Emission regulation: Prices, quantities and hybrids with endogenous technology choice

This paper examines the investment incentives of market-based regulation, with focus on the technology characteristics the different regulatory schemes tend to incentivize. The firms' technology choice is socially optimal if and only if the aggregate emission allowance supply is completely inelastic. Further, in the presence of uncertainty, elastic emission allowance supply and strictly convex environmental damage, it is optimal to tax investment in technologies that induce large variance in emissions. Last, price elastic supply of emission allowances may increase the volatility in the product market, depending on the risk environment the firms face. The results indicate that introduction of permit price stabilizing measures in an emission trading system will come at the cost of suboptimal technology investments. It may also cause increased fluctuations in product prices.

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Introduction
Implementation and development of new technologies have been essential for ameliorating various environmental challenges such as acid rain, climate change and depletion of the ozone layer.
In the longer term, the ability to drive investment and research and development is therefore arguably among the most important criteria when evaluating environmental policy (See, e.g., Kneese and Schultze (1975), Orr (1976) for and Ja¤e and Stavins, 1995 for early presentations of this view).
It is therefore not surprising that a substantial economic literature examines environmental policy and the implementation of new technology (see Ja¤e et al., 2002, Löschel, 2002, or Requate 2005 for literature reviews).One branch of this literature focuses on the investment levels the various regulatory schemes spur, often with a focus on prices versus quantities-based regulation (see, e.g., Denicolo, 1999, andRequate andUnold, 2003).Another part of the literature examines various market failures and rationales for subsidizing technology investment, typically with a focus on subsidies motivated by positive externalities such as technology spillovers or learning by doing (see Arrow (1962) for a seminal paper on learning by doing.See Golombek and Hoel (2005) and Kverndokk and Rosendahl (2007) for its'implications for environmental policy).
In this paper, the focus is not so much on the total level of investment that various types of regulation provide incentives for, but rather on the type of technology that is promoted.This addresses a shortcoming of the literature …rst pointed out by Krysiak (2008), i.e., the literature tends to analyze the amount of investment, but do not consider what type of technology it is invested in.In addition, this paper investigates the rationale for active technology policies that involves taxing investments in certain technology characteristics.More precisely I show that, in the presence of uncertainty, endogenous aggregate emissions and strictly convex environmental damage, it is optimal to tax investment in technologies that induce large variance in emissions.
The reason is that ‡uctuations in emissions creates a negative externality that is not internalized by the …rms.The fact that these ‡uctuations are in ‡uenced by the …rms' technology choice motivates active technology investment policies.Perhaps somewhat counterintuitive, optimal policy may thus involve taxation of investment in certain types of abatement technology.
Emission trading and emission taxes are presently by far the most prominent examples of implemented price or quantity based regulatory policies.The most well-known example of carbon ETS today is perhaps the European Emission Trading Scheme (EU ETS), but the number of ETS is growing fast and currently includes schemes in, e.g., China, Australia, Canada, New Zealand and Japan.Emission trading is also used to reduce other pollutants, such as the U.S. NOx Budget Trading Program (nitrogen oxides) and the U.S. Acid Rain program (sulphur dioxide).Whereas there are currently several international emission trading schemes, emission taxes are predominantly used nationally.Important examples are the U.S. tax on ozone depleting chemicals and taxes on CO2 emissions in several European countries. 1s pointed out by Grull and Taschini (2011), policy makers tend to have di¤erent views about the range of acceptable prices for tradable emission permits.Hence, there has been an interest in mechanisms to keep the permit price from rising too high or falling too low.One important example of such permit price stabilizing mechanisms is the EU ETS Market Stability Reserve (MSR), which aims to absorb historical surplus of allowances and ensure market stability. 2The idea of creating a hybrid system by combining emission trading and taxes was …rst introduced by Roberts and Spence (1976), see Hepburn (2006) for an overview of the literature.
In this paper, I investigate a potentially unwanted side-e¤ect of elastic supply of emission allowances.That is, price elastic supply of emission allowances in ‡uences the volatility in the product prices.For example, the volatility in the price on electricity in the area covered by the EU ETS will be a¤ected by emission allowance stabilizing measures such as the Market Stability Reserve (MSR).Whether the electricity price volatility increases or decreases depends on the characteristics of the demand and supply-side uncertainties.More precisely, price elastic supply of emission allowances involves increased volatility in electricity prices if the shocks to electricity demand and electricity generation costs together are su¢ ciently large as compared with the shock to abatement costs in electricity generation.Weitzman (1974) examined regulation of public goods in the presence of uncertainty.Whereas price-based regulatory instruments …x the price of licenses, but leave the issued quantity uncertain, quantity-based instruments …x the quantity of licenses issued but leave the price uncertain.This trade-o¤ raises the following question for policy design: which type of regulation best help mitigate the cost of uncertainty so as to maximize social bene…ts of the public good?Weitzman (1974) found that price-based instruments are advantageous when the marginal bene…t schedule is relatively ‡at as compared to the marginal cost schedule, and vice versa.This has since been the consensus among most economists (e.g., Kolstad, 2000;Hoel and Karp, 2001;Pizer, 2002;Nordhaus, 2007).The results in the present paper corroborates the results in (Krysiak (2008) and Storrøsten (2014;2015) that, as compared with Weitzman (1974), endogenous technology choice provides a comparative advantage for quantities over price-based regulation.It also extends the result to hybrid regulation with price elastic permit supply and more general uncertainty and cost structures.In particular, the competitive …rms'technology choices remain optimal under (standard) emissions trading also when the …rms can choose their production technology (and not only abatement technology as in Krysiak (2008) and Storrøsten (2014;2015)).Stigler (1939) and Marschak and Nelson (1962) early examined …rms'choice of cost structure and referred to the …rms' ability to change production levels in response to new information as their " ‡exibility".Mills (1984) continued this terminology and showed that an unregulated competitive …rm will invest more in production ‡exibility if demand uncertainty increases.Mendelsohn (1984) examined investment in a prices vs. quantities setting, and showed that quantity-based instruments have an advantage, because price-based regulation induces excessive variation in output.Krysiak (2008) shows that technology choice is socially optimal under quantities, but not so under prices.Further, price-based regulation induces a more ‡exible technology than tradable quantities.Storrøsten (2014) shows that tradable emissions permits and an emissions tax are no longer equivalent when cost structure is endogenous and the product market is taken into account.Storrøsten (2015) derives a criterion comparing prices vs. quantities in terms of expected welfare, given uncertainty, optimal policy and endogenous cost structure.
The present paper adds to the literature on regulation and endogenous technology choice by (i ) examining the role of technology policies in the presence of uncertainty and endogenous cost structure, (ii ) expanding the analysis to hybrid schemes with price elastic supply of emission allowances, including the optimal hybrid policy in the presence of endogenous technology, (iii ) increasing the focus on the product market (of which production causes emissions as a byproduct) with uncertain production costs, and (iv ) allowing for a more general menu of cost structures than previous studies.
Sections 2.1 to 2.3 features the analytical analysis, whereas Section 2.4 presents a brief discussion that puts the results into the well-known analytical framework of Weitzman (1974).Section 3 concludes.

Theoretical analysis
The theory model is divided into three stages.First, the regulator determines the regulatory regime in Stage 1.Then, the …rms invest in production and abatement technology in Stage 2.
Last, consumers choose consumption and …rms choose production and emission levels in Stage 3. The model is solved backwards to …nd the subgame perfect Nash equilibrium.
The cost function of …rm i 2 I is: where x i = f i ; i ; i ; i ; e q i ; e a i g are positive technology parameters, q i is production, a i is abatement, and i and ' i are …rm speci…c stochastic variables with expected values equal to 0 and variances 2 and 2 ' , respectively.That is, we have i 0; 2 i and ' i 0; 2 ' i .Note that this cost structure implies that …rms produce at minimum e¢ cient scale (MES) when q i = e q i .Further, the MES unit cost of production is i + i , while the higher cost following q i 6 = e q i increases in the technology parameter i .The interpretation regarding abatement cost is similar.
The stochastic elements may re ‡ect, e.g., ‡uctuations in input prices or productivity, or a breakdown of equipment.As pointed out by Weitzman (1974), the stochastic variables may stem from genuine randomness or just imperfect information.The stochastic shocks enter the functional form linearly, which is similar to Weitzman (1974), Hoel and Karp (2002), Krysiak (2008) and Storrøsten (2014;2015).3Firm i's emissions are given by " i = kq i a i , where k is a positive constant, kq i is emissions without regulation (business as usual) and a i kq i .Note that this stylized speci…cation corresponds to end-of-pipe abatement.The functional form (1) is chosen primarily because it allows for interpretable analytical results.The …rm pays " i for emissions " i , where is the price on emissions.
The …rms can choose the technology parameters x i in the …rst stage of the game.The investment cost function, (x i ), is convex and decreasing in i , i , i and i , and satis…es ( ) ! 1 as i i i i !0. Further, the investment cost function is convex and increasing in the minimum e¢ cient scale parameters e q i and e a i .
The price on the homogenous good sold by the …rms is p.Firm i's pro…t in Stage 3 is given by: i = max where c( ) is given by equation (1).Note that it is mathematically equivalent whether the …rms maximize with respect to emission or abatement levels.
In this paper I examine standard emissions trading (ETS), an emissions tax (TAX), and a hybrid scheme (HYB) where the supply of emission allowances increases in the price on allowances.This is modelled using the following emission allowance supply function: where v 0, e is the emissions price target and S = P i2I " i is the aggregate emissions target.I assume that these parameters (i.e., v, e and S) are determined by the regulator in Stage 1.
Hence they are exogenous in stages 2 and 3.One special case, particularly relevant if moving from ETS with v = 0 to hybrid regulation HYB, is where S is the …xed emissions cap and e is the expected permit price given this cap.Even tough v is the slope of the permit supply function, I will sometimes refer to v as the price elasticity of permit supply for convenience (this elasticity is actually given by ve =").Equation (3) equals standard emissions trading if v = 0, converges towards a standard emissions tax if v ! 1, and a …nite (positive) v yields a hybrid scheme (see also Lemma 1 below).Equation (3) determines the emission cap as a linear function of S and .Note that this simple scheme does not allow regulation where the price on emissions follows marginal environmental damage, unless the environmental damage function is linear. 4We also observe that HYB encompasses ETS and TAX.Hence, expected welfare will always be higher or equal under HYB as compared with TAX and ETS (given that the value of v maximizes expected welfare).
The utility function of consumer j 2 J is given by: u j (q j ; j ) = (b + j ) q j d 2 q 2 j g j (") ; where b and d are positive parameters, g j (") is harm from aggregate emissions, " = P i " i , and j 0; 2 i is a consumer speci…c stochastic variable.
I assume that the outcomes of the stochastic variables i , ' i and j are determined in between stages 2 and 3. Hence, the …rms invest under uncertainty in Stage 2, but know their cost and the equilibrium prices when they make the production and emission decisions in Stage 3. The consumers only act in Stage 3, when they choose their consumption levels under full information.This implies that the …rms are all equal in Stage 2, because they face the same uncertainty, whereas …rms and consumers are heterogenous in Stage 3. It follows that, for each regulatory regime, all …rms invest in the same technology.That is, we have where superscript g = fets; hyb; taxg denotes the regulatory regime (see also Appendix A).I henceforth omit the …rm speci…c subscript on the technology parameters.

Stage 3: Production and consumption
Firm i 2 I maximizes pro…ts in Stage 3 and solves (2), given equation ( 3) and the technology parameters x.Each …rm's pro…t is a sum of linear and concave functions (cf., equation ( 2)), implying that the objective function in equation ( 2) is concave.It follows that the maximization problem (2) has a unique global maximum.The industry supply function and demand for emissions can be derived from the …rst order conditions associated with (2).Aggregate supply and aggregate demand for emissions are given by: respectively, with P i2I i and ' P i2I ' i .Consumer j 2 J solves max q u j (q j ; j ) pq j , with solution q j = 1 d (b p + j ). 5 Total demand for the good q is then given by: with P j2J j .The economy is assumed to be closed and hence we have the following product market equilibrium condition: The competitive equilibrium solves equations ( 5), ( 6) and ( 7) subject to the market equilibrium conditions (3) and (8).
We see from equations ( 5), ( 6) and ( 7) that the stochastic properties of the aggregate shocks , ' and , and their relations with the individual shocks i , ' i and i , are important for the characteristics of the competitive equilibrium.In this paper I assume that the shocks j , ' i and i are independent, i.e., we have expected values for all i 2 I and j 2 J.I further assume symmetrically correlated shocks, with correlation coe¢ cients given by for all …rms i; i 0 2 I (i 6 = i 0 ) and = E j j 0 = 2 j for all consumers j; j 0 2 J (j 6 = j 0 ).These assumptions are helpful for achieving interpretable theoretical results.Whereas the nature of the shocks is not modelled explicitly, it is reasonable to assume that the correlation coe¢ cients will depend on the origin 5 The alternate model formulation where the consumer maximizes utility subject to a budget constraint leads to a system of second order equations with solutions for q s and " d that do not permit a tractable analytical solution for the whole model (including technology investment in period 2). of the uncertainty.For example, shocks caused by ‡uctuating factor prices will typically be stronger correlated than randomness caused by equipment failure.Note that symmetrically correlated shocks implies that we must have 2 [ 1= (m 1) ; 1], ' 2 [ 1= (n 1) ; 1] and   2 [ 1= (n 1) ; 1] for the covariance matrixes to be valid (i.e., positive semi-de…nite).It can be shown that we have with analogous expressions for the variances, covariances and correlations of the other two shocks ' i and j .Furthermore, we have var( ) 2 , where these expressions reach their lower and upper limits at = 1= (n 1) and = 1, respectively.
The particular case with uncorrelated i 's, characterized by = 0, yields var( ) = n 2 i and cor( ; i ) = 1= p n (and similarly for ' i and j ).
The solutions for the market equilibrium product and emission prices are given by: where the expectations are the non-stochastic parts and the 's are the random elements, see Appendix A for the exact expressions and derivations.Whereas the prices depend on all the three stochastic elements under ETS and HYB, the product price under TAX does not depend on the shocks to abatement costs. 6Moreover, aggregate emissions are exogenous under ETS, while the permit price is exogenous under TAX (cf., equation 3).The expected values depend on the technology parameters x and the utility function parameters b and d (see equation ( 4)) Suppose the emission price target (or tax) is set such that e = E( ets ) in equation (3), i.e., the expected price on emissions is equal across all schemes.Further, let ets , tax , hyb denote an endogenous variable in the case of ETS (v = 0), TAX (v ! 1) and HYB (0 < v < ) for some …nite constant , respectively (e.g., ets 2 q ets ; " ets ; p ets ; ets ).We then have the following result regarding the relation between HYB, TAX and ETS: Lemma 1. Suppose e = E( ets ) and technology x is …xed and equal for all v.Then, the endogenous variables satisfy the following: (iv) hyb is strictly monotonic in v.

Proof. See Appendix A.
The …rst bullet point (i ) states that the expected values of the endogenous variables are all equal across the regulatory regimes.The realized prices and quantities will di¤er, depending on the stochastic shocks, however.Bullet point (ii ) simply states that hybrid regulation equals emissions trading if v = 0, and converges towards an emission tax as v ! 1 (as pointed out in the discussion of equation ( 3) above).The third bullet point (iii ) states that the solutions under hybrid regulation always is somewhere in between the solutions for standard emission trading and an emission tax.This may not be surprising, as HYB is constructed as a linear combination of ETS and TAX (cf., equation ( 3)).Last, point (iv ) states that hyb is either strictly decreasing or strictly increasing in the price elasticity of emission allowances v. So, for example, production and emissions under hybrid regulation will always be in between the outcomes under a tax and standard emissions trading, given equal technology vector x and the outcomes of the stochastic variables.Further, the regulator can choose to increase v to get closer to emission tax regulation, or reduce v to get closer to emissions trading.The assumption of equal technology across regulatory regimes is crucial in the derivation of Lemma 1, and it does not apply to case of endogenous technology in general.The exception is Lemma 1 (ii ), which follows directly from equation (3) and hence remains valid for endogenous technology.
The reduced form solutions in equation ( 9) allow for comparison of how the di¤erent stochastic variables a¤ect the volatility of prices and quantities under the di¤erent regulatory schemes.
Whereas it is rather intuitive that the volatility in the price on emissions ( ) decreases in v, and that the volatility in the aggregate emissions increases in v (see equation ( 3) and Appendix A), the results regarding the product market are less obvious.We have the following result: Lemma 2. The variances in aggregate production and the product price under TAX are larger (smaller) than under ETS unless 2 + 2 is su¢ ciently small (large) as compared to 2 ' .Proof.See Appendix A.
Lemma 2 states that shocks to production costs ( ) and consumer utility ( ) have smaller impact on the product price under emissions trading than under emissions taxes, while shocks to abatement costs (') have lower impact under the emissions tax.Regarding and production, the economic rationale is that the equilibrium allowance price decreases in the shocks to production costs, because the demand for emission allowances decreases when production costs increase and vice versa.That is, the positive shock to costs is partly o¤set by a lower price on emissions under ETS.This counteracts the initial e¤ect of shocks to production costs ( ) under emissions trading.Similarly, the shock to demand also has a smaller impact on aggregate production under emissions trading.The reason is that the allowance price increases in the demand shock , because the …rms must reduce their emission intensity in order to increase production in response to the positive demand shock under emissions trading.Hence, the higher product price following a positive shock to demand is partly o¤set by a higher price on emissions under ETS.
This contrasts with an emission tax, where the …rms pay a …xed price on emissions regardless of the aggregate emissions level.Last, the shock to abatement cost (') has a negative impact on production under emissions trading, because abatement becomes more costly.Aggregate production is independent of the shocks to abatement costs under an emissions tax, because the marginal cost of emissions is equal to the …xed emissions tax which does not depend on '.
The economic intuition regarding the result on the product price in Lemma 2 is very similar.
Remember that the variances of the aggregate shocks, featured in Lemma 2, are functions of the variances of the individual shocks and the correlation coe¢ cients (e.g., n 1 + (n 1) ).It follows that the variance in aggregate production and the product price tends to be larger under an emissions tax (as compared with ETS) if the shocks to the …rms' production costs ( i ) and consumer utility ( j ) have large variances and are strongly positively correlated ( and ).Conversely, the variance in product prices tends to be lower under TAX if the shocks to …rms abatement cost (' i ) have large variance and strong positive correlation ( ' ). 7emma 2 relates to Mendelsohn (1984) and Krysiak (2008), which both shows that quantitybased instruments have an advantage over price-based regulation, because the latter induces excessive variation in output.The result in Lemma 2 di¤ers mainly because the theory framework in the present paper features both a product market and a market for emissions allowances.
We have the following result: Proposition 1.The volatility of aggregate production increases (decreases) in the price elasticity of the supply of emission allowances ( v ) if and only if 2 + 2 is su¢ ciently large (small) as compared with 2 ' .Proof.The proposition follows from Lemma 1 and Lemma 2.
Proposition 1 highlights the role that a price elastic supply of emission allowances may have on the product market.Speci…cally, more price elastic supply of emission allowances involves increased volatility in aggregate production if the shocks to consumer demand and production costs together are su¢ ciently large as compared with the shock to abatement costs.This is relevant, e.g., if allowance price stabilizing measures are introduced to reduce allowance price volatility, and thereby create a less risky environment for investment in abatement technologies. 8he reason is that e¤orts to reduce volatility in the emission allowance market may come at the cost of a more risky investment environment in the product market.For example, Proposition 1 implies that the volatility in the price on electricity in the area covered by the EU ETS will be a¤ected if emission allowance stabilizing measures such as the Market Stability Reserve (MSR) are introduced or expanded.Whether the volatility increases or decreases depends on the characteristics of the uncertainty (as described by Proposition 1).
The volatility in aggregate emissions unambiguously increases in the parameter determining the price elasticity of permit supply v; see equation ( 9) and the reduced form solutions for aggregate emissions in Appendix A. Conversely, the volatility in the price on emissions unambiguously decreases in v. Lemma 2 and and Proposition 1 are valid for any allowance price target e (i.e., they do not require the assumption e = E( ets )).
So far, the focus has been on aggregate production and emissions.What matters for the …rms' investment levels, however, are the …rst and second order moments of their individual production and abatement levels (see below).In order to establish a reasonable basis for comparison, suppose that e in equation ( 3) are set such that the expected price on emissions are equal to the equilibrium price under ETS (v = 0) across the regulatory schemes.I show in Appendix A that the …rms individual production and abatement is then given by: q i ( ; '; ) a i + ets a i ( ; '; ) q i + hyb q i ( ; '; ) a i + hyb a i ( ; '; ) Here, q i and a i are each …rm's expected production and abatement, which is equal across …rms.The 's represent the stochastic elements, which di¤er across …rms (see Appendix A for the exact expressions).Remember that emissions are a linear combination of production and abatement (" i = kq i a i ).Whereas the expected terms are equal across the schemes, the stochastic elements di¤er.In particular, we have: in equation ( 10), with the stochastic elements under HYB being somewhere in between those of ETS and TAX (depending on v), cf.Lemma 1.9 Equations ( 10) and ( 11) highlight the di¤erent environments with respect to risk imposed by the di¤erent regulatory regimes.Speci…cally, …rms are not exposed to the variance in other …rms'abatement costs under TAX, because the allowance price is …xed.Moreover, …rms'production under TAX is independent of the uncertainty in the …rms'own abatement costs, because the marginal cost of emissions is exogenously given by the emissions tax.Last, the …rms'abatement levels (but not emissions) is independent of the shocks to production costs and consumer utility under TAX.Does this mean that …rms are less exposed to risk, interpreted as higher variances in production and abatement levels, under TAX?We have the following result on the relative variances in individual …rms production across the regulatory schemes: Lemma 3. Suppose e = E( ets ) and technology x is …xed and equal for all v. Then we have the following: and 2 ' i > 0 we have var(q tax i ) < (=)var(q hyb i ) < (=)var(q ets i ), given ' > (=) 1=(n 1) and …nite n and m.

Proof. See Appendix A.
Lemma 3 corresponds to the conclusion from Proposition 1, but this time for individual …rms.Lemma 3 implies that the variation in each individual …rm's production can increase in the price elasticity of emission allowances (v in equation ( 3)) if the shocks to consumer demand and production costs together are su¢ ciently large, as compared with the shock to abatement costs.Conversely, the variation in production decreases in v if the shocks to abatement costs are the dominant factor.This in turn has implications for the …rms' risk environment and their choice of production technology in Stage 2 (see Section 2.2 below).If several shocks are present at the same time, the relative size of the variances ( 2' i , 2 i and 2 i ) and the correlation coe¢ cients ( , ' and ) will together be decisive.

Stage 2: The …rms'investment decisions
In Stage 2, any …rm i 2 I maximizes expected pro…ts with respect to cost structure, as determined by the technology parameters x = f ; ; ; ; e q; e ag: with i given by equation ( 2).The maximand in equation ( 12) is a sum of linear and strictly concave functions, implying that the maximization problem is strictly concave and hence has a unique global maximum.The maximization in equation ( 12) is done under uncertainty, which contrasts with the …rms' optimization problem (2) in Stage 3. The interior solution to the maximization problem ( 12) is characterized by the following …rst-order conditions (where I used the envelope theorem in the derivation): where an asterisk ' 'indicates that the level solves the …rms technology optimization problem (12).Note that the expectations E (q i ) and E (a i ) in Stage 2 are equal across …rms, even tough realized abatement and production in Stage 3 di¤er between …rms.This is why the …rms chose equal technology in Stage 2 (across …rms, not regulatory schemes).
We see that investment that reduces abatement and production unit costs ( and ) increases in expected production and abatement.For example, the more you expect to produce, the more you are willing to invest to decrease the unit cost of production.Further, the production ‡exibility parameter is set such that it becomes less expensive to deviate from minimum e¢ cient scale (MES) production, as determined by e q i , if the variances in production increases.
The logic is similar for the abatement ‡exibility parameter , which increases in the variance of the …rm's abatement. 10Last, the MES parameters (e q i and e a i ) are set such that MES are close to (but below) expected production and abatement.The di¤erence between MES and expected production (abatement) increases in investment cost, and decreases in the cost of producing (abating) at a another production level determined by ( ).
We know from equation ( 9) and Proposition 1 that the second order moments of production and abatement di¤ers across the regulatory regimes, even if regulation is rigged such that E( ets ) = E( hyb ) = E( tax ), i.e. by setting e = E( ets ) (cf., Lemma 1).It follows that the regulatory regimes induces di¤erent types of technology.
Suppose the …rms have adapted to a standard ETS (v = 0) with the associated optimal technology.Then introducing elastic permit supply (v > 0) entails investment costs, because the …rms must recon…gure their technology to the risk environment induced by the new type of regulation.This cost, associated with changing regulatory regimes, may be a part of the discussion when considering regulatory changes.

Stages 1 and 2: Welfare
I assume that the social planner …rst determines the regulatory parameters v, e and S in equation ( 3) in Stage 1.These parameters may correspond to optimal policy, but the results in this paper are valid for any parameter values unless otherwise stated.
The optimal policy in Stage 3 under ETS and TAX would be the well-known condition that the emission cap or allowance price equalizes expected marginal environmental damages with expected marginal abatement costs.Note that the product market does not need any interventions from the regulator except for the emissions regulation (cf., perfect competition and the …rst theorem of welfare).The optimal v in equation ( 3) under HYB, i.e. the v that maximizes expected welfare given the …rms'investment decisions, may be less obvious, however.
Suppose a bene…cial social planner maximizes expected welfare with respect to v in equation (3), subject to the actions of the …rms and consumers in stages 2 and 3.The social planner's maximization problem is given by: u j (q j ; j ) X i2I c(q i ; a i ; x i ; i ; ' i; ) + (x) g (") 3 5 ; where g (") = P j2J g j (") and utility, u j (q j ; j ), and production cost, c(q; a; x i ; i ; ' i ), are given by equations ( 4) and ( 1), respectively.It can be shown that the socially optimal interior solution for v in equation ( 3) satis…es (see the Appendix): 11 We …rst observe that aggregate emissions and aggregate abatement moves in opposite directions inside the parenthesis in equation ( 14) (see the derivation of equation 14 in the Appendix).
Suppose emissions increase in v (and abatement decreases).Then the …rst order condition (14) states that v is calibrated such that the expected increase in environmental damage equals the expected savings from less abatement.If this where not true, expected welfare could be increased by changing v.For example, if marginal environmental damages decline more than 1 1 Attempts to derive reduced form solutions for v leads to large analytical expressions that are hard to interpret.
the associated increase in (aggregate) marginal abatement cost following a decrease in v, total welfare could be increased by reducing v. Whereas the requirement that the marginal change in expected welfare followed by a marginal increase in v is zero at optimum may not be that surprising, we observe that investment costs do not enter equation ( 14) directly.Investment costs are indirectly present as they determine the marginal changes in emissions and abatement inside the parenthesis in equation ( 14), however.We last note that a corner solution where HYB collapses to ETS and v = 0 is also possible, and that HYB converges towards a tax as v becomes very large.
To examine socially optimal technology investment in Stage 2, suppose a bene…cial social planner maximizes expected welfare with respect to the technology parameters x = f ; ; ; ; e q; e ag, subject to the competitive equilibrium in Stage 3. The social planner's maximization problem is given by: where utility, u j (q j ; j ), and production cost, c(q; a; x i ; i ; ' i ), are given by equations ( 4) and (1), respectively.Further, assume that S and/or e in equation ( 3) is set such that E (g " ) = E ( ) and assume quadratic environmental damage g (") = g 1 " + g 2 "=2.Then we have the following (see Appendix A): where superscript 'sp'denotes the socially optimal technology investment levels and the variables with asterisk are given by equation ( 13).The above assumption that E (g " ) = E ( ) states that the regulator chooses the policy parameters in Stage 1 such that the expected price on emissions equals expected marginal environmental damage in Stage 3 (optimal policy).If this assumption is violated, the socially optimal technology investment levels di¤er from those of the …rms for all parameters, not only and .The reason is that the social planner then uses the technology to compensate for the suboptimal regulatory policy in Stage 3. The di¤erence between the socially optimal technology and the …rms'investment is given by the covariances Cov g" n ; d" in equation ( 15).Interestingly, these terms are zero in the case of (i ) standard ETS (v = 0), because the …xed and binding emissions cap S implies that d"=d = d"=d = 0, and (ii ) linear environmental damage (given the assumption that E (g " = )).
We have the following result: Proposition 2. Let environmental damage be quadratic and strictly convex in aggregate emissions.Then, the …rms technology choice is socially optimal in the presence of uncertainty if and only if the permit supply is perfectly inelastic ( v = 0).
The term 'uncertainty'in Proposition 2 is interpreted such that at least one of the stochastic shocks , ' and is present, and that not all shocks present are correlated such that they cancel each other out.Proposition 2 generalizes the results of Krysiak (2008) and Storrøsten (2014;2015) to hybrid regulation, stochastic shocks to production costs (as opposed to abatement cost shocks only) and a more general cost structure. 12Proposition 2 is valid also if E (g " ) 6 = E ( ) Proposition 2 entails that endogenous technology choice provides a comparative advantage for ETS over HYB (for any given v > 0), and HYB over TAX (for any …nite v).The strength of this advantage increases in the absolute value of the magnitude of the variances of the shocks to the demand ( ) and supply ( and ') sides in the economy, as well as the correlation between these shocks (the 's).It also increases in the convexity of environmental damages. 13One implication of this is that allowance stabilizing measures such as the Market Stability Reserve (MSR), if implemented in a standard ETS (v = 0), introduces a negative technology externality, and that the strength of this externality is determined by the characteristics of the uncertainty and the convexity of the environmental damage function.In this context, it is important to distinguish between the optimization of welfare and the optimization of technology.Speci…cally, it may increase expected welfare to introduce a price-elastic supply of emission allowances (e.g., moving from ETS to HYB), even if this involves suboptimal investments in technology.The reason is that the bene…ts from elastic supply of emission allowances may very well outweigh the loss from the negative technology externality.
Proposition 2 suggests that technology policies can improve welfare if and only if the regulatory scheme features endogenous aggregate emissions. 14Assume TAX regulation, with the tax set equal to the marginal environmental damage from expected emissions, E (g " ) = tax .Further, let environmental damage be strictly convex and given by g (") = g 1 " + g 2 "=2 with g 1 > 0 and g 2 > 0. We then have the following result on technology investment policies under TAX: 1 2 The current paper has six endogenous technology variables, whereas Storrøsten (2014;2015) and Krysiak (2008) have two endogenous technology parameters.
1 3 Note that these issues are re ‡ected in the expressions for the optimal subsidies in Proposition 3 below.
1 4 Remember that other potential reasons for technology subsidies, like, e.g., technology spillovers or learning by doing, are not present in the analysis.
Proposition 3. Assume E (g " ) = tax and g (") = g 1 " + g 2 "=2.Then, the optimal taxes on investment in and under TAX are given by: respectively.It is not optimal to tax or subsidize investment in the other technology parameters (i.e., we have t tax = t tax = t tax e q = t tax e a = 0).Proof.See Appendix A.
Proposition 3 gives the taxes that induce the …rms to invest in the socially optimal technology, given optimal policy and the assumed quadratic environmental damage function.
A key message from Proposition 3 is that, in the presence of uncertainty, endogenous aggregate emissions and strictly convex environmental damage, it is optimal to tax investment in technology that increases the variance in emissions.The reason is that ‡uctuations in emissions creates a negative externality that is not internalized by the …rms.More precisely, the cost of emissions is strictly convex from the social planner's perspective (by assumption).Hence, the expected damage from emissions is larger than the damage from expected emissions from a welfare perspective, cf.Jensen's inequality and convex environmental damage.The …rms, on the other hand, face a linear cost of emissions, which is simply given by the …xed tax multiplied with their emissions level.Consequently, the …rms do not consider the increased environmental damage that follows from the ‡uctuations of emissions around their mean.The fact that these ‡uctuations are in ‡uenced by the …rms' technology decisions is what constitutes the negative externality that motivates the taxes on investment in the ‡exibility technology parameters and in Proposition 3. Note that, in the case of t tax , Proposition 3 involves taxation of investment in abatement technology.
We observe that the optimal tax to investment in tax is zero if 2 ' i = 0, if ' = 1=(n 1), or if ' = 0 and n ! 1.The explanation is that the part of the variance in emissions that is in ‡uenced by tax collapses to zero in these cases.Hence, there is no need to regulate investment in tax (the rationale for the investment tax is to reduce variance in emissions and thereby environmental damage).By the same reasoning, the optimal tax to investment in tax is zero if the uncertainty parameters are such that the uncertainty that is a¤ected by tax disappears.We also observe that Proposition 3 would prescribe a subsidy to investment on and in the case of concave environmental damage (in which case we have g 2 < 0 in Proposition 3).Whereas this may not be very realistic in the case of environmental damage, it may be relevant for some other public goods.
Proposition 3 indicates that it is also optimal to regulate investment in and under HYB.
That is, even though the emission price is under HYB is endogenous, marginal environmental damage is endogenous and may di¤er from the price on emissions.Derivations of expressions for the optimal subsidies do not lead to interpretable analytical results, however (the expressions become too large).We nevertheless observe that the optimal taxes under HYB satisfy t hyb = Last, it is worth emphasizing that the optimal tax scheme in Proposition 3 depends strongly on the functional form of the cost function (1).For other technology structures, the expressions will change, and may even change sign from a tax to a subsidy.As such, the key lessons from Proposition 3 is that (i ) implementation of the socially optimal cost structure may require technology policies (e.g., taxes, subsidies or technology standards) even without the presence of issues like positive technology externalities or market power.And, (ii ), that these technology policies should aim at reducing variation in aggregate emissions.Weitzman (1974) The theory model in this paper is quite di¤erent from that of Weitzman (1974), but the results are nevertheless closely related to his analysis.They are also perhaps easier to understand and put in perspective if placed into that well-known framework.

Interpretation of the results on investment in the context of
As in Weitzman (1974), the present analysis features a model with polluters (the …rms) with marginal bene…ts (or savings) from emissions that are unknown to the regulator.The bene…ts are stochastic but known to the polluting …rm when it chooses emissions (in Stage 3).We have a non-stochastic marginal environmental damage function that is known to the regulator. 15Weitzman (1974) considers the following research problem: Does price or quantitybased regulation induce the highest expected welfare?
Weitzman's problem is illustrated in Figure 1.Here marginal damages (MD) are increasing in emissions, whereas expected (E(MB)) and realized marginal bene…ts (MB) from emissions are decreasing in emissions.Note …rst that without uncertainty we have MB = E(MB) and the optimal allocation is given by point C in Figure 1 (i.e., no shift to MB). 16 At C the sum of environmental damages (the area BCKA) and emission reductions (KCI) is minimized.This can be achieved with either the TAX or the emission CAP graphed in Figure 1, so price and quantity-based regulation performs equally well.But this changes with uncertainty.In Figure 1 I consider a positive shock to the marginal bene…ts from emissions, but the case is analogous with a negative shock.The optimal allocation is now given by E, but price and quantity-based regulation based on expected marginal bene…ts induces the suboptimal outcomes G and D, respectively.
The optimal allocation E is associated with a total cost (i.e., the cost of damages plus emission reductions) given by the area ABEH.In comparison price and quantity-based regulation induces total costs equal to the areas BFGHA and BCDHA, respectively.The welfare loss associated with quantity-based regulation is given by the shaded area CDE, whereas the loss associated with price-based regulation is given by the area EFG (both as compared with the optimal allocation E).We see that price and quantity-based regulation are no longer equal in the presence uncertainty, and that the relative performance of the instruments depends on the slopes of the marginal damage and emission reduction functions.Speci…cally, Weitzman (1974) …nds that quantity-based regulation (ETS) outperforms price-based regulation (TAX) if marginal damages are more steeply sloped than marginal bene…ts from emissions.Conversely, price-based regulation induces the highest expected welfare if marginal bene…ts from emissions are more steeply sloped than marginal damages.Figure 1 is drawn such that price-based regulation induces highest expected welfare.
One important di¤erence between the present paper and the analysis in Weitzman (1974) is that I examine technology investment.More precisely, the present paper features three stages: (1) choice of regulatory regime, (2) technology investment, and (3) production and emissions, with the stochastic variables determined in between Stage 2 and 3.The Weitzman (1974) by the consumers and thus not internalized by the …rms.Therefore, the …rms' investment is suboptimal in the case of a tax, and socially optimal under ETS.This issue, i.e., whether the welfare loss is burdened on the …rms or the consumers, is not central in Weitzman (1974).
The reason is that it only has a distributional impact and does not a¤ect overall welfare (in the Weitzman (1974) model with exogenous technology).Yet, it is essential for the results on technology investment and active technology policies in the present paper.
We last note that expected welfare may well be higher under ETS than under TAX even if the choice of technology under TAX is suboptimal (from a welfare perspective).An analogue to the well-known Weitzman (1974) criterion on prices vs. quantities would have to take the endogeneity of the marginal cost function into account.Whereas the present paper does not derive such a criterion, the optimal policy is given by the hybrid policy with v as in equation ( 14) (see also Storrøsten, 2015). 17endogenous aggregate emissions and strictly convex environmental damage, it is optimal to tax investment in technologies that induce large variance in emissions.I also examined how elastic supply of emission allowances in ‡uences the volatility of product prices and derived the optimal hybrid policy with price elastic supply of emission allowances (and endogenous technology).I found that price elastic supply of emission allowances involves increased volatility in the product market if the shocks to consumer demand and production costs are su¢ ciently large, as compared with the shocks that hit the abatement part of the …rms cost function.
The results are derived under rather strict assumptions on the functional forms.In particular, the cost function is separable in production and abatement, implying that abatement is modelled as end of pipe technology.Whereas the mechanisms examined would still be present with a more general cost function, the results would also be contingent on the cross derivatives between abatement and production.The results in the present paper would remain valid, unless the cross derivatives where signi…cantly large and with the opposite sign as compared with the direct e¤ect, however.
HYB cannot perform worse than TAX or ETS under optimal policy, since HYB is more ‡exible and can replicate both if appropriate.Nevertheless, the results in the present paper indicate that regulatory schemes with endogenous aggregate emissions have a disadvantage as compared with regulation with …xed aggregate emissions in the presence of endogenous technology choice and convex environmental damage.Speci…cally, introduction of permit price stabilizing measures in an emission trading system will come at the cost of suboptimal technology investments, and may also cause increased ‡uctuations in product prices (which in itself is not necessarily negative).Such considerations may be particularly relevant if the regulator has imperfect information and optimal policy, including the taxes on investment in Proposition 3, is di¢ cult to implement.
are given by: p = K hyb n (bm + dn e q i dn + dkn ) + bmv + dnv e a i dkn 2 + bk 2 mn Sdkn +K hyb ( e q i dnv + dknve ) De…ne K tax = 1= (m + dn).Then the solutions under TAX are given by (note that v ! 1 in equation 3 requires = e ): p tax = K tax (bm + dn e q i dn + e dkn) + K tax (d + ) Hence, HYB converges towards a tax as v ! 1, and equals ETS if v = 0 (given the same technology).In addition, we have Neither of which change sign in v. Hence, the equillibrium product price and emissions price under HYB are both monotonic in v.This proves Lemma 1.
Proof of Lemma 2: The lemma follows from the stochastic elements in the reduced form solutions under the regimes above.Speci…cally, we observe that aggregate production depends on all three stochastic elements under ETS and HYB, but only and under a tax.More Remember that K ets < K tax .It is straightforward to compare the expressions to verify that the derivatives under HYB are in between those of ETS and TAX.Lemma 2 follows.
Solutions for production and abatement for individual …rms under ETS and TAX when emissions tax equals expected price on emissions under ETS: The letters with an overbar refer to the expected values, while the stochastic elements are represented by the 0 s.
Variances for production and emissions for individual …rms when emissions tax (TAX) equals emissions price under ETS: Then we have: , and similar for the other stochastic variables.
Case (i ), only 2 > 0: Consider the case with 2 ' = 2 = 0. Then the derivative of the variance in production under ETS is given by: We further observe that when = 1=(n 1) we have var( (m k 2 +m +dn) 2 .How does this compare with a tax?The variance in production under TAX when = 1=(n 1) is given by var(q Hence, the variance declines faster in under ETS.Conclusion case (i ): Suppose 2 ' = 2 = 0. Then we have var(q ets i ) var(q tax i ), with var(q ets i ) = var(q tax i ) if = 1=(n 1) and var(q ets i ) < var(q tax i ) if > 1=(n 1).Case (ii ), only 2 ' > 0: Consider the case with 2 = 2 = 0. Then we have ets q i = km ' m k 2 n +dn 2 +mn 2 and tax q i = 0. Hence, var ets q i > var tax q i i¤ > 1 n 1 for …nite n Case (iii ), only 2 > 0: Consider the case with 2 ' = 2 = 0. Then we have ets q i = n m k 2 n +dn 2 +mn 2 and tax q i = (m +dn) = 0. Hence, var ets and var tax q i = (m +dn) 2 2 , which yields var ets q i var tax 0 for …nite m and n.
The results that HYB is in between follows from Lemma 1. Lemma 3 follows.
Stage 1: The optimial v We …rst remember the …rms focs in stage 2: , and focs of …rms and consumers in stage 3: With X = p P i2I q i P j2J q j + P i2I (kq i kq i + a i a i ) = 0 and subject to v 0. The Lagrangian is: Maximization of L requires the standards …rst and second order conditions,and the complementary slackness conditions (csc): Note that the csc implies that = 0 if v > 0, and that we must have v = 0 if > 0 (this does not exclude = v = 0 which is allowed).Di¤erentiating the Lagrangian w.r.t v we get: with Y v given by: with 0, with = 0 if v > 0. This yields equation ( 14).
Then we have d" byb dv = km X and da byb dv = (m + dn) X.It follows that the two terms inside the expecation in equation ( 14) have opposite signs.
dn 2 + mn + dnv + mv + k 2 mn 2 K hyb = 1= dn 2 + mn + dnv + mv + k 2 mn dn 2 + mn + dnv + mv + k 2 mn 2 dn 2 + mn + dnv + mv + k 2 mn 2 = 0 Stage 2: Social planner investment: We …rst note that the …rst-order conditions (focs) in stage 3 are given by d i dq i = p i k + (Q i q i ) = 0, d i da i = i ' i + (A i a i ) = 0, and du j dq i = b + j dq j p = 0.The social planner maximizes welfare: (c ( ) (x)) g (") + X 3 5 ; with X = p P i2I q i P j2J q j + P i2I (kq i kq i + a i a i ) = 0. Di¤erentiating w.r.t we get dW d = E [Y ], with Y given by: Y e q = X j2J (b + j dq j ) dq j de q X i2I ( + i + (q i e q)) dq i de q X i2I (e q q i ) X i2I ( + ' i + (a i e a)) da i de q X i2I e q g " d" de q + dX de q : Repeating the steps above, we get Y e q = P i2I (e q q i ) P i2I + ( g " ) d" d which, for each …rm i 2 I, yields = (e q q i ) + (g" ) n d" d .Applying the same procedure for and and summarizing results yields: e a i = i E (a i e a i ) + E g " n d" de a i ; which is identical to the …rms'…rst order conditions, except for the expectation involving environmental damage (which is not present in the …rms'optimization problem).We observe that this term is zero in the case of pure emissions trading (with an exogenously given and binding emissions cap), and optimal policy and linear environmental damage (with g " = ).
Implementation and development of new technologies are essential for tackling environmental challenges such as climate change, acid rain and depletion of the ozone layer.In this paper I examined the investment incentives of market-based regulation, with focus on the technology characteristics the di¤erent regulatory schemes tend to drive forward.The paper also investigated the rationale for active technology policies.I showed that, in the presence of uncertainty, , d" tax dQ = kmn m +dn and d" tax dA = n.Hence, E((g " ) d" tax d ) = E((g "