y > 0, as long
h
i
∗
∗
as the denominator 1 − µ (1 − β) + gβg
= (g +ρ)(1−µ)+µβρ
> 0, which is
∗ +ρ
g ∗ +ρ
true if µ < 1.
(ii)
C
Y
h
∗
< 1 if β− gβg
<
1−µ
(1 − β) +
∗ +ρ
βg ∗
g ∗ +ρ
i
, that is,
βρ−(g ∗ +ρ)+µ[(1−β)(g ∗ +ρ)+βg ∗ ]
g ∗ +ρ
0. Since g ∗ + ρ > 0, the numerator must be negative. After some algebra, it
can be rewritten as βρ − (g ∗ + ρ) < 0. Since β ∈ (0, 1), βρ < ρ holds. Thus,
βρ < g ∗ + ρ also holds because g ∗ > 0.
(iii) Also, 1 − µ YC > 0, because
∗
µβ− gµβg
∗ +ρ
i
h
∗
1−µ (1−β)+ gβg
∗ +ρ
< 1 ⇐⇒ 1 − µ > 0.
µ
¶
C
I
= (1 − β) 1 − µ
Y
Y
(i) This ratio is positive because β ∈ (0, 1) and 1 − µ YC > 0.
(ii) Also,
I
Y
< 1, because β ∈ (0, 1) and 0 < µ YC < 1.
βg ∗
R
= ∗
Y
g +ρ
µ
¶
C
1−µ
Y
βg ∗
C
g ∗ +ρ > 0 and 1 − µ Y >
g ∗ + ρ and 0 < µ YC < 1.
(i) This ratio is positive because
(ii) Also,
R
Y
< 1, because
βg ∗
<
32
0.
<
Appendix B - Proof (Lemma 1)
Assume A =
∂(1−ϕ(κs ))
∂κ
´1
³ ´1 ³
1 β
γ (1 − γ)1−γ β (η Z)1−γ (η L)γ
γ
Z
L
ψ
< 1 ⇐⇒
∂κ(κs )
∂µ
<0
where
1−γ
∂(1−ϕ(κs ))
∂κ
=
"
(1−γ)2 Aβ −1 1−µ
κs β
µ
1−γ
κs β
(1−γ)A 1−µ
µ
−1
ρ
#2
+(1−γ)ρ
and
∂κ(κs )
∂µ
1−γ
=
"
1−γ
(1−γ)A 1−µ
κs β +(1−γ)ρ
µ
Notice that
∂κ(κs )
∂µ
−(1−γ)µ−2 Aκs β ρ
#2
∂(1−ϕ(κs ))
∂κ
−1
ρ
< 1 is equivalent to have the denominator of
positive. Since the numerator of
On the other hand, if
1−γ
−(1−γ)2 Aβ −1 1−µ
κs β
µ
∂κ(κs )
∂µ
∂κ(κs )
∂µ
is negative, then
∂κ(κs )
∂µ
< 0.
< 0, since the numerator is negative, then the
denominator has to be positive, implying that
∂(1−ϕ(κs ))
∂κ
< 1.
Appendix C - Constraint on µ
Recall equation (47):
κ = 1 − ϕ(κ), that is
κ=1−
µρ
(1 − γ) (1 − µ) Aκ
1−γ
β
+ (1 − γ) µρ
´1
³ ´1 ³
1−γ β
1 β
γ
γ (1 − γ)
(η Z Z)1−γ (η L L)γ .
where A = ψ
h
i β
1−γ
µργ
If ∂(1−ϕ(κ))
,
evaluated
at
κ̄
=
, where 1 − ϕ (κ) = 0, is
∂κ
(1−γ)(1−µ)A
smaller than 1, then there is no solution for equation (47). This condition
holds for µ >
1−γ A
γ
γ ρ
³
1−γ
A
1+ 1−γ
γ β
γ ρ
Summarizing, if
equation (47).
1−γ
β
1−γ
β
³
´ 1−γ
β
´ 1−γ
, which is clearly smaller than 1.
1−γ
β
β
´ 1−γ
1−γ ³
1−γ A
1−γ
β
γ β
γ ρ
β
´ 1−γ
1−γ ³
1−γ
β
A
1+ 1−γ
γ β
γ ρ
β
33
< µ < 1, there is no κ that solves
Appendix D - Sensivity of
•
•
•
C
Y
∂( )
∂µ
∂ ( YI )
∂µ
R
∂( Y
)
∂µ
C I
Y, Y
∂g ∗
³
ρ
∗ ´h
∗ i
∂µ
(1−β)+ gβg
(1−µ)+ β− gβg
∗ +ρ
∗ +ρ
2
∗
(g +ρ)
h
h
ii2
∗
1−µ (1−β)+ gβg
∗ +ρ
and
R
Y
to µ
−β
=
= − (1 − β)
∗
=
β ∂g
ρ
∂µ
(g ∗ +ρ)2
¡
µ
C
Y
∂( C )
+ µ ∂µY
1 − µ YC
¢
−
¶
βg ∗
g ∗ +ρ
>0
<0
µ
C
Y
C
∂( Y
)
+ µ ∂µ
¶
<0
Appendix E - Proof (Proposition 6)
Notice that the share of the clean good in the final output is given by the
Ã
µ ∂Y ¶−1 !1−γ
³ ´1−γ
∂YZ
, that is, from (7), by YYL = 1−γ
.
inverse of YYL = YYZL
∂Y
γ
∂YL
After substituting for the efficient marginal rate of transformation between
the two goods, with endogenous technical change (equation (42), and (NZ /NL )
³
´β ¶1−γ
³ ´DE
³ ´∗ µ
1−γ
1−γ ηZ Z
Y
Y
=
κ
.
from equation (44)), we obtain YL = κs γ
s
ηL L
YL
µ
¶
³
´ −γ
³ ´DE
³ ´∗ ³ ´−γ
ηZ Z β
−γ
Y
Likewise, we have YYZ = YYZL
= κs 1−γ
=
κ
,
s
γ
η L
YZ
L
that is, the inverse of the share of the dirty good in the³final
´ output,
³ ´obtained
∗
DE
in a similar way. Since 0 < κs < 1, it follows that YYL
< YYL
and
³ ´∗ ³ ´DE
Y
> YYZ
. Thus, it is optimal to have a higher share of the clean
YZ
good and a lower share of the dirty good in the aggregate output, when
compared to the decentralized equilibrium. By differentiating
Y
YL
and
Y
YZ
with respect to µ, it follows that YL becomes relatively more important on
the production of the final output the higher µ is, and the opposite occurs
with YZ .
34
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36
Proof (Proposition 5) - Only for the referees
Recall that A =
´1
³ ´1 ³
1 β
γ (1 − γ)1−γ β (η Z)1−γ (η L)γ and
γ
Z
L
ψ
1−γ
∂κ
∂µ
=
−(1−γ)µ−2 Aκ β ρ
¸2
∙
1−γ
1−γ
−1
κ β +(1−γ)ρ −(1−γ)2 Aβ −1 1−µ
κ β
ρ
(1−γ)A 1−µ
µ
µ
ρAκ
=
(1−µ)µρ 1−γ
Aκ
β
1−γ
β
∙
1−γ
1−γ
−1
β
−(1−γ) (1−µ)Aκ β
¸2
+µρ
=
<0
Differentiating g ∗ with respect to the relevant variable yields:
∂g ∗
∂L
(i)
∂κ
∂L
=
= βA 1−γ
β κ
+ βAκ
1−γ
β
γ (η L L)−1 η L > 0 because
γ(η L L)−1 η L
∙
¸2
1−γ
(1−µ)Aκ β +µρ
>0
1−γ
−1
(1−µ)µρ 1−γ
Aκ β
−(1−γ)
β
= βA 1−γ
β κ
∂κ
∂Z
=
1−γ
−1
β
∂κ
∂Z
+ βAκ
1−γ
β
1−γ
β
(1 − γ) (η Z Z)−1 η Z > 0 because
(1−γ)(η Z Z)−1 ηZ
∙
1−γ
1−γ
−1
β
(1−µ)µρ 1−γ
Aκ
−(1−γ)
(1−µ)Aκ β
β
−µρ(1−µ)Aκ
(ii)
∂g ∗
∂η L
∂κ
∂η L
=
∂g ∗
∂η Z
= βA 1−γ
β κ
= βA 1−γ
β κ
1−γ
−1
β
∂κ
∂η L
+ βAκ
1−γ
β
1−γ
−1
β
∂κ
∂η Z
γ (η L L)−1 L > 0 because
1−γ
β
+ βAκ
1−γ
β
1−γ
β
(iii)
∂g ∗
∂ρ
= βA 1−γ
β κ
1−γ
−1 ∂κ
β
∂ρ
(1−µ)µρ 1−γ
Aκ
β
∂g ∗
∂µ
1−γ
−1 ∂κ
β
∂µ
+µρ
¸2
1−γ
β
∙
1−γ
1−γ
−1
β
−(1−γ) (1−µ)Aκ β
= βA 1−γ
β κ
¸2
>0
>0
− 1 < 0 because
µ(1−µ)Aκ
=
+µρ
(1 − γ) (η Z Z)−1 Z > 0 because
(1−γ)(η Z Z)−1 Z
∙
1−γ
1−γ
−1
(1−µ)µρ 1−γ
Aκ β
−(1−γ) (1−µ)Aκ β
β
=
−µρ(1−µ)Aκ
>0
¸2
+µρ
γ(η L L)−1 L
∙
1−γ
1−γ
−1
β
(1−µ)µρ 1−γ
Aκ
−(1−γ)
(1−µ)Aκ β
β
−µρ(1−µ)Aκ
∂κ
∂η Z
(iv)
∂κ
∂L
1−γ
−µρ(1−µ)Aκ β
∂g ∗
∂Z
∂κ
∂ρ
1−γ
−1
β
< 0 because
37
+µρ
∂κ
∂µ
¸2
<0
<0