Long-Run Effect of Emissions Trading on Green R&D Naohiko Wakutsu∗ Institute of Innovation Research Hitotsubashi University Tokyo, Japan 186-8603 June 2013

Abstract In this paper, we examine the effect of emissions trading on a firm’s R&D investment in green technology. In a common wisdom, emissions trading schemes, also known as cap-and-trade systems are considered as the most efficient mechanisms to reduce industrial pollution at least costs, since in such systems, given two fundamental choices between reducing their own emissions and buying emission credits, firms are encouraged to take an effort that is in their own interests, so that firms that can easily reduce emissions will do so, while those for which it is harder buy credits. Does this view still hold in the long run? Do they spur technical progress, or rather decrease firms’ green R&D? To address these issues, we consider a firm’s incentives for green R&D in a two-period Cournot model of duopoly under a carbon emission market. We will show that the long-term effect of emissions trading on green R&D interestingly varies according to the level of an allowance price and the degree of the product market competition. Keywords: emissions trading, cap-and-trade system, green R&D, competition

1

Introduction

Today, emissions trading schemes, also known as cap-and-trade systems, have become key policy instruments in many environmental areas. The most prominent examples of such systems are EU ETS, the US REginal CLean Air Incentives Market (RECLAIM) program and Regional Greenhouse Gas Initiatives (RGGI). A stylized emissions trading scheme may be described as follows. At the inception of program, a regulator controls the initial distribution of allowances and sets the level of a penalty for each emission unit not offset by an allowance certificate that is applied at the ∗

E-mail address: [email protected]

1

end of the complied period. Firms then may either reduce their own carbon emissions or buy emission credits from a third party. A common wisdom dated at least to Montgomery (1972) says that this transfer of allowances by trading is the core principle that leads to the minimization of cost for controlling total carbon emissions: companies that can easily reduce emissions will do so, while those for which it is harder buy credits. They are thus widely considered as the most efficient mechanisms to reduce industrial pollution at least costs. Does this view still hold in the long run? In the long run, firms’ green technologies are endogenous, which affect the future emissions to be controlled. Do they spur technological progress, or rather decrease firms’ investments in green R&D? If so, when and why does it happen? We will address these issues in this paper. A market for tradable emission permits could affect a firm’s investment in green R&D in the long run. A simple two-period scenario is illustrated in Figure 1. Suppose that a firm is an innovator with an effective (i.e., including emission cost) marginal cost e of producing one unit of good by using an endowed conventional technology. If it buys emission credits through an emission market, it avoids unwanted penalties and reduces its effective marginal cost to some lower level than e. However, this reduced marginal cost only applies in a current complied period, period 1 and its effective marginal production cost at the beginning of period 2 will be once again equal to e. Suppose that the firm competes with a follower. While the innovator buys emission credits rather than developing a “cleaner” technology, the follower copies the innovator’s endowed technology and lower its production cost from e′ to c′1 . The two firms are then in a fiercer competition due to the reduced technological gap c′1 − e. In contrast, if there were no such emissions trading schemes, the firm conducted green R&D and achieved a lower effective marginal cost c∗1 and enjoyed a larger technical lead c′1 − c∗1 . Among economists, there is a long-standing debate about whether competition increases or decreases innovation. For instance, see Boldrin et al. (2012) for a recent review of the literature. In this figure, a fiercer competition caused by the reduced technical lead c′1 − e results in a less development c2 of clean technology, while the larger technical advantage c′1 − c∗1 induces further innovation and yields a more-developed technology c∗2 . Therefore, emissions trading not only delays a firm’s R&D investment by one period, but also leads to less innovation by c∗2 − c2 . Although this is a highly simplified example, to the extent that this is the case, emissions trading reduces a firm’s long-term green R&D and thus causes a tradeoff between the static efficiency and the dynamic inefficiency. To examine the long-run effect of emissions trading on green R&D, we develop a twoperiod Cournot model of duopoly where in each period, an innovator either invests in 2

Figure 1: A simple two-period scenario

green R&D to reduce its emission cost or buys emission credits under an emission market, and then ask who (a less-developed innovator or a more-developed innovator) has greater incentives to buy emission credits and who and under what conditions increases or decreases R&D investments with the product market competition. The following conclusions emerge. First, emissions trading schemes tend to exhibit a dynamic trade-off between the short-term efficiency in minimizing emission costs and the long-term inefficiency in inducing green investment. Second, the (negative) effect of emissions trading on green R&D could be relatively small, if the firms faces a fierce competition in the product market. Otherwise, the effect may be large. It not only delays a firm’s green R&D by one period, but also induces less innovation both in the short and long run. Lastly, the negative effect may be amplified and become the largest, if emission trading is chosen over green R&D in period 1, since in that case it may be chosen in period 2 as well and no innovation occurs throughout the periods. In summary, the effect interestingly varies according to the level of an allowance price and the degree of the product market competition. The economic literature on technological change and environmental policy is large. A recent review of the literature at the microeconomic level is in Popp (2010), and a more thorough review, in Popp et al. (2010). For a review of the literature on the macroeconomic effect of endogenous technological change for climate policy, see Gillingham et al. (2008). We study how the incentives provided by an environmental policy (emissions trading scheme) affect a profit-maximizing firm’s R&D investments in a two-stage Cournot model. In similar settings, among others, Shittu and Baker (2010) study a firm’s optimal R&D portfolio in response to emission taxes, Hagem (2009) examines how firms’ R&D investments in developing countries differ under the emissions trading scheme and under the clean development mechanism regime, and De Vries (2007) compares emission taxes 3

and emission reduction subsidies in the incentives they create to enhance the diffusion of a clean technology.

2

Model

To examine the long-term effect of emissions trading on green R&D, we consider a firm’s R&D investment in green technology under a carbon emission market in a two-period Cournot model of duopoly. Production Market The two firms are Cournot competitors. One is an innovator and the other, a follower. Each supplies a market with a differentiated product in periods 1 and 2. Let qt , t = 1, 2 be the innovator’s level of production in period t and pt , the level of price for its product in that period. Likewise let qt′ and p′t be the follower’s corresponding variables. The market demands for these products in period t are qt = max{0, a − pt + bp′t },

qt′ = max{0, a − p′t + bpt }

where pt , p′t , a and a′ are positive and finite numbers and 0 < b < 1. Here, 0 < b < 1 implies that two products are imperfect substitutes. Moreover, they become closer substitutes as b approaches to unity. Let p(qt , qt′ ) and p′ (qt′ , qt ) be the associated inverse demand functions. Both firms produce at constant marginal and average costs. Without loss of generality, we take the marginal cost of production for the innovator when it using an endowed technology as zero. The marginal production cost for the follower in period t is then simply the excess of production cost over the innovator’s and denoted by c′t . Cap-and-trade System Production causes carbon emissions. In order to reduce this externality, a regulator uses a cap-and-trade system. To be more specific, the regulator allocates the innovator a certain number of emission credits, called “cap”, and determines the level of a penalty for each emission unit over the emission certificate at the outset of each period t. To avoid such unwanted penalties, the innovator then has two choices: reduce emission through green R&D or buy emission credits from a third party. As in Carmona et al. (2010), the effective marginal cost of production for the innovator in this system, denoted by ct , is the sum of its production cost that is normalized to zero and its emission cost per unit of produced output. The emission cost incurred by the firm is assumed to be the product of its emission factor measuring the volume of carbon dioxide

4

emitted per unit of produced good and a penalty charged for each emission unit not offset by its initial emission credits. That is, ct = 0 + (et − eˆt ) λet , where et is the volume of carbon dioxide emitted per unit of produced good, eˆt , the initial emission certificates, and λet , the penalty charged for each excess emission unit. If we set eˆt = 0 and λet = 1 for both t, then ct = et . In words, the firm’s effective marginal cost in period t is simply its emission factor in that period. We take this simplicity to ease notations. Green R&D Before production, the innovator may conduct green R&D. That is, it chooses the level rt of an R&D expenditure in the hope of reducing its emission factor. Given rt , it receives as an R&D outcome an alternative value zt of the emission factor that arises from a new technology, and selects the lower of the initial emission factor et that is associated to the endowed conventional technology and the new alternative value zt . That is, its finalized effective marginal cost is ct = min{zt , et }. R&D is a gamble that is modeled as a stochastic process described by a cumulative distribution function (cdf) G(z|r) with a density function g(z|r) over a fixed interval I: ∫ G(z|r) = g(z|r)dz. I

(Here, the time index is suppressed in r and z.) Any non-negative R&D expenditure r gives the firm a single purely random draw from I. It may be natural to consider that the likelihood of a lower z-value depends on the level r of the firm’s R&D expenditure. Specifically, we will assume the following stochastic relations between z and r. Assumption 1. For all z ∈ I and for all r > 0, ∂G(z|r) ≥ 0, ∂r where the strict inequality holds in some non-degenerate interval of z. In words, G(z|r) is strictly first-order stochastic dominant with respect to r, as illustrated in Figure 2. So, the advantage of a higher R&D expenditure is that it gives a more favorable G from which to sample. Assumption 2. For all z ∈ I and for all r > 0, ∂ 2 G(z|r) ≤ 0, ∂r2 where the strict inequality holds in some non-degenerate interval of z. 5

Figure 2: A stochastic shift in G(z|r) when r > r′

It says that the stochastic shift caused by a unit increase in r become smaller as the value of r rises. Given the follower’s output level and the own finalized marginal cost, each firm’s production profits in period t are Π(qt ; qt′ , ct ) = [p(qt , qt′ ) − ct ] qt ,

Π′ (qt′ ; qt , c′t ) = [p′ (qt′ , qt ) − c′t ] qt .

Let π(ct , c′t ) be the equilibrium profit to the innovator in period t. Then, its expected gross (i.e., exclusive of R&D expenditure) profit from a given R&D expenditure rt is ∫ ′ V (rt ; et , ct ) = π(min{zt , et }, c′t )dG(zt |rt ). I

Emission Market Alternatively, prior to production, it may choose to buy emission credits called allowance under an emission market, where, for simplicity, no banking is allowed. Suppose that the innovator chooses to buy xt units of emission credits in period t. Then, its finalized effective marginal cost this period is ct = et − xt . Let λ denote the allowance price in periods 1 and 2, which is exogenous and satisfies 0 < λ < 1. Given et , c′t and λ, its profit from emissions trading in period t net of trading cost is then −xt qt λ + π(et − xt , c′t ).

Summary Figure 3 summarizes the timing of moves in period t. At the inception of each period, the regulator decides the levels eˆt and λet of cap and penalty. For simply, we set eˆt = 0 and λet = 1 for both periods. Then, to avoid the unwanted penalties, the innovator chooses 6

Figure 3: Timing of moves in period t = 1, 2

whether to conduct R&D or purchase emissions credits from a third party. Meanwhile, the follower copies the (conventional) production technology the innovator is initially endowed with. Finally, the production begins. To keep matters simple, we assume that c′t is exogenous for any t. Also, let c′1 > c′2 > 0 and e2 = c1 . A proposed interpretation is that the follower is a imperfect copier and both imitation and innovation are cumulative. All these pieces of information are common knowledge. The following section add notations and an detailed analysis of each stage, beginning with the production stage.

3 3.1

Analysis Production

At this stage, the two firms are Cournot competitors. Each knows both firms’ finalized marginal costs. Given ct and c′t , let (q ∗ (ct , c′t ), q ′∗ (c′t , ct )) be an equilibrium profile in period t. Operationally, it is obtained as a solution of the system of equations formed by the best response correspondences of each firm. Substituting this into Π(qt ; qt′ , ct ) yields the equilibrium production profit to the innovator in period t: π(ct , c′t ) = Π(q ∗ (ct , c′t ), q ′∗ (ct , c′t )). Figure 4(a) is a three-dimensional plot of π(ct , c′t ). As can be seen, π is jointly continuous and non-increasing in ct and non-decreasing in c′t . Although it satisfies ∂ 2 π/∂c2t ≥ 0 and ∂ 2 π/∂c′t 2 ≤ 0 almost everywhere, π is neither convex in ct nor concave in c′t , which is less obvious in Figure 4(a) but clearer in Figures 4(b)-(c). Intuitively, these result from the fact that there are four alternative configurations: both firms produce, only one produces and none produces, as described in Figure 4(d). Here, the area labeled Area +, 0, for example, is associated to the set of vectors (ct , c′t ) with which only the innovator produces in an equilibrium. The interpretations of Area +, +, Area 0, + and Area 0, 0 are analogous. Since a monopolist’s profit changes at a different rate from that of a duopolist, the 7

1.2

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equilibrium profit π to the innovator is kinked at the boundary between monopoly by the innovator and duopoly, and this kink results in the non-convexity of π. Also it is noted here that each firm’s equilibrium profile is non-increasing in the own production cost, non-decreasing in the opponent’s production cost and linear in both production costs. For instance, the innovator’s equilibrium profile q ∗ (ct , c′t ) is non-increasing in ct , non-decreasing in c′t and linear in both ct and c′t . Given these, the next subsections proceed backward and consider the innovator’s choice between green R&D and emissions trading. Meanwhile, we restrict our attention to period 2. A detailed analysis of period 1 is relegated to Section 3.4.

3.2

Green R&D

Before production, the innovator can either invest in green R&D or buy emission credits. In period 2, suppose that it has chosen green R&D rather than purchasing emission permits. Then, its problem is choosing the level of r2 to maximize −r2 + V (r2 ; e2 , c′2 ) given e2 and c′2 . As established below, V increases with r2 at a diminishing rate. (All the proofs of the statements are in Appendix.) Lemma 1. V ′ (r; e, c′ ) is strictly positive and decreasing in r, given e and c′ .

1.3

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Figure 5: Net R&D profit

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Since r2 increases at a constant rate that is unity, it is instant that the innovator’s problem is a concave programming, as depicted in Figure 5. Let r2∗ (e2 , c′2 ) be an optimal R&D spending in period 2. Necessarily, it satisfies the first-order condition 1 ≤ V ′ (r2 ; k2 , k2′ ), 9

where r2∗ = 0 if the strict inequality holds and r2∗ > 0 only if the equality holds. The resulting maximized net R&D profit is ϕ(e2 , c′2 ) ≡ −r2∗ (e2 , c′2 ) + V (r∗ (e2 , c′2 ); e2 , c′2 ). As stated below, ϕ exhibits similar properties to π with respect to their own arguments. Lemma 2. ϕ(e, c′ ) is non-increasing in e and non-decreasing in c′ . Figure 6 is a restriction of ϕ(e, c′ ) given c′ . Although it cannot be determined analytically, it is inferred from the figure that ∂ 2 ϕ/∂e2 ≥ 0 holds in many areas. An important problem to our interest is determining how the optimal R&D expenditure for the innovator is affected by each firm’s production costs et and c′t , since it helps us declare who reduces the level of its R&D expenditure and when. This is no doubt a comparative statics question. The results are summarized as follows (with the time index t = 2 being suppressed in r∗ , e or c′ ). Lemma 3. In period 2, suppose r∗ > 0. Then, (i) ∂r∗ /∂e ≥ 0; (ii) ∂r∗ /∂c′ > 0 if c′ is small; (iii) ∂r∗ /∂c′ = 0 if c′ is large; and (iv) otherwise, ∂r∗ /∂c′ < 0 is possible. Lemma 3 says that all else being the same, a less-developed innovator has at least as large incentives in R&D than a more-developed innovator. This also implies that a firm’s R&D investment eventually stops. Second of all, the effect of a unit increase in the opponent’s production cost c′ on r2∗ interestingly varies with the level of c′ . Intuitively, this is because a prospect to be a monopolist for the innovator becomes smaller as c′ falls, as is illustrate in Figure 4. Since the expected return from a given R&D expenditure to the innovator is higher in a monopoly than in a duopoly, how large a prospect to be a monopolist exists to the innovator should affects his choice on R&D both in level and at the margin. Specifically, if c′ is low, there may be little prospect for the innovator to be a monopolist and at best it can can only be a duopolist. In that case, a fiercer competition resulting from a lower c′ decreases the innovator’s R&D. If c′ is large, an innovator may successfully be a monopolist almost certainly. Since the follower is out of the market in this case, the innovator’s investment level should be unaffected by the follower’s production cost. So, competition has no effect on the innovator’s R&D. If it is somewhere in-between so that the innovator’s chance to be a monopolist largely depends on its R&D choice, then it may raise its R&D expenditure and escape from an expected tough duopolistic competition. These implications are summarized as follows. Proposition 1. (i) An innovator’s R&D investment decreases with the development level of its green technology and stops eventually. Moreover, an increase in competition caused by 10

emissions trading (ii) reduces an innovator’s R&D investment if the follower’s production cost is low; (iii) has no effect on an innovator’s R&D investment if the follower’s production cost is high; and (iv) may increase an innovator’s R&D investment if the follower’s production cost is medium.

3.3

Emissions Trading

Suppose instead that the innovator has chosen emissions trading rather than green R&D. Then, its problem is choosing the number x of emissions credits to buy at a given price λ to maximize the net profit −x2 q2∗ (e2 − x2 , c′2 )λ + π(e2 − x2 , c′2 ). Let x∗2 (e2 , c′2 ) be an optimal x2 in period 2 that maximizes this profit. With recalling the property that q2∗ is a linear and non-increasing function of e2 − x as well as Figures 4(a) and 4(c), while x2 q2∗ (e2 − x2 , c′2 )λ increases in x2 at a constant rate, π increases in x2 at an increasing rate, especially for large x2 . Therefore, as depicted in Figure 7, it is inferred that the net profit −x2 q2∗ (e2 − x2 , c′2 )λ + π(e2 − x2 , c′2 ) reaches its maximum at the terminal point e2 in many cases. That is, x∗2 (e2 , c′2 ) = e2 . In this case, the maximized net profit from emissions trading is −e2 q2∗ (0, c′2 )λ + π(0, c′2 ).

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Figure 8: Choice between ET and R&D

Figure 7: Emissions trading

Given these, let us now consider the choice between green R&D and emissions trading by the innovator. The decision rule to be followed is that it chooses emissions trading over green R&D if its value −e2 q2∗ (0, c′2 )λ + π(0, c′2 ) from emission permits is greater than that ϕ(e2 , c′2 ) from green R&D: −e2 q2∗ (0, c′2 )λ + π(0, c′2 ) > ϕ(e2 , c′2 ). 11

(1)

Apparently, the value from emissions trading decreases with e2 at a constant rate λ. while the value from green R&D decreases with e2 at a diminishing rate. (See also Figure 6.) Moreover, if e2 = 0, then no R&D or emissions trading should occur. So, the firm’s maximized net profits are the same at e2 = 0. Combining these together implies that given c′2 , the value from emissions trading tends to outweigh that from green R&D if e2 is relatively small and the value from green R&D tends to outweigh that from emissions trading if e2 is relatively large, unless λ is very high. If λ is very large, the value from emissions trading may be smaller than that from green R&D except for very small e2 . See Figure 8. Since more-developed firms tend to benefit more from emissions trading than from green R&D, more-developed producers tend to choose emissions trading over green R&D compared with less-developed firms. So, emissions-trading firms are more likely to be more-developed producers. The result is intuitive since further innovation is more difficult to obtain for those that has already developed a cleaner technology, as pointed out by Montgomery (1972). These implications may be summarized as follows. Proposition 2. (i) Unless an allowance price is very high, more-developed producers tend to choose emissions trading emissions trading over green R&D compared with less-developed firms; and (ii) if it is very high, only the most-developed firms, or even none, choose emissions trading over green R&D.

3.4

Lock-In

Given the analysis of period 2, we now proceed backwards and consider the innovator’s problem in period 1. Green R&D Given e1 , c′1 and c′2 , the innovator chooses the level of r1 to maximize its net expected lifetime profit r1 +V (r1 ; e1 , c′1 )+β W (r1 ; e1 , c′2 ). Here, β is the discount factor with 0 < β < 1 and W (r1 ; e1 , c′2 ) is the maximized expected profit in period 2: ∫ ′ W (r1 ; e1 , c2 ) ≡ max {ϕ(min{z1 , e1 }, c′2 ), − min{z1 , e1 }q ∗ (0, c′2 )λ + π(0, c′2 )} dG(z1 |r1 ). I

Since ϕ has similar properties to π, it is inferred that W tends to have similar properties to V . So, an optimal R&D expenditure in period 1, denoted by r1∗ (e1 , c′1 ), may be fully characterized by the first-order condition in many cases. Although the comparative statics analysis for r1∗ is rather complicated, the result may be similar to Lemma 3. 12

Emissions Trading Regarding the choice between emissions trading and green R&D, the innovator’s decision rule to be followed this period is that it chooses emissions trading over green R&D if − e1 q ∗ (0, c′1 )λ + π(0, c′1 ) + β max{ϕ(e1 , c′2 ), −e1 q ∗ (0, c′2 )λ + π(0, c′2 )} > −r1∗ (e1 , c′1 ) + V (r1∗ (e1 , c′1 )) + β W (r1∗ (e1 , c′1 ); e1 , c′2 ), (2) and it chooses green R&D over emissions trading otherwise. With the comparison of conditions (1) and (2), the innovator’s choice between emissions trading and green R&D in periods 1 and 2 may have the following tendency. Proposition 3. An innovator tends to choose emissions trading over green R&D in period 2, if he does so in period 1. Proposition 3 suggests the tendency that if the innovator chooses emissions trading this period, then he will do so in period 2 as well. In other words, the innovator is more likely to be locked in by his own choice of emissions trading in period 1. Clearly, this amplifies all the negative effects of emissions trading on green R&D described above, and terminates green innovation.

3.5

Discussion

So, what is the effect of emissions trading on green R&D? From the analyses in Sections 3.2 through 3.4, the following conclusions emerge. First, emissions trading schemes tend to exhibit a dynamic trade-off between the short-term efficiency in minimizing emission costs and the long-term inefficiency in inducing green investment. To see this, suppose first that an allowance price λ is very high. Then, presumably, it is only the most-developed producers that choose emissions trading over green R&D. Since they are those of the least R&D incentives, the reduction in the investment in green R&D caused by emissions trading may be small. Although the (negative) effect of emissions trading on green R&D is likely to be small, there may not be a sufficient number of participants in emissions trading. Hence, the efficiency of the emission market tend to be low as well. As λ falls, more firms come to choose emissions trading over green R&D. Those marginal participants in emissions trading are of larger R&D incentives than the intra-marginal participants. So, the reduction in the investment in green R&D caused by emissions trading becomes larger, the lower is λ. With a larger number of participants, on the other hand, the market for tradable emission permits may become more efficient (at least initially). So, the short-run efficiency of emissions trading tends to be higher, the lower is λ. Thus, in general, there may be a dynamic trade-off between the short-term efficiency and the long-term inefficiency. 13

Second, when λ is not sufficiently large, the effect of emissions trading on green R&D could be relatively small, if the product market is in a fierce competition. Primarily it is because with or without emissions trading, green R&D would be dying out. Otherwise, the (negative) effect may be large. That is, emissions trading not only delays a firm’s green R&D by one period, but induces less innovation both in the short and long run. Lastly, the negative effect may be amplified and become the largest, if emission trading is chosen over green R&D in period 1, since in that case it may be chosen in the subsequent period as well and so no innovation occurs throughout the periods. In summary, the effect interestingly varies according to the level of an allowance price and the degree of the product market competition.

4

Conclusion

In this paper, we studied the long-term effect of emissions trading on green R&D. In a common wisdom, markets for tradable emission permits are considered as the most efficient mechanisms to reduce industrial pollution at least costs, since in such systems, given two fundamental choices between reducing their own pollution and buying emission credits, firms are encouraged to take an effort that is in their own interests, so that companies that can easily reduce emissions will do so, while those for which it is harder buy credits. Does this view still hold true in the long run, where firms’ green technologies are endogenous? Do they spur technical progress, or rather decrease firms’ green R&D? If so, when and why does it happen? To this end, we constructed a two-period Cournot model of duopoly where in each period, an innovator either invests in green R&D to reduce its emission cost or buys emission credits under an emission market, and then asked who (a less-developed innovator or a more-developed innovator) has greater incentives to buy emission credits and who and under what conditions increases or decreases R&D investments with the product market competition. The following conclusions emerged. First, emissions trading schemes tend to exhibit a dynamic trade-off between the short-term efficiency in minimizing emission costs and the long-term inefficiency in inducing green investment. Second, the (negative) effect of emissions trading on green R&D could be relatively small, if the firms faces a fierce competition in the product market. Otherwise, the effect may be large. It not only delays a firm’s green R&D by one period, but also induces less innovation both in the short and long run. Lastly, the negative effect may be amplified and become the largest, if emission trading is chosen over green R&D in period 1, since in that case it may be chosen in period 2 as 14

well and no innovation occurs throughout the periods. In summary, the effect interestingly varies according to the level of an allowance price and the degree of the product market competition.

Appendix: Proofs This appendix contains proofs of the results given in the text. Proof of Lemma 1. Since g(z|r) is a proper pdf, ∫ g(z|r)dz ≡ 1 for all r > 0 I

and so

∫ I

∂g(z|r) dz ≡ 0 ∂r

for all r > 0.

As well, since G is strictly first-order stochastic dominant with respect to r by Assumption 1, necessarily there is a z ′ in the interior of I such that ∫ z′ ∫ ∂g(z|r) ∂g(z|r) dz = − dz > 0. ∂r ∂r z′

(3)

Figure 9 (left) illustrates this point, in which I = [z, z]. The innovator’s marginal expected gross profit from an R&D expenditure r is ∫ ∂g(z|r) ′ ′ V (r; e, c ) = π(min{z, e}, c′ ) dzt ∂rt I ∫ z′ ∫ ∂g(z|r) ′ ∂g(z|r) = π(min{z, e}, c ) dzt + π(min{z, e}, c′ ) dzt . ∂rt ∂rt z′ Since π(min{z, e}) is strictly decreasing for z < e and is constant at π(e) for z ≥ e, ∫

z′

∂g(z|r) dz + V (r; e, c ) > π(e, c ) ∂r ∫ ∂g(z|r) ′ = π(e, c ) dz ∂r I = 0. ′

′

′

This last step is illustrated in Figure 9 (left). Similarly, since g(z|r) is a proper pdf, ∫ 2 ∂ g(z|r) dz ≡ 0 ∂r2 I 15

∫

π(e, c′ ) z′

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∂g(z|r) dz ∂r

Assumption 2 then implies that there must be a z ′′ in the interior of I such that ∫ z′′ 2 ∫ ∂ g(z|r) ∂ 2 g(z|r) − dz = dz > 0. ∂r2 ∂r2 z ′′

(4)

The rate of change of the innovator’s marginal expected profit with respect to the level of an R&D expenditure is ∫ ∂ 2 g(z|r) ′′ ′ V (r; e, c ) = π(min{z, e}, c′ ) dz ∂r2 I ∫ z′′ ∫ 2 2 ′ ∂ g(z|r) ′ ∂ g(z|r) π(min{z, e}, c ) = π(min{z, e}, c ) dz + dz. ∂r2 ∂r2 z ′′ Again, since π(min{z, e}) is strictly decreasing for z < e and is constant at π(e) for z ≥ e, ∫

z ′′

∂ 2 g(z|r) V (r; e, c ) < π(e, c ) dz + ∂r2 ∫ 2 ∂ g(z|r) ′ = π(e, c ) dz ∂r2 I = 0, ′′

′

′

∫

π(e, c′ ) z ′′

∂ 2 g(z|r) dz ∂r2

which is illustrated in Figure 9 (right). This completes the proof.

Figure 9: Calculation of the marginal expected gross R&D profit

Proof of Lemma 2. By the Envelope Theorem, sgn

∂ϕ ∂V = sgn , ∂α ∂α

for α = e, c′ .

The rate of change in the expected gross R&D profit from a unit increase in e is ∫ ∂V (r; e, c′ ) ∂π(min{z, e}, c′ ) = dG(z|r). ∂e ∂e I 16

′

′

′

) ) ) Since ∂π(min{z,e},c = 0 for z < e while ∂π(min{z,e},c is constant at ∂π(e,c ≤ 0 for z > e with ∂e ∂e ∂e ′ ′ the strict equality when π(e, c ) > 0 for the given c , this is negative.

It is analogous to establish ∂V /∂c′ ≥ 0. So, the claim follows. Lemma A1. ∂ 2 V /∂r∂e ≥ 0. Proof of Lemma A1. The rate of change in the marginal expected gross R&D profit by a unit increase in e is ∫ ∂π(min{z, e}, c′ ) ∂g(z|r) ∂ 2V = dz ∂r∂e ∂e ∂r I ∫ z′ ∫ ∂π(min{z, e}, c′ ) ∂g(z|r) ∂π(min{z, e}, c′ ) ∂g(z|r) = dz + dz. ∂e ∂r ∂e ∂r z′ Since

∂π(min{z,e},c′ ) ∂e

= 0 for z < e while

∂π(min{z,e},c′ ) ∂e

′

is constant at

∂π(e,c′ ) ∂e

≤ 0 for z > e with

′

the strict equality when π(e, c ) > 0 for the given c , ∫

z′

∂π(e, c′ ) ∂g(z|r) dz + ∂e ∂r ∫ ∂π(e, c′ ) ∂g(z|r) = dz ∂e ∂r I = 0.

∂2V ≤ ∂r∂e

∫ z′

∂π(e, c′ ) ∂g(z|r) dz ∂e ∂r

Lemma A2. (1) ∂ 2 V /∂r∂c′ > 0 if c′ is small; (2) ∂ 2 V /∂r∂c′ = 0 if c′ is large; and (3) otherwise, ∂ 2 V /∂r∂c′ < 0 is possible. Proof of Lemma A2. The rate of change in the marginal expected R&D profit by a unit increase in the follower’s production cost c′ is ∫ ∂ 2V ∂π(min{z, e}, c′ ) ∂g(z|r) = dz ∂r∂c′ ∂c′ ∂r I ∫ ∫ z′ ∂π(min{z, e}, c′ ) ∂g(z|r) ∂π(min{z, e}, c′ ) ∂g(z|r) dz + dz = ∂c′ ∂r ∂c′ ∂r z′

(5) (6)

where z ′ is as in (3). To determine the sign of this expression, consider three cases: (i) one is when c′ is so small that either a duopoly by the both firms or a monopoly by the follower occurs in an equilibrium; (ii) another is when c′ is so large that either a monopoly by the innovator or none producing occurs in an equilibrium; and (iii) the other is when c′ is moderate so that one of a monopoly by the innovator, a duopoly by the both firms and a monopoly by the follower occurs in an equilibrium. 17

′

) Consider the first case. In this case, ∂π(min{z,e},c is strictly positive initially, mono∂c′ ′) tonically decreases as z rises and eventually becomes constant at ∂π(e,c ≥ 0. Hence, as ∂c′

illustrated in Figure 5 (a), ∫

z′

∂π(e, c′ ) ∂g(z|r) dz + ∂c′ ∂r ∫ ∂π(e, c′ ) ∂g(z|r) = dz ∂c′ ∂r I = 0.

∂ 2V > ∂r∂c′

∫ z′

∂π(e, c′ ) ∂g(z|r) dz ∂c′ ∂r

′

) = 0 for all z ∈ I. So, trivially, the Look at the second case. In this case, ∂π(min{z,e},c ∂c′ entire expression is zero. ′) Consider the last one. In this case, ∂π(min{z,e},c is not monotonic. For z < e, it is zero ∂c′

initially, then jumps up and becomes strictly positive and monotonically decreases as z ′) ′) rises till ∂π(e,c ≥ 0. For z > e, it is constant at ∂π(e,c ≥ 0. So, although the sign of ∂c′ ∂c′ the entire expression cannot be determined in general, the sign tends to be negative if it is in the case like Figure 5. (Instead, if it is in the case like Figure 5 then the sign may be positive.) Hence the claim follows.

Figure 10: Calculation of the rate of change in the marginal expected gross R&D profit by a unit increase in c′ : when c′ is large (above) and when c′ is moderate (below)

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Proof of Lemma 3. The Implicit Function Theorem ensures that the signs of ∂r∗ /∂c and ∂ 2 V /∂r∂e coincide and so do those of ∂r∗ /∂c′ and ∂ 2 V /∂r∂c′ . Applying Lemmata A1 and A2, the claim follows. Proof of Proposition 3. Notice first that for any z1 ∈ I, ϕ(e1 , c′1 ) < ϕ(min{z1 , e1 }, c′1 ) −e1 q ∗ (0, c′1 )λ + π(0, c′1 ) < − min{z1 , e1 }q ∗ (0, c′1 )λ + π(0, c′1 )} holds. Now suppose (2). Then, this implies that −e1 q ∗ (0, c′1 )λ + π(0, c′1 )} > −r1∗ (e1 , c′1 ) + V (r1∗ (e1 , c′1 ); e1 , c′1 ) holds (by a large amount). Note e1 = e2 . So, −e2 q ∗ (0, c′1 )λ + π(0, c′1 )} > −r2∗ (e2 , c′1 ) + V (r2∗ (e2 , c′1 ); e2 , c′1 ). Unless c′1 is significantly close to zero, the innovator may be either a monopolist or a strong duopolist. So, q ∗ should be seldom affected by the level of the follower’s production cost c′1 . And so is π, while ϕ is not decreasing (maybe increasing) in c′1 . By assumption, c′1 > c′2 . These together imply that the inequality is seldom reversed with the replacement of c′1 to c′2 . That is, −e2 q ∗ (0, c′2 )λ + π(0, c′2 )} > −r2∗ (e2 , c′2 ) + V (r2∗ (e2 , c′2 ); e2 , c′2 ) tends to be true. Therefore the claim follows.

References Boldrin, Michele, Juan A. Correa, David Levine, and Carmine Ornaghi (2012) “Competition and Innovation,” Cato Papers on Public Policy, Vol. 1, No. 1, pp. 109–159. Carmona, Ren`e, Max Fehr, Juri Hinz, and Arnaud Porchet (2010) “Market Design for Emission Trading Schemes,” SIAM Review, Vol. 52, No. 3, pp. 403–452. De Vries, Frans (2007) “Market Structure and Technology Diffusion Incentives under Emission Taxes and Emission Reduction Subsidies,” Journal of Institutional and Theoretical Economics, Vol. 163, No. 2, pp. 256–268. Gillingham, Kenneth, Richard G. Newell, and William A. Pizer (2008) “Modeling Endogenous Technological Change for Climate Policy Analysis,” Energy Economics, Vol. 30, No. 6, pp. 2734–2753. 19

Hagem, Cathrine (2009) “The Clean Development Mechanism versus International Permit Trading: The Effect on Technological Change,” Resource and Energy Economics, Vol. 31, No. 1, pp. 1–12. Montgomery, David W. (1972) “Markets in Licenses and Efficient Pollution Control Programs,” Journal of Economic Theory, Vol. 5, No. 3, pp. 395–418. Popp, David (2010) “Innovation and Climate Policy,” Annual Review of Resource Economic, Vol. 2, pp. 275–298. Popp, David, Richard G. Newell, and Adam B. Jaffe (2010) “Energy, the Environment and Technological Change,” in Bronwyn Hall and Nathan Rosenberg eds. Handbook of Economics of Innovation, Vol. 2, Burlington: Academic Press, pp. 873–938. Shittu, Ekundayo and Erin Baker (2010) “Optimal Energy R&D Portfolio Investments in Response to a Carbon Tax,” IEEE Transactions on Engineering Management, Vol. 57, No. 4, pp. 547–559.

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