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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= à à ∂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 à 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 à 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. 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[21] Stokey, N., 1998, “Are There Limits to Growth?”, International Economic Review 39, 1-31. 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